Flavor Physics for the Millennium

L=

(le)L'

R = 6R

(8)

When the electroweak symmetry is spontaneously broken, the electron mass emerges as

me = Cv/V2 .

(9)

The Yukawa couplings that reproduce the observed quark and lepton masses range over many orders of magnitude, from Qe « 3 x 1 0 - 6 for the electron to C,t ~ 1 for the top quark. Their origin is unknown. In one sense, therefore, all fermion masses involve physics beyond the standard model.16 We cannot be sure that finding the Higgs boson, or understanding electroweak symmetry breaking, will bring enlightenment about the fermion masses. Neutrino masses may have a different origin than the masses of the quarks and charged leptons: alone among the known fermions, the neutral neutrino can be its own antiparticle. 17 This fact opens the possibility of several varieties of neutrino masses. It is worth remarking on another manifestation of the logical separation between the origin of gauge-boson masses and the origin of fermion masses. The observation that a fermion mass is different from zero (mf ^ 0) implies that the electroweak gauge symmetry SU{2)L ® U(l)y is broken, but electroweak symmetry breaking is only a necessary, not a sufficient, condition for the generation of fermion mass. The separation is complete in simple

12 technicolor, 1 8 the theory of dynamical s y m m e t r y breaking modeled on the Bardeen-Cooper-Schrieffer theory of the superconducting phase transition. Finally, the electroweak theory we are about to describe does not predict the mass of the Higgs boson, and there is no assurance t h a t finding the Higgs boson will tell us the origin of this mass. Will the discovery of the Higgs boson be stupendously i m p o r t a n t ? Beyond any doubt, as the rest of these lectures will begin to show. But will it explain the origin of all mass? Be careful what you promise! 2

Antecedents of t h e Electroweak Theory

In The Odd Quantum, Sam T r e i m a n 1 9 quotes from the 1898-99 University of Chicago catalogue: "While it is never safe to affirm t h a t the future of the Physical Sciences has no marvels in store even more astonishing t h a n those of the past, it seems probable t h a t most of the grand underlying principles have been firmly established and t h a t further advances are to be sought chiefly in the rigorous application of these principles to all the phenomena which come under our notice . . . . An eminent physicist has remarked t h a t the future truths of Physical Science are to be looked for in the sixth place of decimals." These confident words were written, we now know, j u s t as the classical world of determinism and uncuttable atoms and continuous distributions of energy was beginning to come a p a r t . Much crucial progress did come from precise measurements—not always in the sixth place of the decimals, but precise nonetheless. At the same time, the century we are leaving has repeatedly shown t h a t N a t u r e ' s marvels are not limited by our imagination, and t h a t exploration can yield surprises t h a t completely change the way we think. Before we leap into a discussion of the modern electroweak theory, it will be useful to spend a few m o m e n t s recalling the soil in which the electroweak theory grew. We shall not a t t e m p t anything resembling a full intellectual history, 2 0 ' 2 1 ' 2 2 , 2 3 but only hit a few of the high spots. 2.1

Radioactivity,

(3 decay, and the

neutrino

Becquerel's discovery of radioactivity in 1896 is one of the wellsprings of modern physics. In a short time, physicists learned to distinguish several sorts of radioactivity, classified by Rutherford according to the energetic projectile emitted in the spontaneous disintegration. N a t u r a l and artificial radioactivity includes nuclear /? decay, observed as A

Z-+

A

(Z+l)

+ f3~ ,

(10)

13

en -»—»

c CD >

Expected

0

Observed

£ 13

Beta particle energy Figure 7. Expectations and reality for the beta decay spectrum.

where f3~ is Rutherford's name for what was soon identified as the electron and AZ stands for the nucleus with Z protons and A — Z neutrons. Examples' are tritium /? decay, 3

HX -»• 3 He 2 + /?- ,

(11)

n - > p + /? ,

(12)

214

(13)

neutron (3 decay,

and /3 decay of Lead-214, 214

Pb;82 ->

Bi83 + 0-

For two-body decays, the Principle of Conservation of Energy & Momentum says that the /3 particle should have a definite energy, indicated by the spike in Figure 7. What was observed was very different: in 1914, James Chadwick (later to discover the neutron) showed conclusively24 that in the decay of Radium B and C ( 2 1 4 Pb and 214 Bi), the (3 energy follows a continuous spectrum, as shown in Figure 7. ' I t is a curious fact that /3+-emitters, AZ —>• A(7, — 1) + /?+, are rare among the naturally occurring isotopes. The first example, radio-phosphorus produced in aAl collisions, was found by Irene and Frederic Joliot-Curie in 1934, after the discovery of the positron in cosmic rays. In our time, the decay 1 9 Ne —t- 1 9 F 4- /3+ has been a favorite for the study of right-handed charged currents and time reversal invariance. And, of course, the positron emitters form the technological basis for positron-emission tomography.

14

What could be the meaning of this completely unexpected behavior? Niels Bohr was willing to consider the possibility that energy and momentum are not uniformly conserved in subatomic events. The /?-decay energy crisis tormented physicists for years. On December 4, 1930, Wolfgang Pauli addressed an open letter 7 to a meeting on radioactivity in Tubingen. Pauli could not attend in person because his presence at a student ball in Zurich was "indispensable." In his letter, Pauli advanced the outlandish idea of a new, very penetrating, neutral particle of vanishingly small mass. Because Pauli's new particle interacted very feebly with matter, it would escape undetected from any known apparatus, taking with it some energy, which would seemingly be lost. The balance of energy and momentum would be restored by the particle we now know as the electron's antineutrino. The proper scheme for beta decay is thus A

Z->

A

(Z + 1) + / T +9 .

(14)

Pauli's new particle was indeed a "desperate remedy," but it was, in its way, very conservative, for it preserved the principle of energy and momentum conservation and with it the notion that the laws of physics are invariant under translations in space and time. The hypothesis fit the facts.* After Chadwick's discovery of the neutron in 1932, Fermi named Pauli's hypothetical particle the neutrino, to distinguish it from the neutron, and constructed his four-fermion theory of the weak interaction. Experimental confirmation of Pauli's neutrino had to wait for dramatic advances in technology.' J

Pauli's letter (in the original German) is reproduced in Ref. 2 5 . For an English translation, see pp. 127-8 of Ref. 2 0 . It begins, "Dear Radioactive Ladies and Gentlemen, I have hit upon a desperate remedy regarding . . . the continuous /3-spectrum . . . " Pauli concluded, "For the moment I dare not publish anything about this idea and address myself confidentially first to you. . . . I admit t h a t my way out may seem rather improbable a priori . . . . Nevertheless, if you don't play you can't win . . . . Therefore, Dear Radioactives, test and judge." Pauli's neutrino, together with the discovery of the neutron, also resolved a vexing nuclear spinand-statistic problem. *As you TASI students continue in physics, you will be amazed and delighted to find how quickly we learn—or how little we knew just a short time ago. It is easy to assume that anything we read in textbooks has been known forever, so it is sometimes stunning to learn that something we take for granted actually had to be discovered. An example that has both scientific and touristic interest for students in Boulder is Jack Steinberger's discovery on Mount Evans of the continuous electron spectrum in muon decay. 26 'Detecting a particle as penetrating as the neutrino required a large target and a copious source of neutrinos. In 1953, Clyde Cowan and Fred Reines 2 7 used the intense beam of antineutrinos from a fission reactor AZ -> A(Z + 1) + j3~ + v , and a heavy target (10.7 ft 3 of liquid scintillator) containing about 10 2 8 protons to detect the reaction u + p -> e+ + n. Initial runs at the Hanford Engineering Works were suggestive but inconclusive. Moving their apparatus to the stronger fission neutrino source at the Savannah River nuclear plant,

15 We can now recognize /? decay as the first hint for flavor, the theme of this s u m m e r school. Indeed, neutron b e t a decay is the prototype charged-current, flavor-changing interaction:

W

2.2

The neutron

and

flavor

symmetry

T h e discovery of the neutron m a d e manifest the case for flavor, with two species of nucleon nearly degenerate in mass: M{n)

= 939.565 63 ± 0.000 28 MeV/c 2

M(p)

= 938.272 31 ± 0.000 28 MeV/c 2

A M = 1.293318 ± 0.000 009 MeV/c

(15)

2

so t h a t AM/M « 1.4 x 1 0 - 3 . T h e similarity of the neutron and proton masses makes it plausible to inquire into the charge independence of nuclear forces. T h e two-nucleon system is not particularly informative: among NN states, pp and nn are unbound, and only the isocalar np state, the deuteron, is very lightly b o u n d . Hints for the charge independence of nuclear forces come from m a n y light nuclei. We m a y compare the binding energy of 3

B(ppn)

3

= 8.481 855 ± 0 . 0 0 0 013 MeV

H e ( p n n ) = 7.718 109 ± 0.000 013 MeV

(16)

A(B.E.) = 0.763 46 MeV . T h e difference in binding energy is very close to a primitive estimate for the C o u l o m b repulsion in 3 H e : taking the measured charge radius of r = 1.97 ± 0.015 fm, we estimate the C o u l o m b energy as a/r « 0.731 MeV. More detailed evidence t h a t nuclear forces are the same for protons and neutrons come from the level structures in mirror nuclei. I show in Figures 8 and 9 the kinship between the 7 Li(4p + 3n) and 7 Be(3p + An) level schemes, and between the n B ( 5 p + 6ra) and 1 1 C ( 6 p + 5 n ) level schemes. In b o t h cases, isospin / = 3/2 isobaric analogue levels are present in the 7 He(2p + 5n) and 7 B ( 5 p + 2n) ground states, and the n B e ( 4 p + In) and n N ( 7 p + An) ground states. Cowan and Reines and their team made the definitive observation of inverse /3 decay in 1956. 2 8

16 A=7 20

r

15 -

10 -

5 -

0 -

= 7

He

7

^^Z Li

7

Be

7

B

Figure 8. Simplified isobar diagram for the A = 7 nuclei. The presumed JT = 3/2 isobaric analogue levels are shown in grey. Following the usual practice, the diagrams for individual isobars are shifted vertically to eliminate the n-p mass difference and the Coulomb energy. Data taken from Ref. 2 9 .

"Be

"B

"C

11

N

Figure 9. Simplified isobar diagram for the A = 11 nuclei. The presumed J = 3/2 isobaric analogue levels are shown in grey. Following the usual practice, the diagrams for individual isobars are shifted vertically to eliminate the n-p mass difference and the Coulomb energy. Data taken from Ref. ° .

17 12

A = 14

10

8

6

4

2

0

Figure 10. Simplified isobar diagram for the A = 14 nuclei. The presumed 7 = 1 isobaric analogue levels are shown in grey. Following the usual practice, the diagrams for individual isobars are shifted vertically to eliminate the n-p mass difference and the Coulomb energy. Data taken from Ref. .

Extremely compelling evidence for the charge independence of nuclear forces comes from the systematic study of two-nucleon states in the A = 14 nuclei, which consist of two nucleons outside a closed core: 14 14

0 :

12

C + (pp) h = + 1

N :

12

C + (pn) Iz = 0

14

C:

12

(17)

C + (rm)/3 = - l

We see in Figure 10 t h a t the 1=1 levels are common to all three elements. Nitrogen-14 has m a n y additional 7 = 0 levels. T h e charge independence of nuclear forces led to the first flavor symmetry, isospin invariance. We regard the proton and neutron as two states of the nucleon doublet,

and consider the nuclear interaction to be invariant under rotations in isospin space. In the absence of electromagnetism, which supplies the distinguishing information t h a t the proton carries an electric charge b u t the neutron does not, it is a matter of convention which state—or which superposition of states—we n a m e the proton or neutron." 2 Once we have the notion of If we lived in a world without electromagnetism, how could you determine t h a t there are,

18

isospin invariance in mind, it is a small step to the idea of weak isospin, which we infer from the fact that the charged-current weak interaction transforms one member of the nucleon doublet into another. The notion of weak-isospin families is a cornerstone of our modern electroweak theory. 2.3

Parity violation in weak decays

A series of observations and analyses through the 1950s led to the suggestion that the weak interactions did not respect reflection symmetry, or parity. In 1956, C. S. Wu and collaborators detected a correlation between the spin vector J of a polarized 60 Co nucleus and the direction pe of the outgoing (3 particle. 32 Since parity inversion leaves spin, an axial vector, unchanged: V:J-*J,

(18)

while reversing the electron direction: V:pe^-Pe,

(19) 60

the correlation J • pe is parity violating. Detailed analysis of the Co result and others that came out in quick succession established that the chargedcurrent weak interactions are left-handed. Since parity links a left-handed neutrino with a right-handed neutrino, VL

•Iff

we build a manifestly parity-violating theory with only VL. How can we establish that the known neutrino is left-handed? The simplest experiment to describe—though not the earliest to measure neutrino helicity—is to measure the helicity of the outgoing fi+ in the decay of a spinzero 7T+ -» p+ZV 33 ' 34 v» <—(**)~*

^

By angular momentum conservation, the spin projections of the muon and neutrino must sum to zero, so the helicity of the neutrino is equal to that of the muon: h(i/^) — h(fi+). Note that because the massless neutrino must be left-handed, the /z + is forced to have the "wrong helicity" in pion decay: the antilepton fi+ is naturally right-handed, and can only have a left-handed helicity because it is massive. This "helicity suppression" inhibits the decay 7r+ —> ii^Vp, and it inhibits the analogue decay TT+ -»• e+ue still more. The in fact, two species of nucleons?

19 decay amplitude in each case is proportional to the charged-lepton mass, and this accounts for the dramatic ratio

ff? -* ^

= 1-23 x 10- .

(20)

The classic determination of the electron neutrino's helicity was made by M. Goldhaber and collaborators, who inferred h(ve) from the longitudinal polarization of the recoil nucleus in the electron-capture reaction 35 152 E u m ( J = 0) -> e +

152

Sm*(J = l ) + i/,

U7+152Sm.

(21)

Obviously, this achievement required not only impressive experimental technique, but also a remarkable knowledge of the characteristics of nuclear levels! Recently a group in Zurich has emulated the original achievement in a muon-capture reaction, fi~

12

C(J = 0) ->

12

B(J = 1) Vvk

(22) 36

to determine h(vp) by angular momentum conservation. See the Review of Particle Physics for the most recent determinations of h(isT) in tau decays.9 The fact that only left-handed neutrinos and right-handed antineutrinos are observed also means that charge-conjugation invariance is violated in the weak interactions. Charge conjugation takes a vi, into a nonexistent z/j,:

ft

VL

The consequence of the C violation is very dramatic for muon decay, as the decay angular distributions of the outgoing e* in //* decay are reversed, dNj/i* -> e* + dxdz

=

x2{?>-2x)

(2x - 1)

(23)

l±z (3 - 2x)

where x = J9 e /p? lax and z = s^ -pe. The positron follows the spin direction of the fj,+, but the electron avoids the spin direction of the fi~. 2.4

An effective Lagrangian for the weak interactions

After the observation of maximal parity violation in the late 1950s, a serviceable effective Lagrangian for the weak interactions of electrons and neutrinos could be written as the product of charged leptonic currents, £V-A

-GF _ = —^-*OV(1-75)ee7'i(l-75)z/-|-h.c.

,

(24)

20 where Fermi's coupling constant is GF = 1.16632 x 1 0 - 5 G e V ~ 2 . This is often called the V — A (vector minus axial vector) interaction. It is straightforward to compute the F e y n m a n amplitude for antineutrino-electron scattering,™

M = -

iGF

V2

v(v,qihn(l-~t5)u(e,Pi)u{e,p2)~ff'(l--f5)v(v,q2)

,

(25)

where the c m . kinematical definitions are indicated in the sketch. e; n : P i e o u t : P2

fin : 91

The differential cross section is related to the absolute square of the amplitude, averaged over initial spins and summed over final spins. It is d
-> vt)

_

\Mf

_Gp-

2mEv{\

64TT 2 S

dttcm

-

zf

16TT2

(26)

where z = cos 9*. T h e total cross section is simply av-A(ve

—>• ve) =

G\ • 2mEu 3TT

Eu 0.574 x 1 ( T 4 1 cm 2 """" VI G e V ,

(27)

it is small! for small energies. Repeating the calculation for neutrino-electron scattering, we find d
(28)

and aV-A{ve

-»• ue) =

&F

' 2mEu

« 1.72 x 1CT 41 cm 2

Eu 1 GeV

(29)

It is interesting to trace the origin of the factor-of-three difference between the vt and ve cross sections, which arises from the left-handed nature of the charged current. In neutrino-electron scattering, the initial state has spin projection Jz = 0, because the incoming neutrino and electron are both left -handed. T h e y can emerge in any direction—in particular, in the backward direction denoted by z — + 1 — a n d still satisfy the constraint t h a t J2 = 0. "See Ref.

37

for conventions and tricks for calculating amplitudes.

21

fr incoming

4

-/*=0

outgoing, z — + 1

fr J Z = o

In antineutrino-electron scattering, the situation is different, because the antineutrino is right-handed. The initial angular momentum has spin projection Jz = 1; for backward scattering, the outgoing electron and antineutrino combine to give Jz = - 1 , so scattering at z = + 1 is forbidden by angular momentum conservation.

incoming

2.5

fr

Jz = +i

outgoing, z = +1

J, = - 1

Lepton families and universality

The muon is distinct from the electron; what is the nature of the neutrino emitted in pion decay, 7r+ —»• fi+v? In 1962, Lederman, Schwartz, Steinberger, and collaborators carried out a two-neutrino experiment using neutrinos created in the decay of high-energy pions from the new Alternating Gradient Synchrotron at Brookhaven. 38 They observed numerous examples of the reaction vN —>• fi + X, but found no evidence for the production of electrons. Their study established that the muon produced in pion decay is a distinct particle, v,,, that is different from either ve or ve. This observation suggests that the weak (charged-current) interactions of the leptons display a family structure, (30)

We are led to generalize the effective Lagiangian (24) to include the terms

£v-\ = "rTf"""7^1 ~ 75 ^ ^"t 1 ~ 75^e + hx- >

(31)

in the familiar current-current form. With this interaction, we easily compute the muon decay rate as T(n -» eveVp) =

1927T3

(32)

22 With the value of the Fermi constant inferred from /? decay, (32) accounts for the 2.2-/US lifetime of the muon. The resulting cross section for inverse muon decay, o-(^e -> fj,ve) = (TV-A{vee -> vee) 1

{ml - ml) 2meEl/

2

(33)

is in good agreement with high-energy data in measurements up to E„ as 600 GeV. However, partial-wave unitarity constrains the modulus of an inelastic amplitude to be \Mj\ < 1. According to the V — A theory, the J = 0 partial-wave amplitude is M0

GF • 1meEv TIV2

{ml-ml) 2meE„

(34)

which satisfies the unitarity constraint for Ev < ir/GFme\/2 « 3.7 x 108 GeV. These conditions aren't threatened anytime soon at an accelerator laboratory (though they do occur in interactions of cosmic neutrinos). Nevertheless, we encounter here an important point of principle: although the V — A theory may be a reliable guide over a broad range of energies, the theory cannot be complete: physics must change before we reach a c m . energy y/s fa 600 GeV. A few weeks after my TASI00 lectures, members of the DONUT (Direct Observation of NU Tau) experiment at Fermilab announced the first observation of charged-current interactions of the tau neutrino in a hybrid-emulsion detector situated in a "prompt" neutrino beam. 39 The vT beam was created in the production and decay of the charmed-strange meson D+^ vT + anything.

(35)

Their "three-neutrino" experiment was modeled on the two-neutrino classic: a beam of neutral, penetrating particles (the tau neutrinos) interacted in the hybrid target to produce tau leptons through the reaction vTN —> T + anything.

(36)

Although extensive studies of r decays had given us a rather complete portrait of the interactions of vT, the observation of the last of the known standardmodel fermions gives a nice sense of closure, as well as a very impressive demonstration of the experimenter's art. We have a great deal of precise information about the properties of the leptons, because the leptons are free particles readily studied in isolation. All of them are spin-4, pointlike particles—down to a resolution of a fewx 10""l cm.

23 Table 1. Some properties of the leptons.

Lepton

Mass

Lifetime

e" ve H~ j/„

0.510 999 07 ± 0.000 000 15 MeV/c < 10 - 15 eV/c2 105.658 389 ±0.000 034 MeV/c2 < 0.19 MeV/c2 (90% CL)

T~ i/T

1777.06to 26 MeV/c2 < 18.2 MeV/c2 (95% CL)

2

> 4.3 x 10 23 y (68% CL) 2.197 03 ± 0.000 04 x 10~ 6 s 290.2 ± 1.2 x 10~ 15 s

The kinematically determined neutrino masses are all consistent with zero, though the evidence for neutrino oscillations argues that the neutrinos must have nonzero masses. A brief digest of lepton properties is given in Table 1. An important characteristic of the charged-current weak interactions is their universal strength, which has been established in great detail. We'll content ourselves here with the most obvious check for the lepton sector. Using the generic formula (32) for muon decay, we can use the measured lifetime of the muon to estimate the Fermi constant determined in muon decay as l

G, = fl^lY V Ttirnfi

= L

i638

x

io-5

GeV-2

(37)

J

Similarly, we can evaluate the Fermi constant from the tau lifetime, taking into account the measured branching fraction for the leptonic decay. We find l

3 (T(T^evevT) 1927r „3t /i\2(38) °r = - b TTT1 r= 1-1642 x 10~ 5 GeV~ 2 . V T(T -> all) rrm\ ) Both are in excellent agreement with the best value of the Fermi constant determined from nuclear (3 decay,0

Gp = 1.16639(2) x 10- 5 G e V - 2 .

(39)

The overall conclusion is that the charged currents acting in the leptonic and semileptonic interactions are of universal strength; we take this to imply a universality of the current-current form, or whatever lies behind it. °In this discussion, but not in the number quoted, I'm glossing over the complication that the strangeness-preserving transition is not quite full (universal) strength. We'll encounter "Cabibbo universality" in §3.5.

24

3

The SU(2) L ® U ( 1 ) Y Electroweak Theory

Let us review the essential elements of the SU(2)L ® U(l)y electroweak theory. 37 ' 40,41 The electroweak theory takes three crucial clues from experiment: • The existence of left-handed weak-isospin doublets, ve e

and

;,

;.

(i

The universal strength of the (charged-current) weak interactions; • The idealization that neutrinos are massless. 3.1

A theory of leptons

To save writing, we shall construct the electroweak theory as it applies to a single generation of leptons. In this form, it is neither complete nor consistent: anomaly cancellation requires that a doublet of color-triplet quarks accompany each doublet of color-singlet leptons. However, the needed generalizations are simple enough to make that we need not write them out. To incorporate electromagnetism into a theory of the weak interactions, we add to the SU{2)L family symmetry suggested by the first two experimental clues a U{\)Y weak-hypercharge phase symmetry/ We begin by specifying the fermions: a left-handed weak isospin doublet L = ( - )

L

m

with weak hypercharge YL — — 1, and a right-handed weak isospin singlet R = eR

(41)

with weak hypercharge YR = — 2. The electroweak gauge group, SU{2)L ® U(l)y, implies two sets of gauge fields: a weak isovector 6^, with coupling constant g, and a weak isoscalar p

W e define the weak hypercharge Y through the Gell-Mann-Nishijima connection, Q I3 + hY, to electric charge and (weak) isospin.

25

Ap, with coupling constant g'. Corresponding to these gauge fields are the field-strength tensors Flu = dX, - dX

+ gejkifyl

,

(42)

for the weak-isospin symmetry, and

for the weak-hypercharge symmetry. We may summarize the interactions by the Lagrangian L — -Cgauge "T AWeptons t

V

/

with •£>gauge =

—

~ir p^r

— -ijpvj

,

(45)

and Aeptons = R il" (d, + ijA^Y) + L i 7 " (d,, + i^A^Y

R

(46)

+ i9-f • b^j L.

The SU(2)L ® U(l)y gauge symmetry forbids a mass term for the electron in the matter piece (46). Moreover, the theory we have described contains four massless electroweak gauge bosons, namely _4M, 6* 62 and 6^, whereas Nature has but one: the photon. To give masses to the gauge bosons and constituent fermions, we must hide the electroweak symmetry. The most apt analogy for the hiding of the electroweak gauge symmetry is found in superconductivity. In the Ginzburg-Landau description 42 of the superconducting phase transition, a superconducting material is regarded as a collection of two kinds of charge carriers: normal, resistive carriers, and superconducting, resistanceless carriers. In the absence of a magnetic field, the free energy of the superconductor is related to the free energy in the normal state through ( ^ s u p e r (UJ — ^ n o r m a l

(0) + a H 2 + / ? k / > | 4

,

(47)

where a and /? are phenomenological parameters and \ip\ is an order parameter that measures the density of superconducting charge carriers. The parameter /? is non-negative, so that the free energy is bounded from below. Above the critical temperature for the onset of superconductivity, the parameter a is positive and the free energy of the substance is supposed

26

Order Parameter \\t

Order Parameter \|/

Figure 11. Ginzburg-Landau description of the superconducting phase transition.

to be an increasing function of the density of superconducting carriers, as shown in Figure 11(a). The state of minimum energy, the vacuum state, then corresponds to a purely resistive flow, with no superconducting carriers active. Below the critical temperature, the parameter a becomes negative and the free energy is minimized when rj> = rpo = yj—a/p ^ 0, as illustrated in Figure 11(b). This is a nice cartoon description of the superconducting phase transition, but there is more. In an applied magnetic field H, the free energy is Gsuper(-n

) — Gsuper(O) +

H2

1

+ ^ ^ I

ihVip -

(e*/c)Aip\2

(48)

where e* and m* are the charge (—2 units) and effective mass of the superconducting carriers. In a weak, slowly varying field H m 0, when we can approximate ip « ipo and V ^ « 0, the usual variational analysis leads to the equation of motion,

V2I-i?^o|2I=0,

(49)

the wave equation of a massive photon. In other words, the photon acquires a mass within the superconductor. This is the origin of the Meissner effect, the exclusion of a magnetic field from a superconductor. More to the point for our purposes, it shows how a symmetry-hiding phase transition can lead to a massive gauge boson. To give masses to the intermediate bosons of the weak interaction, we take advantage of a relativistic generalization of the Ginzburg-Landau phase

27

transition known as the Higgs mechanism. 43 We introduce a complex doublet of scalar fields

with weak hypercharge Y$ — + 1 . Next, we add to the Lagrangian new (gaugeinvariant) terms for the interaction and propagation of the scalars, ^scalar = {V)\V^

- Vtfj),

(51)

+ i9-r • b^ ,

(52)

V() + \M(4>^)2-

(53)

where the gauge-covariant derivative is Vll = dll+ i^AMY and the potential interaction has the form

We are also free to add a Yukawa interaction between the scalar fields and the leptons, £ Yukawa = -Ce [R(^L) + (L»R] .

(54)

We then arrange their self-interactions so that the vacuum state corresponds to a broken-symmetry solution. The electroweak symmetry is spontaneously broken if the parameter /i 2 < 0. The minimum energy, or vacuum state, may then be chosen to correspond to the vacuum expectation value

Wo = (

^ ,

(55)

where v = \J—n2/ |A|. Let us verify that the vacuum (55) indeed breaks the gauge symmetry. The vacuum state (^)o is invariant under a symmetry operation exp (iaQ) corresponding to the generator Q provided that exp(iaQ)()o, is., if G{)o = 0. We easily compute that ^ ) o = ( : : ) ( ^ ) = ( ^ ) ^

—

^=(!7)U)-(-^)^broken! M*)o = (1°)

(JK) 0-lJ\v/y/2j

=L,/./o^0

\-v/y/2/

Y()0 = Y^)0 = +l(0)o = (J^\

#0

broken! broken!

(56)

28

However, if we examine the effect of the electric charge operator Q on the (electrically neutral) vacuum state, we find that

QWo

= i ( r 3 + Y)Wo

=

(oo) (v/U)

= ! ( \

=

{I)

+ 1

y

0

° _ i ) w °

unbroken!

(57)

The original four generators are all broken, but electric charge is not. It appears that we have accomplished our goal of breaking SU(2)L ® U{\)y —> C^(l)em- We expect the photon to remain massless, and expect the gauge bosons that correspond to the generators T\, T^, and K = \{T3 — Y) to acquire masses. As a result of spontaneous symmetry breaking, the weak bosons acquire masses, as auxiliary scalars assume the role of the third (longitudinal) degrees of freedom of what had been massless gauge bosons. Specifically, the mediator of the charged-current weak interaction, W± = (b\ =Fi&2)/v2, acquires a mass characterized by MW

= Y

•

( 58 )

With the definition g' = g tan Qw, where 8\y is the weak mixing angle, the mediator of the neutral-current weak interaction, Z — 63 cos 6\y — A sin 9\y, acquires a mass characterized by Mf = M^, / cos2 8\y • After spontaneous symmetry breaking, there remains an unbroken U(l)em phase symmetry, so that electromagnetism, a vector interaction, is mediated by a massless photon, A = A cos 9w + b3 sin 6\y, coupled to the electric charge e = gg'/\/g2 + g'2. As a vestige of the spontaneous breaking of the symmetry, there remains a massive, spin-zero particle, the Higgs boson. The mass of the Higgs scalar is given symbolically as Mjj = — 2fi2 > 0, but we have no prediction for its value. Though what we take to be the work of the Higgs boson is all around us, the Higgs particle itself has not yet been observed. The fermions (the electron in our abbreviated treatment) acquire masses as well; these are determined not only by the scale of electroweak symmetry breaking, v, but also by their Yukawa interactions with the scalars. The mass of the electron is set by the dimensionless coupling constant £e = mey/2/v, which is—so far as we now know—arbitrary.

29 3.2

The W boson

The interactions of the W-boson with the leptons are given by Av-iep = ^ | [*e7"(l -7s)eW+

+ e 7 " ( l -fs)veW-]

.etc.,

(59)

so the Feynman rule for the veeW vertex is

2y/2

The W-boson propagator is ^ \ / v / \ / \ .

7A(1-75J

' ^

=

k

^lMw

Let us compute the cross section for inverse muon decay in the new theory. We find ff4me£„ <7(^e -+ ^ e ) = —-—4

[l-(ml-m2e)/2meE„]2

,

(60)

which coincides with the four-fermion result (33) at low energies, provided we identify 94

16M&

= 2G2F ,

(61)

which implies that 1

r.?. \ 2

6 =»J

•

With the aid of (58) for the VF-boson mass, we determine the numerical value, _! v = (GFV^)

2

« 246 GeV

.

(63)

The high-energy limit of the cross section (60) is ,. hm ^

e

g4 GFM%, = ^rjr- = ~ V ^

, -

x

64

30

independent of energy. The benign high-energy behavior means that partialwave unitarity is now respected for

s < M l exp w

/

TTV2

1

\GFMW

(65)

an immense improvement over the four-fermion theory. Let us now investigate the properties of the jy-boson in terms of its mass, Mw • Consider first the leptonic disintegration of the W~, with decay kinematics specified thus: , . e(p)

/ Mw Mw sin 9 „ Mw cos 9 *"« (^-2-1 2 '°' 2

, ,

/ Mw

Mi)

Mw sin 9

?«(-s-;

n

Mw cos 9

o—>°>

The Feynman amplitude for the decay is l

M = -i(

F Wa /-

\ 2 ) u(e,p)ffi(l-f5)v(i/,q)sl"

,

(66)

where e^ = (0; e) is the polarization vector of the VF-boson in its rest frame. The square of the amplitude is

\M? =

GFM w ^

V2

-tr[/(l-75)«r(l + 7 5 ) ^

8GFMW

~

V2

(67)

[e • q e* • p - e • e* q • p + s • p e* • q + ie^paS^q1's*ppa\

The decay rate is independent of the W polarization, so let us look first at the case of longitudinal polarization eM = (0;0,0,1) = e*M, to eliminate the last term. For this case, we find • A .,2

4GFMW

.

2a

\M\ = —-j=—saSO ,

(68)

so the differential decay rate is dTo _\M\2 S12 d£l 64TT2 MW '

(69)

.

31

where S12 = \f[M^

- (ra e + mv)2}[M'^ - (m e - m„) 2 ] = M^, so that dT0

,„ =

GFM^r3

-

(70)

= sin f ,

and GFM^

r ( W -> ez/)

(71)

6TT\/2

For the other helicities, e ^ = (0; —1, ^ i , 0)/V^, arithmetic that is only slightly more tedious leads us to

f[T±i

=

G£M|

2

(72)

The extinctions at cos0 = ± 1 are, as we have come to expect, consequences of angular momentum conservation:

W

it -

it ,. [9 = 0) forbidden

,, . (0 = 7r) allowed

ve e~ The situation is reversed for the decay of W+ —>• e + i v Overall, the e + follows the polarization direction of W+, while the e~ avoids the polarization direction of W~. This charge asymmetry was important for establishing the discovery of the VF-boson in pp (qq) collisions. 3.3

Neutral Currents

The interactions of the Z-boson with leptons are given by &Z-v

=

4 cos 9\y

^ ( l - T s ) ^

(73)

and C Z-e

=

-9 -e[L ^(l- )+R ^(l+ )}eZ e l5 e l5 l 4 cos &w

\i 1

(74)

where the chiral couplings are Le = 2 sin2 9w — 1 = Ixw + T3 , ; 22 Re = 2 " 9w • ! fsin

(75)

32

By analogy with the calculation of the W-boson total width (71), we easily compute that GFMl I (Z —> VV) =

^r

' 12lTy/2 2 2 (76) T(Z -> e+e-) = T(Z -»• i/P) [L + i? ] . The neutral weak current mediates a reaction that did not arise in the V — A theory, i/^e —>• i/^e, which proceeds entirely by Z-boson exchange:

e

This was, in fact, the reaction in which the first evidence for the weak neutral current was seen by the Gargamelle collaboration in 1973. 44 It's an easy exercise to compute all the cross sections for neutrino-electron elastic scattering. We find

,( v -, v ) = ^

a(vee -» vee) =

[

(

3^L

9ee) = Gr™°E"

^

+

^/3] ,

[{Le + 2f

+ i^/3] ,

[(Le + 2) 2 /3 + R2e] •

(77)

By measuring all the cross sections, one may undertake a "modelindependent" determination'' of the chiral couplings Le and Re, or the traditional vector and axial-vector couplings v and a, which are related through a = \{Le-Re) Le = v + a

v=

\{Le-Re)

Re = v — a

By inspecting (77), you can see that even after measuring all four cross sections, there remains a two-fold ambiguity: the same cross sections result ' I t is model-independent within the framework of vector and axial-vector couplings only, so in the context of gauge theories.

33

if we interchange Re <->• — Re, or, equivalently, v <-> a. The ambiguity is resolved by measuring the forward-backward asymmetry in a reaction like e+e~ —> n+n~ at energies well below the Z° mass. The asymmetry is proportional to (Le — i? e )(I/ /J — Rfi), or to aea^, and so resolves the sign ambiguity for R or the v-a ambiguity. 3.4

Electroweak interactions of quarks

To extend our theory to include the electroweak interactions of quarks, we observe that each generation consists of a left-handed doublet

h

Q Y =

2(Q-h) (79)

+ and two right-handed singlets, h

Q Y =

Ru = uR

0 +|

Rd = dR

0 —g

2{Q~I3) +|

(80)

—3

Proceeding as before, we find the Lagrangian terms for the VF-quark chargedcurrent interaction, iV-quark = ^

[ « e 7 " ( l ~ fs)dW+

+ d7"(l -

l5)u

W~]

,

(81)

which is identical in form to the leptonic charged-current interaction (59). Universality is ensured by the fact that the charged-current interaction is determined by the weak isospin of the fermions, and that both quarks and leptons come in doublets. The neutral-current interaction is also equivalent in form to its leptonic counterpart, (73) and (74). We may write it compactly as £z-quark =

. ~ \ 4 COS fjy

Y— ) qa" ^- ' i=u,d

[Li{l

- 75) +

fl,(l

+ 75)] Qi Z„ ,

(82)

where the chiral couplings are Li = r 3 - 2Qi sin 2 9\y , Ri = -2Qi sin2 9W .

(83)

Again we find a quark-lepton universality in the form—but not the values—of the chiral couplings.

34

3.5

Trouble in Paradise

Until now, we have based our construction on the idealization that the u <-» d transition is of universal strength. The unmixed doublet

does not quite describe our world. We attain a better description by replacing

CD,-*(£),• where de = d cos 6c + s sin 9c ,

(84)

with cos 0c = 0.9736±0.0010. r The change to the "Cabibbo-rotated" doublet perfects the charged-current interaction—at least up to small third-generation effects that we could easily incorporate—but leads to serious trouble in the neutral-current sector, for which the interaction now becomes £ z - q u a r k = ~.

T~

Z> { « 7 M [ - M * ~ 75) + - f t u ( l + J5)] U

4 COS U\v

+ J 7 " [Ld(l - 75) + Rd(l + 75)] d cos2 9C +sf" [Ld(l - 75) + Rd{l + 75)] s sin2 9C +d^ [Ld(l - 75) + Rd{l + 75)] * sin 9C cos 9C + s1'i[Ld(l-l5) + Rd(l+-f5)]d sin 9C cos9 C } , (85) Until the discovery and systematic study of the weak neutral current, culminating in the heroic measurements made at LEP and the SLC, there was not enough knowledge to challenge the first three terms. The last two strangenesschanging terms were known to be poisonous, because many of the early experimental searches for neutral currents were fruitless searches for precisely this sort of interaction. Strangeness-changing neutral-current interactions are not seen at an impressively low level.* Only recently has Brookhaven Experiment 787 47 detected a single candidate for the decay K+ —> ir+i/i>, r T h e arbitrary Yukawa couplings that give masses to the quarks can easily be chosen to yield this result. s F o r more on rare kaon decays, see the TASI 2000 lectures by Tony Barker 4 and Gerhard Buchalla. 4 6

35

and inferred a branching ratio B{K+ -» TY+I/I>) = 1.5;f;| x 1 0 - 1 0 . The good agreement between the standard-model prediction, B(KL —> ^+^~) — 0.8±0.3 x 1 0 - 1 0 (through the process Ki —> 77 —>• p + / i _ ) , and experiment 48 leaves little room for a strangeness-changing neutral-current contribution:

that is easily normalized to the normal charged-current leptonic decay of the K+: .; ><+

s

The resolution to this fatal problem was put forward by Glashow, Uiopoulos, and Maiani. 49 Expand the model of quarks to include two left-handed doublets,

(86)

where

sg = s cos 6c — d sin 9c

(87)

plus the corresponding right-handed singlets, e#, fiR, UR, <1R, CR, and SR. This required the introduction of the charmed quark, c, which had not yet been observed. By the addition of the second quark generation, the flavorchanging cross terms vanish in the Z-quark interaction, and we are left with:

36

4 cos 9w • 7 A [ ( 1 - 7 6 ) £ , -

+ ( 1 + 75)J*,-]

which is flavor diagonal! The generalization to n quark doublets is straightforward. charged-current interaction be

£^_ quark = ^= [*7"(1 -

75)0*

W+ + h.c]

,

Let the (88)

where the composite quark spinor is c

(89)

f = d s

\'-J and the flavor structure is contained in O

'OU 00

(90)

where U is the unitary quark-mixing matrix. The weak-isospin contribution to the neutral-current interaction has the form /•ISO

•'-'Z-quark

-a

4 cos 6w

*7"(l-75)[0,0t]¥

(91)

Since the commutator

[o,o^} =

I 0 0-/

(92)

the neutral-current interaction is flavor diagonal, and the weak-isospin piece is, as expected, proportional to T3. In general, the nxn quark-mixing matrix U can be parametrized in terms of n(n — l ) / 2 real mixing angles and (n — l)(ra — 2)/2 complex phases, after

37

exhausting the freedom to redefine the phases of quark fields. The 3 x 3 case, of three mixing angles and one phase, often called the Cabibbo-KobayashiMaskawa matrix, presaged the discovery of the third generation of quarks and leptons. 50 4 4-1

P r e c i s i o n Tests of t h e Electroweak T h e o r y 5 1 Measurements on the Z° pole

In its simplest form, with the electroweak gauge symmetry broken by the Higgs mechanism, the SU(2)L®U(1)Y theory has scored many qualitative successes: the prediction of neutral-current interactions, the necessity of charm, the prediction of the existence and properties of the weak bosons W* and Z°. Over the past ten years, in great measure due to the beautiful experiments carried out at the Z factories at CERN and SLAC, precision measurements have tested the electroweak theory as a quantum field theory, 52,53 at the one-permille level, as indicated in Table 2. A classic achievement of the Z factories is the determination of the number of light neutrino species. If we define the invisible width of the Z° as 1 invisible — 1 Z

1 hadronic

«1 leptonic j

V*")

then we can compute the number of light neutrino species as NV = r i n v i s i b l e / r S M ( z -». Vi0i).

(94)

A typical current value is Nu = 2.984 ±0.008, in excellent agreement with the observation of light ve, i/,,, and vr. As an example of the insights precision measurements have brought us (one that mightily impressed the Royal Swedish Academy of Sciences in 1999),

Table 2. Precision measurements at the Z pole. (For sources of the data, see the Review of Particle Physics, Ref. 9 and the LEP Electroweak Working Group web server, http://www.cern.ch/LEPEWWG/.)

Mz Tz hadronic 1 hadronic 1 leptonic t invisible

91188±2.2MeV/c 2 2495.2 ± 2 . 6 MeV 41.541 ± 0.037 nb 1743.8 ± 2 . 2 MeV 84.057 ± 0.088 MeV 499.4 ± 1 . 7 MeV

38

1988

1990

1992 1994 Year

1996

1998

Figure 12. Indirect determinations of the top-quark mass from fits to electroweak observables (open circles) and 95% confidence-level lower bounds on the top-quark mass inferred from direct searches in e+e — annihilations (solid line) and in pp collisions, assuming that standard decay modes dominate (broken line). An indirect lower bound, derived from the W-boson width inferred from pp —> (W or Z) + anything, is shown as the dot-dashed line. Direct measurements of rtit by the CDF (triangles) and D 0 (inverted triangles) Collaborations are shown at the time of initial evidence, discovery claim, and at the conclusion of Run 1. The world average from direct observations is shown as the crossed box. For sources of data, see Ref. 9 . Inset: Electroweak theory predictions for the width of the Z° boson as a function of the top-quark mass, compared with the width measured in LEP experiments. (From Ref. u . )

I show in Figure 12 the time evolution of the top-quark mass favored by simultaneous fits to m a n y electroweak observables. Higher-order processes involving virtual top quarks are an i m p o r t a n t element in q u a n t u m corrections to the predictions the electroweak theory makes for m a n y observables. A case in point is the t o t a l decay rate, or width, of the Z° boson: the comparison of experiment and theory shown in the inset to Figure 12 favors a top mass in the neighborhood of 180 GeV/c 2 . T h e comparison between the electroweak theory and a considerable universe of d a t a is shown in Figure 1 3 , 5 4 where the pull, or difference between the global fit and measured value in units of s t a n d a r d deviations, is shown for some twenty observables. T h e distribution of pulls for this fit, due to the L E P Electroweak Working Group, is not noticeably different from a norm a l distribution, and only a couple of observables differ from the fit by as much as about two s t a n d a r d deviations. This is the case for any of the recent fits. From fits of the kind represented here, we learn t h a t the standard-model

39

Osaka 2000 Measurement m z [GeV] r z [GeV] °hadr t n b l R, A0'1 A

fb

Ae A,

sin*C m w [GeV] Rb Re b

< A£C Ab Ac

sinXT sin 2 8 m m w [GeV] m, [GeV]

AaSd(n»z)

91.1875 + 0.0021 2.4952 + 0.0023 41.540 + 0.037 20.767 + 0.025 0.01714 + 0.00095 0.1498 + 0.0048 0.1439 + 0.0042 0.2321 +0.0010 80.427 ± 0.046 0.2165310.00069 0.1709 ±0.0034 0.0990 ± 0.0020 0.0689 ± 0.0035 0.922 ± 0.023 0.631 + 0.026 0.23098 + 0.00026 0.2255 ± 0.0021 80.452 ± 0.062 174.3 ±5.1 0.02804 ± 0.00065

Pull .05 -.42 1.62 1.07 .75 .38 -.97 .70 .55 1.09 -.40 -2.38 -1.51 -.55 -1.43 -1.61 1.20 .81 -.01 -.29

Pull -3-2-1 0 1 2

3

• ——

•

• •

«

i

-3-2-10123

Figure 13. Precision electroweak measurements and the pulls they exert on a global fit to the standard model, from Ref. s 4 .

interpretation of the data favors a light Higgs boson. 55 We will revisit this conclusion in §5.4. The beautiful agreement between the electroweak theory and a vast array of data from neutrino interactions, hadron collisions, and electron-positron annihilations at the Z° pole and beyond means that electroweak studies have become a modern arena in which we can look for new physics "in the sixth place of decimals."

40

4-2

Parity violation in atomic physics

Later in TASI00, you will hear from Carl Wieman 56 about the beautiful tabletop experiments that probe the structure of the weak neutral current. I want to spend a few minutes laying the foundation for the interpretation of those measurements.* At very low momentum transfers, as in atomic physics applications, the nucleon appears elementary, and so we can write an effective Lagrangian for nucleon /? decay in the limit of zero momentum transfer as £p = - ~j=«7A(1 - lh)vpl

(1 - 9Alz)n ,

(95)

where gA ~ 1.26 is the axial charge of the nucleon. Accordingly, the neutralcurrent interactions are £ep - —7=e~i\{l - 4xw - 7s)e P7 (1 - 4xw - 7s)p , ^

e n

=

ITlz^7A(1

- 4xw

- 7 5 )e h-y (1 - j5)n

,

(96)

where xw = sin 2 9\v • For our purposes, we may regard the nucleus as a noninteracting collection of Z protons and ,/V neutrons. We perform a nonrelativistic reduction of the interaction; nucleons contribute coherently to the axial-electron-vectornucleon (ACVN) coupling, so the dominant parity-violating contribution to the eN amplitude is M

W P-*l p v _= -Tl=rQ ePN{v)lse 2V2

,

(97)

where PN{Y) is the nucleon density at the electron coordinate r, and Qw = Z(\ — Axw) — N is the weak charge. Bennett and Wieman (Boulder) have reported a new determination of the weak charge of Cesium by measuring the transition polarizability for the 6S-7S transition. 57 The new value, Q^(Cs) = -72.06 ± 0.28 (expt) ± 0.34 (theory),

(98)

represents a sevenfold improvement in the experimental error and a significant reduction in the theoretical uncertainty. It lies about 2.5 standard deviations above the prediction of the standard model. We are left with the traditional ' T h e book by Commins and Bucksbaum, Ref. world" problems at low Q2.

22

, is an excellent reference for many "real-

41

situation in which elegant measurements of parity nonconservation in atoms are on the edge of incompatibility with the standard model. A number of authors 58 have noted that the discrepancy in the weak charge Qw and a 2-0- anomaly" in the total width of the Z° can be reduced by introducing a Z' boson with a mass of about 800 GeV/c2. The additional neutral gauge boson resembles the Zx familiar from unified theories based on the group E6-59 4-3

The vacuum energy problem

I want to spend a moment to revisit a longstanding, but usually unspoken, challenge to the completeness of the electroweak theory as we have defined it: the vacuum energy problem. 60 I do so not only for its intrinsic interest, but also to raise the question, "Which problems of completeness and consistency do we worry about at a given moment?" It is perfectly acceptable science— indeed, it is often essential—to put certain problems aside, in the expectation that we will return to them at the right moment. What is important is never to forget that the problems are there, even if we do not allow them to paralyze us. For the usual Higgs potential, V(
V«*V>o) = ££ = - ^ 1 < 0.

(99)

Identifying Mfj = —2^i2, we see that the Higgs potential contributes a fieldindependent constant term, QH = - f - .

(100)

I have chosen the notation QJJ because the constant term in the Lagrangian plays the role of a vacuum energy density. When we consider gravitation, adding a vacuum energy density £>vac is equivalent to adding a cosmological constant term to Einstein's equation. Although recent observations1' raise the intriguing possibility that the cosmological constant may be different from zero, the essential observational fact is that the vacuum energy density must "In Erler & Langacker's fit, for example, the number of light neutrino species inferred from the invisible width of the Z° is N„ = 2.985 ± 0.008 = 3 - la. "For a cogent summary of current knowledge of the cosmological parameters, including evidence for a cosmological constant, see Ref. 6 1 .

42 be very tiny indeed,™ Pvac £ 1 ( T 4 6 GeV 4 .

(101) 5

Therein lies the puzzle: if we take v = ( G F \ / 2 ) ~ W 246 GeV and insert the current experimental lower b o u n d 6 2 MJJ ^ 113.5 GeV/c 2 into (100), we find t h a t the contribution of the Higgs field to the vacuum energy density is ejf£108GeV4,

(102)

some 54 orders of m a g n i t u d e larger t h a n the upper bound inferred from the cosmological constant. W h a t are we to make of this mismatch? T h e fact t h a t QH 3> £>vac means t h a t the smallness of the cosmological constant needs to be explained. In a unified theory of the strong, weak, and electromagnetic interactions, other (heavy!) Higgs fields have nonzero vacuum expectation values t h a t m a y give rise to still greater mismatches. At a fundamental level, we can therefore conclude t h a t a spontaneously broken gauge theory of the strong, weak, and electromagnetic interactions—or merely of the electroweak interactions—cannot be complete. Either we must find a separate principle to zero the vacuum energy density of the Higgs field, or we m a y suppose t h a t a proper q u a n t u m theory of gravity, in combination with the other interactions, will resolve the puzzle of the cosmological constant. T h e vacuum energy problem must be an i m p o r t a n t clue. B u t t o w h a t ? 5 5.1

T h e Higgs Boson Why the Higgs boson must

exist

How can we be sure t h a t a Higgs boson, or something very like it, will be found? One p a t h to the theoretical discovery of the Higgs boson involves its role in the cancellation of high-energy divergences. An illuminating example is provided by the reaction e+e~

-> W+W~

,

(103)

which is described in lowest order by the four Feynman graphs in Figure 14. T h e contributions of the direct-channel 7- and Z°-exchange diagrams of Figs. 14(a) and (b) cancel the leading divergence in the J = 1 partialwave amplitude of the neutrino-exchange diagram in Figure 14(c). This is the famous "gauge cancellation" observed in experiments at L E P 2 and the Tevatron. T h e L E P measurements in Figure 15 agree well with the predictions ""For a useful summary of gravitational theory, see the essay by T. d'Amour in §14 of the 2000 Review of Particle Physics, Ref. 9 .

43

v\r

(a)

(b)

w-

(c)

Figure 14. Lowest-order contributions to the e"*" e

—}W+W

scattering amplitude.

of electroweak-theory Monte Carlo generators, which predict a benign highenergy behavior. If the Z-exchange contribution is omitted (dashed line) or if both the 7- and Z-exchange contributions are omitted (dot-dashed line), the calculated cross section grows unacceptably with energy—and disagrees with the measurements. The gauge cancellation in the J = 1 partial-wave amplitude is thus observed. However, this is not the end of the high-energy story: the J = 0 partialwave amplitude, which exists in this case because the electrons are massive and may therefore be found in the "wrong" helicity state, grows as s 1 / 2 for the production of longitudinally polarized gauge bosons. The resulting divergence is precisely cancelled by the Higgs boson graph of Figure 14(d). If the Higgs boson did not exist, something else would have to play this role. From the point of view of S'-matrix analysis, the Higgs-electron-electron coupling must

44

LEP

Preliminary

20 -

10

Racoon WW / YFS WW 1.14 Gentle 2.1 (+0.7%) no ZWW vertex only v exchange

160

170

180

190

200

210

E, m [GeV] Figure 15. Cross section for the reaction e+e — —• W~^W~ measured by the four LEP experiments, together with the full electroweak-theory simulation and the cross sections that would result from ^-exchange alone and from (u + -y)-exchange (from http://www.cern.ch/LEPEWWG/).

be proportional to the electron mass, because "wrong-helicity" amplitudes are always proportional to the fermion mass. Let us underline this result. If the gauge symmetry were unbroken, there would be no Higgs boson, no longitudinal gauge bosons, and no extreme divergence difficulties. But there would be no viable low-energy phenomenology of the weak interactions. The most severe divergences of individual diagrams are eliminated by the gauge structure of the couplings among gauge bosons and leptons. A lesser, but still potentially fatal, divergence arises because the electron has acquired mass—because of the Higgs mechanism. Spontaneous symmetry breaking provides its own cure by supplying a Higgs boson to remove the last divergence. A similar interplay and compensation must exist in

45

any satisfactory theory. 5.2

Bounds on MH

The Standard Model does not give a precise prediction for the mass of the Higgs boson. We can, however, use arguments of self-consistency to place plausible lower and upper bounds on the mass of the Higgs particle in the minimal model. Unitarity arguments 63 lead to a conditional upper bound on the Higgs boson mass. It is straightforward to compute the amplitudes M. for gauge boson scattering at high energies, and to make a partial-wave decomposition, according to M(s,t)

=

16TT^(2J+

(104)

l)aj{s)Pj(cos6)

Most channels "decouple," in the sense that partial-wave amplitudes are small at all energies (except very near the particle poles, or at exponentially large energies), for any value of the Higgs boson mass MH- Four channels are interesting: ZQLZ°L/V2HH/V2HZl

W+W£

(105)

,

where the subscript L denotes the longitudinal polarization states, and the factors of \/2 account for identical particle statistics. For these, the s-wave amplitudes are all asymptotically constant (i.e., well-behaved) and proportional to GpMH. In the high-energy limit, x

lirn (a 0 )

-GFMH 4TTV2

1 \/y/%l/y/% 1/V8 3/4 1/4 I / A / 8 1/4 3/4 0 0 0

0 0 0 1/2.

(106)

Requiring that the largest eigenvalue respect the partial-wave unitarity condition |ao| < 1 yields 1/2

MH <

8TTV2

3GF

= 1 TeV/c2

(107)

as a condition for perturbative unitarity. ^It is convenient to calculate these amplitudes by means of the Goldstone-boson equivalence theorem, 6 4 which reduces the dynamics of longitudinally polarized gauge bosons to a scalar field theory with interaction Lagrangian given by £; n t = — \vh(2w~^~w~~ + z 2 + h2) — (\/4)(2w+w+ z2 + h2)2, with 1/v2 = GF\p2 and A = GFMJj/\/2.

46

If the bound is respected, weak interactions remain weak at all energies, and perturbation theory is everywhere reliable. If the bound is violated, perturbation theory breaks down, and weak interactions among W±, Z, and H become strong on the 1-TeV scale. This means that the features of strong interactions at GeV energies will come to characterize electroweak gauge boson interactions at TeV energies. We interpret this to mean that new phenomena are to be found in the electroweak interactions at energies not much larger than 1 TeV. It is worthwhile to note in passing that the threshold behavior of the partial-wave amplitudes for gauge-boson scattering follows generally from chiral symmetry. 65 The partial-wave amplitudes ajj of definite isospin / and angular momentum J are given by aoo K

GFS/&TT^/2

attractive,

a n w GFS/48TTV2

attractive,

(108)

c.20 ^ —Gfs/167r\/2 repulsive. The electroweak theory itself provides another reason to expect that discoveries will not end with the Higgs boson. Scalar field theories make sense on all energy scales only if they are noninteracting, or "trivial." 66 The vacuum of quantum field theory is a dielectric medium that screens charge. Accordingly, the effective charge is a function of the distance or, equivalently, of the energy scale. This is the famous phenomenon of the running coupling constant. In \4 theory (compare the interaction term in the Higgs potential), it is easy to calculate the variation of the coupling constant A in perturbation theory by summing bubble graphs like this one:

(109)

The coupling constant \(fi) on a physical scale \i is related to the coupling constant on a higher scale A by 1

X(n)

1

A(A)

+ - L l2o g ( A / A I ) .

(110)

2TT

This perturbation-theory result is reliable only when A is small, but lattice field theory allows us to treat the strong-coupling regime. In order for the Higgs potential to be stable (i.e., for the energy of the vacuum state not to race off to —oo), A(A) must not be negative. Therefore

47

we can rewrite (110) as an inequality, (111) This gives us an upper bound, A(/x)<2jr 2 /31og(A//i) ,

(112)

on the coupling strength at the physical scale p. If we require the theory to make sense to arbitrarily high energies—or short distances—then we must take the limit A —> oo while holding p. fixed at some reasonable physical scale. In this limit, the bound (112) forces X(p) to zero. The scalar field theory has become free field theory; in theorist's jargon, it is trivial. We can rewrite the inequality (112) as a bound on the Higgs-boson mass. Rearranging and exponentiating both sides gives the condition

A exp

^ GS!)) •

(u3

>

Choosing the physical scale as p — MH, and remembering that, before quantum corrections, M2H = 2\{MH)v2 ,

/

,

(114)

1/2

where v = (G F v 2)~ « 246 GeV is the vacuum expectation value of the Higgs field times y/2, we find that A
w

)

.

(115)

For any given Higgs-boson mass, there is a maximum energy scale A* at which the theory ceases to make sense. The description of the Higgs boson as an elementary scalar is at best an effective theory, valid over a finite range of energies. This perturbative analysis breaks down when the Higgs-boson mass approaches 1 TeV/c2 and the interactions become strong. Lattice analyses 67 indicate that, for the theory to describe physics to an accuracy of a few percent up to a few TeV, the mass of the Higgs boson can be no more than about 710 ± 60 GeV/c2. Another way of putting this result is that, if the elementary Higgs boson takes on the largest mass allowed by perturbative unitarity arguments, the electroweak theory will be living on the brink of instability. A lower bound is obtained by computing 68 the first quantum corrections to the classical potential (53). Requiring that (<^>)o ^ 0 be an absolute minimum of the one-loop potential up to a scale A yields the vacuum-stability

48

log 1 0 A[GeV] Figure 16. Bounds on the Higgs-boson mass that follow from requirements that the electroweak theory be consistent up to the energy scale A. The upper bound follows from triviality conditions; the lower bound follows from the requirement that V(v) < V(0). Also shown is the range of masses permitted at the 95% confidence level by precision measurements.

condition Ml > ^ ^ ( 2 M ^ + M« - 4mf)log(A 2 / V 2 ) .

(116)

The upper and lower bounds plotted in Figure 16 are the results of full two-loop calculations. 69 There I have also indicated the upper bound on M # derived from precision electroweak measurements in the framework of the standard electroweak theory. If the Higgs boson is relatively light—which would itself require explanation—then the theory can be self-consistent up to very high energies. If the electroweak theory is to make sense all the way up to a unification scale A* = 10 16 GeV, then the Higgs-boson mass must lie in the interval 134 GeV/c2
49

5.3

Higgs-Boson Properties

Once we assume a value for the Higgs-boson mass, it is a simple matter to compute the rates for Higgs-boson decay into pairs of fermions or weak bosons. 71 For a fermion with color Nc, the partial width is

r(J

'-

>/fl

GFm2,MH

(

r

1

4m2f\3/2

= -i^5-' --( -l^j

•

(m)

which is proportional to MJJ in the limit of large Higgs mass. The partial width for decay into a W+W~ pair is hl - xf'2{A - ix + 3x2) , (118) 327rv 2 Similarly, the partial width for decay into a pair of Z°

r ( # - • W+ W~) = where x = \M^ /Mfj. bosons is

GpM

T{H -+ Z°Z°) = 9l^k{i 647r\/2

_ a ./)i/2( 4 _

Ax'

+

3a .'2) ;

(H9)

where x' = 4 M | / M ^ . The rates for decays into weak-boson pairs are asymptotically proportional to M^ and | M ^ , respectively, the factor | arising from weak isospin. In the final factors of (118) and (119), 2x2 and 2a;'2, respectively, arise from decays into transversely polarized gauge bosons. The dominant decays for large MJJ are into pairs of longitudinally polarized weak bosons. Branching fractions for decay modes that hold promise for the detection of a light Higgs boson are displayed in Figure 17. In addition to the / / and VV modes that arise at tree level, I have included the 77 mode that proceeds through loop diagrams. Though rare, the 77 channel offers an important target for LHC experiments. Figure 18 shows the partial widths for the decay of a Higgs boson into the dominant W+W~ and Z° Z° channels and into ti, for mt = 175 GeV/c2. Whether the ti mode will be useful to confirm the observation of a heavy Higgs boson, or merely drains probability from the ZZ channel favored for a heavy-Higgs search, is a question for detailed detector simulations. Below the W+W~ threshold, the total width of the standard-model Higgs boson is rather small, typically less than 1 GeV. Far above the threshold for decay into gauge-boson pairs, the total width is proportional to M ^ . At masses approaching 1 TeV/c2, the Higgs boson is an ephemeron, with a perturbative width approaching its mass. The Higgs-boson total width is plotted as a function of MJJ in Figure 19.

50

240

2

Higgs Boson Mass [GeV/c ] Figure 17. Branching fractions for the prominent decay modes of a light Higgs boson.

1000

i

I

!

I

I

1

I

_ > CD

a

100 =-

^ W

+

W>^

-

.55 Q.

1 -

<"*tt

10 :

-

!

/ /

200

I

400 M

I

1

600

1

1

800

i

1000

Higgs [ Q e V / c 2 ]

Figure 18. Partial widths for the prominent decay modes of a heavy Higgs boson.

51 1 UU

^

—a

^

iiu

^

i

*-•

4

T3

1

£

y^

I

/

r

/

•

/

CD 0 . 1 j ~

X 0.01

"

I I

W

O)

^ . -=

X

1 1 ii

•—•

.c

^ ^

10

\

>" ® O

" I " " I' " j j

:

J

:

I

r

-a

: -

/

n nm 0

100

200

300

400

500

Higgs Mass [GeV/c2] Figure 19. Higgs-boson total width as a function of mass.

5.4

Clues to the Higgs-boson mass

We have seen in §4.1 that the sensitivity of electroweak observables to the (long unknown) mass of the top quark gave early indications for a very massive top. For example, the quantum corrections to the standard-model predictions (5) for Mw and (6) for Mz arise from different quark loops:

t

z°,

w+ z°

w+ t

t

tb for Myy, and tt (or bb) for Mz- These quantum corrections alter the link (6) between the W- and Z-boson masses, so that

M ^ = M | (1 - sin 2 9W) (1 + Ap)

(120)

52

where Ap a A / ^ u a r k s ) = SGlUt

.

(121)

The strong dependence on m 2 is characteristic, and it accounts for the precision of the top-quark mass estimates derived from electroweak observables. Now that mt is known to about 3% from direct observations at the Tevatron, it becomes profitable to look beyond the quark loops to the next most important quantum corrections, which arise from Higgs-boson effects. The Higgs-boson quantum corrections are typically smaller than the top-quark corrections, and exhibit a more subtle dependence on MH than the m 2 dependence of the top-quark corrections. For the case at hand, A/?(Higgs) = c

. m f^jA

j

(122)

where I have arbitrarily chosen to define the coefficient C at the electroweak scale v. Since Mz has been determined at LEP to 23 ppm, it is interesting to examine (see Figure 20) the dependence of Mw upon mt and MH . Also indicated in Figure 20 are the direct determinations of mt and Mw, and the values inferred from a universe of electroweak observables, both shown as one-standard-deviation regions. The direct and indirect determinations are in reasonable agreement. Both favor a light Higgs boson, within the framework of the standard-model analysis. The Mw-fnt-Mn correlation will be telling over the next few years, since we anticipate that measurements at the Tevatron and LHC will determine mt within 1 or 2 GeV/c2 and improve the uncertainty on Mw to about 15 MeV/c2. As the Tevatron's integrated luminosity grows past 10 fb , CDF and D 0 will begin to explore the region of Higgs-boson masses not excluded by the LEP searches. 72 Soon after that, ATLAS and CMS will carry on the exploration of the Higgs sector at the Large Hadron Collider. By itself, the W-boson mass suggests a preference—always within the standard model—for a light Higgs boson. The indication becomes somewhat stronger when all the electroweak observables are examined. Figure 21 shows how the goodness of the LEP Electroweak Working Group's global fit depends upon MH- Within the standard model, they deduce a 95% CL upper limit, MH & 170 GeV/c2. Since the direct searches at LEP have concluded that MH > 113.5 GeV/c2, excluding much of the favored region, either the Higgs boson is just around the corner, or the standard-model analysis is misleading. Things will soon be popping!

53

OU.O"

i

i

i

i

i

i

i

|

i

i

'

I

'

'

' A

— LEP1.SLD vN Data LEP2, ppDatd 68% CL 80.5-

>

/

CD

£ 80.4•'

E /

80.3-

/ •

on o

m H [GeV| 113, 300 1000

1CJO

150

170

'

Preliminary 190

21

mt [GeV]

Figure 20. Comparison of the indirect measurements of Mw and mt (LEP I+SLD+i/AT data, solid contour) and the direct measurements (Tevatron and LEP II data, dashed contour). Also shown is the standard-model relationship for the masses as a function of the Higgs mass. (From the LEP Electroweak Working Group, Ref. 5 4 .)

6

The electroweak scale and beyond

We h ave seen that the scale of electroweak symmetry breaking, v = ( G F \ / 2 ) ~ 2 « 246 GeV, sets the values of the W- and 2-boson masses. But the electroweak scale is not the only scale of physical interest. It seems certain that we must also consider the Planck scale, derived from the strength of Newton's constant, and it is also probable that we must take account of the S77(3)c SU(2)L U{1)Y unification scale around 1 0 1 5 - 1 6 GeV. There may well be a distinct flavor scale. The existence of other significant energy scales is behind the famous problem of the Higgs scalar mass: how to keep the distant scales from mixing in the face of quantum corrections, or how to stabilize the mass of the Higgs boson on the electroweak scale.

54

Cithco:1! j ' -:rtair • 0.02804I0.00065J 0.0275510.00046

CvJ

If

Excluded

Preliminary

10

10

10

m H [GeV]

from a global fit to precision d a t a vs. the Higgs-boson mass, Figure 21. A x 2 MJJ. The solid line is the result of the fit; the band represents an estimate of the theoretical error due to missing higher order corrections. T h e vertical band shows the 95% CL exclusion limit on MJJ from the direct search at LEP. The dashed curve shows the sensitivity to a change in the evaluation of ct(MV). (From the LEP Electroweak Working Group, Ref. 5 4 .)

6.1

Why is the electroweak scale small?

To this point, we have outlined the electroweak theory, emphasized that the need for a Higgs boson (or substitute) is quite general, and reviewed the properties of the standard-model Higgs boson. By considering a thought experiment, gauge-boson scattering at very high energies, we found a first signal for the importance of the 1-TeV scale. Now, let us explore another path to the 1-TeV scale. The SU(2)i U(1)Y electroweak theory does not explain how the scale of electroweak symmetry breaking is maintained in the presence of quantum corrections. The problem of the scalar sector can be summarized neatly as

55

follows.73 The Higgs potential is

v{4>H) = ii2{H) + \\\(H)2.

(123)

With fi2 chosen to be less than zero, the electroweak symmetry is spontaneously broken down to the U(l) of electromagnetism, as the scalar field acquires a vacuum expectation value that is fixed by the low-energy phenomenology, {4>)o = ^ - / i 2 / 2 | A | = {GFV$y1/2

» 175 GeV .

(124)

Beyond the classical approximation, scalar mass parameters receive quantum corrections from loops that contain particles of spins J = 1,1/2, and 0:

m'ip^m2^

^ J=1

+

O

+

J=1/2

Q J=0

(125) The loop integrals are potentially divergent. Symbolically, we may summarize the content of (125) as m2{p2) = m 2 (A 2 ) + Cg2 /

dk2 + • • • ,

(126)

where A defines a reference scale at which the value of m2 is known, g is the coupling constant of the theory, and the coefficient C is calculable in any particular theory. Instead of dealing with the relationship between observables and parameters of the Lagrangian, we choose to describe the variation of an observable with the momentum scale. In order for the mass shifts induced by radiative corrections to remain under control (i.e., not to greatly exceed the value measured on the laboratory scale), either A must be small, so the range of integration is not enormous, or new physics must intervene to cut off the integral. If the fundamental interactions are described by an SU(i)c ® SU(2)L (§) U(1)Y gauge symmetry, i.e., by quantum chromodynamics and the electroweak theory, then the natural reference scale is the Planck mass, y ^It is because Mpi a n c i, is so large (or because Gjjewton ' s s o small) that we normally consider gravitation irrelevant for particle physics. The graviton-quark-antiquark coupling is generically ~ E/Mp^nck, so it is easy to make a dimensional estimate of the branching fraction for a gravitationally mediated rare kaon decay: B(K]J —• ir°G) ~ (Aff 4 -/Mpi anc i;) 2 ~ 1 0 ~ 3 8 , which is truly negligible!

56

( h \ A ~ Mpi anck = ( - — ^ — )

1/ 2

'

w 1.22 x 1019 GeV .

(127)

V ^Newton/

In a unified theory of the strong, weak, and electromagnetic interactions, the natural scale is the unification scale, A ~ Mu « 10 15 -10 16 GeV .

(128)

Both estimates are very large compared to the scale of electroweak symmetry breaking (124). We are therefore assured that new physics must intervene at an energy of approximately 1 TeV, in order that the shifts in m2 not be much larger than (124). Only a few distinct scenarios for controlling the contribution of the integral in (126) can be envisaged. The supersymmetric solution is especially elegant. 74,75,76,77 Exploiting the fact that fermion loops contribute with an overall minus sign (because of Fermi statistics), supersymmetry balances the contributions of fermion and boson loops. In the limit of unbroken supersymmetry, in which the masses of bosons are degenerate with those of their fermion counterparts, the cancellation is exact:

J2 •

Cifdk2 = 0.

(129)

fermions -[-bosons

If the supersymmetry is broken (as it must be in our world), the contribution of the integrals may still be acceptably small if the fermion-boson mass splittings A M are not too large. The condition that g2AM2 be "small enough" leads to the requirement that superpartner masses be less than about 1 TeV/c2. A second solution to the problem of the enormous range of integration in (126) is offered by theories of dynamical symmetry breaking such as technicolor. 78,79,80 In technicolor models, the Higgs boson is composite, and new physics arises on the scale of its binding, ATC — 0 ( 1 TeV). Thus the effective range of integration is cut off, and mass shifts are under control. A third possibility is that the gauge sector becomes strongly interacting. This would give rise to WW resonances, multiple production of gauge bosons, and other new phenomena at energies of 1 TeV or so. It is likely that a scalar bound state—a quasi-Higgs boson—would emerge with a mass less than about 1 TeV/c 2 . 81 We cannot avoid the conclusion that some new physics must occur on the 1-TeV scale/ ^Since the superconducting phase transition informs our understanding of the Higgs mech-

57

6.2

Why is the Planck scale so large?

The conventional approach to new physics has been to extend the standard model to understand why the electroweak scale (and the mass of the Higgs boson) is so much smaller than the Planck scale. A novel approach that has been developed over the past two years is instead to change gravity to understand why the Planck scale is so much greater than the electroweak scale. 86 Now, experiment tells us that gravitation closely follows the Newtonian force law down to distances on the order of 1 mm. Let us parameterize deviations from a 1/r gravitational potential in terms of a relative strength £G and a range AQ, so that V(r) = -JdnJ

G rfr2

NewtonPMp(r2)

{1 + £G e x p ( _ r i 2 / A Q ) ]

>

(130)

where p(r,) is the mass density of object i and T\I is the separation between bodies 1 and 2. Elegant experiments that study details of Casimir and Van der Waals forces imply bounds on anomalous gravitational interactions, as shown in Figure 22. Below about a millimeter, the constraints on deviations from Newton's inverse-square force law deteriorate rapidly, so nothing prevents us from considering changes to gravity even on a small but macroscopic scale. For its internal consistency, string theory requires an additional six or seven space dimensions, beyond the 3 + 1 dimensions of everyday experience. 93 Until recently it has been presumed that the extra dimensions must be compactified on the Planck scale, with a compactification radius /^unobserved * 1/Mpianck « 1 . 6 x l 0 ~ 3 5 m . The new wrinkle is to consider that the SU(3)C ® SU(2)L <8> U(\)Y standard-model gauge fields, plus needed extensions, reside on 3 + 1-dimensional branes, not in the extra dimensions, but that gravity can propagate into the extra dimensions. How does this hypothesis change the picture? The dimensional analysis (Gauss's law, if you like) that relates Newton's constant to the Planck scale changes. If gravity propagates in n extra dimensions with radius R, then GNewton ~ M p , ^ ~ M^^^R"1

,

(131)

where M* is gravity's true scale. Notice that if we boldly take M* to be as small as 1 TeV/c2, then the radius of the extra dimensions is required to be anism for electroweak symmetry breaking, it may be useful to look to other collective phenomena for inspiration. Although the implied gauge-boson masses are unrealistically small, chiral symmetry breaking in QCD can induce a dynamical breaking of electroweak symmetry. 8 2 (This is the prototype for technicolor models.) Is it possible t h a t other interesting phases of QCD—color superconductivity, 8 3 ' 8 4 , 8 5 for example—might hold lessons for electroweak symmetry breaking under normal or unusual conditions?

58 1

e

10

1

o

w

sz D) •1—•

c
• * - •

U)

1

1—I—I

I I I I

n

1

r

-Lamoreaux Moscow

10*

moduli''

itf

dilaton

2 extra /dimensions

tiv

CD

cc

m°

CD

DC 10"2

vacuum energy limits

Ebt-wash <s

10"4

J_

_l

10"*

Irvine

axion I

T---1

'

I '

j

i

i_

10"3

Range XG [meters] Figure 22. 95% confidence level upper limits (heavy line labeled ESt-wash) on the strength £Q (relative to gravity) versus the range \Q of a new long-range force characterized by (130), from Ref, 8 7 . The region excluded by earlier work 8 8 , 8 9 ' 9 0 lies above the heavy lines labeled Irvine, Moscow and Lamoreaux, respectively. The theoretical predictions are adapted from Ref. 9 1 , except for the dilaton prediction, from Ref. 9 2 .

smaller than about 1 mm, for n > 2. If we use the four-dimensional force law to extrapolate the strength of gravity from low energies to high, we find that gravity becomes as strong as the other forces on the Planck scale, as shown by the dashed line in Figure 23. If the force law changes at an energy 1/R, as the large-extra-dimensions scenario suggests, then the forces are unified at an energy M*, as shown by the solid line in Figure 23. What we know as the Planck scale is then a mirage that results from a false extrapolation: treating gravity as four-dimensional down to arbitrarily small distances, when in fact—or at least in this particular fiction—gravity propagates in 3+n spatial dimensions. The Planck mass is an artifact, given by Mpi anck = M*(M*R)n/2. Although the idea that extra dimensions are just around the corner—

59

to CD O

c

CD

(1 mm)-1 1/R M* 1TeV

Planck

Figure 23. One of these extrapolations (at least!) is false.

either on the submillimeter scale or on the TeV scale—is preposterous, it is not ruled out by observations. For that reason alone, we should entertain ourselves by entertaining the consequences. Many authors have considered the gravitational excitation of a tower of Kaluza-Klein modes in the extra dimensions, which would give rise to a missing (transverse) energy signature in collider experiments. 94 We call these excitations provatons, after the Greek word for a sheep in a flock. The electroweak scale is nearby; indeed, it is coming within experimental reach at LEP2, the Tevatron Collider, and the Large Hadron Collider. Where are the other scales of significance? In particular, what is the energy scale on which the properties of quark and lepton masses and mixings are set? The similarity between the top-quark mass, mt « 175 GeV/c2, and the Higgsfield vacuum expectation value, v/^/2 « 176 GeV, encourages the hope that in addition to decoding the puzzle of electroweak symmetry breaking in our

60 explorations of the 1-TeV scale, we might gain insight into the problem of fermion mass. This is an area to be defined over the next decade. 7

Outlook

T h e creation of the electroweak theory is one of the great achievements of twentieth-century science. Its history shows the importance of the interplay between theory and experiment, and the significance of b o t h searchand-discovery experiments and precision, p r o g r a m m a t i c measurements. We owe a special salute to the heroic achievements of the L E P experiments, of the L E P accelerator team, and of the theorists who devoted themselves to extracting the most (reliable!) information from precision measurements. Their work has been inspiring. T h e electroweak story is not done. We haven't fully understood the agent of electroweak symmetry breaking, b u t we look forward to a decade of discovery on the 1-TeV scale. We probably have much to learn about the nonperturbative aspects of the electroweak theory, about which I've said nothing in these lectures. 9 5 And we are j u s t learning to define—then to solve—the essential mystery of flavor, the problem of identity. These are the good old days! For many topics not covered in these lectures, including an introduction to Higgs searches, see my 1998 G r a n a d a lectures. 9 6 Acknowledgments Fermilab is operated by Universities Research Association Inc. under Contract No. DE-AC02-76CH03000 with the United States Department of Energy. It is a pleasure to thank Steve Ellis for gracious hospitality at the University of Washington during the writing of these notes. I would also like to take this opportunity to compliment J o n a t h a n Rosner, Hitoshi Murayama, K. T . M a h a n t h a p p a , and the TASI staff for their efficient organization of "Flavor Physics for the Millennium," and for the pleasant and stimulating atmosphere in Boulder. References 1. F . Wilczek, " W h a t Q C D Tells Us about Nature — and Why We Should Listen," Nucl. Phys. A 6 6 3 , 3 (2000) hep-ph/9907340. 2. R. K. Ellis, W. J. Stirling, and B. R. Webber, QCD and Collider Physics (Cambridge University Press, Cambridge, 1996).

61

3. J. M. Conrad, M. H. Shaevitz, and T . Bolton, "Precision Measurements with High Energy Neutrino Beams," Rev. Mod. Phys. 7 0 , 1341 (1998), hep-ex/9707015. 4. B. Hirosky and F . Bedeschi, preliminary d a t a presented at the XIII Hadron Collider Physics Workshop, J a n u a r y 14-20, 1999, Mumbai, India. 5. G. C. Blazey and B. L. Flaugher, Ann. Rev. Nucl. Part. Sci. 4 9 , 633 (1999), hep-ex/9903058. 6. C. Mesropian and A. A. B h a t t i , uas Measurement from Inclusive Jet Cross Section," Internal C D F Notes 4892, available at http://www-cdf.fnal.gOv/physics/new/qcd/qcd99_blessed_plots.html#alpha. 7. S. Marti i Garcia, "Review of as Measurements at L E P 2," hepex/9704016. 8. I. Hinchliffe and A. Manohar, Ann. Rev. Nucl. Part. Sci. 5 0 , 643 (2000), hep-ph/0004186. 9. D. E. G r o o m , et al. (Particle D a t a G r o u p ) , Euro. Phys. J. C 1 5 , 1 (2000); available at http://pdg.lbl.gov/. 10. F . Wilczek, "Mass without Mass I: Most of Matter," Phys. Today 5 2 (11) 11 (November, 1999). 11. C. Quigg, "Top-ology," Phys. Today 5 0 , 20 (May, 1997); extended version circulated as F E R M I L A B - P U B - 9 7 / 0 9 1 - T , hep-ph/9704332. For more on top physics, see E. H. Simmons, "Top physics," Lectures at TASI 2000, hep-ph/0011244. 12. R. N. C a h n , "The eighteen arbitrary p a r a m e t e r s of the standard model in your everyday life," Rev. Mod. Phys. 6 8 , 951 (1996). 13. F . Butler, et al. (GF11 Collaboration), Nucl. Phys. B 4 3 0 , 179 (1994). 14. S. Aoki, et al. ( C P - P A C S Collaboration), Phys. Rev. Lett. 8 4 , 238 (2000). For progress toward an unquenched s p e c t r u m see A. Ali Khan, et al. ( C P - P A C S Collaboration), "Full Q C D light hadron spectrum and quark masses: Final results from C P - P A C S , " hep-lat/0010078. 15. F . Wilczek, "Mass without mass II: T h e m e d i u m is the mass-age," Phys. Today 5 3 (1) 13 (January, 2000). 16. G r a h a m Ross, "Theories of Fermion Masses," Lectures at TASI 2000. 17. Boris Kayser, "Neutrino Masses and Mixing," Lectures at TASI 2000. 18. S. Weinberg, Phys. Rev. D13, 974 (1976), 1 9 , 1277 (1979); L. Susskind, 2 0 , 2619 (1979). 19. S. Treiman, The Odd Quantum, (Princeton University Press, Princeton, 1999). 20. For an introduction to what was known early on, see A. Pais, "Introducing A t o m s and Their Nuclei," in Twentieth Century Physics, edited by L. M. Brown, A. Pais, and B. Pippard (Institute of Physics Publishing, Bristol

62

21.

22. 23.

24. 25.

26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

40.

& Philadelphia; American Institute of Physics, New York, 1995), vol. II, p. 43. For a comprehensive treatment, see A. Pais, Inward Bound: of matter and forces in the physical world (Clarendon Press, Oxford, 1986). V. L. Fitch and J. L. Rosner, "Elementary particle physics in the second half of the 20th century," in Twentieth Century Physics, edited by L. M. Brown, A. Pais, and B. Pippard (Institute of Physics Publishing, Bristol & Philadelphia; American Institute of Physics, New York, 1995), vol. II, pp. 635-794. Eugene D. Commins and Philip H. Bucksbaum, Weak Interactions of Leptons and Quarks (Cambridge University Press, Cambridge, 1983). Robert N. Cahn and Gerson Goldhaber, The experimental foundations of particle physics (Cambridge University Press, Cambridge and New York, 1989). J. Chadwick, Verh. Deutsch. Phys. Ges. 16, 383 (1914). W. Pauli, "Zur alteren und neueren Geschichte des Neutrinos," in Collected Scientific Papers, edited by R. Kronig and V. F. Weisskopf (Interscience, New York, 1964), volume 2, p. 1313. J. Steinberger, Ann. Rev. Nucl. Part. Sci. 47, xiii (1997). F. Reines, and C. L. Cowan, Jr., Phys. Rev. 92, 830 (1953). C. L. Cowan, Jr., F. Reines, F. B. Harrison, H. W. Kruse, and A. D. McGuire, Science 124, 103 (1956). F. Ajzenberg-Selove, Nucl. Phys. A490, 1-225 (1988). F. Ajzenberg-Selove, Nucl. Phys. A506, 1-158 (1990). F. Ajzenberg-Selove, Nucl. Phys. A523, 1-196 (1991). C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes and R. P. Hudson, Phys. Rev. 105, 1413 (1957). M. Bardon, et al., Phys. Rev. Lett. 7, 23 (1961). A. Possoz, et al., Phys. Lett. 70B, 265 (1977), Erratum B73, 504 (1978). M. Goldhaber, L. Grodzins and A. W. Sunyar, Phys. Rev. 109, 1015 (1958). L. P. Roesch, V. L. Telegdi, P. Truttmann, A. Zehnder, L. Grenacs and L. Palffy, Am. J. Phys. 50, 931 (1981). C. Quigg, Gauge Theories of the Strong, Weak, and Electromagnetic Interactions (Perseus, New York, 1997). G. Danby, et al., Phys. Rev. Lett. 9, 36 (1962); for perspective, see the Nobel Lecture of M. L. Schwartz, Rev. Mod. Phys. 6 1 , 527 (1989). Updates may be found on the DONUT collaboration's web server, http://fn872.fnal.gov/. For extensive background information, see http://www.fnal.gov/pub/donut.html. For other textbook treatments of the electroweak theory, see T.-P. Cheng

63

41.

42.

43.

44. 45. 46. 47.

48. 49. 50. 51.

52.

53.

and L.-F. Li, Gauge Theory of Elementary Particle Physics (Oxford University Press, Oxford, 1984); I. J. R. Aitchison and A. J. G. Hey, Gauge Theories in Particle Physics: A Practical Introduction, second edition (Adam Hilger, Bristol, 1989). M. K. Gaillard, P. D. Grannis and F. J. Sciulli, "The standard model of particle physics," Rev. Mod. Phys. 71, S96 (1999) hep-ph/9812285, included in More Things in Heaven and Earth: A Celebration of Physics at the Millennium, edited by B. Bederson (Springer-Verlag, New York, 1999), p. 161. V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950); English translation: Men of Physics: Landau, Vol. I, edited by D. ter Haar (Pergamon, New York, 1965), p. 138. P. W. Higgs, Phys. Rev. Lett. 12, 132 (1964). The parallel between electroweak symmetry breaking and the Ginzburg-Landau theory is drawn carefully in R. E. Marshak, Conceptual Foundations of Modern Particle Physics (World Scientific, Singapore, 1993), §4.4. For a rich discussion of superconductivity as a consequence of the spontaneous breaking of electromagnetic gauge symmetry, see Steven Weinberg, The Quantum Theory of Fields, vol. 2 (Cambridge University Press, Cambridge, 1996), §21.6. F. J. Hasert, et al. (GargamelleNeutrino Collaboration), Phys. Lett. B46, 138 (1973). A. R. Barker, "Kaon Physics: Experiments," Lectures at TASI 2000. G. Buchalla, "Kaon and Charm Physics: Theory," Lectures at TASI 2000. S. Adler, et al. (E787 Collaboration), Phys. Rev. Lett. 79, 2204 (1997), hep-ex/9708031, and Phys. Rev. Lett. 84, 3768 (00), hep-ex/0002015; for an update, see T. K. Komatsubara [for the E787 Collaboration], "Recent results from the BNL E787 experiment," hep-ex/0009047. D. Ambrose, et al., (E871 Collaboration), Phys. Rev. Lett. 84, 1389 (2000). S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2, 1285 (1970). M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973). For a description of methods and results for many important measurements, see Young-Kee Kim, "Electroweak Experiments," Lectures at TASI 2000. A. Sirlin, "Ten Years of Precision Electroweak Physics," in Proc. of the 19th Intl. Symp. on Photon and Lepton Interactions at High Energy LP99, ed. J. A. Jaros and M. E. Peskin, eConf C990809, 398 (2000), hepph/9912227. M. Swartz, "Precision Electroweak Physics at the Z," in Proc. of the 19th Intl. Symp. on Photon and Lepton Interactions at High Energy

64

54. 55.

56. 57. 58.

59. 60.

61.

62.

63. 64.

LP99, ed. J. A. Jaros and M. E. Peskin, eConf C990809, 307 (2000), hepex/9912026. LEP Electroweak Working Group, http://www.cern.ch/LEPEWWG/. W. J. Marciano, "Precision electroweak parameters and the Higgs mass," in Physics Potential and Development of Muon Colliders and Neutrino Factories, edited by D. B. Cline, AIP Conference Proceedings 542 (American Institute of Physics, Woodbury, NY, 2000), p. 48, hep-ph/0003181. Carl Wieman, "Tests of the Standard Model in Atomic Physics," Lectures at TASI 2000. S. C. Bennett and C. E. Wieman, Phys. Rev. Lett. 82, 2484 (1999). J. Erler and P. Langacker, Phys. Rev. Lett. 84, 212 (2000), hepph/9910315; R. Casalbuoni S. De Curtis, D. Dominici, and R. Gatto, Phys. Lett. B460, 135 (1999) hep-ph/9905568; J. L. Rosner, Phys. Rev. D61, 016006 (2000) hep-ph/9907524. J. L. Hewett and T. G. Rizzo, "Low-Energy Phenomenology Of Superstring Inspired E6 Models," Phys. Rep. 183, 193 (1989). M. Veltman, Phys. Rev. Lett. 34, 777 (1975). A. D. Linde, JETP Lett. 19, 183 (1974). J. Dreitlein, Phys. Rev. Lett. 33, 1243 (1977). S. Weinberg, Rev. Mod. Phys. 6 1 , 1 (1989). M. S. Turner, "Cosmological Parameters," in Proceedings of Conference on Particle Physics and the Early Universe (COSMO 98), edited by D. Caldwell (American Institute of Physics, Woodbury, NY, 1999) AIP conference proceedings 478, pp. 113-128, astro-ph/9904051; M. Fukugita and C. J. Hogan, "Global Cosmological Parameters: HQ, £-M , and A," §17 of Ref. 9 . For a preliminary "end-of-LEP" summary, see Peter Igo-Kemenes, "Status of Higgs Boson Searches," transparencies available at http://lephiggs.web.cern.ch/LEPHIGGS/talks/pik_lepc.nov3_2000.pdf. See also the very useful LEP Higgs Working Group summary of the standardmodel Higgs search for the November 3, 2000, meeting of the LEPC, available at http://lephiggs.web.cern.ch/LEPHIGGS/papers/doc_nov3_2000.ps B. W. Lee, C. Quigg, and H. B. Thacker, Phys. Rev. D16, 1519 (1977). In the high-energy limit, an amplitude for longitudinal gauge-boson interactions may be replaced by a corresponding amplitude for the scattering of massless Goldstone bosons: M(WL,ZL) = M(w,z) +0(Mw /y/s) The equivalence theorem can be traced to the work of John M. Cornwall, David N. Levin, and George Tiktopoulos, Phys. Rev. D10, 1145 (1974), 1 1 , 972E (1975). It was applied to this problem by Lee, Quigg, and Thacker, Ref. 63 , and developed extensively by Chanowitz and Gaillard, Ref. 65 , and others.

65

65. See, for example, M. Chanowitz and M. K. Gaillard, Nucl. Phys. B261, 279 (1985); M. Chanowitz, M. Golden, and H. Georgi, Phys. Rev. D36, 1490 (1987); Phys. Rev. Lett. 57, 2344 (1986). 66. K. G. Wilson, Phys. Rev. D3, 1818 (1971); Phys. Rev. B4, 3184 (1971); K. G. Wilson and J. Kogut, Phys. Rep. 12, 76 (1974). 67. For a review, see Urs M. Heller, Nucl. Phys. B (Proc. Supp.) 34 (1994) 101 hep-lat/9311058. The numerical analysis that leads to these conclusions is reported in Urs M. Heller, et al., Nucl. Phys. B405, 555 (1993) hepph/9303215. 68. Lower bounds on the Higgs mass date from the work of A. D. Linde, Zh. Eksp. Teor. Fiz. Pis'ma Red. 23, 73 (1976) [English translation: JETP Lett. 23, 64 (1976)]; S. Weinberg, Phys. Rev. Lett. 36, 294 (1976). An early investigation of the effects of heavy fermions appears in P. Q. Hung, Phys. Rev. Lett. 42, 873 (1979). For a review of lower bounds deduced from the requirement of vacuum stability of electroweak potentials, see M. Sher, Phys. Rep. 179, 273 (1989). For useful updates in light of the large mass of the top quark (mt ~ 175 GeV/c2), see G. Altarelli and G. Isidori, Phys. Lett. B337, 141 (1994); J. Espinosa and M. Quiros, Phys. Lett. B353, 257 (1995). 69. J. A. Casas, J. R. Espinosa, M. Quiros, and A. Riotto, Nucl. Phys. B436, 3 (1995); Y. F. Pirogov and O. V. Zenin, Eur. Phys. J. CIO, 629 (1999), hep-ph/9808396. 70. L. Maiani, G. Parisi, and R. Petronzio, Nucl. Phys. B136, 115 (1978). 71. Two early studies provide valuable introductions to a light Higgs boson: J. Ellis, M. K. Gaillard and D. V. Nanopoulos, Nucl. Phys. B106, 292 (1976), and A. I. Vainshtein, V. I. Zakharov, and M. A. Shifman, Usp. Fiz. Nauk 131, 537 (1980) [English translation: Sov. Phys.-Uspekhi 23, 429 (1980)]. For a first look at the properties of a heavy Higgs boson, see Lee, Quigg, and Thacker, Ref. 63 . A useful general reference is J. F. Gunion, H. E. Haber, G. L. Kane, and S. Dawson, The Higgs Hunter's Guide (Addison-Wesley, Redwood City, California, 1990). 72. M. Carena, et al., "Report of the Tevatron Higgs working group," hepph/0010338. 73. M. Veltman, Acta Phys. Polon. B12, 437 (1981); C. H. Llewellyn Smith, Phys. Rep. 105, 53 (1984). 74. For an approachable introduction, see Joseph D. Lykken, "Introduction to Supersymmetry," in Fields, Strings, and Duality: TASI 96, edited by C. Efthimiou and B. Greene (World Scientific, Singapore, 1997). p. 85 hep-th/9612114. 75. Stephen P. Martin, "A Supersymmetry Primer" hep-ph/9709356, latest

66

version available at http://zippy.physics.niu.edu/primer.shtml. 76. S. Dawson, "The MSSM and Why It Works," in Supersymmetry, Supergravity and Supercolliders: TASI 97, edited by J. Bagger (World Scientific, Singapore, 1999), p. 261 hep-ph/9712464. 77. H. Murayama, "Supersymmetry," Lectures at TASI 2000. 78. R. S. Chivukula, "An Introduction to dynamical electroweak symmetry breaking," in Advanced School on Electroweak Theory, edited by D. Espriu and A. Pich (World Scientific, Singapore, 1998) p. 77, hepph/9701322. 79. R. S. Chivukula, "Lectures on technicolor and compositeness," Lectures at TASI 2000, hep-ph/0011264. 80. E. Eichten, and K. Lane, Phys. Lett. 90B, 125 (1980); S. Dimopoulos, and L. Susskind, Nucl. Phys. B155, 237 (1979). 81. I will not discuss this alternative further. See M. S. Chanowitz, "Strong WW scattering at the end of the 90's: Theory and experimental prospects" hep-ph/9812215. 82. M. Weinstein, Phys. Rev. D8, 2511 (1973). 83. T. Schafer, "Color Superconductivity," nucl-th/9911017, to be published in the proceedings of 10th International Conference on Recent Progress in Many-Body Theories (MBX), Seattle, Washington, 10-15 Sep 1999. 84. F. Wilczek, "The Recent Excitement in High-Density QCD," Nucl. Phys. A663, 257 (2000), hep-ph/9908480. 85. K. Rajagopal and F. Wilczek, "The condensed matter physics of QCD,"hep-ph/0011333, to appear as Chapter 35 in the Festschrift in honor of B. L. Ioffe, At the Frontier of Particle Physics / Handbook of QCD, ed. M. Shifman (World Scientific). 86. I. Antoniadis, Phys. Lett. B246, 377 (1990); J. D. Lykken, Phys. Rev. D54, 3693 (1996), hep-th/9603133; N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B429, 263 (1998), hep-ph/9803315; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B436, 257 (1998), hep-ph/9804398. 87. C. D. Hoyle, et al., "Submillimeter tests of the gravitational inversesquare law: a search for 'large' extra dimensions," hep-ex/0011014. 88. J. K. Hoskins, R. D. Newman, R. Spero and J. Schultz, Phys. Rev. D32, 3084 (1985). 89. V. P. Mitrofanov and O. I. Ponomareva, Sov. Phys.—JETP 67, 1963 (1988). 90. S. D. Lamoreaux, Phys. Rev. Lett. 78, 5 (1998). 91. G. C. Long, H. Chang and J. Price, Nucl. Phys. B539, 23 (1999). 92. D. B. Kaplan and M. B. Wise, hep-ph/0008116.

67

93. For an introduction, see J. Lykken, "Physics of Extra Dimensions," Lectures at TASI 2000. 94. G. Giudice, R. Rattazzi, and J. Wells, Nucl. Phys. B544, 3 (1999), hepph/9811291. E. A. Mirabelli, M. Perelstein, and M. E. Peskin, Phys. Rev. Lett. 82, 2236 (1999), hep-ph/9811337. J. L. Hewett, Phys. Rev. Lett. 82, 4765 (1999), hep-ph/9811356. 95. For the important application to electroweak baryogenesis, see, for example, A. Riotto and M. Trodden, "Recent progress in baryogenesis," Ann. Rev. Nucl. Part. Sci. 49, 35 (1999), hep-ph/9901362. 96. C. Quigg, "Electroweak symmetry breaking and the Higgs sector," lectures presented at 27th International Meeting on Fundamental Physics, Sierra Nevada, Granada, Spain, 1-5 Feb 1999. Acta Phys. Polon. 530, 2145 (1999), hep-ph/9905369.

This page is intentionally left blank

_. .M««MRims Lincoln Wolfenstein

This page is intentionally left blank

C P Violation Lincoln Wolfenstein Department of Physics, Carnegie Mellon University, PA 15213, USA E-mail: [email protected]

Pittsburgh,

Three possibilities for the origin of CP violation are discussed: (1) the Standard Model in which all CP violation is due to one parameter in the CKM matrix, (2) the superweak model in which all CP violation is due to new physics and (3) the Standard Model plus new physics. A major goal of B physics is to distinguish these possibilities. CP violation implies time reversal violation (TRV) but direct evidence for TRV is difficult to obtain.

1

Introduction

The symmetries P, C, and T have played a large role in the physics of the past 50 years, where P is left-right symmetry, C is particle-antiparticle symmetry and T for time reversal. Originally, these symmetries were not postulated but discovered theoretically as symmetries of the Hamiltonian of QED. In nuclear and particle physics one tried to guess at the Hamiltonian or Lagrangian density without any classical analogue. The first of these was the Fermi theory of the weak interaction governing nuclear beta decays: Hw=GF

{p^ne-fv)

<J(r).

(1)

Indeed this may be said to mark the beginning of particle physics. As a result of many experiments on nuclear beta decays it appeared that additional terms would have to be added to Eq. (1.1) to account for spin-flip (Gamow-Teller) transitions. The major advance, however, followed from the observation in 1956 by Lee and Yang that no experiment had been sensitive to whether parity was conserved in the weak interaction. This led to a number of experiments that showed that P and also C were very much violated. This could be accounted for by replacing the fields in Eq. (1.1) by left-handed chiral fields 1 , or, equivalently, replacing 7M by 7M(1 —75). This became the standard (V - A) theory. Even before Lee and Yang it had been shown that it was easy to construct Hamiltonians that violated C or P or T but that in fact for every relativistic local quantum field theory one could always define the symmetry CPT. Thus the (V — A) theory violated C and P but still had the symmetries CP and T. Thus it came as a great surprise when it was discovered in 1964 that there was a small violation of CP symmetry in K° decays. 71

72

The K° is characterized by an additive strong and electromagnetic interactions but Since S is not a good quantum number K° S = — 1, so one expected the eigenstates to

quantum number S conserved in violated by the weak interaction. with 5 = 1 mixes with K° with be CP eigenstates

IK,) = -L (\K°) + \K0)) , l*2> = ^

(\K°) - | * ° » ,

\K°) = CP \K°). The observed eigenstates were Kg (lifetime rg = 0.9 x 10~ 10 sec) and KL (TL = 5.2 x 10~ 8 sec) with TUL - ms — 0.48 Ts ~ 10~5eV. The primary decays were Ks ->• TT+ -K~~ and KL -> 3TT,

TT°

7r°,

consistent with the CP assignment Ks = ^ i and KL = ^ 2 - The discovery in 1964 was that KL also decayed into w+ n~ with a small branching ratio. There appeared then two alternatives: 1. Modify the AS — 1 interaction in some small way from the standard (V - A) form. 2. Assume there exists a much weaker AS = 2 interaction that violates CP. This would be described by an effective Hamiltonian (in terms of quarks) naw=GswsOdsOd+h.c,

(2)

where O is some Dirac operator. It is sufficient that Gsw ~ 1 0 - 1 0 to l O - 1 1 ^ .

(3)

The superweak idea is that CP violation is confined to K° — K° mixing, which is a AS = 2 process, second order for the standard theory. In the Ki — K2 representation M

l

£ _ / Ml 2 ~ \-im'/2

M2

im'/2\_i(T1 J 2\ T2

73

where m' is the superweak term and the i is required by CPT invariance. Then \Ks) = (\K1)+e\Ki))Hl \KL) = (\K2)+e\K2))/(l

+ \§\*), + \i\2),

(Mi-M 2 )-i(ri-r 2 )/2' Mi - M2 « Ms - ML = -AMjf,

( ) [

'

v

'

ri - r 2 « r s - rL « rs. The observations give e ~ 2 x 1 0 - 3 which then determines m' from which Eq. (1.3) follows. The superweak theory made three predictions: 1. The CP violation is completely described by e; in particular the observables V+- = %o = £> where T)+- = A(KL ->• 7r+ ir~)/A(Ks

-)• TT+ TT~),

IJdO = ^ ( i ^ L -»• 7T° 7 r ° ) / A ( ^ S -> 7T° 7T°).

2. The phases of T]^ and 7700 are equal and determined by Eq. (1.5) to be about 43.5° using the empirical values of AMK and Ts3. The CP violation is so small that it will not be seen anywhere else. These predictions proved all too true for more than 25 years. The alternative to superweak says that CP is violated in the decay amplitude AQ el6° and A2 el52 corresponding to final 1 = 0 and I = 2 irir states, where 6j is the strong phase shift. Prom CPT and unitarity ImAj give the amplitudes for the CP-violating transitions K2 —> TTTT. Since there is still in any model the contribution m' there are now three CP-violating quantities 2 m', ImAo, ImA 2 . However there is a choice of phase convention using the £7(1) transformation s —> se~ia under which the strong and electromagnetic interactions are invariant. For the infinitesimal f/(l) as an example s —> s(l — ia), ImAi -> ImAi - a Re Ai, m' -+ m' + a(Mi - M 2 ).

74

Thus there are only two independent quantities which may be chosen as e' oc ImAo/ReAo - Im A 2 /Re A 2 , e ~ £ + i(ImA 0 /ReAo).

,„> {)

Then e'

1+-=£ +

l+u,Jrf' 2£

(7)

1 + V2w where u = Re A 2 /Re A0 ~ 0.045 and the last numerical result is empirical and a sign of the AI = 1/2 rule. The quantity e' is a measure of CP violation in the decay amplitudes and so not due to a superweak interaction. 2

The Standard Model vs Superweak

For 35 years experiments have sought to determine e' by finding the difference between TJ^ and 7700 and thus detecting CP violation in the decay amplitude. The experiments actually measure V+-

2

~ l + 6Re(e'/e).

(8)

?7oo

By chance the phase of e given to a good approximation by Eq. (1.5) and the phase of e' given by (7r/2 — <5o + $2) a r e both approximately equal to 45° so that Re(e'/e) — |£'/ e l- Recent experiments have left no doubt that \e'/e\ is non-zero and my average of the results is |e'/e| = ( 2 0 ± 6 ) x 10" 3 . The big development in weak interactions was the spontaneously broken gauge theory. Originally given by Weinberg as a theory of leptons it was extended by Glashow, Iliopoulos and Maiani (GIM) to quarks by adding a fourth quark charm. With the discovery of neutral currents in 1973 this became the new Standard Model. A striking feature of this new theory was that it had CP and T invariance and so provided no solution to the CP violation problem. In one paragraph of a paper 3 in 1973 that few people noticed for several years it was proposed that if there were six quarks then CP violation could be allowed in the Standard Model. With the discovery of b quark physics in 1978 this became the standard CKM model of CP violation. The only place one can insert complex phases to give CP violation in the Standard Model is in the Yukawa interaction. This shows up after symmetry

75

breaking in the unitary CKM matrix Vji connecting down quarks d; to up quarks Uj in the interaction with the W bosons: QjVji^il-^diW,,.

(9)

With only four quarks the unitary 2 x 2 matrix has three phases which can be removed by the (7(1) transformations of S, charm C, and electric charge Q. With six quarks the 3 x 3 matrix has six phases but only five can be removed. In a standard phase convention there are two matrix elements with large phases:

Vub^\Vub\e-^~AX3(p~ir,), VM = | ^ | e - ^ ~ A A 3 ( l - p - i q ) ,

, , [W)

where the last equality corresponds to a standard parameterization in powers of A = sin# c ~ 0.22 (see the lectures by Falk). The CP violation is directly proportional to 77. The CP-violating observable e (determined from 77+-) is calculated from the box diagram giving the K — K mixing parameter m'. (Note I m A 0 / R e A 0 is very small compared to e in using Eq. (1.6).) This is directly proportional to 77 and requires that 77 be between 0.2 and 0.6. This is very consistent with the constraint from the magnitude of \Vub\. Note this consistency is significant; when one thought mt was 20 GeV instead of 170 GeV the value of 77 required was larger and the consistency was much less obvious. Given this value of 77 the phase 7 is expected to lie between 45° and 135° and the phase /3 between 15° and 30°. The calculation of e' in the Standard Model is very uncertain due to the difficulty of calculating hadronic matrix elements (see the lectures by Buchalla). The dominant contribution is the so-called penguin diagram in which a loop involving t o r e quarks emits a gluon. The s —> d + g transition contributes only to Im Ao. There is also an electroweak penguin s —> d + (7 or Z) which can contribute to I111A2 and somewhat decreases e' (see Eq. (1.6)). Different theoretical calculations give values of e'/e from 1.3 to 6 x 10_37) with large errors. Given the uncertainties it seems the Standard Model can explain, though not really predict, the value of e'/e. The direct CP violation indicated by e'/e is a major blow to the superweak theory. Before completely abandoning superweak I thought it would be good to define it. There are many possible theories which are effectively superweak; their common feature is that at a low energy scale their effective interaction is given by iieff

= no + nSwt

where "Ho is the Standard Model with the CP-violating parameter 77 chosen to

76

zero so that all CP violation is given by Hsw which contains terms of the form Gijki q~i Qj qk qi

with all G^ki <S Gp- In particular, as noted, to explain e, Ggdsd ~ 1 0 - 1 0 to 10~nGp- In general, the terms in Tisw are derived from diagrams involving heavy particles beyond those in the Standard Model 4 . Because e' is only about 5 x 10~ 6 the possibility arises that in some theories that are effectively superweak such a value of e' could be explained. In fact as long as one is only dealing with the first four quarks one can think of the Standard Model as effectively superweak. Then Ho is the 4-quark Standard Model, which, as we noted, is CP-invariant and %sw corresponds to effective CP-violating interactions due to diagrams involving the heavy t quark which determines e and e'. The really distinctive features of the CKM model appear only when we consider the CP violation involving B mesons. The B — B mixing is dominated by the box diagram involving t quarks giving in the B — B representation M12 oc (Vt*d Vtb)2 ~ A2 A6[(l - p)2 +

rj^e2^.

Thus in this phase convention there is large CP violation in the mixing. The time-dependent B°(B°) decay rate to a CP eigenstate has a term 5 ±rjf sin 2((3 — iff) sin Am t,

(11)

where iff is the CP-violating phase in the decay amplitude, rjf is the CP eigenvalue, and +(—) corresponds to initial B°(B°). The observation of this asymmetry is the major goal of B factories. The first example will be the decay B°(B°) -¥ $ K s which is due to the transition b ->• ccs, for which iff = 0. A result from the hadron collider experiment CDF has given the result sin 20 = 0.8±0.4. By the time these lectures are published more accurate values should be available from the BABAR and BELLE experiments. Large positive value of sin 2(3 will give strong qualitative evidence for the Standard Model. Nevertheless this asymmetry can still be blamed on superweak mixing with Gidid ~ 10~ 7 , although only in special models would one expect such a large effect in B — B mixing. To finally kill all superweaklike theories one must look for the expected large CP violation in the B decay amplitude. To allow for the superweak possibility we call the result of the first experiment sin 2/?; then if a different decay to a final state / ' gives the asymmetry s i n 2 0 —
77

The most obvious choice for / ' is the decay B -> 7r+ w~. In the "tree" approximation this is due to b —> uud and so that the phase n+ TT~ may be only about 5 x l O - 6 . Furthermore if d + gluon via a t quark loop (see the lectures of Rosner). Thus, the decay amplitude is given as A(B^ir+x-)=Te-i'y + Pei0eiA, (12) since the penguin is proportional to Va Vt*d and so has the phase (3. Estimates based on the rate of B —> K -K 6 suggest P/T as high as 0.4. Assuming the strong phase A is small, then \ p+ TT~ and p~ n+. An alternative would be to consider decays that are dominated by the b —> d penguin. In this case $ Ks would then be evidence for a large direct CP violation. Unfortunately the best candidates for such decays might not be very practical; they correspond to b ->• dss yielding B° -»• Ks Ks, B° -» p° 77, etc. Note that even if the decay were not pure penguin one would expect an asymmetry very different from that for 5 ° -» * Ks. Another possibility is a CP-violating effect in which mixing is not involved. One can look for a difference between the rates of B+ —>• / and B~ —> / or in the case of B° looking at time t = 0 before mixing for the difference between B° -> f and B° -» / . In practice this means looking for the cos ATUB t term 5 rather than the sin A m ^ t. As an example, consider B°(B°) —• 7r+ IT~; from Eq. (2.5) _ r ( £ ° -> 7T+ 7T~) - T(B° -> 7T+ 71-) ^ ~ T(B° ->• 7T+ 7T-) + r(J5° -4 7T+ 7T-) '

78

2TPsin(/? + 7)sinA T 2 + P 2 + 2TPcos(/3 + 7 ) c o s A ' For 0 + 7 ~ 90° this gives (with P/T = r) 1r sin A. 1 + rz Thus a very large CP-violating asymmetry is possible if r is greater than 0.3, but it all depends on the strong phase A. It is very difficult to make definite statements about the strong phases in B decays in contrast to K decays. For K -» TTTT the final state interaction can be thought of as elastic scattering with phase shifts S2 and So corresponding to 7T7T states with 1 = 2 and I = 0. However, KIT s-wave scattering at 5 GeV is highly inelastic involving many channels. The phase A arises from the absorptive parts of diagrams corresponding to the strong scattering from other final states into the TTTT state. For any weak interaction operator Oi we can define the real decay amplitude in lowest order An = -

Mti = M% =

(f\Oi\B).

If / were an eigenstate one would then multiply this by el Sf. Going from the eigenstate basis to the states of interest Mfi = £ < / | S 1 / 2 | / ' > < m i - B > , (14) /' where S is the strong-interaction S matrix. The sum in Eq. (2.7) is over a large number (almost uncountable) of states. One can only make some general comments about it: 1. The strong phase depends on the operator 0 , that affects the relative importance of different states / ' . The phase A in Eqs. (2.5) and (2.6) is the difference between the strong phase of the "tree" operator and that of the "penguin". 2. Since the strong scattering is expected to be very inelastic the diagonal element (/IS 11 / 2 !/) has as its major effect the reduction of M / J ; this is a kind of absorption effect. Thus we could write Mfi = M°fiai + i Ri =

\Mfi\ei5i,

where a < 1 is the reduction due to absorption. For a "typical" state, by unitarity, the scattering "in" due to R compensates for the scattering "out" so

Ri = \Jl-a] =sm5i.

(15)

79

3. An estimate can be based on a crude statistical argument 7 in which case one can reduce the multichannel problem to an equivalent 2-channel problem „ _ / cos2<9 i s i n 2 0 \ \i sin20 cos 20 J ' For / = 7r n the diagonal element cos 20 can be estimated by extrapolating data from n N scattering to n n giving cos 2(9 = T] ~ 0.7. Then Eq. (2.7) becomes Mu = cos 6 An + i sin6 A21, tan Si = i tan 6 —^, An where t a n ^ = ( i ^ | f )

1 / 2

(16)

~ 0.42.

The "typical" result corresponds to A2 = A\ and gives a strong phase of 20° to 25°. The quantitative conclusion from Eq. (2.9) is that if the state of interest (labeled 1 here) is a "more probable" final state than the states into which that state scatters (lumped into state 2 here) then the strong phase may be small. In conclusion, after it is clearly shown that CP violation is not superweak the next step is to find quantitative tests of the Standard Model by showing the consistency of a number of different experiments. This is a major program for the next decade. 3

Time Reversal Violation

By the CPT theorem CP violation implies time-reversal violation (TRV). Strong evidence for CPT invariance comes from the phase of e determined from Eq. (1.5). CPT violation would allow a real off-diagonal term m" in the matrix in addition to im' and thus would change the phase. Since the measured phase agrees with theory to about 1° there is a strong limit on m" which corresponds to a limit on [m(K°) —m(K°)]/mK < 10~ 18 . Nevertheless, it is of great interest to look for direct evidence for TRV both as another way to study CP violation as well as a way to demonstrate T violation in a straightforward way 8 . We discuss here four types of direct evidence for TRV; by this I mean a single experiment that by itself is seen to violate T. These are

80

1. A non-zero value of a T-odd observable in a stationary state. The simplest example is the electric dipole moment of an elementary particle or an atom. 2. A violation of the reciprocity condition on the S matrix

Sfi = S-it-f corresponding to comparing a reaction and its inverse. 3. A non-zero value of a T-odd observable in the final state of a weak decay. As discussed below this depends upon the neglect of final-state interactions. 4. In an oscillation a difference in the probability of a —» b from b —¥ a at a given time. It is interesting to note that each example immediately implies a test of CP violation (conceptual if not practical) by going to the anti-particles. In contrast the simplest tests of CP violation have no direct relation to TRV; for example, T(B -4 / ) = T(B —>• / ) involves a rate which has nothing to do with a TRV observable. Experimental limits on the dipole moments of the electron and neutron are dn < 10~ 25 e - cm, de < 4 x 1(T 27 e - cm. In the Standard Model dn is second-order weak and the calculation depends on long-distance effects giving of order 1 0 - 3 2 e-cm; de is third order and perhaps 10~ 38 e-cm. Thus the interest lies in the search for physics beyond the Standard Model (see the lecture by Thomas). Many tests of the reciprocity relations exist for strong interactions although they are of limited accuracy; it is very hard to study the reverse of weak interactions. An interesting proposal by Bowman involves the scattering of slow neutrons from polarized nuclei in the resonance region. One compares the observable < an • I x k > for incident polarization an and final polarization an, where / is the nuclear polarization. T-violating effects in the nuclear wave functions could enhance the result. An example of a "T-odd observable" in the final state of decay is the muon polarization P =< a" •fcMx kv >

81

in the decay K -> it + fi + v. This does not really violate T except in the Born approximation when final state interactions (FSI) can be avoided. As a simple didactic example, consider the scattering from a potential Vo +

Via-L

which certainly is T-invariant. The resulting amplitude is A = fo +

ifia-h

where n is the normal to the scattering plane. In the Born approximation / 0 and / i are real and so < a • h > = 0, but beyond the Born approximation / 0 and / i are complex; for example, if s and p waves dominate / 0 would have a phase el s° and / i the phase el ^. For the case of K° -> ir+ +/x~ + PM there is a Coulomb FSI so that without T violation, P ~ 10~ 3 ; for the case of K+ ->• TT° + ju+ + v^ the FSI involves 2-y exchange so that P ~ 10~ 6 . In the Standard Model the real TRV is expected to vanish in semi-leptonic decays. One would expect a real TRV in non-leptonic decays such as A —> pn~ where there is a defined parameter (3 oc< aA • a* x k > . However, the FSI effect is much larger being proportional to sin(J p — Ss) where 6P, 6S are the n-p phase shifts. If the experiment is also done with A, then /3 + /? is a clear CP-violating effect and is associated with true TRV, but this is hardly "direct evidence" of TRV. A large "T-odd observable" has been found in the decay KL —> n+ ir~ e+ e~ C = < ne x n^ • z > < he • nn >, where n^h^) are the normals to the e+ e~(ir+ ir~) planes and z is the unit vector between the pairs. This was predicted as a result of K - K mixing as an interference between an M l virtual 7 from if2 -> TTTTJ and an Ei virtual bremsstrahlung from K\ —> n tt 7. The theoretical result 9 is C = 0.15 sin(v?e + A), where A ~ 30° comes from mr phase shifts. The experimental result 10 verifies this; the result is so large because for the e + e~ energy considered the El is much larger than M l which compensates for the small admixture |e| of K\. Since A is involved this is not again obvious TRV. It is of didactic interest to consider the limit A —)• 0. In this case C is proportional to sin<£>£. We know

82

that • K° -> 7T+ 7T" e+ e" with 7r+ 7T_ on the mass shell thus giving an absorptive part. Only if A = 0 and A r = 0 would a non-zero C directly show TRV. The possibility of seeing CP violation in neutrino oscillations from the difference between v^ —>• ve and PM -> ue has been discussed in many papers (see the talk by Kayser). The same formula gives the TRV difference between v iL —> ve and ve —> fM. Note that the time dependence (or, equivalently, the distance dependence) of the difference is an odd function of time. The possibility of doing the TRV experiment requires beams of both v^ and ve as has been proposed for "neutrino factories" based on a muon storage ring. In the CP LEAR experiment n a difference has been observed between the transitions K° -> K° and K° -» K°. Here the initial K° (K°) has been identified by its associated production with a K+ (K~) and the final K° (K°) by the charge of the lepton in the semi-leptonic decay. The result in agreement with a simple calculation is

r(jr°->jr°)-r(ir0->jr°) T{K° -»• K°) + T(K° -> K°) independent of time. This seems somewhat strange since we expected an odd function of time. One can also ask from unitarity if K° goes to K° more than K° goes to K° what compensates for this. The answer is that the K° decays to 7T7T more than K°. Thus, decay plays an essential role, rendering this as a direct test of TRV somewhat questionable. As we have noted the phase of e is completely consistent with CPT invariance. There is no reason to doubt CPT invariance, which appears very fundamental, and so we conclude that the observed CP violation is associated with TRV. Nevertheless, unambiguous direct tests of TRV may prove very difficult.

83

Acknowledgments Many of the early papers on CP violation are reprinted in L. Wolfenstein, CP Violation (North Holland) (1989). This work was supported by the U.S. Department of Energy under Grant No. DE-FG02-91ER40682. References 1. In the original paper on the V — A theory, Feynman emphasizes the elegance of using chiral fields, R.P. Feynman and M. Gell-Mann, Phys. Rev. 109 (1958), 193. 2. More exactly there are also CP-violating parameters describing K —> 3ir decays which affect K -» 2ir via the off-diagonal terms in the T matrix. These can affect the phase of e in the Standard Model, probably less than 1°. A more general treatment of the phase e is given by the BellSteinberger relation, Proc. Oxford Conference (1965), 195. 3. M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973), 652. 4. For a general discussion and motivation of effectively superweak theories, see S.M. Barr, Phys. Rev. D 34 (1986), 1567. 5. For a derivation see H. Quinn in Review of Particle Properties, Euro. Phys. Journal 3, pp 555-562 (1998). 6. J.P. Silva and L. Wolfenstein, Phys. Rev. D 49, R1151 (1994); R. Fleischer, hep-ph/0001253. More recent data from BABAR suggests this may be an overestimate. 7. M. Suzuki and L. Wolfenstein, Phys. Rev. D 60 (1999), 074019. 8. See L. Wolfenstein, Int. J. Mod. Phys. E8 (1999), 501. 9. L.M. Sehgal and M. Wanninger, Phys. Rev D 46 (1992), 1035; Erratum, ibid D 46 (1992), 5209. 10. A. Alav-Harati et al., Phys. Rev. Lett. 84 (2000), 408. 11. A. Angelopoulous et al, Phys. Lett. B444 (1998), 43.

This page is intentionally left blank

'¥*3J#SR

Young-Kee Kim

This page is intentionally left blank

PRECISION ELECTROWEAK

PHYSICS

Young-Kee Kim University of California at Berkeley & Lawrence Berkeley National Laboratory Berkeley, California 94720 USA E-mail: [email protected] These three lectures review the experimental state of electroweak interactions and the search for electroweak symmetry breaking.

1

Introduction

One of the central problems in particle physics is to understand the origin and values of the particle masses. In the S t a n d a r d Model, particles acquire their masses by interacting with Higgs scalars. T h u s uncovering the secrets of the Higgs sector is the focus of much present and future experimental research. T h e Higgs sector can be probed by the precision electroweak measurements such as Mz, sin29w, Mw, and M t o p by means of q u a n t u m corrections. However, the n a t u r e of the symmetry breaking sector can only be established by its direct discovery and detailed study. T h e first lecture (Section 2) surveys the accelerators t h a t produce heavy particles such as W, Z, and top quark, and the detectors t h a t identify t h e m and measure their properties. T h e second and third lectures review the precision measurements t h a t probe the Higgs sector (Section 3) and the current and future searches of the Higgs boson (Section 4). 2 2.1

S u r v e y of A c c e l e r a t o r s a n d D e t e c t o r s Colliders

versus Fixed-target

machines

Suppose t h a t we wish to study the Z boson. This requires a centre-of-mass energy of ^ 9 0 GeV. In the e+e~ symmetric collider (e 4 , energy = e~ energy), this would require a beam energy of ~ 4 5 GeV. Assuming a quark in a proton carries a b o u t 1/6 of the proton energy on average (see Figure 1), the proton and anti-proton collider would require a beam energy of ~ 2 7 0 GeV. a In order to achieve the same goal, a proton fixed-target machine would require a proton beam energy of ~ 2 0 TeV. a

N o t e that the energy loss due to synchrotron radiation is much larger for electrons than protons (Esync ~ 1/m 4 in a circular accelerator ring), thus it is easier to accelerate protons than electrons. 87

88

Figure 1: Quark and gluon energy distribution functions at fi2 = 10 GeV 2 where x is momentum fraction carried by a quark or gluon. Figure courtesy of "QCD and Collider Physics" by R.K. Ellis, W.J. Stirling and B.R. Webber.

With the great technological achievements of accelerator physics in colliding one high-energy beam with another, colliders satisfy our need to study elementary processes at high energies better than fixed target machines. 2.2

e+e~ colliders versus Hadron colliders

The highest beam energy achieved in an e + e~ collider is 104 GeV by LEP-2 (LEP stands for Large Electron Positron collider) at CERN, and the highest energy achieved in a hadron collider is 900 GeV b by the Tevatron at Fermilab. The beam energy in LHC (Large Hadron Collider), which will begin in about 2006, will be 7 TeV. Since electrons are elementary particles, all of their energy can be available to produce new particles, thus energy upto 208 GeV is available at LEP-2. The available energy is much larger at the Tevatron hadron collider as demonstrated in Figure 2 although the cross-section rapidly decreases with the energy. With large luminosity, hadron colliders would be good machines to discover massive particles. Note that the 3 heaviest particles, W, Z, and top quark, 6 For the upcoming Tevatron run (Tevatron Run II), the proton and anti-proton beam energy will be 980 GeV.

89 ~n

|

i

|

i

|

i

|

:

CDF d a t a o n j e t ET d i s t r i b u t i o n . 0.1<\r)\<0.7, R=0.7 s t a t i s t i c a l e r r o r s only .Theory-MRS(DO) p a r t o n s

200 ET [GeV]

Figure 2: Jet Er distribution from the CDF collaboration at the Tevatron.

were discovered at hadron colliders (see Table 1). l ' 2 On the other hand, e+e~ colliders have advantages such as their clean environment that makes precision measurements much easier and the fact that the final state particles are mostly detected so that the energy-momentum conservation can be used as a tool to improve the measurement resolution. e+e~ colliders and hadron colliders are thus complementary to each other. 2.3

Production of W, Z, and Top quark

Z production at LEP-1/SLC and W production at LEP-2 The experimental study of the process e + e~ —• Z —> ff :

began in 1989 at LEP and SLC at SLAC. The LEP program completed datataking at the Z resonance in 1995 (LEP-1), and the SLC program finished in

90 Table 1: Discovery of Fermions and Gauge Bosons Type Lepton e e+ M r "e

Mass (GeV) 0.00051 0.00051 0.106 1.777 ~0 ~0

~o

Quark

Gauge Boson

d s c b t 7 9 W Z

0.003-0.009 0.001-0.005 0.075-0.170 1.15-1.35 4.0-4.4 174.3±5.1 0 0

80.419±0.056 91.188±0.002

1898 1931 1937 1975 1953 1962 2000

Discovery J . J . Thomson Anderson Anderson et al Martin Perl Cowan & Reines Lederman, Schwartz, Steinberger DONUT

cosmic ray cosmic ray e + e~ collider fission reactor p accelerator p accelerator

1947 1974 1977 1995

Rochester & Butler Mark I & BNL-E-0598 FNAL-E-0288 CDF & D 0

cosmic ray e + e ~ coll. & p accel. p accelerator pp collider

1979 1982 1982

Tasso / Mark J U A 1 & UA2 U A 1 & UA2

e + e ~ collider pp collider pp collider

1998. The total LEP event sample consists of 15.5 x 106Z -> qq and 1.7 x 10 6 Z —> £+£~ events collected at ~ 7 energies about the Z mass. The total SLC sample consists of 557,000 events collected with left-handed and right-handed electron beams. Typical Z candidates recorded are shown in Figure 3. Starting in 1996, LEP increased the beam energy to produce W boson pairs (LEP-2). Three diagrams contribute to the W pair production: W+ W+

w+

w+ The neutrino exchange diagram dominates near the W pair threshold, yfs ~ 2Mw — 161 GeV. As A/S increases, the contributions of the other two diagrams get larger, but there is a negative interference between them. This is illustrated in Figure 4, where the expected cross-section is shown with the full Standard Model structure, and if one or both of the diagrams with triple vector boson couplings is omitted, unitarity is violated. Figure 5 shows a W+W~ candidate recorded at the LEP-2 experiments. The cross sections in e+e" annihilation with y/s <7_00 GeV are summarized in Figure 6. Dominant processes are e + e~ -> 7* -» / / at *fs < Mz, e+e~ -t Z ->• / / at Vs - Mz, and e+e" -> W+W~ at y/s > 200 GeV.

91

^"••"'"'•"•nF

Figure 3: An e+e

Z —> / i + / i candidate (Top) recorded at the OPAL experiment and candidate (Bottom) recorded at the L3 experiment at LEP-1.

92

02/03/2001

LEP

Preliminary *

*4-T«u

Racoon WW / YFSWW 1.14 no ZWW vertex (Gentle 2.1) onlyv exchange (Gentle 2.1)

160

170

180

190

200

210

ERm [GeV]

Figure 4: LEP-2 average W pair production cross-section, corrected to correspond to the three doubly-resonant W production diagrams.

93

' D E L P H I R I P ' *&£, l.im; I * J . 4 CiV "TH^I DASi M - A p r - I M !

Ijgr^

| | : 43; it

109372 E M . 5483 P r • c ;1 » -Api -1 1uI S e i . • 1P•A pr - ! 0) 8

n • J

n X2oo J

c I

o I

Figure 5: An e + e —> W+W candidate recorded at the DELPHI experiment at LEP-2, where W+ and W~ decays each into a pair of quarks giving a four jet topology.

100 Vs [GeV]

200

Figure 6: Cross sections in e+e

500

annihilation.

94

Z and W production at the Tevatron The primary production of Z and W bosons at the Tevatron arises from the reactions: uu -> Z, dd -» Z, ud -> W+, and ud -t W~, where the up and down quarks (and antiquarks) can be Valence or sea quarks in the proton c :

E xiE u

— - — « -

E X2E •*-=•—

7 d

Although both Z and W± are produced through quark-antiquark annihilation, the dominant contribution is not from the valence-valence collisions but from valence-sea collisions. The typical qq center-of-mass energy %/I for W, Z production is the mass of the boson, %/I ~ Mz,w- Since these boson masses are around 100 GeV, about 1/20 of the pp center-of-mass energy, both valence and sea quarks have a good probability for carrying a sufficient fraction of the proton's energy to produce a gauge boson (see Figure 1). The valence-sea production mechanism is about 4 times larger than the valence-valence and sea-sea production mechanisms. It is coincidental that the valence-valence and sea-sea mechanisms are about equal at this energy. At higher energies such as LHC, the sea-sea mechanism dominates; at lower energies such as SppS where the center-of-mass energy was 560 GeV, the valence-valence mechanism dominates.

Top-quark production at the Tevatron The Tevatron has been (and will be until LHC turns on) the unique place to produce top quarks. The dominant top-quark production mechanism at the Tevatron is the annihilation of the valence quark and the valence antiquark into a gluon, which then decays into a tt pair. One of the tt cadidates recorded at the Tevatron is shown in Figure 7. The dominant top-quark production mechanism at the LHC (proton-proton collider) will be the annihilation of a gluon and a gluon into a gluon, which in turn decays into a tt pair. c T h e proton consists of three valence quarks (uud) which carry its electric charge and baryon quantum numbers, and an infinite sea of light qq pairs.

95

CDF Top Event M

= 79 GeV/c2

Jet 2 Jet 3

Figure 7: A tt candidate recorded at the CDF experiment, where t —> W+b t —• W~b —y qqb.

—• e+ffe and

Summary Table 2 summarizes the number of W, Z and tt events identified at LEP-1, LEP-2, SLC and the Tevatron. 2.4

Survey ofW,

Z, and Top quark Properties

The W, Z, and top particles (and the other elementary particles in the Standard Model) are structureless upto our current resolution, which is about 10~ 16 - 10~ 17 cm; Note that the top-quark mass is enormous, as heavy as a gold atom which consists of 79 protons and 118 neutrons and yet it is structureless. Coincidentally the top-quark mass is about the sum of the Z boson mass and the W boson mass. The top quark and W and Z bosons decay immediately after they are produced; their lifetimes, provided by their width measurements, are

96 Table 2: The number of W's, Z's and top quarks.

Heavy Particle Z

W top

Accelerator LEP-1 SLC Tevatron LEP-2 Tevatron Tevatron

C M . Energy ~91 GeV ~91 GeV 1.8 TeV 132 ~ 208 GeV 1.8 TeV 1.8 TeV

# of events 17,221,000 557,000 9,000 10,000 180,000 100

around 10~ 25 sec. With current date techniques, we can not trace their decays in space. Z's decay into a fermion (a quark or a lepton) and its anti-fermion, and W's decay into a fermion and its weak interaction partner: W+ —> e+ve, n+Vn, T+VT, ud, and cs. Top quarks decay about 100% of time into a W and a b quark, the weak interaction partner of the top quark (t —> W+b). B hadrons (hadrons containing b quarks) live long, 1.5 x 10~ 12 s, at the elementary particle scale. For instance, b quarks from the top decay which have a momentum of ~50 GeV travel about 4 mm before decaying, which is large enough to trace down in space. This is demonstrated in Figure 7 by the displaced vertices (seconary vertices). 2.5

Detectors

In order to identify various types of particles such as electrons, muons, taus, neutrinos, and 6-quarks, which are the products of Z, W, and top decays, a typical experimental detector consists of layers of devices as shown in Figure 8. Imagine you are riding on a particle that was just produced by the collision of a proton and an anti-proton or an electron and a positron. It encounters a thin beam pipe. It then zips through a silicon device, capable of resolving very tiny distances (~100 /xm), thus identifing fo-quarks in the event as shown in Figure 7. After the silicon device, it zips through a gas containing an immense number of very thin gold wires. The particle passes ~100 of these wires. If the particle is charged, each nearby wire records its passage, and the particle's path is determined (see Figures 3,5,7). A measurement of the curvature (this device and the silicon device are typically inside of magnetic field B produced by a solenoid) gives us the momentum of the particle: qBr

97

Figure 8: The beam's view of a typical detector.

where q is the charge of the particle, r is the inverse curvature, and c is the speed of light. If the particle is neutral, there will be no signal on the wire. Next the particle passes through coil of solenoid magnets. It then passes into a calorimeter section, which measures particle energy except muons and neutrinos. Different particles interact differently with matter, thus with calorimeter. If an electron, it fragments on a series of closely spaced thin lead plates and scintillator between the plates, giving up its entire energy in 3 or 4 inches. If a hadron, it penetrates 10 to 20 inches of calorimeter material before exchausting all of its energy through nuclear collisions. If it is a muon, it zips through the calorimeter sections and leaves hits in a gas containing thin gold wires which is located outside of the calorimeter. Neutrinos leave the detector entirely, leaving behind not even a hint of their fragrance. The system stores the data with about one million bits of information for each event.

98

3 3.1

Precision Electroweak Measurements Tree-level formulation

QED can be described by one parameter e or a = e2/(47r) where l/a — 137.03599959(38)(13), most precisely derived from g - 2 measurements. 3 In the electroweak theory, the strengths are specified by three paramters. They are two gauge coupling constants g and g', and v, the vacuum expectation value of the Higgs field (see Chris Quigg's lectures in this proceedings). These three parameters as inputs predict charges, Mw, Mz, and Gp via the following relationships:

sin2 w = 1

°

~M

=

n™ V2GFM^ e2

(1) (2) V ;

(3)

9

2

e

The theory is democratic so that any three variables in the above equations may be chosen as inputs. It makes much more sense to choose the three parameters most accurately measured so that the predictions for any other measurements become as precise as possible, allowing sensitive tests of the theory. In the early 1980s, those were e (or a), GF d and the electroweakmixing parameter s'm29w- With the measured values of these parameters, the masses of the W and Z bosons were prediced to be around 80 GeV and 90 GeV, respectively. This prediction lead to the discovery of the W and Z boson at SppS at CERN. 1 By the end of August 1989, the 4 LEP experiments measured Mz to within 160 MeV. Since then, a, GF and Mz were adopted as the basic input parameters. The current Mz accuracy by LEP is 2 MeV (0.002%). 4 Most likely, this measurement won't be improved in the near future. 3.2

Higher Order Corrections

Once higher order corrections are included, all these equalities are no longer exactly true. In gauge theories, higher order corrections such as the loop diagrams lead to infinities which require renormalization. A general consequence d

GF = (1.16637 ± 0.00001) x 10~ 5 GeV~ 2 is derived from the muon lifetime measurements using the radiative corrections of the V — A Fermi theory.

99

of this is the introduction of Q2 dependent corrections to the parameters of the theory, therefore corrections to the relationships among the parameters. For instance, the Born level relationship GF

- y/MPy, Ml - M ^

W

will be modified by M r - l ™ z ^"l-Ar^M^Ml-M^

(to W

in higher orders. Here Ar ~ Aro — pt/tan#iy Ar 0 ~ 1 - a{0)/a(Mz) ^ 0.06

(7) (8)

where a(Mz) is the electromagnetic coupling constant at the scale of the Z mass and the top quark contributes pt- As shown in Aro, the bulk of the corrections can be observed in "running" (Q 2 -dependent) coupling. The uncertainty in a(Mz) is dominated by the contribution of the light quarks (it through 6), A a ^ ( M f ) 1 2 :

This is evaluated using dispersion relations and a{e+ e~ —> hadrons) measurements at low \ / s , and perturbative QCD calculations at large yfs. The relationship can also be written as Mw MzcosOw

l2

= 1

(10)

in the lowest order, and MW Mzcos0w

l

ia g 1 +

3GFMJ 8\/27r2

~ 1 + 0.0096

( n )

Mt 175GeV

100

with higher order corrections. The second term in this equation (or pt in Eq. 9) is contributed by the top quark in loop diagrams: t t W+-

W+

With Mtop ~ 175 GeV, the correction is about 1% level. Thus precision measurements of My/, Mz, and cos9y/ with much better than 1% accuracy can predict the top-quark mass. The predicted top mass from electroweak measurements is Mtop = 172+Jj GeV. 4 The top mass measured by the CDF and D 0 Tevatron experiments is Mtop — 174.3 ± 5.1 GeV, 5 which agrees with the prediction very well. This is a good example of the successful interplay between theory and experiments. Any inconsistency between the predicted and measured values would have hinted at new physics. Secondary contributions to electroweak observables (or to Eq. 11) are from the Higgs boson in loop diagrams H H W+-

W+

w+ which are proportional to HM2H/M2W) where the Higgs mass M # is unknown in the Standard Model. New particles beyond the Standard Model could also contribute to electroweak observables through loop diagrams. Difficulties arise due to these unknown masses, Mjj and M n e w particle- This could be bad news since the predictions are uncertain and it is hard to test the theory. Or this could be good news since the predictions depend on the unknown masses. Indeed precision measurements provide information about these unknown masses of particles which are too heavy to be produced directly, or whose production cross section is too small to be observed. For example, with ~30 MeV uncertainty in Mw and ~ 2 GeV uncertainty in Mtop, we can predict the Higgs boson mass within ~30%. When precision measurements are inconsistent each other (for example, different sets of precision measurements predict different Higgs masses), this may signal the presence of new physics beyond the Standard Model.

101

3.3

The Experimental Inputs

The last decade has been a remarkable improvement in our knowledge of various electroweak parameters. Much of the improvement is due to the study of the Z resonance at LEP-1 and the SLC, and the study of W and top quark at LEP-2 and the Tevatron. Z boson parameters The precise determination of the Z boson parameters from the measurements at the Z resonance by the four collaborations ALEPH, DELPHI, L3 and OPAL in e + e~ collisions at LEP and by the SLD collaboration in e+e~ collisions at the SLC is a landmarkforprecision tests of the electroweak theory. Cross section at the Z peak The coupling of the Z to a fermion (/) and an anti-fermion (/) pair is described by the following Lagrangian density, L = (^f)1/2*n»(vf

- an5)$fZ»

(12)

= (^M)i/2*/7/i[ff/(i_75)+5/(i+75)]$/^

(13)

where vj and a/ are vector and axial vector coupling constants, and g*L = (vf + a,f)/2 and g^ = (vf — a/)/2 are left-handed and right-handed combinations. The vector and axial vector couplings are related to the quantum numbers of the fermion as follows vf = y/pj(2l(

- 4Q / sin 2 0 / )

(14)

af = y/pj(2l()

(15)

where I[ is the third component of weak isospin, Qf is the electric charge, and the parameters pf ~ 1 and sin 2 0/ ~ 0.23 incorporate electroweak radiative corrections. The cross section for the process e+e~(Pe) -» Z —> ff is described in the center-of-mass frame by the following expression, daf/ dtt

=

9 sTeeTff/Ml 4 (s - M | ) 2 + s 2 r | / M | [(1 + cos26>)(l - PeAe) + 2cos6Af{-Pe

l

+ Ae)]

j

102

where Pe is the polarization of the electron beam, s is the square of the centerof-mass energy, Tz is the total width of the Z, 9 is the angle between the incident electron and the outgoing fermion, Fff is the partial width for Z —• / / , and Af is the left-right coupling constant asymmetry. The partial widths and coupling constant asymmetries are related to the couplings defined in the Lagrangian,

2vIaL_

=

v

=

a

jgtf

- (g/)» 2

(^) + (
)+ )

where 8 ~ 1 + SaQ^/Air + nfas/-rr, (n/ is 1 for quarks and 0 for leptons) accounts for final state QED and QCD radiative effects. The small size of i^/a^(~ 0.08), where £ = e,/i,r, makes the leptonic coupling asymmetries A( particularly sensitive to electroweak vacuum polarization corrections. The leptonic asymmetries are usually parameterized in terms of s'm29y(, = sin 2 ^ (assuming lepton unversality). It follows that small changes in sin 2 #^>' produce large effects on At, =

'

2(l-4sin2^) l + (l-4sin2^/)2

^ - - S - ^ s i n

2

^ ) -

(20)

Electroweak Observables at the Z pole The cross section described in Eq. 16 is only the dominant term in the total s-channel e+e~ cross section which can be expressed as dafJt(s)

dn

=

daff(s)

dft

dcr^js)

dn

daf/(s)

dn

'

[

'

where the first and second terms represent photon exchange, and the interference between photon and Z diagrams, respectively. At ^/s = 30 — 40 GeV (the centre-of-mass range for the PEP and PETRA colliders), the dominant contribution comes from al^ and the a z contribution to the total cross section is about 2 - 3%. At ^fs ~ 60 GeV (the centre-of-mass range for the TRISTAN collider), the a'z contribution to the total cross section increases to about 25%. The contribution from az is negligible in both cases. The cross section

103

10

£?

10'

§ !^3 t*2

P

10

O 13

1 0 " =-

10

I I I I I I I I I I I I I I I I I I I 1I I I I I I I I I

0

20

40

60

80

100

Center of Mass Energy Figure 9: Cross-section of e+e~

120

[GeV]

—• / / .

near yfs = Mz does not differ dramatically from the resonance cross section given in a'z' because the Z — 7 interference term vanishes at the pole, and the photon exchange cross section is approximately 1000 times smaller than the Z-exchange cross section. The cross-section measurements for fermion pair production from e+e~ annihilation at yfs < 120 GeV are summarized in Figure 9; the agreement between the measurements and the Standard Model predictions are excellent. Using the information contained in hadronic and leptonic cross sections around the Z pole, it is possible to define a number of experimental observables: 1. The line shape parameters, which consists of the Mz and Yz whre the definition is based on the Breit-Wigner denominator (s-M^ + isTz/Mz) with s-dependent width, and the peak hadronic cross section a°h (see

104

Figure 10) o°h = 12nTeerhad/(M*r%);

(22)

2. The cross section ratios Re = Te/rhad,

Rb = Tbb/Thad, Rc = Tcc/Thad

(23)

where I — e, n, r; 3. The unpolarized forward-backward asymmetries A

FB = -J f=0.75AeAf, aF + aB

f = £,b,c

(24)

where aJF is the cross section for finding the scattered fermion in the hemisphere defined by the incident electron direction and oL is the cross section for finding it in the positron hemisphere; 4. The left-right asymmetry which is defined as PA P

^

-PA A

L

R

-

P

^

_
f

, , * 6

,„. {25)

where a?(Pe) is the total cross section for the production of / / pairs with an electron beam of helicity Pe; 5. The polarization of final state r-leptons which depends upon the direction of the T, Ae, and AT, PAcose) = -

AT{l + cos29) + 2Aecos6 i + cos2e + 2 A ^ c o s e -

(26)

6. The left-right forward-backward asymmetries {f = FB

4(~\Pe\) - 4HPe\) afF(-\Pe\) + 4(-\Pe\) 4(-\Pe\)+4(~\Pe\)-0.75PAf,

~ 4( + \Pe\) + 4( + \Pe\) + 4(' '+ -\Pe\) " • + 4(f ' +-\Pe\) "

f=i,s,c,b.

The presence of initial-state QED radiation smears the center-of-mass energy. With the initial state QED corrections, ^ i is reduced and a changes. On the Z resonance, the cross-section is reduced by ~30%, as demonstrated in Figure 10.

105 •

|

i

i

i

|

.

i

i

|

I

I

.

1

1

'

1

•

*'\ !* \\ ** * ; »

ALEPH DELPHI L3 OPAL

30

*

:

* / * / * &

20 r

-•

. _ -

\

» /*\ * * / * \ *• i

10

£~*

yv------"'

40

\* \. * -

i

measurements, error bars / / increased by factor 10 / / — a from fit • - - QED unfolded

/

\

;

/ /J / / V

1

86

,

,

,

1

88

.

,

,

1

90

. ,sLMz, .

92

.

,

1

94

E cm [GeV] Figure 10: Average over measurements of the hadronic cross-sections by the four experiments, as a function of centre-of-mass energy. The dashed curve shows the QED deconvol v e d cross-section, which defines the Z parameters.

The net correction to r-polarization and the left-right asymmetry is less than 2%, and that to the leptonic forward-backward asymmetries is about 100%. A summary of the combined LEP-1 results is presented in Table 3 . 4 Note the remarkable precision of these measurements, especially the Z mass which is measured to 2.2 x 10~ 5 . It is the third most precisely determined electroweak parameter and used as an input to the theory. The measurement of the left-right cross section asymmetry (ALR) by SLD at the SLC provides a systematically precise, statistics-dominated determination of the coupling Ae, and is presently the most precise single measurement, with the smallest systematic error, of this quantity. In principle the analysis is straightforward: one counts the numbers of Z bosons produced by left and right longitudinally polarized electrons (Nz(L) and NZ(R)), forms an asymmetry, and then divides by the e~ beam polarization magnitude (the e + is not

106 Table 3: Summary of the Z parameter measurements at LEP-1 and SLC. The middle box shows the combined result of the LEP lineshape analyses. R[ and ApB assumed lepton universality. The bottom box presents the result from the SLC analysis.

Z parameter LEP-1 Mz Tz < Re

Average Value

sm c;

91.1875 ±0.0021 GeV 2.4952 ± 0.0023 GeV 41.540 ± 0.037 nb 20.767 ±0.025 0.0171 ±0.0010 0.1439 ±0.0042 0.1498 ±0.0048 0.2321 ±0.0010

sin%ff

0.23098 ± 0.00026

A

FB

AT Ae 2

SLC

polarised): ALR

=

1 NZ(L)-NZ(R) PeNz(L)+Nz(R)-

(28)

The average electron polarization at the interaction point has been in the range 73% to 77%. The measured value of ALR 4 can be translated into the value of s i n 2 ^ = 0.23098 ± 0.00026.

(29)

The decays of the Z to neutrinos are invisible in the detectors and give rise to the "invisible width", Tinv — NVTVU, where Nv is the number of light neutrino species. The invisible width can be determined from the measurements of the decay widths to all visible final states and the total width, which is given by the sum over all partial widths, L

Z —t

ee

+ 1 mi T 1 T-T T 1 had ± t

inv

(30)

The ratio of the invisible and leptonic widths is found to be I W r « = 5.941 ± 0.016.

(31)

Dividing this number by the Standard Model value for the ratio of neutrino and leptonic widths r „ „ / r « = 1.9912±0.0012 yields the number of light neutrinos, Nv = 2.9835 ± 0.0083

(32)

107

1 I I I I I I 1 I I i I I I 1 1 1 I i I I I I I I I

87

88

89

90

91 ^=Ecm

92 93 (GeV)

94

95

96

Figure 11: The Z lineshape at LEP-1. The three curves represent the predicted lineshapes with the assumption of two (dashed line), three (solid line), and four (dotted line) light neutrinos.

(see Figure 11). Note that N„ is about 2
(33)

The accuracy of the measurements makes them sensitive to the mass of the top quark and to the mass of the Higgs boson through loop corrections as stated in section 3.2. An example of such sensitivity is presented in Figure 12. Predicted values from all the Z parameters are Mw = 80.374 ± 0.034 GeV,

(34)

Mtop = 169^° GeV,

(35)

MH

56+^ GeV.

(36)

108

Total width T,

24SV.f»

J.2Mt*"-T

2499.6 ± 4.3 M e V

2494.6 ± 2.7 M e V common 1.7 MeV not com 2.1 MeV X2/dof= 3.3/3

91 1.86 ± 2 M e V 60-1000GeV

2480

2490

2500

T z [MeV]

Figure 12: Summay of the Z width measurements at LEP-1 and the sensitivity of the Z width to the mass of the top quark.

109 Table 4: Typical W pair event selection efficiencies and purities.

Decay mode WW -¥ qqqq WW -> qqtv WW -+ tvtv

EfRciency 90% 82% 60 - 80%

Purity 80% 90% 90%

W boson parameters The precise determination of the W boson and top-quark parameters was initiated by the CDF and D 0 collaborations at the Tevatron. Since 1996, the LEP-2 program made a significant contribtuion on the W parameters. These make complementary tests of the Standard Model to those of LEP-1 and SLC. Together with the Z parameters, efforts to constrain MH have become increasingly interesting. W branching fractions, W mass and width from LEP-2 at CERN By doubling the e+e~ collision energy, LEP-2 provided a sample of W pair events. The typical selection efficiencies and purities for W pair events at LEP-2 are given in Table 4. The branching fractions for W decays via the electron, muon, tau, and hadronic modes have been measured by all four experiments 4 . The LEP-2 average results are given in Table 5. The results for the individual leptonic channels are consistent with lepton universality, and the average leptonic branching fraction (10.74 ±0.10)% is also consistent with the Standard Model expectation. The leptonic W branching fraction can be re-interpreted in terms of the CKM matrix element Vcs without need for a CKM unitarity constraint, using the relatively well-known values of other CKM matrix elements involving light quarks 6 . These indirect constraints currently lead to a value 4 of |VCS| = 0.989 ± 0.016, much more precise than the value derived from D decays of 1.04±0.16 6 . The W mass can be extracted from the measurement of the W pair cross section at the W pair threshold, \/s ~ 2Mw- See the measurement point at y/s = 161 GeV in Figure 4. The value of y/s = 161 GeV was chosen from the 1995 Tevatron Mw measurement. At center-of-mass energies above the W pair threshold, the technique for measuring the W mass lies in the reconstruction of the directions and energies of the four primary W decay products. These may be either four quarks, approximated by four jet directions and energies, for the qqqq channel; or two quarks (or two jets) and a charged lepton for the

110 Table 5: LEP average W decay braching fraction measurements.

Decay mode WW ->ev WW•->

fit/

WW ->TP WW -+qq

Branching fraction (%) 10.62 ±0.20 10.60 ±0.18 11.07±0.25 67.78 ±0.32

qqiv channel, deducing the neutrino direction and energy from the missing momentum in the event: P(e+)+P(e-)

= 0 = P(W+)+P(W-)

(37)

+

P{£ ) + P{yi) + P{q) + P(q) = 0 Et+ + EVl +Eq + Eq = vrs = 2Ebeam.

(38)

Decay to Iviv are of limited use because at least two neutrinos are undetected. The W decay products are paired up to give reconstructed W mass estimates. A substantial improvement is made in the mass resolution for both qqqq and qqiv channels by applying a kinematic fit, constraining the total energy and momentum in the event to be that of the known colliding electron-positron system as shown in Eq.s 37 and 38, and making a small correction for possible unobserved initial-state radiation. A typical reconstructed mass distribution from the kinematic fit are shown in Figure 13, for the qqiv channel. A clear W mass peak is observed with low background. The W mass is extracted from the measured W masses in each data event using a Monte Carlo technique, the details of which differ from one experiment to another. The Monte Carlo techniques have in common that they use full detector simulations to correct for the effects of finite detector acceptance and resolution, as well as initial-state radiation. An example of the results obtained from the fits are shown in Figure 13 (histogram). For all of these fits, the W width is taken to have its expected Standard Model dependence on the W mass. The combined W mass from direct reconstruction, taking into account all correlations including those between the W+W~ —> qqiv and W+W~ -¥ qqqq channels, gives Mw = 80.428 ± 0.030(stat.) ± 0.036(syst.) GeV,

(39)

with a x 2 /d.o.f of 27.1/29, corresponding to a x 2 probability of 57%. Table 6 summarizes the combined result from each experiment at LEP-2. The results

111 Table 6: Summary of the W mass measurements from direct reconstruction (v/s = 172 — 202 GeV) at LEP-2. Results are given for W+W~ -> qqlv and W+W~ -> qqqq channels. The x2 per d.o.f is 27.1/29.

Experiment ALEPH DELPHI L3 OPAL Total

Mw (GeV) 80.449 ± 0.065 80.380 ±0.071 80.362 ± 0.078 80.486 ± 0.066 80.428 ± 0.047

are consistent with the W mass extracted from the measurement of the W pair threshold cross-section at 161 GeV (see Figure 4): Mw = 80.40 ± 0.20(stat.) ± 0.03(syst.) GeV.

(40)

The overall LEP average W mass measurement obtained is: Mw = 80.427 ± 0.046 GeV.

(41)

The width of the W mass distributions shown in Figure 13 has components from the true W width and from detector resolution. In many events, the mass resultion is comparable to, or better than, the true width. It is consequently possible to measure directly both the W mass and width, and in practice the two results are little correlated. The combined result is Tw = 2.12 ± 0.20 GeV.

(42)

W branching fraction, W mass and width from the Tevatron at Fermilab At the Tevatron, the W bosons are only detected in their decays to eu and [iv because the decay to qq' is swamped by the QCD dijet background whose cross section is over an order of magnitude higher than the mass range of interest. Also one does not know the event s and one can not determine the longitudinal (parallel to the beam direction) neutrino momentum because a significant fraction of the products from the pp interaction are emitted at Small angle to the beam pipe where there is no instrumentation. Consequently, one must determine the W mass from transverse (perpendicular to the beam direction) quantities 7 , i.e. : the transverse mass Mr, the charged lepton

112 100 ^


a

ALEPH Preliminary

40

i—i i-

(i)

Vs = 191.6,195.5,

evqq selection •

XI)

&. ^ 70 a 60

199.5,201.6 GeV

Data (Luminosity = 237 pb"1) MC (m w = 80.60 GeV/c z )

|"L

Non-WW background

w

50 40 30 20 10 'i

50

i.i.iii.i

55

60

•^••••i-

65

70

75

80

85

90

95

M w (GeV/c2) Figure 13: Reconstructed W mass distribution compared to the best fit for the e+e W+W~ -> e+isqq' channel at the ALEPH experiment at LEP-2.

transverse momentum Pr, or the neutrino transverse momentum PT (or the missing transverse energy $,?). $T is inferred from a measurement of PT and the remaining PT in the detector, denoted by u, i.e. gluons

F4f +Pr

= 0

PT + PT + u = 0 or P^ = £T = -{PT + u).

(43)

113

The following cartoon demonstrates kinematics of W boson production and decay, as viewed in the plane transverse to the antiproton-proton beams, where the recoil energy vector u is the sum of the transverse energy vectors E%T of the particles recoiling against the W:

£

1 2

u receives contributions from two sources: first, the so-called W recoil, i.e. the particles arising from initial state QCD radiation from the qq legs producing the hard scatter, and, second, contributions from the spectator quarks (pp remnants) and additinal soft scatters by p'p' which occur in the same crossing as the hard scatter. This second contribution is generally referred to as the underlying-event contribution. Experimentally, these two contributions cannot be distinguished. Owing to the contribution from the underlying-event, the missing transverse energy resolution has a significant dependence on the instantaneous pp luminosity. Transverse mass of W, M™, is defined as M$ = ^ / 2 P | P ^ ( l - c o s 0 )

(44)

where is the angle between PJf and Pj.. M-j- is to first order independent of the transverse momentum of the W (P™) where as Pj. is linearly dependent

114

on Pf. For this reason, and at the Tevatron luminosities where the effect of the # T resolution is not too severe, the transverse mass is the preferred quantity to determine the W mass. However, the W masses determined from the PT and $T distributions provide important cross-checks on the integrity of the MT result because the three measurements have different systematic uncertainties. Figure 14 and 15 show the transverse mass (M-Jf) and lepton transverse momentum (PT) distributions of W events. The W mass at the Tevatron is determined through a precise simulation of the transverse mass line-shape, which exhibits a Jacobian edge at MT ~ MwThe simulation of the line-shape relies on a detailed understanding of the detector response and resolution to both the charged lepton and the recoil particles. This in turn requires a precise simulation of the W production and decay. The similarity in the production mechanism and mass of the W and Z bosons is exploited in the analysis to constrain many of the systematic uncertainties in the W mass analysis. The lepton momentum and energy scales are determined by a comparison of the measured Z mass from Z -» e + e~ and Z —> H+H~ decays with the value measured at LEP. The simulation of the W PT and the detector response to it are determined by a measurement of the Z PT which is determined precisely from the decay leptons and by a comparison of the leptonic (from the Z decay) and non-leptonic ET quantities (u) in Z events. The reliance on the Z data means that many of the systematic uncertainties in the W mass analyses are determined by the statistics of the Z sample. The W and Z events in these analyses are selected by demanding a single isolated high PT charged lepton in conjunction with missing transverse energy (W events) or a second high PT lepton (Z events). Depending on the analyses, the $T cuts are either 25 or 30 GeV and the lepton PT cuts are similarly 25 or 30 GeV. In total, ~84A; events are used in the W mass fits and ~9fc Z events are used for calibration. The Tevatron average W mass measurement obtained is 8 Mw = 80.448 ± 0.062 GeV.

(45)

The Tevatron experiments determine the width by a one parameter likelihood fit to the high end of the transverse mass distribution (see Figure 16). Detector resolution effects fall off in a Gaussian manner such that at high transverse masses (MT > 120 GeV), the distribution is dominated by the Breit-Wigner behavior of the cross section. In the fit region, CDF has 750 events in the electron and muon channels combined. The result 9 is I V = 2.055 ±0.125 GeV.

(46)

115

Transverse Mass (GeV/c )

g 700 > t+H 600 "-

o

S-H

j£ 500 §400

I /

300 ; 200

tiA tr \

/ \

100

0,l^-n-H-l

II

50 55 60 65 70 75 80 85 90 95 100

m T (GeV) Figure 14: W Transverse mass distributions compared to the best fits for the W channel from the CDF experiment (Top) and the D 0 experiment (Bottom).

116

C/3 •(—>

Si 200

4

> CD

"oiooo

M

VH

Oi

M

' § 800 tf

=3

C

600 400 -A t

V

200

\* °25

30

35

40

45

50

55

pT(e) (GeV) Figure 15: The electron transverse momentum distribution compared to the best fit for the W —• ev channel from the D 0 experiment.

117

40

60

80

100

120

140

160

180

200

MT(e,v) (GeV)

140

160

180

200

MT(n,v) (GeV) Figure 16: Transverse mass spectra (filled circles) for W —• ev (Top) and W —> [LV (Bottom) data from the CDF experiment, with best fits superimposed as solid curves. The lower curve in each graph shows the sum of estimated backgrounds. Each inset shows the 50 — 100 GeV region on a linear scale.

118

At LEP-2, the W branching fractions are determined by an explicit cross section measurement, while at the Tevatron they are determined from a measurement of a cross section ratio. Specifically, the W branching fraction can be written as: o.Br{W^ev)=a^T{Z-'ee) oz Tz

-1 R

(47)

where

_ % • Br{W -» ev) ~ az-Br(Z^ee)

R

(48)

is the measurement made at the Tevatron. This determination thus relies on the LEP-1 measurement of the Z branching fractions and the thoretical calculation of the ratio of the total Z and W cross sections. The Tevatron measurement Br(W —> ev) = (10.43 ± 0.25)% is now becoming systematics limited. In particular, the uncertainty due to QED radiative corrections in the acceptance calculation and in aw/o~z contributes 0.19% to the total systematic uncertainty of 0.23%. The corresponding measurement from LEP-2 is Br(W -> ev) = (10.62 ± 0.20)%. The large sample of W events at the Tevatron has also allowed a precise determincation of gT/ge through a measurement of the ratio of W —> rv to W —> ev cross sections. The latest Tevatron measurement of this quantity is gT/ge = 0.99 ± 0.024, in good agreement with the Standard Model prediction of unity and the LEP-2 measurement of gT/ge = 0.101 ± 0.022. W mass prediction from NuTeV at Fermilab Neutrino scattering experiments have contributed to our understanding of electro weak physics for more than three decades. Early determinations of sin29w served as the critical ingredient to the Standard Model's successful prediction of the W and Z boson masses. More precise investigations in the late 1980s set the first useful limits on the top quark mass. The recent NuTeV measurement of the electroweak mixing angle from neutrino-nucleon scattering represents the most precise determination to date. The result is a factor of two more precise than the previous most accurate j/N measurement 10 . In deep inelastic neutrino-nucleon scattering, the weak mixing angle can be extracted from the ratio of neutral current (NC) to charged current (CC) total cross sections; ajv^N -» VltX) - ajv^N -> v^X) a^N ->• fj,-X) - o-^N -> n+X)

K

'

119

R" -rW =

:

1 =

o

„ _

S m

B

W-

1—r 2 where Rv = cr(v^N —> vtlX)/a(vllN -> n~X). Because R~ is formed from the difference of neutrino and anti-neutrino cross sections, almost all sensitivity to the effects of sea quark scattering cancels. This reduces the error associated with heavy quark production (principally due to the imprecise knowledge of the charm quark mass) by roughly a factor of eight relative to the previous analysis. The substantially reduced uncertainties, however, come at a price. The ratio R~ is difficult to measure experimentally because neutral current neutrino and anti-neutrino events have identical observed final states. From the i/N interactions, 386 k NC and 919 k CC events are recorded and from the PN interactions, 89 k NC and 210 k CC events. The extracted value of sin29w (on-shell) = 0.2254 ± 0.0021 which can be translated in to an My/ value of 80.25 ± 0.1(stat.) ± 0.05(syst.) GeV, where the systematic error also receives a contribution from the unknown Higgs mass. Summary of W mass, width, and branching fraction

measurements

The LEP-2 mass values are compared with the Tevatron values in Figure 17. They are in excellent agreement despite being measured in very different ways with widely different sources of systematic error. The systematics at LEP-2 are dominated by the uncertainty in the beam energy (which is used as a constraint in the mass fits) and by the modeling of the hadronic final state, particular for the events where both W bosons decay hadronically. At the Tevatron, the systematics are dominated by the determination of the chareged lepton energy scale and Monte Carlo modeling of the W production, in particular its Px and Pz distribution. At the Tevatron, one can not use a beam energy constraint to reduce the sensitivity of the W mass to the absolute energy (E) and momentum (p) calibration of the detector. Any uncertainty in the detector E, p scales thus enters directly as an uncertainty in the Tevatron W mass. This means that the absolute energy and momentum calibration of detectors must be known to better than 0.01%. By contrast at LEP, an absolute calibration of 0.5% is sufficient. These thus provide welcome complementary determinations of the W mass. These direct measurements are also in good agreement with the indirect measruement from NuTeV and the prediction based on fits to existing, non W, electroweak data (the Z parameters). There is also good agreement between the LEP-2 and Tevatron measurements on the W width and branching fraction. Table 7 summarizes the W parameter measurements at LEP-2 and the Tevatron.

120

W-Boson Mass [GeV] pp-colliders

•— 80.452 + 0.062 'I A 'ff

Average

4-

80.436 ± 0.037 X2/DoR0.1/1

LEP1/SLDA'N/mt -i

1

80

1

1

1

80.386 ± 0.025

1

1

80.2

1

1

1—n

80.4

1

1

1

1

1

r

80.6

m w [GeV] Figure 17: Mw measurements and the prediction from the electroweak measurements including M t o p .

121

Table 7: Summary of the W and top-quark parameter measurements at LEP-2 and the Tevatron. Parameter Mw Br(W -> Iv) Mtop

LEP-2 Tevatron 80.427 ± 0.046 GeV 80.448 ± 0.062 GeV 2.12 ± 0.20 GeV 2.055 ±0.125 GeV (10.74 ±0.10)% (e + M + r) (10.43 ±0.25)% (e) 174.3 ± 5 . 1 GeV -

Top-quark parameters The top quark discovery 2 at the Tevatron in 1995 was the culmination of a search lasting almost twenty years. The top quark is the only quark with a mass in the region of the electroweak gauge bosons and thus a detailed analysis of its properties could possibly lead to information on the mechanism of electroweak symmetry breaking. In particular, its mass is strongly affected by radiative corrections involving the Higgs boson. The emphasis in top quark studies at the Tevatron has been to make the most precisie measurement of the top quark mass. Substantial progress has been made in bringing the mass uncertainty down from > 10 GeV, at the point of discovery, to 5.1 GeV in 1999. In the Standard Model, a top quark decays ~100% of the time to Wb. If both W's decay to qq', the final state from tt is qq'qq'bb and the event sample is referred to as "all-hadronic". Conversely, the "di-lepton" event sample is realized by selecting an £+v£~Dbb final state, where both W's have decayed leptonically (to e or /i). The "lepton+jet" event sample is one in which one W has decayed hadronically and the other leptonically. The precision with which the top quark mass can be measured with each sample depends on the branching fractions, the level of background and how well constrained the system is. The all hadronic sample has the largest cross section, but has a large background from QCD six jet events (Nsignai/Nbackground ~ 0.3), while the dilepton sample has a small background (N S i gna i /Nbackgrounct ~ 4), but suffers from a small cross section and the events are "under-constrained" because they contain two neutrinos. The optimum channel in terms of event sample size, background level and kinematic information content is the lepton+jet channel. Indeed, this channel has a weight of ~80% in the combined Tevatron average. Figure 18 shows the reconstructed mass distribution using the lepton+jet event sample from the CDF experiment. The Tevatron combined value 5 is determined using lepton+jet, di-lepton, and all-hadronic data (see Figure 19) and accounting for all correlations between the measurements. The two experiments have assumed a 100% correlation on all systematic uncertainties related

122

to the Monte Carlo models. The uncertainty in the Monte Carlo model of the QCD radiation is one of the largest systematic uncertainties. This error source will require a greater understanding if the top mass precision is to be significantly improved in the next Tevatron run (Run II). The other dominant systematic error is the determination of the jet energy scale which relies on using in-situ control samples such as Z+jet and 7+jet events. In Run II, due to significant improvements in the trigger system, both experiments should be able to accumulate a reasonable sample of Z —» bb events which will be of great assistance in reducing the uncertainty in the 6-jet energy scale. The top-quark mass measurments from the CDF and D 0 experiments with various channels are summarized in Figure 19. The combined Tevatron mass value 5 is Mtop = 174.3 ± 3.2(stat.) ± 4.0(syst.) GeV = 174.3 ± 5 . 1 GeV.

(50)

This measurement is a supreme vindication of the Standard Model, which, based on other electroweak measurements, predicts a top mass of : Mtop = 172±i? GeV. Of all the quark masses, the top quark is now the best measured.

(51)

123

100

150

200

250

300

350

Reconstructed Mass (GeV/c')

Figure 18: The reconstructed top quark mass distribution compared to the best fit for the tt~+ (W*b) + (W~b) -* (£ + t/6) + (qi?b). The plot also shows the level of background.

124

Tevatron Top Quark Mass Measurements 168.4 ± 1 2 . 8 GeV/c 2 1 73.3 ± 7.8 GeV/cr

Dilepton Lepton+jets

172.1 ± 7.1 GeV/c 1

Combined

167.4 ± 11.4 GeV/cr

Dilepton

176.1 ± 7.4 GeV/c 2

Lepton4-jets

186.0 ± 11.5 GeV/V

All-Hadronic

t D0 I CDF

176.1 ±

6.6 GeV/c 2

Combined

Tevatron combined 174.3 ± 5.1 GeVAr

i""i 150

160

170

1111 11 11111

1S0

190 200

M^ (GeV/c2)

Figure 19: The summary of the top-quark mass measurements from the CDF and D 0 experiments at the Tevatron.

125

Summary and Outlook With the current uncertainties, the measurements of the W mass and topquark mass start to provide an interesting further test of the Standard Model relative to precision electroweak measurements at the Z resonance. This is illustrated in Figure 20; the predicted W and top masses extracted from fits to lower electroweak data are consistent with the direct measurements from LEP2 and Tevatron, and the precision of the measurements is similar to that of the prediction. From the overlaid curves showing the Standard Model expectation as a function of the Higgs boson mass, it is further evident that both the precise lower energy measurements, and the direct W and top mass measurements favor a Standard model Higgs boson in the relatively low mass region (see Figure 21). The 95% confidence level upper limit on MH (taking the band into account) is 165 GeV. The lower limit on MH of approximately 113 GeV obtained from direct searches 13 has not been used in this limit determination. For LEP-1, LEP-2 and NuTeV, no further data are planned. In contrast, the Tevatron has undergone a major luminosity upgrade augmented with substantial improvements in the collider's detectors. At least 2 fb _ 1 is expected per experiment at an increased center of mass energy of 2 TeV in three years. It is expected that both the top quark mass and W mass measurements will become limited by systematic uncertainties. The statistical part of the Tevatron W mass error in the next run will be ~10 MeV, where this also includes the part of the systematic error which is statistical in nature such as the determination of the charged lepton E and p scales from Z events. At present the errors non-statistical in nature contribute 25 MeV out of the total Tevatron W mass error of 60 MeV. A combined W mass error is expected to be 20 — 30 MeV. The W width is expected to be determined with an uncertainty of 20 — 40 MeV. The statistical uncertainty on the top quark mass will be ~ 1 GeV. The systematic error arising from the uncertainties in the jet energy scale and modeling of QCD radiation are expected to be the dominant errors, with a total error value of 2 — 3 GeV per experiment expected.

126

Figure 20: The direct measurements of the W and top quark mass from the Tevatron experiments, and the direct measurement of the W mass from the LEP-2 experiments and the indirect W and top mass measurement from LEP, SLC, and Tevatron neutrino experiments. The curves are from a calculation of the dependence of Mw on Mtop in the Standard Model using several Higgs boson masses. The band on each curve is the uncertainty obtained by folding in quadrature uncertainties on a{M%), a s ( M § ) , and Mz (Reference 11). The dominant contribution to the band width comes from the uncertainty on a ( A f | ) , A a h a d ( M | ) , the contribution of light quarks to the photon vacuum polarisation.

127

V

D

I el 3

'•'

Ac

fe I

Cd =

1 I 11 I

J:

V

\ — 0.02804±0.00065 Y /

V,

\-— 0.02755± 0.00046/ j

I "

\\

4-

I

2\

\

/

Excluded\ \ / / 0J 10 10 \

1 — i — i — i * i

*i i I'I

/

^

Preli ml nary 1—i—i—i—i

i 11

10"

m H [GeV] Figure 21: A * 2 = X2 - X2min vs. MH curve. The line is the results of the fit using all the Electroweak measurements; the band represents an estimate of the theoretical error due to missing higher order corrections. The vertical band shows the 95% CL exclusion limit on MH from the direct search. The dashed curve is the results obtained using the evaluation of A a ^ 5 ) d ( A f | ) from Reference 12.

128

(e)

(f)

(g)

Figure 22: Tree-level diagrams for W+ + W~ —> W+ + W~ scattering. Figure courtesy of "QCD and Collider Physics" by R.K. Ellis, W.J. Stirling and B.R. Webber.

4

Search of Electroweak Symmetry Breaking

As stated in Section 3, the Higgs sector can be brobed by the precision electroweak measurements by means of radiative corrections. However, the nature of the symmetry breaking sector can only be established by its direct discovery and detailed study. The LEP-2 experiments have searched for a light Higgs boson upto ~115 GeV. In March 2001, the CDF and D 0 experiments at the Tevatron will begin a high-luminosity run with considerable sensitivity to new physics, and offer promise for light-Higgs searches in the future. For more massive Higgs bosons, the focus of attention shifts to higher-energy colliders, in particular the LHC, where pp collisions will be studied at A/S = 14 TeV, beginning in about 2006. Future possible e+e~ linear colliders would offer complementary possibilities for the study of electroweak symmetry breaking. 4-1

Theoretical Constraints on the Higgs mass

The Higgs mass can not be too large; as it increases, the amplitude for WW scattering via Higgs exchange (see Figure 22 for the diagrams) becomes large and the contribution from the lowest-order diagrams exceeds the perturbative unitarity limit unless MH < 1 TeV. It is also true that very massive Higgs bosons are wide resonant states (see Figure 23), and therefore their detection

129

0

200

400

600 MH [GeV]

800

1000

Figure 23: The total Higgs decay with for Mtop = 175 GeV. Figure courtesy of "QCD and Collider Physics" by R.K. Ellis, W.J. Stirling and B.R. Webber.

in invariant mass distributions becomes difficult. Another limit on M # comes from consideration of the renormalized A parameter which appears in the Higgs potential V(4>+) = A((/>+0) 2 - n24>+

(52)

where is the single complex doublet of scalar Higgs fields. With the parameters A, /j,2 > 0, this potential has a minimum at <> / ^ 0 and the vacuum expectation value v (= 246 GeV) is equal to \J'/z2/2A. A at low energy (X(v)) can not be too large because it will increase with the scale and eventually a singularity will be reached. This gives an upper bound (Triviality bound) on the Higgs mass, which depends on the scale where new physics comes in. For example, MH < 175 GeV if the new physics is at A p i anck (1019 GeV), and MH < 1 TeV if it is at 0 ( 1 TeV). The Higgs vacuum will be unstable if the coupling strength between the top quark and the Higgs, Yukawa coupling gt, is large so that g2 is comparable to A. This can be avoided if MH > 2{Mtop - 110) GeV or MH > 130 GeV (Vacuum Stability bound), assuming no new physics below the Planck scale. In summary, the theoretical limits on the Higgs mass are 130 < MH < 175 GeV if new physics does not set in until the Planck scale, and MH < 1 TeV

130

if new physics comes in at 0 ( 1 TeV). In other words, the Higgs mass can be almost arbitrary. See a detailed discussion on the Higgs mass constraints in Chris Quigg's lectures in this proceedings. 4-2

Searches for the Standard Model Higgs

Searches at LEP-2 The LEP-2 experiments have searched for the Higgs boson via the process e+e" -> Z* -)• ZH ^ (ff + bb) or (ff + rf):

The final-state Z boson can be produced either on mass-shell if y/s > MH+MZ, or off mass-shell if ,/s < MH + Mz- Figure 24 shows the cross section as a function of v ^ for three different values of MH- The LEP-2 collider went upto the center-of-energy of 208 GeV, higher than the design value. With this machine, the Higgs mass can be discoverd upto ~110 GeV where Higgs decays mainly into bb or TT (see the branching fractions in Figure 25). At their highest energy, the 4 LEP-2 experiments have seen a 2.9 a hint with MH = 115 GeV (the excess is dominated by the ALEPH experiment) 14 . For instance, with a set of cuts which give the signal to background ratio of 2, the number of background events and the number of signal events with 115 GeV Higgs are estimated to be 1.72 and 3.03 (4 experiments combined), respectively. The number of candidates they observed is 4. With the same data, one can set the bound on the Higgs Mass; MH < 113 GeV is excluded at 95% Confidence Level. The LEP-2 was shutdown in October 2000, passing the baton to the Tevatron for the Higgs searches (see the cartoon, Figure 26, made by the ALEPH experiment). Searches at the Tevatron At the Tevatron, the CDF and D 0 experiments will begin a high-luminosity run in March 2001 with 10% increase in the center-of-mass energy, corresponding to ~30% increase in the production cross section. Both experiments have

131

e V - ZH

150

160

170

180 Vs [GeV]

190

200

210

Figure 24: Cross section for ZH production at LEP-2. Figure courtesy of "QCD and Collider Physics" by R.K. Ellis, W.J. Stirling and B.R. Webber.

undergone major upgrades on their detectors, triggers, and data handling systems. These upgraded accelerator and experiments offer discovery potential of the light Higgs boson before the LHC turns on. Figure 27 shows the crosssections as a function of the Higgs mass for various production mechanisms. For the Higgs lighter than ~ 120 GeV, the search strategy is via the process qq -> V* -»• VH -> ffbb,

where V is W or Z. Although many more Higgs bosons are produced via qq -> H -> bb, they will be very difficult to detect due to large QCD background via qq —> g —> bb.

132 i

:

i-•

|

._

bb

~

;

.^i^.

-

cc y y

\'A

'-

v

I

i

,

; --

150 MH [GeV]

1

1

=— —

'

zz

/ /

/ /

200

'

50

1

JJ. J

100

:

—

, ,

~

\ 1, \

150 M„ [GeV]

i

~z

-2

/ / /

\

I

100

'Apu' y ' "^-^

Zy / / '
<

VM

I

1

i '

,'-

r

IN.

'_

'

-

_

A

r

i

r :

^\\ \ \ w \\\ \\\

/

i

:

TT

:

i

•'--

\

200

1

—

WW

zz .2

...It

/

—

—

05 i

.02

.01

200

1

/

i 400

,

i 600 M„ [GeV]

1 800

1

1000

Figure 25: Branching fractions of the Standard Model Higgs boson for light Higgs (top) and heavy Higgs (Bottom). The top quark mass is 175 GeV. Figure courtesy of "QCD and Collider Physics" by R.K. Ellis, W.J. Stirling and B.R. Webber.

133

C

hey •

key*

U S %OU.V /kin. .

X^"^-

»CI>U5 ,' J C

=^v

Figure 26: A cartoon made by the ALEPH experiment after the LEP-2 shutdown.

134

For the Higgs heavier than ~ 130 GeV, qq -»• H -»• W+W~* or ZZ* will be the dominant processes for the searches. 1

x

/

/

Q

ff

^

\

V

Lines in Figure 28 show the total luminosity required per experiment for 2a, 3a, and 5a excesses due to the Higgs as a function of the Higgs mass. The Tevatron is expected to deliver ~2 fb _ 1 per experiment in the first 3 years (called "Run Ha"). With the Run Ha data, the Tevatron experiments will observe a 2a excess if the Higgs mass is 115 GeV. With further machine upgrade, CDF and D 0 expect to have ~10 - 15 fb _ 1 per experiment by year 2006 (called "Run lib"). With this luminosity, they will see upto a 3a excess over most of the region with MH < 180 GeV. Searches at LHC For more massive Higgs bosons, we must wait for the LHC. The LHC will produce light Higgs bosons copiously (see Figure 29) but they will suffer from severe backgrounds (more severe than the Tevatron). One strategy is to look for gg —> H —> 77 events. The braching fraction of H —>• 77 (see Figure 25) is extremely small (less than 1%), but backgrounds will be much smaller. For heavier Higgs bosons, the production cross section is smaller, but backgrounds are more manageable. The best process for this case could be via gg —>• H —» ZZ* —> l+£~£+£~~. The invariant mass distributions for these two cases are shown in Figures 30 and 31. 4-3

Higgs Beyond the Standard Model

The Higgs mechanism, while it has many attractive features, is not without difficulties. There is, for example, a severe fine-tuning ('naturalness') problem associated with preventing the Higgs mass being renormalized up to some very high 'new-physics' or grand unification scale. Supersymmetry (SUSY) is one way of circumventing this difficulty while retaining the essential features of the Standard Model Higgs sector. In the minimal supersymmetric model (MSSM), it is necessary to introduce two complex doublets of Higgs fields, and the masses and couplings of the

135 :

1

1

1

|

1

1

1

|

1

i

i

|

i

i

I

'

'

'

1

'

'

'

rj(pp-+H+X) [pb]

Vs = 2 TeV Mt=175GeV CTEQ4M

r

^^^_Jg-^H

f qq^Hqq

:

:

->HW ~ " ~~- - 1 ~ : - - - -S3

-=

qq->HZ gg,qq-*Htt

:

r

gg,qcH>Hb5 i

80

100

.

120

I

I

I

!

.

140

i

i

160

i

i

>

i

180

i

i

200

M H [GeV] Figure 27: Higgs production cross sections at the Tevatron.

resulting five physical Higgs bosons are expressible (at leading order) in terras of just two parameters. The key feature of the minimal SUSY model is the existence of at least one neutral, scalar Higgs boson with a mass less then O(150 GeV), the exact upper limit depending weakly on the parameters of the model on Mtop- Note that a large fraction of SUSY models predict Muiggs < 130 GeV. In most respects, the lightest minimal SUSY Higgs boson behaves exactly like the Standard Model Higgs, and so the calculations and search strategies described in this section again apply. Via loop diagrams, new particles make contributions to the electroweak parameters. Figure 32 shows the predicted Mw - Mtop region in the MSSM which is consistent with the current electroweak measurements. 5

Future Prospects

We will have possible senarios in this decade. • We will discover the Higgs boson with Mniggs < 130 GeV. This will lead

136

combined CDF/DO thresholds

s

,

10

j

,

(

,

f

,

!—

Q. X

30 fb" 1 J10 fb" 1

110' I c "E

2 fb- i

3

™ 9 5 % CL limit

1.10°

- 3a evidence - 5a discovery

(D

J

80

100

120

140

i

I

160

i

L

180

200

Higgs mass (GeV/c 2 ) Figure 28: Higgs discovery potential at the Tevatron Run II.

to another question; is it the Standard Model Higgs ? If not, this will imply new physics. We will discover the Higgs boson with MHiggs > 130 GeV. This will rule out a large fraction of SUSY models. This will lead to a question, "Will this agree with the indirect predicion from the electroweak precision measurements ?" We will not discover upto LHC. In this senario, we expect that detectable effects appear in the production rate and properties of W boson pairs at ~ 1 TeV. Whatever the outcome, it will be extremely interesting. At present, it is essentially an experimental question.

137

1000

~"

I

r

p+p^H+X

100

Vs=14 TeV

\ -

, rQ

1

r-

L£J

b

^^^^i^L

10 ^ = -•"V. E ~""~^. — \ v\ 1 =~ \ \ \ B \ " \ "-.\ V \

\

~^~--^ ^

~

\

^

^""-^ qq^qqtl

v

V

.1 _ = .01 =—

"

" • ' • - . _

•.

X

xXx X x X g g . q ^ txst xH -x x > \ q q ^ H " --T^-^-..-^

"^""-^ "^*«.qq->ZH~---~.r

.001

.0001 200

400 600 MH [GeV]

800

1000

Figure 29: Higgs production cross section in pp collisions at the LHC. Figure courtesy of "QCD and Collider Physics" by R.K. Ellis, W.J. Stirling and B.R. Webber.

Acknowledgments I would like to thank Jon Rosner who organized the TASI-2000 and edited Lecture Notes. I would also like to thank and acknowledge the collaborators of the CDF, D 0 and NuTev experiments, and the ALEPH, DELPHI, L3, OPAL, and SLD experiments. This work is supported by the U.S. Department of Energy. References 1. G. Arnison et al. (UAl Collaboration), PLB 122, 103 (1983); M. Banner et al. (UA2 Collaboration), PLB 122, 476 (1983) 2. F. Abe et al. (CDF Collaboration), PRL 74, 2626 (1995), PRD 50, 2966 (1994); S. Abachi et al. (D0 Collaboration), PRL 74, 2632 (1995). 3. A. Czarnecki and W.J. Marciano, Report BNL-HET-98/43, hepph/9810512. 4. The LEP Collaborations, the LEP Electroweak Working Group, the SLD

138

,19000

14000

-

13000

12000

128

130

(GeV)

Figure 30: Expected Af77 spectrum as simulated in the ATLAS detector at the LHC for M g = 120 GeV and one year of data. The dotted line is background spectrum, and the solid line is background and signal combined spectrum. Figure courtesy of "QCD and Collider Physics" by R.K. Ellis, W.J. Stirling and B.R. Webber.

Heavy Flavour and Electroweak Working Group, hep-ex/0103048. 5. F. Abe et al. (CDF Collaboration), PRD 63, 032003 (2001); F. Abe et al. (CDF Collaboration), PRL 82, 271 (1999); B. Abbott et al. (D0 Collaboration), PRD 58, 052001 (1998). 6. C. Caso et al, Eur. Phys. J. C3, 1 (1998). 7. V. Barger et al, Z. Phys. C 21, 99 (1983); B. J. Smith et al, Phys. Rev. Lett. 50, 1738 (1983). 8. J. Alitti et al. (UA2 Collaboration), Phys. Lett. B 276, 246 (1992); T. Affolder et al. (CDF Collaboration), hep-ex/0007044 (submitted to Phys. Rev. D); S. Abachi et al. (D0 Collaboration), PRL 84, 222

139

»

10

250

500

750

1000

1250

1500

1750

2000

mass (GeV)

Figure 31: Expected Mzz spectrum as simulated in the ATLAS detector at the LHC for MH = 800 GeV and one year of data. The different histograms repesent different selection cuts. Figure courtesy of "QCD and Collider Physics" by R.K. Ellis, W.J. Stirling and B.R. Webber.

(2000); 9. T. Affolder et al. (CDF Collaboration), submitted to PRL (2000). 10. K. S. McFarland, et al, Eur. Phys. Jour. C31, 509 (1998). 11. M. Swartz, hep-ph/9509248; S. Eidelman, F. Jegerlehner, hepph/9502298 12. B. Pietrzyk, The global fit to electroweak data, talk presented at ICHEP2000, Osaka, July 27 - August 2, 2000. Preprint LAP-EXP-00.06. 13. K. Hoffman, Year 2000 update for OPAL and LEP HIGGSWG results, talk presented at ICHEP2000, Osaka, July 27 - August 2, 2000. 14. ALEPH Collaboration, CERN-EP/2000-138.

140

MSSM

mm LEPLSLD,vN data 1VL„-IVL contours : 68% CL

130

140

150

160

170

180 190 200 M top (GeV/c2)

Figure 32: The direct and indirect measurements of the W and top quark mass from LEP, the Tevatron, and SLC overlaid with the allowed region from the MSSM model.

..,.^' r .?**?•>

^|I|

Pllllliliil

••iffliisiiii

G. Buchalla

This page is intentionally left blank

K A O N A N D C H A R M PHYSICS: THEORY

Theory

G. B U C H A L L A Division, CERN, CH-1211 Geneva 23, E-mail: [email protected]

Switzerland

We introduce and discuss basic topics in the theory of kaons and charmed particles. In the first part, theoretical methods in weak decays such as operator product expansion, renormalization group and the construction of effective Hamiltonians are presented, along with an elementary account of chiral perturbation theory. The second part describes the phenomenology of the neutral kaon system, CP violation, e and e'/s, rare kaon decays (K —> -KVV, KL —> 7 r ° e + e _ , K^ —> /J,+/J,~), and some examples of flavour physics in the charm sector.

1

Preface

These lectures provide an introduction to the theory of weak decays of kaons and mesons with charm. Our main focus will be on kaon physics, which has led to many deep and far-reaching insights into the structure of matter, is a very active field of current research and still continues to hold exciting opportunities for future discoveries. Another, and in several ways complementary source of information about flavour physics is the charm sector. Standard model effects for rare processes are in this case typically suppressed to almost negligible levels and positive signals, if observed, could therefore yield spectacular evidence of new physics. Towards the end of the lectures we will describe a few selected examples in charm physics and contrast their characteristic features with those of the kaon sector. For both subjects we will concentrate on the flavour physics of the standard model. We discuss the phenomenology as well as the theoretical tools necessary to achieve a detailed and comprehensive test of the standard model picture that should eventually lead us to uncover signals of the physics beyond. The experimental aspects of these fields are explained in the lectures by Barker (kaon physics) and Cumalat (charm physics) at this School. Before we start our tour of flavour physics with kaons and charm, we give a brief outline of the contents of these lectures. We begin, in the following section 2, with recalling some of the historical highlights of kaon physics and with an overview of the main topics of current interest in this field. In section 3 we introduce theoretical methods that are fundamental for the computation of weak decay processes and for relating the basic parameters of the underlying theory to actual observables. These important tools are the operator product expansion, the renormalization group, the effective low-energy 143

144

weak Hamiltonians, where we discuss the general AS = 1 Hamiltonian as an explicit example, and, finally, chiral perturbation theory. Our main emphasis will be on an elementary introduction of the relevant ideas and concepts, rather than on more specialized technical aspects. With this background in mind we will then address, in section 4, the phenomenology of the neutral-kaon system and CP violation. We discuss a common classification of CP violation, the kaon CP parameters e and e'/e, and the standard analysis of the CKM unitarity triangle. Section 5 is devoted to the physics of rare kaon decays, in particular the "golden" channels K+ -> -K+VV and KL -t ir°vv, and the processes KL -> ir°e+e~ and Kj_ —> A*+M~In section 6 we discuss the prominent features of flavour physics with charm. We present some opportunities with rare decays of D mesons and describe the phenomenology of D°-D° mixing, which is of current interest in view of new experimental measurements by the CLEO and FOCUS collaborations. Finally, section 7 summarizes the main points and presents an outlook on future opportunities. We conclude these preliminary remarks by mentioning several review articles, which the interested reader may consult for further details on the topics presented here, for a discussion of related additional processes and for a complete collection of references to the original literature. Very useful accounts of rare and radiative kaon decays and of kaon CP violation can be found in I,2,3,4,5,6,7_ rp n e g r s t £ v e a r t i c i e s a i s o discuss the relevant experiments. Nice reviews on flavour physics with charm are 8 ' 9 ' 1 0 ' 1 1 . Further details on theoretical methods in weak decays are provided in 12 - 13 .

2 2.1

Kaons: Introduction and Overview Historical Highlights

The history of kaon physics is remarkably rich in groundbreaking discoveries. It will be interesting to briefly recall some of the most exciting examples here. We do not attempt to give an historically accurate account of the development of kaon physics. This is a fascinating subject in itself. For a more complete historical picture the reader may consult the excellent book by Cahn and Goldhaber 1 4 . Here we will content ourselves with a brief sketch of several highlights related to kaon physics. They serve to illustrate how the observation of unexpected - and sometimes tiny - effects in this field is linked to basic concepts in our theoretical understanding of the fundamental interactions.

145

Strangeness Already the discovery of kaons alone, half a century ago, has had an impressive impact on the development of high-energy physics. One of the characteristic features of the new particles was associated production, that is they were always produced in pairs by strong interactions, for instance as 7T+ + p ->• K+ + K° + p

S

0

0

+1

-1

0

(Alternatively a K+ could be produced along with a A(uds) baryon.) This property, together with the long' lifetime, suggested the existence of a new quantum number, called strangeness, carried by the kaons and conserved in strong interactions. The discovery of strangeness opened the way for the SU(3) classification of hadrons and the introduction of quarks (u, d, s) as the fundamental representation by Gell-Mann. The quark picture, in turn, formed the basis for the subsequent development of QCD. In modern notation the K mesons come in the following varieties K+(su) K°(sd) K-(su) K°(sd) where the flavour content is indicated in brackets. The pairs {K+, K°) and (K°, K~) are doublets of isospin. Parity Violation The new mesons proved to be strange particles indeed. One of the peculiarities is known as the 6-r puzzle. Two particles decaying as 8 —> 27r (P even final state) and r —> Sir (P odd), and hence apparently of different parity, were observed to have the same mass and lifetime. This situation prompted Lee and Yang to propose that parity might not be conserved in weak interactions. This was later confirmed in the famous 60 Co experiment by C.S. Wu. Today the 6+ and r + are known to be identical to the K+ meson and parity violation is firmly encoded in the chiral SU(2)L gauge group of standard model weak interactions. CP Violation After the recognition of parity violation in weak processes the combination of parity with charge conjugation, CP, still appeared to be a good symmetry. The neutral kaons K° and K° were known to mix through second order weak interactions to form, if CP was conserved, the CP eigenstates KL,S = (K° ± K°)/V2 (here CP K° = -K°). Clearly, CP symmetry then forbids the decay

146

of the CP-odd KL into the CP-even TT+IT~ final state. Instead, Christenson, Cronin, Fitch and Turlay showed in 1964 that the decay does in fact occur, establishing CP violation. Compared to the CP-allowed decay of K$ —> 7r+7T~ the amplitude is measured to be A(KL

Arir

-t

TT+TT")

+

^ -2.3-10"3

(1)

CP violation is thus a very small effect, in contrast to P violation, but the qualitative implications are nevertheless far-reaching. As we will discuss later, CP violation defines an absolute, and not only conventional, difference between matter and anti-matter. Also, as we now know, CP violation indirectly anticipated in a sense the need for three families of fermions within the standard model. Finally, CP violation is a necessary prerequisite for the generation of a net baryon number in our universe according to Sakharov's three conditions (the other two being baryon number violation and a departure from thermal equilibrium). F C N C Suppression Another striking property of weak interactions that manifested itself in kaon decays is the suppression of flavour-changing neutral currents (FCNC). While the standard, charged-current mediated process K+ —> JJL+V has a branching fraction of order unity B{K+ -> H+P) = 0.64 (2) the similarly looking neutral-current decay KL —> fi+fi~ is suppressed to a tiny level B{KL-+ /x+/x-)« 7 - H r 9 (3) Naively, a "three-quark standard model" would allow a sdZ coupling at tree level. This would lead to a KL -> A*4fJ>~ amplitude of strength Gp, comparable to K+ -> /J,+ V, in plain disagreement with (3). Even if the tree-level coupling of sdZ were forbidden, the problem would reappear at one loop. This is illustrated in the second diagram of Fig. 1. The loop integral is divergent, where a natural cut-off could be expected at the weak scale ~ M\y • The amplitude should then be of the order GpM^,, which would still be far too large. Of course, the three-quark model is not renormalizable and therefore not a consistent theory at short distances. The introduction of the charm quark by Glashow, Iliopoulos and Maiani (GIM) solves all of these problems, which plagued the early theory of weak interactions. The complete two-generation standard model is perfectly consistent and the tree-level sdZ coupling is automatically eliminated by the orthogonality of the 2 x 2 Cabibbo mixing matrix.

147

Figure 1: GIM Mechanism.

The sdZ coupling can still be induced at one-loop order, but the disturbing G2FMw term is now canceled between the up-quark and the charm-quark contribution. The remaining effect is, up to logarithms, merely of the order G\n?c, which is well compatible with (3), unless mc would be too large. To turn this observation into a more quantitative constraint on the charm-quark mass mc is, however, not easy in this case because KL —>• /x + /i~ is actually dominated by long-distance contributions (we will discuss this further in section 5.3). Another FCNC process, K°-K° mixing, proved to be more useful in this respect. K-K

mixing, GIM and Charm

The following example represents one of the great triumphs of early standard model phenomenology. In the four-quark theory, K-K mixing occurs through AS = 2 W-box diagrams with internal up and charm quarks. This AS = 2 transition induces a tiny off-diagonal element M\i in the mass matrix

»-(£,",?)

<«>

of the K-K system. The corresponding eigenstates are KL,S — {K° ±K°)/y/2 with eigenvalues Mi^s- The difference between the eigenvalues AMK = ML — Ms is related to M\2 and can be estimated from the box diagrams (see Fig. 2). Anticipating a more detailed discussion of the calculation, we simply quote the result: AMK _ G\PK 2 2 _ 15 MK

6TT2

'

cs

cdl

c

~

(

'

where the number on the right is the experimental value. The theoretical expression is approximate since we have taken the required hadronic matrix

148

U,C

K[]±K[]

v

{)

\w Iw R

KL

"~ i2 s

«,c

1 M M12 \ \ M12 M j

tf1 i?1

d

Figure 2: A ' 0 - ^ ' 0 mixing.

element in the so-called factorization approximation, we have neglected the (fairly small) up-quark contribution and assumed that mc AQCD, the up-quark contribution is negligible). Finally there are the obvious CKM parameters. This analysis was performed in 1974 by Gaillard and Lee. The charm quark had just been introduced for the theoretical reasons mentioned above, but had not yet been discovered in experiment. Gaillard and Lee realized that the quadratic dependence on the unknown charm quark mass in (5), resulting from the GIM cancellation of the AS = 2 FCNC amplitude, could be turned into an estimate of this mass. They concluded that mc should be about a few GeV. If you take the formula in (5) and put in numbers, you will find that mc m 1.5 GeV (!). In view of the uncertainties of (5) the accuracy of this result cannot be taken too seriously, but it is in any case amusing that indeed a quite realistic charm mass comes out already from this simplified formula. What is more important, however, is the spirit of the argument, which led Gaillard and Lee to a correct prediction of mc by taking the short-distance structure of the theory seriously. This was certainly one of the most beautiful successes of flavour physics with kaons. All these examples illustrate that a careful analysis of low-energy processes such as kaon decays, can lead to truly profound insights into fundamental physics. Especially remarkable is the circumstance that kaon physics, which "operates" at the ~ 500 MeV scale, carries important information on the dynamics at much higher energy scales. The quark-structure of hadronic matter, the chiral nature of weak gauge interactions, the properties of charm, and

149

Figure 3: (Semi)leptonic K decays.

those of the top quark entering CP violating amplitudes, are prominent examples that illustrate this point. In fact, we see that several of the most crucial pillars of the standard model rest on results derived from studies with kaons. Of course, direct experiments at high energies, that have led for instance to the production of on-shell W and Z bosons or quark jets, are indispensable for exploring the strucutre of matter. However, indirect, low-energy precision observables are equally necessary as a complementary approach. They can yield information that is hardly accessible in any other way, such as the elucidation of the GIM structure of flavour physics or the violation of CP symmetry. It is with this philosophy in mind that studies of rare kaon processes, but also rare decays of b hadrons or charmed particles, continue to be pursued with great interest. The most promising future opportunities with kaons will be the subject of later sections in these lectures. We conclude this introductory chapter with a brief general overview of physics with kaon decays. 2.2

Overview of K decays

We may classify the decays of K mesons into several broad categories, some of which are more determined by nonperturbative strong interaction dynamics, while others have a high sensitivity to short-distance physics both in the standard model and beyond. Tree-Level (Semi-) Leptonic Decays These are the simplest decays of kaons. They typically have large branching ratios and are well studied. Examples are the purely leptonic decay K+ —> \x+v and the semileptonic mode K+ —> 7r°e+z/, which are illustrated in Fig. 3. K+ —• ir°e+i/ is very important for determining the CKM matrix element Vus (the sine of the Cabibbo angle). This is possible because the hadronic matrix

150

element of the vector current (7T°|(su)v|.ftT+) is absolutely normalized in the limit of SU(3) flavour symmetry and protected from first order corrections in the SU(3) breaking (Ademollo-Gatto theorem). One finds \VUS\ = 0.2196 ±0.0023

(6)

This is a basic input for the CKM matrix. Furthermore, knowing |VUS|, K+ —> fi+v may be used to determine the kaon decay constant fx — 160 MeV. Nonleptonic Decays Nonleptonic decays, such a s i f - > nir, are strongly affected by nonperturbative QCD dynamics. Nevertheless they provide an important window on the violation of discrete symmetries, P and CP for example. CP violation is currently of special interest. It enters through K~K mixing via box graphs or through penguin diagrams in the decay amplitudes, with the virtual top quarks playing a decisive role. This is sketched in Fig. 4.

K

S

t

d

C~\ I O

R

v—^^j^y ] d t s i

u

u

7T7T Figure 4: SM origin of CP violation in K —> WK decays.

Long-Distance Dominated Rare and Radiative Decays Examples of this class of processes are K+ -> TT+1+1~ , KL -> 7r°77, Kg -t 77 or KL —> M+M-- A typical contribution to K+ —> ir+e+e~ is illustrated in Fig. 5. These processes are determined by nonperturbative low-energy strong interactions and can be analyzed in the framework of chiral perturbation theory. The treatment of nonperturbative dynamics within a first-principles approach as provided by chiral perturbation theory is of great interest in its own right. In addition, the control over long-distance contributions afforded by chiral perturbation theory can be helpful to extract information on the flavour physics from short distances.

151

Figure 5: Typical contribution to K+ —> 7r+e+e

Short-Distance Dominated Rare Decays The prime examples in this category are the processes K+ —> it+vv and KL —> n°vV. Here short-distance dynamics completely dominates the decay and the access to flavour physics is very clean and direct. For this reason the K —>• -KVV modes are special highlights among the future opportunities in kaon physics. To a somewhat lesser extent also KL —> 7r°e+e~ qualifies for this class. In this case the fact that the process is predominantly CP violating enhances the sensitivity to short-distance physics in comparison to K+ —>• 7r + e + e~. Decays Forbidden in the SM Any positive signal in processes that are forbidden in the standard model would be a very dramatic indication of new physics. A good example are kaon decays with lepton-flavour violation. Stringent experimental limits exist for several modes of interest B{KL -^ fie) < 4 . 7 - l C T 1 2 +

B(K+ - > 7 r V e ~ ) < 4.8 • KT B{KL ->

TTV)

(7)

11

(8)

9

(9)

< 3.2 • 10~

In principle these processes could be induced through loops in the standard model with neutrino masses. However, the smallness of the neutrino masses compared to the weak scale results in unmeasurably small branching fractions of typically below 10~ 25 . Larger values can be obtained within the minimal supersymmetric standard model (MSSM). However, there are strong constraints from direct limits on p. —• e conversion processes (// —> ej decay, or /j, —> e conversion in the field of a nucleus). The disadvantage of KL —> fie is that flavour violation is needed simultaneously both in the lepton sector and in the

152

d

u

Figure 6: QCD effects in weak decays.

quark sector. Interesting effects could however still occur in some regions of parameter space. Systematically larger branching ratios are allowed in scenarios with R-parity violation, where decays such as KL —> fie can proceed at tree level 15 . In very general terms, a scenario where the exchange of a heavy boson X mediates sd —¥ fie transitions at tree level receives strong constraints from the tight experimental bound (7). Assuming couplings of electroweak strength, the bound (7) implies a lower limit of the X mass Mx ~ 100 TeV. Such a sensitivity to high energy scales is very impressive, however one has to remember that the tree-level scenario assumed above is quite simple-minded and in general very model dependent. Generically, one would expect some additional suppression mechanism to be at work in Ki —> fie. In this case the scale probed would be less, but the high precision of (7) still guarantees an excellent sensitivity to subtle short-distance effects. 3

Theoretical Methods in Weak Decays

The task of computing weak decays of kaons represents a complicated problem in quantum field theory. Two typical cases, the first-order nonleptonic process K° -» n+n~, and the loop-induced, second-order weak transition K+ —> 7r+z/z> are illustrated in Fig. 6. The dynamics of the decays is determined by a nontrivial interplay of strong and electroweak forces, which is characterized by several energy scales of very different magnitude, the W mass, the various quark masses and the QCD scale: mt, Mw ^ w c ^> AQCD 3> THU, md, (m s ). While it is usually sufficient to treat electroweak interactions to lowest nonvanishing order in perturbation theory, it is necessary to consider all orders in QCD. Asymptotic freedom still allows us to compute the effect of strong interactions at short distances perturbatively. However, since kaons are bound

153

states of light quarks, confined inside the hadron by long-distance dynamics, it is clear that also nonperturbative QCD interactions enter the decay process in an essential way. To deal with this situation, we need a method to disentangle long- and short-distance contributions to the decay amplitude in a systematic fashion. The required tool is provided by the operator product expansion (OPE). 3.1

Operator Product Expansion

We will now discuss the basic concepts of the OPE for kaon decay amplitudes. These concepts are of crucial importance for the theory of weak decay processes, not only of kaons, but also of mesons with charm and beauty and other hadrons as well. Consider, for instance, the basic W-boson exchange process shown on the left-hand side of Fig. 7. This diagram mediates the

Figure 7: OPE for weak decays.

decay of a strange quark and triggers the nonleptonic decay of a kaon such as —¥ ir+ir . The quark-level transition shown is understood to be dressed with QCD interactions of all kinds, including the binding of the quarks into the mesons. To simplify this problem, we may look for a suitable expansion parameter, as we are used to do in theoretical physics. Here, the key feature is provided by the fact that the W mass M\y is very much heavier than the other momentum scales p in the problem (AQCD, mu, ma, ms). We can therefore expand the full amplitude A, schematically, as follows ^

c

( ^ ,

a

, ) .

( e )

+ 0 (|_)

(10)

which is sketched in Fig. 7. Up to negligible power corrections of 0(p2/Myy), the full amplitude on the left-hand side is written as the matrix element of a local four-quark operator Q, multiplied by a Wilson coefficient C. This expansion in l/Mw is called a (short-distance) operator product expansion because

154

the nonlocal product of two bilinear quark-current operators (su) and (ud) that interact via W exchange, is expanded into a series of local operators. Physically, the expansion in Fig. 7 means that the exchange of the very heavy W boson can be approximated by a point-like four-quark interaction. With this picture the formal terminology of the OPE can be expressed in a more intuitive language by interpreting the local four-quark operator as a four-quark interaction vertex and the Wilson coefficient as the corresponding coupling constant. Together they define an effective Hamiltonian ~Heff = C • Q, describing weak interactions of light quarks at low energies. Ignoring QCD the OPE reads explicitly (in momentum space) o A

i

= -YVusVudk2

_

M 2

(su)V^A(ud)V-A

= -.^C.^cW) + 0 ( ^ ) with C = 1, Q = (su)v-A(ud)v-A He/,

(.i)

and

= ^V:sVud(su)v-A(ud)V-A

(12)

As we will demonstrate in more detail below after including QCD effects, the most important property of the OPE in (10) is the factorization of longand short-distance contributions: All effects of QCD interactions above some factorization scale \x (short distances) are contained in the Wilson coefficient C. All the low-energy contributions below \i (long distances) are collected into the matrix elements of local operators (Q). In this way the short-distance part of the amplitude can be systematically extracted and calculated in perturbation theory. The problem to evaluate the matrix elements of local operators between hadron states remains. This task requires in general nonperturbative techniques, as for example lattice QCD, but it is considerably simpler than the original problem of the full standard-model amplitude. In some cases also symmetry considerations can help to determine the nonperturbative input. For example, the only matrix element relevant for K+ —> ir+vv is (n+\(sd)v\K+)

= V2(7r°\(su)v\K+)

(13)

where the equality with the right-hand side uses isospin symmetry and allows us to obtain the matrix element from measuring the standard semileptonic mode K+ -> TT°1+V. The short-distance OPE that we have described, the resulting effective Hamiltonian, and the factorization property are fundamental for the theory of

155

+

•5Z--
K decays. However, the concept of factorization of long- and short-distance contributions reaches far beyond these applications. In fact, the idea of factorization, in various forms and generalizations, is the key to essentially all applications of perturbative QCD, including the important areas of deepinelastic scattering and jet or lepton pair production in hadron-hadron collisions. The reason is the same in all cases: Perturbative QCD is a theory of quarks and gluons, but those never appear in isolation and are always bound inside hadrons. Nonperturbative dynamics is therefore always relevant to some extent in hadronic reactions, even if these occur at very high energy or with a large intrinsic mass scale (see also the lectures by Soper in these proceedings). Thus, before perturbation theory can be applied, nonperturbative input has to be isolated in a systematic way, and this is achieved by establishing the property of factorization. It turns out that the weak effective Hamiltonian for K decays provides a nice example to demonstrate the general idea of factorization in simple and explicit terms. We will next discuss the OPE for K decays, now including the effects of QCD, and illustrate the calculation of the Wilson coefficients. A diagrammatic representation for the OPE is shown in Fig. 8. The key to calculating the coefficients Ci is again the property of factorization. Since factorization implies the separation of all long-distance sensitive features of the amplitude into the matrix elements of (Qi), the short-distance quantities d are, in particular, independent of the external states. This means that the Ci are always the same, no matter whether we consider the actual physical amplitude where the quarks are bound inside mesons, or any other, unphysical amplitude with on-shell or even off-shell external quark lines. Thus, even though we are ultimately interested in K —> 7T7T amplitudes, for the perturbative evaluation of Ci we are free to choose any treatment of the external quarks according to our calculational convenience. A convenient choice that we will use below is to take all light quarks massless and with the same off-shell momentum p (p2 ^ 0). The computation of the Ci in perturbation theory then proceeds in the following steps: • Compute the amplitude A in the full theory (with W propagator) for

156

1

i

ik sr^

l

^

s

d

u

k

T

T UJ

jl

Figure 9: QCD correction with colour assignment.

arbitrary external states. • Compute the matrix elements (Qi) with the same treatment of external states. • Extract the d from A = Cj (Qi). We remark that with the off-shell momenta p for the quark lines the amplitude is even gauge dependent and clearly unphysical. However, this dependence is identical for A and (Qi) and drops out in the coefficients. The actual calculation is most easily performed in Feynman gauge. To 0(as) there are four relevant diagrams, the one shown in Fig. 8 together with the remaining three possibilities to connect the two quark lines with a gluon. Gluon corrections to either of these quark currents need not be considered, they are the same on both sides of the OPE and drop out in the d- The operators that appear on the right-hand side follow from the actual calculations. Without QCD corrections there is only one operator of dimension 6 Ql = (siUi)v-A(Ujdj)V-A

(14)

where the colour indices have been made explicit. (The operator is termed Qi for historical reasons.) To 0(as) QCD generates another operator Q\ = (SiUj)v-A(Ujdi)V-A

(15)

which has the same Dirac and flavour structure, but a different colour form. Its origin is illustrated in Fig. 9, where we recall the useful identity for SU(N) Gell-Mann matrices

157

It is convenient to employ a different operator basis, defining

Q± = ^ f ^ 1

(17)

The corresponding coefficients are then given by C± = C2 ± Ci

(18)

If we denote by S± the spinor expressions that correspond to the operators Q± (in other words: the tree-level matrix elements of Q±), the full amplitude can be written as A=(l

+ 1+as In ^ | ) S+ + (1 + 7 - a s In ^ | ) S_

(19)

Here we have focused on the logarithmic terms and dropped a constant contribution (of order as, but nonlogarithmic). Further, p2 is the virtuality of the quarks and 7± are numbers that we will specify later on. We next compute the matrix elements of the operators in the effective theory, using the same approximations, and find (Q±)=

(l+7±«s(j+ln^))s±

(20)

The divergence that appears in this case has been regulated in dimensional regularization (D = 4 — 2e dimensions). Requiring A = C+(Q+) + C-(Q-)

(21)

we obtain

M2 C± = 1 + l±as In —f (22) H2 where the divergence has been subtracted in the minimal subtraction scheme. The effective Hamiltonian we have been looking for then reads Heff

= ^V:sVud

(C+(fi)Q+

+ C-MQ-)

(23)

with the coefficients C± determined in (22) to 0(as log) in perturbation theory. The following points are worth noting: • The 1/e (ultraviolet) divergence in the effective theory (20) reflects the Mw —> oo limit. This can be seen from the amplitude in the full theory (19), which is finite, but develops a logarithmic singularity in this limit. Consequently, the renormalization in the effective theory is directly linked to the In Mw dependence of the decay amplitude.

158

We observe that although A and (Q±) both depend on the long-distance properties of the external states (through p 2 ), this dependence has dropped out in C±. Here we see explicitly how factorization is realized. Technically, to 0(as log), factorization is equivalent to splitting the logarithm of the full amplitude according to l n

M | —pz

= l n

M | , fiz

+ l n J

^

—pz

Ultimately the logarithms stem from loop momentum integrations and the range of large momenta, between M\y and the factorization scale JJL, is indeed separated into the Wilson coefficients. To obtain a decay amplitude from 'Kefs m (23), the matrix elements ( / | Q ± | - ^ Q ( M ) have to be taken, normalized at a scale \x. An appropriate value for /i is close to the hadronic scale in order not to introduce an unnaturally large scale into the calculation of {Q). At the same time fi must also not be too small in order not to render the perturbative calculation of C(/x) invalid. A typical choice for K decays is fi ss 1 GeV
159

matrix elements. Of course, both quantities have to be evaluated in the same scheme to obtain a consistent result. The renormalization scheme is determined in particular by the subtraction constants (minimal or nonminimal subtraction of 1/e poles), and also by the definition of 75 used i n D ^ 4 dimensions in the context of dimensional regularization. • Finally, the effective Hamiltonian (23) can be considered as a modern version of the old Fermi theory for weak interactions. It is a systematic low-energy approximation to the standard model for kaon decays and provides the basis for any further analysis. 3.2

Renormalization

Group

Let us have a closer look at the Wilson coefficents, which read explicitly

r

1 , a «M ^i0> ,„ ^

c± = 1 +

jo)

ln

7±

^r^ Aq;

J

4

=\-8

/ 9 =x

( 25)

where we have now specified the exact form of the 0(as log) correction. Numerically the factor as(fi)j±'/(8TT) is about +7% (—14%), a reasonable size for a perturbative correction (we used as(n = 1 GeV) = 0.43). However, this term comes with a large logarithmic factor of ln(/i 2 /M^) = —8.8, for an appropriate scale of fj, = 1 GeV. The total correction to C± — 1 in (25) is then —60% (120%)! Obviously, the presence of the large logarithm spoils the validity of a straightforward perturbative expansion, despite the fact that the coupling constant itself is still reasonably small. This situation is quite common in renormalizable quantum field theories. Logarithms appear naturally and can become very large when the problem involves very different scales. The general situation is indicated in the following table, where we display the form of the correction terms in higher orders, denoting £ = ]n(fj,/Mw) LL NLL as£ as a2£2 a2£

0(1)

a2

0(as)

In ordinary perturbation theory the expansion is organized according to powers of as alone, corresponding to the rows in the above scheme. This approach is invalidated by the large logarithms since as£, in contrast to as, is no longer a small parameter, but a quantity of order 1. The problem can be resolved by

160

resumming the terms (as£)n to all orders n. The expansion is then reorganized in terms of columns of the above table. The first column is of 0(1) and yields the leading logarithmic approximation, the second column gives a correction of relative order as, and so forth. Technically the reorganization is achieved by solving the renormalization group equation (RGE) for the Wilson coefficients. The RGE is a differential equation describing the change of C±(fi) under a change of scale. To leading order this equation can be read off from (25) d

a

n i \

« (o)

c±{n)

(27)

(a s /47r)7j_ J are called the anomalous dimensions of C±. To understand the term "dimension", compare with the following relation for the quantity /in, which has (energy) dimension n: d

dlnn

V

(28)

n-/x"

The analogy is obvious. Of course, the C±(fi) are dimensionless numbers in the usual sense; they can depend on the energy scale /j. only because there is another scale, Mw, present under the logarithm in (25). Their "dimension" is therefore more precisely called a scaling dimension, measuring the rate of change of C± with a changing scale fi. The nontrivial scaling dimension derives from 0(as) loop corrections and is thus a genuine quantum effect. Classically the coefficients are scale invariant, C± = 1. Whenever a symmetry that holds at the classical level is broken by quantum effects, we speak of an "anomaly". Hence, the j±' represent the anomalous (scaling) dimensions of the Wilson coefficients. We can solve (27), using da

s dlnfi

9/?

a2s 4n

Po =

33-2/

C±(M,w)

1

(29)

and find 7

(0)

C±(n)

as(Mw) as((i)

200

1

±'

200

(30)

_l + / 3 o ^ l n ^ _

This is the solution for the Wilson coefficients C± in leading logarithmic approximation, that is to leading order in RG improved perturbation theory. The all-orders resummation of as log terms is apparent in the final expression in (30).

161

K°

7T

)

•<+(

s

u

%

*<>

u d

Figure 10: K°

3.3

-> TT+T^- and

K+

-> TT+TT0.

AI = 1/2 i M e

At this point, and before continuing with the construction of the complete AS = 1 Hamiltonian, it is interesting to discuss a first application of the results we have derived so far. Let us consider the weak decays into two pions of a neutral kaon, Ks —• + TT+TT-, and a charged kaon, K —> 7r+7r°, which are sketched in Fig. 10. The two cases look very much the same, except that the spectator quark is a u quark for the charged kaon and a d quark for the neutral one. Naively one would therefore expect very similar decay rates. The experimental facts are, however, strikingly different: T(KS

- » 7T+7T-)

T(K+ -»

TT+TT°)

2

~

45U

(2L2j

(31)

To get a hint as to where this huge difference in the decay rates may come from we have to analyze the isospin structure of the decays. A kaon state has isospin / = 1/2. Taking into account Bose symmetry, one finds that two pions from the decay of a K meson can only be in a state of isospin 0 and 2. More specifically, \TT+TT~) has both 1 = 0 and 1 = 2 components, while |7r+7r°) is a pure 1 = 2 state. The change in isospin is then as follows K+

- • TT+TT0 AI

= 3/2

K°

-> TT+TT" AI

= 1/2,

(32) 3/2

(33)

In particular, K+ —> TT+TT0 is a pure AI = 3/2 transition. The large ratio in (31) means that AI = 1/2 transitions are strongly enhanced. This empirical feature is refered to as the A7 = 1/2 rule. We next take a closer look at the isospin properties of the effective Hamiltonian. Using the Fierz identities of the Dirac matrices the operators Q± can

162

be rewritten as Q± = (siUi)V-A{ujdj)v-A

± {siUj)v_A(ujdi)v-A

= (SiUi)v-A(Ujdj)V-A

± (Sidi)v-A{UjUj)V-A

= (34)

where now all quark bilinears appear uniformly as colour singlets. Retaining only the flavour structure, but dropping colour and Dirac labels for ease of notation, the Hamiltonian (23) has the form

-as(Mwy6/25

H eff

. "*(M) .

{(su)(ud) + (sd)(uu)) +

CK/r \ 1-12/25

+

QS(M)

((su)(ud) — (sd)(uu))

.

(35)

We can now see that the operator Q-, in the second line of (35), is a pure AI = 1/2 operator: u and d appear in the combination

ud-du=\ U) ~ I -It)

(36)

which has isospin 0. The strange quark is also an isospin singlet. The isospin of <5_ is therefore determined by the factor u, which has isospin 1/2. From the Wilson coefficients we have calculated we observe that the contribution from Q_ receives a relative enhancement over Q+ in (35) by a factor "S(A0

18/25

as{Mw)

i

2.6 3.4 /

n

H = 1 GeV M = 0.6GeV ii

i

i

-»u

\i

(37)

Qualitatively, this is precisely what we need: Q - , which is purely AI = 1/2 and can thus only contribute to K° —> TT+TT~, but not to K+ —> 7r+7r°, is re-inforced by the short-distance QCD dynamics. Quantitatively, however, the RG improved QCD effect falls still short of explaining the amplitude ratio 21.2 in (31) by a sizable factor. We might be tempted to decrease fi, which enhances the effect, but we are not allowed to go much below /i = 1 GeV where perturbation theory would cease to be valid. The remaining enhancement has to come from nonperturbative contributions in the matrix elements. Nevertheless it is interesting to see how already the short-distance QCD corrections provide the first step towards a dynamical explanation of the AI = 1/2 rule. 3.4

AS = 1 Effective

Hamiltonian

In this section we will complete the discussion of the AS = 1 effective Hamiltonian. So far we have considered the operators Q\ = (SiUj)v -A{Ujdi)v

-A

(38)

163 Q2 = (siUi)v-A{ujdj)v_A

(39)

which come from the simple W-exchange graph and the corresponding QCD corrections (Fig. 11). In addition, there is a further type of diagram at 0(as),

d

•U, C

Figure 11: QCD correction to W exchange.

which we have omitted until now: the QCD-penguin diagram shown in Fig. 12. It gives rise to four new operators

d

w

u, c, t q

£

q

Figure 12: QCD-penguin diagram.

Qi = {sidi)v-A'Y^(qjqj)v-A

(40)

Q4 = (sidj)v^A

(41)

^2(qjqi)v-A 1

Qh - (sidi)v_A

^2(q~jqj)v+A

(42)

9

Q6 = (sidj)v-A^2(qjqi)v+A

(43)

i

Two structures appear when the light-quark current (qq)v from the bottom end of the diagram is split into V — A and V + A parts. In turn, each of those comes in two colour forms in a way similar to Qi and Q2The operators Qi, • • • ,Qe rnix under renormalization, that is the RGE for their Wilson coefficients is governed by a matrix of anomalous dimensions,

164

generalizing (27). In this way the RG evolution of Ci i 2 affects the evolution of C 3 , . . . , C6- On the other hand Ci,2 remain unchanged in the presence of the penguin operators Q3,..-,Qe, so that the results for C 1|2 derived above are still valid. For some applications (e.g. e'/e) higher order electroweak effects need to be taken into account. They arise from 7- or Z-penguin diagrams (Fig. 13) and also from W-box diagrams. Four additional operators arise from this

Figure 13: Electroweak penguin. source. They have a form similar to the QCD penguins, but a different isospin structure, and read (eq are the quark charges, eUjC = + 2 / 3 , ed,s,b = —1/3) q

Q? = ^(sid^v-A^egiqjq^y+A

(44)

9 Q

1

Q9 = -(sidi)v_A'^2eq(qjqj)V-A
(46)

o

Q10 = -{sidj)v„AY^eq{qj
(47)

9

The construction of the effective Hamiltonian follows the principles we have discussed in the previous sections. First the Wilson coefficients Ci(nw), i = 1 , . . . , 10, are determined at a large scale nw = 0(Mw,rnt) to a given order in perturbation theory. In this step both the W boson and the heavy top quark are integrated out. Since the renormalization scale is chosen to be nw = 0{Mw,irit), no large logarithms appear and straightforward perturbation theory can be used for the matching calculation. The anomalous dimensions are computed from the divergent parts of the operator matrix elements, which correspond to the UV-renormalization of the Wilson coefficients. Solving the RGE the C; are evolved from nw to a scale /x& = 0(mt,) in a theory

165

with / = 5 active flavours q = u,d, s, c, b. At this point the b quark (which can appear in loops) is integrated out by calculating the matching conditions from a five-flavour to a four-flavour theory, where only q = u,d,s,c are active. This procedure is repeated by integrating out charm at \ic — 0(mc) and matching onto an / = 3 flavour theory. One finally obtains the coefficients C;(/i) at a scale /i < He, describing an effective theory where only q = u,d,s (and gluons of course) are active degrees of freedom. The terms taken into account in the RG improved perturbative evaluation of C»(/x) are, schematically: as\n^-)

,

a

NLO:as(asln^f)

l n ^ ( a

s

l n ^ j

, a(asln^f)

at leading and next-to-leading order, respectively. Here a is the QED coupling, refering to the electroweak corrections. The final result for the AS = 1 effective Hamiltonian (with 3 active flavours) can be written as

ntfr1 = ~\u £ (*(/*) - x ^ ) ) Q* + hx-

(48)

where \p = V*Vvd- In principle there are three different CKM factors, Xu, Xc and Xt, corresponding to the different flavours of up-type quarks that can participate in the charged-current weak interaction. Using CKM unitarity, one of them can be eliminated. If we eliminate Ac, we arrive at the CKM structure of (48). The Hamiltonian in (48) is the basis for computing nonleptonic kaon decays within the standard model, in particular for the analysis of direct CP violation. When new physics is present at some higher energy scale, the effective Hamiltonian can be derived in an analogous way. The matching calculation at the high scale fiw will give new contributions to the coefficients Cj(/ivv), the initial conditions for the RG evolution. In general, new operators may also be induced. The Wilson coefficients z* and yi are known in the standard model at NLO. A more detailed account of T-Lfff1 and information on the technical aspects of the necessary calculations can be found in 12 and 13 . 3.5

Chiral Perturbation Theory

An additional tool for kaon physics, complementary to the OPE-based effective Hamiltonian formalism, is chiral perturbation theory (%PT). The present section gives a brief and elementary introduction into this subject. For other, more detailed discussions we refer the reader to 6 ' 1 6 , 1 7 ' 1 8 .

166

Preliminaries The QCD Lagrangian for three light flavours g = (u,d, s)T can be written in terms of left-handed and right-handed fields, qi>R = (1 =p 75)9/2, in the form £QCD = qhi VQL + W VlR ~ QLMqR - qRM\L

(49)

where M — diag(m u , md,ms). If M is put to zero, CQCD is invariant under a global SU{2>)L SU(3)R symmetry -> LqL -> RqR

QL

qR

(50) (51)

with L and R (independent) SU(3) transformations. The explicit breaking of this chiral symmetry through a nonzero M is a small effect and can be treated as a perturbation. Simultaneously, chiral symmetry is not reflected in the hadronic spectrum, so it must also be spontaneously broken by the dynamics of QCD. For instance the octet of light pseudoscalar mesons 7TU

a

a

$ = T ir =

V2

+

,

V

Ve

7T~

K~

1T+

K+

\

K

-s+^ ° K°

(52)

-fj

is not accompanied in the spectrum of hadrons by a similar octet of mesons with opposite parity and comparable mass. On the other hand the octet $, comprising the lightest existing hadrons, is the natural candidate for the octet of Goldstone bosons expected from the pattern of spontaneous chiral symmetry breaking SU{2)L ® SU{Z)R -> 517(3) (53) down to group of ordinary flavour 517(3), where q^,R —> Vqh,RThe mesons in $ are not strictly massless due to the explicit breaking of chiral symmetry caused by M. and are thus often refered to as pseudoGoldstone bosons. Still they are the lightest hadrons and they are separated by a mass gap from the higher excitations of the light-hadron spectrum. (The masses of the latter remain of order AQCD, while the masses of $ vanish in the limit M ->• 0.) The idea of x ? T is to write a low-energy effective theory where the only dynamical degrees of freedom are the eight pseudo-Goldstone bosons. This is appropriate for low-energy interactions where the higher states are not kinematically accessible. Their virtual presence will however be contained in the coupling constants of xPT. The guiding principles for the construction of %PT

167

are the chiral symmetry of QCD and an expansion in powers of momenta and quark masses. By constructing the most general Lagrangian for $ compatible with the symmetries of QCD, the framework is model independent. By restricting the accuracy to a given order in the momentum expansion, only a finite number of terms are possible and the framework becomes also predictive. A finite number of couplings needs to be fixed from experiment; once this is done, predictions can be made. x P T is a nonperturbative approach as it does not rely on any expansion in the QCD coupling as. Both x P T and the quark-level effective Hamiltonian Heff a r e low-energy effective theories applicable to kaon decays. What is the difference and how are these two approaches related? The essential, and obvious, difference is that x P T is formulated directly in terms of hadrons, H e / / in terms of quarks and gluons. The advantage of %PT is therefore the direct applicability to physical, hadronic amplitudes, without the need to deal with the complicated hadronic matrix elements of quark-level operators. The advantage of Tieff, on the other hand, is the direct link to short-distance physics, which is encoded in the Wilson coefficients. This type of information is important in the context of CP violation or in the search for new physics. In x P T such information is hidden in the coupling constants, which are not readily calculable and need to be fixed experimentally. From these considerations it is clear that W.eff is more useful for applications where short-distance physics is essential (CP violation, e'/e), whereas x P T is especially suited to deal with long-distance dominated quantities, which are hard to come by otherwise. To relate the two descriptions directly is not an easy task and has so far not been accomplished. A calculation of the couplings of x P T from the quark picture, establishing a link between "H e // a n d x P T , requires one to solve QCD nonperturbatively, which is not possible at present.

SU(Z) Transformations Before describing the explicit construction of x P T it will be useful to recall a few important properties of SU(3) transformations and to introduce some convenient notation. We define {ql,q2,q3)

= (u,d,s)

{q1,q2,q3) = {u,d,s)

(54)

and by Ulj the components of a generic SU{3) matrix U. By definition, changing upper into lower indices, and vice versa, corresponds to complex conjugation, thus [/.' = U*) = U^

(55)

168

The unitarity of U implies UklU) = U^U)

= Sij

(56)

U\Uk

= 5^

(57)

= U\U^

Also, det U = eijk U\U2j U\ = 1

(58)

1

The fundamental SU(3) triplet q and anti-triplet qi transform, respectively, as ql -> U^qi (59) Qi ->• Ujqj

(60)

k

It follows from the above that the singlet q qk as well as the Kronecker symbol 6%j and the totally antisymmetric tensor eljk are invariant under SU(3) transformations. Higher dimensional representations can also be built. For example, the traceless tensor Sij=qiqi-lsijqkqk

(61)

is an irreducible representation of SU(3). Its eight components constitute an SU(3) octet, which transforms as S*, -»• U\U]lS\

(62)

r* = eijkqjqk

(63)

j k

(64)

We next define the objects n = eijkq q

They transform in the same way as the fundamental triplet and anti-triplet in (59) and (60), respectively. We show this for (63). From (60) and using (55) and (57), we have rl = eijkqjqk

->•

k

m

= UisenikUn'UjlUkmqlqm

e^ U/Uk qiqm k

]l

U\ e^ U^nU 3U

]m

slm

k

=

q,qm = U\ e (detU^qiqm

= U\rs

(65)

which proves our assertion. Let us consider two simple applications of this formalism. a) The meson field $ in (52) corresponds to the quark-level tensor Slj and both transform as octets under SU(3). The connection can be seen by writing

169 out the components of S l - and comparing with (52). One recovers the quark flavour composition of the meson states. For example $ x 2 =7r+ = S12 =qlq2 $\

= -—r, = S\=q3q3--qkqk

=ud = --{uu + dd - 2ss)

(66) (67)

The transformation law for $1^ is the same as for Slj in (62) or, in matrix notation, $ H> [/*[/* (68) b) In section 3.3 we have seen from the discussion of the A / = 1 / 2 rule that the largely dominant part of the weak Hamiltonian is contributed by the pure AI = 1/2 operator Q_. We will now show the important property that Q- transforms as the component of an octet under SU(3)LTO see this we first note that the operator Q'i =riri-\sijrkrk

(69)

is an SU(3) octet. This follows because the rl in (63), (64) transform as the ql, and Slj in (61) is an octet. We next show that the (2,3) component of Q% • indeed has the flavour structure of Q^\ Q 2 3 = r 2 r 3 = (g3<7i - qiq3)(q1q2 - q2qX) = (su)(ud) - (sd){uu) - (uu)(sd) + (ud)(su)=2Q^

(70)

Since the quark fields in Q_ are all left-handed, we see that Q_ transforms as the (2,3) component of an octet Q4 • under SU(3)LTrivially, it is also a singlet under SU(3)R. Hence, Q_ transforms as a component of a (8^,1^) under SU(3)L ® SU(3)R. The transformation law for Ql • in matrix notation is, with L G SU(3)L, Q -»• LQtf

(71)

Including' the hermitian conjugate we may write 2(Q^ + Ql) = Q23 + Q32=tv\6Q

(72)

where the trace with the Gell-Mann matrix A6 is used to project out the proper components (A6 is the matrix with entry 1 at positions (2,3) and (3,2), and 0 otherwise). It is not hard to see that the penguin operators Q 3 , . . . , Q6 also transform as part of an (8L, 1R), in the same way as Q-.

170

Chiral L a g r a n g i a n We will now construct explicitly the leading terms of the chiral Lagrangian. Since we have to write down the most general form for this Lagrangian to any given order in the momentum expansion, the specific manner in which chiral symmetry is realized does not matter. The most convenient and standard choice is a nonlinear realization where one introduces the unitary matrix E = exp (j$\

(73)

as the basic meson field. Here / is the generic decay constant for the light pseudoscalars (we have used a normalization in which fv = 131 MeV). The field S is taken to transform under 5 ( 7 ( 3 ) L ® SU(3)R as E -» LEflt

(74)

with L £ SU(3)L and R G SU{2>)R. In general, (74) implies a complicated, nonlinear transformation law for the field $. However, for the special case of an ordinary 5(7(3) transformation, where L = R = U, (74) becomes equivalent to (68). We thus recover the correct transformation for the octet $ under ordinary SU(S). We can also see that the vacuum state, which corresponds to $ —> 0, hence E —> 1, is invariant under ordinary 5(7(3) (1 —• U 1 W = 1) as it must be. On the other hand, the vacuum is not invariant under the chiral transformation in (74): 1 —> L1R) ^ 1. This corresponds to the spontaneous breaking of chiral symmetry. The field (73) with the transformation (74) therefore has the desired properties to describe the pseudo-Goldstone bosons. The chiral Lagrangian is constructed as a series in powers of momenta, or equivalently numbers of derivatives £QCD =

CQCD+£QCD+^

CAS=1 = CAS=1 + CAS=1 +

( ? 5 )

^

( 7 6 )

describes strong interactions, £ A S = 1 AS = 1 weak interactions, and the subscripts on the right-hand side indicate the number of derivatives. The lowest order strong interaction Lagrangian has the form QQCD

/ _ t r [D/1E£>"Et + 2 B o ( X S f + EA^)] (77) 8 We have written DM for the derivative, which we later will generalize to a covariant derivative to include electromagnetism. For the moment we may consider D^ as an ordinary derivative. CQCD

=

171

The terms in (77) have to be built to respect the symmetries of the QCD Lagrangian in (49). For M = 0, (49) is chirally invariant and we are thus looking for invariants constructed from E in (74). Only trivial terms are possible with zero derivatives, for example tr(EE T ) = const. The leading term comes with two derivatives, as anticipated. The only possible form is tr(D M ED M E^). Here -D^ED^E* transforms as (8L,1_R) L ^ E Z ^ E 1 -> L £>„££>"£* IS

(78)

and taking the trace gives an invariant. Another possibility would seem to be t r ( D 2 E E t ) , but this term differs from t r ^ E D ^ E * ) only by a total derivative. The second term in (77) breaks chiral symmetry. Its form can be found by noting that the symmetry breaking term proportional to M in (49) would be invariant if M was interpreted as an auxiliary field transforming as M —»• LMB)• To first order in M and to lowest order in derivatives this leads to the mass term in (77). For M —» LMB) it would be invariant. For M fixed to the diagonal mass matrix it breaks chiral symmetry in the appropriate way. We will soon find that this term indeed counts as two powers of momentum. The 1S second order Lagrangian £ j then complete. The factor in front of the first term in (77) is fixed by the requirement that the kinetic term for the mesons be normalized in the canonical way. There are no additional parameters for this contribution. The second term in (77) comes with a coupling £?o- This new parameter is related to the meson masses. To see this more clearly, we can extract the kinetic terms from (77) by expanding to second order in the field $. If we keep, for example, only the contributions with neutral kaons K, K, we find (up to an irrelevant additive constant) C C

2 K°kin = d^Kd^K

- B0(ms + md)KK

(79)

From (79) and similar relations for the other mesons we obtain expressions for the pseudo-Goldstone boson masses in terms of the quark masses and the parameter J5 0 =

O(AQCD)

m2K0 = B0(ms 2

m

K+

+md)

- B0(ms + mu)

(80)

ml+ = B0(mu +md) The meson masses squared are proportional to linear combinations of quark masses. This also clarifies why one factor of M is equivalent to two powers of momenta in the usual chiral counting (see (77)). Expanding (77) beyond second order in $ we obtain terms describing strong interactions among the mesons, such as ir-n scattering.

172

We next need to determine the form of Cfs=1. As we have seen in sec. 3.3, the dominant contribution to the weak Hamiltonian comes from the component of an (8L, 1.R) operator as shown in (72). Since we know empirically from the AI = 1/2 rule that the enhancement of this piece of the weak Hamiltonian is quite strong, we shall here make the additional approximation to keep only this contribution and drop the rest (related to the operator Q+). This is a reasonable approximation for many applications. With the results we have derived so far, it is then easy to write down the correct form for C^s=1. The structure with two derivatives and the correct transformation properties as an (8i, li?) is given in (78). According to (72) we simply need to take the trace with A6 to obtain the right components. Factoring out basic weak interaction parameters we can write AAS=1 = ^\V:sVud\g8^tr\6D^D^

(81)

This Lagrangian introduces one additional parameter, the octet coupling g$. Eq. (81) already contains the usual hermitian conjugate part (see (72)). We have neglected the small CP violating effects and factored out the common leading CKM term V*sVud = VusV*d = \V:sVud\. Ks ->

TT+TT-

from Cfs=1

In order to make predictions, we first need tofixthe constant g$. We can use the dominant nonleptonic decay Ks —>• ir+n~ for this purpose. Expanding the interaction term in (81) to third order in $, and keeping only K, K, ir+ and 7T~, we find tr AedSSEt = - ^ [Tr+dKdir- - dKn-dn+

+ (K - K)dir+dw-]

(82)

Neglecting CP violation we have K\ = Ks, K2 = K^, where CPK\$ ±#1,2, and {CPK = -K)

=

Expressing K, K in terms of Kit2 we obtain for the square brackets in (82) [...] = -!=7r+dTr-{dK2-dKl)-^=n-dTT+{dK2+dK1) v2 v2

+ V2K1dTr+dTr- (84)

173

From (84) the Feynman amplitudes for if1)2 -> TT+-K can be read off. Denoting the momentum of the kaon, n+, n~ by fc, pi, P2, respectively, we get for Kx [...]Ki -^-^=(2p1-p2

+ k-(p1+p2))

= -V2(m2K-ml)

(85)

One may check that the corresponding amplitude for K% —»• ir+ir~ gives zero, as required by CP symmetry. From (85) and (81) we obtain the decay amplitude A(Ki ->• TT+TT") = iGFg8\\u\U(m2K

- ml)

(86)

(ml - ml)' G2Fg28\Xu\2f2

(87)

This gives the branching ratio

B{KS ->

TT+O

= rKs V™lJ™1

Using TKS

= 1.3573 • 1014 GeV" 1

B(KS

-*• TT+TT^) = 0.6861

we find 58 - 5.2

(89)

We have thus determined g% to lowest order in xPT. Using this result, we can make predictions. For instance, expanding (81) to fourth order in <£, we can derive amplitudes for the decays K -> 3TT. In this manner x P T relates processes with different numbers of soft pions. Such relations, also known as the soft-pion theorems of current algebra, are nicely summarized in the framework of %PT in terms of the lowest-order chiral Lagrangians. Other important applications are radiative decays as Ks —> 77, to which we will come back in the following paragraph. So far we have worked at tree level, which is sufficient at 0(p2). At the next order, 0(p4), one has to consider both tree-level contributions of the 0{pA) terms in the Lagrangians (75), (76), and one-loop diagrams with interactions from the 0(p2) Lagrangians. The loop diagrams are in general divergent. The divergences are absorbed by renormalizing the couplings at 0(p4). An example will be described below. Radiative K Decays Electromagnetism and the photon field A^ can be included in x P T in the usual way. The U(l) gauge transformation for the meson fields is £' = UYXJ* => $' = U*U\

U = exp(-ieQQ)

(90)

174

7

+ ... 7 Figure 14: K\ —> 77 in x ? T .

with 0 = 0(x) an arbitrary real function and the electric charge matrix Q = d i a g ( 2 / 3 , - 1 / 3 , - 1 / 3 ) . Writing out (90) for the components of $, one finds that each meson transforms with its proper electric charge as the generator. With A'^ = Afj, — d^Q, the covariant derivative that ensures electromagnetic gauge invariance is D^ = d^-ieAil[Q,T] (91) Using this assignment, (77) and (81) include the electromagnetic interactions of the mesons. At higher orders in the chiral Lagrangian also terms with factors of the electromagnetic field strengths F^v have to be included. In the chiral counting F^ is equivalent to two powers of momentum. We finally give a further illustration of the workings of X^T with two examples of long-distance dominated, radiative kaon decays. We first mention an important theorem for these processes. It says that the amplitudes of nonleptonic radiative kaon decays with at most one pion in the final state start only at 0(p4) in * P T . This means there are no tree-level contributions at 0(p2). Such terms are forbidden by gauge invariance. There can only be tree-level amplitudes from C(p 4 ), and, at the same order, loop contributions generated from 0(p2) interactions. Decays that fall under this category are K -¥ 77, K ->• jl+l~, K ->• 7T77 or K -> nl+l~. A particularly interesting example is K$ —• 77- In this case it turns out that there is no direct coupling even at 0(p4) and hence no counterterm to absorb any divergence from the loop contribution. As a consequence, the oneloop calculation (Fig. 14) is in fact finite. The only parameter involved is g8, which we have already determined. The finite loop calculation then gives a unique prediction 19 . It yields B{KS -> 77) = 2.1 • 10" 6

(92)

This compares well with the experimental result 20 (2.4±0.9) • 10~ 6 , which has recently been improved t o 2 1 (2.6 ± 0.4) • 10^ 6 .

175

Of course this situation with a finite loop result is somewhat special. A more generic case is K+ —> n+e+e~, shown in Fig. 15. Here the loop calcu-

e

e Figure 15: K+ ->• n+e+e'

in x P T .

lation is divergent and renormalized by the counterterm at 0(pi). There is now one additional free parameter, which can be determined from the rate of K+ —> n+e+e~. Other observables as the e+e~~ mass spectrum or the rate + and spectrum of K —> 7T + /J + /J,~ can then be predicted. In the same manner one can also analyze the amplitude for Kg -» 7r°e + e~, which determines the indirect CP violating contribution in Ki —> n°e+e~. Kg -> ir°e+e~ is very similar to K+ —> 7r + e + e~, but the required counterterm is different. For this reason the measurement of K+ -> 7r + e + e~ cannot be used to obtain a prediction for Ks -» 7r°e + e~. A separate measurement of the latter decay will therefore be needed in the future. 4 4.1

The Neutral-K System and CP Violation Basic Formalism

Neutral K mesons can mix with their antiparticles through second order weak interactions. They form a two-state system (K° — K°) that is described by a Hamiltonian matrix H of the form

*=(KH(&£) where CPT invariance has been assumed. The absorptive part T^ of H accounts for the weak decay of the neutral kaon. In Fig. 16 we show the diagrams that give rise to the off-diagonal elements of H. Diagonalizing the Hamiltonian H yields the physical eigenstates KL,S- They are linear combinations of the strong interaction eigenstates K and K and can be written as KL=Ne[{l+E)K

+ {l-e)K\=pK

+ qK

(94)

176

Figure 16: Diagrams contributing to M12 and Ti2 in the neutral kaon system.

Ks = Afg [(1 + E)K - (1 - e)K] = pK - qK

(95)

with the normalization factor A4 = l/-\/2(l + \e\2). Here e is determined by l-e_q 1+ e p

M{2 - i r j 2 (AM + f A r ) / 2

where A M and AP are the differences of the eigenvalues ML,S — i^L,s/^ responding to the eigenstates KL,S AM = ML-Ms>0

A r = T 5 - TL > 0

(96) cor-

(97)

The labels L and S denote, respectively, the long-lived and the short-lived eigenstate so that A r is positive by definition. We employ the CP phase convention CP • K = —K. Using the SM results for M i 2 , r i 2 and standard phase conventions for the CKM matrix (see (121)), one finds in the limit of CP conservation (77 = 0) that e = 0. With (94), (95) it follows that KL is CP odd and Kg is CP even in this limit, which is close to realistic since CP violation is a small effect. As we shall see explicitly later on, the real part of e is a physical observable, while the imaginary part is not. In particular (1 — e)/{l + e) is a phase convention dependent, unphysical quantity. A crucial feature of the kaon system is the very large difference in decay rates between the two eigenstates, the lighter eigenstate decaying much more rapidly than the heavier one (Ts = 579 T^). The basic reason is the small number of decay channels for the neutral kaons. Decay into the predominant CP even two-pion final states 7T+TT~, ir°ir° is only available for K$, but not (to first approximation) for the (almost) CP odd state K^. The latter can decay into three pions, which however is kinematically strongly suppressed, leading to a much longer Ki lifetime.

177

4-2

Classification of CP Violation

The fundamental weak interaction Lagrangian violates CP invariance via the CKM mechanism, that is through an irreducible complex phase in the quark mixing matrix. This leads to a violation of CP symmetry at the phenomenological level, in particular in the decays of mesons. For instance, processes forbidden by CP symmetry may occur or transitions related to each other by CP conjugation may have a different rate. In order for CP violation to manifest itself in this manner, an interference of some sort between amplitudes is necessary. The interference can arise in a variety of ways. It is therefore useful to introduce a classification of the various possibilities. We shall discuss it in terms of kaons, which are our main concern here, but it is applicable also to D and B mesons in a similar way. According to this classification, which is very common in the literature on CP violation, we may distinguish between: a) CP violation in the mixing matrix. This type of effect is based on CP violation in the two-state mixing Hamiltonian H (93) itself and is measured by the observable quantity Im(ri2/Mi 2 ). It is related to a change in flavour by two units, AS = 2. b) CP violation in the decay amplitude. This class of phenomena is characterized by CP violation originating directly in the amplitude for a given decay. It is entirely independent of particle-antiparticle mixing and can therefore occur for charged mesons as well. Here the transitions have AS = 1. c) CP violation in the interference of mixing and decay. In this case the interference of two amplitudes takes place between the mixing amplitude and the decay amplitude in decays of neutral mesons. This very important class is sometimes also refered to as mixing-induced CP violation, a terminology not to be confused with a). Complementary to this classification is the widely used notion of direct versus indirect CP violation. It is motivated historically by the hypothesis of a new superweak interaction which was proposed as early as 1964 by Wolfenstein to account for the CP violation observed in Ki —> ir+ir~ decay (see the lectures by Wolfenstein in this volume). This new CP violating interaction would lead to a local four-quark vertex that changes the flavour quantum number (strangeness) by two units. Its only effect would be a CP violating contribution to Mi2, so that all observed CP violation could be attributed to particleantiparticle mixing alone. Today, after the advent of the three generation SM, the CKM mechanism of CP violation appears more natural. In principle the superweak scenario represents a logical possibility, leading to a different pattern of observable CP violation effects. Now, any CP violating effect that can be entirely assigned to CP violation in

178 Mi2 (as for t h e superweak case) is termed indirect CP violation. Conversely, any effect t h a t can not be described in this way and explicitly requires C P violating phases in t h e decay amplitude itself is called direct CP violation. It follows t h a t class a) represents indirect, class b) direct C P violation. Class c) contains aspects of both. In this latter case t h e magnitude of C P violation observed in any one decay mode (within t h e neutral kaon system, say) could by itself be ascribed t o mixing, thus corresponding t o an indirect effect. On the other hand, a difference in t h e degree of C P violation between two different modes would reveal a direct effect. We illustrate these classes by a few important examples. We will also use this opportunity t o discuss several aspects of kaon C P violation in more detail. a) - Lepton

Charge

Asymmetry

T h e lepton charge asymmetry in semileptonic KL decay is a n example for C P violation in t h e mixing matrix. It is probably t h e most obvious manifestation of C P nonconservation in kaon decays. T h e observable considered here reads (I = e or /i) A =

T{KL

->• n-l+u)

- Y{KL

->• TT+Z"^)

T(KL

->• -K-1+V)

+ T{KL

->•

w+l-9)

2Ree« 7 l m ^ 4 Mi 2

|l + e l 2 - | l - £ | 2 |l+£|2 + |l-e|2 (98)

If C P was a good symmetry of nature, KL would be a C P eigenstate and t h e two processes compared in (98) were related by a C P transformation. T h e rate difference A should vanish. Experimentally one finds however 2 0 Aexp

= (3.27 ± 0 . 1 2 ) - 1CT3

(99)

a clear signal of C P violation. T h e second equality in (98) follows from (94), noting t h a t t h e positive lepton l+ can only originate from K ~ (sd), l~ only from K ~ (ds). This is true t o leading order in SM weak interactions and holds to sufficient accuracy for our purpose. T h e charge of t h e lepton essentially serves t o t a g t h e strangeness of t h e K, thus picking out either only t h e K or only t h e K component. Any phase in t h e semileptonic amplitudes is irrelevant and t h e C P violation effect is purely in t h e mixing matrix itself. In fact, as indicated in (98), A is determined by I m ( r i 2 / M i 2 ) , t h e physical measure of C P violation in t h e mixing matrix. From (99) we see t h a t A > 0. This empirical fact can be used t o define positive electric charge in an absolute sense. Positive charge is t h e charge of

179

the lepton more copiously produced in semileptonic Ki decay. This definition is unambiguous and would even hold in an antimatter world. Also, using some parity violation experiment, this result implies in addition an absolute definition of left and right. These are quite remarkable facts. They clearly provide part of the motivation to try to learn more about the origin of CP violation. b) - CP Violation in the Decay Amplitude Observable CP violation may also occur through interference effects in the decay amplitudes themselves (pure direct CP violation). This case is conceptually perhaps the simplest mechanism for CP violation and the basic features are here particularly transparent. Consider a situation where two different components contribute to the amplitude of a K meson decaying into a final state / A = A{K -> /) = Aie^e^ + A ^ e ^ (100) Here Ai (i = 1,2) are real amplitudes and Si are complex phases from CP conserving interactions. The Si are usually strong interaction rescattering phases. Finally the 4>i are weak phases, that is phases coming from the CKM matrix in the SM. The corresponding amplitude for the CP conjugated process K -¥ f then reads (the explicit minus signs are due to our convention CP • K = —K, (CP-f = / ) ) A = A(K -> / ) = - A i e ^ e - * * 1 - A2ei^e~i^

(101)

Since now all quarks are replaced by antiquarks (and vice versa) compared to (100), the weak phases change sign. The CP invariant strong phases remain the same. From (100) and (101) one finds immediately \A\2 - \A\2 ~ AYA2 sin((Ji - S2) sin(0i -

fa)

(102)

The conditions for a nonvanishing difference between the decay rates of K —> f and the CP conjugate K —• / , that is direct CP violation, can be read off from (102). There need to be two interfering amplitudes A\_, A2 and these amplitudes must simultaneously have both different weak (;) and different strong phases (Si). Although the strong interaction phases can of course not generate CP violation by themselves, they are still a necessary requirement for the weak phase differences to show up as observable CP asymmetries. It is obvious from (100) and (101) that in the absence of strong phases A and A would have the same absolute value despite their different weak phases, since then A = -A*.

180

A specific example is given by the decays K{K) 7r 7r~ = / ) . The amplitudes can be written as

-> 7r+7r~ (here / =

+

A+_ = \^A0eiS° V o

+

yj i

~A2e^

A+- = - y f ^ S e " 0 - ^A*2ei5*

(103)

where AQ^ are the transition amplitudes of K to the isospin-0 and isospin-2 components of the 7r+7r~ final state, defined by (7T7r(/ = 0,2)\% W \K) = A0t2eid^

(104)

They still include the weak phases, but the strong phases have been factored out and written explicitly in (103), (104). Taking the modulus squared of the amplitudes we get T(K -> TT+TT-) - T{K ->• 7T+7T-) = r- . Sm( T(K -> ^ + T T - ) + T ( i ? -»• 7T+7T-) " ° = 2 Re e'

_ 2)

Re,42 (JmA2 ReA 0 VReA2

_ ImA 0 ReA) (105)

The quantity so defined is just twice the real part of the famous parameter e', the measure of direct CP violation in K —> nir decays. The real parts of Aot2 can be extracted from experiment. The imaginary parts have to be calculated using the effective Hamiltonian formalism. We should stress that the quantity in (105) is not the observable actually used to determine e' experimentally. We have discussed it here because it is of conceptual interest as the simplest manifestation of e'. The realistic analysis requires a more general consideration of KL , Kg -> 7T7T decays to which we turn in the following paragraph. c) - Mixing Induced CP Violation in K —> irir: e, e' In this section we will illustrate the concept of mixing-induced CP violation with the example of K —> 7T7T decays. These are important processes, since CP violation has first been seen in KL —> n+n~ and as of today our most precise experimental knowledge about this phenomenon still comes from the study of K -> 7T7T transitions. There are two distinct final states and in a strong interaction eigenbasis the transitions are K°,K° —> irir(I = 0),irir(I = 2), with definite isospin for mr. Alternatively, using the physical eigenbasis for both initial and final states, one has KL,KS —> n+ir~,ir°ir°.

181

Consider next the amplitude for Ki going into the CP even state irir(I = 0), which can proceed via K (~ (1 + e)A0) or via K (~ (1 - E)AQ). Hence (to first order in small quantities) A{KL -»• 7T7r(/ = 0)) ~ (1 + e)A0ei6° - (1 - e)A*0ei5° ~ e + i j ^ = e (106) Kej4o

This defines the parameter e, characterizing mixing-induced CP violation. Note that e involves a component from mixing (e) as well as from the decay amplitude (lmA0/ReA0). Neither of those is physical separately, but e is. Note also that the physical quantity Ree discussed above satisfies Ree = Ree. More generally one can form the following two CP violating observables A(KL

-» 7T+7T-)

A(KS

-> TT+TT")

_ A{KL

-> 7T07r°)

° ~ A{KS

-» 7r°7r°)

Vo

These amplitude ratios involve the physical initial and final states and are directly measurable in experiment. They are related to e and e' through r)+- = e + e'

rj00 = s - 2e'

(108)

The phase of e is given by e = |e| exp(i7r/4). The relative phase between e' and e can be determined theoretically. It is close to zero so that to very good approximation e'/e = Re(e'/e). Both ?7-j and 7700 measure mixing-induced CP violation (interference between mixing and decay). Each of them considered separately could be attributed to CP violation in K-K mixing and would therefore represent indirect CP violation. On the other hand, a nonvanishing difference r)^ —rjoo = 3e' 7^ 0 is a signal of direct CP violation. Experimentally one has 2 0 \e\ = (2.282 ±0.019) • 10" 3

(109)

The quantity e' can be measured as the ratio Re {e'/e) « e'/e using the double ratio of rates V+= 1+ 6 Re(110) »7oo

e

Ten years ago, and until recently, the experimental situation was characterized by the following, somewhat inconclusive results 2 2 , 2 3 : £^ f ( 2 3 ± 7 ) • 10" 4 CERNNA31 e ~ \ (7.4 ± 5.9) • 10" 4 FNAL E731

{

>

In particular the second measurement was well compatible with zero. A new round of experiments, conducted at both CERN and Fermilab, was therefore

182

anticipated with great interest. direct CP violation 24 ' 25 :

The recent results have firmly established

e^ _ f (28.0 ± 4.1) • 10" 4 FNAL KTeV e ~ \ (14.0 ± 4.3) • 10" 4 CERN NA48

"

>

These results rule out the superweak hypothesis, at least in its most stringent form. The analyses of the experiments are currently still ongoing and should eventually settle the value of e'/e to an accuracy of (1 — 2) • 10~ 4 . 4-3

Theory of e and the Unitarity Triangle

Calculation of e In the theoretical expression for e in (106), the term ImAo/KeAo is numerically negligible (in standard phase convention). The value for e is then approximately given by e from (96), and can be written as W 4

MM» V2AM

( n 3 )

Mi2 is related to the first diagram shown in Fig. 16. It is given by Mi 2 = ^(K0\H?ff2\K°)

(114)

Here T-LffjT2 is the effective Hamiltonian for AS = 2 transitions, which is derived from the box diagrams for Mi2 in Fig. 16 by performing an operator product expansion. In this case there is only a single operator Q A S = 2 = {ds)v-A(ds)V-A

(115)

in the effective Hamiltonian. One obtains explicitly £ = e

W 4 ^ / l 2

12TT

rnK ^2AMK

2

•Im [A; S0(xc)r]i + K2S0{xt)m

+ 2\*\*tS0(xc,xt)m]

(116)

Here A; = V*sVid, fx = 160 Mel/ is the kaon decay constant and, at NLO, the bag parameter B^ is defined by 9

W(,A]-V BK=BK(v)[a^(n)}

1 +

_

4 ^

J s

(117)

183 Table 1: NLO results for rn with A I I L = (325 ± 110) MeV,

mc(mc)

= (1.3 ± 0.05) GeV,

mt{mt) = (170 ± 15) GeV. The third column shows the uncertainty due to the errors in Avre- and quark masses. The fourth column indicates the residual renormalization scale uncertainty at NLO in the product of rji with the corresponding mass dependent function from eq. (116). These products are scale independent up to the order considered in perturbation theory. The central values of the QCD factors at LO are also given for comparison.

m m m

NLO (central) 1.38 0.574 0.47

MS'

m

V

±35% ±0.6% ±3%

scale dep. ±15% ±0.4% ±7%

(K°\(ds)V-A(d8)v-A\K0)

NLO ref. 26 27 28

= -BK{y)fKm2K

LO (central) 1.12 0.61 0.35

(118)

The index (3) in eq. (117) refers to the number of flavours in the effective theory and J3 = 307/162 (in the so-called NDR scheme12). The Wilson coefficient multiplying BK in (116) consists of a charm contribution, a top contribution and a mixed top-charm contribution. It depends on the quark masses, Xi = m^/M^, through the functions SQ. The rji are the corresponding short-distance QCD correction factors (which depend only slightly on quark masses). Detailed definitions can be found in 1 2 . Numerical values for rji, T)2 and 773 are summarized in Table 1. Concerning these results the following remarks should be made. • £ is dominated by the top contribution (~ 70%). It is therefore rather satisfying that the related short distance part r)2So(xt) is theoretically extremely well under control, as can be seen in Table 1. Note in particular the very small scale ambiguity at NLO, ±0.4% (for 100 GeV" < \it < 300GeV). This intrinsic theoretical uncertainty is much reduced compared to the leading order result where it would be as large as ±9%. • The rji factors and the hadronic matrix element are not physical quantities by themselves. When quoting numbers it is therefore essential that mutually consistent definitions are employed. The factors 77, described here are to be used in conjunction with the so-called scale- (and scheme-) invariant bag parameter BK introduced in (117). The last factor on the right-hand side of (117) enters only at NLO. As a numerical example, if the (scale and scheme dependent) parameter -BK-(M) is given in the NDR scheme at ti = 2GeV, then (117) becomes BK = BK(NDR, 2 GeV) • 1.31 • 1.05.

184

• The quantity BK has to be calculated by non-perturbative methods. A representative range is BK =0.80 ±0.15 (119) The status of BK is reviewed in 2 9 . Determination of the Unitarity Triangle The source of CP violation in the standard model (SM) is the CabibboKobayashi-Maskawa (CKM) matrix V entering the charged-current weak interaction Lagrangian CCC

=

^ ^

r

(

1

"

75)

^"^+

+ h C

(120)

- -

where (1*1,1*2,1*3) = (u,c,t), (di, cfe, d%) = (d,s,b) are the mass eigenstates of the six quark flavours and a summation over i,j = 1,2,3 is understood. The unitary CKM matrix has the following explicit form (VudVusVub\ V=\Vcd Vcs Vcb)~\

\VtdVtsVtbJ

(

l-A2/2 -A

\AX3(l-g-irj)

A 1 - A 2 /2

-AX2

AX^g-i^X AX2

1

(121)

/

where the second expression is a convenient parametrization in terms of A, A, g and 77 due to Wolfenstein. It is organized as a series expansion in powers of A = 0.22 (the sine of the Cabibbo angle) to exhibit the hierarchy among the transitions between generations. The explicit parametrization shown in (121) is valid through order C(A 3 ), an approximation that is sufficient for most practical applications. Higher order terms can be taken into account if necessary 12 . The unitarity structure of the CKM matrix is conventionally displayed in the so-called unitarity triangle, Fig. 17 (left). This triangle is a graphical representation of the unitarity relation VudV*b+VcdV*b + VtdVt*b = 0 (normalized by —VcdV*b) in the complex plane of Wolfenstein parameters (g, 77), The angles a, /? and 7 of the unitarity triangle are phase convention independent and can be determined in CP violation experiments. The area of the unitarity triangle, which is proportional to rj, is a measure of CP nonconservation in the standard model. We briefly summarize the main ingredients of the standard analysis of the unitarity triangle, where e plays a central role. There are 4 independent parameters A, A, Q and 77 in the CKM matrix, which are determined from 4 measurements as follows:

185

as BK,mt,Vcb

I

Figure 17: The normalized unitarity triangle in the {g, r]) plane (left), and its standard determination (right).

• A = 0.22, from K+ -t n°l+p, or from semileptonic hyperon decay (A ->• pev, Y,~ —» neP, S~ —> hev). • Vcb = AX2 = 0.040 ± 0.002, from b ->• civ transitions. • \Vub/Vcb\ = A y V +rf = 0.09 ± 0.02, from b ->• ulv transitions. . |e| ~ r, ((1 - Q)A2S(mt) + c) A2BK with

BK

= 0.80 ± 0.15, from indirect CP violation in K ->

TTTT.

Under the final item we have indicated the dependence of e (from (116)) on the most important parameters, writing the CKM quantities explicitly in Wolfenstein form. Here c denotes a constant that is independent of A, g, T], nit and BKThe standard determination of the unitarity triangle is illustrated in Fig. 17 (right). The relevant input parameters are BK, rut, Vcb and IK^/Vctl. For fixed BK, rnt and Vcb, the measured |e| determines a hyperbola in the Q-TJ plane of Wolfenstein parameters. Intersecting the hyperbola with the circle defined by l^fc/Vcil determines the unitarity triangle (up to a two-fold ambiguity). There is a simple regularity, which is quite useful and easy to remember: As any one of the four input parameters becomes too small (with the others held fixed), the SM picture becomes inconsistent (see Fig. 17). Using this fact, lower bounds on these parameters can be derived. The large value that has been established for the top-quark mass in fact helps to maintain the consistency of the SM. In principle, once the unitarity triangle is fixed in this manner, any further, independent measurement of a quantity in the (Q,T]) plane provides us with

186

8,
0.6 It 0.4

0.2

-0.6

-0.4

-0.2

0.0 P

0.2

0.4

0.6

Figure 18: The allowed region (shaded) in the (g, fj) plane, combining information from E, \Vub/Vcb\ a n d including the constraint from AM^. The independent constraint from the lower limit on AMs/AMd excludes the region to the left of the curves labeled with A M S in the plot. £ ~ 1.2 measures SU(3) breaking in the hadronic matrix elements of B^-B^ versus Bs-Bs mixing.

an additional standard model test. In practice, however, the accuracy of such a test is limited by hadronic uncertainties, which enter mainly through BK and IVub/Vcbl- Useful additional restrictions come from B-B mixing. Both AM4 and AMd/AMs, where A M , is the mass difference in the Bq-Bq system, constrain y^(l - g)2 + rj2. The ratio AMd/AMs is particularly important, because the hadronic uncertainties cancel in the limit of SU(3) symmetry. Currently, while AMd is well measured, only a lower bound AMS > 14.3 p s _ 1 exists at present. However, already this bound is interesting. Together with AMd, it implies a quite clean upper bound on y^(l — g)2 + rf. This essentially excludes negative values of g, severely restricting the allowed range of g and 77.

The results of a complete analysis of the unitarity triangle are shown in Fig. 18. Here the axes are labeled by g = g(l - A 2 /2) and f) = 77(1 - A 2 /2), instead of g and r\. In this way higher terms in the Wolfenstein expansion ~ A2, which can be neglected to first approximation, are consistently taken into account. The plot is taken from 30 where further details can be found.

187

4-4

Calculating e'' je

The formula for s'/e, which can be derived from the definition in (107), (108), is given by e' u (lmA-2 lmAa\ e y/2\e\ \ReA2 ReA0i where ui = ReA2/ReA0.

This may be compared with (105) using (123)

arg(£ ) = - + S2 - 60 « j The expression (122) for e'/e may also be written as £

V2|£|ReA 0 V

£

ImA o

(124)

lm.A2

ImAo,2 are calculated from the general low energy effective Hamiltonian for AS = 1 transitions (48), which we have described in sec. 3.4. One has ImAo,2 =

G

10

V2

i=3

(125)

-lmXt-y=^2yi(fi){Qi)0t2

Here y,- are the Wilson coefficients and (Qi)02eiS°-2

= (mr(I = 0, 2)IQ^isT0),

A* = v;svtd. For the purpose of illustration we keep only the numerically dominant contributions and write 7 = 2miA0lmXt

(y6{Q6)o - hy*{Q*h+•

(126)

•

Q6 originates from gluonic penguin diagrams and Qs from electroweak contributions, as indicated schematically in Fig. 19. The matrix elements of Q6 and Q8 can be parametrized by bag parameters B6 and Bs as (Qe)o = - 4

1 2

mK .ms(/x)

2 ~ v ^

+md{n)

mK m g (^) +m d (/i)

/

x 2

m2K(fK - fn) • B6 ~ p ^

m x / x • -Bs

BH

56

(127)

(128)

£?6 = ^8 = 1 corresponds to the factorization assumption for the matrix elements, which holds in the large NQ limit of QCD.

188

d

d

9

1,Z q

Q

q

Figure 19: Gluonic and electroweak penguin contributions, which give rise to operators and Q%, respectively.

The numerical importance of the contributions from Q$ and Q% can be understood as follows. Q§$ are particular because they are of the (V — A) ® (V + A) form, which results in a (5 + P) ® (S — P) structure upon Fierz transformation. Factorizing the matrix element of such operators gives for example (TT+TT

\(su)s+p{ud)s-p\K)

-•

-(TT

•

\SU\K)

(TT+ \wy5d\0)

(129)

Taking the derivative (<9M) of {n+(p)\(uj^5d)(x)\0)

=

UV^VX

(130)

and using the equations of motion, we find (7r + |u 75 d|0) = A

mu + md

h

K

=Um

s

+md

(131)

Here the second equality follows from the x ? T relations in (80). A second factor of m2K/(ms + rrid) comes in a similar way from the scalar current matrix element (-K~\SU\K). This explains the quark mass dependence in (127) and (128). Since l

K (132) Bo = 0(AQCD) m s +md we see that the matrix elements are primarily not proportional to (m s +rrid)~2, but to B0, which remains finite in the chiral limit ms, m^ —> 0. However TTIK is precisely known and it is customary to trade the x P T parameter BQ for the quark masses ms + rrid ~ ms. Because B0 is numerically, and somewhat accidentally, quite large (equivalently, the quark masses quite small), the matrix elements of Qe,s are systematically enhanced over those of the ordinary

189

(V — A) ® (V — A) operators. Q$ and Qj have a Dirac structure similar to Qe and Qs, but a different colour structure, which leads to a 1/NQ suppression. In addition their Wilson coefficients are numerically smaller. This implies that Qe and Qg give the dominant contributions. V6{Q6}o and ys{Qs}2 are positive numbers. The value for e'/e in (126) is thus characterized by a potential cancellation of two competing contributions. Since the second contribution is an electroweak effect, suppressed by ~ a/as compared to the leading gluonic penguin ~ (Qe)o, it could appear at first sight that it should be altogether negligible for e'/e. However, a number of circumstances actually conspire to systematically enhance the electroweak effect so as to render it a sizable contribution: • Unlike Qe, which is a pure AI = 1/2 operator, Qs can give rise to the 7T7r(7 = 2) final state and thus yield a non-vanishing ImA2 in the first place. • The 0(a/as) suppression is largely compensated by the factor l/u « 22 in (126), reflecting the AI =1/2 rule. •

—

^ ( Q s h gives a negative contribution to e'/e that strongly grows with rrit- For the realistic top mass value it can be substantial.

In order to estimate e'/e numerically (see (126)), the hadronic matrix elements have to be determined within a nonperturbative framework (e.g. lattice QCD, 1/Nc expansion, chiral quark model), while the coefficients j/j are known from perturbation theory, and Re^lo, w, GF, \S\ are fixed from experiment. Finally, the CKM quantity ImAt ~ r) is obtained from the standard determination of the unitarity triangle described in the previous section. The Wilson coefficients j/j have been calculated at NLO 3 1 ' 3 2 . The shortdistance part is therefore quite well under control. The remaining problem is then the computation of matrix elements, in particular B§ and B%. The cancellation between these contributions enhances the relative sensitivity of e'/e to the anyhow uncertain hadronic parameters which makes a precise calculation of e'/e impossible at present. The order of magnitude of e' can however be understood from (122). The size of ImAi/ReAi is essentially determined by the small CKM parameters that carry the complex phase and which are related to the top quark in the loop diagrams of Fig. 19. Roughly speaking ImAi/ReAi ~ lmVt*Vtd ~ 10~ 4 . Empirically we have, from the AI = 1/2 rule, ReA2/ReA0 ~ 10~ 2 . This leads to a natural size of e' of ~ 10~ 6 , thus e'/e ~ 10~ 3 . A complete analysis gives the result 33,29 1.4-lCT 4 <e'/e<

32.7- 10" 4

(scanning)

(133)

190

Figure 20: The leading order electroweak diagrams contributing to K model.

5.2 • 1CT4 < e'/e < 16.0 • 10~ 4

ixvv in the standard

(Gaussian)

(134)

This is compatible with the experimental results (111), (112) within the rather large uncertainties. The two ranges refer to different treatments of uncertainties in the experimental input: the assumption of Gaussian errors, or flat distributions (scanning). Similar findings are reported by other groups 34 ^ 12 . A detailed review of the theoretical status of e'/e, including recent developments (hadronic matrix elements, final state interactions, isospin breaking corrections etc.), along with further references can be found in 4 3 . The recent experimental confirmation that indeed e' ^ 0 constitutes a qualitatively new feature of CP violation and is as such of great importance. However, due to the large uncertainties in the theoretical calculation, a quantitative use of this result for the extraction of CKM parameters is unfortunately rather limited. For this purpose one has to turn to theoretically cleaner observables. As we will see in the next section, rare K decays in fact offer very promising opportunities in this direction. 5 5.1

Rare K Decays K+

->• -K+VV and KL -> -n°vv

The decays K —> TXVV proceed through flavour changing neutral currents. These arise in the standard model only at second (one-loop) order in the electroweak interaction (Z-penguin and W-box diagrams, Fig. 20) and are additionally GIM suppressed. The branching fractions are thus very small, at the level of 10 - 1 0 , which makes a detection of these modes rather challenging. However, the loop process K —> iri/P, a genuine quantum effect of standard model flavour dynamics, probes important short distance physics, in particular properties of the top quark (mt, Vtd, Vts). It is also very sensitive to poten-

191

tial new physics effects. At the same time, the K -> itvv modes are reliablycalculable, in contrast to most other decay modes of interest. A measurement of K+ —> ir+vv and KL —> -KQVV will therefore be an extremely useful test of flavour physics. Let us discuss the main properties of these decays, concentrating first on the charged mode. The GIM structure of the amplitude can be written as Y,

XlF(xl) = Xc(F(xc)-F(xu))

+ Xt{F(xt)-F{xu))

(135)

with Xi = V*s Vid and Xi = m] /M^. The first important point is the characteristic hard GIM cancellation pattern, which means that the function F depends as a power on the internal mass scale F{xu) ~ % £ ~ 10- 5 « F(xe) ~ ^ f - In ¥"- ~ 10- 3 « F(xt) ~ 1 (136) The up-quark contribution is a long-distance effect, determined by the scale AQCD- AS an immediate consequence, top and charm contribution with their hard scales mt, mc dominate the amplitude, whereas the long-distance part F(xu) is negligible. Note that the charm contribution, Ac F(xc) ~ 10 _ 1 • 10~ 3 , and the top contribution, At F(xt) ~ 10 _ 4 -1, have the same order of magnitude when the CKM factors are included. The origin of the hard GIM mechanism is the fact that the neutrinos only couple to the heavy gauge boson W and Z. It is interesting to contrast the situation with K+ —> ir+e+e~, where photon exchange can contribute as shown in Fig. 21. For simplicity we consider the case of internal quarks that are light compared to Mw The W propagator can then be contracted and the loop reduces essentially to a vacuum polarization diagram. Electromagnetic gauge invariance, which is unbroken, requires the sd-photon vertex to have the form Tu(q) ~ s 7 " ( l - J5)d • (q2g^ - q^qv) In —

(137)

77Zj

where q is the photon momentum and we assumed q2 <JC mj -C M^,. The structure in (137) ensures current conservation, q"Tv(q) = 0. When the vertex is contracted with the electron current e-y"e, the q^qv term vanishes by the equations of motion. The q2 factor of the remaining term is canceled by the photon propagator, which yields a local (sd)v-A(ee)v interaction and a loop function F(Xi) ~ In MUL

(138)

192

d -**•

q 4- ^7 Figure 21: Soft GIM mechanism in the photon penguin.

The logarithmic behaviour of the photon penguin is refered to as the soft GIM mechanism. It is in contrast to the hard GIM structure arising from the W and Z contributions where gauge symmetry is spontaneously broken, leading to the power behaviour ~ (mf /M^) ln(Mw//m;). The unsuppressed sensitivity to the light quark mass m; = mu in (138) is a signal of the long-distance dominance in the K+ —> n+e+e~ amplitude. Of course, xPT is needed for a consistent analysis in this case. The quark-level result in (138), derived in perturbation theory, is strictly speaking not valid, but it is enough to indicate the long-distance sensitivity and the order of magnitude of the contribution. The short-distance dominance of the s —• dvv transition next implies that the process is effectively semileptonic, because a single, local operator (sd)v-A{vv)v-A describes the interaction at low-energy scales. Hence the amplitude has the form A(K+ -> n+vv) ~ GFa{XcFc + \tFt)(Tr+\{sd)v\K+)

(vv)v_A

(139)

The coefficient function \CFC + XtFt is calculable in perturbation theory. The hadronic matrix element can be extracted from K+ —>• 7r°e + ^ decay via the isospin relation (13). The K+ —> ix+vv amplitude is then completely determined, and with good accuracy. The neutral mode proceeds through CP violation in the standard model. This is due to the definite CP properties of K°, ir° and the hadronic transition current (sd)v-A- Using CP\n°) = -|7r°>

CP\K°) = -\K°)

CP (sd)v (CP)-1

=

-{ds)v (140)

193

we have (A(sd)v\K°)

= -{n0\(ds)v\K0)

(141)

(The axial vector currents (sd)A, (ds)A do not contribute to the K —> TT transition because of parity.) With KL = (K° + K°)/y/2 ( the ^-contribution is negligible) we then obtain for the matrix element of the hadronic transition current {n°\\i(sd)v + K{ds)v\KL) ~ ImA, (142) where A; is the appropriate CKM factor. This demonstrates the CP violating character of the leading standard model amplitude for KL -> 7r°i/i>. For simplicity we have given the argument here assuming standard phase conventions. A manifestly phase convention independent derivation of the same result is discussed in 4 4 . The amplitude then has the form A{KL ->

TPVV)

~ ImAt Ft + ImAc Fc

(143)

where ImAt Ft ~ 1(T 4 • 1 > ImAc Fc ~ 1CT4 • 10" 3

(144)

The violation of CP symmetry in KL —> K°VV arises through interference between K°-K° mixing and the decay amplitude. This mechanism is an example of mixing-induced CP violation. In the standard model, the mixing-induced CP violation in KL —> ifivv is larger by orders of magnitude than the one in Ki ~> 7r+7r~, for instance. This is because A(KL -> *°uu) in contrast to the per-mille-size ratios in (107). Any difference in the magnitude of mixing induced CP violation between two K^ decay modes is a signal of direct CP violation. For this reason, the standard model decay KL —> i^Qvv is a signal of almost pure direct CP violation, revealing an effect that can not be explained by CP violation in the K — K mass matrix alone. The K -> iri/9 modes have been studied in great detail over the years to quantify the degree of theoretical precision. Important effects come from short-distance QCD corrections. These were computed at leading order in 4 5 . The complete next-to-leading order calculations 46>47>48 reduce the theoretical uncertainty in these decays to ~ 5% for K+ —• w+vv and ~ 1% for KL —> 7rVz>. This picture is essentially unchanged when further small effects are considered, including isospin breaking in the relation of K -> -KVV to K+ —> TT°l+vi9, long-distance contributions 50 ' 51 , the CP-conserving effect in KL —>

194 Table 2: Compilation

of important properties and results for K —> •KVU. K+

CKM contributions scale dep. (BR) BR (SM) exp.

-> -K+UU

KL

CP conserving Vtd top and charm ±20% (LO) -> ±5% (NLO) (0.8±0.3)-10^ i U (1.5+^) • 10- 1 0 BNL 787 55

- > TT^VV

CP violating lmVt*Vtd ~ J C p ~ ?? only top ±10% (LO) ->• ± 1 % (NLO) (2.8 ±1.1) • 10" 1 1 < 5.9-10- 7 KTeV 56

ir°vv in the standard model 50 ' 52 , two-loop electroweak corrections for large mj 53 and subleading-power corrections in the OPE in the charm sector 54 . While already K+ —> it+vv can be reliably calculated, the situation is even better for KL, —> iPvv. Since only the imaginary part of the amplitude contributes, the charm sector, in K+ —• TT+VV the dominant source of uncertainty, is completely negligible for KT, -> -n^vv (0.1% effect on the branching ratio). Long distance contributions ( ^ 0.1%) and also the indirect CP violation effect ( ^ 1%) are likewise negligible. The total theoretical uncertainties, from perturbation theory in the top sector and in the isospin breaking corrections, are safely below 3% for B(KL -* n°vv). This makes this decay mode truly unique and very promising for phenomenological applications. In Table 2 we have summarized some of the main features of K+ -> ir+ vv and KL —> irQvv. Note that the ranges given as the standard model predictions in Table 2 arise from our, at present, limited knowledge of standard model parameters (CKM), and not from intrinsic uncertainties in calculating the branching ratios. With a measurement of B(K+ -)• ir+i/v) and B{KL ->• TPVV) available very interesting phenomenological studies could be performed. For instance, B{K+ ->• ix+vv) and B(KL ->• -K°VV) together determine the unitarity triangle (Wolfenstein parameters g and rf) completely (Fig. 22). The expected accuracy with ±10% branching ratio measurements is comparable to the one that can be achieved by CP violation studies at B factories before the LHC e r a 4 4 . The quantity B(KL -> ^vv) by itself offers probably the best precision in determining ImVj* V^ or, equivalently, the Jarlskog parameter JCP = lm(V;sVtdVusV:d)

= A ( l - y ) ImAt

(146)

The prospects here are even better than for B physics at the LHC. As an

195

Figure 22: Unitarity triangle from K+ —> ir+vv and KL —> 7r°^f.

example, let us assume the following results will be available from B physics experiments sin 2a = 0.40 ± 0.04

sin 2/3 = 0.70 ± 0.02

Vcb = 0.040 ± 0.002

(147)

The small errors quoted for sin 2a and sin 2/? from CP violation in B decays require precision measurements at the LHC. In the case of sin 2a we have to assume in addition that the theoretical problem of 'penguin-contamination' can be resolved. These results would then imply IrnAt = (1.37 ± 0.14) • 1 0 - 4 . On the other hand, a ±10% measurement B{KL ->• ir°vv) = (3.0 ±0.3) • 1 0 ^ n together with mt(mt) = (170 ± 3)GeV would give ImAt = (1.37 ± 0.07) • 10~ 4 . If we are optimistic and take B{Ki —> -K°VD) = (3.0 ±0.15) • 1 0 ~ n , mt{mt) = (170 ± l)GeV, we get ImAt = (1.37 ± 0.04) • 10~ 4 , a remarkable accuracy. The prospects for precision tests of the standard model flavour sector will be correspondingly good. The charged mode K+ —• ii+vv is still being studied by Brookhaven experiment E787, which will be followed by a successor experiment, E949 57 . Recently, a new experiment, CKM 5 8 , has been proposed to measure K+ —> ix+vv at the Fermilab Main Injector, studying K decays in flight. Plans to investigate this process also exist at KEK for the Japan Hadron Facility (JHF) 5 9 . The neutral mode, KL, —> Tr°vD, is currently pursued by KTeV. For KL —> •K0VV a model independent upper bound can be infered from the experimental result on K+ —> 7r+i/z/, which at present is stronger than the direct experimental limit 60 . It is given by B(KL -» -K°VD) < 4AB(K+ -> TT+UP) < 2 • 10" 9 . At least this sensitivity will have to be achieved before new physics is constrained with B(KL —>• ifivV). Concerning the future of KL, —> ifivv experiments, a proposal exists at Brookhaven (BNL E926) to measure this decay at the AGS with a sensitivity of 0(1O~ 12 ) 61 . There are furthermore plans to pursue this mode with comparable sensitivity at Fermilab 62 and KEK 6 3 . Prospects for

196

KL —> "K°vv at a ^-factory were discussed in 6 4 . 5.2

KL -> 7T°e+e-

The electric charge of the leptons and the resulting interaction with photons make the decay KL —» 7r°e+e~ more complicated than KL —> iftvv. The short-distance dominated part of the KL ->• 7r°e+e~ amplitude can be analyzed using OPE and the renormalization group in analogy to the case of the AS = 1 effective Hamiltonian. This approach is reviewed in 1 2 . Here we would like to give a qualitative discussion, which highlights the characteristic points and also summarizes the main differences between KL —> n°e+e~ and KL —> TT0VP. The basic diagrams for KL —• 7r 0 e + e~ are similar to those for KL —• ir°vv, except that also a photon can be exchanged instead of the Z boson in the penguin diagrams (see Fig. 20). Taking the GIM mechanism into account, the structure of the effective Hamiltonian reads neff

~ At (Ft - Fc) + Xu (Fu - Fc)

(148)

Recalling that we can write KL « K2 + eKi, where K2 (K\) is the CP odd (CP even) neutral kaon state, the decay amplitude takes the form A[KL -> n°ff) ~ ImA t (F t - Fc) + e [Re\t(Ft - Fc) + Re\u(Fu - Fc)] 10"4 • 1

+10"

io-. i

+io-^101

{=

v

1 / =e Here we have kept the expression general to allow for the cases f = e and v. We have assumed the structure of the short-distance Hamiltonian for this exercise, although this is not strictly correct for the parts that are sensitive to long-distance physics. However it is sufficient for the qualitative argument we would like to make. We recall a few points from the discussion in the section o n K —» 7TZ/P.

First, as we have seen, the K2 component yields an amplitude proportional to the imaginary parts of the CKM elements. Correspondingly, the K\ amplitude (opposite CP), which is multiplied by e, gives the real parts. Second, we need to consider that the GIM mechanism is hard for / = v and soft for / = e. This implies Ft ~ 1, Fc ~ 10" 3 , Fu ~ 10" 5 for / = v, and F„, c , t ~ 1 for f = e. Third, we determine the hierarchy of the CKM elements. Putting this information together, we find the order-of-magnitude estimates shown above for the various terms. For / = v we recover what we already know from the previous section: The amplitude is purely from direct CP violation (the e-component is negligible)

197

e

K . i

71

e

Figure 23: CP conserving two-photon contribution to KL —> 7r°e+e

and it is dominated by short-distance physics (only Ft, Fc contribute). For / = e we can read off: Both direct and indirect CP violation contribute at the same order (~ 10~~4) and the latter component is determined by long-distance dynamics (Fu). In addition to what we have discussed so far, a long distance dominated, CP conserving amplitude with two-photon intermediate state can contribute. Although it is of higher order in the electromagnetic coupling, it can potentially compete with the other contributions because those are suppressed by CP violation. Treating Ki —> 7r°e+e~ theoretically one is thus faced with the need to disentangle three different contributions of roughly the same order of magnitude. • Direct CP violation: This amplitude is short-distance in character, theoretically clean and has been analyzed at next-to-leading order in QCD 65 . Taken by itself this mechanism leads to a Ki —» 7r°e + e _ branching ratio of (4.5 ± 2.6) • 10~ 12 within the standard model. • Indirect CP violation: This part is given by ~ e • A(Ks —> ir°e+e~). The Ks amplitude is dominated by long distance physics and has been investigated in chiral perturbation theory 66,67,68,69 r j u e to unknown counterterms that enter this analysis a reliable prediction is not possible at present. The situation would improve with a measurement of B(Kg —>• 7r°e + e~), which could become possible at the CERN experiment NA48 or with KLOE at DA$NE, the Frascati ^-factory. Present day estimates for B(Ki -> 7r°e + e~) due to indirect CP violation alone allow values of 10-12 _ i Q - i o .

• The CP conserving two-photon contribution is again long-distance dominated. It has been analyzed by various authors 6 8 , 7 0 ' 7 1 . The estimates are typically a few 10~ 12 . Improvements in this sector might be possible

198

li

KL- -&1 7

\

Figure 24: The dominant contribution to KL ~> Al+M

by further studying the related decay K L —• 7r°77 whose branching ratio has already been measured to be (1.7 ± 0.1) • 10 - 6 Originally it had been hoped that the direct CP violating contribution would be dominant. It is possible that the CP conserving part is not too important. However, this is unlikely to be true for the amplitude from indirect CP violation. Experimental input on Kg —> 7r°e+e~ will be indispensable to solve this problem. We also mention that the CP violating contributions interfere with a relative phase, which is known up to a sign ambiguity. The CP conserving part simply adds incoherently to the decay rate. Besides the theoretical issues, Ki —> n°e+e~ is also challenging from an experimental point of view. The expected branching ratio is even smaller than for KL —• -K°VD. Furthermore a serious irreducible physics background from the radiative mode KL —> e + e~77 has been identified, which poses additional difficulties l . A background subtraction seems necessary, which should be possible with enough events. Additional information could in principle also be gained by studying the electron energy asymmetry 68 ' 71 or the time evolution 68,72,73

5.3

KL -*• fl+fj,-

KL —> ^+^~ receives a short distance contribution from Z-penguin and Wbox graphs similar to K —• irvv. This part of the amplitude is sensitive to the Wolfenstein parameter Q. In addition Ki ->• H+H~ proceeds through a long distance contribution with the two-photon intermediate state, which actually dominates the decay completely (Fig. 24). The long distance amplitude consists of a dispersive (A dis ) and an absorptive contribution {Aabs). The branching fraction can thus be written B(KL - • n+fi-) = \ASD + Adls\2

+ \Aabs\2

(149)

199 Using B{KL ->• 77) it is possible to extract 1 \Aabs\2 = (7.1 ±0.2) • 10~ 9 . Adis on the other hand cannot be calculated accurately at present 7 4 ' 7 5 , 7 6 ' 7 7 . This is rather unfortunate, in particular since B(Ki —> n+fi~) has already been measured with very good precision B(KL

-> fj+n-) = (7.18 ± 0.17) • 10" 9 BNL 871 7 8

(150)

Interestingly, the absorptive contribution essentially saturates the total rate. For comparison we note that 3 0 B(KL -> H+H~)SD = \ASD\2 = (0.9±0.4)-10 -9 is the expected branching ratio in the standard model based on the shortdistance contribution alone. Because Adis is largely unknown, Ki —> /x + //~ is at present not a very useful constraint on CKM parameters. 6 6.1

Flavour Physics with Charm Rare D Decays

Weak decays of charmed particles, D mesons in particular, can also be used to probe the physics of flavour. Due to the characteristic pattern of standard model weak interactions, rare processes with D mesons are markedly different from their kaon counterparts. First of all, the charm quark mass mc sa 1.4 GeV is considerably bigger than the strange quark mass, and also the QCD scale A-QCD- Sometimes methods from the theory of heavy quarks can be employed for charmed particles, although they are less reliable than in the case of the much heavier B mesons (see the lectures by Falk in this volume for a general discussion in the context of B physics). This situation makes a theoretical treatment of D decays more difficult than for the heavy B mesons on one hand, and for the light kaons on the other. More important, however, is the fact that charm is a quark with weak isospin T3 = + 1 / 2 , in contrast to s and b. For this reason the virtual particles appearing in FCNC loop diagrams are the down-type quarks d, s, 6, rather than u, c, t familiar from K and B physics. Examples of rare D decays are D -» p 7 , D -+ TVI+1~ , D -> n+/d~, D -> 7 7 , D ->• -KVV

(151)

They proceed through c —> u FCNC transitions and their AC = 1 amplitudes have the generic form

v;dvudFd + V;SVUSFS + vc\vubFb = VC*SVUS(FS - Fd) + V;bVub(Fb - Fd)

(152)

where F{ is the amplitude with internal quark i = d, s, b. A potentially large contribution could come from Fb, which can have a quadratic mass dependence

200

~ ml/Myv- This exceeds the contribution from light flavours in the loop, but it is still much smaller than virtual top effects. In addition, the 6-quark contribution is very strongly CKM suppressed since V*bVub ~ A5. On the other hand, V*SVUS ~ A is much larger, but this term is multiplied by Fs — Fd- The latter is non-zero only through the effects of SU(3) breaking and therefore receives a strong GIM suppression. Moreover, the light-quark loops Fs, F^ are sensitive to nonperturbative QCD dynamics. Additional long-distance mechanisms, different from Fs, F^, can also become important. An example is 8 D° -> /J,+fi~- Here the amplitude in (152) yields a tiny branching fraction of ~ 10~ 19 . Alternatively the decay can proceed, for instance, via the long-distance mechanism D° —> K° —• /i + /u~, with an off-shell kaon. Estimates of this and similar sources give together 8 B(D° —> n+n~) ~ 10~ 15 . This still leaves a window for the discovery of potential new physics effects below the current experimental limit of20 B(D° —> yu + /i~) exp < 4-10~ 6 . 6.2

D°-D°

Mixing

A further interesting probe of the flavour sector is D°-D° mixing, a process with AC = 2. Recent measurements from the CLEO and FOCUS collaborations have stimulated the interest in this observable. A detailed discussion of these results and a list of references can be found in 79 . D°-D° mixing can be searched for in the time-dependent analysis of hadronic two-body modes. We may distinguish three types of decays, Cabibbo favoured (CF): D° -» K'-K+ +

singly Cabibbo suppressed (SCS): £1° - • K K~ doubly Cabibbo supressed (DCS): D° -*• K+Tr~

(153) (154) (155)

In powers of the Wolfenstein parameter A, the CF amplitude c —> sud is of order A0, the SCS amplitude c —¥ sus of order A, and the DCS amplitude c —¥ dus of order A2, which establishes a clear hierarchy in the decay rates. The classification applies of course also to the charge-conjugated modes D° —» K+TT- (CF), D° -> K+K(SCS) and D° -4 K~ir+ (DCS). The framework for describing D°-D° mixing is analogous to the case of K°-K° mixing discussed in sec. 4.1. The mass eigenstates are Di,2=pD±qD

CPD

= -D

(156)

We introduce the convenient definitions ( r is the average total decay rate) M2-Mt x

-—f~

r2-r!

y = -2r-

p(K+n-\D)

t = -q{K+7r-lD)

(15?)

201

x, y, £ are small quantities of order A2. The precise calculation of x and y is difficult for the reasons mentioned above (GIM suppression, long-distance sensitivity). Analyses based on hadronic estimates (K, ir intermediate states) or the heavy-quark expansion (quark-level calculation) lead to a standard model expectation of x, y < 1CT4 (158) We then consider the time dependent decay F(D°(t) -> K+TT~). The state D°(t) is obtained by solving the Schrodinger equation for the D°-D° two-state system with the initial condition D°(t = 0) = D°. The solution reads 2

T(D°(t) ^K+n-)

= e-rt

r(£>° ->

K+TT-

|£| 2 + (y Re£ + x I m O r t + \{x2 + y2)(Tt)2

(159)

To obtain this result we have used the following approximations. We expand in the small quantities £, x, y, assumed to be of the same order (of order x, say), and drop terms of 0{xA) and higher. We also require r t to be at most of 0(1). This is the range that is relevant experimentally, since the D mesons will have decayed once Ft becomes too large. The form of (159) is not hard to interpret. If there is no mixing, x = y = 0, only the first term inside the square brackets survives. It simply describes the exponentially decaying rate of the DCS process D° —> K+TT~ . Even if x, y ^ 0, at time t = 0 the DCS decay is the only effect. On the other hand, if the small DCS decay was absent, £ = 0, only the third term contributes. Then the K+7r~ final state can arise only through mixing D° —> D° —> K+n~. The rate vanishes at t = 0 because D°(t = 0) = D° and there was no time yet for mixing. The (xTt)2 behaviour represents the onset of an oscillating trigonometric function of which it is the remnant in our approximation. Finally, the linear term ~ Tt is from the interference of DCS decay and mixing. Without the DCS mode, the observation of K+ir~ from an initial D° would be an unmistakable sign of mixing. The real situation is more complicated. To demonstrate the presence of the mixing term, the three contributions in (159) have to be disentangled, which is possible in principle due to their different time dependence. So far only the DCS component (|£|2 term) has been unambiguously identified. At present the signal for mixing is still compatible with zero. It will be interesting to follow the future experimental results on this issue. The detection of a substantial value of x would indeed be exciting evidence for new physics.

202

7

Summary and Outlook

Kaon decays have played a key role in the development of the standard model. Today kaon physics is a mature field. A whole array of modern field theoretical techniques is at our disposal to help us extract the underlying mechanisms. Among these tools are perturbative quantum field theory, including the perturbative treatment of QCD at short distances, the operator product expansion, the renormalization group and chiral perturbation theory. While an impressive number of crucial insights has been already obtained in the past, excellent opportunities continue to exist for present and future studies: • Chiral perturbation theory constitutes a complementary handle on the elusive nonperturbative dynamics of QCD at long distances. This can be helpful to control long-distance contributions that contaminate the shortdistance physics that is of primary interest. However, chiral perturbation theory, as a model-independent framework for low-energy QCD, is also of considerable interest in its own right. Typical processes that are studied in this context are K+ —• ir+l+l~, KL —¥ 7r°77, or Ks —> 77. • CP violation in K —> TTTT is still an important area of current interest. Indirect CP violation measured by e is well determined experimentally and provides us with a valuable CKM constraint. Direct CP violation, now established, but still under further experimental investigation, gives an important qualitative test of the standard model. • Standard model precision tests will be possible with the "golden" decay modes K+ —» -K+VV and KL —• -K°vi>. • Other opportunities of interest include decays as KL —> ir°e+e~7 fipolarization in K+ —• 7r°jU+i/, among many other rare processes. • Very direct and clean probes for new physics are decays that are forbidden in the standard model. Lepton flavour violating modes as KL —» e/j,, K —> nfj,e are important examples. In parallel to kaon physics many other observables, as provided from decays of hadrons with beauty and charm, will be necessary to get a reliable and complete picture of the physics of flavour and its possible origins. In these lectures, we have discussed selected examples from the phenomenology of mesons with strangeness and charm. We have also emphasized the theoretical framework for an analysis of these processes, which is crucial to interpret the experimental data and to extract the underlying physics. The coming years promise to be very fruitful for the study of flavour physics and important discoveries are possible in the near future.

203

Acknowledgments I thank the organizers of TASI 2000 for inviting me to this very interesting and pleasant Summer School and for their hospitality at Boulder. Thanks are also due to the students for their active participation. I am grateful to Gino Isidori for comments on the manuscript. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

L. Littenberg and G. Valencia, Ann. Rev. Nucl. Part. Sci. 43, 729 (1993). J. L. Ritchie and S. G. Wojcicki, Rev. Mod. Phys. 65, 1149 (1993). B. Winstein and L. Wolfenstein, Rev. Mod. Phys. 65, 1113 (1993). P. Buchholz and B. Renk, Prog. Part. Nucl. Phys. 39, 253 (1997). A. R. Barker and S. H. Kettell, hep-ex/0009024. G. D'Ambrosio and G. Isidori, Int. J. Mod. Phys. A 13, 1 (1998). A. Pich, hep-ph/9610243. S. Pakvasa, hep-ph/9705397. G. Burdman, hep-ph/9508349. E. Golowich, Nucl. Phys. Proc. Suppl. 59, 305 (1997). E. Golowich, hep-ph/9706548. G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996). A. J. Buras, in Probing the Standard Model of Particle Interactions, eds. F.David and R. Gupta (Elsevier Science B.V., 1998), hep-ph/9806471. R. N. Cahn and G. Goldhaber, The Experimental Foundations Of Particle Physics, (Cambridge University Press, UK, 1989). A. Belyaev et al., hep-ph/0008276. H. Georgi, Weak Interactions And Modern Particle Theory, (AddisonWesley, Menlo Park, USA, 1984). A. Pich, hep-ph/9806303. G. Colangelo and G. Isidori, hep-ph/0101264. G. D'Ambrosio and D. Espriu, Phys. Lett. B 175, 237 (1986); J. L. Goity, Z. Phys. C 34, 341 (1987). D. E. Groom et al, Eur. Phys. J. C 15, 1 (2000). A. Lai et al. [NA48 Collaboration], Phys. Lett. B 493, 29 (2000). H. Burkhardt et al. [NA31 Collaboration], Phys. Lett. B 206, 169 (1988); G. D. Barr et al. [NA31 Collaboration], Phys. Lett. B 317, 233 (1993). L. K. Gibbons et al. [E731 Collaboration], Phys. Rev. Lett. 70, 1203 (1993). A. Alavi-Harati et al. [KTeV Collaboration], Phys. Rev. Lett. 83, 22 (1999).

204

25. V. Fanti et al. [NA48 Collaboration], Phys. Lett. B 465, 335 (1999); A. Ceccucci, CERN seminar (29.2.2000), http://na48.web.cern.ch/NA48. 26. S. Herrlich and U. Nierste, Nucl. Phys. B 419, 292 (1994). 27. A. J. Buras, M. Jamin and P. H. Weisz, Nucl. Phys. B 347, 491 (1990). 28. S. Herrlich and U. Nierste, Phys. Rev. D 52, 6505 (1995). 29. S. Bosch et al., Nucl. Phys. B 565, 3 (2000). 30. A. J. Buras, hep-ph/9905437. 31. A. J. Buras et al., Nucl. Phys. B 400, 37 (1993); A. J. Buras, M. Jamin and M. E. Lautenbacher, Nucl. Phys. B 400, 75 (1993). 32. M. Ciuchini et al, Nucl. Phys. B 415, 403 (1994). 33. A. J. Buras et al., Nucl. Phys. B 592, 55 (2000). 34. M. Ciuchini and G. Martinelli, hep-ph/0006056. 35. M. Ciuchini, E. Franco, L. Giusti, V. Lubicz and G. Martinelli, hepph/9910237. 36. S. Bertolini, M. Fabbrichesi and J. O. Eeg, Rev. Mod. Phys. 72, 65 (2000); hep-ph/0002234. 37. T. Hambye et al., Nucl. Phys. B 564, 391 (2000); hep-ph/0001088. 38. S. Narison, Nucl. Phys. B 593, 3 (2001). 39. J. Bijnens and J. Prades, JEEP 0006, 035 (2000); hep-ph/0010008. 40. A. A. Belkov et al., hep-ph/9907335; hep-ph/0010142. 41. E. Pallante and A. Pich, Phys. Rev. Lett. 84, 2568 (2000); Nucl. Phys. B 592, 294 (2000); E. Pallante, A. Pich and I. Scimemi, hep-ph/0010073. 42. Y. Wu, hep-ph/0012371. 43. A. J. Buras, hep-ph/0101336. 44. G. Buchalla and A. J. Buras, Phys. Rev. D 54, 6782 (1996). 45. V. A. Novikov et al, Phys. Rev. D 16, 223 (1977); J. Ellis and J. S. Hagelin, Nucl Phys. B 217, 189 (1983); C. Dib, I. Dunietz and F. J. Gilman, Mod. Phys. Lett. A 6, 3573 (1991). 46. G. Buchalla and A. J. Buras, Nucl. Phys. B 398, 285 (1993); Nucl. Phys. B 400, 225 (1993); Nucl. Phys. B 412, 106 (1994). 47. M. Misiak and J. Urban, Phys. Lett. B 451, 161 (1999). 48. G. Buchalla and A. J. Buras, Nucl. Phys. B 548, 309 (1999). 49. W. J. Marciano and Z. Parsa, Phys. Rev. D 53, 1 (1996). 50. D. Rein and L. M. Sehgal, Phys. Rev. D 39, 3325 (1989). 51. J. S. Hagelin and L. S. Littenberg, Prog. Part. Nucl. Phys. 23, 1 (1989); M. Lu and M. B. Wise, Phys. Lett. B 324, 461 (1994). 52. G. Buchalla and G. Isidori, Phys. Lett. B 440, 170 (1998). 53. G. Buchalla and A. J. Buras, Phys. Rev. D 57, 216 (1998). 54. A. F. Falk, A. Lewandowski and A. A. Petrov, hep-ph/0012099. 55. S. Adler et al. [E787 Collaboration], Phys. Rev. Lett. 84, 3768 (2000).

205

56. A. Alavi-Harati et al. [The E799-II/KTeV Collaboration], Phys. Rev. D 61, 072006 (2000). 57. BNL E949 collaboration, http://www.phy.bnl.gov/e949/. 58. R. Coleman et al. (CKM), FERMILAB-P-0905 (1998). 59. T. Shinkawa, in: JHF98 Proceedings, KEK, Tsukuba. 60. Y. Grossman and Y. Nir, Phys. Lett. B 398, 163 (1997). 61. BNL E926 collaboration, http://sitka.triumf.ca/e926/. 62. E. Cheu et al. [KAMI Collaboration], hep-ex/9709026. 63. T. Inagaki, in: JHF98 Proceedings, KEK, Tsukuba. 64. F. Bossi, G. Colangelo and G. Isidori, Eur. Phys. J. C 6, 109 (1999). 65. A. J. Buras et al., Nucl. Phys. B 423, 349 (1994). 66. G. Ecker, A. Pich and E. de Rafael, Nucl. Phys. B 303, 665 (1988). 67. C. Bruno and J. Prades, Z. Phys. C 57, 585 (1993). 68. J. F. Donoghue and F. Gabbiani, Phys. Rev. D 51, 2187 (1995). 69. G. D'Ambrosio, G. Ecker, G. Isidori and J. Portoles, JEEP 9808, 004 (1998). 70. A. G. Cohen, G. Ecker and A. Pich, Phys. Lett. B 304, 347 (1993). 71. P. Heiliger and L. M. Sehgal, Phys. Rev. D 47, 4920 (1993). 72. L.S. Littenberg, in Proceedings of the Workshop on CP Violation at a Kaon Factory, ed. J.N. Ng, TRIUMF, Vancouver, Canada (1989), p. 19. 73. G. O. Kohler and E. A. Paschos, Phys. Rev. D 52, 175 (1995). 74. D. Gomez Dumm and A. Pich, Phys. Rev. Lett. 80, 4633 (1998). 75. M. Knecht et al., Phys. Rev. Lett. 83, 5230 (1999). 76. G. Valencia, Nucl. Phys. B 517, 339 (1998). 77. G. D'Ambrosio, G. Isidori and J. Portoles, Phys. Lett. B 423, 385 (1998). 78. D. Ambrose et al. [E871 Collaboration], Phys. Rev. Lett. 84, 1389 (2000). 79. S. Bergmann et al., Phys. Lett. B 486, 418 (2000).

This page is intentionally left blank

c?

o^5

(**$

x

1

A. R. Barker

>

s * *K-<% *

- *->

* * :*.^

This page is intentionally left blank

K A O N PHYSICS: E X P E R I M E N T S A. R. Barker Department of Physics, UCB390 University of Colorado, Boulder CO 80309, e-mail: [email protected]

USA

Experiments in the area of kaon physics are discussed. Two main categories of experiments are considered: those aiming to measure the CP-violation parameter e'/e and those devoted to measuring various rare kaon decays. Characteristic aspects of each type of experiment are presented, and the details of individual experiments are described.

1

Overview

These two lectures will be devoted to a discussion of experiments in the kaon sector. Kaon experiments were of course the first to reveal the phenomenon of CP violation 1 and the absence of Flavor-Changing Neutral Currents, both important clues to the structure of the Standard Model. Although experiments in the kaon sector have to some extent been overshadowed by B-factory projects recently, kaon experiments are still providing important information about flavor dynamics, and will continue to do so. Recent kaon-physics experiments can be divided into two categories, according to the physics topics they are investigating. One class consists of experiments that are attempting to measure the CP-violation parameter e'/e. The significance of this number in the Standard Model has been discussed in the lectures by G.Buchalla; I will review it only briefly here. Superweak models in which CP violation in K —> TTTT arises entirely from CP violation in mixing between the K° and the K predict that e' = 0. By contrast, the Standard Model allows for so-called direct CP violation in the decay of the CP-odd eigenstate Ki to two pions; consequently it predicts a non-zero value for e'. In the Standard Model, e', like the mixing CP-violation parameter e, is proportional to TJCKM- Unfortunately, efforts to extract a precise theoretical prediction for e' have so far failed due to the difficulty of calculating the required hadronic matrix elements. There is a long history of attempts to measure e'/e; until fairly recently all the results have been consistent with zero. The two experiments that preceded the present generation were E731 at Fermilab and NA31 at CERN. While E731 found that e'/e differed from zero by only about 1 a, NA31, with a similar experimental uncertainty, got a result approximately 3 a from zero. The significance of the difference between the two experiments was approximately 209

210

1.7cr. This unsettled situation led to proposals for two improved experiments, one (KTeV-E832) at Fermilab, and one (NA48) at CERN. Many of the E731 experimenters joined E832, and many of the NA31 experimenters joined NA48. More recently, a third effort to measure e' has begun at the low-energy e + e~ collider DA$NE in Frascati. DA$NE operates as a 0 factory, and thus provide a source of coherent KsKi pairs. The KLOE experiment there hopes to study kaon decays using these pairs in order to measure a variety of CP-violation parameters, including e'. Experiments searching for rare kaon decays also have a long and interesting history. Recently, Brookhaven National Laboratory on Long Island has been a center for this work. Other notable rare decay experiments in the last few years have operated at KEK, CERN, and Fermilab. These experiments have searched for a variety of different phenomena. One major thrust has for some time been the search for decays like K\ —> fi^e^ that violate lepton-flavor conservation. Such decays are absolutely forbidden in the Standard Model (except for vanishingly small branching fraction levels around 1 0 - 2 4 resulting from neutrino mixing); thus, any observation of them at the experimental sensitivities on the order of 10~ 12 would be an unambiguous signal of physics beyond the Standard Model. A second goal of rare kaon decay experiments has been to look for decays that have interesting Standard Model contributions, for example from penguin operators (describing the Flavor Changing Neutral Current s —> dy, for example). The rate for such decays is in principle sensitive to interesting Standard Model parameters like Vtd that may be very difficult to measure directly. Finally, rare decay experiments can do a wide range of less exciting, but still important measurements of what I call "bread-andbutter physics". Future work in rare kaon decay experiments is coalescing around efforts to measure the difficult but rewarding modes K+ —> -n+vv and K° —» -K°VD, also discussed by Buchalla. 2 2.1

Experiments Measuring e'/e Definition of e1 /e and Measurement Technique

There are two neutral kaon states which are eigenstates of strangeness: K° = sd and K = sd. The operator CP transforms these into each other, and so has two eigenstates

\K°1) = -±=(\K°) + \T?)) with CP\K°) = +\K°), and the corresponding antisymmetric combination K\, with CP\K^) = —\K°). If CP were an exact symmetry of the weak interactions, these would correspond to the mass eigenstates K\ and Kg. Specifically,

211

only the CP-even state K° would be allowed to decay to two pions, and it would therefore be identified with the Ks, which nearly always decays in that channel; while the CP-odd state K° would have to decay to 3TT or semileptonic final states, and would therefore correspond to the longer-lived K^. Since 1964, we have known that this simple picture, in which the K° is identified with the Kg, and the K\ with the K°L, is not correct. In that year, Christenson, Cronin, Fitch, and Turlay l found that the K\ has a small, but non-zero probability of decaying into two-pion final states, which implies that CP symmetry is violated. Unlike the violation of parity symmetry alone, which is maximal, CP violation is a small effect. The violation of CP in neutral kaon decays could be explained by a small CP-violating mixing between the CP eigenstates, so that the mass eigenstates would be

\K%) = ~^—

{\K°)+e\K«))

and

\Kl) = -j=L=

(IK^+elK?)).

This mechanism leads to what is called indirect CP violation, in which the K\ decays to TY+TT~ or 7r°7r° final states through its small K° admixture. The observed rate for K\ —\ irw can be explained if e is a complex number with magnitude |e| = 2.269 x 1(T 3 . Soon after the discovery of CP violation, it was proposed that the mixing between CP eigenstates could be a consequence of a so-called Superweak 2 interaction which would mediate CP-violating transitions with AS — 2. Such interactions would permit K° -» K and K —> K° transitions which could lead to indirect CP violation. They could not, however, lead to CP violation in the AS = 1 decays of the K° or K°. This second type of CP violation is called direct. Other theories, including the Standard Model, that explain CP-violation generally have AS = 1 effects as well as higher-order AS = 2 effects (such as mixing). Direct CP violation can be distinguished from indirect by considering the balance between the two possible TTTT final states. The Ks decays to 7T+7r~ twice as often as it does to 7r°7r°. Since the Ks is dominantly the K° eigenstate, which decays dominantly to irn, the ratio observed in Ks decays is the ratio in K® decays. If CP violation is entirely indirect, then the K\ can decay to TTTT only through its K° component, and the ratio of charged to neutral TVTY final states would be identical to that observed in Ks decays. If, on the other hand, direct CP ciolation is also allowed, then the ratio of charged to neutral

212

final states seen in K\ decays might be altered slightly from that observed in Kg decays because some of the rnr decays of the K\ would occur directly, through its dominant K° component. To make this quantitative, one defines the amplitude ratio _ A{K%

-> TT+TT")

V+

~ ~ A(K°S ->• 7T+7T-)

and its neutral counterpart 7700- If CP violation were entirely indirect, then both of these amplitude ratios would simply be equal to e. If there is direct CP violation as well, then the two amplitude ratios can split apart. With the usual definition of e', the splitting is given to a good approximation by ?7oo — e — 2e'

and

r/^

= e + e'.

To measure the splitting, experiments consider the double ratio of partial widths T(KL -> 7 r ° 7 r ° ) / r ( ^ -+ 7r°7r°) T{KL -> 7T+ir-)/r(KS -> TT+TT") ' in terms 01 the parameters denned above, tnis ratio is given Dy 2

R

??oo

= l-6Re '

£

V+Thus, roughly speaking, one measures all four decay modes and determines whether the double ratio R differs from unity. The difference yields six times the quantity Re(e'/e). Predictions of Re(e'/e) in the Standard Model are quite difficult because of the hadronic matrix elements involved. More information on the theoretical calculation can be found in the TASI-2000 contribution from G. Buchalla. 2.2

KTeV Experiment E832

KTeV (Kaons at the TeVatron) is a project at Fermilab encompassing two distinct experiments: E832, whose goal is a precision measurement of Re(e'/e); and E799, which aims to search for a wide variety of rare K\ decays. The e' experiment, E832, requires two parallel beams of neutral kaons. These are created by having a beam of 800 GeV protons from the TeVatron strike a BeO target. Any charged particles produced are magnetically swept from the beam. The resulting beam of long-lived neutral particles passes through collimators which produce two parallel beams travelling through an evacuated beampipe toward the spectrometer. Additional sweeping magnets are placed just upstream of the spectrometer to remove the decay products

213 ANALYSIS MAGNET TRIGGER HODOSCOPES Cat CALORIMETER Pb WALL HADRON VETO

3

20

MUON TRIGGER 'MUON VETO MUON SHIELDING

COLLAR ANTI (EB32)

105

115

Figure 1: The KTeV apparatus at Fermilab.

of particles like A baryons or Ks mesons, which may have decayed in the beampipe. Since the goal of E832 is to compare K\ and Ks decays, the next step in E832 is to create Ks particles in one of the two beams. This is done by placing a regenerator consisting of a series of plastic scintillator tiles in one of the two beams. As the kaons pass through the regenerator, the mixture of K° and K , originally 50-50, changes gradually, so that the exiting beam contains a regenerated Ks component in addition to an attenuated K\ component. Although the regeneration amplitude is of only modest size, the K% that are produced decay quickly downstream of the regenerator and so dominate the decays in the beam in which the regenerator is placed. More precisely, the regenerator beam is a coherent superposition of Ks and K\ components. The distribution of decay proper times in the regenerator beam exhibits the usual interference phenomena seen in a mixed neutral kaon beam, and its detailed shape depends on parameters of the neutral kaon system such as the KQL-KS mass difference Am and the Ks lifetime TS as well as the regeneration amplitude and the CP-violating amplitude ratios r]0o and ?7+_. The decay distributions seen in the second (vacuum) beam, are almost pure K\. The vacuum beam is used to normalise the decay distributions seen in the regenerator beam and to under-

214

stand the acceptance of the apparatus. Because the two beams (left and right) are not absolutely identical in energy, size, or intensity, the regenerator alternates from one beam to the other every minute in order to average out these differences. This also averages out any left-right asymmetry in the response of the spectrometer used to detect the K° decay products. Since K°L and Kg decays are collected simultaneously, with identical event selection criteria, many possible systematic differences in the acceptance cancel out. KTeV Apparatus The KTeV apparatus is shown in figure 1. The regenerator can be seen at the lefthand (upstream) edge of the schematic. After the regenerator, there is a large vacuum decay volume. The vacuum tank extends somewhat more than 30 meters downstream from the regenerator, and increases gradually in diameter from about 1.5 meters at the regenerator to about 2.0 meters just before the window at its downstream end. Every few meters along this vacuum tank, there are so-called "ring vetoes", lead-scintillator stacks with a square hole in the center, labelled RC6 — RC10 in figure 1. Decay products that are exiting the tank at large angles, so that they would miss the spectrometer elements further downstream, hit the veto detectors and cause the event to be rejected. This helps to minimize backgrounds due to missing particles. At the downstream end of the vacuum tank, there is a large, thin window. Just beyond this window is the first of four multiwire proportional chambers, which together with the analysis magnet, are used to measure the momenta of charged particles. The drift chambers are each surrounded by additional veto detectors in order to reject events with particles escaping the fiducial volume of the detector before reaching the calorimeter. These veto detectors are labelled SA2 - SAA in figure 1. The calorimeter is used to measure the energy of photons as well as the energy deposited by charged particles, which can be electrons, pions, or muons. It consists of 3100 crystals of pure Csl. To prevent particles in the beam from interacting in the Csl, the calorimeter has two beam holes each 15 cm square where the beams pass through. A final veto detector surrounds the outer edge of the Csl array. In addition, the inner edge of the Csl acceptance is precisely defined by a tungsten-scintillator veto detector that surrounds the edges of the two Csl beam holes. The ratio E/p, where E is the measured energy in the Csl, and p is the momentum determined from the drift chambers, can then be used to measure the resolution of the Csl calorimeter, which is found to be about 0.7%. This excellent resolution is extremely important for the neutral-mode (7r°7r°)

215

analysis, since the location of the decay vertex is determined entirely from the calorimeter's photon energy measurements. Analysis Procedure Neutral-mode events must have exactly four energy clusters in the Csl. The four photon clusters can be separated into two distinct pairs in three ways. For each pair in a pairing, the hypothesis is made that the two photons came from the decay of a single n°. The distance of the hypothetical 7r° decay from the Csl calorimeter (Az) can then be found from the equation \jE\E2

Az

Ar,

where Mv is the n° mass; Ex and E2 are the measured photon energies; and Ar is the transverse separation of the two photon clusters at the Csl. Two different Az values are calculated for the two pairs in a pairing, and uncertainties on the two Az values are estimated. The best pairing is chosen and the z decay vertex is determined. After this, the total invariant mass of the event can be calculated; events with total mass between 490 and 505 MeV/c 2 form the final neutral-mode sample. ~

10 10

=s 1 0

b 5 A

10

3

10

2

R e g 1

0.47

0.48 0.49

O.S

0.51

O.S2

vac beam

•

•

•

1

TC"*"TC" •

•

.

T-I

•

0.47 0.48 0.49 0 . 5 0 . 5 1 O.S2 r I V I „ „ K - ->TL*TC~ , r e g b Ge caVm/ c

CcV/c^

—>TZ*Ti

•

IO IO

5 4

a l O *

r

3

I O -' r IO

2

teg , , . . 1 . . .

0.47

0.48 0.49

O.S

o 1 —>7T

7T

0.51

0.52

C.«V/c

vac b e a m

TT'V

T-,

1 . , . . ITWT-

0 . 4 7 0 . 4 8 0 . 4 9 0 . 5 O.S1 0 ^ 2 IVI^^ K—>rc 7z , r e g b e a m

Figure 2: Distribution of invariant mass for the charged-mode and neutral-mode candidate events, in both E832 beams.

K° —> TTTT

216

To identify TT+TT~ events, events were required to have exactly two drift chamber tracks. The tracks had to be matched to energy clusters in the Csl, and, to be sure that they were produced by pions, the E/p ratio had to be less than 0.85. The invariant mass of the two pions had to be between 488 and 508 MeV/c 2 . To reject events with missing particles, a quantity called PT was calculated by finding the component of the total of the two measured pion momenta which was perpendicular to the kaon's line of flight from target to decay vertex. Only events with P%. < 250MeV 2 /c 2 , consistent with resolution effects, were retained in the final charged-mode data sample. The invariant mass distributions for these events are shown in figure 2. Before Re(e'/e) can be extracted from these data, backgrounds must be identified and subtracted, and acceptance corrections must be determined, using a Monte Carlo simulation of the experiment. The backgrounds to the charged mode are very small, since the Pj. cut is very efficient at removing them. In the vacuum beam, there is a small charged-mode background from misidentified Kt$ and K^z semileptonic decays, in which the lepton is misidentified as a pion. In the regenerator beam, the main background comes from regenerator scattering events. The shape of the backgrounds match the data very well at large Pj,, allowing reliable determination of the remaining background under the signal peak near zero. The level of scattering backgrounds required to match the data and Monte Carlo Pj. spectra can also be used to help in the determination of neutral-mode scattering backgrounds. Because the decay vertex cannot be as precisely located in the neutral mode, there is no precise P% quantity available to reduce the backgrounds. As a result, the backgrounds are substantially larger in the neutral mode, because the peak region is less tightly defined. Figure 3 shows the final background determinations for all four KL,S —> nn decay modes. As the figure shows, the charged-mode backgrounds are less than 0.1%, while the neutral mode backgrounds are somewhat less than 1% in the vacuum beam, and somewhat more than 1% in the regenerator beam. Extraction of Re(e'/e) Once the backgrounds have been determined, they are subtracted bin-by-bin from the signals, resulting in the four decay-vertex distributions shown in figure 4. The data are further divided into energy bins. In each bin of kaon energy and decay vertex position, the Monte Carlo simulation is used to determine an acceptance correction. The acceptance-corrected numbers of events are combined in 10 GeV energy bins. The ratio of regenerator to vacuum beam events is then fit to the predicted distribution, which depends on parameters

217 Oa,e-lcg;i'Oi.i-iicl S u b t . r a c t . i o n s fox' J*C —->- 2-zr

O

1 OOO

SOOO

3000

4QOO

O

1 OOO

ZOOO

3000

4000

O

1 OO

200

300

400

O

1 OO

200

300

4QO

p= (MeV a /c a )

Ring Number

p= (MeV 2 /c = )

Ring

Number

Figure 3: Summary of total background levels for all four K^^s —> n modes. Shown as a function of P^, for the charged mode, and of a similar quantity, "Ring Number", in the neutral mode.

including not only Re(e'/e), but also Am, TS, the phases 0oo and 0_| of rjoo and r)^ , and the regeneration amplitude. The fit can be done in a variety of ways, either allowing many parameters to float simultaneously, or fixing all parameters but one. For the final fit, all parameters except e' were fixed to the Particle Data Group averages. There are a variety of systematic uncertainties in the extraction of Re(e'/e), which are summarized in figure 5. The most important charged-mode systematic is the overall acceptance uncertainty. Its origin lies in the fact the the observed z-vertex distribution in the data does not exactly match the predicted distribution in the Monte Carlo. In the neutral mode, there is no single, dominant source of systematic uncertainty. The background uncertainty is much larger in the neutral mode simply because the level of background is about an order of magnitude larger than in the charged mode. The energy scale uncertainty is due to the fact that the calculation of the vertex location depends entirely on the overall Csl energy scale; an error in this scale would cause the whole z-vertex distribution to slide, and since this distribution is very different for K°L and K°s decays, such a shift would impact the fit results for Re(e'/e). Combining all the systematic errors in quadrature, the total is 2.8 x 10"~4. With this dataset, the statistical uncertainty is 3.0 x 10~ 4 . To avoid any bias in developing the analysis cuts or the Monte Carlo simulation based on the result for Re(e'/e), the fit results were "hidden" by

218 x 10 2 180000 <5 70000 rT~"~~^--~, 60000 \ ^ ^ 50000 \ 1 40000 r 30000 ~ J V a c 7i+7i" 20000 - j 10000

t/,,

19000 £8000 7000 ^ 6000 \ 5000 1 4000 \ 3000 1 2000 1000

,....,...]

110 120 130 140 150 160 Vertex Z K—>7T+7i", vac beam 325000

F-

"

..,.!S^_...... +

Vertex Z K—>7t 7c", reg beam x 102500 2000

15000

1500

10000 F-

1000

5000

500

^ \ Reg 7t°7i°

M

0 0

11 K L\ Reg n+n \

110 120 130 140 150 160 m

20000

110 120 130 140 150 160

1

M

I

I

V

,.,\l_,....

110 120 130 140 150 160

m

Vertex Z K—>n n , vac beam

0

0

m

Vertex Z K—>n n , reg beam

Figure 4: Background-subtracted decay-vertex distributions. Notice the sharply falling regenerator beam distributions, dominated by K$ decays, compared to the relatively flat vacuum, beam distributions, dominated by K9 decays.

adding an unknown offset to the reported value of Re(e'/e) until the decision was made to report the result in public. The final result from the fit was quite large: 3 Re(e'/e) = (28.0 ± 3.0 ± 2.8) x 10 - 4 As figure 6 shows, this result is considerably larger than the previous E731 result, but quite consistent with the NA31 result, as well as with the NA48 result 4 . It is also larger than most recent theoretical predictions, but as was mentioned above, the predictions are not yet well constrained considering the ranges of parameters that must be considered. This result, together with the new results from NA48, does, however, rule out conclusively the hypothesis that the observed CP violation in neutral kaon decays might be due exclusively to a new, CP-violating Superweak interaction. Whether or not sources of CP violation beyond the Standard Model will be required to explain such a large value for Re(e'/e) awaits the development of more definite theoretical predictions for the CP violation induced by the CKM matrix.

219

S u m m a r y of S y s t e m a t i c U n c e r t a i n t i e s Systematic Trigger (L1/L2/L3) Detector Resolution Calibration/ Alignment Energy Scale DC simulation Csl non-linearity Apertures (incl Reg Edge) Analysis Cuts Backgrounds Overall Acceptance Monte Carlo Statistics Attenuation Slope Movable Absorber External Parameters

7r+7r Analysis (xlO-4) 0.50 0.35 0.25 0.12 0.63

7r°7r° Analysis (xlO-4) 0.29 <0.10 0.38 0.70

0.26

0.60 0.48

0.59 0.20 1.59 0.50

0.78 0.81 0.68 0.90 0.24 0.20 0.19

Figure 5: Systematic uncertainties in the KTeV-E832 measurement ofKe(e'ft); the largest single uncertainty is due to a mismatch between the data and Monte Carlo z-vertex distributions for the charged mode.

2.3

CERN Experiment

NA48

The N A 4 8 Way The current generation e'/e experiment at CERN is called NA48. This is the successor the the earlier NA31 experiment, which was the first to report a result for e'/e significantly different from zero. The NA48 approach to measuring e' is similar in many respects to that used by E832 at Fermilab, but there are also some important differences. Both experiments emphasize the importance of collecting data for all for Kits -* TTTT decay modes simultaneously. This is important in order to ensure that backgrounds due to accidentals are as similar as possible in the two modes; and so that time-dependent changes in detector response (due to varying gains, malfunctioning channels, and the like) will not affect different decays differently. Both experiments also use ideantical ccuts and identical fiducial regions for KL and Ks decays. The experiments differ, though, in how the Ks are produced. As was discussed above, E832 has a target far upstream of the spectrometer, followed by collimators that produce two parallel beams. Almost all the Ks fproduced at the primary target decay far upstream, so that the two beams are both almost pure KL by the time they reach the detector. One of the E832 beams then hits an active regenerator, where the differing interactions of the K° and K components turn the KL beam into a coherent mixture \KL) + p\Ks), where p is the regeneration amplitude. The Ks is the regenerator beam then decay quickly just downstream of the regenerator.

220

35 O

X

30 25

CD

KTeV 28.0±4.1 7 NA31 !23±6.5

"

1

20 -; I

15 10 -: :

5

,, J

^

NA48 18.5-1-7.3

pDG

M

World Avq. "19.3±2.4

15±8 n NA48 1 2.2±4.9 [~731 7.4±5.9

i

i

90 91 92

,i

i

i

i

i

i

i

93 94 95 96 97 98 99 00 01 02 03

Publication Year Figure 6: Final result for Re(e'/e)> and comparison to earlier experiments.

A schematic of the NA48 beamline is shown in Fig. 7 In NA48, as in E832, there is a KL target far upstream of the detector. While both KL and Ks are produced in this target, the Ks decay away far upstream of the detector, as in E832. However, in the NA48 experiment, a small fraction of the primary proton beam is deflected by a bent crystal just downstream of the Ki target. The deflected beam is then directed onto a Ks target just upstream of the fiducial region. Ks particles prodcued at this downstream target then decay in the same fiducial region used for the KL decays. The advantage of this technique compared to E832 is that NA48 does not suffer from the substantial backgrounds E832 sees due to diffractive and inelastic interactions in the regenerator material. Also the momentum spectrum of primary Ks and KL are the same, whereas in E832 the Ks momentum spectrum differs because of the momentum dependence of the regeneration amplitude. However, there are also some disadvantages associated with the NA48 method for producing Ks. First, the Ks beam is not exactly coincident with the KL beam, so different parts of the detector are illuminated, which can lead to systematic differences between the acceptance for Ks and KL. Second, one must identify whether a given decay seen in the spectrometer is due to a Ks

221 Ke anticounter Ks Target,

#l

^

K Target 2 / • /

Muon sweeping

Bent / crislal Ks tagging station

Last collimator'

( -3.10 7proions per spil)

Decay Region (~ 4u m long)

-120 m

12pm

Figure 7: The NA48 beamline at CERN.

decay or a KL decay. In the case of if —> 7r+7r™~ decays, this is simple, because the decay vertex reconstructed from the two charged tracks clearly points back to either the KL target or the Ks target. However, for neutral decays this technique cannot be used and there is an ambiguity as to whether the decaying particle was a Ks or KL- TO resolve this, NA48 uses timing information to determine which neutral decays are which. The protons deflected by the bent crystal pass through a Ks tagging counter. When a K -> 7r°7r° decay is detected, one looks to see whether there was a signal in the Ks tagger at the right time to have been produced by a proton that could later have hit the Ks target, producing the Ks whose decay was observed. If there was no tagger hit in the appropriate timing window, then the decay is identified as a KL -+ ^°^° event; if there was a tagger hit, then the decay is called a Ks ™> 7r°7r° event. This identification is usually correct, but mistakes are made in which the tagging counter fails to fire in the appropriate window (Ks misidentified as KL) or in which the time of a KL decay is accidentally coincident with a hit in the tagging counter (KL misidentified as if 5). The level of misidentification must be determined and corrected for, and the residual uncertainty in this correction is an important component of the final systematic error in the NA48 result for e'/c Another difference between NA48 and E832 is in the way in which the different ifs and ifL lifetimes are handled in the analysis. Although E832 uses the same fiducial region for Ks and KL decays, the observed decay distributions

222

Figure 8: The NA48 apparatus at CERN.

are very different, with the Ks decays being concentrated just downstream of the regenerator, whereas the KL decays are distributed nearly evenly along the length of the fiducial region. As a result of this, the E832 spectrometer elements are illuminated differently by Ks and KL decays, and a correction is applied from a Monte Carlo simulation to account for the resulting difference in acceptance. The distribution of decay vertices in NA48 is also quite different; with the Ks decays being concentrated near the Ks target. However, rather than correct for the resulting acceptance differences using a Monte Carlo, NA48 chooses to reweight the KL decays so the the weighted KL decay vertex distribution looks the same as the Ks decay vertex distribution. After the weighting is applied, the remaining acceptance differences are extremely small. Effectively, NA48 trades statistical power away (by introducing different weights for differen events) in order to reduce the systematic error. KTeV makes the opposite choice, retaining maximum statistical power at the expense of a systematic error resulting from acceptance differences. The spectrometer elements in NA48 are similar in performance to those in E832. There is a system of drift chambers and an analysing magnet with 250 MeV transverse-momentum kick to measure the trajectories and momenta of charged particles. There is a high-precision liquid-krypton calorimeter, with

223

energy resolution similar to that achieved in E832's Csl crystal calorimeter. As with E832, mass resolution in the charged mode is about 2.5 MeV. The liquid krypton calorimeter has one important job that the Csl does not have in E832: the NA48 calorimeter is also used to measure the time of photon showers in order to make the correlation with the K$ tagging counter described above. A timing resolution of about 250 ps has been achieved by NA48 for neutral events. A slightly better time resolution is obtained for charged events, using planes of scintillator upstream of the calorimeter. NA48 collected data using this apparatus during 1997, 1998, and 1999, for a total of about 352 days of running, and about 3.6 million of the rarest of the four 7T7T decay modes, KL —• 7r°7r°. This is about half of the roughly 8 million of these decays collected during the two E832 runs in 1997 and 1999. One of the more important systematic uncertainties in NA48, and one that is different from E832, is the uncertainty resulting from mistagging. To correct for mistagging, charged events (where the true source of the event is known by looking at the vertex location from the tracks) are subjected to the tagger-timing analysis used in the neutral mode. From these events, the tagger inefficiency for charged decays is measured to be (1.97 ± 0.05) x 10~ 4 , and the probability of accidental tagging for charged decays is measured to be 11.05 ±0.01%. Tagger inefficiency causes Ks decays to be misidentified as KL, while accidental tagging causes KL decays to be misidentifed as Ks- The possibility that the neutral misidentification rates may be different is also studied using runs where the KL beam is blocked and by measuring the accidental tagging rate for purely KL decays such as KL —> 7r07r°7r°. The most important contribution to the final systematic error comes from the uncertainty in the neutral accidental tagging fraction, which contributes about 9 x 10~ 4 to the systematic error on the double ratio R, or 1.5 x 10^ 4 to the systematic error on Re(e'/e). The backgrounds in NA48 are much smaller than in the Fermilab experiment E832, because of the absence of regenerator scattering. However, there is still background in the charged mode from semileptonic decays. This has to be subtracted, leads to a correction of 9.9 x 10~ 4 in the measured double ratio, and a contribution of 0.6 x 10~ 6 to the final systematic error on Re(e'/e)Neutral mode background from KL —> 7r07r07r° similarly leads to a correction of 6.6 x 10~ 4 in the double ratio, and contributes 0.3 x 10~ 4 to the final systematic error. The largest correction to the double ratio (which is still quite small) comes from the residual acceptance corrections that are needed even after the reweighting procedure has been used. The correction to R is (31±6±6) x 10~ 4 . The two uncertainties (statistical and systematic) both contribute to the sys-

224

tematic error on e'/e. It is notable that although the Fermilab experiment makes a much larger acceptance correction (a few percent compared to a few per mil), the uncertainty in the size of the correction quoted by E832 is similar to the uncertainty quoted by NA48, so that the final contribution of acceptance correction uncertainties to the systematic errors in the two experiments is not very different. Another substantial systematic uncertainty in NA48 that is very similar to one from E832 is due to the overall energy scale. This uncertainty in Re(e'/e) is estimated at 1.7 x 10~ 4 for NA48, compared to 0.8 x l(T 4 for E832. Another large systematic uncertainty (2 x 10~4) comes from biases that may result from accidental activity. Combining the results from their 1997 and 1998 runs, NA48 quotes a preliminary result: 4 Re(e'/e) = (14.0 ± 4.3) x 10" 4

(NA48 average)

This result is only half as big as the KTeV result 3 Re(e'/e) = (28.0 ± 4.1) x 10~4

(KTeV)

but has about the same quoted precision. The situation is unsettling, in that the two new results differ from each other with 2.4
(NA48 average)

However, it should be noted that the Particle Data Group's averaging procedure, which inflates the uncertainty when there is poor agreement between results, will give a larger error bar. 2.4

The KLOE Experiment at DA&NE

The DA$NE $ factory in Frascati, is a 1.02-GeV e+e~ collider sitting at the $ resonance. The accelerator began commissioning in early 1999, with a design luminosity goal of L = 5 x 10 32 . To date, the accelerator performance has unfortunately been rather far from this goal. The KLOE experiment 5 was designed to measure e'/e and other CP and CPT parameters, although it will also search for a variety of rare KL, KS and K+ decays using tagged kaons from copious $ decays. For CP physics, KLOE has an advantage compared to KTeV and NA48 in that it produces a

225 100 < Kaon Energy < 110 GeV

K L events are weighted according to the K s / K L ratio of proper time distributions

2 2.5 3 3.5 K decay time from AKS (cr) 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5

KS/ weighted! KL .»r*^4»^i*«>.
2 2.5 3 3.5 K decay time from AKS (cr)

Figure 9: The NA48 reweighting procedure,

pure KLKS quantum state, making tagging and interferometry possible, as in B-factory experiments. The KLOE detector is a 4TT apparatus working in the center of mass frame. It is a cylindrically symmetric general-purpose detector with a low-mass central drift chamber of very large volume (4-m diameter) using helium as the ionizing medium, surrounded by a Pb-scintillating fiber calorimeter. The detector is in a 0.6-T magnetic field, so the calorimeter is read out with high-field fine-mesh PMTs. The calorimeter is designed to have good timing resolution (about 130 ps) and to be highly efficient, in order to reduce backgrounds from KL —> 7r07r°7r° events. KLOE is performing well, haiving already demonstrated mass resolution of about 1 MeV for Ks -> TT+TT- events. Data-taking began in the summer of 1999, but to date not many KL -> 7T7r events have been collected. 3 Rare Kaon Decay Experiments This lecture reviews the status of rare kaon decays, with emphasis on the general thrusts of rare kaon decay experiments and on progress made in the last few years. Several good review articles are available that focus on rare kaon decays 6 ' 7,8 , and theoretical studies of rare kaon decays 9 ' 1 0 ' 1 1 ' 1 2 . There are three primary themes of rare kaon decay experiments. The first what I call "bread-and-butter" physics. This is done by studying a variety of rare and semi-rare processes including semileptonic, radiative, and electromagnetic decays. Detailed measurements of the branching fractions and kinematic distributions in these decays provide information on form factors, as well as tests of QED and chiral perturbation theory calculations. While some of these

226

results are not too fascinating for their own sake, they are often needed in order to understand backgrounds or normalization modes for more interesting decays. The second theme is to use rare kaon decays to test the Standard Model and to determine the values of Standard Model parameters. This usually involves selecting kaon decays that involve suppressed flavor-changing neutral currents with important contributions from electromagnetic penguin operators. Examples include KL ->• /u+/x~, KL -> 7r°e + e~, and particularly K -> -KVV. Measurements of these processes should be sensitive to both the real and imaginary parts of the difficult-to-determine CKM matrix element Vtd- These efforts have been underway for some time, but as yet sensitivities have not reached the level required to make precision measurements. Finally, the long kaon lifetime makes kaon decays an excellent place to search for processes that are forbidden in the Standard Model, which would signal the presence of new physics. Exotic decays of this type that have been the subject of rare decay experiments include KL —> n+±eT, and K —> TTfi±eT. To date, none of these searches have turned up any evidence for exotic decays, despite sensitivites to branching ratios in the lO^ 11 range. 3.1

Bread-and-Butter Rare Decay Physics

In addition to the rare kaon decays that directly probe standard-model parameters and those that are sensitive to exotic physics, there is a wide array of other decay modes on which substantial experimental progress has been made in recent years. Although these results receive less attention, they provide critical information in a variety of areas. For example, there has historically been strong interest in the Kl —> /i+ yT decay as a probe of weak interaction dynamics, specifically R,e{Vtd), through its short-distance amplitude. But the short-distance amplitude is known to be quite small compared with the long-distance part, involving the 77 and 7*7* intermediate states. An accurate determination of the KLJ*J* form factor is needed in order to evaluate the long-distance contribution, which is needed in turn to extract Re{Vtd) from B(K£->/z+/u~ ). The form factor can be determined in part from measurements of electromagnetic kaon decays such as KL ->• e + e~7, KL -> M+/"~7; KL -» e+e~e+e~, and KL -» e + e ~ / i + / j ~ . The measurement of such modes as K1^TT°JJ and K°L^r e+e~jj is imr portant for a different reason. The decayK° L ^tt t~77 is an important background in the search for K°L -^ir°£+£~ . Both the absolute number of events from this process and the kinematic distributions of those events are thus important to the effort to learn about standard-model parameters from K°L —>• 7r°e+e

227

and KI->IT°[I+IJ,~ . In particular, large samples of these events must be studied to determine the effectiveness of kinematic cuts necessary to observe this extremely rare decay. The decay if£-»7r°77 , though not a background to K°L—>-K°e+e~~ , can be used to determine the CP-conserving part of the amplitude. Additional contributions from the 7r°7*7* intermediate state with offshell photons are also important. These can be determined from ChPT models, but again there are undetermined parameters that must be extracted by studying kinematic distributions in K°L -> 7r°77 and the related mode K°L -> ix°e+e~-y Because of their low energy release and wide variety of final states, kaon decays provide an excellent testing ground for the predictions of ChPT. For example, the K —^77 modes and the direct-emission component of radiative semileptonic decay modes have proven to be good testing grounds for comparing 0(p4) to 0(p6) calculations. ChPT calculations of 7T7T scattering can likewise be tested by measuring the form factors of K-^-KITIVI (IQ4) decays. Sometimes the study of these less well-known modes can turn up new phenomena of considerable interest. For example, an interesting observation of a CP-violating and T-odd angular asymmetry in the K°L -^ TT+TV~~e+e~ decay has been made by the NA48 and KTeV experiments. This is the largest CPviolating effect yet seen, and the first CP-violating effect ever observed in an angular distribution. Electromagnetic Decay Modes Like the 7r°, the neutral kaons couple to two photons. The interaction with two real photons can be extended to off-shell photons with nonzero k2. In this case, the coupling /p 7 * 7 * can depend on the two k2 values. The form factor / K L 7 7 is needed to accurately calculate the long-distance contribution to K°L->fi+ yT and K°L~>e+e~ , which both have important contributions from the 7*7* intermediate state. This long-distance contribution must be subtracted from the precise experimental measurement of K°L -^ /i+fi~ in order to determine the interesting short-distance part of the amplitude for that decay, which can be related to the real part of the CKM matrix element Vtd or, equivalently, to the standard-model parameter p. Rare kaon decays can be used to study the If 77 form factors in several regions. For example, the electron and muon Dalitz decays K°L—•e+e~7 and K°L—>/u+^~7 are sensitive to the form factor with k\ = 0 and 4m 2 < k\ < M\, where m2 is the lepton mass. From lepton universality the form factors obtained in the electron and muon modes should be the same. As Table 1 shows, substantial statistics are now available in both of these ££j modes. The rarer double-internal-conversion modes, where the final state consists

228 Table 1: Summary of results of kaon decays to two photons and related modes

Decay Mode ^s°^77 K°L^-yj K°L->[i+V~~ K°L-+e+eKl^-e+e-j Ki^fi+fi-j K°L-+e+e-e+eKl~>n+n~e+eKl-^fl+fi-fjL+fiKl^ii+fiK^e+eKl^e+e^jj

'Ki^n+n~'ry

Branching Ratio (2.6 ±0.4 ±0.2) x 10~G (5.92 ±0.15) x 10~4 (7.24 ± 0.17) .x 10^ 9 (8.7^; Y ) x 10~ 12 (1.06 ± 0.02 ± 0.02 ± 0.04) x 10" 5 (3.66 ±0.04 ±0.07) x 10" 7 (3.77 ± 0.18 ± 0.13 ± 0.21) x 10~8 (2.50 ±0.41 ±0.15) x 10" 9 no limit < 3.2 x 10" 7 < 1.4 x 10~ 7 (5.84 ±0.15 ±0.32) x 10" 7 ( 1 . 4 2 l ^ ± 0 . 1 0 ) x 10" 9

events 148 110000 6200 4 6854 9105 436 38

Experiment NA48-00 13 NA31-87 14 E871-00 15 E871-98 16 NA48-99 17 KTeV-00 18 KTeV-00 18 KTeV-00 18

0 0 1543 4

CERN-73 19 CPLEAR 2U KTeV-00 21 KTeV-00 22

of two lepton pairs, are sensitive to the form factor in the region where both photons are off the mass shell. The largest sample of e + e~e + e~ decays reported to date is from KTeV, with 436 events in the 1997 data sample. KTeV has also reported seeing 38 e + e _ /,t + /i~ events. Two decay modes related to K°L —^77 , though not especially interesting in themselves, have significant implications for the attempt to observe direct CP violation in K°L —>7r°e+e~ and K°L —>7r°/j,+^~ . These are the radiative Dalitz decays K°L—>e+e~j~f and K°L —»/i+/i~77 . With a typical infrared cutoff of 5 MeV for the photon energies in the kaon center of mass, the electron mode, K°L —^e+e~77 , has a branching ratio of about 6 x 10~ 7 , five orders of magnitude higher than the expected rate for K°L-^fK°e+ e~ . Moreover, the peak of the 77 invariant mass spectrum in observed events is near the n° mass. With a good calorimeter, experiments can limit the n° mass range to a few MeV, but the number of e + e~77 events in this range still swamps the expected 7r°e+e~ signal. KTeV has identified a sample of over 1500 e + e~77 events and has verified that their kinematic distributions are generally in agreement with those predicted. The decay K°L —>fi+fi^-yy is likewise a serious background to the measurement of K°L —>TT°/J+IJ~ . The absolute rate for this decay is much less than the rate for the corresponding electron mode (see Table 1). Unfortunately, the part of the phase space where this decay can be a background to i ^ —>-7r°/i+/i~ ,

229

after the M 7 7 and other kinematic cuts, is not particularly suppressed. Thus, the K°L —>ir°[I+IJ,~ mode does not allow experiments to eliminate the radiative Dalitz background. K —>TTTTJ

The radiative K772 decays—K+—»7r+7r°7 ; K£->7r + 7r~7 , and K$ —>-7r+7r~7 — have two contributions. In the inner bremsstrahlung (IB) process, a photon is radiated from one of the charged particles. In the direct emission (DE) process, the photon is radiated from an intermediate state. In the neutral kaon decay, K°L—>TT+TT~'J , the DE part of the decay can be either CP-violating or CP-conserving, but experiments show that the DE decay is consistent with a CP-conserving Ml radiative transition. There is also a CP-odd interference term. These CP-odd and CP-even terms manifest themselves in a CP-violating asymmetry in the polarization of the photon, which is not observable in these experiments. However, a CP- and T-odd angular asymmetry is expected in the related decay K°L —>7r+7r~e+e~ , in which the photon internally converts to an e+e~ pair, since the angular distribution of the leptons preserves information about the photon polarization. This effect, predicted in 1992 by Sehgal & Wanninger 23 , is an asymmetry in the distribution of the angle

0) - iV(sincos0 < 0) * ~ N{sm4>cos(f) > 0) + 7V(sin
^'

It is important to note that the raw asymmetry may be significantly different from the acceptance-corrected asymmetry. This occurs not because of any asymmetry in the detectors but because the asymmetry varies across the phase space for the 7r+ir~e+e~ final state, and in general, acceptance is better in regions of the phase space where the asymmetry is large. The raw asymmetries observed by the two experiments are therefore not directly comparable. Nevertheless, they seem to agree. NA48 finds A^(raw) = (20 ± 5)% while KTeV measures ^ ( r a w ) = (23.3 ± 2.3)%.

230

So far, only KTeV has reported an acceptance-corrected asymmetry 28 . An important ingredient in making the acceptance correction is the form factor in the M l DE amplitude, which is extracted by fitting the Mnn and other kinematic distributions. Using the fitted form factor, the acceptance-corrected average asymmetry is found to be A$(corrected) = (13.6 ± 2.5 ± 1.2)%, in excellent agreement with the theoretical prediction. 3.2

Using Rare Kaon Decays to Measure Standard Model Parameters

The unitarity triangle is most readily expressed for the kaon system as follows:

v:svud + v;svcd + v;svtd = o

(2)

or

\u + \c + \t = 0, with the three vectors A; = V*sVid forming a very elongated triangle in the complex plane, as shown in Figure 10. The first vector, A„ = V*sVud, is (0,lmAt) B(K+^7T+^)1

(-Au+ReXt.O)

A(1-X 2 /2-A 4 /8)

(ReAt,0)

f ( K ->7T e u) Figure 10: Unitarity triangle for the K system (not to scale).

well known. The height will be measured by K°L-^-K°VV and the third vector, \ t = Vt*Vtd, will be measured by the decay K+ -^ir+vV . The theoretical ambiguities in interpreting all of these measurements are very small. It may also be possible to extract additional constraints on the height of the triangle from Kl^ir°£+£~ decays and on Re(Xt) from Kl->n+[j,~ decays.

The decay if£—• fj,+n~ is dominated by the process of AT£->77 with the two real photons converting to a fi+n~ pair. This contribution can be exactly calculated in QED 33 based on a measurement of the ^ £ - ^ 7 7 branching ratio.

231

However, there is also a long-distance dispersive contribution, through off-shell photons. This contribution needs additional input from ChPT 34>35J which may be aided by new, improved measurements of the decays K°L—>e+e~"f , K1->fi+[i~j , K°L—>e+e~e+e~ and K°L —>/x+/i~e+e~ . Most interesting is the short-distance contribution which proceeds through internal quark loops, dominated by the top quark. This contribution is sensitive to the real part of the poorly known CKM matrix element Vtd or equivalently to p 1 1 ' 3 7 . This mode has now been measured very accurately 15 by the BNL-E871 collaboration, who have reported B(K£-)- / u + /i~ ) = (7.18±0.17) x 10~ 9 . This measured ratio is only slightly above the unitarity bound from the on-shell two-photon contribution and consequently limits possible short-distance contributions. Unlike K°L —¥/J,+H~ , which is predominantly mediated by two real photons, the decay K°L—>e+e~ proceeds primarily via two off-shell photons. The relative contribution from short-distance top loops is significantly smaller than in Kj^fx+/j,~ . However, the recent observation by E871 16 of four events, with a branching ratio of B(K£->-e + e~ ) = (8.7t\'7i) x 10~ 12 , is consistent with ChPT predictions 35 ' 36 and is the smallest branching ratio ever measured for any elementary particle decay. K°L->n°l+lThe decays i ^ —»7r°e+e~ and i Q —>7r°/i+/U~ can proceed via s —> dj and s —» dZ processes described by electromagnetic penguin operators in the Standard Model. These processes are calculable with high precision since they are dominated by top-quark exchange. If these were the only contributions to this decay, the branching ratio for K°L —»7r°e+e'~ would be cleanly related to CKM matrix elements. With the current best-fit value of 1.38 x 10~ 4 for Im(Xt) 39 , one would expect from short-distance effects a branching ratio of about 5 x 10" 1 2 . Unfortunately, the decay K°L ->n°e+e~ can occur in two other ways. First, there is an indirect- CP-violating contribution from the CP-even, K° component of K°L. There is also a CP-conserving amplitude involving a ir°j*j* intermediate state, from which the virtual photons materialize into an e+e~ pair. Given these three contributions, it will be difficult to extract CKM matrix parameters from even a precision measurement of K°L —>ir°e+e~ . A still more formidable roadblock to progress on these modes was first pointed out by Greenlee 40 : The radiative Dalitz decay K°L —»e+e~77 has a rather large branching ratio (~ 6 x 10~ 7 ). The two photons may have an invariant mass near that of the ir°, so that the final state is indistinguishable from the 7r°e + e~ mode. Two strategies can be used to deal with this background. First, a high-

232

precision calorimeter can be used to minimize the size of the region in M11 where confusion can occur; second, the difference in the kinematic distributions expected in the radiative Dalitz decay can be used to remove most of the background events, at a cost of some acceptance for the signal mode K°L —» 7r°e + e~ . These techniques reduce, but cannot eliminate, this background, so that the present searches for K°L -^-K°e+e" are background-limited at the level of 10^ 10 . The most recent limit on K°L—»7r°e+e~ comes from the KTeV experiment 4 1 . The analysis selects on the direction of the photons with respect to the electrons to minimize the background from radiative Dalitz decays while preserving as much sensitivity as possible. KTeV found two events that passed all cuts, compared with an expected background level of 1.1±0.4 events. This finding leads to an upper limit B(K°L->ir°e+e~ ) < 5.1 x 10" 1 0 (90% CL). A similar analysis of the related muon mode by KTeV resulted in a slightly smaller upper limit 45 , B(K°L - > 7 r > V ~ ) < 3.8 x 1(T 10 (90% CL). K ->irvT>

The decay modes K+ -+K+VV and K°L^tix°vV are the golden modes for determining the CKM parameters p and r\. The K-^-KVV decays are sensitive to the magnitude and imaginary part of Vtd • From these two modes, the unitarity triangle can be completely determined. The theoretical uncertainty in the branching ratio B(K+—>TT+V'U ) is roughly 7%, while that for K°L —¥Tr0vi7 is even smaller, about 2%. In terms of the Wolfenstein parameters, and based on our current understanding of standard-model parameters, the branching ratios are predicted to be B(Kl->n°vV

) = 4.08 x lO" 1 0 ^?? 2 = (3.1 ±1.3) x 10"

+

B(K+-nr vv

(3)

11

) = 8.88 x 10~u A4[{p0-p)2

+ (orj)2}

(4)

11

= (8.2 ±3.2) x 10" , where a = (1 — 4 r ) - 2 and /50 = 1.4 46 . The decay amplitude K°L^fK°vV is direct- CP-violating, and offers the best opportunity for measuring the Jarlskog invariant JCPK+ -+ix+vV Although the decay K+ -+it+vV is attractive theoretically, it is quite challenging experimentally. Not only is the branching ratio expected to be less than 10~ 10 , it is a three-body decay with two undetectable neutrinos. The key to

233

a convincing measurement of this decay is a thorough understanding of the background at a level of 10 - 1 1 . The E787 experiment previously reported results of the analysis of the 1995 data sample 4 7 . This experiment employs two guiding principles for determining the background. First, the background is measured from the same data as the K+ —w+vv signal. In this manner, hardware problems, changes in rates, and changes in detector performance are automatically taken into account. Second, two independent sets of selection criteria are devised, with large rejection (e.g. typical rejections are R > 100) for a given background source. This allows a measurement of background levels at a sensitivity R times greater than the signal by reversing one set of selection criteria. The three major sources of background, _ftf+—»/i+i/M , K+ ->ir+TT° , and pions from the beam, are all measured, with a total background of 0.08 ± 0.02 events from the analysis of the data collected during 1995-1997. One clean K+ —>TT+VV event was found (see Figure 11), and based on this one event 4 8 ,

1 2 0

130

E

140

ISO

( M e V )

Figure 11: E787: Final data sample collected in 1995-1997 after all cuts. One clean K^^t-n^vV event is seen in the signal box. The remaining events are K+—>7r+7r° background.

which was also seen in the earlier data, the branching ratio is B(K+ ~^/K+VV ) = 1-51^2 x 10~ 10 . The E787 experiment, with all data recorded should reach a factor of two higher sensitivity—to the level of the standard-model expectation for K+ -^-n+vV . A new experiment, E949, is under construction at BNL and will run from 2001 through 2003. E949 should observe 10 standard-model events in a twoyear run. The background is well understood and based on E787 measurements is expected to be 10% of the standard-model signal. A proposal for an experiment which promises a further factor of 10 improvement has been prepared at FNAL. The CKM experiment (E905) is designed to collect 100 standard-model

234

events, with an estimated background of approximately 10% of the signal, in a two-year run starting in about 2005. This experiment will use a new technique, with K+ decay-in-flight and momentum/velocity spectrometers. K°L^nt°vV The decay K°L -^fK°vv is even cleaner theoretically and is purely direct- CPviolating. Unfortunately, it is even more difficult experimentally, because all particles involved in the initial and final states are neutral. Presently, the best limit on K°L-^v;°vv is derived in a model-independent way 49 from the E787 measurement of K+ -)-K+VV : B(K°L^tTX°vv ) < 4.4 x B(K+->ir+vT7) < 2.6 x 10" 9 (90% CL).

(5)

The goal is to observe this mode directly in order to extract a second of the CKM matrix parameters. The K°L -^n°vV decay is identified by two photons from the common decay n° —• 77. KL decays such as K°L ->7r07r°7r° and KaL —»7r07r° can easily produce background if all but two of the final-state photons are unobserved. Background can also arise from 7r°'s produced by A or H° hyperon decays to final states such as mr° with the neutron undetected, if the beam contains large numbers of hyperons. An excellent system of photon veto detectors can substantially reduce these backgrounds, but additional kinematic cuts will also be necessary. In the center of mass experiments, a simple invariant mass cut can be made to reject K1^-K°TT° and K°h —>TT°IT0'K° backgrounds. In experiments like KTeV where the kaon momentum is unknown, one can exploit the fact that the neutrinos recoiling against the 7r° in K°L->t it° vv are massless, so that the transverse momentum of the n° extends to larger values than are possible in the background modes, as shown in Figure 12. KTeV does not measure the kaon momentum; in order to determine the transverse momentum of the 7T°, the decay vertex must be known. The longitudinal position of the vertex can be determined from the invariant mass constraint, but the transverse position can only be known within the size of the kaon beam. Thus a narrow "pencil" beam is needed, which limits the available intensity. KTeV tried this approach in a one-day test run and observed one background event, probably from a neutron interaction. From this special run, a 90%-CL limit 50 of B{K°L -*IT°I>V ) < 1.6 x 10" 6 was determined. An alternative is to use the rarer ir° -> e + e ~ 7 decay. This is a factor of 80 less sensitive but has several advantages. First, the location of the decay vertex can be determined from the charged tracks, so that a high-intensity, wide neutral beam can be used. This allowed the KTeV data for this mode to be taken in the

235

standard configuration with standard triggers. Second, this approach allows determination of the transverse momentum with better precision, reducing the background level. The PT distribution of ir° events passing all other cuts can be seen in Figure 12. The backgrounds nearest the search region come from

Figure 12: KTeV: Final K°L -tn°vv data sample collected during 1996-1997 after all cuts. No K°L^-K°VV events are seen above PT ~ 160 MeV/c.

the decays A ->• mr° and 2° - • A7r°. In this search using the full 1997 KTeV data set, with an expected background of 0.121O!O4J no events were seen, and at the 90% confidence level,51 B(KI^-K°VV ) < 5.9 X 10" 7 , still more than four orders of magnitude from the standard-model prediction. 3.3

Lepton-Flavor Violation as a Probe of New Physics

All experimental evidence to date supports the exact conservation of an additive quantum number for each family of charged leptons. If neutrino masses are nonzero, some very tiny mixing effects could permit such a decay in the standard model, but it would occur at unobservably small levels, many orders of magnitude beyond the present experimental sensitivity. Any observation of a signal for the decays K°L-* lie , K+ -»7r + /i+e~, or K°L ->-7r0/ze would thus be conclusive evidence for new physics beyond the standard model 5 2 . Although this lepton-flavor-nurnber conservation law appears to be respected in the standard model, there is no fundamental reason or underlying symmetry to explain why this should be so. Indeed, many possible extensions to the Standard Model predict new interactions involving heavy intermedi-

236

ate gauge bosons that could mediate the otherwise forbidden lepton-flavorviolating decays. Some of the specific models that lead to such decays include compositeness of quarks and leptons, left-right symmetric models, technicolor, some supersymmetric models, unified theories with horizontal gauge bosons, leptoquarks, and string theories. It is important to look for both K°L —> fie and the modes with an extra pion, K+ -»7r + ^ + e~and i^£->7r°//e , because the K^-^fie decay is sensitive to pseudoscalar and axial vector coupling, whereas the other modes are sensitive to scalar or vector couplings. In both cases, the excellent sensitivity of these experiments probes mass scales that are very large. The sensitivity of the experiments to new interactions depends on the coupling constants involved. If the new coupling for an intermediate vector boson of mass Mx is gx, then the lower bound on Mx implied by an upper limit on B(K°L-*\ie ) is given in terms of the electroweak coupling g by 0 v

m 10-- n

Mx ~ 200TeV/c2 x — x B(Kl^fie) 9

!/4

(6)

K°L^fie The best result on K°L^r\xe comes from BNL E871 53 , and was published in 1998. This experiment featured two analysis magnets for redundant momentum measurements. Electrons and muons were each identified in two different ways to reduce background from particle misidentification. The analysis cuts were then chosen to minimize the remaining backgrounds (involving either accidental coincidences or scattered electrons) while maintaining as much sensitivity as possible to the signal. A single-event sensitivity of about 2 x 10~ 12 was achieved, with an expected background of 0.1 event. Having found no events in their signal box, E871 set a 90%-CL upper limit of B(Kl -+\ie ) < 4.7 x 10~ 12 , the smallest upper limit set to date on any kaon decay mode. For an exotic boson with electroweak coupling strength, Equation 6 then implies a lower bound on its mass of 150 TeV. There are no near-term plans to pursue this decay further, as the background from Kl —>/K±eTi'e with a muon decay and a scattered electron is difficult to reduce below a level of 10~ 13 , which is just beyond the E871 sensitivity. K+

-¥-K+n+e~~

E865 at BNL was designed to search for the lepton-flavor-violating decay K+—>7r+yU+e~. This decay, with an extra pion in the final state, is sensitive to exotic gauge bosons with different quantum numbers from those that

237 400

300 O CD

™dr

200 Exclusion Box 100 Signal B o x

490

Azn

495

i

I

i l i i

500

i

505

510

M , e (MeV/c^) Figure 13: E871: Final data sample with the reconstructed mass M^e vs the square of the transverse momentum relative to the kaon direction, after all cuts there are no events in the signal region. The exclusion box (the larger box enclosing the signal box) was used to set cuts in an unbiased way on data far from the signal region. The shape of the signal box was optimized to maximize signal/background.

E871 could detect. The experiment uses K+ decays in flight, and the detector concept is similar to that of E871, with redundant particle identification by two Cerenkov detectors and an electromagnetic calorimeter, and a muon range stack. The limit on this mode from the 1996 r u n 5 5 , with no events above a likelihood to be K+ -•7r + /i + e _ (L i r / j e ) of 20%, is B(K+ ->7r+ n+e~) < 3.9 x 1 0 ~ n . From the combined results from E777 and the E865 runs in 1995 and 1996, a limit of B(K+ ->TT+/x+e") < 2.8 x 10" 1 1 is obtained. E865 is already close to being limited by background from accidentals; there are no plans to continue with this search. The E865 limit implies a lower bound of several tens of TeV on exotic bosons with electroweak coupling, depending on the exact model used.

i q ^

fie

In addition to the search for K+ —»7r+/i+e~performed by BNL E865, a search for the corresponding neutral mode K°L-+-x° \ie has recently been carried out by KTeV at FNAL. The main background concern was the common decay

238

K°L—>7r±e:F^e , in which the pion is misidentified as a muon, with 0.6±0.6 expected events. After all cuts, two background events were observed in the signal box, and the preliminary 90%-CL limit on K°L^,-K°\ie from KTeV 5 6 is B(AT£->7r>e ) < 4.4 x KT 1 0 . 4

Conclusions and Future Prospects

Tremendous progress has been made over the past decade in measuring Re(e'/e). Both KTeV and NA48 have made precise measurements and increased precision can be expected as both experiments finish analysing data already collected. The two experiments unfortunately do not agree with each other very well; but the most serious difficulties lie on the theoretical side at the moment. In the area of rare decay searches, the remarkable sensitivities of rare kaon decay experiments in setting limits on lepton flavor violation have constrained many extensions of the standard model. In addition, the observation of K+ —>K+VV may soon permit measurements of the unitarity triangle completely within the kaon system. An accurate determinations of the CKM matrix element Vtd could well come from the generation of experiments that is starting now. Comparison with the B meson system will then overconstrain the unitarity triangle and test the standard-model explanation of CP violation. The primary focus for the future of rare kaon decays is on the measurement of the so-called golden modes, K°L-+-K°VV and K+ —\v+vV , at sensitivities sufficient for observation of 100 events. Major initiatives with this goal are underway at BNL, FNAL and KEK. At the same time the study of a number of medium-rare and radiative modes will continue to be pursued, both as byproducts and in dedicated experiments. References 1. J.H.Cristenson, J.W.Cronin, V.L.Fitch, and R.Turlay, Phys. Rev. Lett. 13, 148 (1964). 2. L.Wolfenstein, Phys. Rev. Lett, 13, 562 (1964). 3. A.Alavi-Harati et al, Phys. Rev. Lett. 83, 22 (1999). 4. V.Fanti et al, Phys. Lett. B465, 335 (1999); G.Graziani, Proc. XXXV Rencontres de Moriond, Les Arcs, France, March 2000 J. Tran Thanh Van, ed., Paris: Ed. Frontieres (2000). 5. M.Antonelli et al., Nucl. Phys. B Proc. Suppl. 54, 14 (1997); S.Dell'Agnello et al, Nucl. Phys. B Proc. Suppl. 54, 57 (1997); V.Elia et al, Nucl. Phys. B Proc. Suppl. 54, 66 (1997); S.Spagnolo et al, Nucl. Phys. B Proc. Suppl. 70, (1997); F.Lacava et al, Nucl. Phys. B Proc. Suppl. 54, 327 (1997).

239

6. L.Littenberg and G.Valencia, Annu. Rev. Nucl. Part. Sci. 43, 729 (1993) 7. S.H.Kettell and A.R.Barker, Annu. Rev. Nucl. Part. Sci. 50 (2000). 8. J.Hagelin and L.Littenberg, Prog. Part. Nucl. Phys. 23, 1 (1989); J.Ritchie and S.Wojcicki, Rev. Mod. Phys. 65, 1149 (1993); P.Buchholz and B.Renk, Prog. Part. Nucl. Phys. 309, 253 (1997). 9. G.Buchalla, A.J.Buras, and M.E.Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996). 10. A.J.Buras and R.Fleischer, hep-ph/9704376. 11. A.J.Buras, hep-ph/9806471. 12. G.D'Ambrosio and G.Isidori, Int. J. Mod. Phys. A 1 3 , 1 (1998). 13. V.Kekelidze et al., Proc. XXX Int. Conf. High Energy Phys., Osaka, Japan, July 2000 Singapore: World Sci. (2001). 14. H.Burkhardt et al., Phys. Lett. B199, 139 (1987). 15. D.Ambrose et al., Phys. Rev. Lett. 84, 1389 (2000). 16. D.Ambrose et al, Phys. Rev. Lett. 81, 4309 (1998). 17. V.Fanti et al, Phys. Lett. B458, 553 (1999). 18. B.Quinn, Proc. Meet. DPF, Columbus OH, August 2000 Singapore: World Sci. (2001). 19. S.Gjesdal et al, Phys. Lett. B 4 4 , 217 (1973). 20. A.Angelopoulos et al, Phys. Lett. B 4 1 3 , 232 (1997). 21. A.Alavi-Harati et al.,, hep-ex/0010059 (2000). 22. A.Alavi-Harati et al, Phys. Rev. D 62, 112001 (2000). 23. L.M.Sehgal and M.Wanninger, Phys. Rev. D46, 1035 (1992); (E) Phys. Rev.D46, 5209 (1992). 24. P.Heiliger and L.M.Sehgal, Phys. Rev. D48, 4146 (1993); J.K.Elwood, M.B.Wise, and M.J.Savage, Phys. Rev. D 5 2 , 5095 (1995); Phys. Rev. D 5 3 , 2855 (E) (1996); J.K.Elwood, M.B.Wise, M.J.Savage, and J.W.Walden, Phys. Rev. D 5 3 , 4078 (1996); L.M.Sehgal and J.van Leusen, Phys. Rev. Lett. 83, 4933 (1999). 25. P.Lubrano, Proc. Heavy Flavours 8, Southampton, UK, July 1999 P.Dauncey and C.Sachrajda, eds., J. High Energy Phys. (1999). 26. J.Adams et al., Phys. Rev. Lett. 80, 4123 (1998). 27. A.R.Barker, Proc. Heavy Flavours 8, Southampton, UK, July 1999 P.Dauncey and C.Sachrajda, eds., J. High Energy Phys. (1999). K.Senyo, Proc. Int. EuroPhys. Conf. High Energy Phys., Tampere, Finland, July 1999 K.Huitu et al.„ eds., Bristol, UK: IOP-Publishing (2000). 28. A.Alavi-Harati et al., Phys. Rev. Lett. 84, 408 (2000) 29. M.Contalbrigo, Proc. XXXV Rencontres de Moriond, Les Arcs, France,

240

March 2000 J.Tran Thanh Van, ed., Paris: Ed. Frontieres (2000). 30. A.Alavi-Harati et al.„ hep-ex/0008045 (2000). 31. S.Adler et al, Phys. Rev. Lett., hep-ex/0007021 (2000). 32. G.D.Barr et al, Phys. Lett. B328, 528 (1994); M.Zeller, Kaon Physics, Proc. Chicago Conf. Kaon Phys., J.L.Rosner and B.Winstein, eds., June 1999, Chicago: Univ. Chicago Press (2000). 33. L.M.Sehgal, Phys. Rev. 183, 1511 (1969). 34. G.D'Ambrosio, G.Isidori G, and J.Portoles, Phys. Lett. B423, 385 (1998). 35. D.Gomez-Dumm and A.Pich, Phys. Rev. Lett. 80, 4633 (1998). 36. G.Valencia, Nucl. Phys. B517, 339 (1998). 37. C.Q.Geng and J.N.Ng, Phys. Rev. D 41, 2351 (1990). 38. T.Inami and C.S.Lim, Prog. Theor. Phys. 65, 297 (1981). 39. S.Bosch et al, Nucl. Phys. B565, 3 (2000). 40. H.Greenlee, Phys. Rev. D42, 3724 (1990). 41. A.Alavi-Harati et al, hep-ex/0009030 (2000). 42. A.Barker et al, Phys. Rev. D 41, 3546 (1990). 43. K.E.Ohl et al, Phys. Rev. Lett. 64, 2755 (1990). 44. D.A.Harris, et al, Phys. Rev. Lett. 71, 3918 (1993). 45. A.Alavi-Harati et al, Phys. Rev. Lett. 84, 5279 (2000). 46. G.Buchalla and A.Buras, Nucl. Phys. B548, 309 (1999). 47. S.Adler et al, Phys. Rev. Lett. 79, 2204 (1997). 48. S.Adler et al, Phys. Rev. Lett. 84, 3768 (2000); S.H.Kettell, Proc. 3rd Int. Conf. B Phys. and CP Violation, Taipei, Taiwan, Dec. 1999 H.Y.Cheng and W.S.Hou, eds., World Sci. (2000). 49. Y.Grossman and Y.Nir, Phys. Lett. B398, 163 (1997). 50. J.Adams et al., Phys. Lett. B447, 240 (1999). 51. A.Alavi-Harati et al, Phys. Rev. D 61, 072006 (2000). 52. F.Wilczek and A.Zee, Phys. Rev. Lett. 42, 421 (1979); R.Cahn and H.Harari, Nucl. Phys. B176, 135 (1980). 53. D.Ambrose et al., Phys. Rev. Lett. 81, 5734 (1998). 54. D.R.Bergman, A search for the decay K+ -^ir+u.+e~~. PhD thesis. Yale Univ. (1998); S.Pislak, Experiment E865 at BNL: A Search for the Decay K+^n+n+e'. PhD thesis. Univ. Zurich (1998). 55. R.Appel et al, Phys. Rev. Lett. 85, 2450 (2000). 56. A.Bellavance, Proc. Meet. DPF, Columbus OH, August 2000 Singapore: World Sci. (2001).

J. P. Cumalat

This page is intentionally left blank

T H E STATUS OF MIXING IN T H E C H A R M S E C T O R J. P. CUMALAT Department of Physics, University of Colorado, Boulder, CO 80309-0390, USA E-mail: [email protected] During the past year there has been significant progress in the study of charm mixing. The CLEO Collaboration has reported new results from direct (wrong sign) searches and both CLEO and FOCUS have reported results from lifetime difference (Ar) searches. It seems that the observation of mixing or limits for r m ; x of 5 x 10~ 5 (or better!) will soon be available.

The observation of K° — K° oscillations were crucial in the development of the standard model. Similarly, the observation of Bd — Bd oscillations has had an essential impact on the standard model. So why is mixing in the charm sector so small? The explanation is in the so-called GIM (Glashow, Iliopoulos, and Maiani) suppression 1 . It is the GIM suppression that makes searching for charm mixing so interesting. An observation of mixing in the charm sector, rmiK of greater than 1 x 1 0 - 6 , might provide a signature for new physics. 1

GIM Suppression

One of the box diagrams which represent the lowest order short-distance contribution to D°-D° mixing is presented in figure 1. The mixing amplitude calculated from these diagrams is proportional to 2 : (D°\Hwk\D°)

cc £

V?t Vui Vcj V£ S(mlm])

(1)

i,j=d,s,b

where Vij are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. If the quark masses m; were all equal, the loop functions .S(m?, m 2 ) would all be equal and the amplitude would be zero due to the unitarity (J2 K* K»» = 0) 0I" the CKM matrix. If the mass differences are small, GIM very nearly works and mixing is small. For charm, the CKM factor V*b Vub is insignificant (~ A5 in the Wolfenstein parameterization 3 ) relative to the factors V*d Vud and V*s Vus (both ~ A). Only the i, j = d,s terms contribute in equation 1, and the mass difference between the d and s quarks is relatively small. The D°~D° mixing probability calculated from these box diagrams is ~ 10~ 10 — 10~ 19 . 4 By contrast, neutral b quark mesons exhibit large mixing because the top mass is 243

Figure 1: One of the two box diagrams for mixing. The second diagram is found by exchanging the internal W and quark lines with each other.

so large. For example, for B° the three CKM factors (V*b Vid, i = u,c,t) are roughly equal (~ AX3) so the contribution of the top quark dominates. Because the short distance predictions for charm mixing are so small, long distance contributions may be important. They are more difficult to calculate, and there is significant disagreement over their size 5 ' 6 ' 7 ' 8 . With long distance effects, r m i x may be as large as 10~ 3 . If there is new physics such as a fourth generation of quarks, leptoquarks, etc., it can contribute to the box diagrams for mixing or the penguin diagrams for FCNC decays 9 . Because the standard model predictions are so small, there is a large window to observe the additional contributions of this new physics, unhidden by standard model effects. 2 2.1

Charm Mixing Basic Theory

The neutral D mesons evolve according to: l

-U^[D°

dt {D°J _(

M-iT/2 \M*2-iT*12/2

M12-iT12/2\ M-iT/2

(D_° ){D°

Diagonalizing gives weak eigenstates DH = PD° + qW; q= p

DL = PD° - qD°

\Ml2-iTl2l2V'2 [M12-iT12/2_

245

of definite mass and lifetime: = M± K[(M*2 - i r ; 2 / 2 ) ( M i 2 -

iTu/2)]1/2

rH,L = r T 2Z[(M{2 - iT*12/2)(M12 -

iT12/2)}^2.

MH,L

These eigenstates evolve with time as: = e-i^-it-ir»-itDff,i(0).

DH,L(t)

If iJwk conserves CP, then Du and £>L are CP eigenstates and p = q = 1. If you start with a D°, the probability that it is a D° at time t is: r m «(t) = r(£>°-H>°) 2

|e —TH*

2e -r* cos AMt]

I g—r^t

where A M = (MH - ML) and AT = (TH - TL). Experimental limits on D°-D° mixing 10 ' 11 ' 12 - 13 - 14 indicate that A M < < F and AP < < T, so rt cW = i e -

If we define x = AM/T

(AM 2 + i A r 2 ) i 2 .

and y = A r / 2 r , then 2

cW

4e

(x2+y2)(r2^2)

(2)

(z 2 +2/ 2

(3)

Integrated over all time: 'mix —

For the charge conjugate process (D°—>D°): 2

^*mix^J — 44 ^

(z 2

2 + 2/

)(r 2 * 2 )

(4)

so that r m i x = r m i x only if |g/p| = 1. Expectations from standard model short distance calculations are that x and y are approximately equal. However, if r m i x is large, the source is likely to be x. While long distance effects and/or new physics can increase A M substantially, they do not make significant contributions to A r 5 ' 9 ' 1 5 . In general AM^> receives contributions from virtual intermediate states, while ATD is generated by on-shell transitions. A nice compilation 16 of theory predictions has been made by H. Nelson and they are presented in figure 2.

246

O - O IVtixJmg Predictions10

-1

*-*

&

Figure 2: _D° — D° Mixing Predictions where the triangles are SM predictions of x, squares are SM predictions of y, and circles are NSM predictions of x. The predictions encompass 15 orders magnitude for rm{x.

2.2

Search Strategies

There are two basic methods currently employed to search for charm mixing. In direct or "wrong sign" searches, one looks for D°^D°^f, where / is a Cabibbo favored (CF) mode such as K+-n~ or fsT+/u~fM. The sign of the daughter Kor /j, distinguishes D°->/from D ° - » / . The produced D is identified using a D* 'tag': D*+^D°n+. The sign of the "bachelor" pion from the D* decay tags the produced neutral D as a D° (tr+) or D° (ir~). The signal for mixing is that the bachelor pion and the K daughter have the same charge: D*+-+-K+D°;

D°-+D°^K+TT-.

The second method is to look for a lifetime difference between DH and

247

DL- If DH and DL are C P + and CP 2/CP

eigenstates, then:

= — - — = -^r = V-

(5)

The lifetimes r+ and r_ can be measured using neutral D decays to states of definite CP, such as D°-*K+K(CP+) or D°^K%4> (CP"). Even if one relaxes the "no CP violation" requirement, J/CP ~ V because it is known 17 that CP violation is small in charm decay and therefore DH and DL are approximately CP eigenstates. 2.3

The DCS Interference

For wrong sign hadronic searches ( / — K+ir~, K+ir~Tr+ir~,...) there is an interesting interference which occurs. The interference is pictorially displayed in figure 3. Because mixing is at best a small effect, doubly Cabibbo suppressed (DCS) decay cannot be ignored when looking for a wrong-sign signal 18,19 . Via DCS, D°—>f can occur directly and not just through mixing D0—>JD°—>•/. In this case: Aws = ADCS(D°->f) + Amix(D°->D°^f) and the wrong sign decay rate becomes: —Tt (f\H\D°)2CFCF x{\\\>+ rws = e 2 2 2 \(x + y )TH + (K(A)y + <3{\)x) Tt}

(6)

where A_p
-

g(f\H\D°)Z '

T _ «
By measuring the proper time of the decay of the neutral D meson, one can disentangle the contributions from DCS and mixing: while all terms in equation 6 share a common exponential time dependence, the mixing term (x2 + y2) is proportional to t2, the DCS term (|A|2) has no additional time dependence, and the interference term is proportional to t. The interference term is particularly interesting as it may be observable even if the mixing term isn't (in which case mixing would be observable in hadronic modes but not semileptonic!). Alternatively, if the interference term has the opposite sign of the mixing term (destructive interference) and it is of roughly equal size, then the effects of mixing could be masked 18 . Finally, this interference term could allow experimenters to distinguish between A r and AM contributions to mixing. This is especially true if CP violation in mixing is very small or zero (see section 2.4).

248

Doubly Cabibbo Suppressed

ccsin24

K+tv Figure 3: The interference from the mixed decay and from DCS is pictorially displayed. For the semileptonic decay, there is no interference, but for the hadronic decay one must worry about size and relative phase of the DCS decay.

Note that the CP conjugate rate for wrong sign decays is not the same as equation 6: rWs = e \{x2 + y2)T2t2 2-4

Simplifying

(f\H\D%Fx{\\\2+ + (»(A)j/ + Z(\)x)

Tt}

(7)

Assumptions

The wrong sign rates of equations 6 and 7 can be simplified by making assumptions about the nature of CP violation. For example, it is likely that there is no direct CP violation in CF or DCS decays0 and that CP is not violated in charm mixing 15 , i.e. \v/p\ = i. (8) If we assume that both of the above assumptions are valid, then |A| = |A|.

(9)

°For direct CP violation to appear in a decay mode, there must be two amplitudes that contribute significantly to that final state.

249

Defining the strong (final state interaction) phase 5 and the CP violating phase

: VfiDCS • ei5 = ( / l * l ^ ) n . s ei

=

?

(10)

the wrong sign rates become: rws, r W s = e - r t ( / | t f | ; D 0 ) c F x {fines + | r m i x r 2 i 2 + + [j/v/ifocs cos(S ± ) -X^RDGS

sin(<J ± 0)] Tt} .

(11)

If we assume CP invariance (<j) = 0) then rws = ?ws and: rws = e - r t ( / | # | ^ ° > c F x {fiocs + | r m i x r 2 i 2 + y'VR^H}

(12)

where y' = y cos J — a: sin 5; x' = a; cos S + y sin 5.

(13)

If (5 is small, as has been argued 19,20 , then y'-ty and x'-tx. If one has sufficient proper time resolution, equation 12 shows that the contributions due to x' and y' can be resolved (assuming CP invariance). 2.5

Comments on Assumptions

There may be excellent theoretical motivation for the assumptions made in subsection 2.4. But, as demonstrated by E791 in their hadronic wrong sign search 13 , there are no technical reasons for experimenters to make them. E791 first quoted mixing limits based on equations 6 and 7 (minimal assumptions). They then quoted limits after making the simplifying assumptions of equations 8 and 9, and finally for the case of no mixing (DCS only). This approach avoids two difficulties. First, if experimenters quote results based on only one set of assumptions, it can be difficult and misleading to compare results from different experiments (if past history is any guide, they are unlikely to make similar ones). Second, and more importantly, assumptions mask possibly interesting phenomena (assuming CP invariance precludes searching for CP violation in mixing).

250

The situation in A r searches is slightly different. There, one measures J/CP, and y = yep only if DH and DL are CP eigenstates. An observation of 2/CP 7^ 0 is evidence for mixing but extracting y from this requires additional information. 3

Wrong Sign Mixing Searches

E791 and ALEPH have previously published results from wrong sign searches, but only E791 published results for the most general case and hence, I will restrict my discussion to the E791 results. However CLEO has recently published results with significantly higher precision and performed fits for a variety of different assumptions. 3.1

E791 Wrong Sign Searches

E791 at Fermilab is a fixed-target hadroproduction experiment. Using a 500 GeV 7T~ beam and thin target foils, they logged 2 x 1010 hadronic interactions. Their spectrometer employed silicon microstrip detectors for vertexing, two threshold Cerenkov detectors for particle identification, a muon hodoscope and a lead/liquid scintillator electromagnetic detector for electron identification. They have published results for wrong sign searches using hadronic and semileptonic decay modes. Their hadronic analysis 13 uses the CF decay modes D°-lK~-ir+ir~ir+ and D°^-K~TT+. Figure 4 shows the right sign and wrong sign signals they obtain, where they have combined D° and D° modes for the purpose of making the figure. In the plots, Q = m(irKmr) — m(Kmr) — m(n). There are 5643 and 3469 reconstructed signal events in the right sign Kir and K3ir samples. In their analysis, they keep the D° and D° samples separate, and perform a simultaneous binned maximum likelihood fit to each of the eight resulting data sets. The D° mass and width and the Q peak and width are constrained to be the same in each data set, but most parameters (such as backgrounds) are uncoupled, leading to a 41 parameter fit in the most general case. Making no assumptions about CP violation or mixing, E791 sets the following 90% CL limits: rmix(D°^D°)

< 1.45%

fmix(D°^D°)

< 0.74%.

(14)

Assuming CP violation only in the interference term (see equations 8 and 9), they find a 90% CL limit of: rmix < 0.85%.

(15)

251

Figure 4: E791 right sign (top row) and wrong sign (bottom row) hadronic data samples. Candidates for the decay modes Kir and K3n are shown in the left and right columns, respectively. D° and D° candidates are kept separate in the analysis, but are combined here. Q = m^Kmr) — m(Kmr) — m(n).

If they assume no mixing, they find relative branching ratios (equation 10) for DCS modes: RBCS(KTT)

= (0.68±g;f| ± 0.07)%

i?DCs(^37r) = (0.25±g;12 ± 0.03)%.

(16)

The E791 semileptonic wrong sign search 12 uses the D° decay modes K~fj.+I'/J, and K~e+ve. The missing neutrino gives a two-fold ambiguity in the D° momentum; based on Monte Carlo studies they pick the higher momentum solution. Fixing the D° mass, they then fit the Q and proper time t distributions. The right and wrong sign Q plots are shown in figure 5. The dotted lines show the estimate of the background they get using an event mixing technique (they combine LP candidates from one event with a bachelor pion from another). In this analysis, E791 does not quote a "minimal assumption" result, instead they make the assumption of equation 8 from the outset and fit to a time dependence given by equation 2. The number of right sign decays

252

0.040

Q = M(Klvn) - M(D°) - M(7i) (GeV/c2) Figure 5: E791 right sign (top row) and wrong sign (bottom row) semileptonic data samples. Candidates for the decay modes Kev and Kp,v are shown in the left and right columns, respectively.

they get from their fit is 1237 ± 45 {Kev) and 1267 ± 44 limit they obtain is: r m i x < 0.50%. 3.2

{K/J,V).

The mixing (17)

CLEO Wrong Sign Search

The CLEO collaboration has reported 21 results from a wrong sign search for mixing using data from 9.0 fb _ 1 of integrated luminosity taken with the CLEO II.V detector. This analysis takes advantage of the three layer, doublesided silicon vertex detector (SVX) they installed in 1995. In addition to giving CLEO the ability to measure the proper time of charm decays, the SVX dramatically improves their measurement resolution of Q, the energy release

253

in the D*± decay used to tag the initial state of the neutral D. This improved resolution enhances their sensitivity to mixing by substantially increasing their signal to noise. After cuts designed to suppress backgrounds from other D° decays and "cross-talk" between their right-sign and wrong-sign samples, they obtain the D mass and Q plots for wrong sign candidates shown in figure 6. Superimposed on the data (solid lines) in these plots are colored regions which show the contributions from backgrounds determined by a two-dimensional fit to Q and M. The background shapes were determined from Monte Carlo. Prom their fit they find 44.8ig;7 D°^K+TT~~ events in their wrong sign sample.

Figure 6: CLEO wrong sign Q and M distributions for KIT candidates from D* decay. Backgrounds from various sources are shown, and come from two-dimensional fits to Q and M (shapes determined from simulation).

A similar fit to their right sign sample yields 13527 ± 1 1 6

D°^K~ir+

254

candidates. Using these numbers and assuming no mixing, they find TD

= ( ° - 3 3 2 ™ ± 0-04)%.

= ? T £ T TRS

(18)

(K IT)

With this no mixing assumption, it is worthwhile to compare this result to older rD measurements of E791 1 3 (0.68 ±0.34 ±0.07)% and of CLEO 2 2 (0.77± 0.25 ± 0.25)%. The new CLEO value is within errors of the naive expectation of tan49 and more than a factor of two smaller than the older results. With this assumption of no mixing, they determine the branching ratio of D° -» K+ir~ = (1.28±g;l| ± 0.15 ± 0.03) x 10"*. Using a time dependent fit to differentiate between mixing and DCS contributions they measure RDOS(KW)

= (0.48 ± 0.12 ± 0.04)%. \x'\ < 2.9%

z' = 0 ± 1 . 5 ± 0 . 2 %

(19)

y' = (-2.5+l;| ± 0-3)% - 5 . 8 % < y' < 1.0%

(20)

CLEO has further split its wrong sign sample into D° and D° candidates and they find no evidence for a time-integrated CP asymmetry. In their most general fit they allow for CP violation via state-mixing, direct decay, and the interference between the two processes. In leading order they allow for both x1 and y' to be scaled by ( 1 ± A M / 2 ) , RD ->• (1 ± AD), and 6 ->• 5 ±
(21)

AD = (-0.01±g;J? ± 0.01) -0.36 < AD < 0.30

(22)

sin<j) = 0.00 ± 0.60 ± 0.01) No limit(95%CL).

(23)

255

They further assume CP invariance and use the proper time distribution given by (12) in their fits. In this parameterization the interference term gives independent information on y'. From their fits, CLEO finds one-dimensional intervals at 95% CL of: \x'\ < 2 . 8 % x'=0±1.5±0.2%

(24)

y' = (-2.3il;5 ± 0.3)% -5.2% < y' < 0.2%

(25)

Systematic errors are included when finding the above intervals. It is not possible to determine the sign of x' from the fit, which depends only on (a;')2. With this assumption of CP invariance, and if S = 0 so that y'—>y and x'—>x, the above results give separate limits on the A r and AM contributions to mixing, an interesting and important result (particularly so given their excellent sensitivity). A fit to the time dependence for D° —> K+ir~~ for CLEO is presented in figure 7. The vertical hatched lines below zero in the figure indicate the fit contribution from the destructive interference with mixing. The CLEO group concludes that their data is consistent with no D° — D° mixing. They also conclude that they observe no evidence for CP violation. Nevertheless their results are a substantial advance in sensitivity to mixing beyond E791. 4

Lifetime Difference Searches

E791 has published results of a search for A r , while CLEO and FOCUS searches are in progress. 4.1

E791 AT Search

E791 searched 23 for a lifetime difference between the C P + and CP~ eigenstates of the D°. To do so, they compared the lifetimes of the decays D°^K~K+ (CP+) and D°->K-7r+ ( | CP+ + | C P " ) : T(K-K+) - T{K-TT+) _ T(K-rr+) ~

r + - | ( r + + r_)

= yep-

T(K-ir+) (26)

Their fit to the mass distribution includes Breit-Wigner shapes to account for reflections from misidentified K~ir+ and K~-ir+ir0 decays. After cuts, they have

256

D*+ -> D V , D° -> KV

Figure 7: CLEO wrong sign distribution in proper decay time. The data are shown as the solid points with error bars. The figure displays the fit contributions from the direct D° —> K+TT~ decay, from the destructive interference with mixing, and from backgrounds from the charm and light quark production.

3213 ± 77 signal events in K~K+ and 35,427 ± 206 in K~ir+. They observe no difference in lifetimes, and they quote: yep = 0.008 ± 0.029 ± 0.010 -0.04 < yCp < 0.06 (90%CL).

4.2

(27)

CLEO A r Search

CLEO has presented preliminary results 24 on A r based on data they have collected with the CLEO II.V detector. They plan to compare the lifetimes of D°^-K~-K+ and D°-*7r-7r + (both CP+) to D 0 ->K-7r + to extract a measurement of J/CP via equation 26. Based on partial samples of roughly 1300 K~K+,

257

475 7T-7T+, and 19000 K~-K+ (figure 8) events they find: 2/CP = 0.032 ± 0.034 ± 0.008 -0.076 < 2/CP < 0.012

(90%CL).

D*+ -* D V , D° -> Kn+

(28)

D*+ -» D V , D° -> K K+

Figure 8: CLEO D°—>K~n + and D ° - > K _ K + mass plots, for candidates used in their preliminary A F search.

The CLEO Collaboration estimates an eventual sensitivity for the measurement y of oy RS 0.028(stai) ± O.OlQ(syst) using the channels D°->K~ir+ and rP-*K~K+ They are also investigating CP odd lifetimes via channels K°, K°sp, K°SUJ, and K*~ir+. If they are able to cleanly identify these states, distinguishing these CP odd channels from the CP even backgrounds, such as K°f° in the K°K+K~ mass plot, then it should be possible to obtain an overall ay sensitivity of under 1%. CLEO measures the proper time of decays using only y coordinate information: 2/vtx — J/beamspot

Wco

C

Py

TDo =

(their beam spot has ay « 10/im, ax « 250fim). With much more data and other channels to analyze, CLEO believes their sensitivity will increase substantially. It is obvious that the BABAR and

258

BELLE Collaborations should become major players in the next year in this lifetime difference search area for mixing. If CLEO III can solve their early luminosity difficulties, then they may also remain major competitors in the quest to conclusively observe mixing.

4.3

FOCUS A r Search

FOCUS (E831) is a photoproduction experiment located at Fermilab that is the successor of the E687 experiment 25 The experiment has an upgraded vertex silicon detector and better particle identification. The data taking was done during the 1996-1997 fixed target run. FOCUS searched for A r by comparing the lifetimes of the C P + final states K+K~ to that of K~ir+ (equation 26) 2 6 . Mass plots for their K+K~ and K~ir+ candidates are shown in figure 9.

Figure 9: (a) Reconstructed mass distribution of D° -> K~ir+ and its conjugate decay. There is a total of 119738 signal events, (b) Reconstructed mass distribution of D° -» K~K+ candidates. There is 10331 D° -> K~ K+ signal events. The vertical, dashed lines indicate the signal and sideband regions used for the lifetime and yep fits.

259

Due to excellent proper time resolution ( « 7% of the D° lifetime) the fractional error in the FOCUS lifetime is roughly equal to the fractional error in the K+K~ yield. In order to account for the small D° -¥ K~TT+ reflection in the D° -> K+K~ mass, FOCUS assumes that the time evolution of the reflection is described by the lifetime of D° -> K~ix+ and the reduced proper time distributions of the D° -> K~TT+ and D° ->• K+K~ samples are fit at the same time. The fit parameters are: the D -> Kir lifetime, the lifetime difference yep-, and each number of background events under the signal region in D° -> K~ir+ and in D° -» K~K+. The signal contributions for the D° ->• K~ir+, D° -> K~K+, and the reflection are described by f(t')exP(—t'/T) in the fit likelihood. The value /(£') is a function which corrects for detector acceptance and the absorption of particles in matter. The background parameters are either floated or fixed to the number of events in the mass sidebands. The signal yields as a function of reduced proper time for JD° -> K~7T+ and for D° -> K~K+ are presented in figure 10.

W

o

o w d >

2000

3000

4000

t' (fs) Figure 10: Plots of the signal yield versus the reduced proper time for D° —> K D° —> K~ K+ events. The data is background subtracted.

7r+ and for

260

The FOCUS results are: yCp = (3.42 ±1.39 ±0.74)% T{D ->• Kit) = 409.2 ± 1.3fs The systematic errors on yep are estimated by changing the selection cuts and by trying different fitting methods. Tests of the Cerenkov identification hypothesis for kaons and for the detachment cut between the secondary and primary vertices were performed. FOCUS also varied the number of bins and used two alternate methods for handling the background. The result on the lifetime of D —• Kir has only a statistical error as this is the most accurate measure of the lifetime thus far, but FOCUS expects to do better when the analysis of the D —>• Kiririr channel is completed. It is intriguing to ask with a yep value of 3.42% what the D° lifetime means? There is a 27 fs difference between the D° measured with a CP odd channel and with the K+K~ channel. One must exercise care in the future when quoting the D° lifetime. 5

Summary

The summary of the limits on x and y are presented in figure 11. The large diagonally lined circular region limits are from E791 semileptonic results. The small circular and peanut shaped regions are derived from different fits to the wrong sign hadronic signals from CLEO. The horizontal bands are determined from measurements of the lifetime difference values for D° —> K~ir+ and D° -» K~K+ from E791 and from FOCUS. First, one notes that the CLEO measurement of y', (—5.8% < y' < 1.0%) and the FOCUS measurement of yep = (3.42 ± 1.39 ± 0.74)% are consistent with the earlier results from E791 (yCp = (0.8 ± 2.9 ± 1.0)%). Second, one recognizes that both CLEO and FOCUS observe mixing results of more than 1.5cr, but in opposite directions. The two measurements just barely overlap. Although the data from FOCUS and CLEO have been presented on a single plot, one should exercise caution as FOCUS makes a measurement oiycp while CLEO reports a limit on x' and y', not on x and y. In figure 11 all values are plotted as x or y. Recently, a paper 2 7 by S. Bergmann, et al. has attempted provide a model which encompasses all results. The authors mention that a large value for the strong phase angle difference between the DCS and the CF channels would reduce the discrepancy between the two experiments. While the new results are tantalizing, we need to be patient. CLEO should soon have a measurement for the lifetime difference A r . CLEO's sensitivity

261

DP~DP

n"! 5 I

#

,,,

Mixing Uniits

*

1

L

s

i©

is

K

«

—J

ao>

Figure 11: A plot of 95%CL limits for measurements of x and y mixing values.

should be comparable, if not better, to the uncertainty in the result from FOCUS. We also expect to soon hear about hadronic results from FOCUS. In addition, both FOCUS and CLEO collaborations have large semileptonic samples in which they can search for mixing. The sensitivity from these samples is expected to be on order 2% or better. Finally, we should recognize that BABAR and BELLE will soon have the largest charm samples. We should expect that their future results will help to clarify whether the current indication of mixing is being observed.If not, then we would hope that -these new experiments will push limits on yep to below

icr3.

262

Acknowledgments I would like to thank the TASI Program Co-Directors, the TASI Scientific Advisory Board, and the TASI Local Organizing Committee for an informative and productive school. I would also like to thank members of the CLEO, E791, and FOCUS collaborations who provided me with information and plots. I would like to extend a special thanks to Professor Paul Sheldon of Vanderbilt whose summary talk and proceedings 28 for the Heavy Flavours Conference at Southampton, UK, 1999 were invaluable references and from which I borrowed heavily, particularly in the introductory theory discussion of mixing in the charm sector. References 1. 2. 3. 4. 5.

6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2, 1285 (1970). A. Pich, Nucl. Phys. 66(Proc. Supl.) 456 (1998) [hep-ph/9709441]. L. Wolfenstein, Phys. Rev. Lett. 5 1 , 1945 (1983). A. Datta and D. Kumbhakar, Z. Phys. C 27, 515 (1985); H. Y. Cheng Phys. Rev. D 26, 143 (1982). L. Wolfenstein, Phys. Lett. B 164, 170 (1985); J. Donoghue, E. Golowich, B. R. Holstein, and J. Trampetic, Phys. Rev. D 33, 179 (1986). H. Georgi, Phys. Lett. B 297, 353 (1992) [hep-ph/9209291]; T. Ohl, G. Riccaiardi and E. Simmons, Nucl. Phys. B 403, 605 (1993) [hepph/9301212]. S. Pakvasa, Flavor Changing Neutral Currents in Charm Sector, [hepph/9705397]. A. J. Schwartz, Mod. Phys. Lett. A 8, 967 (1993); P. Singer and D. X. Zhang, Phys. Rev. D 55, 1127 (1997) [hep-ph/9612495]. See for example: J. L. Hewett, T. Takeuchi, and S. Thomas, Indirect Probes for New Physics, [hep-ph/9603391]. W. C. Louis, et al, E615 Collab., Phys. Rev. Lett. 56, 1027 (1986). J. C. Anjos, et al, E691 Collab., Phys. Rev. Lett. 60, 1239 (1988). E. M. Aitala, et al, E791 Collab., Phys. Rev. Lett. 77, 2384 (1996) [hep-ex/9606016]. E. M. Aitala, et al, E791 Collab., Phys. Rev. D 57, 13 (1998) [hepex/9608018]. R. Barate, et al, ALEPH Collab., Phys. Lett. B 436, 211 (1998) [hepex/9811021]. Y. Nir, Nuovo Cimento 109 A, 991 (1996) [hep-ph/9507290]. H.N. Nelson, [hep-ex/9908021].

263

17. P. L. Frabetti, et al, E687 Collab., Phys. Rev. D 50, 2953 (1994); J. Bartelt, et al, CLEO Collab., Phys. Rev. D 52, 4860 (1995); E. M. Aitala, et al, E791 Collab., Phys. Lett. B 403, 377 (1997) [hepex/9612005]. 18. G. Blaylock, A. Seiden, and Y. Nir, Phys. Lett. B 355, 555 (1995) [hep-ph/9504306]. 19. T.E. Browder and S. Pakvasa, Phys. Lett. B 383, 475 (1996) [hepph/9508362]. 20. L. Wolfenstein, Phys. Rev. Lett. 75, 2460 (1995) [hep-ph/9505285]. 21. M. Artuso, et al., CLEO Collab., Search for D°-D° Mixing, [hepex/9908040]. 22. D.Cinabro, et al., CLEO Collaboration Phys. Rev. Lett. 72, 1406 (1994) 23. E.M. Aitala, et al., E791 Collab., Phys. Rev. Lett. 83, 32 (1999) [hepex/9903012]. 24. CLEO Collaboration, talk by Craig Prescott at APS 2000 Meeting in Long Beach, CA, April 29, 2000. 25. P.L. Frabetti et al., E687 Collaboration, Nucl. Instrum. Methods A 320, 519 (1992). 26. J.M. Link, et al, FOCUS Collaboration, Phys. Lett. B 485, 62 (2000). 27. S. Bergmann, Y. Grossman, Z. Ligeti, Y. Nir, A.A. Petrov, FERMILABPub-00/102(May 2000), [hep-ph/0005181]. 28. P.Sheldon, Proceedings of the Heavy Flavours 8 Conference held in Southhampton, UK, 1999 [hep-ex/9912016].

This page is intentionally left blank

li|plil§iplll|pl

•iiiiii

illllB M

v Davison E. Soper

This page is intentionally left blank

B A S I C S OF Q C D P E R T U R B A T I O N

THEORY

DAVISON E. S O P E R Institute of Theoretical Science University of Oregon, Eugene, OR 97403 Email: [email protected] This is an introduction to the use of QCD perturbation theory, emphasizing generic features of the theory that enable one to separate short-time and long-time effects. I also cover some important classes of applications: electron-positron annihilation to hadrons, deeply inelastic scattering, and hard processes in hadron-hadron collisions.

1

Introduction

A prediction for experiment based on perturbative Q C D combines a particular calculation of Feynman diagrams with the use of general features of the theory. T h e particular calculation is easy at leading order, not so easy at next-toleading order and extremely difficult beyond the next-to-leading order. This calculation of Feynman diagrams would be a purely academic exercise if we did not use certain general features of the theory t h a t allow the Feynman diagrams to be related to experiment: the renormalization group and the running coupling; the existence of infrared safe observables; the factorization property t h a t allows us to isolate hadron structure in parton distribution functions. In these lectures, I discuss these structural features of the theory t h a t allow a comparison of theory and experiment. Along the way we will discover something about certain i m p o r t a n t processes: e + e ~ annihilation; deeply inelastic scattering; hard processes in hadron-hadron collisions. By discussing the particular along with the general, I hope to a r m the reader with information t h a t speakers at research conferences take to be collective knowledge knowledge t h a t they assume the audience already knows. Now here is the disclaimer. We will not learn how to do significant calculations in QCD perturbation theory. Three lectures is not enough for t h a t . I hope t h a t the reader may be inspired to pursue the subjects discussed here in more detail. A good source is the Handbook of Perturbative QCD1 by the C T E Q collaboration. More recently, Ellis, Stirling a n d Webber have written an excellent book 2 that covers the most of the subjects sketched in these lectures. For the reader wishing to gain a mastery of the theory, I can recommend the recent books on q u a n t u m field theory by Brown, 3 S t e r m a n , 4 267

268

Peskin and Schroeder, 5 and Weinberg. 6 Another good source, including b o t h theory and phenomenology, is the lectures in the 1995 TASI proceedings, QCD and Beyond.7 I have published a substantially similar set of lectures in the proceedings of the 1996 SLAC Summer school. 8 2

Electron-positron annihilation and jets

In this section, I explore the structure of the final state in Q C D . I begin with the kinematics of e+e~ —> 3 partons, then examine the behavior of the cross section for e + e ~ —> 3 partons when two of the parton m o m e n t a become collinear or one parton m o m e n t u m becomes soft. In order to illustrate better what is going on, I introduce a theoretical tool, null-plane coordinates. Using this tool, I sketch a space-time picture of the singularities t h a t we find in m o m e n t u m space. T h e singularities of perturbation theory correspond to long-time physics. We see t h a t the structure of the final state suggested by this picture conforms well with what is actually observed. I draw a the distinction between short-time physics, for which perturbation theory is useful, and long-time physics, for which the perturbative expansion is out of control. Finally, I discuss how certain experimental measurements can probe the short-time physics while avoiding sensitivity to the long-time physics. 2.1

Kinematics

of e+e~

—» 3 partons

/

V

Figure 1. Feynman diagram for e + e

—tqqg.

Consider the process e + e ~ —> qqg, as illustrated in Fig. 1. Let -y/s be the total energy in the c m . frame and let q^ be the virtual photon (or Z boson) m o m e n t u m , so q^q^ = s. Let pf be the m o m e n t a of the outgoing partons

269 {q,q,g) and let Ej = p° be the energies of the outgoing partons. It is useful to define energy fractions x; by Ei

2pi • q

V*/2

s

(1)

"

Then 0<xi.

(2)

Energy conservation gives

yXi=^CLPi)-9=2m *-^

(3)

s

i

T h u s only two of the x, are independent. Let 6{j be the angle between the m o m e n t a of partons i and j . We can relate these angles to the m o m e n t u m fractions as follows: 2pi • P2 = (pi + P2) 2 = (q ~ Pa) 2 = s - 2q • p 3 , 2£i£2(l-cos012) = s(l-z3).

(4) (5)

Dividing this equation by s / 2 and repeating the argument for the two other pairs of partons, we obtain three relations for the angles 0;J : £12:2(1 - cos#i 2 ) = 2(1 - £3), £22:3(1 -

cos

^ 2 3 ) = 2(1 -

xi),

z 3 zri(l - cos0 3 i) = 2(1 - 3:2).

(6)

We learn two things immediately. First, Xi < 1.

(7)

Second, the three possible collinear configurations of the partons are m a p p e d into Xi space very simply:

0 023 -> 0 031 -> 0 #12 —¥

<=>

X3

«•

XX

<=>

X2

-> 1, -> 1, -> 1.

(8)

T h e relations 0 < X{ < 1, together with £3 = 2 — X\ — X2, imply t h a t the allowed region for (x±, £2) is a triangle, as shown in Fig. 2. T h e edges a;,- = 1 of the allowed region correspond to two partons being collinear, as also shown in Fig. 2. T h e corners s; = 0 correspond to one parton m o m e n t u m being soft

(p?->0).

270

2 & 3 collinear 2 soft

soft

\ -Vis x

•

; o

1 & 2 \ . * • Ic& * collinear ' •ii O

X} e l x

soft —r—

2

Figure 2. Allowed region for (x\, £2). Then X3 is 2 — xi — X2- The labels and small pictures in the right hand diagram show the physical configuration of the three partons corresponding to subregions in the allowed triangle.

2.2

Structure

of the cross

section

One can easily calculate the cross section corresponding to Fig. 1 and the similar amplitude in which the gluon attaches to the antiquark line. T h e result is 1

da

X -t

\

X n

?±CF_ 2n (1 — « i ) ( l — £ 2 ) '


(9)

where CF = 4 / 3 and ao = (47ra 2 /s) £} Q2, is the total cross section for e + e " hadrons at order a ° . T h e cross section has collinear singularities: (l-£i)->0,

(2fe3 collinear);

(1 - x2) -> 0 ,

(1&3 collinear).

(10)

There is also a singularity when the gluon is soft: x3 —> 0. In terms of x\ and X2, this singularity occurs when (l-Xl)->0,

(l-*2)-»0,

(l-*2

const.

(11)

Let us write the cross section in a way t h a t displays the collinear singularity at 03i —> 0 and the soft singularity at E3 —y 0: 1

da

do dE3d cos 031

=

2n

^CF

f(E3,03 £3(1 — cos #31)

(12)

Here f(E3,03i) a rather complicated function. T h e only thing t h a t we need to know a b o u t it is t h a t it is finite for E3 —> 0 and for $31 —> 0.

271 Now look at the collinear singularity, #31 —> 0. If we integrate over the singular region holding E3 fixed we find t h a t the integral is divergent: d c o s 0 3 i -T7T-] W~ = dE3d cos #31

lo

g(°°)-

(13)

Similarly, if we integrate over the region of the soft singularity, holding #31 fixed, we find t h a t the integral is divergent:

[adE3

d

/

=log(oo).

(14)

J0 dE3d cos 03i Evidently, perturbation theory is telling us t h a t we should not take the perturbative cross section too literally. T h e total cross section for e+e~ —> hadrons is certainly finite, so this partial cross section cannot be infinite. W h a t we are seeing is a breakdown of perturbation theory in the soft and collinear regions, and we should understand why.

Figure 3. Cross section for e+e or collinear with the quark.

—± qqg, illustrating the singularity when the gluon is soft

Where do the singularities come from? Look at Fig. 3 (in a physical gauge). T h e scattering m a t r i x element M contains a factor l / ( p i + P3) where (Pi + P 3 ) 2 = 2pi -p3 = 2E1E3{1

- cos03i).

(15)

Evidently, l / ( p i + P3) is singular when 0 3 i —¥ 0 and when E3 —> 0. T h e collinear singularity is somewhat softened because the numerator of the Feynm a n diagram contains a factor proportional to #31 in the collinear limit. (This is not exactly obvious, but is easily seen by calculating. If you like s y m m e t r y arguments, you can derive this factor from quark helicity conservation and overall angular m o m e n t u m conservation.) We thus find t h a t |X|2oc

(16) i3f3l

272 for E3 —> 0 and $31 —> 0. Note the universal nature of these factors. Integration over the double singular region of the m o m e n t u m space for the gluon has the form

j •&**£>*<*„ J tu^v.

(17)

Combining the integration with the m a t r i x element squared gives da

I

2

E3dE3d6l1d(j)

fdE3d6i1

C31 .-^3^31 .

~ J -* W ^

(18)

T h u s we have a double logarithmic divergence in perturbation theory for the soft and collinear region. W i t h just a little enhancement of the argument, we see t h a t there is a collinear divergence from integration over #31 at finite £ 3 and a separate soft divergence from integration over E3 at finite #31. Essentially the same argument applies to more complicated graphs. There are divergences when two final state partons become collinear and when a final state gluon becomes soft. Generalizing further, 9 there are also divergences when several final state partons become collinear to one another or when several (with no net flavor q u a n t u m numbers) become soft. We have seen t h a t if we integrate over the singular region in m o m e n t u m space with no cutoff, we get infinity. T h e integrals are logarithmically divergent, so if we integrate with an infrared cutoff Mm, we will get big logarithms of MjR/s. T h u s the collinear and soft singularities represent perturbation theory out of control. Carrying on to higher orders of perturbation theory, one gets l + a s x (big) + a 2 x (big) 2 + • • •.

(19)

If this expansion is in powers of as(Mz), we have as
Interlude:

Null plane

coordinates

In order to understand better the issue of singularities, it is helpful to introduce a concept t h a t is generally quite useful in high energy q u a n t u m field theory, null plane coordinates. The idea is to describe the m o m e n t u m of a particle using m o m e n t u m components p^ = (p+, p~, p1, p 2 ) where

P± =

(p°±p3)/V2.

(20)

273

Figure 4. Null plane axes in momentum space.

For a particle with large m o m e n t u m in the +z direction and limited transverse m o m e n t u m , p+ is large and p" is small. Often one chooses the plus axis so t h a t a particle or group of particles of interest have large p+ and small p~ and pxUsing null plane components, the covariant square of p^ is p2 = 2p+p~ T h u s , for a particle on its mass shell, p

(21)

-p£. is

p|, + m2 2p+

P

(22)

Note also t h a t , for a particle on its mass shell, p+ > 0 ,

p~ > 0 .

(23)

Integration over the mass shell is (2T)

7

d3p (2<

2

2^fP + m

-3

f*"L

'dP+ 2p+

(24)

We also use the p l u s / m i n u s components to describe a space-time point x^: x± = (x° ± x3)/\/2. In describing a system of particles moving with large m o m e n t u m in the plus direction, we are invited to think of x+ as "time." Classically, the particles in our system follow paths nearly parallel to the x+ axis, evolving slowly as it moves from one x+ = const, plane to another. We relate m o m e n t u m space to position space for a q u a n t u m system by Fourier transforming. In doing so, we have a factor exp(ip • x), which has the form p •x = p

x

x

— px • x y .

(25)

274 T h u s x is conjugate to p + and x+ confusing, but it is simple enough. 2.4

Space-time

picture

of the

is conjugate to p .

T h a t is a little

singularities

Pi P3

Figure 5. Correspondence between singularities in momentum space and the development of the system in space-time.

We now return to the singularity structure of e"1 qqg. Defir Pi + large and kT = 0. Then P3 = k^. Choose null plane coordinates n t h k k2 = 2k+k~ becomes small when , _

..

-

2 P3,T

, 2p+

2 , P3,r

,

, 2p+

(26)

becomes small. This happens when P 3 , T becomes small with fixed p~{ a n d p j , so t h a t the gluon m o m e n t u m is nearly collinear with the quark m o m e n t u m . It also happens when P 3 ? T and p j both become small with p j oc | P 3 , T | , SO t h a t the gluon m o m e n t u m is soft. ( It also happens when the quark becomes soft, but there is a numerator factor t h a t cancels the soft quark singularity.) T h u s the singularities for a soft or collinear gluon correspond to small k~. Now consider the Fourier transform to coordinate space. T h e quark propagator in Fig. 5 is Mk)

= fdz+dX-dxeXp(i[k+*-+k-X+-*•*])

SP(

(27)

When k+ is large and k~ is small, the contributing values of x have small x~ and large x+. T h u s the propagation of the virtual quark can be pictured in space-time as in Fig. 5. T h e quark propagates a long distance in the x+ direction before decaying into a quark-gluon pair. T h a t is, the singularities t h a t can lead to divergent perturbative cross sections arise from interactions t h a t happen a long time after the creation of the initial quark-antiquark pair.

275 2.5

Nature of the long-time

physics

Figure 6. Typical paths of partons in space contributing to e"*"e~ -4 hadrons, as suggested by the singularities of perturbative diagrams. Short wavelength fields are represented by classical paths of particles. Long wavelength fields are represented by wavy lines.

Imagine dividing the contributions to a scattering cross section into long-time contributions and short-time contributions. In the long-time contributions, perturbation theory is out of control, as indicated in Eq. (19). Nevertheless the generic structure of the long-time contribution is of great interest. This structure is illustrated in Fig. 6. Perturbative diagrams have big contributions from space-time histories in which partons move in collinear groups and additional partons are soft and communicate over large distances, while carrying small m o m e n t u m . T h e picture of Fig. 6 is suggested by the singularity structure of diagrams at any fixed order of perturbation theory. Of course, there could be nonperturbative effects t h a t would invalidate the picture. Since nonperturbative effects can be invisible in perturbation theory, one cannot claim t h a t the structure of the final state indicated in Fig. 6 is known to be a consequence of Q C D . One can point, however, to some cases in which one can go beyond fixed order perturbation theory and sum the most i m p o r t a n t effects of diagrams of all orders (for example, Ref. [ 10 ]). In such cases, the general picture suggested by Fig.. 6 remains intact. We thus find t h a t perturbative Q C D suggests a certain structure of the final state produced in e + e ~ —¥ hadrons: the final state should consist of jets of nearly collinear particles plus soft particles moving in r a n d o m directions. In fact, this qualitative prediction is a qualitative success. Given some degree of qualitative success, we may be bolder and ask whether perturbative Q C D permits quantitative predictions. If we want quantitative predictions, we will somehow have to find things to measure t h a t are

276 not sensitive to interactions t h a t happen long after the basic hard interaction. This is the subject of the next section.

2.6

The long-time

problem

We have seen t h a t perturbation theory is not effective for long-time physics. But the detector is a long distance away from the interaction, so i t would seem t h a t long-time physics has to be present. Fortunately, there are some measurements t h a t are not sensitive to longtime physics. An example is the total cross section to produce hadrons in e + e ~ annihilation. Here effects from times At 3> 1/y/s cancel because of unitarity. To see why, note t h a t the quark state is created from the vacuum by a current operator J at some time t; it then develops from time t to time oo according to the interaction picture evolution operator [/(oo,f), when it becomes the final state \N). T h e cross section is proportional to the sum over N of this amplitude times a similar complex conjugate amplitude with t replaced by a different time t1. We Fourier transform this with exp(—iy/s (t — t')), so t h a t we can take At = t — t' to be of order l/y/s. Now replacing ^2 \N)(N\ by the unit operator and using the unitarity of the evolution operators U, we obtain Y,{WW,oo)\N)(N\U{oo,t)J(t)\0)

(28)

N

= (0\J(t')U{t'

,oo)U{oo,t)J{t)\0)

= (0\J(t')U{t'

,t)J(t)\0).

Because of unitarity, the long-time evolution has canceled out of the cross section, and we have only evolution from t to t'. There are three ways to view this result. First, we have the formal argument given above. Second, we have the intuitive understanding t h a t after the initial quarks and gluons are created in a time At of order l/y/s, something will happen with probability 1. Exactly what happens is long-time physics, but we d o n ' t care a b o u t it since we sum over all the possibilities \N). Third, we can calculate a t some finite order of perturbation theory. Then we see infrared infinities at various stages of the calculations, but we find t h a t the infinities cancel between real gluon emission graphs and virtual gluon graphs. An example is shown in Fig. 7. We see t h a t the total cross section is free of sensitivity t o long-time physics. If the total cross section were all you could look at, Q C D physics would be a little boring. Fortunately, there are other quantities t h a t are not sensitive to infrared effects. They are called infrared safe quantities. To formulate the concept of infrared safety, consider a measured quantity

277

Figure 7. Cancellation between real and virtual gluon graphs. If we integrate the real gluon graph on the left times the complex conjugate of the similar graph with the gluon attached to the antiquark, we will get an infrared infinity. However the virtual gluon graph on the right times the complex conjugate of the Born graph is also divergent, as is the Born graph times the complex conjugate of the virtual gluon graph. Adding everything together, the infrared infinities cancel.

t h a t is constructed from the cross sections, -

^

,

(29)

to make n hadrons in e + e ~ annihilation. Here Ej is the energy of the j t h hadron and Qj = (&j,(j>j) describes its direction. We treat the hadrons as effectively massless and do not distinguish the hadron flavors. Following the notation of Ref. [ u ] , let us specify functions Sn t h a t describe the measurement we want, so t h a t the measured quantity is X

dn

=2! I

+ 3! 1 +

4!

da[2] 2(Pl,P2)

dn2

d£l2dE3dtt3

da[3] dQ,2dE3d£l3

Sz{pt,P2,P3)

dQ2dE3d£l3dE4d£l4 da[4] dQ,2dE3dQ3dE4dil4

SI{PI,P2-,P3,PA)

(30)

+ •

The functions S are symmetric functions of their arguments. In order for our measurement to be infrared safe, we need ?n + l ( p i ' ) . . . , ( l - A ) p £ I A p £ )

for 0 < A < 1.

= S„(p$',. - P d )

(31)

278

Y Figure 8. Infrared safety. In an infrared safe measurement, the three jet event shown on the left should be (approximately) equivalent to an ideal three jet event shown on the right.

W h a t does this mean? T h e physical meaning is t h a t the functions S„ and <Sn_i are related in such a way t h a t the cross section is not sensitive to whether or not a mother particle divides into two collinear daughter particles t h a t share its m o m e n t u m . T h e cross section is also not sensitive to whether or not a mother particle decays to a daughter particle carrying all of its m o m e n t u m and a soft daughter particle carrying no m o m e n t u m . T h e cross section is also not sensitive to whether or not two collinear particles combine, or a soft particle is absorbed by a fast particle. All of these decay and recombination processes can h a p p e n with large probability in the final state long after the hard interaction. But, by construction, they don't m a t t e r as long as the sum of the probabilities for something to happen or not to happen is one. Another version of the physical meaning is t h a t for an IR-safe quantity a physical event with hadron jets should give approximately the same measurement as a parton event with each jet replaced by a parton, as illustrated in Fig. 8. To see this, we simply have to delete soft particles and combine collinear particles until three jets have become three particles. In a calculation of the measured quantity 1, we simply calculate with partons instead of hadrons in the final state. The calculational meaning of the infrared safety condition is t h a t the infrared infinities cancel. T h e argument is t h a t the infinities arise from soft and collinear configurations of the partons, t h a t these configurations involve long times, and t h a t the time evolution operator is unitary. I have started with an abstract formulation of infrared safety. It would be good to have a few examples. The easiest is the total cross section, for which

< S n ( K , . . . , p £ ) = l.

(32)

A less trivial example is the thrust distribution. One defines the thrust Tn of

279

an n particle event as r„K,---,P^)=maxEj=nll^:fl . (33) L,i = l\Pi\ Here u is a unit vector, which we vary to maximize the sum of the absolute values of the projections of pi on u. Then the thrust distribution (l/atot) da/dT is defined by taking Sn(rf,...,rt)

= (l/
•

(34)

It is a simple exercise to show that the thrust of an event is not affected by collinear parton splitting or by zero momentum partons. Therefore the thrust distribution is infrared safe. Another example is the energy-energy correlation function d12/dcos(6)12: Sn(p1, . . . ,p£) = £ ^ « $ (cos(0y) - cos(0)) . *—' s J

(35)

J

This measures the correlation between the energies measured by detectors separated by an angle 9 as depicted in Fig. 9. Is this infrared safe? Note that the contribution from a particle with E{ —> 0 drops out. In addition, replacing one particle by two collinear particles doesn't change the thrust: (1 - A) En Ej +XEn Ej = En Ej.

(36)

This works for the autocorrelation term too: (1 - A) 2 El + 2A(1 -X)E2n

+ A2 E2n = E2n.

(37)

%

Figure 9. The energy-energy correlation function

A final example is the cross section to make n jets, crn. Intuitively, a jet is supposed to be a spray of particles all going in approximately the same

280 direction. To make this precise, we need a definite algorithm. There are several algorithms to choose from. Here is the simplest (but not the best) one.

Figure 10. Jet definition

Start with a list of m o m e n t a Pi,p%, ••• ,p%- At the start, these represent the m o m e n t a of particles. (In a perturbative calculation, they are the m o m e n t a of partons.) Choose a parameter j / c u t . Now proceed through the following steps: 1. Find the pair (i,j) such t h a t (pi + Pj)2 is the smallest. 2. If (pi + pj)2 > ycut s, exit. Else continue. 3. Replace the two m o m e n t a pi and pj in the list by their sum p£ = pf+pf • 4. Go to 1. This produces a list of m o m e n t a pi of jets. n + X) dEn This cross section can be written as a convolution of two factors. T h e first factor is a calculated "hard scattering cross section" for e+e~ —¥ quark + X or e + e ~ —>• gluon + X. T h e second factor is a "parton decay function" for quark —> 7r -f X or gluon —> ir + X. These functions contain the long-time sensitivity and are to be measured, since they cannot be calculated perturbatively. However, once they are measured in one process, they can be used for another process. This final state factorization is similar to the initial state

281 factorization involving parton distribution functions, which we will discuss later. (See Refs. f 1 ]^ 2 ]^ 1 4 ] for more information.) 3

T h e smallest t i m e scales

In this section, I explore the physics of time scales smaller t h a n 1/\fs. One way of looking at this physics is to say t h a t it is plagued by infinities and we can m a n a g e to hide the infinities. A better view is t h a t the short-time physics contains wonderful truths t h a t we would like to discover - t r u t h s a b o u t grand unified theories, q u a n t u m gravity and the like. However, q u a n t u m field theory is arranged so as to effectively hide the t r u t h from our experimental a p p a r a t u s , which can probe with a time resolution of only an inverse half TeV. I first outline what renormalization does to hide the ugly infinities or the beautiful t r u t h . Then I describe how renormalization leads to the running coupling. Because of renormalization, calculated quantities depend on a renormalization scale. I look at how this dependence works and how the scale can be chosen. Finally, I discuss how one can use experiment to look for the hidden physics beyond the Standard Model, taking high Ej jet production in hadron collisions as an example. 3.1

What renormalization

does

In any Feynman graph, one can insert perturbative corrections to the vertices and the propagation of particles, as illustrated in Fig. 11. T h e loop integrals in these graphs will get big contributions from m o m e n t a much larger t h a n \/s. T h a t is, there are big contributions from interactions t h a t happen on time scales much smaller t h a n l/y/s. I have tried to illustrate this in the figure. T h e virtual vector boson propagates for a time \/y/s, while the virtual fluctuations t h a t correct the electroweak vertex and the quark propagator occur over a time At t h a t can be much smaller than l/\/s. Let us pick an ultraviolet cutoff M t h a t is much larger t h a n -^/s, so t h a t we calculate the effect of fluctuations with 1/M < At exactly, up to some order of perturbation theory. W h a t , then, is the effect of virtual fluctuations on smaller time scales, At with At < 1/M but, say, At still larger t h a n ^Pianki where gravity takes over? Let us suppose t h a t we are willing to neglect contributions to the cross section t h a t are of order s/s/M or smaller compared to the cross section itself. Then there is a remarkable theorem 1 5 : the effects of the fluctuations are not particularly small, but they can be absorbed into changes in the couplings of the theory. (There are also changes in the masses of the theory and adjustments to the normalizations of the field operators,

282 but we can concentrate on the effect on the couplings.)

Figure 11. Renormalization. The effect of the very small time interactions pictured are absorbed into the running coupling.

T h e p r o g r a m of absorbing very short-time physics into a few parameters goes under the n a m e of renormalization. There are several schemes available for renormalizing. Each of them involves the introduction of some scale parameter t h a t is not intrinsic to the theory but tells how we did the renormalization. Let us agree to use MS renormalization (see Ref. [15] for details). Then we introduce an MS renormalization scale /J,. A good (but approximate) way of thinking of // is t h a t the physics of time scales At ^ l / / i is removed from the perturbative calculation. T h e effect of the small time physics is accounted for by adjusting the value of the strong coupling, so t h a t its value depends on the scale t h a t we used: as — as{fi). (The value of the electromagnetic coupling also depends on //.) 3.2

The running

coupling

Figure 12. Short-time fluctuations in the propagation of the gluon field absorbed into the running strong coupling.

We account for time scales much smaller than 1/// by using the running coupling as(fi). T h a t is, a fluctuation such as t h a t illustrated in Fig. 12 can

283

be dropped from a calculation and absorbed into the running coupling that describes the probability for the quark in the figure to emit the gluon. The fj, dependence of as(/i) is given by a certain differential equation, called the renormalization group equation (see Ref. [15]): d

Qfs

^)

dM^)~^r

at t \\ = p{asM) =

a f^litlY

R f^AtlY ft

^°\~^) " l~J

_i_ +

—


One calculates the beta function f3(as) perturbatively in QCD. The first coefficient, with the conventions used here, is A> = (33 -2Nf)/l2, (40) where Nf is the number of quark flavors. Of course, at time scales smaller than a very small cutoff 1/M (at the "GUT scale," say) there is completely different physics operating. Therefore, if we use just QCD to adjust the strong coupling, we can say that we are accounting for the physics between times 1/M and 1/fi. The value of as at Ho w M is then the boundary condition for the differential equation. See Fig. 13. fixed order ??

log(l/M)

log(l/n)

l°g(^*)

Figure 13. Time scales accounted for by explicit fixed order perturbative calculation and by use of the renormalization group.

The renormalization group equation sums the effects of short-time fluctuations of the fields. To see what one means by "sums" here, consider the result of solving the renormalization group equation with all of the /?,• beyond fto set to zero: a,(ri

« a,(M) - (A,/TT) ln(p 2 /M 2 ) a2s(M) + ((3oM2 In2 ( / i 2 / M 2 ) a ? ( M ) + --=

^

.

(41)

A series in powers of as(M) - that is the strong coupling at the GUT scale is summed into a simple function of fi. Here as(M) appears as a parameter in the solution. Note a crucial and wonderful fact. The value of as(fi) decreases as // increases. This is called "asymptotic freedom." Asymptotic freedom implies

284

t h a t Q C D acts like a weakly interacting theory on short time scales. It is true t h a t quarks and gluons are strongly bound inside nucleons, but this strong binding is the result of weak forces acting collectively over a long time. In Eq. (41), we are invited to think of the graph of as(/j,) versus \i. T h e differential equation t h a t determines this graph is characteristic of Q C D . There could, however, be different versions of QCD with the same differential equation but different curves, corresponding to different boundary values as(M). T h u s the parameter as(M) tells us which version of QCD we have. To determine this parameter, we consult experiment. Actually, Eq. (41) is not the most convenient way to write the solution for the running coupling. A better expression is

a M

' "ftlnJW

^

Here we have replaced as ( M ) by a different (but completely equivalent) parameter A. A third form of the running coupling is , ^

W *

<x,(Mz)

l + (f30/n)as(Mz)ln(^/M^y

,

.

(

^>

Here the value of as(p) at fi = Mz labels the version of Q C D t h a t obtains in our world. In any of the three forms of the running coupling, one should revise the equations to account for the second term in the beta function in order to be numerically precise. 3.3

The choice of scale

In this section, we consider the choice of the renormalization scale \i in a calculated cross section. Consider, as an example, the cross section for e + e ~ —>• hadrons via virtual photon decay. Let us write this cross section in the form

Here s is the square of the c m . energy, a is e 2 /(47r), and Qj is the electric charge in units of e carried by the quark of flavor / , with / — u, d, s, c, b. T h e nontrivial part of the calculated cross section is the quantity A, which contains the effects of the strong interactions. Using MS renormalization with

285 scale /i, one finds (after a lot of work) t h a t A is given by Ref. [ 16 ]:

A

=

^M

+ [ 1 . 4 0 9 2 + 1 . 9 1 6 7 In (M2/s)}

( ^ Y

+ [-12.805 + 7.8186 In (>a2/s) + 3.674 ln2(^/s)]

(SiAtTj (45)

Here, of course, one should use for as (fi) the solution of the renormalization group equation (39) with at least two terms included. As discussed in the preceding subsection, when we renormalize with scale fi, we are defining what we mean by the strong coupling. T h u s as in Eq. (45) depends on fi. T h e perturbative coefficients in Eq. (45) also depend on \±. On the other hand, the physical cross section does not depend on //: /

T

A

= 0.

(46)

T h a t is because [i is just an artifact of how we organize perturbation theory, not a parameter of the underlying theory. Let us consider Eq. (46) in more detail. Write A in the form oo

A ~ £ c

n

M a , M " .

(47)

n= l

If we differentiate not the complete infinite sum but just the first TV terms, we get minus the derivative of the sum from TV + 1 to infinity. This remainder is of order ct^"1"1 as as —> 0. T h u s N c

d n

2J2

^

n(p)

a . M " ~0(a,(n)N+1).

(48)

n= l

T h a t is, the harder we work calculating more terms, the less the calculated cross section depends on fi. Since we have not worked infinitely hard, the calculated cross section depends on /i. W h a t choice shall we make for fil Clearly, In (// 2 /s) should not be big. Otherwise the coefficients cn(fi) are large and the "convergence" of perturbation theory will be spoiled. There are some who will argue t h a t one scheme or the other for choosing ji is the "best." You are welcome to follow whichever advisor you want. I will show you below t h a t for a well behaved quantity like A the precise choice makes little difference, as long as you obey the c o m m o n sense prescription t h a t In (// 2 /s) not be big.

286

3.4

An example

0.06 0.05

0.03 0.02 0.01

° -T

-2

'~

-1

"6

1

2

Figure 14. Dependence of A ( / J ) on the MS renormalization scale [i. The falling curve is A j . The flatter curve is A2. The horizontal lines indicates the amount of variation of A2 when /J, varies by a factor 2.

Let us consider a quantitative example of how A(/i) depends on fi. This will also give us a chance to think about the theoretical error caused by replacing A by the sum A n of the first n terms in its perturbative expansion. Of course, we do not know what this error is. All we can do is provide an estimate. (Our discussion will be rather primitive. For a more detailed error estimate for the case of the hadronic width of the Z boson, see Ref. [17].) Let us think of the error estimate in the spirit of a "1 <x" theoretical error: we would be surprised if | A„ — A| were much less than the error estimate and we would also be surprised if this quantity were much more than the error estimate. Here, one should exercise a little caution. We have no reason to expect that theory errors are gaussian distributed. Thus a 4er difference between A n and A is not out of the question, while a 4 a fluctuation in a measured quantity with purely statistical, gaussian errors is out of the question. Take as{Mz) = 0.117, ^/i = 34 GeV, 5 flavors. In Fig. 14, I plot A(p) versus p defined by H = 2pJ~s.

(49)

The steeply falling curve is the order a) approximation to A(^), Ai(/i) = as(fj,)/7T. Notice that if we change y, by a factor 2, Ai(//) changes by about 0.006. If we had no other information than this, we might pick Ax (y/s) w 0.044 as the "best" value and assign a ±0.006 error to this value. (There is no special

287

magic to the use of a factor of 2 here. The reader can pick any factor that seems reasonable.) Another error estimate can be based on the simple expectation that the coefficients of a" are of order 1 for the first few terms. (Eventually, they will grow like n\. Ref. [17] takes this into account, but we ignore it here.) Then the first omitted term should be of order ± 1 x a2 PS ±0.020 using a s (34 GeV) m 0.14. Since this is bigger than the previous ±0.006 error estimate, we keep this larger estimate: A RS 0.044 ± 0.020. Returning now to Fig. 14, the second curve is the order a2 approximation, A2(/i). Note that A2(p) is less dependent on p. than Ai(/i). What value would we now take as our best estimate of A? One idea is to choose the value of p, at which k-iiji) is least sensitive to p,. This idea is called the principle of minimal sensitivity18: A P M S — A(^PMS') ,

dA(p) d\np,

0.

(50)

This prescription gives A £S 0.0470. Note that this is about 0.003 away from our previous estimate, A RJ 0.0440. Thus our previous error estimate of 0.020 was too big, and we should be surprised that the result changed so little. We can make a new error estimate by noting that A2(/i) varies by about 0.0012 when p changes by a factor 2 from PPMS- Thus we might estimate that A PS 0.0470 with an error of ±0.0012. This estimate is represented by the two horizontal lines in Fig. 14. An alternative error estimate can be based on the next term being of order ±1 xaj(34 GeV) RS 0.003. Since this is bigger than the previous ±0.0012 error estimate, we keep this larger estimate: A & 0.0470 ± 0.003. I should emphasize that there are other ways to pick the "best" value for A. For instance, one can use the BLM method, 19 which is based on choosing the fj, that sets to zero the coefficient of the number of quark flavors in A2(p). Since the graph of A2(/i) is quite fiat, it makes very little difference which method one uses. Now let us look at A(p) evaluated at order a,, Az(p). Here we make use of the full formula in Eq. (45). In Fig. 15, I plot A^(p) along with A 2 (^) and Ai(/i). The variation of Aa(/t) with \i is smaller than that of A2(/^). The improvement is not overwhelming, but is apparent particularly at small p. It is a little difficult to see what is happening in the first graph of Fig. 15, so I show the same thing with an expanded scale. (Here the error band based on the p dependence of A 2 is also indicated. Recall that we decided that this error band was an underestimate.) The curve for As(p) has zero derivative at two places. The corresponding values are A PS 0.0436 and A m 0.0456. If

288

log2(/i/\/s)

log2{fl/y/s)

Figure 15. Dependence of A(fJ.) on the MS renormalization scale n, first with a normal scale and then with an expanded scale. The falling curve is Aj . The flatter curve is A2. The still flatter curve is A3. The horizontal lines represent the variation of A2 when /J, varies by a factor 2.

I take the best value of A to be the average of these two values and the error to be half the difference, I get A Pd 0.0446 ± 0.0010. T h e alternative error estimate is ± 1 x 0^(34 GeV) f» 0.0004. We keep the larger error estimate of ±0.0010. Was the previous error estimate valid? We guessed A ?a 0.0470 ± 0.003. Our new best estimate is 0.0446. The difference is 0.0024, which is in line with our previous error estimate. Had we used the error estimate ±0.0012 based on the fi dependence, we would have underestimated the difference, although we would not have been too far off.

3.5

Beyond

the Standard

Model

We have seen how the renormalization group enables us to account for QCD physics at time scales much smaller than -*/s, as indicated in Fig. 13. However, a t some scale At ~ 1/M, we run into the unknown! How can we see the unknown in current experiments? First, the unknown physics affects a , , aem, sm2(0w)- Second, the unknown physics affects masses of u, d,..., e, fi,.... T h a t is, the unknown physics (presumably) determines the parameters of the Standard Model. These parameters have been well measured. Thus, a Nobel prize awaits the physicist who figures out how to use a model for the unknown physics to predict these parameters. There is another way t h a t as yet unknown physics can affect current experiments. Suppose t h a t quarks can scatter by the exchange of some new particle with a heavy mass M, as illustrated in Fig. 16, and suppose t h a t this mass is not too enormous, only a few TeV. Perhaps the new particle isn't

289

Figure 16. New physics at a TeV scale. In the first diagram, quarks scatter by gluon exchange. In the second diagram, the quarks exchange a new object with a TeV mass, or perhaps exchange some of the constituents out of which quarks are made.

a particle at all, but is a pair of constituents t h a t live inside of quarks. As mentioned above, this physics affects the parameters of the S t a n d a r d Model. However, unless we can predict the parameters of the S t a n d a r d Model, this effect does not help us. There is, however, another possible clue. T h e physics at the TeV scale can introduce new terms into the lagrangian t h a t we can investigate in current experiments. In the second diagram in Fig. 16, the two vertices are never at a separation in time greater t h a n 1/M, so t h a t our low energy probes cannot resolve the details of the structure. As long as we stick to low energy probes, y/s -C M, the effect of the new physics can be summarized by adding new terms to the lagrangian of Q C D . A typical term might be AC=

AP

' ^

^ '

(51)

2

There is a factor g t h a t represents how well the new physics couples to quarks. T h e most i m p o r t a n t factor is the factor 1/M2. This factor m u s t be there: the product of field operators has dimension 6 and the lagrangian has dimension 4, so there must be a factor with dimension —2. Taking this argument one step further, the product of field operators in A £ must have a dimension greater t h a n 4 because any product of field operators having dimension equal to or less t h a n 4 t h a t respects the symmetries of the S t a n d a r d Model is already included in the lagrangian of the S t a n d a r d Model. 3.6

Looking for new terms in the effective

lagrangian

How can one detect the presence in the lagrangian of a term like t h a t in Eq. (51)? These terms are small. Therefore we need either a high precision

290

experiment, or an experiment t h a t looks for some effect t h a t is forbidden in the S t a n d a r d Model, or an experiment t h a t has moderate precision and operates a t energies t h a t are as high as possible. Let us consider an example of the last of these possibilities, p+p —> jet+X as a function of the transverse energy (~ PT) of the jet. T h e new term in the lagrangian should add a little bit to the observed cross section t h a t is not included in the s t a n d a r d Q C D theory. When the transverse energy ET of the jet is small compared to M , we expect D a t a — Theory Theory *

9

_9 E% W-

<52>

Here the factor g2/M2 follows because AC contains this factor. T h e factor ET follows because the left hand side is dimensionless and ET is the only factor with dimension of mass t h a t is available.

• CTEQ3M CDF (Preliminary)* 1.03 DO (Preliminary) * 1.01

E, (GeV)

Figure 17. Jet cross sections from CDF and DO compared to QCD theory. (Data — Theory)/Theory is plotted versus the transverse energy Ej- of the jet. The theory here is next-to-leading order QCD using the CTEQ3M parton distribution. Source: Ref. [20]

In Fig. 17, I show a plot comparing experimental jet cross sections from C D F 2 1 and DO 22 compared to next-to-leading order QCD theory. T h e theory works fine for ET < 200 GeV, but for 200 GeV < ET, there appears to be a systematic deviation of just the form anticipated in Eq. (52). This example illustrates the idea of how small distance physics beyond the Standard Model can leave a trace in the form of small additional terms in the effective lagrangian t h a t controls physics at currently available energies. However, in this case, there is some indication t h a t the observed effect might

291 be explained by some combination of the experimental systematic error and the uncertainties inherent in the theoretical prediction. 2 3 In particular, the prediction is sensitive to the distributions of quarks and gluons contained in the colliding protons, and the gluon distribution in the kinematic range of interest here is rather poorly known. In the next section, we turn to the definition, use, and measurement of the distributions of quarks and gluons in hadrons.

4

Deeply inelastic scattering

Until now, I have concentrated on hard scattering processes with leptons in the initial state. For such processes, we have seen t h a t the hard part of the process can be described using perturbation theory because as{fi) gets small as /i gets large. Furthermore, we have seen how to isolate the hard part of the interaction by choosing an infrared safe observable. But what about hard processes in which there are hadrons in the initial state? Since the fundamental hard interactions involve quarks and gluons, the theoretical description necessarily involves a description of how the quarks and gluons are distributed in a hadron. Unfortunately, the distribution of quarks and gluons in a hadron is controlled by long-time physics. We cannot calculate the relevant distribution functions perturbatively (although a calculation in lattice Q C D might give them, in principle). T h u s we must find how to separate the short-time physics from the parton distribution functions and we must learn how the parton distribution functions can be determined from the experimental measurements. In this section, I discuss parton distribution functions and their role in deeply inelastic lepton scattering (DIS). This includes e + p —»• e + X and v + p —> e 4- X where the m o m e n t u m transfer from the lepton is large. I first outline the kinematics of deeply inelastic scattering and define the structure functions i*\, F2 and F3 used to describe the process. By examining the spacetime structure of DIS, we will see how the cross section can be written as a convolution of two factors, one of which is the parton distribution functions and the other of which is a cross section for the lepton to scatter from a quark or gluon. This factorization involves a scale fip t h a t , roughly speaking, divides the soft from the hard regime; I discuss the dependence of the calculated cross section on /J.p. W i t h this groundwork laid, I give the MS definition of parton distribution functions in terms of field operators and discuss the evolution equation for the parton distributions. I close the section with some comments on how the parton distributions are, in practice, determined from experiment.

292

4-1

Kinematics

of deeply inelastic

lepton

scattering

P Figure 18. Kinematics of deeply inelastic scattering

In deeply inelastic scattering, a lepton with m o m e n t u m k^ scatters on a hadron with m o m e n t u m p^. In the final state, one observes the scattered lepton with m o m e n t u m k'^ as illustrated in Fig. 18. T h e m o m e n t u m transfer q" = A" - k'"

(53)

is carried on a photon, or a W or Z boson. The interaction between the vector boson and the hadron depends on the variables q^ and p M . From these two vectors we can build two scalars (not counting m2 = p2). T h e first variable is Q2 = -q\

(54)

where the minus sign is included so t h a t Q2 is positive. The second scalar is the dimensionless Bjorken variable,

xhi = £ _ .

(55)

(In the case of scattering from a nucleus containing A nucleons, one replaces pf by pV/A and defines *bj = AQ2/(2p • q).) One calls the scattering deeply inelastic if Q2 is large compared to 1 GeV . Traditionally, one speaks of the scaling limit, Q2 —> oo with £bj fixed. Actually, the asymptotic theory to be described below works pretty well if Q2 is bigger t h a n , say, 4 GeV and *bj is anywhere in the experimentally accessible range, roughly 1 0 - 4 < Zbj < 0.5.

293 T h e invariant mass squared of the hadronic final state is W2 = (p + q)2• In the scaling regime of large Q2 one has

W2=m2 + i ^ i

Q2 » m2.

(56)

This justifies saying t h a t the scattering is not only inelastic but deeply inelastic. We have spoken of the scalar variables t h a t one can form from p^ and q^. Using the lepton m o m e n t u m k^, one can also form the dimensionless variable y=

p~k-

4-2

Structure

functions

(5?)

for DIS

One can make quite a lot of progress in understanding the theory of deeply inelastic scattering without knowing anything about Q C D except its s y m m e tries. One expresses the cross section in terms of three structure functions, which are functions of x\,j and Q2 only. Suppose t h a t the initial lepton is a neutrino, v^, and the final lepton is a muon. Then in Fig. 18 the exchanged vector boson, call it V, is a W boson, with mass My — Mw • Alternatively, suppose t h a t b o t h the initial and final leptons are electrons and let the exchanged vector boson be a photon, with mass My = 0. This was the situation in the original DIS experiments at SLAG in the late 1960's. In experiments with sufficiently large Q2, Z boson exchange should be considered along with photon exchange, and the formalism described below must be augmented. Given only the electroweak theory to tell us how the vector boson couples to the lepton, one can write the cross section in the form

^=^w\w^L*VMW^9)>

(58)

where C'v is 1 in the case t h a t V is a photon and 1/(64 sm4 9w) in the case t h a t V is a W boson. T h e tensor L^ describes the lepton coupling to the vector boson and has the form !"" = iTr(fc-77"fe'-77y)

(59)

in the case t h a t V is a photon. For a W boson, one has I"" = T r ( k - 7 r"fc'-7r"),

(60)

294

where 1^ is 7^(1 - 75) for a W+ boson {v -> W+£) or 7^(1 + 75) for a W~ boson (P -> VF-f). See Ref. [*]. The tensor W1"' describes the coupling of the vector boson to the hadronic system. It depends on pM and q^. We know that it is a Lorentz tensor and that Wv,i — W'"'*. We also know that the current to which the vector boson couples is conserved (or in the case of the axial current, conserved in the absence of quark masses, which we here neglect) so that qliWf11' = 0. Using these properties, one finds three possible tensor structures for W^. Each of the three tensors multiplies a structure function, Fi, -F2 or F3, which, since it is a Lorentz scalar, can depend only on the invariants x^] and Q2. Thus

w,v

= - (fiV

Fi(xhhQ2)

^Tj

+ [P, - g „ y J [PV - lo^jjr) —

F

^bhQ2)

-ie^XaPXq"

F3(xhj,Q2). (61) VI If we combine Eqs. (58,59,60,61), we can write the cross section for deeply inelastic scattering in terms of the three structure functions. Neglecting the hadron mass compared to Q2, the result is dCT

dxhi dy

=N(Q2

]

yF1+

(62)

-^-F2+Sv(l-^)F3

Here the normalization factor N and the factor &v multiplying F3 are 4na2 N = ——, Qz N=

j

na202 2

4ain*{0w) [Q + N=

-.

6V=0,

e~+h->e-

,

Sv = l,

v+h->

,

Sv = -l,

v+h^v+

uT

+X, +X,

2

Mw)

--^

+ X.

(63)

In principle, one can use the y dependence to determine all three of Fi, F2, F$ in a deeply inelastic scattering experiment. 4-3

Space-time structure of DIS

So far, we have used the symmetries of QCD in order to write the cross section for deeply inelastic scattering in terms of three structure functions, but we

295

Figure 19. Reference frame for the analysis of deeply inelastic scattering.

have not used any other dynamical properties of the theory. Now we turn t o the question of how the scattering develops in space and time. For this purpose, we define a convenient reference frame, which is illustrated in Fig. 19. Denoting components of vectors vM by (v+, v~, v y ) , we chose the frame in which (g+,«r,q) = ^ = ( - Q , Q , 0 ) .

(64)

We also d e m a n d t h a t the transverse components of the hadron m o m e n t u m be zero in our frame. Then +

-

\

l

i Q

V2v*bj'

x

bimh

Q

n

\

'"''

(65)

Notice t h a t in the chosen reference frame the hadron m o m e n t u m is big and the m o m e n t u m transfer is big.

Figure 20. Interactions within a fast moving hadron. The lines represent world lines of quarks and gluons. The interaction points are spread out in a:* and pushed together in x~.

296 Consider the interactions among the quarks and gluons inside a hadron, using x+ in the role of "time" as in Section 2.3. For a hadron at rest, these interactions happen in a typical time scale Ax+ ~ 1/m, where m ~ 300 MeV. A hadron t h a t will participate in a deeply inelastic scattering event has a large m o m e n t u m , p+ ~ Q, in the reference frame t h a t we are using. T h e Lorentz transformation from the rest frame spreads out interactions by a factor Q/m, so t h a t

Ax+~Ix^ =i

(66)

This is illustrated in Fig. 20. I offer two caveats here. First, I a m treating Zbj as being of order 1. To treat small £bj physics, one needs to put back the factors of £bj, and the picture changes rather dramatically. Second, the interactions among the quarks and gluons in a hadron at rest can take place on time scales Ax+ t h a t are much smaller than 1/m, as we discussed in Section 3. We will discuss this later on, but for now we start with the simplest picture.

Figure 21. The virtual photon meets the fast moving hadron. One of the partons is annihilated and recreated as a parton with a large minus component of momentum. This parton develops Into a jet of particles.

W h a t happens when the fast moving hadron meets the virtual photon? T h e interaction with the photon carrying m o m e n t u m q~~ ~ Q is localized to within Ax+ ~

1/Q.

(67)

297

During this short time interval, the quarks and gluons in the proton are effectively free, since their typical interaction times are comparatively much longer. We thus have the following picture. At the m o m e n t x+ of the interaction, the hadron effectively consists of a collection of quarks and gluons (partons) t h a t have m o m e n t a (pf, p,-). We can treat the partons as being free. T h e pf are large, and it is convenient to describe t h e m using m o m e n t u m fractions £,•: ^=P+/P+,

0<&<1.

(68)

(This is convenient because the £,• are invariant under boosts along the z axis.) T h e transverse m o m e n t a of the partons, p ; , are small compared to Q and can be neglected in the kinematics of the 7-parton interaction. T h e "on-shell" or "kinetic" minus m o m e n t a of the partons, p~ = pj/(2pf), are also very small compared to Q a n d can be neglected in the kinematics of the 7-parton interaction. We can think of the partonic state as being described by a wave function V>(PI~>PI;P2">P2;---),

(69)

where indices specifying spin and flavor q u a n t u m numbers have been suppressed.

£p

r Figure 22. Feynman diagram for deeply inelastic scattering.

This approximate picture is represented in Feynman diagram language in Fig. 22. T h e larger filled circle represents the hadron wave function tp. T h e smaller filled circle represents a s u m of subdiagrams in which the particles have virtualities of order Q2. All of these interactions are effectively instantaneous on the time scale of the intra-hadron interactions t h a t form the wave function.

298 T h e approximate picture also leads to an intuitive formula t h a t relates the observed cross section t o the cross section for 7-parton scattering:

a^-j^EA/^M^+owg).

(TO)

In Eq. (70), the function / is a parton distribution function: fa/h{€, A*F) d£ gives probability t o find a parton with flavor a = g, u, u, d,... in hadron h, carrying m o m e n t u m fraction within d£ of £ = pf /p+. If we knew the wave functions ip, we would form / by summing over the number n of unobserved partons, integrating \i/)n\2 over the m o m e n t a of the unobserved partons, and also integrating over the transverse m o m e n t u m of the observed parton. The second factor in Eq. (70), dcra'/dE' du>', is the cross section for scattering the lepton from the parton of flavor a and m o m e n t u m fraction £. I have indicated a dependence on a factorization scale /ip in both factors of Eq. (70). This dependence arises from the existence of virtual processes among t h e partons t h a t take place on a time scale much shorter t h a n t h e nominal Aai + ~ Q/m?. I will discuss this dependence in some detail shortly. 4-4

The hard scattering

cross

section

T h e parton distribution functions in Eq. (70) are derived from experiment. T h e hard scattering cross sections do~a(p)/dE' duj' are calculated in perturbation theory, using diagrams like those shown in Fig. 23. The diagram on t h e left is t h e lowest order diagram. The diagram on the right is one of several that contributes to da a t order as; in this diagram the parton a is a gluon.

Lowest order.

Higher order.

Figure 23. Some Feynman diagrams for the hard scattering part of deeply inelastic scattering.

299

Figure 24. Kinematics of lowest order diagram.

One can understand a lot about deeply inelastic scattering from Fig. 24, which illustrates the kinematics of the lowest order diagram. Recall t h a t in the reference frame t h a t we are using, the virtual vector boson has zero transverse m o m e n t u m . T h e incoming parton has m o m e n t u m along the plus axis. After the scattering, the parton m o m e n t u m must be on the cone k^k^ = 0, so the only possibility is t h a t its minus m o m e n t u m is non-zero and its plus m o m e n t u m vanishes. T h a t is £p++g+=0.

(71)

Since p+ = Q/(a;bj-\/2) while q+ — —Q/V2, this implies * = *bj.

(72)

T h e consequence of this is t h a t the lowest order contribution to dcr in Eq. (70) contains a delta function t h a t sets £ to Xbj- T h u s deeply inelastic scattering at a given value of x^j provides a determination of the parton distribution functions at m o m e n t u m fraction £ equal to x'bj, as long as one works only to leading order. In fact, because of this close relationship, there is some tendency to confuse the structure functions F„(a:bj) Q 2 ) with the parton distribution functions fa,h((,, HF)- I will try to keep these concepts separate: the structure functions F„ are something t h a t one measures directly in deeply inelastic scattering; the parton distribution functions are determined rather indirectly from experiments like deeply inelastic scattering, using formulas t h a t are correct only up to some finite order in a s . 4-5

Factorization

j'or the structure

functions

We will look at DIS in a little detail since it is so i m p o r t a n t . Our object is to derive a formula at lowest order in perturbation theory relating the measured

300 structure functions for e + h -» e + X via photon exchange and the parton distribution functions. Start with Eq. (70), representing Fig. 22. We change variables in this equation from (E1 ,io') t o {xb-},y). We relate £bj t o the m o m e n t u m fraction £ and a new variable x t h a t is just aibj with the proton m o m e n t u m / replaced by the parton m o m e n t u m £p^: *bj = ^

- = £ ^

- = tx.

(73)

T h a t is, £ is the parton level version of x^ • The variable y is identical to the parton level version of y because p^ appears in both the numerator and denominator: _

PJ_Q_

P

• k

_ ip-q £p • k

(74^

T h u s Eq. (70) becomes

d^bj dy

J 0 JO

daa(fiF) £ [ dxdy ?

*-^ „

+ 0(m/Q).

(75)

We can calculate daa/(dx dy) in perturbation theory. At lowest order this is particularly simple, and we obtain results proportional t o delta functions of £bj/£- Using Eq. (62) t o relate da/'{dx^dy) t o the structure functions F\ and i<2 for 7 exchange, we obtain the simple lowest order results

*i(*bj, Q2) ~ \ J2 Ql f»d*hi) + °(a°) + 0(m/Q), F2{xhi,Q2)~52

Q2axbifa/h(xbi)+0(a,)

+ 0(m/Q).

(76) (77)

a

T h e factor 1/2 between £bj-fi spin 1/2 quarks. 4-6

HF

an

d F2 follows from the Feynman diagrams for

dependence

I have so far presented a rather simplified picture of deeply inelastic scattering in which the hard scattering takes place on a time scale Ax+ ~ 1/Q, while the internal dynamics of the proton take place on a much longer time scale Ax+ ~ Q/m2. W h a t happens when one actually computes Feynman diagrams and looks a t what time scales contribute? Consider the graph shown in Fig. 25. One finds t h a t the transverse m o m e n t a k range from order m to order Q,

301

Figure 25. Deeply inelastic scattering with a gluon emission.

corresponding to energy scales k — k 2 / 2 f c + between k ~ m2 /Q and k — Q2/Q ~ Q, or time scales Q/m2 < Ax+ < 1/Q. T h e property of factorization for the cross section of deeply inelastic scattering, embodied in Eq. (70), is established by showing t h a t the perturbative expansion can be rearranged so t h a t the contributions from long time scales appear in the parton distribution functions, while the contributions from short time scales appear in the hard scattering functions. (See Ref. [24] for more information.) Thus, in Fig. 25, a gluon emission with k 2 ~ m 2 is part of / ( £ ) , while a gluon emission with k 2 ~ Q2 is part of da. Breaking up the cross section into factors associated with short and long time scales requires the introduction of a factorization scale, ^F- W h e n calculating the diagram in Fig. 25, one integrates over k. Roughly speaking, one counts the contribution from k 2 < fiF as part of the higher order contribution to fa/h(S,i HF), convoluted with the lowest order hard scattering function da for deeply inelastic scattering from a quark. T h e contribution from /J,F < k 2 then counts as part of the higher order contribution to da convoluted with an uncorrected parton distribution. This is illustrated in Fig. 26. (In real calculations, the split is accomplished with the aid of dimensional regularization, and is a little more subtle t h a n a simple division of the integral into two parts.) hard scattering

parton di;:;nburioii<:

iog(i/M

i°sP*)

Figure 26. Time scales in factorization.

302 A consequence of this is t h a t both d&a{fip)/dE'dto' and fa/h(£, (J-F) depend on fip. T h u s we have two scales, the factorization scale /up in fj/h[£, HF) a n d the renormalization scale fi in as(fi). (When we expand da in powers of as(fi) then the coefficients depend on fi.) As with p, the cross section does not depend on fip. T h u s there is an equation d(cross section)/d fip = 0 t h a t is satisfied to the accuracy of the perturbative calculation used. If you work harder and calculate to higher order, then the dependence on ftp is less. Often one sets fip = A* in applied calculations. In fact, it is rather common in applications to deeply inelastic scattering to set fip — fi = Q.

4-7

Contour

graphs of scale

dependence

As an example, look at the one jet inclusive cross section in proton-antiproton collisions. Specifically, consider the cross section dajdEfdr] to make a collimated spray of particles, a jet, with transverse energy ET and rapidity n. (Here ET is essentially the transverse m o m e n t u m carried by the particles in the jet and r\ is related to the angle between the jet and the beam direction by T) = l n ( t a n ( 0 / 2 ) ) . We will investigate this process and discuss the definitions in the next section. For now, all we need to know is t h a t the theoretical form u l a for the cross section at next-to-leading order involves the strong coupling a s ( / i ) and two factors fa/h{x> HF) representing the distribution of partons in the two incoming hadrons. There is a parton level hard scattering cross section t h a t also depends on fi and pp. How does the cross section depend on p in as(p) and fip in fa/h{x,MF)? In Fig. 27, I show contour plots of the jet cross section versus p and pp at two different values of ET- T h e center of the plots corresponds to a s t a n d a r d choice of scales, p = fip = ET/2. The axes are logarithmic, representing \og2{2fi/ET) and \og2(2pp/ ET)- T h u s fi and fip vary from ET/S to 2ET in the plots. Notice t h a t the dependence on the two scales is rather mild for the nextto-leading order cross section. The cross section calculated at leading order is quite sensitive to these scales, but most of the scale dependence found at order a2s has been canceled by the a3s contributions to the cross section. One reads from the figure t h a t the cross section varies by roughly ± 1 5 % in the central region of the graphs, both for m e d i u m and large ET- Following the argument of Sec. 3.4, this leads to a rough estimate of 15% for the theoretical error associated with truncating perturbation theory at next-to-leading order.

303

Nuv

Nuv

ET = 100 GeV

ET = 500 GeV

Figure 27. Contour plots of the one jet inclusive cross section versus the renormalization scale /i and the factorization scale fj.p. The cross section is drr/dE^dri at rj = 0 with ET = 100 GeV in the first graph and E-p = 500 GeV in the second. The horizontal axis in each graph represents NJJV = log 2 ^ p / E x ) and the vertical axis represents NQO = log 2 (2/if /ET). The contour lines show 5% changes in the cross section relative to the cross section at the center of the figures. The c m energy is y/s = 1800 GeV.

4-8

MS definition of parton distribution

functions

The factorization property, Eq. (70), of the deeply inelastic scattering cross section states that the cross section can be approximated as a convolution of a hard scattering cross section that can be calculated perturbatively and parton distribution functions fa/A{x, HF)- But what are the parton distribution functions? This question has some practical importance. The hard scattering cross section is essentially the physical cross section divided by the parton distribution function, so the precise definition of the parton distribution functions leads to the rules for calculating the hard scattering functions. The definition of the parton distribution functions is to some extent a matter of convention. The most commonly used convention is the MS definition, which arose from the theory of deeply inelastic scattering in the language of the "operator product expansion." 25 Here I will follow the (equivalent) formulation of Ref. [14]. For a more detailed pedagogical review, the reader may consult Ref. [26]. Using the MS definition, the distribution of quarks in a hadron is given

304 as the hadron m a t r i x element of certain quark field operators:

filhU^F) = lJ^7e-'^+y-(p\M^y-,0)1+Fi>i(Q)\p).

(78)

Here \p) represents the state of a hadron with m o m e n t u m p M aligned so t h a t PT = 0. For simplicity, I take the hadron to have spin zero. T h e operator V>(0), evaluated at x^ = 0, annihilates a quark in the hadron. T h e operator V>;(0,y~,O) recreates the quark at x+ = x y = 0 and x~ = y~, where we take the appropriate Fourier transform in y~ so t h a t the quark t h a t was annihilated and recreated has m o m e n t u m k+ = £p+. The motivation for the definition is t h a t this is the hadron m a t r i x element of the appropriate number operator for finding a quark. There is one subtle point. T h e number operator idea corresponds to a particular gauge choice, A+ = 0. If we are using any other gauge, we insert the operator F=Vexp(-ig

T

dz~A+% z~, 0) ta J .

(79)

T h e V indicates a p a t h ordering of the operators and color matrices along the p a t h from ( 0 , 0 , 0 ) to ( O , y ~ , 0 ) . This operator is the identity operator in A+ = 0 gauge and it makes the definition gauge invariant.

DIS

Parton distribution

Figure 28. Deeply inelastic scattering and the parton distribution functions.

T h e physics of this definition is illustrated in Fig. 28. T h e first picture (from Fig. 21) illustrates the amplitude for deeply inelastic scattering. T h e fast proton moves in the plus direction. A virtual photon knocks out a quark, which emerges moving in the minus direction and develops into a jet of particles. T h e second picture illustrates the amplitude associated with the quark

305 distribution function. We express F as F2F1 where F2 = V exp (+ig

f

F1 = Vexp(-igf

dz~At(0,

z',0)tA

dz~ A+(0, z~ ,0)ta)

, .

(80)

and write the quark distribution function including a sum over intermediate states \N):

1 J

Z7F

N

(81) Then the a m p l i t u d e depicted in the second picture in Fig. 28 is (N\Fi^i(0)\p). T h e operator ip annihilates a quark in the proton. T h e operator F± stands in for the quark moving in the minus direction. T h e gluon field A evaluated along a lightlike line in the minus direction absorbs longitudinally polarized gluons from the color field of the proton, just as the real quark in deeply inelastic scattering can do. T h u s the physics of deeply inelastic scattering is built into the definition of the quark distribution function, albeit in an idealized way. T h e idealization is not a problem because the hard scattering function da systematically corrects for the difference between real deeply inelastic scattering and the idealization. There is one small hitch. If you calculate any Feynman diagrams for fi/hiii A*F)) you are likely to wind up with an ultraviolet-divergent integral. T h e operator product t h a t is part of the definition needs renormalization. This hitch is only a small one. We simply agree to do all of the renormalization using the MS scheme for renormalization. It is this renormalization t h a t introduces the scale \xp into fi/h{£, HF)- This role of fip is in accord with Fig. 26: roughly speaking fip is the upper cutoff for what m o m e n t a belong with the parton distribution function; at the same time it is the lower cutoff for what m o m e n t a belong with the hard scattering function. W h a t a b o u t gluons? T h e definition of the gluon distribution function is similar to the definition for quarks. We simply replace the quark field t\> by suitable combinations of the gluon field A^, as described in Refs. [14] and [ 26 ].

4.9

Evolution

of the parton

distributions

Since we introduced a scale fip in the definition of the parton distributions in order to define their renormalization, there is a renormalization group equa-

306 tion t h a t gives the /J,F dependence

-71

fa/h(x,/jF)

= V" /

— Pab(x/Z,as(nF))

fb/h(t,fiF)-

(82)

This is variously known as the evolution equation, the Altarelli-Parisi equation, and the D G L A P (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) equation. Note the sum over parton flavor indices. The evolution of, say, an up quark (a = u) can involve a gluon {b = g) through the element Pug of the kernel t h a t describes gluon splitting into uu. T h e equation is illustrated in Fig. 29. When we change the renormalization scale fiF, the change in the probability to find a parton with m o m e n t u m fraction x and flavor a is proportional to the probability to find such a parton with large transverse m o m e n t u m . T h e way to get this parton with large transverse m o m e n t u m is for a parton carrying m o m e n t u m fraction £ and much smaller transverse m o m e n t u m to split into partons carrying large transverse m o m e n t a , including the parton t h a t we are looking for. This splitting probability, integrated over the appropriate transverse m o m e n t u m ranges, is the kernel Pab-

d d log nF ' ^ xp

Figure 29. The renormalization group equation for the parton distribution functions.

T h e kernel P in Eq. (82) has a perturbative expansion

P*{*li,«M)

= P%\*lti)^

+ P%\*Ii)

( ^ )

2

+ - - - (83)

T h e first two terms are known and are typically used in numerical solutions of the equation. To learn more about the D G L A P equation, the reader may consult Refs. [x] and [ 26 ].

307 4.10

Determination

and use of the parton

distributions

T h e MS definition giving the parton distribution in terms of operators is process independent - it does not refer to any particular physical process. These p a r t o n distributions then appear in the Q C D formula for any process with one or two hadrons in the initial state. In principle, the parton distribution functions could be calculated by using the method of lattice Q C D (see Ref. [ 26 ])Currently, they are determined from experiment. Currently the most comprehensive analyses are being done by the C T E Q 2 0 and M R S 2 7 groups. These groups perform a "global fit" to d a t a from experiments of several different types. To perform such a fit one chooses a parameterization for the parton distributions at some s t a n d a r d factorization scale HQ. Certain sum rules t h a t follow from the definition of the parton distribution functions are built into the parameterization. An example is the m o m e n t u m sum rule:

E I <%tf°/h(Z,n)

= l-

(84)

Given some set of values for the parameters describing the fa/h{x> f-o), one can determine fa/h{xi^) f ° r a U higher values of fi by using the evolution equation. Then the QCD cross section formulas give predictions for all of the experiments t h a t are being used. One systematically varies the parameters m fa/h{xi A*o) to obtain the best fit to all of the experiments. One source of information about these fits is the world wide web pages of Ref. [ 28 ]. If the freedom available for the parton distributions is used to fit all of the world's d a t a , is there any physical content to Q C D ? T h e answer is yes: there are lots of experiments, so this program won't work unless Q C D is right. In fact, there are roughly 1400 d a t a in the C T E Q fit and only a b o u t 25 parameters available to fit these data. 5

Q C D in h a d r o n - h a d r o n collisions

When there is a hadron in the initial state of a scattering process, there are inevitably long time scales associated with the binding of the hadron, even if part of the process is a short-time scattering. We have seen, in the case of deeply inelastic scattering of a lepton from a single hadron, t h a t the dependence on these long time scales can be factored into a parton distribution function. But what happens when two high energy hadrons collide? T h e reader will not be surprised to learn that we then need two parton distribution functions.

308

I explore hadron-hadron collisions in this section. I begin with the definition of a convenient kinematical variable, rapidity. Then I discuss, in turn, production of vector bosons (7*, W, and Z) and jet production. The theory for the production of heavy quarks is similar and I omit it. 5.1

Kinematics:

rapidity

In describing hadron-hadron collisions, it is useful to employ a kinematic variable y that is called rapidity. Consider, for example, the production of a Z boson plus anything, p+p —> Z + X. Choose the hadron-hadron c m . frame with the z axis along the beam direction. In Fig. 30, I show a drawing of the collision. The arrows represent the momenta of the two hadrons; in the c m . frame these momenta have equal magnitudes. We will want to describe the process at the parton level, a + b —> Z + X. The two partons a and b each carry some share of the parent hadron's momentum, but generally these will not be equal shares. Thus the magnitudes of the momenta of the colliding partons will not be equal. We will have to boost along the z axis in order to get to the parton-parton c m . frame. For this reason, it is useful to use a variable that transforms simply under boosts. This is the motivation for using rapidity. » 4

Figure 30. Collision of two hadrons containing partons producing a Z boson. The c m . frame of the two hadrons is normally not the c m . frame of the two partons that create the Z boson.

Let q** — (q+,q~, q) be the momentum of the Z boson. Then the rapidity of the Z is defined as

y-\m{f).

(85)

The four components (q+,q~, q) of the Z boson momentum can be written in terms of four variables, the two components of the Z boson's transverse

309 m o m e n t u m q, its mass M, and its rapidity: q" = ( e V ( q 2 + M 2 ) / 2 , e " V ( q 2 + M 2 ) / 2 , q ) .

(86)

T h e utility of using rapidity as one of the variables stems from the transformation property of rapidity under a boost along the z axis: 9

+-»eV,

9"^e-wg-,

q ->• q.

(87)

Under this transformation, y^y

+ to.

(88)

This is as simple a transformation law as we could hope for. In fact, it is just the same as the transformation law for velocities in non-relativistic physics in one dimension.

Figure 31. Definition of the polar angle 9 used in calculating the rapidity of a massless particle.

Consider now the rapidity of a massless particle. Let the massless particle emerge from the collision with polar angle 0, as indicated in Fig. 3 1 . A simple calculation relates the particle's rapidity y to 6: y= - l n ( t a n ( 0 / 2 ) ) ,

(m = 0).

(89)

Another way of writing this is t a n 0 = l/sinhy, One also defines the pseudorapidity r] = - l n ( t a n ( 0 / 2 ) )

(m = 0).

(90)

r\ of a particle, massless or not, by or

t a n 0 = l / s i n h rj.

(91)

T h e relation between rapidity and pseudorapidity is sinh r; = J l + m2/q%, sinh y.

(92)

Thus, if the particle isn't quite massless, r\ may still be a good approximation to y.

310

5.2

7*; W, Z production

in hadron-hadron

collisions

Consider the process A + B-+Z

+ X,

(93)

where A and B are high energy hadrons. This process and the corresponding process in which a W boson is produced are historically i m p o r t a n t because they are the processes by which the W and Z bosons were first observed. 2 9 Two features of this reaction are important for our discussion. First, the mass of the Z boson is large compared to 1 GeV, so t h a t a process with a small time scale At ~ \/Mz must be involved in the production of the Z. At lowest order in the strong interactions, the process is q + q —> Z. Here the quark and antiquark are constituents of the high energy hadrons. The second significant feature is t h a t the Z boson does not participate in the strong interactions, so t h a t our description of the observed final state can be very simple. In process (93), we allow the Z boson to have any transverse m o m e n t u m q. (Typically, then, q will be much smaller than Mz-) Since we integrate over q and the mass of the Z boson is fixed, there is only one variable needed to describe the m o m e n t u m of the Z boson. We choose to use its rapidity y, so t h a t we are interested in the cross section da/dy.

Figure 32. A Feynman diagram for Z boson production in a hadron-hadron collision. Two partons, carrying momentum fractions £^ and £#, participate in the hard interaction. This particular Feynman diagram illustrates an order aB contribution to the hard scattering cross section: a gluon is emitted in the process of making the Z boson. The diagram also shows the decay of the Z boson into an electron and a neutrino.

T h e cross section takes a factored form similar to t h a t found for deeply inelastic scattering. Here, however, there are two parton distribution functions:

^ « V fdU

fdiB

dy

JXB

^JXA

fa/A(U,HF) h/B^B^F)

d

^ll. dy

(94)

311 T h e meaning of this formula is intuitive: fa/A(£,A, PF) d£,A gives the probability to find a parton in hadron A\ fb/B^B, Hj) d^B gives the probability to find a parton in hadron B; daab/dy gives the cross section for these partons to produce the observed Z boson. T h e formula is illustrated in Fig. 32. T h e hard scattering cross section can be calculated perturbatively. Fig. 32 illustrates one particular order as contribution to dcrab/dy. T h e integrations over parton m o m e n t u m fractions have limits XA and XB , which are given by xA = eyy/M2/s,

xB = e-yy/M2/s.

(95)

Eq. (94) has corrections of order m/Mz, where m is a mass characteristic of hadronic systems, say 1 GeV. In addition, when dcrab/dy is calculated to order a ^ , then there are corrections of order a ^ + 1 . We could equally well talk about A+B —> f*+X where the virtual photon decays into a muon pair or an electron pair t h a t is observed and where the mass Q of the 7* is large compared t o 1 GeV. For A + B —> / i + + \i~ + X one has the formula

dQHy

d^B fa/A(U,HF)

fb/B^B^F)

^ J ^ J ^ o ' o i " ^ ' ^ , ^ ^ , ^ ,

"

2

,

•

(96)

dQ2dy

This process is historically i m p o r t a n t . Before Q C D , one had partons and Q E D . Partons and Q E D did a good j o b of explaining deeply inelastic scattering. But there were other ways to explain deeply inelastic scattering. High mass dimuon production was investigated experimentally by Lederman et a/. 30 Drell and Y a n 3 1 proposed to explain the experimental results using the lowest order version of the formula above. It worked. T h e alternative m e t h o d s t h a t worked for deeply inelastic scattering did not work here. This helped to establish the parton picture. 5.3

Factorization

is not so obvious

The factorization formula Eq. (94) is supposed to hold up to m2/Q2 corrections. This result is not so obvious, and in fact does not hold graph by graph. A graph for which it does not hold is shown in Fig. 33. Does factorization hold if one sums over graphs? T h e answer is yes, but to show this one needs to use unitarity, causality and gauge invariance. For more information, the reader is invited to consult Ref. [ 24 ]. 5.4

Jet

production

In our study of high energy electron-positron annihilation, we discovered three things. First, QCD makes the qualitative prediction t h a t particles in the final

312

PA

.!---'** PB

'"—•—«, S .

Figure 33. A graph for which factorization does not work. The spectator partons interact softly with the active partons, so that the soft part of the graph does not break up into two factors.

state should tend to be grouped in collimated sprays of hadrons called jets. The jets carry the momenta of the first quarks and gluons produced in the hard process. Second, certain kinds of experimental measurements probe the short-time physics of the hard interaction, while being insensitive to the long-time physics of parton splitting, soft gluon exchange, and the binding of partons into hadrons. Such measurements are called infrared safe. Third, among the infrared safe observables are cross sections to make jets.

Figure 34. Sketch of a two-jet event at a hadron collider. The cylinder represents the detector, with the beam pipe along its axis. Typical hadron-hadron collisions produce beam remnants, the debris from soft interactions among the partons. The particles in the beam remnants have small transverse momenta, as shown in the sketch. In rare events, there is a hard parton-parton collision, which produces jets with high transverse momenta. In the event shown, there are two high Pf jets.

These ideas work for hadron-hadron collisions too. In such collisions, there is sometimes a hard parton-parton collision, which produces two or more jets, as depicted in Fig. 34. Consider the cross section to make one jet plus anything else, A + B -> jet + X.

(97)

313

Let ET be the transverse energy of the jet, defined as the sum of the absolute values of the transverse m o m e n t a of the particles in the j e t . Let y be the rapidity of the jet. Given a definition of exactly what it means to have a jet with transverse energy ET and rapidity y, the jet production cross section takes the familiar factored form ——— Ssl V / dU d£B fa/A(U, dETdr] ^JXA JxB

HF) fb/B(tB,

HF)

,F f • dETdn

(98)

One diagram t h a t contributes to da at nexL-to-leading order is shown in Fig. 35.

Figure 35. A Feynman diagram for jet production in hadron-hadron collisions. The leading order diagrams for A + B —> jet + X occur at order a2s. This particular diagram is for an interaction of order aa. When the emitted gluon is not soft or nearly collinear to one of the outgoing quarks, this diagram corresponds to a final state like that shown in the small sketch, with three jets emerging in addition to the beam remnants. Any of these jets can be the jet that is measured in the one jet inclusive cross section.

W h a t shall we choose for the definition of a jet? At a crude level, high ET jets are quite obvious and the precise definition hardly m a t t e r s . However, if we want to make a quantitative measurement of a jet cross section to compare to next-to-leading order theory, then the definition does m a t t e r . There are several possibilities for a definition t h a t is infrared safe. T h e one most used in hadron-hadron collisions is based on cones. Here I will present a different algorithm t h a t is similar to the algorithms used to define jets in electronpositron annihilation. 5.5

kx

algorithm

The main idea of the kr algorithm 3 2 is to modify one of the algorithms used in e+e~ annihilation so t h a t we use ET, r\ and as variables and to avoid

314

contamination by the many low ET particles in the event. We choose a merging parameter R. Then we start with a list of "protojets" with m o m e n t a Pi,... ,p^N a s illustrated in Fig. 36. We also start with an empty list of finished jets. The end result is a list of m o m e n t a pk of finished jets, ordered in Ex-

i

Figure 36. A two jet event in a proton antiproton collision. The two protojets on the lower left are the first to be combined.

T h e algorithm can be stated very simply. See Fig. 36. 1. For each pair of protojets define dij = mm(ETti,

ETtj)

[fa -

2

Vj)

+ (0,- - <j>j)2]/R2.

(99)

For each protojet define di = ETii.

(100)

2. Find the smallest of all the dij and the d,- Call it dm-m. 3. If dm\n is a dij, merge protojets i and j into a new protojet k with Er,k — Er,i + ET,J r)k = [ET,i Vi + ETJ

Vj]/ET,k

k = [ET,i i + ETJ

4>j]/ETjk

(101)

4. If d m in is a di, then protojet i is "not mergable." Remove it from the list of protojets and add it to the list of jets. 5. If protojets remain, go to 1. Evidently, if two protojets are collinear, they will be merged right away. If one has vanishing m o m e n t u m , it will either get merged with a protojet

315 nearby in angle, or it will become a low ET jet in the final list. Many of the jets have small ET and are really minijets, or just part of low ET debris. For an inclusive cross section to make n high ET jets plus anything else, the m a n y low ET jets do not affect the result. For an exclusive n jet cross section, one would use a cutoff i?T,min- T h u s in either case, low ET particles do not change the result. T h u s the algorithm is infrared safe. 6

Epilogue

Q C D is a rich subject. T h e theory and the experimental evidence indicate t h a t quarks and gluons interact weakly on short t i m e a n d distance scales. But the net effect of these interactions extending over long time and distance scales is that the chromodynamic force is strong. Quarks are bound into hadrons. Outgoing partons emerge as jets of hadrons, with each jet composed of subjets. T h u s Q C D theory can be viewed as starting with simple perturbation theory, but it does not end there. T h e challenge for both theorists and experimentalists is to extend the range of phenomena t h a t we can relate to the fundamental theory. References 1. G. Sterman et al., Handbook of Perturbative QCD, Rev. Mod. Phys. 6 7 , 157 (1995). 2. R. K. Ellis, W. J. Stirling, and B. R. Webber, QCD and Collider Physics, (Cambridge University Press, Cambridge, 1996). 3. L. Brown, Quantum Field Theory, (Cambridge University Press, C a m bridge, 1992) 4. G. Sterman, An Introduction to Quantum Field Theory, (Cambridge University Press, Cambridge, 1993). 5. M. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, (Addison-Wesley, Reading, 1995). 6. S. Weinberg, The Quantum Theory of Fields, (Cambridge University Press, Cambridge, 1995). 7. D. E. Soper, ed., QCD and Beyond, Proceedings of the 1995 Theoretical Advanced Studies Institute, Boulder, 1995, (World Scientific, Singapore, 1996). 8. D. E. Soper, Basics of QCD Perturbation Theory, e-Print Archive: hepph/9702203, in The Strong Interaction, from Hadrons to Partons, XXIV SLAG Summer Institute on Particle Physics Stanford, August 1996, edited by L. DePorcel (SLAC, Stanford, 1997).

316

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

22. 23. 24. 25. 26.

27. 28. 29. 30. 31. 32.

G. Sterman, Phys. Rev. D 17, 2773 (1978); 17, 2789 (1978). J. C. Collins and D. E. Soper, Nucl. Phys. B 1 9 3 , 381 (1981). Z. Kunszt and D. E. Soper, Phys. Rev. D 4 6 , 192 (1992). C. L. Basham, L. S. Brown, S. D. Ellis, and S. T. Love, Phys. Rev. D 19, 2018 (1979). S. Bethke, Z. Kunszt, D. E. Soper, and W. J. Stirling, Nucl. Phys. B 3 7 0 , 310 (1992). J. C. Collins and D. E. Soper, Nucl. Phys. B 1 9 4 , 445 (1982). G. Curci, W. Furmanski and R. Petronzio, Nucl. Phys. B 175, 27 (1980). J. C. Collins, Renormalization, (Cambridge University Press, Cambridge, 1984). L. Surguladze and M. Samuel, Rev. Mod. Phys. 6 8 , 259 (1996). D. E. Soper and L. Surguladze, Phys. Rev. D 54, 4566 (1996). P. M. Stevenson, Phys. Rev. D 2 3 , 2916 (1981). S. J. Brodsky, G. P. Lepage and P. MacKenzie, Phys. Rev. D 28, 228 (1983). H. Lai et al., e-Print Archive hep-ph/9606399, Phys. Rev. D 55, (to be published). C D F Collaboration (F. Abe al), Phys. Rev. Lett. 77, 438 (1996); B. Flaugher, C D F Collaboration, Proceedings of the XI Topical Workshop on ppbar Collider Physics, Padova, Italy, May 1996. DO Collaboration: N. Varelas, Proceedings of the International Conference on High Energy Physics, Warsaw, July 1996. J. Huston et al., Phys. Rev. Lett. 77, 444 (1996). J. C. Collins, D. E. Soper, and G. Sterman, in Perturbative QCD, edited by A. Mueller, (World Scientific, Singapore, 1989). W. A. Bardeen, A. J. Buras, D. W. Duke, and T Muta, Phys. Rev. D 18, 3998 (1978). D. E. Soper, in M. Golterman et al. eds., Lattice '96 International Symposium on Lattice Field Theory, St. Louis, June 1996 (Elsevier Science, A m s t e r d a m , to be published). A. D. Martin, R. G. Roberts, and W. J. Stirling, Phys. Lett. B 387, 419 (1996). P. A n a n d a m and D. E. Soper, "A Potpourri of Partons", http://zebu.uoregon.edu/~parton/. UA1 Collaboration (G. Arnison et al.), Phys. Lett. B 1 2 2 , 103 (1983); UA2 Collaboration (G. Banner et al.), Phys. Lett. B 1 2 2 , 476 (1983). J. C. Christenson et al., Phys. Rev. Lett. 25, 1523 (1970). S. D. Drell and T.-M. Yan, Ann. Phys. 66, 578 (1971). S. D. Ellis and D. E. Soper, Phys. Rev. D 48, 3160 (1993).

•, \ ^ : * v

~^e

Thomas DeGrand

This page is intentionally left blank

LATTICE QCD A N D THE C K M M A T R I X

Thomas DeGrand

University

Department of Colorado,

of Physics Boulder CO

80309-390

These lectures provide an introduction to lattice methods for nonperturbative studies of Quantum Chromodynamics. Lecture 1 (Ch. 2) is a very vanilla introduction to lattice QCD. Lecture 2 (Ch. 3) describes examples of recent lattice calculations relevant to fixing the parameters of the CKM matrix.

1

Introduction

The lattice 1 regularization of QCD has been a fruitful source of qualitative and quantitative information about QCD for many years, especially when combined with Monte Carlo simulation. Lattice methods are presently the only way we know how to compute masses and matrix elements in the strong interactions beginning with the Lagrangian of QCD. My goal in these lectures is to give enough of an overview of the subject that you will be able to make an intelligent appraisal of a lattice calculation. The first lecture will describe how to put QCD on a lattice. This is a long story with a lot of parts. Lattice QCD is full of technicalities, but I will try to make the discussion physical. In Lecture Two I will discuss lattice calculations of matrix elements which are needed to convert experimental numbers to predictions for the CKM matrix. These calculations have a lot of ingredients: a typical one starts with a particular choice of discretization and simulation algorithm, and a choice of operators whose matrix elements are appropriate for one's measurement. After the lattice number is computed, it may have to be converted into a number in a continuum regularization scheme (like MS), which will involve some kind of perturbative or nonperturbative matching calculation. Finally, it might have to be extrapolated in quark mass, to some physical quark mass value or the the chiral limit. None of these parts are simple or obvious (or more precisely, most of the time the simple and obvious idea doesn't work very well). Hopefully you will find the physics in the calculations more interesting than the tables of numbers which result. 319

320

2 2.1

Basics of L a t t i c e Q C D Lattice Variables and Actions

All quantum field theories must be regulated in order to control their ultraviolet divergences while calculations are performed. The lattice is a space-time cutoff which eliminates all degrees of freedom from distances shorter than the lattice spacing a. As with any regulator, it must be removed after renormalization. Contact with experiment only exists in the continuum limit, when the lattice spacing is taken to zero. The lattice is a unique regulator compared to the ones you might already know. Other regularization schemes are tied closely to perturbative expansions: one calculates a process to some order in a coupling constant; divergences are removed order by order in perturbation theory. The lattice, however, is a nonperturbative cutoff. Before a calculation begins, all wavelengths less than a lattice spacing are removed. All regulators have a price. On the lattice we sacrifice all continuous space-time symmetries but preserve all internal symmetries, including local gauge invariance. This preservation is important for nonperturbative physics. For example, gauge invariance is a property of the continuum theory which is nonperturbative, so maintaining it as we pass to the lattice means that all of its consequences (including current conservation and renormalizability) will be preserved. The bill is paid when we take the lattice spacing to zero and try to recover what we have left out. Let's begin by thinking about a lattice version of scalar field theory. One just replaces the space-time coordinate i M by a set of integers nM (xM = an^, where a is the lattice spacing). Field variables (f>(x) are defined on sites 4>{xn) =
/" d 4 ^ - > a 4 ^ •*

£(#„).

(1)

n

and the generating functional for Euclidean Green's functions is replaced by an ordinary integral over the lattice fields

Z= f(l[d
(2)

n

Gauge fields are a little more complicated. They carry a space-time index \i in addition to an internal symmetry index a (A^(x)) and are associated with a path in space x M (s): a particle traversing a contour in space picks up a phase factor %j) -> P(exp ig / dx^A^ip = U(s)ip(x). (3)

321

P is a path-ordering factor analogous to the time-ordering operator in ordinary quantum mechanics. Under a gauge transformation g, U(s) is rotated at each end: U(s) ^ g-1(x^8))U(8)g(xll(0)). (4) These considerations led Wilson 2 to formulate gauge fields on a space-time lattice, in terms of a set of fundamental variables which are elements of the gauge group G living on the links of a four-dimensional lattice, connecting neighboring sites x and x + aii: U^x), with Utl{x + fx)^ = U^x) U^n)=exp(igaTaA«(n))

(5)

for SU(N). (g is the coupling, A^ the vector potential, and Ta is a group generator). Under a gauge transformation link variables transform as Uli(x)^V(x)U^(x)V(x

+^

(6)

and site variables as 4>{x) -> V(x)ip{x)

(7)

so the only gauge invariant operators we can use as order parameters are matter fields connected by oriented "strings" of U's ^(x1)Ull{xl)Uti(x1+Li)...i>(x2)

(8)

or closed oriented loops of U's Tr . . . U^(x)U^x + p.) • •. -> Tr . . . U^x)V\x

+ fi)V{x + fi)U^x + /},).... (9)

An action is specified by recalling that the classical Yang-Mills action involves the curl of AM, F^. Thus a lattice action ought to involve a product of C/M's around some closed contour. Gauge invariance will automatically be satisfied for actions built of powers of traces of U's around arbitrary closed loops, with arbitrary coupling constants. If we assume that the gauge fields are smooth, we can expand the link variables in a power series in gaA' s. For almost any closed loop, the leading term in the expansion will be proportional to F^v. This is not a bug, it is a feature. All lattice actions are just bare actions characterized by many bare parameters (coefficients of loops). In the continuum (scaling) limit all these actions are in the same universality class, which is (presumably) the same universality class as QCD with any regularization scheme, and there will be cutoff-independent predictions from any lattice actions which are simply predictions of-QCD.

322

Let's hold that thought while we do an example: The simplest contour has a perimeter of four links. In SU(N) S

= ^ E

E

Re Tr

(! - U^n)U„{n + £ ) t / > + 0)Ut(n)).

(10)

This action is called the "plaquette action" or the "Wilson action" after its inventor, g2 is the bare lattice coupling, whose associated cutoff is a. The lattice parameter @ = 2N/g2 is often written instead of g2 = 4iras. Let us see how this action reduces to the standard continuum action. Specializing to the U(l) gauge group, and slightly redefining the coupling, S = ^ E E

R

e

(l-exp(i9a[Ali{n)+A„(n

+ p.)-All(x

+ i>)-Av{n)})).

(11)

The naive continuum limit is taken by assuming that the lattice spacing a is small, and Taylor expanding A^n

+ v) = A^n)

+ ad„A^{n) + ...

(12)

so the action becomes PS = \

Yl E

l

~

9

Re

(exp(t5o[a(a„^ - d,LA„) + 0(a2)]))

^ f l 4 E E ^ +l 4

j
(13)

( 14 ) (is)

transforming the sum on sites back to an integral. 2.2

Numerical

Simulations

In a lattice calculation, like any other calculation in quantum field theory, we compute an expectation value of any observable T as an average over a ensemble of field configurations:


(16)

We do this by Monte Carlo simulation: we construct an ensemble of states (collection of field variables), where the probability of finding a particular configuration in the ensemble is given by Boltzmann weighting (i. e. proportional

323

to exp(—S). Then the expectation value of any observable T is given simply by an average over the ensemble: 1

N

1=1

As the number of measurements N becomes large the quantity T will become a Gaussian distribution about a mean value, our desired expectation value. The idea of essentially all simulation algorithms 3 is to construct a new configuration of field variables from an old one. One begins with some initial field configuration and monitors observables while the algorithm steps along. After some number of steps, the value of observables will appear to become independent of the starting configuration. At that point the system is said to be "in equilibrium" and Eq. (17) can be used to make measurements. Dynamical fermions are a complication for QCD 4 . The fermion path integral is not a number and a computer can't simulate fermions directly. However, one can formally integrate out the fermion fields. For n / degenerate fermion flavors Z=

f[dU}[diP][dij}exp(-pSG(U)~^2ijM(U)ip) ^

(18)

i=l

= [[dU}{detM(U))n'exp(-(3S(U)).

(19)

The determinant introduces a nonlocal interaction among the U's: Z=

f[dU]exp(-l3S(U)-nfTr\n(M(U))).

(20)

Generating configurations of the U's involves computing how the action changes when the set of U's are varied. Typically, this involves inverting the fermion matrix M(U) {dlogM/dM = M~l). This is the major computational problem dynamical fermion simulations face. M has eigenvalues with a very large range- from 2ir down to mQa~ and in the physically interesting limit of small mq the matrix becomes ill-conditioned. At present it is necessary to compute at unphysically heavy values of the quark mass and to extrapolate to mq = 0. (The standard inversion technique today is one of the variants of the conjugate gradient algorithm 5 . ) This tremendous expense is responsible for one of the"standard" lattice approximations, the "quenched" approximation. In this approximation the back-reaction of the fermions on the gauge fields is neglected, by setting n/ = 0 in Eq. (18). Valence quarks, or quarks which

324

appear in observables, are kept, but no sea quarks. No one knows how good an approximation this is, in principle. In practice it works very well for spectroscopy. The only way we know how to test it is to compare simulations in the quenched approximation with those from full QCD. 2.3

Spectroscopy Calculations

Masses are computed in lattice simulations from the asymptotic behavior of Euclidean-time correlation functions. A typical (diagonal) correlator can be written as C(t) = (0|O(*)O(0)|0). (21) Making the replacement 0(t) = eHt0e-m

(22)

and inserting a complete set of energy eigenstates, Eq. (21) becomes

C(t) = J2\(0\O\n)\2e-E«t.

(23)

n

At large separation the correlation function is approximately C{t) ~ |(0|O|l)| 2 e- £ '*

(24)

where E\ is the energy of the lightest state which the operator O can create from the vacuum. Fig. 1 shows an example of this. If the operator does not couple to the vacuum, then in the limit of large t one hopes to to find the mass Ei by measuring the leading exponential falloff of the correlation function. If the operator O has poor overlap with the lightest state, a reliable value for the mass can be extracted only at a large time t. In some cases that state is the vacuum itself, in which E\ = 0. Then one looks for the next higher state-a signal which disappears into the constant background. This is hard to do. Most of the observables we are interested in will involve valence fermions. Let's suppose we wanted to measure the mass of a meson. Then we might take C{t) = '£(J(x,t)J(p,0))

(25)

X

where J(x, t) = ${x, t)Y4>{x, t)

(26)

and T is a Dirac matrix. The intermediate states \n) which saturate C(x, t) are the hadrons which the current J can create from the vacuum: the pion,

325

icr 3

10" 5

10~ 10 0

10

20 t

Figure 1: An obviously very nice looking lattice correlator and its fit. Periodic boundary conditions convert the exponential decay into a hyperbolic cosine.

for a pseudoscalar current, the rho, for a vector current, and so on. Now we write out the correlator in terms of fermion fields C(t) = J 3 ( 0 | ^ ( a ; , t ) a r < i ^ ( a ; ) t ) a ^ / t ( 0 ) 0 ) ^ r H ^ ( 0 , 0 ) ^ | 0 >

(27)

X

with a Roman index for spin and a Greek index for color. We contract creation and annihilation operators into quark propagators (0\T(^^t)aM0,0)0)10)

= G«f ( M ; 0 , 0 )

(28)

so

C(t) = J2 TrG(x, t; 0,0)rG(0,0; x, t)Y

(29)

X

where the trace runs over spin and color indices. Baryons are constructed similarly. A good way to think about these correlators is by using a sort of Feynman-diagram language which keeps track of the valence quark lines but ignores all the gluons and sea quarks. 2.4

The Continuum

Limit

When we define a theory on a lattice the lattice spacing a is an ultraviolet cutoff and all the coupling constants in the action are the bare couplings defined with

326

respect to it. When we take a to zero we must also specify how the couplings behave. The proper continuum limit comes when we take a to zero holding physical quantities fixed, not when we take a to zero fixing the couplings. On the lattice, if all quark masses are set to zero, the only dimensionful parameter is the lattice spacing, so all masses scale like 1/a. Said differently, a lattice calculation produces the dimensionless combination am(a). One can determine the lattice spacing by fixing one mass from experiment. Then all other dimensionful quantities can be predicted. Imagine computing some masses at several values of the lattice spacing. (Pick several values of the bare parameters and calculate masses for each set of couplings.) Our calculated mass ratios will depend on the lattice cutoff. If the lattice spacing is small enough, the typical behavior will look like (arriiia))/'(am2(a))

= m 1 ( 0 ) / m 2 ( 0 ) + 0(mia)

+ 0((mia)2)

+ ...

(30)

The leading term does not depend on the value of the UV cutoff, while the other terms do. The goal of a lattice calculation is to discover the value of some physical observable as the UV cutoff is taken to be very large, so the physics is in the first term. Everything else is an artifact of the calculation. We say that a calculation "scales" if the a—dependent terms in Eq. (30) are zero or small enough that one can extrapolate to a = 0, and generically refer to all the a—dependent terms as "scale violations." We can imagine expressing each dimensionless combination am(a) as some function of the bare coupling(s) {g(a)}, am = f({g(a)}). As a —• 0 we must tune the set of couplings {g(a)} so lim -f({g(a)})

—> constant.

(31)

a->0 a

From the point of view of the lattice theory, we must tune {g} so that correlation lengths 1/ma diverge. This will occur only at the locations of second (or higher) order phase transitions. In QCD the fixed point is gc = 0 so we must tune the coupling to vanish as a goes to zero. One needs to set the scale by taking one experimental number as input. A complication that you may not have thought of is that the theory we simulate on the computer is different from the real world. For example, the quenched approximation, or for that matter QCD with two flavors of degenerate quarks, almost certainly does not have the same spectrum as QCD with six flavors of dynamical quarks with their appropriate masses. Using one mass to set the scale from one of these approximations to the real world might not give a prediction for another mass which agrees with experiment.

327

(The glass is always half empty...In the strong coupling limit, lattice regularized QCD automatically confines 2 and chiral symmetry is spontaneously broken 6 . So unless there is some kind of phase transition as the bare couplings are tuned to take the cutoff away, which probably doesn't happen, we are working with a confining theory without doing anything special.) Today's QCD simulations range from 163 x 32 to 32 3 x 100 points and run from hundreds (quenched) to thousands (full QCD) of hours on the fastest supercomputers in the world. The cost of an unquenched Monte Carlo simulation in a box of physical size L with lattice spacing a and quark mass mq scales roughly as (-)4(1)1-2(-L)2-3 (32) a a mq where the 4 is just the number of sites, the 1-2 is the cost of "critical slowing down"-the extent to which successive configurations are correlated, and the 2-3 is the cost of inverting the fermion propagator, plus critical slowing down from the nearly massless pions. Thus it is worthwhile to think about how to do the discretization, to maximize the value of the lattice spacing. The thing to keep in mind is that the lattice action is just a bare action defined with a cutoff. No lattice discretization is any better or worse (in principle) than any other. Any bare action which is in the same universality class as QCD will produce universal numbers in the scaling limit. However, by clever engineering, it might be possible to devise actions whose scaling behavior is better, and which can be used at bigger lattice spacing. An example 7 of a test of scale violations is shown in Fig. 2. The x axis is the lattice spacing, in units of a quantity n, which is defined through the heavy quark potential: rfdV(r)/dr\ri = 1.0, about 0.4 fm. The plotting symbols are for different kinds of discretizations. The flatter the curve, the smaller the scale violations. The simplest organizing principle for "improvement" is to use the canonical dimensionality of operators as a guide. Consider the gauge action as an example. If we perform a naive Taylor expansion of a lattice operator like the plaquette, we find that it can be written as 1 - ^Re TrUpiaQ = r0TiF^v + a2[n £ „ „ TiD^F^D^F^ r3'Elll/^DllFltaDvFva] 4

+0{a )

+

+ (33)

The expansion coefficients have a power series expansion in the coupling, Vj = Aj + g2Bj + ... and the expectation value of any operator T computed using

328

(a)

2.0

„ 1.

1.6

1.4 0.0

I

, I ,

0.2 0.4 (a/rj2

0.

Figure 2: Lattice calculations of the (a) rho and (b) nucleon mass, interpolated to the point mnri — 0.778, as a function of lattice spacing.

the plaquette action will have an expansion (T(a)) = (T(0)> + 0(a) + 0(g2a) + ..

(34)

Other loops have a similar expansion, with different coefficients. Now the idea is to take the lattice action to be a minimal subset of loops and systematically remove the an terms for physical observables order by order in n by taking the right linear combination of loops in the action, S = "}2JCJOJ with Cj = 2 8 9,10 Co ; + S Cc] J ++ .... This method was developed by Symanzik and co-workers ' in the mid-80's. Ordinary perturbation theory (expansions in the bare lattice coupling g) are not very convergent, but clever prescriptions for definitions of couplings n or nonperturbative tuning methods 1 2 have been quite successful in developing improved lattice actions. 2.5

Relativistic Fermions on the Lattice

Defining fermions on the lattice involves yet another problem: doubling. Let's illustrate this with free field theory. The continuum free action is S=

/ d4x[iP{x)-fnd^(x)

+ m$(x)i>(x)).

(35)

329 One obtains the so-called naive lattice formulation by replacing the derivatives by symmetric differences: gnaive

=

^

^„g(^ n + / 1 -

^ . ^

+

n,n

m

^

^ „ .

(36)

n

The propagator is: G(p) = (z7^ sinp^a + ma)

= ——— —L M s l n " Pfia + m2 a2

(37)

We identify the physical spectrum through the poles in the propagator, at Po = iE: sinh 2 Ea = '^2 sin2 Pja + m2a2 (38) 3

The lowest energy solutions are the expected ones at p = (0,0,0), E ~ ± m , but there are other degenerate ones, at p = (IT, 0,0), (0, IT, 0,), . . . (71", 7r, 7r). As a goes to zero, the lightest excitations of the spectrum, the ones whose energy is 0(1), not 0 ( l / a ) , are the relevant ones, and there are sixteen of these, in all the corners of the Brillouin zone. Thus our action is a model for sixteen light fermions, not one. This is the famous "doubling problem." In fact, associated with the "doubling problem" is the Nielsen-Ninomaya 13 theorem, which says that no lattice action can be undoubled, chiral, and have couplings which extend over a finite number of lattice spacings (ultralocality). However, there are three ways to get two out of three. They are (a) Wilson Fermions (undoubled, nonchiral, ultralocal) We can alter the dispersion relation so that it has only one low energy solution. The other solutions are forced to E ~ 1/a and become very heavy as a is taken to zero. The simplest version of this solution, called a Wilson fermion, adds an irrelevant operator, a second-derivative-like term SW = ™

$ ^ n ( V w " 2 ^ « + ».-„) - ari,D2^

(39)

n,fi

to Snaive. The parameter r — 1 is almost always used and is implied when one speaks of using "Wilson fermions." There are two dimension-five operators which can be added to a fermion action. The Wilson term is just one of them. The other dimension-five term is a magnetic moment term Ssw

j-4>{x)ainJF^ip(x)

(40)

330

^

=

=

^

Figure 3: The "clover term".

and if both terms are included, their coefficients can be tuned so that there are no 0(ag2) lattice artifacts. This action is called the"Sheikholeslami-Wohlert" 14 or "clover" action because the lattice version of F^IV is the sum of paths shown in Fig. 3. Wilson-type fermions contain an explicit chiral-symmetry breaking term. This causes a lot of bad things to happen. The most obvious is that the zero bare quark mass limit is not respected by interactions; the quark mass is additively renormalized. The value of bare quark mass rnq which the pion mass vanishes, is not known a priori before beginning a simulation; it must be computed. This is done in a simulation involving Wilson fermions by varying mq and watching the pion mass extrapolate quadratically to zero as m^ ~ mq — mcq. It actually turns out that this is a worse problem than you would think: the Dirac operator D o n a gauge configuration could develop a real eigenmode A at minus the bare quark mass you dialed into the program. Then D + m would be non-invertible! Other nasty things happen (operator mixing, see the next section) and people argue about how serious they are in practice 15

(b) Staggered or Kogut-Susskind Fermions (chiral, doubled, ultralocal) In this formulation one reduces the number of fermion flavors by using one component "staggered" fermion fields rather than four component Dirac spinors. The Dirac spinors are constructed by combining staggered fields on

331 0.9

a \

CO

t

6 CO

Figure 4: An example of flavor symmetry breaking in an improved staggered action. The different 7's are a code for the various pseudoscalar states. Data are from Ref. 1 7 . For an explanation of the splitting, see Ref. 1 8 .

different lattice sites. Staggered fermions preserve an explicit chiral symmetry as mq —> 0 even for finite lattice spacing, as long as all four flavors are degenerate, although it is not the SU(Nf) x SU(Nf) of the continuum, it is a U{\). Thus there is only one Goldstone pion at finite a, plus other non-degenerate pseudoscalar states whose mass goes to zero in the continuum limit (See Fig. 4 for an example of this.) They are preferred over Wilson fermions in situations in which the chiral properties of the fermions dominate the dynamics. They also cheaper to simulate than Wilson fermions, because there are less variables. However, flavor and translational symmetry are all mixed together 16 , (c) Chiral, undoubled, but not ultralocal These actions implement a modified version 19 of the chiral rotation 1 dtp - 75(1 - -aD)ip;

Sip = tp(l -

1

-aD)y,

(41)

which is sufficient to preserve all the interesting features of continuum chiral symmetry. An example of such an action is the "domain wall fermion 20 ." It is a variation on the idea that if you have a fermion coupled to a scalar field, and the scalar field interpolates between two minima (forms a soliton), the fermion will develop a zero-energy chiral mode bound to the center of the soliton. Now we go into brane world, extend QCD into five dimensions,

332

and put ourselves and our four dimensional world on the kink. There is an anti-kink out there in the fifth dimension, and as long as it is far away the mode on the kink doesn't see the anti-kink and the 4-d theory on the kink is chiral. But if the anti-kink is too close (fifth dimension too small) the modes mix and chiral symmetry is broken. How close is "too close" is (yet) another engineering question. There are four dimensional analogs of this - think of integrating out the modes in the fifth dimension in favor of a tower of massive fermions, and get "overlap fermions 21 ," or construct a low energy Wilsonian effective action from an underlying chiral theory and get "fixed point fermions 22 ." The bad feature is that these actions have couplings which reach out to many neighboring sites. Their strength drops exponentially with distance, so they are true local actions in the continuum limit, but they are very expensive to simulate. But stay tuned... 3

Hadronic Matrix Elements from the Lattice

One of the major goals of lattice calculations is to provide hadronic matrix elements which either test QCD or can be used as inputs to test the standard model. 3.1

Generic Matrix Element Calculations

Most of the matrix elements measured on the lattice are extracted from expectation values of local operators J(x) composed of quark and gluon fields. For example, if one wanted (0| J{x)\h) one could look at the two-point function

Cjo(t) = J2(0\J(x>t)°(0>ty\0)-

( 42 )

X

Inserting a complete set of correctly normalized momentum eigenstates -, _ 1 _

1 V^f,

\A,f)(A,p\ 2EA(P)

[

'

A,p

and using translational invariance and going to large t gives

W| = , - ' » H .

(44)

2mA

A second calculation of Coo(t) = £ < 0 | O ( M ) O ( 0 , 0 ) | 0 ) -> e - ^ t l ( 0 | 2 ^ l ) | 2

(45)

333

is needed to extract (0| J\A) by fitting two correlators with three parameters. Similarly, a matrix element (h\J\h') can be gotten from CAB(t,t')

= Y,(Q\OA(t)J(x:t')OB(0)\0)-

(46)

x

by stretching the source and sink operators OA and OB far apart on the lattice, letting the lattice project out the lightest states, and then measuring and dividing out (0\OA\h) and (0\OB\h). These lattice matrix elements are not yet the continuum matrix elements. Typically, one is interested in some matrix element defined with a particular regularization scheme. It is a generic feature of quantum field theory that an operator defined in one scheme (MS) will be a superposition of operators in another scheme (lattice). In principle, the superposition could be all possible operators. So generically an operator of dimension D will mix like

(f\Ocn°nt(n)\i)m

= aDJ2 Znm(f\Olatt(a)m\i)

(47)

m

The only restriction are symmetries: in a theory where parity is conserved a vector operator and an axial vector operator can't mix. This is relevant for lattice calculations because the symmetries of the lattice action are in general different from continuum symmetries. For example, the space-time symmetry of the lattice is given by the group of discrete rotations. A more serious source of mixing for light quark operators is the way lattice fermions treat chiral symmetry. Wilson-type fermions break chiral symmetry (even massless ones do so off-shell) and so nothing prevents mixing into "wrong chirality" operators. In Eq. (47) the "diagonal" term will contain the anomalous dimension of the continuum operator Q2 Znn = 1 + T 7 T T ( 7 n !<>g afi + A) + . . . 1071"-

(48)

(which cancels the mu-dependence of the coefficient functionC(fi)(f\Ocont\i)fl is independent of the renormalization point). In principle the leading log could be summed, but in practice we don't know how much of the constant term A should be absorbed into a change of scale of g, so they are just left there. The mixing terms to other dimension D operators die out in the continuum so they don't have any logs. There are also terms for mixing with higher dimensional operators, which give contributions proportional to positive powers of a. (These are usually benign.) One can also have mixing with lower dimensional operators, with contributions involving negative powers of a. (Four-fermion

334

operators for BK could mix with sd.) These are deadly. They must drop out in the continuum but it is a delicate business, since they look like they are growing as an inverse power of a. This is probably more than you wanted to know, but you need the Z n m 's to produce numbers. People get them in a number of ways. Most straightforward is to compute them in perturbation theory, but lattice perturbation theory in terms of the bare coupling g(a) is not very convergent, and it is a long tricky story to do better. The culprit is the "tadpole graph." The lattice fermion-gauge field interaction is generically ip(x)U^(x),il)(x + ft) and £/ ~ 1 + igaA^ - g2a2/2A2l + .... The ipAfa vertex, not present in any sensible continuum regularization, causes problems when the gluon forms a loop: the quadratic divergence from the loop integral combines with the a2 to give a finite contribution - in fact, it is often the dominant contribution. In perturbation theory one must also choose the momentum scale in the running coupling constant. There are reasonable choices for how to do that 2 3 ' 2 4 . Often one can find Znm's by forcing lattice observables to obey Ward identities 25 . One can also play this game with quark propagators and vertices, by computing analogs of quark vertices on the lattice and matching ones results to a continuum calculation 26 . Besides, the Z's, there are other things that can go wrong. Most lattice actions break down when the quark mass gets heavy. The dispersion relation for Wilson or clover actions is E(p) = mi + p2/(2m2) and the quark magnetic moment is fi = l/m^ with rrii ^ m 2 ^ m^. The residue of the quark propagator at its pole is not 1/(2E) as in the continuum. What to do then is not obvious (meaning that lattice people fight over what to do). 3.2

Heavy quark operators

There are many lattice calculations of fg, fjj, BB , and form factors for semileptonic decay. B — B mixing is parametrized by the ratio Xd = (AM)b^/Tb^: d2

*i = rb^rjQCDF(^) on*

m2

mw

•}

\VtlVtd\2b(n){-(B\hP(l o

- 75)d&7,(l - 75)rf|2?>}

(49) Experiment is on the left; theory on the right. Moving into the long equation from the left, we see many known (more or less) parameters from phase space integrals or perturbative QCD calculations, then a combination of CKM matrix elements, followed by a four quark hadronic matrix element 2T . We would like to extract the CKM matrix element from the measurement of Xd (and its strange partner xs). To do so we need to know the value of the object in the curly brackets, defined as (3/8)M(,d and parameterized as mBf'BdBt>d

335

where Bid is the so-called 5-parameter, and JB is the B-meson decay constant, (0|67o75^|-B> = fBiriB- Vacuum saturation suggests that BB = 1. From the lattice one can try to get a real value. In Eq. (49) b(fi), the coefficient which runs the effective interaction down from the W-boson scale to the QCD scale fi, and the matrix element M(fi) both depend on the QCD scale. One often sees the renormalization group invariant quantities Mbd = b(ij)Mu{^) or B^ = b(fi)Bbd(lJ-) quoted in the literature. Decay constants probe very simple properties of the wave function: In the nonrelativistic quark model / M OC ip(0)/y/mM, where V(0) is the qq wave function at the origin. For a heavy quark (Q) light quark (q) system ij>(0) should become independent of the heavy quark's mass as the Q mass goes to infinity, and in that limit one can show in QCD that ^TUMIM approaches a constant. The decay constant is computed by combining a heavy quark and a light antiquark propagator into Eq. (42). You might think it would be difficult to calculate JB directly on present day lattices with relativistic lattice fermions because the lattice spacing is much greater than the b quark's Compton wavelength (or the UV cutoff is below mt,). But it is better to think of the lattice theory as an effective field theory for the low-momentum excitations in the presence of additional high energy scales - the cutoff (inverse lattice spacing) and the heavy quark mass. As in any effective field theory, the effects of the short distance are lumped into coefficients of the effective theory 28 . As a practical matter, one can use the good old clover action to do the calculations - it contains all the necessary operators. The bare mass mi has nothing direct to do with the results; one tunes it, monitoring the kinetic energy E(p) = mi + p2/(2m-2) + ..., and takes the hadron mass to be miNonrelativistic QCD has also been discretized and used to make very precise calculations of the properties of quarkonia 29 . This formalism can also be used for the heavy quark (again as long as its momentum is small.) The "static" limit (infinite 6-quark mass) is often used as an additional point on the curve. Then one can try to extrapolate all the way from light quarks to heavies and get all the decay constants at once. I will show some pictures from the lattice decay constant calculation of Ref. 30 . These authors (my name is on it but I didn't do anything) did careful quenched simulations at many values of the lattice spacing, which allows one to extrapolate to the continuum limit by brute force. They have also done a set of simulations which include light dynamical quarks, which should give some idea of the accuracy of the quenched approximation. Examples of results of Ref. 30 are shown in Figs. 5 and 6. The simulations

336

with dynamical fermions are not as good quality as the quenched simulations: the lattice spacings are generally larger, the simulations all have two degenerate flavors (what about the strange quark?), and the dynamical quark masses are still a bit large. We think that the Wilson results (crosses in Fig. 6(b)) overestimate the continuum result, and the clover action we are using underestimates it, but we also suspect that quenched JB is a bit too low.

0.6

T

r

\\

w

0.5

\,

CO

T

1

1

r

u s e d for-

CL=0.7!=:

u s e d for-

CL=0.7S'

u s e d for-

CL=0.99

> CD

O

0.4

>

0.3

0.2

& 0.0

D

i

i

0.5

i

1.0 -1

1/Mp (GeV ) Figure 5: Pseudoscalar meson decay constant vs 1/M, from Ref.30. CL is the confidence level of the fit.

Soni 3 1 has presented a summary of data from various collaborations, as of last winter. Again, there is a hint that the Nj = 2 results may be about 30 MeV above the quenched results. Lattice calculations have been predicting quenched /D, — 200 MeV for about twelve years. The central values have changed very little, while the uncertainties have decreased. Experimental determinations of fut all come in higher than the lattice results, though with large error bars. We need to do a good quality unquenched lattice calculation.

337 (a)

0.0

0.2

0.4 a (Gey)"1

(b)

0.6

0.8

0.0

0.2

0.4 a (OeV)-1

0.6

0.8

Figure 6: JB VS. a from Ref. 3 0 , showing extrapolations to the continuum limit of quenched (a) and full (b) QCD data. The scale is set by fw = 132 MeV throughout.

Table 1: Heavy-light decay constants and their ratios.

Quantity /s/MeV fBs/MeV fn/MeV fDs/MeV fBs/fB IDSIID

Quenched (Nf = 0) 170 ± 20 190 ± 20 205 ± 20 225 ± 20 1.14 ± . 0 6 1.10 ±.06

Partially Unquenched (Nf — 2) 200 ± 30 220 ± 30 225 ± 30 245 ± 30 1.14 ± . 0 6 1.10 ± . 0 6

Now back to the B parameter. On the lattice, one could measure the decay constants and B parameter separately and combine them after extrapolation, or measure M directly. In principle the numbers should be the same, but in practice the first technique has produced better numbers so far. That is because the B parameter is measured as the ratio of a correlator with a fourfermion vertex to a product of two current-current correlators (see Fig. 7). A lot of systematics cancel in the ratio. Many groups have visited this problem. Reviews by Draper 3 3 and Soni 31 quote a world summary, which I copy into Table 2. Semileptonic decays involve processes like Eq. (46). On the lattice, one just measures the matrix element of a current and fits it to the expected set of form factors - for B —¥ •nf.v, for example, (7r(p)|^|B(p')> = / + ( g 2 ) b ' +P ~ ^

.^q],

+/ V r

1

ml n2

Q,

(50)

338

71

Figure 7: BK (shown) and BB are computed by taking the ratio of four quark operator and two two-point functions. (Figure from Ref. 3 2 .)

Table 2: Summary of heavy-light B-parameters.

BBd(mb) BBs/BBd

fBd(BnBy/2 /BS(B£';)1/2

WB£';)1/2

Quenched .86(4) (8) 1.00(1)(2) 195 ± 25 MeV

"Unquenched" .86(4)(8) 1.00(1)(2) 230 ± 35

1.14 ±.06

1.14 ± . 0 7

The best signals come when the momenta of the initial and final hadron are small. Then the large B mass forces q2 (q = lepton 4-momentum = ps — Pn) to be large. If the form factor is needed at q2 ~ 0, a large extrapolation is needed, and there will be additional errors and model dependence in the answer. (Lattice people have no advantage over anyone else at guessing at functional forms.) However, finding Vub from experimental data only requires knowing the form factor at one value of q2. This should work so long as the experiment has enough data to measure the differential rate around that region of q2. Two recent approaches try to do this: UKQCD focussed on near the end-point or the zero-recoil region where the lattice data tends to be cleanest and heavy quark symmetry can be used. The FNAL group has measured B —• Dlv form factors at zero recoil 34 . They have a clever technique from removing much of the lattice-to-continuum Z-factors by computing ratios of matrix elements, such as {D\clQb\B){B\bl0c\D) (D\cl0c\D)(B\bl0b\B)

= \h+(vv'

= l)f

(51)

339 T h e denominators are just diagonal matrix elements of the charge density, and they can easily be normalized. They 3 5 are also computing semi-leptonic form factors for B —> -KIV and D —• n(K)lis, by concentrating directly on the differential decay spectrum in an interval with 0.4 < pn/GeV < 0.8 thus avoiding the need for large extrapolation in q2. 3.3

Kaon Matrix

Elements

Lattice calculations of kaon weak interaction matrix elements begin with the full S t a n d a r d Model at high energies and use the operator product expansion, combined with the renormalization group, to construct a low-energy effective field theory valid at scales /j, of a few GeV. T h e effective Hamiltonian basically reduces to a sum of four-fermion interactions

People have expended the most effort on, and have the best results for, B^; there are some results on the AI = 1/2 rule; and last, there is e'/e, with unreliable results so far. BK_

T h e J L Q C D collaboration has the best results on BK, from a calculation using staggered fermions 36 . They have d a t a from many lattice spacings and several choices for the lattice discretization of the operator. (See Fig. 8.) They find quenched BK(MS,II = 2 GeV) = 0.616(5). T h e main limitations of this result are quenching, plus the fact t h a t the lattice calculations are actually done without S t / ( 3 ) flavor breaking (the lattice "kaon" is a pseudoscalar m a d e of degenerate quarks). These effects are believed 3 1 to be 5-10 per cent corrections. J L Q C D 3 7 has also done a calculation with Wilson fermions. This was done not so much to get a number itself but to check the staggered result. T h e systematics are very different and the operator mixing is fierce due t o t h e loss of chiral symmetry inherent in Wilson fermions. For example, S7 M (1 — 7s)d • S7 M (1 — 75)d mixes with s^d • £75 d and t h a t operator has a K — K matrix element ten times greater. A I = 1/2 Rule Can the lattice reproduce the experimentally observed factor of 22 between K° —• (7T7r)/=o and K° —> (TT7T)J=2 amplitudes? The lattice calculations are difficult. In addition to graphs of Fig. 7, which are reasonably straightforward to compute, there are a host of other topologies, some of which involve Computing propagators from many points on the lattice to many other points. But I think the reason there are so few lattice results is because all of the

340

BK(NDR, 2GeV) vs. mpa q =1/a, 3-loop coupling, 5 points

0.9 e 0.8

0.7

0.6

0.5 -0.2

o non-invariant O invariant

0.0

0.2

0.4 m„a

0.6

0.8

Figure 8: B ^ from staggered fermions, as a function of the lattice spacing, for two different choices of lattice operators. (Figure and results from Ref. 3 6 .)

quantities of interest are scheme dependent and one must compute a latticeto-continuum matching factor. In addition, people don't calculate K —> TTTT directly on the lattice; it is difficult38 to extract the phase shifts from the TTTT final state interactions from lattice data (never mind trying to separate the two pions to asymptotically great distances). Instead, they use chiral perturbation theory 39 to relate K —> TTTT amplitudes to K —> IT. In the case of the A7 = 3/2 amplitude there is a factor of two change in the lattice result depending on whether tree level or one loop chiral perturbation theory is used. This is shown in Fig. 9.

Hi The only recent work I am aware of is by Pekurovsky and Kilcup 32 . The calculation is hard. The biggest operators (06 and <38 in the nomenclature) have opposite signs and nearly cancel. But the big problem is the scheme matching. In a perturbative calculation, we saw that {0)j^ ~ Z(0)iatt and Z = 1 + asC + But in this equation, what scheme is used to compute as, and what is the scale q* at which as is evaluated? Pekurovsky and Kilcup found that their numbers for one operator, 06> shifted by a factor of 4 when

341

0.04

T

1

1

r

i

i

i

r

C +
+

r

|K>

>c 0.03 X

<>

<> <>

2 x expj.

-A>

0.02

<> <>-

St

¥ 11

0.01

41

0.0

expt-

0.5 m.

1.0 GeV<

Figure 9: AI — 3/2 K —> irir amplitude with (fancy symbols) and without (plain symbols) one-loop corrections of quenched chiral perturbation theory. Data are crosses and fancy Data are plotted as a function of lattice crosses, ; diamonds and fancy diamonds, meson mass. This figure (and much else) is from'

they were converted into MS, and the factor of 4 could become a factor of 30 (or worse) as q* was varied from ir/a to 1/a. They attempt to guesstimate numbers but since they say plainly that they should be used "with extreme caution" I won't quote them. A nonperturbative approach to matching is clearly needed but does not exist yet. 4

Conclusions

What about the future? Matrix elements are at the end of a long chain involving a large set of both simulation and physics issues. They are the most complicated corner of the lattice game. If all you want are the numbers, Moore's

342

law says that computer speed doubles every eighteen months, and statistics is y/N, so error bars will fall by a factor of 2 every three years for everything we know how to do today and will learn nothing new about how to do better in the future. And there are many projects and proposals to build clusters or dedicated supercomputers at a cost which is "chicken feed" compared to the experimental program. This will enable us to begin to chip away at the biggest systematic in all the calculations I have shown here - the neglect of the quenched approximation. But new hardware is not really where the action is. It is merely "enabling technology," so we can make mistakes faster, learn more about the physics, and test new ideas. The main bottleneck to progress on hadronic matrix elements is just that these calculations are complex and have many parts. Some of us (me, let's not be shy) think that better discretization algorithms will help. The problem with that approach is that many pieces of the puzzle have to be determined from scratch: learning how to optimize the new algorithm, testing spectroscopy, computing the Z's. This takes a couple of years, if the inventor of the algorithm doesn't get tired first. Others of us prefer to live with poorer algorithms (which have already been well calibrated) and try to tweak the parts of them which work the worst. The simulations still take a couple of years. Believe it or not, even though lattice QCD is a mature field, there are still many questions about QCD which lattice people do not know how to answer, and an outsider might. Maybe you would enjoy thinking about them. Acknowledgements I would like to thank S. Gottlieb, Y. Kuramashi, A. Kronfeld, M. Ogilvie, S. Sharpe, A. Soni, and D. Toussaint for their help preparing these lectures. This work was supported by the U . S . Department of Energy. 1. Some standard reviews of lattice gauge theory are M. Creutz, "Quarks, Strings, and Lattices," Cambridge, 1983. M. Creutz, ed., "Quantum Fields on the Computer," World, 1992, and I. Montvay and G. Munster, "Quantum fields on a Lattice," Cambridge, 1994. The lattice community has a large annual meeting and the proceedings of those meetings (Lattice 'XX, published so far by North Holland) are the best places to find the most recent results. Other reviews, that I have particularly enjoyed reading, include G. P. Lepage, Schladming 1996: Perturbative and nonperturbative aspects of quantum field theory, H. Latal and W. Schweiger, eds., Springer 1997. hep-lat/9607076; M. Liischer, Les Houches Summer School lecture, 1997. hep-lat/9802029. My 1996 TASI lectures have more

343

2. 3.

4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15.

detail and a different focus than these: T. DeGrand, TASI-96, B. Green, ed; hep-th/9610132. Sharpe's 1994 TASI lectures are still quite relevant: S. R. Sharpe, TASI-94, J. Donoghue, ed.; hep-ph/9412243. The MILC code, one of the modern packages of lattice QCD codes, is available at h t t p : / / c l i o d h n a . c o p . u o p . e d u / ~ h e t r i c k / m i l c / . There is even documentation, but these codes are not designed to be run as black boxes. K. Wilson, Phys. Rev. D 10, 2445, 1974. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953); F. Brown and T. Woch, Phys. Rev. Lett., 58, 2394, 1987; M. Creutz, Phys. Rev., D36, 55, 1987. H. C. Andersen, J. Chem. Phys., 72, 2384, 1980; S. Duane, Nucl. Phys. B257, 652, 1985; S. Duane and J. Kogut, Phys. Rev. Lett. 55, 2774, 1985; S. Gottlieb, W. Liu, D. Toussaint, R. Renken and R. Sugar, Phys. Rev. D35, 2531, 1987; S. Duane, A. Kennedy, B. Pendleton, and D. Roweth, Phys. Lett. 194B, 271, 1987. For a recent review, see A. Frommer, Nucl. Phys. B(Proc. Suppl.) 53, 120 (1997); hep-lat/9608074. Cf. H. Kluberg-Stern, A. Morel, and B. Petersson, Phys. Lett. 114B, 152,1982. C. Bernard, et al. (MILC Collaboration), Nucl. Phys. B(Proc. Suppl.) 73 (1999) 198. K. Symanzik, in "Recent Developments in Gauge Theories," eds. G. 't Hooft, et al. (Plenum, New York, 1980) 313; in "Mathematical Problems in Theoretical Physics," eds. R. Schrader et al. (Springer, New York, 1982); Nucl. Phys. B226 (1983) 187, 205. P. Weisz , Nucl. Phys. B212 (1983) 1. M. Luscher and P. Weisz, Comm. Math. Phys. 97, 59 (1985). M. Luscher and P. Weisz, Phys. Lett. B158, 250 (1985). M. Alford, W. Dimm, G. P. Lepage, G. Hockney, and P. Mackenzie, Phys. Lett. B361, 87 (1994); G. P. Lepage, 1 . K. Jansen, et al., Phys. Lett. B372, 275 1996 (hep-lat/9512009); M. Luscher, S. Sint, R. Sommer and P. Weisz, Nucl. Phys. B 478, 365 (1996) [hep-lat/9605038]. M. Luscher, S. Sint, R. Sommer, P. Weisz, H. Wittig and U. Wolff, Nucl. Phys. Proc. Suppl. 53, 905 (1997) [heplat/9608049]. H. B. Nielsen and M. Ninomiya, Phys. Lett. B105, 219 (1981); Nucl. Phys. B193, 173 (1981); Nucl. Phys. B185, 20 (1981); B. Sheikholeslami and R. Wohlert, Nucl. Phys. B259, 572 (1985). For technical reasons, lattice people using Wilson or clover fermions like to work with the "hopping parameter," universally labeled K. It is related

344

to the bare quark mass by K = l/(2mg

+ 8). The pion is massless at

16. See M. Golterman and J. Smit, Nucl. Phys., B255, 328, 1985. 17. K. Orginos, D. Toussaint and R. L. Sugar [MILC Collaboration], Phys. Rev. D60, 054503 (1999) [hep-lat/9903032]. 18. W. Lee and S. R. Sharpe, Phys. Rev. D60, 114503 (1999) [heplat/9905023]. 19. M. Liischer, Phys. Lett. B428, 342 (1998) [hep-lat/9802011]. 20. For a recent review see T. Blum, Lattice 98, Nucl. Phys. B (Proc. Suppl.) 73 (1999) 167, hep-lat/9810017. 21. For a review, see H. Neuberger,Lattice '99, Nucl. Phys. B (Proc. Suppl.) 83-84 (2000) 67. 22. P. Hasenfratz and F. Niedermayer, Nucl. Phys. B414 (1994) 785; T. DeGrand, Phys. Rev. D60, 094501 (1999) [hep-lat/9903006]. 23. S. Brodsky, G. P. Lepage and P. Mackenzie, Phys. Rev. D28, 228 (1983). 24. G. P. Lepage and P. Mackenzie, Phys. Rev. D48, 2250 (1993). See also G. Parisi, in High Energy Physics-1980, XX Int. Conf., Madison (1980), ed. L. Durand and L. G. Pondrom (AIP, New York, 1981) 25. L. Maiani and G. Martinelli, Phys. Lett. B178 (1986) 285. 26. G. Martinelli, et al., Nucl. Phys. B445 (1995) 81. 27. See the lectures of J. Rosner at this summer school. 28. A. X. El-Khadra, A. S. Kronfeld and P. B. Mackenzie, Phys. Rev. D55, 3933 (1997); A. S. Kronfeld, Phys. Rev. D 62, 014505 (2000) [heplat/0002008]. 29. C. Davies et al., Phys. Rev. D52 (1995) 6519, Phys. Lett.B 345 (1995) 42; Phys. Rev. D50 (1994) 6963; Phys. Rev. Lett. 73 (1994) 2654; and G.P. Lepage et al., Phys. Rev. D46 (1992) 4052. 30. C. Bernard et al, Phys. Rev. Lett. 81, 4812 (1998) [hep-ph/9806412]. 31. A. Soni, to be published in the Third International Conference on B Physics and CP Violation, hep-ph/0003092. 32. D. Pekurovsky and G. Kilcup, hep-lat/9812019. 33. T. Draper, Nucl. Phys. Proc. Suppl. 73, 43 (1999) [hep-lat/9810065]. 34. S. Hashimoto, et al, Phys. Rev. D61, 014502 (1999). 35. S. Ryan, A. El-Khadra, S. Hashimoto, A. Kronfeld, P. Mackenzie and J. Simone, Nucl. Phys. Proc. Suppl. 73, 390 (1999) [hep-lat/9810041]. S. M. Ryan, A. X. El-Khadra, A. S. Kronfeld, P. B. Mackenzie and J. N. Simone, Nucl. Phys. Proc. Suppl. 83, 328 (2000) [hep-lat/9910010]. 36. S. Aoki, et al., Phys. Rev. Lett. 80, 5271 (1998). 37. S. Aoki et al. [JLQCD Collaboration], Phys. Rev. D60, 034511 (1999) [hep-lat/9901018].

345

38. L. Maiani and M. Testa, Phys. Lett. 235B, 585 (1990). For recent developments, see L. Lellouch and M. Luscher, hep-lat/0003023. 39. C. Bernard, et al., Phys. Rev. D32, 2343 (1985). 40. C. Bernard, in the Proceedings of the 1989 TASI Summer School, eds. T. DeGrand and D. Toussaint, World Scientific, Singapore, 1989. 41. JLQCD Collab., Phys. Rev. D58 (1998) 054503. 42. Y. Kuramashi, Nucl. Phys. B(Proc. Suppl.) 83-84 (2000) 24; heplat/9910032.

This page is intentionally left blank

illl^lililllllllll^llSIllBllll^lilllillllll

•Hflii

i

;7

'

s

T|J^« iA^s^m^&^^^

^'•mmm

Michael Dine

This page is intentionally left blank

THE STRONG CP PROBLEM

Michael Dine Stanford Linear Accelerator Center, Stanford CA 94309 Santa Cruz Institute for Particle Physics, Santa Cruz CA 95064 Physics Department, University of California, Santa Cruz CA 95064

These lectures discuss the 6 parameter of QCD. After an introduction to anomalies in four and two dimensions, the parameter is introduced. That such topological parameters can have physical effects is illustrated with two dimensional models, and then explained in QCD using instantons and current algebra. Possible solutions including axions, a massless up quark, and spontaneous C P violation are discussed.

1

Introduction

Originally, one thought of QCD as being described a gauge coupling at a particular scale and the quark masses. But it soon came to be recognized that the theory has another parameter, the 6 parameter, associated with an additional term in the lagrangian:

where

K** = \^-Fe°a-

(2)

This term, as we will discuss, is a total divergence, and one might imagine that it is irrelevant to physics, but this is not the case. Because the operator violates CP, it can contribute to the neutron electric dipole moment, dn. The current experimental limit sets a strong limit on 9, 9
C = —^Flv +qf +q fmqq + m*?q*.

(3)

Here, I have written the lagrangian in terms of two-component fermions, and noted that a priori, the mass need not be real, m = \m\eid.

(4)

C = Re m qq + Im m qq^q.

(5)

In terms of four-component fermions,

349

350 In order to bring the mass term to the conventional form, with no 75's, one would naively let -ie/2n

-i»/2-

(6)

However, a simple calculation shows that there is a difficulty associated with the anomaly. Suppose, first, that M is very large. In that case we want to integrate out the quarks and obtain a low energy effective theory. To do this, we study the path integral:

z = jldA^jldqWdqy8

(7)

Again suppose m = e'sM, where S is small and M is real. In order to make m real, we can again make •iS/275 5).) The the transformations: q -¥ qe ,e/2;q -> qe , s / 2 (in four component language, this is < result of integrating out the quark, i.e. of performing the path integral over q and q can be written in the form:

Z = J[dA„]JeiS-»

(8)

Here 5 e / / is the effective action which describes the interactions of gluons at scales well below M.

>

Figure 1: The triangle diagram associated with the four dimensional anomaly.

Because the field redefinition which eliminates 8 is just a change of variables in the path integral, one might expect that there can be no ^-dependence in the effective action. But this is not the case. To see this, suppose that 6 is small, and instead of making the transformation, treat the 9 term as a small perturbation, and expand the exponential. Now consider a term in the effective action with two external gauge bosons. This is given by the Feynman diagram in fig. 1. The corresponding term in the action is given by SCeff =

-i-g2MTr(T"Tb)

r d*k

J (2^>

Tr 7 5

1

1

fl+tf1-M"ll-M"ll-42-M

(9)

Here, as in the figure, the qi's are the momenta of the two photons, while the e's are their polarizations and a and b are the color indices of the gluons. To perform the integral, it is convenient to introduce Feynman parameters and shift the k integral, giving: SCeff

= -i8g2MTT(TaT»)

J daida2

J J ^ r ^ n ^ ~ « i 4x + »2 f/2+ «i + M) rf

(V - « i <jx + «2 4i + M) ^(V-ai

(10)

1

q ! + a2 42~ 4i + M)

(k2 - M2 + 0( 9 , 2 )) 3

(11)

351 For small q, we can neglect the g-dependence of the denominator. The trace in the numerator is easy to evaluate, since we can drop terms linear in k. This gives, after performing the integrals over the a's,

sccff = j » M t o ( r r ' ) e ^ g f ^ 4 / ^ y . ' ^ , .

(12)

This corresponds to a term in the effective action, after doing the integral over k and including a combinatoric factor of two from the different ways to contract the gauge bosons:

6C

°"

=

zkieTT{Fp)-

(13)

Now why does this happen? At the level of the path integral, the transformation would seem to be a simple change of variables, and it is hard to see why this should have any effect. On the other hand, if one examines the diagram of fig. 1, one sees that it contains terms which are linearly divergent, and thus it should be regulated. A simple way to regulate the diagram is to introduce a Pauli-Villars regulator, which means that one subtracts off a corresponding amplitude with some very large mass A. However, we have just seen that the result is independent of A! This sort of behavior is characteristic of an anomaly. Consider now the case that m -C AQCD- In this case, we shouldn't integrate out the quarks, but we still need to take into account the regulator diagrams. For small m, the classical theory has an approximate symmetry under which q -> eiaq (in four component language, q —> eiaibq).

q -> eiaq

(14)

In particular, we can define a current: 3s = 9757^9-

(15)

and classically, dJs

= mq-y^q.

(16)

Under a transformation by an infinitesmal angle a one would expect SL = adrfg

= maq-yaq.

(17)

But what we have just discovered is that the divergence of the current contains another, m-independent, term: d„j£ = mql5q

+ ^ F F .

(18)

This anomaly can be derived in a number of other ways. One can define, for example, the current by "point splitting,"

iS = g(x + U)eiS.'*'dt'A'q{x)

(19)

Because operators in quantum field theory are singular at short distances, the Wilson line makes a finite contribution. Expanding the exponential carefully, one recovers the same expression for the current. A beautiful derivation, closely related to that we have performed above, is due to Fujikawa, described i n 1 . Here one considers the anomaly as arising from a lack of invariance of the path integral measure. One carefully evaluates the Jacobian associated with the change of variables q —*• q(l + ij^a), and shows that it yields the same result 1 . We will do a calculation along these lines in a two dimensional model shortly. The anomaly has important consequences in physics which will not be the subject of the lecture today, but it is worth at least listing a few before we proceed:

352 • TT° decay: the divergence of the axial isospin current, 0 1 ) " = J»T57"« " hil"d

(20)

has an anomaly due to electromagnetism. This gives rise to a coupling of the ir° to two photons, and the correct computation of the lifetime was one of the early triumphs of the theory of quarks with color. • Anomalies in gauge currents signal an inconsistency in a theory. They mean that the gauge invariance, which is crucial to the whole structure of gauge theories (e.g. to the fact that they are simultaneously unitary and lorentz invariant) is lost. The absence of gauge anomalies is one of the striking ingredients of the standard model, and it is also crucial in extensions such as string theory. Our focus in these lectures will be on another aspect of this anomaly: the appearance of an additional parameter in the standard model, and the associated "Strong CP problem." What we have just learned is that, if in our simple model above, we require that the quark masses are real, we must allow for the possible appearance in the lagrangian of the standard model, of the 6-terms of eqn. 1. ° This term, however, can be removed by a B + L transformation. What are the consequences of these terms? We will focus on the strong interactions, for which these terms are most important. At first sight, one might guess that these terms are in fact of no importance. Consider, first, the case of QED. Then

/'

d4xFF

(21)

is a the integral of a total divergence, FF = EB=l-d^l>aA-'F»°.

(22)

As a result, this term does not contribute to the classical equations of motion. One might expect that it does not contribute quantum mechanically either. If we think of the Euclidean path integral, configurations of finite action have field strengths, F^ which fall off faster than 1/r 2 (where r is the Euclidean distance), and A which falls off faster than 1/r, so one can neglect surface terms in Cg. (A parenthetical remark: This is almost correct. However, if there are magnetic monopoles there is a subtlety, first pointed out by Witteri 2 . Monopoles can carry electric charge. In the presence of the S term, there is an extra source for the electric field at long distances, proportional to 9 and the monopole charge. So the electric charges are given by: (23) where nm is the monopole charge in units of the Dirac quantum.) In the case of non-Abelian gauge theories, the situation is more subtle. It is again true that FF can be written as a total divergence: F i ? = a«if M a

^

=

W

( A ^ _ | / ° ^ ^ A < ) .

(24)

I n principle we must allow a similar term, for the weak interactions. However, B + L is a classical symmetry of the renormalizable interactions of the standard model. This symmetry is anomalous, and can be used to remove the weak 8 term. In the presence of higher dimension B + L-violating terms, this is no longer true, but any effects of 6 will be extremely small, suppressed by e~27r^aw as well as by powers of some large mass scale.

353 But now the statement that F falls faster than 1/r 2 does not permit an equally strong statement about A. We will see shortly that there are finite action configurations - finite action classical solutions - where F T-, but A -» -, so that the surface term cannot be neglected. These terms are called instantons. It is because of this that 9 can have real physical effects.

2

A Two Dimensional Detour

Before considering four dimensions with all of its complications, it is helpful to consider two dimensions. Two dimensions are often a poor analog for four, but for some of the issues we are facing here, the parallels are extremely close. In these two dimensional examples, the physics is more manageable, but still rich.

2.1

The Anomaly In Two

Dimensions

Consider, first, electrodynamics of a massless fermion in two dimensions. Let's investigate the anomaly. The point-splitting method is particularly convenient here. Just as in four dimensions, we write: jg = i>(x + ie)e'f'

'A'ix'~f''4i(x)

(25)

For very small e, we can pick up the leading singularity in the product of ^{x + e)'t/> by using the operator product expansion, and noting that (using naive dimensional analysis) the leading operator is the unit operator, with coefficient proportional to 1/e. We can read off this term by taking the vacuum expectation value, i.e. by simply evaluating the propagator. String theorists are particularly familiar with this Green's function:

$(x + eW(x)) = l-£

(26)

Expanding the factor in the exponential to order e gives d^

= naive piece + ^ ^ e ^ t r - ^ V •

(27) 2

Taking the trace gives e ^ e " ; averaging e over angles (< eMe„ > = | r / ^ e ) } yields d,f,

= ^ F " " .

(28)

E x e r c i s e : Fill in the details of this computation, being careful about signs and factors of 2. This is quite parallel to the situation in four dimensions. The divergence of the current is itself a total derivative: d„#

= ^ ^ A " .

(29)

So it appears possible to define a new current, J»=ft-^d»A»

(30)

However, just as in the four dimensional case, this current is not gauge invariant. There is a familiar field configuration for which A does not fall off at infinity: the field of a point charge. Indeed, if one

354

has charges, ±6 at infinity, they give rise to a constant electric field, Foi = e9. So 8 has a very simple interpretation in this theory. It is easy to see that physics is periodic in 8. For 8 > q, it is energetically favorable to produce a pair of charges from the vacuum which shield the charge at oo.

2.2

The CPN Model: An Asymptotically

Free Theory

The model we have considered so far is not quite like QCD in at least two ways. First, there are no instantons; second, the coupling e is dimensionful. We can obtain a theory closer to QCD by considering the CPN model (our treatment here will follow closely the treatment in Peskin and Schroeder's problem 13.2?). This model starts with a set of fields, zit i = 1 , . . . N + 1. These fields live in the space CPN. This space is defined by the constraint:

£ N 2 = i;

(3i)

in addition, the point Zi is equivalent to exaZi- To implement the first of these constraints, we can add to the action a lagrange multiplier field, \{x). For the second, we observe that the identification of points in the "target space," CPN, must hold at every point in ordinary space-time, so this is a U(l) gauge symmetry. So introducing a gauge field, A^, and the corresponding covariant derivative, we want to study the lagrangian: C = ±[\D„zi\2-\(x)(\zt\2-l)]

(32)

Note that there is no kinetic term for A^, so we can simply eliminate it from the action using its equations of motion. This yields

^ ^ • V Z + I^^-I2]

(33)

It is easier to proceed, however, keeping A^ in the action. In this case, the action is quadratic in z, and we can integrate out the z fields: Z = [[dA^dXlidz^expl-C]

-/

[dA\[d\]eS

d2xT

: J[dA][d\]exp{-Nti\og{-D2

2.3

The Large N

(34)

-"[A'x]

(35)

- A) - -i- f

d2x\]

Limit

By itself, the result of eqn. 35 is still rather complicated. The fields A,, and A have complicated, nonlocal interactions. Things become much simpler if one takes the "large A^ limit", a limit where one takes N -> oo with g2N fixed. In this case, the interactions of A and A^ are suppressed by powers of N. For large N, the path integral is dominated by a single field configuration, which solves

^f=°

(36)

355 or, setting the gauge field to zero, N


J (2^P

1

1

+X

g2'

l

'

giving A = m2 = M e x p [ - ^ ] .

(38)

Here, M is a cutoff required because the integral in eqn. 37 is divergent. This result is remarkable. One has exhibited dimensional transmutation: a theory which is classically scale invariant contains non-trivial masses, related in a renormalization-group invariant fashion to the cutoff. This is the phenomenon which in QCD explains the masses of the proton, neutron, and other dimensionful quantities. So the theory is quite analogous to QCD. We can read off the leading term in the /3-function from the familiar formula:

m = Me~Svk

(39)

P(g) = ~y3b„

(40)

so, with

Z7T

we have b0 — 1. But most important for our purposes, it is interesting to explore the question of ^-dependence. Again, in this theory, we could have introduced a 9 term:

Ce = -^Jd\^F»\

(41)

where FM„ can be expressed in terms of the fundamental fields ZJ. As usual, this is the integral of a total divergence. But precisely as in the case of 1 + 1 dimensional electrodynamics we discussed above, this term is physically important. In perturbation theory in the model, this is not entirely obvious. But using our reorganization of the theory at large N, it is. The lowest order action for A^ is trivial, but at one loop (order 1/iV), one generates a kinetic term for A through the usual vacuum polarization loop: -

N

2ivm2

-

(42) 2

2

At this order, the effective theory consists of the gauge field, then, with coupling e = ™ , and some coupling to a dynamical, massive field A. As we have already argued, 8 corresponds to a nonzero background electric field due to charges at infinity, and the theory clearly can have non-trivial ^-dependence. There is, in addition, the possibility of including other light fields, for example massless fermions. In this case, one can again have an anomalous U(l) symmetry. There is then no ^-dependence, since it is possible to shield any charge at infinity. But there is non-trivial breaking of the symmetry. At low energies, one has now a theory with a fermion coupled to a dynamical U(l) gauge field. The breaking of the associated U(l) in such a theory is a well-studied phenomenon 3 . E x e r c i s e : Complete Peskin and Schroeder, Problem 13.3.

2.4

The Role of

Instantons

There is another way to think about the breaking of the f/(l) symmetry and (J-dependence in this theory. If one considers the Euclidean functional integral, it is natural to look for stationary points of

356 the integration, i.e. for classical solutions of the Euclidean equations of motion. In order that they be potentially important, it is necessary that these solutions have finite action, which means that they must be localized in Euclidean space and time. For this reason, such solutions were dubbed "instantons" by 't Hooft. Such solutions are not difficult to find in the CPN model; we will describe them briefly below. These solutions carry non-zero values of the topological charge, — / cfxe^Ff,,,

=n

(43)

and have an action 27m. As a result, they contribute to the ^-dependence; they give a contribution to the functional integral:

j[dSzi\e-"i^"'+...

^e^

(44)

It follows that: • Instantons generate ^-dependence • In the large N limit, instanton effects are, formally, highly suppressed, much smaller that the effects we found in the large N limit • Somewhat distressingly, the functional integral above can not be systematically evaluated. The problem is that t h e classical theory is scale invariant, as a result of which, instantons come in a variety of sizes. J[dSz] includes an integration over all instanton sizes, which diverges for large size (i.e. in the infrared). This prevents a systematic evaluation of the effects of instantons in this case. At high temperatures 4 , it is possible to do the evaluation, and instanton effects are, indeed, systematically small. It is easy to construct the instanton solution in the case of CP1. Rather than write the theory in terms of a gauge field, as we have done above, it is convenient to parameterize the theory in terms of a single complex field, 0. One can, for example, define tp — z\/z2- Then, with a bit of algebra, one can show that the action for <j> takes the form:

£

= (w*>r^-(TT»

(45)

One can think of the field <j> as living on the space with metric given by the term in parenthesis, g^-. One can show that this is t h e metric one obtains if one stereographically maps the sphere onto the complex plane. This mapping, which you may have seen in your m a t h methods courses, is just: ... *!+«». 1 - x3

(46)

The inverse is

*(1+ z )

1+ z

2

|z| - 1 X3 •

N 2 + i'

It is straightforward to write down the equations of motion -~) = 0

(48)

357 Now calling the space time coordinates z = n + ix2, z* = xx - 1x2, you can see that if is analytic, the equations of motion are satisfied! So a simple solution, which you can check has finite action, is (z) = pz.

(49)

In addition to evaluating the action, you can evaluate the topological charge,

^Jd2xe^F""

=l

(50)

for this solution. More generally, the topological charge measures the number of times that <j> maps the complex plane into the complex plane; for <j> = zn, for example, one has charge n. Exercise: Verify that the action of eqn. 45 is equal to £ = 9w9d09e0*

(51)

where g is the metric of the sphere in complex coordinates, i.e. it is the line element dx\ + dx\ + dx\ expressed as gZiZdz dz + gZiZ,dz dz" + gz-zdz" dz + gd^dz-dz'dz*. A model with an action of this form is called a "Non-linear Sigma Model;" the idea is t h a t the fields live on some "target" space, with metric g. Verify eqns. 45,47. More generally, = j f ^ j is a solution with action 27r. The parameters a,...d are called collective coordinates. They correspond to the symmetries of translations, dilations, and rotations, and special conformal transformations (forming the group SL(2, C)). In other words, any given finite action solution breaks the symmetries. In the path integral, the symmetry of Green's functions is recovered when one integrates over the collective coordinates. For translations, this is particularly simple. If one studies a Green's function, < (x) « / d2x0cl{x - x„)cl{y - y0)e~S" The precise measure is obtained by the Fadeev-Popov method. parameter p yields a factor / dpp~le~^M

...

(52)

Similarly, the integration over the

(53)

Here the first factor follows on dimensional grounds. The second follows from renormalization-group considerations. It can be found by explicit evaluation of the functional determinant?. Note that, because of asymptotic freedom, this means that typical Green's functions will be divergent in the infrared. There are many other features of this instanton one can consider. For example, one can consider adding massless fermions to the model, by simply coupling them in a gauge-invariant way to A^. The resulting theory has a chiral U(l) symmetry, which is anomalous. In the presence of an instanton, one can easily construct normalizable fermion zero modes (the Dirac equation just becomes the statement that if is analytic). As a result, Green's functions computed in the instanton background do not respect the axial (7(1) symmetry. But rather than get too carried away with this model (I urge you to get a little carried away and play with it a bit), let's proceed to four dimensions, where we will see very similar phenomena.

3

Real Q C D

The model of the previous section mimics many features of real QCD. Indeed, we will see that much of our discussion can be carried over, almost word for word, to the observed strong interactions. This analogy is helpful, given that in QCD we have no approximation which gives us control over the theory comparable to that which we found in the large N limit of the CPN model. As in that theory:

358 • There is a 6 parameter, which appears as an integral over the divergence of a non-gauge invariant current. • There are instantons, which indicate that there should be real ^-dependence. However, instanton effects cannot be considered in a controlled approximation, and there is no clear sense in which ^-dependence can be understood as arising from instantons. • There is another approach to the theory, which shows that the S-dependence is real, and allows computation of these effects. In QCD, this is related to the breaking of chiral symmetries.

3.1

The Theory and its

Symmetries

While it is not in the spirit of much of this school, which is devoted to the physics of heavy quarks, it is sufficient, to understand the effects of S, to focus on only the light quark sector of QCD. For simplicity in writing some of the formulas, we will consider two light quarks; it is not difficult to generalize the resulting analysis to the case of three. It is believed that the masses of the u and d quarks are of order 5 MeV and 10 MeV, respectively, much lighter than the scale of QCD. So we first consider an idealization of the theory in which these masses are set to zero. In this limit, the theory has a symmetry SU(2)i x SU(2)R. This symmetry is spontaneously broken to a vector SU(2). The three resulting Goldstone bosons are the w mesons. Calling

'l)

>

<*>

the two SU(2) symmetries act separately on q and q (thought of as left handed fermions). The order parameter for the symmetry breaking is believed to be the condensate: M„ = (qq).

(55)

This indeed breaks the symmetry down to the vector sum. The associated Goldstone bosons are the n mesons. One can think of the Goldstone bosons as being associated with a slow variation of the expectation value in space, so we can introduce a composite operator

M = qq = M^"-*^ (I

°)

(56)

The quark mass term in the lagrangian is then (for simplicity writing mu = ma = m , ; more generally one should introduce a matrix)

m„M.

(57)

Expanding M in powers of ir/f„, it is clear that the minimum of the potential occurs for 7rn = 0. Expanding to second order, one has

mlfl

= mqM0.

(58)

But we have been a bit cavalier about the symmetries. The theory also has two [/(l)'s; q-^eiaq ia

q -> e q

q -> eiaq q -+ e-iaq

(59) (60)

The first of these is baryon number and it is not chiral (and is not broken by the condensate). The second is the axial C/(l)s; It is also broken by the condensate. So, in addition to the pions, there should

359 be another approximate Goldstone boson. The best candidate is the r/, but, as we will see below (and as you will see further in Thomas's lectures), the 77 is too heavy to be interpreted in this way. The absence of this fourth (or in the case of three light quarks, ninth) Goldstone boson is called the U(l) problem. The f7(l)5 symmetry suffers from an anomaly, however, and we might hope that this has something to do with the absence of a corresponding Goldstone boson. The anomaly is given by ^

= ^ F F

(61)

Again, we can write the right hand side as a total divergence, FF = d^K"

(62)

where K» = ^VPM%F$.

~ \rbcAlAbpA%).

(63)

So if it is true that this term accounts for the absence of the Goldstone boson, we need to show that there are important configurations in the functional integral for which the rhs does not vanish rapidly at infinity.

3.2

Instantons

It is easiest to study the Euclidean version of the theory. This is useful if we are interested in very low energy processes, which can be described by an effective action expanded about zero momentum. In the functional integral, : j[dA}[dq}[dq}e

(64)

it is natural to look for stationary points of the effective action, i.e. finite action, classical solutions of the theory in imaginary time. The Yang-Mills equations are complicated, non-linear equations, but it turns out that, much as in the CPN model, the instanton solutions can be found rather easily5. The following tricks simplify the construction, and turn out to yield the general solution. First, note that the Yang-Mills action satisfies an inequality: f(F ± Ff

= f(F2

+ F2 ± 2FF) = f (2F2 + 2FF) > 0.

(65)

So the action is bounded / FF, with the bound being saturated when F = ±F

(66)

i.e. if the gauge field is (anti) self dual. 6 This equation is a first order equation, and it is easy to solve if one first restricts to an SU{2) subgroup of the full gauge group. One makes the ansatz that the solution should be invariant under a combination of ordinary rotations and global SU(2) gauge transformations: A„ = f(r2) b

+ h(r2)x-f

(67)

This is not an accident, nor was the analyticity condition in the CPN case. In both cases, we can add fermions so that the model is supersymmetric. Then one can show that if some of the supersyrnmetry generators, Qa annihilate a field configuration, then the configuration is a solution. This is a first order condition; in the Yang-Mills case, it implies self-duality, and in the CPN case it requires analyticity.

360 where we are using a matrix notation for the gauge fields. One can actually make a better guess: define the gauge transformation: 9(x) = -^—r

(68)

A„ = fir^gd^g-1

(69)

and take

Then plugging in the Yang-Mills equations yields: r2 / = -5 7 (70) v r 2 + p2 ' where p is an arbitrary quantity with dimensions of length. The choice of origin here is arbitrary; this can be remedied by simply replacing x —> x — x0 everywhere in these expressions, where x0 represents the location of the instanton. E x e r c i s e Check that eqns. 69,70 solve 66. iProm this solution, it is clear why f d^IC does not vanish for the solution: while A is a pure gauge at infinity, it falls only as 1/r. Indeed, since F = F, for this solution, [F2

=

[F2

= 32TT2

(71)

This result can also be understood topologically. g defines a mapping from the "sphere at infinity" into the gauge group. It is straightforward to show that

h!dixFp

32vr2

(72)

counts the number of times g maps the sphere at infinity into the group (one for this specific example; n more generally). We do not have time to explore all of this in detail; I urge you to look at Sidney Coleman's lecture, "The Uses of Instantons" 7 . To actually do calculations, 't Hooft developed some notations which are often more efficient than those described above 5 . So we have exhibited potentially important contributions to the path integral which violate the £7(1} symmetry. How does this violation of the symmetry show up? Let's think about the path integral in a bit more detail. Having found a classical solution, we want to integrate about small fluctuations about it: eie f[d6A][dq][dq]eis2s

(73)

Now 5 contains an explicit factor of 1/g 2 . As a result, the fluctuations are formally suppressed by g2 relative to the leading contribution. The one loop functional integral yields a product of determinants for the fermions, and of inverse square root determinants for the bosons. Consider, first, the integral over the fermions. It is straightforward, if challenging, to evaluate the determinants 5 . But if the quark masses are zero, the fermion functional integrals are zero, because there is a zero mode for each of the fermions, i.e. for both q and q there is a normalizable solution of the equation: Jf)u = 0

J/lu = 0

and similarly for d and d. It is straightforward to construct these solutions:

(74)

361 where ( is a constant spinor, and similarly for u, etc. This means that in order for the path integral to be non-vanishing, we need to include insertions of enough q's and q's to soak up all of the zero modes. In other words, non-vanishing Green's functions have the form {uudd)

(76)

and violate the symmetry. Note that the symmetry violation is just as predicted from the anomaly equation: A(

2 s = 4 ^ fd4xFF = 4

(77)

However, the calculation we have described here is not self consistent. The difficulty is that among the variations of the fields we need to integrate over are changes in the location of the instanton (translations), rotations of the instanton, and scale transformations. The translations are easy to deal with; one has simply to integrate over x0 (one must also include a suitable Jacobian factor 7 ). Similarly, one must integrate over p. There is a power of p arising from the Jacobian, which can be determined on dimensional grounds. For our Green's function above, for example, which has dimension 6, we have (if all of the fields are evaluated at the same point),

Jdpp~7.

(78)

However, there is additional ^-dependence because the quantum theory violates the scale symmetry. This can be understood by replacing g2 —> g2(p) in the functional integral, and using e-s*V(p)

~ (pM)bo

(79)

for small p. For 3 flavor QCD, for example, b0 = 9, and the p integral diverges for large p. This is just the statement that the integral is dominated by the infrared, where the QCD coupling becomes strong. So we have provided some evidence that the U(l) problem is solved in QCD, but no reliable calculation. What about ^-dependence? Let us ask first about (9-dependence of the vacuum energy. In order to get a non-zero result, we need to allow that the quarks are massive. Treating the mass as a perturbation, we obtain E{0) = CA9QCDmumdcos(6)

f dpp~3p9.

(80)

So again, we have evidence for ^-dependence, but cannot do a reliable calculation. That we cannot do a calculation should not be a surprise. There is no small parameter in QCD to use as an expansion parameter. Fortunately, we can use other facts which we know about the strong interactions to get a better handle on both the U(l) problem and the question of ^-dependence. Before continuing, however, let us consider the weak interactions. Here there is a small parameter, and there are no infrared difficulties, so we might expect instanton effects to be small. The analog of the £/(l) 5 symmetry in this case is baryon number. Baryon number has an anomaly in the standard model, since all of the quark doublets have the same sign of the baryon number, 't Hooft realized that one could actually use instantons, in this case, to compute the violation of baryon number. Technically, there are no finite action Euclidean solutions in this theory; this follows, as we will see in a moment, from a simple scaling argument. However, 't Hooft realized that one can construct important configurations of nonzero topological charge by starting with the instantons of the pure gauge theory and perturbing them.

362 If one simply takes such an instanton, and plugs it into the action, one necessarily finds a correction to the action of the form SS = \v2p2. (81) 92 This damps the p integral at large p, and leads to a convergent result. Affleck showed how to develop this into a systematic computation 9 . Note that from this, one can see that baryon number violation occurs in the standard model, and that the rate is incredibly small, proportional to e - 2 " " 7 .

3.3

Real QCD and the U(l)

Problem

In real QCD, it is difficult to do a reliable calculation which shows that there is not an extra Goldstone boson, but the instanton analysis we have described makes clear that there is no reason to expect one. Actually, while perturbative and semiclassical (instanton) techniques have no reason to give reliable results, there are two approximation methods techniques which are available. The first is large JV, where one now allows the JV of SU(N) to be large, with g2N fixed. In contrast to the case of CPN, this does not permit enough simplification to do explicit computations, but it does allow one to make qualitative statements about the theory, available in QCD. Witten has pointed out a way in which one can at relate the mass of the 7} (or rj' if one is thinking in terms of SU(Z) x SU(3) current algebra) to quantities in a theory without quarks. The point is to note that the anomaly is an effect suppressed by a power of JV, in the large JV limit. This is because the loop diagram contains a factor of g2 but not of JV. So, in large JV, it can be treated as a perturbation, and the the rj is massless. d^j^ is like a creation operator for the r), so (just like d^jt? is a creation operator for the IT meson), so one can compute the mass if one knows the correlation function, at zero momentum, of ( « W « ( i / ) ) « ^(F(x)F(x)F(yMy))

(82)

To leading order in the 1/JV expansion, this correlation function can be computed in the theory without quarks. Witten argued that while this vanishes order by order in perturbation theory, there is no reason that this correlation function need vanish in the full theory. Attempts have been made to compute this quantity both in lattice gauge theory and using the AdS-CFT correspondence recently discovered in string theory. Both methods give promising results. So the U(l) problem should be viewed as solved, in the sense that absent any argument to the contrary, there is no reason to think that there should be an extra Goldstone boson in QCD. The second approximation scheme which gives some control of QCD is known as chiral perturbation theory. The masses of the u, d and s quarks are small compared to the QCD scale, and the mass terms for these quarks in the lagrangian can be treated as perturbations. This will figure in our discussion in the next section.

3.4

Other Uses of Instantons:

A Survey

In the early days of QCD, it was hoped that instantons, being a reasonably well understood nonperturbative effect, might give insight into many aspects of the strong interactions. Because of the infrared divergences discussed earlier, this program proved to be a disappointment. There was simply no well-controlled approximation to QCD in which instantons were important. Indeed, Witten 10 stressed the successes of the large JV limit in understanding the strong interactions, and argued that in this limit, anomalies could be important but instantons would be suppressed exponentially. This reasoning (which I urge you to read) underlay much of our earlier discussion, which borrowed heavily on this work.

363 In the years since Coleman's "Uses of Instantons" was published, many uses of instantons in controlled approximations have been found. What follows is an incomplete list; I hope this will inspire some of you to read Coleman's lectures and develop a deeper understanding of the subject. • ^-dependence at finite temperature: Within QCD, instanton calculations are reliable at high temperatures. So, for example, one can calculate the ^-dependence of the vacuum energy in the early universe, and other quantities to which instantons give the leading contribution 11 . • Baryon number violation in the standard model: We have remarked that this can be reliably calculated, though it is extremely small. However, as explained in 7 , instanton effects are associated with tunneling, and in the standard model, they describe tunneling between states with different baryon number. It is reasonable to expect that baryon number violation is enhanced at high temperature, where one has plenty of energy to pass over the barrier without tunneling. This is indeed the case. This baryon number violation might be responsible for the matter-antimatter asymmetry which we observe 12 . • Instanton effects in supersymmetric theories: this has turned out to be a rich topic. Instantons, in many instances, are the leading effects which violate non-renormalization theorems in perturbation theory, and they can give rise to superpotentials, supersymmetry breaking, and other phenomena. More generally, they have provided insight into a whole range of field theory and string theory phenomena.

4 4-1

The Strong C P Problem 6-dependence

of the Vacuum

Energy

The fact that the anomaly resolves the 17(1) problem in QCD, however, raises another issue. Given that J d^xFF has physical effects, the theta term in the action has physical effects as well. Since this term is CP odd, this means that there is the potential for strong CP violating effects. These effects should vanish in the limit of zero quark masses, since in this case, by a field redefinition, we can remove 6 from the lagrangian. In the presence of quark masses, the ^-dependence of many quantities can be computed. Consider, for example, the vacuum energy. In QCD, the quark mass term in the lagrangian has the form: Cm = muuu + mddd + h.c.

(83)

Were it not for the anomaly, we could, by redefining the quark fields, take mu and vtid to be real. Instead, we can define these fields so that there is no 8FF term in the action, but there is a phase in mu and md. Clearly, we have some freedom in making this choice. In the case that m„ and mi are equal, it is natural to choose these phases to be the same. We will explain shortly how one proceeds when the masses are different (as they are in nature). We can, by convention, take 8 to be the phase of the overall lagrangian: Cm = (muuu + rnddd) cos(#/2) + h.c.

(84)

Now we want to treat this term as a perturbation. At first order, it makes a contribution to the ground state energy proportional to its expectation value. We have already argued that the quark bilinears have non-zero vacuum expectation values, so E(8) = (mu + md)eie(qq).

(85)

364 While, without a difficult non-perturbative calculation, we can't calculate the separate quantities on the right hand side of this expression, we can, using current algebra, relate them to measured quantities. A simple way to do this is to use the effective lagrangian method (which will be described in more detail in Thomas's lectures). The basic idea is that at low energies, the only degrees of freedom which can readily be excited in QCD are the pions. So parameterize qq as qq = Z=

e'" *'°

(86)

We can then write the quark mass term as Cm = eieTrMqY,.

(87)

Ignoring the 6 term at first, we can see, plugging in the explicit form for £ , that

mlfl

= (m„ + md) < qq > .

(88)

So the vacuum energy, as a function of 9, is: E(6) = mlficos(6).

(89)

This expression can readily be generalized to the case of three light quarks by similar methods. In any case, we now see that there is real physics in 9, even if we don't understand how to do an instanton calculation. In the next section, we will calculate a more physically interesting quantity: the neutron electric dipole moment as a function of 9.

4.2

The Neutron Electric Dipole

Moment

As Scott Thomas will explain in much greater detail in his lectures, the most interesting physical quantities to study in connection with C P violation are electric dipole moments, particularly that of the neutron, d„. It has been possible to set strong experimental limits on this quantity. Using current algebra, the leading contribution to the neutron electric dipole moment due to 9 can be calculated, and one obtains a limit 8 < 10 _ 9 e . The original paper on the subject is quite readable. Here we outline the main steps in the calculation; I urge you to work out the details following the reference. We will simplify the analysis by working in an exact SC/(2)-symmetric limit, i.e. by taking mu = md = m. We again treat the lagrangian of [84] as a perturbation. We can also understand how this term depends on the IT fields by making an axial SU(2) transformation on the quark fields. In other words, a background n field can be thought of as a small chiral transformation from the vacuum. Then, e.g., for the T$ direction, q —> (1 + iTT3T3)q (the 7r field parameterizes the transformation), so the action becomes: ^ • T 3 ( ? 7 5 9 + Sqq)

(90)

In In other words, we have calculated a CP violating coupling of the mesons to the pions. This coupling is difficult to measure directly, but it was observed in 8 that this coupling gives rise, in a calculable fashion, to a neutron electric dipole moment. Consider the graph offig.2. This graph generates a neutron electric dipole moment, if we take one coupling to be the standard pion-nucleon coupling, and the second the coupling we have computed above. The resulting Feynman graph is infrared divergent; we cut this off at mw, while cutting off the integral in the ultraviolet at the QCD scale. Because of this infrared sensitivity, the low energy calculation is reliable. The exact result is:
(NftiT'qmMMN/mJ-^Mn.

(91)

The matrix element can be estimated using ordinary SU(3), yielding dn = 5.2 x 1O~160 cm. T h e experimental bound gives 6 < 1 0 - 9 - 1 0 . Understanding why CP violation is so small in the strong interactions is the "strong CP problem."

365

N

N

Figure 2: Diagram in which CP-violating coupling of the pion contributes to dn.

5

Possible Solutions

What should our attitude towards this problem be? We might argue that, after all, some Yukawa couplings are as small as 10~ 5 , so why is 1CT9 so bad. On the other hand, we suspect that the smallness of the Yukawa couplings is related to approximate symmetries, and that these Yukawa couplings are telling us something. Perhaps there is some explanation of the smallness of 8, and perhaps this is a clue to new physics. In this section we review some of the solutions which have been proposed to understand the smallness of 6.

5.1

Massless u Quark

Suppose the bare mass of the u quark was zero (i.e. at some high scale, the u quark mass were zero). Then, by a redefinition of the u quark field, we could eliminate 9 from the lagrangian. Moreover, as we integrated out physics from this high scale to a lower scale, instanton effects would generate a small u quark mass. In fact, a crude estimate suggests that this mass will be comparable to the estimates usually made from current algebra. Suppose that we construct a Wilsonian action at a scale, say, of order twice the QCD scale. Call this scale A0. Then we would expect, on dimensional grounds, that the u quark mass would be of order: mdms mu = —

(92)

Now everything depends on what you take A 0 to be, and there is much learned discussion about this. The general belief seems to be that the coefficient of this expression needs to be of order three to explain the known facts of the hadron spectrum. There is contentious debate about how plausible this possibility is. Note, even if one does accept this possibility, one would still like to understand why the u quark mass at the high scale is exactly zero (or extremely small). It is interesting that in string theory, one knows of discrete symmetries which are anomalous, i.e. one has a fundamental theory where there are discrete symmetries which can be broken by very tiny effects. Perhaps this could be the resolution of the strong CP problem?

C P as a S p o n t a n e o u s l y B r o k e n S y m m e t r y A second possible solution comes from the observation that if the underlying theory is CP conserving, a "bare" 0 parameter is forbidden. In such a theory, the observed CP violation must arise spontaneously, and the challenge is to understand why this spontaneous CP violation does not lead to a 9 parameter. For example, if the low energy theory contains just the standard model fields, then some high energy breaking of C P must generate the standard model CP violating phase. This must not generate a phase

366 in d e t m , , which would be a 8 parameter. Various schemes have been devised to accomplish this 13 . Without supersymmetry, they are generally invoked in the context of grand unification. There, it is easy to arrange that the 8 parameter vanishes at the tree level, including only renormalizable operators. It is then necessary to understand suppression of loop effects and of the contributions of higher dimension operators. In the context of supersymmetry, it turns out that understanding the smallness of 8 in such a framework, requires that the squark mass matrix have certain special properties (there must be a high degree of squark degeneracy, and the left right terms in the squark mass matrices must be nearly proportional to the quark mass m a t r i x . / 4 . Again, it is interesting that string theory is a theory in which CP is a fundamental (gauge) symmetry; its breaking is necessarily spontaneous. Some simple string models possess some of the ingredients required to implement the ideas of 13 .

The Axion Perhaps the most popular explanation of the smallness of 8 involves a hypothetical particle called the axion. We present here a slightly updated version of the original idea of Peccei and Quinif 5 . Consider the vacuum energy as a function of 8, eqn. [85]. This energy has a minimum at 8 = 0, i.e. at the CP conserving point. As Weinberg noted long ago, this is almost automatic: points of higher symmetry are necessarily stationary points. As it stands, this observation is not particularly useful, since 8 is a parameter, not a dynamical variable. But suppose that one has a particle, a, with coupling to QCD: £„*,„„ = ( 9 , a ) 2 +

{al + 9) l; 2 FF

(93)

/„ is known as the axion decay constant. Suppose that the rest of the theory possesses a symmetry, called the Peccei-Quinn symmetry, a-t

a+a

for constant a. Then by a shift in a, one can eliminate 8. E(8 is now V(a/fa), the axion. It has a minimum at 8 = 0. The strong CP problem is solved.

(94) the potential energy of

One can estimate the axion mass by simply examining E{8).

2 _ mlf**

(95)

Ja

If fa ~ TeV, this yields a mass of order KeV. If /„ ~ 10 16 GeV, this gives a mass of order 10~ 9 eV. As for the ^-dependence of the vacuum energy, it is not difficult to get the factors straight using current algebra methods. A collection of formulae, with great care about factors of 2, appears i n 1 6 Actually, there are several questions one can raise about this proposal: • Should the axion already have been observed? In fact, as Scott Thomas will explain in greater detail in his lectures, the couplings of the axion to matter can be worked out in a straightforward way, using the methods of current algebra (in particular of non-linear lagrangians). All of the couplings of the axion are suppressed by powers of fa. So if fa is large enough, the axion is difficult to see. The strongest limit turns out to come from red giant stars. The production of axions is "semiweak," i.e. it only is suppressed by one power of fa, rather than two powers of raw', as a result, axion emission is competitive with neutrino emission until /„ > 10 10 GeV or so.

367 As we will describe in a bit more detail below, the axion can be copiously produced in the early universe. As a result, there is an upper bound on the axion decay constant. In this case, as we will explain below, the axion could constitute the dark matter. Can one search for the axion experimentally 17 ? Typically, the axion couples not only to the FF of QCD, but also to the same object in QED. This means that in a strong magnetic field, an axion can convert to a photon. Precisely this effect is being searched for by a group at Livermore (the collaboration contains members from MIT, University of Florida) and Kyoto. The basic idea is to suppose that the dark matter in the halo consists principally of axions. Using a (superconducting) resonant cavity with a high Q value in a large magnetic field, one searches for the conversion of these axions into excitations of the cavity. The experiments have already reached a level where they set interesting limits; the next generation of experiments will cut a significant swath in the presently allowed parameter space. The coupling of the axion to FF violates the shift symmetry; this is why the axion can develop a potential. But this seems rather paradoxical: one postulates a symmetry, preserved to some high degree of approximation, but which is not a symmetry; it is at least broken by tiny QCD effects. Is this reasonable? To understand the nature of the problem, consider one of the ways an axion can arise. In some approximation, we can suppose we have a global symmetry under which a scalar field, (f>, transforms as 0 —> elct as:

= Saeia'1'

\{)\=h

(96)

If this field couples to fermions, so that they gain mass from its expectation value, then at one loop we generate a coupling aFF from integrating out the fermions. This calculation is identical to the corresponding calculation for pions we discussed earlier. But we usually assume that global symmetries in nature are accidents. For example, baryon number is conserved in the standard model simply because there are no gauge-invariant, renormalizable operators which violate the symmetry. We believe it is violated by higher dimension terms. The global symmetry we postulate here is presumably an accident of the same sort. But for the axion, the symmetry must be extremely good. For example, suppose one has a symmetry breaking operator AH+4

V

(97)

Such a term gives a linear contribution to the axion potential of order fan . If fa ~ 10 1 1 , this swamps the would-be QCD contribution ( " * ; " ) unless n > 1218!

/«

This last objection finds an answer in string theory. In this theory, there are axions, with just the right properties, i.e. there are symmetries in the theory which are exact in perturbation theory, but which are broken by exponentially small non-perturbative effects. The most natural value for fa would appear to be of order MQUT — Mp. Whether this can be made compatible with cosmology, or whether one can obtain a lower scale, is an open question.

6

T h e A x i o n in C o s m o l o g y

Despite the fact that it is so weakly coupled, it be copiously produced in the early universe 19 . The point is the weak coupling itself. In the early universe, we know the temperature was once at lease 1 MeV, and

368 if the temperature was above a GeV, the potential of the axion was irrelevant. Indeed, if the universe is radiation dominated, the equation of motion for the axion is:

For t - 1 S> ma, the system is overdamped, and the axion does not move. There is no obvious reason that the axion should sit at its minimum in this early era. So one can imagine that the axion sits at its minimum until t ~ m " 1 , and then begins to roll. For /„ ~ Mv, this occurs at the QCD temperature; for sm.aller /„ it occurs earlier. After this, the axion starts to oscillate in its potential; it looks like a coherent state of zero momentum particles. At large times, = fl3/2(t) cos(rrU)-

(99)

so the density is simply diluted by the expansion. The energy density in radiation dilutes like T 4 , so eventually the axion comes to dominate the energy density. If fa ~ Mp, the axion energy density is comparable when oscillations start. If fa is smaller, oscillation starts earlier and there is more damping. Detailed study (including the finite temperature behavior of the axion potential) gives a limit /„ < 10 11 GeV.

7

Conclusions: O u t l o o k

The strong CP problem, on the one hand, seems very subtle, but on the other hand it is in many ways similar to the other problems of flavor which we confront when we examine the standard model. 8 is one more parameter which is surprisingly small. The smallness of the Yukawa couplings may well be the result of approximate symmetries. Similarly, all of the suggestions we have discussed above to understand the smallness of 6 involve approximate symmetries of one sort or another. In the case of other ideas about flavor, there is often no compelling argument for the scale of breaking of the symmetries. The scale could be so high as to be unobservable, and there is little hope for testing the hypothesis. What is perhaps most exciting about the axion is that if we accept the cosmological bound, the axion might well be observable. T h a t said, one should recognize t h a t there are reasons to think t h a t the axion scale might be higher. In particular, as mentioned earlier, string theory provides one of the most compelling settings for axion physics, and one might well expect the Peccei-Quinn scale to be of order the GUT scale or Planck scale. There have been a number of suggestions in the literature as to how the cosmological bound might be evaded in this context. At a theoretical level, there are other areas in which the axion is of interest. Such particles inevitably appear in string theory and in supersymmetric field theories. (Indeed, it is in this context that PecceiQuinn symmetries of the required type for QCD most naturally appear). These symmetries and the associated axions are a powerful tool for understanding these theories. Acknowledgements: This work was supported in part by a grant from the U. S. Department of Energy. 1. The text, M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison Wesley (1995) Menlo Park, has an excellent introduction to anomalies (chapter 19).

369 2. E. Witten, "Dyons of Charge §£", Phys. Lett. 8 6 B (1979) 283. For a pedagogical introduction to monopoles, I strongly recommend J.A. Harvey, "Magnetic Monopoles, Duality and Supersymmetry", hep-th/9603086. 3. J. Kogut and L. Susskind, Phys. Rev. D l l (1976) 3594. 4. I. Affleck, Phys.Lett. B 9 2 (1980) 149. 5. G. 't Hooft, Phys. Rev. D 1 4 (1976) 3432. 6. A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin, Phys. Lett. 5 9 B (1975) 85. 7. S. Coleman, "The Uses Of Instantons," in Aspects of Symmetry, Cambridge University Press, Cambridge, 1985. 8. R.J. Crewther, P. Di Vecchia, G. Veneziano and E. Witten, Phys. Lett. 8 8 B (1979) 123. 9. I. Affleck, "On Constrained Instantons," Nucl. Phys. B 1 9 1 (1981) 429. 10. E. Witten, Nucl.Phys. B 1 4 9 (1979) 285. 11. D. Gross, R. Pisarski and L.G. Yaffe, "QCD and Instantons at Finite Temperature," Rev. Mod. Phys. 53 (1981) 43. 12. A.G. Cohen, D.B. Kaplan and A.E. Nelson, "Progress in Electroweak Baryogenesis," Ann. Rev. Nucl. Part. Sci. 43 (1993) 27, hep-ph/9302210. 13. A. Nelson, Phys. Lett. 1 3 6 B (1984) 387; S.M. Barr, Phys. Rev. Lett. 53 (1984) 329. 14. M. Dine, R. Leigh and A. Kagan, Phys. Rev. D 4 8 (1993) 4269, hep-ph/9304299. 15. R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38 (1977) 1440; Phys. Rev. D 1 6 (1977) 1791. 16. M. Srednicki, "Axion Couplings to Matter: CP Conserving Parts," Nucl. Phys. B 2 6 0 (1985) 689. 17. P. Sikivie, "Axion Searches," Nucl.Phys.Proc.Suppl. 87 (2000) 41, hep-ph/0002154. 18. M. Kamionkowski and J. March-Russell, "Planck Scale Physics and the Peccei-Quinn Mechanism," Phys. Lett. B 2 8 2 (1992) 137, hep-th/9202003. 19. M. Turner, "Windows on the Axion," Phys.Rept. 197 (1990) 67.

This page is intentionally left blank

I^IIIIIIIIBII

Carl E. Wieman

This page is intentionally left blank

A B I B L I O G R A P H Y OF ATOMIC P A R I T Y VIOLATION A N D ELECTRIC DIPOLE M O M E N T E X P E R I M E N T S

JILA,

CARL E. WIEMAN University of Colorado and National Institute of Standards and Technology, 440 UCB, Boulder, CO 80309-0440, USA E-mail: [email protected]

This bibliography contains references in regards to searches for electric dipole moments of atoms and neutrons, and also for atomic parity violation experiments and theory. [The author felt that a separate set of lecture notes would duplicate material already published in these references - Editor.]

Electric dipole moments of atoms and neutrons 1. E.D. Commins, CP Violation in Atomic and Nuclear Physics, Proceedings of the XXVII SLAC Summer Institute on Particle Physics, 1999). An excellent review of the experimental and theoretical aspects of the subject in the Proceedings of the XXVII SLAC Summer School. This also has an extensive list of relevant references. See http://www.slac.stanford.edu/gen/meeting/ssi/1999/index.html. 2. Y. Nir, CP Violation in and beyond the Standard Model, (Proceedings of the XXVII SLAC Summer Institute on Particle Physics, 1999). A review of the general theory of CP violation and its connection with EDMs is covered by Y. Nir in Proceedings of the XXVII SLAC Summer School. See http://www.slac.stanford.edu/gen/meeting/ssi/1999/index.html 3. L.I. Schiff, "Measurability of nuclear electric dipole moments," Phys. Rev. 132, 2194 (1963). Presents Schiff's Theorem about the EDM in an atom (or lack thereof) due to an EDM of the nucleus or the electron. 4. P.G.H. Sandars, "The electric dipole moment of an atom," Phys. Lett. 14, 194 (1965); P.G.H. Sandars, "Enhancement factor for the electric dipole moment of the valence electron in an alkali atom," Phys. Lett. 22, 290 (1966). Discusses how Schiff's Theorem does not apply to heavy atoms so the atomic EDM is actually enhanced over the EDM of the electron. 5. J.P. Jacobs, W.M. Klipstein, S.K. Lamoreaux, B.R. Heckel, and E.N. Fortson, "Limit on the electric-dipole moment of HG-199 using synchronous optical-pumping," Phys. Rev. A 52, 3521 (1995). Provides best limits on T violating nuclear-nuclear interactions by search373

ing for EDM of the mercury atom. 6. S.A. Murthy, D. Krause, Z.L. Li, and L.R. Hunter, "New limits on the electron electric-dipole moment from cesium," Phys. Rev. Lett. 63, 965 (1989). Provides the best limit on electron EDM that has been obtained in a vapor cell experiment. 7. E.D. Commins, S.B. Ross, D. DeMille, and B.C. Regan, "Improved experimental limit on the electric-dipole moment of the electron," Phys. Rev. A 50, 2960 (1994). Provides best current limit on the EDM of the electron, although improvements are expected soon. 8. P.G. Harris, C.A. Baker, K. Green, P. Iaydjiev, S. Ivanov, D.J.R. May, J.M. Pendlebury, D. Shiers, K.F. Smith, M. van der Grinten, P. Geltenbort, "New experimental limit on the electric dipole moment of the neutron," Phys. Rev. Lett. 82, 904 (1999). Provides the best limits on the EDM of the neutron. 9. E.A. Hinds, "Testing time reversal symmetry using molecules," Physica Scripta T70, 34 (1997). Discusses the possibilities for using molecules to measure electron and nuclear EDMs.

Atomic parity violation 10. S.A. Blundell, J. Sapirstein, and W.R. Johnson, "Relativistic all-order calculations of energies and matrix-elements in cesium," Phys. Rev. A 43, 3407 (1991). Discusses the atomic structure calculations used in connecting atomic PV experiments to the standard model. 11. S.A. Blundell, W.R. Johnson, and J. Sapirstein, "3rd-order many-body perturbation-theory calculations of the ground-state energies of cesium and thallium," Phys. Rev. A 42, 3751 (1990). 12. S.A. Blundell, W.R. Johnson, and J. Sapirstein, "High-accuracy calculation of the 6S1/2 to 7S1/2 parity-nonconserving transition in atomic cesium and implications for the standard model," Phys. Rev. Lett. 65, 1411 (1990). 13. S.A. Blundell, J. Sapirstein, and W.R. Johnson, "High-accuracy calculation of parity nonconservation in cesium and implications for particle physics," Phys. Rev. D 45, 1602 (1992). 14. V.A. Dzuba, V.V. Flambaum, and O.P. Sushkov, "Polarizabilities and parity nonconservation in the Cs atom and limits on the deviation from

375

the standard electroweak model," Phys. Rev. A 56, R4357-R4360 (1997). 15. V.A. Dzuba, V.V. Flambaum, and O.P. Sushkov, "Summation of the high orders of perturbation theory in the correlation correction for the parity violating El-amplitude of the 6s to 7s transition in the cesium atom," Phys. Lett. A 141, 147 (1989). 16. V.A. Dzuba, V.V. Flambaum, and O.P. Sushkov, "Summation of the perturbation theory high order contributions to the correlation correction for the energy levels of the cesium atom," Phys. Lett. A 140, 493 (1989). 17. V.A. Dzuba, V.V. Flambaum, A. Ya. Kraftmakher, and O.P. Sushkov, "Summation of the high orders of perturbation theory in the correlation correction to the hyperfine structure and to the amplitudes of Eltransitions in the cesium atom," Phys. Lett. A 142, 373 (1989). 18. C.S. Wood, S.C. Bennett, D. Cho. B.P. Masterson, J.L. Roberts, C. Tanner and C. E. Wieman, "Measurement of parity nonconservation and an anapole moment in cesium," Science 275, 1759 (1997). Provides the best experimental measurement of atomic parity violation. 19. S.C. Bennett and C.E. Wieman, "Measurement of the 6S to 7S transition polarizability in atomic cesium and an improved test of the standard model," Phys. Rev. Lett. 82, 2484 (1999). Provides the calibration needed to interpret the parity violation measurement in terms of fundamental coupling constants, and also reexamines the accuracy of the earlier atomic theory. 20. C.S. Wood, S.C. Bennett, J.L. Roberts, D. Cho and C.E. Wieman, "Precision measurement of parity nonconservation in cesium," Canad. J. Phys. 77, 7 ((1999). Provides a very extensive discussion of the potential systematic errors of the cesium PNC experiment. 21. M.A. Bouchiat and C. Bouchiat, "Parity violation in atoms," Reports on Progress in Physics 60, 1351 (1997). Provides a general review of atomic PNC experiments. 22. J. Erler and P. Langacker, Constraints on extended neutral gauge structures, Phys. Lett. B 456, 68 (1999). J. Erler and P. Langacker, "Indications for an extra neutral gauge boson in electroweak precision data," Phys. Rev. Lett. 84, 212 (2000). Provides an analysis of precision weak interaction measurements and the constraints they set on extensions of the standard model. 23. A. Derevianko, "Reconciliation of the measurement of parity nonconservation in Cs with the standard model," Phys. Rev. Lett. 85, 1618 (2000). Provides an update of the atomic theory calculations of 1 and shows how they are shifted by the Breit correction.

This page is intentionally left blank

k\\

Adam F. Falk

This page is intentionally left blank

T H E C K M M A T R I X A N D T H E HEAVY Q U A R K E X P A N S I O N ADAM F. FALK Department of Physics and Astronomy The Johns Hopkins University 3400 North Charles Street, Baltimore, Maryland

21218

These lectures contain an elementary introduction to heavy quark symmetry and the heavy quark expansion. Applications such as the expansion of heavy meson decay constants and the treatment of inclusive and exclusive semileptonic B decays are included. The use of heavy quark methods for the extraction of \Vcf,\ and |Vuf,| is presented is some detail.

1

Heavy Quark Symmetry

In these lectures I will introduce the ideas of heavy quark symmetry and the heavy quark limit, which exploit the simplification of certain aspects of QCD for infinite quark mass, TUQ —>• 00. We will see that while these ideas are extraordinarily simple from a physical point of view, they are of enormous practical utility in the study of the phenomenology of bottom and charmed hadrons. One reason for this is the existence not just of an interesting new limit of QCD, but of a systematic expansion about this limit. The technology of this expansion is the Heavy Quark Effective Theory (HQET), which allows one to use heavy quark symmetry to make accurate predictions of the properties and behavior of heavy hadrons in which the theoretical errors are under control. While the emphasis in these lectures will be on the physical picture of heavy hadrons which emerges in the heavy quark limit, it will be important to introduce enough of the formalism of the HQET to reveal the structure of the heavy quark expansion as a simultaneous expansion in powers of A Q C D / ^ I Q and as{mo). However, what I hope to leave you with above all is an appreciation for the simplicity, elegance and coherence of the ideas which underlie the technical results which will be presented. The interested reader is also encouraged to consult a number of excellent reviews,1 which typically cover in more detail the material in these lectures.

1.1

Introduction

We begin by recalling the properties of charged current interactions in the Standard Model. They are mediated by the interactions with the W * bosons, 379

380

which for the quarks take the form. (u

c

f)7M(l-75)VCKM ( s J W ^ + h . c .

(1)

The 3 x 3 unitary matrix VCKM is

(

Vud Vcd

Vus Vub \ Vcs Vcb .

VU

Vu

(2)

Vtb J

The elements of VCKM have a hierarchical structure, getting smaller away from the diagonal: Vud, Vcs and Vtb are of order 1, Vus and VCd a r e of order 1 0 - 1 , Vcb and Vts are of order 10~ 2 , and Vub and Vtd are of order 10~ 3 . By contrast, except for small effects associated with neutrino masses, the interaction of the W± with the leptons is flavor diagonal. The CKM matrix is of fundamental importance, because it is the low energy manifestation of the higher-energy physics which breaks the global flavor symmetries of the Standard Model. In the absence of the Yukawa couplings, the quark sector of the Standard Model may be characterized by its gauge symmetry 517(3) x 5*7(2) x U{1), and its global symmetry U(i)Q x U{2)u * *7(3)D which rotates the triplets of colored fields QlL, UR and DlR among each other. The global symmetry group has a total of 3 x 3 2 = 27 generators. With the addition of fields <j> and (possibly composite) carrying Higgs and conjugateHiggs quantum numbers, one may write Yukawa-type interactions, Aj? (Qi 4>UR) + A« (Q'L <j>DR)+ h.c.

(3)

which break the flavor symmetries explicitly. The complex matrices \%J and XlJ correspond to 2 x (2 x 32) = 36 independent parameters. They break the global flavor symmetries completely, except for a remaining conserved baryon number U{\)B- The 27 - 1 = 26 broken generators may be used to rotate 26 of the Yukawa couplings to zero, leaving 36 — 26 = 10 physical parameters. When <j> and <j> get vacuum expectation values, one may go to the mass eigenstate basis for the quark fields, in which case the ten parameters are six quark masses and four parameters characterizing VCKM- An independent examination of VCKM confirms this counting. A 3 x 3 unitary matrix has 9 parameters, of which (3) = 3 are angles and the remaining 6 are complex phases. However, one may adjust 5 relative phases of the mass eigenstate quark fields, leaving 6 — 5 = 1 physical phase in VCKM- Thus VCKM is indeed characterized by four parameters, one of which is a CP-violating phase.

381

It is an instructive exercise, left to the reader, to perform the analogous counting for the general case of U(Nf)Q x U(Nf)u x U(Nf)o global flavor symmetry. One finds that the Yukawa couplings contain Nj + 1 physical parameters, of which 2Nf are quark masses. The remaining (Nf — l ) 2 parameters characterize VCKM, with ^Nf(Nf — 1) being angles and \(Nf — l)(Nf — 2) being complex phases. In particular, one notes that for Nf = 2 there is no CP violation in the weak interactions. Why is an understanding of QCD crucial to the study of the properties of VCKM? As an example, consider semileptonic b decay, b -^ civ, from which one would like to extract |VC&|. This process is mediated by a four-fermion operator, Obc

= ^ f ^ l

- 7 5 ) ^ 7 M ( 1 - 7 5 )^-

(4)

The weak matrix element is easy to calculate at the quark level, Ajuark = {c£i>\ Obc\b) = — - ~ u c ( p c ) ^ ( l

- j5)ub(pb)

ui{p()'ylj,(l - 7 5 )tt„(p„) •

(5) However, *4quark is only relevant at very short distances; at longer distances, QCD confinement implies that free b and c quarks are not asymptotic states of the theory. Instead, nonperturbative QCD effects "dress" the quark level transition b —> c Iv to a hadronic transition, such as B -)• Dtv

or

B -> D*lv

or

...

(6)

(In these lectures, we will use a convention in which a B meson contains a b quark, not a b antiquark.) The hadronic matrix element *4hadron depends on nonperturbative QCD as well as on GFVCI>, and is difficult to calculate from first principles. To disentangle the weak interaction part of this complicated process requires us to develop some understanding of the strong interaction effects. There are a variety of methods by which one can do this. Perhaps the most popular, historically, has been use of various quark potential models.2 While these models are typically very predictive, they are based on uncontrolled assumptions and approximations, and it is virtually impossible to estimate the theoretical errors associated with their use. This is a serious defect if one builds such a model into the experimental extraction of a weak coupling constant such as Vcb, because the uncontrolled theoretical errors then infect the experimental result. These are issues which are important for the extraction of all the elements of VCKM- Let us pause now to review our current experimental knowledge of

382

each of the magnitudes. The results are taken from the Particle Data Book.3 (The phases of the matrix elements must be extracted from CP violating asymmetries, as discussed elsewhere at this school.) We start with the submatrix describing mixing among the first two generations. The parameter \Vud\ is measured by studying the rates for neutron and nuclear (3 decay. Here the isospin symmetry of the strong interactions may be used to control the nonperturbative dynamics, since the operator ^7^(1 — j5)u which mediates the decay is a partially conserved current associated with a generator of chiral SU(2)L x SU(2)R. The current data yield \Vud\ = 0.9735 ±0.0008,

(7)

so \Vud\ is known at the level of 0.1%. The parameter \VUS\ is measured similarly, via K -> TT£U( and A -> pl&£. Here chiral SU(3)L x SU(3)R must be used in the hadronic matrix elements, since a strange quark is involved; because the ms corrections are larger, |y u s | is only known to 1%: \VUS\ =0.2196 ±0.0023.

(8)

The VCKM elements involving the charm quark are not so well measured. One way to extract | V^s | is to study the decay D -> K£+V(. Unfortunately, there is no symmetry by which one can control the matrix element (ii'|s7' i (l — 75)c|-D), since flavor SU(4:) is badly broken. One is forced to resort to models for these matrix elements. The reported value is \VC.\ = 1.04 ± 0 . 1 6 ,

(9)

but it must be said that this error estimate is not on very firm footing, and should probably be taken to be substantially larger. An alternative is to measure Vcs from inclusive processes at higher energies. For example, one can study the branching fraction for W+ —> cs, which can be computed using perturbative QCD. The result of a preliminary analysis is \VCS\ = 1.00 ± 0 . 1 3 ,

(10)

consistent with the model-dependent measurement. In this case, however, the error is largely experimental, and is unpolluted by hadronic physics. Similarly, one extracts \Vcd\ from deep inelastic neutrino scattering, using the process u^ + d ->• c + yT. This inclusive process may be computed perturbatively in QCD, leading to a result with accuracy at the level of 10%, |Vcd| = 0.224 ±0.016.

(11)

383

The elements of VCKM involving the third generation are, for the most part, harder to measure accurately. The branching ratio for t ->• bl+v can be analyzed perturbatively, but the experimental data are not very good. The present bound on |14b| is

SFM

=

"

i , a

(I2)

If one imposed the unitarity constraint \Vtd\2 + \Vts\2 + \Vtb\2 = 1, then this would amount to a 15% measurement of |Vtj|, but this unitarity constraint is one of the properties of VCKM which one is trying to test. More generally, in fact, one should be wary of constraints on VCKM which impose unitarity as part of the analysis; while one often obtains tighter constraints in this way, these constraints have a different meaning than do direct measurements. What the comparison of a constrained and unconstrained "measurement" really tells you is whether the direct determination may be used to test the unitarity of VCKM • Unfortunately, there are as yet no direct extractions of |V*d| o r \Vts\- One often speaks of these elements being measured in Ba — B& and B$ — Bs mixing, but again, the hypothesis that the Standard Model is responsible for these processes is something that one really wants to check. The correct way to view this part of the experimental program is to say that the Standard Model, including the unitarity of VCKMJ constrains Vts and Vtd severely enough that testable predictions can be made for the mixing parameters Am<j and A m s . This leaves us with the matrix elements Vub and Vcb, for which we need an understanding of B meson decay. In these lectures we will discuss an approach to understanding the relevant hadronic physics which exploits the fact that the b and c quarks are heavy, by which we mean that mb,rnc S> AQCD- The scale AQCD is the typical energy at which QCD becomes nonperturbative, and is of the order of hundreds of MeV. The physical quark masses are approximately mb w 4.8 GeV and m c « 1.5 GeV. The formalism which we will develop will not make as many predictions as do potential models. However, the compensation will be that we will develop a systematic expansion in powers of AQCD /wi(,)C, within which we will be able to do concrete error analysis. In particular, we will be able to estimate the error associated with the fact that m c may not be very close to the asymptotic limit mc ^§> AQCD • Even where this error may be substantial, the fact that it is under control allows us to maintain predictive power in the theory.

384

1.2

The heavy quark limit

Consider a hadron HQ composed of a heavy quark Q and "light degrees of freedom", consisting of light quarks, light antiquarks and gluons, in the limit TTIQ —> oo. The Compton wavelength of the heavy quark scales as the inverse of the heavy quark mass, XQ ~ I/TRQ. The light degrees of freedom, by contrast, are characterized by momenta of order AQCD, corresponding to wavelengths Xe ~ 1/AQCD- Since Xi ~S> XQ, the light degrees of freedom cannot resolve features of the heavy quark other than its conserved gauge quantum numbers. In particular, they cannot probe the actual value of XQ, that is, the value of mQ. We may draw the same conclusion in momentum space. The structure of the hadron HQ is determined by nonperturbative strong interactions. The asymptotic freedom of QCD implies that when quarks and gluons exchange momenta p much larger than AQCD, the process is perturbative in the strong coupling constant as (p). On the other hand, the typical momenta exchanged by the light degrees of freedom with each other and with the heavy quark are of order AQCD, for which a perturbative expansion is of no use. For these exchanges, however, p < TUQ, and the heavy quark Q does not recoil, remaining at rest in the rest frame of the hadron. In this limit, Q acts as a static source of electric and chromoelectric gauge field. The chromoelectric field, which holds HQ together, is nonperturbative in nature, but it is independent of TUQ. The result is that the properties of the light degrees of freedom depend only on the presence of the static gauge field, independent of the flavor and mass of the heavy quark carrying the gauge charge." There is an immediate implication for the spectroscopy of heavy hadrons. Since the interaction of the light degrees of freedom with the heavy quark is independent of TUQ, then so is the spectrum of their excitations. It is these excitations which determine the spectrum of heavy hadrons HQ. Hence the splittings A; ~ AQCD between the various hadrons HQ are independent of Q and, in the limit TTIQ —> oo, do not scale with rriQ. For example, the bottom and charmed meson spectra are shown schematically in Fig. 1, in the limit mb,mc > AQCD- The light degrees of freedom are in exactly the same state in the mesons Bi and Di, for a given i. The offset Bi — Di = rrib — mc is just the difference between the heavy quark masses; in no way does the relationship between the spectra rely on an approximation mb m mc. We can enrich this picture by recalling that the heavy quarks and light a

T o p quarks decay too quickly for a static chromoelectric field to be established around them, so the simplifications discussed here are not relevant to them.

385

1 i 1

B4

> ,

B3

'A3

B2

i

1>

" A4

By

1

A:

"A3

B{ o mb

£>4 £>3

D2

-mc

Dx

a

Ai

D{o Figure 1: Schematic spectra of the bottom and charmed mesons in the limit mi,,mc 2> A Q C D The offset of the two spectra is not to scale; in reality, m j — mc 3> Aj ~ A Q C D -

degrees of freedom also carry spin. The heavy quark has spin quantum number SQ = | , which leads to a chromomagnetic moment MQ

(13)

2mQ

Note that /XQ —> 0 as JTIQ —>• oo, and the interaction between the spin of the heavy quark and the light degrees of freedom is suppressed. Hence the light degrees of freedom are insensitive to SQ; their state is independent of whether SQ = | or SQ = — | . Thus each of the energy levels in Fig. 1 is actually doubled, one state for each possible value of SQ . To summarize, what we see is that the light degrees of freedom are the same when combined with any of the following heavy quark states: Qi(t),

Qi(i);

Q2(t),

Q2U);

QNh(t),

QNAI),

(14)

where there are Nh, heavy quarks (in the real world, Nh, = 2). The result is an SU(2Nh) symmetry which applies to the light degrees of freedom. 4 ' 5 ' 6 ' 7 ' 8 A new symmetry means new nonperturbative relations between physical quantities. It is these relations which we wish to understand and exploit.

386

The light degrees of freedom have total angular momentum Ji, which is integral for baryons and half-integral for mesons. When combined with the heavy quark spin SQ = | , we find physical hadron states with total angular momentum J=\Jt±\\. (15) If S( ^ 0, then these two states are degenerate. For example, the lightest heavy mesons have S( = | , leading to a doublet with J = 0 and J = 1. In the charm system we find that the states of lowest mass are the spin-0 D and the spin-l D*\ the corresponding bottom mesons are the B and B*. The heavy quark spin operator SQ exchanges these two states. Writing the spin wave function \mQ,me), w e have

|M*(j3 = o)> = -^(in> + ut». (16)

|M> = - ^ ( i u > - u t » , Then it is easy to show that

S Q |M) = i | M * ( J 3 = 0)),

SQ\M*(J3=0))

= ±\M).

(17)

When effects of order I/TUQ are included, the chromomagnetic interactions split the states of given St but different J. This "hyperfine" splitting is not calculable perturbatively, but it is proportional to the heavy quark magnetic moment /J,Q . This gives its scaling with TUQ : mo* — rn,D ~ l/mc m,B' — TUB ~ l/mi,.

(18)

From this fact we can construct a relation which is a nonperturbative prediction of heavy quark symmetry, m2B.

-mB

= m2D. - m2D .

(19)

Experimentally, m2B, - m2B — 0.49 GeV2 and m2D, - m2D = 0.55 GeV 2 , so this prediction works quite well. Note that this relation involves not just the heavy quark symmetry, but the systematic inclusion of the leading symmetry violating effects. Generally, the mass of a heavy hadron HQ may be expanded in inverse powers of TUQ m(HQ) =mQ + AH + 0(l/mQ), (20) where A H is independent of Q and is associated with the energy of the light degrees of freedom in the hadron HQ. For the lowest lying Jt = | doublet,

387

this quantity is usually just referred to as A. Prom dimensional considerations, one expects A to be of order of a few hundred MeV. So far, we have formulated heavy quark symmetry for hadrons in their rest frame. Of course, we can easily boost to a frame in which the hadrons have arbitrary four-velocity v^ = 7(1,v). For heavy quarks Qi and Q2, the symmetry will then relate hadrons Hi(v) and Hi{v) with the same velocity but with different momenta. This distinguishes heavy quark symmetry from ordinary symmetries of QCD, which relate states of the same momentum. To remind ourselves of this distinction, henceforth we will label heavy hadrons explicitly by their velocity: D(v), D*(v), B(v), B*(v), and so on. 1.3

Semileptonic decay of a heavy quark

Now let us return to the semileptonic weak decay b —> civ, but now consider it in the heavy quark limit for the b and c quarks. Suppose the decay occurs at time t = 0. For t < 0, the b quark is embedded in a hadron H\,; for t > 0, the c quark is dressed by light degrees of freedom to H'c. Let us consider the lightest hadrons, Hi, = B(v) and H'c = D(v'). Note that since the leptons carry away energy and momentum, in general v ^ v'. What happens to the light degrees of freedom when the heavy quark decays? For t < 0, they see the chromoelectric field of a point source with velocity v. At t — 0, this point source recoils instantaneously 6 to velocity v'\ the color neutral leptons do not interact with the light hadronic degrees of freedom as they fly off. The light quarks and gluons then must reassemble themselves about the recoiling color source. This nonperturbative process will generally involve the production of an excited state or of additional particles; the light degrees of freedom can exchange energy with the heavy quark, so there is no kinematic restriction on the excitations (of energy ~ AQCD) which can be formed. There is also some chance that the light degrees of freedom will reassemble themselves back into a ground state D meson. The amplitude for this to happen is a function only of the inner product w = v • v' of the initial and final velocities of the color sources. This amplitude, £(w), is known as the Isgur-Wise function.4 Clearly, the kinematic point v = v1, or w — 1, is a special one. In this corner of phase space, where the leptons are emitted back to back, there is no recoil of the source of color field at t = 0. As far as the light degrees of freedom are concerned, nothing happens! Their state is unaffected by the decay of the heavy quark; they don't even notice it. Hence the amplitude for them to remain in the ground state is exactly unity. This is reflected in a 6

The weak decay occurs over a very short time St ~ 1/Mw

<S 1 / A Q C D -

388

nonperturbative normalization of the Isgur-Wise function at zero recoil,4 £(!) = !•

(21)

As we will see, this normalization condition is of enormous phenomenological use. It will be extremely important to understand the corrections to this result for finite heavy quark masses rrn, and, especially, m c . The weak decay b —> c is mediated by a left-handed current c~7M(l — 7 5 )6. Not only does this operator carry momentum, but it can change the orientation of the spin SQ of the heavy quark during the decay. For a fixed light angular momentum J(, the relative orientation of SQ determines whether the physical hadron in the final state is a D or a D*. However, the light degrees of freedom are insensitive to SQ , so the nonperturbative part of the transition is the same whether it is a D or a D* which is produced. Hence heavy quark symmetry implies relations between the hadronic matrix elements which describe the semileptonic decays B —> Dtv and B -» D*lv. It is conventional to parameterize these matrix elements by a set of scalar form factors. These are defined separately for the vector and axial currents, as follows: (D{v')\ C7"6 \B{v)) = h+(w){v + v'Y + h-{w){v - v'Y (D*(v',e)\c^b\B(v))

=

hv(w)ie^aPelv'avp

{D(v')\&f-y6b\B{v))=Q

(22)

5

h

(D*(v', e)| c 7 " 7 6 \B(v)) = hM (w)(w + l)e*" - e* • v[hAa («>K + A3

Hv'11].

The set of form factors hi(w) is the one appropriate to the heavy quark limit. Other linear combinations are also found in the literature. In any case, the form factors are independent nonperturbative functions of the recoil or equivalently, for fixed m;, and mc, of the momentum transfer. However, in the heavy quark limit they correspond to a single transition of the light degrees of freedom, being distinguished from each other only by the relative orientation of the spin of the heavy quark. Hence they may all be written in terms of the single function £(w) which describes this nonperturbative transition. As we will derive later, the result is a set of relations,4 h+(w) = hv(w) = hAl(w) h-(w)=hA2(w)=0,

= hAs{w) = £{w) (23)

which follow solely from the heavy quark symmetry. Of course, all of the form factors which do not vanish inherit the normalization condition (21) at zero recoil. This result is a powerful constraint on the structure of semileptonic decay in the heavy quark limit.

389 1.4

Heavy meson decay constant

As a final example of the utility of the heavy quark limit, consider the coupling of the heavy meson field to the axial vector current. This is conventionally parameterized by of a decay constant; for example, for the B~ meson we define fB via (0\urj5b\B-(PB)) = ifBP%. (24) What is the dependence of the nonperturbative quantity fB on m s ? To address this question, we rewrite Eq. (24) in a form appropriate to taking the heavy quark limit, mB —> oo (which is equivalent to mj, —> co). This entails making explicit the dependence of all quantities on mB. First, we trade the B~ momentum for its velocity, pB = mBV» .

(25)

Second, we replace the usual B~~ state, whose normalization depends on TUB, (B(Pl)\B(p2))

= 2EB6^(p1-p2),

(26)

by a mass-independent state, \B{v)) = -±=\BbB)),

(27)

VmB

satisfying (B(vi)\B(v2))

= 27<5(3)(pi - p 2 ) .

(28)

Then Eq. (24) becomes y/m^{0\ U7"756 \B~(v)) = ijBmBv^

.

(29)

The nonperturbative matrix element (0| u^^b \B~(v)) is independent of mB in the heavy quark limit. Hence, we see that in this limit fB takes the form fB = mB ' x (independent of mB).

(30)

This makes explicit the scaling of fB with ran • It is more interesting to write this as a prediction for the ratio of charmed and bottom meson decay constants. We find 4-5-7

£ = / " £ +0( AQ^.W). ID

V mB

\

TUD

(31)

TUB J

For the physical bottom and charm masses, of course, the correction terms proportional to A Q C D / T I Q could be important.

390

Figure 2: The nonleptonic decay of a 6 quark.

2 2.1

Effective Field Theories General Considerations

A central observation which underlies much of the theoretical study of B mesons is that physics at a wide variety of distance (or momentum) scales is typically relevant in a given process. At the same time, the physics at different scales must often be analyzed with different theoretical approaches. Hence it is crucial to have a tool which enables one to identify the physics at a given scale and to separate it out explicitly. Such a tool is the operator product expansion, used in conjunction with the renormalization group. Here a general discussion of its application is given. Consider the Feynman diagram shown in Fig. 2, in which a b quark decays nonleptonically. The virtual quarks and gauge bosons have virtualities fi which vary widely, from AQCD to M\y and higher. Roughly speaking, these virtualities can be classified into a variety of energy regimes: (i) /u ^> Mw\ (ii) Mw > A* > m&; (iii) mb > V > AQCD; (iv) (J. « AQCD- Each of these momenta corresponds to a different distance scale; by the uncertainty principle, a particle of virtuality /i can propagate a distance x « 1/fi before being reabsorbed. At a given resolution Ax, only some of these virtual particles can be distinguished, namely those that propagate a distance x > Ax. For example, if Ax > 1/Mw, then the virtual W cannot be seen, and the process whereby it is exchanged would appear as a point interaction. By the same token, as AIE increases toward 1/AQCD, fewer and fewer of the virtual gluons can be seen explicitly. Finally, for /i « AQCD, it is inappropriate to speak of virtual gluons at all, because at such low momentum scales QCD becomes strongly interacting and an expansion in terms of individual gluons is inadequate. It is useful to organize the computation of a diagram such as is shown in

391 Fig. 2 in terms of the virtuality of the exchanged particles. This is important both conceptually and practically. First, it is often the case that a distinct set of approximations and approaches is useful at each distance scale, and one would like to be able to apply specific theoretical techniques at the scale at which they are appropriate. Second, Feynman diagrams in which two distinct scales /ii S> fi2 appear together can lead to logarithmic corrections of the form a"ln n (;Ui//U2), which for ln(//i//i2) ~ l / a s can spoil the perturbative expansion. A proper separation of scales will include a resummation of such terms. 2.2

Example I: Weak b Decays

As an example, consider the weak decay of a 6 quark, b -» cud, which is mediated by the decay of a virtual W boson. Viewed with resolution Ax < I/My/, the decay amplitude involves an explicit W propagator and is proportional to {i92? P* M2w

c^{l-j5)bd^{l-j5)ux

(32)

where pM is the momentum of the virtual W. Since mi, -C! Mw, the kinematics constrains p2 -C Mw, so the virtuality of the W is of order Mw, and it travels a distance of order 1/Mw before decaying. Viewed with a lower resolution, Ax > 1/Mw, the process b —> cud appears to be a local interaction, with four fermions interacting via a potential which is a 6 function where the four particles coincide. This can be seen by making a Taylor expansion of the amplitude in powers of p2/Mw, cry"(l - 7 5 ) ^ 7 M ( 1 - 7 5 ) "

Mw

1+

P Mw

P

+ Mw + ...

(33)

The coefficient of the first term is just the usual Fermi decay constant, GF/V%The higher order terms correspond to local operators of higher mass dimension. In the sense of a Taylor expansion, the momentum-dependent matrix element (32), which involves the propagation of a W boson between two spacetime points, is identical to the matrix element of the following infinite sum of local operators: £ | C7"(l -7 5 )&

,

{id)2 Mw

{id)4 Mw


(34)

where the derivatives act on the entire current on the right. This expansion of the nonlocal product of currents in terms of local operators, sometimes known

392

as an operator product expansion, is valid so long as p2
Radiative Corrections

At tree level, the effective theory is constructed simply by integrating out the W boson, because this is the only particle in a tree level diagram which is off-shell by order Mw. When radiative corrections are included, gluons and light quarks can also be off-shell by this order. Consider the one-loop diagram shown in Fig. 3. The components of the loop momentum k^ are allowed to take all values in the loop integral. However, the integrand is cut off both in the ultraviolet and in the infrared. For k > Mw, it scales as dik/k6, which is convergent as k ->• oo. For k < rrn,, it scales as dik/kzmbMw, which is convergent as k —> 0. In between, all momenta in the range mj, < k < Mw contribute to the integral with roughly equivalent weight. As a consequence, there is potentially a radiative correction proportional to as ln(Mw/mb). Even if as(fi) is evaluated at the high scale y, = Mw, such a term is not small in the limit Mw -> oo. At n loops, there is potentially a

393

Figure 3: The nonleptonic decay of a 6 quark at one loop. term of order a™ lnn(Mw/mb). For asln(Mw/mb) ~ 1, these terms need to be resummed for the perturbation series to be predictive. The technique for performing such a resummation is the renormalization group. The renormalization group exploits the fact that in the effective theory, operators such as 0 / = c i 7 M (l - lb)V

J,-7M(1

- 7V

(35)

receive radiative corrections and must be subtracted and renormalized. (Here the color indices i and j are explicit.) In dimensional regularization, this means that they acquire, in general, a dependence on the renormalization scale \i. Because physical predictions are independent of fi, in the renormalized effective theory the operators must be multiplied by coefficients with a compensating dependence on \x. It is also possible for operators to mix under renormalization, so the set of operators induced at tree level may be enlarged once radiative corrections are included. In the present example, a second operator with different color structure, On = ci7M(l - lh)V dj^(\

-

7

V

,

(36)

is induced at one loop. The interaction Hamiltonian of the effective theory is then fteff = C/(/i)0/(/i) + Cn(n)On{jx),

(37)

and it satisfies the differential equation MA^eff = 0.

(38)

394

By computing the dependence on fi of the operators Oj(/it), one can deduce the /i-dependence of the Wilson coefficients Cj(/u). In this case, a simple calculation yields

*-«-i[(^r*(^ri

<39)

For fj, = rrib, these expressions resum all large logarithms proportional to ans\nn{Mwlmb). The decays which are observed involve physical hadrons, not asymptotic quark states. For example, this nonleptonic b decay can be realized in the channels B —> Dir, B -> D'irir, and so on. The computation of partial decay rates for such processes requires the analysis of hadronic matrix elements such as {Dir\ crf(l

- 7 5 ) 6 U 7 A I ( 1 - j5)d \b).

(40)

Such matrix elements involve nonperturbative QCD and are extremely difficult to compute from first principles. However, they have no intrinsic dependence on large mass scales such as M\y Because of this, they should naturally be evaluated at a renormalization scale fi -C Mw, in which case large logarithms ln(Mw/?7ib) will not arise in the matrix elements. By choosing such a low scale in the effective theory (37), all such terms are resummed into the coefficient functions Ci{rrn,). As promised, the physics at scales near Mw has been separated from the physics at scales near m^, with the renormalization group used to resum the large logarithms which connect them. In fact, as we will see in the next section, nonperturbative hadronic matrix elements are usually evaluated at an even lower scale // « AQCD -C rrib, explicitly resumming all perturbative QCD corrections. 3

Heavy Quark Effective Theory

We have already extracted quite a bit of nontrivial information from the heavy quark limit. We have found the scaling of various quantities with TUQ, we have studied the implications for heavy hadron spectroscopy, and we have found nonperturbative relations among the hadronic form factors which describe semileptonic b —• c decay. However, all of these results have been obtained in the strict limit TUQ -> oo. If the heavy quark limit is to be of more than academic interest, and is to provide the basis for quantitative phenomenology, we have to understand how to include corrections systematically. There are actually two types of corrections which we would like to include. Power corrections are subleading terms in the expansion in A Q C D / ^ Q i those proportional to

395 A Q C D / W C are the most worrisome, because of the relatively small charm quark mass. Logarithmic corrections arise from the implicit dependence of quantities on rriQ through the strong coupling constant as(mo) ~ l/ln(mg/AQCD)- For the physical values of mj and mc, either of these could be important. What we need is a formalism which can accommodate them both. In short, we need to go from a set of heavy quark symmetry predictions in the TUQ -> oo limit, to a reformulation of QCD which provides a controlled expansion about this limit. The formalism which does the job is the Heavy Quark Effective Theory, or the HQET. The purpose of the HQET is to allow us to extract, explicitly and systematically, all dependence of physical quantities on rriQ, in the limit TUQ ^> AQCD- In these lectures, we will develop only enough of the technology to treat the dominant leading effects, providing indications along the way of how one would carry the expansion further. The HQET, as formulated here, was developed in a series of papers going back to the late 1980's, 4 ' 5 ' 7,9 ' 10 ' 11 ' 12 ' 13 ' 14 ' 15 ' 16 which the reader who is interested in tracking its historical development may consult.

3.1

The effective Lagrangian

Consider the kinematics of a heavy quark Q, bound in a hadron with light degrees of freedom to make a color singlet state. The small momenta which Q typically exchanges with the rest of the hadron are of order AQCD -C TUQ, and they never take Q far from its mass shell, p ^ = m L Hence the momentum p Q can be decomposed into two parts, p Q = mQv» + As" ,

(41)

where rriQV^ is the constant on-shell part of PQ, and fcM ~ AQCD is the small, fluctuating "residual momentum". The on-shell condition for the heavy quark then becomes m

Q = (mQV^ + k^f

=m2Q + 2mQv • k + k2 .

(42)

In the heavy quark limit we may neglect the last term compared to the second, and we have the simple condition v •k =0

(43)

for an on-shell heavy quark. Here the velocity v^ functions as a label; since soft interactions cannot change uM, there is a velocity supers election rule in the heavy quark limit, and v>* is a good quantum number of the QCD Hamiltonian.

396

We find the same result by taking the TUQ —>• oo limit of the heavy quark propagator, i> — m' Q + it -

^ 2 ^v • k. + ie

(44)

In this limit the propagator is independent of TUQ . The projection operators P

±

= ^

(45)

project onto the positive (P+) and negative (P_) frequency parts of the Dirac field Q. This is clear in the Dirac representation in the rest frame, in which P+ and P_ project, respectively, onto the upper two and lower two components of the heavy quark spinor. In the limit TUQ -» oo, in which Q remains almost on shell, only the "large" upper components of the field Q propagate; mixing via zitterbewegung with the "small" lower components is suppressed by l/2mQ. Hence the action of the projectors on Q is P+Q{x) = Q[x) + 0(l/mQ),

P-Q(x)=0

+ O{l/mQ).

(46)

More precisely, these relations should be understood as pertaining to those modes of the field Q(x) which annihilate heavy quarks and antiquarks in a heavy meson. The momentum dependence of the field Q is given by its action on a heavy quark state, Q(x) \Q(p)) = e-*"* |0>. (47) If we now multiply both sides by a phase corresponding to the on-shell momentum, eimQvxQ(x)

|

Q ( p ) )

=

g-ik.x

|Q)

t

( 4 8 )

the right side of this equation is independent of TUQ. Hence the left side must be, as well. Combining this observation with the argument of the previous paragraph, we are motivated to define a THQ-independent effective heavy quark field hv(x), hv(x)=eim^vxP+Q(x). (49) Note that the effective field carries a velocity label v and is a two-component object. The modifications to the ordinary field Q(x) project out the positive frequency part and ensure that states annihilated by hv{x) have no dependence on rriQ. Hence, these are reasonable candidate fields to carry representations of the heavy quark symmetry. Of course, the small components cannot be

397 neglected when effects of order represented by a field

I/TUQ

are included. In the HQET they are

Hv(x) = eimc>vx P-Q(x).

(50)

The field Hv(x) vanishes in the TUQ —> oo limit. The ordinary QCD Lagrange density for a field Q(x) is given by £QCD = Q(x)(ip-mQ)Q{x),

(51)

where D^ = d^ — igA^T0, is the gauge covariant derivative. To find the Lagrangian of the HQET, we substitute Q{x)=e-im^vxhv(x)

+ ...

(52)

into £ Q C D and expand. With the aid of the projection identity P+j>iP+ = v^, we find A J Q E T = hv(x)iv • Dh(x). (53) This simple Lagrangian leads to the propagator we derived earlier, (54)

v • k + ie and to an equally simple quark-gluon vertex, igTav»Al.

(55)

Note that both the propagator and the vertex are independent of TUQ, reflecting the heavy quark flavor symmetry. They also have no Dirac structure, reflecting the heavy quark spin symmetry. Our intuitive statements about the structure of heavy hadrons have been promoted to explicit symmetries of the QCD Lagrangian in the limit TUQ —> oo. It is straightforward to include power corrections to £ H Q E T - Write Q(x) in terms of the effective fields, Q(x) = e-im*v-x

[hv(x) + Hv(x)} ,

(56)

and apply the classical equation of motion (ip — rriQ)Q(x) = 0: iphv(x)

+ (ip - 2mQ)Hv{x)

=0.

(57)

Multiplying by P_ and commuting $ to the right, we find (iv-D + 2mQ)Hv(x)

=ip_i_hv{x),

(58)

398

where D1^ = D** — v^v • D. We then substitute Q(x) into eliminate Hv{x) and expand in I/TUQ to obtain £HQET

= hviv • Dhv + hvip±_

£QCD

as before,

ip±hv iv • D + 2mQ

= /i„w • -D/i„ +

1

[

2mQ

hv{Wx)2hv

+ | hv
+....

(59)

The leading corrections have a simple interpretation, which becomes clear in the rest frame, wM = (1,0,0,0). The spin-independent term is

2m,Q

>K

2m,Q

hv{iD±)2hv

-J-

2TTIQ

hv(iD)2hv,

(60)

which is the negative of the nonrelativistic kinetic energy of the heavy quark. Because of the explicit factor of l/2mQ, this term violates the heavy flavor symmetry. The spin-dependent part is 1

-

2m,Q

1

oT

afi„ « „

•a 'a = ^— % Ka Gaphv 2rriQ 2

.

1 -> -±- h^T^xgG^ 4TTIQ

= g^-Ba,

(61)

which is the coupling of the spin of the heavy quark to the chromomagnetic field. Because it has a nontrivial Dirac structure, this term violates both the heavy flavor symmetry and the heavy spin symmetry. For example, OQ is responsible for the D — D* and B — B* mass splittings. These correction terms will be treated as part of the interaction Lagrangian, even though OK has a piece which is a pure bilinear in the heavy quark field. 3.2

Effective currents and states

The expansion of the weak interaction current C7M(1 - j5)b is analogous. However, here we must introduce separate effective fields for the charm and bottom quarks, each with its own velocity: b->hb,

c -> / ^ •

(62)

Then a general flavor-changing current becomes, to leading order, cTb^hcv,Thbv,

(63)

where T is a fixed Dirac structure. With the leading power corrections, this is

cv b -> hcv, rftj+ - ^ hcvlr(ip±)hbv + ~ hcv, (-i^±)rhbv

+ ....

(64)

399

Q

9

Q

q

Q

q

Q

q

Figure 4: Tree level plus the one loop renormalization of the current qTQ in QCD. The box represents the current insertion.

The effective currents, and other operators which appear in the HQET, may often be simplified by use of the classical equation of motion, iv-Dhv(x)=0.

(65)

However, it is only safe to apply these equations naively at order I/ITIQ; at higher order the application of the equations of motion involves additional subtleties. 17 - 18 ' 19 To complete the effective theory, we need mQ-independent hadron states which are created and annihilated by currents containing the effective fields. For example, there is an effective pseudoscalar meson state \M(v)) which couples to the effective axial current
(67)

from which we immediately find the relationship (31) between fo and }B3.3

Radiative corrections

We can use the effective Lagrangian £HQET to compute the radiative corrections to the matrix element (66). In particular, we would like to extract the dependence of FM on InmQ. This dependence comes through the one-loop renormalization of the current q-y^^Q. At lowest order, of course, the renormalization is straightforward: we simply compute the set of graphs found in Fig. 4. The result is finite, because the current is (partially) conserved, and we extract a result of the form qn5Q

( l + 7 o | ^ ln(m Q /m g ) + . . . ) .

(68)

400

hv

q

hv

q

hv

q

hv

Figure 5: Tree level plus the one loop renormalization of the current qi^y^hv The double line is the propagator of the effective field hv.

q in H Q E T .

Note that there is no explicit dependence on the renormalization scale /u, since there is no divergence to be subtracted. The same result may be obtained in the effective theory. In this case we must match the currents in full QCD onto HQET currents of the form q^^^hhv. This step will induce a matching coefficient containing the explicit dependence on TTIQ, which is absent, by construction, from the operators and Lagrangian of the HQET. C In addition, the effective current will not necessarily be conserved, since the ultraviolet properties of QCD and the HQET differ. Hence the form of the matching, once radiative corrections are included, is

q-f-fQ -» C(mQ, /i) x ^ V M M ) •

(69)

We can deduce the form of C(rriQ,n) by considering the renormalization of the effective current qj^^5hv, shown in the last three terms of Fig. 5. These diagrams, computed in the effective theory, are independent of TUQ. However, in general they are divergent, so they depend on the renormalization scale n; the renormalization takes the form 97 V M M ) = q^5hv(mq)

x (l +

7o

g ; ln(/i/m,) + . . . ) ,

(70)

where here mq acts as an infrared cutoff. The \x dependence in the second term must be canceled by C(m,Q, /j). Since the logarithm depends on a dimensionless ratio, C(m,Q, /x) must be of the form C(m Q ,/i) = l + 7 o ^ ln(m Q //i) + . . . .

(71)

Comparing the dependence on TUQ of C(mQ, fi) and the expansion (68), we see that 70 = 7oc In general, the matching procedure at order a3 can also induce new Dirac structures They do not affect the leading logarithms discussed here.

qThv.

401

However, the effective theory allows us to go beyond leading order, and to resum all corrections of the form a™ In" rriQ. We do this with the renormalization group equations, which express the independence of physical observables on the renormalization scale /z. In this case, they require that the /i dependence of C(m,Q, fj.) cancel that of the one loop diagrams in Fig. 5, under small changes in fi:

The logarithms are resummed because the partial derivative is promoted to a total derivative with respect to /i, including the implicit dependence on /i of the coupling constant a s (/x):

2 2^ 0O = 11 - - t y = — f o r t y = 4 ,

(73)

where ty is the number of light flavors. We compute the anomalous dimension 7o from the ultraviolet divergent parts of the one loop diagrams shown in Fig. 5. It is instructive to perform the radiative correction to the current in detail, since this is different from the diagrams one is used to in ordinary QCD. With the HQET Feynman rules, the diagram may be written in Feynman gauge as Cf (iff) V

[ 7^7 J (27r)4

e

vela -, 7 V —vauhx=±, i vq q2

(74)

where Cf = | is the color factor. This expression may be simplified to

-Cf ig'v^n5

uh M« J ^ L _ ^ _ ,

(75)

which by Lorentz invariance is simply

-Cfig*va^uhsj^L±

r,*

(76)

Rotating to Euclidean space, performing the integral and extracting the pole in e, we find m M 7 5 uh x (2Cf) ^ -

e

+ finite.

(77)

402

Since / / = 1 + eln/i + ..., the one loop contribution to the matrix element then depends on ln/x as vtr15uhx(2Cf)^lnfi.

(78)

The contribution to the anomalous dimension is then 2C/ = | . The calculation of the wavefunction renormalization of the heavy quark (the third diagram in Fig. 5) is similar, but requires a novel version of the Feynman trick, dA

- = r

(79) 2

ab

{

J0 (a + A6) '

'

One also has to pick out the term which cancels the 1/v • k pole in the heavy quark propagator. With the factor of | which accompanies the contributions from wavefunction renormaliaztion, the result is I x va"l5

uh x (4Cf)^

In^.

(80)

Including the usual QCD renormalization of the light quark field, we find from the three terms in Fig. 5, respectively,5,7 8

1 /

8\

1 /16\

„

The solution of the renormalization group equation is

This then yields the leading logarithmic correction to the ratio /B_

JD

[rnp m

V B

/B//D:

fas{mc)Y/25

\as(mb)J

The radiative correction is approximately a ten percent effect. In fact, it has a simple physical interpretation. For virtual gluons of "intermediate" energy, mc < Eg < rrib, the bottom quark is heavy but the charm quark is light. Such gluons contribute to the difference between / # and fo even in the heavy quark limit. In summary, then, the purpose of the HQET is to make explicit all dependence of observable quantities on TUQ . The logarithmic dependence, through as{m,Q), arises from intermediate virtual gluons with m c < Eg < m&. We

403

obtain these corrections by computing perturbatively with the HQET Lagrangian, then using the renormalization group to resum the logarithms to all orders. The power dependence, I/TUQ, is extracted systematically in the heavy quark expansion. We have seen how to expand the Lagrangian and the states to subleading order; the application of the expansion to a physical decay rate will be presented in the next section. These lectures are meant to be pedagogical, so we will only treat the leading corrections to a few processes. However, the state of the art goes significantly beyond what will be presented here. For many quantities, not only the leading logarithms, a" In" TTIQ, but the subleading (two loop) logarithms, of order a " + 1 ln n mQ, have been resummed. Similarly, many power corrections are known to relative order I/TUQ. It is particularly important phenomenologically to take into account the corrections of order \jrr?c. 4

Exclusive B Decays

We now have the tools we need for an HQET treatment of the exclusive semileptonic transitions B —)• D Iv and B —> D* Iv. Earlier, we argued on physical grounds that in the heavy quark limit all of the hadronic matrix elements which appear in these decays are related to a single nonperturbative function £(u>). Now we will sharpen this analysis to actually derive these relations, and to include radiative and power corrections. In fact, almost all of our effort will go into the power corrections, since the radiative corrections to the transition currents are computed just as in the previous section. 4-1

Matrix element relations at leading order

The transitions in question require the nonperturbative matrix elements (D(v')\ &r"b \B(v)),

<£>>', e)| afb \B(v)),

{D*(v', e)| c W 6

\B(v)), (84) parameterized in terms of form factors as in Eq. (22). Our first task is to derive the relations between these form factors, as promised earlier. These relations depend on the heavy quark symmetry, that is, on the fact that the spin quantum numbers of Q and of the light degrees of freedom are separately conserved by the soft physics. Hence we need a representation of the heavy meson states in which they have well defined transformations separately under the angular momentum operators SQ and Ji. In particular, the representation must reflect the fact that a rotation by SQ can exchange the pseudoscalar meson M(v) with the vector meson M*(v,e).

404

The solution is to introduce a "superfield" M(v), denned as the 4 x 4 Dirac matrix 1 2 ' 2 0 M(v) = ^ ± i [ y » M ; ( « , e) -

5 7

M( V )] = V(v, e) + P(v).

(85)

Under heavy quark spin rotations SQ, M.{V) transforms as M{v) ^ D{SQ)M{v),

(86)

and under Lorentz rotations A, as M(v) -»• D(A)M{A-1v)D-1

(A).

(87)

Here D{- • •) is the spinor representation of 50(3,1). The superfield satisfies the matrix identity P+ M{v) P_ = M{v), (88) so it transforms the same way as the product of spinors hv q, representing a heavy quark and a light antiquark moving together at velocity vM. It is straightforward to verify the transformation properties of the superfield in the rest frame, in which v*1 = (1,0,0,0). In this frame, the spinor representation of the angular momentum operator J has components Sl = | 7 5 7 ° 7 l . It acts on the superfield by JlM = [Sl,Ai]. Defining the polarization vectors e£ = (0,1/V%, ±i/V%, 0) and e£ = (0,0,0,1), it is easy to check that J 2 P = J 3 P = 0,

J2V{e) = 2V{e),

J3V(e) = 0, (89) so P has spin zero and ^(e) spin one. On the other hand the heavy quark spin operator S Q has the same component representation but acts only on the left, l (SQ)1M. — S M.. One may then check that (SQ)3P=\v(e3),

J3V(e±) = ±V(e±),

(SQfV(e3)

= ±P.

(90)

As promised, heavy quark spin tranformations exchange the pseudoscalar and vector mesons. A current which mediates the decay of one heavy quark (Q) into another (Q1) is of the form hv' Thv. Under a rotation by SQ, the effective field hv transforms as hv->D(SQ)hv, (91) while hv' is unchanged. The current would remain invariant if we took T to transform as r^TD-l{SQ). (92)

405

On the other hand, the matrix element of superfields {M'{v')\hv,Thv\M(v)) is invariant if we rotate both hv formation law (86) for M(v), it must be proportional to TM(v). we find that the matrix element (M'(v')\hv,

(93)

and M(v) by the same SQ. With the transfollows that the Sg-invariant matrix element When we also consider rotations under SQ*, is restricted to the general form

Yhv \M(v)) = -y/MMMM,

Tr \M {v')YM{v)F{v,v')\

. (94)

The product of masses in front is a convention which restores the relativistic normalization of the states. Note that the heavy quark symmetry allows an arbitrary 4 x 4 Dirac matrix F(v,v') to act on the "light quark" part of the superfields. Its presence reflects the fact that only Lorentz symmetry constrains the spin of the light degrees of freedom during the decay. A general expansion of F(v, v') in terms of scalar functions Fi(w = v • v') takes the form F{v,v') = Fiiw) + F2(w)i> + F3(w)f

+ F4(w)1>f .

(95)

However, the identity (88) applied to the matrix element (94) yields F{v,v') = P_ F{v,v') PL = [Fiiw) - F2{w) - F3(w) + F4(tu)] P- PL . (96) In other words, F(v, v') actually may be taken to be a scalar, which we identify with the Isgur-Wise function, F{v,v') = t{w).

(97)

As an exercise, let us apply this formalism to the matrix elements for B —> (D, D*) Iv. For a given matrix element, we pick out the part of the superfield M(v) which is relevant. Hence we find (D(v')\crb\B(v))

= (MD(v')\hcv,rhbv

\MB(v) 5

= - v ^ ^ T r [ 7 ? ; 7 " P + ( - 7 5 ) ] $(w) = yJmDmB £(w) {v + v'Y (D*(v',e)\cnsb\B(v)}

=

(98)

(M*D(v',e)\hcv,rj5hbv\MB(v))

= -^mD.mB = y/mD.mBS{w)

Tr [ ^ + / 7 5 P + ( - 7 B ) ] £(w)

(99) M

[{w + l j e ' " - e* • (v + v') ]

406

(D*(v',e)\crb\B(v))

= (MD(v',e)\hcv,rhbv\MB(v)) = -^mD.mB TV [ / P | 7 " P + ( - 7 5 ) ] £(«,) = ^nn,.mflfW ^ ^ e » ^ ,

(100)

reproducing explicitly the relations (23) between the independent form factors hi(w). We can also derive the normalization condition at w = 1. Consider the matrix element of the b number current bj^b between B meson states. In QCD, the matrix element of this current is exactly normalized, (B{v)\h"b\B{v))

= 2pB = 2mBvfl.

(101)

But in HQET, we have (B(v)\h»b\B(v))

= (MB(v)\h„rhv |MB(«)> = ms£(u • v)(v + vY = 2ro B u" £(1).

(102)

Hence the normalization condition at zero recoil, £(1) = 1,

(103)

follows directly from the conservation of the heavy quark number current. 4.2

Power corrections to the matrix elements

The matrix elements we have derived are computed in the strict limit mb}C —> 00. How are they affected by corrections of order l/m& and l / m c ? There are two sources of I/TTIQ corrections in the effective theory: the corrections (64) to the heavy quark currents, and the corrections (59) to the Lagrangian. When radiative corrections are included, the expansion of the heavy quark current cYb in terms of HQET operators has a form which is somewhat more general than Eq. (64),

cTb -+ a0(as) hcv,Thbv + ^ ^ K,T?iDahbv + ^ ^ K H X ^ X + • • zrrib

Imc

(104) The matrix elements of the power corrections are constrained by heavy quark symmetry in a manner completely analogous to the leading current. In terms of traces over the superfields, we have 14 (M'{v')\hv, TaiDa hv \M(v)) = -yjMMMM< Tr [ A * V ) Va M(v) Ga{v,«')] , (105)

407

where Ga(v,v')

is another arbitrary 4 x 4 Dirac matrix. The matrix element (M'(v')\ hv. H ^ a ) r i X \M(v))

(106)

may also be written in terms of Ga(v,v'), using charge conjugation. The l/m,Q corrections OK and OG to the Lagrangian contribute somewhat differently. In order to apply heavy quark symmetry, the matrix elements of the local currents, both leading and subleading, must be written in terms of the effective states \M(v)). However, these states are not eigenstates of the Hamiltonian, once OK and OG are included in the Lagrangian. Hence, for example, we must allow for the possibility that if an effective state \M(v)) is created at time t — —oo, then OK or OG could act on the state before its decay at t — 0. This possibility is accounted for by including time-ordered products in which OK or OG is inserted along the incoming or outgoing heavy quark line. If we are keeping terms of order I/TUQ, only one insertion of OK or OG needs to be included. The time-ordered products are of the form 14 {DM(v)\ cTb\B(v)) •••

+

=

^(M'WlijdyTfarhlOK

+ ^-b(M'(v')\iJdyT{hcv,Thbv,0K

+ O^lMiv)) + OG}\M(V)),

(107)

where the ellipses include the current corrections computed earlier. The evaluation of the matrix elements of the time-ordered products will lead to still more nonperturbative functions like F(v,v') and Ga(v,v'). 4-3

Corrections at zero recoil

It is straightforward, but not very illuminating, to expand all of the new nonperturbative functions which arise at order I/TUQ in terms of scalar form factors. In the end, the corrections may be parameterized in terms of four functions of the velocity transfer w, and a single nonperturbative parameter A, all proportional to the mass scale AQCD- 14 The new parameter has a simple interpretation as the "energy" of the light degrees of freedom, and is given by A=

lim (rriB - m&).

(108)

mi-s-oo

Instead of a general treatment, however, we will consider the I/TUQ corrections at the zero recoil point w = 1. This is clearly the most important case, because it is at this point that the nonperturbative matrix elements are

408

absolutely normalized in the heavy quark limit. What happens to this normalization condition when I/TUQ corrections are included? Let us study the corrections to the current in detail. They are described by the nonperturbative function Ga(v,v'). At v = v', Ga{v,v) may be expanded as Ga{v,v) = Giva + G27a + Gzvai> + G47aJ*. (109) But Ga(v,v) is subject to the same constraint as F(v,v'), Ga(v,v)

= P-Ga(v,v)P-

= (Gi - G2 - G3 + G 4 )««P_ = GvaP-,

(110)

and it, too, is equivalent to a Dirac scalar (the same is not true of the general function Ga(v, v')). Now consider the matrix element where we take Ta = va. Then we have (M'(v)\ hv, iv-Dhv

\M(v)) = - s/MMMM, = - Gs/MMMM,

Tr

\M'(V)

Tr

va

M(v) Gva

\M\V)M{V)

(111)

But this matrix element vanishes by the classical equation of motion in the effective theory, v-Dhv(x)=0. (112) Hence G = 0 = Ga(v,v). There are no 1/rriQ corrections from the current to the normalization condition at zero recoil.14'21 The same is true of insertions of the corrections OK and OG to the Lagrangian: their contribution vanishes at w = 1. To show this requires the imposition of the conservation of the b number current at order 1/m;,, much as we derived the normalization of the Isgur-Wise function at leading order. This part of the argument is analogous to the classic nonrenormalization theorem of Ademollo and Gatto. 22 In the end, we have the result known as Luke's Theorem}* There are no corrections at zero recoil to the hadronic matrix elements responsible for the semileptonic decays B —> D Iv and B —>• D* Iv. The leading power corrections to the normalization of zero recoil matrix elements are only of order 1/m 2 . Given that A Q C D / ^ C ~ 30% and A Q C D / ? 7 1 2 ~ 10%, the implication is that the leading order predictions at w = 1 are considerably more accurate than one might have expected. In addition, away from zero recoil the l / m c corrections must be suppressed at least by (w — 1). On closer inspection, this result is more interesting for B ->• D* Iv than for B -> D Iv. This is because the leading order matrix element for B -> D Iv vanishes kinematically at zero recoil for a massless lepton in the final state. Hence, in this case the l / m c corrections are not suppressed as a fractional correction to the lowest order term. 23

409 4-4

Extraction of \Vcb\ from B ->• D* tv

An immediate application of these results is the extraction of |VC&| from the exclusive decay B —)• D* tv. This process is mediated by the weak operator Obc (4), whose matrix element factorizes as {D* tv\ Obc \B) = ^ ^

(D*| C7"(l -

5 7

)6 |J3> (lu\ llfi(l

-

7

> |0).

(113)

The leptonic matrix element may be computed perturbatively, while we treat the hadronic matrix element in the heavy quark expansion. The result is a differential decay rate of the form 24 £

= ± ^ \Ychf (mB - mD.fml.{w

+ 1)3V/^:T

4sw m2B — 2wmbrriD* + m2D, 2 F (w). w+1 {rriB — mi)*)2

(114)

All of the HQET analysis goes into the factor F(w), which has an expansion F(w) = £(w) + (radiative corrections) + (power corrections).

(115)

We extract \Vcb\ by studying the differential decay rate near w = 1, where the hadronic matrix elements are known. Of course, this requires extrapolation of the experimental data, since the rate vanishes kinematically at w = 1. For massless leptons, only the matrix element {D*\ C7M756 \B) of the axial current contributes at this point. The analysis of this quantity in the HQET yields an expansion of the form

n i ) = VA

1 + — + — + S1/m2 + mc mb

(116)

The correction <$i/m2, which contains terms proportional to 1/m2, 1/m2 and l/m c mj,, is intrinsically nonperturbative. It has been estimated from a variety of models to be small and negative, 18 ' 25,26 51/m2 « -0.055 ± 0.035 .

(117)

Note that the model dependence in the result has been relegated to the estimation of the sub-subleading terms. The radiative correction r\A has now been computed to two loops, 27 ' 28 rjA = 0.960 ± 0.007.

(118)

410

The result is a value for F ( l ) with errors at the level of 5%, F ( l ) =0.91 ± 0 . 0 4 .

(119)

This is the theory error which the experimental determination of |Vcf,| will inherit. It is dominated by the uncertainty in the nonperturbative corrections, and it is difficult to see how this can be improved much in the future. All that is left experimentally is to extrapolate the data to w = 1 and extract lim

,

l

^ .

(120)

«>-t-i y/w — 1 d w

Once the kinematic factors in Eq. (114) have been included, this amounts to a direct measurement of the combination |Vy.F(l). Both CLEO and LEP have reported results for this quantity. 29,30 They have taken the slightly different value F(l) = 0.88 ± 0.05, and I have scaled up the theory error of the CLEO result to make it consistent with LEP. Then we have quite consistent results for Vcb: CLEO: LEP average :

(39.4 ± 2.1 ± 2.0 ± 2.2) x 10" 3 (38.4 ± 1.1 ± 2.2 ± 2.2) x 1 0 - 3 .

(121)

This value of \Vcb\ has almost no dependence on hadronic models. In contrast to model-based "measurements", here the theoretical error is meaningful, in that it is based on a systematic expansion in small quantities. 5

Inclusive B Decays

An exclusive semileptonic B decay, such as B —>• D (.P, is one in which the final hadronic state is fully reconstructed. An inclusive decay, by contrast, is one in which only certain kinematic features, and perhaps the flavor, of the hadron are known. In this case, we need a theoretical analysis in which we sum over all possible hadronic final states allowed by the kinematics. Fortunately, this is possible within the structure of the HQET. As in the case of exclusive decays, the key theme is the separation of short distance physics, associated with the heavy quark, from long distance physics, associated with the light degrees of freedom. We will also rely on heavy quark spin and flavor symmetry. However, the new ingredient will be the idea of "parton-hadron duality", which, as we will see, also relies on the heavy quark limit nib » AQCD-

411

5.1

The inclusive decay B —> Xc Iv

Let us consider the inclusive decay B(PB)

-> XC(PX)

Hpt)v(pv),

(122)

where all that is known about the state Xc are its energy and momentum, and the fact that it contains a charm quark. This decay is mediated by the weak operator 0\,c. It is easy to generalize our discussion to inclusive decays of other heavy quarks, such as b —>• u Iv and c —>• s iv, by replacing Obc with the appropriate weak operator. The treatment of exclusive decays required both the b and c quarks to be heavy. For inclusive decays we can relax this condition on the c quark, requiring only mj S> AQCD- What does the weak decay of the b, at time t = 0, look like to the light degrees of freedom? For t < 0, there is a heavy hadron composed of a point-like color source and light quarks and gluons. At t = 0, the point source disappears, releasing both its color and a large amount of energy into the hadronic environment. Eventually, for t > 0, this new collection of strongly interacting particles will materialize as a set of physical hadrons. The probability of this hadronization is unity; there is no interference between the hadronization process and the heavy quark decay. There are subleading effects in powers of A Q C D / " ^ 6 , but they do not alter the probability of hadronization. Rather, they reflect the fact that the b quark is not exactly a static source of color: it has a small nonrelativistic kinetic energy and it carries a spin, both of which affect the kinematic properties of its decay. As in the case of exclusive decays, we will compute the inclusive semileptonic width T(B —> Xclv) as a double expansion in powers of as(mi)) and AQCD/»TI&- 1 9 ' 3 2 ' 3 3 ' 3 4 The expansion in as(mb) reflects the applicability of perturbative QCD to the short distance part of the process. The heavy quark expansion will be continued to relative order 1/mj, as there is an analogue of Luke's Theorem which eliminates power corrections to the rate of order l / m j . These corrections will be written in terms of three nonperturbative parameters. The first, A, is defined in Eq. (108). It is essentially the mass of the light degrees of freedom in the heavy hadron, but we will see that it is plagued by an ambiguity of order AQCD in the definition of the b quark mass. The other two parameters are the expectation values in the B meson of the leading corrections OK and OQ to £HQET- They are defined a s 1 8

X2

= -6LW°GW>

< 123 >

412

where we take the usual relativistic normalization of the states. Hence, Ai may be thought of roughly as the negative of the b quark kinetic energy, and A2 as the energy of its hyperfine interaction with the light degrees of freedom. Now let us outline the computation. The inclusive decay involves a sum over all possible final states, which is actually a sum over exclusive modes (such as D, D*,Dir,...), followed by a phase space integral for each mode. We write F(B -> XclD) = J2 /d[P.S.] \{XC19\ Obc \B)\2 .

(124)

There is an Optical Theorem for QCD, which follows from the analyticity of the scattering matrix as a function of the momenta of the asymptotic states. Its content is that a transition rate is proportional to the imaginary part of the forward scattering amplitude with two insertions of the transition operator, T(B -* Xclu)

= -21mi f dxeikx

(B\T {olc{x),Obc(0)}

\B) = 2 I m T .

(125) In what follows, we will write the time-ordered product T{0'bc, Obc} as a series of local operators, using the Operator Product Expansion. As we will see, the applicability of this expansion, and its computation in perturbation theory, will rest on the limit nib S> AQCD- We will then use this limit again to expand the matrix elements of these local operators in the HQET. The first step is to factorize the integration over the lepton momenta, which can be performed explicitly. Written as a product of currents, Obc takes the form Obc=q^JbUcJi»,

(126)

where j^c =

cr(i-j5)b

J^=^7"(l-75)i/.

(127)

Then T can be decomposed as an integral over the total momentum q^ = Pi + Pv transferred to the leptons, T = \02F\Vcb\2 jdqT^(q)

L^(q).

(128)

Here the lepton tensor is L^(q)

= j d[P.S.] <0| Jl \£9) (19\ Jtv |0) = -o^ Wiv - q2g»v) :

(129)

413

and the hadron tensor is T^(q)

= -ijdxe**

(B\T{J£(X),J£C(0)}

\B).

(130)

We will need the imaginary part, Im T^v. Where is it nonvanishing? In quantum field theory, a scattering amplitude develops an imaginary part when there can be a real intermediate state, that is, the intermediate particles can all go on their mass shell. Whether this is possible, of course, depends on the kinematics of the external states. In this case, there are two avenues for creating a physical intermediate state. 32 The first is to act on the external state \B) with the transition current J^c. The state which is created has no net b number and a single charm quark; the simplest possibility is the decay process b —> c. The momentum of the intermediate state is px = PB — Q', the condition that it could be on mass shell is simply Px = (PB - qf > m2D . (131) If we define scaled variables PB = mBv^ ,

gM = q*jmB ,

mD = mD/mB

,

(132)

this condition becomes v • q < \ (1 + q2 ~ m2D) .

(133)

Another possibility is to act on \B) with the conjugate operator ,/£.'. This operation would produce an intermediate state with two b quarks and one c. For this state to be on shell, the momentum transfer has to satisfy Px = (PB + qf > {2mB + mD)2 ,

(134)

v • q > r (3 - q2 + 4m D + m2D) .

(135)

that is,

The physical intermediate states are shown as cuts in the v • q plane in Fig. 6. Also shown is the contour corresponding to the phase space integration over the lepton momentum q. For physical (massless) leptons which are the product of a heavy quark decay, this integral runs over the top of the lower cut, for the range y/q^+ ie
-(l + q2-m2D)+ie.

(136)

414

integration contour I

(extended)

Figure 6: Analytic structure of T>"/ in the complex v • q plane, for fixed, real q2. The integration contour is over the "physical cut", corresponding to real decay into leptons. The unphysical cuts correspond to other processes.

As indicated by the dotted line, we can continue this contour around the end of the cut and back along the bottom, to v • q = y/q^ - ie. Since TM"(w • q *) = —T^v(v • q) for real q2, we compensate for extending the contour by dividing the new integral by two. We now encounter our central problem. The integral over v • q runs over physical intermediate hadron states, which are color neutral bound states of quarks and gluons. Hence the integrand depends intimately on the details of QCD at long distances, which is intrinsically nonperturbative. A perturbative calculation of T M ", which is all we have at our disposal, would appear to be of no use. The solution is to deform the contour away from the cut, into the complex v • q plane, as shown in Fig. 7. Since the scale of momenta is set by m&, the contour is now a distance of order rrib away from the resonances.32 Since mb ^ AQCD, it is reasonable to hope that a perturbative treatment in this region is valid. Essentially, we are saved because we do not need to know TliU(q) for every value of q, just suitable integrals of T Mi/ . That we can use such arguments to compute perturbatively the average value of a hadronic quantity, where at each point the quantity depends on nonperturbative physics, is known as (global) parton-hadron duality. Parton-hadron duality has the status of being somewhat more than an assumption, since it is known to hold in QCD in the limit m;, —> oo, but somewhat less than an approximation, since it is not known how to compute systematically the leading corrections to it. In any case, the limit mi, > AQCD

415

Figure 7: The deformation of the integration contour into the complex v • g plane.

plays a crucial role here. By deforming the integration contour a distance of order mb away from the resonance regime, we find the correspondence in QCD of our earlier intuitive statement: the probability of the decay products materializing as physical hadrons is unity, independent of the kinematics of the short distance process. The local redistribution of probability in phase space due to the presence of hadronic resonances is irrelevant to the total decay. Finally, we should note that since we do not have control over the corrections to local duality, it might work better in some processes than in others, for reasons that need not be apparent from within the calculation. Hence one must be particularly wary of drawing dramatic conclusions from any surprising results of these inclusive calculations.35 Let us perform the operator product expansion at tree level, and for decay kinematics. The Feynman diagram is given in Fig. 8, which yields the expression T"" = B y (1 -

5 7

)

*"~i + m ° . 7"(1 - 75)& • (Pb ~ q)2 -m2 + ie

(137)

We now write

p£ = mhV» + kfi = mb(v" + &) q» = q»/mb rhc = mc/mb b{x) = e-imhVX hv (x) + 0(1/mb),

(138)

and expand in powers of l/m^. Since the operator product expansion is in terms of the effective field hv, a factor of k11 corresponds to an insertion of the covariant derivative iD^. Operator ordering ambiguities are to be resolved by considering graphs with external gluon fields. As an example of the procedure, let us expand the propagator to order \/m\. (There are also corrections to the currents at this order, which are

416

Pb-q •

Figure 8: The operator product expansion at tree level.

included in a full calculation.) It is convenient to define the scaled hadronic invariant mass, s = (mbV*1 - q^)2/m2b = 1 - 2v • q + q2 . v

Then we find a contribution to T^

(139)

of the form

2k-q-k2 1 hv . rh2. + ie + {s — fh2,+ it)2+ ... '(140) From this expression we can read off the operators which appear in the operator product expansion. Since T"" = — ^ 7 < ^ - | ) 7 " ( l - 7 5 ) mb

Im

(s — rh2 + ie)n

= 7r(n-l)!(-l)n5("-1)(s-m^),

(141)

we see that the effect of taking the imaginary part in each term is to put the charm quark on its mass shell. The leading term is a quark bilinear,

—

mi,

hvl,1{i>-i)lv{l-l5)hv,

(142)

It is straightforward to compute its matrix element in the HQET using the trace formalism, (B\hvr^-4)Y(l-J5)hv\B)

= 2mB (2v*vv - g^ - v»q" - vv
(1 - 8m2c + ml

- ml - Urht lnmc2)

(144)

417

Of course, if we only intended to reproduce the free quark decay result, we would never have introduced so much new formalism. The value of the HQET framework is that it allows us to go beyond leading order and compute the next terms in the series in 1/m™. For example, consider the operators induced by the expansion of the propagator (140). The correction of order \jm\, comes from the operator m 2 (s — m 2 + it)2

hvl"(i,-4)Y(l-jb)q-iDhv.

(145)

However, the matrix element of this operator is of the form

(B\hvTa(v,q)iDahv\B),

(146)

which, as we have seen, vanishes by the classical equation of motion. (In writing Eq. (140), we have already dropped terms explicitly proportional to v • k, for the same reason.) In fact, since all l/m& corrections, from any source, have a single covariant derivative, they all vanish in the same way. This is the analogue of Luke's Theorem for inclusive decays.32 The correction of order 1/m2, in Eq. (140) is

J

1 2

mf (s - m

+it)

2

^7M(^-|)7"(l-75)(^)2^-

(147)

The matrix element of this operator is related by the heavy quark symmetry to Ai, the expectation value of OK- The full expansion of j f " also induces operators with explicit factors of the gluon field, whose matrix elements are related to A2. We now present the result for the inclusive semileptonic decay rate, up to order 1/m2, in the heavy quark expansion, and with the complete radiative correction of order as. We also include that part of the two loop correction which is proportional to PoQ.^. Since @o w 9, perhaps this term dominates the two loop result. In any case, it is interesting for other reasons, as we will see below. Let us first consider the decay B —• Xu £p, for which the decay rate simplifies since mu = 0. We find33,34,36,37,40 r(B4i„ft)

G2F\Vub\2 „ 6 1 + [ 25 _ 2 ^ ^ 1 1927T3 mh 6 3 - (2.98/30 + Cu)

as{mb)

as{mb) A!-9A2 2m 2

(148)

+

'

418

When we include the charm mass, it is convenient to write the unknown quark masses in terms of the measured meson masses and the parameters of the HQET. In terms of the spin averaged mass TUB = ( m j + 3ms* )/4, we have mb = mB - A + -±-

+ ... ,

(149)

and analogously for mc. We then find33-34'38*39

T(B^XclD)=

°^cb[

|2

mB x 0.369 1 _ 1 . 5 4 S ^ 1

- (1.43/3o + Cc) (Si^Bl)2 \

-K

J

_ 1.65 A U _ o.87as{mb) mB

\

0.95-^- - 3.18—|- + 0 . 0 2 ^ mB mB mB

(150)

All the coefficients which appear in this expression are known functions of rap/mB, and are evaluated at the physical point WIDI^B = 0.372. In both B —> Xu tv and B —> Xc Iv, the power corrections proportional to Ai and A2 are numerically small, at the level of a few percent. 5.2

Renormalons and the pole mass

The inclusive decay rate depends on the heavy quark mass rrn,, either explicitly, as in Eq. (148), or implicitly through A, as in Eq. (150). At tree level, mj is just the coefficient of the b b term in the QCD Lagrangian, but beyond that we are faced with the question of what exactly we mean by mb- Should we take an MS mass, such as m\i{mb)l Or should we take the pole mass m^° e , or maybe some other quantity? The various prescriptions for m\, can vary by hundreds of MeV, and, since the total rate is proportional to mjj, the question is of practical importance if we hope to make accurate phenomenological predictions. At a fixed order in QCD perturbation theory, the answer is clear. The heavy quark masses which appear come from poles in quark propagators, so we should take m£ ole (and mP° le ). This is also the prescription for the mass which cancels out the on-shell part of the heavy quark field in the construction of £HQET- Hence the difference of heavy quark pole masses is known quite well,

„}*-„**_ (m,-J+5*L + . . . ) - ( m D - A + 5 ^ = 3.34GeV + 0(H2QCC/m%).

+

.. (151)

419

+

+

fy , & J ^+

Figure 9: The radiative corrections to m£°

e

of order a 3 ( m j , ) n + 1 j3^.

Since T(B ->• X c £p) depends approximately as m\{rrn, — m c ) 3 , the uncertainty due to quark mass dependence is reduced. The problem, of course, is that there is no sensible nonperturbative definition of m\°e, since due to confinement there is no actual pole in the quark propagator. Hence a direct experimental determination of a value for m£° e to insert into the theoretical expressions (148) and (150) is not possible. How, then, can we do phenomenology? One approach would be to define m £ ° e to be the pole mass as computed in perturbation theory, truncate at some order, and then estimate the theoretical error from the uncomputed higher order terms. However, it turns out that even within perturbation theory the concept of a quark pole mass is ambiguous. Consider a particular class of diagrams which contribute to m£° e , shown in Fig. 9. The perturbation theory is developed as an expansion in the small parameter as(mb), so we hope that it will be well behaved. Each of the bubbles represents an insertion of the gluon self-energy, which is proportional at lowest order to as(m;,)/3o- Of course, the infinite sum of the graphs in Fig. 9 can be absorbed into the one loop graph, with a compensating change in the coupling from as(mb) to as(q), where q is the loop momentum. The result is an expansion for m£° e of the form pole pole

Tnvb

— / —

\

= mb(mb)

1 + a i a s + (a2/3o + b2)a2s + (a3ffi + b3/30 + c3)a\ +.

(152)

where as = as(mb). The graphs in Fig. 9 contribute the terms proportional to a"+1P^. Since /30 « 9 these terms are "intrinsically" larger than ones with fewer powers of /?o, and we might hope that their sum approximates the full series. However, it is important to realize that the only limit of QCD in which

420

such terms actually dominate is that of large number of light quark flavors, in which case the sign of /30 is opposite to that of QCD. Although this is a physical limit of an abelian theory, we are certainly not close to that limit here. The ansatz of keeping only the terms proportional to a"+1(m;,)/?Q is known as "naive nonabelianization" (NNA).43 What is most interesting about the series of terms shown in Fig. 9, which takes the form J^a„a"/3J _ 1 , is that it does not converge. Already in the graphs kept in the NNA ansatz, we are sensitive to the fact that QCD is an asymptotic, rather than a convergent expansion. For large n the coefficients an diverge as n!, much stronger than any convergence due to the powers a™. The series can only be made meaningful if this divergence is subtracted. As with many subtraction prescriptions, there is a residual finite ambiguity.1* This ambiguity, known as an "infrared renormalon", leads to an ambiguity in the pole mass of order 4 3 ' 4 4 ' 4 5 <5mP° le ~100MeV.

(153)

By the definition (108), A also inherits this ambiguity. The expressions (148) and (150) are plagued by two problems. The first is the renormalon ambiguity in m£° e and A. The second is that the perturbative expansion for the rate T is itself divergent, and also has an infrared renormalon. In the expansion r = T0 [ ^ a ^ a " ( m & ) / 3 o _ 1 + (power corrections)

,

(154)

the coefficients a'n also diverge as n!. However, it turns out that these two problems actually cure each other, because the infrared renormalons in m^° e and in the perturbation series for T cancel.44'46 We can exploit this cancelation to improve the predictive power of the theoretical computation of the rate. Without this improvement, the infrared renormalons render the expressions (148) and (150) of dubious phenomenological utility. The most reliable approach, theoretically, is to eliminate m£° or A explicitly from the rate by computing and measuring another quantity which also depends on it. For example, let us consider the charmless decay rate T(B —y Xulv~) and the average invariant mass (SH) of the hadrons produced in the decay. Each of these expressions suffers from a poorly behaved perturbation series in the NNA approximation. Ignoring terms of relative order 1/ml and writing the rate in terms of A instead of m£° e , we find to five loop In a formal treatment, this ambiguity arises from a choice of contour in the Borel plane.

421

order 43

1927T3

m.

- 2 . 4 1 ^ - 2.98 (^-) •K

\

/J0-4.43 ( ^ )

7T /

ft

V 7T /

+... 5 A + G>|VUJ,|

1927T 3

5

•771,

1 - 0.061 - 0.120 - 0.107 - 0.111 - 0.136 + . . . (155)

- 5A/m B + . . .

for as(mb) = 0.21 and /?o = 9- As we see, not only does the perturbation series fail to converge, it does not even have an apparent smallest term, where one should truncate to minimize the error of the asymptotic series. The series for {SH) exhibits a similar behavior,40 (sH) = m%

0.20^ + 0.35 (—) 2 A> + 0.64 ( ^ ) Vo + 1-29 ( ^ ) Vo +

2.95p)Vo4 + -- + ^ — + -." \ 7T /

10 771R

= m B [0.0135 + 0.0141 + 0.0156 + 0.0189 + 0.0261 + . . . -7A/10mB + ...l .

(156)

However, the situation improves dramatically if we eliminate A and write T directly in terms of (SH), 1

~

192*3

^

1 - 7.14

(SH)

m%

0.064 - 0.020 - 0.0002 - 0.022 - 0.047

•

(157)

By truncating this series at its smallest term, 0.0002, we obtain a new expression in which the theoretical errors are under control. The price is that we must now measure a second quantity, (SJJ), in the same decay. In principle, the same procedure works for decays to charm.40 In practice, it is best to combine a number of determinations of A. This has been done by CLEO, 41 which has performed a comparison of measurements of the moments {Ei), {E}), {SH -fn2D) and {{SR - ^ D ) 2 ) - The results, reproduced in Fig. 10,

422

Figure 10: The CLEO determination of A and Ai from B —>• Xctv.

are somewhat disappointing, in that one does not obtain a very consistent determination of A and Ai. Perhaps this is just a fluctuation, or perhaps this is a sign that parton-hadron duality is failing in these days. An alternative approach is to express the width V in terms of the running mass mb{mt,) instead of another inclusive observable.44'47 Since the MS mass is a short distance quantity, this also eliminates the infrared renormalon, which is associated with long distance physics. However, from a phenomenological point of view, it raises the question of how the running mass is to be determined from experiment. Possibilities include quarkonium spectroscopy, QCD sum rules, and lattice calculations, but in all of these cases it is important to determine reliably the accuracy of the method, and how to deal with renormalon ambiguities in a manner that is consistent with their treatment in the calculation of T. Nevertheless, such an approach, particularly one based on

423

analyzing quarkonium and production near threshold, should eventually prove fruitful.48 5.3

Phenomenology of VCb and Vub

Despite this ambiguous situation, groups have presented extractions of V^ based on inclusive semileptonic b decays by simply inserting "reasonable" values for A and Ai. Such an approach clearly has its dangers! At any rate, the quoted result is 4 2 \Vcb\ = (40.0 ± 0 . 4 ±2.4) x 1 0 - 3 .

(158)

The lack of controversy about this procedure is no doubt due in part to the fact that this number is quite consistent with that determined from the analysis of exclusive decays. There are additional, much more interesting, problems with extracting \VU(,\ from the inclusive decay B —> Xulv. They arise from the problem that the process B —• Xctv presents an overwhelming background to 5 -> Xulv. The only way to avoid this background is to restrict oneself to a corner of phase space in which charmed final states are kinematically inaccessible. Existing experimental analyses isolate the charmless decays by imposing the requirement Et > (m2B — m2D)/2mB or SH < m2D. Unfortunately, the OPE can be shown to break down in these restricted corners of the phase space. 49 ' 50 ' 51 I do not have space here to explore this issue in much detail, but it is easy to appreciate the essence of the problem. For massless final states, the OPE is an expansion in powers of the light quark propagator, l/m&(l — v • q + q2). Over most of the final state phase space, the denominator is of order mi, and the OPE is well behaved. But there exist configurations for which both the denominator vanishes and the operator matrix elements which appear in the OPE are nonzero. It turns out that the dangerous region is when v • q —> \ and q2 —> 0. The precise form of the divergence depends on the kinematic distribution being studied. The general form, however, is universal. In this "endpoint region", let y be a scaled variable such that y —> 1 at the kinematic endpoint. For example, for dT/dEi, we take y = 2Ei/mi,, and for dT/dsn we take y = SH /Ami,. Then near y = 1, the OPE takes the general form dr ~ An ^-oc> c n ~ —, dy f^0 m£(l - y)n

(159)

where c„ are coefficients of order one, and the An are moments defined by (B{v)\ hv W^

... iD^ hv \B(v))/2mB

= Anv^

...v^

+ ... .

(160)

424

The ellipses represent terms involving factors of the metric tensor g^iflj, which are subleading. Since the An are associated with totally symmetric combinations of the covariant derivatives, they may be interpreted roughly as the moments of the heavy quark momentum. Note that as defined, AQ = 1, A± = 0 and A2 = Ai. The divergences in Eq. (159) can be controlled only if one integrates over a large enough region near the endpoint, 1 - 8 < y < 1. For <5 ~ A/mt, one finds a series for T which does not converge, since the individual terms are of order An/An ~ 1. This situation reflects a dependence of the shape of dT/dy on the entire b quark momentum distribution in the B meson. Since, for example, the window 2.3 GeV < E{ < 2.6 GeV corresponds to 8 ~ 0.1, this problem pollutes the extraction of \Vub\ from dT/dEi. One is forced to introduce a model for the b wavefunction, with the attendant uncontrolled theoretical uncertainties. The same is true, it turns out, for dT/dsn• The current "best" measurement of Vub from LEP, based on an inclusive analysis, is 52 \Vub\ = [4.05tlil(sUt.)^fxt00f7(sys.)

± 0.02(r6) ±0.16(HQE)] x 1 0 " 3 , (161) or approximately IV^/V^I = 0.104+QoJg. While these analyses are experimentally very sophisticated, they rely intensively on a two-parameter model of the b quark wavefunction. Essentially, in such a parameterization all moments of the b momentum distribution are correlated with the first two nonzero ones, a constraint which is unphysical. Even if the two parameters are varied within "reasonable" ranges, it is doubtful that such a restrictive choice of model captures reliably the true uncertainty in \Vub\ from our ignorance of the structure of the B meson. While the central value which is obtained in these analyses is reasonable, the realistic theoretical error which should be assigned is not yet well understood. A recent analysis by CLEO of the exclusive decay B —>• pi v yields 53 |K ft | = [3.25 ± 0.14(stat.)t£;^(syst.) ± 0.55(theory)] x 1CT3 ,

(162)

or approximately |V^b/V^i,| = 0.083^o'oi6> essentially consistent with the LEP result. In this case the reliance on models is quite explicit, since one needs the hadronic form factor {p\ U7M(1 — j5)b \B) over the range of momentum transfer to the leptons. The CLEO measurement relies on models based on QCD sum rules, which have uncertainties which are hard to quantify. Hence, just as in the case of the LEP measurement, the quoted errors should not be taken terribly seriously. All of the current constraints are consistent with |Vui,/Vci,| = 0.090 ± 0.025, where I strongly prefer this more conservative estimate of the theoretical errors. The problem lies not in the experimental analyses, but in our insufficient understanding of hadron dynamics.

425

Recently it has been pointed out 5 4 that the one may alternatively reject the charm background by studying the distribution dT/dq2 and restricting oneself to q2 > (m^ — mo)2- Not only does this cut eliminate charmed final states, but it also avoids the troublesome region near q2 = 0. Hence such a determination would not be polluted by the divergence described above. Whether the neutrino reconstruction algorithms of the B Factories will be up to the task of this measurement yet remains to be seen. 6

Concluding Remarks

Unfortunately, we have had time in these lectures only to introduce a very few of the many applications of heavy quark symmetry and the HQET to the physics of heavy hadrons. Since its development less than ten years ago, it has become one of the basic tools of QCD phenomenology. Much of the popularity and utility of the HQET certainly come from its essential simplicity. The elementary observation that the physics of heavy hadrons can be divided into interactions characterized by short and long distances gives us immediately a clear and compelling intuition for the properties of heavy-light systems. The straightforward manipulations which lead to the HQET then allow this intuition to form the basis for a new systematic expansion of QCD. The deeper understanding of heavy hadrons which we thereby obtain is increasingly important as the B Factory Era begins. Acknowledgements It is a pleasure to thank the organizers of TASI-2000 for the opportunity to present these lectures, and for arranging a most interesting and pleasant summer school. This work was supported by the National Science Foundation under Grant No. PHY-9404057, by the Department of Energy under Outstanding Junior Investigator Award No. DE-FG02-94ER40869, and by the Research Corporation under the Cottrell Scholarship program. References 1. There are many excellent reviews of heavy quark symmetry and its applications. In particular, see A. V. Manohar and M. B. Wise, Heavy Quark Physics (Cambridge University Press, Cambridge, 2000); M. Neubert, Phys. Rep. 245, 259 (1994); M. Shifman, preprint TPI-MINN-95-31-T, to appear in QCD and Beyond, Proceedings of TASI-95. Much of what appears in these lectures is similar to my 1996 SLAC Summer Institute lecture notes, A.F. Falk, hep-ph/9610363. Material has also been taken

426

2.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29.

from A.F. Falk, hep-ph/9812217, an introductory chapter written for T i e BaBar Physics Book, SLAC-R-504. For example, see M. Wirbel, B. Stech and M. Bauer, Z. Phys. C29, 637 (1985); J.G. Korner and G.A. Schuler, Z. Phys. C38, 511 (1988); N. Isgur et ai, Phys. Rev. D39, 799 (1989). C. Caso et ai, Eur. Phys. J. C 3 , 1 (1998); update at http://pdg.lbl.gov. N. Isgur and M. B. Wise, Phys. Lett. B232, 113 (1989); Phys. Lett.B237, 527 (1990). M.B. Voloshin and M.A. Shifman, Yad. Fiz. 45, 463 (1987); Yad. Fiz. 47, 511 (1988). S. Nussinov and W. Wetzel, Phys. Rev. D36, 130 (1987). H.D. Politzer and M.B. Wise, Phys. Lett. B206, 681 (1988); Phys. Lett. B208, 504 (1988). E.V. Shuryak, Phys. Lett. B93, 134 (1980). E. Eichten and B. Hill, Phys. Lett. B234, 511 (1990); Phys. Lett. B243, 427 (1990). H. Georgi, Phys. Lett. B240, 447 (1990). B. Grinstein, Nucl. Phys. B339, 253 (1990). A.F. Falk et al, Nucl. Phys. B343, 1 (1990). A.F. Falk and B. Grinstein, Phys. Lett. B247, 406 (1990). M. Luke, Phys. Lett. B252, 247 (1990). A.F. Falk, B. Grinstein and M. Luke, Nucl. Phys. B357, 185 (1991). T. Mannel, W. Roberts and Z. Ryzak, Nucl. Phys. B355, 38 (1991). H.D. Politzer, Nucl. Phys. B172, 349 (1980). A.F. Falk and M. Neubert, Phys. Rev. D47, 2965 (1993) A.F. Falk, M. Luke and M.J. Savage, Phys. Rev. D49, 3367 (1994). A.F. Falk, Nucl. Phys. B378, 79 (1992). P. Cho and B. Grinstein, Phys. Lett. B285, 153 (1992). M. Ademollo and R. Gatto, Phys. Rev. Lett. 13, 264 (1964). M. Neubert and V. Rieckert, Nucl. Phys. B382, 97 (1992). M. Neubert, Phys. Lett. B264, 455 (1991). M.A. Shifman, N.G. Uraltsev and A.I. Vainshtein, Phys. Rev. D 5 1 , 2217 (1995); Erratum, D52, 3149 (1995). M. Neubert, Phys. Lett. B338, 84 (1994). A. Czarnecki, Phys. Rev. Lett. 76, 4124 (1996). For earlier one-loop and one-loop improved calculations, see A.F. Falk and B. Grinstein, Phys. Lett. B247, 406 (1990); Phys. Lett. B249, 314 (1990); M. Neubert, Phys. Rev. D46, 2212 (1992); Phys. Rev. D 5 1 , 5924 (1995); Phys. Lett. B341, 367 (1995). B. Barish et al. (CLEO Collaboration), Phys. Rev. D51, 1041 (1995).

427

30. The LEP results are combined by the LEP Vct Working Group. Details can be found at the web address h t t p : / / l e p v c b . w e b . c e r n . c h . 31. M. Neubert, Int. J. Mod. Phys. A l l , 4173 (1996). 32. J. Chay, H. Georgi and B. Grinstein, Phys. Lett. B247, 399 (1990). 33. I.I Bigi, N.G. Uraltsev and A.I. Vainshtein, Phys. Lett. B293, 430 (1992); I.I Bigi et al, Phys. Rev. Lett. 71, 496 (1993). 34. A.V. Manohar and M.B. Wise, Phys. Rev. D49, 1310 (1994). 35. A. Falk, I. Dunietz and M. B. Wise, Phys. Rev. D51, 1183 (1995). 36. B. Guberina, R.D. Peccei and R. Ruckl, Nucl. Phys. B171, 333 (1980). 37. M. Luke, M.J. Savage and M.B. Wise, Phys. Lett. B343, 329 (1995). 38. Y. Nir, Phys. Lett. B221, 184 (1989). 39. M. Luke, M.J. Savage and M.B. Wise, Phys. Lett. B345, 301 (1995). 40. A.F. Falk, M. Luke and M.J. Savage, Phys. Rev. D53, 2491 (1996); Phys. Rev. D53, 6316 (1996). 41. J. Bartelt et al. (CLEO Collaboration), CLEO-CONF-98-21. 42. R. Poling, proceedings of Lepton-Photon 1999, hep-ex/0003025. 43. M. Beneke and V.M. Braun, Nucl. Phys. B426, 301 (1994); M. Beneke, V.M. Braun and V.I. Zakharov, Phys. Rev. Lett. 73, 3058 (1994). 44. I.I. Bigi et al, Phys. Rev. D50, 2234 (1994). 45. M. Neubert and C.T. Sachrajda, Nucl. Phys. B438, 235 (1995). 46. M. Luke, A.V. Manohar and M.J. Savage, Phys. Rev. D51, 4924 (1995). 47. P. Ball, M. Beneke and V.M. Braun, Phys. Rev. D52, 3929 (1995). 48. For the state of the art, see M. Beneke and A. Signer, Phys.Lett. B471, 233 (1999). 49. M. Neubert, Phys. Rev. D49 (1994) 3392; D49 (1994) 4623; I.I. Bigi et al., Int. J. Mod. Phys. A9 (1994) 2467. 50. A.F. Falk, Z. Ligeti and M.B. Wise, Phys. Lett. B406, 225 (1997); R.D. Dikeman and N.G. Uraltsev, Nucl. Phys. B509, 378 (1998). 51. A.F. Falk et al, Phys. Rev. D49, 4553 (1994). 52. D. Abbaneo et al (LEP Vub Working Group), LEPVUB-99/01, 1999. 53. B.H. Behrens et al (CLEO Collaboration), hep-ex/9905056. 54. C. Bauer, Z. Ligeti and M. Luke, Phys. Lett. B479, 395 (2000).

This page is intentionally left blank

J. L. Rosner

This page is intentionally left blank

CP VIOLATION IN B DECAYS

Enrico

J O N A T H A N L. R O S N E R Fermi Institute and Department of Physics, University 5640 South Ellis Avenue, Chicago, IL 60637 E-mail: [email protected]

of

Chicago

The role of B decays in the study of CP violation is reviewed. We treat the interactions and spectroscopy of the b quark and then introduce CP violation in B meson decays, including time-dependences, decays to CP eigenstates and non-eigenstates, and flavor tagging. Additional topics include studies of strange B's, decays to pairs of light pseudoscalar mesons, and the roles of gluonic and electroweak penguin diagrams, and final-state interactions.

1

Introduction

Discrete symmetries such as time reversal (T), charge conjugation (C), and space inversion or parity (P) have provided both clues and puzzles in our understanding of the fundamental interactions. The realization that the chargechanging weak interactions violated P and C maximally was central to their formulation in the V — A theory. The theory was constructed in 1957 to conserve the product CP, but within seven years the discovery of decay of the long-lived neutral kaon to two pions x showed that even CP was not conserved. Nearly twenty years later, Kobayashi and Maskawa (KM) 2 proposed that CP violation in the neutral kaon system could be explained in a model with three families of quarks, at a time (1973) when no evidence for the third family and not even all evidence for the second had been found. The quarks of the third family, now denoted by b for bottom and t for top, were subsequently discovered in 1977 3 and 1994, 4 respectively. Decays of hadrons containing b quarks now appear to be particularly fruitful ground for testing the KM hypothesis and for displaying evidence for any new physics beyond this "standard model" of CP violation. A meson containing a b quark will be known generically as a B meson, in the same way as a K meson contains an (anti-) strange quark s. The present lectures are devoted to some tests of CP violation utilizing B meson decays. (Baryons containing b quarks also may display CP violation but we will not discuss them here.) We first deal with the spectroscopy and interactions of the b quark. In Section 2 we describe the discovery of the charmed quark, the tau lepton, the b quark, and B mesons. Section 3 is devoted to the spectroscopy of hadrons containing the b quark, while Section 4 treats its weak interactions. Neutral mesons containing the b quark can mix with their_antiparticles (Section 5), 431

432

providing important information on the weak interactions of b quarks. We then introduce CP violation in B meson decays. After general remarks and a discussion of decays to CP eigenstates (Section 6) we turn to decays to CP-noneigenstates (Section 7) and describe various methods of tagging the flavor of an initially-produced B meson (Section 8). Some specialized topics include strange £?'s (Section 9), decays to pairs of light mesons (Section 10), and the roles of penguin diagrams (Section 11), and final-state interactions (Section 12). Topics not covered in detail in the lectures but worthy of mention in this review are noted briefly in Section 13. The possibility that the Standard Model of CP violation might fail at some future time to describe all the observed phenomena is discussed in Section 14, while Section 15 concludes. Other contemporary reviews of the subject 5 ' 6 may be consulted. 2 2.1

Discovery of the b quark Prelude: The charmed quark

During the 1960's and 1970's, when the electromagnetic and weak interactions were being unified by Glashow, Weinberg, and Salam,7 it was realized 8 that a consistent theory of hadrons required a parallel 9 between the then-known two pairs of weak isodoublets of leptons, (i/ e ,e~), (f M ,/x~), and a corresponding multiplet structure for quarks, (u,d), (c,s). The known quarks at that time consisted of one with charge 2/3, the up quark u, and two with charge —1/3, the down quark d and the strange quark s. The charmed quark c was a second quark with charge 2/3 and a proposed mass of about 1.5 to 2 GeV/c 2 . 10 ' 11 The parallel between leptons and quarks was further motivated by the cancellation of anomalies 8 ' 12 in the electroweak theory. These are associated with triangle graphs involving fermion loops and three electroweak currents. It is sufficient to consider the anomaly for the product I^LQ2, where I^L is the third component of left-handed isospin and Q is the electric charge. The sum ^2i(hL)iQ2 over all fermions i must vanish. If a family of quarks and leptons consists of one weak isodoublet of quarks and one of leptons, this cancellation can be implemented within a family, as illustrated in Table 1. The first hints of charm arose in nuclear emulsions 13 and were recognized as such by Kobayashi and Maskawa.2 However, more definitive evidence appeared in November, 1974, in the form of the 3Si cc ground state discovered simultaneously on the East 1 4 and West 15 Coasts of the U. S. and named, respectively, J and xp. The East Coast experiment utilized the reaction p + Be —» e+e~ +... and observed the J as a peak at 3.1 GeV/c 2 in the effective e+e~ mass. The West Coast experiment studied e+e~ collisions in the SPEAR storage ring and

433 Table 1: Anomaly cancellation in the electroweak theory. Family Neutrino Charged lepton Q = 2/3 quark Q = - 1 / 3 quark

1

2

e~ u d

c s

3

Contribution per family

T~

(-l/2)(-l)2 = - l / 2 3(l/2)(2/3) 2 = 2/3 3(-l/2)(-l/3)2 = - l / 6

t b

saw a peak in the cross section for production of e+e~, n+'/j,~, and hadrons at a center-of-mass energy of 3.1 GeV. Since the discovery of the Jftp the charmonium level structure has blossomed into a richer set of levels than has been observed for the original "onium" system, the e+e~ positronium bound states. The lowest charmonium levels are narrow because they are kinematically unable to decay to pairs of charmed mesons (each containing a single charmed quark). The threshold for this decay is at a mass of about 3.73 GeV/c 2 . Above this mass, the charmonium levels gradually become broader. The charmed mesons, discovered in 1976 and subsequently, include D+ = cd (mass 1.869 GeV/c 2 ), D° = cu (mass 1.865 GeV/c 2 ), and Ds = cs (mass 1.969 GeV/c 2 ). These mesons were initially hard to find because the large variety of their possible decays made any one mode elusive. For example, the two-body decay D° -> K~ir+ has a branching ratio of only about 3.8%; 16 higher-multiplicity decays are somewhat favored. 2.2

Prelude: The r lepton

About the same time as the discovery of charm, another signal was showing up in e+e~ collisions at SPEAR, corresponding to the production of a pair of 17 new leptons: e + e -> 7* —> The r signal had a number of features opposite to those of charm: lower- rather than higher-multiplicity decays and fewer rather than more kaons in its decay products, for example, so separating the two contributions took some time.18 The mass of the r is 1.777 GeV/c 2 . Its favored decay products are a tau neutrino, vT, and whatever the charged weak current can produce, including ePe, fiiy^, 7T, p, etc. It thus contributes somewhat less than one unit to

R^Y,Q>

a(e+e —>• hadrons) or(e+e~ —> /u+/x_)

(1)

which would have risen from the value of 2 for u, d, s quarks below charm threshold to 10/3 above charm threshold if charm alone were being produced, but was seen to rise considerably higher.

434

One problem with accepting the r as a companion of the charmed quark was that the neat anomaly cancellation provided by the charmed quark, mentioned above, was immediately upset. The anomaly contributed by the T lepton would have to be cancelled by further particles, such as a pair of new quarks (t, b) with charge 2/3 and —1/3. Such quarks had indeed already been utilized two years before the r was established, in 1973 by Kobayashi and Maskawa 2 in their theory of CP violation. The names "top" and "bottom" were coined by Harari in 1975,19 in analogy with "up" and "down." 2.3

Dilepton spectroscopy

One reason for the experiment which discovered the J particle 14 was an earlier study, also at Brookhaven National Laboratory, by L. Lederman and his collaborators, of /J.+fi~ pairs produced in proton-uranium collisions.20 The m(fi+fi~) spectrum in this experiment displayed a shoulder around 3.5 GeV/c 2 . It was not recognized as a resonant peak and was displaced in mass from the true J/ip value because of the poor mass resolution of the experiment. After the discovery of the J/ip, Lederman's group continued to pursue dilepton spectroscopy. In 1977 a search with greater sensitivity and better mass resolution turned up evidence for peaks at 9.4, 10.0, and possibly 10.35 GeV/c 2 . 3 These were candidates for the IS, 2S, and 3S 3 5i levels of a new QQ system. Several pieces of evidence identified the heavy quark Q as a b quark. (1) The T(1S) and T'(2S) were produced in 1978 by the electron-positron collider DORIS at DESY and their partial widths to e+e~~ pairs were measured.2 It was shown 22 that if the QQ system was bound by the same quantum chromodynamic force as as the cc (charmonium) system, one could use the cc states to gain some idea about the details of the QQ binding. Since T(QQ) oc €Q, where eQ is the charge of the quark Q, it was possible to conclude from the data that \CQ\ = 1/3 was favored over |eg| = 2/3. (2) The Cornell e+e~ ring CESR began operating in 1979 23 reaching a fourth T(4S) peak and finding it broader than the first three. This indicated that the meson pair threshold lay below M[T(4S)]= 10.58 GeV/c 2 . Farther above this threshold, wiggles in the total cross section for hadron production averaged out to indicate a step in R of 1/3, confirming that \CQ\ = 1/3. (3) The possibility that Q was an isosinglet quark of charge —1/3, and thus not the partner of some quark t with charge 2/3, was ruled out by the absence of significant flavor-changing neutral current decays such as b —> S/u+/x-.24'25 The structure of the T levels is remarkably similar to that of the charmonium levels except for having more levels below flavor threshold. For example, the fact that the 3S level is below flavor threshold allows it to decay to the

435

2P levels via electric dipole transitions with appreciable branching ratios; the transitions between the S and P levels are well described in potential models which reproduce other aspects of the spectra. Several reviews treat the fascinating regularities of the spectroscopy of these levels.26 2.4

Discovery of B mesons

The lightest meson containing a b quark and each flavor of light antiquark is expected to decay weakly. The allowed decays of b are (c or u) + (virtual W~), with the c giving rise to lots of strange particles while the u gives few strange particles. The virtual W~ can decay to ud, cs, e~i>e, /x~i/M, and T"VT. In e+e~ collisions above BB threshold, several signals of B meson production were observed by the CLEO Collaboration starting around 1980: 27 • Prompt leptons (signals of semileptonic decay) • An abundance of kaons (a signal that b —> c + W^irt is preferred over b -> u + WT rt ) • "Daughter" (lower-momentum) leptons from c semileptonic decays. These indirect signals were followed by reconstruction of B+ and B° decays,28 e.g., B+ = bu -» cudu -> D0n+

,

B° = bd-> cudd =

D~TT+

.

(2)

Typical branching ratios for these final states 16 are (5.3 ± 0.5) x 1 0 - 3 for 3 D°-K+ and (3.0 ± 0.4) x 10" for D~TT+. B° ->• £»°7r° is also allowed but not yet observed. These small branching ratios mean that reconstruction of exclusive final states is even harder for B mesons than for charmed particles. 3 3.1

The known B hadrons B mesons

The nonstrange ground-state B (pseudoscalar) and B* (vector) mesons are compared with the corresponding charmed mesons in Table 2. Evidence for the B* exists in the form of a photon signal for the decay B* —> Bs"/?9 The photon energy, 46 MeV, is expected to be the same as that seen in B*° —• _B°7.30 Since the B* and B states are separated by only 46 MeV, a B* should always decay to a B of the same flavor and a photon. This is in contrast to the case of the D* and D states, whose separation is just about a pion mass. The electromagnetic mass splittings are such that D*+ —> D°n+, D*+ —>

436 Table 2: Ground-state heavy-light (Qq) pseudoscalar mesons and the corresponding vector mesons. Here the spectroscopic notation 2L+lLj is used to denote the spin, orbital, and total angular momenta of the Qq state.

Quark content cu cd cs bu Id bs

Pseudoscalar ( 1 5o) meson Name Mass (MeV/c 2 ) 1864.5 ± 0 . 5 D+ 1869.3 ± 0.5 Ds 1968.6 ± 0.6 B+ 5279.0 ± 0.5 B° 5279.4 ± 0 . 5 5369.6 ± 2.4

Vector ( 3 5i) meson Name Mass (MeV/c 2 ) 2006.7 ± 0 . 5 B*+ 2010.0 ± 0 . 5 2112.4 ± 0 . 7 B*+ 5325.0 ± 0.6 B*° 5325.0 ± 0.6 ~5416

D+ir°, and D*° -+ D°ir° are just barely allowed, while D*° ->• D+ir~ is forbidden. The low-momentum TT+ in D*+ —» D°n+ acts as a "tag," useful both for signalling the production of a charmed meson 3 1 and, by its charge, distinguishing the D° from a D°. Since B* decays are not useful for this type of "flavor" tag, one must resort to the decays of heavier excited bq states (Section 8). The hyperfine splitting of B mesons is smaller than that in charmed mesons because the chromomagnetic moments of the heavy quarks scale as the inverse of their masses: mB* - mB l/m6 mc 1 . ~ —— = — ~ - . (3) m c . — mo l/Tnc nib 3 3.2

The A;, baryon

The lightest baryon containing a b quark is the Aj = b[ud]i=o- Its mass is 5624 ± 9 MeV/c 2 . 16 The ud system must be in a color 3* (antisymmetric) state, since the b is a color triplet and the Ab is a color singlet. The spin-zero state of ud is favored over the spin-one state by the chromodynamic hyperfine interaction. By Fermi statistics, the ud pair must then be in an (antisymmetric) isospin-zero state. For similar reasons, the 1 = 0 state of a strange quark and two nonstrange quarks, the A = s[ud]i=o with mass 1116 MeV/c 2 , is lighter than the £ = s(uu,ud,dd)r=i states with average mass 1193 MeV/c 2 . The charmed analogue of the Aj, is the A c = c[ud]i=0 with mass 2284.9±0.6 MeV/c 2 . The difference in mass of the two particles is M(Ab) — M(AC) = 3339 ± 9 MeV/c 2 . This provides an estimate of mi — mc since there are no hyperfine terms involving the heavy quark; the light-quark system has zero spin in both baryons. There will be a correction of order m~l — m^1 due to

437

possible differences in kinetic energies. One can perform a similar estimate for Qq mesons by eliminating the hyperfme energy, performing a suitable average over vector ( 3 Si) and pseudoscalar (1S'o) meson masses. The expectation value of the relevant interaction term is ( cl~i>i and b —> ccs conserve isospin leads to the requirement that the final states have zero isospin in both cases, restricting the number of additional pions that can be produced. 4

Interactions of the b quark

In this Section we shall discuss the way in which the interactions of the b quark provide information on the pattern of charge-changing weak interactions of quarks parametrized by the Cabibbo-Kobayashi-Maskawa (CKM) matrix y 2,32 More details on determination of the CKM matrix are included in the lectures by Buchalla,33 DeGrand,34 Falk,35, Neubert,36 and Wolfenstein,37 4-1

The b lifetime: indication of small \Vcb\

The long b quark lifetime (> 1 ps) indicated that the CKM element Vc\, was considerably smaller than \VUS\ ~ \Vcd\ ~ 0.22. One can estimate Vcb using a free-quark method. The subprocess b —> cW*~ —> cl~9e has a rate T(b^cri>e)

G2 = —f^m5b\Vcb\2f(mb,rnc,me)

,

(4)

where GF = 1.16637(2) x 10~ 5 GeV~ 2 is the Fermi coupling constant. In the limit in which the lepton mass can be neglected, f(mt,,mc,me) = f{m2/ml), 3 4 2 with f(x) = 1 — 8x + 8x — x — Ylx lnx. The uncertainty in the prediction for T(b —> cl~D() due to that in mj is mitigated by the constraint noted above on rrib — mc ~ 3.34 GeV/c 2 . Taking a nominal range of quark masses around mi, = 4.7 GeV/c 2 (and hence a range around m c = 1.36 GeV/c 2 , f(m2c/ml) = 0.54), rj — 1.6 x 10~ 6

438

s,

and the branching ratio B(b -t clvt) ~ 10.2%, one finds

Thus if mj is uncertain by 0.3 GeV/c 2 (my guess), \Vcb\ is uncertain by ±0.0024. Recent averages 16,35 give rise to values of \Vcb\ somewhat above 0.040 with errors of ±0.002 to ±0.003. A new report by the CLEO Collaboration 38 finds \Vcb\ = 0.0462 ± 0.0036 based on the exclusive decay process B° —• D*~t+vi. This new determination bears watching as it would affect many conclusions regarding predictions for CP-violating asymmetries in B decays. We shall take \Vcb\ = 0.041 ± 0.003 as representing a conservative range of present values. 4-2

Charmless b decays: indication of smaller \Vub\

Although the u quark is lighter than the c quark, its production in b decays is disfavored, with T(b —> u(.v)/T(b —> civ) only about 2%. Since the phase-space factor / ( m 2 / m 2 ) is very close to 1, while f{m2c/m\) ~ 1/2, this means that |^u(,/Vc(,|2 ~ 1%, or |K,&/Vcf,| — 0.1. The error on this quantity is dominated by theoretical uncertainty 35 ; detailed studies 39 indicate |FU6/VC;,| = 0.090±0.025. 4-3

Pattern of charge-changing weak quark transitions

The relative strengths of charge-changing weak quark transitions are illustrated in Fig. 1. Why the pattern looks like this is a mystery, one of the questions (along with the values of the quark masses) to be answered at a deeper level. The interactions in Fig. 1 may be parametrized by a Cabibbo-KobayashiMaskawa (CKM) matrix of the form 40 1-

A2 2

--A A

VCKM = 3

AX3(p-irj)

A \2

]1 - ^ -

L - 2—

_AX (1 - P--irj) -AX L

2

A\2

(6)

1

The columns refer to d, s,b and the rows to u,c,t. The parameter A = 0.22 represents sin# c , where 9C is the Gell-Mann-Levy-Cabibbo 32,41 angle. The value |Vc6| = 0.041 ±0.003 indicates A = 0.85±0.06, while \Vub/Vcb\ = 0.090± 0.025 implies (p2 + T? 2 ) 1 / 2 = 0.41 ± 0.11. Further information may be obtained by assuming that box diagrams involving internal quarks u, c, t with charge 2/3 are responsible for both the CP-violating contribution to K°-K° mixing and to mixing between neutral B

439

Q=-l/3 3 2 > O

1 0 -1

QO

o

-2 -3

Figure 1: Pattern of charge-changing weak transitions among quarks. Solid lines: relative strength 1; dashed lines: relative strength 0.22; dot-dashed lines: relative strength 0.04; dotted lines: relative strength < 0.01. Breadths of lines denote estimated errors.

mesons and their antiparticles. The parameter \CK\ = (2.27±0.02) x 10 Buchalla's lectures 33 ) then implies a constraint 42 r)(l - p + 0.39) = 0.35 ±0.12

3

(see (7)

where the 1 — p term in parentheses arises from box diagrams with two internal top quarks, while the correction 0.39 is due to diagrams with one charmed and one top quark. The error on the right-hand side is due primarily to uncertainty in the Wolfenstein parameter A = |V^j,|/A2, which enters to the fourth power in the ti contribution to e^. A lesser source of error is uncertainty in the parameter BK describing the quark box diagram's matrix element between a K° and a K°. We have chosen 43 BK = 0.87 ± 0.13. Present information on B°~B mixing, interpreted in terms of box diagrams with two quarks of charge 2/3, leads to a constraint on \Vtd\2 which implies 42 \l-p-ir)\ = 0.87±0.21 for the parameter range fsV^B = 230±40 MeV describing the matrix element of the short-distance 4-quark operator taking bd into db between B° and B states. The best lower limit on B°-Bs mixing 44 AMS > 15 p s _ 1 , when compared with the corresponding value for

440

0.6

0.4 —

P 0.2 —

0.0

-0.50

-0.25

0.00

0.25

0.50

P Figure 2: Region of (p,rf) specified by ±1
B°-B

mixing, Amd = 0.487 ± 0.014 ps" 1 , leads to the bound PB.BB.

Vts

S%BB

vtd

>29

(8)

This may be combined with the estimate 45 /B, \/BBS < 1-25JBVB~B based on quark models. (Lattice gauge theories 34 estimate this coefncent more precisely, generally giving values between 1.1 and 1.2.) One finds |Vis/Vtd| > 4.4 or |1 — p — ir)\ < 1.01. The constraints may be combined to yield the allowed range in (p,rj) space illustrated in Fig. 2. Smaller regions are quoted in other reviews 46 which view the theoretical sources of error differently. 4-4

Unitarity triangle

The unitarity of the CKM matrix implies that to the order we are considering, V*b + Vtd = AX3. If this equation is divided by AX3, one obtains a triangle in the (/j, 77) complex plane whose vertices are at (0,0) (internal angle 7), (p,r]) (internal angle a), and (1,0) (internal angle /?).

441

CP-violating asymmetries in certain B decays can measure such quantities as sin 2a and sin 2/3. The former, measured in B° —> ir+n~ with some corrections due to "penguin" diagrams, may occupy a wide range, as illustrated in Fig. 2. The latter, measured in the "golden mode" B° -» J/tpKs with few uncertainties, is more constrained by other observables. The goal of measurements of CP violation and other quantities in B decays will be to test the consistency of this picture and to either restrict the parameter space further, thus providing a reliable target for future theories of these parameters, or to expose inconsistencies that will point to new physics. Hence part of the program will be to overconstrain the unitarity triangle, measuring both sides and angles in several different types of processes. While we discuss such measurements based on B mesons, Buchalla 33 describes how, for example, K+ —> TT+I/D constrains the combination |1 — p — in + 0.44|, where the last term is a charmed quark correction to the dominant top quark contribution, and the purely CP-violating process KL —• ir°vv constrains n. 5 5.1

Mixing of neutral B mesons Mass matrix formalism

We shall work in a two-component basis utilizing the states (B° ,B ). [It is also sometimes useful to consider a basis 37 (£?+, £?_), where B± = (B° ± B )/\/2.) The time-dependence of these states is described via a mass matrix M = M — i r / 2 , where M and T are Hermitian by definition:

\B°] -=fi

B

= M

\B°] -F=fl

B

The requirement of CPT invariance, which we shall assume henceforth, implies Mn — M22, or equal transition amplitudes for K° ->• K° and K -> K . Exercise: (a) Show this. Remember time reversal is an antiunitary operator. (b) Show that a similar argument applied to M12 or M 21 leads to no constraint. (c) Relate the result in (a) to the result quoted by Wolfenstein37 for the (B° ± S 0 ) / ^ basis. [Answer to Part (a): Insert the unit operator (CPT)-1 CPT before and after M in Mn = (B°\M\B°). Note that CPT(M - iT/2)(CPT)-x =M + iT/2. Then Mn =

(B

0

= (CPTB°\M

+

Y\CPTB0)*

|M + y | S 0 ) * = ( 5 ° | M - y | B 0 ) = A 4 2 2

,

(10)

442

where the antiunitarity of T has been used in the first step. For a discussion of antiunitary operators see, e.g., Sakurai's book on quantum mechanics.47] The eigenstates of M may be denoted by BH ("heavy") and BL ("light"): \BH,L)

= PH,L\B°)

The corresponding eigenvalues

M

Pi

[I,H,L

= Mi

+ qH,L\B^)

.

(11)

= rriH,L — ^H,L

satisfy

Pi

,

(12)

Hi

(13)

(i = H,L)

[qi\

[
M21-+

Pi

Mu

Qi

so that Mi2{qi/Pi) - M2i(Pi/qi), or (vijqif - M12/M21 For B°~B-^ mixing, in contrast to the situation for neutral kaons, the scarcity of intermediate states accessible to both B° and i=fi B and the presence of a large top-quark contribution to Mi2 means that \Y\2\
M12 M,

Pi_ qi

(14)

We may choose pi = pu = p, qh — ~qH = <7- Normalizing \p\2 + \q\2 — 1, we then write \BL) = p\B°) + g|B°> ,

\BH) = p\B°) - q\B*)

.

(15)

The sign ambiguity in (14) may be resolved as follows. Since q

p (16) = Mu + M21, p q q p (17) HH = Mu-Mi2-= M11-M21, p q then p,H - VL = ~2Mi2(q/p), which in the limit jr 1 2 j < |Mi 2 | is \IH ~ ML = / ( - , + ) v M i 2 M 2 i for the choice of (+,-) in (14). Since M 2 i = M*2 (M is Hermitian), we must take the - sign in (14) in order that the "heavy" mass ran be greater than the "light" mass m^. Then (J-L

= Mu +M12-

EL

Mi 2

qi

M-21

(18)

443

Neglecting Ti2 in comparison with IM12I, we then find Am = mH-mL^2\M12\

,

AT =

rH-TL~0

(19)

If one keeps Ti2 to lowest order, one can show 48 that q p

Mf2 \MX

1

- ^Ira (w,

(20)

In the limit that Ti 2 is negligible and A r = 0, q/p is a pure phase, determined by the phase of Mi 2. Now, Mi2 takes B = bd into B° = db, so its phase is that of (VtbVt*d)2, 2t/3 or e . Thus in this limit we find q/p ~ e~2l!3. More specifically, in the phase convention in which (CP)\B°) = +\B ), we find M 12 = -^{VtbV; )2M2wmBf2BBBnBS 2 d

(^-\

MwJ

12TT

(21)

Here fB is the B meson decay constant, BB is the vacuum saturation factor, r\B = 0.55 is a QCD correction factor,49 and 5 0

S(x) = \

+

3 — 9a; + (x-1)2

6x2 In x (x-1)3

(22)

The appropriate top quark mass for this calculation 51 is mt(mt) — 165 GeV/c 2 . The BaBar Physics Book 48 may be consulted for further conventions and details. 5.2

Time dependences

We would like to know how states which are initially B° or B time. The mass eigenstates evolve as Bi —> Bie~l^il (i = L,H). eigenstates are expressed in terms of them as

t = 0: \B°)

\BL)

+ \BH) 2p

|5"> =

t > 0 : \B°(t)) = (\BL)e-l,iLt i

+ t

|B°(t)> = ( | B i > e - ' " -

~ \BH) 2q

\BL)

\BH)e-^Ht)/2p i

t

\BH)e- »" )/2q

evolve in The flavor

(23)

(24) (25)

444

Now substitute back for BL,H\B°(t)) = \B0)U(t)

+ p-(t)\B°)

,

(26)

\B°(t)} = \B°)f+(t)

+ Pqf-(t)\B0)

,

(27)

,

(28) (29)

f+(t)=e-imte-n^cos(Afit/2) /_(*) =e-imte-rt/2ism{Afit/2)

,

A// = HH - HL = Am - i ( A r / 2 ) , Am = m # - m i , A r = TH - FL, m = (mH + mL)/2,T = (rH + rL)/2. Again, for simplicity, we shall neglect A r in comparison with Am. A lowest-order quark model calculation (for which QCD corrections change the answer) gives 52 r ^ _ M 12

3^ m\lMl ml ( 1 2 5 K / M 2 , ) m2 V

8 mg T ^ A , 3m^ttyf*r

"^ o ™ 2 T / T / *

1 180 ion

'

l

0

^

where S(x) was defined in Eq. (22). The intermediate states dominating the loop calculation of Ti2 have typical mass scales mj, whereas loop momenta of order mt give rise to the main contributions to MyiNeglecting A r and performing time integrals, one finds

r

r^<'>|2=2^i) • rr^-wf=wh>

• <3i)

where Xd = Am,Bd/TBd, and Bd is another name for B° = bd to distinguish it from Bs — bs. The sum of the two terms is 1. The first term is 1 for xd — 0, approaches 1/2 for Xd —> oo, and is about 0.82 for the actual value Xd = 0.754 ± 0.027. The second term is 0 for xa = 0, approaches 1/2 for Xd —> oo, and is about 0.18 in actuality. Thus a neutral non-strange B of a given flavor (B° or B ) has about 18% probability of decaying as the opposite flavor. 6 6.1

CP violation Asymmetry:

general remarks

We wish to compare (f\B°=Q(t)) / = {CP)f. Now define

I = $g>

and (f\Bt=0(t)),

ss<^>, (f\W)

where / is a final state and

^ u , V

-XOSE,
.

(32)

445

Using the time evolution derived earlier for B°=0 and Bt=0, {f\B°=0(t))

one then finds

= (f\B°) [/+(*) + Ao(t)/_(t)]

,

<7|B?=0(t)> = <7l^°> [/+(*) + Ao(*)/-(*)]

(33) •

(34)

This result can be simplified under several circumstances, (a) If there is a single strong eigenchannel, final-state strong interaction phases in x or x cancel, since the numerator and denominator refer to the same final state. Then x = x*, since weak phases flip sign under CP. (b) Recall that \q/p\ is nearly 1 for B mesons. (For non-strange B's, we found q/p ~ e~2t/3.) Combining (a) and (b), we find A0 = XQ for these cases. 6.2

Time-dependent

asymmetry

According to Eq. (33), the rates for a (B°,B ) produced at t — 0 to evolve to the respective final states (/, / ) at a time t are dr(B?=0^f)/dt~\f+(t) dr(B°t=0-+7)/dt~\f+(t)

+ \0Ut)\2

,

(35)

2

,

(36)

+ \0f-(t)\

with the coefficients of proportionality identical if there is a single strong eigenchannel. Now consider the case of a CP-eigenstate / such that / = ± / . Then we have not only x = x* (see above), but also x = x~x, so |x| = 1. In that case, when \q/p\ = 1 as is the case for neutral B's, we have |Ao| = 1 and Ao = AQ. Then Ami ., . Amt h iAo u sin 2 2 -rt e - - [1 - ImAo sin Amt] , rt cos

dT{B°=0 -> f)/dt ~ e~rt [1 - ImAo sin Amt] rt

dT(B°t=0 -> J)/dt ~ e~

[1 + ImAo sin Amt]

(37) (38) , .

(39)

The second term in each of these equations consists of an exponential decay modulated by a sinusoidal oscillation. The time-dependent asymmetry is then A - dr^B°=o -> Mdt ~ dT(^t=o - » D l d t . . . ..n. T . At = zzn = = —ImAo sin Amt . (40) dT(B°=0 -> f)/dt + dT(Bt=0 -> f)/dt When A m / r 3> 1, the wiggles in Eqs. (39) average out, and not much timeintegrated asymmetry is possible, while when Am/T
446

before there is time for oscillations. The maximal time-integrated asymmetry occurs when Am/T = 1. When more than one eigenchannel is present, the condition |Ao| = 1 need not be satisfied, so that the terms cos2 (Ami/2) and sin 2 (Am£/2) in (37) need not have the same coefficients, and a cos Ami term is generated in the rates. This is the signal of "direct" CP violation, as will be discussed below. Its presence for B —>• mr was pointed out by London and Peccei 53 and by Gronau.54 6.3

Time-integrated

asymmetry

If one integrates the rates for B°=0 —> f and Bt=0 time-integrated asymmetry 55

—> / , one can form the

_r(j?° = 0 ->/)-r(flt 0 ->7) w = rr^ :=o =: • r(Bt°=0->/)7.+ r(s f = 0 -»/)

,,n (41)

,-^o* If we consider the cases (as above) in which |(/|B°)| = \(f\B )|, we just need the integral

and we then find

//o Jo

dtsm(Amt)e-rt = ^ - ^ ^ J- 1 + xd C

f = -7Tl2lmX°

,

(42)

(43)

when \x\ = 1. This is indeed maximal when Xd = 1; the coefficient of —ImAo is 1/2. For the actual value of Xd — 0.75, the coefficient is 0.48 instead, very close to its maximal value. 6.4

Specific examples in decays to CP eigenstates

When / is a CP eigenstate, a CP-violating difference between the rates for B° -> / and B —> / arises as a result of interference between the direct decays and those proceeding via mixing (i.e., B° —» B —> / and B —»• B° —> / ) . The second term in Eqs. (39) is the result of this mixing. As mentioned, the rate asymmetry goes to zero when Xd -» 0 or Xd -> 00. We now illustrate the calculation for two specific examples, B° —>• J/ifiKs and B° —> TTTT. The "golden mode": J/ipKs The quark subprocess governing B° -t J/tpKs is b -> ccs, whose CKM factor is V*bVcs. The Ks is produced through its K° component. The cor0 _ —0 responding decay B -> J/tpKs proceeds via b -> ccs and involves the K component of K$-

447

For a CP-eigenstate, we defined x = (f\B )/(f\B°) and A0 = (q/p)x, but what we actually calculate is (f\B )/(f\B°) where / = r)}Cpf with X]'CP = ± 1 . For / = J/tpKs, vfcp = - 1 - To show this, note that CP\KS) = \KS) and CP\J/ip) = \J/tp) (since J/ip has odd C and P). The decay of the spin-zero B° to the spin-one J/ip and the spin-zero Ks produces the final particles in a state of orbital angular momentum £ = 1 and hence odd parity, introducing an additional factor of —1. Then

(KS\K°)(K°\B°)

V

'

'

(A good discussion of the sign is given by Bigi and Sanda.56) Now \Ks) = 0 so that (KS\K°) = q*K and (KS\K°) = p*K. These numbers PK\K )+qK\K°), are very close to l / \ / 2 . If the loop calculation of M\2 for K°-K mixing is dominated by the charmed quark, then (QK/PK) — {Vcd.V*s) / (V*dVcs), and 0

vcdvc*sv;bvcsv;dvtb

'

>

We assumed a specific quark to dominate the calculation of Mi2 to illustrate the self-consistency of the expression for Ao with with respect to redefinition of quark phases. Note first of all that the denominator is the complex conjugate of the numerator. Then note that each quark is represented by the same number of V's and V*'s in the numerator: 2 for the charmed quark and 1 each for d,s,b, and t. Thus any phase rotation of a quark field leaves the expression invariant. (Bjorken and Dunietz have introduced a nice representation of this invariance.57) The same cancellation would have occurred if we were to say another quark dominated K°~K mixing. For the final state / = J/ipKs we thus find Ao = —e~2l/3 and Im Ao = sin 2/3, leading to the time-integrated rate asymmetry CJ/^KS — ~xd sin 2/3/(1+ x%). In practice the experiments often select events occurring for a proper time t > to > 0 in order to enhance the signal/noise ratio, so that analyses are usually based on the time-dependent asymmetry mentioned earlier. Some recent results on sin 2/3 58 ~ 62 are quoted in Table 3, and ±1<7 limits from the average are plotted in Fig. 2. While the central value is somewhat below that favored by other observables, there is no significant discrepancy. The 7r+7r~ mode and its complications The main subprocess in B° -> 7r+7r~ is the "tree" diagram in which b —> ir+u, with the spectator d combining with the u to make a ir~. Let us temporarily assume this is the only important process and compute the

448 Table 3: Values of sin 20 implied by recent measurements of the CP-violating asymmetry in

B° -> J/i>Ks.

Experiment OPAL 5 8 CDF 5 9 ALEPH 6 0

Value 3.2+^ ±0.5 0.79i°^ 0 . 8 4 ™ ±0.16

Belle"

0.58±g;S±g;~

BaBar 6 2 Average

0.34 ± 0.20 ± 0.05 0.48 ±0.16

CP-violating rate asymmetry. We shall return in Section 11 to the important role of "penguin" diagrams. Since the (spin-zero) ir+ir~ system in B° decay has even CP, we find

x A0 = -x P

{ir+n-\W) V„,V ub^ud (ir+ir-lB*) V:bVud VtdV* VubV, -2i/3 ud

(46) -2i

7

(47)

v;dvtb v*bvud

Exercise: Check the invariance of this expression under redefinitions of quark phases. Since j3 + j = n - a, we have An = e2ia, Im(A0) = sin 2a, and Cn+V- = —xdsm2a/(l + x2d). [Remember that our asymmetries are defined in terms of (B° - B )/{B° + B ).] This result is limited in its usefulness for several reasons. (a) Our neglect of penguin diagrams will turn out to make a big difference. (b) The range of sin a is large enough that early asymmetry measurements are unlikely to expose contradictions with the standard prediction. (c) An even larger range of negative sin 2a turns out to be allowed if Vcb is larger than assumed in Sec. 4. An interesting exercise (whose result would, of course, be modified by penguin contributions) is to suppose that the asymmetries in B° -> J/tpKs and B° -> 7r+7r~ are due entirely to mixing (i.e., to a "superweak") interaction.63 In this case, since J/ipKs and IV+IT~ have opposite CP eigenvalues, one has C„+n- = -Cj/^KsWhat range of parameters in the standard CKM picture would imitate this relation? In other words, for what p and r\ would one have sin 2a = - sin 2/3? [The answer is rj = (1 - p)yJp/{2 - p).]

449

(a)

Figure 3: Contributions to B° —> K+TT "penguin" amplitude ~ V*bVts-

7

. (a) Color-favored "tree" amplitude ~ V*bVua; (b)

Decays to CP-noneigenstates

If the final state / is not a CP eigenstate, i.e. if / ^ ± / , as in the case / = K+n~, f = K~TT+, then a CP-violating rate asymmetry requires two interfering decay channels with different weak and strong phases: A(B -> / ) = A i e ^ V * 1 + A2ei
1 iSl —> 7 ) = A i e - * *'e™' "

,

1 ^i<52 +- -A 24e„ 0 - "^l2"e"

(48) (49)

Here the weak phases = (pi —
Af) Then

7../

\A(B ^ f)\* - \A(B ^ f)\* |A(B->/)P + |^(B->/)|2

(50)

-2AiA 2 sinA0sinA(5 A? + A\ + 2AXA2 cos A(f) cos A<5

(51)

Examples of interesting channels

B°-±K+TT-

vs.

B°

K-TT" 1

We illustrate two types of contribution to B° -t K+ir~ in Fig. 3. The "tree" contribution, which in this case is color-favored since the color-singlet current can produce a quark pair of any color, has weak phase 7 = Avg(V*bVus) and strong phase ST, while the "penguin" contribution has weak phase 7r = Aig(Vt*bVts) and strong phase Sp.

450

u u

s u Figure 4: Color-suppressed tree diagram contributing to B+ ->

K+TT°.

Even though ST — Sp is unknown, and may be small so that little CPviolating asymmetry is present in B —> K^TT^, it will turn out that one can use rate information for several processes, with the help of flavor SU(3) (which can be tested) to learn weak phases such as 7. B+ ->• K+n° vs. B~ -> K~n° Exercise: Identify the main amplitudes which contribute. What are the differences with respect to B —• K^-K^ ? Answer: There are two "tree" amplitudes, one color-favored [as in Fig. 3(a)] and one color-suppressed (Fig. 4). Both have weak phases 7 = Krg{y^bVus) There is a penguin amplitude [as in Fig. 3(b)] with weak phase IT = Arg(Vt*bVts)• Since TT° = (dd-uu)/V2 in a phase convention in which ir+ = ud, the colorfavored tree and penguin amplitudes are the same as that in B° -> K+TT~, but divided by \f2. Thus the overall rate for 5 ± -» if ± 7r° is expected to be 1/2 that for B —• K±nzf if the penguin amplitude dominates or if the color-suppressed amplitude is negligible. In that case one expects similar CPviolating asymmetries for B° -> K+n~ and B+ -> K+n°. 64 ' 65 B+ -» K°TT+ vs. B~ -»

Tfir"

Exercise: Show that there is no tree amplitude and hence no CP-violating asymmetry expected. This process is expected to be dominated by the penguin amplitude and thus provides a reference for comparison with other processes in which tree amplitudes participate. Small contributions to B+ -> K+n° and B+ ->• K°ir+ are possible from the process in which the hu pair annihilates into a weak current which then produces su. A qq pair is produced in hadronization, giving K+ir° if q = u and K°n+ if q = d. These contributions are expected to be suppressed by a factor

451

of JB/TUB if the graphs describing them can be taken literally. However, they can also be generated by rescattering from other contributions, e.g., (B+ —• K+7r°)tree —> K°n+. We shall mention tests for such effects in Sec. 12. 7.2

Pocket guide to direct CP asymmetries

We now indicate a necessary (but not sufficient) condition for the observability of direct CP asymmetries based on the interference of two amplitudes, one weaker than the other. The result is that one must be able to detect processes at the level of the absolute square of the weaker amplitude?6 This guides the choice of processes in which one might hope to see direct CP-violating rate asymmetries. Suppose the weak phase difference A
w = ° ($&)*%

***«*• •

(52)

Imagine a rate based on the square of each amplitude: Ni = const. |Aj| 2 . Then

\A\ ~

2y^jN[.

The statistical error in A is based on the total number of events. For A2
.

(53)

Thus (aside from the factor of 2) one must be able to see the square of the weaker amplitude at a significant level in order to see a significant asymmetry due to A1-A2 interference. 7.3

Interesting levels for charmless B decays

Typical branching ratios for the dominant B decays to pairs of light pseudoscalar mesons are in the range of 1 to 2 parts in 105. Some recent data are summarized in Table 4. Here the average between a process and its charge conjugate is quoted. These data are based on results by CLEO, 67 ' 68 ' 69 [including a value for B(B+ —> TT+TT°) extracted from an earlier CLEO report 7 0 ], Belle 71 and BaBar.72 The averages are my own. The relative K-K rates are compatible with dominance by the penguin amplitude, which predicts the rates involving a neutral pion to be half those

452 Table 4: Branching ratios, in units of 10

Mode 7T + 7r~ 7T+TT0

K+TTK°n+ K+n° K°TT° K+r)1 K°ri'

C L E 0

11

fj+3.0+1.4

14 «+5.9+2.4

for B° or B + decays to pairs of light pseudoscalar

Belle 71

67,68,69,70

4.3±i;t ± 0.5 5.4 ±2.6 17.2i^±1.2 18.21^ ± 1.6

6

5.6+^± 0 .4 7 O+3.8+0.8 '•°-3.2-1.2 1Q Q + 3 . 4 + 1 . 5 1».0_3 2-0.6 i o 7+5.7+1.9 i o - ' -4.8-1.8

16.311:111:2 2

i6.ol^l 2 :

5 7

BaBar72 4.1 ±1.0 ±0.7 5.ll2°±0.8 16.7 ± 1.61J:? i o 9+3.3+1.6 10 z - -3.0-2.0

io.8i? : JU:3 o 9 + 3 . 1 + l.l °-z-2.7-1.2

62 ±18 ± 8

80lJ° ± 7 89tie ± 9

Average 4.4 ±0.9 5.6 ±1.5 17.4 ±1.5 17.3 ±2.4 12.2 ±1.7 10.4 ±2.6 75 ±10 78 ± 9 (a)

(a) Average for K+r\' and K°r]' modes. with a charged pion. This conclusion is supported by an estimate of the tree contribution via the decay B —> -KO,V and factorization. One then needs some idea of the form factor at m(lv) = m„ or VHK- The result is that one estimates Btree(B° -»• 7r+7r-) ~ IO- 5 , or Btree(B°

->

K+n~

!_K_

K

u

Vud

x 10"

(54)

With j K = 161 MeV, / „ = 132 MeV, fK/U = 1-22, Vus/Vud = tan9c = 0.22/0.975 = 0.226, the coefficient of 10~ 5 on the right-hand side is 0.076. Thus in order to see a significant CP-violating rate asymmetry in B —> Kir one needs at least 13 times the sensitivity that was needed in order to see all the B -» Kn modes. This would correspond to about 100 fb _ 1 at e+e~ colliders, or samples of about IO8 identified B's at hadron machines. In other words, one needs to be able to see branching ratios of a few parts in IO7 with good statistical significance. This is within the capabilities of experiments just now getting under way. 8 8.1

Flavor tagging States of BB with definite charge-conjugation

The process e+e~ -» BB is typically studied at the mass of the T(45) resonance, Ec.m. = 10.58 GeV, above the threshold of 2MB - 10.56 GeV for BB production but below the threshold for production of one or two B*'s:

453

MB + MB- = 10.605 GeV, 2MB- = 10.65 GeV. Now, the T(45) has C = - 1 . It is produced via a virtual photon 7* (which has odd C). It is a 3 5 i 66 state, where the superscript 2Sbi + 1 denotes the total spin S6j = 1. The 66 pair has orbital angular momentum L = 0 and total angular momentum J = 1. A QQ state 2 S + 1 L j in general has C = (-1)L+S. The BB pair produced at the T(45) thus has a definite eigenvalue of charge-conjugation, C{BB) = —1, correlating the flavor flavor of the neutral B whose decay (e.g., to J/ipKs) is being studied with the flavor of the other B used to "tag" the decay, e.g., via a semileptonic decay 6 —> cl~i>i or b —> c^ + t^. Let B°B be in an eigenstate of C with eigenvalue r\c — ±1- (To get a state with 7)c — +1 it is sufficient to utilize the reaction e+e~ —> B°*B or B°B —> B°B 7 just above threshold.) In the B°B center-of-mass system, the wave function of the pair, * c = - L [B°(p)B°(-p) + r)cB°(p)B°(-p)\

(55)

may be expressed in terms of the mass eigenstates BL,H in order to study its time evolution. Since B° = - ~ [BL + BH]

, fi° = -L= [BL - BH]

,

(56)

we have *c = ^

^

{[BL(P)

+ B H ( P ) ] [ S L ( - P ) - B/r(-p)]

+Vc[BL(p) - BH{p)][BL{-p)

+ BH(-p)}}

.

(57)

For rjc = — 1 the LL and -ffiJ terms cancel (this is also a consequence of Bose statistics) and one has *c(lc

= - 1 ) = ^ = - [ ^ ( P ) S L ( - P ) - BL{p)BH(-p)]

.

(58)

.

(59)

For 77c — + 1 the HL and LH terms cancel and one has Vc(ric = +1) = ~^~

[BL(p)BL(-p)

~ BH{p)BH{-p)\

Define t and t to be the proper times with which the states p and — p evolve, respectively: BLMP)

-> BLM(P)e-l^'Ht

,

BL
->• BLiH{-p)e-iln-H~t

.

(60)

454

Project the state with p into the desired decay mode (e.g., J/tpKs) and the state with — p into the tagging mode (which signifies B° or B at time t, e.g., £~ -H- B , £+ -f4 £?°). Then, for a CP-eigenstate, it is left as an Exercise to show in the limit A r = 0 that

d*r[f(t),e-(t)] dt dt d2T[f(t),£+(t)} dt dt

~e-r(t-K)[l-sinAm(*T£)ImA]

^

^

~ e - r ( f + ' ) [ l + sinAm(<=Fi)ImA]

.

(62)

'7C=:fl

(Hints: Recall that A = ( 7 / p ) ( / | B ° ) / ( / | B 0 ) , A = (p/q)((f\B0)/(f\B°), |A| = 1, A = A*, Am = mff - mi. For T?C = - 1 , write the decay amplitude as a function of t and £._ It will have two terms, one ~ e-l(mnt+mLt) ancj the other ~ e~l^rnLt+rn"i\ whose interference in the absolute square of the amplitude gives rise to the sin Am(t — t) terms.) These results have some notable properties. (1) For either value of rjc, the sum of the £+ and £~ results is as if one didn't tag, and the oscillatory terms cancel one another. (2) For r\c — —1) note the antisymmetry with respect to t — t. This is a consequence of the Bose statistics and the C = — 1 nature of the initial state. If one integrates over all times, the CP-violating asymmetry vanishes. Thus in order for the tagging method to work in a C-odd state like T(4S) one must know whether t or t was earlier. An asymmetric -B-factory like PEP-II or KEK-B permits this by spreading out the decay using a Lorentz boost. Exercise: Show for rjc = —1 that if one subdivides the t,t integrations according to t t, then

JJdtdtjcfT/dtdt)^-^ Jfdtdt{d2r/dtdt)[(£-,t

> t) - (£~,t < t) - (£+,t > t) + (£+,t < t)} > t) + {£-,t t) + (£+ ,t < t)} Xd

1+ 4

ImA

.

(63)

In practice the BaBar and Belle analyses will probably fit the time distributions rather than simply subdividing them, since background rejection and signal/noise ratio are functions of t — t. (3) For rjc = + 1 the oscillatory term behaves as ~ Am(t + t), so it is not necessary to know whether t or t was earlier, and the asymmetric collision geometry is not needed. However, as shown above, in order to produce a B°B state with rjc = + 1 in e + e~ collisions one must work at or above BB* threshold, thereby losing the cross section advantage of the T(4S') resonance.

455

8.2

Uncorrelated BB pairs

Pairs of S's produced in a hadronic environment are likely to arise from independent fragmentation of b and b quarks, so that it is unlikely that they are produced in a state of definite rjc- (The interesting case of partially-correlated B-B pairs can be attacked by density-matrix methods. 73 ) Thus, one must resort to either the fact that a 6 is always produced in association with a b by the strong interactions ("opposite-side tagging"), or the fact that the fragmentation of a b into a B favors one particular sign of charged pion close to the B in phase space ("same-side tagging"). "Opposite-side" methods The strong interactions qq —> bb or gg —• bb (g = gluon) conserve beauty, so that a b can be identified if it is found to be produced in association with a b. The opposite-side b can be identified in several ways. (1) The jet-charge method makes use of the fact that a jet tends to carry the charge of its leading quark 74 , since the average charge of the fragmentation products is zero in the flavor-SU(3) limit. (There is some delicacy if strange quark production is suppressed, since Q{u) ^ — Q(d)J5) (2) The lepton-tag method uses the charge of the lepton in the semileptonic decays b —> ct~vi and b —• cl+f£ to signify the flavor of the decaying oppositeside b quark. The signal is diluted since semileptonic decays occur after the b quark has been incorporated into a meson. Sometimes this meson is a neutral B, in which case information on its flavor is nearly completely lost if it is a strange B and partially lost if it is a nonstrange B. An initial Bs will decay half the time as a B s , while an initial B will decay about 18% of the time as a B°. (See Section 5.2.) (3) The kaon-tag method uses the fact that the signs of kaons produced in b decays are correlated with the flavor of the decaying b. A b gives rise to c, whose products have more K~ than K+. This method is subject to the same dilution as the lepton-tag method. In order to utilize the above methods, one needs the relative probabilities of production of B°, B+, Bs, and A&, The CDF Collaboration 76 has measured these in high-energy hadron collisions to be in the ratios 0.375:0.375:0.16:0.09, while LEP Collaborations find 0.40:0.40:0.097:0.104 for Z° -> bb decays.77 Taking account of P(B° -* B°) ~ 18% and P{BS -» B~s) ~ 1/2, the probability of a "wrong" tag is (3/8)(0.18) + (0.16)(l/2) ~ 0.15, which dilutes the efficacy of the tag by a factor (right - wrong)/(right + wrong) ~ 0.70. For an extensive study of the first two tagging methods, see recent papers by the CDF Collaboration.78 These methods were a key ingredient in obtaining

456

(a)

(b)

Figure 5: Fragmentation of a b or b quark into a B° or B

the CP-violating asymmetry in B° —> J/IJJKS mentioned in Sec. 6.4. "Same-side" methods: Fragmentation and B** resonances The fragmentation of a b quark into a neutral B meson is not chargesymmetric. This was noted quite some time ago in the context of strange £?'s.79 A Bs contains a b and an s. This s must have been produced in association with a s. If that s is incorporated into a charged kaon, the kaon must be a K+ = us. A similar argument applies to non-strange neutral B's and charged pions.80 A B° is then found to be associated more frequently with a 7r+ nearby in phase space, while a B tends to be associated with a ir~. This correlation is the same as that found in resonance decays: B° resonates with 7r+ but not 7r~, while B resonates with ir~ but not 7r+. The fragmentation of a b or b quark is illustrated in Fig. 5. If one cuts the diagrams to the left of the pion emission, one finds either a 6M or a bu state. Thus, a positively charged resonance can decay to B°n+, while a negatively charged one can decay to B TT~ . Recall the case of D*+ -> ir+D° mentioned in Sec. 3.1. The soft pion in that decay may be used to tag the flavor of the neutral D at the time of its production, which is useful if one wants to study D°-D mixing or Cabibbodisfavored decays. The difference between M(D*+) and M(TT+) + M(D°) (the "Q-value") is only about 5 MeV, so the pion is nearly at rest in the D*+ c.m.s., and hence kinematically very distinctive. In the case of JB's, however, the lowest vector meson B*+ cannot decay to n+B° since the B*-B mass difference is only about 46 MeV. One must then utilize the decays of highermass i?*'s, collectively known as 5**'s. The lightest such states consist of a 6 quark and a light quark (u, d, s) in a P-wave.

457 Table 5: Quantum numbers of B** resonances (P-wave resonances between a b quark and a light quark q).

h 1/2 1/2 3/2 3/2

J 0 1 1 2

Decay prods. B-K B*TT

B*ir BIT, B*TT

Part, wave S wave S wave D wave D wave

Width Broad Broad Narrow Narrow

Many key features of the spectroscopy of a heavy quark and a light one were first pointed out in the case of charm, 81 ' 82 and codified using heavy-quark symmetry.83 The heavy quark spin degrees of freedom nearly decouple from the light-quark and gluon dynamics, so it makes sense to first couple the relative angular momentum L = 1 and the light-quark spin sq = 1/2 to states of total light-quark angular momentum jq = 1/2 or 3/2 and then to couple j q with the heavy quark spin SQ = 1/2 to form total angular momentum J. For j q = 1/2 one then gets states with J = 0,1, while for j q = 3/2 one gets states with J = 1,2. The j q = 1/2 states decay to B-K or B*-K only via S-waves, while the jq = 3/2 states decay to BIT or B*ir only via D-waves. These properties are summarized in Table 5. The j q = 3/2 resonances, decaying via D-waves, have been seen, with typical widths of tens of MeV and masses somewhere between 5.7 and 5.8 GeV/c 2 in all analyses. 84 ' 85 The j q = 1/2 resonances are expected to be considerably broader. There is no unanimity on their properties, but evidence exists for at least one of their charmed counterparts. 86 More information on £?**'s would enhance their usefulness in same-side tagging of neutral B mesons. 9 9.1

The strange B Bs-Bs

mixing

The limit 44 Ams > 15 p s _ 1 mentioned in Sec. 4.4 was a significant source of constraint on the (p,rj) plot. What range of mixing is actually expected? One may place an upper bound by noting that

Amd

Vts Vtdt

BB. / 7 B . BB

V/B

(64)

Let us review the estimate 4 5 JBSI!B < 1-25. The upper limit (larger than the lattice range 3 4 ) comes from the nonrelativistic quark model, which implies 8 7 | / M | 2 = 12|*(0)| 2 /M M for the decay constant fM of a meson M of

458

Figure 6: Mixing of Bs and Bs as a result of shared sees intermediate states.

mass MM composed of a quark-antiquark pair with relative wave function \&(r). One estimates the ratios of |*(0)| 2 in D and Ds systems from strong hyperfme splittings. Since M{D*+) - M(D+) ~ M(D*+) - M (£>+), one expects |^>(0)|Qj/rrid ~ | * ( 0 ) | g s / m s for mesons containing a heavy quark Q. In constituent-quark models 88 m,i/ms ~ 0.64, so fQg/fQs — V0.64 = 0.8. An upper limit on |Vt»/Vtd| (see Sec. 4.3) is then < [A(0.66)]_1 = 6.9, implying Ams/Amd < 74 or Ams < 36 p s - 1 . A recent prediction 89 based on lattice estimates for decay constants is Ams = 16.2 ± 2.1 ± 3.4 ps" 1 ; there is a hint of a signal at ~ 17 ps"- 1 4 4 9.2

Bs lifetime

The mass eigenstates of the strange B are expected to be nearly CP eigenstates. Their mass splitting is expected to be correlated with their mixing; large values of Ams imply large values of A r s . (In the calculation of AT/ Am, the values °f | / M | 2 cancel.) The value of A r s is much greater than that of A I ^ because of the shared intermediate states in the transition Bs — bs —> sees —> sb = Bs illustrated in Fig. 6. Specific calculations 90 imply that this quark subprocess may be dominated by CP = + intermediate states, implying a shorter lifetime for the even-CP eigenstate of the Bs-Bs system by 0(10%). The imaginary part of the diagram in Fig. 6 is due to on-shell states, which do not include those involving the top quark. This is the reason for the factor of (mllm2w)IS(m\jM'^) in Eq. (30). The lowest-order ratio, A r s / A m s ~ -1/180, implies that if | A m s / r s | = (20 ps _1 )(1.6 ps) = 32 (a reasonable v a l u e ) , t h e n | A r s / r s | ~ 1/6. Recent calculations predict A r s / r s = ( 9 . 3 ^ 6 ) % 91 or (4.7±1.5±1.6)% 9 2 . The latter group 93 also finds fBd ^/B~B~d = 206(28)(7) MeV, fB,yrB^/(fBVB^) - 1.16(7), fB, y/B^ = 237(18)(8) MeV.

459

9.3

Measuring A r s / r s

The average decay rate of the two mass eigenstates BH and Bi can be measured by observing a flavor-specific decay, e.g., Bs(= bs) -» DJ(= cs)t+vt

or Bs -*• Djir+

,

DJ -> ^TT"

.

(65)

The flavor of Ds labels the flavor of the Bs and then we note that \BS)~±={\BL)

+ \BH))

•

(66)

Such a flavor-specific decay then gives a rate f = ( I ^ + r # ) / 2 . One can also look for a decay in which the CP of the final state can be easily identified.94 Bs -4 J/ip