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.

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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.

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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.

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_. .M««MRims Lincoln Wolfenstein

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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.

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'¥*3J#SR

Young-Kee Kim

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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

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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

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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

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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

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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.

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

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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

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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

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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.

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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

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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

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150

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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

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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.

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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 *

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0.48 0.49

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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

^

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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).

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J. P. Cumalat

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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].

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li|plil§iplll|pl

•iiiiii

illllB M

v Davison E. Soper

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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