Dynamics of the Standard Model (Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology)

—/i 2 with /i2 > 0), we enter a different phase of the system. Show that the ground state is now (p = const* = v, A^ = 0. What is the photon mass in this phase? Calculate the potential between two static point charges each of value Q. What sets the scale of the screening length? d) Let us now add an external field to the system, 1 M) ~~* M) T ^rfii/rext

'

To see that F£x\ indeed acts like an applied field, show that if one disregards the field

•{ °

Again, these correspond to unscreened and screened phases of the electromagnetic field. f) Calculate the energy of the two phases if F^t describes a constant magnetic field. Show that the phase in part (e) with

5Criticai whereas for B < JBCriticai it is the phase with

II Interactions of the Standard Model

A gauge theory involves two kinds of particles, those which carry 'charge' and those which 'mediate' interactions between currents by coupling directly to charge. In the former class are the fundamental fermions and nonabelian gauge bosons, whereas the latter consists solely of gauge bosons, both abelian and nonabelian. The physical nature of charge depends on the specific theory. Three such kinds of charge, called color, weak isospin, and weak hypercharge, appear in the Standard Model. The values of these charges are not predicted from the gauge symmetry, but must rather be determined experimentally for each particle. The strength of coupling between a gauge boson and a particle is determined by the particle's charge, e.g. the electron-photon coupling constant is —e, whereas the M-quark and photon couple with strength 2e/3. Because nonabelian gauge bosons are both charge carriers and mediators, they undergo selfinteractions. These produce substantial nonlinearities and make the solution of nonabelian gauge theories a formidable mathematical problem. Gauge symmetry does not generally determine particle masses. A fermion mass term, although violating chiral symmetry, is consistent with gauge invariance. Although gauge-boson mass would seem to be at odds with the principle of gauge symmetry, the Weinberg-Salam model contains a dynamical procedure, the Higgs mechanism, for generating mass for both gauge bosons and fermions alike. I I - l Quantum Electrodynamics Historically, the first of the gauge field theories was electrodynamics. Its modern version, Quantum Electrodynamics (QED), is the most thoroughly verified physical theory yet constructed. QED represents the best introduction to the Standard Model, which both incorporates and extends it. 24

II-l

Quantum Electrodynamics

25

U(l) gauge symmetry Consider a spin 1/2, positively charged fermion represented by field ip. The classical lagrangian which describes its electromagnetic properties is Cem = -^F2

+ ^(Hp-m)i/j

.

(1.1)

Here, the covariant derivative is D^ = (d^+ieA^ip^ m and e are respectively the mass and electric charge for ^ , A^ is the gauge field for electromagnetism, F^ is the gauge-invariant field strength, and F2 = F^VF^V. This lagrangian is invariant under the local t/(l) transformations

tl){x) -> e- fo (*ty(x) , Ap{x)-+All{x)

l

+ e- dl]La{x) .

(1.2) (1.3)

The associated equations of motion are the Dirac equation

(ip-m-e4)^ = 0,

(1.4)

and the Maxwell equation e$Yip .

(1.5)

It is worthwhile to consider in more detail the important subject of C/(l) gauge invariance, addressing both its extent and its limitations. (i) Universality of electric charge: The deflection of atomic and molecular beams by electric fields establishes that the fractional difference in the magnitude of electron and proton charge is no larger than 0(1O~ 20). Likewise, there is no evidence of any difference between the electric charges of the leptons e, /x, r. Whatever the source of this charge universality may be, it is not the U(l) invariance of electrodynamics. For example assume that in addition to ^ , there exists a second charged fermion field ip' with charge parameter /3e. It is easy to see that gauge invariance alone does not imply j3 = 1. The electromagnetic lagrangian for the extended system is Cem - ~

F2 + ^ {ilf) - m) ' ,

(1-6)

where DJI/J f = (d^ + i/3eA^(x))^ . The above lagrangian is invariant under the extended set of gauge transformations ,

i//(x) -> e - ^ < * y (x) ,

(17)

This demonstration of gauge invariance is valid for arbitrary /?, and thus says nothing about its value. The f/(l) symmetry is compatible with, but does not explain, the observed equality between the magnitudes of the electron and proton charges. We shall return to the issue of charge

26

/ / Interactions of the Standard Model

quantization in Sect. II—3 when we consider how weak hypercharge is assigned in the Weinberg-Salam model. (ii) A candidate quantum lagrangian: The quantum version of Cem is in fact the most general Lorentz-invariant, hermitian, and renormalizable lagrangian which is C/(l) invariant. Consider the seemingly more general structure

ZF2 + iZ^\Tp^

+ iZ^Ip^

M$l

M*tfl (1.8)

where Z, ZR^ are constants,^ is the covariant derivative of Eq. (1.1), and M can be complex-valued. This lagrangian not only apparently differs from £em, but seemingly is CP-violating due to the complex mass term. However, under the rescalings

4

e' = Z-^e ,

<

L

= 4'SW ,

(1.9)

we obtain

C> = -\F* + 09 V - M^l where Mf = {ZRZL)~1^2M.

- M'> L VH ,

(1.10)

A subsequent global chiral change of variable e

il>L,R = ~*a75V>L,i2

( " = constant)

(1.11)

does not affect the covariant derivative term but modifies the mass terms,

^Yi>l^

. (1.12)

Choosing the parameter a so that Im (M'e2ta) = 0 and defining m = Re(M'e2ta), we see that £gen reduces to Cem which appears in Eq. (1.1). (iii) Renormalizability and [7(1) - Renormalizability plays a role in the preceding discussion because [7(1) symmetry by itself would admit a larger set of interaction terms. In principle, J7(l) invariant terms like ^a^F^, WF^F^, ^YYl^^FnvFap, etc. could appear in the QED lagrangian. However, they do not because the condition of renormalizability admits only those contributions which have dimension d < 4. As discussed in App. C-3, the canonical dimension of boson and fermion fields is d = 1, 3/2 respectively, and each derivative adds a unit of dimension. Accordingly, the above candidate operators have d = 5, 7, 7 and ^a^F^Fa^, thus are ruled out. There remains an operator, F^F**" = which is gauge-invariant and has dimension 4. A noteworthy aspect of this quantity is that, unlike the other operators encountered thus far, it is odd under CP. This follows from writing it as —8E • B and realizing that under CP, E —• E and B —• —B. However, a simple exercise shows that we can identify this operator as a four-divergence F^VF^V = where K» = £e» UOL(3AvdaAp. Thus a contribution proportional to F

II-l Quantum Electrodynamics

27

can be of no physical consequence. Upon integration over spacetime, it becomes a surface term evaluated at infinity. There is nothing in the structure of QED which would cause such a surface term to be anything but zero. QED to one loop The perturbative expansion of QED is carried out about the free field limit, and is interpreted in terms of Feynman diagrams. Two distinct phenomena are involved, scattering and renormalization. The latter encompasses both an additive mass shift for the fermion (but not for the photon) and rescalings of the charge parameter and of the quantum fields. To carry out the calculational program requires a quantum lagrangian £QED t° establish the Feynman rules, a regularization procedure to interpret divergent loop integrals, and a renormalization scheme. One can develop QED using either canonical or path-integral methods. In either case a proper treatment necessitates modification of the classical lagrangian. As we have seen, the [7(1) gauge symmetry implies a certain freedom in defining the A/i(x) field. Regardless of the quantization procedure adopted, this freedom can cause problems. For canonical quantization, the procedure of selecting a complete set of coordinates and their conjugate momenta is upset by the freedom to gauge transform away a coordinate at any given time. For path integration, the integration over gauge copies of specific field configurations gives rise to specious divergences (cf. App. A-6). In either case, superfluous gauge degrees of freedom can be eliminated by introducing an auxiliary condition which constrains the gauge freedom. There are a variety of ways to accomplish this. The one adopted here is to employ the following gauge-fixed lagrangian, £QED

=

-]F2

- - L ( 0 • A)2 + ^ {ip -eofi-

mo)ip ,

(1.13)

where eo and mo are respectively the fermion charge and mass parameters. The quantity £o is a real-valued, arbitrary constant appearing in the gauge-fixing term. This term is Lorentz-invariant but not U(l) invariant. One of its effects is to make the photon propagator explicitly dependent on £o- The value £o — 1 corresponds to Feynman gauge, whereas the limit £o —> 0 defines the Landau gauge. The zero subscripts on the mass, charge, and gauge-fixing parameters denote that these bare quantities will be subject to infinite renormalizations, as will the quantum fields. This process is characterized in terms

28

/ / Interactions of the Standard Model

of quantities Zi and <Sra,

1> = Zxfyr ,

A^ = Z\I2AI ,

e 0 = ZiZ 2 - 1 Z 3 " 1/2 e ,

m0 = m - 6m ,

(1.14)

where the superscript V labels renormalized fields. The renormalization constants Zi, Z2, and Z3 (associated respectively with the fermion-photon vertex, the fermion wavefunction, and the photon wavefunction) and the fermion mass shift 6m are chosen order by order to cancel the divergences occurring in loop integrals. For vanishing bare charge eo = 0, they reduce to Zi?2,3 = 1, 6m = 0. The Feynman rules for QED are: fermion-photon

vertex.

p

a

-

(1.15)

fermion propagator iSap(p): P

i (p + rao)Qjg

p

p2 — ra|j + ie

photon propagator

-

»»

a

(1.16) iD

fJ>u

(q):

(1.17) In the above e is an infinitesimal positive number. The remainder of this section is devoted to a discussion of the one-loop radiative correction experienced by the photon propagator.* Throughout, we shall work in Feynman gauge.

Fig. II—1 The full photon propagator as an iteration. * We shall leave calculation of the fermion self-energy to Prob. II—3 and analysis of the photonfermion vertex to Sect. V-l.

II-l Quantum Electrodynamics

29

Let us define a proper or one-particle irreducible {IPI) Feynman graph such that there is no point at which only a single internal line separates one part of the diagram from another part. The proper contributions to photon and to fermion propagators are called self-energies. The point of finding the photon self-energy is that the full propagator iD'^v can be constructed via iteration as in Fig. II-l. Performing a summation over self-energies, we obtain iD' = —i

where the proper contribution

V

2 0

(1.19)

is called the vacuum polarization tensor. It is depicted in Fig. II-2(a) (along with corrections to the photon-fermion vertex and fermion propagator in Figs. II-2(b)-(c)), and is given to lowest order by

=-He)2 / A T TV y ^

10!-^1

1 •

(1.20) This integral is divergent due to singular high momentum behavior. To interpret it and other divergent integrals, we shall employ the method of dimensional regularization [BoG 72, 'tHV 72, Le 75]. Accordingly, we consider UaP(q) as the four-dimensional limit of a function defined in d spacetime dimensions. Various mathematical operations, such as summing over Lorentz indices or evaluating loop integrals, are carried out in d dimensions and the results are continued back to d = 4, generally expressed as an expansion in the variable* e = 4 — d. Formulae relevant to this procedure are collected in App. C-4. For all theories

p-q (a)

(b)

(c)

Fig. II—2. One-loop corrections to (a) photon propagator, (b) fermion-photon vertex, and (c) fermion propagator. We shall follow standard convention is using the symbol e for both the infinitesimal employed in Feynman integrals and the variable for continuation away from the dimension of physical spacetime.

30

// Interactions of the Standard Model

described in this book, we shall define the process of dimensional regularization such that all parameters of the theory (such as e2) retain the dimensionality they have for d = 4. In order to maintain correct units while dimensionally regularizing Feynman integrals, we modify the integration measure over momentum to (1.21) The parameter ji is an arbitrary auxiliary quantity having the dimension of a mass. It appears in the intermediate parts of a calculation, but cannot ultimately influence relations between physical observables. Indeed, there exist in the literature a number of variations of the extension to d ^ 4 dimensions. These are able to yield consistent results because one is ultimately interested in only the physical limit of d = 4. Let us now return to the photon self-energy calculation to see how the dimensional regularization is implemented. The self-energy of Eq. (1.20), now expressed as an integral in d dimensions, is

-2

e

J (2?r)d

]

)

(1.22) where we retain the same notation 11°^(g) as for d — 4 and we have already computed the trace. Upon introducing the Feynman parameterization, Dirac relations, and integral identities of App. C-4, we can perform the integration over momentum to obtain

f I

x(l - x) /

9

9/1

\\*/9

'

[ml - qlx(\ - x)flA (1.23) We next expand Tla^(q) in powers of e and then pass to the limit e —• 0 of physical spacetime. In doing so, we use the familiar ae = l + eln a + ... , (1.24) /o Jo

and take note of the combination r(e/2)

2

^

where 7 = 0.57221... is the Euler constant. The presence of e~l makes it necessary to expand all the other e-dependent factors in Eq. (1.23) and to take care in collecting quantities to a given order of e. To order e2, the

II-1 Quantum Electrodynamics

31

vacuum polarization in Feynman gauge is then found to be

6TT 2

(1.26) The above expression is an example of the general property in dimensional regularization that divergences from loop integrals take the form of poles in e. These poles are absorbed by judiciously choosing the renormalization constants. Renormalization constants can also have finite parts whose specification depends on the particular renormalization scheme employed. One generally adopts a scheme which is tailored to facilitate comparison of theory with some set of physical amplitudes. In the minimal subtraction (MS) renormalization, the Z{ subtract off only the e-poles, and thus have the very simple form, ^T

(* = 1.2,3).

(1-27)

n=l

Because the < Z\ H have no finite parts, they are sensitive only to the ultraviolet behavior of the loop integrals, and the c^n are independent of mass. The simple appearance of the MS scheme is somewhat deceptive since further (finite) renormalizations are required if the mass and coupling parameters of the theory are to be asociated with physical masses and couplings. A related renormalization scheme is the modified minimal subtraction (MS) in which renormalization constants are chosen to subtract off not only the e-poles but also the omnipresent term ln(47r) — 7 of Eq. (1.25). Minimal subtraction schemes are typically used in QCD where, due to the confinement phenomenon (cf. Sect. II—2), there is no natural renormalization scale that could naturally be associated with the mass of a freely propagating quark. Yet another approach is the onshell (o-s) renormalization, where the renormalized mass and coupling parameters of the theory are arranged to coincide with their physical counterparts.

32

/ / Interactions of the Standard Model On-shell renormalization of the electric charge

The renormalization scale for electric charge is set by experimental determinations typically involving solid state devices like Josephson junctions. These refer to probes of the electromagnetic vertex — eT J/(p2,pi) of Fig. II2(b) with on-shell electrons (p2 = Pi = m e 2 ) an<^ with q2 = (jp\ — P2)2 — 0. The value of the electromagnetic fine structure constant a = e2/4?r obtained under such conditions is given in rationalized units by a" 1 = 137.0359895(61) .

(1.28)

To interpret this in the context of the theoretical analysis performed thus far, recall from Eq. (1.18) how the photon propagator is modified by radiative corrections, 2 ~>

ie2D

'

=

- ^

We display only the g^v piece since, in view of current conservation, only it can contribute to the full amplitude upon coupling the propagator to electromagnetic vertices. The above suggests that we associate the physical, renormalized charge e with the bare charge parameter eo by

4

-

(1JW

>

In this on-shell renormalization prescription, the g^v part of the photon propagator ID' (q) is seen to assume its unrenormalized form in the physical limit q2 —• 0. The appellation 'on-shell' means that the physical kinematic point q2 = 0 is used to implement the renormalization condition and by absorbing the singular vacuum polarization in the electric charge, one ensures that the photon has zero mass. Likewise, in the onshell renormalization approach fermion propagators have poles at their physical masses. Next, we show how to infer the form of the renormalization constant Z% i n the on-shell scheme. There is a relation, called the Ward identity, that implies Z\ = Z2 as a consequence of the gauge symmetry of the theory. Prom Eq. (1.14), this gives (1.31) Use of the relation e2 = Z3 tion constant to be

e§ then specifies the on-shell renormaliza-

II-l

Quantum Electrodynamics

33

One can similarly absorb the e-pole in either the MS or MS schemes by adopting

(1 33)

Eqs. (1.32), (1.33) display how the various renormalization constants differ by finite amounts. The e-poles in the fermion self-energy and the fermion-photon vertex can be dealt with in the same manner and we find, e.g. in MS renormalization (c/. Prob. II—3 and Sect. V-l),

^ m -

+ O(e4) .

(1.35)

Electric charge as a running coupling constant The concept of electric charge as a 'running' coupling constant is motivated by the following consideration. In the perturbative Feynman expansion for a given theory, the hope is that corrections to the lowest order amplitudes will be small. However, potentially large corrections of the form In Q2/QQ can arise if the theory is renormalized at scale QQ but then applied at a very different scale Q2. It is convenient to deal with this problem by absorbing such logarithms into scale-dependent or 'running' renormalized coupling constants and masses. To see why scale-dependent charge is not an unreasonable concept, consider the vacuum polarization process of Fig. II—3, which depicts virtual production of a fermion of charge Qie together with its antiparticle near a charge source. Due to the source, each such vacuum fluctuation is polarized, and thus the source becomes screened. All charged fermion species contribute to the screening, and the larger the mass of the virtual pair, the closer they lie to the source. The effect is somewhat akin to concentric onion skins, with each virtual pair forming a layer, resulting in an effectively scale-dependent source charge.

Fig. II—3

Virtual pair production in the vicinity of a charge.

34

// Interactions of the Standard Model

Let us seek a method for specifying a running fine structure constant a(q) for nonzero momentum transfers, with a(0) to be identified with the a of Eq. (1.28). The interpretation of eg/(l + Re H(q)) as a running charge is appealing since it would maintain the simple —i/q 2 structure of the lowest order photon exchange amplitude. The fact that H(q) is divergent (see Eq. (1.26)) can be circumvented by subtracting off its value at q2 = 0 to define a finite quantity tl(q) = H(q) — n(0) and defining

so that a(q) = e2(g)/47r. It is not difficult to deduce the behavior of from the integral representation of Eq. (1.26), and we find

£{*,

""

(137)

Observe that the arbitrary energy scale // is absent from ft(#), as would be expected since fi(q) is a physically measureable quantity. The above formulae correspond to the loop correction of one fermion of mass m. Generally, loops from all available fermions must be included, although contributions of heavy (m2 » q 2) fermions are seen to be suppressed. Important modern applications of the Standard Model engender phenomena at scales provided by the gauge boson masses M\y,Mz. To obtain an estimate for a(Myy), we can apply Eq. (1.36) to find

- ! ) + - ] • MS) If a sum over quark-loops (each being accompanied by the color factor Nc = 3) and lepton-loops is performed, then the mass values in Table 1-1 yield the approximate determination a~l(M^) ~ 128. The main uncertainty in this approach arises from quarks. It is possible to perform a more accurate evaluation of a(M^r) (cf Sect. XVI-5) which avoids this difficulty. Let us return to the question of how to define a momentum-dependent coupling. To emphasize the fact that a 'running fine structure constant' is after all a matter of definition, let us consider a somewhat different derivation (and definition) of a(q2). One is able to renormalize the electric charge in a mass-independent scheme [We 73] by calculating renormalization constants with m = 0. If we return to the vacuum polarization

II-l

Quantum Electrodynamics

35

diagram, but with m = 0, we find e/2

(1.39)

In order to apply the renormalization program, we must specify the value of the coupling at some renormalization point*, which we choose to be q2 = - / x | , identifying

(1.40) However, if we had chosen a different renormalization point fi2R , we would have obtained a different value,

The functional dependence of the charge on the renormalization scale is embodied in the so-called beta function of electrodynamics [GeL 54],

It can be shown [Po 74] that the leading and next-to-leading terms in a perturbative expansion of /3QED are independent of both renormalization and gauge choices. The quantity e2(/i^) defined by integrating the beta function,

^TT "— • #QED(e)

(143>

fiR

is not exactly the same quantity as the running coupling constant defined in Eq. (1.36), differing by a (small) finite renormalization. For example, the electron contribution to the running coupling in the range ml < ji 2R < M$y is

°-\A)\^mi

- «" V S )l i = M s, = ^ In ^ f ,

(1.44)

which contains the dominant logarithmic dependence, but differs from Eq. (1.38) by a small additive term. However, complete calculations of * Note that the renormalization point fiR and the scale factor /x in dimensional regularization need not be identical. They are sometimes confused in the literature, and hence we use a different notation for the two quantities.

36

/ / Interactions of the Standard Model

all corrections to physical observables using the two schemes will yield the same answer. Since the running coupling constant is but a bookkeeping device, one's choice is a matter of taste or of convenience. Regardless of the specific definition employed for a(# 2 ), we see that as the energy scale is increased (or as distance is decreased), the running electric charge grows, as is expected on physical grounds from vacuum polarization. The use of a mass-independent scheme is convenient for identifying the high energy scaling behavior of gauge theories. One useful feature is in the calculation of the one-loop beta function. Dimensional analysis requires that the one-loop charge renormalization be of the form, 9 = 90

- 9ob ( Z~2 )

(- +

finite t e r m s

(1.45)

)

where g is the 'charge' associated with the gauge theory being considered. Choosing the renormalization point as q2 = —^ 2R and forming the beta function as in Eq. (1.42), we see that (3 = bg3. This allows the beta function to be simply identified with the coefficient of e" 1 to this order. II—2 Quantum Chromodynamics Chromodynamics, the nonabelian gauge description of the strong interactions, contains quarks and gluons instead of electrons and photons as its basic degrees of freedom [PrG 72, MaP 78]. A hallmark of quantum chromodynamics is asymptotic freedom [GrW 73a,b, Po 73], which reveals that only in the short-distance limit can perturbative methods be legitimately employed. The necessity to employ approaches alternative to perturbation theory for long-distance processes motivates much of the analysis in this book. SU(3) gauge symmetry Chromodynamics is the SU(3) nonabelian gauge theory of color charge. The fermions which carry color charge are the quarks, each with field t 3L m , where a = u, d, s,... is the flavor label and j = 1,2,3 is the color index. The gauge bosons, which also carry color, are the gluons, each with field Aa, a = 1,..., 8.* Classical chromodynamics is defined by the lagrangian

£color = ~\ F?F%, + J2 ^VttVjk ~ ™

(Q)

Wia) ,

(2-1)

* In this section, it will be particularly important to explicitly display color indices. We shall reserve indices which begin the alphabet for gluon color indices (e.g., a,b,c — 1,...,8 ), use mid-alphabetic letters for quark color indices (e.g., j,k,£ = 1,2,3 ), and employ greek symbols for flavor indices.

II-2 Quantum Chromodynamics

37

where the repeated color indices are summed over. strength tensor is F% = d^Al - dvAl - 9ifahcA\Al

The gauge field

,

(2.2a)

gs is the SU(3) gauge coupling parameter, and the quark covariant derivative is T>^ = (dfl + igsAa^W

.

(2.2b)

The lagrangian of Eq. (2.1) is invariant under local SU(3) transformations of the color degree of freedom, under which the quark and gluon fields transform as given earlier in Eqs. (1-4.11), (1-4.17). Equations of motion for the quark and gluon fields are

(2 3)

*

In its quantum version, the gs —• 0 limit of £coior describes an exceedingly simple world. There exist only free massless spin one gluons and massive spin one-half quarks. However, the full theory is quite formidable. In particular, accelerator experiments reveal a particle spectrum which bears no resemblance to that of the noninteracting theory. The group SU{3) has an infinite number of irreducible representations R. The first several are R = 1, 3, 3*, 6, 6* 8, 10, 10*, . . . , where we label an irreducible representation in terms of its dimensionality. Quarks, antiquarks, and gluons are assigned to the representations 3, 3*, 8 respectively. We denote the group generators for representation R by {Fa(R)} (a = 1 , . . . , 8). The quantities A/2 are group generators for the d = 3 fundamental representation, i.e., F(3) = A/2. They have the matrix representation 0 0 A4 = . 0 0 1 0 A2 = \ i

0

0

A5 =

0 0 0 0 i 0 0

A3 = I 0

-1

0 I

A6 = | 0 u 0

0 0u

i> 0 .

^

=

/ —i 0 0 1I j

73 0 0

.

1 0 /

0

73 0

0 0 -2

73(2 4^ y

'

J

As generators, they obey the commutation relations [Aa, A6] = 2i/ a6c A c

(a, b, c = 1 , . . . , 8)

(2.5a)

38

/ / Interactions of the Standard Model

where the /-coefficients are totally antisymmetric structure constants of SU(3). There exist corresponding anticommutation relations {Aa, Xb} = pab I + 2dabc\c

(a, 6, c = 1,..., 8)

(2.5b)

with d-coefficients which are totally symmetric. Values for fabc and dabc are given in Table II—1. Useful trace relations obeyed by the {Aa} are Tr Aa = 0

(a = l , . . . , 8)

(2.6)

from Eq. (2.4) and TV \aXb = 26ab (a, b = 1,..., 8) from Eq. (2.5). The statement of completeness takes the form, KjKi = —SijSid + 26aSjk

(t, j ,fc,J = 1,2,3) ,

(2.7) (2.8)

where a = 1,..., 8 is summed over. Useful labels for the irreducible representations of SU(3) are provided by the Casimir invariants. For any representation R, the quadratic Casimir invariant C2(R) is defined by squaring and summing the group generators {Fa(i?)}5 8

o=l

There is also a third-order Casimir invariant, 8

CS(R)I=

] T dabcFa(R)Fb(R)Fc(R) a,6,c=l

Table II—1. Nonvanishing /, d coefficients abc

123 1 147 1/2 156 -1/2 246 1/2 257 1/2 345 1/2 367 -1/2 458 V3/2 678 V3/2

dabc

abc

dabc

118 l/>/3 355 1/2 146 1/2 366 -1/2 157 1/2 377 -1/2 228 1/VS 448 -l/2>/3 247 -1/2 558 -l/2>/3 256 1/2 668 -l/2\/3 338 1/V3 778 -l/2\/3 344 1/2 888 -1/V5

.

(2.10)

II-2 Quantum Chromodynamics

39

The quark and antiquark states form the bases for the smallest nontrivial irreducible representations of SU(3). It is possible to use products of them, say p factors of quarks and q factors of antiquarks, to construct all other irreducible tensors in SU(S). Each irreducible representation R is then characterized by the pair (p, q). For example, we have the correspondences 1 ~ (0,0), 3 ~ (1,0), 3* ~ (0,1), 8 ~ (1,1), 10 ~ (3,0), etc. The (p, q) labeling scheme provides useful expressions for the dimension of a representation, dip, q) = (p + l)(q + l)(p + q + 2)/2 ,

(2.11)

and of the two Casimir invariants, C2(p,
3)/l8.

From Eq. (2.12) we find C 2 (3) = C2(3*) = 4/3 for the quark and antiquark representations. Equivalently, upon setting j — k and summing in Eq. (2.8) we obtain

Kj^i = fh

= ^C2(3)6il.

(2.13)

Generators for the d = 8 regular (or adjoint) representation are determined from the structure constants themselves, (Fa(8))bc = -ifabc

(a, 6, c = 1 , . . . , 8) .

(2.14)

It follows directly from Eq. (2.14) and from using Eq. (2.12) to compute C 2 (8) = 3 that (2-15)

facdfbcd = ^2(8) 8ah = 3 6ab • This result, in turn, enables us to determine = ifabcfbcd^d = ^ 2 ( 8 ) Aa .

^

(2.16)

As a final example involving 5C/(3), we evaluate the quantity a b 6b A6] - [A0a, A"]A \b]Xb& + + A Xb6XAb6XAa a + + AXaA"A XX A6AaA" = - ( A6[Aa , A"] 2 V

a

6

c

y

= 4C 2 (3)A + i/ a6c [A , A ] = 4 (C 2 (3) - -C 2 (8)) Aa .

(2.17)

Shortly, we shall see how such combinations of color factors arise in various radiative corrections.

40

II Interactions of the Standard Model

Including only gauge-invariant and renormalizable terms, we can write the most general form for a chromodynamic lagrangian as

(2.18) where the flavor matrices ZL,R are hermitian, color and flavor indices are as before, except that for simplicity we suppress quark color notation. The final contribution to Eq. (2.18) is called the 0-term. We can reduce £gen to the form of £Color by first rescaling, A>a = z A

>

£

gz ?

(

)

and then diagonalizing and rescaling with respect to quark flavors, IPL,R = UL,RTPL,R,

UL>RZL,RUIR = AL,R,

<

f l

= A ^ ' ,

(2.20)

where AL,R are diagonal. Finally we diagonalize the mass terms

Anass = -J'LaM'a^

- SSWfy? ,

(2-21)

where M' = A^ ' ULMU^A^ ' , by means of yet another set of unitary transformations on the quark fields. Aside from the 0-term, this results in the canonical expression for £coior °f Eq. (2.1). We shall demonstrate later in Sect. IX-5 that the above quark mass diagonalization procedure induces a modification in the ^-parameter, M' .

(2.22)

This does not imply 0 = 0 because both 0 and the original quark mass matrices are arbitrary from the viewpoint of renormalizability and SU(3) gauge invariance. In fact, the 0-term cannot be ruled out by any of the tenets which underlie the Standard Model. Moreover, although the 0term can be expressed as a four-divergence (2 23)

^ ,

' (2.24)

analysis demonstrates that K^1 is a singular operator and that its divergence cannot be summarily discarded as was done in electrodynamics. This is a curious situation because the 0-term is CP-violating. Thus, one is faced with the spectre of large CP-violating signals in the strong interactions. Yet such effects are not observed. Indeed, it has been estimated that the 0-term generates a nonzero value for the neutron electric dipole moment de(n) ~ 5 x 10~16 0 e-cm, but to date no signal has been

II-2 Quantum Chromodynamics

41

observed experimentally, de(n) < 10~25 e-cm. This provides the upper bound 9 < 2 x 10"10. Perhaps Nature has dictated 0 = 0, albeit for reasons not yet understood. QCD to one loop To develop Feynman rules for QCD, we must first obtain an effective lagrangian which properly addresses the issue of SU(3) gauge freedom. For the £7(1) gauge invariance of QED, this was accomplished by adding a gauge-fixing term to the classical lagrangian. The situation for SU(3) is analogous, but somewhat more complicated due to its nonabelian structure. If we continue to use a Lorentz-invariant gauge-fixing procedure, the effective QCD lagrangian (for simplicity, consider just one quark flavor) can be expressed as

Ca + gsfi

Bare quantities carry the subscript '0' and the field strengths and covariant derivative are defined as in Eqs. (2.2a), (2.2b). The quantities {ca(x)} (a — 1,..., 8) are called ghost fields. As explained in App. A-6, they are anticommuting c-number quantities (i.e., Grassmann variables) which couple only to gluons. Ghosts occur only within loops, and never appear as asymptotic states. Each ghost field loop contribution must be accompanied by an extra minus sign, analogous to that of a fermionantifermion loop. Their presence is a consequence of the Lorentz-invariant gauge-fixing procedure. In alternative schemes such as axial or temporal gauge, ghost fields do not appear, but compensating unphysical singularities occur in Feynman integrals instead. The Feynman rules for QCD are three-gluon vertex: v,b

P^n^r

f

- p)v]

q

S \,c

(2.26)

quark-gluon vertex. n,a

y)jfc

S P,k —»

2

•_

a ,j

(2.27)

42

/ / Interactions of the Standard Model

four-gluon vertex: - Wlfl[(fabefcde(9fi\9v
(2.28) ghost-gluon vertex.

•

I

*. _ c

(2.29)

J

quark propagator iS ap(p): itijk U> + mo)ap

P

p2 -ml + ie

(2.30) b

gluon propagator iD" u(q):

q2

(2.31)

ghost propagator.

b

m-

a

(2.32)

The above rules involve a total of four distinct interaction vertices. Of these, the three-gluon and four-gluon self-vert ices, and the ghost-gluon coupling have no counterpart in QED. That all four vertices are scaled by a single coupling strength g$ is a consequence of gauge invariance. Also, chromodynamics exhibits a certain coupling constant universality, called flavor independence, in the quark-gluon sector. All fields which transform according to a given representation of the SU(3) of color have the same interaction structure, e.g., all triplets couple alike, all octets couple alike but differently from triplets, etc. Quarks are assigned solely to the color triplet representation. Thus, the quark-gluon interaction is independent of flavor.

//-# Quantum Chromodynamics

43

The renormalization constants of QCD are defined by 53,0 = ^ 1 ^ 3 ^—

Zf

4A

53 , o CI'X ^° *

JLJ O

2-

€o = ^ 3 € 5

1

Z 3 - 1/2 < 73 ,

= ^ 1 ^ 3 ^3

(2.33)

53 j

mo = m — 6m . In the following, working in £o — 1 gauge we shall compute the one-loop contributions to the gluon self-energy and to the quark-gluon vertex, and by absorbing the e-poles, thereby obtain expressions for Z3 and Z\F to leading order. Determination of the remaining renormalization constants, which can be computed from loop corrections to the quark and ghost propagators and the three-gluon, four-gluon and ghost-gluon vertices will be left as exercises. However, it is clear from the definition of #3,0 in Eq. (2.33) that the relations Z\ Z\

Z\ Z3

Z\ Z3

must hold in any consistent renormalization scheme. These are the analogs of the Ward identities in QED. Physically, they ensure that the coupling constant relations which appear in the QCD lagrangian (as a consequence of gauge invariance) are maintained in the full theory. The QCD one-loop contribution to the quark-antiquark vacuum polarization amplitude of Fig. II-4(a)*, quark

2

J (2?T) (Z.60)

*

differs from the QED self-energy only by the group factor (Aa)jfc(A6)jtj = Tr (AaA6) = 26ab (cf. Eq. (2.7)). Comparing with Eq. (1.42), we obtain

+ . . . ] • (2.36) This must be multiplied by the number of quark flavors rif which contribute in the vacuum polarization loops. To avoid notational clutter, we shall not put subscripts on the bare coupling for the remainder of this subsection.

44

77 Interactions of the Standard Model

(a)

(b)

(c)

Fig. II—4. One-loop corrections to the gluon propagator: (a) quark-antiquark pair, (b) gluon pair, and (c) ghosts.

The contribution from the gluon-gluon intermediate state of Fig. II4(b) can be written y-74 7

A/*^

with N$ = gsf^i-g^q + *), + ^(2fc - q){, + 9v0{2q - fc)M] acd t x g3f W a(q + k)v +
]

(2.39)

with Nap = ("5
+ (2d - 3) (qakf3 + 9/jfca) + (6 - 4d)fck Integration of Eq. (2.39) yields

The final contribution to the gluon propagator is the ghost loop amplitude of Fig. II-4(c),

=

~ J (2 — [93f

acd

ka]

.

II-2 Quantum Chromodynamics

45

The bracketed quantities arise from the gluon-ghost vertices, and the minus prefactor must accompany any ghost loop. Following the standard steps to a d-dimensional form, we arrive at

J (2^

ghost

which becomes to leading order in e,

The sum of gluon and ghost contributions takes the gauge-invariant form

(2.45) Finally, adding the quark contribution for n/ flavors gives the total result

Renormalizing at q2 = — ji

(2.46)

2

R,

we find

Proceeding next to the quark-gluon vertex, written through first order as

= -if

(a)

•••

(b)

Fig. II—5 One-loop corrections to the quark-gluon vertex.

,

(2.48)

46

// Interactions of the Standard Model

we see from Fig. II—5 that there are radiative corrections from both quark and gluon intermediate states. The quark contribution is 3

/»

J4

7

*

OL3

(2 49)

Aside from the replacement e —• g% and a color factor AbAaAb/8, which is evaluated in Eq. (2.17), the remaining expression is the QED vertex which will be analyzed in detail in Sect. V-l. Thus we anticipate from Eq. (V-l.24) that at p\ = p2 = p and \p\2 » ra 2,

= (C2(S) - \C2(8)) & 1

(^j

(2.50) The two-gluon intermediate state, which has no counterpart in QED, has the form

fJ - k - pi)Q + gpa(2k - pi - pi)v + g w (2pi - k [k? - m2 + te][(pi - A;)2 + ze]2[(p2 - A;)2 + ie]

(2.51) By a now-standard set of steps, it is not difficult to extract the e-pole from the extension of the above to d dimensions, and we find

J^

\

... , (2.52)

implying a total vertex correction of the form,

(K(P,P)hLt = (A7%7, [C2(3) + C2(8)] ^

(^j

(2.53) We thus determine the renormalization constant for the quark-gluon vertex at pf = —i^\ to be

Z1F = 1- [C2(3) + C 2 (8)]i ( ^ y i + ...) .

(2.54)

There remains the task of determining Z2- We shall leave this for an exercise (c/. Prob. II.3) and simply quote the result

i

+

...).

(2.55,

II-2 Quantum Chromodynamics

47

Asymptotic freedom and renormalization group A striking property of QCD is that of asymptotic freedom. This is the statement that, unlike the electric charge, the coupling constant QZ{IJLR) of color decreases as the scale of renormalization fiR is increased. To demonstrate this, we proceed just as we did for QED, using the relation .

(2.56)

Prom the coefficient of e"1, we learn that

| ^ = /3QCD = - [jC 2 (8) - ^C 2 (3)] ^

+ O{gl) ,

(2.57a)

which implies

/?QCD = - [ y C 2 ( 8 ) - ^

|

^ (2.57b)

The sign of the leading term in /?QCD is negative for the six-flavor world rif = 6, becoming positive only if the number of quark flavors exceeds sixteen. As we have already seen, the QED vacuum acts as a dielectric medium with dielectric constant CQED > 1 because spontaneous creation of charged fermion-antifermion pairs results in screening (i.e., vacuum polarization) of electric charge. The dielectric property CQED > 1 means that the QED vacuum has magnetic susceptibility HQED < 1? &nd thus is a diamagnetic medium. The QCD vacuum is the recipient of similar effects from virtual quark-antiquark pairs, but these are overwhelmed by contributions from virtual gluons. As a result, the QCD vacuum is a paramagnetic medium (//QCD > 1) and antiscreens (CQCD < 1) color charge [Hu 81]. The effect of asymptotic freedom can be displayed most clearly by performing a renormalization group (RG) analysis on the IP I amplitudes of the theory. A connected* renormalized Green's function is defined in coordinate space as G^n»\{x})

= (0\T (f{xx)...

Ar{xn)) |0)conn

(2.58)

where the numbers of quark and gluon fields are rip, TIB respectively, and for convenience we suppress color and Lorentz indices. We employ the * All the fields participating in a connected Green's function are affected by interactions; in a disconnected Green's function, one or more of the field quanta propagate freely.

48

/ / Interactions of the Standard Model

same symbol G^nF'UB^ for the momentum Green's function (2TT)464(PI

+..

i (2.59) where n = np + % . The \PI amplitudes r(" FiTlB) are obtained by removing the external-leg propagators from G^/jr' , (2.60) where unprimed (primed) momenta represent initial (final) states. The relations of Eq. (2.33) imply for any renormalization scheme, which we need not specify yet, that G

Z ~Z2 * _CT° ' (2.61) D — Z3 Do; S = Z2 So , where the zero subscript denotes unrenormalized quantities. Prom this, we have

and the combination of terms ^

(2-63)

is therefore independent of the renormalization scale //#. Let us now ascertain the behavior of r( n F ' n B ) in the deep Euclidean kinematic limit where all momenta {p} are both spacelike (in order to avoid singularities) and very large compared to any other mass scale in the theory. To keep the situation as simple as possible, we omit the dependence of T^nFTlB^ on both the quark-mass m(/x^) and gauge £(/J>R) parameters. Then from Eq. (2.59) we find in response to a scale transformation p —> Xp that

This behavior is almost that of a homogeneous function occurring in a scale invariant theory. Canonical dimensions of the fields appear in the exponent of the scaling factor along with an additive factor of four arising from the four-momentum delta function in Eq. (2.59). However in G^nF'nB\ there is also an implicit dependence on A due to the presence of the renormalization scale /i#. The corresponding scaling property of

II-2 Quantum Chromodynamics

49

the IP I amplitude is found from Eqs. (2.63), (2.64) to be (2.65)

or

(2.66) B

( ' This functional relationship can be converted to a differential RG equation by taking the A-derivative of both sides and then setting A = 1,

(2.67)

where ^

In Zl212 ,

1B

= / i * / - In Z\12

(2.68)

are called the anomalous dimensions of the respective fields and /3QCD is as in Eq. (2.57). Let us now see how to obtain leading order estimates for the anomalous dimensions and the beta function. To this order, the result for /?QCD is both gauge and renormalization scheme-independent. To start, we can use the result of Eq. (2.55) to determine 7^, ' < 2 - 69) and analogously for 75. To solve the RG equation, we introduce the scaling variable t = In A and the running coupling constant g3 (£), ^f

,

93(0)=93-

(2-70)

Then it is straightforward to verify that the solution to Eq. (2.67) is where

V(t) = exp (- jT dtf [nBlB (gs(t')) + nFlF {g^t1))} )

(2.72)

is the anomalous dimension factor. The scaling behavior of the IP I amplitude is seen to have field dimensions with anomalous contributions in addition to the canonical values.

50

77 Interactions of the Standard Model

Despite naive expectations, the interaction strength at the scaled momentum is not #3, but rather the running coupling constant ~g% whose magnitude decreases as the momentum is increased. Employing the lowest order contribution Eq. (2.57) for /3QCD> w e c a n integrate Eq. (2.70) over the interval t\ < t < t^ to obtain - 2 (11 - 2n//3) (t2 - *I)/16TT2 ,

(2.73)

where nj is the number of quark flavors having mass less t h a n yft^. It is conventional t o express this relation in a somewhat different form. Defining a scale A at which ~g% diverges and letting as(q2) = g^(2 we have ( 2^

12?r

(33 - 2nf) 6(153 - 19n/) In (ln(g2/A2)) 1(33 - 2n/) 2

where rif is the number of quark flavors with mass less than y q2, and for completeness we have included the next-to-leading term. If as{q2) continues to grow as q2 is lowered, any perturbative representation of /3QCD ultimately becomes a poor approximation, and we can no longer integrate Eq. (2.70) with confidence. Although unproven, a popular working hypothesis is that the QCD coupling indeed continues to grow as the energy is lowered, leading to the phenomenon of quark confinement. In QED, the free parameter a(q2 ~ 0) ~ 1/137 is quite small and expansions in powers of a converge rapidly. However QCD behaves differently. In particular it is clear from Eq. (2.74) that as is not really a free parameter, but is instead inexorably related to some mass scale, e.g., A. This phenomenon, called dimensional transmutation, means that an energy such as A can effectively serve to replace the dimensionless quantity as in the formulae of QCD. Specifying QCD operationally requires not only a lagrangian but also a value for A. For example, QCD perturbation theory is useful only if large mass scales M2 ((A/M) 2 << 1) are probed. Because the complexity of low energy QCD has thus far prevented direct analytic solution of the theory, there have been substantial efforts to develop alternative approaches. These include attempts to solve QCD numerically (lattice-gauge theory), phenomenological study of various theoretical constructs (potential, bag, Skyrme models), exploitation of the invariances contained in £QCD (notably chiral and flavor symmetries), and consideration of the infinite color limit iVc —• oc as a first approximation to QCD (N~* expansion). The first of these topics is beyond the scope of this book [Cr 83], but the others will form the basis for much of our discussion.

II-3 Electroweak interactions

51

Table II—2. Determinations of as Experiment

as (34 GeV)

0.14 ±0.02 0.127 ±0.006 DILS T decay 0.123 ±0.009 e+e~ —> jets 0.135 ±0.015 0.120 ±0.016 7-7 Re+e~

Ag^(MeV)

A^(MeV)

370 ± 350 220 220 ± 60 180 ± 80 330 ± 200 175 ± 125

240 ± 230 140 140 ± 40 120 ± 50 215 ± 130 115 ± 80

Attempts to infer as(q2) from experimental data are typically carried out under kinematic conditions for which a perturbative analysis of QCD presumably makes sense. Systems commonly used for this purpose include deep inelastic lepton scattering (DILS) structure functions, T decay, e+e~ scattering data (both total hadronic cross section (oc R) and jet structure), and photon (77) structure functions. Suppose, as is often the case, a given process is computed to leading and next-to-leading order in QCD perturbation theory and regularized in the MS scheme. If such a theoretical expression is then used to fit the data with a q2 characteristic of the given process employed, an expression such as Eq. (2.74) can be used to infer a value of A and as(q2) can be evolved to different q2. Since this operation depends on both the regularization procedure and the number of quark flavors rif used in Eq. (2.74), a notation like A—^ would be precise. Unfortunately there is no uniformity in the rate of convergence of QCD perturbation theory from process to process. Thus determinations of as(q2) are affected by both theoretical and experimental uncertainties, and a scatter of quoted values results. We display in Table II—2 a collection of values inferred from data collected at moderate energies [Al 89]. Additional determinations from higher energy LEP data have more recently become available [He 92], [ 0.115 ± 0.008

= { 0.141 ±0.017

(jet structure)

(rrViT)

•

(2J5)

Collectively, the LEP data imply as = 0.120 ± 0.007, which corresponds To summarize, determinations of as(q2) are qualitatively in accord consistent with the q2 dependence of Eq. (2.74). Taken over the full range of available data, values in the range 0.1 < A(GeV) < 0.4 are not uncommon.

52

77 Interactions of the Standard Model II—3 Electroweak interactions

The Weinberg-Salam-Glashow model [Gl 61, We 67, Sa 69] is a gauge theory of the electroweak interactions whose input fermionic degrees of freedom are massless spin one-half chiral particles. It has the group structure SU{2)L x C/(l), where the SU(2)ii U(l) represent weak isospin and weak hypercharge respectively. The subscript 'L' on SU(2)L indicates that among fermions, only left-handed states transform nontrivially under weak isospin.

Weak isospin and weak hypercharge assignments First we shall discuss how the fermionic weak isospin (Tw, Tw%) and weak hypercharge (Yw) quantum numbers are assigned. The fermion generations are taken to obey a 'template' pattern - we assume that each succeeding generation differs from the first only in mass. Thus, it will suffice to consider just the lightest fermions for the remainder of this section. The first generation electroweak assignments are displayed in Table II-3. For weak isospin, experience gained from charged weak current interactions such as nuclear beta decay dictates that left-handed fermions belong to weak isodoublets while right-handed fermions be placed in weak isosinglets, as in e

leptons :

£L = (

quarks:

qL = I : I

j

eR u R dR

.

Observe that by assumption there is no right-handed neutrino. Each of the degrees of freedom displayed above must be assigned a weak hyperTable II-3. SU(2)L x assignments Particle Tw ue eL eR UL

dL UR

dR

1/2 1/2 0 1/2 1/2 0 0

Tw3

Yw

1/2 -1 -1/2 -1 -2 0 1/2 1/3 -1/2 1/3 4/3 0 0 -2/3

II-3 Electroweak interactions

53

charge. There are a priori five in all,*

Y(qL) = Yq, Y(uR) = Yu, Y{dR) = Yd, Y(eR) = Ye.

(3.2) Shortly, we shall see that the {Yi} are related to the fermion electric charges. One might therefore attempt to proceed by using the latter to determine the former. However, given the weak isospin assignments already made, it turns out to be both possible and worthwhile to use the theoretical apparatus of the Weinberg-Salam model to predict the {Yi}. Several constraints can be obtained by employing the anomaly cancelation conditions alluded to previously in Sect. 1-2 and to be discussed in detail in Sect. Ill—3 (see especially Eq. (III-3.60b) and subsequent discussion). In particular, the conditions TrF 3 2 F w = 0 , 2 = 0, = 0 ,

(3.3a) (3.3b) (3.3c)

which arise from the cancelation requirement, imply respectively the relationships 2Yq = Yu + Yd ,

6Yq3-3Yu3 = 3Yd3-2Ye3 + Ye3 ,

(3.4a)

(3.4c)

where in Eq. (3.3a), F% is the third generator of the octet of color charges, and "Tr' represents a sum over fermions. To obtain further constraints, we note the linear relation between the electric charge Q and the SU(2)L X U(l)y quantum numbers TW3 and Yw, aQ = TwS + bYw ,

(3.5)

where a, b are constants. We can use the freedom in assigning the scale of the electric charge Q to choose a = 1. Ultimately, of course, the lefthanded and right-handed components of the charged chiral fermions must unite to form the physical states themselves. Consistency demands that the electric charges of the chiral components of each such charged fermion * The reason that weak hypercharge engenders so many free parameters in contrast to weak isospin lies in the difference between an abelian gauge structure (like weak hypercharge) and one which is nonabelian (like weak isospin). Thus all doublets have the same weak isospin properties irrespective of their other properties, analogous to flavor independence in QCD. For the abelian group of weak hypercharge, the group structure by itself provides no guidelines for assigning the weak hypercharge quantum number. Like the electric charge, it is a priori an arbitrary quantity.

54

77 Interactions of the Standard Model

be the same, whatever value that charge might have. From this, we learn Yq Yu

Yq Yd+ Ye Ye + (3 bj - ~2b' - 2b' ~ 2b • Then, for example, insertion of the above into Eqs. (3.4b), (3.4c) yields

'+ ^ 1 =0 •

(3-7)

Thus all products bY{ (c/. Eq. (3.5)) are determined, resulting in the observed electric charge values (c/. Table I.I). The above analysis is instructive in several respects. First, it reveals that once the weak isospin is chosen as in Eqs. (3.1), (3.2) and all possible chiral anomalies are arranged to cancel, one obtains the observed quantization of electric charge. Also, we see that any attempt to determine weak hypercharge values from the known fermion electric charges is affected by an arbitrariness associated with the value of 'b\ This accounts for the variety of conventions seen in the literature. For definiteness, we have taken b = 1/2 in Table II—3, so that .

(3.8)

SU(2)ixU(l)y gauge-invariant lagrangian Having assigned quantum numbers, we turn next to the electroweak interactions. The Weinberg-Salam lagrangian divides naturally into three additive parts, gauge (G), fermion (F), and Higgs (J5T), Avs = CG + CF + CB. (3.9) Throughout this section we shall concentrate on establishing the general form of the electroweak sector, referring at times to only a few tree-level amplitudes. We shall return in Chap. V to the subject of electromagnetic radiative corrections, and present the electroweak Feynman rules along with various radiative corrections in Chap. XVI. The gauge boson fields which couple to the weak isospin and weak hypercharge are respectively W^ = (W^, Wj*, Wj*) and By,. These contribute to the purely gauge part of the lagrange density as

\

J

\

,

(3.10)

where F^v (i = 1,2,3) is the SU(2) field strength,

Flv = dliWl-duWJl-g2eiikWiW* and B^, is the U(l) field strength, B^ = d^Bv - d^

.

,

(3.11) (3.12)

IIS

Electroweak interactions

55

The fermionic sector of the lagrange density includes both the lefthanded and right-handed chiralities. Summing over left-handed weak isodoublets ipL and right-handed weak isosinglets ipn, we have

Since right-handed chiral fermions do not couple to weak isospin, their covariant derivative has the simple form D^R

= (d^i^-YwB^R

.

(3.14)

This expression serves to define the U(l) coupling g\. Its normalization is dictated by our convention for weak hypercharge Yw. The corresponding covariant derivative for the SU(2)L doublet ipL is )

L,

(3.15)

given in terms of the SU{2) gauge coupling constant gi and the 2 x 2 matrices I, r . We shall not display the quark color degree of freedom in this section for reasons of notational simplicity. However, it is understood that all situations in which quark internal degrees of freedom are summed over, as in Eq. (3.13), must include a color sum. Similarly, relations like Eq. (3.14) or Eq. (3.15) hold for each distinct internal color state when applied to quark fields. The above equations define a mathematically consistent gauge theory of weak isospin and weak hypercharge. However, it is not a physically acceptable electroweak theory of Nature because the fermions and gauge bosons are massless. A Higgs sector must be added to the above lagrangians to arrive at the full Weinberg-Salam model. Thus we introduce into the theory a complex doublet

(£)

(3-16)

of spin-zero Higgs fields with electric charge assignments as indicated. The quanta of these fields then each carry one unit of weak hypercharge. The Higgs lagrange density Cn is the sum of two kinds of terms, £HG and £HF> which contain the Higgs-gauge and Higgs-fermion couplings respectively. The former is written as (3.17) where

56

/ / Interactions of the Standard Model

and V is the Higgs self-interaction, (3 1 9 ) 2

The parameters fi and A are positive but otherwise arbitrary. For simplicity, we write the Higgs-fermion interaction in this section for just the first generation of fermions. Denoting the left-handed quark and lepton doublets respectively as qi and ^£, we have £HF = -ML^R

~ fdqL^dR - feh$eR + h.c. ,

(3.20)

where the coupling constants fui /^, and fe are arbitrary and we employ the charge conjugate to 3>, $ = 2T2$* .

(3.21)

There is no term in Eq. (3.20) containing a right-handed neutrino because we assume such particles do not exist. In a sense the Higgs potential V and Higgs-fermion coupling £HF lie outside our guiding principle of gauge invariance because neither contains a gauge field. However, there is no principle which forbids such contributions, and their presence is phenomenologically required. Moreover, note that each is written in SU(2)L x [/(I) invariant form. Spontaneous symmetry breaking Mass generation for fermions and gauge bosons proceeds by means of spontaneous breaking of the SU(2)L X [/(I) symmetry. To begin, we obtain the ground state Higgs configuration by minimizing the potential V to give $ ( - / i 2 + 2A$ f $) = 0 .

(3.22)

We interpret this ground state relation in terms of vacuum expectation values, denoted by a zero subscript. Eq. (3.22) has two solutions, the trivial solution ($)o = 0 and the nontrivial solution, (3.23) with

v= ][^ •

(3.24)

Let us consider the latter alternative. A nontrivial vacuum Higgs configuration which obeys the constraint Eq. (3.23), respects conservation of electric charge, and describes the spontaneous symmetry breaking of the original SU(2)L x [7(1) symmetry is

°)

.

(3.25)

II-3 Electroweak interactions

57

In one interpretation, it is the order parameter for the Weinberg-Salam model, playing a role analogous to the magnetization in a ferromagnet. Group theoretically, it is seen to transform as a component of a weak isodoublet. The energy scale, v, of the effect is not predicted by the model and must be inferred from experiment. The fermion and gauge boson masses are determined by employing Eq. (3.25) for the Higgs field everywhere in the lagrange density CH. We first define chargedfieldsWjjf,

j

\

.

(3.26)

corresponding to the gauge bosons W±. By substitution, we find for the mass contribution to the lagrange density

= -j=(fuuu + fddd + feee)+(^f^ -9l92\ ' " ' " N 9\ )

'

(3 27)

'

The fermion masses are given by v ma = —f=fa {OL = u, d, e, ...) . (3.28) V2 Although the theory can accommodate fermions of any mass, it does not predict the mass values. Instead, the measured fermion masses are used to fix the arbitrary Higgs-fermion couplings. The charged VF-boson masses can be read off directly from Eq. (3.27), Mw = ^g2

,

(3.29)

but the symmetry breaking induces the neutral gauge bosons to undergo mixing. Their mass matrix is not diagonal in the basis of VF3, B states. Diagonalization occurs in the basis Z^ = cos0w Wl - sin0w B^ , A^ = sin 0W W* + cos 9W B^ ,

where the weak mixing angle (or Weinberg angle) 6W is defined by tan0 w = ^

.

(3.31)

92

The neutral gauge boson masses are found to be (3.32)

58

/ / Interactions of the Standard Model

and the fields A^ and Z^ correspond to the massless photon and massive Z°-boson respectively. Observe that the W±-to-Z° mass ratio is fixed by .

(3.33)

Electroweak currents Now that we have determined the mass spectrum of the theory in terms of the input parameters, we must next study the various gauge-fermion interactions. The traditional description of electromagnetic and low energy charged weak interactions of spin one-half particles is expressed as /4 /eh

fr> QA\

where Jem is the electromagnetic current

JL = -e^e + | « y « - idVd + • • • ,

(3.35)

J^h is the charged weak current (ignoring quark mixing) J

ch = ^ 7 ^ ( 1 + 75)e + 5 ^ ( 1 + 75)d + • • • ,

(3.36)

and GF ^ 1.166 x 10"5 GeV"2 is the Fermi constant (cf. Sect. V-2). Alternatively, we can use Eqs. (3.13)-(3.15) to obtain the charged and neutral interactions in the SU(2)L X U(1)Y description, - gi-D^yJem — ^

w3J

,

[6.6()

where J^ 3 is the third component of the weak isospin current, "T

'

,

(3.38)

summed over all left-handed fermion weak isodoublets. Substituting for B^ and W^ in Eq. (3.37) in terms of A^ and Z^ yields ^ J ^ + £ntl-wk ,

(3.39)

77-5 Electroweak interactions

59

where J ^ = 2 J^ 1 + i 2 is given in Eq. (3.36) and the neutral weak interaction £nti_wk for fermion / is*

= T(f)

_

— ^w3

2

. 2Q

Z S m

Specifically, we have

.(u,c,t) _ I _ f

gin2

OU)

^w V e l

g

(/) ,

5a

= T (/)

—

(tx,c,*) _

" " 2

w3

I

1 2 v

i

1 3

~~

(3'41)

2 '

9v

Observe the structure of the neutral weak couplings g^l. If 0W were to vanish, neutral weak interactions would be given strictly in terms of TW3, the third component of weak isospin. However in the real world, phenomena like low energy neutrino interactions, Mw/Mz, deep inelastic lepton scattering data, etc. all depend on the value of 0W. In addition, we note that because sin2 #w ~ 0.23, the leptonic vector coupling constants g{e,H,T) a r e substantially suppressed relative to the axial-vector couplings. Comparison of Eq. (3.34) with Eq. (3.39) yields e = (7icos#w = <72sin#w .

(3.42)

The Fermi interaction of Eq. (3.34) corresponds in the Weinberg-Salam model to a second-order interaction mediated by VF-exchange and evaluated in the limit of small momentum transfer (1 » q 2

Together these relations provide a tree-level expression for the W-boson mass, *,2 1 ML = —o

/ 37.281 G e V \ 2 .

™ •=— ~

One should be careful not to confuse Eq. (3.40) with the alternate form

Vf = -J^

W

2 sin ^ w cos ^ w which also appears in the literature.

el

£w3 2 sin ^ w cos 0^

(3.44)

60

// Interactions of the Standard Model

Also, Eqs. (3.29), (3.43) imply v = 2" 1/4G~1/2 ~ 246.221(2) GeV .

(3.45)

It is the quantity v which sets the scale of spontaneous symmetry breaking in the SU(2)L X [ / ( 1 ) theory, and all masses in the Standard Model are proportional to it, although with widely differing coefficients. We shall resume in Chaps. XV, XVI discussion of a number of topics introduced in this section, among them the Higgs scalar, the W± and Z° gauge bosons, and phenomenology of the neutral weak current. Also included will be a description of quantization procedures for the electroweak sector, including the issue of radiative corrections. First however, in the intervening chapters we shall encounter a number of applications involving light fermions undergoing electroweak interactions at very modest energies and momentum transfers. For these it will suffice to work with just tree-level W± and/or Z° exchange, and to consider only photonic or gluonic radiative corrections. We shall also neglect the gauge-dependent longitudinal polarization contributions to the gauge boson propagators (analogous to the q^qu term in the photon propagator in Eq. (1.21)), as well as effects of the Higgs degrees of freedom. For photon propagators, the q^q" terms do not contribute to physical amplitudes because of current conservation. Although current conservation is generally not present for the weak interactions, both the q^qv propagator terms and Higgs contributions are suppressed by powers of (rrif/Mw)2 for an external fermion of mass rrif. II-4 Fermion mixing In our discussion of the Weinberg-Salam model, we limited the number of fermion generations to one. We now lift that restriction and consider the implication of having n generations. Although the existing experimental situation supports the value n = 3, we shall take n arbitrary in our initial analysis. Diagonalization of mass matrices To begin, it is necessary to generalize the Higgs-fermion lagrangian of Eq. (3.20) to

-£HF = ff qtj*ip + ff qitfidju + ff ll^h

+ h'c- '

II~4 Fermion mixing

61

where we employ the summation convention a, (3 = 1,..., n , and adopt the notation d' = ( d ' , s ' , b ' ,

...),

vo

The primes signify that the states which appear in the original gaugeinvariant lagrangian are generally not the mass eigenstates. That is, there is no reason why the n x n generational coupling matrices fw, f^, fe should be diagonal. Following spontaneous symmetry breaking, we obtain the generally nondiagonal n x n mass matrices m j , mj, m / from the analog of Eq. (3.28), mQ' = — f

Q

(a = u, d, e)

(4.3)

Although not diagonal in the gauge basis, these matrices can be brought to diagonal form in the mass basis. The transformation from gauge eigenstates to mass eigenstates is accomplished by means of the steps —A\ mass = u£ mj^i? + d£ nijd£ + e£ mje£

+ h.c. ,

= UL mu UR + di md dji + ex, m e en -\- h.c. = u mu u + d m.d d + e m e e . The n x n unitary matrices *S£ R (a = u, d, e) relate the basis states, U

'L

— ^L UL 5

d/ = S^ di ,

e/ = S^ e^ ,

d ^i? = S1RUR i R = s i rfi? » and induce the biunitary diagonalizations

e

i? = s /? eR 5

= SUL mu

, (4.6)

thus yielding the diagonal quark mass matrices / mu 0 0 ... \ / rrid 0 0

\

mc 0

0 mt

... I ... '

0 0

m d =

\

0 ms 0

0 0 mi,

(4.7a)

62

/ / Interactions of the Standard Model

and the diagonal lepton mass matrix

/me me =

0 0

0

0

ra» 0

0 mT

\ : : : Although the Weinberg-Salam model is first written down in terms of the gauge basis states, actual calculations which confront theory with experiment are performed using the mass basis states. We must then transform from one to the other. This turns out to have no effect on the structure of the electromagnetic and neutral weak currents. One simply omits writing the primes which would otherwise appear. The reason is that (aside from mass) each generation is a replica of the others, and products of the unitary transformation matrices always give rise to the unit matrix in flavor space. Thus, at the lagrangian level, there are no flavor-changing neutral currents in the theory. As an example of this, consider the leptonic contribution to the electromagnetic current, (4.8) where we sum over family index a = 1 , . . . , n and invoke the unitarity of matrices S ^ . Note that there is no difficulty in passing the SeL R through 7^ because the former matrices act in flavor space whereas the latter matrix acts in spin space. The charged weak current of leptons behaves analogously. Transformation from gauge basis to mass basis leaves the form of the charged current essentially unchanged. However, the reasoning differs from that used above, and involves the neutrino sector in an essential way. Let us denote neutrino states (u^) in the gauge basis and (V®) in the mass basis,

H = ("e', < , "T')L ,

*i° =iyl, i/J, u% ,

(4.9)

with a transformation S£ relating the two, "£, a = (SLUVIP

(a,0 = l,...,n).

(4.10)

Upon writing down the charged leptonic current we encounter J&(lept) = 2%>"e L ' i a = 2(P £ °SL t ) a y(Sie) a ,

(4.11)

which we re-express as J^(lept) - 2vL^ea

.

(4.12)

II~4 Fermion mixing

63

That is, we employ a third neutrino basis (Z?L), vL,a = (S^SDapulp

(a, (3 = 1,..., n) ,

(4.13)

which serves to define the set of physical neutrino states. We are allowed mathematically to perform this transformation on the original choice of mass eigenstates provided the neutrinos are assumed to be mass degenerate. Existing data is consistent with all neutrinos having zero mass. Quark mixing Thus far, the distinction between gauge basis states and mass eigenstates has been seen to have no apparent effect. However, mixing between generations does manifest itself in the system of quark charged weak currents. By convention, the mixing is assigned to the Q = —1/3 quarks by )X

<

,

(4.14)

where < a = Va(,dLt(,

(a, (3 = 1,..., n) ,

(4.15)

and

V = S£fS£ .

(4.16)

Thus the Q — —1/3 quark states participating in transitions of the charged weak current are linear combinations of mass eigenstates. The quark-mixing matrix V, being the product of two unitary matrices, is itself unitary. The Standard Model does not predict the content of V. Rather, its matrix elements must be phenomenologically extracted from data. For the two generation case, V is called the Cabibbo matrix [Ca 63]. For three generations, it is called the Kobayashi-Maskawa matrix [KoM 73] (or sometimes simply the weak mixing matrix) and is often denoted by the abbreviations KM or CKM. We shall analyze properties of such mixing matrices for the remainder of this section. An nxn unitary matrix is characterized by n2 real-valued parameters. Of these, n(n — l)/2 are angles and n(n + l)/2 are phases. Not all the phases have physical significance, because 2n — 1 of them can be removed by quark rephasing. The effect of quark rephasing uL,a -> eiB«u L,* ,

dL,a - • e^d^a

(a = 1 , . . . , n)

(4.17)

on an element of the mixing matrix is VcH-^Vape^-W

(<*,/?= l , . . . , n ) .

(4.18)

Since an overall common rephasing does not affect V, only the 2n — 1 remaining transformations of the type in Eq. (4.17) are effective in

64

II Interactions of the Standard Model

removing complex phases. This leaves V with (n — l)(n — 2)/2 such phases. One must be careful to transform the left-chirality and rightchirality fields of a given flavor in like manner to keep masses real. If so, all terms in the lagrangian other than V are unaffected by this procedure. For two generations, there are no complex phases. The only parameter is commonly taken to be the Cabibbo angle OQ and we write

cos0cJ * A common notation for the n = 2 mixed states is

Within the two generation approximation, weak interaction decay data imply the numerical value, sin 9c — 0.22. The three generation case involves the 3 x 3 matrix /Kd V = \Vcd

\vtd

Ks Vub\ Vcs Vch ,

(4.21)

vts vthj

written here in a form which emphasizes the physical significance of each matrix element. Current experimental data place the matrix element magnitudes in the 90% confidence intervals [RPP 90] ' 0.9738 - 0.9750 0.218 - 0.224 0.001 - 0.007 \ 0.218 - 0.224 0.9734 - 0.9752 0.030 - 0.058 0.003 - 0.019 0.029 - 0.058 0.9983 - 0.9996 /

.

(4.22)

The n = 3 mixing matrix can be expressed in terms of four parameters, of which one is a complex phase. The presence of a complex phase is highly significant because it signals the existence of CP violation in the theory. We shall return to this point shortly. The Kobayashi-Maskawa (or KM) representation employs three mixing angles 9a (a — 1, 2,3) and a complex phase 6. It can be viewed as an Eulerian construction of three rotation matrices and a phase matrix,

where sa = s i n ^ , ca = cos9a (a — 1,2,3). In combined form this becomes /

C\

— S1C3

— S1S3

\

(4.23)

II~4 Fermion mixing

65

By means of quark rephasing, it can be arranged that the KM angles #1,2,3 all he in the first quadrant. In the limit 62 = #3 = 0, KM mixing reduces to Cabibbo mixing with the identification 9\ = —6cAn alternative approach for describing quark mixing, the Wolfenstein parameterization [Wo 83], assumes that the mixing matrix is endowed with a hierarchical structure. That is, one defines A = |Kis| — 0.22, and expands a given matrix element of V in powers of A according to the perceived magnitude of the matrix element. One finds to order A3 (A4) in the real (imaginary) couplings, l-A2/2 V= -A \X3A(l-p-ir)) /

A 1 - A2/2 - irjA2X4 -X2A

X3A(p-iri(l X2A(1 + ir)X2) 1

(4.24) where A, p, 77 are real-valued parameters anticipated to be of order unity. B meson data (cf. Sect. XIV-1) imply the values A = 0.95 ± 0.14 and v/p2 + T72 = 0.45 ±0.14. CP-violation and rephasing-invariants It is clear from the preceding discussion that there is no unique parameterization for three generation quark mixing. Any scheme which is convenient to the situation at hand may be employed as long as it is used consistently and adheres to the underlying principles. There is, however, a somewhat different logical position to adopt, that of working solely with rephasing-invariants. After all, only those functions of V which are invariant under the rephasing operation in Eq. (4.18) can be observable. An obvious set of quadratic invariants are the squared moduli A^j = l^ a/?|2 where a, (3 = 1, 2,3. The unitarity conditions V^V = VV* = I constrain the number of independent squared-moduli to four. They are of course all real-valued. In addition there are quartic functions Aap = V^V^rVXV^, where we suspend the summation convention for repeated indices and, to avoid redundant factors of squared-moduli, take a, /?, 7 (p, a, r) cyclic. These nine quantities are generally complex-valued. A permutation on either index cycle, e.g. a, /?, 7 —>• a, 7, /?, maps a quartic invariant into its complex conjugate. The Aap are not all independent. If we multiply the unitarity relation V

*rVs*

(v ± r)

by V*V7T and recall the cyclic nature of the indices, we obtain A(4) - -A (4) * - W \2\V I2

(4.25)

66

/ / Interactions of the Standard Model

From this line of reasoning, we can determine that there is just one independent quartic term and that all nine have the same imaginary part. There are yet higher order invariants, but they are all expressible in terms of the quadratic and quartic functions. An important consequence of our discussion thus far is that a unique measure of CP violations for three generations is given by the rephasinginvariant ImA^i . It is independent of its indices (which we therefore suppress) and is given in the KM and Wolfenstein parameterizations by Im A' 4 ' - s21c1s2c2sscss6

= X6A2rj + O(XS) .

(4.27)

These combinations of mixing parameters will always appear in calculations of CP violating phenomena. To have nonzero CP violating effects, the KM angles must avoid the values #1,2,3 = 0, TT/2, and 6 = 0, TT. The CP violating invariant ImA^4^ achieves its maximum value for c\ — 1/^/3, c2 = c3 = l/>/2, s6 = 1 at which it equals 1/6A/3. This set of circumstances is very unlike the real world where current estimates place Im A ^ - I O " 4 . The consideration of rephasing-invariants need not involve just the mixing matrix V, but can also be applied to the Q = 2/3, —1/3 nondiagonal mass matrices m!u, m^ themselves. In particular, the determinant of their commutator is found to provide an invariant measure of CP violations [Ja 85]. If, for simplicity, we work in a basis where m u ', m^ are hermitian, it can be shown that S^ = Su/ = S M . Thus we have m'd] = Su [m,, VmdVt] Swt - SUV [ V W V , md] V+S^ , (4.28) from which it follows that det [m^rn^] = det [mu, VmdV^

= det [V^niuV.md]

.

(4.29)

The two commutators on the right hand sides of this relation are skewhermitian and each of their matrix elements is multiplied by a Q — 2 / 3 , - 1 / 3 quark mass difference respectively. The determinant is thus proportional to the product of all Q = 2/3, —1/3 quark mass differences, and explicit evaluation reveals

det [m^,my = 2i ImA^ JJ(m M?a - mu^)(md^a - md,p) -

(4.30)

This provides a more extensive list of necessary conditions for CP violations to be present. Not only are the mixing angles constrained as discussed above, but also the quark masses within a given charge sector must not exhibit degeneracies. Our discussion of the n = 2, 3 generation cases suggests how larger systems n — 4, . . . can be addressed, although the number of parameters becomes formidable, e.g. four generations require six mixing angles

Problems

67

and three complex phases. However, existing data indicates the existence of just three fermion generations - measurements from Z°-decay fail to see additional neutrinos, and there is no evidence from e + e" or pp collisions for additional quarks or leptons lighter than roughly 50 GeV. Moreover, if there were more than three quark generations the full quarkmixing matrix would be unitary, but one would expect to see violations of unitarity in any submatrix. Yet to the present level of sensitivity, the 3 x 3 KM mixing matrix obeys the unitarity constraint. The most accurate data occurs in the (V^y)n = 1 sector, where using the values (cf. Chaps. VIII, XII) Kd = 0.9737 ± 0.0010, Ks = 0.220 ± 0.003, and |Kb| < 0.01, we find )n = |Kd| 2 + |Ks| 2 + |Kb| 2 - 0.9965 ± 0.0032 .

(4.31)

Problems 1) SU(3) a) Starting from the general form A^A^ = A6ij (a = 1 , . . . , 8 is summed), determine A, S, C by using the trace relations of Eqs. (2.5a-b), etc. b) Determine TrAaA6Ac. c) Determine e^ewA^Aj^A^A^.. d) Consider the 8 x 8 matrices (Fa)&c = —if abc, where the {fabc} are SU(3) structure constants (a,6,c = 1,...,8). Show that these matrices (the regular or adjoint representation) obey the Lie algebra of 577(3), and determine Tr FaFb. 2) Gauge invariance and the QCD interaction vertices a) Define constants / , g such that the covariant derivative of quark q is D^q = (d^ + ijA^)q and the QCD gauge transformations are A^ —• UA^U~l + ig^UdyJJ'1 and q —> Uq, where A^ are the gauge fields in matrix form (c/. Eq. (1-4.15)). Show that qJp q is invariant under a gauge transformation only if / = g. b) Define a constant h such that the QCD field strength is F^v — d^Av — dyA^ + ih[A^ Av], Let the gauge transformation for A^ be as in (a). Show that F^v transforms as F^v —» UF^U~l only if h = g. 3) Fermion self-energy in QED and QCD a) Express the fermion QED self-energy, —iS(p), of Fig. II-2(b) as a Feynman integral and use dimensional regularization to verify the forms of Z{2MS), 6m,(Msr> appearing in Eqs. (1.37), (1.38). b) Proceed analogously to determine Z^ for QCD and thus verify Eq. (2.55).

68

/ / Interactions of the Standard Model

4) Gravity as a gauge theory The only force which remains outside of the present Standard Model is gravity. General relativity is also a gauge theory, being invariant under local coordinate transformations. The full theory is too complex for presentation here, but we can study the weak field limit. In general relativity the metric tensor becomes a function of spacetime, with the weak field limit being an expansion around flat space, g^v{x) — g^v + Wv\x), with 1 ^> Wv. Let us consider weak field gravity coupled to a scalar field, with lagrangian C = £grav + ^matter defined as

£matter =

\

where h^p = h^v — g^vh\j2^ all indices are raised and lowered with the flat space metric g^v and GN is the Cavendish constant. a) Show that the action is invariant under the action of an infinitesimal coordinate translation, x^ —• x/fl — x^ + e^{x) (1 ^> e /i(x)), together with a gauge change on (p'(x') =
Ill Symmetries and anomalies

Application of the concept of symmetry leads to some of the most powerful techniques in particle physics. The most familiar example is the use of gauge symmetry to generate the lagrangian of the Standard Model. Symmetry methods are also valuable in extracting and organizing the physical predictions of the Standard Model. Very often when dealing with hadronic physics, perturbation theory is not applicable to the calculation of quantities of physical interest. One turns in these cases to symmetries and approximate symmetries. It is impressive how successful these methods have been. Moreover, even if one could solve the theory exactly, symmetry considerations would still be needed to organize the results and to make them comprehensible. The identification of symmetries and near symmetries has been considered in Chap. I. This chapter is devoted to their further study, both in general and as applied to the Standard Model, with the intent of providing the foundation for later applications. III-l Symmetries of the Standard Model The treatment of symmetry in Sects. 1-4, 1-6 was carried out primarily in a general context. In practice, however, we are most interested in the symmetries relevant to the Standard Model. Let us briefly list these, reserving for some a much more detailed study in later sections. Gauge symmetries: As discussed in Chap. II, these are the SU(3)C x SU(2)L x U(l)y gauge invariances. It is interesting to compare their differing realizations. SU(3)C is unbroken but evidently confined, whereas SU(2)L x U(l)y undergoes spontaneous symmetry breaking, induced by the Higgs fields, leaving an unbroken U(l)em gauge invariance. Fermion number symmetries: There exist global vector symmetries corresponding to both lepton and quark number. These are of the form ^ - e-^Va 69

(LI)

70

777 Symmetries and anomalies

for fields of each chirality. The index a refers to either the set of all leptons or the set of all quarks, and the conserved charges Qa are just the total number of quarks minus antiquarks and the total number of leptons minus antileptons. In addition, if the neutrinos are all massless, there are separate conserved numbers for each type of lepton (e, i/e), (/i, i/^), (r, ur). Conservation of baryon number B is violated due to an anomaly in the electroweak sector, but B — L remains exact. Global vectorial symmetries of QCD: If the quarks were all massless, there would be a very high degree of symmetry associated with QCD. Even if m ^ 0, symmetries are possible if two or more quark masses are equal. Three of the quarks (c, 6, t) are heavy compared to the confinement scale AQCD and widely spaced in mass, so they cannot be accommodated into a global symmetry scheme.* However the ix, d, and s quarks are light enough that their associated symmetries are useful. The best of these is the isospin invariance, which consists of field transformations

where {T1} (i = 1,2,3) are SU(2) Pauli matrices and {#*} are the components of an arbitrary constant vector. Associated with the SU(2) flavor invariance are the three Noether currents Jf=i>i^

•

(1-3)

Isospin symmetry is broken by the up-down mass difference, £mass =

^

\uu + dd)

(uu - dd) ,

(1.4)

and by electromagnetic and weak interactions. Inclusion of the strange quark extends isospin to SU(3) flavor transformations %j)' = exp(-i0 • A)^ ,

(1.5)

where {Aa} (a = 1,2,...,8) are the SU(3) Gell-Mann matrices. The SU(3) flavor symmetry is broken significantly by the strange quark mass, and to a lesser extent by other effects. Predictions of isospin symmetry work at the 1% level, whereas SU(3) predictions hold only to about 30%. It is occasionally convenient to employ a particular SU{2) subgroup of See, however, the discussion of the dynamical heavy-quark symmetries in Sects. XIII-3, XIV-2.

III-l Symmetries of the Standard Model

71

*SC/(3), called U-spin, which corresponds to the transformations .

(1.6)

[/-spin is also a symmetry of the electromagnetic interaction, since its generators commute with the electric charge operator. The [/-spin symmetry is broken by the large d-quark, s-quark mass difference. Approximate chiral symmetries of QCD: The vectorial symmetries are valid if quark masses are equal. If the masses vanish, there are additional chiral symmetries, because in this limit the left-handed and right-handed components of the fields are decoupled (cf. Sect. 1-3), m=0

A 4

;

,

(1.7)

i.e. the left-handed and right-handed fields have separate invariances. For massless up and down chiral quarks, the symmetry operations are ipL -+ exp(-WL • r)ipL = L^L ,

^R -+ exp(-iOR • r)ipR = RipR , (1.8)

where ^L,R are chiral projections of the ij) doublet in Eq. (1.2). These can also be expressed as vector and axial-vector isospin transformations, , (1.9) with Oy = ( 0 £ + 0 R ) / 2 , and 0A — {0L—0 R)/2. This invariance is variously referred to as chiral-#C/(2), SU{2)L x SU{2)R or SU(2)V x SU(2)A. In QCD, it is broken by quark mass terms, J

Anass = -muuu-mddd

= -mu(uLuR+uRUL)-rnd(dLdR+dRdL).

(1.10)

Thus if mu = rrid ^ 0, separate left-handed and right-handed invariances no longer exist, but rather only the vector isospin symmetry. The generalization to three massless quarks defines chiral 5/7(3) (or SU(3)L X SU(3)R) and is a straightforward extension of the above ideas. More discussion of these chiral symmetries will be given in Chaps. VI,VII. Discrete symmetries: Since the Standard Model is a hermitian and Lorentz-invariant local quantum field theory, it is invariant under the combined set of transformations CPT. Both QCD (given the absence of the 0-term) and QED conserve P, C, and T separately. By contrast, the electroweak interactions have maximal violation of P and C in the charged-current sector. If a nonzero phase resides in the quark-mixing matrix, there will exist a breaking of CP, or equivalently of T, invariance. Otherwise the weak interactions are invariant under the product CP. In addition to the above exact or approximate symmetries of the Standard Model, there are some important 'non-symmetries' of QCD. By

72

/ / / Symmetries and anomalies

these we mean invariances of the underlying lagrangian which might naively be expected to appear as symmetries of Nature but which, for a variety of reasons, do not. These include the following. Axial U(l): The QCD lagrangian would have an axial U(l) invariance of the form

if the u, d, s quarks were massless. However, this turns out not to be even an approximately valid symmetry, as it has an anomaly. We shall return to this point in Sect. Ill—3. Scale Transformations'. If quarks were massless, the QCD lagrangian would contain no dimensional parameters. The lagrangian would therefore be invariant under the scale transformations

^X) _> A3/fy(As) ,

A%{x) - \Al{\x)

,

(1.12)

where ip and A^ are respectively the quark and gluon fields. This invariance is also destroyed by anomalies (see Sect. Ill—4). 'Flavor Symmetry': Because the gluon couplings are independent of the quark flavor, one often finds reference in the literature to a flavor symmetry of QCD. Unless the specific application is reducible to one of the above true symmetries, one should not be misled into thinking that such a symmetry exists. For example, flavor symmetry is often used in this context to relate properties of the pseudoscalar mesons 7/(549) and 7/(960) (or analogous particles in other nonets). However the result is rarely a symmetry prediction. Rather, this approach typically pertains to specific assumptions about the way quarks behave, and is dressed up by incorrectly being called a symmetry. In group theoretic language, this may arise by assuming that QCD has a U(3) symmetry rather than just that of SU(3).

Ill—2 Path integrals and symmetries The transition from classical physics to quantum physics is in many ways most transparent in the path integral formalism. In this chapter we use these techniques to provide a quantum description of symmetries, complementing the treatment at the classical level of Sects. 1-4, 1-6. A brief pedagogical introduction to those path integral techniques which are important for the Standard Model is provided in App. A.

III-2 Path integrals and symmetries

73

The generating functional In order to implement a quantum description of currents and current matrix elements, one studies the generating functional, W, of the theory. For a generic field (p, we have W[j] = eiZ^ = J[dtp] expij

d4x (£(, dtp) - j(p) ,

(2.1)

where j(x) is an arbitrary classical source field whose presence allows us to probe the theory by studying its response to the source. The symbol [dip] indicates that at each point of spacetime one integrates over all possible values of the field
6n In

6j(xk)...6j(xp)

(2.2) 3=0

where n is the number of fields in the matrix element. If there is more than one field, i.e. the set {
(2.3)

In this case all matrix elements involving J^ can be obtained by functional derivation with respect to v^, J»{x) = i

6\nW

(2.4)

where the bar in J^ indicates that it is a functional describing matrix elements of the current J^. Specific matrix elements are obtained by further derivatives, as in (2.5) This device allows one to discuss all possible matrix elements of the current J*\ As an example, consider the vector and axial-vector currents of QED. We define

= AdVWH^K^^^

. (2.6)

74

/ / / Symmetries and anomalies

A three-current (connected) matrix element is obtained then as Tfuxpix, y, z)conn = (0 \T (4>(x)jtl'y5ip(x) $(y)-yatp(y) tp(z)^0ip(z)) | 0)

lnW

[ J

^

)

(2.7) where the axial-vector quantity J^ is defined in analogy with Eq. (2.4).

Noether's theorem and path integrals Returning to the general case, let us consider an infinitesimal transformation of a set offields\jp%} (2.8) such that the current under discussion is

If this is a symmetry transformation, one has up to a total derivative, C {tp\ dip') = C (
.

(2.10)

If e(x) is a constant, the lagrangian is invariant under the transformation. This is the statement of the classical symmetry condition. In order to study the consequences of this situation, we rewrite our previous definition of the current matrix elements ]

(2.11)

0Vfi{X)

in integral form by noting

8\nW[v^] = lnW[v^ + 6vfl}-lnW[v^\

= -i fd4xJ^{x)6v^x)

, (2.12)

which is just the inverse of Eq. (2.11). Now choosing the particular form for Svp, 6v^x) =-d^(x)

,

we have 6€ ln W[v»] = ln W[v^ - d^e] - ln W[v^}

= i f dAx J»(x)d^(x) = -i f d4x e(x)dfiJfi(x) .

(2.13)

777-3 The U(l) axial anomaly

75

With this procedure we can isolate a divergence condition for J^. If v^ — d^e] = Wfr;^], then d^J^{x) = 0. To check this, consider

expijd4x

(£fo, d
. (2.15)

If we can change integration variables so that

J[dhpi] =

(2.16)

with $ given by Eq. (2.8), then we obtain W[v^-d^e)

= Jid^expiJ'd*x(£{
and therefore dPJ^x) = 0 .

(2.18)

This change of variables seems reasonable and in most cases is perfectly legitimate. After all, the symbol [d(fi(x)] means that we integrate over all values of the field cpi separately at each point in spacetime. Shifting the origin of integration at point x by a constant, (fi(x) = (f[(x) — e(x)/j, and then integrating over all values of (p[ should amount to the original integration. Given this shift, we have obtained in Eq. (2.18) by Noether's theorem a quantum conservation law involving matrix elements. The expression dfJ/JfI(x) = 0 means that all matrix elements of JM, obtained via further functional derivatives (as in Eq. (2.5)), satisfy a divergenceless condition, i.e. the current J^ is conserved in all matrix elements. It was Fujikawa who first pointed out the consequences if the change of variables, Eq. (2.16), is not a valid operation in a path integral [Fu 79]. Certainly, many procedures involving path integrals need to be examined carefully in order to see if they are well-defined. We shall explicitly study some examples in which the change of variable is nontrivial and can be calculated. In such cases one finds dflJfJ'(x) ^ 0, which implies that the classical symmetry is not a quantum symmetry. In these situations it is said that there exists an anomaly. III-3 The U(l) axial anomaly For massless quarks mu = rrid = ms = 0, the lagrangian of QCD contains an invariance £QCD —> £QCD under the global U(l) axial transformations (3.1) In this limit, which we shall adopt until near the end of this chapter, Noether's theorem can be applied to identify the classically conserved

76

77/ Symmetries and anomalies

axial current,

4°]=uWu

+ dWd + sw5s,

d*40J=0 ,

(3.2)

where the superscript on J^J denotes an SU(3) singlet current. We shall see that this is not an approximate symmetry of the full quantum theory in that the current divergence has an anomaly. This can be demonstrated in various ways. For a direct 'hands-on' demonstration, the early discussion [Ad 69, BeJ 67, Ad 70] of Adler and of Bell and Jackiw, which we recount below, has still not been improved upon. However for a deeper understanding, Fujikawa's path integral treatment [Fu 79], also described below, seems to us to be the most illuminating. The effect of an anomaly is simply stated, although one must go through some subtle calculations to be convinced that the effect is inescapable. An anomaly is said to occur when a symmetry of the classical action is not a true symmetry of the full quantum theory. The Noether current is no longer divergenceless, but receives a contribution arising from quantum corrections. It is this contribution which is often loosely referred to as the anomaly. The Ward identities relating matrix elements no longer hold, but rather, are replaced by a set of anomalous Ward identities which take into account the correct current divergence. There are two applications of the axial anomaly which have proved to be of importance to the Standard Model. One is in connection with the SU(3) singlet axial current described above. Here the anomaly will end up telling us that the current is not conserved in the chiral limit, but rather that

d"JjS = ^Fa^F^

{F^ = d^FS0)

.

(3.3)

This will serve to keep the ninth pseudoscalar meson, the 7/, from being a pseudo-Goldstone boson. The other application is in the decay TT° —> 77, which is historically the process wherein the anomaly was discovered. The quantity of interest here is an isovector axial current J^ which transforms as the third component of an SU(3) flavor octet,

(3.4)

J

Without the anomaly, one would expect that the current J§ } would be conserved in the chiral 5/7(2) limit even in the presence of electromagnetism. This follows from the apparently correct procedure

= u\(f+ iQ4) 75 + 75 (fi +

} (3.5)

III-3 The U(l) axial anomaly

77

However explicit calculation shows that the current has an anomaly, such that

43 J

^

,

(3.6)

where F^v is the electromagnetic field strength. This will be important in predicting the TT° —> 77 rate and serves as a test for the number of quark colors.

Diagrammatic analysis To review the work of Adler and of Bell and Jackiw, we first consider the Ward identities for the coupling of the C/(l) axial current to two gluons. We define

, q)=ij

d4x d4y e*fcV™ (o \T (4°J(z) J°(y) J>(0))| o)

(3.7)

where J^ is a flavor-singlet (color-octet) vector current coupled to gluons

l = £

«7«7a y<7

•

(3-8)

q=u,d,s

It is important to understand that the SU(3) matrices pertain here to the color degree of freedom and should not be confused with analogous matrices which operate in flavor space. The amplitude T°^Qis related to the vacuum-to-digluon matrix element by a

{\u q) G\X2, -k - q)

, q) .

(3.9)

There are two Ward identities, representing the conservation of axial and vector currents. The vector Ward identity, corresponding to color current conservation, is

<7 a O*;,
(3.10)

The axial Ward identity is derived in a similar fashion using the assumed conservation of the 17(1) axial current in the massless limit,

Jx) = 0 ,

(3.11)

to yield b

,q)=0

.

(3.12)

In order to reveal the anomalous behavior of this coupling, we calculate the vertex in lowest order perturbation theory via the triangle diagram of Fig. Ill—1. With the momenta as labeled in the figure, this produces

III Symmetries and anomalies

78 the amplitude

Aa

\ bl

(3.13)

where the prefactor of 3 arises from the three massless quarks, each of which contributes equally. Observe that these integrals are linearly divergent, and so may not be well-defined. In particular, there exists an ambiguity corresponding to the different possible ways to label the loop momentum. An example will prove instructive, so we consider the integral

-J

P

[P4

(3-14)

(p-

This is evaluated by transforming to Euclidean space, where po = ip± and p2 — —p\ — p 2 — —p\- In order to perform the integration, one may note that for a general function, F(p), whose four-dimensional integral is linearly divergent (i.e. one with p3F(p) ^ 0, but psF'(p) = p3F"(p) = ... = 0 for p —> oc), one finds by Taylor expanding and using Gauss' theorem that

J dAPE [F(p) - F(p - £)] =

+ ...

dzS»F{p) (3.15)

-k-q

-k-q

Fig. Ill—1 Triangle diagram associated with the axial anomaly.

III-3 The U(l) axial anomaly

79

where dsS^ indicates integration over a three-dimensional surface at p —• oo.* Applying this result to the case at hand, we obtain a surface integral 7

\P

\P ~ £) J

V P (3.16) Note that from euclidean covariance we can replace p^p^ by ^ 7 p 2 / 4 , to yield o

r

J

1

P

J

^2o

(3.17) where the last step uses the surface area of a three-dimensional surface in four-dimensional euclidean space, S4 = 2ir2R3. In the case of T ^ , consider the effect of shifting the integration variable of the first term in Eq. (3.13) from p to p + b\q + b2(—k — q). In order to maintain the Bose symmetry of TJ^Q (i.e. symmetry under the interchange a <-» (3 at the same time as q <-> (—k — q)) we must shift the second integration from p to p+b\(—k — q) + b
(3.18) induced by the shift of the original integration variable jf. This is an indication that there may be trouble in the calculation of this diagram, but it is not yet proof of any violation of the Ward identities. Let us now check the Ward identities. In both cases, use can be made of identities similar to qa = pa — (pa — qa) in order to change the result into a difference of integrals. We find for the vector Ward identity

36ab r d4P

J

* Note that this is just the four-dimensional generalization of the one-dimensional formula

f

dx [f(x + y)- f(x)] = ^

dx \yf(x) + \y2f"{x)

= y[/(oo)-/(-oo)

valid for /(±oo) ^ 0 but /'(±oo) = /"(±oo) = ... = 0.

+ .. .1

III Symmetries and anomalies

80

= -6i6a

J (2*)_

k)2(p-q)2

Y

(3.19)

{p

while for the axial-vector case, d4p

ab

= -*S ea0paj

f

757/?"

k)P(p-q)c

dd4p

1

-^

qfp* {p-q)pp°

(3.20)

It is easy to see that if one could freely shift the integration variable, each expression would separately vanish. However, direct calculation using Eq. (3.14)-Eq. (3.17) yields ) = -Y^ellPpak^

and WT£0(k,q) = g ^ - ^ f c V • (3.21) If, on the other hand, the original integration variable were shifted as in Eq. (3.18) one would obtain S6ab 36ab 8^"

(3.22)

Thus either one of the original Ward identities may be regained by a particular choice of 61 — 62? but both expressions cannot vanish simultaneously. The discussion of the manipulations of the Feynman diagrams should not obscure the main physical fact illustrated above, i.e., despite the claim of Noether's theorem that there are two sets of conserved currents (vector 5(7(3) of color and axial-vector (7(1)), one-loop calculations indicate that only one can in fact be conserved. On physical grounds, we know that in Nature the vector current is conserved, as its charge corresponds to QCD color charge. Thus it must be the axial current which is not conserved. This phenomenon is at first sight quite surprising and it deserves the name 'anomaly' by which it has come to be called. Noether's theorem has misled us, and it is only by direct calculation of the quantum corrections that the true symmetry structure of the theory has been "ex-

1/7-3 The U(l) axial anomaly

81

posed. Note that the situation is not the same as spontaneous symmetry breaking, where the symmetry is hidden by dynamical effects. There the currents remain conserved, as demonstrated in Sect. 1-6. Here current conservation has been violated. In particular, the calculation described above (with b\ — b
^4} = ^ ^ ^

( 3 - 23 )

•

Both sides of this equation have the same two-gluon matrix elements. It is clear from this that the apparent Z7(l) symmetry predicted by Noether's theorem is not a symmetry of the quantum theory after all. Path integral analysis In a path integral treatment [Pu 79], the symmetry of the theory can be tested by considering the generating functional, as described in Sect. Ill— 2. In particular, if we consider a functional of the gluon field Ab^ and an axial current source a^,

J

(3.24)

then the steps leading to Eq. (2.14) produce

-i f dAxP(x)&*J$(x)

= InW [a^ - 8^(3, Ab^ -\nW [aM, Aj] (3.25)

where JL (x) denotes the matrix elements of the current J§ ,

J }

S

J k l n w K ^ ] L =° •

(3 26)

-

In particular, the two-gluon matrix described above is given by

A< =0 au=0

\

(3.27)

In order to solve for d^J^J, we note that the 8^(3 term can be absorbed into a redefinition of the fermion fields. This can be seen from the identity (for infinitesimal /?), $ifhl> + 8^(3 ^ 7 5 ^ = ^ (1 - i/?75) ip (1 - Wis) ^ • The following quantities are invariant under this transformation:

(3.28)

82

III Symmetries and anomalies

Mass terms would not be invariant, but we are presently working in the massless limit. Therefore, if we define ^ ' = (1 - i / W = e - ^ —.

—

—

+ O(f32) ,

a

l

n

if/ = V(l - i07i>) = V'e^ 7 5 + C(/32) , we see that the lagrangian can be written in terms of ip', £QCD(, & AD + &>0 J<J> = £QCD(V>', ^', A%)

.

(3.31)

1

Furthermore, we would like to change from ip to if) in the path integration. To be general, we allow for the possibility of a jacobian J accompanying this change of variables, viz.,

JW][dfi}J .

(3.32)

If, as will be shown later, the jacobian J is independent of xp and ?/>, it can be taken to the outside of the path integral, resulting in J

Thus the test for the symmetry, Eq. (3.25), depends entirely on J, lnj=-i

I d4x f3{x)d^jf^{x) .

(3.34)

The lesson learned is that if the lagrangian and the path integral measure are invariant under the U(l) transformation, then there exists a [/(I) symmetry in the theory, with d^JU = 0. However if the lagrangian is invariant, as it is in this case, but the path integral is not (i.e. J / 1), then the [/(I) transformation is not a symmetry of the theory, i.e., We shall show below that the jacobian, when properly regularized, has the form

J = exp(-2itr/3 75) = exp \-i Jd4xf3(x)3^F^Fa^]

,

(3.35)

so that the current divergence has the form given in Eq. (3.3),

Functional differentiation using Eq. (3.27) yields the same result for q^Tf^p as obtained in ordinary perturbation theory. The nontrivial transformation of the path integral measure has prevented the axial U(l) transformation from being a symmetry of the theory. We now turn to the calculation of the jacobian.

III-S The U(l) axial anomaly

83

The jacobian in fact diverges, and a regularization is needed in order to make it finite. In Fujikawa's original calculation the regularizer was introduced early into the procedure, allowing each step to be well-defined. We will be slightly less rigorous by introducing the regularizer somewhat later. In order to calculate the jacobian we need to review the properties of integration over Grassmann numbers (which are described in more detail in App. A-6). The anticommuting nature of the variables requires that any function constructed from them terminates after linear order in each variable. Thus a function of two Grassman numbers z\, z2 {z\z2 = —z 2z\, zf = 0) becomes f(zuz2)

= /o + fizi + h*2 + fi2Ziz2 ,

(3.36)

where /o, / i , /2? /12 are real numbers. The primary property of an integral to be transferred to Grassmann numbers is completeness, i.e.

fdzf(z)=

fdz f(z + z') .

(3.37)

where zf is a constant Grassmann number. Expanding both sides we have

J dz (/o + hz) = J dz (/0 + hz + flZ')

.

(3.38)

For this to be true, the condition

fdz = 0

(3.39)

is required. Now consider a change of variables z\ = cnz[ + c12z'2 ,

z2 = C2\z[ + C22Z2 ,

(3.40)

involving a matrix of coefficients C. The jacobian is defined by / dzx dz2 f(z) = jfdz[

dz'2 /(Cz') .

(3.41)

Application of Eq. (3.36) leads to the consideration of only the f\2 term, /12 / dzi dz2 z\z2 = Jf 12 / dz[ dz2 (cnz[ + ci2z2)(c2iz[ + c22z2) = Jfi2{cnc22

- C12C21) / dz[ dz2 z[z2 , (3.42)

and hence the identification of the jacobian, ^^[detC]"1

.

(3.43)

Although derived in the simple 2 x 2 case, Eq. (3.43) generalizes to arbitrary dimension. Note that this is the inverse of the usual jacobian, due to the Grassmann nature of the variables.

84

777 Symmetries and anomalies

Turning now to the path integral, we temporarily consider ip(x) as a finite number of Grassmann variables corresponding to four Dirac indices at each point of spacetime (i.e., imagine that the spacetime label is discrete and finite). At each point, the transformation is from \p —> xj)f rp(x) = el^x^^{x)

,

$(x) = $'(x) e*^ 75 ,

(3.44)

so that the overall jacobian has the form

J = [det (e i/375 )] ~* [det (e^ 7 6 )] ~*

(3.45)

with one factor from each of the ip and jp variables. The determinant runs over the 4 x 4 Dirac indices, the three flavors, colors and also the spacetime indices. This is a rather formal object, but can be made more explicit by using = etrlnC ,

(3.46)

valid for finite matrices, to write The symbol tr denotes a trace acting over spacetime indices plus Dirac indices, flavors and colors, tr/3 7 5 - Tr' f d*x (x\/3^\x)

,

(3.48)

with Tr7 indicating the Dirac, color and flavor trace. This will become clearer through direct calculation below. The jacobian still is not regulated. Fujikawa suggested the removal of high energy eigenmodes of the Dirac field in a gauge-invariant way. Consider, for example, the simple extension

J=

lim exp[-2itr f/?75 e~^^2)]

,

(3.49)

where Ip is the QCD covariant derivative. The insertion of a complete set of eigenfunctions of Ip exponentially removes those with large eigenvalues. There has been an extensive literature demonstrating that other regularization methods produce the same results as Fujikawa's, provided that the regulator preserves the vector gauge invariance. In order to complete the calculation we employ the following identity:

(3-50)

£>„£>"

III-3 The U(l) axial anomaly

85

In this case the expression

(x\exp-(ff)/M)2\x)

(3.51)

has the same form as given in Eqs. (B-l.l), (B-1.9), (B-l.17-18) with the identifications <*„ = !>„,

a = ^XaF^

,

r = ^

.

(3.52)

Applying the calculation done there to our present situation yields

J= = lim

lim

M —00

ei^/

M-+OO

The notation is defined in App. B-l. The first two traces vanish, leaving only the factor with two a^v matrices in (22. Prom the result Tr ( 7 5 ^ ^ ) = - Tr 757 V T V = -^^uap

,

(3.54)

it is easy to calculate J exp

=

= exp (j±;Jd*x0{x) 3 • 2<5a6 • 4 ie^&F^F^

(3.55)

where the trace Tr' has produced factors for three flavors, color and the Dirac trace. Although the calculation of the jacobian has been somewhat involved, we have succeeded in making sense out of what seemed to be a rather abstract object. The fact that it is not unity is an indication that the U(l) transformation is not a symmetry of the theory. Applying Eq. (3.34) we see that

In J = -iJd4xP(x)d»J^(x) = -iJd4xf3(x)^F^Fr

, (3.56)

or once again

The choice of a regulator which preserves the vector SU(3) gauge symmetry is important. Whereas in the Feynman diagram approach, we had the apparent freedom to shift the integration variable to preserve either the vector of axial-vector symmetries, the corresponding freedom in the path integral case is in the choice of regularization.

86

III Symmetries and anomalies

If quark masses are included, the operator relation becomes

4°J

mdd^d + mss75s) + ^F^F^

.

(3.58)

Masses do not modify the coefficient of the anomaly, basically because it arises from the ultraviolet divergent parts of the theory, which are insensitive to masses. One does not have to go through these lengthy calculations for each new application of the anomaly. The anomalous coupling for currents

4 where Ty\T^

l

(3.59)

are matrices in n the space of quark flavors, is of the th form r\bcd

=j Dbcd

^ eF^FZp

_^

+ mass terms ,

£rW | r W , T f } )

,

(3.60a) (3.60b)

where Nc is the number of colors. In particular, for the electromagnetic coupling to the isovector axial current we have (3-61) leading to the result already quoted in Eq. (3.6). The full content of the anomaly was given by Bardeen [Ba 69]. Consider a fermion with rj internal degrees of freedom (flavor or color) coupled to vector and axial-vector currents v^a^

C = ^{ip-1)-fa)xP

.

(3.62)

These currents are in an rj x 77 representation Vfi

= vp + v*\k,

aM = a0/ + ajAfc .

(3.63)

Thus the axial current is J5 = ^7^75 A ^ , and the anomaly equation becomes

2i

2i

8

+ i[v^ vv\ + i[a^ av\ ,

+ i[vfJLi av\ + i[vv, aM] .

(3.64)

III-S The U(l) axial anomaly

87

This may also be expressed in terms of the left-handed and right-handed field tensors i^ and r^v by using the identities,

\

±a^aaf3 = 1 ( V ^ + 7>rn + 1 ( ^ + *

(3.65) In the language of Feynman diagrams, one encounters the anomaly contributions not only in the triangle diagram, but also in square and pentagon diagrams (e.g. from the a^a^^a^ term). Our previous result, Eq. (3.57), is obtained for aM = 0, v^ = g^A^Xk/2, with three flavors and three colors of quarks. We have seen that symmetries of the classical lagrangian are not always symmetries of the full quantum theory. This is the general situation when there are anomalies. These appear in perturbation theory and are associated with divergent Feynman diagrams. This sometimes gives the mistaken impression that the dynamics has 'broken' the symmetry, and hence one might expect a massless particle through the application of Goldstone's theorem. In the path integral framework the impression is different. There the symmetry never exists in the first place, as the calculation performed above is simply the path integral test for a symmetry, generalizing Noether's theorem. Hence there is in general no expectation for a Goldstone boson. Can anomalies cause problems? When the anomaly occurs in a global symmetry, such as the above U(l) example, the answer is 'no'. They just need to be taken properly into account, e.g. as in Eq. (3.61). Given the specific form of the anomaly operator relation, there exist 'anomalous Ward identities' which contain terms attributable to the anomaly [Cr 78]. These anomalies can even be associated with a variety of specific phenomena. For example, in Sect. VI-3 we shall see how the decay TT°—> 77 is attributed to the axial anomaly. Another example, the anomaly in the baryon-number plus lepton-number current */B+L, c o u ld in principle lead to an even more dramatic kind of experimental signal. However, the proper physical conditions, if any, for it to be observed have not yet been agreed upon at this time. The presence of anomalies in gauge theories is far more serious because they destroy the gauge invariance of the theory and wreak havoc with renormalizability. Thus, one attempts to employ only those gauge theories which have no anomalies. In some cases this can be arranged by ensuring, through the group or particle content of the theory, that the coefficient Dbcd of Eq. (3.60b) vanish. For example, in the Standard Model it must be checked that this occurs for all combinations of the

88

/ / / Symmetries and anomalies

SU(3)C x SU(2)L x U(l)y generators. These were already compiled in Eqs. (II-3.3a-c) and were seen to lead to the quantized fermion charge values observed in Nature.

Ill—4 Classical scale invariance and the trace anomaly If the fermion masses were zero in either QED or QCD, these theories would contain no dimensional parameters in the lagrangian, and they would exhibit a classical scale invariance. The associated quark and gluon scale transformations would be ip{x) —» A3/2/0(Ax) and Aa(x) —• \Aa^{\x) for arbitrary A. We saw in Sect. 1-6 that this leads to a traceless energymomentum tensor, with conserved dilation current ^ ^

=0 ,

(4-1)

where 6^ is the energy-momentum tensor. Such a situation would have drastic consequences on the theory, since all single particle states would be massless. This can be seen as follows. For any hadron H, the matrix element of the energy-momentum tensor at zero momentum transfer is (H(k)\e^\H(k))=2k»k 1/

,

(4.2)

where the normalization of states is chosen in accordance with the conventions defined in App. C-2. A vanishing trace would imply zero mass, i.e. (H(k)\e^\H(k))

= 0 = 2M2H .

(4.3)

This is most obviously a problem in QCD where the quark masses are small compared to most composite particle masses.* We would not expect the proton mass to vanish if the quark masses were set equal to zero, yet the scale invariance argument implies that it must. A resolution is suggested by the method which is used to renormalize the theory. In practice, renormalization prescriptions introduce dimensional scales into the theory. Most commonly, there is the momentum scale at which one specifies the running coupling constant to have a particular value, e.g. as(9l GeV) = 0.12. This in turn defines a scale A which enters the formula for the running coupling constant, Eq. (II—2.74). Thus, to fully specify QCD one needs to specify not only the lagrangian, but also a scale parameter, and the full quantum theory is not scale invariant. Although this argument does not, at first sight, seem to nullify the reasoning based on Noether's theorem, it turns out that the trace of the energy-momentum tensor has an anomaly [Cr 72, ChE 72, CoDJ 77], and the specification of a scale and the coefficient of the anomaly are in fact related. * As can be justified, we neglect here the existence of very heavy quarks, c, b and t.

III-4 Classical scale invariance and the trace anomaly

89

In the following, let us start directly with the path integral treatment [Fu 81], again in the framework of QCD, concentrating on the effect of a single quark. We can introduce an external source coupled to 9^ into the generating functional

l

J

J

[

l

)

+ h(x)e^] , (4.4)

where

e^ = yh^rl) .

(4.5)

As in the case of the chiral anomaly, we can use this as a starting point to explore the nature of the trace 9^. The key is that if one makes the change of variables ip(x) = e~ a ( : r ) / y(z)

,

(4.6)

one obtains for infinitesimal a

r

= / dAx [£QCD (', Al) + a()mlj'l'

(4.7)

+

ifi^'d^a]

The last term vanishes after an integration by parts. The focus of our calculation can thus be shifted to a jacobian J ' by a change of variable,

(4 g)

=f Thus we obtain the identity i f d4x 0^a{x) =lnj

+ i f d*x m^

a(x) .

(4.9)

The form of the jacobian which follows from the work done in Sect. Ill—3 is 1-2

J = [det ( W 2 ) l

= Km e*'!** <*l««P-W/JO a|*>

(4 . 10 )

where we have adopted the same regulator as used previously. The final result is easily obtained from the general heat kernel calculation of App. B-l, again using the identity of Eqs. (B-1.17), (B-1.18).

90

/ / / Symmetries and anomalies

After some algebra this becomes TV'(z|exp

4TT2

-Qp/M)2\x) «A*

48TT 4 8 22

""

a

[D^ ° r *fi + a0 PF F

b

a

(4.11) Here we have found both a term which is a divergent constant, and one which involves two-gluon field strengths. The divergent constant corresponds to the infinite zero-point energy of the vacuum. This can be seen by noting that if the zero point energy is defined by the vacuum matrix element (4.12) then Lorentz covariance requires a non-zero trace 0*01 0) = ^ gT = > (0 \d»(x)\ 0) = 4 ^ .

(4.13)

Thus, a constant in the vacuum matrix element of the trace is just four times the zero point energy density. It is standard practice to subtract off this zero point energy, and we shall do so by dropping the constant term. This is similar to the procedure of normal ordering the energy-momentum tensor. If we now combine these results using Eq. (4.9), we obtain

i f d*x 0>(z) = ijd4x ^-A^F^Fr + m^] a(x)

(4.14)

which is equivalent to the operator relation ^

.

(4.15)

One may also derive the trace anomaly via the calculation of a Feynman diagram, the triangle diagram of Fig. Ill—1 but with the axial current replaced by the energy-momentum tensor. The trace anomaly is different from the chiral anomaly in that it also receives contributions from gluons. In the Feynman diagram approach, this arises from the replacement of quark lines by gluons, while in the path integral context it occurs when one considers scale transformations of the gluon field. A full calculation yields 0" = P9™F"F^ + muuu + mddd + msss + ... ,

(4.16)

7/7-5 Chiral anomalies and vacuum structure

91

where /?QCD is the beta function of QCD (cf. Eq. (II-2.58)). The result of our previous calculation, Eq. (4.15), corresponds to the lowest order contribution of a single quark to the beta function. A feeling of why the beta function enters can be obtained from an extremely simple, but heuristic, derivation of the trace anomaly. Let us rescale the gluon field to Aa^ = g^A^, such that the massless action becomes

£ = -^»Fr

+ iWD^

.

(4.17)

The coupling constant g% now enters only as an overall factor in the first term. However in renormalizing the coupling constant, we need to introduce a renormalization scale. If we interpret this coupling as a running parameter, the action is no longer invariant under scale transformations. Instead, taking A = 1 + £A, we find

ML - [d4x— (-—*—]2 F a F^ - fdf 4x a SX ~ J dX V 4g 3(X)J ^ " J

2gs where we have changed back to the standard normalization of A^ in the final term. By Noether's theorem, the scale current is no longer conserved, and Eq. (4.16) is reproduced. The need to specify a scale in defining the coupling constant has removed the scale invariance of the theory. The trace anomaly occupies a significant place in the phenomenology of hadrons because it is the signal for the generation of hadronic masses. Returning to the discussion of masses which began this section, we see that the mass of a state is expressible as a matrix element of the energymomentum trace. For example, we find for the nucleon state that mNu(p)u(p) = msss + muuu + mddd\N(p)) The terms containing the light quark masses mu,md are expected to be small, and indeed the 'cr-term' determined in TTN scattering (cf. Sect. XII3) implies that they contribute about only 45 MeV. This leaves the bulk of the nucleon's mass to the gluon and s-quark terms in Eq. (4.19), of which the Ftf1'F^v part is expected to be dominant. Although this presents a conceptual problem for the naive quark model interpretation of the proton as a composite of three light quarks, it is nevertheless a central result of QCD. Ill—5 Chiral anomalies and vacuum structure There is a fascinating connection between the axial anomaly described previously in this chapter and the vacuum of QCD. This has important

92

777 Symmetries and anomalies

phenomenological consequences for both the rj mass and the strong CP problem. Here we present an introductory account of this topic [Pe 89].

The 0-vacuum One is used to considering the effect on gluon fields of 'small' gauge transformations, i.e. those which are connected to the identity operator in a continuous manner. There also exist 'large' gauge transformations which change the color gauge fields in a more drastic fashion. For example the gauge transformation [JaR 76] generated by A

,

x

Al(x)

x 2 -d2

+

2idr -x

/r_N

(o)

=^T^ ^T^ '

where d is an arbitrary parameter and r is an SU(2) Pauli matrix in any SU(2) subgroup of SU(3), transforms the null potential A(x) = 0 into

2X,-(T • x) - 2d(x x T)i]

.

(5.2) Here, we are using the matrix notation A

^

This potential lies in an SU(2) subgroup of the full color 577(3) group, and is 'large' in the sense that it cannot be brought continuously into the identity. The r • x factor couples the internal color indices to the spatial position such that a path in coordinate space implies a corresponding path in the SU(2) color subspace. All gauge potentials AM carry a conserved topological charge called the winding number,

n=^Jd3xTr

(M*)Aj(*)A*(*))^ •

(5-4)

As can be demonstrated by direct substitution, the gauge field of Eq. (5.2) corresponds to the value n = 1. Fields with any integer value of the winding number n can be obtained by repeated applications of Ai(x), viz., A (V) = fAi (x)]n

(5 5)

All gauge potentials can be classified into disjoint sectors labeled by their winding number. The existence of these distinct classes has interesting consequences. For example, consider a configuration of the gluon field that starts off at t = — oo as the zero potential A(x) = 0, has some interpolating A{x, t) for

III-5 Chiral anomalies and vacuum structure

93

intermediate times, and ends up at t = +oc lying in the gauge equivalent configuration A(x) = A^\x).* Then the following integral can be shown to be nonvanishing: /

.

(5.6)

This is surprising because the integrand is a total divergence. As noted previously in Eq. (II-2.23), FF can be written as

and thus the integral can be written as a surface integral at t — ±00. For the field configuration under consideration, this reduces to the winding number integral 2

f

^3

2 A

/ d TF

64TT^ J

a

F

afJll/

— ^

**

f

3

/ d4rd

647T^ J ^—

2

r

On

I I

"^7r

K^ t=o

°

o CL X -TYf)

^

<=-oo

9l i I dzx eijk Tr (Af}(x)A*1'(a;)A^(a;))

24TT 2

= 1 . (5.8) More generally, the integral of F F gives the change in the winding number

64TT 5

f r14r Fa Fa^ -

9

*

f

t=oo

(5.9)

between asymptotic gauge field configurations. Thus, the vacuum state vector will be characterized by configurations of gluon fields which fall into classes labeled by the winding number. Moreover, there will exist a corespondence between the gauge transformations {An} and unitary operators {Un} which transform the state vectors. For example, a vacuum state dominated by field configurations in the zero winding class ('near' to A^ = 0) would be transformed by U\ into configurations with a dominance of n = 1 configurations, or more generally, Ui\n) = |n + l) . * Such configurations are known to exist [Co 85].

(5.10)

94

III Symmetries and anomalies

This implies that a gauge-invariant vacuum state requires contributions from all classes, such as the coherent superposition \6) = Ye~ine\n)

,

(5.11)

n

where 9 is an arbitrary parameter. It follows from Eq. (5.10) that this 9-vacuum is gauge-invariant up to an overall phase U1\e)=e»\d) .

(5.12)

The QCD vacuum must contain contributions from all topological classes. The 0-term

Given this nontrivial vacuum structure, one requires three ingredients to completely specify QCD: (1) the QCD lagrangian, (2) the coupling constant (i.e. AQCD), and (3) the vacuum label 9. How can we account for the diflFerent vacua corresponding to different choices of 91 In a path integral representation, the 9 = 0 vacuum would imply generic transition elements of the form out<0

= 0\X\9 = 0) in - [ ^

n,m

The presence of a nonzero 9 leads to an extra phase, n)

* ont{m\X\n)in .

(5.13) (5.14)

n,m

However, this phase can be accounted for in the path integral by the addition of a new term to SQCD- In particular we have, through the use of Eq. (5.9), X out(m|X|n) out n,m

where X is some operator. We see that the quantity (ra — n) given by the winding number difference of the fields contributing to the path integral is equivalent to a new exponential factor containing F^Fa^v. Thus a correct procedure for doing calculations involving 9-vacua is to follow the ordinary path integral methods but with a QCD lagrangian containing the new term {

°$

^

.

(5.16)

7/7-5 Chiral anomalies and vacuum structure

95

The parameter 6 is to be considered a coupling constant. Since the operator FF is P-odd and T-odd, a nonzero 0 can induce measurable T violation. In Sect. IX-5, we shall show how to connect 6 to physical observables. There is an important distinction between the various 9 vacua of QCD and the many possible vacuum states of a spontaneously broken symmetry such as the Higgs sector of the electroweak theory. In the latter case, the various possible vacuum expectation values of the Higgs field label different states within the same theory. In contrast, each value of 6 corresponds to a different theory, just as each value of AQCD would label a different theory. Specifying 0 and AQCD then specifies the content of the version of QCD used by Nature. Connection with chiral rotations There is a connection between the axial anomaly and the presence of a #-vacuum ['tH 76a,b]. It involves the matrix element of FF as follows. Consider the limit of Nf massless quarks. The U(l) axial current Nf

is not conserved due to the anomaly,

PJ$!! = ^FSJ°r

.

(5.18)

However, because of the fact that FF is a total divergence, one can define a new conserved current

While J^^ does form a conserved charge, Q5 = fd3x

J5,o(z) ,

(5.20)

neither Q5 nor J^ is gauge-invariant. In fact, under the gauge transformation Ai of Eq. (5.1), it follows from Eq. (5.8) that the operator Q5 changes by a c-number integer UxQhU{x = Qh-2Nf

.

(5.21)

This tells us that in the world of massless quarks, the different 0-vacua are related by a chiral 17(1) transformation, ^

^

^

)

e

^

5

|

0

)

,

(5.22)

96

/ / / Symmetries and anomalies

or from Eq. (5.12), eia^\6) = \0-2Nfa)

,

(5.23)

where a is a constant. Therefore, in the limit of massless quarks, when Q5 is a conserved quantity, all of the 6-vacua are equivalent and one can transform away the 6 dependence by a chiral U(l) transformation. The same is not true if quarks have mass, as the mass terms in £QCD are not invariant under a chiral transformation. We shall return to this topic in Sect. IX-5. To summarize, one finds that the existence of topologically nontrivial gauge transformations, and of field configurations which make transitions between the different topological sectors of the theory, leads to the existence of nonvanishing effects from a new term in the QCD action. Chiral rotations can change the value of 0, allowing it to be rotated away if any of the quarks are massless. However for massive quarks, the net effect is a measurable CP violating term in the QCD lagrangian. Problems 1) Currents and anomalies a) Show that all currents coupled to gauge bosons in the Standard Model are anomaly free. b) Show that the baryon number current B^ has an anomaly, but that the current B^ — L^ (L^ is the lepton number current) is free from anomalies. 2) Trace anomaly in QED In d dimensions, the trace of the energy-momentum tensor does not vanish classically, except at d = 4. For example, in massless QED the energy-momentum tensor,

has trace 0^ = ^^FXcTF\a. In the renormalization of the operator FX(J F\a, one encounters a renormalization constant which diverges as d —• 4. Use this feature to calculate the QED trace anomaly using dimensional regularization.

IV Introduction to effective lagrangians

The purpose of the effective lagrangian method is to represent in a simple way the dynamical content of a theory in the low energy limit, where effects of heavy particles can be incorporated into a few constants. The basic plan of attack is to write out the most general set of lagrangians consistent with the symmetries of the theory. At sufficiently low energies only one, or perhaps a few, of the lagrangians are relevant, and it is straightforward to read off the predictions of the theory. Effective lagrangians are used in all aspects of the Standard Model and beyond, from QED to superstrings. Perhaps the best setting for learning about them is that of chiral symmetry. Besides being historically important in the development of effective lagrangian techniques, chiral symmetry is a rather subtle theory which can be used to illustrate all aspects of the method, viz. the low energy expansion, non-leading behavior, symmetry breaking and loops. In addition, the results can be tested directly by experiment since effective lagrangians provide a framework for understanding the very low energy limit of QCD. IV-1 Nonlinear lagrangians and the sigma model The linear sigma model, introduced in Sects. 1-3, 1-5, provides a 'user friendly' introduction to effective lagrangians because all the relevant manipulations can be explicitly demonstrated. The Goldstone boson fields, the pions, are present at all stages of the calculation. By contrast, low energy QCD is far less transparent, involving a transference from the quark and gluon degrees of freedom of the original lagrangian to the pions of the physical spectrum. Nevertheless, we expect the low energy properties of the two theories to have many similarities. 97

98

IV Introduction to effective lagrangians

First let us illustrate what an effective lagrangian is by simply quoting the result. Recall the sigma model of Eq. (1-3.14), C = ipipip + \dpir

^ (L1)

1 A 2 a - IT • 7r75) \j) + -/x2 {a2 + TT2) - - (a 2 + TT2)

.

This is a renormalizable field theory of pions, and from it one can calculate any desired pion amplitude. Alternatively, if one works at low energy, then it turns out that all matrix elements of pions are contained in the rather different looking 'effective lagrangian' £eff = ^ - Tr (dpUdvU^ ,

U = expzr • n/F ,

(1.2)

where F = v = \/fi2/\ at tree level (cf. Eq. (1-5.9)). This effective lagrangian is to be used by expanding in powers of the pion field ...

, (1.3)

and taking tree-level matrix elements. This procedure is a relatively simple way of encoding all the low energy predictions of the theory. Moreover, with a modest increase in effort, corrections to the low energy results can also be treated in the effective lagrangian formulation.

Representations of the sigma model In order to embark on the path to the effective lagrangian approach, let us rewrite the sigma model lagrangian as + ^ - Tr (^Y) - — \ C = \ Tr (dj:d^\ 4 V M / 4 V / 16 L + ^hip^L + ^RifiipR - 9 {^PL^R + $

2

with E = a + ir • Tr. The model is invariant under the SU(2)L transformations

X

SU(2)R

(1.5) for L,i? in SU{2). This is the linear representation.* After symmetry breaking and the redefinition of the a field,

* A number of distinct 2 x 2 matrix notations, among them E, U, and M, are commonly employed in the literature for either the linear or the nonlinear cases. It is always best to check the definition being employed and to learn to be flexible.

IV-1 Nonlinear lagrangians and the sigma model

99

the lagrangian reads^ £ = i (dJfr&'a - 2fj2a2) + \d^ A 2 - - (a2 + 7T2) +i>{ip-

• d»n - Xva (a 2 + TT2) gv) ip - gi\)(a - ir • 7775) ^ ,

(L7)

indicating massless pions and a nucleon of mass gv. All the interactions in the model are simple nonderivative polynomial couplings. There are other ways to display the content of the sigma model besides the above linear representation. For example, instead of a and TT one could define fields 5 and <£>, S=\/(a

+ V) +7T2-V

— C7 + . . . ,

(fi =

= = 7T + . . . ,

V (a + vf + ^ (1.8) where one expands in inverse powers of v. For lack of a better name, we can call this the square root representation. The lagrangian can be rewritten in terms of the variables S and cp as c =

„ 9^1 . 1 fv + S^2

1 [,n ^2 2

" " • • " • •

-

g

+

()

(1.9) Although this looks a bit forbidding, no longer having simple polynomial interactions, it is nothing more than a renaming of the fields. This form has several interesting features. The pion-like fields, still massless, no longer occur in the potential part of the lagrangian, but instead appear with derivative interactions. For vanishing 5, this is called the nonlinear sigma model. Another nonlinear form, the exponential parameterization, will prove to be of importance to us. Here the fields are written as E = a + iT'n such that nf — 7T +

c = \ Ud.sf

= (v + S)U ,

U = exp (ir • n'/v)

Using this form, we find

2ns} +

!i

- XvS3 - - S 4 + ikpt/; - g(v + S) (i>LUi(>R + Here, and in subsequent expressions for C, we drop all additive constant terms.

(1.10)

100

IV Introduction to effective lagrangians

The quantity U transforms under SU(2)L does £, i.e.

X SU(2)R

in the same way as (1.12)

This lagrangian is reasonably compact and also has only derivative couplings.

Representation independence We have introduced three sets of interactions with very different appearances. They are all nonlinearly related. In each of these forms the free particle sector, found by looking at terms bilinear in the field variables, has the same masses and normalizations. To compare their dynamical content, let us calculate the scattering of the Goldstone bosons of the theory, specifically TT+TT0 —> TT+TT0. The diagrams that enter at tree level are displayed in Fig. IV-1. The relevant terms in the lagrangians and their tree-level scattering amplitudes are as follows. 1) Linear form: b =~

4

2 2 vI* )7

- \vair2 ,

{-2i\vf

2

q

o

= ~2i\

T 1

%

g

(1.13)

m(J 2 2\v 1 iq2 =z 2 1 = \ 2 + ... , q -2\v \ v\

where q = p+ — p + = po — Po and the relation ml — 2Xv 2 = 2/J,2 has been used. The contributions of Figs. IV-1 (a), l(b) are seen to cancel at q2 = 0. Thus, to leading order, the amplitude is momentumdependent even though the interaction contains no derivatives. The vanishing of the amplitudes at zero momentum is universal in the limit of exact chiral symmetry. 2) Square root representation:

(a) Fig. IV-1 Contributions to TT+TT0 elastic scattering

IV-1 Nonlinear lagrangians and the sigma model

101

For this case, the contribution of Fig. IV-l(b) involves four factors of momentum, two at each vertex, and so may be dropped at low energy. For Fig. IV-l(a) we find

./,

^ .2

a-15)

3) Exponential representation: d = (V + ^

Tr (dpUWU^ + ... .

(1.16)

Again Fig. IV-l(b) has a higher order (C?(p4)) contribution, leaving only Fig. IV-l(a),

r ' ' *

The lesson to be learned is that all three representations give the same answer despite very different forms and even different Feynman diagrams. A similar conclusion would follow for any other observable that one might wish to calculate. The above analysis demonstrates a powerful field theoretic theorem, proved first by R. Haag [Ha 58, CoWZ 69, CaCWZ 69], on representation independence. It states that if two fields are related nonlinearly, e.g. ip = xF(x) with ^(0) = 1, then the same experimental observables result if one calculates with the field

102

IV Introduction to effective lagrangians

one uses the nonlinear representations, whereas for the linear representation more complicated calculations involving assorted cancelations of constant terms are required to produce the correct momentum dependence. In addition, the nonlinear representations allow one to display the low energy results of the theory without explicitly including the massive a (or S) and xj) fields.

IV-2 Integrating out heavy fields When one is studying physics at some energy scale i£, one must explicitly take into account all the particles which can be produced at that energy. What is the effect of fields whose quanta are too heavy to be directly produced? They may still be felt through virtual effects. When using an effective low energy theory, one does not include the heavy fields in the lagrangian, but their virtual effects are represented by various couplings between light fields. The process of removing heavy fields from the lagrangian is called integrating out the fields. Here, we shall explore this process.

The decoupling theorem There is a general result in field theory, called the decoupling theorem, which describes how the heavy particles must enter into the low energy theory [ApC 75, OvS 80]. The theorem states that if the remaining low energy theory is renormalizable, then all effects of the heavy particle appear either as a renormalization of the coupling constants in the theory or else are suppressed by powers of the heavy particle mass. We shall not display the formal proof. However, the result is in accord with physical expectations. If the heavy particle's mass becomes infinite, one would indeed expect the influence of the particle to disappear. Any shift in the coupling constants is not directly observable because the values of these couplings are always determined from experiment. Inverse powers of heavy particle mass arise from propagators involving virtual exchange of the heavy particle. In the Standard Model, the most obvious example of this is the role played in low energy physics by the W± and Z gauge bosons. For example, while VF±-loops can contribute to the renormalization of the electric charge, the effect cannot be isolated at low energies. Also, the residual form of W^-exchange amplitudes is that of a local product of two weak currents (Fermi interaction) with coupling strength GF- Its effect is suppressed because GF OC M^ 2 . However, in the Standard Model there is an example where the heavy particle effects do not decouple. For a heavy top quark, there are many loop diagrams which do not vanish as mt —* oc, but instead behave as

IV-2 Integrating out heavy

fields

103

m2 or \n(m2). This can occur because the electroweak theory with the t-quark removed violates the SU(2)i symmetry, as the full (£) doublet is no longer present. Without the constraint of weak-isospin symmetry, the theory is not renormalizable and new divergences can occur in flavor changing processes. These would-be divergences are cut off in the real theory by the mass m*. Note that at the same time as mt —• oc, the top quark Yukawa coupling also goes to infinity, and hence induces strong coupling which can also lead to a violation of decoupling. In the sigma model, all the low energy couplings of the pions are proportional to powers of 1/v2 oc 1/raj*, the simplest example being Eq. (1.9). Hence the effective renormalizable theory is in fact a free field theory, without interactions. The interactions have been suppressed by powers of heavy particle masses. We shall use the energy expansion of the next section to organize the expansion in powers of the inverse heavy mass. Integrating out heavy fields at tree level The name of this procedure comes from the path integral formalism, where the process of integrating out a field H and leaving behind light fields £i is defined in terms of an effective action Ze^[£i], eiZev[ti]

= I

[dH] eif#xC(H(x)A(*))/

I

[dH]

However, the procedure is equally familiar from perturbation theory, in which the effect of the path integral is represented by a sum of Feynman diagrams. Let us proceed with a path integral example. Consider a linear coupling of H to some combination of fields J, with the lagrangian C = \ (d^Hd^H - m2HH2) + JH .

(2.2)

One way to integrate out H is to 'complete the square', i.e. we write

fd4xC(H,J)=

f d4x \~HVH

+ JH\

= - i fd4x [(H - V-1 J) V(H-

V~1J) - JV~1J]

= - i I d4x [H'VH' - JV~XJ] , (2.3)

104

IV Introduction to effective lagrangians

where we have used the shorthand notations, V = H + m2H ,

V~lJ = - J dAyAF{x-y)J{y) , (Dx + m2H) AF{x -y) = -8*(x - y) ,

H\x) = H(x) + Jd4y AF(x - y)J(y) ,

f d*x JV~lJ

= - j dAxdAy J(x)AF(x - y)J(y) ,

and have integrated by parts repeatedly. Since we integrate in the path integral over all values of the field at each point of spacetime, we may change variables [dH] = [dHf] so

= f[dH'}ei _Afd4xJV-1J from which we obtain

ZeS[J] = -^Jd4xd4yJ(x)AF(x-y)J(y)

.

(2.6)

Finally one may obtain a local lagrangian by noting that the heavy particle propagator is peaked at small distances. Over these scales, we may Taylor expand J(y) as „=*+ •••

•

(2-7)

Keeping the leading term and using

we obtain r

i

+ ... ,

(2.9)

where the ellipses denote terms suppressed by additional powers of Outside the path integral context, this result is familiar from VF-exchange in the weak interactions. We can apply this procedure to the lagrangian for the sigma model where the scalar field 5 is heavy with respect to the Goldstone bosons. Thus, considering the theory in the low energy limit, we may integrate out

IV-3 The low energy expansion

105

the field S. Referring to Eq. (1.11) and neglecting the 5 2 interactions, it is clear that we should make the identifications H —• S and J —• vTv{dyJJd^U^)/2. The effective lagrangian then takes the form

£eff = j IV (dpUVrf) + -^

[Tr (dlxUd^)]\

...

,

(2.10)

where the second term in Eq. (2.10) is the result of integrating out the 5-field and gives rise to the diagram of Fig. IV-l(b). Additional tree-level diagrams are implied by the sigma model when one includes the S3 and 5 4 interactions. Since these carry more derivatives, the above result is the correct tree-level answer with up to four derivatives. Upon including the S2 Tr (d^Ud^U^) terms, loop diagrams become possible. Our purpose is not to calculate these explicitly, but merely to indicate the nature of the effective lagrangian. The net result will be a low energy effective lagrangian which is written as an expansion in powers of Tr (d^Ud^W). In calculating transitions of pions, this is used by expanding the U matrix in terms of the pion fields and taking matrix elements. Interested readers may verify that this procedure applied to Eq. (2.10) reproduces the first two terms in the TT+TT0 scattering amplitude previously obtained in Eq. (1.13).

IV-3 The low energy expansion The expansion in energy is the essential feature which allows the effective lagrangian framework to be useful. We have already seen some aspects of it in the sigma model example. Let us now explore this topic in more generality.

Expansion in energy What would happen if, instead of having a straightforward theory like the linear sigma model, we were dealing with an unknown or unsolvable theory with the same SU{2)L X SU(2)R chiral symmetry? In this case there would exist some set of pion interactions which, although not explicitly known, would be greatly restricted by the SU(2) chiral symmetry. Once again we could choose to describe the pion fields in terms of the exponential parameterization U, with a symmetry transformation (3.1) for L, R in SU(2). Not having an explicit prescription, we would proceed to write out the most general effective lagrangian consistent with the chiral symmetry. In view of the infinite number of possible terms contained in such a description, this would appear to be a daunting process. However, the energy expansion allows it to be manageable.

106

IV Introduction to effective lagrangians

It is not difficult to generate candidate interactions which are invariant under chiral SU{2) transformations. For the purpose of illustration, we list the following two-derivative, four-derivative and six-derivative terms in the exponential parameterization,

There can be no derivative-free terms in a list such as this because Tr (C/C/t) = 2 is a constant. It is clear that one can generate innumerable similar terms with arbitrary numbers of derivatives. The general lagrangian can be organized by the dimensionality of the operators, C = C2 + £4 + A, + Cs + . . . +ai[Tr (d^Ud^U^]2

(3.3)

a2Tr (dpUdvU^ • Tr The important point is that, at sufficiently low energies, the matrix elements of most of these terms are very small since each derivative becomes a factor of the momentum q when matrix elements are taken. It follows from dimensional analysis that the coefficient of an operator with n derivatives behaves as 1/Mn~4, where M is a mass scale which depends on the specific theory. Therefore the effect of an n-derivative vertex is of order qn/Mn~4, and at an energy small compared to M, large-n terms have a very small effect. At the lowest energy, only a single lagrangian, the one in Eq. (3.3) with two derivatives, is required. We shall call this an 'O(E2y contribution in subsequent discussions. The most important corrections to this involve four derivatives, and are therefore 'C?(i?4)\ In practice then, the infinity of possible contributions is reduced to only a small number. The coefficients of these terms are not generally known, and must thus be determined phenomenologically. However, once fixed they can be used to allow predictions to be made for a variety of reactions.

Loops It would appear that loop diagrams could upset the dimensional counting described above. This might happen in the calculation of a given loop diagram if, for example, two of the momentum factors from an O(E4) lagrangian are involved in the loop and are thus proportional to the loop momentum. Integrating over the loop momentum apparently leaves only two factors of the 'low' energy variable. It would therefore seem that for certain loop diagrams, an O(E4) lagrangian could behave as if it were

IV-3 The low energy expansion

107

O(E2). If this happened, it would be a disaster because arbitrarily high order lagrangians would contribute at O(E2) when loops were calculated. As we shall show, this does not occur. In fact, the reverse happens. When O{E2) lagrangians are used in loops, they contribute to O(E4) or higher. Before we give the formal proof of this result, let us demonstrate how it works by example. Consider a loop diagram for TT+TT0 —• TT+TT0, as in Fig. IV-2. Using the lowest order vertex for this process, we find

de

- /

^) 2 i

a2

i

(e-P+-po)2

A*-i/)2

{3A)

where / is the loop integral with the factor v 4 extracted. Counting powers of energy factors is most easily done in dimensional regularization. The loop integral contains no dimensional factors other than p+, po and j / + . Since, in four dimensions it has the overall energy unit E4, it must therefore be expressible as fourth order in momentum. Despite the loop integration, the end result is expressed only in terms of the external momenta. These momenta are small, and hence all the energy factors involved in power counting are taken at low energy. In dimensional regularization, there can also be a dependence on the arbitrary scale //,

f d4£^ fi4~d I dd£ ,

(3.5)

but in the limit d —• 4 this occurs only in dimensionless logarithms such as ln(E2/fji2). Thus, the order of momentum can be found by counting the factors of 1/v2 which occur for every vertex from the lowest order lagrangians. We shall soon see (cf. Eq. (5.3)) that the parameter u, defined in the sigma model, is equal to the pion decay constant Fn. Each factor of 1/v2 must be accompanied by momenta in the numerator in order to produce a dimensionless amplitude. Each vertex in a diagram contributes powers of 1/v2, and higher order loop diagrams require more vertices. Thus every time a loop is formed, the overall momentum power of the amplitude must increase rather than decrease.

Fig. IV-2 Loop contribution to TT+TT0 elastic scattering

108

IV Introduction to effective lagrangians

Weinberg's power counting theorem To prove this result [We 79], consider some diagram with a total of Ny vertices. Then letting Nn be the number of vertices arising from the subset of effective lagrangians which contain n derivatives (e.g. N4 is the number of vertices coming from four-derivative lagrangians), we have Ny = T,nNn. The overall energy dimensionality of the coupling constants is thus MNc with

where M is a mass scale entering into the coefficients of the effective lagrangian (e.g. the quantity v in the sigma model). Each pion field comes with a factor of 1/v, so that associated with NE external pions and Ni internal pion lines is an energy factor (1/M)2NI+NE . (Recall that two pions must be contracted to form an internal line.) However, the number of internal lines can be eliminated in terms of the number of vertices and loops (NL), n

-l

.

(3.7)

Any remaining dimensional factors must be made up of powers of the energy E times a dimensionless factor of E/fi where // is the scale employed for renormalizing the coupling constants. (When using dimensional regularization, these factors of E/fi enter only in logarithms.) Thus the overall matrix element is composed of energy factors M ~ (M)£n^-4) K

]

L—

EDF(E/fi)

MNE+2NL+2^N"-2

K

'

}

(3.8)

4

~ (Mass or E n e r g y ) " ^ , where the second line is the overall dimension of an amplitude with external bosons. The renormalization scale \i can be chosen of the order of E so no large factors are present in F(E/[/,). Overall the energy dimension is then

^2

.

(3.9)

A diagram containing NL loops contributes at a power E2NL higher than the tree diagrams. This theorem is of great practical consequence. At low energy, it allows one to work with only small numbers of loops. In particular, at O(E4) only one-loop diagrams generated from £2 need to be considered. The astute reader will have noticed that the loop example used above raises another difficulty, viz., the loop integral is badly divergent. Of

IV~4 Symmetry breaking

109

course divergences are generally present in a quantum field theory, and here they can be handled in the usual way, by renormalizations of the parameters in the theory. If the original effective lagrangian which we have written down is indeed the most general one consistent with the given symmetry, then it must have enough parameters of the right form to encompass any divergences which occur. In particular, our power counting argument tells us that when £2 is used in one-loop diagrams, the divergences are of order E4 and should be capable of being absorbed into the parameters of that order. Since the parameters are generally unknown and are to be determined phenomenologically, the only difference this makes is to cast physical results in terms of the renormalized parameters instead of the bare ones. The end result is a very simple rule for counting the order of the energy expansion. The lowest order (E2) behavior is given by the two-derivative lagrangians treated at tree level. There are two sources at the next order (E4): (i) the O(E2) one-loop amplitudes, and (ii) the tree-level O(E4) amplitudes. When the coefficients of the E4 lagrangians are renormalized, finite predictions result. IV—4 Symmetry breaking Effective lagrangians can be used not only in the limit of exact symmetry but also to analyze the effect of small symmetry breaking. Let us first return to the sigma model for an illustration of the method, and then consider the general technique. The SU(2)L X SU(2)R symmetry of the sigma model is explicitly broken if the potential V(a, TT) is made slightly asymmetric, e.g. by the addition of the term breaking = CL<J = | TV (E + E + )

(4.1)

to the basic lagrangian of Eq. (1.4). To first order in the quantity a, this shifts the minimum of the potential to

and produces a pion mass

"4 = 2 .

(4-3)

Although the latter result can be found by using the linear representation and expanding the fields about their vacuum expectation values, it is

110

IV Introduction to effective lagrangians

easier to use the exponential representation, taking = \{v + S)Tr(U + &) = \{v + S)TV (2 - ( ^ )

2

+ ...

2

= a(v + S)- -^n - 7T + ... = a(y + S) - ^ n •

+ ... . (4.4) The chiral SU(2) symmetry is seen to be slightly broken, but the vectorial SU(2) isospin symmetry remains exact. As we have seen, the O(E2) Lagrangian is obtained by setting 5 = 0, 2

TT

2

C2 = V- Tr (dJJ&lfi) + ^-v2 Tr(U + rf) .

(4.5)

Higher order terms will contain products like l Tr(U + [/+)]\ ml Tr (U + /7f) • Tr (d,AUdtAtf), ... ,

(4.6)

and can be obtained by integrating out the field 5 as was done in Sect. IV2. It is important to realize that the symmetry breaking sector also has a low energy expansion, with each factor of m^ being equivalent to two derivatives. If ra^ is small, the expansion is a dual expansion in both the energy and the mass. If we encounter a theory more general than the sigma model, the effect of a small pion mass can be similarly expressed in low orders by, ^breaking = axm\ TV (U + tf) + a2 [ml Tr(U l Tr(U + U]) TV (d^Ud^)

+ a4ml TV \(U

(4.7) with coefficients that are generally not known. An important consideration is the symmetry transformation property of the perturbation. The symmetry breaking term of Eq. (4.1) is not invariant under separate left-handed and right-handed transformations but only under those with L — R. All the terms in Eq. (4.7) have this property. Other symmetry breakings can be analyzed in a manner analogous to the treatment just given of the mass term. One identifies the symmetry transformation property of the perturbing effect and writes the most general effective lagrangian with that property. Most often the perturbation is treated to only first order, but higher order behavior can also be studied. We shall encounter another example of this in our study of weak decays.

IV-5 PCAC

111

IV-5 PCAC In addition to effective lagrangian techniques for dealing with chiral symmetry, there also exist current algebra methods which go by the name of Partial Conservation of the Axial Current or PCAC [AdD 68]. These yield the same results but are often more cumbersome. Let us again use the sigma model to illustrate this subject, now with a pion mass included and the 5-field integrated out. The lagrangian of Eq. (4.5) gives rise to the vector and axial-vector currents V

2

Vj = -i - TV \rk (u%U + Ud^uA] , 4

L

4 = i"-"TV [[r [ 2

= i"-2 TV

V

n

(5-1)

)]

(

with k = 1, 2, 3. The equation of motion is found to be

d^ (ifidJJ) H \

/

^ (C/- C/+) — 0 . 2 V

(5.2)

/

and two important matrix elements are

(p)) = ivp^

(o\ dMj | it* (p)) = vml6ki .

,

(5.3)

0 The former allows the identification v = Fn, where Fn is the pion decay constant Fn ~ 92 MeV, while the latter follows either from Eq. (5.1) directly, or by use of the equation of motion for A*, v2m2

0M£ = -i

r

k

/

f

M

- Tr \ r \ U - C/ J = F ^ m 2 ^ + ...

.

(5.4)

This last equation forms the heart of the PCAC method. It describes a situation covered by Haag's theorem (recall Sect. IV-1), and says that we may use either ?rfc or d^A^ (properly normalized) as the pion field. It is more general than the sigma model which we used to motivate it. This, plus certain smoothness assumptions, gives rise to a soft-pion theorem for the following matrix element of a local operator O, lim (7rk(c[)(3\O\a) = ~4r (p\ \Q\IO\ qf-+0 \

I

Fn \

L

\<X) J /

,

(5-5)

k

where /?, a are arbitrary states and Q = f d?x AQ(X) is an axial charge. The soft-pion theorem The proof of this starts with the LSZ reduction formula. We consider the matrix element for the process a —> (3+7r k(q) as the pion four-momentum

112

IV Introduction to effective lagrangians

q is taken off the mass-shell,

Uk{q)j3\O{Q)\a\ = i Id4xeiqx

( • + m\) ((3\T Uk{

= i fd4xeiqx(-q2

+ ml) ((3\T (7r*(x)0(O)) \a) ,

(5.6) The pion field can be replaced by using the PC AC relation (valid in the sense of the Haag theorem),

"* leading to

(7rk(q)(3\O(0)\a) = i{ml~J*] f dAx e^x (f3\T (&>A*(x)O(0)) |a) . (5.8) The derivative can be extracted from the time-ordered product by using (ilJ(x)O(O)) | a )

a) + 6(x0) (/3\ [4(x), O(0)] \a) ,

(5 9)

where the last term arises from differentiating the functions 6(±xo) which occur in the time-ordering prescription. Upon integrating by parts, we find

x | - ( 0 | |An(x),O(0)] \a} 6(x0) - itf (/3\T (A*( (5.10) Up to this stage all the formulae are exact for physical processes, even if appearing rather senseless, since d^A^ has the same singularity for q2 —• ml- as does the field 7rfc. However, to obtain the soft-pion theorem one assumes that the matrix element does not vary much between its on-shell value and the point where the pion's four-momentum vanishes. In that circumstance, we have [NaL 62, AdD 68] lim {7rk(q)/3\O\a) = —— (/3\ \QLO(0)\ q»-+o \ v J ' ' / Fn \ L °

la) + lim iq^R* M, J / q^-^o

(5.11) v y

where i*

r

.

.

i

/

\

\

|a>

.

(5.12)

The remainder term of Eq. (5.11) vanishes unless Rk has a singularity as q11 —> 0. Such a singularity can occur if there are intermediate states in

IV-6 Matrix elements of currents

113

R^ which are degenerate in mass with either a or f). This last statement can be proven by inserting a complete set of intermediate states in the time-ordered product in i2*, and taking the q** —• 0 limit. This caveat should be kept in mind as it is sometimes relevant. The soft-pion theorem relates to the intuitive picture for dynamically broken symmetries mentioned in Sect. 1-6. Since a chiral transformation corresponds in the symmetry limit to the addition of a zero energy Goldstone boson, we expect the states (f3\ and (TT£ = O /?| to be related by the symmetry, and indeed, the soft-pion theorem expresses this. Although the soft-pion theorem is exact in the symmetry limit, a smoothness assumption is needed in the real world to pass from q^ — 0 to q 2 = ra^, implying that corrections of order q^ or of order m%. can be expected. In the Standard Model, the charge commutation rules are commonly abstracted from those of the quark model. Upon expressing charge operators in terms of quark fields,

Qk = J tfx jno^

, Q\ = jdzxi>lQl^,

(5.13)

one obtains the algebra

[Q\ V*] = if**Vt , [Ql V£\ = if*A* , i k [Q ,AJl]=if* A$, [Ql^]=ifjkVk . These commutation rules can be extended to equal-time commutators which contain a charge density, e.g.

[V$(x), 4(y)] lO=y o = ifikAk6®(x

- y) .

(5.15)

However, commutators which involve two spatial components can be more problematic [AdD 68]. Sometimes, in the PC AC approach, if the matrix element is assumed to be strictly constant, the various soft-pion limits turn out to be contradictory. If so, the amplitude must be extended to include momentum dependence, as happens in nonleptonic kaon decay. By contrast, the effective lagrangian approach automatically gives the appropriate momentum dependence, and its predictions follow in a straightforward manner. Moreover, effective lagrangians are especially useful in identifying and parameterizing corrections to the lowest order results. They allow a systematic expansion in terms of energy and mass. IV—6 Matrix elements of currents There is an elegant technique which allows one, at a minimal increase in complexity, to calculate matrix elements of currents from a chiral effective lagrangian [GaL 84,85a]. The idea is to add to the lagrangian

114

IV Introduction to effective lagrangians

terms containing external sources coupled to the currents in question. Construction of the effective lagrangian, including source terms, then allows the current matrix elements to be easily identified. We shall explain this technique here, and use it extensively in our discussion of QCD in subsequent chapters. First consider how current matrix elements are identified in a path integral framework. We have seen in Chap. Ill (see also App. A) that by adding a source coupled to the desired current, matrix elements can be obtained from differentiation of the path integral, e.g. Eqs. (Ill—2.1), (III— 2.4). For example, we can modify three-flavor QCD by adding sources to obtain

C = -\F%F\? + $Hp1> ^ p ^

^lj^r^

where £^ rM, s, p are 3 x 3 matrix source functions expressible as

^ = ^+^A a ,

r^rl+r^W

s = s°+sa\\

p = p°+pa\a , (6.2)

with a = 1,..., 8. The lagrangian in Eq. (6.1) reduces to the usual QCD lagrangian in the limit l^ — r^ = p = 0, s = m, where m is the 3 x 3 quark mass matrix. The electromagnetic coupling can be obtained with the choice l^ = r^ = eQA^ where A^ is the photon field and Q is the electric charge operator defined in units of e. Various currents can be read off from the lagrangian, such as the left-handed current J {X) =

^

"a||)= ^)^^P^V(z)

(6-3)

or the scalar density ^

•

(

6

-

4

)

Moreover, matrix elements of these currents can be formed from the path integral by taking functional derivatives. The simplest example is i=r=p=O

while other examples appear in Sect. Ill—2. Matrix elements and the effective action A low energy effective action for the Goldstone bosons of this theory will be a functional of the external sources. One way to define the connection of the effective action with QCD is to consider the effect of the sources

IV-6 Matrix elements of currents

115

in QCD,

At low energy, all heavy degrees of freedom can be integrated out and absorbed into coefficients in the effective action Z. However, the Goldstone bosons propagate at low energy, and they must be explicitly taken into account. One then writes a representation of the form

_ f

(6.7)

where as usual U contains the Goldstone fields. This form then allows inclusion of all low energy effects while maintaining the symmetries of QCD. The lagrangian of Eq. (6.1) has an exact local chiral SU(3) invariance if we have the external fields transform in the same way as gauge fields. In particular, the transformations ipL - •

if>R -• R{x)^R , + id,L(x)L\x) , + idfiR{x)R\x) , 0 + ip) -^ L(x)(s + ip)R\x) provide an invariance for any L{x), R(x) in SU(3). In constructing the effective action, these invariances must be included. This is easy to do if l^ and r^ enter in the same way as gauge fields. In particular, upon defining a covariant derivative DJJ = O^U + i^U - iUr^ , (6.9) L(X)I/JL

,

and field strength tensors V = d^ - dj^ + %[ip, 4 ] , c a a , -r l

^6 °^

we obtain the following covariant responses to local transformations: U -> L(x)UR\x) ,

D,U -v L{x)D,UR\x) ,

L^ -^ L(x)L^(x) , R^ -f JR(x)i?^i?t(a;) . The effective action is then expressed in terms of these quantities. At order E2, there are only two terms in the effective lagrangian, £ 2 = ^L TV (DpUiyrf}

+ ^ Tr (xU] + UX])

(6.12)

+ ip),

(6.13)

where

116

IV Introduction to effective lagrangians

and Bo is a constant with the dimension of mass. In the limit l^ = r^ — p = 0, s = ra, this is the same effective lagrangian with which we have been dealing in the SU(2) examples, with the identification m^ — (mu + rrid)Bo. Note that this usage requires Bo to be positive. Having constructed the effective action, we can obtain a number of interesting matrix elements. For example, use of Eq. (6.5) provides the identification of the vacuum scalar-density matrix element as (6.14) to this order in the effective lagrangian. Similarly, use of Eq. (6.3) reveals the left-handed current to be LJ =

_i^L ^ ^\kUd^

.

(6.15)

One other advantage of the source method is to allow the use of the equations of motion. The standard Noether procedure for identifying currents does not work if the equations of motion are employed in the lagrangian. To become convinced of this, one can consider the following exercise. We examine the response of the two trial lagrangians, A =
IV-1 Heavy particles in effective lagrangians

117

We shall proceed by first working out an example, the fermionic sector of the linear sigma model, £f = $[ift-

g(a -ir

- TT75)] I/;

We shall drop reference to the scalar field S in the following. The above lagrangian is invariant under the chiral transformations ^L^L^L,

II>R-+RII>R,

U-^LUrf

,

(7.2)

with L in SU(2)L and R in SU(2)R. AS always, we are free to change variables via contact transformations. In this instance, a useful choice of field redefinitions turns out to be

£/ = ££ ,

(7.3)

where £ = exp(zr • ir/2Fw). This is seen, after some algebra, to convert the fermion lagrangian to

which is a theory of fermions of mass M = gv having pseudovector coupling. The new fields transform as

M

- idpV* • V) V* ,

Ap -

^

,

M

^

(7.5) For purely vector transformations we have L = R = V. For L ^ i?, the property of V is more complicated, and Eq. (7.5) implies that it cannot be a simple global transformation, but must be a function of TT(X) and hence a function of x. At first sight, the need to express an 5(7(2) transformation matrix like V as a function of TT(X) appears unnatural. However, it is in fact consistent with physical expectations. Recall from the general discussion of dynamical symmetry breaking in Sect. 1-6 that in the symmetry limit, axial transformations mix the proton not with the neutron (as in isospin transformations), but rather with states consisting of nucleons plus zero momentum pions. Mathematically, the important point is that iV^ and NR transform in an identical fashion. This corresponds to the fact that heavy fields do not transform chirally, but have a common vectorial 5/7(2) transformation. It can be directly verified that Eq. (7.5) is a symmetry of the lagrangian. Thus we have obtained the expected result that the baryons can have a vectorial SU(2) invariance, while maintaining a chiral invariance for pion couplings.

118

IV Introduction to effective lagrangians

We see in the above example the ingredients of a general procedure for adding heavy fields to effective chiral lagrangians. The heavy fields are assumed to have an SU(2) (or SU(n), if desired) transformation described by the matrix V. A derivative d^ acting on a heavy field must be incorporated as part of a covariant derivative V^ in order to maintain this invariance. Couplings to pions are described by the matrices £ and [/, with £ having the same transformation as in Eq. (7.5). It is usually straightforward to combine factors of £ and U in such a way that the overall lagrangian is invariant. In the general case, each invariant term will have an unknown coefficient which must be determined phenomenologically. For example, the ip fi ip term in Eq. (7.4) would be expected to have a coefficient different from unity; the unit coefficient is a prediction specific to the linear sigma model. Effects which break the symmetry in an explicit fashion, like mass terms or electroweak interactions, can be added by using appropriate external sources. To date, heavy field lagrangians have been used in applications primarily at tree level. The feature which is essential for their application is that the pion momenta are small, and hence the heavy fields are essentially static. We conclude this discussion with an application to B meson decay. Consider the quark semileptonic decay, b —> u + e + ve, induced by the weak current J» = h»(l + 7b)u .

(7.6)

This quark process appears at the hadronic level in decays like i?(+'°) —>• (TT'S) + ez/e, etc. For most of the kinematically allowed region, the pions have too much energy to contemplate using chiral symmetry methods. However, in that corner of phase space where the lepton pair eve carries off most of the available energy, chiral symmetry can be used to restrict the allowed couplings, and an effective lagrangian analysis is justified. Let us obtain an expression for the hadronic weak current in B decay by constructing a chirally-invariant lagrangian which contains a weak bquark current as an external source. We first define an isospin-doublet B of heavy-meson fields and an isospin-doublet source current J^,

Then noting that the source currents are purely left-handed, we have the transformation laws ^

,

,

U -> LUB) ,

B^VB, <£ - L£V]

V£R)

IV-8 Effective lagrangians in QED

119

A chirally invariant effective lagrangian, containing up to one derivative acting on the pion fields, is then

£ = faJl&rB +foJl {dPU) &B +foJl (dyU) £V»V»B .

(7.9)

Finally, the hadronic weak current can be found by differentiating with respect to the source current J^ as described in Sect. IV-6. Observe that a derivative acting on a heavy field is not a small quantity in the energy expansion, but in this corner of phase space a derivative acting on the pion fields is small. An additional example of this approach, but applied instead to baryons, appears in Sect. XII-3. IV-8 Effective lagrangians in QED We have explored in some detail the structure of effective lagrangians by using chiral symmetry as an example. However, this is not meant to imply that effective lagrangians are useful only in that one context. In fact, they can be applied to a wide variety of situations. Here, we apply the technique to QED. Consider situations in which the photon's four-momentum is small compared to the electron mass. In such cases, the electron and other fermions cannot be produced directly, but instead influence the physics of photons only through virtual processes. The lowest order diagrams, i.e. those which contain a single electron loop, with increasing numbers of external photon legs are shown in Fig. IV-3. Note that the one-loop diagram containing three photons, or indeed any odd number of photons, vanishes by virtue of charge conjugation invariance. This is true to all orders in the coupling e, and is refered to as Furry's theorem. Diagrams like those in Fig. IV-3 have effects at low energy which are typically calculated in perturbation theory. The associated amplitudes have coefficients which scale as some power of the inverse electron mass. They can be generated by means of an effective lagrangian, as we shall now discuss. Let us seek a description which eliminates the electron degrees of freedom. That is, we wish to write a lagrangian which involves only photons, but nevertheless includes effects like the ones in Fig. IV-3. The result

(a)

< (b)

Fig. IV-3 Photon amplitudes containing a single fermion loop.

120

IV Introduction to effective lagrangians

must of course be gauge invariant. The procedure may be defined by the path integral relation / [dAp] exp \i / dAx £eff(Au) p][di/;][dx/j] exp [i JdAx £ Q E D ( A ^ VO] exp [i f d±x

(8 1)

where CQED is the full QED lagrangian, and Co is the free fermion lagrangian. Thus £eff has precisely the same matrix elements for photons as does the full QED theory. Specifically, it includes the virtual effects of electrons. The techniques described in App. A-5 enable us to formally express the content of Eq. (8.1) as [Sc 51],

J

In [ ^ ^ 1 •

(8.2)

This form, although formally correct, does not readily lend itself to physical interpretation. However, we can determine various interesting effects directly from perturbation theory. For example, the vacuum polarization of Fig. IV-3(a) modifies the photon propagator, i.e. the two-point function. From Eqs. (II—1.26), (II—1.29), we determine the result for a photon of momentum q to be 2

^

^

+ ...

.

(8.3)

The essence of the effective lagrangian approach is to represent such information as the matrix element of a local lagrangian. In the present example, we find that the term in Eq. (8.3) corresponds to the interaction

where • = d^d^. The calculation of Fig. IV-3(b) is a somewhat more difficult, but still straightforward, exercise in perturbation theory. We shall lead the reader through a calculation using path integrals in a problem at the end of this chapter. It too can be represented as a local lagrangian, and is usually named after Euler and Heisenberg [ItZ 80]. One finds the full result to one-loop order to be 4

607rm 2 J

^ a2

r

. 7

_

.1

(8-5)

IV-9 Effective lagrangians as probes of new physics

121

where F^v = t^apF - Corrections to this effective lagrangian can be of two forms, (i) even at one loop there are additional terms of higher dimension

involving either more fields or more derivatives, or (ii) the coefficients of these operators can receive corrections of higher order in a through multiloop diagrams. We see here an example of the energy expansion which we have discussed at length earlier in this chapter. In this case it is an expansion in powers of q2/m2. The effective lagrangian of Eq. (8.5) can be used to compute aspects of low energy photon physics such as the low energy contribution of the vacuum polarization process or the matrix element for photon-photon scattering.

IV-9 Effective lagrangians as probes of new physics One of the most common and important uses of effective lagrangians is to parameterize how new physics at high energy may influence low energy observables. The general procedure can be abstracted from our earlier discussion. Remember that one is trying to represent the low energy effects from a 'heavy' sector of the theory. This is accomplished by employing an effective lagrangian (9.1) where the {Oi} are local operators having the symmetries of the theory and are constructed from fields that describe physics at low energy. There need be no restriction to renormalizable combinations of fields. Most often the operators can be organized by dimension. The lagrangian itself has mass dimension 4, so that if an operator has dimension d{ the coefficient must have mass dimension a~MA-di

.

(9.2)

The mass scale M is associated with the heavy sector of the theory. It is clear that operators of high dimension will be suppressed by powers of the heavy mass. To leading order, this allows one to keep a small set of operators. Some applications will involve phenomena for which the dynamics is well understood. If so, the coefficients of the effective lagrangian can be determined through direct calculation as in the preceding sections. Other examples occur in the theory of weak nonleptonic interactions (c/. Sect. VIII-3) and in the interactions of W-bosons (c/. Sect. XV-4). Even more generally, effective lagrangians can also be used to describe the

122

IV Introduction to effective lagrangians

effects of new types of interactions. In these cases, dimensional analysis supplies an estimate for the magnitude of the energy scales of possible new physics. We shall conclude this section by using effective lagrangians to characterize the size of possible violations of some of the symmetries of the Standard Model. Given certain input parameters, the Standard Model is a closed, selfconsistent description of physics up to at least the mass of the Z°, and is described by the most general renormalizable lagrangian consistent with the underlying gauge symmetries. What would happen if there were new interactions having an intrinsic energy scale of several TeV or beyond? In general, such new theories would be expected to modify predictions of the Standard Model. The modifications would be described by non-renormalizable interactions, organized by dimension in an effective lagrangian description as £eff = £SM + jCb + ^ £

6

+ ...

(9.3)

where Cn has mass dimension n and A is the energy scale of the new interaction. Consider the contribution £5 of dimension 5. There is, in fact, only a single operator of this dimension which contains the fields of the Standard Model and is consistent with its symmetries, i.e., C5 = c s e ^ f g ) ^
(i,j,k,e= 1,2) ,

(9.4)

where C5 is a constant, (^L)& and $& are respectively members of the lepton and Higgs doublets, and (4C')fe = ^ ) * 7 2 7 o

(9.5)

is the conjugate lepton representation. Replacing the Higgs fields by the vacuum expectation value v/y/2 produces neutrino Majorana masses of order

^-fx •

(9 6)

-

Taking C5 = O(l) and A = (9(lTeV) would yield a neutrino mass of about 60 GeV. Phrased differently, existing upper bounds on Majorana masses of order 10 eV require that A > 1010 TeV. Turning next to possible contributions of dimension 6, there are eighty distinct operators consistent with the gauge symmetries of the Standard Model [BuW 86]. These can generate a variety of effects which deviate from the Standard Model. For example, the operator (*t*) W W^ ,

(9.7)

Problems

123

containing the Higgs field
a2

The current level of precision, p = 1.003 ± 0.004 (for mt = 100 GeV), requires A > 4.0 TeV for d = 1. Another possibility concerns the violation of flavor symmetries in the Standard Model. The operator c" C = ^ ^ ( 1 + 75 )/i 57^(1 + 7s)d + h.c. conserves generational or family number but violates the separate lepton number symmetries. It leads to the transition KL —> e~/x+ such that (99)

The present bound, B r ^ o ^ g < 2.2 x 10~10 at 90% confidence level, requires A > 250 TeV for c" ~ 1. In a similar manner, constraints on other physical processes imply bounds on their corresponding energy scales A, generally in the range 5 —> 3000 TeV. Of course, if there is new physics in the TeV energy range, it need not generate all eighty possible effective interactions. The ones actually appearing would depend on the couplings and symmetries of the new theory. In addition, the coefficients of contributing operators could contain small coupling constants or mixing angles, diminishing their effects at low energy. However, the effective lagrangian analysis indicates that the continued success of the Standard Model is quite nontrivial and places meaningful bounds on possible new dynamical structures occurring at TeV, and even higher, energy scales.

Problems 1) Effective lagrangian for fi —» e + 7 In describing the decay /i —* e + 7, one may try to use an effective lagrangian £3,4 which contains terms of dimensions 3 and 4, £3,4 = as(efi + fie) + ia± where D^ = d^ + ieQ^A^ and as, a^ are constants. a) Show by direct calculation that £3,4 does not lead to /x —• e + 7. b) If £3,4 is added to the QED lagrangian for muons and electrons, show that one can define new fields / / and e1 to yield a lagrangian which is diagonal in flavor. Thus, even in the presence of £3,4, there are two conserved fermion numbers.

124

IV Introduction to effective lagrangians c) At dimension 5, // —> e + 7 can be described by a gauge-invariant effective lagrangian containing constants c, d, £ 5 = e
2) The Euler-Heisenberg lagrangian: Constant magnetic field Consider a charged scalar field

= det(C 2 + m2)/det(D2 + m2) ,

The operation 'Tr In' applied to a differential operator is not a trivial one and the purpose of this problem is to evaluate this quantity for the case at hand. a) Demonstrate that SeS(B) = i b) In order to evaluate the trace we require a complete set of solutions to the equations D2fin(x, y, z, t) = \n
SeS(B) = t £ f ° e'- m n

Jo

2s

(e—

s

S

c) With the gauge choice A^ = (0, Bxj ) show that the eigenstates are (f(x, y, z, t) = exp i(kxx + kyy + kzz - ktt) ,
,

where ^n(^) is an eigenstate of the harmonic oscillator hamiltonian, and the eigenvalues are Kn = —k 2 + k2 + k2 + k2, Xn = -k2 + k2z + eB(2n + l)

Problems

125

d) Rotate to euclidean space and evaluate the trace using box quantization. Taking a box with sides Li, 1/2,1/3 and a time interval T, we have , r°° d*k (2TT)4 '

J—c

L2L3T /

A:

dky I

JO

J-oo

n=Q

where the integration on ky is over all values with x' = x — ky/eB positive, e) Evaluate the effective action

2n+l)s _ f°° dk xdky ,-(*S+*j))« 2TT

loo W2

n=0

and show that Seff(B) = 1

/o

eBs -1 \sinh eBs

s°

Expand this in powers of 5 , finding the (divergent) wavefunction renormalization and the B4 piece of the Euler-Heisenberg lagrangian. f) Show that the corresponding result of a constant electricfieldcan be found by the substitution B —> iE so that roo

ds

- l

C3

g) Demonstrate that, although Im Setf(B) = 0, one nonetheless obtains Im n=\

and discuss the meaning of this result [Sc 51].

V Leptons

From the viewpoint of probing the basic structure of the Standard Model, leptons constitute an attractive starting point. Since effects of the strong interaction are generally either absent or else play a secondary role, the theoretical analysis is relatively clean. Moreover, a great deal of high quality data has been amassed involving these particles. Thus leptons serve as an ideal system for defining our renormalization prescription, and for investigating the effects of various radiative corrections. V - l The electron Some of the most precise tests of the Standard Model (or more exactly of QED) occur within the elementary electron-proton system. The renormalization program for the theory has been introduced in Sect. II—1, where it was shown how ultraviolet divergent contributions to such calculations can be removed by means of subtraction from a finite number of suitably constructed counterterms. Here we examine the finite pieces which remain after such subtractions and compare theory with experiment. Breit-Fermi interaction The electromagnetic properties of the electron are studied by use of a photon probe. To lowest order, the eej vertex has the structure (e(p'e,X'e)\J^m\e(pe,Xe)) = -eu(p'e,X'e)-fu(pe,\e)

,

(1.1)

and the interaction between two charged particles is governed by the exchange of a single virtual photon. An important example is the electron126

V-l The electron

127

proton interaction, which has the invariant amplitude* MeP

= e2u{p'e1 A'e

(p e , Xe)^u(Pp, \phpu(

p,

Xp) ,

(1.2)

where p e , pfe and p p , p^, are respectively electron and proton momenta and q = pe—p' e is the four-momentum transfer. In the following, we shall demonstrate how the above single photon exchange amplitude is associated with well-known contributions in atomic physics. Denoting proton two-spinors with tildes, we begin by reducing the amplitude of Eq. (1.2) in the small momentum limit to

p^+p?

1 - Pp+Pp

X

Pe

XX

'

2mp

-x

2me

(1.3) where me,mp are respectively the electron and proton masses. The various terms in the above expression can be interpreted physically by recalling that in Born approximation the transition amplitude and interaction potential are Fourier transforms of each other,

VeP(v) = J - ^

(1.4)

where r = r e — ry From the relation 5)

we recognize the leading (velocity-independent) term, (1-6)

X']X

as the Coulomb interaction between electron and proton. The identity f ddzzqq ee22 iia • p'e x pe 3 2

J (27r) q

4 m

2

e

_iq,r

e2 a • r x p e 4m2

4?rr3

^ -

We work temporarily with spinors normalized as u^ (p, s) w(p, s)=l and consider only ideal 'Dirac' protons.

128

V Leptons

allows us to recognize an additional piece of Eq. (1.3) as the spin-orbit potential, which is often expressed as

but evaluated in this instance with Vo — — e2/47rr. Combining the remaining 0(p 2 /ra 2 ) terms in Eq. (1.3), we can cancel the q2 term in the denominator to obtain the so-called Darwin potential, ^ X ^ X X

4

X

•

(1-9)

This term has its origin in the electric interaction between the particles, and by employing the Gauss' law relation, ,

(1.10)

it can be re-expressed in the equivalent form VD = ^

V • E C o u l x'jX X']X •

(l.H)

The spin-orbit and Darwin potentials, together with the 0(p 2 /ra 2 ) relativistic corrections to the electron kinetic energy, give rise to atomic fine structure energy effects. The remaining terms in the photon exchange interaction of Eq. (1.3) are effects produced by electron and proton current densities, the terms (Pe + Pe)/2me a n d ~i°"x (Pe — Pe)/<^rne representing convection and magnetization contributions respectively. In particular, the interaction between magnetization densities is equivalent to the dipole-dipole potential

£

|

( ^ f l ) .

(1.12)

Recognizing that the magnetic field produced by the magnetic dipole moment of a (point) proton is Bproton = V X (— * ' t ^ X

XV-rM ,

(1.13)

we can interpret the hyperfine energy as the interaction between the electron magnetic moment and the spin-induced proton magnetic field. Upon dropping the proton and electron spinors and using the identity ,

,U4)

the dipole-dipole interaction may be written as a sum of hyperfine and tensor terms, Vdple-dple — ^hyp + Vtensor j

V-l The electron

Censor =

129

^ ^ [3(se - f)( Sp • r) - Se - Sp]

Denoting the total electron-proton spin as Stot^Se + Sp, it follows that the hyperfine interaction splits the hydrogen atom ground state into components with Stot — 1 and stot = 0- The frequency associated with this splitting is one of the most precisely measured constants in physics and is the source of the famous 21 cm radiation of radioastronomy. As seen in Table V-l, the experimental determination is about six orders of magnitude more precise than the theoretical value. Precision in the latter is limited by the nuclear force contribution (about 3 parts in 105). Let us gather all the terms discussed thus far. In addition, we treat the proton and electron on an equal footing, since it will prove instructive when we discuss models of quark interactions in Chaps. XI-XIII. We then obtain the full one-photon exchange potential (Breit-Fermi interaction) for the electron-proton system, a -

a [se-rxpe r 3 [ 2m|

sp -r x pp 2m^

(

s p • r x p e — s e • r x pp m e mp

pe • p p + r(r • p e ) • pp] ,

(1.16)

where we recall r = r e — rp and note that a spin-independent orbitorbit interaction has been included as the final term. The single-photon exchange interaction is seen to include a remarkable range of effects, all of which are necessary to understand details of atomic spectra. QED corrections Also important in precision tests of atomic systems are the higher order QED corrections. We have just demonstrated how the simple q2 piece of the photon propagator leads to the Breit-Fermi interaction between electron and proton. The vacuum polarization correction discussed in Sect. II—1 produces an additional component of the e-P interaction called the Uehling potential. From Eq. (II—1.37), we recall that in the on-shell renormalization scheme the subtracted vacuum polarization ft behaves in the small momentum limit m^ ^> q2 as

130

V Leptons

By the process of Fig. II—3, this yields the contribution

) - f -fS_e-^r - x — - J ^ ^ e

£- - -—6^(r)

^ Xg 0 7 r 2 ^ -

15ma

d

(1 17)

W •

(1-17J

The presence of the delta function implies that S-wave states of the hydrogen atom are shifted by this potential while other partial waves are not. Contributions from the Uehling potential have been observed in recent scattering experiments despite its O(a) suppression relative to the dominant Coulomb scattering [Ve et al. 89]. The photon-electron vertex is also affected by radiative corrections. Let us write the proper (IPI) electron-photon vertex through first order in oc as ieTv{p'e,pe) = iejv + ieK(p'e,Pe) + . . . ,

(1.18)

where, referring to Fig. II-2(b), we have in Feynman gauge

Note that a small photon mass A has been inserted to act as a cutoff in the small momentum domain, and we take both incoming and outgoing electrons to obey p2e — p^ = m2e. With a modest effort, the integral in Eq. (1.19) can be continued to d spacetime dimensions,

dx J ——

(2TT)«

(£

O l jf»/^/ lr

I A.lr(T)

I nrJ \

Am

d

i»

4lT>

I /ri

|

] (1.20) where px = xpe + (1 — x)p'e, and the result of performing the fc-integration can be expressed as p'e,Pe) = (h)u + (h)u ,

(1-21)

where (ii)t, is singular in the e —> 0 limit, .

e3

r(e/2) (e-2) 2 /x e Z"1

Z"1

j/ (1.22)

V-l The electron

131

and (h)v is not, ""

(4TT)-/2

\^y 3

y0

2

y0 f

V J « + A 2 (I-

^ — 2/ (e — 2)ftxjl/'fix + 4y [(pe + p e)v$x — Tne(px)u — (pe

(1.23) The singular term (h)i,, which arises from the # 7 ^ term in Eq. (1.20), is infrared finite, and thus the photon mass A can be dropped from it. Upon expanding (Ii)v in powers of e and performing the y-integral, we obtain

{h)v = ieiu^

[ \ + ln(47r) --y-1-Jdxhi

{^j

+ O{e)\ . ((1.24)

Because (I2)v is not multiplied by any quantity which is singular in e, we can immediately take the e —• 0 limit to cast it in the form dx

JO

JO

dy

y2p 2 + \2{l-y) V

' *"

(1.25)

- rne(px)v - (pe + Pg) • p ^ ] • The photon mass A can be dropped from the terms in Nv proportional to y2 and ys since they are nonsingular even if A = 0. Performing the y-integration then yields the result, O3

ri rx

V * % ]v"x

i~ ^h^c

re, J

+4[(p e + pfe)vj)x — rne(px)v — (pe + Pg) • p^7i/]) • The identities VxivVx — 2m 2

Je{px)u

2

— Vr ^iv i

2

p j = m e-

q x(l

\Ve + Pf>) ' Px — 2vn

p

—a

2 ,

- x)

allow (/2)i/ to be expressed in terms of #2 = (pe — Pg)2, and the dependence of p2 on the symmetric combination x{l — x) implies / dx (Px)J(p2) = \(pe +p'e), / dx f(p2) Jo * Jo

.

(1.28)

These steps, plus use of the Gordon decomposition of Eq. (C-2.8) finally lead to the expression

r PP

5 Pe)

~

^OV

e2 1

' ~A o

4TT 2

r i i 7T~

2e

2+7-

132

V Leptons

'ml - q2x(l - x) \

2

f

dx

2

1 f1

\

/i

~9

3m 2 - q2

T^,

d •

(1-29)

Jo

In the on-shell renormalization program, the electron-photon vertex

is constrained to obey 11 TY"1 1 & I

11111

ICl

^

' [ 1T\

j.

1 7^^ 1 &^S1

ITk

\irP^ir€,)

*"^ /l^

I

5

I

•{ I

I

V -*••*-'•*• 1

Q—*0

so that

(o_s) 1

e 2 [" 1

7 — ln(4?r)

2

4TT [26

y(MS)

e

2

f

1

{™>l\ 1 2

4

4

\ /i /

2

7 — ln(4?r)

1

fml\

1,

]

[ml

\ A2

fmV '(1.32)

The on-shell renormalized vertex is thus given by (1.33) where F\{q2) is given by a complicated expression which we do not reproduce here, and

8TT2 y 0

dx

2— -

In addition to its original spin structure 7^, the electromagnetic vertex is seen in Eq. (1.33) to have picked up a contribution proportional to (JvpqP. The 7^ and GV^ contributions are called the Dirac and Pauli terms respectively, and F\(q2) and F2(q2) are the Dirac and Pauli form factors of the electron. The vertex correction turns out to have several important experimental consequences.

V-l The electron

133

Consider the interaction of an electron with a classical electromagnetic field for very small q2. Using Eq. (1.33) and the Gordon identity, we have nmt = eAv{x){e(p'e)\ J» m(x)\e(pe))

= -eAu{x)u(p'e) ]^f - %^^\

u(pe)e-^ + O(q2)

(1.35) The first term describes the coupling of the photon to the convective current of electron, but it is the second term which interests us here. Ignoring the convective term and integrating by parts, we obtain to lowest order in q, P2

T)

i

^

. (1.36) Noting that in the nonrelativistic limit a^vFpv/2 —> — a • B, we see that this is the coupling of a magnetic field to the electron magnetic moment. The result is usually expressed in terms of the gyromagnetic ratio ge\, where /xe = —eg e\se/2me, and to order e2 we have (1 +

) M ( } ) e ) e

Clearly, the radiative corrections have modified the Dirac equation value, g^ irac) = 2. The factor a/27r, which arises from the Pauli term, is but the first of the anomalous contributions. Terms of higher order in a are sufficiently large to be measurable [KiY 90], 2(^1 y - 2 ) = d =- ,

C2 = -0.328 478 965 ... ,

C4 = -1.472(152) ,

6 ~ 4.46 x 10~12 ,

C3 = 1.175 62(56) ,

(1.38) where uncertainties in those coefficients which were determined numerically are placed in parentheses and 6 represents contributions from sources like muon loops, tau loops, and hadrons. The theoretical values of the g-factors for both electron and muon are displayed in Table V-l, and are seen to be in accord with the experimental determinations. These represent an even more stringent test of QED than the hyperfine frequency in

134

V Leptons

hydrogen. In this case, theory is far less influenced by hadronic effects, and is thus about a factor of 104 more accurate. Radiative corrections also modify the form of the Dirac coupling. One effect of this vertex correction is to contribute to the Lamb shift which lifts the degeneracy between the 2£ 1 / 2 and 2P\j2 states of the hydrogen atom. Recall that the fine structure corrections, computed as perturbations of the atomic hamiltonian, give a total energy contribution (A-E)fine str — (AE')Darwin + (A£% p i n _ O rbit + (AE)Te\

_

kin en

4

7.245 x !Q- eV / 1 _ _3_\ rfi \j + 1 / 2 4nJ

(1.39)

which depends only upon the quantum numbers n and j . Thus the 2Si/2 and 2Pi/2 atomic levels are degenerate to this order, and in fact to all orders. However, the vertex radiative correction breaks the degeneracy, lowering the 2Px/2 level with respect to the 2Si/2 level by 1010 MHz. When the anomalous magnetic moment coupling (+68 MHz), the Uehling vacuum polarization potential (—27 MHz), and effects of higher order in a/ir are added to this, the result agrees with the experimental value (c/. Table V-l). Since the entire Lamb shift arises from field-theoretic radiative corrections, one must regard the agreement with experiment as strong confirmation for the validity of QED and of the renormahzation prescription.

The infrared problem Viewed collectively, the results of this section point to a remarkable success for QED. Yet there remains an apparent blemish - the theory still contains an infinity. When the photon 'mass' A is set equal to zero, the vertex modification of Eq. (1.29) diverges logarithmically due to the presence of terms logarithmic in A2. The resolution of this difficulty lies in realizing that any electromagnetic scattering process is unavoidably accompanied by a background of events containing one or more soft photons Table V-l. Precision tests of QED Experiment i/£yp [geY - 2}a [0M - 2]a AS (Lamb)6 a

1420.405 751 767(1) 1 159 652 193.(10) 11 659 230.(84) 1057.851(2)

Actually O.b(g - 2) x 1012. In units of MHz.

6

Theory 1420.403(1) 1 159 652 133.(29) 11 659 194.7(143) 1057.867(11)

V-l The electron

135

whose energy is too small to be detected. For example, consider Coulomb scattering of electrons from a heavy point source of charge Ze. The spinaveraged cross section for the scattering of unpolarized electrons in the absence of electromagnetic corrections is dtt

" | P e |2/32 s i n 4| '

4

where (3 — \pe\/E is the electron speed. Radiative corrections modify this result. Using the on-shell subtraction prescription and neglecting the anomalous magnetic moment contribution, one has in the limit m 2e^> q2, LIU

,

~~^

_^~

^

.

# • • "C

1

v

-'

from the QED vertex correction. This diverges if we attempt to take A->0. However, we must also consider the bremsstrahlung process, in which the scattering amplitude is accompanied by emission of a soft photon of infinitesimal mass A and four-momentum fc^. Forfcosufficiently small, the inelastic bremsstrahlung process cannot be experimentally distinguished from the radiatively corrected elastic scattering of Eq. (1.41). To lowest order in the photon momentum k the invariant amplitude MB for bremsstrahlung is* KA -MB

Ze

~l

'\\l

i

•

ie

— ~T (PP) \\~ i)~r, U

q2

•

Ti

& + 1k-me

L

\}

(~2e7oJ

J

(

u

\

\Pe)

_ r i ] 1 (-ief)\ u(Pe) + u(Pe) (~ie70)-i j L p e — p — me »* K

pe -

and has the corresponding cross section da1 = da(0> e2 / — - ^ — > 7 J (2TT)3 2fc0 ^

^—- - ^—. \p'e • k pe-kj

(1.43)

v

J

The prime on the integral sign denotes limiting the range of photon energy, A < fco < A£*, where AE is the detector energy resolution. The polarization sum in Eq. (1.43) is performed with the aid of the complete* For simplicity, we shall take the photon as massless in this amplitude, and at the end indicate the effect of this omission.

136

V Leptons

ness relation for massive spin-one photons 1

+

^ ,

(1.44)

pol

t o yield f ' dzk dH ~

cM

e

1 /

m2e

2pe-p'e

m2e

(2TT) 33 2k 2k~QQ \j/ee • k pee • k ~ (p e • k)2

JJ

{p'e • k)2 )

(1.45) Performing the angular integration in Eq. (1.45) with the aid of

"*°

a

4

"i

<146)

rh-f!* I we find

Adding this to the nonradiative cross section of Eq. (1.41), we obtain the finite result, da

+

da^

da® I",

1+

dn -£ = ~dn i

2a q2 / ,

ln

/

me \

+

5\1

uk ( ( 2 ( A E ) j 8)\ •

,

JoN

(L48)

Thus, the net effect of soft photon emission is to replace the photon mass A by the detector resolution 2A2?, leaving a finite result.* V-2 The muon The analysis just presented for the electron can just as well be repeated for the muon. However, the muon has the additional property of being an unstable particle, and in the following we shall focus entirely on this aspect. The subject of muon decay is important because it provides a direct test of the spin structure of the charged weak current. It is also important to be familiar with the calculation of photonic corrections to muon decay, as they are part of the process whereby the Fermi constant Gn is determined from experiment. * As anticipated, the result quoted in Eq. (1.48) is not quite correct, since although we have given the photon an effective mass A we have not consistently included it, as in Eq. (1.42). In a more careful evaluation the constant | is replaced by the value ^ .

V-2 The muon

137

Muon decay at tree-level The transition /i(pi,s) —• ^(P2) + e(p3) +Ve(p4) is the dominant decay mode of the muon. In the Standard Model, this process occurs through W^-boson exchange between the leptons. However, since the momentum transfer is small compared to the W-boson mass, it is possible to express muon decay in terms of the local Fermi interaction, ^Fermi = ~^ ^

^ ^ ( 1 + T s ) ^ ^7a(l ^

a

( l +

) ^ ^ ( l

+T*)^

(2.1)

+

(2.2)

)/) ,

where the coupling constant G^ is to be considered a phenomenological quantity determined from the muon lifetime. At tree-level, G^ is related to basic Standard Model parameters as in Eq. (II-3.43). The orderings in Eqs. (2.1)-(2.2) are called respectively the charge exchange and charge retention forms of the interaction, and are related by the Fierz transformation of Eq. (C-2.11). Let us consider the decay of a polarized muon, with rest-frame spin vector s, into final states in which spin is not detected. For simplicity, we set the electron mass to zero. The muon decay width is given in terms of a three-body phase space integral by

5

2^3,S4

(2.3)

where in charge-exchange form,

The muon polarization is described by a four-vector s^ which equals (0, s) in the muon rest frame. In computing the squared matrix element, we employ ^(pi,5i)U a (pi,5i) = - [(ra^+^iXl -75^)]0 a

(2.5)

to obtain \M\2 = 64GJ (pi -P4P2-P3-

m^pt • s p2 • p 3 ) •

(2.6)

The neutrino phase space integral is easily found to be n Pi)P

-

^

=

l l ^

2

+ 2QV), (2-7)

138

V Leptons

where Q = p\ — p%. For the electron phase space, it is convenient to define a reduced electron energy x = Ee/W, where W = ra^/2 is the maximum electron energy in the limit of zero electron mass. The standard notation for the electron spectum involves the so-called Michel parameters p, <5, £ whose values depend on the tensorial nature of the beta decay interaction,

(—

- 1J J j j lx2dxsin0d9

. (2.8)

For the V-A chiral structure of the Fermi model, we find p = <5 = 0.75 ,

f = 1.0 ,

(2.9)

in good agreement with the current experimental values, p = 0.7518 ± 0.0023 ,

6 = 0.755 ± 0.009 ,

£P^/p

> 0.99677 , (2.10)

where P^ is the longitudinal muon polarization from pion decay (P^ — 1 in V-A theory). In making comparisons between Eq. (2.8) and Eq. (2.10), one shoud first subtract from the data corrections due to radiative effects. Additional possible terms in Eq. (2.8), which would arise from taking electron mass into account or allowing for electron polarization, are also found to support the Fermi model. Upon integration over the electron phase space, Eq. (2.10) gives rise to the well-known formula,

This relation is often used to provide an estimate for decay rates of heavy leptons and quarks.

Photon radiative corrections Thus far, we have worked to lowest order in the local Fermi interaction and have assumed massless final state particles. A more precise determination which includes lowest order corrections for the small electron mass, the large VF-boson mass, and O{a) QED radiative effects yields 2+

5

G 2 ra 5 ^ | [1 + (-4203.85 - 187.12 + 1.05) x 10~6 + . . . ] , (2.12) where the magnitude of each correction is exhibited in the second line. The QED radiative correction is by far the largest of these, whereas the

V-2 The muon

139

Table V-2. Determinations of Fermi-model couplings Factor

Determination

GJJ, G(3

Muon decay GM plus QED theory f Nuclear beta decay \ f Hyperon beta decay Pion beta decay (TT^) Kaon beta decay (K12)

W-boson mass correction is near the present limit of sensitivity. Taking these corrections into account one obtains from the muon lifetime the value GM = 1.16637(2) x 10"5 GeV"2. One can improve the O(a) calculation by incorporating a summation of leading-log photonic effects. This again yields Eq. (2.12), but with a replaced by the 'running' fine structure constant a{mfy. In particular, the electron-positron loop contribution to Eq. (II-1.36) leads to oTl[ml) - 136. The above analysis serves to define G^ as the Fermi model coupling constant extracted from muon decay, including photon radiative corrections but not other possible electroweak radiative contributions. The latter are certainly present, and will be discussed in Sect. XVI-5. The coupling G^ is also commonly used to describe weak leptonic transitions of the r lepton, provided lepton universality is assumed. However, for weak semileptonic transitions of hadrons {e.g. nuclear beta decay) the photonic corrections are not identical to those in muon decay because quark charges differ from lepton charges. Such processes define instead a quantity called G/?, and we shall present in Sect. VI-1 a calculation of Gp for the case of pion decay. As seen in Table V-2, determinations involving G/3 generally contain quark mixing factors and also meson decay constants.

r\ (a)

(b)

(c)

(d)

(e)

Fig. V-l. Contributions to muon decay from (a) vertex, (b-c) wavefunction renormalization, and (d-e) bremsstrahlung amplitudes.

140

V Leptons

Computation of the lowest order electron and W-boson mass corrections appearing in Eq. (2.12) is not difficult, and is left to Prob. V-l. However, the QED radiative correction is rather more formidable, and it is to that which we now turn our attention. Rather than attempt a detailed presentation, we summarize the analysis of [GuPR 80]. We shall work in Feynman gauge, and employ the charge retention ordering for the Fermi interaction. There is an advantage to performing the calculation as if the muon existed in a spacetime of arbitrary dimension d. Working in d = 4 dimensions entails factors which are logarithmic in the electron mass and which would forbid the simplifying assumption me = 0. Dimensional regularization frees one from this restriction, and such potential singularities become displayed as poles in the variable e = 4 — d. Although there would appear to be difficulty in extending the Dirac matrix 75 to arbitrary spacetime dimensions, this turns out not to be a problem here. The set of radiative corrections consists of three parts which are displayed in Fig. V-l, (i) vertex (Fig. V-l (a)), (ii) self-energy (Fig. V-l(b-c)), and (iii) bremsstrahlung (Fig. V-l(d-e)). We shall begin with the bremsstrahlung part of the calculation and then proceed to the vertex and self-energy contributions. The amplitude for the bremsstrahlung (B) process /i(pi) —• e(P3) + ^e(p4) + 7(P5) is given by

x«(P3) [-hd + 7,)^ _ I _ mJ

+ ih +1

_ ^7.(1 + 75)] u(Pl) ,

(2.13) where e is the photon polarization vector. The spin-averaged bremstrahlung transition rate for d spacetime dimensions in the muon rest frame is then given by* 1

f

5

2«V J A j=2l

x

J\

'

'

spins

(2.14)

* Since the result that we seek is finite and scale independent, in this section we shall suppress the scale parameter /i introduced in Eq. (II—1.21).

V-2 The muon

141

where Q' = Q — p§ = p\ — ps — p$. A lengthy analysis yields a result which can be expanded in powers of e = 4 — d to read

_
r(2-f)

I92TT3 4

2> 2

(2.15)

w

6

Observe that singularities are encountered as e —> 0. The radiative correction (R) contribution to the muon transition rate is given by

r

«=AI n ( j=2

x

J^

'

'

spins

(2.16) where |A^|?nt is the interference term between the Fermi model amplitudes which are respectively zeroth order (M^) and first order (M^) in e 2, \M&t = M<®*MW + MW*M<® .

(2.17)

The first-order amplitude can be written as a product of neutrino factors and a term (MR) containing radiative corrections of the charged leptons,

i^

(2.18)

The quantity M^ is itself expressible as the sum of vertex (V) and selfenergy (SE) contributions, MaR = u(p3)(M$

+ MaSE)u(Pl)

.

(2.19)

The vertex modification of Fig. V-l(c) (27T)4

A;2 (p, - fc)2 [ ( p i - k)2 _

TO2]

'

^ ^

has the same form as the electromagnetic vertex correction for the electron discussed previously (c/. Eq. (1.19)) except that it contains the weak vertex 7 a ( l + 75). Upon employing the Feynman parameterization in Eq. (2.20) and using the muon equation of motion, the extension of the vertex amplitude to d dimensions can ultimately be expressed in terms of hypergeometric functions,

(2.21)

142

V Leptons

where

_F(3-f,l;f;Q 1

(d-3)(d-2)

(d- 3 ) F ( 2 - j , 1; j ; Q (d-4)(d-2)

f,l;g-l,fl '(d-3)

=

2F(3 -1,2; 1 + 1;fl

'

( 2 2 2 )

2F(3 - f, 1; f;Q

d(d2) (d-2)(d-3) " For the muon self-energy (SE) amplitude of Fig. (V-l(d)), we write

remembering that a factor of e2 has already been extracted in Eq. (2.18). Implementing the Feynman parameterization and integrating over the virtual momentum yields an expression,

;

2 ;

f / dx [md+x(2d)^1

^ ( 2 . 2 4 )

which with the aid of App. C-4 can be written in terms of hypergeometric functions,

f) i — d

d+ 2 p

When the self-energy is expanded in powers of j£ — m^, the leading term is just the mass shift, which is removed by mass renormalization. We require the /^derivative of T,(p) evaluated at jj = mM. Being careful while carrying out the differentiation to interpret p2 factors as /£/£, we find 1

r(2-|)

i-d

—IZA-,

I2"26)

n •

It is this quantity multiplied by the vertex 7 a (l + 75) which ultimately gives rise to M^. However, in addition to mass renormalization there is also wavefunction renormalization, whose effect is to reduce the above quantity by a factor of 2, yielding 1

^ » )

7

°(

l + i > )

-

<2 27>

-

V-3 The r lepton

143

In principle, there also exists the electron self-energy contribution. As can be verified by direct calculation, this vanishes because the electron is taken as massless. Thus, we conclude that

V

1 d-1 2 "~4(4-d)(d-3) *

'

The net effect of the self-energy contribution is to replace A\ in the vertex amplitude of Eq. (2.21) by A = Ax + A2. Insertion of the radiatively corrected amplitudes into Eq. (2.16) leads to a transition rate TR which expanded to lowest order in e = 4 — d, has the form

_Gjm» 3a (( ml \ " € r(2 - f) * I92TT3 4 4^ ^3 322^^ rr(|-|)r(|-f)r(5-|) (|-|)r(|-f)r(5-|) 24

10-127

,.

571 2

o2

"

(229)

5

Like the bremsstrahlung contribution, the radiatively corrected decay rate is found to be singular in the e —> 0 limit. However, the final result which is gotten by adding the radiative correction of Eq. (2.29) to that of the bremsstrahlung expression of Eq. (2.19) is found to be free of divergences,

^s- -

1

^

- - j

,

(2.30)

and we obtain the finite QED correction term appearing in Eq. (2.12). V-3 The r lepton The heaviest known lepton is r(1784), having been discovered in e + e~ collisions in 1975. There exists also an associated neutrino vr with current mass limit mVr < 35 MeV. Like the muon, the r can decay via purely leptonic modes, M" + Pfj, + vT _> e~ + ue + uT T

(31)

However a new element exists in r decay, for numerous semileptonic modes are also present [Ts 71],

144

V Leptons Inclusive decays

It is possible to obtain a simple estimate of the relative amounts of the leptonic and semileptonic decays as follows. Because the r is lighter than any charmed state, semileptonic decay amplitudes must involve the weak quark current J& = Kd d-fil + -%)u + Ks sY(l + 75)M ,

(3-3)

where VU(\ and VQS are KM mixing elements. Neglect of both all final state masses and also effects associated with quark hadronization (an assumption only approximately valid at this relatively low energy) implies the simple estimates

r emileptonic s

_

^

1 leptonic

^

B

B

2

~ j—j^

NC=S

^

,~ , x

0.2 ,

where Nc is the number of quark color degrees of freedom. The predicted ratio of semileptonic-to-leptonic rates is somewhat smaller than the existing experimental value, J- semileptonic 1 leptonic

= 1.85 ±0.02.

(3.4b)

expt

In order to improve upon the above naive analysis, one must include additional effects. We shall limit our discussion to a consideration of electroweak and QCD perturbative corrections on the predicted semileptonicto-leptonic ratio in Eq. (3.4a). Although there exists an argument that QCD perturbation theory is justified at the relatively low scale of mT [Br 88], this claim is controversial. In the following, we shall simply quote the result of working respectively to second and third order in the electroweak and strong perturbations,

r,semileptonic -*- leptonic

theo

l

L

7T

\mT 2

X

(3.5)

V-3 The r lepton

145

where the term containing the Z° mass accounts for the different electroweak radiative corrections of leptonic and semileptonic processes* and the QCD correction is computed with the aid of Eq. (II-2.74). The qualitative effect of such corrections is seen to improve agreement between theory and experiment. Exclusive leptonic decays The r affords an opportunity to test the principle of lepton universality, i.e. the premise that the only physical difference among the charged leptons is that of mass. In particular, all the charged leptons are expected to have identical weak couplings. Thus, the muon and electron branching ratios should be equal, aside from a relative phase space suppression for the former,

^r _ei> T

eJ / T

- 1-8 f^V-12 feV In 4m\+ - ••- 0.973 . (3.6) \m ) \m ) T

T

From the measured leptonic branching ratios [Dan 92], 0.1789 ±0.0014

(r-> ePci/T) ,

we obtain i?expt = 0.969 ± 0.019, which is consistent with the prediction of Eq. (3.6). Another noteworthy consequence of lepton universality is the observation that the rates for r-^e+ue+uT and /i—>e+P e+z^ must have the same form (cf. Eq. (2.12)). Thus, their lifetimes should obey the scaling relation

Using the experimental value for BrT^e^eZ,r given in Eq. (3.7) as input, this can be used to predict the tau lifetime, which we display together with the measured value, r r | thy = (2.942 ± 0.024 ± 0.038) x 10"13 s , i9

(3.8b)

r r | expt = (2.95 ± 0.03) x 10"13 s , where the two uncertainties which we associate with r T | thy arise respectively from BrT-+ejyel/T and mT. Agreement between theory and experiment * This point is considered in detail in Sect. VI-1.

146

V Leptons

in Eq. (3.8b) is somewhat unsatisfactory. Since the data base continues to expand, a sharper test of universality should be forthcoming. The momentum spectra of the electron and muon modes also probe the nature of r decay. The Michel parameter p of Eq. (2.8) should equal 0.75 for the usual V-A currents, zero for the combination V+A and 0.375 for V or A separately. The observed value p — 0.70 ± 0.05 is in accord with the V-A structure. Exclusive semileptonic decays Matters are somewhat more complex for the hadronic final states, due in part to the large number of modes. However, for many of these we can make detailed confrontation of theoretical predictions with experimental results. We begin by noting that the semileptonic decay amplitude factorizes into purely leptonic and hadronic matrix elements of the weak current J^k = J£ pt + j£ adr ,

a -Msemilept = "7= ^

Hfr

,

p = (hadrons|j£ adr |0) - (hadrons|Cl where the flavor SU(3) components l+i2 (4-M5) induce AS = 0 (AS = 1) transitions respectively. It is easiest to analyze the modes containing a single meson, T~-+Meson + */r

(Meson = TT, K~, p"(770), if*-(892)) . (3.10)

Weak-current matrix elements which connect the vacuum with Jp = 0~,l + hadrons are sensitive to only the axial-vector current, whereas Jp = 0 + ,l~ states arise from the vector current. In each case, the vacuum-to-meson matrix element has a form dictated up to a constant by Lorentz invariance, (7r-(q) |Ajr i2 (0)| 0) = -iyftF^

,

(3.11a)

(if"(q) |4~ i 5 (0)| 0) = -iV2FKq,

,

(3.11b)

2

(p"(q,A) | ^ ( 0 ) | 0 ) = V 2 ^ ( q , A) ,

(3.11c)

iT-(q, A) |^fwhere the quantities gp and QK* are the vector meson decay constants. These quantities contribute to the transition rates for pseudoscalar (P)

V-3 The r lepton

147

and vector (V) emission, and we find from straightforward calculations

(3-12) 2

mT

where rap, ray are the meson masses, 77KM — c? for AS = 0 decay and rfcM = c^s\ for AS = 1 decay. It is possible to imagine using the above formulae to extract the quantities F ^ , . . . , gx* from tau decay data. However, they are obtained more precisely from other processes and in practice, one employs them in tau decay to make branching ratio predictions. In Chaps. VI, VII, we shall show how the values Fn = 92 MeV and FK/F^ = 1.22 are found from a careful analysis of pion and kaon leptonic weak decay. Interestingly, the hadronic matrix elements which contribute there are just the conjugates of those appearing in Eqs. (3.11a), (3.11b). By contrast, the quantity gp is obtained not from weak decay data, but rather from an electromagnetic decay such as p°—• ee. That the same quantity gp should occur in both weak and electromagnetic transitions is a consequence of the isospin structure of quark currents. That is, the electromagnetic current operator is expressed in terms of octet vector current operators by J^m — Vjf + -7?^?Since the latter component is an isotopic scalar whereas the p meson carries isospin one, it follows from the Wigner-Eckart theorem that

<0|J£V(P)> = (0|^>0(P)> = ^ = 9PeM

• (3.13)

The transition rate for the electromagnetic decay p°—> ee is given, with final state masses neglected, by (3.14) from which we find gp/mp = 0.198 ± 0.009. The K* vT mode can be estimated by using the flavor SU(S) relation gK*—9p- The predictions for single hadron branching ratios are collected in Table V-3, and are seen to be in satisfactory agreement with the observed values. A somewhat different approach can be used to obtain predictions for strangeness conserving modes with Jp = 1~. Since matrix elements of the vector charged current can be obtained through an isospin rotation from the isovector part of the ee annihilation cross section into hadrons,

148

V Leptons Table V-3. Some hadronic modes in tau decay. Mode T~ —• 7T~

Hadronic input + VT

r

T" -> K~ + I/T r " -> p " 4- vT

A:

T~ —•

K*~ -\- Vr T~ —• 7T~7r~7r 7r°iyr T~

a

—> 7T~7T 7T 7r°i/r

C%Sl9l

a(e+e~ —^ • h a d r ) cr(e + e~ —J • h a d r )

Br[thy]°

Br [expt]a

11.4 0.8 23.4 ± 0.8 1.1 ±0.1 4.9 0.98

10.8 ± 0.6 0.78 ± 0.08 22.2 ±1.0 1.33 ±0.13 4.8 ± 0.4 0.98 ±0.14

Branching ratios are given in percent,

we can write for a given neutral 7 = 1 hadronic final state / ° ,

where we have defined / The r

(3.16) transition into the isotopically related charged state / ,

in

(h

is governed by the same function Uf(q2) of hadronic final states as occurs in Eq. (3.15). Including the lepton current and relevant constants, and performing the integration over vT phase space yields a decay rate

tf {ml - q2)Hml + 2
f*

(m 2 - S ) 2 (m 2 + 2 s ) s ^^ / O ( S ) . (3.19)

Thus, we find, e.g. for 4?r final states, the results listed in Table V-3. Yet another hadronic channel of interest is that with Jp = 1 + , which is expected to be dominated by the axial-vector meson ai(1260). In view of the decay chain a{

- > p°n~ - • 7T+7r~7r" ,

(3.20)

this resonance should appear in the 7r+7r~7r~ spectrum in r decay. In this instance, r decay data is being put to the somewhat novel use of helping

V~4 The neutrinos

149

to pin down the a\ mass and width values. The determination of these quantities has been a long-standing problem in hadron spectroscopy due in part to the presence of substantial background in the data, whereas r decay data is expected to relatively free of such problems. The current world averages are [RPP 90] mai = 1260 ± 30 MeV ,

r fll = 330 -> 500 MeV ,

(3.21a)

to be compared with the values inferred from r decay [Dan 92], mai = 1211 ± 70 MeV ,

r a i = 446±? MeV ,

(3.21b)

There exist numerous additional hadronic decay modes of the r lepton. Examples include final hadronic states containing KK,KKTT, etc., and it is possible to analyze each of these with various degrees of theoretical confidence. Another interesting use of the r semileptonic decay has been to confirm by inference the fundamental structure of the weak quark current from the absence of the mode r~ —> 7r~r]uT. This mode, proceeding through the vector current, would violate G-parity invariance. Here Gparity refers to the product of charge conjugation and a rotation by TT radians about the 2-axis in isospin space, G = Ce~i7rl2 .

(3.22)

A weak current which could induce a AG ^ 0 transition is referred to as a second class current. Such currents do not occur naturally within the quark model. The 7r~r) VT mode has not been detected, with an existing sensitivity of Br r _>,,.-^ < 0.9%. This result, consistent with the absence of second class currents, fits securely within the framework of the Standard Model. V—4 The neutrinos Throughout this book, we assume that neutrino mass is zero. However, if neutrinos do have mass [KaGP 90, BoV 87, MoP 91], the phenomenon of neutrino mixing will arise, analogous to the quark mixing of Sect. II—4. For simplicity, let us consider the case of just two neutrino generations, v e and i/r, which are defined by the weak interaction. The weak interaction eigenstates are related to mass eigenstates v\ and v
\vr)\

V-sin0 v

where 6y is the vacuum mixing angle. How is it possible to determine the mass of neutral and weakly interacting particles like neutrinos? One probe involves measurement of the electron energy spectrum in nuclear beta decay, (A, Z) —» (A, Z + l) + e~ + ve.

150

V Leptons

If, for the sake of argument, we assume rai < 7712, then the maximum electron energy (end-point of the electron energy spectrum) in nuclear beta decay occurs at ml

.

(4.2)

Studies of tritium beta decay yield the best bound, mVf, < 9 eV. With the birth of neutrino astronomy via detection of a neutrino burst from supernova 1987A, an independent check of the ve mass limit has yielded a consistent, although slightly weaker limit, mVe < 18 eV. Useful information from nuclear beta decay data is also contained in the electron energy spectrum N(Ee) away from the end-point. It follows from Eq. (4.1) that N(Ee) should be the sum of two components, corresponding respectively to detection of v\ and v
Neutrino oscillations Despite the lack of direct evidence for neutrino mass, there is a vigorous ongoing experimental effort to search indirect signals like neutrino oscillations. In the two-generation description, an electron-type neutrino created at time t = 0 and propagating with momentum p will have at subsequent time t the state vector \ue(t)) = |i/i) cos0 v e~iElt + \u2) sin0 v e~iE2t = |i/ e )(cos 2 0 V e~iElt + sin 2 6 V e~iE^) + |i/r) sin(9v cos(9v (e~

iE2t

iElt

- e~ )

(4.3) ,

where E\ = >/p 2 + m 2 and E
PUe(t) = 1 - PVT(t) . (4.4)

Since neutrino mass is small compared to neutrino momentum in typical situations, we approximate

^i

... ,

(4.5)

so that PVT (t) ~ sin2 20V sin 2 [Am 2 L/4^] ,

P^ (t) = 1 - PVT (t) .

(4.6)

where Am 2 = ml — mf. This is the expression that is usually employed in two-flavor analyses of neutrino oscillation experiments. It contains the

V~4 The neutrinos

151

two theoretical parameters 6y and Am 2. Observe that we have written the expression in terms of the path length L ~ t/c instead of the time t. Sometimes, L is expressed in units of a 'vacuum length' Lv, PUr(t) ~ sin2 2<9V sin2[7rL/Lv]

(Ly = AnE^/Am2)

.

(4.7)

For Am2 = 2.48 eV2 and Ev = l MeV, one has L v = l m . Terrestial searches for neutrino mixing There are two categories of terrestial searches for neutrino oscillations. The first consists of 'disappearance' experiments in which detectors are placed outside a reactor that emits an intense ve flux, and reaction rates for processes like uep —> e+n are measured as a function of the reactordetector separation L. If no oscillation is present, the measured rates should scale like L~2 as L is varied. However, neutrino mixing superimposes the additional factor P^e(L), and it is this modulation which signals the existence of the mixing phenomenon. The second category involves 'appearance' experiments. Suppose an intense 7r± flux is generated at a proton accelerator by directing the proton beam onto a massive target (such an operation is called a 'beam-dump'). Escaping from the target will be a high flux of v^, v^ and ve particles resulting from the decays TT+ —> /i + v^ /i + —• e + z/ e ^. Where are the ue? They are in fact present, but since most of the \i~ flux from TT~ decay will be absorbed by the target before decay, there will be a suppression of ve particles leaving the target. Thus, one probes the generation of ve from v^-ve mixing by measuring the rate for ve interactions at the detector. Results of disappearance and appearance experiments are compiled in [RPP 90]. No positive signals from either type of experiment have yet been observed, and the region of parameter space with sin2 26y &nd Am2 both 'large' has been ruled out, e.g. ve-f+ve disappearance experiments imply the constraints sin2 20V^O.14 (for large Am2) and Am2^0.014 eV2 (for sin2 2#v — !)• Of course, this need not indicate that neutrino mixing does not occur, since regions of very small Am 2 and #v can never be ruled out. In particular, taking (Eu) ~ 10 MeV and L ~ 100 m, we estimate a sensitivity for terrestial experiments at the level of Am 2 > 10"1 eV2 for reasonable values of the mixing angle. This can be improved with the study of solar neutrinos, where L ~ 1.5 x 1011 m. In addition, one encounters the interesting phenomenon of neutrino propagation in matter. Solar neutrinos As a result of various nuclear processes in the Sun's core, solar neutrinos are created with a spectrum of energies. For the sake of simplicity, we

152

V Leptons

can classify energies as high (H), intermediate (/), and low (L), EH>6

MeV :

Ei ~ 0.8 -+ 2 MeV :

Bs -> (5e 8 )* + e+ + ue J5e7 + e" -> Li7 + i/e

0 15

EL ~ 0.2 -+ 0.5 MeV :

_^ ^15

+

e+ +

^

+

p + p - * d + e + z/e ,

and solar nuclear reactions which contribute to each class are also shown. There are a number of existing and proposed solar neutrino detectors which probe different parts of the neutrino spectrum. Those which have published solar neutrino data are the Homestake mine chlorine detector [Da 89], the Kamiokande water Cerenkov detector [Hi et.al. 90], and the SAGE gallium detector [Ga 90]. It is common to summarize the data in terms of R = /£ b s //^ S M , the ratio of observed solar neutrino flux to that predicted on the basis of the Standard Solar Model [Bah 90], and to express R in terms of the survival probability PUe for solar neutrinos as measured on Earth. Letting P^e (k = H, / , L) represents the survival probability of z/e's in the high, intermediate, and low parts of the energy spectrum, we can approximate the energy dependence of each type of detector as ifamst ^ 0.78 P ^ + 0.22 Ple

[ye + Cl37 -> Ar37 + e~)

ifaam ^ 0.14 + 0.86 P%

(i/c + e~ -> ve + e~)

{ye + Ga71 -> Ge71 + e~) , (4.9) where reactions used in each detector are also shown. To date, the detected flux of high energy neutrinos is found to be less than that predicted, i?Gai ^ 0.1 P? + 0.36 Pi + 0.54 P£

#Hmst = 0.27 ± 0.05 ,

#Kam = 0.46 ± 0.08 .

(4.10)

Taken at face value, Eqs. (4.9-4.10) indicate the suppression of high energy neutrinos is accompanied by an even stronger suppression of intermediate energy neutrinos. However, because of the strong sensitivity to experimental uncertainties and details of solar structure, the interpretation of these results is not yet decisive. Fortunately, gallium detectors have begun to observe events, and if one assumes values P^ ~ 0.38, P j ~ 0, sensitivity to the less model dependent flux of lower energy neutrinos is predicted, i?Gai ^ 0.04 + 0.54 P£ .

(4.11)

In addition, heavy water [Ea et. al. 86] and borex [RaP 88] detectors will provide sensitivity to charged and neutral current neutrino scattering

V~4 The neutrinos

153

events, „

i

x

Heavy water:

-

^

•- •-

(charged)

(neutral) V ; Borex:

.

<

(charged)

(4.12)

(neutral) , where i/a is a neutrino of arbitrary flavor. Comparison of neutral and charged current events within a given detection is insensitive to calculations of absolute solar fluxes, thus permitting a direct test of the oscillation scenario. Several possibilities for attributing the solar neutrino deficit to neutrino properties are: 1) 2) 3) 4)

neutrino neutrino neutrino resonant

decay into noninteracting ('sterile') particles, oscillation in the vacuum, spin-flip induced by magnetic effects within the Sun, neutrino oscillation within the Sun.

Of these, let us consider the last, the MSW effect [Wo 78, MiS 86], since it is perhaps the most natural explanation and has attracted a good deal of attention. Considering the two-flavor model and absorbing any common phase factors into the state vectors, we have for neutrino propagation within the Sun l

,d_ I" |i/c) 1 _ Am2 / - cos 20V + 4 ^ p / A m 2

dt [|i/ T )J ~~ AEV V

sin20 V

sin 20V \ I We)'

cos20 v y [\vT)

- (4.13) where p = \/2G^Ne and Ne = Ne(r) is the solar electron number density. The factor p arises from the vee charged current interaction,* and modifies the ve propagation relative to any other neutrino flavor, thus giving rise to an additional phase in \ve(t)). The vacuum mixing of Eq. (4.1) is thus altered to

K)l

\vT)

=

/ cos0M

\ — sin0M

where 0M, the mixing angle in matter, obeys tan 20M =

cos 20 V -

* Neutral weak current effects are common to all neutrino flavors.

154

V Leptons

with the critical electron number density N£ given by ^rr

N s

Ara 2 cos20 v

_ Am2(eV2)

3

- -JTiG^r - - ' " s s W -

94

*

on

10 cm

o

•

Since Ne will decrease from the solar center to the surface, provided Ne ~ 1026 cm" 3 ^> N£ at or near the solar center, interesting effects are expected within the Sun at Ne ~ iVec [RoG 86], [Hax 87], [BaB 90]. Two solutions having the required suppression of high energy solar neutrinos are found [GeR 86], an adiabatic solution* corresponding to large mixing (sin2 20 ~ 1) with little suppression at intermediate and low energy, and a nonadiabatic solution which can be approximated as log10 Am2(eV2) + log10 sin2 20V ^ -7.5 ,

(4.17)

with stronger suppression at lower energies. Although existing results appear to support the latter solution, data from low energy neutrinos will be needed to provide a stringent test of the MSW mechanism. Dirac mass and Majorana mass In a theory containing both left-handed and right-handed massless neutrinos of a given flavor, the neutrino fields are expressible as tp^/ = F L V ^ \ R

R

where the projectors FL = ^(1 ± 75) ensure that the masslessfieldsij)\/ R R . have just two nonzero components. It is possible to construct a Hermitian, Lorentz-invariant, lepton-conserving interaction which couples the two chiralities £Dira, = - m i ) ( # # + # # ) = - m D ^ ^ , (4.18) where the field ip^ is now a four-component entity.^ Such an interaction is said to give rise to a Dirac contribution to fermion mass, and in the Standard Model it is this mechanism (via the Higgs effect) which gives rise to fermion mass. Of course, if the field ip^ does not exist, then no such construction is possible. This is why neutrinos are massless in the Standard Model. A logically independent source of neutrino mass, called a Majorana contribution, is available even in the absence of right-handed neutrinos. In this instance, a chiral field such as ^L *S c o u Pl e d to its conjugate ^ * Neutrino propagation within the Sun is 'adiabatic' if the fractional change in Ne is small per neutrino oscillation cycle. t Note that lepton conservation is ensured by the invariance of Z^Dirac under the field transformations ?/>L —• e~iotipL and ^R, ~> ^ R e i a .

Problems

155

(cf. Eq. (IV-9.5)), ^Majorana = -™>M{$^

C

^Y, + h x - ) *

( 4 -19)

Although ^Majorana is Lorentz invariant and Hermitian, it does not conserve lepton number. One can use this feature to seek experimental evidence for Majorana mass in nuclear double beta decay, A,Z + 2) + 2e-+2Pe A,Z + 2) + 2e"

(m,.=0) (Majorana) .

The first case in Eq. (4.20), which occurs in the usual description with massless neutrinos, corresponds to a second-order weak interaction transition, and has been observed in Se82 [E1HM 87] with half-life T 1 / 2 = 1.1^0*3 x 1020 yr- This constitutes the slowest transition which has been detected by a laboratory measurement. The second case (neutrinoless double beta decay) can occur for Majorana neutrinos, which upon being emitted from a beta decay vertex can propagate to initiate a second beta transition. Neutrinoless double beta decay has not been observed in Ge76 at the level of T 1 / 2 > 5 x 1023 yr [Ca etal. 87]. This places the limit ^M < 0.7 —• 1.8 eV on the effective Majorana mass. The range of values indicates the uncertainty in evaluation of nuclear matrix elements and 'effective' mass is a linear combination of Majorana mass eigenstates, each weighted by the absolute square of the corresponding mixing-matrix element. Problems

1) Muon Decay a) Obtain the leading (9(ra 2 /ra 2 ) correction to the Fermi model expression Eq. (2.17) for the muon decay width. b) Do the same for the leading O{m2tl/M^) correction. 2) Vacuum polarization and dispersion relations The vacuum polarization II(g2) associated with a loop containing a spin one-half fermion-antifermion pair, each of mass m, can be written as the sum of a term containing an ultraviolet cutoff A and a finite contribution ft2

U(q2) = - \l In ^ - 2 f dx x(l - x) In (1 - ^x(l - x) a , A2 i

156

V Leptons a) Show that flf(q2) is an analytic function of q2 with branch point at q2 = 4m2 and with Im tlf(q2) = aRf(q2)/3, where

is related to the rate for radiative pair creation via

-p/)(f\JZaWWL\o)

= (-flV + 9 ^

b) Use Cauchy's theorem and the result of (a) to express 3TT J4m2

s(s - q

2

- le)

2

c) The form of tif{q ) given in part (a) can be re-expressed in a dispersion representation. First change variables in (a) to y = 1 — 2x and integrate by parts to obtain

(V " I") Then, change variables again to s = 4m2/(1—y 2) and demonstrate that the dispersion result of (b) obtains.

VI Very low energy QCD — pions and photons

At the lowest possible energies, the Standard Model involves only photons, electrons, muons and pions, as these are the lightest particles in the spectrum. We give a separate discussion of this portion of the theory because it can be done with a higher level of rigor than most other topics. The main tool is chiral symmetry, in particular the effective chiral lagrangians. Pion physics gives us a chance to see how these techniques work and to test them in a relatively clean setting. VI-1 QCD at low energies The SU(2) chiral transformations AA i>L,R =

,

-> exp (-iOL,R' T)I\)L,R

(1.1)

WL,R

almost give rise to an invariance of the QCD lagrangian for small rau, m^, but do not appear to induce a symmetry of the particle spectrum. This is because the axial symmetry is dynamically broken (i.e. hidden) with the pion being the (approximate) Goldstone boson. Vectorial isospin symmetry, i.e., simultaneous SU(2) transformations of ipL a n d ipR, remains as an approximate symmetry of the spectrum. Isospin symmetry is seen from the near equality of masses in the multiplets (T^TT 0 ), (K+,K0), (p, n), etc. In the language of group theory, we say that SU(2)L X SU(2)R has been dynamically broken to SU(2)y> What is the evidence that such a scenario is correct? Ultimately it comes from the predictions which result, such as those which we detail in the remainder of this chapter. There is as yet no rigorous proof that QCD does indeed undergo dynamical symmetry breaking with the pattern observed in Nature. Lattice Monte Carlo techniques [Cr 83] are coming close to such a proof, and perhaps need only the inclusion of dynamical fermions to become convincing. However, our confidence in this symme157

158

VI Very low energy QCD - pions and photons

try realization has historically been developed mainly through theoretical self-consistency and phenomenological success. The effective lagrangian for pions at very low energy has already been developed in Chap. IV. In particular we recall the formalism of Sect. IV6 which includes couplings to left-handed (right-handed) currents ^ ( x ) (r^x)), and scalar and pseudoscalar densities s(x) and p(x), with the resulting O(E2) lagrangian,

C2 = *f TV (D,UD^) U = exp(ir • TT/F^) ,

+ *f TV \ = 2B0(s + ip) ,

where D^U = d^U + ilJU — iUr^ and BQ is a constant. QCD in the absence of sources is recovered with l^ = r^ — p = 0 and s = m, where m is the quark mass matrix.

Vacuum expectation values and masses With a dynamically broken symmetry, the lagrangian is invariant but the vacuum state does not share this symmetry. A useful measure of this noninvariance in QCD is the vacuum expectation value of a scalar bilinear, ( 0 | ^ | 0 > = <0|ViLVi?|0> + <0|^VL|0>

.

(1.3)

Up to small corrections, isospin symmetry implies (0|im|0) = (0|
.

(1.4)

Such matrix elements, if nonzero, cannot be invariant under separate left-handed or right-handed SU(2) transformations. Indeed it is evident from Eq. (1.3) that the vacuum expectation value couples together the left-handed and right-handed sectors. One way that the vacuum expectation values of Eq. (1.4) affect phenomenology is through the pion mass. If u and d quarks were massless, the pion would be a true Goldstone boson with m^ = 0. The part of the QCD lagrangian which explicitly violates chiral symmetry is the collection of quark mass terms Hm = —C

m

= muuu + mddd .

(1.5)

To first order in the symmetry breaking, the pion mass is generated by the expectation value of this hamiltonian, ml = (n \muuu + rriddd\ TT) .

(1.6)

This quantity can be related to a vacuum expectation value by using soft-pion techniques (see Sect. IV-4) to remove both pions, resulting in

VI-1 QCD at low energies

159

where qq represents either uu or dd. Equivalently, the effective chiral lagrangian can be used to find the vacuum expectation value ( 0 | ^ | 0 ) . Using the notation of Sect. IV-6, we have ml = (mu + md)B0,

(0 \qq\ 0) = —^

= -F^B

0

,

(1.8)

which has the same content, to this order in the energy expansion, as Eq. (1.7). Thus since both BQ and the quark masses are required to be positive, consistency requires that (0|gd independently. As an aside, we note that for Goldstone bosons there is a clear answer to the perennial question of whether one should treat symmetry breaking in terms of a linear or quadratic formula in the meson mass. For states of appreciable mass, the two procedures are equivalent to first order in the symmetry breaking since 6(m2) = (mo + 6m) — TTIQ = 2rriQ 6m + . . . .

(1-9)

However when the symmetry expansion is about a massless limit, the m vs m2 distinction becomes important. Because pions are bosonic fields we require their effective lagrangian to have the properly normalized form, (0^7r^7rra£7r-7r)+... .

(1.10)

The prediction for the pion mass must then have the form m\ = (mu + md)B0 + (mu + md)2C0 + {mu - md)2D0 + . . . .

(1.11)

In principle, Nature could decide in favor of either m\ ex mq orra^ex m2 depending on whether the renormalized parameter BQ vanishes or not. However, the choice Bo = 0 is not 'natural' in that there is no symmetry constraint to force this value. Since one generally expects a nonzero value for Bo, the squared pion mass is linear in the symmetry breaking parameter mq. There is every indication that BQ ^ 0 in QCD. When we add the kaons and the eta to the analysis, this will imply that SU(3) symmetry breaking and the Gell-Mann-Okubo relation must also be treated with squared masses.

Pion leptonic decay and Fn Throughout our previous discussion of chiral lagrangians, the pion decay constant Fn has played an important role. It is defined by the relation »Fi r p^* ,

(1.12)

160

VI Very low energy QCD - pions and photons

where the axial vector current A^ is expressible in terms of the quark

fields V = 0 as

4 = V^75y •

(1-13)

This operator is probed experimentally in the decays n —> eve and TT —> • ^ which are induced by the weak hamiltonian

(1.14)

The decay TT+ —• /i"^^ has invariant amplitude

where the Dirac equation has been used to obtain the second line. An analogous expression holds for TT4" —> e+ve. We see here the well-known helicity suppression phenomenon. That is, the weak interaction current contains the left-handed chiral projection operator (1 + 75) which in the massless limit produces only left-handed particles and right-handed antiparticles. However, such a configuration is forbidden in the decay of a spin-zero particle to massless fiP^ or eve because the leptons would be required to have combined angular momentum Jz = 1 along the decay axis. Thus the amplitudes for n —> /iP^, eve must vanish in the limit m^ = me = 0. Since the neutrino is always left-handed, the /x+, e + in pion decay must have right-handed helicity to conserve angular momentum. It is helicity flip which introduces the factors of m^, me. The decay rate is found to be

^

[j

.

(1.16)

However, before using this expression to extract the pion decay constant, one must include radiative corrections. We shall do this in some detail as similar considerations will be used to obtain the kaon decay constant in Sect. VII-2 and in the analysis of nuclear beta decay in Sect. XII4. Since a complete analysis would be overly lengthy, we present a simplified argument which stresses the underlying physics. A more formal presentation, with identical results, appears in [Si 72]. In Chap. V we found that the radiative correction to the muon lifetime is ultraviolet finite even in the approximation of a strictly local weak interaction. However, this is not the case for semileptonic transitions, as

VI-1 QCD at low energies

161

Fig. VI-1 Photonic radiative corrections to the weak quark-quark interaction can be easily demonstrated. Consider the photon loop diagrams shown in Fig. VI-1. We divide the photon integration into hard and soft components. The former, which determine the ultraviolet properties of the diagrams, have short wavelengths A < < i?, where R is a typical hadronic size, and are sensitive to the weak interaction at the quark level. In Landau gauge (i.e. £=0), the ultraviolet divergences arising from the wavefunction renormalization and vertex renormalization diagrams depicted in Fig. VI-1 vanish. For example, the vertex term is r(u.v.)

iGF

f d4k 1

2

k ^7A(1 + 7

where Qie is the electric charge of the ith particle. Using J

(1.17)

k* ~ 4 J (27r)4fc4

(2TT)4

we find that /Vertex = 0 as claimed. It is clear, employing a Fierz transformation, that photon exchange between particles 4,1 and 2,3 is also ultraviolet finite. This result simply represents the nonrenormalization of the vertex of a conserved current found in Chap. V. The only ultraviolet divergences then arise from final state and initial state interactions, i.e. photon exchange between particles 4,2 and 3,1 , for which d k1 r(u.V.) —^G JP 2/-I

f *

(

j (2^^ ( x «4y^ ^F

C

7

A

( l + 7 5 ) ^ 2 7 ^ 7 A ( 1 + 75)«3 + (2,4 -> 1,3)

/^ yr^ ! _

A r_

_

/.,

, _ x

_ _.a

a X

V2 327r 2 ^ 4 ^ ' l + 7 5 ) u i ] + (2,4-> 1,3) .

(1.19)

162

VI Very low energy QCD - pions and photons

Using the identity in Eq. (C-1.5) for reducing the product of three gamma matrices, Eq. 1.19 becomes 1 ^

= -M{0) x —(Q4Q2 + QsQi) ln(A/// L) ,

(1.20)

where M.^ is the lowest-order vertex. However, the full calculation of the radiative corrections must include the propagator for the H^-boson as well. When the contact weak interaction is replaced by the ^-exchange diagram and is added to that with the photon-exchange replaced by Zexchange, one obtains a finite result at the ultraviolet end with A = mzThe integral is cut off at the lower end at some point /J,L ~ rri£ below which the full hadronic structure must be considered. In the case of muon decay we have

Thus as found in Chap. V, there is no divergence. On the other hand, for beta decay we obtain QeQu + Qv Qds =

(1.22)

We observe that there exists an important difference between the betadecay effective weak coupling (Gp) and the muon-decay coupling (G^) (1.23) This hard photon correction must be added to the soft-photon component which can be found be evaluating the radiative corrections to a structureless ('point') pion with a high energy cutoff //#. These were calculated long ago with the result [Be 58, Ki 59],

'-) .

r (0)

a-24)

where

^z l n x l l x — 1

\\n(x2-l)-2\nx--A

J[ , ) \nx

4 3 lOz -?, 15z2 - 2 1 + 4 ^ L ( l x + z\wx + ^ , 4 x2 — 1 4 (x2 — I)2 4(x2 — 1) (1.25) with L(z) = JQ J- ln(l — t) being the Spence function and x = m^/m^. Adding the hard and soft-photon contributions with //# = JIL — mp, we 2

VI-2 Chiral perturbation theory to one loop

163

find the full radiative correction,

l + f (B{x) + 3In ™* + In ^ m

2?r V

Taking Vud

7r

- 6In ^

rnp

m^

(1.26) from Sect. XII-4, and T ^ + i / = 3.841 x 10 s" , we find Fv = 92.4 ± 0.2 MeV , (1.27) {

7

1

where we have appended an uncertainty associated with possible radiative effects O(a/27r) which are not included in Eq. (1.26). For chiral symmetry applications in this book we shall generally employ the value Fv~92MeV

.

(1.28)

A clear indication of the importance of radiative corrections can be seen in the ratio R = I 7+

which is strongly suppressed by the helicity mechanism discussed earlier. Application of the lowest-order formula given in Eq. (1.16) leads to a prediction

in disagreement with the measured value Rexpt = (1.218 ± 0.014) x 10"4 .

(1.31)

However, when the full radiative correction given in Eq. (1.26) is employed, the theoretical prediction is modified to become Rth

= R
+ .. ) = 1.235 x 10"4 ,

(1.32)

which is consistent with the experimental value.

VI-2 Chiral perturbation theory to one loop Let us summarize the development thus far. Interactions of the Goldstone bosons can be expressed in terms of an effective lagrangian having the correct symmetry properties. To lowest order in the energy expansion, i.e. to order E2, it suffices to use the minimal lagrangian of Eq. (1.2) at treelevel. In the SU(2) theory, this involves just the known constants Fn and m^. At the next order, one encounters both the general O(E4) lagrangian, given below, and also one-loop diagrams [ApB 81, GaL 84,85a]. The O(E4) lagrangian introduces new parameters, which must be determined

164

VI Very low energy QCD - pions and photons

from experiment. It is also necessary to give a prescription which allows one to handle the loop calculations. The general method is described in this section. The program is called chiral perturbation theory. If one works to order E4 in the energy expansion (this is the present state of the art), there are typically three ingredients: 1) the general lagrangian £2 (of order E2) which is to be used both in loop diagrams and at tree-level, 2) the general lagrangian £4 (of order E4) which is to be used only at tree-level, 3) the renormalization program which describes how to make physical predictions at one-loop level. The general O(E2) lagrangian has already been given in Eq. (1.2). Now we shall turn to the construction of the chiral SU(n) lagrangian to order E4. The order E4 lagrangian The O(E4) lagrangian can involve either four-derivative operators or twoderivative operators together with one factor of the quark mass term, X ~ 2mqBo (which itself is of order ra^ or m2K) or products of two quark mass factors. There are four possible chiral invariant terms with four separate derivatives, TV (

V

Tr

D

^

U

( D ^ U D V U * )

D

^

U

]

J

D

' T r( D ^ U D ^ U ^

V

U

V

,

[Tr

D

V

U (

^

^

^

J

]

) , T r ^ (2.1) 2

Other structures, such as

[Tr (x'tfDpU) Tr (Vt/tZ^f/)]

,

(2.2)

can be expressed in terms of these by using SU(n) matrix identities. For the case of SU(3), the operators in Eq. (2.1) are not linearly independent. The identities quoted in Eq. (II—2.17) can be used to show that Tr + Tr IDUUDVIP) • Tr iLPUirW)

- 2 TV

(2.3) leaving only three independent operators in this class. In 5C/(2), a further identity, 2Tr [D^UD^U]DVUDVU^

= [Tr

(D^UD^U^)2

,

(2.4)

V

)

VI-2 Chiral perturbation theory to one loop

165

leaves us with only two independent O(E^) terms. Another conceivable class of operators could have at least two derivatives acting on a single chiral matrix, such as • Tr (u^DuDvU^

Tr

.

(2.5)

However, since the E 4 lagrangian is to be used only at tree-level, all states to which it is applied obey the equation of motion,

D" (U^D^U) + i (xfC/ - tfX) = 0 •

(2.6)

This can be used to eliminate all the double derivative operators in favor of those involving four single derivatives or with factors of \. The remaining operators are reasonably straightforward to determine, and the most general O(E4) 5/7(3) chiral lagrangian is,* 10

= c*i [Tr (D^UD^U^ + a3 Tr

+ a2Tv (D^UD^

• Tr

(D^UD^D (

^

Tr

UX]))

a7[Tr +ai0Tr

^

(2.7) where L ^ , R^v are the field-strength tensors of external sources given in Eq. (IV-6.10). This is a central result of the effective lagrangian approach to the study of low energy strong interactions. Much of the discussion in the chapters to follow will concern the above operators and involve a phenomenological determination of the {a^}. In chiral 5C/(2), three operators become redundant. * We are using the operator basis first set down by Gasser and Leutwyler [GaL 85a]. However, in order to avoid confusion with the symbol for lagrangian, we shall call the coefficients ai instead of ^ .

166

VI Very low energy QCD - pions and photons Table VI-1. Renormalized coefficients in the chiral lagrangian £ 4 given in units of 10~3 and evaluated at renormalization point ji = nirj.

Coefficient a\

«S

Origin

0.65 ± 0.28 1.89 ±0.26 -3.06 ± 0.92

7T7T scattering

2.3 ±0.2

«5 r

Value

r

2a 7 + a 8 "10

K)° K)6 («S) 6

and Ki4 decay FK/FK

0.4 ±0.1

meson masses

7.1 ±0.3 -5.6 ±0.3

rare pion decays

-0.4±? -0

~0

a

Determined only with the additional assumption of 77-7/ mixing. b Vanishes in the 7VC —> 00 limit.

For completeness, we note that there may also exist two combinations of the external fields, £ext = Pi TV (L

+ R^BT) + 02 TV (x f x) ,

which are chirally invariant without involving the matrix U. These do not generate any couplings to the Goldstone bosons and hence are not of great phenomenological interest. However, if one were to use the effective lagrangian to describe correlation functions of the external sources, these two operators can generate contact terms. Finally, we summarize in Table VI-1 the values of the low energy constants {cti} as obtained phenomenologically. The determination and application of these coefficients will be discussed in subsequent sections of the book. They provide a characterization of the low energy dynamics of QCD. The renormalization program The renormalization procedure is as follows. The lagrangian, £2, when expanded in terms of the meson fields, specifies a set of interaction vertices. These can be used to calculate tree-level and one-loop diagrams for

VI-2 Chiral perturbation theory to one loop

167

any transition of interest. This result is added to the contribution which comes from the vertices contained in the O(E4:) lagrangian £4, treated at tree-level only. At this stage, the result contains both bare parameters and divergent loop integrals. One needs to determine the parameters from experiment. The first step involves mass and wavefunction renormalization, as well as renormalization of Fn. In addition, the parameters entering from £4 need to be determined from data. If the lagrangian is indeed the most general one possible, relations between observables will be finite when expressed in terms of physical quantities. All the divergences will be absorbed into defining a set of renormalized parameters. This fundamental result is demonstrated explicitly in App. B-2. There exists always an ambiguity of what finite constants should be absorbed into the renormalized parameters a\. This ambiguity does not affect the relationship between observables, but only influences the numerical values quoted for the low energy constants. Similarly, the regularization procedure for handling divergent integrals is arbitrary.* We use dimensional regularization and the renormalization prescription, •

(2 8)

-

where the constants 7$ are numbers given in Table B-l. When working to O(E4:) the following procedure is applied. One first computes the relevant vertices from £2 and £4. There are too many possible vertices to make a table of Feynman rules practical. In practice, the needed amplitudes are calculated for each application. The vertices from £2 are then used in loop diagrams, including mass and wavefunction renormalizations. The results may be expressed in terms of the renormalized parameters of Eq. (2.8). If these low energy constants can be determined from other processes, one has obtained a well-defined result. Including loops does add important physics to the result. The low energy portion of the loop integrals describes the propagation and rescattering of low energy Goldstone bosons, as required by the unitarity of the 5-matrix. One-loop diagrams add the unitarity corrections to the lowest order amplitudes and in addition contain mass contributions and other effects from low energy. The effective lagrangian may be used in the context either of chiral 5*7(2) or of chiral SU(3). Because SU(2) is a subgroup of 517(3), the general SU(3) lagrangian of Eq. (2.7) is also valid for chiral SU(2). However, the SU(2) version has fewer low energy constants, so that only * Care must be taken that the regularization procedure does not destroy the chiral symmetry. Dimensional regularization does not cause any problems. When using other regularization schemes, one sometimes needs to append an extra contact interaction to maintain chiral invariance [GeJLW 71]. The problem arises due to the presence of derivative couplings which imply that the interaction Hamiltonian is not simply the negative of the interaction lagrangian. The contact interaction vanishes in dimensional regularization.

168

VI Very low energy QCD - pions and photons

certain combinations of the a\ will appear in pionic processes. If one is dealing with reactions involving only pions at low energy, the kaons and the eta are heavy particles and may be integrated out, such that only pionic effects need to be explicitly considered. This procedure produces a shift in the values of the low energy renormalized constants OL\ such that the a\ of a purely 5/7(2) chiral lagrangian and an 5/7(3) one will differ by a finite calculable amount. In this book, we shall use the SU(3) values as our basic parameter set. The SU{2) coefficients can be found by first performing calculations in the SU(3) limit and then treating ra^ra^ as large. Equivalently, all may be calculated at the same time using the background field method [GaL 85 a]. The results are

3 = 2«\ + a\ - \t lK 2

,

«)r

- \lK - ±fa ,

=

- _

"if = cTw + tK

^ ^ l n 4 >

(2-9)

where we use the superscript (2) to indicate constants in the SU(2) theory and define £, = [ln(mf//x2) + l] /384TT2. In practice these shifts are much smaller than the magnitude of the low energy constants, so that v/e always simply quote the SU(3) value. Let us now calculate the mass and wavefunction renormalization constants to O(E4) in chiral 5(7(2). Setting % = (mu + VTI^BQ = m2,, we may expand the basic lagrangian as

(2.10)

where Fo denotes the value of Fn prior to loop corrections. When this lagrangian is used in the calculation of the propagator, the terms of O((p4) in £2 will contribute to the self-energy via one-loop diagrams which in-

VI-2 Chiral perturbation theory to one loop

169

volve the following d-dimensional integrals, 6ikI(m2) =iA F j k (0) = T,

2N

4-

4-

f ddk

m

y (a^M

2

(2.11) These contributions can be read off from £2 by considering all possible contractions among the O((p4) terms, and result in the quadratic effective lagrangian,

£ eff = I

8a»]] - ^

• *»f [32a? .

(2.12)

To one-loop order, there are no other contributions to the self energy. Observe that we have changed mo, FQ into ra^, Fn in all of the O(E4) corrections, as the difference between the two is of yet higher order in the energy expansion. If we expand in powers of d — 4 and define the —1/2

renormalized pion field as (pr = Zn '

]

(2.13) then the lagrangian assumes the canonical form £eff = \dtf>T • &><pr ~ \m\ipr • ifr

.

(2.14)

Note that, using the definitions of the renormalized parameters, the physical pion mass is identified as (2)r

A

(2)r

o

(2)rl .

^4

m (2.15)

170

VI Very low energy QCD - pions and photons

The quantity £eff i n Eq. (2.14) is the quadratic portion of the one-loop effective lagrangian. Since loop effects have already been accounted for, it is to be used at tree level. This is a simple application of the background field renormalization discussed in App. B-2. An alternative approach to the renormalization is the subject of Prob. VI-5.

VI-3 Interactions of pions and photons In order to understand the nature of chiral predictions and the range of validity of the energy expansion, let us work out several examples. At first, these will seem to be rather obscure processes, but they are the simplest hadronic reactions of QCD. As the bosonic interactions of the Goldstone bosons of the theory, they are the cleanest processes for demonstrating the dynamical content of the symmetries and anomalies of QCD.

The pion form factor The electromagnetic form factor of charged pions is required by Lorentz invariance and gauge invariance to have the form* where q** = (pi — pif and (7^(0) = 1. The electromagnetic current may be identified from the effective lagrangian of Eq. (1.2) by setting fji _ r/i _ eQAy^ x — 2i?om, where Q is the quark charge matrix and m is the quark mass matrix. To (9(i?4), we then find

8ai2>l^ (3.2)

0 71

>Y > (a)

(b)

Fig. VI-2 Radiative corrections to the pion form factor The neutral pion form factor is required to vanish by charge conjugation invariance.

VIS

Interactions of pions and photons

171

The renormalization of this current involves the Feynman diagrams in Fig. VI-2. That of Fig. VI-2(a) is simply found using the integral previously defined in Eq. (2.11), (3.3) (2a)

Evaluation of Fig. VI-2(b) is somewhat more complicated. Using the elastic TT+TT" scattering amplitude given by £2? (7r + (k 1 ) 7 r-(k 2 )|7r + (p 1 ) 7 r-(p 2 )) (3.4) we compute the vertex amplitude to be

*_ f

A k ddk

- ^ J {2ix)

1 (k

1 it (k - \q) - ml 2

(3.5) Upon integration, most terms drop out because of antisymmetry under k^ —> — fc^,and we find

+ hf - ml) ((& - h)2 ~ ™l)

(3.6)

We can evaluate this integral using dimensional regularization, 2 fj,A~d f , \ 1 (Pi + P 2 ) T (l — 2) Wem>(26) = ~J^ ^ d / 2 J dx\~2 ( m 2 _ 2X n _ x\) l-d/2

"(3.7) where as usual fi is an arbitrary scale introduced in order to maintain the proper dimensions. On-shell, we can disregard the term in q^ since q • (pi +P2) = ml — ml = 0. For the remaining piece, we expand about d = 4 to obtain (Pi

(3.8) x

172

VI Very low energy QCD - pions and photons

and the x-integration then yields (Jem) (2b) —

d-4

+ 7 — 1 — In 4TT

(3-9) where

H(a) = - /

Jo

dx\n(l-ax(l-x))

2-2\--l V a

(0 < a < 4)

= < In

m^(a - 4)

(otherwise) .

(3.10) Now we add everything together. The tree-level amplitude is modified by wavefunction renormalization, Sml ,„ (2) C2)^

ml

f2

4TT2F2 \ d - .

7 — 1 — In 4?r + In —^ (2)

(3.11) while Figs. VI-2(a),2(b) contribute as

f 2

^ < -—r + 7 - 1 - In47r + In 2

(2a)

(26)

16^2^2

{d-4

^~lq

) lrf-4

7-1 -

m2

(3.12)

VI-3

Interactions

of pions and

photons

173

respectively. Summing Eqs. 3.11,3.12 we see t h a t all terms independent of q2 cancel, a^ ' becomes a$ and the final result is

(3-13) T h e divergences have been absorbed in a^ , while the imaginary part required by unitarity is contained in H(q2/m^). Note t h a t the loops also induce a non-power-law behavior in Gn(q2). However, numerically this t u r n s out to be small and is unobservable in practice. A simple linear approximation,

2a {2)r

TTI"

+ ...

(3.14)

is obtained by Taylor expanding about q2 = 0. The corresponding result for chiral SU(3) is

The pion form factor is generally parameterized in terms of a charge radius, G,{q2) = 1 + \{rl)q2 + • • • •

(3.16)

Thus, for any given value of the energy scale //, the parameter a^ can be determined from the experimental charge radius. Prom the present experimental value (r£) = (0.44 ± 0.02) fm2, we obtain ag(fj, — m v) = (7.1 ±0.3) x 10- 3 . The scale fj, enters the calculation in such a way that, had we used a different scale // but kept the physical result invariant, we would have had

V' 2

(3-17)

In fact, if we look back to the origin of In fi in the transition from Eq. (3.7) to Eq. (3.8) using 2

2

'

°

~"

-4) ,

(3.18)

174

VI Very low energy QCD - pions and photons

we see that the scale dependence is always tied to the coefficient of the divergence.* The general result is then (3.19) where {7^} are the constants of Table B-l of App. B, used in the renormalization condition of Eq. (2.8). This calculation also nicely illustrates the range of validity of the energy expansion. The pion form factor is well-described by a monopole form, G

*(q2) " 1 -

2/m2

= 1 +

^

+

'"

^ 3 ' 20 )

'

with m ~ ™> p(77o)- The energy expansion is then in powers of q2/m2.

Rare pion processes The calculation above is clearly non-predictive as it contains a free parameter, ag, which must be determined phenomenologically. However, predictions do arise when more reactions are considered because relations exist between amplitudes as a consequence of the underlying chiral symmetry. In particular there exists a set of reactions which are described in terms of two low energy constants. These pionic reactions are shown in Table VI-2. With the additional input of FK/FK, the kaonic reactions shown there are also predicted. Each case contains hadronic form factors which need to be calculated. This section briefly describes the procedure for relating such reactions in chiral perturbation theory. All calculations follow the pattern described above, so that we shall only quote the results [GaL 85 a, DonH 89]. In the processes involving photons (?r+ —» e+^ e 7, TT+ —• e+i/ee+e~ and 7?r+ —> 77r+), there are always Born diagrams where the photon couples

« \ (a)

(b)

(c)

Fig. VI-3. Born diagrams for (a) TT+ —> e + i/ e 7, 7r+ —> e^~uee^e~, and (b)-(c) +

+

We have chosen to keep the low energy constants {a^} dimensionless in the extension to d dimensions. In [GaL 85a], the constants have dimension / i d ~ 4 . However, the resulting physics is identical in the limit d —>• 4.

VI-3 Interactions of pions and photons

175

Table VI-2. The radiative complex of pion and kaon transitions. Pions 7 —•

Kaons

7 -> K - ^ +

7T + 7T~

^ + _7e+^7

TT"*" —* e~^~i/e'y +

n+ -»7r°e i/ e e~

K -> ite+ve K+ -+ e+vee+e~

to hadrons through the known 7T7T7 coupling. These are shown in Fig. VI3. In addition, there can be direct contact interactions associated with the structure of the pions. These introduce new form factors. For the decays TT+ —• e+z/e7, e+vee+e~, the matrix elements are

^

j (

P

, q ) - ^

(3.21)

X u{p2)l^v{pi)u{pv)Y{^-]rlb)v{Pe)

,

where the hadronic part of the quantity M^v has the general structure

,q) = J ((p - ?)M9i/ - g^q - (p - q)) a p^ .

(3.22)

The first line represents the Born diagram and in subsequent lines the subscripts V and A indicate whether the vector or axial-vector portions of the weak currents are involved. The form factor rA in Eq. (3.22) can only contribute with virtual photons, i.e., as in TT+ —» e + i/ e e + e~. The 7?r+ —» 7TT+ reaction is analyzed in terms of the pion's electric and magnetic polarizabilities, C*E and (3M, which describe the response of the pion to electric and magnetic fields. In the static limit, electromagnetic fields induce the electric and magnetic dipole moments, PE = 4naEE

,

» M = 4TT/3MH ,

(3.23)

which correspond to an interaction energy V =-2n

(aEE2 + (3MH2)

.

(3.24)

176

VI Very low energy QCD - pions and photons

These forms emerge in the non-relativistic limit of the general Compton amplitude

,p',qi) = - i J d*X € (2pf + g l ) ^(2p - q2) ~mi

(P-Q2)

-mi

- g^qi ' 92) + • • • , (3.25) where <J is a coefficient proportional to the polarizability and q^q2 are the photon momenta, taken as outgoing, with p = pr + q\ + q2. The first three pieces are the Born and seagull diagrams. The last contains the extra term which emerges from higher order chiral lagrangians, and the ellipses indicate the presence of other possible gauge invariant structures which we shall not need. The chiral predictions are obtained in the same manner as used for the pion form factor. The results at q2 ~ 0 are hy =

N, 12y/2 ir2 F* Nc=3

= 0.027 m;; '

(3.26) where t = (q\ + qz)2 and •^*****

X

i V "^ / "" /

7

-i r» ~T

l»a;

* * *

*

lO.^I

I

12

The prediction for hy is especially interesting since hy is related by an isospin rotation to the amplitude for TT0 —• 77 (c/. Prob. VI-2). As shown in Sect. VI-5, this is absolutely predicted from the axial anomaly. The presence of a[0 implies that one of the above measurements must be used to determine it. We use the precisely known value for hAJhy to yield a[0(fjL = rrtri) = -(5.6 ± 0.3) x 1(T3 .

(3.28)

The results are compared with experiment in Table VI-3. We see that with one exception, the chiral predictions are in agreement with experiment. That exception, the electric polarizability in 7TT+ — > • 7TT+, comes from a single difficult experiment using a pion beam on a heavy Z atom [An et al. 85]. The coulomb exchange in n+A —> ^n+A is used to provide the extra photon (this is called the Primakoff effect),

VI~4 Pion-pion

scattering

177

Table VI-3. Chiral predictions and data in the radiative complex of transitions. Reaction 7 —> 7r

+

7r~

7 -» * : + # K+ -> e + ^ e 7 • e-

Quantity

Theory

Experiment


0.44a

0.44 ± 0.02

<^> (fm2) ^(m-1) hA/hv (hy + hA)(rn~1)

0.44 0.027 0.46a 0.038

0.34 ± 0.05 0.029 ± 0.017 0.46 ± 0.08 0.043 ± 0.003 2.3 ±0.6 1.4 ±3.1

rA/hv

(a£+/?M)(10- 4 fm) a£;(10-4fm)

X _>

7re+Zye

f = /_(0)// + (0) A+ (fm2) Ao (fm2)

a

2.6 0 2.8

-0.13 0.067 0.040

6.8 ±1.4 -0.20 ± 0.086 0.065 ± 0.005fe 0.050 ± 0.0126

Used as input. There axe experimental inconsistencies in these measurements. We use average values.

b

and the Born diagram must be carefully subtracted off. Before being concerned with this discrepancy, it would be preferable to have the experiment carefully redone. We have also listed the known results on kaonic processes predicted by the same constants. The analyses for 7 —» K+K~ and K —> e+uej are identical to the above results. For K —* n°e+uei we defer a presentation until the next chapter. VI-4 Pion-pion scattering The process na + TT^ —> TT7 + TT6 is, of course, not directly measurable in the laboratory. However, information about it can be inferred indirectly [MaMS 76]. For example, in the decay K —• 7T7reue the rescattering of the pions in the final state leads to an observable interference between the 5wave and P-wave TTTT amplitudes. A somewhat more complex procedure can be used in TTN —> TTTTTV. Here the longest range scattering will be due to pion exchange, as in Fig. VI-4. If the amplitude, defined as a function of q2, were to be measured at the pion pole (q2 = ra2), the pion scattering amplitude could be unambiguously identified. Unfortunately, the pion pole cannot be reached in this reaction, because q2 is negative. In principle, however, the data can be extrapolated outside the physical region to determine the value at the pion pole. This is the procedure used in most studies of TTTT scattering. In practice, some caution should be exercised in that additional theoretical assumptions are often introduced

178

VI Very low energy QCD - pions and photons

in the extrapolation procedure. These make the experimental error bars smaller than is justified by experiment alone, and this has led to the situation where experiments are often in conflict outside of their quoted errors. However, the general trend of these experiments is reasonably agreed upon. In terms of the Mandelstam variables S= (Pa+ Ppf ,

P7)2 ,

t=(pa-

U= (pa- Psf

,

(4.1)

the 7T7T scattering amplitudes are determined in terms of a single function A(s,t,u) as = A ( s , t, u ) 6 a p 6 7 s + A ( t , s , u ) 6

a i

6p

6

+ A ( u , t,

s

)

^

(4.2) Moreover, they can be decomposed into amplitudes of definite isospin, T°(s, t, u) = 3A(s, t, u) + A(t, s, u) + A{u, t, s) , Tl(s, t, u) = A{t, s, u) - A(u, t, s) ,

(4.3)

2

T (s,t,u) =A(t,s,u) + A{u,t,s) , and the partial wave amplitudes can be projected out, 7

«,t,u)

.

(4.4)

Below the threshold for inelastic reactions such as TTTT —• 4TT, KK, etc., the 5-matrix is given in terms of the phase shift 6j by Sj = e2tSt for each isospin and partial wave. In the following, we work with T-matrix elements, defined by

In practice this form is useful up to about 5 = 1 GeV2, by which point inelastic effects have become sizeable J.

_

_

_

_

_

_

_

A

K

N

Fig.

n N

VI-4 Pion-pion scattering from irN —> TTTTN.

VI~4 Pion-pion scattering

179

Table VI-4. The pion scattering lengths and slopes. Lowest Ordera First Two Ordersa

Experimental

a°0

0.26 ± 0.05

0.16

0.20

b°o

0.25 ± 0.03

0.18

0.26

al

-0.028 ± 0.012

-0.045

-0.041

bl

-0.082 ± 0.008

-0.089

-0.070

a\

0.038 ± 0.002

0.030

0.036

b\

—

0

0.043

0

20 x 10"4

0

3.5 x 10"4

al al a

(17 ±3) x 10"

4

(1.3 ±3) x 10"4

Predictions of chiral symmetry.

At low energies, the partial wave amplitude can be expanded in terms of a scattering length a\ and slope b|, defined by

where q2 = (s — 4ra^) /4. Since the chiral expansion is similarly a power series in the energy, a\ and b\ provide a useful set of quantities to study. In practice they are extracted from data by using dispersion relations and crossing symmetry to extrapolate some of the higher energy data down to threshold. The only accurate very low energy data is that from K —> 7T7reue. The experimental values are given in Table IV-4. At lowest order in the energy expansion, the amplitude for TTTT scattering [We 66b] can be obtained from £2 with the result A(8,t,«) = — ^

•

(4.7)

This produces the scattering lengths and slopes 0

"

2

m

l

u2 _

m

l

*

(4.8)

with the numerical values shown in the table. It is remarkable that the lowest energy form of a scattering process may be determined entirely from symmetry considerations.

VI Very low energy QCD - pious and photons

180

The nature of the chiral expansion also becomes evident from these formulae. The lowest order predictions are real and grow monotonically. As such, they must eventually violate the unitarity constraint at some point. The worst case is the / = £ = 0 amplitude Is-ml)

,

(4.9)

which violates the simplest consequence of unitarity, a-4ro* ,

Z| 2

(4.10)

<1 ,

below y/s = 700 MeV. In addition, there are no imaginary terms, which must be present due to the unitarity constraint, (4.11)

Im T =

These drawbacks are remedied order by order in the energy expansion. Note that since |T/| starts at order £"2, Im T/ starts at order E4. When one works to order J54, loop diagrams generate an imaginary piece given by Eq. (4.11) with the lowest order predictions for T/ inserted on the right-hand side. This process proceeds order by order in the energy expansion.* The O(E4) predictions modify the scattering lengths slightly [GaL 84]. Aside from the renormalization of mn and Fn, the corrections depend only on the low energy constants (2a\ + ajj) and ar2. We shall determine the low energy constants in Chaps. VII, VIII, with the most important

1.00.8T1 1

i

0.6-

0.40.20.0

200

400

600

800

1000

Fig. VI-5 Pion-pion scattering data. One can invent a variety of schemes for imposing unitarity to all orders, but these all involve additional assumptions and depart from the controlledframeworkof chiral perturbation theory.

VI-5 The axial anomaly and TT°—> 77

181

influence being the K —> 7T7reue form factors. The formulae are not illuminating, and we quote the results in Table VI-4. The overall agreement is good. More instructive is a pictorial representation of the result. One may see the order EA improvement and the nature of the chiral expansion by considering the I = 1, £ — 1 channel, where some of the higher energy data are shown in Fig. VI-5. The resonance structure visible is the p(770). The chiral prediction is

with loops having a negligible effect. The lowest order result is given by the dashed line. It clearly does not reflect the presence of the p(770) resonance. The solid line represents the result at order E4 and starts to reproduce the low energy tail of the p(770). It is, of course, impossible to represent a full Breit-Wigner shape by two terms in an energy expansion - all orders are required. The chiral predictions at O(E4) may reproduce the first two terms, with the resulting expansion being in powers of q2/m^ in agreement with Eq. (3.20).

VI-5 The axial anomaly and TT0

77 The description of pions and photons presented thus far does not include the decay TT° —> 77. This process is important in QCD, because to understand it one must include the anomaly in the axial current. The TT°—> 77 amplitude has the general structure fkv^y0

,

(5.1)

as required by Lorentz invariance, parity conservation and gauge invariance, and leads to the decay rate

g

\2

(5.2)

.

Prom the experimental value, T = 7.7 ± 0.6 eV, we find A 77 - 0.025 ± 0.001 GeV"1 . The effective lagrangians that we have been using thus far do not contain the totally antisymmetric tensor e^VOL^', so it is clear that they cannot generate TT°—> 77. However, there exists another class of contributions. These are associated with the anomaly content of the theory. Let us defer to Chap. VII the derivation of such contributions because there are important features which first appear only for chiral SU(3). Here we simply quote the restriction to Srf/(2), CA = 4 ^ 2

[ ^

p

p

^

p

(5.3)

182

VI Very low energy QCD - pions and photons

with

Tp=Tr (Q2LP + Q2RP + ^ where Aa is the photon field, F^v is the photon field strength, and Nc = 3 is the number of colors. A crucial aspect of this expression is that it has a known coefficient. In this respect, it is unlike other terms in the effective lagrangian, which have free parameters that need to be determined phenomenologically. This is because it is a prediction of the anomaly structure of QCD. A corollary of this is that CA must not be renormalized by radiative corrections. This was proven at the quark-gluon level by Adler and Bardeen [AdB 69]. The 7T° —• 77 amplitude is found by expanding CA to first order in the pion field, yielding

(5.5) where we have integrated by parts in the second line. This produces a Tj-0 _> ^j matrix element of the form rvN

t

°^GV-> ,

(5.6)

in excellent agreement with the experimental value. This is widely recognized as an important test of QCD, both as a measurement of the number of colors and also as a reflection of the symmetries and anomalies of the theory. It is a remarkable result. What would have happened if the axial anomaly were not present? The decay TT0 —> 77 could still occur, but it would be suppressed. The 7T°—> 77 transition must be at least of order i? 4 , as it must involve the dimension 4 operator FF. The anomaly occurs at this order. However, non-anomalous lagrangians leading to this transition can be constructed at order E6. This result was first derived by Sutherland and Veltman using a soft-pion technique [Su 67, Ve 67]. Following the steps employed in Sect. IV-4, one uses the PC AC identification dxJ§x = rrv^F^ip^o to calculate the matrix element

Id'x 2

_

m

2\

J

(5.7)

VI-6 The physics behind the QCD chiral lagrangian

183

In going from the first to the second line, we have integrated by parts and have used the fact that the commutator terms vanish. In the third line, the form which appears for the matrix element is required by the gauge invariance of the electromagnetic currents together with parity. Because of the overall factor of q2, the two-photon amplitude A 77 would seem to vanish in the soft-pion limit. It could exist at order ra2, but this would lead to a strong suppression of the decay, roughly of order (ml/1 GeV2) ~ 3 x 10~4 in rate. However, the flaw in this argument is the use of the PC AC relation, as will be demonstrated in Chap. VIII. At what level would we expect corrections to the anomaly prediction for 7T°—> 77? It has been checked that ml In ml corrections, which in principle can occur when meson loops are present, do not in fact modify the lowest order result when it is expressed in terms of the physical decay constant F^. However, there still could be corrections of order m 2 /A 2 , where A is the scale in the energy expansion, i.e. A ~ mp —> 1 GeV. These could amount to modifications of order a few percent or perhaps ±0.0008 in A 77. VI-6 The physics behind the QCD chiral lagrangian For the most part, we have been using chiral lagrangians as our primary tool for making predictions based on the symmetry structure of QCD. In this section, we pause to examine which features of QCD are important in determining the structure of chiral lagrangians. The general strategy can perhaps be appreciated by a comparison of low energy and high energy QCD methodology. At high energies, due to the asymptotic freedom of QCD, hard scattering processes can be calculated in a power series expansion in the strong coupling constant. However, some dependence on 'soft' physics remains in the form of structure functions, fragmentation functions, etc. These are not calculable perturbatively and must be determined phenomenologically from the data. At high energy then, the predictions of QCD are relations among amplitudes parameterized in terms of various phenomenological structure functions and the strong coupling constant. At very low energies, because of the symmetries of QCD, low energy scatterings and decays can be calculated in a power series expansion in the energy. However, some dependence on 'harder' physics remains in the form of the constants {a£}. These are not calculable from the symmetry structure and must be determined phenomenologically from the data. At low energy then, the predictions of QCD take the form of relations among amplitudes whose structure is based on symmetry constraints but which are parameterized in terms of empirical constants. Nevertheless, QCD should in principle also predict the very structure functions and low energy constants which are employed by these techniques. The trouble at present is that we do not have techniques of

184

VI Very low energy QCD - pions and photons

comparable rigor with which to calculate these quantities. Nevertheless, by using models plus phenomenological insight we can learn a bit about the physics which leads to the chiral lagrangian. The low energy constants Fn and m^ which occur at order E2 do not reveal much about the structure of the theory. All theories with a slightly broken chiral 5/7(2) symmetry will have an identical structure at order E2. The pion decay constant Fn will be sensitive to the mass scale of the underlying theory, while the pion mass m^ will be determined by the amount of symmetry breaking. However, approached phenomenologically, these are basically free parameters and do not differentiate between competing theories. The situation is different at order E 4 . Here the chiral lagrangian contains many terms, and the pattern of coefficients is a signature of the underlying theory. The linear sigma model without fermions provides us with an example of how one can compare a theory with the real world. In Sects. IV-2,4 we calculated the tree-level terms in £4 which would be present in the linear sigma model, and obtained a result expressible as F2 1 2ai + as = 2a 4 + a 5 = 8a 6 + 4a 8 = — ^ = r-r , ^2,7,9,10 = 0 . (6.1) This pattern is quite different from the structure obtained phenomenologically. It appears that the linear sigma model is not a good representation of the real world. Unfortunately, it is harder to theoretically infer the {c^} directly from QCD. However, a look at phenomenology indicates that we should consider the effects of vector mesons, in particular the p(770). This is the most clear in the pion form factor which shows a dramatic p resonance in the timelike region. Indeed, the whole form factor can be well understood in a simple model as being a Breit-Wigner shape due to the p resonance

G G

tf)

*tf)

=

m2 P 2 - 4m2) ' q2 _ m 2 + impTp(q)0(q

(62) (6 2) *

where the normalization is chosen to enforce the condition Gn(0) = 1. This works even in the timelike region. Comparison with the chiral lagrangian approach implies that this model would predict 17.2

a9 = -£=• - 7.2 x 10" 3 , (6.3) 2 2ra in good agreement with the value obtained earlier, a$ = (7.1 ±0.3) x 10~3. This analysis can be extended to aio- This enters into the W+TT+J vertex which occurs in TT+ —• e+ue^f. Here both vector and axial-vector mesons can generate corrections to the basic couplings. Explicit calcula-

VI-6 The physics behind the QCD chiral lagrangian

185

tion yields [EcGPR 89] = -5.8 x 1(T3 .

(6.4)

Here a\ refers to the lightest axial-vector meson ai(1260) (c/Sect. V3), and Fai and Fp are the couplings of a\ and p to the W+ and the photons respectively. Again the result is close to the empirical value aio = (-5.6±O.3) x 1(T3. The phenomenological low energy constants are scale dependent, and their analysis includes loop effects, while those in Eqs. (6.3), (6.4) are constants, to be used at tree level. Nevertheless, there is some sense in comparing them. The effect of loops in processes involving #9,0:10 is small, and the scale dependence only makes a minor change, ag(fi = 300 MeV) = 7.7 x 10~3 vs ar9(n = 1 GeV) = 6.5 x 10"3. Presumably the appropriate scale is near /i = 771^770). The p(770) provides a much more important effect here than any other input. Finally it also turns out that the use of vector meson exchange leads to a good description of TTTT scattering [DoRV 89, EcGPR 89]. This is not too surprising in light of the need for the chiral lagrangians to reproduce the tail of the p(770), as described in Sect. VI-4. As a consequence of crossing symmetry, the p(770) must also influence the other scattering channels. To a large extent, the chiral coefficients 0^,0:2,03 are dominated by the effect of p(770) exchange. We see from these examples that phenomenology indicates that the exchange of light vector particles is the most important physics effect behind the chiral coefficients which we have been discussing. The idea that vector mesons play an important dynamical role is not new. It predates the Standard Model, originating with Sakurai [Sa 69], in a form called vector dominance. The vector dominance idea has never been derived from the Standard Model, but nevertheless enjoys considerable phenomenological support. Put most broadly, vector dominance states that the main dynamical effect at energies less than about 1 GeV is associated with the exchange of vector mesons. The use of a chiral lagrangian with parameters described by p-exchange, is compatible with this idea and puts it on a firmer footing. These considerations suggest that for chiral lagrangians the prime ingredient of QCD is the spectrum of the theory. The linear sigma model has a quite different spectrum, with a light scalar and no p, and hence does not agree with the data. QCD, however, seems to predict that deviations from the lowest order chiral relation must be in such a form as to reproduce the low energy tails of the light resonances, in particular the p. At present, we cannot rigorously prove this connection. However, it remains a useful picture in estimating various effects of chiral lagrangians.

186

VI Very low energy QCD - pions and photons

Problems

1) Radiative corrections and TT^ decay To bring Tir^eVe /FV-^i^ into agreement with experiment requires a radiative correction whose dominant contribution is the so-called seagull component (y/^F^g^) of M^ (cf. Eq. (3.22)). a) Verify that gauge invariance requires

and show that the seagull term is required in this regard to cancel the pion pole contribution. b) Use the seagull term in Feynman gauge to calculate the radiative correction to TT^ decay. Introduce a photon cutoff via 1 k2

1

-A 2

k2 k2 - A 2

so that

X w(Pi/)7A(l + 75)

—fr e + fa — rr

and show that \

2?r rri£

where

\

L

is the lowest order amplitude for the 71^2 process. This then is the origin of the lepton mass dependent radiative correction. 2) Radiative pion decay and the anomaly Writing the TT°decay amplitude as

= -ie^*e? J

Problems

187

and the vector current amplitude in radiative TT+ decay as

^Lt^a

= -*< J d*xe^

demonstrate that isotopic spin invariance requires y/2hy = A77. 3) Pion polarizability and pion decay The relationship given in Eq. (3.26) between the charged pion static electric polarizability and the axial structure function for radiative pion decay HA was first obtained in [Te 73], not via chiral lagrangian methods, but rather using soft-pion PC AC techniques. a) Take the limit p1 —• 0 in the Compton amplitude of Eq. (3.25) and show using current-algebra/PC AC methods that lim

T^(p,p',qi)

where M^i/(p, g) is the axial-vector component of the radiative pion decay amplitude given in Eq. (3.22) and p — pf + q\ + ^2b) Using the explicit kinematic forms of these amplitudes demonstrate that the relation a(t = 0) = y/2hA/Fn obtains, and that this is equivalent to the relationship between pion polarizability and radiative pion decay given in Eq. (3.26). 4) Unitarity and the pion form factor a) Verify that the pion form factor given in Eq. (3.13) obeys the strictures of unitarity, i.e.

- «&) P«7r|T|7r+TT

)

where T is the two-derivative (tree-level) pion-pion scattering amplitude and is given in Eq. (3.4). b) How does this result change if the K+K~ intermediate state is added to TT+TT"? 5) Background field renormalization Verify that the effective lagrangian developed in Eqs. (2.10)-(2.15) yields the same renormalization constant and renormalized mass for the propagator as a standard on-shell renormalization prescription carried out in momentum space.

VII Introducing kaons and etas

As the energy is increased, the next degree of freedom which becomes excited is the strange quark, s. The new mesons encountered are kaons, which carry the extra quantum number strangeness of the s quark, and even heavier mesons 7/(549) and 7/(960) with no net strangeness. The kaons are stable against strong and electromagnetic decays, and must decay weakly. In this chapter we describe the basic structure of the TT, K, 77, 7/ system, deferring a discussion of weak decays to Chaps. VIII, IX. Our treatment will continue to be based on effective lagrangians, although quark model ideas enter into the discussion of the rf'. Important results include information on quark mass ratios and the derivation of the WessZumino-Witten anomaly action. VII-1 Quark masses The addition of an extra quark adds to the number of possible hadrons. If the strange quark mass is not too large, there are additional low mass particles associated with the breaking of chiral symmetry. Including the quark mass terms, 'mu 0 0 Anass = fami/jR + $RPMI>L ,

m=[

0

md

0 I ,

(1.1)

the QCD lagrangian has an approximate SU(3)L X SU(3)R global symmetry. If the u, d, s quarks were massless, the dynamical breaking of x SU(3)L SU(3)R to vector £77(3) would produce eight Goldstone bosons, one for each generator of £1/(3). These would be the three pions TT*1, TT0, four kaons IT*, K°, K°, and one neutral particlerjs with the quantum numbers of the eighth component of the octet. Due to nonzero quark masses, these mesons are not actually massless, but should be light if the quark masses are not 'too large'. 188

VII-1 Quark masses

189

What should the K, rjs masses be? Unfortunately, QCD is unable to answer this question, even if we were able to solve the theory precisely. This is because the quark masses are free parameters in QCD, and thus must be determined from experimental input. This means that the TT, K and 778 masses can be used to determine the quark masses rather than vice versa. The discussion is somewhat more subtle than this simple statement would indicate. Quark masses need to be renormalized, and hence to specify their values one has to specify the renormalization prescription and the scale at which they are renormalized. Under changes of scale, the mass values change, i.e. they 'run.' However, quark mass ratios are rather simpler. The QCD renormalization is flavor independent, at least to lowest order in the masses. In this situation, mass ratios are independent of the renormalization. There can be some residual scheme dependence through higher order dependence of the renormalization constants on the quark masses. However to first order, we can be confident that the mass ratio determined by the TT, K, % masses is the same ratio as found from the mass parameters of the QCD lagrangian. The content of chiral 577(3) is contained in an effective lagrangian expressed in terms of U = exp[i(A •
0

i

1 ~>

4-

TS+

K

\

°

(1.3)

as expressed in terms of the pseudoscalar meson fields. If we choose the parameters in Eq. (1.3) to correspond to QCD without external sources, viz. s=m ,

p=0,

D^U = d^U ,

(1.4)

the meson masses obtained by expanding to order cp2 are ml = Bo(mu + md) , m2K0 = B0(ms + md) ,

m2K± = Bo(ms + mu) , m28 = -B0(4ms + mu + md) . o

Upon temporarily neglecting isospin breaking, we obtain from Eq. (1.5) the mass relations,

m_ =

ms

ml 2m\-m\

=

J_ 26 '

y

' '

190

VII Introducing kaons and etas

where rh = (mu + rrid)/2. Eq. (1.6a) demonstrates the extreme lightness of the u, d quark masses. Despite the difficulties of defining an absolute magnitude for quark masses, most estimates put that of the strange quark in the range ms ~ 150-300 MeV. This in turn suggests rh ~ 6-12 MeV, a range of values significantly smaller than the scale AQCD- Of course, the existence of very light quarks in the Standard Model is no more (or less) a mystery than is the existence of very heavy quarks. Both are determined by the Yukawa couplings of fermions to the Higgs boson, which are unconstrained (and not understood) input parameters of the theory. In any case, the small values of the u, d masses are responsible in QCD for the light pion, and for the usefulness of chiral symmetry techniques. The mass relation of Eq. (1.6b) is the Gell-Mann-Okubo formula as applied to the octet of Goldstone bosons [GeOR 68]. As mentioned in Sect. VI-1, this relation is quadratic in the meson masses. It predicts mVs — 566 MeV, not far from the mass of the 77(549). The small difference between these mass values can be accounted for by second order effects in the mass expansion. In particular, mixing of the r)% with an SU(3) singlet pseudoscalar produces a mass shift of order (ms — rh)2. The difference between the predicted and physical masses is then an estimate of accuracy of the lowest order predictions. The use of the full pseudoscalar octet allows us to be sensitive to isospin breaking due to quark mass differences in a way not possible using only pions. This is because, to first order, the A / = 2 mass difference mn± — ra^o is independent of the A / = 1 mass difference m^ — mu. In contrast, the kaons experience a mass splitting of first order in ra^ — ra u. In particular, the quark mass contribution to the kaon mass difference is

(1.7) In addition there are electromagnetic contributions of the form (m2Ko-m2K+)em = mlo-ml+

.

(1.8)

This result, called Dashen's theorem [Da 69], follows in an effective lagrangian framework from (i) the vanishing of the electromagnetic selfenergies of neutral mesons at lowest order in the energy expansion, (ii) the fact that K+ and ?r+ fall in the same [/-spin multiplet and hence are treated identically by the electromagnetic interaction, itself a [/-spin

VII-2 Higher order analysis of decay constants and masses

191

singlet.* By isolating the quark mass and electromagnetic contributions to the kaon mass difference, we can write a sum rule ™>d ~ mu 1 2 n md + mu\ / =

2

V^K0

2 ~

m

K+)

\

(1.9) /

~

2

V^TT0 ~~ m-K+

2\ )

'

which yields md - mu = Q Q 2 3 m d - mu = Q ^ ra m
VII—2 Higher order analysis of decay constants and masses The purpose of this section is to describe what is presently known about the analysis of masses and decay parameters at the next order in the expansion in energy and/or 5/7(3) breaking. Our emphasis will be on the * Recall that {/-spin is the SU(2) subgroup of 5(7(3) under which the d and s quarks are transformed.

192

VII Introducing kaons and etas

effects of quark mass, as it is important to document the level of understanding of these basic parameters. The pion mass (or equivalently the mass of u, d quarks) is not sufficiently large to modify most observables in a major way. However, the kaon mass (influenced by the strange quark mass) has a more significant impact. It is important to examine the effect of this mass on low energy observables. Working to O(E^), the effective lagrangian consists of the O(E2) terms already given in Eq. (VI-1.2), the Wess-Zumino-Witten anomaly lagrangians to be derived in Sect. VII-3 plus a set of fourth order terms given in Eq. (VI-2.7). In Chap. VI on pionic physics, we saw that there were several reactions sensitive to the four-derivative terms and to the photon couplings. This allowed us not only to determine the parameters but also to make predictions. However, there was no phenomenological information on the mass terms in the effective lagrangian. With the addition of the K and 77 particles, we are now sensitive to such terms. Unfortunately, the data set is not sufficiently rich to allow use of these new parameters in a predictive way. Ambiguities in mass parameters Before proceeding, we wish to point out an important feature of the chiral SU(3) effective lagrangian, and indeed of all effective lagrangians. We have constructed the form of the mass terms from symmetry considerations alone, using the fact that \ (= 2i?o(s + w)) and m transform as & ( 3 L 5 3 # ) representation under chiral transformations. However, there exists a continuous family of forms having the same transformation property [KaM86, DoW91], X -* XA = X + A det[xf]x ( x ^ ) " 1 ,

m -> mA = m + A det[m] m 1 , (2.1) where A is an arbitrary parameter and A = 2i?oA. This means that in any effective lagrangian based on symmetry considerations, one can equally well use xx o r mA instead of \ or m. This algebraic reparameterization invariance can be explicitly seen in the chiral lagrangian with the aid of an identity involving any 3 x 3 matrix M, detM = M 3 - M2TTM

- -M [Tr (M2) - (TrM) 2 ]

,

(2.2)

which can be derived [Go 87] through use of the Cayley-Hamilton theorem. In our case this yields

Tr (XAf/f + UX\) = Tr (xtf* + UX]) - \ Tr ( V X W + tf W x )

c/xt-C/tx)]2} • (2.3)

VII-2 Higher order analysis of decay constants and masses

193

The additional term of Eq. (2.1) is equivalent to a combination of O(E4) operators. Any piece of phenomenology obtained with \ o r m a n ( i set {#6, a*?, a$} can be obtained identically with x\ o r mA a n d the set 0 6 0 6 + ^ , ID

«7

a

7 +

T7T ,

O$

ID

<*8

.

(2.4)

O

Thus, the family m^ of quark mass matrices leads to the same phenomenology, and one cannot distinguish between any individual members by symmetry techniques. The same ambiguity can occur whenever quark masses occur in other effective lagrangians. The reparameterization invariance corresponds in terms of the quark masses to mu = mu + Xrudrris , m^ — rrid + Xmums , ms = ms + Xmurrid . (2.5)

Note that this change is of second order in the quark mass and hence is important only if one analyzes quark masses beyond leading order. The above ambiguity is an algebraic feature of the effective lagrangian. However, it also represents a possible form of mass renormalization in QCD. To see this, consider a set of 'bare' quark mass values having mu = 0, but morris ^ 0. In such a world, the chiral 5/7(2) and SU(3) symmetries are explicitly broken (by rad,ms), and axial U(l) is not a global symmetry (because of the axial anomaly and of the vacuum structure of QCD). For an up-quark moving in the vacuum fields of QCD, the value mu — 0 is not 'protected', and thus it can be renormalized to a nonzero value. Presumably mu will then be given by some function of rrid and ms. Since the original vanishing ix-quark mass would be protected by a chiral SU(2) symmetry in the limit rrid —> 0 or ms —> 0, the dependence must be proportional to the product rridms to accommodate such limits properly. An implication of this discussion is that masses need not be multiplicatively renormalized beyond leading order, but may mix in a way consistent with the underlying symmetries. This is what the transformations of Eq. (2.5) are expressing. In principle, different renormalization schemes could involve differing mass matrices m\. As regards phenomenology, the ambiguity has a simple resolution - one can equally well work with any of the family of HIA since all yield the same phenomenology. Most applications of quark masses occur at first order in the mass, and these must use the lowest order mass matrix of Eq. (1.1). In order to define the mass matrix to next order, one must propose an extra constraint which fixes the definition of the mass. We shall encounter an example of this shortly. Decay constants One effect of higher order lagrangians is to distinguish between the various pseudoscalar decay constants. Thus the decay constants are no longer

194

VII Introducing kaons and etas

equal for the full octet of pseudoscalar mesons. Of the various terms in £4, the one proportional to 0:4 leads only to an overall shift common to Fin FK, FV8. The only tree level contribution to the axial current which differentiates between Fn, FK and Fm comes from a§. In addition, the loop diagram of Fig. Vll-l(a) can introduce some mass dependence, as well as renormalizing the tree level parameters. The result is [GaL 85a] FK

,

I

3

5

F

S o re N

16

So

where 9

mf

9

, mf

or numerically, ^

= 1 - 0.01

-^ = -f

+ 0.02 .

The experimental value of FK can be determined from the measured K+ —> / i + ^ decay rate, analogous to the extraction of Fn in Sect. VT-1. The experimental ratio Y~ = 1.23 ±0.02

(2.9)

implies the value ag (mv) = (2.3 ± 0.2) x 10"3. Masses The Gell-Mann-Okubo mass relation is also modified at O(E4) due to higher order quark mass effects. A calculation of the masses of the pseudoscalars including the loop diagrams of Fig. Vll-l(b) results in

n (a)

(b)

Fig. VII-1 Loop diagram contributions to (a) decay constants, (b) masses.

VII-2 Higher order analysis of decay constants and masses mz,

ml K

195

(2.10)

3 ms + m

with

8 (m\ - mj)

128

T

(2.11)

{ml-ml?

After some rearrangement, the violation of the Gell Mann-Okubo relation can be cast in the form K

,

* ,

2 = 0.21 ,

(2.12a)

compared to

(

2

2\ r r

r

(2.12b)

+2 At this stage, one cannot make a prediction for m, because of the inclusion of new low energy constants. Instead, we can reverse the process to find 2ar7 + ar8 = 0.4 x 10 - 3

(2.13)

evaluated at the scale /i = m^. Since this combination of coefficients is expressible in terms of physical data, it is naturally invariant under the reparameterization discussed at the start of this section. Finally, we return to the quark mass ratio, m 77i, + m

1-0.49 + 0.17 0.4 x 10- 3 26 1 1 - 0.32 - 0.17 x 26

(2.14)

196

VII Introducing kaons and etas

The result depends on the relative contributions of 07, a% to Eq. (2.13),* and illustrates the ambiguity of the quark mass beyond leading order, as given in Eqs. (2.1)-(2.5). If we insist on obtaining a mass ratio, we need extra input. One possibility is to assume that a? is given by the 77 — rf mixing model as described in Prob. VII-3 which results in

ar7 = - 0 . 4 x 10" 3 ,

_A^ = -I • ms + m 26

(2-15)

This choice, which goes beyond the limits of pure chiral symmetry, serves to define a mass matrix at second order in the masses. However, other definitions of mass are possible [DoW 92]. We are not presently able to calculate the u — d mass difference beyond leading order due to a lack of knowledge of the electromagnetic contributions to kaon and pion masses at next order in the quark mass. VII-3 The Wess-Zumino-Witten anomaly action At this stage one must also include the effect of the axial anomaly. The anomaly influences not only processes involving photons, such as ir® —> 77, but also purely hadronic processes. For example the reaction KK —> 7r+7r~7r°,allowed by QCD, is not present in any of the chiral lagrangians appearing in previous sections. Its absence is easy to understand because the hadronic part of the lagrangian, with external fields set equal to zero, has the discrete symmetry C/t) which forbids the transition of an even number of mesons to an odd number. However this is not a symmetry of QCD. More importantly, there are a set of low-energy relations, the Wess-Zumino consistency conditions [WeZ 71], which must be satisfied in the presence of the anomaly and which involve hadronic reactions. The effect of the anomaly was first analyzed by Wess and Zumino who noted that the result could not be expressed as a single local effective lagrangian, and gave a Taylor expansion representation for it.* Witten [Wi 83a] subsequently gave an elegant representation of the Wess-Zumino contribution as an integral over a five-dimensional space whose boundary is physical four-dimensional spacetime. Since the considerations leading to the Wess-Zumino-Witten action can be rather formal, it is best to adopt a direct calculational approach. Fortunately, we are able to employ the familiar sigma model (with fermions) because it contains the same anomaly structure as QCD. That is, it is the presence of fermions having the same quantum numbers as quarks which ensures that the anomaly will occur. The absence of gluons in * Note that if Eq. (2.13) is used to express Eq. (2.14) solely in terms of «7, the result is explicitly scale-independent because otj does not depend on /z. * For a textbook treatment, see [Ge 84].

VII-3 The Wess-Zumino-Witten anomaly action

197

the sigma model is not a problem since, according to the Adler-Bardeen theorem[AdB 69], the inclusion of gluons would not modify the result. Since the sigma model involves coupling between mesons and fermions, we can also observe directly the influence of the anomaly on the Goldstone bosons. Although somewhat technically difficult, our approach will clearly illustrate the connection with treatments of the anomaly based on perturbative calculations. Consider as a starting point the lagrangian, Eq. (IV-1.11), of the linear sigma model

C = &pip - gv (i>LUipR + ^ C / V L ) + ... .

(3.1)

We have displayed neither the term containing Tr (d^Ud^U^) and nor any term containing the scalar field S. Such contributions are not essential to our study of the anomaly and will be dropped hereafter. In order to simulate the light quarks of QCD, we shall endow each fermion with a color quantum number (letting the number Nc of colors be arbitrary) and assume there are 3 fermion flavors, each of constituent mass M — gv. Although the original linear sigma model has a flavor SU(2) chiral symmetry, Eq. (3.1) is equally well defined for flavor SU(3). Our analysis begins by imposing on Eq. (3.1) the change of variable

$'L = £tyL , ^ = WR, tf = U ,

(3.2)

like that appearing already in Eq. (IV-7.3). This yields C = $'(i lp -

(?d»z - td^) .

(3 3)

'

For this change of variable the jacobian is not unity, and thus we must write the effective action as eiT(U)

=

f[

= f[d^][df]J

elfd4x P'W-MW

(3.4)

= exp(lnj r ) exp(tr \n(iI/> - M)) . For large M, it can be shown that the tr ln(z Ip — M) factor does not produce any terms at order EA that contain the e^aP dependence char-

198

VII Introducing kaons and etas

acteristic of the anomaly.* Hence the effect of the anomaly must lie in the jacobian J, and it is this we must calculate. It is possible to determine the jacobian by integrating a sequence of infinitesimal transformations. Thus we introduce the extension £ —• £ r , gr = expi T

^ = expiry

,

(3.5)

where r is a continuous parameter and £ = £r=i- Transformations induced by the infinitesimal parameter 6r will give rise to the infinitesimal quantities £$r and 6J, (3-6)

Prom Eqs. (III-3.44), (III-3.47), we find 6J to be 8J = exp(—2i<5rtr ^75) ,

(3-7)

or dlnj dT

(3.8)

r=0

This result should be familiar from our discussion of the axial anomaly in Sect. Ill—3. There remain two steps, first to calculate the regularized representation of tr (^75), and then to integrate with respect to r. To regularize the trace, we employ the limiting procedure tr (^75) = lim tr (^75 exp [-e fT fT\)

{V» = d"+iV^T+iA^)

, (3.9)

with AT and V1^ as in Eq. (3.3), except now constructed from £T and £}. For arbitrary r, we make use of the identities

vT =
{

„i m j

to express Tf)r Tf>T in the form

Tprfr = d^

+a ,

dli = dlt + iVTll + a^Xis = d^ + FTfi , a = -2ATIX + i[(dM + iVTtl) ,X] 75 •

(3.11)

This can be verified by expanding as tr ln(i# - M) = tr ln(-M(l - Up/M)) = tr ln(-M) - tr (z#) 2 /2M 2 + ... . The first term can be regularized as in the text and directly calculated using the techniques described in App. B. The remaining terms vanish for large M.

VII-3 The Wess-Zumino-Witten

anomaly action

199

Prom the heat kernel expansion of App. B, we have*

Carrying out the 'TV' operation, which involves some application of Dirac algebra, yields

= 2iNcTr (^e^lpXXXX)

+•• • ,

(3.13)

where the ellipses denote contributions not involving e^va^ and the factor Nc comes from the sum over each fermion color. Combining the above ingredients, we have for the regulated action

where we recall that Tp = A • (p/(2Fn). This result expresses the effect of the anomaly on the Goldstone bosons. Unfortunately there is no simple way to integrate the entire expression of Eq. (3.14) in closed form. In principle, we could represent each of the axial-vector currents therein (e.g. A^) as a Taylor series expanded about r = 0 and perform the integrations to obtain a series of local lagrangians. Alternatively, however, one can simply express Eq. (3.14) as an integral over a five-dimensional space provided we identify r with a fifth coordinate x§ (defined to be timelike). In this case, we use i *

•

(3-15)

plus the cyclic property of the trace to write )

,

(3.16)

where i , . . . , m = 5,0,1,2,3 with e50123 = +1. This is Witten's form for the Wess-Zumino anomaly function. The r = 1 boundary is our physical spacetime, and the fifth coordinate is just an integration variable. Since each term in the Taylor expansion can be integrated, the result depends * Note the distinction between 'tr' and 'Tr', as in Eq. (Ill—3.48).

200

VII Introducing kaons and etas

only on the remaining four spacetime variables. Observe that vanishes for U in SU(2) due to the properties of Pauli matrices. For chiral 5/7(3), the process K+K~ —• 7r+7r~~7r° is the simplest one described by this action and after expanding Fwzw? it is described by the lagrangian, ^

,

(3.17)

with if = A • cp given in Eq. (1.3). The above discussion has concerned the impact of the anomaly on the Goldstone modes. We must also determine its proper form in the presence of photons or W± fields. For this purpose, we can obtain the maximal information by generalizing the fermion couplings to include arbitrary left-handed or right-handed currents t^r^ 1 ^

(3.18)

-r The calculation of the jacobian then involves the operator V

-d

+U

1 + 75

I ir

1 - 7 5

which generalizes Eq. (3.3). It is somewhat painful to work out the full answer directly, but fortunately we may invoke Bardeen's result of Eq. (Ill— 3.64) for the general anomaly. Using the identities

where ^v,r^v

are given in Eq. (Ill—3.65), we obtain

Fwzw = -^J

drjd4x

+ Va V) 0) ++ (l +^M M^ + 2i,

— —{a^ajyVa^

f^lap)

(3.21)

+ a^Vvadp + v

Note that the first term corresponds to our previous calculation of Eq. (3.14). The WZW anomaly action contains the full influence of the anomalous low energy couplings of mesons to themselves and to gauge fields. By construction, it is gauge invariant. The r integration can be explicitly performed for all terms but the first in Eq. (3.21). However, in the general nonabelian case the result is extremely lengthy [PaR 85]. For

VII-4 The rf (960)

201

the simpler but still interesting example of coupling to a photon field A^, the result is Twzw (U, A^) = Twzw(f^)

+

a/3 d4a; iv(Q(

^^ / K

^

— ie FuVAa Tr I Q (LR + R3) + — \ 2

' (3.22) where R^ = (d^U^)U, L^ = Ud^U^.* We have already used the twophoton portion of this result in Sect. VI-5 for the decay TT° —• 77. We have seen here that whereas the anomalous divergence of the axial current represents the response to an infinitesimal anomaly transformation, the WZW lagrangian represents the integration of a series of infinitesimal transformations. In our analysis of the sigma model, the anomaly has forced the occurrence of certain couplings, among them Tr0 —> 77, 7 —> 3TT and KK —• 3TT. AS noted earlier, although these results are based on an instructional model, the result has the same anomaly structure as QCD because the answer must depend on symmetry properties alone. Indeed, such conclusions were originally deduced from anomalous Ward identities [WeZ 71] without any reference to an underlying model. We regard such predictions as among the most profound consequences of the Standard Model. VII-4 The

T/(960)

Thus far, we have been dealing with hadrons whose presence could be inferred from the symmetry properties of QCD. We have not needed to invoke the quark model to describe the meson spectrum. This procedure is less clear for the rf (960), which is not a Goldstone boson for any of the symmetries of QCD. The least complicated description of the rf is with the quark model. Here one notes that the eight particles (TT"1", TT~, TT°, K+, K~, K°, K° and rjs) havethe same quantum numbers as the respective quark-antiquark pairs (ud, du, (uu—dd), us, su, ds, sJ and (uu+dd—2ss)). However, such QQ pairing suggests a ninth state, the SU(3) singlet 770 ~ (uu+dd+ss). A look through the Particle Data Tables reveals that the lightest candidate for a ninth pseudoscalar is 7/(960). In fact, the 7/(549) and 7/(960) turn out to involve mixtures of 7/8 and 770 (and in principle of other states as well), with the 7/(960) being treated as predominantly the SU(3) singlet. * Witten's original result did not conserve parity, and this was subsequently corrected [PaR85, KaRS84].

202

VII Introducing kaons and etas

One immediate puzzle involving the rf would appear to be its mass, which is almost twice that of the kaon. Of course, the TT, K, r)$ masses must vanish in the limit of zero quark mass, while because of the axial U(l) anomaly there is no such requirement for the TJQ. Moreover, there is the presence of the quark annihilation diagram, Fig. VII-2. This can keep the 770 mass finite in the chiral limit. The exact value of this mass contribution is difficult to calculate within the quark model. At any rate, the proper expectation for the lightest pseudoscalar masses is ml = 2mBo , 2 m 5 m 3 ((22m* + + ™) ) 5 o ,

m2K = (ms + m)Bo , 2 m200 + << = m + -(ms + 2m)B' ,

(4 1)

where rao is the 770 mass in the chiral limit and B1 is in general a constant different from Bo. Here we have temporarily ignored 770-778 mixing. The discussion becomes more interesting when the connection to the axial anomaly is considered. The SU(3) singlet axial current

= ^F^F^

+ 2imuybu + 2imdd7bd + 2ims^s

where Fa" = e ^ ^ i 7 ^ , has the 770-to-vacuum matrix element * ,

(4.3)

in which F^ is the 770 decay constant. By taking the divergence of this equation we obtain

| ^

J, .

(4.4)

i=u,d,s

If the anomaly were not present, there would exist an almost conserved current for the £^4(1) symmetry. If this were dynamically broken along with the SU(3) chiral symmetry, the (pseudo-) Goldstone boson would be the 7/o. The above relation, without the FF term, would then express the result that miQ = O(mq). However, the anomaly contribution (O\FF\r)o) does not in general vanish, so that m^Q remains nonzero in the chiral limit of zero quark mass.

u,d,s

u,d,s u,d,s •

u,d,s G

Fig. VII-2 Annihilation diagram

VII-4 The rjf (960)

203

In fact, it can be argued that the largest contribution to the 770 mass is due to the axial anomaly. The argument is rather subtle, and the 770 mass was long considered to be a problem despite the anomaly. The difficulty arises because there does exist a conserved J7(l) current in the chiral limit. FF is expressible as a total divergence, FF = d^K^1 with K*1 given in Eq. (Ill—5.7). The conserved current is formed by subtracting K^ from the usual current T(0) _

T(0)

K

B

T(0)/Z _

n

(

,

-x

where quark masses are being neglected here. The associated charge will generate UA(X) transformations, and, since UA(1) is not an observable symmetry of the spectrum, it must be dynamically broken. Such reasoning leads one to expect a Goldstone boson. In the matrix element analysis of Eqs. (4.3), (4.4), the same difficulty arises because the identification of FF as a total divergence would seem to imply, by integrating by parts and discarding the surface term at infinity, that fd4xFF

= 0 or

f d4x (0\FF |r?o(p)) = 0 .

(4.6)

The latter matrix element would imply the possibility of a zero momentum state (if the quark masses vanished), apparently producing a Goldstone boson despite the anomaly. The resolution of this apparent paradox is twofold. In the first place, the conserved current JL is gauge dependent due to K^. While this does not invalidate either its use as a U(l) generator or the need for a Goldstone boson, it does allow the possibility that the Goldstone boson need not appear in the physical gauge-invariant spectrum. It can appear rather as a gauge-dependent effect [KoS 75] and need not couple to gauge-invariant operators. (However, neither is it known to be forbidden to do so.) In the matrix element argument, the resolution occurs because there exist topological effects in QCD such that the surface term at the spacetime boundary can prevent Eq. (4.6) from vanishing ['tH 76b] (see the discussion in Sect. Ill—5). This allows the simple analysis of the 7/ mass, which was given in the preceding paragraphs, to be valid. While the proof that precisely these phenomena do occur in QCD is not without controversy, it would appear that this is the route by which Nature has chosen to avoid a ninth Goldstone boson. As a result, the couplings and decays of the rjf are not related by symmetry conditions to those of the octet of pseudoscalars. That is, in QCD there is no 'nonet symmetry' in the standard context of what is meant by symmetry.

204

VII Introducing kaons and etas 770 - rjs mixing

If the u, d, s quarks were massless, the picture developed thus far would imply an octet of massless pseudoscalars plus the massive SU(3) singlet 7/0- When quark mass is introduced in perturbation theory, each squaredmass of the pseudoscalar octet becomes linear in quark masses, and the mass of the T/O is shifted slightly. In addition, SU(S) breaking in the quark mass matrix induces a mixing between the singlet and octet, i.e. between rjo and rjg. This mixing yields the physical eigenstates rj and rf \rj) = cos0\r]g) - sin0 |r/o) ,

|T/) = sin0 |T/8) + cos(9 |r/o) .

(4.7)

The goal is then to calculate 6 theoretically and measure it experimentally. There are two crucial assumptions which go into the mixing scheme described here. One is that it is possible to neglect the other pseudoscalar 7 = 0 states which could mix with both r/g and 770. A check with [RPP 90] shows that the next possible candidate is 77(1279). This is distant from the 77s mass, and hence might not mix significantly with it, but is not extremely far from the 770 mass. The second, somewhat related assumption is that the mixing parameters do not depend significantly on the energy of the state. If they did, a more complicated scheme for finding the mass eigenstates would be required. These assumptions have not yet been justified in QCD. The utility of this scheme seems to be founded on its success in phenomenological applications [GiK 87]. The 77-7/ mixing angle may be determined from experiment. In our opinion, the cleanest phenomenological analysis occurs in the two photon decays of TT0, 77, 7/. All of these decay amplitudes have the same Lorentz structure,

7(k2)

= -§

(4.8)

where the convenient normalization CM — (l> l/>/3, 2>/2/3 ) has been chosen for the unmixed states M = (TT°, 7/8, 7/0). The parameters FM are not a priori known. However, a prediction of the axial anomaly analysis is that Fn = F^, where Fn is the usual pion decay constant. Similarly, the Wess-Zumino-Witten lagrangian of Sect. VII-3 predicts Fm = Fn. We shall explore the sensitivity of this parameter to SU(3) breaking. Finally Fm is not predicted by any symmetry. Quark model prejudice, described below, also suggests Fm ~ Fn. Allowing for 77 — 7/ mixing as in Eq. (4.7),

205

VII-4 The 77'(960)

the decay rates become

1 m* fi^-costf 3^4 Fm

vo

a svn.9

(4.9)

Fvcos6

+

3 ml

The data is very sensitive to 77-77' mixing, largely because of the factor y/8 in the 77 decay amplitude, and in fact requires its presence. In particular, the 'reduced' rates are -3.0 ±0.3 ,

3 ml I V /V = 8

i^sinfl n0

|

F^cosO

(4.10) = 0.61 ±0.05 .

Using the Wess-Zumino-Witten value for FVs, Eq. (4.10) implies 6 = - 1 7 ° ±3° ,

F o = (1.08 ±0.04)2^ .

The dependence on SU(3) breaking in Fm is not large. If we use instead the value JF^ = Fm, where Fm is the usual axial current decay constant calculated in Eq. (2.8), one obtains the compatible values 6 = -22° ±3° ,

~F0 = (1.05 ± 0 . 0 4 ) ^ .

(4.11)

Two-photon decays then require a mixing angle which is around —20°. What is the effect of this mixing? In the quark model, the 778 and 770 states are octet and singlet qq states respectively. The physical mixtures would then be 77

/ _

-

_x

sin^/ _

-

_x

—(uu + dd + ss) v 6 (uu + da — 2ss) ~ 0.58(uu + dd) - 0.5755 , 3 j^[uu + da — 2ss) 2ss) HH ^ v6 v3 ~ 0.40(im + dd) + 0.8255 .

77 =

(4.12)

+ dd + ss)

The mixing is seen to decrease the 55 content of the 77 and increase it for the 77'. It is the added uu content of the 77 which enhances the transition V ~^ 77 due to the larger u quark charge. It is interesting that the parameter FVo is found to be so nearly equal to Fn. In the quark model, if one assumes that the spatial wavefunctions of the quarks in the TT, 778 and 770 are identical, the two-photon amplitudes would be proportional

206

VII Introducing kaons and etas

to the squared quark charge, i.e., + Q2d + Q2s\M°) .

(4.13)

This would produce relative amplitudes TT°: 778 : 770 = 1 : l / \ / 3 : 2^/2/3, which corresponds in the normalization of Eq. (4.8) to Fn = Fm — Fm. This need not work as well as it seems to. The theoretical aspect of mixing is addressed by considering the mass matrix, with off-diagonal element mls = (r/o \Hmass\ Vs) •

(4.14)

This can be estimated by considering the model where we use the quark model description of 770 and 773, with identical wavefunctions. In that case we would calculate 5

Bo =

—rm\

~ -0.9m2K

.

(4.15)

Although this is not a definitive analysis, it does serve to illustrate how mass mixing can be generated. In an obvious notation, the overall 2 x 2 mass matrix is then m 08 m 0 The mass rn^ is completely unknown, but m\ should be close to the GellMann-Okubo value (4mKml)(l

+ 6) ,

(4.17)

where 6 parameterizes those deviations from the Gell-Mann-Okubo relations which can occur even in the absence of 77-7/ mixing. The dependence on 6 is rather strong. If 6 = 0, diagonalization of the matrix gives the values 0 = -10°, mg8 = -0.44ra^, ra0 = 0.95 GeV, while for 6 = 0.25, one finds 6 = -24°, mg8 = 0.93 m\, m0 = 0.90 GeV. These results lie in a reasonable range, consistent with theoretical expectations. Alternatively, one can work backwards from the observed 7]-r]f mixing to determine the parameters in the mass matrix. The choice 9 = —20° ± 2° produces 6 = 0.16 ± 0.04, and mg8 = -(0.81 ± 0.05)ra^. Again, the quark model prediction Eq. (4.15) is found to work surprisingly well.

Problems 1) Other worlds Describe changes in the macroscopic world if the quark masses were slightly different in the following ways:

Problems

207

a) mu = rrtd = 0, b) mu > ra^, c) mu = 0, rrid — m8. 2) r] decay a) Prove that the amplitude for rj —> 3TT vanishes in the limit of isospin symmetry. b) Using the lowest order chiral lagrangian, calculate the rj —• 3TT decay amplitude and slope, A A

K A

1

i

I

u

yv477_).7r+7r-7ro = Alo 1 + ^ I in terms of the quark mass difference rrid ~ mu, where To is the TT°kinetic energy and Q = m^ — Zm^. Compare your results with experiment. This decay remains in disagreement even when a full O(E4) calculation is performed [GaL 85c].

3) T/-?/ mixing and chiral lagrangians

Although the 5/7(3) singlet 770 does not occur directly in the SU(3) chiral lagrangian, its effect can appear indirectly through virtual processes. a) Show that the only lowest order chiral coupling of an 770 to the chiral octet is -iV6Fnm20S

_

° X)

with a normalization constant chosen to reproduce the 77 — 7/ mixing of Eq. (5.10). b) Integrate out the 7/0 field and show that one thereby obtains a term in the chiral lagrangian with 7

64m20(r4_m2)2 •

Evaluate this numerically. Thus the effect of rj — 7/ mixing is represented in the chiral lagrangian by the value of a.7.

VIII Kaons and the AS = 1 interaction

The kaon is the lightest hadron having a nonzero strangeness quantum number. Since the weak interactions do not conserve strangeness, the kaon is unstable and decays weakly into states with zero strangeness, containing pions, photons and/or leptons. There are in fact dozens of allowed modes, and many of these have been carefully studied. In lieu of detailing all of these possibilities, we shall instead concentrate on some of the primary decay channels. VIII-1 Leptonic and semileptonic processes Leptonic decay The simplest weak decay of the charged kaon, denoted by the symbol K^-, is into purely leptonic channels K+ —» /x + ^, K+ —• e+z/e. Such decays are characterized by the constant FK, As discussed previously, because of SU(3) breaking FK is about twenty percent larger than the corresponding pion decay constant Fn. As with the pion, but even more so because of the larger kaon mass, helicity arguments require strong suppression of the electron mode relative to that of the muon. The ratio of e+z/e to / i + ^ decay rates, as in pion decay, provides a test of lepton universality [RPP 90],

r*K . e ^

v

*

= (2.44 ±0.11) x 10"5 ,

(1.2)

K+-+H+V14 expt

in good agreement with the suppression predicted theoretically, m

ml

1 - ml/ml •til •"•K 208

(1 + 8) = 2.41 x 1(T5 ,

(1.3)

VIII-1 Leptonic and semileptonic processes

209

where 6 = —0.04 is the electromagnetic radiative correction including the bremsstrahlung component.* The notation F' indicates that the experimenters have subtracted off the large structure dependent components of > £+i/£^y but have included the small bremsstrahlung component.

Kaon beta decay and V^ The kaon beta decay reactions K+ —> n o£+iy£ and K° —• TT~^ + I^, called g} and K^ respectively, also are important in Standard Model physics. They are each parameterized by two form factors,

[ (1-4) Isospin invariance implies f± = f± * = f±. SU(3) symmetry can be invoked to relate these matrix elements to the strangeness conserving transition TT+ -> iPl+vi, resulting in /+(0) = - 1 and /_(0) = 0. The deviation of /+(0) from unity is predicted to be second order in SU(3) symmetry breaking, i.e. of order (ms — rh)2. This result, the AdemolloGatto [AdG 64] theorem, is proved by considering the commutation of quark vector charges, n

[Q^iQ™]

=QUu~~ss

,

(1.5)

where

j

(

1

.

6

)

Taking matrix elements and inserting a complete set of intermediate states gives

(

2

r

)

•

d-7)

Finally, we isolate the single n~ state from the sum and note that in the SU(3) limit the charge operator can only connect the kaon to another state within the same SU(3) multiplet. This implies^ (1.8) * The dominant term here is the simple contact contribution —3(a/7r)ln(m Sect. VI-L t This is easiest to obtain in the limit p ^ —• oo

At/me)

discussed in

210

VIII Kaons and the AS — 1 interaction

where e is a measure of SU(3) breaking, and we thus conclude that ,

(1.9)

which is the result we were seeking. It is interesting that the SU(2) mass difference mu ^ rrid can modify f+ n (0) in first order despite the Ademollo-Gatto theorem. This can be seen by considering a K+ in the formulae of Eq. (VII-1.1). Now there exist two intermediate states in the same octet as the kaon, i.e. TT°and 7?°,and it is their sum which obeys the Ademollo-Gatto theorem, 1 (0) =1 + O(e2) . (1.10) 4 In the isospin limit, each term must separately obey the theorem because n ~. However, when mu ^ rrid each of the isospin relation f+*n° = f+° form factor in Eq. (VII-1.10) can separately deviate from unity to first order in mu — rrid as long as the first order effect cancels in the sum. Indeed this is what happens, yielding (cf. Prob. VIII. 1)

/f-(0) where IK-K — 0.004 arises from chiral corrections at O(E 4) [GaL 856]. This number can also be easily extracted from experiment by using the ratio of K+ and K° beta decay rates, with the result -^ = 1.029 ± 0.010 , J+

"

(1.12)

\y)

in agreement with the prediction. One of the first applications of current algebra/PC AC techniques was made to K& systems, yielding [CaT 66] <7r°(p) |57Mii| tf+(k)) _ , - - L (o I [Ql

(1.13)

The corresponding condition on the form factors, [f+(q2) + f-(q%2=m2K

=- ^

,

(1.14)

K

is called the Callan-Treiman relation. Interestingly, this does not imply that /-(0) = 1 — FK/FK, as was originally assumed. We can demonstrate this by using the chiral lagrangian of Eq. (VI-2.7) at tree level (loop effects

VIII-1 Leptonic and semileptonic processes

211

are small and will be added later). Expansion of the vector current matrix element yields

(1.15)

This form explicitly satisfies the Callan-Treiman relation off-shell, yet at q2 = 0 yields a positive value for /-(0),

= f+(m2K,ml,0)=-l

,

mj) - 0.13 ,

^ K

§F

(L16)

where we have used the previously determined value Og = (7.7±0.2) x 10~3 (which includes the effects of loops). This prediction is in agreement with the results of K^ experiments

f (0)

f

a35 ± 15 = uw 1 "-0.20 °' ± 0.08 =

{K ]

^' (combined) ,

(L17)

which clearly require the presence of ag contributions as contained in the chiral lagrangian formalism. The q2 variation of the form factors ,2\ _

/o( 9 2 ) =

£ /J2\

,

9 171

K

/

/.2\ ..

-.

A

2

(1*18)

~

follows from the same input parameter as was used to predict rare pionic processes, with the addition of FR- The agreement with experiment as displayed in Table VI-2 is good. The prime importance of the K& process is that it provides the best determination of the weak mixing element V^. Because of the AdemolloGatto theorem, the reaction is protected from large symmetry breaking corrections. In addition, the use of chiral perturbation theory allows a reliable treatment of the reaction. The above study of the form factors indicates that the theory is under control within the limits of experimental precision. The value [LeR 84] Vus = 0.2196 ± 0.0023 +

follows from an analysis of the K° and K decay rates.

(1.19)

212

VIII Kaons and the AS = 1 interaction The decay K —>

7nreue

The final semileptonic process which we consider is K —> TTTT^, labeled X^4. This reaction has the particular interest in providing the only known constraint on the large Nc assumptions sometimes made concerning chiral lagrangians, and is important in determining the low energy constants. As will be explained more fully in Chap. X, the large Nc limit imposes certain relationships between some of the terms in the chiral lagrangian. The particular relationship that is able to be tested in the K& system is the prediction OL
x [(p+ + p_)M h + (P+ - P-V h + (k - P+ - P-)M /a] , <7r + (p + )7r(p_) | l 7 ^ l ^ + ( k ) > = g ^ W ^ ^ + Z

(1.20)

9 •

The form factor f% is essentially unobservable because its effect is proportional to me in ife4, and we shall not consider it further. We have chosen the normalization such that the lowest order chiral predictions are f1 = f2 = g = l [We 66a]. The vector current form factor g = 1 is a prediction of the axial anomaly, as it is related by an SU(3) rotation to 7 —> 3?r. Experimentally, the magnitudes and slopes of the form factors at threshold

+ Ar^]

with fc2 = i ( ( p + + p _ ) 2 - 4 n 4 )

Table vra-i. Experiment and theory for Ku decays. Data /i(0)

/ 2 (0) Ai

A2 9(0) a

1.47 ± 0.04 1.25 ± 0.07 0.08 ± 0.02 0.08 ± 0.02 0.96 ± 0.24

Lowest order0 Order (E4)a 1.00 1.00 0.00 0.00 0.00

1.45 1.24 0.08 0.06 1.00

The lowest order and second order predictions of chiral symmetry.

(1.21)

VIII-2 The nonleptonic weak interaction

213

have been measured with the results [Ro et al 77] given in Table VIII-1. No dependence on the variable q2 = (k — p+ — p-)2 was observed. The enhancement in the threshold form factors and slopes is determined at O(E4:) primarily by 0:1,2,3, which also enters into TTTT scattering. A very consistent pattern emerges. The chiral KM predictions become [Bi 90, RiGDH 91] /i(0) = 1 + Xi + ^

+ {m2K

[32mjal + 4(m2K + 4ml

l

l

\

/2(0) = 1 + X2 - A [(m2. *IT

Ai = n + 8 ^ | [16aJ + 4a£ + 5ar3] ,

X2 = Y2- 8^ar3

,

where the quantities X\ = 0.127, X2 = 0.00, Fi = 0.076, Y2 = 0.045 describe the net effects of loops when evaluated at \x = m^. Since the dependence on a^ is small because of the factor of m^, we shall proceed with c*4 = 0. Phenomenologically, one may either fit /i(0),/2(0), Ai to determine ai?2,3 and then predict A2 and the seven TTTT scattering lengths and slopes, or provide the best determination of ai?2,3 by simultaneously considering all eleven of the TTTT and K^ observables. The results are very similar either way, and therefore we present only the second case. The parameters are those shown in Table VI-1, and the predictions for the observables are contained in Table VI-4 for TTTT physics and in Table VIII-1 for Ku decay. The overall description of the two systems is excellent. We note also that the analysis results in the constraint

which verifies the suppression predicted by the large Nc limit. VIII-2 The nonleptonic weak interaction Thus far in this chapter we have discussed leptonic and semileptonic processes. For these, at most one hadronic current is involved. There exist also nonleptonic interactions, in which two hadronic charged weak currents are coupled by the exchange of W± gauge bosons,

= f f
214

VIII Kaons and the AS = 1 interaction

with V being the KM matrix, given in Eq. (II-4.26). Such interactions are difficult to analyze theoretically because the product of two hadronic currents is a complicated operator. If one imagines inserting a complete set of intermediate states between the currents, all states from zero energy to Mw are important, and the product is singular at short distances. Thus one needs to have theoretical control over the physics of low, intermediate, and high energy scales in order to make reliable predictions. Because this is not the case at present, our predictive power is substantially limited. Let us first consider the particular case of AS = 1 nonleptonic decays. These are governed by the products of currents dT^u uT^s ,

dT^c cT^s ,

dTH iT^s ,

(2.2)

where F^ = 7^(1 + 75) and color labels are suppressed. The first of these would naively be expected to be the most important, because kaons and pions predominantly contain u, d, s quarks. However, the others also contribute through virtual effects. Some properties of the AS = 1 nonleptonic interactions can be read off from these currents. The first product contains two flavor-ST/(3) octet currents, one carrying I = 1/2 and one carrying / = 1, 27 , ( 8 0 8)symm 1 1 ^ 3 1(5 = *2" 2 "2 '

SU(3) : Isospin :

(2.3a) (2.3b)

where the symmetric product is taken because the two currents are members of the same octet. The singlet SU(3) representation is excluded from Eq. (2.3a) because a AS = 1 interaction changes the SU(3) quantum numbers and hence cannot be an 5(7(3) singlet. The other two products in Eq. (2.2) are purely SU(3) octet and isospin 1/2 operators. The currents are also purely left-handed. Thus the nonleptonic hamiltonian transforms under separate left-handed and right-handed chiral rotations as (8L,1R) and (27 L,1R)- These symmetry properties, valid regardless of the dynamical difficulties occurring in nonleptonic decay, allow one to write down effective chiral lagrangians for the nonleptonic kaon decays. The hamiltonian is a Lorentz scalar, charge neutral, AS = 1 operator, and has the above specified chiral properties. At order E2, there exist two possible effective lagrangians for the octet part, viz. £Octet = C$ + C%, where in the notation of Sect. IV-6, £ 8 = g8 Tr (^DJJD»U^

,

£ 8 = 98 Tr (A 6X t/t) + h.c. .

(2.4)

It can easily be checked that both £% and C% are singlets under righthanded transformations, but transform as members of an octet for the left-handed transformations. The barred lagrangian in Eq. (2.4) can in

VIII-2 The nonleptonic weak interaction

215

fact be removed, so that it does not contribute to physical processes. This is seen in two ways. At the simplest level, direct calculation of K —• 2TT and K —> 3?r amplitudes using Cs, including all diagrams, yields a vanishing contribution. Alternatively, this can be understood by noting that in QCD the quantity x appearing in Eq. (2.4) is proportional to the quark mass matrix, x — 2B$mq. Thus the effect of £g is equivalent to a modification of the mass matrix, mq —• mq = mq + g%\§mq

.

(2-5)

This new mass matrix can be diagonalized by a chiral rotation Tr (m'qU) -+ Tr (Rm'qLU) = Tr (mDU)

,

(2.6)

with mo diagonal. The transformed theory clearly has conserved quantum numbers, as it is flavor diagonal. This means that the original theory also has conserved quantum numbers, one of which can be called strangeness. When particles are mass eigenstates, even in the presence of £8? the kaon state does not decay. Hence this Cs can be discarded from considerations, leaving only Cs as responsible for octet K decays [Cr 67]. This octet operator is necessarily A / = 1/2 in character. Another allowed operator, transforming as (27^,1^), contains both A / = 1/2 and A / = 3/2 portions,

where /»(l/2)

jL/97

27

: =

(1/2) x-^1/2 rri

5^97

= ^27

^

/> ^

| \ Ot O r r TT\ \bcs TT TT^ I

i "U

\ ^ OurJ U *• Ou*J C/ I "+" n.C. ,

^ a 6 ^^ l ^ ^ ^ C/' A a^C/ (7 ' ) + h.C. \ /

/O O \

iZ.odi)

(z.ODJ

The coefficients are given by C ly/2

=1

^3/2

_-

°fi-i-«7/9 3 —

x

J

C1/2

= — \/2

^3/2

_ J_

°4+t5/2,l-t2/2 ~~

/o

^1//2 *

The complete classification at order E4 is difficult, but has been obtained [KaMW 90]. We shall apply these lagrangians to the data in Sects. VIII4, XII-6. There we shall see that gs » g^, whereas naive expectations would have octet and 27-plet amplitudes being of comparable strength. This is part of the puzzle of the A / = 1/2 rule. The reliable theoretical calculation of the nonleptonic decay amplitudes, which is tantamount to (1/2)

(3/2)

predicting the quantities gs, #27 and g^7 , is one of the difficult problems mentioned earlier. It has not yet been convincingly accomplished.

216

VIII Kaons and the AS = 1 interaction

The best we can do is to describe the theoretical framework of the short distance expansion, to which we now turn. VIII-3 Short distance behavior At short distances, the asymptotic freedom property of QCD allows a perturbative treatment of the product of currents. The philosophy is to use perturbative QCD to treat the strong interactions for energies Mw > E > /i. The result is an effective lagrangian which depends on the scale fi. Ultimately matrix elements must be taken which include the strong interaction below energy scale /i and the final result should be independent of //. Short distance operator basis The outcome of the short distance calculation can be expressed as an effective nonleptonic hamiltonian expanded in a set of local operators with scale dependent coefficients [Wi 69], .

(3.1)

As in any effective lagrangian, those operators of lowest dimension should be dominant. If the operator O{ has dimension d, its coefficient obeys the scaling property Q ~ M^d. Let us first see how this hamiltonian is generated in perturbation theory. We can later use the renormalization group to sum the leading logarithmic contributions. The lowest order diagrams renormalizing the current product are given in Fig. VIII-1. The process in Fig. VIII-1 (a) corresponds to a left-handed, gaugeinvariant operator of dimension 4, O (d=4) = dlp{\ + 75)5 .

(3.2)

This operator can be removed from consideration by a redefinition of the quark fields (cf. Prob. IV-1). The remaining operators are of dimension 6. Simple ^-exchange with no gluonic corrections gives rise in the short distance expansion to the local operator OA = d7fi (1 + 7 5 ) ^ 7 ^ (1 + 75) s ,

(3.3)

u,c,t

W W (a)

s

u (b)

u

Q

Q (O

Fig. VIII-1 QCD Radiative corrections to the AS = 1 nonleptonic hamiltonian

VIII-3 Short distance behavior

217

with a coefficient CA = 2 in the normalization of Eq. (3.1). The gluonic correction of Fig. VHI-l(b) generates an operator of the form J7 M (l+75)A a u«7 / '(l+75)A o s ,

(3.4)

a

where the {A } are color SU(3) matrices. However, use of the Fierz rearrangement property (see App. C) and the completeness property Eq. (II2.8) of SU(3) matrices allow this to be rewritten in color-singlet form h» (1 + 75) Aa« «

7

M

(1 + 75) Xas = -\oA

+ 20B ,

+ jb)ud^(l

.

where OB = u^(l

+ j5)s

(3.5)

The strong radiative correction is seen to generate a new operator OBPerturbative analysis Consider now the one-loop renormalizations of the four-fermion interaction Fig. VIII-l(b). In calculating Feynman diagrams we typically encounter integrals such as (neglecting quark masses) -

J

16TT2M^

K2

-r iv±w

(3.6)

where we evaluate the integral at the lower end using a scale /x. Clearly presents a natural cutoff in the sense that

The modification of the matrix element to first order in QCD is then ,

(3.8)

where gs is the quark-gluon coupling strength. The gluonic correction to OB must also be examined, and a similar analysis yields OB - OB - JL

In (^)(3OA-

O B) .

(3.9)

We observe that the operators, O± = ^(OA±OB)

,

are form-invariant, O± —• c±O±, with coefficients c±,

(3.10)

218

VIII Kaons and the AS = 1 interaction

where e/+ = —2 and GL = +4. The isospin content of the various operators can be determined in various ways. Perhaps the easiest method involves the use of raising and lowering operators [Ca 66, Li 78], /_i_d — u ,

I+u = —d ,

I-U = d ,

I-d = —u ,

(3.12)

to show that I+O- = 0, implying that O_ is the Iz = 1/2 member of an isospin doublet. With repeated use of raising and lowering operators, one can demonstrate that O_ is purely A / = 1/2 whereas O+ is a combination of A / = 1/2 and A / = 3/2 operators. Prom Eq. (3.11), we see that under one-loop corrections the operator O- is enhanced by the factor

fL^!2.1

(3.13)

where we use a8(fi) ~ 0.4 (AQCD — 0.2 GeV) at // ~ 1 GeV. Similarly O+ is accompanied by the suppression factor

Renormalization group analysis Choosing an even smaller value of // would lead to an even larger correction. However, maintaining just the lowest order perturbation in the QCD interaction would then be unjustified. It is possible to do better than the lowest order perturbative estimate by using the renormalization group to sum the logarithmic factors [GaL 74, AIM 74]. In a renormalizable theory physically measurable quantities can be written as functions of couplings which are renormalized at a renormalization scale /iR. Physical quantities calculated in the theory must be independent of /iR. Denoting ail arbitrary physical quantity by Q, this may be written Q = /(03(MR)>MR)

>

(3.15)

where / is some function of //R and 53 is the strong coupling constant of QCD. Differentiating with respect to /XR, we have O ,

(3.16)

which is the renormalization group equation. It represents the feature that a change in the renormalization scale must be compensated by a modification of the coupling constants, leaving physical quantities invariant. In order to see how this program can be carried out for the effective weak hamiltonian, consider the following irreducible vertex function

VIII-3 Short distance behavior

219

which represents a typical weak nonleptonic matrix element, (0 \

( ^

(y^f (0 where the {<&} are quark fields carrying momenta {pi}. Z2 is the quark wavefunction renormalization constant for the fermion field, and subscripts 'ren', 'unren' denote renormalized and unrenormalized quantities. Choosing the subtraction point yy\ = —/J*^ we require that unrenormalized quantities be independent of //R, r

unren

= 0.

(3.18)

This implies ) \ o)™ = 0,

0

where g$r is the renormalized strong coupling constant, /?QCD is the QCD beta function of Eq. (II-2.57) and JF is the quark field anomalous dimension of Eq. (II-2.69). As we have seen, QCD radiative corrections generally mix the local operators appearing in the short distance expansion, (0 \T (Onqiq2q3q4)\ 0)£n - £

Xnn, (0 \T (On

and the mixing matrix can be diagonalized to obtain a set of multiplicatively renormalized operators (0 \T (OkQlq2q3q4)\ 0>™ = Zk (0 \T {Om
•

(3-20)

If Zk has anomalous dimension 7^, Z f c ~ l + 7fclnMR + -.. ,

(3.21)

then the coefficient functions Ck(^Rx) must satisfy — + /SQCD^— + Ik ~ 47F I ck(fjLRx) = 0 . /iR dgsr ) Prom the above, we have for the operators O± &±

•

(3.22)

(3 23)

-

Having specified the anomalous dimensions of the operators O±, we can solve Eq. (3.22) with methods analogous to those employed in Sect. II—2. That is, because QCD is asymptotically free and we are working at large

220

VIII Kaons and the AS = 1 interaction

momentum scales, we can use the perturbative result (cf. Eq. (II-2.57)),

where 6 = 1 1 — | n / , rif being the number of quark flavors. Upon inserting the leading term in the perturbative expression for as (cf. Eq. (II-2.74)), 12TT M

(325)

one can verify that the solution to Eq. (3.22) is given by / C±(»R)/C ±(MW)

n2

M2

\d±/b

= ^1 + ^ f t l n - f J

.

(3.26)

Note that in the perturbative regime where 1 ^> as, we have C±(/*R)

.

93

M ]n W

c±(Mw) which agrees with our previous result, Eq. (3.11). It is the renormalization group which has allowed us to sum all the 'leading logs'. Of course, at scale Mw one must be able to reproduce the original weak hamiltonian, implying c+(Mw) = c-(Mw) = 1. Taking //R ~ 1 GeV and as = 0.4 as before, we find with c_(/iR) ~ 1.5 ,

c+(/iR) ^ 0.8 .

(3.29)

We observe then a A / = 1/2 enhancement of a factor of 2 or so, which is encouraging but still considerably smaller than the experimental value of A0/A2 ~ 22 discussed in the next section. Two additions to the above analysis must now be addressed. One is the proper treatment of heavy quark thresholds. In reducing the energy scale from Mw down to /iR, one passes through regions where there are

Fig. VIII-2

Penguin diagram.

VIII-3 Short distance behavior

221

successively six, five, four or three light quarks, the word 'light' meaning relative to the energy scale //R. The beta function changes slightly from region to region. A proper treatment must apply the renormalization group scheme in each sector separately. This is a straightforward generalization of the procedures described above. The other addition is the inclusion of penguin diagrams of Fig. VIIIl(c) [ShVZ 77, ShVZ 79b, BiW 84], whimsically named because of a rough resemblance to this antarctic creature. The gluonic penguin is noteworthy because it is purely A/ = ^, thus helping to build a larger A/ = | amplitude, and because it is the main source of CP violation in the AS — 1 hamiltonian. The electroweak penguin, wherein the gluon is replaced by a photon or a Z° boson, also enters the theory of CP violation. The CP conserving portion of the penguin diagrams involves a GIM cancelation between the c, u quarks and hence enters significantly at scales below the charmed quark mass. On the other hand, in the CP violating component, the GIM cancelation is between the t, c quarks and thus this piece is short distance dominated. At lowest order, before renormalization group enhancement, one obtains the following effective interactions for the diagram of Fig. VIII-2, ^Vus In ^

L ^

ln

+ V;dVts In ^ |

m? + ^

^ r^

^R

m

^

^

)

^

c\

(3.30) We have used a scale /iR instead of the up quark mass and have quoted only the logarithmic mt dependence. The quarks q = u,d,s are summed over and Qq is the charge of quark q. Note that since the vector current can be written as a sum of left-handed and right-handed currents, this is the only place where right-handed currents enter Hw. The gluonic penguin contains the right-handed current in an SU(3) singlet, hence retaining the (8L, lfl) property ofHw. However, the electroweak penguin introduces a small (8^,8^) component. The full result can be described with the four-quark AS = 1 operators,

2HC + 2HD , O 5 = d^(l

+

7b)X

2HC - 3HD , O 6 = J 7 / i ( l + 75)5 O7 = |g>y^(l + 75)d K)Qqq >

(3.31)

222

VIII Kaons and the AS = 1 interaction

3 O8 = - 2 ^ 7 / i ( l + 75)4/ qj'fil

- ^fb)QqQi i

where q = u,d,s are summed over in 05,6,7,85 * a n d J is the charge of quark q and HA = <*7/i(l + 75)^ ^ ( 1 + 75)5 ,

are

color labels, Qq

# c = d7/x(l + 75)5 ^ ( 1 + 75 K

rf7/i(l + 75)5 vr/Hl + 75)ix , JETD = ^7/i(! + 75)5 57^(1 + 75)5. (3.32) The operators are arranged such that Oi52,5,6 have octet and A / = \ quantum numbers, 03(04) are in the 27-plet with A7 = ^(A7 = | ) , while OY$ arise only from the electroweak penguin diagram. The full hamiltonian is (3 33)

•

-

A renormalization group analysis of the coefficients [BuBH 90] yields = 1.90 - 0.62r ,

c5 = -0.011 - 0.079r ,

c2 = 0.14 - 0.020r ,

ce = -0.001 - 0.029r ,

c3 = c 4 /5 ,

c7 = -0.009 - (0.010 - 0.004r)a ,

c4 = 0.49 - 0.005r ,

c8 = (0.002 + 0.160r)a ,

Cl

(3.34)

with A - 0.2 GeV, /i R ~ 1 GeV, mt = 150GeV and r = -V^Vts/V*^. The number multiplying r has a dependence on mt whereas (within the leading logarithm approximation) the remainder does not if mt > MwThe r dependence in C4 arises only because of the electroweak penguin diagram. This hamiltonian summarizes the QCD short distance analysis and is the basis for estimates of weak amplitudes.

VIII-4 The A/ = 1/2 rule

Phenomenology In the decays K —• TTTT, the 5-wave two-pion final state has a total isospin of either 0 or 2 as a consequence of Bose symmetry. Thus, such decays can be parameterized as

^ o

= Ao ei6° - V2A2ei6*

,

(4.1)

VIII-4 The AI = 1/2 rule

223

where the subscripts 0,2 denote the total TTTT isospin and the strong interaction 5-wave TTTT phase shifts <5j enter as prescribed by Watson's theorem (c/. Eq. (C-2.15). There are, in principle, two distinct 1 — 2 amplitudes A2 and Af2. These are equal if there are no AI = 5/2 components in the weak transition, as is the case in the Standard Model if electromagnetic corrections are neglected. The experimental decay rates themselves imply r+Tr-1 = 5.56 x W-7mK , ^O«O\ = 5.28 x W-7mK , \AK+-+n+*o\ = 3.72 x 10-smK .

(4.2)

The test to see whether Af2 = A2 depends on the imprecisely known TTTT phase difference 80 — 82- For the value 8$ — 82 — 45°favored by fits to 7T7T scattering there appears to exist some AI = 5/2 effect, which would presumably arise from electromagnetic corrections, while for 80 — 82 — 57° there is none. We shall neglect this possibility from now on, and employ just the isospin amplitudes \A0\ = 5.46 x 10-7mK ,

\A2\ = 0.25 x 10" 7 m^ .

(4.3)

The ratio of magnitudes, \A2/Ao\~l/22 ,

(4.4)

indicates a striking dominance of the AI = 1/2 amplitude (which contributes to Ao) over the AI = 3/2 amplitude (which contributes only to A2). This enhancement of Ao over A2, together with related manifestations to be discussed later, is called the AI — 1/2 rule. As we have seen in previous sections, a naive estimate (and even determinations which are less naive!) do not suggest this much of an enhancement. However, the factor 20 dominance of AI = 1/2 effects over those with AI — 3/2 is common to both kaon and hyperon decay. A similar enhancement of AI — 1/2 is found in the K —• TTTTTT channel. In this case, it is customary to expand the transition amplitude about the center of the Dalitz plot. For the decay amplitude K(k) —• 7r(pi) n(p2) ft(ps), the relevant variables are Si

= (k-

(4.5) x

y so

, s

o

where S3 labels the 'odd' pion, i.e. the third pion in each of the final states n+n~n°,nonO7T+,ir+Tr+n~. The large A / = 1/2 amplitudes are

224

VIII Kaons and the AS = 1 interaction

considered up to quadratic order in these variables while the A / = 3/2 amplitudes contain only constant plus linear terms, O^+V-KO

=

OI -

2a3

+

(6i

|

( W +

^ -

= 2(ai + a 3 ) -

^j iS

(4.6)

, i6

+ 2c (V + ^

where ai,6i,c,d are A/ = 1/2 amplitudes, 03,63,623 are A7 = 3/2 amplitudes, and the phases {61} in <5MI (= &M — S\) and 621 (= 62 — f)\) refer to final-state phase shifts in the 7 = 1 , 2 and mixed symmetry 7 = 1 states respectively. Because of the relatively small Q-value for such decays (QnTTTT = ™
a3 = 0.37 ± 0.018 , 63 = -0.646 ± 0.125 , d = -3.39 ± 0.33 .

623 = -2.3 ± 0.31 ,

(4.7) Dominance of the A7 = 1/2 signal is again clear in magnitude and in slope terms, e.g. we find at the center of the Dalitz plot, las/ail-1/26 .

(4.8)

In SU{3) language, the dominance of A7 = 1/2 effects over A7 = 3/2 implies the dominance of octet transitions over those involving the 27plet. This is a consequence, within the Standard Model, of the fact that the A7 = 1/2 27-plet operator contributes relative to the A7 = 3/2 27plet operator with a fixed strength given by the scale-independent ratio of coefficients C3/C4 ~ 1/5 (viz. Eq. 3.34). The 27-plet operator then gives only a small contribution to the A7 = 1/2 amplitudes, with the major portion coming from the octet operators. We shall therefore ignore the A7 = 1/2 27-plet contribution henceforth.

VIII-4 The A/ = 1/2 rule

225

Chiral lagrangian analysis The left-handed chiral property of the Standard Model may be directly tested by the use of chiral symmetry to relate the amplitudes in K —> TTTTTT to those in K —> TTTT. We have already constructed the effective lagrangians for (8/,, 1#) and (27^,1#) transitions. Dropping g^7 , the nonleptonic decays are described by the two parameters gs and g^ at O(E2). Let us see how well this parameterization works, and afterwards add O(E4) corrections. The two free parameters may be determined from AQ and A^ in K —> TTTT decays. Prom the chiral lagrangians of Eqs. (2.4), (2.8b), we find (3/2) (

m2}

<

(4

g)

which yields upon comparison with Eq. (4.3), gs ~ 7.8 x l O " 8 ^ ,

5g

/2)

- 0.25 x l ( T 8 i ^ .

(4.10)

The K —• TTTTTT amplitude may be predicted from these. Because there are only two factors of the energy, no quadratic terms in Eq. (4.6) are present in the predictions, 1 (1/2)

v

U---A

,(3/2)

_

^ 4

(3/2)

_

A 2 m^

'"A

i^

fi

'

7T-

5

[

which correspond to the numerical values (again in units of 10~ 7),

+7r+jr _

= 0.47 + 2 . 3 y .

These are to be compared to the experimental results, T+n_n0

= 9.15 + 14.1 y - 4.85 f + 0.88 x2 ,

^ ^ - ^ = -0.71 + 1 . 3 y , +7r+7r _

(4.12)

= 0.71 + 2.9 y ,

with error bars given previously in Eq. (4.7). This comparison can be seen in Fig. VIII-3, where a slice across the Dalitz plot is given. Also

226

VIII Kaons and the AS = 1 interaction

shown are the extrapolations outside the physical region to the 'soft-pion point' where either p+ —> 0 or j£ —> 0. Predictions at these locations are obtained by using the soft-pion theorem. The chiral relations clearly capture the main features of the amplitude and demonstrate that the K —• 3?r A/ = 1/2 enhancement is not independent of that observed in K —• 2?r decay. However, for the A/ = 1/2 transitions, knowledge of the parameters c and d which accompany the quadratic kinematic terms (cf. Eqs. (4.6),(4,7)) allows us to do somewhat better. The kinematic dependence of x2 or y2 can come only from a chiral lagrangian with four factors of the momentum, and only two combinations are possible: Aquad = 7i& • PoP+ • P- + 72 (k • k+Po ' P- + k • p-p0 - p+)

•

(4.13)

Such behavior can be generated from a variety of chiral lagrangians, (4.14)

g'lTr

+

However the predictions in terms of 7$ are unique. Fitting the quadratic terms to determine 71,72 yields the full amplitude, -*O = (9.5 ± 0.7) + (16.0 ± 0.5) y-4.85 y +0.88 x2 , (4.15) which provides an excellent representation of the data. Final state interaction effects also provide an important contribution and must be included in a complete analysis [KaMW 90]. Note that in the process of determining the quadratic coefficients, the constant and linear terms have also become improved. This process cannot be repeated for A/ = 3/2 amplitudes due to a lack of data on quadratic terms.

-5-4-3-2-1

0

1 2

3 4

5 6

Fig. VIII-3 Dalitz plot

VIII-5 Rare kaon decays

227

Vacuum saturation The discussion of direct calculations of the nonleptonic amplitudes is beyond the scope of this book. Suffice it to say that no treatment is presently adequate. Let us give the simplest estimate, called vacuum saturation, as a convenient benchmark with which to compare the theory. For simplicity we consider only O\ (the largest A/ = 1/2 operator) and O4 (the A/ = 3/2 operator), ,

(4.16)

with c\ ~ 1.9 and C4 ~ 0.5. The vacuum saturation approximation consists of inserting the vacuum intermediate state between the two currents in any way possible, e.g. <7r+(p+)7r-(p_) \drf (1 + 75) uu-f (1 + 75) s = <7T-(P_) \d>fw\ 0) <7T+(P+) \u*f8 + <7r+(p+)7r-(p_) \upfual 0> (0 14 n(p_ - P + In obtaining this result the Fierz rearrangement property den* (1 + 75) uaup^ (1 + 75) sp = daY (1 + 75) spup^ (1 + 75) ua has been used, where a, j3 are color indices which are summed over. In addition, the color singlet property of currents is employed, (0 \dal^sp\ K°(k))= iy/2FKk^

.

(4.18)

Within the vacuum saturation approximation, we see that the amplitudes are given completely by known semileptonic decay matrix elements. Putting in all of the constants, we find that AQ = —V* dVwFn ( m i - ml) cx = 0.84 x 10~7ra# , (4.19) ^2 =

5—-KdV^F,. (m^ - m£) c

4

7

= 0.42 x 10" r

We see that the above estimate of A2 works reasonably well, but that Ao falls considerably short of the observed A/ = 1/2 amplitude. This demonstrates that vacuum saturation is not a realistic approximation. However, it does serve to indicate how much additional A/ = 1/2 enhancement is required to explain the data.

228

VIII Kaons and the AS = 1 interaction VIII-5 Rare kaon decays

Thus far, we have discussed the dominant decay modes of the kaon. There are, however, many additional modes which, despite tiny branching ratios, are the subject of intense experimental activity. We can divide these into three main categories. 1) Forbidden decays - These include tests of the flavor conservation laws of the Standard Model. Positive signals would represent signals of physics beyond our present theory. 2) Rare decays within the Standard Model - These include decays which occur only at one-loop order. Such processes can be viewed as tests of chiral dynamics as developed in this and preceding chapters (e.g., radiative kaon decays) or as particularly sensitive to the particle content of the theory (e.g., K+ —• TT + Z/^ (£ = e^fi^r) probes the top quark mass and the number of light neutrinos). 3) CP violation studies - As will be discussed in the next chapter, the kaon system has thus far provided the only positive information on CP violation.* Any of these have the potential to yield exciting physics. We shall content ourselves with discussing only a small sample of the many possibilities. Consider first the rare decay K+ —• n+U£U£. This mode is often called 'K+ to TT+ plus nothing', in reference to its unique experimental signature. This process can take place only through loop diagrams, such as those in Fig. VIII-4. Calculation of these Feynman diagrams leads to an effective lagrangian of the form [ImL 81, HaL 89]

< • ( ) y^

()••

Fig. VIII-4 The decay The observed baryon-antibaryon asymmetry of the universe also requires the existence of CP violation within the standard cosmological model.

VIII-5 Rare kaon decays

229

and

where Xj =

(4-z) 3 x x D(x) = (5.2) 2 4 4 1— x (1-x) {l-xf 8 In this formula we have neglected mj/rriyy (£ = e,/x, r). The matrix element in this case is simple, being related by isospin to the known charged current amplitude

(?r+(p) [sj^d] K+(k)) = V2

(k)) = f+{q2) {k +p)^ (5.3) with /+(0) = —1, and yields a straightforward prediction for the branching ratio in terms of the number Nu of light neutrinos, Brf

+

(TT°(P) \S^^U\ K

7

(5.4)

W'

= 0.7 x 10T 6 D{xc) + s2 The precise branching ratio depends on the KM angle and top quark mass. Although present estimates suggest only a range of branching ratios, 0.5-8.0 x 10~10, future determinations of the KM angles and mt can provide a very nontrivial test of the Standard Model at one-loop order. This rate seems to be reasonably insensitive to long distance physics. A different class of rare decays consists of the radiative processes K$ —> 77 and KL —» TT°77. These transitions provide interesting tests of chiral perturbation theory at one-loop order. In this case, the long distance process, Fig. VIII-5, is dominant. An important feature is that there is no tree level contribution at order E2 or E4 from any of the strong or weak chiral lagrangians because all of the hadrons involved are neutral. Thus the decays can only come from loop diagrams, or from lagrangians at O(EG). There is also an interesting corollary of this result concerning the renormalization behavior of the loops. Since there are no tree level counterterms at O(E4) with which to absorb divergences from the loop diagrams, and recalling that we have proven all divergences can be han-

a*

(a)

(b)

Fig. VIII-5 Long distance contributions to radiative kaon decays.

VIII Kaons and the AS — 1 interaction

230

died in this fashion, it follows that the sum of the loop diagrams must be finite. This is in fact born out by direct calculation. For Ks —> 77, the prediction of chiral [D'AE 86, Go 86] loops is given in terms of known quantities such as a2m\ i j S f _> 7 7

1-

167T 3 2

Tnz,

mK

mK

In2 Q{z) - 2m In Q(z)]

F(z) = l-z[ir 1 Q(z) =

(5.5)

where g$ is the nonleptonic coupling defined previously in Eq. (2.4b). The theoretical one-loop branching ratio,

g S n = 2-0 x 1(T6 ,

(5.6a)

compares favorably with the recent measurement +77

=(2.4±1.2)xlO-6

(5.6b)

without the need to consider possible contributions at O(Ee). The case of KL —> n0/yj is also instructive. Again, one-loop contributions are finite and [EcPR 88] unambiguous. Indeed, we know that KL —• TT°77 and KL —* 77 are related by the soft-pion theorem in the limit"pfc—»• 0. Explicit evaluation yields ,1/2

dT dz / X

where z =

z—

ml

mK

V

m2

\

rn?KJ

(5.7)

F(z)

and X(a, b) = 1 + a2 + b2 - 2(o + b + ab)

(5.8)

Integration over 2 yields the branching ratio r

(loop)

- 6.8 x 10

(5.9)

For this reaction, there is also an O(E6) correction which could be important [Se 90], viz. the diagram of Fig. VIII-5(c) which shows the effect of p-exchange. Such a contribution would have a very different spectrum from that of Eq. (5.7). However, experiment [Ba et al. 90] reveals a photon energy distribution dT/dz which matches that of Eq. (5.7), but with the rate B r ^ o ^ o ^ ^ (2.1 ± 0.6) x 10~6. No evidence exists for a p-exchange contribution.

Problems

231

Problems 1) Ke3 decay The ratio f+^n° (0)/f+O/ir (0) of semileptonic form factors is a measure of isospin violation. Part of this quantity arises from 7r°-r/g mixing. a) By diagonalizing the pseudoscalar meson mass matrix, show that rrid ^ mu induces the mixing |TT°) = cose\(fs) + sine\(fs) where e ~ V3(md — mu)/[4t(m8 — TO)]and TO = (mu + TO^)/2. b) Demonstrate that this leads to the result (c/. Eq. (1.11))

2) Current algebra/PCAC and K -> 3?r decay The results derived in Sect. VIII-4 with effective lagrangians can also be obtained by means of current algebra/PCAC methods. a) Using PC AC, show that the soft-pion limit of the K —> 3?r transition amplitude is given by qa-*0

a

c

b {

Fn

where Q^ = f d3x Afifa t) is the axial charge, b) Demonstrate that this may be also written as

)h^(rfyqbK°c\Hw(0)\K2)

= y(7riKcqc\[Qa,Hw(0)]\K%) ,

where Qa is an isotopic spin operator, and hence that

1 where / = 5, | signifies question. c) Use a linear expansion Eq. (4.6) with c=d=0) to corrections of order

+ 0)

the isospin component of the quantities in of the K -+ 3n transition amplitude, (i.e. to reproduce the results of Eq. (4.11), up ml.

IX Kaon mixing and CP violation

In our discussion of the electroweak interactions in Chap. II, we saw that the KM matrix contains imaginary couplings which have the potential to violate CP invariance. These arise originally in the Yukawa couplings of quarks to the Higgs bosons, but after the definitions of mass eigenstates, they appear only as a single phase in the W± gauge couplings. In this chapter, we focus on the system of neutral kaons to describe how this phase gives rise to CP violating observables. IX-1 K°-K° mixing It is clear that K° and K°should mix with each other. In addition to less obvious mechanisms discussed later, the most easily seen sourceof mixing occurs through their common TTTT decays, i.e. K°<-> TTTT <-> K°. We can use second-order perturbation theory to study the phenomenon of mixing. Mass matrix phenomenology Writing the wavefunctions in two-component form \

,

)

we have the time development

232

(1.1)

IX-1 K°-K° mixing

233

where, to second order in perturbation theory, the quantity in parentheses is called the mass matrix and is given by*

2 Uj

2mK

(K?\Hw\n (1.3) Here the absorptive piece F arises from use of the identity ^—^

= p(-^-=r)-in6(En-u;)

u — t,n + xe

\u) —

,

(1.4)

bjn)

and hence involves only physical intermediate states 1

n> (n\Hw\K°) 2n8{En - mK)

.

(1.5)

Because M and T are hermitian, we have M21 = M*2 and F2i = T\2> The diagonal elements of the mass matrix are required to be equal by CPT invariance, leading to a general form A p2 where A, p2 and g2 can be complex. The states K° and K° are related by the unitary CP operation,

CP\K°)=tK\K°)

(1.7)

with \£K\2 = 1. Our convention will be to choose £# = — 1. The assumption of C P invariance would relate the off-diagonal elements in the mass matrix so as to imply p = q,

{K°\neS\K°) = (K°KCPJ^CP HeS (CP)-1CP| ^°> = (K°|Weff| K°), (1.8) where (K° \Hes\ K°) is defined in Eq. (1.3). Combined with the hermiticity of M and F, this would imply that Myi and Fi2 are real. In the actual CP-noninvariant world, this is not the case and we have for the eigenstates of the mass matrix, \KL ) = —=j s

V\P\

2

+ kl

2

\p\K°)± q\K0)} ,

(1.9)

* The factors l/2ra/<- are required by the normalization convention of Eq. (C-2.7) for state vectors.

234

IX Kaon mixing and CP violation

where from the above discussion, we have p_

Mn - |Fi2

The difference in eigenvalues is given by 2qp = (mL - ms) - ~{TL - Ts) ( i \1/2/ i \1/2 = 2 ( M12 - ^ r i 2 Mx*2 - - H 2 ~ 2ReM12 - i

(1.11) where the final relation is an approximation valid if CP violation is small (1 » ImMi2/ReMi2). The subscripts in KL and Ks, standing for 'long' and 'short', refer to their respective lifetimes, which differ by a factor of 580. To understand this large difference, we note that if CP were conserved (p = q), these states would become CP eigenstates K± (not to be confused with the charged kaons K±\), \Ks) ^

\K°+) ,

]KL)

0

\K°±) = - L [\K°)T |^ )] ,

_^

]Ko_} f

P=Q

CP\Kl) = ±\K%) .

n 22)

In this limit, which well approximates reality, Ks would decay only to CP-even final states like TTTT, whereas KL would decay only to CP-odd final states. Since the phase space for the former considerably exceeds that of the latter at the rather low energy of the kaon mass, Ks has much the shorter lifetime. The states KS,L, expanded in terms of CP eigenstates, are

- Ts)

(1-13) K°-K° mixing can be observed experimentally from the time development of a state which is produced via a strong interaction process, and therefore starts out at t = 0 as either a pure K° or K°,

(1.14)

9±(t) = I

IX-1 K°-K° mixing

235

where AF = T$ — TL and Am = m^ — 7715, each defined to be a positive quantity. From such experiments, the very precise value Am = (3.522 ± 0.016) x 10"12 MeV has been obtained. Box diagram contribution The Standard Model predicts K°-K° mixing at about the observed level. However, it is difficult to precisely calculate the mass difference Am. There are two main classes of effects, short-distance box diagrams of Fig. IX-1 (a),(b) and long-distance contributions like those in Fig. IX-2, which can generate the mixing amplitude. We turn now to a study of the first of these. When calculated at the quark level, one can envision several Feynman diagrams which mix K° and K° (see Fig. IX-1). The box diagrams in Fig. IX-1 (a),(b) lead to an effective hamiltonian dKs)2 H(xc)m%

+ (V^Vts)2

H(xt)m2m

+ 2 (V&VtaV&Vn) G(x« x t)m2cm] OAS=2 + h.c. , where X{ = mf/M^ and [InL 81] l

9

1

3

1

,

4 + 4T^-2(l^J-2(T^ l n a ; ' (1.16)

v

* ' 4(l-s)(l-y) The factors 771, 772 and 773 are QCD short-distance renormalization factors analogous to the renormalization group coefficients for the AS = 1 hamiltonian discussed in the previous chapter. For rrtt > Mw,

u.c.t

u,c,t

u,c,t

W (a)

<j

s

w s

u,c,t d (b)

Fig. IX-1 Box (a),(b) and other contributions to CP violation.

236

IX Kaon mixing and CP violation

0.2 GeV, they have the values [BuBH 90], 7/i ~ 0.85 ,

% ^ 0.62 ,

773 ~ 0.36 .

(1.17)

Note that the GIM mechanism is at work here, in that the u, c, t intermediate states would exactly cancel each other if their masses were equal. The mass of the ?x-quark has been neglected with respect to heavy quark masses in writing Eq. (1.15). Given present bounds on the KM elements and the t quark mass, the most important contribution to the real part of H^ox is that of the c-quark. In view of this, and noting that H(xc) ~ 1, we then have Re H^T * ^2^Re(F c * d F c s ) 2

Vl

OAS=2 .

(1.18)

At this stage one can provide a rationale for neglecting Fig. IX-l(c). In this diagram the GIM cancelation is logarithmic rather than power law, so no factor of m2 appears. Evidently, all matrix elements of other diagrams should be suppressed compared to the box diagram by factors like fJ?/rn% (where /i is a hadronic scale) or m^/m2. This has been verified for Fig. IX-l(c), and will be discussed below as a 'long-distance' effect. The double penguin diagram of Fig. IX-l(d) has a long-distance part and a short-distance component, obtained when the loop momenta are high. Separating the two is difficult, but the short-distance piece has been studied [EeP 87]. To make contact with phenomenology, one must evaluate the matrix element of OAS=2 between K° and K° states. It is conventional to express the results in terms of the so-called B-parameter, ^

,

(1.19)

where B = 1 corresponds to the simple vacuum saturation approximation described in Sect. VIII-4. A wide variety of models have been applied to determine this amplitude, with estimates for B which vary considerably [PaT 89]. One approach attempts to extract B directly from experiment by using chiral symmetry and SU(3) [DoGH 82]. The point is that both OAS=2 and the A/ = 3/2, AS = 1 operator O4 (see Eq. (VIII-3.31)) are formed from a pair of SU(3) octet left-handed currents, and because of their quantum numbers, each belong to the same 27-dimensional representation of SU(3). This implies the SU(3) relation, = v^2 .

(1.20)

The right-hand side of the above can be related to the K —• 2n matrix element with the aid of chiral symmetry. Using either soft pion techniques

IX-1 K°-K° mixing

237

or the chiral lagrangian given in Eq. (VIII-2.7), we find

(n°(k)\O4\K0(k))=2VlF«

/

+ + 2(n n°\OAI=3/2\K )

, (1.21)

or with on-shell kaons (k2 = rrv^-)^ 8y/2F,

ml

where rji is any one of the QCD factors 771, 772, 773. Expressed this way the result is independent of the scale chosen in QCD short-distance corrections, as it cancels in the ratio of rji/c^. This occurs because the QCD corrections are SU(3) symmetric and treat all members of the 27-plet in an equal fashion. Written in terms of the 5-parameter the result is* B = 0.33[as(fi)]2/9

.

(1.23)

We stress that any model which respects both chiral symmetry and 5f/(3), and also obtains the correct K+ —• TT+TT0 amplitude must lead to this result. Often, model dependent estimates get an answer different from Eq. (1.23) because they incorrectly predict K+ —> TT+TT0. The only significant issues which can affect the above determination of B are the possible breaking of SU{3) symmetry and the contribution of higher-order chiral corrections. One attempt [BiSW 84] to calculate the higher-order effects has examined the leading logarithmic corrections obtained when one calculates loops in chiral perturbation theoryt. In the language of Chap. VI, this consists of keeping the quantities m In(ra 2//i2) which occur in loop contributions, but dropping all factors of ra4 and all effects of the low energy constants occurring at O(E4). The results depend strongly on the 'chiraF scale, which we may write as /xchin that occurs in the chiral logarithmic factor - no corrections are obtained if Mchir — 1/2 GeV whereas a 50% enhancement obtains if /xchir — 1 GeV. Unfortunately, this logarithmic approximation has not been accurate in other settings, and a full chiral perturbation theory calculation cannot yet be done because one does not have sufficient information to pin down the needed low-energy constants at O(E4). A final resolution will require a better understanding of the origins and structure of the weak chiral lagrangian. Lattice QCD computer studies in the quenched approximation (i.e. no fermion loops) presently indicate a value B ~ 0.7, but are also presently obtaining values for the K+ —• TT+TT0 amplitude which are about twice as large as the empirical value. * Here a slight correction has been included for the effects of 77°-7r°mixing which can lead to a simulated A/ = 3/2 contribution to the K —> 2ir decay process. This is discussed in Sect. IX-3. t The paper [BiSW 84] contains an error in the normalization of the B factor, which we have corrected for in the quoted result.

238

IX Kaon mixing and CP violation

If we piece together all of the ingredients to the short-distance predictions, we obtain from the charm contribution to the box diagrams the mass difference, (Am)^ o r y ~r 7 l B(Am) e x p t .

(1.24)

This result is clearly of the right order of magnitude, but there also exist potentially sizable but incalculable long-distance corrections, which preclude a clean prediction of the mass difference. The long-distance contributions are generically of the form of Fig. IX2, with the AS — 1 weak hamiltonian acting twice. As discussed above, a naive analysis suggests that such contributions should be suppressed by a factor ~ m2K/m2c ~ 1/10 with respect to the short-distance effect. However, empirically the AI = 1/2 portion of H^s=1 has a factor of twenty enhancement over the simplest estimates (see Sect. VIII4), such that the naive factor should perhaps be modified to be about (20 7n#/ra c ) 2 ~ 40. Although the effect is not quite that large in reality, estimates of K° <-• (TT0, TJ) <-> K° and K° <-»TTTT <-• K° intermediate states yield long-distance effects comparable to the experimental mass difference. It is unfortunately not possible to be precise about these, but they can be large. For example, the K —• rj —> K° intermediate state by itself yields a contribution to Am of about 1.1 times the experimental result! This is estimated by using SU(3) plus chiral symmetry, and if one allows a 30% uncertainty in the prediction, it produces a large uncertainty by itself in the predicted mass difference. The TTTT contribution also has significant uncertainty. We can only conclude that, while a precise prediction of Am is not possible, the observed size is roughly compatible with that expected in the Standard Model. IX-2 The phenomenology of kaon CP violation The TTTT final state of kaon decay is even under CP provided the strong interactions are invariant under this symmetry. For the 7r°7r° system, this is clear since TT0 is itself a CP eigenstate, CP|TT0) = — |TT°), and the two pions must be in an 5-wave (£ = 0) state, CPITTV)

= ( - l f ( - l ) V V ) = +|7r°7r°).

K°

(a)

(b)

Fig. IX-2 Long-distance contributions.

(2.1)

IX-2 The phenomenology of kaon CP violation

239

The corresponding result for charged pions reflects the fact that ?r+ and 7T~ are CP-conjugate partners, CP|TT ± ) = — |TTT). We have seen that if CP were conserved, the two neutral kaons would organize themselves into CP eigenstates, with only Ks decaying into TTTT. Alternatively, KL decays primarily into the TTTTTT final state, which is CP-odd if the pions are in relative 5-waves. The observation of both neutral kaons decaying into 7T7T is then a signal of CP violation. There can be two sources of CP violation in KL —> TTTT decay. We have already seen that KK mixing can generate a mixture of the CP eigenstates in physical kaons due to CP violation in the mass matrix. There also exists the possibility of direct CP violation in the weak decay amplitude, such that the CP-odd kaon eigenstate \K^_) makes a transition to 7T7T. These two mechanisms are pictured in Fig. IX-3. The Kirn decay amplitudes have already been written down in Eq. (VIII-4.1) in terms of real-valued moduli Ao, A2- This decomposition is a consequence of Watson's theorem, and relies in part upon the assumption of time reversal invariance. However, if direct CP violation occurs, Ao and A
,

A2 = \A2\e^

,

(2.2)

with CP violation in the decay amplitude being characterized by the phases £o and £2- Consequently, the KQ —• TTTT and KQ —> TTTT decay amplitudes assume the modified form

+r- = \A0\e*°eiS°+ i ^ e ^ e * ,

Using the definitions of KL and Ks in Eq. (1.13), a straightforward calculation leads to the following measures of CP violation: _

KL

K+

(a)

/

K,

^

(b)

Fig. IX-3 Mechanisms for CP violation.

240

IX Kaon mixing and CP violation

where e — e + z£o ? ~~

A/2

iei(62-*o)

A2 Ao

-v/2

A2

/1I m A2

Im Ao

Ao VRe A2

Re A o

(2.5) The expression for e can be simplified by approximating the numerical value Ara/Ar = 0.478 ± 0.003 by Am/AT ~ 1/2. This yields the approximate relation, e i7r / 4 1 e y/2 Am '

i 1 Am+^AT

which we shall use repeatedly in the analysis to follow. In addition, since the rate for K —• TTTT is much larger than that for K —> TTTTTT, and K° —• TTTT is in turn dominated by the 7 = 0 final state because of the A / = \ rule, we have .

(2.7)

The above relations allow us to write i

Im M12

\m M12 , ^ _ e'f ^ Im M12 , Im Ap\ Am + *>) ~ y/2 V2Re M 12 + Re Ao) '

l

'

where UJ = Re^/ReAo ~ 1/22. We see that e is sensitive to CP violation in the mass matrix, while ef is a unique manifestation of direct A 5 = 1 CP violation. All CP-violating observables must involve an interference of two amplitudes. In Eq. (2.8), the quantity e expresses the interference of K° —> 7T7T with K° —• K° —• 7T7T, while ef involves interference of the 7 = 0 and 7 = 2 final states. The formulae for e and ef exhibit an important theoretical property. Since the choice of phase convention for any meson M is arbitrary, its state vector may be modified by the global strangeness transformation \M) - • e iA5 |M). For the K° and K° states, this becomes \K°) - • eiX\K°) ,

iX \K°) -> e\K°)

,

(2.9)

which has the effect,

Im Ar _^ Im A/ Re A/ ""* Re A/

'

Im Mu _^ Im M i2 _ Re M i2 ~" Re M i2

'

l

. . ''

We see that the values of e and e' are left unchanged. Various phase conventions appear in the literature. In the Wu-Yang convention, A is

IX-2 The phenomenology of kaon CP violation

241

chosen so that the Ao amplitude is real-valued. This is always possible to achieve by properly choosing the phase of the kaon state. However, it is inconvenient for the Standard Model, where the Ao amplitude naturally picks up a CP-violating phase. We shall therefore employ the convention in which no such phases occur in the definitions of the kaon states. It was in the K —» TTTT system that CP violation was first observed. At present, the status of measurements is \c\ = (2.263 ± 0.023) x 10~3 , f 0.0023 ± 0.0007 [Ba 92] ,

(2.H)

={

[ 0.0006 ± 0.0007 [Wi 92] , f (46.9 ± 2.2)° h_ = phase(r7+_) = < [(43.2 ±1.5)°

[Wi92] ,

= phase(r/00) = (47.1 ± 2.8)°

[Ca et al 90] .

[Ca et al 90] ,

As this book goes to print, there remains a discrepancy as to whether a nonzero value of e; has been found. This is a critical issue in the study of CP violation whose resolution is eagerly awaited. A violation of CP symmetry has also been observed in the semileptonic decays of KL and Kg. These are related to matrix elements of the weak hadronic currents. Since K° must always decay into e^ue7r~ while K° goes to e~Pe7r+, we have

1-6

If the semileptonic decays proceed as in the Standard Model, there is no direct CP violation in the transition amplitude, so that ~ l + 4Ree .

(2.13)

Since Re e = Re e, the above asymmetry is sensitive to the same parameter as appears in the KL —• TTTT studies. Here, measurement gives Re e = (1.635 ± 0.060) x 10"3 = |e| cos(44 ± 3)° ,

(2.14)

which is consistent with the experimental values from K —> TTTT. Finally, although one ordinarily emphasizes the phenomenon of CPviolation in studies of kaon mixing, precision experiments also probe the CPT transformation. For example, a prediction of CPT invariance, the equality of phases <^+_ = <^oo (up to very small corrections from e;), is

242

IX Kaon mixing and CP violation

seen to be consistent with existing data (0.2 ±2.9)°

[CaetoZ.90],

(2.15)

In addition, kaon data can be used to provide a sensitive test of the wellknown CPT prediction that the mass of a particle must equal that of its antiparticle. Specifically, the impressive bound \(mKo —m^o)/m Ko\ < 4x 10~18 has been obtained [Ca et al. 90]. Further study of CPT invariance is left to Prob. XI-3. IX-3 Kaon CP violation in the Standard Model After diagonalization, there can remain a single phase in the KM matrix. This phase generates the imaginary parts of amplitudes which are required for CP violation. It is a physical requirement that results be invariant under rephasing of the quark fields. As a consequence, all observables must be proportional to Im A(4) = A2\er) = cic2c$s\s2szs8 ,

(3.1)

written in the notation of Sect. II—4. In particular, all CP-violating signals must vanish if any of the KM angles vanish. We shall now study the path whereby this phase is transferred from the lagrangian to experimental observables. For kaons, we have seen that the relevant amplitudes are those for AS = 2 K°-K° mixing and for AS = 1 K—> TTTT decays. Tree level amplitudes in kaon decay can never be sensitive to the full rephasing invariant, so that one must consider loops. Typical diagrams are displayed in Fig. IX-4. Experiment can help in simplifying the theoretical analysis. Note that e' is sensitive to AS = 1 physics through the penguin diagram [GiW 79], while e is sensitive to AS = 2 mass matrix physics as well as to AS = 1 effects. However, since experiment tells us that e » e r, it follows that the AS = 1 contributions to e must be small. Likewise, the long-distance contributions of Fig. IX-2 and the contribution of Fig. IX-l(d) must both be small because each also involves the AS = 1 interaction. This

(a)

(b)

Fig. IX-4. (a) Penguin and (b) electroweak-penguin contributions to CP violation in AS = 1 transitions.

IX-3 Kaon CP violation in the Standard Model

243

leaves the box diagrams of Fig. I X - l ( a ) , ( b ) as t h e dominant component of e. Moreover, since t h e KM phase 6 is associated with t h e heavy quark couplings, only t h e heavy quark parts of t h e box diagrams are needed. Hence e is very clearly short-distance dominated.

Analysis of e The evaluation of e follows directly from the treatment of the box diagram in Sect. IX-1, and we find (c/. Eq. (1.15)), G2 —^%

Im M12 = x

2

^-^-r)2m2H(xt)

c \r)iml-

\ p )

- rjsm2cG(xc,

x c)

%

s\

(3.2)

The above relations depend on three principal quantities, viz. mt, the KM angles, and the B parameter. At present, we have only the bound mt > 89GeV, so that not much can be inferred about the influence of the t-quark mass. The information on KM angles allows the range A2X6r] = sls2s3ss < 0.7 x 10~4 ,

(3.3)

where no firm value can yet be given because there is no independent measure of 6. However, by taking t h e representative values mt = 150 GeV, mc — 1.5 GeV, A = 1, p = 1/2, we can express e in t h e useful form

[G(xc, xt)~

(3 4)

'

In this way, the magnitude of e is naturally obtained. Penguin contribution to ef The analysis of ef is rather more involved because one must confront A/ = 1/2 amplitudes in K —• TTTT. These are not yet under theoretical control, so that the predictions of ef must necessarily be uncertain. However, we can review the main ingredients of the analysis, which by now have become clear. As can be seen from Eq. (2.8), the most natural scale for the magnitude of ef/e is just u = Re A2/Rje A$ ~ 1/22. Yet, the experimental value of e'/e is found to be suppressed by at least another order of magnitude. This suppression is a consequence of the CP-violating phase 6 which appears

244

IX Kaon mixing and CP violation

only in the heavy quark sector. Its contribution to the direct K —> TTTT process occurs only as a result of virtual oquark and t-quark loops in the penguin diagram of Fig. IX-4(a) and the electroweak-penguin diagram of Fig. IX-4(b), from which we can extract some general features. The gluonic penguin diagram involves the s —• d transition and is purely A / = 1/2. It therefore generates a phase only in A$. The electroweak penguin would seem at first sight to be suppressed when compared to the usual penguin because of the factor of a/as. However, as will be described below, the electroweak penguin is relevant because it can generate a phase for A2. The full prediction emerges only when we have calculated the phases of both AQ and A*i. Let us first give estimates of various contributions to e' using QCD perturbation theory, but without renormalization group improvements. This serves to identify the important physics and can be adjusted afterward to incorporate the effects from short-distance renormalization. The penguin and electroweak-penguin diagrams yield respectively the CP-odd operators given in Eq. (VIII-3.30). If one wishes to explicitly check the rephasing invariance, it is necessary to recognize that CP-odd interference with other diagrams in general involves quantities like Im (V^VtsV^Vtid) which are invariant under rephasing the quark fields. The other feature to extract from the above operatorsis the symmetry structure. The gluonic penguin involves a lefthanded (ds) current which is a member of an SU(3) octet. The remaining current is a flavor singlet, so that overall the interaction transforms as an octet and hence carries A / = 1 / 2 . More precisely, under chiral SU(3) it transforms as (8^,1^). By contrast the electromagnetic coupling in the electroweak penguin is itself an octet, having both left-handed and right-handed portions,

^

^

(U, sR),

(3.5)

and is a mixture of A / = 1 and A / = 0, __ fQq=

2uu — dd — ss uu — dd = ^ ~ + §

uu + dd — 2ss g •

(3-6)

Thus the symmetry structure of the electromagnetic penguin is more complicated, involving SU(3) octet and 27-plet representations, isospin 1/2 and 3/2 portions, and the chiral properties ( 8 L , 1 # ) , (27^,1^) and

(0 In order to complete the estimate of e', one needs to compute hadronic matrix elements. To illustrate this procedure, we shall use the vacuum saturation method, with the understanding that this approximation could be seriously flawed. One interesting new ingredient, beyond those described in Sect. VIII-4, is the presence of matrix elements of scalar and

^x

IX-3 Kaon CP violation in the Standard Model

245

pseudoscalar operators which arise as a consequence of the Fierz relations, rfn)i(l + 75)sj<&7/i(l - 75)% = -2dj(l - 75)Mfc(l + 75)5j .

(3.7)

For matrix elements of such operators, we have

= Bo ( +

^ )

V2SF (l

,

§)

(3.8)

= -iV2B0F U + J2 where So is a constant, F is F^ in the chiral limit and A defines an energy scale in the momentum dependent terms. To obtain these expressions, we have assumed chiral SU(3) symmetry and, for simplicity, have omitted loops. The momentum dependence is required to obtain a nonzero result, and the constant Bo can be identified from our previous chiral lagrangian studies, where the lowest order relations (ra = (mu + m^)/2), m% = (n\m{uu + rfrf)|7r) ,

m2K — (K\m{uu + dd) + msss\K)

, (3.9)

reveal that Bo = $• = ~^r

•

(3.10)

m +m

2m

However, Bo and the quark masses are not separately obtained from chiral symmetry - only the product rhBo or the mass ratio rh/ms is well-defined. In Chap. XII, we shall obtain a more model dependent estimate of ms from hyperon mass splittings, where a first order treatment yields m\ — mp — (A|ras5s|A) — (p\msss\p) + O (m/ms) = msZm .

(3.11)

The matrix element Zm is estimated to lie in the range 0.5 < Zm < 0.8 within a variety of quark models, and so Bo *

m2 TTlA

*— Z ™

m

.

(3.12)

Note also that the momentum dependence is related to FK/FK, since ms + m

ms + m

(3.13) where the equations of motion have been used in the final relation. From Eq. (3.8), we find _

246

IX Kaon mixing and CP violation

With these ingredients now defined, we can piece together the structure of e'/e. Using the physical values of e and Re Ao, we find Im

(TV+Trf°

Inserting as ~ 0.2 (as is appropriate for heavy quarks), mt ~ 150 GeV, mc — 1.5 GeV and fixing A with the FR/F^ analysis, we obtain finally - = 0.0002 A2r) U + 3.9 Z2m

.

(3.16)

Remember that this result still lacks the renormalization group enhancement. However, it does indicate that one expects a small but non-zero value of e'/e in the Standard Model. Additional contributions to e' In the limit of isospin invariance, the gluonic penguin contributes only to Ao. Since e' is proportional to the difference between the phase of Ao and the phase of A%, we need to be certain also about the phase of Ai- A little thought will convince one that A2 is capable of picking up a sizeable phase from small effects because Re A2 is itself a small number. That is, if there is a small CP-violating process which contributes to A2, the phase IDQLA2/R&A2 can be enhanced because of the smallness of Re^2. We can estimate such an effect. For example, if isospin is broken by quark mass effects, then the decomposition into Ao and A2 will not uniquely represent the A/ = 1/2 and A/ = 3/2 transitions. Some A/ = 1/2 effects will appear in the parameter A2. An example of this is given below. The size of such effects would be roughly yiiso—brk

A

*

A

A

m

d

~~ ™"u

~ °-—^T~ '

/ o i v\

(3>17)

where the second factor is a measure of isospin breaking. In this case the phase of A2 is of order Im A2 Re A2

I m Ao Tiid — TRU Re A2 ms

Im Ao Re Ao vrtd — TUU Re A$ Re A2 ms A 22 = 22 • — ~ 00 7 30 Re Ao 30 Re Ao '

(3-18)

where we have used a first order chiral estimate of the quark mass ratio. In Fig. IX-4(b) we display a second example, the electroweak penguin diagram. This can contribute directly to A2 with an estimated size

IX-3 Kaon CP violation in the Standard Model

247

relative to the penguin of Im A™»

~ — Im ^ e n g ,

lmA2 Re ^2

aImAlengReA0 a s Re Ao Re A2

Im Ao n o ImA 0 Re Ao ~ ' Re ^4o (3.19) We see that because of the extra factor of 22 in Eqs. (3.18), (3.19), even suppressed effects can modify ef markedly if they contribute to A^ The above results are order-of-magnitude estimates only, not real calculations. However, they do indicate that a careful study of isospin breaking and the electroweak penguin is required to fully understand e1. It has become conventional to express e'/e in the form e

l

Re An

1/137 0.2

nn

- '

(3.20)

22

r^rT" Im AQ Re A
Im

To determine the correction factor ft, let us first estimate the isospin breaking correction due to the u — d quark mass difference. One such effect is clearly calculated from chiral perturbation theory. The u—d mass difference induces 7r°-r/ mixing,and this in turn influences the transitions K -» 7r°7r°and K+ - • TT+TT0 through the diagram of Fig. IX-5. The ingredients of the calculation are (i)the weak transition,

^K0^m

= ^=Af

,

(3.21)

which is found using the weak lagrangian Eq. (VIII-2.4b) and where the superscript (0) has been added to indicate that one uses AQ before 7r°-77 mixing, (ii) the amplitude for TTO-778 mixing,

^ ^ ^ K - mD -

K+

Fig. IX-5 Effect of 7r°-rj mixing

(3-22)

248

IX Kaon mixing and CP violation

which is found from the chiral lagrangian of Eq. (IV-5.10). Using these to calculate Fig. IX-5 yields )

~ft > 2

\

2

<

(3.23) We see that, subsequent to mixing, this is equivalent to the modified isospin decomposition A - 4<°)+ lmd-mu Is A

_

4 (0)

1

(Q) m

m d - mu

(3.24)

(Q)

v

^

The change in A$ is small, but the effect in A% is relatively larger, with

^

= _ l ! ^ ^ 4 2 ^ _Q M

A2 3\/2 m s - m A2 For our purposes it is most important that a phase has been induced into A2, with magnitude O? = 0.14 .

(3.26)

There will also be a correction for the rf. Although this cannot be calculated in chiral perturbation theory, it can be estimated from quark model relations plus rj — r( mixing that ^iso-brk ^ % +
(3.27)

We are also interested in the contribution of the electroweak penguin operator to Im A
(3.28)

whereas the (8/,, 1R) and (27^, 1#) operators each require two derivatives. Due to the presence of the electric charge operator Q, the matrix element vanishes for the neutral mode K° —> 7r°7r° but not for the charged mode K° —> 7r+7r~. From this, we can determine the relative contributions of the electroweak penguin to A2, AQ in terms of the K° —> TT+TT" amplitude, 22^2 3

Im (^-\Hewp\K) Im (TT+TTiWlKO) '

(

]

IX-3 Kaon CP violation in the Standard Model

249

Again, the need to calculate a hadronic matrix element hinders one's ability to make a firm prediction. However, in the approximation of vacuum saturation we estimate _

22x/2 a

A2

2B% - {m\ - m2.)

Interestingly, the electroweak penguin operator is chirally enhanced. At the effective lagrangian level this follows because its operator is O(E°) in the energy expansion, while the penguin operator is O(E2). This is reflected in the above matrix element by the factor h? /{m2K — rnfy. Note also that in this approximation fiewp is negative so that it enhances e'. Inserting the numerical factors as ~ 0.2, A = 1 GeV, Zm ~ 3/4, we estimate that fiewp — —0.30. This is a surprisingly large correction for an electroweak contribution, and indicates that it can be an important part of the prediction for e'\ The final ingredient is the calculation of short-distance effects in QCD using the renormalization group. This proceeds along the lines described in Sect. VIII-3. The usual penguin can generate a phase primarily in Oi, O5 and OQ to yield Im Ao = ^ K d K s (TT+TT-I £

Tma Oi \K°) ,

(3.31)

i=l,5,6

and the electroweak penguin contributes to A
ImC

* Oi \K°) .

(3.32)

Present estimates [BuBH 90] yield the values Im c7 ~ -0.004as2sss6 = -0.004aA4A2rj , Im c$ ~ -0.16as2Ssss = -0.16a\4A2r) ,

(3-33)

for mt - 150 GeV, A ~ 0.2 GeV, and fi - 1.0 GeV. In addition, the renormalization group analysis including the electroweak penguin induces a phase in the coefficient of the A/ — 3/2 operator O4. This arises from the purely left-handed portion of the electroweak penguin, with transformation property (27^,1^). The result is a phase in A2, of magnitude /C4

Re c4

This result is independent of the QCD scale /i in the short-distance analysis. Note that by use of Eq. (2.8) and the experimental value of e, this

250

IX Kaon mixing and CP violation

piece by itself generates

°0 1 *'/" = - 0 . 3 X 1 0 - ^ ,

U

(3.35)

C1

which can be significant given the size of the leading contribution in Eq. (3.16). The isospin breaking effect discussed above is unchanged by QCD short-distance corrections. We see then that there are several important contributions to e'. Because of the possibility of cancelations and because we cannot yet calculate the matrix elements with acceptable precision, the only safe statement is that e1 is expected to be nonzero, and to occur in the range probed by present experiments. To summarize, we have outlined the main ingredients for the predictions of e and e'. As this is being written, the experimental results have not settled down, and important theoretical uncertainties lie in the KM angles, the t-quark mass, and the hadronic matrix elements. There is cause for optimism that more will be known about all of these in the near future.

IX—4 Electric dipole moments The condition of T invariance forbids the existence of an electric dipole moment for any elementary particle. This can be seen heuristically in the following manner. The only vectorial quantum numbers associated with a particle are its momentum p and spin s. Thus, for a particle at rest the electric dipole moment must be proportional to the spin, dc = d e l ^ s

ll

.

(4.1)

The dipole interacts with an electric field as = -de-E

=-de^

.

(4.2)

Under time reversal E is unchanged (i.e. think of the E field of capacitors or static charges, which do not change under T), but all angular momenta reverse sign. Hence, i^edm is °dd under T.* For a relativistic spin-1/2 particle, the electric dipole moment contribution has the matrix element (p'\j;ia\p)\edm

= ideu(p')afiUql/l5u(p)

,

(4.3)

with qu = (p1 — p)u, which is equivalent to an interaction density ^

.

(4.4)

* One might be concerned that, since T is an antiunitary operator, this conclusion could be changed by adding a factor of i to He^m. However, hermiticity requires that no additional factors of i appear.

IX~4 Electric dipole moments Since FOi = -E{

251

and

this interaction attains the hamiltonian form in the nonrelativistic limit, #edm = [cPxHedm "> ~de((r) • E .

(4.6)

Note that an electric dipole moment also violates parity. The best existing experimental limits on electric dipole moments are

{

1.2 x 10~25e-cm

(neutrons [RPP 90]) ,

(-2.7 ± 8.3) x 10"27e-cm (electrons [Ab et al. 90]) , (4.7) 23 -3.7 x 10" e-cm (protons [RPP 90]) . The extreme sensitivity of these limits can be seen from the observation that if neutrons were expanded to the size of the earth, the limit quoted would correspond to a dipole with a unit electric charge separated by only a micron! In practice, the dipole moment is a sensitive probe of CP violation, especially for theories beyond the Standard Model. The Standard Model can produce an electric dipole moment either through the 0 parameter of QCD or through W-exchange. The discussion of 6 and the strong CP problem is given in the next section. Some of the possible diagrams involving the electroweak interactions are shown in Fig. IX-6. There emerges the very important property that no electric dipole moment is produced at first order in the weak interaction (i. e.

O(Gp)) in the Standard Model This is because the process does not change flavor. Tree-level processes which do not change flavor always involve the combination of KM matrix elements V^V^ for some flavors i, j , and the loop diagrams of Fig. IX-6 (a) involve ]T\ ^f^ij- Hence these are real and cannot involve the CP-violating phase. Theories which do have first order CP violation tend to produce neutron dipole moments close to the experimental bounds. At second order in the weak interactions, there is enough structure to produce a dipole moment. Interestingly, the single quark line diagrams, of which one example appears in Fig. IX-6(b), sum to zero net effect [Sh 78].

(a)

(b)

(c)

Fig. IX-6 Contributions to an electric dipole moment

252

IX Kaon mixing and CP violation

However, with an extra gluon loop a dipole moment occurs at O{G2Fas). Since as ~ 1 at hadronic scales, this provides no extra suppression. The multiquark interactions of Fig. IX-6(c) can also lead to a neutron dipole moment [GaLOPRP 82]. For the electron, CP violation vanishes if the neutrinos are massless. One can obtain a reasonable estimate of the neutron dipole moment from dimensional analysis alone. There is always a factor of Im A^4^ (cf. Eq. (3.1)) associated with any CP-violating process. In addition, the GIM mechanism would cancel the contributions of degenerate mass quarks, so a contribution of the form (mh — m\)/Myy is also expected. Altogether then, we have de(neutron) - e ^ f - ^ - I m A^/i 3 ~ 10"31e - cm ,

(4.8)

where // is a typical hadronic scale (we use /i ~ 0.3 GeV) which has been included to make the dimensions correct, and we have incorporated factors of n as anticipated from loop diagrams. This estimate is in the middle of the range found in model calculations [BaM 89], which span from 10~33 to 10~30e — cm. It is well out of the reach of experiments in the foreseeable future. IX-5 The strong CP problem The possibility of a 0-term in the QCD lagrangian raises potential problems (see Sect. Ill—5). For 0 ^ 0, QCD will in general violate parity, and even worse, time reversal invariance. The strength of T violation (and hence by the CPT theorem, CP violation) is known to be small, even by the standards of the weak interaction. This knowledge comes from both the observed KL —• 2?r decay and bounds on electric dipole moments. From these it becomes clear that QCD must be T invariant to a very high degree. However, there is nothing within the Standard Model which would force the ^-parameter to be small; indeed, it is a free parameter lying in the range 0 < 6 < 2TT. The puzzle of why 6 ~ 0 in Nature is called the strong CP problem. One is tempted to resolve the issue with an easy remedy first. If QCD were the only ingredient in our theory, we could remove the strong CP problem by imposing an additional discrete symmetry on the QCD lagrangian, the discrete symmetry being CP itself. This wouldn't really explain anything but would at least reduce a continuous problem to a discrete choice. In reality, this will not work for the full Standard Model since, as we have seen, the electroweak sector inherently violates CP. It would thus be improper to impose CP invariance upon the full lagrangian. Moreover, even if one could set #bare = 0 in QCD, electroweak radiative corrections would generate a nonzero value. These turn out to

IX-5 The strong CP problem

253

occur only at high orders of perturbation theory, and are expected to be divergent by power counting arguments, although they have not been explicitly calculated. This divergence is not a fundamental problem because one could simply absorb #bare plus the divergence into a definition of a renormalized parameter 9ren, which could be inferred from experiment. However we are then back to an arbitrary value of 0ren and to the problem of why 9Ten is small. The parameter 0 The situation is actually worse than this in the full Standard Model, as the quark mass matrix can itself shift the value of 6 by an unknown amount. Recall that CP violation in the Standard Model arises from the Yukawa couplings between the Higgs doublet and the fermions. When the Higgs field picks up a vacuum expectation value, these couplings produce mass matrices for the quarks which are neither diagonal nor CP invariant. The mass matrices are diagonalized by separate left-handed and right-handed transformations, and CP violation is shifted to the weak mixing matrix. However, because different left-handed and right-handed rotations are generally required, one encounters an axial U(l) rotation in this transformation to the quark mass eigenstates, and as discussed in Sect. Ill—5, this produces a shift in the value of 0. Let us determine the magnitude of this shift. Denoting by primes the original quark basis, one has the transformation to mass eigenstates given by (cf. Eqs. 11-4.5,4.6) m = S* m'S* ,

tl>L = Slt//L,

1>R = S\tl/R .

(5.1)

Here, we have combined the u and d mass matrices into a single mass matrix. Expressing SL,R as products of U(l) and SU(N) factors, i

^

,

(5.2)

with SL, SR in SU(N), one obtains an axial U(l) transformation angle of (fR — (fL- From the discussion of Sect. Ill—5, this is seen to lead to a change in the 6 parameter, e^0

= O + 2Nf(
(5.3)

where Nf = 6 for the three-generation Standard Model. However, noting that the final mass matrix m is purely real, we have arg(det m) = 0 = arg (det 5^ det m' det SR)

= arg (det sty + arg (det m') + arg (det SR) = 2N (
arg (det m') ,

(5-4)

254

IX Kaon mixing and CP violation

where we have used the SU(N) property, detSn = det SL = 1. The resultant ^-parameter is then 0 = 0 + arg(detm')

,

(5.5)

with m' being the original nondiagonal mass matrix. The real strong CP problem is to understand why 6 is small. One possible solution to the strong CP problem occurs if one of the quark masses vanishes. In this case, the ability to shift 6 by an axial transformation would allow one to remove the effect of 0 by performing an axial phase transformation on the massless quark. Equivalently stated, any effect of 8 must vanish if any quark mass vanishes. Unfortunately, phenomenology does not favor this solution. The u quark is the lightest, but a value mu ^ 0 is favored. Connections with the neutron electric dipole moment The 6 term is not the source of the observed CP violation in K decays. This can be seen because it occurs in a AS = 0 operator, and while this may ultimately generate effects in AS = 1 processes, its influence is stronger in the AS = 0 sector. In particular, it generates an electric dipole moment de for the neutron. Since no such dipole moment has been detected, one can obtain a bound on the magnitude of 6. To determine the effect of 0, it is most convenient to use a chiral rotation to shift the 6 dependence back into the quark mass matrix. A small axial transformation produces the modified mass matrix md

\ ^+ir)^T^

j

= ^LM^R+^RM^L

, (5.6)

where rj is a small parameter proportional to 0 having units of mass, and T is a 3 x 3 hermitian matrix. Consistency requires T to be proportional to the unit matrix. If this were not the case, and instead we wrote T = 1 + \{Ti/2, the effective lagrangian would start out with a term linear in the meson fields, £eff ~ iV IV (rt/T - t/Tt) = 2-^- (T37ro + TsV8 + ...) ,

(5.7)

rather than the usual quadratic dependence. The vacuum would then be unstable because it could lower its energy by producing nonzero values of, say, the TTO field. Thus to incorporate ^-dependence without disturbing vacuum stability, one chooses T = 1. The act of rotating away any depen-

IX-5 The strong CP problem

255

dence on 9 produces a nonzero value of argdet M, and also determines 77, 9 = arg det M = arg \{mu + irj) (rrid + ir)) (ms + irj)] , mumdms r/ ~ 0 (for small 77) Tnvrid + Tnums + rridm such that the mass terms become TUUUU -

msSS

mdmdms

,_

-

_

,

(5.8) v '

(5.9)

murrid + mums The last term is the CP-violating operator of the QCD sector. Note that, as expected, 6 vanishes if any quark is massless. A nonzero neutron electric dipole moment de requires both the action of the above CP-odd operator and that of the electromagnetic current,

(5

*10)

where q — p'—p and we have inserted a complete set of intermediate states {/} in the neutron-to-neutron matrix element. For intermediate baryon states, the matrix elements of Tp^tp are dimensionless numbers of order unity and magnetic moment effects are of order the nucleon magneton, \in. Thus we find for de, de * 9

^ ^ - ^ , (5.11) murrid + mums + mdms AM where AM is some typical energy denominator. Using AM = 300 MeV, we obtain de~0x

10"15 e-cm .

(5.12)

Far more sophisticated methods have been used to calculate this, with results that unfortunately have a spread of a factor of 50 [Pe 89]. Our simple estimate is near the average. In explicit calculations, some subtlety is required because one must be sure that the evaluation correctily represents the U(1)A behavior of the theory [AbGLOPR 91]. However, the precise value is not too important; the significant fact is that bounds on d e ^10~ 2 5 e-cm requires 9 to be tiny, 0<1O~ 10. The strong CP problem does not have a good resolution within the Standard Model. It would appear that the abnormally small value of 0, and of the cosmological constant as well, are indications that more physics is required beyond that contained in the Standard Model.

256

IX Kaon mixing and CP violation Problems

1) CP violation and K-> Sn The Standard Model makes definite predictions for the existence of CP violation within the K —• 3TT system. One such effect is the existence of the decay Ks —• 7r +7r~7r° which one would expect to be forbidden in the limit of CP conservation. a) Actually this expectation is not quite correct. Calculate the amplitude for Ks —• 7T+7r~7r° in the limit of CP conservation and show, using the results of Prob. VIII-2, that

Thus there exists a CP-even piece of the amplitude, but it is small, arising from the A/ = | component of the hamiltonian, and vanishes at the center of the Dalitz plot. b) Now consider the CP-violating component of the amplitude. Using Prob. VIII-2 together with the fact that in the Standard Model, CP violation is purely A/ = \, show that + 7T-7T0

—

V2 c) Defining =

demonstrate that 77+-0 = ^?oo = € — 2ef as first given in [LiW 80]. In this approximation, any deviation of r/+_o — e from zero is small due to the same mechanism which suppresses r/+_ — e (This result can be modified by electroweak-penguin contributions, which have a different chiral structure, and higher order energy dependence, but the result remains discouragingly small). d) Why is experimental measurement of 7/+_o much more difficult than that of r/ + _? 2) Strangeness gauge invariance a) Physics must be invariant under a global strangeness transformation \M) —• exp(iA5)|M), where A is arbitrary. Explain why this is the case. b) Demonstrate that such a transformation has the effect ImMi2 ImMi2 ' ReMi 2 ~* ReM i 2 ~

Problems

257

as claimed in Eq. (2.9b), and that, while unphysical quantities such as e, £o are affected by such a change, physical parameters such as e, ef are.not. 3) Neutral kaon mass matrices and CPT invariance Some of the ideas discussed in this chapter can be addressed in terms of simple models of the neutral kaon mass matrix M which appears in Eq. (1.2). a) Consider the following CP-conserving parameterization as defined in the (If0, If0) basis: A m0 where A is real-valued. Determine the basis states (if_,if+) in which Mo —> M± becomes diagonal and obtain numerical values for 777-0,

A.

b) Working in the (if_,if + ) basis, extend the model of (a) to allow for CP violation by introducing a real-valued parameter <S, rri0

, , , _ / ra_ —id

0 \ 777-4- /

y 2(5

771-1-

and assume there is no direct CP violation. This mass matrix corresponds to the superweak (SW) model. By expressing M± in the (if0, K°) basis, use the analysis of Sects. IX-1,2 to predict (fe ' = phase e and determine 6 from the measured value of |e c) Finally, extend the model in (b) to M±"=(m; * V X * m+ where Re x is a T-conserving, CP-violating and CPT-violating parameter. Show that the states which diagonalize M±" are |*r5} ^

\K+)

-

||K_)

,

\KL) ~ \K.) + £\K+) , where V = {TRL — rns)/2 + iYs/^> Then evaluate r?+_ and 7/00, allowing for the presence of direct CP violation (i.e. e' ^ 0), and derive the following relation between phases,

The result \m^o —m Ko\/mKo < 5 x 10~18 which follows from this relation is the best existing limit on CPT invariance.

X The 1/N C expansion

The Nc * expansion is an attempt to create a perturbative framework for QCD where none exists otherwise. One extrapolates from the physical value for the number of colors, Nc = 3, to the limit iVc —> oo while scaling the QCD coupling constant so that g%Nc is kept fixed ['t H 74]. The amplitudes in the theory are then analyzed in powers of N'1. The hope is that the Nc —* oo world bears sufficient resemblance to the real world to yield significant dynamical insights. There is no magical process which makes the Nc —• oo theory analytically trivial; nonlinearities of the nonabelian gauge interactions are present, and the theory is still not solvable. However any consistent approximation scheme for QCD is welcome, and the large Nc expansion is especially useful for organizing one's thoughts in the analysis of hadronic processes. X—1 The nature of the large N c limit In passing from SU(3) to SU(NC), the quark and gluon representations, originally 3 and 8, become N c and N^ — 1 respectively. The analysis of Feynman graphs at large Nc is simplified by modifying the notation used to describe gluons. As usual, quarks carry a color label j , with ,7 = 1,2,..., Nc. Gluons can be described by two such labels, i.e. A;

-> A*

{A*, = 0) ,

(1.1)

where a = 1,..., N% — 1 and j , k = 1,..., Nc. In doing so, no approximation is being made. The new notation in simply an embodiment of the group product N c x N c —> (Nj? — 1) ^ 1. The quark-gluon coupling is then written

fi , 258

(1.2)

X-l

The nature of the large Nc limit

259

Fig. X-l. Double line notation: (a) quark and (b) gluon propagators, (c) quarkgluon, (d) three-gluon and (e) four-gluon vertices.

and the gluon propagator is

Jd'x e** (0 |r (A*j(x)Aj}(0)) I 0) = (1.3) The term proportional to Nc 1 must be present to ensure that the color singlet combination vanishes, A3- — 0. However, as long as we avoid the color singlet channel, this term will be suppressed in the large Nc limit and may be dropped when working to leading order. Using this new notation, the Feynman diagrams for propagators and vertices are displayed in Fig. X-l. A solid line is drawn for each color index, and each gluon is treated as if it were a quark-antiquark pair (as far as color is concerned). In this double line notation, certain rules which are obeyed by amplitudes to leading order in 1/NC emerge in an obvious manner. Although general topological arguments exist, we shall review these rules by examining the behavior of specific graphs. The power of Feynman diagrams to build intuition is rather compelling in this case. We consider first the familiar quark and gluon propagators. The quark propagator, unadorned by higher order corrections, is O(l) in the Nc —• oo limit. Fig. X-2 depicts two radiative corrections. Fig. X-2(a), the one-gluon loop, is O(l) in powers of Nc because the suppression from the squared coupling g\ is compensated for by the single closed loop, which corresponds to a sum over a free color index and thus contributes a factor of Nc. The graph then is of order g\Nc which is taken to be constant. The graph Fig. X-2(b) with overlapping gluon loops is O(N~2) because, with no free color loops, it is of order g% = (glNc) N~2 ~ N~2. The terms planar and nonplanar are used respectively to describe Figs. X-2(a),2(b), because the latter cannot be drawn in the plane without at least some internal lines crossing each other.

Fig. X-2. Radiative corrections to the quark propagator: (a) planar, (b) nonplanar.

260

X The 1/NC expansion

Four distinct contributions to the gluon propagator are exhibited in Fig. X-3. Figs. X-3(a),3(b) depict in double line notation the quarkant iquark and two-gluon loop contributions. It should be obvious from the above discussion that these are respectively O(N~1) and 0(1). A new diagram, involving the three-gluon coupling, appears in Fig. X-3(c). With three color-loops and six vertices, it is of order (glNc) = O(l). Figure X-3(d) is a nonplanar process with six vertices and one color sum, and is thus O{N~2). The discussion of the gluon propagator indicates why we constrain . to be fixed when taking the large Nc limit. The beta function of QCD is determined to leading order by Figs. X-3(a),3(b). If g\ were held fixed, the beta function would become infinite in the large Nc limit, leading to the immediate onset of asymptotic freedom. The choice g\Nc ~ constant leads to a running coupling constant and is compatible with the behavior for the realistic case of Nc = 3. To summarize, there are several rules which can be abstracted from examples such as these: (i) the leading order contributions are planar diagrams containing the minimum number of quark loops; (ii) each internal quark loop is suppressed by a factor of A^T1; and (iii) nonplanar diagrams are suppressed by factors of N~2. The suppressions in rules (ii), (iii) are combinatorial in origin. Quark loops and nonplanarities each limit the number of color-bearing intermediate states, and consequently cost factors of N~l. X—2 Spectroscopy in the large N

c

limit

In order for the large Nc limit to be relevant to the real world, it must be assumed that confinement of color singlet states continues to hold. In this case, we expect the particle spectrum to continue to be divided into mesons and baryons. Let us treat the mesons first, as the baryons are more problematic. One can form color singlet meson contributions from QQ pairs. To form a color singlet, one must sum over the quark colors. In order to produce a properly normalized QQ state one must therefore include a —

I/O

normalization factor of Nc ' into each QQ meson wavefunction, such

(a)

(b)

(c)

Fig. X-3 Various radiative corrections to the gluon propagator.

X-2 Spectroscopy in the large Nc limit

(a)

261

(b)

Fig. X-4 Mesons in the double line notation. that |Q(*)Q(0)) color

singlet

i^6( Q )t^)t|o) y/Nc

5

(2.1)

where a, /3 are flavor labels, i = 1,..., Nc is the color label, and tf (rf) are the quark (antiquark) creation operators. Meson propagators, as represented in Fig. X-4(a), are then 0(1) in Nc since the factors of (JV«T ' )2 from the normalization of the wavefunction are compensated by a factor of Nc from the quark loop. This leads to the prediction that meson masses are of (9(1) in the large Nc limit, i.e. they remain close to their physical values. Multiquark intermediate states, as in Fig. X-4(b), are suppressed by 1/NC, indicating a suppression of mixing between QQ and Q2Q2 sectors. That is, large Nc plus confinement implies the existence of QQ mesons which contain an arbitrary amount of glue in their wavefunction, but which do not mix with Q2Q2 states. What about the decay widths of QQ mesons? The decay amplitude is pictured in Fig. X-5 (other possibilities involve the suppressed quark loops). This diagram contains three meson wavefunctions and one quark loop and hence is of order (iVc~ ' )3-/Vc = Nc in amplitude or N~l in rate. The large Nc limit thus involves narrow resonances, i.e. T/M —> 0, where F is the meson decay width and M is the meson mass. This is reasonably similar to the real world, where most of the observed resonances have T/M ~ 0.1-0.2 [RPP 90]. Color singlet gluonic states, called glueballs, may also exist. The normalization of a glueball state can be fixed by means of the following argument. Suppose, as will be defined in a gauge invariant manner in Sect. XIII-4, that a neutral meson can be created from two gluons. Then in normalizing this configuration, one must sum over the Nc2 gluon color labels. As a consequence, a normalization factor A^"1 is associated with

Fig. X-5 Strong interaction decay of a QQ meson.

262

X The 1/NC expansion

(a)

(b)

Fig. X-6 Meson-meson scattering. each glueball state. Glueball propagators also emerge as being O(l). There is no physical distinction between two-gluon states, three-gluon states, etc., because all are mixed with each other by the strong interaction. As a result, there need not be any simple association between a specific physical state and gluon number, and thus the concept of a 'constituent gluon' need not be inferred. In glueball decays, however, one must distinguish between glueballs decaying to other glueballs, and those decaying to QQ mesons. Where kinematically allowed, the decay of glueballs to glueballs is 0(1), while that to QQ states is O (1/NC). The lowest lying glueball(s) will then be narrow, while those above the threshold for decay into two glueballs will be of standard, nonsuppressed width. Meson-meson scattering amplitudes are also restricted by large Nc counting rules. Consider the diagrams of Fig. X-6. That of Fig. X-6(a) is of order (N~1/2)4NC ~ TV"1, whereas Fig. X-6(b) is O(N~2) because of the extra quark loop. The scattering amplitudes thus vanish in the large Nc limit, and the leading contributions are connected, planar diagrams. The large Nc limit also predicts that neutral mesons (i.e. Q^Q^ composites with a = j3) do not mix with each other. The possible mixing diagram is given in Fig. X-7, and includes any number of gluons. However, because of the extra quark loop, it is of order N~l, and thus vanishes in the infinite color limit. This means that uu states do not mix with dd or 55, nor do the latter two mix. The large Nc spectrum thus displays a nonet structure with the uu and dd states degenerate (to the extent that electromagnetism and the mu-md mass difference are neglected) and the ss states somewhat heavier. This pattern is reflected in Nature, except that the uu and dd configurations now appear as states of definite isospin, uu ± dd. For example, let us consider the JPC — 1 L 2 + + mesons. For the former, p(770) and a;(783) are interpreted as uu, dd isospin 1 = 1 and

Fig. X-7 Meson-meson mixing.

X-3 Goldstone bosons and the axial anomaly

263

7 = 0 combinations, while y>(1020) is the ss member of the nonet. Including the if*(892) doublet as the us, ds combinations, a simple additivity in the quark mass would imply m

v?(i020) -

m

p(770) = 2 (m#*(892) - m p ( 7 7 0 )) ,

(2.2)

+

which works well. A similar treatment of the 2 + mesons, identifying a2(1320) and /2(1270) as the corresponding uu, dd states and /2(1525) as an ss composite, predicts -

m

a2a(1320) 2(1320) =

22

( mm/q(1430) ~

mm

a2(1320)

which is also approximately satisfied. The fact that p(770), a;(783), /2(1270) and a2(1320) decay primarily to pions, and
(Pj(q)\dllAU0)\0) = Fjmfak .

(3.2)

For the octet of currents, the divergence vanishes for zero quark mass, and as usual leads to the identification of TT, K, rjs as Goldstone bosons. However, for the singlet current the anomaly is present. Even in the

X The 1/NC expansion

264

limit of vanishing quark mass, the current divergence has nonzero matrix elements, in particular, (3.3)

If one repeats the calculation of the anomalous triangle diagram as in Sect. Ill—3 but now allows N c to be arbitrary, one sees that it is proportional to Tr (AaAfe) = 26ab and is therefore independent of Nc. However, by using large Nc counting rules, the matrix element in Eq. (3.3) is seen to be of order g\Nc ~ Nc .* This implies that the gluonic contribution to the axial anomaly vanishes in the large Nc limit. When we take into account the behavior of jfy, we conclude that m2, ~ 1/NC —> 0. The rf is thus massless in the large Nc limit, and we end up with a nonet of Goldstone bosons. To illustrate what happens when the number of colors is treated perturbatively, let us consider the 1/NC corrections to the meson spectrum together with the effects of quark masses. If we first add quark masses, we have, in analogy with the results of Sect. VII-1, the mass matrix \m(uu + dd) + msss\

(3.4)

where we have taken mu = rrid =TO.This leads to a squared-mass matrix (2m ins

0

0

0

0

0

+m

0

0

0

\

(3.5)

§(2m8 \{ rns

0

in the basis (TT, if, r)g,r)o). If this were diagonalized, one would find an isoscalar state degenerate with the pion. This is a manifestation of the U(l) problem which arises when there is no anomaly. However, at the next order in large Nc, the matrix picks up an extra contribution in the SU(3) singlet channel due to the anomaly, yielding /2m 2

m = B0

0

0 ms

i

I

m

0

0

0

0

0

0

| (2ms + m)

\ 0

0

mm-ms)

{m-

\(

\

(3.6)

2rh)

This result depends on the assumption that topologically nontrivial aspects of vacuum structure are smooth in the 7VC —> oo limit.

X-4

The OZI rule

265

where e = O(N®). This mass matrix yields an interesting prediction. The quantities Born and Boms are fixed as usual by using the n and K masses. Also the trace of the full matrix must yield rn^+rn\+mi+m?,, which fixes e = 2.16GeV2. The remaining diagonalization then predicts m^ = 0.98 GeV, 77?^ = 0.50 GeV with a mixing angle of 18°. This is a remarkably accurate representation of the situation in the real world. Although e/N c is suppressed in a technical sense, note how sizeable it actually is. One is hard pressed to imagine any sense in which the physical rj mass can be taken as a small parameter. X - 4 The OZI rule In the 1960's, an empirical property, called the Okubo-Zweig-Iizuka (OZI) rule [Ok 63, Zw 65, Ii 66], was developed for mesonic coupling constants. Its usual statement is that flavor disconnected processes are suppressed compared to those in which quark lines are connected. In the language which we are using here, flavor disconnected processes are those with an extra quark loop. Unfortunately, the phenomenological and theoretical status of this so-called rule is ambiguous. We briefly describe it here because it is part of the common lore of particle physics. The empirical motivation for the OZI rule is best formulated in the decays of mesons. Let us accept that <^(1020) and /2(1525) are primarily states with content ss whereas a;(783) and /2(1270) have content (uu + dd) / \ / 2 . Mixing between the ss and nonstrange components can take place with a small mixing angle, such that

Amp (ss) Amp (\uu + dd)/\/2)

EE

tan 0 ,

(4.1)

with 9 = 9y for the vector mesons and 0 = 6T for the tensor mesons. In both cases, 9 is small. Experimentally, the