PROBLEM
George Ạ CHRISTOS
Trang 2CHIRAL SYMMETRY AND THE U(1) PROBLEM George Ạ CHRISTOS* Department of Theoretical Physics, Research School of Physical Sciences, The Australian National University, G.P.Ọ Box 4, Canberra, ẠC.T 2601 Australia Received | July 1984 Contents:
1 Introduction 253 4.3 Solution to the @ periodicity puzzle 306 2 Chiral symmetry 259 5 Applications of the large N expansion to the U(1) problem 307
2.1 The exact chiral limit 259 5.1 The large N, philosophy of the U(1) problem 308
2.2 Explicit breaking of chiral symmetry 266 5.2 Phenomenological applications 315
3, The anomaly 281 5.3 Effective chiral Lagrangians with effects of the axial
3.1 The VVA triangle diagram 28] anomaly 320
3.2 The U(1) axial anomaly in QED 284 5.4 Proofs of spontaneous chiral symmetry breaking 327
3.3 The extension to OCD 286 5.5 Some other developments and conclusion 329
4 The U() problem 290 Acknowledgements 330
4.1, Preanomaly exposition of the U(1) problem 29 References 330 4.2 Inclusion of the anomaly 294
Abstract:
+a
This review gives a detailed account of recent progress in the U(1) problem from the point of view of the anomalous Ward identities and the large N, expansion Two important ingredients that go into the formulation of the U(1) problem chiral symmetry and the QCD anomaly, are
extensively discussed The basic concepts and techniques of chiral symmetry and chiral perturbation theory, as realized in the Gell-Mann-—Oakes— Renner scheme, are reviewed The physical meaning of the anomaly is clarified and its effects are consistently implemented through the anomalous spectrum of topological charge are discussed in the framework of chiral perturbation theorỵ Other aspects of the U(1) problem such as: the means by which the 7’ obtains its large mass, the details of the required (modified) Kogut-Susskind mechanism, phenomenological applications {such as
Trang 3
1 Introduction
The purpose of this report is to review recent developments in the U(1) problem This problem, like quantum chromodynamics itself, has its origins in current algebra and chiral symmetry, subjects which were fashionable during the sixties It is therefore natural to begin with a historical survey of developments from the first use of chiral symmetry in strong interactions to the modern formulation of the U(1) problem I hope that the result will be a useful guide for physicists of my generation whose research experience began long after the heyday of current algebrạ
The idea of a conserved axial-vector current in the context of the strong interactions dates back to Schwinger in 1957 [1.1] The application of chiral symmetry to phenomenology has its roots in the Goldberger-Treiman relation [1.2] The Goldberger-Treiman relation was originally discovered in 1958 by assuming pole-dominance for certain dispersion relations It was not until 1960 that Gell-Mann and Lévy [1.3] and Nambu [1.4] were able to explain it in terms of a (partially) conserved axial-vector current Nambu [1.4] also explained how such a symmetry could exist almost undetected in nature, with the appearance of only a massless pseudoscalar particle, to be identified with the pion As the pion is not massless, the idea of an approximate symmetry came into being (PCAC - partially conserved axial-vector current) Through the efforts of Gell-Mann [1.5] and others, the concept of an approximate symmetry became more precisẹ SU(2) vector (isospin) symmetry had been extended to an approximate SU(2) x SU(2) chiral symmetrỵ
In the meantime Gell-Mann [1.6], and independently Ne’eman [1.7], had further proposed the extension of SU(2) isotopic spin symmetry to SU(3) symmetry — the eightfold waỵ Gell-Mann [1.5] also suggested the extension to chiral SUG) x SU(3) to incorporate both chiral SU(2) x SU(2) and SUG) symmetries
precise the notion of | realizing a symmetry dynamically [I Al “They found models with a - physical
spectrum containing ess excitations as well a assive sta which did not betong-to irreducible
representations of the Hamiltonian symmetrỵ This symmetry realization was later termed spontaneous
symmetry breaking The result of Goldstone et al was that there should be a massless scalar excitation with every spontaneously broken generator of the group (ịẹ for every symmetry charge operator Q
that did not annihilate the vacuum, Q)0) # 0)
Chiral SU(2) x SU(2) symmetry became popular and received widespread recognition with the
discovery of the Adler-Weisberger relation [1.10] and Weinberg’s [1.11] analysis of a—7 scattering
lengths SU(2) x SU(2) was understood as an approximate symmetry, spontaneously broken down to
SU) vector symmetry with three light Psendoscalars the Pons | 7`, ` and - - Simllarly, due to the
that SUG) x SU(3) was spontaneously broken down to SUG) vector c symmetrỵ The pseudoscalar octet
Trang 4SU(3) vector symmetry might also be spontaneously broken However, the resulting Nambu—Goldstone bosons would be spin ()*, isospin 5 strange mesons, and would acquire a mass of not more than [1.13] 670 MeV through explicit breaking of SU(3)x SU(3) symmetrỵ The experimental absence of such particles now rules out spontaneous breaking of SU(3)y symmetrỵ
Of fundamental importance for subsequent developments was the introduction of quarks by Gell-Mann [1.15] and Zweig [1.16] in 1964 The idea was to imagine hadrons being built from constituent quarks such that the SU(3) scheme previously proposed by Gell-Mann [1.6] and Ne’eman [1.7] could be given a natural explanation Quarks were assumed to form an SU(3) triplet (qi, q2, qs) = (u, d,s) consisting of a non-strange isodoublet (u, d) and a strange isosinglet s The observed spectrum could then be understood if baryons were made of three quarks qqq and mesons of a quark~antiquark qq (Nowadays, heavy SU(3) singlet quarks c, b, t, must be included in order to account for observations of charm and other new ‘flavours’.) Gell-Mann [1.15] also suggested that a quark mass term could be used to monitor explicit breaking of chiral SU(3) x SUG) symmetrỵ
In the late 60’s, interest in chiral symmetry was further stimulated by the development of techniques for systematic expansions [1.17] about the chiral limit In 1971 Li and Pagels [1.18] discovered that chiral perturbation theory did not involve just simple powers of the symmetry breaking parameter ¢ but that there were also non-analytic corrections ~¢ In ¢ or V ¢ that arise from internal loops involving the light pseudoscalars whose mass*~ ¢ Langacker and Pagels [1.19] then developed techniques for extracting these non-analytic corrections and used them to improve some of the previous results,
In 1972, Gell-Mann et al [1.20] proposed quantum chromodynamics (QCD) as the theory of strong interactions In QCD, each quark q; carries a new quantum number r called ‘colour’:
qi > (qi)-: r=1,2.3
The index r labels the triplet representation of a local SU(3) colour symmetrỵ The locality of colour
V CITy Calis dt OTO ary QÖ VALIV d p Uy aVe lO D aqaCcad DY COVarld UCTIVdtives
D,, = d, — 1gA,,
Trang 5
where
Fuy = [D,., D,Ì = „Âu — 0,A, — ig[A, A,|
is the field-strength tensor, the index i labels quark flavours u, d, s, c, b, , and colour indices are suppressed It is assumed that all observable hadronic states are colour singlets (‘colour confinement’) In order to quantize this theory properly, we should ađ gauge-fixing and Fađe’ev—Popov ghost terms to Locp [1.21] However these technicalities are unimportant for the present discussion, and further reference to them is not necessarỵ
QCD is regarded as an excellent candidate for the theory of strong interactions because of its renormalizability [1.22], asymptotic freedom [1.23], and many other reasons [1.24] In particular, QCD possesses a global SU(L) x SU(L) chiral symmetry in the limit m; = 0, i= 1,2, , LD N;ỵ This means that Locp is invariant under the global flavour transformations
gi > (exp(iaara)) is G (SU(L) vector)
(1.3) gi (exp(iBarays))yq (SU(L) axial)
where i and j run from 1 to L This is no accident, indeed QCD was constructed [1.20] with this very property in mind in order that the successes of current algebra of the 60’s could be taken over
The symmetry (1.3) is explicitly broken by a quark mass term
However, (1 3) is is not ‘the maximal chiral symmetry sgodated with the limit m; =0, 7= 1,2, ,E<
Vẹ A L7) x L7 Ụ etry Ca be Extemrdedc O ĐIODa LÝ L7 :
U(J) vector symmetry corresponding to the transformation đ;—> €'”đ; represents baryon number con- servation and poses no problem On the other hand, the extra U(1) axial symmetry corresponding to the transformation
gi > exp(iP ys) 4 (1.4)
presented a serious problem If it was to be realized manifestly, the physical spectrum would have to consist of equal mass parity partners, and if it was realized spontaneously, there would have to be an
T, LK, n (L = 3) As neither of these alternatives are seen in nature, there is a “U(1) problem” — how is this -extra-axial U(1) symmetry realized? —
The U(1) problem is a characteristic feature of quark models It was originally encountered in 1967
AR CaN A ^
h ach « C hxPe¬an c ard 2 Ha r^ he
Trang 6solution that the flavour singlet (” meson, to be identified with the 7’, had a mass near that of the pion Two years later, a sharper version of the problem was obtained [1.27] from U(2) x U(2) invariance in the SU(2)x SU(2) limit Then Brandt and Preparata [1.28] found a related difficulty: the decays n> mam seemed to vanish in the SU(2) X SU(2) limit, even if explicit isospin violating effects (m, # Ma) were included The deep connection between the y-decay problem and the U(1) problem will be elucidated in subsequent sections
At that time (1969-70), most people thought that quark models should not be taken so seriouslỵ As a result, almost everybody forget about the U(1) problem for a few years
It was the advent of QCD which prompted Fritzsch, Gell-Mann and Leutwyler [1.20] to revive the U(1) problem They emphasized the necessity of solving the problem if QCD was to be the theory of strong interactions They also suggested that the anomaly [1.29] associated with axial U(1) trans- formations might be used to obtain such a solution However, they were unable to proceed further because of uncertainty about identifying the axial U(1) current operator in the presence of the anomalỵ A definitive analysis of the anomalous axial U(1) current was then provided by Bardeen [1.30] Because of the anomaly, the naive Noether operation of multiplying đ,(x) Đy y,ys q(x) to form a current is necessarily invalid Instead, renormalized composite operators have to be considered When the vertex y,ys couples to gluons via fermion-loop triangle and box diagrams, there is a divergence proportional to the operator counterterm
> <“
chr«B Aa Fa, — nef Ae Aj} (€o123 — 1 — —e1t 3), (1.5)
Consequently, renormalization induces « an ambiguity proportional to K, When one > attempts to define a
malization prescription to the definition of the composite operator The two most interesting pos-
Trang 7The standard form of the anomaly equation in the SU(L) x SU(L) limit follows from the conservation of the symmetry current J 775, sym!
2
§ ö*J„s= 2L
Jus=2 3277 FăF°y" (1.7)
The crucial point of Bardeen’s analysis was the observation (generalizing remarks of Adler for the Abelian case [1.31]) that it is the symmetry current J'c5, sym Which is associated with axial U(1) transformations, ịẹ whose charge
Qš — | Pe I,m
produces commutators with correct chiralities independent of the gluon coupling constant g Asa result, the solution of the U(1) problem remained obscurẹ It was not possible (nor is it possible now!) to argue that the F-F term in (1.7) represents an explicit breaking of axial U(1) symmetry, because J%; has nothing to do with axial U(1) transformations (Unfortunately, this subtlety was completely overlooked by most subsequent authors That is why the U(1) problem has become so controversial.)
This negative result led to proposals for ađitional terms in the divergence equation (1.7) produced by ađitional anomalies [1.32] or fictitious scalars [1.33] None of these alternatives have been or should be taken seriouslỵ The U(1) problem received widespread attention when Weinberg [1.34] obtained the bound Miavour S 311, singlet (1.8)
and Kogut and Susskind [1.35] proposed a solution of the U(1) problem based on analogies with QED in 2 dimensions Rogut and Susskind L0 0805100 that the ninth Nambu-Goldstone boson (for FUG
insufficient a 36] to resolve the U() problem We will discuss this i in more e detail later
O74” 0 ĐƠ ° ehecected tha LÍ Q CG netantanc* aLlit© JI Q W aga ahla tna give the R av } PỊÌY L) ¬ .¿} 3 io a nnn
value and hence explicitly break U(1) axial invariancẹ He further obtained the chirality equation AQE=2Lv — (9 (19) where v denotes possible values of the ‘topological charge’ operator (g?/3277) f d*x F- F However,
’t Hooft had misused equation (1.7) in obtaining (1.9) He had wrongly indentified the U(1) axial charge as f,Jos(x), instead of J, Jos, sym(x) This was pointed out by Crewther [1.40] who further examined,
for the first time, the anomalous Ward identities associated with the U(1) axial symmetrỵ Under the assumption that there is no U(1) problem, Crewther [1.40] derived the equation
Trang 8
AQš=-—2L„ (1.10) in place of (1.9) The wrong sign in (1.9) resulted from ‘t Hooft’s incorrect identification of the U(1) axial chargẹ Further it was shown that ‘t Hooft’s calculation with instantons was an example of spontaneous symmetry breaking
Of=0, AOKI)
as opposed to explicit breaking Besides dismissing *t Hooft’s claim of explicit breaking of U(1) axial symmetry, Crewther also showed that instantons were insufficient to resolve the U(1) problem Instantons (gauge field configurations with integer topological charge) were unable to give (gq) a non-zero value in the chiral limit and at the same time avoid producing physical U(1) bosons; fractional topological charge configurations were required
The U(1) problem had now been precisely formulated This inspired a totally new approach to the problem based on the large N (N.=number of quark colours) expansion [1.41] Witten [1.42] suggested that the »’ (for L = 3) was as much a relic of U(1) axial symmetry as the pions were of SU(2) xX SU(2) symmetrỵ The argument relied on the fact that the anomaly could be turned off as N.>~ and then there would be nine genuine Nambu—Goldstone bosons (L = 3) With finite N Witten suggested that the mass squared of the 7’ was proportional to 1/N, Following Witten, Veneziano [1.43] proposed a more explicit mechanism which was consistent with Crewther’s anomalous Ward identities Veneziano introduced a massless vector gluonic ghost particle that could not only solve the nạ => r7 problem but was also able to give the n’ its exceptionally large mass
The consistency of this approach then opened the way for new physics concerning the CP-violating parameter 6 of QCD*-
#ocp(Ø) = #oco(0)~ 6(g7/3277°) F- F (1.11)
In his earlier analysis of the U(1) problem, Crewther [1.40] found that he was unable to give a consistent value for the @ period of Green’s functions as different SU(L) x SU(L) limits were considered This difficulty was simultaneously resolved by Crewther [1.44], Di Vecchia and Veneziano [1.45], and Witten [1.46] They observed, both on general grounds [1.44] and within the large-N approach [1.45, 1.46] that two CP- -conjugate ground states appear at@=7 (mod 2) for certain values m; # 0 of the nent quark 8= mod a po ~ ry “3 Onne tn AePTyaeaan he BH t^ f1? on £ an ht Y = 4 a wo, ` 7 J S4 U 7 Yc p7 C LÝ Y operator H950) 5 F the 6 period is om, but v cannot be restricted to > integer values
expansion Shortly afterwards, Veneziano II 48] used the large-N, picture to give a very convincing argument as to why chiral U(£)x U(E) symmetry must be spontaneously broken
It is not possible to review the U(1) problem without taking a controversial stand somewherẹ The
literature contradicts itself too often to allow an account which is both democratic and logical The cause of the trouble is bad anomaly theory-—far too many authors still think that canonical methods
Trang 9
such as the Noether construction remain valid in the presence of renormalization In compiling this review, my guiding principle has been the requirement that correct anomaly theory be used at all times 2 Chiral symmetry
In this section we will be principally concerned with the theoretical aspects of chiral symmetry and not so much with the variety of its phenomenological applications and the accompanying experimental results The latter are extensively reviewed elsewhere [2.1] Also, anomalies and other failures of canonical theory are ignored throughout this section Changes in the formulation required by renor- malization are postponed to the next section
2.1 The exact chiral limit
2.1.1 Transformations, currents and commutators
As mentioned in the introduction, QCD possesses a global U(L)x U(L) flavour symmetry [2.2] when* L of the quark mass parameters m; vanish In other words LYocp (m; = 0; i= L) is invariant under the colour independent global flavour transformations [q is both a colour 3-vector and a flavour L-vector Below the SU(L) A matrices are in flavour space and act on (qi, q2, G3, , qr) @*A= Tey atA7] SU(L) vector: q>exp(ia:A)q; 4(1F ys)q =qụr->exp(ia‘A)4qịr SU(L) axial: q>exp(iB-Ays)g; qu, r>exp(FiB:A)gire U(1) vector: q > exp(ia) q; Gur explia) qịr 5 › L,R L,R or equivalently in terms of the left and right transformations SU(L) left: q > exp(il À2(1 — y:))4; Q.>exp(il-A)qi; đâ>đ 5 5 › L L> R I R
U(1) left: q—exp(2(1- ys))q; gị > exp(il) qu; Qn > Gr
U(1) right: q>exp(r2(1+ ys))q; tr > 93 đqa>exp(r)đạ (2.2)
Trang 10
With every Lagrangian symmetry there is a conserved Noether current ¥, and an associated charge generator, defined by t-2 tạo — 2= | Px Sox, 0) ( which is also conserved, d2/dt = 0 The Noether currents associated with the QCD chiral symmetry are Aa SU(L)v: #2 = — aỵ 2 2 “4; SU(L)a: Fas = q⁄.s 5 4 a=1/2, ,1”—1
U()y: J¿=4y44: U()A: J¿s= -y„}s4 (2.4)
The respective chiral charge generators are denoted by ƑF“, F$ Ó*“ and Qš Using the general formula*
5(X0) [141 g(x), QP 242 q(0)] = 28°) (Ur Pol, t2) + 0, Pal Le) a0), (2.5)
where /°, and /’, are any Dirac matrices and 1, and 12 are any flavour matrices, it is easy to show that the chiral charges satisfy the algebra associated with the (equal-time) commutation relations [2.3]
[F°, F° — 1E [ƑF“.F 2| = 1F [Fz.F s]= i abe Fre
2.6
[Ó.G]=[O,G]=0: — G=F*“.Fš.Q',QẸ (2.6)
In terms of the combinations Ff, p = (F* + F$), (2.6) separates into two commuting U(L) algebras, corresponding to the left-right transformations (2.2)
It is worth noting that the chiral transformation properties (2.1) of any operator can be elegantly
Trang 11That these equations encompass (2.1), for Op = q, is easily seen by utilizing the identity 1
êABe ^=B+{A,PB]+ x [A, [A, B]]+ 31 [A,[A,LA, B]lJ+ :
and the equal-time commutators Aa — — q, Àa 2 LFš, qÌ = —?5'2 [G", 4Ì = -đq, [Oš q] — —7sq [F*, q]= = 1 between every pair of quark fields in the operator Op 2.1.2 Manifest symmetry Consider some particle eigenstate |n) of the Hamiltonian H|n(q)) = Vg? + mĩ |n(q)) (2.8) and some conserved charge generator 2 3 =-i[2, H]=0 (2.9) Together (2.8) and (2.9) imply that H2|n(q))= Vạ?+ mĩ 2|n(q))
That is, for every state [n) and conserved generator 2, there is another state 2|n) with the same energy (or mass) as |n) If we had an algebra of charges this would imply a multiplet of equal mass particles, generated by continual application of all the charge generators This type of symmetry realization is
ermed mantest-as naS a direct CORSECGCUECHCE OF ne pnysicar Specirum, TOrcing O-segre gate Into
equal mass multiplets
believed to be of this typẹ The physical spectrum consists of approximately equal mass irreducible
F[CDTCSC ALIOT D U C.Ũ OdOUDICt5, OtHDICIS, CiC.}) an0 U C.2 OCICS, de neis, €
observed intramultiplet mass splittings reveal the approximate nature of the chiral symmetries This is
reflected In turn by allowing non-vanishing quark mass parameters in the QCD Lagrangian (see section 2.2)
If the axial symmetries of QCD were also realized manifestly, the physical spectrum would further be paired into equal mass parity partners This is because the axial charges are ođ under paritỵ There is
little empirical evidence for this
The hypothesis for manifest symmetry requires 2|n) to be a one particle eigenstate of H (perhaps different from |n)) Consider
Trang 12
If 2/0) = 0 then 2|n(q)) = [2, a3.(q)]|0) is likely to represent a one particle state because the commutator [2, a}(q)] is of the same form as a}(q) This is most transparent in the quark model
2.1.3 Spontaneous symmetry breaking; Goldstone’s theorem
If 2|0) # 0 the procedure of generating new one particle eigenstates of H is upset by the presence of the second term on the RHS of (2.10) As [92, H] = 0, it follows that the set of states defined by
|B) = exp(—128)|0) (2.11)
are all zero energy eigenstates of H and hence candidates for the physical vacuum This vacuum degeneracy for 2\0) # 0* can effectively hide the Lagrangian symmetry from appearing in the physical spectrum In order to properly quantize the theory it is necessary to select one of the states (2.11) as the ground statẹ This type of symmetry realization is usually termed spontaneous symmetry breaking The use of the word ‘breaking’ is very misleading The symmetry is not really broken at all; any one of the vacua (2.11) can be chosen; 2 = 0 A more appropriate name would be ‘hiđen symmetry’ [2.4]
As all of the eigenstates (2.11) have zero energy, one may suspect that the charge operators 2 can create (or annihilate) particles of zero energy and momentum (and hence zero mass) It then seems plausible that this type of symmetry realization is associated with the appearance of zero mass bosons [2.5] The proof [2.6] of this conjecture is known as Goldstone’s theorem
An elegant proof [2.7] of Goldstone’s theorem can be given by utilizing the Ward identities, whose derivation [2.8] we now briefly review Consider the action of 6/dx, on the time-ordered product
TỜ, (œ) Op(0)) = O(%0) F(x) Op(0)) + O(— x0) (Op) F,.(x))
The derivative can act on either ¥,(x) or the @ functions, 4% 6(+Xx)) = £6*" 8(%) As $,(x) Is conserved, the result is the equal-time commutator
Trang 13Integrating by parts gives
lim | d*x (-ig) é** T(Y, (x) Op(0)) + lim | d*x a¢{é** TY, (x) Op(0))} 4 0 (2.15)
q>0 q>0
Because of the damping (oscillatory) factor é*'* the second term in (2.15) is a surface term that can be neglected (unlike (2.14)) Note that the integral is supposed to be evaluated for g # 0 before taking q >0 Therefore
lim | d*x (—ia“)e** T{Z„(x) Op(0)) = {O, Op(0)]) z 0 (2.16)
q>0
The LHS of (2.16) vanishes unless there exists a zero mass boson that couples to ¥,(x), which by
Lorentz covariance gives an extra factor of q,/q’
If 2|0) z 0 there necessarily exists some operator B=[2, A] which has a non-zero vacuum expectation value (v.ẹv.) (B) = 42, A]) 40 (2.17) Conversely, the existence of any such operator B is sufficient to enforce spontaneous symmetry breaking 2.1.4 Chiral symmetry realization and current algebra
The question of how the chiral symmetries of QCD are realized is now evident As already noted the vector symmetries appear to be realized in the manifest modẹ On the other hand, the axial symmetries
show little evidence of being manifest; there is no indication of approximately equal mass parity partners or near massless baryons.* It then appears that the axial symmetries are realized spon-
Trang 14
vector symmetries The condensate [2.8]
(44) # 0 (2.20)
spontaneously breaks U(L), xX U(L)g down to U(L)ỵ This is because gq is invariant under U(L), transformations but not under U(L), —see (2.1) or (2.7)
Consider the SU(L),4 Ward identities | d*x 4% T(F4 s(x) Op(0)) = (FS, Op(0))) (2.21) rewritten as lim | d*x e** (iq“) T42; :(x) Op(0)) = (LF, Op(0))) (2.22) q>0 Denote the massless Nambu-Goldstone (NG) pseudoscalars generically by 7° and define the pion decay constant (F„) by (0|Z¿z(x)|?”(4)) = 14„F„õ¿, e— "45 (2.23) The LHS of (2.22) can be saturated** with the NG pseudoscalar pole, fig |, lim (~i4*) 0|Z2s(0)lz“(4)) = (z^(4)|Op|0) = (F2 Op]) (2.24) q—>0 On using (2.23) this gives 1 i aft_Nlify Qy\ ( lim (z“(4)|Op|Ð) = ( 40 2 -_—— — ¬ Ab 3 — —” ~ t42 to an
The result (2.25) is known as a soft-‘pion’ theorem (2.9]: the effect of contracting a ‘pion’ state out of
ry pPlAm Ar a Ar ¬ ry Ary ryt OO Alaahea ˆ of ry ne Cary (` ry re Fenaroathy
a a oO piroud wad Oo “tat Cl a - Gre 5 a 7
should be noted that the proof of Goldstone’s theorem also implies that F, # 0
Extending the above arguments to the Wardidentity
Trang 15where (| and |@) are any hadronic states (perhaps multiparticle), does not necessarily derive the equation
lim (8, 7*(gOpla) = (=) GILFS, Oplia) (2.27)
particularly if |@) and |8) contain nucleon states [2.10] In this case the LHS of (2.26) has contributions
Trang 16states Assuming SU(2), symmetry*
a
(N( pr) Fị s(0)|N(p1)) = U (pr) (=) [yrs C19") + 4,7: C2(4”)] U(m) (q = Pr~ pi)- (2.30)
Together (2.29) and (2.30) imply that
2Mn C4(q*) + q? C3(q*) = 0 (2.31)
As C4(q?=0)= ga # 0 (ga is the axial-vector nucleon coupling constant, which is measured from B-decay to be ~ 1.260 + 0.007), there are only two ways of satisfying eq (2.31) at q° = 0; either
(i) My = 0
or (ii) there is a massless pseudoscalar (the pion) that couples to ¥{.5 and produces a pole to cancel the
q° factor in front of C$(q’)
Alternative (i) is a hallmark of manifest chiral symmetry while (ii) corresponds to spontaneous breaking of SU(2), In the latter case we derive the Goldberger-Treiman relation
MNEA = 8zNNŸzx (2.32)
where g,nn = 13.4 is the pion—nucleon coupling constant, defined by the interaction term igannNysAqTN Experimentally, (2.32) works very well [2.11]
The presence of the quark mass term in (1.2) explicitly breaks the chiral invariance (2.1) and the
Trang 17following cases are also interesting for a theoretical appraisal of the more realistic situation (a) m, = m2 = m3= 0; exact chiral limit, SUG), x SU(3)p (b) m,=m,=0 4 m3; SUQ@), x SUQ@)a (c) m,= m,=m # m3 # 0; SU(2)y (d) m,=m,=m3=m 4 0; SUB) (2.35) The remaining unbroken (ịẹ explicitly) symmetry invariances associated with each case is also given in (2.35)
Whenever one of the currents (2.4) is not conserved, its corresponding charge generator is also not conserved Nevertheless the equal-time commutators (2.5) and (2.6) and the transformation laws (2.7) remain intact This is what makes the notion of an approximate symmetry still useful [2.3]
With the chiral symmetry explicitly broken, we no longer expect to have equal mass multiplets and zero mass NG bosons The NG bosons are expected to acquire a* mass? of the order of the breaking of the axial symmetries, ịẹ O(m quark) On the other hand, intramultiplet mass splittings are induced by violations of the vector symmetries and are expected to be proportional to quark mass differences
In the chiral limit, the vector charges annihilate the vacuum, (2.18) With explicit symmetry breaking this conclusion is modified Coleman’s theorem [2.12] states that, if 2|0)=0 then the corresponding current, from which 2 is derived, is conserved If the vector charges were to annihilate the vacuum this would imply that @“¥% = 0, in obvious contradiction to (2.34) Therefore, for every vector current that is not conserved, the corresponding generator does not annihilate the vacuum Also, with explicit chiral mmetry breaking, the v.ẹv.’s of diagonal quark bilinez Nn as (GA3q), (GAsq), € need not vanish Instead we expect that (du) # (đ) # (Ss), etc
() di IOn_O E nhe hi Fnne hreakino VI ale PY PO ATO ng ne imple Pry} ne
Trang 18
As an illustration of how this comes about, consider (0|¥} </7') Using” F“jz”)= ¡ƒ““|mr*) and ín particular F*|z’)= (i/2)|7') we can rewrite this as —2i{0/F,sF la’) Since (O|F*= 0, this equals 2i(0|F*, F), s]l’) = (O|F,, s|7’) Hence F,' = F,” By iterating this procedure, the remaining identities in (2.39) can be established In the SU(2)y limit
Fo = Ft and F yo zo = Pye [SU(@)v limit] (2.40)
With the inclusion of explicit symmetry breaking effects, the relations (2.40) are violated by terms O((m2— m,)/A) while the remaining relations in (2.39) by O((m 3 — m,,2/.1)) The so-called wave function renormalization constants, VZ,», VZ,: etc., also satisfy relations like (2.39) and (2.40) and undergo a similar fate under explicit symmetry breaking
2.2.2 Chiral Ward identities
A quantitative study of broken chiral symmetry begins with the chiral Ward identities, which are exact equations
As before, the chiral Ward identities are derived by differentiating the 7T-ordered product TŒ,(x) Op(0)) where Y, (x) = F4(x) or F2,5(x) As a*¥,.(x) no longer vanishes, eq (2.12) is replaced by
3š TỢ,(x) Op(0) = 7(2„(x) Op(0)) + (8 (Xo) [Fo(x), Op(O)))
It is usual to use the equations of motion to replace 0“¥, by its classical value, (2.34) (This procedure can be justified in the path integral method of quantization — see section 3.)
Intecratineg aver A4sy_euee £ fc | #x9:T0,6)Op(0)= | dÝx T(2/, @) Op(0) + (20), OpO)}e-r)- G441) With explicit symmetry breaking we do not expect to have any zero mass (physical) particles, in which
case the LHS of (2.41) vanishes This assumption is based on the intuition that a zero mass particle would necessarily develop a non-zero mass through radiative corrections in the absence of any symmetry to restrain it from doing sọ We then have 0= | dx TOS, () Op) + (2, OPOle:r) (2.42) ` *
ar Ne moande a 7 y Mme a 4 Cl k7 Vr a2 C ˆ = c non 2 C Aa L3,
Ward identity (2.42) can further be generalized to include non-local operators such as a product k=l k\Yk}- N k-1 N T{S {T] Opz@»)[2.Op.)lez II Op,@)}) “k=1 “m=1 n-k+l at
*This relation is a consequence of the definition lr? (q)) = a„s(4)l0) = (—i) ƒ dx exp(—iq-x) 2od„(x)0) where qo= VạZ+ mỹ and
Trang 19Some particularly useful identities are derived from the axial Ward identity
| ax T(o"F%.s(x) Op) + (FS, Opler) = 0 (2.43)
with Op(0) = 2 7#2z(0)= ¡ D5; Gi(m; + m;) ys(A-/2); q;(0) Evaluating the commutators with (2.5) derives the equations | d*x Ta" Fix) °F 30) = (ig It 2w) (2.44a) | d*x T(2“#132(x) ð*#1s s50) = (ig (“;”\( 1 g4) (2.44b) | dx Ta" FX (x) a" F43*(0)) = (ig 7X 0 J#) (2.44c) | d*x T(2“#Ø5%”(x) 2”Ø5z”(0)) = (ig (™™)(' 1 ,w) (2.44d) | d^x T(2“#8 4(x) 2“#8z(0)) = (ig : (™" mạ " (2.44e) "" „5 v5 = (il V3 2 0
There are some other equations other than those listed above but these are uninteresting because the V.ẹV.’s Of the associated commutators vanish identicallỵ
Another set of useful relations follow from (2.43) with Op(0) = ãy:À 4(0): - _/1 „5 5A3 = 0 [ ote 7 (a3) gy = 4(0))= ụ ("1 " (2.45b) | d ‘xT (a F530) Gỵ 5 “Ai = ụ (’ 0 1)4) (2.45)
ro TỆ 2" 6+17(v) Alay À6 — lÀ¿ ¬/a\\ _/-/0 1 Nà (2 A\
Trang 202.2.3, Intermediate state insertions
The technique of intermediate state insertions is extremely useful in extracting information from the chiral Ward identities This relies on inserting the field theory analogue of the quantum mechanics identity
> |n)(n| = 1 (2.46)
between operators in matrix elements
In quantum field theory, the sum in (2.46) also extends over multiparticle states Furthermore, every state should be physical, ịẹ on-shell and positive energỵ In field theory, the correct identity, in place of (2.46), is* 00|+ > | d*p,, > 3 5 Ôn) oP mĩ) 8(Ppa)|n(p„)) (n(pa )| d*p, dp, *> | ozy903- m) đợt) [ P= a¢pis— ms) (95) mtn) mPa) (n,) 0u) +: ; (2.47)
Trang 21Only the one particle contribution has been given in (2.49) To obtain (2.49), extract the x dependence
out of Ăx) by
(O|Ăx)|n(p,)) = Ole’? * AO)ẽ?*|n(p,)) = O|AO)In(p,)) e"”"™ ,
write the @ functions as co -l rd +] 9 (x0) = ƒ œ exp(FiwXxo) i w tie —œ
and then integrate over Xo, w, x, Dy and Dn
The sum in (2.49) extends over all possible states with the correct quantum numbers so that both
(0|A|n) and (n|B|0) are non-vanishing The mediating factor is just the usual Feynman propagator
associated with fig 3ạ It is very important to realize that all matrix element states are on-shell In other words the 4-momentum of the state |n(q)) in (2.49) is (q, Vq? + m2)
By following much the same procedure as above, the two particle contribution to (2.48) can also be evaluated The result ( | d*y ei T(Ăx) B())) 2 ptlẹ contribn = a (@m# dpm _ m mj/ Y)
x ((q— pm) m(p„)|B(0)|0) 5 2 — mm +ìe (@— Pa} I — mà +ie (2.50)
is equivalent to the usual Feynman rules associated with the diagram of fig 3b with the further clarification that all matrix element states are on-shell This general philosophy of using Feynman rules with on-shell matrix elements can also be extended to higher particle multiplicity contributions
2.2.4, Leading order results
Applying the results of the previous section to the Ward identity (2.44a) gives p m n - UC ^ Lên ——=— => B Ca 7 n nm — a-P ta) (b]
o 3 The intermediate ate contnibution_comin
with the diagrams (a) and (b) respectivelỵ
Trang 22lim {(01an# s(0)|zrˆ(4)) | (7°(q)\a”F > 5(0)|0)
q—0 ự T ~ mm TT£ ˆ
+S (02*#240Jln@)z—m>-rịy (n(4)|27#34(0)0) H # 7
+ multiparticle insertion contributions | = (ig(™ Ha 04): (2.51)
Consider the matrix element (0|a”¥2 5(0)|7°(q)) = m7oF, As 0“ ¥3 5 = O(m,2) and F,» = O(1)* then mo= O(m,2), as expected On the other hand (0|d"¥3 <(0)|\n(q)) = m23F, n# 7°, but now as m2 = O(1) (ịẹ does not vanish in the chiral limit) we must have F,, = O(m,,>) Rewrite (2.51) as lim \m 2 oF 0 q>9 2 i my Tri mMEy+ 3 mẠR,S———mậE,+:-:Ì= gq’ — mit ie h # 2m T7 q Ì~ mã +1e (04Í 4 2 0 1 là Lẹ m?F?+ 3 mộF2+cc= =(g(n mạ 04): (2.52) n# n°
Trang 23
im (x(q lids (ON) =~ F-(G(* 1 g)a) + OCm.a) (2.54a)
(figys S22 gha=(@)=-=—(ă" 1 )a)+OCm.2 (2.54b)
(ligyssSH2 gia) = -2—(ẳ 0 )a)+ OCms) (2.54)
0iáy,Ê< atm, RO)= - ——{ă" 1, )a)+ OCs) (2.544)
((0)li4ysÀsa|0) = — E (a5 Ữ 1 nN) + O(ms) (2.54e)
From the qualitative discussion in section 2.2.1 we expect that
F9 >*,K®, KP, KP ạ = F, (2.55)
and
(itu) = (đ) = (5s) (2.56)
to leading order Therefore, to the order we are presently considering, we should not really distinguish
petween 0 + € n equations ` and 4 V1ith th n mind, equation an De used [fo
solve for the quark mass ratios* in terms of the pseudoscalar mass squares** [2.16] (m2— m,)/(m2+ m,) = (m?+ + m°— mắ+ — m2)/m20= 0.29 (2.57a) 2m3/(m, + m2) = (myot met — m2+)im2o= 25.8 (2.57b) The error in using either one of these equations is of order 8m%/m2, where 5m is the error in using the lowest order expressions for me and mx: obtained by setting Fx = F,, etc in equations (2.53c) and
(2.53d) We are assuming that 8m, being an SU(2) breaking effect, is considerably smaller and can be
neglected If, for example, we take 8m ~5% mx then 8m\/m2 ~0.5 This has little effect on the
estimate (2.57b) but makes (2.57a) somewhat unreliablẹ
Inserting a complete set of states into the general Ward identity (2.43), and keeping the lowest
selves-with the precise definition of qua * The ratios of quark masses are renormalization group invariants in the absence of electromagnetic interactions [2.13] We need not concern nasses {2.14} ** These relations are carefully chosen so that the electromagnetic mass contribution to 7* and K* is not included Dashen’s theorem [2.15]
Trang 24(1>(0)}Op|0) = (=) (FS, Opl) + Om), (2.58)
This equation, like most current algebra results, is approximate and only becomes exact in the chiral limit
Before concluding this section it should be emphasized that pion pole dominance is not simply a consequence of the fact that the pion is light in comparison to other hadrons This is evident from eq (2.52) In general, pion pole dominance is a consequence of symmetry and symmetry alone [2.15] Models in which the pion is accidentally light (ẹg the quark model) cannot account for the successful results of chiral symmetrỵ
2.2.5, Chiral perturbation theory
The QCD Lagrangian (1.2) can arbitrarily be separated into two parts
#= #,+#' (2.59)
and any v.ẹv (or Green’s function)
[tua 44] léA,Jop exofi | #09}
| {aq a9) fa, Jexpfi | 769] (2.60)
(Op) =
calculated with the full Lagrangian %, can be expressed in terms of Green’s functions calculated with the Lagrangian 4 by writing f (_ƒ <2 ( — f | Idg dq dẠ) Opexpy | Fox) > Win f Sx) ] 2) | On) XI Xn (Op) = — ———¬ TỶ : | [dq dg] [4A,,] exp}i J Z4(x)} S„_„/n9 Ị F(x) | Fla) | #G„) x l = (Op,+ | TOPO HeMs+5, | [TOPO LE)i Hea) l tại [ | [ T(Op0)i2G))ị205)i 20981 G61) XI X2 X3
The superscript c denotes that the Green’s functions are connected and the subscript 0 that they are calculated with the Lagrangian ⁄⁄%
Trang 25
For brevity we will limit the rest of the discussion in this section to chiral SU(2)x SU(2) with m,= m= m We will study this limit by expanding around the chiral limit, m = 0 For our purposes
L= Lo- mG 1)q = Lo— mgq, where Lo is SU(2) x SU(2) invariant From (2.61)
(Op)= (Opbưm [ T(Op(0)-04ø0))i+5r | [ T(Op)C0) aac) C¡) 44G)
tai [ [ [ T(OpC0)460)C1)840) 8405): 2.63)
where the subscript 0 denotes evaluation in the chiral limit, m, = m,=0 Consider eq (2.63) with Op = Gq
(aa) = Ga)ơm { TGq(0) i) aalerns+ 3 ff T@aO)(-i)aaee)(-i)gaeay+ 069
XI XI X2
The dominant contribution to the second term on the RHS of (2.64) is expected to come from the two pion insertion diagram of fig 4 The integral involved ~J d*p/p* is both logarithmically ultraviolet and infrared divergent The ultraviolet divergence poses no problem* and can suitably be removed by imposing a cutoff at some typical mass scale of QCD, A ~ 1 GeV The infrared singularity is removed
by summing the set of diagrams in fig 5 The net contribution is
(-1m =im) ) ( 199|77 05 | ÔaÝ p2(pP~ m(nlãq|m)) ca T44] do- Olqglmo> | 225 i (ara gq|0 (2.05) 2.65
The combinatorial factor of 3 comes about because the pion in the loop can be either a 7°, 7* or ạ
Trang 26- - 3m - mG (4á) = 44) [L~ s22 (800,lox(A° (=9) | + O(m) (2.66) or in terms of (m2)eọ = —m(q)ol(F)o 3(m * Yeọ (4a) = (4q)o {! +
These logarithmic terms were first discovered by Li and Pagels [2.17] They clearly demonstrate the non-trivial nature of chiral perturbation theory, which is not a simple expansion in powers of m These logarithms are widely believed to be important and to constitute the leading order corrections In general, chiral perturbation theory gives [2.18]
A = A,(1+ O(m log m)+ O(m)) (2.68)
As log m is singular as m > 0 it is clear that these m log m terms are more important than the O(m) terms near the chiral limit,
milogm|>m, m<1
The relevance of this assumption to the physical world (with m,, mz and m3 # 0) is not immediately obvious As the O(m) corrections are practically impossible to calculate there is nothing lost in going along with this assumption Ultimately the comparison with experiment will decide whether the
approximation is reasonable or not
The extension of these techniques to matrix elements is somewhat more complicated One possible
procedure [2.18] for handling matrix elements is to reduce them to Green’s functions by the LSZ reduction formulạ For example
The m dependence of matrix elements can now come from either the Green function, the factor of m7 he on-sh onstraint or go Altern e Ol one could use the chiral Ward iden and
intermediate state insertions to successively relate matrix elements to Green's functions and inductively
` /
afe¥a A ne L†] Atm A he (1Ø hrn" (` Pp Or (` h rìe alwe AN ic 2< Arm
WU iu `7 Cl ° k7 `7 `7 L7 MUI 7 eure wre Cl 74 V 7
and m2 have been calculated The results are [2.18] F, = (Fz}(1 + 2X) VZ„ — (VZ„) + x): mĩ = (m Jeo ~ x) * (2.69) where y = [(m2)zọ/327 (F*)] log(4”/(m?) ) -
2.2.6, Leading order corrections
In this section we assume that the Li-Pagels logarithms constitute the most important corrections to the leading order results and reanalyse the results of section 2.2.4 with due respect to these terms
Trang 27“7 ~
⁄ "N
yal x ` ST} va
Fig 6 The mơ and azz intermediate state insertion contributions to the LHS of (2.44a)
deriving (2.53a) the contribution coming from only the one pion insertion was kept This was O(m) Other one particle states were shown to give an O(m’) contribution and can be neglected to the order we are presently considering Logarithmic terms may arise from two and three particle insertions involving pions Potentially there are two possible diagrams that can produce such logarithms, fig 6 The contribution coming from the first of these diagrams (a is some scalar particle) is
(02“82.|m0) | 2 pz- mg mg (ơl278340
= |0l#? Jnỏ mì {In n(A re) Me in(A tm) (2.70)
As m2 = O(1) and (0|“#2) s|70) = O(m) this contribution is O(m7) and is thus of no interest
The other diagram of fig 6 gives TT ` _đếp ƒ d*k i i i (ar Faden) | a | Oa GF ma) = ma) PEEPS we) (zrzr|ð”#?:|0) (2.71) Since (0|2“#2 :|7r7) = O(m), a m“ log m term can only arise if the integral i 4 4 1 Im’) = [ep [otk or HT KP =HỆNG TK - mà) (2.72) 2 _ ° ° 2
It then follows that eq (2.53a) is true as stated; in other words, with logarithmic corrections included +
e definition a ° Ot C m3 ? € and (Ga > Thị ^2lSO aĐlHIe 3© = Othe C = other = equations ïctHdc 1n ae a ana a ä Nie
therefore have two © systems of equations — the decay constants F a mK, K,R 3 the e pseudoscalar
equations relate the logarithmic coefficients ir in sin the pe chiral expansions of these parameters and consider-
ably simplify the task of calculatin m Note resu a wi
equations
Trang 28
Using the SU(3) x SU(3) chiral perturbation theory results of Langacker and Pagels [2.18]
Fx>121F, VZx>1023VZ,
F¿e>10065F,¿:, VZ»>1.00072V Z4: (2.75)
one finds that [2.21]
(5s)= 1.48 (au), (đ)~ 1.018 (au) (2.76)
These results can also be calculated more directly as in the previous section Note that the reason why
\(Ss)| is greater than |(au)| is simply because Fy > F,
Equations (2,53) can be used to solve for the quark mass ratios Using (2.75) and (2.76) this leads to the values* (2.13, 2.21]
(m;— mì)/(m;+ m,)= 0.55 (2.77a)
2maj(mì + M2) = 30.8 (2.77b)
Results like (2.75), (2.76) and (2.77) clearly demonstrate that chiral logarithmic corrections can be quite appreciablẹ For applications to other than the pseudoscalar sector see ref [2.23] 2.2.7 Vacuum stability illustrated by the model with potential V(b) =Ả ~ °Y (2.78)
which is invariant under the transformation @ > —@ There are two minima at ¢ = +c, both of which are eligible candidates for the ground statẹ Suppose we were to ađ a term e¢ to V(¢) which explicitly breaks the ¢ > —@ symmetrỵ In this case the double well potential (2.78) becomes lopsided and there is only one minimum at @ =—c (Â = tc) if ô >0 (« <0) The degeneracy of the vacuum has thus been lifted
The same phenomenon occurs with chiral symmetrỵ The presence of the quark mass term in (1.2) explicitly breaks chiral invariance and selects one of the states (2.11) as the ground statẹ The massless excitations (ịẹ the NG bosons) which are associated with group translations around the degenerate minima are now massive
To help facilitate the study of this problem Dashen [2.24] devised the following theorem**: If |vac) is the ground state of WME 2cCtC f ft H= Hot eH’ = | Px Ho(x) + « | dex #'(x)
H đlsœơ-a mm asom a are aiu O Ồ K+ jem b ’ ~ nan
Trang 29then
$(w) = (vaclexp(iw - 2) « H’'(0) exp(—iw - 2)vac) (2.79)
has a local minimum at w = 0 Proof: HN = (vaclie[2*, #'(O)]ivac) = (vac| - a* $5 (0)vac) w=0 and hence vanishes as the vacuum cannot carry any momentum a E(w) dw? dw? = (vac| — [27,[2°, « #'(0)||\vac) = (vac| — i[ 2°, a" £2 (0)]|vac) w=0
= (vacli { dtx To" F(x) a*°F20)\vac)>0
(For the axial currents this equals F2 6,,m 72+ O(m4-).)
If we started with some false vacuum |vac); there would be pseudoscalars with a negative mass? that would be produced in proliferation to make |vac); unstable and force it to decay into the true vacuum, |vac) In this case, the minimum of €(w)saie would occur at some non-zero value of w = wọ The true vacuum would be given by \vac) = exp(—i@o: 2)|vac), This theorem will prove extremely useful in analyzing the U(1) problem 2.2.8 Alternative schemes of spontaneous chiral symmetry breaking For simplicity we will again restrict ourselves to chiral SU(2) x SU(2) Consider the equation defining
the pion decay constant (0|2“2#2:(0)|r°}= ô°°“Fazm? (2.80)
In the usual scheme of spontaneous chiral symmetry breaking we have
F,=O(1), m2=O(m) and VZ, = (OligysAag|77) = O(1) (2.81)
It is a logical possibility to also have [2.25]
Trang 30characterized by the condensates [2.25]*
(iu) = cm,, (đ) = cmg (2.83)
On the face of it, this looks absurd; how can the condensates (2.83) spontaneously break chiral symmetry if they vanish in the chiral limit?
The answer to this question lies in the observation** that (i is not summed over)
am (Op(0)) = (-i) | T(Op(0) Gigi(X)) conn - (2.84)
x
Equation (2.84) follows directly from the path integral representation (2.60) It may then be expected that this alternative scheme is associated with a non-vanishing 4-quark condensate [2.26]
J Tau) tu(x))conn = | Td (0) dăx))conss = O(1) 2.85)
The usual Goldstone theorem then follows from the Ward identity
| T(" Fs) Gq Gysq) ~ (Gq: Gq) # 0: (2.86)
x
This sort of alternative scheme is motivated by some apparent inabilities of the usual scheme to
Trang 31m2=O(m’), F,=O(1) and VZ,=O(m?") (2.89)
In the language of chiral perturbation theory, these alternative schemes correspond to the situation where, for some reason or another, all of the terms in the expansion (2.61) up to O(m?~*) vanish identically and the leading order correction (to the chiral limit) is O(m?) Even though such a situation may be unlikely and may not be necessary to account for existing phenomenology, the existence of such schemes should nevertheless be kept in mind from a theoretical point of view
3 The anomaly
The appearance of anomalies [1.29] in theories with fermions is well known Although it would be inaccurate to say that anomalies are well understood, there are many aspects of them which are well established and have been from the very outset [3.1] In this section we aim to highlight those features of anomaly theory which are of crucial importance to understanding the U(1) problem Their misconception, even in the present literature, has been the cause of most of the controversy surrounding the U(1) problem
Our discussion will begin by analysing the basic triangle diagram that leads to the anomalỵ Before considering QCD we will discuss the conspicuous features of anomaly theory, such as the role of renormalization and the definition of the U(1) axial charge generator, for quantum electrodynamics
(QED)
3.1 The VVA triangle diagram
[2.2] involving an ođ number of axial- vector couplings B 3] The simplest of these diagrams (given i in associated with this diagram i is (m denotes the mass of the fermion), we = {-1) le xư— @m}*L7~m ˆ” ýƒ+ =m Y„ —- _ ⁄-———ỵÌ: Y-q-m""} (3.1) Consider (p + g¥ Turẹ Using (9 + 4) = (pf + / — m)- (/- 4 — m) this can be written as _ * 1 1 ƒ đÝ? 1 1 (0+ 0) Tow =i | eZ ZH gom || Gap Tm ew | đ? [ 1 1 1 tí đt ae FENG =| (3.2)
Trang 32| (p+q) AN WG p+F rq C B “1É, 7 » p
Fig 7 The anomalous VVA triangle diagram
Consider now q’T,,,, Using 4 = (7 —- m)- (7-4 - m) this becomes d‘r Te =i | Geet ey Se || Qn at [*zzx.z< a MIs | | (3.4
The last term in (3.4) vanishes for the same reason as abovẹ The first term in (3.4) on the other hand can only be recast into such a form by shifting the momentum integration variable to r’ = r+ p This step is however illegitimate, as we shall see, because the integral (3.4) is linearly divergent
[note that tr(y„7y„yz) = 0]
The translation of the integration variable in a linearly divergent integral involves a surface term that may not vanish [3.4] Consider
For a linearly divergent integral f(r)~ l/r’ the first term on the RHS of (3.5) is a surface term Rotating into Euclidean space and performing the integration over a 4-sphere with r= R enables us to rewrite (3.5) as
a | ony Ga (1+ 0(5 )) /0)|= "gol Ms lim RR *(1+ O(@)) AR) (3.6)
Trang 33-j ° 4 1u = gạ2P 4 EuAer- 3.8) Similarly, it follows that 1 P° Tro = 2 PT" E wre ' (3.9) In QED, the Lagrangian of which is Loev = ~4/„„ƒ“” + j(Lế — e4 — m)ÿ (3.10)
where ƒ„„ = ó„đ„ — 0„đ„, the vector and axial-vector currents (respectively j„ = Wy, and j,s= Wy, ys),
associated with the transformations y>é*y and w—>exp(iBys)y respectively, have the naive diver-
gences
aj, = 0 (3.11)
0" Jus = 2impysy (3.12)
Our considerations of the basic triangle diagram (fig 7) suggest that (3.12) is satisfied at the axial-vector vertex A but that (3.11) is not satisfied at the vector vertices B and C On the other hand there is nothing to stop us from rerouting the momentum in fig 7 in such a way that (3.11) remains valid —————————— ạt BandC.]f we do this however we will be unable to maintain (3.12) at vertex Ạ In this case we
would have, in place of (3.3), (3.8) and (3.9), {n+ reƠR = 1 _ nxađ + Pé \f ` 1) JÀurơ Dả ~ TOE W ` 2\5rơ; TT} fy om Lr (3.13) q KRyuro — Y; P Ryro = VU Here we have denoted the basic triangle diagram by* R,,, in place of T,,, to distinguish it from the
expression (3.1) and the momentum routing assumed therẹ The two are related by T;„+—== “OT 4z? 4 \w Ê mxơăƑƒ 4) Dp L Nyro > Tyre T AS Ts (defined in (3.3)) and its counterpart R;,, are convergent ~ f d*r(i/r°) they are related by a
————— ]legittimate shift in the integration variable Rs,.,= Ts5rq + Tsq0-— SSeS
Trang 343.2 The U(1) axial anomaly in QED
Consider the case where two photons are attached to the vector vertices B and C The two most interesting renormalization prescriptions are:
(i) where we impose vector current conservation (gauge invariance) at B and C In this case it can be shown that (3.12) is modified to [3.5]*
— 2 ~
2*j„s= 2imÿÙysử + se th, (3.14)
where „„=3£„„„sƒ° Since gauge invariance has been preserved ¡in the renormalization leading to (3.14) the current j,,5 appearing there is gauge invariant
(ii) where we preserve U(1) axial symmetry (for m = 0) at the vertex Ạ In this case the U(1) axial current is conserved in the chiral limit
0" 5, sym = 2imypysw (3.15)
As we must violate gauge invariance at B and C this current is necessarily gauge dependent The subscript ‘sym’ refers to the fact that we have preserved axial symmetry in the renormalization prescription leading to (3.15) The RHS of (3.14) is actually a total divergence Q Om nh an NHíT 7 (3.16) sO we Can rewrite eq (3.14) as eg Le oe” (ins — hua) = dimpysip (3.17) It then follows on comparison with (3.15) that [3.5] 2 Jus, gym = Jus 73 hud” : (3.18)
The fact that f“’f,, is a total divergence and that j,,5, ym and j,; are related by (3.18) should not come as
^ Cp cane A Ane ry pon An A ry Ayes An AY me An nro Ata
the triangle diagram ambiguity ~(e2/4z?)/„„4"
A 1 e vauge Invarial L P , IfT ISUä 0a OCrTate-]1 WÌ EO
5 Đ Đ 1t DETALO WV VSW e gauge
dependent symmetry current would then correspond to j,.5, ym = „sử — (e”/4r?)f„„a” Thịs is very
appealing because j,,s,.ym then looks gauge dependent, as (e7/47)f,,a” is itself gauge dependent This however is only a convention that is completely misleading if taken literallỵ
Trang 35
The question now arises as to which of the currents j,5 Or J,5, sym is the symmetry current, ịẹ whose associated charge (defined by an equation like (2.3)) generates the U(1) axial transformations Define
Xsã— | d?x jos(x), qs= | d°x jos, eym(X) ‹ (3.19)
In what follows we will take m = 0 The arguments given are not affected by m # 0
Because j,s,sym iS conserved it is not renormalized gs is then renormalization group invariant On
the other hand, j,5 is multiplicatively renormalized [3.5] (j,,s)en,= Zăj,s)oare SO Xs is also mutli-
plicatively renormalized and hence is not renormalization group invariant
In the fully renormalized theory it is g; which has the commutators with ý and ÿ” that one would
expect the symmetry charge generator to havẹ Namely, [gs, ý(x)] = ys W(x), [4s, ÿˆ(z)] = —/'(x) ys
Furthermore, it can be shown (see below) that g; commutes with the photon field operators, while x; does not This is an essential requirement for the U(1) axial charge generator; the photon field and its derivatives are invariant under U(1) axial transformations
That xs does not generate U(1) axial transformations is evident from the anomalous commutators
[3.7] (Latin indices run from 1 to 3):
[a, (x), js(y)] ô(xo— Yo) =0
[đođo(x), /,s(y)] ô(xo— Yo) = 0
[ doai(x), jos(y)| ô(Xo— yo) = “7 ie can fulx) 8 “(x- y)
— = je* 4 —
47?
These equations cannot be derived by canonical reasoning; they are inconsistent with taking /„,:=
wy, ysw and working out its commutators with the fermion anticommutation relations From (3.18) and (3.20) it follows that [4s a, | = 0, (qs, AoA] = 0, (9s, da; | = 0 (3.21) confirming that Lá»: is the li, axial charge generator d d u5,sym Ù ave C dl d
(2l4m2ƒ.,a": contrary to expectations based on gauge invariancẹ All of this i is part and parcel of the anomaly —a breakdown of canonical reasoning The most logical thing to do is to avoid conventions and think in terms of renormalized operators {1 40]
Bs = (Úy„sÚ)s inv.» Tus, sym = (Úy„ysÚ).n inV + (3.22)
Trang 36elements has also been violated This is because Wy, ysw is an external operator that does not have a direct bearing on the dynamics of QED This is also evident from the fact that although j,,,5, ym iS gauge dependent qs is still gauge invariant [3.5]
3,3 The extension to OCD
In QCD the U(1) axial symmetry (2.1) is anomalous because of ambiguities associated with the triangle and box diagrams of fig 8 The outgoing particles are gluons Once again this ambiguity can only be removed by appending a renormalization prescription As before the two most interesting cases are where we either preserve gauge invariance at the gluon vertices or U(1) axial symmetry at the y, ys; vertex In the former case we have a non-conserved gauge invariant current (denoted by J%;) which satisfies* [3.8]
2
3ì Fa, pene (3.23)
0“J1+= 2L
where F#"? =te#® F2, and F4, is the OCD field-strength tensor In the latter case, when we impose B a £ p
U(1) axial symmetry at the y,ys vertex, we have a conserved gauge dependent U(1) axial current
(denoted by J7 s ,v„)
ONT Ts sym — () ‘ (3.24)
Jiis.sym 1S gauge dependent because in preserving U(t) axial symmetry at the axial-vector vertex we
must relinquish gauge invariance at the gluon vertices In so doing gauge invariance in the full theory has not been affected; /~, Gy, ysq; is an external operator This is unlike the case where three gauge
bosons interact via a triangle diagram (fig 9) These sort of anomalies can destroy gauge invariance [3.9] (and renormalizability) and must be removed (by cancellation) in order to have a sensible theorỵ Such
anomalies are known to occur in the Weinberg-Salam model and many grand unified theories
Trang 37Just as in QED, the U(1) axial charge generator is derivable from the symmetry current [3.8, 3.10] and not the gauge invariant current The charge
Qs = | dỶx J 5s, sym(X) (3.26)
is conserved (in the chiral limit), is renormalization group invariant and produces ‘chiralities’, ịẹ commutators, which are gauge invariant Although Q5 is not gauge invariant, its commutators are [3.10] It is the commutators which actually occur in Ward identities and are the correct physical objects to consider.* Furthermore, the charges (2.3) defined by limỵ Jy d°xfo(x, f) diverge as V>~ [3.11] This divergence is absent in commutators because of the commutativity at space-like separations.** Also, Q§ produces all of the correct commutators with the fermion and gauge fields In particular,
[O4, H(A,, A,,)] =0, as required
On the other hand X§=f d’xJ6,(x) is not renormalization group invariant and thus produces chiralities which are not independent of the strong coupling constant g Further X§ has non-vanishing commutators with the gluonic field operators The generalization of the anomalous commutators (3.20) are obtained by replacing a,, j,s and f2, by AZ, Jos and F%, respectively and multiplying the RHS of (3.20) by L
Once again the temptation of associating Ji5 with Gy, ysq should be avoided Such an identification is at ođs with the anomalous commutators, which are actually more suggestive that Js, ym behaves like Gy, ysq The best philosophy to adopt is to think in terms of renormalized or composite operators As Q£ =0, the U(1) axial symmetry is not explicitly broken Claims that the anomaly breaks U(1) axial symmetry are necessarily based on misidentifying the U(1) axial charge with X5 and observing
that Xš # 0 The U(1) axial symmetry is a symmetry at both the classical and the quantum level By repeating the argument given in section 2.2.2, we can easily derive the U(1) axial Ward identity [3.10] f ft | d1x AST is, sym(x) Op(O)) = | d'x TDi (x) Op) + (Os, Oper.) (3.27) where OSs, sym = 21 > MiGi¥sqi = DL ¿=1 The equation for J„s can then be obtained on using (3.25) (3.28) Note that the anomaly term J d*x a2 T(K, (x) Op(0)) is not the same as f d*x T((g?/327)F « F(x) Op)
If we had attempted to bring the derivative inside the time-ordering, equal-time commutators of Op(0)
with f d°x Ko(x) would have been produced
* Goa allen cactinn 29 7 we aso Secuon 22a
Trang 38In the remainder of this section we briefly comment on Fujikawa’s recent derivation [3.12] of the anomaly and the anomalous Ward identities from the path integral method of quantization [3.13] The starting point is the QCD generating functional*
Z2 .1)= | (d3 44] ldA,]exp|t | dẺx (#aco(x)+8n)3 ñ4@)+ 7 At@)l] 629
The idea is to perform a change of variables in the functional integral corresponding to the infinitesimal transformations (B <1)
q(x) expti B(x) ys} g(x) = (1+ i B(x) ys) g(x)
G(x) > G(x) expti B(x) ys} = G(x) (1 +i B(x) ys) (3.30)
These transformations resemble local versions of the U(1) axial transformations (2.1) but any direct correspondence with them should be avoided; for the present purposes they are nothing more than a change of variables
Under the transformations (3.30)
ocp(#)—> ZQcp(x)~ 21 8(x) a mÑ¡ys G(X) — (0"B(X)) Gyu¥s 4(x) (3.31) Ận(x)+ 1 4(x)> 4 n(x) +t 74(x)+ i B(x) Gys n(x) + i B(x) Hrs Q(x) (3.32) while the measure [3.12] m x
where the ¢,, are some complete set of fermionic eigenfunctions used to define the measurẹ The
negative sign in the exponent of G is a consequence of the definition o rassmann integratior
[3.14] The result of Fujikawa is that the anomaly is contained in the exponential factor of (3.33)
On using the completeness of the ¢,, the “anomaly factor’ > bm(X) 5 bm (X) (3.34) becomes
(tr yz) lim ô*(x - y) (3.35)
must regularize and then renormalizẹ
Trang 39
Fujikawa [3.12] does this by suppressing the sum over the large eigenvalues of ị The result of
doing this is that under (3.30)
{ tag da) { {ag dq) exp{-3E% { atx ate) Fa, Fes} 6.36)
Under the change of variables (3.30) in the functional integral (see (3.31), (3.32) and (3.36)) the generating functional (3.29) becomes
ZU, in) = | [6g dq) [AZ] exp(i | Ax { aco) - BX) Dil)
(BUD drarsals)* Has) + 418) + 18 AMC) +i BLS Asal)
+i B(x) dysn0)~ B(x) 585 Fs F(a) 637
Since this is only a change of variables this expression is independent of 8 The vanishing of the first functional derivative of (3.37) w.r.t B(y), at 8 =0, derives the equation
0=(—D/)¡a,a + 9547.754) an — 2LAS (KY) i,5.0 + ấy: 1†(y) + ifs 9) j.4.0 (3.38) where the source subscripts mean that the amplitudes are calculated in their presencẹ We have also made use of the general property of path integrals | ld] oy (oY) et | #@)}= a | Id¿] /(@0)) et] #@)}= dy Fo) (3.39) in deriving (3.38) The result (3.38) is an alternative form of the general anomalous Ward identity (3.28) The simplest Case 7 = n = 9 = 0 gives ala — y\i*u A 2>^2â n P ơ- he 412
U LẺ s arOu VV dig GC U ODta Ca DY U s
respect to j n and n The Ward identity G 28) with some operator Op(0) = Opl2(0) 200) À2 “(0)] ‹ can be obtained by acting on (3.38) with
oojL_Š_ 1 § 1 7 PL in) 1 SH) 1 8)70)!
Trang 40
is the anomalous Ward identity (3.28) with the commutator explicitly evaluated Comparing this with (3.28) confirms that QS is indeed the U(1) axial charge generator Furthermore, the path integral approach automatically replaces the term with d“J,,; m inside the time-ordering with the divergence obtained on using the naive equations of motion
The other chiral Ward identities (2.41) can also be derived in this approach by simply replacing (3.30) with the corresponding local transformations For vector transformations q and @ transform with opposite sign exponentials, in which case the fermionic measure is invariant and there are no anomalies For the SU(L) axial symmetries, anomalies do not appear because of the tracelessness of the A matrices By taking 6 = constant in (3.37) it appears that one can derive (3.28) without the term on the LHS This is however incorrect Suppose we were to do the same thing for the other SU(L) axial currents In the chiral limit we would then derive (2.21), without the term on the LHS But we know that this term is necessarily non-vanishing due to the presence of massless pions in the chiral limit In other words results based on Ø = constant are untrustworthy; important surface terms are obviously lost
This is not quite the end of the storỵ It is important to understand how one can derive (3.27) directly, without recourse to (3.25) From the arguments in perturbation theory we know that this should be possiblẹ Fujikawa’s insistence on using the eigenfunctions of 19 and cutting off its large eigen- values in the sum (3.34) is arguably a gauge invariant renormalization As iY transforms covariantly under gauge transformations its eigenvalues are gauge invariant If on the other hand we were to expand the fermion fields in terms of the eigenfunctions of Lế and renormalize (3.34) by suppressing its large eigenvalues, the anomaly factor would vanish [3.15,3.12] Because 14 does not transform covariantly under gauge transformations its eigenvalues are not expected to be gauge invariant, making this procedure gauge dependent The reason why this approach is chirally invariant is only known a posteriori and not a priorị The same is true in perturbation theory and the point splitting method
LO}, Cr appro One TEQuire JED C WINE aClOr O € WTOIE Ờ
The repetition of Fujikawa’s argument with the bare Lagrangian in (3.29) suggests the validity of the
result (3.38) fo all orders in g [3.12] This is a very formal argument that is compatible with the Adler-Bardeen theorem [3.6] Fujikawa’s approach is also very interesting because it allows a simple
derivation of the anomaly that is In essence non-perturbativẹ For some recent applications of this method to other anomalies see ref [3.17] 4 The U(1) problem 4.1 Preanomaly exposition of the U(1) problem
Under the assumption that there is no anomaly, the consequences of the U(1) axial symmetry are
devastating; the accompanying theoretical predictions are in total disagreement with experime TIS 1
the preanomaly version of the U(1) problem
Let us suppose that there is no anomaly and see what these disasters arẹ We assume that chiral U(L) x U(L) symmetry is spontaneously broken down to U(L)y symmetry by the condensate (2.20) On
the L = 2 level this would imply the existence of a fourth NG boson (in the chiral limit: m, = m2 =0) in ađition to the pion isotriplet Non-zero quark masses would then give this isoscalar a mass compatible