The U(1) problem and translational invariance

Volume 95B, number 3,4

PHYSICS LETTERS

6 October 1980

THE U(1) PROBLEM AND TRANSLATIONAL INVARIANCE S. CECOTTI Scuola Normale Superiore, INFN, Pisa, Italy Received 20 June 1980 We show that the equivalence of a chiral U(1) transformation with a change in 0 is independent of the fate of the U(1) boson, being a consequence of the translational invariance. Assuming that there is no accidental degeneracy the U(1) problem is solved.

Recently, Crewther [1 ], Di Vecchia and Veneziano [2] and Witten [3] have shown that it is possible to saturate the Ward identities relevant for the U(1) problem [4] without introducing an U(1) "Goldstone boson" with a mass o f order O(m~). In particular, the last three authors [2, 3] have constructed, using symmetry requirements and 1IN reasoning, an effective lagrangian giving explicitly the current algebra required for the solution o f the U(1) problem. However, also after these fundamental contributions, the fact that an eventual U(1) boson has not an O ( m . ) mass seems to be an independent assumption, based only on phenomenological considerations [5], corresponding to a rather particular realization o f the anomalous chiral U(1) symmetry. This hypothesis is explicit in the papers b y Crewther [1 ] while in the papers o f Di Vecchia and Veneziano [2] it is assumed that a chiral U(1) rotation is equivalent to a change in the vacuum angle 0 [6] ; i.e. it is supposed that a U(1) transformation changes the QCD action by f d 4 x L Q c D ( x ) -*

f d4xLeCD(X)

327r 2

fd4xFu~,FU"(x),

*1 The corresponding condition in the language of the ref. [2] is a/N ~ 0 where a is defined by the equation [2] f d4x T(O = 01Q(x)Q(0)I0 = 0)[pur e Yang-Mills

g2 + Lob

o f this identification (in a sense to be specificated b y eq. (18)) is guaranteed b y the translational invariance o f the theory and hence it is not necessary to assume a priori that the U(1) particle, if it exists, has a 0 ( 1 ) mass in order to construct the effective lagrangian o f refs. [2] and [3]. Moreover, if we suppose that there is no accidental degeneracy o f the chiral symmetry breaking parameters ,1, then Crewther's argument can be inverted and we can show explicitly that there is no pole at p2 = O(m 2) in the channel o f the would-be U(1) boson. In this way, the conclusions o f refs. [ 1 - 3 ] are confirmed and reinforced from a more general point of view. We consider the super-potential cl~ (ui, v i ; 0) [7] as a function o f the independent variables u i and v i equal in value to * 2

(1)

where ~ is the chiral U(1) angle and L is the number o f the light quark flavours. But, as Crewther [1,4] has emphasized, this identification o f the 0 angle with the chiral U(1) angle is i t s e l f a consequence o f the assumption that there are no O(mTr ) mass bosons in the singlet channel. The point o f this letter is to show that the validity

= i(F~/2N)a. .2 Note that u i and v i are ill-defined by eq. (2) because of the singularity in the product (0 rqi(x)qi(x + e)l O) for e ~ 0. In order to give a precise definition of u i and oi one must identfficate the singular part for e --. 0 (using asymptotic freedom) and then subtract this contribution (this procedure does not change the Ward identities). This subtraction is formally implemented by a particular choice of the chiral sources in the functional integral; it is in terms of this subtracted functional integral that our quantities are defined. Note that the previous procedure does not alter the leading order in chiral expansion, the only relevant for our discussion 401

Volume 95B, number 3,4

PHYSICS LETTERS where

u i = (0 IqiqilO> IL(Mi,Ni), oi = (0 I~iTSqi IO>IL(M¢,Nz),

(2)

where the symbol IL(Mi,Ni) means that the 0-vacuum expectation values are calculated from the lagrangian

L(Mi, Art.; O) = -~FuuFUV + #iOq

- ~ (li(Mi + Ni75)loqi - Og2 Fuvffuv i=4

~

'

i

qiT~/5qi(x))IGauge_invarian t

~rl 2-1

Oc))(u,o;O)_Ni 0 Ooi .

XI2-1

~,s (x) = ~,y~,~s ~ j - q ( x ) , l, 0 ..... 0).

The translational invariance of 0-vacuum expectation values (of gauge-invariant operators) for the theory specified b y t h e lagrangian (3) requires [8] (4)

As is well known [7], cy is defined in terms of the Legendre transform P[Ui(x), Vi(x); 0] of the generating functional of the Green's functions of the form *a T<0I .[I ~ijqi/(xj) ~k ~TikT5qik(Xk)lO) ]

(9)

(3)

where M i and N i are given by

Ocl)(u,v;O)=Mi 0 Oui ,

J~5(x)=(~i and

x,~_1=[__~2 11/2diag(1 ..... 1, 1 Ll(l - 1).]

L

i

6 October 1980

(5)

= 0

(10)

and

(OlaUTt~-l(x)lO> = 0 .

(11)

We take suitable linear combinations of eqs. (10) and (11), and use eqs. (2), (7) and (8) to obtain

- <0 INj~/qjlO> by the equation

P[ Ui(x), Vi(x); 0] = -(27r)454(pf-Pi) cl) (ui, °i ; 0). (6) It is convenient to study the chiral properties of the lagrangian (3) for arbitrary values of the mass-parameters M i and N i because our independent variables u i and oi are (in value) equal to the expectation values (2) calculated from this lagrangian. The equations of motion for the axial currents, diagonal in the flavour indices, deduced from the lagrangian (3) are n = 2ng___~ 2 ,- ~,.z,- - + 2 i ~ Mi?li75qi ~uJ~5(x) 327r2 % v r ~ ~x) i=l

= -- (0 INk~kq~lO>, j , k = 1 ...L

(12)

and L


_

Lg 2

(OlfuvffuvlO>.

(13)

32~r2 We define ~bi = arg[ui + ivi]

(14)

n

- 2 ~ Ni~iqi, i=1

n = 1 ... L ,

(7)

and, using eq. (4), we get a~(u,o;0)

and l-1 12-1 = [2/l(1- 1)] 1/2 { ~1 [iMi#iT5qi-Ni~iqi] u~t F~S

-(l-

1)[iMtYtl'rSqt-Nlgltqt] } ,

l = 2 . . . L , (8)

+a We adopt a mass-independentrenormalization procedure. Cf. the discussion in the second paper of ref. [4]. 402

(° iu i

-o i-

(u i,vi;O)

= N/uf - ]V~ o] .

(15)

Comparing the r.h.s, of eq. (15) with eqs. (12) and (13) we have the differential equations a ~ ( u , o; 0)

~C~(u, o; 0) = 0 ,

j,k = 1 ...L,

L L OW(u, v; 0) + ~ aq~(u, o; 0) _- 0 ,

(16)

Volume 95B, number 3,4

PHYSICS LETTERS

where we have used that, as a consequence of eqs. (4) and (6) • 3 c ) ) ( u , v; 0 ) _

]

g2

~

327r2 ( O I F u v F u v I O ) L ( M i , N i ) . (17)

30

Integration of eq. (16) yields c~(ff i COS ~bi , t~i sin ~bi ; 0) = Q~ [ffi cos(~bi + ai) , ffi sin(~bi + oti) ; 0 + ~_,ioti] . (18)

Eq. (18) has a simple meaning: the chiral U(1) angle equals the 0 parameter in the chiral limit. It is well known [1,4] that this statement is a typical consequence of the absence of the U(1) boson; however, here it has been derived on general grounds, without making such an hypothesis. The actual values of ffi and ~bi in 0-vacuum (indicating the chiral U(L) × U(L) symmetry breaking) are the solution to the equations • aCP(u, v; O)p 1

~tlj

Ui=t~iCOSqbi = % ' ' oi = 0 i sin ~i

i OqY(u,__o;O) ~o]

= u i = a i cos 4~i [ui = ui sin 0i

0

(19)

We consider first the case of an exact chiral symmetry, i.e. M i = 0 for all i. In this case the chiral symmetry breaking parameters (0 [qiqi [0) and (0 [~iTSqi [0) are simply the values of u i and v i that minimize the function q.~ (ui, vi; O) for the 0 considered. For 0 = 0, because o f CP-invariance of the vacuum (which, in turn, implies that the non-chiral SU(L) group leaves the vacuum unchanged) the breaking parameters are (up to a chiral SU(L) X SU(L) transformation) ui = ffi = C :~ 0 ,

~bi = 0 ,

all i .

(20)

For 0 4= 0 we consider u i = C cos ~i,

vi = C sin ~i,

(and also to an absolute minimum, if the solution (20) is an absolute minimum for 0 = 0). Hence, eqs. (21), (22) are a solution of eq. (19) with M i = 0. Note that the condition (22) does not fix uniquely the solution; this multiplicity in the solution corresponds to determining the breaking parameters up to a chiral SU(L) × SU(L) transformation. Indeed, the first of eqs. (16), which fixes the direction of the chiral symmetry breaking (Dashen's theorem [8] ), is identically satisfied for M i = 0 because of eq. (19). In general, there may be further accidental degeneracy of the solutions of eqs. (19). For example, the u i and v i o f e q . (21) may not depend at all on 0 ; we shall suppose for the rest of the paper that this is not the case for QCD. If the masses M i do not vanish, the argument goes on exactly in the same way; with the previous hypothesis we can show, using standard arguments, that the corrections to the eqs. (21), (22) are of order O(M). There is only a subtlety to take into account; we must define correctly the direction of the 0-vacuum in the chiral group at the zero order in the chiral perturbation theory [1,8,9]. The correct identification of the unperturbed vacuum is given by Dashen's theorem [8], i.e. by the first of eqs. (16). Using eq. (19), we obtain the system of equations M i sin 4~i = independent o f / = X ,

i

with 0 = ~ Oi • (22) i Because o f eq. (18), the value of c-y(ui, vi;O ) calculated from the u i and u i of eqs. (21), (22) is equal to the value ofC~(ui, u i ; 0 = 0) as calculated from the u i and v i of eq. (20); then the values (21), (22) of the breaking parameters correspond to a minimum of qY

(23a)

cki = 0 + O [ M ( a 2 qY/aO2to=o, Mi=O)-I ] .

(238)

Eqs. (23) are identical to those obtained by Crewther [1 ] and others [2,3] in the hypothesis that there is no U(1) boson pole contribution in the leading order of chiral expansion. In refs. [ 1 - 3 ] there is a complete analysis of eqs. (23) and of their consequences. Here we limit ourselves to show how the non-existence of the U(1) boson follows from the eqs. (23) and the no-degeneracy assumption a2~/302

(21)

6 October 1980

4= 0 , 4

,4 There is a more direct, but less elegant proof. Indeed, because of eq. (18), a2C')~I _ a2z a'z aNjl

1.2m~'l

LMi 0

.

Mi=O

(z is the connected Green's function generator with constant sources). This equation shows directly that there is no massless particle in the isoscalar channel. Note that in the leading 1/N order [2, 3] m~/L = a/N and hence we reobtain the condition of papers [2,3]. 403

PHYSICS LETTERS

Volume 95B, number 3,4

I thank P. Menotti and G. Veneziano for discussions.

Let AR be A R = ~ ei~bi(qLqR)i" (24) i We note [1] that AR is, for every O, an invariant under the group SU(L)an n, i.e. the SU(L) subgroup of the chiral SU(L) × SU(L) group, which leaves the 0-vacuum unchanged. Consider the anomalous Ward identity

0 = 2Lg2

fd4x r~OIF"f(x) ~R (0)10> - 2(,OIZ~e iO>

327r2

+ fd4x T(OIDL(x)aR(0)10).

(25)

Then, using eqs. (21), (22) and the analogous for Green's functions o f eqs. (17) [1,5] we obtain

fd4x T(O ]DL (x) AR (0) 10)

corm

while, in presence o f a U(1) boson with a mass squared of order O(M) the 1.h.s. o f the eq. (26) would be o f order O(1). This completes our argument.

404

6 October 1980

References [1] R.J. Crewther, Chiral properties of QCD, preprint TH. 2791 -CERN (1979); Quark-mass dependence of gluonic boundary conditions, Bern preprint (1980). [2] P. Di Vecchia and G. Veneziano, Chiral dynamics in the large N limit, preprint TH.2814-CERN (1980). [3] E. Witten, preprint Harvard HUTP-80/A005. [4] R.J. Crewther, Phys. Lett. 70B (1977) 349; Status of the U(1) problem, in: Instantons in field theory, Riv. Nuovo Cimento 2 (1979) No. 8, p. 63; S. Cecotti, Nuovo Cimento 55A (1980) 101. [5] S. Weinberg, Phys. Rev. D l l (1975) 3583. [6] C.G. CaUan, R.F. Dashen and D.J. Gross, Phys. Lett. 63B (1976) 334; R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 8; second paper of ref. [4]. [7] G. Jona-Lasionio, Nuovo Cimento 34 (1964) 1740; for reviews see E.S. Abers and B.W. Lee, Phys. Rep. 9C (1973) 1, and references therein. [8] R. Dashen, Phys. Rev. D3 (1971) 1879; H. Pagels, Phys. Rep. 16C (1975) 219, and references therein, [9] S.P. De Alwis, Phys. Lett. 86B (1979) 67.