Volume 85B, number 4 PHYSICS LETTERS 27 August 1979

AN EFFECTIVE LAGRANGIAN WITH NO U(1) PROBLEM IN CP n - 1 MODELS AND OCD

P. DI VECCHIA CERN, Geneva, Switzerland

Received 11 June 1979

We write an effective lagrangian which gives two-point Green's functions satisfying the anomalous Ward identities for the U(1) axial vector current with a singlet particle that has a non-vanishing mass in the chiral limit. We show that the mech- anism that has been postulated by Witten and Veneziano for solving the U(I) problem in the framework of the 1/n expan- sion in QCD is fully active in the two-dimensional CP n-1 model where the 1In expansion can be explicitly performed.

The U(1) problem is one of the long standing prob- lems in hadron physics [ 1 ]. Recently, in two very in- teresting papers by Witten [2] and Veneziano [3] it has been shown that under certain assumptions in the framework of the 1/n expansion o f QCD the singlet acquires a nonvanishing mass in the chiral limit. More- over, the existence of a singlet with a nonvanishing mass is also needed in order to eliminate the dependence on the 0 angle in massless QCD [2]. An algebraic saturation of the anomalous Ward identities without a Goldstone boson in the singlet channel hasalso been achieved [3]. Finally, low-energy theorems for the emission of the singlet have been derived in the limit n -+ ~o [2,3].

In this letter we write down an effective lagrangian giving two-point Green's functions that satisfy the anomalous Ward identities without a massless singlet. We then analyse in some detail the U(1) problem in cpn - 1 models where the 1/n expansion has been ex- plicitly performed [4,5] and we show that this problem is solved in these models in exactly the same way as it has been postulated for QCD [2,3].

It is well known that in QCD the singlet axial vec- tor current has an anomaly proportional to the density of topological charge. One finds that the divergence of the axial vector current is given by * 1 :

ouaLu5 = 2iLq(x) + DL(X), (1)

,1 We follow the notations of ref. [1] except that we work in euclidean space (eq. (3) differs by a factor 2L).

where

q(x) = (g2/321r2)Fuv(x)ff uv(x) , (2)

and D L is a soft term that is vanishing when the quark masses are zero. L is the number of light quark fla- yours (assumed to have a common mass m -+ 0). The anomaly term can be written as the divergence o f the following current:

K u = 2L(g2/167r2)euat~Aa(0~Aa 7 + ~gfabcA~AcT). (3)

One gets:

O uKu = 2Lq(x) . (4)

The absence of a massless singlet implies the following Ward identities:

f d4x (2iLq(x)D L (0))

= - f d4x (DL(X)DL(O)) + 4(AL) , (5)

f d4x (19 L (x)2iLq(O))

= - f d 4 x (2iLq(x)2iLq(O)). (6)

The satisfaction o f these Ward identities is a necessary condition for the resolution o f the U(1) problem [1]. Introducing an axial four-vector ghost Veneziano has proved that eqs. (5) and (6) are satisfied.

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It is possible to write down an effective lagrangian involving the fields appearing in the Ward identities that summarizes Veneziano's result. One gets the fol- lowing effective lagrangian:

.t? = (2F2(m 2 - mn2s)) -1 (auKu) 2

+ ½ [(OuS) 2 + m2s $2] + ( i /Fs)SbuKu, (7)

where S is the singlet field which is related to D L by the following relation:

D L (x) = m2sFsS(x) , (8)

m s (mns) is the mass of the singlet (nonsinglet) and in the case o f L = 2, runs is equal to the pion mass. F s is the decay constant of the singlet defined by the following relation:

(SIJL510) = - iq u F s . (9)

From lagrangian (7) one can easily compute the two- point Green's functions. One gets:

~S(x)S(y) )=f dap eip(x-Y) 1 (10)

2 2 (S(x)Ku(y))= f d4p e ip(x_y)Fs(ms - mns)P ~

(2tr)4 p2 + m 2 p 2 ( l l )

(12) p2 + m2ns 2 Pu pv = f d4p e ip(x-y) 2 2 _ _ p2 m ~ + Fs (ms - runs) p4 "

~(27 t ) 4

It is easy to check that those Green's functions satu- rate the Ward identities (5) and (6). The vacuum ex- pectation value of A L can be computed from the Ward identity for the non-singlet current:

f d4x a b _ (auJtt 5 (x)[ lJu 5 (0)) - 8ab(A2 ) , (13)

which gives:

(A2) = 2 2 mnsFns , (14)

after having used the SU(L) symmetry that implies:

(AL)= {L(A2)= 1_ 2 ~2 (15) Lm nsrns .

Notice that the propagators (10), (11) and (12) have a pole at the mass of the singlet. Those involving Ku have also a pole at p2 = 0 which is a consequence of

a modified Kogut-Susskind mechanism [6] as has been shown in ref. [3]. This pole at p2 = 0, however, only appears in Green's functions involving fields which are gauge variant like K u. auK ~ is gauge in- variant and Green's functions involving it do not have any pole at p2 = 0. Finally as has been shown in refs. [2,3], consistency in the framework of the 1/n ex- pansion requires that the parameters F s and m 2 - m2s appearing in eq. (7) must be related to fd~lx × (q(x)q(0)) of the pure Yang-Mills theory. The pre- cise relation which can be immediately obtained from eqs. (4) and (12) is the following:

f dax (q(x)q(O))Yang_MiUs = (F2s/4L 2)(ms2 - m2s) . (16)

In the following we review some of the relevant properties of the CP n - 1 models and we show that the U(1) problem, that was already analysed in refs. [4,5], is solved in these models in exactly the same way as it has been postulated in QCD in refs. [2,3].

The CP n - 1 model with quarks is defined by the following lagrangian:

.12 = D uzD uz + -~ (1~ - MB) ~ + (e2 /2n)f( ~'y~ ~b ) 2

_ (gv/2n)[(~rit~)2 + (~r&/5 ~)2] , (17)

where D u = a u + ( i /x /~X u when acting on z and D~ = a u + (i/;v/n)eXu when acting on ~. One has in addition also the constraint:

Izl 2 = n/2 f . (18)

We follow the notations of ref. [4] but we do not give any "colour" index to the fermions and there- fore the constraint ~ • ~ = 0 is not imposed. The field z a has a "colour" index a that runs from 1 to n, while the quark field ~a has only a flavour index a that goes from 1 to L. In this way as in QCD a fer- mion loop is down by a factor L/n with respect to a boson loop in the limit L, n ~ oo and L/n fixed and small.

The interest for the CP n - 1 model in two dimen- sions stems from the fact that it has many properties in common with QCD in four dimensions. We list some of them:

(i) Lagrangian (17) contains no dimensional cou- pling constant. It is conformal invariant at the clas- sical level.

(ii) Asymptotic freedom and dimensional trans-

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mutation. (iii) Non-trivial topology for any value of n. In addition one can prove that there is a spontane-

ous mass generation for both the parton field z and the quark field ~b and that chiral symmetry is spon- taneously broken. The vacuum expectation value of ~ is given by:

~gv f ¢) = L M , (19)

where M is the spontaneously generated quark mass. The properties of the CP n - 1 model can be studied

by expanding the Green's functions for large values of n and L. This expansion has been already perform- ed in refs. [4,5]. One gets an effective lagrangian in- volving some composite fields made up of the funda- mental z and qJ fields. In the low energy region the effective lagrangian becomes the following:

= ~.F2 + 1 [(0zS) 2 + m2sS2]

1 [(azrri)2 + mn2s(rri)2] + ieF" Sx/L/rrn, (20) +~

where F = euvbu~ v, S is the singlet and 7r i is the non- singlet. The constant a is given by

ot = (1 + 2(L/n)e2m2/M2)/Z4rrm 2 , (21)

m 2 (M 2) is the mass spontaneously generated for t h e z (~k) field. As has been shown in ref. [4] the low en- ergy effective lagrangian contains also other fields, that we omit here because they are not relevant for the analysis of the U(1) problem. As in the case of QCD the divergence of the U(1) axial vector current has an anomaly that is proportional to the density of topological charge. One gets that:

aUaL5 = 2ieLq(x) + m2sFs S , (22)

where

q(x) = (27rx/h--)- 1F. (23)

In terms of the following field:

K u = 2eL(2rrx/~- 1 eu vXv , (24)

the effective lagrangian (20) takes exactly the same form as eq. (7) except for the "pion" term which has been omitted in eq. (7). In addition one gets an ex- plicit expression for the parameters F s and m 2 - m2s in terms of the fundamental parameters of the theory. They are given by

F s = x / ~ , (25)

ms2 _ m2s=(12e2m2L/n)[1 + (2L/n)e2m2/M2]_l

Remember that in two dimensions F s is dimensionless. One gets therefore the same two-point functions

as eqs. (10), (11) and (12) and they satisfy of course the anomalous Ward identities with a singlet, that is not massless in the chiral limit. Therefore, the mech- anism for avoiding a Goldstone boson in the singlet channel is quite the same in the CP n - 1 model as in QCD.

Also relation (16), which has been postulated in QCD for satisfying the Ward identities and for elimi- nating the dependence on the 0 angle in the massless theory, is valid in the CP n - 1 model. One finds that

f d2x (q(x)q(0))without quarks

= 3 m 2 = ~ F 2 ( m 2 - m 2 s ) , (26) mr 4L2e 2 s

in the limit L/n ,~ 1. This is a relation that gives the product F2(m2 s - m2ns)in terms of the theory with- out quarks. Another interesting relation that can be obtained from eq. (26) is the following:

2 f (q(x)q(O))with quarks d2x = mns (27)

f (q(x)q(O))without quarks d2x m2

For the sake of completeness it would be nice to check the saturation of the anomalous Ward identi- ties for the general N-point Green's function and in particular to check the 0 dependence of certain Green's functions as (~qJ)0 and (F) 0 . Work in this di- rection is in progress and we hope to report on this problem in a future publication.

We thank G. Veneziano for communicating his re- suits prior to publication and for many discussions that strongly stimulated the present analysis.

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References

[1] For a review of the difficulties in solving the U(1) prob- lem see: R. Crewther, Status of the U(1) problem, CERN preprint TH 2546 (1978); see also R. Crewther, Phys. Lett. 70B (1977) 349.

[2] E. Witten, Harvard Univ. preprint (1979). [3] G. Veneziano, CERN preprint TH. 2651 (1979). [4] A. D'Adda, P. Di Vecchia and M. Liischer, Nucl. Phys.

B146 (1978) 63; DESY preprint 78•75 (1978). [5] E. Witten, Nucl. Phys. B149 (1979) 285. [6] J. Kogut and L. Susskind, Phys. Rev. Dl l (1975) 3594;

see also S. Weinberg, Phys. Rev. Dl l (1975) 3583.

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