Z. Phys. C 62, 511-520 (1994) ZEITSCHRIFT FOR PHYSIK C �9 Springer-Verlag 1994

Dirac fermion in Euclidean metric and the

H. Banerjee 1, P. Mitra 1, D. Chatterjee 2

1Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India 2Vijaygarh Jyotish Ray College, Calcutta 700 032, India

Received: 5 October 1992/In revised form: 1 June 1993

U(1) and strong CP problems

Abstract. In Euclidean metric the usual description of a Dirac fermion needs two independent Dirac fields for O(x) and ~(x). In the instanton sector this leads to the first option for the fermion path integral which vanishes in the chiral limit due to the zero eigenvalues of the Euclidean Dirac operator. We propose here a novel representation for the conjugate field ~(x) which depends on 0(x), has the correct chiral properties, and reproduces all the fermionic Green functions of the Euclidean theory. In the alternative scenario, the chiral limit of FPI coincides with the product of only the non-zero eigenvalues of the singular Dirac operator in instanton sector, and the QCD action is in- variant under global chiral rotation. The anomaly term in effective theory conforming to the alternative scenario breaks chiral symmetry spontaneously and solves the U(1)-problem. A chiral phase in quark mass, which is a possible source of CP violation in the first option, does not have any observable effect in the second. The strong CP problem is a legacy of the first option. In the alter- native scenario CP invariance is ensured in QCD if 0 = 0.

1 Introduction

In Euclidean metric Yang-Mills fields reveal interesting mathematical structures. The structure which seems to have had profound effect on the development of QCD over the years is the instanton configuration characterised by a non-trivial winding number v(~ 0)

9 2 v = l - ~ n 2 ~ Tr(V,, ffu~) d4x. (1.1)

Effects of instanton configurations manifest directly in the QCD action

SQCD---= I ~(49--irn)~k d4x+�88 F~) d4x+OAS (1.2)

through the 0-term

AS= i92 fTr(Fu~/~ ) d4x (1.3) - 167z2 J

If v = 0, i.e., in the absence of instantons, the 0-term drops out. The 0-term is odd under CP and T transformations, and, therefore, a potential source of CP violation in strong interactions.

In the fermion sector the signature of a non-trivial winding number of Yang-Mills fields is the emergence of normalisable zero modes of the Euclidean Dirac operator. In the kernel space, i.e., in the space spanned by eigenfunc- tions corresponding to zero eigenvalues, the Dirac oper- ator D and the 75 matrix can be simultaneously diagonal- ised. This means that the normalisable zero modes are all of definite chirality. The precise relation between the winding number v and the zero modes is given by the Atiyah-Singer index theorem

v=n+ - n _ , (1.4)

where n+ (n_) is the number of normalisable zero modes of positive (negative) chirality.

The fermionic weight of instanton configuration of gluon fields is represented by the fermion path integral (FPI)

Zy (m) = ~ d# exp [ - ~ ~ (49 - ira) 0 d4x], (1.5)

where d/~ denotes the integration measure for Dirac fer- mions. The first and the popular option is to identify the FP! in chiral limit m = 0 with the determinant of the Dirac operator [1-3]:

Zj.(m = 0) = det 49. (1.6)

In non-trivial sector the Euclidean Dirac operator is sin- gular and non-invertible and as a result the fermion propagator is undefined. What is puzzling is that in rela- tivistic Minkowski metric there is no identifiable signature of the zero modes of the Euclidean Dirac operator. The second option proposed sometime ago by Gamboa- Saravi et al. [4] gets around this problem by prescribing the value of the FPI in chiral limit as the product of only the non-zero eigenvalues of the singular Dirac operator

Zi(m = 0) = lim det ( D ? el) = det' 49. (1.7) e ~ 0 /3

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In v=0 sector the two options, (1.6) and (1.7) coincide. Differences manifest only when v r 0. A resolution of the ambivalence between the two options, therefore, holds the key to the physics emanating from instanton configura- tions of gluon fields.

To resolve this ambivalence from first principles the natural starting point is to parametrise the Dirac fields O(x) and O(x) in terms of Grassmann generators [5]. Precisely at this stage, one is confronted with the second ambivalence. Question is whether the Grassmann gener- ators needed to parametrise ~(x) should depend on or be independent of those employed for ~ (x). Stated differently, must we employ two independent Dirac fields for ~ (x) and its conjugate ~(x) to describe a single Dirac fermion in Euclidean metric, or, as in relativistic field theory, the fields ~(x) and ~(x) associated with the Dirac fermion must be related to and derivable from each other? The only guideline, thanks to Osterwalder and Schrader [-6] (OS), is the injunction that in Euclidean metric

?p(x)#~,*(x). (1.8)

This is dictated by the hermiticity properties of the Dirac propagator in Euclidean metric. To implement the injunc- tion (1.8), one possible way, advocated by the authors of [6], is to assume that tp(x) is a new Dirac field indepen- dent of O(x). Although this was mooted originally as a suggestion, it propagated in the literature [7] as the only available option. Deviation [8] from this orthodox atti- tude is a recent phenomenon. An extra Dirac field, in principle, suggests that in the passage from relativistic theory to Euclidean metric the degrees of freedom of a Dirac fermion has doubled [6,8]. This is not only unnatural but unnecessary.

A precise description of a Dirac fermion is funda- mental to the study of FPI. A resolution of the first ambivalence, therefore, depends in a large measure on the position one adopts vis-a-vis the second, viz., whether only one or two Dirac fields are employed to describe a Dirac fermion. We show in Sect. 2 that if O(x) is assumed to be independent of O(x) then the natural result for FPI is the first option (1.6). But this is achieved at a price. The hermitian conjugates of the Green functions are not calcu- lane in the framework of this option. If, on the other hand, the conjugate field ~(x) is required to depend on and be derivable from O(x), and if both O(x) and tp(x) are to transform unitarily under global chiral rotation

~//(x)-+ei~lp(x), ~(x)~ff(x)e i~ (1.9)

then the result for the FPI is, as shown in Sect. 3, the second option (1.7). This alternative scenario for the Dirac fermion reproduces correctly all the fermionic Green func- tions in the Euclidean metric.

An artifice to which recourse is taken almost univer- sally to extract the CP violating effects of AS, is the Baluni trick [9, 10] of implementing a global chiral U(1) rotation (1.9) of fermi fields in FPI. We show in Sect. 4 that the two options respond differently to this innocuous procedure. In the first option the coefficient 0 of AS in (1.2) can be changed at will and for a special choice of the parameter

of chiral rotation, AS can be traded for the Baluni term [9, 10]

A'S = rnS~Ts o d4x. (1.10)

In the second option, however, AS remains unaffected, with the Baluni term (1.10) appearing in the fermion sector as an extra piece. The fact that in the first option the 0-term in QCD transforms anomalously under chiral rotation is the genesis of what 't Hooft [11] calls the U(1)- dilemma, and of the unabated controversy [11, 12] regarding the representation of the "anomaly term" in effective theory. In the alternative scenario for Dirac fer- mion the QCD action is chirally invariant. The breaking of chiral symmetry in effective theory conforming to this scenario is, therefore, spontaneous as recommended by the authors of [12], and not explicit. What is interesting is that the chirally invariant anomaly term, which in effec- tive theory realises the anomaly in axial Ward identity, gives, as shown in Sect. 4, a non-zero mass for the flavour singlet Goldstone boson in the chiral limit and thus solves the U(1)-problem.

The Baluni term (1.10) is unphysical in the framework of the second option and cannot constitute the basis of any observable effect like CP violation. What prevents it from being unphysical in the first option is the presence of fermion zero modes [-13], and not as is often argued [14], the ABJ anomaly in the divergence of axial current. This is elucidated in Sect. 5. Neither perturbative treatment nor current algebraic method [13] not even the state of the art technology of lattice calculation [14] are equal to the task of unravelling the effects of the fermion zero modes which are essentially non-perturbative. This recognition, pre- sumably, has led several authors [15] to discover ambi- guities in the derivations of a large neutron electric dipole moment (NEDM) in effective Lagrangian framework [16]. In any case, the strong CP problem is a legacy of the first option. In Sect. 5 it is argued that in the alterna- tive scenario, QCD with vacuum parameter 0--0 is CP invariant.

2 The Euclidean Dirac fermion and the first option for FPI

In Euclidean metric the Dirac operator defined as

0 - 7.( i0.-gA.) , (2.1)

with 7-matrices obeying the algebra {7., 7v} = 26.v, is her- mitian with real eigen-values 2. and orthonormal eigen- functions 05.(x):

/2)05. (x) = 2. (x) qS. (x), $05*. (x) 05m (X) d4x = 6m.. (2.2)

Each non-zero eigenvalue 2. has its chirally conjugate partner - 2. with eigen-function 05_=

005_,= -2,05_,, 05-,=7505,. (2.3)

In non-trivial sector there will be zero eigenvalues. The normalisable eigenfunctions corresponding to these eigen- values are all of definite chirality. Assume, for definiteness, v = 1 so that there is only one zero eigenvalue with eigen- function 05o of positive chirality

D05o =0, 050 =�89 + 7s)05o �9 (2.4)

The set {qS,(x)} together with q~o(X) constitute a com- plete basis in function space. The Dirac field can be expanded in this basis as

q/(x) = ~ (a, + a_,?5)On(x) + ao(Oo(X), (2.5) ~ > 0

where the sum is over only positive eigenvalues. The four degrees of freedom corresponding to each mode of a Dirac field is accounted for if we split [2] a+n as

a+, = (c~+n + ifi+,) (2.6)

with e+,, fl_+, real-valued Grassmann generators. The mode qSo, which is essentially a Weyl mode, has only two degrees of freedom:

ao = ~o + iflo. (2.7)

The hermitian conjugates a_+, - (e+ n - 1/3+,), ao - (% - i3o) are the involutions [5] of a+, and ao, respectively. The generators, therefore, define a Grassmann algebra with involution [5].

It is convenient to substitute )C(x) for ~(x) in the fermion action

St(m)=-~Zt(x)(D-im)O(x) d4x, Zt(x)=-~(x). (2.8)

The second ambivalence is about whether to make an ansatz of the form

Z(x)=O(x)O(x), (2.9)

where the only constraint imposed by the OS injunction (1.8) is that the operator O(x) should be non-trivial. SO(4) and gauge invariance are implemented by an O(x) inva- riant under these transformations. In [6], Osterwalder and Schrader recommended the alternative scenario wherein Z(x) is independent of ~k (x) and a relation like (2.9) does not exist. In the OS scenario [6] the conjugate Dirac field admits of an expansion exactly analogous to (2.5):

Z(x)= ~ (b,+b-nTs)gPn(x)+boOo(X), (2.10) ~n>0

with a new set of complex-valued generators {b}, unre- lated to the set {a} employed for ~,(x). The hermitian conjugates b*,, b* are, as before, the involutions in the Grassmann algebra [5]. The formula (2.8) for the fermion action yields the result

St(m)= ~' [(2,-im)b*a,

+ ( - An - ira)b*_, a_ ,] - imb*ao, (2.11)

where the subsets {b} and {a*} are absent. To integrate over the subsets of generators {a} and {b*} which appear in the action one introduces the corresponding truncated measure

d# = db* dao l-[ db* da, db*_ n da_ n, A~>0

-= l-I db* da (2.12)

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and obtains

d/~ exp [ - St(m)] = det(0 - ira). (2.13)

In chiral limit this agrees with the first option (1.6) for FPI. In the OS scenario [6] the number of effective degrees

of freedom is reduced by a factor of half by excluding essentially the operation of hermitian conjugation in the fermion sector. This is done in two steps. First, only the component St(m) instead of the full hermitian form [St(m)+S~(m)] is considered as the fermion action. The second step consists in integrating with the truncated measure (2.12) instead of the full measure 1-[ db* db da* da. In the absence of these prescriptions of truncating both the action and the measure the result would be det(O z -I-m 2) instead of (2.13) as expected for the FPI for two independent Dirac fields with requisite number of dynamical degrees of freedom. An obvious fall-out of the loss of the operation of hermitian conjugation is that the two-point function <)~(x)~ t(y)>, which is hermitian conju- gate to the fermion propagator <O(y)zt(x)>, is incalcu- lable in the OS scenario. The degrees of freedom, i.e., the Grassmann generators, associated with Z(x) and ~ t(y) are not dynamical. Recall that the injunction (1.8), which is the genesis of the OS scenario [6], has its origin in the mismatch of the hermiticity properties of the Green func- tion @(y)~t(x)>, and the perturbative Dirac propagator in Euclidean metric. It is, therefore, rather strange that precisely this hermitian conjugate, <X(x)0*(y)), of the fermion propagator should be incalculable in the OS scenario.

3 The Euclidean Dirac fermion and the second option for FPI

In the OS scenario [61 the unsatisfactory features arising from the definition of the conjugate Dirac field manifest already in the trivial v = 0 sector. They have nothing to do with fermion zero modes. Let us first assure ourselves that there exists a representation for )~(x) free from these blem- ishes.

First observe that an infinitesimal unitary chiral rota- tion

O(x)-,(1 +i0~)~(x) (3.1)

induces among the generators for O(x) a transformation

an~an+iOa-,, a_,~a_n+iOa,. (3.2)

Conversely, the transformation (3.2) of the generators implements a unitary chirat rotation (3.1) for O(x). It is now easy to verify from (3.2) that the combination (an - a_nT~) transforms as

(an - a _,Ts)--*(a, - a _ ,Ts)(1 - i0? 2).

This means that the representation [17]

Z(x)= ~ (a,-a_nTs)c~,(x) (3.3a) An>O

realises for the conjugate field the desired transformation Z(x)--*(1-i075))~(x) for infinitesimal chiral rotation.

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One can motivate the representation (3.3a) also from an analogy with the formula X(x)=7oO(X) valid in Minkowski metric. The opposite chiral transformation property of Z(x) is realised through the 7o matrix. In Euclidean metric, where no distinction is to be made between time and space components, the opposite chiral property of the conjugate field is achieved by substituting for Yo a matrix of the form Pu~, with P~ a suitably nor- malised four vector. The twin requirements of SO(4) and gauge invariance lead to the unique choice PuocDu, the covariant derivative, and hence to the representation [17]

1 X(x) = lira ~ (02 + ~2)1/2 O0(x). (3.4)

for the conjugate field. The infinitesimal parameter e en- sures that the normalisation factor makes sense even in the kernel space of 0- One easily verifies the representa- tion (3.3a) from (3.4).

Substituting (3.3a) for 7~(x) in the formula (2.8) for fermion action, one obtains

SE(m)= ~ {2,(a*a,+a*_,a_,) ~,,~ >- 0

- i m ( a * a , - a*_,a_,)}. (3.5a)

Integration with the full fermion measure appropriate for the single Dirac field ~k(x)

d# = 1-] da* da, da*, da_, (3.6) 3,n>O

gives the desired result for FPI in the trivial v--0 sector

Z~=o(m)=-~d#exp(-S~(m))= 1-[ (2,2+mZ) �9 (3.7a) ~,n>0

In non-trivial v C0 sector the central question is whether the zero modes can appear in the representation for )~(x). In the present framework, where Z(x) depends on and is derived from O(x), and, further, has the opposite chiral transformation property there is no way for zero modes to appear in Z(x). A priori, this should be evident from the formula (3.4). The Dirac operator 0 annihilates the zero modes, if any, present in ~,(x). To probe further, note that each zero mode is really a Weyl mode endowed with just two degrees of freedom. For the positive chirality Weyl mode q~o(X), for example, these two degrees of free- dom are already accounted for by ao, the complex-valued generator (2.7) in the representation (2.5). Under chiral rotation ao acquires a phase ao-~ao(1 +it0. If ~bo(X) were to appear in )~(x) then, according to (2.9), we must have

(1 - ie) O (x) = O (x) (1 + ic 0 . (3.8)

to implement the opposite chiral transformation property for X(x). This, obviously, is not possible. In the case of the contribution from a Dirac mode qS,(x), however, reversal of chiral property is not a problem. This is because each Dirac mode comprises two Weyl modes of opposite chirality. The Grassmann generators corresponding to the two Weyl modes can be swapped to realise the chiral property appropriate for the contribution from the Dirac mode ~b,(x) to the conjugate field Z(x). In the OS scenario [6], this problem is a non-issue, simply because the conju-

gate field is independent of O(x) and one is free to invent the generators for Z(x) without any prejudice for the generators employed for O(x).

The virtue of the principles underlying the construc- tion of the conjugate field, viz., (a) t:(x) must depend on O(x), but (b) have chiral transformation properties oppo- site to that of O(x), is now transparent. Even in the presence of the mass term the degrees of freedom ao, a* associated with fermion zero modes do not qualify as dynamical, and Z(x) is represented by a formula, like (3.3a), devoid of zero modes

Z(x)~eo = ~ (a,-a-,Ts)(o,(x). (3.36) s

As a result, the fermion action (2.8) depends, as in (3.5a), on the degrees of freedom associated with modes corres- ponding only to non-zero eigenvalues

S~(m)~eo= ~ {2,(a*a,+a*_.a_,) ~,n>0

- im(a*a,- a*,a ,)}. (3.5b)

The fermion measure corresponding to this action is given again by a formula similar to (3.6) and yields for FPI the result

Zv,o(m)=--Sd#exp(--SE(m))= I-I (2,Z+m2), (3.7b) 2 . > 0

which depends only on non-zero eigenvalues of the singu- lar Dirac operator. In the chiral limit (m=0) the result coincides with the second option (1.7).

To convince oneself that our construction of the con- jugate field (3.3a) yields the standard Feynman rules in perturbation theory it is sufficient to compute the two- point Green function in v = 0 sector. One obtains

(~(x) z t (yG=o

= E t / .,lr > 0 ',, r - -

[ = 2 (x],~,) 4, 4, ( 2 , l y ) + ( x l - 2 ~ ) /~r> O

with (x l2~) -G(x) . The formula assumes the familiar form ( 1 ) (0(x))((y))~=o= x ~ m y (3.10)

if one uses the completeness relation in v = 0 sector,

[ - I L > < ~ r I + I - L > < - L I ] = 1. (3.11) ~.r>O

In perturbative treatment 0~ i r and in momentum rep- resentation the formula (3. l 0) coincides with the standard Dirac propagator in the Euclidean metric [6]

1 , .4 i0+im --.(x-y) (0(x)z,(y))===7, , j O ( 2 ~ 1 ~ pp2~_~m2e w (3.12)

We are thus assured that with the construction (3.3a) of the conjugate field, the usual Feynman rules, indeed, fol- low in perturbation theory.

Formula (3.9) holds for the two-point function in non- trivial sector as well, but in the completeness relation (3.11) the identity operator on the right-hand side is to be replaced by 1 - P where P is the projector on the kernel space of ~. Thus, one obtains

Ix 1 I ~ C~o~(X)(So~(y) (3.13) (O(x)){~(Y))'=~ ~ Y -- _17m '

where qSo~(x) are the zero modes. This is in agreement with the formula for fermion Green function in non-trivial sector given by Nielsen and Schroer, by Dowker, and by Corrigan et al. [18]. The second term on the right-hand side of (3.13) is cancelled exactly by the zero mode contri- butions present in the first term. As a result, the chiral limit is well defined. This is not true in the first option, where the second term is absent in the two-point Green function, thus making it singular in the m - 0 limit [18].

All the remaining fermionic Green functions may be obtained with the help of the formula

( ~ ( x l ) . . . ~ ( X m ) Z t ( y l ) . . . Z* (Yn)) = ~mn det [(~(xi) zr (3.14)

which, in path integral framework, is a consequence of the anticommutation of O(x~) and Z~(yj). The perturbative Dirac propagator (3.12), therefore, ensures that the full set of fermionic Green functions is reproduced correctly with the definition (3.3a) or equivalently (3.4). From this follow [19] the usual axioms in the fermion sector of the Euclid- ean theory, viz., Euclidean invariance, reflection positivity and cluster property.

The definition (3.3a) (or (3.4)) is intrinsically non-local, and yet this non-locality is not reflected in the Green functions calculated in path integral framework. The rea- son is that in function space both Z and O are paramet- rised by the set of Grassmann generators {a,} which are coefficients of the complete set of eigenfunctions of the Dirac operator. In the space of these parameters the relation between Z and ~ is local as is evident from a comparison of (3.3a) and (2.5). Since the spectrum of the Dirac operator is both Euclidean and gauge invariant, the definition (3.3a) (or 3.4) naturally admits of ultraviolet regularization through a gauge and Euclidean invariant cut-offl 2, [ > M. This, according to Osterwalder and Seiler [20], is a key element for preserving the positivity of metric of the Hilbert space of physical states and the symmetries of the theory.

Recently, Kupsch [21] has given a construction of Euclidean Dirac fields as transforms of white noise, which reproduces all Schwinger functions and has the correct behaviour under Euclidean and gauge transformations. It is remarkable that this construction is similar to the representation (3.4) for conjugate Dirac field which was motivated (see also [17]) essentially by intuitive arguments.

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4 Chiral anomalies and the U(1) problem

4.1 Local chiral anomalies in the two options

In the path integral framework, the key to the chiral anomaly is Fujikawa's result [22] for the Jacobian

In J [c~ (x)] = - 2i S c~ (x) ~ ~bt, (x) 7 s q~, (x) d4x (4.1)

induced in the fermion measure by a space-time depend- ent (local) chiral transformation

O(x)+ei'(~)" O(x), ~(x)--r~(x)e i=(~)'5 . (4.2)

In the integrand in (4.1) the sum extends only over dynam- ical modes, i.e., those whose generators appear in the fermion measure (a,, b* in the first option and a,, a* in the second). The anomaly defined by

A (x) = 6@(x) In J [c~(x)]

= - 2i ~ 4;,(x)ys(a,(x) (4.3)

can be expressed in terms of the density of the topological charge of the background gauge field with the help of the identity [22]

92 ~ qS*, (x)ysq~, (x) = ~ Tr Fur F,~. (4.4)

The point to remember is that the sum on the left-hand side of the identity extends over the complete set of eigen- modes of the Dirac operator, including zero modes, if any.

In the v = 0 sector there are no zero modes of the Dirac operator and in either option the dynamical modes consti- tute the complete set. As a result, the anomaly equation in either option assumes the familiar ABJ form

i 2 (OuJus)~'I-_Io=2m(~750)v=O+log~f2Treu~ff,~. (4.5)

The zero modes which occur in v r sectors are not dynamical in the second option. Therefore, in this option, the anomaly is given according to (4.3) and (4.4) by

II Tr Fu~/~- 2~bto~75~bo~ (4.6) A(x)v~o = - 2 i \ 1 0 7 ~

where ~boi'S are the zero modes. This leads to the anomal- ous Ward identity [4, 18]

1I

. g2

+ 21 ( ~ - ~ Tr Fu~ ffuv - ~ q~*o/75 q~ o/) . (4.7)

In the first option, however, zero modes in v r 0 sectors are dynamical and the anomaly A(x) has the same form as in the v=0 sector. As a result, the anomalous Ward identity

1 - i g 2 T ~ @ , J , 5 ) ~ o = 2m (~k75~b)~ o + 8-~7~2 r (F~F~) (4.8)

looks exactly like the identity (4.5) in the v=0 sector. There is, however, an important difference. In the v r

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sector of the background potential, (~75~) has a pole at m = 0. In the chiral limit, the mass term in (4.8) does not vanish but coincides precisely with the last term [23] of (4.7).

4.2 Global chiral anomaly in the two options

In either scenario for Dirac fermions, the Jacobian in the v = 0 sector for space-time independent (global) chiral ro- tations (1.9) is trivial:

jr- ~l n In t~) 7= o = c~A (x)~ = o d4x = 0. (4.9)

This follows from the fact that the dynamical modes correspond to non-zero eigen-values and in the v = 0 sec- tor they constitute a complete set of modes.

The situation is different in v r 0 sectors. There in the OS [6] scenario the Jacobian for global chiral rotations is non-trivial:

In J(a)~ ~ o = a ~A(x)~ ~ o d4x

i92c~ ~ - ~ z I T r ( e , ~ F , ~ ) d ' x . (4.10)

Thus, by implementing a global chiral rotation (1.9) in the QCD vacuum functional

ZOcD -- ~ NA d# exp [SocD ] (4.11)

one obtains a new QCD action SbcD in v r 0 sectors I SQCD = ~ [O(D --im)O + �88 (F,, ff~)] d4x + 2aA'S

+ ( 0 - 2c~) A S, (4.12)

which, in the OS [6] option, is equivalent to SQco in (1.2). This equivalence is usually exploited with the specific choice e=�89 to get rid of the topological term AS and trade it [-9, 10] for the Baluni term (1.10) A'S.

In contrast, in the alternative scenario in v ~ 0 sectors, the Jacobian is trivial [4]. This follows directly from the anomaly (4.6) and the index theorem:

lnJ(a)~o" = - 2 i c ~ I 1-~2TrFu,Fu~--2Cb~o~TsOo, d 4 x = 0 .

(4.13)

Implementing a global chiral rotation (1.9), one now ob- tains yet another QCD action

S~co = ~ [~(D -ira) 0 + �88 d4x

+ 2a A' S + O AS , (4.14)

where the topological term remains unaltered, and a Baluni term is added. The Baluni term in (4.14) can be transformed away with impunity. It cannot constitute the basis of any observable effect like CP violation.

4.3 Solution of the U(1)-problem in the second option

In the effective Lagrangian framework there is no fer- mion measure to yield a non-trivial Jacobian. Breaking of chiral symmetry and the anomaly in the axial Ward identity, all arise from the so-called anomaly term in the effective action. In order that chiral properties of the

underlying QCD realised with the alternative scenario for Dirac fermions are reflected faithfully in the anomaly term

II ASeff, the latter should satisfy the following criteria:

(i) its variation under global U(1) chiral rotation must vanish:

~(AS~ff) = 0. (4.15)

This corresponds to the fact that the Jacobian for global chiral rotations, according to (4.9) and (4.13), is trivial in the second option.

6x(ASeff) under local chiral transform- (ii) its variation i1 ations, which gives the anomaly in the effective Lagran- gian framework, must be chirally invariant and indepen- dent of the 0-parameter:

6 [6x (A SIelff)] = 0 , (4.16)

d dO [6x (A Snff)] = 0. (4.17)

These criteria reflect the properties of the anomaly (4.6) in the second option.

Let us consider the ansatz [12]

A SIIff = c 2 ~ ~#~ (x) G ( x - y) Ourl (y) d4x d4y (4.18a)

for the anomaly term in the Goldstone boson sector of the effective action. Here r/represents the field of the flavour singlet Goldstone boson, c is a constant with the dimen- sion of mass and G ( x - y ) is the Green function satisfying

Ou~?u G (x - y) = 64 (x - y). (4.19)

Under U(1) chiral symmetry, the field of the Goldstone boson translates

q(x) ~t/(x) + e, (4.20)

where e is the parameter of transformation. Anomaly in the axial Ward identity is obtained from AS~f by a local variation, q(x)~q(x) + fl(x),

6x (A S~Iy) = - 2c 2 S ~u G (x - y) ~3, q (y) d4y. (4.2 la)

It is now a simple matter to verify that the ansatz (4.18a) satisfies all the criteria (4.15)-(4.17).

Integrating (4.18a) by parts one obtains

II __ __ C 2 I q 2 (X) d4x + surface terms, (4.22) ASeff-

which has the familiar form of a local mass term. Neglect of the surface terms is allowed if t/(x) falls off rapidly in the asymptotic domain. Note that even if one starts with an asymptotically well behaved q(x), the translation (4.20) will spoil this asymptotic property. The same translation property also shows that the local form (4.22) breaks U(1) chiral symmetry explicitly.

In order that a local mass term like (4.22) qualifies as the anomaly term, t/(x) should be replaced by the field ~]phys(X) for the physical Goldstone boson obtained by subtracting from r/(x) its expectation value in 0-vacuum

/~phys (X) ~ / '~ (X) - - ( /~ (X)~ 0 �9 ( 4 . 2 3 )

A non-zero value for (q(X))o is the signature of sponta- neous breaking of U(1) chiral symmetry. Transformations of the spontaneously broken chiral symmetry simply

translates [11] the 0-vacuum to the (0 + cr

(tl(X) )o +~ = (tl(X) )o + or (4.24)

This shows that t/phys(x) does not translate, and

( t ] p h y s ( X ) ) 0 + e = ( ~ ] p h y s ( ( X ) ) 0 = 0 . (4.25)

The chiral properties of the underlying QCD with the alternative scenario for Dirac fermion are, therefore, realised faithfully by the local form for the anomaly term in the effective action

1I A S~ff = - c 2 S t/2hy~ (x) d4x, (4.18b)

where t/vhy~(X), the field for the physical Goldstone boson, is defined in (4.23). The anomaly in axial Ward identity (4.21a) is also expressed in terms of ~phy~(X)

~)x(ASIleff) = - - 2C2/Tphys(X) . (4.21b)

The anomaly term (4.18b) realises a non-zero mass for the physical flavour singlet Goldstone in the chiral limit and solves the U(1)-problem in the second option.

To bring out the similarity of (4.18b) with the form familiar in literature we substitutef~(�89 log U/U*) for t/(x) in (4.23) where U is the meson matrix [16] and f . is the pion decay constant. One thus obtains from (4.18b)

ASnff = - c':f2 ~ [�89 log(U/U*)

-- (�89 Tr log(U/Ut) )o] 2 d4x (4.18c)

to be compared with anomaly term in the OS [6] option which appears in literature [16]

i _ c , 2 f z j [ � 8 9 (4.26) A S e f f =

Chiral symmetry is broken explicitly by the anomaly term ASIff. This is the genesis of what 't Hooft defines [11] as the U(1)-dilemma. An unpleasant fall-out of this in effective theory is that the anomaly in axial Ward identity depends explicitly on 0 [11] - a feature which is not shared by the anomalous Ward identities (4.5 and 4.7) in underlying QCD even in the first option. The anomaly term ASIIe (4.18a, b, or c) is, on the other hand, chirally invariant and the symmetry is broken spontaneously in the second option. The parameter 0 appears only as the label for the vacuum which is continuously degenerate.

5 Absence of strong CP problem in the second option

The term A'S (1.10) named after Baluni was used by him [9] and by others [10] to estimate the electric dipole moment of the neutron. The estimate came out to be 0 times a rather large number. Since a dipole moment has not been observed in practice, either 0 is essentially zero, or the calculations are in error.

Let us consider the first alternative to start with. The interpretation of 0, and the connotations of its vanishing are different in the two options for Euclidean fermions. The coefficient 0 coming from the gluon sector is supple- mented, in the first option, by an additional quark contri- bution to the chiral mass term: 0eff=0OcD+0OF D. The second piece on the right comes from the CP-non-con- serving electroweak sector which provides the quark

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mass, so that there are infinite renormalizations. To have the effective value nearly equal to zero requires the fine tuning of 0oco to the unrelated and incalculable renor- malized value of -0QVD. This is highly unnatural and is referred to as the "Strong CP Problem". Attempts to solve this problem by introducing symmetries [24] have led to the prediction of undetected particles [25].

The situation is very different with the second option. Here the 0OCD-term does not get converted to a Baluni form, so that any possible dipole moment calculated from the Baluni term refers to the electroweak 0QFD. Our dis- cussion in Sect. 4.2 shows that such a term can be gener- ated in this option without affecting any other term, including the 0QcD-term. Hence OQvo must be unphysical, i.e., it cannot by itself give rise to CP violation or any other effect and, in particular, its contribution to the dipole moment is precisely zero. Thus, in the second option the dipole moment can come only from 0QcD. Setting this equal to zero is far more natural than setting it equal to the incalculable quantity - 0QFD. In this way, it is possible, in the second option, to preserve CP symmetry in QCD. The strong CP problem disappears.

It is also a fact that direct calculations of the neutron dipole moment from 0oc o without conversion to a Baluni- like form are not available at the moment. Nowadays strong CP is studied through effective Lagrangians [15, 16]. Even in this approach, the second option yields CP conservation as is clear from the representation in Sect. 4.3 where there is no explicit 0ocD-dependence in the effective Lagrangian when it is expressed in terms of qphys. Thus, there really is no basis even for saying that 0QCD must be small if the second option is adopted. It is therefore doubly true that there is no strong CP problem.

How does one reconcile the CP invariance of QCD having 0Qc D = 0 in the second option with CP violation in the first option? To examine this difference, we consider the definition of parity in the presence of the Baluni term. The usual free Dirac action

So = Jd4x ~(i~ - m)O (5.1)

is known to be invariant under the parity transformation

0(x, xo)~'/~ ~ ( - x , xo), ~(x, xo) - - ,~( - x, xo)v ~ . (5.2)

Under this transformation, ~ transforms as a scalar, ~i~5~ as a pseudoscalar, ~'?"0 as a (four-) vector and so on. We have to consider a theory where the mass term is not a scalar but involves a scalar piece as well as a pseudoscalar one.

Sj3 = Sd4x ~'(i# - m - i 7 sm')O' (5.3)

Under (5.2), the pseudoscalar mass term picks up a minus sign, while the other terms stay unchanged, indicating that S; is not invariant. However, this does not mean that the theory breaks parity. Just as the transformation (5.2) was found in an attempt to keep So invariant, one should attempt to find a transformation that replaces fields at x by fields at - x and leave (5.3) invariant [13]. After all, symmetry transformations in field theory are determined by the symmetry properties of the Lagrangian in the

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asymptotic domain and the pseudoscalar mass term, which is quadratic in the fields, must be taken into ac- count in the construction of the symmetry transformation. Let this transformation be

r Xo)-~ vO'(-x, Xo), ~'(x, Xo)~7'(- x, Xo)7 ~ Vt~ ~ (5.4)

This will leave (5.3) invariant if

V=7~ ~ , (5.5)

where

m !

f l=tan - 1 - (5.6) m

Thus, the transformation (5.4) leaves S; unchanged, show- ing that this theory does have reflection invariance. Note that V satisfies

Vt= V= V-1 (5.7)

as usual, and there is no conflict with the fact that the square of a parity transformation is the identity. In fact, the transformation (5.4) can be seen to correspond to the usual parity transformation in terms of fields ~ which have a real mass term, i.e., Vmay be obtained from the 7 o of the standard parity transformation by using the chiral trans- formation used by Baluni. It is important to note that along with the definition of parity, the transformations of the fermion bilinears also change. Clearly, tp'O' is no longer a scalar and t7/'75 ~' no longer pseudoscalar:

~'~'(x, x0)~(cos 2fl) t~'~'(- x, Xo)

+ i(sin 2fl)~'75ff'(- x, Xo), (5.8)

~'75~'(x, x o ) ~ - (cos 2fl)~,%r Xo)

- i(sin 2fl)~'~'(-x, x0). (5.9)

The expected transformation rules are obeyed by ~'e i ~ 0' (scalar) and t~'75elP~ ' (pseudoscalar), respectively. Thus, the mass term ~'(m+i75m')~', which can also be written as (rn2+rrt'2)l/2~'eifl~@/, is actually a scalar under the transformation (5.4). Note that ~'7"r ' continues to trans- form like a vector and ~'TuTs~ ' like an axial vector.

Having found that the presence of 75 in the mass term in a free fermionic theory does not spoil reflection inva- fiance, one has to ask what happens when interactions are switched on. The transformation defined above leaves the pure quark part of the QCD action invariant. The intro- duction of gluons necessitates the definition of a parity transformation for them. We adopt the usual definition:

Ao(x, Xo)~Ao(-X, Xo), Ai(x , X o ) - - ~ - A i ( - x , Xo). (5.10)

The integral of the kinetic term for gluons is then invariant as usual. What is more non-trivial is the invariance of the interaction piece. But this does hold even though V differs from the usual 7~

t~'yuAU~'(x, Xo)--* ~'TuAU~'(-- x, Xo). (5.11)

We conclude then that the action of QCD with a Baluni term but without a gluonic 0-term is invariant under the

modified parity transformation in which 70 is replaced by V. If the gluonic 0 term is present, this invariance is broken.

What implication does this modified symmetry have for the quantum theory corresponding to QCD with the Baluni term? One would be tempted to say that the quantum theory too has a reflection symmetry. First of all, it is easy to see that perturbative calculations cannot detect any violation of reflection invariance. This is be- cause the vertices are standard and involve 7", and only the quark propagator involves 0. However, this propaga- tor can be written as

[ l~-rn- iTsm'O]- l=e- iP~ -Ip~ (5.12)

and the wavefunction as

u(p, m + iTsOm')=eiO~ m), (5.13)

while

]ju = eipO~,,/2 7u ei/~o~'s/2, (5.14)

Thus, the 0-dependent factors in the internal quark lines cancel out at the vertices, leaving the propagators and the vertices of the theory without 0. If there are external quark lines, the remaining factors in the propagators cancel out with those in the wave functions. Consequently, diagram- matic techniques will never detect the presence of 0.

However, there may be non-perturbative effects. Al- though the action is invariant under a classical reflection symmetry, this symmetry will be broken in the quantum theory if the chiral piece of the transformation alters the fermion measure. Note that we are concerned here with constant chiral transformations rather than space-time- dependent ones: the distinction is important because only the latter yield perturbatively non-trivial Jacobians. We have seen however that in the second option the Jacobian corresponding to a constant chiral transformation is unity. This means that the reflection symmetry is not broken by non-perturbative effects. This confirms the earlier conclusion that there is no CP violation in the second option due to any possible Baluni terms.

In case these arguments appear somewhat uncomfort- able, the reader should think of the situation with abelian gauge fields. In this case there is no admissible topological 0-term and no fermionic zero modes, so that the Baluni term must be capable of being rotated away and there is no CP violation, in agreement with the above results. CP violation, when it occurs, has to be much more subtle.

All these arguments are for the second option for Euclidean fermions. What about the first option? In this option, the Baluni term is equivalent to the original 0-term. However, the equivalence arises precisely because of the fermionic zero modes. Calculations which are not sensitive to zero modes can be thought of as being ignor- ant of the choice of one of the two options that has to be made. Accordingly, such calculations cannot get a non- zero CP violation because the second option is bound to yield zero. A zero-mode-sensitive calculation can however detect a CP violation through the Baluni term in the first option, as we shall now argue.

Recall the arguments given above for the second op- tion. As far as perturbation theory is concerned, there is

no difference in the two options. However, for the con- stant chiral transformation (1.9), the measure of the path integral now changes by an anomalous Jacobian exp( ~(i92/87: z) (Tr ~)yd4x Tr(F/~)). While expressions like Tr(FF) appear in perturbative calculations, it is an exact four-divergence, so that y d4x Tr(F/~) vanishes for pertur- bative gluon configurations. That is why there is no per- turbative parity violation. The only possible effects must be non-perturbative, coming through topologically non- trivial gluon configurations where ~ d4x Tr (Fff) does not vanish even though Tr(FF) is a four-divergence, i.e., where fermionic zero modes occur. This is exactly the condition for the gluonic 8-term to be non-trivial and CP violating. Calculations which take such topologies into account are not available, and we do not know to what extent CP is violated in the first option or indeed in the second option with a n o n - z e r o 8QC o.

In this connection, we would like to point out that following our mention of the anomaly in [13], some lattice-based authors [14] stated that lattice calculations, which take disconnected diagrams (with quark loops) into account and hence see the perturbative gluon anomaly, should be able to detect an NEDM, in contrast to calcu- lations which neglect such diagrams. But this is untenable. Lattice methods would need to include instanton-like gluon configurations in their purview. While this is pos- sible in principle, topological calculations on the lattice are in practice beset with such severe ambiguities [26] that there is no hope of any reliable lattice calculation of the NEDM in the near future.

As regards the effective Lagrangian approach, we feel that they are at the moment only ad hoc and not solidly based on QCD. The baryon sector, in particular, is highly model dependent. CP violation effects come from terms breaking chiral invariance which vary from author to author [15, 16].

Finally, we discuss an approach which has some his- torical importance. Some current algebraists [10] appar- ently thought that they would be able to see CP violating effects of 0~ff because current algebra is non-perturbative. But this method, which is blind to even the perturbative anomaly, cannot hope to see the non-perturbative CP violation caused through the integral of the anomaly by the Baluni term. The fact is that they did think they had identified a CP violating quantity. We shall demonstrate that they were wrong in their interpretation. (See also [15].)

The crux of the calculation of the electric dipole mo- ment of the neutron by the current algebraic method [10] is a CP violating "scalar coupling" of pions and nucleons, represented by

Here O(oc0) is the scalar coupling invented by the said authors, O~r is the nucleon field and re" the ath mem- ber of the pion (Nambu-Goldstone boson) octet. A term of the form (5.15) has been alleged to occur in the effective theory of pions and nucleons. This was thought to arise from the Baluni term and thus to be related to the non-

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zero matrix element

i(-i8)m'(Tr"N l ~'75O' l N ) . (5.16)

Let us try to understand the error [13] in the current algebra calculation [10] of the neutron electric dipole moment. The coupling coming from (5.16) was identified as a scalar coupling because the soft pion theorem yields

/ - i O m ' \

2i ' a Om -, 2 , = ( - ~ - ) ( N , ~ ~-~p IN) . (5.17)

But the pion also has a conventional pseudoscalar coup- ling to the nucleon. This was not treated in any detail by the current algebraists [10], who simply wrote down the phenomenological value for the coupling constant. The pseudoscalar coupling in the conventional theory with 8=0 is related to the matrix element m, z (Sire"IN), which, _bY PCAC, can be written as (1/f~) (NlOu(~u~5(R"/2)tp)lN). By virtue of the equation of motion, this is proportional to m(Nl~ff75(2a/2)OIN). Now this expression is not invariant under the chiral transformation taking 0 to O':

- 2 " - 2 " =2imO'~;5 ~ tp ' -28m 'O '~ tp ' . (5.18)

Thus, the pion coupling is related to

The previous authors [10] identified the first piece on the RHS as a pseudoscalar coupling and obtained the dipole moment from the interference between these two. But this identification is inappropriate, for these two terms by themselves do not have well-defined transformation prop- erties. Only the combination has such a property: it is pseudoscaIar. In other words, there is no scalar coupling, and no dipole moment in the current algebra approxima- tion [13].

We end with a table summarizing the conclusions regarding dipole moment calculations in the two options (see Table 1).

Table 1. Possibility of obtaining dipole moment from Baluni term with 0QCD = 0

Calculationat method Option 1 Option 2

Perturbation theory NO NO Current algebra NO NO Effective Lagrangian YES NO Exact (hypothetical) UNKNOWN NO

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6 Concluding remarks

The traditional and popular OS scenario [6] needs two independent Dirac fields 0(x) and t~(x) to describe a Dirac fermion in Euclidean metric. This leads to the first option (1.6) for FPI but at the price of making the hermitian conjugates of Green functions of the theory incalculable. In the alternative scenario for Dirac fermion presented here the conjugate field ~(x)depends on and is derivable from 0(x). Both O(x) and O(x) transform unitarily (1.9) under chiral rotation. The scenario leads to the second option (1.7) for FPI and reproduces all the fermionic Green functions of the Euclidean theory.

The distinctive feature of profound physical signifi- cance in the alternative scenario is that the Q C D action is invariant under global chiral rotations (1.9). The anomaly term in the effective theory conforming to this scenario breaks chiral symmetry spontaneously, and not explicitly as in models [16] based on the OS scenario, and yet solves the U(1)-problcm. The chiral phase in the quark mass matrix, which is a possible source of CP violation in models [9, 10, 16] based on the OS scenario, is unphysical in the alternative scenario and cannot constitute the basis of any observable effect. Thus, the strong CP problem is a legacy of the OS scenario for Dirac fermions. In Q C D with 0ocD=0 in the alternative scenario CP is a good symmetry and the neutron has no electric dipole moment.

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