Can confinement ensure natural CP invariance of strong interactions?

Can confinement ensure natural CP invariance of strong interactions?

Nuclear Physics B166 (1980) 4 9 3 - 5 0 6 © North-Holland Publishing C o m p a n y CAN CONFINEMENT ENSURE NATURAL CP INVARIANCE OF STRONG INTERACTION...

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Nuclear Physics B166 (1980) 4 9 3 - 5 0 6 © North-Holland Publishing C o m p a n y

CAN CONFINEMENT ENSURE NATURAL CP INVARIANCE OF STRONG INTERACTIONS? M.A. SHIFMAN, A.I. VAINSHTEIN* and V.I. Z A K H A R O V Institute of Theoretical and Experimental Physics, Moscow 117259, USSR Received 19 July 1979

P - and T-invariance violation in q u a n t u m chromodynamics due to the so-called 0-term is discussed. It is shown that irrespectively of how the confinement works there emerge observable P- and T - o d d effects. The proof is based on the assumption that Q C D resolves the U(1) problem, i.e., the mass of the singlet pseudoscalar m e s o n does not vanish in the chiral limit. We suggest a modification of the axion scheme which restores the natural P and T invariance of the theory and cannot be ruled out experimentally.

1. Introduction The discovery of instantons [1] has brought the question of natural parity conservation in strong interactions [2]. It was argued that one can add to the standard QCD lagrangian the so-called 0-term 2 A ~ = 0 gs a 32,,72 G~,~Gu~,

(1)

a

~

l

.,.~ t-i

where G,~ is the gluon field strength tensor, c , , , = 5 e , ~ ¢ u ~ and gs is the quarkgluon coupling constant. The 0-term preserves renormalizability of the theory but is P- and T-odd. A very specific feature of the extra term (1) is that it is a full divergence: a

a

~fl, x --

a.

a

1

- 2e~,vAo-(A~,o,~Ao-+~gsf

abc

a

b

c

A,~A,~Ao-).

The common wisdom says that total divergences can be safely omitted from the lagrangian and that is why the 0-term evaded attention for some time [3]. However, instanton solutions demonstrate that a total divergence can be * P e r m a n e n t address: Institute for Nuclear Physics, Novosibirsk 90, USSR. 493

(2)

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meaningful. Indeed, the term (1) can be transformed into a surface integral which does not vanish for the instanton solution: I d3S~, K~lo,ei,st # 0.

(3)

The overwhelming reaction to this observation is that the example of the oneinstanton solution does prove CP violation in the presence of the 0-term. Our final result complies with the conclusion on CP violation so that most readers can, justifiably, loose interest in the paper at this point. Still, there exists an alternative line of reasoning which calls for further and more thoughtful analysis. The point is that the non-vanishing surface integral (3) could be an artifact of the dilute instanton gas approximation which completely ignores the most profound feature of QCD-confinement of color. In more detail, the surface integral (3) does not vanish since the instanton field falls off too slowly at large distances. This fact is due to the presence of massless gluons in the original lagrangian which do not show up, however, in the physical spectrum. Therefore, one might hope that the confinement effects would make the surface integral vanish. Such a viewpoint has been strongly advocated by Polyakov ever since the discovery of the problem. Moreover, the picture is realized in the Polyakov 2 + 1 compact electrodynamics [4] which exhibits both asymptotic freedom and confinement: the 0-term can be added, does not vanish in the oneinstanton approximation but produces no effect whatsoever in the confining phase [5]. A similar solution within a different context was proposed in ref. [6]. An attempt is made in this paper to show that large-distance dynamics eliminates the effect of the 0-term through spontaneous breaking of the chiral symmetry. In view of all this, it is desirable to start with the original lagrangian and to evaluate some observable (at least in principle) CP-odd effects directly. This is just the aim of the present paper. We prove that the 0-term does imply P and T violation in OCD. It is essential that we can prove the fact without going into details of how confinement works. We will present a computation of two matrix elements: the vacuum expectation value a

=--Of~m~ ( ~ ) 2 ,

(4)

and the transition amplitude m2 4toured A(r/~ ~'+Tr-) ~ 0 - ~/-6[~ (mu + ma) 2 '

(5)

where mu, md are the current quark masses, 4mumd/(mu+ ma) 2~- 0.9 and f~ is the ~r~/zv decay constant, f~ --- 133 MeV. a Although the vacuum average (Ol(as/zr)G~,~G~,v]O) is not directly measurable, it can be extracted indirectly. We will show that (01G~10)~o does imply the CP

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violation in physical transitions and estimate, say, the mixing of the charmonium levels with JP= 0 + and JP= 0 in terms of (0IG(~I0). Only dispersion relations are used to translate (0IGGI0)¢ 0 into a proof of CP violation in physical effects. Eqs. (4), (5) are more or less direct consequences of the current algebra. The crucial point in their derivation is the absence of the ninth light pseudoscalar meson. Let us remind the reader that, naively, the QCD lagrangian with three light quarks possesses approximate U(3) ® U(3) symmetry which would imply nine (pseudo) Goldstone mesons [7]. However, the rl' mass apparently does not vanish in the chiral limit. We heavily use this observation. If otherwise, the matrix elements (4), (5) would vanish in the approximation considered. Thus, if QCD resolves the so-called U(1) problem [8] it cannot resolve the 0-problem and vice versa. Apart from demonstrating the fact of P and T violation in the presence of the 0-term, eqs. (4), (5) give the scale of the CP-odd effects and can be used to bound the value of 0 starting from experimental data. The ratio O~s

a

~a

°~s

(6)

can serve as a measure of the CP violation. It is worth noting that the ratio (6) is of first order in small quark masses, although it vanishes both in the limit mu = 0 and m d = 0.

Since QCD is not self-protected against the CP violation induced by the 0-term, the question arises whether it is possible to kill it somehow. The celebrated solution is the existence of a nearly massless pseudoscalar particle [2, 9], the so-called axion. We will add a few comments on the properties of the axion. Namely, we will show that there is no difficulty in introducing such a model that has practically no observable consequences although it ensures 0err = 0. The axion is sterile in this model, unlike the particle proposed in the original papers [9]. The model discussed here is in a sense extreme and does not pretend to be a true one, but it is intended to demonstrate that a solution can depend crucially on the structure of the theory at very short distances. The organization of the paper is as follows. In sect. 2 we derive eq. (4). In sect. 3 we use this equation to estimate the CP-odd mixing of the charmonium levels in the presence of the 0-term. The estimate of r/--> 27r decay is also presented here. In sect. 4 we comment on the axion properties.

2. Vacuum expectation value

a

~a

(OJG.vG~IO)

For simplicity we assume 0 << 1. Then one can expand in 0, and, in particular, in first order

( 0 --~ ° G~.vG.~ $

a

a

0 = ½KO,

(7)

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where K is the low-energy limit of the following two-point function:

as . ~ as b K = { i I d x e iqx(ot T [~-~Gu~(x)G.~(x),~-~Go~(O)Go~(O)]lO)}q= o.

(83

The relation arises immediately, if, e.g., one represents the vacuum expectation value (7) in the form of a functional integral and expands the integrand in 0. Proceed now to the evaluation of K and start with two light quarks with masses mu and md ~. The generalization to the case of three light quarks is straightforward and essentially does not change the results. The operator G~.~G~.. is related to the divergence of the isosinglet axial current [10]: -

2as

a

Ova. = 2imu~ysu + 2imadysd +--~ G ~ G .~,

(9)

where

a~, = ffy~,ysU + dyg ysd. It is convenient for this reason to introduce along with the two-point function specified in eq. (8) the correlators II~(q) and l'I,(q): II.~(q) = i f dx e~q~(OlT{a~(x), a~(0)}t0), l'I.(q) = ; f dx eiq~(0lT { au(x),~-~ Gu~(0)G.~(0)}I0).

(10)

These correlators satisfy the rather evident conditions

q.q~II..(q)]q~o~O,

q.fI~(q)lq~o~O,

which assume that there are no massless physical states coupled to a.. In particular, there is no axion so far. As for the pion, its mass is kept finite since we account for small quark masses explicitly. Integrating by parts transforms the amplitudes q~q~II.~ and %1"I. to the T products of the current divergences (9) plus the contact term. The latter is determined by the equal-time commutator

[(2muff (x )ysu(x ) + 2 m j ( x )ysd (x )), ao(O)]8(Xo) = (4muau + 4mddd)8(4~(x ) . * Hereafter we assume, if not stated to the contrary, that the mass matrix has the standard 3'5 free form, - ~ mqgtq with mq> 0. The mass matrix can always be reduced to this form by performing the 3"5 rotation and redefining the parameter 0 [2].

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In this way we get 0 = (q~.q~,Flf.,.)q=o = (0]4muSU +4mdddlO)+K

+ i f dx(OlT{(2imua3,su + 2imddysd):,, (2imuffys u + 2imddysd)o}lO) +2i f dx (0[T{ (2irnuSysu+ 2imdd-ysd)x, (~-~ G~Gu2"-a~)o}[0), 0

=

(11)

(qt. I~t~.)q =o

= K +i f dx(01T { (2imuSysu + 2imddysd)x, i/ asc. ac. ~U)oll \ I, 0).

(12)

Moreover, for small quark masses one has (012muSu + 2 m j d ] 0 ) = -f2m 211 + O(rno,d)] (for a derivation see, e.g., the review paper [11]). Eliminating the extra correlator from eqs. (11) and (12) we get the following relation for K: K = -2f~m 2 2 2 ~ + O(mu.d) +

i I dx (O[T{(2imuaysu + 2imddysd)x, (2imuaysu + 2imddysd)o}lO).

(13)

The T product on the r.h.s, of eq. (13) is apparently proportional to mZ,d while we are interested in terms linear in the quark masses. This means that only such intermediate states whose masses squared are proportional to mu.d must be kept in the T product. It is crucial that there is only one such s t a t e - the rr-meson (with the quark content ~ u - dd). Let us remind the reader that so far we are working within the theory with two light quarks. The mass of the singlet state ~au + aTd does not vanish in the limit m~,d-+ 0. Otherwise we would have kept its contribution as well (see below). As for the pion, its residue can be readily expressed in terms of the ~r ~ tzz, decay constant f= and the quark masses:

(Oliaysu[rr °) = -{01id75dlrr °) =

(14) 42 (mu + rod)"

Thus, eq. (13) gives

2 2 4muma K = -2f,~m = (mu + md)2'

(15)

and its accuracy is O(m2q). Let us notice that for mu = rnd eq. (15) was first derived by Crewther [12] within a different context. Substituting eq. (15) into eq. (7) we come to the announced result for (0[G(~[0) [see eq. (4)]. Two remarks are now in order concerning its derivation.

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First we would like to elaborate the remark made above on the close connection between the so-called U(I) problem and the 0-term effects. Indeed, forget for a moment that the isoscalar axial vector current is not conserved because of the triangle anomaly [10]. Then for two nearly massless quarks there would be two nearly massless electrically neutral pseudoscalar mesons called, hereafter, U and D, with the quark content flu and dd, respectively. (To avoid confusion, experimentally there is only one such state, ¢r° - f l u - d d and this is the essence of the U(1) problem [7].) Parallelizing our previous derivation we obtain, instead of eq. (15), g = (0]4mutiU +4mddd]O)

/

+ ~l(OI2imua~,su+2imdJTsdlg)12+(g~w)

/

,

and the following current algebra results for the masses and residues

1-2 ](OI2im~aysU + 2 i m j y s d l U)I 2 = -4mu(0lfful0), mu

1

2 mD

1(OI2imuff3,su+ 2imddysdlD)l 2 = --4md(01dd 10).

Combining these relations immediately gives K = 0 (at least to the first order in quark masses). Thus, if the U(1) symmetry of the lagrangian were not fictitious, the effect of the 0-term could be eaten up by strong interactions. Our second remark concerns with the generalization of eq. (15) to the case of three light quarks (u, d, s). Proceeding in the same way as above one comes to 2 2 2 mumdms K = -6f=(m ~ + m ,) (m~ + md + m~)(m~rnd + rn~m~+ mdmO"

(16)

Note that in the limit mdmu,d >>1 eq. (16) coincides with eq. (15) up to terms of order mu,d/m~ which represents negligible corrections. Thus, the accuracy of eq. (4) is evidently of order mu.d/m~. To derive eq. (16) it is again essential to keep in mind that there is no ninth light pseudoscalar meson. Moreover, for comparable m~, md, m~ the diagonal states are not proportional to ((tu- dd) and (flu + d d - 2gs) but are reduced to linear combinations of (flu -dd) and (flu +dd - 2~s). 3. Observable effects induced by the O-term The result of sect. 2 reduces to the computation of (01GG]0). It is rather evident that the ratio a

~

a

a

(ol~s/ ~)o .~o .~ho)/ (ol(~s/ ~)o .~o ,,vlo) -- -o.o25o characterizes the CP-odd effects induced by the 0-term*. a * T o c o m p u t e the r a t i o w e use eq. (4) for 0 . 0 1 2 G e V 4 w h i c h follows f r o m Q C D s u m rules [13].

(O[(o
(O](as/'n')G~G~]O)=

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The suppression of the matrix element by the factor - 10 -2 is due to the smallness of the quark masses. Indeed, is proportional to m~ which is small in the scale of hadronic masses. Moreover, eq. (4) demonstrates explicitly that (0]G(~]0) vanishes in the limit of either mu = 0 or md = 0. Thus, eq. (4) agrees with the general theorem [2, 9] according to which there is no violation in the presence of a massless quark. It is amusing, however, that (01G010)is of first order in the quark masses and is not of order m2d as one would conclude from the fact that it vanishes for mu or md equal to zero. The appearance of the terms linear in mu,~ is due to the observed asymmetry of the spectrum of the pseudoscalar mesons: the isoscalar meson is much heavier than is required by the naive U(1) symmetry of the lagrangian. Just at this point QCD deviates from the (2 + 1) model mentioned in sect. 1. In the latter model there is no U(1) problem and the 0-term is fictitious in the confining phase. In QCD the resolution of the U(1) problem implies that the O-term is actually operative. Taking the ratio (6) as representative of the scale of violation manifested in the electric dipole moment of the neutron one can estimate (electric dipole moment) -2.5.10-20 or /gexp<10 s . (magnetic moment of neutron) To be completely convincing at this point we must be able to convert the knowledge of the asymmetry of the vacuum state into reliable predictions for observable quantities, not simple estimates. We will compute the CP-odd effects in r/decays and in the mixing of the charmonium levels. Unfortunately, we cannot calculate the electric dipole moment of the neutron since it is difficult in this case to account for strong interactions*. But there seems nothing special, say, in the 77~ 2~decay, and we can take it as a trial example. In this case we can handle strong interactions more or less accurately by means of the soft meson technique. Let us start with the mixing of the charmonium levels. Our aim here is to translate the result for (01GG[0) into the prediction for the mixing of the 0 + and 0states of charmonium. Introduce to this end the polarization operator Fl(q2):

(0JG010)

CP

CP

CP

Fl(q 2)

=if dx eiqX(olw{e(x)c(x),e(0)&sc(0)}[0),

(17)

where c is the charmed quark. If is conserved there is no physical state contributing to the imaginary part of l-l(q 2) and [I(q 2) reduces to a polynomial. On the other hand, it is a trivial matter for us now to prove that I-[(q2) is nOt a polynomial. Consider ]-l(q 2) far off the physical cut, i.e., 4m~ _q2 >>//,2, where mc is the charmed quark mass and tz is a typical hadronic mass. Then II(q 2) is calculable by means of the so-called heavy quark expansion. The result is

CP

1 g2

~

2

d ~ : ' ~2

I-I(q2)=-3-~2(OIG~G~lO)-~¢ [l+~cc f (l_~(l_~2/m2)2J, o * See note added in proof.

(18)

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1

1 i

~'c~.~.~

~'~c + PERM.

I I I I

quork

..g [ u o n

Fig. 1. The diagrams for the coefficient of (01GGI0) in the two-point function (17).

where we neglect terms of higher order in inc. The heavy quark expansion used to derive eq. (18) is actually a direct generalization of the Wilson operator expansion. The expansion coefficient is determined by the graph in fig. 1. The appropriate technique was elaborated in detail in refs. [11, 13]. As compared to the standard way of calculation, there is a circumstance that calls for a special care. The point is that G~G~ vanishes for gluons of equal momenta. Therefore, we cannot use usual perturbation theory to extract the expansion coefficient. We used the external field technique introduced in QED by Schwinger [14]. The generalization to the case of QCD was made by us in collaboration with Novikov [15]. The representation for I~(q2) provided by QCD [see eq. (18)] can be confronted with the general dispersion relation for the same quantity: 17(q2) = I f

Im II(s) as s-q

.

As was already mentioned, the imaginary part here is due to the mixing of levels with opposite parity. If the distance to the physical cut is moderate, say, q 2 0, it is natural to approximate the dispersion integral by the contribution of two lowestlying levels of charmonium: pseudoscalar r/c and scalar X (3.45). A similar approximation in other cases usually works well [11]. Referring the reader to the paper [11] for details, we give here only the final result for the mixing parameter: e(r/c-X(3.45)) ~ 10 20, where we assume that m,Tc= 3.0 GeV. Note that the result is indeed close to the estimate (6). It might be worth emphasizing that the very fact of CP violation does not depend, of course, on approximating the polarization operator by two levels. Indeed, eq. (18) shows that II(q 2) does not reduce to a polynomial and this is enough for our purposes. If we apply similar sum rules to currents constructed from light quarks, the non-vanishing (0JG(~J0) would imply a cr-~ mixing which manifests itself, say, through rl ~2zr decay. However, we can find the r/-~2zr decay amplitude in a more direct way without invoking the sum rule technique.

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Indeed, using eq. (9) we can represent the matrix element of the ~ ~ 2~- decay in the form A ( n -~ 27r) =

(2,rlA2~ln)

= ¼0(2~rlO.a.

2imuaysu

-

-

2imaffysd]~7).

The term O~,a. drops off as usual. As for the rest of the matrix element, we would like to evaluate it to the first order in quark masses. There is a close analogy with the case of the correlator K found in sect. 2 and we just sketch the derivation. There are two kinds of graph which contribute to the matrix element considered, namely the contact and pole terms (see figs. 2a and 2b, respectively). The latter formally contains m2u,d, but actually, due to the o propagator, one power of mu,d goes away. Both contributions are readily found in the limit of vanishing ~r and r/ 4-momenta: graph 2a: 2 1

+

-

,

-

,

-

~O(rr ¢r ]- 2 z m , u y s u - 2tmddysd]~)

m,, ,/6 f~

graph 2b: / d4x T{(- 2imu~ysu - 2imddysd)o, ( - imufcu - i m j d ) x } l n ) m 2 ( m u - md) 2 --

0

-

-

~/6f~ (mu+md) 2' Here we have accounted for the fact that 7/~ 7r+~"- decay proceeds in the S-wave; the ~r+rr - wave function is symmetric and therefore only the piece proportional to (mu+md) is operative in the vertices 2a, while in the vertices 2b it is the piece (mu-ma) that works. Finally, m2 4mumd A ( n ~ ~r+~r-) = Ox/-6 f~ (mu + md) 2"

Note that the result looks very similar to the expression for <0]GG]0) found above.

mo~Lu+m,:td~'~d

muG#' ~

u

~

-rra b Fig. 2. The r/~ ~-+~- amplitude to the leading approximation (linear in the quark masses).

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The effect of the finite r/mass tends to increase the matrix element. Introducing ~r-r/mixing one can estimate the correction of first order in m2n, it reduces to the factor (1 + mn/m~,)~ 2 2 1.5. Comparing the prediction for the decay rate F(rl ~7r+zr ) =02



4 4 2 2 --75-m~ [ - m- uzm d] ( l + m n2/ m ,2~ )2( 1 - 4 m ; ~ 96zrf,,m n I-(mu+ma) J \ m2 ]

l

with the experimental data [16], we get a rather modest bound 0 < 2 . 1 0 -3. It might be worth mentioning that A(rl -~ 27r) can be directly related to the -q ~ 3~r amplitude known experimentally. Indeed, r / + 37r decay is due to the mass term in the lagrangian ½(md-mu)(ftu- dd). Using the PCAC hypothesis to reduce a pion we come to the matrix element determining rl -~ 27r decay in the presence of the 0-term. Thus, we have demonstrated that the 0-term does lead to observable effects. Confinement does not kill it. We must look for another explanation of natural parity conservation in QCD.

4. Heavy quarks and the axion In fact, the problem of the 0-term cannot be solved by QCD alone since it is intimately related to the origin of the quark masses [2] and the mass generation is considered to be a prerogative of weak interactions. The interrelation between the quark sector and the 0-term is obvious. Thus, in a theory with a single quark described by the lagrangian

~q = Clil)q mqt](cos d -

+

iy5 sin d)q

(19)

a "/5 rotation of the quark field eliminates the y5 mass term but simultaneously changes the coefficient of the G G piece of the gluon lagrangian, 0 -~ 0err = 0 + c~. Thus, no matter how heavy the quark is it influences the 0-term. This fact indicates that the whole problem is intimately connected with the structure of the theory at very short distances, regularization and renormalization program and so on, a conclusion which is not surprising for those who keep in mind the role of the triangle anomaly. To make the point explicit let us consider a particular scheme. One convenient way of regularization is to introduce the PauliVillars regulator fields. In the case considered it is sufficient to introduce a single regulator field R which is described by a lagrangian similar to (19) (just replace q by R, m q ~ m R and d ~ dR). The metric of the regulator field is opposite to normal. Usually one assumes that dR = 0 although there is no a priori reason to make such an assumption. Indeed, the R field can regulate equally well at any dR. If dR ¢ 0 then the effective value of 0 is equal to 0e~r= 0 + ~ - dR.

(20)

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Since 0en is in principle directly observable (see above) it seems to be a unique example of how a laboratory experiment can shed light on regulator properties, Alternatively, one can say that it is very suspicious, A few words on the derivation of eq. (20). Integrating over the fermionic degrees of freedom one obtains

q

Ff

[ il)-mq(COSa+iyssina)]

exp [i j d4x (LCq(X)+ LPR(X))j ~ d e t lil)--MR (C0S aR+ iy5 sin aR)-J ' 1 •

(21)

--a--a

where D~, is the covariant derivative, D . = 0.-~tgsa .a.. Moreover, the determinant can be rewritten identically as

[ il)-mq(COSa+iyssina)] det 1_i / ) - MR(cos aR + iv5 sin aR)J

[ i19 - mq ] i19 - MR =detkiD-MRJ det [i/5 - MR[COS (CeR--a) + iy5 sin (aR - a ) ] ] which demonstrates that the dependence on a and aR enters only through the difference a - dR. The entire dependence on a - aR is now concentrated in the regulator and can be calculated explicitly at MR ~ oo. Thus we come to the recipe (20). The sensitivity of 0err to the regularization procedure displayed above emphasizes once more the necessity of neutralizing the 0-term. Since the whole problem of CP non-conservation is born at short distances and depends on how to handle ultraviolet divergences, it is natural to resolve it by coupling an axion only to heavy particles. The axion mechanism was discussed in detail in the original papers of Weinberg and Wilczek [9]. However, in these papers the axion is coupled to the ordinary u, d, s quarks and is involved into the usual weak interactions. As a result, some predictions arise which do not seem to be met by experimental data. To our mind, there are no special reasons to include an axion in the Weinberg-Salam model. We would like to suggest an alternative realization. It is constructed from a (very) heavy quark which does not participate in weak interactions at all and an axion which is (nearly) sterile with respect to usual hadrons. Denote this hypothetical heavy quark by ~ and introduce a complex scalar field ¢(x). The lagrangian describing these fields is of the standard form: ,~,~,p = ~iJO~ - h (¢gTR~L+ ~o+~ffL,I//R)

+(O~,~+ )(Ou~o)+m2 ¢ + ~o-a(¢ + ¢) 2 .

(22)

The ~o field develops a non-vanishing vacuum expectation value, I( >l -=

= m/~/~,

so that the lagrangian (22) describes a scalar state with mass m~/2 and a pseudoscalar one - - the axion - - whose mass vanishes in the classical approximation. The

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quark mass is equal to h¢o and we take it large, ¢o ~ co. Since such a quark is not observable, the only interesting thing is the effective lagrangian which governs the axion-gluon interaction. It is determined by the heavy quark loop and in this approximation the effective lagrangian is of the form

c~ a(x)G~(x)G~dx) , 8~-45 ¢o

(23)

where the axion field a(x) is related to the original field ¢(x) in the following way:

/. a(x)

~;(x) = ¢o exp ~t---o-o-~). At this point we neglect quantum fluctuations of the modulus of ¢, considering the mass of the corresponding particle to be large (these fluctuations describe the scalar particle with mass m~/2 mentioned above). As was first argued within the dilute instanton gas approximation [2, 9], the interaction (23) eliminates all CP-odd effects. We give a somewhat more general derivation of the same effect in the appendix. Clearly enough, interaction (23) results in substitution of the parameter 0 by Gfffx)= 0 + a(x)/¢o~/2 and the vacuum expectation value of a(x) can be seen to cancel the original 0. Iterating interaction (23) one can evaluate also the axion mass (see the appendix):

f~m~f 4toured ma=~

]1/2

[ (mu~--~md)2J

[l+O(mu,d/ms)].

(24)

The above formula for the axion mass coincides in its essence with the Weinberg result [9] except for one point. The vacuum expectation value of the ¢ field, ¢o, in our approach is a free parameter while in the Weinberg-Wilczek model it is related to the Fermi constant GF. If ¢o becomes arbitrarily large, the axion interaction with normal hadrons vanishes. (The same is true for its mass.) The new quark also becomes unobservable. This theoretical phantom still restores the natural P and T invariance of QCD. Although the model discussed is evidently a toy one, one might hope that something of the kind happens in unified theories. The authors are grateful to A.A. Anselm, V.N. Gribov, V.A. Novikov and A.M. Polyakov for valuable discussions.

Appendix In this appendix we present in brief a computation of the axion field expectation value and the axion mass (see sect. 4).

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The lagrangian describing the axion field and its interaction with gluons has the form

5fa(a,A)=l(O~,a)(O+,a)+(o+ a(~)2) -~ as G.~G~,~ a 4"~OCD,

(A.1)

where we have neglected terms of higher order in ~o I and m -1 (for definitions see sect. 4). To determine the vacuum expectation value and the mass of the a field we need to integrate over the gluon fields and find the effective lagrangian for the a field governing its interactions at momenta much lower than the hadronic masses: exp [ i f

d<)xZa(a)]= I

Note that the axion field

0 +a(x)/~oX/2.

~ A exp [ / I

a(x) enters

d4x~a(a,A)].

z,~fr(a) only through the combination

(A.2)

Oe~r(x)=

The first terms of the expansion of 2 ' ~ in this combination are readily obtained a

2

f"(a)=½(O~a)(Oua)+½(0+ o-~) I~K + O((0e~)4),

(A.3)

where the correlator K is defined in eq. (8) (see sect. 2). From eq. (A.3) we conclude that the a field actually develops an expectation value and it is adjusted automatically in such a way that O + (a}figoX/2 = 0 where (a} is the vacuum expectation value. As a result, there are no P and T non-invariant effects left since they are just proportional to the same O+(a)/¢ox/2. Moreover, eq, (24) for the axion mass follows immediately from the low-energy theorem derived for K [see eq. (15)].

Note added in proof Recently there appeared a paper by Crewther et al. [17] who managed to apply current algebra for a calculation of the electric dipole moment of the neutron. Numerically their estimate is close to ours.

References [1] A. Belavin, A. Polyakov, A. Schwartz and Yu. Tyupkin, Phys. Lett. 59B (1975) 85. [2] R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38 (1977) 1440; Phys. Rev. D16 (1977) 1791. [3] S. Weinberg, Phys. Rev. Lett. 31 (1973) 494; Phys. Rev. D8 (1973) 4482. [4] A. Polyakov, Nucl. Phys. B120 (1977) 429. [5] S. Vergeles, Nucl. Phys., B152 (1979) 330. [6] P. Minkowski, Phys. Lett. 76B (1978) 439.

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[7] S. Weinberg, Phys. Rev. D l l (1975) 3583. [8] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8. [9] S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279. [10] S. Adler, Phys. Rev. 177 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cim. 60A (1969) 47. [11] V.A. Novikov et al., Phys. Reports 41 (1978) 1. [12] R.J. Crewther, Phys. Lett. 70B (1977) 349. [13] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1978) 385,448. [14] J. Schwinger, Particles, Sources and fields (Addison-Wesley, New York, 1973). [15] V. Novikov, M. Shifman, A. Vainshtein and V. Zakharov, to be published. [16] J.J. Thaler et al., Phys. Rev. D7 (1973) 2569. [17] R. Crewther, P. Di Vecchia, G. Veneziano and E. Witten, Phys. Lett. 88B (1979) 123.