Neutral pseudoscalar masses in lattice QCD

Nuclear Physics B284 (1987) 234 252 North-Holland, Ambterdam

NEUTRAL PSEUDOSCALAR MASSES IN LATTICE QCD Jan SMIT and Jeroen C VINK In,mute of Theorett~al Phvstc ~. 1/aldvemer~traat 65. 1018 ~ L" 4m~terdam, The Netherlands

Received 1 September 1986

A lattice derivation is given of the Wltten-Venezlano mass formulas for the neutral pseudoscalar borons

1. I n t r o d u c t i o n

Some time ago a remarkable formula was derived [1,2] which could give a quanntatlve resolution of the "U(1) problem" in QCD [3 5], ,)

m;,

-

21 m~; -

6 ~1m ; , ,"

=

f7 7~ X ,

(1 1)

where X is the "topological susceptlblhty" in the gauge theory without quarks.

x= f dx
(1 2)

1

q= 327r~e,~ootrG,~G~,.

(1 3)

with G~,~ the gluon field strength (we use the euchdean metric) The formula (1 1) was derived for large N (N = number of colors) and small quark masses m Questions of regulanzatlon which can plague discussions of the "' U(1) problem" in the continuum formulation were beneficially ignored In this paper we seek a derivation of (1 1) within the lattice regularlzation using Wllson's ferlmon method Staggered fermlons will be considered in another article

[6] The original derivation of (1 1) was based on the observation that for massless quarks the 0-parameter of QCD should be unobservable [1] In Wllson's fermlon formulation the possible chlral angles of the mass terms naturally combine into a 0-parameter when the gauge fields are smooth [7.8] An understanding of (1 1) based on 0-independence of the massless lattice theory may be possible [9] 0550-3213/87/$03 50 ~,Else~ler Science Pubhsher~ B V (North-Holland Physics Publabhmg Dl,dSlon)

235

J Shut, J C Vmh / Neutralpseudo~(alal ma~es

We follow the standard approach of studying the lmphcatlons of Ward-Takahash~ (WT) relations for the pseudoscalar mass spectrum In sect. 2 we introduce the axial currents A" and their divergence D, the normalized flu and D and the normahzation constants xa and ~v entering m the latter construction The WT relauons are used to obtain a mass formula for the neutral pseudoscalars (sect 3) This formula appears to be useful only in the quenched approxlmatmn The latter has some obscuring features and therefore we call upon the effectwe action descrlptmn [10] for clarlficatmn (sect 4). Large N supports the quenched approximation, but so does large d ( d = space-time &menslon), and the quenched approxamatlon with N = 3 and d = 4 may be quite reasonable With a further approxlmatmn of neglectmg terms of order m/N or told compared to order m we recover the lattice analogue of (1 1) (sect. 5). The normallzatmn constant xe enters in this expression and therefore we describe in sect 6 a non-perturbative method for the computatmn of xe and XA- This method also g~ves a determination of the critical hopping parameter, Sect 7 contains notatmnal differences with other authors and rewews results for XA and Kp from weak coupling perturbatmn theory (WCPT) [11-14] 2. Axial current divergence relations The action for Wdson fernuons is given by 1

- S = ~a ~_. [~f(x)y~U~(x)~y(x +a~)-~/f(x +a~)y~U2(x)~t(x) ] x,~f + EM:~:(x)~:(x) ~c,f r

E [~f(x)U~(x)~y(x+a,)+~:(x+a,)U~(x)~f(x)]

(21)

f - E,~Wr+

MjS~(~)-

(x)

,

(2 2b)

f where f = u. d.s.c, denote the n flavors and Mj is related to Wllson's hopping parameter Kf by 2aM/= 1/Kf We use the notations

(2 3)

1% ( x ) = +j( :, ),v5%( x ), P % ( x ) = ¼E

( [~f( x )'ysU,(x )d/,( x + a~.) + ~f(x + a,)tysUf ( x )+.( x )]

+[x-->x-a,]},

sjg(x) = Pj g(x)

l,,<. a ,

EssW(x)x.= E( XvjW) g~ t

(24) ,3,, ~

i

(2 5)

J S m t t J C ftnA /

236

,~eutralp~eudo~calat m a ~ e s

Expressions for axial currents can be obtained by gauging the O part of the action The divergence relations for these currents [11] can be obtained from the response to a local chlral variation,

(O/A~(x))=(Dt~(A)+st O;f(x)

),

(26)

= 1 [f(~c) - f ( . ~ - a . ) ] ,

(2 7)

A~.( x ) = ½ [ ~, , (.,c ),V.y,U.( a ) ~,~( :, + a~, ) + ~/ t ( t + a~,),y.ysU~t( x ) ¢ ~( x ) ] ,

(2 8) 2r

Df~( x ) = ( Mj + Mg)Pr~( x ) - - - P/W(.x ).

(2 9)

a

where s t stands for the contributions of source terms in the action The flavor non-slnglet currents need to be renormahzed by multlpllcanon with a fimte factor KA(g, r) [11-13] and the slnglet current EIA~j(x) needs to be renormalized with an additional factor ZA(g, r, ln a/x) [15, 16], winch is divergent m flmte orders of W C P T (# is some renormahzatlon scale) The renormahzed dwergence relations m a y be written as

(O.Af~(x))=(T)fx(x)+st

(2 10)

), 1

~(x)

= KAA~g(X ) + 3j~(Z a - 1)~¢A - E A ~ < ( x ), ?l

Ojg(.x ) =

Djg(.~ ) -~- (gA--

1) O~A~(x)

(2 11)

L

1

+ a[,4( Z A -

1)~A-E c?;A~e(.x) tl (2 12)

In the scahng region one expects [11,13] that the renormahzed divergence Ds~(x) becomes proportional to the quark masses, except for the U(1) anomaly, Dfg(a) = ~e( mf + m ~) Pt~( x ) + 3t,2,~I( x } + O( a )

(213)

Here m r is the bare quark mass parameter defined by m r = M / - M n,,

(2 14)

with Mcn t the critical value of M, gp(g, r) 1s a renormahzatlon factor similar to ~

J Start, J C Vmk / Neutralpseudoscalar masses

237

and ~(x) is the renormahzed anomaly (see below) The notation O(a) is used to indicate that terms of order am and ap are neglected, for typical momenta and mass values of interest (a Fourier transformation to momentum space being performed first) The relation (2 13) should be valid for matrix elements of the operators in Hllbert space The relation (2 13) expresses the fact that in the scahng region the operator P ~ ( x ) becomes eqmvalent to a linear combination of chosen standard operators with the same quantum numbers and dimension ~ 4 [11,13] 2r

- ~ P ~ ( x ) = 2MP/g(x) + (KA -- 1) cg;A/~g(x) - 8fg2Kqtq(x) + O ( a )

(2 15)

Here q(x) is some definmon of the "topological charge density". For example [17], 1

a4q(x)

5127r2

+--4

e~ootr[U~.(x)Upo(x)],

E y.vpa=

(2 16)

+1

U~(x) = Ul,(x)U,,(x + a , ) U J ( x + a t ) U J ( a ) .

(2 17)

(of [17] for notation) In the classical continuum limit

Vu~(x)-N[1-m2G~,~(x)+

],

(218)

and q(x) reduces to (13) Another choice for q(x) would be one of the construetions using fibre bundle concepts [18,19] (These are not continuous functions of the hnk variables U~,(x).) The factor % in (215) depends of course on the choice of q(x). We assume it to be finite, Comparison of (215) w~th (2 % (212) and (213) shows that

~pmf = M y - M, tel(x) = Xq,q(x) + ( Z A - -

(2 19) 1 1)KAy--E Z/'/

O;A~ee(X)

(2 20)

e

The operator q(x) mixes with ) 2 0 At ~ e# ( X ) under renormahzation, the combination ?/(x) IS finite m WCPT [16] The renormahzation factor ~a can be calculated in W C P T by imposing normalized quark matrix elements of the currents The divergent part of Z A enters first in two loop order [16] Imposing some normallzanon c o n d m o n would deterlmne Z a completely See the appendix for WCPT results on the x's Non-perturbative detern'unations of x A may be based on imposing current algebra or anomaly constraints [11-13] A non-perturbative deterrmnatlon of ~e would simply follow from a comparison of the left-hand side and right-hand side of (2.13), assuming that ~A hidden in D is known In sect 6 we elaborate upon the method using anomaly constraints

238

J Start, J C ~ ml, ,/ N'eutral p~eudos(alat ma~w~

3. W T identities and pseudosealar mesons

With a source term eD~d(y) in the action the renormallzed &~ergence relanons (2 10) to order e turn into the Ward-Takahashl ldentmes (31) where c t denotes the contact terms coming from the chiral *armtlon of ~ O ~ d ( l ' ) In m o m e n t u m space the W T relations simplify after neglecting terms of order ap. ,a lp.( A.hD d) = ( D.t,D,d) + ( s . . + %,,)3.d3t, ~ + Cp ~-

(32)

(when there is no danger for confusion with the lattme &stance a we also use a as a flavor label), with

(3 3)

a

We use the abbrewated notataon

( .,,D~d ) = E e

(A.,,(X)Dd(O)).

(3 4)

\

etc The contact term Cp 2 comes from the chlral variation ol the O/,A ~' terms m eD, a().') (it is unimportant for our purpose) In the following we assume that at least the limit of an mfimte time domain has been taken, such that P4 is a continuous variable and p a m c l e poles in this variable can be identified In the spatial domain we assume p e n o d m b o u n d a r y conditions for gauge l n v a n a n t quantmes, such that we mNy set p = 0 The contribution of the pseudo scalar meson lnterme&ate states to (3 2) can be taken into account in the usual way For a 4= b a n d / o r c ¢ d only charged particle states c o n t n b u t e (1 e 13 ¢ 0, Y4= 0 in case of three flavors) Denoting these states b~ [pq) ([pq) and (Pql have the quantum numbers of ~p~,l[0) and (O[~q@. respectively), we write (pqlX.~h(0)]0) = -tp~f.~,6~p3~,,~,

(3 5)

(pqlD.h(O)[O) = d.b3w, Sh,, ,

(3 6)

fud= ~[2f,,, f~ = 93 MeV For the neutral states [1)

( j = vr °, 7, ~/', n = 3) we write

(llY~.A0)[0) = -,p%°3.,,.

(3 7)

(jlD.t,(O)lO) = d.fi.[,

(3 8)

239

J Shut, J C Vmh / Neutralpseudo~alar masse7

With these parametnzatlons, the WT relauon (3 2) can be written as (p2fab+d~b)db a ( p 2fj. + da,a) d: a m27 ~ - p2 3~a3~t,(1- 3a*') + ~ m T - +~p5 3,~b3,,:l

= -- (Saa -4- Sbb)~ad~b¢ -- ( D a b B e d )

r

/L

,

"~- tpg - - @ 2 ,

(3 9)

where mab 2 = rn2a and my"~ are the pseudoscalar masses and the primes indicate that the pseudoscalar meson poles have been subtracted According to (3 9), the residues of the poles have to cancel, i e f.hmZab = dab,

(3 10)

fjam~ = d j . ,

(3 11)

and substltutmn back into (3 9) leads to the relations f~2hm2ab = -- ( G . + shh) -- (DabDb~)', Z f j a f j ~ m ; = - (DaoDcc)',

a 4 b,

(3.12a)

a 4= c,

(3 12b)

d

E f j a f. j a m j 2 = - 2 s . . -

(Do~D..) t ,

(3 12c)

d

where we put p = 0 The right-hand side of (3 12a) evidently has to be identified with - 2 m ( ~ + ) of the continuum formulation Apart from the expected expectauon value s . . + Shb of the chlral symmetry breaking terms in the action, there is also the contribution (DahDb.)', which is less easy to interpret In the cinral limit D--, 0 it becomes a contact term which cancels divergent contributions to s ~ + s~h It is obwous that the right-hand side of (3 12a) ~s finite m WCPT because the left-hand side vamshes in WCPT More discussion on the identification of the (fq~) condensate can be found m ref [13] To proceed, we note that the correlation function (D.bDca) consists of two types of contributions, according to fig. 1,

(DohD~a) =A~(P)6ad6b~ + B.~(p)6~b3~d

(3 13)

Diagrams of the type of fig la have also meson poles for a = c, with corresponding f.a and m]a These decay constants and masses are fictitious, since the diagrams of the type of fig. l b have to be taken into account as well However, fa~ and m2.. are well defined quantities winch can be determined for example in computer simula-

J Shut, J C Dnl~ /

240

.

.

.

,~eutralp~eudo~calar ma~ae~

.

(A)

(B)

F~g 1 Illustration of the decomposition (3 13) The hnes represent quark propagator~ m the fluctuating gluon field Vacuum bubble quark loop~ are not exphcatl'v m&cated

tlons Hence. we may extend eq (3 12a) to the case a = b and use th~s relation to eliminate s . . in (3 12c) Then the following relations are obtained,

fj.jj~m]-J.2.m-~ *. . . . 8.( = - B L (0)

(3 14)

I

with again the prime indicating the subtraction of the pseudoscalar meson poles Note that these poles can only be identified when the momentum p is general and only after their subtraction p is set to zero Although the 8~A ~ terms in D do not contribute In B.~(0), they do influence B'(O) (for example, for B ( p ) = p 2 ( m 2 + p2) i. B'(O) = B'(p) = 1) The exact relation (3 14) IS useful only if ~e can find a method for eliminating the pole contributions to B.~(p) To get a feehng for the problem, let us first look at the effective action description

4. E f f e c t i v e a c t i o n illustration

The relations of the previous section are illustrated by the effective action model [101 Self= f a x

=

dx

- ~ f E [ t r 0 , U 0 ~ . U * + t r / * ( U + U ~)] + l ~ t r l n U +

-½tr0.er07-tr/*~r2-~trv+

2

~2~-

+O(~r 3) .

,t

/

(4l)

where N , is the pseudoscalar matrix field ( U = exp(12vr/J )). and X = a/N of ref [10] The diagonal matrix /2 parametrlzes the explicit chlral symmetry breaking Flavor symmetry breaking In f is neglected for simplicity (it is of order/,2) The response of the action to a local chlral transformation shows that

J O,O~r.a-fm~aTr~a + 28{a~ + O( 7r3 ) = source terms.

(4 2)

m2a= /,( +/,j,

(4 3)

J Start,J C Vmk / Neutralpseudo~alarmasres

241

so we may identify

A~d =fO.%a+ O(rr3),

(4 4)

D
(4 5)

For c = d this ldennficanon lmphes a normalization convention for the flavor slnglet current, with q= q

(4 6)

the renormahzed "topological charge density" Treating the latter as an independent field, it is straightforward to obtain various correlation functions In pamcular, in matrix notation,

1

(D.aD<<> = f 2 ( ( m 2 + Xo) m 2 + )to +p2 ( m2 + ~to) -- )to

)

(4 7a)

ac

t/)_L

m 4

=f2

(

m 2- + p 2 + m2+ p2 /,=l

-

1) rn 2 +p2

(m2)~< = rn2~.3~<, (o)~=1,

(4 7b) (4 8)

o2=no.

(49)

The matrix o projects onto the flavor slnglet state The correlation function (4 7) is illustrated in fig 2 through its expansion in powers of X The pole (m 2 +p2)-x, which comes with each power of X, is Identified in fig. 2 with a random walk of paired quark-antlquark hnes From order X2 onwards these pairs correspond to vacuum bubble quark loops We may identify A.<(p) in (3 13) with the first term in (4 7b) and B . t ( p ) with the rest. It follows from (4 7a) that

B'<(p) = -f2X

(4 10)

We may further identify

i<<=i, with Osc an orthogonal matrix dlagonahzmg the mass matrix m 2 + Xo

-<-7>">

.. +--<--=->-V C 7 > - > - -

Fig 2 Illustrationof (4 7b)

(4

II)

J Start, J C VmL / Neutralp~eudos~alar ma~se~

242

These relations illustrate the content of (3 14) It is a peculiarity of the approximarion embodmd m S~ff that B ' is independent of p For example, tins IS no longer so if we substitute )t ~ R/(m{; +p2). where m~. is the mass of a pseudoscalar glueball (before mixing with the flavor slnglet meson) The quenched approximation corresponds to keeping only the first two terms m (4 7b) (with quenched values for f, X and m understood), such that m: B~<(p)lq . . . .

hed:J

2

m2

m2 + p2

~kO

me .~ -1- ~ k O - -

me + p~

m2 + p2

me -1- --

m2 + p_, ~ k ( I - -

~kO

)

.,

(412) Here the various terms correspond to the decomposition D =Ira e~r + 2g/ Hence, in the quenched approximation B ' is given by

B'=4fdx£q(.~)q(O))

(4 13)

noquark,

5. Quenched approximation After tins effective action intermezzo we continue with the lattice derivation For the quenched approximanon there are no vacuum bubble type quark loops and fig lb shows explicitly the two quark propagators contributing to B.<(p) As tllustrated by (4 12), B.~(p) has poles at p2 = - m 5~, , and p2 = - m ~ < For a = there is also a double pole, winch is an amfact of the quenched approximation We are interested m the non-pole part B.'<(p) Its most important contribution can be found by sphttlng D..(x) into two terms,

T)..(x)= 2N.P..(x) + X..(:~),

(51)

with m . chosen such that

Iq. . . . hod =

{aalX..(0)]0)]q ....

(aal2U~,,P,.,

(0)[0)[q . . . .

hed"

hed= 0

(52a) (5 2b)

Then. in the decomposition

(D.,,D,~)tq .... bed = [4N.~<(P..P<<) + 2~,,(P<,,,2<,) +2FiT<(X~.P) + (X<,<,X,<)]q..... h~a,

(5 3)

the last term has no poles at p2= -rn~., -m~ There are also non-pole contributions in the first three terms on the right-hand side of (5 3). for example, from

J Start,J C Vmk / Neutralpseudos~alarmasses

243

radmlly excited pseudoscalar particles We assume that they can be neglected in c o m p a n s o n with the ( X X ) term, because they are suppressed by at least one power of the quark mass We even assume that they can be neglected in comparison to the O(mquark) terms on the left-hand side of (3 14), because relative to these they are suppressed by the increased number of quark loops ( " l / N " , " l / d " ) In this approximation*,

= E:,(L°(x);.(o))

(5 4)

Another form can be obtained in case of periodic boundary conditions for gauge Invarlant quantmes, for which we may invoke translation lnvariance and write 1 Batc(0) = h lnoo v E ( X a a ( x ) X ~ ( Y ) ) '~v

(5 5a) fig l b

1 hm - - E ( ( . ~ . ( x ) ) ~ ( X . ( y ) ) ~ ) v~ooV

4rrtam c hm l/~oO

o

E((P~(x)),(P,~(y)),)c,

(5 5b)

(5 5c)

r

4mJn~ = -- hm ~oo

(Tr(.lsG~.)Tr(ysG.))c ,,

V

(5 5d)

where V is the space-time volume In step (5 5c) the relation (5 1) was used and the W T relation (2 10) with s t = 0 (which holds for each gauge field configuration separately) E(D..(x))+=

E(O.A,,~' T. ( x ) ) ~ = 0

~c

(5 6)

x

The trace in (5 5d) IS over all indices except flavor and G.. is the quark propagator in a given gauge field.

G~.(x, y ) = ( ~ b ~ ( x ) ~ b . ( y ) ) , ; = ( D + M , , - W),, 1

(5 7)

Let us now look at the definition of X/j and mf m o r e closely In the quenched approximation we replace Z A--* 1 (cf (2 11), (2 12)), because Z A needs to be Introduced only when a non-minimal number of quark loops is taken into account, as is clear from WCPT From the condition (5 2a) the mass parameter nNf can be -

-

m

* From now on we omit for convemence the label "quenched"

J Stall, J { [ +nl~ / Neutral pseudmcalat masaes

244

determined in a quenched simulation, for instance bly comparing the residues of the pseudoscalar particle poles in the correlation funcUons (Ptl L). ) and ~ Pt/Pt; ) The correlation functions are to be computed from fig la only In fact. in the scaling region this procedure amounts to a check on the behavior (2 13). with the identification 2lc7 = Xfr+ O ( a }

(quenched),

(5 8) (5 9)

~l[ = I'CpDZ!-I- 0 ( a )

The behavior (2 13) should not only hold for the matrix elements m {5 2) but also between excited laa) states and 10). ~¢p computed lrom (5 9) should come out independent of m I and also B~'{ given by (5 5) should be flavor Independent (in the scaling region, neglecting O(a)) As is clear from (5 5), B.+(0) is s y m m e m c in the indices a and c It ts therefore consistent to interpret the quenched approx~maUon such that in (3 14) (compare with (4 11)) f].

=

(5 to)

,

where Oj~ is an orthogonal matrix (recall that Z A --* 1) Putting things together we obtain the quenched predmtion

{

m j : g O;.OA, m2+,.8,,b+

~.6-

4

)

¼E (( X..(+~ )5 ¢(~,6(0)) ¢) +

(5 ]1) (5 12a)

g p m atTZb

hm ~c

( T r ( T s G . . ) T r ( vsG;,,, ))ph~

V

(5 12b)

These expressions have the same form as the large N mass formulae of refs [1,2] and lead directly to (1 1). with X.h playing the role of the topological susceptlblhty In fact, X,,b as given by (5 12b) is formally equal to the topological subceptlbilat~ through the index theorem [20] In the continuum

Tr 75 G..

-

,,i + - n m.

Q

- m. ,

(5 13)

where n + ( n _ ) is the number of zero modes of the Dlrac operator in the external gauge field U with chlrahty + 1 ( - 1) and Q is the topological charge On the lattice. in the extreme continuum limit where g -~ 0 and t¢v ~ 1. the gauge field configura-

J Smlt. J C Vmk / Neutralpseudos~alar masses

245

tIons contributing to the path integral become smooth such that the assignment of Q's becomes unambiguous [18,19], and we may expect that 1

2~b-

hm v~v

--(Q2)u,

g~O

(5 14)

This also follows more directly from the verification of (5 8) in WCPT [11, 21,7] However, (5 12) is supposed to make sense also for g2 > 0, in the scaling region, where xp 4= 1 We defer the results of a study of the properties of (5 12) to another article [22] We end this section with some remarks on the boundary conditions In the infinite volume lllrnt the physics should be independent of the boundary conditions One expects this to be true of the form (5 12a) for 2 (assuming x = 0 not near the boundary) The convergence of the summation over x is presumably controlled by a non-zero pseudoscalar glueball mass m o, the coupling to quark-antlquark states being cancelled in X. Still, it could be dangerous to interchange ) 2 with the hmit g ~ o~. as in (5 12b) and (5 14) To simplify the discussion, let us assume that we are in a regime of very weak coupling where we may replace ( ~ ( x ) ) + by 12q(x), with well defined charge F~,q ( x ) The susceptibility x =

C(x)

(515)

-t

= hm

(q(x)q(O)) c

(5.16)

should be independent of the boundary conditions on the U-field (If the "naive" q(x) of (2 16) is used, then a divergent contact term (the "perturbatlve tall") should be subtracted first from C(x) before summing over x in (5 15) ) For finite V there could be power corrections V i V 2, to C(x), and a V 1 term might obviously be a problem. An example is provided by the condition of restricting the U-average to a sector of fixed topological charge Q [24] Then

]C._,(q(x)q(O))Q=Q(q(O))Q.'-~QQ

V'

x

(5 17)

where the latter equality should hold for sufficiently large V An ansatz satisfying these requirements IS given by

(q(x)q(O))

x

= C(x)-

- - -}-

v

Q2

7

-t- o ( g - 3 )

(5 18)

Hence, mterchanglng the limit V--+ oo with ~ , produces erroneous results (X =

246

J Smlt, J ( ~mh / Neutralpseudo~alat masw~

Q 2 / V ~ 0). m this example The proper hrmting order can be Imposed approximately by choosing V l/4 much larger than the correlation length and restricting the summation over .t to a region of order of a few correlation lengths Howe~er. the restriction to fixed Q is a non-local condition The effect of true boundar'~ conditions should be limited to distances ol order 1/m~. from the boundary and become negligible as V ~ oc The form (5 12b) exphcatly refers to periodic boundary condmons On a periodic lattice, integrating over all U ' s and ~ ' s generates all possible field configurations on the toms T 4 (for fermlons periodic or anupenodlc does make no difference m (5 12)) Hence all Q ' s will occur in (5 14), as it should This may seem surprising from the classical point of view, since classical gauge potentials for gauge fields with non-zero Q are not periodic However, they may be considered peno&c modulo gauge transformations (the transition functions), which allows for their restrlct~on to the periodic lattice See ref [22] for a simple example 6 Determination of xA and top

For the evaluation of ~ m (5 12) the value of xp is needed If XA IS known, D .... can be constructed and Xp can be found from (5 9) and the determination of h 5 via (5 1), (5 2) In this section we shall gwe another method for obtaining Kp by rephrasing the suggestion [11,12] of using the known normahzatIon of the triangle anomaly ("Adler-Bardeen theorem") tor the determination of ~A m a form suitable for numerical simulations We are interested in the response of the theory to ~eak external gauge fields L,~(x) coupling to (vector-hke) flavor Let v~(.~) be a continuum gauge field on a torus T 4 of size L 4 and

W,,( x ) = P e x p [ - l f " ~"" d : . v~(z)]

(6 1)

be the corresponding lattice gauge field, on an N4at lattice, ulth a = L/NL~ t (P is the path ordering operator) The lattice field W~ ina3 belong to the flavor groups SU(n), U(1)" i or U(1) For a topologically non-trivial gauge field one needs to specify how to treat various charts and transition functions of the continuum gauge field in order to obtain a unique (up to gauge equivalence) configuration of ~ ( : , )'s See ref [22], for example The field W~(a) is coupled to the theor?y by making the replacement U,(.x)8,~ ~ /.).(x)/~ = U . ( . , ) W . ( x ) / ~

(6 2)

Consider now the axial current divergence relation (2 10) It 15 natural to include the source terms "s t " corresponding to t,~ in the A-" and D operators via U~,(A) Then the ordinary derivative 0~ gets replaced by a covarlant lattice derivative D~' In

J Smlt, J C Vmk / Neutralpseudoscalar masses

a gauge where we may write W.(x)= 1 under consideration. D2 takes the form

mv.(x)+

•

247

, m the space-time region

D2A~(x) = O~A~(x) - l~(X)g~A~(x) + wu(x)~fA~g(x) +

(6 3)

We are interested in the scaling behavior of (2 10) as the lattice distance a approaches zero Projecting out the flavor non-singlet combinations w~th traceless flavor matrices )t a as in A~ = ~(X~)g/A~g(x),

(6 4)

one expects [11] (2 10) to take the form

rA( D;A:(x)) - ~p( 2mP~(x))

= ,2q2(x ) + O( a )

(6 5)

Here the mass m a m x is taken flavor symmetric and q2 as the flavor anomaly

1 q~ = 32~r2 e~ootr (~x,~v.,,vo~,),

(6 7)

or.= O.v.- O~v.- ,[vu, v~]

(6 S)

Recall that " O ( a ) " lmphes a transformation to m o m e n t u m space The only difference with the Adler-Bardeen theorem in the continuum theory is In the deviation of KA and ~p from 1 The K's, which in W C P T represent high m o m e n t u m renormahzatlons, should be independent of v., Slnnlar to their mass independence The relation (6 5) shows the difference of two non-local functlonals of t~ to be a local polynomial in the field strength v.~ A suggestive illustration m second order in v. of non-local contributions containing particle poles as in fig 3a Such contributions depend on matrix elements of the operators between intermediate states and the vacuum state, similar to (2.13), and KA and ~v in (6 5) have to be the same as the ~'s introduced in (2 11)-(2 13) By measuring (A~(x)) and (P~(x)) in a simulation. KA and Kp can be determined from (6 5) In case of the flavor diagonal group U(1) n-1 (1 e c~(x) and X~ are diagonal matrices) the covarlant derivative D2 reduces to the ordinary derivative O~ If we only want to know ~p we may sum over ~ in (6 5) to obtain - ~ P E (mPo~(x)) = t E q ~ , ( x ) = tQ2

(6 9)

(assuming periodic boundary conditions on A~), where Q~ can be conveniently chosen. For example, for n = 3 we may take vu(x) of the form v z ( x ) = v~(x)X~.

248

J Start, J C ~ ml~ -k

' Yeutralpseudus~alal

-

ma~es

_x

-

A,D

A,D

i

i

V/.,/

VZ)

V/l

(ct)

Vl) (b)

Fig 3 Somc &agrams lor ,' 4,,) ~ and Diagram (b) contnbute~ only to stogie! operator~forwhlch~D/~2m~p,P>+2teq;

with c)i(a) a U(1) gauge field over T 4 with topological charge Q, then Q3 = 0, Q~ = ~,"~ Q (using the convention X 3 = dlag(1, - 1 , 0 ) . X~ = ~,,1 dlag(1,1, - 2 ) ) A b o v e we used flavor non-singlet currents in order not to have to deal with the topological charge density q(.x) of the gluon field (hg 3b) Of course, m a simulation it is possible to exclude contributions of the type of fig 3b '" by h a n d " This allo~s one to use simply one flavor with a U(1) external gauge field c~ We m a y generate gluon field configurations without the external gauge field, quenched or unquenched, and then compute quark propagators in these configurations ~lth the external gauge field added, to study the axial current divergence relation The analogue of (6 9) for this case is

mgp(Tr(ysG))c, = Q~

(6 10)

H e r e G is the quark propagator in the gauge field L~. the average is over the gluon field configurations and Q" is the topological charge of the l'~ field Note that (6 10) also provides us with a method of obtaining the critical mass M n t which is usually determined in spectrum calculations by the condition m . = 0 (6 10) states that the quantity ( T r ( y s G ) ) ~ has a pole in the variable M at M = M~r,t, with residue Q'/~p 7. Discussion

Suggestions for using the explicit form of the axial current divergence in constructing a lattice version of q ( a ) tend to be somewhat ad hoc [7, 8, 13, 23.24] The

J Smlt, J C Vml~ / Neutralpxeudoaealar maases

249

obvious ansatz

2tq(x)--+T)(X)[M=M~n,

(7 1)

(with only the explicit M appearing m D set equal to M ~ t ) differs from our result (5 12) by the factor x e Fortunately, this factor has a clear xnterpretatlon Consider for example the form (5 12b) which may be written as =

hm

(Q~-)Ph
0 = Tr~pmys(D +

(7 2a)

m c n t - W-}- DI) -1 ,

(7 2b)

suppressing flavor labels The m In the denominator stems from the ~(x)~b(x) mass term in the action, whereas the m in the numerator corresponds to the operator m~(x)t'ys~,(x ) The factor Kp is necessary to compensate the different renormallZatlon properties of these fields, i e m~(x)+(x) and Kpm~(x)t~,5+(:~) transform as a chiral multlplet It is now straightforward to generalize (7 2) to staggered fermlons, 1 : - T r K p m F s ( D + m) ?/

1

(7 3)

where n is the number of flavors (usually n = 4), D and F 5 are the staggered constructions of O and "/5, and Xp IS the corresponding normalization factor A derivation of (7 3) along the hnes in this paper wdl be given in [6] A preliminary study of the properties of the Q ' s in (7 2) and (7 3) is recorded in ref [22] The Q may be thought as giving a solution to the "lattice U(1) problem" (1 e how to give a definition of topological charge for any gauge group that relates to the neutral pseudoscalar masses on the lattice) We like to thank M F L. Golterman, M Teper and K Johnson for discussions This work is financially supported by the Stlchtlng voor Fundamenteel Onderzoek der Materie (FOM) Note added Recently a non-perturbatlve calculation of the ~'s has been done using the current algebra method [25] Note added in proof Below eq (5 16) we remarked in passing that a divergent contact term has to be subtracted from the qq correlation function C(x) when the naive q of (2 16) is used If this would be implemented by evaluating C(x) for x >> a and extrapolating to

250

J Start,J C' lima / Neutralpseudo~calarmasses

a = 0, then the resulting X may come out negaUve This is because the euclidean q(x) is an antlhermltlan operator in Halbert space and for x > 0 (non-overlapping q ' s ) we can Insert intermediate states and find a negahve spectral ~elght funcUon To obtain the correct X. a 8(x) type contact term has to be added w~th a posture coefficient According to ref [26] this contact term is unambiguous and can be calculated in terms of the spectral ,~elght funcUon Thus it ma~ m principle be possible to subtract the much discussed "perturbatl~e tail" [17] unambiguously

Appendix N o t a t i o n herrmtlan g a m m a matrices, m and g are the bare parameters Otherwise we prefer to stay with the notation of [11], winch uses •'s lor finite latuce artifact renormahzatlons and Z ' s for renormahzatlon factors which are familiar f r o m c o n t i n u u m W C P T The difference m notaUons used by various authors can be confusing, a partial list of translaUons is g~ven by KA V = ZA V

[13],

M=Mo+4r/a

M:M+4,/a

[13],

[13],

m(=m 0 [11])=(M 0-M)(I+C)

= ,~

[13],

[13]

(A 1)

The constants x A and Kp can be calculated m W C P T from the quark-quark ~ertex function D a ( p ' p ) of the unrenormahzed divergence D t~(x) To one-loop order it can be expressed as [11]

D 4( p ', p) = g,z,~l( r )2mtT5 + fi, 2d 4( r )t( p - p' )tV5 + the vertex funchon of2mPt~(.x) + O(g 4 ) + O ( a ) ,

(A 2)

(cf (5 35) in ref [11], m is taken flavor symmetric), with

(A3)

tCA(g2, r ) = l - - d A ( r ) , ~ 2 + O ( g 4 ) ,

and ~2 = Cvg_ ~g- Comparison with (2 12). (2 13) sho~s that ~,x( p , p) = (1 + j~2~/1 + ) × (vertex function of 2mPj~(x)), or ~p = 1 + S J l ( r ) ~ 2 +

O ( g 4)

(A 4)

( Z . - 1 and q(x) enter only in order g4) Alternatively. we can express the content of (A 2). (A 3) in the form (2 15) using (2 9). which leads to an expression for M. M:

m - (1 -1-J~/l~r2 Jr-

)tH

=Ment - (J~'ig2+ = (4, -s~/.~2 +

)m )a

1 -- ( ~.Zl,~2 @

)DI,

(A5)

where the a ~ terms m M are explicit An expression for M ~s g~ven m eq (A 2) of

J Shut, J C Vml~ / Neutralpseudos¢alarmavses

251

ref [13], from which -~'1 can be identified as z~'I ( r ) = r

2 fer

j

d4p

f

--4~-~A~+ -,~ (2~r) - "

Esm2p~

+r-)~2cos"

A2= ~sen2pt~+4r2(~sln2(½p~)) 2

Tp~

A2-, ( A 6 a )

(A 6b)

The quantity ~p takes into account the mismatch in "strength" between the scalar and pseudoscalar densities, such that mS/g(x) and ~emP/g(x) fit into the same chlral multlplet Such a "strength" compensating factor was denoted (1 + C) 1 in ref [13], so we should expect that ~p=(l+C)

I

[131

(A7)

Indeed, It lS straightforward to bring the O ( g 2) expression (A 12) in ref [13] for - C into the above form (A 6) for ag I From this "strength" viewpoint Kp can also be obtained from the ratio of the finite renormahzatlon factors for the scalar and pseudoscalar densities between lattice and continuum, as calculated in ref [14] Numerical evaluation of (A 6) gives ..~'1(1) = 0.061065

(A.8)

As a function of r 2, d I rises with infinite slope from 0 at r 2 = 0, reaches a m a x i m u m of 0 0730 near r 2 = 0 22 and then drops slowly to the value (A 8) in r 2 = 1 The coefficient d A is calculated by several authors, see [12] for references For completeness we quote the result [12] 4dA(1) = 0 073143(2)

(A 9)

Fig 1 In ref [12] illustrates the r dependence of dA(r)

References [1] [2] [3] [4]

E Wltten, Nucl Phys B156 (1979)269 G Venezlano, Nucl Phys B159 (1979) 213 G "t Hooft, Phys Rev Lett 37 (1976) 8, Phys Re~ D14 (1976) 3432, Ph~s Reports 142 (1986) 357 S Coleman, The use~ of mstantons, m The wh~'5 of subnuclear physlc~ (Ence 1977) ed A Zlchlchl (Plenum 1979) [5] R Crewther, Rav Nuovo Clm 2 (1979) 63 G A Chnstos, PhTys Report~ 116 (1984) 251

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J Shut J C ~ ml, /

~euttalpseudo~alat pna~se~

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