Lattice QCD with exponentially small chirality breaking

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Nuclear Physics B (Proc. Suppl.) 83-84 (2000) 645-647 www.elsevier.ni/locate/npe

Lattice QCD with exponentially small chirality breaking A.A.Slavnov ~* ~Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina st. 8, GSP-1, l17966,Moscow, Russia A new multifield formulation of lattice QCD is proposed. The model is free of spectrum doubling and preserves all nonanomalous chiral symmetries up to exponentially small corrections. It is argued that a small number of fields may provide a good approximation making computer simulations feasible.

1. I n t r o d u c t i o n . In the present talk I describe a new version of a multifield formulation of QCD which has no spectrum doubling and provides exponential suppression of all chirality breaking effects [1]. The model does not contain any dimensional parameter except for a lattice spacing and does not require a fine tuning of any chiral noninvariant counterterms. Although the model include an infinite number of fields, we present the arguments indicating that truncation of the series by a finite number of terms produces exponentially small corrections. So this model may be a competitive alternative to other approaches like truncated overlapp [2] or domain wall models [3]. 2. E f f e c t i v e a c t i o n f o r l a t t i c e Q C D q u a r k determinant. In this section I present an effective action which includes an innfinite number of auxilliary fields ¢ , having the same spinorial and internal structure as the quark field but posessing an alternating Grassinanian parity ( - 1 ) " . We shall show that integrating over these auxilliary fields one gets a chiral invariant quark determinant up to to correction terms of the order e x p { - ( a q ) -1} where q is a typical external momenta and a is a lattice spacing. In the same way one may compute correlation functions of arbitrary gauge invariant operators which include an even number of Fermi fields and therefore may be presented as

fermion determinants. Our effective action looks as follows oo

I = ~

1

-

,

- ~ ¢ , ( x ) [ % , ( D t ` + Ot`) -

~

x ,p n:--oo,n?~O

-n~D*~Dt`)]¢n(z), Dt`¢ = l [ u , ( z ) ¢ ( x + at, ) - ¢(x)]

(1) (2)

The first term in the eq.(1) represents the chiral invariant part of the action and the second term is the Wilson term multiplied by n. The dimensionless parameter t¢ is chosen in the interval 0<~<1. Let us consider some fermion loop diagram with L vertices, which may be presented by the following integral: ~ra-1

HL = f t J - r:a-1

d4P

( - 1)" n=-c~,n¢O

Tr [G(p)r+(p, ql)G(p + q l ) . . . ]

(3)

Here G are the Green function of the fields ¢ , , 1"i are the interaction vertices and qt is a total momentum entering the corresponding vertex. Taking the trace we can present HL as the following SUITI: ~ra-- 1

c~

•R(~-- i

n=-oc,ny~O

Z

F0(p, q) + nZ Fl (p, q) + . . . ( - 1 ) n ( - ~ T ~-~-2)(-~ "+-~m~i .-:.

(4)

where * s u p p o r t e d in p a r t by R B R F INTAS-9670

grant 99-01-00190 a n d

st,' ----a -1 sin[(p + ql + ... + qt)t,a]

0920-5632/00/$ - see front matter © 2000 Elsevier Science B.V All rights reserved. PII S0920-5632(00)00402-3

(5)

A.A. Slavnov/Nuclear Physics B (Proc. Suppl.) 83--84 (2000) 645-647

646 mt = ~ a - l ~ ( 1 -

cos(p+ q l + . . . + qt),).

(6)

Let us consider this equation for the case when all the momenta qt are external. In this case we can assume that Iqtl < ea-1,

Le < 1.

(7)

The integration domain in the eq.(4) may be separated into two parts: V1 : [p[ < e a - 1 ;

v2: Ipl >

(s)

In the domain V2 the integrand may be expanded in terms of qta. Therefore integration over this domain produces only local counterterms. To calculate the integral over V1 we replace the sum in the eq. (4) by the following integral in the complex plane / c lr (Fo + z2F1 +...)F(P,q)dz sin(z~r)(s02 + m2z2)...

(9)

Here the contour C encloses the real axis except for a vicinity of the point z = 0. By Cauchi theorem this integral is equal to the integral over the large circle plus the sum of the residues at the poles situated on the imaginary axis at the points

which coincides with the corresponding diagram in massless QCD with "naive' fermion action and cutted Brillouin zone [qla[ < e plus local counterterms. As our procedure is obviously gauge invariant, the amplitude obtained in this way is also gauge invariant. Global chiral invariance of multiloop diagrams requires a special study. Individual fields en generate chiral noninvariant terms like fermion mass counterterm. However it was shown in our paper [1] that after summation over all these fields the noninvariant counterterms cancel. The idea of the proof may be illustrated by the two loop diagram. One can show that individual noninvariant counterterms have the following structure:

nf(n2)(p,~K¢,~

(13)

• Here K is some local operator, which may depend on the gauge fields, but has no dependence on n. The function f(z) is analytic in the complex z plane cuffed along the imaginary axis from 2i to 2i The integrand of the diagram with the counterterm insertion may be presented in the

form similar to eq.(9):

fc

(Po+ z2Pl +...)

(s0 + m0 z2) .-- ( 4 - 1 +

×

~rz2f(z2)gF(P' q) dz

(14)

sin(zTr) z=0,

z2__

s~

m 2"

(10)

The residue at z = 0 gives

FoF(p, qy

(11)

which is the manifestly chiral invariant expression for the integrand of a massless fermion diagram. The integral over large circle vanishes and the residues at the remaining poles are proportional to [sinh(~-~t)]-I ~-" exp{-[(p+ql+...+ql)at¢]-l}.(12) One sees that for external momenta much smaller than the cut-off these terms are exponentially suppressed. So we proved that any one loop quark diagram is equal to a manifestly chiral invariant expression

The integration contour again encloses the real axis except for a vicinity of zero. Transforming the integration contour as before we shall have again exponentially small contribution from the poles at the imaginary axis and also the contribution from thr intrgral enclosing the cut from 2.3_/to ~ . However due to the symmetry properties of the integrand this integral is equal to zero, and we arrive to the same conclusion as in the one loop case. Two loop diagrams posess global chiral invariance up to exponentially small corrections. This proof is easily generalized to diagrams with arbitrary number of loops. It was also shown in our paper [1] that in anomaly free theories chiral Ward identiies are satisfied by the Green functions generated by the action(I) whereas the U(! ) anomaly may be present.

A.A. Slavnov/Nuclear Physics B (Proc. Suppl.) 83-84 (2000) 645-647

3. Discussion It was shown in the previous section that the path integral of the exponent of the effective action (1) in the continuum limit reproduces exactly the quark determinant of massless QCD without any chiral noninvariant counterterms. For a finite lattice spacing chirality breaking corrections are exponentially small O(exp{-(,ce)-l}),

e ~ Iq]a

Finite quark masses may be easily incorporated into our scheme. One should simply add to the effective action (1) a common mass term. It will lead to the following modification of the sum in the eq.(4):

~

n=-~,n#O

(-I)" rIl(s~+ (m,n + too)~

(15)

This sum may be transformed as before to the integral in the complex plane which may be calculated by Cauchi theorem. The residue at z = 0 gives the amplitude for the quark of mass too: H,(s~ + m~) -1

(16)

and the remaining terms produce exponentially small corrections. Of course in this case the chiral invariance is explieitely broken by the bare quark 1TIaSS.

For practical simulations one has to truncate the sum over n by some finite number N. It is easy to see that choosing N ~ e -1 one gets the corrections of order e x p { - e - 1 } . Indeed, cutting the series by a finite N results in the changing of tile integration contour in the eq.(9). Now the contour does not extend to +oo but is cutted by i N . Therefore applying the Cauchi theoren i we have to take into account the integral over large circle. However on the large circle the integrand is ~ e x p { - e - 1 } , which proves our statement. It shows that a small number N < 10 gives a very good accuracy, which makes computer simulations feasible. It is important to note that to compute the contribution of individual ¢ , field one has to deal with the usual Wilson QCD action and the well known tecnique may be used.

647

I would like to indicate that the model is well suited for simulation of quantities depending on "small" external momenta Iql < a - l - Some care should be taken if one applies it to mass spectrum calculations. The standard procedure is based on the study of asymptotics of correlation functions < J(t)J(O) > ~ e x p { - m t }

(17)

All momenta, including tq] ~ ca-1 contribute to these functions and therefore our procedure is not applicable in a straight forward way. To use our model one has to consider a smeared correlation functions E e x p { ( x 0 - t ) 2} < J(xo)J(O) >

(18)

50

Choosing properly a parameter a on can cut all "large" momenta in the Fourier transform of this amplitude and at the same time preserve the asymptotic behaviour (17).

Acknowledgements. I am grateful to the organizers of the Conference "Lattice-99" in particular A.Di Giacomo for hospitality and financial support.

REFERENCES 1. A.A.Slavnov, Nucl.Phys. B544 (1999) 759. 2. H.Neuberger, Phys.Rev. D57 (1998) 5417. 3. V.Furman and Y.Shamir, Nucl.Phys. B439 (1995) 54.