Nuclear Physics B (Proc. Suppl.) 26 (1992) 363-365North-Holland

FROM THE LATTICE WORLD TO THE REAL WORLD

A. I . Sanda

Rockefefer University New York,NY 10021,

U S.A .

1 . Introduction

Computational limitations prevent present daylattice simulations with pion mass having realis-tically light values . So, often the lattice resultsare in a world which is rather far from the chi-ral regime (i.e . large pseudoscalar masses), andtherefore, require extrapolation to a world witha small pion mass . It may then be convenientto have a model which can be used to interpretthe lattice results and relate them to physicalobservables .

Such chiral lagrangian approach is not suit-able for analyzing this problem . High masses forthe psedoscalar mesons must be generated by alarge explicit chiral symmetry breaking . The chi-ral symmetry breaking can not be treated per-turbatively. In this paper, we shall investigatethe linear o, model which allows one to introducea large explicit chiral symmetry breaking [1] .(a) The linear o, model

In this section, we describe the crucial differ-ence between the linear and the non-linear chiralLagrangilans .We record here the linear a model Lagrangian :

fC = Tr(a,,Mc1,,M+) .-- 4(Tr(hYM+))2

0920-5632/92305.00 01992- Elsevier Science Publishers B.V

All ridhtç reserved_

(b) The nonlinear chiral lagrangian

Using the linear o model, we study the apprcach to chiral symmetry as the pseudoscalar meson mass (M) is variedfrom ti 500MeV down to the physical value - M,r . We show that, even at a tree graph level, the model implies ahighly non-trivial M dependence of the decay constant (FM), the constituent quark mass (my) and the scalar (c)mass (MQ) .

-4TT((MM+)2) -f- p2Tr(MM+)+pc(detM + det M+)

+-l Tr(m°(M + M+))

Here M = S + iP, S and P being nonetsof U(3) scalar and pseudoscalar mesons, respec-tively ; m° is a diagonal current quark mass ma-trix with diag(mû, má, m°). Indeed, the pres-ence of gin° generates an explicit chiral symme-try breaking . When p2, ai and A2 > 0, andthus < S >96 0, the chiral symmetry is brokenspontaneously. Due to the presence ofan explicitchiral symmetry breaking term with m°, the vac-uum is not SU(3) symmetric, i.e < S >= Sibij .The explicit m° dependence of < S > can becomputed .

Contrary to the linear a, model, m° is nowtreated as a perturbation . The ground state isdefined by:

< S >= Fbij .

(2)

When < S > is proportional to the unit ma-trix, eq.(1-) simplifies greatly[1] :

£NL = Tr(a� UagU+) + K(det U + det U+)

+ -LTr(m°(U + U+))

(3)

where

=ezp(iP/ )F.e nonlinear approach gives a very good ac-

count of the pion dynamics provided one is inter-ested in the region where the expansions in m°and the number of derivatives make sense.

2 . h dependence of F, m. and M,,,

n either of these approaches one has :_ 1 m®

2G F.where we denote F, = 2 < Si >.

In the lattice computations, physical observ-ables are studied as functions of the psuedoscalarmass , The explicit chiral symmetry break-ing parameter m° must be increased, in orderto consider large M . But remember that in thenon-linear chiral Lagrangian approach, a powerseries expansion in m° is made. Thus one hasto be careful to stay within the radius of con-vergence . A psuudoscalar mass of 500MeV maynot be tolerated - making the use of chiral la-grangian approach invalid . This problem can beavoided by the use of the linear o, model . Sincethe vacuum expectation value < M r is com-puted including thr explicit symmetry breakingterm in the lagrangian, the m° dependence canbe computed to all orders .

For definiteness, we shall use the linear sigmamodel studied in detail by Chang and Ilay-maker[2] . The Lagrangian, given in eq.(1), hasthe following parameters : A a = 74.77 ; m.3/m° =30; A2 = 110.4 ; 1`2 = 0.15GeV2 ; m°/G =21i,í.2®/F, ; 'c = 4.45GeV are set so that Mme , MK,

1,7 are reYroduced at the tree level .In order to simplify the initial analysis, we take

the following SU(3) limit :

< Sl >=< S3 >=< S >

FA=FK=F=2<S>

.l S

IFrom thelattice world to the realworld

(a) The Decay ConstantTo obtain the functional dependence of the

decay constant on the psuedoscalar mass M,we start from the mass obtained from the lagrangian, eq.(1) :

M2 = 1 [A2F2 + Ai 3F2 - 21S2 -UF] .

Now we can solve eq.(5) for F in terms of M.In Fig . 1 this result is compared with the latticeresult . It should be noted that the higher ordercorrections are e cpccted to become increasinglyimportant at hig%er M .

57 200m

(b) The Constituent Quark Mass

F-mq,

a

Figure 1 . The psuedoscalar decay constant as afunction of the psuedoscalar mass[3] .

To understand the origin of the sigma model,one can consider the Nambu-Jona Lasinio modelwhich posesses chiral symmetry . The sigma modeleq.(1) can be derived by integrating out thefermion fields . In such a derivation, we obtain[4] :

where mq is a costituent quark mass . Figure 2shows m q as a function of the psuedoscalar massin the SU(3) limit . Here the normalization fac-tor is chosen so that rPq = 300 Mei1 in the chi-

ral limit . The result is compared with the latticecomputation of rn. in the Landau gauge.

0.6

E 0.4

0.20

Figure 2 . The constituent quark mass as a func-tion of the pseudoscalar mass[5] .(c) The o, Mass

In the linear a model, nonvanishing al orte mixes o, and Q' . The mass matrix elements

and MQ,o , can be obtained formeq.(1) . Figure 3 gives Mó, the smaller eigenvalueof the mass matrix as a function of M.

LANDAU 24'x4Oß-6.0

a"- 1.7GeV

2 (GeV z)

A.I. Sandy /From the lattice world to the real world

3. Conclusion

The nonlinear chiral lagrangian is the correctlow energy limit of QCD. It is the leading termof the power series expansion in both the nurdof derivatives, and the current quark m M0 .

However rigorous this approach may be, it is notuseful for guiding the lattice simulation efforts.The computational requirements forces the pionmass to be large (> 500MeV). The dynamics ofthe world with such a large pion mass is outsidethe region of validity of the chiral lagrangian ap-proach .

In this paper, we have considered the linearsigma model to describe the dynamics of the lat-tice world . We have studied the pseudoscalar de-cay constant, constituent quark mass, and thea mass as fm:ctions of the pseudoscalar mesonmass .AcknowledgmentsThis work is supported in part by U.S . De-

partment of Energy, Grant number DOEAC02-87ER-40325.TASKH, and the NSF Contract Grantnumber INT90-16750 .

r

Referencesmv0 [i] For the full account of this work including the con-

sideration of weak matrix elements, and also for re-lated references see: T . Morozumi, A. I. Sanda, A.Soni, to be published .

[21 1-H Chan and R . Haymaker Phys . Rev . D10 (1974). 4143 .

0 .2 .4 .6 .8[3) For the data points, see C . Bernard,et.al., Proc . Int.

M (GeV) Conf. on Field Theory, Tallahassee (19W), Nucl.Figure 3. The o, meson mass as a function of the Phys. B(Proc. Suppl.)20 (1991) 410 .psudoscalar mass . [4] See for example, D. Ebert e.nd H . Reinhardt, NucL

Phys. B271 (1986) 188 .

[51 C. Bernard, et.al . Phys. Rev. D38 (1988) 3540 givesdata points .