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Nuclear Physics B (Proc. Suppl.) 106 (2002) 581-583

PROCEEDINGS SUPPLEMENTS www.elsevier.com/locate/npe

Physical observables from lattice QCD at fixed topology* R. Brower a, S. Chandrasekharan b j . W. Negele c and U.-J. Wiese ct

aDepar tment of Physics, Boston University, Boston, Massachusetts 02215, USA

bDepar tment of Physics, Duke University, Durham, North Carolina 27708, USA

cCenter for Theoretical Physics, MIT, 77 Massachusetts Ave., Cambridge, MA 02139, USA

Because present Monte Carlo algorithms for lattice QCD may become trapped in a given topological charge sector when one approaches the continuum limit, it is important to understand the effect of calculating at fixed topology. In this work, we show that although the restriction to a fixed topological sector becomes irrelevant in the infinite volume limit, it gives rise to characteristic finite size effects due to contributions from all 0-vacua. We calculate these effects and show how to extract physical results from numerical data obtained at fixed topology.

1. T O P O L O G I C A L C H A R G E S E C T O R S A N D 0 - V A C U A

In principle, one should calculate observables in QCD with finite quark masses in a fixed 8- vacuum. However in practical lattice calcula- tions, algorithms that change the gauge field con- figuration in small steps tend to become t rapped in a fixed topological charge sector because they cannot overcome the action barriers between sec- tors. One sees concrete evidence of this prob- lem, for example, from the large equilibration times for the topological charge required in hy- brid Monte Carlo calculations, which increase as the quark mass is decreased [1]. In addition, im- proved pure gauge actions, such as DBW2, re- duce the low eigenmodes tha t are undesirable for overlap and domain wall fermions by suppress- ing small instantons and thereby reduce tunneling between sectors. Since we know tha t the restric- tion to fixed topology becomes irrelevant in the infinite volume limit, we have calculated the the finite volume dependence of this restriction and thereby show how to extract physical results from practical calculations at fixed topology.

The part i t ion function of the QCD Hamilto- nian H with eigenstates In, 8) in a given sector may be writ ten in tems of the lowest eigenvalue Eo(8) = Veo(8) in the large volume, low temper- ature limit as follows:

Zo = Tro e -BH = E e -BE"(O) ~ e -fWe°(O).

n

The part i t ion function at fixed topological charge Q is an integral over all 8-vacuum sectors. Ex- panding the ground state energy

1 02 1 84 eo(8)=eo(0)+ ~xt + ~ 'y +...,

expansion to and performing a saddle-point 0(I/~2V2),

i f ZQ = 2-~ , d O Zoe i°Q (1)

-+ - ~ , d8 eiOQe-~Ve°(°)

~. e-,V~0(0). ~ - ~ - V -vTo

x I 24~Vx2 3 + - - - ~ Z x, Zv x~

Using Zo = f T)A:D~:D@ e-S[A'C~,~']-iOQ[A], the parameters in the expansion of eo(8) are

*Talk presented by J. W. Negele. Work supported in part by the U.S. Department of Energy (DOE) under coop- erative research agreement # DE-FC02-94ER40818, DE- FG02-96ER40945 and DE-FG02-91ER40676. tpresent address: University of Bern, Sidlerstrasse 5, CH- (Q2) < Q4 } _ 3( Q2 }2 3012 Bern, Switzerland X.t = ~ V ' "f ~- - ~ V

0920-5532/02/$ - see front matter © 2002 Published by Elsevier Science B.V. PII S0920-5632(01)01784-4

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582 R.C. Brower et aL /Nuclear Physics B (Proc. SuppL) 106 (2002) 581-583

The topological susceptibility, Xt, and fourth moment, 7, are parameters characterizing the vacuum that can be calculated in lattice QCD. Once they are known, we may use eq. (1) to cal- culate any observable at fixed Q and obtain finite volume corrections.

2. MASS S P E C T R U M IN A F I X E D T O P O L O G I C A L S E C T O R

As a concrete example, consider the calculation of hadron masses from the large time behavior of two-point correlation functions. At sufficiently low temperature, the two-point correlation func- tion at fixed Q of operators (9 with the appropri- ate quantum numbers to create the hadron states of interest may be written

, f (O(h)O(t2))Q = Z-'-Q dO e ~°Q-¢Ve°(°)

× I(O, OfOfl,0)l%-M(°)(t '- t~),

where M(O) is the lowest hadron mass. Perform- ing a saddle-point expansion, the effective mass used to measure the hadron spectrum is

M~lf _ d log((O(tl)O(t2))Q) dtl

f dOe-½(Q2)°2+i°Qf(O) M(O) f doe- ½ (Q2)o2+i°Qf(O)

where f(O) = 1(0, 81011, 0) 1% -M(o)(tl-'~) contains all the time dependence. Expanding M(O) = M(O) + 1MU(O)O2 + 1,.M""(O)O4 + . "

0 a " 0 02+ 1 ",,,, 0 04+ and f ( O ) = f ( ) + ~ f ( ) ~.f ( ) . . .we obtain the desired expansion,

M~ f f = M(O) + ~M"(O) '~ + ~.M'"'(O)-O -4

6f"(0)M"(0) + 4!f(0) L[ g~ - ( ~ ) 2 ] , (3)

where 0--- ff = f dO exp(- 1(Q2)02 + iOQ)On

f dO exp(- ½ {Q2)02 + ieQ)

The moments required in eq. (3) are

0-- ~- 1 { Q(_~)} - ( Q 2 ) 1 -

0---~- - 1 { 6 Q 2 Q4 } (Q2)2 3 - (Q2---~ +

where (Q2) = /3Vxt by eq. (2). Note that to leading order, all dependence of the mass shift on f(0) exactly cancels and therefore the result is independent of tl - t2.

Our leading order result for the Q-dependence of the mass is thus

IM"(0 ) [1 ] . (4) M~ I f -- M(O) + 2 flVxt /~-~-Xt

Therefore, if a calculation in a volume flV is trapped in a Q sector, the error is of order 1/(flV). Furthermore, one can measure MQ in several sectors and at several space-time volumes and thereby extract M(0) , M"(O), and Xt us- ing eq. (4). Also note that when averaged over Q

with the distribution e-:(Q-~>, (~-) = 0.

3. 0 D E P E N D E N C E IN T H E CHIRAL A N D L A R G E Arc L I M I T S

It is possible to make some general argu- ments concerning the O-dependence in QCD. The transformation ¢ ' --- exp(i2--~/75)¢ shifts the 0- dependence to the mass term,

m - ~ -+ -~ {m cos(O/Nf ) + i m sin(O/NI)75 }

and the contribution to the fermionic measure cancels the FlU-term. By parity, the sin(O/Nf) term only contributes in even orders, so at lin- ear order in the quark mass, the only effect of 0 enters through the change of the mass term to m cos(O/Nf). Hence, for example, to leading or- der in the chiral limit, the pion, nucleon and 7?' masses must have the forms:

M2,~(O) = M~(O) + M2.(O)[cos(O/Nf) - 1]

MN(O) = MN(O) + ClM2(O)[cos(O/Nf) - 1]

+ c2M3(O)[cos3/2(O/Nf) - 1]

M~,(O) = M2,(O) + blM~(O)[cos(O/Nf) - 1]

from which the Q-dependence follows directly from eq. (4). In the large Nc limit, it can be shown that bl -- 1 so that the rr and ~' have the same 0-dependence and thus the same Q- dependence.

Using chiral perturbation theory, one can ex- plore the chiral and large Arc limits more system- atically. Using the effective Lagrangian proposed

R.C Brower et at/Nuclear Physics B (Proc. Suppl.) 106 (2002) 581-583 583

by Witten [2] to treat the UA(1) anomaly, we ob- tain the following results for the r and y~ masses:

. Ns°) ( . ; / 2 0 )

M~, m~ + _ cos

where eo(O) = - m ( ~ ) cos [ " ' ~ 0 + k F~ V N! N! 2No "

Solving explicitly for ~ , we also note that in the

. 0 wo ecov r limit a n d in the

( limit Nc -~ co we obtain cos k:,n(¢¢)N¢] ~ 1 as

Nc --+ c¢ as expected.

4. Q - D E P E N D E N C E I N A N I N S T A N T O N G A S

Recently, the dependence of the 7r and rf masses on Q has been studied in lattice calcu- lations [1], which for practical reasons, are suffi- ciently far from the chiral limit that the previous results from chiral perturbation theory are inap- plicable. Each mass was evaluated in two sectors, one with [QI < 1.5 and the other with [QI > 1.5. Whereas the pion mass in both sectors agreed within statistics of a few percent, the ~/~ mass was of the order of 15 % heavier in the IQI > 1.5 sec- tor than in the [QI < 1.5 sector. Motivated by the success of the Veneziano-Witten formula [3] [4] re- lating the ~f mass to fluctuations in the topologi- cal charge, we obtain a qualitative understanding of the Q dependence from the fluctuations in the topological charge arising in an instanton gas.

The vertex generating the shift in the rf mass is proportional to the number of instantons, N, plus the number of antiinstantons,/V. Assuming inde- pendent Poisson distributions with {N) = (N) = A, the probability of having N instantons and antiinstantons with the constraint N - N = Q is:

A(N+~) f d9 e-2X-iO(Q-N+FI) PQ(N, IY) = ] 2~ N!fil! J

Summing over N and N and distinguishing A and A ~ for subsequent convenience, we write the gen-

erating function

f dO AI(N+~)e_2A_iO(Q_N+N) ZQ(A,A') = E 27r N!/~r--------~

N,N

e_2A f dO e_iOQ+2A, cos(0)

. 1

1 , 2 = e2( ~ -~)e- ~"

2v7 Differentiation of In ZQ(A, A') with respect to A ', setting A = A t, and noting 2A = {Q2) = xt~V yields the desired result for the density of instan- tons plus antiinstantons at fixed Q:

(N + N)Q 1 - 1 + (5)

Eq. 5 is a simple and physically appealing re- suit. Since M~, = M~ + p and /~ c( (N+N)o which equals Xt when averaged over the ~Vtri]

_9Z bution P(Q) = ~ e ,~, the r/' mass is con- sistent with the Veneziano-Witten formula. The finite volume corrections provide the desired Q dependence.

To compare with the lattice results of ref [1], we note that ~V = 6.81fm a, Xt = 0.70fm -a, and (Q2) = l~Vxt = 4.75. In the chiral limit,

MgH(Q) - - s o 1" ] _ ~ \ ' 1

4N f L , - . ~ , - , i d

the shift at Q2 = 0 is -M,~/(4N~(Q2)) = 0.013M.. Hence, the effect of calculation at fixed topology is of the order of 1%, consistent with the lattice 7r results. The analogous chiral estimate for the ~' is of order (M,/Mo,) 2 x 1%, in strong disagreement with lattice results. However, us-

ing the instanton gas result, f d q P ( Q ) ~ is 0.65 for Q < 1.5 and 0.75 for Q > 1.5, yielding 5M,,/M, 7, = 8%, which is of the order of magni- tude of the observed Q dependence.

R E F E R E N C E S

1. G. S. Bali et al. [SESAM Collabora- tion], Phys. Rev. D64 (2001) 054502 [hep- lat/0102002].

2. E. Witten, Annals Phys. 128 (1980) 363. 3. G. Veneziano, Nucl. Phys. B159 (1979) 213. 4. E. Witten, Nucl. Phys. B156 (1979) 269.