The U(1) problem and topological charge on a lattice

366

Nuclear Physics B (Proc. Suppl.) 4 (1988) 366-370 North-Holland, Amsterdam

THE U(1) PROBLEMAND TOPOLOGICAL CHARGE ON A LATTICE Tomoteru YOSHIE I n s t i t u t e of Physics, University of Tsukuba, Ibaraki 305, Japan The U(1) problem is investigated by taking Wilson fermions as quarks. F i r s t we show that the ~ - n s p l i t t i n g is caused both by the existence of t o p o l o g i c a l l y n o n - t r i v i a l gauge configurations and by the fact that the u and d bare quark masses are very small: This is based on the fact that quark propagator has a pole at some hopping parameter close to Kr for a t o p o l o g i c a l l y n o n - t r i v i a l configur a t i o n because of the Atiyah-Singer theorem. Then we cal#ulate the propagators of the pion and the q meson in the quenched approximation. When the quark masses are small, the ~-n s p l i t t i n g is clearl y seen. We also estimate the masses of f l a v o r s i n g l e t mesons and f i n d that the estimated masses are in accord with the real world. We take Wilson's fermion action II for quarks:

I . INTRODUCTION There are two a l t e r n a t i v e ways to investigate the U(1)

problem 1'2 in l a t t i c e QCD3.

One is to

calculate the topological s u s c e p t i b i l i t y in pure gauge

theory

Witten-Veneziano There

and

to

check

r e l a t i o n 4'5

have been several

whether is

works

because i t

one for

D(n,m;K) = 6n, m - K ~ { ( I - ~

)g (n,~) n+~,m

the

satisfied. How-

ever we should keep in mind that the r e l a t i o n is an approximate

(3)

+ (l+~a)U+(m,u)

m+~a,n} . (4)

computing the

topological s u s c e p t i b i l i t y on a l a t t i c e 6. only

S = - ~ ~(n)D(n,m;K)¢(m) q n,m with

the real

world,

is derived in the I/N expansion and

2. ATIYAH SINGER THEOREMON A LATTICE First Wilson's

Here we would rather prefer to use the other

summarize

Dirac

operator

the which

properties will

be

of used

later. (I)

in the chiral l i m i t .

we

The spectral

representation

of

D- l

is

given by

method, that i s , to calculate d i r e c t l y the propD-l(n,m;K) = ~

agators of f l a v o r s i n g l e t mesons7.

i

For s i m p l i c i t y we consider QCD with two equal mass quarks and call

the f l a v o r SU(2) s i n g l e t

pseudo-scalar meson the q meson. We take

a

renormalization

group

improved

gauge action

1

~i(n)~(m)~5

1-K/p i

(¢i~5¢i)

and ¢i are eigen functions of the Dirac ¢i operator with eigen values l - K / p i and l-K/p~ respectively.

¢i'

When Pi is real,

~i and Pi are K independent. the eigen value of the Dirac

operator becomes a zero eigen value at K=pi . Sg = g ~ { C o Z T r (simple plaquette loop)

In

t h i s case we call the eigen function a zero mode and the Pi a c r i t i c a l

+ C l Z T r (Ix2 rectangular loop) }

(5)

(I)

hopping parameter KC.

(2) There e x i s t exact zero modes on some of

with

Monte Carlo generated gauge configurations.

C1 = -0.331 , CO = I - 8CI. (2) I t s form has been determined by a block spin RG

state of the c h i r a l i t y .

study 8 and by analyses of instantons on a l a t t i c e 9 ' I 0 .

(4) KC for each configuration scatters around ensemble ensemble the KC . Here the KC is the

the

existence

0920-5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

of

(3)

Each zero mode is an approximate eigen

367

T. Yoshi$ / The U(1) problem hopping parameter at which the pion mass vanishes. (5) I f there exists one zero mode around the KCensemble, there also exist fifteen associated zero modes in the unphysical hopping parameter region (I/K<4).

• Gc • =GDC =

K=OA566

101

We have investigated the Atiyah Singer theorem12 for smooth gauge configurations.

For all

100

configurations we have prepared by cooling configurations generated by Monte Carlo simula-

10-t

tions, we find that the relation Q = . .n+ ..... is satisfied.

n_"

(6)

zero modes around the KCensemblewith positive (negative)

10-2.

Here "n+" ("n ") is the number of

approximate c h i r a l i t y and Q is the

topological charge. (Topological

lO-S.

charge on a

l a t t i c e can be unambiguously defined for con-

10-4-

figurations smooth enough.) The relation between the topological charge and the number of zero modes becomesmore subtle

TIME

at f i n i t e B, because the concept of topology has no intrinsic meaning on rougher configurations.

FIGURE l

However the number of zero modes has definite

The CC and GDC without Fourier transformation on a topologically non-trivial gauge configuration

meaning even on a rough configuration. Hereafter we say that a configuration is topologically non-trivial when i t has an exact zero mode. (See

same as the pion propagator,

ref.13 for related works.)

much smaller than the GC for r e l a t i v e l y heavy

3. U(1) PROBLEMON A LATTICE

ty to solve the U(1) problem.

quarks.

The propagators of

the

n

meson and the

This has been regarded as the d i f f i c u l -

We have verified that only for a topologically non-trivial configuration and only when the

meson are given by Gq(n,O) = GC(n,O) - Nf GDC(n,O)

(7)

hopping parameter is very close to the KC of a

g (n,O) = gC(n,O)

(8)

the GC.

zero mode, the GDC is large and comparable to

and ,

Let me recall

where

gC(n,O) =

(9)

(eq.5).

the

spectral

representation

For a topologically non-trivial config-

uration and when the K is close to the KC, the

and

GDC(n,O) = . Here Nf i s the number of f l a v o r s . account f o r the ~-n s p l i t t i n g , at

Howeverthe DDC is

least

connected

comparable part

of

the

to q

the

quark propagator is well

(lO) In o r d e r t o

of K at KC in the spectral representation. The

the GDC should be GC

approximated by the

contribution from the zero mode, due to the pole

because

the

p r o p a g a t o r GC i s

the

contributions of the zero mode ¢ are identical both for the GC and the GDC and are given by

T. Yoshi~ / The U(1) problem

368

GC(n,O) ~ GDC(n,O)=1 ¢+(n)¢(n)¢+(O)¢(O) (I-K/Kc)2 (¢+~50) 2

G has an n pole and therefore the GDC has both n a pion pole and an n pole. We here assume as

(11)

working hypothesis that the behavior GC~cIe-m~t

Therefore the GDC is large and comparable to the

and. GDC~c2e-m~t+C3e-mnt

is v a l i d even in the

GC in t h i s case.

quenched approximation.

The c o e f f i c i e n t s

In f i g . l

we show a typical

propagator on a t o p o l o g i c a l l y n o n - t r i v i a l figuration

obtained

on

an

83x16

con-

lattice

at

Ci's

are in general d i f f e r e n t from those in f u l l QCD.

find that the GDC is always much smaller than

Gq is obtained by replacing Nf in eq.7 by ~=CI/C2: G = GC - ~GDC (12) q We determine the { by equating ~=GC (t=8)/G Dc

the GC even i f the quark mass is small.

(t=8),

8=2.4. For a t o p o l o g i c a l l y t r i v i a l

configuration, we

In t h i s way we have shown that the ~-n s p l i tting

is

caused

by t o p o l o g i c a l l y

non-trivial

If

the assumption is v a l i d ,

because Gq should be very small at t=8. (When the ~ thus determined is larger than Nf, t h i s is indeed the case for heavy quarks, we set

gauge configurations.

~=Nf.

4. PROPAGATORSOF FLAVOR SINGLET MESONS

quarks. ) Our assumption GDC~c2e-m~t+C3e-mqt

The

q

propagators

propagators

are

approximation.

as

well

as

in

the

calculated

The results I w i l l

the

pion

quenched

The ~'s are smaller than Nf for

from that of the I/N expansion.

light

differs

I f the l a t t e r

is v a l i d , GDC has a double pion pole.

However

report here

the propagators we have obtained are consistent

are based on I0 configurations separated by 500

with our assumption and the GDc does not seem to

sweeps at

behave as the prediction of the I/N expansion at

~=2.4 on an 83x16 l a t t i c e .

We f i n d

that nine of them are t o p o l o g i c a l l y n o n - t r i v i a l

least for r e l a t i v e l y heavy quarks.

and only one of them is t o p o l o g i c a l l y t r i v i a l .

for d e t a i l s . )

When we discuss

the

U(1)

problem

in

the

(See ref.7

I f we calculate the GC and the

GDC on a larger l a t t i c e ,

i t would be possible to

quenched approximation, we should be concerned

judge which assumption is v a l i d .

with the v a l i d i t y of the approximation.

that our assumption is v a l i d and calculate the

remind you that not

so

bad

Let me

the quenched approximation is

for

flavor

non-singlet

Here we assume

n propagators by the method explained above.

hadrons.

Fig.2 shows the propagators of the pion and

Fukugita, Oyanagi and Ukawa14 have indeed shown

the n meson at various hopping parameters.

that the ground state masses of f l a v o r non-sin-

~-n

glet hadrons in f u l l QCD are almost i d e n t i c a l to

quarks

as shown in f i g . 2 a .

those

to deviate from that of the pion at K=

the

relatively

heavy

The n propagator

starts 0.154.

Moreover we have recently shown that the masses

sponds to the strange quark mass. When the quark

non-singlet

if

absent for

one

flavor

quenched approximation,

is

renormalizes the bare gauge coupling constant. of

in

splitting

The

hadrons in the quenched

approximation agree with experiment with at most 15% error 15. effects

However, generally speaking, the

of dynamical quarks may be crucial for

the ~ - n s p l i t t i n g .

Therefore the issue is how

This hopping parameter

masses get smaller, the clearer,

and when the

almost corre-

~-q splitting

becomes

quarks have masses of

almost the u and d quark masses (K=0.1564), the ~-n

splitting

is c l e a r l y seen (Fig.2c).

This

shows that mn is much larger than m~ when the

much we can absorb the dynamical quark effects

quarks have physical u and d quark masses. This

by renormalizing B.

is the most important conclusion of t h i s work.

If

we calculate

expect that

the G

G

and G

correctly,

has a pion pole

we

and the

Fig.3 shows the propagators of the p meson and

the

m

meson.

There

is

no n o t i c i a b l e

T.

369

Yoshib / The U(I) problem

102

102

a

101

• •

K : 0.1400

= Gr : G'q

b

101

100

100

10 -t _

10-I _

10-2-

10-2_

10-3_

I0-3_

10-4-

10 -4_

•

K:0.1540

-'o 10-5-

I

10-6_

10-6_

I'4

o

10 -7

4

0

16

~

¢

•

10~

K:0.1564

the f i n i t e As I

•

O

+

16

are commom t o a l l

have a l r e a d y p o i n t e d difference

out,

between

of the

there

the

p r o p a g a t o r and t h e m meson p r o p a g a t o r .

+

is

p

no

meson Thus we

estimate that

I0-3.

m ~ mp (13) This e s t i m a t e is c o n s i s t e n t w i t h e x p e r i m e n t .

I 0 -4-

We n o t i c e

i0-5_

K=0.1564 ~0-6.

physical

10-7

4

~

cal

The ~ and q propagator without Fourier t r a n s f o r m a t i o n f o r the ensemble average

This i m p l i e s t h a t the

p and t h e m a r e almost degenerate.

Our l a t t i c e

reduce t h e f i n i t e

singlet

s i z e is not l a r g e enough t o

s i z e e f f e c t s c o m p l e t e l y and to

determine the masses from the a s y m p t o t i c behavthe p r o p a g a t o r s .

and

d

quark

the

q

propagator

corresponds masses)

to

and t h a t

at the at

strange quark mass) are almost i d e n t i c a l

to

Thus we e s t i -

mate t h a t m ~m , nSU(2 ) P

(14)

m ~m nstrang e P

(15)

( q s u ( 2 ) = I / , / 2 " ( u x 5 u + d ~ 5 d ) and nstrange=~X5 s.) These e s t i m a t e s are in accord w i t h t h e r e a l

5. MASSES OF FLAVOR SINGLET MESONS Let me e s t i m a t e t h e masses o f f l a v o r

u

both almost

t h e p(~) propagators at K=0.1564.

FIGURE 2

between them.

that

(which

K=0.154 (which almost corresponds t o t h e p h y s i -

6 I'o I'2 i~ TIME

of

size effects

noticiable

10 -2 .

ior

12 I~

meson p r o p a g a t o r s .

10-t

mesons.

Ib

t h e e s t i m a t e s f o r meson masses by assuming t h a t

= G~

• = G~

10o

difference

8 TIME

TIME I02

,"

I 0 -s-

I 10-7

= Gw

• = G~

However we can o b t a i n

world,

because i f

we assume t h a t the t r a n s i t i o n

mass m a t r i x between the nSU(2) and t h e q s t r a n g e is 200MeV, we would o b t a i n t h e c o r r e c t masses of t h e n' meson and the q meson.

T. Yoshi~ / The U(I) problem

370

ACKNOWLEDGEMENTS This 10z

work

Y.lwasaki

10~

• = G#

K=0"1564

tions

• : Gw

was done in

and S.Itoh.

reported

HITAC $810/I0

collaboration

with

The numerical calcula-

here have been performed with at KEK.

I would l i k e to thank

10o

members

10-t

T.Yukawa for t h e i r warm h o s p i t a l i t y and strong

of

KEK, p a r t i c u l a r y

H.Sugawara

and

support. 10 -z

REFERENCES

10-3_

I. S. Weinberg, Phys. Rev. DII(1975) 3583.

*+f+*

10 -4_

2. G. 'tHooft, Phys. Rev. Lett. 37(1976) 8.

10 -5_

3. K.G. Wilson, Phys. Rev. D14(1974) 2445.

10-6.

4. E. Witten, Nucl. Phys. B156 (1979) 269. 5. G. Veneziano, Nucl. Phys. B159 (1979) 213.

10-~

6. J. Hoek, M, Teper and J. Waterhouse,Nucl. Phys. B288 (1987) 589; M. G~ckeler, A.S. Kronfeld, M.L. Laursen, G. Schierholz and U.-J. Wiese, Nucl. Phys. B292 (1987) 349; Y. Arian and P. Woit, Phys. Lett. 183B (1987) 341; J. Smit and J.C. Vink, Phys. Lett. 194B (1987) 433.

TIME

FIGURE 3 The p and m propagators for the ensemble average 6. CONCLUSIONS We have shown that the qSU(2 ) is much heavier than the pion in the quenched approximation. We have also c l a r i f i e d

the important role of the

topological e x c i t a t i o n s for the ~ - n s p l i t t i n g . In future calculations we have to include the effects

of

account

the

dynamical mixing

quarks and to of

the

nSU(2 )

take into and the

7. S. Itoh, Y. lwasaki and T. Yoshie, Phys. Lett. 184B (1987) 375; Phys. Rev. D36 {1987) 527. 8. Y. lwasaki, Nucl. Phys. B258 (1985) 141; preprint UTHEP-II8. 9. Y. lwasaki and T. Yoshie, Phys. Lett. 131B {1983) 159. IO.S. Itoh, Y. lwasaki and T. Yoshie, Phys. Lett. 1478 (1984) 141. II.K.G. Wilson, in New Phenomena in Subnuclear Physics, Erice, 1975, ed. by A. Zichichi (Plenum, New York, 1977).

nstrange" In the quenched c a l c u l a t i o n ; we have adjusted

12.M. Atiyah and I. Singer, Ann. Math. 87 (1968) 484.

the ~ factors to obtain the n propagators.

In

full

If

13.F. Karsh, E. Seiler and 1.0. Stamatescu, Nucl. Phys. B271 (1986) 349; E.-M. l l g e n f r i t z , M.L. Laursen, M. MUllerPreussker, G. Schierholz and H. S c h i l l e r , NucI. Phys. B268 (1986) 693; I. Barbour and M. Teper, Phys. Lett. 175B (1986) 445; J. Smit and J.C. Vink, preprint ITFA-87-7 and references therein.

QCD, there is no room to adjust the 4.

we calculate the n propagators by eq.7, that i s , by setting the ~ the number of the f l a v o r s , the q propagators would be well behaved. in t h i s d i r e c t i o n is in progress.

The work

We would l i k e

to report the results of the propagators in f u l l QCD in the near future. The d e t a i l s

of

the analyses presented here

have been reported in r e f . 7 .

14°M. Fukugita, Y. Oyanagi and A. Ukawa, Phys. Rev. Lett. 57 (1986) 953; Phys. Rev. D36 (1987) 824. 15.S. Itoh, Y. lwasaki and T. Yoshie, Phys. Lett. 183B (1987) 351.