E L S EV I ER

UCLEAR PHYSIC~

Nuclear Physics B (Proc. Suppl.) 47 (1996) 374-377

PROCEEDINGS SUPPLEMENTS

M o n o p o l e s a n d h a d r o n s p e c t r u m in q u e n c h e d Q C D

Tsuneo Suzuki ~ , Shun-ichi Kitahara ~ , Tsuyoshi Okude ~ , Fumiyoshi Shoji ~ and Osamu Miyamura b

~Department of Physics, Kanazawa University, Kanazawa 920-11, Japan

bDepar tment of Physics, Hiroshima University, Higashi Hiroshima 739, Japan

, Kazuya Moroda ~

We report the preliminary results of the studies of hadron spectrum under the background of abelian and Inonopole gauge fields in quenched Wilson SU(3) QCD. Abelian gauge fields reproduce the same chiral limit as in the full case. Critical hopping parameter ~¢ and mp are the same in both cases. We need more time to get a definite result in the case of Inonopole background. The photon contribution do not produce any mass gap in the chiral limit (n = n¢ ~ 0.17). The behavior is similar to those in the free photon case for n¢ = 0.125.

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

Recent Monte-Carlo simulations suggest that abelian monopoles after abelian projection are a key quantity of confinement mechanism in QCD [1-3]. Especially the abelian projection in the maximally abelian (MA) gauge is interesting. Then mass generation from scale invariant QCD may be explained by the abelian monopoles alone. The expectation is supported by the preliminary data of recent simulations in quenched SU(2) QCD with Kogut-Susskind ferlnions[4]. The pur- pose of this s tudy is to report preliminary results of the studies of hadron spectrum under the back- ground of abelian and monopole gauge fields in quenched SU(3) QCD with Wilson fermions.

2. M a x i m a l l y a b e l i a n g a u g e in SU(3) Q C D

We adopt the usual SU(3) Wilson action. The MA gauge ~ given by performing a gauge trans- formation, U(s ,p ) = V ( s ) U ( s , / ~ ) l / - l ( s + ~ ) , such that

R = ~ (](/11(s,/~)12 + IU22(s,#)[ ~ + IU33(s,p)l ~) s , #

is maximized. Then a matrix

[F(s,.)AoFt(s,.) # , a

+Ft(s - p., p)A~ F(s - Z, ~), Ao]

is diagonalized.Here

A1 = diag(1, - 1 , 0), A2 = d iag( -1 ,0 , 1),

A3 -- diag(0, 1 , - 1 ) .

After the gauge fixing is over, we can extract an abelian link field,

U(s, #) = A(s , tO'u(s, p),

where

u(s, p) = diag(e i°(~)(s'~) , e ~°(~)(' '~'), e ie(~)(~''))

1 # ) ( s , p ) = arg(U.(~, p)) - ~¢ ( s , p )

¢(s, p) = ~ " arg(Uii (s,/~)) Imod 2~ " i

u(s, #) is a diagonal abelian gauge field.

3. A b e l i a n g a u g e fields f r o m m o n o p o l e D i r a c s t r i ngs

a p l a q u e t t e v a r i a b l e i s g i v e n b y =

0 0Y)(s) - where

O(i)(s) : - Z D(s - s ')[O,f~,,(s ~' (i) ,) + O,(O~ O(i)(s,))] 3,

and D(s - s') is the lattice Coulomb propagator. Define abelian gauge filed as

= - D ( s - ' ' ( ') ' s )O~f~# ( s ) . 3 /

The gauge fields automatically satisfy the Landau gauge condition.

Extract the Dirac strings from the field

strength, satisfying ~ i (i) ~ . ~ ( s ) = O:

s2;,(s) = f ; ; ( s ) +

0 9 2 0 - 5 6 3 2 / 9 6 / $ 1 5 . 0 0 © 1996 Elsevier Science B.V. All rights reserved. Plh S0920-5632(96)00077-1

T. Suzuld et al./Nuclear Physics B (Proc. Suppl.) 47 (1996) 374-377 375

Then an abelian gauge field from Dirac string is

= - s ),

whereas an abelian gauge field from the photon is

o P h ( i ) l x ---- I I t s j n ( s - s S I

Abelian link fields from Dirac string and pho- ton are constructed as

u n " ( s , p ) = diag(eio,~'(')(~) eiO,°~(2)(s) eio~(3)(.~)), • i O P h ( i ) ( 1 o P h ( 2)" " " ' t t P h ( s , ~ ) = dlag(e . . . . . , eZ, ts~, ezo,Pa(3)(s)).

The criterion of the MA gauge condition is ~ loft diagonal part of X I < 10 -7 where (X is the operator to be diagonalized.) Over-relaxation method with w = 1.93 is used.

The criterion of the solution of Wilson ma- trix inversion is [residue[ < 10 -5. The hopping parameter expansion and the CG method with red-black preconditioning is used.

The mass fitting function is c c o s h ( m ( t - Nt/2)) and the mass fitting range is 4 _< Nt < 28.

4. I n v e r s e of W i l s o n q u a r k m a t r i x

Now let us evaluate the inverse of Wilson fermion matrix

[ (1 - 7, )U(s , , )6 ,+a. . , , #

+

for

• U(s , ~)

• **(s, p)

( f r i l l )

(a.belian)

• uD~(s ,p) (Dirac string)

• uPh(s, tt) (photon),

and then measure hadron mass spectrum. The simulations are performed as follows:

• Our calculation is usually done on a 8PE's system of a vector parallel supercomputer Fujitsu VPPS00. Lattice size adopted is 163 x 32. 3 = 5.7.

• Antiperiodic (Periodic) boundary condi- tions in the time (space) direction.

• Number of sweeps for thermalization is 3000.

• Number of sweeps to get independent con- figurations is 1000.

• Number of configurations for average is 20.

The CPU time , the number of terms of the hopping parameter expansion and the number of CG iterations in a quenched propagator compu- tation are listed in Table 1.

su(3): time(see.) HPE CG

0.160 230 1200 240 0.165 551(16PE) 1200 3000 abelian:

time(see.) HPE CG 0.150 482 360 1560 0.155 910 300 3600 0.1575 1188 120 4440 0.160 2074 300 8040 0.1625 3000(16PE) 300 20400 monopole: t~ time(sec.) HPE CG 0.135 345 1200 600 0.140 429 180 1560 0.1425 346(16PE) 120 2160

Table 1 Various parameters of the simulations.

5. R e s u l t s

Examples of hadron propagators with abelian and monopole backgrounds are shown in Fig. 1 and Fig. 2.

376 T. Suzuki et al./Nuclear Physics B (Proc. Suppl.) 47 (1996) 374-377

1 0 ,2

1 0 -s

10 "s

10"

10 -~4

abel ian propagator ( K:= 0 .150 )

~,.~. ........ p J ! " pro ~-~ . • " ~ ' ~ , - - ton ,,~..,. '-

\,% .. =

e a

. i~ II t l

$

i i , J i i

0 10 20 30

t i m e d i s t a n c e

10 -~

10 -3

10 "5

10 "7

I 0 "g

1 0 -11

1 0 -is

m o n o p o l e p r o p a g a t o r ( K = 0 .140 )

¢ ~...

'- '~ proton .~.~"

-\ li

0 10 20 30

t i m e d i s t a n c e

Figure 1. Hadron propagators with abelian b a c k -

g r o u n d for x = 0.150. Figure 2. Hadron propagators with monopole background for ~ = 0.140.

The est imated masses of % p and nucleon are listed in Table 2[5].

The squared masses of r and p and the mass of nucleon and p are plotted versus 1/x in Fig. 3 and Fig. 4. Abelian gauge fields alone reproduce the same chiral limit as in the full case. Critical hopping parameter x~ and mp are the same in both cases as expected. The abelian case gives a smaller ratio m~/ma at the same ~. At present we have obtained the ratio as small as m~/rnp ~ 0.48 at t~ = 0.1625.

The abelian case gives a smaller ratio mp/mp at the same x. Namely the O(a) correction of this ease is smaller in the abelian ease. At present we have obtained the ratio as small as mp/mp ,,~ 1.32 for mpa -- 0.55 ( at ~ = 0.1575).

We need more time to get the monopote con- tribution clearly and it is in progress.

Fig. 5 shows the squared masses of ~r and p ver- sus 1/x under the photon and free backgrounds. The photon contribution do not produce any mass gap in the chiral limit (~ = n¢ ,-~ 0.17). The behavior is similar to those in the free pho- ton case, although x~ = 0.125 in the free c a s e .

In summary, the above preliminary data sug- gest abelian and also monopole gauge fields are essential for the mass generation of hadrons in QCD. More da ta are needed to get a definite con- clusion. However the data are consistent with the

results of recent simulations in quenched SU(2) QCD with Kogut-Susskind fermions[4].

This work is financially supported by JSPS Grant-in Aid for Scientific Research (B) (No.06452028) (T.S.) and (C) (No.07640411) (O.M.). We are grateful to At- sushi Nakamura for useful discussions and espe- cially for helping us in writing an efficient code in the early stage of the work.

R E F E R E N C E S

1 T. Suzuki, Nucl. Phys. B(Proc. Suppl.) 30, 176 (1993) and references therein.

2 H.Shiba and T.Suzuki, Phys. Lett. B351, 519 (1995) and references therein.

3 T. Suzuki, Monopole Dynamics and Confine- ment in SU(2) QCD, Kanazawa Univ. Report No.KANAZAWA 95-02,April, 1995 and ref- erences therein. Talk at 'QCD on Massively Parallel Computers ' held at Yamagata, Japan fi'om March 16 till March 18, 1995.

4 O M i y a m u r a and S.Origuchi, in this proceed- ings. See also "QCD monopoles and chiral symmetry breaking on SU(2) lattices" Hi- roshima Univ. Report hep-lat 9508015.

5 D.Weingarten, Nucl.Phys. B(Proc. Suppl.) 34, 29 (1994).

T. Suzuki et al./Nuclear Physics B (Proc. Suppl.) 47 (1996) 374-377 377

su(3):

0.160 0.165 0.169(~,)

m l r

0.689(3) 0.459(4)

0.0

mp 0.813(4) 0.676(1)

0.543

m N

1.256(18) 1.645(25)

abefian:

0.150 0.155 0.1575 0.160 0.1625 0.170 (~,)

m i -

0.545(6) 0.442(5) 0.364(4) 0.300(4) 0.244(4)

0.0

mp 0.630(5) 0.543(4) 0.514(6) 0.492(9)

0.5o3(23) 0.553

I n N

1.00(2) 0.81(3) 0.68(7)

monopole: /g

0.135 0.140 0.1425 0.150 0.1575

11"/1=-

0.671(5) 0.533(3) 0.500(5) 0.314(6)

0.139(18)

r n p

o.713(5) 0.606(3) 0.581(6)

0.442(14) o.2a(lO9)

m N

1.29(7) 1.04(3) 0.91(5)

Table 2 Hadron masses.

2

1/~ vs. ~,p mass

' Kc('abeli 'an;ffi0"17;(14) . . . . ~ ~ F '

2 .0 K=(full)=0.169 , , #

:'; p fulI(GFI 1 ('94)) .//":'/ :-J p abel ian ..:y

1.5 .: p monopo le . . , ? A l t ful l (GF11('g4)) ~ ,.,'~/ .< x abel ian ..~/ ../

' ~ r:,, x monopo le ../~,P .,/ ,* 1.o . . , / / .. E ~Ko '"'Z , ...... /

~::.- / - .... / . / .... . " ~ / . / . . - " / . / ..

0.5 ,- I e.x ":<" / ~ .h--'"/- f . I~

I / ¢ . . . . . . . . . . . . . . . 5.8 6.0 6.2 6.4 6.6 6.8 7 .0 7.2 7.4

1 / ~ Figure 3. The squared masses of a- and p versus 1/~.

m

3.0

2.5

2.0

1.5

t .0

0.5

0.0 5.5

Pro ton and p masses

• Proton (full), GF 11 ('94) "-" p (full), GFI 1('94) • Proton (abelian) , ,

p (abelian) ~' Proton (monopole) I • i 0 (monopole)

t • x " , - . . ~ [

. , . - " r

[ , , 6.0 6.5 7.0 7.5

1/x:

Figure 4. The masses of nucleon and p versus 1/~.

e,,

,,I

E

3.0

2.0

1.0

0.0

I O ~ ( ~ l e e } ' / . . . . . .

--i-p (photon)/ /'

.~. ~

i i i i i i

5.0 6.0 7.0 8.0 g.O 10.0

1/~:

Figure 5. The squared masses of 7r and p versus