V01ume 226, num6er 3,4 PHY51C5 LE77ER5 8 10 Au9u5t 1989

0 2 1 RULE A N D 1 N 5 7 A N 7 0 N 5

M.A. N 0 W A K ~1, J.J.M. V E R 8 A A R 5 C H 0 7 6 and 1. 2 A H E D ~

a Phy51c5 Department, 5UNY, 5t0ny 8r00k, NY 11794, U5A ~ CERN 7he0ry D1v1510n, CH-1211 6eneva 23, 5w1t2er1and

Rece1ved 17 May 1989

we 1nve5t19ate the effect5 0f 5tran9ene55 m1x1n9 1n the QcD vacuum de5cr16ed a5 a 4uantum d150rdered pha5e 0f 1n51ant0n5 and ant1-1n5tant0n5, we f1nd that the c0n5t1tuent ma55 0f the 119ht 4uark5 15 a decrea51n9 funct10n 0f the current 4uark ma55e5, and that 1t5 ma55 15 a1m05t 1n5en51t1ve t0 the va1ue 0fthe 5tran9e 4uark ma55.0ur re5u1t5 cann0t acc0unt f0r a 6reach 1n the 021 ru1e 1n the vacuum. 7he re1evance 0f th15 re5u1t f0r the am0unt 0f 59 pa1r5 1n the nuc1e0n and the p10n-nuc1e0n 519ma term 15 d15cu55ed.

7he 155ue 0f f1av0r m1x1n9 1n QCD at 10w ener9y depend5 fundamenta11y 0n 0ur under5tand1n9 0f the Q C D vacuum. 7he 1atter 15 kn0wn t0 c0nta1n n0n- tr1v1a1 9au9e c0nf19urat10n5 re5p0n5161e f0r the f0r- mat10n 0 f a 4uark and 91u0n c0nden5ate. Wha t 151e55 kn0wn 15 whether the5e c0nf19urat10n5 are c1a551ca1 9au9e c0nf19urat10n5 5uch a5 1n5tant0n5 0r m0n0- p01e5, 0r pure1y 4uan tum 91u0n f1uctuat10n5. 7here 15 50me 4ua11tat1ve ev1dence, ma1n1y fr0m the Q C D 5um ru1e5 [ 1 ], 1n fav0r 0 f t h e 4ua51-c1a551ca1 appr0ach.

50 far, there have 6een var10u5 at tempt5 t0 de- 5cr16e the QCD vacuum 1n term5 0f c1a551ca1 9au9e c0nf19urat10n5 dre55ed w1th 5ma11 4uan tum f1uctua- t10n5. 7he 1n5tant0n appr0ach t0 the QCD vacuum, wh11e 1t d0e5 n0t c0mp1y w1th the 9enera1 re4u1re- ment 0f c0nf1nement at 1ar9e d15tance 5ca1e5, 5eem5 t0 capture the e55ence 0f ch1ra15ymmetry 6reak1n9 at 5h0rt d15tance 5ca1e5 [2 -5 ]. Due t0 4uan tum d150r- der, 1n5tant0n5 and ant1-1n5tant0n5 1n the vacuum 5ta61112e 1n a d11ute 114u1d 5tate, where 119ht 4uark5 are de10ca112ed 0ver a ran9e 0 f t h e 0rder 0fthe1r c0n- 51tuent ma55. 7he 1atter 15 dynam1ca11y 9enerated 1n the 114u1d 5tate (5ee e.9. ref5. [ 6 - 9 ] ).

1n a recent w0rk [ 9,10 ] we 1nve5t19ated the depen- dence 0f the ch1ra1 c0nden5ate 0n the 5tran9e current 4uark ma55 f0r a f1xed avera9e den51ty and 512e 0 f the

0n 1eave 0f a65ence fr0m 7he 1n5t1tute 0f Phy51c5, Ja9e110n1an un1ver51ty, crac0w, P01and.

1n5tant0n5 1n the vacuum. We c0nc1uded that the re- 5u1t5 0fth15 m0de1 are 1n a9reement w1th the 021 ru1e and a pr10r1 1nc0n515tent w1th a 1ar9e p10n-nuc1e0n 519ma term [ 1 1,12]. 7he purp05e 0f th15 1etter 15 t0 further 1nve5t19ate the effect5 0f f1av0r m1x1n9 1n 9en- era1, and 5tran9ene55 m1x1n9 1n part1cu1ar. We w111 5tudy the 5en51t1v1ty 0f the avera9e den51ty and the 512e 0f the 1n5tant0n5 1n the vacuum 0n the current 4uark ma55e5 and eva1uate the effect5 (1f any) 0n the c0n5t1tuent 4uark ma55.7he phen0men01091ca1 re1e- vance 0 f 0 u r re5u1t5 w111 6e d15cu55ed.

7he vacuum v1ewed a5 a d11ute 114u1d 0f1n5tant0n5 and ant1-1n5tant0n515 t0p01091ca11y neutra1.7he 9rand part1t10n funct10n 0f a neutra1 5y5tem 0f N+ 1n5tan- t0n5 and N• ant1-1n5tant0n5 can 6e appr0x1mated 6y [13,14,6]

1 2 = ~ N+•N•v

A~+ ,N

" N

X F[ d42• d U~ dp, d(p,) exp [ -- f1(p) U,,• ] 1

X d e t ( -1y, ,V~1m), ( 1 )

where the 1nte9rat10n 15 0ver the c011ect1ve var1a61e5 den0t1n9 the p051t10n (21) 1n f0ur-d1men510na1 eu- c11dean 5pace, the c010r 0r1entat10n (U~) 1n 5U (Nc) and the 512e (p1) 0 f t h e p5eud0part1c1e5.7he 512e d15- tr16ut10n d(p) 15 91ven 6y

382 0370-2693/89/$ 03.50 • E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)

V01ume 226, num6er 3,4 PHY51C5 LE77ER5 8 10 Au9u5t 1989

5 / 8x 2,2N~ d(p)=Cp ~ 7 ) (MP)2/~/3 exP[-f1(P) ] ~

(2)

where f1 15 the 0ne-100p 6e11-Mann-L0w funct10n eva1uated at a 5ca1e that 15 0 f the 0rder 0f the 512e p 0f the 1n5tant0n5

f1(p) = - 6 109 (pA), •• 6 = 7 N c - ~ N r , (3)

and C = 4 . 6 0 exp( - 1.68Nc) /~2(Nc - 1 )•( N,.- 2 )•.A 15 the u1tra-v101et cut-0ff1n the Pau11-V111ar5 5cheme, and M 15 a ren0rma112at10n ma55. 7he ar9ument 0f the 6eta funct10n 1n ( 1 ) 15 the avera9e 512e/~ 0f the 1n5tant0n5.1n ( 1 ) um, 15 5h0rt f0r the tw0-60dy 1nter- act10n 6etween an 1n5tant0n and an ant1-1n5tant0n. 1n 9enera1, 1t depend5 0n the1r re5pect1ve 512e5, c010r 0r1entat10n5 and 5eparat10n5.0n the avera9e, 1t 15 re- pu151ve [ 15 ].

7he ferm10n determ1nant 1n ( 1 ) 1nv01ve5 the c0- var1ant der1vat1ve V 1n the c1a551ca16ack9r0und 0f1n- 5tant0n5 and ant1-1n5tant0n5 (p5eud0part1c1e5), and a d1a90na1 ma55 matr1x m=d1a9(m1 .. . . , m 0 f0r Nr f1av0r5.1t can 6e fact0r12ed 1n a 10w m 0 m e n t u m fac- t 0 r ( - d e h 0 w ) and a h19h m 0 m e n t u m fact0r ( - detn~9h ). 80th fact0r5 are re9u1ar12ed 6y a ma55 M a n d n0rma112ed 6y the n0n-1nteract1n9 ferm10n determ1- nant [ 6 ]. 5pec1f1ca11y,

deth19h =17 (Mp2) --2/3.1.339M1p1, M1p1 << 1, 1, f

(4)

where ]~115 an ar61trary ma55 5Ca1e 5eparat1n9 the 10W m 0 m e n t u m C0mp0nent5 1n the ferm10n determ1nant fr0m the h19h m 0 m e n t u m 0ne5 .7he 10W m 0 m e n t u m part 0f the ferm10n determ1nant Can 6e 5atUrated 6y the ferm10n1C 2er0 m0de5 [ 6 ]

det ( 7+ 1rnrK- 1mf) det10w det( 7+1mrK-1M, ) • (5)

where the matr1x e1ement5 0f herm1t1an matr1ce5 7 and K are def1ned a5 f0110w5:

7u = fd4x 0117,, 0,,0.,, (6)

K1j = f d4X * 0~0,1 - ~u . (7)

N0te that the determ1nant51n e4. ( 5 ) are rea1. % rep- re5ent5 the ferm10n1c 2er0 m0de 1n the f1e1d 0f 0ne-

1n5tant0n 0r 0ne-ant1-1n5tant0n. 7he c0n515tency 0f th15 appr0ach re4u1re5 that the f1na1 re5u1t5 are 1nde- pendent 0fM~, 1.e. 1ndependent 0f the 5eparat10n 1nt0 10w and h19h m0menta . 7h15 p01nt w111 6e ver1f1ed 6e10w.

We w111 5tudy the 11m1t 1n wh1ch the 0ver1ap matr1x K can 6e ne91ected. 7h15 c0rre5p0nd5 t0 the 11m1t 0f 5ma11 current 4uark ma55e5 and a d11ute 1n5tant0n 114u1d [6]. 1n th15 ca5e the d15tr16ut10n 0f the e19en- va1ue5 0f the 0ver1ap matr1x 7 15 5em1c1rcu1ar [6] t0 1ead1n9 0rder 1n N~.. 7h15 9reat1y 51mp11f1e5 the eva1- uat10n 0f the ferm10n determ1nant. 1n the therm0dy- nam1c 11m1t (N/V4=c0n5tant, N ~ , V4-,00) the avera9e 0f the determ1nant 1n (5) can 6e appr0x1- mated 6y a 1ead1n9 0rder cumu1ant expan510n. 7he re5u1t 15

(109 det( 77*+m 2) ) 2tt- 1/2 N ~ / 1 d,~(1 - 2 2 ) 109(m2--1-2 2) (8)

-7rK _ ~ 4K2J • 2K

where ( ) 15 5h0rt f0r avera91n9 0ver the d15tr16ut10n 0f the c011ect1ve c00rd1nate5. F0r a 1ar9e num6er 0f c010r5 N,., the var1ance K 0f the e19enva1ue5 0f 7 can 6e eva1uated u51n9 a mean f1e1d appr0x1mat10n [ 16 ]

~¢2=6.62V~N p2 . (9)

Perf0rm1n9 the 1nte9ra1 0ver the e19enva1ue d15tr16u- t10n 1n (7) and c011ect1n9 a11 0ther fact0r5 y1e1d the f0110w1n9 re5u1t f0r the fu11 ferm10n determ1nant.

H( rnD ~(1.34M~p1)xf(p,M)-2x~/311r H(M, ) , (10)

where

H(m) = [ 1 m + • ( rn2+ 4K 2) ,/2]

1 m - ( m 2 +4~c2)~/2~ × e x p ~m+(rn2+4K2)1/2j . (11)

N0t1ce that 1n the 11m1t 1//~>> M~ >> 1¢ the Mj depen- dence 1n e4. (10) cance15 and the re5u1t 15 1ndepen- dent 0f the 5p11tt1n9 1nt0 10w and h19h m0menta . 1n the 10w-den51ty 11m1t, th15 re5u1t repr0duce5 the ma55 dependence 0r191na11y f0und 6y •t H00ft [ 13 ].

F0r 1ar9e va1ue5 0f the current 4uark ma55 we have t0 take 1nt0 acc0unt the effect5 due t0 the 0ver1ap ma- tr1x K. A150 the m1x1n9 0f the 2er0 m0de5 w1th the

383

V01ume 226, num6er 3,4 PHY51C5 LE77ER5 8 10 Au9u5t 1989

c0nt1nuum 5tate5 6ec0me51mp0rtant. 7echn1ca11y, we are una61e t0 hand1e th15 m1x1n9. H0wever, H ( m ) 1n (10) eva1uated f0r 5ma11 current 4uark ma55e5 ha5 the c0rrect m-dependence f0r 1ar9e m. 7heref0re we expect e4. (10) t0 6e a 900d 1nterp01at1n9 f0rmu1a f0r a11 va1ue5 0fthe 4uark ma55e5.

An0ther ar9ument 1n fav0r 0f e4. (10) 15 06ta1ned 6y n0t1n9 that t0 10we5t 0rder 1n the current 4uark ma55e5 H ( m ) 15 91ven 6y

H(m)~m+1c, m-+0. (12)

7h15 re5u1t 15 4ua11tat1ve1y 51m11ar t0 the re5u1t adv0- cated 6y 5h1fman et a1. [ 17 ] u51n9 the 0perat0r pr0d- uct expan510n. 7he1r re5u1t can 6e 51mp1y 06ta1ned 1n 0ur framew0rk 6y eva1uat1n9 the n0n-pertur6at1ve part 0f the ferm10n determ1nant u51n9 a 2er0 m0de appr0x1mat10n [9]. 1n th15 appr0ach, the current ma55 1n the 512e d15tr16ut10n 0f the 1n5tant0n5 15 re- p1aced 6y a determ1nanta1 ma55 Ma

1 f d4k M a = r n + ~ ( -~)4k20~2(k)(4/~(k)6~(-k)) .

(13)

Here the vacuum expectat10n va1ue 15 den0ted 6y ( ) , and the funct10n ¢• (k) 15 re1ated t0 the F0ur1er tran5f0rm 0f the ferm10n1c 2er0 m0de 1n the 0ne-1n- 5tant0n 6ack9r0und [ 6 ]. At 10w m0mentum (k-~ 0) th15 funct10n appr0ache5 - 2np/1k[. Rep1ac1n9 ~ (k) 6y th15 11m1t repr0duce5 the re5u1t 0f 5h1fman et a1. [171,

27~2p 2 M a = m + ~ 1(~U*41) , (14)

Where (~U*~U) 15 the eUC11dean Ch1ra1 C0nden5ate. 1n 0rder t0 5ee the C0nneCt10n 6etWeen e45. (12)

and (14), We have t0 eva1Uate the m0mentum 1nte- 9ra1 1n e4. (14). 7h15 can 6e ach1eved 6y u51n9 the exp11c1t expre5510n f0r the 4uark pr0Pa9at0r 1n the 1n- 5tant0n med1um,

( ~ . ( k ) ~ ( - - k ) ) = [ ~ . k . - ~ ( k ) ] , ~ / ~ . (15)

N0t1ce that the ferm10n pr0Pa9at0r 15 d1a90na11n f1a- v0r 5pace. Here M(k) 15 a m0mentum dependent ma55 (••c0n5t1tuent ma55••) n0t t0 6e c0nfu5ed w1th the determ1nanta1 ma55 Ma. 7he ma55 M(k) 15 91ven 6y

~N M ( k ) - 2 ~ k 2 ~ 2 ( k ) , (16)

and 5at15f1e5 the f0110w1n9 9ap e4uat10n

f d4k M2(k) N (2~2)4k2+M2(k) - 4V4N~ ( 1 - m e ) . (17)

U51n9 ( 16 ) - ( 18 ), the determ1nanta1 4uark ma55 Ma 15 91ven 6y

Ma=•m+• (m2+4x 2) 1/2 (18)

wh1ch 15 exact1y the pre-exp0nent1a1 fact0r 91ven 1n e4. (1 1 ). 7he determ1nanta1 ma55 06ta1ned 1n th15 way ( .~ 80 MeV) 15 r0u9h1y a fact0r 3 5ma11er than the 2er0 m0mentum determ1nanta1 ma55 06ta1ned 6y 5h1fman et a1. [ 17 ]. 7h15 w111 very 11ke1y m0d1fy the1r e5t1mate5 0n the ran9e 0f va11d1ty 0f the 1n5tant0n 114- u1d m0de1.

7he avera9e 512e and den51ty 0f the 1n5tant0n5 1n the pre5ence 0f the ferm10n determ1nant 15 eva1uated acc0rd1n9 t0 a pr0cedure pr0p05ed 6y Dyak0n0v and Petr0v [ 15 ]. 7hey e5t1mate the tw0-60dy 1nteract10n u1n~ 6etween 1n5tant0n51n ( 1 ) 6y u51n9 the Feynman var1at10na1 pr1nc1p1e under the a55umpt10n that the 6ack9r0und f1e1d 15 91ven 6y a 11near 5uperp051t10n 0f 0ne-1n5tant0n and 0ne-ant1-1n5tant0n c0nf19ura- t10n5.1n th15 way the tw0-60dy 1nteract10n 1n ( 1 ) can 6e taken 1nt0 acc0unt 1n term5 0f an effect1ve 0ne- 60dy d15tr16ut10n/2 (p). 1n the var1at10na1 appr0ach the 1nteract10n enter5 0n1y a5 an avera9e 0ver the c01- 1ect1ve c00rd1nate5 (den0ted 6y a 6ar) 1ead1n9 t0 a c0n51dera61e 51mp11f1cat10n 0f the ana1y515. 5pec1f1ca11y,

277c 2 N~ u1~t=72p~p2, 72• . (19)

4 N~--1

7he 501ut10n 0f the var1at10na1 e4uat10n f0r/~ 15 [ 18 ]

1~(p)=d(p)exP(~ ~1(m,~) -

• f1(p)y2np~ (p2 ~ 5 ) ) ,

0H(mr) (p-p) 0p

(20)

where dha5 6een def1ned 1n (2) and 1nv01ve5 a fact0r 0r191nat1n9 fr0m the ferm10n determ1nant. 7he aver- a9e 54uared 512e 15 determ1ned 5e1f-c0n515tent1y 6y

384

v01ume 226, num6er 3,4 PHY51C5 LE77ER5 13 10 Au9u5t 1989

p~ fdpp21•t(P) (21) - . f d p ~ ( p ) •

and p 15 def1ned 6y (p2) /2. 7he den51ty N/V4 0f p5eud0part1c1e5 1n the vacuum 15 den0ted 6y n. 1n term5 0f ~t the free ener9y def1ned a5 F----109 2 a5- 5ume5 a part1cu1ar1y 51mp1e f0rm

F=N -1092A4

+109 fdp/1(p)14eXp[ 1 ~ ~ 2 ] ) . -~f1(p)~-np- (22)

8y max1m121n9 the free ener9y w1th re5pect t0 N+ and N• we f1nd the e4u1116r1um den51ty 0f the p5eud0par- t1c1e5. 1t 15 c1ear fr0m (22) that f0r 2er0 0 an91e we have N+ = N . f0r 2er0 ma55e5 the var1ance 0f the t0- ta1 num6er 0fp5eud0part1c1e515 e4ua1 t0 4/6~ [ 6 ] 1n a9reement w1th the 10w-ener9y the0rem5 der1ved 6y N0v1k0v et a1. [19,20]. N0t1ce that the var1ance 1n the num6er 0f p5eud0part1c1e5 1n the vacuum 15 5ta- 61e 1n the 1ar9e N~, 11m1t.

1n 9enera1 the avera9e 512e p and den51ty r~ 0f the p5eud0part1c1e5 have t0 6e determ1ned numer1ca11y. 1n f195. 1 and 2 we 5h0w the1r dependence 0n the 5tran9e ma55 1n the ca5e 0f three f1av0r5.7he ma55 0f the 119ht (u, d) 4uark5 15 0.024A. We f1nd that the avera9e den51ty 0f the p5eud0part1c1e5 1ncrea5e5 a5 a funct10n 0f rn~, wherea5 the1r avera9e 512e decrea5e5 fa5t en0u9h 50 that the pr0duct np 4 decrea5e5 a func- t10n 0f m5. 51nce at the 0ne 100p 1eve1 the c0up11n9 c0n5tant 82c2/9 2 15 n0t ren0rma112ed we re501ve th15 am619u1ty 6y ch0051n91t5 va1ue e4ua1 t0 - 6 109 (0.1).

0.5

1c

0.25

0 1 1 1 r 1 J 1 1 1 0 0,5 m~/A

F19. 1. 7he m~A dependence 0f the avera9e den51ty t1A 4 0f the p5eud0part1c1e5.

0.75

10

0.5 : ~

0.25

0.5 m , / A

F19. 2.7he d1men510n1e55 avera9e 512e #A a5 a funct10n 0f the 5tran9e ma55 m51n un1t5 0fthe cut-0ffA.

(7he 5ame appr0ach wa5 f0110wed 1n ref. [21 ]. ) W1th a cut-0ff A 0f 250 MeV th15 91ve5 r15e t0 phen0me- n01091ca11y accepta61e num6er5. M0re 5pee1f1ca11y, f0r the va1ue5 0f the three current 4uark ma55e5 e4ua1 t0 6 MeV, 6 MeV and 150 MeV, we f1nd (190 MeV) 4 f0r the 91u0n c0nden5ate. 7he w1dth 0f the e19enva1- ue5 15 x/A = 0.7 and the 512e 0f the p5eud0part1c1e5 15 /~A = 0.45, mak1n9 a 5eparat10n 6etween 10w and h19h m0menta p055161e. 7he reader can ver1fy that 0ur 5y5tem 15 d11ute and that the u5e 0f a 5em1-c1a551ca1 de5cr1pt10n 15 ju5t1f1ed.

N0w, we are ready t0 eva1uate the 5en51t1v1ty 0f the ma55 Mu [def1ned 1n e45. (16) and (17); the 5u6- 5cr1pt den0te5 the f1av0r] t0 var1at10n5 0f the current 4uark ma55e5 1n the 1n5tant0n vacuum, 1n part1cu1ar, t0 var1at10n5 0f the 5tran9e 4uark ma55. F0r that, we need the 501ut10n 0f the 9ap e4uat10n (17). 7h15 e4uat10n depend5 0n the d1men510n1e55 parameter5 r/p 4 and mrp. N0t1Ce, h0Wever, that at the mean f1e1d 1eve1 there 15 n0 C0Up11n9 6etween the var10U5 f1av0r5 0ther than thr0U9h the 1mp11C1t 4Uark ma55 depen- dence 0f the vacuum parameter5 (den51ty and 512e 0f the 1n5tant0n5). 7h15 exp1a1n5 why the numer1ca1 va1- ue5 0f - 0 . 1 1 and - 0 . 3 4 we f1nd f0r 0,,,~Mu and 0,,,dMu, re5pect1ve1y, are an 0rder 0f ma9n1tude 5ma11er than the numer1ca1 va1ue 0f - 2 . 7 we f1nd f0r ~,,,0Mu. 7he m05t 5urpr151n9 feature 0f the5e re5u1t5 15 the1r 519n. 1n part1cu1ar, the 519n 0f ~,,,0Mu w111 n0t 6e a1tered 6y tak1n9 1nt0 acc0unt c0rrect10n5 due t0 the f1n1te 4uark5 ma55e5.70 10we5t 0rder the c0n5t1t- uent ma55 w111 6e c0rrected 6y the va1ue 0f the cur-

385

v01ume 226, num6er 3,4 PHY51C5 LE77ER5 8 10 Au9u5t 1989

rent ma55, add1n9 1 t0 the der1vat1ve. 1f we a55ume that the nuc1e0n ma55 15 91ven 6y the 5um 0f three c0n5t1tuent ma55e5, th15 re5u1t 1ead5 t0 a ne9at1ve va1ue f0r 2~N, 1n d15a9reement w1th the exper1ment.

1n a m0re rea115t1c m0de1 0f the nuc1e0n the c0n5t1t- uent ma55e5 are 60und. 7ak1n9 1nt0 acc0unt the m0-

men tum dependence 0f M(k) we 06ta1n fr0m e45. ( 1 6 ) - ( 1 8 ) a c0n5t1tuent ma55 0f ~ 500 MeV. 1n 0r-

der t0 de5cr16e the nuc1e0n 1n 0ur m0de1 we need a

61nd1n9 ener9y 0f ~ 500 MeV, Apparent1y, the cur- rent 4uark ma55 dependence 0f the 61nd1n9 ener9y 15 1mp0rtant. H0wever, 1n the pre5ent framew0rk we are

n0t a61e t0 e5t1mate th15 effect. 0 u r ma1n c0nc1u510n 15 that the c0n5t1tuent ma55 15 near1y 1n5en51t1ve t0

the 5tran9e 4uark ma55.7h15 re5u1t 15 t0 6e c0ntra5ted w1th re5u1t5 fr0m the ch1ra1 6a9 m0de1 [22], the 5kyrme m0de1 re5u1t [ 12 ], and the NJL re5u1t 0f ref. [23]. 1n part1cu1ar, the f1r5t tw0 m0de15 91ve r15e t0 a

1ar9e 5tran9ene55 adm1xture 1n the nuc1e0n 5tate. 7he determ1nanta1 ma55 M2x 5h0w5 a d1fferent de-

pendence 0n the current 4uark ma55. F1r5t, the 519n 0f MA w1th re5pect t0 the current 4uark ma55 15 p051- t1ve. 5ec0nd, the va1ue5 0f the der1vat1ve5 are 5ma11er.

F0r the der1vat1ve5 0fMAu w1th re5pect t0 mu, md and m5 we f1nd 0.54, 0.028 and 0.022, re5pect1ve1y. H0w-

ever, the va1ue 0fM~u 15 0n1y 80 MeV. 0 u r re5u1t5 were 06ta1ned under the a55umpt10n

that the c011ect1ve c00rd1nate5 0f the p5eud0part1c1e5 are d15tr16uted acc0rd1n9 t0 the1r 1nvar1ant mea5ure. Actua11y, the1r d15tr16ut10n 5h0u1d 6e we19hted 6y the ferm10n determ1nant there6y 1ntr0duc1n9 an add1- t10na1 ma55 dependence. Numer1ca1 51mu1at10n 0f the ferm10n determ1nant f0r f1xed avera9e 512e and av-

era9e den51ty were carr1ed 0ut 1n ref5. [ 8,10,9 ]. 7he re5u1t5 pr0v1ded ev1dence f0r the hyp0the515 that 1n

the therm0dynam1c 11m1t the ch1ra1 c0nden5ate 15 1n-

5en51t1ve t0 var1at10n5 0f the 4uark ma55e5. 1n c0nc1u510n, we f1nd that the ma55 dependence 0f

the avera9e 512e and den51ty 0f the 1n5tant0n5 91ve5 r15e t0 50me m1x1n9 6etween 4uark5 0f d1fferent f1a- v0r5 1n the QCD vacuum de5cr16ed a5 a 114u1d 0f 1n- 5tant0n5 and ant1-1n5tant0n5.7he m1x1n9 15 an 0rder 0f ma9n1tude 1e55 than the 0ne adv0cated 6y the 5kyrme m0de1. H0wever, the ne9at1ve va1ue 0f the

p10n-nuc1e0n X term 1nd1cate5 that the 61nd1n9 en-

er9y 0f the c0n5t1tuent 4uark5 ha5 a 5tr0n9 ma55 de- pendence wh1ch we h0pe t0 5tudy 1n a future pu611- cat10n. 0 v e r a11, 0ur re5u1t5 1nd1cate that there 15 n0 5tr0n9 v101at10n 0f the 021 ru1e 1n the 1n5tant0n vacuum.

7h15 w0rk wa5 5upp0rted 1n part 6y the U5 De- par tment 0f Ener9y under C0ntract N0. DE-F602- 88ER40388. We w0u1d 11ke t0 thank E.V. 5huryak and

E.M. 119enfr1t2 f0r u5efu1 d15cu5510n5.

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