Pseudoscalars in the instanton liquid model

Volume 233, number 1,2

PHYSICS LETTERS B

21 December 1989

P S E U D O S C A L A R S IN T H E I N S T A N T O N L I Q U I D M O D E L ~ R ALKOFER

a I 2,

M.A N O W A K a.3, j j M V E R B A A R S C H O T b and I Z A H E D

a

a Phvstcs Department, SUNY, Stony Brook, N Y 11794. USA b Theory Dmston, CERN, CH-1211 Geneva 23, Swuzerland

Received 8 September 1989

The masses and decay constants of the nonet of pseudoscalarmesons are investigated In the instanton hquld model of the QCD vacuum using the long wavelengthapproximation The lnstanton induced interactions betweenthe light quarks yield conservation laws for the flavor currents that are consistent with QCD mcludlng anomalies We show that the induced anomaly in the flavor singletcurrent givesrise to an r/' mass that is in agreementwith the Wttten-Veneziano relation The g-r/' mixing matrix coincides with the result derived previouslyby Veneziano

Recently there has been a considerable interest in the m s t a n t o n picture of the Q C D vacuum. Due to q u a n t u m disorder interacting lnstantons and anti-instantons in the v a c u u m stabilize in a dilute liquid state free of the long standing infrared problem [ 1,2 ] The liquid state consists of an amorphous network o f l n s t a n t o n s and antiinstantons constantly absorbing and emitting light quarks of different flavors The amorphous structure leads to a delocahzatlon of the fermlonic zero modes and to the spontaneous breakdown ofchlral symmetry. Detailed numerical investigations have been carried out to assess quantitatively these effects [3,4] For a v a c u u m state with a realistic n u m b e r of flavors and quark masses the instantons are in a liquid state [5] for which the chlral symmetry IS spontaneously broken For decreasmg m s t a n t o n density (e g by increasing the temperature [ 6,7 ] ) the amorphous structure breaks apart in favor of sparse molecular configurations. This state lacks flavor coherence and preserves the chiral symmetry The spontaneous breakdown of chiral symmetry in the instanton liquid phase is reminiscent to the metal-insulator transition (Mott transition) tn solids In his pioneering work on lnstantons 't Hooft [ 8 ] pointed out that at scales larger than the instanton size, the effects of a single lnstanton on the hght quarks can be well described in terms of an effective albeit nonlocal interaction Following this suggestion we were able to derive an effective action that describes light quarks in the instanton llqmd picture [4 ]. Our construction was based on a generalization of the one m s t a n t o n p a r t m o n function, and at first we ignored fluctuations in the lnstanton density. Recently, by including bulk correlations in the liquid state and proper coarse graining, we have shown that in the long wavelength limit our effective action is consistent with generic effective actions obtained from large Nc arguments [9] In our case, the scalar glueball field is identified with the local variations in the density of instantons in the vacuum, and the pseudoscalar glueball field is generated by allowing for local variations in the topological susceptibility Our effective action satisfies both the chlral and scale anomaly The latter is directly related to the compressibility of the lnstanton liquid state In this letter we will derive the masses and decay constants of the lowest pseudoscalar excitations using our effective action In particular, we will show that in the chlral limit the z~, K and q particles are Goldstone modes in the hquid state, as first noted by Diakonov and Petrov [ 10], whereas the r/' particles are heavy due to the Supported in part by the US Department of Energy under Grant No DE-FG02-88ER40388 1 Supported by a DFG postdoctoral fellowship 2 On leave of absence from Physlk-Department der TU Munchen, T30, D-8046 Garching, FRG 3 On leave of absence from Institute of Physics, Jagellonian University, PL-30059 Cracow, Poland 205

Volume 233, number 1,2

PHYSICS LETTERS B

21 December 1989

exphclt breaking o f the UA( 1 ) a n o m a l y by m s t a n t o n s In the presence o f current masses, the bulk p a r a m e t e r s o f the llqutd state yield a pseudoscalar nonet spectrum in good agreement with the data A f e r m m n propagating m the h q u l d state can be exther t r a p p e d m a zero m o d e state (~1) o f a given m s t a n t o n ( a n t l - m s t a n t o n ) or undergo b a c k g r o u n d field scattering to the next a n t l - m s t a n t o n ( m s t a n t o n ) Since the lnstanton field is short ranged the scattering part can be a p p r o x i m a t e d by a free propagator (So) If we were to assume that m s t a n t o n s and antl-lnstantons in the h q m d state are distributed according to their m v a r m n t measure ~ then the euchdean v a c u u m p a r t i t i o n function Z [ 0 ] associated to the p r o p a g a t i o n o f a fermlon can be written as follows

Z[O]= f dgtd~t*exp(_ ~ ~'Sff~)l~f (1--2~m f ~'Sff'fbif ~,Sfflgt)) ,

(1)

where m ~s the current quark mass matrix, and the average is m e a n t to be over the collective coordinate (position, color and size) o f the m s t a n t o n s and antl-lnstantons By using a p r o p e r coarse graining for the product over the pseudopart~cles a n d by invoking a c u m u l a n t expansion for the averages, the fermlonlc actmn can be written as

S= f

d4z [ ~136-11ff-n + (z) log det+ ( z ) - n - ( z ) log det_ (z) ] ,

(2)

where the smeared 't Hooft d e t e r m i n a n t s are given by ~2 det_+

= Nf--1~detfg(mfi- P

gtg)up)

(3)

The brackets ( )up denote averaging over the gauge group and the n o r m a h z e d size distribution. Since the zero modes have definite chlrallty, the d e t e r m m a n t a l factors break explicitly the UA ( 1 ) s y m m e t r y (leaving l n v a r m n t the remaining flavor t r a n s f o r m a t i o n s ) and reduce nonlocal interactions between fermlons on the scale o f the size o f the m s t a n t o n s At the mean-field level they also reduce the spontaneous b r e a k d o w n o f chlral s y m m e t r y Notice that for an equal density n+ (z) o f m s t a n t o n s and density n_ (z) o f a n t l - l n s t a n t o n s the partition function ( 1 ) revolves exclusively the UL ( 3 ) × UR ( 3 ) m v a r l a n t c o m b i n a t i o n d e t + d e t _ . By allowing for fluctuatmns in the n u m b e r o f pseudopartlcles (as m a grand canomcal d e s c n p U o n for instance) the UA( 1 ) breaking c o m b l n a t m n d e t + / d e t _ appears leading to the explicit breaking o f the s y m m e t r y in the v a c u u m To describe the long wavelength pseudoscalar excltauons, we can either study s o u r c e - s o u r c e correlators in the v a c u u m described by ( 1 ), or equivalently bosonlze the fermlonlc degrees o f freedom in ( 1 ) To achieve this we use the completeness relation

l= ~ dn+-f dP+-exp(-~f P+-(n+--(q/tSff'O+_Ot+_Sff'q)J)),

(4)

where P -+ are the auxiliary fields associated to the pseudoscalar fields n +. Notice that ~u* a n d ~, are i n d e p e n d e n t g r a s s m a n m a n fields. Therefore the P • need not to be h e r m l t e a n and in general they are N f × Nf complex matrices The fields n -+ are e h m l n a t e d with the help o f the m e a n field equations for fixed values o f P x In this way the gausslan fluctuations in n -+ are included in the partition function. After bosonlzatlon the fermlonlc part o f the effective action takes the following form ttl Ignoring reduced fermlonlc correlations and pseudopartlcleinteractions ~2 Note that there is a printing error m eq (24) ofref [9] The averaging brackets should be around the second term reside the determmant only 206

Volume 233, number 1,2

PHYSICSLETTERSB

Sv=-Nc~fTriog[l~+im+lP+(k-l) +tP-(k-1)

1+ k2]~,~

1+

21 December 1989

Mx/MkkM~

( 1 + tm•'] kZ j y ~ ( 1 + ~ _ 2 / ) Mx/MkM~].

(5,

The trace is over the Dlrac lndtces and the four-momenta. In order to distinguish the Goldstone modes from the massive modes we consider eq. (5) for zero quark masses Then the action is mvarlant under the global axial (at this point we do not consider the lnvartance under global vector transformations) transformatton #3 P+ ~ e x p ( + ½ix)P + exp( + ½ix),

P - ~ e x p ( - ½ix)P- exp( - ½IX).

(6)

This suggests the parametrlzatlon P-+ = e x p ( + ½ix) a e x p ( + ½ix) ,

(7)

where x and a are Nf×Nfhermltean matrices. The x variables can be identified as the pseudoscalar Goldstone modes The a variables are massive and can be eliminated with the help of the saddle point equations. To lowest order the saddle point solutton ~ does not depend on the quark masses We can set t~= 1 after a suitable redefinition of the consmuent mass Mk tn (5) To extract the mass term in (5) we choose to redefine P-+~ exp(-T-½1x)P -+ exp(-T- ½tx) As a result the contribution to (5) to order x 2 and m depends solely on the contribution T r [ m ( 1 - ~x2)] .

(8)

Thts expression can be written as

2 1 ) 2 m + m s - ~ - 1 1 4 3 ( Xo+--xsx/~

2ms ( x ° - x / ~ x s ) 2 + 2 r n k=,,2,3~x~,+(m+ms) + --if-

k=4,5,6,7E/~ 1 '

where we have used the decomposition x = S~~= o Xk2k (Tr 2 ~2b = 2dab) For zero momentum pseudoscalar excitattons (constant xf) we can immediately write down the relations (to order m and x 2)

OzSv OSv Ox~, - - m ~m ' k = 1 , 2 , 3 ,

02SF

Ox2 -

( m + ms) OSv k = 4, 5, 6, 7 2 Om'

( 1O)

and stmllar relations involving the Ko and K8 fields The derivatives with respect #a to m are related to the euclidean condensate by

OSv Om -

- 1 (Ifft~ll)

( 11 )

from which we immediately obtain the PCAC relation Note that to order m the chIral condensate is flavor independent To mvest~gate the momentum content of the pseudoscalars constder first the Xo and x8 excitations, which can be identified as the r/' and the r/field, respecttvely Inserting (7) in (5) and expanding around zero momentum gives ( m o m e n t u m integration understood)

½f Zp2 (r]2 + r]'2) + ½1( ~ttl//)E[ r]'2( ] m + imp) + q2( ~m+ ]ms) +qq' Iv~2 ( m - m s ) ] ,

(12)

#3 N o anomaly xs expected m the absence o f gauge couplings ~4 W h e n we dlfferentmte with respect to m, we take into account that m ~s the average o f r n . and m a By the derivative with respect to m we m e a n the derivative e~ther w~th respect to mu or with respect to ma

207

Volume 233, number 1,2

PHYSICS LETTERS B

21 December 1989

where the quark condensate and the pseudoscalar decay c o n s t a n t f

f d4k / - 2 = q2~ohm 4Nc

[M(k+)K_-M(k_)k+] 2

(27~)4 [ k ~ + M Z ( k + ) ] [ k 2 _ + M 2 ( k _ )

],

k+=k+½q,

(13)

are evaluated m the choral limit A slmdar result has been derived in ref [ 10] For n and K excitations we have the following m 2= -2m(~,)/f2

m 2= -- ( m + m s ) ( ~ l > / f

(14)

2

(15)

We have exphcltly used the relation ( ~ ) = - 1 ( ~ * ~ , ) E between euclidean and Mmkowskl condensates. To account for the mass of the q' we have to add to (12) the gluomc part due to the fluctuations m the topological susceptlbdlty In ref. [9] we have shown that the terms proportional to the difference n + ( z ) - n - ( z ) in the pseudopamcle densmes are given by SG= ~l'fd42

[ n + ( z ) - n - (z) ]2-1 2N//2Nf~ d 4 z [ n + ( z ) - n - ( z ) ] q

'

(16)

This gives rtse to a gluonlc contrlbutmn to be added to the fermionic contnbutton (5) pseudopamcle difference gives the term

Integrating out the

N,f]3q N ,2

(17)

After addmg this to ( 12 ) and substituting (14) and ( 17 ) into ( 12 ), we obtain the following result for the second order expansion m the q, q' fields of our effective action ( m o m e n t u m lntegratton understood) 2 1 rw 2, ..1__~c 27]t2 ( 32_ty/2 _[_ gl m .2) + ~'2r/q ' zl f 2-2 p ( r / 2 + r / , z ) + ~ 2l N f ~ r N / t2 + _~f2~214 ~gmK~,,,n;

2 4x/2 (m,~2 --mK)

(18)

This result comctdes with the result obtained by Venezlano [ 11 ] In particular, for current zero quark masses, we obtain the Witten-Venezmno [ 11,12] formula At thts point, we disagree wtth the analysts o f ref [ 10] concernmg the U ( 1 ) problem #5 The present analys~s shows the importance of the fluctuatmns m the d~fference between the number o f mstantons and antl-mstantons (topological susceptlbthty) m generating the mass of the q' and thus solving the U ( 1 ) problem Using the standard mstanton hqutd model parameters N / V = 1 f m - 4 and/~= ½ fm we obtain for the constttuent mass M ( 0 ) = 345 MeV, for the condensate ( ¢ ~ ) = (255 MeV) 3 and for the decay constant f = 91 MeV The current masses m = 5 MeV and m, = 120 MeV reproduce the experimental values o f m= and mK. Dlagonahzatmn of the mass m a m x given m (18) ymlds rn~=527 MeV (expertment' 549 MeV) and rn, = 1172 MeV (experiment 958 MeV) The corresponding mixing angle comes out to be 0= - 11 5 ° (experiment ~ - 2 0 ° [ 13] ) We have shown that the effective actmn for light quarks derived from the mstanton hqutd ptcture of the Q C D v a c u u m leads to the proper descnptmn of low m o m e n t u m pseudoscalar exmtattons. In parttcular, the spontaneously broken vacuum state yields the correct PCAC relatmns to leadmg order m the current quark masses The gaussmn fluctuatmns m the number difference between mstantons and antl-mstantons lead to an r/' mass m agreement with the Wttten-Veneziano relation In the long wavelength limit and to leading order m the current mass our effective action reduces to the effective action used by Venezlano The bulk mstanton h q m d parameters ymld masses and decay constants that are m good agreement with the data ~6 It all seems to lndmate #5 Indeed, for Nf> 1 thmr result leads to tachyomc smglet exmtauon They circumvented the problem by taking Nf-~0 #6 We have used here the value of the decay constant in the choral limit Taking into account the finite current masses for the r/' decay constant would improve the agreement with the data Details will be given elsewhere

208

Volume 233, number 1,2

PHYSICS LETTERS B

21 December 1989

t h a t a n m s t a n t o n l i q u i d d e s c r i p t i o n o f t h e Q C D v a c u u m p r o v i d e s a f a i r a c c o u n t o f low m o m e n t u m p s e u d o s c a l a r excitations.

References [ 1 ] D I Dlakonov and V Yu Petrov, Nucl Phys B 245 (1984) 259 [2] D I Dlakonov and VYu Petrov, Nucl Phys B272 (1986)457 [ 3 ] E V Shuryak, Nucl Phys B 302 ( 1988 ) 574 [4] M A Nowak, J J M Verbaarschot and I Zahed, Nucl Phys B 324 (1989) 1 [5] E V Shuryak and J J M Verbaarschot, CERN preprlnt TH-5492/89 (1989), submitted to Nucl Phys B [6] E M Ilgenfntz and E V Shuryak, Novoslblrsk prepnnt 88-85 (1988) [7] M A Nowak, J J M Verbaarschot and I Zahed, CERN preprlnt TH-5275/89 (1989), Nucl Phys B, m press [8] G 't Hooft, Phys Rev D 14 (1976) 3432 [9] M A Nowak, J J M Verbaarschot and I Zahed, Phys Lett B 228(1989) 251 [ 10] D I Dlakonov and V Yu Petrov, Leningrad preprmt LNPI-1153 (1986) [ 11 ] G Venezlano, Nucl Phys B 159 (1979) 461 [12] E Wltten, Nucl Phys B 156 (1979) 269 [ 13] F J Gdman and R Kauffman, Phys Rev D 36 (1987) 2761

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