Is the nucleon strange?

Volume 217, number 1,2 PHYSICS LETTERS B 19 January 1989

IS T H E N U C L E O N STRANGE? ~

M.A. N O W A K a,l, J.J.M. VERBAARSCHOT b,2 and 1. ZAHED a Physics Department, State University of New York at Stony Brook, Stony Brook, NY 11794, USA

b TheoryDivision, CERN, CII-1211 Geneva 23, Switzerland

Received 9 September 1988

The issue of the strangeness content of the proton in relation to a large X~N term is examined using the instanton-antiinstanton description of the QCD ground state. Modulo plausible assumptions, our results indicate no strangeness admixture in the nucleon state at zero momentum transfer.

I f we were to believe the presently extrapolated value of the Z~N term, the naive picture o f the nucleon made of constituent up and down quarks with no strange quarks seems to have difficulty. Indeed, the experimental low-energy nN scattering amplitude

1 3 S~N(t )= ~ , ~ ( N ( p ' ) [ [Q~, [Q~, H ] ] I N ( p ) )

(1)

extrapolated at the Cheng-Dashen point s = M 2 and t=2rnZ~ is about 60 MeV [ 1 ]. (Here Q~ isthe SU(2) f axial charge.) Use of chiral perturbation theory indicates that at t = 0 , Z~N decreases by about 5 MeV so that overall S~N is about 5 5 MeV [ 2 ].

Theoretically, X~N measures the shift in the nucleon mass due to the explicit breaking ofchiral symmetry. More specifically, in the (3, 3 ) ~3 ( 3, 3 ) chiral symmetry breaking scheme (ignoring isospin breaking) [3,4]

•rcN = m ( N [ a u + a d l N ) , (2)

where m = ½ (m, + ma) is the average current mass of the light quarks. If we denote by rn~ the current mass of the strange quark, then current algebra and chiral

~" Supported in part by the US Department of Energy under Grant No. DE-FG02-88ER40388, and No. DE-AC02-76ER13001.

J On leave of absence from the Institute of Physics, Jagellonian University, PL-30059 Cracow 12, Poland.

2 On leave from State University of New York at Stony Brook, Stony Brook, NY 11794, USA.

perturbation theory indicates that m / m s ~ 0.04 [5]. It is straightforward to rewrite (2) in terms of the octet and singlet scalar densities. I f we were to as- sume that the hyperon mass shifts are linear in the current quark masses (strong assumption), then for the octet scalar density we have

½ (ms - m ) (NI o u + a d - 2gs lN)

= ½ ( M = - M N ) + ~ ( M z - M A ) . ( 3 )

Using this result together with the current quark mass ratio, we can solve for the gs content in the nucleon as a function o f S~N

( N l g s l N ) ( N I 0 u + a d I N ) ~ ( X ~ N - 2 5 ) / 2 X ~ N .

(4)

In the naive constituent quark model we expect no gs pairs in the nucleon, i.e. X~N ~ 25 MeV, almost half its empirical value at t=0 . On the other hand, the empirical value suggests that the ratio o f strange to non-strange quark pairs (4) is about 0.25. This is far beyond the naive expectation and would very likely mean that almost 300 MeV of the nucleon mass is carried by gs pairs. I f true, this result would have dra- matic effects on low energy hadronic physics [6-8 ].

While the extraction of S~N is not without its own problems, there have been recently several attempts to try to assess the possibility of non-vanishing gs pairs in the nucleon as a way of explaining a large 27~N term. Part of the arguments recently put forward were based on the bag model [ 9 ], the Skyrme model [ 10 ] or the

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157


Nambu-Jona-Lasinio model [ 11,12 ]. The bag model violates chiral symmetry explicitly as well as Lorentz invariance. The Skyrme model violates Lorentz invariance and has difficulties of its own in the SU (3)c sector. The Nambu-Jona-Lasinio model has problems with chiral perturbation theory [13] and can- not be probed beyond the mean-field level. In gen- eral, most of these models do not relate immediately to the QCD degrees of freedom.

In this letter we would like to address the above issue, using a microscopic picture &the QCD ground state as a liquid of instantons and antiinstantons. The problem of constructing a vacuum state from a gas of instantons has been recently reexamined in great de- tail by Dyakonov and Petrov [14,15] and also by Shuryak [ 16-19 ]. In particular, they were able to cir- cumvent the conventional infrared problems and obtain a reasonable description of the bulk properties of the QCD ground state.

In the naive quark model, the nucleon is composed of three constituent quarks each of mass MQ (ignoring isospin breaking). The latter are a complicated admixture of all the light QCD degrees of freedom including strange quarks. In terms of the Hellmann- Feynman theorem we have

(NI~s lN) =OMN/Oms'~ OMq/Om~ , (5)

where My is the nucleon mass. The appearance of MQ is a direct consequence of the non-vanishing of (Ou), i.e. the spontaneous breakdown of chiral symmetry. This mass can be obtained in the framework of the instanton model of the QCD vacuum (see e.g. Shur- yak [20 ] and Dyakonov and Petrov [ 1 5 ] ). One ob- tains the relation MQ~p2( fm) , where p is the average size of the instantons. Independent of the proportionality constant we find that

(N l gs lN) ~MN ~,~, log l ( f m ) l • (6)

Below we show to evaluate (6) in the instanton model of the QCD vacuum using SU (2)~ gauge fields as described by Dyakonov and Petrov [ 15 ] and also by Shuryak [ 16 ]. The SU (3) c case has been discussed recently by Shuryak [21 ]. Overall, his results are consistent with the SU (2) ~ case.

In the instanton description of the QCD ground state the SU (2)-color gauge fields are approximated by a linear superposition of instantons and antiinstantons of constant scale parameter p = ~ fm. Each

instanton is described by seven collective coordinates, four for its position (z~) and three for its ori- entation [ SU (2) matrix U]. The Hilbert space of the fermions is truncated to the space of zero modes as- sociated with the instantons. For a system with an equal number ½ N of instantons and antiinstantons the following partition function can be derived [ 16 ]:

Z = j dg2 d~ l-~ det(m~ + T ' T )

- -

where Tis an N× Nmatrix with matrix elements given by

T I j = i F ( l z t - z j [ ) T r ( U t r + ( e t - z j ) z U j ) , (8)

corresponding to the overlap of the zero modes asso- ciated to instantons and antiinstantons. The collective coordinates of the instantons (antiinstantons) are denoted by g2(g2), T + = (~, +_il) and Fis a known scalar function of the interparticle distance. The chiral condensate is obtained by differentiating with respect to the current mass of the up-quarks and can be expressed in the average eigenvalue density ~ (2 )= ( y~ u ~ ~(2--2i) ) of the overlap matrix T,

- 1 f 2m (ua)= ~4 dA~(2)22+m ~, (9)

where V4 is the euclidean four-volume with periodic boundary conditions. The averaging ( ) is over all collective coordinates with the fermion determinant in (7) as a weight. The partial derivative can also be expressed in terms of the eigenvalues of T,

1 f t,-~(2, 2' ) 0m~(taU)=~44 d2d2' ( 2 + i m ) ( 2 ' + i m 0 ' (10)

where Vc (2, 2' ) is a connected two-point function defined by

~c(2, 2' ) = ( ~ ~(2-2 i )~(2 ' - 2 j ) )

#(2, 2' ) measures the correlation between the eigenvalues. The variance of the eigenvalue density follows from the diagonal part of & (2, 2' ). Notice that

158

Volume 2 1 7, number ! ,2 PHYSICS LETTERS B 1 9 January 1 989

the above two-point function carries a non-trivial mass dependence via the statistical average.

The averaging over the collective variables amounts to the calculation of a 7N-dimensional integral in- volving a fermion determinant. As is well known from lattice QCD, such integrals can be evaluated with the Langevin equation. The solution of this stochastic equation produces a statistical distribution of the collective variables with weight given by the fermion determinant. The most costly numerical problem in this algorithm is the inversion of a J J ~N× ~N matrix for each iteration step. We will perform an exact inversion. Our numerical methods have been tested in many different ways. At the moment we only want to mention that in the case of one instanton and one antiinstanton, several observables can be calculated an- alytically. Within our numerical accuracy these results have been reproduced. For more details we refer to ref. [22].

Before we present our numerical results we evaluate the derivative in two extreme cases, the limit of low instanton density and the limit of high instanton density. The first case is characterized by molecules of instantons and antiinstantons [ 16 ]. The full fermion determinant factorizes into the product of two by two determinants. The eigenvalues of each of them depend solely on the internal state of each molecule and, to a good approximation, are statistically independent. In the limit of a high instanton density the instantons overlap. Instead of a gas of molecules we have an instanton-antiinstanton medium in a liquid state, The distribution of the collective coordinates can be approximated by a uniform distribution. Con- sequently, the matrix elements Ttj will be distributed randomly and the eigenvalues will be strongly corre- lated according to correlation functions obtained by random matrix theory [23,24].

In the low density limit, the eigenvalues in (l 1 ) are uncorrelated and all terms with 2i~2j cancel. From the diagonal terms we obtain

2ms ( ~ u ) - 2 m ( ~ s ) 0,,,~ ( f lu) = ( m ~ _ m 2 )

2V4 ( g s > ( O u ) . ( 1 2 ) N

To leading order in ~ )/rn~ the terms that are inversely proportional to the density of instantons can-

cel. Notice that in this case the connected two-point function corresponding to ( 11 ) is

t Y , . ( 2 , ) / ) = ~ ( 2 ) ~ ( 2 - 2 ' ) - ( 2 / N ) v ( 2 ) ; ( 2 ' ) (13)

and satisfies the consistency condition fd2~c(2, 2 ' ) = 0 .

In the high-density limit the correlation function vc (2, 2' ) is given by

V c ( 2 , 2 ' ) = v ( 2 ) d ( 2 - 2 ' ) - v ( 2 ) v ( 2 ' ) Y z ( r ) , (14)

where Y: (r) is Dyson's two-point cluster function for the gaussian orthogonal ensemble [25,26 ],

sin 2 zcr zr cos zcr--sin gr Y z ( r ) - - ~2r2 ;¢r2

× ( i ds s in~s 1 d ) rrs 2 d r r r l , (15)

o

and r is the distance between eigenvalues in units of the average level spacing

2'

r = | O(x) dx. (16) 2

Asymptotically, Y2 (r) behaves as (7rr) - 2. Its value at r = 0 is equal to 1. The correlation function in (10) satisfies the consistency condition fd2 Vc (2, 2' ) =0. With this in mind (10) can be rewritten as follows:

0m~(171U) ----- V44 d 2 d 2 ' 2 m m s ( j [ 2 - - ~ . ' 2 ) 2/7c1(2, ). ' )

X [ (22+m 2)(2 '2+m~ 2) ( 2 ' 2+m 2) (22+m~)] -J ,

(17)

where vd is defined as ~c(2, 2' ) - o (2 )~ (2 -2 ' ). From (12) it is clear that the integrand can be approximated by the asymptotic result for large values of 2 - 2 ' . For constant level density and by using the asymptotic result for 112 (r) the integrals can be evaluated trivially. The result is

0m~ < flU> = -- 1/Vn(m+ms) 2 . (18)

We conclude that at high instanton densities 0,,s ( f lu) vanishes in the thermodynamical limit. From (6) it follows that the strangeness content of the nucleon is z e r o .

Our numerical results for the correlation function 0(2, 2 ' ) will show whether the rather low realistic

159


0

0

Yz(r) -0.4

-0.8

r ~

0.8 1.6 2.4 I I I

+: × / ~ 3

/ ox ~=8.33E-4

Fig. 1. The two-point cluster function Y2(r) at low instanton density. The quantity r is the distance between the eigenvalues in units of the average level spacing. The full line represents the random matrix result given in eq. (15). The circles and the crosses show results for an instanton density of 32/144p-4 (=8.33× 10-4p 4) with 32 and 64 particles, respectively. The current quark masses are equal to (0.02p- ~, 0.02p- ~, 0.3p-~). Results for massless quarks at a density of 64/204/) -4 ( =4.0X 10 4/)-4) are given by the pluses.

value of the ins tanton density is sufficient for repro- ducing the volume dependence shown in (18) . Nu- merically we obtain P(2, 2' ) in the form of a histo-

gram with bin size equal to 0.003 p - ~ at low instanton density and 0.01 p-~ at high ins tanton density. Dy-

son's two-point cluster funct ion Y2 (r ) is obtained by averaging the result for the eigenvalues below 0.4 p - ~. The results for the low-density l imit are given in fig. 1. The circles and the crosses show the case in which the density is equal to 32/144 p - 4 ( = 8 . 3 3 X 1 0 - 4

p -~ ) and the number of particles equal to 32 and 64, respectively. The pluses show results for a density of 64/204 p --4 ( =4.0;< 10-4p - - 4 ) . In the latter case the

current masses are zero. In all other cases they are (0.02 p - l , 0.02 p-~, 0.3 p - ~ ) . The full line repre-

sents the two-point cluster function given in ( 15 ). For

samples of N particles the two-point cluster function is - 2 / N . From this figure it is clear that this result

gives a fair description of the numerical data at low

density and for zero masses. For finite masses chiral

symmetry is explicitly broken. This results in the dis-

sociation of the instanton molecules and gives rise to

stronger correlations between the eigenvalues. In-

deed, this is what is shown by the points in this figure

for non-zero quark masses.

Results for a high and a realistic value of the in-

stanton density are given in fig. 2. Here, the circles

and the crosses show results for a density of 0.5 p-4 ,

and a number of particles equal to 32 and 64, respec-

tively. The full circles and the pluses show results for a density of 32/84p -4 ( =7.81 × 1 0 - 3 p - 4 ) , and the

number of particles equal to 32 and 64, respectively.

The latter density reproduces the value of the gluon

condensate as obtained in the QCD sum rules [27].

The random matrix correlation function is close to

the simulated data. The higher the density the better

the fit. In view of these results we expect that the de-

rivative 0,,~ log I ( O u) I is inversely proportional to the volume. As can be seen from table l, at high in-

s tanton density we reproduce this volume depen-

dence. Most remarkably, we find an even stronger

volume dependence at low density. On the other hand,

the chiral condensate shows only very little volume

dependence. The derivative 0 , , l o g l ( f m ) l is also

strongly volume dependent. However, the ratio of the

two derivatives,

Table 1 Dependence of I (0u) I, O-,s log I (flu) l, ~,,, log I (flu) I and R on the instanton density N/V4. N is the number of instantons plus antiinstantons in a box of size V 4 = L 4. Results for 32 and 64 particles are given for each density. The quark masses are equal to (0.02, 0.02, 0.3 ). For convenience, all numbers are given in units where the size p of the instantons is 1.

N L N/V4 I( uu ) I am, log I (tiu) I OmlOg I (aU) I R

64 16.73 8.33 X 10 -4 8 . 2 2 × 10 -3 --0.109 -4.86 32 14.00 8 . 3 3 X 10 - 4 9 . 2 9 X 10 -3 -0.331 - 13.58

64 9.51 7.81X 10 -3 4.18 X l0 2 -0.208 - 5.62 32 8.00 7.81X 10 -3 3.97X 10 2 -0.499 - 11.87

64 3.36 0.5 0.991 -0.130 -9.10 32 2.83 0.5 0.998 -0.270 -17.50

0.0224 0.0244

0.0370 0.0420

0.0143 0.0154

160


0

0

T Y2(r)

-0.4

-0.8

r ~ 0.8 1.6 2.4

< • o N f : 5

ox ~=05

e+ ~ = 7 8 1 E - 5

Fig. 2. The two-point cluster function Y2(r) at "high" instanton density. The circles and the crosses show results for an instanton density of0.5p 4 with 32 and 64 particles, respectively. Results at the realistic density of 32/84p-4( = 7.81 × 10 3p-4) are given by the full circles (32 particles) and the pluses (64 particles). The quark masses are equal to (0.02p -l , 0.02p l, 0.3p ~). For further explanation see the caption of fig. 1.

R : ( N I g s l N S / ( N l f u I N )

~0,,,, log I ( 0 u 5 1 / 0 , , log I ( f l u5 I , ( 1 9 )

has a m u c h weake r v o l u m e dependence . At the real-

istic ins tan ton dens i ty o f 7.82 X 10-3p--4 this ra t io is

equal to 0.0420 and 0 .0370 for a total n u m b e r o f par-

t icles equal to 32 and 64, respect ively .

In conc lus ion , ou r ca lcu la t ions show that the chiral

condensa te o f the light quarks is ve ry insens i t ive to

the mass o f the s t range quark. In t e rms o f (6 ) and

unde r the a s s u m p t i o n that the Q C D g round state can

be desc r ibed in t e rms o f ins tan tons and an t i ins tan-

tons, we expec t no m o r e than a negl igible s t rangeness

a d m i x t u r e in the nuc leon state and the re fore a small

Z~N term. T h e r e b y we ques t ion the va l id i ty o f e q . ( 3 )

and with it the range o f appl icab i l i ty o f the na ive

SU (3) qua rk model . Th is conc lus ion we bel ieve, is

gener ic to any n o n - p e r t u r b a t i v e and i n h o m o g e n e o u s

desc r ip t ion o f the Q C D ground state.

We wou ld l ike to thank G.E. Brown and T.H.

Hans son for useful discussions. O n e o f us ( J . J .M.V. ) wou ld like to thank H.W. Wyld for i n t roduc ing h im

to the n u m e r i c a l t echn iques o f l a t t i c e - Q C D sim-

ula t ions .

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Is the nucleon strange?