Medium effects on nucleon properties

Volume 225, number 1,2

PHYSICS LETTERS B

13 July 1989

MEDIUM EFFECTS ON NUCLEON PROPERTIES E. R U I Z ARRIOLA, Chr.V. CHRISTOV ~ and K. G O E K E

lnstitut f~r TheoretischePhysik II, Ruhr-UniversitdtBochum, D-4630 Bochum, Fed. Rep. Germany Received 7 March 1989; revised manuscript received 3 May 1989

The modifications of the nucleon structure due to the presence of an external baryon medium are investigated in a chiral nonlinear quark-meson theory. To this end the Nambu-Jona-Lasinio approach is used to evaluate the pion decay constant and the pion and sigma masses at finite density. Those meson properties serve to fix the parameters of the linear chiral sigma model, which is then solved in a variational projected mean field approach in order to obtain nucleon properties. The proton radius shows an increase of 22% and the nucleon mass a decrease of 17% if the medium reaches nuclear matter density. The axial vector couplingconstant is reduced by about 10% and the electric form factors show remarkable changes at q~ 1 GeV/c.

In the last years it has been suggested several times that nucleons might suffer a change of their internal structure when they are embedded into a m e d i u m with finite baryon density. Ideas like the "swelling" of the nucleon and the reduction of the nucleon mass have become popular to explain various experimental facts seen e.g. at the nuclear structure function in deep-inelastic lepton scattering [ 1 ], the quenching of the axial vector coupling constant, gA, in nuclear [3decay [2], the strength function in nuclear quasielastic electron scattering [ 3 ], or the peculiarities of the ratio of the K+-12C to K + - d elastic scattering cross section [4]. In the present paper we investigate those m e d i u m effects in the framework of the relativistic and chiral invariant N a m b u - J o n a - L a s i n i o model [ 5 ] with scalar and pseudoscalar quark-quark couplings and solve it in the c o n t i n u u m approximation for a finite Fermi m o m e n t u m kF = ( 3 n 2,0) i/3 u n d e r the condition that for kv= 0 the spontaneously broken chiral v a c u u m is reproduced with proper pion decay constant f~--93 MeV and proper pion mass m~= 139.6 MeV. In a second step the m e d i u m values f*, m* are used to define a modified linear chiral sigma model [6]. This is solved to obtain nucleon properties by well-known projected mean-field techniques [7]. Such a particPermanent address: Institute for Nuclear Research and Nuclear Energy, Sofia 1784, Bulgaria. 22

ular scheme is consistent in the sense that the two models are related by a well-converging heat-kernel and gradient expansion, at least for classical fields and hedgehog structures [ 8 ]. The lagrangian of the N a m b u - J o n a - L a s i n i o model is used in the form ~ = ~Piy*'Oj,~ - mo ~PqJ

+ ~C[ ( ~o~u)2+ ( ~ e ) 2 ] . Introducing sigma

and

(1) pion

fields

by

a=

_g~t~j/~2 and n = -g~t75"f~t/f12and assuming them to be classical, one can prove [8,9] that the total energy density of the v a c u u m reads A

E

n-

_4Nc fJ

d3k x/k2 +g2 (t72 + 7t 2 )

o -~- 1/.~ 2 ( 0"2 "~- I[' 2 ) -- OL0" '

(2)

if a t h r e e - m o m e n t u m cut-off is used for simplicity. The a is related to the bare mass mo by c~= -l~2mo/g and the G is expressed by G=gZ/It 2. Dem a n d i n g PCAC yields that c~= - f ~ m 2. D e m a n d i n g further that E/g2 shows a m i n i m u m at the v a c u u m values of a and ~t yields immediately g v = 0 and av(/22-4Ncg2j~/2(A))=f~m 2. In the limit of vanishing pion mass the latter is the well-known gap equation. The requirement that 02E/c~Ttzl,~v=o = m 2

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Volume 225, number 1,2

PHYSICS LETTERS B

is fulfilled by av =f~ which allows then to obtain the yet unknown sigma mass as m .2= O2E/aa21 . . . . r~ = m~ + ( 2 g £ ) 2. F o r f ~ a n d m~ we assume their experimental values. Thus besides the q u a r k - m e s o n coupling constant g the only p a r a m e t e r i n d e t e r m i n e d so far is the cut-off A, which appears in the regularized integral

13 July 1989

m.2 = m ~2f J f

J , ( A ) = "~ dsk

(k2+g2f

2 )-"

.

(7)

~,

(8)

m .2 -m,~- .2 + 4Ncg2 (giC~)2J~/2 (A) , where A

J*(A) = f

A

*

(3)

d3k 1 (2z0 3 ( k 2 + g 2 f .21,~"

(9)

kv

0

F o r a given g the A is fixed such that the pion decay constantf~, evaluated by the corresponding F e y n m a n diagram [ 8 - 1 0 ], agrees with the v a c u u m value o f a which is by construction av=f~ as well. This results in

N~g2J3/2(A) = 1 .

(4)

All those conditions together leave g as the only free parameter. It will be fixed later to g = 4 . 9 8 such that the nucleon mass is M N = 938 MeV. This value o f g leads to a constituent quark mass m =gf~ = 463 MeV. The value of the quark condensate is then ( flu ) = ( d d ) = - ( 242 MeV ) 3, which corresponds to a bare quark mass m o = 5.8 MeV. A p p a r e n t l y these values are in good agreement with the n u m b e r s extracted from Q C D sum rules [ 11 ]. The present N a m b u - J o n a - L a s i n i o a p p r o a c h can easily be extended to a m e d i u m with a finite density. The symmetry-breaking term and the above v a c u u m values of/z, A a n d g are kept fixed because they define the lagrangian. The total energy density, however, becomes m o d i f i e d by introducing the F e r m i m o m e n t u m kv:

F o r g = 4.98 the dependence o f f * , m=* and m~ on the d e n s i t y p = k 3 / 3 n 2 is shown in fig. 1. One realizes a clear decrease o f f * and m* a n d an increase o f m* with increasing m e d i u m density. At about three times nuclear m a t t e r density (Pnm= 0.16 f m - 3 ) a phase transition from the chirally broken G o l d s t o n e phase to the restored Wigner phase takes place b e y o n d which f * ~ 0 and m* ~ m*, both increasing with med i u m density p. Actually these curves are close to those o b t a i n e d by Bernard et al. [ 12 ] in the same model with a different formalism. In the present p a p e r we are interested only in the region 0 ~

1,000-

~

g:4.98 A=577 MeV

8oo- ~

/

A

E* 12 - - 4 N c

(" d3k J

400-

Y

hT

+ ½ / z 2 ( a . 2 + n . 2 ) - f ~ m ~2a . .

j~/~.

(5)

The m e d i u m reduced pion decay constant, f * , is given by the m i n i m u m o r E * with respect to a*. This yields, besides lr* = 0, the equation

-- 4NcJT/2 ( A ) g 2 f . + It 2 f . = m ~¢'~.

(6)

The second derivative o f E* with respect to a* a n d n* at a~* =f~* and at zero, respectively, are used to d e t e r m i n e the masses rn~* a n d m~* in the m e d i u m . They are given by

200-

. . . .

-- -- ~

~

/

~

--

f,~

-.

sigma mass

--

pion m a s s

0 •0 0

.10

.20

.30 .40 .50 .60 medium baryon density (fm -~)

.70

.80

Fig. 1. Medium effects on meson properties. The effective pion decay constant,f* (solid line), the pion mass, m~* (dashed line), and the sigma mass m* (dash-dotted line), are plotted versus the baryon density of the medium. The calculations are done for a quark-meson coupling constant g= 4.98 and a corresponding ultraviolet cut-offA = 577 MeV. The values for f* are enlarged by a factor of ten. 23

Volume 225, number 1,2

PHYSICS LETTERS B

~ = ~oiT~,Ol' ~V_gtp(a.+iys,rn. ) ~v

0.100

medium d e n | l t y (fro-3):.

- ( ½,~*)~(~*~ + n * ~ - v *~) 2 + ½O"o*O,,~* . 0.080l p zt'O~Tt* - f ~rn~ * " 2 a* , + ~0

(10)

with a* = f * , / l ' 2 = (m .2 - m .2 ) / 2 f .2 and v*2 = f . 2 _ rn.2/2.2. For the nucleonic solution of this lagrangian we employ the variational procedure of Fiolhais et al. [ 7 ] based on mean field states with generalized hedgehog structure and projection techniques for spin and isospin. These methods are well established for vanishing medium density and will be identically used at finite kv. To be specific, the trial function for the nucleon is assumed to be Io/) = Iq3) i]~) II I ) . Here l Z ) and IH ) are quantal coherent Fock states representing the sigma and pion cloud and the Iq3) corresponds to three quarks of different colour in a 1sorbit with the spin-flavour structure of a generalized hedgehog: [u{) sin ~/- [ d t ) cos q. The actual nucleon state is obtained from [~P) by application of projection operators: = _J T pT pT , . I~PJT) ~ g~i,K, ,,1~ M~x,-I ~g) ( 11 ) K,K?

The gx.s>, z y the flavour mixing coefficient r/and the orbital degrees of freedom in Iq 3 ), IE > and 117> are determined variationally as described in details by Fiolhais et al. [7]. Needless to say that consistency checks like the Goldberger-Treiman relation and virial theorems are fulfilled in the vacuum as well as at finite density. Following the lines of Alberto et al. [13,14] the form factors can be evaluated and fig. 2 shows the neutron electric form factors. As known [13] the model gives for this very sensitive quantity a good agreement with the experimental data at vanishing medium density. For growing kr the form factors show a clear trend to decreasing values yielding at nuclear matter density and finite m o m e n t u m transfers a reduction of about 50%. The proton electric form factor shows the similar effects. Trends of comparable clarity are encountered at all observables as one can see in table 1. All square radii increase between 20% (magnetic), 40% ( a x i a l ) a n d 50% (proton charge). Only the neutron square charge radius decreases its absolute value by 20%. The trends are in accordance with the behaviour of the form factors at zero as well as at finite m o m e n t u m numbers. 24

13 July 1989

--

0.0o

- - 0.08 •~ 0,060-

-- 0.16

o

.~ 0.040-

~ 0.020-

0.000 .000

.050

.100

.150

.200

.25o

..500 .350

qZ (6eVz)

.400

.450

.500

Fig. 2. The neutron electric form factor. It is plotted versus q2 for various baryon densities of the medium. The experimental data are taken from ref. [ 15 ].

The nucleon energy decreases resulting in an m * / m = 0 . 8 3 at nuclear matter density. The g~ and the magnetic moments are only little affected, the ga decreases slightly and /tp and I/tn I show a small increase. To contrast these trends the pion-nucleon coupling constant g~NN is very little affected and stays practically constant at increasing medium density. It fulfills the Goldberger-Treiman relation with medium values. Actually the trends of the radii are easily explained. Due to the reduction off~ to f * the scale of the quark distribution is bound to increase. This and the increase of the pion mass cause the pion field to decrease its magnitude and to reduce its tail, because its source is spatially more extended. Since most of the observables are dominated to 70% by the quark contribution, its "swelling" causes an increase of the total radius. The only exception is the neutron square charge radius. It results from a very detailed balance between positive charge from quarks and negative one from pions and the pion field tail is responsible for the negative sign of ( r ~ ) n . Since the magnitude of the tail is reduced with increasing medium density and the quarks show "swelling" the [ ( r ~ ) . [ gets reduced as well. At densities of p > 2pnm the ( r ~ ) n becomes even slightly positive. It is interesting to compare the above values with estimates originating from different models as well as with some experimental data. For example a "swelling" of the nucleon is used [4] as an explanation of the problems associated with the analysis o f K +-~2C

Volume 225, number 1,2

PHYSICS LETTERS B

13 July 1989

Table 1 Medium effects on nucleon properties. The medium values of the meson sector are given together with the observables and square radii of the nucleon in dependence on the medium density p. For finite medium densities the values are given relative to the theoretical numbers at p = 0. Quantity

Absolute values

Relative values

experiment

p=0.0fm 3

p=0.08fm 3

p=0.16fm 3

f~ (MeV) m~ (MeV) mo (MeV)

93 139.6

93 139.6 937.7

0.89 1.06 0.93

0.77 1.14 0.85

EN (MeV) Ea-EN (MeV) ( r ~ ) p (fm ~) ( r ~ ) n (fm 2) ( rZm)p (fm 2) ( r m ) n (fm 2 ) ( r 2 ) (fm 2) /zo (n.m.) ~un (n.m.) gA g~Nr~

938 295 0.65 -0.12 0.706 0.757 0.507 2.79 - 1.91 1.23 13.6

938 156 0.662 -0.094 0.765 0.802 0.477 2.82 - 2.44 1.77 17.1

0.92 0.92 1.21 0.92 1.11 1.10 1.19 1.05 1.03 0.97 1.00

0.83 0.79 1.41 0.80 1.28 1.25 1.48 1.11 1.07 0.94 1.00

and K + - d scattering data and indeed the increase of the present proton proton radius by 20% provides the right order of magnitude. A reduction of the nuclear mass by (15-20)% is predicted by Mahaux et al. [ 16,17 ] quite in agreement with outer numbers. Furthermore a phenomenological analysis of Mulders [18] shows a noticeable sensitivity of the isoscalar magnetic moment in contrast to the isovector one, a trend also to be seen in our numbers. However, the magnetic radii stay about constant in his analysis whereas they show some variations in the present approach. Actually after completing the calculations the present authors were confronted with preprints of Meissner [19]. There a generalized Skyrme model including dynamical vector mesons was used with medium modified meson values evaluated by means of a suitably formulated Nambu-Jona-Lasinio approach. However, none of the parameter sets used shows a shrinking of the neutron charge radius, as our numbers do, although the corresponding form factors behave similar to ours at finite momentum numbers. The other trends are qualitatively similar in both approaches. Recent calculations by Krewald [20] suggest that the "swelling" of the nucleon, caused by the pion cloud polarization due to the A-hole excitation, is masked by Pauli blocking. The effects of Pauli block-

ing can roughly be estimated in the present model by removing from the valence orbit those Fourier components which are occupied by the medium quarks. This Pauli-modified orbit, properly normalized, serves then as a source for new meson fields. The resulting proton charge distributions are plotted in fig. 3. Apparently in our model the medium effects seem not to be dominated by Pauli blocking. Nevertheless lO Lmedlum

density

(fro-3):

C

!

t' --

.00

~---~ .20

.40

~

~

~-%

.60

.80

1,00

with Pauli b l o c k i n g

--

7 1.40

,

~ 1.80

~

~ 2.20

R (fro)

Fig. 3. The effect of Pauli blocking on the charge distribution. For medium densities of p = 0 , p=p,,m/2 and P=P,,m the charge distribution of the proton is plotted versus the distance from the centre. Dashed curves are with Pauli blocking corrections, solid curves are without.

25

Volume 225, number 1,2

PHYSICS LETTERS B

a m o r e precise t r e a t m e n t is presently u n d e r investigation. We can s u m m a r i z e o u r points: U s i n g the N a m b u J o n a - L a s i n i o m o d e l for the d e s c r i p t i o n o f the m e s o n sector o f the m e d i u m w i t h a finite b a r y o n d e n s i t y we are able to e v a l u a t e the solitonic sector o f a n u c l e o n e m b e d d e d into this c o n t i n u u m . We find a n o t i c e a b l e increase o f all n u c l e o n radii except o f the n e u t r o n charge radius w h i c h shows a decrease. T h e f o r m factors get strongly r e d u c e d at finite m o m e n t u m n u m bers whereas the m a g n e t i c m o m e n t s and axial v e c t o r c o u p l i n g c o n s t a n t stay nearly constant. P a u l i blocking effects t u r n to be not v e r y i m p o r t a n t . T h e c o n t r i b u t i o n s o f P. A l b e r t o a n d M. F i o l h a i s in p r e p a r i n g the c o m p u t e r codes are gratefully a c k n o w l edged. T h i s w o r k is partially s u p p o r t e d by the Bund e s m i n i s t e r i u m ftir F o r s c h u n g u n d T e c h n o l o g i c (Int e r n a t i o n a l e s BiJro a n d C o n t r a c t 0 6 - B 0 - 7 0 2 ) , the K F A Jfilich ( C O S Y - P r o j e c t ) a n d the B u l g a r i a n M i n istry o f Culture, Science a n d E d u c a t i o n u n d e r C o n tract No. 325.

References [ 1] EM Collab., J.J. Aubert et al., Phys. Lett. B 123 ( 1983 ) 275; R.G. Arnold et al., Phys. Rev. Lett. 52 (1984) 727.

26

13 July 1989

[ 2 ] T.E.O. Ericson and W. Weise, Pion in nuclei (Oxford, 1988 ). [ 3 ] R. Barreau et al., Nucl. Phys. A 402 ( 1983 ) 515. [4] G.E. Brown, C.B. Dover, P.B. Siegel and W. Weise, Phys. Rev. Lett. 60 (1988) 2723. [ 5 ] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 ( 1961 ) 354. [ 6 ] M. Gell-Mann and M. Levi, Nuovo Cimento 16 (1960) 705. [ 7 ] M. Fiolhais, K. Goeke, F. Grfimmer and J.N. Urbano, Nucl. Phys. A 481 (1988) 727. [8] Th. Meissner, E. Ruiz Arriola, F. Griimmer, H. Mavromatis andK. Goeke, Phys. Lett. B 214 (1988) 312. [9] Th. Meissner, F. Grtimmer and K. Goeke, to be published. [ 10] V. Bernard, Phys. Rev. D 34 (1986) 1601. [ 11 ] M.A. Shifman, A.J. Vainstein and V.I. Zakharov, Nucl. Phys. B 147 (1979) 385;B 163 (1980) 43. [ 12 ] V. Bernard, U.-G. Meissner and I. Zahed, Phys. Rev. Lett. 59 (1987); Phys. Rev. D 36 (1987) 819. [ 13 ] P. Alberto, E. Ruiz Arriola, M. Fiolhais, F. Griimmer, J. Urbano and K. Goeke, Phys. Lett. B 208 (1988 ) 75. [14]P. Alberto, E. Ruiz Arriola, M. Fiolhais, K. Goeke, F. Grfimmer and J. Urbano, to be published. [ 15 ] G. H6hler, E. Pietarinen, I. Sabba-Stefanescu, F. Borkowski, G.G. Simon, V.H. Walter and R.D. Wendling, Nucl. Phys. B 114 (1976) 505; G.G. Simon et al., Z. Naturforsch. 35A (1980) 1. [ 16 ] C. Mahaux and R. Sartor, Nucl. Phys. A 475 ( 1987 ) 247. [ 17 ] C.H. Johnson, D.J. Horen and C. Mahaux, Phys. Rev. C 36 (1987) 2252. [ 18] P.J. Mulders, Phys. Rev. Len. 54 (1985) 2560. [19] U.-G. Meissner, Phys. Rev. Lett. 62 (1989) 1013; MIT preprint CTP:~ 1679, to be published. [ 20 ] S. Krewald, Jfilich preprint ( 1988 ), submitted to Phys. Lett.