Hadron bags in nuclear matter

Nuclear Physics @ North-Holland

A435 (1985) 669-676 Publishing Company

HADRON

BAGS IN NUCLEAR

MATTER+

C.W. WONG Department

of Physics, University of California, Received

28 August

Los Angeles, CA 90024, USA 1984

Abstract: The idea of Celenza, Rosenthal, and Shakin for studying nucleon structure in nuclear matter by having the nuclear mean fields act directly on the quarks in nucleons is applied to the MIT bag model with center-of-mass correction. An attractive scalar potential makes quarks more relativistic by decreasing their masses. The probability for populating the lower Dirac component is thereby increased, the axial charge is reduced, and the A -N mass difference is increased. Nucleon magnetic moments become larger, while nucleon bag and matter radii tend to decrease. These results are rather different from those obtained in the CRS quark-diquark model. The limitations of the calculated results are briefly discussed.

1. Introduction

Recently Celenza, Rosenthal, and Shakin ‘) (CRS) have pointed out that nucleon structure in nuclear matter (NM) can be studied directly by applying the nuclear mean fields on the quarks themselves. Using a quark-diquark model ‘) they have previously constructed, they find that the following nucleon properties are all increased in NM: the scalar density radius by about 60%) the baryon density radius by 40%, charge radii by 30%, magnetic moments by 20%, and the axial charge by 10%. Based on these results, they ls3) have suggested that the A-dependence of the structure function for lepton inelastic scattering “) [the so-called EMC effect ‘)I could be accounted for by this radial expansion of the nucleon in NM. Since these calculated medium effects on nucleons are so large, it appears useful to try to understand how they occur. The purpose of this paper is to point out that this can be done quite easily in the MIT (Massachusetts Institute of Technology) bag model 6-8), which is a much more transparent model of hadrons. The main feature (sect. 2) turns out to be a reduction of the quark mass by the attractive nuclear scalar potential. The probability for populating the lower Dirac component is increased, causing a reduction of the axial charge. Both bag and nucleon radii shrink. Unfortunately, the bag models of ref. “) do not support a very strong nucleon scalar potential. Stronger scalar potentials can be supported in bag models with massive quarks. One such model is studied in sect. 3. The axial charge still decreases ’ This work is supported

in part by NSF contract 669

March

1985

PHY 82-08439.

C. W Wong / Hadron bags

670

in NM,but bag and nucleon radii might increase slightly depending on the treatment. The limitations of the model are pointed out. The conclusions drawn from this study are briefly summarized in sect. 4. 2. Bags in nuclar matter We follow ref. ‘) by approximating the nuclear mean field on a baryon in NM by constant scalar and vector (4th component) potentials S and U, respectively. Thus its invariant mass (squared) in NM is (M+S)2=(H-

U)2-P,

(1)

where M is the mass in free space. The numerical values of the mean fields used in ref. ‘) for nucleons in normal NM are S = -285 MeV,

U=270MeV.

(2)

The quarks themselves also interact with the nuclear mean field: h=a*p+/3(m+s)+u=ho+u,

where by the conservation

(3)

of the vector current u= U/n,

(4)

n being the number of quarks in the baryon. Hence the vector potential U disappears completely from the quark analog of eq. (1):

=n(m+~)~+2

C (hoih,-pi*pj) i
for n independent further to

(5)

quarks in the nuclear field. For a (1 s)” configuration, this reduces M&,=n(m+s)2+n(n-l)w~,

where w. is the eigenvalue of hw If the quarks interact via an additional baryon mass becomes M:,=((H,+

perturbation

(5) V, the quark part of the

V)‘-I”)

=(E,o+(V))2-n[o~-(m+s)2],

(7)

where the dispersion (V’) - ( V)2 has been neglected and EQo stands for nwo. This result actually holds for mesons also. Eq. (7) is a simple transcription of the Lan-Wong result *) to massive quarks in hadrons in NM. We must next relate the scalar potential s on a quark to the scalar

C. W. Wong / Hadron bags

671

potential S on a hadron. This is easy to do numerically: If M(s) is the hadron mass in NM where each quark sees a scalar potential s, then the scalar potential on the hadron itself is just S(s) = M(s) - M(s = 0) .

(8)

In the model of CRS, these scalar potentials are simply related: S(s)=np,(q2=O)s,

(9)

where ps(q2=O)=l-3s

(10)

is the quark scalar density [which can be expressed in terms of the Bogoliubov ‘) parameter S giving f of the probability that the quark is in the lower Dirac component]. The bag model result is more complicated, and it is simplest to leave it in the form of eq. (8). We expect from eq. (9) that under normal circumstances S and s have the same sign. An attractive S normally implies an attractive s, so that both nucleon and quark masses in NM would decrease. Now it can be shown readily that the Bogoliubov parameter S increases monotonically from 0 at the quark mass mq = cc (NR limit) to 3 at mq = -00, the latter corresponding to that relativistic limit in which the quark is entirely in the lower component. As 6 increases, the nucleon axial charge gA=$(l-2s)

(11)

decreases (through zero to a negative value!); the dimensionless quark 1s eigenmomentum x decreases from T to 0; the dimensionless hadron or quark rms radius rh/R=((r/R)2)1’2=[(1-~6)((r/R)2)u+f~((r/R)2)L11/2 looks more and more like that for the lower component; radius of the scalar density r,lR =

(12) the dimensionless

{((rlR)2),-ts[((rlR)2)u+((rlR)2)~l}1’2

rms

(13)

decreases rapidly [eventually to -((r/ R)2),]. In this connection, it is interesting to note that in the lowest 1s state, the dimensionless ms radii have the following relativistic and NR bounds:

(14) where the upper bounds are the relativistic limits obtained when x + 0. As a result, the more relativistic system is more surface peaked with a larger dimensionless rms radius.

672

C. W. Wong j Hadron bags

The shell-model quark magnetic moment

where wo=[(m+s)2+(~/R)2]1’2

(16)

is the quark energy, shows a more complex behavior. As the effective quark mass mq = m + s decreases, it will increase rapidly because of the increase in the magneton unit. Eventually it will reach a maximum at a negative value of m, and decrease rapidly to zero as mq+ -CO. These formulas show that if the quark mass decreases in NM, the axial charge is sure to decrease. This disagrees with the CRS result. The rms radii and the quark magnetic moment depend also on R, so that the bag part of the problem is also involved. The results are likefy to be model dependent. Those for the model of Lan and Wong labeled “OSC-2” are shown in fig. 1 in units of the free-space value for a range of attractive scalar potentials s. The ud quarks of this model are massless in free space. They develop negative masses in NM. As a result, 6, pa/R, and rh/R increase with decreasing s, while g, and r,/R decrease. However, the bag radius R actually shrinks, so that r, itself actually decreases. This is opposite to what CRS find in their model. The nucleon mass in NM shows a complex behavior: It decreases to a minimum of 0.8M, (free space) and then increases again with decreasing s because the single quark energy increases again when the quark mass becomes more negative. The

Fig. 1. Variation of nuclear properties (in units of the free-space values) in nuclear matter with the scalar potential s for the MIT bag model OX-2 of ref. *).

C. W. Wong / Hadron bags

673

maximum decrease of MN turns out to be less than 200 MeV. This behavior is probably due to the fact that the model is inadequate (especially in its treatment of quark self-energies) when the quark mass becomes large and negative.

3. Massive quark model Stronger scalar potentials on nucleons can be supported by using massive quarks in free space. A quark mass of 0.34 GeV is chosen in order to study relativistic effects in the constituent quark model at the same time. Except as noted below, the model is similar to the “oscillating” models of Lan and Wong in which quarks can oscillate against the bag. The bag mass and total quark energies are taken to be

E,=

Cwi(l+f)+a, [i

1 a,,&+ne,

i
R, I/

where av and Iii are known matrix elements and radial integrals of the color magnetic (CM) interaction. The quark self-energy will be assumed to be proportional to the single-quark energies wi themselves, i.e. eq. (18) will be used with f# 0 and e, = 0. This makes the model better behaved over a range of quark masses. The model parameters B, J; and CY,will be fitted to the masses of N, A, and w. The major problem turns out to be that the (Y,needed is too large. Typically the nucleon mass has a local minimum at a reasonable radius, but it will eventually collapse at small radii where the CM interaction dominates. It is probably unrealistic to insist that the full A -N mass difference should come from the CM interaction, but it is desirable at this stage to keep the model comparable to other bag models. For these reasons, the following procedure is used. The model is first studied by using the nucleon mass at the local minimum at a reasonable radius. This will allow the input of the N-, A-, and w-masses as usual. After the model parameters are determined, results on baryon properties in NM will again be calculated after switching off half and then all the CM interaction. The range of results gives an indication of the uncertainties in the model. They will hopefully motivate more realistic studies of the problem in the future. The resulting model, called OSC-3a, is compared with the model OSC-2 in table 1. We see that relativistic effects, so well characterized by S itself, are reduced by the increased quark mass, but they are still substantial. Indeed, g, turns out to be close to the experimental value of 1.25, rather than the NR value of 2. The nucleon magnetic moments, calculated with the Lan-Wong center-of-mass corrections, are smallish. The properties of a nucleon in NM having a mass of -653 MeV (corresponding to a nucleon scalar potential of -285 MeV) are also shown. We see that 6 is increased by 20% and N - A by 35%) while g, and r,, are decreased.

C. W. Wong / Hadron bags

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TABLE 1

Comparison

between

various quark bag models of hadrons on quarks s = 0) and in nuclear osc-2

in free space (with the nuclear matter (with s # 0)

OSC-3a

scalar potential

OSC-3b

model parameters B’14 [GeV]

0.1688 0 0.315 -0.454 -

m, [GeVl ec es

f s[GeYl ~NPWl

0 938

R [GeV-‘1

3.63 1233 4.21 783 3.39

MA

R M, R

0.11 0.34 0.749 -

0.11 0.34 0.375 -

-0.2628 0 939 3.69 1232 6.28 782 5.12

-0.128 652 3.03 1052 6.47 669 5.30

-0.2628 0 1045 4.98 1179 5.96 721 4.59

-0.158 760 5.02 944 6.17 568 4.77

0.115 1.28 0.59 0.49 0.35 1.80 -1.17

(1.23) (0.93) (0.92) (0.85) (0.78) 1.93 -1.53

0.099 1.34 0.73 0.65 0.49 1.94 -1.20

(1.30) (0.93) (1.17) (1.05) (0.96) 2.19 -1.58

nucleon properties s

0.174 1.09 0.73 0.52 0.3 1 2.04 -1.32

g.4

DW’I rh WI rs lfml pLplnml ICYlnml PQ

The numbers

within

parentheses

are given in free-space

units.

Part of the decrease in R could be due to the tendency of the nucleon to collapse under the strong CM attraction of the model. This can be avoided by reducing this attraction sufficiently. Model OSC-3b with half of the CM interaction strength gives stable nucleon solutions. Relativistic effects (increased S and decreased g,&) still increase on going into NM, but both bag and hadron radii now increase, although by only 1% and 5%, respectively. Even a complete removal of the color magnetic interaction does not change these results much more (with R increasing by 3% and r,, by 8%). Since these modified models are not necessarily more realistic, I believe that in most MIT bag models, the bag radius and perhaps even the hadron radius tend to decrease. The detailed s-dependences of nucleon properties in NM are shown in fig. 2 in units of the respective free-space values. The results shown are roughly independent of the CM interaction except for MN and R. The bag radius expands a little when the CM interaction is absent (model 3c) or at half strength (model 3b), but it contracts with the full CM interaction. The contraction is so strong that eventually

C. W. Wong / Hadron bags 1.4

--3b *.... 3c 1.2

0.8

D.6

Fig. 2. Variation

of nuclear properties (in units of the free-space values) in nuclear potential s for the MIT bag models OSC-3 of this paper.

matter with the scalar

even the local minimum of MN in R disappears as the system collapses. This collapse could probably be prevented by adding a repulsive term such as that arising from the coupling to the w-meson lo). The addition of other physical processes might also change the behavior of R. It would be interesting to study this question further. Solutions for A are stable, because the CM interaction is repulsive here. The A-mass also decreases in NM, but less rapidly than the N-mass. As a result, the A - N mass difference increases. Finally, I should point out that the quark mass dependence of baryon masses has also been calculated in the Friedberg-Lee ‘I) (FL) soliton model by Saly and Sundaresan 12). From their table. IV, one can see that the k(N+ A) mass can be decreased from 2045 to 1760 MeV (a difference of 285 MeV) by decreasing the quark mass from 340 to 170 MeV. The axial charge gA then decreases by 7%) while the hadron radius increases by 12%. The corresponding results for our model OSC-3c are the decrease of $(N + A) from 1120 to 835 MeV by decreasing the quark mass from 340 to 160 MeV, the decrease of gA by 9% and the increase of rh by 8%. Our results are thus in rough agreement with those of ref. 12). 4. Conclusions In many respects the bag models only give crude pictures of hadrons, but they are very simple and easy to work with. Quarks in bags immersed in an attractive

616

C. W. Wong / Hadron bags

scalar potential become less massive and more relativistic. This increases its occupation probability in the lower Dirac component, reduces its axial charge, and increases the A -N mass splitting. The reduction of g, should contribute to the quenching of the Gamow-Teller strength 13), while the increased A -N mass splitting could become a signature of the attractive nuclear scalar potential. It should be pointed out that the calculated g, has not yet been corrected for center-of-mass effects. While these effects are expected to be small, additional studies are needed to pin them down. Nucleon bag radii, and possibly also nucleon radii, may not change drastically on immersion in nuclear matter. These properties appear to be sensitive to the presence of other physical processes not included in the present study, especially the possible partial deconfinement of quarks from nucleons in nuclei 14). In summary, bags in nuclear matter appear to behave differently from the hadrons of the CRS model I,‘). However, the CRS idea of applying the nuclear mean fields directly on the quarks themselves appears to be a useful one. References 1) L.S. Celenza, A. Rosenthal and C.M. Shakin, Many-body solution dynamics, Brooklyn College preprint, BCINT 84/041/123, unpublished 2) L.S. Celenza, A. Rosenthal and C.M. Shakin, Covariant soliton dynamics, Boooklyn College preprint, BCINT 84/031/122, unpublished 3) L.S. Celenza, A. Rosenthal and C.M. Shakin, Phys. Rev. Lett. 53 (1984) 892 M. Jiindel and G. Peters, Phys. Rev. D30 (1984) 1117 4) R.G. Arnold et al., Phys. Rev. Lett. 52 (1984) 727 5) J.J. Aubert et al., Phys. Lett. 123B (1983) 275 6) T. DeGrand et aL, Phys. Rev. D12 (1975) 2060 7) C.E. Carlson, T.H. Hansson and C. Peterson, Phys. Rev. D27 (1983) 1556 8) I-F. Lan and C.W. Wong, Nucl. Phys. A423 (1984) 397 9) P.N. Bogoliubov, Ann. Inst. Henri PoincarC 8 (1967) 163 10) V. Vento, Phys. Lett. 107B (1981) 5 11) R. Friedberg and T.D. Lee, Phys. Rev. D15 (1977) 1694; D16 (1977) 1096 12) R. Saly and M.K. Sundaresan, Phys. Rev. D29 (1984) 525 13) C. Goodman, Nucl. Phys. A374 (1982) 241~ 14) D. Nachtmann and H.J. Pimer, Z. Phys. C21 (1984) 277; Heidelberg preprint HD-THEP-84-7