Strong interactions at all densities in a chiral soliton model coupled to quarks

Volume 152B, number 3,4 PHYSICS LETTERS 7 March 1985

STRONG INTERACTIONS AT ALL DENSITIES

IN A CHIRAL SOLITON MODEL COUPLED TO QUARKS ¢r

Vikram SONI Nuclear Science Division, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA and Department of Physics, Syracuse University, Syracuse, NY 13210, USA

Received 11 October 1984

In the context of a linear chiral o-model coupled to quarks we argue that the single baryon and high density ground state occurs in a topological chiral symmetry (TCB) broken phase and not the usual space uniform (UCB) chiral symmetry broken phase. We expect this to transit directly to a QCD chirally restored phase at higher density. New, but plausible features of dense matter in this phase are highlighted and it is shown that known features of nuclear matter can be recovered.

The Skyrme [1] topological model for the nucleon enjoys a belated but, now, substantial, stand- ing. A feature of topological models, e.g., vortices in superfluid 4He, where the mapping is from the boundary of space to the field or order parameter space, is that the energy goes as the second power of the topological charge n, E ~ n 2 . This induces a repulsive core which results in a periodic array of vortices as the ground state, above a certain critical rotation.

Though, the Skyrme model is a mapping of the

whole of compactified space into the field SU 2 space such a feature persists, but not quite exactly. This is responsible for a repulsive "core" as two in- finitely separated skyrmions are made coincident. This informs us that in a many skyrmion system with high density of skyrmions we may see a phenomenon of a periodic lattice in analogy to superfluid 4He. It is thus expected that such a periodic array of skyrmions is likely to be the ground state of such matter at high density. Such a model has been, following Skyrme [2], suggested by Kutschera, Pethick and Ravenhall [3] . A feature of the Skyrme lagrangian,

¢~ This work was supported by the Director, Office of Energy Research, Division of Nuclear Physics of the Office of High Energy and Nuclear Physics of the US Department of Ener- gy under Contract DE-AC03-76SF00098.

extrapolated to high densities, is that the energy density goes as the 4/3 power of the density, i.e. like the QCD based quark model at high densities. This is in- deed noteworthy. However, there are two differences between the Skyrme model and the QCD quark model: (i) the Skyrme model, based on the nonlinear e- model, has spontaneous chiral symmetry breaking (for all densities) unlike the QCD quark model; (ii) the Skyrme model is based on a low energy effective lagrangian which is not expected to be correct at high densities when the QCD quark gluon degrees of freedom are explicit.

Recently, a linear chiral model for quark meson (pion) interactions was proposed by the author, Pdpka and Kahana [4] and by Birse and Banerjee [5] witlr no quartic Skyrme-like term. In this model the gluon degrees of freedom are completely ignored except as the driving force for rr, o fields. A similar lagrangian was considered by Rajeev and Bhattacharya [6] and others, where they integrated out the quark degrees of freedom to obtain the anomalY and the quartic Skyrme-like term *~. This suggests that the Skyrme lagrangian is an effective lagrangian for the model considered in refs. [4,5], though, probably QCD will introduce some, hopefully minor, modifications at short distances. Also, in this sense the quark-pion

,1 Of course higher derivative terms are also obtained.

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lagrangian is an intermediate scale lagrangian between Skyrme and QCD where the quark degree of freedom is still retained, i.e. within the confinement scale, but pions are still not broken down to q?q and glue. This would be a more reasonable lagrangian to work with than the Skyrme model at smaller distances. Let us review the main features of this model.

The model lagrangian density is given by (SU2) L X (SU2) R chiral o-model,

d? = 1(~O)2 + l ( ~ r r 2 ) 2 -- 1 ~k [O2 + (rr2) 2 - f2 ] 2

+ ~ ~q [~ +g(o + i'151raTa)l t~q, (?,.) color

with f . = 93 MeV, the pion decay constant. For our present purpose we can approximate ,2 this by a nonlinear model [4,7], i.e.

o2/f 2 + rr2/f 2 = 1.

If we then choose off , = cos 0 and lra/f~ = (dr)a sin O, we have (i) for 0 = 0 everywhere the usual space uniform spontaneously broken chiral symmetry (UCB) model with the quark spectrum being entirely con- tinuous beginning at m = gf~r; (ii) for O(r = O) = -lr at the origin and O(r = oo) = 0 we have the Skyrme background field with a space varying and topologi- cally non-trivial classical meson field configuration. We shall call this topological chiral symmetry breaking (TCB).

In such a background a soluble model that approx- imates [4,7] the spectrum with fair accuracy is

o = . ( r / R - 1), r<~R,

0 = 0 , r > R , (2)

where R is a size parameter, determined by minimizing the energy with respect to it. A valence quark bound state solution is found for case (ii). These are eigenstates of I = (at + ~), with l lbound state) = 0, giving only a color degeneracy of 3.

Further , the conserved topological [1 ] number of the familiar S 3 -+ SU 2 mapping is given by T = (0 - sin 20/2)10 and can be identified with the fermion

¢2 Calculations are done in this approximation as it is found that the energy of the soliton, in what follows changes only marginally even ifh = 10. For what follows, however, keep in mind that, actually, we have a linear o-model.

number [8] of the vacuum state ,a for each color species of quark, which implies, for QCD quarks, a baryon number equal to 1 for the new vacuum.

The total energy of the color singlet, occupied, 3 quark bound state configuration is composed of quark orbital energy and meson gradient energy minimized with respect to R [eq. (2)] .

It is found [4,5,7] that the above TCB (ii) configuration exists with an energy obviously lower than the 3 plane wave quark UCB (i) color singlet configuration o f E = 3m.

Occupation of the TCB bound state of 3 quarks forming a color singlet adds a unit of baryon number to the vacuum. Since the Skyrme vacuum already carries a unit of baryon number this state was expected to carry B = 2 as maintained in refs. [4,7]. However, the identification of T with the baryon number of the vacuum is valid only in the slowly varying approximation. This is equivalent to taking the parameter R -+ oo in the parameterization of 0 [eq. (2)] . One finds [4,7] that in this limit the valence quark bound state recedes to negative energy. The new vacuum in the R -+ oo limit, then, has this state occupied and has baryon number B = T = I . As R is reduced this same state pops out of the Dirac sea into a positive energy, valence, quark bound state. Thus, the baryon number of the occupied bound state 3 quark solution above is actually 1 and 2 as suggested in refs. [4,7] ,4

The connection between the Skyrme model is now most transparent. At Rmi n the bound state sits in the "negative" energy sea and we have a skyrmion. As the skyrmion is squeezed the quark bound state moves up to positive energy and the solution acquires a quark model structure. Consider now the generic expression for the energy of the (3 quark) singlet or B = 1 state in the TCB model ,s

EB/gf, = 3(a/X - D) + 27X/g 2. (3)

,a The vacuum shall designate the state in which all negative energy states are occupied.

¢4 A correction to this effect is now in preparation. ,s This parameterizationbreaksdown for small X, for X ~ 0

corresponds to (o) = f~r, Or) = 0 everywhere. However, this is for the isolated soliton and ignores the tunnelling of the quark wavefunction which will rectify the situation by giving rise to band structure. This will be reported sepa- rately.

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Above, the term in parentheses is the quark energy for a profile given as in eq. (2) with X =gf~R, a di- mensionless size parameter. The other term is the meson field gradient energy. For the single nucleon (/3 = 1) solution this expression must be minimized with respect to X. The constants a and D are specified later.

Xmi n = ~gv~. (4)

This immediately provides a new feature of universali- ty associated with this model: Rmi n = Xmin/gfTr = lx /~ / f ~ is independent ofg .

To identify the B = 1 state above with the nucleon

we set

EB/gf , = 18X/~/g - - 3 D = 1000/93g, (5)

where the energy is fitted to M B = 1 GeV. In the orig- inal model [4] the parameters a and D were obtained by fitting the single bound state orbital which crosses zero energy (as X is increased) to be a = 3 .12 ,D --- 0.94. These will obviously be modified on inclusion o f the Dirac sea effects which include other *~ negative energy bound states and the new negative energy continuum. With a little algebra we find D ~< 0.093 × 18x/~ - 1, for a stable TCB solution. We shall look at two cases ,v :

a = 3 . 1 2 , D = 0 . 9 4 ,

which gives g = 7.5, R min = 1.28 fm, (6a)

a = 1 . 5 3 , D =0 .94 ,

which gives g = 4.1, Rmin = 0.89 fm. (6b)

In (6a) the numbers are those from fitting the single bound state [4] ; (6b) is fi t ted to reproduce an Rmi n

= (Xmin/gf~r) such that (1/2Rmin) 3 = n B = 0.18 f m - 3 , i.e. where the size of the soliton (not necessari- ly quark density) Rmin, corresponds to n B = 0.18 fm - 3 .

At this point we are ready to consider the anti- cipated ground state in this model. Just as for the pure Skyrme model, we find an even more substantial energy barrier when two B = 1 objects are mode coincident from an infinite separation. Clearly, making

4:6 That there are only two negative E bound states is suggested in ref. [91.

4:7 Actually, D is expected to be 1 (to be explained in later work [101).

a giant "skyrmion" like object with many cycles in 0 centred at some point is a most unfavorable/unstable configuration. The expectation is that a periodic lattice of these objects centred at each site, will be the favoured ground state at high density just as for superfluid 4He.

(1) We shall consider the simplest of such ground states for X < X m i n . We consider a cubic lattice of our B = 1 configurations with lattice spacing 2R (2X) ,8. The periodic potential is provided by the meson field with O(r), the chiral angle, going from - r r at the centre of such a configuration to 0 at the boundary of the cell. The quarks sit in the potential well provided by the meson field. The energy per unit cell is then sim- ply given by ,s

EB/gfTr = 3(a/X - D) + 27X/g 2 , (7)

the first term being the quark energy for a meson field prof'de which can be scaled with X (or density);

the second term is the meson energy. As pointed out earlier a, D and g are already determined from the single nucleon solution considered earlier. The quali- tative and quantitative limitations of the above will be considered in the discussion.

The baryon density is given by n B = [(1/2R)] 3 fm - 3 = 1/8X 3 in our units.

Clearly, in this model, for low density, the TCB state will be favored over the UCB state of the o model, since the quarks now appear in bound states, the energy per quark then being obviously less than gf~, the energy of a UCB quark.

(2) Let us now consider a possible model for UCB which has been considered by many authors [11,12] in the context of "nucleon" a-model. This is the mean field o-model in which the quark mass is dynamically generated from the vacuum expectation value o f the o field. In this model we have (o) = o 0 and (rr) = 0. (Charged pion condensation in the o model admits the possibility of (rr(space uniform)) :/= 0 and (o) 4= 0 with o 2 + 7r 2 = f 2 , but will not be considered here as it has been dealt with mainly as a non-relativistic model, see refs. [11-13] .)

The energy per nucleon composed of 3 chiral QCD quarks, is for neutron matter n B = n u (= k~3/rr 2) ---

,a This is equivalent to having a spherical Wigner-Seitz-like construction in each cubic unit cell with going from -Tr at the centre to 1 at the boundary.

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~ n d (= kd3/lr2),

EB((o), k~) = Ep/n B

- [ ~ {xi(1 + xi:)l /2(2x i; 1) -~ i=u,d m4(3/8rr2) +

-- ln[x / + (1 + xi2) 1/2 ] }

+ ¼~k((o)2 _ f 2 ) 2 + O(m2zr))nB 1 ' (8 )

with

m = g(o), xi = k~/m, X = 2 g 2,

n B = (k~)2/Tr 2 = 1/(2R) 3.

In our units

gf =1, R = X , nB=I /8X3 , k~=Or2)l/3/2X.

Substituting n B, k~, k d in terms of X, the energy per baryon is obtained by minimizing EB((o),kD with respect to (o) for each density.

The results ,9 are plotted for the two values o f g = 7.5 and 4.1, and as expected, yield a higher energy per baryon than our model (topological chiral breaking) up to well above nuclear density (figs. 1 and 2).

Note that in our model, the TCB phase, by construction chiral symmetry is spontaneously broken for all density.

The model (UCB) has symmetry breaking at lower density but goes to a restored phase ,lo at high density at which the quarks begin to resemble a massless QCD quark gas.

(3) Finally, we must compare with QCD quark matter model, or rather, to a simple realization of this in terms of a large bag of massless quarks; for this is what we expect for the state of very dense matter.

For a quark gas derived from neutron matter we have the same constraints,

nB= nu = 1 n d = (k~)3/lr2.

,9 The author thanks Norman Glendenning and John Boguta for providing these results.

,10 The property of the o model that saturation and lowest energy minimum occurs in the abnormal or chirally restored phase at density much higher than nuclear, has been pointed out and also modifications attempted by the authors in refs. [11-13].

11f t 11 i g = 7.5

1 i(1) fn = 93 MeV

6 ' \

\~ - Chiral Restoration / \ . /

31- ~ _ ~ " , ' N , . \ / ............................ / / E B \ .....

11- r/B( - )" z ~ 0.24 " " ~ 0 5.4 0.674 &2 0.108

0 1 2 3 4

X

Fig. 1. E B, the energy per baryon plotted against X (nB, baryon density) for models (1), (ours TCB), (2) (UCB) and (3) A and B, quark model. The dashed line connecting A and B is interpolating (3). The cut in (2) is the point of chiral restoration. The markers on the bottom line indicate the coexistence from the tangent construction. Quark matter (3) and (1) TCB.

To order tx =g2ff/4~r (the QCD fine structure constant) in gluon interactions, for massless QCD quarks, the energy per baryon simplifies [12,14] to

= ~u, 3 i 4 Ep/nB ,'= d ~(k~) [1 + 2ot(p)/31r]/n B + bin B

= ~(1 + 24/3)lr2k~(1 + 2a/3n) + bin B. (9)

In terms of our unitsgfn = 1, n B = 1/8X 3 as before. Thus

EB/gf n = C 2.846/X + bX 3 , (10)

where

b = 0 . 0 1 5 for g = 7 . 5 ,

b = 0 . 1 6 7 for g = 4 . 1 .

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12

11

10

9

8

7

6

(3)1

E B

gf~

I I

(3)A

I

g =4.1 f= = 93

I

MeV

Chiral Restorat ion (2)

(2) • T/B(F-3 )

p, 0.294

692 o ~ 02~ 0~08 I I ( I t I

0.5 1.0 144/1.5 2.0 X ~ _

Fig. 2. Same as fig. 1, but UCB and (1) TCB.

l

The second term follows from the bag pressure b = 55 MeV f m 3 which has been recast in our units. We take a = 2.2, the MIT bag value. The constant C is then 1.45. I f all interactions are dropped, C = 1. The actual situation for quark matter is likely to be C = 1.45 (curve A) at around nuclear density (0.16 fm -3 ) going to C = 1 at very high density (curve B). Such a curve may be termed interpolating ,11 QCD(3).

Results. (i) As expected, we find that for low density, i.e.

close to and above nucleons, the TCB model provides the ground state over the UCB model. At higher density depending on the value o f g there is cross-over to the quark moffel/UCB model. At high densities, however, (a) is close to zero and model (2) and (3) (with a -* small) are indistinguishable.

(ii) For g = 7.5, there is small density discontinuity between model (1) and interpolating (3). The tan-

,11 Interpolat ing (3) is one in which a goes f rom 2.2 around 0.1 fm -3 gradually changing [14] to 0.36 at 13 fm -3.

gent construction would give a mean coexistence density ~ 2 times nuclear matter density.

(iii) g = 4.1. There is a density discontinuity between model (1) and (2) which then merges into interpolating (3). The tangent construction gives a mean coexistence density of ~ 2 times nuclear density, between (1) and (2) [but, see following remarks, espe- cially discussion (x)] .

Discussion. (i) The scenario that emerges from the foregoing

considerations is the following. Below nuclear matter density the skyrmions behave as a nucleon gas on quantization [14] ; the n - n skyrmion-skyrmion force can be calculated [15-17] and it is found to give qua- litative agreement with the observed n - n potential including rr, p, co, etc. Thus, the equation of state at low density which follows from quantized skyrmions is in close agreement with the usual nuclear physics equation of state. At densities close to and above Pnuc. there should be a tendency, due to topological repul- sion, to form an ordered structure, e.g. lattice/liquid crystal. We have made a most rudimentary estimate of this.

As is well known [12] the a model in the UCB phase goes from a chiral breaking phase to the "abnormal" or chirally restored phase at high density, with an unnatural lowest energy minimum (E B versus nB) in the latter phase. Many efforts have at tempted to variously correct this by means of a hard core, quantum corrections, etc. In our model there is a smooth interpolation from the TCB phase at nuclear density to a UCB chirally restored phase at high density with saturation around nuclear density (g = 4.1). It is this new feature of the TCB phase, introduced in this paper, that clears up all the irksome irregulari: ties of the UCB o-model equation of state.

Actually, the above calculations [14] have been carried out in the nucleonic a-model. This model does not even have the complication ofQCD gluons. I f such an exercise is carried out for the nucleonic a-model [7], again all the problems of the previous UCB for- mulation are obviated.

All three models have E ° ~ n 4/3 and are distin- guishable only in the coefficient of the 1/X term in EB/gf~. Moreover, the coefficient for model (2) and (3) is identical. It is expected and consistent with chiral symmetry restoration that the coefficient of

235


1/Xin our model (1), 3a, be more than 2.846 for (2) and (3). This is the case fo rg = 7.5 and 4.1, when lower energy bound state solutions exist.

(ii) The main imponderable and handicap of our model lagrangian is that it neglects the effect of the direct gluon coupling to quarks. It also does not provide a clue to confinement. Such effects are under consideration [18]. Since this model does integrate out to the Skyrme-type model ,12 one is led to be- lieve that at the length scales considered, only the chiral symmetry order parameter, (zr) and (o), which involves gluons indirectly, is important. It is the topological pion (o) condensate which controls the dynamics in baryon matter. The gluon-quark interactions are through the condensate of 7r, and o fields. The gluon condensate itself is presumably important only at larger length scales, e.g. through the bag pressure b. This clearly distinguishes this model from others, e.g. MIT and other bags.

(iii) The presence of the TCB pion condensate is the underpinning of this model and therefore it is small wonder that the much sought UCB charged pion condensate is not seem.In many respects the TCB condensate is similar to the 7r 0 condensate which is also real and periodic and imposes a lattice structure [19]. Of course, a difference still exists - topolo- gy.

(iv) One may ask when will the confinement mech- anism be important. On examining the bag pressure term one finds that it ceases to become important at just above nuclear density and thus at higher density. The formation of the topological background field lattice will result in relativistic band structure (Bloch structure) arising in the quark wavefunction ,13. This is under investigation. This is qualitatively important in finding the correct ground state and may also lower the energy, pushing up the phase transition in density. Also, the spreading of the quark wavefunction may provide a natural explanation for the EMC effect [28] (in a highly distorted pre-lattice).

(v) Since we must have charge neutrality and Q = ~3/2 + B/2 we have r 3 = --B. If all the baryons were to be an ! = (~ + S) = 0 state one might expect a ferro-

,12 Plus higher derivative terms. ~13 Actually, this is more subtle. In our crude estimate we

neglected the spiUover of the quark wavefunction out- side of the background solution size X for small X.

magnetic structure (Tr 0 condensation) and a possible connection with neutron star magnetic fields. How- ever, the quantization of the skyrmion and the calculation of n - n potential indicates [21 ] nfnl" (n~nJ,) is more repulsive than n]'n4,(n4,nl") and thus an antiferromagnetic structure may be favored; the effect in (iv) which gives rise to a Fermi sea and would also favor this. The question of quantization of this model needs investigation for an answer for this prob- lem.

(vi) The parameterization for 0 we have used will obviously yield only an upper bound to E B. This must be corrected self consistently. Also the "Wigner- Seitz" construction and cubic lattice certainly will not select the lowest energy state. All these effects will move the energy per baryon down in this model, consequently moving the cross over to the chiral restored phase up in density. Also the topological background may be modified. The simplified TCB calculation is an illustrative upper bound.

(vii) This should give us a rough idea of the pion size as it is expected the pion will dissociate around the restoration density, which gives a lower bound on the pion size. An upper bound on the pion size is avail- able from the one-baryon TCB configuration. If the pion is a "q~t" Cooper pair, then for a pion condensate to exist, its size must be < R m i n. The pion, in- cidentally, the lowest pseudo scalar (Goldstone boson) of QCD with fermions and not just a qCq pair). For ex- ample in (ii),g = 4.1, gives an upper bound of 0.89 fm.

(viii) The transition to quark matter is softened in this model due to energy density going as (nB)4/3 in both cases.

(ix) We have not considered an SU 3 flavor exten- sion where the situation could be rather different with, B = S = 2, H [22] particle or B = S = 4 particle ,14 re- placing the B = 1 as a lattice site.

(x) The results seem to indicate a chiral restoration density rather too small ~2Pnu c. This will be modified by gluon corrections. At least, once chiral symmetry is restored (mg = 0) in (2) we should add 1 gluon ex- change as in the quark model, which drives the transition density up to (~8Pnuc). Also (vi) will raise the chiral restoration density.

If, even then, the restoration density is low, the lattice will not last into high density. Lattice forma-

,14 Work in progress with A.P. Balachandaran.

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tion is then far more delicate and less conclusive. Also low chiral restoration ,as density will give an

early transition to quark matter indicated possibly by copious production of the H particle. The crude picture just presented already seems to produce a telling plausibility and clears up most of the erstwhile problems of the o-model.

The auth or thanks Sarada Rajeev and Aiylam Balachandran for discussions and K.C. Wali and the High Energy Theory Group, Syracuse University, where most of this work was done, for their hospitality. Norman Glendenning, John Boguta and B. Banerjee are thanked for discussions and the Nuclear Science Division, Lawrence Berkeley Laboratory for their hospitality. This work was supported by the Director,

Office of Energy Research, Division of Nuclear Physics

of the Office of High Energy and Nuclear Physics of the US Department of Energy under Contract DE- AC03-76SF00098.

,15 In this model the transition is weakly first order; it may even become second order in an exact calculation.

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[17] R. Vinh Mau, M. Lacombe, B. Loiseau, W.N. Cottinghann and P. Lisboa, The static baryon-baryon potential in the Skyrme model, Stony Brook preprint (1984).

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Strong interactions at all densities in a chiral soliton model coupled to quarks