Pion condensation and the σ-model in liquid neutron matter

1.C

[

Nuclear Physics A 3 1 9 (1979) 3 2 3 - 3 4 8 ; ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

PION C O N D E N S A T I O N IN

LIQUID

A N D THE

NEUTRON

w-MODEL

MATTER*

FRANCOIS DAUTRY

Service de Physique Th(orique, Centre d'Etudes Nucl~aires de Saclay, BP no. 2, 91190 Gif-sur- Yvette, France and Department of Physics, State University of New York, Stony Brook, New York II 794, USA and E B B E M. N Y M A N * *

Department of Physics, State University of New York, Stony Brook, New York 11794, USA Received 24 April 1978 (Revised 12 D e c e m b e r 1978)

Abstract: T h e

nature of the ground state of neutron matter at neutron star densities is discussed, starting from the linear ~ - m o d e l Lagrangian. It is found that there is a possibility of a new, previously unknown, type of condensation, which involves coherent, non-vanishing expectation values of the neutral m e s o n fields of the theory, the o, and ~r° fields. T h e o'*r° condensate would, like normal neutron matter, develop its own *r- condensate. It is shown that the most general, translationally invariant, condensate is a combined ~r~r° and ~r condensate with arbitrary, independent, wave numbers. T h e wave vectors of the condensates are determined by a minimization process, and are found to be non-vanishing and perpendicular. T h e o'zr ° condensate corresponds to a state which is very different than previously considered states of neutron matter: all neutron spins are aligned, presumably with some macroscopic domain structure. Thus, criteria for the occurrence of this condensate depend on the energy difference between very different states. This m e a n s that any prediction as to whether or not the state actually occurs in nature must at the m o m e n t be regarded as uncertain. However, using available hypernetted-chain calculations of the contribution to the energies from the direct n e u t r o n - n e u t r o n forces, it is demonstrated that a ~r~"° condensate (with its accompanying ~r- condensate) might well occur at neutron densities above perhaps 0.5 particles/fm 3. This paper leans heavily on t h e linear o,-model. However, the neutral condensate is a general consequence of chiral symmetry, and can thus also be obtained e.g. from Weinberg's Lagrangian.

1. Introduction A great deal of attention neutron Brown

star matter

has been paid in the last few years to the possibility that

might develop

and Weisel).]

A charged

a pion condensate. pion

condensate

[See, e.g., the review article by would,

if it e x i s t s , e n h a n c e

* Work supported in part by U S D O E under contract no. EY-76-S-02-3001. ** O n leave of absence from Physics D e p a r t m e n t , ~.bo Akademi, Abo, Finland. 323

the

324

F. DAUTRY AND E. M. NYMAN

cooling rate of hot neutron stars through neutrino emission. The effect might m a k e it possible to determine empirically whether or not these theoretical ideas are correct2). A t the m o m e n t this question has not been settled. A popular starting point for theoretical analysis of pion condensation has been the g - m o d e l of Gell-Mann and Levy [see, e.g., Lee3)]. This model summarizes a n u m b e r of successes involving ~r-mesons and nucleons into a renormalizable Lagrangian field theory. The theory is approximately invariant under so-called chiral rotations, which in general mix the fields of the pseudoscalar w-mesons with a scalar field, that of the ~r-meson. The chiral-symmetry approach to pion condensation was introduced by Campbell, Dashen and Manassah4). These authors study a ~r- condensate (which in general has an admixture of ~r÷), comparing its ground-state energy to the corresponding non-condensed state. If the condensate has the lower energy, one concludes that pion condensation does take place at low temperatures. Because of the underlying chiral symmetry, the states in question are almost degenerate in energy, and the occurrence of a condensate depends only on symmetry-breaking correction terms. For the same reason, the energy gain is not very large. Theoretical studies of r~- condensation generally predict that above some critical density pure neutron matter as well as real neutron star matter (which also contains protons and negative leptons) do develop condensates. Estimates of the critical density are somewhat uncertain, however. The sigma-model approach, modified to account for nuclear correlations and intermediate-state resonances, gives a threshold density of two or three times nuclear matter densitiesS). This is in agreement with the results of phenomenological calculations provided one includes correlations between nucleons and d-resonances6). Migdal and his collaborators who do not include these correlations get systematically lower values for the threshold density [see ref. 7)]. Furthermore, in the particular case of charged pion condensation in neutron matter they have argued that only pair condensation would occur, excluding the possibility of a ~r- condensate. In this p a p e r we do not attempt to improve on existing methods for dealing with the usual rr- condensate. Instead, we want to discuss the possibility of other types of condensation. In the ~ - m o d e l approach of Campbell, Dashen and Manassah4), the classical ~r- (and rr +) fields which one uses as a candidate for the wave function of the condensate have, in fact, the nature of an a n s a t z wave function. Our approach is somewhat m o r e ambitious: W e regard all the meson fields of the theory as unknown quantities, to be determined by the calculations, and m a k e no a priori assumption about their functional forms. What we would like to do is to compute the energy as a functional of these unknown, condensed, meson fields and then obtain the functional forms of the fields through a minimization process. Unfortunately, this appears too difficult; here we only deal with the easier problem of translationally invariant (liquid) condensates, where the nucleonic charge and n u m b e r densities are constant.

PION CONDENSATION

325

The main result of our study is that a coherent condensation of ~r and ,r ° mesons can be of liquid type, [Note that zr° condensation alone always leads to a solid-type condensateS).] However, the new ~r~"° condensate is of liquid type only when fully developed or accompanied by a ~-- condensate. Therefore, it is not possible, within the limited function space we set out to search in this paper, to deal with any situation where the ~r~r° condensate alone is only partially developed. The transition between the two phases can be studied within our model at best only if it is a first-order transition. It may well happen that there is a transition region where the meson fields develop more gradually through a solid-type condensate. Even within the scope of the present investigation the question whether the ~rzr° condensate has a lowered energy is more difficult than for the n'- condensate - there is no threshold condition involving, say, the occurrence of poles in a meson propagator. One distinctive feature of neutron matter in presence of a ~r~ ° condensate is that neutrons will tend to be fully polarized. This makes the comparison between the energies of the normal phase (or ~'- condensate) and the ~r~ ° condensate a difficult problem, and we are, in fact, not at the m o m e n t able to state definitely whether the tr~r° condensate will actually ever be present in nature. It is clear that this aspect will require a further study. This paper is organized as follows: We start in sect. 2 by determining those meson fields which correspond to a liquid state of the neutron matter. In this section we also discuss chiral rotations, and demonstrate that all our solutions can be reached by such transformations (apart from changes in the effective nucleon mass). The next step is to determine the single-particle energies and wave functions of nucleons which interact only with the condensate. As we only deal with meson fields which correspond to chiral rotations and changes in the n u c l e o n mass, we could follow the approach of Campbell, Dashen and Manassah 4) and apply the rotations to the Hamiltonian instead. However, for pedagogical reasons, we want to demonstrate that the problem can as easily be handled without the rotations. The single-nucleon wave functions are more complicated in the unrotated frame, but they are still obtainable as eigenfunctions of a m o m e n t u m operator which we determine. Once the spatial dependence of the wave functions is known, the x-dependence drops out of the Dirac equation and it only remains to find the eigenvalues of an 8 × 8 matrix in the relativistic case. In the non-relativistic limit, simple, explicit solutions are given in sect. 3. Fully relativistic solutions are found in some cases, and this allows us to discuss the validity of the non-relativistic approximation. Having obtained the ingredients, we proceed to discuss special types of condensates in sect. 4. We start by considering the new ~r~ ° condensate alone. We first evaluate the contribution to the energy of otherwise noninteracting neutron matter from such a condensate. Then, we discuss the occurrence of ,r- condensation within this state, and the resulting additional energy gain. In the absence of nuclear forces, the tr~"° condensate is energetically favored above some critical density. However, it is clear that nuclear forces eneter in a different way here than in normal neutron matter. Thus, we report, in sect. 5, on a calculation of

326

F. DAUTRY AND E. M. NYMAN

such effects made for us by R. A. Smith. We then combine the results in order to determine whether ~r~"° condensation will occur also in the presence of other nuclear forces. We will find that if certain conditions are met, this will indeed be the case. In sect. 6, we m a k e some comments about the Weinberg Lagrangian. H e r e we also discuss some problems which in our opinion still have to be solved before the final word about this subject can be told.

2. Possible meson fields in liquid neutron matter

Our starting point for the determination of the fields is the o'-model Lagrangian. (In addition to this, we will, in the end of this paper, incorporate also nuclear forces as observed in scattering experiments.) The or-model has been discussed m a n y times in the literature and there should be no need for further discussion here. With very few exceptions we use the notation of Campbell, Dashen and Manassah4). In general, our paper is quite similar in approach to that one and we feel justified in assuming that the reader is somewhat familiar with this work. For this reason we only mention very briefly the steps necessary to obtain an effective Hamiltonian from the ~r-model Lagrangian. In order to implement constraints corresponding to zero charge density and a fixed baryon n u m b e r density p, one introduces Lagrange-multiplier terms, where the multipliers themselves are the chemical potentials for charge and baryon number, Further, one minimizes functionally with respect to the meson m o m e n t a . These steps give the effective Hamiltonian [see eq. (3.21) of ref. 4)], H~ff = - i / ~

• ~rN + gA~(o"+ i~" • n"ys)N

+ ½(Wr. Wr + V ~ • V~,) + ~ ~ ( ~ + ~ 1

-i~

2z

2

2

1

--

~a + ~2)+~N~

0

v 2)~

z 3 N - ~'~7°N+H~b,

(2.1)

where the nucleon and meson fields are denoted N, ~" and or. The chemical potential for charge is /~, and we use the abbreviation u ' = u-½tz where u is the baryonn u m b e r chemical potential. The symmetry-breaking term H ~b will be discussed further below, in sect. 4. It is our intention to deal with the meson fields in the mean-field (tree) approximation. On the other hand, we want to improve the tree approximation by introducing, whenever possible, experimentally known numbers which in general incorporate contributions from processes which are outside the scope of the mean-field approach. This is a delicate matter because the tree approximation implies relations which are not exactly satisfied by the experimental numbers. It is clear that the constant g in eq. (2.1) may be identified as the well-known ~rN coupling constant, and we are somewhat reluctant to use other than the experimental value, g2/4rr ~ 13.4. Further, the nucleon mass m is in the or-model generated by the

PION CONDENSATION

327

vacuum expectation value o-0 of the scalar field. Therefore, one must require go-0 = m, and this corresponds numerically to o-0 ~ 72 M e V for the vacuum value of the scalar field. The customary approach to the o--model is not the one outlined in the previous paragraph, however. The reason for this lies in the deep connections between the Hamiltonian (2.1) and the weak axial vector current. In particular, one wants the (rr-meson decay constant f,~ to be reproduced exactly by the semiclassical approximation. One must then require o-0 =/~,~ ~ 95 M e V as well as a choice of g which keeps the nucleon mass at its experimental value. In this approach there are then loop corrections associated with the physical zrN coupling constant, enhancing it by a factor gA (the axial-vector coupling constant), according to the G o l d b e r g e r - T r i c u n a n relation, g~Nr~ gkm/f.,. The factor gg is in this case a genuine renormalization (loop) effect; it cannot be incorporated into the Hamiltonian. The coupling constant h in eq. (2.1) determines the mass of the scalar meson as well as the self-interactions of the meson fields. The actual values of h matter in the mean-field approximation only if o-2+ ,/.l. 2 deviates from o-o2. In the mean-field approximation, one may immediately neglect the operator nature of the meson fields, and replace them by classical values. For any given set of functions o- and ~r the Hamiltonian (2.1) then gives a Dirac equation for the nucleons, which determines their single-particle energies. In the present approximation, which is to be thought of as giving an unperturbed energy, one simply fills the appropriate Fermi seas, adding single-nucleon energies to the energy in the fields. This gives, at least in principle, the energy of the system as a functional of the fields. The next problem, still in the mean-field approximation, is to determine those meson fields which minimize the energy. If this produces a result other than ~r = 0, o- = const, we shall say that there is a meson condensate. The prescription just outlined appears to be too ambitious to be carried out in practice. The main reason is that it requires being able to solve the Dirac equation for the nucleons with an arbitrary, non-constant, potential. Thus, one must study simpler cases. As we shall see, the requirement that the single-nucleon density be constant in space gives the desired simplification, and is sufficient to allow us to determine the functional forms of the classical meson fields, giving a " m o s t general" liquid condensate with a few parameters. The previously [e.g., ref. 4)] studied ~r- condensate which is related to the non-condensed state through a chiral rotation is within this class. In order to be able to solve the p r o b l e m at hand fully, we need one (probably harmless) additional assumption: that the single-nucleon wave function t# has at most a finite n u m b e r of Fourier components. It is straightforward to show that the requirement of a constant It#]2 t h e n prohibits the n u m b e r of Fourier c o m p o n e n t from exceeding the n u m b e r of spinor components in t# (in this case eight, as in general we deal with neutron-proton mixtures). Further restrictions come, of course, from the requirement that t# be obtainable from the Hamiltonian (2.1). =

328

F. DAUTRY AND E. M. NYMAN

From these considerations, it is clear that the space of single-particle wave functions within which we solve the problem consists only of functions of the following form: ~b~(.r)=

8 ~ t3=l

M~aAi3eik~'x,

(2.2)

where the spinor index t~ is given explicitly. The vectors ka are the wave numbers for the various Fourier components. If there are less than eight of them, some of the coefficients A~ will vanish. The requirement that I~bl2 be independent of the coordinates x means that the matrix M + M of the coefficients in (2.2) must be diagonal; by appropriate choice of the coefficients A~ the matrix M can be made unitary. Next, we apply the m o m e n t u m operator - i V on the functions (2.2), and have

- i V y , , = ~ M~At~k~e ik~'~ = p~b,~+ ~ K~d/#, t~

(2.3)

~

where in the last equality we have subtracted a constant m o m e n t u m p, such that the vector matrix K~a is traceless. Below, p will serve as a d u m m y index labeling the single-nucleon energies. It will in fact, be useful later to add some constant to the m o m e n t u m label p, such that the Fermi seas are centered around p = 0. This is then not quite the same as K being traceless, but we return to this point below, when the needed shifts can be calculated. It follows from (2.3) that the single-neutron wave functions are also eigenfunctions of a generalized m o m e n t u m operator P :

P= -iV-K,

(2.4)

with the eigenvalue p. This means that P must commute with the Dirac Hamiltonian, and gives further constraints on the v e c t o r / f , which is a constant matrix in spin and isospin space. We require [P, ~/°(-i'g • V + g(o" + i~" ~"/s)) + ~ r 3 ] = 0,

(2.5)

and consider possible forms of K which satisfy this condition. Writing the m a t r i x / f as a multilinear combination of y- and r-matrices (this can always be done), it is not difficult to see that the only such matrices which are consistent with eq. (2.5) are 3'5 and r3. Further, a term simply proportional to y5 (without r~) requires all meson fields to vanish and will not be studied here. (This would give the so-called L e e - W i c k abnormal state.) We have established that there exists a m o m e n t u m operator P which commutes with the Hamiltonian, and is of the following form: •

1

1

P = - i V + ~kr3 + iqr3 ys,

(2.6)

where the vectors k and q are (so far) arbitrary. The factors ½are introduced in order

PION CONDENSATION

329

to maintain the same notation as the general literature. T h e c o m m u t a t i o n requirem e n t (2.5) n o w leads to differential equations for the fields; we obtain by straightf o r w a r d calculations: V~-~ = -k~-2,

VTr3 = +qtr,

V~r2 = +kzrl,

Vtr

(2.7) =

--q'/7"3.

O n e can immediately solve eqs. (2.7) for the fields, obtaining 7./-1 :i: i7r2 = re±i~'x sin 0, (2.8) COS 0,

O" d: i7r3 = r e ± i q ' x

w h e n r and 0 are integration constants. In each of eqs. (2.8) we have left out an overall phase factor, which does not m o d i f y the physics of the system. W e recognize the first of eqs. (2.8) as the ansatz for the ~r- c o n d e n s a t e used by Campbell, D a s h e n and Manassah4). T o otain this as a special case of o u r general solution, o n e simply sets q = 0: "lrl ± iqr2 : r e ± i k ' x

sin 0, (2.9)

7r3 = 0,

tr -- r cos 0.

T h e o t h e r special case of interest is o b t a i n e d when the ~r- c o n d e n s a t e is absent, and is o b t a i n e d by setting 0 = 0: z r l = ~rz = O,

~r + izr~

=

re ±iq'x

COS O.

(2.1 O)

In this case we deal with a neutral condensate, which leaves n e u t r o n s and p r o t o n s distinct. W e call this a o'zr ° condensate. O n e can also obtain a tr~"° c o n d e n s a t e of variable amplitude by setting k = O: 7rx ± i7r2 = r e ± i ~

sin O, (2.11)

o- + i-/r3 = re ±~'x cos 0. A l t h o u g h this type of wave function offers the possibility of having a continuous transition with an infinitesimal o-Tr° c o n d e n s a t e we shall not discuss it because it uses as a starting point (0 = ~Tr) a static 7r- c o n d e n s a t e which is never the g r o u n d state of n e u t r o n matter. In general, w h e n considering the mixed c o n d e n s a t e [eq. (2.8)], the a r g u m e n t s that lead to the search for zr- condensation only at finite m o m e n t u m still apply and one expects the minimization of the e n e r g y to give k different f r o m zero. Finally, we discuss the c o n n e c t i o n b e t w e e n the general solution (2.8) and the chiral transformations. In the m e s o n sector the chiral transformations reduce to the f o u r - d i m e n s i o n a l rotation g r o u p R(4) and all the states satisfying the condition (or (x))2 + ~ (Trl (x))2 = constant, i

can be d e d u c e d f r o m o n e a n o t h e r by a chiral transformation.

(2.12)

330

F. DAUTRY AND E. M. NYMAN

Since the usual vacuum of the ~ - m o d e l (i.e., in the absence of nucleons) is such that (tr) = tr0,

(zri) = 0,

(2.13)

it is possible to describe the general solution (2.8) as the result of a chiral transformation on the usual ground state, provided r = tr0. Using the notations of ref. 4), the chiral notation is

U=exp(ilq

"xQ~d3x)exp(-ilk

.xV~dax)exp(iOQ51).

(2.14)

H e n c e all the liquid condensates can be obtained from the usual ground state by the combination of a chiral transformation and a change in the chiral radius. The chiral radius r is related to the effective mass of the nucleons and a discussion of its value requires the inclusion of the nuclear forces (see sect. 5). One advantage of describing condensates in terms of chiral transformations is that through the approximate invariance of the effective Hamiltonian it provides a simple form for the Dirac equation which describes the nucleons interacting with the condensate. In this p a p e r we shall use the equivalent information which is contained in the fact that the nucleon wave functions are eigenfunction of a generalized m o m e n t u m operator.

3. Single-nucleon wave functions and energies A first step in the determination of the energy of the system in the semiclassical approximation is the determination of the shifts given by the condensate to the energies of otherwise noninteracting nucleons, in other words, to solve Dirac equations for the various cases. This task is in our case made easy by the fact that the general spatial dependence of the wave functions is determined by the m o m e n t u m operator P introduced in the previous section. In the case of the general, mixed condensate we have, on account of eq. (2.6) -iV~b = [p

-- ~7"3(k q-

vsq)]~b,

(3.1)

where p is the eigenvalue of the operator P. Eq. (3.1) can be integrated immediately, giving

~bo(x ) = e-~ i'~3(k+vs'~)'XX(p )e iP'x,

(3.2)

where X is an undetermined spinor in spin and isospin space, consisting of integration constants of the differential equation (3.1). The requirement that 4~p be an eigenfunction of H gives constraints on X, along with the energy levels. These are obtained by solving the following Dirac equation: [-iot • V+ fig(o" + i3~5¢" ~r) + ½"r3~]O = (E + v')O.

(3.3)

Using the general expressions (2.8) for the meson fields, one can obtain the following formula:

(o- + iys7 • ~r) = r(e i'3v~q'xcos 0 + irl~/se i*3k'xsin O).

(3.4)

PION CONDENSATION

331

Eqs. (3.2) and (3.4) are easiest to understand in a representation where y5 (as well as ~'3) is diagonal - one can then represent the matrices 75 and "/'3 by one of their eigenvalues (i.e., +1) - but they are valid in other representations as well. We now insert the wave function (3.2) into the Dirac equation (3.3). Some straightforward manipulations using eq. (3.4) lead to the following formula: fl (o- + i~/5~" " ~ ' ) e -~ i'r3(k'x+v5tl'x) = e -~ ir3(k'x +v~q'x) flreiZ~ vs°.

(3.5)

Because of this it is clear that the plane-wave exponential functions will cancel from the Dirac equation, and that one is left with an x - i n d e p e n d e n t eigenvalue equation for the energy and the spinor X. One knew in advance, of course, that this must happen, as the x - d e p e n d e n c e was determined using the conserved operator P. W e obtain the following eigenvalue equation: (~

• p - ~ (1~

• k+~

•

q y ~ ) + g r f l e ' ~" * ~ ° + ~ a ~ - E - ~ ' ) X

=0..

(3.6)

If one ignores the contribution from the ~ o condensate (i.e., sets q = 0) this equation is equivalent to a corresponding one obtained by Campbell, Dashen and Manassah 4) using a chiral rotation. In order to put our equation into precisely the same form, one performs the unitary transformation ("global chiral rotation") generated by the matrix exp (~ir~y~O), called U~ by the above authors. A more compact equation, suitable for manipulations in order to determine the eigenvalues E, is obtained in a four-vector notation: p " = (E + ~', p),

k " = (~, k),

q " = (0, q).

(3.7)

This gives the following equation: 1

[ ~ - ~(~r3 cos # + ffz2 sin #) + ~ 5 ( ~ r 2 sin 0 + gr3 cos 0) - gr]x = 0.

(3.8)

The product gr will play the role of the nucleon mass. In order to have the correct mass m for free nucleons in the vacuum, one must have g~o = m.

(3.9)

In principle, the quantity gr may be different from m ; we call it the effective mass, m*:

gr = m * .

(3.10)

We discuss below, in sect. 5, effects which arise when m* # m. As was already mentioned in sect. 2, one cannot in the tree approximation obtain an equation where the axial-vector coupling constant gA enters. If one wants to maintain the simple link to weak interactions, however, and put ~0 =f~, it is necessary to introduce gA at this stage, anticipating effects of neglected higher-order terms. The value of gA depend in principle on the nucleonic density as well as other p a r a m e t e r s of the theory; the experimental value of gA refers to the physical vacuum

332

F. DAUTRY AND E. M. NYMAN

only. When we treat gA as a constant, independent of the chiral angle 0 as well as the density, we have actually made assumptions about the nature of the breaking of the chiral symmetry. The well known zr- condensate depends crucially on the introduction of gA, due to the delicate balance between s-wave repulsion and p-wave attraction in the zrN system. When gA = 1 the two cancel exactly and the condensate does not develop. On the other hand, the neutral crzr° condensate does not suffer from such cancellations. In fact, we shall see that the value of gn cancels out precisely in this case; results for the energy are identical for the two approaches outlined in the beginning of sect. 2. We have derived eq. (3.8) in a way which obscures the relation to the vector and axial-vector currents. In fact, without the chiral rotations one would not uniquely be able to introduce gn. However, because of the work of ref. 4) it is clear that the interactions between the nucleons and the condensate is mediated by these currents. This makes it straightforward to introduce the axial-vector enhancement gn in (3.8), and we have:

[p-½(J('r3cosO+¢[rasinO)+~gA~/ (~(r2s~nO+c[r3cosO)-m*]x=--Mx=O. 1

5

•

(3.11)

We return now to the main line of the development, which is to determine the single-particle eigenenergies. Here, we shall deviate from the standard treatment and improve on the lowest-order non-relativistic approximation which has become usual. A n important reason why we must deal with relativistic equations is that we want to be able to discuss, at least qualitatively, the general situation when the effective nucleon mass is allowed to change [i.e., other values than r = or0 in eq. (2.8)]. In general one gets attraction by lowering the mass somewhat, but we must demonstrate that this attraction is present also relativistically. We know, of course, that the non-relativistic theory becomes grossly inaccurate sooner or later when the mass decreases. A t nuclear-matter density, one expects effective masses in the range 9) 0.8<~ m*/m ~ 0.9, and the deviation from unity certainly increases with the density. Thus the range, say 0.6 ~
[U-1MU] = 0,

(3.12)

PION CONDENSATION

333

where U is any non-singular matrix. It now follows that the following determinant also vanishes: Det [ U - 1 M U M ] = 0.

(3.13)

This result will be useful if we can find transformations U such that the matrix of (3.13) is simpler than the original, i.e., is in a reduced form. The method can then be repeated until there are no more matrices left. A first reduction of eq. (3.11) is obtained by the simple choice U = U 1 = ys. This produces an expression where (apart from ys) the y-matrices only occur in commutators [3',,, y,] =-2i~r,v. Such matrices can be brought into a reduced form by choosing a representation where ys is diagonal. As we shall see below, the dominating p h e n o m e n o n in the combined condensate is the ~rr ° part. It is useful to regard the ~r- condensate as a (small) perturbation in this state. We shall therefore not obtain the relativistic energy for the general case. The relativistic kinematics in the ~rr ° condensate alone can be solved explicitly. We obtain from eq. (3.13), dropping the rr- condensate, i.e., setting k = 0 and 0 = 0 in eq. (3.11), the following equation: Det {p2_ m.2 --agAq ~ 2 2 +ggAr3"Ys[q,/~]} 1 = 0.

(3.14)

It is now easy to obtain the eigenvalues. The final reduction is in this case obtained by taking U = rl, and using eq. (3.13) again. This gives a fourth-order equation for the energy: ( p 2 _ m , 2 _ aggq 1 2 2 ~)2 +g2A(pEq2-- (pq)2) = 0,

(3.15)

where scalar products are four-dimensional. The explicit solution to eq. (3.15) is

Ep + v' = +Ira .2 +P 2 +agAq 1 2 2 ~+gA(rn . 2 q 2 + ( p ' q ~ x2x-~l ~J~,

(3.16)

where we are to keep only the positive-energy solutions. (Note that q2 = _l/2.) It would be inconvenient below to use the exact, relativistic formula, especially since the Fermi sea is not spherical in this case. We therefore consider the nonrelativistic approximation to eq. (3.16):

Ep + v, = m . + p 2 / 2 m . + ~gAq 1 + O(1/m*2).

(3.17)

Note that there is no term proportional to q2/m. Here, q denotes the magnitude of q. Having access to the exact solution, it is straightforward to determine the range of validity of the non-relativistic approximation. First, we observe that when p and q are parallel, the relativistic equation (3.16) simplifies to Ep + p ' = Ira*2 +p2] ~+½gAq,

(3.18)

provided that q is not so large that the smaller energy is negative. Thus, the only error in the non-relativistic approximation therefore comes from the usual approximation (p2 + m2)~ ~ m + p i / 2 m for the nucleons. This means that when p and q are parallel

334

F. DAUTRY AND E. M. NYMAN

0.2~ q/m ~

0.0 ~ 0 : 5 E ~E ->i +

I~0

I;,5

210

~-

--~.. -0.4

"

-0.6

~ ""-... ~

~ ~.~.

-0.8

Nelativistic

~..~ x~.~.

~'~-.~ Non- relativistic \-~.

-I.O

Fig. 1. The single-particleenergy of a nucleon of momentum p = 0.4 m* as a function of the wave number q of the o'er° condensate. The dashed line c.orresponds to the non-relativistic approximation, eq. (3.17), while the solid line gives the result of the exact calculation for the case when p and q are perpendicular. t h e e r r o r t e r m in eq. (3.17) cancels o u t c o m p l e t e l y in t h e c o m p a r i s o n of c o n t r i b u t i o n s to t h e e n e r g i e s of t h e c o n d e n s e d a n d u n c o n d e n s e d states. T o get a feeling for t h e o v e r a l l a c c u r a c y in g e n e r a l , we s i m p l y c o m p a r e the e x a c t a n d a p p r o x i m a t e e q u a t i o n s n u m e r i c a l l y . W e show, in fig. 1, results for t h e s m a l l e r p o s i t i v e r o o t w h e n p a n d q a r e p e r p e n d i c u l a r (the w o r s t case) at a fixed, fairly large v a l u e of the n u c l e o n m o m e n t u m , p = 0 . 4 m * . A s seen, n o n - r e l a t i v i s t i c k i n e m a t i c s is sufficiently a c c u r a t e u p to q ~ rn*. (It is a n o t h e r s t o r y t h a t at such m o m e n t a t h e r e m a y b e f o r m f a c t o r a n d o t h e r effects which r e d u c e t h e c o n t r i b u t i o n s f r o m t h e c o n d e n s a t e o r o t h e r w i s e m a k e o u r results i n a c c u r a t e . ) W e o b s e r v e that t h e angle b e t w e e n p a n d q o n l y occurs in a t e r m w h i c h is n e g l e c t e d n o n - r e l a t i v i s t i c a l l y . Thus, in this a p p r o x i m a t i o n all F e r m i seas b e c o m e s p h e r i c a l a n d a r e a u t o m a t i c a l l y c e n t e r e d a r o u n d t h e origin, p = 0. T h e s i n g l e - n u c l e o n e n e r g i e s in t h e 7r- c o n d e n s a t e can b e o b t a i n e d with t h e s a m e m e t h o d as a b o v e . W e discuss this case also b e c a u s e it has s o m e g e n e r a l i n t e r e s t , a l t h o u g h it is slightly off t h e m a i n line of o u r discussion. W e h a v e the f o l l o w i n g equation: D e t I - p + rn* + ~ r 3 ( c o s 0 - igA3,57"1 sin 0)] = 0.

(3.19)

A first r e d u c t i o n is o b t a i n e d , as a b o v e , with U = 3'5, giving in a f o u r - v e c t o r n o t a t i o n : ~ , 1.2

2

D e t [ - p 2 + m * 2 _ ~k a cos a v ~ ~K g A sin e 0 + 1

r3pk cos 1.

0 2

+ ~r2gAT5[/~,,k ~] sin O+~lrlgAk T5 sin 0 cos O ] = 0 .

(3.20)

PION CONDENSATION

335

We observe that the matrix ~5 c o m m e n t e s with the matrix [p, )(]. Both can therefore be assumed to be diagonal, with eigenvalues +1 and ±2[(pk) 2 -p2k2]~, respectively. We thus simply need a transformation U which changes the signs of that combination of r-matrices which occurs in (3.20). This leads, through eq. (3.13), to an equation where the coefficients of the various r-matrices are squared separately, eliminating the ~,-matrices in the process. After some manipulations, one can put the resulting equation in the following compact form: [(p+½ak)Z_m,Z][(p_l

2 sin20=O, ~ c ~ k2) - m , 2 ] + m , 2 k 2g A

(3.21)

where • 2 = COS 2 0 +

g~ sin 2 0,

(3.22)

and the vector squares are still four-dimensional. It m a y seem strange that in the limit 0 = 0 eq. (3.21) does not simply reduce to free nucleons with m o m e n t u m p although eq. (2.9) quite clearly corresponds to this case for 0 = 0, no matter what values k takes. The reason for this lies in our preliminary k - d e p e n d e n t choice of m o m e n t u m label p for the single-nucleon wave functions, see eq. (2.3). In our final equations we will explicitly recover the property that the dependence on k disappears at 0 = 0. It is easy to see already here that at 0 = 0 one simply needs to shift the m o m e n t u m label p by the amounts + ~k for the various roots of eq. (3.21). Eq. (3.16) is a fourth-order equation for the energy. However, to a good approximation it is merely a second-order equation for p~ = (Ep + u') 2. The only term which is odd in p0 is p0/z (p • k), and occurs in the term (pk) 2. Thus, we obtain the following equation, still exact: po2=p2+

1 2 I~ 2 _-r~ct 1 ~ 21.2 .~...2. _ ~ _ ~a ~ +[m*2~2cos2

-b Og2~/, 2ff 2 --

2aZpolzp

•k

0 + m * 2 k 2 g 2 A sin2 0

+ o~2(~ ° k ) 2 -]- ¼~4~[.~2/~2]~.

(3.23)

A fast-converging sequence of approximations is obtained by iterating this equation. The first approximation, putting p0 = m* in the right-hand side, is already good enough even to study low-order relativistic corrections. Performing a non-relativistic expansion, and keeping all terms of order l / m * but not 1/m .2, we get

E v + v' = m * + p Z / 2 m * + (,k 2 cos 2 0 +/a,2g2A sin 2 0 ) / 8 m * ±½(/z 2 cos 2 0 + kEg2A sin 2 0 - 2aEl~p • k / m * ) ~ + O(1/m*2).

(3.24)

This equation gives correctly the terms of order l / m * provided that/z and 0 do not both vanish. In that case a term which is otherwise of order 1 / m .2 becomes +p • ka/2rn* and one recovers the shifted free Fermi seas as we knew from the relativistic result (eq. (3.21)). The l / m * expansion differs from the eorresporiding result of Campbell, Dashen and Manassah4), because they d r o p p e d the t e r m

336

F. DAUTRY AND E. M. NYMAN

1

sin 0 in the F o l d y - W o u t h u y s e n transformation. [Since the missing term in ref. 4) vanishes with/~, their result agrees with ours in symmetric nuclear matter.] K e e p i n g in mind that we are interested in n e u t r o n matter, we shall now express the single-particle e n e r g y in terms of a shifted m o m e n t u m so that the F e r m i seas are c e n t e r e d at the origin. Ignoring 1 / m .2 terms one can obtain the following: /d, (~'/'2)'YSgA

Ep+v'=m

*

1 2 2 1 +p t 2 /2m . +~(tz cos O+g2~k2sin2 0)~+AE;

p ' = p :t: ½t~2/zk (/z ~ cos 2 0 + g~,k 2 sin 2 0)-~.

(3.25)

where d E is given by the following expression: d E = g ~ (/z 2 -- k 2)2 sin 2 0

COS 2 0 / [ 8

m* (/z 2 c°s 2 0 + k 2g2A sin 2 0)].

(3.26)

Thus, the p • k term can, to the n e e d e d accuracy, be r e m o v e d f r o m the square r o o t by a shift in the origin for p. W e observe again that when all contributions of o r d e r 1 / m * are c o m b i n e d [eqs. (3.25) and (3.26)], the sum simplifies at 0 = 0, and we recover non-interacting nucleons (with the shifted m o m e n t u m label p'). T h e numerical i m p o r t a n c e of the true recoil correction (3.26) for 0 ~ 0 deserves s o m e c o m m e n t . It is straightforward e n o u g h to insert typical values for/~ and k in (3.26); o n e then finds that in neutron matter the recoil term is entirely insignificant. T h e situation is quite different in symmetric nuclear m a t t e r (/~ = 0), where eqs. (3.25) and (3.26) are still approximately valid if sin 2 0 > IPl [k]/m*2. In that case the term AE = k 2 cos 2 0 / S m * is not negligible c o m p a r e d to the o t h e r terms and adds a repulsive contribution to the single-particle energies. In s u m m a r y a c o m p l e t e t r e a t m e n t of the nonrelativistic limit for the c h a r g e d c o n d e n s a t e in n e u t r o n matter enables one to deal with spherical Fermi seas and a constant e n e r g y shift. This result is precisely the one used by C a m p b e l l et al. after they d r o p p e d their incomplete recoil terms, thus improving their nonrelativistic expression for the eigenvalues. A b o v e , we have o b t a i n e d the full non-relativistic expressions for the singleparticle energies in both the ~rTr° and the rr- condensate, taken separately. W e f o u n d in both cases that the only significant contribution of o r d e r l / m * was the usual kinetic enrgy, pZ/2m*. W e shall therefore, when dealing with the c o m b i n e d condensate, immediately leave out all other contributions in this order. W e p e r f o r m again the first reduction ( U = ys) on eq. (3.11) and then drop contributions which are to be neglected. W e first leave out terms of o r d e r q2, k 2 and (qk) and obtain the following equation: D e t {p2 _ m*2

--

1 , ~'3pk cos 0 - ~'2Pq sin 0 + ~gA~'2Ys[J(,/~] sin ~

+ ½gAr33'5[e/, p ] COS 0} = 0,

(3.27)

where the vectors are still four-dimensional. A c c o r d i n g to what was said, we m a y still d r o p the t h r e e - v e c t o r p e v e r y w h e r e except in the p2 term. Putting p,~ = (m, 0) every-

PION CONDENSATION

337

w h e r e except the first t e r m of the d e t e r m i n a n t (3.27) gives D e t [ p 2 _ m , 2 _ ~'3rn*p, cos 19+ rn*gAz2~/5'y • k'ro sin 0 + m*gAr3"Ys'Y • q'Y0 cos 19] = 0. /3.28) T h e matrices ~/5~/~/0 take simple forms in the usual n o t a t i o n : ~5~'~0 =

--

0

°)

~

~

a n d this e l i m i n a t e s the ~ - m a t r i c e s in favor of t w o - d i m e n s i o n a l Pauli matrices. W e n o w o b s e r v e that the d e t e r m i n a n t (3.28) v a n i s h e s if p~ - m ~ is an e i g e n v a l u e e~ of the matrix 2 m e , w h e r e 1

1

e = -~r3

1

cos 0 - ~gAr2~ " k sin 0 - ~gAr3~

"

q COS 0.

(3.30)

T h e n o n - r e l a t i v i s t i c e n e r g y is t h e n E p + ~' = p 2 / 2 m *

+ e~.

(3.31)

T h e e i g e n v a l u e s e~ of the m a t r i x e are easy e n o u g h to find - o n e o b t a i n s a s e c o n d - o r d e r e q u ~ i o n for e ~. W e find the following four solutions: ei = ~ [ ( ~ 2 + g ~ q 2 )

cos 2 0 + k 2 g ~ sin 2 0

• 2 g A q ( ~ 2 COS2 0 + g ~ k 2 sin 2 0 sin 2 ~)~ cos 0] ~,

(3.32)

w h e r e & is the angle b e t w e e n k a n d q. T h e smallest root has its m i n i m u m w h e n the two c o n d e n s a t e m o m e n t a are p e r p e n d i c u l a r , a n d is t h e n simply the s u m of the 1 s e p a r a t e c o n t r i b u t i o n s . P u t t i n g & = ~ gives 1

e ~ = - ~ g A q COS 0 -- ~(~ 2 COS2 0 + g ~ k ~ sin e 0)~.

(3.33)

This t h e n gives o u r final a p p r o x i m a t i o n to the s i n g l e - n u c l e o n energies in the c o n d e n s e d state of n e u t r o n m a t t e r .

4. The energies o| the unperturbed condensates 4.1. THE crTr° CONDENSATE W e n o w p r o c e e d to calculate the e n e r g y of n e u t r o n m a t t e r in the p r e s e n c e of the various c o n d e n s a t e s . W e deal first with a simplified situation, w h e r e the o n l y i n t e r a c t i o n s are those b e t w e e n the c o n d e n s a t e a n d the n u c l e o n s . W e r e t u r n to the direct N N i n t e r a c t i o n s below, in sect. 5. In o r d e r n o t to be misleading, however, we w a n t to e m p h a s i z e a l r e a d y here that these o t h e r i n t e r a c t i o n s are different in the cr~"° c o n d e n s e d t h a n in the o r d i n a r y (or 7r- c o n d e n s e d ) state. In case of r r - c o n d e n s a t i o n there is an a r g u m e n t according to which, at least in s o m e zeroth a p p r o x i m a t i o n , the c o n t r i b u t i o n f r o m the n u c l e a r forces are the same as in the n o n - c o n d e n s e d case. No such a r g u m e n t is applicable here.

338

F. DAUTRY AND E. M. NYMAN

In order to simplify the calculations, we consider the case of pure neutron matter.. In reality, there are always small admixtures of other particles (protons, electrons, etc.), but not enough to modify any of our conclusions. Thus, we may simply fill neutron Fermi seas to some fixed density at which we are working. (It would not m a k e sense to require stability against density variations here, as we have not yet included the nuclear short-range repulsion.) Let us first recall that in the mean-field approximation the energy density of the normal phase of neutron matter is given by E°=~

3 (3"n'2)32- ~ ~ m ./9 + E ~ "b',

(4.1)

where the interaction between the nucleons and the " i t - c o n d e n s a t e " generates the nucleon mass. (We leave out the rest energy of the neutrons.) In presence of acrcr ° condensate the two spin projections correspond to different single-particle energies, split by the term + in eq. (3.18). W e start by studying the case where q is large so that there is only one Fermi sea. The nucleon energies then give E . . . . 3 (6¢r2) ~ ~_ 1 (4.2)

½ggq

5 ~m P~--~PagAq,

where the first term is the energy density of a free Fermi gas with only one spin state (i.e., the neutrons are fully polarized). We use here the free nucleon mass, m* = m. The energy of the field is [see eqs. (2.1) and (2.8)] Efield

1 2 = ~q or02 + E s'b,

(4.3)

where we must still insert an expression for the symmetry-breaking term E s'b'. We shall first use the m o r e popular symmetry breaking, which is linear in the scalar field. We have in this case ~,~.b. = tr0m2~cr, (4.4) and observe that the symmetry-breaking term averages out in the condensate 2 2 (assuming q # 0), whereas in the normal state it gives a contribution - c r o m =, the energy density of the normal state (and the vacuum). Thus, the net repulsion in the condensate for this ('cos 0') symmetry breaking is

ES.b._E~b. = ~rom 2 2 ~,.

(4.5)

The other possibility, ~ . b . = - ~m ~ 2~¢r • ~" gives an averaged contribution to E s b E~ b which is four times smaller than eq. (4.5). W e combine the equations of this section to obtain the difference in energy A E ni between the crTr° condensed and normal Fermi gases, with a superscript "ni" to indicate no (other) interactions: 2

AEni= 3 5

½

(3~r~)~(2~-l)p~--~o~g~q+ q ~ ' o ~ + ~ b ' - E ~ 2m

"b'.

(4.6)

PION CONDENSATION

~339

We are to minimize with respect to the condensate m o m e n t u m q. This is not difficult and gives (4.7)

q = laagA/2Cr2o.

It is straightforward to determine the density beyond which the assumption of only one Fermi sea is correct. This is simply the following condition: 2

2

(4.8)

p ~ / 2 m <=g A q = gA (pa/2cr0),

which gives ~

2

2

2

(4.9)

pF=6zr cro/gAm ~330 MeV/c,

corresponding to a baryon n u m b e r density of approximately t9 ~ 0.08 fm -3. In all numerical results we use the value O'o/gA = m / g = 72.4 MeV, combined with gn = 1.26. We now briefly describe the behavior of the system at lower densities, i.e., when both spin orientations are present. T h e energy density is given by 3

(2m) ~r,

5

1

5

~_ 2 2 +ES.b..

E = - 15 ¢r 2 "t~z, + ~ gAq)3 + (U -- ~ g A q ) ~ + ~ (aa)Pa + 2q ~0

(4.10)

The requirement that the derivative with respect to q vanish, can be written as 2 ~ ~m

2 tkz

gA

g /

2 XZ

gA

~ /

J

by using the equation 8 E / S v = - p a . The singular point in eq. (4.11) at q = g A p a / 2 ~ corresponds to the o n e - F e r m i - s e a limit. The appropriate solution to eq. (4.11) is ~ 64 4/ 2~ 2 x3 plotted in fig. 2. For low densities, p ~ ~ g ~ o / g A m ) , the wave n u m b e r q vanishes; in this case the solution to the equation for the ~ o condensate coincides with the 64 4~ 2t 2 ~3 ~ 4 2 2 3 normal phase. In the region T ~ / g ~ o / g A m ) ~ p < 3 6 r ( ~ O / g A m ) , the magnitude of q increases from zero to the one-Fermi-sea value. The energy densities for the normal phase and the ~ 0 condensate are plotted in fig. 3. It is immediately apparent that at this level of approximation (neglecting the short-range interaction between the nucleons) the ~ 0 condensate is unstdble against collapse towards high density. One also observes that the one-Fermi-sea approximation is always valid when the ~ 0 condensate is lower in energy than the normal phase. Within our variational space there is no way to connect continuously the ~ o condensate and the normal phase. ~ e r e f o r e , the phase transition would have to be of first order and would require a Maxwell (double tangent) ~ n s t r u c t i o n . However, if we were to consider arbitrary wave functions for the condensates it is possible that there would be continuous transition from the normal phase to the ~ o condensate

340

F. DAUTRY AND E. M. NYMAN q

I

(MeV) I00

-

5o.

0

~P~/~I/ o

~

0.05

o.~o

o.~5

~

o.zo

ps(fm -3) Fig. 2. Value of the condensate momentum q as a function of baryon number density. For p~ _-<0B N 02 neutron with both spin orientations are present while for o > 02 only one spin state is occupied.

E

(MeWfm5)

6o b

40 C 20

0

0

I

0.2

I

I

0.4 P (fm -~)

I

I

0.6

I

I

0.8

B

Fig. 3. Energy densities for the normal phase (a) and the tr~-° condensate (b and c) in the semi-classical approximation. Curve (b) corresponds to the symmetry breaking .~,.b. =O'om~r and curve (c) to ~s.b. 1 2 = -~m,~" .,r, for the case where gA is introduced with the value 1.26. (if it is r e a c h e d at all). F u r t h e r m o r e , t h e l a c k of s t a b i l i t y of t h e tr~r ° c o n d e n s a t e p r e v e n t s a n y M a x w e l l c o n s t r u c t i o n at this stage. W e s h o u l d at this s t a g e also c h e c k t h e v a l i d i t y o f t h e n o n - r e l a t i v i s t i c k i n e m a t i c s . U s i n g fig. 1, w e m a d e (sect. 3) t h e o b s e r v a t i o n t h a t r e l a t i v i s t i c e f f e c t s s t a r t at v a l u e s o f q of t h e o r d e r of t h e n u c l e o n m a s s . U s i o g eq. (4.7) w e find t h a t this a p p r o x i m a t i o n is

341

PION CONDENSATION

acceptable up to perhaps ten times nuclear matter density. This upper limit on the density is quite far from the lower limit obtained above, giving the present, simplified approach to the kinematics validity in a long density range, including a region where the u n p e r t u r b e d o'er ° condensate is lower in energy than the free Fermi gas. Finally, we observe that the energy gain from the trTr° condensate when only one spin orientation is present [see eqs. (4.6) and (4.7)], depends on gA only through the ratio O'o/gA. In both choices of p a r a m e t e r s this ratio equals g,,yr~/rn~. This is due to the fact that in the two cases the G o l d b e r g e r - T r e i m a n relation is satisfied with the same values for g,mr~ and mN. In conclusion, with our choice of parameters we get the same result in the simplest renormalized tree approximation as by introducing renormalized matrix elements of the axial current. One should emphasize that although the energy gain is exactly unchanged, the condensate m o m e n t u m varies like gA/0"2o. However, the general properties of the trTr° condensate do not depend on the precise value of gA. This is an important difference with respect to the 7r- condensate of ref. 4) which required gA > 1.

4.2. COMBINED CONDENSATES We now study the energy of neutron matter in the presence of the " m o s t general" (liquid) condensate calculated in sect. 2. We view this as a o-~r° condensed ground state developing also a charged (~r-) condensate. The situation is analogous to the way in which the 7r- condensate is built on the ordinary ground state in the familiar chiral-symmetry approach. The nucleon-energy contribution to the energy density becomes EnUC

3 (67rE) ~ ~_

-5

-~

-

p --~l-oa[fgq cosO+(lzEcosE O+gE k~sinE O)~--Ix]' (4.12)

and the field energy is Efield = 5qaErr02 COS2 0 +~(kE-/xE)~r~ sin E O+E sb.

(4.13)

H e r e we have used the fact that the two condensates will be perpendicular. In this case the energy changes from the two condensates are simply additive. We can determine the uncharged-condensate m o m e n t u m in the same manner as above, obtaining

q = gApa/20"~ COS 0.

(4.14)

When this value is inserted into eq. (4.12), the energy density becomes 2 2

E-5

3 (6~")3 ~ ~ m Pa

2

2

1 PBgA 8 ~o

1

2

1

~pB[(/z COS2

o+kEgEAsin 2 0)~-/~]

+~(kE--/xE)o'o2 sin 2 0 + E s'b'.

(4.15)

342

F. DAUTRY AND'E. M. NYMAN

Thus, in the mixed c o n d e n s a t e the e n e r g y gain due to the trzr ° c o n d e n s a t e is i n d e p e n d e n t of 9 (i.e., of the amplitude of the condensate). This is possible because q goes to infinity when the amplitude of the c o n d e n s a t e vanishes. H o w e v e r , the non-relativistic e q u a t i o n for the single-particle energies is inaccurate for large q; there are corrections which restore the 9 - d e p e n d e n c e of the e n e r g y gain. T h e minimization with respect to k, the ~r- c o n d e n s a t e m o m e n t u m , as well as the charge-neutrality condition (OE/cgtz = 0) are unaffected by the presence of the tr~"° condensate. O n e obtains the following expression for the e n e r g y density when these r e q u i r e m e n t s are taken into account: 3 ( 6 ~ r 2) ~2 ~

g2Apg

-

E - 5 ~

tgB

8tr02

2 2 ( ~ -- 1) sin 2 9 1 + E S b " 8o~02 (g~ - 1) sin 2 9 +

gAPB

(4.16)

where the third term on the right-hand side is the e n e r g y gain due to the ~rcondensate. T h e minimization with respect to 9 d e p e n d s u p o n the choice of s y m m e t r y breaking. In the case .L~'b" = trorn~r 2 the contribution is o-om,~, 2 2 i n d e p e n d e n t of 9. 1 Thus, if gA is greater than o n e the m i n i m u m would c o r r e s p o n d to 9 = ~r. H o w e v e r , relativistic corrections w o u l d p r e v e n t 9 f r o m reaching this point. T h e o t h e r symmetry breaking, ~L~s'b = - ~m,~" 1 2 • ~', ES.b. = gives a contribution 1 2 2zl 2 ~m,~rot~ cos 9 + s i n 2 9). In this case there is a m i n i m u m in the e n e r g y density for non-triviaJ 9 in the density range

4-~m,,o-2o gg(gEg _ 1) ~ < p <

gA x/~m~o'~ (g2A_ 1) ~

(4.17)

T h e great similarity of these results with those concerning the ~r condensate alone 4) is a c o n s e q u e n c e of the constant e n e r g y gain of the o'Tr° condensate. F r o m eq. (4.16) one can estimate the extra attraction due to the zr- condensate. In the unphysical limit 9 = ½7r the ~r- contribution has an extra factor (g2g -- 1)/gEg ~- 0.37 relative to the o-~r° contribution. By considering a limit o n 9 such as cos 9 = 0.5 the relative n u m b e r for the zr- contribution is r e d u c e d to 0.31. In any case, the qualitative b e h a v i o r of the mixed c o n d e n s a t e is the same as that of the ~ r ° c o n d e n s a t e alone. A s we shall see in the next section, the nuclear forces m o d i f y the e n e r g y density m u c h m o r e strongly.

5. Effects of nuclear forces and correlations It n o w remains to discuss the influence of nuclear forces on the c o n d e n s e d and u n c o n d e n s e d states. E q u a t i o n s of state for u n c o n d e n s e d n e u t r o n m a t t e r have b e e n calculated by m a n y authors, and within the models used, one has confidence in these calculations. In the case of ~r- condensation, one obtains an e q u a t i o n of state by simply adding an e n e r g y gain f r o m the condensate. Since zr- c o n d e n s e d n e u t r o n matter also contains p r o t o n s this prescription obviously requires that the forces

PION CONDENSATION

343

involving p r o t o n s a r e t h e s a m e as t h e o r i g i n a l n e u t r o n - n e u t r o n i n t e r a c t i o n , i.e., t h a t t h e r e is n o isospin d e p e n d e n c e . T h e m o s t difficult p a r t d o e s n o t lie h e r e , h o w e v e r , b u t in t h e fact t h a t t h e p r o p a g a t o r s for t h e m e s o n s t h a t give rise to t h e f o r c e a r e d i f f e r e n t in t h e p r e s e n c e of a c o n d e n s a t e . T h e r e f o r e , o n e d o e s n o t a priori h a v e t h e right to a s s u m e t h a t a n y n u c l e a r f o r c e in t h e c o n d e n s a t e is t h e s a m e as o b s e r v e d in f r e e space. C a m p b e l l , D a s h e n a n d M a n a s s a h 4) give a r g u m e n t s a c c o r d i n g to w h i c h this a s s u m p tion s h o u l d b e a r e a s o n a b l e s t a r t i n g p o i n t for t h e s h o r t - r a n g e p a r t of t h e i n t e r a c t i o n . H o w e v e r , o n e k n o w s v e r y well t h a t t h e p i o n p r o p a g a t o r is m o d i f i e d b y t h e c o n d e n sate. T h e r e f o r e , at least t h a t c o n t r i b u t i o n to t h e g r o u n d s t a t e e n e r g y w h i c h has to d o with t h e o n e - p i o n - e x c h a n g e p o t e n t i a l s h o u l d r e c e i v e s o m e a t t e n t i o n b e y o n d t h e e a r l y e s t i m a t e s of S a w y e r a n d S c a l a p i n o l ° ) . It w o u l d r e q u i r e a m a j o r effort to t a k e into a c c o u n t fully t h e s e o b s e r v a t i o n s . W e shall t h e r e f o r e , as a s t a r t i n g a p p r o x i m a t i o n , a s s u m e t h a t t h e usual n u c l e o n - n u c l e o n p o t e n t i a l gives a g o o d e n o u g h d e s c r i p t i o n of t h e n e u t r o n - n e u t r o n i n t e r a c t i o n also in t h e p r e s e n c e of t h e m e s o n c o n d e n s a t e s . In t h e presen.t a p p l i c a t i o n it w o u l d b e w r o n g to s i m p l y a d d in t h e e n e r g y f r o m t h e c o n d e n s a t e (as in fig. 3). T h e least w e h a v e to d o is a d d it to a c a l c u l a t i o n of t h e e n e r g y for a s y s t e m of t h e right type, i.e., n e u t r o n m a t t e r with o n l y o n e n e u t r o n spin p r o j e c t i o n o c c u p i e d . R e s u l t s for such a n o v e l s i t u a t i o n a r e n o t a v a i l a b l e in t h e l i t e r a t u r e . H o w e v e r , R. A . S m i t h has m a d e s o m e c a l c u l a t i o n s for us for e x a c t l y this s i t u a t i o n , using t h e B e t h e - J o h n s o n p o t e n t i a l in a h y p e r n e t t e d chain v a r i a t i o n a l a p p r o a c h . S o m e of his results a r e s h o w n in fig. 4. In t h e o n e - F e r m i - s e a case, ./ ./

E/A (MeV

CONDENSED

./'

.~

~..-'"

15C \ 100

50

°o.o

3 P

~=-'-----_~o.~

.~"

.~'~

~

././

" ~

oi~

p(fr~ ~)

c~s

~..

~ ~

ALIGNED . ~ / _ ~ ./ 'D2+~'__~---" // ~ NORMAL "~----~ IS+3P o'J

~

~ ~

0.4

'~

Fig. 4. The dashed lines show results corresponding to the energy per particle in neutron matter as calculated using HNC techniques by Smith. The nuclear interaction is the Bethe-Johnson potential [simplified in two different ways (see text)]. Shown is also the energy of neutron matter with occupation only in one spin state, the dash-dotted line. The solid lines give the energy in the presence of the o'~"° condensate, using various assumptions for the effective nucleon mass, as described in the text.

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interactions take place only in odd angular m o m e n t u m states, and in this case the potential is the same for all I. In the normal state of neutron matter, one has interactions in even-/ partial waveg also. In the calculations of Smith it was impractical to introduce more than one even-state potential. Therefore there are two curves for the normal state, depending on whether the 1S or 1D potential was used for all even l-values. The s-state curve is expected to be more accurate at low densities, whereas at high densities the l > 0 partial waves are more important. Thus, the truth should be closer to the other curve in this case. We observe that the spin-aligned case has a much higher energy than the unaligned one. The main reason for this is that one has lost the s-state attraction here. Kinetic energies are more repulsive also, and the higher Fermi m o m e n t u m causes deeper penetration into the short-range repulsion of the interaction. The prescription we use now is to add the condensate energy to the appropriately calculated potential and kinetic energy. The amount to be added in does not correspond to the results shown in fig. 3, as these already included the kinetic-energy difference. We must, however, now take a standpoint versus the effective neutron mass. According to what was said above, in sect. 3, we could treat the effective nucleon mass as a variational parameter and determine it within our model. This approach was used by Lee and Wick 11) in nuclear matter and lead to the speculative prediction of highly bound, unobserved, abnormal nuclei. It has, however, been argued that this approach, if used too naively, includes a strong three-body force which is not present in nature and in the model prevents ordinary nuclei from being stable12). Nevertheless, Siemens and Chanowitz 13) have made such a study of ~rcondensation. The intrinsic instability of the model showed up in their calculation as a tendency of the mass to take strongly reduced values. Because the non-relativistic kinematics then became inaccurate, no definite conclusions could be made. With the exception of some r e c e n t work14As), most authors have studied pion condensation without allowing the meson fields and nuclear forces to change the nucleon mass. This also is somewhat unreasonable, as it is in fact well established that the effective mass is reduced in neutron matter. The Fermi-liquid theory of Landau gives, when applied to neutron matter, a reduction of the order of 20% in the mass. Some explicit results, obtained from ref. 9), are given in fig. 5. ~It would, of course, be incorrect to include in our effective mass any aspect of the critical behavior associated with a second-order phase transition to a charged pion condensate, such as the strong reduction of m* found in ref. 15).] It is necessary, of course, that we establish that our effective mass may be approximately identified with the parameter bearing the same name in a nonrelativistic many-body theory. Two aspects must then be checked. The first, and obvious one, is how the single-particle energy depends on the momentum. Our non-relativistic expansion gives a term p2/2m*, and such a term is generally used to

PION C O N D E N S A T I O N

345

mm'/m 1.0

0.8

0.6 ¸

i

o.o

o.I

i

i

i

02

0.3

o.,~

p ( f m °~')

Fig. 5. Shown are calculated values for the effective mass of neutrons in neutron matter in Landau's Fermi-liquid theory. Curves (a) and (b) correspond to results of Sj6berg 9) using Reid's soft-core potential, while curve (c) was obtained by B~ickman, K~illman and Sj6berg 9) for the same force.

define the effective mass, e.g., in the Laundau theory. (Values shown in fig. 4 relate to momenta p near the Fermi surface.) The kinetic-energy term is only one aspect of our relativistic effective mass. The other, more important, aspect of it is the change in rest energy. In our case, when the mass goes down by some amount, this corresponds also to an attractive contribution to the ground-state energy. Such an energy contribution is actually present and well known in non-relativistic theories of nuclear matter. It simply corresponds, in the tree approximation, to the Hartree term in the single-particle energy, for the potential corresponding to the exchange of the scalar o--meson. As we have seen, the energy coming from changes in the nucleon mass corresponds non-relativistically to part of the contribution from nuclear forces. For this reason it is not in the tree approximation justifiable to treat the mass as a variational parameter. The nuclear force also contains components which are grossly misrepresented or totally absent in such an approximation. Instead, we prefer to use value of rn* obtained from other calculations (in the normal state) which deal with the nuclear forces in a much more accurate manner. In this case the meson self-interaction constant A will never play a role in our discussion. In the non-relativistic limit, the pseudoscalar 7rN coupling of the tr-model is equivalent to a gradient coupling proportional g/2m*. When the effective nucleon mass m* is allowed to decrease, this quantity is enhanced relative to the usual coefficient/c//z of the non-relativistic coupling. For this reason results are quite sensitive to the choice of effective mass. Therefore, we show, in fig. 4, results with and without effective masses, according to the approximate parametrization m * / m = 1-cp/po,

(5.1)

346

F. DAUTRY AND E. M. NYMAN

where p0 is the nuclear-matter density (0.17fm-3). The p a r a m e t e r c thus simply gives the reduction at p = p0. We used two non-zero values, c = 0.1 and c = 0.2, corresponding approximately to values obtained in ref. 9). As seen, the modifications are drastic. With c = 0.2 one cannot escape the prediction that the tr~-° condensed state goes below the normal one, but with c = 0 (i.e., rn* = m) the conclusion is the opposite. Both the condensed and uncondensed phases in fig. 4 correspond to 0 = 0, i.e., no ~r- condensate. As discussed in sect. 4, one should (for gA > 1) lower both curves by considering other values of 0. However, those values of 0 which would be obtained through a minimization would depend crucially on several uncertain aspects of our calculation~ As the actual energy gain from the ~r- condensate is insignificant (although not equal) in both cases, we have therefore simply neglected it.

6. Discussion The trTr° condensate appears at first sight to be strongly dependent upon the particular representation of the chiral symmetry in the meson sector. This is, in fact, not the case. A neutral condensate can also be obtained, e.q., from the usual Weinberg Lagrangian16), either through a chiral rotation, using the o p e r a t o r (2.14) with k = 0 = 0, or directly from the requirement that the system be in a liquid state. In the limit of non-relativistic nucleons, one would, in fact, recover the same energy as we have determined here (apart from effects from the effective nucleon mass). The general case, containing both neutral and scalar condensates, appears to be m o r e complicated in the Weinberg model than in the it-model. W e finally return briefly to various aspects of our work and pion condensation in general, where further studies in our opinion are most obviously needed. First, the axial-vector coupling constant gA- W e feel somewhat uneasy about introducing this coefficient into the equations without a dynamical model in terms of some set of loop diagrams. Therefore, we have also considered the simplest possible semiclassical calculation, where one automatically has gA = 1. We observed that the trrr ° condensate persists in this simple minded approach, while it is crucial for the ~rcondensate that gA exceed unity. We have obtained our results using the assumption that the nucleon-nucleon interaction is the same in the presence of a condensate as without it (but we have not assumed that it is spin-independent). This is an approximation of unknown accuracy. It will not be easy to obtain conclusive results here, although some rough estimations have been performed1°). One aspect of our work where it would be straightforward to obtain m o r e accurate results is the kinematics of the single nucleons. H e r e we have always used the non-relativistic expansion in lowest order, which allowed results to be obtained analytically. However, at the expense of performing a computer calculation, one could maintain the full, relativistic formalism throughout the mean-field approxima-

PION CONDENSATION

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tion. H e r e , we h a v e d e t e r m i n e d ( a n d f o u n d small) t h e m a g n i t u d e of relativistic c o r r e c t i o n s o n l y for t h e b a r e ~rzr° c o n d e n s a t e , a s s u m i n g t h a t t h e a d d i t i o n a l zrc o n d e n s a t e gives a r e l a t i v e l y m i n o r p e r t u r b a t i o n , as i n d e e d it d o e s w i t h o u t t h e ~r~r° c o n d e n s a t e . It is q u i t e clear t h a t t h e c o r r e c t i o n s g r o w with 0. Thus, it is h a r d l y useful to d e t e r m i n e t h e a m p l i t u d e of t h e c h a r g e d p i o n field t h r o u g h a m i n i m i z a t i o n with r e s p e c t to t h e chiral angle O. A l l t h e a p p r o x i m a t i o n s m e n t i o n e d h e r e h a v e the n a t u r e of a l o w - d e n s i t y e x p a n sion. Thus, t h e y can b e u s e d at l i m i t e d d e n s i t i e s only, at m o s t a few t i m e s n u c l e a r m a t t e r density. O u r g r a p h s t e r m i n a t e at t h r e e to five t i m e s t h a t d e n s i t y ; it is o b v i o u s t h a t o n e c a n n o t rely on o u r results b e y o n d s o m e d e n s i t y of r o u g h l y this m a g n i t u d e . T h e ~r~r° c o n d e n s a t e d e v e l o p s , if i n d e e d it d e v e l o p s at all, at a d e n s i t y w h e r e t h e a p p r o x i m a t i o n s u s e d a r e at t h e v e r g e of b r e a k i n g d o w n . It is t h e r e f o r e p e r h a p s p r e m a t u r e to d e t e r m i n e the e q u a t i o n of state for n e u t r o n m a t t e r . F i n a l l y , w e r e p e a t t h a t we h a v e in this p a p e r o n l y s t u d i e d l i q u i d - t y p e c o n d e n s a t e s w h e r e t h e s i n g l e - n u c l e o n w a v e f u n c t i o n s o n l y c o n t a i n a few F o u r i e r c o m p o n e n t s . This is c e r t a i n l y t h e m o s t restrictive of o u r a s s u m p t i o n s . T h e r e is no r e a s o n at all to b e l i e v e t h a t p i o n c o n d e n s a t i o n in n a t u r e is within this f r a m e w o r k . H o w e v e r , it s e e m s a p p r o p r i a t e to p r o c e e d in a s y s t e m a t i c w a y a n d p e r f o r m a full s e a r c h for liquid c o n d e n s a t e s b e f o r e c o n t i n u i n g to t h e m u c h m o r e difficult q u e s t i o n of t h e m o r e g e n e r a l cases. O n e of us (F.D.) has b e n e f i t e d f r o m s t i m u l a t i n g discussions with G . B a y m , R. D a s h e n a n d A . B. M i g d a l . H e also a c k n o w l e d g e s t h e c o n s t a n t i n t e r e s t in this w o r k of M. R h o . B o t h of us a r e g r a t e f u l to G. E. B r o w n for t h e exciting a t m o s p h e r e a n d financial s u p p o r t of t h e N u c l e a r T h e o r y G r o u p , S U N Y at S t o n y b r o o k a n d w o u l d like to t h a n k p a r t i c u l a r l y R. A . S m i t h for his n e u t r o n m a t t e r calculation.

References 1) G. E. Brown and W. Weise, Phys. Reports 27C (1976) 1 2) J. Kogut and J. T. Manassah, Phys. Rev. Lett. 41A (1972) 129; O. Maxwell, G. E. Brown, D. K. Campbell, R. F. Dashen and J. T. Manassah, Astrophys. J. 216 (1977) 77; G. E. Brown, Comments Astrophys. 7 (1977) 67 3) B. W. Lee, Chiral dynamics (Gordon and Breach, New York, 1972) 4) D. K. Campbell, R. F. Dashen and J. T. Manassah, Phys. Rev. !)12 (1975) 979, 1010 5) G. Baym, D. K. Campbell, R. F. Dashen and J. T. Manassah, Phys. Lett. 58B (1975) 304 6) S.-O. B~ickman and W. Weise, Phys. Lett. $5B (1975) 1 7) A. B. Migdal, Rev. Mod. Phys. 50 (1978) 107 8) G. Baym and E. Flowers, Nucl. Phys. A222 (1974) 29 9) O. Sj6berg, Nucl. Phys. A265 (1976) 511; S.-O. B~ickman, C.-G. K~illman and O. Sj6berg, Phys. Lett. 43B (1973) 263 10) R. F. Sawyer and D. J. Scalapino, Phys. Rev. 1)7 (1973) 953 11) T. D. Lee and G. C. Wick, Phys. Rev. 1)9 (1974) 2291

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12) S: Barshay and G. E. Brown, Phys. Lett. 34 (1975) 1106; E. M. Nyman and M. Rho, Phys. Lett. 60B (1976) 134; E. M. Nyman, Nucl. Phys. A285 (1977) 368 13) P. J. Siemens and M. Chanowitz, Phys. Lett. 70B (1977) 175 14) Y. Futami, H. Toki and W: Weise, Phys. Lett. 77B (1978) 37 15) G. E. Brown and B. Friman, Nordita preprint (1978) 16) S. Weinberg, Phys. Rev. 166 (1968) 1568