Strange goings on in dense nucleonic matter

Volume 175, number 1 PHYSICS LETTERS B 24 July 1986

STRANGE GOINGS ON IN DENSE NUCLEONIC MATTER

D.B KAPLAN and A.E. NELSON Lyman Laboratory of Phystcs, Harvard Untverstty, Cambridge, MA 02138, USA

Recewed 2 April 1986

It has been suggested that charged pIons form a Bose-Einstein condensate In baryonlc matter at zero temperature and about twice nuclear density In this letter It IS shown that at somewhat higher densities one finds a charged kaon condensate, driven to a large extent by the "sigma term" interaction with baryons Using the SU(3) × SU(3) chlral lagranglan to model meson-baryon interactions it is found that baryonic matter acquires a strangeness-per-baryon ratio approaching one at several times nuclear density The relevance of kaon condensation as a route to strange matter and its role In neutron stars are discussed

1. Introduction. The appearance of a Bose-Einstein

condensate of charged pions m dense nuclear matter has been extensively discussed since it was first suggested m the early seventies [1 ]. Such a condensate is thought to occur at about twice nuclear density, where

attractive p-wave n - - n e u t r o n interactions make the appearance of n - ' s energetically favorable. A simple and elegant treatment of plon condensation using an SU(2) × SU(2) sigma model is given m ref. [2].

In this letter we use the SU(3) × SU(3) charal lagrangmn to consider the possibility of charged kaon condensatmn, and argue that it may occur at densities as low as 2.3 times nuclear density. It is very surprising that kaons can be in equilibrium m dense baryomc matter when pions are so much lighter. In an Ideal gas model one would not expect kaons to appear at any density However, the strongly attractwe interactions between kaons and baryons, especially from the so- called "sigma term", cause the effectwe kaon mass to drop below the effective plon mass in dense baryomc

matter. This effect is simply modeled by a kaon -ba ryon

mteractaon of the form H I = m 2 IKI 2- BB/ncfi t. At high baryon density the kaon mass is reduced by the

fraction n/ncrlt. When the classical kaon field has a non-zero expectation value (Le. when there is a Bose- Einstein condensate of kaons) the baryon mass is like- wise reduced, lowering the ground state energy. The

chtral lagranglan value for ncrit IS ncrit ~ 3n 0, where

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V (North-Holland Physics Publistnng Division)

n O is the nuclear density. A kaon condensate could affect the nature of neu-

tron stars in several ways. One possibility is that the

condensate softens the equation of state for dense baryonlc matter, leading to a lower critical mass for gravitational collapse. However, the more dramatic possibility exists that kaon condensation will provide a pathway to three-flavor quark matter, or "strange matter", which may be stable at pressures encountered in these stars [3] * 1. The fact that a kaon condensate is energetically favored implies that the usual barner of reqmrlng simultaneous multiple weak interactions to build up a critical value of strangeness is no longer present. This strongly suggests that when the strangeness-per-baryon ratio due to kaon condensation reaches roughly one, then, if strange matter is stable at the corresponding density, it will be reahzed without having to breach any energy barrier.

In the next section we show how to use the ma- chinery of the SU(3) × SU(3) chlral lagranglan to discuss the formation of a kaon condensate, giving special attention to the validity of the chiral perturbation expansion in dense matter.

In the following section we present the results of a computer analysis of the ground state of this system, which exhibits kaon condensation at densities as low

,1 We are not assuming that strange matter must be stable at zero pressure, as considered by Wltten [4] See also ref [5]

57


as 2.3 times nuclear density. We conclude with a few speculations on the role o f kaon condensation :n neu- t ron stars. We present various aspects of the validity of our approximations in two appendices.

2. The S U ( 3 ) × S U ( 3 ) lagrangian. Our basic tool for investigating kaon condensation is the SU(3)L X SU(3)R ctural lagrangmn , z . The usual non-linear sigma field may be wri t ten as

£ = ~_ f2 Tr 3E3Y~ + + ½ f 2 A [ T r M(G - 1) + h.c.]

+ Tr g(i¢ - mB)B + 1 Tr g3'" [V~, B]

+ D Tr BTU75 {Au , B}

+ F T r BTu3"5 [Au, B] + a 1 Tr g(~M~ + h.c . )B

+ a 2 Tr BB(~M~ + h.c.)

= exp (2iTra G / f ) , (1) + a 3 [Tr M ( E - 1) + h.c.] Tr BB + . . . . (4)

where zr a is the meson octet, f ~ f ~ = 93 MeV, and the T a are SU(3) generators with Tr T a T b = ½ lab . transforms as L N R + under SU(3)L X SU(3)R.

The baryon octet Is gwen by B = Ba(Vt-2Ta). The baryon fields transform nonlinearly under SU(3)L × SU(3)R as B -+ U(zr) BU+ (Tr) , 3 . To couple the mesons to baryons in this representation it IS conve- nient to define

= N1/2 = exp(izra T a f t ) , (2)

which transforms as ~ -+ L ~ U + (zr) = U(rO~R +. One may then construct the mesomc axml and vector cur- rents

1 V. = : (~+a.~ + ~0 .~ +) A . - i (3)

These fields transform as V - * U V U + + U3U +, A -+

UA U + . SU(3) X SU(3) symmetry-brealang effects due to the bare quark masses are accounted for by the ma- trix M = dlag(mu, rod, ms) taken to transform as M ~ L M R + .

The phi losophy of chiral per turbat ion theory is to write down all operators invarlant under SU(3) × SU(3) In an expansion m powers of A -1 , where A is the chlral symmetry breaking scale A ~ 1 GeV. Using the dimensional analysis of ref. [7], we parame- tnze the charal lagrangian as

,2 We follow the conventions of refs. [6,7]. ,a This is not a God-given basis. For example, one could

define B' = ~B~ +. These two bases will look very different when (~) 4: 0. For example, a neutron in one basis could look like a E- in another. Quantities such as the charge, strangeness and energy density of the ground state are basis independent.

The ellipse represents terms we are ignoring which are higher order in (M/A) or (0/A), or which have four or more baryon fields.

The terms with more baryon fields will cause a break- down of perturbat ion theory at hagh density. In appendix A we &scuss how to esttmate the size of these effects and conclude that one can probably go to baryon densities n B ~ A f 2 ~- 7n0, where n o is the nuclear den- sitS', before the neglected terms with four or more baryon fields Introduce ~30% corrections.

One might think that the expansion m (O/A) is an expansion m m K / A ~-- 0.5, which would render perturbat ion theory unreliable. This would indeed be the case for ka on -nuc l e on scattering. We will see, however, that the charged meson condensates have a time dependence exp(-i /~t) , where/a ~ m~r. Our expansion in ~0/A is therefore very good. The meson condensate we find is spatially varying as well, and so higher-order terms become important when 3/A approaches one. At sufficiently low density, however, ~ /A < 0.3 and so our calcula- t ion of the ground state energy is reliable to 30%.

We will also keep only the leadmg nonrelativistic con- mbut lons to the m e s o n - b a r y o n system with actions, dropping the ¥. V and 70~/5A0 terms in eq. (4), as well as neglecting the baryon kmetIc energy and the 75M terms (which we did not bother writing down.) This approximatxon breaks down at lower densities than one might expect, since the meson condensate can substan- tially reduce the baryon rest mass and render relativistic effects more important .

Finally, we have the usual expansion m the SU(3) breaking parameter m s / A ~ m 2 / A ~" 25%. We wall treat our expansxon parameters 3]A and M ] A as being of comparable unportance.

The coefficients in eq. (4) have all been measured ,4

For footnote see next page.

58


and are given by [6,8,9]

D = 0 81 , F = 0.44,

a 1 = - -0 .28 , a 2 = 0 . 5 6 , a 3 = - 1 . 1 + 0 . 3 ,

m B = 1.2 GeV,

m u = 0 0 0 6 G e V , m d = 0 0 1 2 G e V , m s =0 .24GeV. (5)

The quark masses rnu, md, and m s are actually determined in terms of the product Amq from fitting meson

masses, and even then are rather poorly known [11 ] However, errors that we may incur by using the above values are presumably comparable to hrgher effects we are neglecting.

The coefficient a 3 deserves specml mention since it is Important for kaon condensation. It is related to the so-called "sigma term" and is determined (rather poorly, as one can see) from l r -N scattering. Unfortunately,

K - N scattering data does not help determine a 3. As previously mentioned, the chlral perturbation expan- stun is very poor in this case since (0/A) ----- mK/A ~-- 0.5. This means one must include terms whrch are higher order in (0/A) in eq. (4) to analyze K - N scattering, wbach makes a more premse determination of a 3 im- possible. (See footnote 4.)

The conclusion from this section are that chlral per- turbatmn theory provides a means to investigate kaon condensation in systems where the baryon density n B is less than A f 2 ~ 7n0, provided that the baryons are sufficiently nonrelatlvistlc and the expansion parameter 0/A is small. At densities with appreciable plon and kaon condensation, the effective expansmn parameter is numerically ~<0.3, which is tolerable but not ti-

ny. In practice it causes more problems for a discus-

,4 Although these measurements may be quite accurate, there will always be errors of ~25% m matching these numbers to parameters in the chtral lagrangmn. The 30% error quoted for an a 3 is an additional experimental uncertainty. Since large negative values for a 3 are conduc- tive to kaon condensation, we have considered the values a 3 = -1.1 anda 3 = -0.6 m this paper, to be on the con- servative side. Although a 3 = -1 1 IS the experimentally preferred value, at gives a large contribution to the proton mass from the bare strange quark mass [10] a3 = -0.6 may be more reasonable, because then amproton/Om s --~ 0 m the chtral lagrangian.

stun of plon condensation than for kaon condensation, since in the former there is a near cancellation of first- order terms The subtleties of flus problem are discussed in appendix B.

3. The ground state. We wish now to find an ex- pression for the ground state energy of a system with a given baryon density and zero charge density We will effect the latter constraint by introducing a Lagrange multiplier - the charged-particle chemical potentml So we construct the thermodynamic potential

= H - / 2 p (-)

= ( 0 t / o f f ) fl + - - / 2 p ( - ) , ( 6 )

where p(-) IS taken to be the negative charge density, due to both mesons and baryons. We also include elec- trons, although their effects are only important at the condensation threshold. At zero charge density ~2 IS just the energy of the system.

Eq. (6) may be slmphfied. The tlme dependence for a scalar field O, with negative electric charge qt (i.e., q = i for the 7r- and K - ) is [12]

q~i (t) = exp (-1/2qi t) ~b t (0) (7)

One may derive eq. (7) by noting that in the mesonic

sector p(M ) = --lqi~ilIt, H t being the momentum con- jugate to q~t" Then the equation dH =/2dp(-) in con- junction with Hamilton's equations yield the result m eq. (7). This in turn lmphes that (O£/O~);r = gp(M ), which allows us to simphfy eq. (6)

a = (OZl0 ) - ( 8 )

where p(B - ) and p~-) are the usual baryomc and elec- tronic contributions to the negative charge denslty.

p(-) = _ ~ 7 0 p _ 2+70N + + 2 - 3 , 0 N - + ~ 70~ - ,

=/23/3 2 . (9)

The task that now faces us is as follows. (1) gwen a background field 7r a (x, t) and chemical potential/2, dlagonahze the baryon hamiltonian in ~ ; (2) fill the resultant elgenstates consistent with fixed baryon density and minimal energy; (3) rmnirmze ~ with respect to ~a, and extremlze with respect to/2.

The eight meson fields have a priori arbitrary spati- al dependence, which makes solving the complete prob-

59


lem difficult. In this letter we make the ansatz

x (x , t) = Ut n(O, O) U, (10)

where

U = dlag (1, exp [1(/~ t - p~" x ) ] , exp [i(/at - pK" x)]),

with P~r and PK cohnear In other words, we have assumed that the charged plons and charged and neutral kaons are in plane waves, with the K 0 momentum a linear combination of the zr + and K + momenta. We make this simplifying assumption so that we may rid the baryon hamlltonian of space-t ime dependence through the transformation

B~F -+ Ut BU . (11)

This leaves the baryon hamiltonian unchanged to the

order in which we are working. We now numerically

minimize the thermodynamic potential with respect to n (0, 0), PTr and PK, while maximizing with respect to ~. Because of the number of variables involved we have used the Metropohs algorithm to find the mini- mum of 12 ,s

Our task is further comphcated by the fact that to

the order we are working, the energy levels of the two spin states of the baryons get split in the presence of the meson condensate, with the baryon spin preferring to antl-afign with the meson momentum. The splitting gets as large as ~102 MeV/baryon, comparable to the

kinetic energy, which we are ignoring m our expansion. Furthermore, there are spin dependent nuclear effects which we are ignoring. We have dealt with the splitting by finding the ground state energy both for a system with spins paired, as well as for the case where only the lowest energy spin state is filled. It seems plausible to us that the true ground state may be partially spin polarized.

The characteristics of the ground states we find are plotted in figs. 1 -4 , where figs 1 and 3 represent a completely spin polarized state, and figs. 2 and 4 a completely unpolarlzed state. In both cases we have performed the calculation for two different values of

,s At the pion (kaon) condensation threshold, the energy is fairly insensmve to the plon (kaon) wavevector, and so our results for pn near the plon threshold and PK near the kaon threshold are not very accurate. We hope to im- prove the accuracy of these results with more computer time.

1 2 5

I 00

0 7 5

0 50

0 2 5

0 O0

6

4

2

0

- 2

2

0

~ a

? 2 3 4 5 n o

nB/( f r r ) a

~ b

PK/f~

. . . . i . . . . i . . . . I . . . . I . . . .

I . . . . I . . . . I . . . . I . . . . .

" " ~ 2 3 4 5 6 n 0

n B / ( f ~ ) a

Fig. 1. Results of mlmmizatlon of the thermodynamic potential with the baryons assumed to be completely spin polar- ]zed and with a 3 = -1.1. The dotted part of the graph indi- cates that some of our expansmn parameters have become larger than 0.3 (a) The total strangeness/baryon of the ground state (b) The chemical potential for negative charge, and the plon and kaon wave vectors, in umts off~ r = 93 MeV

the sigma term, a 3 = - 1 . 1 and a 3 = - 0 . 6 . In these figures the solid lines change to dotted when our ap- proximation of the ground state energy to ~30% accuracy is rendered invalid due to large momenta of the mesons, or relatlvXStlC corrections for the baryons. Note that the strangeness/baryon ratio becomes large before our approxmaatlons break down, except in fig. 4a. With unpolarlzed baryons and a 3 = - 0 .6 , the pIon momentum becomes shghtty too large to claim knowl- edge of the condensate energy to within 30% at the kaon threshold.

60


1 25

I O0

0 75

0 50

0 25

0 O0

8

4

2

0

-2

2

0

a3= - t . 1 a UNPOLARIZED

P~/PB

/ . . . . . I . . . . . . . ! . . . . . . . v , , , . I . . . .

? 2 - - 3 4 5 n O

n B / ( f T r ) s

aS= - I 1 b UNPOLAR ZED

PK/r" /. /

. . . . I . . . . I . . . . I . , . , I . . . .

i

. . . . I . . . . I . . . . I . . . . I . . . ,

? 2 S 4 5 n o

nB / ( f r r ) 3

Fig. 2. (a), (b). Minimization results with the baryons completely unpolarized and a 3 = - 1 1.

1 25

I O0

0 75

0 50

0 25

0 O0 . . . . I . . . .

? 2 n o

B

PS/p B /

t . . . . I . / . I . . . .

3 4 5

nB/(ITT) s

- 2 2

b

PK/f~

. . . . t . . . . I . . . . I . . . . I . . . .

~ I f ~ ~ , , :

. . . . I , , , , I . . . . I , , , ,

3 4 5

n B / ( f r r ) s

. . . . I

0 ? 2 n O

Fzg. 3. (a), (b) Mmlmizatzon results wzth the baryons completely spin polarized anda 3 = 0.6 (see footnote 4)

1.25

1.00

0 .75

0 .50

0 .25

0.00

.3= -.8 1 a b UNPOLARIZEDJ

4 a3= - e P T r / f ~

[ UNPOLAR]ZED I ~ /

2

0 P K / f ~

P"/PB - - ~ . . . . . f . . , . I . . . . I . . . . I . , , ,

. . . . I . . . . r . . . . I . . , . I . . . . 0 . . . . I . . . . I . . . . t . . . . I , , , .

? ~' 3 4 5 ? 2 3 4 5 no n o

n B / ( f ~ ) 3 nB / ( f r r ) s

F~g 4. (a), (b) Mimmizatlon results wzth the baryons completely unpolarlzed and a 3 = -0.6.

61


O I.

[.~

-100

- 2 0 0 0

-I00

"•__•UNPOLARIZED POLARIZED /

. . i . . . . i . . . . i

'~ 2 3 4 n o

n B / ( f ~ ) a

U ~ t Z E D x , N

POLAR[ZED /

5

Fig 5 The condensate energy/baryon for the four cases in- vestigated, neglecting baryon-baryon interactions

4. Conclusions. We have demonstrated that there is strong quahtative evidence for kaon condensation m matter with baryon density 2.3n 0 ~<n B ~< 3n 0, where n o is the nuclear density. We cannot specify the critical density accurately due to experimental uncertainties about the sigma term, 30% corrections from neglected higher-order operators, as well as dif- ficulties In prechctmg the existence or strength of pion condensation (as discussed m appendix B)

Kaon condensation could prove to be the route taken from nuclear matter to a three-flavor quark state. Building up strangeness in this fashion obviates the need for multiple simultaneous weak interactions. In any case, a kaon condensate will certainly affect the equation of state and coohng rate of neutron stars, even if strange matter is unstable at the relevant pressures.

We thank Sekhar Chlvukula, Andrew Cohen, Howard Georgi, and Aneesh Manohar for useful rater- actions. This research is supported in part by the National Science Foundat ion under Grant No. PHY- 82-15249 and also by the Harvard Society of Fellows.

Appendix A. In (4) we only considered the effects of terms containing a single power of ~ 4. For instance, we neglected the effects of the short-range nuclear forces, which are important in the prevention of neutron star collapse We also neglected the effects of operators which are sinular to those in (4) but with

extra powers of ~ / A f 2 appended. In the Hartree approxnnat lon at nonrelatlvistlc densities, each power of ~ is just a factor of density n, and so one might expect that our perturbat ion expansion would yield poor results for n / A f 2 > 0.5 (n ~ 3n0). The effective hamlltonian also contains short-range nucleon nucleon interaction terms of order ( ~ ) 2 / f 2 , ~vhich in the Hartree approxmlat lon would be as impor tant as the meson-nuc leon interactions, even at nuclear density.

When nuclear correlation effects are considered, the cbaral perturbation expansion looks much better. The repulsive short-range interactions do not allow the baryons to approach each other, and the importance of terms with extra powers of ~ / A f 2 is di- minished. One should really perform a variational calculation including nuclear correlations to calculate the contribution of these terms to the ground state energy.

As an estimate of the importance of all terms with higher powers of ~ , we turn to conventional nuclear matter theory. Bethe and Johnson (BJ) have performed a variational calculation of the ground state energy of pure neutron matter [13] , s . They find the energy density to be

e = mnn +2 .86 fn (n /A f2 ) 1 54 , (12)

for densities up to 3 0 f 3 , where nucleons become relativistic The second term In the BJ formula is less than

0.27n2/f 2 for n < A f 2 , which is less than 27% of the value given by naive dimensional analysis. We tentative- ly conclude that it is justifiable to use the chlral lagranglan at densities below A f 2 , and assume multi- baryon terms are less than 30% as important as terms with a single power of ~ff.

The above arguments should be valid only as long as the nuclear correlations are unaffected by a meson condensate. The short-range nuclear forces could still have a large effect when the meson condensate is spatially varying. For instance, it IS well known that the p-wave p lon -nuc leon attraction is reduced m baryonlc matter [15], and so the nuclear forces inhibit plon condensation (the Ericson-Ericson, Lorenz -Loren tz effect). This effect should be of little importance for kaon condensation which is not driven by attractive p-wave couplings, but is mainly produced by the sagma term.

+6 See ref. [14] for a good revaew of these issues.

62


Appendix B. As exp la ined In sec t ion 2, any discus-

slon o f m e s o n c o n d e n s a t i o n in dense m a t t e r relies on

a p e r t u r b a t i o n e x p a n s i o n where the re levan t p a r a m e t e r

is (rnn/A ~ 2mTr/A ) ~< 0 3 The f i rs t -order t e rms in th is

expans ion ough t to be rel iable enough to give a gener-

al idea o f w h a t is going on , so long as there is no t any

consp i r acy t h a t causes t h e m to a lmos t cancel . In fact ,

t h o u g h , there ts such a consp i racy for p lons , a l t h o u g h

no t for kaons Thus we are in the unusua l pos i t ion o f

having charal p e r t u r b a t i o n t h e o r y w o r k b e t t e r for

k a o n s t h a n p lons

The p r o b l e m w i t h p lons is t h a t the vec tor in terac-

t ions w i t h nucle i are repulsive, whi le the axial vector

i n t e r ac t i ons are a t t rac t ive . At lowes t order , p lon con-

densa t i on will occur i f (D + F - 1) = (gA 1) is

grea ter t h a n zero. In fact , (gA -- 1) ~-- 0 25 , w h i c h is

the size o f our e x p a n s i o n p a r a m e t e r Thus the leading

effects in our e x p a n s i o n w h i c h drive p lon condensa-

t i on are ac tua l ly the same size as two der ivat ive me-

s o n - b a r y o n in t e r ac t i ons (e.g ( I / A ) Tr BAuA UB)

w h i c h we are d r o p p m g * 7. Smce the size and sign o f

these ope ra to r s are n o t measurab le , it is imposs ib le

to say w i t h con f idence w h e t h e r or n o t p lon condensa-

t i on ac tua l ly occurs . There are add i t iona l comphca -

t lons due to shor t - range nuc lear forces and isobar ex-

change, a l ready discussed in the l i t e ra tu re [16] .

Wha t does this m e a n for k a o n c o n d e n s a t i o n 9 The

graphs in figs. 1 - 4 m a k e it clear t h a t t e rms h igher

o rder m PTr are i m p o r t a n t at dens i t ies above where

k a o n c o n d e n s a t i o n has appea red , sunp ly because one

f inds P,r >> 2rn~ if one ignores t h e m . However , whi le

these t e rms m a y af fec t how fast a k a o n c o n d e n s a t e

t u rns on , t h e y do n o t a f fec t the t h r e sho ld , since PTr

on ly grows af te r there is an apprec iab le k a o n conden-

sate

,7 This should be no surprise In the hnear sigma model one must go to one sigma loop to generate gA :# 1, and one- loop diagrams will also generate the two derivative nucleon couphngs, so these two effects are both formally second order.

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Strange goings on in dense nucleonic matter