kaon condensation in “nuclear star” matter

ELSEVIER

8 September 1994

Physics Letters B 335 (1994) 266-272

PHYSICS LETTERS B

Kaon condensation in "nuclear star" matter

Chang-Hwan Lee ~, G.E. Brown b, Mannque Rho c "Department of Physics and Center fi>r Theoretical Physics, Seoul National University, Seoul 151-742, South Korea

h Department of Physics, State University of New York. Stony Brook. NY 11794, USA Sen,ice de Physique Thiorique, CEA Saclay 91191 Gif-sur-Yvette Cedex, France

Received 19 April 1994; revised manuscript received 21 June 1994 Editor: G.F. Bertsch

Abstract

The critical density for kaon condensation in "nuclear star" matter is computed up to two-loop order in medium (corresponding to next-to-next-to-leading order in chiral perturbation theory in free space) with a heavy-baryon effective chiral Lagrangian whose parameters are determined from KN scattering and kaonic atom data. To the order considered, the kaon self-energy has highly non-linear density dependence in dense matter. We find that the four-Fermi interaction terms in the chiral Lagrangian play an important role in triggering condensation, predicting for "na tura l" values of the four-Fermi interactions a rather low

critical density, p,. < 4p~.

In a recent paper, Brown and Bethe [ 1 ] suggested that if kaon condensates develop at relatively low matter density in the collapse of large stars, then low-mass black holes are more likely to form than neutron stars of the mass greater than 1.5 times the solar mass M. Ever since the seminal paper of Kaplan and Nelson [2], there have been numerous investigations on kaon condensation in dense neutron star matter as well as in nuclear matter based both on effective chiral Lagran- gians [3-7] and on phenomenological off-shell meson-nucleon interactions [8,9]. The results have been quite confusing: while the chiral Lagrangian cal- culations generally predict a relatively low critical density, Pc~ (2-4)po, the phenomenological approaches have indicated that a kaon condensation at such a low density may be incompatible with kaon-nucleon data and in some versions seem to exclude any condensation at all. It is now understood [ 10,11 ] that the main dif- ference in the two approaches lies in terms higher than lincar in density in the energy density of the matter.

0370-2693/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD10370-2693(94 )00893-0

In this paper, we report the first higher-order chiral perturbation calculation of the critical density with a chiral Lagrangian that when calculated to one loop order (i.e., to @(Q2) relative to the leading order), correctly describes s-wave kaon-nucleon amplitude near threshold and that includes four-Fermi interactions constrained by kaonic atom data. Our prediction for critical density is p~ = (3-4)Po.

To implement spontaneously broken chiral symmetry in the computation, we take the Jenkins-Manohar heavy-baryon chiral Lagrangian [12] as extended in [ 131 to 6¢(0 3) tO describe s-wave kaon nucleon scattering to one-loop order in chiral perturbation theory (ChPT). In addition to the usual octet and decuplet baryons and the octet pseudo-Goldstone fields, the A(1405) was found to figure importantly in the kaon- nucleon process. This field which provides repulsion at threshold in K - p scattering was introduced in [ 131 as an elementary field. By fitting the coefficients of the resulting chiral Lagrangian by empirical kaon nucleon

C.-H. Lee et al. / Physics Letters B 335 (1994) 266-272 267

s-wave scattering data at low energy ~, it was shown there that higher order chiral corrections can systemat- ically be calculated while preserving the "naturalness" condition for on-shell scattering amplitudes. By a straightforward off-shell extension, we have predicted an off-shell kaon-nucleon amplitude that could be applied to kaonic atom [151 as well as kaon condensation phenomena. The predicted off-shell amplitude was lbund to be in fair agreement with the phenomenological fit obtained by Steiner [ 16]. A simple way of understanding the result so obtained is to use the chiral counting appropriate for the meson-baryon system. In heavy-baryon formalism (HBF), we can order the relevant observables as a power series in Q, say, Q ~ where Q is the characteristic energy or momentum scale we are interested in and ~, an integer. Thus to leading order, the kaon-nuclcon amplitude T Ku goes as G ( Q ~ ), to next order but involving only tree graphs as eg(Q 2) and to next-to-next order (or N2LO) at which one-loop graphs enter as G(Q3) . The off-shell amplitudc calculated in [13] contains therefore all terms up to f f (Q~). This amplitude could be used in impulse approximation for in-medium processes. The corresponding contribution to the kaon self-energy given by Fig. la is

= - - (P t , ' f f / [ree ( ( ' / ) ) -~-rt')n '~ '- free ( / ) ) ) , ( ! )

where 3 KS is the off-shell s-wave KN transition matrix 2. This can provide an optical potential for kaonic atom and the linear density approximation for kaon condensation. We will shortly discuss what critical density is obtained in this approximation.

To go beyond the linear density approximation as required for a more reliable treatment of both kaon condensation and kaonic atom, we need to compute the effective action (or effective potential for uniform matter). For this the first obvious correction to the self-

d f

(a)

~, i l I i l I

~, I 1 I

(h) (,t)

% % % ~ J S' %%% f f f

@ (e) (f)

Fig. 1. (a) : The linear density approximation to the kaon self-energy in-medium, Ilx. The square blob represents the off-shell K " N amplitude calculated to ~(Q-~); ( b ) - ( f ) : medium corrections to T xN of Fig. (a) with the free nucleon propagator indicated by a double slash replaced by an in-medium one, gq. (2). The loop labeled ,oN repre-

sents the in-medium nucleon loop proportional to density, N ~ the nucleon hole (n - ~ and/or p - ~ ), the external dotted line stands for the K - and the intemai dotted line for the pseudosc~Jar octets ~, 7, K.

The present situation with low-energy kaon -nuc l eon scattering is

summar ized in [ 141. The values used in Ref. 1131 are essenti',dly

the same as those quoted in this reference. Al though the experimental K - N scat ter ing lengths are given with error bars, the avai lable K ' N

data are not very well determined. Since both are used in titling the parameters o f the Lagrang ian , we did not quote the error bars in [ 13 I and shall not u ~ them here for fine-tuning. For our purpose, as will be clear soon, we do not need great precision in the data a.s the results

are extremely robust agains t changes in the parameters . 2 The ampl i tude .7 rA, taken on-shell , that is, to = Mr , and the scat-

tering length a KN are related by a K,v= [ 1 / 4 ~ ( I + Mr/m A) IT xN.

energy ( I ) comes from the influence of the medium on the amplitude .y-rN which is readily taken into account by replacing the heavy-nucleon propagator in Figs. l b - l f b y the in-medium one,

i G°(k) - vk+i-----~e -27r6(ko)O(k,..- I k l ) , (2)

where kr is the nucleon Fermi momentum related to density puby the usual relation p,v = (y/6"n "2) k3u with

268 C.-H. lee et al. / Physics Letters B 335 (1994) 266-272

the d e g e n e r a c y factor 3 '= 2 for neu t ron and pro ton in

nuc lea r matter . W e shall call this c lass o f cor rec t ions

&F- K-N It is c lear that it will g ive rise to non- l inea r .

dens i ty dependence . F u r t h e r m o r e we expec t it to be

repuls ive as it c o r r e s p o n d s to the Pauli exc lus ion effect.

The second c o r r e c t i o n - w h i c h is a lot more impor t an t

- c o m e s f rom " p a r t i c l e - h o l e " exc i t a t ions that do not

figure in K N sca t t e r ing but can con t r ibu te impor tan t ly

in med ium. T h e s e are dep ic ted in Figs. 2. S ince we are

dea l ing wi th s -wave kaon in terac t ion , the mos t impor -

tant con f igurat ion that K - can coup le to is the A ( 1 4 0 5 )

p a r t i c l e - n u c l e o n hole ( d e n o t e d as A N ' wi th N e i ther

a p ro ton ( p ) or neu t ron ( n ) ) . T h u s Fig. 2a invo lves

the A N - ' - A N - ' in te rac t ion whereas Figs. 2b can

A A

p- , p - t

p - t "

(b)

N

p-I

(c) Fig. 2. Two-loop diagrams involving A(1405) contributing to the kaon self-energy. Diagrams (a) and (b) involve four-Fermi interactions describing the AN - ,_AN - t vertex. Diagram (c) does not involve four-Fermi interactions and hence is unambiguously determined by on-shell parameters. Here the internal dotted line represents the kaon.

invo lve bo th the A N - ' - A N - ' and N N - ' - N N -

in teract ions . In wha t fo l lows we will no t specificall

cons ide r the lat ter wh ich invo lves no s t r angeness fl~

vor: we will a s sume it to be g iven by wha t is de te rmine

in the non- s t r ange ( n u c l e a r ) sector , i.e., s y m m e t t

energy, etc 3. Now for the s -wave i n - m e d i u m kaon sel

energy, the r e l evan t four -Fermi in te rac t ions th;

invo lve a A ( 1 4 0 5 ) can be reduced to a s imp le fort

i nvo lv ing two u n k n o w n cons t an t s

4--~4- fel'~ion : c'~ f l A Tr 1fiB+ C ] A a k A Tr B¢rkB ,

(3

where B is the ba ryon (he re n u c l e o n ) field, C'~/r al

the d i m e n s i o n - 2 ( M 2) pa ramete r s to be fixed empi

ically and o "k acts on ba ryon spinor .

Addi t iona l ( i n - m e d i u m ) two- loop g raphs th:

invo lve A N - ' exc i ta t ions are g iven in Fig. 2c. The

do not, however , invo lve con tac t four -Fermi intera~

t ions, so are ca lcu lab le u n a m b i g u o u s l y .

W e shall deno te the sum of these con t r i bu t i ons froJ

Figs. 2 to the se l f -energy by HA 4.

If one is on ly in teres ted in cri t ical dens i ty for kac

c o n d e n s a t i o n and in proper t ies o f kaonic a tom, it su

rices to c o m p u t e the kaon se l f -energy. For the equat ic

o f state to wh ich we will re turn in a separa te public~

, ion, nonl inear i t i es in the c o n d e n s a t e fields will t

.t In the sense of Fermi liquid, this part of interactions should be givt in terms of the standard Landau-Migdal interactions which assume, as in condensed maUer physics 1171. to be a fixed-poi theory. As such, one can take this part of interactions to be accurate given by nuclear matter properties. 4 There is one class of potentially important in-medium two-lo~ graphs that we have not taken into account in this paper and that the set of graphs which could screen the leading ~ (Q) term assoc ated with the vector-meson ( coand p) exchange [ 13 I. To be specifi one can visualize it as Fig. 1 a with the box replaced by an w-exchan; vertex and with the nucleon loop attached to another nucleon Io~ through a four-Fermi interaction. In the co-exchange channel, tl screening will go like ~ 1/(1 + Fo), where the Fo is the Landal Migdal parameter [ 18]. In dense matter, one expects Fo > 0. Hen one might fear that there could be substantial loss of attraction. V believe this will not be serious here for two reasons. Firstly the gau: coupling of the vector meson is predicted to scale down as densi increases as in the hidden-gauge symmetry Lagrangian theory [ I c so that/~o will increase less rapidly than in the absence of the scalin Secondly the vector-exchange attraction is scaled up by the E scaling 1201 by the factor (mo/m'~)z> 1 where m o is the vectc meson mass with the star indicating the in-medium quantity. The two opposing effects are expected to more or less cancel out. V leave both effects out in this paper. They will be treated in detail a longer paper in preparation.

C.-H. Lee et aL / t 'hys ics l .etters B 335 (1994) 2 6 6 - 2 7 2 269

course have to be taken into account. This can be done in a straightforward way as described in [6]. Putting all graphs up to two loops in medium, we have the complete in-medium two-loop kaon self-energy

f f l I (( O)) = -- ( p l , , ~ Klret,-,,( O.)) -~ pn .Y fKrc~n(O)) )

(pv&,y K-p K - , ~,N (tO) + p , 6.~ I IA(w) -- ,,~ (O9)) +

(4)

- - K N is the scattering amplitude obtained in where ,~" f~e LJMR [ 13], PN ( N = p , n) is the nucleon density and

(97- K N 6.~ ~,~, are the medium modifications, by Figs. l b - I f, K - N to ,Uf~,. ' and

,( IIA( W) = _ g5 o9 U o9 +mn - m A

C t _z']- ( p , , + ~pp - ~ ,,~,,, f

o g + m : - m A o92(.yp + 2~¢) (5)

C s X .~pp

-4 gA ~ PP

with

i Ikl ~- 1 d[kl o92 , ~, (6) Xx u ( w ) = 27r2 - M k [k["

where ,qa is the renormalized KNA(1405) coupling constant determined in [ 131. In Eq. (5), the first term comes from the diagrams of Figs. 2a and 2b and the second term from the diagrams of Figs. 2c. While the second term gives repulsion corresponding to a Pauli quenching, the first term can give either attraction or

1 repulsion depending on the sign of (CSA[p, + ~p~,] -- ~CAp, ) with the constants C~ 'r being the only parameters that are not determined by on-shell scattering data. We can fix them from kaonic atom data [15] which require that there be an effective attraction. In condensed matter physics [ 17 ], such an attractive four- Fermi interaction in a particular kinematic situation turns by renormalization into a "marginally relevant" interaction that causes instability of the system, tbr example, pair condensation in superconductivity. It is tempting to conjecture that something similar happens here, leading to a strangeness condensation. However, what appears to be different here from a generic case in condensed matter physics is that the four-Fermi

The explicit formulae will be given in a longer paper in preparation.

interaction contribution (5) has a quadratic w dependence which makes it increasingly less effective toward the regime o f small w where the condensation takes place.

We have now all the ingredients needed to calculate the critical density. For this, we will follow the proce- dure given in [6]. As argued in [3], we need not consider pions when electrons with high chemical potential can trigger condensation through the process e - ~ K - re. Thus we can focus on the spatially uniform condensate

( K ) = V x e- iu~, (7)

where p. is the chemical potential which is equal, by Baym's theorem [21 ], to the electron chemical potential. We shall parametrize the proton and neutron den- sities by the proton fraction x and the nucleon density u = plpo as

pp=xp, p , , = ( l - x ) p , p = u p o . (8)

Then the energy density g - which is related to the effective potential in the standard way - is given by

g(u, x,/z, vK)

3 r.d)) uS/3po + V( u ) 4- upo ( l - 2x) 2S( u ) 5L, F

- [ # 2 - M Z x - IlK( t z, u, x)]vex + tY(v~)

+ tzupox+ ge + 0( I/z[ - m ~ , ) g u , (9)

where E ~!" = ( p ~.!" ) 2 ~ 2ran and p~2~ = ( 3 7r2 po/ 2 ) I / 3 are, respectively, Fermi energy and momentum at nuclear density. The V(u) is a potential for symmetric nuclear matter as described in [22] which we believe is subsumed in contact four-Fermi interactions (and one-pion-exchange - nonlocal - interaction) in the non-strange sector as mentioned above. It will affect the equation of state in the condensed phase but not the critical density, so we will drop it from now on. The nuclear symmetry energy S(u) - also subsumed in four-Fermi interactions in the non-strange sec tor -does play a role, as we know from Ref. [ 22] : protons enter to neutralize the charge of condensing K - ' s making the resulting compact star "nuclear" rather than neutron star as one learns in standard astrophysics text- books. We take the form advocated in [22]

, ~ , , ~ 2/3 S ( u ) = ( 2 2 / 3 - . j ~ . ~ F ~u - F ( u ) ) + S o F ( u ) .

( I0 )

270 C.-H. Lee et al. /Physics Letters B 335 (1994) 266-272

Table 1 Critical density in linear density approximation corresponding to Fig.

la.

F(u)

u 2u2/(1 +u) &

4 3.90 3.77 4.11

where F(u) is the potential contributions to the symmetry energy. Three different forms of F( u) were used in [22]:

F(u) =z4, F(u) = E, F(u) =&. (11)

It will turn out that the choice of F( u) does not signif- icantly affect the critical density.

The contributions of the filled Fermi seas of electrons

and muons are [ 61

=- 8”j* ((2t2+ l)#X

- ln(P+@X)) - pp$$ (12)

wherepFH=dp2--rn$ is the Fermi momentum and

t =pF,JmF. The ground-state energy prior to kaon condensation is obtained by extremizing E’ with respect to x, I_L and vK:

x ax “KxO =

0, a’ = air. UK=0

0, 3 =o, K UK=0

(13)

from which we obtain three equations corresponding, respectively, to beta equilibrium, charge neutrality and dispersion relation:

p=4(1-2x)S(u),

0= -xupo+ $ +O(I*--mJp$,

0=~-1(~,U,“)=~2--~-~K(~,U,n)

= P=-M;~( p, u, x) . (14)

We have solved these equations using for the kaon self- energy (a) the linear density approximation, Eq. ( 1) , and (b) the full two-loop result, Eq. (4). Table 1 shows the case (a) for different symmetry energies, Eq. ( 11) . We see that the precise form of the symmetry energy does not matter quantitatively, so we will simply take F(U) = u from now on. The corresponding “effective kaon mass” M: is plotted versus u in Fig. 3 in solid line. Note that even in this linear density approximation kaon condensation does take place, albeit at a bit higher density than obtained before.

We now turn to the (in-medium) two-loop calcula-

tion. For this we need to fix the parameters C;*. This

could be done with kaonic atom data. The presently available data [ 151 imply that the optical potential for the K _ in medium has an attraction of the order of

AV= - (180+20) MeV at u=O.97, (15)

which implies approximately for x = 1

(16)

Fig. 3. Plot of the effective kaon mass Mi obtained from the disper-

sion formula D -’ (p, u) = 0 versus the chemical potential p prior to

kaon condensation, with F(u) = u. The solid line corresponds to the linear density approximation and the dashed lines to the in-medium

two-loop results for (C:-C~)_f~=lO and C”,f:=lO, 5, 0, respectively, from the left. The point at which the chemical potential

p intersects Mi corresponds to the critical point.

C.-H. Lee et aL / P h y s i c s Let ters B 335 (1994) 2 6 6 - 2 7 2 271

Table 2 Self-energies for kaonic atom in nuclear matter (x = 0.5) in units of M~, for (C'~ - CrX¢,4,~ 2,~ -- 10 and I"(u) = u. AV=--M*x - M x is the attraction (in units of MeV) at given density.

U M x A V x - t ~ f '~ - p6.9 rr~'~ 171 t 112

0.2 424.6 - 70.37 0.5 - 0.0673 0.0034 - 0.1998 0.007607 0.4 390.0 - 105.0 0.5 - 0.0920 0.0084 - 0.3024 0.00664 I 0.6 364.3 - 130.7 0.5 - 0.1173 0.0143 - 0.3610 0.005635 0.8 342.6 - 152.4 0.5 - 0.1462 0.0207 - 0.3996 0.004794 1.0 323.5 - 171.5 0.5 - 0.1789 0.0274 - 0.4250 0.004088 1.2 306.2 -- 188.8 0.5 - 0.2150 0.0343 - 0.4404 0.003483 1.4 289.9 - 205.1 0.5 - 0.2540 0.04 13 - 0.4460 0.002945

This leaves one parameter free for x~: ½. It will, however, turn out that this freedom docs not d iminish sig- nificantly the predict iveness of the theory. We shall not attempt to fine-tune these constants in this work but take (16) to be some mean value in what follows. The result turns out be pretty much insensi t ive to the precise values of the constants. W h e n one can pin them down (through, for example, isotope effects) from kaonic atom data, our prediction could be made considerably more precise.

In Table 2, we list the predicted density dependence of the real part of the kaonic atom potential for x = 0.5 obtained for ( C s - C ] ) f ~ ~ 10. To exhibit the role of A ( 1405 ) in the kaon sel l -energy, we list each contribution of H. Here /-/free-- K-N - - P u , Y - ~ e e , 6 I I = -

P N 6 , ~ x - N, I l i a corresponds to the first term of Eq. (5) which depends on C'~/~ and / / z to the second term independent of C',S~ "r. We observe that the C~l:r-dcpend -

cnt term plays a crucial role for attraction in kaonic atom. For the value that seems to be required by the kaonic atom data, Eq. (16 ) , the critical density comes out to be about u,.-~ 3, rather close to the original Kap- l an -Ne l son value.

In Table 3 and Fig. 3 are given the predictions for a wide range of values for C'~ f ~.. What is remarkable here is that while the C ' ] r -dependen t four-Fermi interactions are e s s e n t i a l for tr iggering kaon condensat ion,

Table 3 Critical density u, in in-medium two-loop chiral perturbation theory for (C'~ - C!~)f ~ = 10 and F(u) =u.

c",f~

I0 5 0

u, 3.13 3.33 3.69

the critical density is quitc insensi t ive to their strengths. In fact, reducing the constant (C ~ - 7 2 CA ) f ,. that represents the kaonic atom attraction by an order of mag- nitude to 1 with C ~ e 2,.= I0, 0 modifies the critical

density only to u,. = 3.3, 4.5, respectively. In conclusion, we have shown that chiral perturba-

tion theory at order N2LO predicts kaon condensat ion in "nuc lea r s tar" matter at a density p , . < 4 ~ with a large fraction of protons - x = 0.1 ~ 0.2 at the critical point and rapidly increasing afterwards - neutral iz ing

the negative charge of the condensed kaons. For this to occur, four-Fermi interactions involv ing A (1405) are found to play an important role in tr iggering the con-

densat ion but the critical density is surprisingly insen- sitive to the strength of the four-Fermi interaction. This makes the condensat ion phenomenon even more robust

than thought before. We have not taken into account the BR scaling [201 - which we should, to be fully consistent. An approximate account of the BR scaling is found to lower the critical density below 3pc~, prac-

tically independent ly of the magni tude of C~ '7. As

remarked above, there are compensa t ing effects such as the screening of the vector channel which has to be considered on the same footing, so a fully consistent treatment would require more work. What is fairly cer- tain is that although the detailed mechanism appears quite different here from that in the low-order treat-

ments, once the condensat ion sets in, the rest of the star properties are expected to resemble closely the struc- ture obtained in [6,7] . A detailed analysis of the "nuc lea r s tar" that we obtained in this paper will be made elsewhere.

CHL is grateful for valuable discussions with H. Jung and D.-P. Min. He would also like to thank V.

272 C.-H. Lee et al. / Physics Letters B 335 (1994) 266-272

Thorsson for informative discussions at the 1993 Chiral Symmetry Workshop at ECT* in Trento. We have benefited from conversations with N. Kaiser, K. Kubodera, A. Manohar and H. Yabu. Part of this work was done while the two of us (CHL and MR) were attending the 94 Winter Workshop in Daejeon (Korea) sponsored by the Center for Theoretical Physics of Seoul National University. The work of CHL was sup- ported in part by the Korea Science and Engineering Foundation through the CTP of SNU and in part by the Korea Ministry of Education and the work of GEB by the US Department of Energy under Grant No. DE- FG02-88ER40388.

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kaon condensation in “nuclear star” matter

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