Kaon condensation in nuclear matter

Physics Letters B 273 North-Holland

( 199 I ) 156- I62

PHYSICS

Kaon condensation

Received

30 September

LETTERS

B

in nuclear matter *

I99 1

Using the leading operators m the chiral lagrangian for meson-baryon mteractions (at low momentum) and treating the meson fields classically, Kaplan and Nelson have observed, using numerical methods. that a charged kaon condensate forms at several times nuclear density. We derive an analytical expression for the critical density. The effects of virtual mesons and higher dimcnsion operators are discussed. The possibility of a neutral kaon condensate IS also considered.

The nature of the ground state at fixed baryon density is of interest for the study of neutron stars and large nuclei. Recently Kaplan and Nelson have suggested that at a few times nuclear density a charged kaon condensate forms and the ground state develops a net strangeness [ 1.21. In this paper we also explore this issue. Assuming the transition to the state where charged kaons condense is smooth, we are able to find an analytical expression for the critical density (using approximations similar to those of Kaplan and Nelson). The interactions of mesons with baryons are described at low momentum by the chiral Lagrange density [ 31 +Tr Bti &,B+i Tr B’[ r’(,, B] -D Tr B+oB[A, B)

Y’= i,f’ Tr a,,Zt a,‘C+r’Tr(,n,~+h.c.) -FTr

B+a[A, B] +a, Tr Bt(&n,<+h.c.)B+a,

In eq. ( 1 ) B is a 3 x 3 matrix that contains

Tr BtB(&~r,,<+h.c.)+a,Tr(rn,C+h.c.)

Tr BtB.

(1)

the baryon fields P

B=

-1p+L/i

z-

Jz

.Y is a 3 x 3 unitary E= exp ( 2%/j)

n

(2)

$

matrix that contains

the meson fields

.

(3)

where

(4)

* Work supported

156

in part by the US Department

of Energy under Contract

0370.2693/91/$

no. DEAC-03-8

I ER40050.

03.50 0 199 I Elsewer Science Publishers

B.V. All rights reserved.

Volume 273, number 1.2

PHYSICS LETTERS B

12 December 1991

is the 3 × 3 unitary matrix that squares to X, i.e., ~2 =X, and V~,=½(~* 0/,~+d0~,d*),

A,,=~i(~* 0/,~-~0~,~*) .

(5a,b)

In eq. (1) the two-component baryon fields have been rescaled, B - - , e x p ( - i M t ) B ( M is the c o m m o n SU ( 3 ) t . × SU ( 3 ) r symmetric mass), so that derivatives on the baryon fields give factors of the residual four, m o m e n t u m [4]. Neglecting the small up and down quark masses, we take the quark mass matrix to be

mq =

0

0

(i 0° m~/°/

.

(6)

The values of the parameters in the chiral lagrangian are determined from experiment. Weak leptonic pion decay gives f ~ 135 MeV, and weak nuclear beta-decay and semileptonic hyperon decay give F-~ 0.44 and D-~ 0.81. The pseudoscalar masses are m ~ = 3m ~ = 4m~ t,/f 2 ,

(7)

and the experimental values of the kaon and eta masses imply that m¢,-~ 182 MeV) 4. The baryon mass splittings are related to alm~ and a2m~. Explicitly M z - M N =2a2m~,

:$IA--MN = ~ (a2 -2a~ )m~,

M=_-MN = 2 ( a 2

--a

I

)DI s

(8a,b,c)

The experimental values of the masses of the baryons give a~m~-~ - 6 7 MeV and a2m~-~ 134 MeV. The coefficient a3 is determined by measuring the pion-nucleon sigma term, G~N. This involves a delicate extrapolation in momentum, and consequently the value of a3 has a large uncertainty associated with it [ 5 ]. Experimentally, a 3 m ~ = - (310_+ 50) MeV. According to the lagrangian (1) the dependence of the nucleon mass MN on the strange quark mass is 0MN m~ 0m~ -- --2(a2 + a 3 ) m ~ ,

(9)

so the central value a 3 1 r / s = - 3 1 0 MeV corresponds to a very large dependence of MN on m~, i.e., trlsOMN/ 0m~-~ 350 MeV [6 ]. As m ~ 0 chiral perturbation theory is valid and ms 0MN/0ms~0. Furthermore as m s ~ o o perturbative Q C D gives that msOMN/Oms~gMN~--65 MeV. (For m~ much greater than the scale of the strong interactions it is appropriate to integrate out the strange quark, going over to an effective two-quark theory. The relationship between AQcD in the full theory and its value, A'QcD, in the effective theory is: A~)cD = ( m J Aocu ~Jt 2 / 2 9 A~'~CD.Since MN ~ A beD, we have that as m ~ oo, m~ OMN/Om~-, ~-~M N . ) We view 350 MeV as a very large value for m~ OMN/Oms because it does not lie between these two extremes. We are interested in studying the zero charge ground state at fixed baryon density d. At low baryon densities this consists of spin paired neutrons filled up to a Fermi m o m e n t u m pv= (3~r2d) 1/3 and vanishing expectation values for the meson fields. If at some higher critical baryon density the ground state changes in a smooth way to one where meson fields acquire expectation values, then it is possible to study the onset of this transition by treating the meson field expectation values as small quantities. Expanding the ground state energy to quadratic order in the meson field expectation values, we perform a stability analysis, finding the critical density above which a non-zero expectation value is energetically favored. This procedure identifies the mode that first goes unstable with increasing density. However, to determine properties of the actual condensed ground state one must include higher powers of the meson fields. The zero charge ground state is found by solving for the baryon density d g r o u n d state of H--H+/~Q,

(10)

157

Volume 273. number 1,2

PHYSICS LETTERS B

12 December 1991

where H is the h a m i l t o n i a n corresponding to lagrangian ( 1 ) and Q is the electromagnetic charge. The chemical p o t e n t i a l / t is chosen so that the energy o f this state, E, satisfies

OE 0 31z

(II)

where/~ is viewed as a function of the expectation values of the meson fields and their canonical momenta and of the chemical potential ~t. The energy E is m i n i m i z e d with respect to the expectation values o f the fields and their canonical m o m e n t a and stationary with respect to/z. This is the same procedure as was previously used ~ to consider charged pion condensation {7,8] and so we shall just sketch the derivation of E. M i n i m i z a t i o n with respect to the canonical m o m e n t u m expectation values if trivial. Essentially it sets

( 3 ) ) = i g [ O , ( M ) ],

(2=

-~

.

0

12a,b)

l

For densities less than about five times nuclear density there is no instability in the expectation values of the 7r° and ~/fields. The ground state expectation values of the 7r , K ° and K + fields are written as (~z-) =exp(-iltt)

exp(ik.x)t,,:,

( K °) =exp(iq'x)VKO,


(13a,b,c)

In the presence o f these expectation values the lowest mass baryon states are i(F+D)

IX(p,s) >=ln(p,s) > + - -flt +

i3 3F+D (a'q)VKo]A(p,s) ) (a'k)GlP(P's))+f2xfl6--- (a2-Za,)m~

i(D-F) (a.q)~,KolSO(p,s)) + f 2xf2a: m~

i(D-F)

(a'l)t'K+lS

(p,s)).

(14)

j[ 2a2 m~ - ~ ]

The mass of these states is

Ins ln~ :14x--MN=-~2(IGI2--II'K+I2)+lL'K+I2(4a3+2a2)~- +ll:K°12(4a3+2a2+2al) f2 ( 3F+D)2

-4(a~-2a,)m~f" _

q2

Ir'K°12

. ~

(D-F)2 ]2 . q21Z,KO]2-- , ( D - F ) 2 r , 12lVK+ 4a2 m s f 2 ( za2 Ins --/I ).J-

T h e ground state at baryon n u m b e r density d is o b t a i n e d by filling s = + ½ states

( F + D ) 2 k2 l{f2

I t'~[2

(15)

IX(p, s) ) up to the Fermi

m o m e n t u m pv = (392d) ~/3. It has an energy per unit volume E / { ' = i n K ( It;KO 12+ IL'K+ 12)+k2iV,~12+q2[t'K o 12"[-12]UK + 12--]d2( I/'K + I 2q- It'll 2)+d~'llx •

(16)

If the up and down quark masses were included, E / / / w o u l d also contain a term m~ I~'~12. When the zero charge constraint, 3E/O~t = 0, has been implemented, the "energy density" E / V in eq. (16) is the same as the true energy density El V. Since the coefficient of ] t'KO I2 is It independent, it is straightforward to show that the m i n i m u m energy E occurs at either q2 = 0 or q2 = oo depending on whether the density is less than or greater than the critical density

t"(

(3F+D)Z

dK°=--\4(a,m-2a~l)nT~ +

(D-F)2~ l 4a2nl~ /

~ Our analysis follows closely that ofref. [7 ]. 158

(17a) "

Volume 273, number 1,2

PHYSICS LETTERS B

12 December 1991

With the values o f the parameters in the chiral lagrangian m e n t i o n e d previously dK,,--~3.2dnu,, where dnu~~ - lf2m~ is the n u c l e a r density. For d dKo the m i n i m u m energy occurs at q2 = ~ a n d the coefficient of [ VKO[ 2 in E is negative. W h e n d = ( 1 + e)dKo (where e is a small positive q u a n t i t y ) the coefficient o f IVKO[2 in E is negative only for q2> [m K + (dKo/.1"2) (4a~ + 2a2 + 2a~ )ln~ ] /e. T h i s makes c o n c l u s i o n s a b o u t the sign o f the coefficient of [VKO]2, for densities n e a r dKo, particularly sensitive to higher derivative terms in the chiral lagrangrian for m e s o n b a r y o n interactions. At q2=O the coefficient of ]VKo [2 in E is negative for densities greater t h a n

(

+

d' - -+ -m~: K o - - f (4a~+~a2~-2al)trl~

)"

(17b)

For a3m~ = - 310 MeV, dko -~ 3.2dn~¢. However, were [a3[ s o m e w h a t smaller, dko would be larger t h a n dKo. Next we c o n s i d e r the d e p e n d e n c e of the energy E on ] t,= [ e a n d ] z,K+ [ 2. Since the coefficients of these q u a n tities d e p e n d on ~ we find the values o f k e a n d ! 2 that m i n i m i z e E (at fixed Iv~l a n d IrK+ [) subject to the c o n s t r a i n t 3 E / 0 l l = 0. T h e m i n i m a with respect to the m o m e n t a m a y occur at zero, infinity, or an i n t e r m e d i a t e s t a t i o n a r y point. Hence, there are several cases. W h i c h case gives the lowest critical density d e p e n d s on the n u m e r i c a l values of the p h e n o m e n o l o g i c a l p a r a m e t e r s ,3. For reasonable values, the first t r a n s i t i o n occurs with ,u=(Fq-D)2¢,

k2=(J2_)e(Fq-D)e{[2(F+D)2-I]+[2(F+D)2+I]lVK+/I;~I2],.

12=0.

(18a,b,c)

The d e p e n d e n c e o f E o n IrK+ t a n d Iv~[ is

+lvK+l

-+ m K - ( F + D )

[(F+D)-'+I]+

e

~2 ( 4 a 3 + 2 a 2 ) 1 n ~

.

(19)

In eq. ( 19 ) we have i n c l u d e d the c o n t r i b u t i o n m~ [v~ [2 to E~ I ' c o m i n g from up a n d d o w n q u a r k masses. As has previously been n o t e d for d g r e a t e r t h a n [ 7 ]

G =f2 ( F + D ) [ ( Fm~ + D ) 2 - 1 ]'/++ "

(20)

the coefficient o f ] v~ [ 2 in E b e c o m e s negative (gA = F + D). Numerically, G-~ 2. I d ...... Since ( F + D ) -~ 1.25 is close to u n i t y this c o n c l u s i o n is very sensitive to the form o r E in eq. ( 1 6 ) . We shall discuss corrections to eq. ( 16 ) towards the end of this paper. For d greater t h a n

f2I~ls(2a3 +a2) dK+ = - - ( F + D ) 2 [ 1 + ( F - D )

e]

(,/1

+

_ m;?(2a3 + a ~ ) e

,)

,

the coefficient of IrK+ I 2 in the energy b e c o m e s negative. The value ofdK+ d e p e n d s on a3, b u t this is not a very sensitive d e p e n d e n c e . F o r a3m+ = - 3 1 0 MeV, dk+ -~ 2.2 d ~ . while for "~ a3m+ = - 140 MeV, dK+ --~3.1 d.oc. This lack of sensitivity occurs because in the chiral l i m i t / n ~ 0 , dK+ ~ f 2 m K / ( F + D ) [ 1 + ( F + D ) ~] ,/2 i n d e p e n d e n t o f the value o f a3. A s s u m i n g that corrections to eq. ( 16 ) keep the coefficient of [ v= [ 2 positive (for d~< dK+ ), o u r analysis indicates that the charged kaon field acquires a v a c u u m expectation value at the critical density dK+. Because o f the difference in sign b e t w e e n the terms (,u/f e) [ G [ -~a n d - (p/j.e) [t'K+ [ e in 3Ix, charged kaon c o n d e n s a t i o n is not ~2 Here we assume dKo/j2~< --~nK/(4a3+2a2+2a~ )m~.This is true throughout most of the range of reasonable values for a 3. ,3 For example, for densities greater than dk+ =f2lmK/14a3+2a2[ms} at ke=12=O the energy Egoes negative when [VK+] = IV=140. For a31ns=-- 310 MeV, dk+ - 3.5 d..... which is greater than dK+ in eq. ( 21 ). ~4 a3m~= -- 140 MeV corresponds to a small value for ms OMN/Oms. 159

Volume 273, number 1,2

PHYSICS LETTERS B

12 December 1991

nearly as sensitive to corrections to/2 as charged pion condensation is. For example, reducing the P-wave p i o n nucleon attraction [9] can remove the charged pion instability while only increasing somewhat the critical density for charged kaon condensation. When the density exceeds dK+ (assuming the coefficient of It'= I -~in E is positive) our analysis indicates that the energy is minimized by having I v~ I zero, and k 2 infinite (see eq. (18b) ). O f course higher derivative operators become i m p o r t a n t for large k e, and it seems reasonable to imagine that k 2 remains finite and I z,'=l non-zero. X is a linear c o m b i n a t i o n of the neutron and the proton. The electromagnetic charge o f the baryons compensates that o f the mesons giving a ground state with zero charge but non-zero strangeness. At a density d~o the coefficient o f I ~'KOI 2 in E goes negative. Since this density is greater than dK+ (for most of the range o f reasonable values of a3) we cannot properly conclude that the neutral kaon field acquires a ground state expectation value. For densities somewhat greater than dK+, we should include the interactions which stabilize the value of t'K+ before exploring further instabilities. (Recall, however, that higher derivative terms in the chiral lagrangian will affect somewhat both dK0 and dK+ and may alter this conclusion. ) O u r conclusions are similar to those o f K a p l a n and Nelson who previously examined the possibility of a charged kaon condensate using numerical methods. The approach presented here has the limitation o f only being sensitive to phase transitions that are smooth, but it has the advantage of being tractable analytically ~5. Also we note that K a p l a n and Nelson make an ansatz concerning the spatial dependence of the meson field expectation values which sets q = l + k , while we are able to allow a more general spatial dependence. Corrections to the expressions for d=, dK+ and dKo come from higher order terms in the chiral lagrangian and from F e y n m a n diagrams involving virtual mesons. The critical densities d=, dK+ and dKo vanish in the chiral limit where the light up, down and strange quark masses go to zero. Naively one might hope that our expressions for these densities become increasingly more accurate as the light quark masses go to zero. Unfortunately, we do not find this to be the case. Consider first the correction to/2 involving virtual pions coming from the F e y n m a n diagrams in fig. 1. Their contribution t o / 2 is d e t e r m i n e d by evaluating these diagrams at k°=/z, and we find that they give a contribution (~-]+1) ,_

(F+D)4k2(d) 2 ~_ f 2 Iv~l ~+...

(22)

to/2/V. The ellipses denote terms that become unimportant in the chira] limit ~6. THe three terms in the parentheses of eq. (22) denote respectively the contributions of figs. l a, l b and l c. Note that the additional factor of (dZf 2) in the numerator is compensated by an additional factor o f y in the denominator so this term does not diminish in importance, compared with the term proportional to k 2 in eq. ( 16 ). The factor of/~ in the denominator arises because of an infrared divergence that occurs since it costs no energy to excite a neutron hole-proton pair. Higher order terms in the chiral lagrangian, that were omitted from eq. ( 1 ), also contribute in the chiral limit. Consider, for example, adding the term 1

~5 T r ( B * B ) T r ( B * B )

(23)

to the lagrangian ( 1 ). Naturalness arguments for a strongly coupled theory suggest that A ~ f [ 3, l 1 ]. Evaluating the F e y n m a n diagram in fig. 2, at k°=/~, we find that the operator in eq. (23 give the contribution

k2(F+D)2( d) 2 f2 #5

\~5]

~5_ Iv~l 2

(24)

to F2/k: Again the additional factor o f ( d / [ 2 ) in the n u m e r a t o r is a c c o m p a n i e d by an additional factor of/L in ~5 For a qualitative discussion ofkaon condensation using analytic methods see ref. [ 10]. ~6 We take it of order d/[ "2and k 2of order (d/f 2)2 as is indicated by' eqs. ( 18 ). Eq. ( 21 ) implies that near the critical density for charged kaon condensation (d/f 2) is of order mR, which goes to zero as ms~0. It is anausing to note that in the chiral limit dKo<< dK+ since dKo vanishes as ms while NK+ vanishes as \.'J~ s 160

Volume 273, number 1,2

PHYSICS LETTERS B

JI-

12 December 1991

JI"

(a) Ft

n

J'c

~

p

Ji

p (b)

p

---,c--

R

F1

n

.(---

p

n

(c) Fig. 1. Feynman diagrams responsible for the correction to eq. (16) given in eq. (22).

p

p

Fig. 2. Feynman diagram responsible for the correction to eq. ( 16 ) given in eq. (24). Here the shaded square denotes an insertion of the cperator in eq. (23).

t h e d e n o m i n a t o r so t h i s c o r r e c t i o n to E / I " d o e s n o t d i m i n i s h m i m p o r t a n c e as t h e light q u a r k m a s s e s (,and c o n s e q u e n t l y d ) go to zero. T h e v i r t u a l m e s o n c o r r e c 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 in eq. ( 2 2 ) a n d c o n t r i b u t i o n o f a h i g h e r o r d e r t e r m in t h e c h i r a l g i v e n in eq. ( 2 4 ) are n o t s m a l l c o m p a r e d w i t h w h a t was p r e s e n t e d in eq. ( 1 6 ) . T h a t t h e r e ' a r e v a r i o u s i m p o r t a n t a d d i t i o n s to eq. ( 16 ) h a s b e e n e x t e n s i v e l y d i s c u s s e d in t h e c o n t e x t o f c h a r g e d p i o n c o n d e n s a t i o n ~7. H e r e we h a v e e m p h a s i z e d t h a t s o m e o f t h e s e c o r r e c t i o n s d o n o t d i m i n i s h in i m p o r t a n c e as t h e light q u a r k m a s s e s go to zero. E v e n t h o u g h c h a r g e d k a o n c o n d e n s a t i o n is m u c h less s e n s i t i v e to c o r r e c t i o n s to eq. (16) than charged pion condensation, any serious conclusions concerning charged kaon condensation must a w a i t t h e s y s t e m a t i c i n c l u s i o n o f t h o s e c o r r e c t i o n s t h a t are i m p o r t a n t in t h e c h i r a l limit. W o r k o n t h i s is in progress.

~7 Fora review see refs. [7,12].

References [1 ] D.B. Kaplan and A.E. Nelson, Phys. Lett. B 175 (1986) 57. [2] D.B. Kaplan and A.E. Nelson, Talk presented at the 1987 Intern. Syrup. on Strangeness in hadronic matter (Bad Honnef, FRG), Nucl. Phys. A 479 (1988) 273c. [ 3 ] H. Georgi, Weak interactions and modern particle theory ( Benjamin/Cummings, Menlo Park, CA, 1984 ). [4] E. Jenkins and A. Manohar, Phys. Lett. B 255 ( 1991 ) 562; H. Georgi, Phys. Lett. B 240 (1990) 447: E. Eichten and B. Hill, Phys. Lett. B 234 (1990) 511. [ 5 ] J. Gasser, H. Leutwyler and M.E. Sainio, Phys. Left. B 253 ( 1991 ) 252. 161

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[6] J.F. Donoghue and C.R. Nappi, Phys. Lett. B 168 (1986) 105. [7] G. Baym and D.K. Campbell, in: Mesons in nuclei, Vol. 3, eds. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979 ). [8] A.B. Migdal. Zh. Eksp. Theor. Fiz. 61 (1971) 2210 [Soy. Phys. JETP 36 (1973) 1052]: Phys. Rev. Left. 3l (1973) 247: R.F. Sawyer, Phys. Rev. Lett. 29 (1972) 382: D.J. Scalapino, Phys. Rev. Left. 29 (1972) 386. [9] M.O. Ericson and T.E.O. Ericson, Ann. Phys. 36 (1966) 323: G. Baym and G.E. Brown, Nucl. Phys. A 247 (1975) 395: G.E. Brown, S.-O. B~ickman, E. Oset and W. Weise, Nucl. Phys. A 286 ( 1977 ) 191 : G. Baym and E. Flowers, Nucl. Phys. A 222 (1974) 29: G. Baym, D. Campbell, R. Dashen and J.T. Manassah, Phys. Lett. B 58 ( 1975 ) 304. [10] G.E. Brown, K. Kubodera and M. Rho, Phys. Lett. B 192 (1987) 273. [ 1 I ] A. Manohar and H. Georgi, Nucl. Phys. B 234 ( 1984 ) 189. [ 12] S.-O. B/ickman and W. Weise, in: Mesons in nuclei, Vol. 3, eds. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979).

162