an effective chiral lagrangian approach to kaon-nuclear interactions kaonic atom and kaon...

ELSEVIER Nuclear Physics A 585 (1995) 401-449

N U C L E A R P H Y S I C S A

An effective chiral lagrangian approach to kaon-nuclear interactions.

Kaonic atom and kaon condensation

Chang-Hwan Lee a, G.E. Brown b, Dong-Pil Min a, Mannque Rho c a Department of Physics and Center for Theoretical Physics, Seoul National University, Seoul 151-742,

South Korea b Department of Physics, State University of New York, Stony Brook, NY 11794, USA

c Service de Physique Th~orique, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France

Received 24 June 1994; revised 10 November 1994

Abstract

On-shell kaon-nucleon scattering, kaonic atom and kaon condensation are treated on the same footing by means of a chiral perturbation expansion to the next-to-next-to-leading order ("N2LO"). Constraining the low-energy constants in the chiral lagrangian by on-shell KN scattering lengths and kaonic atom data, the off-shell s-wave scattering amplitude up to one-loop order corresponding to N2LO and the critical density of kaon condensation up to in-medium two-loop order are computed. The effects on kaon-proton scattering of the quasi-bound A(1405) and on kaonic atoms and kaon condensation of A( 1405)-proton-hole excitations through four-fermi interactions are studied to all orders in density within the in-medium two-loop approximation. It is found that the four-fermi interaction terms in the chiral lagrangian play an essential role in providing attraction for kaonic atoms, thereby inducing condensation but the critical density is remarkably insensitive to the strength of the four-fermi interaction that figures in kaonic atoms. The prediction for the critical density is quite robust against changes of the parameters in the chiral lagrangian, and gives - for "natural" values of the four-fermi interactions - a rather low critical density, Pc < 4po. When the BR scaling is suitably implemented, the condensation sets in at pc ~-- 2po with loop corrections and four-fermi interactions playing a minor role.

1. Introduction

In a series of recent short papers [ 1-3 ], we have discussed kaon-nucleon and kaon-

nuclear interactions in terms of a chiral perturbation expansion with the objective to

0375-9474/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 037 5 -9474 ( 94 ) 0062 3-7

402 C.-H. Lee et al./Nuclear Physics A 585 (1995) 401-449

predict within the framework of chiral effective lagrangians the onset of kaon condensation in dense hadronic matter relevant to compact stars that are formed from the gravitational collapse of massive stars. This research was given a stronger impetus by a recent suggestion of Brown and Bethe [4] that if kaon condensates develop at a matter density p < 4p0 (where p0 ~ 0.16//fm3 is the normal nuclear matter density) in the collapse of large stars, then low-mass black holes are highly likely to form in place of neutron stars of mass greater than 1.5 times the solar mass M e. The purpose of this paper is to provide the details of our previous publications, such as the approximations made, inherent uncertainties involved etc. with complete and explicit formulae that we were unable to supply in the letter papers because of the space limitation. In addition, some additional justifications that have in the meantime been uncovered and understood

better will be discussed. Ever since the first paper of Kaplan and Nelson [5], there have been numerous

investigations on kaon condensation in dense neutron-star matter as well as in nuclear matter based both on effective chiral lagrangians [ 1,6-9] and on phenomenological off- shell meson-nucleon interactions [ 10,11 ]. The two ways of addressing the problem gave conflicting results with the chiral lagrangian approaches generally predicting a relatively low critical density, Pe ~(2-4)p0 , while the phenomenological approaches based more or less on experimental inputs giving results that tend to exclude condensation at a sufficiently low density to make it relevant in the collapse process.

We should stress that it is not our purpose here to clarify what makes the phenomenological approach differ from the chiral lagrangian approach. Instead, our objective in this paper is to do as consistent and systematic a calculation as possible in the context of the chiral perturbation theory. This effort is largely motivated from the strong-interaction point of view by the recent success in confronting, in terms of chiral dynamics, the classic nuclear physics problems such as nuclear forces [12,13] and exchange currents [ 14-16]. This paper addresses the problem of applying chiral perturbation theory to multi-hadron systems that contain the strange-quark flavor. While the standard problems of nuclear physics such as nuclear forces and exchange currents involve the chiral quarks u and d for which the mass scale involved is small compared with the typical QCD chiral symmetry breaking scale A x ,-~ 1 GeV so that a simultaneous expansion in derivatives and quark mass matrix is justified, here the strange-quark mass which is not small renders the expansion in the quark mass a lot more delicate and hence the low-order expansion highly problematic. This caveat has to be kept in mind in assessing the validity of the procedure we will adopt.

Another problem of potential importance is that both kaonic atom and kaon condensation introduce an additional scale, namely the matter density p or more precisely the Fermi momentum kF. So far in low-order chiral perturbation calculations, the result de- pended on the order of density dependence included in the calculation. In fact, one of the important differences between the chiral lagrangian approach and the phenomenological model approach arose at the order p2. It is thus clear that one has to be consistent in the chiral counting, not only with respect to the usual expansion parameters practiced in free space, but also with respect to the density expansion. This then raises the question

C.-H. Lee et al./Nuclear Physics A 585 (1995) 401--449 403

of how to treat the dynamics involved in the nonstrange sector as well as in the strange

sector. So far the dynamics in the nonstrange sector is assumed to be given by what

we know from nuclear phenomenology that is mostly given in terms of meson-theoretic approaches combined with many-body techniques, and perturbations in the strange di-

rection are treated in terms of chiral lagrangian at tree order or at most one-loop order. The problem with this is that there is no consistency between the two sectors as regards chiral symmetry and other constraints of QCD. Indeed so far nobody has been able to

describe correctly nuclear ground state (including nuclear matter) starting from chiral lagrangians, so one is justified to wonder how a low-order chirai lagrangian calculation of kaon condensation without regard to the normal matter can be trusted. We cannot

offer a solution to this problem here but we will make an effort to point out the salient

points that are closely related to this issue.

Following the recent development of nuclear chiral dynamics [ 12-16], we incorporate

spontaneously broken chiral symmetry using the Jenkins-Manohar heavy-baryon chiral

lagrangian [ 17] as extended in Ref. [2] to O(Q 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 is because as is well known, the A(1405) (which we denote A* for short) influences strongly the amplitude of the K - p scattering near threshold and hence kaon-nuclear interactions in kaonic

atom [ 18 ] and kaon condensation involving protons as in "nuclear stars". We introduce this state as an elementary field as discussed in Ref. [2]. The reason for this is that

first of all, the A* is a bound state and hence cannot be described by a finite chiral perturbation expansion and secondly in the Callan-Klebanov skyrmion description [ 19],

it is a configuration of a K - wrapped by an SU(2) soliton and hence is as "elementary"

as the even-parity A(1115) of the octet baryon. In addition to these terms operating in the single-baryon sector, we need terms

that involve multi-baryon fields in the lagrangian for describing many-body systems. There have been discussions of four-fermi interaction terms in nonstrange sectors [12,13,15,20]. We find that in the s-wave kaon-nuclear sector, two such four-fermi interaction terms involving A* can intervene. In p-wave kaon-nuclear interactions, there can be more four-fermi interactions as they can involve the entire battery of the octet

and decuplet but we will not be concerned with them in this paper. By a straightforward extension of an amplitude whose parameters are fixed by on-

shell kaon-nucleon scattering, we are able to almost (but not quite) uniquely predict an

off-shell kaon-nucleon amplitude relevant for kaonic-atom as well as kaon-condensation phenomena. The predicted off-shell amplitude was found to be in fair agreement with the phenomenological fit [21]. This off-shell amplitude provides the kaon self-energy in linear density approximation, equivalent to the usual optical potential approximation. The critical density obtained in this approximation is a bit higher than that obtained before at tree order but still in the regime quite relevant to the stellar collapse. If one goes beyond the linear density approximation which would be required in a simultaneous expansion in all the scales involved, four-fermion interactions come into play. For the

404 C-H. Lee et al./Nuclear Physics A 585 (1995) 401-449

s-wave kaon condensation process, there are two independent four-fermi interactions with arbitrary constants. In order to fix these parameters, we appeal to the recent data on kaonic atoms [ 18]. We are able to fix unambiguously only one of the two constants with the presently available data. While the other constant remains free, its sign and magnitude can, however, be constrained by a "naturalness" condition and furthermore the physical quantities that we are interested in turn out to be rather insensitive to the free parameter.

The four-fermi interactions - which are higher order in density - play an important role for giving rise to an attraction for kaonic atoms. This attraction certainly comes in for pushing the system toward condensation. However, they remain "irrelevant" and become suppressed at the kinematic regime in which condensation occurs. As a consequence, their influence on the critical density is quite weak: The strength of the four-fermi interactions, which cannot be pinned down precisely at present, does not figure importantly in the condensation phenomena.

We do not consider six-fermi and higher-fermi interactions as they would be further suppressed by the scale set by the chiral symmetry breaking scale A x ,~ 1 GeV.

The paper is organized as follows. The effective chiral lagrangian to O(Q 3) in the chiral counting, consisting of the octet pseudo-Goldstone bosons and the octet and decuplet baryons that figure in our calculation, is given in Section 2. In Section 3, we calculate to one-loop order, corresponding to N2LO, both on-shell and off-shell KN scattering amplitudes. Some issues regarding Adler's soft-meson conditions in chiral perturbation theory are also discussed. The kaonic atom is treated in Section 4 and kaon condensation in Sections 5 and 6. In Section 7, we mention some of the unsolved open issues in the problem. Detailed formulas are collected in the appendices.

2. Effective chiral lagrangian

We start by writing down the effective chiral lagrangian that we shall use in the calculation. Let the characteristic momentum-energy scale that we are interested in be denoted Q. The standard chiral counting orders the physical amplitude as a power series in Q, say, Q~, with z, an integer. To leading order, the kaon-nucleon amplitude T grq goes as O(QI) , to next order as O(Q 2) involving no loops and to next-to-next order (i.e., N2LO) at which one-loop graphs enter as O(Q3). Following Jenkins and Manohar [ 17], we denote the velocity-dependent octet baryon fields By, the octet meson fields exp( icraTa/ f ) - ( , the velocity-dependent decuplet baryon fields T~, the velocity four-vector o~, and the spin operator S~ (o . So = 0, ~ = _ 3 ) , the vector current V~ = ½ [ ( t , 0~,(] and the axial-vector current A~ = ½i{~ :t, a~:}, and write the lagrangian density to order Q3, relevant for the low-energy s-wave scattering I , as

L ( l ) = Tr By(iv. 7~)Bv + 2 D Tr-BvS~{A~,Bo} + 2 F T r - B v S ~ [ A ~ , , B v ]

l The relevant terms in component fields useful for calculating Feynrnan diagrams are given in Appendix A.


--~ " C(~AjzBv-F-BvA~Tv~) + 2~T v (Sv" A)To,~, (1) - T o ( w . D - g~r)Tv,~ + -~

/:(2) = al Tr BvX+Bv + a2 Tr -BvBvX+ + a3 Tr -BvBv Tr X+

+dl Tr-B~A2Bv + d 2 Tr Bv(v . A)2Bv +d3 Tr BvB~A 2

+d4 Tr B--'vBv(v. A) 2 + d5 Tr BvB~ Tr A 2 + d6 Tr BvB~ Tr (v . A) 2

+d7 Tr BvA~, Tr BoA ~ + d8 Tr Bv(v . A) Tr Bo(v. A), (2)

~<3) =Cl Tr -~v~iv. Z~)38v + gl Tr-~A,~(iv. ~)A"8~ + 82 Tr 8oA,~(iv. ~)A'~-~,,

+g3 Tr B--vv. A(iv. ~))v. ABv + g4 Tr Bvv. A(iv. D ) v . A-By

+g5 Tr-B~A~ Tr (iv. D)A~B~ - Tr-BvA~(iv. ~D) Tr A~Bv

+86 (Tr eo . a Tr 8v(i . - Tr A Tr A)

+g7 Tr By[v-A , [iD ~ ,A~l lBv + g8 Tr Bv[v. A, [iD ~,Aul]Bv

+hi Tr Bvx+(iv . D)Bv + h2 Tr By(iv. D)Bvx+

÷h3 Tr B~(iv .D)Bv T r x + + l l Tr B v [ x _ , v . A]B~

+12 Tr BvB~[x_ ,v . A] +/3[ Tr BvX- , Tr Bvv. A], (3)

where the covariant derivative D~ for baryon fields is defined by

~D~B~, = O~Bv + [ V~, B~ 1,

Z)l~Tv,abc_ ~ d ~ d 1, d --O,T~,ab c + (V~)aT~,db c + (V~)bT~,ad c + (V~)cT:.ab d, (4)

8r is the SU(3)-invariant decuplet-octet mass difference, and

X ~ - - ~-A'/~:-F~:t.A'I~ :t, (5)

with .A//= diag (mu, md, ms) the quark mass matrix that breaks chiral symmetry explicitly, There are many other terms involving the decuplet that one can write down but we

have written only those that enter in the calculation. Among the many parameters that figure in the lagrangian, a few can be fixed right away. For instance, we will simply fix the constants F and D at tree order since to O(Q 3) that we will be interested in, they are not modified. We shall use D = 0.81 and F --- 0.44. The constant C can also be

fixed at this stage from the decay process A(1230) ~ NTr. We shall use ICI2(~ 2.58). Of course, the flavor SU(3) can be substantially broken as we will discuss later, so one cannot take this value too seriously. The determination of all other constants ai . . . . . l i

(or more precisely the combinations thereof) will be described below. The number of parameters that seem to enter may appear daunting to some readers

but the situation turns out to be much simpler than what it looks. As we will see later, once the constants are grouped into an appropriate form, there remain only four parameters for on-shell K±N amplitudes. These parameters can be fixed on-shell by the four s-wave scattering lengths. Off-shell, however, one parameter remains free but the off-shell amplitude turns out to be rather insensitive to the one free parameter. This drastic simplification can be understood easily as follows. First of all, the heavy-fermion


- " 3 " " "

(~) (b)

~ s ~ ~ ~ P

?-_, -=, / f %

(c) (d)

, , - :.-

(~-l) (~-2)

(f-l) (f-2)

Fig. 1. One-loop Feynman diagrams contributing to K+N scattering: The solid line represents baryons (nucleon for the external and octet and decuplet baryons for the internal line) and the broken line pseudo-Goldstone bosons (K + for the external and K, ~r and r /for the internal line). There are in total thirteen diagrams at one loop, but for s-wave KN scattering, for reasons described in the text, we are left with only six topologically distinct one-loop diagrams.

formalism (in short HFF) makes those subleading terms (i.e., terms with v /> 2) involving the spin operator S/~ vanish, since they are proportional to S. q, S . q', or S. qS. q', all of which are identically zero. As a consequence, there are no contributions to the s-wave meson-nucleon scattering amplitude from one-loop diagrams in which the external meson lines couple to baryon lines through the axial vector currents. This leaves only six topologically distinct one-loop diagrams, Fig. 1 (out of thirteen in all), to calculate for the s-wave meson-nucleon scattering, apart from the usual radiative corrections in external lines. Since we are working to O(Q3), only /~(l) enters into the loop calculation. Loops involving other terms can contribute at O(Q 4) or higher. The next term £(2) contributes terms at order v = 2, that is, at tree order. These will be determined by the KN sigma term and terms that could be calculated by resonance saturation. There are some uncertainties here as we shall point out later, but they turn out to be quite insignificant in the results. The next terms in £(3) remove the divergences in the one-loop contributions and involve two finite counter terms - made up of two

C.-H. Lee et al . /Nuclear Physics A 585 (1995) 401-449 407

linear combinations of the many parameters appearing in the lagrangian - that are to be determined empirically. As we will mention later, these constants are determined solely by isospin-odd amplitudes, the loop contribution to isospin-even amplitudes being free of divergences.

3. KN scattering amplitudes

3.1. On-shell amplitudes

The complete on-shell s-wave KN scattering amplitudes calculated to N2LO (O(Q3)) [ 2 ] read

Kip mB [q=MK+(-~s.q_-~v)M2 +{(Ls+Lv)+(~s+_~v)}M3K] a° = 47rf 2 (rob + MK)

K:t:p +t)aA, ,

a~:n = mB [q:½MK + (ds - dv)M 2 + {(Ls - Lv) -4- (gs - gv)} M3] 4~ ' f 2 ( ms + MK)

(6)

where ds is the t-channel isoscalar contribution of O(Q2), and dv is the t-channel isovector one of O(Q 2) :

1 ds = -~--~ (al + 2a2 + 4a3) + l (d l + d2 + d7 + d8) + ½(d3 + d4) -a t- d5 + d6,

1 dv = - :--:-~ a~ + ¼ ( d l + d2 + d7 + d8) (7)

z/~0

with Bo = M~/(ff~ + ms) where MK is the kaon mass and ~ = ½(mu + md). Here 8K+p aa. is the contribution from the A* to be specified below and Ls (Lv) is the finite crossing-even t-channel isoscalar (isovector) one-loop contribution

-0.109 fm,

1 ( - ½ ( D + F ) ( D -3F)(M~+3M2n)(M,r+M,1) LvMK = 1287rfZM ~

2 2 -3MK~/M ~ M 2 - l (D + F ) ( D - 3F)(M2 + 3M2)(M2 + M 2)

1

0 ~ / ( 1 - x)M~ + xM2 n

+0.021 fm, (8)

408 C.-H. Lee et al./Nuclear Physics A 585 (1995) 401--449

where f = 93 MeV and physical masses are used to obtain the numbers. The quantity

gs (gv) is the crossing-odd t-channel isoscalar ( isovector) contribution from one-loop

plus counter terms which after the dimensional regularization specified in Appendix C,

takes the form

gs,v = cr~,v + fl~,v + Ys,v + ~ ~,v In (9) i=¢r,K,7/

where /Z is the arbitrary scale parameter that enters in the dimensional regularization

and cr and fl are contributions from the counter terms in O ( Q 3) 2, and y and d/are

finite loop contributions. The explicit forms o f at, fl, y and ~5 are given in Appendix C.

It should be noted that while a and fl are /z-dependent, ~ is scale independent. ( y

and 8 are scale-independent numbers.) Thus if one fixes ~ from experiments, then for

a given /z, one can fix a + fl at a fixed /z. Equivalently, we can separate out the

specific /z-dependent terms so as to cancel the ln /z term in Eq. (9 ) , thereby defining

/z-independent constants a I and fll

1 1 M 2 a~,v + fl~,v = a's,v + fl~,v 327r 2 f 2 M 2 ~ t~v In (10)

i=Ir, K,v 1 " - ' ~

and determine Otis, v and fl~,v from experiments. From now on when we go off-shell (i.e.,

in Appendix E) , we will drop the primes understanding that we are dealing with the

/z-independent parameters. (On-shell , this subtlety is not relevant since we can work

directly with ~ of Eq. ( 9 ) . )

To understand the role o f the A*, we observe that the measured scattering lengths are

repulsive in all channels except K - n [22,23] 3 :

a K÷p = - 0 . 3 1 fm, a0 K-p = - 0 . 6 7 + i0.63 fm

a0 K+n = - 0 . 2 0 fm, a0 ~-n = +0 .37 + i0.57 fm. (11)

The repulsion in K - p scattering cannot be explained from Eq. (6) without the A*

contribution. In fact it is well known that the contribution of the A(1405) bound state

gives the repulsion required to fit empirical data for s-wave K - p scattering [2,24]. As

mentioned, we may introduce the A* as an elementary field. To the leading order in the

chiral counting, it takes the form

2 On-shell, one can combine ot and fl into one set of parameter to be determined from experiments. Off-shell, however, they are multiplied by a different power of the frequency to as indicated in Appendix E and hence represent two independent parameters. This introduces one unfixed parameter in the off-shell case. However, it turns out that the off-shell amplitudes are rather insensitive to the precise values of these constants, so we set (somewhat arbitrarily) ar,v ~ fl~,v in our calculation, Fig. 2.

3 Although the experimental K-N scattering lengths are given with error bars, the available K+N data are not very well determined. Since both are used in fitting the parameters of the lagrangian, we do not quote the error bars here and shall not use them for fine-tuning. For our purpose, we do not need great precision in the data as the results are extremely robust against changes in the parameters.

C.-H. Lee et a l . / N u c l e a r Physics A 585 (1995) 401 -449 409

F~A,=-~vv(iv.O--ma,+mB)av+(x/2gz, Tr(- '~v .aBv)+h.c . ) . (12)

The coupling constant gA* can be fixed by the decay width A* ~ ~ r [2] if one ignores SU(3) breaking

g2a.(pK-) ~ g2a.(2cr) ~ 0.15. (13)

This is what one would expect at tree order. If one wants to go to one-loop order [25] corresponding to O(Q 3) at which SU(3) breaking enters, then we encounter two counter terms h~, 2,

* -"* • h~x/2A-~v Tr • A) + (14) /~,=3 = hlx/~A v Tr (X+V ABv) + (x+Bov h.c.

and the renormalized coupling to O(Q 3) will take the form

gra* (Xqr) = gA* + ~/.w in M~..~_5_ + /3z~ + 2(h~r + h~)~ , i='n',K,'q

grA*(pK-) =gA* -1- ~ ~/pK- l n ~ +/3pK- +2(h~rms -k- h~r~), (15) i=¢r,K,r/

where a i and /3 are calculable loop contributions. In Ref. [25], Savage notes that if one ignores the counter terms, then the finite log terms would imply that gA*(pK-) would come out to be considerably smaller than ga* ( ~ ) , presumably due to an SU(3) breaking. In our approach we choose to pick the constants from experiments since we see no reason to suppose that the counter terms are zero and furthermore the presently available data [23] give

(g2a. (pK-) )exp "~ 0.25, (16)

which is bigger than the SU(3) value (13). We will take this value in our calculation. It should also be mentioned that the Callan-Klebanov skyrmion predicts a value close to (13) [26]. The A* contribution to the kaon-proton scattering amplitude is now completely determined,

K+p= mB [ g2A.M~: ] (17) aa* 4¢rf 2 ( ~ ' + MK) mB q: MK - - m a * "

To one-loop order, the A* mass picks up an imaginary part through the graph A* ---, 27r ~ A*. In our numerical work we will take ma* to be complex. The presence of the imaginary part explains that the empirical coupling constant (16) is bigger than the SU(3) value (13).

We are left with four parameters in (6), ds, dr, gs and ~v, which we can determine with the four experimental (real parts of) scattering lengths ( 11 ). The results are

ds ~ 0.201 fm, d~ ~ 0.013 fro,

~sMK ~ 0.008 fm, ~vMK ,~ 0.002 fro. (18)

410 C.-H. Lee et aL/Nuclear Physics A 585 (1995) 401--449

Table 1 Scattering lengths from three leading-order contributions for the empirical value of the constant gZa, = 0.25. Also shown is the contribution from A*

g2a,=0.25 O(Q) O(Q 2) O(Q 3) A*

a K+p (fm) --0.588 0.316 --0.114 0.076

a K-p (fm) 0.588 0.316 --0.143 --1.431 a K÷n (fm) --0.294 0.277 --0.183 0.000

a K-n (fm) 0.294 0.277 --0.201 0.000

The scattering amplitudes in each chiral order are given in Table 1. One sees that while the order-Q and order-Q 2 terms are comparable, the contribution of order Q3 is fairly

suppressed relative to them. As a whole, the subleading chiral corrections are verified

to be consistent with the "naturalness" condition as required of effective field theories.

Using other sets of values of f , D and F does not change Ls and Lv significantly and leave unaffected our main conclusion. As expected, the A* plays a predominant role in K - p scattering near threshold. This indicates that it will be essential in describing

kaon-nuclear interactions, e.g., kaonic atoms.

3.2. Off-shell amplitudes

We now turn to off-shell s-wave K - forward scattering off static nucleons. The kinematics involved are t = 0, q2 = q,2 = a~2, s = (mB +a~) 2 with an arbitrary (off-shell)

(see Appendix E).

In going off-shell, we need to separate different kinematic dependences of the constant

ds,v of Eq. (7) which consists of what would correspond to the KN sigma term 2KN at

tree order,

O'KN( "~ "~KN) -- - -½(~ + ms)(a l +2a2 +4a3 )

involving the quark mass matrix .At, and the di terms containing two time derivatives.The o'lcr~ term, which gives an attraction, is a constant independent of the kaon frequency ca, whereas the d i terms which are repulsive are proportional to ~o 2 in s-wave. In kaonic atoms and kaon condensation, the oJ-value runs down from its on-shell value MK. This means that as a~ goes down, the attraction stays unchanged and the repulsion gets suppressed. Thus while for on-shell amplitudes, they can be obtained independently of any assumptions, they need to be separated for off-shell amplitudes that we are interested in. If we could determine the KN sigma term from experiments, there would be no ambiguity. The trouble is that the sigma term extracted from experiments is not precise enough to be useful. The presently available value ranges

"~KN "~ (200-400) MeV. (19)

C.-H. Lee et al./Nuclear Physics A 585 (1995) 401-449 411

Here we choose to separate the two components by estimating the contributions to

di from the leading 1/mB corrections with the octet and decuplet intermediate states

in the relativistic Born graphs. As suggested by the authors of Ref. [27] for 7rN

scattering, we might assume that the counter terms di could equally be saturated by such intermediate states. This strategy is discussed in detail in Appendix D. This is somewhat like saturating the dimension-four counter terms Li in the chiral lagrangian by resonances, a prescription which turns out to be surprisingly successful. The reason for believing that this might be justified in the present case is that the O(Q 2) terms are not

affected by chiral loops, so must represent the degrees of freedom that are integrated out from the effective lagrangian. But there are no known mechanisms that would contribute

to di other than the baryon resonances. When computed by resonance saturation, the contributions go like 1/mB. However, this is not to be taken as 1/mB corrections that

arise as relativistic corrections to the static limit of a relativistic theory. The HFF as used

in chiral perturbation theory does not correspond merely to a nonrelativistic reduction although at low orders, they are equivalent. To be more specific, imagine starting with the following relativistic lagrangian density:

/~rel . . . . + ei Tr [-By~A~y~A~B] + . . . + f i Tr [ DIz-BA~AgD~B] + . . . .

In going to the heavy-baryon limit, we get the O(Q 2) terms of the form

(20)

f~v . . . . + di Tr [ B--vV . A 2 B . ] + . . . (21)

with di = (dUm + ei + mg f i + . . . ) where dl/m is the calculable 1~rob correction from the relativistic leading-order lagrangian. Clearly the ei and f i terms cannot be

computed [ 25 ]. Thus if one imagines that the constants di are infested with the terms of the latter form, there is no way that one can estimate these constants. While this

introduces an element of uncertainty in our calculation, it does not seriously diminish the predictivity of the theory: Much of the uncertainty are eliminated in our determination of the parameters by experimental data. From Eq. (18) and the di terms estimated in Appendix D, we can extract the parameter 4

o'r~ ~ 2.83M,~. (22)

This parameter will be used for off-shell KN scattering amplitudes and kaon self-energy.

The predicted off-shell K - p and K - n scattering amplitudes are shown in solid line in Fig. 2 for the range of x/~ from 1.3 to 1.5 GeV with g2a, = 0.25 and Fa* = 50 MeV

taken from experiments. (The dotted lines in Fig. 2 are explained in Appendix D.) The explicit formulas are listed in Appendix E. The K - n scattering is independent of the A* and so the amplitude varies smoothly over the range involved. [2] Our predicted K - p amplitude is found to be in fairly good agreement with the empirical fit of Ref. [ 21 ]. The striking feature of the real part of the K - p amplitude, repulsive above and attractive

4 This is not the sigma term 2~KN = ½ (~+ ms)(PF~u +-gslP). There are loop corrections to be added to this value.


a -N (fro)

-1

aK-n )

~ T Z e ( a K-~ )

f 1.3 1.4 1.5

v ~ (GeV)

Fig. 2. K-N amplitudes as function of v~: These figures correspond to Eqs. (23) and (24) with ~2a. = 0.25, FA* = 50 MeV and Otp,n = tip.n, fixed in the way described in the text. The first kink corresponds to the KN threshold and the second around 1.5 GeV to v G = mB + M~ for M~ ~ 547 MeV. The solid line of the real part correspond to Z = -0.5 and the dashed line correspond to Z = 0.15, and the imaginary parts are independent of the Z-value.

below ma* (1405 MeV) as observed here, and the to-independent attraction of the K - n

ampli tude are relevant to kaonic atoms [ 18] and to kaon condensation in "nuclear star"

matter. The imaginary part of the K - p ampli tude is somewhat too high compared with

the empirical fits. This may have to do with putting the experimental A* decay width

for the imaginary part of the mass. Self-consistency between loop corrections and the

imaginary part o f the mass would have to be implemented to get the correct imaginary part of the K - p amplitude.

3.3. Adler soft-meson conditions

The off-shell ampli tude calculated here does not satisfy Adler ' s soft-meson conditions

that fol low from the usual PCAC assumption that the pseudoscalar meson field ~r

interpolate as the divergence o f the axial current. The chiral lagrangian used here does

not give the direct relation 7r / ~ auJ~ u where J~U is the axial current with flavor index


i. Therefore, in the soft-meson limit which corresponds in the present case to setting o~ equal to zero, the ~rN amplitude is not given by - , ~ r ~ / f 2 as it does in the case of Adler's interpolating field [28]. In fact, it gives ,~rN/f 2 which has the opposite sign to Adler's limit. This led several authors to raise the possibility that a different physics might be involved in the chiral perturbation description of the off-shell processes that take place near w = 0 [ 10,11 ].

A simple answer to this issue is that physics should not depend upon the interpolating field for the Goldstone bosons 7r [29]. The physics is equivalent whether one uses the w-field as defined by the chiral lagrangian used here or the ~ oc O~,J~ field that gives Adler's conditions. Both are interpolating fields and they are just field redefinitions of each other. This is natural since the Goldstone boson field is an auxiliary field in QCD. An explicit illustration of the equivalence in the case considered in this paper is given in Appendix F, to which skeptics are referred. Let it suffice here to say that if one wishes, one could rewrite the effective lagrangian in such a way that Ward-Takahashi identities, to which Adler's conditions belong, are satisfied, without changing the physics involved. See Ref. [30].

4. Kaon self-energy and kaonic atom

If one were to limit oneself to linear density approximation which would be reliable in dilute systems, then what we have obtained so far is sufficient for studying kaon-nucleon interactions in many-body systems. The off-shell KN amplitude calculated to 0 ( 0 3) in impulse approximation gives the optical potential, which, expressed as a self-energy, is depicted by Fig. 3a,

u mP( ) PnTfr~ee

( ~:-p K-n (~o)) (23) = - ppTf ( , o ) + ,

where T I~ is the off-shell s-wave KN transition matrix 5. For the purpose of studying kaon condensation, the linear density approximation may not be reliable enough and one would have to study the effective action (or effective potential in translationally invariant systems).

To go beyond the linear density approximation, there are two major effects to be considered. The first is the Pauli correction and the second is many-body correlations.

The Pauli effect can be most straightforwardly taken into account in the self-energy by modifying the nucleon propagator in the loop graphs contributing to the KN scattering

amplitude to one appropriate in medium

GO(k ) ~_ i -2~rS(ko)O(kF - I k [ ) , (24) v . k + i e

where kF is the nucleon Fermi momentum related to density PN by the usual relation

5 The ampli tude 7 -r'N taken on-shell , i.e., to = MK, and the scattering length a r ~ are related by a r'~ =

{4zr( 1 + MK/mB)}--IT KN.

414 C.-H. Lee et aL/Nuclear Physics A 585 (1995) 401-449

% J

(~)

~'% J / l I

(b) (c) (d)

% J % f %%. ~, f "%% j J

(e) (f)

Fig. 3. (a): The linear density approximation to the kaon self-energy in medium, HK. The square blob represents the off-shell K-N amplitude calculated to O(Q3); (b)-(f): medium corrections to "7" KN of (a) with the free nucleon propagator indicated by a double slash replaced by an in-medium one, Eq. (24). The loop labeled pN represents the in-medium nucleon loop proportional to density, N - l the nucleon hole (n - l and/or p- t ) , the external dotted line stands for the K- and the internal dotted line for the pseudoscalar octet ~-, r/, K.

PN = ~'~2 FN

with the degeneracy factor "y = 2 for neutron and proton in nuclear matter. The resulting

correction denoted 8Tp~ -N and given explicitly in Appendix G is clearly nonlinear in

density and repulsive as befits a Pauli exclusion effect.

For the second effect, the most important one is the correlation involving "particle-

hole" excitations. This is of typically many-body nature. There are two classes of cor-

relations one would have to consider. One involves nonstrange-particle-hole excitations

and the other strange-particle-nonstrange-hole excitations. All these can be mediated by

four-fermi interactions described above.


A* A*

p-1 p -1

(~)

PN

_ A * @ _ .

p-1 (b)

N

p-1

(¢)

. . . .

p-1 p-1

(d)

Fig. 4. Two-loop diagrams involving A* contributing to the kaon self-energy. The diagrams (a) and (b) involve four-fermi interactions, while the diagrams (c) and (d) do not involve four-fermi interactions and hence can be unambiguously determined by on-shell parameters. Here the internal dotted line represents the kaon.

We first consider the latter. These are depicted in Fig. 4. Since we are dealing with s-wave kaon interaction, the most important configuration that K- can couple to is the

A* particle-nucleon hole (denoted as A*N -1 with N either a proton (p) or neutron (n) ) . We shall return to nonstrange particle-hole correlations when we treat density

effects on the basic constants of the lagrangian (i.e., "BR scaling"). Here we focus on the former type. Now for the s-wave in-medium kaon self-energy, the relevant four-fermi interactions that involve a A* can be reduced to a simple form involving two unknown constants

S "'r* * - - T - ' ~ k * ~4--fermion = C a. AoAv Tr B vBv + CA*ArtY A~, Tr-BotrkBv, (25)

where cS: r are the d i m e n s i o n - 2 (M -2) parameters to be fixed empir ica l ly and trk acts


on baryon spinor. Additional (in-medium) two-loop graphs that involve A*N -1 excitations are given

in Figs. 4c and d. They do not however involve contact four-fermi interactions, so are calculable unambiguously.

We shall denote the sum of these contributions from Figs. 4 to the self-energy by Hi*. A simple calculation gives

¢O {CAS*pp (/On n t- lpp) __ 3~T 2" - ~ca.p{,i Ha. (to) = - to+mB -ma*

+ ~ P P ( t o + m ~ _ m A . ) t°2

x { (2~K(t°) + X~(t°)) -- g2a* (tO + m ~ _ m a . ) (~K(t°) + ~ ( t ° ) ) } ,

(26)

where gA* is the renormalized KNA* coupling constant determined in Ref. [2] and XN(to) is given by

1 / ]k] 2 . v~ ( , o ) = ~ dlkl ,02 _ M ~ - Ikl 2 ( 2 7 )

0

In Eq. (26), the first term comes from the diagrams of Figs. 4a and b and the second term from the diagrams of Fig. 4c and d. While the second term gives repulsion corresponding to a Pauli quenching, the first term can give either attraction or repulsion depending on the sign of

(CAS [pn + 21_pp] _ 3 T ~C:l*Pp)

with the constants cS~ r being the only parameters that are not determined by on-shell data.

The complete self-energy to in-medium two-loop order is then

The additional parameters cS; r that are introduced at the level of four-fermi interactions in the strange particle-hole sector require experimental data involving nuclei and nuclear matter. We shall now discuss how these constants can be fixed from kaonic atom data. In order to fix both of these constants, we would need data over a wide range of nuclei. One sees in (26) that for the symmetric matter, what matters is the combination (cS. - CT.). At present, this is the only combination that we can hope to pin down from kaonic atom data. That leaves one parameter unfixed. We shall pick C s. for the reason to be explained later. We shall parametrize the proton and neutron densities by the proton fraction x and the nucleon density u = P/Po as

C-H. Lee et al./Nuclear Physics A 585 (1995) 401-449

Table 2 K- self-energies in symmetric nuclear matter (x = 0.5) in units of M 2 at u = 0.97 for ~a- = 0.25. /iV = M~ -- MK is the attraction (in units of MeV) at a given (cS, - Cra,)f 2

417

( cSa. - c~.) f ~ M~ 4v - p T ~ -p~7 -r"= u~. U~A.

1 402.5 -92.49 --0.2511 0.0476 --0.2506 0.1151 5 353.3 -141.7 --0.1821 0.0315 --0.3684 0.0286

10 327.3 -167.7 -0.1740 0.0265 --0.4304 0.0156 50 254.1 -240.9 -0.1843 0.0184 -0.5738 0.0034

100 218.8 --276.2 -0.1929 0.0162 -0.6293 0.0017

pp = x p , Pn = (1 -- x ) p , p = upo. (29)

Now what we know from the present ly available kaonic atom data [ 18] is that the

optical potent ia l for the K - in m e d i u m has an attraction of the order of

A V e - ( 1 8 0 ± 2 0 ) Me V at u = 0.97. (30)

This impl ies approximate ly for x = ½

( CSA . -- C~. ) f 2 ~ 10. (31)

Table 2 gives details o f how this value is arrived at. It also lists the contr ibut ions of each

chiral order to the self-energy (28 ) , A V = M ~ - MK and M ~ which we shall loosely

call "effective kaon mass" 6

/ _

(32)

To exhibi t the role of A* in the kaon self-energy, we list each cont r ibut ion o f / / . Here

//free = --pNTfrKe~ N, 811 = - - p N S T K-N, //la, corresponds to the first term of Eq. (26)

which depends on cS; r and H i , to the second term independent o f CAS; r . We observe

that the CaS;r-dependent term plays a crucial role for attraction in the kaonic atom.

In Table 3 and Fig. 5, we list the predicted densi ty dependence of the real part o f the

K - optical potent ia l for x = 0.5 obta ined for ( c S , - Ca r , ) f 2 ~ 10 7

To unders tand what the r emain ing parameter cS , is physically, we consider the mass

shift o f the A* in medium. To one- loop order, there are two graphs given in Fig. 6. A

s imple calcula t ion gives

8mA* = E S~(Ai) ( tO = mA* -- m a ) , (33) i=a,b

where

6 This is not, strictly speaking, a mass but we shall refer to it as such in labeling the figures. 7 If one solves the K - dispersion equation with Eq. (28), there are two spectral branches, a real K- branch

and a A*-hole branch. The results shown in Fig. 5 and Figs. 8-13 correspond to the lowest branch which goes over to the K - branch when the A*KN coupling is turned off. Since the s-wave condensation is driven by a mass flow, collective "phonons" do not seem to play an important role in contrast to the p-wave condensation (e.g., pion condensation) driven by the flow of a Yukawa coupling.

M* ( M e V ) K + 700

600

500

400

300

x~

\ \ \

\ \

200

I00

J J

J


0 0 I I I I I I 1 2 3 4

u = P/Po

Fig. 5. Plol of the K + effective potential in symmetric nuclear matter (x = 0.5). The upper solid line corresponds to K + "effective mass" without the BR scaling, and the lower solid (dashed) line to K - "effective mass" without (with) the BR scaling.

=-7,o 8X~A b) (,o) = - c S . (Pp + Pn). (34)

The supersc r ip t ( a ) , ( b ) s tands for the f igures (a ) and ( b ) o f Fig. 6. The cont r ibu t ion

Table 3 K - self-energies in symmetric nuclear matter (x = 0.5) in units of M 2 for g2a. = 0.25 and (cS. - C A T . ) f 2 = 10. AV ---- M~¢ - MK is the attraction (in units of MeV) at the given density

u g~, AV _p~-f~e --p87"f~ n~a. Hi.

0.2 425.5 -69.51 -0.0680 0.0035 -0.2044 0.0140 0.4 390.5 -104.5 -0.0924 0.0085 -0.3066 0.0154 0.6 365.0 -130.0 -0.1177 0.0144 -0.3668 0.0159 0.8 343.6 -151.4 -0.1466 0.0208 -0.4079 0.0159 1.0 324.6 -170.4 -0.1791 0.0276 -0.4338 0.0155 1.2 307.2 -187.8 -0.2151 0.0345 -0.4484 0.0149 1.4 2 9 1 . 1 -203.9 -0.2539 0.0415 -0.4556 0.0142


PN

A* ~,% .. / A*

A* N

A*

(~) (b)

Fig. 6. One-loop diagrams contributing to the A* mass shift in dense medium. The diagram (a) involves the

intermediate states of K - p and K-°n and (b) the four-fermi interaction caS, with both protons and neutrons.

from Fig. 6a is completely given with the known constants. The dependence on the unknown constant caS, appears linearly in the Fig. 6b. For the given values adopted

here, the mass shift is numerically

8 r n a . ( u , x , y ) = [ r ( u , x ) - 150.3 × u × y] MeV, (35)

where y = c S . f 2 and r ( u , x ) - 8 ~ a) with the numerical values given in Table 4.

Table 4 Numerical values in MeV of r as function of x and u

r(u,x) x = 0 . 0 x = 0 . 5 x = l . 0

u = 0.5 30.78 38.25 30.78 u = 1.0 47.52 61.55 47.52 u = 1.5 60.20 79.75 60.20

One can see from Eq. (35) and Table 4 that the shift in the A* mass in medium is

primarily controlled by the constant cS.. For symmetric matter (x = ½, u = 1 ), the shift is zero for y = 0.41 and linearly dependent on the y-value for nonsymmetric matter. At

present we have no information as to whether the medium lowers or raises the mass of the A*, so it is really a free parameter but it is reasonable to expect that if any the shift

cannot be very significant. We consider therefore that a reasonable value for y is O( 1 ). In Table 5 and Fig. 5 are given the properties of K + in nuclear matter. The self-

energy of K + is simply obtained from that of K - by crossing to --, - to . One can note here that the interaction of K + with nuclear medium is quite weak as predicted by phenomenological models and supported by experiments. The M~: grows slowly as a function of density 8. This is a check of the consistency of the chiral expansion approach

to kaon-nuclear interactions.

8 Recent heavy-ion experiments at Brookhaven [31] find that K+'s come out of quark-gluon plasma with a temperature of order of 20 MeV which is much lower than the freeze-out temperature of ,-~ 140 MeV. To understand this phenomenon, there has to be a mechanism at high temperature and density that reduces the K + mass considerably [32]. Within the framework adopted here, this can happen only if the strong repulsion due to the to-exchange lodged in the first term of/~(1), Eq. (1) , is strongly suppressed so that the attraction coming from the sigma term becomes operative, thereby reducing the mass. As discussed in a recent paper by Brown and Rho [33,34], this can indeed happen if the vector mesons to and p decouple at high temperature and density. In this paper we are not concerned with this regime.

420 C.-H. Lee et al. /Nuclear Physics A 585 (1995) 401--449

Table 5 Self-energies for K + in nuclear matter 0.25 and (CSA . -- Cra.) f 2 = 10. /IV = of MeV) at given density

(x = 0.5) in units of M 2 for g2a. = /~r - MK is the repulsion (in units

U M~ AV --p"2 "free --p8"1 "free HI , II2.

0.2 505.6 10.6 0.0252 0.0210 -0.0009 -0.0023 0.4 515.0 20.0 0.0479 0.0396 -0.0038 -0.0047 0.6 522.5 27.5 0.0685 0.0593 -0.0087 -0.0071 0.8 529.4 34.4 0.0869 0.0771 -0.0157 -0.0094 1.0 535.0 40.0 0.1037 0.1017 -0.0247 -0.0125 1.2 540.5 45.5 0.1183 0.1249 -0.0359 -0.0156 1.4 545.5 50.5 0.1312 0.1507 -0.0493 -0.0190

5. Critical density for kaon condensation

We have now all the ingredients needed to calculate the crit ical densi ty for negat ively

charged kaon condensa t ion in dense nuclear star matter. For this, we wil l fo l low the

p rocedure g iven in Ref. [9 ] . As argued in Ref. [6 ] , we need not cons ider p ions when

e lec t rons wi th h igh chemica l potent ia l can t r igger condensat ion through the process

e - ~ K - r e . Thus we can focus on the spatial ly un i fo rm condensate

( K - ) = VK e - i # t , (36 )

x(u)

0.4

F(u) 2u2 - - l + u

0.3

0.2 = u

I , I ~ I I I t I t I 0"00 1 2 3 4 5 6

u = P/PO

Fig. 7. Plot of the proton fraction x(u) prior to kaon condensation for different forms of F(u).

C-H. Lee et al./Nuclear Physics A 585 (1995) 401-449 421

Mk (uev) 500

400 \x x ~ X \ \

N \ \ \ N \

\ N \

300 ., .." .. ".. \ \ \

\ \ x

200

100

O ; n I n I I I I I , I n I 1 2 3 4 5 6

U = P/Po

Fig. 8. Plot of the quantity M[ obtained from the dispersion formula D - 1 (/z, u) = 0 versus the chemical potential/z prior to kaon condensation for g2A. = 0.25 and F(u) = 2u2/( 1 + u). The solid line corresponds to the linear density approximation and the dashed lines to the in-medium two-loop results for (cAS. -- CaT. )f2 = 10 and cS, f 2 = 10, 5, 0, respexzfively from the left. The point at which the chemical potential/z intersects corresponds to the critical point.

where /x is the chemical potential which is equal, by Baym's theorem [35] , to the

electron chemical potential. The energy density ~ - which is related to the effective

potential in the standard way - is given by

3 ~.(o),,5/3,,^ _ V ( u ) + upo( 1 - 2 x ) 2 S ( u ) C'(U'X'lZ'UK) = 5 " ' F '~ / - ' o - -

- [ / z 2 - M 2 _ I Ix( lX , U,X) ]V 2 + E an(tZ, U,X)V~

n>/2

+ixupox + ~e + 0(I/zl - m~)~#, (37)

where E(F °) = (p~° ) )2 /2mB and pF {°) = (3~r2p0)t/3 are, respectively, Fermi energy and

momentum at nuclear density. The V ( u ) is a potential for symmetric nuclear matter as

described in Ref. [36] which is presumably subsumed in contact four-fermi interactions

(and one-pion-exchange - nonlocal - interaction) in the nonstrange 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 nonstrange sector - does play a role as we

422 C-H. Lee et al./Nuclear Physics A 585 (1995) 401-449

M~. ( M e V )

5OO

400

300

200

100

x • \ \ \

\ \ \ \ \ \ \

\ \ \ \ \ \ \

\ \ \ \ \ \ \ \ \

\ \ \ N \ \ \ \ \ \ \ \

\ x \

\ \ \ \

, - ) . .

i I t I i I ~ I i I i I l 2 3 4 5 6

U = P/Po

Fig. 9. The same as Fig. 8 for F(u) = u.

know from Ref. [36] : 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 textbooks. We take the form advocated in Ref. [36]

S ( u ) =(22/3 1) 3~,(0) ( - 2 / 3 - F ( u ) j + S o F ( u ) , (38)

where F ( u ) is the potential contributions to the symmetry energy and So ~ 30 MeV is the bulk symmetry energy parameter. We use three different forms of F ( u ) as in Ref, [36]

2u 2 F ( u ) = u, F ( u ) = 1 + u F ( u ) = v/ft. (39)

It will turn out that the choice of F ( u ) does not significantly affect the critical density. The contributions of the filled Fermi seas of electrons and muons are 9 [9]

/[.6 4

ee = 127r 2 , 4

m~ ( + + + t 2 ~ / ~ - ~ ) -/.z~ 2 , (40) ~ = e~ --/zplz = ~ 2 (2t2 l ) t v / ~ 1 -- ln( t 2 pF3~

9 We ignore hyperon Fermi seas in this calculation. Their effects are under investigation.

C.-H. Lee et al./Nuclear Physics A 585 (1995) 401~149

M k ( M e V )

5O0

423

400

300

200

100

N \ ? \ \ \ \

\ \ \

\ \ \ \

\ \

, . . \ \

i I J I i I J I i I i I 1 2 3 4 5 6

u = P/Po

Fig. 10. The same as Fig. 8 for F(u) = x/'u.

F------ w h e r e PF~ = X//x2 -- m~ is the Fermi m o m e n t u m and t = pF, , /mu.

The ground-s ta te energy prior to kaon condensat ion is obtained by extremizing the

energy densi ty g: with respect to x, /z and VK:

~X a~ ae VK--O ae = O, ~ = O, 8v 2 = 0 (41)

VK=O vK=O

from which we obta in three equat ions corresponding, respectively, to beta equi l ibr ium,

charge neutra l i ty and dispers ion relation:

Table 6 Critical density uc in in-medium two-loop chiral perturbation theory for g], = 0.25. (a) correspond to linear density approximation of Figs. 3a and b to the full two-loop result for (caS, - Car,)f 2 = 10

F(u) (a) (b)

cS./2 =1o c~./:= 5 cS./2 =o

2u--~-2 3.77 2.81 2.98 3.25 I+U u 3.90 3.13 3.34 3.71 x,/'ff 4.11 3.72 3.97 4.42


Table 7 Critical density Uc in in-medium two-loop chiral perturbation theory for g2a. = 0.25 and F(u) = u

(C~. - C ~ . ) f 2 C ~ . f 2 = 100 C~ . f 2 = 10 C ~ . f 2 = 0

1 2.25 3.25 4.63 10 2.24 3.13 3.71

100 2.18 2.66 2.77

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

/~3 , p 3 0 = - x u p o + ~ + O(Iz - m~,) -~--~2'

O = D - l (l~,U,X) = iz 2 - M~ - l lK(ix, u , x ) =- tz 2 - h,UK2(/x, u, x). (42)

The proton fractions x ( u ) and chemical potentials/z prior to kaon condensation are plotted in Fig. 7 and Figs. 8-10 for various choices of the symmetry energy F ( u ) .

We have solved these equations using for the kaon self-energy (a) the linear density approximation, Eq. (23) and (b) the full two-loop result, Eq. (28). Table 6 (a) shows the case (a) for different symmetry energies Eq. (39). We see that the precise form of the symmetry energy does not matter quantitatively. The corresponding "effective kaon mass" M~ is plotted versus u in Figs. 8-10 with solid lines. Note that even in this linear density approximation kaon condensation does take place, albeit at a bit higher density than obtained before.

For the value that seems to be required by the kaonic atom data, (31), the critical density comes out to be about uc ,-~ 3, rather close to the original Kaplan-Nelson value.

In Table 6b and Figs. 8-10 are given the predictions for a wide range of values for cS, f2. What is remarkable here is that while the cS,r-dependent four-fermi interactions are essential for triggering kaon condensation, the critical density is quite insensitive to their strengths. In fact, as one can see in Table 7, reducing the constant (cS. - Car.)f2 that represents the kaonic atom attraction by an order of magnitude to 1 with cS, f 2 =

10, 0 modifies the critical density only to ue ~ 3.3, 4.5, respectively. To see how robust kaon condensation is with respect to the A*KN coupling constant,

Table 8 K - self-energies in symmetric nuclear matter (x = 0.5) in units of M~: at u = 0.97 for g2A. = 0.05. AV = M~ -- MK is the attraction (in units of MeV) at a given (caS. - Car.) f2

( cJ. - p AV - p a t Wa.

10 386.4 -108.6 --0.1142 0.0410 --0.3243 0.0092 20 364.4 --130.6 -0.1158 0.0342 -0.3786 0.0054 50 330.3 -164.7 --0.1278 0.0270 -0.4566 0.0026 70 316.1 -178.9 --0.1346 0.0248 --0.4849 0.0019

100 300.0 --195.0 -0.1428 0.0227 -0.5139 0.0014

C.-H. Lee et al . /Nuclear Physics A 585 (1995) 401--449

Table 9 Critical density uc in in-medium two-loop ChPT for s r (Ca. -CA.) f2 = 70, g2a. = 0.05

F (u ) caS.f 2 = 70 c S . f 2 = 35 CSA.f 2 = 0

2u 2 l+'---'ff 2.68 2.86 3.14

u 2.99 3.21 3.59 x,/'ff 3.56 3.83 4.35

425

let us take the extreme value gZa. ,~ 0.05 USed in Ref. [25] which gives the wrong sign to the K - p amplitude at threshold. In Table 8, AV of the kaonic atom (x = 0.5) is given for u = 0.97 and for various choices of (cS, - Cr, )f2. Constraining to the kaonic atom data implies approximately - within the range of error involved - (cS. - Car. ) f2 ~ 70.

In Table 9, the resulting critical densities are given for (CAS. -- Ca r. ) f2 ~ 70, and those for other sets of cS. and Car, in Table 10. We see that given the constraint from the kaonic atom data, the resulting critical densities are sensitive neither to the value of cS; r

nor to gA*.

6. Four-fermion interactions and "scaled" chiral lagrangians

So far we have ignored four-fermi interactions in the nonstrange sector with the

understanding that the effects not involving strangeness are to be taken from what we know from nuclear phenomenology. From the chiral lagrangian point of view, this is

not satisfactory. One would like to be able to describe both the nonstrange and strange sectors on the same footing starting from a three-flavor chiral lagrangian. While recent

developments indicate that nuclear forces may be understood at low energies in terms

of a chiral lagrangian [ 12,13], it has not yet been possible to describe the ground-state property of nuclei including nuclear matter starting from a lagrangian that has explicit chiral symmetry. So the natural question is: What about many-body correlations in the

nonstrange sector, not to mention those in the strange sector for which we are in total

ignorance? Montano, Politzer and Wise [ 20] addressed a related question in pion condensation in

the chiral limit and found that four-fermi interactions in the nonstrange sector played an important role in inhibiting p-wave pion condensation in dense matter. It has of course

Table 10 Critical density uc in in-medium two-loop ChPT for g2a. = 0.05 and F(u)=u

(CA* -- C ~ . ) p C ~ . ~ = 100 C~ . I 2 -- 10 c S . g = 0

1 2.95 4.07 5.47(?) 10 2.94 3.92 4.45

100 2.83 3.33 3.44


been known since some time that the mechanism that quenches gA in nuclear matter to a value close to unity banishes the pion condensation density beyond the relevant regime. As shown in Ref. [37], this mechanism can be incorporated by means of a four-fermi interaction in a chiral lagrangian in a channel corresponding to the spin-isospin mode (that is, the Landau-Migdal g' interaction).

In this section, we discuss a simple approach to including the main correlations in kaon condensation. We cannot do so in full generality to all orders in density but we can select what we consider to be the dominant ones by resorting to the scaling argument (which we shall call "BR scaling") introduced by Brown and Rho [34,38].

How to implement the BR scaling in higher-order chiral expansion with the multiple scales that we are dealing with has not yet been worked out. It has up to date been formulated so as to be implemented only at tree order with the assumption that once the BR scaling is incorporated, higher-order terms are naturally suppressed. If this assumption is valid, which we can check ~ posteriori, by taking the BR scaling into account at tree order, that is, at O(Q 2) in our case, we will be including most of higher-order density dependences through the simple scaling.

We first consider the leading-order (O(Q) ) term, which for s-wave kaon-nucleon interactions is given by the first term of Eq. (1)

~KtOoKBtvBv. (43)

In terms of vector-meson exchanges, it is equivalent to an w-meson exchange, attractive in the K - channel and repulsive in the K + channel. Four-fermi interactions in the nonstrange channel can modify this interaction, the most important one being

D-~ Kt OoK( BtvBv) 2 (44)

with D an unknown constant. One may try to estimate the constant D using dynamical models. For instance, one can have a four-fermi interaction that arises when a massive scalar (or in the linear o--model) is integrated out. This would increase the attraction of Eq. (43) in the K - channel, which is equivalent to increasing the magnitude of D. One can also have a four-fermi interaction that arises from integrating out a massive vector exchange of the w-meson quantum number, responsible for the screening of the attraction by a factor (1 + F0) -1 where F0 is the Landau-Migdal parameter > 0 in the density regime we are interested in Ref. [39]. This would decrease the magnitude of D. There are of course other terms but if we add them all up and write an effective two-fermi interaction with correct symmetries by taking the mean-field (BvBv) ~ p, then the higher-density dependence is expected to modify (43) to

--F.2 Kt OoKB~Bv, (45)

where the asterisk denotes density dependence f* = f (p ) . Later we will assume this to be given by the BR scaling [34,38]. Whether or not this assumption is viable will be tested ?t posteriori.


Table 11

K - self-energies with BR scaling in symmetric nuclear matter (x = 0.5) in units of M 2. at u = 0.97 for g2a. = 0.25. AV = M~ - MK is the attraction (in units of MeV) at a given (CSa. - Cra.) f ~"

(cS. _ CA~.) I2 M;, 4v - p ~ = - p ~ = n~. //~a-

1 392.3 - 1 0 2 . 7 --0.3092 0.0432 -0 .1891 0.0833 5 344.9 - 1 5 0 . 1 - 0 . 2 5 9 8 0.0297 - 0 . 3 0 8 0 0.0234

10 319.2 --175.8 - 0 . 2 5 5 1 0.0253 - 0 . 3 6 7 5 0.0130 50 246.5 - 2 4 8 . 5 - 0 . 2 6 7 8 0.0179 - 0 . 5 0 4 2 0.0029

100 211.5 - 2 8 3 . 5 - 0 . 2 7 6 1 0.0158 --0.5551 0.0014

Similar four-fermi interaction terms can also be written down for the O(Q 2) terms. However, the effect here is expected to be less important for the following reason. First of all, the term involving the KN sigma term is associated with the strange-quark property, in particular with the kaon decay constant fK and we expect that the fK is not modified significantly since the s-quark condensate does not change much as the density and/or temperature is increased [40]. This implies that four-fermi interactions will be less effective in modifying this term. The same must be the case with the counter terms di that are associated with the octet and decuplet of strange-quark flavor. We may assume them to be also unaffected by the renormalization.

In summary, our proposition is that the net effect of multi-fermi interactions in the nonstrange sector can be summarized by the scaling f --* f* in the leading term in (1). As stated, we propose this scaling to be given by the BR scaling l0

f f ~ m~, ~ 1 cu (46) f mv

with c ~ 0.15. Since the multi-Fermi interactions are higher order in the chiral counting in terms of an expansion kF/A x, it is consistent to leave them out in the loop corrections, that is at O(Q3).

10 We use this linear density formula for getting a rough idea. It could in principle be a highly nonlinear function of density, the large-density behavior of which is totally unknown at present.

Table 12 K - self-energies with BR scaling in symmetric nuclear matter (x = 0.5) in units of M 2 for g2a. = 0.25 and (CSA . -- CTA.)f 2 = 10. AV ~ M~ -- MK is the attraction (in units of MeV) at given density

M~ AV - p T f~ -p~T ~ n~. //~.

0,2 424.3 --70.66 --0.0699 0.0034 --0.1982 0.0135

0.4 389.6 -- 105.4 --0.1038 0.0084 - 0 . 2 9 9 9 0.0150 0.6 362.3 -- 132.7 --0.1447 0.0141 --0.3448 0.0148

0.8 338.6 -- 156.4 --0.1984 0.0202 --0.3677 0.0141 1.0 315.7 --179.3 --0.2663 0.0262 --0.3658 0.0128

1.2 292.8 --202.2 --0.3504 0.0320 --0.3448 0.0110 1.4 267.6 --227.4 --0.4531 0.0371 --0.3013 0.0088


Table 13 Critical density uc with BR scaling in in-medium two-loop ehiral perturbation theory for g2a, = 0.25. (a) correspond to the linear density approximation of Figs. 3a and b to the full two-loop result for (CaS, - Car,)f 2 = 10

F(u) (a) (b)

c .i = 1o c .i = 5 c J . i = o

~_..~2 2.11 2.08 2.11 2.14 l+u u 2.16 2.15 2.17 2.20

2.22 2.23 2.25 2.26

The self-energy for kaonic atoms with the BR scaling is given in dashed line in Fig. 5

and tabulated in Tables 11 and 12. Comparing with Tables 2 and 3, we see that the BR

scaling gives only a slightly more attractive potential at low densities. The attraction,

however, increases significantly at higher densities. From Table 11, we find for kaonic

atoms, approximately,

(CAS. -- CAr.)f 2 ~ 10. (47)

This is roughly the same as without the BR scaling since the scaling effect is not impor-

tant at u = 0.97. The BR scaling becomes important in the region where condensation sets in. The corresponding critical density is given in Tables 13 and 14. The character-

istic feature o f the effect o f the BR scaling is summarized in Figs. 11, 12 and 13. A

remarkable thing to notice is that the critical density lowered to uc ~ 2 is completely

insensitive to the parameters such as the form of the symmetry energy, the constants

CAS; r etc. in which possible uncertainties of the theory lie.

In order to verify the key hypothesis of the BR scaling - that the scaling subsumes

higher-order effects, thus suppressing higher chiral order effects o f the scaled lagrangian,

we keep all the parameters f ixed at the values determined at O(Q3) , BR scale as

described above, then ignore all terms of O ( Q 3) (i.e. loop corrections) and (A) set

cS, = CAT, = 0 and (B) set (CAS. -- CAr, ) f 2 = 10 and calculate the critical density. The

results are given in Table 15. We see that the effects of loop corrections and four-fermi

interactions amount to less than 10%. This may be taken as an h posteriori verification

of the validity of the assumption made in deriving the scaling, although it is difficult

to assess how reliable the absolute value of the predicted critical density (and the

Table 14 Critical density uc with BR scaling in in-medium two-loop chiral perturbation theory for g2a, = 0.25 and F(u) = u

w J , - c .)p c .p = lOO c .i 2 = lO c a.P = 0

1 1.92 2.16 2.21 10 1.92 2.15 2.20

100 1.89 2.09 2.12


M k ( M e V )

5OO

400

300

200

100

• \ \ \ " \N~ \ \ \

\ \ \ \ \

\ \ \ \ \ N , \ \

\ X \ \

\ \ \ \

\ \ \ \

~ ~ I ~ i i I ~ , i I 1 2 3

u = P/Po

Fig. 11. The same as Fig. 8 with the BR scaling for F(u) = 2u2/(1 +u) .

cor responding equat ion o f state of the condensed phase) is.

7. Discussion

In conc lus ion , we have shown that chiral per turbat ion theory at order N2LO predicts

kaon condensa t ion in "nuclear star" matter at a densi ty 2 < uc < 4 with a large fraction

o f protons - x = 0.1 ~ 0.2 at the critical poin t and rapidly increasing afterwards

- neut ra l iz ing the negative charge of the condensed kaons. For this to occur, four-

Table 15 Critical density Uc with BR scaling and without O(Q 3) terms for (A) caS. = Car. = 0 and (B) (caS. - c T . ) f 2 = 10. This should be compared with the results of Table 13

F(u) (A) (B)

C~.=100 C~.=10 C ~ . : 0

2u----~2 1.89 1.68 1.85 1.88 l+u u 1.94 1.76 1.91 1.94

2.00 1.87 1.98 2.00


M~. ( M e V )

500

400

\ \ \ \

\ \ 300 \ \ ~x

x . \ ~ \ \ ~ \ \

200 x ~ ~ #

100

0 i i i I I I I I J J I I 1 2 3

tt = P/Po

Fig. 12. The same as Fig. 8 with the BR scaling for F(u) = u.

fermi interactions involving A* are found to play an important role in driving the condensation but the critical density is negligibly dependent on the strength of the

four-fermi interaction. It is found that the BR scaling [34,38] favors a condensation at a density as low

as twice the matter density and that when the BR scaling is operative, higher chiral

corrections coming from the scaled lagrangian are insignificant, justifying the basic

assumption that goes into the derivation of the scaling relation. This suggest that at least for kaon condensation, the tree approximation with the scaled lagrangian is consistent with the basic idea of the BR scaling.

Given the relatively low critical density obtained in the higher-order calculation of this paper, we consider it reasonable to assume that the compact-star properties will be qualitatively the same as in the tree-order calculation of Ref. [9]. We will report on this matter in a future publication.

Our treatment is still far from self-consistent as there are many nuclear correlation effects that are still to be taken into account. How to incorporate them in full consistency with chiral symmetry is not known. In particular the role of four-fermi interactions in the nonstrange channel, e.g., short-range nuclear correlations which involve both interaction terms in the lagrangian and nuclear many-body effects, is still poorly understood. A problem of this sort may have to be addressed in terms of renormalization group flows


M~. (MeV) 5OO

400

300

200

100

\ \ \

\ \ \ \ ~ , \ \ \

\ \

J i i I i i J I I i I I

I 2 3

u = P/Po

Fig. 13. The same as Fig. 8 with the BR scaling for F(u) = x/u.

as in condensed matter physics [41]. A work is in progress along this line [42].

Acknowledgements

We acknowledge useful discussions with H. Jung, K. Kubodera, A. Manohar, F. Myhrer, V. Thorsson, A. Wirzba, W. Weise and H. Yabu. The work of CHL and DPM were supported in part by the Korea Science and Engineering Foundation through the CTP of SNU and in part by the Korea Ministry of Education under Grant No. BSRI- 94-2418 and the work of GEB by the US Department of Energy under Grant No.

DE-FG02-88ER40388.

Note added

After this paper was submitted, two papers [45,46] appeared in the literature in which a criticism was raised regarding the use in this paper of the vector density B t By instead of the scalar density BB (without the subscript v) which would appear in relativistic lagrangians. The authors of Refs. [45,46] thought that an approximation of replacing


the scalar density by a vector density has been made in the calculation of the KN sigma (i.e., Kaplan-Nelson) term, an approximation which would be invalidated at higher densities.

We would like to point out that there is a misunderstanding in this criticism. In the heavy-fermion formalism that is used in this paper - which is what is required for a consistent chiral expansion in the presence of baryon fields, the expansion is made in terms of B--~(-..)By which is equivalent to Bt~( . . . )Bo as one can easily verify by the projection operator that defines the subscripted baryon field. There is no approximation made in the calculation. Of course this quantity differs from the scalar density B B

(without the projection operator), particularly at high density. One might then wonder where the difference shows up in the HFF. It is easy to see where. The difference between the scalar density and the vector density is O(Q 2) in the chiral counting and since the KN sigma term containing the quark mass matrix is of O(Q2), the difference enters at O(Q4). Our calculation to the one-loop order is valid to O(Q 3) and hence the

difference does not figure in our theory to the order considered. A consequence of the above observation is that it is inconsistent with the chiral

expansion to use the scalar density in the Kaplan-Nelson term unless one includes at the same time all other counter terms of O(Q 4) (there are many with varying physical inputs, some known and some unknown) as wel l as all finite two-loop terms. Since chiral symmetry is preserved order by order of the chiral expansion, taking into account only one such term as is done in Refs. [45,46] using the Walecka-type model (which has no chiral symmetry) would not preserve chiral Ward identities. The large effect seen in Refs. [45,46] for increasing density could simply be an artifact of this defect. While one cannot conclude that the approach of Refs. [45,46] is wrong, it is certainly not a justifiable procedure for our scheme.

Schaffner et al. [45] suggest associating the square of the vector interaction with an increase in the kaon effective mass. This quadratic vector interaction was discarded by Brown et al. [47] on the ground that it meant using to quadratic order a term which had been derived only to linear order. The one-loop corrections in our present paper evaluate all terms to the given order - i.e. 0 ( 0 3) - and all terms of O(Q 2) are correctly included in the lagrangian L (2). Furthermore the authors of Ref. [47] argued that effective masses should involve scalar fields and the vector mean field should only give a shift in the chemical potential, as in Walecka mean-field theory.

Appendix A. Vertices and Feynman rules

In this appendix, the chiral lagrangian used is given explicitly in terms of the component fields (meson octet, baryon octet and decuplet) we are interested in. The vertices of the Feynman graphs can be read off directly. The subscript n in Ln denotes the number of lines attached to the given vertex. In Fig. 1, the three-point vertex L3 enters in the diagrams (c), (d), (f) , the four-point vertices L4 g (4-meson) and L4 B (2-meson + 2-baryon) in the diagrams (b), (d) and (b), (c), (e), respectively and the higher-point

C.-H. Lee et al./Nuclear Physics A 585 (1995) 401-449

vertices /25 and /~6 figure in the diagrams (f) and (a), respectively.

<o+.,,: :<o 1 [v~fD + F)pvS~v- /~noO/.,#+ + x,/'2fD - F)p.S~- /~.Z; +auK ° f

+X/2(D - ~ - + - F)..S~ Z; O.l~

- ~ g ( D + 3F)~vSUvAvO~K + - V~g(D + 3F)fioS~vAvOuK °

- i . t + +( D - F)p .S~ ~ O g K - ( D - F)nvS~ ~ O u K ° + h.c.]

- ~ o + - ~ + o V ~ f i v O K "Yv,~ C [v/2poO ¢r A.,s, + + pro ¢r Av, ~ 1 - ~ + .o ,/gf

- ~, - + + - /-, * + v , ~ n . a ~ r O a O , ~ - l , - + -v'~p.a "Jr A,,,~. - p,,a K°Z'~,u + -n.a ~ A,,.~,

_X/~,,oa, - ,, ro z;,,~,o + V%va,, #+ a;,,, + r,,O" K+ Z~.; + h.c.],

433

(A.I)

+ - ( K + ~ K-)( r r + a ~ 7r-) + (K + 0~, ~ ' - ) (K- a # zr +)

- ( K + 0~ K - ) ( K + a ~ K - )

BO + - I V~3!(mu ms)zrOr/ +~-'-~ [K K (~(3mu +ms)('rr°) 2 +

+½(ran + 3ms)r/2) + (2mu + m d + m s ) r + z r - K + K -

+(mu + m s ) K + K - K + K - + mu + m d + 2 m s ) K + K - K ° K ° ] , (A.2)

+ - . v ~ . - 2pvV~pv - r~.v~' n . )

4 3 4 C.-H. Lee et al. /Nuclear Physics A 585 (1995) 401-449

- (,,- ,,)

.,[( ) - t ' ~ - ~ K + v • O u K - {'A++ A ++'~ [-g "-~-+ *+'~ ~-"'~ -" + V g zaoy.a,

+ /7--+-*+ ,~.+., F ~ ,o.~ ~ - ~ . o ~,.o.,

- V g"%'~"% - ' - V 3'-v.~--, :j' (A.3)

1 [K+K_{v/-~fvS~pv2F(O.rrO + v/-~O/.,~) £5 - 6v~f 3

- f f2fzoS~n,( D - F)(Ot ,~ "° + v'30ur/) }

+ K + K - { ( D + F ) p , S ~ n , Ot, qr + + ( D - F )p ,S~X+OuK °

+ 2 ( D - F ) R v S ~ X v O u K + - V~g(D + 3V)p , s~aoog K+

- ~ g ( D + 3V)fivs~vavc)uK ° + v ~ ( D - F)fvSv~vvO~,K +

+

+ -- 2 - /~ 0 + - /z +

12~f3 + :::..Ao + .: _ ::,:

_:::,,o.= + :,::,:.

.//~'_~ aUK+V.- "t-V~nvOP"qAOO,lS " -t- ~tvOl~q'g+'4~,.u " + V 3 v ~'v,l.~

(A.4)

t2 6 = [P,# 'Pv{( K - 0~ K - ) (5¢r+~ "- +5K-°K °

+ 8 K + K - + ('7/'0) 2 -.{- 2 V ~ 0 ' q + 3"q 2)

+ 7 K + K - ('rr + ~# "rr- ) - 7 ( K -° ~ K ° ) K + K - }

+n:"n,{(K + ~',, K-) (½(:)2 + v%.on --I- 3 ~ 2 - 2.rr+~ - + 4 K + K - + 7K-°K °)

- 5 ( K -° ~ K ° ) K + K - + 2 K + K - (~r + Ou o r - )} ] , (a.5)

C.-H. Lee et al•/Nuclear Physics A 585 (1995) 401-449 435

where the quark masses are related to the meson masses as

2 = 2 B o ( ~ + 2ms), M 2 = B0(mu + md) = 2B0~, M,7

M2=Bo(~ + ms). (A.6)

The propagator for the baryon octet in HFF is i/(v • k), where ks ̀ is the residual momentum of the baryon as defined by ps` = mBVS ̀+ k s ̀ [ 17]• The propagator for the decuplet is also simplified to the form

i d - 3 . ,,'~ • 4 ~ - ~ $ 2 S ~ ) .

In the HFF, the spin operator takes the form

S` v 1 : _ s ` v a f l . . c

Given the vertices and the propagators, one can immediately write down the integral entering in the diagram (e) of Fig. 1,

li E = 1 f dnk V. ( 2 q - k) v. (q + q' - k) 7 (2~r)" vTk-'-+~g ( q - k ) 2 - M 2 i + i e

= - ( v • q~ + 2v. q)Ai + 2v. qv. (q + q t ) ~ i ( V • q ) , (A.7)

where

1 f dnk , 1 M2 In M2

a / = 7 - + - 7 M~2L,

,,vi(to) = _1 / dnk 1 1 i (2¢r)" k 2 - M2i + iev. k + w + ie

= 8rr2 ~o 1 - In + -fi(w) - 4~oL. (A.8)

Here the divergent term L and the finite loop term f i (w) are given by

L = l - ~ 2 / - t - e ( - 2 + ' y - l - l n 4 ¢ r ) ' •

- f i ( w ) = ~ f - ~ - M 2 ( l n [ w - - - ~ ( o T ~ +i2"n'O(w-Mi)) f o r l w l > M i ,

= - - w 2 ~ '+2 t an -~ ~ _ w 2

(A.9)

Note that f i ( o ~ ) has a kink at to = Mi, which explains the nontrivial behavior of the KN scattering amplitude seen in Fig. 2. In order to get one-loop results, we also need the following well-known integrals


1 f dnk ~ 1 1 J i (q 2) = -f J (-~--~)ntz k2 _ M2i + iE (k - q)2 _ M2i + ie

1 ( M ~ ) - 16~r 2 1 + I n - 2 L + J i ( q 2 ) , (A.10)

where

1 [ V / I _ 4 M 2 / t l n v / I _ 4 M 2 i / t _ ~ ] " (A.11) J i ( t ) = ~ 2 - V/1 - 4M2i/t +

All the integrals needed for the diagrams in Fig. 1 can be obtained from Ai, J i ( q 2) and ~i(to) •

Appendix B. Mass and wave-function renormalizations

B

At one-loop order, we need one counter term,/:e = --SmB Tr BB, to renormalize the nucleon mass. It does not affect KN scattering amplitude, but it is needed to absorb the divergences coming from one-loop self-energy graphs. The nucleon self-energy including the counter terms is

.,~rq (~ ) = 8rob - 2a1~ - 2a2ms - 2a3 (ms + 2~) - 2toh - clto 3

Ai ,~_/ to 4 AD,i ~./i(to "~- l ~ . ~ i( ) ' ~ - ~ i ~'- '~- --t~T)' (B.1)

where to = v • p - may and

h = hlff~ + h2ms + h3(2~ + ms),

~i ( to) =--¼ [toAi + (M/2 -- to2).,vi(to)],

A i = M2 In M2i - M~2L, - 16'n "2

-~i(to) = ~21 [ t o ( 1 - I n .-~/t~d.r..,/M? _ f i ( t o ) ] - 4toL,

fi(to)=~//--A~i - - t o2 (~r+2s in - l -~ - l i i ) ,

l . ( ~ ) L=l--~21Xd.r. -- + y - - 1 --ln4zr (B.2)

and the coefficients Ai'S are

A,r = 3(D + F) 2,

AK = ½ ((D + 3F) 2 + 9 ( D - F )2 ) ,

a,, = ½ ( n - 3 F ) L

AD,rr = 4,~D,K = 2C 2 • (B.3)


Here/.td.r. stands for the arbitrary mass scale/z that arises in the dimensional regulariza-

tion. One can compute Zo and Z3 from 2~N(tO) or write down the corresponding counter terms as in renormalizable theory, (Zo - 1)mB Tr BB and (Z3 - 1 ) Tr-Biv. OB l, However, in a nonrenormalizable theory such as ours, many counter terms enter at order by order, so it is more convenient to use the constraints on ZN(tO) directly. Using a physical scheme, the convenient constraints are

"~N(tO) to--0 ---- 0, 3"~N(t'O)0tO ~o=o = 0. (B.4)

In this case, the roB, that figures in the leading-order lagrangian, can be identified as the physical baryon mass. The constant 8rob is determined by the first constraint to absorb the divergence and h, relevant for KN scattering, can be fixed by the second constraint. The latter is explicitly given by

Cg~N(tO) o__O=-2-h-b ~i Ai ( -1Aid-M20"i( t° ) ) Ow . -~ ao) ,o--o '

2 2 aZi(O)) + 4 ~__.~/ _ lAi .~ 2aT~(_~T ) -~ ( g i _ _ ¢ ~ T ) T ~o- 'T;

_ _ "

(B.5)

The constraints

ZN -- 1 = ~ '~N(09~) = 0 &o o)::::0

(B.6)

completely determine h as a function of/-td.r.. One convenient choice is h = hr(Izd.r. ) + ~ i °tiL where L is the divergent piece, and ~r is the finite part that includes the chiral log terms ln(M2//z2.r.). The renormalized parameter ~(/xa.r.) is related with each other

at different scales/Za.r, through the relation

--r h (A/,~d.r.----/.LI) ~--~/~ (B.7) .--r i( ~T) In #1 h ( ~d.r. = /-/,2 ) ~2

Consider now kaon self-energy which we can write as

/ /K(q 2) = ~ - ~ A i + c.t., (B.8)

where ai are given by

ce~ : -3M~, aK = - 6 M 2, a n = M 2. (B.9)

In the physical scheme, the constraints are

n In standard notation, Zo (Z3) corresponds to the mass (wave- funct ion) renormalization. The counter term (Zo - 1)mB Tr B B corresponds to 8rob Tr BB, and (Z3 - 1 ) Tr -Biv • OB to the hi terms in the v = 3 lagrangian.


HK(q 2) q 2=M2 = 0, ZK -- 1 - aHK(q2)aq 2 q2=M] = 0, (B.10)

where MK is the physical kaon mass 12. Note however that the counter terms in kaon self-energy do not affect the KN scattering directly. So we shall not need the explicit magnitudes of these counter terms for our purpose.

Finally, the scattering amplitudes can be obtained simply by calculating the six topologically distinct diagrams of Fig. 1 using physical masses

7-KN= ~ Ti KN. (B.11) i=a,...,f

This is because there is no contribution from the mass and wave-function renormalizations,

7-r.KNr = a ( 1 - - Z N + I - - Z K ) 23¢Z V. (q + q'), (B.12)

where a = 2 for K+p and a = 1 for K+n.

Appendix C. Renormalization of O(Q 3) counter terms

The quantity gs (gv) is the crossing-odd t-channel isoscalar (isovector) contribution from one-loop plus counter terms which after the standard dimensional regularization, takes the form

1 , ( 1 gs,v=as,v+fls,v+32zr------Sf2M-----~K rs ,v+ L.., s,v--~--ffT L+3--~-~21n ,(C.1)

i=Ir,K,r/ J ""K

where L contains the divergence, and

l aS=-M---~ ( 3 ~ + ( 1 1 _ 2 / 2 + 1 3 ) ( ~ + m s ) ) ,

fls = _ 1 ( (g2 -- 2g3 + g6) -4- (g4 -- 2g5 + g7) + (g8 - 2g9) ) ,

1 av =--M---~K ( - ¼ h + (ll + / 3 ) ( r e + m s ) ) ,

fly = 1 ((g2 + g6) + (g4 + g7) + g8) (C.2)

with

"h = hlff~ + h2ms + h3(2~ + ms). (C.3)

Here we have separated as,v and fls,v because in off-shell amplitudes, the constant as,v is multiplied by to while the fls,v is multiplied by to 3, thus behaving differently for

12 Here we pick the renormalization point at/ZK = MK for the kaon wave-function renormalization. For lr and r/, we take M~r and M, 1 in the tree-order lagrangian as physical masses. This physical scheme is independent of the arbitrary mass scale/Xd.r, figuring in the dimensional regularization.

C-H. Lee et al./Nuclear Physics A 585 (1995) 401-449 439

eo 4= MK. The constants Ys,v - coming from finite loop terms - and ~,v _ multiplying the divergence - are given by

Ys =--3M2K -- 9 MKf~r( - -MK) -- 9MKfn(--MK) -- 9( D + F) 2M2

_ l 3F)2M2 + _ + 4 ( D - ( 3 ( D F) 2 I ( D + 3 F ) 2) M 2

((1 + +

= 1 2 4 2 I I ( D + F ) 2 M ~ yv ~M,~ ~MK + IMKf,~(-MK) - 3MKf~(-MK) +

- h ( D - 3F)2Mg + ( -~(O - F) 2 + ~(O + 3F) ~) M~

+1C12((1 + ~ v ~ ) C , ~ ( - S r ) + v / ~ C K ( - S r ) + ] F ~ ( - S r ) + ~-6FK(-Sr)),

~ s = 9 M 2 - 3 2 ~ ( D + F ) 2 M ~ + l C 1 2 ( 3 + 3 v / 3 ) ( 2 8 2 M 2 ) ~M~r - - ,

K_ 3.,f2 _ ( ~ ( D - F) 2 3F) 2) M 2 - - - - ~ ' " K + 3 ( D +

+lcl ~ (~ + ~ % (~8~ - M~), 8sr/=9~M K2 _ .~M n 3 2 _ 3 ( 0 _ 3F)ZM2,

1 2 I 2 9(D + F ) 2 M Z + iCl2(~ + V/~)(262 M 2), ~v = - ~ M K -- ~ M ~ r -

8v K=I~M K2 - (9(D - F)2+~(D+3F)2)M 2

+1cl2 (_ l + iv5 ) (28~- M~,), 8vn=~M K 3 2 _ ~ M n l 2 _ 8 1 _ ( 0 _ 3 F ) 2 M ~ , (C.4)

where C i ( -T r ) , F i ( -T r ) and f i ( (o) are

Ci(- -8T ) = --( M 2 -t- 82 ) --

F i ( - S r ) = 4 2 682 - g M i + +

V / ~ - M/2, In

36r f i (--8T ) ,

68r f i(--ST ) ,

o , _ + ~ for O) 2 ~ M 2, f/(o~) =

Note that f i ( to ) result from diagram of Fig. le as explained in Appendix A. We write the counter terms as

r div fl~,v = fl~,v + "/5~i~,'v ~ (C.6) O~s, v = O~s, v -~- O~s, v ,

div and /5~ are to cancel the divergence part L in Eq. (C.1) where the divergent parts %,v and the finite counter terms a r and fir are to be fixed by experiments. After removing the divergences, the constant gs,v can be written as


I 1 ~ gs,v = ds,v + fl~,v + 3 2 7 f2M---~ K ~'s,v + ~ t~s,v In . (C.7)

i=~',K,7/

Note that gs,v is/z-independent and hence can be fixed from experiments independently of/z. How to do this for off-shell amplitudes is described in the main text.

Appendix D. l i m B corrections

In writing down the off-shell amplitudes with the parameters determined on-shell, we have assumed that the O(Q 2) terms quadratic in to can be calculated by saturating the Born graphs with the octet and decuplet intermediate states. In calculating the decuplet contributions in the lowest order in limb from the Feynman graphs in relativistic formulation, one encounters the usual off-shell nonuniqueness characterized by a factor Z in the decuplet-nucleon-meson vertex when the decuplet is off-shell:

L=C~tz (ga~ _ (Z + ½) .y~),v) A~B + h.c. (D.1)

This gives the Z-dependence in the limb corrections to ds,v:

l/m=--~8 ((O "q- 3F) 2 + 9(D - F) 2) _~1 _ l l c i21 (z + ~) (1 _ z ) , d--s ' mB

- 1 _ _ + l l c l 2 _ ~ _ ( z + ~ ) ( ½ _ z ) " dv,1/m = - ~ ((O + 3F) 2 - 3 ( O - F) 2) 1 mB B

(D.2)

We shall fix Z from 7rN scattering ignoring SU(3) breaking which occurs at higher chiral order than we need. The precise value of Z turns out not to be important for both kaonic atoms and kaon condensation. Now our chiral lagrangian gives, at tree order, the isoscalar 7rN scattering length

a(+) 1 (2/),rN M2 + -~,rN), (D.3) • rN -- 4,r f2(1 + m~r/ma)

where ~,~N is the 7rN sigma term (..~ 45 MeV) and /),rN = l (d l +d2) + ½(d5 +d6) . If we take the empirical value of -'(+) -0.01M~. 1 , we obtain

/~emp • rN ~ --1.29/rob ~ --0.27 fm. (D.4)

The 1/m correction to 1rN scattering is

/~l/m ~N = - - l ( O + F) 2"~-1 -- 2-1C 2_.L (Z + ~) (½ - Z ) . (D.5) mB 9 mB

For the constants D, F and C used in this paper, this formula reproduces the experimental value (D.4) for Z ~ -0.5. This will be used in our calculation of the off-shell amplitudes.


Alternatively one could fix Z at one-loop order as in Ref. [27]. This however affects our results insignificantly. To see this, take Z ~ 0.15 obtained in Ref. [27]. The resulting off-shell KN amplitudes are given by the dotted lines in Fig. 2. This represents less than 1% change in the kaon self-energy, an uncertainty that can be safely ignored in the noise of other uncertainties inherent in heavy-baryon chiral perturbation theory.

Appendix E. Off-shell amplitudes

In terms of the low-energy parameters fixed by the on-shell constraints, the off-shell K-N scattering amplitude is given by

1 / ) aK-P=4cr(l+o)/mB) TK-P(O)=MK)--ff- o)+m--B---ma..

( O'KN ( ~ + m s ) a , ) + ~ ( o ) - M K ) + ~ ( o ) Z - M 2) ds M2 + d r + 2M 2

1 + 1 } + T ( L p (o)) - L~ (MK)) - T(L~- (O) ) - Lp-(M~:)) , (E.1)

aK- n = 1 (T~-n(o) = MK) 4zr(1 + O)/mB)

O'KN (m + ms)al + 2-~ (o) -- MK) + ~ ( o ) 2 - M2)(ds M2 dv 2M 2 )

1 + _ - ~(L~-(o)) - Ln(MK)) + ~ ( L n (o)) L+ (MK)) }. (E.2)

Here

L+(w) - 64wf 2 [ 2 ( D - F) 2 + ] (D + 3F)ZlMK + ~(D + F)2M,~

+ I ( D - 3F)ZMn - ½ (O + F)(D - 3F) (M,~ + M n) l

0 d (1 - x ) M 2 + x M 2

o) 2 +~-'-~ (4~.,v(~+)(-o))+ 5~.,V(K+)(O))+ 22~+)(o))+ 32'(n+)(o))) ,

_ 3 y ( - ) (o))}, Lp (MK) = (gs + gv)M3K ,

L+ (o)) _ 1 { 64,rrf.2 o) 2 [~(D - F) 2 + ~(D + 3F)2IMK + 3(D + F)2M~r

+½(D - 3F)ZMn + ½(D + F)(D - 3F)(M~. + M n)


1

/ ' } + ~ ( D - t - F ) ( D - 3 F ) ( M 2 + M2) o dx v / ( l _ x ) M 2 + x M2

1 2{ ½ Z ( - ) ( t o ) _ 45_~(-)(to)_ 3Z(1-)(to) } , Ln- (to) = anM~to + flnto 3 + 4 -F to

L2 ( MK) = ( ~ - ~v ) g 3, (E3)

where

~ V ~ i 2 i / ~ Z~+)(w) = - - 0) 2 x O(Mi - Itol) + - M/2 x O(to - Mi),

1 2 to + ~ × o(Itol - Mi)

- 1 2 V ~ ? to2sin - ! to - - - - × O ( M i - Itol)- 2 7 r Mi

(E.4)

Note that X/i (to) result f rom diagrams of Fig. l e (see Appendix A) , where intermediate

states can be real particle and result in imaginary amplitude. The functions L~,n(to )

contain four parameters ap,n and flp,n. Owing to the constraints at w = MK, L~,n(MK), they reduce to two. These two cannot be fixed by on-shell data. However, since the off-shell amplitudes are rather insensitive to the precise values of these constants, we

will somewhat arbitrarily set ap,n ,-~ flp,n in calculating Fig. 2.

A p p e n d i x E In te rpo la t ing fields

In this appendix, we show explicitly that to the chiral order we are concerned with,

physics does not depend on the way the kaon field K (or in general the Goldstone boson field 7 ) interpolates. There is nothing new in what we do below: It is a well-known theorem 13. But to those who are not very familiar with the modern notion of effective

field theories, it has been a bit of a mystery that an off-shell amplitude which does not obey Adler 's soft-pion theorem such as in the case of our lagrangian could give the same physics with the amplitude which does in a situation which involves the equation of state, not just an on-shell S-matrix. To have a simple idea, we start with the toy model o f Manohar 14

El. Adler's conditions in a toy model

Consider the simple lagrangian,

13 A general discussion closely related to this issue is given in a recent paper by Leutwyler [43]. 14 We wish to tlumk A. Manohar who showed us how to understand the problem using this toy model.


/~ = 1 (0K)2 - 1 2 ~MK(1 + e-BB)K 2, (F.1)

where K is the charged kaon field which will develop a condensate in the form (K- ) =

VK e -Ou. As written, this lagrangian can give only a linear density dependence in the kaon self-energy at tree order. The effective energy density for s-wave kaon condensation linear in v 2 is

g = (/z 2 - M2( 1 + EpN))V 2. (F.2)

Thus the chemical potential in the ground state is

/x 2 = M2( 1 + epN). (F.3)

We shall now redefine the kaon field. To do this, we have, from the Noether construction and the equation of motion,

j~ = fO~K,

a~,j~ = fM2K( 1 + e-BB)K. (F.4)

This invites us to redefine the kaon field as

= K( 1 + eB--B) = aaj~ fm~' (F.5)

thus giving the divergence of the axial current as the interpolating field of the kaon field. This kaon field will then satisfy the Adler soft-meson theorem. To see this, we rewrite the original lagrangian in terms of the new field K,

l ( 1 ( 8 ~ ) 2 l 2 ) + O ( # B ) (F.6) / ~ - (1 + e-BB) 2 - 2MK(1 + eBB)K2

=½(1 2eBB) (0K') 2 l 2 ( ) - - ~MK(1 -- EBB)~ "2 + O ~'2(BB)n/>2 + O(OB).

We can immediately read off the KN scattering amplitude and verify that it satisfies the Adler theorem. Note, however, that to obtain the same kaon self-energy that enters into the energy density of the matter system as the one given by Eq. (F.1), it would be necessary to keep multi-baryon terms, which means that in medium, higher density dependence needs to be taken into account.

Now to see that the interpolating fields K and K give the same physics in medium, consider the critical point at which condensation sets in. From Eq. (F.5), we have

I(K)I =I~K = OK(I -a t- ~PN). (F.7)

Substituting this into Eq. ( E l ) or directly from F_,q. (F.6), we get the effective energy

density

~ CF.8) g = (/1. 2 -- M2K(1 + epN)) (1 + epN) 2"


Clearly (E2) and (E8) give the same critical point given by the vanishing of the energy density. This is of course quite trivial. If one keeps terms up to linear in pN, Eq. (E8) reads

= (/.t 2 - M~ - (2/z 2 - M2K)epN)~2K . (F.9)

This is the kaon self-energy in linear density approximation, which satisfies Adler's consistency condition. As it stands, it looks very different from Eq. (E2) , but to the linear order in density, this gives the same pole position,

(/z 2 - M~ - (2/z 2 - M~)epN) = 0 , (1 - 2eptq)/z 2 = M2K(1 -- epN)

, /z 2 = M2K( 1 + epN). (F.10)

Beyond the linear-density approximation, one must use Eq. (F.8) instead of Eq. (F.9) for the physics of the toy-model lagrangian: Consistency requires that all kaon-multi- nucleon scattering terms (that is, terms higher order in PN) he included.

We now turn to the real issue. We shall compare the approach used here (called "Kaplan-Nelson" (KN)) to the Gasser-Sainio-Svarc (GSS) approach which imple- ments Adler's conditions by means of external fields [30].

E2. The KN approach

The KN approach corresponds to introducing the source field Pi in the lagrangian

/~source = p + K - + p - K + . . . (E l 1)

The self-energy is then obtained from

(7(XI, X2)) = ( r ( g + ( x 2 ) g - ( x l ) ) ) p

= - 8 p + ( x l ) S p - ( x 2 ) 7~[K,B] p + =O,...=O p

=iA(0) (Xl -- x2) -t--i~K./A(0) (Xl -- Z ) 4 ( O ) ( Z -- x2) + O ( S ~ ) ,

(F.12)

where (-- .}p represents the expectation value in the ground state of dense matter. Here A(K °) is the free kaon propagator

(2~) 4 1 f e - i k ' ( x - y ) A(KO) ( x -- y ) -- ki--_ -~K -~ i~ d4 k (F.13)

and 2K is the kaon self-energy calculated to order v = 3 with the lagrangian

£o = aK+ OK - - M2 K+ K - ,

/~int =/~i~nt I -{" /~iVnt 2 + ' ' " • ( E l 4 )

Summing the series (F.12), we have

C.-H. Lee et al./Nuclear Physics A 585 (1995) 401--449

i f eik.(xl-x2) 7 " ( X l , X 2 ) - ( 2 ~ . ) 4 . , k 2 _ M 2 K _ X K + i E d 4 k .

The propagator to order v = 3 in momentum space is

i A K =

k 2 - M 2 - .~K -I'- ie

and the effective energy density to order v 2 is

• = --iAffltVKI 2

= - ( ~ = - M~¢ - ZK)IVKI 2.

445

(F.15)

(E16)

( E l 7 )

E3. The GSS approach

The GSS approach corresponds to introducing into the lagrangian the source field Pi of the form

/:source =p+K- ( 1 + otp/~p + otn~n) + p-K+( 1 + ~ppp + t~nhn) + . ' . , ( E l 8 )

where

1 ap - f 2 M ~ (trraq + CKN),

1 O~n- f2m2K (trKN - CKN)

are related with the parameters of the lagrangian (2) by

trKN = - - ½ ( ~ + ms) (a l + 2a2 + 4a3) ,

CKN = - - ½ ( ~ + ms)a l .

As shown in Ref. [30] , this way of introducing the source field reproduces Adler 's conditions (soft-pion, self-consistency, etc.). Now the self-energy is obtained from the

two-point function

('/ '(X1, X2)) ---- -- B p + ( X l ) S p _ ( x 2 ) p~=O,...=OIp

= (T(K+( 1 + appp + Ctnhn) ( x 2 ) K - ( 1 + appp + a.rm) (xl))p

= (1 + appp + anPn)2(T(K+(x2)K-(xl)))p

2 i f e ik'(xl-x2) d4k, (F.19) = ( 1 + Otppp + anpn) ~ k 2 - M2K -- .~r + ie

where 2K is identical to that o f the KN approach since the t~p and an are v = 2 terms that are unaffected by loop corrections (o f order v = 3). Finally, the full propagator is

given by


d~ = ( 1 + Otppp + OtnPn) 2 i k2 - M 2 - .~K + i e ' (E20)

which can be rewritten in the form of Eq. ( E l 6 ) ,

i ArK = k 2 - M 2 - . ,~ + ie (F.21)

with a redefined self-energy

, 1 { ( k 2 - M 2 ) [ ( l + o l p p p + a n P n ) 2 1 ] + X K } (F.22) "~K = ( 1 + O~ppp + a n P n ) 2 - - "

One can verify that to the linear order in density, this has the structure mentioned in the main text, namely, it changes from an attraction on-shell to a repulsion off-shell.

Finally, the effective energy density linear in 0~ is

I OK 2 g = --i(A~;) --1 [/~K]2 = _iAKI 1 + O~ppp + O~np n ' (F.23)

where OK is the kaon expectation value in terms of the GSS field.

E4. Effective energy density and physical observables

As in the toy model, we can write OK in terms of UK:

~K = VK (1 + ax + b), (F.24)

where a = (Otp - O~n)p and b = Otnp. Thus the energy densities to all orders in OK and

0K can be written as

e-- y~ a,,(~, x, p) IOKI" -t-~(/z,x,p), (F.25) n~>2

I 0._K_K b n g = Z an(/ ' t ' x ' P) l + a x + + / 3 ( I * , x , p ) , (E26)

n/>2

where an (/*, x, p) and/3(/z, x, p) are given in Eq. (37). By differentiating the effective energy density with respect to l~,X, VK (~K), one can get the equation of state as a function of density p. In the GSS approach, we have three equations of state,

( g~K ) n O/3(l"t,x,P) 0 = Og. Oan( i~ ,x ,p ) 1 + Op, o , , + '

n>:2

Og: Z c?°tn(l~,X,P) ( O__K.K ) n O/3(I-t',x,P)

0 = ~ = n>~2 OX 1 + ax + b + Ox

VK ) n

a l + T x + b ' 1 + ax + b ,,>2


O- O~_ 1 ZnOZn(IZ, x ,p ) l + ~ x x + b (F.27) OUK OK n~2

Now using the third equation, we see that the last term in the second equation vanishes. To see that the two ways give the same physics, it suffices to note that the KN approach gives the same set of equations except for the replacement VK = VK( 1 + ax ÷ b) which does not affect anything.

An identical conclusion is reached by Thorsson and Wirzba [44] by a slightly different reasoning.

Appendix G. In-medium modifications

In this appendix we write down the explicit forms of the quantities that enter in Eq. (28).

1{ 12f4 (D + F) 2 [(M 2 + M 2 - w2)D~ar~r - D~,~-zr ]

+ l ( D + F) 2 [(M 2 + M2K -- O)2)DP ~.~. - D~,,~.]

_ _ - -

- ( D ÷ F ) ( D - 3 F ) [ ( I M 2 1 2 _ D p _ D p

+ ~ - ~ (8zP(o)) + Z~(o))) + 2(D + F)FM2g(O)

+ ( 0 + F)2 M2 X~r( O) - 2 ( 0 - 3F) FM~XP~(O) }, (G.1)

~ ' T K - n = l l f f { (D+F )2 [ (M2 +M2 -O)2 )DP4 , ~ -Dg , ~ , ]

÷ ½ ( D + F)2 2 2 , 6:rzr] [(M~ + M K - o)2)D~ ~.r - Dn

+~(0 3F) 2[(M~-l 2 o" -0~,,,] - gM,r - - 0)2) 4,*p/

2 o n + ( D + F) (D - 3F) 1 2 gMK - - 0)2) 4,1r~

o) 2 + ~ - g (2..,Y~(o)) + 2:P(o))) + 12-~{ ( 0 2 - F2)M2~(O)

+(D + F)2U2rrg(O) + (D - 3F) (D - F ) M 2 g ( 0 ) }, (G.2)

kF N

N 1 f d l k I 1 ik[21+M~jlk[,~, D,~,i j = 2,n. 2 Ikl 2 + M 2 o

kF N

1 / dill Ikl2 (G.3) xN(o)) = ~ o)2 _ M/2 _ ikl 2"

0

K-p a:T~ -


Here the subscripts 7r, ~/and K are the octet Goldstone bosons, the superscripts n and p stand for neutrons and protons.

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