Thermal and quantum fluctuations in the kaon-condensed phase

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 653 (1999) 133-165 www.elsevier.nl/locate/npe

Thermal and quantum fluctuations in the kaon-condensed phase T o s h i t a k a T a t s u m i I, M a s a t o m i Y a s u h i r a 2 Department of Physics, Kyoto University, Kyoto 606-8502, Japan Received 23 October 1998; revised 19 March 1999; accepted 27 April 1999

Abstract

A new formulation is presented to treat thermal and quantum fluctuations around the kaon condensate on the basis of chiral symmetry. Separating the zero-mode from the beginning and using the path-integral method, we can formulate the method of taking in the fluctuations in a transparent way. Nucleons as well as kaons are treated in a self-consistent way to one-loop order. The effects of the Goldstone mode, stemming from the spontaneous breakdown of V-spin symmetry in the condensed phase, are figured out. A procedure is discussed to renormalize the divergent integrals properly up to one-loop order. Following this procedure the thermodynamic potential is derived. It is pointed out that the zero-point fluctuations by nucleons give a sizable effect, as compared with that by kaons. @ 1999 Published by Elsevier Science B.V. All rights reserved.

PACS: l l.30.Rd; 12.39.Fe; 13.75.Jz; 21.65.+f; 26.60.+c; 97.60.Jd Keywords: Chiral symmetry;Non-linearLagrangian; Kaon condensate; Fluctuations;Thermodynamic potential; Renormalization

1. I n t r o d u c t i o n

It has been extensively discussed that kaons, the lightest hadrons with strangeness, show characteristic features through k a o n - n u c l e o n interactions once immersed in nuclear medium. The scalar interaction (called the sigma term) and the vector interaction (called the Tomozawa-Weinberg term) are the leading ones, and as a net effect, they bring about a large attraction for the excitation energy of K - mesons at low momentum, and l E-mail: [email protected] 2E-mail: [email protected]

0375-9474/99/$ - see frontmatter @ 1999 Published by Elsevier Science B.V. All rights reserved. PIIS0375-9474(99)00170-0

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a repulsion for K ÷ mesons. There are many proposals to see the consequences through these interactions experimentally; K - enhancement in the subthreshold kaon production or the anomaly around the ~b meson peak in the dilepton production from the relativistic heavy-ion collisions are some of them [ 1-3]. In 1986 Kaplan and Nelson suggested the possibility of kaon condensation in dense matter by using a chiral Lagrangian [4]. Since then this subject has been studied by many authors [5]. The s-wave attractions between kaons and nucleons remarkably reduce the excitation energy of K - mesons in neutron-star matter and thereby kaons should condense when the degeneracy energy of electrons exceeds it. It is well known that the kaon condensation results in the large softening of the equation of state (EOS) and the inducement of new processes due to the presence of the condensate. Since it may occur in neutron-star matter at relatively low densities, (3-4)p0 with Po being the nuclear density _~ 0.16 fm -3, it is believed to have many implications for neutron-star physics; the fast cooling mechanism, rapid rotation or the low-mass black hole scenario are typical examples [6-8]. We have studied this subject from the point of view of chiral rotation on the chiral manifold G/H with G = SU(3)L × SU(3)R and H = SU(3)v [9]. We have defined the kaon-condensed state by letting a unitary operator 0K act on the normal matter 10) (the meson vacuum); I K ) = / ) x ( ( 0 ) ) 1 0 ) with (Jx((O)) = exp(i(0)F45), where F45 is the fourth component of the axial-vector charge of the SU(3)L x SU(3)R algebra and (0) is the chiral angle. 3 Since the kaon-condensed state is a chiraily rotated one, the change of the kaon-nucleon dynamics from that in normal matter arises from the symmetry breaking terms in the Hamiltonian. Thus we have succeeded in extracting the essential features of kaon condensation in this formalism. So far almost all of the studies have been done using the mean-field theory, the classical approximation without any loop. Recently there start to appear studies about kaon condensation in protoneutron stars [ 10], where thermal effects or the roles of neutrino trapping become important and we must treat it beyond the tree level. Low-mass black holes, if they exist, may be produced in this era [8]. A numerical simulation based on general relativity has already been done about the delayed collapse of protoneutron stars to low-mass black holes due to kaon condensation [1 1 ]. However, they used a cold EOS for the kaon-condensed phase, while the temperature (T) rises to several tens of MeV there. Prakash et al. treated the kaon-condensed phase at finite temperature and discussed the properties of protoneutron stars within the meson exchange model [ 10], since there is no consistent theory based on chiral symmetry. In a recent paper, Thorsson and Ellis have tried to include quantum or thermal fluctuations within heavy-baryon chiral perturbation theory (HBCPT) [ 12], freezing dynamical degrees of freedom of nucleons to the thermodynamic potential [ 13]. However, they have discussed only the effect of the zero-point fluctuations of kaons at T = 0 to oneloop order. The most interesting result in their paper may be the dispersion relations of kaonic modes in the condensed phase, which is a key ingredient for the discussion 3The notation (O) implies the thermal averageof 0.

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135

Fig. 1. Chiral rotations on the SU(3) manifold. Circles C, CK indicate the fluctuation "areas" around the vacuum (O) and kaon-condensed state (K), respectively. The symbols F and FK stand for the fluctuation points around the vacuum and kaon-condensed state, respectively. of thermal properties of the condensed phase. They have used the Kaplan-Nelson Lagrangian, where the basic variable is the SU(3) matrix specified by the eight Goldstone fields ~b~, U(fb,,) = exp(2iT~,dp~/f) with f the pion decay constant and T~ the generators of SU(3) Lie algebra. They picked up only the degrees of freedom for the charged kaon fields K +, namely ~b4 and ~bs, and separated the classical fields, given by the condensate, and the fluctuation fields around them in the standard manner, K 4- = (K ±) + k ±. The resulting dispersion relations for kaonic modes have a very complicated form due to the interaction between the condensate and fluctuations. Accordingly, other thermodynamic quantities also become too complicated to be tractable. Thus there is no consistent calculation at finite temperature for the chiral case. In Ref. [ 14] we have presented a formalism to treat the kaon condensation at finite temperature within HBCPT. We have introduced fluctuations around the condensate by applying the chiral transformation on the chiral manifold successively; after generating fluctuations around the meson vacuum by a chiral transformation (r/) we transform this state further by another chiral rotation ( ( ) , the latter of which is the same as that in the classical approximation (see Fig. 1). Since the constant condensate can be considered as a kind of zero-mode, this procedure can be regarded as a separation of the zero-mode from the original matrix U(qba). A similar procedure has been applied in a different context to see the finite-volume effects of the chiral symmetry restoration [ 15 ]. Then we can obtain the dispersion relations for the kaonic excitations around the condensate. We have seen that the dispersion relations show a feature similar to the one by Thorsson and Ellis, whereas its form is very different. We have also proposed an approximation which simplifies the expressions of the thermodynamic quantities considerably. We have checked numerically that this approximation works very well [ 14]. In this paper, we treat the dynamical degrees of freedom of kaons and nucleons beyond HBCPT; we take into account the modification of the nucleon single-particle energies not only by the condensate but also by the fluctuations. For this purpose we use the Hartree approximation for the four-point vertices among kaons and nucleons; the nucleon Green function contains kaon loops and the kaon Green function contains

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nucleon loops. Therefore we sum up the infinite series of bubble diagrams composed of the one-loop diagrams by kaons and nucleons in a self-consistent way. We shall see that our formalism makes the analysis of the structure of kaon-nucleon ( K N ) dynamics transparent and clarifies the physics included there. Generally these loop diagrams include divergences, which should exist even at T = 0. We discuss how finite contributions can be extracted in our formalism. Analyzing the structure of the self-energy terms, we show that these divergences are properly renormalized to one-loop order by introducing relevant counterterms corresponding to the nucleon and kaon masses, and the K N and K K interaction vertices. Then no more counterterms are needed to renormalize the thermodynamic potential. The effects of the zero-point (quantum) fluctuations by kaons have been shown to be small [ 13,14]. We shall see that the zero-point fluctuations by nucleons have sizable effects on the ground-state property of the condensed phase. In Section 2, we present our formalism to treat fluctuations around the condensate by way of the path integral approach. In Section 3, we develop our approach and perform the one-loop calculation. There we get the dispersion relations for kaonic modes which are key objects to discuss the properties of the kaon-condensed phase at finite temperature. We discuss in Section 4 how the thermodynamic potential is renormalized by analyzing the self-energy terms. We also give some numerical results about the zeropoint fluctuations by nucleons. Section 5 is devoted to some discussions and concluding remarks. In Appendix A we discuss the separation of zero-modes, one of which is the would-be condensate. Some results within the mean-field approximation are summarized in Appendix B. In Appendix C, we give some properties of the kaon Green functions and expressions of the loop contributions.

2. Path integral formulation 2.1. Partition function

We start with the Kaplan-Nelson Lagrangian as a chiral Lagrangian [ 16,17], Echiral : E0 + £SB,

(2.1)

where /~o is the symmetric part, 3,.2

£o = ~ - tr(O~Uta~U) + i tr{/~(~ - m e ) B } + D tr(BTuys{a~', B})

+ F tr(/~y#y5 [ A'~, B] ),

(2.2)

and ESB the symmetry breaking part, ESB = X tr rh,l(U + Ut - 2) + al tr/~((&q~ ÷ b.c.) B + a2 tr/~B((rhq~ + h.c.) +a3{tr/~B} tr(l~z,:U + b.c.)

(2.3)

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137

with the quark mass-matrix ~q ~_ diag(0, 0, m,.). The coefficient g implies the vacuum expectation value of the quark bilinear, - X = (01Oql0), and the coefficients ai measure the strength of the explicit symmetry breaking: the kaon mass is given as m 2 ~_ g m ~ / f 2 and the K N sigma terms as 2:Kp = - m s ( a l + a2 + 2a3) and ~Kn = - m , . ( a 2 + 2a3). U is the SU(3) matrix parametrized by the eight Goldstone fields ~,,, U = exp[2iT~,dpa/f] E SU(3) ~ G / H

(2.4)

and V#, A~ are the vector and axial-vector fields constructed by U: vu

= ½(fray( + fa#ft), i

(2.5)

with ( = U I/2. The baryon octet B can be represented by the SU(3) matrix, so

Ao

~?+

_ _

so B

p

Ao

=

~-

so

-2A 0

Then the covariant derivative D~ is defined with the vector field V~, (2.6)

D u B =_ a~B ÷ [ V ~ , B ] .

The transformation properties of the fields U and B under the group G are found in, e.g., Refs. [ 16,17]. To treat thermal and quantum fluctuations consistently we use here the path-integral formulation. It is well known that chemical potentials /za, which should be taken into account to ensure various conservation laws in nuclear matter, can be introduced by "gauging" the Lagrangian with artificial "gauge fields" A~, =- TaA~ with Aau = (/xa, 0). Here we must take into account conservation of two quantities: electromagnetic charge and baryon number. Accordingly we consider two chemical potentials: with the charge chemical potential/zx, which, we shall see, is identified as the kaon chemical potential, the "covariant derivative" D~U for the field U then reads Du=oU = OU/Ot + ilZK[ Tem, U],

79iU = OiU

for others,

(2.7)

with the charge operator, Tem ~ 7".3-}- I/V/'3T8 = d i a g ( 2 / 3 , - 1 / 3 , - 1 / 3 ) . With the baryon chemical potential /z,,, which, we shall see, is the neutron chemical potential, the "covariant derivative" ~Tu=oB for the baryon field reads Vu=oB = OB / Ot - itz,,B + itzx [ Tem, B] + [ Vo, B ] ,

Vi B = DiB

for others. (2.8)

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Therefore the partition function in the imaginary-time formalism can be represented as follows (7" = it, fl = l / T ) : gchira, =

xftdu] [dB][d[~]

(2.9)

exp[ Schiral ' ],

/3

Stchiral=

/3

J/

d3XCchiral(T)#U, U, V#B, B) =fdrfd3x[£chiral + 6 £ ] ,

dT

o

(2.10)

0

where the "time derivative" should be read as the derivative with respect to imaginary time. We can see that there appears an additional term 6 £ besides the original chiral Lagrangian/~chiral, which is in general non-invariant under the chiral transformation. For the Lagrangian (2.1), we find ~_. -

f2['ZK tr{ [ Tern, U] OU~; + OU

~-7

-ff~rITem' U*]}

/~K t r { B , [ ( ( t [ T ~ m , ~ ] + ([T~m,~t]), B] } 2

"~ 2

f-#Xtr{[Tem, U][Tem, U*]}+tx,,tr{BtB}-IzKtr{B*[Tem, B]}. 4

(2.11)

Here the first term corresponds to the mesonic charge, and the last two terms indicate nothing but the baryon number and the electromagnetic charge of baryons.

2.2. Transformation of the coordinates We show our formulation to treat thermal and quantum fluctuations around the condensate in the partition function (2.9). Before that let us recall the tree case. As already mentioned before, the kaon-condensed state tK) can be represented as a chiral-rotated state [ 9 ] IK) = 0K((O)) ]0)

(2.12)

with the SU(3) operator, Ux((O)) --- exp(iF45{0)). We can specify this state on the manifold G/H coordinated by the Goldstone fields, where the meson vacuum 10) corresponds to the fiducial point (the origin, q~,, = 0). Then the kaon-condensed state (2.12) corresponds to the SU(3) matrix Ux = exp[2i{T4(~b4) + Ts(cbs)}/f], and can be represented as a chiral rotated one from the vacuum on this manifold,

Ux( (O) ) = ¢Uv(= where Uv = U(cb,, = ( = exp(i(M)/x/2f)

M=

0 0 K-0

0 0

(2,

(2.13)

0) -- 1 corresponds to the meson vacuum and ( is a group element, with the constant matrix

,

K±=(~4+i~5)/x

'/~

and

02-2K+K-/f

2.

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

139

q O:

~ [ U f = q. q =exp(2iTatSa / 0 ] r[

l=¢xp(2iT a - 0If) [ -. %%% % ~

-, TE %%% % %

K:

U K = ~ " I "~ =cxp(2iTa ,~ >If) ]

I

//

u=;

uf;

]

:FK

Fig. 2. Diagram showing the relation among the transformations on the chiral manifold (cf. Fig. 1). The symbol TE corresponds to the one by Thorsson and Ellis 113]. Thus we have seen that the classical kaon field (K +) induces the chiral rotation on the manifold [9]. When we introduce the quantum or thermal fluctuations, there may be several ways (Figs. 1 and 2). Thorsson and Ellis introduced the kaon fluctuation fields K+ as deviations from the classical kaon fields in the standard manner [ 13], K :~ =
(2.14)

Instead, we introduce the fluctuation fields by developing the idea of chiral rotation mentioned above. First, we notice that any fluctuation of the Goldstone fields can be introduced by a chiral rotation r/ from the vacuum Uv = 1, "

U f = ~TUvrl = .q2

(2.15)

with ~/ = exp(iT,,da,,/f). 4 The subsequent chiral rotation by ( transforms Uf into the form

( :

U((O),dp,,) = ( U t ( ~ , , ) ( .

(2.16)

We can easily see that in the limit ~b,, --~ 0, which corresponds to the previous classical approximation, the matrix U is reduced to U,v in Eq. (2.13). Thus we have introduced the fluctuation fields around the condensate by the two-step procedure. 5 It may be also worth noting that we can easily take into account fluctuations of any meson field in the kaon-condensed state by the form of Eq. (2.16). 6 Accordingly, ge operator, which is defined through U = (2, can be obtained by solving the subsidiary equation ( = (r/u t = ur/(,

(2.17)

where the matrix u is defined by the second equality. It depends on t~a and (0) nonlinearly in a complicated way, so that it is very difficult to find u for the general form 4 Hereafter, we omit the tilde on the fluctuationfields for the sake of simplicity. 5 The similar decomposition was also used in the context of the loop expansion within chiral perturbation theory 1181 or the bound-state approach to the Skyrme model [ 19]. 6 Alternatively,we can regard this procedure as a separation of the "zero-mode" from the full SU(3) matrix U(~ba) (see Appendix A). Using Eq. (A.16) we can see that the value of the thermal average (0) should be determined by imposing the extremum condition for the resultant thermodynamicpotential at the end.

T. ~ttsumi. M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

140

of UU [ 16]. A systematic way to find the form of u may be the perturbative method: expanding the matrices "r/ and u with respect to the fluctuation fields ~ba, we can solve Eq. (2.17) order by order. Hence we can find the relevant form of u up to any order of &,,, depending on the problem or the approximation we are interested in. On the other hand, if we restrict ourselves to the fluctuations of the kaon sector such that U f = exp(iv/2M/f),

(2.18)

we can find u in the closed form u = diag(K*/lK[, I, K/IKI),

(2.19)

with K = cos((0)/2) cos(0/2)

K~(K-~) s i n ( ( 0 ) / 2 ) sin(0/2).

I(K)tlKI

(2.20)

Here the following relation is to be noted, u~rTemtt = Ten>

(2.21)

Defining a new baryon tield B' by the use of the matrix u,

B' = ut Bu,

(2.22)

we can see that the chiral-invariant pieces of the Lagrangian are not changed and only the symmetry breaking pieces play essential roles: /~chiral(U, B) = £ o ( U [ , B ' ) + £ s u ( ( U / ( , u B ' u t ) ,

g £ ( U , B) = 6£((Uu(, uB'ut).

(2.23)

Thus all the dynamics of kaons and baryons in the condensed phase are completely prescribed by the non-invariant terms under chiral transformation. It is to be noted that this feature is quite similar to the one within the classical approximation at T = 0 [9]. Using Eqs. (2.19), (2.22), we find ~ £ . ( ( U f ( , uBtlt ¢ ) = - ( c o s ( P ) - l)#,v tr{B'¢ [ V~, B'] } + f21x2/2 sin2(0)

- / z x cos(O)(K +c)K- /ar - aK+ /OrK - )

,÷txx cos(O)K+K-/,[-2 tr{B'* [ V3, B'] } + cos2(O)~K+K - - ~x2/4 sin2(0)(exp(i'y) K - + e x p ( - i T ) K+) 2 +p.,, tr{B'*B'} -/XK tr{B'* [Tern, B ' I } + . . . .

(2.24)

with the V-spin operator, 1/~ = ½(T3 + ',/3T8). The phase 3' is defined by exp(4-iy) = (K+)/I(K)], which can be absorbed into the fluctuation fields by redefining it, exp(::Fi.y)K ± -~ Km. In other words, the phase of the condensate is arbitrary, reflecting V3 symmetry in the chiral Lagrangian, and we can choose y = 0 without loss of generality. As a consequence,

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141

I/3 symmetry is spontaneously broken in the condensed phase, and thereby we shall see the Goldstone mode associated with this symmetry (see Section 4). 7 If £sB originates from the quark mass term, £mass = --{lmqq, we can further explore the transtormation property of £SB in a model-independent way: first we can see that /~mass ----+/~mass --

isinO[f'~,£mass] + (cos0 - I)--YK

(2.25)

with the sigma operator Sx = [ F45, [ P,~, £.1a~] ], under the chiral transformation corresponding to r/ in Eq. (2.15). Similarly, £mass transforms like (2.25) with (0) instead of 0 under the chiral transformation corresponding to ( in Eq. (2.16). Hence £mas~ eventually transforms, under the successive chiral transformations corresponding to (2.15) and (2.16), like £ma,~ ~ £ma~s --/(sin ((0) + 0) -- sin 0) [k~, £mass] + (cos((0) + 0) - cos 0) Zx, (2.26) where we have used the property [P~, [F45, [F45,£ma.~] ] ] = [F45,Emass]. The second term is a parity-odd operator and irrelevant due to the fact, (ClYsq) = 0. The term proportional to sin(0) implicitly included in the last term is also irrelevant in the following calculation because of the thermal-equilibrium condition (cf. Eq. (A.16)). Therefore there is essentially left the term /~mass --+ /~mass q- (COS(0) -- l ) C O S 0~'K.

(2.27)

Accordingly, in terms of the effective Lagrangian EsB, it should transform like

£SB(fUr(, uB'ut) = £sB(U/, B') + (cos(0) - 1) cos 02'x( 1, B')

(2.28)

besides the irrelevant terms, where XK(1, B') has the standard form

~ x ( 1, B') = f2 m x - Zxpp'p' -- .SKnh'n',

(2.29)

for nuclear matter, s In Eq. (2.29) we have used the following notations: f2m:k = x [ m , + m~],£I¢pO,) = (P(n)l~x[P(n)). This consequence is rather general and, of course, consistent with that in the Kaplan-Nelson Lagrangian (2.3). The integration measure (N -- 3) is given as N2 - I

[dU] = M(4~) U

[d4~],

(2.30)

i=1

where M ( ~ ) is the Lee-Yang term [ 2 t ] ; M(q~) = exp[ ½~5(4) (0) fo~ dr f d3x In g(q~) ] with the Cartan-Killing metric on the SU(N) manifold from the invariant line element, ds 2 = gik(c/a)ddpiddpk = ½tr[dUdUt]. It is to be noted that the Lee-Yang term is important to make the integration measure chiral invariant. 9 The measure (2.30) is 7 We met the similar situation in the context of pion condensation [ 20]. 8 We, hereafter, omit the prime on the nucleon fields for simplicity. 9III Refi [13], their treatment of the Lee-Yang term is obscure: 3A/2 in Eq. (13) in Ref. [t3], which corresponds to g in our notation, should read expl t 6 ( 4 ) ( O ) f ~ d~"f d3x In Y l.

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T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

invariant under the transformation (2.16) by (,

[dU] = [d((Uf() ] = [dUll,

(2.31)

tr[dU:dU)],

because the chiral group G is the isometry, tr[fdUffftdUtf( t ] = so that the geometry around the condensed point is the same as in the vicinity of the vacuum. Since we are concerned with the fluctuations, the measure can be approximated by the one with flat curvature [15], [dUf] ~- [IiNel-' [dcbi]. In this case, we are not worried about the Lee-Yang term because of g = 1 + 0(052/f2). Thus we find Zchiral "-- /

N H I [dc~i][dB/][dB']expl./dr /3 i= 1

peff

.

f d3x"eff Z--chiral k"~" (:,, O r f , Bt)I,

(2.32)

0

where ~chi,-al(g, Ut, B') Eqs. (2.24) and (2.28).

=- £o(Uf, B') + £sB(fUf(, uB'u t) + 8E((Uf(, uB'u t)

with

3. Thermodynamic potential

3.1. Partition function for a chiral Lagrangian Using the formulation given in Section 2 we evaluate the partition function in the kaon-condensed state. J0 In the following calculation we only consider the one-loop diagrams, and thereby we retain only the quadratic terms with respect to the kaon field in ~chiral'/-eft"Moreover, we are concerned here with only charged-kaon fluctuations and nuclear matter without hyperons, l~ Then we can write Zchiral = N

/

[d~b4] [d~5] [dO t ] [ d ~ ] exp[~tl,.at]

(3.1)

with the spinor

by the use of Kaplan-Nelson Lagrangian (2.1). The effective chiral action /3

t, = /. ~hiral

dr jf

d3x Z::e. chir.l('L U t,

B'),

o

can be written as ff
(3.2)

m We briefly discuss the case of the classical approximation in Appendix B, where we can see that our previous results given in Ref. 191 are recovered. 11 An extension to include hyperons may be straightforward, while there is still a large ambiguity about their coupling strengths with hadrons, see, e.g., Ref. [ 10l.

T. Tatsumi.M. Yasuhira/NuclearPhysicsA 653 (1999) 133-165

143

where Sc is the classical kaon action

[1 2~2

]

Sc =/3V [~P'KJ sin2(O) - f2m2x(l - cos(O)) ,

(3.3)

SK the kaon action Sx = / dr f d3x EaaK+O.K- - cos(O) ( mZx -1.t2x cos(O) ) K+ K o --tZKCOS(O)K+ ~ K- - tze--~rsin2(0)(K + + K-) 2 + 4

with the shorthand notation OaK+OaK- -~ -arK+OrK - nucleon action

SN=

//E(

dT" d3x ~

- y ~ +iy.~7-M+tzuy

O(IK[4)]

(3.4)

~TK+XTK-, and SN the

0

0

+/ZK( 1 - COS(0))'y0 ~-~73 3 + ( I - COS(0)) .~KN) ,~/t].

(3.5)

The interaction term, Sint, mainly consists of the two terms, the sigma-term and the Tomozawa-Weinberg (TW) term,

Sint= / dT"J d3x[~cos(O)~.,~KN~K -K+ 0 1 -

3+7"3.

'--*

cos(O)K+K - } +O(IKI4)]

(3.6)

with the KN sigma term 2?my. The sigma-term 2m¢ and nucleon chemical potential/-to are 2 x 2 diagonal matrices in the isospin space:

"SKN=( 2KpO XXn0)

and

I z N = ( tzp ) 0 / x0n

with/1, t, - / x n - / Z X , respectively. ff I is already bilinear with respect to the nucleon field, Since the effective action S~chira we can easily integrate out the nucleon degree of freedom in Eq. (3.1) to leave Zchiral = g I t d t ~ 4 ] [dq~5] exp[Sc + Sx] DetGo

=

N/Ida4][d4,s]

exp[Sc + Sx + Trln Go],

(3.7)

with the Dirac operator Go. The capital letter for determinant or trace is used for the Dirac operator Go of infinite dimension over space-imaginary-time. The Dirac operator Go is composed of two parts, Go = Go + 3G, with the "free" part

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T. Tatsumi M. Yasuhira/NuclearPhysics A 653 (1999) 133-165

(a)

(b)

(c)

Fig. 3. Three-loop diagrams. (a) and (b) are the direct (bubble) diagrams, which are taken into account by the Hartree approximation. (c) is the exchange diagram, which should be suppressedby the power of density or temperature in comparison with (a) or (b). 0

Go=--~-~T+iyo'y'lT--Myo+lXU+tZX(l

-cos(O))

3 +7 3 4

+ ( 1 - cos(0) ) ~/O~KN,

(3.8)

and the interaction part

8G = ~

cos(O)yoK- K + 1 3 + T3

f2

8

{K + O~ K - - 2ttKCOS(O)K+K - } + O(IK[4).

(3.9)

A naive treatment may proceed with treating Go as the zeroth-order operator and expanding Tr In Go in terms of 6G, Trln Go

=

Trln Go + Tt'[GoI 6G] + O([KI4).

(3.10)

The first term on the r.h.s, of Eq. (3.10) gives the one-loop contribution of nucleons to the partition function, and the second term to the kaon self-energy term. The latter, together with SK, gives the one-loop contribution of kaons to the partition function. However, this procedure is not adequate for treating the KN interactions in a consistent way; a simple analysis of the loop counting shows that the kaon one-loop contribution includes an infinite series of nucleon loops, while the nucleon one-loop contribution does not include any kaon loop. In the following we use the Hartree approximation to take into account all the bubble diagrams by both nucleons and kaons non-perturbatively, while the exchange diagrams are discarded. Generally speaking, the exchange diagrams are suppressed in the high-density or high-temperature limit, compared with direct diagrams (see Fig. 3 for the three-loop case) [22,23]. We define a new operator GH, GH = Go + (~G), by taking the thermal average of t~G for kaons, 0

3+7" 3

Gt4=--~-~T+iyo'y'V+tZN+----~(n~+2tXK(l--Cos(O))) with the effective-mass matrix of the nucleon,

M* =

(')

MI, 0 OM* 7

•

-M*y °

(3.11)

T. Tatsumi, M. Yasuhira/NuclearPhysics A 653 (1999) 133-165

145

The effective mass M T is given as

M* = M - Xxi/ f 2 cos(0)n~ -- Xxi(1 --cos{0)),

(3.12)

where n ~ - - n ° + 21zxcos(0)n~ with n ° = (K + aT K - ) , and n~ - (K+K-). The quantities n~ and n ° mean the thermal kaon-loop contributions for the self-energy of nucleons and they can be represented in terms of the thermal Green function of kaons (see Appendix C). 12 Using GH we can rewrite Eq. (3.10) as Trln GD = T r l n G H + Tr[GHt3G] - Tr[G~41(BG)] + O(]KI4).

(3.13)

Substituting Eq. (3.13) into Eq. (3.7), we find Zchil-al

N./[dd?4] [d095] exp[Sc + ffKff -- Tr[G~4I(fG)] + O([K]4)] D e t G n , (3.14)

with the effective kaon action, fixf~ = Sx + Tr[GHI~G], ~e

S~xo= f dr f d 3 x [ - O~K+O~K - + cos{O)(tzZ cos(O) - mZ + ~r + 2bl~K)K+K o - ( t z x cos(O) + b ) g + a~ K - - #2---~-K sin2(O)(g + + g - ) z + o ( [ g l 4 ) ] 4 (3.15) where o" _-- f-2Tr[GHITo~YKN] and b - f - 2 T r [ G ~ l (3 + 7-3)/8] = ( 4 f 2)-1 (Pn + 2pp) with nucleon number densities pi(i = n,p). It may be worth noting that the selfenergy term o- contains some divergences (see Section 4), and is reduced to O'tree f-2(~KpPp + ~KnPn) in the static limit of nucleons [ 14]. Expanding the fluctuation fields according to

~4,5 = ~//-~7~Z v v

ei(p.r+

. . . . ) ~b4,.~ . . t ' .~,,, v .,j

(3.16)

n,p~0

with the Matsubara frequency o~,, = 27rnT and substituting it into Eq. (3.15), we find ~Kff• = - - ~, ~

,,p~o

( q ~ 4 ( - - n , - - p ) , q 5 5 ( - - n , - - p ) ) D eff (q~4(n,p)

\~bs(n,p)

)

(3.17)

"

It is to be noted that the zero-mode (n = 0 , p = 0) should be removed to avoid the double counting (see Eq. (2.16)). Here the inverse thermal Green function D en can be written as 12The quantity n}¢ includes the divergence and should he properly renormalized. We shall meet other quantities that also include some divergences in this section. We leave the renormalization problem in Section 4 and hereafter treat them schematically in this section.

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

146

Delf = 132

( o)~ + (6J_ --/2K) (&+ +/2r) +/x2 sin2(0) --2(/2K + b)o~.

2([xx+b)~on ~o~ + (go_ - ~K)(go+ + [zr)

/

,

(3.18) where /2K =/xh- cos(0) and (5:t: = + b ÷

(p2 ÷ C2)1/2

(3.19)

with C 2 = &~2 + b 2 and the effective mass, gnU:2 -- c o s ( 0 ) [ m } - o']. It is to be noted that the effects of the condensate come in through the modifications of the chemical potential,/zK ~ /xX cos(0) except the additional term,/x 2 sin2(0) and the effective mass, m~.2 = [rn~ - o'] --, c o s ( 0 ) m ) 2. Then, the partition function can be simply represented as In Zchiral =

Sc

-

Tr [ G~41 (fiG) ] - 1/2Tr' In D

eft. ÷

Tr In Gl4,

(3.20)

where the symbol T ( means that the zero-mode should be removed. 3.2. Dispersion relations f o r kaonic modes The excitation energy of the kaonic modes are given by the solutions o) =-- iwn, which satisfy IDeff[ - det(D en) = 0 (see Appendix B) w 4 - 2(cl + c3 + 2c~)w 2 + 2clc3 + c~ = 0,

(3.21)

where cj = ((5_ - / i x ) ((5+ + / i x ) , c2 = / i x + b and c3 = l/2/x2x sin2(0). Then we have two solutions which correspond to the K :k modes, E2~ = (cl + c3 + 2c~) + ~/(c3 ÷ 2c2) 2 + 4clc~.

(3.22)

In the limit (0) = O, cl ---+ ( w _ - Izx)(o~+ + ILK), C2 ---' #K + b and c3 ~ 0, with the dispersion relation in the normal phase, w + ( p ) = :tzb + (p2 + m.x2 + b2)l/z. Then E~ --+ (w:k -t-/zx) 2, which naturally recovers the previous results. In the condensed state ((0) 4= 0), the mode corresponding to E_ is the Goldstone mode as a consequence of the breakdown of V-spin symmetry; we can choose such a form for the condensate as (K +) = f ( O ) / x / 2 , e x p ( + i y ) with an arbitrary phase y. However, once a phase is chosen, the kaon-condensed state is a symmetry-broken state with respect to the rotation around the third axis in V-spin space by the U ( I ) operator, U v ( u ) = exp(iVa~,) with an arbitrary parameter ~,, while the effective Lagrangian is still invariant under the transformation, U --~ UvUUtv . Thus we observe a spontaneous symmetry breaking (SSB) there. In the classical (tree-level) approximation, cl ( p = 0) = 0 (see Appendix B), which directly means E_ ( p = 0) = 0. Actually we can expand E_, E2 c3 - --~ ~ 2 P

2

p4 + ~ + ....

(3.23)

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

147

for low momentum, [p[ << C. This dispersion relation may remind us of the Bogoliubov spectrum for the phonon mode in superfluidity or Bose-Einstein condensation [25]. When we consider the thermal kaon loop, E_ directly enters into the Bose-Einstein distribution function, f B ( E ) = [exp(flE) - l ] - l , and it should diverge at p = 0. The other is the massive mode, E+ >> 100 MeV, and we may discard it for temperature we are interested in (T ~< 100 MeV). Thus only the thermal K - loops play an important role. When we take into account the fluctuation effects (kaon loops) in the thermodynamic potential, their order of magnitude is O(h) at T = 0. Accordingly, this relation should be modified in O ( h ) , because any kaon loop is not included in Eq. (3.22).13 Hence we should find cl (p = 0) = 0 + O ( h ) , which means that once fluctuations are included in the thermodynamic potential, the soft mode loses the Goldstone-boson nature. This situation is inevitable in the perturbative treatment [22]: the one-loop diagrams are taken into account in our thermodynamic potential, whereas the self-energy diagrams for kaons never include any kaon loop. However, we may expect its deviation to be small. Actually, we have found that the self-energy contribution from kaon loops is very small compared with that in the tree level [ 13,14]. If we can safely neglect the c3 term, we get a simple expression for E+, E+ = ~ ± + ~K

=

V/P ~ q- C2 ± (b + ~K).

(3.24)

There are several reasons to support the validity of these formulae, especially for E_. First, we can see from Eq, (3.23) that in the high momentum limit, IPl >> ~ = /xK sin{0), the difference of E_ between Eq. (3.22) and Eq. (3.24) is trifling. 14 On the other hand, in the low momentum limit, [Pl << C, E_ in Eq. (3.24) can be expanded as p2 E_ ~_ - - + . . . 2C

(3.25)

within the tree approximation, which just corresponds to the second term in Eq. (3.23). Therefore, the approximated formula (3.24) may be well justified for the momentum, Ipl >> ~K sin(0), while it spoils the Goldstone-boson behavior (E_ ~x Ipl) near p = 0. So we must carefully treat the smali-p region, when we use this approximation. Comparing (3.22) and (3.24) we can expect that the "non-relativistic" dispersion relation (3.24) holds under the condition, v2 - c3/2C 2 << 1, namely the small velocity limit. Typically P.r "" O(100 MeV) and C ~,, O ( m x ) [5], so that we find v2 ,-- O(10 -2) << 1. At finite temperature, we can find a more meaningful criterion for the validity of Eq. (3.24). Consider a typical integral at finite temperature, 1=f

d3p ~ I -:-;----~.~j~,:,_(p)),

(3.26)

13 On the other hand, the nucleon loops can be treated consistently (see Section 4). 14This property is very important for the renormalization of kaon loops, where ultraviolet divergences become relevant, see Section 4.

148

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

which is the particle number of thermally excited K - . Using Eq. (3.24) we find (CT)3/2

1 ,-~ (27r)3/------~((3/2)

(3.27)

for T ~< C. However, as mentioned above, Eq. (3.24) is no longer good for Ipl < /xK sin(0). We can estimate the integral for this region Ix~:sin(0)

I~ =.

j

f

p2dp 27.r2 fs( E-(p) )

Ctzx sin(0) ~

(3.28)

q7.2t~

o

Therefore, the approximation (3.24) is verified, if 1~/I << 1, which requires the following condition for c3 o r / z x sin(0) as the function of density and temperature, c3 <<

7"(2(3/2)

or

# x s i n ( 0 ) << ~ ( 2 1 r C T ) J / 2 ( ( 3 / 2 ) .

(3.29)

We can expect Eq. (3.29) for relevant densities and temperatures [ 14], except for very low temperature, where thermal effects should be trivially unimportant. Secondly, we can also expect the smallness of c3 qualitatively: in the limit, (0) --* 0 or /xx ~ 0, c3 gives no contribution. We also know that (0) and /zx are inversely proportional to each other [5]. Hence we might expect that the c3 term becomes relatively small.

3.3. Thermodynamic relations The effective thermodynamic potential J?chiral = - T l n Zchiral reads from Eq. (3.20), J?chiral = d?c -~- 6Q.u" -~- J~K -~- ~(~N,

(3.30)

where /2,, is the classical kaon contribution,

12,.= V[-- f2nIK(COs(O) -- 1) -- ~/X7¢ I "~,.2 J sine(0)],

(3.31)

12sc the subtraction term to avoid the double-counting of the interaction terms within the Hartree approximation, 12,~ = TTr[G h' (6G)I = V [o-cos(0)n}¢ + bnx].

(3.32)

The kaon contribution can be written as (3.33)

12K = f~K P -~- 12~,

where the zero-point energy (ZP) contribution s2zp and the thermal one 12~ are given as follows:

v/ ( 2d~p) 3 !2 [ E + ( p )

+ E_(p)],

12~ =TV f ~ d3P ln(l - e -/3e' (v) )(1 - e - ~ e - I v ) ) .

(3.34) (3.35)

149

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

The thermodynamic potential for nucleons, s2u = - T In Zu, can be also written as ~"2u = - T T r l n GH = ~ N p + $"2~,

(3.36)

where S"~up denotes the ZP contribution

S2ZuP = - 2 V Z

~

~ ( e i + e i) ,

(3.37)

n,p

and /2F the thermal contribution ~ d3P

,O~, = - 2 T V

[ln(1

+ In( l +

.

(3.38)

tl ,p

Here, ~i(~7), i = p, n are the single-particle energies of the nucleons (anti-nucleons), Ep,p = :v

::;:/.,K(1

- cos(o))

+

g,

1 2 _~ • e,,,r, = qz-~Fnx q: - - ( I - cos(0)) + E,,,

(3.39)

with the kinetic energy, E,* = ~/p2 + M 2 . Using the thermodynamic relations Schiral -

OJ?chiral - - ,

aT

O,.('2chi,.al

Qi -

--,

o~/.t,i

Echiral = ~chiral Jr- TSchiral -k Z

(3.40)

tziQi'

i

we can find charge, entropy and internal energy. We easily see that these quantities become too complicated to be tractable if we use the exact dispersion relation in Eq. (3.22); e.g. consider the kaonic charge given by way of the relation, QK = --Oqg~2chiral/Old~K, which results in a complicated form. Besides this, the quantity, n}¢ or 9 nx., which can be written by the use of the thermal Green's functions, has a complicated form as well (see Appendix B). Hence a useful approximation is desirable. We have already discussed the validity of the approximated form for the dispersion relations in Eq. (3.24) and we can see that this approximation allows us to write down the thermodynamic quantities in clear forms; for the kaonic charge it immediately gives

QK/V

= I ~ K f 2 sin2(0) + cos(0)nK + (1

+ x) sin2( (O}/Z)pB,

(3.41)

where nx is the number density of thermal kaons,

d3p nx = f ~ [fs(E-(p)) - fB(E+(p))].

(3.42)

The resultant form (3.41) with (3.42) suggests that the charges of kaonic modes are screened by the condensate; in fact, we can see that their effective charges are given by 4-eeff with eeff = cos(0)e by the use of the effective Lagrangian (3.15). Accordingly the

150

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

modified chemical potential /2r = cOS(0)/zK gets the meaning of the chemical potential for the number of thermally excited kaons.

4. Effects of zero-point fluctuations 4.1. Renormalization

In our formalism the thermodynamic potential (3.30), as well as the quantities n~ and o-, is constructed by many one-loop bubbles, which include intrinsic divergences. Hence we must renormalize them to extract finite contributions. The building units of these quantities are graphically one-loop bubbles of kaons and nucleons, so that it suffices to renormalize these diagrams [24]. Then all the diagrams with many loops have no more divergences. Since the temperature-dependent part never induces a new divergence, we renormalize the temperature-independent part. Using the similar method proposed in Ref. [ 13 ], we can properly renormalize the thermodynamic potential. However, we must carry out the renormalization program by including not only the kaonic contributions but also the nucleonic one. The counterterms are introduced without spoiling the original symmetry-structure of the Lagrangian; the symmetric terms are also invariant under the transformation with ( and the symmetry-breaking terms should transform like Eq. (2.23). Since we have already known the structure of the effective Lagrangian, given in Eqs. (3.3)-(3.6), it is easy to find the proper counterterms. It is to be noted that the vector-type contributions, 6/2, induced by introducing the chemical potentials as well as the original Tomozawa-Weinberg term, never suffer from divergences (cf. Eq. (C.13), see also Ref. [24] ). Thereby it is almost clear the introduction of chemical potentials never generates new divergences. In the following we use the approximated formulae (3.24) for the dispersion relations of kaonic modes for simplicity. First, we renormalize the kaon effective mass, m ~ 2 = m 2 -- o'.

(4.1)

The self-energy term o- is given by the nucleon loop, o- = f - Z T r [G~IyoXKN] = o'tree + o'loop with the inverse propagator Gn (3.1 ! ). We only consider the loop correction o'loop in the following. To one-loop order we do not care about the n}¢ that is given by the kaon loop in the effective mass M [ (3.12). Then the temperature-independent part Oqooplr=0 reads trlooplr=o = - 4 ~

d3k M? (2~) 3 2E*

S ~ i f -2

i

k~
Mi.~K,f &-My&+

=-4 Z

*

-2

Myln\4K~Jj

(4.2)

i

with the cut-off (A) regularization, where di's are the quadratically and logarithmically divergent integrals,

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

151

....... • ...... X

O~2

g

. . . . . . . . . . .

-ii

......

a0 (a)

(b)

Fig. 4. Divergent diagrams (a) and the counterterms (b) to one-loop order for the kaon propagator (dashed line). Solid line denotes the nucleon propagator without nit and the cross ( x ) denotes the condensate (0). The black blob stands for the contribution from the counterterm.

d3k d2 =

1

A2

(9'rg~ 3 2k - 8¢r2' k<~A

f do =

d3k

1

1 --In(A),

(4.3)

(9,./r.~34k 3 - 87r2 k<~A

with an arbitrary momentum-scale K. To renormalize the effective mass, we redefine the kaon mass and the KK interaction vertex. Then the counterterms should be given as

O-c, = 4 Z

M, Xxif -2 [ or2 -

M*2ao].

(4.4)

i

The meaning of the counterterms ai is graphically clear (Fig. 4): oe2 renormalizes the kaon mass and ao the KK vertex at momentum p = 0. The wavefunction renormalization is not necessary to one-loop order. Then they are determined by the following conditions: (O%oplz=0 + oct)Ivac = 0,

oq 2 o3(0)2 (O"looplT_-0 -[- Orct) vac = 0.

(4.5)

From Eqs. (4.2), (4.4) we find

a2 = d2

167r2,

ao = do + ~

In

-

,

(4.6)

and consequently the finite contribution 0 "ren loopT---O=--Y~Mi

"~Kif--------~2 477.2

M72 In

~ -M-~-J

+ M2

M .2

-

(4.7)

i

en Note that Thus the self-energy term o- is renormalized to be trren, O'ren = O'tree + or,,looplT--0" the dispersion relation is modified by the replacement of tr by O'ren, compared with the

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

152

/ I

x

~2

g

I

/~

x

}(

f% (a)

(b)

Fig. 5. Divergent diagrams (a) and the counterterms (b) to one-loop order for the nucleon propagator. Notations are the same as in Fig. 4.

tree approximation where o- is given by o- = O'tree [ 14]. However, we can see that the Goldstone-boson nature of E_ is never spoiled by such a procedure (see Eq. (4.27)). Next, we renormalize the nucleon effective mass, M~'(i = n , p ) , which is given by Eq. (3.12). It includes the kaon loop correction n~:, t5 the temperature-independent part of which is given by

n}¢lr=° =

f

d3p

1

= d2 -

(2"rr) 3 E+ + E ~

C2do +

72 67r

C 2 In

( ~x2x C22 ) '

(4.8)

p<~A

(see Eq. (C.11)). Accordingly we introduce the counterterms, which renormalize the nucleon mass and the K N interaction vertex at momentum p = 0 (see Fig. 5), M~.t = f _9-Xxi cos(O)[/32 - C2/30] -

(4.9)

Here C 2 is given by C2

=

( m 2 - o-) cos(O) + b e,

(4.10)

but the nucleon loop in o" is irrelevant up to one-loop order as well. The overall factor c o s ( 0 ) in Eq. ( 4 . 9 ) stems from the transformation ( (see Eqs. ( 3 . 4 ) and ( 3 . 6 ) ) .

Then renormalization conditions are given as follows: (

M* i =M, ilr=o+Mc,)lvac

i

~92 ( M*i[r=o+Mc,) ¢9(0)2

=--.~Ki.

vac

(4.11)

From Eqs. ( 3 . 1 2 ) , ( 4 . 8 ) , ( 4 . 9 ) we find

/32 = d2

1m2x 6 ~ 2'

/30 = do + ~

In

-

'

(4.12)

15Another kaon-loop contribution 172 never includes divergence (see Eq. (C. 13)), because it originates from the vector-type KN interaction, while n}¢ from the scalar-type interaction.

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165 x

,, ,,

~<

x

,, ,,

x

xx

xx

X

xx

xx

153

",,,,/,,"

© @© x

X

;,-'..

x

;,,',:,':;

..,:'.

/

,

.....

F(

.....

(a)

(b)

(c)

Fig. 6. Divergent diagrams to one-loop order for the thermodynamic potential. Notations are the same as in Fig. 4. The diagrams in (a) are quadratically divergent and renormalized by the counterterms given in Figs. 4 and 5, while the diagrams given in (b) and (c), which are renormalized by redefining KK interaction vertex at momentum p = 0, give the quadratic and logarithmic divergences, respectively; the counterterms for (b) and (c) are already given in Figs. 4 and 5. and thereby the finite contribution to n I reads

l

,,ren

[m2K_CZ_C21n(m2K~]

(4.13)

\cijj

It is to be noted that the structure of (4.12) is the same as that of (4.6). Once tr and n I are thus renormalized, we can use the renormalized ones given by Eqs. (4.7) and (4.13) in the thermodynamic potential (3.30) instead of the unrenormalized ones. Although the thermodynamic potential (3.30) still includes the divergences through the ZP contributions in Eq. (3.34) and Eq. (3.37), the total ZP contribution S2zp = s2zP + / 2 zP should be renormalized by the use of the counterterms ai and ,8i without introduction of another counterterms (see Fig. 6), except the trivial divergence coming from the vacuum energy. Thus we find the renormalized ZP contribution, s 2 ~ , =

,t,~en

zP,K +

(4.14)

n

with the kaon-loop contribution

y~Z~,,K/V = 647"r 21

[

(C2 _ m~)(m2x _ 3C2) + 2C41n

(4.15)

and the nucleon-loop contribution

J~zT,I V / V = - E

~

, [( M * Z - M 2 ) ( M 2 - 3 M / 2 )

+2MT41n\

t'M M 22)q JJ "

(4.16)

i=n,p

Here we have subtracted the vacuum energy which is quartically divergent, M4 m4 Ovac/V = - 3 d 4 - 167r-----2 + --64,rr2

(4.17)

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

154

with the divergent integral, d3k k =

d4 =

A4

(4.18)

8zr2 " k<~A

Then we can easily verify the relations f_2XxiO(

ren

O('Qrzp.K/V) c~C2 - nx,ren T=O,

E i=n,p

oren / V ) ~"ZP,N/ 3M7

= O'~l~np n It_-0.

(4.19)

The magnitude of ZP contributions (4.15) and (4.16) are bounded from above,

IS~Z?,K/VI<

12 M e V . fm -3

and

la~,N/Vl

< 620 M e V . fm -3.

(4.20)

Comparing both limit values, we can see that the effect of the zero-point fluctuations by nucleons should be more pronounced than that by kaons. In the infinite-mass limit for nucleons, M ---, oo, there is only left the kaon contribution and f 2 ~ is reduced to a simple form [14], Or~z~ ~ / ~ , K " It may be worth noting the relations f12 - d2 = f 2 and (/30 - do) + (/32 - d 2 ) / m ~ = f o in terms of f i given in Ref. [ 13], which is graphically understood in Fig. 6. Finally, we make a remark about the use of the approximated formulae (3.24) for the dispersion relations of the kaonic modes. As has been already mentioned in Section 3, the difference between the approximated formula (3.24) and the exact formula (3.22) converges to zero for the high-momentum limit, so that the structure of the ultraviolet divergences is the same. Therefore, even if we use the exact formulae, we find the same counterterms in Eq. (4.12). 4.2. N u c l e o n - l o o p contribution

We investigate the effects of zero-point fluctuations in neutron-star matter at T = 0, since the negative-sea effect is dominant there. At T = 0 the thermodynamic potential is reduced into a simple form f2cnirallT_-0 = ~(2c -~- f2sclT= 0 -{- f2NIT_-O + ~ i ~ ,

(4.21)

where Osclr=o = VlO'renn .,r Kl,ren Ir_-ocos(0)]. The nucleon contribution /'2NIT---0can be written as I-3 n /

3 P

/2NIT--O--~ V L3eFt 1 -- x ) + -geFX -- IX,( 1 -- X) -- tzpx] ptJ --(2bld~K Jr O ' t r e e ) f 2 ( l

-- C O S ( 0 ) )

(4.22)

with the baryon number density Ps, the proton mixing ratio x = P p / P a , and Fermi energies for nucleons, e~- = ( 3 z r 2 p B ( 1 - x)) 2/3/2M, ePF = ( 3 1 r 2 p s x ) 2 / 3 / 2 M , within the non-relativistic approximation. Since we already know that the kaon contribution /-/~,K gives only a tiny effect [ 13,14], we, hereafter, concentrate on the nucleon one

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

155

/2~,N by simply dropping kaon loops. After the calculation, we will confirm it as a consistency check. It is well known that the nuclear symmetry energy plays an important role for the ground-state properties of the condensed phase. Hence we take it into account to get a realistic result [ 14]. Since we have already included the kinetic energy for nucleons, it suffices to consider only the potential energy contribution. Following Prakash et al. [26] we effectively introduce a potential energy contribution for the nuclear symmetry energy,

.Qsymm N = WpB ( I

(4.23)

- 2 x ) 2 SP°t ( u )

with the relative density, u = p s / p o . The coefficient Sp°t(u) is given as Sp°t(u) = (So- (22/3- 1)(3/5)e°v)F(u) with the constraint F ( I ) = 1, to reproduce the empirical symmetry energy So -~ 30 MeV, where e ° is the Fermi energy at the nuclear density po, e ° = ( 3 7 r 2 p o / 2 ) 2 / 3 / 2 M , and F ( u ) is a simulated function [26], for which we take here F ( u ) = u for simplicity. Then the total thermodynamic potential, we consider here, is given by further adding the leptonic one -O~lr=0,

osymm +

Otot = S2chirallr=0+ "~N where

~tlr=0

Otlr=0,

(4.24)

can be written in the standard non-interacting form

s2tlr=0 = V I - l/Z---~27r 2+

0(l~xl -

m~)

( 4 [ x

m~,2 (2t 2 4- 1 )tV/~ 4- 1 - l n ( t + 2x/t-~-~ ~-~ t )

]

nl3[~K[t3}J 37r2

, (4.25)

with t = l//x 2 - m ~ / m u. Here the first term is due to electrons and the second term w

due to ( l = e, The charge

muons, and we have used the relation/zt = / z x for lepton chemical potentials/zt tz). parameters (0), x and tzK are determined by the equilibrium conditions and neutrality of the ground state [5,9,13]. First, we get

f2 sin(O)(m 2

_ O.tree_

2tZK b _ / . z 2 Cos(O))

+

O(f'~Z~,N/V) .= 0

a(0)

(4.26)

from the condition 0f2tot/0(0 ) = 0. Recalling the relation in Eq. (4.19), we find it is further written as f : sin(0) (m~2 - 2tZKb -- I,L2 cos(0)) = 0,

(4.27)

with the effective kaon mass m~2 = m~ - O'ren. Then the critical density is unchanged even if the zero-point fluctuation D.~,N is included, since o'ren is reduced to trm,e as (07 --~ 0. It is to be noted that Eq. (4.27) ensures the Goldstone-boson nature of K mode (see Eqs. (3.22), (3.24)), even if the ZP contribution is included. Secondly, we get

T. Tatsumi. M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

156

Table 1 Optimum values of the parameters for the tree case and the one-loop case. We use the value, a3ms = - 134 MeV tree u 4.19 4.40 4.80 5.20 5.60 6.00 6.40 6.80 7.20

tree + loop

(0)tdeg. I

/xK [MeVl

0.00 18.85 31.97 40.85 47.62 52.92 57.09 60.36 62.9 l

255.8 244.0 22 I. I 197.7 174.4 151.9 130.6 110.8 92.6

e%-e~-4Sp°t(u)(l-2x)

x 0.193 0.231 0.293 0.343 0.384 0.417 0.442 0.462 0.477

(0)ldeg. I

/zx [MeVl

0.00 18.74 31.49 39.91 46.26 51.26 55.25 58.46 61.05

255.8 244.2 222.6 201.6 181.3 161.9 143.5 126.0 109.7

x 0.193 0.231 0.290 0.338 0.376 0.407 0.431 0.450 0.465

{ 2 ( S x , - XKt,) - I,Zx} + tzx = 0 + l - cos(0) 2 (4.28)

from the condition O~$'2tot/(~X ---- 0, which is equivalent to the chemical equilibrium condition a m o n g kaons and nucleons,/z,, - ~t, = / z x . Finally, charge neutrality d e m a n d s the relation

Q x + Qt = Q/,( = x V p e ) ,

(4.29)

which implies OS2tot/,)l,Zx = 0 by definition, where the lepton charge is given by

Qt/V=

tz3

IzK m3t3 + O(ltzx I - m , ) ]l,zxl 3 7 2 '

(4.30)

and the kaonic charge by

QK / V = l,z x f 2 sinZ(0) + (1 + x) sinZ ( (o) /Z ) pt~.

(4.31)

Then the total energy is given by way of the t h e r m o d y n a m i c relation Et°t = d'2t°t + Z

IziQi.

(4.32)

n.l.,,K,I Note that these equations are the same as the previous ones in the h e a v y - n u c l e o n ren limit [ 14] if we simply put S2Zr,,U = 0. In Tables 1 and 2 we present the o p t i m u m values of/zK, x, and (0) for given densities, in c o m p a r i s o n with those in the tree case. We use the values, alms = - 6 7 MeV, a2ms = 134 M e V and a3ms = - 1 3 4 ( - 2 2 2 ) M e V in Table 1 (Table 2). j6 We can observe a sizable effect of the zero-point fluctuations by nucleons; it shifts the o p t i m u m values o f x and (0) d o w n w a r d , while that of/Xx upward, compared with the values in the tree case. The effect b e c o m e s p r o m i n e n t as the value of the K N sigma term is increased, since 16Recent lattice simulations favor the lalger values for ]a3m.~[ 1271.

157

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

Table 2 Optimum values of the parameters for a3 m. = -222 MeV tree U 3.08 3.20 3.60 4.00 4.40 4.80 5.20 5.60 6.00

(O)ldeg. I

tree + loop

~KI MeVI

x

218.8 206.4 160.0 109.2 59.1 14.4 -23.3 -54.9 -81.5

0.156 0.198 0.326 0.425 0.494 0.538 0.565 0.581 0.591

0.00 18.28 39.75 53.71 63.74 71.05 76.13 79.67 82.19

50

,

,

:315

~,

,

,

(0) Ideg. I

/Zr [MeV]

0.00 17.42 33.60 42.20 48.33 53.12 57.04 60.30 63.08

218.8 207.9 181.3 161.8 144.7 128.5 112.3 95.8 79.2

,

,

,

51.5

I 6

61.5

x 0.156 0.195 0.286 0.344 0.386 0.419 0.444 0.465 0.482

0

>

cQ ~U

-50

-100

-150

-2003

415 ; U

Fig. 7. Energy differences per baryon, AE/NB =- (Et(~t - Etot((O) = 0))/NB, and ZP contributions for a3ms = -134 MeV (thin lines) and a3ms = -222 MeV (thick lines). Solid lines show the results with the zero-point fluctuation by nucleon, while dotted lines those in the tree level. Dashed lines show the ZP contribution. Kaon ZP contributions are estimated by the use of optimum values given in Tables I and 2 (dash-dotted lines). the reduction rate of the n u c l e o n effective mass becomes high. As m e n t i o n e d above, it never affects the critical density, while it suppresses the growth of the condensate. In Fig. 7 the energy difference between the c o n d e n s e d phase and the normal one is depicted as a function of density. We can see that the energy gain is reduced along with the suppression o f the condensate; the zero-point fluctuations o f n u c l e o n s reduces the energy gain by shifting the o p t i m u m values of parameters, while the resultant Z P contribution is small in magnitude. We also estimate the m a g n i t u d e o f the effects o f the kaon loops by the use of the values given in Tables 1 and 2 to be tiny for a consistency check. We notice that these c o n s e q u e n c e s originate from the replacement of the K N interaction term O'tree in the tree a p p r o x i m a t i o n by O-ren in the effective mass of kaons m~<; the zero-point fluctuations work to reduce the attraction. It m a y be worth n o t i n g that m~:2 stays to be in positive value and / z r as well, once the zero-point fluctuations are included. It r e m i n d s us of the relativistic effect by nucleons [ 2 8 ] , which also gives rise

158

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

to the same effects. Hence, if we further take into account the relativistic effect in our formalism, it may be milder than before. The dispersion relations are also checked to be fairly good by comparing the results given by Eqs. (3.22) and (3.24).

5. Summary and concluding remarks In this paper we have formulated the method of treating fluctuations around the condensate within the framework of the path integral for the non-linear sigma model. Our approach is based on the group theoretical argument; Goldstone fields are regarded as coordinates on the chiral manifold SU(3)L × S U ( 3 ) R / S U ( 3 ) v , and each state corresponding to the condensed phase or the normal phase with fluctuations can be represented as a chiral rotated point from the origin (the vacuum). We introduced the fluctuations in the vicinity of the condensate by way of the successive chirai transformations by sr and ~7 from the origin. This procedure can be regarded as the introduction of the local coordinates around the condensed point on the chiral manifold. Since the chiral transformation is the isometry, the geometry around the condensed point is the same as in the vicinity of the vacuum, where we might use the flat curvature. Hence this method has the advantage of avoiding the Lee-Yang term in the path integral. More interestingly, this method corresponds to the separation of the zero-mode; since there should appear the Goldstone mode as a result of the symmetry breaking of V-spin beyond the critical density, we must treat the zero-modes non-perturbatively even in the normal phase. In the infinite-volume limit the condensate only contributes to the thermodynamic potential among zero-modes in the condensed phase. Therefore we have put U into the form given in Eq. (2.14) from the beginning. This method might have a wider applicability for studying phase transitions, e.g. in finite volume, if the system has some symmetry. We have constructed the thermodynamic potential by gathering the one-loop diagrams of kaons and nucleons. The K N interactions have been treated self-consistently within the Hartree approximation. Consequently it consists of an infinite series of diagrams with many one-loops. Then the dispersion relations for kaonic excitations, which are fundamental objects to examine the thermal properties of the condensed phase, can be obtained. There appear two modes: a Goldstone-like soft mode and a very massive mode, which correspond to K - - and K+-mesonic excitations, respectively, in the condensed phase. Hence, the thermal loops due to the soft mode play an important role. We have seen that the form of the dispersion relations can be reduced to simple ones, if we can take an approximation, c3 = 0. It should be worth noting that once this approximation works, the expressions of other quantities like charge, entropy, internal energy become very simple. The thermodynamic potential as well as the self-energy terms for kaons and nucleons contains some divergences. We have shown that these divergences are properly renormalized by way of the cut-off regularization; counterterms are introduced to redefine the masses of kaons and nucleons and the K N and K K interaction vertices. Once the self-

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

159

energy terms are renormalized, we need not require more counterterms to renormalize the thermodynamic potential. We have discussed the effect of the zero-point fluctuations by nucleons at T = 0. It provides a sizable effect for the optimum values of the parameters, so that the energy gain is reduced from the classical value. This is because the K N sigma term remarkably reduces the nucleon effective mass in the condensed phase, while the critical density is not affected. On the other hand, the contribution by the kaon loops is still tiny. It may be interesting to study thermal effects in the condensed phase, which is closely related with the problem of protoneutron stars, especially the effects of thermal loops of n K i ,ren, n~ [29]. In a subsequent paper [30] we discuss the thermal properties of the kaon-condensed phase on the basis of the formalism given in this paper. We shall obtain the phase diagram in the density-temperature plane and the equation of state for the kaon-condensed phase. Also, in protoneutron-star matter, various time scales concerning collapsing, neutrinotrapping or initial cooling may be relevant. Hence we cannot assume chemical equilibrium a priori for such hot matter since it may be achieved through weak interactions. The problem of how the condensate appears and grows to establish chemical equilibrium is an interesting subject to study [31].

Acknowledgements We thank T. Muto for his interest in this work. We also thank K. Matsuyanagi for useful discussions about the finite volume effect on the phase transition. This work was supported in part by the Japanese Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Science, Sports and Culture (08640369).

Appendix A. Separation of zero-modes Here we reconstruct our formalism from the viewpoint of separation of the zeromode, following Ref. [15]. To make our argument clear we first consider the finite volume V = L 3 and put in infinity at the end. Since we are interested in the kaon condensation, we restrict ourselves to the kaon and nucleon degrees of freedom for simplicity. Considering the partition function (2.9) and integrating out the baryonic degrees of freedom, we get the partition function for U, Zchiral =

N' f [ dU] exp[ SK (U) + Tr In (~(U) ],

(A.I)

where S t ( U ) stands for the pure mesonic action and the second term stems from the integration over the baryon field. The capital letter for trace is used for the Dirac operator O(U) of infinite dimension over space-imaginary time. Substituting U = e x p ( 2 i ~ b / f )

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160

with ~b = given as

T,,dpa in

Eq. (A. 1), the action has the following structure: the kaon action is

' K : i dT i d3x [-`3"K+O°:K- - (m2K-- tZ2K)K+K- - ]~KK+ `3~K-J-O(IKI4)] 0

.

(A.2) The Dirac operator 0 reads

O = Go + 6G

(A,3)

with the free part `3 Go = -O-rr + iyo'Y • V -

yoM + tXN

(A.4)

and the interaction part

~O = ~ ? / o K+ K -

f213 +r38

( K+ `3~K- - 21,,KK+K-)

+ O(IK{4).

(1.5)

For the purpose of the Hartree approximation, we define OH as OH = G0 + (6G),

(A.6)

by taking the thermal average tbr the kaon field. Then the effective action of the nucleonic part can be written as Trln O = Trln OH + Tr[GF.,16G] - Tr[ G~,,! (6G)]

+ O(iKi4).

(A.7)

Eventually the effective action S eft can be written as seff = SK + Trln GH + T r [ O ~ ' ~ G ] - T r [ O ~ ( S G ) ] +

o(Ig14).

(1.8)

Substituting Eq. (A.5) into Eq. (A.8), we find /3

s~ft'= Trln OH - Tr[ O~fI (6G) ] + S d-r / d3x[-`3~K+ `3~K- - M2xK+K0

- ( b + p , x ) g + ,3, K -

+ o ( I g l 4) ,

(A.9)

where M2x is the "mass" term for the kaon field, M2x =-- m~ - //KN, with the selfenergy term, I1KN = --,5- -- 2/zxb. Here O- and b stand for 0" = f-2Tr[yoXxuG~-4 j ] and b = f - Z T r [ ( 3 + 7-3)/8G_q I ], respectively. It is to be noted that the "mass" term should include the contributions from the meson-baryon interactions (//my) and the chemical potential besides the genuine mass of kaons, m~.

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

161

To perform the integral in Eq. (A.I) we expand the Goldstone fields in terms of periodic plane waves like Eq. (3.16), n~exp[2rri( n" r/L +n4rT)]~b~.

~a = ~

(A.10)

Inserting Eq. (A.10) into Eq. (A.9), we can see that the magnitude of ~b,] is at most O(T/~/M2K + L-2). Hence if T << MK, the Gaussian integral should give the dominant contribution to the partition function. However, if the phase transition occurs and the "mass" term M~ tends to vanish at the critical point, the standard chiral perturbation fails, because there is no quadratic term for the zero-mode (n~ = 0); in fact, the "mass" term takes negative values before the phase transition, while it vanishes at the critical density. Hence we need to reorder the perturbation series by treating the zero-mode non-perturbatively to get a consistent formulation valid even at the critical point. Following Gasser and Leutwyler, we treat the zero-modes as collective variables [ 15],

U = vUfv,

(A.I 1)

with Uf = exp[2i(T4~?4+Ts~Ts)/f] and the U(1) matrix v = exp(iOoT4), v is the constant unitary matrix representing the zero-mode, and r/a sums up the non-zero-modes, r/a = ~/-~flzZ

exp[2zri(n - r/L + n4rT) ] &','~.

(A.12)

nv~0

After integrating out with respect to the non-zero-modes, we find the integral over the zero-mode (U(1) C SU(3)). 17 Substituting Eq. (A.11) into Eq. (A.1), we can see that the form of the effective action is the same as Eq. (3.7) with 0o instead of (0). Hence the integral over the zero-mode becomes Zchiral = f dOoexp [ -

$'~chiral (00)/T

],

(A. 13)

where ~chi~al(00) is given by Eq. (3.30) with replacing the thermal average (0) by 0o. Separating the trivial volume dependence from the thermodynamic potentials, we write Eq. (A.13) as Zchiral :

(A. 14)

J dO0exp[ --Vflgo],

with & = ~chiral(Oo)/V. In the large volume limit or the low temperature limit, we can apply the steepest descent method to evaluate the integral in Eq. (A.14): Zchiral '~

exp[-V/3&(0~')]

V/3&"'

(A. 15)

17 In the general case the zero-modes cover the SU(3) manifold, and the partition function is given by the group integral over SU(3) 115 I.

162

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

where to" stands for c~2&/O02too=0~;,and the saddle point 0~' is determined by the equation

ago(Oo)/aOo]oo=o,,; = 0.

(A. 16)

This is nothing but an extremum condition for the thermodynamic potential with respect to the order parameter. Hence 0~' is equivalent to the thermal average of 8, 0~' = (0). Accordingly, the zero-mode matrix Vm is reduced to (. In conclusion, when some degrees of freedom should condense, it is enough to separate them from the beginning in the form (A.11) in the limit, V --+ oo. Here it is also interesting to observe that there is a logarithmic singularity in the thermodynamic potential at the critical point as long as V 4= oo.

Appendix B. Classical approximation (model independent) In the classical approximation, ~b~,= 0, the operator u becomes trivial,

Uf = u = 1.

(B.1)

Furthermore, replacing the bilinear operator of the nucleon field by its expectation value, e.g., q;g, ~ ( ~ ) , then we get

~f~= ~-tr{B¢[((t[Tem,(] + ([Tem,(t]),B]} + ~ =/~K(COs(O) -- l ) ( l

+x)pe

--

tr{[Tem,( 2] [Tem, (t2] }

2~KJ1 2 ,-2sin2(0),

(B.2)

which stems from the kinetic terms of the Goldstone bosons and is therefore model independent [9]. The symmetry breaking part is also transformed, £SB ( UK -- (2, B ) ---/~SB ( 1, B ) -[- (COS(0> - 1) (~?K( 1, B))

= (cos(0> - l)(-f2m2x + ,'~KpPp q- ~KnPn) + £ s B ( 1 , B ) , (B.3) according to the octet dominance hypothesis (see Eq. (2.28)), where we have used the non-relativistic approximation for nucleons. Therefore the result is also model independent once we employ this hypothesis [9]. Thus we can recover the previous result: for the energy difference between kaon condensed phase and normal one,

aE/V = Ec,/V + IzKQ~/V,

(B.4)

where the "classical" energy Ecl, which is nothing but the classical thermodynamic potential at T = 0, is given by

Ec,/V = (cos(0) - 1) [/zK( 1 + x)ps/2 - f2m2x + •KpPp -[- XKnPn) ] __~I.LK J 12 .~2 sin2(0).

(B.5)

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

163

The charge of the condensed kaons, QXel is given by Q K / v = tz2Kf 2 sin2(0) +/ZK(1 + x ) p o ( l -- COS(0))/2.

(B.6)

It is to be noted that the result (B.4) has been obtained without recourse to the definite Lagrangian [9]. The optimum values of the parameters, (O), x,/zK, are determined by the extremum conditions for Eel, OEcl/O(O) = O,

OEcl/OX = O,

OEcl/OtzK = 0.

(B.7)

In particular, we find the field equation for the condensate from the first equation of (B.7), f2 sin(0) (m*K2 -- 2tzKb -- Iz2XCOS(0)) = 0

(B.8)

with m~¢2 = m~: - ~rtree, which implies cl (p = 0) = 0 for (0) 4: 0.

A p p e n d i x C. K a o n G r e e n functions

The Green functions for kaons can be extracted from the matrix D eft. Define the inverse propagator matrix 79 - I =- ,8-2Dell: D~l I = w 2n + cl + 2c3, 79~1 = o)n2 + Cl, and 79721 = -79211 = -2c2¢o~. Then the propagators are given by (~b4~4) = 7911 = 79~1 179-'1-'

(C. l )

(~bs~bs) = D22 = D ~ l[D - I I- l ,

(C.2)

and

with d e t ( D - t ) =

179-11. Recalling

the Dzyaloshinski relation [ 32], (C.3)

79(o~n) = D R ( i w n ) ,

with the retarded (real-time) Green function D R , we find the excitation energy of the kaonic modes are given by D(I.I~-l = 0 or d e t ( D -1) = 0. Similarly (~b4~5) = ~D12 = -~)211 [~D-I 1-1 ,

(~5~4) = 792, = --79~21179-11 - l .

(C.4)

C. 1. Calculation of n,,, o ,,,,, I

Consider the following quantities: n~ = (K+ K - ) , n ° =- ( ( K+ K - - K+ K - ) ) and n~ = n ° + 2/2n~o Hence it is sufficient to evaluate n ° and n~. By the use of the kaon Green functions they are represented as

T. Tatsumi, M. Yasuhira/Nuclear Physics A 653 (1999) 133-165

164

n}¢= ( K + K - ) = ½B-' Z

/ (-2-~).~(79|, + 7922)

II

= ½/~-' Y~.

(zrr)-

n

(c.5)

[ 179-' I-' <797,' + ~5') ]

and

n ° = ((g+[( - - K ~ K - ) ) = fl-=

d"P,2co, lD-11-~79[-z I. (2rr) -~

(C.6)

Generally it is rather complicated to evaluate these quantities, but we shall see that we can easily do them under the approximation c3 = 0. In this case, 79~1 = 79~)1, thereby the determinant ID-t[ is factorized into (79~t + i79~21) (79~11 _ i79~1). Hence we find

(

'

det(79-]) ---4 79~1 +i79~t + 79~I _ (

1

, ) -i79~] 1

79~1 +i79~]

(279H l ) - ]

)(-2iDa)')-'.

(C.7)

79~] i79~

Note that 79~] 4- i79~ I = (w,, :L i E _ ) ( w , qz iE+). Then n °, n I can be written as n1=/3-'~

~

E++E_

;~+E

~_

+ o.,~¥ ~

'

(c.8)

tl

1E2+) ]

o~2 -1-

(C.9)

/1

Using the relation 1

x2 + n2

- ~" coth ~'x,

(C.IO)

x

n=--O~

we finally get nl =

/

d3p

1

( 2 i f ) 3 E+ + E _

no=/

1+ - -

E

'

+ - -

e~ e - - I

d3p 1 [-(E_-E+)/2 (27"r) 3 (E+ + E _ ) / 2

'1

e~e+ - 1

(C.ll) '

E+ E_] eClE~-~ + ePE-~_ 1 "

(c.12) The first terms in Eqs. (C.I1) and (C.12) are the contributions from the zero-point fluctuations to be renormalized. Finally n 2 = n ° + 2/2rn}¢ can be written as n~ =

d3p 1 (27r) 3 (e+ + e _ ) / 2

E+ - [XK + . eP e, - 1 7~-- --- i-J

(C.13)

It is to be noted that there is no contribution from the zero-point fluctuations in Eq. (C.13) due to the vector nature of the KN interaction.

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