Pionic contribution to relativistic Fermi liquidparameters

Kausik Pal

Abstract: We calculate pionic contribution to the relativistic Fermi liquid parameters (RFLPs) using a chiral effective La-grangian. The RFLPs so determined are then used to calculate chemical potential, exchange and nuclear symmetry ener-gies due to pN interaction. We also evaluate two loop ring diagrams involving s, u, and p meson exchanges and compareresults with what one obtains from the relativistic Fermi liquid theory (RFLT).

PACS Nos: 21.65.–f, 13.75.Cs, 13.75.Gx, 21.30.Fe

Resume : Nous calculons la contribution des pions aux parametres du liquide de Fermi relativiste (RFLPs), en utilisant unLagrangien effectif chiral. Les parametres ainsi obtenus sont alors utilises pour calculer le potentiel chimique, les energiesd’echange et de symetrie nucleaire dues a l’interaction pN. Nous evaluons egalement deux diagrammes en boucle impli-quant l’echange de mesons s, u et p et nous comparons nos resultats avec ceux obtenus de la theorie du liquide de Fermirelativiste (RFLT).

[Traduit par la Redaction]

1. IntroductionFermi liquid theory (FLT) provides us with one of the

most important theoretical schemes to study the propertiesof strongly interacting Fermi systems involving low-lyingexcitations near the Fermi surface [1]. Although developedoriginally in the context of studying the properties of 3He, ithas widespread applications in other disciplines of many-body physics such as superconductivity, super fluidity, nu-clear and neutron star matter, etc. [2, 3].

In nuclear physics, FLT was first extended and used byMigdal [4] to study the properties of both unbound nuclearmatter and finite nuclei [5]. FLT also provides a theoreticalfoundation for the nuclear shell model [5] as well as nucleardynamics of low-energy excitations [2, 6]. In particular,ref. 7 reveals the connection between Landau, Brueckner–Bethe, and Migdal theories, while refs. 8 and 9, on the otherhand, calculate the Migdal parameters using one-boson ex-change models of the nuclear force and show how these pa-rameters are modified if nuclear matter is considered withinthe context of the relativistic Brueckner–Hartree–Fock(RBHF) model. Some of the recent work that examinesFermi liquid properties of hadronic matter also incorporatesBrown–Rho (BR) scaling, which is very important for thestudy of the properties of hadrons in dense nuclear matter(DNM) [10, 11].

Most of the earlier nuclear matter calculations that in-volved Landau theory were done in a nonrelativistic frame-work. The relativistic extension of FLT was first developedby Baym and Chin [3] in the context of studying the proper-

ties of DNM. In [3], the authors invoked the Walecka model(WM) to calculate various interaction parameters (fpp’) butdid not consider mean fields (MFs) for the s and u mesons,for which the FLPs are calculated perturbatively.

Later, Matsui revisited the problem in [12] and starts fromthe expression of energy density in the presence of scalarand vector meson MFs and uses the functional derivativesto determine the FLPs. The results are qualitatively differentthan the perturbative results (see [3, 12]). A comparison ofrelativistic and nonrelativistic calculations were performedin [8, 9], which also discuss how the FLPs are modified inthe presence of the s and u MFs and contrast those withperturbative results.

Besides s and u mesons, ref. 12 also includes the r and p

mesons, and the model adopted was originally proposed bySerot to incorporate pions into the WM. It is to be noted,however, that the FLPs presented in [12] are independent ofthe p meson. This is because p, being a pseudoscalar, fails tocontribute at the MF level. Hence to estimate the pionic con-tribution to FLPs, it is necessary to go beyond MF formalism.

It is to be noted that, to our knowledge, such relativisticcalculations that include p exchange do not exist, despitethe fact that the pion has a special status in nuclear physics,as it is responsible for the spin–isospin dependent long-range nuclear force. Furthermore, there are various nonrela-tivistic calculations, including the celebrated work of Mig-dal, which show that the pionic contribution to FLPs areimportant [10, 11, 13] and most dominant for low-energyexcitations. We mention here that ref. 11 discusses how toincorporate relativistic corrections to Fp

1 in the static poten-tial model calculation. We, however, take the approach of[3], in which all fields are treated relativistically.

The other major departure of the present work fromrefs. 3 and 12 resides in the choice of model for the descrip-tion of the many-body nuclear system. Unlike previous cal-culations [3, 12], here we use the recently developed chiral

Received 20 May 2010. Accepted 20 June 2010. Published onthe NRC Research Press Web site at cjp.nrc.ca on 14 August2010.

K. Pal. High Energy Physics Division, Saha Institute of NuclearPhysics, 1/AF Bidhannagar, Kolkata 700064, India (e-mail:kausik.pal@saha.ac.in).

585

Can. J. Phys. 88: 585–590 (2010) doi:10.1139/P10-044 Published by NRC Research Press

effective field theory (chEFT) to calculate the RFLPs. Sucha choice is motivated by the following earlier works that wenow briefly review.

The first attempt to include p and r mesons into the WMwas made in [14], where pseudoscalar (PS) coupling wasused to describe the pN dynamics. Such straightforward in-clusion of a p meson into the WM has serious difficulties.

In particular, at the MF level it gives rise to a tachyonicmode in matter at densities as low as 0.1r0, where r0 is thenuclear saturation density [15]. Inclusion of exchange dia-grams removes such unphysical modes but makes the effec-tive mass unrealistically large. If one simply replaces the PScoupling of ref. 14 with the pseudovector (PV) interaction,these difficulties can be avoided. This, however, makes thetheory non-renormalizable [16].

In [16] it was shown how, starting from a PS couplingone can arrive at a PV representation that preserves renor-malizibility of the theory and at the same time yields realis-tic results for the pion dispersion relations in matter. But thismodel was also not found to be trouble free, particularly ithad several shortcomings in describing pN dynamics inmatter (not discussed here but see [17–22]).

Furthermore, the WM itself has several problems in rela-tion to convergence that prevent the use of a systematic ex-pansion scheme to perform perturbative calculations. Thiswas first exposed in ref. 23.

The most recent model that provides us with a systematicscheme to study the dense nuclear system is provided bychEFT [21, 23–25], where the criterion of renormalizibilityis given up in favor of a successful relativistic descriptionof DNM and the properties of finite nuclei. The chEFT,apart from s and u mesons, also includes pions and there-fore is best suited for the present purpose.

In this paper we estimate the contributions of pion ex-change to the FLPs within the framework of RFLT, and sub-sequently use the parameters to calculate various quantitiessuch as pionic contribution to the chemical potential, energydensity, symmetry energy (b), etc. For completeness and di-rect comparison with the two-loop results, we also calculatethe exchange energy due to the interaction mediated by thes, u, and p mesons.

2. Formalism

We will quickly outline the formalism. In FLT the energydensity E of an interacting system is the functional of occu-pation number np of the quasi-particle states of momentump. The excitation of the system is equivalent to the changeof occupation number by an amount dnp. The correspondingenergy of the system is given by, [2, 3]

E ¼ E0 þXs

R d3p

ð2pÞ3 30psdnps

þ 1

2

Xss0

R d3p

ð2pÞ3d3p0

ð2pÞ3 fps;p0s0 dnps dnp0s0 ð1Þ

Here, s is the spin index. The quasi-particle energy can bewritten as,

3ps ¼ 30ps þ

Xs0

R d3p0

ð2pÞ3 fps;p0s0 dnp0s0 ð2Þ

where superscript 0 denotes the ground state [2]. It is to beremembered that although the interaction (fps,p’s’) betweenthe quasiparticles is not small, the problem is greatly simpli-fied because it is sufficient to consider only pair collisionsbetween the quasiparticles [5].

Fig. 1. Dimensionless Landau parameters in symmetric nuclearmatter for pion exchange in relativistic theory. Solid and dashedlines represent F0 and F1, respectively.

Table 1. Dimensionless Landau parameters and che-mical potential at r = r0. Note that F0 and F1 are thedimensionless isoscalar LPs.

Meson F0 F1

s –5.04 0.875u 5.44 –0.93p 0.68 –0.20

Fig. 2. Two-loop contributions to the nuclear matter energy density.The solid line represents the baryon propagator. s, u, and p me-sons are denoted by dotted, wavy, and dashed lines, respectively.

Table 2. Exchange energy in MeVfrom FLT at r = r0.

Meson Eex

s 40.48u –23.41p 12.49

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Since quasi-particles are well defined only near the Fermisurface, one assumes

3p ¼ mþ vf ðp� pf Þp ’ p0 ’ pf

g ð3Þ

Then the LPs fls are defined by the Legendre expansion offps,p’s’ as [2, 3],

fl ¼2lþ 1

4

Xss0

R dU

4pPlðcos qÞfps;p0s0 ð4Þ

where q is the angle between p and p’, both taken to be onthe Fermi surface, and the integration is over all directionsof p. We restrict ourselves to l £ 1, i.e., f0 and f1, as contri-butions decrease rapidly at higher l[3, 12].

Now the Landau Fermi liquid interaction fps,p’s’ is relatedto the two particle forward scattering amplitude via [3],

fps;p0s0 ¼M

30p

M

30p0Mps;p0s0 ð5Þ

where the Lorentz invariant matrix Mps;p0s0 consists of theusual direct and exchange amplitude, which may be evalu-ated directly from the relevant Feynman diagrams. Thespin-averaged scattering amplitude (fpp’) is given by [3],

fpp0 ¼1

4

Xss0

M

30p

M

30p0Mps;p0s0 ð6Þ

The dimensionless LPs are Fl = N(0)fl, where N(0) is thedensity of states at the Fermi surface, defined as [12]

Nð0Þ ¼ gsgIpf 3f2p2

ð7Þ

Here gs and gI are the spin and isospin degeneracy factors,respectively.

3. Landau parametersBy retaining only the lowest order terms in the pion

fields, one obtains the following Lagrangian from the chir-ally invariant Lagrangian [24, 25]

L ¼ �J gmði@m � guumÞ � igA

fpgmg5@m �

p� ðM � gsFsÞ� �

J

þ 1

2@mFs@mFs �

1

2m2

sF2s �

1

4umnumn þ

1

2m2

uumum

þ 1

2@mFp � @mFp �

1

2m2

pF2p þ LNL þ dL

ð8Þ

where

umn ¼ @mun � @num;�p ¼ 1

2ðt �FpÞ

and t is the isospin index. Here J is the nucleon field, andFs, um, and Fp are the meson fields (isoscalar-scalar, iso-scalar-vector, and isovector-pseudoscalar, respectively). Theterms LNL and dL contain the nonlinear and counterterms, re-spectively (for explicit expressions see [25]).

Because of the presence of pion fields in the chiral Lagran-

gian, we have a component in the interaction that acts on theisospin fluctuation. One can derive the isospin-dependentquasi-particle interaction along the lines of ref. 3. For pions, asmentioned earlier, the direct term vanishes, and it is only theexchange diagram that contributes to the interaction parameter,

f ex;ppp0 jp’p0¼pf ¼

3

2

g2AM�2

4f 2p3

2f

p2f ð1� cos qÞ

2p2f ð1� cos qÞ þ m2

p

( )ð9Þ

where 3f ¼ ðp2f þM�

2Þ1=2 and g2A ¼ 1:5876, fp = 93 MeV,

mp = 139 MeV [25]. The factor 3/2 arises because the iso-spin factor is 3/2 in the isoscalar channel [11].

The effective nucleon mass M* is determined self-consis-tently from the following equation [12],

M� ¼ M � g2s

m2s

Xi

niM�

ðp2i þM�2Þ1=2

ð10Þ

Using (4) and (9), we can derive isoscalar LPs f ex;p0 and f ex;p

1

f ex;p0 ¼ � 3g2

AM�2

32f 2p3

2f

�2þ m2p

2p2f

ln 1þ4p2

f

m2p

!" #ð11Þ

and

1

3f ex;p1 ¼ � 3g2

AM�2m2

p

64f 2p3

2f p

2f

� �2þ m2p

2p2f

þ 1

!ln 1þ

4p2f

m2p

!" #ð12Þ

Using (11) and (12) we find that

f ex;p0 � 1

3f ex;p1 ¼ 3g2

AM�2

16f 2p3

2f

� m4p

8p4f

ln 1þ4p2

f

m2p

!� m2

p

2p2f

þ 1

" #ð13Þ

It is this combination, i.e., f0 – (1/3)f1, which appears in thecalculation of chemical potential and other relevant quanti-ties. For the massless pion, (13) turns out to be finite

f ex;p0 � 1

3f ex;p1

� �jmp!0 ¼

3g2AM�2

16f 2p 3

2f

ð14Þ

It is to be noted that in the massless limit for the s and u

mesons, f ex0 and f ex

1 diverge as shown in [3], in contrast tothe pion, for which even in the massless limit these are fi-nite. This is due to the presence of (1 – cos q) in the numer-ator of (9), unlike for s and u mesons.

The dimensionless LPs can be determined by equating theequation F0 = N(0)f0 and F1 = N(0)f1, where N(0) is the den-sity of states at the Fermi surface defined in (7). Thus thedimensionless parameters are

Fex;p0 ¼ �gsgI

3g2ApfM

�2

64p2f 2p3f

�2þ m2p

2p2f

ln 1þ4p2

f

m2p

!" #ð15Þ

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Published by NRC Research Press

and

1

3Fex;p

1 ¼ �gsgI3g2

Am2pM�2

128p2f 2ppf 3f

� �2þ m2p

2p2f

þ 1

!ln 1þ

4p2f

m2p

!" #ð16Þ

In Fig. 1, we show the density dependence of F0 and F1on pionic interaction. Numerically, at nuclear matter densityr0 = 0.148 fm–3, Fex;p

0 ¼ 0:68 and Fex;p1 ¼ �0:2. In the non-

relativistic limit ef ? M*, one obtains the same expressionfor Fp

1 that was reported in [10, 13].

4. Chemical potential

We will now proceed to calculate the chemical potential,which in principle will be different for the neutron and pro-ton in asymmetric nuclear matter.

Although in the present work we deal with symmetric nu-clear matter (SNM), for which mp = mn, to calculate the gen-eral expression for chemical potential with arbitraryasymmetry (a = (nn – np)/(nn + np)), we use the distributionfunction with explicit isospin index (b or b’), so that the var-iation of distribution function gives [2],

dnb ¼ �Nbð0Þ½Xb0f bb

0

0 dnb0 � dmb� ð17Þ

where Nb(0) is the isospin dependent density of states at thecorresponding Fermi surface. Equation(17) yields

@mb@nb¼ 1

Nbð0ÞþXb0f bb

0

0

@nb0

@nbð18Þ

In our case b (or b’) = n, p. Separately for neutron and pro-ton we have,

ð@mn

@mpÞ ¼

1

Nnð0Þþ f nn

0 fnp0

fpn0

1

Npð0Þþ f pp

0

0BBB@

1CCCA

@nn

@np

!ð19Þ

where the superscripts nn and np denote neutron–neutronand neutron–proton quasi-particle interaction. Nn(0) andNp(0) denote the density of states for the neutron and protonFermi surfaces, respectively [26, 27]. For pion exchange inSNM, f npðpnÞ

l ¼ 2fnnðppÞl [13], and from (19), we have

@mn ¼1þ 3Fnn

0

Nnð0Þ@nn ð20Þ

where Fnn0 ¼ Nnð0Þf nn

0 . Similarly, one can determine qmp.Motivated by [5], we define

f bl ¼1

2

Xb0f bb

0

l ð21Þ

Evidently in SNM, f nnl ¼ f

ppl , and therefore we write mn =

mp = m [3]

mdm ¼ pf þ gsgImpf

2p2f0 �

1

3f1

� �� �dpf ð22Þ

Here f0 and f1 are given by (11) and (12).To calculate m, it is sufficient to let m = ef on the right-

hand side of (22). With the constant of integration adjustedso that at high density pf ’ 3f , (22) upon integration, to-gether with (13) yields

m ¼ 3f � gs3g2

AM�4

128p2f 2p3f

� �2y3p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�y2

p þ 4

qtan�1 x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�y2

p þ 4pyp

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

p !"

þ y4p

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

px

ln 1þ 4x2

y2p

� �

� 4xffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

p� 2ðy4

p � 2y2p � 2Þ ln ðxþ

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

pÞ#ð23Þ

where x = pf/M* and yp = mp/M*.The calculations of LPs for other mesons is straightfor-

ward. However, for brevity, we do not present the corre-sponding expressions for s and u mesons but quote theirnumerical values in Table 1. The numbers cited above arerelevant for normal nuclear matter density, r0 = 0.148 fm–3.For the coupling constants, we adopt the same parameter setas designated by M0A in [25].

Interestingly, the individual contribution to LPs of s andu mesons are large, while the sum of their contributions toFtot

0 is small because of the sensitive cancellation of Fs0 and

Fu0 , as can be seen in Table 1. Such a cancellation is respon-

sible for nuclear saturation dynamics [8, 9]. Numerically,Fsþu

0 is approximately 3/2 times smaller than Fp0 .

5. Exchange energy

Once m is determined, one can readily calculate the en-ergy density due to p –N interaction in SNM as [3, 28]

Eexp ¼

Rdrðm� 3f Þ ¼ �g2

s

3g2AM�6

128f 2pp

4

� Ip þ y4p �

x2

2þ y

2p þ 4x2

8ln 1þ 4x2

y2p

� �� ��

� x4 þ y4p

2� y2

p � 1

� �fxh� lnðxþ hÞg2 � x4

�ð24Þ

where

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

pand

Ip ¼ �2y3p

ffiffiffiffiffiffiffiffiffiffiffiffiffi4� y2

p

q R x2

htan�1 x

ffiffiffiffiffiffiffiffiffiffiffiffiffi4� y2

p

pyph

!dx ð25Þ

For the massless pion this reads as

Eexp jmp!0 ¼

3g2AM�6

32f 2pp

4½xh� ln ðxþ hÞ�2 ð26Þ

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The contribution arising from pion exchange from the di-rect evaluation of Fig. 2c reads as [25]

Eexp ¼ gsgI

3g2A

32p4f 2p

M�2Z pf

0

jpj2dp

3p

Z pf

0

jp0j2dp03p0

�Z 1

�1

dðcos qÞ 3p3p0 � jpjjp0jcos q �M�223p3p0 � 2jpjjp0jcos q � 2M�2 þ m2

p

� �ð27Þ

Similarly, from Figs. 2a and 2b one can determine exchangeenergy due to s and u meson interaction [25]. In Table 2, wegive the exchange energy results calculated from RFLPs, whichis in agreement with two-loop results [25]. It might be men-tioned here that in the limit mp ? 0, (27) can be evaluated ana-lytically, which reproduces (26) derived in the FLT approach.

6. Symmetry energyKnowing the ‘‘isovector’’ combination of the LPs, one can

determine nuclear symmetry energy. The symmetry energyis defined as the energy difference between the neutron mat-ter and symmetric nuclear matter and is given by the follow-ing expression [12, 29]

b ¼ 1

2r@2E

@r23

jr3¼0 ð28Þ

In terms of LPs, the symmetry energy can be expressed as[5]

b ¼p2f

63fð1þ 2F 00Þ ð29Þ

where

F 00 ¼1

2ðFnn

0 � Fnp0 Þ

is the isovector combination of the dimensionless Landauparameters F0 [5, 29, ]. Deriving F 00, one can find symmetryenergy. Numerically at saturation density (r = r0), we obtainb = 14.57 MeV. So, a relatively small contribution to bcomes from one pion exchange diagram [30].

7. Summary and conclusionIn this paper, we calculate RFLPs within the framework of

RFLT. For the description of a dense nuclear system, chEFTis invoked. Although our main focus was to estimate the con-tribution of pions to RFLPs, for comparison and complete-ness we also present results for the s and u mesons. It isseen that the pionic contribution to the FLPs are significantlylarger compared with the combined contributions of s and umesons. Thus, any realistic relativistic calculation for theFLPs should include p mesons, which necessarily impliesgoing beyond the MF calculations. The LPs that we deter-mine here are subsequently used to calculate exchange andsymmetry energy of the system. Finally, we evaluate two-loop ring diagrams with the same set of interaction parame-ters and show that the numerical results are consistent withthose obtained from the FLT. It might be mentioned herethat in the present calculation we have ignored nucleon–nucleon correlations, which might be worthwhile to inves-tigate. It should, however, be noted that the inclusion of

correlation energy would require a readjustment of the cou-pling parameter to reproduce the saturation properties ofnuclear matter.

AcknowledgmentsThe author would like to thank G. Baym and C. Gale for

their valuable comments. I also wish to thank A.K. Dutt-Mazumder for the critical reading of the manuscript and S.Biswas for his help with the numerical calculations.

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