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ω meson propagation in dense nuclear matter and collective excitations
Nuclear Physics A 713 (2003) 119–132 www.elsevier.com/locate/npe
ω meson propagation in dense nuclear matter and collective excitations Abhee K. Dutt-Mazumder TRIUMF Theory Group, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3 Canada Received 21 August 2002; received in revised form 18 September 2002; accepted 26 September 2002
Abstract The bosonic excitations induced by the ω meson propagation in dense nuclear matter is studied within the framework of random phase approximation. The collective modes are then analyzed by finding the zeros of the relevant dielectric functions. Subsequently we present closed form analytical expressions for the dispersion relations in different kinematical regime. Next, the analytical behaviour of the in-medium effective propagator for the ω meson is examined. This is exploited to calculate the full spectral function for the transverse (T ) and longitudinal (L) mode of the ω meson. In addition, various sum rules are constructed for the ω meson spectral density in nuclear medium. Results are then discussed by calculating the residues at the poles and discontinuities across the cuts. 2002 Elsevier Science B.V. All rights reserved. PACS: 25.75.-q; 21.65.+f Keywords: ω meson; Dispersion; Sum-rule; Nuclear matter
1. Introduction One of the cardinal challenges facing the nuclear physics today is to determine the properties of hadrons at finite temperature and/or chemical potential. Such studies are important both in the context of high energy heavy ion collisions (HIC) and fixed target electron scattering experiments [1,2]. Medium effects have also been found to be important to explain pion induced pion production data. It is seen that a clear enhancement of π + π − pair in the low invariant mass region could be accounted for by invoking the in-medium σ meson spectral function [3,4]. Similarly it is observed that the photoabsorption data E-mail address: [email protected] (A.K. Dutt-Mazumder). 0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 2 ) 0 1 2 9 7 - 6
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in γ + A system can be explained by invoking the medium modified ρ meson spectral function [1]. In heavy ion collision, among others, the light vector mesons have acquired particular interest. This is because of their dileptonic decay channels. Dileptons, once produced leaves the medium without suffering further scattering and hence provide a penetrating probe to study the properties of the dense nuclear medium as has been discussed at length in the recent literatures [1]. Experimentally the dilepton invariant mass spectra measured at CERN/SPS suggests that in-medium properties of the vector mesons, in particular that of ρ meson, are crucial to understand the low mass enhancement of the lepton pairs. In the theoretical arena such studies were stimulated by the pioneering work of Brown and Rho [5] where it was proposed that, in hot and/or dense matter, vector meson masses would drop from their values in free space as a precursor phenomenon of chiral symmetry restoration. Since then, several theoretical models have been invoked to study the inmedium properties of ω, ρ or φ meson which include QCD sum-rule, Nambu–Jona-Lasinio model, Chiral perturbation theory or simple hadron effective Lagrangian approach [6–10]. In the present work we discuss the vector meson propagation in nuclear matter which could be visualized as meson waves traveling through dispersive medium. This concept is usually invoked to study the pionic collective effects in nuclear matter [11]. A characteristic feature of meson propagation in matter is that there exists two distinct scales depending upon its momenta; if the momenta of the meson is small compared to the Fermi momenta of the nuclear matter, the meson propagation becomes inseparable from the macroscopic collective oscillation of the system induced by the mesonic field, while on the other hand, for large momenta, as we shall see, the meson propagation can still be regarded as usual particle propagation with its dispersion characteristics very similar to that of free space. In particular we study the propagation of the ω meson in dense nuclear matter in the long-wave length limit, i.e., in the regime when the typical characteristic length over which the meson field varies (1/q) is large compared to the de Broglie wave-length of the nucleons 1/pF . Such a study was initially pursued by Chin [12] who showed that, in nuclear matter, ω meson picks up the collective oscillation of the system and becomes massive. However, in Ref. [12], the vacuum contribution (Dirac sea) was neglected on the ground that it only renders the coupling constant momentum dependent and in general, in the long-wave length limit (i.e., for low momentum transfer), the effect is only marginal. Later, in a seminal paper by Jean et al. [6] it was shown that, in presence of a mean scalar field (Walecka model), the vacuum part contributes substantially with opposite sign. This, therefore, plays a crucial role in determining the over all sign and magnitude of the net polarization insertion [6]. In the current work we include both the effects and present new and more complete results for the ω meson dispersion relations in nuclear matter by considering various kinematical limits of the ω meson momentum compared to the density characterized by the Debye mass or the plasma frequency of the system. In addition to the calculation of the dispersion relations, we also examine the analytical properties of the effective ω meson propagator for the longitudinal and transverse excitations and construct various sum rules for the relevant spectral densities (ρT ,L ). Later, the full spectral functions (ρT ,L ) are expressed in terms of the residues at the poles and cuts of the propagators. Such studies, as we shall see, provide insight to understand and interpret the collective excitations of the nuclear matter via the sum rules and the residues
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in different kinematic domains of the momenta. It is worthwhile to mention here that the present scheme allows, as we shall see, not only to examine the collective modes in different kinematic regimes, compared to the characteristic plasma frequency of the system, but also helps to explore the analytic structure of the effective ω meson propagator in matter. This in turn permits to calculate the relevant spectral function and various sum rules as we report here. Such studies to the best of our knowledge has not been pursued before. The paper is organized as follows. We first outline the formalism and describe the collective modes in terms of the zeros of the dielectric functions involving density dependent ω meson polarization tensor. In Section 3, the concept of hard nucleon loop (HNL) approximation is introduced which (i) calculates ΠL and ΠT relevant for the collective oscillation, (ii) allows direct comparison with familiar results of classical plasma physics and hot QED/QCD. In Section 4, several sum rules for the ω meson spectral functions are constructed by exploiting the properties of dispersion relation techniques of complex variables. Finally the conclusion and summary is presented in Section 5.
2. Formalism It is well-known that the collective modes corresponding to density fluctuations are determined from the poles of the meson propagators. In Ref. [12] it has been demonstrated that in nuclear matter such collective modes are dominated solely by the vector interaction in the high-density limit. This, in turn, implies that the in-medium properties of the vector mesons are directly linked with the collective excitation set by their propagation. The dielectric functions relevant for the ω meson propagation in dense nuclear matter can be calculated within the frame work of random phase approximation (RPA) which essentially implies repeated insertion of the ω meson self-energy involving nucleon– nucleon loop as described in Refs. [12,13]. ¯ µ ψωµ , Considering the interaction of the ω meson with the nucleon field, LI = gV ψγ the second order polarization tensor Πµν , can be written as −i αβ (1) Πµν = d 4 p Tr iγµα iG(p + q)iγνβ iG(p) , 4 (2π) where (α, β) are the isospin indices and G(p) is the in-medium nucleon propagator [13] G(p) = GF (p) + GD (p), where GF (p) = pµ γ µ + M ∗
and GD (p) = pµ γ µ + M ∗
1 p2 − M ∗ 2 + i
(2)
iπ ∗ δ p − E (p) θ p − | p| . 0 F E ∗ (p)
(3)
(4)
The first term in G(p), namely, GF (p), is the same as the free propagator of a spin arises from Pauli 1/2 fermion, while the second part, GD (p), involving θ (pF − |p|),
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blocking, describes the modification of the same in the nuclear matter at zero temperature. E ∗ (p) = p2 + M ∗ 2 and M ∗ denotes the in-medium mass of the nucleon, which in the present context might be determined from the mean scalar density [13]. In a similar vein the polarization insertions can also be written as sum of two parts: F D Πµν = Πµν + Πµν .
(5)
From the above set of equations we can write the real part of the density dependent ω-meson self energy as [10,12] D Πµν
g2 = v3 π
pF 0
d 3 p Pµν q 2 − Qµν (p · q)2 , E(p) q 4 − 4(p · q)2
(6) q q
q )(pν − p.q q ) and Qµν = (−gµν + µq 2 ν ). It is evident that the where Pµν = (pµ − p.q q2 µ q2 ν form of the polarization tensor conforms to the requirement of current conservation, i.e., D = Π D q ν = 0. In order to evaluate Π D conveniently, we choose q q µ Πµν to be along µν µν the z axis, i.e., q = (q0 , 0, 0, | q |), and p.q = E(p)q0 − |p||q|χ , where χ is the cosine of the angle between p and q . On account of integration over the azimuthal angle, only D = Π D and six components survive. Then again for isotropic nuclear matter we have Π11 22 D D Π03 = Π30 . Furthermore, the current conservation condition puts additional constraints D , linear combinations of leaving only two non-vanishing independent component of Πµν D , namely, Π D (q) = which gives us the longitudinal and transverse components of Πµν L D D D D D −Π00 + Π33 (q) and ΠT (q) = Π11 = Π22 The free part is obtained by putting G(p) = GF (p) in Eq. (1) which is divergent and can be regularized using appropriate renormalization scheme [12,14]. The regularization procedure we adopt here is the following: ∂ n Π F (q 2 ) = 0, (7) ∂(q 2)n Mn∗ →M,q 2 =m2ω where, (n = 0, 1, 2, . . ., ∞). This yields F Πµν
g2 = Qµν v2 q 2 π
1 0
∗2 M − q 2 x(1 − x) dx x(1 − x) log . M 2 − m2ω x(1 − x)
(8)
It is evident that free part does not distinguish between the longitudinal and transverse mode and we can take ΠLF = ΠTF = Π22 = Π11 . In order to study the collective modes induced by the ω meson propagation in nuclear matter, there is an additional effect which one should take into account. The longitudinal ω meson mode gets coupled with the scalar mode corresponding to the σ meson propagation in matter. This is due to the presence of mixed polarization function d 4k Πµ q0 , | q | = igs gv Tr G(k)γµ G(k + q) . (9) 4 (2π)
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This is exclusively density dependent effect. With the evaluation of the trace and after a little algebra Eq. (9) could be cast into a suggestive form: gv gs Πµ q0 , |q| = 3 q 2 M ∗ π
kF 0
q
µ d 3 k kµ − q 2 (k · q) . E ∗ (k) q 4 − 4(k · q)2
(10)
This immediately leads to two conclusions. First, it respects the current conservation condition, viz. q µ Πµ = 0 = Πν q ν . Secondly, there are only two components which would survive after the integration over azimuthal angle. In fact this guarantees that it is only the longitudinal component of the ω meson which couples to the scalar meson while the transverse mode remains unaltered. Furthermore, current conservation implies that out of the two non-zero components of Πµ , only one is independent. In presence of mixing the combined meson propagator might be written in a matrix form where the dressed propagator would no longer be diagonal: D = D0 + D0 ΠD .
(11)
It is to be noted that the free propagator is diagonal and has the form
0 Dµν 0 D0 = . 0 ∆0
(12)
In Eq. (12) the non-interacting propagator for the σ and ω are given respectively by ∆0 (q) =
1 , q 2 − m2σ + i
0 Dµν (q) =
−gµν +
qµ qν q2
q 2 − m2ω + i
(13) ,
(14)
The mixing is characterized by the polarization matrix which contains non-diagonal elements
Πµν (q) Πν (q) Π= . (15) Πµ (q) Πσ (q) It should be mentioned that similar to the ω meson polarization tensor, Πσ has also two parts which can be written as Πσ = ΠσF + ΠσD . The free part is divergent which can be regularized to yield: 2 3gs2 ∗ 2 F 3 M − M2 − 4 M∗ − M M − M∗ 2 − M2 Πσ q = 2 2π 1 × 0
1 − 0
∗2 M − x(1 − x)q 2 dx ln M2 ∗2 2 M − x(1 − x)q 2 2 . dx M − x(1 − x)q ln M 2 − x(1 − x)m2σ
(16)
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The density dependent part is evaluated in the next section in the long wave length approximation. To determine the collective modes, one defines the dielectric function as [12]: q |) = det 1 − D0 Π = T2 × mix , (17) (q0 , | where T corresponds to two identical transverse (T ) modes and mix correspond to the longitudinal mode with the mixing. The latter, of course, also characterizes the mode relevant for the σ meson propagation T = 1 − d0 ΠT ,
d0 =
q2
1 , − m2ω + i
mix = (1 − d0 ΠL )(1 − ∆0 Πs ) −
q2 ∆0 d0 (Π0 )2 | q |2
= (1 − d0 ΠL )(1 − ∆0 Πs ) − ∆0 d0 (Πmix )2 ,
(18)
q2
2 = (Π0 )2 . where Πmix | q |2 In order to study collective response of the nuclear system and to obtain the dispersion relation of the induced oscillation, one needs to find zeros of the dielectric function defined through q0 , |q| = T2 × mix = 0. (19)
It is to be noted that the free part ΠTF,L,σ depends only on the Lorentz scalar q 2 while the density dependent part ΠTD,L,σ involves both q0 and |q| individually. The equations (q0 , |q|) = 0 in general have to be solved numerically. However, in the next section, we discuss appropriate approximation scheme which admits close form solutions.
3. Collective modes and dispersion relations In this section we discuss first the approximation scheme to derive the polarization tensors analytically. Once the polarization insertions are determined one can solve Eq. (19) to find the dispersion relations for the ω meson in nuclear matter taking both the effect of free and dense part of the polarization function in the various kinematic limits as mentioned in the introduction. In addition, this also provides us a way to interpret the results in terms of the classical plasma oscillations. First we recall that for collective excitations, the wavelength of the oscillations must be greater than the interparticle spacing. This means the meson momenta must be small compared to the nucleon momenta. Therefore, quantitatively, it is legitimate to assume that all the loop momenta (nucleon) are hard and the external momenta (meson) are soft i.e., p ∼ pF , |q| pF . We term this as hard nucleon loop (HNL) approximation. This allows us to drop q 4 compared to 4(p.q)2 in Eq. (6). Such an approximation, is, in effect, similar to what has been adopted in Ref. [12]. Physically this corresponds to long-wave length (low momentum transfer) excitations near the Fermi surface. Under this assumption the
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integrations of Eq. (6) relevant for the longitudinal and transverse modes are performed easily to give following results:
F ω Log ω+kv ω2 ω−kvF D 2 ΠL (ω, k) = −3Ω 1 − −1 + (20) 2kvF k 2 vF2 and ΠTD (ω, k) =
ω+kvF
3 2 ω2 ω2 ω Log ω−kvF Ω , + 1− 2 2kvF k 2 vF2 k 2 vF2
where, q0 = ω, |q| = k, vF = pF / F is the Fermi velocity with F = is identified as the plasma frequency given by Ω 2 =
gV2 π2
1 3
pF3 F
(21)
pF2 + M 2 and Ω 2
. This is related to the Debye
screening mass as = [15,16]. It is revealing to observe that in the limit vF → 1, with the appropriate redefinition of the plasma frequency Ω 2 , Eqs. (20) and (21) reduce to that of photon polarization functions [17] in hot QED. It might be mentioned here that the full analytical expression for the ΠT (ω, k, pF ) can be found in Ref. [18]. Before proceeding further, few comments are in order. It is evident that for k = 0 in matter, the longitudinal and transverse self-energies are non-degenerate. This gives rise to splitting between these two collective modes in nuclear matter which could be attributed to the presence of Pµν in Eq. (6). However, they become identical in the static limit, i.e., ΠL (ω, k → 0) = ΠT (ω, k → 0) = Ω 2 . Next we consider the mixed polarization vector Πµ given by Eq. (10). This can also be evaluated within the HNL approximation scheme to yield
F 2 ωLog ω+kv gs2 ∗2 2 ω2 − k 2 ω−kvF 2 . (22) Πmix = 2 M 3Ω −1 + π k2 2kvF Ω2
2 2 3 mD
Similarly ΠσD , can also be evaluated within the HNL scheme, which in the high density g2
limit (when M ∗ < kF ) reduces to ΠσD 32 gs2 Ω 2 . For the mixed polarization, the exact v expression can be found in Ref. [18]. The collective excitations, relevant for the longitudinal and the scalar modes are determined by the zeros of mix which could be rewritten as 2 . (23) mix = 1 − d0 ΠL − ∆0 Πσ − d0 ∆0 ΠL Πσ − Πmix In the high-density limit Πσ and ΠL goes as Ω 2 while Πmix is proportional to Ω implying 2 . It is to be noted that in the high density limit M ∗ Ω and with increasing ΠL Πσ Πmix ∗ density M decreases while Ω increases [13]. Hence the scalar and the longitudinal modes, to a first approximation, decouple from each other. This is particularly so for the low meson momentum when the mixing amplitude is small [18,19]. Thus we write mix (1 − d0 ΠL )(1 − ∆0 Πσ ) = 0.
(24)
The fact that the effect of mixing on the dispersion curves is only marginal has also been observed numerically in Ref. [19]. However, with increasing momentum, as shown in Ref. [18], the mixing effect might become important.
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Eq. (19) in conjunction with Eq. (21), (20), (27) and (24) define the dispersion characteristics for the collective excitations gv2 M ∗ − M 2 mω M 3π 2 ω+kvF
3 2 ω2 ω2 ω Log ω−kvF =0 − Ω + 1− 2 2kvF k 2 vF2 k 2 vF2
ω2 − k 2 − m2ω −
(25)
for T mode and gv2 M ∗ − M 2 mω M 3π 2 ω+kvF
2 ω Log ω ω−kvF −1 + = 0, + 3Ω 2 1 − 2 2 2kv k vF F
ω2 − k 2 − m2ω −
(26)
for L mode. It should be noted that in the above equations, the free part of the ω meson self-energy (also known as Dirac sea contribution) given by Eq. (8), following Ref. [6] has been simplified by making a Taylor series expansion of Eq. (8) around M ∗ . To the leading order the contribution is found to be [6,20]
2 g2 M ∗ − M 2 mω F mω + O ΠL,T . (27) (ω, k) = v2 M 3π 4M 2 As in nuclear matter effective nucleon mass is smaller than its free space value (M ∗ < M), it is clear from the last expression, that the free part reduces the effective ω meson mass in the medium. Generally the transcendental equations, viz. Eq. (25) and (26), have to be solved numerically, however, analytical solutions could be obtained in two limiting cases viz., when k < Ω and k > Ω by expanding above equations in powers of k and solving the 2 equations by successive iterations. To the first order, when k Ω one gets ωL(T ) = ∗2 2 mω + Ω . We note that the modes are degenerate in this limit and oscillations are independent of the wave vector k. In the case of electron–ion plasma, this mode was identified as the ‘plasma waves’ or ‘Langmuir’ oscillation and Ω 2 there is given by e2 ρ/m2 where ρ is the density of electron gas and m is the electron mass [21]. This justifies identifying Ω as the plasma frequency in the present pretext. It should be mentioned that here we have absorbed the contribution of the free part as
g2 M ∗ − M m∗ω2 = m2ω 1 + v2 . (28) 3π M Once the leading order solution is known, we can solve Eqs. (25) and (26) iteratively to obtain 1 Ω2 + ···, ωT2 = m∗ω2 + Ω 2 + k 2 + k 2 vF2 ∗ 2 5 mω + Ω 2
(29)
Ω2 3 2 ωL = m∗ω2 + Ω 2 + k 2 vF2 ∗ 2 + ···. 5 mω + Ω 2
(30)
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Evidently, the transverse mode lies above the longitudinal one and it requires more energy for excitation. However, the splitting in the low k region is very marginal. In the limit mω → 0 we recover the results presented in [12], i.e., 1 ωT2 = Ω 2 + k 2 + vF2 k 2 + · · · , 5 3 2 ωL = Ω 2 + vF2 k 2 + · · · . 5
(31) (32)
Furthermore, this taken with vF → 1 results in dispersion relations arising out of the photon propagation at finite temperature or density (where electron is considered to be ultrarelativistic) with the appropriate definition of the plasma frequency [16,17]. It is worthy to note that the contribution of the density dependent part is opposite to that of the Dirac part of the polarization tensor. This is consistent with Ref. [6]. Next we consider the case when k > Ω but still ω > k and in fact ω ∼ kvF . The successive approximation for this asymptotic value of k yields solutions of the form 3 2 ωT2 = k 2 + m∗2 ω + Ω + ···. 2
(33)
The corresponding longitudinal frequency takes little bit complicated form 2 ωL
= k 2 vF2
vF2 m∗2 2 k 2 vF2 ω −2 . + 1 + 4 exp − 3 Ω2 3Ω 2 (vF2 − 1)
(34)
Evidently, the longitudinal mode approaches the line ω = kvF and exponentially suppressed for very large values of k. We have solved Eqs. (25) and (26) numerically and checked that for meson momenta k ∼ 1 GeV dispersion results are in good agreement with Eqs. (29) and (30). It is worthwhile to note that the dispersion relations presented both in Eqs. (29), (30) and Eqs. (33), (34) coincide with the ones relevant for the classical plasma in the limit m∗ω → 0 as may be verified from Ref. [21].
4. Spectral functions and sum-rules Equipped with Eqs. (20) and (21), we can proceed to calculate the spectral functions of the ω meson. As demonstrated already, the ω meson in matter has two modes. Confining our attention on the transverse sector first, we write the propagator as ∆T (ω, k) =
ω2
− k2
− m∗2 ω
−
3 2 ω2 2 Ω (kvF )2
−1 + 1−
ω2 ω (kvF )2 2kvF
. (35) F Log ω+kv ω−kvF
Next chain of steps would be to construct the sum rules by exploiting the analytic properties of this effective propagator. First we note that the function ∆T is analytic in the complex ω-plane having a cut from −kvF to +kvF ; in addition to the poles ω = ±ωT . Having
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observed this one can make use of Cauchy’s theorem for ∆ = ∆T dω ∆(ω , k) ∆(ω, k) = 2πi ω − ω Γ
∞ = −∞
dω ∆(ω + iη, k) − ∆(ω − iη, k) + 2πi ω − ω
Γ
dω ∆(ω , k) . 2πi ω
(36)
In the above equation Γ is a circle with its radius pushed to infinity. This equation can be rewritten in terms of the spectral density ρ(ω, k) containing both the discontinuities across the cuts and the residues at the poles ρ(ω, k) = 2 Im ∆(ω + iη, k)
(37)
as ∞ ∆(ω, k) = −∞
dω ρ(ω , k) + 2π ω − ω
Γ
dω ∆(ω , k) . 2πi ω
(38)
It is evident from Eq. (35) that ∆T (ω , k) → ω12 as ω → ∞, therefore, the integration over the contour Γ fails to contribute in this region. First sum rule is then immediately obtained by setting ω = 0 in Eq. (38) ∞ −∞
dω ρT (ω, k) 1 = ∆T (0, k) = 2 . 2π ω k + m∗ω2
(39)
Other sum rules are obtained by examining the asymptotic behaviour of ∆T (ω, k) when ω → ∞; by taking the parity property of the spectral density, i.e., ρT (ω, k) = −ρT (−ω, k) into account, one can write from Eq. (38) for ω → ∞ ∞
1 ∆T (ω, k) = − ω
∞
n=0 −∞
dω ω 2n+1 ρT (ω , k). 2π ω
(40)
On the other hand, one can expand Eq. (35) in powers of ω−1 ; comparing this with Eq. (40) one writes for n = 0 ∞ −∞
ω dω ρT (ω, k) = 1 2π
(41)
and similarly for n = 1 one observes ∞ −∞
ω3 dω ρT (ω, k) = k 2 + m∗ω2 + Ω 2 . 2π
(42)
Thus we obtain other two sum rules. Of particular importance is Eq. (41). This does not depend on k which could be shown to be a consequence of the canonical commutation
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relation. Therefore, it is natural to expect that such a relation will also be satisfied by the transverse ω meson in nuclear matter. The longitudinal propagator is denoted as ∆L (ω, k) =
ω2 − k 2 −1 2 ω2 2 k −1 + ω2 − k 2 − m∗2 2 ω + 3Ω 1 − k2 v
F
ω 2kvF
. F Log ω+kv ω−kvF (43)
Contrary to the case of transverse propagator, we now have non-vanishing contribution from the contour denoted by Γ in Eq. (38) as in the limit ω → ∞ the logarithmic term → 1. Hence in this limit the longitudinal propagator goes like ∆L (ω, k) → −1/k 2 . Therefore, a subtraction, unlike the transverse propagator, is necessary. The subtracted dispersion relation gives the following sum rule essentially in the same manner as in the transverse case: ∞ 1 3Ω 2 + m∗2 dω ρL (ω, k) ω = 2 + ∆L (0, k) = 2 2 . (44) 2π ω k k (k + m∗2 + 3Ω 2) ω −∞
Similarly to the asymptotic behaviour of the longitudinal propagator one can write, ∞
1 1 ∆L (ω, k) = − 2 − ω k
∞
n=0−∞
dω ω 2n+1 ρT (ω , k). 2π ω
(45)
This is identical with Eq. (40) except here we have an additional factor of −1/k 2 for reason mentioned in the previous paragraph. Likewise, we now can derive the sumrules by first setting n = 0 to get ∞ −∞
ω dω m∗2 + Ω 2 ρL (ω, k) = ω 2 2π k
and n = 1 to obtain ∞ 3 2 2 2 2 2 ∗2 2 (Ω 2 + m∗2 ω dω ω )(k + mω + Ω ) − 5 k Ω vF ρL (ω, k) = . 2π k2
(46)
(47)
−∞
We can write down the full spectral function of the transverse and longitudinal modes for the ω meson in nuclear matter in terms of the poles (and residues at the poles) together with the cuts of the propagator 1 ρT ,L (ω, k) = ZT ,L (k) δ(ω − ωT ,L ) − δ(ω + ωT ,L ) 2π + βT ,L (ω, k)θ k 2 − ω2 ,
(48)
where, βT ,L is given by βT ,L (ω, k) =
2 Im ΠT ,L ∆T ,L (ω, k) . π
(49)
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The imaginary part of the self-energies ΠT ,L correspond to Landau damping and relevant only for the space-like momenta of the meson. Physically this refers to the energy loss due to scattering of the ω meson with hard nucleons in the medium. They are given by the imaginary part of Eqs. (20) and (21). The residues at the poles which are used to determine the full spectral function of the ω meson in nuclear matter can be determined from the one-loop effective propagators. For instance the residue at the pole for the transverse excitation is determined by −1 (50) ZT = − ∂∆−1 T /∂ω ω=ω T (k)
=
ωT (k 2 vF2 − ωT2 ) ωT2 (ω2 − 3(ΩT2 + k 2 + m∗ω2 )) + k 2 (k 2 + ωT2 + m∗ω2 )vF2
.
(51)
In a similar way we also calculate the reside at the pole for the longitudinal oscillation which is found to be ZL = −
2 )(−ω3 + ω k 2 v 2 ) (k 2 − ωL L F L 2 (−3k 2 − 3m∗ 2 + ω2 − 3Ω 2 ) + k 2 (k 2 + m∗ 2 + ω2 + 3Ω 2 )v 2 ] k 2 [ωL ω ω L L F
. (52)
We now present the limiting values of these residues at the poles given in Section 3. First considering k < Ω, we have for the transverse ω
1 k 2 5m∗ω2 + (5 + 3vF2 )Ω 2 − (53) ZT (k) (Ω 2 + m∗ω2 )5/2 2 Ω 2 + m∗ω2 20 while for large k, i.e., for k > Ω we have 1 . ZT (k) 2 k 2 + m∗ω2
(54)
Similar results are obtained for the longitudinal mode also. They are given by, for k < Ω Ω 2 + m∗ω2 25(m∗ω2 + Ω 2 ) − 22vF2 Ω 2 ZL (k) − (55) 2k 2 20(Ω 2 + m∗ω2 )3/2 and for k > Ω ZL (k)
vF2 4kvF 2 k 2 vF2 m∗2 ω − 2 . exp − + 3 Ω2 3Ω 2 3Ω 2 (vF2 − 1)
(56)
It is now evident from Eqs. (28) and (54) that the transverse spectral function for large meson momentum reduces to that of the ω meson propagation in free space, except its mass is modified because of the presence of the mean scalar field. Longitudinal mode in this limit decouples from the system. For small momenta, the sum rules are almost saturated by the pole contributions. This could be checked numerically.
5. Summary and conclusion To conclude and summarize, in the present work, we investigate the collective modes induced by the ω meson propagation in dense nuclear matter. We then determine the
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dispersion relations for the ω meson by considering both the effect of the Fermi sea and Dirac vacuum within the scheme of HNL approximation. It is observed that in the appropriate limit, i.e., for meson momenta small compared to the plasma frequency of the system, the results are consistent with the previous calculations [6,12]. The ω meson mass, in presence of a mean field, decreases even at normal nuclear matter while for free Fermi gas the effect of nucleon loop is to increase the ω meson mass proportional to the characteristic plasma frequency of the system. We also present analytical results for the dispersion relation in the limit k > Ω which shows that for the large value of the momenta the medium effects die off and the transverse mode propagates like a free meson. The longitudinal mode in this limit decouples from the system. We also construct various sum rules satisfied by the ω meson spectral density in nuclear matter by investigating the analytic properties of the relevant effective propagator at finite density. It is observed that sum rules are mostly saturated by the pole contributions for small momenta. The residues are also presented in the different limit of the meson momenta to understand the collective modes and their behavior. Similar studies can also be pursued for other light vector mesons. For instance, it is known that the ρ nucleon interaction is very similar to the ω nucleon interaction, where, in addition to the vector interaction, one has to include tensor interaction as well. We, therefore, believe that the ρ meson propagation in cold nuclear matter can also be studied within the framework HNL approximation. This would allow for an analytical derivation for the spectral density of the ρ meson. Hence , sum rules similar to what we present here, relevant for the ρ meson spectral function at finite density can also be constructed in a parallel approach. Such studies are in progress and shall be reported elsewhere.
Acknowledgements The author would like to thank A.G. Williams and J. Piekarewicz for useful private communications.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
R. Rapp, J. Wambach, Adv. Nucl. Phys. 25 (2000) 1. B.M. Preedom, G.S. Blanpied, CEBAF report PR-89-001 (1989). T. Hatsuda, T. Kunihiro, H. Shimizu, Phys. Rev. Lett. 82 (1999) 2840. M. Buballa, W. Norenberg, B.-J. Schaefer, J. Wambach (Eds.), Proceedings of the International Workshop XXVIII on Gross Properties of Nuclei and Nuclear Excitations, 2000. G.E. Brown, M. Rho, Phys. Rev. Lett. 66 (1991) 2720. H.-C. Jean, J. Piekarewicz, A.G. Williams, Phys. Rev. C 49 (1994) 1981. H. Shiomi, T. Hatsuda, Phys. Lett. B 334 (1994) 281. A. Hosaka, Phys. Lett. B 244 (1990) 363. K. Saito, K. Tsushima, A.W. Thomas, A.G. Williams, Phys. Lett. B 433 (1998) 243. A.K. Dutt-Mazumder, B. Dutta-Roy, A. Kundu, T. De, Phys. Rev. C 53 (1996) 790. T. Ericson, W. Weise, Pions and Nuclei, Oxford Sci. Publication, Oxford, 1998. S.A. Chin, Ann. Phys. 108 (1977) 301. B.D. Serot, J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1.
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[14] T. Hatsuda, S.H. Lee, H. Shiomi, Prog. Theor. Phys. 95 (1996) 1009. [15] M. Le Bellac, Thermal Field Theory, in: Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, 1996. [16] J.I. Kapusta, Finite Temperature Field Theory, Cambridge Univ. Press, Cambridge, 1989. [17] J.P. Blaizot, E. Iancu, Phys. Rep. 359 (2002) 355. [18] K. Lim, C.J. Horowitz, Nucl. Phys. A 501 (1989) 729. [19] O. Teodorescu, A.K. Dutt-Mazumder, C. Gale, Phys. Rev. C 61 (2000) 051901; O. Teodorescu, A.K. Dutt-Mazumder, C. Gale, Phys. Rev. C 63 (2001) 034903. [20] A.G. Williams, private communication. [21] E.M. Lifshitz, L.P. Pitaevskii, in: Physical Kinetics, Vol. 10, Pergamon, 1981.
ω meson propagation in dense nuclear matter and collective excitations Abhee K. Dutt-Mazumder TRIUMF Theory Group, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3 Canada Received 21 August 2002; received in revised form 18 September 2002; accepted 26 September 2002
Abstract The bosonic excitations induced by the ω meson propagation in dense nuclear matter is studied within the framework of random phase approximation. The collective modes are then analyzed by finding the zeros of the relevant dielectric functions. Subsequently we present closed form analytical expressions for the dispersion relations in different kinematical regime. Next, the analytical behaviour of the in-medium effective propagator for the ω meson is examined. This is exploited to calculate the full spectral function for the transverse (T ) and longitudinal (L) mode of the ω meson. In addition, various sum rules are constructed for the ω meson spectral density in nuclear medium. Results are then discussed by calculating the residues at the poles and discontinuities across the cuts. 2002 Elsevier Science B.V. All rights reserved. PACS: 25.75.-q; 21.65.+f Keywords: ω meson; Dispersion; Sum-rule; Nuclear matter
1. Introduction One of the cardinal challenges facing the nuclear physics today is to determine the properties of hadrons at finite temperature and/or chemical potential. Such studies are important both in the context of high energy heavy ion collisions (HIC) and fixed target electron scattering experiments [1,2]. Medium effects have also been found to be important to explain pion induced pion production data. It is seen that a clear enhancement of π + π − pair in the low invariant mass region could be accounted for by invoking the in-medium σ meson spectral function [3,4]. Similarly it is observed that the photoabsorption data E-mail address: [email protected] (A.K. Dutt-Mazumder). 0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 2 ) 0 1 2 9 7 - 6
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in γ + A system can be explained by invoking the medium modified ρ meson spectral function [1]. In heavy ion collision, among others, the light vector mesons have acquired particular interest. This is because of their dileptonic decay channels. Dileptons, once produced leaves the medium without suffering further scattering and hence provide a penetrating probe to study the properties of the dense nuclear medium as has been discussed at length in the recent literatures [1]. Experimentally the dilepton invariant mass spectra measured at CERN/SPS suggests that in-medium properties of the vector mesons, in particular that of ρ meson, are crucial to understand the low mass enhancement of the lepton pairs. In the theoretical arena such studies were stimulated by the pioneering work of Brown and Rho [5] where it was proposed that, in hot and/or dense matter, vector meson masses would drop from their values in free space as a precursor phenomenon of chiral symmetry restoration. Since then, several theoretical models have been invoked to study the inmedium properties of ω, ρ or φ meson which include QCD sum-rule, Nambu–Jona-Lasinio model, Chiral perturbation theory or simple hadron effective Lagrangian approach [6–10]. In the present work we discuss the vector meson propagation in nuclear matter which could be visualized as meson waves traveling through dispersive medium. This concept is usually invoked to study the pionic collective effects in nuclear matter [11]. A characteristic feature of meson propagation in matter is that there exists two distinct scales depending upon its momenta; if the momenta of the meson is small compared to the Fermi momenta of the nuclear matter, the meson propagation becomes inseparable from the macroscopic collective oscillation of the system induced by the mesonic field, while on the other hand, for large momenta, as we shall see, the meson propagation can still be regarded as usual particle propagation with its dispersion characteristics very similar to that of free space. In particular we study the propagation of the ω meson in dense nuclear matter in the long-wave length limit, i.e., in the regime when the typical characteristic length over which the meson field varies (1/q) is large compared to the de Broglie wave-length of the nucleons 1/pF . Such a study was initially pursued by Chin [12] who showed that, in nuclear matter, ω meson picks up the collective oscillation of the system and becomes massive. However, in Ref. [12], the vacuum contribution (Dirac sea) was neglected on the ground that it only renders the coupling constant momentum dependent and in general, in the long-wave length limit (i.e., for low momentum transfer), the effect is only marginal. Later, in a seminal paper by Jean et al. [6] it was shown that, in presence of a mean scalar field (Walecka model), the vacuum part contributes substantially with opposite sign. This, therefore, plays a crucial role in determining the over all sign and magnitude of the net polarization insertion [6]. In the current work we include both the effects and present new and more complete results for the ω meson dispersion relations in nuclear matter by considering various kinematical limits of the ω meson momentum compared to the density characterized by the Debye mass or the plasma frequency of the system. In addition to the calculation of the dispersion relations, we also examine the analytical properties of the effective ω meson propagator for the longitudinal and transverse excitations and construct various sum rules for the relevant spectral densities (ρT ,L ). Later, the full spectral functions (ρT ,L ) are expressed in terms of the residues at the poles and cuts of the propagators. Such studies, as we shall see, provide insight to understand and interpret the collective excitations of the nuclear matter via the sum rules and the residues
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in different kinematic domains of the momenta. It is worthwhile to mention here that the present scheme allows, as we shall see, not only to examine the collective modes in different kinematic regimes, compared to the characteristic plasma frequency of the system, but also helps to explore the analytic structure of the effective ω meson propagator in matter. This in turn permits to calculate the relevant spectral function and various sum rules as we report here. Such studies to the best of our knowledge has not been pursued before. The paper is organized as follows. We first outline the formalism and describe the collective modes in terms of the zeros of the dielectric functions involving density dependent ω meson polarization tensor. In Section 3, the concept of hard nucleon loop (HNL) approximation is introduced which (i) calculates ΠL and ΠT relevant for the collective oscillation, (ii) allows direct comparison with familiar results of classical plasma physics and hot QED/QCD. In Section 4, several sum rules for the ω meson spectral functions are constructed by exploiting the properties of dispersion relation techniques of complex variables. Finally the conclusion and summary is presented in Section 5.
2. Formalism It is well-known that the collective modes corresponding to density fluctuations are determined from the poles of the meson propagators. In Ref. [12] it has been demonstrated that in nuclear matter such collective modes are dominated solely by the vector interaction in the high-density limit. This, in turn, implies that the in-medium properties of the vector mesons are directly linked with the collective excitation set by their propagation. The dielectric functions relevant for the ω meson propagation in dense nuclear matter can be calculated within the frame work of random phase approximation (RPA) which essentially implies repeated insertion of the ω meson self-energy involving nucleon– nucleon loop as described in Refs. [12,13]. ¯ µ ψωµ , Considering the interaction of the ω meson with the nucleon field, LI = gV ψγ the second order polarization tensor Πµν , can be written as −i αβ (1) Πµν = d 4 p Tr iγµα iG(p + q)iγνβ iG(p) , 4 (2π) where (α, β) are the isospin indices and G(p) is the in-medium nucleon propagator [13] G(p) = GF (p) + GD (p), where GF (p) = pµ γ µ + M ∗
and GD (p) = pµ γ µ + M ∗
1 p2 − M ∗ 2 + i
(2)
iπ ∗ δ p − E (p) θ p − | p| . 0 F E ∗ (p)
(3)
(4)
The first term in G(p), namely, GF (p), is the same as the free propagator of a spin arises from Pauli 1/2 fermion, while the second part, GD (p), involving θ (pF − |p|),
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blocking, describes the modification of the same in the nuclear matter at zero temperature. E ∗ (p) = p2 + M ∗ 2 and M ∗ denotes the in-medium mass of the nucleon, which in the present context might be determined from the mean scalar density [13]. In a similar vein the polarization insertions can also be written as sum of two parts: F D Πµν = Πµν + Πµν .
(5)
From the above set of equations we can write the real part of the density dependent ω-meson self energy as [10,12] D Πµν
g2 = v3 π
pF 0
d 3 p Pµν q 2 − Qµν (p · q)2 , E(p) q 4 − 4(p · q)2
(6) q q
q )(pν − p.q q ) and Qµν = (−gµν + µq 2 ν ). It is evident that the where Pµν = (pµ − p.q q2 µ q2 ν form of the polarization tensor conforms to the requirement of current conservation, i.e., D = Π D q ν = 0. In order to evaluate Π D conveniently, we choose q q µ Πµν to be along µν µν the z axis, i.e., q = (q0 , 0, 0, | q |), and p.q = E(p)q0 − |p||q|χ , where χ is the cosine of the angle between p and q . On account of integration over the azimuthal angle, only D = Π D and six components survive. Then again for isotropic nuclear matter we have Π11 22 D D Π03 = Π30 . Furthermore, the current conservation condition puts additional constraints D , linear combinations of leaving only two non-vanishing independent component of Πµν D , namely, Π D (q) = which gives us the longitudinal and transverse components of Πµν L D D D D D −Π00 + Π33 (q) and ΠT (q) = Π11 = Π22 The free part is obtained by putting G(p) = GF (p) in Eq. (1) which is divergent and can be regularized using appropriate renormalization scheme [12,14]. The regularization procedure we adopt here is the following: ∂ n Π F (q 2 ) = 0, (7) ∂(q 2)n Mn∗ →M,q 2 =m2ω where, (n = 0, 1, 2, . . ., ∞). This yields F Πµν
g2 = Qµν v2 q 2 π
1 0
∗2 M − q 2 x(1 − x) dx x(1 − x) log . M 2 − m2ω x(1 − x)
(8)
It is evident that free part does not distinguish between the longitudinal and transverse mode and we can take ΠLF = ΠTF = Π22 = Π11 . In order to study the collective modes induced by the ω meson propagation in nuclear matter, there is an additional effect which one should take into account. The longitudinal ω meson mode gets coupled with the scalar mode corresponding to the σ meson propagation in matter. This is due to the presence of mixed polarization function d 4k Πµ q0 , | q | = igs gv Tr G(k)γµ G(k + q) . (9) 4 (2π)
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This is exclusively density dependent effect. With the evaluation of the trace and after a little algebra Eq. (9) could be cast into a suggestive form: gv gs Πµ q0 , |q| = 3 q 2 M ∗ π
kF 0
q
µ d 3 k kµ − q 2 (k · q) . E ∗ (k) q 4 − 4(k · q)2
(10)
This immediately leads to two conclusions. First, it respects the current conservation condition, viz. q µ Πµ = 0 = Πν q ν . Secondly, there are only two components which would survive after the integration over azimuthal angle. In fact this guarantees that it is only the longitudinal component of the ω meson which couples to the scalar meson while the transverse mode remains unaltered. Furthermore, current conservation implies that out of the two non-zero components of Πµ , only one is independent. In presence of mixing the combined meson propagator might be written in a matrix form where the dressed propagator would no longer be diagonal: D = D0 + D0 ΠD .
(11)
It is to be noted that the free propagator is diagonal and has the form
0 Dµν 0 D0 = . 0 ∆0
(12)
In Eq. (12) the non-interacting propagator for the σ and ω are given respectively by ∆0 (q) =
1 , q 2 − m2σ + i
0 Dµν (q) =
−gµν +
qµ qν q2
q 2 − m2ω + i
(13) ,
(14)
The mixing is characterized by the polarization matrix which contains non-diagonal elements
Πµν (q) Πν (q) Π= . (15) Πµ (q) Πσ (q) It should be mentioned that similar to the ω meson polarization tensor, Πσ has also two parts which can be written as Πσ = ΠσF + ΠσD . The free part is divergent which can be regularized to yield: 2 3gs2 ∗ 2 F 3 M − M2 − 4 M∗ − M M − M∗ 2 − M2 Πσ q = 2 2π 1 × 0
1 − 0
∗2 M − x(1 − x)q 2 dx ln M2 ∗2 2 M − x(1 − x)q 2 2 . dx M − x(1 − x)q ln M 2 − x(1 − x)m2σ
(16)
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The density dependent part is evaluated in the next section in the long wave length approximation. To determine the collective modes, one defines the dielectric function as [12]: q |) = det 1 − D0 Π = T2 × mix , (17) (q0 , | where T corresponds to two identical transverse (T ) modes and mix correspond to the longitudinal mode with the mixing. The latter, of course, also characterizes the mode relevant for the σ meson propagation T = 1 − d0 ΠT ,
d0 =
q2
1 , − m2ω + i
mix = (1 − d0 ΠL )(1 − ∆0 Πs ) −
q2 ∆0 d0 (Π0 )2 | q |2
= (1 − d0 ΠL )(1 − ∆0 Πs ) − ∆0 d0 (Πmix )2 ,
(18)
q2
2 = (Π0 )2 . where Πmix | q |2 In order to study collective response of the nuclear system and to obtain the dispersion relation of the induced oscillation, one needs to find zeros of the dielectric function defined through q0 , |q| = T2 × mix = 0. (19)
It is to be noted that the free part ΠTF,L,σ depends only on the Lorentz scalar q 2 while the density dependent part ΠTD,L,σ involves both q0 and |q| individually. The equations (q0 , |q|) = 0 in general have to be solved numerically. However, in the next section, we discuss appropriate approximation scheme which admits close form solutions.
3. Collective modes and dispersion relations In this section we discuss first the approximation scheme to derive the polarization tensors analytically. Once the polarization insertions are determined one can solve Eq. (19) to find the dispersion relations for the ω meson in nuclear matter taking both the effect of free and dense part of the polarization function in the various kinematic limits as mentioned in the introduction. In addition, this also provides us a way to interpret the results in terms of the classical plasma oscillations. First we recall that for collective excitations, the wavelength of the oscillations must be greater than the interparticle spacing. This means the meson momenta must be small compared to the nucleon momenta. Therefore, quantitatively, it is legitimate to assume that all the loop momenta (nucleon) are hard and the external momenta (meson) are soft i.e., p ∼ pF , |q| pF . We term this as hard nucleon loop (HNL) approximation. This allows us to drop q 4 compared to 4(p.q)2 in Eq. (6). Such an approximation, is, in effect, similar to what has been adopted in Ref. [12]. Physically this corresponds to long-wave length (low momentum transfer) excitations near the Fermi surface. Under this assumption the
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integrations of Eq. (6) relevant for the longitudinal and transverse modes are performed easily to give following results:
F ω Log ω+kv ω2 ω−kvF D 2 ΠL (ω, k) = −3Ω 1 − −1 + (20) 2kvF k 2 vF2 and ΠTD (ω, k) =
ω+kvF
3 2 ω2 ω2 ω Log ω−kvF Ω , + 1− 2 2kvF k 2 vF2 k 2 vF2
where, q0 = ω, |q| = k, vF = pF / F is the Fermi velocity with F = is identified as the plasma frequency given by Ω 2 =
gV2 π2
1 3
pF3 F
(21)
pF2 + M 2 and Ω 2
. This is related to the Debye
screening mass as = [15,16]. It is revealing to observe that in the limit vF → 1, with the appropriate redefinition of the plasma frequency Ω 2 , Eqs. (20) and (21) reduce to that of photon polarization functions [17] in hot QED. It might be mentioned here that the full analytical expression for the ΠT (ω, k, pF ) can be found in Ref. [18]. Before proceeding further, few comments are in order. It is evident that for k = 0 in matter, the longitudinal and transverse self-energies are non-degenerate. This gives rise to splitting between these two collective modes in nuclear matter which could be attributed to the presence of Pµν in Eq. (6). However, they become identical in the static limit, i.e., ΠL (ω, k → 0) = ΠT (ω, k → 0) = Ω 2 . Next we consider the mixed polarization vector Πµ given by Eq. (10). This can also be evaluated within the HNL approximation scheme to yield
F 2 ωLog ω+kv gs2 ∗2 2 ω2 − k 2 ω−kvF 2 . (22) Πmix = 2 M 3Ω −1 + π k2 2kvF Ω2
2 2 3 mD
Similarly ΠσD , can also be evaluated within the HNL scheme, which in the high density g2
limit (when M ∗ < kF ) reduces to ΠσD 32 gs2 Ω 2 . For the mixed polarization, the exact v expression can be found in Ref. [18]. The collective excitations, relevant for the longitudinal and the scalar modes are determined by the zeros of mix which could be rewritten as 2 . (23) mix = 1 − d0 ΠL − ∆0 Πσ − d0 ∆0 ΠL Πσ − Πmix In the high-density limit Πσ and ΠL goes as Ω 2 while Πmix is proportional to Ω implying 2 . It is to be noted that in the high density limit M ∗ Ω and with increasing ΠL Πσ Πmix ∗ density M decreases while Ω increases [13]. Hence the scalar and the longitudinal modes, to a first approximation, decouple from each other. This is particularly so for the low meson momentum when the mixing amplitude is small [18,19]. Thus we write mix (1 − d0 ΠL )(1 − ∆0 Πσ ) = 0.
(24)
The fact that the effect of mixing on the dispersion curves is only marginal has also been observed numerically in Ref. [19]. However, with increasing momentum, as shown in Ref. [18], the mixing effect might become important.
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Eq. (19) in conjunction with Eq. (21), (20), (27) and (24) define the dispersion characteristics for the collective excitations gv2 M ∗ − M 2 mω M 3π 2 ω+kvF
3 2 ω2 ω2 ω Log ω−kvF =0 − Ω + 1− 2 2kvF k 2 vF2 k 2 vF2
ω2 − k 2 − m2ω −
(25)
for T mode and gv2 M ∗ − M 2 mω M 3π 2 ω+kvF
2 ω Log ω ω−kvF −1 + = 0, + 3Ω 2 1 − 2 2 2kv k vF F
ω2 − k 2 − m2ω −
(26)
for L mode. It should be noted that in the above equations, the free part of the ω meson self-energy (also known as Dirac sea contribution) given by Eq. (8), following Ref. [6] has been simplified by making a Taylor series expansion of Eq. (8) around M ∗ . To the leading order the contribution is found to be [6,20]
2 g2 M ∗ − M 2 mω F mω + O ΠL,T . (27) (ω, k) = v2 M 3π 4M 2 As in nuclear matter effective nucleon mass is smaller than its free space value (M ∗ < M), it is clear from the last expression, that the free part reduces the effective ω meson mass in the medium. Generally the transcendental equations, viz. Eq. (25) and (26), have to be solved numerically, however, analytical solutions could be obtained in two limiting cases viz., when k < Ω and k > Ω by expanding above equations in powers of k and solving the 2 equations by successive iterations. To the first order, when k Ω one gets ωL(T ) = ∗2 2 mω + Ω . We note that the modes are degenerate in this limit and oscillations are independent of the wave vector k. In the case of electron–ion plasma, this mode was identified as the ‘plasma waves’ or ‘Langmuir’ oscillation and Ω 2 there is given by e2 ρ/m2 where ρ is the density of electron gas and m is the electron mass [21]. This justifies identifying Ω as the plasma frequency in the present pretext. It should be mentioned that here we have absorbed the contribution of the free part as
g2 M ∗ − M m∗ω2 = m2ω 1 + v2 . (28) 3π M Once the leading order solution is known, we can solve Eqs. (25) and (26) iteratively to obtain 1 Ω2 + ···, ωT2 = m∗ω2 + Ω 2 + k 2 + k 2 vF2 ∗ 2 5 mω + Ω 2
(29)
Ω2 3 2 ωL = m∗ω2 + Ω 2 + k 2 vF2 ∗ 2 + ···. 5 mω + Ω 2
(30)
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Evidently, the transverse mode lies above the longitudinal one and it requires more energy for excitation. However, the splitting in the low k region is very marginal. In the limit mω → 0 we recover the results presented in [12], i.e., 1 ωT2 = Ω 2 + k 2 + vF2 k 2 + · · · , 5 3 2 ωL = Ω 2 + vF2 k 2 + · · · . 5
(31) (32)
Furthermore, this taken with vF → 1 results in dispersion relations arising out of the photon propagation at finite temperature or density (where electron is considered to be ultrarelativistic) with the appropriate definition of the plasma frequency [16,17]. It is worthy to note that the contribution of the density dependent part is opposite to that of the Dirac part of the polarization tensor. This is consistent with Ref. [6]. Next we consider the case when k > Ω but still ω > k and in fact ω ∼ kvF . The successive approximation for this asymptotic value of k yields solutions of the form 3 2 ωT2 = k 2 + m∗2 ω + Ω + ···. 2
(33)
The corresponding longitudinal frequency takes little bit complicated form 2 ωL
= k 2 vF2
vF2 m∗2 2 k 2 vF2 ω −2 . + 1 + 4 exp − 3 Ω2 3Ω 2 (vF2 − 1)
(34)
Evidently, the longitudinal mode approaches the line ω = kvF and exponentially suppressed for very large values of k. We have solved Eqs. (25) and (26) numerically and checked that for meson momenta k ∼ 1 GeV dispersion results are in good agreement with Eqs. (29) and (30). It is worthwhile to note that the dispersion relations presented both in Eqs. (29), (30) and Eqs. (33), (34) coincide with the ones relevant for the classical plasma in the limit m∗ω → 0 as may be verified from Ref. [21].
4. Spectral functions and sum-rules Equipped with Eqs. (20) and (21), we can proceed to calculate the spectral functions of the ω meson. As demonstrated already, the ω meson in matter has two modes. Confining our attention on the transverse sector first, we write the propagator as ∆T (ω, k) =
ω2
− k2
− m∗2 ω
−
3 2 ω2 2 Ω (kvF )2
−1 + 1−
ω2 ω (kvF )2 2kvF
. (35) F Log ω+kv ω−kvF
Next chain of steps would be to construct the sum rules by exploiting the analytic properties of this effective propagator. First we note that the function ∆T is analytic in the complex ω-plane having a cut from −kvF to +kvF ; in addition to the poles ω = ±ωT . Having
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observed this one can make use of Cauchy’s theorem for ∆ = ∆T dω ∆(ω , k) ∆(ω, k) = 2πi ω − ω Γ
∞ = −∞
dω ∆(ω + iη, k) − ∆(ω − iη, k) + 2πi ω − ω
Γ
dω ∆(ω , k) . 2πi ω
(36)
In the above equation Γ is a circle with its radius pushed to infinity. This equation can be rewritten in terms of the spectral density ρ(ω, k) containing both the discontinuities across the cuts and the residues at the poles ρ(ω, k) = 2 Im ∆(ω + iη, k)
(37)
as ∞ ∆(ω, k) = −∞
dω ρ(ω , k) + 2π ω − ω
Γ
dω ∆(ω , k) . 2πi ω
(38)
It is evident from Eq. (35) that ∆T (ω , k) → ω12 as ω → ∞, therefore, the integration over the contour Γ fails to contribute in this region. First sum rule is then immediately obtained by setting ω = 0 in Eq. (38) ∞ −∞
dω ρT (ω, k) 1 = ∆T (0, k) = 2 . 2π ω k + m∗ω2
(39)
Other sum rules are obtained by examining the asymptotic behaviour of ∆T (ω, k) when ω → ∞; by taking the parity property of the spectral density, i.e., ρT (ω, k) = −ρT (−ω, k) into account, one can write from Eq. (38) for ω → ∞ ∞
1 ∆T (ω, k) = − ω
∞
n=0 −∞
dω ω 2n+1 ρT (ω , k). 2π ω
(40)
On the other hand, one can expand Eq. (35) in powers of ω−1 ; comparing this with Eq. (40) one writes for n = 0 ∞ −∞
ω dω ρT (ω, k) = 1 2π
(41)
and similarly for n = 1 one observes ∞ −∞
ω3 dω ρT (ω, k) = k 2 + m∗ω2 + Ω 2 . 2π
(42)
Thus we obtain other two sum rules. Of particular importance is Eq. (41). This does not depend on k which could be shown to be a consequence of the canonical commutation
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relation. Therefore, it is natural to expect that such a relation will also be satisfied by the transverse ω meson in nuclear matter. The longitudinal propagator is denoted as ∆L (ω, k) =
ω2 − k 2 −1 2 ω2 2 k −1 + ω2 − k 2 − m∗2 2 ω + 3Ω 1 − k2 v
F
ω 2kvF
. F Log ω+kv ω−kvF (43)
Contrary to the case of transverse propagator, we now have non-vanishing contribution from the contour denoted by Γ in Eq. (38) as in the limit ω → ∞ the logarithmic term → 1. Hence in this limit the longitudinal propagator goes like ∆L (ω, k) → −1/k 2 . Therefore, a subtraction, unlike the transverse propagator, is necessary. The subtracted dispersion relation gives the following sum rule essentially in the same manner as in the transverse case: ∞ 1 3Ω 2 + m∗2 dω ρL (ω, k) ω = 2 + ∆L (0, k) = 2 2 . (44) 2π ω k k (k + m∗2 + 3Ω 2) ω −∞
Similarly to the asymptotic behaviour of the longitudinal propagator one can write, ∞
1 1 ∆L (ω, k) = − 2 − ω k
∞
n=0−∞
dω ω 2n+1 ρT (ω , k). 2π ω
(45)
This is identical with Eq. (40) except here we have an additional factor of −1/k 2 for reason mentioned in the previous paragraph. Likewise, we now can derive the sumrules by first setting n = 0 to get ∞ −∞
ω dω m∗2 + Ω 2 ρL (ω, k) = ω 2 2π k
and n = 1 to obtain ∞ 3 2 2 2 2 2 ∗2 2 (Ω 2 + m∗2 ω dω ω )(k + mω + Ω ) − 5 k Ω vF ρL (ω, k) = . 2π k2
(46)
(47)
−∞
We can write down the full spectral function of the transverse and longitudinal modes for the ω meson in nuclear matter in terms of the poles (and residues at the poles) together with the cuts of the propagator 1 ρT ,L (ω, k) = ZT ,L (k) δ(ω − ωT ,L ) − δ(ω + ωT ,L ) 2π + βT ,L (ω, k)θ k 2 − ω2 ,
(48)
where, βT ,L is given by βT ,L (ω, k) =
2 Im ΠT ,L ∆T ,L (ω, k) . π
(49)
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The imaginary part of the self-energies ΠT ,L correspond to Landau damping and relevant only for the space-like momenta of the meson. Physically this refers to the energy loss due to scattering of the ω meson with hard nucleons in the medium. They are given by the imaginary part of Eqs. (20) and (21). The residues at the poles which are used to determine the full spectral function of the ω meson in nuclear matter can be determined from the one-loop effective propagators. For instance the residue at the pole for the transverse excitation is determined by −1 (50) ZT = − ∂∆−1 T /∂ω ω=ω T (k)
=
ωT (k 2 vF2 − ωT2 ) ωT2 (ω2 − 3(ΩT2 + k 2 + m∗ω2 )) + k 2 (k 2 + ωT2 + m∗ω2 )vF2
.
(51)
In a similar way we also calculate the reside at the pole for the longitudinal oscillation which is found to be ZL = −
2 )(−ω3 + ω k 2 v 2 ) (k 2 − ωL L F L 2 (−3k 2 − 3m∗ 2 + ω2 − 3Ω 2 ) + k 2 (k 2 + m∗ 2 + ω2 + 3Ω 2 )v 2 ] k 2 [ωL ω ω L L F
. (52)
We now present the limiting values of these residues at the poles given in Section 3. First considering k < Ω, we have for the transverse ω
1 k 2 5m∗ω2 + (5 + 3vF2 )Ω 2 − (53) ZT (k) (Ω 2 + m∗ω2 )5/2 2 Ω 2 + m∗ω2 20 while for large k, i.e., for k > Ω we have 1 . ZT (k) 2 k 2 + m∗ω2
(54)
Similar results are obtained for the longitudinal mode also. They are given by, for k < Ω Ω 2 + m∗ω2 25(m∗ω2 + Ω 2 ) − 22vF2 Ω 2 ZL (k) − (55) 2k 2 20(Ω 2 + m∗ω2 )3/2 and for k > Ω ZL (k)
vF2 4kvF 2 k 2 vF2 m∗2 ω − 2 . exp − + 3 Ω2 3Ω 2 3Ω 2 (vF2 − 1)
(56)
It is now evident from Eqs. (28) and (54) that the transverse spectral function for large meson momentum reduces to that of the ω meson propagation in free space, except its mass is modified because of the presence of the mean scalar field. Longitudinal mode in this limit decouples from the system. For small momenta, the sum rules are almost saturated by the pole contributions. This could be checked numerically.
5. Summary and conclusion To conclude and summarize, in the present work, we investigate the collective modes induced by the ω meson propagation in dense nuclear matter. We then determine the
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dispersion relations for the ω meson by considering both the effect of the Fermi sea and Dirac vacuum within the scheme of HNL approximation. It is observed that in the appropriate limit, i.e., for meson momenta small compared to the plasma frequency of the system, the results are consistent with the previous calculations [6,12]. The ω meson mass, in presence of a mean field, decreases even at normal nuclear matter while for free Fermi gas the effect of nucleon loop is to increase the ω meson mass proportional to the characteristic plasma frequency of the system. We also present analytical results for the dispersion relation in the limit k > Ω which shows that for the large value of the momenta the medium effects die off and the transverse mode propagates like a free meson. The longitudinal mode in this limit decouples from the system. We also construct various sum rules satisfied by the ω meson spectral density in nuclear matter by investigating the analytic properties of the relevant effective propagator at finite density. It is observed that sum rules are mostly saturated by the pole contributions for small momenta. The residues are also presented in the different limit of the meson momenta to understand the collective modes and their behavior. Similar studies can also be pursued for other light vector mesons. For instance, it is known that the ρ nucleon interaction is very similar to the ω nucleon interaction, where, in addition to the vector interaction, one has to include tensor interaction as well. We, therefore, believe that the ρ meson propagation in cold nuclear matter can also be studied within the framework HNL approximation. This would allow for an analytical derivation for the spectral density of the ρ meson. Hence , sum rules similar to what we present here, relevant for the ρ meson spectral function at finite density can also be constructed in a parallel approach. Such studies are in progress and shall be reported elsewhere.
Acknowledgements The author would like to thank A.G. Williams and J. Piekarewicz for useful private communications.
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