Role of Magnetic Interaction in Dense Plasma

Abstract

Quasiparticle excitations and associated phenomena of energy and momentum transfer rates have been calculated in terms of the drag and the diffusion coefficients exposing clearly the dominance of the magnetic interaction over its electric counterpart. The results have been compared with the finite temperature results highlighting the similarities and dissimilarities in the two extreme regimes of temperature and density. Non-Fermi-liquid behavior of various physical quantities like neutrino mean free path and thermal relaxation time due to the inclusion of magnetic interaction has clearly been revealed. All the results presented in the current review are pertinent to the degenerate and ultradegenerate plasma.

1. Introduction

Understanding the properties of the hot and dense ultrarelativistic plasma has been at the forefront of contemporary research for the past few decades. Interests in this regime cover a broad area starting from the laboratory based heavy ion collisions (HIC) to the wider domain of naturally occurring astrophysical sites like neutron stars, supernovae, and white dwarfs and so forth.

In the present review we mainly focus on the properties of plasma with high chemical potential (𝜇) and zero or low temperature (𝑇). It is to be noted that the theoretical calculations so far involving both the quantum electrodynamics and quantum chromodynamics (QED/QCD) have been confined largely to the domain of high temperature particularly to address the issues related to heavy ion collisions. The main motivation for this has been the possibility of quark-hadron phase transition which might occur in HIC mimicking the conditions of microsecond old universe. Studies with dense plasma on the other hand find their applications mainly in astrophysics.

Several calculations have recently been performed exposing subtle departure of the characteristic behavior of the quasiparticle excitations in dense plasma than their high temperature counterparts. For example, the role of magnetic interaction in dense plasma, as we shall see, modifies the fermion self-energy in a nontrivial way involving fractional powers of the excitation energy [1] leading to phenomena not seen in the high temperature plasma. In fact the transverse or magnetic interaction seems to dominate over the corresponding electric or longitudinal interactions [1] in degenerate electron or quark matter. It is to be noted that such departures can only be significant in the relativistic plasma as in the nonrelativistic case magnetic interactions are 𝛽=𝑣/𝑐 suppressed [2–5]. The introduction of magnetic interaction changes some of the characteristic behaviours of dense plasma which show departure from the normal Fermi liquid case. In this review this particular aspect will be illuminated upon further.

Our discussion starts with the quasiparticle damping rate (𝛾). In the relativistic case 𝛾∼(𝐸−𝜇) [6] unlike the known non-relativistic scenario where 𝛾∼(𝐸−𝜇)². Here, 𝐸 is the energy of the quasiparticle and 𝜇 is the chemical potential [6–8]. More importantly the infrared behavior of this quantity differs dramatically from the high temperature case [6–8]. In a plasma the other quantities of interest have been the drag (𝜂) and the diffusion coefficients (ℬ) which eventually determine the equilibration time scale of the plasma. These two quantities are actually related to the energy loss and the momentum transfer due to the scattering of the constituent particles of the plasma. In high temperature case they are known to be connected via the Einstein relation. We address this issue to see the scaling behavior of these quantities in the domain of zero and high temperature plasma [2].

Recently the implications of the medium modified quasiparticle self-energies have been discussed by several authors showing their significance in the cooling behavior of neutron stars. The appearance of logarithmic terms in the specific heat, emissivity, and thermal relaxation time of the degenerate matter has been found to be of crucial importance. How the medium modified dispersion relation changes the cooling behavior of neutron star has been first exposed in [9] and later in [3, 5, 10].

The plan of the review is as follows. In Section 2 we have shown the behavior of quasiparticle damping rate in ultradegenerate plasma. In this section we compare the results of quasiparticle lifetime in the extreme limits of high temperature and high density. For the former it is known that the damping rate for the exchange of static gauge boson remains unscreened even when one uses hard thermal loop corrected propagator for the gauge bosons [11, 12]. It has been shown in the literatures that a further resummation is required to resolve this issue by invoking Bloch-Nordseick techniques as discussed in great detail in [13, 14]. For the degenerate case, as we shall see this problem does not appear. In this context however we shall expose the relative importance of the magnetic interaction over the electric part unlike the hot relativistic plasma.

In this review we also calculate the drag and the diffusion coefficients for both degenerate and ultradegenerate cases. Different properties of 𝜂 and ℬ in the zero temperature and low temperature limit have been reported in Section 3. In Section 4 explicit analysis has been presented to exhibit how the nature of neutrino mean free path changes with the inclusion of the transverse mode. One distinguishing feature of the present work is to show the structure of the fermion self-energy near the Fermi surface. Of particular importance is the vanishing group velocity at the Fermi surface which in turn is responsible for the non-Fermi liquid (NFL) corrections. Section 5 describes the implication of the neutrino mean free path vis-a-vis cooling of the neutron star specially focusing on NFL corrections to the various physical quantities. We calculate the thermal relaxation time exposing the role of magnetic interaction further in Section 6 where we also discuss NFL corrections.

In the last two sections of this review we estimate the neutrino mean free path and thermal relaxation time of degenerate matter. At very high density nowadays it is well known that quark matter is expected to form a color superconducting color-flavour locked (CFL) phase [15]. In this phase all quark excitations are gapped, and the mean free path or the thermal relaxation time is modified by exponential factor involving gap parameter [16–18]. However, in order to highlight particular aspects of NFL behaviour of the relativistic plasma we deal with only ungapped quark matter. The extension of the present calculation for the gapped quark matter is straightforward.

2. Damping Rate

The propagation of a particle immersed in plasma gets modified through interaction with the medium. In case of ultradegenerate (𝑇=0, 𝜇≠0) plasma these medium modified particles (quasiparticles) scatter with the other particles close to the Fermi surface. Information about the quasiparticle lifetime is obtained from the retarded propagator. The damping rate is expected to follow exponential decay in time 𝑆_𝑅(𝑡,𝑝)∼𝑒^−𝐸𝑡𝑒^−𝛾𝑡, so that |𝑆_𝑅(𝑡,𝑝)|²∼𝑒^−Γ𝑡, where Γ=𝛾/2. The life time of the single-particle excitation is then 𝜏(𝑝)=1/Γ(𝑝). The concept of quasiparticles becomes meaningful only if their life time is long or in other words the damping width is much smaller than its energy. In a plasma the damping rate of the quasiparticle can be written in terms of the imaginary part of the fermion self-energy or the scattering rate (see Figure 1). Mathematically one writes [19] ^{𝛾(𝐸)1=−𝑝}_4𝐸^TrImΣ₀^{+𝑖𝜎𝐩}/^𝑃_𝑝0=𝐸^, (1) where Σ is the fermion self-energy, 𝜎→0⁺, and we have used 𝑃_𝜇=(𝑝₀,𝐩). The aforementioned expression is true for both the high temperature and the zero temperature plasmas. The one-loop electron self-energy is given by Σ(𝑃)=𝑒²𝑑⁴𝑞(2𝜋)⁴𝛾_𝜇𝑆_𝑓(𝑃−𝑄)𝛾_𝜈Δ_𝜇𝜈(𝑄). (2) In the previous equation 𝑆_𝑓(𝑃−𝑄) is the free fermion propagator and Δ_𝜇𝜈(𝑄) is the photon propagator.

Figure 1 

Fermion self-energy with resummed photon propagator.

Focusing on (2) we can see that it involves the boson propagator (Δ_𝜇𝜈) which is well known to diverge in the infrared regime. This is a generic feature of both the high temperature and the zero temperature plasmas. In case of hot QED or QCD plasmas one handles this problem by employing the hard thermal loop (HTL) prescription which was originally developed in [11, 12]. In this method the exchanged momentum (𝐪) integration is performed in two domains by introducing an intermediate cut-off parameter (𝑞^∗), below which the one-loop corrected dressed boson propagator has to be used and the bare propagator can be used to perform the integration above 𝑞^∗ (Braaten and Yuan's prescription) [20].

For electric interaction this is sufficient to remove the infrared singularity associated with the exchange of massless bosons (photons or gluons). The interaction mediated by the transverse bosons on the other hand still poses a problem at high temperature [13, 14]. We shall discuss this further at the end of this section where we compare the zero temperature case with the corresponding high temperature calculations. It might be mentioned that to construct the one-loop corrected photon propagator one needs to evaluate the photon self-energy in dense plasma [21]. In the ultradegenerate case, the dominant contribution to the photon self-energy comes from the loop momentum ~𝜇, and the external momentum for the soft modes is assumed to be 𝐪∼𝑒𝜇 (Figure 2). This, in principle, runs similar to the HTL approximation where the loop momentum is ~𝑇 and the external momentum is ~𝑒𝑇. Likewise for the degenerate case one can construct hard dense loop (HDL) propagator where loop momentum is assumed to be ~𝜇 [21].

Figure 2 

Feynman diagram for 𝑒−𝑒−→𝑒−𝑒− scattering process with effective photon propagator.

The structure of the dressed photon propagator in the Coulomb gauge can be written as [19] Δ_𝜇𝜈(𝑄)=𝛿_𝜇0𝛿_𝜈0Δ_𝑙(𝑄)+𝑃^𝑡_𝜇𝜈Δ_𝑡(𝑄), (3) with 𝑃^𝑡_𝑖𝑗=(𝛿_𝑖𝑗−𝑞_𝑖𝑞_𝑗),𝑞^𝑖=𝐪^𝑖/|𝐪|, 𝑃^𝑡_𝑖0=𝑃^𝑡_0𝑖=𝑃^𝑡₀₀=0, and Δ_𝑙, Δ_𝑡 are given by [19] Δ_𝑙𝑞₀,𝑞=−1𝑞²+Π_𝑙,Δ_𝑡𝑞₀,𝑞=−1𝑞²₀−𝑞²−Π_𝑡. (4)(5)

Δ_𝑙,𝑡 are the longitudinal and the transverse components of the boson propagator, and Π_𝑙,𝑡 are the longitudinal and transverse parts of the self-energy [19]: Π_𝑙𝑞₀,𝑞=𝑚²_𝐷𝑞1−⁰𝑞2𝑞ln⁰+𝑞𝑞₀,Π−𝑞_𝑡𝑞₀,𝑞=𝑚²_𝐷𝑞²₀2𝑞²+𝑞₀1−𝑞²₀/𝑞²𝑞4𝑞ln⁰+𝑞𝑞₀,−𝑞 (6) where the Debye mass is 𝑚_𝐷=𝑒𝜇/𝜋.

The bosonic spectral functions are the imaginary part of the propagators: 𝜌_𝑙,𝑡𝑞₀,𝐪=2ImΔ_𝑙,𝑡𝑞₀+𝑖𝜎,𝐪, (7) where 𝜌_𝑙,𝑡 in the HDL approximations are given by [6] 𝜌_𝑙𝑞₀=,𝑞2𝜋𝑚²_𝐷𝑥Θ1−𝑥²2𝑞²+𝑚²_𝐷(1−(𝑥/2)ln|(𝑥+1)/(𝑥−1)|)²+𝑚⁴_𝐷𝜋²𝑥²,𝜌/4_𝑡𝑞₀,𝑞=2𝜋𝑚²_𝐷𝑥1−𝑥²Θ1−𝑥²×2𝑞²𝑥²−1−𝑚²_𝐷𝑥²1+1−𝑥²2𝑥ln𝑥+1𝑥−1²+𝑚⁴_𝐷𝜋²𝑥²1−𝑥²²4⁻¹, (8) with 𝑥=𝑞₀/𝑞.

To calculate 𝛾 in the ultradegenerate plasma the energy of the quasiparticle is therefore considered to be hard, that is, 𝐸∼𝜇. Hence, for the quasiparticle with momentum close to the Fermi momentum the electron-photon vertex can be replaced by the bare one and the one-loop self-energy is dominated by the photon with soft momentum (𝐪∼𝑒𝜇). So, the calculation of imaginary part of Σ can be performed with the help of the free fermion spectral function 𝜌_𝑓(𝑃−𝑄)=2𝜋𝜀(𝑘₀)𝛿(𝑘²₀−𝐸²_𝑘) (𝜀(𝑘₀)=𝑘₀/|𝑘₀|) and the medium modified 𝜌_𝑙,𝑡 [6]. Explicitly, 𝛾(𝐸)=𝜋𝑒²𝐸𝑑³𝑞(2𝜋)^3∞−∞𝑑𝑘₀𝜌2𝜋^𝑓𝑘₀^∞_−∞𝑑𝑞₀×𝑞2𝜋1+𝑛₀−𝑛𝑘₀𝛿𝐸−𝑘₀−𝑞₀×𝑝₀𝑘₀+𝐩⋅𝐤×𝜌_𝑙𝑞₀,𝑞+2𝑝₀𝑘₀−(𝐩⋅𝐪)(𝐤⋅𝐪)𝜌_𝑡𝑞₀,𝑞. (9)

In the previous equation 𝑛(𝑞₀) and 𝑛(𝑘₀) are the boson and fermion distribution functions. In case of the cold plasma these distribution functions can be replaced by (1+𝑛(𝑞₀))=Θ(𝑞₀) and 𝑛(𝑘₀)=Θ(𝜇−𝐸+𝑞₀), where Θ represents the step function. These theta functions, as we shall see, restrict the phase space of the 𝑞₀ integration severely. This eventually with the help of the HDL propagator makes 𝛾 finite [13, 14, 21]. To proceed further an approximation in the region (𝐸−𝜇)/𝑚_𝐷=𝜈≪1 has been made. This is due to the fact that scatterings close to the Fermi surface are only relevant as mentioned earlier. To perform the 𝑞 integration the integration domain is divided into two sectors. In the soft sector (𝑞<𝑞^∗) with the HDL corrected spectral function mentioned in (8) one obtains [6] 𝛾_soft(𝐸)≃𝑒²𝑚_𝐷𝜈+𝑒24𝜋²𝑚⁴_𝐷𝜈²𝜋32𝜋2𝑚³_𝐷−1𝑞^∗3. (10) Only the leading order terms in (𝐸−𝜇)/𝑚_𝐷 have been retained in writing the previous equation. For the hard momentum region 𝑞>𝑞^∗ bare propagator is sufficient to calculate the damping rate: 𝛾_hard(𝐸)≃𝑒²𝑚⁴_𝐷𝜈²132𝜋𝑞^∗3−1𝑞³_max, (11) where 𝑞_max≃𝜇 is the maximum momentum transfer that is allowed by kinematics.

Two points are to be noted here. One that 𝑞^∗ dependent term appear only at the higher orders that is, the leading order result is independent of 𝑞^∗. Second upon summation the 𝑞^∗ terms cancel out exactly. The total damping rate is then given by [6] 𝛾(𝐸)≃𝒢₁𝜈+𝒢₂𝜈²+⋯, (12) where 𝒢₁=(𝑒²𝑚_𝐷)/(24𝜋) and 𝒢₂=(𝑒²𝑚_𝐷)/32. In the last equation the first term corresponds to the magnetic interaction, and the second term comes from the electric interaction revealing the fact that the magnetic interaction here dominates over the electric one. In this connection we can recall the results of the damping rate in case of the high temperature plasma with vanishing chemical potential. These are given by [13, 14] 𝛾_𝑡=𝑒²𝑇2𝜋^𝜔𝑝₀𝑑𝑞𝑞,𝛾_𝑙∼𝑒²𝑇, (13) where 𝜔_𝑝 is the plasma frequency related to the Debye mass (𝜔_𝑝=𝑚_𝐷/√3).

The last equations have been obtained by using the one-loop corrected dressed propagator which makes 𝛾_𝑙 finite. The transverse part, on the other hand, remains divergent that can be cured by another resummation as discussed in [13, 14]. In the case of dense QED or QCD matter interestingly the second resummation is not required. This happens, as indicated above, due to the restrictions imposed by the Pauli blocking which severely cuts off the phase space. A comparative study between high and zero temperature damping rates has been listed in Table 1 [22].

Table 1 

Comparison of high and zero temperature damping rates.

In the nonrelativistic case the excitation energy dependence of 𝛾 can be derived from the simple phase-space factor without performing any rigorous calculation. A particle with energy 𝐸>𝜇 interacts with the particle of energy 𝐸_𝑘<𝜇. The scattered particles are now in the energy state 𝐸_𝑝′>𝜇 and 𝐸_𝑘′>𝜇. The total probability of the scattering process is proportional to ∫𝛿(𝐸+𝐸_𝑘−𝐸_𝑝′−𝐸_𝑘′)𝑑³𝑘𝑑³𝑝^′. For 𝐸−𝜇≪𝜇, the permissible regions of variation for the scattered particles are 𝜇<𝐸_𝑝′<𝐸+𝐸_𝑘−𝜇 and 2𝜇−𝐸<𝐸_𝑘<𝜇. The integral over 𝑑³𝑘 and 𝑑³𝑝^′ can be computed with the approximation 𝐸_𝑘≈𝐸_𝑝′≈𝜇, that is, scattering occurs near the Fermi surface. The scattering probability is then (𝐸−𝜇)² [23]. Similar arguments can be made for the transverse sector which will render the damping rate of the particle ~(𝐸−𝜇) at the leading order for relativistic case.

3. Drag and Diffusion Coefficients

3.1. Zero Temperature Plasma

Next we review the drag and diffusion coefficients in zero temperature relativistic plasma and see how the magnetic interaction changes the behavior of these coefficients. These two are the important quantities to study the equilibration properties of the plasma as mentioned earlier and are related to the energy and the momentum transfer per scattering, respectively. 𝜂 can thus be defined as follows [24]: 1𝜂=𝐸−𝑑𝐸𝑑𝑥. (14) The energy loss −(𝑑𝐸/𝑑𝑥) can be calculated by averaging over the interaction rate (Γ) times the energy transfer per scattering 𝜔 [25]: 𝑑𝐸=𝑑𝑥𝑑Γ𝜔, (15) where Γ=𝛾/2. Inserting the energy exchange 𝜔 in the expression of 𝛾 to calculate 𝑑𝐸/𝑑𝑥 from the previous equation we obtain (Figure 2) −𝑑𝐸=𝑑𝑥𝜋𝑒²𝐸𝑑³𝑞(2𝜋)^3∞−∞𝑑𝑘₀𝜌2𝜋^𝑓𝑘₀^∞_−∞𝑑𝑞₀𝑞2𝜋⁰×1+𝑛𝑞₀−𝑛𝑘₀𝛿𝐸−𝑘₀−𝑞₀×𝑝₀𝑘₀+𝐩⋅𝐤𝜌_𝑙𝑞₀𝑝,𝑞+2₀𝑘₀−(𝐩⋅𝐪)(𝐤⋅𝐪)𝜌_𝑡𝑞₀,𝑞. (16) The basic assumption behind the calculation of 𝜂 lies in the fact that (𝐸−𝜇)/𝑚_𝐷=𝜈≪1. Taking into consideration the previous assumption in case of the hard photon exchange we get [2], −𝑑𝐸≃𝑒𝑑𝑥²𝜈³𝑚⁵_𝐷16𝜋𝑑𝑞𝑞⁴. (17) The above expression is infrared divergent with algebraic dependence. Here, it would be worthwhile to comment that in case of the high temperature plasma the divergence is logarithmic (𝑒⁴𝑇²/24𝜋𝐸)∫(𝑑𝑞/𝑞). The divergence is even worse in case of the ultradegenerate plasma. But we show below that with the HDL propagator and Pauli blocking one can render 𝜂 finite. In the soft sector 𝜂 is given by the following expression: 𝜂_soft(𝐸)≃𝑒²8𝜋²𝐸×_𝐷𝑑𝑞𝑞𝑑𝑞₀𝑞₀𝜌_𝑙𝑞₀+𝑞,𝑞1−²⁰𝑞²𝜌_𝑡𝑞₀,𝑞. (18)

The integration domain (𝐷) above is restricted by the Θ functions: 𝐷∶0≤𝑞₀𝑞≤𝐸−𝜇;₀≤𝑞≤𝑞^∗. (19) After explicit calculation, the electric and the magnetic contributions to the expressions of 𝜂 take the following form [2]: 𝜂|^𝑙_soft≃𝑒²𝑚²_𝐷𝜈³−𝑒96𝐸²𝑚⁵_𝐷𝜈³72𝜋𝑞^∗3𝐸,𝜂|^𝑡_soft≃𝑒²𝑚²_𝐷𝜈²−𝑒48𝜋𝐸²𝑚⁵_𝐷𝜈³144𝜋𝑞^∗3𝐸. (20)

In the last two equations the leading order terms are independent of the cut-off parameter as they appear only at 𝑂(𝑒⁷). In the region of hard photon momentum exchange we obtain [2] 𝜂|_hard≃𝑒²𝑚⁵_𝐷𝜈³148𝜋𝐸𝑞^∗3−1𝜇³. (21) On addition of the soft and the hard sectors it can be seen that the expression of the drag coefficient becomes [2], 𝜂≃ℋ₁𝜈³+ℋ₂𝜈³+⋯, (22) where ℋ₁=(𝑒²𝑚²_𝐷)/(48𝜋𝐸) and ℋ₂=(𝑒²𝑚²_𝐷)/(96𝐸). In the previous equation the first term corresponds to the magnetic interaction, and the second term has come from the longitudinal interaction. From (21) it is important to note that the hard sector involves even higher power. The entire contribution at the leading order then comes from the soft sector alone, and Braaten and Yuan's prescription is not required in case of the ultradegenerate plasma [2]. The final expression of 𝜂 reveals that with the magnetic interaction 𝜂 changes significantly from the Fermi liquid result. In the Fermi liquid theory, since the magnetic interaction is suppressed in comparison with the electric sector 𝜂 goes with (𝐸−𝜇)³. Here, we can recall the result of the drag coefficient in case of high temperature plasma and compare the same with the previous result. The coefficient at high temperature looks like [25–29] 𝑒𝜂≃⁴𝑇²−136𝜋𝐸2+32𝑞log^max𝑚_𝐷+122log√𝜋. (23) From the previous result it is clear that there is no splitting between electric and magnetic modes, and in hot plasma both contribute at the same order. A comparative study between high temperature and zero temperature 𝜂 can be found in [22]:

Another important quantity to study the equilibration property of the plasma is the momentum diffusion coefficient 𝐵_𝑖𝑗. The coefficient is related to the interaction rate via the following relation [24]: 𝐵_𝑖𝑗=𝑑Γ𝑞_𝑖𝑞_𝑗. (24) Decomposing 𝐵_𝑖𝑗 into longitudinal (𝐵_𝑙) and transverse components (𝐵_𝑡) we get the following expression, 𝐵_𝑖𝑗=𝐵_𝑡𝛿_𝑖𝑗−𝑝_𝑖𝑝_𝑗𝑝²+𝐵_𝑙𝑝_𝑖𝑝_𝑗𝑝². (25)𝐵_𝑙,𝑡 are the longitudinal, transverse squared momenta acquired by the particle through collision with the plasma. The longitudinal momentum diffusion coefficient is related to the drag coefficient via the Einstein relation in case of equilibrating plasma. Like 𝜂 longitudinal momentum diffusion coefficient (𝐵_𝑙=ℬ, suppressing the index 𝑙) can be expressed as ℬ=𝜋𝑒²𝐸𝑑³𝑞(2𝜋)^3∞−∞𝑑𝑘₀𝜌2𝜋^𝑓𝑘₀^∞_−∞𝑑𝑞₀𝑞2𝜋^2‖×1+𝑛𝑞₀−𝑛𝑘₀𝛿𝐸−𝑘₀−𝑞₀×𝑝₀𝑘₀+𝐩⋅𝐤𝜌_𝑙𝑞₀𝑝,𝑞+2₀𝑘₀−(𝐩⋅𝐪)(𝐤⋅𝐪)𝜌_𝑡𝑞₀,𝑞. (26) Here, 𝑞_‖ is the longitudinal exchanged momentum transfer in a scattering and 𝑞_‖=𝑞cos𝜃. Explicitly ℬ is then given by [2] ℬ≃ℬ₁𝜈³+ℬ₂𝜈⁴+⋯, (27) where ℬ₁=(𝑒²𝑚³_𝐷)/(72𝜋) and ℬ₂=(𝑒²𝑚³_𝐷)/128. In this case also the electric and magnetic modes split as the magnetic mode is ~(𝐸−𝜇)³ whereas electric mode is ~(𝐸−𝜇)⁴. One can relate the leading order terms of 𝜂 and ℬ from (22) and (27) via a common relation, namely, Einstein's relation as in case of high temperature plasma. In ultradegenerate plasma the relation can be written as ℬ=3𝐸(𝐸−𝜇)𝜂/4.

In the high temperature and zero chemical potential regime ℬ has the following expression [25–29]: 𝑒ℬ≃⁴𝑇³−118𝜋2+32𝑞log^max𝑚_𝐷+122log√𝜋. (28) In case of high temperature plasma from (23) and (28), Einstein's relation becomes ℬ=2𝑇𝐸𝜂.

From (22), (23), (27), and (28) it can be seen that zero temperature 𝜂 and ℬ show entirely different characteristics from the high temperature case. The logarithmic dependence in high temperature takes an algebraic form in the ultradegenerate limit [22].

3.2. Low Temperature Plasma

In this section we extend our calculations to include finite temperature effects in the domain where |𝐸−𝜇|∼𝑇≪𝑒𝜇≪𝜇. Moreover, we also go beyond the leading order calculations as previously reported above. Finally we shall see how the leading order zero temperature results appear as a limiting case from our present analysis including higher order correction terms.

For the soft sector contribution we take 𝑇/𝑚_𝐷 as an expansion parameter and 𝛼=(𝐸−𝜇)/𝑇∼1. We start with the following expression for the soft sector [4]: 𝜂^soft(𝐸)≃𝑒²8𝜋²𝐸^𝑞∗₀𝑑𝑞𝑞³¹₋₁𝑑𝑥𝑥1+𝑛𝑞𝑥−𝑛×𝜌𝐸−𝜇−𝑞𝑥_𝑙𝑞𝑥,𝑞+1−𝑥²𝜌_𝑡𝑞𝑥,𝑞. (29)

The previous equation can easily be derived from (16). After subtracting the energy independent part from the above equation we get, 𝜂^soft(𝐸)−𝜂^soft_𝐸=𝜇𝑒=−²8𝜋²𝐸^𝑞∗₀𝑑𝑞𝑞³¹₋₁𝑑𝑥𝑥𝑛𝐸−𝜇−𝑞𝑥−𝑛×−𝑞𝑥1−𝑥²𝜌_𝑡𝑞𝑥,𝑞+𝜌_𝑙𝑞𝑥,𝑞. (30) We substitute 𝑞 and 𝑞₀ by introducing dimensionless variables 𝑧 and 𝑣 in the above equation: 𝑞=2𝑞_𝑠𝑧(𝜋𝑣)^1/3,𝑞₀=𝑇𝑣, (31) where 𝑞_𝑠 is the screening length in the magnetic sector and given by (𝜋𝑚²_𝐷𝑞₀/2)^1/3. This fractional power in the transverse sector will eventually show up in different coefficients as we show below. The previous substitutions actually yields, 𝑞=𝑚_𝐷𝑎^1/3𝑎𝑧,𝑥=^2/3𝑣𝑧. (32) After performing the 𝑞₀ and 𝑞 integrations we obtain [4]: 𝜂^soft_𝑡(𝐸)−𝜂^soft_{𝑡,𝐸=𝜇}=𝑒²𝑚²_𝐷𝐸×1𝑇48𝜋𝑚_𝐷ℎ₁𝐸−𝜇𝑇²−3×2^1/372𝜋^7/3𝑇𝑚_𝐷ℎ₂𝐸−𝜇𝑇^8/3−6×2^2/39𝜋^11/3𝑇𝑚_𝐷ℎ₃𝐸−𝜇𝑇^10/3, (33) where ℎ₁𝐸−𝜇𝑇=Γ(3)Li₂(−𝑒^−𝛼)−Li₂(−𝑒^𝛼)^1/2,ℎ₂𝐸−𝜇𝑇=Γ113Li_8/3(−𝑒^−𝛼)−Li_8/3(−𝑒^𝛼)^3/8,ℎ₃𝐸−𝜇𝑇Γ133Li_10/3(−𝑒^−𝛼)−Li_10/3(−𝑒^𝛼)^3/10. (34) In case of the longitudinal sector we substitute 𝑞=𝑞_𝑠𝑦 and 𝑞₀=𝑇𝑢/𝑦 or 𝑞=𝑚_𝐷𝑦 and 𝑥=𝑎𝑢/𝑦. Since screening length is different in electric and magnetic sectors the substitutions therefore involve different coefficients of 𝑚_𝐷 and 𝑇 for the transverse and the longitudinal cases [1] as can be seen from the structure of 𝜌_𝑙,𝑡 (8). For the leading term in the longitudinal sector one writes, 𝜂^soft_𝑙(𝐸)−𝜂^soft_{𝑙,𝐸=𝜇}=𝑒²𝑚²_𝐷𝑇96𝐸𝑚_𝐷𝑔₁𝐸−𝜇𝑇³, (35) where 𝑔₁𝐸−𝜇𝑇=Γ(4)Li₃−𝑒^−𝛼−Li₃−𝑒^𝛼1/3. (36) The final expression for drag coefficient then becomes [4] 𝑒𝜂=²𝑚²_𝐷𝐸1𝑇48𝜋𝑚_𝐷ℎ₁𝐸−𝜇𝑇²−3×2^1/372𝜋^7/3𝑇𝑚_𝐷ℎ₂𝐸−𝜇𝑇^8/3−6×2^2/39𝜋^11/3𝑇𝑚_𝐷ℎ₃𝐸−𝜇𝑇^10/3+𝑒²𝑚²_𝐷𝑇96𝐸𝑚_𝐷𝑔₁𝐸−𝜇𝑇³. (37) From the previous expression it is evident that 𝜂 is polylogarithmic in nature and contains fractional powers in (𝐸−𝜇). The fractional powers indicate the nonanalytic behavior of 𝜂. Significant it is to note that here the subleading transverse term is greater than the leading order longitudinal one. Thus we see that here too the transverse contribution dominates over the longitudinal one. In the zero temperature limit the functions behave as ℎ_𝑖(𝛼)→|𝛼| and 𝑔_𝑖(𝛼)→|𝛼|. Hence, 𝜂 in the limit becomes [4] 𝜂=ℋ₁𝜈²−ℋ₃𝜈^8/3+ℋ₂𝜈³+⋯. (38) In the last equation ℋ₃=(3×2^1/3𝑒²𝑚²_𝐷)/(72𝜋^7/3𝐸), and ℋ₁, ℋ₂ are the same as before. We can infer from the previous observations that the inclusion of transverse interaction for a relativistic particle changes the nature of 𝜂. In the nonrelativistic case considering only the electric interaction 𝜂 goes as (𝐸−𝜇)³, but with the inclusion of the transverse sector we see that anomalous fractional powers appear. Like 𝜂 one can also compute the longitudinal momentum diffusion coefficient in the low temperature limit. The expression for the longitudinal momentum diffusion coefficient as defined in (26) can be written as [4] ℬ=𝑒²𝑚³_𝐷1𝑇72𝜋𝑚_𝐷ℎ₄𝐸−𝜇𝑇³−3×2^1/399𝜋^7/3𝑇𝑚_𝐷ℎ₅𝐸−𝜇𝑇^11/3−20×2^2/339𝜋^11/3𝑇𝑚_𝐷ℎ₆𝐸−𝜇𝑇^13/3+𝑒²𝑚³_𝐷𝑇128𝑚_𝐷𝑔₂𝐸−𝜇𝑇⁴, (39) where ℎ₄𝐸−𝜇𝑇=Γ(4)Li₃(−𝑒^−𝛼)−Li₃(−𝑒^𝛼)^1/3,ℎ₅𝐸−𝜇𝑇=Γ143Li_11/3(−𝑒^−𝛼)−Li_11/3(−𝑒^𝛼)^3/11,ℎ₆𝐸−𝜇𝑇=Γ163Li_13/3(−𝑒^−𝛼)−Li_13/3(−𝑒^𝛼)^3/13,𝑔₂𝐸−𝜇𝑇=Γ(5)Li₄−𝑒^−𝛼−Li₄−𝑒^𝛼1/4. (40) The final expression for ℬ in the extreme zero temperature limit becomes [4] ℬ=ℬ₁𝜈³−ℬ₃𝜈^11/3+ℬ₂𝜈⁴+⋯, (41)ℬ₃=(2^1/3𝑒²𝑚³_𝐷)/(33𝜋^7/3), ℬ₁, and ℬ₂ are the same as earlier. It is to be noted that, for the low temperature case, so far, no reference has been made to the case of hard momentum regime. This is justified as we have seen before that up to the order of our interest the hard sector does not contribute, and the entire contribution comes from the soft sector alone.

Furthermore, it is worthwhile to mention that the NFL behaviour of these coefficients is related to the dynamical screening involving “𝜔” in the static limit when one evaluates Π_𝑡. To see this one can recall the expressions for the polarization functions in (6). In the static limit (𝑞₀/𝑞∼0) the functions become Π_𝑙=𝑚²_𝐷,Π_𝑡=𝑖𝑚²_𝐷𝜋𝜔.4𝑞 (42) The “𝜔” that appears in the denominator of (5) from the Π_𝑡 above along with the Pauli blocking is responsible for this NFL behavior.

In Figures 3 and 4  𝜂 and ℬ versus energy of the incoming fermion in the small temperature (𝑇/𝑇_𝑓≪1) region (𝑇_𝑓=𝜇/𝑘_𝐵 is the Fermi temperature) have been plotted. From the figures, it is evident that with increasing 𝑇/𝑇_𝑓, both 𝜂 and ℬ decrease. This nature is consistent with what one finds for the fermionic damping rate at small temperature [1].

Figure 3 

Energy dependence of the drag coefficient at 𝑇/𝑇𝑓=1/3 (dotted curve), 𝑇/𝑇𝑓=1/2 (dashed curve), and 𝑇/𝑇𝑓=3/5 (dash dotted curve) [10].

Figure 4 

Energy dependence of the diffusion coefficient at 𝑇/𝑇𝑓=1/3 (dotted curve), 𝑇/𝑇𝑓=1/2 (dashed curve), and 𝑇/𝑇𝑓=3/5 (dash dotted curve) [10].

4. Neutrino Mean Free Path

In this section we calculate the neutrino mean free path (MFP) for cold and warm QCD matters. When a new star is born following a supernova explosion, it starts emitting neutrinos via the direct or the modified Urca processes. Emission of these neutrinos is responsible for the initial cooling of the star so produced. For quark matter, the dominant contribution to the emission of neutrinos is given by the 𝛽 decay and the electron capture [9]. These reactions are named as “quark direct Urca” processes which have been studied in detail by Iwamoto [30, 31]. For quark matter, the MFP was previously derived in [30] where the calculations were restricted to the leading order by assuming free Fermi gas interactions. In this context we recall two important works on the neutrino MFP in QED plasma. One is due to Tubbs and Schramm [32], and the other is done by Lamb and Pethick [33]. In [32], MFP was calculated in the neutranized core and just outside the core. On the other hand, it has been shown in [33] that neutrino degeneracy reduces the neutrino MFP which suggests that neutrino may flow out of the core rather slowly. Here, we revisit the calculation and go beyond the leading order results to incorporate the NFL corrections both for degenerate and nondegenerate neutrinos.

In the interior of a neutron star, there are two distinct phenomena for which the neutrino mean free path is calculated: one is absorption, and the other involves scattering of neutrinos [31]. The corresponding mean free paths are denoted as 𝑙^abs_mean and 𝑙^scatt_mean, and combine them to obtain the total mean free path [34]: 1𝑙^total_mean=1𝑙^abs_mean+1𝑙^scatt_mean. (43)

In our model the Lagrangian density is described by [30] ℒ_𝑊𝑥(𝑥)=𝐺√2𝑙_𝜇(𝑥)𝒥^𝜇_𝑊(𝑥)+H.C., (44) where the weak coupling constant is 𝐺≃1.166×10⁻¹¹ in MeV units, and 𝑙_𝜇 and 𝒥^𝜇_𝑊 are the lepton and hadron charged weak currents, respectively. The weak currents are 𝑙_𝜇(𝑥)=𝑒𝛾_𝜇1−𝛾₅𝜈_𝑒+𝜇𝛾_𝜇1−𝛾₅𝜈_𝜇𝒥+⋯,_𝜇𝑊(𝑥)=cos𝜃_𝑐𝑢𝛾^𝜇1−𝛾₅𝑑+sin𝜃_𝑐𝑢𝛾^𝜇1−𝛾₅𝑠+⋯, (45) where 𝜃_𝑐 is the Cabibbo angle (cos²𝜃_𝑐≃0.948) [35].

The mean free path is determined by the quark neutrino interaction in dense quark matter via weak processes. We consider the simplest 𝛽 decay reactions: the absorption process 𝑑+𝜈_𝑒⟶𝑢+𝑒⁻ (46) and the other is its inverse relation 𝑢+𝑒⁻⟶𝑑+𝜈_𝑒 (47)

The neutrino mean free path is related to the total interaction rate due to neutrino emission averaged over the initial quark spins and summed over the final state phase space and spins. It is given by [31] 1𝑙^abs_mean𝐸_𝜈=𝑔,𝑇2𝐸_𝜈𝑑³𝑝_𝑑(2𝐩)³12𝐸_𝑑𝑑³𝑝_𝑢(2𝐩)³12𝐸_𝑢𝑑׳𝑝_𝑒(2𝐩)³12𝐸_𝑒(2𝜋)⁴𝛿⁴𝑃_𝑑+𝑃_𝜈−𝑃_𝑢−𝑃_𝑒,|𝑀|²𝑛𝑝_𝑑1−𝑛𝑝_𝑢1−𝑛𝑝_𝑒−𝑛𝑝_𝑢𝑛𝑝_𝑒1−𝑛𝑝_𝑑, (48) where 𝑔 is the spin and color degeneracy, which in the present case is considered to be 6. Here, 𝐸, 𝑝, and 𝑛_𝑝 are the energy, momentum, and distribution function for the corresponding particle. |𝑀|² is the squared invariant amplitude averaged over initial 𝑑 quark spin and summed over final spins of 𝑢 quark and electron as given by [31] |𝑀|²=12_{𝜎𝑢,𝜎𝑑,𝜎𝑒}𝑀_𝑓𝑖²=64𝐺²cos²𝜃_𝑐𝑃_𝑑⋅𝑃_𝜈𝑃_𝑢⋅𝑃_𝑒. (49) Here, we work with the two flavor system as the interaction involving strange quark is Cabibbo suppressed [9, 36].

4.1. Degenerate Neutrinos

In this section we consider the case of degenerate neutrinos. When the neutrino chemical potential (𝜇_𝜈) is considered to be much larger than the temperature (𝑇), the neutrinos become degenerate. This is also termed as trapped neutrino matter. In this case both the direct (46) and inverse (47) processes can occur, and both terms in (48) under curly brackets are retained [31]. Consequently, the 𝛽 equilibrium condition becomes 𝜇_𝑑+𝜇_𝜈=𝜇_𝑢+𝜇_𝑒. Neglecting the quark-quark interactions and by using (48) and (49), the mean free path becomes 1𝑙^abs,𝐷_mean=34𝜋⁵𝐺²cos²𝜃_𝑐𝑑³𝑝_𝑑𝑑³𝑝_𝑢×𝑑³𝑝_𝑒1−cos𝜃_𝑑𝜈1−cos𝜃_𝑢𝑒×𝛿⁴𝑃_𝑑+𝑃_𝜈−𝑃_𝑢−𝑃_𝑒1+𝑒^{−𝛽(𝐸𝜈−𝜇𝜈)}×𝑛𝑝_𝑑1−𝑛𝑝_𝑢1−𝑛𝑝_𝑒. (50) In the square bracket, the second term 𝑒^{−𝛽(𝐸𝜈−𝜇𝜈)} is due to the inverse process (47). One can neglect the mass effect on the neutrino MFP as the masses of 𝑢, 𝑑 quark and electron are very small. To perform the momentum integration we define 𝑝≡|𝑝_𝑑+𝑝_𝜈|=|𝑝_𝑢+𝑝_𝑒| as a variable. Following the same procedure as described by Iwamoto [31] one has sin𝜃_𝑑𝜈𝑑𝜃_𝑑𝜈=𝑝𝑑𝑝𝑝_𝑓(𝑑)𝑝_𝑓(𝜈),1−cos𝜃_𝑑𝜈1−cos𝜃_𝑢𝑒≃𝑝⁴−2𝑝²𝑝²_𝑓+𝑝⁴_𝑓4𝑝_𝑓(𝑑)𝑝_𝑓(𝜈)𝑝_𝑓(𝑢)𝑝_𝑓(𝑒),𝑑³𝑝_𝑑=2𝜋sin𝜃_𝑑𝜈𝑑𝜃_𝑑𝜈𝑝²_𝑑𝑑𝑝_𝑑𝑝=2𝜋^𝑓(𝑑)𝑝_𝑓(𝜈)𝑝𝑑𝑝𝑑𝑝_𝑑𝑑𝐸_𝑑𝑑𝐸_𝑑𝑝=2𝜋^𝑓(𝑑)𝑝_𝑓(𝜈)𝑝𝑑𝑝𝑑𝑝_𝑑𝑑𝑑𝜔𝑑𝜔,₃𝑝_𝑢𝑝=2𝜋^𝑓(𝑢)𝑝_𝑓(𝑒)𝑝𝑑𝐸_𝑒𝑑𝑝_𝑢𝑑𝜔𝑑𝜔, (51)(52)(53)(54) where we denote the single particle energy 𝐸_𝑑(𝑢) as 𝜔. For the free case 𝑑𝑝/𝑑𝜔 is the inverse quark velocity. It is well known that this slope of the dispersion relation changes in matter due to the scattering from the Fermi surface and excitation of the Dirac vacuum. The modified dispersion relation can be obtained by computing the on-shell one-loop self-energy. For quasiparticles with momenta close to the Fermi momentum, the one-loop self-energy is dominated by the soft gluon exchanges [8]. The quasiparticle energy 𝜔 satisfies the relation [8, 21] 𝜔=𝐸_𝑝(𝜔)−ReΣ𝜔,𝑝(𝜔), (55) where we have retained only the real part of self-energy since the imaginary part of Σ turns out to be negligible compared to its real part [1, 37]. For the detailed analysis we refer the reader to [11, 12, 21].

The analytical expressions for one-loop quark self-energy can be written as [1, 8, 37–40] 𝑔Σ=²𝐶_𝐹12𝜋²𝑚𝜔−𝜇ln^𝐷𝑔𝜔−𝜇+𝑖²𝐶_𝐹.24𝜋𝜔−𝜇 (56) It exhibits a logarithmic singularity close to the Fermi surface, that is, when 𝜔→𝜇.

𝑑𝑝(𝜔)/𝑑𝜔 can be obtained by differentiating the dispersion relation (55) with respect to 𝑝. At leading order in 𝑇/𝜇 this yields 𝑑𝑝(𝜔)≃𝜕𝑑𝜔1−(𝜔)𝐸𝜕𝜔ReΣ^𝑝(𝜔)𝑝(𝜔)=𝐶1+^𝐹𝛼_𝑠𝑚3𝜋ln^𝐷𝑇𝐸_𝑝(𝜔)𝑝(𝜔), (57) where 𝛼_𝑠 is the strong coupling constant, 𝐶_𝐹=(𝑁²_𝑐−1)/(2𝑁_𝑐), and 𝑁_𝑐 is the color factor. Using (57), (50), and (52)–(54), the neutrino mean free path can be determined for two conditions. For |𝑝_𝑓(𝑢)−𝑝_𝑓(𝑒)|≥|𝑝_𝑓(𝑑)−𝑝_𝑓(𝜈)|1𝑙^abs,𝐷_mean=4𝜋³𝐺²cos²𝜃_𝑐𝜇²_𝑢𝜇³_𝑒𝜇²_𝜈11+2𝜇_𝑒𝜇_𝑢+1𝜇10𝜇^𝑒_𝑢²×𝐸_𝜈−𝜇_𝜈²+𝜋²𝑇²𝐶1+^𝐹𝛼_𝑠𝑚3𝜋ln^𝐷𝑇². (58)

To derive (58), we use the chemical equilibrium condition 𝑝_𝑓(𝑢)+𝑝_𝑓(𝑒)=𝑝_𝑓(𝑑)+𝑝_𝑓(𝜈) neglecting the masses for quarks and electrons. For the phase-space integral we have [31, 41] ^∞₀𝑑𝐸_𝑑^∞₀𝑑𝐸_𝑢×^∞₀𝑑𝐸_𝑒1+𝑒^{−𝛽(𝐸𝜈−𝜇𝜈)}𝑛𝑝_𝑑×1−𝑛𝑝_𝑢1−𝑛𝑝_𝑒𝛿𝐸_𝑑+𝐸_𝜈−𝐸_𝑢−𝐸_𝑒≃12𝐸_𝜈−𝜇_𝜈²+𝜋²𝑇². (59)

Similarly, for |𝑝_𝑓(𝑑)−𝑝_𝑓(𝜈)|≥|𝑝_𝑓(𝑢)−𝑝_𝑓(𝑒)|, the corresponding expression for mean free path can be obtained by replacing 𝜇_𝑢↔𝜇_𝑑 and 𝜇_𝑒↔𝜇_𝜈 in (58).

The other contribution to the mean free path comes from the quark-neutrino scattering. The neutrino scattering process from degenerate quarks is given by 𝑞_𝑖+𝜈_𝑒𝜈_𝑒⟶𝑞_𝑖+𝜈_𝑒𝜈_𝑒 (60) for each quark component of flavor 𝑖(=𝑢or𝑑). The scattering mean free path of the neutrinos in degenerate case can be calculated similarly as evaluated by Lamb and Pethick in [33] for electron-neutrino scattering. Assuming 𝑚_𝑞𝑖/𝑝_𝑓𝑖≪1 and including the suitable modifications of the phase space, the mean free path is given by 1𝑙^scatt,𝐷_mean=3𝑛16^𝑞𝑖𝜎₀𝐸_𝜈−𝜇_𝜈²+𝜋²𝑇²𝑚²_𝑞𝑖×𝐶1+^𝐹𝛼_𝑠𝑚3𝜋ln^𝐷𝑇²Λ𝑥_𝑖. (61) Here, 𝑚_𝑞𝑖 is the quark mass. 𝐶_𝑉𝑖 and 𝐶_𝐴𝑖 are the vector and axial vector coupling constant given in Table II of [31]. The expression of (61) has been found to be in agreement with the results reported in [33] for dense and cold QED plasmas by making suitable changes for the color factors and by dropping the second square bracketed term. In (61) the constants 𝜎₀≡4𝐺²𝑚²_𝑞𝑖/𝜋 [32], and 𝑛_𝑞𝑖 is the quark number density: 𝑛_𝑞𝑖𝑑=2³𝑝(2𝐩)³1𝑒^{𝛽(𝐸𝑞𝑖−𝜇𝑞𝑖)}, (62) where 2 is the quark spin degeneracy factor. Explicit form of Λ(𝑥_𝑖) can be written as [31, 33] Λ𝑥_𝑖=43Min𝜇_𝜈,𝜇_𝑞𝑖𝜇_𝑞𝑖×𝐶²_𝑉𝑖+𝐶²_𝐴𝑖12+5𝑥²_𝑖+2𝐶_𝑉𝑖𝐶_𝐴𝑖𝑥_𝑖, (63) and 𝑥_𝑖=𝜇_𝜈/𝜇_𝑞𝑖 if 𝜇_𝜈<𝜇_𝑞𝑖, and 𝑥_𝑖=𝜇_𝑞𝑖/𝜇_𝜈 if 𝜇_𝜈>𝜇_𝑞𝑖.

4.2. Nondegenerate Neutrinos

The mean free path has been also derived for nondegenerate neutrinos, that is, when 𝜇_𝜈≪𝑇. This is the case for untrapped neutrino matter. For this case the inverse process (47) does not contribute to the MFP. Hence we neglect the second term in the curly braces of (48). Here, only those fermions take part in the reaction whose momenta lie close to their respective Fermi surfaces. It is to be mentioned here, if quarks are treated as free, as discussed in [31, 42, 43], the matrix element vanishes since 𝑢, 𝑑 quarks and electrons are collinear in momenta. The inclusion of strong interactions between quarks relaxes these kinematic restrictions resulting in a nonvanishing squared matrix amplitude. Since the neutrinos are produced thermally, we neglect the neutrino momentum in energy-momentum conservation relation [31]. This is not the case for degenerate neutrinos where 𝑝_𝜈≫𝑇, and therefore such approximation is not valid there. By doing angular average over the direction of the outgoing neutrino, from (49) the squared matrix element is given by [9]: |𝑀|²=64𝐺²cos²𝜃_𝑐𝑝²_𝑓𝑉_𝑑⋅𝑃_𝜈𝑉_𝑢⋅𝑃_𝑒=64𝐺²cos²𝜃_𝑐𝑝²_𝑓𝐸_𝜈𝜇_𝑒𝐶_𝐹𝛼_𝑠𝜋, (64) where 𝑉=(1,𝑣_𝑓) is the four velocity. To calculate (64) we have used the chemical equilibrium condition 𝜇_𝑑=𝜇_𝑢+𝜇_𝑒 and also the relations derived from Fermi liquid theory [9] 𝑣_𝐹𝐶=1−^𝐹𝛼_𝑠𝐶2𝜋;𝛿𝜇=^𝐹𝛼_𝑠𝜋𝜇. (65)

Putting |𝑀|² in (48) we have 1𝑙^abs,𝑁𝐷_mean=3𝐶_𝐹𝛼_𝑠4𝐩⁶𝐺²cos²𝜃_𝑐𝑑³𝑝_𝑑𝑑³𝑝_𝑢𝑑³𝑝_𝑒𝛿⁴𝑃_𝑑+𝑃_𝜈−𝑃_𝑢−𝑃_𝑒𝑛𝑝_𝑑1−𝑛𝑝_𝑢1−𝑛𝑝_𝑒. (66) Neglecting the neutrino momentum in the neutrino momentum conserving 𝛿 function, the integrals can be decoupled into two parts. Following the procedure described by Iwamoto [31], the angular integral is given by 𝒜=𝑑Ω_𝑑𝑑Ω_𝑢𝑑Ω_𝑒𝛿𝑝_𝑑−𝑝_𝑢−𝑝_𝑒=8𝜋²𝜇_𝑑𝜇_𝑢𝜇_𝑒 (67) and the other part 𝒟=^∞₀𝑝²_𝑑𝑑𝑝_𝑑𝑑𝐸_𝑑𝑑𝐸_𝑑^∞₀𝑝²_𝑢𝑑𝑝_𝑢𝑑𝐸_𝑢𝑑𝐸_𝑢^∞₀𝑝²_𝑒𝑑𝐸_𝑒𝛿𝐸_𝑑+𝐸_𝜈−𝐸_𝑢−𝐸_𝑒𝑛𝑝_𝑑1−𝑛𝑝_𝑢1−𝑛𝑝_𝑒. (68)

Changing the variables to 𝑥_𝑑=(𝐸_𝑑−𝜇_𝑑)𝛽, 𝑥_𝑢=−(𝐸_𝑢−𝜇_𝑢)𝛽, and 𝑥_𝑒=−(𝐸_𝑒−𝜇_𝑒)𝛽 and denoting the single particle energy 𝐸_𝑢(𝑑) as 𝜔 we have from (68) 𝒟=^∞_−∞𝑑𝑥_𝑑𝑑𝑥_𝑢𝑑𝑥_𝑒𝑑𝑝_𝑑(𝜔)𝑑𝜔𝑑𝑝_𝑢(𝜔)𝑝𝑑𝜔^2𝑑𝑝²_𝑢𝑝²_𝑒𝛿×𝑥_𝑑+𝑥_𝑢+𝑥_𝑒+𝛽𝐸_𝜈𝑛𝑥_𝑑𝑛−𝑥_𝑢𝑛−𝑥_𝑒. (69) As the contribution dominates near the Fermi surfaces, extension of lower limit is reasonable [41, 44].

The integration of (69) can be performed using (57) and following the procedure defined in [37, 41, 44] to give 𝒟=𝜇²_𝑑𝜇²_𝑢𝜇²_𝑒𝐸²_𝜈+𝜋²𝑇²21+𝑒^{−𝛽𝐸𝜈}𝐶1+^𝐹𝛼_𝑠𝑚3𝜋ln^𝐷𝑇². (70)

Using (66), (67), and (70), the mean free path at leading order in 𝑇/𝜇 is given by 1𝑙^abs,𝑁𝐷_mean=3𝐶_𝐹𝛼_𝑠𝜋⁴𝐺²cos²𝜃_𝑐𝜇_𝑑𝜇_𝑢𝜇_𝑒×𝐸²_𝜈+𝜋²𝑇²1+𝑒^{−𝛽𝐸𝜈}𝐶1+^𝐹𝛼_𝑠𝑚3𝜋ln^𝐷𝑇². (71) The first term is known from [31], and the additional terms are higher order corrections to the previous results derived in the present work.

For the scattering of nondegenerate neutrinos in quark matter, the expression of mean free path was given by Iwamoto [31]. We incorporate the anomalous effect which enters through phase space modification giving rise to 1𝑙^{scatt,𝑁𝐷}_mean=𝐶²_𝑉𝑖+𝐶²_𝐴𝑖𝑛20^𝑞𝑖𝜎₀×𝐸_𝜈𝑚_𝑞𝑖²𝐸_𝜈𝜇_𝑖𝐶1+^𝐹𝛼_𝑠𝑚3𝜋ln^𝐷𝑇². (72) Here, we have assumed 𝑚_𝑞𝑖/𝑝_𝑓𝑖≪1. The constants 𝜎₀ and number density 𝑛_𝑞𝑖 have been defined earlier.

We now estimate the numerical values of the neutrino mean free paths. Here, 𝐸_𝜈 is set to be equal to 3𝑇 and 𝑚_𝑞=10 MeV [31, 34]. For the quark chemical potential, following [9], we take 𝜇_𝑞≃500 MeV corresponding to densities 𝜌_𝑏≈6𝜌₀, where 𝜌₀ is the nuclear matter saturation density. The electron chemical potential is determined by using the charge neutrality and beta equilibrium conditions which yields 𝜇_𝑒=11 Mev. The other parameters used are the same as [9].

From Figure 5, we find for degenerate neutrinos at the anomalous logarithmic terms reduce the value of the mean free path appreciably both in the low and high temperature regimes. Figure 6 shows that for nondegenerate neutrinos NFL correction is quite large at low temperature while at higher temperature it tends to merge. It is interesting to see from these two plots that NFL correction to the MFP in degenerate neutrinos is less than that of nondegenerate neutrinos [3]. This reduced mean free path is expected to influence the cooling of the compact stars.

Figure 5 

Neutrino mean free path in quark matter for degenerate neutrinos [3].

Figure 6 

Neutrino mean free path in quark matter for nondegenerate neutrinos [3].

5. Cooling of Neutron Star

To analyze the cooling of the compact star [45–47], the specific heat capacity of the quark matter core needs to be taken into consideration along with the emissivity via the cooling equation [9, 36] 𝜕𝑢=𝜕𝑡𝜕𝑢𝜕𝑇𝜕𝑇𝜕𝑡=𝐶_𝑣(𝑇)𝑑𝑇(𝑇),𝑑𝑡=−𝜖 (73) where 𝑢 is the internal energy, 𝐶_𝑣 is the specific heat at constant volume and baryon number, 𝜖 is the emissivity, 𝑡 is time, and 𝑇 is the temperature, and we have assumed that there is no surface emission.

Emissivity under different scenarios both for quark and nuclear matters has been studied extensively. For the specific heat however earlier literatures were concerned only with the kinetic energy contribution without paying much attention to the QCD interaction. Recently several authors have shown that the interaction can lead to a significant correction to 𝐶_𝑣. It has been shown that this interaction mediated by the gluons can give rise to a leading 𝑇ln𝑇⁻¹ term in the specific heat [48, 49]. This logarithmic enhancement comes from long-range chromomagnetic fields which eventually leads to non-Fermi-liquid behavior. The calculation of 𝐶_𝑣 including NFL correction has been done in [48, 49] which we quote here for further use: 𝐶_𝑣=𝑁_𝑐𝑁_𝑓3𝜇²𝑁𝑇+^𝑐𝑁_𝑓𝜇9𝜋²𝐶_𝐹𝛼_𝑠𝑇ln0.28𝑚_𝐷𝑇. (74)

On the other hand, the neutrino emissivity of degenerate quark matter was first derived by Iwamoto [31]. Latter Schäfer and Schwenzer [9] calculated the neutrino emissivity including the corrections for NFL effects. The calculation has been made by considering the energy loss due to neutrino emission from the core of the neutron star. Alternatively, one can also determine the neutrino emissivity from the mean free path by evaluating the integral [31] 𝑑𝜖=³𝑝_𝜈(2𝐩)³𝐸_𝜈𝑙^abs,𝑁𝐷_mean−𝐸_𝜈≈,𝑇457𝐺630²cos²𝜃_𝑐𝛼_𝑠𝜇_𝑢𝜇_𝑑𝜇_𝑒𝑇⁶𝐶1+^𝐹𝛼_𝑠𝑚3𝜋log^𝐷𝑇². (75) This is the same as given in [9]. In this context, it is to be mentioned that without such non-Fermi liquid effects we have 𝜖∼𝑇⁶ and 𝐶_𝑣∼𝑇. In this case the temperature scales turn out to be 𝑇∝1/𝑡^1/4 [9]. With logarithmic corrections included, no simple analytic solution is possible, and we solve (73) numerically.

We observe from Figure 7 that the cooling of neutron star is significantly faster with NFL corrections as compared to the Fermi liquid result [9].

Figure 7 

The cooling behavior of neutron star with core as neutron matter and degenerate quark matter. The dotted line represents the Fermi liquid result; the solid line gives the result including non-Fermi liquid correction. The dash-dotted line gives the cooling behavior of the neutron star core made up of purely neutron matter.

6. Thermal Relaxation Time

In this section we address the issue of thermal conduction that follows the initial cooling by the neutrino emission. This phenomenon is intimately connected to the thermal relaxation mechanism. A new born star emits a large amount of neutrinos to cool the core, but the crust remains hot. Hence, a temperature gradient is set up between the crust and the core. Immediately after the neutrino emission discussed earlier in the paper thermal energy gradually flows from outer crust to the inner core by heat conduction which, alternatively, might be viewed as the propagation of the cooling waves from the center towards the surface leading to thermalization. We in this section estimate the thermalization time scale (𝜏_𝜅) and reveal the NFL behavior of the quantity in the context of the neutron star crust. In the latter the electrons constitute an almost ideal Fermi gas and scatter between themselves. The kinetic theory definition of 𝜏_𝜅 is 𝜏_𝜅=3𝜅𝐶_𝑣. (76) In case of the strongly degenerate electron gas the electron thermal conductivity (𝜅_𝑒) can be expressed in terms of the thermal current 𝐽_𝑇 as follows: 𝜅_𝑒=𝐽_𝑇𝑇𝜈_𝑒,𝜈_𝑒=𝜈_𝑒𝑖+𝜈_𝑒𝑒, (77) where 𝜈_𝑒 is the total effective collision frequency. In the neutron star crust the main components that contribute to different transport coefficients are electrons and ions. Hence, the total collision frequency is the sum of the partial collision frequencies of electron-ion (𝜈_𝑒𝑖) and electron-electron (𝜈_𝑒𝑒). Evidently 𝜅_𝑒 is related to 𝜅_𝑒𝑒 and 𝜅_𝑒𝑖 via the following expression [50, 51]: 1𝜅_𝑒=1𝜅_𝑒𝑖+1𝜅_𝑒𝑒,𝜅_𝑒𝑖=𝐽_𝑇𝑇𝜈_𝑒𝑖,𝜅_𝑒𝑒=𝐽_𝑇𝑇𝜈_𝑒𝑒. (78) Thus the derivation of 𝜅_𝑒𝑒 involves the calculation of 𝜈_𝑒𝑒. To derive 𝜈_𝑒𝑒 we start with the Boltzmann equation which describes the kinetics of the particles: 𝜕𝜕𝑡+𝐯_𝐩⋅∇_𝐫+𝐅⋅∇_𝐩𝑛_𝑝=−𝒞𝑛_𝑝, (79) where 𝐩 is the momentum of the quasiparticle, 𝐅 is the external force, 𝐯_𝐩 is the velocity of the heat carrier, and 𝑛_𝑝 is the distribution function of electrons. The right hand side of the Boltzmann equation is the collision integral describing the rate of scattering of fermions. In the presence of a weak stationary temperature gradient Boltzmann equation takes the following form: 𝐯_𝐩.∇_𝐫𝑛_𝑝=−𝒞𝑛_𝑝. (80) The Fermi-Dirac distribution functions deviate from equilibrium distribution functions 𝑛_𝑖 due to the presence of a weak temperature gradient, which we write as 𝑛_𝑖=𝑛_𝑖+𝜕𝑛_𝑖𝜕𝜖_𝑖Φ_𝑖∇𝑇𝑇, (81) where 𝑛_𝑖=𝜖exp^𝑖−𝜇𝑇+1⁻¹, (82)𝜖 is the particle energy, 𝜇 is the chemical potential, and 𝑇 is the temperature. The term with Φ measures the deviation from equilibrium. The collision integral can be written as follows: 𝒞𝑛_𝑝=𝜈𝜈^′_{𝑝′,𝑘,𝑘′}𝑛_𝑝𝑛_𝑘1±𝑛^′_𝑝1±𝑛^′_𝑘−𝑛^′_𝑝𝑛^′_𝑘1±𝑛_𝑝1±𝑛_𝑘(2𝜋)⁴𝛿⁴×𝑝+𝑘−𝑝^′−𝑘^′|𝑀|². (83) In the previous equation 𝜈, 𝜈^′ are the degeneracy factors and |𝑀|² is the squared matrix element for the 2→2 scattering process. The ± sign is meant for stimulated emission or Pauli blocking. Since we consider only the electron-electron scattering, the negative sign will remain in the phase space factor. We write the previous equation in the form |𝑋⟩=𝐼|Φ⟩, where |𝑋⟩=(𝜖_𝑝−𝜇)𝑣_𝑝 and 𝐼 is the integral operator. The thermal conductivity 𝜅_𝑒𝑒 is then given by the maximum of the following equation [52, 53]: 𝜅_𝑒𝑒=⟨𝑋∣Φ⟩²𝑇⟨Φ|𝐼|Φ⟩; (84)⟨⋅∣⋅⟩ denotes an inner product; for Ψ=Φ the quantity ⟨𝑋|Ψ⟩²/𝑇⟨Ψ|𝐼|Ψ⟩ is minimal with the minimal value 𝜅_𝑒𝑒. With the help of this definition one can write 1𝜅_𝑒𝑒≥𝜈_𝑝𝜖_𝑝−𝜇𝑇𝑣_𝑧𝑛_𝑝1−𝑛_𝑝Ψ_𝑝⁻²×𝜈𝜈^′_{𝑝,𝑝′,𝑘,𝑘′}𝑛_𝑝𝑛_𝑘1−𝑛^′_𝑝1−𝑛^′_𝑘×(2𝜋)⁴𝛿⁴𝑝+𝑘−𝑝^′−𝑘^′|𝑀|²×Ψ_𝑝+Ψ_𝑘−Ψ_𝑝′−Ψ_𝑘′²4; (85) the first bracketed term in the denominator is the thermal current 𝐽_𝑇. In principle Φ should be determined by minimizing (84) using the variational principle. But in the present scenario we consider the simplest trial function as [50, 52] Ψ_𝑝∝𝜖_𝑝−𝜇𝑣_𝑧. (86) The term in the bracket can be averaged over the 𝑧 axis keeping 𝑥 and 𝜙 fixed using the previous trial function, where, 𝜙 is the azimuthal angle between 𝑣_𝑝 and 𝑣_𝑘. After averaging we obtain [52] Ψ_𝑝+Ψ_𝑘−Ψ_𝑝′−Ψ_𝑘′²=23𝜔²1−𝑥²1−cos𝜙. (87) For small energy transfer the electron-electron scattering squared matrix element is given by [52] |𝑀|²=32𝑒⁴1𝑞²+Π_𝑙+(1−𝑥²)cos𝜙(𝑞²−𝜔²+Π_𝑡)². (88) The medium modified photon propagator in the previous equation contains the polarization functions Π_𝑙(𝑞₀,𝑞) and Π_𝑡(𝑞₀,𝑞), which describe plasma screening of interparticle interaction by longitudinal and transverse plasma perturbations, respectively. Now, we first analyze the denominator of (85). The denominator is the thermal current as defined earlier and is given by 𝐽_𝑇=𝜈_𝑝𝜖_𝑝−𝜇𝑇𝑣_𝑧𝑛_𝑝1−𝑛_𝑝Ψ_𝑝=𝜈𝜇²𝑇²6. (89) Here, the degeneracy factor for electrons is 𝜈=2.

The fermionic dispersion relation gets modified due to the inclusion of the fermion self-energy in the presence of the medium. Hence, energy momentum relation changes from the vacuum. For the momentum integration in the phase-space integral in (85) we will use the medium modified dispersion relation given in (55). For this one needs to know the fermion self-energy. The fermion self-energy for the QCD matter has already been quoted in (56). In case of electrons at low temperature with NLO correction it is given by the following [1]: Σ(𝜖,𝑘)=𝑒²𝑚×𝜖12𝜋²𝑚4√log2𝑚+𝜋𝜖+1𝑖𝜖+224𝜋𝑚^1/3√345𝜋^7/3𝜖𝑚^5/3sgn(𝜖)−√+𝑖3𝑖64√2𝜖𝑚²2−20^2/3√3189𝜋^11/3𝜖𝑚^7/3sgn(𝜖)+√−3𝑖6144−256𝜋²+36𝜋⁴−9𝜋⁶864𝜋⁶𝜖𝑚³×log0.928𝑚𝜖−𝑖𝜋sgn(𝜖)2𝜖+𝒪𝑚^11/3, (90) where 𝑚²=𝑚²_𝐷/2 and 𝜖 is chosen to be (𝜖_𝑘−𝜇). The phase-space correction due to the medium modified dispersion relation can now be written as [10] 𝑑𝑘𝑑𝜖_𝑘𝑒≃1+²12𝜋²4log+2𝜋𝜆_2/3𝑒²𝜆^2/39√3𝜋^7/3−40×2^1/3𝑒²𝜆^4/327√3𝜋^11/3⋯=1+𝛽, (91) where 𝜆=𝑇/𝑚_𝐷. The expression for 𝜅⁻¹_𝑒𝑒 now becomes 1𝜅_𝑒𝑒=𝒫𝑇²𝑑𝑘𝑑𝜖_𝑘𝑑𝑝𝑑𝜖_𝑝𝐼_𝜅𝑒𝑒𝑇𝑚_𝐷, (92) where 𝒫=3𝑒⁴/4𝜋⁵ and 𝐼_𝜅𝑒𝑒𝑇𝑚_𝐷≡^∞₀𝑑𝜔𝜔𝜔²/4𝑇²(sinh(𝜔/2𝑇))²×¹₀𝑑𝑥₀^2𝜋𝑑𝜙𝑥2𝜋²1−𝑥²×11−cos𝜙1+𝑥𝑚_𝐷/𝜔²𝜒_𝑙(𝑥)−cos𝜙1+𝑥𝑚_𝐷/𝜔²𝜒_𝑡(𝑥)/(1−𝑥²)². (93) In the previous equation we have used the following expression [10, 52]: 𝑑𝜖_𝑝𝑑𝜖_𝑘𝑛_𝜖𝑝𝑛_𝜖𝑘1−𝑛_{(𝜖𝑝+𝜔)}1−𝑛_{(𝜖𝑘−𝜔)}=1+𝛽𝑇²𝜔²/4𝑇²(sinh(𝜔/2𝑇))²; (94) while writing the previous equation we have considered the medium modified phase-space factor only for the particle with momentum 𝑘. One can in principle take the correction for both the particles with momentum 𝑝 and 𝑘, but that will contribute at higher order in coupling constant.

The final expression for the electron thermal conductivity now takes the following form [10]: 𝜅_𝑒𝑒=𝒫𝑇²×1+𝛽2𝜆²𝜁(3)+(2𝜋)^2/33𝜆^8/3𝜁113Γ143+𝜋⁵𝜆15³⁻¹. (95) Unlike the Fermi-liquid result where 𝜅_𝑒𝑒∝1/𝑇, here the temperature dependence is nonanalytical and anomalous in nature. This is reminiscent of other quantities involving ultradegenerate plasma presented in the previous sections [1, 3–5]. 𝜅_𝑒𝑒 involves fractional powers in (𝑇/𝑚_𝐷) coming from the medium modified phase space factor too.

For the estimation of relaxation time the other quantity which we require is the specific heat. For the degenerate electron gas it can be written as [48, 49] 𝐶_𝑣=𝜇²𝑇3+𝑚²_𝐷𝑇436ln𝜋𝜆+𝛾_𝐸−6𝜋²𝜁^′(2)2−3−40^2/3Γ(8/3)𝜁(8/3)𝑚³_𝐷27√3𝜋^7/3𝜆^5/32+560^1/3Γ(10/3)𝜁(10/3)𝑚³_𝐷81√3𝜋^11/3𝜆^7/3. (96) With (95) and (96) the relaxation time for thermal conduction can be written as [10] 𝜏_𝜅𝑒𝑒𝒫=3𝑇²×1+𝛽2𝜆²𝜁(3)+(2𝜋)^2/33𝜆^8/3𝜁113×Γ143+𝜋⁵𝜆15³⁻¹×𝜇²𝑇3+𝑚²_𝐷𝑇436ln𝜋𝜆+𝛾_𝐸−6𝜋²𝜁^′(2)2−3−40^2/3Γ(8/3)𝜁(8/3)𝑚³_𝐷27√3𝜋^7/3𝜆^5/32+560^1/3Γ(10/3)𝜁(10/3)𝑚³_𝐷81√3𝜋^11/3𝜆^7/3−1. (97) The thermal relaxation time up to NLO terms contains some anomalous fractional powers originated from the transverse interaction. This in turn changes the temperature dependence of 𝜏_𝜅𝑒𝑒 nontrivially. The departure from the Fermi liquid result (𝜏_𝜅𝑒𝑒∝1/𝑇²) is evident from the final expression of 𝜏_𝜅𝑒𝑒.

In Figures 8, 9, 10, and 11 we have plotted 𝜅_𝑒𝑒 and 𝜏_𝜅𝑒𝑒 with 𝑇 using (95) and (97). From the plots it can be seen that the inclusion of both the medium modified propagator and 𝛽 decreases the value of 𝜅_𝑒𝑒 and 𝜏_𝜅𝑒𝑒. It shows strong deviation from the Fermi liquid in both cases. In Figures 9 and 11 it has been shown that 𝛽 reduces both the thermal conductivity and the thermal relaxation time. This has a serious implication on the total electron conductivity 𝜅_𝑒. In [50] it has been shown that magnetic interaction decreases 𝜅_𝑒𝑒 which in turn increases the electron-electron collision frequency. Thus to the total electron thermal conductivity electron-electron scattering dominates over electron-ion scattering. The phase space correction due to the medium modified electron dispersion relation further enhances the electron-electron collision frequency [10].

Figure 8 

Temperature dependence of the electron thermal conductivity and comparison between the Fermi liquid and the non-Fermi-liquid NLO results where 𝛽≠0.

Figure 9 

The reduction of 𝜅𝑒𝑒 with the inclusion of 𝛽.

Figure 10 

Temperature dependence of the thermal relaxation time and comparison between the Fermi liquid and the non-Fermi-liquid NLO results where 𝛽≠0.

Figure 11 

The reduction of 𝜏𝜅𝑒𝑒 with the inclusion of 𝛽.

7. Conclusion

Investigation of the quasiparticle excitation in ultradegenerate plasma has been the cardinal focus of the present review. The main interesting feature which has been exposed here is the role of magnetic interaction mediated by the transverse gauge bosons leading to the phenomena very different from its high temperature counterpart. In particular the NFL behaviour of various quantities like 𝐶_𝑣, 𝜅_𝑒𝑒, and 𝑙^total_mean involving excitations near the Fermi surface is a very special characteristic behaviour of degenerate or ultradegenerate plasma. We also show clearly how the dynamical screening leads to finite damping rate emanating from the restricted phase space driven by the Pauli blocking. Besides the damping rate we also present results for the drag and the diffusion coefficients, that is, the energy and the momentum relaxation of the quasiparticle excitation in such a plasma revealing further the domination of the magnetic interaction over the electric interactions in ultradegenerate plasma. This separation between the longitudinal and the transverse sectors is a very special feature of the cold and dense plasma which is not seen in the finite temperature case.

Another aspect of the present work has been the modification of the quasiparticle dispersion relation which changes the phase-space in a nontrivial way leading to modification of 𝜏_𝜅𝑒𝑒 and 𝑙_mean. Further investigations along this line are pertinent to understand various phenomena of astrophysical interests. Some of these extensions have recently been undertaken by various authors showing the importance of this emerging area particularly due to the possibility of experiments in the domain of high chemical potential and low temperature in addition to its relevance in the astrophysical context.