Revisiting equation of state for white dwarfs within finite temperature quantum field theory Golam Mortuza Hossain∗ and Susobhan Mandaly Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur - 741 246, WB, India (Dated: September 25, 2019) The effects of fine-structure constant on the equation of state of degenerate matter in the white dwarfs are computed in literature using non-relativistic considerations ab initio. Given special rel- ativity plays a key role in the white dwarf physics, such computations are therefore unsatisfactory. After reviewing the existing literature, here we employ the techniques of finite temperature rela- tivistic quantum field theory to compute the equation of state of degenerate matter in the white dwarfs. In particular, we compute the leading order corrections due to the finite temperature and the fine-structure constant. We show that the fine-structure constant correction remains well-defined even in the non-relativistic regime in contrast to the existing treatment in the literature. Besides, it involves an apriori undetermined length scale characterizing the electron-nuclei interaction. I. INTRODUCTION transformations. Unfortunately, the methods employed in computing the Coulomb effects to the equation of state In the study of degenerate matter within the white of the white dwarfs use electrostatic considerations which dwarfs, as pioneered by Chandrasekhar [1, 2], the degen- are non-relativistic ab initio. Therefore, from the fun- erate electrons are treated as free particles which follow damental point of view these computations should be the Fermi-Dirac statistics. The effect of electromagnetic viewed as an approximation of the corrections that one interaction on the degenerate matter in the white dwarfs, would expect from a Lorentz invariant computation. Fur- by the means of `classical' Coulomb energy, was first con- ther, these computations are usually performed at the sidered by Frenkel [3], and was followed up by Kothari zero temperature [6] (however see [14{16]). Besides, it [4], Auluck and Mathur [5]. However, a more accurate has been noted that the future detection of low-frequency study on various corrections to the equation of state due gravitational waves from the extreme mass-ratio merger to the so-called Coulomb effects was done by Salpeter [6]. of a black hole and a white dwarf could determine the The implications of these corrections on the mass-radius equation of state of the degenerate matter within the relation of the white dwarfs were carried out by Hamada, white dwarfs with very high accuracy [17]. Such an accu- Salpeter [7] and Nauenberg [8]. racy could probe the extent of Coulomb effects and hence One may classify the total Coulomb effects that are provides additional motivation to revisit the corrections considered by Salpeter into following broad components to the equation of state of the white dwarfs. (see also [9, 10]). (a) The `classical' Coulomb energy in- cludes the electrostatic energy of uniformly distributed degenerate electrons within the Wigner-Seitz cells, each surrounding a positively charged nucleus within a rigid lattice. It includes the electron-nuclei interaction and the In order to compute the equation of state for the white self-interaction of electrons. (b) The Thomas-Fermi cor- dwarfs, a natural arena which respects Lorentz sym- rection arises due to the radial variation of electron den- metry, is provided by the finite temperature relativis- sity within a Wigner-Seitz cell. (c) The `exchange energy' tic quantum field theory. Following the pioneering work and the `correlation energy' arise due to the transverse of Matsubara [18], the techniques of finite-temperature interactions between two electrons, essentially due to the quantum field theory was employed in the context of Lorentz force between them apart from its electrostatic quantum electrodynamics (QED) by Akhiezer and Pelet- component which is already included in (a). We may minskii [19], and later by Freedman and McLerran [20], to compute the ground state energy of the relativistic arXiv:1904.09174v2 [astro-ph.HE] 24 Sep 2019 mention here that the Thomas-Fermi model is a non- relativistic model and relativistic corrections to it have electron gas that includes corrections due to the fine- been considered for the white dwarfs in Ref. [11{13]. structure constant. However, these treatments are insuf- The special relativity plays a key role in the white ficient to describe the degenerate matter in the white dwarf physics. In particular, the existence of the Chan- dwarfs as they do not describe the dominant interac- drasekhar upper mass limit for the white dwarfs arises es- tion, as seen in non-relativistic computations, between sentially due to the special relativity which demands that the degenerate electrons and positively charged heavier the physical results should be invariant under the Lorentz nuclei. Therefore, in this article to describe the degen- erate matter in the white dwarfs using the framework of finite temperature quantum field theory, we consider an additional interaction between the electrons and pos- ∗Electronic address: [email protected] itively charged nuclei, described by a Lorentz invariant yElectronic address: [email protected] action, along with the quantum electrodynamics. 2 II. WHITE DWARF STARS is governed by the Maxwell action Z Z In order to specify the associated scales, let us consider 4 4 1 µν SA = d xLA = d x − Fµν F ; (6) the white dwarf star Sirius B which has observational 4 mass M = 1:0 M , radius R = 0:008 R and the effec- tive temperature T = 25922 K [21]. Therefore, its mass where the field strength Fµν = @µAν − @ν Aµ. density is ρ ≈ 2:8 × 106 g=cm3. In natural units that we follow here (i.e. speed of light c and Planck constant ~ are set to unity), a fully degenerate core implies that the elec- C. Field interactions 15 3 tron density is ne ≈ 6:4 × 10 (eV) . The corresponding 2 1=3 5 Fermi momentum is kF ' (3π ne) ≈ 5:7 × 10 eV. The degenerate electrons within a white dwarf ex- The associated temperature scale of the white dwarfs perience two kinds of interactions, namely the self- −1 β ≡ kBT = 2:2 eV then leads to a dimensionless pa- interaction between the electrons and the interaction be- rameter tween electrons and the positively charge nuclei. The 5 self-interaction between the electrons are mediated by βkF ≈ 2:6 × 10 ; (1) gauge fields Aµ and governed by the interaction term of the quantum electrodynamics which plays an important role in characterizing the white dwarf star. Z Z − 4 − 4 ¯ µ SI = d xLI = d x [e γ Aµ] ; (7) A. Fermionic matter field where the parameter e is the dimensionless coupling con- stant. In order to compute the equation of state of the de- We may recall that the conserved 4-current corre- generate matter, we consider the spacetime within a sponding to the action (2) is given by jµ = ¯γµ which white dwarf star to be described by the Minkowski metric represents the contribution of the electrons. Similarly, ηµν = diag(−1; 1; 1; 1) i.e. we ignore the corrections from we may consider a background 4-current, say J µ, to rep- the general relativity (as also done by Salpeter [6]). The resent the contributions from the nuclei. Therefore, we degenerate electrons are fermionic degrees of freedom and may model the attractive interaction between the elec- are represented by the Dirac spinor field along with the trons and the positively charged nuclei by a Lorentz in- action variant action containing the current-current interaction Z Z 4 4 p µ Z Z S = d xL = − d x −η [iγ @µ + m] ; (2) + 4 + 4 2 2 ¯ µ SI = d xLI = d x [−Ze d Jµ γ ] ; (8) where η = det(η ) = −1. The Dirac matrices γµ satisfy µν 2 the anti-commutation relation where the coupling constant contains the term −Ze which signifies the strength of the attractive interactions µ ν µν fγ ; γ g = −2η I : (3) between an electron and a positively charged nucleus with atomic number Z. The parameter d is introduced The minus sign in front of ηµν in the Eq. (3) is chosen in order to make the action (8) dimensionless. It has the such that for given metric signature, the Dirac matrices dimension of length and and it represents an effective satisfy the usual relations (γ0)2 = I and (γk)2 = −I for length scale associated with the current-current interac- k = 1; 2; 3. In Dirac representation, these matrices can tion. be expressed as Therefore, the total action that describes the dynamics of the degenerate electrons within a white dwarf can be 0 0 σk γ0 = I ; γk = ; (4) written as 0 −I −σk 0 + S = S + SA + SI = SQED + SI ; (9) where Pauli matrices σk are given by − + where SI = S + S . The inclusion of the additional 0 1 0 −i 1 0 I I σ1 = ; σ2 = ; σ3 = : (5) interaction term (8) preserves the symmetry of the ac- 1 0 i 0 0 −1 tion of quantum electrodynamics SQED. In other words, apart from being Lorentz invariant, the total action (9) is also invariant under local U(1) gauge transformations B. Gauge field 1 iα(x) Aµ ! Aµ − e @µα(x) and (x) ! e (x) with α(x) being an arbitrary function. Given the coupling constant The electromagnetic interaction between the fermions e is small, we can study the interacting theory by using are mediated by the gauge fields Aµ whose free dynamics perturbative techniques. 3 III. PARTITION FUNCTION It is convenient to express the partition function using the Matsubara frequencies and wave-vector by transforming In a spherically symmetric star, the pressure and the the field in Fourier domain as mass density both vary along the radial direction.