Revisiting Equation of State for White Dwarfs Within Finite Temperature Quantum Field Theory

Revisiting equation of state for white dwarfs within ﬁnite temperature quantum ﬁeld theory

Golam Mortuza Hossain∗ and Susobhan Mandal† 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 relativity 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 relativistic 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 includes 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 degenerate 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 †Electronic 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 effective 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 ψ[e γ Aµ]ψ , (7)

A. Fermionic matter field where the parameter e is the dimensionless coupling constant. 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 = ψγ¯ µψ 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 ψ 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 √ µ Z Z Sψ = d xLψ = − d x −η ψ[iγ ∂µ + m]ψ , (2) + 4 + 4 2 2 ¯ µ SI = d xLI = d x [−Ze d Jµψγ ψ] , (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 {γ , γ } = −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 = Sψ + 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 ψ(x) → e ψ(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. On 1 X the other hand, in order to apply the techniques of finite ψ(τ, x) = √ ei(ωnτ+k·x)ψ˜(n, k) , (15) V temperature quantum field theory we need to consider a n,k spatial region which is in thermal equilibrium at a given temperature T . Within such a region thermodynamical where V denotes the spatial volume of the box. The Eq. quantities such as the pressure and the density are uni- (14) then implies that the Matsubara frequencies are form. Therefore, in order to deal with both these aspects ω = (2n + 1)π β−1 , (16) we consider here a finite spatial box at a given radial co- n ordinate. The box is assumed to be sufficiently small so where n is an integer. The Eqs. (12) and (15) then lead that the pressure and the density remain uniform within to the box yet it is sufficiently large to contain enough degrees of freedom to achieve required thermodynamical β X ˜¯ ˜ Sψ = ψ β p/ − m ψ , (17) equilibrium. The corresponding partition function, de- n,k scribing the degrees of freedom within the box, can be µ expressed as where pµ = (p0, ~p) = (−iωn + µ, k) and p/ = γ pµ. The h ˆ ˆ i spinor fields ψ and ψ¯ satisfy the same algebra as the Z = T r e−β(H−µQ) , (10) Grassmann variables. Using Dirac representation of the gamma matrices and the result of Gaussian integral over where β = 1/kBT with kB being the Boltzmann constant, µ refers to chemical potential and Q is the con- Grassmann variables we get served charge of the system. The Hamiltonian operator βV 48µk Hˆ represents the matter fields described by the action 3 2¯2 F ln Zψ = 2 2µkF − 3m kF + 2 , (18) (9). The trace operation is carried out over the degrees 24π β of freedom contained within the specified spatial region. where µ + k k¯2 ≡ µk − m2 ln F . (19) A. Partition function for free fermions F F m

The action (2) of free spinor field is invariant under In the Eq. (18), we have also ignored higher order tem- a global U(1) gauge transformations ψ(x) → eiαψ(x) perature corrections which are at least O((βµ)−2). where α is an arbitrary constant. Consequently, there exists an associated conserved current jµ = ψγ¯ µψ such µ that ∂µj = 0. Then the corresponding conserved charge B. Partition function for photons can be expressed as Q = R d3x j0(x) = R d3x ψγ¯ 0ψ. 0 Additionally, the conjugate field momentum correspond- Due to the gauge symmetry Aµ(x) and Aµ(x) = ing to the spinor field ψ(x) can be expressed as π(x) = 1 Aµ(x) − e ∂µα(x) represent the same physical configu- ∂Lψ = −iψγ¯ 0 = −iψ†. Therefore, the partition func- ∂(∂0ψ) ration. Therefore, in order to avoid over-counting in tion which contains contributions only from free spinor evaluating the partition function using functional inte- field can be expressed using path integral method [22] as gral methods, it is convenient to introduce the Faddeev- Popov ghost fields C and C¯ [23, 24]. These Grassmann- Z β ¯ −Sψ Zψ = DψDψ e , (11) valued fields effectively cancel the contributions from two gauge degrees of freedom. Therefore, the thermal parti- where tion function containing contributions from the physical Z β Z photons can be expressed as Sβ = dτ d3x LE + µψ¯(τ, x)γ0ψ(τ, x) . (12) ψ ψ Z 0 −Sβ −Sβ Z = DA e A DC¯ DC e C ≡ Z 0 Z , E A µ A C The Euclidean Lagrangian density Lψ is obtained by sub- stituting t → iτ in Lagrangian density Lψ and can be (20) β R β R 3 1 µν 1 µ 2 expressed as where SA = 0 dτ d x [− 4 Fµν F − 2ξ (∂µA ) ]|t=iτ E ¯ 0 k β Lψ = −ψ(τ, x)[γ ∂τ + iγ ∂k + m]ψ(τ, x) , (13) with gauge-fixing parameter ξ and SC = R β dτ R d3x [∂µC∂¯ C] . Unlike the spinor field, where k = 1, 2, 3. In the functional integral (11), the 0 µ |t=iτ spinor field is subject to anti-periodic boundary condi- both Aµ(x) and C(x) fields are subject to the periodic tions given by boundary conditions ¯ ¯ ψ(τ, x) = −ψ(τ +β, x); ψ(τ, x) = −ψ(τ +β, x) . (14) Aµ(τ, x) = Aµ(τ + β, x); C(τ, x) = C(τ + β, x) . (21) 4

β β β β R β R 3 − As earlier, we evaluate the partition function in Fourier where SI = S− + S+ with S− = 0 dτ d x [LI ]|t=iτ domain by transforming the ﬁeld as β R β R 3 + and S+ = 0 dτ d x [LI ]|t=iτ . Using perturbative r method, the total partition function (29) can be ex- β X A (τ, x) = ei(ωnτ+k·x)A˜ (n, k) . (22) pressed as µ V µ n,k ln Z = ln Zψ + ln ZA + ln ZI , (30) The deﬁnition (22) ensures that the Fourier modes A˜µ(n, k) are dimensionless and the Eq. (21) implies where the contribution due to the interactions is −1 ωn = 2nπβ with n being an integer. By choosing Feyn- ∞ ! man gauge ξ = 1 and dropping the boundary terms, we X 1 β l β ln ZI = ln 1 + h(−SI ) i . (31) can express SA as l! l=1 β X ¯ 1 S = A˜µ − β2(ω2 + k2) A˜ . (23) Including only the leading order terms we can express A 2 n µ n,k the Eq. (31) as

Using the identity for Riemann integrals 1 β 2 β 3 R −xiDij xj N/2 −1/2 ln Z = h(S ) i − h(S )i + O(e ) , (32) x1 . . . xN e = π (det(D)) , we can I 2 − + evaluate the contributions from the gauge field by where the symbol h.i denotes the ensemble average. X 2 2 2 ln ZA0 = −2 ln β (ωn + k )} (24) n,k where the gauge field is Wick rotated as Ak → iAk for 1. Finite-temperature propagators k = 1, 2, 3 to make the integral convergent. Similarly, one can define the Fourier modes of the ghost field as In order to compute ln ZI , one needs the finite- r temperature propagators for the spinor field and the β X C(τ, x) = ei(ωnτ+k·x)C˜(n, k) , (25) Maxwell’s field. In particular, the finite-temperature V propagator for spinor field in real space is defined as n,k

0 where the modes C˜(n, k) are again dimensionless and G (∆τ, ∆x) = hψ(τ1, x1)ψ(τ2, x2)i , (33) −1 ωn = 2nπβ with n being an integer. By dropping the boundary terms, we can the express Sβ as where ∆τ = τ1 − τ2, ∆x = x1 − x2. The corresponding C propagator in Fourier space is defined as X Sβ = C˜¯ β2(ω2 + k2) C.˜ (26) C n Z β Z n,k 3 −i(ωnτ+k·x) 0 G(ωn, k) = dτ d x e G (τ, x) . (34) 0 The ghost fields C and C¯ being Grassmann-valued field, we can use the same identity as used for fermions in order Using the Eq. (17), the propagator for the free spinor to evaluate their contributions as field in Fourier space can be obtained as

X 2 2 2 ln ZC = ln β (ωn + k )} . (27) 1 p/ + m G(ωn, k) = = − , (35) n,k p/ − m p2 + m2

By combining the contributions (24) and (27) one can µ write the partition function for the physical photons as where pµ = (p0, ~p) = (−iωn + µ, k), p/ = γ pµ and 2 µν p = η pµpν . Similarly, the finite-temperature prop- V π2 agator for Maxwell’s field in real space is defined as ln Z = . (28) A 45β3 Dµν (∆τ, ∆x) = hAµ(τ1, x1)Aν (τ2, x2)i. Following the Eq. (34), the propagator for free Maxwell fields in Fourier space can be obtained using the Eq. (23) as C. Contributions from the interactions 0 −ηµν Dµν (ωn, k) = 2 2 . (36) Including both kinds of interaction for the degenerate ωn + k electrons, we can express total partition function as We may emphasize here that in the Eq. (35), the Mat- Z −1 β β β β subara frequencies are ωn = (2n+1)πβ whereas in the ¯ −(Sψ +SA+SC +SI ) Z = DψDψDAµDC¯ DC e , (29) −1 Eq. (36), they are ωn = 2nπβ with n being an integer. 5

2. Electron-electron interaction In order to carry out the summation over Matsubara frequencies, we have used the identity Using the Eq. (7), we can express leading order con- ∞ tributions due to the self-interaction of the electrons as X z coth z = . (47) Z β Z n2π2 + z2 β 2 2 3 3 n=−∞ h(S−) i = −e dτ1dτ2 d x1d x2 Dµν (∆τ, ∆x) 0 × Tr γµG0(∆τ, ∆x)γν G0(−∆τ, −∆x) , (37) The summation over k is carried out by converting it to an integral as earlier. Subsequently, by using the Rie- where the trace is carried over the Dirac indices. Here R ∞ t mann zeta function identity ζ(2) = 0 dt t/(e − 1) = we have dropped the divergent diagrams that arise from π2/6 and dropping the divergent part, we have expressed the usage of the Wick’s theorem. The Eq. (37), can be the ﬁnite part of I1. In order to evaluate I0 we can ex- expressed in terms of the propagators in Fourier space as press it as 2 β 2 e X h(S−) i = − Dµν (∆ωn, ∆k) βV X 1 1 1 + − n1,n2,k1,k2 I0 = + ≡ I0 + I0 , (48) 2ω p0 + ω −p0 + ω µ ν n,k × Tr [γ G(ωn1 , k1)γ G(ωn2 , k2)] , (38) where ∆ωn = ωn1 − ωn2 and ∆k = k1 − k2. Using the −1 Eqs. (35, 36) one can simplify the Eq. (38) as where p0 = −i(2n + 1)π β + µ and 2 2 4e 4m + 2p1 · p2 β 2 X ± X β z± h(S−) i = − 2 2 2 2 2 , I = , (49) βV (p + m )(p1 − p2) (p + m ) 0 2 2 2 n1,n2 1 2 4ω n π + (z±) n,k k1,k2 (39) where p = (−iω + µ, k ) and p = (−iω + µ, k ). 1 1 n1 1 2 n2 2 with z± = 2 {β(ω ± µ) ∓ iπ}. By using the identity (47), Here we have used the trace identities for the Dirac ma- the summation over n can be carried out as trices Tr(γµγν ) = −4ηµν , Tr(γµγν γργσ) = 4(ηµν ηρσ − µρ νσ µσ νρ η η + η η ) and the fact that (p1 − p2)0 = i∆ωn. X β 1 1 I± = − . (50) The Eq. (39) can be written in four parts as 0 2ω 2 eβ(ω±µ) + 1 k 2 4e2 h(Sβ ) i = − (S + S + S + S ) , (40) − βV 1 2 3 4 The anti-particle contributions are contained in the term + where I0 . So by ignoring the anti-particle contributions, the divergent zero-point energy and by using the approxima- X 1 ¯2 2 tion βµ 1, I0 can be evaluated as I0 = −βV k /8π . S1 = 2 2 2 , (41) F (p1 + m )(p1 − p2) n1,n2,k1,k2 Therefore, the ensemble average becomes X 1 S2 = , (42) 2¯2 ¯2 2 2 2 β 2 βV e k k 1 (p2 + m )(p1 − p2) h(S ) i = F F + . (51) n1,n2,k1,k2 − 4π2 4π2 3β2 X −1 S3 = 2 2 2 2 , (43) (p1 + m )(p2 + m ) n1,n2,k1,k2 We note that the Eq. (51) diﬀers from an analogous ex- 2 pression, describing the contributions from the electron- X 2m S4 = 2 2 2 2 2 .(44) electron interaction, given in the textbook by Kapusta (p1 + m )(p1 − p2) (p2 + m ) n1,n2,k1,k2 and Gale (Eq. 5.59) [22]. However, the expression in the textbook is erroneous as it implies that electromag- It can be shown that the term S4 is infrared divergent and hence ignored. Further, using the symmetries of the netic repulsion between the electrons causes a reduction of pressure for a system of degenerate electrons in the expressions, we may note that S1 = S2 = I0 I1, S3 = −(I )2 where ultra-relativistic regime. In particular, if one ignores 0 temperature corrections, in the ultra-relativistic limit 1 1 4 X X (kF m) rhs of the Eq. (51) varies as kF whereas the I0 = 2 2 ,I1 = 2 . (45) 4 p + m (p − p2) textbook expression varies as −k . On the other hand, n,k n,k F in non-relativistic limit (kF m), the Eq. (51) varies 6 2 4 Despite the appearance of p2 in its expression, the eval- as kF /m whereas the textbook expression varies as kF . uated I1 does not depend on p2 and is given by We note that the textbook expression which describes 2 pressure corrections due to repulsive electron-electron in- X β /4 X β 1 1 V I1 = = ( + β|k| ) = . teraction, changes sign as one goes from relativistic to 2 2 βk 2 |k| 2 e −1 12β n,k n π + ( 2 ) k non-relativistic regime. This aspect itself signals internal (46) inconsistency of the expression given in the textbook. 6

3. Electron-nuclei interaction where we have ignored the finite temperature corrections within the parenthesis as the coupling constant e and the −1 The leading order contribution due to the electron- term (βkF ) both are small. nuclei interaction can be expressed as Z β Z β 2 2 3 µ IV. EQUATION OF STATE hS+i = −Ze d dτ d xJµ(τ, x)hψ(τ, x)γ ψ(τ, x)i . 0 (52) Using the evaluated partition function we can compute In order to evaluate the integral (52), it is convenient to the pressure and the mass density within the considered express it in the Fourier domain as box located at the given radial coordinate. Subsequently, we may read off the corresponding equation of state of β 2 2 ˜ X µ hS+i = −Ze d Jµ(β) Tr [γ G(ωn, k)] , (53) the degenerate matter at the given radial location. For n,k later convenience, we now define following dimensionless where the average background 4-current density parameters ¯ Z β Z m µ kF kBT ˜ 1 3 σ ≡ , σµ ≡ , σk ≡ , σT ≡ . (60) Jµ(β) = dτ d x Jµ(τ, x) . (54) kF kF kF kF βV 0 −1 Within the given box, the spatial motion of the heav- We note that σ , as defined here, can be identified with ier nuclei can be neglected. So we may assume that the so called ‘relativity parameter’√ in the literature [6]. 2 2 2 the average background 3-current density J˜k(β) = 0 for We also note that σµ = 1 + σ , σk = σµ − σ log((1 + −1 k = 1, 2, 3. By identifying the average background charge σµ)/σ) and σT = (βkF ) . For typical white dwarfs, the 0 Eq. (1) implies σ 1. For a system of ultra-relativistic density n+ = J˜ (β) = −J˜0(β) and by using the trace T identity of Dirac matrices, we can express the Eq. (53) degenerate electrons σ 1 which leads to σµ ' 1 and as σk ' 1. X p0 hSβ i = −4Ze2d2n I ,I = . (55) + + 2 2 p2 + m2 n,k A. Mass density Similar to the Eq. (48), I can be expressed as 2 The number density of the electrons can be com- X 1 1 1 − + puted from total partition function as ne ≡ hNi/V = I2 = − ≡ I2 − I2 , (56) (βV )−1(∂ ln Z/∂µ). Given the partition function due to 2 −p0 + ω p0 + ω n,k the interaction terms (59) itself depends on the electron and the summation over the Matsubara frequencies can number density, it leads to an algebraic equation for ne be carried out as as given below 3 2 2 2 X β 1 1 k 2 + σ 3α 8π d neσµ I± = − . (57) n = F 1 + + σ2 − . 2 β(ω±µ) e 2 2 −1 k 2 2 e + 1 3π (6σT ) 2π kF k (61) + As earlier, by ignoring the anti-particle contributions I2 , In order to arrive at the Eq. (61), we have used two very ¯2 the divergent zero-point energy and by using the approx- useful relations (∂kF /∂µ) = (µ/kF ) and (∂kF /∂µ) = imation βµ 1, finite part of I2 can be expressed as 2kF . The Eq. (61) can be solved in a straightforward 3 2 β manner to result I2 = −βV kF /12π . The ensemble average hS+i then becomes 3 " 2 2 2 # kF 1 + 6σT (2 + σ ) + (3α/2π)σk 2 2 3 ne = , 1 (62) β βV Ze d k n+ 2 2 −1 hS i = F . (58) 3π 1 + d+(α/π)σµ + 3π2 where the fine structure constant is α = e2/4π in natural units. By using the chemical potential µ which 4. Total contributions from the interactions comes naturally in the partition function (10), we have defined a dimensionless parameter d+ ≡ 2dµ. The pa- The number density of positively charged nuclei must rameter d+ characterizes the associated length scale with satisfy Zn+ = ne as system is overall electrically neutral. the electron-nuclei interaction and it needs to be fixed by Therefore, by combining the contributions from the self- separate consideration (see FIG. 1). Now the mass den- interaction of the electrons (51) and the electron-nuclei sity of the system is given by interaction (58), we can express the partition function (32) due to the total interaction as ρ = µemune , (63)

βV e2 where mu is the atomic mass unit. The parameter µe ≡ ln Z = 3k¯4 − 32π2d2n k3 , (59) I 96π4 F e F (A/Z) is deﬁned so that µemu speciﬁes ‘the average mass 7

1.01 1 ) d = 0.1 µ-1 2 ) d = 0.5 µ-1 2 /3 π 0.8 3 -1 F d = 1.5 µ /12 π

1 4 F 0.6

0.4 0.99 d = 0.1 µ-1 µ-1 0.2 d = 0.5 Pressure ( in k -1 Number density ( in k d = 1.5 µ 0.98 0 0 2 4 6 0 2 4 6

m/kF m/kF

FIG. 1: The number density of electrons at zero temperature FIG. 2: The degeneracy pressure at zero temperature for dif- for diﬀerent values of length scale d. ferent values of length scale d. per electron’ where A is the atomic mass number. For a reduces to 4 white dwarf with pure Helium He core µe is 2. 3 2 2 2 2 kF 6m kBT α(1 − d+)kF ne ' 2 1 + 4 + , (65) 3π kF πm

B. Pressure and the pressure (64) reduces to

k5 30m2k2 T 2 2α(1 − 2d2 )k Using the expression of pressure for a grand canoni- F B + F P ' 2 1 + 4 + . (66) cal ensemble, we may read off the pressure due to the 15π m kF 3πm degenerate electrons as P = (βV )−1 ln Z and due to ψ ψ If one disregards the corrections due to the finite tem- the interactions as P = (βV )−1 ln Z . One may check I I perature and the fine-structure constant, the Eqs. (65, that the radiation pressure P = 1 π2β−4 is insignifi- A 45 66) represent the standard non-relativistic expressions. cant even compared to P , as for white dwarfs βk 1. I F However, one may note that in the non-relativistic regime Therefore, by ignoring the radiation component, we can the effects of finite temperature become important. Nev- express the total pressure which includes leading order ertheless, these equations are valid in non-relativistic corrections due to the finite temperature and the fine- regime as long as corresponding chemical potential µ sat- structure constant, as isfies βµ 1. 4 2 2 2 2 2 kF 1 + 24σT 3σ σk 3α 4 8π d+ne P = 2 −1 − + (σk − 3 2 ) . 12π σµ 2 2π 3kF σµ D. Temperature corrections (64) We may again note that the degeneracy pressure depends For non-interacting, zero temperature degenerate elec- on the parameter d+ which characterizes the electron- tron gas, the number density of electrons is given by nuclei interaction length scale (see FIG. 2). Using the 3 2 ne = (kF /3π ). However, the effect of finite temper- Eqs. (62, 63, 64), in principle, one can express the ature causes this relation to be modified even for non- equation of state for the degenerate matter within white interacting electrons as dwarfs as P = P (ρ) which includes the corrections due to the fine structure constant α and the finite temperature. 2 2 2 ne(T ) 6(2 + σ )kBT = 1 + 2 . (67) ne(T = 0) kF Analogously, the effect of finite temperature on the pres- C. Non-relativistic limit sure of non-interacting degenerate electron gas can be expressed as We have the considered matter field actions to be man- ifestly Lorentz invariant here. Consequently the studied P (T ) 48σ k2 T 2 = 1 + µ B . (68) equation of state is well suited for describing the relativis- 2 2 2 P (T = 0) k (2σµ − 3σ σ ) tic regime. However, for the consistency, the equation of F k state must also have correct non-relativistic limit when Clearly, the finite temperature causes the pressure to in- 1 −1 1 −3 kF m. In such limit σ 1, σµ = σ + 2 σ − 8 σ + crease for a given Fermi momentum kF . However, the −5 2 2 −1 1 −3 −5 O(σ ) and σk = 3 σ − 5 σ + O(σ ). Therefore, increase in pressure is very small given it is of the order −2 −10 in the non-relativistic regime, the number density (62) ∼ (βkF ) ∼ 10 for typical white dwarfs (1). 8

E. Fine-structure constant corrections 1 No interaction: α=0 ) Relativistic QFT (d=1.18 µ-1) 2 0.8 The effects of the electromagnetic interaction i.e. the Salpeter (Z=6) /12 π 4 Coulomb effects on the equation of state are expressed F 0.6 using the fine-structure constant α ' 1/137 which is a small number. However, theses corrections are much 0.4 larger compared to the temperature corrections. At the 0.2 zero temperature, the leading order effect of the fine- Pressure ( in k structure constant on the electron number density can 0 be expressed as 0.1 1 10 m/kF ne(α) α 2 2 −1 = 1 + 3σk − 2d+σµ . (69) ne(α = 0) 2π FIG. 3: A comparison of the pressure in a broadly relativistic Similarly, at the zero temperature the leading order ef- domain. fect of the fine-structure constant on the pressure can be expressed as there is a modification to the expression of the electron P (α) α (9σ4 − 8d2 σ−2) number density due to the electromagnetic interactions. = 1 + k + µ . (70) 2 2 In turns, it would imply a difference in equation of state P (α = 0) 3π (2σµ − 3σ σk) even if the pressure expressions considered by Salpeter We note that the number density and the pressure both and here, were to agree. contain an undetermined dimensionless parameter d+ = 2dµ in the corrections involving the fine-structure constant. As mentioned earlier, the length scale d is asso- 1. No interaction ciated with the current-current interaction between the electrons and the nuclei. In the partition function, a nat- The expressions of the number density (62) and the ural length scale is provided by the chemical potential µ. pressure (64) agree exactly with the Salpeter’s expres- Therefore, intuitively one would expect that the dimen- sions when one ignores the fine-structure constant cor- sionless parameter d+ to be an O(1) number for the white rections by setting α = 0 and identifies σ−1 as the ‘rela- dwarfs. However, determination of its exact numerical tivity parameter’ x√along with the mathematical identity values can only be done by using separate considerations, sinh−1 x = ln(x + 1 + x2). possibly by using observations. In the standard literature, this one-parameter uncertainty is often overlooked as usually there one fixes the lattice scale associated with 2. Relativistic domain positively charged nuclei by heuristic arguments. How- ever, we have argued that this length scale is associated with the electron-nuclei interaction and its independent In the ultra-relativistic limit, kF m, we can express determination in principle can allow one to understand the total pressure which includes leading order correc- the property of the underlying lattice structure formed tions due to the fine-structure constant, as by the nuclei within the degenerate matter of the white k4 3 4d2 dwarfs. P = F 1 + α − + . (71) 12π2 2π 3π

F. Comparison with Salpeter’s corrections On the other hand, analogous expression for pressure with leading order corrections considered by Salpeter can 4 kF 1 6 4 1/3 2/3 In order to compare the number density (62) and the be expressed as P = 12π2 1 + α( 2π − 5 ( 9π ) Z ) 2 3 81π2 1/3 2/3 pressure (64) with that of Salpeter’s we need to set [6]. Therefore, if one chooses d+ = 4 + ( 250 ) Z σT = 0 as these are studied at zero temperature by then one would get the same pressure corrections in the Salpeter [6]. Further, for comparison we consider the ultra-relativistic limit. In particular, if one chooses the terms up to leading order in ﬁne structure constant α atomic number Z = 2 (Helium) or Z = 6 (Carbon) from the combined expressions of non-interacting de- then the Salpeter’s corrections would correspond to the −1 −1 generacy pressure P0, classical Coulomb corrections PC , length scale d being 0.88 µ , 1.18 µ respectively. This Thomas-Fermi corrections PTF , exchange corrections Pex is in agreement with the intuitive expectation that d+ and correlation corrections Pcor as described in [6]. should be an O(1) number. The pressure comparison in In the treatment by Salpeter, the relation between the a broadly relativistic domain is given in the FIG. 3. number density of electrons ne and the Fermi momentum Nevertheless, we emphasize that the length scale d is kF is assumed to be ﬁxed. On the other hand, the usage undetermined apriori in the approach that we have con- of grand canonical partition function here implies that sidered here. For a given system of degenerate electrons 9

0.1 the electron-nuclei interaction, does not aﬀect the mass No interaction: α=0

) limit for the white dwarfs. In contrast, the modiﬁed 0.08 Relativistic QFT (d=1.18 µ-1) 2 Salpeter (Z=6) Chandrasekhar mass limit which uses the Salpeter’s cor- /12 π

4 0.06

F rections, can be expressed up to leading order in α, as 0 3 6 4 1/3 2/3 1 0.04 Mch/Mch = 1 − 2 α 5 ( 9π ) Z − 2π [6–8]. So the reduction of the Chandrasekhar mass limit there is non- 0.02 universal as it depends on the atomic number of the con-

Pressure ( in k 0 stituent nuclei.

-0.02 10 100

m/kF V. DISCUSSION

FIG. 4: A comparison of the pressure in a broadly non- In the literature, the effects of the electromagnetic in- relativistic domain. The pressure becoming negative signals teraction on the equation of state of the degenerate mat- the breakdown of the underlying assumptions in Salpeter’s ter within the white dwarfs are computed by consider- treatment. ing the so-called classical Coulomb energy, the Thomas- Fermi effect, the exchange and correlation energy at zero temperature. These computations rely on the electro- and nuclei, in principle, it may be possible to derive an static considerations which are non-relativistic ab ini- effective action corresponding to Eq. (8) where ∼ d+e tio. In this article, after reviewing the existing litera- may be viewed as the renormalized coupling constant be- ture, we have presented a computation of the equation tween the electrons and the nuclei at the energy scale set of state of degenerate matter for the white dwarfs by by the chemical potential µ. employing the techniques of finite temperature relativistic quantum field theory. The corresponding equation of state includes the leading order effect due to the fine- 3. Non-relativistic domain structure constant and the effect of finite temperature. The correction to the equation of state due to the fine- The corrections to the pressure expression, considered structure constant has two components. The first compo- by Salpeter, are known to become unreliable in a fairly nent arises from the self-interaction between the degen- non-relativistic domain. Salpeter noted that with such erate electrons and described by the action of quantum corrections the total pressure could become negative, sig- electrodynamics. For the second component we have con- naling the breakdown of the underlying assumptions [6]. sidered a Lorentz invariant interaction term to describe In contrast, the non-relativistic expression (66) here re- the interaction between electrons and positively charged mains well defined. A comparison of the pressure in a nuclei. Further, we have argued that a fully relativis- broadly non-relativistic domain is given in the FIG. 4. tic consideration leads to an apriori undetermined length scale in the corrections to the equation of state involving the electron-nuclei interaction. This aspect of the G. Modified mass limit of white dwarfs equation state is overlooked in the literature. Instead there one fixes the associated scale by using heuristic ar- In order to understand the effect of the fine-structure guments. An independent determination of this length constant on the Chandrasekhar mass limit for white scale may shed light on the underlying lattice structure dwarfs, we need to find the corrections to the equation of formed by the nuclei within the degenerate matter of the state in the ultra-relativistic limit. In such limit, the Eqs. white dwarfs. Besides, the effect of fine-structure con- (62, 63, 64) together lead to a polytropic equation of state stant reduces the Chandrasekhar mass limit of the white of the form P ∝ ρ4/3 where fine-structure constant mod- dwarfs by a universal factor which is independent of the ifies the proportionality constant. Such a modification atomic number of the constituent nuclei and the electron- in turn leads to the modified Chandrasekhar mass limit nuclei interaction length scale. Mch which including up to the leading order correction In order to describe the background geometry within in α, is given by [25] the white dwarfs, here we have used the Minkowski spacetime. In other words, we have ignored the effects of gen- M 3α eral relativity as one expects the effect of gravity on the ch = 1 − , (72) 0 equation of state to be smaller than the effect of the fine- Mch 4π structure constant for the observed white dwarfs. How- 0 where Mch is the Chandrasekhar mass limit without fine- ever, it would be interesting in its own right to consider structure constant corrections. We note that the effect of the effects of general relativity within the considered ap- the fine-structure constant α reduces the Chandrasekhar proach as done for other approaches [13, 26] or even for mass limit for white dwarfs by a universal factor [25]. different applications [27]. In particular, the length scale d which is associated with Finally, we note that the future detection of low- 10 frequency gravitational waves from the extreme mass- that of Salpeter’s equation of state for white dwarf is rela- ratio merger of a black hole and a white dwarf could tively larger in the non-relativistic regime, as can be seen determine the equation of state of the degenerate matter in the FIG. 4. Clearly, such deviations can be confronted within the white dwarf with an accuracy reaching up to with the expected accuracy of the future low-frequency 0.1% [17]. It is based on the expectation that the tidal gravitational wave detectors. The possible effects of the disruption of a white dwarf, during the final phase of modified equation of state, as studied here, on the mass- inspiral around a massive black hole, could be measured radius relation of the white dwarfs is being investigated. very accurately through low-frequency gravitational wave Besides, the corrections to the equation of state as stud- signals. The properties of the tidal disruption of a white ied here may be relevant also for neutron star physics dwarf would necessarily depend on the equation of state where neutron star core is surrounded by an envelope of of the degenerate matter present within the white dwarf, degenerate electrons and ions. rather than its maximal mass limit. On the other hand, we may note from the Eqs. (69, 70) that the corrections to the number density and the pressure due to the fine-structure constant are of the order ∼ α/π ∼ 0.2%. Acknowledgments Therefore, the effects of the fine-structure constant as studied here could well be within the detection realm of the future gravitational wave detectors. We may mention SM would like to thank CSIR, India for supporting this here that the deviation of equation of state compared to work through a doctoral fellowship.

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