Gradient expansion technique for inhomogeneous, magnetized quark matter

Abstract

A quark-magnetic Ginzburg–Landau (qHGL) gradient expansion of the free energy of two-flavor inhomogeneous quark matter in a magnetic field H is derived analytically. It can be applied away from the Lifshitz point, generalizing standard Ginzburg-Landau techniques. The thermodynamic potential is written as a sum of the thermal contribution, the non-thermal lowest Landau level contribution, and the non-thermal qHGL functional, which handles any arbitrary position-dependent periodic modulation of the chiral condensate as an input. The qHGL approximation has two main practical features: (1) it is fast to compute; (2) it applies to non-plane-wave modulations such as solitons even when the amplitude of the condensate and its gradients are large (unlike standard Ginzburg-Landau techniques). It agrees with the output of numerical techniques based on standard regularization schemes and reduces to known results at zero temperature ($T = 0$) in benchmark studies. It is found that the region of the $\mu$-T plane (where $\mu$ is the chemical potential) occupied by the inhomogeneous phase expands, as H increases and T decreases.

Appendices

Appendix A: Neutron star applications

In this appendix, we discuss briefly extensions of our calculations to typical neutron star conditions. For simplicity, we focus on the CDW modulation. In Appendix A.1 we consider $\beta$-equilibrated quark matter. In Appendix A.2 we discuss nonzero quark bare masses ($m_q \ne 0$). The latter affect the condensate parameters and reduce the size of density region where inhomogeneous phases are thermodynamically favored [44, 45], i.e. the region where neutrino emission due to quark beta decay is allowed.

1.1 Appendix A.1: Quark matter in $\beta$-equilibrium

To apply our technique to neutron star matter, both $\beta$-equilibrium and electrical neutrality should be considered, strictly speaking [19, 43]. Additionally, one should consider a mixture of quarks and leptons [19]. Broadly speaking, however, our conclusions do not change qualitatively, when these features are included. The case of asymmetric, $\beta$-equilibrated quark matter with electrons has been considered already in other works (see [19, 21] for example). Considering a mixture of up and down quarks as well as electrons, $\beta$-equilibrium amounts to having two different chemical potentials for up and down quarks ($\mu _u$ and $\mu _d$ respectively), which are related by

$$\begin{aligned} \mu _u = \mu - \frac{2}{3}\mu _e, \qquad \mu _d = \mu + \frac{1}{3}\mu _e \end{aligned}$$

(A.1)

where $\mu$ is the quark number chemical potential and $\mu _e$ denotes the electron chemical potential, determined by the condition of electrical neutrality. In our case, this amounts to rewriting the total free energy as

$$\begin{aligned} {\mathcal {F}} = {\mathcal {F}}_e + {\mathcal {F}}_{\mathrm{{qHGL}}} \ , \end{aligned}$$

(A.2)

where ${\mathcal {F}}_e$ denotes the electron free energy and the qHGL approximation reads

$$\begin{aligned} {\mathcal {F}}_{\mathrm{{qHGL}}} = {\mathcal {F}}_{\mathrm{{qHGL}}}(\varDelta _u, K_u, \mu _u, H) + {\mathcal {F}}_{\mathrm{{qHGL}}}(\varDelta _d, K_d, \mu _d, H) \ , \nonumber \\ \end{aligned}$$

(A.3)

where $\varDelta _u, K_u$ denote the condensate parameters for up quarks, and $\varDelta _d, K_d$ the corresponding ones for down quarks. To calculate the condensate parameters for the up and down quarks, one has to minimize Eqs. (A.2) and (A.3) with the condition in Eq. (A.1), imposing that the system is electrically neutral.

1.2 Appendix A.2: Expansion away from the chiral limit

Direct Urca neutrino emission due to quark beta decay is prohibited by kinematic arguments in free quark matter, but can activate in inhomogeneous quark matter. Nonzero bare quark masses reduce the size of the density region where quark matter is inhomogeneous, and hence reduce the stellar volume where direct Urca emission due to quark beta decay is possible. For nonzero bare quark masses, Eq. (2) becomes

$$\begin{aligned} M(\mathbf{x} ) = m_q - 2 G_S \Big [\langle {{\bar{\psi }}} \psi \rangle (\mathbf{x} ) + \langle {{\bar{\psi }}} i \gamma ^5 \tau ^3 \psi \rangle (\mathbf{x} )\Big ] \ , \end{aligned}$$

(A.4)

where $m_q \ne 0$ denotes the bare quark mass. In the qHGL approximation, one needs to modify both the Taylor expansion to obtain the gradient terms and the homogeneous term in Eq.  (16). The homogeneous term in the qHGL approximation for nonzero $m_q$ can be written as [44]

$$\begin{aligned} {\mathcal {F}}_{\mathrm{{hom}}}(M^{\prime }, m_q) = {\mathcal {F}}^{\mathrm{{vac}}}_{\mathrm{{hom}}}(M^{\prime }) + {\mathcal {F}}^{\mathrm{{med}}}_{\mathrm{{hom}}}(M^{\prime }) + \frac{|M^{\prime } - m_q|^2}{4 G_S},\nonumber \\ \end{aligned}$$

(A.5)

with $M^{\prime } = m_q + \varDelta$. In the chiral limit, the Taylor expansion in Eq. (7) is performed by calculating the derivatives of the free energy for $\varDelta = 0$, i.e. close to the chiral restoration transition, with $M(\mathbf{x} ) = 0$. On the other hand, as studied in [45], the Taylor expansion cannot be performed around $M(\mathbf{x} ) = 0$ for $m_q \ne 0$, because chiral restoration is not reached due to the presence of the nonzero bare quark mass $m_q$. Accordingly, the authors in [45] find (via standard Ginzburg–Landau techniques) that the free energy density contains odd powers of M proportional to some coefficients $\gamma _{2j + 1}$ that do not appear for $m_q = 0$. In our case, if the Taylor expansion in Eq. (7) is not centered at $M (\mathbf{x} ) = 0$ the term $\partial {\mathcal {F}}/\partial (\varDelta ^2)$ cannot be calculated with the technique developed in this work, and the coefficients $\gamma _{2j + 1}$ proportional to odd powers of the condensate parameters cannot be obtained.

In summary, the extension of our results to the case of finite masses is interesting but nontrivial, and will be addressed in future work.

Appendix B: Expansion for small magnetic fields

The free energy of the higher Landau levels (Eq. (9)) is a function of $|e_f H|$. The expansion of Eq. (15) for small magnetic fields requires caution, since the absolute value is not differentiable at $e_f H = 0$. Hence, a mathematical prescription is necessary to ensure the validity of the Taylor expansion for small magnetic fields.

Fig. 4

Smoothing functions Eqs. (B.1) and (B.2) versus x for different various values of the steepness parameter k

We replace the absolute value |x| function with

$$\begin{aligned} |x| \mapsto f(x, k) = \frac{2}{k}\ln (1 + e^{k x}) - x - \frac{2}{k}\log (2) \end{aligned}$$

(B.1)

where the k parameter determines the steepness of the curve and the constant term ensures that $f(0, k) = 0$. The derivative is given by the logistic sigmoid function

$$\begin{aligned} \frac{d|x|}{dx} \mapsto f^{\prime }(x, k) = \frac{2}{1 + e^{-k x}} - 1 \end{aligned}$$

(B.2)

which is continuous in $x = 0$.

Figure 4 shows the plots of f(x, k) and $f^{\prime }(x, k)$ as functions of x for different values of k. For $k \approx 100$ both f and $f^{\prime }$ reproduce well the absolute value and the sign function of x respectively, with f differentiable in the origin. Hence, when expanding the quark free energy for small magnetic fields, one has to replace $|e_f H|$ with $f(e_f H, k)$, so that f(x, k) and $f^{\prime }(x, k)$ reproduce the absolute value of the magnetic field and its derivative, and the Taylor expansion centered in $e_f H = 0$ is mathematically consistent. Additionally, we notice that when performing the Taylor expansion of Eq. (15) (and similarly for the vacuum term), the terms proportional to odd powers of the magnetic field vanish, since they are proportional to $f^{\prime }(0, k) = 0$. This ensures that the qHGL functional is independent of the magnetic field sign and therefore preserves the rotational symmetry of the system [11, 18].

Fig. 5

Accuracy of the qHGL approximation: comparison of CDW $\varDelta$ and K obtained with the qHGL approximation and ${\mathcal {F}}_{\text {num}}$ for two typical magnetic field strengths ($eH = 1000$ $\hbox {MeV}^2$ and $eH = 2000\,\hbox {MeV}^2$). a $eH = 1000\,\hbox {MeV}^2$. b $eH = 2000\,\hbox {MeV}^2$. In both panels, we include corrections in the qHGL functional up to $\gamma _{10}$ (orange, dashed lines) or $\gamma _{16}$ (red solid lines)

Appendix C: Coefficients of the qHGL expansion

We report here the coefficients $\gamma _j$ introduced in Eq.  (16). Since $\gamma _j$ is obtained by expanding the free energy for small magnetic fields, it is given by the sum of the magnetic field-independent contribution plus corrections proportional to even powers of the magnetic field [18], as discussed in Appendix B. For $H = 0$, the formulas below for $\gamma _j$ reproduce the coefficients found in [31]. Including only corrections up to $(e_fH)^2$, the coefficients are given by

$$\begin{aligned} \gamma _4(\mu ,\varLambda , H) =&-\frac{N_c}{16\pi ^2}\sum _{f}\bigg [\log \Big (\frac{32\mu ^2}{3\varLambda ^2}\Big ) \nonumber \\&+ (2 e_f H)^2\Big ( \frac{85}{72 \varLambda ^4} + \frac{1}{16 \mu ^4}\Big )\bigg ] \end{aligned}$$

(C.1)

$$\begin{aligned} \gamma _6(\mu ,\varLambda , H) =&\frac{N_c}{32\pi ^2}\sum _{f}\bigg [\Big (\frac{11}{9\varLambda ^2}+\frac{1}{3\mu ^2}\Big ) \nonumber \\&+ (2 e_f H)^2\Big ( \frac{575}{324 \varLambda ^6} - \frac{1}{24 \mu ^6}\Big )\bigg ] \end{aligned}$$

(C.2)

$$\begin{aligned} \gamma _8(\mu ,\varLambda , H) =&\frac{N_c}{256\pi ^2}\sum _{f}\bigg [\Big (\frac{1}{2\mu ^4} - \frac{85}{27\varLambda ^4}\Big ) \nonumber \\&- (2 e_f H)^2\Big ( \frac{18305}{1620 \varLambda ^8} + \frac{5}{32 \mu ^8}\Big )\bigg ] \end{aligned}$$

(C.3)

$$\begin{aligned} \gamma _{10}(\mu ,\varLambda , H) =&\frac{N_c}{1024\pi ^2}\sum _{f}\bigg [\Big (\frac{230}{567\varLambda ^6}+\frac{1}{21\mu ^6}\Big ) \nonumber \\&+ (2 e_f H)^2\Big ( \frac{3233}{1215 \varLambda ^{10}} - \frac{1}{40 \mu ^{10}}\Big )\bigg ] \end{aligned}$$

(C.4)

$$\begin{aligned} \gamma _{12}(\mu ,\varLambda , H) =&\frac{N_c}{4096\pi ^2}\sum _{f}\bigg [\Big (\frac{1}{36\mu ^8} - \frac{1046}{3645\varLambda ^8}\Big ) \nonumber \\&- (2 e_f H)^2\Big ( \frac{137845}{45927 \varLambda ^{12}} + \frac{1}{48 \mu ^{12}}\Big )\bigg ] \end{aligned}$$

(C.5)

$$\begin{aligned} \gamma _{14}(\mu ,\varLambda , H) =&\frac{N_c}{16384\pi ^2}\sum _{f}\bigg [\Big (\frac{1}{55\mu ^{10}} + \frac{25864}{120285 \varLambda ^{10}}\Big ) \nonumber \\&+ (2 e_f H)^2\Big (\frac{1666750}{505197 \varLambda ^{14}} - \frac{1}{56 \mu ^{14}}\Big )\bigg ] \ , \end{aligned}$$

(C.6)

$$\begin{aligned} \gamma _{16}(\mu ,\varLambda , H) =&\frac{N_c}{65536\pi ^2}\sum _{f}\bigg [\Big (\frac{1}{78\mu ^{12}} - \frac{1102760}{6567561 \varLambda ^{12}}\Big ) \nonumber \\&- (2 e_f H)^2\Big (\frac{912622}{255879 \varLambda ^{16}} + \frac{1}{64 \mu ^{16}}\Big )\bigg ],\nonumber \\ \end{aligned}$$

(C.7)

where the sum over the index f runs over the quark flavor.

Appendix D: Numerical accuracy

We can compare the qHGL approximation against “more exact” quantities. Our work involves two approximations for the free energy: (i) Taylor expansion in the condensate parameters; and (ii) continuum approximation for the discrete energy levels. For the latter, we follow the literature and expand also for small magnetic fields (cf. [18] for example). Let us denote with $\mathcal {F_{\mathrm{{num}}}}$ the free energy obtained applying only the approximation in point (ii). Let us denote with $\mathcal {F^{\mathrm{{PT}}}_{\mathrm{{exact}}}}$ the exact free energy density regularized with the proper-time regularization scheme, calculated with neither the approximation in point (i) nor (ii) [11, 19, 34]. We test quantitatively the accuracy of the qHGL approximation by comparing the condensate parameters obtained with the qHGL expansion with $\mathcal {F_{\mathrm{{num}}}}$, and qualitatively with $\mathcal {F^{\mathrm{{PT}}}_{\mathrm{{exact}}}}$.

In Figures 5a, b we compare the amplitude $\varDelta$ and parameter K of the CDW modulation obtained with the qHGL functional including gradient terms up to $\gamma _{10}$ or $\gamma _{16}$ (solid red and dashed orange lines) and with ${\mathcal {F}}_{\mathrm{{num}}}$ (blue, dashed-dotted line) calculated for $eH = 1000\,\hbox {MeV}^2$ and $eH = 2000$ $\hbox {MeV}^2$. The qHGL approximation results show a good agreement with ${\mathcal {F}}_{\mathrm{{num}}}$ (either including terms up to $\gamma _{10}$ or $\gamma _{16}$) for $300\, \text {MeV} \lesssim \mu \lesssim 340$ MeV. For $\mu \lesssim 310.5$ MeV, the blue, orange and red curves are degenerate. Close to the first-order phase transition, the condensate parameters are determined to within $\lesssim 3$ per cent. For $eH = 1000$  $\hbox {MeV}^2$ and $\mu \lesssim 340$ MeV we find that the typical relative error in $\varDelta$ is within $\lesssim 10$ per cent and in K to within $\lesssim 1$ per cent. For $\mu \lesssim 340$ MeV and stronger fields ($eH = 2000\,\hbox {MeV}^2$) the relative error in $\varDelta$ is within $\lesssim 13$ per cent and in K to within $\lesssim 2$ per cent. An improved version of the qHGL expansion will be presented in future work.

It is interesting to compare the results obtained with the Pauli-Villars and proper-time regularization schemes, as discussed briefly in Sects. 4.1 and 4.2. The two schemes correspond to related but different physics [1], so one should not expect detailed quantitative agreement. However, one does hope for broad qualitative agreement, to confirm that the main conclusions depend weakly on regularization, and indeed this is what we find. We compare the results obtained with the qHGL approximation with $\mathcal {F^{\mathrm{{PT}}}_{\mathrm{{exact}}}}$ calculated in the literature, where the case of ultra-strong magnetic fields is usually considered [11, 19, 21, 34]. For the CDW, the typical values of $\mu _c, \varDelta$ and K are similar for $eH = 0$. For example, using the parameters adopted in [34], one finds that the effective quark mass in vacuum is $\approx 330$ MeV, and $\mu _c \approx 350$ MeV, in contrast to our case (with $M \approx 300$ MeV and $\mu _c \approx 345$ MeV, [31]). However qualitative features in $\mathcal {F^{\mathrm{{PT}}}_{\mathrm{{exact}}}}$ are preserved by the qHGL approximation for $eH > 2000\,\hbox {MeV}^2$ and $T = 0$ , such as the suppression of chiral restoration. For strong magnetic fields, one can neglect the magnetic corrections of the higher Landau levels in the qHGL approximation and include magnetic effects only via the LLL (i.e. the leading-order contribution). For example, we find that for $eH \gtrsim 8000\,\hbox {MeV}^2$ (not shown here), the CDW parameters obtained with the qHGL approximation are in qualitatively agreement with the parameters obtained with $\mathcal {F^{\mathrm{{PT}}}_{\mathrm{{exact}}}}$ [11], showing that chiral restoration is not reached up to $\mu \approx 400$ MeV.