Vacuum Birefringence in a Supercritical Magnetic Field and a Subcritical Electric Field

When dealing with the effective Lagrangian of the vacuum in a constant electromagnetic field, the following Lorentz- and gauge-invariant parameters are convenient for analysis [29]:

where $F^{\mu\nu}$ and $F^{*\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ ($\varepsilon^{0123}=1$) are the field-strength tensor and its dual, respectively [30]. Then the Maxwell scalar an and pseudoscalar are given as

where $\sigma$ denotes the sign of $G$. As will be shown later, the formulation in terms of $a$ and $b$, instead of $F$ and $G$ or $\mathbf{E}$ and $\mathbf{B}$, has the advantage to facilitate the expansion of the effective Lagrangian as a series of $b$.

The parameters $a$ and $b$ can be used to classify the cases of constant electromagnetic fields, as shown in Fig. 1. The sign of the Maxwell scalar $F$ determines which field is stronger between the electric and magnetic fields, dividing the $ab$-plane into two regions: the upper left where the electric field is stronger and the lower right where the magnetic field is. The condition $G=0$ shrinks each region to its attached coordinate axis. The cases with $G=0$ are called wrenchless, while the fields with $G\neq 0$ are said to have an electromagnetic wrench [4]. In the wrenchless case, an appropriate Lorentz transformation can remove the weaker field between the magnetic field and the electric field [30]. Thus, the $a$-axis ($b$-axis) in Fig. 1 represents the condition of essentially being under a magnetic (electric) field; of course, it includes the case of a pure magnetic (electric) field. In studying the vacuum birefringence of astrophysical relevance, the magnetic field is much stronger than the electric field, and thus the region of $a\gg b$ in Fig. 1 is of our interests.

The physics of the vacuum in intense electromagnetic fields has been studied with the effective Lagrangian, which is obtained by integrating out the matter field degrees of freedom in the complete Lagrangian [31]. In the effective Lagrangian, the term other than the free-field part, the Maxwell theory ($\mathcal{L}^{(0)}(a,b)=(b^{2}-a^{2})/2$), is responsible for the phenomena such as pair production, vacuum birefringence, photon splitting, etc. As the term is dominantly contributed from the one-loop [32, 33, 34] at least for the magnetic field strengths of astrophysical relevance, we consider the effective Lagrangian up to the one-loop:

The one-loop contribution $\mathcal{L}^{(1)}(a,b)$ for the spinor QED is given as a proper-time integral [1, 2, 3]:

where $m$ is the electron mass, and $1+(es)^{2}\left(a^{2}-b^{2}\right)/3$ is subtracted to remove the zero-point energy and renormalize the charge and fields for yielding a finite physical quantity [3]. This expression can be rewritten in a form convenient for the case of $a\gg b$, i.e., $\tilde{b}=b/a\ll 1$:

where the dimensionless variable and parameters are

For a pure magnetic field, $\bar{a}=(B_{c}/B)/2$ and $\bar{b}=\infty$, where $B_{c}=m^{2}/e=4.4\times 10^{13}\,\mathrm{gauss}/\sqrt{4\pi}$ is the critical magnetic field strength.

This integral can be numerically evaluated as described in App. A, but an explicit closed expression can be favored for theoretical analysis. For the case of $b=0$, an explicit expression was obtained either by the dimensional regularization of (4) [25] or by the Schwinger-DeWitt in-out formalism combined with $\Gamma$-function regularization [27]:

where $\zeta(s,\bar{a})$ is the Hurwitz zeta function, and $\zeta^{\prime}(-1,\bar{a})=d\zeta(s,\bar{a})/ds|_{s=-1}$ [35] (See App. B for more details.). This expression perfectly matches the numerical evaluation of (5) with $\tilde{b}=0$, as shown in Fig. 2. We use this analytic expression of $\mathcal{\bar{L}}^{(1)}(\bar{a},0)$ to express $\mathcal{\bar{L}}^{(1)}(\bar{a},\tilde{b})$ as a series of $\tilde{b}=b/a$, as shown in the next section.

In the magnetosphere of pulsars, the magnetic field is much stronger than the electric field, and thus $a\gg b$, or equivalently $\bar{a}\ll\bar{b}$, from (1) and (2). For example, a pulsar model [21] gives $\tilde{b}$ ($=b/a$) as a function decreasing with the distance from the pulsar center: $\tilde{b}(R)\leq 0.2$ and $\tilde{b}(10R)\leq 0.02$, where $R$ is the radius of the pulsar. For such a condition, the expansion of $\mathcal{\bar{L}}^{(1)}(\bar{a},\tilde{b})$ in $\tilde{b}$ is useful for analysis.

In the integral expression of $\mathcal{\bar{L}}^{(1)}(\bar{a},\tilde{b})$ (5), $\cot(\tilde{b}z)$ has poles at $z=n\pi/\tilde{b}$ ($n=1,2,\dots$), which may apparently make the integral diverge. However, only the Cauchy principal value of the integral is relevant for vacuum birefringence, and the symmetric behavior of the cotangent function around the poles prevents the principal value from diverging. Furthermore, if $\exp(-2\bar{a}z)$ suppresses the integrand sufficiently much before the first pole, i.e., $1/(2\bar{a})\ll\pi/\tilde{b}$, or equivalently $\bar{b}\gg 1/(2\pi)$, an asymptotic expression valid for $\bar{b}\gg 1/(2\pi)$ can be obtained. To proceed, we substitute the series form of $(\tilde{b}z)\cdot\cot(\tilde{b}z)$ in (5) (4.19.6 of [35]):

where $B_{2n}$ are the Bernoulli numbers. The series is convergent only for $|\tilde{b}z|<\pi$ due to the nearest poles at $\tilde{b}z=\pm\pi$. Upon substitution, the contribution outside the region of convergence asymptotically vanishes as $\bar{b}\rightarrow\infty$. Then, the integral in (5) is written as

To evaluate the integral, we use the integral representation of $H(\bar{a})$ in (7), which is derived by comparing (7) with (46):

Differentiating (10) $2n$ times by $\bar{a}$ yields a useful formula:

of which closed form is given as (See App. C for the derivation)

where $\psi^{(m)}(\bar{a})$ is the polygamma function. By using (11) and (12) and the integral representation of the $\Gamma$ function, one can integrate (9) term-by-term to obtain an asymptotic series of $\mathcal{\bar{\mathcal{L}}}^{(1)}(\bar{a},\tilde{b})$:

in which all the terms are given in terms of special functions. Thus, (13) provides a systematic explicit expression of $\mathcal{\bar{L}}^{(1)}(\bar{a},\tilde{b})$ in powers of $\tilde{b}=b/a$ for an arbitrary value of $\bar{a}$. For example, the first three leading orders are given as

A similar expansion in $K=-16a^{2}b^{2}$ was made by Heyl and Hernquist to yield the explicit expressions of the lowest few orders [36].

When both electric and magnetic fields are highly subcritical, i.e., $\bar{a}\gg 1$ and $\bar{b}\gg 1$ hold, the lowest order can be obtained from the expansion up to $O(\tilde{b}^{4})$, and the second lowest from the expansion up to $O(\tilde{b}^{6})$:

The term with the first bracket is written in terms of $\mathbf{E}$ and $\mathbf{B}$ as

which was obtained by Heisenberg and Euler [1], and Schwinger [3].

It needs to be mentioned that the one-loop effective Lagrangian $\mathcal{L}^{(1)}(a,b)$ (4) can be expressed as a convergent series of which terms are some special functions of $a$ and $b$ [37, 38, 39]. The expansion may be exact but is not very convenient for the condition of an arbitrarily strong magnetic field combined with a weaker electric field because the expansion needs an infinite sum. The convergence of the sum is slow, and thus its evaluation needs acceleration techniques [38]. In this regards, the expansion (II.3) is more convenient for theoretical analysis.

The exact dependence of $\mathcal{\bar{\mathcal{L}}}^{(1)}(\bar{a},\tilde{b})$ with $\tilde{b}$ can be investigated by numerically evaluating the integral expression (5), as described in App. A. As $\mathcal{\bar{\mathcal{L}}}^{(1)}(\bar{a},0)$ is completely known, the ratio $\mathcal{\bar{\mathcal{L}}}^{(1)}(\bar{a},\tilde{b})/\mathcal{\bar{\mathcal{L}}}^{(1)}(\bar{a},0)$ represents the dependence on $\tilde{b}$ alone. In Fig. 3, the offset difference of $\mathcal{\bar{\mathcal{L}}}^{(1)}(\bar{a},\tilde{b})$ from $\mathcal{\bar{\mathcal{L}}}^{(1)}(\bar{a},0)$ increases with $\tilde{b}$. Furthermore, additional significant differences appear as $\bar{a}$ decreases below certain onset values. For example, when $\tilde{b}=0.2$, the ratio is close 1.2 until $\bar{a}$ decreases to 1, but, as $\bar{a}$ decreases below 1, it significantly increases and then decreases . When $\tilde{b}$ is smaller, the behavior is similar except that the offset difference is smaller, and the onset value of $\bar{a}$ decreases. This additional difference is considered to be attributed to the pair production. When $1/(2\bar{a})\gtrsim\pi/\tilde{b}$, or equivalently $\bar{b}\apprle 1/(2\pi)$, the contribution from the first pole of the cotangent function in (5) becomes significant. As $\bar{b}=E_{\mathrm{cr}}/(2E)$ in the parallel field configuration, the condition $\bar{b}\apprle 1/(2\pi)$ implies the emergence of the pair production by the strong electric field. In such a case, the effective Lagrangian develops an imaginary part, which is accompanied by a variation in the real part, i.e., the additional difference in Fig. 3. It is similar to the anomalous dispersion near a resonance region in the linear optical response [40]. As the imaginary part gives the pair production probability, the plasma effects appears when the probability is sufficiently high. Then, it is no more pure vacuum birefringence. In our analysis, we focus on real part of the Lagrangian to study the pure vacuum birefringence.

By using the numerical evaluation of $\mathcal{\bar{\mathcal{L}}}^{(1)}(\bar{a},\tilde{b})$, we can find the parameter range in which the expansion (13) is accurate. The ratio $\mathcal{\bar{\mathcal{L}}}_{\mathrm{exp}}^{(1)}(\bar{a},\tilde{b},n)/\mathcal{\bar{\mathcal{L}}}_{\mathrm{int}}^{(1)}(\bar{a},\tilde{b})$ is plotted for $n=0,1,2,3$ and $\tilde{b}=0.2,0.05$, where $\mathcal{\bar{\mathcal{L}}}_{\mathrm{exp}}^{(1)}(\bar{a},\tilde{b},n)$ is the expansion (13) up to $O(\tilde{b}^{2n})$, and $\mathcal{\bar{\mathcal{L}}}_{\mathrm{int}}^{(1)}(\bar{a},\tilde{b})$ is the numerical evaluation of (5). In Fig. 4(a), $\mathcal{\bar{\mathcal{L}}}_{\mathrm{exp}}^{(1)}(\bar{a},\tilde{b}=0.2,n\geq 1)$ is accurate within 1% for $\bar{a}\geq 0.3$. The 1%-accuracy threshold of $\bar{a}$, denoted by $\bar{a}_{1\%}$, decreases slightly as $n$ increases, but a higher-$n$ expansion blows up faster below $\bar{a}_{1\%}$; a typical behavior of the Taylor expansion. The $n=0$ expansion, corresponding to the effective Lagrangian with $\tilde{b}=0$, has an error larger than 17%. At a lower value of $\tilde{b}=0.05$, the overall behavior is similar to the case of $\tilde{b}=0.2$, but $\bar{a}_{1\%}$ is lowered to 0.025, and the minimum error of the $n=0$ expansion decreases to 2%, as shown in Fig. 4(b). In practice, the $n=2$ expansion can be a reasonable choice for analysis, showing $\bar{a}_{1\%}=0.1$ for $\tilde{b}=0.2$ ( $B\leq 5B_{\mathrm{cr}}$ and $E\leq E_{\mathrm{cr}}$) and $\bar{a}_{1\%}=0.02$ for $\tilde{b}=0.05$ ( $B\leq 25B_{\mathrm{cr}}$ and $E\leq 1.25E_{\mathrm{cr}}$). Noting that the expansion is not accurate when the pair production is significant, one may roughly estimate a threshold value of $\bar{a}$ like $\bar{a}_{1\%}$ by requiring $\tilde{b}/(2\bar{a}_{\mathrm{thres}})\simeq 1$. As $\tilde{b}/(2\bar{a})=E/E_{\mathrm{cr}}$ in the parallel field configuration, the expansion would not be accurate when this ratio is the order of unity. In Fig. 4, the threshold values $\bar{a}_{1\%}$ are close to the estimation $\bar{a}_{\mathrm{thres}}=\tilde{b}/2$.

The effective Lagrangian is a function of $a$ and $b$ (1) and (2), and thus we consider the variation of a general differentiable function $f(a,b)$:

As $a$ and $b$ are functions of $F$ and $G$, the variations $\delta a$ and $\delta b$ can be written in terms of the variations $\delta F$ and $\delta G$:

In turn, the variations of $\delta F$ and $\delta G$ can also be written in terms of $\delta\mathbf{E}$ and $\delta\mathbf{B}$:

Thus, usign the chain rules, the coefficients in these variations can be calculated by using (1) and (2) to express $\delta f$ in terms of $\delta\mathbf{E}$ and $\delta\mathbf{B}$:

where

The operators $\hat{S}$ and $\hat{A}$ are symmetric and antisymmetric under the parity inversion, respectively. Note that $(a^{2}+b^{2})\hat{S}$ measures the difference of homogeneity of polynomials such that $(a^{2}+b^{2})\hat{S}(a^{m}b^{n})=(m-n)(a^{m}b^{n})$, while $(a^{2}+b^{2})\hat{A}$ measures the mixed homogeneity of polynomials such that $(a^{2}+b^{2})\hat{A}(a^{m}b^{n})=(mb^{2}+na^{2})(a^{m-1}b^{n-1})$. In addition, $(a^{2}+b^{2})\hat{S}$ preserves the polynomial order $(m,n)$, while $(a^{2}+b^{2})\hat{A}$ changes it to $(m-1,n+1)$ and $(m+1,n-1)$ while maintaining the sum of the orders of $a$ and $b$. Replacing $f$ with $\mathcal{L}^{(1)}$ and using (17), one can obtain $\mathbf{P}$ and $\mathbf{M}$ from $\mathcal{L}^{(1)}(a,b)$:

Despite the appearance, this relation is not linear in $\mathbf{E}$ and $\mathbf{B}$, as $\hat{S}\mathcal{L}^{(1)}$ and $\hat{A}\mathcal{L}^{(1)}$ are nonlinear functions of $\mathbf{E}$ and $\mathbf{B}$.

The material-like vacuum with the nonlinear response relation (23) affects the propagation of photons. When the photon energy is much smaller than the electron’s rest mass energy ($\omega\ll m=0.5\,\mathrm{MeV}$), the photon can be treated as a weak perturbation field (probe field) added to the strong background field [44, 45]. Then, $\mathbf{E}$, $\mathbf{B}$, $\mathbf{D}$, $\mathbf{H}$, $\mathbf{P}$, and $\mathbf{M}$ can be decomposed as

where $\mathbf{A}_{0}$ ($\delta\mathbf{A}$) refers to the quantities of the background (probe) field. Furthermore, as $\mathbf{A}_{0}$ is uniform, while $\delta\mathbf{A}$ is varying, the relation (17) should be satisfied separately between the uniform and varying quantities:

Then, $\delta\mathbf{D}$ and $\delta\mathbf{H}$ can be obtained by varying the relation (23) around $(\mathbf{E}_{0},\mathbf{B}_{0})$, for which the formula (21) is convenient. A straightforward but lengthy calculation yields the linear relations among the varying quantities relevant to the probe field:

where $\boldsymbol{\epsilon}_{E}$, $\boldsymbol{\epsilon}_{B}$, $\bar{\boldsymbol{\mu}}_{B}$, and $\bar{\boldsymbol{\mu}}_{E}$ are 3-by-3 tensors. Their components are as follows:

where

Here, the subscript 0 means that $\hat{S}$ and $\hat{A}$ are evaluated at $(\mathbf{E}_{0},\mathbf{B}_{0})$. Note that these formulas are valid for any effective Lagrangian. From now on, we shall focus on the Heisenberg-Euler and Schwinger effective Lagrangian (4).

The tensors $\boldsymbol{\epsilon}_{B}$ and $\bar{\boldsymbol{\mu}}_{E}$ represent the magneto-electric response to the probe field, in which a magnetic (electric) field induces polarization (magnetization) [46], unlike usual dielectric and magnetic materials (See [47, 48] for the recent studies of the magneto-electric response in condensed matter). Such response disappears as the background electric field vanishes: when $\mathbf{E}_{0}=0$, $\mathcal{L}_{A}^{(1)}$ and $\mathcal{L}_{SA}^{(1)}$ in (29,31) vanishes for the Lagrangian (4) to yield $\boldsymbol{\epsilon}_{B}=\bar{\boldsymbol{\mu}}_{E}=0$.

The quantities in (32) have the complete information for the analysis of vacuum birefringence because they immediately lead to the permittivity and permeability for an arbitrary background field configuration through (28,29,30,31). By using the expansion (II.3), we can obtain the formulas of (32) contributed from each order of $\tilde{b}$. For example, the contributions from the lowest three orders are shown in Tabs. 1, 2, and 3. In Tab. 1, the dependence on $\bar{b}$ appears due to the $b\cdot\partial_{a}$ term in $\hat{A}$ although $\mathcal{\bar{\mathcal{L}}}^{(1)}(\bar{a},0)$ does not depend on $\bar{b}$. As the combined field of the background field and the probe field is not wrenchless in general, the expression of $\mathcal{\bar{\mathcal{L}}}^{(1)}(\bar{a},\tilde{b}\neq 0)$ is necessary for wrenchless background fields.

For the case of $b=0$ with an arbitrary value of $a$, i.e., the wrenchless case, the exact expressions of $\mathcal{L}_{S}^{(1)}$, $\mathcal{L}_{A}^{(1)}$, $\mathcal{L}_{SS}^{(1)}$, $\mathcal{L}_{SA}^{(1)}$, and $\mathcal{L}_{AA}^{(1)}$ can be obtained first by combining the results in Tabs. 1 and 2 and then by taking the limit of $\bar{b}=\infty$. Higher order contributions are absent in this case. The results are shown in Tab. 4. The same results were obtained by Karbstein et al. [43], who expanded the one-loop effective action up to the second order of the probe field and specified the calculation to the wrenchless case.