PHYSICAL REVIEW D 98, 016006 (2018)
Vacuum stabilized by anomalous magnetic moment
Stefan Evans and Johann Rafelski Department of Physics, The University of Arizona, Tucson, Arizona 85721, USA
(Received 9 May 2018; published 10 July 2018)
An analytical result for Euler-Heisenberg effective action, valid for electron spin g−factor jgj < 2, was extended to the domain jgj > 2 via discovered periodicity of the effective action. This allows for a simplified computation of vacuum instability modified by the electron’s measured g ¼ 2.002319. We find a strong suppression of vacuum decay into electron positron pairs when magnetic fields are dominant. The result is reminiscent of mass catalysis by magnetic fields.
DOI: 10.1103/PhysRevD.98.016006
I. INTRODUCTION The here presented results are a step towards under- We explore the effect of anomalous electron spin g-factor standing of nonperturbative QED vacuum structure in E ð 2 − 1Þ on vacuum instability in Euler Heisenberg (EH) effective ultrastrong fields at the scale EH= g= . Here we action [1–3]. In the wake of theoretical development of EH accommodate the nonperturbative character of modifica- ≠ 2 action, studies of the nonlinear QED vacuum have almost tions arising from g explicitly and quantify how for ≠ 2 always relied on an electron spin g-factor of exactly 2. a given value of g modifications of the instability of Since both real and imaginary parts of the effective action the EH effective action arise in presence of external fields. g ≠ 2 In order to complete QED vacuum structure study in are modified by an anomalous -factor, the rate of “ ” α ≃ 1 137 vacuum decay in strong fields by pair production is two loop order: second order in = but to all affected. We study this effect here. orders in external (constant) fields, it is necessary to Employing an anomalous g ≠ 2-factor is an improvement consider (i) magnetic field dependent g, along with (ii) direct self-energy modifications of the particle mass on the EH effective theory in its current form, serving as an – introduction of higher order corrections into the theory. EH m by (constant) external fields [24 27]. Our results j j 2 – presented in this work allow to insert the field dependence action has been extended to g < [4 7]. A finite regular- → ð Þ → ð Þ ized expression for all g was obtained by exploiting perio- g gf a; b ;m mf a; b (subscript f reminds of field dicity of the action [8]. While a magnetic field added to an dependence) to obtain from the one loop EH effective electric field enhances the vacuum decay rate at g ¼ 2 [9,10], action the corresponding two loop result accounting in we will show that for strong magnetic fields a significant our approach for the nonperturbative behavior of EH suppression to particle production arises for g ≠ 2,aneffect effective action as a function of g. reminiscent of magnetic mass catalysis [11–15] in that it can be rather precisely described by modification of particle mass II. MODIFIED FORM OF EFFECTIVE ACTION induced by the magnetic field. A. jgj < 2 Strong magnetic fields are found on surfaces of magnet- B The Schwinger proper time method at g ¼ 2 was ized neutron stars (magnetars), suggested to possess fields j j 2 ∼102 E ¼ 2 extended to g < and produces effective action [3,7] above EH critical field ( EH m =e in Lorentz- Z ∞ Heaviside natural units, where m is electron mass) 1 ds 2 L ¼ −im s – EH 2 3 e [16 18]. Nonlinear QED phenomena in such scenarios have 8π 0 s been studied extensively, see [19,20] and references therein. eas cosh½g eas ebs cos½g ebs Even stronger B fields are produced for ultrashort time 2 2 − 1 ð Þ × ½ ½ ; 1 intervals in relativistic heavy ion collisions [19,21–23]. sinh eas sin ebs where a2 − b2 ¼ E2 − B2 ¼ 2S; ab ¼ E · B ¼ P ð2Þ Published by the American Physical Society under the terms of and α ¼ e2=4π (as used in [3]). We note that charge the Creative Commons Attribution 4.0 International license. renormalization subtraction which modifies in Eq. (1) Further distribution of this work must maintain attribution to 2 2 the author(s) and the published article’s title, journal citation, the term −1 → −1 þða − b Þ=3 is modified for g ≠ 2 and DOI. Funded by SCOAP3. to read −1 → −1 þða2 − b2Þðg2=8 − 1=6Þ.
2470-0010=2018=98(1)=016006(5) 016006-1 Published by the American Physical Society STEFAN EVANS and JOHANN RAFELSKI PHYS. REV. D 98, 016006 (2018) L ð Þ¼L ð Þ ð Þ For a pure electric field, meromorphic expansion and EH gk EH gk−1 ; 9 regularization produces temperature representation [7,28] for any real integer k where Z 2 ∞ X m T 2 2 πg − −2 þ 4 2 þ 4 ð Þ L ¼ ½ − ½1 þ i 2 E=T k 2 2 We can now write the effective action for all values of gk as m− ¼ m − iϵ ð5Þ Z ∞ 1 ds 2 L ¼ −im s EH 2 3 e to offset poles. These two representations of EH action are 8π 0 s only valid for jgj < 2: Eq. (1) diverges if jgj > 2, readily −4 −4 eas cosh½gk k eas ebs cos½gk k ebs shown when writing cosh and cos terms as exponentials, 2 2 − 1 × ½ ½ ; and meromorphic expansion in Eq. (3) fails for jgj > 2. sinh eas sin ebs ð12Þ B. jgj > 2 avoiding any divergences encountered if jg j > 2 appeared j j 2 k To extend to g > , we take advantage of EH action alone in the arguments of cos and cosh. In addition, being periodic in g. We repeat in abbreviated format the meromorphic expansion can now be applied to obtain B arguments and derivations seen in [8]. For a constant temperature representation for any g: Eq. (3) is also field pointing in zˆ the Landau orbit spectrum is obtained generalized to arbitrary gk. Whenever using in the EH using the so called Klein-Gordon-Pauli generalization of action an anomalous magnetic moment, of any magnitude, the Dirac equation and any B-dependence, the periodic reset is implied qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi according to 2 2 En ¼ m þ pz þ QjeBj½ð2n þ 1Þ ∓ g=2 ; ð6Þ g → g − 2 ð Þ 2 2 k; 13 where Q ¼ 1. Summing orbit quantum number n, we see that a shift so that the value of g is always in the principal domain −2 016006-2 VACUUM STABILIZED BY ANOMALOUS MAGNETIC MOMENT PHYS. REV. D 98, 016006 (2018) B. Imaginary part of action: General form We compute the imaginary part of effective action for nonzero a, b. Using Eqs. (1) and (13), which now produce a nondivergent action for the electron anomalous g-factor, we plug in a normalized expansion of the csch functions, typically used in computing scalar QED action [10]: Z ∞ −1 ds 2 g g L ¼ −im s EH 3 3 e cosh x cos y 4π 0 s 2 2 ∞ X ð−1Þn csch½nπy=x csch½nπx=y x3y − xy3 ; × 2 þ 2π2 2 − 2π2 n¼1 n x n y n ð15Þ plus renormalizing terms, and where FIG. 1. Ratio ℑ½L ða; b; g ¼ 2.002319Þ =ℑ½L ða; b; g ¼ 2Þ ¼ ¼ ð Þ EH EH x eas; y ebs: 16 is plotted against a, at fixed values of b in units EEH. We can neglect the second term as it does not contribute to the imaginary part of action, and write the first part as: enough b=a dominates the conventional g ¼ 2 vacuum Z decay enhancement by magnetic fields: the eþe− produc- −1 ∞ L ¼ dsð ½ 2 − ½ 2 Þ tion is maximized for b=a ∼ 102.Atb=a ∼ 103, all effects EH 3 3 cos m s isin m s 4π 0 s ∼1 E cancel and we are left with a ratio of (pure field X∞ n ∼ 100E g g ð−1Þ 3 csch½nπb=a result). For fields of order a EH there is further ×cosh x cos y x y þ ; g 2 2 2 þ 2π2 suppression, lowering the peak in Fig. 2 (cos½2 nπ con- ¼1 n x n n tribution discussed in Sec. III A). ð17Þ To better understand the results shown in Fig. 2, we look at the asymptotic limit b=a → ∞ in Eq. (18): where the second term gives after summing residues X∞ ð Þð Þ X∞ ð−1Þn ðeaÞðebÞ ð−1Þn g ℑ½L ¼ ea eb g π ℑ½L ∼ cos nπ EH 2 cos n EH 8π2 2 8π n 2 ¼1 n n¼1 n g X∞ cosh½ nπb=a 2 ˜ 2 2 −nπm =ea ð Þ × 1 þ 2 e−2rnπb=a e−nπm =ea ð20Þ × ½ π e : 18 sinh n b=a r¼1 → 0 In the limit b we recover Eq. (14): X∞ n ðeaÞðebÞ ð−1Þ g 2 2 2 X∞ n −nπm˜ =ea e E ð−1Þ g 2 ∼ cos nπ e ; ð21Þ ℑ½L ¼ cos nπ e−nπm =eE 8π2 n 2 EH 8π2 2π 2 n¼1 n¼1 n ∞ m2T2 X ð−1Þn g ¼ π −nm=T ð Þ 8π 2 cos 2 n e ; 19 n¼1 n the result for pure E fields from [7]. C. Evaluation and asymptotic behavior In contrast to the result for a pure E field, Sec. III A, the result for arbitrary a and b obtained evaluating Eq. (18) exhibits vacuum stabilization by the anomalous magnetic moment, see Fig. 1, significant for large b=a. Pair pro- duction is completely suppressed as b=a → ∞. Figure 2 offers the same result as in Fig. 1, but with B field dependence of pair production made more visible by FIG. 2. ℑ½LEHða; b; gÞ =ℑ½LEHða; b ¼ 0;g¼ 2Þ is depicted as taking the ratio with imaginary action for a pure E field. We a function b=a. Dashed line shows rescaled mass function from see that the anomalous g-induced suppression at large Eq. (21) with g ¼ 2.002319. 016006-3 STEFAN EVANS and JOHANN RAFELSKI PHYS. REV. D 98, 016006 (2018) where rescaled mass We note that for field strengths b=a ∼ 103, the vacuum decay is suppressed by factor ∼ × 10−3. In limit b=a → ∞, j − 4 j 2 2 gk k pair production is completely removed. Our result is a step m˜ ¼ m þ − 1 eb; ð22Þ 2 toward accounting for extreme magnetic field physics effects with b=a ≫ 1 arising in heavy ion collisions and where we have written explicitly the transformation to g0, on magnetars. Eq. (13), to demonstrate the periodicity as a function of g While not the focus of this work, we recognize a growing of our result. The outer absolute value in Eq. (22) assures experimental interest in the high-intensity laser frontier that the mass always increases as is seen in Eq. (20). The [33–35]. Seen our results, the modification of vacuum inner absolute value is due to evenness of the cos and cosh polarization for anomalous g-factors [36–38] should be terms in Eq. (18). The result obtained using rescaled mass reconsidered: our results differ due to recognition of the according to Eq. (21) is shown as a dashed line in Fig. 2 and periodicity in g, potentially resulting in experimentally is valid for b=a > 1. discernible differences between the cited results, and a This form of mass rescaling shares similarities with mass periodic in g result for the effect of vacuum polarization. – catalysis computations [11 15], since the leading correc- Another concern we have is that in the prior work on the α tion to g is linear in fine-structure constant . self-energy in leading OðαÞ, B-dependent corrections were B E found to be negative within the range < EH, while being IV. CONCLUSION B ≫ E – positive for EH [24 27]. These results will need to be An extension of the anomalous moment modification to reinspected to account for the nonperturbative character of two-loop effective action is of interest [32]. We have made g-factor periodicity. here a step in this direction and have demonstrated that To conclude: we presented QED vacuum instability in due to the electron’s anomalous magnetic moment, vac- strong magnetic fields allowing for g ≠ 2. Our work is the uum instability by decay into eþe− pairs experiences large first step toward a complete characterization at two loop modification for large b=a. This effect can be described in order of strong B-field behavior, valid to all orders in compact form by rescaling mass, Eq. (21). We have shown magnetic field. We recognize new nonperturbative effects that the effect of the gyromagnetic anomaly compensates related to periodicity as function of g. Our results apply in and overwhelms the Schwinger g ¼ 2 result where B straightforward fashion to consideration of B-dependence → ð Þ → ð Þ enhances pair production. We thus believe that an evalu- in g gf a; b and m mf a; b . 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