Vacuum Stabilized by Anomalous Magnetic Moment
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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 Æ πg − −2 þ 4 2 þ 4 ð Þ L ¼ ½ − ½1 þ i 2 E=T k<gk < k; 10 EH 2 dE ln E m− ln e e ; 16π 0 Æ with cusps at the boundaries. Thus the electron anomalous ð3Þ moment, slightly greater than 2 (within domain g1), can be transformed into the domain jg0j < 2 by where temperature parameter E L ð ¼ 2 þ 0 002319Þ¼L ð ¼ −2 þ 0 002319Þ ¼ e ð Þ EH g . EH g . : T π ; 4 m ð11Þ and mass 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 ¼1. Summing orbit quantum number n, we see that a shift so that the value of g is always in the principal domain −2 <g<2.A“raw” value jgj > 2 may never be used in g → g þ 4k ð7Þ Eq. (1) as the EH effective action is ill defined in that case corresponds to a shift in n, leaving the summed states [8]. Equation (13) explains how any value of g [including (Casimir energy, [2]) unchanged. Noting this periodicity gðBÞ] has to be periodically reset, and only the reset value is and using the usual method of discrete summation [1,2] to be used in all expressions involving the EH effective to obtain an integral representation, a periodic function action. introduces the Bernoulli polynomials [29,30]. With this periodic extension of the Bernoulli polynomials, a modified III. STABILIZATION OF THE VACUUM meromorphic expansion is used (Eq. (9) in [8]), agreeing in E limit g → 2 with [10,28,31]. A. Vacuum decay in a pure field At magnetic field strengths beyond We first show the standard case: vacuum decay in a pure electric field. The imaginary part of action (Eq. (14) in [7]) m2 jeBj < ∼ 862m2;g¼ g ð0Þ¼2.002319; ð8Þ becomes jg=2 − 1j f ∞ m2T2 Xð−1Þn g the energies in Eq. (6) become imaginary, that is the ℑ½L ¼ π −nm=T ð Þ EH 8π 2 cos 2n e : 14 associated eigenstates disappear from the spectrum and n¼1 n self-adjointness is lost. Since magnetic fields cannot do work, a compensating modification of mass may stabilize Comparing this result at g ¼ 2.002319 to that at g ¼ 2, the 1% 60E 1% the vacuum in presence of ultrastrong magnetic fields. We two are different by <: for a< EH, and < for 60 100E postpone this question to future study and limit the present <a< EH. Thus pair production in a pure electric discussion by condition Eq. (8). field is modified by an insignificant amount when cor- We exploit the g-periodicity, which applies even for recting the electron’s gyromagnetic ratio. We now move to B-dependent anomalous moment, to write vacuum instability in both E and B fields. 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.