Electron Electromagnetic-Mass Melting in Strong Fields

PHYSICAL REVIEW D 102, 036014 (2020)

Electron electromagnetic-mass melting in strong fields

Stefan Evans and Johann Rafelski Department of Physics, The University of Arizona, Tucson, Arizona 85721, USA

(Received 21 November 2019; accepted 29 July 2020; published 14 August 2020)

We study the response of the electron mass to an externally applied electrical field. As a consequence of nonlinear electromagnetic (EM) effective action, the mass of a particle diminishes in the presence of an externally applied electric field. We consider modification of the muon anomalous magnetic moment g − 2 due to electron loop insert in higher order. Since the virtual electron pair is in close proximity to the muon, it experiences strong field phenomena. We show that the current theory-experiment muon g − 2 discrepancy could originate in the (virtual) electron mass nonperturbative modification by the strong muon EM field. The magnitude of the electron mass modification can be also assessed via enhancement of eþe−-pair production in strong fields.

DOI: 10.1103/PhysRevD.102.036014

I. INTRODUCTION The electron mass response to external EM fields has – The Higgs minimal coupling mass generating mecha- been studied before [2 4]. We revisit the topic with the aim nism of heavy, beyond GeV mass scale, elementary to better understand the relation between the two dominant particles is well established. The origin of the lightest contributions to electron mass, the EM and Higgs portions. standard model (SM) electron mass is not expected to be The EM mass in external fields can be studied in QED resolved experimentally in the near future: the LHC pp- either as EM-self-energy or in the EM-energy-momentum collider is capable of constraining the minimal coupling to tensor analysis: We show this in Sec. II, where we also factor ∼100 above the predicted SM value, and next propose an effective method allowing exploration of field- generation eþe−-colliders are limited to factor ∼10 above dependent mass in the strong field nonperturbative regime. the required sensitivity [1]. However, given the small value We show how the nonlinear mixing between fields suppresses the total field to an amount that can be smaller than of electron mass a significant electron mass contribution ’ from electromagnetic (EM) self-energy in the realm of their linear superposed sum. For this reason the electron s QED can be expected. EM field mass portion diminishes in an externally applied We propose here a complementary probe of electron electric field, an effect we call in the following mass mass, the mass modification by a strong external field. melting. In Sec. III using a specific limiting EM field strength This is based on the observation that a negligible in model we explore the supercritical field regime [5,6]. The magnitude beyond SM (BSM) component is irrelevant, model is tailored to match closely to the QED effective while the Higgs mass component can respond to external Euler-Heisenberg-Schwinger (EHS) action [7–10] for qua- EM field strengths of electro-weak natural strength, inac- siconstant fields up to the EHS field strength, and contains cessible today. However, the electromagnetic (EM) mass an adjustable parameter that allows for probing of the component of the electron is susceptible to modification relation between EM and non-EM components of electron by an applied strong external field, measured in electron mass. In Sec. IVA we compute the model mass melting mass m natural units characterized by the Schwinger e effect in supercritical fields, and in Sec. IV B show that in (EHS) field, subcritical fields this effective modification has the right α 2 0 structure (powers in ) and order of magnitude to be E með Þ 1 323 1018 EHS ¼ ¼ . × V=m: ð1Þ consistent with evaluation of perturbative self-energy e corrections in QED. Most high precision QED experiments probe fields below the EHS critical field, in which the mass modification can be explored using perturbative QED. Fields Published by the American Physical Society under the terms of beyond the EHS strength appear in the muon g − 2: The the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to anomaly in the muon magnetic moment is sensitive to the the author(s) and the published article’s title, journal citation, melting of the (virtual) electron mass entering the vacuum and DOI. Funded by SCOAP3. polarization experienced by the virtual photon, induced by

2470-0010=2020=102(3)=036014(11) 036014-1 Published by the American Physical Society STEFAN EVANS and JOHANN RAFELSKI PHYS. REV. D 102, 036014 (2020) the muon-sourced EM field. Due to the small effective size field driven mass (mass catalysis [3], and references of the muon, as a quantum wave localized at its Compton therein). The perturbative approach is assumed to be valid wave length scale, the virtual electrons entering the muon for weak fields well below the EHS critical field scale. The g − 2 consideration experience much stronger fields than supercritical regime requires extending the computation to that in the electron g − 2 case. We show in Sect. VA via a nonperturbative summation in self-energy diagrams. This computation of the induced vacuum polarization displace- includes a self-consistency requirement in Eq. (2): On the ment current charge density that the polarized virtual right-hand side (rhs) the mass scale accompanying a, b electron pairs lie close enough to the muon to experience dependence can be interpreted as being affected by the EM strong field phenomena. We argue that therefore a pertur- response. bative QED evaluation of g − 2 is unreliable. We show that To explore this mass response in the nonperturbative our model mass melting of the electron by the field of the regime we propose to consider a model formulation of muon is capable to explain the observed theory-experiment mEM, the EM field mass portion of electron mass me.How − 2 g discrepancy [11]. a finite mEM with a stable EM stress configuration arises Another experimental process highly sensitive to the remains an open question today [6,20,21]. Electron mass value of the electron mass is the QED vacuum decay into cannot be entirely EM in origin due to lepton mass electron-positron pairs, see Sec. VB. For uniform homog- differences: to resolve the EM portion is not possible enous EM fields, pair production in strong fields is inherent within perturbative QED, but may arise in a self-consistent in the EHS effective action obtained for quasiconstant and nonperturbative approach, see [22]. By EM field mass fields. For inhomogeneous fields the nonperturbative proc- we refer to the energy contained within the EM field of a ess of pair production has been explored in the context of particle: this quantity is strongly dependent on the particle heavy ion collisions [12]. The localization of strong field charge distribution—in QED the “electron size” is gov- phenomena has facilitated consideration of the local vac- erned by the Compton wavelength, while in classical EM uum structure and instability [13]. Pair production in the theory the α−1 ¼ 137 times smaller so-called classical EHS action seen in perspective of the development of novel electron radius is the appropriate scale. ultraintense-pulsed-laser technologies [14] has been spur- ring a renaissance of strong field physics [15–18]. We close this paper with an outline of future research opportunities. B. EM field mass We evaluate meða; bÞ by integrating EM field mass II. MASS MELTING ARISING IN NONLINEAR density. The Lorentz invariant field mass density U ELECTROMAGNETISM obtained from the field 4-momentum density Pν, see for A. Effective scalar potential example Eq. (28.11) in [23]: The mass response to external EM fields is recognized ν μν P ¼ uμT ; ð5Þ exploring the position of the electron propagator pole, see pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e.g., [19]. The field-dependence of electron self-energy [2] ≡ ν μν α allows us to write U P Pν ¼ uμT Tναu ; ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≡ ϕ ; 00 2 0 2 meða; bÞ me þ ðme a; bÞ: ð2Þ U ¼ ðT Þ þðT kÞ ; ð7Þ ϕ Only in the limit of quasi-constant fields can in Eq. (2) be where T is the EM energy-momentum tensor and u is the expressed as a function of both Lorentz and gauge invariant relative 4-velocity between observer and the source of the m 0 ≡ m quantities. Here eð Þ e is the physical electron mass; field. The last relation follows in the comoving (relative the invariants rest) reference frame of the field source u ¼ð1; 0⃗Þ. L a2 − b2 ¼ 2S ¼ E2 − B2;a2b2 ¼ P2 ¼ðE · BÞ2; ð3Þ T for any nonlinear effective EM Lagrangian is obtained by varying L with respect to the metric gμν, are appearing in the eigenvalues of the EM field tensor. The resulting in [24]: scalar ϕ itself now enters the Dirac equation since ∂L ∂L ∂L M Tμν ¼ Tμν − gμν L − S − P ; ð8Þ ∂S ∂S ∂P ½γ · Π − meða; bÞÞψ ¼ 0; ð4Þ where TM is the Maxwell energy momentum tensor in view of Eq. (2). M α gμν αβ The scalar potential ϕ has been studied applying per- M − TμνðxÞ¼TμνðxÞjL¼L ¼ FμαFν 4 FβαF : ð9Þ turbative QED methods: The leading self-energy diagram consisting of the dressed (by external fields) electron g is the space-time (here Minkowski) metric, F is the EM propagator along a virtual photon loop [2,4]; and magnetic field tensor, and the Maxwell Lagrangian

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1 ˜ D D D − D LM E2 − B2 Uð Þ¼Uð p þ exÞ Uð exÞ: ð17Þ ¼ 2 ð Þ: ð10Þ Equation (17) is integrated to obtain the external field- For a pure electric field, Eq. (8) may be written as dependent mass: Z D E2 M 3 Tμν Tμν gμν − gμνL; E ˜ D ¼ E þ 2 ð11Þ mEMð exÞ¼ d rUð Þ: ð18Þ where we introduced the displacement field This expression for EM field mass is applicable to both linear and nonlinear actions. ∂L Only in the linear Maxwell theory is the mass unaffected D ¼ : ð12Þ ∂E by the presence of external fields: the Maxwell Lagrangian

Plugging Eq. (11) into Eq. (5) produces LM ¼ E2=2; ð19Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E2 2 gives superposable fields M − L U ¼ E T00 þ g00 2 g00 E E E D D D qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p þ ex ¼ ¼ p þ ex; ð20Þ ¼ ðE · D − LÞ2: ð13Þ (only in Gauss-type units can we set E ¼ D, in SI units there is a further vacuum dielectric constant factor). The The same result is obtained by performing a Legendre mass density becomes transform of L [5]. Note that U and E are to be evaluated as functions of D which is sourced by some applied charge D2 ˜ D p D D distribution. Uð Þ¼ þ p · ex: ð21Þ E 2 Consider the electric field p sourced by the charge of the electron (we postpone for now the magnetic field Evaluating the field mass according to Eq. (18) sources by the magnetic dipole moment), in the presence Z of an external constant and homogeneous electrical field D2 3 p E ’ m E M d r D · D ex. First we write the EM field mass for the electron s EMð exÞjL¼L ¼ 2 þ p ex electric component in the absence of external fields Z D2 Z 3 p E 0 ¼ d r ¼ mEMð ex ¼ Þ; ð22Þ E 0 3 D 2 mEMð ex ¼ Þ¼ d rUð pÞ; ð14Þ where the mixing term integrates to zero for any radial where for a point electron (or, outside of charge distribution Coulomb field in presence of a constant and homogeneous of an electron) we have external field. Our intuitive chain of argument using classical fields shows clearly that only when the super- ∂L D ˆ e position principle of fields holds can one expect to describe p ¼ ∂E ¼ r 2 : ð15Þ p 4πjr⃗j the particle interaction with electron mass unchanged by the external field. To include an external field, a closer look at the super- Upon field quantization, that is in QED, the situation position principle is required. Only for Maxwell action may becomes more complex since any interacting quantum field E E E be written as a linear superposition of p and ex.Inany theory is intrinsically a nonlinear theory. However, to nonlinear theory, the superposition of electric fields is lowest order the Maxwell Lagrangian would then include violated, and only the displacement fields generated by the vacuum polarization contribution, described diagrammatically by an electron loop coupled to two photon lines inhomogeneous Maxwell equations may be superposed. 2 We use displacement fields to distinguish the external and (of order E ). Such a contribution though being nonlocal, is particle field in the rest frame: still linear and thus does not introduce external field corrections to the EM field mass. e The correction we anticipate appears at higher order. For D ¼ D þ D ¼ rˆ þ D : ð16Þ p ex 4πjr⃗j2 ex the case of (quasi-)constant fields, an exact result was obtained by Euler and Heisenberg, and illuminated by The superposed displacement fields enter the expression Schwinger [7–9]. This EHS field-dependent action is for mass density [Eq. (13)], from which we subtract the giving the vacuum state the properties of a nonlinear contribution from the external field alone: dielectric and introduces light-light scattering diagrams,

036014-3 STEFAN EVANS and JOHANN RAFELSKI PHYS. REV. D 102, 036014 (2020) beginning with order E4. We have as action to lowest order A. EHS effective action the Maxwell term complemented by nonlinear effective In QED, in the quasiconstant (local) field approximation, QED term: the EHS effective action arises [10] E2 E4 Z 1 ∞ L ¼ þ þ …; ð23Þ 1 ds − 2 0 EHS 2 E2 L LM − með Þs cr EHS ¼ 2 3−δ e 8π 0 s here expanded to leading fourth order contribution. E is a e2abs2 cot½eas cr × − 1 ; ð28Þ “critical” field scale we recognize in magnitude in quanti- tanh½ebs tative consideration of QED effects. The new nonlinear term is mixing the particle and the depending implicitly on the Schwinger (EHS) field, Eq. (1). external field in a more complex fashion due to violation of The argument “0” seen above for the electron mass reminds the EM field superposition principle. In order to use the that the result was obtained without allowing for a superposition principle for the displacement fields we dependence of electron mass on the external applied field. evaluate the relation between field and displacement field, The field invariants a and b in Eq. (28) are given by and its inversion, in the weak field limit Eq. (3), and the pre-factor 1=8π2 in Eq. (28) follows units α 2 4π E3 D3 used by Schwinger in which ¼ e = , see Ref. [10]. The D E 4 … → E D − 4 …; function in Eq. (28) has been subtracted to remove the zero- ¼ þ E2 þ ¼ E2 þ ð24Þ cr cr point energy, and the logarithmically divergent contribution to be absorbed by charge renormalization is regularized by see [25]. The electric field is suppressed by the light-light infinitesimal δ. scattering response; hence the negative contribution to E on In the perturbative regime, weak field expansion the rhs of Eq. (24). The total electric field is thus smaller (E , B ≪ E ) produces the light-light scattering than the superposed fields encountered in the linear theory ex ex EHS contribution in Eq. (20), reducing the EM field mass density: 2 4 2α 7 D D L LM 2 2 O 3 D − − … EHS ¼ þ 2 S þ P þ ðS Þ : ð29Þ Uð Þ¼ 2 E2 : ð25Þ 45πE 4 cr EHS D D D Writing ¼ p þ ex explicitly and subtracting the Understanding of the magnitude of the mass melting external field contribution far from the Coulomb field effect requires a finite computable EM field mass, see source, Eq. (18), yet the EHS action, without improvements, leads to a divergent result. We demonstrate this divergence D2 1 in two steps: first considering the divergent electric U˜ D p D D − D4 2D2D2 ð Þ¼ 2 þ p · ex E2 f p þ p ex component of the Maxwell EM field mass and later cr EHS. For the Maxwell case we find: 4D2D · D 4D2 D · D þ p p ex þ ex p ex Z 4 D D 2 − … þ ð p · exÞ g : ð26Þ 0 3 E D − LM E mEMð ÞjL¼LM ¼ d rð p · p ð pÞÞ Z D D E D Plugging Eq. (26) into Eq. (18), odd powers p · ex 3 p · p D2 ¼ d r integrate to zero and we obtain, to order p, 2 2 Z Z 2 2 2 2 e 1 D 2D D þ4ðD ·D Þ ¼ dr r2 · → ∞: ð30Þ m E d3r p − p ex p ex : 8π 4 EMð exÞ¼ 2 E2 ð27Þ r cr To remedy the divergent Maxwell expression requires an In Eq. (27) the leading correction to the EM field mass effective action which suppresses E, where at the origin density is quadratic in external fields, and due to its E ≪ D E D ∝ 1 2 p p, for example such that product p · p =r , negative sign, mass decreases (melts) in an external field. 4 instead of 1=r as in the Maxwell case. Now if the original We now turn to a nonperturbative formulation, essential to a EHS action were applied instead, quantitative study of EM field mass. Z m 0 d3r E · D − L E ; 31 III. NONLINEAR EFFECTIVE ACTION EMð ÞjL¼LEHS ¼ ð p p EHSð pÞÞ ð Þ We study mass melting in context of the following EM E ≫ E actions: where at the origin the strong field limit ( p EHS)gives

036014-4 ELECTRON ELECTROMAGNETIC-MASS MELTING IN STRONG … PHYS. REV. D 102, 036014 (2020) E2 α g − 2 contribution becomes a field-dependent quan- L p 1 − 2 E 2 0 Re½ EHS¼ 2 3π ln½ e p=með Þ : ð32Þ tity [4] which must enter into EHS action in a self consistent manner. For incorporation of g − 2 into E D E EHS action see [34–37], recently shown by the Differentiating Eq. (32) by p to obtain pð pÞ, we find E D E D authors to have a significant influence on pair that p > p, causing the product p · p to be even more divergent than in the Maxwell case. production in strong magnetically dominated Four here relevant additional corrections to EHS result fields [38]. are known and we believe can influence the divergent behavior of the EM self-fields above: B. Generalized Born-Infeld model (1) In the original EHS result, the fermions are non- Practical use of EHS action considering the above dynamical since they are integrated out in order to described self-consistency extensions is difficult as con- obtain an effective action: self-energy corrections to siderable improvement of our understanding of the (elec- the mass can only appear in higher order corrections. tron) mass response to external fields is required. Therefore Since we know that the EHS effective action is we explore a model of the Born-Infeld (BI) type created nonlinear, we need to account for the possibility that expressly to describe EM-mass of the electron. We recall the electron mass is melted in strong fields. This EM that BI model also introduces a limit to the achievable field field mass correction described in Sec. II enters the strength. We do not consider the model as an extension to Dirac equation at the start of derivation of EHS QED, which has in large part been constrained [39–41]. action, Eq. (4). One can understand this considering Instead we fine-tune it so that it will allow to make such a self-consistent correction applied to the predictions in lieu of the EHS effective action accounting Landau energy levels summed in the Weisskopf perhaps for the required EHS improvements we discussed. EHS evaluation [8]. The result thus is We fine-tune a BI-like model introduced in Ref. [6] Z ∞ 1 ds − 2 2 2 2 2 2 L LM − meða;bÞs E − n EHSþmelt ¼ 2 3−δ e crðnÞ a b a b 8π 0 s L ¼ − 1 − − 7 − 1 : ð35Þ eff 2 E2 E4 2 2 n crðnÞ crðnÞ e abs cot½eas − 1 × : ð33Þ 1 2 E 89 72E tanh½ebs Choosing n ¼ = and cr ¼ . EHS we obtain the E original BI result [42]. cr provides a limit on the maximum Eq. (33) contains a higher order correction to the strength that an electric field may reach, and we inserted the EHS action via field-dependent mass, hence sub- a2b2 coefficient so that the model more closely tracks in its script “+melt.” Interestingly, the Schwinger field is functional form the EHS action in presence of both electric now an external field-dependent quantity: and magnetic fields. We further choose the value of the critical field such that m2ða ;b Þ Emelt ¼ e ex ex : ð34Þ an expansion of Eq. (35) generates exactly the EHS light- EHS e light scattering result, Eq. (29) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2) Nonlocal corrections to the action are required in 45πð1 − nÞ order to account for nonlinear mixing between the E E crðnÞ¼ EHS 2α : ð36Þ external fields and the inhomogeneous fields of the particles [26–29]. (3) Another effect has been pointed out by Gies and Karbstein [30,31]: in addition to the well-studied internal photon line corrections to the EHS action [32,33], reducible connecting photon line corrections to the EHS action have been shown to be nonvanishing. This contribution may also be ac- counted for in a self-consistent manner. We continue this work in an upcoming paper, following a suggestion by Weisskopf that the EM fields entering effective action are themselves screened by the vacuum response [8]. (4) The modification to effective action due to QED

induced anomalous magnetic moment has recently FIG. 1. Nonlinear contribution of Leff field for all −1=2 ≤ n ≤ 1 2 L B 0 attracted attention and we refrain from in depth = alongside the real part of EHS ( ¼ ). Choice of n does not L discussion here. We note that in addition to mass, the affect eff until stronger EM fields.

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L L This also assures that eff and the original EHS agree in E shape up to EHS field EHS, see Fig. 1. Because the action Eq. (35) has two free parameters, different choices of n ≤ 1=2 in Eq. (35) allow to choose the magnitude of finite EM field mass, as opposed to the original BI model where all electron mass was attributed to the EM field. A finite EM- mass arises in the range n ≤ 1=2 since the product E · D ∝ 1=r2 at the origin.

IV. EVALUATION OF EFFECTIVE MASS A. Model computation We compute the electric component of EM field mass via the displacement field, differentiating the model for effective action, Eq. (35): ∂L E2 n−1 D D D eff E 1 − : ¼ p þ ex ¼ ∂E ¼ E2 ð37Þ crðnÞ

D is numerically inverted to obtain an expression for E D E D D electric field ð Þ¼ ð p þ exÞ. We separately define the externally applied electric field E as the limit far from ex FIG. 2. Top: dashed lines plot mass-melting models with the particle: parameters n ¼ 1=2 and n ¼ 0, solid lines for the non-EM mass mH þ mBSM. Bottom: zoom-in detail of mass behavior below the E2 n−1 EHS field. lim D ¼ D ¼ E 1 − ex ; ð38Þ ⃗→∞ ex ex E2 jrj crðnÞ B. Perturbative self-consistent corrections E D E E numerically inverted to obtain exð exÞ. and ex are then The leading EM self-energy correction is known via the plugged into the mass density given by Eqs. (17) and (18): perturbative QED computation of the internal photon line correction to EHS action. To ensure that the effective form Z studied here is consistent, we compare the two and show E 3 D − D mEMð exÞ¼ d rðUð Þ Uð exÞÞ that the model effect is of the right order. Z To understand the weak field behavior of the mass ¼ d3rfðEðDÞ · D − L ðEðDÞÞÞ melting effect we expand the integrand in Eq. (39) in eff E “ ” powers of ex (from hereon we drop subscript ex ) and − E D D − L E D ð exð exÞ · ex effð exð exÞÞÞg: ð39Þ integrate to obtain the light-light scattering contribution. This is the quadratic in EM field mass modification: Eq. (39) is computed numerically and plotted in Fig. 2 for 2 the examples of n ¼ 1=2 and n ¼ 0. α E m ðEÞj 1 2 ¼ m ð0Þ 1 − − … ; The plots in Fig. 2 are a model example. We see the EM n¼ = EM 30π E2 EHS significant nonperturbative behavior and in the zoom-insert α E2 m E m 0 1 − − … : at bottom the domain which explains why one can often EMð Þjn¼0 ¼ EMð Þ 36π E2 ð40Þ argue that mass-melting is a minor effect. Total melting EHS of the EM field mass occurs at the critical fields E ≫ E E 0 crðnÞ EHS: p is suppressed as the externally applied The EM field mass mEMð Þ makes up the following E E 1 2 0 field ex approaches crðnÞ, due to the violation of linear portions of total electron mass for n ¼ = and n ¼ : superposition (see Sect. II). That is, when the total field 0.8703með0Þ and 0.9754með0Þ, Fig. 2 top. Counting E α αE2 E2 cannot exceed crðnÞ and the external field approaches powers in , the mass modification of order = EHS this limit, there is no room left for the particle field. At corresponds to the leading perturbative QED treatment, this point only the (model n-dependent) non-EM mass described diagrammatically by the electron propagator (Higgs þ BSM) components remain, flat solid lines in connected to a virtual photon loop that encloses two Fig. 2. external photon lines [2].

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We track how the effective action is modified by mass computation. Repeating this procedure forces us to con- melting in a self-consistent manner. Eq. (40) is input into sider higher orders in the semi-convergent perturbative the mass entering EHS action for a pure electric field: EHS series, along with higher order radiative corrections Z described above in Sec. III. Thus a full self-consistent ∞ 1 ds − 2 E computation is in principle nonperturbative and the radia- L LM − með Þs EHSþmelt ¼ 2 3−δ e 8π 0 s tive corrections have to be built in. × ðeEs cot½eEs − 1Þ: ð41Þ V. METHODS FOR MEASURING MASS Writing explicitly the bare charge in the Maxwell contri- MELTING MODEL bution to demonstrate renormalization procedure and A. Muon magnetic moment expanding to order E4, Turning our attention to precision QED experiments, we − 2 L consider the muonic g anomalous magnetic moment EHSþmelt Z (μAMM). Mass melting affects the electron mass entering 2 2 ∞ 2 4 e E 1 ds 2 ðeEsÞ ðeEsÞ the vacuum polarization contributions to the μAMM. The −meðEÞs ¼ 2 þ 2 3−δ e þ B 2e0 8π 0 s 3 45 muon sees ex, the external magnetic field applied in μ 2E2 2 2 E measuring the AMM, while the virtual electron sees both e e0 −1 2 með Þ B E B 1 δ − γ − m 0 − ex and the electromagnetic fields of the muon μ, μ. ¼ 2 2 þ 12π2 E ln½ eð Þ ln 2 0 e0 með Þ Figure 3 shows the contribution to the μAMM in which αE4 the electron loop is ‘doubly-dressed’: zoom-in shows the þ 90πE2 : ð42Þ propagator dressed by the muon’s fields (two-line), and EHS summation of its self-energy corrections—the two-line We apply renormalized charge propagator sums all photons connecting the electron loop to the muon, and the self-energy correction sums virtual 1 1 2 photons that only couple to the electron. 1 e0 δ−1 − γ − 2 0 2 ¼ 2 þ 2 ð E ln½með ÞÞ ; ð43Þ The electron vacuum polarization correction to the e e 12π 0 μAMM is given by Eq. (77) in [11] and use effective mass model Eq. (40) to write the 2 2 α 1 mμ 25 remaining finite logarithmic expression to order E , Δaμðm ðEÞÞ ∼ ln − : ð47Þ e π2 3 m ðEÞ 36 e 2 E 0 8703α E2 með Þ − . ln 2 ¼ 2 ; We have extended this well known expression by allowing m ð0Þ 1 2 30π E e n¼ = EHS the electron mass to be function of the EM field, i.e., to 2 2 m ðEÞ 0.9754α E melt. The ln½mμ=me term is sensitive in nonperturbative ln e ¼ − : ð44Þ 2 36π 2 fashion to value of me that is subject to mass melting. We með0Þ 0 E n¼ EHS did not suggest that the muon mass melts—we expect that To order E4 the effective action becomes the electron proportionally has more EM field mass than the muon, and the muon requires much stronger fields (on 2 αE4 0.8703α the order of mμ=e) for noticeable EM field mass modifi- L LM 1 ; EHSþmeltjn¼1=2 ¼ þ90πE2 þ 2π cation to occur. EHS The electron and its mass is impacted by the strong muon αE4 0.9754·5α L LM 1 ; EM field; we consider this remark in quantitative manner EHSþmeltjn¼0 ¼ þ90πE2 þ 12π ð45Þ EHS by estimating the relative virtual electron pair location with respect to the muon. To illustrate that the virtual electrons aligning in order of α well with the known 2-loop QED cannot resolve the muon to distances smaller than the correction to EHS action [10,32,43], αE4 40α M L 2 L 1 : EHSþ loop ¼ þ 90πE2 þ 9π ð46Þ EHS

We see that the EM field effective mass correction Eq. (45) is smaller compared to QED perturbative result Eq. (46). For higher order in α and E effects, the self-consistency requirement means that the corrected “EHS þ melt” effec- FIG. 3. Electron vacuum polarization contributions to μAMM: “ ” B tive action must again be plugged back into EM field mass x denotes ex.

036014-7 STEFAN EVANS and JOHANN RAFELSKI PHYS. REV. D 102, 036014 (2020) muon Compton wavelength, we assign a charge distri- strong Coulomb fields was also noted by Sikora and 1=2 2 2 −1 −3 bution ρμðrÞ¼2α exp½−r =ƛμπ ƛμ . This assumes collaborators [44]. However these authors considered the that the muon is localized to the distance of one muon electron AMM only. Compton wavelength, much smaller compared to that of To estimate the order of magnitude of the nonperturba- the electron: ƛμ ¼ðme=mμÞ·ƛe ¼ð1=206Þ·386fm¼1.87fm. tive effect we evaluate as an example the electron mass The muon does not possess such a charge distribution, subject to a field strength of 4 times the EHS field, giving but this provides an estimate for the field strengths that meðEÞ¼0.99876með0Þ and meðEÞ¼0.99897með0Þ, for the virtual ‘quantum sized’ particles experience. We com- n ¼ 1=2 and n ¼ 0 respectively. The μAMM increases by pute the induced perturbative current derived by Schwinger [9] and repeat this computation replacing the muon Δ E − Δ 0 2 23 10−9 aμðmeð ÞÞjn¼1=2 aμðmeð ÞÞ ¼ . × by an electron localized to its Compton wavelength. In Δ E − Δ 0 1 86 10−9 Figs. 4 and 5 we plot the muon and respective electron aμðmeð ÞÞjn¼0 aμðmeð ÞÞ ¼ . × ; ð48Þ 1=2 2 2 −1 −3 (ρeðrÞ¼2α exp½−r =ƛeπ ƛe ) charge distributions, and their induced (virtual electron) polarization charge close to the discrepancy between experimental and theo- clouds. retical values of 2.9 × 10−9 [11]. The induced charge is much closer to the muon than the We do not discuss in comparable depth the electron- electron: we mark the muon Compton wavelength at which AMM, as it is less affected than the μAMM. As noted Eμ is 309 times the EHS field, and further out the radius at already, due to the difference in Compton wavelengths, which Eμ is equal to the EHS field. We see that the muon’s virtual electrons exist much farther from a real electron than induced polarization charge occupies the domain of field from a muon, experiencing weaker mass melting fields: at ƛ 1 137 strengths in which perturbative treatment of the polarization e the electric field is only = of the EHS field, at which loop contribution to g − 2 cannot be trusted. The electron’s perturbative QED is valid and higher order loop diagrams induced charge lies in the perturbative regime and thus the have been highly constrained [45]. The model gives a −9 electron’s g − 2 avoids this difficulty. The importance of maximum melting effect of meðEÞ ∼ með0Þð1 − 10 Þ, nonperturbative AMM computation in the presence of modifying the electron g − 2 on the order of one part per 1012, two orders of magnitude below the experimental uncertainty [46].

B. Pair production in strong fields As ultrashort pulse laser fields approach strengths capable of probing vacuum instability, the EHS action has been a focus of much theoretical attention [10,12,47]. We explore how the mass melting modifies the imaginary part of EHS action describing the pair production

2 X∞ − π 2 e ab coth½nπb=ae n meða;bÞ=ea Im½L ¼ : EHSþmelt 8π3 n n¼1 FIG. 4. Solid: muon charge distribution, dashed: induced electron vacuum charge. ð49Þ

FIG. 6. Enhancement of pair production for effective models FIG. 5. Solid: electron charge distribution, dashed: induced with n ¼ 1=2, 0, normalized to the original result obtained with electron vacuum charge. constant electron mass.

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From Eq. (49) the rate of pair production and vacuum decay hydrogen: for a long time there has been a Lamb-shift time is obtained [48]. In Fig. 6 the enhancement of pair discrepancy [56], which in part motivated this work. production according to Eq. (49) with mass modified by the However this may have been resolved recently [57]. model computation, Sec. IVA, is shown. Since the muon and proton fields cancel at least in part, We find that the mass melting model does not signifi- we expect the mass melting effect to be smaller than in the cantly modify pair production until EM field strengths case of the muon g − 2 discrepancy. E beyond EHS are reached, that is when a significant portion Regarding future theoretical developments, we return to of electron mass has melted. While high intensity lasers are Sec. III where we have discussed how we plan to improve unlikely to be suitable for measuring mass melting until the EHS action for quasiconstant fields accounting for they reach the EHS field strength regime, stronger fields field-dependent mass and the recent development in reduc- capable of probing the effect are generated in heavy ion ible diagram summation [30,31]. Another modification is – collisions [49 52]. the nonlocal correction to account for nonlinear mixing between the external fields and the inhomogeneous fields VI. OUTLOOK of the particles [26–29]. These improvements must be We proposed an effective formulation of the EM field incorporated in a self-consistent manner: the nonlinear mass in the strong field regime not amenable to perturbative effective action, used to compute the external field-depen- QED evaluation. We discuss the possible experimental dent mass via our effective formulation, also contains the implications. corrected mass built in. The external field-dependent electron mass can be Another topic discussed in Sec. III concerns how non- obtained considering nonlinear EM theory: The violation linear EM model theories may be used to model QED of superposition in nonlinear EM theories is described in effective action. The model action is a monotonically Sec. II. The effective EM field mass given in Eq. (13) increasing function that matches the known EHS light- causes mass modification, Eq. (27). We have explored this light scattering, yet the two actions differ in stronger fields. effect using a model limiting field effective action adjusted A questionable sign flip occurs in the real part of EHS E to match the EHS light-light scattering, Eq. (35). In our action beyond EHS, which causes the EM field mass to be model mass melting aligns in magnitude with perturbative divergent. This sign flip is in the regime where higher order QED for weak fields (up to field-gradient corrections), and corrections are capable of significantly altering the action. produces a large degree of melting in the regime of Since mass melting reduces the EHS field Eq. (34),we supercritical fields where an exact QED evaluation is not expect a self-consistent computation will produce a com- available, Fig. 2. pounding effect that becomes important in strong fields, We explored observable consequences of the mass possibly producing a further change in sign. melting in the μAMM. The EM fields sourced by the Study of mass melting should also consider the muon at the location of virtual polarized eþe−-pairs are magnetic moment’s contribution to EM field mass, which in the regime in which nonperturbative strong field QED may affect the predicted rate of mass melting. Such a computation is required, Fig. 4. Applying the mass computation is more involved: so far the generalized melting model prediction we found a narrowing of the model limits only the E field [6], and an extension of the experimental result discrepancy with the theory Eq. (48). model to limit both E and B has not yet been invented: We relate to a proposed resolution to the μAMM the BI-like models are singular upon inclusion of a point- discrepancy that involves an external scalar field like magnetic dipole. Since the dipole field is important [53,54], since the external field-dependent shift in mass at a smaller radius than the Coulomb field, whether the can be described in terms of an effective scalar potential. electric or magnetic mass component is the dominant Our nonperturbative result influences the μAMM in a contribution depends on the nonzero effective size of the manner not visible to perturbative QED evaluation. The electron that arises [20]. This size lies in between the two model prediction has a negligible effect on the electron physically relevant length scales: the classical electron AMM, due to the weaker EM fields experienced by the radius (2.82 fm) and the electron Compton wave- virtual electrons. We have also evaluated a mass melting length (386 fm). enhancement of pair production in fields relevant to The work we presented is a tip of an iceberg of many heavy ion collisions, Sec. VB. questions that our insight about the effect of EM field A future area of interest is that of high Z atoms, which nonlinearity presents, inherent in QED. It is possible that also probe mass melting of the electron: An electron in the there is not a smooth mass melting but a mass discontinuity hydrogen-like uranium 1S-state is likely subject to mass that will appear in the presence of both electric and melting since the atomic nucleus with Z ¼ 92 provides magnetic self-mass sources. Our work could also revive E strong EM fields, on the order of EHS, and there is ample effort to conduct numerical study of QED leading to model experimental measurement of the 1S-state binding energy independent nonperturbative evaluation of EM field mass to probe mass melting [55]. We also comment on muonic melting in the presence of strong fields.

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The methods we presented should allow the develop- ACKNOWLEDGMENTS ment of theoretical description of the EM mass compo- The authors would like to express their gratitude to nent of the electron in presence of strong external fields. Dr. Tamás Biró and Dr. P´eter L´evai for their hospitality at A natural consequence of this QED based reconsidera- the Wigner Research Center for Physics, Budapest, and at tion of electron mass is the differentiation between the Balaton Workshop, where in part this work was carried intrinsic material mass (due to Higgs field) and the out. Johann Rafelski was a Fulbright Fellow during this EM electron mass, otherwise inaccessible to present day period. experiment.

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