]>PLB34490S0370-2693(19)30162-510.1016/j.physletb.2019.03.008The AuthorsPhenomenologyFig. 1EHS complements: vacuum fluctuation contributions fourth order in the photon field. Left: four photon interactions via virtual ALP; and Right via PCAC pion anomaly.Fig. 1Fig. 2We adapt Figure 111.1 from Ref. [6], adding purple lines to denote a new region labeled ‘virtual effect’, where according to Eq. (20) the ALP supplement to effective action surpasses the first nonlinear EHS effect.Fig. 2Virtual axion-like particle Complement to Euler-Heisenberg-Schwinger actionStefanEvans⁎evanss@email.arizona.eduJohannRafelskiDepartment of Physics, The University of Arizona, Tucson, AZ 85721, USADepartment of PhysicsThe University of ArizonaTucsonAZ85721USA⁎Corresponding author.Editor: A. RingwaldAbstractWe modify action in an external electromagnetic field to include effects of virtual axion-like particle (ALP) excitations. A measurable addition to QED-Euler-Heisenberg-Schwinger (EHS) action is obtained and incorporated into experimental constraints placed on ALP mass and coupling to two photons. The regime of these constraints in which the ALP vacuum effect surpasses the EHS effect is characterized. We show that probing of the virtual vacuum effect offers an alternative method in search for physics related to ALPs.1IntroductionWe show that a pseudoscalar coupling between two photons and axion-like particles (ALP) produces a nonlinear in EM field addition to the QED vacuum effect akin to a specific term in the Euler-Heisenberg-Schwinger (EHS) action [1,2]. We report on this additional effect and the conditions required for it to be of or above in magnitude to the QED vacuum fluctuation EHS result.The Adler-Bell-Jackiw anomaly is often used in study of pseudoscalar decay process π0→γγ [3,4]. This was followed by proposed Axion-Like-Particle (ALP) ↔γγ processes. Searches for these processes focus on possible flux of on-mass shell ALP particles from astronomical sources [5,6]. Another consequence of the ALP↔γγ interaction is a possible modification of the vacuum by virtual fluctuations which we describe here.The virtual EHS effect of the electron loop on electromagnetic processes has motivated measurement of vacuum birefringence in strong magnetic fields [7,8]. We complement this effective QED-EHS action including effects on vacuum structure by two other effects, the virtual ALP excitations and pions – the ALP couples to two photons with point particle form factor while the pion via quark triangle loop, see Fig. 1.The ALP supplement to EHS action is obtained by rescaling ALP fields to include photon-loop corrections. This produces a fourth-order in EM field addition to EHS action, a consequence of ALP fluctuations introducing an extra diagram contributing to vacuum polarization shown in the left-hand-side of Fig. 1.The strength of the ALP fluctuation supplement to EHS action is characterized by ALP-γγ coupling involving the ratio of coupling and mass GA/mA. Observations of astrophysical ALP sources have provided constraints on GA, mA: see [5,6] and references therein. Unlike in the case of axions, for ALPs GA/mA is not fixed. While the fixed value of pseudoscalar-γγ coupling to mass ratio (or product of mass and decay constant) excludes significant additions to EHS action in the case of axions and pions, the ALP contribution does not have this restriction. We obtain an ALP action supplement to the EHS result valid for external field photon frequency ω<mA, that is for external fields varying slowly over the ALP Compton wavelength. Ongoing search for ALP effects in vacuum birefringence in quasi-constant magnetic fields is carried out by PVLAS [9–13] and BMV [14,15]. While these experiments use laser wavelengths outside of domain ω<mA, they probe mA and GA in a regime where the here obtained ALP supplement surpasses EHS action. Existence of an ALP with parameters in this regime would significantly modify nonlinear vacuum effects, manifest as light-light scattering by long wavelength lasers: see [16–18] and references therein.In the following we evaluate the size of the ALP contribution complement to the EHS effective action in section 2. We describe the experimental constraints in section 3, where the domain of GA/mA is shown in which the ALP contribution surpasses the corresponding contribution from the EHS action.2Pseudoscalar coupling input to effective EHS action2.1ALP complement to EHSWe consider the effective QED action using EM field invariants(1)S=12(E2−B2),P=E⋅B, in the respective terms(2)L=S+LEHS+Lφ+Lint. We see first the Maxwell action, complemented by renormalized effective EHS action, since in the interaction term bare (e0E0)(e0B0) is equivalent to renormalized (eE)(eB):(3)LEHS=me4f(Sme4,P2me8), where me is electron mass and the function f is well known [1,2]. Note that for reason of parity conservation by the QED vacuum, a P-term must have even powers in Eq. (3). The low field expansion generates the well known term(4)LEHS(1)=e4(4π)2245me4(4S2+7P2), where superscript (1) denotes action up to the leading nonlinear EM contribution.The two supplemental and here relevant ALP terms in Eq. (2) include the pseudoscalar mass contribution(5)Lφ=−mA22φ2, where φ is the ALP field. We neglect the kinetic energy term, as we are interested in the infrared limit of vacuum fluctuation contribution. For the effective ALP with two photon interaction term [5,6](6)Lint=±GAφP, where ALP to two photon coupling GA[GeV−1] is constrained by experimental observation. Note that the sign in the effective action Eq. (6) requires further consideration: its value will not appear in our computations which are quadratic in this term, thus we choose in further considerations the positive sign.The total action we thus consider comprises three contributions: EHS Eq. (4) and the two ALP terms(7)L(1)=S−mA22φ2+GAφP+e4(4π)2245me4(4S2+7P2). The ALP degrees of freedom can be ‘rotated’ to diagonalize the ALP-γγ interaction. Completing the square,(8)−mA22φ2+GAφP=−mA22(φ−GAPmA2)2+GA22mA2P2=−mA22φ˜2+GA22mA2P2, where the ALP-field is configuration mixed with electromagnetic γγ-contribution(9)φ˜=φ−GAPmA2. After configuration mixing there is an additional contribution, the last term in Eq. (8), which like the considered term of EHS action (second term in Eq. (4)) is proportional to P2 but only depends on ALP properties and electron charge, and is thus independent of electron mass.The total action up to 4th order in EM fields is(10)L=S−mA22φ˜2+L˜EHS, where the first term is the QED Maxwell field action, followed by ALP mass contribution and(11)L˜EHS=LEHS+GA22mA2P2, independent of the sign of the ALP interaction term in Eq. (6), as mentioned earlier. The ALP supplement to EHS action is valid for EM fields characterized by external photon frequency ω<mA. EM fields must be quasi-constant over the ALP Compton wavelength, a distance greater than the electron-positron Compton wavelength for the ALP masses we consider in the section 3.The ALP contribution is written as an addition to the lowest order term in EHS action (Eq. (4)),(12)L˜EHS(1)=e4(4π)2245me4(4S2+7P2)+GA22mA2P2. This is equivalent to modification of the P2 coefficient in Eq. (4) by(13)P2→P2(1+45me428α2GA2mA2), where α=e2/4π. In order for the ALP effect to be equal or larger than the QED effect, the ratio of ALP-γγ coupling to mass mA amounts to(14)GA2mA2≥28α245me4. This condition will be used to characterize presence of a virtual pseudoscalar effect on electromagnetism comparable to electron loop effects, in context of constraints from astronomical observation and vacuum birefringence experiment. We first discuss the differences between ALPs, axions and pions in context of Eq. (14).2.2Pion and axionEvaluation of the P2 rescaling in Eq. (13) is repeated for the pion, a result of supplementary diagram to QED vacuum polarization shown in the right-hand-side of Fig. 1. Eq. (6) is replaced with(15)Lint=α2πfπφP, where pion decay constant fπ=93MeV [19]. This is reminiscent of Schwinger's result characterizing pion decay, which was missing a phase factor enforcing gauge invariance, summarized and reconciled with PCAC in [20–22].Repeating steps in section 2.1, with φ now denoting a pion field and mA→mπ∼135MeV,(16)P2→P2(1+45me4112π21fπ2mπ2)=P2(1+O(10−10)), producing a negligible addition to EHS action. Using instead an axion-field, the result is still characterized by a product fama on the order of fπmπ [5,6]: the virtual axion effect does not produce a significant contribution to L˜EHS either.Even though the result for pions is very small, we note that the decay constant fπ is measured in timelike kinematic domain with pion momentum Qπ2→mπ2. Whether application of fπ=93MeV to the infrared domain Qπ2→0 of external EM fields is valid remains an open question. While this issue of kinematic domains awaits resolution, we only consider vacuum fluctuations of ALPs and not pions nor axions. We proceed to experimental constraints for ALPs.3Evaluated ALP vacuum fluctuation effect3.1Constraints on GA and mAWe wish to add our result Eq. (14) into constraints on GA and mA, based on results provided in Figure 111.1 from [6]. All these constraints are based on real axions, with the exception of virtual effects probed by the vacuum birefringence measurements carried out by the PVLAS [10–13] collaboration. Our addition is a virtual effect, a supplement to EHS effective action.For clarity we describe in detail how the new domain arises: Taking the square root of Eq. (14) and writing units explicitly,(17)GA[GeV−1]mA[eV]≥2845α1(0.511MeV)2. We write(18)GA[GeV−1]=10yGeV,mA[eV]=10xeV, where the coefficients y, x serve as variables log10GA[GeV−1], log10mA[eV] in Fig. 2. Eq. (17) becomes(19)10y−x−9(eV)2≥2845α10−12(0.511)2(eV)2=2.21⋅10−14(eV)2. Taking log10 and keeping only y on the left hand side,(20)y≥x+log102.21⋅10−5=x−4.66. With this we define boundary at which ALP and QED effects equal:(21)yQED=xQED−4.66→GA(QED)mA(QED)=10−4.66(GeV)(eV).In Fig. 2, the region of Figure 111.1 of [6] where y,x satisfy Eq. (20) is shaded. The boundary of the shaded region, at which the ALP and QED effects are equal is given by Eq. (21). This boundary is parallel to the KSVZ and DFSZ models for the axion shown in the figure, which have fixed coupling to mass ratios (denoted axion):(22)yaxion∼(xaxion−4.66)−5→GA(axion)2mA(axion)2=10−10GA(QED)2mA(QED)2. This is in agreement with the ∼10−10 suppression of virtual pion vs QED effect in section 2.2, recalling the similar (differences are model-dependent) products of decay constants and masses of axions and pions.3.2PVLAS resultThe summary of observational data presented in Figure 111.1 in Ref. [6] and copied into our Fig. 2 includes an update from PVLAS, measuring vacuum birefringence and dichroism in external magnetic fields. PVLAS 1992 work excludes massless ALPs: mA<10−3 eV at coupling GA>3.6⋅10−7GeV−1 [10]. PVLAS 2006 results suggested a coupling stronger by factor ∼10, subsequently revised by more recent results [10–13]. Even without factor ∼10 enhancement of coupling, ALP constraints lie within the ‘virtual effect’ region in Fig. 2, where an ALP virtual contribution surpasses the EHS effect.We find that the updated PVLAS range plotted, see [13], is included within the shaded region denoting prominent virtual effects:(23)GA(PVLAS)2mA(PVLAS)2∼(10−7(GeV−1)10−3(eV))2=101.32GA(QED)2mA(QED)2, using Eq. (21) in the last line. Thus the PVLAS constraint suggests a possible virtual pseudoscalar effect as much as 20 times the strength of the QED-EHS effect.4ConclusionThe ALP vacuum effect as shown in Fig. 2 covers a large domain provided by the astronomical observation constraints. These results rely on propagation of real ALPs, either over large distances from astronomical sources, or length scales probed by resonant cavities and light-through-wall experiments, see [5,6] and references therein.In the virtual ALP experiments EM fields must be strong, preferably as near as possible to the critical field for the electron loop: Ec≡me2c3/eħ=1.3⋅1018V/m, in order to probe the effects inherent to EHS action and virtual ALPs. We note that as EM fields exceed the domain in which low field expansion of action to order P2 is valid, higher order (photon number) ALP interactions must be evaluated: another paper is required to study this limit. The low field expansion studied here behaves differently for the ALP contribution than in EHS action. In the latter subcritical fields must satisfy E2/Ec2<1 and B2/Ec2<1, while the ALP term requires PGA2/αmA2<1. Thus a supercritical EM field in context of EHS action may be subcritical for ALP effects if P is small. Even for small mass mA<10−5 eV, subcritical requirement for low field expansion is satisfied for appropriate ratio GA/mA. However in considering the possibly small ALP mass range in Fig. 2, we further need EM fields quasi-constant over the range of the Compton wavelength of the ALP: ω<mA<10−5 eV. Study of light-light scattering ([16–18]) in this long wavelength limit offers a probe for significant nonlinear vacuum effects: the regime where virtual ALP modification of Maxwell equations surpasses that from EHS action.The virtual effect region of Fig. 2 includes the domain of GA, mA obtained by PVLAS, which we benchmark at mA=10−3 eV and GA=10−7GeV−1. The PVLAS experiment probes virtual ALP and e+e− effects via vacuum birefringence in an external magnetic field [10–13]. The external field driving the 4-photon interaction in Fig. 1 consists of a magnetic field of 2.5T constant over ƛc, and most recently a 1064 nm laser [13], similar to BMV parameters [14,15]. While external fields strengths satisfy condition for low field expansion in P2 given above, the laser wavelength is smaller by a factor ∼10−2 than Compton wavelength ƛc=ħ/mAc=2×10−4m for mA=10−3 eV. A larger mass mA∼10−1 eV is required in order for ω<mA to be satisfied. Constraints on this larger mass via study of vacuum effects on external EM fields await resolution [5]. However, we note that in context of EHS action virtual behavior of periodic fields may correspond to that of constant fields [23,24]. Whether this argument can be extended to ALP effects awaits resolution in future work: it is possible that the PVLAS experiment [10–13] using a standing laser wave and an external magnetic field probes virtual effects on the required length scale.We note ongoing effort to study the very strong field environments with high intensity lasers at ELI [25,26], and the long lasting study of supercritical fields in relativistic heavy-ion collisions [27,28]. Both methods offer encouraging prospects as probing methods for QED + ALP action though the length and mass scale is today quite different, with EM fields varying over a shorter range than the usually applied ƛc range.We conclude that our consideration of virtual ALP effects adds a new method for detection of ALPs that does not rely on propagation of real ALPs. Should ALPs exist only virtually in the vacuum (like quarks and gluons), they will never be discovered as free-streaming particles, but could via vacuum fluctuations we evaluated in this work. This observation adds to the findings we presented in Fig. 2 an additional motivation to relevant experiments, such as PVLAS and BMV.References[1]W.HeisenbergH.EulerFolgerungen aus der Diracschen Theorie des PositronsZ. Phys.98193671410.1007/BF01343663[2]J.S.SchwingerOn gauge invariance and vacuum polarizationPhys. Rev.82195166410.1103/PhysRev.82.664[3]S.L.AdlerAxial vector vertex in spinor electrodynamicsPhys. Rev.1771969242610.1103/PhysRev.177.2426[4]J.S.BellR.JackiwA PCAC puzzle: π0→γγ in the σ modelNuovo Cimento A6019694710.1007/BF02823296[5]J.JaeckelA.RingwaldThe low-energy frontier of particle physicsAnnu. Rev. Nucl. Part. Sci.60201040510.1146/annurev.nucl.012809.104433[6]A. Ringwald, L.J. Rosenberg and G. Rybka, Axions and other similar particles, Review of Particle Physics, Chapter 111 in:M.TanabashiParticle Data GroupPhys. Rev. D983201803000110.1103/PhysRevD.98.030001[7]Z.Bialynicka-BirulaI.Bialynicki-BirulaNonlinear effects in Quantum Electrodynamics. Photon propagation and photon splitting in an external fieldPhys. Rev. D21970234110.1103/PhysRevD.2.2341[8]E.IacopiniE.ZavattiniExperimental method to detect the vacuum birefringence induced by a magnetic fieldPhys. Lett. B85197915110.1016/0370-2693(79)90797-4[9]L.MaianiR.PetronzioE.ZavattiniEffects of nearly massless, spin zero particles on light propagation in a magnetic fieldPhys. Lett. B175198635910.1016/0370-2693(86)90869-5[10]R.CameronG.CantatoreA.C.MelissinosG.RuosoY.SemertzidisH.J.HalamaD.M.LazarusA.G.ProdellF.NezrickC.RizzoE.ZavattiniSearch for nearly massless, weakly coupled particles by optical techniquesPhys. Rev. D471993370710.1103/PhysRevD.47.3707[11]E.ZavattiniG.ZavattiniG.RuosoE.PolaccoE.MilottiM.KaruzaU.GastaldiG.Di DomenicoF.Della ValleR.CiminoS.CarusottoG.CantatoreM.BregantExperimental observation of optical rotation generated in vacuum by a magnetic fieldPhys. Rev. Lett.96200611040610.1103/PhysRevLett.99.129901arXiv:hep-ex/0507107Erratum:Phys. Rev. Lett.99200712990110.1103/PhysRevLett.96.110406[12]E.ZavattiniG.ZavattiniG.RaiteriG.RuosoE.PolaccoE.MilottiV.LozzaM.KaruzaU.GastaldiG.Di DomenicoF.Della ValleR.CiminoS.CarusottoG.CantatoreM.BregantNew PVLAS results and limits on magnetically induced optical rotation and ellipticity in vacuumPhys. Rev. D77200803200610.1103/PhysRevD.77.032006[13]F.Della ValleA.EjlliU.GastaldiG.MessineoE.MilottiR.PengoG.RuosoG.ZavattiniThe PVLAS experiment: measuring vacuum magnetic birefringence and dichroism with a birefringent Fabry-Perot cavityEur. Phys. J. C76120162410.1140/epjc/s10052-015-3869-8[14]R.BattestiB.Pinto Da SouzaS.BatutC.RobilliardG.BaillyC.MichelM.NardoneL.PinardO.PortugallG.TrénecJ.M.MackowskiG.L.J.A.RikkenJ.ViguéC.RizzoThe BMV experiment: a novel apparatus to study the propagation of light in a transverse magnetic fieldEur. Phys. J. D46200832333310.1140/epjd/e2007-00306-3[15]M.T.HartmanR.BattestiC.RizzoCharacterization of the vacuum birefringence polarimeter at BMV: dynamical cavity mirror birefringencearXiv:1812.10409 [physics.ins-det][16]M.SoljacicM.SegevSelf-trapping of electromagnetic beams in vacuum supported by QED nonlinear effectsPhys. Rev. A62200004381710.1103/PhysRevA.62.043817[17]F.BrisceseLight polarization oscillations induced by photon-photon scatteringPhys. Rev. A965201705380110.1103/PhysRevA.96.053801arXiv:1710.03338 [physics.optics][18]F.BrisceseCollective behavior of light in vacuumPhys. Rev. A973201803380310.1103/PhysRevA.97.033803arXiv:1710.07703 [physics.optics][19]K.HuangQuarks, Leptons and Gauge Fields1992World ScientificSingapore333 pp[20]B.ZuminoAnomalous properties of the axial vector currentConf. Proc.C6901141969361[21]S.B.TreimanE.WittenR.JackiwB.ZuminoCurrent Algebra and Anomalies1985World ScientificSingapore537 pp[22]S.L.AdlerAnomalies to all ordersG.'t Hooft50 Years of Yang-Mills Theory2005World ScientificHackensack, USA487 pp[23]T.N.TomarasN.C.TsamisR.P.WoodardBack reaction in light cone QEDPhys. Rev. D62200012500510.1103/PhysRevD.62.125005arXiv:hep-ph/0007166[24]J.AvanH.M.FriedY.GabelliniNontrivial generalizations of the Schwinger pair production resultPhys. Rev. D67200301600310.1103/PhysRevD.67.016003arXiv:hep-th/0208053[25]H.GiesStrong laser fields as a probe for fundamental physicsEur. Phys. J. D55200931110.1140/epjd/e2009-00006-0[26]G.V.DunneNew strong-field QED effects at ELI: nonperturbative vacuum pair productionEur. Phys. J. D55200932710.1140/epjd/e2009-00022-0[27]R.RuffiniG.VereshchaginS.S.XueElectron-positron pairs in physics and astrophysics: from heavy nuclei to black holesPhys. Rep.4872010110.1016/j.physrep.2009.10.004[28]J.RafelskiJ.KirschB.MüllerJ.ReinhardtW.GreinerProbing QED vacuum with heavy ionsS.SchrammM.SchäferNew Horizons in Fundamental PhysicsFIAS Interdisc. Sci. Ser.201721125110.1007/978-3-319-44165-8_17