JHEP12(2019)162 Springer May 22, 2019 : November 5, 2019 December 30, 2019 November 26, 2019 : : : Received Revised Accepted Published , c Published for SISSA by https://doi.org/10.1007/JHEP12(2019)162 [email protected] , and G. Gregori c [email protected] , K.A. Beyer b . 3 1905.05201 The Authors. c Beyond Standard Model, Precision QED

The decay of a massive pseudoscalar, scalar and U(1) boson into an electron- B.M. Dillon, , 1 [email protected] a, Corresponding author. 1 Jamova 39, 1000 Ljubljana, Slovenia Department of Physics, UniversityParks of Road, Oxford, Oxford, OX1 3PU, U.K. E-mail: [email protected] Centre for Mathematical Sciences, UniversityPlymouth, of PL4 Plymouth, 8AA, U.K. Jo˘zefStefan Institute, b c a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: probability. A strong-field limitIn is modelling identified in the which fall-off the ofties beyond threshold a the quasi-constant-crossed is constant magnetic found limit field, toas and we disappear. locally investigate calculate constant. when probabili- the decay probability can be regarded netic field. Theproduct total of field probability and depends lightfront momentumdifferent on invariants. decay Depending the regimes on are quantum the identified.depends seed nonlinearity particle In on parameter mass, the a — below-threshold non-perturbative case, a and tunneling we the exponent particle find depending mass. the on probability Ineter the the is above-threshold quantum varied case, parameter linearly, we the find that probability when oscillates the nonlinearly quantum around param- the spontaneous decay Abstract: positron pair inparticular the interest presence is the of constant-crossed-fieldmeasure strong limit, high-energy relevant electromagnetic axion-like-particle for conversion backgrounds experiments into that is electron-positron aim calculated. to pairs in Of a mag- B. King, Axion-like-particle decay in strong electromagnetic backgrounds JHEP12(2019)162 ], 10 4 ]) to convert laser 9 ], and the 3.55 keV galaxy 5 – ], which has a weak coupling 2 1 13 – 1 – 16 11 9 12 1  ]. k 8 – χ 6 8 1 18 A promising route to detecting ALPs is through their coupling to SM particles in the 3.1 Below threshold3.2 decay Above threshold3.3 decay Strong fields, photons in a magnetic backgroundinto into a ALPs, low-noise which detection then propagate regionof through with this a a background “wall” background field and magnetictal are field. then signal. ALPs reconverted into in The photons,however the current upgrades which presence state-of-the-art to provide LSW the this experiment experimen- set-up is and the other more ALPS advanced I LSW experiment experiments [ are planned have subsequently been suggested totransparency explain of various astrophysical the phenomena universecluster such to emission as the line high [ energy gamma-rays [ electromagnetic sector. The couplingLight-Shining-through-the-Wall (LSW) of experiments ALPs (for to a the review electromagnetic see field [ is exploited in term to be aa pseudoscalar dynamical Nambu-Goldstone parameter boson thatto called can photons the relax and axion toModel electrons, [ zero scenarios predict as and the well predicts existenceand of the as light electrons, existence bosonic other which states of SM that are couple particles. referred weakly to to photons Other collectively beyond-the-Standard- as Axion-Like-Particles (ALPs). They “Naturalness” seems incompatiblethe with “strong-CP” the problem, Standard whichviolated Model asks so (SM) why little charge-parity whenLagrangian in (CP) one would the conjugation considers induce strong invarianceattractive a is sector solution large is even the but though Peccei-Quinn unobserved (PQ) the mechanism neutron CP-violating which electric promotes term the dipole in CP-violating moment. the QCD An 1 Introduction 4 Edge effects of static5 constant fields Example experimental scenarios 6 Summary 3 Constant fields Contents 1 Introduction 2 Derivation of pseudoscalar decay probability in a plane-wave pulse JHEP12(2019)162 ] ]), for 25 27 χ ], (more 23 , ]. 22 16 ]). Depending on the and corresponding an- 24 , which we are studying l φ ] and the proposed IAXO 13 ] and for the decay of a neutral 28 (for lepton l , for axions ν − ], to suppress particle production. Also, l e 29 + → e ]. When studying neutral pion decay, it was ]). Whilst we note some similarities between 30 10, a sizeable contribution of the decay width → W 32 – 2 – φ ≥ ] fields, the decay lifetime of a muon has also been χ 21 – ], is being measured. This means that the signal in he- 19 were calculated, where it was found that in the limit of vector bosons in a CCF was investigated, where it was l 15 l  → W ]) that a CCF can accelerate the decay into a muon but slow 0 Z , which is forbidden in vacuum. Our work complements these 31 , t t 22 ]. Helioscope experiments, i.e. CAST [ → ], a weak magnetic field has been found to slightly enhance the decay 12 0 , 26 Z ] and plane-wave [ 11 ], also use a similar detection set-up, but since the generation stage occurs in ] the decay 18 , ), was a nonmonotonic function of the quantum nonlinearity parameter ], the decay of 14 and pion decay studies. We will derive the process in a plane wave of arbitrary l 34 ν 0 17 33 Z 1. In [ In the current paper we study the process of an ALP decaying to an electron-positron previous pulse form, hence extending theworks. treatment of We the will background usethat beyond the decay this can CCF arbitrary occur of form previous outsidelocally-constant-field to of approximation the model (LCFA). field, the This which edge result is not of will a a be result magnetic shown that field, is to obtainable and be using show due the to the found that the partialtineutrino width of theχ decay < large quantum nonlinearity parameter originates from studies, as we see that axion decay to a pair is most closely related kinematically to the down the decay into anours electron and Ritus’ (see results also for [ and pion of decay in a a general plane-wavekinematics background, nature. these of are Moreover, the rather the process limited here. processes as studied the In by process [ Ritus do not have the same but for parallel Sauter typescalar electric to and two charged magnetic scalars fieldsthe in [ effect a thermal of bath a [ form constant factors magnetic has field recently on beenfound the investigated (first [ neutral by and Ritus charged [ pion masses and decay recent speculations to the contraryset-up were criticised of in background the fields, literature aFor [ magnetic example, field for may constant enhanceand or homogeneous suppress scalar parallel particle QED electric production. [ andof magnetic the fields vacuum in into electron-positron QED pairs [ (more details can be found in the review [ which there are very fewon results altering in the particle literature. decay propertiesbe This using study of external adds interest to electromagnetic for the (EM)magnetic overall future fields [ discussion and lab-based could ALP searches.investigated, and shown In to addition be only to slightly the changed in decay an of EM background photons in in [ and a bound on this coupling has been derivedpair by in the a high-intensity CAST electromagnetic collaborationThe field [ motivation via for a direct ourground; coupling work ii) of is: to the look ALP i) to atALPs; to the electrons. iii) effect consider learn of the something the decay about pulse-shape field-dressed of on processes the ALPs background in in on strong-field a detection QED of plane-wave (SFQED), these back- of experiment [ complex astrophysical environments such ascoupling in the in sun, e.g. both the productione.g. Primakoff via axionic-Compton the process, emission di-photon and [ lioscope production experiments, via unlike the LSW, electron-ALP is coupling also in sensitive to the coupling of ALPs to electrons, for the future [ JHEP12(2019)162 ]), ]. In 46 ] also – ], with 56 34 – 43 39 , – 53 31 ] or through 36 51 – decay. ([ ] laser pulses, or 49 0 37 Z ] for 34 ]. 39 , we present an example derivation of 2 – 3 – ] and intermediate many-cycle [ 38 ]. The production of ALPs in the interactions between elec- ], strong [ 35 38 ]. Recent calculations have begun exploring the possibility of using ]. Due to the immense number of laser photon “probe” particles, ]. These papers demonstrated how lab-based experiments using 52 37 48 , 47 ]. Solutions to the Dirac equation in a plane-wave electromagnetic background, we analyse the constant crossed field (CCF) result, which is integrated over the 41 ]. ]. As such, the derivation of ALP decay rates in quasi-constant electromagnetic 3 40 42 The paper is structured as follows. In section To perform calculations in strong electromagnetic backgrounds, we employ the Furry This process is relevant for both terrestrial experiments utilising strong magnetic fields The production of ALPs via their coupling to photons in a circularly-polarised laser pulse), focusing on the decay oftions a for massive pseudoscalar a to scalar an and electron-positroncases a pair. vector are Deriva- boson presented. follow awhat very is We similar often then format referred and perform tosection final as a results the for local “locally-constant-field-approximation” these (LCFA) expansion [ of the probability and obtain parametric excitation [ the collision of electronfor bunches example with using laser weak pulsesleveraging [ to collective measure effects such the as ALP-electron coherent coupling, emission [ ALP decay in a plane-wave electromagnetic background with finite support (e.g. a laser fields has much in commonwith with high-intensity the QED decay (reviews can ofE144 be experiment photons found [ in in [ amost interest laser in background extensions being ofThis high-intensity measured can QED manifest has experimentally itself been in in in the the the polarisation ALP-diphoton properties coupling. of a photon probe [ jects [ picture [ so-called “Volkov” states, represent the fermionsfield “dressed” [ in the external electromagnetic extends this work byconstant considering magnetic the fields, decay derivedplane-wave of as electromagnetic ALPs background. a to limit electron-positron of pairs the in case quasi- infor which ALP the conversion and process searches occurs for in extraterrestrial a ALPs from strongly-magnetised ob- trons and high-intensity electromagnetic fieldsALP-seeded has been electron-positron studied pair previously production inbeing [ in considered a in monochromatic laserhigh-intensity [ lasers background and also electrons may providepling, lab-based complementary bounds on to the those ALP electron derived cou- from helioscope experiments. The current paper for axions.) Onefree new parameter. feature Therefore, introduced in atunneling by region CCF, studying for we axions axion were lighter able decay, thandependency to the is on pair identify that rest an the mass exponentially-suppressed the axion and mass the mass transition for is to axions a a heavier oscillatory thanbeam the has pair been rest studied mass. in [ and therefore the usualstudy Taylor expansion axion of decay the inrate LCFA a on misses the CCF, this quantum and nonlinearity phenomenon. parameterfield” find for region We an weak for also fields analogous large as quantum oscillatory wellstudies parameter, as dependence the just a angular as of monotonic was dependence “strong- the found of decay in the [ decay process, which is beyond our current analysis fact that the contribution from outside the field is vanishing at small interference phase, JHEP12(2019)162 , φ 0), we , m 2 (2.1) (2.2) , has 5 is the . The V , x 0 1 1 m k , through 1) so that , the vector 2 , x δ ) , µ ϕ 0 √ ( = 0, and we , / q A and mass 0 x − a · = (0 , dψ, k · iS ik ) ⊥ + − κ x x ψ e · 2)(1 ( , / 2 iq 3 = e a + x p κ φ ) V · −  0 q ) ( q κ 0 0 = ( ψ 2 r x 2 ( v κ a p = · ) of the background. The lightlike p is the polarisation and ϕ  , 2 ( x x + q q µ · ϕ ψ − κ 0) 5 E Z , γ p = ⊥ , with four-momentum = ε ε ψ φ ϕ ,ε ) = + q ) is the local classical intensity parameter, ϕ ( ϕ we discuss the results and conclude. x φ is the charge of a positron and p ( 4 = (0 e 6 d ξg – 4 – µ ε Z ) = , ψ ) 1), φe ϕ ϕ ( ( we consider the effect of the detector’s magnetic field ig ξ p ), where 4 is transverse to the vector potential, iS ϕ | ≤ = ( ) + fi µ κ x ϕ · S ( ip g eA | e , where µ = V ) ,S ε 0 . We will define the seed particle’s mass parameter p µ ) ) p ( p ⊥ a r ϕ 2 · ϕ x , and use lightfront co-ordinates: ( ( u / a κ − p − x / 2 ) κ mξ 0 ϕ = is a normalisation volume. κ ( + p ⊥ 4 ) = V E x ϕ . Feynman diagram for pseudoscalar-seeded electron-positron pair production. ( 2 = . = µ / 2, 2 is the electron-pseudoscalar coupling, with Volkov states: a ) = 1 p / − , and ϕ x ∓ ψ φe φ /m ( x g + 2 p m k κ E Figure 1 To calculate the probability of decay in the electromagnetic background, we assume = ) is the pulse envelope ( = = ϕ  ( 2 where we assume thatmass the pseudoscalar is in a plane-wave state, where δ the produced fermions are solutionsbackground to (Volkov the states). Dirac equation Thisscattering-matrix in is a element depicted plane-wave for by electromagnetic a the pseudoscalar double is fermion then line in figure wavevector of the background represent g electron mass. We chooseϕ a system of co-ordinatesx in which We begin by considering adecaying pseudoscalar to particle, an electron-positronscaled vector pair potential in apotential, plane-wave depends electromagnetic on a background. single variable, the The phase consider two experimental scenarios involvingtection a region magnetic respectively, field and and taketypical a the experimental laser upcoming parameters. pulse LUXE in In experiment the at section de- DESY to provide 2 Derivation of pseudoscalar decay probability in a plane-wave pulse and above-threshold decay arepair-creation presented, tunneling as and is oscillation a exponentsergy description depend and of upon field how the strength. ALP thebeyond In mass, non-perturbative the section particle LCFA en- and showterpretation of how a large local field production gradients of at pairs the and can detector influence edge the change total the yield. in- In section non-constant background to form the LCFA. Asymptotic and perturbative limits for below- JHEP12(2019)162 , , 6 )] ) 0 , π ) ϕ (2.5) (2.6) (2.7) (2.3) (2.4) ( , q q ( (2 0 # − r S q/ v 3 − ) irθ ) ϕ ϕ ( p d ( + q q 3 . − ) d E S dφ p 2 5 [ ( | i  γ s fi ) q u S )]+ | · ϕ ) 0 0 − ( ϕ k p k ( tr ϕ κ 0 together with the on- · p 3 ( E ) S p p ) κ − · π V k , − spin E p ) ( 5 ⊥ ϕ (2 κ r p ( P γ ) p ) u 0 R S = φ − [ ) ( ϕ i 2 2 ϕ ( 2 for scalar decay to reflect their − ( − , dt q V )+ a q 0 ⊥ . − + E R ϕ dx 0 k = ) − iS P − ϕ q - R V . ϕ ) ( = ( . )+ 0 P 3 φ − + q s ϕ ) q ir ( ( k · v − · e 2 φ 1 π ϕ , a p ) · T ⊥ κ q − iS m κ (2 0 ( = 0 q q 0 2
+ + ” co-ordinates we have r κ θ q − = v 2 − ⊥ − q iϕr ) ) e + ) l ϕ q ; – 5 – p ( dϕ dϕ ( 0 θ φ · p · + q ( +
· ϕ ) p Z dϕ a k δ l κ E − 2 2 ) ( · 5 κ + l p Z # = + ” and “ ( γ p = ) ) ) ϕ δ r −  k − θ k r , ϕ 2 ( ⊥ = − − 2 − p 3 δ θ/ q q σ θ/ 0 + E − V dp + ) − σ 0 p → + p σ p l k ⊥ − ( ( Z 0 p p q r ( for pseudoscalar decay and q i − 2 , u 0 d " - − ⊥ . In preparation for eventually performing a local expansion, we p , ) over the “ P tr δ 2 8 ⊥ (0) = lim 0 Z 3 δ m − exp ) p 2.1 , 3 ,s,s 0 π V 0 ⊥ T V (0) = X 0 δ κ (2 0 3 r,r k 2 − ) k , 2 φe q 0 ) π (we use ⊥ = g q 0 δ - 0 (2 " dσ dθ κ T p P 2 φe φe 8 = 8( g Z 2 ig p | 0 fi = S κ - | P Integrating eq. ( tr = fi S spin X We also note have defined the average and difference phase variables The probability can be written in a much neater way by observing that where momentum conservation isshell enforced condition via probability behaviour under parity transformation) isleading defined as to Since the electromagnetic backgroundof is integration), of it is a morebe finite useful extent spacetime-dependent to (disappears consider as probabilities at they than the cross-sections depend boundaries (which on would the background field strength). The decay where the trace terms are included in the factor ground is given by To obtain the probability we must square the matrix element where a measure of the lightfront momentum absorbed from the electromagnetic back- JHEP12(2019)162 . , q q · 2 · ) p #) (2.8) (2.9) k κ (2.11) (2.10) · dφ p . Then · κ 2 , ( )) ) that the  κ m φ q ( · ) 2 φ 0 2 κ 2.5 = m 2 ϕ a , ( 2 − a  q −

2 · ) q Π 0 · p · ) · κ k φ is the proper time, ) ϕ ( ( 2 φ  τ ( a a iθ )) a 0 e 2 − φ (2 ( T ) + q p k , a k φ · · · · ( 2 φ , κ 2 2 κ κ κ 2 m a q ) + p q · dσ dθ · q φ (which only remains in the · − + ( κ · κ ). This would seem to lead to a q a 2 2 p κ ) Z q ( )] )) + a · 2 2 φ 0 · 0 φ
· q p ( dφ. · k ϕ k φ − m ) − · p a ( ( 2 q κ · q k φ a a p 2 · · − κ ( k · = 2 θ f κ κ (in q k − a
· · 2 ) + ) + 2 2 θ 2 integrals. We note from eq. ( + − ⊥ κ φ κ θ/ ϕ 2 θ p ( p + ( θ/ φ 2 p + κ − σ ⊥ a a − σ · [ ( σ – 6 – p m θ σ + · Z k q Z k q, λ − · a p − · + θ 1 = κ p k − + − κ − ! dp 2 · · q  p p q p · := ⊥ · 2 φ − κ p κ + p p · 2 i q = κ 2 ) θ m p f κ 2 d k + h · κ q · Π = k∂ · 2 λ · − )] κ Z is a constant in a plane wave, where 0 κ ( m κ + q p 2 ϕ 3 i · · ( ) k + k a κ π 2

(2 → − 0 θ dϕ/dτ ) + 2 φe k " q ϕ p 2 g · ( · i ) a 0 k κ [ ( κ k · + 4( κ q · = p - )] = exp was found from the first equation using the on-shell condition ) also contains terms quadratic in P + 2 2 λ ... ( m i First of all, we can remove explicit dependence on To proceed, we wish to perform the Then we have = 4 exp [ T we can make the following replacement in the pre-exponent the pre-exponent starts to look like the exponent. Writing this explicitly as where making the replacement an integral over surface terms, which must disappear. term) by using the trick: exponent is of thein form of eq. a ( Gaussiandivergence, oscillation however, in we these will variables, show but that the the pre-exponent divergent contribution can be reinterpreted as Performing the spin-sum and the trace we find and we define the phase-window-average allowing us to cancel the integrals and introduce corresponding momentum factors). where the classical plane-wave momentum of the electron is (where we note that JHEP12(2019)162 . , ]. + ⊥ ]) 2 k 2 p k P η η 58 2 57 iθδ 2 iθδ − (2.17) (2.15) (2.16) (2.12) (2.13) (2.14) − ) . (The ) t 2 θ ) − ( θ ( (1 ih t k − iθµ k/m η e 2 · , e o 2 k κ )] η . 2 2 iθδ θ k ) = (  η − 2 ... iθδ ) ) ( k t t ih i − η [ ) e − ) θ − t 0 ( (1 1 ) # t − θ K θ (1 k ( iθµ (1 i t η t 2 2 2 k iθµ  k ∂ η e · ) 2 0 − e κ ) φ i 1 ( 2 , ) a  )] − E 2 0 2 ) 2 − φ 2 )] -integral analytically (see e.g. [ t  ), we state that the probability 0 ( t )  ) ) )] 0 φ a 0 t − φ ( a ϕ m φ ( ( a − 2.12 ( − a (1 a for the decay of a pseudoscalar into an a  ) t , − 2 - D  i 2 (1 φ − I ) t − ( P ) 2 k − + φ a ) 1 [ ( η ϕ 2 φ 2 a ( ( δ 2 [ E π + a ) a g 4 . Selecting a linearly-polarised background, we [  – 7 – θ a 2 - m + ( , and the energy parameter δ p I − · 2 = D 2 h ) − ) δ h  k − κ k P ) q /k · ( 4 t · ) is the modified Bessel function of second kind [ πη − κ x ( 2 4 − κ p ( iε )] + ( δ / 2 θ n 2 iε ( dθ + = ) = 1 + g 2(1 K + θ t θ dθ ih + ( [ 2 φ − θ µ 1 = ( 2 1 m and - K 2  " P k dσ dt δ 2 η ) for the decay of an unpolarised massive U(1) boson is proportional = dσ dt 2 θ  Z ( ) / γ ) h Z π ... i P iε θ ( 4 i n ( π i ) will prove to be the more useful form of the probability for numerical e dθ + 4 = iε 4 θµ T θ - dθ = + − I 2.13 . Then we define the probability θ + I I dσ dt ) = dσ θ Z ( Z h π i π 1 4 8 Without further derivation, in the spirit of eq. ( Let us write probabilities in the following way: = = - γ I I and the probability to the integral: for the decay of a scalar into an electron-positron pair is proportional to the integral: where However eq. ( evaluation. energy parameter can bethe thought pair of rest as energy, thephoton for to squared the produce ratio case a of when pair.) the the It is seed centre-of-mass possible photon energy to collides to perform with the a single background where we define the Kibble mass factor lightfront momentum fraction integration, electron-positron pair as then eventually arrive at integrals without encountering a divergence. where the coupling and flux prefactors have been separated from the process-dependent If we assume thatinfinite there future, can we can be discard no the contribution derivative term to in the the probability pre-exponent in and perform the the infinite past or which simplifies the pre-exponent considerably such that, in the end, we just have JHEP12(2019)162 · ] = ) ϕ 59 , ( (3.2) (3.3) (3.4) (3.1) µν ) and T ε T 53 · · a magnetic p ( h ].) ε = for spacelike − ) and taking 31 2 = ϕ ξ 2.17 a ∼ , ) ) which corresponds ) , 3 ϕ σ , ( θ  ( ( , a 2 3  0 1, where / ) f O 1 z ; 2 z ) ( θ r  3 √ 0  2 c ) 3 , stress-energy tensor δ / ξ σ Ai p 1 (3 ( ) 0 up to k → −  r 3 z 0 χ c θ 2 √
) Ai (3 ) − 0 σ  3 implies a cycle-average over the phase 0 φ ( ].) (The LCFA is sometimes explained / 1 ( z k 2 ϕ , satisfies ) a χ πi 61 Ai 3 ξ = , 2 c h·i − − z πi ) 55 (3 ) 2 φ z ( − − ( ; 1 a 3 – 8 – / = = 2 Ai ) ) 3 3 2  2 θ θ δ ) 3 3 c c σ 3  ( ) + + θ q σ ) rθ rθ χ ( ( ( ) σ i i k ( e e σ 12 dσ dt 2 χ ( f p iε Z χ dθ θ dθ + +  = θ 0, leads exactly to the QED expression for photon-seeded pair- θ ∞ = −∞ ), by expanding the exponent in is the Faraday tensor and ∞ Z → → 0 −∞ LCFA z - F Z 2 I 2.13 δ θµ 4, ). Using the results: ) and the linearly-polarised background can be written / σ σ 2 . (Applying the above procedure to the U(1) case eq. ( ( ( , for massive seed particle four-momentum 0 k F 2 ξ η ) ) mf p tr ]. Furthermore the mass-dependent part of the Airy argument has the same σ · ( ) = µν , we then find 62 f η σ κ ) = R ( ( 0) wavevector, is relativistically inequivalent to a constant crossed field (CCF), σ − f 2 ( ∈ 0 ].) ]. (However, recent analyses of nonlinear Compton scattering hint that the infra-red ) = < µν a 3 Typically, we are interested in lab-based detection of ALPs using constant One can acquire the LCFA result from the probability for a process in a plane-wave ) c σ · 63 /m 60 2 2 ( [ ϕ ε k i κ F form as expected from e.g.in the [ second step of electron-seeded pair-creation, given explicitly fields. Clearly, a( constant magnetic field, which can be written χ the massless limit creation [ for where we define where − in the particle’s rest-frame is approximately that of apulse, constant-crossed such field as [ eq.to ( the highest power contributingreplacements to the constant field case. This amounts to making the p ( ϕ behaviour is badly approximated by the LCFAby [ reference to when a massive seed particle is highly relativistic, the electromagnetic field The Locally Constant Field Approximation (LCFA) allowsprocesses one in to non-trivial calculate plane-wave probabilities electromagnetic for backgroundsexpansion by performing of a the local background, field non-trivial and form integrating of the the plane-wave. resultingwhen It the constant has intensity field been parameter result shown of to over the be the background a good approximation [ 3 Constant fields JHEP12(2019)162 4, π/L > (3.6) (3.5) 2 ]. The δ = 2 67 , one can 0 2 κ δ ]. This fact, 64 to be as 0 ), then we see that 2 κ . F ¯ λ L 4, where the process is − m k ], was recently explicitly ), can be written < ϕ → 2 66 ( ; ii) above threshold, , δ is the amplitude of the field 1 , a ) 65 dσ 0 ϕ, ϕ , F 1, then from their rest frame, a ( ) 0 Q 0 Z 2 F F 1 k by first writing a  1 , δ C η L k λ F χ ∂t ∂ϕ 0 ( − /m k e 1 k field-induced − CCF - ; k m R − ) = m k ¯ λ L → ϕ Q – 9 – ( ) − F for different axion mass parameter, 1 F m k ϕ ( A 2 2 t π = g ∂ 4 k mξ and can proceed in the zero-field limit, and iii) strong = fη ) = ) = ϕ = CCF - , we use the same reasoning as in eq. ( ( ϕ in figure P ( k F 1 , to the field-strength k for lightlike wavevector is then the constant crossed field limit. χ a a χ is the Schwinger limit. Other quantities can then be written ϕ is the Compton wavelength, → ∂ϕ/∂t ∼ on ) /e 2 ) σ ( ϕ field-assisted π/m m k CCF - ( χ R a = = 2 Q ], and so we will not analyse it in great detail. However, we highlight the C F λ , which is the rate per unit detector length (measured in units of the reduced 68 , 1, where decay is so likely, there is no threshold behaviour anymore. Plotting CCF - 59 R being the longitudinal spatial extent of the constant field (formally infinite in the  L χ In relation to the ALP mass and the field strength we can identify three distinct regimes To approximate the probability in a constant magnetic field using a CCF, we can relate 3.1 Below thresholdBelow-threshold, decay pair creationponentially can suppressed. only occur Thiscreation situation as [ is a very tunneling similar process to and studies hence on photon-seeded is pair- ex- for creation of electron-positronforbidden in pairs: the i) limitwhere below of the zero threshold, process field and is field hence is the dependency of clearly see where these three regions occur. where all the non-trivialfunction dependency on experimental parametersCompton is wavelength). contained within the Then we see: where we have introduced awith nominal “frequency” of the constantCCF field limit), strength, and independent of any external field frequency: To proceed in evaluating the non-trivial component of the reduced vector potential, the scaled vector potential, where we pick the background to be polarised in the 1-direction without loss of generality. to highly relativistic seedconstant magnetic particles, field i.e. will with which appear underlies to the approximation be Weizs¨acker-Williams shown [ well-approximated to by hold a forconstant-field CCF limit nonlinear [ Thomson scattering in a constant magnetic field [ which has equal magnitude electric and magnetic fields. However, if we restrict our analysis JHEP12(2019)162 1 � χ  (3.9) (3.7) (3.8) k 10 χ 6 5 4.4 4 5 = = = = above it. b) 2 2 2 2 2 δ δ δ δ δ . 2 / 3  , as: 1 2 4 δ − , ϑ/ξ 1 2 2 )), can be written, with , allowing us to use the t  0.5 3 2 k 0. Then in this case, the 8 χ → F 3 cϑ . 2.17 2 > θ − e − e )  t + ϑ )  t 2 − 2 ∼ 0.1 δ − ) and ( m 2 (1 δ t − + (1 2 t 4 manifest, . 2 δ 2.16 2 b k p k 0.04 δ k 2 χ − χ χ p 2 1 − 0.5 0.2 2 = 3 1 + ϑ 2 ], recently applied to high-intensity laser-based  m  2 – 10 – 69 δ / 1 + 1 + ) (and eqs. ( 5  1: − 2 ) )  t  2 A 2.13 8 δ e − k iϑ χ tunneling exponent has been shifted by the seed parti- − (1 4, we note that 1 k 1 + p χt ( χ <  3  q 2 2 / 4.5 4 and 3 δ 8 √ √ exp < − 4 , the distinction between above- and below- threshold is lost and the ) = k k 2 t for various seed-particle masses. a) the effect of the threshold when ( χ δ χ , yields a real exponent of the form p E CCF - ∼ iϑ R 4. We can understand this in an intuitive way by using arguments . 4 → - CCF CCF - → R 0 0.03 0.05 0.07 0.1 ϑ R 2 k k k k k , when the decay is above-threshold, the probability oscillates about the vacuum values δ χ χ χ χ χ k ]. The energy of the produced electron is: . Plots of χ 70 a We then find for In a CCF, the exponent in eq. ( 0.2 0.8 0.6 0.4 1.2 1.0 cle mass, where thethreshold tunneling behaviour clearlybased on disappears energy-momentum conservation as [ theQED [ mass approaches the We notice the familiar In the subthreshold case, turning points of thecontour exponent with always lie on theLaplace imaginary method. axis. There Rotating are the two turning integration points and one is subdominant. difference that a massive seedtive particle limits brings, of by calculating the asymptotic and perturba- a change of variable to make the dependency on for small (shown as grey horizontal lines)suppressed. and in the For below-threshold large regime,dependence the on probability particle is mass exponentially become disappears. more In accurate b), for black heavier masses solid and lines smaller are the asymptotic results, which Figure 2 — exponential suppression below the threshold and an oscillatory dependence on JHEP12(2019)162 1, . 4)  , with (3.12) (3.10) (3.11) − t k 2 0. Then χ δ ( < 2 . ) δ t  p 2 − 4 δ 1 2 (1 t − + 2 ) = 0. The smallest ) δ # it p 2 a, where we plot this , where the threshold ( / − 2 3 − E 2 2 δ k  k ( 2 4 δ p 4 . .  # 1 + 2 2 CCF - ) / ) appear, with opposite sign, on − 2 3 R   m 3.7 , we acquire a final integral in ∼ eF t 2 k , 4 ( ϑ δ  2 8 χ 0 4 δ 3 − dt " 1 + ) − 1 0 t 1  (  cos k E r 2 8  χ ∆ k 3 4 m – 11 – t m 2 -integration exists where 1 eF 0 2 − t − on ALP mass parameter, Z δ " 2 ≈ i = δ  k ∗ CCF - t exp R is the shortest time for which ∆ exp − E 4 + ∼ ∗ t )  t ∼ limit, the probability for decay oscillates as the field-strength 2 ( CCF - = / p P k 1 R t − dt d χ − k E  2 8 δ 4, a region of the ) + where t > ( ∗ p ), we indeed find that: t 1 + 2 E δ  3 3.11 ) = √ t 8 k ( ]: E χ axis. ∆ 71 ϑ 2, from which it follows that the tunneling time is: ∼ − 4, that: We see that in this low- Then applying the method of stationary-phase in k/ CCF - > = R 2 is varied. This istransition, demonstrated and for the various dependencyeffect axion of can masses clearly in be figure seen. which is where theδ zero-field contribution originates. Altogether we find when in this case, twothe turning real points of the exponent eq. ( an oscillating exponent which also haspoints turning points conspire on the to real produce axis. Of a those, cosine, two turning and the third turning point gives a constant term, Therefore, we can be somewhat confident that we3.2 have the correct tunneling Above exponent. threshold decay Above threshold, and using eq. ( For a tunneling process, we canand approximate this integrating integral to by using the saddle-pointenergy method, difference corresponds to anp equal distribution of the initial energy and momentum We can approximatemethod the [ form of the rate for the process to occur using the WKB Assuming all particles involvedenergy in change is: the decay are highly relativistic, we can see that the where in the last equality, we have used the fact that the background field is constant. JHEP12(2019)162 . 2 2 δ  δ 3 ∼ / is the 2 k (3.13) (3.14) χ q t . ) 3 ) in the small / 2 − k 3.3 χ ( . ) O 3 / 2 + − k 2 . 3 χ δ should not be significant.) (  ) O t , m 4).) ) depends on the mass as ) = + − 4) / 0 k = 2  χ to show that, in the limit ), we find the same result, which δ − (1 2 ( cl 3.13 t 4 3 2 t δ 2 − / O ) δ δ 2 k ( 2.13 cl − 2 (1 χ t 1. − δ ( 1 ) can be understood by using intuitive (because of dimensions), where  1 p   eF q ∼ − ); 3 ∼ k 0 k 3.13 / 2 /t ) χ 2 χ mT 0 k (  0 in eq. ( , we find: χ 2 π ) O t ( g – 12 – 4 t → O + − 1 , and the one real root leads to the relation: given by → ) a q ) from the uncertainty relation. (Since the process 7 6 t q (1 ) must, of course, be independent of the form of the 3 cl ); t t 0) / t ( 0 k 2 k  )Γ( . So in the perturbative limit, we must also assume that E χ → ( 3 1 6 3.12 3 πχ / ∆ / ξ 3 2 k O 1 / 2 k ( Γ( / - χ 4 3 + /χ / P 2 1 1 2 increases towards 3 = 1 = δ / 3 k 2 k  q z t χ χ ∼ ≈ k ) the scalar mass term is z χ ) ∼ 2 4) as in the intuitive method, it is not entirely surprising that this / (because one vertex) and 1 , δ CCF - 1, this perturbative expansion decreases in accuracy for increasing 2.16 2 k 2 δ R since the Airy argument is given by: ]. In this case, the process is above-threshold and so the rate is simply χ g ( .  ) generates a cubic in k − 1, then dt k 70 CCF - , /χ (1 χ  R R 3.10 69 2 . ∼ δ = 2 δ T  Again, the functional dependence of eq. ( After a straightforward integration in The zero-field result in eq. ( 3 / , the mass of the seed particle ceases to play a role, and in general, the concept of a 2 k 2 and not term differs since ita originates different from form the whetherhave parity-dependent dealing we term inserted with in the e.g. fact the a(Indeed that trace, from photon, we eq. which scalar are ( takes dealing or with pseudoscalar a and pseudoscalar nowhere in this intuitive picture. Also here, weAlthough include the the pseudoscalar mass next-to-leading term order, in our to result show eq. ( the dependency on the mass. methods [ proportional to quantum time fulfilling is quantum, the classicalUsing timescale eq. ( We have included the next-to-leading-orderδ term in threshold disappears as χ At a given Suppose 3.3 Strong fields, In this parameterparameter region, 1 one can simply perturbatively expand eq. ( background field. By takingcan the be limit written as: where JHEP12(2019)162 , ) Q /F (4.1) 2.13 ) 1 and 5, and . = 1 m, ϕ , (  ), whence L B = 0 k )  η C 1 3.5 ϕ ϕ − Φ L/λ 5, 1 . , for a terrestrial )( 0 ϕ 1, there may be a − 2 −  /k = 6 0 0 k ( tanh ϕ Φ φ . The potential is then de- ξ m  3 0 1 = 1 T and length 1, so in this case ϕ 0.01 0.05 0.2 ) = 0 Φ = = = −  ϕ Φ Φ Φ F ( 0 ϕ a  ¯ λ/L ) is non-trivial, because the constant- π 0.5 2.13 = 2 1 + tanh /m  0 – 13 – κ 1.0 0.8 0.6 0.4 0.2 0.0 Φ) / 1 ϕ 0.5 - ) are the phase positions of the two “edges” of the field and Φ 0 Φ) tanh( ). For a magnetic field of 0 gives the top-hat function. 0 / Q F 1 0 → - ϕ ) in the constant-field limit. /F > ϕ 0 − F 1 3.5 for various sharpness parameters in figure ϕ )( 1. Therefore, following from the results in figure ¯ λ B  L/ (and k 1 1 + tanh( , ξη = ( 0 ϕ ξ ), however the frequency scale = and to numerically evaluate the resulting, absolutely convergent integral. ) = 2 . Magnetic part of a constant crossed field, scaled so the maximum is unity. For highly- = ϕ k θ ( χ ϕ (10 B O In this case, the numerical evaluation of eq. ( Upon comparison with the LCFA, we expect that when To apply the result for the probability for axion decay in a plane-wave field eq. ( ∼ field part is notonce absolutely in convergent. We found it was sufficientdiscrepancy. to integrate by parts hence magnetic field, we expect theto, pair-decay or to already only above occurcase for threshold. as axions it Still, with gives a a wea clearer mass will rapid demonstration that begin of drop-off is the by at close effect the analysing of end the field of gradients below-the-threshold the introduced magnetic by having field. which reduces to eq. ( to a quasi-static magneticit field follows in the highly-relativisticξ regime, we use eq. ( where is a sharpness parameter.exhibit We the choose, form without of loss ofrived from generality, this form of the field numerically by solving approximated by replacing aof constant a field quasi-constant with magnetic a field, CCF. we Therefore, choose to a represent field the of edge plane-wave form: magnetic field. The limit Φ 4 Edge effects of static constantWe fields recall that we are working in the highly-relativistic regime, where processes are well- Figure 3 relativistic axions, the crossed field produces an equivalent effect to entering a constant homogeneous JHEP12(2019)162 0 φ 1 . 05 κ . (4.3) (4.2) = 0 0.6 k ) as the η 2, Φ = 0 ) to give: 2.6 . = 0. θ 0.4 .  = 10, 2.13 2 δ ξ 1, and for these − in eq. ( ) ) t ξ y 0.2 σ −  — for example the ( for (1 µ t in eq. ( 4 ))  σ = 1 and pair-creation is k y/ξ ( χ iy 0 k 2 y/ξ . 2 χ e /a 1 for the LCFA to be valid. 0.25 0.20 0.15 0.10 0.05 0.00 0.05 ) → m , -      σ 4 1 θ  2 ( φ − 2 000 i 1, for three cases: Φ = 0  a − . 10 ξ ( y 2 2 2 Φ) ) 0.6   = 0 ξ − 0 Q 2 k F F σ η y/ξ ) t  C ∆ λ a 0.4 − 1, [(  − = 10, (1 (we recall the definition of t  ξ  2 ξ – 14 – y 2 2 0.2 /∂σ 1; + = 0, ))] P 2 are also seen to oscillate around zero outside of the σ σ ∂ δ  (  0 4 2 a /a h  can be discarded — and hence the LCFA used — include ) 0.15 0.10 0.05 0.00 0.25 0.20 0.05 3 L σ ∆ - + — to be the basis of a good approximation. The conditions ] that ( y ξ 1 2 00 µ φ 55 δ a  )(      y 0.6 dy y/ξ for a field when − I higher than dσ dt 0.4 y 1, we expect a Taylor expansion of functions in Z π i 4  , if the ALP collides head-on with the wavevector of the magnetic field, Φ then between 0 and 1, which were distributed with a cubic weighting towards 01 respectively from left to right. The dashed line is the LCFA, the dotted line is the is the Compton wavelength of an electron. Therefore, the weaker the field, the 0.2 . . Plots of ξ θ = C - λ π/L I The numerical curves in figure We demonstrate the effect of the sharpness of the field in figure We can justify the LCFA by applying the substitution 0.05 0.00 0.25 0.20 0.15 0.10 0.05 which would require a 100appreciable. TeV ALP in a 1 T magnet), so that magnetic field. It is knownseed [ particle’s average phase position) does not have to be positive, as long as the total where more important its shape.and This so is the somehow approximation intuitive:this that a corresponds it weaker to is field a has locally field a constant edge lower of should intensity approximately be ∆ (a worse. much For higher a 1 value T than magnet, from an ALP in a homogeneous magnetic field in the lab, represents some length orLCFA is duration, valid ∆, when: over which the field falls off at its edges. Then such inequalities as [( terms to make athese difference, inequalities the are violated. probabilityFor must It these not then parameters, follows be where that alreadywith ( we 2 vanishingly have small chosen when to associate the external-field frequency Then if dimensionless Kibble Mass, that powers of points in Figure 4 and Φ = 0 mean numerical result andnumerics the were run gray five area times marks for out each plot, one and standard each deviation numerical from evaluation randomly the allocated mean. 1000 The - JHEP12(2019)162 ), m L /∂σ 0 = 0, = 0, P κ 2 θ ∂ r δ with a ( 5 1 and how 4 sinc( is increased.  ∼ θ θ ]. ), given by 72 a how the integrand . b) The corresponding σ, θ 575. According to the 5 ( . C /∂σ − = 0 I σ ∂ b makes the field formally constant dy, 1) for . ) positive. Therefore in general, 0). The convergence of the integral is = 0 σ, y -integral originates from , we plot in figure is ( → ∞ k ≥ θ θ − η ∂σ∂y θ I 2 3 (as the problem is symmetric around . ∂ . It is instructive to compare figure , we note that the introduction of strong field θ θ 0 = 10, 5) of the magnetic field, to illustrate differences – 15 – = 0 4 . Z ξ 0 σ  ) = 01, . = σ σ, θ ( C 0, which would give a frequency spectrum ]. Although homogeneous magnetic field strengths are limited → = 0, Φ = 0 21 , 1) evaluated at , with the same parameters as the central plot in figure . 2 θ δ in the lab, there is increasing interest in the quasi-static fields of the 20 = 0 T b where the cumulative distribution function k 5 η 10) (where T that are generated in intense laser-plasma collisions [ − 5 4 shows the major contribution to the (1 = 10, 10 5 O near this edge. One way this can be understood is by considering the Fourier ) depends on ξ is a real number. The opposite limit Φ . a) Example integrand to be integrated over to calculate a ∼ 4.2 r 05, . /∂σ Figure To demonstrate the numerical integration in P ∂ the frequency of oscillationpoint increases outside with the sharpbetween edges the (at full result and theplot LCFA. Consider of the figure sharper magnetic field in the right-hand indicated in figure is plotted and tends towards a constant as the upper integration bound in to around order of in eq. ( Φ = 0 we have simply calculated points in the region the larger the contributionpair-creation from threshold, higher thereby frequencies, reducing whichprobability. the This can necessary is tunneling bridge similar time the toof and gap the increasing a situation to the of plane-wave the pair-creation laser 2 creation by pulse, a probabilities where photon [ in shorter the pulses background were found to drastically increase pair- gradients through a sharperof magnetic field edge,transform allows for of an the increasewhere limit in Φ the amplitude resulting in a delta frequency spectrum at the origin. Thus, the higher the field gradient, Figure 5 cumulative distribution function. probability, which is defined for asymptoticcannot states, be interpreted as a rate. In figure JHEP12(2019)162 ) 5, . 0 does 2.14 1 and . < 2 2). = 0 θ/ k θ/ η −  σ σ = 10, ξ 2, as this was where the 575 it is not sensitive to . . , the full probability 4 = 0 σ b 5 GeV electron beam that drives the . (recall that the Kibble mass in eq. ( θ , we see that indeed in the full probability ]. The set-up we envisage is a variation of 6 4), to the point where the magnetic field only 74 ) evaluated for phases between ≤ and therefore at – 16 – 575 for the magnetic field with . 2 θ δ 2.10 5, even if the field changes a substantial amount at = 0 . in figure σ = 0, however, at precisely the value that = 0 θ at σ ∂P/∂σ /∂σ∂θ 1), the effect of the magnetic field enhances pair decay. As the . − ], which will combine the 17 I 2 = 4 , we give some examples of the LCFA probability for pair-creation 73 ∂ 2 7 δ a we note that, as the axion mass is increased, the effect of the magnetic 5, i.e. the distance away from the edge of the magnetic field. b, we see that as the energy parameter is increased for a typical above-the- . 7 0 7 − . a) A plot of a σ 01. b) The corresponding cumulative distribution function. The vertical line in the plots is . 2 = In figure Finally in figure The LCFA is an expansion in small θ/ The results of the previousthat section can could be used be toexperiment measured provide at an at estimate DESY for [ typical exclusionEuropean upcoming bounds XFEL experiments. laser, withused One a as example thin-foil a is target probe to the in produce LUXE photon-laser Bremsstrahlung, collisions which [ can be axion impinges from the vacuum invacuum the magnetic decay field, to the transition field-assisted incan the decay also probability is from lead nontrivial, to and a suppression depending of on the the decay axion’s probability. 5 energy, Example experimental scenarios field is one from increasingslightly axion modifies decay ( the probability, andit. in fact In in figure some regionsthreshold of scenario parameter ( space suppresses contributions from the fieldthis at point. in a magnetic field innumerics the and above-threshold LCFA agreed case. well We for choose the Φ massless = case. 0 actually contribute. Plotting there is zero contribution around the integral begins to contribute withis increasing defined using a window average eq. ( Figure 6 Φ = 0 at LCFA, there is no contribution here, but as is clear from figure JHEP12(2019)162 2, 2, = . x 0 2.5 2.0 1.5 3.5 3.0 ], of Q ≥ (5.1) B B δ 10 e γ 2, there X/X =2 k χ δ <

) = 10, Φ = 0 , t 2 ξ bremsstrahlung in , δ 1, k . . For values of χ ∂t 9 ( = 0 . k CCF − # η axionic 2 = 10 R  ∂ e − − N dt p k 1 , with  0 Z 4 +  2 − − p x k 4 3 b + x − 4 3 4 3 ), as long as the field decays over a distance " − 4.3 – 17 – 0 4 3 X X  2 2 2 e g 6 4 2 0 10 8 x dx ≈ 1 0 ) to acquire an approximate rate of axionic bremsstrahlung: Z − 2 − ¯ λ k L ( e /e dk 2 γ 1 g 2 dN edges for the first case in figure 0 − X X k is the relativistic gamma factor of the electrons hitting the foil, an LCFA 2 - 21 m then using eq. ( 4 I .  2 10 π g 4 = 4 ×  5 L . 3 shots ≈ . Plots for N ], and multiply by e e 1 (unless these parameters are varied in the plot). Left: the transition from below- to is the average number of radiation lengths undergone in the foil (we take γ . N 74 0 We assume the collision is head-on with the magnetic field wavevector. To estimate In the first scenario, we consider the regeneration to occur in a magnetic field. If we = = 4 = 5 T and a φ 1). For the number of shots, we assume 24 hours of continuous operation and a frequency . 2 X/X 0 of 1 Hz for collisions, andthe rate take of the regeneration number is of almost electronsis identical a to to that small be without region a of magnetic parameter field. space where For the magnetic field field can cause regeneration to N where The number of axions regenerated is then: B greater than 2 mm, which we will assume, the LCFAthe can regeneration be rate used. offrom axions, [ we take the thin-target approximation to bremsstrahlung a typical LSW experiment, wherethe the thin-foil generation target, mechanism and is distance regeneration away. of the axions occurs behind sometake thick as shielding, parameters a for the magnet, those using in the ALPs experiment at DESY [ δ above-the-threshold behaviour. Right:the magnetic the field role region.alongside of each (The plot.) the colours energy represent the parameter vertical in axis increasing height as decay denoted in by the scale Figure 7 JHEP12(2019)162 ] 75 δ 1 2 - ). So with this set- is partially bridged 10 8 m 4 - 10 6 - 4 5 6 7 8 10 - - - - - 10 10 10 10 10 0.001 in the magnetic set-up (left) and laser set-up (right). δ – 18 – - . 2 φe 2 g 8 - 10 3 - 10 4 - 10 5 - 10 5 - 10 7 - 10 = 10 and pulse duration Φ = 50, using a similar analysis, we find the exclusion ξ 5 6 4 - - - . Projected exclusion bounds for 10 10 10 0.001 0.100 0.010 In below-threshold decay, a new mass-dependent tunneling exponent was identified, In the second scenario, we take the regeneration to occur in an intense laser pulse, highly-relativistic seed ALP particles.“edges” Using and a hence phenomenological strong model, field the gradients effect was of investigated. field which shows how the gapby to the the mass threshold of pair-creation the energy axion. of This 2 is reminiscent of Schwinger pair production catalysis [ parameters. Three distinct regions wereof identified: decay i) was below threshold via decay, tunnelingcan where the through proceed rate the in background the field; absenceof of ii) a a threshold above background threshold disappears. field; decaywe With calculated iii) which interest the strong-field in decay limit, probability lab where inThis measurement the a in is quasi-constant concept a crossed expected field constant of to magnetic finite field, be spatial extent. a good approximation to decay in a constant magnetic field for was presented, and thethough results the for pre-exponents massive of scalars scalarsnonpertubative and and exponential vector vectors dependency bosons are and were different kinematicsresults to also are to the identical given. have and pseudoscalar Al- significance so case, we foran the expect these example our cases background as to well. investigate how A the constant decay crossed depends field on was experimental chosen and as axion 6 Summary The decay of aground massive field ALP has into been an investigated. electron-positron An pair example in derivation a for high the intensity case EM of back- massive axions which is synchronised with theaxions firing arrives of at the the centre electron of beam theparameter laser such pulse. that the Assuming a pulse plane-wavebounds of laser pulse generated in of the intensity right-hand plot of figure occur where it is forbiddenup, by although vacuum the (see detection the region left-handunless is plot the of always axion “on”, figure is pair-creation above-threshold. isthe This exponentially di-photon set-up suppressed coupling. is more suitable for axion regeneration via Figure 8 JHEP12(2019)162 k χ – 19 – 1, which would be challenging to arrive at in the  k χ ), which permits any use, distribution and reproduction in the probability for ALP decay, due to it inducing an oscillation in CC-BY 4.0 ]. This article is distributed under the terms of the Creative Commons 70 , 69 and hence the field strength and ALP lightfront momentum. An asymptotic decrease k χ and To investigate a quasi-constant crossed field of finite spatial extent in the lab, we intro- In the strong-field regime, The case of above-threshold decay shows an interplay between the two channels of: i) thank AWE plc for support. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. Acknowledgments The authors thank Subiracknowledges computational Sarkar resources for supplied suggestedknowledge by improvements funding A. to from Ilderton. the Grant manuscript. No. B. K. B. EP/P005217/1. and K. B. K. M. A. D. B. ac- and G. G. would like to presented the nontrivial dependency ofas the decay the probability mass in of a the magnetic decaying field particle with edge, crosses the vacuum decay threshold. from a top-hat shapeproximation at of Φ taking = thefield probability 0. rate shape We for (the found a so-calledstill that constant contributions locally-constant field the from and field sharper outside integrating of approximation). theover it the the field, over field, In trajectory the the which of particular, we worseFinally, the found the there identifying to seed ap- are a be particle, region traceable which where to are interference the absent locally from constant the field simple approximation approximation. was valid, we disappears, and the nonperturbativeto asymptotic be result depends expected only —negligibly on small, eventually the and if field. so the loses This field meaning. is is strong enough,duced the a vacuum field decay with channel dimensionless is “sharpness” parameter, Φ, parametrising the departure the background is notgeneral a characteristic constant of crossed the field, above-thresholdplane-wave decay fields. it of is ALPs expected into electron-positron that pairs such in oscillations are a lab, but may have significance in some astrophysical scenarios, the concept of a threshold (a combination of external-field and ALPincrease particle parameters). The fieldthe was found probability to around both the vacuumbut value. also in This oscillationformula is in for the the ALP oscillations mass was parameter, found using a stationary phase analysis, and so, even if the ALP mass plays thethe role tunnelling of exponent the higher-frequency wasarguments background. arrived in at The [ same independently expression by for using simple energy-momentum vacuum decay and ii) field-stimulated decay through a nonperturbative dependence on by a second, higher-frequency background overlaid on a constant background, where here, JHEP12(2019)162 ] ] , , (2010) Phys. 13 8 , (2014) Axion-like 113 B 689 (2011) 211 Phys. Rev. JINST , (2013) 010 Hints of the 52 05 arXiv:1510.06892 Nature Phys. arXiv:0707.4312 2013 , [ , [ ]. Phys. Lett. ]. , JCAP , Phys. Rev. Lett. SPIRE , (2016) 37 ]. IN SPIRE Unidentified Line in X-Ray ][ IN Contemp. Phys. 12 , ][ (2007) 121301 ]. 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