Weak decay of magnetized pions
Gunnar S. Bali, Bastian B. Brandt, Gergely Endrodi, Benjamin Glaessle
WWeak decay of magnetized pions
G. S. Bali,
1, 2
B. B. Brandt, G. Endrődi, and B. Gläßle Institute for Theoretical Physics, Universität Regensburg, D-93040 Regensburg, Germany Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India. Institute for Theoretical Physics, Goethe Universität Frankfurt, D-60438 Frankfurt am Main, Germany Zentrum für Datenverarbeitung (ZDV), Universität Tübingen, Wächterstr. 76, D-72074 Tübingen, Germany
The leptonic decay of charged pions is investigated in the presence of background magnetic fields.In this situation Lorentz symmetry is broken and new fundamental decay constants need to beintroduced, associated with the decay via the vector part of the electroweak current. We calculatethe magnetic field-dependence of both the usual and a new decay constant non-perturbatively on thelattice. We employ both Wilson and staggered quarks and extrapolate the results to the continuumlimit. With this non-perturbative input we calculate the tree-level electroweak amplitude for thefull decay rate in strong magnetic fields. We find that the muonic decay of the charged pion isenhanced drastically by the magnetic field. We comment on possible astrophysical implications.
Introduction . Strong (electro)magnetic fields bear asignificant impact on the physics of various systemsranging from off-central heavy-ion collisions through theevolution of the early universe to magnetized neutronstars (magnetars). In particular, many novel phenom-ena emerge from the competition between electromag-netism and color interactions if the magnetic field B becomes similar in magnitude to the strong interactionscale: eB ∼ Λ . If the time-scale of the fluctuationsin B is larger than other relevant scales of the problem,it is reasonable to treat the magnetic field classically asa background field. For reviews on this subject, see forexample Refs. [1, 2].Such a background magnetic field is known, for in-stance, to affect the phase diagram of quantum chromo-dynamics (QCD) [3–5]. For cold astrophysical environ-ments, the low-temperature (hadronic) phase of QCDis particularly relevant. In this regime, a prime roleis played by the lightest hadrons, i.e. pions and kaons.Specifically, their masses appear in the nuclear equationof state within compact stellar objects and, thus, influ-ence their mass-radius relations. For stability and equi-librium analyses, the respective decay rates are equallyimportant. Dominant cooling mechanisms for magne-tars [6] involve (inverse) β -decay, photo-meson interac-tions and pion decay [7]. Pions radiate energy via in-verse Compton scattering until they decay, imprintingthe spectrum of the subsequently produced neutrinos [8].Strong electromagnetic fields are also created in violentastrophysical processes such as neutron star mergers andsupernova events, where weak nuclear reactions and de-cays govern cooling mechanisms and affect the neutrinospectrum [9].The B -dependence of pion masses has been investi-gated in various settings, ranging from chiral perturba-tion theory [10, 11] through numerical lattice QCD sim-ulations [3, 12–14] to model approaches [15–20]. Less isknown about the decay rates for nonzero magnetic fields.The decay constant for the neutral pion has been studiedin chiral perturbation theory [10, 11, 21] and in modelsettings [15–19, 22]. The decay constant of the chargedpion has only been discussed so far in chiral perturbation theory [11].In this letter we investigate the magnetic field-dependence of the decay rate of charged pions at zerotemperature. We demonstrate that the previous stud-ies in this direction are incomplete: in the presence ofthe magnetic field both neutral and charged pions havetwo independent decay constants, of which only one hasbeen investigated up to now. We determine both de-cay constants for charged pions non-perturbatively on thelattice, employing two different fermionic discretizations.Using this QCD input, we proceed to calculate the weakdecay rate using leading-order electroweak perturbationtheory. For this calculation we employ the lowest Lan-dau level (LLL) approximation for the outgoing chargedlepton state, which is a viable simplification for strongbackground magnetic fields. Our preliminary results us-ing Wilson fermions on a reduced set of lattice spacingswere presented in Ref. [23]. Pion decay constants . The pion decay constant is re-lated to the hadronic matrix elements H µ of the weakinteraction current between the vacuum and a pion statewith momentum p µ . For B = 0 , parity dictates thatthe matrix element (cid:104) | ¯ dγ µ u | π − (cid:105) vanishes, since the onlyLorentz-structure available is p µ : H µ ≡ (cid:10) | ¯ d ( x ) γ µ (1 − γ ) u ( x ) | π − ( p ) (cid:11) = − (cid:10) | ¯ d ( x ) γ µ γ u ( x ) | π − ( p ) (cid:11) = − ie ipx f π p µ . (1)The coefficient f π is the pion decay constant, which coin-cides for negatively and positively charged pions due tocharge conjugation symmetry. Throughout this letter weuse the normalization where f π ≈ MeV for a physicalpion in the vacuum.In the presence of a background electromagnetic field F µν the relation (1) takes a more general form. Exploit-ing Lorentz-covariance, using the tensor F µν and the vec-tor p µ , additional vector and axial vector combinationscan be formed: (cid:10) | ¯ d ( x ) γ µ γ u ( x ) | π − ( p ) (cid:11) = ie ipx (cid:20) f π p µ + f (cid:48)(cid:48) π eF µν p ν (cid:21) , (cid:10) | ¯ d ( x ) γ µ u ( x ) | π − ( p ) (cid:11) = ie ipx (cid:20) i f (cid:48) π (cid:15) µνρσ eF νρ p σ (cid:21) , (2) a r X i v : . [ h e p - l a t ] A ug where e denotes the elementary charge and we follow theconvention (cid:15) = +1 . Charge conjugation implies thatthe decay rate is the same for positively and negativelycharged pions and is also independent of the direction ofthe magnetic field. This is ensured by the ratios f π /f (cid:48) π and f π /f (cid:48)(cid:48) π being real, as we will see below. In our con-ventions, all three decay constants are real and positive.We remark that the new Lorentz structures also exist formatrix elements involving neutral pions.We consider a background magnetic field B that pointsin the z direction, implying F = − F = B . For apion of mass M π with vanishing momentum along themagnetic field, p = 0 , H = − ie iM π x f π M π , H = e iM π x f (cid:48) π eBM π . (3)For charged states that are in the LLL, only these twocomponents of H µ contribute to the decay rate. For de-tails on this and further elements of the perturbative cal-culation, we refer to Secs. I and II of the SupplementalMaterial. The decay constants f π and f (cid:48) π depend on theLorentz-scalars F µν F µν / B and p µ p µ = M π ( B ) .The matrix element of the vector current can also beinterpreted from a different perspective: the magneticfield mixes the pion with the ρ -meson having zero spinprojection along the magnetic field (i.e. s = 0 ) [14].Since the latter has the same quantum numbers as the µ = 3 component of the vector part of the electroweakcurrent, this mixing gives rise to a nonzero value for thevector matrix element, the second relation of Eq. (2).We mention that for nonzero temperature an addi-tional vector u µ describing the thermal medium ( u (cid:54) = u i )appears and leads to a splitting between spatial and tem-poral decay constants (see, for example, Ref. [22]). Herewe work at T = 0 , where this effect is absent. Further-more, note that the presence of the two terms f π and f (cid:48)(cid:48) π in the first relation of Eq. (2) implies that the axialvector matrix element is different for indices µ = 0 , and µ = 1 , , as was also found in Ref. [22]. However, fora purely magnetic background the term involving f (cid:48)(cid:48) π isabsent from H and H . Pion decay rate . The weak interaction matrix ele-ment (3) enters the rate of the leptonic decay process π − ( p ) → (cid:96) − ( k ) ¯ ν (cid:96) ( q ) , where p , k and q denote the four-momenta of the pion, the charged lepton (cid:96) and theantineutrino ¯ ν (cid:96) , respectively. The decay into a muon (cid:96) = µ is the dominant channel, with a decay fractionof . [24] at B = 0 .We work at the tree level of electroweak perturba-tion theory and employ the effective, four-fermion in-teraction with Fermi’s constant G as coupling. Due tothe current-current structure of the effective electroweakLagrangian [25, 26], the decay amplitude factorizes intoleptonic and hadronic parts, M = G/ √ · cos θ c L µ H µ ,where the Cabibbo angle θ c entered due to the mix-ing between the down and strange quark mass eigen-states. The relevant hadronic components H µ are shownin Eq. (3). Moreover, the leptonic component reads L µ ≡ ¯ u (cid:96) ( k ) γ µ (1 − γ ) v ν ( q ) in terms of the bispinor solu-tions u (cid:96) and v ν .The decay rate Γ involves the modulus square of theamplitude, integrated over the phase space and summedover the intrinsic quantum numbers of the outgoingasymptotic states. To find the latter for the charged lep-ton, we need the bispinor solutions of the Dirac equationfor B > . These are the so-called Landau levels —orbits localized in the spatial plane perpendicular to B with quantized radii. The Landau levels come with amultiplicity proportional to the flux Φ = | eB | · L of themagnetic field. In order to regulate this multiplicity, weneed to assume that the outgoing states are defined in afinite spatial volume V = L . For the decay rate suchvolume factors will cancel.For strong fields the dominant contribution stems fromthe lowest Landau level. The sum over the multiplicityof the LLL states gives [27] (cid:88) LLL u (cid:96) ( k )¯ u (cid:96) ( k ) = Φ2 π · ( /k (cid:107) + m (cid:96) ) · − σ , (4)where /k (cid:107) = k γ − k γ and σ = iγ γ is the relativis-tic spin operator. Eq. (4) reflects the fact that the LLLsolutions have their spin anti-aligned with the magneticfield (since the lepton has negative charge) and are char-acterized only by the momentum along the z direction(i.e. along B ). Due to angular momentum conservationthe antineutrino spin is also aligned with the magneticfield. Moreover, the right-handedness of the antineutrinoalso sets the direction of its momentum to be parallel tothe magnetic field.Having determined |M| , we finally need to integrateover the phase space for the outgoing particles. The re-sulting decay rate reads Γ( B ) = | eB | G π cos θ c | f π + if (cid:48) π eB | m (cid:96) M π . (5)As anticipated above, the decay rate only depends on themagnitude of B , due to the absence of an interferenceterm in | f π + if (cid:48) π eB | = f π + [ f (cid:48) π eB ] . Dividing by the B = 0 result [25], the dependence on G and θ c cancels: Γ( B )Γ(0) = f π + [ f (cid:48) π eB ] f π (0) · (cid:20) − m (cid:96) M π (0) (cid:21) − · | eB | M π (0) M π ( B ) . (6)We stress that this result was obtained using the LLLapproximation, which is in general valid for strongfields [2, 28]. For the leading-order perturbative decayrate, higher Landau-levels turn out to give zero contri-bution for eB > M π (0) − m (cid:96) . Lattice setup . Eq. (6) contains three non-perturbativeparameters, which describe the response of the pionto the background field: M π ( B ) , f π ( B ) and f (cid:48) π ( B ) .We calculate these via two independent sets of latticeQCD simulations. First, we work with quenched Wil-son quarks. The zero-temperature ensembles generatedand analyzed in Ref. [14] are supplemented by a fourth,finer lattice ensemble, so that the lattice spacing spans . fm ≤ a ≤ . fm. The B = 0 pion mass is setto M π (0) ≈ MeV. To remove B -dependent O ( a ) effects on quark masses, the bare mass parameters aretuned to fall on the magnetic field-dependent line of con-stant physics determined in Ref. [14].In the second set of simulations we work with N f =2 + 1 flavors of dynamical staggered fermions, using theensembles of Refs. [3, 29]. The employed lattice spac-ings lie in the range . fm ≤ a ≤ . fm, and thequark masses are set to their physical values [30] suchthat M π (0) ≈ MeV. For both formulations we per-form a continuum extrapolation based on the availablefour lattice spacings. This enables us to quantify thesystematics related to the differences between the twoapproaches: heavier-than-physical versus physical pionmass and quenched versus dynamical quarks. We re-mark that simulations with dynamical Wilson quarks inthe presence of a background magnetic field would requirecomputational resources that are by orders of magnitudelarger than those used for the current study.The general measurement strategy involves the analy-sis of the matrix elements H and H of Eq. (3). Theseare encoded in the spatially averaged Euclidean correla-tors C O P ( t ) = (cid:10) (cid:80) x O ( x , t ) P † ( , (cid:11) with O being eitherof P = ¯ uγ d , A = ¯ uγ γ d or V = ¯ uγ d . In the large- t limit, the dominant contribution to the spectral repre-sentation of all three correlators comes from a pion state.We fit the three correlators using C O P ( t ) = c O P (cid:104) e − M π t ± e − M π ( N t − t ) (cid:105) , (7)where the positive sign is taken for C P P and the negativefor C AP and C VP due to the time reversal properties ofthe correlators. The decay constants are extracted via f π = Z A · √ c AP √ M π c P P , if (cid:48) π eB = Z V · √ c V P √ M π c P P , (8)where Z A and Z V are the multiplicative renormalizationconstants of the axial vector and vector currents.For Wilson quarks we employ smeared pseudoscalarsources (for more details, see Ref. [14]) and fit all threecorrelators simultaneously. For the staggered analysis wework with point sources and fit the C AP and C P P cor-relators to find M π and f π . In a second step, volumesources are employed for C VP /C AP to enhance the sig-nal in f (cid:48) π /f π . The staggered discretization of A and V requires operators nonlocal in Euclidean time and hasbeen worked out in Ref. [31]. For staggered quarks andthe currents we use the renormalization constants aretrivial, Z A = Z V = 1 . For Wilson quarks this is notthe case, nevertheless, these ultraviolet quantities areexpected to be independent of the magnetic field. Weemploy the B = 0 non-perturbative results of Ref. [32](see also Ref. [33]) and fit these in combination with theasymptotic perturbative two-loop results of Ref. [34] (seealso Ref. [35]) to a Padé parametrization. Results . Inspecting the
B > correlation functions, wesee clear signals for C VP (see Sec. III of the SupplementalMaterial), which vanishes at B = 0 . The mass and thedecay constants are extracted using the fits described inEqs. (7) and (8). For the complete magnetic field range,the pion mass is found to be described within by theformula M π /M π (0) = (cid:112) | eB | /M π (0) , (9)which assumes pions to be point-like free scalars. Thishas been observed many times in the literature, bothusing dynamical staggered [3], quenched Wilson [12, 14]and quenched overlap quarks [13].The normalized combinations f π /f π (0) and f (cid:48) π eB/f π (0) are shown for four lattice spacings inFig. 1, both for staggered and for Wilson fermions. FIG. 1. Continuum extrapolation (gray bands) of the de-cay constants for staggered (upper panel) and Wilson quarks(lower panel). Both panels include results for f π /f π (0) (up-per points) and for f (cid:48) π eB/f π (0) (lower points). The staggeredresults were obtained at the physical point, while the Wilsonresults correspond to a B = 0 pion mass of M π (0) = 415 MeV.For f π /f π (0) we also compare the two continuum extrapola-tions after a rescaling of the magnetic field for the staggeredcurve (purple band; see the text for details). To parameterize the B -dependence, we found that itis advantageous to consider polynomial fits for theamplitudes f π M π and f (cid:48) π eBM π of the matrix elementsof Eq. (3). The continuum extrapolation is carried outby including lattice artefacts of O ( a ) (for Wilson) and O ( a ) (for staggered) in the coefficients. Specifically, theparameterizations of the individual decay constants read f π /f π (0) = [1 + c | eB | ] · M π (0) /M π ,f (cid:48) π /f π (0) = [ d + d | eB | + d | eB | ] · M π (0) /M π , (10)and M π /M π (0) is taken from Eq. (9). The quality ofthe staggered data for f (cid:48) π /f π (0) only allows for a fit with d = d = 0 . For larger magnetic fields we also include asystematic error estimated using the uncertainties of thedata at high B . Ideally, the analysis in this region shouldbe complemented by additional finer ensembles to makethe continuum extrapolation of f (cid:48) π more robust. Withinour range of B fields, however, the decay rate and itsuncertainty are dominated by f π .Motivated by the dependence of M π /M π (0) on thescaling variable eB/M π (0) , we compare the continuumextrapolated Wilson results (obtained for M π (0) =415 MeV) to the staggered data (obtained for physicalpion masses), after rescaling the magnetic field for thelatter. In particular, we take the staggered results for f π /f π (0) at the magnetic field eB · (415 / . Theresulting curve is also included in the lower panel ofFig. 1, revealing nice agreement between the two ap-proaches. In particular, the slope at the origin is foundto be − . GeV − for staggered and − . GeV − for Wilson — the ratio of which is consistent with thesquared pion mass ratio.For low magnetic fields the ratio f (cid:48) π /f π (0) approaches aconstant so that in this case the two discretizations can becompared to each other without a similar rescaling. Weindeed find consistent results: f (cid:48) π /f π (0) = 0 . GeV − for staggered and . GeV − for Wilson, respectively.We note that the errors of the staggered data for thisdecay constant increase quickly as B grows, rendering acomparison for higher magnetic fields inconclusive. Wemention moreover that due to the different treatmentof sea quark loops in the two approaches, the observedagreement is rather surprising and calls for a better un-derstanding of the role of dynamical quarks in the Wilsonsetup.To determine the decay rate (6) we employ the con-tinuum extrapolated staggered results. On the basis ofthe above comparisons, we also consider the Wilson re-sults, using a rescaling to the physical point as explainedabove. For the pion mass we use the analytic depen-dence (9), including a systematic error. The so ob-tained curves for the muonic decay rate are shown inFig. 2 for magnetic fields eB ≤ . GeV , where bothstaggered and (rescaled) Wilson results are available.The decay rate is enhanced drastically by the magneticfield: for eB ≈ . GeV we observe an almost fifty-foldincrease with respect to B = 0 . We note that while theordinary decay mechanism dominates in our study, the FIG. 2. The muonic decay rate in units of its B = 0 value us-ing the continuum extrapolated staggered results with physi-cal quark masses (green). For comparison, the continuum ex-trapolated Wilson data at higher-than-physical quark massesare also included after a rescaling of the magnetic field bythe squared pion mass (yellow). The LLL approximation weemployed for the decay rate is valid for eB > M π (0) − m µ ,marked by the dashed vertical line. contribution of the new vector decay constant f (cid:48) π grows toabout of the total decay rate at the largest magneticfield of Fig. 2.We remark that Eq. (6), supplemented by our stag-gered lattice results, suggests the decay rate K − → µ − ¯ ν µ to be enhanced only by a factor of about two at eB ≈ . GeV . This is mainly due to the larger mass of thekaon. Finally, the pion decay rate into electrons under-goes an enhancement by a factor of about ten. In fact,according to Eq. (5) the ratio of muonic and electronicdecay rates becomes independent of the magnetic field, Γ( B ) π → e ¯ ν e (cid:14) Γ( B ) π → µ ¯ ν µ = ( m e /m µ ) ≈ . · − , (11)and is by about a factor of . smaller than the corre-sponding fraction . · − at B = 0 . Conclusions . In this letter we computed the rate forthe leptonic decay of charged pions in the presence ofstrong background magnetic fields. The result is givenby Eq. (6), for which we employed electroweak perturba-tion theory and the lowest-Landau-level approximationfor the outgoing charged lepton (cid:96) , valid for strong fields.Including higher-order terms in the electroweak calcula-tion (see, e.g., Refs. [36, 37]), as well as going beyondthe lowest Landau level is possible, allowing to systemat-ically improve this result. In this case also the constant f (cid:48)(cid:48) π , that we have not determined here, may enter.We demonstrated that — besides the ordinary piondecay constant f π — the decay rate depends on an addi-tional fundamental parameter f (cid:48) π . The latter decay con-stant characterizes a new decay mechanism that becomesavailable for nonzero magnetic fields. We calculated bothdecay constants, together with the pion mass, using lat-tice simulations employing dynamical staggered quarkswith physical masses, and also compared to the results ofquenched Wilson simulations with heavier-than-physicalquarks. For both cases, continuum extrapolations werecarried out to eliminate lattice discretization errors. Forlow magnetic fields, we obtain for the new decay constant f (cid:48) π = 0 . GeV − + O ( B ) .Our final result for the full decay rate is visualized inFig. 2, revealing a dramatic enhancement of the rate or,correspondingly, a drastic reduction of the mean lifetime τ π = 1 / Γ . A typical B > lifetime is τ π = 5 · − s for B ≈ . GeV /e ≈ · T . Since lifetimes of magnetic fields in off-central heavy-ioncollisions are by − orders of magnitude smaller [1],it is clear that this effect will not result in any observ-able predictions for heavy-ion phenomenology. How-ever, the B -dependence of weak decays is expected tobe essential in astrophysical environments. (Notice thatthe upper limit for magnetic field strengths in the core of magnetized neutron stars is thought to be around B = 10 − T [38, 39].) Indeed, for B = 0 the pionmean lifetime and the time-scale for cooling via inverseCompton scattering are roughly comparable [8]. Thus, areduction in τ π will inevitably decrease radiation energyloss of pions and result in a harder neutrino spectrum.Similarly to the pion decay rate, the magnetic fieldwill have an impact on (inverse) β -decay rates andnucleon electroweak transition form-factors. Indeed,an enhancement by the magnetic field is expected forprocesses involving nucleons as well [40–42], see, e.g.,the review [43]. The tools developed in the present letterwill also be useful to study these effects that are relevantfor cold and magnetized environments. Acknowledgments . This research was funded by theDFG (Emmy Noether Programme EN 1064/2-1 andSFB/TRR 55). GB thanks Basudeb Dasgupta for dis-cussion. The computations were carried out on the GPU,iDataCool and Athene 2 clusters of Universität Regens-burg.
Supplemental Material
I. DIRAC BISPINORS FOR
B > We work with the metric g µν = diag (1 , − , − , − and use Minkowskian Dirac matrices with the conven-tions σ = iγ γ and γ = iγ γ γ γ .To write down the bispinor states for the outgoingcharged lepton, we need to solve the Dirac equation for B > . The solutions are the so-called Landau levels,indexed by the Landau index n . These have the energies E n,k ,s = ± (cid:113) (2 n + 1 + 2 s ) eB + k + m (cid:96) , (S1)where k is the momentum along the magnetic field —which is chosen to lie in the positive z direction — and s is the spin projection on the z axis. Note that the lowestenergy is n = k = 0 with s = − / , reflecting the factthat the lepton spin (due to its negative charge − e ) tendsto align itself anti-parallel to B . Higher-lying levels haveenergies of at least √ eB and are in general not expectedto be relevant at low temperatures and densities, as longas eB (cid:29) m (cid:96) . Here we employ the lowest Landau level(LLL) approximation ( n = 0 , s = − / ), which simpli-fies the discussion considerably. The calculation can begeneralized to take into account all levels.We start from the LLL modes calculated in Ref. [27]in the gauge A = − Bx and in the infinite volume (seealso Ref. [2]). Here we consider the volume to be largebut finite to regulate the number of modes. In a volume V = L this degeneracy factor is Φ / (2 π ) with Φ = | eB | L being the flux of the magnetic field through the area of the system. The modes in coordinate space read, u (cid:96) ( k ) = (cid:112) k + m (cid:96) · (cid:18) eBL π (cid:19) / · (cid:18) , , , − k k + m (cid:96) (cid:19) (cid:124) · exp (cid:34) ik x − ik x − ik x − eB (cid:18) x + k eB (cid:19) (cid:35) , (S2)where the energy is k ≡ E ,k , − / = (cid:112) k + m (cid:96) . Notethat while k is a momentum, k is a quantum numberindexing the degeneracy of the Landau-modes and has nomomentum interpretation. Also note that the LLL canonly have negative spin so that σ u (cid:96) ( k ) = − u (cid:96) ( k ) . (S3)The normalization of the state (S2) is (cid:90) d x u † (cid:96) ( p ) u (cid:96) ( k ) = (2 π ) δ (2)1 , ( p − k ) 2 k L . (S4)Thus, the states with different k quantum numbers areorthogonal to each other. Regulating the δ -functions ina finite volume , we have | u (cid:96) ( k ) | = (cid:90) d x u † (cid:96) ( k ) u (cid:96) ( k ) = 2 k V , (S5) Compared to Ref. [27] we insert a factor √ k + m (cid:96) · √ L for amore convenient normalization. The Fourier-transform of the δ -function in a finite linear size L is (2 π ) δ ( p ) = (cid:82) d x e ipx so that δ (0) = L/ (2 π ) . which coincides with the usual normalization of bispinors(cf. Ref. [26]).Having specified the normalization of the charged lep-ton state, we can calculate the sum over the intrinsicquantum numbers. For our LLL states this does not in-volve a spin-sum, since only s = − / is allowed. How-ever, we have to sum over the unspecified quantum num-ber k . In a finite volume, this index takes Φ / (2 π ) dif-ferent values and can be approximated as L/ (2 π ) timesthe integral over k , (cid:88) LLL ≡ (cid:88) k = L (cid:90) d k π . (S6)After some algebra we obtain (see also Ref. [27]) the re-sult that we quoted in Eq. (4) in the body of the text. II. DECAY RATE FOR
B > As explained in the main text, the antineutrino hasfixed spin (in the positive z direction) as well as fixed chi-rality. The corresponding bispinor solution v ν ( q ) neces-sarily has vanishing perpendicular momenta q = q = 0 ,satisfies the on-shell relation q = | q | and fulfills v ν ( q )¯ v ν ( q ) = P σ /q P σ , P σ = 1 − σ , (S7)keeping in mind that the physical spin for antiparticles isthe opposite of the eigenvalue of P σ . Inserting (S7) andthe spin sum (4) into the modulus square of the summedamplitude, we obtain A = (cid:88) LLL MM ∗ = G Φ4 π cos θ c H µ H ∗ ρ T µρ , (S8)where T µρ ≡ tr (cid:104) P σ /qP σ γ ρ (1 − γ )( /k (cid:107) + m ) P σ γ µ (1 − γ ) (cid:105) . (S9)The trace is simplified using standard identities, revealingthat T µρ is only nonzero if the indices µ and ρ assumethe values or . Thus, only H and H contribute tothe decay rate, as anticipated in the main text. Makinguse of Eq. (3) we arrive at A = G Φ π cos θ c M π ( k + k )( q + q ) | f π + if (cid:48) π eB | . (S10)We proceed by specifying the kinematics in the framewhere the pion has vanishing momentum along the mag-netic field (and is in the LLL). Together with the con-servation rules, the on-shell conditions for the outgoingleptons fully specify the magnitude of q = − k in termsof the masses M π and m (cid:96) , the decay only being possiblefor M π > m (cid:96) . Due to the right-handedness of the an-tineutrino and the fixed spins of the leptons, k must in fact be negative, also visible from the q + q = | q | + q = | k | − k factor of Eq. (S10). Thus we have k = M π + m (cid:96) M π , k = m (cid:96) − M π M π . (S11)Inserting this into Eq. (S10), we obtain A = G Φ π cos θ c | f π + if (cid:48) π eB | ( M π − m (cid:96) ) m (cid:96) . (S12)Finally, we need the phase space integration factor toobtain the full decay rate Γ . This is the integral of thedifferential probability d P over a time interval T [26], Γ = 1 T (cid:90) d P , (S13)where, working with k = − k > ,d P = A · M π V q V k V · V (2 π ) d q L π d k · T V (2 π ) δ ( M π − q − k ) δ ( q + k ) δ ( q ) δ ( q ) . (S14)Here the normalization of the one-particle states (forthe charged lepton, see Eq. (S5)) was used, and the δ -functions ensure that the perpendicular momenta of theantineutrino vanish. Note that the phase space for thecharged lepton is one-dimensional due to its LLL nature.Performing the integral over q and expressing the ener-gies through k we obtain Γ = A M π L (cid:90) ∞ d k k (cid:112) k + m (cid:96) δ (cid:18) M π − k − (cid:113) k + m (cid:96) (cid:19) . (S15)After the variable substitution y = k + (cid:112) k + m (cid:96) , weinsert A from Eq. (S12), to finally arrive at Eq. (5) ofthe main text. As anticipated, all volume factors havecanceled. We note that while the inclusion of higher Lan-dau levels is considerably more involved, their contribu-tion is constrained via energy conservation. Indeed, forthe charged lepton to be created with quantum num-bers n and s , it is required that M π > E n, ,s =(2 n + 1 + 2 s ) eB + m (cid:96) , see Eq. (S1). Thus, for mag-netic fields eB > M π (0) − m (cid:96) , only the LLL ( n = 0 , s = − / ) can contribute. For higher orders in elec-troweak perturbation theory soft photons become rele-vant and this simple picture changes. We note that the same expression can also be obtained by start-ing with an incoming pion wave function in the LLL (analogousto the spacetime-dependence of Eq. (S2)), expanding it in planewaves and finding the contribution of each plane wave to M .Care must be taken in this case to normalize the δ -function forthe perpendicular antineutrino momenta with the correct B → limit. Our preliminary result in Ref. [23] was missing a factor / . The decay rate may be compared to the B = 0 re-sult [25], Γ(0) = G π cos θ c f π (0) [ M π (0) − m (cid:96) ] m (cid:96) M π (0) , (S16)where we explicitly indicated that M π and f π are to beunderstood at B = 0 . The ratio of Eqs. (5) and (S16)gives Eq. (6) in the body of the text. For charged kaondecay the calculation is completely analogous, only thesubstitutions M π → M K , f π → f K , f (cid:48) π → f (cid:48) K and cos θ c → sin θ c have to be made. III. CORRELATORS ON THE LATTICE
The non-perturbative determination of the decay con-stants involves the analysis of the correlators C P P , C AP and C VP defined in the main text. These are plotted inFig. 3 for both fermion formulations for a high magneticfield and meson masses approximately equal in latticeunits (for the staggered case a kaon correlator is shown).A clear signal is visible for the C VP correlator, whichwould vanish on average at B = 0 . FIG. 3. The correlators using Wilson (upper points) andstaggered (lower points) quarks. The employed magneticfields are eB ≈ . GeV (Wilson) and eB ≈ . GeV (staggered). For the axial vector and vector operators theabsolute value of the correlator is shown. The Wilson data isshifted vertically for better visibility.[1] D. Kharzeev, K. Landsteiner, A. Schmitt, and H.-U.Yee, “Strongly Interacting Matter in Magnetic Fields,” Lect. Notes Phys. (2013) pp.1–624.[2] V. A. Miransky and I. A. Shovkovy, “Quantum fieldtheory in a magnetic field: From quantumchromodynamics to graphene and Dirac semimetals,”
Phys. Rept. (2015) 1–209, arXiv:1503.00732[hep-ph] .[3] G. Bali, F. Bruckmann, G. Endrődi, Z. Fodor, S. Katz, et al. , “The QCD phase diagram for external magneticfields,”
JHEP (2012) 044, arXiv:1111.4956[hep-lat] .[4] G. Endrődi, “Critical point in the QCD phase diagramfor extremely strong background magnetic fields,”
JHEP (2015) 173, arXiv:1504.08280 [hep-lat] .[5] J. O. Andersen, W. R. Naylor, and A. Tranberg, “Phasediagram of QCD in a magnetic field: A review,” Rev.Mod. Phys. (2016) 025001, arXiv:1411.7176[hep-ph] .[6] R. C. Duncan and C. Thompson, “Formation of verystrongly magnetized neutron stars - implications forgamma-ray bursts,” Astrophys. J. (1992) L9.[7] E. Waxman and J. N. Bahcall, “High-energy neutrinosfrom cosmological gamma-ray burst fireballs,”
Phys.Rev. Lett. (1997) 2292–2295, arXiv:astro-ph/9701231 [astro-ph] .[8] B. Zhang, Z. G. Dai, and P. Mészáros, “High-energyneutrinos from magnetars,” Astrophys. J. (2003)346–351, arXiv:astro-ph/0210382 [astro-ph] .[9] B. D. Metzger, T. A. Thompson, and E. Quataert, “Amagnetar origin for the kilonova ejecta in GW170817,”
Astrophys. J. no. 2, (2018) 101, arXiv:1801.04286[astro-ph.HE] . [10] N. O. Agasian and I. Shushpanov,“Gell-Mann-Oakes-Renner relation in a magnetic fieldat finite temperature,”
JHEP (2001) 006, arXiv:hep-ph/0107128 [hep-ph] .[11] J. O. Andersen, “Chiral perturbation theory in amagnetic background - finite-temperature effects,”
JHEP (2012) 005, arXiv:1205.6978 [hep-ph] .[12] Y. Hidaka and A. Yamamoto, “Charged vector mesonsin a strong magnetic field,”
Phys. Rev.
D87 no. 9,(2013) 094502, arXiv:1209.0007 [hep-ph] .[13] E. V. Luschevskaya, O. E. Solovjeva, O. A. Kochetkov,and O. V. Teryaev, “Magnetic polarizabilities of lightmesons in SU (3) lattice gauge theory,” Nucl. Phys.
B898 (2015) 627–643, arXiv:1411.4284 [hep-lat] .[14] G. S. Bali, B. B. Brandt, G. Endrődi, and B. Gläßle,“Meson masses in electromagnetic fields with Wilsonfermions,” arXiv:1707.05600 [hep-lat] .[15] S. Fayazbakhsh, S. Sadeghian, and N. Sadooghi,“Properties of neutral mesons in a hot and magnetizedquark matter,”
Phys. Rev.
D86 (2012) 085042, arXiv:1206.6051 [hep-ph] .[16] S. S. Avancini, R. L. S. Farias, M. Benghi Pinto, W. R.Tavares, and V. S. Tim´teo, “ π pole mass calculation ina strong magnetic field and lattice constraints,” Phys.Lett.
B767 (2017) 247–252, arXiv:1606.05754[hep-ph] .[17] R. Zhang, W.-J. Fu, and Y.-X. Liu, “Properties ofMesons in a Strong Magnetic Field,”
Eur. Phys. J.
C76 no. 6, (2016) 307, arXiv:1604.08888 [hep-ph] .[18] S. Mao and Y. Wang, “Effect of Quark DimensionReduction on Goldstone Mode in Magnetic Field,” arXiv:1702.04868 [hep-ph] . [19] D. Gómez Dumm, M. Izzo Villafañe, and N. N.Scoccola, “Neutral meson properties under an externalmagnetic field in nonlocal chiral quark models,” Phys.Rev.
D97 no. 3, (2018) 034025, arXiv:1710.08950[hep-ph] .[20] M. Coppola, D. Gómez Dumm, and N. N. Scoccola,“Charged pion masses under strong magnetic fields inthe NJL model,”
Phys. Lett.
B782 (2018) 155–161, arXiv:1802.08041 [hep-ph] .[21] J. O. Andersen, “Thermal pions in a magneticbackground,”
Phys. Rev.
D86 (2012) 025020, arXiv:1202.2051 [hep-ph] .[22] S. Fayazbakhsh and N. Sadooghi, “Weak decay constantof neutral pions in a hot and magnetized quark matter,”
Phys. Rev.
D88 no. 6, (2013) 065030, arXiv:1306.2098[hep-ph] .[23] G. S. Bali, B. B. Brandt, G. Endrődi, and B. Gläßle,“Pion decay in magnetic fields,”
EPJ Web Conf. (2018) 13005, arXiv:1710.01502 [hep-lat] .[24]
Particle Data Group
Collaboration, K. A. Olive et al. , “Review of Particle Physics,”
Chin. Phys.
C38 (2014) 090001.[25] L. Okun,
Leptons and Quarks . North-Holland PersonalLibrary. Elsevier Science, 2013.[26] M. Schwartz,
Quantum Field Theory and the StandardModel . Cambridge University Press, 2014.[27] K. Bhattacharya, “Solution of the Dirac equation inpresence of an uniform magnetic field,” arXiv:0705.4275 [hep-th] .[28] F. Bruckmann, G. Endrődi, M. Giordano, S. D. Katz,T. G. Kovács, F. Pittler, and J. Wellnhofer, “Landaulevels in QCD,”
Phys. Rev.
D96 no. 7, (2017) 074506, arXiv:1705.10210 [hep-lat] .[29] G. S. Bali, F. Bruckmann, G. Endrődi, Z. Fodor, S. D.Katz, and A. Schäfer, “QCD quark condensate inexternal magnetic fields,”
Phys. Rev.
D86 (2012)071502, arXiv:1206.4205 [hep-lat] .[30] S. Borsányi, G. Endrődi, Z. Fodor, A. Jakovác, S. D.Katz, et al. , “The QCD equation of state withdynamical quarks,”
JHEP (2010) 077, arXiv:1007.2580 [hep-lat] .[31] G. W. Kilcup and S. R. Sharpe, “A Tool Kit forStaggered Fermions,”
Nucl. Phys.
B283 (1987) 493–550.[32] V. Gimenez, L. Giusti, F. Rapuano, and M. Talevi,“Nonperturbative renormalization of quark bilinears,”
Nucl. Phys.
B531 (1998) 429–445, arXiv:hep-lat/9806006 [hep-lat] .[33] M. Göckeler, R. Horsley, H. Oelrich, H. Perlt,D. Petters, P. E. L. Rakow, A. Schäfer, G. Schierholz,and A. Schiller, “Nonperturbative renormalization ofcomposite operators in lattice QCD,”
Nucl. Phys.
B544 (1999) 699–733, arXiv:hep-lat/9807044 [hep-lat] .[34] A. Skouroupathis and H. Panagopoulos, “Two-looprenormalization of vector, axial-vector and tensorfermion bilinears on the lattice,”
Phys. Rev.
D79 (2009)094508, arXiv:0811.4264 [hep-lat] .[35] G. S. Bali, F. Bursa, L. Castagnini, S. Collins,L. Del Debbio, B. Lucini, and M. Panero, “Mesons inlarge-N QCD,”
JHEP (2013) 071, arXiv:1304.4437[hep-lat] .[36] V. Lubicz, G. Martinelli, C. T. Sachrajda, F. Sanfilippo,S. Simula, and N. Tantalo, “Finite-Volume QEDCorrections to Decay Amplitudes in Lattice QCD,” Phys. Rev.
D95 no. 3, (2017) 034504, arXiv:1611.08497 [hep-lat] .[37] A. Patella, “QED Corrections to HadronicObservables,”
PoS
LATTICE2016 (2017) 020, arXiv:1702.03857 [hep-lat] .[38] D. Lai and S. L. Shapiro, “Cold equation of state in astrong magnetic field - Effects of inverse beta-decay,”
Astrophys. J. (Dec., 1991) 745–751.[39] E. J. Ferrer, V. de la Incera, J. P. Keith, I. Portillo, andP. L. Springsteen, “Equation of State of a Dense andMagnetized Fermion System,”
Phys. Rev.
C82 (2010)065802, arXiv:1009.3521 [hep-ph] .[40] J. J. Matese and R. F. O’Connell, “Neutron Beta Decayin a Uniform Constant Magnetic Field,”
Phys. Rev. (1969) 1289–1292.[41] L. Fassio-Canuto, “Neutron beta decay in a strongmagnetic field,”
Phys. Rev. (1969) 2141–2146.[42] S. Shinkevich and A. Studenikin, “Relativistic theory ofinverse beta decay of polarized neutron in strongmagnetic field,”
Pramana (2005) 215–244, arXiv:hep-ph/0402154 [hep-ph] .[43] C. Giunti and A. Studenikin, “Neutrino electromagneticinteractions: a window to new physics,” Rev. Mod.Phys. (2015) 531, arXiv:1403.6344 [hep-ph]arXiv:1403.6344 [hep-ph]