Cosmological Chirality and Magnetic Fields from Parity Violating Particle

arXiv:2101.06344v1 [hep-ph] 16 Jan 2021 tddet hrleet nSc I.Orcnlsosare conclusions Our III. Sec. gener- in be effects can an chiral that find we to helicity We due where magnetic ated II, the currents. Sec. to electric in bound lepton induced upper tau the the estimate of also context the in tail fields magnetic helicity. too their cosmological is of and affect helicity decays significantly particle to net for small the only particles, mag- accounting model generate However, standard can that leads helicity. CME principle the netic in via helicity currents cosmolog- particle electric the net to to The helicity particle medium. net ical chiral weak a are provides the products this Since decay and the interactions. parity, weak epochs, violate unsta- the interactions cosmological of various to number at due a decay primarily has that model basic particles standard The ble the quarks. that and is leptons idea heavy parity- the of to cosmol- decays due standard violating how interactions in particle discuss standard arise we with effectively paper ogy can this CVE In and CME 27–33]. been [5, also astrophysical fields and have magnetic cosmological CVE of evolution and and the to CME (CME chiral- applied The effects the [26] respectively). namely chiral-vortical renewed CVE plasmas, and been in [25] has effects magnetic chiral there in [24], interest experiments collision ion 23]. no [22, with the yet though measure as 21] par- to detections [20, a used significant spectrum of be helicity can field claims magnetic that are signature direct there violating more Concurrently ity and 18], [12– [19]. [17, debate strength evidence lower without field indirect not direct magnetic though an and cosmological 16], for the 10] claims on [9, are bound arguments There well analytical quite [11]. of simulations understood basis now 5–8]. the is on [1, cos- fields extensions of magnetic evolution possible mological (MHD) its magneto-hydrodynamical and par- The known [2–4] to tied physics in ideas directly fields ticle are Several magnetic which generate of [1]. some can cosmology, that fronts proposed several been on have fields magnetic mic esatb eciigtepyia ffc nsm de- some in effect physical the describing by start We heavy- by motivated developments, these to parallel In cos- in interest growing seen have decades few last The omlgclciaiyadmgei ed rmprt viol parity from fields magnetic and chirality Cosmological oe atce,fcsn nteeapeo a etn.The cosmological the leptons. to tau contribution p significant of to effect. a example due make the medium to on cosmological small focusing the particles, of chirality model the estimate We .INTRODUCTION I. ∗ hsc eatet rzn tt University, State Arizona Department, Physics amyVachaspati Tanmay ut nvriy efr,M 25,USA. 02155, MA Medford, University, Tufts eateto hsc n Astronomy, and Physics of Department ep,A 58,USA. 85287, AZ Tempe, † nttt fCosmology, of Institute ∗ lxne Vilenkin Alexander , umrzdaddsusdi e.IV. Sec. in discussed and summarized ieecs h hsc are vrt h aeo decays of small case for the particles. to Except other over background of vorticity. carries the fluid physics in and the leptons differences, field tau magnetic of a decay of the discuss will trinos n nituit oirn n neutrinos and positrons into anti-tau and u opril neatos esalaotti iwin view this adopt shall as follows. We change electrons what interactions. only final can particle Helicity the to of state. due helicity ultrarelativistic think definite to a for in better helicity being is to it stars, so equal neutron particles, and nearly supernovae is of context chirality not the in in is emphasized [34] chirality as Ref. Instead, but particles. interactions, massive for weak preserved the by produced decay. tau’s as chirality cosmological universe left-handed the the net and anti-taus, gains of than medium asymmetry taus lepton more are the right-handed pre- there to relativistic tau Due a into the positron. decays Therefore anti-tau an electron, while left-handed interactions. relativistic primar- a weak into proceed decays dominantly the they by since parity ily violate decays These etfaeis frame rest asaecntnl erdcdb ecin nes to inverse reactions by reproduced constantly are taus oilsrt h hsc fteeet nti eto we section this in effect, the of physics the illustrate To a etn ea noeetos(rmos n neu- and muons) (or electrons into decay leptons Tau The are they since chiral are electrons product decay The I HRLT RDCINI TAU IN PRODUCTION CHIRALITY II. τ antcfil i h chiral-magnetic the via field magnetic asis mass o-rva hrlt shwvrtoo however is chirality non-trivial rt iltn easo standard of decays violating arity τ τ 2 = m .Pril decay Particle A. τ τ τ † − . + 77MVadthe and MeV 1777 = 9 → → × DECAYS tn atcedecays particle ating 10 e e − + − ¯ + + 13 ν ν .Hwvr h decaying the However, s. τ τ + ¯ + ν ν e e τ ieiei its in lifetime (1) (2) 2

1 eq (1), (2), so the number density of taus, nτ , remains time), and the approximate solution of (7) is δnτ ≈ δnτ . eq comparable to that of photons until the time Inserting (7) in (4) and then using δnτ ≈ δnτ we have −7 d d a3n tτ ∼ 10 s, (3) (a3n )= − a3δneq − χ . (8) dt χ dt τ τ
χ when the cosmic temperature drops to T ∼ m . At later τ With a given δneq, this equation can be solved in quadra- times n decreases exponentially. τ τ tures, but we shall instead find an approximate solution. The particle helicity density, n , is produced in tau de- χ Since we are in the regime where τ ≪ t, the term on the cays and is dissipated by helicity flipping scatterings and χ left-hand side can be neglected and we obtain due to the chiral anomaly. The latter is proportional to 3 2 eq α B [35], where α ≈ 1/137 is the fine structure constant 3 d δnτ nχ ≈−τχT , (9) and B the magnetic field strength, and is much slower dt  T 3  than helicity flipping scatterings for vanishing or weak magnetic fields. We will ignore the anomalous flipping where we have used aT ≈ const. for now but will discuss it in Sec. ?? when we consider Once we determine the equilibrium excess of anti-taus eq the effect of particle chirality on the generation of mag- over taus, denoted by δnτ , we can estimate the chirality netic fields. The evolution of particle helicity density can density of the universe due to tau decays using (9). be described by the kinetic equation in the relaxation time approximation, B. Equilibrium density 3 3 d 3 a eq a nχ (a nχ)= (δnτ − δnτ ) − , (4) eq dt τd τχ The equilibrium density δnτ is given by ∞ where eq 1 2 E − µτ E + µτ δnτ = 2 dpp f − f , 2π Z0   T   T  + − (10) δnτ = nτ − nτ , (5) where f(x) = (ex + 1)−1 is the Fermi distribution, E = − + 2 2 nτ and nτ are the densities of tau and anti-tau particles, p + mτ , and µτ is the chemical potential of τ particles. eq respectively, δnτ is the equilibrium value of δnτ , τd ∼ pAt T ≫ mτ ,µτ we can expand the integrand in Eq. (10) 2 2 (T/mτ )ττ is the decay time of taus (assuming that T > in powers of mτ /p and µτ /T . The integrations are then −1 mτ and with time dilation taken into account) and τχ easily performed and we find is the electron helicity flipping rate. At T ≫ me, the 2 2 helicity flipping rate is suppressed by a factor m2/T 2 eq µτ T 3mτ e δnτ ≈ 1 − 2 2 . (11) compared to the scattering rate αT [36] (earlier estimates 6  2π T  of the scattering rate were suppressed by another factor We assume that the baryon and/or lepton asymmetry of α [34]), of the universe was generated at T ≫ TEW by some in- 2 teractions beyond the Standard Model, for example by 1 T (B − L)-violating leptoquark decays. This asymmetry τχ ∼ 2 . (6) αT me was then redistributed between the Standard Model lep- tons and quarks by sphaleron processes, so at T ≪ TEW The excess of anti-tau’s over tau’s, δnτ , decreases due we expect the chemical potentials of light baryons and to tau decay and is described by the equation, leptons to be of the order µ/T ∼ ηB [37, 38], where η ∼ 10−9 is the observed baryon to photon ratio. In d a3 B (a3δn )= (δneq − δn ). (7) the high-temperature regime, when T is large compared dt τ τ τ τ d to all relevant particle masses, we have µτ /T ≈ const, with a mass correction O(m2/T 2) [39]. Then Eq. (11) At temperatures below the elecroweak phase transi- becomes tion, T . TEW ∼ 100 GeV, we have τd ≪ t, where t is the cosmic time2. This means that the equilibrium density δneq m2 τ ≈ Cη − Kη τ , (12) of taus establishes very quickly (compared to the Hubble T 3 B B T 2 where C and K are O(1) numerical constants3. The mass correction term in (12) can be qualitatively understood 1 Tau-particles are also produced and destroyed in scattering re- actions like τ − + νe e− + ντ . We disregard them in what follows, since they do→ not change the order of magnitude of the effect. 3 This estimate assumes that taus are the heaviest particles present 2 2 This is easily verified using the relation t mP/√NT , where in equilibrium at temperature T . If a heavier particle is present ∼ mP is the Planck mass and N is the number of particle species in equilibrium, it too will contribute to the mass correction and in equilibrium. may change the estimate. 3 as follows. As the temperature decreases, it becomes where we have used the known cosmic baryon number energetically favorable to transfer the conserved τ-lepton density and are using natural units. Then number from τ-particles to τ-neutrinos. The excess τ- lepton number is also decreased as a result [39]. 2 Substituting Eq. (12) in (9) we obtain αmτ T −10 hmax ∼ 2 hB ∼ 10 hB (18) 4πm mP 2 ˙ e nχ ≈−3KηBτχmτ T. (13)

2 With T˙ = −T/2t, t ∼ mP/T and τχ from Eq. (6), this where we have used T ∼ 100GeV in the numerical es- gives (omitting numerical factors) timate. Even if the decay of top quarks with mass ∼ 175GeV to down quarks with mass ∼ 1 MeV is consid- 2 −6 ηB mτ T ered, hmax ∼ 10 hB. Comparing to observations, even nχ ∼ 2 nγ, (14) −19 αme mP with the most conservative lower bound of 10 G on Mpc scales, we get an estimate for the observed helicity 3 −38 2 where nγ ∼ T is the photon number density. ∼ 10 G Mpc. This estimate was derived assuming T ≫ mτ , but it IV. CONCLUSIONS still applies at T ∼ mτ . Reactions (1), (2) remain in equilibrium when T drops well below mτ . In this regime, standard δnτ and nχ decrease exponentially. We have shown that the decays of certain Similar formulae can be written down for the decay of model particles can lead to a chiral cosmological medium other unstable particles. The largest helicity is injected around the epoch of the electroweak phase transition. by the decay of the heaviest particle into the lightest The final result for the chiral asymmetry due to tau- particle and at the highest temperature. lepton decays is given in (14). However, the asymmetry −9 is suppressed by the baryon to entropy ratio (ηB ∼ 10 ) and the effect on magnetic field helicity generation is very III. MAGNETIC HELICITY weak as we have shown in Sec. III. Nonetheless it is of interest that the cosmological medium was chiral at the As noted in Ref. [32], the maximum magnetic helicity earliest moments even within the standard model of par- that can be obtained from particle helicity can be derived ticle physics. from the chiral anomaly equation, which can be written as a conservation law, 4π n + h = constant. (15) χ α V. ACKNOWLEDGEMENTS where h = hA · Bi is the magnetic helicity. Assuming that the initial magnetic helicity and the final particle helicity vanish, we get We thank the participants of the Nordita workshop on “Quantum Anomalies and Chiral Magnetic Phenomena”, 2 α ηB mτ T especially Axel Brandenburg and Kohei Kamada for feed- hmax = nχ ∼ 2 nγ (16) 4π 4πme mP back. We also thank Matt Baumgart, Cecilia Lunardini, Igor Shovkovy, and Tracy Slatyer for discussions. TV’s where we have used (14). We compare hmax to magnetic work is supported by the U.S. Department of Energy, helicity that could be induced due to baryogenesis [3, 4], Office of High Energy Physics, under Award No. DE- η SC0019470 at Arizona State University. AV is supported h ∼ B n ∼ 10−5 cm−3 ∼ 10−45 G2 Mpc (17) by the National Science Foundation Award No. 1820872. B α γ

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