PHYSICAL REVIEW D 103, 103528 (2021) Cosmological chirality and magnetic fields from parity violating particle decays Tanmay Vachaspati 1 and Alexander Vilenkin2 1Physics Department, Arizona State University, Tempe, Arizona 85287, USA 2Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA (Received 1 February 2021; accepted 30 April 2021; published 20 May 2021) We estimate the chirality of the cosmological plasma due to parity violating decays of Standard Model particles, focusing on the example of tau leptons. The nontrivial chirality is however too small to make a significant contribution to the cosmological magnetic field via the chiral-magnetic effect. DOI: 10.1103/PhysRevD.103.103528 I. INTRODUCTION particle helicity to the cosmological plasma. The net particle helicity in principle leads to electric currents via The last few decades have seen growing interest in the CME that can generate magnetic helicity. However, cosmic magnetic fields on several fronts [1]. Several ideas accounting only for decays of Standard Model particles, the have been proposed that can generate magnetic fields in net particle helicity is too small to significantly affect cosmology, some of which are directly tied to known cosmological magnetic fields and their helicity. particle physics [2–4] and its possible extensions [1,5–8]. We start by describing the physical effect in some detail The magneto-hydrodynamical (MHD) evolution of cos- in the context of the tau lepton in Sec. II, where we also mological magnetic fields is now understood quite well on estimate the induced electric currents. We find an upper the basis of analytical arguments [9,10] and direct simu- bound to the magnetic helicity that can be generated due to lations [11]. There are claims for an indirect lower bound chiral effects in Sec. III. Our conclusions are summarized on the cosmological magnetic field strength [12–16], and discussed in Sec. IV. though not without debate [17,18], and more direct evidence [19]. Concurrently there are claims of a parity violating signature that can be used to measure the II. CHIRALITY PRODUCTION IN TAU DECAYS magnetic field helicity spectrum [20,21] though with no To illustrate the physics of the effect, in this section we significant detections as yet [22,23]. will discuss the decay of tau leptons in the background of a In parallel to these developments, motivated by heavy- magnetic field and fluid vorticity. Except for small ion collision experiments [24], there has been renewed differences, the physics carries over to the case of decays interest in chiral effects in plasmas, namely the chiral- of other particles. magnetic [25] and chiral-vortical [26] effects (CME and CVE respectively). The CME and CVE have also been applied to the evolution of cosmological and astrophysical A. Particle decay magnetic fields [5,27–33]. In this paper we discuss how Tau leptons decay into electrons (or muons) and CME and CVE can effectively arise in standard cosmology neutrinos with standard particle interactions due to the parity-violat- − − ing decays of heavy leptons and quarks. The basic idea is τ → e þ ντ þ ν¯e ð1Þ that the Standard Model has a number of unstable particles that decay at various cosmological epochs, primarily due to and antitau into positrons and neutrinos the weak interactions. Since the weak interactions violate parity, the decay products are chiral and this provides a net þ þ τ → e þ ν¯τ þ νe: ð2Þ These decays violate parity since they proceed primarily by Published by the American Physical Society under the terms of the weak interactions. Therefore the tau predominantly the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to decays into a relativistic left-handed electron, while an the author(s) and the published article’s title, journal citation, antitau decays into a relativistic right-handed positron. Due and DOI. Funded by SCOAP3. to the lepton asymmetry of the Universe there are more taus 2470-0010=2021=103(10)=103528(5) 103528-1 Published by the American Physical Society TANMAY VACHASPATI and ALEXANDER VILENKIN PHYS. REV. D 103, 103528 (2021) than antitaus, and the cosmological plasma gains net left- 1 T2 τ ∼ : ð Þ χ 2 6 handed chirality as taus decay. αT me The decay product electrons are chiral since they are produced by the weak interactions, but chirality is not The excess of antitaus over taus, δnτ, decreases due to preserved for massive particles. Instead, as emphasized in tau decay and is described by the equation Ref. [34] in the context of supernovae and neutron stars, 3 chirality is nearly equal to helicity for ultrarelativistic d 3 a eq ða δnτÞ¼ ðδnτ − δnτÞ: ð7Þ particles, so it is better to think of the final electrons as dt τd being in a definite helicity state. Helicity can only change due to particle interactions. We shall adopt this view in At temperatures below the electroweak phase transition, T ≲ T ∼ 100 τ ≪ t t what follows. EW GeV, we have d , where is the 2 The τ mass is mτ ¼ 1777 MeV and the τ lifetime in its cosmic time. This means that the equilibrium density of −13 rest frame is ττ ¼ 2.9 × 10 s. However, the decaying taus establishes very quickly (compared to the Hubble eq taus are constantly reproduced by reactions inverse to (1), time), and the approximate solution of (7) is δnτ ≈ δnτ . 1 eq (2), so the number density of taus, nτ, remains comparable Inserting (7) in (4) and then using δnτ ≈ δnτ we have to that of photons until the time a3n d 3 d 3 eq χ −7 ða nχÞ¼− ða δnτ Þ − : ð8Þ tτ ∼ 10 s; ð3Þ dt dt τχ eq when the cosmic temperature drops to T ∼ mτ. At later With a given δnτ , this equation can be solved in quad- times nτ decreases exponentially. ratures, but we shall instead find an approximate solution. The particle helicity density, nχ, is produced in tau Since we are in the regime where τχ ≪ t, the term on the decays and is dissipated by helicity flipping scatterings and left-hand side can be neglected and we obtain 3 2 due to the chiral anomaly. The latter is proportional to α B eq [35], where α ≈ 1=137 is the fine structure constant and B is 3 d δnτ nχ ≈−τχT ; ð9Þ the magnetic field strength, and is much slower than dt T3 helicity flipping scatterings for vanishing or weak magnetic fields. We will ignore the anomalous flipping for now but where we have used aT ≈ const. will discuss it in Sec. III when we consider the effect of Once we determine the equilibrium excess of antitaus eq particle chirality on the generation of magnetic fields. The over taus, denoted by δnτ , we can estimate the chirality evolution of particle helicity density can be described by density of the Universe due to tau decays using (9). the kinetic equation in the relaxation time approximation, B. Equilibrium density 3 3 d a a nχ eq ða3n Þ¼ ðδn − δneqÞ − ; ð Þ The equilibrium density δnτ is given by dt χ τ τ τ τ 4 d χ Z 1 ∞ E − μ E þ μ δneq ¼ dpp2 f τ − f τ ; ð Þ τ 2 10 where 2π 0 T T þ − x −1 δnτ ¼ nτ − nτ ; ð5Þ wherepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifðxÞ¼ðe þ 1Þ is the Fermi distribution, 2 2 E ¼ p þ mτ , and μτ is the chemical potential of τ − þ nτ and nτ are the densities of tau and antitau particles, particles. At T ≫ mτ; μτ we can expand the integrand in eq 2 2 respectively, δnτ is the equilibrium value of δnτ, τd ∼ Eq. (10) in powers of mτ =p and μτ=T. The integrations are ðT=mτÞττ is the decay time of taus (assuming that T>mτ then easily performed and we find −1 and with time dilation taken into account) and τχ is the T ≫ m μ T2 3m2 electron helicity flipping rate. At e, the helicity δneq ≈ τ 1 − τ : ð Þ 2 2 τ 2 2 11 flipping rate is suppressed by a factor me=T compared to 6 2π T the scattering rate α2T and a correction factor of α−1 due to infrared effects [36] (earlier estimates of the flipping rate We assume that the baryon and/or lepton asymmetry of T ≫ T were suppressed by another factor of α [34]), the Universe was generated at EW by some inter- actions beyond the Standard Model, for example by 1Tau particles are also produced and destroyed in scattering pffiffiffiffi τ− þ ν → e− þ ν 2 t ∼ m = NT2 reactions like e τ. We disregard them in what This is easily verified using the relation P , where follows, since they do not change the order of magnitude of the mP is the Planck mass and N is the number of particle species in effect. equilibrium. 103528-2 COSMOLOGICAL CHIRALITY AND MAGNETIC FIELDS FROM … PHYS. REV. D 103, 103528 (2021) (B − L)-violating leptoquark decays. This asymmetry was where h ¼hA · Bi is the magnetic helicity. Assuming that then redistributed between the Standard Model leptons and the initial magnetic helicity and the final particle helicity quarks by sphaleron processes. Let us denote the lepton-to- vanish, we get T ≪ T η photon ratio at EW by L. Generically we expect the α η m2 T chemical potentials of light baryons and leptons to be of the h ¼ n ∼ L τ n ð Þ μ =T ∼ η ∼ η η ∼ 10−9 max 4π χ 2 m γ 16 same order, and τ L B [37,38], where B 4πme P is the observed baryon-to-photon ratio.