Cosmological Chirality and Magnetic Fields from Parity Violating Particle

Cosmological chirality and magnetic fields from parity violating particle decays Tanmay Vachaspati∗, Alexander Vilenkin† ∗Physics Department, Arizona State University, Tempe, AZ 85287, USA. †Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA. We estimate the chirality of the cosmological medium due to parity violating decays of standard model particles, focusing on the example of tau leptons. The non-trivial chirality is however too small to make a significant contribution to the cosmological magnetic field via the chiral-magnetic effect. I. INTRODUCTION summarized and discussed in Sec. IV. The last few decades have seen growing interest in cosmic magnetic fields on several fronts [1]. Several ideas II. CHIRALITY PRODUCTION IN TAU have been proposed that can generate magnetic fields in DECAYS cosmology, some of which are directly tied to known particle physics [2–4] and its possible extensions [1, 5–8]. To illustrate the physics of the effect, in this section we The magneto-hydrodynamical (MHD) evolution of cos- will discuss the decay of tau leptons in the background mological magnetic fields is now understood quite well of a magnetic field and fluid vorticity. Except for small on the basis of analytical arguments [9, 10] and direct differences, the physics carries over to the case of decays simulations [11]. There are claims for an indirect lower of other particles. bound on the cosmological magnetic field strength [12– 16], though not without debate [17, 18], and more direct evidence [19]. Concurrently there are claims of a par- A. Particle decay ity violating signature that can be used to measure the magnetic field helicity spectrum [20, 21] though with no Tau leptons decay into electrons (or muons) and neu- significant detections as yet [22, 23]. trinos In parallel to these developments, motivated by heavy- − − ion collision experiments [24], there has been renewed τ → e + ντ +¯νe (1) interest in chiral effects in plasmas, namely the chiral- magnetic [25] and chiral-vortical [26] effects (CME and and anti-tau into positrons and neutrinos CVE respectively). The CME and CVE have also been + + applied to the evolution of cosmological and astrophysical τ → e +¯ντ + νe (2) magnetic fields [5, 27–33]. In this paper we discuss how CME and CVE can effectively arise in standard cosmol- These decays violate parity since they proceed primar- ogy with standard particle interactions due to the parity- ily by the weak interactions. Therefore the tau pre- violating decays of heavy leptons and quarks. The basic dominantly decays into a relativistic left-handed electron, arXiv:2101.06344v1 [hep-ph] 16 Jan 2021 idea is that the standard model has a number of unsta- while an anti-tau decays into a relativistic right-handed ble particles that decay at various cosmological epochs, positron. Due to the lepton asymmetry of the universe primarily due to the weak interactions. Since the weak there are more taus than anti-taus, and the cosmological interactions violate parity, the decay products are chiral medium gains net left-handed chirality as tau’s decay. and this provides a net particle helicity to the cosmolog- The decay product electrons are chiral since they are ical medium. The net particle helicity in principle leads produced by the weak interactions, but chirality is not to electric currents via the CME that can generate mag- preserved for massive particles. Instead, as emphasized in netic helicity. However, accounting only for decays of Ref. [34] in the context of supernovae and neutron stars, standard model particles, the net particle helicity is too chirality is nearly equal to helicity for ultrarelativistic small to significantly affect cosmological magnetic fields particles, so it is better to think of the final electrons as and their helicity. being in a definite helicity state. Helicity can only change We start by describing the physical effect in some de- due to particle interactions. We shall adopt this view in tail in the context of the tau lepton in Sec. II, where we what follows. also estimate the induced electric currents. We find an The τ mass is mτ = 1777 MeV and the τ lifetime in its −13 upper bound to the magnetic helicity that can be gener- rest frame is ττ = 2.9 × 10 s. However, the decaying ated due to chiral effects in Sec. III. Our conclusions are taus are constantly reproduced by reactions inverse to 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 leptons 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 reactions 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.

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