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provided by Caltech Authors - Main PHYSICAL REVIEW D 94, 043515 (2016) Sterile neutrino dark matter: Weak interactions in the strong coupling epoch
Tejaswi Venumadhav,1,5 Francis-Yan Cyr-Racine,1,2,6 Kevork N. Abazajian,3 and Christopher M. Hirata4 1California Institute of Technology, Mail Code 350-17, Pasadena, California 91125, USA 2Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA 3Center for Cosmology, Department of Physics and Astronomy, University of California, Irvine, Irvine, California 92697, USA 4Center for Cosmology and Astroparticle Physics (CCAPP), The Ohio State University, 191 West Woodruff Lane, Columbus, Ohio 43210, USA 5School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, New Jersy 08540, USA 6Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 14 August 2015; published 15 August 2016) We perform a detailed study of the weak interactions of standard model neutrinos with the primordial plasma and their effect on the resonant production of sterile neutrino dark matter. Motivated by issues in cosmological structure formation on small scales, and reported x-ray signals that could be due to sterile neutrino decay, we consider 7 keV-scale sterile neutrinos. Oscillation-driven production of such sterile neutrinos occurs at temperatures T ≳ 100 MeV, where we study two significant effects of weakly charged species in the primordial plasma: (1) the redistribution of an input lepton asymmetry; (2) the opacity for active neutrinos. We calculate the redistribution analytically above and below the quark-hadron transition, and match with lattice QCD calculations through the transition. We estimate opacities due to tree-level processes involving leptons and quarks above the quark-hadron transition, and the most important mesons below the transition. We report final sterile neutrino dark matter phase space densities that are significantly influenced by these effects, and yet relatively robust to remaining uncertainties in the nature of the quark- hadron transition. We also provide transfer functions for cosmological density fluctuations with cutoffs at k ≃ 10h Mpc−1, that are relevant to galactic structure formation.
DOI: 10.1103/PhysRevD.94.043515
I. INTRODUCTION plasma [10–14], or (ii) that sterile neutrinos are massive enough to form the inferred population of dark matter (DM) Deep in the radiation dominated epoch of the Universe, in the Universe (see e.g. Ref. [15]). Sterile neutrinos with the three neutrinos present in the standard model (SM) of masses in the keV range act as DM in the CMB era, but are particle physics [1] make up a significant population of relativistic in the BBN era, when they do not significantly relativistic species within the primeval cosmic plasma. We impact the expansion rate due to their negligible energy have strong evidence of their existence at these early density (compared to the Fermi-Dirac value). epochs from probes of the primordial Universe such as Early works in this direction studied right-handed sterile the cosmic microwave background (CMB) (probing tem- ≈ 0 1 − 100 ∼ 0 25 neutrinos with masses ms . keV, produced by perature T . eV) [2], and the synthesis of light the oscillation of left-handed SM neutrinos [16–20]. The elements during the epoch of big bang nucleosynthesis mixing angle between the SM and sterile neutrinos is fixed (BBN) which depends on the neutron-to-proton ratio set by the present day DM abundance. In the original ∼ 1 5 at Tdec . MeV [3], the temperature of weak neutrino Dodelson-Widrow scenario [17], sterile neutrinos are decoupling. Above this temperature, SM neutrinos interact produced with a momentum distribution reflecting that with species that carry weak charge, through which they of the active neutrino species, and thus constitute “warm” remain coupled to the primordial plasma [4]. DM [21–24]. However, small-scale structure formation There is a long history of speculation about additional [25–32] and x-ray observations [33–37] appear to be in neutrino species (see Ref. [5] for a recent review). Owing to significant conflict with the fiducial Dodelson-Widrow the precise measurement of the invisible decay width of the scenario, hence prompting the search for alternative sterile SM Z boson [1], any extra neutrino species must be “sterile” neutrino production mechanisms [18,38–55]. (i.e. electroweak singlets) [6]. Furthermore, precise mea- In this paper, we examine in detail the resonant production surements of the CMB [2,7] and of the primeval abundance of sterile neutrinos in the presence of a small primordial of light elements [8,9] strongly constrain the presence of lepton asymmetry. Originally proposed by Shi and Fuller extra relativistic species in the early Universe. These [38], this production mechanism makes use of a small lepton constraints indicate that (i) unlike SM neutrinos, light sterile asymmetry to modify the plasma’s interaction with SM neutrinos never fully thermalize with the rest of the cosmic neutrinos in such a manner as to resonantly produce sterile
2470-0010=2016=94(4)=043515(28) 043515-1 © 2016 American Physical Society TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) neutrinos at particular momenta [18,41,56]. This generically redistribution modifies the dynamics of the resonant sterile results in a “colder” DM distribution which improves neutrino production, resulting in a modified final PSD. consistency with models of cosmological structure forma- Secondly, we incorporate several new elements to the tion [57–65], while requiring a modest primordial lepton calculations of the neutrino opacity (i.e. the imaginary part asymmetry, which is relatively poorly constrained [66–70]. of the self-energy) at temperatures 10 MeV ≤ T ≤ 10 GeV. Sterile and active neutrino mixing, which is needed for Accurate neutrino opacities are needed since they basically the former’s production, also leads to their decay [71,72]. control the production rate of sterile neutrinos through cosmic For typical values of the sterile neutrino mass this predicts epochs. Early works on neutrino interactions in the early an x-ray flux from the DM distribution in the low redshift Universe [4,18,84] assumed that neutrinos largely scattered off Universe [18,33]. This has been the subject of much recent relativistic particles and thus scaled their cross sections with interest, due to hints of an excess flux at ∼3.5 keV in the center-of-mass (CM) energy. In addition, these calcula- stacked x-ray spectra of several galaxy clusters [73] and in tions also neglected the effects of particle statistics. Under Γ observations of M31, the Milky Way, and Perseus [74,75]. these two simplifying assumptions, the opacity ðEνα Þ for an There is currently an active debate on the existence, input neutrino of energy Eνα is of the form significance and interpretation of this excess [76–83].In Γ λ 2 4 the present work, we use this tentative signal as a ðEνα Þ¼ ðTÞGFT Eνα ; ð2Þ motivation to study in detail the physics of sterile neutrino λ production in the early Universe, but the machinery we where GF is the Fermi coupling constant, and ðTÞ develop is more generally applicable to the broader is a constant that depends on the number and type of available parameter space of the Shi-Fuller mechanism. relativistic species in the cosmic plasma. References [20,85] We present here an updated calculation of resonantly subsequently developed a framework to include particle produced sterile neutrinos and relax several simplifications masses, loop corrections, and particle statistics in the neutrino that had been adopted previously in the literature. opacity calculation. In the present work, we add previously Furthermore, we leverage recent advances in our under- neglected contributions to the opacity such as two- and three- standing of the quark-hadron transition in order to include a body fusion reactions, and also use chiral perturbation theory more realistic treatment of the strongly interacting sector. to compute the hadronic contribution to the opacity below the Our motivation is twofold: (a) improve the treatment of quark-hadron transition. We find both quantitative and quali- lepton asymmetry, which is a crucial beyond-SM ingredient tative modifications to the form of Eq. (2).Whereverwe − in the mechanism, and (b) provide realistic sterile neutrino present matrix elements, we use the þþþmetric signature. phase space densities (PSDs) and transfer functions for Thirdly, we fold the asymmetry redistribution and matter fluctuations, which are starting points for studying opacity calculations into the sterile neutrino production cosmological implications on small scales. Our improve- computation, and provide updated PSDs for the range of ments to the sterile neutrino production calculation can parameters relevant to the x-ray excess. As part of this broadly be classified in three categories. process, we carefully review and correct the numerical Firstly, we study how the cosmic plasma reprocesses a implementation of the sterile neutrino production used in primordial lepton asymmetry. For models that can explain Ref. [63]. Our sterile neutrino production code is publicly ‑ the above x-ray excess, the majority of sterile neutrinos are available at https://github.com/ntveem/sterile dm.We produced at temperatures above 100 MeV [56]. At these finally use the updated sterile neutrino PSDs in a standard temperatures, there is a significant population of either cosmological Boltzmann code [86] and provide new dark quarks or mesons, depending on whether the temperature is matter transfer functions. above or below the quark-hadron transition. Since these We organize the paper such that the beginning sections hadronic species are coupled to neutrinos and charged deal with SM physics, while the later ones apply their leptons through weak processes, the establishment of results to sterile neutrino DM. We introduce the production chemical equilibrium among the different constituents of mechanism in Sec. II. We then study the asymmetry the cosmic plasma will automatically redistribute the redistribution in Sec. III and active neutrino opacities in primordial asymmetry within the charged and neutral Sec. IV. Finally, we apply these results to sterile neutrino leptons, and lead to a charge asymmetry within the production in Sec. V, evaluate transfer functions for matter hadronic sector. An illustrative example is the reaction fluctuations in Sec. VI, and finish with a discussion of our assumptions and uncertainties in Sec. VII. We collect þ þ νμ þ μ ⇌π ; ð1Þ technical details into the appendixes. which can redistribute an initial neutrino asymmetry into II. OVERVIEW OF RESONANT STERILE asymmetries of the charged leptons and hadrons (keeping NEUTRINO PRODUCTION the net baryon number constant). At lower temperatures, the asymmetry is redistributed to a lesser degree between the In this section, we briefly review the resonant production leptonic flavors. As we discuss in the body of the paper, this of sterile neutrinos in the early Universe. We first present
043515-2 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) the specific scenario that we consider in this work, and then discuss the Boltzmann formalism used to compute the out- of-equilibrium production of sterile neutrinos. We finally discuss how the presence of the thermal bath and lepton asymmetry change the neutrino self-energy and govern the sterile neutrino production. We refer the reader to Refs. [18,38,56] for more details.
A. Assumptions In our current study, we focus on the following scenario. (1) We consider an extra sterile neutrino species, νs, that is massive compared to the SM neutrinos νe=μ=τ, which we take to be effectively massless. The propagation (light/heavy) and interaction (active/ sterile) eigenstates are related by a unitary trans- formation, the most general version of which is a FIG. 1. Sterile neutrino DM parameter space: shaded regions are 4 × 4 matrix. We assume that the sterile neutrino consistent with the x-ray signal at 1,2 and 3σ. The blue contours mixes with only one of the SM ones, which we take show the best determined parameters from the analysis of Ref. [73], to be the muon neutrino, i.e., who use the stacked x-ray spectrum of 73 clusters obtained with the MOS detectors on XMM Newton. Statistically consistent Ψ θ θ Ψ νμ cos sin 0 signals are found in their core-removed Perseus spectrum, and ¼ : ð3Þ M31 [74]. The lines show constraints at the 90% level from Ψν − sin θ cos θ Ψ s ms Chandra observations of M31 (H14) [87], stacked dwarf galaxies (M14) [88], and Suzaku observations of Perseus (T15) [89]. Stars The fields on the left- and right-hand sides are mark the models that we study in the body of the paper. interaction and mass (ms) eigenstates, respectively, and θ is the active-sterile mixing angle. The choice of a muon neutrino is arbitrary, and reflects the with constraints from Chandra observations of M31 [87], choice of previous work [18,63]. stacked dwarf galaxies [88], and Suzaku observations of (2) We assign a nonzero lepton asymmetry to the Perseus [89]. The stars show a range of mixing angles at a primordial plasma. In the general case, each SM specific value of ms, and mark models that we study in flavor has its own asymmetry, but we assume a Secs. V and VI. nonzero value only for the mu flavor (i.e. the one that For all these models, the bulk of the sterile neutrinos are mixes with the sterile neutrino): produced at temperatures well below the masses of the weak gauge bosons (∼80 GeV), but above weak decou- Δˆ Δˆ − ≡ Lˆ δ Lˆ ∼ 1 5 nνα þ nα α ¼ αμ μ; ð4Þ pling at T . MeV [56]. Active-active neutrino oscil- lations in the primordial plasma are suppressed at these ˆ where the dimensionless asymmetry ΔnA in species temperatures [90]; hence it is consistent to assign individual A is the temperature-scaled difference between asymmetries in Eq. (4) and neglect electron and tau the particle and antiparticle densities, Δnˆ A≡ neutrino mixing in Eq. (3). − 3 δ ðnA nAÞ=T , and αμ is the Kronecker delta. In general, entropy-scaled asymmetries are preferable, B. Boltzmann formalism since they are conserved through epochs of annihi- In its full generality, out-of-equilibrium sterile neutrino lation. However, the definition used in Eq. (4) production (via oscillations) is best described by the simplifies the process of comparison with lattice evolution of the two-state density matrix of the neutrinos QCD calculations in Sec. III. We fix by hand the mu in the active-sterile (interaction) basis [91–94]. lepton asymmetry at high temperatures to produce For the parameter range in Fig. 1, most sterile neutrinos Ω 2 0 1188 the canonical DM density, DMh ¼ . in the are produced above temperatures T ≳ 100 MeV. At these current epoch [2]. temperatures, the two-state system is collision dominated; In the rest of the paper, we use a hat to indicate i.e. the interaction contribution dominates the vacuum temperature scaled quantities. We choose to study the oscillations. In this regime, the evolution of the density parameter space of interest for resonantly produced sterile matrix separates out and yields a quasiclassical Boltzmann neutrino DM consistent with the recent x-ray signal. transport equation for the diagonal terms, which are the 2 2θ Figure 1 shows a section of the ms and sin plane with PSDs of the active and sterile components [95–97]. The contours for the unidentified lines of Refs. [73,74], along Boltzmann equation for the sterile neutrino PSD is
043515-3 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) Z X 3 3 ∂ ∂ d pa d pi 4 4 fν ðp; tÞ − Hp fν ðp; tÞ¼ ð2πÞ δ ðp þ p þ − p − Þ ∂ s ∂ s 3 3 a i t p ð2πÞ 2Ea ð2πÞ 2Ei νxþaþ →iþ X 1 2 × hP ðνμ → ν ; p; tÞið1 − fν Þ jMj → ν f ð1 ∓ f Þð1 − fν Þ 2 m s s iþ aþ μþ i a μ X − ν → ν ; 1 − M 2 1 ∓ hPmð s μ p; tÞifνs ð fνμ Þ j jνμþaþ →iþ fa ð fiÞ : ð5Þ
We can write an analogous equation for the antineu- quantum-damped and collisionally driven sterile neutrino trinos. Here, the fðpÞ are PSDs for particles with three- production is [18] momentum p and energy E, and H is the Hubble expansion rate. The right-hand side sums over all reactions that ∂ ∂ fν ðp; tÞ − Hp fν ðp; tÞ Pconsume or produce a muon neutrino. The symbol ∂t s ∂p s jMj2 denotes the squared and spin-summed matrix Γ νμ ðpÞ element for the reaction, and the multiplicative factors of ≈ hP ðνμ ↔ ν ; p; tÞi½fν ðp; tÞ − fν ðp; tÞ ; 2 m s μ s ð1 ∓ fÞ implement Pauli blocking/Bose enhancement respectively. The factor of 1=2 accounts for the fact that ð8Þ only one (i.e. the muon neutrino) state in the two-state – with a related equation for antineutrinos. There are subtleties system interacts [91 93]. The Pm are active-sterile oscil- lation probabilities in matter, which depend on the vacuum with the effects of quantum damping in the case of resonance mixing angle θ, and are modified by interactions with the [98], but tests with the full density matrix formalism find that medium. The latter are parametrized by the neutrino self- the quasiclassical treatment is appropriate [56]. energy [84], and the “quantum damping” rate for active neutrinos. In terms of these quantities, the oscillation C. Asymmetry and thermal potentials probabilities are [96,97] We now expand on the origins of the asymmetry and thermal potentials appearing in Eq. (6). These potentials ν ↔ ν ; 1 2 Δ2 22θ hPmð μ s p; tÞi¼ð = Þ ðpÞsin encapsulate the self-energy of propagating active neutrinos × fΔ2ðpÞsin22θ þ D2ðpÞ due to interactions with the plasma. Under the conditions we are interested in, there are three contributions to the 2 −1 þ½ΔðpÞ cos 2θ − VL − VthðpÞ g : neutrino self energy: (a) an imaginary part proportional to ð6Þ the net neutrino opacity, (b) a real part due to finite weak gauge boson masses (Vth), and (c) a real part proportional to L We have introduced the symbol ΔðpÞ for the vacuum asymmetries in weakly interacting particles (V ). We Δ ≡ δ 2 2 follow the treatment in Ref. [84], and recast it in terms oscillation rate, ðpÞ mνμ;ν = p, and split the neutrino s of the quantities that we compute later. self-energy into the lepton asymmetry potential VL, and the th Figure 2 shows lowest-order contributions to active thermal potential V [the asymmetry contribution enters neutrinos’ self-energy. Thick red lines are thermal propa- with the opposite sign in the version of Eq. (6) for gators of weakly charged species in the background antineutrinos]. The quantity DðpÞ is the quantum damping plasma. There are two corrections—bubbles and tadpoles, rate, and equals half the net interaction rate of active neutrinos [the factor of half enters for the same reason as it does in Eq. (5)]. The net interaction rate for a muon neutrino is Z X 3 3 Γ d pa d pi νμ ðpÞ¼ 3 3 ð2πÞ 2Ea ð2πÞ 2Ei νxþaþ →iþ × ð2πÞ4δ4ðp þ p þ − p − Þ X a i M 2 1 ∓ × j jνμþaþ →iþ fa ð fiÞ ð7Þ
We simplify the phase space integrals in Eq. (5) by using FIG. 2. Lowest-order contributions to a propagating active detailed balance to equate the forward and backward neutrino’s self-energy. Red lines are thermal propagators. In − reaction rates. The resulting Boltzmann equation for (a), f is any species with weak charge. In (b), f ¼ να, α .
043515-4 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) θ shown in Figs. 2(a) and 2(b) respectively. The background In the above equations, W is the weak mixing angle, and fermion is a lepton of the same flavor in the former, and any MZ=W are the masses of the weak gauge bosons. The δ ρ ρ weakly charged species in the latter. symbol αβ is a Kronecker delta; the quantities α and να A massless active neutrino’s “dressed” propagator is are net energy densities of charged and neutral leptons, respectively; and Δn and Δn are densities of the baryon −1 − 1 − γ 2 B Q Gνα ðpνα Þ¼pνα bνα ðpνα Þuð 5Þ= ; ð9aÞ number B and electric charge Q, respectively. The standard model baryon number asymmetry is small compared to the ð0Þ ð1Þω ω − bνα ðpνα Þ¼bνα þ bνα να ; να ¼ pνα · u: ð9bÞ lepton asymmetry of interest [3]; hence it can be set to zero for the purposes of this calculation. ’ Here, pνα and u are the neutrino and plasma s four- According to the assumptions in the first part of this γμ section, the plasma starts out with a net lepton asymmetry momenta, v is shorthand for vμ, and bνα is the left- handed neutrino’s self-energy. Equation (9b) divides this in the mu flavor. As we showed in Sec. I, this asymmetry is self-energy into two contributions that affect the particle redistributed between muons and muon neutrinos. and antiparticle poles of Eq. (9a) differently. Figure 3 Moreover leptons of other flavors acquire asymmetries illustrates their association with asymmetry and thermal that respect Eq. (4), and the strong fluid acquires a net Δ potentials: electric charge density nQ to maintain overall neutrality. Equation (12a) shows how the asymmetry potential L ð0Þ depends on the redistributed asymmetries, which we study Vν ¼ bν ; ð10Þ α α in the ensuing section.
th ð1Þ Vν ðEν Þ¼bν Eν : ð11Þ α α α α III. REDISTRIBUTION OF AN INPUT ASYMMETRY The authors of Ref. [84] computed these terms by summing over all species in Fig. 2. Both kinds of diagrams contribute Weak processes couple leptonic and hadronic degrees of to the asymmetry potential, while only bubble diagrams freedom in the primordial plasma. In this section, we study 1 contribute to the thermal potential. We write the answer in this coupling’s effect on lepton asymmetries. We define terms of the leptons’ asymmetries, and the densities of the relevant susceptibilities in Sec. III A, and compute them strong fluid’s conserved quantities: over a range of temperatures in Sec. III B. pffiffiffi X 1 L 2 A. Definitions and parametrization Vν ¼ 2G δαβ − þ 2sin θ Δnβ− α F 2 W β∈fe;μ;τg Let us consider the primordial plasma at temperatures X 1 above the quark-hadron transition temperature, TQCD. The þ ð1 þ δαβÞΔnν − Δn following reactions couple leptons of different flavors, and β 2 B β∈fe;μ;τg the quark and lepton sectors: 1 − 2 2θ Δ − − þð sin WÞ nQ ; ð12aÞ να þ β ⇌νβ þ α ; ð13aÞ
pffiffiffi ν αþ⇌ 8 2 ρ ρ α þ a þ b; ð13bÞ th GF να α Vν ðEν Þ¼− þ Eν : ð12bÞ α α 3 2 2 α MZ MW where a and b are quarks with charges of þ2=3 and −1=3 respectively. Free quarks no longer exist at temperatures below TQCD, and the reactions in Eq. (13b) transition to ones involving mesons, like Eq. (1). In principle, we could study the effect of all these reactions on input asymmetries, but it is a daunting task, one that is further complicated by the quark-hadron transition. The following consideration of the relevant time
1During this preparation of this manuscript, we became aware of Ref. [99], which points out the relevance of this effect to sterile neutrino production, and estimates it under the simplifying Stefan-Boltzmann approximation for free quarks, along with a modified number of colors as a phenomenological correction for FIG. 3. Matter potentials for massless neutrinos in the plasma’s the effects of confinement. Wherever possible, we will try to rest frame: filled and unfilled circles are poles at finite and zero compare our results to those of Ref. [99], which we will refer to as temperature, respectively. See Sec. IV for the imaginary shift. “GL15” in the rest of the paper.
043515-5 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016)
μˆ − μˆ − − μˆ 0 α ∈ μ τ scales suggests a solution. In the radiation dominated να α Q ¼ ; fe; ; g: ð16Þ ≈ 2 105 −1 1=2 2 era, the Hubble rate is H × s g ðT=GeVÞ . μˆ At temperatures above the quark-hadron transition, the Here Q is the chemical potential for adding a unit of Γ ≃ 2 5≈ rates of reactions in (13) are ðTÞ GFT electric charge. The conserved quantities’ densities are 2 × 1014 s−1ðT=GeVÞ5, while the relevant rates below related to their chemical potentials by their susceptibilities: the transition are those of pion decays. The most important ! ! ! þ þ channel for the latter is the muonic decay, π → μ þ νμ, Δˆ χˆQ χˆQB μˆ nQ 2 11 Q which is faster than the Hubble rate (Γπþ→μþ ν ¼ : ð17Þ þ μ ¼ Δˆ BQ B μˆ nB χˆ χˆ B 3.8 × 107 s−1). Thus, a significant number of the reactions 11 2 in Eq. (13) are faster than the Hubble rate.2 Equation (17), along with net charge and baryon number This has two primary consequences. Firstly, high reaction conservation, yields the constraint equations rates enforce kinetic equilibrium; i.e. all active species’ PSDs approach the Fermi-Dirac or Bose-Einstein forms. Secondly, Δˆ χˆBQμˆ χˆBμˆ ≈ 0 forward and backward reactions are in chemical equilibrium, nB ¼ 11 Q þ 2 B ; ð18Þ one effect of which is to equate the chemical potentials for X Q QB 0 χˆ μˆ χˆ μˆ − Δˆ − both sides (the Saha equation). However, it has another ¼ 2 Q þ 11 B nα implication—the plasma’s complicated internal dynamics α∈fe;μ;τg X can be abstracted into a few parameters or susceptibilities that Q QB ¼ χˆ μˆ þ χˆ μˆ − χˆα− μˆ α− : ð19Þ completely specify its response to small external “forces,” or 2 Q 11 B α∈ e;μ;τ in this case, input asymmetries. All that remains is to compute f g the susceptibilities relevant to our problem. Equations (15), (16), (18) and (19) are eight linear We now define a few useful quantities and notation. equations for eight unknowns. The resulting asymmetries Given any conserved quantity F, the symbol μ denotes its F (obtained via their chemical potentials) are the “redistrib- chemical potential. The asymmetry Δnˆ A, in a particle A,isa uted” input lepton asymmetries Lα. function of its chemical potential μˆ ≡ μ =T. The quan- A A We symbolically represent the solutions as tities Δnˆ A and μˆ A are small, and in the linearized limit, related by X ∂μˆ μˆ A Lˆ A ¼ ˆ α; ð20Þ ∂Lα Δnˆ A ¼ χˆAμˆ A; ð14Þ α∈fe;μ;τg 2 ∂μˆ ∂Lˆ where χˆA ≡ χA=T is the number-density susceptibility. where the coefficients ð A= αÞ depend on the suscep- The lepton asymmetries in the three flavors are tibilities of both the leptons and the strong fluid. We also express the redistributed asymmetries as ˆ Lα ¼ Δnˆ α− þ Δnˆ ν α X ∂Δˆ X ∂μˆ nA ˆ A ˆ ¼ χˆα− μˆ α− þ χˆν μˆ ν ; Δnˆ ¼ Lα ¼ χˆ Lα: ð21Þ α α A ∂Lˆ A ∂Lˆ α∈fe;μ;τg α α∈fe;μ;τg α α ∈ fe; μ; τg: ð15Þ At the temperatures of interest, the lepton susceptibilities The strong fluid is described by the densities of its are essentially given by the free particle, or Stefan- conserved quantities: the charge and baryon-number den- Δˆ Δˆ 3 Boltzmann, formula: sities nQ and nB, respectively. The chemical equilib- Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rium of the reactions in Eq. (13) implies g ∞ χˆ ˆ − A ˆ ˆ 2 ˆ 0 ˆ 2 ˆ 2 AðmAÞ¼ 2 dpp nFð p þ mAÞ: ð22Þ π 0 2 þ þ The electronic channel for the pion decay, π → e þ νe is 3 −1 helicity suppressed (Γπþ→ þ ν ¼ 4.7 × 10 s ) and of the order e þ e In this equation, g and mˆ ≡ m =T are the spin ≃ 50 A A A of the Hubble rate at temperatures T MeV; hence one might nˆ 0 x worry that leptons with electronic flavor depart from equilibrium. degeneracy and mass respectively, and Fð Þ¼ This is resolved by the observation that they are coupled to ðd=dxÞf1=½expðxÞþ1 g is the derivative of the Fermi- muonic species by other nonhelicity-suppressed, and conse- Dirac distribution. The strongly interacting fluid’s suscep- þ ν ↔ μþ ν quently faster, reactions such as e þ e þ μ and tibilities are considerably more complicated, especially þ þ ¯ μ ↔ e þ νe þ νμ. 3 near the quark-hadron transition. We evaluate them using We do not follow the strangeness, S, since it is not conserved a number of techniques: perturbative quantum chromody- in weak reactions. Above the transitions, it disappears at the Γ ≃ 2 2 5 ≈ 1013 −1 5 namics (QCD) at high temperatures, matching to lattice Cabbibo-suppressed rate S jVusj GFT s ðT=GeVÞ , while below the transition the relevant rate is the kaon inverse QCD results near the transition, and a hadron resonance gas Γ 8 1 107 −1 lifetime, K ¼ . × s . (HRG) approximation at low temperatures.
043515-6 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) B. Susceptibilities of the strongly interacting plasma g4 ~2 2 s αMS g3 ¼ gs þ 7 ; ð27Þ In this section, we compute the strongly interacting ð4πÞ2 E plasma’s susceptibilities to baryon number and electric αMS charge fluctuations. The susceptibilities are the following and the functions En are given by derivatives of the QCD pressure, π2 XNf MS 2 2 α d 4C F1 mˆ ; μˆ ; ∂ pˆ E1 ¼ A 45 þ A ð i iÞ ð28Þ χˆX QCD 1 2 ¼ 2 ; ð23Þ i¼ ∂μˆ μˆ 0 X X¼ XNf 1 MS 2 2 2 α −d F2 mˆ ; μˆ 1 6F2 mˆ ; μˆ ∂ pˆ E2 ¼ A 6 ð i iÞ½ þ ð i iÞ χˆXY QCD 1 11 ¼ ; ð24Þ i¼ ∂μˆ ∂μˆ μˆ μˆ 0 2 X Y X; Y ¼ ˆ μ mi 3 2 ˆ 2 μˆ þ 2 ln þ F2ðmi ; iÞ 4π mi where X; Y ∈ fB; Qg; μˆ X ≡ μX=T is the chemical potential ˆ d C of the conserved charge X; and the pressure pQCD is given − 2 ˆ ˆ 2 μˆ − A A miF4ðmi ; iÞ 144 ; ð29Þ by the logarithm of the QCD partition function ZQCD, γ 1 22C μe E 1 pQCD αMS A pˆ ≡ ¼ ln Z ðV;T;μ ; μ Þ; ð25Þ E7 ¼ ln þ QCD T4 VT3 QCD Q B 3 4πT 22 2 XNf μ − 2 ˆ 2 μˆ where V is the volume. In Eq. (24), the off-diagonal term ln þ F3ðmi ; iÞ : ð30Þ XY 3 mi χˆ11 encodes the correlation between the fluctuations of i¼1 conserved charges X and Y. Note that the susceptibilities in ≡ 2 − 1 ≡ Eqs. (23) and (24) are dimensionless. Here, dA Nc and CA Nc stand for the gauge-group The sterile neutrino production calculation carried in the constants for the adjoint and fundamental representation of present work requires knowledge of these susceptibilities SUðNcÞ, respectively. In this work, we adopt the standard 3 over a broad range of temperatures, both above and below value of Nc ¼ . In the above, Nf is the number of quark μ the quark-hadron transition. At very high temperatures flavors, is the energy scale at which the masses and the T ≫ T , the QCD pressure can be computed using a coupling constant are evaluated (not to be confused with QCD γ standard perturbative approach, while at intermediate tem- the chemical potentials), and E is the Euler-Mascheroni … peratures T ≃ T , perturbative techniques become inad- constant. The functions F1; ;F4 are given in QCD Appendix A. equate; we must rely on lattice calculations (see e.g. We also need a prescription for the running of the [100,101]) to compute susceptibilities through the quark- coupling constant g and of the quark masses with the hadron transition. At low temperatures T 9 first separately calculate it both above and below the quark- CF ln μ =Λ 11C −4T hadron transition using either perturbative or HRG tech- μ μ ð ref MSÞ A F mið Þ¼mið refÞ μ Λ ; ð32Þ niques, and then smoothly join the results with those from ln ð = MSÞ lattice QCD computations in the regions of overlap. 2 2 − 1 2 Λ where TF ¼ Nf= , CF ¼ðNc Þ=ð NcÞ, and MS is the μ 2 1. High-temperature limit: Perturbative approach MS renormalization scale. Here, we take ref ¼ GeV. We follow Ref. [105] and use the criterion of minimal We follow the approach of Ref. [104] to perturbatively sensitivity to set the scale μ: compute the QCD pressure and its derivative up to order 2 − 4 4 O Ncþ Nf ln ðgs Þ, where gs is the standard QCD gauge coupling −γ 22 −4 μ ¼ 4πTe E e Nc Nf : ð33Þ constant. The starting point is to write the QCD pressure as To compute the susceptibilities, we substitute Eq. (26) into ˆ αMS ~2αMS pQCD ¼ E1 þ g3 E2 ; ð26Þ Eqs. (23) and (24), remembering that the relation between the quark chemical potentials and those of the conserved where g~3 is the effective gauge coupling baryon number and electric charges is 043515-7 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) μˆ 1 2 μˆ 1 X u 3 3 B ˆ M μ ¼ ; ð34Þ pHRG ¼ 3 ln Zi ðV;T; QÞ μˆ 1 1 μˆ VT d 3 − 3 Q i∈mesons X B μ μ þ ln Zj ðV;T; Q; BÞ ; ð35Þ where μˆ u and μˆ d are the chemical potentials for up- and j∈baryons down-type quarks, respectively. We numerically evaluate … the integrals in the functions F1; ;F4. The temperatures where relevant to sterile neutrino production are well below the Z top quark mass; we therefore adopt N ¼ 5. We use quark 3 ∞ pffiffiffiffiffiffiffiffiffiffiffi f VT 2 − ˆ 2 ˆ 2 M − ˆ ˆ 1 − p þmi masses evaluated at the reference scale μ from Ref. [1]. ln Zi ¼ 2 di dpp ln ð zie Þ; ð36Þ ref 2π 0 The only free parameter in the perturbative calculation is Z the MS renormalization scale, which we set so as to 3 ∞ pffiffiffiffiffiffiffiffiffiffiffi VT 2 − pˆ 2þmˆ 2 ZB d dpˆ pˆ 1 z e i ; match with the lattice calculation in the regime where ln i ¼ 2π2 i ln ð þ i Þ ð37Þ Λ 0 both approaches are valid. This entails us setting MS ¼ 105 MeV; the results are only logarithmically dependent where di denotes the degeneracy factor of species i and zi is on the particular value we choose. the fugacity: Biμˆ þQiμˆ 2. Intermediate temperatures: Lattice zi ¼ e B Q : ð38Þ calculations Here B and Q are the baryon number and electric charge Susceptibilities at temperatures close to the quark-hadron i i of species i, respectively. We construct the partition transition have been previously studied in the context of function given in Eq. (35) by summing over all hadron heavy ion collision experiments [106], where the crossover is resonances with mass below 2 GeV from the particle data signaled by fluctuations in the plasma’s quantum numbers group [1]. We then compute the susceptibilities using [107,108]. At zero chemical potential, these are studied in Eqs. (23) and (24). We numerically perform the integrals lattice QCD by mapping to the expectation values of traces. in Eqs. (36) and (37). Their auto- and cross-correlations are directly related to the We find that the HRG results are in good agreement diagonal and off-diagonal elements of the susceptibility with the lattice QCD calculations for temperatures matrix of Eq. (17) [109]. 125 MeV ≲ T ≲ 150 MeV, and we smoothly match the We directly use the susceptibilities so computed at HRG-derived susceptibilities to those from the lattice intermediate temperatures. Specifically, we use the results technique in this regime. from the Wuppertal-Budapest (WB) lattice QCD collabo- ration [100] and the HotQCD Collaboration [101].Even 4. Susceptibilities at all temperatures though the groups use different staggered fermion actions on the lattice, their results are broadly consistent with each We combine results from the three regimes into a single Q B QB smooth susceptibility tensor, valid over the range of temper- other. They report the susceptibilities χˆ2 , χˆ2 , and χˆ11 , together with their estimated errors, in (2 þ 1)-flavor QCD atures relevant to the production of sterile neutrinos O 10 extrapolated to the continuum limit.4 The lattice QCD with masses of order ð keVÞ.Figures4(a), 4(b), χˆQ χˆB χˆQB results are in good agreement with the perturbative calcu- and 4(c) display the susceptibilities 2 , 2 ,and 11 for lations described above for the temperature range temperatures satisfying 10 MeV 043515-8 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) FIG. 4. Panels (a)–(c): Components of the quadratic susceptibility tensor for the primordial plasma’s electric charge and baryon number. In all panels, the thick solid black line shows our smooth fit used in the computation of sterile neutrino production. At low temperatures, we illustrate the HRG results with dashed red lines, while the high-temperature perturbative results are shown with dotted cyan lines. We also show the results from the WB lattice QCD collaboration [100] with green error bars. For comparison, we also display the Stefan-Boltzmann approximation to the susceptibilities assuming free quarks at all temperatures. Panel (d): Effective populations of all leptonic degrees of freedom after the redistribution of an infinitesimal mu leptonic asymmetry at all temperatures. Light lines show the effects of redistribution according to GL15’s prescription. an accidental cancellation in the off-diagonal susceptibility, (1) At temperatures T>2 GeV, the redistribution is QB ≃60% χˆ11 , in the (2 þ 1)-flavor model which does not appear in efficient and of the mu leptonic asymmetry 5 the Nf ¼ theory. This arises because the sum of the ends up in the muons. All the charged leptons are electric charges of the up, down, and strange quarks exactly effectively massless at this epoch; hence the pop- QB vanishes. Hence, we expect χˆ11 → 0 for temperatures ulations of the electron and tau flavors are identical. above the strange quark mass in the (2 þ 1)-flavor model. (2) The rise in the mu and tau lepton populations above 5 temperatures of ≃25 MeV and 300 MeV reflects, in In the Nf ¼ model however, the charm quark becomes rapidly important at T ≳ 300 MeV, leading to a sharp part, the rise in their particle number susceptibilities QB as the temperature becomes comparable to their turnover in χˆ11 near this temperature. Given a set of infinitesimal lepton asymmetries, we solve masses [see Eq. (22)]. However, the largest contri- for the chemical potentials using the above susceptibilities bution to the former is from the disappearance of the in Eqs. (15), (16), (18) and (19), We obtain the redistributed hadronic degrees of freedom below the quark- asymmetries in all the constituent species by using these hadron transition, and the associated drop in the chemical potentials, along with the appropriate susceptibil- strongly interacting fluid’s susceptibilities. ities in Eq. (21). Figure 4(d) plots the redistributed (3) The “kink” in all the redistributed asymmetries asymmetries for an infinitesimal input mu leptonic asym- close to temperatures T ≃ 170 MeV is a signature metry. We note the following features: of the sharp change in the strongly interacting fluid’s 043515-9 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) the potential per unit physical μ lepton asymmetry using pffiffiffi these solutions; this quantity is constant and equals 2 2 ¼ 2.83 in the absence of redistribution. As shown in the figure, asymmetry redistribution corrects the potential at the 10% level above temperatures T ≳ 100 MeV, which is where the bulk of the sterile neutrinos are produced. This contribution changes the resonant momenta, and the resultant sterile neutrino dark matter’s phase space densities; we explore this further in Sec. V. IV. NEUTRINO OPACITY In this section, we outline our calculation of muon neutrino opacities in the early Universe. Initial work in FIG. 5. Asymmetry potential VL per unit unscaled mu lepton this area focused on reactions involving leptons, in the asymmetry. The solid line shows the effect of redistribution using context of neutrino decoupling, active-active neutrino a combination of perturbative QCD, lattice calculations and the oscillations and supernova calculations [4,111,112].In HRG approximation. Dashed and dotted-dashed lines show the particular, Ref. [111] lists a number of relevant matrix result using the Stefan-Boltzmann approximation for free quarks, elements. Our calculations apply to earlier epochs, with a and with a modified number of colors as in GL15, respectively. pffiffiffi larger number of reactions due to the population of 2 2 2 83 The value is constant (¼ ¼ . ) in the absence of redis- hadronic species above the quark-hadron transition. ≃10% tribution [see Eq. (12a)]. The redistribution is a correction Early work on sterile neutrino production used simple at the most relevant temperatures for sterile neutrino production prescriptions for the resultant increase in reaction rates (T ≳ 100 MeV). [18,56]. Recent work in Refs. [20,85] provides a theoretical framework to include particle masses and statistics in the susceptibilities at the quark-hadron transition [see neutrino opacity calculation, and formalism for loop Figs. 4(a), 4(b) and 4(c)]. corrections. We include a number of additional contribu- (4) At lower temperatures, T ≲ 30 MeV, the redistrib- tions to the neutrino opacities that are significant at the ution is inefficient and most of the asymmetry ends temperatures relevant to sterile neutrino production. We up in the muon neutrinos. Moreover, the electron adopt the following simplifying assumptions: neutrino and the muon have identical (small) pop- (1) We neglect small asymmetries in the participating ’ ulations. This is characteristic of inelastic neutrino species populations (as for the thermal potential). − − This is justified since the scattering rates are nonzero scattering, νμ þ e → νe þ μ , which is the most important channel at these temperatures (the had- even in a CP symmetric plasma. Moreover, we ronic susceptibilities are negligible at this epoch). assume thermal and kinetic equilibrium, due to These redistributed asymmetries impact sterile neutrino which the populations of all active species are L Fermi-Dirac/Bose-Einstein distributions. production via the asymmetry potential, Vνμ . Equation (12a) expresses this potential in terms of the (2) We integrate out the massive gauge bosons, Z and asymmetries in the populations of the individual charged W , and approximate the weak interaction by a four- and neutral leptons, along with those in the charge and fermion contact term. Consequently, the reactions baryon number of the strongly interacting fluid. As earlier, separate into leptonic and hadronic processes, de- pending on the species involved. Moreover, we for an infinitesimal input mu leptonic asymmetry, the 0 individual asymmetries are formally represented by the neglect the thermal populations of Z and W . functions in Eq. (21); the solutions for the charged and These steps are valid at low temperatures and momentum transfers, i.e., T; s=t=u ≪ M 0 ≈ neutral leptons are as plotted in Fig. 4(d). The light lines in W =Z 80 this figure show the redistributed asymmetries according to GeV. We operate in the temperature and energy GL15’s prescription. The latter was obtained by rescaling ranges the Stefan-Boltzmann result by an “effective” number of 10 043515-10 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) TABLE I. Two-particle to two-particle leptonic reactions contributing to the muon neutrino opacity; antineutrinos ð−Þ are similar. Symbol α runs over the other leptonic flavors, i.e. α ∈ fe; τg, and ν stands for ν=ν¯. s-channel t-channel Mixed þ þ − − νμ þ μ → να þ α νμ þ α → νμ þ α ð Þ ð Þ νμ þ ν μ → νμ þ ν μ − − νμ þ ν¯μ → να þ ν¯α ð Þ ð Þ νμ þ μ → νμ þ μ νμ þ ν α → νμ þ ν α − þ − − − þ νμ þ ν¯μ → α þ α νμ þ α → μ þ να νμ þ ν¯μ → μ þ μ − þ νμ þ ν¯α → μ þ α (3) We assume incoming and outgoing particles to be similar to the list in Ref. [111], albeit with tau leptons, which noninteracting within two limits—below and above are important at temperatures T ≳ 400 MeV. We also the quark-hadron transition (see Sec. III. 3 of include “three-body fusions,” since they arise at the same Ref. [113]). Below the transition, we include had- level of approximation. These are generated by omitting in ronic channels with pseudoscalar and vector mes- turn the products in the reactions of Table I, adding their ons5 and neglect the small population of baryons. charge conjugates to the reactants, subject to the constraint Above the transition, we include reactions with free that the product’s rest mass is strictly greater than the sum of quarks; i.e. we neglect the strong coupling constant. the reactants’. An example is tau lepton production þ þ This approximation fails at temperatures T ≃ via νμ þ μ þ ντ → τ . TQCD [104]. We show opacities interpolated through Figure 6 shows the leptonic contribution to the muon the transition using a few prescriptions, whose neutrino opacity at a temperature T ¼ 100 MeV, using the consequences for sterile neutrino production we matrix elements for reactions in Table I, and related three- explore in Sec. V. body fusions. For convenience, we only show reactions in The collision integral for a massless muon neutrino is the top five at any particular momentum bin. In the numerical implementation, we evaluate the dimensionless −Γ 2 5 C½fνμ ðEνμ Þ ¼ ðEνμ Þfνμ ðEνμ Þ Γ quantity ðEνμ Þ=GFT (proportional to the unscaled rates) − Γ Eνμ =T 1 − 10−6 þ ðEνμ Þe ð fνμ ðEνμ ÞÞ; ð41Þ to an accuracy of after simplifying the collision integrals in Eq. (B7) and (B25). Γ where and fνμ are the interaction rate (opacity) and PSD, The quark-hadron transition considerably complicates respectively (this expression satisfies detailed balance; see the hadronic reactions. We appeal to assumption 3 and assumption 1). The interaction rate is given by a sum over evaluate their rates in two limits: at low and high temper- all reactions that consume the muon neutrino [see Eq. (7)]. atures, i.e. T ΓðEν Þ Γ~ μ ðEνμ Þ¼ 2 4 : ð42Þ GFT Eνμ In the limit where all the particles involved are relativistic, weak cross sections are proportional to the squared energy in the CM reference frame. If we ignore particle statistics, reaction rates follow the scaling of Eq. (2); hence the scaled rate is proportional to the number of relativistic degrees of freedom involved [84]. We present the scaled rates in the rest of this section, in order to contrast our results with this intuition. In the rest of the section, we enumerate reactions contributing to the opacity and present final rates under the above approximations. We elaborate the calculation of matrix elements in Appendix B. Table I lists the purely leptonic two-particle to two-particle reactions contributing to the muon neutrino opacity. It is FIG. 6. Scaled muon neutrino opacities through leptonic reactions vs energy at T ¼ 100 MeV. Only reactions in the 5 top five at any particular momentum bin are shown. The symbol We also include quark production in s-channel reactions at ð−Þ high CM energies. See Appendix B1a for details. ν stands for ν=ν¯. 043515-11 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) TABLE II. Two-particle to two-particle hadronic reactions contributing to the μ neutrino opacity; antineutrinos are similar. Symbols a and b run over quarks with charge þð2=3Þe and ð−Þ −ð1=3Þe, respectively, and a stands for a=a¯. s-channel t-channel T>TQCD þ ¯ − − νμ þ μ → a þ b ð Þ ð Þ νμ þ a → νμ þ a νμ þ ν¯μ → a þ a¯ ð−Þ ð−Þ νμ þ b → νμ þ b ¯ − νμ þ ν¯μ → b þ b νμ þ b → μ þ a − ¯ νμ þ a¯ → μ þ b a T 043515-12 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) FIG. 8. Scaled muon neutrino opacities for a range of energies and temperatures. Panel (a): Total, leptonic, and hadronic opacities vs 100 20 energy at T ¼ MeV and 2 GeV. Panel (b): Opacities at high energies (Eνμ ¼ T) vs temperature. Black lines are two interpolations through TQCD. They are cubic splines labeled by their cutoff temperature, Tc, as defined in the text. Colored lines are numbers of 210 relativistic degrees of freedom: g ; KT under assumption 3, i.e., that of Kolb and Turner [113] with TQCD ¼ MeV, chosen to match 2006 Ref. [104] whose results are g ,LSð Þ. Panels (c) and (d): Interpolated opacities vs energy and temperature. Blue dashed lines mark ranges where the two values of g differ by more than 10%, and are a rough guide to where these rates can be trusted. Red dotted lines mark the cutoff temperature. − − three-body collision νμ þ e þ νe → μ or the scattering which implies that the modulus squared is þ þ process νμ þ μ → νe þ e . We illustrate this by calculat- 2 2 2 hjMj i¼−64G mμðpν · p þ ÞþOðEν Þ: ð45Þ ing the cross section for the latter, while neglecting the F μ e μ positron’s rest mass and Pauli blocking for simplicity. The Hence the cross section for the μ neutrino, integrated over squared and spin-summed/averaged matrix element for this outgoing particles’ directions, is process is 2 2 2 2 G mμ M 128G pν · p þ pμþ · pν : 43 F hj j i¼ Fð μ e Þð e Þ ð Þ σν ¼ þ OðEν Þ: ð46Þ μ π μ In the limit of zero neutrino energy 2 Such nonzero limiting values are responsible for the rise in ðpμþ · pν Þ¼−mμ=2 þ OðEν Þ; ð44Þ e μ the scaled rates for soft neutrinos in Figs. 6 and 7. 043515-13 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) 250 Tc ¼ MeV and 1000 MeV, which we expect to bracket the range of rates. GL15 With this caveat, Figs. 8(c) and 8(d) show interpolated μ neutrino opacities for a range of energies and temperatures, 250 with Tc ¼ MeV and 1000 MeV, respectively. In the rest of the paper, we use these scattering rates and the potentials defined in Sec. II to study sterile neutrino production via oscillation in the early Universe. We use both interpolations through the quark-hadron transition in order to illustrate the results’ sensitivity to the scattering rates. Figure 9 compares the scaled opacities with the earlier computations of Ref. [20], which are used in GL15. For comparison, we have shown their opacities computed with negligible sterile neutrino masses, which is an excellent FIG. 9. Comparison of neutrino opacities with existing calcu- approximation for all the momenta shown. The most lations. Lines of different colors show scaled muon neutrino ~ significant differences are at low temperatures due to opacities for different scaled energies, Eν . Solid and dashed lines μ two-body fusions (the resonances), and at low momenta are two interpolations through TQCD. Thin dotted lines indicate the calculations of Ref. [20], which are used in GL15. due to the limiting cross sections for scattering processes and three-body fusions involving massive particles. Thirdly, the hadronic opacities at low temperatures, i.e. V. STERILE NEUTRINO PRODUCTION T 043515-14 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π2 d 8π ρ 4 ln a ¼ H ¼ ðρ þ ρν Þ; ð47Þ SM ¼ g T ; ð54Þ 3 2 SM s 30 dt mP ρ P 2π2 1 2 1019 SM þ SM 3 where a is the scale factor, mP ¼ . × GeV is the sSM ¼ ¼ 45 g ;sT ; ð55Þ ρ ρ T Planck mass, and SM and νs are energy densities in standard model particles and sterile neutrinos, respectively. The latter we have the final form of the time-temperature relation 2 is given by an integral over the PSDs, ρν ¼ð1=2π Þ R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 2 2 4 2 p dp p þ ms ½fν ðpÞþfν ðpÞ . During Hubble dT 4Hg T þð30=π Þð∂ρν =∂tÞ s s ¼ − ;s s : ð56Þ expansion from a to a þ δa, (a) the sterile neutrino PSDs dt d½g T4 =dT δ evolve to fνs=νs ðpÞþ fνs=νs ðpÞ due to a combination of mixingwithmuonneutrinosandtheirmomentumredshifting We use numbers of relativistic degrees of freedom g and as pa ≡ const.. and (b) due to large neutrino opacities, all g ;s from Ref. [104] in our numerical implementation. active species maintain equilibrium PSDs with a common temperature, whoseevolutionis affectedbytheproductionof sterile neutrinos. B. Time evolution of asymmetry ˆ The continuity equation for the total stress-energy The temperature-scaled muon asymmetry, Lμ, evolves tensor is both from the depletion of relativistic degrees of freedom due to annihilations and from the production of sterile d d −1 neutrinos. There are subtleties in dealing with the latter in 3 ln a ¼ − ðρ þ ρν Þðρ þ P þ ρν þ Pν Þ ; dT dT SM s SM SM s s the case of resonant production [114], but for the semi- ð48Þ classical approach outlined in Sec. II B, we can write down the contribution in terms of the evolution of sterile neutrino where PSM=ν are SM and sterile neutrino pressures, PSD. Keeping in mind the definition of the lepton asym- respectively. The sterile energy density evolves according metry in Eq. (4), the asymmetry evolution due to both to contributions together is Z ρ ∂ρ ∂ρ Lˆ ˆ ˆ 2 d νs νs d ln a νs dt d μ d dpp ¼ þ : ð49Þ ¼ ½fν ðpÞ − fν ðpÞþ2fμ− ðpÞ − 2fμþ ðpÞ dT ∂ ln a dT ∂t dT dt dt 2π2 μ μ Z ˆ ˆ 2 ∂ The two terms on the right-hand side are the free-streaming −3 d ln T Lˆ − dpp − ¼ H þ μ 2 ½fν ðpÞ fν ðpÞ ; and oscillation contributions, respectively: dt 2π ∂t s s ∂ρ ð57Þ νs ¼ −3ðρν þ Pν Þ; ð50Þ ∂ ln a s s Z 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂ρ ∂ 8.11 νs dpp 2 2 ¼ p þ m ½fν ðpÞþfν ðpÞ : ð51Þ ∂t 2π2 s ∂t s s 4.64 2.66 We substitute Eqs. (49) and (50) into Eq. (48) and solve for 1.52 the relation between the scale factor and temperature 0.87 0.50 0.28 d ln a dρ ∂ρν dt 3 − SM s ρ −1 0.16 ¼ þ ð SM þ PSMÞ : ð52Þ dT dT ∂t dT 0.09 0.05 Substituting Eq. (47), we obtain the time-temperature 0.03 6 relation 0.02 0.01 dT 3H½ρ þ P þð∂ρν =∂tÞ ¼ − SM SM s : ð53Þ dt dρ =dT SM FIG. 10. We illustrate the temperature evolution of the ’ Defining the number of SM relativistic degrees of freedom sterile neutrino s PSD for the central model of Fig. 1 with 2 2θ 7 1 4 10−11 for the energy and entropy densities via ðms; sin Þ¼ð . keV; × Þ. Solid and dashed lines distinguish results with neutrino opacities from Figs. 8(c) and 8(d), respectively. Thin dotted lines show results using the 6We note that we correct here an error introduced in Ref. [18]. calculations of GL15. 043515-15 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) 20.0 10.5 5.5 2.9 1.5 0.8 20.0 10.5 5.5 2.9 1.5 0.8 FIG. 11. Sterile neutrino production mechanism. Panels (a) and (c): The entropy scaled mu lepton asymmetry and the net sterile number density with temperature. For each model with a given mass and mixing angle, the mu lepton asymmetry at high temperatures is tuned by hand to produce the right relic abundance. Panels (b) and (d): Sterile neutrino and antineutrino PSDs, respectively, at T ¼ 10 MeV. Colors differentiate models in Fig. 1, and solid and dashed lines distinguish results with neutrino opacities from Figs. 8(c) and 8(d) respectively. Thin dotted lines show results using the calculations of GL15. Note the different numerical factors multiplying the y-axis of panels (b) and (d). The dotted line in panel (b) is a massless Fermi-Dirac distribution with degeneracy g ¼ 0.003. where the symbol pˆ is the temperature-scaled momentum, neutrinos with specific momenta. Through Eq. (8), the pˆ ≡ p=T. The first term in the square bracket in the resonant momenta at a particular temperature satisfy last line above can be evaluated with the help of Eqs. (47) Δ 2θ − L − th 0 and (56), while the second term can be evaluated using ðpÞ cos V V ðpÞ¼ : ð58Þ Eq. (8). Our large number of momentum bins (1000) Substituting the definition of ΔðpÞ and the potentials from allows us to use spline integration at every time step in Eq. (12), we obtain order to perform the momentum integrals in Eqs. (57) and (51). We set up our Lagrangian momentum bins such 2 L th −3 ms dV dV ðpÞ that 5 × 10 ≤ p=T ≤ 20 at temperature T ¼ 10 GeV. − Lμ − p ¼ 0: ð59Þ 2p dLμ dp We have checked that this range is more than sufficient to accurately capture the most relevant range of the sterile There are two roots, i.e. two momenta resonant at any neutrino PSDs. temperature [18]. Consideration of the terms’ approximate temperature scaling shows that each scaled root (pˆ ≡ p=T) C. Resonant production sweeps to larger values at lower temperatures (ignoring As described in Sec. II, the presence of a lepton changes in the numbers of relativistic degrees of freedom). asymmetry leads to a resonant production of sterile This is reflected in Fig. 10, which shows the sterile neutrino 043515-16 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) PSD’s evolution with temperature for the central model in TABLE III. Parameters for the models marked in Fig. 1, with 7 1 2 2θ 4 10−11 7 1 Ω 2 0 119 Fig. 1 with ms ¼ . keV and sin ¼ × .We ms ¼ . keV and DMh ¼ . [2]. The ranges displayed in observe that most of the neutrinos are produced at the lower the three last columns account for the uncertainties in the neutrino 7 opacities near the quark-hadron transition. resonance and at temperatures close to TQCD. This is also illustrated by Figs. 11(a) and 11(c), which show the L L ð μ=sSMÞi at ð μ=sSMÞf at 8 μ 2 evolution of the entropy-scaled lepton asymmetry and sin 2θ T ¼ 10 GeV T ¼ 10 MeV the net sterile neutrino and antineutrino density for the ×10−11 × 10−5 × 10−5 hp=Tia range of models marked by stars in Fig. 1. The latter is also – – – sensitive to thermal (nonresonant) production, which oper- 0.800 13.0 13.1 6.95 7.03 2.60 2.61 1.104 10.80–10.88 4.74–4.81 2.45–2.47 ates at all temperatures, but is subdominant for the mixing 1.523 9.57–9.64 3.51–3.58 2.28–2.32 angles of interest. Note that the production as computed 2.101 8.81–8.88 2.76–2.83 2.12–2.16 according to the calculations of GL15 requires very differ- 2.899 8.32–8.39 2.27–2.34 1.95–2.01 ent input asymmetries, and produces different output 4.000 7.96–8.03 1.93–2.00 1.80–1.87 asymmetries, for lower mixing angles. 5.519 7.69–7.76 1.68–1.74 1.66–1.74 Figures 11(b) and 11(d) show the sterile neutrino and 7.615 7.45–7.53 1.47–1.54 1.53–1.62 antineutrino PSDs at T ¼ 10 MeV for these models. We 10.506 7.20–7.29 1.28–1.36 1.43–1.52 note that the sterile antineutrinos are produced off- 14.496 6.95–7.05 1.09–1.18 1.35–1.44 – – – resonance for the positive lepton asymmetries we consider 20.000 6.7 6.8 0.9 1.0 1.29 1.38 here, and their abundance is thus significantly suppressed aThe sterile DM distributions are nonthermal; we compute compared to that of the sterile neutrinos. Solid and dashed hp=Ti using the active neutrino temperature. Below the epoch lines in Fig. 11 show results for the two interpolations of the of e annihilation, the latter is related to the CMB temperature by 4 11 1=3 0 714 μ neutrino opacities through T presented in Fig. 8,which the factor ð = Þ ¼ . . We note that for a Fermi-Dirac QCD distribution hp=Ti ≃ 3.15. differ in the temperature range 150 MeV 043515-17 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) 20.0 thermal warm DM transfer function given in Refs. [23,124] “ ” 2 2 with an equivalent thermal mass of mth ¼ . keV; fits 10.5 for models marked with stars in Fig. 1 have a range of 1.6 to 3.2 keV. 5.5 However, the strong difference in shape with the thermal WDM transfer function warrants use of the exact 2.9 sterile neutrino dark matter transfer functions. The thermal warm DM PSDs relevant to the fit are rescaled versions of the Fermi-Dirac distribution; as can be seen from 1.5 Fig. 11(b), the resonantly produced DM’s PSD has an excess at low momenta that cannot be reproduced by such 0.8 a rescaling. Hence, our DM transfer functions do not exhibit the fits’ steep ∼k−10 dependence at large wave numbers and the resultant severe suppression of power on FIG. 12. Suppression of the transfer functions of overall density small scales. This indicates that the models considered in fluctuations relative to the ΛCDM ones for sterile neutrino the present work are more likely to be in agreement with models in Fig. 1, as a function of wave number. Dashed and small-scale structure formation constraints, as recently μ dotted lines show results for the interpolated neutrino scattering pointed out in Refs. [57–62,65]. rates of Figs. 8(c) and 8(d), respectively. The solid black line is the numerical fit for a thermal warm DM transfer function as given in Ref. [124]. VII. DISCUSSION AND CONCLUSIONS Sterile neutrinos are a well-motivated extension of the The main effect of such a collisionless component on standard model of particle physics, and offer a promising matter fluctuations is suppression due to free streaming in candidate for the inferred DM population of the Universe. the epochs where it is relativistic [118,119]. Previous works In this paper, we performed a detailed study of the resonant extensively studied this in the context of warm and/or production of sterile neutrinos with masses and mixing neutrino DM models (see Refs. [120,121] and references angles relevant to the recent x-ray excess. In doing so, we therein), and identified the characteristic scales at which the explored the rich phenomenology associated with the active ’ suppression set in as a function of the neutrinos’ mass and neutrinos weak interaction with the primordial plasma. mean momentum [18]. These interactions efficiently redistribute primordial lepton In order to obtain the suppression’s detailed form, we need asymmetries among all the available degrees of freedom, to incorporate the PSDs of the sterile neutrinos and anti- and impact the temperature and momentum dependence of neutrinos into the Boltzmann equation for the DM compo- neutrino opacities. We incorporated these effects into the nent. This entails solving a perturbed form of Eq. (8),with sterile production calculation, corrected and extended the additional terms due to inhomogeneities, but without the existing numerical implementation, and obtained revised source (production) terms. The scales of interest are non- DM phase space densities. We finally computed transfer linear in the current epoch, but we only provide the linear functions for fluctuations in the matter density, which transfer functions at z ¼ 0, which can be used as initial can be used as starting points for N-body simulations of conditions for cosmological N-body simulations. cosmological structure formation. We use the publicly available CLASS solver [86] to For the parameters relevant to the x-ray excess, resonant integrate the perturbed linear Boltzmann equation.9 We sterile neutrino production coincidentally occurs in the initiate the solver with the Planck background parameters vicinity of the quark-hadron transition [see Fig. 11(c)]. [3], except with the CDM component replaced by colli- Strongly interacting degrees of freedom affect the pro- sionless components with PSDs as shown in Figs. 11(b) duction in two ways: (a) they influence both asymmetry and 11(d). Since we are interested in the detailed shape of redistribution and neutrino opacities through their the transfer function, we turn off the default fluid approxi- interaction with the weak gauge bosons (Z and W ), mation for noncold relics [123]. and (b) the transition from free quarks to hadrons at Figure 12 shows the resulting suppression as a function TQCD influences the time-temperature relation [Eq. (56)]. of the comoving wave number. We illustrate the suppres- We now consider the robustness of each of these elements sion in the fluctuations’ transfer functions relative to their to the remaining uncertainties in the quark-hadron values in ΛCDM. Also shown is the commonly used fit to a transition. The asymmetry redistribution among the strongly inter- acting degrees of freedom depends on the susceptibility of 9Our choice was motivated by the availability of well- documented modules to deal with noncold relics. We have the quark-hadron plasma to baryon number and electric checked our results against those obtained from a modified charge fluctuations. At high temperatures, we used tree- version of the publicly available CAMB solver [122]. level perturbative QCD to compute the susceptibilities. 043515-18 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) There are uncertainties concerning the exact values of the two-state density matrix, rather than phase space densities. quark masses, loop corrections, and the exact implementa- The validity of the semiclassical approach rests on the tion of the MS renormalization scheme. We expect these to assumption that collisions dominate the off-diagonal element have little effect on the final sterile neutrino PSDs since the of the Hamiltonian that is responsible for vacuum oscillations bulk of the production occurs at lower temperatures, where [95–97]. For typical momenta at the temperatures of interest, the lattice QCD- and HRG-derived susceptibilities are most the ratio of these terms is ΔðTÞsin22θ=DðTÞ ≃ 0.6 × ðT= 100 −6 7 2 22θ 10−11 1=2 relevant. Thus, uncertainties in the asymmetry redistrib- MeVÞ ðms= keVÞ ðsin = Þ . The produc- ution are likely dominated by systematic errors in the lattice tion of sterile neutrinos happens at temperatures above, calculations [100], measurement errors in the hadronic but close to where these terms become comparable (note resonances’ masses, and inaccuracies inherent in the HRG the ratio’s steep temperature dependence). Thus, we expect approach near the quark-hadron transition. Our confidence that the results in this paper are relatively unaffected by this in the fit we used in this work is bolstered by the facts that approximation, but further work in this direction can settle (a) an independent lattice QCD calculation [101] found this question. very similar susceptibilities to those we used, and (b) the Finally, we have examined the assumptions underlying HRG approach—without any free parameter—is in very the model itself, which were enumerated in Sec. II A.If good agreement with the lattice calculation for there is indeed an extra neutrino that is an electroweak T ≲ 150 MeV. It is therefore unlikely that uncertainties singlet, it is not restricted to mix with only one flavor. in the susceptibilities will lead to dramatic changes in the However, the general case where the sterile neutrino mixes sterile neutrino PSDs. with all flavors introduces extra mixing angles, which The validity of our neutrino opacities is much less cannot be constrained as easily from observations. The clear—we have attempted to calculate them in as much same can be said about the assumption of a lepton detail as possible, but the hadronic parts still retain asymmetry in a single flavor. We briefly remark on the significant uncertainties due to the quark-hadron transition. possibility of the sterile neutrinos mixing with electron or We expect that opacities at high and low temperatures are tau flavors instead. The redistribution of Sec. III is almost well described by the rates of reactions involving free identical for the cases with electronic and muonic lepton quarks, and the lightest pseudoscalar and vector mesons, asymmetries, but is different in the tauonic case. This is due which are shown in Sec. IV. For temperatures near T , to the significantly larger mass of the corresponding QCD 1 77 we have considered two interpolation schemes (shown in charged lepton (mτ ¼ . GeV [1]), which annihilates Fig. 8) that we expect might bracket the range of possibil- away at higher temperatures. Thus most of an input tau ities. We have computed the sterile neutrino PSDs for both asymmetry ends up in the tau neutrino below T ≲ 400 MeV, and the quark hadron transition does not cases and have shown that they are fairly robust to the impact the redistribution. The electron and tau neutrino choice used, especially for models with larger values of the opacities are different from the muonic case, and so is the p=T average momentum h i. We leave the calculation of balance between the thermal and asymmetry potentials, self-consistent opacities through the transition to future which affects the resonant momenta and ultimately the final work. Yet another approximation we have made is that of dark matter PSDs—we leave for future work the possibility equilibrium distributions for all active species, which has of sterile neutrinos mixing with those flavors. been studied in a different context in Ref. [125]. We expect Also worth considering is active-active neutrino mixing, this to be valid at the temperatures relevant to the models which does not conserve asymmetries in the individual we study. flavors. This was studied in Ref. [90], which showed that To compute the Hubble expansion rate and time- such asymmetries are frozen in at the temperatures of temperature relation, we have used the plasma’s equations interest. An interesting possibility is to revisit this study and of state provided in Ref. [104], which are obtained by use the redistributed asymmetries of Sec. III to calculate the matching to the lattice QCD results of [126].Asthe active neutrino self-energies at this epoch. former’s authors point out, this result is still uncertain at In conclusion, we find it remarkable that sterile neutrino temperatures close to the quark-hadron transition. It models that are in agreement with the x-ray excess have would interesting to update their result with the latest transfer function shapes that can significantly impact struc- lattice QCD computations, which suggest a lower tran- ture formation on subgalactic scales. Fixing the leptonic sition temperature [127]. We expect the uncertainties asymmetry to produce the right DM relic density, the associated with the plasma’s equation of state to be at resonantly produced sterile neutrino transfer function goes most similar in magnitude to those coming from the from warm to cold as the mixing angle is increased from neutrino opacity [20]. small to large values. This indicates that upcoming x-ray Another simplification we adopted is the semiclassical observations [128–130] and ongoing efforts to study small- Boltzmann equation, which greatly facilitates our study of scale structure can together cover all of the allowed mixing the oscillation-driven production. As mentioned in Sec. II B, angle parameter space, and consequently confirm or disfavor the most general analysis considers the evolution of a the model. 043515-19 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) ACKNOWLEDGMENTS APPENDIX A: FUNCTIONS FOR HIGH-TEMPERATURE QCD We thank Olivier Doré and Roland de Putter for fruitful PRESSURE discussions. We are grateful to Mikko Laine for providing us the data for the plasma’s equation of state. T. V. and The functions defined in Sec. III B 1 are [104] C. M. H. are supported by the David and Lucile Packard Z 1 1 ∞ 2 Foundation, the Simons Foundation, and the U.S. μˆ ≡ x F1ðy; Þ 2 dxx Department of Energy. The work of F.-Y. C.-R. was 24π 0 x þ y performed in part at the California Institute of pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi × ½nˆ ð x þ y − μˆÞþnˆ ð x þ y þ μˆÞ ; ðA1Þ Technology for the Keck Institute for Space Studies, which F F is funded by the W. M. Keck Foundation. Part of the Z 1 1 ∞ 2 μˆ ≡ x research described in this paper was carried out at the Jet F2ðy; Þ 2 dx 8π 0 x þ y Propulsion Laboratory, California Institute of Technology, ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ˆ p − μˆ ˆ p μˆ under a contract with the National Aeronautics and Space × ½nFð x þ y ÞþnFð x þ y þ Þ ; ðA2Þ Administration (NASA). K. N. A. is partially supported by Z ∞ 1 NSF CAREER Grant No. PHY-1159224 and NSF Grant 1 x 2 F3ðy; μˆÞ ≡ − dx No. PHY-1316792. K. N. A. acknowledges support from 0 x x þ y the Institute for Nuclear Theory program “Neutrino ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ˆ p − μˆ ˆ p μˆ Astrophysics and Fundamental Properties” 15-2a where × ½nFð x þ y ÞþnFð x þ y þ Þ ; ðA3Þ part of this work was done. Z Z 1 ∞ ∞ 1 μˆ ≡ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi F4ðy; Þ 4 dx1 dx2 p p ð4πÞ 0 0 x1 þ y x2 þ y ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ p − μˆ ˆ p μˆ ˆ p μˆ ˆ p − μˆ × ½nFð x1 þ y ÞnFð x2 þ y þ ÞþnFð x1 þ y þ ÞnFð x2 þ y Þ pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi x1 þ y x2 þ y þ y − x1x2 ×ln pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi x1 þ y x2 þ y þ y þ x1x2 ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ˆ p − μˆ ˆ p − μˆ ˆ p μˆ ˆ p μˆ þ½nFð x1 þ y ÞnFð x2 þ y ÞþnFð x1 þ y þ ÞnFð x2 þ y þ Þ pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi x1 þ y x2 þ y − y þ x1x2 ×ln pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ; ðA4Þ x1 þ y x2 þ y − y − x1x2 ˆ 1 1 where nFðxÞ¼ =½exp x þ is the Fermi-Dirac distribution. to the same generation. We classify reactions as leptonic or hadronic based on the nature of the remaining two particles. APPENDIX B: NEUTRINO OPACITIES: MATRIX We now describe our calculation of these reactions’ ELEMENTS AND COLLISION matrix elements, and the associated contributions to the INTEGRALS neutrino opacity. In this appendix, we expand on the details underlying the neutrino opacities that were presented in Sec. IV. Under the a. Matrix elements for two-particle to set of assumptions presented therein, we add contributions two-particle reactions from a large number of reaction rates, which we present in It is a lengthy, but straightforward, task to enumerate all an organized manner in the rest of this section. leptonic reactions that contribute to the neutrino opacity. Reference [111] lists a complete set of reactions at temper- 1. Rates for neutrinos to go to two-particle atures of a few MeV. Our calculations extend to higher final states temperatures; hence we also include reactions involving tau We compute reaction rates for momenta and temper- leptons. The reactions are enumerated in Table I. It is harder atures where we can integrate the weak gauge bosons out to study hadronic reactions in a consistent manner through and approximate the weak interaction by a four-particle the quark-hadron transition temperature, TQCD. We adopt vertex. For tree-level processes under this approximation, if assumption 3 of Sec. IV: we neglect the strong coupling one of the ingoing particles is a neutrino, one of the other constant and its running at temperatures T>TQCD, and particles is either a neutrino or a charged lepton belonging hence calculate opacities with free quarks. We enumerate 043515-20 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) TABLE IV. Representative examples of two-particle to two-particle reactions that contribute to the neutrino opacity, along with their matrix elements written using a four-fermion vertex for the weak interaction (for reactions involving leptons and quarks) and three-quark chiral perturbation theory (for reactions involving the pseudoscalar meson octet). For a reaction νμ þ A → B þ C, the momenta pi∶ i ¼ 1, 2, 3, 4 within matrix elements are mapped to νμ, A, B and C respectively. P Reaction S jMj2 Remarks Reactions involving leptons ν τ− → μ− ν 128 2 μ þ þ τ GFðp1 · p2Þðp3 · p4Þ Reactions involving quarks ν μþ → ¯ 384 2 2 μ þ u þ d jVudj GFðp1 · p4Þðp2 · p3Þ T>TQCD Reactions involving the pseudoscalar meson octet ν μþ → πþ π0 8 2 2 2 − − − − 2 a μ þ þ jVudj GF½ p2 · ðp4 p3Þp1 · ðp4 p3Þ ðp1 · p2Þðp4 p3Þ T 3 3 1 8 1 8 where fπ is an energy scale associated with the breaking eAμ ¼ lμ þ rμ þ pffiffiffi lμ þ pffiffiffi rμ: ðB5dÞ 3 a 3 3 3 of the SUð ÞA symmetry, and T are generators of SUð Þ. 043515-21 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) 2 θ 3χ Here g2 is the SUð Þ coupling constant, and e ¼ g2 sin W. for which we calculate neutrino opacities using PT [the 3 The overlap of Aμ with the anomalous Aμ is an artifact of total energy range is shown in Figs. 8(a) and 8(b)]. the omission of the charm quark from our analysis. If the We do not self-consistently compute these corrections to charm quark were included (as it has to be to make the SM the currents, as it is beyond the scope of this paper. Instead, nonanomalous) then there is no coupling to an anomalous we modify the s-channel reaction rates in a phenomeno- current. As long as there are no charm quarks (always true logical manner: we apply a cutoff in the CM energy at at the temperatures where 3χPT is applied) this current 1 GeV with a width of 50 MeV, below which we use the 1 3χ looks like that associated with the anomalous Uð ÞA. PT currents, and above which we use the SM free quark We read off the currents that couple to the gauge fields currents. We do not incorporate any corrections to t-channel ðaÞμ ∂L ∂ a a V A reactions; this would involve some knowledge of the parton using the definition Jl=r=V=A ¼ = ðl =r = = Þμ, and the Lagrangian of Eq. (B3). We transform the results to distribution functions for the mesons involved. obtain the currents that couple to the SM electroweak gauge We observe that the squared and spin-summed matrix 0 fields, Zμ, Wμ and Aμ: elements for tree-level processes, be they for processes involving leptons and free quarks (computed using the SM 1 ↔ ↔ þμ μ þ 0 μ þ i 0 μ þ currents), or for those involving pseudoscalar mesons J ¼ pffiffiffi V fπ∂ π þ iπ ∂ π − pffiffiffi K ∂ K 2 ud 2 (computed using 3χPT), are at-most quadratic functions pffiffiffi of the Mandelstam variables. This greatly facilitates a i ↔ i ↔ 3i ↔ − ffiffiffi 0∂μπþ π0∂μ þ η∂μ þ semiautomated computation of the two-particle to two- þ Vus p K þ 2 K þ 2 K ; 2 particle reactions’ contribution to the neutrino opacity, ðB6aÞ which we very briefly describe next. 1 ↔ ↔ −μ ffiffiffi μ − 0 μ − iffiffiffi 0 μ − b. Rates for two-particle to two-particle reactions J ¼ p Vud fπ∂ π − iπ ∂ π þ p K ∂ K 2 2 Consider a general two-particle to two-particle reaction, pffiffiffi i ↔ i ↔ 3i ↔ να þ A → B þ C, that consumes a massless input neutrino, þ V pffiffiffi K0 ∂μ π− − π0∂μK− − η∂μK− ; ν us 2 2 2 α. The particles A, B and C can all be fermions (leptons or quarks), or contain a pair of bosons (pseudoscalar mesons). ðB6bÞ We expand Eq. (7) to write down the following expression for the scattering rate as a collision integral: μ μ − 2θ μ Jz ¼ J3 sin WJEM; ðB6cÞ Z 1 Γ 3p~ 3p~ 3p~ 2π 4 1 1 ↔ ↔ ðEνα Þ¼ d Ad Bd Cð Þ μ μ 0 μ þ μ − þ μ − 2Eν J ¼ fπ ∂ π þ pffiffiffi ∂ η þ iπ ∂ π þ iK ∂ K ; α 3 2 3 × δðpν þ p − p − p ÞS X α A B C ðB6dÞ 2 × jMj fAðEAÞð1 ∓ fBðEBÞÞð1 ∓ fCðECÞÞ; ↔ ↔ μ πþ∂μπ− þ∂μ − ðB7Þ JEM ¼ i þ iK K : ðB6eÞ 3 These currents agree with the leading-order parts of the where the symbol d p~ is shorthand for the Lorentz invariant 3 3 functionals computed in Ref. [132]. The lower half of phaseP space volume element d p=½ð2πÞ 2EðpÞ ; the sym- 2 Table II enumerates the allowed reactions involving pseu- bol jMj is the absolute value of the matrix element doscalar mesons. Table IV shows the squared and spin- squared and summed over all spin states; S is a symmetry summed matrix elements for two such reactions, computed factor for identical particles in the initial and/or final states; using the currents listed in Eq. (B6). and the fðEÞ’s are appropriate Bose-Einstein/Fermi-Dirac A final complication is that 3χPT, and the currents phase space distributions depending on the statistics of derived from it, are valid only when the momentum in the the particles, with plus and minus signs for bosons and intermediate weak gauge bosons is low compared to the fermions respectively. energy scale 4πfπ ∼ 1 GeV [131]. The physical currents We follow the treatment in Ref. [111] to reduce the that couple to the SM electroweak gauge fields are nine-dimensional phase space integral of Eq. (B7) to a continuous functions of this momentum; they approach numerically manageable three-dimensional integral over p p μ pˆ pˆ the SM free quark currents for large momentum values. the variables j Aj, j Bj and B ¼ B · να . This procedure This manifests as the production of quarks in the large CM involves using the delta function to perform the integral energy limit in s-channel reactions, and as “deep-inelastic over pC, and using the form of the matrix elements for tree- scattering” off the mesons’ quark content in the large level processes to analytically perform the integral over μ pˆ pˆ momentum-transfer limit in t-channel reactions. These A ¼ A · να . We refer the reader to Ref. [111] for more limits are important to consider at the higher energies details. The form of the matrix elements also lends itself to 043515-22 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) easy parametrization in terms of a small number of classes; mA=EA is the inverse-Lorentz factor that accounts for the along with the procedure described above, this enables a longer lab-frame lifetime of Aþ at high energies. The simple numerical implementation of the calculation of degeneracy factor is gA ¼ 1 for pions and kaons, but we these reactions’ contributions to the neutrino scattering rate. include it for later use with heavy mesons. With the help of relativistic kinematics, we see that the 2. Rates for neutrinos to go to one-particle lab-frame neutrino energy is final states 1 2 We must also consider the contribution to the neutrino mα 0 Eν ¼ 1 − ðE þ p μ Þ; ðB10Þ Γ α 2 2 A A interaction rate, ðEνα Þ, from interactions with two-particle mA final states (“fusion” or inverse decay). A four-fermion interaction such as the weak interaction (at E ≪ mW;mZ) so that Eq. (B9) can be rewritten in terms of a rate of decays can produce such a final state in two ways. One, applicable per unit volume per unit neutrino energy: at T 043515-23 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) Γvac 2 2 gAmA þ→ν αþ where υ ¼ 1 − mα=m . Note that the numerical calculation Γ A α A fusionðEνα Þ¼ 2 2 2 − 1 − of the logarithm must be treated carefully since for EA;min ð mα=mAÞEνα Z Eν ≫ T ∞ 1 − α we are taking the logarithm of a number that is fνα ðEνα Þ × fAðEAÞ very close to 1. For calculational purposes, we replace the fν ðEν Þ EA;min α α logarithm in Eq. (B19) by a truncation of its Taylor × ½1 − fαðEαÞ dEA: ðB16Þ expansion at the fifth order wherever the argument deviates from unity by less than ϵ ¼ 10−3. Next we substitute in the Bose-Einstein or Fermi-Dirac − phase space distributions, and note that Eα ¼ EA Eνα , b. Rates for two-body fusion processes yielding The rate parameters for the key two-body fusion reac- tions are shown in Table V. vac g m Γ þ þ A A A →ναα Some parameters were not available in the Review of Γ ðEν Þ¼ fusion α 1 − 2 2 2 Particle Properties [1]. Key among these are the decay ð mα=mAÞEνα Z parameters for the weak decays of the ρ, ωð782Þ,and ∞ Eνα =T e K ð892Þ vector mesons. These mesons are actually extremely × − dEA EA=T − 1 EA=T Eνα =T 1 EA;min ðe Þðe e þ Þ broad resonances, and their principal decay mode is into Γvac gAmA þ→ν αþ T E E − Eν lighter mesons. Electromagnetic and especially weak decay A α Φ A;min ; A;min α : ¼ 1 − 2 2 2 modes are less common. Note that the helicity suppression ð mα=mAÞEνα T T 0 arguments that forbid e.g. π → νeνe do not apply to the ðB17Þ vector mesons. The relevant decays of the charged vector mesons ρþ and Here we have defined the dimensionless integral K ð892Þþ can be obtained by noting that the virtual-W diagram results in an effective vertex Z ∞ ea−bdx Φ a; b ð Þ¼ xþa −x−b e 1 e 0 ðe − 1Þðe þ 1Þ L ∋ ffiffiffi l γμ 1 − γ5 ν ffiffiffi eff i p X ð Þ X 2 p ∞ ∞ Z X X ∞ 2 2 sin θW mW 2 2 sin θW a−b k − 1 j x a −k x b ¼ e ð−1Þ e ð þ Þð þ Þe ð þ Þdx ϵ ρ− 0 × Vudmρfρ½ ð Þ μ þ H:c:; ðB20Þ j¼0 k¼0 X∞ X∞ −1 k −ð1þjÞa−kb a−b ð Þ e − ¼ e where fρ is the ρ decay constant, and ½ϵðρ Þ μ is the 1 þ j þ k j¼0 k¼0 polarization of the “on-shell” ρ meson. The advantage of ∞ −1 τþ → ρþν X e−ma Xm this Lagrangian is that the well-measured decay τ ¼ ea−b ð−ea−bÞk is related to the decay of ρþ to a charged lepton and a m m¼1 k¼0 neutrino (predicted branching fraction ∼2 × 10−11). From a X∞ − − e ma 1 − ð−ea bÞm tree-level calculation with this Lagrangian, we infer a ratio ¼ ea−b m 1 a−b m¼1 þ e Γ ρþ → μþν 2 3 2 − 2 2 2 2 2 1 X∞ −ma X∞ −mb ð μÞ mτ ðmρ mμÞ ð mρ þ mμÞ e −1 e ¼ : ðB21Þ ¼ þ ð−1Þm Γ τþ → ρþν 3 3 2 − 2 2 2 2 2 b−a 1 m m ð τÞ mρðmτ mρÞ ð mρ þ mτ Þ e þ m¼1 m¼1 1 ¼ ½− lnð1 − e−aÞþlnð1 þ e−bÞ These rates are included in Table V. b−a 1 e þ The rate for ρ0 → ν ν can be obtained by replacing the − X X 1 1 þ e b terms in Eq. (B20) with the Z couplings and propagator: ¼ ln : ðB18Þ eb−a þ 1 1 − e−a 1 L ∋ e ν γμ 1 − γ5 ν ffiffiffi e Recall that E is a function of Eν and is given by eff i X ð Þ X 2 p A;min α 4 sin θ cos θ m 2 sin θ cos θ Eq. (B13). We then achieve the final simplification: W W Z W W 1 2 0 × − sin θ mρfρ½ϵðρ Þ μ þ H:c: ðB22Þ vac 2 W g m Γ þ þ T A A A →ναα Γ ðEν Þ¼ fusion α −Eν =T 2 υð1 þ e α ÞEν α ρ0 − υ2 2 4 2 4υ Here the coupling of the vector Z current to the meson 1 Eνα =T ð mAþ Eνα Þ=ð Eνα TÞ þ e e was related to the coupling of the vector W current to the ρþ ×ln 2 2 2 ; ðB19Þ −ðυ m þ4Eν Þ=ð4υEν TÞ 1 − e A α α using isospin SUð2Þ symmetry. The conclusion is that 043515-24 STERILE NEUTRINO DARK MATTER: WEAK … PHYSICAL REVIEW D 94, 043515 (2016) TABLE V. The parameters for reactions that go into Eq. (B19). Reactions relevant for the neutrino opacity are shown; antineutrinos are similar. Particle masses are obtained from the Particle Data Group. Decay partial widths are obtained from the sources indicated. All reactions in which a neutrino can produce a hadronic resonance below 1 GeV are included. vac Reaction mA MeV gA υ Γreverse MeV Rate method Reactions involving the pseudoscalar meson octet þ þ −18 νe þ e → π 139.57 1 0.999987 3.110 × 10 PDG þ þ −14 νμ þ μ → π 139.57 1 0.4269 2.528 × 10 PDG þ þ −19 νe þ e → K 493.68 1 0.9999989 8.41 × 10 PDG þ þ −14 νμ þ μ → K 493.68 1 0.95419 3.38 × 10 PDG Reactions involving vector mesons with nonzero isospin 0 −12 τ þ − → ρ0 2 a νX þ ν¯X → ρ 775.26 3 1 9.78 × 10 Average of decay and e e ; assumed isospin SUð Þ þ þ −11 τ νe þ e → ρ 775.26 3 0.9999996 7.00 × 10 decay þ þ −11 νμ þ μ → ρ 775.26 3 0.98143 6.80 × 10 τ decay þ þ −12 τ νe þ e → K ð892Þ 891.66 3 0.9999997 5.45 × 10 decay þ þ −12 νμ þ μ → K ð892Þ 891.66 3 0.98596 5.33 × 10 τ decay Reactions involving vector mesons with zero isospin pffiffiffi ν ν¯ → ω 782 −13 þ − ¯ X þ X ð Þ 782.65 3 1 7 × 10 e e → ωð782Þ; assumed quark content ðuu¯ þ ddÞ= 2 aThe τ decay gives 1.01 × 10−11 and the eþe− → ρ0 computation gives 9.5 × 10−12. 0 8 3 3 Γðρ → νeνeÞ mτ mρ and similarly for the other two flavors. In this relation, we þ þ ¼ 2 2 2 2 2 m m θ Γðτ → ρ ντÞ 3ðmτ − mρÞ ð2mρ þ mτ Þ have used that W ¼ Z cos W. The same decay rate can be obtained by taking the 1 2 − 2θ 2 ð = sin WÞ ratio × 2 ; ðB23Þ jVudj 0 2 2 2 i i i 0 2 Γðρ → ν ν Þ 1 1 mρ 1 h0jðg uγ u þ g dγ d þ g sγ sÞjα i e e ¼ u;V d;V s;V : ðB24Þ Γ ρ0 → þ − 2 2 θ θ 2 θ θ 2 i 1 i 1 i 0 ð e e Þ sin W cos W mZ sin W cos W h0jð3 uγ u − 3 dγ d − 3 sγ sÞjρ i 0 − pffiffiffi Note that the reaction ρ → eþ þ e is mediated princi- approximate it as a pure ðuu þ ddÞ= 2 state (i.e. with 1 pally by the photon instead of the Z. The 2 is due to the fact no strange quark), and repeat the argument used for the that the photon couples to right-handed as well as left- ρ0 → eþe− calculation with Eq. (B24). This result is shown handed electrons. The factor of mα=mρ is a combination of in the table; it is much more uncertain than the calculation kinematic factors appropriate if the decay constant is the for the ρ0 since mixing with ss is allowed. Nevertheless, the same for all octet members. The factor of small rate for ωð782Þ production (as compared with ρ0) 1=ð2 sin θW cos θWÞ is the ratio of the Z coupling to νe;L suggests that it leads to an overall small correction to 2 2 to the photon coupling to eL (or eR). The factor of mρ=mZ is neutrino opacities. the ratio of Z to photon propagators. The last term is the 0 0 ratio of Z coupling to α to γ coupling to ρ , with gu;V ¼ c. Three-body fusion processes 1 2 2 1 1 2 0 4 − 3 sin θW and gd;V ¼ gs;V ¼ − 4 þ 3 sin θW. For the ρ , The final set of reactions that contribute to the neutrino the last term can be computed using isospin symmetry. opacity are three-body fusions. As earlier, these reactions The agreement between τ decay and eþe− → ρ0 is can be either leptonic or hadronic in nature. We adopt the good: the former predicts a partial width for ρ0 → ν ν of prescription outlined in Sec. B1a for the hadronic reac- e e tions. Given the hadronic and leptonic currents coupling to 1 01 10−11 9 5 10−12 . × MeV, and the latter predicts . × MeV, the SM electroweak gauge bosons, we can enumerate all a difference of only 6%. The average is shown in the table. three-body reactions that contribute to the neutrino opacity No similar decay is allowed (i.e. it is not possible with a in the same manner as earlier. We must keep in mind the single intermediate vector boson propagator) for the kinematic constraint that the rest mass of the product must K ð892Þ0 or K ð892Þ0 mesons because the current that be greater than that of the reactants. couples to the Z cannot change strangeness. The matrix element for any three-body fusion reaction is It is less clear how this procedure should be applied to related to one for a two-particle to two-particle reaction by the ωð782Þ meson, which has no isospin. One might crossing symmetry. Thus, we do not need to compute any 043515-25 TEJASWI VENUMADHAV et al. PHYSICAL REVIEW D 94, 043515 (2016) new matrix elements for this section. However, we need to All the symbols are defined identically to Eq. (B7). The modify the treatment of the kinematics from the previous procedure to reduce the dimensionality of this integral is case. Consider a general three-body fusion reaction, exactly analogous to that in Sec. B1band Ref. [111], with να þ A þ B → C. The scattering rate for an input neutrino one important difference. The variables finally left to p p energy Eνα is given by the collision integral: numerically integrate over are, as earlier, j Aj, j Bj and Z μ ¼ pˆ ν · pˆ . If we consider the integration domain for the 1 B α B 3 3 3 4 two-particle to two-particle case, for a given value of jp j, ΓðEν Þ¼ d p~ d p~ d p~ ð2πÞ A α 2 A B C p Eνα energy constraints allow a maximum value of j Bj. For a three-body fusion, jp j has no upper bound, which greatly × δðpν þ p þ p − p ÞS B X α A B C expands the allowed phase space. 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