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Unitarity Bounds of Astrophysical Neutrinos

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Unitarity Bounds of Astrophysical Neutrinos Unitarity Bounds of Astrophysical Neutrinos 1, 1, 2,† 3,‡ Markus Ahlers, ∗ Mauricio Bustamante, and Siqiao Mu 1Niels Bohr International Academy & Discovery Centre, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark 2DARK, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark 3California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA The flavor composition of astrophysical neutrinos observed at neutrino telescopes is related to the initial composition at their sources via oscillation-averaged flavor transitions. If the time evolution of the neutrino flavor states is unitary, the probability of neutrinos changing flavor is solely determined by the unitary mixing matrix that relates the neutrino flavor and propagation eigenstates. In this pa- per we derive general bounds on the flavor composition of TeV–PeV astrophysical neutrinos based on unitarity constraints. These bounds are useful for studying the flavor composition of high-energy neutrinos, where energy-dependent nonstandard flavor mixing can dominate over the standard mix- ing observed in accelerator, reactor, and atmospheric neutrino oscillations. PACS numbers: 14.60.Pq, 14.60.St, 95.55.Vj I. INTRODUCTION property of these models is that the flavor transitions between sources and Earth are entirely determined by a The high-energy astrophysical neutrinos discovered new unitary mixing matrix that connects neutrino flavor by IceCube [1–7] are key to revealing the unknown ori- and propagation eigenstates [38]. gin of high-energy cosmic rays and the physical condi- We will discuss the regions in flavor space that can be tions in their sources [8]. They can escape dense envi- expected from this class of models. The unitarity of the ronments, that are otherwise opaque to photons, and new mixing matrix allows us to compute the boundary travel cosmic distances without being affected by back- of the region that encloses all possible flavor composi- ground radiation or magnetic fields. They also provide tions at the Earth, in spite of not knowing the values of a unique opportunity to study fundamental neutrino the matrix elements. Previous work [20] derived a set properties in an entirely new regime: their energy and of unitarity bounds for specific choices of flavor com- baseline far exceed those involved in reactor, accelera- tor, and atmospheric neutrino experiments. Effects of nonstandard neutrino physics — even if they are intrin- 0.0 sically tiny — can imprint themselves onto the features 1.0 (0 : 1 : 0) Unitarity S of astrophysical neutrinos, including their energy spec- (1 : 2 : 0)S trum, arrival directions, and flavor composition, i.e., the bounds 0.2 (1 : 0 : 0)S proportion of neutrinos of each flavor. 0.8 At the sources, high-energy neutrinos ( GeV) are ν produced by cosmic-ray interactions with gas and radi- ) µ 0.4 fraction ,⊕ ation. These neutrinos are flavor eigenstates from the f τ 0.6 ( weak decay of secondary particles. The initial compo- sition of neutrino flavor states is determined by details 0.6 ( f of the production process. After emission, oscillations 68% C.L. 0.4 µ fraction , ⊕ modify the composition en route to Earth [11–18]. As- τ ) ν 95% C.L. suming standard oscillations, we can predict the ob- 0.8 servable flavor composition from a given source com- IceCube 2015 0.2 arXiv:1810.00893v2 [astro-ph.HE] 31 Dec 2018 position. However, nonstandard neutrino oscillations can alter the composition drastically [19–25]. Nonstan- 1.0 dard effects can originate, e.g., from neutrino interac- 0.0 tions with background matter [26], dark matter [27, 28] 0.0 0.2 0.4 0.6 0.8 1.0 or dark energy [28, 29] or from Standard Model ex- νe fraction ( fe, ) ⊕ tensions that violate the weak equivalence principle, Lorentz invariance, or CPT symmetry [30–37]. A key FIG. 1. Unitarity bounds of astrophysical neutrino flavors for three source compositions indicated by filled symbols. The corresponding open symbols indicate the expected composi- tion at Earth under standard oscillations using the best-fit mix- ing parameters for normal mass ordering [9]. We include the ∗ [email protected]; ORCID: 0000-0003-0709-5631 † [email protected]; ORCID: 0000-0001-6923-0865 best-fit flavor composition from IceCube [10] as a black star ‡ [email protected] and the 68% and 95% confidence levels as grey-shaded areas. 2 position at the sources. We extend this work by pro- the Hamiltonian, including kinetic terms and effective viding a refined and explicit formalism to derive unitar- potentials [38]. In general, the 3 3 unitary mixing ma- ity bounds that are easily applicable to arbitrary source trix U has nine degrees of freedom.× However, neutrino compositions. oscillation phenomena only depend on four indepen- Figure1 shows our results for physically motivated dent parameters, which can be parametrized by three choices of source flavor composition. The ternary plot rotation angles and one phase. Unitarity ensures that shows the source and Earth flavor fractions, i.e., the rela- the total number of neutrinos of all flavors is conserved. tive contribution of neutrino flavors to the total neutrino Neutrino flavor oscillations of pure or mixed states can flux. Assuming that the accessible flavor space is con- be described in terms of the evolution of the density vex, i.e., that every intermediate flavor fraction between matrix r, following the Liouville equation r˙ = i[H, r] any two accessible fractions is also accessible by a suit- with Hamiltonian H. − able unitary matrix, our unitarity bounds are maximally In the case of standard neutrino oscillations and neu- constraining and completely characterize the accessible trino propagation in vacuum, the propagation eigen- flavor space. states are identical to the neutrino mass eigenstates ni The paper is organized as follows. In Sec.II we dis- (i = 1, 2, 3). The mixing matrix between flavor and mass cuss the astrophysical processes of neutrino production eigenstates is the so-called Pontecorvo-Maki-Nagakawa- and the corresponding flavor composition at the source. Sakata (PMNS) matrix [39–41]. In the relativistic limit, We discuss the resulting flavor composition at Earth af- standard oscillations in vacuum can be introduced via ter flavor oscillation with nonstandard neutrino mixing. the Hamiltonian In Sec. III we derive general flavor boundaries for the 2 mi flavor composition at Earth based on the unitary of the H0 ∑ ni ni + ni ni , (2) nonstandard mixing matrix. We conclude in Sec.IV. ' i 2En j ih j j ih j Throughout this paper we will work in natural units where E is the neutrino energy and the sum runs over = = n withh ¯ c 1. projectors onto neutrino and antineutrino mass eigen- states. The solution of the Liouville equation with H = H II. ASTROPHYSICAL NEUTRINO FLAVORS 0 describes the oscillation of neutrino flavors due to the nontrivial mixing and mass splitting, Dm2 m2 High-energy astrophysical neutrinos are products of ≡ i − m2 = 0, for i = j. The oscillation phases are given cosmic-ray collisions with gas and radiation. The flux j 6 6 2 of neutrinos at production can be described as a mixed by Dm `/4En where ` is the distance to the neutrino state of neutrinos na and antineutrinos na where the in- source. In the case of astrophysical neutrinos these os- dex a = e, m, t refers to the neutrino flavor eigenstate cillation phases are much larger than unity. Consider- produced in weak interactions. The relative number of ing the wide energy distribution of neutrinos at their initial neutrino states (Ne : Nm : Nt)S (summed over neu- sources and the limited energy resolution of neutrino detectors, flavor transitions from n to n (or from trinos and antineutrinos) is determined by the physical j ai j bi conditions in the source. In the simplest case, pions (or na to n ) can only be observed by their oscillation- j i j bi kaons) produced in cosmic-ray interactions decay via averaged transition probability given by + + + + p m + nm followed by m e + ne + nm (and 2 2 the charge-conjugated! processes).! This pion decay chain Pab = ∑ Uaa Uba . (3) a j j j j results in a source composition of (1 : 2 : 0)S. However, in the presence of strong magnetic fields it is possible In the following, we will discuss nonstandard neu- that muons lose energy before they decay and do not trino oscillations that can be described by additional ef- contribute to the high-energy neutrino emission [12]. In fective Hamiltonian terms He in the Liouville equation, this muon-damped scenario the composition is expected so that the total Hamiltonian is H = H0 + He. These to be closer to (0 : 1 : 0)S. On the other hand, neutrino effective terms can be generated in various ways, in- production by beta-decay of free neutrons or short-lived cluding nonstandard interactions with matter and Stan- isotopes produced in spallation or photo-disintegration dard Model extensions that violate the weak equiva- of cosmic rays leads to (1 : 0 : 0)S. lence principle, Lorentz invariance, or CPT symmetry. After production, astrophysical neutrinos travel over Concretely, we will study the effect of additional terms cosmic distances before their arrival at Earth. The ob- in the Hamiltonian that can be parametrized in the servable flavor composition is significantly altered by form [42] neutrino oscillations, which are due to neutrino flavor En states being superpositions of propagation states na, n He = n ∑ ea na na + ea na na , (4) L a j ih j j ih j na = ∑ Ua∗a na . (1) j i a j i with n an integer and L the energy scale of the nonstan- dard effects. The eigenvalues of this additional Hamil- These propagation states are defined as eigenvectors of tonian, ea and ea, are required to be nondegenerate, 3 De ea eb = 0, for a = b, in order to induce neu- trino≡ oscillations.− 6 For simplicity,6 we will consider CP- even Hamiltonians with ea = ea that affect neutrinos and antineutrinos equally.
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