PoS(LeptonPhoton2015)017 http://pos.sissa.it/ † ∗ [email protected] Speaker. This work was supported in part by the Innovation Program of the Institute of High Energy Physics under Grant In this talk, I first summarizenos our and emphasize current the knowledge remaining about unsolved the problemsresults in fundamental neutrino on properties physics. neutrino of Then, mass neutri- recent models theoretical aretrino introduced. masses, Different lepton approaches flavor to mixing understanding and tinynew CP neu- progress in violation the are studies presented. of astrophysical Finally,neutrinos neutrinos, I and including report ultrahigh-energy keV cosmic briefly sterile neutrinos. some neutrinos, supernova ∗ † Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). No. Y4515570U1, by the NationalPhysics Youth Thousand (CCEPP). Talents Program, and by the CAS Center for Excellence in Particle XXVII International Symposium on Lepton Photon17-22 Interactions August at 2015 High Energies Ljubljana, Slovenia Shun Zhou Theoretical Results on Neutrinos Institute of High Energy Physics, ChineseCenter Academy for of High Sciences, Energy Beijing Physics, 100049, PekingE-mail: University, China Beijing 100871, China PoS(LeptonPhoton2015)017 ]. 6 , in µ ν Shun Zhou and is relatively µ 13 ν θ and non-electron e ]. In 1957, after the ] experiments. ν ] and the theoretical 3 13 10 from nuclear fusions in e ν from nuclear reactors for the first ]. Later, in 1939, Wendell Furry no- e ν 12 ] and Double Chooz [ 9 2 ]. In 1987, the neutrino burst from SN 1987A, a core- that there are many peaks along the gradually increasing 4 1 could happen only in the former case [ − e 2 ) + A , and provided strong evidence for atmospheric neutrino oscillations [ , µ 2 ν + ]. In 2012, the Daya Bay experiment observed the disappearance of reactor ]. In 1998, the Super-Kamiokande experiment confirmed the disappearance 7 5 Z ( and N µ . Therefore, it is also worthwhile to mention a few seminal works, some of which ν from nuclear reactors, Bruno Pontecorvo made an interesting analogue between ]. In 1937, Ettore Majorana considered the possibility that particles could be their 1 → e ) 11 ν A , which indicate that neutrinos are massive particles and lepton flavors are significantly Z 1 ( ]. In 1968, Raymond Davis, Jr. discovered solar neutrinos 2 ], which was later confirmed by both RENO [ ], signifying the birth of neutrino physics. In 1962, Leon Lederman, Melvin Schwartz N 8 1 The 2015 Nobel Prize in Physics was awarded jointly to Takaaki Kajita and Athur B. McDonald for the discovery In 1930, Pauli proposed the neutrino in order to rescue the law of energy-momentum conserva- The most important progress in the last two decades should be the discovery of neutrino os- In 1956, Clyde Cowan and Fredrick Reines detected Although the neutrino was first conjectured in 1930 by the great theorist Wolfgang Pauli, thus 1 of neutrino oscillations in theFundamental Super-Kamiokande Physics has and been SNO shared experiments. byKam-Biu the Luk In major for addition, neutrino Daya oscillation Bay, the Atsuto experiments 2016 Suzuki andfor for Breakthrough their SNO, KamLAND, leaders: Prize Koichiro Takaaki Nishikawa Kajita Yifang in for Wang and K2K and Yoichiro and Suzuki T2K, for Arthur Super-Kamiokande. B. McDonald time [ ticed a remarkable difference between Majoranadecays and Dirac neutrinos that neutrinoless double-beta tion and explain the continuous energyPauli’s spectrum neutrino of hypothesis, electrons Enrico from Fermi beta putand decays. suggested forward In a his possible 1933, famous determination based effective of on theorythe neutrino of endpoint masses from beta [ the decays, shape of energy spectrum near mixed. The experimental results are crucially importantbut for the the whole research area activities to advance are considerably, as actually peaks dominated in by Fig. theoretical studieswere on inspried by neutrinos, experimental showing results up and some not. own antiparticles, which are now called Majorana particles [ cillations, the Sun, and found the discrepancy between the experimental observation [ of upward-going In 2002, the Sudbury Neutrino Observatoryneutrino (SNO) fluxes, experiment and measured demonstrated both neutrino flavor conversionslar as neutrino a problem solution [ to the long-standingneutrinos so- at a short baseline and foundlarge that the [ smallest leptonic flavor mixing angle discovery of curve, which are stimulated byevents should groundbreaking be discoveries mentioned. in neutrino physics. Some important and Jack Steinberger proved the existence of a different flavor of neutrino, namely far every important step forwardother in authors neutrino did physics before, I hastheir search been titles, in led and the plot by a INSPIRE-HEP experimentalists. curve database ofIt for the As is number the straightforward many of papers to neutrino observe with papers from with "neutrino" Fig. respect in to the year of publication. Theoretical Results on Neutrinos 1. Introduction nature [ prediction from the standard solarcollapse model [ supernova in theBaksan Large experiments Magellanic [ Clound, was observed in Kamiokande-II, IMB and PoS(LeptonPhoton2015)017 , t ]. ]. m 15 14 [ e  ν c m Shun Zhou ↔ e  ν u m ]. In 1985, Stanislav Mikheyev and 16 ]. The Mikheyev-Smirnov-Wolfenstein (MSW) HEP: find t neutrino and date date xx and t neutrino find HEP: - 3 17 in a theoretical model of elementary particles [ µ in 1962, Ziro Maki, Masami Nakagawa and Shoichi ν µ INSPIRE ↔ ν e systems, and postulated a possible transition ν e . As the running masses of quarks and charged leptons are ν τ - e m ν  Drivenexperiments(peaks)by µ and 0 m 1. 2. Theoristshardworking(papers) (Patience)discoveredbeto3. More K Number of Papers Number -  0 e K 0 m 800 600 400 200 1000 1800 1600 1400 1200 and b m  The number of neutrino papers has been shown as a function of the year of publication, where the s m Most of the above theoretical ideas were indispensable for us to explain the experimental In the Standard Model of elementary particles (SM), there are three generations of massless Both quarks and charged leptons take on strong mass hierarchies, namely,  d Nowadays, the lepton flavor mixing matrix1978, is named Lincoln after Wolfenstein those realized four that theoristscould the as significantly PMNS coherent modify matrix. neutrino forward In oscillation scattering phenomenaAlexei of Smirnov [ neutrinos discovered that with such matter aing modification could and enhance lead resonantly to neutrino a flavor mix- substantial flavor conversion [ m observations and understand the fundamental properties ofmany neutrinos. recent In ideas this about talk, the I origin shall ofefforts summarize are neutrino now masses being and devoted flavor to mixing. answeringsome While a of great number experimental those of ideas basic will questions prove in to neutrino be physics, important I and hope more new ideas are2. to come. Fundamental Properties of Massive Neutrinos neutrinos, participating in the weak interactions,charges. and Now they we are know, of the spin onlyexists. one-half difference and Hence, is have I that no first neutrinos electric present are our massive current and knowledge on lepton neutrino flavor masses mixing 2.1 and some Neutrino related Mass issues. Odering matter effects help us solve the solar neutrino problem via neutrino oscillations. Even without knowing the discovery of Figure 1: data have been obtained from INSPIRE-HEP by searching for the keyword "neutrino" in titles. the mixings of neutral Theoretical Results on Neutrinos Sakata predicted the flavor conversion PoS(LeptonPhoton2015)017 | ). 3 ≈ × 2 m 3 5 . m 3 (2.1) 2 e m 2 , / < U 2 GeV. 2 . 2 = 1 3 are the 1, which + , − m 2 1 Shun Zhou m 2 91 2 m ≈ , 10 m < 1 = · 2 3 ]. In the near 2 − ]. For a recent e 3 7 5 Z . . m 2 3 = 0 and 0 0 U m 20 / i ], one can obtain 24 M m + − 2 + ≈ 0 18 1 . m ≡ for 2 2 m m ≈ ei 1 2 31 2 e ≈ / 2 U m 1 U , where the uncertainties s b m ∆ 2 m m m / − ≡ | 1 Xe nuclei. In addition, the m 10 , and from Ref. [ ββ · 136 2 , where 2 Z 9 m 2 3 − . M m 5 eV 2 10 | 5 · ≈ 3 − 8 4 e eV at the same C.L. [ . . τ 1 2 ) U | 10 m − + ) or inverted (i.e., IO for / 9 64 3 + . × . µ 4 2 2 m 0 5 m . m 7 ≈ < 2 ··· | 2 s d 2 = and e 22 m m 4 m . 2 1 3 U 0 | m − < ( , + 1 − 10 3 eV at the 90% C.L., while that by the KamLAND-Zen < 2 1 m · 2 2 − ) m 7 m eV, where the wide ranges are caused by the uncertainties 2 ββ . 10 | 2 eV at the 95% confidence level (C.L.) [ 69 ) 4 . . 1 · m ≡ e 2 0 5 3 . . 52 ≈ U . 0 0 2 21 | < 0 µ ··· − + m for IO) is still allowed to be zero, the neutrino mass hierarchy in 7 β q m . ∆ 20 ··· / 3 m 3 . e ≡ 0 m 15 ( A and LBNF/DUNE), reactor (JUNO and RENO-50), atmospheric m β , in which the effective neutrino mass ≈ . ν − t 0 m c < e ( m m 2 ]. < ββ , 19 17, where )+ m . ββ 3 A ]. 0 − , for NO or m 2 ] is 1 25 ≈ ]. 10 + · 2 m 22 ] is / 21 8 1 Z 1 . . ( 0 1 ) 23 − + N 2 31 extracted from neutrino oscillation experiments have been used. Although the lightest 2 . m 2 → 2 2 eV [ ∆ . ) / 0 eV ≈ A Second, if neutrinos are Majorana particles, useful constraints on the absolute scale of neutrino It seems plausible that neutrino mass ordering is normal, as we have seen in the case of charged Currently there are three practical ways to constrain absolute neutrino masses. The first one 2 21 , 3 c u < Z m − m m Ge-based experiment GERDA gives ( β ∆ experiment [ future, the KATRIN experiment will improvem this upper limit by one order of magnitude, namely, represents the contributions from light-neutrino exchanges.Collaboration The [ upper bound reported by the EXO review, see Ref. [ masses can also be obtainedN from the experimental searches for neutrinoless double-beta decays Using the latest values of running masses of charged fermions at and the charged-lepton ones in the evaluation of76 nuclear matrix elements assosciated with the fermions. Furthermore, if neutrino masses are also hierarchical, we get Theoretical Results on Neutrinos directly related to the Yukawamake coupling a comparison constants among in them the at a fundamental common theory, energy it scale, is e.g.,the the meaningful quark Fermi to mass scale ratios for charged-lepton mass ratios aretheir negligible. mass ordering But is for normal neutrinos, (i.e., NO it for is not yet determined whether The accelerator (T2K, NO (PINGU, ORCA, INO and Hyper-Kamiokande) neutrinoresolve oscillation this experiments are problem able [ to finally ( first-row elements of the PMNS matrix. Theon Mainz the and effective Troitsk experiments neutrino have mass set an upper limit either NO or IO caseIt is is not following also the possible strongly that hierarchial neutrino pattern of masses charged-fermion are masses. nearly degenerate, i.e., 10 neutrino mass ( is to measure precisely thethe energy region spectrum close of to electrons the from endpoint.the tritium effective From beta neutrino the mass decays, observed particularly spectrum, in one can extract the information on are completely different from the charged-fermion massmass ratios. ordering and Thus, the the absolute determination mass of scale neutrino of are fermion of mass crucial importance spectra to and achieve reveal a the unified underlying description symmetry between quarks and2.2 leptons. Absolute Neutrino Masses PoS(LeptonPhoton2015)017 23 eV . 2 0 Shun Zhou < Σ

Cosmological Bound [eV] ퟑ 풎 and neutrinoless double- decays + . The latest observation β β 3 ퟐ - [eV] , then the rate of neutrino- m 풎 푖 O ββ I + + 푚 2 m 2 ퟏ 푒푖 double m 푈 풎 푖 O . The lepton number is accidental- Σ + = N R 1 ν 휮 = C m R 훽 ν 훽 ≡ R 푚 M Σ Neutrinoless 5 ], which is accurate enough to observe a positive signal CDM cosmology, the future experiments will be able to Λ 28

Cosmological Bound [eV] ퟑ ]. In the 풎 016 eV [ . [eV] + 27 0 2 푖 O ퟐ N 푚 O 풎 I 2 ) = + Σ 푒푖 ( ퟏ 푈 σ 푖 풎 ]. However, the cosmological bound actually depends on the chosen data sets decays (KATRIN) decays Σ = β 26 휮 = , where current and future experimental constraints are indicated by shaded areas. 훽 ββ 푚 Tritium Tritium The allowed ranges of the effective neutrino masses in beta decays m are singlets under all the SM gauge symmetries, so one has to enforce the conservation of R ν In the Majorana case, lepton number is no longer a good quantum number, and it is unneces- At present, only the neutrinoless double-beta decays are a feasible way to demonstrate the Third, the precise measurements of cosmic microwave background and large-scale structure Massive neutrinos can be either Dirac or Majorana particles. In the Dirac case, both neutrinos sary to distinguish between neutrinosneutrinos and take antineutrinos. part in Both the left-handedanti-leptons weak via and the interactions, right-handed charged-current and light interaction, are respectively.rana As produced neutrinos we together can shall with be discuss realized later, charged light in leptons Majo- a and class of seesaw models. Majorana nature of massive neutrinos.in Nevertheless, a if special region neutrino such mixing that parameters significant happen cancellation to takes be place in ly conserved in the SMneutrinos, at the the smallness classical of level, but theirbe is masses over anomalously requires twelve violated. orders extremely of tiny Furthermore, magnitude Yukawa forthe couplings, smaller Dirac strong than which hierarchy the should puzzle top-quark Yukawa of coupling. fermion This masses. exaggerates less double-beta decays will bedecisive highly conclusion suppressed. on Dirac In or this Majorana nature case, of it neutrinos becomes in impossible the to near future. make a of nonzero neutrino masses and even sensitive to neutrino mass ordering, as shown in Fig. lepton number in order to forbid a Majorana mass term formation are sensitive to the sum of neutrino masses and the model of cosmology [ 2.3 Type of Neutrino Masses and antineutrinos have left-handed and right-handedtrinos components. However, the right-handed neu- beta decays m Figure 2: reach a sensitivity of Theoretical Results on Neutrinos by the Planck Collaboration, together with Baryon Accoustic Oscillations, leads to at the 95% C.L. [ PoS(LeptonPhoton2015)017 ] ◦ τ ]. - 34 45 µ 36 , (2.2) = are the 14 ]. If the 23 , δ θ Shun Zhou 35   ]. Although ]. If the 29 symmetry with 30 , and 158 793 776 ]: (1) . . . τ 23 0 0 0 - 3 [ θ 31 , µ 2 ··· ··· ··· . Moreover, they can , j 1 m dipole moments can be 137 614 590 = . . . , which is very promising i > i j ◦ is the Bohr magneton [ i µ can be extracted from the d- e m for 270 m δ | i 2 0 has already been excluded by 580 0 699 0 713 0 τ ∼ for / . . . e U 0 0 0 γ δ = | , the best fit to oscillation data points ≡ ]. + ··· ··· ··· σ = 13 j B | θ 32 i ν µ µ 514 441 464 . . . and magnetic U → | i i j ν ε symmetry. Although a maximal mixing angle . Since τ , where ◦ - 6 845 0 517 0 529 0 B . . . µ µ 0 0 0 ) , their results are perfectly consistent with each other symmetry ··· ··· ··· δ ]. The Dirac CP phase or 270 τ - 1 eV ◦ 33 . µ 801 225 246 0 . . . 90 / 0 0 0 ] i = m   29 ( δ = 20 − exists in the PMNS matrix [   | and | | | 10 2 coincides with the 3 3 3 τ ◦ e τ µ ◦ × U U | U U 45 3 = 270 = = | | | | | | | i 2 2 ∼ 2 2 ν 23 µ e τ µ µ θ δ U U U U | . However, the transitional electric or C i | | | | | | | . If such a deviation is indeed confirmed, we expect that a partial 1 ν 1 1 1 ◦ e τ 0; (2) µ τ level. The determinations of neutrino mass ordering, the octant of is compatible with oscillation experiments within 1 U = U U 45 U | | | and the Dirac CP-violating phase ◦ = | i σ ν 6=   45 23 = 13 Another issue associated with massive neutrinos is the electromagnetic properties [ Apart from neutrino mass ordering, leptonic CP violation in neutrino oscillations is the main With the help of the global-fit results, one can figure out the allowed ranges of the absolute One can find the latest global-fit analysis of neutrino oscillation data in Refs. [ θ | θ 23 = 1 θ µ 23 U ifference between neutrino andeffects antineutrino induce oscillation fake CP probabilities, violation although inments. the the For long-baseline Majorana MSW neutrinos, accelerator one matter and ultimately atmosphericCP-violating unavoidable neutrino question phases. is experi- how Neutrinoless to double-beta probe two decays,and Majorana other neutrino-antineutrino related oscillations lepton-number-violating [ processes couldMajorana provide nature a of clue. massive But neutrinos robustexperiments, should evidence rendering first for the be determination found of Majorana in CP the phases neutrinoless necessary. double-beta decay 2.5 Electromagnetic Properties of Neutrinos SM is extended to accommodate massive Diraccan neutrinos, the be magnetic calculated dipole moment of neutrinos For Majorana neutrinos, the magnetic momentsdition turn out to be vanishing due to the neutrality con- interact with external electromagnetic fields,electromagnetic and vertex. electrons with additional contributions from the task for future oscillation experiments [ θ values of PMNS matrix elements [ three different groups have found distinctular best-fit values of neutrino mixing parameters, in partic- primary goals of ongoing and forthcoming neutrinothat oscillation there experiments. exists It a is weak worth mentioning hintfor a on direct a discovery maximal of CP-violating leptonic phase CP violation in the foreseeable future. from which we can observesymmetry a is possible preserved, the flavor mixing angles are restrictivelythe constrained reactor [ neutrino experiments, we arethe left weak with hint only on the second possibility. It is interestingto that | nonzero for both Diracmoments, and neutrinos Majorana become neutrinos. unstable and Because decay of via transitional electromagnetic dipole at the 3 Theoretical Results on Neutrinos 2.4 Flavor Mixing and CP Violation and PoS(LeptonPhoton2015)017 from B µ Shun Zhou 11 − 10 × . First, the elastic 9 . 2 | 2 i j ε gauge symmetry, there < | Y + eff ) 2 µ 1 | ( i j U µ | × q L to generate tiny neutrino masses of ) 2 = ) ( L C ` eff · SU µ T ˜ H )( ˜ H · ]. L , for a partial list of neutrino mass models. Since 7 ` 3 39 ( ]. 38 [ representation of the ) B 1 µ − 12 ]. Second, the plasmon decay into neutrino-antineutrino pairs dom- , − 2 37 ( 10 × 0 . 3 . eff µ belong to the ˜ H An incomplete list of neutrino mass models, where the generated neutrino mass matrix has been and L ` Now that neutrinos are massive particles, one immediate question is how to extend the SM As pointed out by Weinberg, one can regard the SM as an effective theory at the electroweak The implications of neutrino electromagnetic interactions for particle physics and astrophysics are three distinct ways to construct a renormalizable full theory at a superhigh-energy scale: Majorana type. In this case,new the high smallness of energy neutrino or masses mass canboth scale. be See, ascribed to e.g., the Fig. existence of a and generate tiny neutrinomixing masses angles in in the a lepton sector naturalI are introduce way. so neutrino different mass from models Another and those important the in approaches the question implemented quark to is sector. explain In3.1 why lepton the flavor flavor Canonical mixing. following, Seesaw Models scale, and introduce a dimension-five operator 3. Origin of Neutrino Masses and Flavor Mixing inates the neutrino productionrequirement in of white no dwarfs extra or energyrestictive limit losses the via cores neutrinos of in globular-cluster the red globular giants. cluster gives rise The to the most Figure 3: given below the corresponding Feynman diagram [ the GEMMA Collaboration [ lead to stringent bounds on the effective dipole moment Theoretical Results on Neutrinos (anti)neutrino-electron scattering receives the electromagneticone. contribution in The addition best upper to limit the from SM this kind of laboratory experiments is PoS(LeptonPhoton2015)017 . , ] R 3 T Σ ν 46 Y 1 − Σ M plays the Σ Shun Zhou Y R 2 i Σ H ], Type-II [ 45 −h . Light neutrinos , which is coupled ) = are coupled to both H 174 GeV. 2 ∆ ν ) − 2 gauge theory of scalar 0 , M , σ i ≈ ]. 3 ) i 1 ( 1 T H ( 43 ( ∼ H ∼ U ∆ ∆  h µ R ) are coupled to both lepton and ν R ) M 0 ( , 3 O ( GeV. Current data require at least two after the spontaneous symmetry breaking is allowed. The Majorana mass matrix of ∼ 16 for ∆ R R T v ]. The main motivation to consider a scale- ν 10 ν Σ is the Yukawa coupling matrix. Now that ∆ Y C R 1 Y 49 ν ∼ ν − R Y R is the Higgs triplet mass. = 8 M ν M ∆ ν GUT M Y M Λ 2 i and to Higgs doublets , where H L C R ]. Possible solutions to those problems are either to seek ` ν 2 −h 44 ]. In the latter case, new heavy particles running in the loops σ , where H i i = L 48 ∆ ` H ν , and light neutrino masses are given by L ν ˜ ` Y M H ∆ R Y  h Σ 1 2 2 ∆ L ` ]: Three triplet fermions M Σ / ]: The SM is extended with a Higgs triplet ]: Three right-handed neutrino singlets Y 2 42 i 41 40 in the type-I seesaw model. H h R ∆ ν µ is the fermion triplet mass. In this case, the neutral component of ] seesaw models, and set restrictive limits on the model parameters. ≈ Σ 47 ∆ M v = ], if all the parameters of mass dimension are eliminated, the theory becomes classically i ∆ 50 Type-II Seesaw [ light neutrinos is given by Higgs doublets via Type-III Seesaw [ to lepton doublets via Type-I Seesaw [ lepton and Higgs doublets are gauge singlets, a Majorana mass term acquire a tiny Majorana mass term with h where same role as In canonical seesaw models, the suppression of neutrino masses is achieved by the introduction However, such a high-scale seesaw model suffers from two potential problems: (1) The new Instead of presenting a concrete example, I make some comments on radiative neutrino mass • • • and Type-III [ invariant extension is to solve the gaugetroweak hierarchy scale problem gets of the huge SM. correctionsthe The Higgs from Planck mass new scale. or physics the elec- As atfields demonstrated a [ by superhigh-energy Coleman scale, and for Weinberg instance, in a 3.2 Radiative Mass Models of a high energy or mass scale.in a In class contrast, of tiny radiative neutrino mass masses models can [ arise from loop corrections, as a seesaw scenario in the TeVIn region, fact, or the to ATLAS generate and neutrino CMS massesdedicated collaborations in searches at a for completely the lepton-number-violating different CERN signals way. Large in Hadron the Collider low-scale have Type-I performed [ can be around or evensmall below rather the than TeV finely scale, tuned. when the Two typical relevant examples coupling are constants given are in reasonably the last two diagrams in Fig. particles introduced in canonical seesaw modelsterrestrial are experiments; so (2) heavy The radiative that corrections they tothe cannot the hierarchy be Higgs directly or mass tested turn naturalness in out problem to [ be large, causing models in the scale-invariant extension of the SM [ Now neutrino masses are actually suppressedAn by intriguing feature a of loop radiative factor mass and modelssymmetries, is small collider the dimensionless signals possible couplings. and connection dark to matter, neutrino implying masses, a flavor very rich phenomenology. Theoretical Results on Neutrinos It seems natural that the massscale scale of of grand new unified particles theories insinglet (GUT’s), the (triplet) i.e., canonical fermions seesaw in models the is Type-I closeintegrate (Type-III) to seesaw heavy the models. particles At out the and low-energymodels obtain scale, is the one to Weinberg can explain operator. the Another matter-antimatter asymmetry salient via feature leptogenesis of [ seesaw PoS(LeptonPhoton2015)017 τ C ]. τ of - ν and µ r 52 (3.1) ) → x . More µ − 13 , namely, ν and other , ]. θ Shun Zhou d t , 4 M C 56 e A ν = ( that leads to a is satisfied, the 0 ν 1 x → − G r e X ν ) → 0 g f are arbitrary. The . Regarding this point, ( is fixed by the Clebsch ∗ δ r G , where 13 ) ρ 0 12 r θ x α X ( and ∗ l ϕ and r ) = G , g X ]. To be explicit, given the fields 12 ( where θ → r        55 → ρ f C 2 2 ) 0 1 1 θ G x √ √ ( 12 ϕ α , the symmetry group of a tetrahedron [ , while that is invariant under 3 4 ◦ 3 3 ≈ 1 ]. In the quark sector, strong mass hierarchies ]. In the basis where the charged-lepton mass ν A 1 1 √ √ √ d M 12 57 54 − θ 9 ]. Hence the flavor models based on 12 or 270 8 6 α ◦ 6 6 1 2 1 √ ≈ 0 and trivial CP-violating phases. The essential idea is 90 √ √ − on the generic Lagrangian, and the symmetry is broken ` 12 = = 0 from the TB mixing pattern has already been excluded f and the Dirac CP-violating phase        θ δ 13 G = in the charged-lepton and neutrino sectors, respectively. It is = 13 θ θ , ν 13 ◦ and TB G θ U ◦ . If the consistency condition 45 f can be established [ , have been implemented to explain the observed lepton flavor mix- ) 45 = C G and θ l 27 is a prerequisite for leptonic CP violation, so it is intriguing to construct = 23 ( ] G θ ∆ 13 23 , 53 θ ◦ θ 3 . . On the other hand, the down-type quark mass matrix is related to the charged- and C 0 35 θ T ]. , = ≈ in GUT’s, so we have 5 s leads to 49 ` A 12 m , θ C µ M / 4 is diagonal, the neutrino mass matrix ν d S ` , m reflection symmetry has to be mentioned [ 4 → M ]. A paradigm for model building is based on to the Cabibbo angle A τ p τ - , , the generalized CP transformation is defined by Finally, when the flavor symmetry is brought into the GUT framework, a possible connection The mechanisms for neutrino mass generation are usually not responsible for the lepton flavor However, the prediction of 51 ν 3 13 ) µ ≈ is the matrx of transformations associated with the fields in the irreducible representation S x θ ( d r 12 and reflection symmetry is actually a special caseordinary of CP the transformation generalized CP and symmetry, which a combines discrete the flavor symmetry [ lepton one ϕ indicate that the dominant mixing angleθ stems from the down-type quark mass matrix matrix of This was particularly interesting whenmaximal neutrino (TB) oscillation mixing [ data were favoring the so-called tribi- discrete symmetries need to be considerablyimportantly, changed a to nonzero accommodate a relatively large 3.3 Discrete Flavor Symmetries mixing pattern. In the past decade,as a great number of non-Abelian discrete flavor symmetries, such a flavor model to account for both ing [ Theoretical Results on Neutrinos scale-invariant and the spontaneous gaugerections. symmetry breaking Consequently, a can mass beparameters scale triggered is left. emerges by Along and radiative this cor- only line,dimensionless one a couplings can logarithmic and construct the scale a vacuum radiative dependence expectation neutrinoin values mass of of the model mass scalar theory by fields [ using come only in as mass scales resulting in the difference between the symmetrynontrivial breaking PMNS patterns matrix. of the X the discrete flavor symmetry to impose a global flavorinto residual symmetry symmetries by the Daya Bay reactor neutrino experiment [ generalized CP symmetry and discrete flavor symmetrypredicitions can for be CP integrated violation into are a full then symmetry. dependent The on how the full symmetry is broken [ PoS(LeptonPhoton2015)017 , TB U ]. can be = 60 s [ ν θ Shun Zhou U denotes an 11 − a ν 10 × 9 . 4 , where = γ s θ + 2 a 2 ν . One possible interpretation , which is in perfect agreement s σ → as predicted by the TB mixing. m s ◦ / ν 3 d . m 35 p = ≈ ν 12 C θ θ 10 possesses six independent complex elements, which ], where ν M 57 ]. The first evidence comes from the stacking X-ray spectra ]. However, as we have mentioned before, the mass hierarchy 60 58 ]. As a consequence, one can determine the absolute neutrino mass 59 can also be derived [ δ 2. In addition, a correlation among neutrino mixing angles and CP-violating phase cos √ 13 / θ C θ + ≈ ν 12 θ 13 It should be noticed that the approaches of texture zeros are intimately related to those of dis- In the last part of my talk, I turn to the recent progress in the studies of astrophysical neutrinos, In 2014, two independent groups discovered an X-ray line around 3.55 keV in the dark-matter In most models of non-Abelian discrete flavor symmetries, the flavor mixing angles are in In the flavor basis where the charged-lepton flavor eigenstates coincide with their mass eigen- θ in the PMNS matrix receives the contributions from both charged-lepton and neutrino sectors, = 12 13 active neutrino. These decays takeneutrinos place decay if radiatively there at exists an adetermined sterile-active extremely from neutrino small the mixing observed rate. and strength of active The the sterile-active X-ray mixing line, angle implying sin is the radiative decays of a sterile neutrino of 7.1 keV mass of central parts of 81telescopes. galaxy clusters, The which second havethe one been new observed is by blank-sky found XMM-Newton dataset in andthe observed the Chandra global by nearby significance XMM-Newton. Andromeda of galaxy, For the the each Perseus X-ray group cluster line with and observation different is datasets, above 4 crete flavor symmetries. Both Abelianzeros and in non-Abelian the symmetries lepton can mass be matrices used in to the realize cases texture either with or4. without seesaw Recent mechanisms. Progress on Astrophysical Neutrinos including the keV sterile neutrinos, supernovaparticular neutrinos emphasis and will ultrahigh-energy be cosmic placed neutrinos. onastrophysical A how environments. to investigate the fundamental properties of neutrinos in 4.1 keV Sterile Neutrinos dominated astrophysical objects [ scale and three CP phasesneutrino mass-squared in differences. terms Currently, of seven two-zero preciselydata, textures and measured survive will three neutrino be oscillation neutrino tested by mixing the angles precision and measurements of two neutrino mixing parameters. general decoupled from the lepton masses.usually imply In the contrast, relationship the texture amongthe zeros flavor texture in mixing zeros fermion angles in mass and quark matrices fermion mass mass matrices ratios. give rise In to particular, 3.4 Texture Zeros can be reconstructed from three mixing angles,es. three mass If eigenvalues and two three independent CP-violating phas- matrixneutrino elements mixing are parameters taken [ to be zero, there are four real constraints among states, the Majorana neutrino mass matrix of neutrinos is not asbe strong associated as with that large neutrino of mixing charged angles. fermions. The weak neutrino mass hierarchy may θ Theoretical Results on Neutrinos factors. If the flavor symmetry is furtherθ utilized to gurantee the TB mixing fori.e., neutrinos with experimental observations [ PoS(LeptonPhoton2015)017 and a flux Shun Zhou ]. α 64 − E ]. Suppose that 68 ]. In the supernova 63 1 : 2 : 0 and modified by can exchange their spectra = 0 τ τ ν φ ]. In this case, active neutrino : 0 µ 61 and φ ]. Here I focus on how to constrain : e is mediated by a light scalar particle ν 0 e 67 φ g 3 for the energy spectrum . level. Assuming a diffusive astrophysical 0 ± σ 7 3 . . 2 11 = ], or by some other mechanisms. α 62 ]. For now, it remains to see whether the flavor instability is 65 per flavor. The flavor composition at the detector is consistent with 1 − sr ]. 1 66 − s varies from keV to GeV. Requiring that the ultrahigh-energy cosmic neutrinos 2 φ − m 1 : 1 : 1, as predicted by an initial flavor ratio -scale neutrino telescope, IceCube, has recently reported the detection of 37 neutrino 3 = ]. For instance, in the two-flavor approximation, τ 63 GeV cm φ : 8 µ − The origin of ultrahigh-energy cosmic neutrinos could be astrophysical objects, such as Active The km In a dense neutrino gas, neutrinos of different energies are coupled together via neutrino- The latest development in this area is the discovery of spontaneous symmetry breaking, which If this X-ray line is confirmed by future observations, one has to embed the keV sterile neutrino The flavor conversions of supernova neutrinos are complicated by the interaction of neutrinos φ : , whose mass e the neutrino-neutrino interaction with aφ coupling constant Galactic Nuclei, Gamma-Ray Bursts and Starbust Galaxies [ reached in a real supernova environment. 4.3 Ultrahigh-energy Cosmic Neutrinos events in the energy range frommagnetic 28 or TeV hadronic to cascades 2 while PeV, among 9neutrino which events 28 as background events muon are is tracks. identified already The as hypothesissource, exlcuded electro- of one at only can atmospheric obtain the a 5 power-law index of neutrino interaction and they couldtive oscillate oscillations cooperatively of over a supernova wide neutrinosspectra range have [ of been energies. found to Collec- considerably modify neutrino energy means that the inital conditionsfor respect symmetry-breaking a solutions presumed [ symmetry but the equations of motion allow envelope, where the matter density iscorresponding appropriate to for two neutrinos neutrino to mass-squared experience differences. thedensities MSW Further become resonances inward, very both large. neutrino While and theneutrino matter dense gas matter induces tends nonlinear to refraction suppress effects the via the mixing self-interaction angle, of the neutrinos dense [ above a critical energy, below whichinitial neutrino neutrino flavor spectra, conversions multiple never happen. spectralstability splits Depending have are on been also the explored possible. in The assumptionaxial conditions symmetry of about for spherical the the symmetry radial flavor about direction. in- the supernova center and neutrino oscillations [ the secret neutrino interactions with the help of these high-energy neutrinos [ φ into the theory of massive neutrinos.can In the be type-I a seesaw model, good the candidate lightest for right-handed neutrino the 7.1 keV-mass sterile neutrino [ with dense background particles, which has attracted a lot of attention [ Theoretical Results on Neutrinos masses receive the contributions only from twoangle much heavier between right-handed keV neutrinos, as sterile the neutrinos mixing the and production of active keV neutrinos sterile is neutrinosremarkable extremely should primordial small. be lepton aided asymmetry On by [ resonant the MSW other effects, hand, which require a 4.2 Supernova Neutrinos of 10 PoS(LeptonPhoton2015)017 , et al. Shun Zhou , 1562 (1998) , 161 (1934). 81 88 , 1490 (1987); R. M. Bionta , 984 (1968). , 1205 (1968). 58 26 20 , 1209 (1968). , 171803 (2012) [arXiv:1203.1669]. , 052008 (2012) [arXiv:1207.6632]. , 011301 (2002) [nucl-ex/0204008]. , 870 (1962). 20 , 191802 (2012) [arXiv:1204.0626]. 86 89 ]. 28 108 108 69 , 1 (1934); Z. Phys. 11 12 , 36 (1962). , 429 (1957); Sov. Phys. JETP 9 6 , 589 (1987). , 103 (1956). 45 , 171 (1937). 124 , 1184 (1939). 14 56 MeV. These limits complement those from Supernova 1987A, Big Bang , 491 (1933); Nuovo Cim. [SNO Collaboration], Phys. Rev. Lett. 4 , 1494 (1987); E. N. Alekseev, L. N. Alekseeva, V. I. Volchenko and ] , Science [Super-Kamiokande Collaboration], Phys. Rev. Lett. 58 10 [RENO Collaboration], Phys. Rev. Lett. [Kamiokande-II Collaboration], Phys. Rev. Lett. , Phys. Rev. Lett. , [Daya Bay Collaboration], Phys. Rev. Lett. et al. 1 et al. [Double Chooz Collaboration], Phys. Rev. D [ et al. ∈ et al. et al. et al. et al. φ et al. m over Phys. Rev. Lett. I. V. Krivosheina, JETP Lett. [hep-ex/9807003]. Over half a century, our knowledge about the fundamental properties of neutrinos has been I would like to thank Prof. Peter Krizan and Prof. Marko Mikuz for their kind invitation to g [5] K. Hirata [1] C. L. Cowan [2] G. Danby [3] R. Davis, Jr., D. S. Harmer and K. C. Hoffman, Phys. Rev. Lett. [4] J. N. Bahcall, N. A. Bahcall and G. Shaviv, Phys. Rev. Lett. [8] F. P. An [9] J. K. Ahn [6] Y. Fukuda [7] Q. R. Ahmad greatly extended. The most important informationflavors is that are neutrinos are mixed, massive particles as andfuture, firmly lepton neutrino established mass in ordering, neutrinoabsolute leptonic oscillation neutrino CP mass experiments. violation, scale the will Ininformation Majorana provided hopefully the by nature be astrophysics foreseeable of fixed. and cosmology, neutrinos weneutrinos With and expect and this that lepton the tremendous a flavor complete mixing progress theory will and of emerge further massive eventually. Acknowledgements give a review talk atZhao this and Dr. conference, Ye-Ling and Zhou for Prof. their great Zhi-zhong help Xing, in preparing Dr. thisReferences talk. Jue Zhang, Dr. Zhen-Hua 5. Summary and Outlook Nucleosynthesis, and Cosmic Microwave Background [ Theoretical Results on Neutrinos are not remarkably absorbed byon the cosmic neutrino background, one can derive stringent limits [10] Y. Abe [11] E. Fermi, Ric. Sci. [12] E. Majorana, Nuovo Cim. [13] W. H. Furry, Phys. Rev. [14] B. Pontecorvo, Sov. Phys. JETP [15] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. PoS(LeptonPhoton2015)017 , et al. , 3986 , JHEP Shun Zhou 57 , 17 (1986). 9 et al. [Troitsk et al. , 062502 (2013) , 515 (2012) [arXiv:1111.5332]. 110 , 115 (2013) [arXiv:1306.6879]. , 122503 (2013) [arXiv:1307.4720]. , 2856 (1990); G. Raffelt and 84 , 442 (2010) [arXiv:0810.3281]. 64 111 1311 , 608515 (2012) [arXiv:1212.6154]. , 093004 (1998) [hep-ph/9802383]; 623 , 913 (1985); Nuovo Cim. C 58 , 229 (2014) [arXiv:1402.6956]. 42 2012 , 1530001 (2015) [arXiv:1411.4791]. 510 , 013013 (2012) [arXiv:1112.3112]; Phys. Rev. D 30 , 531 (2015) [arXiv:1403.6344]; C. Giunti 86 , 350150 (2012). , 1666 (1981); L. F. Li and F. Wilczek, Phys. Rev. D , 963 (1980); B. W. Lee and R. E. Shrock, Phys. Rev. 13 87 23 45 , 5153 (2014) [arXiv:1311.3846]. 2012 5 , 66 (2015) [arXiv:1309.5383]. , 196 (2014) [arXiv:1404.7021]. S. Luo and Z. Z. Xing, , 166 (2008) [arXiv:0804.3512]. 63 , 052 (2014) [arXiv:1409.5439]; J. Bergstrom 737 666 , in the proceedings of this conference. 1411 , 447 (2005) [hep-ex/0412056]; V. N. Aseev , 093018 (2014) [arXiv:1312.2878]; D. V. Forero, M. Tortola and , 536 (1992). , 2369 (1978). , 112003 (2011) [arXiv:1108.5034]. , 559 (1990); Phys. Rev. Lett. 40 89 , 093006 (2014) [arXiv:1405.7540]; M. C. Gonzalez-Garcia, 264 17 84 90 365 , 053019 (2013) [arXiv:1301.7654]; Z. Z. Xing and Y. L. Zhou, Phys. 87 , Astropart. Phys. [Planck Collaboration], arXiv:1502.01589. [GERDA Collaboration], Phys. Rev. Lett. [EXO-200 Collaboration], Nature , Adv. High Energy Phys. et al. , Phys. Rev. D , 073005 (2014) [arXiv:1408.5005]; Y. L. Zhou, arXiv:1409.8600. [KamLAND-Zen Collaboration], Phys. Rev. Lett. , Eur. Phys. J. C 90 et al. et al. Future Neutrino Program et al. et al. et al. et al. , 033002 (2013) [arXiv:1305.5718]. et al. 88 , 200 (2015) [arXiv:1507.04366]. , 1444 (1977). , 143 (1982); P. Langacker and J. Wang, Phys. Rev. D , 113016 (2008) [arXiv:0712.1419]; See, also, H. Fusaoka and Y. Koide, Phys. Rev. D 16 arXiv:1506.05387. D A. Weiss, Astron. Astrophys. Rev. D 25 Z. Z. Xing, Phys. Rev. D Phys. Rev. D 1509 J. W. F. Valle, Phys. Rev. D M. Maltoni and T. Schwetz, JHEP Collaboration], Phys. Rev. D [arXiv:1211.3863]. 77 (1998) [hep-ph/9712201]; S. Antusch and V. Maurer, JHEP [38] G. G. Raffelt, Astrophys. J. [39] T. Ohlsson and S. Zhou, Nature Commun. [36] K. Fujikawa and R. Shrock, Phys. Rev. Lett. [35] C. Giunti and A. Studenikin, Rev. Mod. Phys. [37] A. G. Beda [34] J. Schechter and J. W. F. Valle, Phys. Rev. D [33] G. C. Branco, R. G. Felipe and F. R. Joaquim, Rev. Mod. Phys. [32] Z. Z. Xing and S. Zhou, Phys. Lett. B [30] T. Fukuyama and H. Nishiura, hep-ph/9702253. [31] Z. Z. Xing and S. Zhou, Phys. Lett. B [25] S. M. Bilenky and C. Giunti, Int. J. Mod. Phys. A [28] K. N. Abazajian [29] F. Capozzi [27] J. Lesgourgues and S. Pastor, Adv. High Energy Phys. [26] P. A. R. Ade [24] M. Agostini [18] Z. Z. Xing, H. Zhang and S. Zhou, Phys. Rev. D [19] K. R. Long, [21] J. Wolf [KATRIN Collaboration], Nucl. Instrum. Meth.[22] A J. B. Albert [23] A. Gando [20] C. Kraus Theoretical Results on Neutrinos [16] L. Wolfenstein, Phys. Rev. D [17] S. P. Mikheev and A. Yu. Smirnov, Sov. J. Nucl. Phys. PoS(LeptonPhoton2015)017 , , 45 , 203 et al. et al. Shun Zhou , 061 (2007) , 077301 , 031901 73 70 0712 , 051805 (2009) , edited by P. van , 87 (2003) , 056201 (2013) 102 560 76 , JHEP , 64 (2005) et al. , 165 (1981). Supergravity 720 23 , 2860 (1980); M. Magg and , 113001 (2012) [arXiv:1209.4051]; , 167 (2002) [hep-ph/0202074]; 22 86 , 144 (2015) [arXiv:1501.05566]; , 112 (2009) [arXiv:0905.3903]; , 2189 (2012) [arXiv:1207.2666]; G. Aad Proceedings of the Workshop on Unified 530 72 748 180 , 073009 (2015) [arXiv:1502.01352]. , 105 (2008) [arXiv:0802.2962]. , 461 (1980)]; K. S. Babu, Phys. Lett. B , 045018 (2014) [arXiv:1402.4271]; S. F. King, 91 95 , 177 (2014) [arXiv:1405.6204]. 466 16 , 1888 (1973). , 108 (2015) [arXiv:1506.01291]; G. Aad , 162 (2015) [arXiv:1506.06020]. , 433 (1977); J. Schechter and J. W. F. Valle, Phys. 7 , 163 (2002) [hep-ph/0203209]; Z. Z. Xing, Phys. , 2244 (2012) [arXiv:1210.5070]. , 2701 (2010) [arXiv:1002.0211]; H. Ishimori , 45 (1986); For a recent review on leptogenesis, see, , edited by O. Sawada and A. Sugamoto (KEK, 70 14 1410 (Plenum, New York, 1980), p. 707; R. N. Mohapatra 72 82 535 1509 1507 174 , 113012 (2001) [hep-ph/0106291]; K. S. Babu, E. Ma and , 032001 (2015) [arXiv:1506.01839]. 64 et al. 92 , 1171 (1998) [hep-ph/9805219]; Phys. Rev. D , 912 (1980). 81 44 , New J. Phys. , 207 (2003) [hep-ph/0206292]; E. Ma, Phys. Rev. D , 421 (1977); T. Yanagida, in et al. , 61 (1980); G. Lazarides, Q. Shafi, and C. Wetterich, Nucl. Phys. B , 1 (2010) [arXiv:1003.3552]; W. Grimus and P. O. Ludl, J. Phys. A 67 552 , 441 (1989). , 091801 (2014) [arXiv:1311.3213]. 94 , 7027 (1998) [hep-ph/9709409]; A. Abada 183 44 57 , 389 (1980) [Phys. Lett. B 112 [CMS Collaboration], Phys. Lett. B 93 [CMS Collaboration], Eur. Phys. J. C , edited by M. Levy et al. et al. [ATLAS Collaboration], JHEP [ATLAS Collaboration], JHEP , 123001 (2015) [arXiv:1510.02091]. , Z. Phys. C , 85 (2002) [hep-ph/0204049]; X. G. He and A. Zee, Phys. Lett. B 42 , 2227 (1980); T. P. Cheng and L. F. Li, Phys. Rev. D et al. et al. et al. 533 22 [ATLAS Collaboration], Eur. Phys. J. C , 287 (1981); R. N. Mohapatra and G. Senjanovic, Phys. Rev. D [hep-ph/0301092]. Prog. Theor. Phys. Suppl. 233001 (2012) [arXiv:1110.6376]; S. F. King[arXiv:1301.1340]; and S. C. F. Luhn, King Rept. Prog.J. Phys. Phys. G [hep-ph/0504165]. Lett. B P. F. Harrison and W. G. Scott, Phys. Lett. B J. W. F. Valle, Phys. Lett. B (2004) [hep-ph/0404199]; G. Altarelli and F. Feruglio, Nucl. Phys. B 132 (1988). E. Ma, Phys. Rev.(2006) Lett. [hep-ph/0601225]; M. Aoki, S. Kanemura[arXiv:0807.0361]. and P. S. O. B. Seto, Dev Phys. and Rev.E. Lett. A. Ma, Pilaftsis, Phys. Phys. Rev. Rev. Lett. D et al. [ATLAS Collaboration], Phys. Rev. D G. Aad Theory and the Baryon Number of the Universe [arXiv:0707.4058]; Z. Z. Xing, Prog. Theor.J. Phys. D. Suppl. Clarke, R. Foot and R. R. Volkas, Phys. Rev. D Tsukuba, 1979), p. 95; M. Gell-Mann, P. Ramond, and R. Slansky, in e.g., S. Davidson, E. Nardi and Y. Nir, Phys. Rept. Nieuwenhuizen and D. Z. Freeman (North-Holland,Quarks Amsterdam, and 1979), Leptons p. 315; S. L. Glashow, in Rev. D C. Wetterich, Phys. Lett. B and G. Senjanovic, Phys. Rev. Lett. 181 [53] P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B [52] E. Ma and G. Rajasekaran, Phys. Rev. D [50] S. R. Coleman and E. J.[51] Weinberg, Phys. Rev.G. D Altarelli and F. Feruglio, Rev. Mod. Phys. [49] M. Lindner, S. Schmidt and J. Smirnov, JHEP [47] G. Aad [48] A. Zee, Phys. Lett. B [46] S. Chatrchyan Theoretical Results on Neutrinos [40] P. Minkowski, Phys. Lett. B [45] V. Khachatryan [44] F. Vissani, Phys. Rev. D [41] W. Konetschny and W. Kummer, Phys. Lett. B [42] R. Foot [43] M. Fukugita and T. Yanagida, Phys. Lett. B PoS(LeptonPhoton2015)017 , , 594 74 , 569 Shun Zhou , 136 (2012) 60 1 , 436 (1977). , 255 (2013) , 038701 (2011) 70 , Phys. Rev. Lett. 84 866 , 191 (2009) , Phys. Rev. D , 85 (2002) et al. 59 , 199001 (2013) , 113 (2004) , 1462 (1993); S. Pastor, 539 et al. 111 48 579 , 046 (2014) [arXiv:1311.3873]. , 083 (2011) [arXiv:1108.4534]; , Nucl. Phys. B , 091101 (2013) [arXiv:1305.7140]. , 101101 (2014) [arXiv:1405.5303]. , 79 (2002) [hep-ph/0201008]; 1407 1109 111 113 536 et al. , 1541008 (2015) [arXiv:1506.08629]. , 2753 (2014) [arXiv:1303.7178]; 24 74 , 009 (2011) [arXiv:1108.0614]; S. F. King, 1111 , 073007 (2012) [arXiv:1208.2875]; F. Feruglio, , 053011 (2002) [hep-ph/0109035]; S. Pastor and , 219 (2002) [hep-ph/0210197]; Phys. Lett. B , 189 (1979); 86 65 , 2832 (1999) [astro-ph/9810076]. 15 , 065035 (2014) [arXiv:1404.2288]; K. Ioka and , 2895 (1987); S. Zhou, Phys. Rev. D 547 , 185 (1977); H. Fritzsch, Phys. Lett. B 82 90 155 , 021001 (2015) [arXiv:1312.5222]; G. J. Ding and 36 , 117301 (2011) [arXiv:1107.3728]; D. Marzocca, 38 39 84 , 027 (2013) [arXiv:1211.5560]; M. Holthausen, M. Lindner , 1 (2014) [arXiv:1312.6587]. E. Waxman, arXiv:1511.00815. 1307 , 128 (1992); S. Samuel, Phys. Rev. D , 13 (2014) [arXiv:1402.2301]; A. Boyarsky 1-2 , 122 (2013) [arXiv:1211.6953]. 287 789 , 191101 (2002) [astro-ph/0207281]; H. Duan, G. M. Fuller and , 159 (2002) [hep-ph/0201151]; Phys. Lett. B , 123004 (2006) [astro-ph/0511275]; H. Duan 89 , 093020 (2014) [arXiv:1403.5846]; G. J. Ding and Y. L. Zhou, JHEP 1304 530 74 89 , JHEAp , 061E01 (2014) [arXiv:1404.2279]; [IceCube Collaboration], Phys. Rev. Lett. et al. 2014 , 136 (2013) [arXiv:1205.0506]; S. Antusch , 317 (1978); Nucl. Phys. B , arXiv:1508.00785. , Astrophys. J. et al. 73 718 et al. et al. , 023 (2014) [arXiv:1404.0592]. , 251301 (2014) [arXiv:1402.4119]. [arXiv:1309.0991]; M. Archidiacono and S. Hannestad, JCAP [arXiv:1106.3880]; B. Ahlgren, T. Ohlsson and S. Zhou, Phys. Rev. Lett. K. Murase, PTEP [arXiv:1306.4954]. A. Mirizzi 105014 (2006) [astro-ph/0606616]. G. Raffelt, Phys. Rev. Lett. Y. Z. Qian, Phys. Rev. D 113 [arXiv:0901.0011]; A. Boyarsky, D. Iakubovskyi and O. Ruchayskiy, Phys. Dark Univ. (2010) [arXiv:1001.2799]; H. Duan, Int. J. Mod. Phys. E G. G. Raffelt and D. V. Semikoz, Phys. Rev. D [hep-ph/0205032]; H. Fritzsch, Z. Z. XingS. and Zhou, S. arXiv:1509.05300. Zhou, JHEP Z. Z. Xing, Phys. Lett. B Phys. Lett. B [arXiv:1205.1051]. S. F. King, Phys. Rev. D S. T. Petcov, A. Romanino andPhys. M. Lett. Spinrath, B JHEP 1406 G. J. Ding and Y. L. Zhou, Chin. Phys. C [hep-ph/0305309]. C. Hagedorn and R. Ziegler, JHEP and M. A. Schmidt, JHEP 324 (2004) [hep-ph/0403278]; W. Grimus and L. Lavoura, Phys. Lett. B [69] E. W. Kolb and M. S. Turner, Phys. Rev. D [67] L. A. Anchordoqui [68] K. C. Y. Ng and J. F. Beacom, Phys. Rev. D [64] J. T. Pantaleone, Phys. Lett. B [62] X. D. Shi and G. M. Fuller, Phys. Rev. Lett. [65] G. Raffelt, S. Sarikas and D. de Sousa Seixas, Phys. Rev. Lett. [66] M. G. Aartsen [63] For recent reviews, see H. Duan, G. M. Fuller and Y. Z. Qian, Ann. Rev. Nucl. Part. Sci. [61] A. Boyarsky, O. Ruchayskiy and M. Shaposhnikov, Ann. Rev. Nucl. Part. Sci. [60] E. Bulbul [59] P. H. Frampton, S. L. Glashow and D. Marfatia, Phys. Lett. B [58] S. Weinberg, Trans. New York Acad. Sci. [57] S. Antusch and V. Maurer, Phys. Rev. D [56] F. Feruglio, C. Hagedorn and R. Ziegler, Eur. Phys. J. C [55] R. N. Mohapatra and C. C. Nishi, Phys. Rev. D Theoretical Results on Neutrinos [54] P. F. Harrison and W. G. Scott, Phys. Lett. B