We explore the earth matter effects on the energy spectra of neutrinos from a
supernova. We show that the observations of the energy spectra of $\nu_e$ and
$\bar{\nu}_e$ from a galactic supernova may enable us to identify the solar
neutrino solution, to determine the sign of $\Delta m^2_{32}$, and to probe the
mixing matrix element $|U_{e3}|^2$ to values as low as $10^{-3}$. We point out
scenarios in which the matter effects can even be established through the
observation of the spectrum at a single detector.Comment: 8 pages LaTeX, 2 eps figures, uses Rinton-P9x6.cls. Talk given at
  CICHEP '2001, Cairo, Egypt, January 200

Dighe, Amol S.

English

arXiv

We explore the earth matter effects on the energy spectra of neutrinos from a supernova. We show that the observations of the energy spectra of $\nu_e$ and $\bar{\nu}_e$ from a galactic supernova may enable us to identify the solar neutrino solution, to determine the sign of $\Delta m^2_{32}$, and to probe the mixing matrix element $|U_{e3}|^2$ to values as low as $10^{-3}$. We point out scenarios in which the matter effects can even be established through the observation of the spectrum at a single detector.We explore the earth matter effects on the energy spectra of neutrinos from a supernova. We show that the observations of the energy spectra of $\nu_e$ and $\bar{\nu}_e$ from a galactic supernova may enable us to identify the solar neutrino solution, to determine the sign of $\Delta m^2_{32}$, and to probe the mixing matrix element $|U_{e3}|^2$ to values as low as $10^{-3}$. We point out scenarios in which the matter effects can even be established through the observation of the spectrum at a single detector

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arXiv:hep-ph/0106325v1  28 Jun 2001EARTH MATTER EFFECTS ON THE SUPERNOVANEUTRINO SPECTRAAMOL S. DIGHEDiv. TH, CERN, CH-1211 Geneva 23, Switzerland∗.E-mail: amol.dighe@cern.chWe explore the earth matter effects on the energy spectra of neutrinos from asupernova. We show that the observations of the energy spectra of νe and ν¯efrom a galactic supernova may enable us to identify the solar neutrino solution,to determine the sign of ∆m232, and to probe the mixing matrix element |Ue3|2 tovalues as low as 10−3. We point out scenarios in which the matter effects can evenbe established through the observation of the spectrum at a single detector.1 IntroductionIn the solutions of the solar and atmospheric neutrino anomalies through theoscillations between the three active neutrino species, three types of ambi-guities remain to be resolved: (i) the solution of the solar neutrino problem,(ii) the mass hierarchy – normal (m3 > m1,m2) or inverted (m3 < m1,m2),and (iii) the value of |Ue3|2. The energy spectra of neutrinos from a TypeII supernova contain signatures that can resolve some of these ambiguitiesirrespective of the model of supernova dynamics 1. Here we concentrate onthe effects due to the passage of these neutrinos through the earth which maygive spectacular signatures of some of the neutrino mass and mixing schemes.The neutrinos from the supernova arrive at the earth as an effectivelyincoherent mixture of the mass eigenstates in vacuum. Depending on the rel-ative position of the supernova and the neutrino detector at the time of arrival,the neutrinos have to travel different distances through the earth. The masseigenstates in vacuum then get intermixed and the observed energy spectraof neutrinos are affected. Since the energy range of the supernova neutrinosoverlaps with the energy range for the MSW resonance with ∆m2⊙, the mat-ter effects can become large. In that case, we have distinctive signatures ofsome of the neutrino mixing schemes. The comparison of the neutrino spectraobserved at two detectors can be used to measure the earth matter effects,but in certain situations even the spectrum observed at a single detector canperform the task.2 Neutrino Conversions in the star and the earthThe neutrino transitions between different matter eigenstates inside the su-pernova take place mainly in the resonance regions H and L, which are char-∗Current address: Max-Planck-Institut fu¨r Physik, Fo¨hringer Ring 6, DE-80805Mu¨nchen, Germany.Talk given at CICHEP ’2001, Cairo, Egypt, January 2001. 1acterized by (∆m2atm, 4|Ue3|2) and (∆m2⊙, sin2 2θ⊙) respectively. Due to the∆m2-hierarchy (∆m2atm ≫ ∆m2⊙), the dynamics in each of the two resonancelayers can be considered independently as a 2ν transition 2. The final neutrinofluxes can then be written in terms of survival probabilities p and p¯ of νe andν¯e respectively (apart from the geometrical factor of 1/R2):[FeFe¯4Fx]=[p 0 1− p0 p¯ 1− p¯1− p 1− p¯ 2 + p+ p¯][F 0eF 0e¯F 0x], (1)where Fe(F0e ), Fe¯(F0e¯ ) and Fx(F0x ) are the final (initial) fluxes of νe, ν¯e and νx(each of the non-electron neutrino or antineutrino species) respectively.Let PH(P¯H) and PL(P¯L) be the probabilities that the neutrinos (an-tineutrinos) jump to another matter eigenstate in the resonance layersH and L respectively. Adiabaticity in the resonance layer H dividesthe possible range of |Ue3|2 into three regions 1,3: (I) completely adia-batic transitions, PH <∼ 0.1, |Ue3|2 >∼ 10−3, (II) partially adiabatic transitions,0.1 <∼ PH <∼ 0.9, 10−5 <∼ |Ue3|2 <∼ 10−3, (III) completely adiabatic transitions,PH >∼ 0.9, |Ue3|2 <∼ 10−5. (For inverted hierarchy, PH needs to be replaced byP¯H . The ranges of |Ue3|2 do not change much, however.)The values of p and p¯ are determined by these flip probabilities and themixing matrix elements |Uei|2:p =∑ai|Uei|2 , p¯ =∑a¯|Uei|2 , (2)where ai and a¯i are as given in Table 1.Table 1. The values of the coefficients ai and a¯i for normal and inverted hierarchiesHierarchynormal inverteda1 PHPL PLa2 PH(1 − PL) 1− PLa3 1− PH 0a¯1 1− P¯L P¯H(1− P¯L)a¯2 P¯L P¯H P¯La¯3 0 1− P¯HThe mass eigenstates arriving at the surface of the earth oscillate in theearth matter. Let Pie be the probability that a mass eigenstate νi enteringthe earth reaches the detector as a νe. The flux of νe at the detector isFDe =∑iPieFi = F0e∑aiPie + F0x (1 −∑iaiPie) , (3)where we have used the unitarity condition∑i Pie = 1. Thus, the νe flux atthe detector can be written asFDe = pDF 0e + (1 − pD)F 0x with pD =∑iaiPie . (4)Talk given at CICHEP ’2001, Cairo, Egypt, January 2001. 2Similarly, for the antineutrinos, we can writeFDe¯ = p¯DF 0e¯ + (1 − p¯D)F 0x with p¯D =∑ia¯iP¯ie . (5)The difference in the νe fluxes at the detector due to the propagation in earthequalsFDe − Fe = (pD − p)(F 0e − F0x ) , FDe¯ − Fe¯ = (p¯D − p¯)(F 0e¯ − F0x ) . (6)2.1 NeutrinosIn the case of normal hierarchy, using Table 1 we getpD − p = PH(P2e − |Ue2|2)(1− 2PL) + (P3e − |Ue3|2)(1− PH −PHPL) . (7)The second term in (7) can be neglected. Indeed, inside the earth, ν3 oscillateswith a very small depth:P3e − |Ue3|2 <∼(2EVearth∆m2atm)sin2 2θe3 , (8)where Vearth is the effective potential of νe in the earth. For neutrino energiesof 5 – 50 MeV, we have 2EVearth/∆m2atm<∼ 10−2. Moreover, sin2 2θe3 ≤ 0.1,so that P3e − |Ue3|2 ≤ 10−3. Then (7) becomespD − p ≈ PH(P2e − |Ue2|2)(1− 2PL) . (9)In general, when the signals from two detectors D1 and D2 are compared, weget the difference of fluxesFD1e − FD2e ≈ PH · (1− 2PL) · (P(1)2e − P(2)2e ) · (F0e − F0x ) , (10)where P(1)2e and P(2)2e are the νe ↔ ν2 oscillation probabilities for the detectorsD1 and D2 correspondingly.For inverted hierarchy, using Table 1 we getpD − p ≈ (P2e − |Ue2|2)(1− 2PL) , (11)so thatFD1e − FD2e ≈ (1− 2PL) · (P(1)2e − P(2)2e ) · (F0e − F0x ) . (12)2.2 AntineutrinosIn the case of normal hierarchy, Table 1 leads top¯D − p¯ = (P¯1e − |Ue1|2)(1− P¯L) + (P¯2e − |Ue2|2)P¯L . (13)Then we obtain the difference in the fluxes at two detectors D1 and D2 asFD1e¯ − FD2e¯ ≈ (1− 2P¯L) · (P¯(1)1e − P¯(2)1e ) · (F0e¯ − F0x ) , (14)where we have neglected the oscillations of ν¯3 inside the earth.Talk given at CICHEP ’2001, Cairo, Egypt, January 2001. 3For the inverted hierarchy,p¯D− p¯ = (P¯1e− |Ue1|2)(1− 2P¯L)P¯H +(P¯3e− |Ue3|2)(1− P¯H − P¯HP¯L) . (15)Since ν¯3 oscillates inside the earth with a very small depth (inequality (8) isvalid with P3e replaced by P¯3e), the second term in (15) can be neglected togetp¯D − p¯ = (P¯1e − |Ue1|2)(1− 2P¯L)P¯H . (16)Therefore, finally we getFD1e¯ − FD2e¯ ≈ P¯H · (1− 2P¯L) · (P¯(1)1e − P¯(2)1e ) · (F0e¯ − F0x ) . (17)3 “Factorized” matter effectsNote that the equations (10), (12), (14) and (17) have significant similarities.All of these may be written in the factorized formFD1 − FD2 = fflux · fstar · fosc . (18)In this section, we shall discuss these factors and the insights they offer onthe extent of the observable earth matter effects.3.1 fflux: the difference in initial fluxesThe relevant difference in initial fluxes isfflux ={(F 0e − F0x ) for νe(F 0e¯ − F0x ) for ν¯e.Since the νe and ν¯e spectra are softer than the νx spectrum, and the luminosi-ties of all the spectra are similar in magnitude 4, the term fflux is positiveat low energies and becomes negative at higher energies where the νx fluxoverwhelms the νe (ν¯e) flux. Therefore, the earth effect has a different signfor low and high energies. The observed effect can be significant in the energyrange where σ× fflux is maximum (here σ is the cross section of the neutrinointeractions).3.2 fstar: the factor characterizing conversions inside the starThe value of this factor depends only on the neutrino conversions occuringinside the star, and hence it also characterizes the different scenarios:fstar =PH(1− 2PL) normal hierarchy νe(1− 2PL) inverted hierarchy νe(1− 2P¯L) normal hierarchy ν¯eP¯H(1− 2P¯L) inverted hierarchy ν¯e. (19)Talk given at CICHEP ’2001, Cairo, Egypt, January 2001. 4Consider the νe conversions in the scenario of normal hierarchy. In thelimit of Ue3 → 0 and PH → 1, (9) reduces to the expression for the eartheffects in the case of two neutrino mixing :[pD − p]2ν = (P2e − |Ue2|2)(1 − 2PL) , (20)which is equivalent to the one used in literature in the context of day-nighteffect for solar neutrinos. Therefore (9) can be looked upon as the earthmatter effect due to the two neutrino mixing (20) suppressed by a factor ofPH . The mixing of the third neutrino thus plays a major role, making theexpected earth effects in the case of supernova neutrinos smaller than thoseexpected in the case of solar neutrinos for the same mixing scheme and inthe same energy range. If the H-resonance is completely adiabatic, the eartheffect vanishes: all the νe produced are converted to ν3 in the star, and theearth matter effect on ν3 is negligibly small (as we have established through(8)).Notice that the expressions for the earth matter effects in the case ofthe inverted mass hierarchy (12, 17) have the same form as the expressionsin the case of the normal mass hierarchy (10, 14), except for the factors ofPH and P¯H . Indeed, since PH , P¯H ≤ 1, these act as suppression factorsfor the earth matter effects on νe and ν¯e, in the case of normal and invertedhierarchy respectively. Then the observation of significant earth matter effectson νe for normal hierarchy would imply PH ≈ 1, i.e. |Ue3|2 <∼ 10−3. Similarly,significant earth matter effects on ν¯e for inverted hierarchy would imply P¯H ≈1, i.e. |Ue3|2 <∼ 10−3.3.3 fosc: the difference in earth oscillation probabilitiesThis factor represents the difference in the neutrino conversions inside theearth due to the different distances travelled by the neutrinos inside the earthbefore reaching the detectors.fosc ={P(1)2e − P(2)2e for νeP¯(1)2e − P¯(2)2e for ν¯e.If the neutrino trajectory crosses only the mantle of the earth, one canuse a constant density approximation which givesP(1)2e − P(2)2e ≈ sin 2θme2 sin(2θme2 − 2θe2)[sin2(pid1lm)− sin2(pid2lm)]. (21)Here θme2 and lm are the mixing angle and the oscillation length in the earthmatter respectively, and di is the distance travelled by the neutrinos inside theearth before reaching the detector Di. The first two terms on the right handside of (21) are positive for the scenarios with the SMA and LMA solutions,so that the sign of fosc is the same as the sign of the term inside the squarebracket in (21). For the scenario with the VO solution, the earth matterTalk given at CICHEP ’2001, Cairo, Egypt, January 2001. 5effects are negligible: for ∆m2 ∼ 10−10 eV2, the mixing in the earth matteris highly suppressed, since sin 2θme2 is very small.In the case of antineutrinos, the constant density approximation givesP¯(1)1e −P¯(2)1e ≈ − sin 2θ¯me2 sin(2θ¯me2−2θe2)[sin2(pid1lm)− sin2(pid2lm)], (22)where θ¯me2 is the mixing angle inside the earth for the antineutrinos. For theantineutrino channel θ¯me2 < θe2 ≪ 1 for SMA solution and θ¯me2 is stronglysuppressed by matter in the VO case. Therefore the earth matter effects onthe ν¯e spectrum can be significant only for the scenario with the LMA (aswell as LOW) solution. In this scenario, sin 2θ¯me2 > 0 and sin(2θ¯me2−2θe2) < 0,so that the sign of fosc is the same as the sign of the oscillation term insidethe square bracket in (22).If neutrinos cross both the mantle and the core, the parametric enhance-ment of oscillations may occur, which leads to the appearance of parametricpeaks apart from the peaks due to the MSW resonances in the core and themantle 5. Correspondingly the factor fosc will be a more complicated functionof the neutrino energy.To summarize, the earth matter effects on the νe spectrum can be sig-nificant only for the scenarios with the SMA or LMA or LOW solutions.Moreover, if the hierarchy is normal, |Ue3|2 <∼ 10−3 is needed in addition.The effect on the ν¯e spectrum can only be significant for the LMA scenario.Let us consider these cases in detail below.In the case of SMA, the factor of fosc can be as large as 0.25 in the energyrange of 20 – 40 MeV, where the term fflux is also significant. However, evenin this scenario, and with optimistic values of PH , PL and θe2, the net effectis only <∼ 10%, which would be difficult to disentangle from the uncertaintiesin the original fluxes (see Fig. 7 in 1).In the case of LMA, the factor fosc can be as large as 0.3 in the energyrange of 20 – 50 MeV both for νe and ν¯e. Also, the transitions in the L reso-nance layer are completely adiabatic 1 for both neutrinos and antineutrinos,so that the factor (1 − 2PL) as well as (1 − 2P¯L) is equal to 1. Indeed, theearth matter effects can be large in both channels. In Fig. 1, we show the νeand ν¯e spectra for different distances travelled by the neutrinos through theearth.Note that in the case of LMA, in some region of parameter space thefinal νe spectrum shows a spectacular dip1,6, which cannot be mimicked byany uncertainty in the original energy spectrum. Thus, the earth mattereffects can be demonstrated in a clean way, independent of any model ofsupernova dynamics and by the observation of signals in only one detector.We demonstrate this case further through a Monte carlo simulation of thefinal νe spectrum in LMA scheme with 300 events in Fig. 2. The dip is clearlyobservable in the spectrum with d = 6000 km.Talk given at CICHEP ’2001, Cairo, Egypt, January 2001. 620 40 60 80 1000.0050.010.0150.0220 40 60 80 1000.0050.010.0150.020.025(a) (b)N NE EFigure 1. The earth matter effects on (a) the νe spectrum and (b) the ν¯e spectrum forPH = 1 in the scheme with the LMA solution (∆m2 = 2 · 10−5 eV2, sin2 2θ⊙ = 0.9) andnormal hierarchy. The dotted, dashed and solid lines show the spectra of the number ofν − N charged current events when the distance travelled by the neutrinos through theearth is d = 0 km, d = 4000 km, and d = 6000 km respectively.4 ConclusionsWe have indicated how the observation of earth matter effects on the νe and ν¯eenergy spectra from a supernova can help resolve ambiguties in the neutrinomass spectrum in a model independent manner.The observation of any earth matter effects rules out the scenario withthe VO solution. If the earth matter effects are observed in the neutrinochannels but not in the antineutrino channels, we either have the invertedmass hierarchy, or the normal mass hierarchy with |Ue3|2 <∼ 10−3.The observation of earth matter effects in the antineutrino channel identi-fies the LMA solution. In addition, if the effects are significant in the neutrinochannel also, |Ue3|2 <∼ 10−3 may be established.The qualitative features of the earth matter effects may be able to explaincertain peculiar features of the SN1987A supernova neutrino energy spectra7, but the number of events is too small to come to any definite conclusion.However, a galactic supernova may provide us with a sufficient number ofevents 8 to enable us to reconstruct the energy spectra of νe and ν¯e.I would like to thank the organising committee of CICHEP’2001 for theirhospitality during the Workshop. I would also like to thank A. Yu. Smirnovfor discussions and insights during the collaboration for 1.This talk points out the salient features of the earth matter effects on theTalk given at CICHEP ’2001, Cairo, Egypt, January 2001. 7Figure 2. The number of events (y-axis) of νe as a function of energy in MeV (x-axis) for theLMA solution (parameters as in Fig 1). The three spectra correspond to d = 0, 4000, 6000km. The spectra are normalized so that the number of events for each spectrum is 300.supernova neutrino spectra. For a more recent detailed study of the spectraexpected at various detectors, the reader is referred to 9.References1. A. S. Dighe and A. Yu. Smirnov, Phys. Rev. D 62 (2000) 033007 [hep-ph/9907423].2. T. K. Kuo and J. Pantaleone, Phys. Rev. D 37 (1988) 298. S. P.Mikheyev and A. Yu. Smirnov, Prog. Part. Nucl. Phys. 23 (1989) 41.3. A. S. Dighe, Nucl. Phys. Proc. Suppl. 87 (2000) 327 [hep-ph/9912414].4. H. -T. Janka, Astropart. Phys. 3 (1995) 377 [astro-ph/9503068].5. E. Kh. Akhmedov, A. Dighe, P. Lipari and A. Yu. Smirnov, Nucl. Phys.B 542 (1999) 3 [hep-ph/9808270].6. K. Takahashi, M. Watanabe and K. Sato, Phys. Lett. B 510 (2001) 189.7. C. Lunardini and A. Yu. Smirnov, Phys. Rev. D 63 (2001) 073009 [hep-ph/0009356].8. A. Burrows, D. Klein and R. Gandhi, Phys. Rev. D 45 (1992) 3361.9. C. Lunardini and A. Yu. Smirnov, hep-ph/0106149.Talk given at CICHEP ’2001, Cairo, Egypt, January 2001. 8

Earth matter effects on the supernova neutrino spectra

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