Detection of Supernova Neutrinos on the Earth for Large θ 13

Transcription

1 Commun. Theor. Phys. 61 (2014) Vol. 61, No. 2, February 1, 2014 Detection of Supernova Neutrinos on the Earth for Large θ 13 XU Jing (Å ), 1, HUANG Ming-Yang (á ), 2, HU Li-Jun ( ), 1, GUO Xin-Heng (À ), 1, and YOUNG Bing-Lin ( ) 3,4, 1 College of Nuclear Science and Technology, Beijing Normal University, Beijing , China 2 Institute of High Energy Physics, Chinese Academy of Sciences, Beijing , China 3 Department of Physics and Astronomy, Iowa State University, Ames, Iowa 5001, USA 4 Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing , China (Received September 13, 2013) Abstract Supernova (SN) neutrinos detected on the Earth are subject to the shock wave effects, the Mikheyev Smirnov Wolfenstein (MSW) effects, the neutrino collective effects and the Earth matter effects. Considering the recent experimental result about the large mixing angle θ 13 ( 8.8 ) provided by the Daya Bay Collaboration and applying the available knowledge for the neutrino conversion probability in the high resonance region of SN, P H, which is in the form of hypergeometric function in the case of large θ 13, we deduce the expression of P H taking into account the shock wave effects. It is found that P H is not zero in a certain range of time due to the shock wave effects. After considering all the four physical effects and scanning relevant parameters, we calculate the event numbers of SN neutrinos for the Garching distribution of neutrino energy spectrum. From the numerical results, it is found that the behaviors of neutrino event numbers detected on the Earth depend on the neutrino mass hierarchy and neutrino spectrum parameters including the dimensionless pinching parameter β α (where α refers to neutrino flavor), the average energy E α, and the SN neutrino luminosities L α. Finally, we give the ranges of SN neutrino event numbers that will be detected at the Daya Bay experiment. PACS numbers: Pq, g, Pt Key words: supernova neutrinos, Daya Bay experiment, θ 13, shock wave effects, MSW effects, P H 1 Introduction In the universe, core-collapse supernovas (SNs) are among the most energetic explosions. [1 3] They not only mark the catastrophic end of some stars, which turn into neutron stars or black holes after explosions, but also are responsible for the richness of heavy elements. [4 6] SN1987A has attracted worldwide interests and has been studied extensively since it came into our sight several decades ago. [7 8] During the explosions of the type II SN, most of the binding energy is released as neutrinos, which are very useful for acquiring information about the intrinsic properties and the explosion dynamics of SN. [3,7] For the past few decades, in most theoretical models, it had been believed that the neutrino mixing angle θ 13 was smaller than 3, or even smaller than 1.5. In Refs. [9 10], the authors studied possible methods to measure this neutrino mixing angle while θ 13 < 3. However, in recent years, some new experimental results indicated a large θ 13 (by large θ 13 we mean θ 13 9 ). [11 16] Last year the Daya Bay experiment measured the value of θ 13 to 5.2σ accuracy and obtained the result θ 13 = 8.8 ± 0.8, [15] which is much larger than 3. Table 1 shows a summary of the recent experimental results about θ 13. In this paper, we will study detection of SN neutrinos on the Earth in the case of large θ 13. SN neutrinos are produced from the core-collapse of SN and propagate outward to the surface of SN. Then they travel a long cosmic distance to reach the detector on the Earth. In this process, they pass through the SN matter and the Earth matter. During their propagation, SN neutrinos are subject to the Mikheyev Smirnov Wolfenstein (MSW) effects, [17 18] the shock wave effects, [19 20] the neutrino collective effects, [21] and the Earth matter effects. [9 10,22] Different from the case in vacuum, the behavior of neutrino oscillation changes when neutrinos propagate in matter. The neutrino matter effects, resulting from the interaction between matter and neutrinos, was found by Wolfenstein, Mikheyev, and Smirnov, then named as MSW effects. Inside the SN, the large mixing angle solution of neutrinos Supported by the National Natural Science Foundation of China under Grant Nos , , , and and the Fundamental Research Funds for the Central Universities in China xj2012@mail.bnu.edu.cn huangmy@ihep.ac.cn hulj@sina.cn Corresponding author, xhguo@bnu.edu.cn young@iastate.edu c 2013 Chinese Physical Society and IOP Publishing Ltd

2 No. 2 Communications in Theoretical Physics 227 results in the crossing probability, P L 0, at the low resonance region. Therefore, we only need to consider the crossing possibility, P H, at the high resonance region. [18,23] Since the shock wave effects may change the density profile of SN as well as the position of the high resonance, and θ 13 is not as small as expected before, the expression of P H for large θ 13 under the shock wave effects needs to be further developed. [10,18,24] By scanning the ranges of relevant parameters in the neutrino spectra, it is possible to obtain the maximum and minimum event numbers of SN neutrinos detected on the Earth. Table 1 Summary of experimental results about θ 13. Numbers with (without) brackets are for normal (inverted) mass hierarchy. Experiment Result Accuracy (Sensitivity) Year Reference Double Chooz 0.03 < sin 2 2θ 13 < [11] MINOS 0 < sin 2 2θ 13 < 0.12(0.19) 1σ 2009 [12] T2K 0.03(0.04) < sin 2 2θ 13 < 0.28(0.34) 2.5σ 2010 [13] MINOS 2sin 2 2θ 23 sin 2 2θ 13 < 0.12(0.20) 90% 2011 [14] Daya Bay sin 2 2θ 13 = ± 0.016(stat) ± 0.005(syst) 5.2σ 2012 [15] RENO sin 2 2θ 13 = ± 0.041(stat) ± 0.03(syst) 4.9σ 2012 [16] This paper is organized as follows. In Sec. 2, we present a brief overview of SN explosions and the production of SN neutrinos. In Sec. 3, we describe the four physical effects on the detection of SN neutrinos including the MSW effects, the shock wave effects, the neutrino collective effects, and the Earth matter effects. In this section, how to obtain the expression of P H in the high resonance region for large θ 13 is presented in detail. In Sec. 4, using the latest result of the Daya Bay experiment, θ 13 = 8.8 ± 0.8, we take into account the four physical effects and calculate the event numbers of SN neutrinos detected on the Earth. The results illustrate how the relevant SN neutrino parameters impact on the variation of event numbers and how many SN neutrinos can be detected at the Daya Bay experiment. Finally, in Sec. 5 we give the summary and discussions. 2 SN Explosions and SN Neutrino Spectra According to the presence or absence of hydrogen lines in their spectra, supernovas can be classified into two types, type I and type II. In this paper, we only pay attention to the type II supernovas, one main source of neutrinos in the universe, which are also named as core-collapse supernovas having hydrogen lines in their spectra. [25] The explosion process of core-collapse supernovas can be divided into several phases, and more details about the scenario of explosion can be found in Ref. [1]. In recent years, some impressive progresses have been made on the simulations of the post-bounce core-collapse SN, indicating how the luminosities and energies of SN neutrinos change in the accretion and cooling phases. [26 27] An SN explosion approximately releases a total energy of E B = erg, about 99% of which is radiated away as SN neutrinos. [28] The relation between the total SN energy and the luminosity of different flavor neutrinos is given by [29] L νe (t) + L νe (t) + L νx (t) = E B τ e t/τ, (1) where ν x represent ν µ, ν τ, ν µ, and ν τ. The luminosity flux of the SN neutrinos L α (α = ν e, ν e, ν x ) decays in time as L α (t) = L 0 α e t/τ. (2) The range of τ was obtained by fitting the experimental data of SN1987A: τ = s. [28,30] Basically, the SN neutrino spectra are parameterized to match the result of Monte Carlo simulations. Here, we use the alpha fit distribution, which was given by the Garching group (we will call it Garching distribution later in this paper). It can be expressed as [31] F α (0) (E) = L α β ( α βα E ) (βα 1) E α Γ(β α ) E α ( E ) exp β α, (3) E α where E α is the average energy of neutrino and β α is the dimensionless pinching parameter. For different neutrinos, their values are typically [1,32] E νe = E νe = MeV, E νx = MeV, β α = (4) The ranges of luminosity ratios for different flavor neutrinos are the following in this model: L νe L νx = , L νe L νx = (5) 3 Four Physical Effects on SN Neutrinos When neutrinos propagate outward to the surface of SN, they can be subject to the SN shock wave effects, the MSW effects, and the collective effects. Before arriving at the detectors, they travel through the Earth matter and are affected by the Earth matter effects. In this section, we will consider all the above four physical effects on SN neutrinos. 3.1 MSW Effects and Conversion Probability The MSW effects are caused by neutrino interactions with matter, which are determined by the matter den-

3 228 Communications in Theoretical Physics Vol. 61 sity profile and the mixing angles. By using the Landau s method, the conversion probability P H for neutrinos to jump from one mass eigenstate to another at the high resonance layer can be expressed as [18] 1, (n e r), (1 tan 2 θ) 2 /(1 + tan 2 θ) 2, (n e r 1 ), F = 1 tan ( 2 θ, ) [ ] (n e e r ), 2 1/n 1 1/2 (tan2θ) 2m, (n e r n ), m=0 2m m + 1 where n e is the electron density, r is the distance to the center of SN, and the adiabaticity parameter γ is defined as [18] γ m 2 sin 2θtan 2θ, (8) E lnn e / r with m 2 being the mass square difference of two mass eigenstates. For the SNs, n 3, the expression of F in the case of n e r n in Eq. (7) is ( ) [ ] 1/n 1 1/2 F = 2 (tan2θ) 2m. (9) 2m m + 1 m=0 In Eq. (9) ( ) 1/n 1 = 2m [ ] 1/2 m + 1 (1/n 1)! (1/n 1 2m)!(2m)!, (10) = ( 1) m J m J m+1 π/4, (11) with π/2 J m = (sin φ) 2m (2m 1)!! π dφ = 0 (2m)!! 2, (12) Eq. (9) can be expressed as a hypergeometric function: ( n 1 F = 2 F 1 2n, 2n 1 ) 2n ; 2; tan2 2θ. (13) In the case of θ [0, π/8], using the Euler integral representation, [33] one has Γ(c) 2F 1 (a, b; c; z) = Γ(b)Γ(c b) 1 0 t b 1 (1 t) c b 1 (1 tz) a dt. (14) We make the Taylor expansion for F near the point 1/n = 0, ( 1 ( 1 2 F = F(0) + F (0) + F n) (0) n) ( 1 ) m + F (m) (0) + (15) n The first two coefficients in Eq. (15) can be obtained straightforwardly, F(0) = 1 tan 2 θ, [ F (0) = (1 tan 2 θ) ln (1 tan 2 θ) tan2 θ tan 2 θ ] ln(1 + tan 2 θ). (16) P H = exp[ πγf/2] exp[ πγf/2sin2 θ] 1 exp [ πγf/2sin 2. (6) θ] The factor F is given by Comparing with the numerical result of the right-hand side of Eq. (14) in the case 1/n 0, we find that the first two terms in Eq. (15) give dominant contributions and other items are negligible, so F can be approximately written as the following equation, Eq. (17), which is identical to the expression in Ref. [18]. ( F = (1 tan 2 θ) 1 1 n ln(1 tan2 θ) + 1 ) [(1 + tan 2 θ)/ tan 2 θ] ln(1 + tan 2 θ). (17) (7) For the case of n = 3, the comparison between the numerical result of the right-hand side of Eq. (14) and that given by Eq. (17) is shown in Fig. 1. It can be seen that for n = 3, Eq. (17) is a very good approximation to 2 F 1 in Eq. (14). Fig. 1 Numerical result of F as a function of θ for n = 3. The solid and dashed curves represent the results when F takes the expression of Eqs. (14) and (17), respectively. 3.2 SN Shock Wave Effects and P H The SN shock wave effects play an important role in the SN neutrino oscillations. As pointed out in Ref. [34] and further studied in Refs. [35 37], after the core bounce, the shock wave propagates inside the SN during the period of neutrino emission and modifies the density profile of the star. In several seconds, the forward shock wave may reach the resonance region where the conversion of different flavor SN neutrinos maximize, thus affecting the transition probability P H in the high resonance region. [19,23] Recent simulations indicate that reverse shock waves can be formed when the velocity of the

4 No. 2 Communications in Theoretical Physics 229 material becomes larger than the local sound speed, therefore it can effect the density as well. [38 39] It was also pointed out that the turbulence effects should be included in the supernova simulations. [40] In the following, we will consider the main character of shock wave effects by taking into account the forward shock wave effects, which are analytically approximated and characterized by a density jump as shown in Ref. [37]. In general, the density distribution of SN might be divided into two phases roughly by time: t < 1 s and t 1 s. The changes of the density profile and the calculation expressions of P H under the influence of shock wave effects were discussed in Refs. [10, 37] in detail, and we refer the reader to these references. For post-bounce time t < 1 s, without shock wave effects, the density profile can be approximated by its static limit ρ 0 as given by [34] ( r ) n ρ 0 (r) g/cm 3. (18) 1 km In Eq. (18) n = 2.4. It has been proven that in this case Eq. (17) is a very good approximation to Eq. (14). The difference between the numerical results from these two equations are negligible when θ 13 = 8.8. Because the Daya Bay experimental result, θ 13 = 8.8 ± 0.8, is quite different from the condition θ 13 < 3 in Refs. [9 10], it is necessary to make clear the behavior of P H for large θ 13. In Eqs. (6) (8), it can be found that P H depends on F and γ, which are related to θ 13 and the neutrino energy E. In Figs. 2, 3, and 4, it is shown how the crossing probability P H changes as a function of the neutrino energy E, the time t, and the mixing angle θ 13, respectively. Fig. 2 The crossing probability at the high resonance region P H as a function of the neutrino energy E for three neutrino mixing angles at t = 6 s. The solid, dashed, and dotted curves correspond to θ 13 = 3,6, 9, respectively. Fig. 3 The crossing probability at the high resonance region P H as a function of the time t for three neutrino energies at θ 13 = 9. The solid, dashed, and dotted curves correspond to neutrino energy E = 11, 16, 25 MeV, respectively. Fig. 4 The crossing probability at the high resonance region P H as a function of the neutrino mixing angle θ 13: (a) for three different times at E = 11 MeV. The solid, dashed and dotted curves correspond to t = 2 s, 4 s, 6 s, respectively; (b) for three different neutrino energies at t = 6 s. The solid, dashed, and dotted curves correspond to E = 11, 16, 25 MeV, respectively.

5 230 Communications in Theoretical Physics Vol. 61 In Fig. 2, it can be seen that the value of P H depends on the energy of SN neutrino. Whatever value θ 13 takes, the value of P H has a great jump approximately at E = 10 MeV. For θ 13 = 3, the curve of P H still has obvious continuous fluctuations from about 15 MeV to higher energy. For θ 13 = 6 and θ 13 = 9, the value of P H changes smoothly and decreases slowly when E 30 MeV. Figure 3 shows the curves of P H for three typical neutrino energies when the time ranges from 0 s to 10 s. We can see that as the energy increases the curve becomes fatter. In other words, the greater the neutrino energy, the longer time P H keeps at high values. In general, the value of P H reaches the maximum value when the time is between 4 6 s. In Fig. 4(a), it is found that for a certain neutrino energy, at different times, the value of P H changes smoothly in the range of θ 13 = However, the curve for t = 6 s has rapid fluctuations between 0 and 5. In Fig. 4(b), all the curves corresponding to three neutrino energies have obvious fluctuations when θ 13 is between 0 and 5. This is far from the real θ 13 value. Figures 2, 3, and 4 illustrate that P H is zero near the real value of θ 13 (8.8 ) when there are no shock wave effects. However, when the shock wave effects turn on P H is not zero in a range of time. 3.3 Collective Effects and Earth Matter Effects The neutrino collective effects, which mechanism is totally different from that of the MSW effects, are caused by the neutrino-neutrino interactions inside the core-collapse SN. Recently, it has been realized to be a crucial feature at very high densities of the core and can change the emitted fluxes of different flavor neutrinos substantially. [21,32] Up to now, there have been a significant amount of studies on the neutrino collective effects. [21,29,32,41 45] In Ref. [32], it was shown that the collective effects depend on the inherent features of SN neutrinos, such as their initial total energy, relative luminosities of different flavors, and the neutrino mass hierarchy. [1,46 47] In this paper, in order to study the collective effects quantitatively, we set P νν as the survival probability that the neutrinos (antineutrinos) ν( ν) remain as their original states after the collective effects. In Ref. [21] the authors introduced a simplified picture to describe the characteristics of the collective effects: { 1 (E < EC ), P νν = 0 (E > E C ), for neutrinos and P νν = 1 for antineutrinos, where E C is a critical energy. We take E C = 10 MeV in our later calculation. [21] When reaching the Earth, the neutrinos are mass eigenstates, then they oscillate in flavors while going through the Earth matter. In Refs. [9 10], the authors studied Earth matter effects in detail. For simplicity, we do not repeat it here but refer the reader to these references. 4 Detecting SN Neutrinos on the Earth There are three reaction channels with which one may detect SN neutrinos at the Daya Bay experiment: the inverse beta decay, neutrino-electron reactions, and neutrino-carbon reactions. The reaction formulas and the means of calculation were discussed in Refs. [9 10] and the relevant effective cross sections mentioned in Refs. [48 49] are still applicable when θ 13 is large. 4.1 Calculation of Event Numbers A detailed description of the Daya Bay experiment can be found in Refs. [50 51]. There are eight detectors located at different sites of the Daya Bay experiment. The total detector mass is about 300 tons and the depth of the detectors h 400 m. The Daya Bay Collaboration uses Linear Alkyl Benzene (LAB) as the main part of the liquid scintillator. LAB has a chemical composition including C and H with the ratio of the number of C and H being about 0.6. Then, the total numbers of target protons, electrons, and 12 C are N (p) T = , N (e) T = , N (C) T = With all of the four physical effects being taken into account, the SN neutrino fluxes at the detector are expressed as F D ν e = pf (0) ν e + (1 p)f (0) ν x, F D ν e = pf (0) ν e + (1 p)f (0) ν x, 2F D ν x = (1 p)f (0) ν e + (1 + p)f (0) ν x, 2F D ν x = (1 p)f (0) ν e + (1 + p)f (0) ν x, (19) where the survival probabilities p and p are given by p = P 2e [P H P νν + (1 P H )(1 P νν )], p = (1 P 2e ) P νν, (20) for the normal mass hierarchy and p = P 2e P νν, p = (1 P 2e )[ P H Pνν + (1 P H )(1 P νν )], (21) for the inverted mass hierarchy. P 2e is the probability that a neutrino mass eigenstate enters the surface of the Earth and arrives at the detector as an electron neutrino ν e. [52] The event numbers N(i) of SN neutrinos in the reaction channel i can be calculated by integrating over the neutrino energy, the product of the target number N T, the cross section of the given channel σ(i), and the neutrino flux function at the detector F D α, N(i) = N T de σ(i) 1 4πD 2 F D α, (22) where α stands for the neutrino or antineutrino of a given flavor, and D is the distance between the SN and the Earth. [9] 4.2 Scanning over the Relevant Parameters In Sec. 2, we gave the ranges of the average energies of neutrinos E α and ranges of dimensionless pinching parameters β α in the parametrization of SN neutrino fluxes as listed in Eq. (4). It is expected to obtain the maximum and minimum values of neutrino event numbers in

6 No. 2 Communications in Theoretical Physics 231 the Daya Bay experiment from our calculation results. To achieve this objective, scanning over the ranges of all the parameters related to the calculation of four physical effects on detecting SN neutrinos is necessary. Notice that the luminosity ratios of different flavor neutrinos in Eq. (5) should be considered as well. In fact, as will be shown in the next subsection, the luminosity ratios do have effects on neutrino event numbers. Comparing the three reaction channels, it can be seen that the cross sections for the neutrino-electron scattering channel ( ) are much smaller than the other two reaction channels ( ). [9] Hence, we will only consider the inverse beta-decay and the neutrino-carbon reactions in the following analysis. It also can be seen that the inverse beta-decay does not involve any parameters about ν e since ν e is not involved in the inverse beta-decay. Based on simulation results the luminosity of ν e and ν e can be taken to be equal, [31] so we can define L νe = L ν e = 1 L νx L νx M. (23) Scanning over the ranges of all the parameters in Eqs. (4) and (5), we obtain the following parameter values for the distribution in Eq. (3): (Max) E νe = 12 MeV, E νx = 18 MeV, β νe = 3.5, β νx = 3.5, M = 2, (24) (Min) E νe = 12 MeV, E νx = 15 MeV, β νe = 3.5, β νx = 6, M = 1.25, (25) for the inverse beta decay, and (Max) E νe = E νe = 12 MeV, E νx = 15 MeV, β νe = β νe = 3.5, β νx = 6, M = 2, (26) (Min) E νe = E νe = 15 MeV, E νx = 18 MeV, β νe = β νe = 3.5, β νx = 3.5, M = 1.25, (27) for the neutrino-carbon reactions. From Eq. (5) it can be seen that M varies between two extreme values, 1.25 and 2. In order to see the influence of the luminosity ratio itself on event numbers, we only change the values of M in Eqs. (24) (27) to the other extreme values, with keeping E α and β α unchanged, then we obtain the following comparison groups of parameters for Eqs. (24) (27), respectively: (Max-C) E νe = 12 MeV, E νx = 18 MeV, β νe = 3.5, β νx = 3.5, M = 1.25, (28) (Min-C) E νe = 12 MeV, E νx = 15 MeV, β νe = 3.5, β νx = 6, M = 2, (29) for the inverse beta decay, and (Max-C) E νe = E νe = 12 MeV, E νx = 15 MeV, β νe = β νe = 3.5, β νx = 6, M = 1.25, (30) (Min-C) E νe = E νe = 15 MeV, E νx = 18 MeV, β νe = β νe = 3.5, β νx = 3.5, M = 2, (31) for the neutrino-carbon reactions. It is noted that no matter how the value of M changes, the total energy of all flavor neutrinos is a constant and the results with parameters in comparison groups are always between the maximum and minimum event numbers. 4.3 The SN Neutrino Event Numbers under the Influence of Four Physical Effects In this subsection, we give the numerical results of SN neutrino event numbers detected at the Daya Bay experiment. Consider a standard supernova at a distance D = 10 kpc from the Earth, which releases total energy E B = erg, and take τ = 3 s as the decay time of its luminosity. [28 30] The values of relevant parameters have already been given in the previous subsections. Given the Daya Bay experimental result, we take θ 13 = 8.8 in our calculations. With the influence of all the four physical effects being taken into account, supposing a neutrino reaches the detector with the incident angle α, [9 10] we calculate the neutrino event numbers with the neutrino spectra of the Garching distribution and plot the neutrino event numbers N as a function of α in Figs. 5 and 6 for the inverse beta-decay and the neutrino-carbon interactions, respectively. In order to show the influence of the luminosity ratio conveniently, the results of two groups with the same E α and β α but different M are plotted in the same figures. For example, in Fig. 5(a) there are four styles of numerical curves representing the results of Max (M = 2) and Max-C (M = 1.25) for both normal and inverted mass hierarchies. Generally speaking, the maximum variation of neutrino event numbers appears at α 93 when α changes due to the Earth matter effects for both reactions. Meanwhile, the variations in the inverse beta-decay are more obvious than those in the neutrino-carbon reactions. Also for both reactions, the numerical results show significant differences between the normal hierarchy and the inverted hierarchy even for the same values of E α and β α and the same luminosity ratio M. For example, comparing the solid curve (normal) and the dashed curve (inverted) in Fig. 5(a), it can be seen that the distance between the two curves are roughly 60 although they have the same parameter values as listed in Eq. (24). This point shows that event numbers largely depend on the mass hierarchy. On the other hand, it is found that even with the same values of E α and β α and with the same mass hierarchy, event numbers still change with the luminosity ratio M. For instance, in Fig. 6(a), the solid curve (M = 2) is about 15 more than the dotted curve (M = 1.25) although they both correspond to the normal hierarchy and the same E α and β α as listed in Eqs. (26) and (30). However, we should note that in Figs. 5(a) and 5(b), the solid curves (normal) are very close to the dotted curves (normal) and the difference between them is less than about 3. By contrast, the difference between the dashed curves (inverted) and dot-dashed curves (inverted) in Figs. 5(a) and 5(b) is significantly greater, which is more than 20. This shows that for the inverse beta-decay reaction, if the mass hierarchy is normal, event numbers change in a relatively

7 232 Communications in Theoretical Physics Vol. 61 narrow range over the values of M, while if the mass hierarchy is inverted, this is not the case. For the neutrinocarbon interaction in Figs. 6(a) and 6(b), it is shown that the variations of event numbers are remarkable with the changes of luminosity ratio no matter which mass hierarchy it is. Based on the above analysis, it is obvious that the luminosity ratio M plays an important role in determining the event numbers. Fig. 5 The event numbers for the inverse beta-decay with the parametrization form in Eq. (3). α is the incident angle of the SN neutrino reaching the detector. N(I) represents normal (inverted) mass hierarchy. In Fig. 5(a) Max and Max-C correspond to parameter values listed in Eqs. (24) and (28), respectively; in Fig. 5(b) Min and Min-C correspond to parameter values listed in Eqs. (25) and (29), respectively. Fig. 6 The event numbers for the neutrino-carbon interactions with the parametrization form in Eq. (3). α is the incident angle of the SN neutrino reaching the detector. N(I) represents normal (inverted) mass hierarchy. In Fig. 6(a) Max and Max-C correspond to parameter values listed in Eqs. (26) and (30), respectively; in Fig. 6(b) Min and Min-C correspond to parameter values listed in Eqs. (27) and (31), respectively. The summary of event numbers for the two reaction channels with the Garching distribution are given in Tables 2 and 3. The numerical results illustrate that the event numbers and the change rates due to the Earth matter effects depend on the parameters E α and β α, as well the mass hierarchy. Furthermore, the luminosity ratio M has an important influence on the event numbers and the change rates while other parameters remain unchanged. Comparing the results with the Min and Min-C groups of parameters for the inverse beta-decay, for example, in the case of inverted mass hierarchy, when M = 1.25 the event numbers at the incipient angle and the change rates are and 0.18%, respectively; while when M = 2 these two corresponding numbers are and 0.30%, both of which are significantly more than the former two. For another example, focusing on the results of the Max and Max-C groups of the neutrino-carbon interactions, in the case of normal mass hierarchy, when M = 1.25 the event numbers at the incipient angle and the change rates are and 0.15%, respectively; while when M = 2 these two numbers are 77.4 and 0.22%, respectively, which are also much larger.

8 No. 2 Communications in Theoretical Physics 233 Table 2 Summary of event numbers of the inverse beta-decay for the parametrization form given in Eq. (3). Max and Max-C correspond to parameter values in Eqs. (24) and (28), respectively; Min and Min-C correspond to parameter values in Eqs. (25) and (29), respectively. N(I) represents normal (inverted) mass hierarchy. The numbers in the columns Incipient and Min are the event numbers when the SN neutrino incident angle is zero and is the angle in the column Angle, respectively. The column Angle gives the angles at which the event numbers are the minimum and the Earth matter effects are the strongest. Ratio gives the percentages of the Earth matter effects. Reaction Conditions Hierarchy Incipient Min Angle Ratio Max N % I % Max-C N % ν e + p I % Min N % I % Min-C N % I % Table 3 Summary of event numbers of the neutrino-carbon interaction for the parametrization form given in Eq. (3). Max and Max-C correspond to parameter values in Eqs. (26) and (30), respectively; Min and Min-C correspond to parameter values in Eqs. (26) and (30), respectively. N(I) represents normal (inverted) mass hierarchy. The numbers in the columns Incipient and Min are the event numbers when the SN neutrino incident angle is zero and is the angle in the column Angle, respectively. The column Angle gives the angles at which the event numbers are the minimum and the Earth matter effects are the strongest. Ratio gives the percentages of the Earth matter effects. Reaction Conditions Hierarchy Incipient Min Angle Ratio Max N % I % Max-C N % ν+ 12 C I % Min N % I % Min-C N % I % Table 4 The event number ranges in the Daya Bay experiment with all the uncertainties taken into account. N(I) represents normal (inverted) mass hierarchy. Reaction Hierarchy Max Min Range ν e + p ν+ 12 C N I N I In Table 4 we give neutrino event numbers detected at the Daya Bay experiment when all the uncertainties are taken into account in the cases of normal and inverted mass hierarchies, respectively. We can see that the event numbers range from and for the inverse beta-decay and the neutrino-carbon interactions, respectively. 5 Summery and Discussions Given the new experimental result about θ 13 from the Daya Bay Collaboration, we deduce the expression of the neutrino conversion probability in the high resonance region inside SN, P H, in the case of large θ 13 ( 8.8 ), by applying the available knowledge for P H. P H is expressed in the form of hypergeometric function. In the derivation, we take the shock wave effects into account. Furthermore, we give numerical results of P H as functions of θ 13, t, and E. It is found that P H is zero near the real value of θ 13 when there are no shock wave effects. However, it is not zero in a certain region of time (roughly 3 s 8 s depending on neutino energies) if the shock wave effects are considered. Our work is different from previous studies which were usually done in the case of small θ 13 (< 3 ). [9 10] We consider the influence of all the four physical effects on the detection of SN neutrinos, including the MSW effects, the SN shock wave effects, the neutrino collective effects, and the Earth matter effects. Scanning over all the relevant parameters in the form of Garching neutrino energy distribution, we calculate the event numbers for two reaction channels, the inverse beta-decay and the neutrino-carbon reactions, both of which can be detected at the Daya Bay experiment. It is found that the event numbers depend on the parameters E α, β α, and L α,

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