Physics Letters B 702 (2011) 209–215

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Physics Letters B

www.elsevier.com/locate/physletb

On the observability of collective flavor oscillations in diffuse supernova neutrinobackground

Sovan Chakraborty a,∗, Sandhya Choubey b, Kamales Kar a

a Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, Indiab Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 October 2010Received in revised form 3 June 2011Accepted 29 June 2011Available online 6 July 2011Editor: S. Hannestad

Keywords:SupernovaeSupernovae distribution with luminosityRelic neutrinosCollective effectHierarchy sensitivity

Collective flavor oscillations are known to bring multiple splits in the supernova (SN) neutrino andantineutrino spectra. These spectral splits depend not only on the mass hierarchy of the neutrinos butalso on the initial relative flux composition. Observation of spectral splits in a future galactic supernovasignal is expected to throw light on the mass hierarchy pattern of the neutrinos. However, since theDiffuse Supernova Neutrino Background (DSNB) comprises of a superposition of neutrino fluxes from allpast supernovae, and since different supernovae are expected to have slightly different initial fluxes, it ispertinent to check if the hierarchy dependent signature of collective oscillations can survive this averagingof the flux spectra. Since the actual distribution of SN with initial relative flux spectra of the neutrinosand antineutrinos is unknown, we assume a log-normal distribution for them. We study the dependenceof the hierarchy sensitivity to the mean and variance of the log-normal distribution function. We findthat the hierarchy sensitivity depends crucially on the mean value of the relative initial luminosity.The effect of the width is to reduce the hierarchy sensitivity for all values of the mean initial relativeluminosity. We find that in the very small mixing angle (θ13) limit considering only statistical errorseven for very moderate values of variance, there is almost no detectable hierarchy sensitivity if the meanrelative luminosities of νe and νe are greater than 1.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Neutrinos coming from supernova explosions can give rich in-formation about the explosion mechanism as they are the onlyparticles that come from regions deep inside the core. They arealso extremely useful for determining neutrino properties as neu-trinos from SN1987A have amply demonstrated [1–3]. In particular,it has been shown that the neutrino mass hierarchy and small-ness of θ13 can be probed using SN neutrinos [4,5]. In the last fewyears, focus has been on the effect of neutrino–neutrino interac-tion in the central regions of the core of supernova, giving rise tothe so-called collective effects [6–43].

The ‘collective’ nature of simultaneous flavor conversions ofboth neutrinos and antineutrinos give rise to ‘splits’ in the spectraof the neutrinos and antineutrinos. These splits occur due to sud-den change in the oscillation probability, causing spectral swapswhich may end up in observable effects. Interestingly the impactof collective oscillations on the spectra are different for the Nor-mal Hierarchy (NH) and the Inverted Hierarchy (IH). This opens up

* Corresponding author.E-mail address: sovan.chakraborty@saha.ac.in (S. Chakraborty).

0370-2693/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2011.06.089

the possibility of identifying the neutrino mass hierarchy via ob-servation of collective effects in the neutrino signal from a futuregalactic supernova event [30].

However with the very small rate of occurrence of galactic su-pernova events (a few per century) one is forced to think of strate-gies of detecting the above mentioned effect otherwise. One of thepromising possibilities is the detection of Diffuse Supernova Neu-trino Background (DSNB) in the near future [44–46]. The cumula-tive number of neutrinos and antineutrinos produced by all earlierSN events in the universe result in a cosmic background knownas DSNB. Though these neutrinos are yet to be detected, one hasreasons to believe that it may be possible to observe them in thenear future [47–66]. Present day upper limits are 1.2νe cm−2 s−1

with energies above 19.3 MeV for Super-Kamiokande (SK) waterCerenkov detectors [67] and 6.8 × 103 cm−2 s−1 with energies be-tween 25 MeV and 50 MeV for Liquid Scintillator Detector (LSD)[68] at 90% C.L. for both.

The two key ingredients in the calculation of DSNB are (i) theSN rate which is proportional to cosmic star formation rate and(ii) the ν and ν energy spectra. Whereas reliable estimates arenow available for the star formation rate and the SN rate [69,70],the prediction for the SN neutrino spectra has gone through anevolution over the years. Earlier considerations of matter induced

210 S. Chakraborty et al. / Physics Letters B 702 (2011) 209–215

resonances were followed by incorporating the ‘collective’ effectsdue to interaction amongst the neutrinos themselves in the highdensity central regions of the core. The first study of the effectof the collective flavor oscillations on DSNB fluxes and the corre-sponding predicted number of events in terrestrial detectors werecarried out in [65] where it was demonstrated that the event rategets substantially modified by collective effects. The results alsoshowed that observation of the DSNB fluxes at earth could shedlight on the neutrino mass hierarchy. However, since then substan-tial progress has been made in the understanding of the collectiveeffects. In particular, it is now clear that the neutrino and antineu-trino survival probability and hence different split patterns dependcrucially on the relative luminosities of the initial neutrino fluxesproduced inside the exploding star [36–38]. Therefore, one canpredict the final neutrino and antineutrino spectra from a given SNwith reasonable accuracy only if one already has access to the ini-tial flux conditions. This complication is further compounded forthe DSNB, as the DSNB flux comes from a superposition of thefluxes from all past SNe. Since the initial flux conditions are ex-pected to be sensitive to the properties of the progenitor star andsince we have a whole distribution of stars which end up being aSN, it is a complicated business to accurately estimate the DSNBspectra after accounting for the collective effects, which are boundto happen in almost every SN.

In this work we incorporate the observation of different splitpatterns in the spectra for the calculation of DSNB and do not takethe relative (anti)neutrino fluxes to have fixed values. The mainfocus of this work is to check the effect of a distribution of super-novae with initial flux on the measurement of the neutrino masshierarchy via the observation of the DSNB signal. Since the dis-tribution of the initial fluxes over all past SNe are not availableto us, we parametrize this by a log-normal distribution. The log-normal distribution has two parameters which define the meanand width of the distribution. Since they are also unknown, wechoose various plausible values for them. We calculate the DSNBevent rate averaged over these distributions. We study how the hi-erarchy measurement is affected when one takes the distributionof initial relative fluxes into account and find situations where thehierarchy determination may be possible.

2. The diffuse supernova neutrino background

The differential number flux of DSNB is

F ′ν(Eν) = c

H0

zmax∫0

RSN(z)Fν(E)dz√

Ωm(1 + z)3 + ΩΛ

, (1)

where Eν = (1 + z)−1 E is the redshifted neutrino energy observedat earth while E is the neutrino energy produced at the source,Fν is the neutrino flux for each core collapse SN, RSN(z) thecosmic SN rate at redshift z, and the Hubble constant taken asH0 = 70h70 km s−1 Mpc−1. For the standard �-CDM cosmology, wehave matter and dark energy density Ωm = 0.27 and ΩΛ = 0.73,respectively [71]. As Eq. (1) suggests the DSNB flux at earth de-pends on two factors: (i) the cosmic SN rate and (ii) the initial SNneutrino spectrum from each SN.

The cosmic SN rate is related to the star formation rateRSF(z), through a suitable choice of Initial Mass Function (IMF) asRSN(z) = 0.0132 × RSF(z)M−1� [72,73]. The IMF takes into accountthat only stars with masses larger than 8M� result in supernovaexplosion. For the cosmic star formation rate per co-moving vol-ume we take

RSF(z) = 0.32 fSNh70e3.4z

3.8z

√Ωm(1 + z)3 + ΩΛ

3/2, (2)

e + 45 (1 + z)

where fSN is normalization of the order of unity and RSF(z) is inunits of M� yr−1 Mpc−3 [54,74]. The initial SN neutrino spectrumemitted from the neutrinosphere is parametrized in the form [75]

F 0ν(E) =

(L0ν

E

)×

((1 + α)1+α

Γ (1 + α)E

(E

E

)α

e−(1+α)E/E)

= φ0ν × ψ(E), (3)

where φ0ν is the total initial flux estimated for the initial luminosity

L0ν and average energy ( E). The spectral shape also depends on the

energy distribution ψ(E), which is parametrized by the pinchingparameter α. In this study we use Eνe = 12 MeV, E νe = 15 MeV,Eνx = Eνy = 18 MeV with ανx = ανx = ανy = ανy = 4 and ανe =ανe = 3. Here νx is a linear combination of νμ , ντ and νy is thecombination orthogonal to νx; in our case νx and νy has sameflux hence Fνx = Fνy . The average energies of the different fluxtypes will also vary from SN to SN. However for simplicity, in thiswork we choose to keep the average energies fixed. We assumethat 3 × 1053 erg of energy is released in (anti)neutrinos by allSNe.

The emitted spectrum Fν(E) is processed by collective flavoroscillation and MSW oscillation effects over the huge drop of mat-ter density inside SN. The collective oscillations are over withina few 100 km from neutrinosphere whereas the MSW oscillationtakes place in the region 104–105 km [76] for the solar and at-mospheric mass squared differences. As the collective and MSWoscillations are widely separated in space, they can be consideredindependent of each other [35]. Thus the flux reaching the MSWresonance region already has the effects of the collective oscilla-tions. This assumption may not hold in SN models [15], where atlate times the MSW resonance and the collective effects can besimultaneous as there the matter density falls substantially com-pared to the early time. In the time integrated DSNB flux it cangive rise to some corrections. However we have ignored theseeffects as in this work we followed matter profile from studies[35] with larger matter density [76,77] finding the two oscillationsregimes to be mutually exclusive even at late times.

It has been seen that collective oscillations can give rise todifferent split patterns of the neutrino spectra depending on theinitial relative flux of νe and νe with respect to flavor νx or νy , so

we define φrνe

=φ0

νe

φ0νx

and φrνe

=φ0

νe

φ0νx

as measures of the relative fluxes

[38]. The electron antineutrino flux beyond the collective regioncan swap to x flavor above some energy (single split) or can swapin some energy interval (double split) or even can remain un-changed (no split) depending on the initial relative flux φr

νeand

φrνe

[36–38]. DSNB is affected differently with these different oscil-lation scenarios. To incorporate the effect of collective oscillationswe work in a effective two flavor scenario with single angle ap-proximation.1

Recent papers [39,42] have explored the effect of three flavorson the outcome of the split patterns in collective oscillations. Thethree flavor results differ a bit from the two flavors only for IHand that too in a small region of the initial flux parameter space(φr

νe,φr

νe), where single split (in 3 flavor) appears instead of the

double splits (in 2 flavor). However the observed single split inNH for this region remains unchanged in the 3 flavor treatment[42]. Thus in a small part [37,38] of the parameter space (φr

νe,φr

νe),

where 2 and 3 flavor results differ, both IH and NH have similar

1 Multi-angle effects can give rise to kinematical decoherence among angularmodes and smear the spectral splits [17], however for spherical symmetry the sin-gle angle approximation i.e. neutrino–neutrino interactions averaged along a singletrajectory comes out to be a fine approximation as the multi angle decoherence insuch a case is weak against the collective features [18,36,78].

S. Chakraborty et al. / Physics Letters B 702 (2011) 209–215 211

Table 1Electron neutrino and antineutrino spectra emerging from a SN.

Normal hierarchy

Fνe = s212 Pc(φ

rνe

, φrνe

, E)(2P H − 1)(F 0νe

− F 0νx

) + s212(1 − P H )(F 0

νe− F 0

νx) + F 0

νx

F νe = c212 P c(φ

rνe

, φrνe

, E)(F 0νe

− F 0νx

) + F 0νx

Inverted hierarchy

Fνe = s212 Pc(φ

rνe

, φrνe

, E)(F 0νe

− F 0νx

) + F 0νx

F νe = c212 P c(φ

rνe

, φrνe

, E)(2P H − 1)(F 0νe

− F 0νx

) + c212(1 − P H )(F 0

νe− F 0

νx) + F 0

νx

split pattern (single split) for the 3 flavor evolution. In other partsof the initial flux parameter space the split patterns remain thesame. Therefore while averaging over the whole parameter spacethe corrections in the 3 flavor treatment compared to the 2 flavorone coming from the small region would not be appreciable. Themain conclusions of this Letter – as we later see – therefore re-main largely independent of these small corrections. Moreover thethree flavor effects are also found to be sensitive to the neutrino–neutrino interaction potential, the evolution of a three flavor sys-tem effectively behaves like a two flavor one with the reductionof the neutrino–neutrino interaction potential by only one order[42]. Hence for such a smaller neutrino–neutrino potential the twoflavor treatment of the collective evolution is reasonable and forlarger values the two flavor treatment can be again be consideredas a reasonable approximation when averaging over the whole ini-tial flux parameter space is taken into account.

In this effective two flavor treatment the collective oscillationsinvolve only two flavors (νe, νx), while the other flavor (νy) doesnot evolve. Thus the Flux (F c

να) after the collective oscillations are

given by

F cνe

= Pc F 0νe

+ (1 − Pc)F 0νx

, F cνe

= P c F 0νe

+ (1 − P c)F 0νx

.

Similarly the quantities F cνx

and F cνx

can be estimated from therelations

F cνx

+ F cνe

= F 0νe

+ F 0νx

and F cνx

+ F cνe

= F 0νe

+ F 0νx

.

As for the other flavor νy(νy) there is no collective evolution,hence F c

νy(νy)= F 0

νy(νy). The quantities Pc and P c are the neutrino

and antineutrino survival probability after the collective effect, re-spectively. The only way νy can affect the final neutrino spectrumis by MSW transition. As we consider self-induced neutrino oscil-lations to happen independent of the MSW, so the pre-processedflux F c

ναand the unchanged ‘y’ flavor will undergo the traditional

MSW conversions.In NH the MSW resonances affect the νe flux, while the νe

flux remains almost unaffected. However for IH, MSW affect the νe

flux, and not the νe flux. Then these neutrinos get redshifted whiletraveling independently as mass-eigenstates until they reach earth,here they are detected as flavor eigenstates before or after havingundergone regeneration inside the earth. The fluxes (Fνe and F νe )arriving at earth after both the collective and MSW oscillation aregiven in Table 1.

In Table 1, P H is the effective jump probability between theneutrino mass eigenstates due to the atmospheric mass squareddriven MSW resonance. It takes a value between 0 and 1 depend-ing on the value of the mixing angle θ13. For θ13 large (sin2 θ13 �0.01) P H � 0, while for small θ13 (sin2 θ13 � 10−6) P H � 1 [79].For the smaller limit of θ13 the effective jump probability has anon-trivial dependence on energy and time, due to multiple res-onances [14,19] and turbulence [16–18]. Since these effects occurin the small time-window when the shock wave is in the reso-nance region one can neglect this sub-leading effect in the timeintegrated observables like DSNB events. The quantities s2

12 andc2 stand for sin2 θ12 (taken to be 0.3 for numerical studies) and

12

Fig. 1. The four specimen distributions used in the calculations. The y-axis gives thedistribution of supernova in arbitrary units.

cos2 θ12, respectively. The neutrino and antineutrino survival prob-ability after the collective effect are calculated numerically taking�m2 = 3 × 10−3 eV2 and a small effective mixing angle of 10−5.However it should be noted that the collective probabilities Pc andP c are almost independent of the specific non-zero values of themixing angle [16,23,36]. Thus the effective mixing angle in thematter for both the limits sin2 θ13 � 0.01 and sin2 θ13 � 10−6 willgive rise to similar collective features. We checked explicitly thatthe results remain the same when one varies the effective mixingangle by orders of magnitude. Hence for the collective evolutionswe consider a representative value of 10−5 as the effective mix-ing angle for both the limits. For further details of analysis of thecollective effects we refer the reader to [38].2 As discussed before,Pc and/or P c , which is a function of the neutrino energy E , showpattern of sudden change between 0 and 1, leading to suddenchange in the neutrino and/or antineutrino spectra. Depending onthe number of times the value of Pc (and/or P c) changes, we canhave more than one sudden swap of the neutrino (and/or antineu-trino) spectra, referred to in the literature as multiple split [36–38].The split patterns are now known to be crucially dependent onthe initial relative flux densities of the neutrino and antineutrino.The initial relative neutrino and antineutrino flux densities are ex-pected to vary among different SN.

This variation in φrνe

and φrνe

in different SNe might have ori-gin in difference of progenitor mass, different luminosity etc. Infact even different simulations allow wide variation of φr

νeand φr

νe

within the 2 fold uncertainty around equipartition ( Eνx2Eνe

� φrνe

�2EνxEνe

; Eνx2E νe

� φrνe

� 2EνxE νe

) [5]. Most models predict lνe � lνe , where

lνe and lνe are the relative luminosity ( LναLTotal

) of νe and νe respec-tively [37]. However, the combined luminosity of νμ , ντ , νμ andντ is seen to be rather disparate between the different model re-sults.3 The most reliable way to reconstruct the relative luminositydistribution function in principle would be from direct observationof SN events along with their neutrino signal. However, as yet onlySN1987A has been observed along with the detection of its neutri-nos/antineutrinos. We require to know neutrino fluxes of differentflavors from a number of galactic SN with a range of stellar massand initial conditions before collapse to have information aboutthe possible variation in φr

νeand φr

νe. This might take decades and

2 Note that Pc = P c = 1 gives back the SN flux without collective oscillation [4].3 See for e.g. the compilation of model results in [80].

212 S. Chakraborty et al. / Physics Letters B 702 (2011) 209–215

is clearly not possible in the near future. So we propose in thisLetter that the SN events contributing to DSNB have various dif-ferent values of φr

νeand φr

νe. For quantitative estimates we assume

specific distributions for them. We take φrνe

and φrνe

distributedlog-normally, defined by the parameters (μ1, σ1) and (μ2, σ2) i.e.,

Dν(μ1,σ1) = e−(

log(φrνe )−μ1

2πσ1)2

√2πσ1φr

νe

;

D ν (μ2,σ2) = e−(

log(φrνe

)−μ22πσ2

)2

√2πσ2φ

rνe

. (4)

We choose a range of values for μ such that the expectation(mean) values of φr

νeand φr

νeare compatible with either Lawrence

Livermore (equipartition) or Garching simulations. The parameterσ determines the width of the distribution. Since the variation ofφr

νeand φr

νeis expected to be within the 2 fold uncertainty around

equipartition [5], the σ ’s for the distributions are chosen such thatthe distribution is well within the 2 fold uncertainty of the ex-pectation value. In order to show the effect of the choice of thedistribution function we simulate our results for four widely dif-ferent distributions for φr

νeand φr

νe. They are chosen as follows:

1. The expectation is the equipartition value, i.e. 1.5 for φrνe

and1.2 for φr

νeand the variation is within 2 fold uncertainty of

1.5 and 1.2 respectively. The corresponding values of (μ,σ )

turn out to be (0.39,0.21) for neutrinos and (0.16,0.24) forantineutrinos. We denote this distributions by D1

νeand D1

νe.

2. The expectation is same for both φrνe

and φrνe

and is takenas 1, the σ is chosen as stated above. Hence the distributionsfor νe and νe are identical and is defined by the parameters(μ,σ ) ≡ (−0.03,0.28). We denote this distribution by D2

νeand

D2νe

.3. The expectation is taken to be the same as the Garching sim-

ulations [75], i.e. 0.8 for both the νe and νe and the distri-butions parametrized by (μ,σ ) ≡ (−0.23,0.20) are the same.We denote this distribution by D3

νeand D3

νe.

4. We consider the distribution to be constant between 0.45–2.55for φr

νeand 0.45–2.05 for φr

νe, with a normalization factor of

12.1 and 1

1.6 respectively. We denote this case by D4νe

and D4νe

.

The four specimen distributions for φrνe

and φrνe

are shown in

Fig. 1. By definition the log-normal distribution functions D1, D2

and D3 are normalized to unity. The choice of our normalizationfactors for the uniform distribution functions in D4 ensures thatthey are normalized to unity as well. We can see from the figurethat D3 has the narrowest spread in the relative flux while the uni-form distribution D4 has the widest. For simplicity we have chosendistribution functions that are independent of the redshift z.

The differential number flux of DSNB with the ith distributionsis given by

F ′iν (Eν) = c

H0

φrνe max∫

φrνe min

φrνe max∫

φrνe min

zmax∫0

RSN(z)Fν

(E, φr

νe, φr

νe

)

× Diνe

(φr

νe

)Di

νe

(φr

νe

) dz dφrνe

dφrνe√

Ωm(1 + z)3 + ΩΛ

, (5)

where Diνe

(φrνe

) and Diνe

(φrνe

) is the number of supernovae in be-tween the interval φr

νeto φr

νe+ dφr

νeand φr

νeto φr

νe+ dφr

νe, respec-

tively. Fν(E, φrν ,φr ) is the neutrino flux of each SN with initial

e νe

Fig. 2. νe fluxes for NH (upper panel) and IH (lower panel) with small θ13, i.e.P H = 1.

relative flux combination φrνe

and φrνe

at a redshift z. zmax is takenas 5, where as φr

αmin and φrαmax are chosen from the 2 fold uncer-

tainty around expectation of the respective distribution. The upperpanel in Fig. 2 shows the antineutrino DSNB flux (in logarithmicscale) for NH and the lower panel shows the flux for IH withP H = 1. In both panels flux is shown for all of our different distri-bution (D1–D4) of φr

νeand φr

νe. In Fig. 2 we also plotted flux for

the case without any distribution of φrνe

and φrνe

but specific value(0.8,0.8) for the initial relative fluxes (φr

νe, φr

νe). From the flux fig-

ures it is evident that the different distributions give similar DSNBflux. To check whether the small differences in their profile aremeasurable or not we next calculate the corresponding event ratein different detectors.

3. DSNB event rates and hierarchy measurement

Since the DSNB fluxes are very small, they are expected tobe observed in either very large detectors or in reasonable sizedetectors with very large exposure times. The only reasonablesize detector running currently is the Super-Kamiokande (SK) [67].Amongst the proposed large detectors which have low energythreshold are the megaton water Cherenkov detectors [81–83],Gadolinium enriched water Cherenkov detectors [84] (which couldbe SK or possibly an even larger water detector), very large liquidscintillator detector (LENA) [53] or very large liquid Argon detector[85–87]. Apart from the liquid Argon detector, none of the otherdetectors hold much promise for the detection of DSNB νe . There-fore, in rest of this discussion we will focus only on the detec-tion possibilities for the DSNB νe fluxes. All detector technologiesmentioned above use the inverse beta decay νe + p → n + e+ in-teraction for detecting DSNB fluxes. The only difference between

S. Chakraborty et al. / Physics Letters B 702 (2011) 209–215 213

Table 2Number of expected events per 2.5 Megaton-year of a water Cherenkov with SK likeresolution and in a similar detector with Gadolinium loaded.

Model Hierarchy 2.5 Mton-yr(19.3–30.0) (MeV)

GD + 2.5 Mton-yr(10.0–30.0) (MeV)

D1 NH 76±9 280 ± 17IH (P H = 0) 92±10 319 ± 18IH (P H = 1) 80±9 288 ± 17

(1.5,1.2) NH 75±9 281 ± 17IH (P H = 0) 98±10 332 ± 19IH (P H = 1) 75±9 280 ± 17

D2 NH 88±10 307 ± 18IH (P H = 0) 103±11 350 ± 19IH (P H = 1) 76±9 280 ± 17

(1.0,1.0) NH 102±11 340 ± 19IH (P H = 0) 103±11 341 ± 19IH (P H = 1) 77± 295 ± 18

D3 NH 98±10 334 ± 19IH (P H = 0) 113±11 379 ± 20IH (P H = 1) 70±9 258 ± 16

(0.8,0.8) NH 112±11 378 ± 20IH (P H = 0) 113±11 378 ± 20IH (P H = 1) 70±9 260 ± 17

D4 NH 82±9 297 ± 18IH (P H = 0) 97±10 335 ± 19IH (P H = 1) 79±9 283 ± 17

(1.5,1.25) NH 77±9 286 ± 18IH (P H = 0) 97±10 330 ± 19IH (P H = 1) 76±9 285 ± 17

them would be in terms of their energy window of sensitivity forthe DSNB. Water detectors use the neutrino energy (Eν ) window19.3 MeV to 30.0 MeV [67], whereas for Gadolinium loaded waterdetectors the detection window is between 10 MeV and 30 MeV[84]. Liquid scintillator on the other hand has a sensitivity rangebetween 10 MeV and 25 MeV [53]. We show the total number ofevents for 2.5 Mton-yr exposure in water detectors and 2.5 Mton-yr exposure in Gadolinium loaded water detectors in the third andfourth columns of Table 2. The number of events are shown forthe four different specimen distributions. Results are shown sepa-rately for P H = 1 and P H = 0 when the neutrino mass hierarchyis inverted (IH). For the NH, there is no dependence of the νe fluxon P H , as one can see from Table 1. In each case we also show forcomparison the number of events expected when φr

νeand φr

νeare

fixed at their respective mean values of the distribution.4 Observ-ing a variation of the number of events changing the φr

νeand φr

νefor these examples without any distribution is an important indi-cator of how inclusion of distribution can change the DSNB eventnumbers.

We next turn our attention to the possibility of measuring theneutrino mass hierarchy using DSNB detection. For the distributionexamples D1, D2 and D4 the difference of number of events be-tween NH and IH (both P H = 1 and P H = 0) seems to be withinstatistical uncertainty. Whereas for the distribution D3, though thedifference in number of events between NH and IH (with P H = 1)decreases significantly compared to the without distribution case,the difference is still greater than the statistical uncertainty. Toprobe this variation of hierarchy sensitivity with the distributionfurther let us define the quantity

η = |NNH − NIH|NNH

, (6)

4 For the uniform distribution D4 we took the central values of the widths astheir representative mean.

Fig. 3. Upper panel shows the number of expected DSNB events in 2.5 Mton-yr ofGadolinium loaded water detector for normal (NH) and inverted (IH) hierarchies(for IH, P H = 1) as a function of φr

νe(= φr

νe). The lower panel shows the change of

η as a function of φrνe

(= φrνe

).

which gives a measure of the hierarchy sensitivity of the exper-iment. The quantities NNH and NIH are the number of expectedDSNB events when the hierarchy is normal and inverted, respec-tively. The quantity η is definitely not an observable as it dependson both NNH and NIH. However it helps describing the variationof hierarchy sensitivity of DSNB in a systematic manner. There aretwo ingredients of interest in Table 2. Firstly, we can see that therelative hierarchy difference η depends on the mean value of φr

νeand φr

νe. Secondly, for a given 〈φr

νe〉 and 〈φr

νe〉 it also depends on

the distribution function involved.For a better understanding of the first issue, we show in the

upper panel of Fig. 3 the variation of the number of events forboth the NH and IH cases, as a function of the relative luminosityfactor. The analysis is again done for the small mixing angle limit(P H = 1) for IH, as it is the more challenging limit from the exper-imental point of view. We show this for 2.5 Mton-yr of Gadoliniumdoped water detector.5 For simplicity we have taken φr

νe= φr

νein

this figure and we do not take any distribution function into ac-count. The lower panel shows the corresponding η as a functionof φr

νe= φr

νe. We see that η has a very complicated dependence on

the relative initial luminosity functions. The variation of η withφr

νs actually depends on several factors like difference betweensplit pattern of NH and IH, the split/swap energies, the initial rel-ative flux of νe , νe and νx . It is rather high for very low values ofφr

νeand φr

νe. It starts decreasing as the value of φr

νeand φr

νein-

crease until it becomes zero around φrνe

= φrνe

� 1.05, thereafterit increases for a short while until it reaches a (local) maximumat around φr

νe= φr

νe� 1.25. Beyond that the value of η decreases

again reaching a second zero at around φrνe

= φrνe

� 1.5. After thatit increases monotonically.

The most noteworthy thing in the upper panel of this figure isthat the number of events for inverted hierarchy has almost lineardependence on φr

νe= φr

νeon both sides of the maximum which

comes around φrνe

= φrνe

� 1.25. For the normal hierarchy one seesdeparture from linearity around φr

νe= φr

νe� 1.1. This feature is

crucial in determining the effect of the distribution function onthe hierarchy sensitivity. The effect of taking the distribution func-tion into account boils down to creating a weighted average of thenumber of events, where the weights are determined by the dis-

5 Most of the results shown in this Letter is for megaton class Gadolinium dopedwater detector to show the impact of the relative initial luminosity and its distribu-tion on the hierarchy measurement using DSNB fluxes. The corresponding sensitivi-ties for smaller scale detectors can be calculated trivially using these numbers.

214 S. Chakraborty et al. / Physics Letters B 702 (2011) 209–215

Fig. 4. Relative difference (η) of number of expected events in normal and invertedhierarchy, per 2.5 Mton-yr in a Gadolinium loaded SK like detector. For the IH casewe consider small mixing angle i.e. P H = 1. The above panels show η for four pos-sible cases with σ being 0.00, 0.05, 0.10, 0.15 and 0.20. The x axis denotes theexpectation of φr

νeand φr

νefor the chosen sigma. Here expectation of φr

νeand φr

νeare taken to be same i.e. μ1 = μ2. The errors shown are the statistical errors only.

tribution itself. For the log-normal case the weights are driven bythe mean and width of the distribution, which we parametrize interms of μ and σ . The effect of any distribution can thus be un-derstood with the help of Fig. 3.

To show the effect of the distribution function we continue tostick to the simplified scenario where φr

νe= φr

νeand show in Fig. 4

the relative difference η as a function of the expectation value〈φr

νe〉(= 〈φr

νe〉). Here again we take P H = 1 for IH. We consider log-

normal distribution for the relative luminosities6 and show η forfive different values of σ which controls the width of the distri-bution. The values and error bars on η correspond to a statisticsof 2.5 Mton-yr data in Gadolinium loaded water detector. The low-est panel with σ = 0 corresponds to the case where we keep φr

νe(φr

νe) fixed and for this case there is no effect of the distribution.

This case is similar to that in the lower panel of Fig. 3. We see thatwithout the effect of the distribution function almost all valuesof φr

νe(= φr

νe) would give hierarchy sensitivity to at least 1σ C.L.,

while for lower values of the relative luminosity the sensitivitycan be seen to be rather good. As we increase σ the sensitivityis seen to go down for all value of the relative luminosity. Wefind that even for very small values of σ = 0.05, there is almostno hierarchy sensitivity for 〈φr

νe〉(= 〈φr

νe〉) � 1. As σ increases this

pattern remains the same, though the sensitivity keeps falling forall values of 〈φr

νe〉(= 〈φr

νe〉). In the above analysis, σ is considered

in a small range. The idea was to avoid deviating too much fromthe simulated values of the relative initial luminosities. If the ac-tual variations of φr

νs for all past supernovae are much larger thanthe range considered here, then the difference between predictedevents for NH and IH would decrease even further, and this couldwashout the hierarchy sensitivity even for the low φr

ν cases.Finally in Fig. 5 we display how we can make sure quantita-

tively that the observed number of events belong to one of thehierarchies and not the other one and see its dependence on theexposure in Mton-yr. Fig. 5 plots the ratio |NNH−NIH|√

NNH+√NIH

for four dif-

ferent values of the exposure i.e. 2.5, 5.0, 7.5 and 10.0 Mton-yr as afunction of the expectation values of the relative flux, 〈φr

νe〉 taken

6 Our conclusions remain fairly robust against the choice of the distribution func-tion. We have explicitly checked this by repeating Fig. 4 with uniform distributionand normal distribution. However, we do not present those results as they followthe same pattern that we get for the log-normal distribution.

Fig. 5. Plot of |NNH−NIH |√NNH+√

NIHwith 〈φr

νe〉 (= 〈φr

νe〉) for σ = 0.20. The ratio is shown for

a Gadolinium loaded SK like detector with different values of the exposure i.e. 2.5,5.0, 7.5 and 10.0 Mton-yr. Horizontal line at ratio 1 denotes that statistical erroris of the same order as the difference between two hierarchies. Higher values like2, 3 will be of more prominence as other uncertainties from flux, luminosities anddetector systematics should be considered along with statistical error.

as equal to 〈φrνe

〉. This is for the case P H = 1 and with σ = 0.20.This shows that with 2.5 Mton-yr, the relative flux expectationswith values below 0.95 (ratio ∼ 1) can distinguish NH from IH (atleast in principle) as the statistical error is smaller than the num-ber of event difference. However to achieve a better sensitivity i.e.ratio ∼ 2 the upper limit for the relative flux expectations is about0.77. For ratio ∼ 3 the value is even lower. However with largervalues of exposures like 5.0 or 7.5 Mton-yr one can still make adistinction between hierarchies when the centroid is below 0.75and 0.82 respectively. But it is clear from the figure that by in-creasing the Mton-yr one gets only a slow increase in the allowedparameter space.

4. Conclusion

In this work we studied the prospects of measuring the neu-trino mass hierarchy from observation of the DSNB signal in ter-restrial detectors. While such studies have been performed beforeby different groups including ours, our work is unique as this isthe first time that distribution of the source SN with initial rela-tive neutrino and antineutrino fluxes has been taken into account.It is natural that different SN would emit neutrino and antineu-trino fluxes with slightly different initial conditions, depending onthe properties of the progenitor star. This is particularly relevantin the context of collective oscillations, where the multiple splitpatterns depend crucially on the initial relative fluxes. Since the ac-tual distribution function of SN with the initial relative fluxes areunknown, we chose four specimen distribution functions, whichhave a mean corresponding to the value from SN simulations anda width such that almost all the values are within a factor oftwo of the mean value. We worked with three log-normal andone uniform distribution. We presented the DSNB fluxes for allthe four distributions for both normal and inverted hierarchies.We calculated the total predicted number of νe events in waterdetectors, both with and without Gadolinium. The log-normal dis-tribution is characterized by its mean value and its variance. Theseare parametrized in terms of the variables μ and σ . We studiedthe dependence of the hierarchy sensitivity to the mean and vari-ance of the log-normal distribution function. We concluded thatthe hierarchy sensitivity in this experiment had a crucial depen-dence on the mean value of the relative initial luminosity φr

νeand

φr . The sensitivity has a predominantly non-linear dependence on

νe

S. Chakraborty et al. / Physics Letters B 702 (2011) 209–215 215

〈φrνe

〉 and 〈φrνe

〉, being higher for lower values of these quantities.The effect of the variance parametrized by σ is to reduce the hi-erarchy sensitivity for all values of the mean 〈φr

νe〉 and 〈φr

νe〉. We

found that even for very moderate values of σ � 0.05, there is al-most no hierarchy sensitivity in the very small mixing angle limitfor 〈φr

νe〉 = 〈φr

νe〉 � 1.

Finally we discuss the issue of using an effective two flavor evo-lution followed in this Letter. As mentioned earlier if the analysisis done using all three flavors only the ‘double splits’ for IH arisingin a small part of the (φr

νe, φr

νe) parameter space seen in the two

flavor case change to ‘single split’ while things remain unchangedfor NH. This effect would have been important for calculations fora single SN event if the initial flux ratios fell in this part of theparameter space. However for DSNB with averaging done over theSN distributions (Di ) the three flavor treatment will result in smallcorrections to the effective two flavor results presented here.

To summarize, using some reasonable simplifying assumptionswe for the first time take into account, for calculating DSNB, a dis-tribution in the relative fluxes in SN neutrinos and study its effecton the ability to distinguish the two hierarchies through a futureobservation of DSNB. A more detailed and rigorous study involv-ing multi-angle effects and three flavors will be able to give moreexhaustive answers to many of the aspects and should be pursuedin future. But the model calculation carried out in this Letter as-serts that for DSNB which is a blend of many SN, not all identical,it is more difficult to distinguish IH from NH compared to a sin-gle SN or the hypothetical case of DSNB made of supernovae withidentical flux ratios. A few favorable scenarios for this hierarchydistinguishability are indicated. We feel that the central finding ofthis first study is robust and will survive more detailed analysis

Acknowledgements

The authors would like to thank Basudeb Dasgupta for valuablecomments and suggestions. S. Chakraborty acknowledges hospital-ity at Harish-Chandra Research Institute during the developmentstage of this work. S. Choubey acknowledges support from theNeutrino Project under the XI Plan of Harish-Chandra Research In-stitute. K. Kar and S. Chakraborty acknowledge support from theprojects ‘Center for Astroparticle Physics’ and ‘Frontiers of Theoret-ical Physics’ of Saha Institute of Nuclear Physics.

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