1. (Xin-Heng Guo, Bing-Lin Young) Beijing Normal University
A Possible Method to Measure Small 13 in the Detection of Supernova Neutrinos
Ming-Yang Huang
(Xin-Heng Guo, Bing-Lin Young)
Beijing Normal University
(hep-ph/ )

2. Contents Detection of SN neutrinos.
Possible method to measure very small 13 at Daya Bay.
Apply this method to some other current experiments.
Summary

3. I. Detection of SN neutrinos
Supernova explosion: natural laboratory to study fundamental issues of physics and astrophysics.
Observation SN neutrinos can serve as an early warning for the optical emission of a type II SN.
SN neutrinos have valuable information of deep inside the core; can be helpful to study of intrinsic properties of neutrinos.

4. Energy spectrum
Since cross section of interactions depend on neutrino energy, the energy spectra of neutrinos are not simple blackbodies. As a result, the energy spectrum has a pinched shape compared with the Fermi-Dirac distribution.
Lα(0) (Tα ): luminosity (temperature) of να ; ηα : pinching parameter; ESN(0) : total energy release. Typical values from numerical simulations are ( < E α > ~ 3.15 Tα):

5. SN neutrinos’ propagation in Earth
Neutrinos from SN explosion will go through some portion of the Earth before reaching detector. Earth effects have to be taken into account.
For Earth matter effects, we use the realistic matter profile:

6. Earth Matter Effect
The treatment of the Earth matter effect has been discussed in detail in Dighe and Smirnov, Phys. Rev. D62 (2000)
Ioannisian and Smirnov, Phys. Rev. Lett. 93 (2004)
Ioannisian, Kazarian, Smirnov, and D. Wyler, Phys. Rev. D71 (2005)
Incident angle
Detector
R: radius of Earth
h: depth of detector AD
Centre of Earth
X=AB OB

7. Neutrino flux at detector
Consider the MSW effect at SN and the matter effects on Earth, one can obtain neutrino flux as follow (PH is the crossing probability at the high resonance region inside the SN and P2e is the probability that a neutrino mass eigenstate ν2 enters the surface of the Earth and arrive at the detector as an electron neutrino):
(Normal hierarchy)
(Inverted hierarchy)

8. the electron number density: and we defined:
The probability P2e has been calculated in Ioannisian and Smirnov, Phys. Rev. Lett. 93 (2004) by applying Schroedinger Eq. in low density matter (neglecting ):
the potential in the Earth:
the electron number density:
and we defined:

9. Event number
Now we calculate the predicted numbers of event that can be observed through various reaction channels at current neutrino experiments.
Integrating the product of the target number NT, the cross section of each channel σ(i) , and the neutrino flux function over the neutrino energy,
D: distance between SN and Earth, i: different channels.
At Daya Bay, liquid scintillator is Linear Alkyl Benzene (LAB), C6H5 – CnH2n+1, n=9~14. Let ratio of C and H to be 0.6, then for total detector mass 300 tons,

10. Cross sections Inverse beta decay has the largest cross section:
neutrino-electron elastic scatterings:
Charged-current capture of electron neutrino on Carbon:

11. Charged-current capture of anti-electron neutrino on Carbon:
Neutral-current inelastic scatterings on Carbon :

12. The event numbers observed at the Daya Bay experiment as a function of the incident angle  when

13. It can learn from the above plot that:
Earth matter effect depends on the incident angle of the neutrino, the mass hierarchy, and the flip probability PH .
When the incident angle of SN neutrino is smaller than a value 0 ~ 90o (L < 100km), Earth matter effect can be ignored for all reactions.
Earth matter effect becomes large and reaches a maximum for  ~ 92o -- 95o .
When  is larger than about 100o, the Earth matter effect is insensitive to .
The inverse beta decay could have the largest Earth matter effect among all the channels: 7%.
For neutrino-electron elastic scattering and reactions with 12C , the maximum Earth matter effect could be as large as 2%.

14. II. Possible method to measure very small 13 at Daya Bay
Using Landau-Zener formula, the crossing probability at high resonance region depends on neutrino energy, neutrino mass difference, mixing angle 13 and density profile of SN:
Take the matter density profile of SN as (the constant C depends on the amount of electron capture during the star collapse and its value between 1 and 10):
For small 13, F≈1. Then

15. From the plot, when 13=0, PH=1.
When 13 varies between 0o and 1.5o, PH varies between 0 and 1.
When 13 > 1.5o, PH is nearly 0.
The variation of PH depends on neutrino energy and parameter C.
In the following, we will consider a possible method to measure small 13 by detecting the event numbers of SN neutrinos at Daya Bay. As an example, we consider

19. From the above figures, it is known that:
(1), when 13 is smaller than 1.5o, N is very sensitive to 13; when 13 is larger than 1.5o, N is nearly independent of 13. Therefore, very small 13 can be measured by detecting the event numbers of SN neutrinos.
(2), At the Daya Bay experiment, the event numbers of SN neutrinos can be measured in the inverse beta-decay. This will help us to get information about 13 smaller than 3o.
　　 (3), If the actual case is the normal hierarchy, we need to detect SN neutrino event numbers in the processes of neutrino-electron scattering and neutrino-carbon scattering. This, however, will be difficult for the Daya Bay experiment.
In order to reduce the uncertainties of the luminosity and distance of an SN, we define Re which represents the ratio of the event numbers of electron and anti-electron neutrino. Therefore, the mixing angle 13　can be determined by Re .

21. III. Apply this method to some other current experiments
Similar to Daya Bay, for Super-k, Kamland, LVD, MiniBooNe, Double Chooze, Borexino, SNO (H2O), the main neutrino reactions are inverse beta-decay. Therefore, vary small 13 can be measured by these experiments while the hierarchy is inverted.

22. D2O
It can be seen that in the cases of both normal hierarchy and inverted hierarchy, event numbers of reactions of neutrinos with deuteron change obviously when 13 changes between 0o and 1.5o.
So in both cases one can get information about small 13 by observing SN neutrinos.
This is an unique advantage of D2O.

23. Summary of current detectors
Summary of current detectors. N (I) represents normal (inverted) hierarchy. We list types of liquid scintillator, detector masses, total numbers of targets (proton or deuterium), depth of detectors, location of detectors, and event numbers for inverse beta-decay process except for D2O at SNO (in this case we list the result for the heavy water reaction).

24. IV. Summary
In our work,
we have calculated the realistic Earth matter effects on the detection of type II SN neutrinos at Daya Bay experiment;
using the relation between the event number of SN neutrinos and the mixing angle 13 , we propose a method to determine very small 13 in the range of 0o and 1.5o by detecting the event number of SN neutrinos;
we also apply this method to some current neutrino detectors.
Therefore, very small 13 can be measured by detecting the event numbers of SN neutrinos.