Numerical Methods and Approximations for the Calculation of Solar Neutrino Fluxes in Three ...

119 Pages·2012·14.43 MB·English

Numerical Methods and Approximations for the
Calculation of Solar Neutrino Fluxes in Three
Flavours Focused on Results for the Sudbury
Neutrino Observatory
by
Ryan Martin
A thesis submitted to the Department of Physics in conformity with the requirements for the
degree of Master of Science
Queen’s University
Kingston, Ontario, Canada
January 2006
Copyright cRyan Martin, 2006
(cid:13)
Abstract
Inthisthesis,wepresentanoverviewofnumericalmethodsandapproximationsforcalculatingsolar
neutrino(cid:13)uxesattheSudburyNeutrinoObservatoryinthree(cid:13)avours. We(cid:12)ndthatthedaytime(cid:13)ux
is well approximated by a three (cid:13)avour adiabatic propagation with a two-(cid:13)avour jump probability.
We also introduce variable step size algorithms for the numerical propagation of neutrinos through
the Sun. A formula for the survival probability of solar neutrinos that have traversed the Earth is
presented in three (cid:13)avours and the dependence on (cid:18) is demonstrated. It is then shown that an
13
experiment for low energy solar neutrinos would provide the best constraints on (cid:18) and that there
13
is a theoretical dependence of the day-night asymmetry on (cid:18) but that the e(cid:11)ect is too small to
13
be measuredwith currentexperiments. Finally, weshowthe e(cid:11)ect of matter aboveanunderground
detector,suchastheSudburyNeutrinoObservatory,onameasurementoftheday-nightasymmetry
and conclude that the night bin should be longer than is implied in the literature.
i
Acknowledgments
I would like to start by thanking my supervisors, Dr. Mark Chen and Dr. Ian Towner, without
whom this work would have been impossible. Their cordiality and willingness to always help me
are much appreciated. I am grateful for the opportunity they gave me to do theoretical research
amongst the Sudbury Neutrino Observatory group. I would like to thank Dr Chen for the insight
thathehasgivenmewhentryingtounderstandtheresultsofmycalculations. Iwouldliketothank
Dr. Towner for all the help in deriving and understanding theory as well as teaching me an overall
frame of mind for doing theoretical research.
It was a pleasure to work with the SNO group at Queen’s as they have provided a very amicable
and professional environment in which to do my work. In particular, I would like to thank Dr.
Aksel Hallinand Dr. PeterSkensvedforthe di(cid:11)erent perspectivesthat they provided,fromwhich I
learned how to defend my work. It is with great pleasurethat I preparemyself to start a doctorate
with the SNO group.
I would also like to thank my fellow graduate students and colleagues for everything they have
taught me and for creating a positive work atmosphere. In particular, I would like to thank Ryan
MacLellan, Mark Kos and Jose Maneira for all the computer related help, as well as discussing my
work with me. I would also like to acknowledge Alex Wright for the many insightful theoretical
discussions we have had together.
Finally, I would like to thank my parents, Claudia and Reginald, as well as my sister, Cristina, for
all the support they have given me over the years and for always believing in me. This work is
dedicated to them.
ii
Contents
Abstract i
Acknowledgments ii
List of Tables v
List of Figures vi
List of Symbols xi
1 Introduction 1
1.1 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Theory 4
2.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Two-FlavourVacuum Oscillations . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Three-FlavourVacuum Oscillations. . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 Matter Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Numerical Methods and Approximations Through Matter . . . . . . . . . . . . . . . 15
2.2.1 PropagationThrough Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Adiabatic Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 Deviations from Adiabaticity-Jump Probability . . . . . . . . . . . . . . . . . 23
2.2.4 Three-FlavourCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.5 Three-FlavourCase - Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.6 Boundary Between Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.7 The E(cid:11)ect of the Phase on PropagationAcross Boundaries . . . . . . . . . . 33
2.2.8 Flux at SNO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Neutrino Propagationin the Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.1 The Solar Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3.2 Numerical Survival Probabilities: Overview with Mixing Parameters Consis-
tent with Experimental Data (Two Flavours) . . . . . . . . . . . . . . . . . . 41
2.3.3 Numerical Survival Probabilities: Dependence on Mixing Parameters (Two
Flavours) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
iii
2.3.4 Numerical Survival Probabilities: Dependence on Mixing Parameters (Three
Flavours) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.5 Numerical Algorithms for Propagationthrough the Sun . . . . . . . . . . . . 50
2.3.6 Two-FlavourApproximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3.7 Three-FlavourApproximations . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.4 Neutrino Propagationin the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.4.1 Incoherent Beam of Neutrinos arriving at Earth. . . . . . . . . . . . . . . . . 68
2.4.2 Survival Probability in the Earth : Dependence on Mixing Parameters . . . . 69
2.5 Solar Neutrinos PropagatingThrough the Earth . . . . . . . . . . . . . . . . . . . . 73
2.5.1 Introduction to PSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.5.2 Derivation of PSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.5.3 Interlude: Two-FlavourCase . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.5.4 Back to Three Flavours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.5.5 Calculating and Choosing P and P . . . . . . . . . . . . . . . . . . . . . 83
E1 E2
2.5.6 Dependence of P on the Mixing Parameters (Two Flavours) . . . . . . . . 84
SE
2.5.7 Dependence of P on (cid:18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
SE 13
3 Applications 94
3.1 Measuring (cid:18) with Solar Neutrinos. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
13
3.1.1 With the Day-Night Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.1.2 With Fluxes at Di(cid:11)erent Energies . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2 De(cid:12)ning the Night Bin for Underground Detectors . . . . . . . . . . . . . . . . . . . 99
4 Summary and Conclusions 102
References 104
iv
List of Tables
1 Fitparametersfortheeigthorderpolynomialinequation(119)forthecurveinFigure
(64) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2 Day-night asymmetry for two di(cid:11)erent sizes of the night bin and corresponding day-
night asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
v
List of Figures
1 Survival probability below and above the critical density as a function of distance in
a medium of constant density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Resonance of matter mixing angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Survival Probabilitythrough amedium with density varyinglinearlyasa function of
distance (two (cid:13)avours) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Survival Probability through a medium with density increasing exponentially as a
function of distance (three (cid:13)avours) . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Matterenergyeigenvaluesasafunctionofdistanceinamediumwithdensityincreas-
ing exponentially (three (cid:13)avours) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6 Matter mass eigenstate content as a function of distance in a medium with density
increasing exponentially (three (cid:13)avours) . . . . . . . . . . . . . . . . . . . . . . . . . 21
7 Comparison of the adiabatic approximation in two (cid:13)avours with a numerical phase
average through a medium with density increasing linearly (two (cid:13)avours) . . . . . . 23
8 Comparisonofadiabaticapproximationintwo(cid:13)avourswithanumericalphaseaverage
through a medium with density increasing exponentially (three (cid:13)avours) . . . . . . . 26
9 Probabilities as a function of distance, through two slabs of constant density and
length 0:325R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
(cid:8)
10 Probabilities as a function of distance, through two slabs of constant density and
length 0:425R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
(cid:8)
11 Probabilitiesasafunctionof distance(inunitsof the Earthradius), throughvacuum
then two di(cid:11)erent slabs of constant density and (cid:12)nally through vacuum again for a
coherent beam of neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
12 Probabilitiesasafunctionof distance(inunitsof the Earthradius), throughvacuum
then two di(cid:11)erent slabs of constant density and (cid:12)nally through vacuum again for a
coherent beam of neutrinos, out of phase compared to Figure (11) . . . . . . . . . . 35
13 Probabilitiesasafunctionof distance(inunitsof the Earthradius), throughvacuum
then two di(cid:11)erent slabs of constant density and (cid:12)nally through vacuum again for a
incoherent beam of neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
14 Diagram showinga possible trajectoryfor a neutrino that originatedin the Sun asis
detected in the Earth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
15 Solar neutrino (cid:13)uxes with uncertainties as a function of energy at 1AU . . . . . . . . 39
vi
16 Fraction of 8B neutrinos produced at each solar radius . . . . . . . . . . . . . . . . . 40
17 Logarithm (base 10) of the electron density (in cm(cid:0)3) in the Sun as a function of
solar radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
18 Survival probability in two (cid:13)avours through the Sun for a 10MeV neutrino with
(cid:1)m =7 10(cid:0)5eV2 and tan2((cid:18) )=0:42 . . . . . . . . . . . . . . . . . . . . . . . 42
21 12
(cid:2)
19 Survival probability in two (cid:13)avours through the Sun for a 5MeV neutrino with
(cid:1)m =7 10(cid:0)5eV2 and tan2((cid:18) )=0:42 . . . . . . . . . . . . . . . . . . . . . . . 42
21 12
(cid:2)
20 Mattermass-squareddi(cid:11)erenceasafunctionofdistancethroughtheSunforneutrino
energies of 5MeV and 10MeV with mixing parameters (cid:1)m = 7 10(cid:0)5eV2 and
21
(cid:2)
tan2((cid:18) )=0:42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
12
21 MattermixingangleasafunctionofdistancethroughtheSunforneutrinoenergiesof
5MeV and10MeV with mixingparameters(cid:1)m =7 10(cid:0)5eV2 andtan2((cid:18) )=0:42 43
21 12
(cid:2)
22 Second matter-mass eigenstate content as a function of distance through the Sun for
neutrinoenergiesof5MeV and10MeV withmixingparameters(cid:1)m =7 10(cid:0)5eV2
21
(cid:2)
and tan2((cid:18) )=0:42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
12
23 SolarneutrinoLMAenergyspectraldistortioncalculatednumericallywithtwo(cid:13)avours
with (cid:1)m =7 10(cid:0)5eV2 and tan2((cid:18) )=0:42. . . . . . . . . . . . . . . . . . . . . 44
21 12
(cid:2)
24 Survival probability through the Sun for a very small vacuum mixing angle for a
10MeV neutrino with (cid:1)m =7 10(cid:0)5eV2 and sin2(2(cid:18) )=0:039 . . . . . . . . . . 46
21 12
(cid:2)
25 Phase-averagedsurvivalprobabilitythroughtheSunfor10MeV neutrinoswith(cid:1)m =
21
7 10(cid:0)5eV2 and di(cid:11)erent values of the mixing angle. . . . . . . . . . . . . . . . . . 46
(cid:2)
26 Phase-averagedsurvivalprobabilitythroughtheSunfor10MeV neutrinoswithsin2(2(cid:18) )=
12
0:833 and di(cid:11)erent values of the mass-squareddi(cid:11)erence . . . . . . . . . . . . . . . . 47
27 Contourplot of the locationof the MSW resonance(in units of the solarradius) asa
function of log (sin2((cid:18) )) and log ((cid:1)m21) . . . . . . . . . . . . . . . . . . . . . . . 48
10 12 10 E
28 Contour plot of the phase-averagedP for neutrinos originating in the center of the
ee
Sun as a function of log (sin2((cid:18) )) and log ((cid:1)m21) . . . . . . . . . . . . . . . . . . 48
10 12 10 E
29 Phase-averaged survival probability as a function of distance through the Sun for
di(cid:11)erent values of sin2(2(cid:18) ) at an energy of 10MeV and (cid:1)m =1 10(cid:0)3eV2. . . . 49
13 31
(cid:2)
30 Phase-averaged survival probability as a function of distance through the Sun for
di(cid:11)erent values of (cid:1)m at an energy of 10MeV with sin2(2(cid:18) )=0:4 . . . . . . . . 50
31 13
31 Contour plot of the phase averagedP for neutrinos originating in the center of the
ee
Sun as a function of log (sin2((cid:18) )) and log ((cid:1)m21) . . . . . . . . . . . . . . . . . . 51
10 13 10 E
vii
32 Contour plot of the phase-averaged P with sin2(2(cid:18) ) = 0:5 (left) and di(cid:11)erence
ee 13
with P when sin2(2(cid:18) )=0:5forneutrinosoriginatingin the centerofthe Sun asa
ee 13
function of log (sin2((cid:18) )) and log ((cid:1)m21). . . . . . . . . . . . . . . . . . . . . . . 51
10 12 10 E
33 Step size (in units of R ) as a function of distance through the Sun as calculated by
(cid:12)
an algorithm similar to LOGSTEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
34 HistogramshowingthenumberofstepstakenateachradiusbytheLOGSTEPalgorithm 55
35 Step size (in units of R ) as a function of distance through the Sun as calculated by
(cid:12)
an algorithm similar to EIGENSTEP . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
36 Histogram showing the number of steps taken at each radius by the EIGENSTEP
algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
37 Step size (in units of R ) as a function of distance through the Sun as calculated by
(cid:12)
an algorithm similar to VARSTEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
38 HistogramshowingthenumberofstepstakenateachradiusbytheVARSTEPalgorithm 59
39 Adiabadicity parameter in the Sun plotted as a function of log (sin2((cid:18) )) and
10 12
log ((cid:1)m21) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
10 E
40 Contour plots of P calculated with the adiabatic approximationand relative di(cid:11)er-
ee
ence from P calculated with a phase average. . . . . . . . . . . . . . . . . . . . . . 61
ee
41 Contour plot of P calculated with the two jump probabilities (in two (cid:13)avours) as a
ee
function of log (sin2((cid:18) )) and log ((cid:1)m21) . . . . . . . . . . . . . . . . . . . . . . . 62
10 12 10 E
42 Percentdi(cid:11)erencebetweensolarsurvivalprobabilitycalculatedwithPetcovandParke
jump probabilities as a function of log (sin2((cid:18) )) and log ((cid:1)m21). . . . . . . . . . 63
10 12 10 E
43 Contour plot of P in the adiabatic approximation (left) and relative di(cid:11)erence
ee
with the result calculated with a phase average (three (cid:13)avours) as a function of
log (sin2((cid:18) )) and log ((cid:1)m21) for tan2((cid:18) )=0:42 . . . . . . . . . . . . . . . . . 64
10 13 10 E 12
44 ContourplotofP inthethree-(cid:13)avouradiabaticapproximationusingthetwo-(cid:13)avour
ee
Parkejumpprobabilityandrelativedi(cid:11)erencewiththe resultcalculatedwithaphase
average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
45 Contour plot of P using the (cid:12)rst-order correction for non zero (cid:18) and the two-
ee 13
(cid:13)avourParkejumpprobability(left)andrelativedi(cid:11)erencewiththeresultcalculated
with a phase average (right), as a function of log (sin2((cid:18) )) and log ((cid:1)m21) for
10 12 10 E
tan2((cid:18) )=0:42 and sin2(2(cid:18) )=0:5. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12 13
viii
46 Contour plot of P using the second-order correction for non zero (cid:18) and the two-
ee 13
(cid:13)avourParkejumpprobability(left)andrelativedi(cid:11)erencewiththeresultcalculated
with a phase average (right), as a function of log (sin2((cid:18) )) and log ((cid:1)m21) for
10 12 10 E
tan2((cid:18) )=0:42 and sin2(2(cid:18) )=0:5. . . . . . . . . . . . . . . . . . . . . . . . . . . 66
12 13
47 Electron density pro(cid:12)le for the Earth based on the PREM model . . . . . . . . . . . 67
48 Survival probability(two (cid:13)avours) for a neutrino that started one Earth radiusaway
fromtheEarth,thentraversestheEarth,and(cid:12)nallytravelsonemoreEarthradiusin
vacuum. Theenergyissetto10MeV, with (cid:1)m =2 10(cid:0)5eV2 andtan2((cid:18) )=0:42. 68
21 12
(cid:2)
49 Survival probability and matter mass eigenstate content for a neutrino that started
oneEarthradiusawayfromtheEarth,thentraversestheEarth,and(cid:12)nallytravelsone
moreEarthradiusinvacuum. Theenergyissetto10MeV,with(cid:1)m =2 10(cid:0)5eV2
21
(cid:2)
and tan2((cid:18) ) = 0:42 and the neutrino started in the state de(cid:12)ned by equation (75)
12
with the phase set to 0 (left) and (cid:25) (right). . . . . . . . . . . . . . . . . . . . . . . . 70
50 sin2(2(cid:18) ) and (cid:1)M as a function of distance for the case illustrated in Figure (49). 70
m 21
51 Survival probability and matter mass eigenstate content for a neutrino that started
oneEarthradiusawayfromtheEarth,thentraversestheEarth,and(cid:12)nallytravelsone
moreEarthradiusinvacuum. Theenergyissetto10MeV,with(cid:1)m =2 10(cid:0)5eV2
21
(cid:2)
andtan2((cid:18) )=0:42andtheneutrinostartedinthestatede(cid:12)nedbyequation(75)but
12
the probabilities are averagedoverthe phase. One notes that the Earth reintroduces
coherence in the neutrino beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
52 Survival probability, matter-mass eigenstate content, mixing angle and matter mass
squareddi(cid:11)erenceasafunctionofdistancethroughtheEarthfora10MeV neutrino,
with (cid:1)m =7 10(cid:0)5eV2 and tan2((cid:18) )=0:42. The Earth has a small in(cid:13)uence on
21 12
(cid:2)
the (cid:13)avour content. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
53 Relative di(cid:11)erence (in percent) between vacuum classical survival probability and
phase averaged survival probability for a trajectory through the Earth’s core as a
function of log ((cid:1)m21) and log (sin2((cid:18) )) . . . . . . . . . . . . . . . . . . . . . . . 72
10 E 10 12
54 Survival probability of electron solar neutrinos as a function of energy at SNO with
(cid:1)m = 4 10(cid:0)5eV2 and tan2((cid:18)) = 0:42 during the day, night and averaged over a
21
(cid:2)
year and corresponding value of the day-night asymmetry in percent . . . . . . . . . 85
55 Zenith angle exposure function for SNO as a function of Nadir angle (in radians),
taken from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
ix