The search of Dark Matter annihilation signals in
neutrino detectors
Universidad de los Andes, Colombia
Sebastian Enrique Sanchez Herrera
Director:
Juan Carlos Sanabria
Acknowledgements
First, I would like to thank Proffesor Juan Carlos Sanabria for all the support he has given me during the elaboration of this project. Adittionally, I want to thank my Mother. Without her constant support during my entire life, none of this would have been possible. I really cannot thank her enough for all that she has done for me. Adittionally, I would like to express my appreciation to my family and relatives, especially to my sister, for always being there for me and for all the things that they have taught me. Lastly, I would like to thank Diana Carolina Lozano, for encouraging me to always give my best in these last few months.
Contents
Contents 3
1 Introduction 4
2 Supersymmetric Dark Matter 7
2.1 Minimal Supersymmetric Standard Model . . . 7
2.2 R-parity conservation . . . 8
2.3 Neutralinos as LSPs . . . 9
2.4 Neutralinos as SUSY dark matter . . . 10
2.5 Neutralino Interactions . . . 11
3 Neutralino annihilation in the Sun 13 3.1 Neutralino relic density in the galactic halo . . . 13
3.2 Absorption of Neutralinos in the Sun . . . 14
3.3 Neutralino-neutralino annihilation rates in the solar core . . . 15
3.4 Production of High-Energy Neutrinos . . . 16
4 Neutrino oscilation 19 4.1 Neutrino Oscilation theory . . . 19
4.2 Analitic expression for the time evolution matrix . . . 22
4.3 Oscilations through the solar medium . . . 24
5 High-energy neutrino fluxes 26 5.1 Initial neutrino energy spectrum . . . 26
5.2 Methodology . . . 29
5.3 Results . . . 38
6 Conclusion 43
Appendices 44
A Validation graphs for the neutrino oscilation software developed in
this project 45
Chapter 1
Introduction
The fact that a high percentage of the mass content of our universe is in the form of a new type of matter, known as dark matter, has been generally accepted by the sci-entific comunity. Dark matter does not emit or absorb radiation, which means that it cannot be observed with telescopes, but its existence has been induced from its gravitational effect on visible matter, and several other physical phenomena, which requires its presence in order to be explained. Nowadays, one can even find models for the dark matter density in our galaxy, our universe, etc. With that being said, it is still unclear what the microscopic composition of dark matter is. Currently, this is one of the most important problems to be solved in physics. As has been pointed out in the past, it is unlikely (although still possible) that the constituents of dark matter are particles within the Standard Model (SM).
There is evidence that the origin of dark matter is non-baryonic. Big bang nu-cleosynthesis predicts a baryonic density much lower than the lowest accepted value for dark matter density in the universe [1]. Furthermore, the fact that the dark matter constituents have not been discovered in our planet implies that they should not participate in the strong or electromagnetic interactions, which are dominant at the atomic and nuclear levels. As a result of this, the neutrinos would be about the only elementary particles in the SM that could be considered valid candidates. The problem is that, as pointed out in [1], the computational simulations conducted in the past suggest that the formation of the current structure of the universe seems really difficult to achieve assuming neutrino dark matter.
Nevertheless, there are other theoretical models beyond the SM that could pro-vide valid candidates for the constituents of dark matter. These “valid candidates” should be weakly interacting particles sufficiently massive to account for the dark matter density in our universe, or WIMPs (Weakly Interacting Massive Particles) and among the many different possibilities, the one that we are interested in for the purpose of this project is the neutralino; a WIMP predicted by the supersymmetric theories.
The neutralino is the LSP (Lightest Supersymetric Particle) in some supersymmet-ric models, and is also considered stable in models containing a particular property known as R-Parity conservation, which prevents decays from supersymmetric part-ners to ordinary particles. It has a non-zero coupling with ordinary matter, as the
supersymmetric models predict that it participates in the weak interaction. As a re-sult, the neutralinos (χ) should conduct neutralino-neutralino annihilation processes of the formχχ→ll, wherel is an ordinary, lighter particle. Thesel particles should subsecuently decay into lighter particles, including highly energetic neutrinos, that could be detected afterwards, providing a distinct method to experimentally test the validity of the supersymmetric dark matter hypothesis. The high energy of these neutrinos would be a direct consecuence of the theoretical accepted values for the neutralino mass, which are thought to be in the range between 10GeV to a few TeV [1].
In order to obtain a clear signal of the existence of supersymmetric dark matter using neutrino detection, one could try to detect high energy neutrinos coming from the Sun. Observational data suggests that our galaxy is inmersed in a dark matter halo. Asumming that the halo is mainly composed by neutralinos, then these par-ticles should be scattered by the nuclei in the Sun, as neutralinos have a nonzero coupling with ordinary matter. These scattering processes could sometimes cause enough energy loss so that the neutralinos in the Galactic halo crossing through the Sun end up trapped on its gravitational field. The Neutralinos trapped in the Sun will then accumulate in the solar core (as a result of successive scatterings whithin the solar medium) and once in there will participate in annihilation processes, pro-ducing neutrinos several times more energetic than the regular solar neutrinos. The signals obtained in this case should be clean as the neutrinos do not interact much with matter. [1]
The objective of this project is to study how it would be possible to obtain evidence of supersymmetric dark matter annihilation in the Sun using neutrino detectors. This particular problem is really interresting today because the comprobation of the existence of supersymmetric dark matter would not only explain the origin of dark matter, which is already outstanding, but would also provide solid scientific verification of supersymmetry (SUSY), which would be of major importance in par-ticle physics. Unfortunately, the time for preparation was not enough to cover all the aspects involved and thus the project had to be reformulated. Only the problem of estimating the neutrino fluxes that would arrive to the earth as a result of neu-tralino annihilation in the Sun will be addressed here, leaving the details concerning the detection of the neutrinos for future analisys.
To find the neutrino fluxes several factors have to be taken into account. First, the particular supersymetric model that would be used to define the properties of the neutralino has to be specified. Using this model, it is then possible to determine the corresponding neutralino masses, interaction cross sections with the nuclei in the Sun and annihilation channels that would produce neutrinos, with their corre-sponding cross sections. Is important to clarify here that any model contains a given number of undetermined free parameters, so that there is no unique description of the model. The neutralino properties depend on the choices made for such param-eters.
The interaction cross sections can be used to study the capture rate of neutrali-nos in the Sun. For this purpose, it is also necessary to research the properties of
the Galactic dark matter halo. This capture rate can then be used, together with the annihilation cross sections to describe the annihilation rate of neutralinos in the solar core. Subsequently, the energy distributions for the number of neutrinos of a given flavor i, associated to each possible neutralino annihilation channel, have to be researched. The total number of neutrinos of each flavor produced per second in the solar core can then be determined and from there the total neutrino fluxes arriving to the Earth.
Nevertheless, the total neutrino fluxes could not be used to account for neutralino annihilation in the Sun. There is an aditional complication which arises from the capacity of the neutrino detectors to estimate the direction from where the neutri-nos came from, but that must nevertheless be accounted for in this project, as the fluxes studied here had to be relevant for future neutrino detection analysis. The complication arises from the experimental fact that only muon neutrinos produce signals in neutrino detectors that can be pointed back to the place from where the neutrino came with a sufficiently high degree of accuracy [2], in other words, from the fact that only the muon neutrino fluxes arriving to the earth could provide solid evidence of neutralino annihilation in the Sun. This becomes a problem because the estimation of the total muon neutrino fluxes arriving to the earth requires to take into account the effect of neutrino oscilations. This effect is simply the physical phenomena of transitioning between neutrino flavors, which is possible because the three neutrino flavors do not represent quantum states asociated to a definite mass and are in fact linear conmbinations of another three quantum states of mass. The problem is that the initial flavor ratios of the neutrinos produced in the Sun, as a result of neutralino annihilation, would change as the neutrinos propagate through the Sun as a result of neutrino oscilations. Thus, the theory of neutrino oscilation also has to be studied, to take into account the change in the initial flavor ratios of the neutrinos.
All these factors were accounted for in this project and the differential energy distri-butions of the high-energy muon neutrinos fluxes that would arrive to the earth were estimated for three different SUSY models, with neutralino masses of 73.33 , 78.27 and 92.25 GeV. The idea was to illustrate how these fluxes could be determined for a particular SUSY model. In the future, it would be interesting to cover a greater range of SUSY models and to calculate the fluxes using this information.
Finally, the method for finding signals of SUSY dark matter annihilation in the Sun using these estimated fluxes is clear. If one could find muon neutrino signals in a particular detector that can be pointed back to the Sun, with energies much higher that those of ordinary neutrinos, it should be tested whether these are compatible with the fluxes estimated using any of the particular SUSY models. If so, this would provide solid evidence of the existance of neutralino-based dark matter.
Chapter 2
Supersymmetric Dark Matter
In this chapter, the Minimal Supersymmetric Standard Model (MSSM) is presented. A brief review of this model will be given in order to introduce the theoretical frame-work under which the neutralino is defined. The main properties of the neutralinos will then be studied, and also the main reasoning behind the hypothesis of the neu-tralinos as the dark matter constituents. Finally, the neutralino interactions which would be relevant for this project, namely neutralino annihilation and scattering from ordinary matter, will be reviewed. The information given in this chapter will afterwards be used in chapter 5 when describing the methodology that was followed in order to calculate the neutrino fluxes arriving to the Earth as a result of dark matter annihilation in the Sun.
2.1
Minimal Supersymmetric Standard Model
The success of the SM in explaining the physical phenomena concerning elemen-tary particle interactions has placed it as one of the most important theoretical achievements in the history of physics. The theory is able to predict a wide range of experimental results in particle physics with a great degree of accuracy and thus is thought to describe satisfactorily the different interactions among the currently known elementary particles.
However, the theory is far from being a complete theory. The list of important phenomena that escapes the scope of the model includes, among others, the gravi-tational interaction and the accelarated expansion of the universe. Additionally, the model has some theoretical problems at the energy scales of the Grand Unification Theory (∼ 1013 TeV) and it fails to explain the microscopic composition of dark
matter, as explained in the introduction.
As a result of these problems, several models beyond the SM have been devel-oped. Some of the most promising are the Supersymmetric models, as they provide some solutions to the theoretical problems in the SM, contain a formalism that gives some insight in the unification of gravitation to the other interactions and are able to provide valid candidates for the constituents of dark matter.
The main idea behind SUSY is the introduction of an extension of the space-time symmetry, known as supersymmetry. SUSY would be a fundamental symmetry in
our universe, that relates every boson field to a fermion field in such a way that every known particle of a given type has a superpartner of the other type.
Among the many supersymetric models, the one that interests us for the problem of SUSY dark matter is the MSSM. This model consists of all the interaction fields of the SM, with the addition of some new Higgs fields, plus the superpartners predicted by supersymmetry. It is, as its name suggests, the minimal possible extension of the SM, when one introduces the mathematical formalism of SUSY. There is no argument to think that this minimal extension is more valid than other extensions of the SM using SUSY, but it is frequently used as it is the simplest model. The SM particles contained in the MSSM, together with their corresponding superpartners are listed in table 2.1, which was taken from [1].
Table 2.1: Particles in the MSSM
: Normal particles
Name Symbol
up quarks q=u, c, t
down quarks q=d, s, b
leptons l=e,µ,τ
neutrinos ν
gluons g
W boson W±
charged Higgs H±
photon γ
Z boson Z0
Heavy Scalar Higgs H10
Light Scalar Higgs H20
Pseudoscalar Higgs H30
: Superpartners
Name Symbol
up squarks q˜u1, ...,q˜u6
down squarks q˜d1, ...,q˜d6
sleptons ˜l1, ...,˜l6
sneutrinos ν˜1,...,˜ν3
gluinos ˜g
charginos χ˜1±, ˜χ2±
neutralinos χ˜10,..., ˜χ40
Finally, it is important to mention that the MSSM has a great deal of free parameters, without even including the non-supersymetric free parameters found in the SM, so that it is still a very complicated model to study phenomenologi-cally. A summary of these parameters can be found in [1], which includes the ratio tan(β) of the vacuum expectation values of the neutral Higgs bosons, three gaugino mass parameters M1, M2 and M3, the higgsino mass parameter µ, the mass of the
pseudoscalar higgs mA, among many others (there are approximately 63 of such
parameters). Nevertheless, the actual description of the free parameters used in this project will be an estimated simplification, that contains a reduced number. The details of such simplification can be found in [3], and the only parameters involved would be: tan(β), M2, µ, mA, together with m0, At, Ab, which would account for
all the remaining parameters through a series of simplifying assumptions.
2.2
R-parity conservation
A very important concept, of great relevance in the SUSY formalism, is R-parity. It is defined by the quantity R = (−1)3B+L+2j, where B is the baryonic number,
L is the Leptonic number and j is the spin. Usually R-parity would be conserved, as B and L are conserved in the SM. Nevertheless, in the SUSY models, L ans B are no longer conserved in all cases, which means that R parity might be broken. For the problem of SUSY dark matter, the most relevant models are those with R-parity conservation. In such models, R=(+1) for SM particles and R=(-1) for the supersymetric partners. As R is conserved, there will be no possibility that a supersymetric particle decays to a SM particle, making it possible to ensure the stability of the LSP. [4]
2.3
Neutralinos as LSPs
Within all the particles involved in the model we are particularly interested in the Neutralinos, which are linear combinations of the supersymmetric partners of the photon and Z0 ( ˜B,W˜
3) as well as the supersymmetric partners of the neutral Higgs
bosons ( ˜H10, ˜H20) [3].
The four neutralino states ˜χi0 are given by the eigenvectors of the mass matrix
of the neutralinos in the basis ( ˜B,W˜3, ˜H10, ˜H20). This matrix is given by the
follow-ing expresion [3]:
Mχ˜i0 =
M1 0
−g0v1
√
2
g0v2
√
2
0 M2
g√0v1
2
−√g0v2
2
−g0v1
√
2
g0v1
√
2 δ33 −µ
g0v2
√
2
−g0v2
√
2 −µ δ44
(2.1)
where hv1i and hv2i are the vacuum expectation values of the neutral Higgs bosons,
g and g’ are the gauge coupling constants of SU(2) and U(1) and δ33, δ44 are the
most important one loop corrections. These are quantities that depend on some of the free parameters in the model [5]. The masses of the four neutralino states will be given by the eigenvalues of Mχ˜i0.
A general result of the MSSM is that the LSP is the lightest neutralino and from now on, the name “neutralino” will be used to refer to that particular state. If we limit ourselves to the MSSM with R-parity conservation, then the Neutralino would also be stable and all other superpartners would eventually decay into them. Fi-nally, from the theory considerations, we know that this particle could only interact weakly (and gravitationally) and that its mass could take any value from 10GeV to a few TeV. [1]
All these properties, as will be discussed in the following section, make the Neu-tralino of the MSSMs with R-parity conservation (an stable WIMP) a very good
candidate for the constituent of dark matter.
2.4
Neutralinos as SUSY dark matter
As has been stated before in the introduction, the WIMPs are among the best can-didates to account for the contituents of dark matter. Beyond the fact that they do not interact strongly or electromagnetically, in agreement with the ”dark” be-haviour of dark matter, calculations of the current density of the universe performed with WIMPs generally produce results suggestively close to the currently measured universal density.
The mechanism that would allow a current presence of WIMPs in the universe is presented in [1], and has the following characteristics. Suppose that there is a new particleχ. The particle is stable and has a well-defined mass mχ. In the early
stages of the universe, at a point where the temperature exceeded the value mχ,
there was a vast population of such particles in thermal equilibrium. It is assumed that those particles annihilated into ligher particles and that the inverse process also happened so that there was an equlibrium state.
When the universe had a temperature high enough so that mχ ≪ T, the number
ofχ particles mantained an equilibrium value, as the processes of both annihilation and production from lighter particles were possible. As T decreased in the early stages of the universe, it reached a point when T ≪ mχ. At that point there was
not enough available thermal energy so that the process of production ofχparticles from lighter particles could take place and thus the number ofχparticles decreased very fast. Thus, if the expansion of the universe had been slow enough as to main-tain equilibrium always, the number density ofχ particles would have decreased to a negligible value.
Nevertheless, the rate of the annihilation processes of the χ into lighter particles Γ =hσvinχ(wherehσviis the total cross section ofχannihilation times the relative
velocity v and nχ is the number density of χ particles) decreased with density so
that there was a point where the probability of two χ particles finding each other to annihilate became negligible as the universe expanded. That moment was char-acterized by the relation Γ< H, where H is the expansion rate of the universe. At that point, the equilibrium condition was lost leaving a finite number density of χ
particles.
Taking into account this mechanism, the density of χ particles that would have remained in the universe would be given by the following expresion [1]:
Ωχh2 = (3×10−27cm3s−1/hσvi) (2.2)
where h is the Hubble constant (in units of 100km s−1 M pc−1), Ω
χ is the density
parameter of WIMPs, defined as ρx/ρc, with ρχ the density of WIMPs in the
uni-verse and ρc ≈ 10−5h2GeV cm−3 is the density corresponding to a plane universe.
On the other hand, using astronomical data, a value of Ωch2 = 0.1198±0.0026 has
As can be seen, the particle χ could account for the density of dark matter in our universe if σ is in the order of magnitude of the cross sections for weak inter-actions (∼ pb), and v is non relativistic. The conclusion is that if there was a vast population of massive, stable particles χ in the early stages of the universe (which seems valid for the neutralinos as LSPs, given that all other supersymetric particles would decay into them), then these particles could have a relic density today that matches that of dark matter if they interact weakly. Evidently, the neutralinos are excelent candidates for the constituents of dark matter.
2.5
Neutralino Interactions
The two main interactions governing the cosmological behaviour of the Neutralinos are Neutralino-Neutralino annihilation and scattering of Neutralinos from ordinary matter.
The fact that Neutralinos (denoted by χ) can annihilate into lighter particles in
χχ¯ → l¯l processes is a result of their coupling with ordinary matter through weak interactions. The total annihilation cross section times relative velocity for neutrali-nos hσvi are very important quantities for the analysis that will take place in this project because the interest is centered in the neutrino fluxes produced as a result of annihilation of neutralinos. These quantities also determine the plausability of neutralinos being dark matter candidates, as described in the previous section. Nev-ertheless, there is other detail concerning these interactions, that will be important later on, and is the description of the possible annihilation channels, according to the MSSM. The complete list can be found in table 2.2, taken from the manual found in [7] for the package darkSUSY [3], designed for calculations related to su-persymmetric dark matter in the framework of the MSSM.
Table 2.2: List of Annihilation channels
Id. Channel Id. channel
1. χχ¯→H10H10 17. χχ¯→µ+µ−
2. χχ¯→H10H20 18. χχ¯→ν¯τντ
3. χχ¯→H20H20 19. χχ¯→τ+τ−
4. χχ¯→H30H30 20. χχ¯→uu¯
5. χχ¯→H10H30 21. χχ¯→dd¯
6. χχ¯→H20H30 22. χχ¯→¯cc
7. χχ¯→H−W+ 23. χχ¯→ss¯
8. χχ¯→H10Z0 24. χχ¯→¯tt
9. χχ¯→H20Z0 25. χχ¯→¯bb
10. χχ¯→H30Z0 26. χχ¯→gg
11. χχ¯→H−W+ andχχ¯→H+W− 27. χχ¯→qqg
12. χχ¯→Z0Z0 28. χχ¯→γγ
13. χχ¯→W+W− 29. χχ¯→Zγ
14. χχ¯→ν¯eνe
15. χχ¯→e+e− 16. χχ¯→ν¯µνµ
The neutralino annihilation channels are important because every channel will pro-duce neutrinos at different rates, which in turn depend on the decay channels of the annihilation products. It should be noticed that the overall effect of a particular channel in the description of the total flux of neutrinos should depend on the an-nihilation cross section for that given channel. The relative contributions would be accounted for by the branching ratios of the annihilation channels, which would be calculated by dividing the individual cross section by the total cross section. The values of the cross sections would depend on the parameters of the MSSM and their calculation is very complicated, so it will not be adressed here. Nevertheless, the details can be found in [1].
The other important interaction is the scattering of neutralinos from ordinary mat-ter. The neutralino annihilation rate to quarks is non-zero and this quantity is related to the amplitude of elatic scattering from quarks as a result of crossing sym-metry. [1]
The cross sections σχi associated to neutralino scattering from a given nuclei i can
be calculated, but would depend on the free parameters of the MSSM as well. These are important in order to calculate the capture rate of neutralinos in the solar core, taking into account the process explained in the introduction. In this case, the physics involved in such calculations are again left undescribed as they escape the scope of this project, but can nevertheless be found in [1]. These particular cross sections can be spin dependent or spin independent.
With this, the description of the neutralinos and their properties has been com-pleted. It is now time to examine how these properties can be used to calculate the capture and annihilation rates of neutralinos in the Sun, which will be done in the following chapter.
Chapter 3
Neutralino annihilation in the Sun
In this chapter, the details concerning production of high enegy neutrinos in the Sun as a result of neutralino annihilation are reviewed. First, the mechanism of accumu-lation of neutralinos in the solar core is studied and expresions for the calcuaccumu-lation of the annihilation rates of these particles in the Sun are presented. Afterwards, it is reviewed how to estimate the production rates of high energy neutrinos that would result from these annihilations. This information will be crucial later on, when the method used in this project for calculation of the neutrino fluxes that would arrive to the Earth is presented.
3.1
Neutralino relic density in the galactic halo
The presence of dark matter can be inferred in spiral galaxies by studying their rotational dynamics. As most of the visible matter in the galaxies is concentrated in their center, it would be expected that the stars in the galactic disk rotated with velocities given by the net effect of a unique body in the center of the galaxy. Nevertheless, this is not what can be observed, instead the stars in the disk rotate with constant velocities [1]. The velocity of a star rotating around the galaxy would be given by
v(r) =
r
GM(r)
r (3.1)
where
M(r) =
Z
ρ(r)d3~r (3.2) is the mass included inside the orbit and ρ(r) is the density of matter of the galaxy. Expresion 3.1 can be deduced from the relation between the gravitational force and the centripetal aceleration of the star.
The observational result of v(r) being constant implies that M(r) is proportional to r, which would happen if ρ is proportional to r−2 (as d3~r = r2drdΩ). This in-dicates that the stars in the disk of an spiral galaxy must be inmersed in a halo of non-luminous matter with a density profile ρ∝r−2. This halo should be esentially
stationary with respect to the velocities of rotation in the disk.
density that could be estimated using models that can be found in the literature. These models propose exact expressions for the dark matter density that take into account the relation ρ ∝ r−2. This allows calculations of ρ from the rotational
di-namics of the stars in the galactic disk. Nevertheless, the case of our Galaxy is more complicated that might be expected because, as explained in [8], it is not possible to directly determine the rotation velocities of the stars inmersed on its galactic disk. Many different observational techniques must be used to find these velocities, which escape the scope of this project.
For the project, the description of the density profile of dark matter in the Galaxy was therefore omitted. Nevertheless, as will be presented in the following section, the capture rates of neutralinos in the Sun could be calculated by only using the value of the local density of dark matter at the Sun location. This value was researched in the literature, and was found in [9] to be approximately 0.43 GeV/cm3.
3.2
Absorption of Neutralinos in the Sun
The neutralinos in the galactic halo, passing through the Sun, have a minimal proba-bility of scattering elastically off the differnt nuclei. After the scattering, the velocity
vχof the neutralino could decrease to a point where it is less than the escape velocity
of the Sunvesc(≈1156kms−1). In that case, the neutralino is gravitationally bound
and will conduct subsequent elastic scatterings with the nuclei in the Sun, until it ends up bound in the solar core.
A first estimate of the form of the capture rate of neutralinos in the Sun should be proportional to the elastic cross section of the neutralinos with the different nuclei species in the Sun, to the flux of neutralinos arriving to the Sun and to the condi-tional probability that a neutralino is scattered to a velocity vχ < vesc, etc [1], [2].
The complete calculation of the capture rate of neutralinos in the Sun is performed in [10]. The details of the calculation are beyond the scope of this project and the following aproximate result, derived from there, will actually be used in this project [2]:
C≈1.3×1021s−1(0.3GeV cmρlocal −3)(
270kms−1 ¯
vlocal )(
100GeV
mχ )
P
(Ai(σχi,SD+σχi,SI)S(mχ/mi)
10−6pb )(3.3)
where ρlocal is the dark matter density at the Sun location, ¯vlocal is the local rms
velocity of dark matter particles in the galactic halo (≈270kms−1, according to [1]),
σχi,SDandσχi,SI are the spin-dependent and spin-independent cross sections for
elas-tic scattering with a given nuclei typei,mi is the mass of the nuclei species i andAi
is a dimensionless factor containing information on the form factor and the relative abundance of the nuclei i in the Sun. The only relevant scattering cross sections in the case of the Sun are σχH,SD, σχH,SI, σχHe,SI and σχO,SI according to [2] with
associated factors AH ≈1.0, AHe ≈0.07 andAO ≈0.0005.
Finally, the dinamical information involved in the capture process (such as the fact that the flux of dark matter through the Sun is focused towards the solar center as
a result of its gravitational field, etc) is contained in the function S, given by:
S(x) =
A(x)3/2 1 +A(x)3/2
2/3
(3.4)
with
A(x) = 3 2
x
(x−1)2
vesc ¯ vlocal 2 (3.5)
3.3
Neutralino-neutralino annihilation rates in
the solar core
The neutralinos accumulated in the solar core can conduct neutralino-neutralino annihilation processes (χχ→ll), producing pairs of ordinary particles. It is certain that the annihilation rate will depend on the number of neutralinos accumulated in the core and thus, on the capture rate. Therefore, it is useful to express the rate of change of the number of neutralinos N(t) in the solar core as a function of both the capture and annihilation rates, in order to obtain a mathematical relation between them. This rate of change is given by [2]:
˙
N(t) =C−AN(t)2−EN(t) (3.6)
with
A = hσvi
Vef f
(3.7)
where hσvi is the total cross section of χ annihilation times the relative neu-tralino velocity v, E is the inverse time for the WIMP to escape from the Sun by
evaporation and Vef f is the effective volume of the solar core, given by:
Vef f = 5.7×1027cm3(100GeV /mχ)3/2 (3.8)
The last term on equation (3.6) represents the rate at which the number of neutrali-nos in the core decreases as a result of evaporation. This process consists in ejection of neutralinos from the Sun as a result of heavy elastic scattering with the nuclei therein and is insignificant for WIMPS if they have energies higher than 10GeV [1]. As such values will be used in our project, we neglect this term.
The remaining terms on the right side of equation (3.6) are the rates of increment of N as a result of WIMP capture and of decrease of N as a result of
annihila-tion processes in the solar core. If we solve the differential equaannihila-tion (neglecting the evaporation term), the result is:
N(t) =
r
C
Atanh(
p
CAt) (3.9)
Therefore, the current rate of neutralino annihilation in the Sun would be:
Γ=
1 2A
N(t)2 =
1
2Ctanh
2(p
where the factor of 12 accounts for the fact that a decrease of 2 WIMPs in the solar core is equivalent to a single annihilation process and t ≈ 4.5 billion years is the
age of the solar system.
As can be seen, the annihilation rate depends both on the capture rate and the total annihilation cross section, as expected. Nevertheless, when 1<<<pCAt
the capture rate is independent of the latter (Γ ≈ 12C). This particular case is
the equilibrium solution, as the annihilation rate would have reached its maximum value. Once the values of the free parameters in the MSSM are especified, it can be tested whether or not this condition is satisfied for the corresponding neutralinos.
3.4
Production of High-Energy Neutrinos
The particles of ordinary matter produced from neutralino annihilation will each have the mass of the neutralino (as neutralinos move relatively slowly once they are captured into the solar core) and will subsequently decay. Some of this decays will generate high energy neutrinos with typical energies between 1/2 to 1/3 of the neutralino mass, much higher than the typical energies of ordinary solar neutrinos, together with many other ordinary particles, which will eventually be absorbed in the Sun.
The exact energies of the high energy neutrinos resulting from dark matter an-nihilation will actually be given by an specific energy spectrum (per anan-nihilation) (dN/dE)f,i, where idenotes the neutrino flavor andf indicates a particular
annihi-lation channel. The total number of neutrinos of the flavoriproduced per unit time through the annihilation channel f will be given by the energy distribution [1]:
4πr2(dφ/dE)f,i(Eν) = ΓBf(dN/dE)f,i(Eν) (3.11)
where Bf is the branching ratio (normalized cross section times relative velocity of
the neutralinos) of the annihilation channel f and Γ is the total neutralino
anni-hilation rate in the Sun, discussed in the previous section. The term (dφ/dE)f,i is
the energy distribution of the high-energy neutrino fluxes, defined so that division by 4πr2 of the neutrinos produced by unit time, where r is the distance from the
center of the Sun, provides the fluxes at that particular distance. The dependance onBf is clear, as the number of neutrinos generated through a given channelf will
increase with the probability of annihilating through that channel.
In fact, in order for this distribution to be correctly defined, the total number of neutrinos produced per unit time, given by the integration over all possible energies, should simply be
Z
4πr2(dφ/dE)f,i(Eν) dEν = ΓBfΓl→νi (3.12)
where l denotes the annihilation product associated to channel f and Γl→νi is the
Aditionally, notice that the total differential energy spectrum per unit time associ-ated to the neutrino flavor i will be the sum of 4πr2(dφ/dE)
f,i over all the possible
channels f.
4πr2(dφ/dE)i(Eν) = Γ
X
f
Bf(dN/dE)f,i(Eν) (3.13)
In order to calculate the energy spectra (per annihilation) (dN/dE)f,iseveral factors
must be taken into account. In a particular annihilation channel f there might be several different decay channels which produce neutrinos of a particular flavori. Let
l be the annihilation product of a particular channel f, with mass ml . In its rest
frame, lcould decay into neutrinos through two main different types of direct decay. The first type is of the form l¯l → νν¯. In this case, each neutrino has an energy
Eν = ml/2. Thus, the rest energy spectrum of such neutrino will be proportional
to a delta function, centered in ml/2. The second type of decay is of the form
l¯l →ν(or ¯ν)+ anything, and usually are three body decays. In this case the energy distribution of the neutrinos are more complicated and such calculations will be left unexplained in this text. Information on the calculation methods involved can be found in [11] and [12].
Once the energy spectra in the rest frame of the decaying particle (dN/dE)rest i,f
has been calculated, the actual neutrino spectra obtained from its decay can be found using the relation [1], [13]:
(dN/dE)i,f(Ed, Eν) =
1 2
Z E+(Eν)
E−(Eν)
1
γβ(dN/dE)
rest i,f ()
d
(3.14)
with
E±(Eν) =
Eν
γ(1∓β) (3.15) where Ed (≈ mχ) is the energy of the decaying particle, md is its mass, β is its
velocity and γ = (1−β2)−1/2 =E
d/md.
Additionally, if the decaying particles are quarks, then there will be hadronization before the actual decay. Usually, this particles would lose energy in this process in such a way that their energy at the moment of decay would actually be Ed=Eizf,
where Ei ≈mχ is the initial energy of the decaying particle and zf is a factor that
depends on the type of quark. [13] In the Sun, however, there is also stopping of the resulting hadrons by the solar medium and Ed will be given by a decay
distri-bution [1], [14]:
1 N dN dEd hadron
(E0, Ed) =
Ec E2 d exp Ec 1 E0 − 1 Ed (3.16)
where Ec is a factor that depends on the type of quark and E0 =Eizf.
The total spectra from quark decay is then found multiplying this relation to the usual spectra, given by equation (3.14) and integrating in the range of the possible values for Ed [13]:
(dN/dE)i,f(Ei, Eν) =
Z E0
0
(dN/dE)i,f(Ed, Eν) (
1
N dN dEd
Finally, the high energy neutrinos produced will be absorbed in the solar medium through charged current interactions and will lose energy through neutral current interactions [1]. The latter interactions will be discussed in the following chapter, as these are of relevance for the problem of neutrino oscilation. On the other hand, the absorption of neutrinos will not affect the neutrino flavor ratios, and thus, will be discussed here.
According to [13], the total probability for a neutrino not being absorbed by the solar medium will be:
Pf =
1 1 +Eτj
αj
(3.18)
where E represents the initial energy of the neutrinos, τν = 1.01 ×10−3GeV−1,
τν¯ = 3.8× 10−4GeV−1, αν = 5.1 and α¯ν = 9.0. The total neutrino production
rates (dN/dE)total
i found after neutrino oscilation through the solar medium should
afterwards be multiplied by these probabilities.
At his point, the elements that had to be studied in order to estimate the pro-duction rates of neutrinos in the Sun as a result of dark matter annihilation have been completely described. It is now possible to address the problem of the oscila-tion of those neutrinos in their path from the solar core to the Earth. This will be described in the following chapter.
Chapter 4
Neutrino oscilation
In this chapter, the theoretical framework of neutrino oscilation is presented. The procedure that should be followed to take into account this effect for the neutrinos produced in the Sun as a result of neutralino annihilation will also be reviewed here.
4.1
Neutrino Oscilation theory
The theory of neutrino oscilation is based on the supposition that the three neutrino flavor states described in the SM, which do not have a particular mass associated to them, are in fact superpositions of another fundamental neutrino states with definite masses, known as mass eigenstates. Neutrinos would evolve as flavor states when participating in weak interactions and as mass eigenstates when propagating through space and as a consecuence, there would be a finite probability of transitioning from one flavor to another. These oscilations between flavors provided an explanation for the so called solar neutrino problem, which consisted in a discrepancy between the theoretical and experimental values for the flux of electron neutrinos coming from the Sun. The number of neutrinos detected was lower than expected as result of the oscilation of the electron neutrinos into the other flavors in their path from their production point in the Sun to the detector. The general features of the theory will be described next in the so called plane wave approximation for propagation of neutrinos, using the general information found in [15].
Let Bf = {|νei,|νµi,|ντi} and Bm = {|ν1i,|ν2i,|ν3i} be the neutrino flavor and
mass eigenbases respectively. They have the same number of elements as both must describe the same Hilbert space of possible neutrino states. The relation between the elements of the two bases in the plane wave approximation is:
|νii=
3
X
j=1
Uij|νji, i=e.µ, τ (4.1)
where theUij coeficients are the entries of the so called neutrino Cabibbo–Kobayashi–Maskawa
(CKM) matrix U, which is simply a change-of-basis matrix. This matrix is therefore unitary (U† =U−1), and for the case in which CP violation is not considered (the case that will be used in this project) is a real valued matrix (Uij=Uij∗) given by
U =
C2C3 S3C2 S2
−S3C1−S1S2C3 C1C3−S1S2S3 S1C2
S1S3−S2C1C3 −S1C3−S2S3C1 C1C2
(4.2)
if parametrized as a function of some coefficients θi which are defined as ”vacuum
mixing angles”. Here Ci =cos(θi) and Si =sin(θi).
Any arbitrary neutrino state |φi can be represented as a linear combination of ele-ments in any of the two eigenbases. These representations of the state |φi will be denoted as |φmi and |φfi. The relation between them would be given by
|φfi=U|φmi (4.3)
On the other hand, the time evolution of a particular neutrino state |φ(t)i would be given by theSchrodinger¨ equation:
id
dt|φ(t)i=H|φ(t)i (4.4)
whereH is the Hamiltonian, with representationsHmandHf in the mass and flavor
eigenbases respectively.
The relation between the initial and final states of neutrinos propagating through space is obtained solving this equation, and is found to be
|φ(t)i=U(t)|φ(0)i (4.5) where
U(t) = e−iHt (4.6) is the time evolution operator. It is useful to sett=L, the distance traveled by the neutrino (it is possible because neutrinos esentially move at the speed of light, and c=1) and from now on only this space evolution operatorU(L) will be used. As can be seen, the description of the evolution of neutrinos as they propagate resumes to the problem of finding the operator U(L).
With that being said, notice that any initial flavor state|φf(0)i=|νji (j ∈ {e, µ, τ})
produced in a weak interaction, will evolve to a final state|φf(L)i=Uf(L)|νjiafter
traveling through a distance L, where Uf(L) is the representation of the evolution
operator in the flavor eigenbasis. The probability of transitioning from an initial flavor state |νji to a different final flavor state|νii after propagation would then be
given by
as this is the probability of finding the eigenvalue associated to |νji when
mea-suring a system in the state |φf(L)i=Uf(L)|νji and thus, in the case when Uf(L)
is not diagonal, there is a finite probability of transitioning from an initial state |νji
to a final state |νii after propagation. This is the mechanism that allows neutrino
oscilation.
For propagation in vacuum the calculation of U(L) is straightforward. The Hamil-tonian Hm would be diagonal in this case, as the mass states should be eigenstates
of the Hamiltonian, with eigenvalues Ei =
p
m2
i +p2, where the values mi are the
masses associated to each neutrino mass state (the momentum p is assumed equal for all eigenstates ). As a result, the mass states are also eigenvalues of the evolution operator (a function of Hm) and thus this operator can be expressed analitically in
the basis Bm as a diagonal matrix Um(L), with eigenvalues e−iEit. The
representa-tion of the evolurepresenta-tion operator in the flavor basis would then be given by
Uf(L) = U Um(L)U−1 (4.8)
and would not be diagonal as expected (neutrinos oscilate in vacuum).
For propagation through matter the calculation ofU(L) becomes a difficult task, as the Hamiltonian is non diagonal in both the mass and flavor bases as a result of an aditional term in the hamiltonian to account for the interaction between the neu-trinos and the medium. The problem of finding U(L) in this case will be adressed in the next section. For now is sufficient to mention that the result depends on the mixing of the flavor states given by the CKM-Matrix, on the energy differences
Eab =Ea−Eb (a, b∈ {1,2,3}) of the neutrino mass eigenstates and on the so called
matter density parameter A(r), given by
A(r)≈ ±√1
2Gf 1
mN
ρ(r) (4.9)
wheremN ≈938.3M eV is the nucleon mass a in the media, Gf = 1.16639×1023eV2
is the fermi weak coupling constant andρ(r) is the matter density. The sign of A(r) will be (+) for neutrinos and (-) for antineutrinos.
The energy differences in the relativistic limit will be considered in this project. These are given by:
Eab ≈
∆m2
ab
2E (4.10)
where the values ∆m2ab = m2a −m2b are the mass squared differences and E is the total energy associated to the original neutrino state that was produced in a given interaction.
Finally, is important to point out here that there are still no generally accepted values for the mass squared differences and the mixing angles of the CKM-matrix. Correct values should account for the solar neutrino problem, as well as many other experi-mental data. We will use the values found in [16], which are the values that best fit the currently known data. These values are given by: sin2(2θ
1)>0.95, sin2(2θ2)≈
0.095, sin2(2θ3) ≈0.857, ∆m221 = 7.5×10−5eV2 and ∆m232 = 2.32×10−3eV2 (the
other mass squared differences can be inferred from this ones). The lowest value for
sin2(2θ
4.2
Analitic expression for the time evolution
matrix
In this section, the method used to calculate a closed expresion for the time evolu-tion operator (t=L) for the case of propagaevolu-tion through matter is described. The procedure presented for its calculation was taken from [15] and [17].
In the case of neutrino propagation through medium of constant density, the to-tal hamiltonian would beH=H+V, where H denotes the hamiltonian in vacuum and V would be an interaction term between the neutrinos and the medium, given by
Vf =A
1 0 0 0 0 0 0 0 0
(4.11)
where A is the constant matter density parameter of the medium.
The corresponding hamiltonian in the mass basis would therefore be the matrix
Hm = Hm+U−1VfU, which is non diagonal. Evidently, its representation in Bf
would not be diagonal as well. The problem of finding Uf(L) is then finding an
analitic expression for the exponential of the non diagonal matrix (−iHfL). As
shown in [15], this is actually possible:
The expresion for the exponential of an N ×N matrix M is usually given in non closed form by
eM =
∞
X
n=0
Mn
n! (4.12)
Although this is an infinite sum, it can be reduced to a finite sum using the Caley-Hamilton theorem.
The theorem states that the matrixM satisfies its own characteristic equation
det(M −λI) =λN +cN−1λN−1 +· · ·+c1λ+c0 = 0 (4.13)
which implies that
MN =−cN−1MN−1+· · ·+c1M+c0I = 0 (4.14)
for some coeficientsci. Therefore any matrix Ml, with l≥N is expresable as a sum
of the operators MN−1, ..., M and the N×N identity matrixI, and as a result, the
infite sum becomes a finite sum ofN terms, given by
eM =
N−1
X
n=0
anMn (4.15)
where the coeficients an must be determined.
To do so, the eigenvalues λ1, ..., λN of M must be calculated and also its
also be diagonal, with eigenvalues eλ1, ..., eλN. Therefore, the following relations are
obtained from equation ( 4.15)
eλ1 =a
0+a1λ1+· · ·aN−1λ1
eλ2 =a
0+a1λ2+· · ·aN−1λ2
.. .
eλN =a
0+a1λN +· · ·aN−1λN
(4.16)
The values of the coeficients a0, a1,...,aN−1 will be the solutions of this system of
equation and it will be possible to find them. Nevertheless, it will not be always possible to find analitical expresions for the N eigenvalues of M because the solution to the characteristic equation for N >4 does not have a closed form in general.
Returning to the case of the evolution operator, it is fortunate that it has dimension
N = 3, so that its characteristic equation will always have an analitical solution (the solution to a cubic equation has a closed form). Therefore, the operator Uf(L) is
calculated in [15] using the method previously described. The calculation will not be reproduced here as it is quite lenghty (the determination of the eigenvalues and the coeficients a0, a1,a2). The resulting expresion is given by:
Uf(L) = φ
3
X
a=1
e−iLλa 1
3λ2
a+c1
[(λ2a+c1)I+λaT˜+ ˜T2] (4.17)
where ˜T = Hf −(trHf)I/3 is a matrix defined to facilitate the calculation of the
eigenvalues,φ=e−iL(trHf)/3and both the the valuesc
1 and the eigenvaluesλa(which
are actually associated to ˜T) are functions of the matter density A and the squared energy diferences Eab, that can be consulted in [15].
Finally, for matter of varying density the evolution operator can be inferred from the expresion found for the constant density case. The general description of such calculation, found in [17], is the following. Suppose propagation through an space interval of lenght L and matter density ρ(r). If this interval is divided in N parts, found at the distances r =r1, r2, ..., rN (with ri < ri+1 and rN =L) from the point
where the neutrinos where produced, then for N sufficiently large the density in each intervalri−1 ≤r < riwould be esentially constant (ρr ∼const). Therefore, the total
evolution operator would be given by the conposition of N constant density opera-tors Ui(ri+1−ri), each calculated in a particular interval using the corresponding
constant density parameter Ai ≈ A(ri). It is important to take into account that
this composition should be done in the correct order, so that
4.3
Oscilations through the solar medium
In this project the case of neutrino oscilation resulting from traveling through the solar medium was studied. In this section, the values of the parameters used for the calculation of the varying density evolution operator U(L) for propagation in the solar medium are summarized and the problem of oscilation with varying neutrino energy as a result of energy loss in the Sun by neutral current interactions is intro-duced.
First, neutrinos are assumed to be produced in the exact center of the Sun (al-thought this is not the actual case; neutrinos are produced in a particular point in the solar core). Therefore the propagation lenght L is taken as R, the radius of
the Sun. The density parameter A(r), on the other hand, would be given by the density profile of the Sun. To calculate A(r), a table with values of the Sun density as a function of the distance from its center, found in [18], was used.
As the intervals found in this table were not small enough to consider a constant density value, an approximated continuous density profile was obtained by inter-polating the data using the cubic spline method. The resulting profile, given in figure 4.1, was then used to obtain the matter density parameteras A(r) of the Sun with equation (4.9). The number of divisions of the propagation lenght N necessary
Figure 4.1: Sun Density Profile as a function of the relative distance from the center of the Sunr/R
to obtain intervals of constant density was considered in this project to be 1000. Thus, the evolution operator U(R) would be given by the composition of 1000
constant matter operators. Finally, the values associated to the mixing angles and mass squared differences that were considered were those presented in section 4.1.
In order to study the oscilation of neutrinos in the solar medium there is an aditional factor to be taken into account, which is the energy loss of neutrinos via neutral cur-rent interactions within the Sun. According to [13], a neutrino originally produced
with an energy E by neutralino annihilation in the solar core will have a final energy
Ef =
E
1 +Eτi
(4.19)
where the coefficients τi are those presented in equation (3.18). A simple model for
the differential energy loss of a neutrino with initial energy E(0) through the Sun was estimated in this project. It was assumed that dE/dx=−Cρ(x)E (where x is the distance from the center of the Sun, ρ(x) =200g/cm3 e−10.54x/R is an
aproxi-mated exponential Sun density profile, found in [17], and C is a unknown constant), so that
E(x) =E(0)eC2(e−10.54x/R−1) (4.20)
whereC2 ≈ −ln(1 +E(0)τi) is a constant determined by setting E(R) = (1+EE(0)(0)τ
i).
As the evolution matrixU(R) would be given by the composition of many constant
matter evolution operators, each of them would be calculated for a different neutrino energy, corresponding to the the energy of the neutrino in that particular interval, and this could affect the overall evolution of the neutrinos in a way that must be studied.
All the necessary information in order to estimate the high energy neutrino fluxes that would arrive to the Earth as a result of neutralino annihilation in the Sun have been reviewed at this point. It is now the time to address this particular problem, and this will be done in the following chapter.
Chapter 5
High-energy neutrino fluxes
In this chapter, the estimation of the energy distributions for the high-energy neu-trino fluxes that would arrive to the Earth as a result of neutralino annihilation in the solar core is finally made. First, the details corresponding to the neutrino spec-tra per annihilation that would be inittially produced after neuspec-tralino annihilation in the Sun will be reviewed. Afterwards, a complete description of the methodol-ogy that should be followed in order to calculate the fluxes is presented, step by step, using the information that has been provided in the previous chapters. This methodology will be used to estimate the fluxes associated to three particular SUSY models, with associated nutralino masses of 73.33 , 78.27 and 92.25 GeV. The results obtained for these models are afterwards presented and discussed.
5.1
Initial neutrino energy spectrum
In this section, the neutrino energy spectra (dN/dE)i,f associated to the channels
bb¯, c¯c, ττ¯, ZZ, W+W−, will be studied. The reason why the other possible
anni-hilation channels are not presented here will be explained in the following section. The expresions that will be presented where derived in [13] (only for the case of muon neutrinos, but generalization of these results is accomplished by replacing the branching ratios associated to a process generating muon neutrinos for another ones, associated to any of the other neutrinos), using results presented in [19] for the neutrino momentum distributions in the rest frame of the decaying particles; only the main features associated to these calculations will be discussed here.
In the case of theττ¯channel, the τ leptons can decay to a ντ plus a pair of leptons
(e, νe or µ, νµ) or plus a pair of quarks (u,d) [19]. The resulting neutrino energy
spectra in the rest frame (dN/dE)resti for such decays will be given by:
(dN/dE)restττ ,i¯ (Eν) =
96Γτ→νi m4
τ
Eν2 (mτ −2Eν) (5.1)
where mτ = 1.78 GeV is the tau lepton mass, 0 ≤Eν ≤ 12mτ and the factor Γτ→νi
is the approximate total branching ratio of allτdecay processes with final productνi.
Using the expression (3.14) on equation (5.1) the spectra in the frame of the moving decaying particle is then obtained, in the relativistic limit (β →1):
(dN/dE)ττ ,i¯ (Eν, Ed) = (2Γτ→νi/Ed)(1−3x
where 0≤Eν ≤Ed, x=Eν/Ed and Ed≈mχ. Notice that this final distribution is
correctly normalized, as the integral of it over the range of possible energies would simply be Γτ→νi, as would be expected.
In the case of the b¯b channel, the b quark can decay to a c quark plus a lepton pair (e, νe orµ, νµ orτ,ντ) [19]. The resulting neutrino energy spectra (dN/dE)resti
for such decays is:
(dN/dE)restb¯b,i(Eν) =
96Γb→νi m4
b
Eν2 (mb−2Eν) (5.3)
where mb = 4.19 GeV is the b-quark mass, 0≤Eν ≤ 12mb, and Γb→νi is the
approx-imate total branching ratio of all b decay processes with final product νi.
The spectra in the frame of the moving decaying particle in this case is calcu-lated using the results discussed in section 3.4 for neutrinos coming from quark decays. First, equation (3.14) is used in the relativistic limit to obtain the following distribution, in the frame of the moving decaying particle:
(dN/dE)b¯b,i(Eν, Ed) = (2Γb→νi/Ed)(1−3x
2
+ 2x3) (5.4) where 0≤Eν ≤Ed,x=Eν/Ed.
Then, equation (5.4) is integrated over the range of posible decay energies using relation (?? to take into account hadronization and the stopping of heavy quark hadrons by the solar media. The resulting expresion is the complete neutrino spec-tra associated to this channel(the following result is slightly different than the result obtained in [13], as absoption and energy loss of neutrinos in the Sun does not apply to the initial neutrino distributions, this will also be the case when calculating in the cc¯channel):
(dN/dE)b¯b,i(Eν, E0) = (2Γb→νi/Ec) exp Ec
E0
−x
(1−18y2+ 24x2y3+x+ 48y3
+ 8y3x3−9y2x2+ 48y3x+ 2y3x4−18y2x−3y2x3) (5.5) where x = max(Ec/E0, Ec/Eν), zf = 0.73, Ec = 470GeV and y = Eν/Ec.
Never-theless, there were some difficulties with the normalization of such expression, as the energy range of the neutrinos is not specified in [13], and assumptions of E0 as the
maximum value did not account for correct normalization. A simple aproximation was then used, in which equation (5.4) was considered instead, taking the average value obtained from the distribution of decay energies as Ed (which depends on
Ei =mχ). In that case, it is easy to verify that the integration over all the possible
neutrino energies for the distribution associated to this channel would be Γb→νi, as
expected.
In the case of the c¯c channel, a c quark can decay to an s quark plus a pair of leptons (e, νe orµ, νµ) [19]. The neutrino energy distribution (dN/dE)resti for such
decays is
(dN/dE)restc¯c,i(Eν) =
16Γc→νi m4
c
wheremc= 1.27 GeV is the c-quark mass, 0≤Eν ≤ 12mc, and Γc→νi is the
approx-imate total branching ratio of allc decay processes with final productνi.
Using equation (3.14) in the relativistic limit, the distribution in the frame of the moving particle is found to be:
(dN/dE)c¯c,i(Eν, Ed) = (Γc→νi/Ed)(
5 3 −3x
2
+ 4 3x
3
) (5.7)
where 0≤Eν ≤Ed, x=Eν/Ed and Ed ≈mχ.
The final distribution taking into account hadronization and stopping of heavy quarks, would be :
(dN/dE)b¯b,i(Eν, E0) = (Γb→νi/3Ec) exp Ec
E0
−x
(5−54y2+ 48x2y3+ 5x+ 96y3
+ 16y3x3−27y2x2+ 96y3x+ 4y3x4−54y2x−9y2x3) (5.8) where x = max(Ec/E0, Ec/Eν), zf = 0.58, Ec = 250GeV and y = Eν/Ec. Again,
there were problems in the correct normalization of this distribution, and thus the distribution in equation (5.7) was used instead, takingEdas the mean energy of
de-cay given by the distribution of possible dede-cay energies (which depends onEi =mχ).
Integration over all the possible neutrino energies, for the distribution associated to this channel would then give a total number of neutrinos generated per decay of Γc→νi, as expected.
The relativistic limit used to calculate the expressions (5.2), (5.4) and (5.7) is valid in the context of this project because β = p1−(mi/mχ)2 is approximately 1 for
50 GeV < mχ, as in this case m2i ≪m2χ (fori∈ {τ, b, c}).
For the channelZZ, each Z-boson decays with an energyEd =mχ into a
neutrino-antineutrino pair. The neutrino spectra in the rest frame of the decaying particle was calculated assuming that the width of the mass distribution of the Z-boson was esentially cero so that the neutrinos are always produced with an energyEν = 12mz,
with mZ = 91.2 GeV. Therefore, the neutrino distribution (dN/dE)resti from the
decays of the ZZ pair is
(dN/dE)restZZ,i(Eν) = 2ΓZ→νiδ(E−mz/2) (5.9)
Using equation (3.14) the distribution in the frame of the moving particle is found to be:
(dN/dE)restZZ,i(Eν, Ed) =
(2ΓZ→
νi
Edβ , if
Ed
2 (1−β)< Eν <
Ed
2 (1 +β)
0, otherwise (5.10)
Finally, the distribution for the W W channel would be the same expresion as in the ZZ case (according to [13]), but dividing by 2 the result (just one neutrino antineutrino pair per annihilation into this channel) and exchanging the Z-boson mass by the W mass mW = 80.4 GeV.
5.2
Methodology
The main interest of this project was to study how it would be possible to obtain evidence of supersymmetric dark matter annihilation using neutrino detectors. Un-fortunaly, the time for preparation was not enough to cover all the aspects involved and thus, the details concerning the detection of the neutrinos once these have ar-rived to the earth had to be left for future analisys. In this project the problem was addressed up to the point of estimating the high energy neutrino fluxes arriving to the Earth as a result of neutralino annihilation at the solar core, as explained in the introduction. All the necessary theoretical information associated to the production and behaviour of these neutrinos fluxes has already been introduced previously and thus, it is now possible to describe the procedure that was followed in order to esti-mate the differential energy distributions for these fluxes.
First of all, as described in Chapter 2, neutralinos are particles with masses given by the free parameters of the MSSM. The other properties of the neutralinos, like their annihilation cross sections and interaction cross sections with ordinary matter would also be given by these choices. Finding of values compatible with the mathe-matical formalism of SUSY was beyond the scope of this project and thus a different approach had to be taken.
The solution was the Software package for calculations involving SUSY dark matter in the framework of the MSSM, known as DarkSUSY [3]. This package can be found in [7], and it comes with a list of sample versions of the MSSM available to be used, which are based on the reduced set of MSSM free parameters obtained from the simplification mentioned in Chapter 2. and explained in [3].
From these versions (which will be referred to as “models” from now on), three models with neutralino masses of 73.33, 78.27 and 92.25 GeV, were chosen for the purpose of this project. These models had the particularity that they did not include annihilation channels containing higgs bosons. The reason was that the masses of these neutralinos were not high enough to include them. The study of versions of the MSSM with neutralino masses sufficiently high to account for annihilation into these bosons was left for future analysis.
For the three models chosen in this project, the annihilation cross sections hσvif
associated to each annihilation channel f, as well as the total cross section hσvitotal
were determined in the limit of cero relative velocityv(which is valid for the neutrali-nos trapped in the solar core) using the DarkSUSY routine dssigmav.f (see manual in [7]). The values of the branching ratios of the annihilation channels were then calculated using
Bf =hσvif/hσvitotal (5.11)
On the other hand, the values of the interaction cross sections of the neutralinos with the hidrogen, helium and oxigen nuclei (σχH, σχHe, σχO) for the three models
were also calculated using DarkSUSY, as these were important for the calculation of the capture rate of neutralinos in the Sun, as described in equation (3.3). The darkSUSY routine used for this purpose was dsddsigma.f (see manual in [7]).
The results obtained, as well as the main SUSY parameters for the three mod-els employed, are shown in tables 5.1 and 5.2. The relic densities of SUSY dark matter corresponding to each models can also be found in those tables. These seem very close to the actual dark matter density in our universe, shown in Chapter 2. This validates the initial hypothesis of the neutralinos as good candidates for con-stituents of dark matter.
Table 5.1: The neutralino masses, relic densities, neutralino-nuclei cross sections and parameters associated to each of chosen SUSY models obtained using DarkSUSY
Models model 1 model 2 model 3
µ(GeV) -.10981755E+03 0.12222682E+03 0.15379393E+03 M2 (GeV) 0.20156435E+03 0.28204398E+03 -.21332589E+03
tan(β) 0.13373882E+02 0.32563794E+01 0.38359151E+02 mA (GeV) 0.85693475E+03 0.58788072E+03 0.70341785E+03
m0(GeV) 0.45114744E+04 0.14045032E+04 0.42496189E+04
At/m0 0.63229229E+00 0.23170287E+01 -.10963316E+01
Ab/m0 0.47241413E+00 -.18082485E+01 -.16247167E+01
mχ (GeV) 73.33 78.27 92.25
Ωχh2 0.102589 0.118513 0.101063
σχH,SD (pb) 2.74E-003 7.32E-008 9.32E-004
σχH,SI (pb) 7.41E-009 9.23E-004 1.83E-009
σχHe,SI (pb) 1.73E-006 1.74E-005 4.74E-007
σχO,SI (pb) 3.37E-004 3.45E-003 9.73E-005
Table 5.2: List with the total neutralino annihilation cross section times relative ve-locity and relevant branching ratios for each particular model: the missing channels correspond to those with negligible values in all models
Model model 1 model 2 model 3
hσvif (cm3s−1) 5.441052E-028 6.278614E-028 1.389746E-026
Bχχ¯→τ+τ− 1.12924E-002 4.99497E-002 1.02772E-002
Bχχ¯→cc¯ 0.195201 5.37760E-002 5.37760E-002
Bχχ¯→¯bb 0.613461 0.75992E-002 4.45511E-002
Bχχ¯→gg 0.142683 4.46243E-002 2.69697E-003
Bχχ¯→Z0Z0 0 0 7.43726E-003
Bχχ¯→γγ 1.19308E-002 2.95188E-002 2.48551E-004
Bχχ¯→Zγ 2.52754E-002 6.15288E-002 1.15800E-003
Bχχ¯→W+W− 0 0 0.930911
Table 5.3: Neutralino capture and annihilation rates in the Sun calculated for each model
Model model 1 model 2 model 3
C(s−1) 2.20371×1024 1.88233×1024 5.97389×1024 Γ(s−1) 1.10185×1024 9.41116×1024 3.48694×1024
Once all the quantities with dependence in the MSSM model parameters were cal-culated, the capture and annihilation rates of neutralinos in the Sun were computed using the relations presented in Chapter 3. The results are presented in table 5.3.
Notice that for all models, but the third, the relation for equilibrium between cap-ture and annihilation of neutralinos in the Sun Γ= 12C was esentially satisfied.
The next step was to determine the neutrino spectra for the annihilation channels. To accomplish this, it was important to first determine the annihilation channels that would be relevant. The branching ratios presented in table 5.2 corresponded to those with a probability of ocurrence per annihilation of the order of 10−3 or higher. Only those channels were taken into account. As can be seen, these corresponded to the channels despicted in chapter 5.1, with the exception of the gluon-gluon, γγ
and Zγ channels.
The lack of the distributions associated to those channels in section 5.1 deserves some mentioning. It proved to be very difficult to find the neutrino spectra asso-ciated to those channels in the literature. Adittionally, the annihilation braching fractions of the γγ and Zγ channels were never particularly high ina any of the models and it was safe to discard them. Nevertheless, the gluon-gluon channel was important for the first channel, because its branching ratio was sufficiently high. Therefore, it would be important to account for this channel in a future analysis. Perhaps the only adittional information that should be given about the expresions in section 5.1 are the values of the decay braching fractions Γl→νi. This values are
listed in table 5.4, using the information found in [16], [19].
Table 5.4: Decay branching ratios asociated to neutrino production
Final Neutrino Stateνi νe νµ ντ
Γτ→νi 0.20 0.20 1.00
Γb→νi 0.11 0.11 0.11
Γc→νi 0.20 0.20 0.00
ΓZ→νi 0.067 0.067 0.067
ΓW→νi 0.1075 0.1057 0.1124
Once the energy distributions had been determined, it was also important to ex-plore up to what point the neutrino energy distributions per annihilation used in this project could resemble a real physical situation. In order to do this, the dis-tributions were plotted for each particular model. Their behaviour was exactly the same in all models, except that the energies avaliable for the neutrinos increased with the mass of the neutralino. As a result, only the graphs corresponding to the third model (figures 5.1 and 5.2) needed to be discussed here (the decay to elec-tron neutrinos was used to generate the plots). This model had the advantage of including the W W and ZZ channels, not available in the first two models (as the mass of the neutralino is lower than the mass of both Weak bosons in those models). It is important to mention here that the maximum values found for the neutrino energies on the graphs of the quark channels were the mean values of the possible decay energies given by the distribution found in equation (3.16), these values were 36.98 GeV, 39.16 GeV and 45.17 GeV for the cc¯channel and 48.49 GeV, 51.46 GeV and 59.68 GeV for the b¯b channel, for models 1, 2 and 3 respectively.