Here is the final report on "Dark matter and particle Physics". This is the project I worked with in IISER, Kolkata during summer, 2012.

1. DARK MATTER
Fig1: A GALAXY
In astronomy and cosmology, dark matter is a type of matter hypothesized to account for a large
part of the total mass in the universe. Dark matter cannot be seen directly with telescopes;
evidently it neither emits nor absorbs light or other electromagnetic radiation at any significant
level.
Evidence
1. Galactic Rotation:
Much of the evidence for dark matter comes from the study of the motions of galaxies.
Consider our Milky Way galaxy, which is a spiral galaxy with Sun in one of the spiral arms
about 2/3rd
of the way from the centre. The bright central region contains most of the mass
(visible). If we apply Kepler’s third law to the rotation of star we get,
V= (GM/r) ½
Where, G is the gravitational constant
M is the mass at the centre of galaxy
R is the distance of the star from centre of galaxy
V is the tangential velocity of the star.
So, we expect V to decrease as the r increases. But the experimental fact suggests that V
almost remains constant with increasing mass.

2. Fig 2: Velocity of star w.r.t the distance from the galactic centre
“A” is the predicted and “B is the observed.
2. Gravitatinal Lensing:
When a light passes through a distribution of masses, it bends as predicted by Einstein’s
theory of General Relativity. When we observe light from a background galaxy, the mass
distribution of the galaxy in between can be found out by the bending of light which passes
through it. Experimental evidences show that the visible matter does not account for the
observed bending.
Fig3: Bullet Cluster
In this figure, the blue region corresponds on the dark matter as inferred from gravitational
lensing. The red region corresponds to the matter emitting Xrays and the white to the matter
emitting optical rays.

3. As it can be seen, during the collision of two galaxies, the spherical shape of the dark matter is
not disturbed. Thus it could be Weakly Interacting Massive Particles (WIMP).
Explanation of the discrepancies.
1. Newton law’s of gravitation does not apply at such large scale.
2. Additional Matter exist which are not visible but which exerts gravitational force
necessary to account for the data. (Dark Matter)
Laws of gravitation as stated by Newton has helped us to understand various phenomena which
give the exact predictions, so there is little chance that this could all be wrong.
So the strong possibility seems to be Dark matter.
Candidates for Dark Matter:
1. Massive Compact Halo Objects. (MACHOs)
2. Neutrinos.
3. WIMP, such as predicted by supersymmetric theory (as discussed below)
Astronomical surveys did not find enough evidence of MACHOs which led to eliminating that
option.
After the discovery of neutrinos oscillation which proved that they have tiny masses, they were
considered as a likely candidate. But Neutrinos travel at relativistic speed, thus it would have
high thermal energy which forms a class of dark matter called “HOT” dark matter, these cannot
explain the binding of cluster of galaxies.
WIMPs move with non-relativistic speed at the time of freeze-out. Heavier particles decay into
the light, stable SUSY particle called Neutralino, a neutral fermion which arises from linear
combination of photino, Zino and two higgsinos(described in pages below). Since these
particles don’t form atoms, they interact only via weak forces. They seem to be the most likely
candidate.

4. Brief review of Particle Physics
The idea that all matter is composed of fundamental constituents dates back to 6th
century BC.
The quest of physicists since then was to find these constituents. During the early 19th
century,
atoms were discovered and they were thought to be the building blocks of matter and that it was
indestructible. This was disproved by the works of various scientists who discovered the
protons, neutrons and electrons. But today in the beginning of 21st
century, (with all the particle
accelerators and colliders) we have found hundreds of particles which are all unstable. It was
thus necessary to see whether this large zoo of particles can be explained in terms of smaller
particles. In this way a Standard Model of particle physics gradually developed.
Forces of nature:
1. Strong force – Nuclear binding
2. Weak force – Radioactive decay
3. Electromagnetism – Interaction of charged particles.
4. Gravitation – Free fall.
Fig 4: Fundamental Forces
CLASSIFICATION OF ELEMENTARY PARTICLES.
Based on spin number:
1. Fermions: They are particles with half integer spin which are subject to Pauli’s exclusion
principle.
Eg: Electron, proton, neutron.
2. Bosons: They are particles with integral spin which obey Bose-Einstein statistics.
Eg. Photon,gluon,w &z boson,graviton

5. Fermions are further divided into two elementary particles
1. Quarks: These are fundamental constituents of Hadrons (pg 5) .These are of six types:
up,down,strange,charm,bottom,top.
Fig 5: Quarks
2. Leptons: They do not interact via strong interactions. They form 3 generation which has
similar properties but only increasing mass.
First generation: Electron and Electron neutrino.
Second generation: Muon and muon neutrino.
Third generation: Tau and tau neutrino.
The heavier muons and tauns will rapidly undergo particle decay. All neutrinos have
negligible mass and zero* charge.
Fig 6: Leptons
*as previously thought.
Hadrons: These are composite particles of quarks.
a) Baryons: They have 3 quarks.
Eg: Protons u(2/3), u(2/3), d(-1/3)
Neutron d(-1/3), d(-1/3), u(2/3)
Charges are mentioned in the bracket.
Mesons: They have a quark and an anti quark pair.
Eg: Pion(u,d antiquark) and Kion (u,s antiquark)

6. Conservation laws
In addition to the charge, momentum and energy conservation laws, in particle physics we also
have :
1. Baryon number conservation (conserved by all interactions: strong, weak and EM )
All the baryons have baryon no. +1 and their antiparticles have baryon no. -1
All the mesons have baryon no. 0
Eg:
-
+ p+
 no
+ -
+ +
0 1 1 0 0
2. Strangeness conservation (conserved by strong and EM interactions but not by weak
interactions)
Strange baryons
S= -1(∑+,∑0,∑-,λ)
S= -2(cascades)
Strange mesons
S=-1(k+,k0)
S=+1(k-,k0 antiparticle)
Therefore pair of strange particles can be produced by collision of non strange particles.
Eg. -
+ p+
 λ + k0
B 0 1 1 0
S 0 0 -1 +1
Strange particles decay via weak interactions which does not conserve strangeness
Eg. λ  -
+ p+
B 1 0 1
S -1 0 0
From, study of all these particles, the Standard model was developed.

7. THE STANDARD MODEL
Fig 7 : Table of the standard model
Standard model mainly consists of two types of fields.
1. Matter field: These are the particles which form matter. Quarks (Violet) and Leptons
(Green) belong to this category.
2. Gauge field: These are also called the force carriers. Matter interacts through the
exchange of these particles and experiences the different kinds of forces.
Strong interactions: Gluon (g)
Weak Interactions: W+
, W-
, Z
Electromagnetism: Photon (γ)
Gravitation: (graviton*
)
Fig 7:Types of Interactions
Except for photon and gluon, all other particles have mass. This is attributed to their interaction
with the Higgs field. When we write down the interactions of various particles in the table, it
turns out that the Higgs Boson do not couple with photon and gluon.
The search for this Higgs boson has been one of the primary goals of Large Hadron Collider
(LHC) which is a particle accelerator. According to the latest outcome, they might have found it.
(*Yet to be discovered.)

8. Lepton Number Conservation.
Initially:
It was observed that when an electron neutrino decays, electron is one of the products.
And when a muon neutrino decays, muon is one of the products. So it was assumed that
each flavor has a separate lepton number and they should also be conserved in a reaction.
Eg:
e + n  p + e-
le 1 0 0 1
lµ 0 0 0 0
n  p+µ-
le 0 0 0 0
lµ 1 0 0 1
A strange quantum mechanical phenomena
Neutrino Oscillations
The standard solar model gives the flux density of the electron neutrino that would reach the
earth. But it was found that, only one third to one half of predicted numbers of electron neutrinos
were detected. Then it was discovered that neutrinos oscillate from one flavor to another (Eg:
electron to muon)
Fig 8: Neutrinos from the Sun
By reverse beta decay process, the proton and an electron combine to give a neutron with
electron neutrino as one of the products. This electron neutrino interacts with the matter
(neutron) and electron and proton are produced. Detectors on earth detect the electron from this
interaction.

9. Their mathematical explanation is as follows.
There are two different types of Eigen States.
Flavor Eigen states: e
Mass Eigen states:
In weak decays, the flavor Eigen states are produced. It does not have a definite mass. They
are represented by Dirac bracket notation.
| e> = | > + | >
| > = - | > + | >
Approximation that there are only two flavors of Neutrino. There are actually three.
The probability of a electron neutrino at t=0 converting into a muon neutrino at time t is given by
P( e, 0; , t) = |< e(0)| (t)>|2 = 4 ( )^2
Where = ((2πm2
2 – m1
2)C3t)/ 4ph
Note:
1) If m1=m2=0 then the probability will be zero which contradicts experiments.
2) We can only determine m1- m2 and never the definite mass.
Neutrinos oscillations have been observed and from that the difference in mass calculated.
These were thought to be a likely dark matter candidate but as explained in the beginning, they
are now overruled.

10. Even though the standard model explains most observed phenomena, it does not have any Dark matter
candidate. Supersymmetric model was established. And as discussed below, this has dark matter
candidates due to symmetry.
Supersymmetry (SUSY)
Corresponding to every Fermion in the standard model, there must be a Boson and vice versa. They are
called super partners. Supersymmetry is broken in nature, i.e all super partners are heavier than SM
particles.
Lepton (Spin ½)  Slepton (spin 0)
Quark (Spin ½)  Squark(Spin 0)
Similarly,
Gauge Boson (spin 1)  Gaugino (spin ½)
Higgs boson (Spin 0)  Higgisnos (spin ½).
Fig 8: Standard Model and Supersymmetric particles
χ0
1,2,3,4 are the Neutralinos which are a mixture of Zino, Photino and neutral higgsinos
χ0
1 is the lightest stable particle. This stability is due to certain symmetry (R parity). These interact only
via gravitation and weak forces. Thus, it is a possible candidate for dark matter.

11. Dark Matter from Big bang:
Just a few seconds after the big bang, the temperature was very high, hence kT was large.
Large unstable particles decay into lighter particles and also collision of lighter particle also
produces heavy particles. Equilibrium is achieved. Under this equilibrium, the number of heavy
particles is proportional to the Boltzman constant e-m/kT
As the universe expands, it cools down and hence kT decreases. Thus, the heavy particles are
not produced, and the existing unstable ones decay.
But consider weakly interacting stable particles. As the universe expands, the annihilation may
become rare as its density is low. Thus there is a departure from the equilibrium and they
remain forever. Moreover, if they do not interact electromagnetically, they become a very strong
dark matter candidate. Neutralino is one such particle.
Fig 9: Big bang expansion and dark matter
It shows the expansion of the universe and the point at which the W-map collaboration
calculated the relic density of dark matter
Dark matter producing interactions:
χ0
1 + χ0
1 SM particles (Annihilation)
χ0
1 + A  SM particles (Co- Annihilation)
Where A is another SUSY particle whose mass is comparable to that of Neutralino.
The dark matter relic density is calculated by numerical solution of the Boltzsman equation. The
W-map collaboration has noted the observed values as 0.09-0.13 h2
. Where, h is the Hubbles
constant.
By varying parameters like the mass of Slepton and Neutralino, the relic densities are calculated
by a standard package. This is then compared with the calculated value.

12. Neutralino-Neutralino Annihilation :
Two Neutralinos can annihilate into SM particles and this process contributed to Dark matter
relic densities. This occurs via the exchange of a virtual Slepton. It turns out that this relic
density (Ω) increases as the annihilation cross section (σ) decreases.
As we increase the Mass of Slepton, σ decreases and hence relic density Ω increases.
This is because in this interaction, Slepton is a virtual particle which disobeys the law of
conservation of energy, with increase in mass, this process becomes more improbable.
Graph 1: variation of Ω with M_eR keeping M1 constant
As seen from the figure, for each mass of Neutralino, there is a range of Slepton mass in which
Ω falls within the experimentally observed range.
Z- Resonance
When the mass of Neutralino is approximately half of Z boson (91Gev) Resonance occurs. This
leads to Neutralino annihilate into the real Z boson which decays into fermion-antifermion pair.
At resonance, Ω decreases. The quark-antiquark pair dominate over e-E and m-M.
Higgs Resonance.
When the mass of Neutralino is approximately half of Higgs. Resonance occurs. The Neutralino
annihilate to into Higgs which decays into fermion-antifermion pair. During resonance, Ω
decreases.
Top quark cannot be produced as it is heavier than higgs (172 Gev) . And so the next heaviest
particle which is the bottom quark dominates.
Also, the fraction of bB is much larger in Higgs resonance than in Z resonance. (Graph 3)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
120130140150160170180190200
Ω
Mass of slepton M_eR
Omega at M1=47
Omega at M1=57
Omega at M1=45

13. As we increase the mass of Neutralino, σ increases and Ω decreases. This is because,
with increase in mass, we have higher energy (E=mC2
) and hence more annihilation.
Graph 2: variation of Ω with M1 keeping M_eR constant
As it can be seen, when M1 is approximately half of that of the mass of Z boson, resonance
effect takes place. σ increases and thus Ω decreases. Also, when M1 is half of that of Higgs
boson, Higgs resonance effect takes place.
Graph 3: Change in contribution % of bB wrt M1
In the above graph, 2 peaks are found. One at M1=47 and the other at M1= 59. It is due to Z
and Higgs resonance respectively.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
43 45 47 49 51 53 55 57
Ω
M1
M_eR = 120
M_eR=150
M_eR = 200
0
20
40
60
80
100
41 43 45 47 49 51 53 55 57 59 61
%ofbB
M1
Z and Higgs resonance

14. Co annihilation:
When the Neutralino interacts with other SUSY particles of comparable masses ( like,Sneutrino)
the pair annihilates and standard model particles are produced. The other particles could be
Chargino, Stop or Sneutrino.
When the Sneutrino mass becomes smaller than Neutralino mass, then Ω drops. As then,
neutralino is no more the LSP. With increase in Slepton mass, sneutrino mass increases.

15. CONCLUSION
Supersymmetry (or SUSY) is an attractive concept because it offers a solution to the "hierarchy
problem" of particle physics, provides a way of unifying the strong and electroweak forces, and
even contains a dark-matter particle. An important prediction of the theory is that every known
particle has at least one superpartner particle – or "sparticle". The familiar neutrino, for example,
is partnered with the yet-to-be discovered sneutrino. These sparticles are expected to have
masses of about one teraelectronvolt (TeV), which means that they should be created in the
LHC.
A possible Dark Matter candidate is the Neutralino. As seen above, we have obtained the dark
matter relic density by varying parameters. For a range of these values, the relic density falls
within the observed range. The search for these particles in supersymmetry is carried out at the
LHC.
It would be interesting to see whether the discovered particles have mass which can explain the
dark matter or else, we will have to look for alternative candidates.