1.5 Outline of the thesis
2.1.1 A brief introduction to Supersymmetry
There are several motivations behind introducing supersymmetry. One of the strongest moti- vation is related to the fact that it provides a solution to the hierarchy problem [234–238]. It is known that the quantum corrections drag the mass parameter associated with the SM Higgs field to a very high value close to the cut off scale of the theory. This is known as hierarchy problem. The problem is even more prominent when the SM Higgs couples to any new physics.
On the other hand, fermions in the SM are protected by the chiral symmetry and gauge boson masses are so by the presence of gauge symmetry. Introduction of supersymmetry can alleviate this problem. As it is a special kind of symmetry which connects the properties of a fermion to that of a scalar (and vice versa), the same chiral symmetry protecting fermion’s mass is also responsible to protect scalar’s mass. Apart from this, within supersymmetric framework, gauge coupling unify at high scale. It also provides a natural candidate of dark matter.
•Supersymmetry algebra: Supersymmetry (for review see [239–248]) is an extended version of Poincare symmetry. In addition to generators of Poincare symmetry (Mµν for Lorentz booss and rotations and Pµ for translations), the supersymmetry includes fermionic generator (Q) which relates fermionic and bosonic states:
Q|fermioni=|bosoni, Q|bosoni=|fermioni, (2.1) where Q is a spin-12 operator. In four dimensional space time, they can be considered as Weyl spinors. According to the standard definition, the fermionic generators are taken to be of the form Qα and ¯Qα˙ with α,α˙ = 1,2 are the Lorentz spinor indices. They satisfy the following commutation and anti-commutation relations [248].
{Qα,Q¯β˙}= 2σµ
αβ˙Pµ, {Qα, Qβ}={Q¯β˙,Q¯β˙}= 0
[Qα, Pµ] = [ ¯Qβ˙, Pµ] = 0, (2.2) where σ is the pauli matrix and Pµ represents the generator of translation symmetry.
2.1. Introduction 39
• Superspace and Superfields: Before entering into the supersymmetric field theory and Lagrangian formalism, we need to build the idea of superspace. Superspace is obtained by extending the 4-component spacetime coordinates (xµ) with four fermionic coordinates. The additional fermionic coordinates are labelled by Grassmann variables. Hence any point in su- perspace is defined by [248]
xµ, θα, θ¯α˙, (2.3)
whereα ,α˙ = 1,2 are Lorentz indices. The new fermionic coordinates satisfy anti commutation relations
{θα,θ¯β˙}={θα, θβ}={θ¯α˙,θ¯β˙}= 0. (2.4) Let us also define the integral over superspace [248]
Z dθ=
Z dθ¯=
Z
dθθ¯= Z
dθθ¯ = 0, Z
dθαθβ =δβα, Z
dθ¯α˙θ¯β˙ =δβα˙˙, Z
d2θθ2 = Z
d2θ¯θ¯2, Z
d4θθ2θ¯2 = 1. (2.5)
At this stage, it is also important to mention that the supercharges (Qα, Q¯α˙) are differential operators in superspace [248].
Qα = ∂
∂θα −iσαµα˙θ¯α˙∂µ, Q¯α˙ =− ∂
∂θ¯α˙ +iθασµαα˙∂µ, (2.6) where Pµ =−i∂µ is the momentum operator in bosonic space. Now we can write the general expression of a superfield which depends on superspace coordinates.
Φ(xµ, θ,θ) =φ(x¯ µ) +θη(xµ) + ¯θχ†(xµ) + ¯θσ¯µθVµ(xµ)
+θ2F(xµ) + ¯θ2F¯(xµ) +θ2θ¯¯λ(xµ) + ¯θ2θC(xµ) +θ2θ¯2D(xµ). (2.7) As evident from Eq.(2.7), a general superfield contains four complex scalar fields (φ, F, F , D),¯ four Weyl spinors (η, χ†, λ, C) and one vector field (V¯ µ). From Eq.(2.7), one can also obtain the form of chiral superfield using the constraints ¯Dα˙Φ = 0 and DαΦ† = 0 where ¯Dα˙ and Dα
40 Chapter 2. Modified sneutrino chaotic inflation and dynamical Supersymmetry breaking are covariant derivatives in superspace defined by [248]
Dα = ∂
∂θα −iσµαα˙θ¯α˙ ∂
∂µ, D¯α˙ =− ∂
∂θ¯α˙ +iθασµαα˙ ∂
∂µ. (2.8)
The standard expression for chiral superfield turns out to be [248]
Φ =φ(x)−iθσµθ∂¯ µφ(x)−1
4θ2θ¯2∂2φ(x) +√
2θη+ i
√2θ2∂µησµθ¯+√
2θ2F(x). (2.9) Hence we see from Eq.(2.9) that the chiral superfield contains two complex scalars (φand the auxiliary field F) and a Weyl spinor, known as supersymmetric chiral multiplet. This also explores one of the most important features supersymmetry theory that any supermultiplet contains equal number of bosonic and fermionic degrees of freedom (nB =nF). Similarly there exists other forms of supermultiplet like gauge supermultiplets and vector supermultiplets.
• SUSY Lagrangian: The Lagrangian for a general supersymmetric theory involving only chiral multiplets is conventionally written as [248]
L= Z
d4θK(Φ) + Z
d2θW(Φ) + Z
d2θ¯W¯(Φ†), (2.10) where K(Φ) is a real function of the chiral field Φi known as K¨ahler potential which provides the kinetic term of Φ. The minimal form of K¨ahler potential can be expressed asK(Φi) = Φ†Φ.
The functionW(Φ) is known as superpotential of the theory which provides the mass term and interaction Lagrangian of the theory. Note that superpotential is a holomorphic function of Φ.
Provided W(Φ) is known, the F-term scalar potential of a supersymmetric theory is written as VF =|FΦ|2=
∂W(Φ)
∂Φ
2
. (2.11)
If the theory is protected by a gauge theory, then there will be another contribution to the scalar potential which reads as
VD = 1 2
X
α
DαDα= 1 2
X
α
gα2(Φ∗TαΦ)2, (2.12) where gα and Tα are the gauge coupling constant and generators of the corresponding gauge symmetry of Φ. VD is called as D-term potential.
• Supersymmetry breaking: Existence of exact supersymmetry in the Universe implies that all properties, except the spin, of particles in a supermultiplet have to be uniform including
2.1. Introduction 41 masses of the individual components. However, supersymmetry cannot be the governing sym- metry of the Nature. Else we should already observe SUSY particles at collider experiments.
Therefore supersymmetry has to be broken at some high energy scale such that the superpart- ners are heavier than the SM particles to a great extent.
In order to construct a proper model of spontaneous supersymmetry breaking, the guiding principle should be the vacuum energy state |0i is not invariant under supersymmetry trans- formations, i.e. Qα|0i 6= 0 and Q†α˙|0i 6= 0. It turns out that the supersymmtetric Hamiltonian (HS) is a function of these generators:
HS = 1
4(Q1Q†1+Q†1Q1+Q2Q†2+Q†2Q2). (2.13) Hence the SUSY breaking criteria can be expressed in terms of the Hamiltonian ash0|HS|0i>0 This implies that for a supersymmetric model ifFandDterm vanish, supersymmetry at ground will be preserved havingVS=VF+VD = 0. On contrary, inability to find simultaneous solutions of all Φi’s considering FΦi = 0 and Dα = 0, directly hints towards spontaneous breakdown of supersymmetry. For the moment let us focus on F-term supersymmetry breaking. Below as an exercise we present a simple example of F-term supersymmetry breaking (named as O’Raifeartaigh model [249]) for illustration purpose. Let us write the superpotential below
WO =−κ1φ1+mφ2φ3+κ2φ1φ23, (2.14) assuming all φi’s are gauge singlet. The superpotential has a additional global symmetry (U(1)R) with R-charges ofφ1,2,3 are two, two and zero respectively. The scalar potential can be otained as
VS=|Fφ1|2+|Fφ2|2+|Fφ3|2, (2.15) where
Fφ1 =κ1−κ2φ˜23, Fφ2 =−mφ˜3, Fφ3 =−mφ˜2−2κ2φ˜1φ˜3, (2.16) where ˜φi is the scalar partner ofφisupermultiplet. Note that, there is no simultaneous solution for Fφ1 = 0 and Fφ2 = 0. Therefore supersymmetry is spontaneouly broken. One can find the minima of the relevant fields by minimizing the VS. The minimum of VS turns out to be at φ˜2 = ˜φ3 = 0 while value of ˜φ1 remains undetermined. Hence the vacuum has nonzero energy with with VS =κ21. Theφ1 field direction is identified as a “flat direction”. This can be lifted by taking into account quantum loop corrections.
42 Chapter 2. Modified sneutrino chaotic inflation and dynamical Supersymmetry breaking One important observation is the correlation of R-symmetry and spontaneous supersymme- try breaking. The renowned Nelson-Seiberg theorem [250] states any theory having spontaneous supersymmetry breaking minimum, must be protected by an exact U(1)R symmetry. However for metastable supersymmetry breaking,U(1)R could be an approximated symmetry [251–258]
with an explicit U(1)Rbreaking term in the superpotentialWO[259]. In addition, the common well motivated perception behind supersymmetry breaking is that it occurs in a separate hidden sector other than the visible world. The effect of supersymmetry breaking is mediated to the visible sector. There are few well known mediation mechanisms: (i) gravity mediated or Planck scale mediated, (ii) gauge mediated and (iii) anamoly mediated. However it is very challenging task to predict the supesymmetry breaking scale. Theoretically it could be arbitrary. Any the- ory where supersymmetry breaks dynamically [260–262] would be very appealing. This means the supersymmetry breaking scale can be naturally generated from the strong coupling scale of the theory through small exponential suppression [260].
Now, as the inflationary energy scale is close to MP, one should consider the local version of SUSY,i.e. Supergravity. The first hurdle one has to deal generally to accommodate inflation in supergravity framework is the famous η problem. This is caused by the field value of the inflaton (χ) during inflation, which exceeds the reduced Planck scale MP ' 2.4×1018 GeV as in case of chaotic inflation. Thereby it could spoil the required degree of flatness of the inflationary potential through the Planck-suppressed operators. In [263–268], chaotic inflation model with shift symmetric K¨ahler potential associated with the inflaton field was proposed to cure this problem. Another interesting aspect of a supersymmetric model of inflation is its relation with supersymmetry breaking. From the completeness point of view, a supersymmetric structure of an inflationary scenario demands a realization of supersymmetry breaking at the end of inflation. Though during inflation, the vacuum energy responsible for inflation breaks supersymmetry (at a large scale of order of energy scale of inflation), as the inflaton field finally rolls down to a global supersymmetric minimum, it reduces to zero vacuum energy, and thereby no residual supersymmetry breaking remains.