If supersymmetry were unbroken, the superpartners of the Standard Model particles would have the same mass as the Standard Model particles themselves14. Since no such particles have been observed, it is clear that supersymmetry has to be broken at low energies.
Since we do not want to break supersymmetry by brute force, we have to investigate if and how supersymmetry can be spontaneously broken. A symmetry is spontaneously broken if the Lagrangian is invariant under that particular symmetry, but the ground state of the theory is not. For the case of supersymmetry this means that we must have
Qα|0i 6= 0 , Q¯α˙|0i 6= 0. (1.106) Furthermore, in the case of global supersymmetry the Hamiltonian can be related to the generators of supersymmetry transformations using Eqn. (1.22) and ¯Qα˙ = (Qα)†
4H = 4P0 =Q1Q†1+Q†1Q1+Q2Q†2+Q†2Q2. (1.107) Thus, for unbroken global supersymmetry with Qα|0i = ¯Qα˙|0i = 0, the vacuum has zero energyh0|H|0i= 0, whereas for spontaneously broken SUSY the vacuum must have positive energy
h0|H|0i= 1 4
Q†1|0i
2+ Q1|0i
2+ Q†2|0i
2+ Q2|0i
2
>0. (1.108)
13 Note that dθ has U(1)R charge −1, since by definition R
dθ θ = 1 etc. Once again, one has to be careful when dealing with Grassmann valued expressions.
14 The reason for this is that the mass-squared operatorPµPµ commutes with the generatorsQα,Q¯α˙
of supersymmetry transformations.
In the absence of spacetime-dependent effects and fermion condensates we haveh0|H|0i= h0|V|0i, with the scalar potential given by Eqn. (1.92). Therefore, supersymmetry is spontaneously broken if eitherFi or Da do not vanish in the ground state15. We discuss both cases, referred to as F-term breaking and D-term breaking respectively, in the next sections.
1.5.1 F-Term SUSY Breaking
Let us start with a discussion of breaking supersymmetry by a non-vanishing F-term. The canonical example of F-term SUSY breaking is given by the O’Raifeartaigh model [59]
with the following superpotential
WOR =−aΦ1+mΦ2Φ3+y
2Φ1Φ23. (1.109)
Here Φ1,Φ2 and Φ3 are gauge-singlet chiral superfields, such that VD = 0. (A possible Fayet-Iliopoulos term is set to 0 in this model.) The scalar potential arising from this superpotential is given by
VOR =VF =X
i
Fi†Fi =X
i
∂WOR(φi)
∂φi
2
= y
2φ23−a
2
+|m φ3|2+|m φ2+y φ1φ3|2 . (1.110) Looking at the first two terms we see that VOR >0, which is what we wanted. Assuming a < m2/y, the absolute minimum of the potential is located at hφ2i = hφ3i = 0 with hφ1i undetermined at tree-level 16. Such a direction in field-space along which the scalar potential is constant (at tree-level) is called a flat direction. These flat directions play an important role in the construction of viable supersymmetric inflationary models as we will see later on.
Coming back to the O’Raifeartaigh model and additionally assuminghφ1i= 0 we can compute the mass spectrum to explicitly see the breaking of supersymmetry as a mass splitting between the masses of some of the bosonic and fermionic component fields. The resulting mass spectrum is summarised in Table 1.1.
As we can see, SUSY breaking is manifest in the mass splitting between the real and imaginary parts of the complex scalar field φ3 and the third fermionic mass eigenstate i (ψ2 −ψ3)/√
2. In the SUSY conserving limit a → 0 this mass splitting vanishes, as it should be the case17.
Another interesting observation is that one Weyl spinor ψ1 remains massless. This is to be expected since we are breaking a global fermionic symmetry. Thus, this massless fermion, called the Goldstino, can be seen as the analogon of the massless Goldstone boson we get from the breaking of a global bosonic symmetry.
15It is also possible that SUSY breaking is realised as a combination of both F-term and D-term breaking.
16A mass term for theφ1direction is generated at the loop-level. We say that the flat direction is lifted by quantum corrections.
17 In this model the order parameter of SUSY breaking is given byh0|F1|0i=−a.
Mass Eigenstate Squared Mass Re(φ1),Im(φ1) 0 Re(φ2),Im(φ2) m2
Re(φ3) m2−ay Im(φ3) m2+ay
ψ1 0
(ψ2+ψ3)/√
2 m2
i (ψ2−ψ3)/√
2 m2
Table 1.1: Mass spectrum of the O’Raifeartaigh model [59] of F-term SUSY breaking in the minimumhφii= 0 withφi=√1
2(Re(φi) + i Im(φi)).
Unfortunately it turns out that although the O’Raifeartaigh model seems quite ap-pealing, it does not break SUSY in the way we want from a phenomenological point of view: we can see from Table 1.1 that we have one scalar field Re(φ3) with a mass that is smaller than the mass of the corresponding fermionic mass eigenstate. If these fermionic mass eigenstates are to represent the Standard Model particles then we should already have seen some scalar particles with masses smaller than their corresponding Standard Model partners. This is clearly not the case. Also, this is no coincidence but follows from the general tree-level result for theories that contain only chiral superfields
STrM2=X
j
(−1)2j(2j+ 1)m2j = 0, (1.111) where the supertrace STr denotes the trace of the mass-squared matrix over the real fields of the theory. In the presence of SUSY breaking, however, this result gets modified at loop-level18. One can therefore try to break SUSY via the O’Raifeartaigh mechanism in what is called the hidden sector, which then communicates the SUSY breaking via loop effects to the visible sector in such a way that all SUSY partners end up heavier than the corresponding Standard Model particles.
1.5.2 D-Term SUSY Breaking
Let us now turn to the second possibility, which is breaking SUSY spontaneously via non vanishing D-terms [58]. Let us start by considering the case of aU(1) vector superfield V coupled to a chiral superfieldΦwithU(1)-chargeq and vanishing superpotentialW(Φ) = 0. In this case the scalar potential looks like the following
V =VD = 1
2(g q φ†φ+ξ)2, (1.112)
18 It can also get modified in theories that contain vector superfields with some non-vanishing D-term.
where we have included a non-zero Fayet-Iliopoulos term ξ 6= 0. We want VD > 0.
However, what can happen is that the scalar component fieldφdevelops a non-zero vacuum expectation value. In particular, if
h0|φ†φ|0i=− ξ
g q , (1.113)
then h0|V|0i = 0, i.e. supersymmetry is unbroken. The U(1) gauge symmetry, on the other hand, is broken by the non-zero VEV of φ. This is not what we want. What we have to do in order to ensure V >0 is to prevent the scalar field from acquiring a VEV.
This can be done by introducing a large mass for the scalar component field φ through suitable terms in the superpotential. However, since φ is charged under the U(1) gauge symmetry and the superpotential has to be a holomorphic, gauge invariant function of the chiral superfields, we need at least two such chiral superfields with opposite charges q1 =−q2. Then we can write
W =mΦ1Φ2, (1.114)
where we take m to be real for simplicity. The resulting scalar potential is given by V =VF +VD =m2
2
X
i=1
φ†iφi
+1 2 ξ+g
2
X
i=1
qiφ†iφi
2
. (1.115)
If we choose m2 > g|qi|ξ, the minimum of the potential is indeed given by h0|φi|0i = 0, which gives
h0|V|0i=ξ2/26= 0. (1.116)
We find that the order parameter of D-term SUSY breaking is given by the Fayet-Iliopoulos term ξ. The mass spectrum of this model of D-term SUSY breaking is summarised in Table 1.2.
Mass Eigenstate Squared Mass Gauge fieldvµ 0
Gauginoλ 0
Re(φ1),Im(φ1) m2+g q ξ Re(φ2),Im(φ2) m2−g q ξ
(ψ1+ψ2)/√
2 m2
i (ψ1−ψ2)/√
2 m2
Table 1.2: Mass spectrum of the Fayet-Iliopoulos model [58] of D-term SUSY breaking in the minimumhφii= 0 withφi=√1
2(Re(φi) + i Im(φi)).
Again, we find that some of the scalar particles ends up lighter than the corresponding fermion which means that this type of SUSY breaking, too, can only be applied indirectly to a supersymmetric extension of the Standard Model. Also, since the gauge symmetry remains unbroken, the gauge field remains massless. Its superpartner, the gaugino, also remains massless in this scenario and can be identified as the Goldstino arising from the breaking of global SUSY.