RG and Supersymmetry
QFTs with supersymmetry (SUSY) have some remarkable properties that make them particularly interesting to study from the point-of-view of the RG.
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RG and Supersymmetry
QFTs with supersymmetry (SUSY) have some remarkable properties that make them particularly interesting to study from the point-of-view of the RG. For instance, one can prove the existence of many non-trivial fixed points of the RG in such theories in d = 4. Our discussion of SUSY theories will be geared towards describing some of these fixed points and the associated RG structure at the expense of many other aspects of SUSY theories. In particular, we shall use a description in terms of component fields rather than introduce the whole paraphernalia of superspace and super-fields. In a SUSY theory, fields are collected into multiplets of supersymmetry which consist of fields with different spin. In d = 4, which we will stick to exclusively, there are 2 basic multiplets: (i) a chiral multiplet consisting of a complex scalar, a Weyl fermion and an auxiliary complex scalar field Φ = (φ, ψα , F),
(5.1)
(ii) a vector multiplet consisting of a gauge field, an adjoint-valued Weyl fermion and an adjoint-valued auxiliary real scalar field V = (Aμ , λα , D).
(5.2)
All our conventions are taken from the text-book by Wess and Bagger (1992).
5.1 Theories of Chiral Multiplets: Wess-Zumino Models Wess-Zumino models are constructed from chiral multiplets Φi with a basic SUSY invariant kinetic term of the form1
1
The kinetic term can be generalized to all the terms with two derivatives:
T. J. Hollowood, Renormalization Group and Fixed Points, SpringerBriefs in Physics, DOI: 10.1007/978-3-642-36312-2_5, © The Author(s) 2013
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5 RG and Supersymmetry
Lkin = −|∂μ φi |2 + i ψ¯ i σ¯ μ ∂μ ψi + |Fi |2 .
(5.4)
Note that the auxiliary fields Fi do not have kinetic terms and their only rôle in the theory is to simplify the SUSY structure. The interactions are determined by the super-potential W (φi ), a function of the fields φi , but not their complex conjugates: Lint = Fi
∂W 1 ∂2W − ψi ψ j + c.c. ∂φi 2 ∂φi ∂φ j
(5.5)
(c.c.=complex conjugate.) Notice that since the auxiliary fields Fi only appear quadratically in the Lagrangian, they can trivially be “integrated out” exactly by using their equations-of-motion Fi = −
∂ W (φ ∗j )
(5.6)
∂φi∗
and substituting this back into the action to leave a net potential on the scalar fields of the form ∂ W 2 ∗ V (φi , φi ) = (5.7) ∂φ . i
i
The theory is invariant under the infinitesimal SUSY transformations involving a space-time constant Grassmann spinor parameter ξα : δφi = ξ ψi , δψi = iσ μ ξ¯ ∂μ φi + ξ Fi , δ Fi = i ξ¯ σ¯ μ ∂μ ψi .
(5.8)
There is an important distinction between the kinetic and interaction terms: Lkin are “D terms”, which are real functions of the fields and their complex conjugates, while Lint are “F terms”, consisting of the sum of a part which is holomorphic in the fields and the couplings and the complex conjugate term which has all quantities replaced by the corresponding anti-holomorphic ones. In particular, the superpotential W (φi ) is a holomorphic function of the φi and also the couplings gn , i.e. W (φi ) does
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