[Paper Review] SUSY Jumps Out of Superspace in the Supersymmetric Standard Model

John A. Dixon|arXiv (Cornell University)|Dec 21, 2010

Black Holes and Theoretical Physics1 references5 citations

TL;DR

This paper demonstrates that in chiral supersymmetric theories in 3+1 dimensions, including the supersymmetric standard model (SSM), physically important composite operators—termed 'Outfields'—require explicit dependence on Grassmann odd $ heta$-parameters, breaking superspace invariance. These Outfields arise from a BRST cohomology mechanism where symmetry violation is proportional to field equations, and they are revealed through spectral sequence analysis of the BRST cohomology, showing that SUSY constraints are tied to the superpotential rather than the full action.

ABSTRACT

The supersymmetric standard model (SSM) appears to be firmly grounded in superspace. For example, it would be natural to assume that all the physically important composite operators can be made by combining superfields and superspace derivatives. But even for the simplest possible, free, massless and unbroken SUSY theory in 3+1 dimensions, this is not true. This paper shows that there is a large set of physically important composite operators in the SSM that require explicit factors of the Grassmann odd `$θ$' parameters of superspace. These explicitly break superspace invariance. These composite operators will be called `Outfields', because they are intrinsically `outside' of superspace. It is not possible to write the Outfields using only superfields and superspace derivatives. The Outfields can be found in the BRST cohomology space of the theory. The calculation of the BRST cohomology space for these theories is performed in this paper using a spectral sequence analysis, starting with the free massless theory, and then adding interactions, and then masses.

Motivation & Objective

• To identify and characterize physically important composite operators in chiral SUSY theories that cannot be constructed from superfields and superspace derivatives alone.
• To show that these operators, termed 'Outfields,' explicitly break superspace invariance due to explicit $ heta$-dependence, despite being invariant under BRST symmetry.
• To demonstrate that the BRST cohomology of chiral SUSY theories reveals a new class of operators whose symmetry breaking is proportional to the field equations, not zero.
• To analyze the cohomology structure of the SSM using spectral sequences, starting from free massless theory and incorporating interactions and mass terms.
• To illustrate how Outfields mix with elementary superfields upon spontaneous symmetry breaking, altering their cohomological status.

Proposed method

• Use of BRST cohomology to classify local composite operators that are invariant modulo field equations, with the BRST operator combining symmetry variation and field equation parts.
• Application of spectral sequence analysis to compute the cohomology of chiral SUSY theories, with nine nested differentials and ten cohomology spaces.
• Identification of Outfields as combinations of a field part violating symmetry (proportional to field equations) and a ghost part involving Zinn sources, ensuring BRST invariance.
• Computation of cohomology starting from the free massless theory, then adding interactions and mass terms, with constraints arising from the superpotential's symmetries.
• Use of grading operators $N_{ m Ghost}$, $N_{ m Zinn}$, and $N_{ m Grading}$ to analyze differential structures and prove spectral sequence collapse at $E_3$ for the free massless case.
• Analysis of the CSSM (chiral SSM with right-handed neutrinos and a Higgs singlet) to illustrate Outfield structures and their mixing with elementary superfields upon VEV development.

Experimental results

Research questions

• RQ1 Can physically important composite operators in chiral SUSY theories be constructed solely from superfields and superspace derivatives?
• RQ2 What is the origin of operators that explicitly break superspace invariance despite being BRST-closed?
• RQ3 How does the BRST cohomology of chiral SUSY differ from that of gauge theories, particularly in the role of Zinn sources and field equations?
• RQ4 Why have these operators, termed 'Outfields,' remained undetected for so long in SUSY field theories?
• RQ5 How do mass terms and spontaneous symmetry breaking affect the cohomology structure and the status of Outfields in the SSM?

Key findings

• Outfields are composite operators in chiral SUSY theories that require explicit dependence on Grassmann odd $ heta$-parameters, breaking superspace invariance.
• These Outfields are BRST-closed but not BRST-exact; their symmetry violation is proportional to the field equations, which are equivalent to zero but not identically zero.
• The spectral sequence for the free massless theory collapses at $E_3 = E_ty$, proving that the cohomology is fully determined at this stage.
• For the interacting and massive cases, the spectral sequence is conjectured to collapse at $E_5$ and $E_9$ respectively, based on structural constraints and sector-by-sector analysis.
• Outfields in the CSSM arise from SU(2) doublet and singlet leptons and Higgs fields, with their cohomological status altered when the Higgs VEV breaks internal symmetry.
• The mixing of Outfields with elementary superfields upon spontaneous symmetry breaking implies that these operators are removed from the cohomology space when the VEV turns on.