The EP Model and its Completion Terms (E4)

Here we present the simple example of an Exotic Invariant with just two chiral electron supermultiplets E and P. In this example we include a mass term, and that means that there is a constraint on the Exotic Invariant. The constraint is easily solved for this simple case. Here we also exhibit a simple conjecture for the Completion Terms. This simple example is very useful, because the constraint that arises in the case of the Exotic Model, presented in E6, is just as easy to solve, and the Completion Terms there are also very similar to those here. So this simple EP model is very useful for understanding the Exotic Model, which is what results from adding an Exotic Invariant to the rather complicated Supersymmetric Standard Model.

💡 Research Summary

The paper “The EP Model and its Completion Terms (E4)” by John A. Dixon presents a minimal supersymmetric construction that illustrates how an Exotic Invariant (EI) can be incorporated into a theory with only two chiral electron supermultiplets, denoted E and P. The central aim is to demonstrate, in a tractable setting, how the presence of a superpotential (mass term) generates a constraint in the BRS cohomology, how that constraint can be solved, and how one may conjecture the form of higher‑order “completion terms” that preserve the master equation.

Model definition
The action is split into a field part A_fields and a pseudo‑field part A_pseudoFields. The field part contains kinetic terms for the two chiral multiplets (Eqs. 4‑5), the CDSS sector (Eqs. 6‑8) which is carried over unchanged from the earlier E3 paper, and a mass term derived from a superpotential (Eqs. 9‑10). The mass term couples the auxiliary fields F_E, F_P and the fermions ψ_E, ψ_P with a common mass parameter m. Because a superpotential is present, the BRS spectral sequence produces a second‑order constraint d² EX = 0 (Eq. 21), which in component form reads a linear combination of the fields multiplied by m.

Pseudo‑fields and structure
Corresponding ghost‑like pseudo‑fields (Γ, Y, Λ, etc.) are introduced for each physical field (Eqs. 13‑19). Their role is to generate the nilpotent BRS operator δ via functional differentiation of the action. A simple structure term K α β C α C β (Eq. 19) completes the BRS algebra.

Exotic Invariant AX
The exotic invariant is defined as the difference of two similar expressions, one built from the E multiplet and one from the P multiplet (Eq. 22). Each side contains eleven monomials with coefficients b_i (Eqs. 23‑26). The coefficients are chosen to satisfy the BRS invariance condition δ AX = 0 (Eq. 27). Explicitly, the BRS variation of each side yields expressions (29) and (30); the minus sign in Eq. 22 forces a cancellation between the two sides, which fixes the signs of the b_i to be ±1 as listed.

Master equation and BRS transformations
The master equation M = 0 (Eq. 32) is written as the sum of four parts (E, P, CDSS, Structure). Functional derivatives of the action give the BRS transformations for all fields and pseudo‑fields (Eqs. 37‑42). These transformations are nilpotent (δ² = 0) and reproduce the constraint (21) when acting on the exotic invariant. The CDSS sector transformations are also displayed (Eq. 41).

Completion terms and conjecture
Having established that the first‑order exotic invariant is BRS‑closed, the author proposes to add it to the original action with a coupling constant g, together with second‑order and mixed terms proportional to g² (Eq. 43). The second‑order pieces A_{X,2} and A_{X,2}^{Mixed} are written in terms of new coefficients b_{12}…b_{20} (Eqs. 45‑49). The conjecture is that the completed action

A_completed = A + g A_X + g A_X + g² A_{X,2} + g² A_{X,2} + g g A_{X,2}^{Mixed}

still satisfies the master equation with the same BRS operator. The author notes that determining the new coefficients will require a computer‑algebra implementation (see reference