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Heavy-Chiral Threshold and a Spurion Soft Trilinear

Physical scope

One formal rigid N=1 light-singlet EFT below a declared heavy-singlet threshold, with one fixed higher-dimensional chiral-spurion insertion. No phenomenological value, numerical smallness, loop protection after SUSY breaking, continuum result, or empirical observation is asserted.

Example

Take one light gauge-singlet chiral superfield $Φ$ and one heavy gauge-singlet chiral superfield $X$ , with canonical quadratic D-terms, in the rigid flat-space $N = 1$ convention of the Wilsonian action definition. Let $M > 0$ be the heavy threshold mass, let $g, y, κ \in C$ be dimensionless, and choose

$W_{+} (Φ, X) = \frac{1}{2} M X^{2} + \frac{g}{2} X Φ^{2} + \frac{y}{6} Φ^{3} .$

Adopt full Axiom of Choice as a sufficient ambient assumption for the action data. Use the tree-level zero-supercovariant-derivative matching prescription at external energies $E_{e x t} ≪ M$ . Select high-energy holomorphic representatives containing $X^{2}, X Φ^{2}, Φ^{3}$ and a low-energy span containing $Φ^{3}, Φ^{4}$ , with a linear projection $P_{-}$ fixing both latter monomials. Supply action-to-scheme identifications with these same holomorphic coefficients, suppression factors and unrescaled light field; no term of $W_{L}$ or $J^{2} / (2 M)$ is discarded. The matched holomorphic Wilsonian superpotential coefficient is

$W_{-} (Φ) = \frac{y}{6} Φ^{3} - \frac{g^{2}}{8 M} Φ^{4} .$

In the low-energy EFT, choose a fixed nondynamical chiral spurion background $X = 2 θ^{1} θ^{2} F_{X}$ with $[X] = 1$ and constant $[F_{X}] = 2$ , and add the single local operator

$Δ W = \frac{κ}{6 Λ_{o p}} X Φ^{3},$

where $Λ_{o p} > 0$ is an operator-suppression scale distinct from $M$ . After fixing this background, the added component term is $\frac{A_{s o f t}}{6} ϕ^{3} + h . c .$ with $A_{s o f t} = κ F_{X} / Λ_{o p}$ and mass dimension one. Here a soft scalar trilinear means a scalar cubic interaction with a positive-mass-dimension coefficient. This is an explicit spurion insertion, not an observed value or a consequence of the supersymmetric threshold theorem. No numerical smallness bound on $A_{s o f t}$ is asserted; omitting additional spurion insertions requires separate power-counting input.

Example references

Rigid $N = 1$ Wilsonian action, field content, and scale conventions

Leading heavy-chiral Wilsonian superpotential match

Smooth chiral superfields and component expansion

Ordered Berezin integration by coefficient extraction

Martin, §5.1 and §7.6

Manohar, Chapter 5, §§5.5–5.7

Facts & Assumptions

Given: Full AC; the fixed high-energy superpotential and matching regime in the Example; the tree-level threshold prescription; and the fixed chiral spurion background and operator truncation.

[F1]

Supersymmetric Wilsonian EFT action and threshold data: light and heavy chiral multiplets are smooth classical $N = 1$ superfields with fixed engineering dimensions; the Wilsonian F-term is distinct from real D-term data, and operator-suppression and threshold scales are separate.

[F2]

Leading Wilsonian Superpotential Match at a Supersymmetric Chiral Threshold: for $W_{+} = W_{L} + \frac{1}{2} M X^{2} + X J (Φ)$ with the stated leading derivative-free tree prescription, and with a retained projection fixing $W_{L}$ and $J^{2} / (2 M)$ and compatible action-to-scheme holomorphic coordinates, the matched holomorphic coefficient is $W_{L} - J^{2} / (2 M)$ .

[F3]

Four-dimensional $N = 1$ chiral superfield on flat superspace: $Φ$ and $X$ are smooth classical chiral superfields; their lowest components are the scalar fields $ϕ$ and $x$ .

[F4]

Four-dimensional $N = 1$ superspace Grassmann algebra and Berezin integration: $\int d^{2} θ$ extracts the coefficient of the ordered monomial $θ^{1} θ^{2}$ , and $[θ^{α}] = - \frac{1}{2}$ .

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function. Full AC is adopted as a sufficient ambient assumption for the selected action data; the regulated equivalence supplier requires no additional full-Choice premise; the arithmetic and Berezin extraction below use no choice.

[A2]

The spurion is nondynamical and fixed after writing the supersymmetric chiral operator; its scalar and spinor components vanish and its physical auxiliary component is the constant $F_{X}$ , with raw top coefficient $2 F_{X}$ .

[A3]

The operator is retained as one specified term in the finite EFT truncation. Any claim that omitted spurion insertions are small requires an additional bound and is not assumed.

[F5]

Wilsonian Superpotential Nonrenormalization in a Finite Supersymmetric EFT Basis: its perturbative holomorphic-coefficient conclusion assumes a rigid-supersymmetry-preserving regulator and Ward identities; it does not assert protection after a fixed supersymmetry-breaking background is imposed.

Verification

1.1givenF2algebra

In the threshold theorem set $W_{L} (Φ) = y Φ^{3} / 6$ and $J (Φ) = g Φ^{2} / 2$ . The selected $P_{-}$ fixes $Φ^{3}$ and $Φ^{4}$ , so it fixes $W_{L}$ and $J^{2} / (2 M)$ ; the supplied coordinate identifications use their same coefficients. Then $J^{2} / (2 M) = g^{2} Φ^{4} / (8 M)$ , which gives the displayed retained $W_{-} (Φ)$ .

1.2givenF1F3F4A2algebra

Since $X = 2 θ^{1} θ^{2} F_{X}$ , the raw coefficient of $θ^{1} θ^{2}$ in $X Φ^{3}$ is $2 F_{X} ϕ^{3}$ : [F3] gives the lowest component $ϕ$ of $Φ$ , while every positive- $θ$ component of $Φ^{3}$ multiplies a repeated odd generator and vanishes. The physical F bracket supplied by [F1] is one-half the raw ordered integral in [F4], and therefore gives $[Δ W]_{F} = κ F_{X} ϕ^{3} / (6 Λ_{o p})$ .

2.1F1F5A1A2A3step 1.2algebra∎

With $[ϕ] = 1$ , $[F_{X}] = 2$ , and $[Λ_{o p}] = 1$ , the coefficient $A_{s o f t} = κ F_{X} / Λ_{o p}$ has dimension one and the scalar cubic term has dimension four. For $F_{X} \neq 0$ , the left derivative gives $Q_{1} (2 θ^{1} θ^{2} F_{X}) = 2 θ^{2} F_{X} \neq 0$ , so fixing this background breaks rigid supersymmetry. The Ward-identity hypothesis of [F5] then fails, and that theorem supplies no protection claim for the fixed soft coefficient. Full AC is the adopted sufficient ambient assumption [A1]; no choice is used in this component calculation. No estimate of the soft coefficient's size follows from its dimension.

Remarks

The Wilsonian tree-level F-term and the fixed soft spurion insertion are separate pieces of the example. For $F_{X} \neq 0$ , fixing the spurion violates the rigid-supersymmetry premise used by the nonrenormalization theorem. If $F_{X} = 0$ , the insertion vanishes and this supersymmetry-breaking conclusion does not apply.

Heavy-Chiral Threshold and a Spurion Soft Trilinear

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources