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Holomorphic and Canonically Normalized Wess–Zumino Couplings

Physical scope

A formal perturbative rigid N=1 Wess–Zumino Wilsonian example at nonzero scale. It distinguishes a holomorphic coefficient from its canonically normalized value and makes no empirical or continuum claim.

Example

Take one chiral superfield $Φ$ with mass dimension one and tree superpotential $W_{0} (Φ) = \frac{1}{2} m Φ^{2} + \frac{y}{3!} Φ^{3}$ , where $m$ has mass dimension one and $y$ is dimensionless. Use a formal perturbative Wilsonian prescription satisfying the $N = 1$ theorem at a nonzero scale $λ$ , and suppose the full real quadratic kinetic coefficient at that scale is a positive function $Z (λ)$ . Then the holomorphic coefficients $m$ and $y$ in the unrescaled field $Φ$ receive no perturbative Wilsonian corrections, while the canonically normalized field $Φ_{c} = Z (λ)^{1 / 2} Φ$ has coefficients $m_{c} (λ) = m / Z (λ)$ and $y_{c} (λ) = y / Z (λ)^{3 / 2}$ . Thus the canonically normalized parameters can run if $Z (λ)$ varies, even though the holomorphic Wilsonian coefficients do not.

Example references

Martin, §6.5

Weinberg, equations (3)–(12)

Facts & Assumptions

Given: The formal $N = 1$ Wilsonian action datum and theorem hypotheses, including the coefficientwise local background-spurion extension with independent unconstrained real spurion jets and chirality before constant-background evaluation; the stated one-field superpotential; and a positive full quadratic kinetic coefficient $Z (λ)$ .

[F1]

Local $N = 1$ Wilsonian Superspace Action Data: the chiral superfield has a holomorphic F-term integrand and a real quadratic kinetic term in the Wilsonian D-term data; real-spurion selection is interpreted in its coefficientwise formal background-jet algebra before evaluating constant backgrounds.

[F2]

$N = 1$ Wilsonian F-Term Nonrenormalization: the prescribed gauge/ghost completion is empty in this ungauged model; under its formal cutoff, symmetry, anomaly-free, formal local background-spurion hypotheses, every positive-field-degree coefficient of the remaining holomorphic integrand is unchanged after collecting matter-independent quadratic gauge-kinetic terms. Equality of its numerical constant additionally uses the supplier's fixed-representative convention; it is not needed for the coefficients $m, y$ . In this ungauged Wess–Zumino example the ultraviolet remainder is $W_{0}$ .

[A1]

The single-field model has no gauge interactions; it is the trivial-gauge-group specialization of the stated action data.

[A2]

$Z (λ) > 0$ , so the real field rescaling to $Φ_{c}$ is invertible.

Verification

1.1givenF1F2A1

By [F2], the coefficients $m$ and $y$ of $W_{0} (Φ)$ are unchanged in the unrescaled holomorphic variable at every formal perturbative order.

2.1A2algebrastep 1.1∎

Since $Z (λ) > 0$ , substitute $Φ = Z (λ)^{- 1 / 2} Φ_{c}$ into $W_{0}$ . This gives $W_{0} = \frac{1}{2} (m / Z) Φ_{c}^{2} + (y / Z^{3 / 2}) Φ_{c}^{3} / 3!$ , hence the stated canonical coefficients. For a nonzero bare parameter these expressions vary whenever $Z$ does; a zero bare parameter remains zero.

Holomorphic and Canonically Normalized Wess–Zumino Couplings

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources