[Paper Review] Effective action in general chiral superfield model

A. Yu. Petrov|ArXiv.org|Feb 2, 2000

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper calculates the leading quantum corrections to the chiral and Kählerian effective potentials in a general N=1 supersymmetric chiral superfield model with arbitrary Kähler potential $K(\bar{\Phi},\Phi)$ and superpotential $W(\Phi)$. Using supergraph techniques and loop expansion in the path integral formalism, it demonstrates that the two-loop chiral correction to the effective action is finite despite the theory being non-renormalizable, providing a universal finite result for $W_{\text{eff}}^{(1)}$ in terms of $W^{\prime\prime}$, $\bar{W}^{\prime\prime}$, and $K_{\Phi\bar{\Phi}}$. This extends known results in the Wess-Zumino model to general models.

ABSTRACT

The effective action in general chiral superfield model with arbitrary kählerian potential $K(\barΦ,Φ)$ and chiral (holomorphic) potential $W(Φ)$ is considered. The one-loop and two-loop contributions to kählerian effective potential and two-loop (first non-zero) contribution to chiral effective potential are found for arbitrary form of functions $K(\barΦ,Φ)$ and $W(Φ)$. It is found that despite the theory is non-renormalizable in general case two-loop contribution to holomorphic effective potential is always finite.

Motivation & Objective

• To compute the leading quantum corrections to the chiral and Kählerian effective potentials in a general N=1 supersymmetric chiral superfield model with arbitrary $K(\bar{\Phi},\Phi)$ and $W(\Phi)$.
• To determine whether quantum corrections to the effective action remain finite in non-renormalizable theories with arbitrary potentials.
• To generalize known results from the Wess-Zumino model to the most general chiral superfield model.
• To establish the universality and finiteness of the two-loop chiral correction across arbitrary superpotential and Kähler potential functions.

Proposed method

• Employing the path integral representation of the effective action with background superfields $\Phi$, $\bar{\Phi}$ and quantum superfields $\phi$, $\bar{\phi}$.
• Expanding the action in powers of quantum superfields to extract quadratic and higher-order terms for propagators and vertices.
• Using supergraph techniques to compute loop contributions, focusing on diagrams with equal numbers of $D^2$ and $\bar{D}^2$ factors for the Kählerian potential.
• Applying $D$-algebra transformations and momentum-space integration to evaluate loop integrals.
• Subtracting divergences via renormalization to isolate finite quantum corrections.
• Evaluating the chiral correction $W_{\text{eff}}^{(1)}$ by considering only massless propagators and chiral vertices, leading to a universal finite expression.

Experimental results

Research questions

• RQ1 Is the two-loop chiral correction to the effective action finite in a general N=1 chiral superfield model with arbitrary $K(\bar{\Phi},\Phi)$ and $W(\Phi)$?
• RQ2 Can a universal finite expression for the one-loop chiral correction $W_{\text{eff}}^{(1)}$ be derived that depends only on $W^{\prime\prime}$, $\bar{W}^{\prime\prime}$, and $K_{\Phi\bar{\Phi}}$?
• RQ3 Does the non-renormalizability of the theory affect the finiteness of the leading chiral quantum correction?
• RQ4 How do the one-loop and two-loop corrections to the Kählerian effective potential $K_{\text{eff}}$ depend on higher derivatives of $K$ and $W$?
• RQ5 Can the known finite result in the Wess-Zumino model be generalized to arbitrary chiral superfield models?

Key findings

• The one-loop chiral correction to the effective superpotential is finite and given by $W_{\text{eff}}^{(1)} = -\int d^{6}z \frac{1}{32\pi^{2}} \frac{W^{\prime\prime}\bar{W}^{\prime\prime}}{K_{\Phi\bar{\Phi}}^{2}} \ln\left(\frac{W^{\prime\prime}\bar{W}^{\prime\prime}}{\mu^{2}K_{\Phi\bar{\Phi}}^{2}}\right)$, independent of the renormalizability of the model.
• The one-loop correction to the Kählerian potential is $K^{(1)} = -\int d^{4}\theta \frac{1}{32\pi^{2}} \frac{W^{\prime\prime}\bar{W}^{\prime\prime}}{K_{\Phi\bar{\Phi}}^{2}} \ln\left(\frac{W^{\prime\prime}\bar{W}^{\prime\prime}}{\mu^{2}K_{\Phi\bar{\Phi}}^{2}}\right)$, showing the same logarithmic dependence.
• The two-loop correction to the Kählerian potential $K^{(2)}$ is finite and expressed as a combination of logarithmic and constant terms involving $|W^{\prime\prime}|^2$, $K_{\Phi\bar{\Phi}}$, and derivatives of $K$ and $W$.
• The result for $W_{\text{eff}}^{(1)}$ reduces to the known finite expression in the Wess-Zumino model when $W = -\lambda \Phi^3 / 3!$ and $K = \bar{\Phi}\Phi$, confirming consistency.
• The finiteness of the chiral correction persists even in non-renormalizable theories, due to the structure of chiral supergraphs and the cancellation of divergences via $D$-algebra and momentum integration.
• The normalization scale $\mu$ is fixed by a suitable normalization condition, and the final results are independent of this choice in the physical content of the effective action.