[Paper Review] On effective Kähler potential in N=2, d=3 SQED

I. L. Buchbinder, Б. С. Мерзликин|arXiv (Cornell University)|May 28, 2015

Black Holes and Theoretical Physics22 references5 citations

TL;DR

This paper computes the two-loop effective Kähler potential in three-dimensional ${\cal N}=2$ supersymmetric quantum electrodynamics (SQED) with a Chern-Simons term for the gauge superfield using the background field method and a nonlocal field redefinition to diagonalize propagators. The key result is that the two-loop effective Kähler potential is gauge-independent and breaks classical superconformal invariance, revealing quantum corrections analogous to one-loop results in 4D ${\cal N}=1$ SQED.

ABSTRACT

We compute the two-loop effective Kähler potential in three-dimensional N=2 supersymmetric electrodynamics with Chern-Simons kinetic term for the gauge superfield. The effective action is constructed on the base of background field method with one parametric family of gauges. In such an approach, the quadratic part of quantum action mixes the gauge and matter quantum superfields yielding the complications in the computations of the loop supergraphs. To avoid this obstacle and preserve dependence on the gauge parameter we make a nonlocal change of quantum matter superfields after which the propagator is diagonalized, however the new vertices have appeared. We fix the suitable background and develop the efficient procedure of calculating the two-loop supergraphs with the new vertices. We compute the divergent and finite parts of the superfield effective action, find the two-loop effective Kähler potential and show that it does not depend on the gauge parameter.

Motivation & Objective

• To compute the two-loop effective Kähler potential in 3D ${\cal N}=2$ SQED with a Chern-Simons kinetic term for the gauge superfield, a model with non-trivial quantum corrections despite its simplicity.
• To address the challenge of non-diagonal propagators arising from gauge-matter mixing in the background field method by introducing a nonlocal redefinition of quantum matter superfields.
• To develop a systematic procedure for computing two-loop supergraphs with new vertices introduced by the field redefinition, ensuring gauge independence of the final result.
• To demonstrate that the two-loop effective Kähler potential is independent of the gauge parameter, confirming its physical relevance.
• To show that quantum corrections break the classical superconformal invariance of the model, analogous to holomorphic effective actions in 4D ${\cal N}=2$ theories.

Proposed method

• The background field method is employed with a one-parameter family of gauges to compute the effective action, preserving gauge dependence during intermediate steps.
• A nonlocal field redefinition is applied to quantum matter superfields to diagonalize the propagator, eliminating mixing with the gauge superfield in the quadratic action.
• The new vertices introduced by the redefinition are systematically included in the calculation of two-loop supergraphs using superspace techniques.
• The computation is performed in ${\cal N}=2$ superspace with dimensional regularization in $d=3-2\varepsilon$ to handle divergences.
• Momentum integrals are evaluated using standard formulas, with divergent parts extracted via $\Gamma$-function expansion in $\varepsilon$.
• The gauge parameter dependence is explicitly tracked and shown to cancel in the final effective Kähler potential, ensuring gauge independence.

Experimental results

Research questions

• RQ1 How do two-loop quantum corrections affect the effective Kähler potential in 3D ${\cal N}=2$ SQED with a Chern-Simons term?
• RQ2 Can the effective Kähler potential be computed in a gauge-independent manner despite the gauge dependence of the effective action?
• RQ3 Does the two-loop effective Kähler potential break the classical superconformal symmetry of the model?
• RQ4 What is the role of the nonlocal field redefinition in simplifying the computation of two-loop supergraphs with mixed propagators?
• RQ5 How does the structure of the two-loop effective Kähler potential compare to known results in 4D ${\cal N}=1$ SQED?

Key findings

• The two-loop effective Kähler potential is computed explicitly and found to be independent of the gauge parameter, confirming its physical significance.
• The divergent part of the effective Kähler potential is proportional to $\frac{1}{\varepsilon}$, with the coefficient involving $\frac{1}{k^2\alpha^2}$, where $k$ is the Chern-Simons level and $\alpha$ is a coupling constant.
• The finite part of the effective Kähler potential contains a logarithmic term $\ln 2$, arising from momentum integrals involving massless and massive propagators.
• The effective Kähler potential breaks the classical superconformal invariance of the model, indicating quantum-scale symmetry breaking.
• The structure of the two-loop correction resembles the one-loop effective Kähler potential in 4D ${\cal N}=1$ SQED, suggesting a deeper analogy between 3D ${\cal N}=2$ and 4D ${\cal N}=1$ theories.
• The computation confirms that the effective Kähler potential in this model is unambiguously defined due to the ability to fix the background for chiral superfields satisfying classical equations of motion.