超对称Chern-Simons模型的重整化组改进与对称性破坏研究

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本文主要探讨了在（2+1）维时空中，超对称Chern-Simons-matter模型中的重整化群改进（Renormalization Group Improvement, RGE）以及其对动态对称破坏（Dynamical Symmetry Breaking, DSB）的影响。研究对象是N-1个超对称理论，其中包含一个与N个标量超场耦合的Abelian Chern-Simons超场。初始模型基于经典拉格朗日量，展现出尺度不变性，然而这一性质会在量子效应，特别是辐射修正（radiative corrections）作用下被有效超电势所打破。作者通过应用重整化群方程（RGE）和beta函数，着重计算了最多两个环的阶次有效超电势。在这个过程中，他们关注了对反常维度的理解，这是计算高阶效应的重要组成部分。通过系统地处理前导日志（Leading Logarithms, LL）、次前导日志（Next-to-Leading Logarithms, NLL）以及其他相关的量子贡献，作者展示了如何利用RGE方法来精细化这些计算。尽管重整化群改进在超对称模型中的应用相较于非超对称理论可能会有所不同，但文中表明，这个改进过程确实有助于提高对模型物理性质的理解。研究结果显示，即使在超对称环境中，RGE的考虑对于理解DSB的行为仍然具有重要价值，尽管可能不像在非超对称理论中那样显著地改变结果。此外，文章还包含了接收日期、修订和接受日期，以及在线发表的链接，这表明它是在核物理学领域的一个知名期刊《核物理学B》上发表的开放获取文章，由Stephan Stieberger编辑。整个研究过程严谨细致，旨在深化我们对超对称Chern-Simons-matter模型在量子场论背景下动态对称破缺行为的认识。

666 A.G. Quinto et al. / Nuclear Physics B 907 (2016) 664–677

this theory are exhausted by bosons and physical fermions making its gauge sector to have no

on-shell degrees of freedom. These requirements are satisﬁed by a Chern–Simons-matter theory

called BLG theory [30–34], which describes two M2-branes. Relaxing the requirement of man-

ifest N = 8 supersymmetry, this approach can be generalized to a N = 6

Chern–Simons-matter

theory with the gauge group U

(N) × U

−k

(N) (k and −k are CS levels) [35,36], which is ex-

pected to be enhanced to N = 8for k = 1or k = 2 [37–39]. The quantization of such model was

thoroughly studied in [40–48]. Also, detailed calculations of the effective superpotential within

N = 2 superﬁeld theories in three dimension have been reported in [49,50].

The ﬁrst part of our wo

rk is the computation of the effective superpotential of a generic su-

persymmetric CS theory coupled to matter superﬁelds, up to two-loops. To this end, we use the

RGE and the β and γ function calculated in [17,51], thus avoiding the direct computation of

any supergraph. With this result in hand, we discuss ho

w we can reorganize the expansion of

the effective superpotential in terms of Leading Logs (LL), Next-to-Leading Logs (NLL) con-

tributions, and so on, in a way that allows us to calculate coefﬁcients arising from higher orders

corrections, thus improving the two-loop evaluation of the effective superpotential. We are thus

able to ﬁnd an i

mproved effective superpotential, which will be used to study DSB in our model.

We will show that, contrary to what happens in the non-supersymmetric case [24], here the RGE

improvement leads only to slight modiﬁcations in the DSB scenario.

In this wo

rk, we shall focus on calculations done in the superﬁeld language [52,53], in which

supersymmetry is manifest in all stages of the calculations. This paper is organized as follows:

in Sec. 2, we present our model and calculate the effective superpotential with the knowledge

of the renormalization group functions found in the literature, together with the RGE. Section 3

views the standard approach to RGE improvement of the effective potential in four dimensional

models, and section 4 is devoted to adapt this procedure to the supersymmetric three-dimensional

case. The resulting improved effective superpotential is used in Sec. 5 to study DSB in our model.

Section 6 presents our conclusions and perspecti

ves. Some explicit results and the Mathematica

code used to obtain them is given as a Supplementary Material to this work.

2. Calculation of the effective superpotential

Our starting point is the classical action in N = 1 superspace of a Chern–Simons superﬁeld



coupled to N massless complex scalars superﬁelds 

, with a quartic self-interaction,

S =





−



−

∇



∇











, (3)

where W



is the gauge superﬁeld strength, ∇

(

− ig 

)

is the gauge su-

percovariant derivative, and a = 1, ... N. We follow the basic conventions for three-dimensional

supersymmetry found in [52].

The main object we shall be interested in studying is the three dimensional effective superpo-

tential [54]. To deﬁne this object, we consider a shift in the N-th component of 



= 

+ σ, (4)

by the background (constant) superﬁeld σ = σ

− θ

. On general grounds, the effective super-

potential can be cast as



[

]



(

)





σ, D



. (5)

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