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JHEP03(2017)081

Published for SISSA by Springer

Received: October 10, 2016

Accepted: March 2, 2017

Published: March 15, 2017

Origin of soft limits from nonlinear supersymmetry in

Volkov-Akulov theory

Renata Kallosh, Anna Karlsson and Divyanshu Murli

Stanford Institute for Theoretical Physics and Department of Physics, Stanford University,

Stanford, CA 94305, U.S.A.

E-mail: kallosh@stanford.edu, annakarl@stanford.edu,

divyansh@stanford.edu

Abstract: We apply the background ﬁeld technique, recently developed for a general class

of nonlinear symmetries, at tree level, to the Volkov-Akulov theory with spontaneously

broken N = 1 supersymmetry. We ﬁnd that the background ﬁeld expansion in terms of

the free ﬁelds to the lowest order reproduces the nonlinear supersymmetry transformation

rules. The double soft limit of the background ﬁeld is, in agreement with the new general

identities, deﬁned by the algebra of the nonlinear symmetries.

Keywords: Global Symmetries, Supersymmetric Eﬀective Theories

ArXiv ePrint: 1609.09127

Open Access,

 The Authors.

Article funded by SCOAP

doi:10.1007/JHEP03(2017)081

JHEP03(2017)081

Contents

1 Introduction 1

2 The VA model 2

2.1 The VA model and the background ﬁeld method 3

3 Iterative solution for the background ﬁeld in the VA model 5

4 Properties of the VA model background ﬁeld 7

4.1 The supersymmetry rules 7

4.2 The double soft limit 8

5 Discussion 8

1 Introduction

In models with linear symmetries there are well-known Ward identities constraining the

physical observables, both at tree level and regarding quantum corrections, in the absence

of anomalies. In the case of nonlinear symmetries, often associated with some coset space

G/H, it is known that certain soft theorems relate on-shell amplitudes with diﬀerent num-

ber of legs to each other. However, until recently there was no general analogue of the

Ward identities for the case of nonlinear, spontaneously broken symmetries, for example

in Volkov-Akulov (VA) theory [1]. Such a general analysis of nonlinear symmetries was

performed at tree level in [2], where the relevant Ward type identities were derived through

a generalization of the background ﬁeld method in the abstract formalism of DeWitt [3–5].

The construction provides an initial method for analyses, with a possible, later continuation

to loops. The new identities in [2] should be applicable to any model with nonlinear sym-

metry

where the action and its nonlinear symmetries are known, and the transformation

rule has a constant ﬁeld-independent part as well as various powers of the ﬁelds beyond

the linear dependence. Examples for which the new identities should be valid include the

VA theory with fermionic, nonlinear supersymmetry [1], N = 8 supergravity with bosonic,

nonlinear E

7(7)

symmetry [9], and the Dirac-Born-Infeld-Volkov-Akulov (DBI-VA) model

with 16 linear + 16 nonlinear supersymmetries, as presented in [10].

The purpose of this article is to test these new, general identities, applicable to such

diﬀerent examples, in the simplest setting possible: the VA model. We will study the

nonlinear supersymmetry and the relation between the symmetry and the soft limits for

The background ﬁeld method for NSM (nonlinear sigma models) was developed in [6]. General models

with single and multiple Nambu-Goldstone bosons were studied recently in [7, 8].

– 1 –

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