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JHEP11(2017)203

Published for SISSA by Springer

Received: October 5, 2017

Accepted: November 22, 2017

Published: November 30, 2017

Soft pion theorem, asymptotic symmetry and new

memory eﬀect

Yuta Hamada

a,b

and Sotaro Sugishita

Department of Physics, University of Wisconsin-Madison,

Madison, WI 53706, U.S.A.

KEK Theory Center, IPNS, KEK,

Tsukuba, Ibaraki 305-0801, Japan

Department of Physics, Osaka University,

Toyonaka, Osaka 560-0043, Japan

E-mail: [email protected], [email protected]

Abstract: It is known that soft photon and graviton theorems can be regarded as the

Ward-Takahashi identities of asymptotic symmetries. In this paper, we consider soft the-

orem for pions, i.e., Nambu-Goldstone bosons associated with a spontaneously broken

axial symmetry. The soft pion theorem is written as the Ward-Takahashi identities of the

S-matrix under asymptotic transformations. We investigate the asymptotic dynamics, and

ﬁnd that the conservation of charges generating the asymptotic transformations can be

interpreted as a pion memory eﬀect.

Keywords: Global Symmetries, Scattering Amplitudes, Spontaneous Symmetry Breaking

ArXiv ePrint: 1709.05018

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP11(2017)203

JHEP11(2017)203

Memory

Effect

Soft

Theorem

Asymptotic

Symmetry

Figure 1. Triangular relation among soft theorem, asymptotic symmetry and memory eﬀect.

number of conserved charges. One can show that the charge conservation is equivalent to

the scalar memory eﬀect discussed in [38, 43]. However, the theories in [45] suﬀer from the

infrared divergences, and in general, the scalar boson acquires ﬁnite mass at loop level.

In this paper, we consider the case that the massless scalar is a NG boson. Unlike the

model considered in [45], we consider a theory without infrared divergences, and massless-

ness of scalar is ensured from the NG theorem.

The behaviors of scattering amplitudes in the soft limit of a NG boson are diﬀerent

from those in QED and QG since the NG bosons interact only through derivative couplings.

Let ω be the energy of a soft particle. In QED and QG, the scattering amplitudes with soft

particle, ω → 0, are factorized into the product of O(ω

−1

) soft factor and the amplitudes

without soft particle. For the soft limit of NG boson, the O(ω

−1

) factors are absent, and

the soft factors start from O(1). Moreover, even the O(1) factors often vanish due to

so-called Adler’s zero [46, 47].

We consider a speciﬁc model that avoids Adler’s zero. The model contains a complex

scalar and a Dirac fermion, and there is a global axial U(1) symmetry, which is sponta-

neously broken by choosing a vacuum. This model may be regarded as a toy model of

real pions or axions in the beyond standard model, and therefore we call the associated

NG bosons pions. In this paper, we ﬁrst review that the scattering amplitudes with a soft

pion give universal O(1) factors as in [48]. Then, we rewrite the soft pion theorem as the

Ward-Takahashi identity of S-matrix by identifying an inﬁnite number of charges which

generate an asymptotic symmetry. Furthermore we show that the charge conservation can

be interpreted as a pion memory eﬀect, where the information of hard particles is memo-

rized in a shift of 1/r

coeﬃcient of pion ﬁelds in future or past null inﬁnity. Therefore,

the triangular relation in ﬁgure 1 is established for a theory with pions.

The remainder of this paper is organized as follows: in section 2, we specify a model

that we consider in this paper, and the soft pion theorem is presented. Then, the soft

theorem is rewritten in a form of the Ward-Takahashi identity, and we ﬁnd an inﬁnite

number of charges generating asymptotic transformations. In section 3, we investigate

the asymptotic behaviors of ﬁelds near null and timelike inﬁnities, and see the asymptotic

– 2 –

JHEP11(2017)203

transformations. In section 4, we argue that the charge conservation is interpreted as a

pion memory eﬀect. We conﬁrm that the memory eﬀect is consistent with the classical

asymptotic dynamics. Section 5 is devoted to the summary and discussion. We summarize

our conventions in appendix A and coordinate systems in appendix B.

2 Soft theorem in U(1)

model

2.1 Model

We consider a system of a complex scalar Φ and a Dirac spinor Ψ interacting as follows:

L= −

Ψ/∂Ψ−

√



1+γ

Ψ+Φ

∗

1−γ



−|∂

Φ|

−

|Φ|

−



√



, (2.1)

where y is the real Yukawa coupling, and λ and v are also real couplings. This Lagrangian

possesses the chiral symmetry,

Φ → e

iθ

Φ, Ψ → e

−iθγ

Ψ. (2.2)

The scalar potential in (2.1) leads to the spontaneous breaking of this U(1) symmetry. We

choose the vacuum conﬁguration as Φ

=v/

√

2, and expand the ﬁelds around the vacuum as

Φ(x) =

√

(v + φ(x)) e

iπ(x)/v

, Ψ(x) = e

−iπ(x)γ

/(2v)

ψ(x), (2.3)

where φ(x) and π(x) are real scalar ﬁelds, and ψ(x) is the redeﬁned Dirac ﬁeld. The

Lagrangian is then given by

L = −

ψ/∂ψ − y(v + φ)

ψψ −



(∂

φ)

+ λ



−



1 +



(∂

π)

(∂

π)

ψγ

ψ −

−

= −

ψ (/∂ + m) ψ − yφ

ψψ −



(∂

φ)

+ m



−



1 +



(∂

π)

(∂

π)

ψγ

ψ −

−

. (2.4)

Here m = yv and m

= λv are mass of ψ and φ respectively. π is the NG boson associated

with the chiral symmetry (2.2), and we call it pion.

2.2 Soft pion theorem

We now investigate the soft theorem for the NG boson π(x). We will see that the soft

limit of our NG boson does not lead to the divergence unlike the leading soft theorems for

photons [15, 16], gravitons [17] and massless scalars [45], but it has a universal behavior at

the subleading order O(1).

See appendix A for conventions in our paper.

In the absence of Ψ, O(1) contributions also vanish due to Adler’s zero [46, 47].

– 3 –

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