Ward identity for loop level soft photon theorem for massless QED coupled to gravity
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Springer
Received: January 1, Revised: July 21, Accepted: September 18, Published: October 16,
2020 2020 2020 2020
Sayali Atul Bhatkar Indian Institute of Science Education and Research, Homi Bhabha Rd, Pashan, Pune 411 008, India
E-mail: [email protected] Abstract: Motivated by Campiglia and Laddha [1], we show that the Sahoo-Sen soft photon theorem [2] for loop amplitudes is equivalent to an asymptotic conservation law. This asymptotic charge is directly related to the dressing of fields due to long range forces exclusively present in four spacetime dimensions. In presence of gravity, the new feature is that photons also acquire a dressing due to long range gravitational force and this dressing contributes to the asymptotic charge. Keywords: Gauge Symmetry, Global Symmetries, Scattering Amplitudes ArXiv ePrint: 1912.10229
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)110
JHEP10(2020)110
Ward identity for loop level soft photon theorem for massless QED coupled to gravity
Contents 1 Introduction and result
1
2 Preliminaries 2.1 Asymptotic conservation laws 2.2 Outline of the paper
4 6 7
3 Dressing of massless scalar field
8 10
5 The asymptotic charge
12
6 Expressions for h1rr and A1r 6.1 Classical part 6.2 Quantum part
14 14 15
7 The Ward identity 7.1 The Sahoo-Sen soft theorem
17 19
A Calculating the 1/u mode in A0A
20
B Quantum modes in h1rr and A1r
23
C Maxwell’s equations in presence of gravity
25
D Herdegen like representation for graviton
26
1
Introduction and result
Asymptotic symmetries strongly constrain low energy physics of gauge theories [3–8]. Leading soft theorems are manifestations of asymptotic symmetries. Soft theorems are statements about universal properties of amplitudes in the limit when energy of some of the interacting massless particles is taken to be small [9–12].1 The equivalence between the two was first demonstrated in the seminal paper [5]. Similar analysis was carried out for QED in [6–8]; it was shown that leading soft photon theorem is equivalent to the Ward identity of the so called large gauge transformations. Large gauge transformations constitute an infinite dimensional subgroup of U(1) gauge transformations. Analogous investigations have been carried out to understand the possible symmetry origins of tree level subleading soft theorem. Ward identity corresponding to Low’s subleading photon theorem has been studied in [14–16]. The symmetry underlying this Ward identity or its relation to U(1) gauge group is not clear. In [17], the authors proved an infinite hierarchy of asymptotic conservation laws for classical electromagnetism and showed 1
Interested readers can look up the references of [2] and [13] for recent literature on Soft theorems.
–1–
JHEP10(2020)110
4 Dressing of gauge field
S0 Mn (pi ) + Slog log ωMn (pi ) + . . . , ω is the leading soft factor and
Mn+1 (pi , k) = here, S0 =
P
.pi i ei pi .q
Slog =
µ qρ ρ µ ig X .pi X ig X ei (pj pi − pρi pµj ) − ei q.pj 4π pi .q 4π pi .q i
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