Diamond of infrared equivalences in abelian gauge theories
Temple He ,
1
Prahar Mitra ,
2
and Kathryn M. Zurek
1
1
Walter Burke Institute for Theoretical Physics
California Institute of Technology
,
Pasadena, California 91125 USA
2
Institute for Theoretical Physics,
University of Amsterdam
Science Park 904,
Postbus 94485, 1090 GL Amsterdam, The Netherlands
(Received 6 August 2024; accepted 4 November 2024; published 25 November 2024)
We demonstrate a tree-level equivalence between four distinct infrared objects in (
d
þ
2
)-dimensional
Abelian gauge theories. These are (i) the large gauge charge
Q
ε
where the function
ε
on the sphere
parametrizing large gauge transformations is identified with the Goldstone mode
θ
of spontaneously
broken large gauge symmetry; (ii) the soft effective action that captures the dynamics of the soft and
Goldstone modes; (iii) the edge mode action with Neumann boundary conditions; and (iv) the Wilson line
dressing of a scattering amplitude, including a novel dressing for soft photons, which have local charge
distributions despite having vanishing global charge. The promotion of the large gauge parameter to the
dynamical Goldstone and the novel dressing of soft gauge particles give rise to intriguing possibilities for
the future study of infrared dynamics of gauge theories and gravity.
DOI:
10.1103/PhysRevD.110.105018
I. INTRODUCTION
Although soft theorems were discovered more than half a
century ago
[1
–
3]
, their full potential was not appreciated
until relatively recently, when they were shown to be Ward
identities associated with asymptotic symmetries
[4
–
14]
.
Advancement in our study of asymptotic symmetries in
asymptotically flat spacetimes has led to a deeper under-
standing of the infrared (IR) sector of quantum field
theories (QFTs), celestial holography, and new memory
effects (see
[15
–
19]
for a review and a complete list of
references).
It has long been known that in theories with massless
particles, scattering amplitudes admit a soft-hard factori-
zation
[20,21]
. More precisely, a scattering amplitude with
m
soft (low energy) particles and
n
hard particles (high
energy) factorizes as
[22]
A
m
þ
n
¼
e
−
Γ
S
m
̃
A
n
;
ð
1
Þ
where
S
m
is the contribution of real soft particles,
Γ
the
contribution of virtual soft particles, and
̃
A
n
the amplitude
involving only hard states (real and virtual). IR divergences
in four-dimensional theories with massless particles are
captured by the factor
e
−
Γ
, which is formally zero. The
precise form of the soft factorization Eq.
(1)
is typically
determined by explicitly evaluating the associated
Feynman diagrams, which involves a rather convoluted
calculation. However, given the infinite-dimensional
asymptotic symmetries that constrain the IR sector of a
QFT, one can hope that the soft factor
e
−
Γ
S
m
can be
obtained more directly from symmetry principles. In
[23]
,it
was shown that this is the case for Abelian gauge theories.
In these theories, the asymptotic symmetries are gauge
transformations that act nontrivially at infinity, which we
refer to as
large gauge transformations
(LGTs). Unlike
gauge transformations that act trivially at infinity (which
are redundancies of the theory), LGTs are
“
real,
”
and
physical states are generally not annihilated by the corre-
sponding Noether charge
Q
ε
, where
ε
is the gauge
parameter. Indeed, such a nontrivial charge implies that
gauge theories have an infinite degeneracy of vacua, all of
which are related by
Q
ε
. Moreover, the charge is conserved,
and its Ward identity is precisely Weinberg
’
s leading soft
photon theorem
[7,10,11,24,25]
. By exploiting these infin-
ite-dimensional large gauge symmetries, the authors of
[23]
showed that the soft factor
e
−
Γ
S
m
in (
d
þ
2
)-dimensional
Abelian gauge theories is exactly reproduced by a
d
-dimensional action (which we refer to as the
soft effective
action
) that describes the effective dynamics of the soft
modes and their interaction with the hard modes.
In this paper, we uncover a tree-level equivalence among
four distinct objects that appear in the study of IR physics in
Abelian gauge theories. These are depicted as the four
corners of what we refer to as the
infrared diamond
in
Fig.
1
. The top corner is the aforementioned soft effective
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
’
s title, journal citation,
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3
.
PHYSICAL REVIEW D
110,
105018 (2024)
2470-0010
=
2024
=
110(10)
=
105018(8)
105018-1
Published by the American Physical Society
action
[26]
. The left corner is the large gauge charge with
the gauge parameter
ε
elevated to the Goldstone operator
θ
.
The right corner is the edge mode action
[27]
, which is the
soft limit of the gauge theory action (with appropriate
boundary terms) evaluated on-shell. Finally, the bottom
corner is a soft Wilson line dressing. This includes a
dressing factor for both the hard charged particles, pre-
viously discussed in
[21,36
–
39]
, and (crucially) for soft
photons, which carry local, but not global, electric charge.
The rest of this paper is organized as follows. In Sec.
II
,
we review each of the corners of the diamond. In Sec.
III
,
we prove the equivalence of the soft effective action to the
other three corners of the diamond. Finally, in Sec.
IV
,we
comment on the implications of these results and future
directions.
II. PRELIMINARIES
A. Coordinate conventions
In this paper, we study
U
ð
1
Þ
gauge theories in
D
¼
d
þ
2
dimensional Minkowski spacetime
M
. We work in
flat null coordinates
X
μ
¼ð
u; x
a
;r
Þ
, related to Cartesian
coordinates
X
A
via
X
A
ð
u; x; r
Þ¼
r
ˆ
q
A
ð
x
Þþ
un
A
, where
ˆ
q
A
ð
x
Þ¼
1
þ
x
2
2
;x
a
;
1
−
x
2
2
n
A
¼
1
2
;
0
a
;
−
1
2
:
ð
2
Þ
In flat null coordinates, the metric of Minkowski spacetime
takes the form
d
s
2
¼
−
d
u
d
r
þ
r
2
δ
ab
d
x
a
d
x
b
:
ð
3
Þ
The asymptotic null boundaries
I
of Minkowski space-
time are located at
r
→
∞
. This is coordinatized by
ð
u; x
a
Þ
and has the topology of
R
×
S
d
. The past (future)
boundary of
I
þ
(
I
−
) is located at
u
→
−∞
(
u
→
þ
∞
)
and is denoted by
I
þ
−
(
I
−
þ
). The future and past timelike
boundaries of
M
are denoted by
i
. The asymptotic
past and future Cauchy slices of the spacetime are
Σ
¼
I
∪
i
, and their corresponding boundaries are
∂
Σ
¼
I
∓
. In these coordinates, the relevant future-point-
ing area elements are given by
d
Σ
μ
j
I
¼
−
δ
μ
u
j
r
j
d
d
u
d
d
x
d
S
μν
j
I
∓
¼
4
δ
½
μ
u
δ
ν
r
j
r
j
d
d
d
x:
ð
4
Þ
B. Scattering amplitudes
We denote by
O
k
the operator that inserts the
k
th external
state in a scattering amplitude. For conciseness, we sup-
press all of its arguments, which include the momentum
p
A
k
(satisfying
p
2
k
¼
−
m
2
k
), electric charge
Q
k
∈
Z
, Lorentz
spin indices, flavor indices, etc. An
n
-point scattering
amplitude evaluated in the standard Lorentz-invariant
QFT vacuum can then be written as
[40]
A
n
¼h
O
1
O
n
i
:
ð
5
Þ
We define the
soft photon operator
as
FIG. 1. The IR diamond above captures the tree-level equivalence among four distinct IR objects. These objects are, beginning on
the left corner of the diamond and moving clockwise, (i) the large gauge charge
Q
ε
with
ε
¼
θ
[Eq.
(10)
] so that it is parametrized by the
dynamical Goldstone field
θ
; (ii) the soft effective action [Eq.
(15)
] which captures the dynamics of the soft photon
N
a
and the
Goldstone mode
C
a
¼
∂
a
θ
; (iii) the (soft limit of the) edge mode action [Eq.
(27)
] previously analyzed in
[32]
; and (iv) a product of tree-
level Wilson line dressing of matter
W
tree
M
and a novel soft photon Wilson line dressing
W
tree
S
[Eq.
(39)
or Eq.
(45)
]. The label on each
edge denotes the section where we prove the equivalence between the two endpoints.
HE, MITRA, and ZUREK
PHYS. REV. D
110,
105018 (2024)
105018-2
N
a
ð
x
Þ
≡
1
2
e
ð
lim
ω
→
0
þ
þ
lim
ω
→
0
−
Þ½
ω
O
a
ð
ω
ˆ
q
ð
x
ÞÞ
;
ð
6
Þ
where
O
a
ð
ω
ˆ
q
ð
x
ÞÞ
inserts a photon with momentum
ω
ˆ
q
A
ð
x
Þ
and polarization
ε
A
a
ð
x
Þ¼
∂
a
ˆ
q
A
ð
x
Þ
, and
e
is the gauge
coupling constant.
C. Soft factorization
Insertions of the soft photon operator Eq.
(6)
are
universal and are described by Weinberg
’
s leading soft
photon theorem
[2]
h
N
a
ð
x
Þ
O
1
O
n
i¼
J
a
ð
x
Þh
O
1
O
n
i
;
ð
7
Þ
where
J
a
ð
x
Þ
≡
∂
a
X
n
k
¼
1
Q
k
ln
j
p
k
·
ˆ
q
ð
x
Þj
:
ð
8
Þ
Here, we are working in the standard convention where
outgoing particles have charge
Q
k
while incoming particles
have charge
−
Q
k
.
D. Large gauge symmetries
Abelian gauge theories are invariant under gauge trans-
formations that act on the gauge field
A
via
[41]
A
μ
ð
X
Þ
→
A
μ
ð
X
Þþ
∇
μ
ε
ð
X
Þ
;
ε
ð
X
Þ
∼
ε
ð
X
Þþ
2
π
:
ð
9
Þ
When
ε
j
∂
Σ
¼
0
, Eq.
(9)
has no physical effect and is the
usual small gauge redundancy. When
ε
j
∂
Σ
≠
0
, the trans-
formations Eq.
(9)
are the physical large gauge trans-
formations. The Noether charge that generates these
transformations is
[7]
Q
ε
¼
∓
1
e
2
I
I
∓
ε
⋆
F
¼
∓
2
e
2
I
I
∓
d
d
x
j
r
j
d
ε
F
ur
;
ð
10
Þ
where
F
μν
¼
∂
μ
A
ν
−
∂
ν
A
μ
. The Ward identity associated
with these large gauge symmetries is the leading soft
photon theorem Eq.
(7) [7,10,11,24,25]
. To show this,
we use Maxwell
’
s equation, namely
d
⋆
F
¼ð
−
1
Þ
d
e
2
⋆
J
⇔
∇
μ
F
μν
¼
e
2
J
ν
;
ð
11
Þ
to rewrite the charge Eq.
(10)
as
Q
ε
¼
1
e
2
Z
Σ
d
ð
ε
⋆
F
Þ
¼
1
e
2
Z
Σ
d
ε
∧
⋆
F
þð
−
1
Þ
d
Z
Σ
ε
⋆
J:
ð
12
Þ
We denote the first term by
Q
S
ε
and the last term by
Q
H
ε
,
so that
Q
S
ε
¼
1
e
2
Z
Σ
d
ε
∧
⋆
F
¼
1
e
2
Z
Σ
d
Σ
μ
∇
ν
ε
F
μν
Q
H
ε
¼ð
−
1
Þ
d
Z
Σ
ε
⋆
J
¼
−
Z
Σ
d
Σ
μ
ε
J
μ
:
ð
13
Þ
The Ward identity
Q
þ
ε
¼
Q
−
ε
then implies
hð
Q
þ
S
ε
−
Q
−
S
ε
Þ
O
1
O
n
i¼
−
hð
Q
þ
H
ε
−
Q
−
H
ε
Þ
O
1
O
n
i
:
ð
14
Þ
It was shown in
[7,25,42]
that the soft charge
Q
S
ε
is the
soft photon operator Eq.
(6)
smeared over
S
d
. With a clever
choice of the gauge parameter
ε
(see
[25]
for details), the
left-hand-side of Eq.
(14)
can be made identical to that of
Eq.
(7)
. On the other hand, the hard charges are quadratic in
the charged matter fields, so they act on the hard states and
transform them. It was shown in
[7,10,11,24,25]
that this
precisely reproduces the right-hand-side of Eq.
(7)
.
E. Soft effective action
In
[23]
, Kapec and Mitra identified the relevant soft
degrees of freedom in Abelian gauge theories and, using the
large gauge symmetry of the system, constructed an
effective action for these modes. At tree-level (see footnote
[22]
), this so-called
soft effective action
is given by
S
tree
eff
¼
−
i
2
c
1
;
1
I
S
d
d
d
x
̃
C
a
ð
x
Þ½
N
a
ð
x
Þ
−
J
a
ð
x
Þ
;
ð
15
Þ
where
N
a
ð
x
Þ
and
J
a
ð
x
Þ
are respectively defined in Eq.
(6)
and Eq.
(8)
, and
C
a
ð
x
Þ
is the boundary value of the gauge
field, i.e.,
C
a
ð
x
Þ
≡
A
a
j
I
þ
−
ð
x
Þ¼
A
a
j
I
−
þ
ð
x
Þ
[43]
. In the
absence of magnetic charges, we have
C
a
ð
x
Þ¼
∂
a
θ
ð
x
Þ
;
θ
ð
x
Þ
∼
θ
ð
x
Þþ
2
π
:
ð
16
Þ
Now, in the perturbative Lorentz invariant vacuum, we have
h
θ
ð
x
Þi ¼
0
[7,14]
. However, under an LGT given in
Eq.
(9)
,
θ
ð
x
Þ
→
θ
ð
x
Þþ
ε
ð
x
Þ
for
ε
ð
x
Þ
≡
ε
j
∂
Σ
ð
x
Þ
, implying
that the Lorentz invariant vacuum state spontaneously
breaks the large gauge symmetry
and
that
θ
ð
x
Þ
is the
corresponding Goldstone mode.
Finally, the tilde on
C
a
denotes the shadow transform.
For a vector field of dimension
Δ
¼
1
, this is defined to be
̃
C
a
ð
x
Þ
≡
Z
d
d
y
δ
ab
−
2
ð
x
−
y
Þ
a
ð
x
−
y
Þ
b
ð
x
−
y
Þ
2
½ð
x
−
y
Þ
2
d
−
1
C
b
ð
y
Þ
:
ð
17
Þ
Up to normalization, the shadow transform is its own
inverse:
̃
̃
V
a
ð
x
Þ¼
c
1
;
1
V
a
ð
x
Þ
:
ð
18
Þ
DIAMOND OF INFRARED EQUIVALENCES IN ABELIAN GAUGE
...
PHYS. REV. D
110,
105018 (2024)
105018-3
The constant
c
1
;
1
appears in the action Eq.
(15)
, and it was
shown in
[23]
that the action Eq.
(15)
correctly reproduces
the soft theorem Eq.
(7)
.
III. EQUIVALENCE RELATIONS
In the following three subsections, we will demonstrate
the equivalences of the IR diamond shown in Fig.
1
.
A. Large gauge
charge
=
soft
effective action
We first prove the upper left edge in Fig.
1
by
demonstrating
Q
þ
θ
−
Q
−
θ
¼
iS
tree
eff
:
ð
19
Þ
To this end, we begin by using Eq.
(10)
to write
Q
þ
θ
−
Q
−
θ
¼
−
2
e
2
I
S
d
d
d
x
θ
½ðj
r
j
d
F
ur
Þj
I
þ
−
þðj
r
j
d
F
ur
Þj
I
−
þ
:
ð
20
Þ
Following
[25]
, we decompose the field strength into a
radiative part (that solves free Maxwell
’
s equations) and a
Coulombic part (that, via a Green
’
s function, describes the
interaction of the gauge field with charged matter fields), so
that
ðj
r
j
d
F
ur
Þj
I
∓
¼ðj
r
j
d
F
R
ur
Þj
I
∓
þðj
r
j
d
F
C
ur
Þj
I
∓
:
ð
21
Þ
The
superscript distinguishes the Green
’
s function that is
used to evaluate the Coulombic part of the field:
þ
for the
advanced Green
’
s function and
−
for the retarded one.
It was shown in
[42]
that the radiative part of the field
strength can be written in terms of the soft photon operator
Eq.
(6)
as
[44]
ðj
r
j
d
F
R
þ
ur
Þj
I
þ
−
þðj
r
j
d
F
ð
R
−
Þ
ur
Þj
I
−
þ
¼
e
2
4
c
1
;
1
∂
a
̃
N
a
:
ð
22
Þ
Additionally,
[25]
showed that the Coulombic part of the
field strength satisfies the equation
ðj
r
j
d
F
C
þ
ur
Þj
I
þ
−
þðj
r
j
d
F
C
−
ur
Þj
I
−
þ
¼
−
e
2
2
Z
R
d
u
½ðj
r
j
d
J
u
Þj
I
þ
−
ðj
r
j
d
J
u
Þj
I
−
þ ½ðj
r
j
d
F
C
þ
ur
Þj
I
þ
þ
þðj
r
j
d
F
C
−
ur
Þj
I
−
−
:
ð
23
Þ
Utilizing an identify proven in
[32]
, the above expression is
related to the soft factor given in Eq.
(8)
via
[45]
ðj
r
j
d
F
C
þ
ur
Þj
I
þ
−
þðj
r
j
d
F
C
−
ur
Þj
I
−
þ
¼
−
e
2
4
c
1
;
1
∂
a
̃
J
a
ð
x
Þ
:
ð
24
Þ
Substituting Eqs.
(22)
and
(24)
into Eq.
(20)
, we obtain
Q
þ
θ
−
Q
−
θ
¼
−
1
2
c
1
;
1
I
S
d
d
d
x
θ
ð
x
Þ
∂
a
½
̃
N
a
ð
x
Þ
−
̃
J
a
ð
x
Þ
¼
1
2
c
1
;
1
I
S
d
d
d
x
̃
C
a
ð
x
Þ½
N
a
ð
x
Þ
−
J
a
ð
x
Þ
;
ð
25
Þ
where in the second equality, we first integrated by parts
and then used the shadow transform property
I
S
d
d
d
xC
a
ð
x
Þ
̃
C
0
a
ð
x
Þ¼
I
S
d
d
d
x
̃
C
a
ð
x
Þ
C
0
a
ð
x
Þ
:
ð
26
Þ
Recalling Eq.
(15)
, we see from Eq.
(25)
that indeed
Eq.
(19)
holds.
B. Soft effective
action
=
edge
mode action
Next, we consider the upper right edge of the diamond in
Fig.
1
. Donnelly and Wall showed in
[28]
that in a gauge
theory, the entanglement entropy receives an extra con-
tribution from the edge modes that live on the codimension-
two entangling surface. The dynamics of these edge modes
is described by a so-called edge mode action, which is the
Maxwell action with appropriate boundary terms evaluated
on-shell. An analogous edge mode action was considered
in
[32]
, albeit with different boundary conditions compared
to
[28]
, and is given by
S
edge
¼
−
Z
M
d
d
þ
2
X
ffiffiffiffiffiffi
−
g
p
1
4
e
2
F
μν
F
μν
þ
A
μ
J
μ
þ
1
e
2
Z
Σ
þ
−
Σ
−
d
Σ
μ
A
ν
F
μν
:
ð
27
Þ
The boundary terms are required to ensure that the
variational principle with radiative boundary conditions
(
n
μ
δ
F
μν
j
Σ
¼
0
) is well-defined, and
J
μ
is the conserved
background matter charge current determined by integrat-
ing out the matter fields in the theory. It was shown in
[32]
that the on-shell and soft limit of the above action precisely
reproduces the soft effective action, which we shall now
review.
The edge modes whose action we are trying to construct
are the soft photon mode Eq.
(6)
and the Goldstone mode
Eq.
(16)
. To ensure they appear in the action, it is necessary
to decompose the gauge field as
A
μ
ð
X
Þ¼
ˆ
A
μ
ð
X
Þþ
∇
μ
θ
ð
X
Þ
;
ð
28
Þ
where we have separated the pure gauge mode
θ
ð
X
Þ
of the
gauge field so that
ˆ
A
μ
ð
X
Þ
is large gauge invariant
[47]
.
Substituting this decomposition into Eq.
(27)
and by taking
the on-shell and soft limit, it was determined in
[32]
that at
tree-level,
S
edge
⟶
on-shell
þ
soft
iS
tree
eff
:
ð
29
Þ
HE, MITRA, and ZUREK
PHYS. REV. D
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105018-4
Indeed, it was shown in
[32]
that this equivalence extends
even to loop-level, but for our current purposes, Eq.
(29)
is
the desired equivalence corresponding to the upper right
edge in Fig.
1
.
C. Soft effective
action
=
Wilson
line dressing
The final equivalence we prove is the relationship
between the soft effective action and a Wilson line dressing
at tree-level, namely
exp
½
−
S
tree
eff
¼
W
tree
S
W
tree
M
;
ð
30
Þ
where
W
S
is a dressing we introduce below for soft
photons,
W
M
the usual Wilson line dressing for charged
matter particles, and the
“
tree
”
superscript indicates we are
only evaluating their tree-level and soft contributions.
There are two ways to interpret the Wilson line dressing
above. The first interpretation is the conventional one,
where the Wilson lines extend from a bulk point (say, the
origin) to the celestial sphere on
I
. The second one is to
view the Wilson lines as living wholly on the celestial
sphere. We will see the equivalence Eq.
(30)
holds for both
interpretations.
We begin with the first interpretation. In
[21]
,itwas
shown that the matter operator insertions
O
k
appearing in
the scattering amplitude are dressed with Wilson lines as
O
k
¼
U
k
ˆ
O
k
, with
ˆ
O
k
being the
“
bare
”
(undressed) oper-
ators, and
U
k
¼
exp
iQ
k
Z
γ
k
d
X
μ
A
μ
ð
X
Þ
;
ð
31
Þ
where
γ
k
is the worldline of the
k
th particle beginning from
the origin
[48]
. The full Wilson line dressing of the
scattering amplitude is given by
U
1
U
n
. The quantity
that appears in Eq.
(30)
is the soft contribution to this
Wilson line, namely
W
M
¼
U
1
U
n
j
soft
:
ð
32
Þ
To evaluate this, we rewrite the line integral in Eq.
(31)
in a
more suggestive way. We parametrize the worldline
γ
k
as
X
A
k
ð
X
0
Þ¼ð
X
0
;
⃗
X
k
ð
X
0
ÞÞ
and write
U
k
¼
exp
i
η
k
Q
k
Z
∞
0
d
X
0
d
X
μ
k
d
X
0
A
μ
ð
X
k
ð
η
k
X
0
ÞÞ
¼
exp
−
i
Z
M
d
d
þ
2
X
ffiffiffiffiffiffi
−
g
p
A
μ
ð
X
Þ
J
μ
k
ð
X
Þ
;
ð
33
Þ
where
η
k
¼þ
1
(
−
1
)ifthe
k
th particle is outgoing
(incoming), and
J
μ
k
ð
X
Þ¼
−
η
k
Q
k
θ
ð
η
k
X
0
Þ
d
X
μ
k
d
X
0
δ
ð
d
þ
1
Þ
ð
⃗
X
−
⃗
X
k
ð
X
0
ÞÞ ð
34
Þ
is the current corresponding to a charged point particle with
θ
ð
X
0
Þ
being the Heaviside step function. Including the
contribution from all the particles, we find
W
M
¼
exp
−
i
Z
M
d
d
þ
2
X
ffiffiffiffiffiffi
−
g
p
A
μ
J
μ
soft
;
ð
35
Þ
where
J
μ
¼
P
n
k
¼
1
J
μ
k
is the
total
conserved charged matter
current. To further simplify this, we use the decomposition
Eq.
(28)
.Asin
[32]
, the
ˆ
A
μ
J
μ
term in the soft limit only
contributes to the quantum piece of the soft effective action.
Hence, only the
∇
μ
θ
J
μ
term contributes at tree-level, and
we have
W
tree
M
¼
exp
−
i
Z
M
d
d
þ
2
X
ffiffiffiffiffiffi
−
g
p
∇
μ
θ
J
μ
¼
exp
−
i
Z
Σ
þ
−
Σ
−
d
Σ
μ
θ
J
μ
;
ð
36
Þ
where we used integration by parts and the current
conservation equation
∇
μ
J
μ
¼
0
, as well as the fact the
matter current vanishes at spatial infinity
i
0
.
We next turn to the soft photon Wilson line dressing in
Eq.
(30)
. At first glance, it seems that soft photons should
not be dressed with any Wilson lines as they are uncharged,
i.e.,
Q
photon
¼
0
. However, as noted in
[7]
, while soft
photons are globally neutral, they have a local charge
distribution, which plays a crucial role in enforcing the
local charge conservation law implied by the large gauge
symmetry. It follows that soft photons
must
be dressed
not with a single Wilson line but with a
“
Wilson line
distribution.
”
Indeed, analogous to Eq.
(34)
, we define a conserved
“
soft
”
current
J
S
μ
¼
−
1
e
2
ð
⋆
d
⋆
F
Þ
μ
¼
1
e
2
∇
ν
F
μν
:
ð
37
Þ
We note the current is trivially conserved since
F
μν
is
antisymmetric. The soft limit of the soft photon dressing is
then just Eq.
(36)
with
J
μ
replaced with
J
S
μ
, so that
W
tree
S
¼
exp
−
i
Z
M
d
d
þ
2
X
ffiffiffiffiffiffi
−
g
p
∇
μ
θ
J
S
μ
¼
exp
−
i
e
2
Z
Σ
þ
−
Σ
−
d
Σ
ν
∇
μ
θ
F
μν
;
ð
38
Þ
where we integrated by parts and used the fact that
F
vanishes on
i
0
to obtain the second line. Combining with
the matter dressing Eq.
(36)
and using Maxwell
’
s equation
Eq.
(11)
only on
Σ
and not on
i
0
(where the matter current
is already assumed to vanish) to ensure the hard modes
associated with the matter current are on-shell but the soft
modes are off-shell
[49]
, we obtain
DIAMOND OF INFRARED EQUIVALENCES IN ABELIAN GAUGE
...
PHYS. REV. D
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105018-5
W
tree
M
W
tree
S
¼
exp
−
i
2
e
2
I
I
þ
−
þ
I
−
þ
d
S
μν
θ
F
μν
¼
exp
−
2
i
e
2
I
I
þ
−
þ
I
−
þ
d
d
x
j
r
j
d
θ
F
ur
:
ð
39
Þ
Comparing this with Eq.
(10)
, we find
W
tree
S
W
tree
M
¼
exp
½
i
ð
Q
þ
θ
−
Q
−
θ
Þ
:
ð
40
Þ
Finally, using Eq.
(19)
, we immediately prove Eq.
(30)
.
We now present the second interpretation of the Wilson
line dressing given in Eq.
(30)
, which relies on only
considering Wilson lines given in Eq.
(31)
to be localized
on the celestial sphere. Letting
̄
γ
k
be such Wilson lines, the
analog of Eq.
(31)
is
U
k
¼
exp
iQ
k
Z
̄
γ
k
d
x
a
A
a
ð
x
Þ
:
ð
41
Þ
Utilizing the decomposition Eq.
(28)
again, the
ˆ
A
a
term in
the soft limit does not contribute at tree-level, so it suffices
to replace
A
a
with
∂
a
θ
, yielding
U
k
j
soft
¼
exp
iQ
k
Z
̄
γ
k
d
x
a
∂
a
θ
ð
x
Þ
¼
exp
½
iQ
k
ð
θ
ð
x
k
Þ
−
θ
ð
x
0
ÞÞ
;
ð
42
Þ
where
x
0
and
x
k
are the endpoints of the Wilson line
̄
γ
k
.If
all the particles in the scattering amplitude are massless,
then the corresponding Wilson line dressing is simply
W
tree
M
¼
U
1
U
n
j
soft
¼
exp
i
X
n
k
¼
1
Q
k
θ
ð
x
k
Þ
;
ð
43
Þ
where the dependence on the base point
x
0
has dropped out
completely due to charge conservation. This means the
dressing factor
W
tree
M
does not depend on the initial point
from which the Wilson lines originate. Indeed, the celestial
Wilson lines
̄
γ
k
can therefore be related to the bulk Wilson
lines
γ
k
by moving the initial point of
γ
k
(namely, the
origin) to
x
0
on the celestial sphere.
The generalization of Eq.
(43)
to the massive
and
soft
Wilson line dressing is rather straightforward. Unlike
massless particles, which pierce the celestial sphere at a
single point, massive particles and soft photons have a
nonlocal charge distribution. The corresponding distribu-
tional Wilson line is
exp
i
I
S
d
d
d
x
ρ
ð
x
Þ
θ
ð
x
Þ
;
ð
44
Þ
where
ρ
ð
x
Þ
is the charge density on the celestial sphere
associated to massive particles and soft photons. For
massless particles, the charge density is
ρ
massless
k
ð
x
Þ¼
Q
k
δ
ð
d
Þ
ð
x
−
x
k
Þ
, which reproduces Eq.
(43)
. For massive
particles, it is given by
ρ
massive
k
ð
x
Þ¼
Q
k
K
d
ð
m
k
=
ω
k
;x
k
;
x
Þ
,
where
K
Δ
is the bulk-to-boundary propagator in AdS
d
þ
1
[23]
. Further including the charge density for the soft
photons, the total Wilson line distribution becomes
W
tree
S
W
tree
M
¼
exp
i
I
S
d
d
d
x
ρ
total
ð
x
Þ
θ
ð
x
Þ
;
ð
45
Þ
where
ρ
total
ð
x
Þ
is the total charge density. The argument of
the exponential is the integral of the charge density-
weighted by
θ
ð
x
Þ
, which, by our charge convention, is
precisely
i
ð
Q
þ
θ
−
Q
−
θ
Þ
. Hence, we see that Eq.
(40)
holds
once more.
IV. DISCUSSION
The primary objective of this paper is to show that there
are four different objects in the IR sector of Abelian gauge
theories that are identical, given that the large gauge charge
Q
ε
is parametrized by the Goldstone mode so that
ε
¼
θ
,
and that soft photons are dressed. While our analysis is
purely in the context of Abelian gauge theories and at tree-
level, we suspect due to the universality of IR dynamics that
these equivalences extend to non-Abelian gauge theories
and, most importantly, gravity. Furthermore,
[32]
proved
that the equivalence between the edge mode action and soft
effective action survives loop corrections, and it would be
worthwhile to explore if the other sides of the IR diamond
survive loop corrections as well.
In our analysis, there are two important novel features we
emphasize, which will be especially interesting in the
context of gravity. The first is the promotion of the large
gauge charge parameter
ε
to the dynamical Goldstone
mode. As a result, the large gauge charge involves a mixing
between the (shadow transform of the) Goldstone
̃
C
a
ð
x
Þ
and the soft photon
N
a
ð
x
Þ
. The large gauge charge is now a
composite operator, and in general its vacuum expectation
value will be nonvanishing. In gravity, the analogous
charge is the supertranslation charge, and it would be
interesting to compute its expectation value and variance to
verify the results of
[51
–
53]
obtained using other means.
Related ideas were explored in
[54]
.
The second novel feature above is the introduction of a
Wilson line distributional dressing for the soft photon. Such
a dressing is quite natural for dressing massive charged
particles, whose charge distributions are not delta function
localized on the celestial sphere. Therefore, because soft
photons similarly have a nontrivial charge distribution
on the celestial sphere, we expect them to also have a
distributional Wilson line dressing. It is not immediately
obvious how to interpret such a dressing from a conven-
tional field theoretic perspective, and we leave a better
understanding of this for future work.
HE, MITRA, and ZUREK
PHYS. REV. D
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105018-6
Ultimately, we would like to extend our above analysis
to non-Abelian gauge theories and (perturbative) gravity.
Unlike soft photons, soft gluons and (at least at subleading
order) soft gravitons carry nontrivial global charge in
addition to local charge, and it would be interesting to
explore the consequences of this difference. Ironically,
because the leading soft behavior of Abelian gauge
theories and gravity is similar, we expect the case with
gravity to be possibly simpler. For instance, the relation-
ship between the soft effective action and the super-
translation charge is nearly identical to our analysis above
[55]
. On the other hand, for non-Abelian gauge theories,
the soft effective action is not even known, and has only
been conjectured in
[23,57,58]
. We hope to pursue a
satisfying understanding of the four corners of the
diamond for these cases.
ACKNOWLEDGMENTS
We would like to thank Laurent Freidel and Ana-Maria
Raclariu for productive conversations, and Adam Ball
for useful feedback regarding our preprint. We would
especially like to thank Allic Sivaramakrishnan for col-
laborating in the early stages of this work. T. H. and K. Z.
are supported by the Heising-Simons Foundation
“
Observational Signatures of Quantum Gravity
”
collabo-
ration grant 2021-2817, the U.S. Department of Energy,
Office of Science, Office of High Energy Physics, under
Award No. DE-SC0011632, and the Walter Burke Institute
for Theoretical Physics. P. M. is supported by the European
Research Council (ERC) under the European Union
’
s
Horizon 2020 research and innovation program (grant
agreement No. 852386). The work of K. Z. is also sup-
ported by a Simons Investigator award.
[1] F. E. Low,
Phys. Rev.
110
, 974 (1958)
.
[2] S. Weinberg,
Phys. Rev.
140
, B516 (1965)
.
[3] F. A. Berends and W. T. Giele,
Nucl. Phys.
B313
, 595
(1989)
.
[4] A. Strominger,
J. High Energy Phys. 07 (2014) 151.
[5] A. Strominger,
J. High Energy Phys. 07 (2014) 152.
[6] T. He, V. Lysov, P. Mitra, and A. Strominger,
J. High Energy
Phys. 05 (2015) 151.
[7] T. He, P. Mitra, A. P. Porfyriadis, and A. Strominger,
J. High
Energy Phys. 10 (2014) 112.
[8] V. Lysov, S. Pasterski, and A. Strominger,
Phys. Rev. Lett.
113
, 111601 (2014)
.
[9] T. He, P. Mitra, and A. Strominger,
J. High Energy Phys. 10
(2016) 137.
[10] M. Campiglia and A. Laddha,
J. High Energy Phys. 07
(2015) 115.
[11] D. Kapec, M. Pate, and A. Strominger,
Adv. Theor. Math.
Phys.
21
, 1769 (2017)
.
[12] M. Campiglia and A. Laddha,
J. High Energy Phys. 12
(2015) 094.
[13] M. Campiglia and A. Laddha,
J. High Energy Phys. 11
(2016) 012.
[14] T. He and P. Mitra,
J. High Energy Phys. 03 (2021) 015.
[15] A. Strominger,
Lectures on the Infrared Structure of Gravity
and Gauge Theory
(Princeton University Press, Princeton,
2017).
[16] A.-M. Raclariu,
arXiv:2107.02075
.
[17] S. Pasterski,
Eur. Phys. J. C
81
, 1062 (2021)
.
[18] S. Pasterski, M. Pate, and A.-M. Raclariu, in
Snowmass
2021
(2021);
arXiv:2111.11392
.
[19] T. McLoughlin, A. Puhm, and A.-M. Raclariu,
J. Phys. A
55
, 443012 (2022)
.
[20] A. Bassetto, M. Ciafaloni, and G. Marchesini,
Phys. Rep.
100
, 201 (1983)
.
[21] I. Feige and M. D. Schwartz,
Phys. Rev. D
90
, 105020
(2014)
.
[22] Strictly speaking, this factorization has been proven to hold
for Abelian gauge theories without massless charged par-
ticles.
[23] D. Kapec and P. Mitra,
Phys. Rev. D
105
, 026009 (2022)
.
[24] D. Kapec, V. Lysov, and A. Strominger,
Adv. Theor. Math.
Phys.
21
, 1747 (2017)
.
[25] T. He and P. Mitra,
J. High Energy Phys. 10 (2019) 213.
[26] The full soft effective action appearing in
[23]
contains
another term
α
R
d
d
x
ð
2
π
Þ
d
½
N
a
ð
x
Þ
2
, which arises from one-loop
effects. In this paper, we restrict our discussion to tree-level
and consequently ignore this term.
[27] Dynamics of edge modes in gauge theory was first dis-
cussed in
[28
–
31]
. The relationship between the edge mode
action and the soft effective action has been discussed
recently in
[32
–
35]
.
[28] W. Donnelly and A. C. Wall,
Phys. Rev. Lett.
114
, 111603
(2015)
.
[29] W. Donnelly,
Phys. Rev. D
85
, 085004 (2012)
.
[30] W. Donnelly,
Classical Quantum Gravity
31
, 214003
(2014)
.
[31] W. Donnelly and A. C. Wall,
Phys. Rev. D
94
, 104053 (2016)
.
[32] T. He, P. Mitra, A. Sivaramakrishnan, and K. M. Zurek,
Phys. Rev. D
109
, 125016 (2024)
.
[33] H. Z. Chen, R. C. Myers, and A.-M. Raclariu,
Phys. Rev. D
109
, L121702 (2024)
.
[34] H. Z. Chen, R. Myers, and A.-M. Raclariu,
arXiv:
2403.13913
.
[35] A. Ball, A. Law, and G. Wong,
J. High Energy Phys. 09
(2024) 032.
[36] E. Laenen, G. Stavenga, and C. D. White,
J. High Energy
Phys. 03 (2009) 054.
[37] C. D. White,
J. High Energy Phys. 05 (2011) 060.
[38] K. Nguyen, A. Rios Fukelman, and C. D. White,
Proc. Sci.
,
CORFU2022 (2023) 289 [
arXiv:2304.01250
].
[39] D. Bonocore, A. Kulesza, and J. Pirsch,
J. High Energy
Phys. 03 (2022) 147.
DIAMOND OF INFRARED EQUIVALENCES IN ABELIAN GAUGE
...
PHYS. REV. D
110,
105018 (2024)
105018-7
[40] Gauge theories have infinitely many vacuum states, and
it is possible to consider scattering amplitudes with different
in
and
out
vacua. We do not explore this possibility here.
[41] We normalize the gauge field so that the pure Maxwell
action is
S
Maxwell
¼
−
1
2
e
2
R
M
F
∧
⋆
F
.
[42] T. He and P. Mitra,
SciPost Phys.
16
, 142 (2024)
.
[43] The second equality holds due to the matching condition
given in
[7]
.
[44]
N
a
as defined in this paper is
N
þ
a
−
N
−
a
in
[42]
.
[45] The appearance of the shadow transform ensures the scaling
dimensions on both sides of Eq.
(24)
to match while
preserving Lorentz symmetry
[46]
.
[46] D. Kapec and P. Mitra,
J. High Energy Phys. 05 (2018) 186.
[47] This decomposition is uniquely specified by imposing a
gauge-fixing condition on
A
μ
ð
X
Þ
.
[48] In the soft limit, the location of the point from which the
Wilson line begins is inconsequential.
[49] Because the soft effective action to which we are matching
the Wilson line dressing is an off-shell quantity involving the
soft modes, we do not impose equations of motion for the soft
modes on the celestial sphere at
I
∓
, or equivalently
i
0
.
Indeed, if we impose equations of motion everywhere, we get
W
tree
S
W
tree
M
¼
1
, which implies the matching condition
Q
þ
θ
¼
Q
−
θ
[see Eq.
(39)
]. This is precisely the result of
[50]
.We
thank Adam Ball for pointing this out.
[50] M. Campiglia and R. Eyheralde,
J. High Energy Phys. 11
(2017) 168.
[51] E. Verlinde and K. M. Zurek,
J. High Energy Phys. 04
(2020) 209.
[52] T. Banks and K. M. Zurek,
Phys. Rev. D
104
, 126026
(2021)
.
[53] E. Verlinde and K. M. Zurek,
Phys.Rev.D
106
, 106011 (2022)
.
[54] D. Kapec, A.-M. Raclariu, and A. Strominger,
Classical
Quantum Gravity
34
, 165007 (2017)
.
[55] The connection between the supertranslation charge and the
“
modular Hamiltonian
”
studied in
[53]
is much less obvious
and is the focus of
[56]
.
[56] T. He, A.-M. Raclariu, and K. M. Zurek,
arXiv:2408.01485
.
[57] L. Magnea,
J. High Energy Phys. 05 (2021) 282.
[58] N. Agarwal, L. Magnea, C. Signorile-Signorile, and A.
Tripathi,
Phys. Rep.
994
, 1 (2023)
.
HE, MITRA, and ZUREK
PHYS. REV. D
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