aainstitutetext: Centre for Theoretical Physics, Department of Physics and
Astronomy,
Queen Mary University of London, E1 4NS, United Kingdombbinstitutetext:
Department of Physics and Astronomy, Uppsala University,
Box 516, 75120 Uppsala, Sweden

# On the anomaly interpretation of amplitudes in self-dual Yang-Mills and
gravity

George Doran a Ricardo Monteiro b Sam Wikeley<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the integrability anomalies arising in the self-dual sectors of
gravity and Yang-Mills theory, focusing on their connection to both the chiral
anomaly and the trace anomaly. The anomalies in the self-dual sectors generate
the one-loop all-plus amplitudes of gravitons and gluons, and have recently
been studied via twistor constructions. On the one hand, they can be
interpreted as an anomaly of the chiral U(1) electric-magnetic-type duality in
the self-dual sectors. On the other hand, due to the self-duality condition,
they also resemble the trace anomaly, sharing the same type of non-local
effective action. We highlight the role of a Weyl-covariant fourth-order
differential operator familiar from the trace anomaly literature, which (i)
explains the conformal properties of the one-loop amplitudes, and (ii)
indicates how this story may be extended to non-trivial spacetime backgrounds,
e.g. with a cosmological constant. Moving beyond the self-dual sectors, and
focusing on the gravity case, we comment on an intriguing connection to the
two-loop ultraviolet divergence of pure gravity, whereby cancelling the
anomaly at one-loop eliminates the two-loop divergence for the simplest
helicity amplitudes.

QMUL-PH-23-26

UUITP–38/23

##  1 Introduction

The main motivation for this paper is to explore the similarities between the
integrability anomalies of the self-dual sectors of Yang-Mills and gravity,
and the more familiar stories of the chiral and trace anomalies, particularly
regarding non-local quantum corrections to the action.

The self-dual sectors are well known to have simple scattering amplitudes,
non-vanishing only at one-loop for all-plus external helicities
$(++\cdots+)$.111In our conventions, particles in a scattering amplitude are
incoming (so that momentum conservation reads $\sum_{i}p_{i}=0$), and a
positive-helicity polarisation is associated to a self-dual field. We note
that the 3-point tree amplitudes with helicities $(++-)$ also arise, and play
an important role if one relaxes the restriction of real kinematics in
Lorentzian signature. Behind the vanishing of the tree-level amplitudes, lies
the integrability of the classical self-dual theories. The one-loop all-plus
amplitudes are also special: they are rational functions with simple soft and
collinear behaviour, in both Yang-Mills and gravity. These properties provided
the guidance for constructing explicit $n$-point formulas for the amplitudes
Bern:1993qk ; Mahlon:1993si ; Bern:1996ja ; Bern:1998xc ; Bern:1998sv . The
formulas are strongly suggestive of the existence of a simple one-loop quantum
correction to the action, incorporating the loop-level effects. The idea that
the one-loop amplitudes should be interpreted as signalling an anomaly of the
classical integrability goes back to Bardeen:1995gk ; see also Chalmers:1996rq
; Cangemi:1996rx . Indeed, the amplitudes arise from an $\epsilon/\epsilon$
contribution in dimensional regularisation. This idea has been the subject of
several investigations over the years, e.g. Brandhuber:2006bf ;
Brandhuber:2007vm ; Boels:2008ef ; Krasnov:2016emc ; Nandan:2018ody ;
Chattopadhyay:2020oxe ; Chattopadhyay:2021udc . It has recently seen a
beautiful realisation in the twistor-space constructions of Costello:2021bah ;
Bittleston:2022nfr ; see also Costello:2022wso ; Costello:2022upu ; Bu:2022dis
; Bittleston:2022jeq ; Bu:2023vjt . The goal of Costello:2021bah ;
Bittleston:2022nfr  was to obtain quantum-integrable ‘improved models’ of
self-dual Yang-Mills and self-dual gravity that uplift naturally to twistor
space. In these models, the loop amplitudes vanish via an anomaly cancellation
mechanism, analogous to the Green-Schwarz mechanism in superstring theory
Green:1984sg . Building on these works, ref. Monteiro:2022nqt  described the
quantum-corrected actions for the usual (‘non-improved’) theories of self-dual
Yang-Mills and self-dual gravity, and described also how the integrability
anomaly in these theories may be expressed as the quantum non-conservation of
the tower of currents that underlie the classical integrability. The quantum-
corrected actions, which are non-local, were shown to admit remarkably simple
expressions in light-cone gauge, neatly exhibiting crucial properties of the
amplitudes. Surprisingly, it was found that a version of the BCJ double copy
for amplitudes Bern:2008qj ; Bern:2019prr  applies directly to the one-loop
(loop-integrated) amplitudes; this is in addition to the expected BCJ double
copy for loop integrands Bern:2010ue , which for the self-dual sectors is well
known to be straightforwardly realised in light-cone gauge Boels:2013bi ;
Monteiro:2011pc .

These recent developments in interpreting the amplitudes in the self-dual
theories as being related to an anomaly are in line with prescient old
suggestions Chalmers:1996rq , particularly regarding a connection to the non-
local effective actions that appear in the context of the trace anomaly. More
recently, a comment reported as footnote [11] of ref. Costello:2022upu  also
pointed in this direction.

The trace anomaly, also known as the Weyl or conformal anomaly, represents the
failure of theories that are conformally invariant at the classical level to
preserve this property at loop level, as revealed by quantum contributions to
the trace of the stress-energy tensor. This subject has a long history
Capper:1974ic ; Deser:1976yx ; Duff:1977ay ; Birrell:1982ix ; Duff:1993wm . We
are particularly interested in the non-local type of action introduced in
Riegert:1984kt ; Fradkin:1983tg  to incorporate this quantum correction; see
Barvinsky:2023exr  for a recent overview of this mechanism and its
applications. We will see that this is precisely the type of non-local quantum
correction appearing in self-dual gravity and self-dual Yang-Mills theory. In
fact, the emphasis on conformal symmetry in the study of the trace anomaly
allows us to understand better the conformal properties of the quantum self-
dual theories, connecting to the results of Henn:2019mvc . Furthermore, the
properties of self-dual fields lead to a close relation between the trace
anomaly and another important type of anomaly, the chiral (or axial) anomaly,
with an even longer history Adler:1969gk ; Bell:1969ts ; Bardeen:1969md ;
Delbourgo:1972xb ; see e.g. Jackiw:2008 ; Fujikawa:2004cx . Refs.
Chattopadhyay:2020oxe ; Chattopadhyay:2021udc  observed already a close
connection between the chiral anomaly and the amplitudes in self-dual Yang-
Mills theory, and the chiral anomaly also plays an important role in the
twistor constructions of Costello:2021bah ; Bittleston:2022nfr . We will
discuss the connection to the chiral anomaly.

In recent years there has been great interest in developing new tools for
classical and quantum field theory on non-trivial backgrounds, including the
application of modern ideas from the study of scattering amplitudes. The self-
dual sectors provide a toy model for extremely challenging loop calculations
on such backgrounds. In fact, they are not just a toy model, because they
correspond to well-defined physical sectors of gravity and Yang-Mills theory.
The connection between the one-loop amplitudes in these sectors and the trace
anomaly will provide clues for extending the one-loop results to curved
spacetimes.

While most of our focus is on the self-dual theories, there is an intriguing
relation between the one-loop amplitudes of self-dual gravity and the two-loop
ultraviolet divergence of pure general relativity Bern:2015xsa ; Bern:2017puu
, which we will comment on in light of the new understanding of the one-loop
amplitudes.

The paper is organised as follows. In section 2, we review the basic aspects
of the trace and chiral anomalies. In section 3, we discuss the construction
of the quantum-corrected actions for self-dual gravity and self-dual Yang-
Mills theory, relating them to recent work on their interpretation as
following from an anomaly. The connection to the trace anomaly highlights the
conformal properties of the quantum self-dual theories, as we will see in
section 4 (and appendix A). This leads to proposals for the quantum-corrected
actions on non-trivial backgrounds, discussed in section 5. In section 6, we
will briefly comment on the relationship between the anomaly in self-dual
gravity and the two-loop divergence of pure gravity. We conclude with a final
discussion in section 7.

##  2 Review of the trace and chiral anomalies

###  2.1 Trace anomaly and non-local actions

In four dimensions, a classically Weyl-invariant theory generically sees this
invariance broken by loop effects on a curved spacetime background. The trace
of the quantum-corrected (q.c.) stress-energy tensor may be expressed as
Capper:1974ic ; Deser:1976yx ; Duff:1977ay ; Birrell:1982ix ; Duff:1993wm

$\displaystyle T_{(\text{q.c.})}{}^{\mu}{}_{\mu}=\alpha
C^{2}+\beta(E-\tfrac{2}{3}\square R)+\gamma F^{2}\,,$ (1)

where we have also allowed for a background Yang-Mills field. We denote
$\square=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$, with $\nabla_{\mu}$ the
covariant derivative, and define

$\displaystyle E$
$\displaystyle=R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho}-4R_{\mu\nu}R^{\mu\nu}+R^{2},$
(2) $\displaystyle C^{2}$
$\displaystyle=C_{\mu\nu\lambda\rho}C^{\mu\nu\lambda\rho}=R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho}-2R_{\mu\nu}R^{\mu\nu}+\frac{1}{3}R^{2},$
(3) $\displaystyle F^{2}$
$\displaystyle=\operatorname{Tr}F_{\mu\nu}F^{\mu\nu},$ (4)

where $\sqrt{|g|}E$ is the Euler density, $C_{\mu\nu\lambda\rho}$ is the Weyl
tensor, and $F_{\mu\nu}$ is the field strength tensor. The coefficients
$\alpha$, $\beta$, $\gamma$ are theory-dependent parameters. The quantum
effect represented by this trace anomaly (also called Weyl anomaly or
conformal anomaly) can be incorporated into a quantum-corrected action via the
addition of a non-local contribution Riegert:1984kt ; Fradkin:1983tg ,

$\displaystyle\Gamma_{\text{tr}}[g,A]=\frac{1}{4}\int
d^{4}x\sqrt{|g|}\left(\alpha C^{2}+\frac{\beta}{2}\Big{(}E-\frac{2}{3}\square
R\Big{)}+\gamma F^{2}\right)\frac{1}{\Delta_{4}}\left(E-\frac{2}{3}\square
R\right)\,,$ (5)

such that

$T_{(\text{q.c.})}{}^{\mu}{}_{\mu}=\frac{2}{\sqrt{|g|}}\,g_{\mu\nu}\,\frac{\delta}{\delta
g_{\mu\nu}}\Gamma_{\text{tr}}[g,A]\,.$ (6)

The non-locality arises from the inverse of the Paneitz operator Paneitz_2008
, a fourth-order differential operator given by

$\displaystyle\Delta_{4}=\square^{2}+2R^{\mu\nu}\nabla_{\mu}\nabla_{\nu}-\frac{2}{3}R\,\square+\frac{1}{3}(\nabla^{\mu}R)\nabla_{\mu}\,.$
(7)

This operator is the unique ‘completion’ of $\square^{2}$ such that
$\sqrt{|g|}\Delta_{4}$ is Weyl invariant, acting on a scalar of zero Weyl
weight. The inverse operator acts in the usual manner, via the Green’s
function $G(x,x^{\prime})$ defined from
$\sqrt{|g|}\,\Delta_{4}\,G(x,x^{\prime})=\delta(x,x^{\prime})$, that is,

$\displaystyle\frac{1}{\Delta_{4}}f(x)=\int
d^{4}x^{\prime}\sqrt{|g|(x^{\prime})}\,G(x,x^{\prime})f(x^{\prime}).$ (8)

The non-local quantum correction (5) can be recast as a local action via the
introduction of new scalar degrees of freedom Riegert:1984kt ; Fradkin:1983tg
; Barvinsky:1994cg ; Deser:1996na ; Deser:1999zv ; Mazur:2001aa ;
Meissner:2007xv ; Coriano:2017mux ; Barvinsky:2023exr . Due to the asymmetry
in the factors on either side of the inverse Paneitz operator, this would
generically require two scalar fields. However, because arbitrary Weyl
invariant terms can be added to (5) without modifying the form of the anomaly,
it is possible to construct a local formulation of the quantum-corrected
action in terms of a single scalar field. To this end, consider adding the
following three non-local Weyl-invariant terms to eq. (5):

$\frac{\alpha^{2}}{8\beta}\int
d^{4}x\sqrt{|g|}C^{2}\frac{1}{\Delta_{4}}C^{2}\,,\quad\frac{\gamma^{2}}{8\beta}\int
d^{4}x\sqrt{|g|}F^{2}\frac{1}{\Delta_{4}}F^{2}\,,\quad\frac{\alpha\gamma}{4\beta}\int
d^{4}x\sqrt{|g|}F^{2}\frac{1}{\Delta_{4}}C^{2}\,.$ (9)

The result is a non-local action that is symmetric in the factors on either
side of the inverse Paneitz operator

$\displaystyle\Gamma_{\text{tr}}[g,A]=$ $\displaystyle\frac{1}{8\beta}\int
d^{4}x\sqrt{|g|}\left(\alpha C^{2}+\beta\Big{(}E-\frac{2}{3}\square
R\Big{)}+\gamma F^{2}\right)\frac{1}{\Delta_{4}}\left(\alpha
C^{2}+\beta\Big{(}E-\frac{2}{3}\square R\Big{)}+\gamma F^{2}\right).$ (10)

Due to the Weyl invariance of the added terms, eq. (6) continues to hold.
Given this ambiguity of adding Weyl-invariant terms, the choice (2.1) is
noteworthy because it can be obtained by integrating out a single auxiliary
scalar field $\varphi$ from the following local action Riegert:1984kt ;
Fradkin:1983tg :

$\Gamma_{\text{tr}}[g,A,\varphi]=\int
d^{4}x\sqrt{|g|}\left(-2\beta^{\prime}\varphi\Delta_{4}\varphi+\varphi\left[\alpha^{\prime}C^{2}+\beta^{\prime}\Big{(}E-\frac{2}{3}\square
R\Big{)}+\gamma^{\prime}F^{2}\right]\right).$ (11)

The new field $\varphi$ is a dimension-zero scalar, which has previously been
referred to as the “conformalon”, due to the fact that linear shifts in
$\varphi$ are related to Weyl transformations of the metric Mottola:2016mpl
.222An alternative scalar action for the Weyl anomaly was used to prove the
$a$-theorem in Komargodski:2011vj . As we will discuss in the following
section, similar dimension-zero scalars have appeared elsewhere in the
literature, relating to anomalies in the self-dual sectors of Yang-Mills
theory and gravity. It is important to note that the parameters appear
‘primed’ in (11) with respect to (2.1). If the conformalon is integrated out
naively, the coefficients would be the same, but in fact there is a change in
the parameters, due to the contribution from the determinant of the operator
$\Delta_{4}$. This determinant affects the measure of the path integral over
the metric, which here can be incorporated back into the action (2.1) as a
redefinition of the parameters, possibly up to Weyl-invariant contributions
(and we have ignored also a local $R^{2}$ term in the Lagrangian density)
Riegert:1984kt .

To begin to make the connection to the story of the self-dual theories, we
note two special cases of the local and non-local versions of the quantum
correction. One special case is to consider a gauge field in flat spacetime,
such that

$\Gamma_{\text{tr}}[g_{\mu\nu}=\eta_{\mu\nu},A,\varphi]=\int
d^{4}x\left(-2\beta\,\varphi\,\square_{0}^{2}\,\varphi+\gamma\,\varphi\,F^{2}\right)$
(12)

and, after integrating out $\varphi$,

$\displaystyle\Gamma_{\text{tr}}[g_{\mu\nu}=\eta_{\mu\nu},A]=\frac{\gamma^{2}}{8\beta}\int
d^{4}x\,\operatorname{Tr}F_{\mu\nu}F^{\mu\nu}\frac{1}{\square_{0}^{2}}\operatorname{Tr}F_{\mu\nu}F^{\mu\nu}\,.$
(13)

Here, the subscript in $\square_{0}$ denotes the flat background. There is no
need for primed parameters, because the operator $\square_{0}^{2}$ is
independent of the gauge field, and therefore the effect on the measure of the
path integral is trivial.

The other special case is that of a vanishing gauge field and a Ricci-flat
metric, such that

$\Gamma_{\text{tr}}[g|_{R_{\mu\nu}=0},A=0,\varphi]=\int
d^{4}x\sqrt{|g|}\left(-2\beta^{\prime}\,\varphi\,\square^{2}\,\varphi+(\alpha^{\prime}+\beta^{\prime})\,\varphi\,C^{2}\right)\,.$
(14)

In this case, the determinant of the operator $\square^{2}$ affects the
measure of the remaining metric path integral, an effect which is taken into
account by a redefinition of the parameters when integrating out $\varphi$:

$\displaystyle\Gamma_{\text{tr}}[g,A=0]=\frac{(\alpha+\beta)^{2}}{8\beta}\int
d^{4}x\sqrt{|g|}\,C^{2}\frac{1}{\square^{2}}C^{2}\,.$ (15)

We will later see the appearance of these functionals in the context of the
self-dual theories.

Notice that non-local actions like (2.1) aim to incorporate the quantum
effects of the conformal fields into the gravitational action. This is in
general a rather complicated problem. However, simple expressions such as
(2.1) are possible because we are looking specifically at contributions that
reproduce the trace anomaly, as opposed to those that capture the full quantum
effects. One might hope, however, that the effect of integrating out a
conformal field is much simpler if we consider a self-dual background. We will
see that this is the case in the following sections.

###  2.2 Chiral anomaly

The chiral anomaly is the quantum violation of the classical conservation of a
chiral current; see Jackiw:2008  for a basic introduction. The standard
example is that of a Dirac fermion coupled to a gauge field,

$\displaystyle S=\int
d^{4}x\;\bar{\psi}\,\gamma^{\mu}(\partial_{\mu}-iA_{\mu})\,\psi\,.$ (16)

The Noether current $j^{\mu}=\bar{\psi}\gamma^{\mu}\gamma^{5}\psi$ ,
associated to the chiral symmetry of the action under $\psi\mapsto
e^{i\theta\gamma^{5}}\psi$, is classically conserved, but this property is
anomalous. In the quantum-corrected theory, we have Adler:1969gk ; Bell:1969ts
; Bardeen:1969md

$\displaystyle\partial_{\mu}\,j^{\mu}_{\text{\,(q.c.)}}=\frac{1}{16\pi^{2}}\,\varepsilon^{\mu\nu\lambda\rho}\,\text{tr}\,F_{\mu\nu}\,F_{\lambda\rho}\,.$
(17)

Similarly, for a Dirac field minimally coupled to gravity, we have
Delbourgo:1972xb

$\displaystyle\partial_{\mu}\,j^{\mu}_{\text{\,(q.c.)}}=-\frac{1}{384\pi^{2}}\,\varepsilon^{\mu\nu\lambda\rho}\,R_{\mu\nu\alpha\beta}\,R_{\lambda\rho}{}^{\alpha\beta}\,.$
(18)

Suppose that we are interested in a quantum-corrected action. We want to know
what is the effect, say for a gauge theory action, of integrating out the
fermion. This is a very challenging problem in general. It turns out to be
simple in two spacetime dimensions, leading to a non-local quantum correction
somewhat reminiscent of (13), from which the anomaly can be computed
Johnson:1963vz ; Jackiw:1983nv .333For an Abelian gauge field, the non-local
functional is $\int d^{2}x\,\ast\\!F\,\frac{1}{\square_{0}}\,\ast\\!F$, where
$\,\ast F=\varepsilon^{\mu\nu}F_{\mu\nu}\,$. The analogous functional for
SU(N) gauge theory turns out to vanish, because $\text{tr}\ast F=0\,$. In four
dimensions, we may hope that the problem is relatively simple in the case of
self-dual gauge theory, and in fact the precise functional (13) will suffice
for certain choices of gauge group.

##  3 Anomalies in the self-dual theories

A recent instance of the appearance of non-local actions associated to
anomalies is the quantum description of self-dual gravity (SDG) and self-dual
Yang-Mills theory (SDYM). These theories are well-known to be classically
integrable, and have the associated property of vanishing tree-level
scattering amplitudes. In fact, the only scattering amplitudes are at one-loop
with positive-helicity external states Bern:1993qk ; Mahlon:1993si ;
Bern:1996ja ; Bern:1998xc ; Bern:1998sv . For Yang-Mills theory, we have in
spinor-helicity language the following colour-ordered amplitudes Bern:1993qk :

$\displaystyle A^{(1)}_{\text{YM}}(1^{+}2^{+}\cdots
n^{+})=-\frac{i}{(4\pi)^{2}\cdot 3}\sum_{1\leq i_{1}<i_{2}<i_{3}<i_{4}\leq
n}\frac{\langle i_{1}i_{2}\rangle[i_{2}i_{3}]\langle
i_{3}i_{4}\rangle[i_{4}i_{1}]}{\langle 12\rangle\langle 23\rangle\cdots\langle
n1\rangle}\,,$ (19)

which correspond to the self-dual sector of the full theory. For the analogous
case in gravity, the $n$-point expression is more involved, but is also known
in closed form Bern:1998xc . The 4-point case can be written as

$\displaystyle
A^{(1)}_{\text{G}}(1^{+}2^{+}3^{+}4^{+})=-\frac{i}{(4\pi)^{2}\cdot
120}\,(s_{12}^{2}+s_{23}^{2}+s_{31}^{2})\left(\frac{s_{12}s_{23}}{\langle
12\rangle\langle 23\rangle\langle 34\rangle\langle 41\rangle}\right)^{2}\,.$
(20)

In both SDYM and SDG, the amplitudes are rational functions of the external
data, i.e. they possess poles but not branch cuts, because the tree amplitudes
vanish. The simplicity of the amplitudes was argued in Bardeen:1995gk  to be a
remnant of the integrability of the classical theories, with the amplitudes
resulting from an anomaly. Recent work, to be discussed in the following, has
provided an elegant realisation of this idea. Let us stress that the
particular anomaly we are considering here is not a pathological feature of
quantum SDYM or SDG. It merely represents the failure of the quantum theories
to exhibit the integrability that is a property of the classical theories, as
signalled by the non-vanishing one-loop amplitudes.

Twistor space has long been a convenient setting to study the classical self-
dual theories, a fact that is closely related to their integrability; see e.g.
lionel1996integrability ; Dunajski:2010zz ; Adamo:2017qyl . Based on a twistor
construction, ‘improved’ non-anomalous versions of quantum SDYM and SDG have
been introduced recently in Costello:2021bah ; Bittleston:2022nfr ; see also
Costello:2022wso ; Costello:2022upu ; Bu:2022dis ; Bittleston:2022jeq ;
Bu:2023vjt . These are improved in the sense that the quantum theories can
still be formulated in twistor space, unlike ordinary quantum SDYM and SDG.
The resulting integrability of the ‘improved’ theories is associated to the
vanishing of their loop amplitudes. In practice, the loop amplitudes (19) and
(20) are cancelled via a Green-Schwarz-type mechanism, analogous to the one in
superstring theory Green:1984sg : one-loop diagrams of gravitons/gluons are
cancelled by diagrams involving an auxiliary axion field. This is represented
in Figure 1.

Figure 1: The ordinary SDYM loop diagrams with gluons (left) are cancelled in
the ‘improved’ theory of ref. Costello:2021bah  by tree-level exchanges of the
scalar field (right). In the ‘improved’ SDG theory of ref. Bittleston:2022nfr
, there is a similar cancellation, involving also an additional type of
contribution (centre).

In this section, we will discuss the quantum correction to ordinary (i.e.
anomalous) SDYM and SDG, with a particular focus on the similarity of this
structure to the trace anomaly. This relies crucially on results presented in
Costello:2021bah ; Bittleston:2022nfr , and at the end we will connect our
discussion to the quantum-integrable ‘improved’ versions of SDYM and SDG
presented in those works. The logic of our discussion of non-local quantum
corrections builds on Monteiro:2022nqt .

###  3.1 Anomaly in self-dual Yang-Mills theory

Let us start with SDYM. The usual action is

$\displaystyle S_{\text{SDYM}}[B,A]=\int B\wedge F_{\text{ASD}}\,,$ (21)

with $B$ a Lagrange-multiplier 2-form that eliminates the anti-self-dual part
of the field strength $F$ Chalmers:1996rq . The non-vanishing amplitudes in
this theory are the all-plus one-loop amplitudes; they correspond to the
contributions on the left of Figure 1, where all lines are gluons.444Fadeev-
Popov ghosts should be included here if appropriate. They are not required in
light-cone gauge, which is particularly convenient for the self-dual theories.
Now, there is a well-known property that these contributions are exactly
reproduced by considering a complex minimally-coupled massless scalar in the
loop, instead of the gluon. In both cases, there are two real bosonic degrees
of freedom running in the loop.555This is easy to see from the Feynman rules
in light-cone gauge. It can also be thought of as a consequence of the so-
called supersymmetric Ward identities Grisaru:1977px ; Grisaru:1979re , which
imply that the non-supersymmetric all-plus (and one-minus) amplitudes are
proportional to the number of bosonic minus fermionic degrees of freedom
running in the loop. Consider the action

$\displaystyle S_{\text{SDYM}}-\int
d^{4}x\,(D_{\mu}\vartheta)^{\dagger}D^{\mu}\vartheta\,,$ (22)

like scalar QCD, but with a self-dual gauge field. The self-duality condition
imposed by the Lagrange-multiplier-type part $S_{\text{SDYM}}$ means that the
massless scalar only couples to one helicity of the gluon. The amplitudes for
external gluons will receive two contributions: one from gluons in the loop
and one from the complex scalar in the loop, with the two contributions being
equal, as already mentioned. Therefore, if we integrate out the complex
scalar, we obtain an exact effective action for the gluons, with one-loop
effective vertices arising from the determinant of the operator acting on the
scalar. Integrating out the scalar is a complicated problem on a generic gauge
field background, requiring regularisation, but one can hope that it has a
simple solution if that background is self-dual. Indeed, in this case, it
generates one-loop vertices for UV-finite amplitudes without branch cuts,
namely those given in (19). Still, a closed-form solution to this problem is
at present available only for a particular set of gauge groups, namely SU(2),
SU(3), SO(8) or one the exceptional groups; this solution follows from the
results of Costello:2021bah . The groups mentioned are identified by the fact
that the Lie algebra $\mathfrak{g}$ satisfies the following property relating
the trace in the adjoint representation to double traces in the fundamental
representation:

$\displaystyle\operatorname{Tr}_{\text{ad}}(T^{(a_{1}}T^{a_{2}}T^{a_{3}}T^{a_{4})})=C_{\mathfrak{g}}\,\operatorname{Tr}(T^{(a_{1}}T^{a_{2}})\operatorname{Tr}(T^{a_{3}}T^{a_{4})})\,.$
(23)

The proportionality constant is

$\displaystyle C_{\mathfrak{g}}=\frac{10\,({\bf
h}^{\vee})^{2}}{2+\text{dim}\,\mathfrak{g}}\,,$ (24)

where ${\bf h}^{\vee}$ is the dual Coxeter number. For these gauge groups and
a self-dual $F$, the results of Costello:2021bah  imply (we will see later the
precise relation) that we obtain the quantum-corrected action

$\displaystyle
S_{\text{q.c.SDYM}}[B,A]=S_{\text{SDYM}}-\frac{a_{\mathfrak{g}}^{2}}{8}\int
d^{4}x\;\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})\,\frac{1}{\square_{0}^{2}}\,\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})\,,$
(25)

where

$\displaystyle
a_{\mathfrak{g}}=\sqrt{-\frac{iC_{\mathfrak{g}}}{12\pi^{2}}}\,.$ (26)

For clarity, by quantum-corrected we mean that all loop effects are accounted
for explicitly in the action, so that loop amplitudes can be calculated using
tree-level-type rules. In particular, the new term introduces one-loop
effective vertices.666This is equivalent to explicitly solving the loop
integrals that are usually implicit when incorporating the functional
determinants associated to loop effects into an action (see Lee:2022aiu ;
Gomez:2022dzk ; Kakkad:2022ryl  for recent examples of integrand-level —
rather than integral-level — approaches). This simple non-local action
produces the expected one-loop amplitudes (19), though this is not immediately
obvious. Notice that the restriction on the gauge group leads to linear
relations between colour traces that would be independent for SU(N). Hence, it
is important to consider the colour-dressed amplitude when comparing with
(19).

We note that the quantum correction in (25), valid for the restricted choice
of gauge groups, is precisely of the type (13) that we encountered in the
context of the trace anomaly. Integrating out $\vartheta$ in the action (22)
has, for the gauge groups considered, precisely the same effect as integrating
out $\varphi$ — which played the role of the conformalon in the trace anomaly
— in the action

$\displaystyle S_{\text{SDYM}}+\int
d^{4}x\left(\frac{1}{2}\,\varphi\,\square_{0}^{2}\,\varphi+\frac{a_{\mathfrak{g}}}{2}\,\varphi\,\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})\right)\,.$
(27)

Here, integrating out $\varphi$ is a simple algebraic operation, because
$\square_{0}^{2}$ does not depend on the gauge field. We are now in a position
to consider an ‘improved’ model of SDYM, which we may call $\varphi-$SDYM;
improved in the sense that it has vanishing amplitudes, and is therefore
quantum-integrable. Its (non-quantum-corrected) action is

$\displaystyle S_{\varphi-\text{SDYM}}[B,A,\varphi]=S_{\text{SDYM}}+\int
d^{4}x\left(\frac{1}{2}\,\varphi\,\square_{0}^{2}\,\varphi+\frac{ia_{\mathfrak{g}}}{2}\,\varphi\,\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})\right)\,.$
(28)

Due to the change of coefficient in the last term, with respect to (27),
integrating out $\varphi$ now leads to (25) with a different sign for the last
term. That is, the effect of the scalar is now to cancel the one-loop gluon
diagrams arising from $S_{\text{SDYM}}$. In terms of the Figure 1, the gluon
diagrams on the left are cancelled by the diagrams on the right, with tree
exchanges of $\varphi$. The diagrams in the centre of Figure 1 are absent for
the action (28), which is why integrating out $\varphi$ is straightforward.
There is one last step before recognising the equivalence of this ‘improved’
theory of SDYM to the one obtained in Costello:2021bah  from a twistor-space
construction. That is to notice that the Lagrange-multiplier-type action
$S_{\text{SDYM}}$ is imposing self-duality of the gauge field, such that

$\displaystyle\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})=-\frac{i}{2}\operatorname{Tr}(\varepsilon^{\mu\nu\rho\lambda}F_{\mu\nu}F_{\rho\lambda})\,.$
(29)

Hence, (28) can be written equivalently as

$\displaystyle
S_{\varphi-\text{SDYM}}[B,A,\varphi]=S_{\text{SDYM}}+\int\left(d^{4}x\;\frac{1}{2}\,\varphi\,\square_{0}^{2}\,\varphi+a_{\mathfrak{g}}\,\varphi\operatorname{Tr}(F\wedge
F)\right)\,.$ (30)

This is the theory introduced in Costello:2021bah . The scalar $\varphi$ has
an axionic coupling to the gauge field here, whereas it had a ‘dilatonic’
coupling in (28). This ambiguity arises naturally from the self-duality
condition.

It would be interesting to work out explicitly the extension of this story to
SU(N) SDYM. The first step in this direction, reproducing the parity-invariant
part of the quantum correction to standard SDYM, i.e. generalising (25), was
taken in Monteiro:2022nqt . This functional corresponds to

$\displaystyle\int
d^{4}x\;\left(\frac{1}{D^{2}}(F_{\mu\nu}F^{\mu\nu})\right)^{ab}\left(\frac{1}{D^{2}}(F_{\mu\nu}F^{\mu\nu})\right)^{ab}\,,$
(31)

where $a,b$ are Lie algebra indices; see Monteiro:2022nqt  for the definition
of the $1/D^{2}$ operation. Still missing is a part carrying a Levi-Civita
symbol $\varepsilon^{\mu\nu\alpha\beta}$ that breaks parity, and contributes
to scattering amplitudes at multiplicity $n>4$.

Let us return to the question of the similarity between the quantum correction
in (25) and the type of non-local expression found in the context of the Weyl
anomaly. This does not mean that the anomaly is of Weyl type. The appearance
of the functional in the Weyl context came via the introduction in (9) — in
particular, the term in the centre — of Weyl-invariant terms to complete the
perfect square in (2.1). The anomaly in SDYM is more closely related to the
chiral anomaly. Indeed, the quantum correction in (25), valid for the
restricted choice of gauge groups, is of a form that is a 4D analogue of the
quantum correction for the 2D chiral anomaly Jackiw:1983nv ; see footnote 3.
Interestingly, refs. Chattopadhyay:2020oxe ; Chattopadhyay:2021udc  presented
a construction analogous to the chiral anomaly that reproduces the loop
amplitudes for SU(N). This construction was based in ‘region-momenta’ space
(adapted to planar amplitudes) instead of position space. One drawback is that
the application of a similar construction to gravity (which has no planar
sector) is unclear.

Regarding the connection to the chiral anomaly, notice that the quantum
correction in (25) can also be seen as the result of integrating out a
massless fermion — up to a sign. Indeed, a well-known procedure to obtain an
‘improved’ SDYM theory, alternative to the bosonic model with the axion
introduced in Costello:2021bah , is to introduce sypersymmetry. The sign of
the quantum correction in (25) being flipped can be achieved by integrating
out a fermion with two real degrees of freedom (the gaugino), instead of
integrating out the scalar in (28) or (30). All-plus amplitudes are absent in
super-Yang-Mills theories, and therefore also in their self-dual sectors; the
same property applies to all-plus amplitudes of gravitons in supergravity
theories (and to one-minus amplitudes in both cases).

We will see later, in section 3.3, another way to think of the quantum
correction as resulting from a chiral anomaly, without resorting to additional
massless fields. Nevertheless, thinking of the trace anomaly is helpful,
because it is better studied from the point of view of non-local functionals,
particularly regarding the appearance of a fourth-derivative Weyl-covariant
operator that will be useful in later sections.

###  3.2 Anomaly in self-dual gravity

The story for SDG is broadly similar to that of SDYM. The usual action is

$\displaystyle S_{\text{SDG}}[\Gamma,e]=\int\Gamma_{\alpha\beta}\wedge
d(e^{\alpha\dot{\alpha}}\wedge e^{\beta}{}_{\dot{\alpha}})\,,$ (32)

where we have
$ds^{2}=\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}e^{\alpha\dot{\alpha}}e^{\beta\dot{\beta}}$,
and the three 1-forms $\Gamma_{\alpha\beta}=\Gamma_{\beta\alpha}$ play the
role of Lagrange multipliers Plebanski:1977zz ; Ashtekar:1987qx ;
Smolin:1992wj ; Capovilla:1991qb .777Plebanski’s two heavenly equations for
SDG Plebanski:1975wn  follow from this action: (i) the Lagrange multipliers
enforce that the three self-dual 2-forms $e^{\alpha\dot{\alpha}}\wedge
e^{\beta}{}_{\dot{\alpha}}$ are closed, and (ii) each of the heavenly
equations corresponds to a choice of Darboux coordinates allowed by that
closure; see e.g. Adamo:2021bej  for details. See also Krasnov:2021cva  for
other choices of SDG action. The non-vanishing amplitudes in this theory are
the all-plus one-loop amplitudes; they correspond to the type of diagram on
the left of Figure 1, where all lines are gravitons. Just as in the SDYM case,
these amplitudes are exactly reproduced by having a complex minimally-coupled
massless scalar running in the loop, instead of the graviton; both have two
bosonic real degrees of freedom. Consider the action

$\displaystyle S_{\text{SDG}}+\int
d^{4}x\,\sqrt{|g|}\;\bar{\sigma}\,\square\,\sigma\,.$ (33)

The self-duality condition imposed by $S_{\text{SDG}}$ means that the massless
scalar only couples to one helicity of the graviton. The amplitudes for
external gravitons receive two equal contributions, corresponding to the
graviton or the complex scalar running in the loop.888The dilaton and axion
fields of ‘$\mathcal{N}=0$ supergravity’ would be another way to provide an
equal contribution to that of gravitons in the loop. Therefore, if we
integrate out the complex scalar, we obtain an exact quantum-corrected action
for the gravitons, with one-loop effective vertices arising from the
determinant of the scalar kinetic operator. A consequence of the results of
Bittleston:2022nfr  (we will see later the precise relation) is that the
quantum-corrected action for SDG can be written as

$\displaystyle
S_{\text{q.c.SDG}}[\Gamma,e]=S_{\text{SDG}}-\frac{a_{\text{G}}^{2}}{8}\int
d^{4}x\,\sqrt{|g|}\;C^{2}\,\frac{1}{\square^{2}}\,C^{2}\,,$ (34)

where

$\displaystyle a_{\text{G}}=\sqrt{-\frac{i}{960\pi^{2}}}\,.$ (35)

The quantum correction is precisely of the form (15) encountered in the
context of the trace anomaly. Let us discuss further the integrating out of
the complex scalar, from (33) to (34). We can consider in (33) the addition of
the following 2-point vertex for the scalar,

$\displaystyle S_{\text{SDG}}+\int
d^{4}x\,\sqrt{|g|}\Big{(}\bar{\sigma}\,\square\,\sigma-\frac{1}{2}\bar{\sigma}^{2}\Big{)}\,.$
(36)

Importantly, on the support of the self-duality condition imposed by the
Lagrange-multiplier-type action $S_{\text{SDG}}$, this does not introduce a
new interaction with the metric (this can be seen in light-cone gauge, where
$g$ is a constant) Bittleston:2022nfr . Integrating out $\bar{\sigma}$, we
obtain

$\displaystyle S_{\text{SDG}}+\int
d^{4}x\,\sqrt{|g|}\;\frac{1}{2}\,\sigma\,\square^{2}\,\sigma\,.$ (37)

Therefore, integrating out the complex scalar, from (33) to (34), is
equivalent to integrating out the ‘conformalon’-like scalar $\sigma$, from
(37) to (34).999Notice that we have, by the procedure (36), assigned a mass
dimension of 2 to $\bar{\sigma}$ and 0 to $\sigma$. This equivalence is also
consistent with the fact that, on a self-dual background, these fields give
precisely the same contribution to the trace anomaly. The contributions to the
trace (1) (see e.g. Antoniadis:1992xu  for the coefficients used here) are
$(\alpha,\beta)=\frac{1}{(4\pi)^{2}}(\frac{1}{60},-\frac{1}{180})$ for the
complex minimally-coupled scalar, which has two real degrees of freedom, and
$(\alpha,\beta)=\frac{1}{(4\pi)^{2}}(-\frac{1}{15},\frac{7}{90})$ for the
scalar with fourth-derivative kinetic term. Self-dual backgrounds are Ricci-
flat, and therefore we have $E-\frac{2}{3}\square R=C^{2}$. In both cases, we
obtain $\alpha+\beta=\frac{1}{(4\pi)^{2}\;90}\,,$ so their contributions to
the trace anomaly are equivalent. From the discussion of the trace anomaly in
section 2, particularly the integrating out of the conformalon, it is natural
to expect the functional appearing in the quantum correction in (34). The
coefficient of that quantum correction cannot, however, be determined from the
trace anomaly, because the effective action for the latter is only defined up
to Weyl-invariant terms, which can take the form of the sought functional —
see the first term in (9). Instead, the coefficient in (34) can be fixed by
comparing to a one-loop amplitude of SDG, say at 4 points. This is an indirect
way of computing the quantum-correction to the SDG action, or at least of
finding a natural candidate and checking the result.

We may proceed, as for SDYM, to obtain an ‘improved’ quantum-integrable
upgrade of SDG, with vanishing amplitudes; we call it $\varphi$-SDG. The idea
is to flip the sign of the quantum correction in (34), and interpret that as a
(non-quantum-corrected) action, where the last term results from integrating
out a scalar field. Given that we know the effect of integrating out $\sigma$
in (37), we can conclude that the following action works,

$\displaystyle S_{\varphi-\text{SDG}}[\Gamma,e,\varphi]=S_{\text{SDG}}+\int
d^{4}x\sqrt{|g|}\left(\;\frac{1}{2}\,\varphi\,\square^{2}\,\varphi-\frac{ia_{\text{G}}}{2}\,\varphi\,C^{2}\right)\,.$
(38)

We called the scalar $\varphi$ to emphasise the similarity to the ‘improved’
SDYM action (38) and to the conformalon of the trace anomaly. The amplitudes
of gravitons in this action vanish as a cancellation of the contributions
represented in Figure 1: on the left, the loops diagrams of gravitons from
$S_{\text{SDG}}$; in the centre, the loop diagrams with $\varphi$ running in
the loop, resulting from the scalar-graviton interactions included in its
kinetic term; and on the right, the tree exchanges of $\varphi$, resulting
from the last term in (38). We have now the theory introduced in
Bittleston:2022nfr  from a twistor-space construction. This becomes clear when
we note that, on the self-duality constraint imposed by $S_{\text{SDG}}$, we
have

$\displaystyle
C^{2}=\frac{i}{2}\varepsilon^{\rho\lambda\sigma\omega}R^{\mu}{}_{\nu\rho\lambda}\,R^{\nu}{}_{\mu\sigma\omega}\,.$
(39)

Hence, we can write the theory with the axionic-like coupling as in
Bittleston:2022nfr ,

$\displaystyle
S_{\varphi-\text{SDG}}[\Gamma,e,\varphi]=S_{\text{SDG}}+\int\left(d^{4}x\sqrt{|g|}\;\frac{1}{2}\,\varphi\,\square^{2}\,\varphi+a_{\text{G}}\,\varphi\,\mathsf{R}^{\mu}{}_{\nu}\wedge\mathsf{R}^{\nu}{}_{\mu}\right)\,.$
(40)

Just as we mentioned for SDYM, an alternative to the bosonic model of
Bittleston:2022nfr  is to cancel the one-loop amplitudes of SDG using a
massless fermion with two real degrees of freedom — as in supergravity.
Integrating out this fermion then leads precisely to the second term in (34),
with the opposite sign. Furthermore, as was the case for SDYM, there is an
interpretation of the anomaly as a chiral anomaly, without resorting to
additional massless fields to be integrated out, as we will soon discuss.

Just before the submission of our paper, ref. Karateev:2023mrb  appeared,
discussing the application of the trace anomaly to graviton-dilaton
amplitudes. There should be a connection to our discussion.

Finally, we note that a relation between conformal self-dual gravity and the
Weyl anomaly was pointed out in Schmidhuber:1992dg . It would be interesting
to understand how this connects to our discussion for ordinary self-dual
gravity.

###  3.3 Quantum-corrected actions in light-cone gauge

The fact that the quantum-corrected actions for SDYM and SDG, namely (25) and
(34), lead to loop amplitudes that are non-vanishing only for one-loop and for
all-plus external states may not be immediately obvious from those
expressions. This is, however, clearly seen in light-cone gauge
Monteiro:2022nqt . We just quote the results here for illustration, and refer
the reader to that work for further details. Consider double-null coordinates
such that

$\displaystyle
ds^{2}_{0}=2(-dudv+dwd{\bar{w}})\,,\qquad\square_{0}=2(-\partial_{u}\partial_{v}+\partial_{w}\partial_{\bar{w}})\,,$
(41)

and define

$\displaystyle\buildrel{\leftrightarrow}\over{P}\;=\;\buildrel{\leftarrow}\over{\partial}_{u}\buildrel{\rightarrow}\over{\partial}_{w}-\buildrel{\leftarrow}\over{\partial}_{w}\buildrel{\rightarrow}\over{\partial}_{u}\,,$
(42)

where the null plane $(u,w)$ is picked up by the gauge choice. Then, the
actions for a particular definition of off-shell positive-helicity fields
($\Psi$ and $\phi$) and negative-helicity fields ($\bar{\Psi}$ and
$\bar{\phi}$) can be written as

$\displaystyle S_{\text{q.c.SDYM}}=\int
d^{4}x\;\;\left(\text{tr}\big{(}\bar{\Psi}\big{(}\square_{0}\Psi+i\,(\Psi\buildrel{\leftrightarrow}\over{P}\Psi))\big{)}-\frac{a_{\mathfrak{g}}^{2}}{32}\Big{(}\,\frac{1}{\square_{0}}\,\text{tr}(\Psi\buildrel{\leftrightarrow}\over{P}\\!{}^{2}\,\Psi)\Big{)}^{2}\right)$
(43)

and

$\displaystyle S_{\text{q.c.SDG}}=\int
d^{4}x\;\left(\,\bar{\phi}\Big{(}\square_{0}\phi-\frac{1}{2}(\phi\buildrel{\leftrightarrow}\over{P}\\!{}^{2}\,\phi))\Big{)}-\frac{a_{\text{G}}^{2}}{8}\left(\,\frac{1}{\square_{0}-(\phi\buildrel{\leftrightarrow}\over{P}\\!{}^{2}\;\cdot)}\,(\phi\buildrel{\leftrightarrow}\over{P}\\!{}^{4}\,\phi)\,\right)^{2}\,\right)\,.$
(44)

The Feynman rules for these actions make it clear that the only loop-level
diagrams that can be constructed are those at one loop with external positive-
helicity fields.

Notice that the presence of $\frac{1}{\square^{2}}$ in the actions above does
not lead to unphysical properties. It is actually essential to recover the
one-loop amplitudes (19) and (20). These have at most simple poles
$\frac{1}{p^{2}}$ in the kinematic data, instead of the poles
$\frac{1}{p^{4}}$ naively associated to $\frac{1}{\square^{2}}$. The reduction
of the order of the pole is due to kinematic identities. In the long history
of the trace anomaly, there have been doubts about the suitability of
$\frac{1}{\square^{2}}$-type operations that appear in the non-local effective
action, as reviewed in Barvinsky:2023exr . In the present instance, at least,
such operations are perfectly legal.

This formulation also makes it easier to understand the soft and collinear
behaviour of the amplitudes from an action/Feynman-rules perspective. Such
behaviour has been recently explored in the context of celestial holography,
in terms of the celestial soft operator-product-expansion algebras; see e.g.
Guevara:2021abz ; Strominger:2021lvk ; Jiang:2021ovh ; Jiang:2021csc ;
Adamo:2021lrv ; Ball:2021tmb ; Mago:2021wje ; Ren:2022sws ; Monteiro:2022lwm ;
Costello:2022jpg ; Bu:2022iak ; Bhardwaj:2022anh ; Guevara:2022qnm ;
Ball:2022bgg ; Adamo:2022lah ; Monteiro:2022xwq ; Garner:2023izn ; Ren:2023trv
; Bittleston:2023bzp ; Costello:2023hmi ; He:2023lvk ; Brown:2023zxm ;
Taylor:2023ajd ; Himwich:2023njb . Historically, it was precisely this special
behaviour that led to the $n$-point formulas for the all-plus one-loop
amplitudes, obtained in Bern:1993qk ; Bern:1998xc  via a kind of ‘soft and
collinear bootstrap’.

It is worth noting that all interactions in (43) and (44), including the one-
loop vertices, exhibit the appearance of the ‘kinematic algebra’ of the self-
dual sector Monteiro:2011pc , which underlies the BCJ colour-kinematics
duality and double copy Bern:2008qj  in the case of the self-dual
theories.101010It would be interesting to see whether the quantum corrections
fit nicely into the formalisms of Borsten:2023paw ; Bonezzi:2023pox , which
also aim to expose the kinematic algebra in the self-dual theories. The
antisymmetrised derivatives with respect to $u$ and $w$ lead in Fourier space
to the appearance of the structure constant

$\displaystyle X(k_{1},k_{2})=k_{1w}k_{2u}-k_{1u}k_{2w}$ (45)

of the Lie algebra of area-preserving diffeomorphisms:

$\displaystyle=X(k_{1},k_{2})\,L_{k_{1}+k_{2}}\,,\qquad L_{k}:=(e^{i\,k\cdot
x}\buildrel{\leftrightarrow}\over{P}\\!\;\cdot\;)\,.$ (46)

This Lie algebra is equivalent, after a change of basis corresponding to a
soft expansion Monteiro:2022nqt , to the wedge algebra of the Lie algebra of
$w_{1+\infty}$, which was identified in the context of celestial holography in
Strominger:2021lvk . In the celestial OPEs, one factor of the $X$ structure
constant enforces the chiral nature of the OPE of positive-helicity gluons or
gravitons, whereas only for gravity there is an additional factor of $X$ in
the 3-point vertex that leads to the celestial $w_{1+\infty}$ algebra
Monteiro:2022nqt . So the $(++-)$ ‘tree’ vertex of self-dual gravity, which is
picked up in full non-chiral gravity by taking two positive-helicity
gravitons, manifestly exhibits the $w_{1+\infty}$ algebra in light-cone gauge.
Natural generalisations of this algebra, via deformations or extensions to
theories with higher (spacetime) spins, follow in a simple manner from light-
cone gauge formulations Monteiro:2022xwq . Very recently, ref. Himwich:2023njb
presented the first study of $w_{1+\infty}$ in the context of gravity with
minimally-coupled massive particles. The light-cone gauge is insightful also
in this context, because of the universal appearance of the algebra $X$ in
soft factors associated to a positive-helicity graviton. In particular, for a
positive-helicity graviton polarisation tensor, we have the appearance of
$\,\varepsilon^{+}_{\mu\nu}(k)p^{\mu}p^{\nu}=-\frac{1}{k_{u}^{2}}\,X(k,p)^{2}$
in the soft factors, where $k$ is the soft graviton momentum, and $p$ is the
momentum of an external hard particle, be it massless or massive.

Regarding the anomaly interpretation of the one-loop amplitudes of the self-
dual theories, the quantum-corrected actions (43) and (44) exhibit the
interpretation of a chiral anomaly. Let us consider the chiral U(1)
transformations

$\displaystyle(\Psi,\bar{\Psi})$
$\displaystyle\mapsto(e^{i\theta}\Psi,e^{-i\theta}\bar{\Psi}),$ (47)
$\displaystyle(\phi,\bar{\phi})$
$\displaystyle\mapsto(e^{i\theta}\phi,e^{-i\theta}\bar{\phi}),$ (48)

in (43) and (44), respectively. These transformations of positive- and
negative-helicity degrees of freedom leave an action invariant when there is
an ‘electric-magnetic’ duality in the theory; see Monteiro:2023dev  for a
detailed discussion, which builds on amplitudes literature Rosly:2002jt ;
Carrasco:2013ypa ; Bern:2013uka ; Bern:2017rjw ; Novotny:2018iph ;
Elvang:2019twd ; Elvang:2020kuj ; Pavao:2022kog ; Carrasco:2022jxn ;
Carrasco:2023qgz . In the present case, we see that not only do the loop
corrections violate this symmetry maximally (because they only exhibit the
positive-helicity fields), but that the symmetry is broken even by the ‘tree’
3-point vertices. One may think, therefore, that the symmetry is non-existent
even at tree-level. However, notice that one may perform the following non-
local field redefinitions in (43) and (44), respectively:

$\displaystyle(\tilde{\Psi},\bar{\tilde{\Psi}})$
$\displaystyle=\Big{(}\Psi+\frac{i}{\square_{0}}(\Psi\buildrel{\leftrightarrow}\over{P}\Psi),\bar{\Psi}\Big{)},$
(49) $\displaystyle(\tilde{\phi},\bar{\tilde{\phi}})$
$\displaystyle=\Big{(}\phi-\frac{1}{2\,\square_{0}}(\phi\buildrel{\leftrightarrow}\over{P}\\!{}^{2}\,\phi),\bar{\phi}\Big{)}.$
(50)

After these redefinitions, the ‘tree’ part of the action corresponds to a
kinetic term that is invariant under the chiral transformation of the new
fields. Recall that these theories have vanishing tree amplitudes.111111The
3-point $(++-)$ amplitudes have no support on on-shell real kinematics in
Lorentzian signature. For our purpose, we can ignore the divergence of
$\frac{1}{\square_{0}}$ for 3-point on-shell kinematics. The loop amplitudes
are unaltered, of course, because this redefinition of the standard action to
that of a ‘free theory’ affects the measure of the path integral, leading in
this way to the appropriate one-loop quantum corrections, as first argued in
Brandhuber:2006bf . In the quantum-corrected action, the redefinitions lead to
new all-plus one-loop vertices that are truly off-shell one-loop amplitudes:
the $n$-point amplitude is computed by simply putting the new $n$-point vertex
on-shell, considering the appropriate symmetry factors; notice that the
propagator cannot connect all-plus vertices. That is, the redefinitions
eliminate the explicit 3-point $(++-)$ vertex but also dress the one-loop
effective vertices with trees built with the 3-point vertex, so that no
interaction is lost. Our point here is simply that the loop amplitudes are
indeed the manifestation of a chiral anomaly, namely the anomaly of the U(1)
chiral symmetry that is best seen in terms of the redefined fields (49) and
(50).

###  3.4 Anomalies in self-dual Einstein-Yang-Mills theory

For completeness, we can briefly discuss the coupling of SDG to SDYM. Ref.
Bittleston:2022nfr  introduced the ‘improved’ theory of self-dual Einstein-
Yang-Mills:

$\displaystyle S_{\varphi-\text{SDEYM}}=S_{\text{SDEYM}}+\int
d^{4}x\sqrt{|g|}\,\,\frac{1}{2}(\square\,\varphi)^{2}+\int\varphi\,\big{(}a_{\mathfrak{g}}\operatorname{Tr}{F\wedge
F}+a_{\text{G},\mathfrak{g}}\,\mathsf{R}^{\mu}{}_{\nu}\wedge\mathsf{R}^{\nu}{}_{\mu}\big{)}\,,$
(51)

where
$a_{\text{G},\mathfrak{g}}=a_{\text{G}}\sqrt{1+\frac{1}{2}\,\text{dim}\,\mathfrak{g}}$.
Here, $\text{dim}\,\mathfrak{g}$ is the dimension of the Lie algebra, with the
gauge group still obeying the restriction mentioned above for SDYM. This
modification of the coupling $a_{\text{G}}$ of $\varphi-\text{SDG}$ occurs due
to the need to cancel diagrams with external gravitons that have gluons
running in the loop. Along the lines of the SDYM and SDG cases, the action
above implies that we can write down a quantum-corrected action for the
ordinary SDEYM theory as

$\displaystyle S_{\text{q.c.SDEYM}}=$ $\displaystyle\;S_{\text{SDEYM}}$
$\displaystyle-\int
d^{4}x\,\sqrt{|g|}\,\Big{(}\frac{a_{\mathfrak{g}}^{2}}{8}\,F^{2}\,\frac{1}{\square^{2}}\,F^{2}+\frac{\tilde{a}_{\text{G},\mathfrak{g}}^{2}}{8}\;C^{2}\,\frac{1}{\square^{2}}C^{2}\;-\;\frac{a_{\text{G},\mathfrak{g}}\,a_{\mathfrak{g}}}{4}\;C^{2}\frac{1}{\square^{2}}\,F^{2}\,\Big{)}\,,$
(52)

where $\,F^{2}=\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})$ . The only subtlety
here is the substitution of the coefficient
$a_{\text{G},\mathfrak{g}}=a_{\text{G}}\sqrt{1+\frac{1}{2}\,\text{dim}\,\mathfrak{g}}$
by the coefficient
$\tilde{a}_{\text{G},\mathfrak{g}}=a_{\text{G}}\sqrt{\frac{1}{2}+\frac{1}{2}\,\text{dim}\,\mathfrak{g}}$
in the second line. This arises from the functional determinant when
integrating out $\varphi$, as explained in the SDG case. This quantum-
corrected action reproduces the one-loop scattering amplitudes of self-dual
Einstein-Yang-Mills theory. For a brief discussion of how the amplitudes in
the self-dual sector relate to those in the full Einstein-Yang-Mills theory
Faller:2018vdz ; Nandan:2018ody , see Bittleston:2022nfr .

##  4 Conformal properties from the Paneitz operator

The trace anomaly gives us clues into the appearance of a fourth-order
(inverse) differential operator in the quantum-corrected actions for SDYM and
SDG. The Paneitz operator $\Delta_{4}$ defined in (7) is such that
$\sqrt{|g|}\Delta_{4}$ is Weyl invariant, and satisfies
$\Delta_{4}=\square^{2}$ on Ricci-flat backgrounds. This operator exposes the
conformal properties of the theories.

One important property of quantum SDYM is the conformal invariance of its loop
amplitudes, given in (19). We recall that these are the all-plus one-loop
amplitudes of full Yang-Mills theory. The conformal invariance was observed
empirically in Henn:2019mvc , by noting that the generators of the conformal
algebra — conveniently expressed in spinor-helicity notation Witten:2003nn  —
eliminate the amplitudes. A brief review of this story is presented in
Appendix A.1.121212Ref. Henn:2019mvc ’s full proof of the conformal invariance
of the amplitudes, not included in the appendix, noted that the amplitudes can
be written as a sum of simple terms that are separately conformally invariant.
Interestingly, these same terms had appeared as BCJ numerators of one-loop MHV
amplitudes in ${\mathcal{N}}=4$ super-Yang-Mills theory He:2015wgf . Indeed, a
connection between the maximally supersymmetric theory and the self-dual
sector of the non-supersymmetric theory has long been known Bern:1996ja ; see
Britto:2020crg  for a recent proof of this relationship. This in turn suggests
that the structures studied in our paper play some role in ${\mathcal{N}}=4$
super-Yang-Mills theory.

The form of the quantum-corrected action for SDYM in (25) provides an
explanation for the conformal invariance of these amplitudes — though only for
the gauge groups where this action is valid, e.g. SU(3). The relation of that
action to the twistor-space theory of Costello:2021bah  already indicates the
presence of conformal symmetry, but let us focus here on the spacetime
formulation. Consider the following action on a general background:

$\displaystyle S_{\text{SDYM}}-\frac{a_{\mathfrak{g}}^{2}}{8}\int
d^{4}x\,\sqrt{|g|}\;\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})\,\frac{1}{\Delta_{4}}\,\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})\,.$
(53)

This action is clearly invariant under a Weyl transformation,131313Notice that
$\chi=\frac{1}{\Delta_{4}}\,\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})$ is Weyl
invariant, as implied by
$\Delta_{4}\,\chi=\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})$.

$\displaystyle g_{\mu\nu}\to e^{2\Omega}g_{\mu\nu}\,,\quad
F_{\mu\nu}F^{\mu\nu}\to e^{-4\Omega}F_{\mu\nu}F^{\mu\nu}\,,\quad\Delta_{4}\to
e^{-4\Omega}\Delta_{4}\,.$ (54)

Moreover, it reduces to (25) in the flat background case. This
straightforwardly explains the conformal invariance of the quantum-corrected
theory on flat spacetime, and hence of the scattering amplitudes. To see this,
we recall that a conformal transformation $x^{\mu}\to x^{\prime\mu}(x)$ is
such that

$\displaystyle\eta_{\mu\nu}dx^{\prime\mu}dx^{\prime\nu}=e^{2\Omega(x)}\,\eta_{\mu\nu}dx^{\mu}dx^{\nu}\,.$
(55)

Here, unlike in (54), $\Omega$ is a restricted function, defined by the
parameters of 4D conformal transformations. Now, Weyl invariance implies that
the flat-spacetime action takes the same form (25) whether the coordinate
choice is $x^{\mu}$ or $x^{\prime\mu}$; that is, the factor $e^{2\Omega(x)}$
in (55) cancels out in the action. This is the statement of conformal
invariance of a theory living on flat spacetime. In fact, one does not need to
appeal to the full Weyl-invariance of the action (53) on a curved background.
The action (25) is already invariant under conformal transformations. Since
$\Delta_{4}=\Box_{0}^{2}$ for flat spacetime, $\Box_{0}^{2}$ has the
restricted Weyl covariance associated to the allowed choices of $\Omega$ in
(55); that is, it is conformally covariant. Explicitly, acting on a scalar
$\phi^{\prime}(x^{\prime})=\phi(x)$, we have under the conformal
transformation:

$\displaystyle(\partial_{x^{\prime}}^{2})^{2}=e^{-4\Omega}(\partial_{x}^{2})^{2}\,,$
(56)

where we denote $\partial_{x^{\prime}}^{2}=\frac{\partial}{\partial
x^{\prime\mu}}\frac{\partial}{\partial x^{\prime}_{\mu}}$ , etc; the
restriction on $\Omega$ allows the conformal factor to ‘factor out’ on the
right-hand side.

While the operator $\Delta_{4}$ is not strictly required to explain the
conformal invariance of the amplitudes, as we have just explained, it is
important when considering the extension to curved backgrounds, to be
discussed below. Furthermore, as mentioned previously, the complete SDYM
quantum-corrected action is not known explicitly for SU(N), despite partial
progress in Monteiro:2022nqt . It is likely that conformal invariance will be
a useful guide to obtaining the complete quantum correction.

It is worth it to mention that, in the gravity case (34), we have

$\displaystyle\int
d^{4}x\,\sqrt{|g|}\;C^{2}\,\frac{1}{\square^{2}}\,C^{2}\;=\;\int
d^{4}x\,\sqrt{|g|}\;C^{2}\,\frac{1}{\Delta_{4}}\,C^{2}$ (57)

under the self-duality constraint imposed by $S_{\text{SDG}}$, which implies
Ricci flatness. Indeed, as we describe in the Appendix A.2, the four-point
gravity amplitude (20) is almost conformal invariant (almost because of the
dimensionful coupling) due to the Weyl invariance of the right-hand side of
(57). The higher-point amplitudes do not share this property, as a result of
$S_{\text{SDG}}$ not being conformally invariant.

##  5 Non-trivial backgrounds

The study of field theory on curved backgrounds has received a new push in
recent years, inspired by novel methods, not least those originating in the
amplitudes literature. In the previous section, we mentioned already the
possibility of extending the non-local quantum-corrected action for SDYM to
non-trivial background spacetimes, in the form

$\displaystyle
S_{\text{q.c.SDYM}}=S_{\text{SDYM}}-\frac{a_{\mathfrak{g}}^{2}}{8}\int
d^{4}x\,\sqrt{|g|}\;\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})\,\frac{1}{\Delta_{4}}\,\operatorname{Tr}(F_{\mu\nu}F^{\mu\nu})+\ldots\,.$
(58)

This a natural conjecture for such an extension, because (i) Weyl invariance
is a natural upgrade for the conformal invariance of quantum SDYM exhibited by
its flat-spacetime scattering amplitudes, and (ii) the Paneitz operator
$\Delta_{4}$ makes an appearance already in the quantum correction associated
to the trace anomaly, also when considering a curved background. The action
above is subject to the restriction on the gauge group mentioned earlier, e.g.
SU(3). The ellipsis accounts for possible terms that have no counterpart in
the flat spacetime limit. Examples where this conjecture could be tested are
backgrounds with significant symmetry, such as plane waves and (anti-)de
Sitter; see e.g. Adamo:2017nia ; Adamo:2017sze ; Adamo:2018mpq ; Adamo:2019zmk
; Adamo:2020qru ; Armstrong:2020woi ; Gomez:2021ujt ; Armstrong:2022mfr ;
Lipstein:2023pih  for recent work along these lines. Another natural setting
is provided by SDG backgrounds Adamo:2020syc ; Adamo:2021bej ;
Bittleston:2023bzp ; Adamo:2023zeh ; Adamo:2023fbj .

In the case of SDG, one may ask how (34) would extend to SDG-$\Lambda$, i.e.
allowing for a cosmological constant. In SDG-$\Lambda$, the self-duality
condition applies to the Weyl tensor only, while the Ricci tensor satisfies
the Einstein-$\Lambda$ equations, $R_{\mu\nu}=\Lambda g_{\mu\nu}$. See
Lipstein:2023pih  for a recent discussion of the second Plebanski equation
with non-zero $\Lambda$. The $\Lambda$-self-duality condition is imposed by
the ‘tree-level part’ of the action, $S_{\text{SDG-}\Lambda}$. For the quantum
correction, a natural candidate is suggested by (57), namely

$\displaystyle
S_{\text{q.c.SDG-}\Lambda}=S_{\text{SDG-}\Lambda}-\frac{a_{\text{G}}^{2}}{8}\int
d^{4}x\,\sqrt{|g|}\;C^{2}\,\frac{1}{\Delta_{4}}\,C^{2}+\ldots\,.$ (59)

Recalling the definition of the Paneitz operator in (7), under
$R_{\mu\nu}=\Lambda g_{\mu\nu}$ we have

$\displaystyle\Delta_{4}=\square^{2}-\frac{2}{3}\Lambda\square.$ (60)

The light-cone version of $S_{\text{SDG-}\Lambda}$ obtained in
Lipstein:2023pih  provides a starting point for testing this proposal, by
performing explicitly a loop calculation, though this is not straightforward
on the non-trivial background.

Another question is how this story extends to self-dual conformal gravity (see
e.g. Dunajski:2013zta ). A natural expectation is that the quantum correction
in (59) is again present. Yet another class of related theories are the chiral
higher-spin theories of Metsaev:1991mt ; Metsaev:1991nb ; Ponomarev:2016lrm ;
Ponomarev:2017nrr , which generalise SDG and SDYM to include massless fields
of higher spins. For some theories in this class, there exist non-vanishing
one-loop amplitudes Skvortsov:2018jea ; Skvortsov:2020wtf ; Skvortsov:2020gpn
, and the same quantum correction occurs, involving only amplitudes with
external gravitons/gluons (no higher spins); the coefficient of the correction
is then easily adjusted Adamo:2022lah ; Monteiro:2022xwq . Ref.
Monteiro:2022xwq  includes a summary of the status of loop amplitudes in
chiral higher-spin theories.

##  6 Relation to UV divergence in two-loop pure gravity

In this section, we will move beyond the self-dual sectors to comment on an
intriguing feature of the ultraviolet behaviour of pure gravity. The well-
known two-loop divergence Goroff:1985sz ; Goroff:1985th ; vandeVen:1991gw  has
been investigated more recently using modern unitarity methods Bern:2015xsa ;
Bern:2017puu ; Abreu:2020lyk . It was argued in Bern:2015xsa  that the
coefficient of the dimensionally-regularised divergence $\frac{1}{D-4}$, which
can be cancelled with a (Riemann3) counterterm, is not directly physical; the
reason being that it depends on a possible coupling to 3-form fields, which
should have no physical consequence as they do not propagate in strictly four
spacetime dimensions. Instead, the physical effect of the divergence is the
renormalisation-scale dependence, signaled by a $\log\mu^{2}$ term in the two-
loop amplitude, which in the case of gravity is not tied to the
$\frac{1}{D-4}$ ultraviolet divergence. Noticing this fact, refs. Bern:2015xsa
; Bern:2017puu  computed the kinematic coefficient of the $\log\mu^{2}$ term
via unitarity cuts, for the four-point two-loop amplitude. They focused on the
all-plus helicity case, which is easier to tackle with the unitarity method
due to the fact that only two-particle cuts contribute to $\log\mu^{2}$. To
obtain these cuts, one glues a four-point all-plus one-loop amplitude to a
four-point tree-level (MHV) amplitude. From the reasoning of the cuts, it is
clear that the coefficient of $\log\mu^{2}$ does not depend on the number of
3-form fields. In addition, they observe that, beyond pure gravity (i.e. with
any number of bosons and fermions in the loop of the one-loop side of the
cut), the coefficient is proportional to
$N_{\text{bosons}}-N_{\text{fermions}}$. An obvious consequence is that the
$\log\mu^{2}$ term is absent for supergravity theories; indeed, supersymmetry
guarantees ultraviolet finiteness at two loops Grisaru:1976nn .

In section 3, we discussed the twistor-derived bosonic models of
Costello:2021bah ; Bittleston:2022nfr  for ‘improved’ SDYM and SDG. These
models achieve the cancellation of the one-loop amplitudes of ordinary SDYM
and SDG, in a very different manner from the cancellation achieved in the
supersymmetric extensions of SDYM and SDG (where the one-loop amplitudes are
also absent). A natural question, given the relevance of the all-plus one-loop
amplitudes to the ultraviolet behaviour in pure gravity, is whether there is
an analogous bosonic mechanism that ultimately cancels the two-loop
divergence, as an alternative to supersymmetry. Taking the perspective of the
non-local action (but one could also consider the local form with an auxiliary
scalar), our observation is that the action

$\displaystyle S_{\text{EH}}+\frac{a_{\text{G}}^{2}}{8}\int
d^{4}x\,\sqrt{|g|}\;C^{2}\,\frac{1}{\Delta_{4}}\,C^{2}+\;\ldots$ (61)

for pure gravity provides such a cancellation of the $\log\mu^{2}$ term. It
leads to the vanishing (by cancellation between the Einstein-Hilbert term and
the non-local term) of the one-loop all-plus amplitudes. Hence, via the
unitarity-cuts argument of Bern:2015xsa ; Bern:2017puu , it leads to a
vanishing coefficient for $\log\mu^{2}$ in the two-loop all-plus four-point
amplitude. That is, the one-loop non-local term is fixing the two-loop
behaviour. The ellipsis (…) above indicates the fact that the non-local term
written here is certainly not sufficient for eliminating the $\log\mu^{2}$
term in all two-loop four-point amplitudes; it is sufficient for the all-plus
two-loop case, but the one-minus case also requires attention (while the two-
plus/two-minus case has no $\log\mu^{2}$ term Abreu:2020lyk ). The non-local
term we have proposed in (61) cannot do the job for the one-minus two-loop
amplitude, because it clearly vanishes when evaluated on the one-minus one-
loop amplitude141414We can see this as follows. At linearised level for $\pm$
helicity and on-shell momentum $k$, we have
$C^{\pm}_{\mu\nu\rho\lambda}\sim\varepsilon^{\pm}_{[\mu}k_{\nu]}\varepsilon^{\pm}_{[\rho}k_{\lambda]}$.
Hence,
$C_{\mu\nu\rho\lambda}(1^{-})\,C^{\mu\nu\rho\lambda}(2^{+})\sim(\varepsilon^{-}_{1}{}_{[\mu}k_{1}{}_{\nu]}\varepsilon^{+}_{2}{}^{[\mu}k_{2}{}^{\nu]})^{2}$,
which vanishes because $\varepsilon^{+}_{2}{}^{[\mu}k_{2}{}^{\nu]}$ and
$\varepsilon^{-}_{1}{}^{[\mu}k_{1}{}^{\nu]}$ are self-dual and anti-self-dual
tensors, respectively. that would appear in the 2-particle cuts. In addition,
for the $\log\mu^{2}$ term of the one-minus two-loop amplitude, there are also
3-particle cuts, involving only tree amplitudes. We refer the reader to
Bern:2017puu  for details. One would also have to consider whether a relation
between the two-loop $\log\mu^{2}$ term and the one-loop all-plus and one-
minus amplitudes persists at higher points, and of course at higher loops.

This is all a long shot, but we see no obstacle of principle that prevents us
from following this procedure for at least the four-point two-loop gravity
amplitudes, though we have not determined the precise form of the required
terms for the ellipsis in (61). We have no further insight into why such a
non-local term should be present in the action. We merely note its potential
consequence.

##  7 Conclusion

The main goal of this paper was to connect the non-local quantum-corrected
actions of self-dual Yang-Mills theory and gravity, to the non-local actions
that have long appeared in the context of anomalies. With the benefit of
hindsight, the quantum corrections to SDYM and SDG Costello:2021bah ;
Bittleston:2022nfr ; Monteiro:2022nqt  could have been determined by simple
analogy with the trace anomaly: by trying a natural non-local functional, and
then fixing the coefficient by matching to an amplitude. In addition, we also
discussed how the quantum corrections in the self-dual theories may be thought
of as an anomaly of the chiral U(1) symmetry normally associated to the
‘electric-magnetic’ duality.

One clue from the trace anomaly literature is the appearance of the Paneitz
fourth-order differential operator $\Delta_{4}$, which allows us to (i)
understand the conformal properties of the loop-level scattering amplitudes in
the self-dual theories, and (ii) present natural conjectures for extending the
quantum-corrected actions to non-trivial spacetime backgrounds. This aspect of
our work is part of ongoing efforts to bring modern amplitudes-related
constructions to the study of quantum field theories on curved backgrounds.

Regarding Yang-Mills theory, we proposed a Weyl-invariant formulation of the
quantum-corrected self-dual sector on curved spacetimes. This is, of course, a
part of the full Yang-Mills theory. It would be interesting to explore whether
the rational parts of loop amplitudes beyond the self-dual sector can be
written in terms of analogous non-local quantum corrections. Such rational
terms can be efficiently computed using unitarity methods, at least at one-
loop Badger:2008cm . We also remarked on the relationship between the anomaly
of the self-dual sector in pure Yang-Mills theory, and the one-loop MHV sector
of ${\mathcal{N}}=4$ super-Yang-Mills theory, pointed out long ago Bern:1996ja
. It would be nice to understand the full extent of this relationship, e.g.
whether it extends to non-trivial backgrounds.

We briefly discussed how the anomaly of the self-dual sector in gravity
suggests a one-loop effective mechanism for altering the pathological
ultraviolet behaviour of pure gravity, at least at two loops — a mechanism
that is an alternative to supersymmetry. This suggestion relies on an apparent
connection between rational one-loop amplitudes and the divergent behaviour of
two-loop amplitudes. Our observation builds on modern studies of the two-loop
divergence in Bern:2015xsa ; Bern:2017puu ; Abreu:2020lyk , and also on what
appears to be a close connection between anomalies and ultraviolet divergences
in pure (super)gravity more generally Carrasco:2013ypa ; Bern:2013uka ;
Bern:2017rjw ; Bern:2019isl ; Carrasco:2022lbm . It would be important to
understand better this intriguing connection.

Partly based on this observation, and partly based on the role of a fourth-
order differential operator, it is natural to ask whether there is an
unexplored connection to the vast topic of higher-derivative gravity.
Quadratic gravity, recently reviewed in Donoghue:2021cza ; Salvio:2018crh ,
has been proposed as a renormalisable theory of quantum gravity, despite the
fact that it seems at odds with common expectations about quantum field
theory, regarding e.g. causality and the appearance of $1/p^{4}$ propagators,
in apparent violation of unitarity. Such propagators arise explicitly from the
non-local quantum corrections we have discussed here. Nevertheless, the loop
amplitudes in the self-dual sector turn out to have only simple poles in the
Mandelstam variables, instead of the quadratic poles naively expected from
$1/p^{4}$, due to kinematic identities. While the self-dual theory is non-
unitary on its own (as it corresponds to a complex truncation of the
Hamiltonian of the full theory), it is also a sector of a unitary theory —
certainly in the Yang-Mills case. The quantum correction in self-dual gravity,
with its $1/p^{4}$ propagator, should also be a part of the quantum
corrections in general relativity, seen as an effective theory. This suggests
that such propagators may not always represent a terminal pathology. Finally,
it is worth it to mention that conformal gravity has been argued to be a limit
of quadratic gravity Salvio:2017qkx . It is tempting to ask whether there is a
common ground between these topics and the aforementioned relation of the
ultraviolet behaviour of pure gravity to the anomaly of the self-dual sector,
the latter being reminiscent of the trace anomaly.

## Acknowledgements

We thank Tim Adamo, Chandramouli Chowdhury, Anton Ilderton, Henrik Johansson,
Arthur Lipstein, Lionel Mason, Silvia Nagy and Kajal Singh for related
discussions. RM is supported by the Royal Society via a University Research
Fellowship. GD is supported by the Royal Society via a studentship grant. The
work of SW is funded by the Knut and Alice Wallenberg Foundation grant KAW
2021.0170 and the Olle Engkvists Stiftelse grant 2180108. We also received
support from the Science and Technology Facilities Council (STFC) Consolidated
Grants ST/T000686/1 “Amplitudes, Strings & Duality” and ST/X00063X/1
“Amplitudes, Strings & Duality”. No new data were generated or analysed during
this study.

##  Appendix A Conformal properties of the scattering amplitudes

In this appendix, we first recall the action of the generators of the
conformal algebra on scattering amplitudes; then review the result in
Henn:2019mvc  regarding the conformal invariance of the all-plus one-loop
amplitudes in Yang-Mills theory. We also address the analogous question in
gravity.

The generators of the four-dimensional conformal group for an $n$-particle
amplitude, expressed in spinor-helicity language, are as follows Witten:2003nn
: the Lorentz generators,

$\displaystyle J_{\alpha\beta}$
$\displaystyle=\frac{i}{2}\sum_{i=1}^{n}\left(\lambda_{i\alpha}\frac{\partial}{\partial\lambda^{\beta}_{i}}+\lambda_{i\beta}\frac{\partial}{\partial\lambda^{\alpha}_{i}}\right),$
(62) $\displaystyle\tilde{J}_{\dot{\alpha}\dot{\beta}}$
$\displaystyle=\frac{i}{2}\sum_{i=1}^{n}\left(\tilde{\lambda}_{i\dot{\alpha}}\frac{\partial}{\partial\tilde{\lambda}^{\dot{\beta}}_{i}}+\tilde{\lambda}_{i\dot{\beta}}\frac{\partial}{\partial\tilde{\lambda}^{\dot{\alpha}}_{i}}\right),$

the momentum generator,

$P^{\alpha\dot{\alpha}}=\sum_{i=1}^{n}\lambda^{\alpha}_{i}\tilde{\lambda}^{\dot{\alpha}}_{i},$
(63)

the dilatation generator,

$D=i\sum_{i=1}^{n}\left(\frac{1}{2}\lambda^{\alpha}_{i}\frac{\partial}{\partial\lambda^{\alpha}_{i}}+\frac{1}{2}\tilde{\lambda}^{\dot{\alpha}}_{i}\frac{\partial}{\partial\tilde{\lambda}^{\dot{\alpha}}_{i}}+1\right),$
(64)

and the special conformal generator,

$K_{\alpha\dot{\alpha}}=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial\lambda^{\alpha}_{i}\partial\tilde{\lambda}^{\dot{\alpha}}_{i}}.$
(65)

Consider an amplitude of the form

$\mathcal{A}(\\{p_{i}\\})=\delta^{4}\left(\sum_{i=1}^{n}p_{i}\right)A(\\{p_{i}\\}).$
(66)

The Poincaré generators act on this expression as

$\displaystyle J_{\alpha\beta}\mathcal{A}$
$\displaystyle=\delta^{4}\left(P\right)J_{\alpha\beta}A,$ (67)
$\displaystyle\tilde{J}_{\dot{\alpha}\dot{\beta}}\mathcal{A}$
$\displaystyle=\delta^{4}\left(P\right)\tilde{J}_{\alpha\beta}A,$
$\displaystyle P^{\alpha\dot{\alpha}}\mathcal{A}$ $\displaystyle=0.$

Thus, Poincaré invariance follows from the Lorentz invariance of the ‘delta-
stripped’ amplitude. This, in turn, follows from the fact the stripped
amplitude can be written as a function of spinor brackets
$\lambda_{i\alpha}\lambda_{j}{}^{\alpha}$ and
$\tilde{\lambda}_{i\dot{\alpha}}\tilde{\lambda}_{j}{}^{\dot{\alpha}}$. The
dilatation operator acts on the amplitude as

$D\mathcal{A}=\delta^{4}(P)(-4iA+DA),$ (68)

where the first term arises from differentiating the delta function as a
distribution. Dilatation invariance requires $DA=4iA$, which holds for a
dimensionless coupling. This is because $-iDA=(d_{A}-d_{c}+n)A$, where $d_{A}$
is the mass dimension of $A$, which is $4-n$ in four dimensions, $d_{c}$ is
the mass dimension of the coupling constant factor in $A$, and $n$ arises from
the $+1$ in the summand defining the generator $D$; hence, $-iDA=(4-d_{c})A$
and

$\displaystyle D\mathcal{A}=-i\,d_{c}\,\mathcal{A}.$ (69)

That is, dilatation invariance corresponds to a dimensionless coupling.
Imposing Lorentz invariance,
$J_{\alpha\beta}A=0=\tilde{J}_{\dot{\alpha}\dot{\beta}}A$, the special
conformal generator acts on the amplitude as

$\displaystyle K_{\alpha\dot{\alpha}}\mathcal{A}$
$\displaystyle=\delta^{4}(P)K_{\alpha\dot{\alpha}}A+\frac{\partial\delta^{4}(P)}{\partial
P^{\alpha\dot{\alpha}}}(-4A-iDA)$
$\displaystyle=\delta^{4}(P)K_{\alpha\dot{\alpha}}A-d_{c}\,\frac{\partial\delta^{4}(P)}{\partial
P^{\alpha\dot{\alpha}}}\,A.$ (70)

We see that special conformal invariance requires that
$K_{\alpha\dot{\alpha}}A=0$, in addition to dilatation invariance.

###  A.1 All-plus one-loop Yang-Mills amplitudes

Ref. Henn:2019mvc  showed that the all-plus one-loop Yang-Mills amplitudes,
first obtained in closed form in Bern:1993qk , and which lie in the self-dual
sector, exhibit conformal invariance. That is,

$\displaystyle
P^{\alpha\dot{\alpha}}\mathcal{A}^{(1)}_{\text{YM}}=0,\;J_{\alpha\beta}\mathcal{A}^{(1)}_{\text{YM}}=0,\;\tilde{J}_{\dot{\alpha}\dot{\beta}}\mathcal{A}^{(1)}_{\text{YM}}=0,\;D\mathcal{A}^{(1)}_{\text{YM}}=0,\;K_{\alpha\dot{\alpha}}\mathcal{A}^{(1)}_{\text{YM}}=0.$
(71)

Only the special conformal invariance is non-trivial. It is most easily seen
in the very simple four-point case, where

$\mathcal{A}^{(1)}_{\text{YM}}(1^{+},2^{+},3^{+},4^{+})=\delta^{4}(p_{1}+p_{2}+p_{3}+p_{4})A^{(1)}_{\text{YM}}(1^{+},2^{+},3^{+},4^{+}),$
(72)

where

$A^{(1)}_{\text{YM}}(1^{+},2^{+},3^{+},4^{+})=-\frac{i\,g^{4}}{48\pi^{2}}\;\frac{[12][34]}{\langle
12\rangle\langle 34\rangle}$ (73)

for the colour-ordered amplitude.

###  A.2 All-plus one-loop gravity amplitudes

The all-plus one-loop gravity amplitudes, which lie in the self-dual sector,
were first obtained in closed form in Bern:1998xc . The four-point case is

$\mathcal{A}^{(1)}_{\text{G}}(1^{+},2^{+},3^{+},4^{+})=\delta^{4}(p_{1}+p_{2}+p_{3}+p_{4})A^{(1)}_{\text{G}}(1^{+},2^{+},3^{+},4^{+}),$
(74)

with

$A^{(1)}_{\text{G}}(1^{+},2^{+},3^{+},4^{+})=-\frac{i}{120(4\pi)^{2}}\left(\frac{\kappa}{2}\right)^{4}(s_{12}^{2}+s_{23}^{2}+s_{31}^{2})\left(\frac{s_{12}s_{23}}{\langle
12\rangle\langle 23\rangle\langle 34\rangle\langle 41\rangle}\right)^{2},$
(75)

where we made the coupling explicit. The Poincaré invariance of the amplitudes
is clear,

$\displaystyle
P^{\alpha\dot{\alpha}}\mathcal{A}_{\text{G}}=0,\;J_{\alpha\beta}\mathcal{A}_{\text{G}}=0,\;\tilde{J}_{\dot{\alpha}\dot{\beta}}\mathcal{A}_{\text{G}}=0.$
(76)

The $n$-point amplitude goes like $\kappa^{n}$, so that the dimensionful
coupling breaks dilatation symmetry,

$\displaystyle D\mathcal{A}_{\text{G}}=i\,n\,\mathcal{A}_{\text{G}}\,.$ (77)

We then have also

$K_{\alpha\dot{\alpha}}\mathcal{A}_{\text{G}}=\delta^{4}(P)K_{\alpha\dot{\alpha}}A_{\text{G}}+n\,\frac{\partial\delta^{4}(P)}{\partial
P^{\alpha\dot{\alpha}}}\,A_{\text{G}}.$ (78)

It turns out that the four-point amplitude (75) is special because

$\displaystyle
K_{\alpha\dot{\alpha}}\,A^{(1)}_{\text{G}}(1^{+},2^{+},3^{+},4^{+})=0,$ (79)

a property that is not shared by higher-point amplitudes. Therefore, it is
almost conformal invariant, with full invariance failing due to the
dimensionful coupling. In terms of the discussion in the main body of the
paper, we interpret this as being due to the following fact: in the action
(34), while the part $S_{\text{SDG}}$ is not conformally invariant, the
quantum correction is; in fact, the latter can be written in a Weyl-invariant
manner as in (57). The quantum correction provides the one-loop effective
vertices, whose explicit form in light-cone gauge we wrote in (44), taken from
Monteiro:2022nqt . It is helpful to think of the Feynman rules resulting from
the light-cone action. The four-point amplitude uses only the four-point one-
loop effective vertex, with legs put on-shell, and therefore it is not
surprising that there is a remnant of conformal symmetry in this particular
amplitude. The five-point amplitude uses both the five-point one-loop
effective vertex and, in a different set of diagrams, the four-point one-loop
effective vertex with a three-point ‘tree vertex’ attached, with the latter
coming from the $S_{\text{SDG}}$ part of the action. We interpret the failure
of the almost-conformal invariance for $n>4$ (that is,
$K_{\alpha\dot{\alpha}}\,A^{(1)}_{\text{G}}\neq 0$) as due to the appearance
of the three-point vertex from the conformally-non-invariant piece
$S_{\text{SDG}}$.

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