aainstitutetext: LAPTh, Université Savoie Mont Blanc, CNRS, B.P. 110, F-74941
Annecy-le-Vieux, Francebbinstitutetext: Max-Planck-Institut für Physik,
Werner-Heisenberg-Institut, D-80805 München, Germanyccinstitutetext:
Dipartimento di Fisica and Arnold-Regge Center, Università di Torino, and
INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy

# Anomalous Ward identities for on-shell amplitudes at the conformal fixed
point

Dmitry Chicherin b Johannes Henn c Simone Zoia<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Conformal symmetry underlies many massless quantum field theories, but little
is known about the consequences of this powerful symmetry for on-shell
scattering amplitudes. Working in a dimensionally-regularised $\phi^{3}$ model
at the conformal fixed point, we show that the on-shell renormalised
amplitudes satisfy anomalous conformal Ward identities. Each external on-shell
state contributes two terms to the anomaly. The first term is proportional to
the elementary field anomalous dimension, and thus involves only lower-loop
information. We show that the second term can be given as the convolution of a
universal collinear function and lower-order amplitudes. The computation of
the conformal anomaly is therefore simpler than that of the amplitude at the
same perturbative order, which gives our anomalous conformal Ward identities a
strong predictive power in perturbation theory. Finally, we show that our
result is also of practical importance for dimensionally-regularised
amplitudes away from the conformal fixed point.

††preprint: LAPTH-036/22, MPP-2022-83

##  1 Introduction

Four-dimensional quantum field theories such as Yang-Mills, massless quantum
electrodynamics or quantum chromodynamics, Yukawa and $\phi^{4}$ theory have
conformal symmetry at the level of the Lagrangian. To date, the consequences
of this powerful symmetry have been widely studied and exploited for off-shell
correlation functions, and mostly in position space Todorov:1978rf ;
DiFrancesco:1997nk ; Fradkin:1997df ; Braun:2003rp . Motivated by problems in
cosmology, several groups have studied conformal symmetry for off-shell
momentum-space correlation functions Coriano:2013jba ; Bzowski:2013sza ;
Bzowski:2019kwd ; Bzowski:2020kfw . However, surprisingly little is known
about the implications of conformal symmetry for on-shell scattering
amplitudes. The latter are fundamental ingredients for theoretical predictions
at particle colliders. The goal of the present paper is to contribute to our
understanding of the implications of this powerful symmetry for such objects.

At tree level, scattering amplitudes are annihilated by the conformal
generators, as was discussed in Witten:2003nn  for gluon scattering
amplitudes. The resulting differential equations are second-order in momentum
space. Very interestingly, the tree-level gluon amplitudes possess a bigger
symmetry group when they are embedded into the supermultiplet of
${\mathcal{N}}=4$ super Yang-Mills. It was shown that the conformal symmetry,
together with enhanced symmetries of ${\mathcal{N}}=4$ super Yang-Mills, is
enough to uniquely fix the form of the tree-level scattering amplitudes
Korchemsky:2009hm ; Bargheer:2009qu .

At loop level, to our knowledge, conformal symmetry predictions are mostly
restricted to certain finite objects in scattering amplitudes (see however
Bargheer:2009qu ; Beisert:2010gn  for interesting work in the context of
maximally supersymmetric Yang-Mills theory). Firstly, it is known that certain
coefficients of transcendental functions at loop level, such as for example
the leading singularities Arkani-Hamed:2010pyv , are conformally invariant.
The reason is that the latter are obtained via generalised unitarity
Bern:1994zx  from tree-level amplitudes in a way that preserves conformal
symmetry. For example, conformally-invariant expressions have been found to
play an important role in all-plus scattering amplitudes Badger:2019djh ;
Henn:2019mvc ; Chicherin:2022bov , but they also appear in more general
helicity configurations. Secondly, it was shown that conformal symmetry
constrains infrared- and ultraviolet-finite loop integrals Chicherin:2017bxc .
The authors found anomalous conformal Ward identities for an $L$-loop
integral, where the anomaly term is expressed as an integral over the
collinear region of a simpler, $(L-1)$-loop integral. This insight can
therefore be useful for practical calculations, as was demonstrated in
references Zoia:2018sin ; Chicherin:2018ubl ; Chicherin:2018rpz .

A key difficulty in generalizing these results to full scattering amplitudes
(in classically conformally-invariant theories) are divergences, which naïvely
break the symmetry. In this paper we wish to generalise the results of
Chicherin:2017bxc  to ultraviolet-divergent (but infrared-finite) scattering
amplitudes, which makes dimensional regularisation necessary. We study the
amplitudes at a conformal fixed point where the $\beta$ function vanishes
Wilson:1971dc .

Although our ultimate aim is to apply conformal methods to Yang-Mills
scattering amplitudes at loop level, we find it convenient for this first
study to use a $\phi^{3}$ toy model in $d=6-2\epsilon$ dimensions, where
$\epsilon$ is the dimensional regularisation parameter Gracey:2020tkk . For
our purposes we find it insightful to have a further parameter, and therefore
we take the scalar field to be matrix-valued, in the fundamental
representation of $su(\textsc{n})$.

We show that on-shell renormalised amplitudes satisfy anomalous conformal Ward
identities (CWIs) at the conformal fixed point. The anomaly has a local
nature, with each external on-shell leg giving a separate additive
contribution made of two terms. The first term is proportional to the
anomalous dimension of the elementary field. The second term is given by the
convolution of a universal collinear function with lower-loop amplitudes. The
anomaly for an $\ell$-loop amplitude is therefore determined by $(\ell-1)$
information. Because of this, our result is not only interesting from a
theoretical point of view, but also useful for computing amplitudes
perturbatively. The amplitude away from the fixed point can then be recovered
from the conformal result by supplying lower-loop information, which makes our
anomalous CWIs a new valuable tool for computing amplitudes in perturbation
theory.

The starting point of our derivation are the conformal properties of the off-
shell correlators, which we review in section 2. After setting the notation in
section 2.1, in section 2.2 we recall that the UV-renormalised correlators
satisfy conformal Ward identities with an anomaly term which is proportional
to the $\beta$ function. At the conformal fixed point, thus, the renormalised
correlators are conformally invariant. From a perturbative point of view, this
is an all-order statement which requires a very intricate conspiracy among
infinitely many Feynman diagrams. We elaborate on this in section 2.3. Section
3 is devoted to the on-shell amplitudes. We define them in section 3.1 by
applying the Lehmann–Symanzik–Zimmermann (LSZ) reduction procedure to the off-
shell correlators. The non-commutativity of the generator of conformal boosts
with the on-shell limit in the LSZ reduction generates an anomaly in the CWIs
for the on-shell amplitudes. We pinpoint the origin of the anomaly to certain
terms of the asymptotic expansion of the amputated correlators in the on-shell
limit. We discuss this thoroughly in section 3.2 for the three-point
correlator with one leg put on-shell, and then generalise to the $n$-particle
case in section 3.3. The anomaly receives contributions of two types. One is
proportional to the elementary field anomalous dimensions, and thus needs to
be computed at a lower loop order. In section 3.4 we show that the second type
of contribution to the conformal anomaly can be expressed as the convolution
of a universal function with lower-loop amplitudes, so that the whole anomaly
is entirely determined by lower-loop information. In section 3.5 we present
the calculation of the conformal anomaly in a few examples. We conclude in
section 4 with a discussion of our results, and an outlook on the future
studies.

We provide useful technical details in several appendices. In appendix A we
review the proof of the Ward identities for the renormalised correlation
functions. In appendix B we show how the knowledge of a correlator/amplitude
at the conformal fixed point constrains it away from the fixed point. In
appendix C we prove that the generator of conformal boosts commutes with the
momentum-conservation delta function. Appendix D is devoted to an instructive
comparison between the perturbative results and the expression fixed by
conformal symmetry for the three-point correlator. The latter is well known
and very simple in position space. Its Fourier transform to momentum space is
given by a one-loop triangle Feynman integral with non-integer powers of the
propagators. We show in appendix E how to efficiently use the differential
equations method in this novel setting.

##  2 Correlation functions at the conformal fixed point

In this section we study the conformal properties of off-shell correlators in
momentum space as a preparatory step before considering on-shell amplitudes.
We define the $su(\textsc{n})$-matrix $\phi^{3}$ theory and recall its
conformal properties at tree level in section 2.1. In section 2.2 we review
that the renormalised correlators satisfy CWIs with an anomalous term
proportional to the $\beta$ function. At the conformal fixed point the $\beta$
function vanishes, and the correlators are annihilated by the conformal
generators. We discuss this explicitly in section 2.3 for the three-point
correlator. The latter gives a spectacular example of the constraining power
of conformal symmetry, and is the perfect toy example for the derivation of
our anomalous CWIs for on-shell amplitudes in section 3.

###  2.1 Conformal invariance at the classical level

In this section we review the conformal invariance of the
$su(\textsc{n})$-matrix $\phi^{3}$ theory at the tree level. The bare action
in $d=6$ dimensions is given by

$\displaystyle
S_{0}=\int\mathrm{d}^{d}x\left[\frac{1}{2}(\partial\phi_{0}^{a})^{2}+\frac{g_{0}}{6}d^{abc}\phi_{0}^{a}\phi_{0}^{b}\phi_{0}^{c}\right]\,,$
(1)

where $\phi_{0}=\sum_{a}\phi_{0}^{a}t^{a}$, the matrices $t^{a}$ are the
$su(\textsc{n})$ generators in the fundamental representation normalised as
$\text{tr}\,t^{a}t^{b}=\delta^{ab}/2$,
$d^{abc}=2\text{tr}\,t^{a}\\{t^{b},t^{c}\\}$ with the curly brackets denoting
anti-commutation, i.e. $\\{A,B\\}:=AB+BA$, and
$a,b,c=1,\ldots,\textsc{n}^{2}-1$. The $0$ subscript denotes bare quantities.

The action in eq. (1) is invariant under conformal transformations. On top of
the usual Poincaré group, in fact, it is invariant also under the following
infinitesimal variation of the fields,

$\displaystyle\phi_{0}^{a}(x)\to\phi_{0}^{a}(x)+\textup{i}\omega
D_{\Delta_{\phi}}\phi_{0}^{a}(x)+\textup{i}\varepsilon_{\mu}K^{\mu}_{\Delta_{\phi}}\phi_{0}^{a}(x)\,,$
(2)

where $\omega$ and $\varepsilon^{\mu}$ are infinitesimal parameters. Here,
$D_{\Delta}$ is the generator of infinitesimal _dilatations_ ,

$\displaystyle
D_{\Delta}=-\textup{i}\left(x^{\mu}\partial_{\mu}+\Delta\right)\,,$ (3)

which are a rescaling of the spacetime coordinates,

$\displaystyle x^{\mu}\to e^{\omega}\,x^{\mu}\,.$ (4)

$\Delta$ is the conformal weight (or scaling dimension) of the field
$\phi_{0}$, and its canonical value — namely the value for which the action in
eq. (1) is invariant with $d=6$ — is $\Delta_{\phi}=2$. $K^{\mu}_{\Delta}$ in
eq. (2) is instead the generator of _conformal boosts_ (or special conformal
transformations),

$\displaystyle
K^{\mu}_{\Delta}=\textup{i}\left(x^{2}\partial^{\mu}-2x^{\mu}x^{\nu}\partial_{\nu}-2\Delta
x^{\mu}\right)\,,$ (5)

which at the level of the spacetime coordinates corresponds to

$\displaystyle
x^{\mu}\to\frac{x^{\mu}-\varepsilon^{\mu}x^{2}}{1-2\varepsilon\cdot
x+\varepsilon^{2}x^{2}}\,.$ (6)

The latter can be more intuitively viewed as the composition of an inversion
($x^{\mu}\to x^{\mu}/x^{2}$), followed by a translation $x^{\mu}\to
x^{\mu}-\varepsilon^{\mu}$, and finally another inversion.

The conformal invariance of the bare action implies that the correlation
functions are conformally invariant at tree level, namely that the tree-level
correlators of $n$ fields are annihilated by the generators of conformal
symmetry,111We consider only correlators of elementary fields of the theory
with action (1), which are (renormalised) $n$-point Green functions. We always
tacitly imply the time ordering in the notation $\langle{\ldots}\rangle$.

$\displaystyle
K^{\mu}_{\Delta_{\phi}}\,\langle\phi_{0}(x_{1})\ldots\phi_{0}(x_{n})\rangle\biggl{|}_{\text{tree}}=0\,,$
(7)

and similarly for the other generators.

The representation of the generators acting on a function of the scalar field
$\phi_{0}$ at $n$ points $x_{i}$ in position space is obtained by summing over
all points the single-point generators in eqs. (3) and (5),

$\displaystyle
D_{\Delta}=-\textup{i}\sum_{j=1}^{n}\left(x_{j}^{\mu}\frac{\partial}{\partial
x_{j}^{\mu}}+\Delta\right)\,,$ (8) $\displaystyle
K^{\mu}_{\Delta}=\textup{i}\sum_{j=1}^{n}\left(x_{j}^{2}\frac{\partial}{\partial
x_{j\,\mu}}-2x_{j}^{\mu}x_{j}^{\nu}\frac{\partial}{\partial
x_{j}^{\nu}}-2\Delta\,x_{j}^{\mu}\right)\,.$ (9)

For example, it is straightforward to show that the two- and three-point tree-
level correlators are annihilated by the generators of dilatations and
conformal boosts,

$\displaystyle\langle\phi^{a}_{0}(x_{1})\phi^{b}_{0}(x_{2})\rangle\biggl{|}_{\text{tree}}=\frac{1}{4\pi^{3}}\frac{\delta^{ab}}{\left(x_{12}^{2}\right)^{2}}\,,$
(10)
$\displaystyle\langle\phi^{a}_{0}(x_{1})\phi^{b}_{0}(x_{2})\phi^{c}_{0}(x_{3})\rangle\biggl{|}_{\text{tree}}=-\frac{g}{(2\pi)^{6}}\frac{d^{abc}}{x_{12}^{2}x_{23}^{2}x_{31}^{2}}\,,$
(11)

where $x_{ij}=x_{i}-x_{j}$.

We are interested in the implications of conformal symmetry for scattering
amplitudes, which are defined in on-shell momentum space. We work in Minkowski
spacetime to describe light-like momenta. The (off-shell) momentum-space
realisation of the conformal generators can be obtained by Fourier
transforming the position-space one, obtaining222To avoid the proliferation of
symbols, we denote with the same symbol the conformal boost generator in both
momentum and position spaces.

$\displaystyle
D_{\Delta}=\textup{i}\sum_{j=1}^{n}\left(p_{j}^{\mu}\frac{\partial}{\partial
p_{j}^{\mu}}+d-\Delta\right)\,,$ (12)

and

$\displaystyle
K^{\mu}_{\Delta}=\sum_{j=1}^{n}\left(-p_{j}^{\mu}\frac{\partial}{\partial
p_{j\,\nu}}\frac{\partial}{\partial
p_{j}^{\nu}}+2p_{j}^{\nu}\frac{\partial}{\partial
p_{j}^{\nu}}\frac{\partial}{\partial
p_{j\,\mu}}+2(d-\Delta)\frac{\partial}{\partial p_{j\,\mu}}\right)\,.$ (13)

While in position space all conformal generators are first-order differential
operators, the conformal boost generator is realised in momentum space by a
second-order operator. This is one of the reasons why studying conformal
symmetry in momentum space is more challenging than in position space. Indeed,
in section D we will review how conformal symmetry allows us to draw sweeping,
all-order conclusions about the correlation functions in position space,
whereas the corresponding results in momentum space are substantially more
involved beyond the tree level.

The Fourier transform of the tree-level two- and three-point functions (10)
and (11) gives the following momentum-space correlators,

$\displaystyle\langle\phi^{a}_{0}(p_{1})\phi^{b}_{0}(p_{2})\rangle\biggl{|}_{\text{tree}}=\frac{\textup{i}\delta^{ab}}{p_{1}^{2}}(2\pi)^{6}\delta^{(6)}(p_{1}+p_{2})\,,$
(14)
$\displaystyle\langle\phi^{a}_{0}(p_{1})\phi^{b}_{0}(p_{2})\phi^{c}_{0}(p_{3})\rangle\biggl{|}_{\text{tree}}=\frac{g\,d^{abc}}{p_{1}^{2}p_{2}^{2}p_{3}^{2}}(2\pi)^{6}\delta^{(6)}(p_{1}+p_{2}+p_{3})\,.$
(15)

One can show that the conformal generators (12) and (13) annihilate these
expressions.

The dilatation and Lorentz symmetry generators are first-order in momentum
space, and hence their constraints on the correlators are easy to take into
account. For this reason we will focus in this paper on the conformal boost
generator.

In addition to this technical complication, two conceptual issues need to be
addressed: the appearance of divergences in the correlators beyond the tree
level, and the definition of an on-shell scattering amplitude starting from
the off-shell correlator. We tackle the latter problem in section 3, and
devote the next section to the former.

###  2.2 Conformal invariance for renormalised correlation functions

Beyond tree level the theory is ultraviolet divergent and needs to be
renormalised. This obscures the conformal properties of the correlators. In
this section we review that the renormalised correlators satisfy CWIs with a
different conformal weight and an anomalous term proportional to the $\beta$
function.

We work in dimensional regularisation, namely we analytically continue the
spacetime dimension to $d=6-2\epsilon$ with $\epsilon>0$. The canonical
conformal weight of $\phi$ in $d$ dimensions is

$\displaystyle\Delta_{\phi}=\frac{d}{2}-1\,.$ (16)

The theory is multiplicatively renormalisable. The renormalised field $\phi$
and coupling $g$ are defined as

$\displaystyle\phi_{0}=Z_{1}^{\frac{1}{2}}\phi\,,\qquad\qquad
g_{0}=\mu^{\epsilon}Z_{g}\,g\,,$ (17)

introducing the renormalisation scale $\mu$. The renormalisation factors
$Z_{1},Z_{g}$ are calculated in perturbation theory as series in the coupling

$\displaystyle u=\frac{g^{2}}{(4\pi)^{3}}\,.$ (18)

Their two-loop expressions, given in ref. Braun:2013tva , are reproduced in
Appendix A. The $\beta$ function and the anomalous dimension $\gamma$ of the
elementary field are

$\displaystyle\beta=\frac{du}{d\log\mu}=-2\epsilon
u-\frac{\textsc{n}^{2}-20}{2\textsc{n}}u^{2}-\frac{5360-496\textsc{n}^{2}+5\textsc{n}^{4}}{72\textsc{n}^{2}}u^{3}+\mathcal{O}(u^{4})\,,$
(19) $\displaystyle\gamma=\frac{1}{2}\frac{d\log
Z_{1}}{d\log\mu}=u\frac{\textsc{n}^{2}-4}{12\textsc{n}}\left(1+u\frac{\textsc{n}^{2}-100}{36\textsc{n}}\right)+\mathcal{O}(u^{3})\,.$
(20)

The four-loop approximation of $\beta$ and $\gamma$ can be found in
Gracey:2015tta .

Renormalisation introduces the scale running of the coupling. As a result,
conformal symmetry is obscured, but not entirely lost. The renormalised
correlators in fact satisfy CWIs with the conformal dimension shifted by the
anomalous dimension $\gamma$,

$\displaystyle\Delta_{\gamma}=\Delta_{\phi}+\gamma(u)\,.$ (21)

The dilatation generator acts on the renormalised correlation function as
follows,

$\displaystyle
D_{\Delta_{\gamma}}\langle\phi(x_{1})\ldots\phi(x_{n})\rangle=-\frac{\beta(u)}{2u}\int\mathrm{d}^{d}x\,\left\langle\phi(x_{1})\ldots\phi(x_{n})\,{\cal
O}(x)\right\rangle\,,$ (22)

where ${\cal
O}(x):=g\mu^{\epsilon}\left[d^{abc}\phi_{a}\phi_{b}\phi_{c}\right]_{R}(x)/6$
is the renormalised local operator which is the UV-finite counterpart of the
theory interaction, and we recall that $\phi$ denotes the renormalised field.
The dilatation Ward identity is equivalent to the renormalisation group
Callan–Symanzik equation for the correlation function, which describes the
evolution of the correlator with the scale $\mu$. The conformal boost Ward
identity has an analogous form,

$\displaystyle
K^{\mu}_{\Delta_{\gamma}}\langle\phi(x_{1})\ldots\phi(x_{n})\rangle=-\frac{\beta(u)}{u}\int\mathrm{d}^{d}x\,x^{\mu}\left\langle\phi(x_{1})\ldots\phi(x_{n})\,{\cal
O}(x)\right\rangle\,.$ (23)

The CWIs for multi-point renormalised correlation functions are well-known and
date back to the early days of the conformal symmetry, see e.g. Parisi:1972zy
. They were also studied in non-abelian gauge theories, see Braun:2003rp  for
a review. For the convenience of the readers, we outline the derivation of the
CWIs (21) and (23) in Appendix A.

The invariance of the action (1) under conformal transformations generates
currents which are conserved at classical level. Beyond the classical
approximation dilatations and conformal boosts are not exact symmetries, and
the conservation of the corresponding currents becomes anomalous. The right-
hand sides of eqs. (22) and (23) provide the quantum anomalies of the
dilatation and conformal boost currents. In section 3 we consider conformal
anomalies of the on-shell amplitudes, which are not to be confused with the
quantum anomalies of the currents originating from the running of the
coupling.

The fact that the right-hand sides of eqs. (22) and (23) are proportional to
the $\beta$ function suggests to use the notion of _conformal fixed point_
(alias Wilson–Fisher fixed point Wilson:1971dc ), i.e. a special value of the
coupling constant $u^{*}$ such that the $\beta$ function vanishes,

$\displaystyle\beta\left(u^{*}\right)=0\,.$ (24)

The coupling at the conformal point is a function of the dimensional regulator
$\epsilon$. Up to order $\epsilon^{2}$ it is given by

$\displaystyle
u^{*}=\frac{4\textsc{n}\epsilon}{20-\textsc{n}^{2}}+\frac{4\textsc{n}\epsilon^{2}}{9}\frac{5\textsc{n}^{4}-496\textsc{n}^{2}+5360}{(20-\textsc{n}^{2})^{3}}+\mathcal{O}(\epsilon^{3})\,.$
(25)

Note that the real-valuedness of the coupling $g$ implies that $u>0$ and,
since $\epsilon>0$, this is possible only for sufficiently small values of the
rank n.333We are mostly interested in the perturbative implications of
conformal symmetry at the conformal fixed point, and therefore do not study
the nature of the latter Giombi:2015haa ; Gracey:2020tkk .  The renormalised
correlators are therefore conformally invariant at the conformal fixed point,

$\displaystyle
K^{\mu}_{\Delta_{\phi}+\gamma}\,\langle\phi(x_{1})\ldots\phi(x_{n})\rangle\biggl{|}_{u=u^{*}}=0\,,$
(26)

though with a conformal dimension shifted by the anomalous dimension $\gamma$.
This is the starting point for the derivation of our anomalous CWIs for the
on-shell amplitudes, which we discuss in section 3. It is important to stress
that the CWIs (26) give very strong constraints on the form of the
renormalised correlators also away from the conformal fixed point. Indeed, in
appendix B we show that the finite part of a renormalised correlator at a
given loop order is entirely determined by its expression at the conformal
fixed point and lower loop information.

###  2.3 A toy example: the three-point correlator

Seen from the point of view of a perturbative computation, having exact
conformal invariance at the conformal fixed point implies a remarkable
conspiracy of infinitely many Feynman diagrams at all orders. It is
instructive to illustrate this for the three-point correlator, which will be a
useful toy example for deriving the anomalous Ward identities for the on-shell
amplitudes in section 3. We give here only the salient points, and refer to
appendix D for a thorough discussion.

It is well known that conformal symmetry fixes the three-point correlators up
to an overall constant (see e.g. ref. Polyakov:1970xd ). The expression is
extremely simple in position space,444We recall that we are working in the
Minkowski space-time signature. The subtraction of a small positive imaginary
part $\textup{i}0$ from each $x_{ij}^{2}$ corresponds to Feynman’s
prescription for the propagators in momentum space.

$\displaystyle\langle\phi^{a}(x_{1})\phi^{b}(x_{2})\phi^{c}(x_{3})\rangle\biggl{|}_{u=u^{*}}=\frac{d^{abc}\,c_{123}(u^{*})}{\left(x_{12}^{2}-\textup{i}0\right)^{\frac{\Delta_{\gamma}}{2}}\left(x_{23}^{2}-\textup{i}0\right)^{\frac{\Delta_{\gamma}}{2}}\left(x_{31}^{2}-\textup{i}0\right)^{\frac{\Delta_{\gamma}}{2}}}\,,$
(27)

where $x_{ij}=x_{i}-x_{j}$, but becomes more complicated in momentum space. We
recall that we have to work directly in momentum space in order to study on-
shell scattering amplitudes. Expressions for the Fourier transform of the
three-point conformal correlator have been obtained in terms of Appell’s
hypergeometric function $F_{4}$ or triple-$K$ integrals Barnes:2010jp ;
Coriano:2013jba ; Bzowski:2013sza ; Bzowski:2015yxv ; Isono:2019ihz ;
Bautista:2019qxj ; Gillioz:2019lgs ; Bzowski:2020lip . We adopt a different
approach, and use of the technology developed for computing Feynman integrals.
The Fourier transform of the three-point correlator can in fact be written as

$\displaystyle\langle\phi^{a}(p_{1})\phi^{b}(p_{2})\phi^{c}(p_{3})\rangle\biggl{|}_{u=u^{*}}\propto
d^{abc}\,\delta^{(d)}(p_{1}+p_{2}+p_{3})\,I\left(2-\frac{\epsilon+\gamma}{2},2-\frac{\epsilon+\gamma}{2},2-\frac{\epsilon+\gamma}{2}\right)\,,$
(28)

up to an overall kinematic-independent factor, where $I(e_{1},e_{2},e_{3})$ is
a Feynman integral of the one-loop “three-mass triangle family,”

$\displaystyle
I\left(e_{1},e_{2},e_{3}\right)=e^{\epsilon\gamma_{\text{E}}}\int\frac{\mathrm{d}^{d}k}{\textup{i}\pi^{\frac{d}{2}}}\frac{1}{\left[-k^{2}-\textup{i}0\right]^{e_{1}}\left[-(k+p_{1})^{2}-\textup{i}0\right]^{e_{2}}\left[-(k+p_{1}+p_{2})^{2}-\textup{i}0\right]^{e_{3}}}\,,$
(29)

whose defining graph is shown in figure 1.

Figure 1: Graph representing the one-loop triangle integral family defined by
eq. (29). The arrows denote the momentum flow.

In contrast with the usual Feynman integrals, the propagators in eq. (28) are
raised to non-integer, $\epsilon$-dependent powers. Given this aspect of
novelty, we discuss thoroughly the computation using the method of
differential equations KOTIKOV1991158 ; Bern:1993kr ; Remiddi:1997ny ;
Gehrmann:1999as  in canonical form Henn:2013pwa  in appendix E.1. As a result,
we obtain the analytic expression of the three-point conformal correlator in
momentum space expanded around $\epsilon=0$.555The expansion is truncated by
the limited perturbative knowledge of the anomalous dimension $\gamma$. The
latter is given up to four-loop order in Gracey:2015tta , which suffices to
expand the correlator up to order $\epsilon^{4}$. It is written in terms of
algebraic functions and two-dimensional harmonic polylogarithms
Gehrmann:2001jv . Remarkably, conformal symmetry captures entirely this
complexity, in an expression which in position space is ridiculously compact.

This is even more remarkable if contrasted with a perturbative computation of
the correlator. The latter in fact involves infinitely many Feynman diagrams
(see e.g. figure 2), which evaluate to complicated transcendental functions.
All these pieces fit together, to all orders, to reproduce the conformally
invariant expression at the conformal fixed point, fixing the normalisation
factor $c_{123}$ in eq. (27). We check this explicitly with a perturbative
computation of the two- and three-point correlators up to two-loop order in
appendix D.2.

|  |  |  |  |
---|---|---|---|---|---
|  |  |  |  |
Figure 2: Representative Feynman diagrams contributing to the three-point
correlator $C^{(3)}$ up to two-loop order.

The analysis of the three-point correlator gives two important lessons. First,
it gives a paradigmatic example of the constraining power of conformal
symmetry. Second, it shows that conformal symmetry underlies the perturbative
results in a very non-trivial way. This is due to the second-order
differential nature of the conformal constraints in momentum space. A precise
understanding of how these constraints are implemented is therefore imperative
in order to unveil the underlying conformal symmetry and exploit it.

In the next section we investigate how this extends to on-shell scattering
amplitudes. The latter are obtained from the correlators by amputating the
external legs and taking the light-like limits. These operations have a non-
trivial interplay with conformal symmetry, and generate an anomaly in the
CWIs. The full analytic control over the three-point correlator (28) achieved
through Feynman integral techniques will be instrumental to derive the
conformal anomaly in the on-shell limit.

##  3 Anomalous conformal Ward identities for scattering amplitudes

In the previous section we have shown that the renormalised correlators are
annihilated by the conformal boost generator at the conformal fixed point,

$\displaystyle
K^{\mu}_{\Delta_{\gamma}}\,\langle\phi(x_{1})\ldots\phi(x_{n})\rangle\biggl{|}_{u=u^{*}}=0\,.$
(30)

In this section we discuss how this translates into a constraint on the
corresponding scattering amplitude. The latter is obtained from the correlator
through the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula Lehmann:1954rq
(see ref. Gillioz:2020mdd  for a formulation of the LSZ reduction procedure in
a conformal field theory).

We begin in section 3.1 by discussing the interplay between the LSZ reduction
formula and conformal symmetry. We show that the conformal boost generator
does not commute with the on-shell limit of the reduction, and how we overcome
this obstacle by analysing the asymptotic expansion of the correlator in the
light-like limit. We do this explicitly for the three-point case in section
3.2. We amputate one of the external legs and work out the anomalous CWIs
satisfied by the resulting amplitude. We then extend this procedure to a
multi-point correlator with several LSZ-reduced legs in section 3.3. The
anomaly in general comprises two terms. One is proportional to the elementary
field anomalous dimension, and thus involves only lower-loop information. In
section 3.4 we argue that the second term is governed by collinear regions of
loop momentum, and that it can be represented as the convolution of a
universal function and lower-order amplitudes. This gives a handle on the
computation of the conformal anomaly, which this way becomes a lower-order
problem. In section 3.5 we give a fewWe recall from examples of computation of
the conformal anomaly: for the three-point case with one on-shell leg, both
exactly to all orders and perturbatively up to one loop, and for the fully on-
shell four-point case perturbatively up to one loop.

###  3.1 Definition of scattering amplitudes via LSZ formula and conformal
symmetry

The LSZ reduction procedure takes from the correlator to the scattering
amplitude by removing the bubble corrections from the external legs and
setting the external momenta on shell. In this section we show that the
conformal boost generator can be pulled through the amputation of the external
bubbles, but does not commute with the on-shell limit.

We define the renormalised momentum-space correlators at the conformal fixed
point by Fourier transforming the position-space ones as

$\displaystyle
C^{(n)}\left(p_{1},\ldots,p_{n}\right)\,(2\pi)^{d}\,\delta^{(d)}\left(p_{1}+\ldots+p_{n}\right)=\int\left(\prod_{j=1}^{n}\mathrm{d}^{d}x_{j}e^{\textup{i}p_{j}\cdot
x_{j}}\right)\langle\phi(x_{1})\ldots\phi(x_{n})\rangle\biggl{|}_{u=u^{*}}\,.$
(31)

We strip off the overall momentum-conservation $\delta$ function and a factor
of $(2\pi)^{d}$ to simplify the expressions. In appendix C we show that the
conformal boost generator commutes with the momentum-conservation $\delta$
function when acting on a correlator. We will thus neglect the $\delta$
function hereinafter, and act with the conformal boost generator directly on
$C^{(n)}$.

We begin with the amputation of the bubble corrections on each external leg.
The latter sum up to the renormalised two-point correlator. Their amputation
therefore amounts to

$\displaystyle
G^{(n)}(p_{1},\ldots,p_{n}):=\left[\prod_{j=1}^{n}C^{(2)}(p_{j},-p_{j})\right]^{-1}C^{(n)}(p_{1},\ldots,p_{n})\,.$
(32)

We call this intermediate object, $G^{(n)}$, the amputated correlator. The
two-point renormalised correlator is given to all orders by (see appendix D.1)

$\displaystyle
C^{(2)}(p_{j},-p_{j})=\tilde{c}_{12}\left(u^{*}\right)\left(-p_{j}^{2}\right)^{-1+\gamma}\,,$
(33)

where $\tilde{c}_{12}\left(u^{*}\right)$ is a kinematic-independent factor
given by eq. (149). We can pull the conformal boost generator through the
inverse two-point correlation functions thanks to the intertwining relation

$\displaystyle
K^{\mu}_{\Delta}\left(p^{2}\right)^{\Delta-\frac{d}{2}}=\left(p^{2}\right)^{\Delta-\frac{d}{2}}K^{\mu}_{d-\Delta}\,.$
(34)

Setting $\Delta=\Delta_{\gamma}$, this relation, together with the invariance
of the correlator (30), implies that the amputated correlator $G^{(n)}$
satisfies the CWI

$\displaystyle K^{\mu}_{d-\Delta_{\gamma}}G^{(n)}(p_{1},\ldots,p_{n})=0\,.$
(35)

Therefore, pulling the conformal boost generator through the inverse two-point
correlators simply amounts to changing the conformal weight from
$\Delta_{\gamma}$ to its “shadow” $d-\Delta_{\gamma}$. Equivalently, instead
of the correlation function $C^{(n)}$ of elementary fields $\phi$, we could
have considered a correlator of the shadow operators Ferrara:1972uq , which
carry the shadow conformal dimension. The latter coincides with $G^{(n)}$, and
the amputation procedure (32) relates the two pictures. In the representation
theoretic language, representations of the conformal algebra with weights
$\Delta_{\gamma}$ and $d-\Delta_{\gamma}$ are equivalent Knapp1971 .

The scattering amplitude, $M^{(n)}$, is then given by the on-shell limit of
the amputated correlator,666In the following, it will also be convenient to
consider analogous objects where only some of the external legs are taken on-
shell, which we will also denote by the letter $M$.

$\displaystyle M^{(n)}(p_{1},\ldots,p_{n})=\lim_{p_{1}^{2}\to
0}\ldots\lim_{p_{n}^{2}\to 0}G^{(n)}(p_{1},\ldots,p_{n})\,.$ (36)

The conformal boost generator, however, does not commute with this limit. In
order to prove this, let us consider a toy example: a generic scalar function
of a single squared momentum, $f(p^{2})$, which is finite at $p^{2}\to 0$. The
conformal boost generator acting on it is

$\displaystyle K^{\mu}_{\Delta}=-p^{\mu}\frac{\partial}{\partial
p_{\nu}}\frac{\partial}{\partial p^{\nu}}+2p^{\nu}\frac{\partial}{\partial
p^{\nu}}\frac{\partial}{\partial p_{\mu}}+2(d-\Delta)\frac{\partial}{\partial
p_{\mu}}\,.$ (37)

The commutator of the conformal boost generator and the on-shell limit acting
on $f$ is then given by

$\displaystyle\left[K^{\mu}_{\Delta},\lim_{p^{2}\to
0}\right]f\left(p^{2}\right)=-4\lim_{p^{2}\to
0}p^{\mu}\left[p^{2}f^{\prime\prime}\left(p^{2}\right)+\left(\frac{d}{2}+1-\Delta\right)\,f^{\prime}\left(p^{2}\right)\right]\,.$
(38)

The two terms on the right-hand side of the previous equation correspond to
two different physical effects that leads to an on-shell conformal anomaly:

  * •

The first term on the right-hand side of eq. (38) also appeared in ref.
Chicherin:2017bxc  in the context of finite conformal integrals. There it was
shown that the action of the conformal boost leads to an anomaly that is
localised on collinear regions of the loop integration (more on this in
section 3.4).

  * •

The second term on the right-hand side of eq. (38) vanishes for
$\Delta=d/2+1$, or equivalently $\Delta=d-\Delta_{\phi}$. However, we see that
there is a mismatch with the conformal weight $\Delta=d-\Delta_{\gamma}$ that
is needed to annihilate the amputated correlator, see eq. (35). The
discrepancy is proportional to the anomalous dimension $\gamma$. This
generates a new type of conformal anomaly, which we investigate in section
3.2. Since it is related to the anomalous dimension, one could say that it is
of ultraviolet origin.

The amputated correlator $G^{(n)}$ is finite at $p_{i}^{2}=0$, but its
derivatives do not vanish in general. To see this, consider for example the
one-loop triangle integral defined by eq. (29) with unit propagator powers. In
a perturbative computation of $G^{(3)}$, this is the only diagram contributing
at one-loop order. In the light-like limit $s_{1}\to 0$, the triangle integral
has the following asymptotic expansion,

$\displaystyle I(1,1,1)=\sum_{m\geq
0}(-s_{1})^{m}I^{\text{hard}}_{[m]}(1,1,1)+\sum_{m\geq
0}(-s_{1})^{1-\epsilon+m}I^{\text{coll}}_{[m]}(1,1,1)\,.$ (39)

The terms of the expansion can be computed using the method of the _expansion
by regions_ Beneke:1997zp ; Smirnov:1999bza ; Jantzen:2011nz . One splits the
loop integration domain into regions in which the expansion in the small
parameter ($s_{1}$ in this case) and the integration commute, and it is thus
possible to expand in series the integrand under the integral sign. The terms
with integer exponents of $s_{1}$ in the expansion (39) stem from what is
typically called the _hard region_ in the literature, which corresponds to
expanding the loop integrand as is under the integral sign, and integrating
term by term. The terms in eq. (39) with $\epsilon$-dependent powers originate
instead from the _collinear region_ of the loop integration. Computing them
requires re-scaling appropriately the loop momenta so as to catch the
contributions from this region, before we expand and integrate. We discuss
this thoroughly for the triangle integrals in appendix E.2. In a perturbative
expansion around $\epsilon=0$, the collinear region generates terms of the
form $s_{1}\log(-s_{1})$, which are finite at $s_{1}=0$ but have singular
derivative. In general, the terms of order $p^{2}$ and $p^{2}\log(-p^{2})$ in
the asymptotic expansion of $f(p^{2})$ in eq. (38) give non-vanishing
contributions to the commutator at $p^{2}=0$.

These terms in the asymptotic expansion of the amputated correlator $G^{(n)}$
prevent us from simply pulling the conformal boost generator through the on-
shell limit. The amplitude $M^{(n)}$ therefore does not inherit the conformal
invariance from the amputated correlator $G^{(n)}$ in general. Instead, it
satisfies CWIs with an anomaly term,

$\displaystyle
K^{\mu}_{d-\Delta_{\gamma}}M^{(n)}(p_{1},\ldots,p_{n})=\mathcal{A}^{(n)\,\mu}(p_{1},\ldots,p_{n})\,,$
(40)

where we recall that the external momenta are on-shell, $p_{i}^{2}=0$. The
anomaly on the right-hand side originates from the non-commutativity of the
conformal boost generator and the on-shell limit. Given that the terms on the
right-hand side of eq. (38) have a specific physical origin, we aim to find a
universal description for them.

Before we present our idea to reach this goal, a word of caution is in order.
The non-commutativity of the conformal boost generator with the on-shell limit
not only prevents us from effortlessly deriving the implications of conformal
symmetry for the scattering amplitude. It also leads to a practical
inconvenience: Regardless of how we obtain the anomalous CWI (40), the
external momenta are put on-shell and the amplitude, the anomaly, and —
through the chain rule — the conformal generator are written in terms of some
set of independent variables. The latter are chosen so as to implement
momentum conservation and the on-shellness of the momenta, but are otherwise
arbitrary. If the conformal generator and the on-shell limit commuted, we
would be free to change the variables at any step. Since this is not the case,
deriving the CWI (40) with a first choice of variables and changing it to a
second one gives a different result than using the second from the beginning.
The difference stems from the non-vanishing commutator in eq. (38) with
$\Delta=d-\Delta_{\gamma}$, and is hence proportional to the anomalous
dimension $\gamma$. This unusual feature may be uncomfortable, but causes no
harm provided we choose the independent variables once and for all at the
beginning of the derivation.

Let us now lay out our plan to derive the implications of conformal symmetry
for the scattering amplitude from the invariance of the correlator. The basic
idea is to study its asymptotic expansion in the light-like limits
$p_{i}^{2}\to 0$, identify which terms contribute to the anomaly, and find a
convenient method to compute them. In the next section we discuss this
explicitly in the $n=3$ case. The procedure and the conclusions can be
straightforwardly generalised to a generic number of particles, as we show in
section 3.3.

###  3.2 Anomalous Ward identity for a three-point amplitude

In this section we study what happens to the conformal properties of the
three-point correlator when amputating external legs and taking the on-shell
limit. As discussed above, the crucial step is the on-shell limit. We will
study the case where one of the external legs goes on-shell. Studying this
case will allow us to understand the mechanism which generates the anomaly in
the CWI, and to generalise it to the general case.

Let us start from the amputated correlator $G^{(3)}$, which satisfies the CWI

$\displaystyle K^{\mu}_{d-\Delta_{\gamma}}G^{(3)}\left(s\right)=0\,.$ (41)

We choose as independent variables $s=(s_{1},s_{2},s_{3})$, and use the chain
rule to express the conformal boost generator as a differential operator in
the $s_{i}$,

$\displaystyle
K^{\mu}_{\Delta}=\sum_{i=1}^{3}p_{i}^{\mu}\hat{K}^{(i)}_{\Delta}\,,$ (42)

where

$\displaystyle\hat{K}^{(i)}_{\Delta}=4s_{i}\frac{\partial^{2}}{\partial
s_{i}^{2}}+2(d+2-2\Delta)\frac{\partial}{\partial s_{i}}\,.$ (43)

By substituting eq. (42) into eq. (41) and using momentum conservation we
obtain the following scalar constraints,

$\displaystyle\left(\hat{K}^{(2)}_{d-\Delta_{\gamma}}-\hat{K}^{(3)}_{d-\Delta_{\gamma}}\right)G^{(3)}\left(s\right)=0\,,$
(44)
$\displaystyle\left(\hat{K}^{(2)}_{d-\Delta_{\gamma}}-\hat{K}^{(1)}_{d-\Delta_{\gamma}}\right)G^{(3)}\left(s\right)=0\,.$
(45)

Next, we take the on-shell limit $s_{1}\to 0$ of the amputated correlator and
define an amplitude-like object,

$\displaystyle M^{(3)}\left(s_{2},s_{3}\right)=\lim_{s_{1}\to
0}G^{(3)}\left(s\right)\,.$ (46)

We choose $(s_{2},s_{3})$ as independent on-shell variables. In order to
establish a link between the amputated correlator and the amplitude in such a
way that we can translate the constraints (44) and (45) into conditions on the
latter, we consider the asymptotic expansion of the former in the $s_{1}\to 0$
limit. The result of this analysis, which we discuss in appendix E.2, is that
the amputated correlator admits the following asymptotic expansion,

$\displaystyle G^{(3)}\left(s\right)=\sum_{m\geq
0}(-s_{1})^{m}\,G^{(3),\text{hard}}_{[m]}\left(s_{2},s_{3}\right)+\sum_{m\geq
0}(-s_{1})^{1-\gamma+m}\,G^{(3),\text{coll}}_{[m]}\left(s_{2},s_{3}\right)\,.$
(47)

The series with integer powers of $s_{1}$ originates from the hard region of
the loop integration defining the conformal correlator $C^{(3)}$ in eq. (27).
It corresponds to expanding the integral around $s_{1}=0$ at the integrand
level. The zero-th term gives the amplitude as one would obtain it from a
perturbative computation, setting $s_{1}=0$ from the beginning,

$\displaystyle
M^{(3)}\left(s_{2},s_{3}\right)=G^{(3),\text{hard}}_{[0]}\left(s_{2},s_{3}\right)\,.$
(48)

The series with powers of $(-s_{1})^{1-\gamma+m}$ in the asymptotic expansion
(47) stems instead from the collinear region of the loop integration.

Upon perturbative expansion around $\epsilon=0$, the terms from the collinear
region generate logarithms of $s_{1}$. For our purpose it is convenient to
spell out these logarithms, by rewriting eq. (47) as

$\displaystyle
G^{(3)}\left(s\right)=M^{(3)}\left(s_{2},s_{3}\right)+\sum_{m\geq
1}\sum_{k\geq
0}s_{1}^{m}\,\log^{k}\left(-s_{1}\right)G^{(3)}_{m;k}\left(s_{2},s_{3}\right)\,.$
(49)

We then substitute this expansion into eqs. (44) and (45). Since
$\hat{K}^{(2)}_{\Delta}$ and $\hat{K}^{(3)}_{\Delta}$ do not act on $s_{1}$,
we can set $s_{1}=0$ in eq. (44), and get a first constraint on the amplitude
$M^{(3)}$,

$\displaystyle\left(\hat{K}^{(2)}_{d-\Delta_{\gamma}}-\hat{K}^{(3)}_{d-\Delta_{\gamma}}\right)M^{(3)}\left(s_{2},s_{3}\right)=0\,.$
(50)

On the other hand, we cannot set $s_{1}=0$ naïvely in eq. (45), because of the
operator $\hat{K}^{(1)}_{\Delta}$ on the right-hand side. Instead, we act with
$\hat{K}^{(1)}_{\Delta}$ on the asymptotic expansion of the amputated
correlator (49), and set $s_{1}=0$ after differentiation. The result is

$\displaystyle\hat{K}^{(2)}_{d-\Delta_{\gamma}}M^{(3)}\left(s_{2},s_{3}\right)=4\gamma
G^{(3)}_{1;0}\left(s_{2},s_{3}\right)+4G^{(3)}_{1;1}\left(s_{2},s_{3}\right)\,.$
(51)

This simple computation, the details of which can be found in Appendix E.3,
also leads to an interesting bonus result. All terms $G_{m;k}^{(3)}$ in the
asymptotic expansion (49) are entirely determined by a sequence of conformal
boost generators acting on just two terms: the amplitude $M^{(3)}$ and
$G^{(3)}_{1;1}$, namely the coefficient of $s_{1}\log(-s_{1})$ in the
asymptotic expansion of the amputated correlator in eq. (49). Let us also note
that the sum of the two terms in the expression of the anomaly on the right-
hand side of eq. (51) is equal to $-4\gamma
G_{[1]}^{(3),\text{hard}}\left(s_{2},s_{3}\right)$, so that the collinear
region contributes neither to the amplitude nor to the conformal anomaly.

Finally, eqs. (50) and (51) can be combined into the desired anomalous CWI,

$\displaystyle
K^{\mu}_{d-\Delta_{\gamma}}M^{(3)}\left(s_{2},s_{3}\right)=\mathcal{A}^{(3)\mu}\,,$
(52)

where the anomaly is given by

$\displaystyle\mathcal{A}^{(3)\mu}=-4p_{1}^{\mu}\left(\gamma
G^{(3)}_{1;0}\left(s_{2},s_{3}\right)+G^{(3)}_{1;1}\left(s_{2},s_{3}\right)\right)\,,$
(53)

and the conformal boost generator,

$\displaystyle
K^{\mu}_{\Delta}=\sum_{i=2,3}p_{i}^{\mu}\hat{K}^{(i)}_{\Delta}\,,$ (54)

is exactly the generator (13) restricted to a function of $s_{2}$ and $s_{3}$,
with $s_{1}=0$. Several comments are in order.

  * •

What predictive power does the anomalous CWI (52) hold? In section 3.5.1 we
write down a closed-form expression for the anomaly, and solve the anomalous
CWI for the three-particle amplitude $M^{(3)}$ exactly.

  * •

In section 3.3 we discuss how the anomalous CWI generalises to the $n$-point
correlator. Although in this case an all-order result is out of reach, the
anomalous CWI (52) retains a strong predictive power in a perturbative
approach, i.e. for computing the amplitude up to a fixed order in $\epsilon$.
To this end, the computation of the anomaly must be simpler than that of the
amplitude at the same order. Let us then look at the two terms of the anomaly
in eq. (52) from a perturbative point of view.

  * •

The term with the coefficient $G^{(3)}_{1;0}$ is proportional to the anomalous
dimension. Since the latter starts at order $\epsilon$, $G^{(3)}_{1;0}$ is
needed at a lower order in $\epsilon$ than the amplitude.

  * •

The second term of the anomaly, containing $G^{(3)}_{1;1}$, is instead needed
at the same order as the amplitude on the left-hand side. In section 3.4 we
propose a method to compute $G^{(3)}_{1;1}$ which relies on lower-loop
information only.

  * •

The tree-level amplitudes cannot contribute any logarithms, and therefore
$G^{(3)}_{1;1}$ starts at order $\epsilon$. The entire anomaly then starts at
order $\epsilon$, which is in agreement with the well-known conformal
invariance at tree level for $\epsilon=0$,

$\displaystyle
K^{\mu}_{d-\Delta_{\gamma}}M^{(3)}\left(s_{2},s_{3}\right)=\mathcal{O}\left(\epsilon\right)\,.$
(55)

In this section we have proposed a procedure to extract the conformal anomaly
associated with amputating and putting on-shell one of the external legs in
the three-point correlator. In the next section we will iterate this procedure
on multiple legs of an $n$-point correlator, and generalise the anomalous CWI
to the multi-point case.

###  3.3 Conformal Ward identity for $n$-particle amplitudes

In the previous sections we considered in detail the three-point correlator at
the conformal fixed point, and studied its conformal anomaly generated by
putting one of its legs on-shell. Many of the observed properties hold for
$n$-point correlators when putting a subset $\Lambda$ of its legs on-shell,

$\displaystyle s_{i}\equiv p_{i}^{2}=0\;,\qquad
i\in\Lambda\subseteq\\{1,\ldots,n\\}\,.$ (56)

As we have already mentioned above, the precise form of the conformal anomaly
equation depends on the choice of independent kinematic variables.
Nevertheless, some generic features, which we develop below, hold for any
choice of the variables.

We complement $\\{s_{i}\\}_{i\in\Lambda}\equiv s_{\Lambda}$ with a set $v$ of
Mandelstam variables in order to parameterise the kinematics of the $n$-point
correlator. Putting $\Lambda$ of the $n$ legs of the amputated correlator
$G^{(n)}$ on-shell results in an amplitude-like object $M^{(n)}$,

$\displaystyle M^{(n)}\left(v\right)=\left(\prod_{i\in\Lambda}\lim_{s_{i}\to
0}\right)G^{(n)}\left(v,s_{\Lambda}\right)\,.$ (57)

The amputated correlator $G^{(n)}$ is conformally invariant with conformal
weight $d-\Delta_{\gamma}$ with respect to each leg. Among the conformal
algebra generators, only the conformal boost symmetry is broken by an anomaly
in the on-shell limit. We rewrite the momentum-space conformal boost generator
(13) in the Mandelstam variables $v,s_{\Lambda}$ using the chain rule as

$\displaystyle K^{\mu}_{\Delta}=\sum_{i=1}^{n}p_{i}^{\mu}K^{(i)}_{\Delta}\,.$
(58)

The scalar operators $K^{(i)}_{\Delta}$ have the following form,

$\displaystyle
K^{(i)}_{\Delta}=\mathbb{K}^{(i)}_{\Delta}+\hat{K}^{(i)}_{\Delta}+\sum_{j\in\Lambda}s_{j}\partial_{s_{j}}A^{(ij)}+\sum_{j\in\Lambda}s_{j}B^{(ij)}\;,\qquad
i\in\Lambda\,,$ (59)

where $\hat{K}^{(i)}_{\Delta}$ defined in eq. (43) acts only on the $s_{i}$
variables. As we will see below, only this piece of the conformal boost
generates the anomaly on the light-cone. The differential operators
$\mathbb{K}^{(i)}_{\Delta},A^{(ij)},B^{(ij)}$ act only on Mandelstam
invariants $v$, and do not involve any of the $s_{\Lambda}$. Their explicit
form can be easily worked out for any given choice of the Mandelstam variables
$v$. Similarly, the operators referring to the remaining legs are as follows,
777The three-point correlator and the amplitude considered in section 3.2
correspond to $\Lambda=\\{1\\}$, $s_{\Lambda}=\\{s_{1}\\}$,
$v=\\{s_{2},s_{3}\\}$ in the notation of the current section. Also, we have
$A^{(ij)}=B^{(ij)}=\mathbb{K}_{\Delta}^{(1)}=0$,
$\mathbb{K}_{\Delta}^{(2)}=\hat{K}^{(2)}_{\Delta}$ and
$\mathbb{K}_{\Delta}^{(3)}=\hat{K}^{(3)}_{\Delta}$.

$\displaystyle
K^{(i)}_{\Delta}=\mathbb{K}^{(i)}_{\Delta}+\sum_{j\in\Lambda}s_{j}\partial_{s_{j}}A^{(ij)}+\sum_{j\in\Lambda}s_{j}B^{(ij)}\;,\qquad
i\in\\{1,\ldots,n\\}\backslash\Lambda\,.$ (60)

In the asymptotic expansion of the amputated correlator in the on-shell limit
$s_{i}\to 0$ for all $i\in\Lambda$, each of the legs $\Lambda$ contributes
either $s_{i}^{m_{i}}$ with $m_{i}\geq 0$, or
$s_{i}^{m_{i}}\log^{k_{i}}(-s_{i})$ with $m_{i}\geq 1$ and $k_{i}\geq 1$,

$\displaystyle G^{(n)}\left(v,s_{\Lambda}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}m_{i}\geq k_{i}\geq 0\\\ k_{i}\geq
m_{i}\geq 1\\\ \text{for all
}i\in\Lambda\end{subarray}}\Biggl{(}\prod_{j\in\Lambda}s_{j}^{m_{j}}\log^{k_{j}}(-s_{j})\Biggr{)}G^{(n)}_{\\{m_{l};k_{l}\\}_{l\in{\Lambda}}}(v)\,.$
(61)

The amplitude $M^{(n)}$ is the term of the asymptotics with $m_{i}=0$ and
$k_{i}=0$ for all $i\in\Lambda$.

In order to formulate the conformal anomaly equation for the amplitude
$M^{(n)}$, only $2|\Lambda|$ coefficients in the asymptotic expansion (61) are
required. They are those with one pair $(m_{i},k_{i})=(1,1)$ or
$(m_{i},k_{i})=(1,0)$ and all remaining pairs $(m_{j},k_{j})=(0,0)$. Given
their importance, let us introduce a notation for them,

$\displaystyle M^{(n)}\left(v\right):=\ $ $\displaystyle
G^{(n)}\left(v,s_{\Lambda}\right)|_{s_{i}=0}\,,$ (62) $\displaystyle
G^{(n)}_{i;\text{pow}}(v):=\ $
$\displaystyle\bigl{[}G^{(n)}\left(v,s_{\Lambda}\right)\bigr{]}_{s_{i}}\,,$
(63) $\displaystyle G^{(n)}_{i;\text{coll}}(v):=\ $
$\displaystyle\bigl{[}G^{(n)}\left(v,s_{\Lambda}\right)\bigr{]}_{s_{i}\log(-s_{i})}\,.$
(64)

Acting on the asymptotic expansion (61) with the conformal boost generator
(58) (with eqs. (59) and (60)), and putting the legs $\Lambda$ on shell
($s_{i}\to 0$), we immediately conclude that the anomaly comes from the
$\hat{K}^{(i)}_{d-\Delta_{\gamma}}$ piece (43) of the conformal boost
generator. It picks out the $\sim s_{i}$ and $\sim s_{i}\log(-s_{i})$ terms of
the asymptotic expansion of the amputated correlator (61).

Putting everything together, we find the following anomalous conformal Ward
identity for the at the conformal fixed point,

$\boxed{\Biggl{(}\sum_{i=1}^{n}p_{i}^{\mu}\,\mathbb{K}^{(i)}_{d-\Delta_{\gamma}}\Biggr{)}M^{(n)}\left(v\right)={\cal
A}^{(n)\mu}}$ (65)

with the anomaly given by

$\boxed{{\cal A}^{(n)\mu}\equiv-4\sum_{i\in\Lambda}p_{i}^{\mu}\left(\gamma
G^{(n)}_{i;\text{pow}}(v)+G^{(n)}_{i;\text{coll}}(v)\right)}$ (66)

This, together with the specific form of the conformal anomaly terms to be
derived in the next section, is the main result of this paper. It generalises
a result of Chicherin:2017bxc  for finite conformal integrals to amplitudes at
a conformal fixed point in $d$ dimensions. Let us discuss the properties of
our anomalous CWIs.

  * •

The anomaly in eq. (66) has a local nature: for each external leg
$i\in\Lambda$ that is put on-shell, there is a separate contribution,
proportional to the momentum $p_{i}^{\mu}$.

  * •

The first anomaly term is proportional to the field anomalous dimension.
Therefore constraining the amplitude $M^{(n)}$ up to a certain order in
$\epsilon$ requires the computation of $G^{(n)}_{i;\text{pow}}$ to one order
lower only.

  * •

The second anomaly term, $G^{(n)}_{i;\text{coll}}$, has a collinear origin. We
will show in the next subsection that it is described by a convolution of
lower-loop amplitudes with a universal collinear kernel, which we determine
here up to the two-loop order.

  * •

The last two points imply that determining the anomaly in perturbation theory
is easier compared to computing the amplitudes directly. This makes the Ward
identities useful in practice.

###  3.4 Collinear anomaly from analysis of regions

In this section we use the method of the analysis of regions of Feynman
integrals Beneke:1997zp ; Smirnov:1999bza ; Jantzen:2011nz  to find a formula
which allows us to compute efficiently the collinear part of the conformal
anomaly, $G^{(n)}_{i,\text{coll}}$ in eq. (66), and to unveil its origin from
the region of the loop integration where some of the loop momenta become
collinear with one of the external on-shell momenta. We find that
$G^{(n)}_{i,\text{coll}}$ is given by the convolution of a universal function
and lower-loop, higher-point amplitudes. This generalises the previous result
of ref. Chicherin:2017bxc  to non-integer spacetime dimensions.

####  3.4.1 Review of the collinear anomaly in integer dimensions

The authors of ref. Chicherin:2017bxc  studied finite Feynman integrals in
integer dimensions in a number of quantum field theories with classically
conformal Lagrangian. They discovered that, whenever one of the external legs
is on shell, the conformal invariance of the integral is broken by a contact
term localised on the configurations where the loop momentum is collinear to
the external on-shell momentum. For scalar $\phi^{3}$ theory in $d=6$
dimensions, the master formula which gives the contribution to the conformal
anomaly from each on-shell corner (i.e. a trivalent vertex with an external
on-shell leg) is

$\displaystyle
K^{\mu}_{d-\Delta_{\phi}}\frac{1}{[q^{2}+\textup{i}0][(p+q)^{2}+\textup{i}0]}=4\textup{i}\pi^{3}p^{\mu}\int_{0}^{1}\mathrm{d}\xi\,\xi(1-\xi)\delta^{(6)}(q+\xi
p)\,,$ (67)

where $p$ is the external on-shell momentum ($p^{2}=0$), $q$ is off-shell
($q^{2}\neq 0$), and $d-\Delta_{\phi}=4$. Whenever the on-shell corner is part
of a loop, the $\delta$ function localises the adjacent loop integration on
the collinear configuration $q=-\xi p$.

Consider now some (finite) $\ell$-loop Feynman diagram ${\cal
F}^{(n+1)}_{d=6}$ in the $su(\textsc{n})$ $\phi^{3}$ theory. We focus on the
corner around the on-shell momentum $p$,

$\displaystyle{\cal
F}^{(n+1)}_{d=6}(p^{a},p_{1}^{a_{1}},\ldots,p_{n}^{a_{n}})=-\textup{i}gd^{abc}\int\frac{\mathrm{d}^{6}q}{(2\pi)^{6}}\frac{1}{q^{2}(p+q)^{2}}{\cal
G}^{(n+2)}_{d=6}(q^{b},(p-q)^{c},p_{1}^{a_{1}},\ldots,p_{n}^{a_{n}})\,.$ (68)

Equation (67) then gives

$\displaystyle K^{\mu}_{d-\Delta_{\phi}}{\cal
F}^{(n+1)}_{d=6}(p^{a},p_{1}^{a_{1}},\ldots,p_{n}^{a_{n}})=4p^{\mu}\frac{gd^{abc}}{(4\pi)^{3}}\int\limits^{1}_{0}\mathrm{d}\xi\,\xi\bar{\xi}\,{\cal
G}^{(n+2)}_{d=6}(\xi
p^{b},\bar{\xi}p^{c},p_{1}^{a_{1}},\ldots,p_{n}^{a_{n}})\,,$ (69)

where we use the short-hand notation $\bar{\xi}:=1-\xi$. According to eqs.
(67) and (69), the only contribution comes from the region of loop
integrations where the collinear light-like momenta $\xi p$ and $(1-\xi)p$
flow through the two propagators in eq. (68). In other words, all momenta
entering the subdiagram $({\cal F}_{d=6}\backslash{\cal G}_{d=6})$ are
collinear in the relevant region of loop integrations. We give a pictorial
representation of this relation in figure 3.

As a result of eq. (69), the anomaly is entirely determined by lower-loop
information. This allows for the computation of suitable integrals through the
solution of the corresponding anomalous CWIs Chicherin:2018rpz ; Zoia:2018sin
; Chicherin:2018ubl . An analogous collinear mechanism is also responsible for
the anomalies of the Yangian symmetry of the on-shell fishnet graphs
Chicherin:2022nqq .

$K^{\mu}_{d-\Delta_{\phi}}\int\mathrm{d}^{6}q$ $\sim
p^{\mu}\int_{0}^{1}\mathrm{d}\xi\,\xi\bar{\xi}$ … $\xi p$ $\bar{\xi}p$ … $p$
$q$ Figure 3:  Pictorial representation of the collinear anomaly mechanism.
The conformal boost generator, acting on a UV-finite diagram in six dimensions
with an on-shell external leg ($p^{2}=0$), produces a contact anomaly
localised on the configuration where the loop momentum is collinear to
$p^{\mu}$. The contact anomaly freezes the adjacent loop integration, giving a
finite anomaly entirely determined by lower loop information. The collinear
anomaly receives one such contribution for each of the external on-shell legs.

####  3.4.2 Collinear anomaly from logarithmically enhanced terms of the
correlator

Let us now formulate the collinear anomaly mechanism in a way which allows for
the extension to non-integer $d$. Recall from eq. (38) that in general we
expect two terms in the conformal anomaly equation. In integer dimensions,
however, we put $\gamma$ to zero, and we focus on the other term. On the one
hand, we can see from eq. (38) that the conformal boost generator picks out
only the term $p^{2}\log(-p^{2})$ of the amputated correlator as $p^{2}\to 0$.
On the other hand, eq. (69) gives a precise formula for the conformal anomaly.
Comparing the two expressions yields

$\displaystyle\left[{\cal
F}^{(n+1)}_{d=6}(p^{a},p_{1}^{a_{1}},\ldots,p_{n}^{a_{n}})\right]_{p^{2}\log(-p^{2})}=\frac{gd^{abc}}{(4\pi)^{3}}\int\limits^{1}_{0}\mathrm{d}\xi\,\xi\bar{\xi}\,{\cal
G}^{(n+2)}_{d=6}(\xi
p^{b},\bar{\xi}p^{c},p_{1}^{a_{1}},\ldots,p_{n}^{a_{n}})\,.$ (70)

This provides a practical recipe to evaluate the term $p^{2}\log(-p^{2})$ in
the asymptotic expansion of ${\cal F}_{d=6}$, based on the knowledge of the
lower-loop subdiagram ${\cal G}_{d=6}$.

Figure 4: An example of a four-point three-loop Feynman diagram ${\cal
F}^{(4)}_{d=6}$ having the tennis-court topology. The only region of loop
integration which contributes to the asymptotics $p^{2}\log(-p^{2})$ at
$p^{2}\to 0$ is the one of collinear momenta, as indicated in the figure.

Let us now reproduce and understand eq. (70) in a different way. We are
interested in the kinematic limit $p^{2}\to 0$ of a correlation function, and
in particular in the term proportional to $p^{2}\log(-p^{2})$ in that limit.
We use the region expansion analysis Beneke:1997zp ; Smirnov:1999bza ;
Jantzen:2011nz  of Feynman diagrams to analyse this limit.

When an external momentum $p^{\mu}$ is taken to be on-shell, we find different
relevant contributions from the region expansion analysis. In addition to the
so-called hard region, which corresponds to the naïve limit, one finds that
collinear regions, where one or more loop momenta become collinear to
$p^{\mu}$, are relevant.

For example, for the diagram shown in figure 4, we find that the only term
contributing to $p^{2}\log(-p^{2})$ in the limit comes from the one-loop
collinear region indicated in the figure. In fact, for (ultraviolet-)finite
Feynman diagrams (as those considered in ref. Chicherin:2017bxc ), one finds
that only one-loop collinear regions can produce logarithmically enhanced
terms in the limit. In this way, one recovers eq. (70). In this paper, we do
not want to make any assumptions on ultraviolet power counting, and perform an
analysis in arbitrary dimension $d$. As we will see in the next subsection,
this means that in addition to one-loop collinear regions, also multi-loop
collinear regions are relevant to the conformal anomaly.

####  3.4.3 Region analysis in non-integer dimensions

Let us now reproduce the previous results from a region analysis, and
generalise the six-dimensional formula (67) to the $d$-dimensional case. To
make the discussion more accessible we begin with explicit examples at one-
and two-loop order, before we present the general result.

One-loop example: We start with the one-loop triangle diagram which gives the
one-loop term of the amputated correlator $G^{(3)}$. We put on-shell $p_{1}$
($s_{1}=0$) and we want to compute the $s_{1}\log(-s_{1})$-term of the
asymptotic expansion in the limit $s_{1}\to 0$. We express this diagram in
terms of the triangle integral from the family defined in eq. (29) as

$\vbox{\hbox{ \leavevmode\hbox to98.43pt{\vbox
to51.48pt{\pgfpicture\makeatletter\hbox{\hskip
48.50504pt\lower-25.92433pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.76505pt}{-13.36995pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.13}]{figures/triangle_collinear.png}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-45.17203pt}{-1.78333pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{1}^{a}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{38.76352pt}{-20.64688pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{2}^{b}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{38.76352pt}{16.71078pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{3}^{c}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\textup{i}gu\dfrac{\textsc{n}^{2}-12}{2\textsc{n}}d^{abc}I(1,1,1)\,.$
(71)

From the expansion in eq. (39), we see that the $s_{1}\log(-s_{1})$-term of
the triangle integral is given by $\epsilon$ times the zeroth term from the
collinear region. We compute the latter using the method of the expansion by
regions in appendix E.2. Using eq. (198), where we replace $\alpha_{1}$ with
$\xi$ to make contact with ref. Chicherin:2017bxc , we find

$\displaystyle\begin{aligned}
\left[I(1,1,1)\right]_{s_{1}\log(-s_{1})}&=\epsilon
I^{\text{coll}}_{[0]}(1,1,1)\\\
&=e^{\epsilon\gamma_{\text{E}}}\epsilon\Gamma\left(\epsilon-1\right)\int_{0}^{1}\mathrm{d}\xi\frac{(\xi\bar{\xi})^{1-\epsilon}}{\xi(s_{2}-s_{3})-s_{2}}\,.\end{aligned}$
(72)

The denominator of the integrand can be viewed as a tree-level four-point
diagram with momenta (all incoming) $\left(\xi
p_{1},\bar{\xi}p_{1},p_{2},p_{3}\right)$ in the $(p_{2}+\xi p_{1})^{2}$
channel. The $s_{1}\log(-s_{1})$-term of the asymptotic expansion of the
triangle diagram can then be expressed as

$\Biggl{[}\vbox{\hbox{ \leavevmode\hbox to98.43pt{\vbox
to51.48pt{\pgfpicture\makeatletter\hbox{\hskip
48.50504pt\lower-25.92433pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.76505pt}{-13.36995pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.13}]{figures/triangle_collinear.png}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-45.17203pt}{-1.78333pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{1}^{a}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{38.76352pt}{-20.64688pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{2}^{b}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{38.76352pt}{16.71078pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{3}^{c}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\Biggr{]}_{s_{1}\log(-s_{1})}=\frac{g\,d^{ah_{1}h_{2}}}{(4\pi)^{3}}\int_{0}^{1}\mathrm{d}\xi\,\Omega^{\text{1L}}\left(\xi,\epsilon\right)\vbox{\hbox{
\leavevmode\hbox to71.06pt{\vbox
to72.11pt{\pgfpicture\makeatletter\hbox{\hskip 38.2014pt\lower-36.23021pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-13.36995pt}{-19.5129pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.13}]{figures/triangle_collinear_anomaly.png}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-34.5559pt}{-30.95276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\xi p_{1}^{h_{1}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-34.8684pt}{26.29999pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\bar{\xi}p_{1}^{h_{2}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.69168pt}{-30.60553pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{2}^{b}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.69168pt}{26.66943pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{3}^{c}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\,,$ (73)

where the one-loop collinear function $\Omega^{\text{1L}}$ is given by

$\displaystyle\begin{aligned}
\Omega^{\text{1L}}\left(\xi,\epsilon\right)&=\left[-\epsilon\Gamma(\epsilon-1)e^{\epsilon\gamma_{\text{E}}}\right]\,(\xi\bar{\xi})^{1-\epsilon}\\\
&=\xi\bar{\xi}+\epsilon\,\xi\bar{\xi}\left[1-\log(\xi\bar{\xi})\right]+{\cal
O}(\epsilon^{2})\,.\end{aligned}$ (74)

This is in agreement with the six-dimensional eq. (70), and generalises it by
also taking into account $\epsilon$-corrections which are necessary to go to
higher orders in $\epsilon$ at the conformal fixed point.

Two-loop example: As a more nontrivial example, we consider a two-loop
triangle which is one of several Feynman diagrams contributing to the
correlator $G^{(3)}$ at the two-loop order,

$\displaystyle\vbox{\hbox{ \leavevmode\hbox to109.81pt{\vbox
to60.02pt{\pgfpicture\makeatletter\hbox{\hskip 54.1955pt\lower-30.1925pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.3}]{figures/fig_2loop_triang.pdf}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-50.86249pt}{-1.78333pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{1}^{a}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{44.45398pt}{-24.91505pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{2}^{b}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{44.45398pt}{20.97896pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{3}^{c}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=\textup{i}gu^{2}\left(\dfrac{\textsc{n}^{2}-12}{2\textsc{n}}\right)^{2}d^{abc}J$
(75)

where $J$ is the following two-loop scalar Feynman integral,

$\displaystyle
J=e^{2\epsilon\gamma_{\text{E}}}\int\frac{\mathrm{d}^{d}k\,\mathrm{d}^{d}l}{\bigl{(}\textup{i}\pi^{\frac{d}{2}}\bigr{)}^{2}}\frac{1}{k^{2}\,l^{2}\,(k+l)^{2}(k+p_{1})^{2}(l-p_{1})^{2}(l+p_{2})^{2}}\,.$
(76)

The propagators are assigned small imaginary parts as in eq. (29). The
asymptotics of $J$ at $p_{1}^{2}\equiv s_{1}\to 0$ is similar to that of the
one-loop triangle given in eq. (39). The new feature is that there are two
regions of loop integrations which are responsible for the collinear terms. We
refer to them as one-loop-collinear and two-loop-collinear,

$\displaystyle J=\sum_{m\geq 0}(-s_{1})^{m}J^{\text{hard}}_{[m]}+\sum_{m\geq
0}(-s_{1})^{1-\epsilon+m}J^{\text{1L-coll}}_{[m]}+\sum_{m\geq
0}(-s_{1})^{1-2\epsilon+m}J^{\text{2L-coll}}_{[m]}\,.$ (77)

Both of them contribute to the $s_{1}\log(-s_{1})$-term of the asymptotics,

$\displaystyle\left[J\right]_{s_{1}\log(-s_{1})}=\epsilon
J^{\text{1L-coll}}_{[0]}+2\epsilon J^{\text{2L-coll}}_{[0]}\,,$ (78)

which is responsible for the collinear part of the amplitude conformal
anomaly. The expansion by regions analysis yields the following expressions
for the one-loop-collinear and the two-loop-collinear contributions,

$\displaystyle\textup{i}gu^{2}\left(\dfrac{\textsc{n}^{2}-12}{2\textsc{n}}\right)^{2}d^{abc}\,J^{\text{1L-coll}}_{[0]}=\frac{g\,d^{ah_{1}h_{2}}}{(4\pi)^{3}}\int^{1}_{0}\mathrm{d}\xi\,\Omega^{\text{1L}}(\xi,\epsilon)\vbox{\hbox{
\leavevmode\hbox to93.82pt{\vbox
to66.42pt{\pgfpicture\makeatletter\hbox{\hskip
49.58232pt\lower-33.38475pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.42}]{figures/fig_box_coll.pdf}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-45.93681pt}{-28.1073pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\xi p_{1}^{h_{1}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-46.24931pt}{23.45453pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\bar{\xi}p_{1}^{h_{2}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{33.07304pt}{-27.76007pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{2}^{b}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{33.07304pt}{23.82397pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{3}^{c}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$ (79)

where $\Omega^{\text{1L}}(\xi,\epsilon)$ is the same as in the one-loop
example (74), and

$\displaystyle\textup{i}gu^{2}\left(\dfrac{\textsc{n}^{2}-12}{2\textsc{n}}\right)^{2}d^{abc}J^{\text{2L-coll}}_{[0]}=\frac{gu\,d^{ah_{1}h_{2}}}{(4\pi)^{3}}\int^{1}_{0}\mathrm{d}\xi\,\Omega^{\text{2L}}_{\rm
dbl\;triang}(\xi,\epsilon)\vbox{\hbox{ \leavevmode\hbox to71.06pt{\vbox
to72.11pt{\pgfpicture\makeatletter\hbox{\hskip 38.2014pt\lower-36.23021pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-13.36995pt}{-19.5129pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.13}]{figures/triangle_collinear_anomaly.png}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-34.5559pt}{-30.95276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\xi p_{1}^{h_{1}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-34.8684pt}{26.29999pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\bar{\xi}p_{1}^{h_{2}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.69168pt}{-30.60553pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{2}^{b}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.69168pt}{26.66943pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p_{3}^{c}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$ (80)

with the two-loop collinear function

$\displaystyle\Omega^{\text{2L}}_{\rm
double\;triangle}(\xi,\epsilon)=\dfrac{\textsc{n}^{2}-12}{4\textsc{n}}\left[\frac{2\xi\bar{\xi}}{\epsilon}+\left(10\xi\bar{\xi}-3\xi\bar{\xi}\log(\xi\bar{\xi})+\xi\log(\xi)+\bar{\xi}\log(\bar{\xi})\right)+{\cal
O}(\epsilon)\right]\,.$ (81)

There are two different contributions to the $\epsilon$-pole in the previous
expression. One comes from the UV-divergence of the one-loop triangle
subdiagram of (75), and the other stems from the collinear divergence in the
one-loop box subdiagram of (75). The UV-divergence is cancelled out by the
corresponding counter-term diagram, which is the one-loop diagram (71) with
the counter-term vertex. The collinear function $\Omega$ for the counter-term
diagram contains an $\epsilon$-pole which cancels out the UV-contribution to
the pole in eq. (81). The cancellation of the collinear contribution to the
pole is discussed below.

Equations (73), (79), and (80) are very suggestive. The term
$s_{1}\log(-s_{1})$ in the asymptotics of a Feynman diagram comes from
subdiagrams with all internal momenta localised on a collinear configuration,
which turn into a function $\Omega$ of the collinear splitting parameter $\xi$
and $\epsilon$.

Figure 5: Subdiagrams with five propagators which contribute to the two-loop
collinear function $\Omega$. They are to be complemented with the
corresponding counter-term diagrams. In the relevant region of loop
integrations all internal momenta are collinear with $p$.

Besides the Feynman diagram (75), there are several other Feynman diagrams
(including those with counter-term vertices) contributing to the three-point
correlator $G^{(3)}$ at the two-loop order. Their asymptotics at $s_{1}\to 0$
has the same form as in (77). Calculating the $\Omega$ function for each of
them and taking their sum, we obtain the full two-loop collinear function (see
figure 5),

$\displaystyle\Omega^{\text{2L}}(\xi,\epsilon)=\,$
$\displaystyle\frac{\textsc{n}^{2}-12}{4\textsc{n}}\frac{\xi\bar{\xi}}{\epsilon}+\frac{59\textsc{n}^{2}-884}{36\textsc{n}}\xi\bar{\xi}-\frac{5\textsc{n}^{2}-68}{12\textsc{n}}\xi\bar{\xi}\log(\xi\bar{\xi})$
$\displaystyle+\frac{\textsc{n}^{2}-12}{4\textsc{n}}\left[\xi\log(\xi)+\bar{\xi}\log(\bar{\xi})\right]+{\cal
O}(\epsilon)\,.$ (82)

The $\epsilon$-pole in the two-loop $\Omega$ is of collinear origin.

Power counting in the collinear regions: A simple power-counting argument
enables us to determine the asymptotics of a given collinear region of loop
integrations. Let us consider a connected subdiagram ${\cal H}$ with $E$
external points and involving $N$ propagators. The subdiagram ${\cal H}$ is
adjacent to the corner with inflowing momentum $p^{\mu}$, which approaches the
light-cone $p^{2}\sim 0$ (see the one- and two-loop examples in figures 4 and
5). We keep in mind that ${\cal H}$ is to be attached to a diagram ${\cal G}$
such that all momenta flowing through ${\cal H}$ can be collinear with
$p^{\mu}$ whereas the momenta flowing through ${\cal G}$ are generic, and
$\ell$ independent momenta flowing through ${\cal H}$ are integrated
over.888Let us note that $\ell$ is not equal to the number of loops in ${\cal
H}$, i.e. the independent nontrivial one-cycles. There are $\ell+2-E$
independent nontrivial one-cycles in ${\cal H}$. The region of loop
integrations over $\ell$ momenta of ${\cal H}$, where the momenta flowing
through all $N$ propagators are collinear with $p^{\mu}$, has the asymptotics

$\displaystyle\left(\frac{1}{p^{2}}\right)^{N}\left(\mathrm{d}^{d}p\right)^{\ell}\sim(p^{2})^{E-2-\ell\epsilon}\,.$
(83)

Here we applied standard graph relations to obtain $3\ell-N=E-2$. Expanding in
$\epsilon$ the right-hand side of eq. (83), we see that the $p^{2}\log(p^{2})$
terms come from the subdiagrams with $E=3$. The subdiagrams in figures 4 and 5
have $E=3$ external points, corresponding to momenta $p,\,\xi p,\,\bar{\xi}p$.
The subdiagrams with $E>3$, where $p^{\mu}$ is split into $E-1$ collinear
fractions, contribute to $(p^{2})^{E-2}\log(p^{2})$, and so are more
suppressed as compared to $p^{2}\log(p^{2})$ and do not contribute to the
conformal anomaly.

UV-finite Feynman diagrams in $d=6$ do not contain two- and three-point
subdiagrams with nontrivial one-cycles. We therefore conclude that only
subdiagrams with $\ell=1$ of finite Feynman integrals have $E=3$. See e.g.
figure 4. As a result, in $d=6$ the collinear anomaly is a “short-range”
effect, namely it receives contribution only from the collinear region of the
loop integration adjacent to the on-shell momentum. However, generic Feynman
graphs in the $d=6-2\epsilon$ dimensional model (1) contain multi-loop UV-
divergent subgraphs with $E=3$ and $\ell>1$. Figure 5 shows examples with
$\ell=2$. Thus, to study the collinear part of the conformal anomaly we have
to take into account “long-range” effects as well.

Finally, let us note that the counting of eq. (83) changes for UV-finite
Feynman integrals in $d=4$ Yukawa theory. As a result, the “long-range”
effects impact the collinear anomaly, namely subdiagrams with $\ell>1$ have to
be taken into account as well.

General structure: Based on the previous one-loop and two-loop examples, and
the power counting argument, let us summarise the outcome of expansion by
region analysis for a generic multi-loop Feynman diagram. In order to describe
the $p^{2}\log(-p^{2})$ contribution to the asymptotics of a Feynman diagram
${\cal F}$ at $p^{2}\sim 0$, we split ${\cal F}$ into a subdiagram ${\cal G}$
with $n+2$ inflowing momenta $q,p-q,p_{1},\ldots,p_{n}$, and the complementary
subdiagram $({\cal F}\backslash{\cal G})$ with three inflowing momenta
$p,-q,-p+q$.999In the notations of the previous paragraph ${\cal H}=({\cal
F}\backslash{\cal G})$ and $E=3$. The subdiagram ${\cal G}$ is
$\ell^{\prime}$-loop ($0\leq\ell^{\prime}<\ell$), and $(\ell-\ell^{\prime})$
independent loop momenta flow through $({\cal F}\backslash{\cal G})$. The
subdiagram $({\cal F}\backslash{\cal G})$ contains $3(\ell-\ell^{\prime})-1$
propagators. The terms $p^{2}\log(-p^{2})$ come from the region of loop
integrations of the subdiagram $({\cal F}\backslash{\cal G})$ where all
propagators carry momenta collinear with $p$, so its inflowing momenta are
also collinear: $p,-\xi p,-\bar{\xi}p$. This collinear region of loop
integrations results in a function $\Omega_{{\cal F}\backslash{\cal G}}(\xi)$,
and summing over all possible ways of splitting ${\cal F}$ into ${\cal G}$ and
$({\cal F}\backslash{\cal G)}$ as described above we find

$\displaystyle\left[{\cal
F}(p^{a},p_{1}^{a_{1}},\ldots,p_{n}^{a_{n}})\right]_{p^{2}\log(-p^{2})}=\frac{gd^{abc}}{(4\pi)^{3}}\sum_{{\cal
G}\subset{\cal F}}\int\limits^{1}_{0}\mathrm{d}\xi\,\Omega_{{\cal
F}\backslash{\cal G}}(\xi,\epsilon,u)\,{\cal G}(\xi
p^{b},\bar{\xi}p^{c},p_{1}^{a_{1}},\ldots,p_{n}^{a_{n}})\,.$ (84)

So far we have considered the asymptotics of individual Feynman diagrams. It
is insightful to sum eq. (84) over all Feynman diagrams ${\cal F}$
contributing to the amputated renormalised $(n+1)$-point correlator
$G^{(n+1)}$, obtaining

$\displaystyle\left[G^{(n+1)}(p^{a},p_{1}^{a_{1}},\ldots,p_{n}^{a_{n}})\right]_{p^{2}\log(-p^{2})}=\frac{gd^{abc}}{(4\pi)^{3}}\int^{1}_{0}\mathrm{d}\xi\,\Omega(\xi,\epsilon,u)\,G^{(n+2)}_{\rm
reg}(\xi p^{b},\bar{\xi}p^{c},p_{1}^{a_{1}},\ldots,p_{n}^{a_{n}})\,.$ (85)

The function $\Omega(\xi,\epsilon,u)$ is the sum of $\Omega_{{\cal
F}\backslash{\cal G}}$ over all three-point collinear subdiagrams. The
“regularised” amputated correlator $G^{(n+2)}_{\rm reg}$ is the $(n+2)$-point
amputated correlator $G^{(n+2)}(q_{1},q_{2},p_{1},\ldots,p_{n})$ calculated in
the collinear configuration $q_{1}=\xi p,\,q_{2}=\bar{\xi}p$. Moreover, the
Feynman diagrams contributing to $G^{(n+2)}$ which are trivially singular as
$q_{1}$ and $q_{2}$ are collinear to $p$ are dropped out.101010The singular
Feynman diagrams are those involving the propagator $1/(q_{1}+q_{2})^{2}$. Let
us stress that usually we cannot take the expression for the renormalised
correlator $G^{(n+2)}$ in generic kinematics and substitute the collinear
configuration $q_{1}=\xi p,\,q_{2}=\bar{\xi}p$ into it. That would result into
collinear divergences which should be properly regularised. Thus, there are
$\epsilon$-poles in $G_{\rm reg}$ which are due to collinear divergences. They
are compensated by the collinear $\epsilon$-poles in $\Omega$. The terms of
order $u^{0}$ and $u^{1}$ of $\Omega$, required to compute the collinear
anomaly up to two-loop order, are given by eqs. (74) and (3.4.3),
respectively.

In conclusion, we find that the $p^{2}\log p^{2}$-term of the asymptotic
expansion of the (renormalised) amputated correlator $G^{(n)}$ is given by the
convolution of a universal function, $\Omega$, with a $(n+1)$-point
regularised correlator with two of the incoming momenta carrying fractions of
the on-shell momentum $p$. Because of the overall factor of $g$ on the right-
hand side of eq. (85), and of the additional particle in the correlator under
convolution, computing the collinear part of the conformal anomaly for an
amplitude at a given loop order requires lower-loop information only.

###  3.5 Applications of the conformal Ward identity

In this section we give three instructive examples of the computation of the
conformal anomaly (66). First we consider the three-point amplitude with one
on-shell leg, to all orders using the method of expansion by regions in
section 3.5.1, and perturbatively at one loop in section 3.5.2. Then we study
the fully on-shell four-point amplitude perturbatively at one loop in section
3.5.3.

####  3.5.1 Exact solution of the anomalous Ward identity for the three-point
amplitude

In this section we derive the exact, closed-form expressions of the anomaly
$\mathcal{A}^{(3)\mu}$ and of the amplitude $M^{(3)}$ in eq. (52). We recall
that we put on-shell only the leg with momentum $p_{1}$. Remarkably, the
anomalous CWI (52) constrains both the amplitude and the anomaly up to two
kinematic-independent coefficients. The computation of the anomaly is then
required only to fix these coefficients.

The starting point are the two scalar anomalous CWIs given by eqs. (50) and
(51). We denote by $A^{(3)}$ the anomaly on the right-hand side of the latter,
namely

$\displaystyle A^{(3)}=4\gamma
G^{(3)}_{1;0}(s_{2},s_{3})+4G_{1;1}^{(3)}(s_{2},s_{3})\,.$ (86)

It is convenient to define dimensionless functions of the ratio
$y=s_{2}/s_{3}$ as

$\displaystyle
M^{(3)}(s_{2},s_{3})=(s_{2}s_{3})^{\frac{\epsilon-3\gamma}{4}}\tilde{M}^{(3)}(y)\,,$
(87) $\displaystyle
A^{(3)}(s_{2},s_{3})=-s_{2}(s_{2}s_{3})^{\frac{\epsilon-3\gamma}{4}-1}\tilde{A}^{(3)}(y)\,.$
(88)

The dimensionful factor multiplying $\tilde{M}^{(3)}(y)$ is chosen so as to be
invariant under swapping $p_{2}$ and $p_{3}$. As a result,
$\tilde{M}^{(3)}(y)$ is symmetric under $y\to 1/y$. Substituting these
definitions into the scalar Ward identities gives the following system of DEs,

$\displaystyle\begin{cases}\Bigl{[}16(1-y)y^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}y^{2}}+8y\left[\epsilon-\gamma+y(\epsilon-\gamma-4)\right]\frac{\mathrm{d}}{\mathrm{d}y}+(1-y)(\epsilon-3\gamma)(\epsilon+\gamma-4)\Bigr{]}\tilde{M}^{(3)}(y)=0\,,\\\\[10.00002pt]
\Bigl{[}16y^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}y^{2}}+8y(\epsilon-\gamma)\frac{\mathrm{d}}{\mathrm{d}y}+(\epsilon-3\gamma)(\epsilon+\gamma-4)\Bigr{]}\tilde{M}^{(3)}(y)=-4y\tilde{A}^{(3)}(y)\,.\\\
\end{cases}$ (89)

We can combine these two equations so as to eliminate the second derivative,
obtaining an equation relating the derivative of the amplitude
$\tilde{M}^{(3)}(y)$ directly to the anomaly $\tilde{A}^{(3)}(y)$,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}y}\tilde{M}^{(3)}(y)=\frac{(1-y)}{4y(\epsilon-\gamma-2)}\tilde{A}^{(3)}(y)\,.$
(90)

The second independent equation can then be chosen to be a second-order ODE
for either the amplitude alone (the first equation of the system (89)), or the
anomaly alone,

$\displaystyle\biggr{\\{}$ $\displaystyle
16(1-y)y^{2}\frac{\mathrm{d}^{2}}{\mathrm{d}y^{2}}+8y\left[\epsilon-\gamma+y(\epsilon-\gamma-8)\right]\frac{\mathrm{d}}{\mathrm{d}y}+$
(91) $\displaystyle
y(3\gamma+4-\epsilon)(\epsilon+\gamma-8)+(\epsilon-3\gamma)(\epsilon+\gamma-4)\biggr{\\}}\tilde{A}^{(3)}(y)=0\,,$

which can be viewed as a consistency condition for the anomaly. Remarkably,
thus, conformal symmetry puts very strong constraints also on the anomaly.

Both the consistency condition (91) for the anomaly and the second-order ODE
for the amplitude in the system (89) can be solved in terms of ordinary
hypergeometric functions, as

$\displaystyle\tilde{A}^{(3)}(y)=y^{\frac{3\gamma-\epsilon}{4}}\biggl{\\{}$
$\displaystyle
a_{1}(\epsilon,\gamma)\,{}_{2}F_{1}\left(2+\frac{\gamma-\epsilon}{2},1+\frac{3\gamma-\epsilon}{2},4+\gamma-\epsilon;1-y\right)+$
(92)
$\displaystyle+a_{2}(\epsilon,\gamma)\,(1-y)^{\epsilon-\gamma-3}\,{}_{2}F_{1}\left(\frac{\epsilon-\gamma}{2}-1,\frac{\gamma+\epsilon}{2}-2,\epsilon-\gamma-2;1-y\right)\biggr{\\}}\,,$

and

$\displaystyle\tilde{M}^{(3)}(y)=y^{\frac{3\gamma-\epsilon}{4}}\biggl{\\{}$
$\displaystyle
b_{1}(\epsilon,\gamma)\,{}_{2}F_{1}\left(1+\frac{\gamma-\epsilon}{2},\frac{3\gamma-\epsilon}{2},2+\gamma-\epsilon;1-y\right)+$
(93)
$\displaystyle+b_{2}(\epsilon,\gamma)\,(1-y)^{\epsilon-\gamma-1}\,{}_{2}F_{1}\left(\frac{\epsilon-\gamma}{2},\frac{\epsilon+\gamma}{2}-1,\epsilon-\gamma;1-y\right)\biggr{\\}}\,,$

where $a_{i}(\epsilon,\gamma)$ and $b_{i}(\epsilon,\gamma)$ are arbitrary
kinematic-independent coefficients. By plugging these solutions into eq. (90)
we can then relate the coefficients of the amplitude to those of the anomaly,

$\displaystyle
b_{1}(\epsilon,\gamma)=\frac{4(3-\epsilon+\gamma)}{(3\gamma-\epsilon)(2+\gamma-\epsilon)(\epsilon+\gamma-4)}\,a_{1}(\epsilon,\gamma)\,,$
(94) $\displaystyle
b_{2}(\epsilon,\gamma)=\frac{a_{2}(\epsilon,\gamma)}{4(2+\gamma-\epsilon)(\epsilon-\gamma-1)}\,.$

The term proportional to $b_{2}(\epsilon,\gamma)$ on the right-hand side of
eq. (93) is singular at $y=1$, or equivalently at $s_{2}=s_{3}$. The amplitude
is free of this singularity, then

$\displaystyle b_{2}(\epsilon,\gamma)\equiv 0\,,$ (95)

and also $a_{2}(\epsilon,\gamma)\equiv 0$. We thus see that conformal symmetry
— supplemented with a simple physical argument — constrains $M^{(3)}$ up to
the kinematic-independent normalisation without requiring any knowledge of the
anomaly. The latter fixes the overall factor.

We compute the closed-form expression of the anomaly using the method of the
expansion by regions Beneke:1997zp ; Smirnov:1999bza ; Jantzen:2011nz . We
discuss this in appendix E.2, and give here the resulting expressions for the
anomaly coefficients:

$\displaystyle
a_{1}(\epsilon,\gamma)=\frac{\tilde{c}_{123}}{(\tilde{c}_{12})^{3}}(3\gamma-\epsilon)(\gamma+\epsilon-4)\frac{e^{\epsilon\gamma_{\text{E}}}\Gamma\left(1-\gamma\right)\,\Gamma^{2}\left(2-\frac{\epsilon}{2}+\frac{\gamma}{2}\right)}{\Gamma\left(4-\epsilon+\gamma\right)\,\Gamma^{2}\left(2-\frac{\epsilon}{2}-\frac{\gamma}{2}\right)}\,,$
(96) $\displaystyle a_{2}(\epsilon,\gamma)=0\,.$

Finally, by putting eqs. (96), (94), (93) and (87) together we obtain the
closed-form expression of the amplitude:

$\displaystyle M^{(3)}(s_{2},s_{3})=$ $\displaystyle
4\frac{\tilde{c}_{123}}{(\tilde{c}_{12})^{3}}\frac{\epsilon-\gamma-3}{\epsilon-\gamma-2}\frac{e^{\epsilon\gamma_{\text{E}}}\Gamma\left(1-\gamma\right)\,\Gamma^{2}\left(2-\frac{\epsilon}{2}+\frac{\gamma}{2}\right)}{\Gamma\left(4-\epsilon+\gamma\right)\,\Gamma^{2}\left(2-\frac{\epsilon}{2}-\frac{\gamma}{2}\right)}(-s_{3})^{\frac{\epsilon-3\gamma}{2}}\times$
(97)
$\displaystyle\qquad{}_{2}F_{1}\left(1-\frac{\epsilon}{2}+\frac{\gamma}{2},\frac{3\gamma}{2}-\frac{\epsilon}{2},2-\epsilon+\gamma;1-\frac{s_{2}}{s_{3}}\right)\,.$

We validated this result with an independent computation done with the method
of the expansion by regions, as discussed in appendix E.2. Using the explicit
expressions for the normalisation factors $\tilde{c}_{12}$ and
$\tilde{c}_{123}$ — given in eqs. (163) and (166) up to order $\epsilon^{2}$ —
we can expand the amplitude around $\epsilon=0$.111111To expand the
hypergeometric functions we used the Mathematica package HypExp Huber:2005yg ;
Huber:2007dx . Up to order $\epsilon$ it is given by

$\displaystyle
M^{(3)}(s_{2},s_{3})=\textup{i}g^{*}\biggl{[}1+u^{*}\frac{\textsc{n}^{2}-12}{4\textsc{n}}\frac{3s_{2}-3s_{3}-s_{2}\log(-s_{2})+s_{3}\log(-s_{3})}{s_{2}-s_{3}}+\mathcal{O}\left((u^{*})^{2}\right)\biggr{]}\,,$
(98)

which we cross-checked against a direct one-loop perturbative computation.

In this subsection we have shown that conformal symmetry fixes both the three-
point amplitude at the conformal fixed point and its conformal anomaly to all
orders in $\epsilon$ up to two kinematic-independent coefficients, which we
computed in closed form.

####  3.5.2 Conformal Ward identity for the one-loop three-point amplitude

We now re-calculate the one-loop approximation of the conformal anomaly of the
three-point amplitude $M^{(3)}$ (see eq. (52)). We label the momenta following
the notation of section 3.4, and consider the three-point correlator
$G^{(3)}(p,p_{1},p_{2})$ at $p^{2}\sim 0$. We would like to apply eq. (85).
Since we aim for the one-loop approximation, we need the lowest order
approximation of $\Omega$, namely $\Omega^{\text{1L}}$ from eq. (74), and the
lowest order approximation for $G^{(4)}$, namely the tree-level approximation,

$\displaystyle G^{(4)}_{\rm
reg}(q_{1}^{b},q_{2}^{c},p_{1}^{a_{1}},p_{2}^{a_{2}})=-\textup{i}g^{2}\left[\frac{d^{a_{1}bf}d^{a_{2}cf}}{(q_{1}+p_{1})^{2}}+\frac{d^{a_{1}cf}d^{a_{2}bf}}{(q_{2}+p_{1})^{2}}\right]\,,$
(99)

where we discarded the contribution from the third crossing channel since it
is singular where $q_{1}$ and $q_{2}$ are collinear with $p$. Substituting all
these ingredients into eq. (70) gives

$\displaystyle\left[G^{(3)}(p^{a},p_{1}^{a_{1}},p_{2}^{a_{2}})\right]_{p^{2}\log(-p^{2})}$
$\displaystyle=-\textup{i}\frac{g^{3}d^{aa_{1}a_{2}}}{(4\pi)^{3}}\frac{\textsc{n}^{2}-12}{2\textsc{n}}\int^{1}_{0}\mathrm{d}\xi\,\frac{2\xi\bar{\xi}}{\xi
s_{1}+\bar{\xi}s_{2}}+{\cal O}(g^{3}\epsilon)$ (100)
$\displaystyle=-\textup{i}gud^{aa_{1}a_{2}}\frac{\textsc{n}^{2}-12}{2\textsc{n}}\frac{s_{1}^{2}-s_{2}^{2}-2s_{1}s_{2}\log(s_{1}/s_{2})}{(s_{1}-s_{2})^{3}}+{\cal
O}(gu\epsilon)\,.$

At the conformal fixed point $u=u^{*}$ (25), the $p^{2}\log(-p^{2})$ term in
the asymptotics of $G^{(3)}$ is the collinear contribution in the conformal
anomaly, which is denoted as $G^{(3)}_{1;1}$. As for the second contribution
to the anomaly, $G^{(3)}_{1;0}$ in eq. (53), note that the amputated three-
point correlator is a constant at the tree-level,

$\displaystyle
G^{(3)}(p^{a},p_{1}^{a_{1}},p_{2}^{a_{2}})=\textup{i}gd^{aa_{1}a_{2}}+{\cal
O}(g^{3})\,.$ (101)

Consequently, the coefficients $G_{m;k}^{(3)}$ with $m\geq 1$ and $k\geq 0$ in
the asymptotic expansion (49) at $p^{2}\sim 0$ are of order ${\cal O}(g^{3})$.
This is in agreement with the explicit expression for $G_{1;1}^{(3)}$ in eq.
(100). The perturbative expansion of the coefficient $G_{1;0}^{(3)}$ also
starts at order ${\cal O}(g^{3})$. The contribution $\gamma\,G_{1;0}^{(3)}$ in
the conformal anomaly (53) at the conformal fixed point is therefore of order
${\cal O}(\epsilon^{2})$, and is thus neglected in the one-loop approximation.
Finally, we find the conformal anomaly (53),

$\displaystyle\mathcal{A}^{(3)\mu}=\textup{i}g^{*}p^{\mu}\biggl{\\{}u^{*}\frac{\textsc{n}^{2}-12}{\textsc{n}}\frac{s_{1}^{2}-s_{2}^{2}-2s_{1}s_{2}\log(s_{1}/s_{2})}{(s_{1}-s_{2})^{3}}+\mathcal{O}\left(\epsilon^{2}\right)\biggr{\\}}\,,$
(102)

which agrees with the $K^{\mu}_{d-\Delta_{\phi}}$-variation of the one-loop
expression of $M^{(3)}$ given in eq. (98), and with the all-order result
obtained in section 3.5.1 (given by eqs. (92) and (96)).

####  3.5.3 Conformal Ward identity for the one-loop four-point amplitude

As a simple illustration of the CWI beyond three points, we consider the four-
point amputated correlator $G^{(4)}$. As independent kinematic variables, we
choose the standard bi-particle Mandelstam variables,

$\displaystyle s=(p_{1}+p_{2})^{2},\quad t=(p_{2}+p_{3})^{2}\,,$ (103)

complemented by $p_{i}^{2}$ with $i=1,\ldots,4$. We define the four-point
amplitude $M^{(4)}$ by putting on-shell all four legs of the amputated
correlator,

$\displaystyle M^{(4)}(s,t)=\left(\prod_{i=1}^{4}\lim_{p_{i}^{2}\to
0}\right)G^{(4)}\left(s,t,p_{1}^{2},p_{2}^{2},p_{3}^{2},p_{4}^{2}\right)\,.$
(104)

Rewriting the conformal boost generator with conformal weight
$\Delta=d-\Delta_{\gamma}$ in the Mandelstam variables as in eqs. (59) and
(60), we find the second-order scalar differential operators in the variables
$s,t$ which play the role of the conformal boost generators for the amplitude,

$\displaystyle\mathbb{K}^{(1)}=4s\partial_{s}^{2}+4t\partial_{t}\partial_{s}+4(2+2\gamma-\epsilon)\partial_{s}\,,$
$\displaystyle\mathbb{K}^{(3)}=\mathbb{K}^{(1)}|_{s\leftrightarrow
t}\,,\qquad\mathbb{K}^{(2)}=\mathbb{K}^{(1)}+\mathbb{K}^{(3)}\,.$ (105)

Since the momentum $p_{4}^{\mu}$ does not appear in the chosen on-shell
Mandelstam variables, $\mathbb{K}^{(4)}\equiv 0$. In other words,
$p_{4}^{\mu}$ is eliminated by the total momentum conservation. Then the
anomalous CWI for the amplitude $M^{(4)}$ (104) takes the following form,

$\displaystyle\left(\sum_{i=1}^{3}p_{i}^{\mu}\,\mathbb{K}^{(i)}\right)M^{(4)}\left(s,t\right)={\cal
A}^{(4)\mu}\,.$ (106)

The situation of the four-point amplitude with all legs on shell is special.
In this case the anomaly in fact vanishes, ${\cal A}^{(4)\mu}=0$. Indeed, it
is easy to check that, for any function $f$,

$\displaystyle\mathbb{K}^{(i)}\,s^{-1+\epsilon-2\gamma}\,f\left(\frac{s}{t}\right)=0\,,\qquad
i=1,2,3\,,$ (107)

where the overall factor of $s$ carries the energy dimensions of the
amplitude. We verify explicitly up to one-loop order that all nontrivial
contributions to the anomaly cancel out.

According to eq. (65) there are several contributions to the anomaly ${\cal
A}^{(4)\mu}$. They are the terms in the asymptotic expansion of the correlator
at $p_{i}^{2}\to 0$,

$\displaystyle
G^{(4)}=M^{(4)}\left(s,t\right)+\sum_{i=1}^{4}\left(p_{i}^{2}\,G^{(4)}_{i;\text{pow}}(s,t)+p_{i}^{2}\log\left(-p_{i}^{2}\right)\,G^{(4)}_{i;\text{coll}}(s,t)\right)+\ldots\,.$
(108)

Despite the fact that the coefficients $G^{(4)}_{i;\text{coll}}$ and
$G^{(4)}_{i;\text{pow}}$ are nontrivial, the anomaly is zero. Indeed, the
contributions to the asymptotics (108) from each of the four legs are
identical, and the anomaly vanishes due to the total momentum conservation.
For the sake of illustration, we present here the coefficients
$G^{(4)}_{i;\text{coll}}$ and $G^{(4)}_{i;\text{pow}}$. Expanding the tree-
level correlator in small $p_{i}^{2}$ we find

$\displaystyle\frac{\textup{i}}{g^{2}}G^{(4)}_{i;\text{pow}}(s,t)=-\frac{d^{acf}d^{bdf}}{(s+t)^{2}}+{\cal
O}(u)\,,\quad i=1,\ldots,4\,.$ (109)

There are no $p^{2}_{i}\log(-p^{2}_{i})$ terms in the asymptotic expansion of
the tree-level correlator. Such terms appear in the one-loop correlator. They
are captured by the collinear anomaly mechanism of section 3.4. For example,
the one-loop contribution from leg $1$ is

$\displaystyle
G^{(4)}_{1;\text{coll}}(s,t)=\frac{\textup{i}gd^{ahg}}{(4\pi)^{3}}\int^{1}_{0}\mathrm{d}\xi\,\xi\bar{\xi}\,M^{(5)}_{\rm
reg}\left(\xi
p_{1}^{h},\bar{\xi}p_{1}^{g},p_{2}^{b},p_{3}^{c},p_{4}^{d}\right)+g^{2}{\cal
O}(u^{2})\,,$ (110)

where $M^{(5)}_{\rm reg}$ is the five-point tree-level amplitude with the
Feynman diagrams which diverge in the forward limit dropped. Pictorially we
have

$\displaystyle\left[G^{(4)}_{1;\text{coll}}\right]_{\text{1-loop}}=\frac{\textup{i}gd^{ahg}}{(4\pi)^{3}}\int^{1}_{0}\mathrm{d}\xi\,\xi\bar{\xi}\left[\begin{array}[]{c}\includegraphics[width=79.6678pt]{figures/5ptampl_fig1.pdf}\end{array}+\begin{array}[]{c}\includegraphics[width=79.6678pt]{figures/5ptampl_fig2.pdf}\end{array}+\text{cross
terms}\right],$ (113)

where the cross-terms are permutations of the legs $2,3,4$, i.e. permutations
of $p_{2}^{b},p_{3}^{c},p_{4}^{d}$. Carrying out the one-fold integration
gives

$\displaystyle\frac{\textup{i}}{g^{2}}G^{(4)}_{i;\text{coll}}(s,t)=u\left[-\frac{\textsc{n}^{2}-12}{4\textsc{n}}\frac{d^{abf}d^{cdf}}{s^{2}}-d^{aef}d^{bfg}d^{cgh}d^{dhe}\frac{1}{st}+\text{cross
terms}\right]+{\cal O}(u^{2})\,.$ (114)

We see from eqs. (109) and (114) that the contributions to the anomaly from
the $i$-th on-shell leg are independent of $i$, and hence sum to zero in the
anomaly $\mathcal{A}^{(4)}$ because of momentum conservation. This completes
the check.

##  4 Discussion and outlook

In this paper we analysed quantum field theories at a conformal fixed point in
dimensional regularisation. We derived conformal symmetry constraints for on-
shell scattering amplitudes, in the form of anomalous conformal Ward
identities.

Previously, studies of implications of conformal symmetry in momentum space
were mostly restricted to off-shell quantities. It turns out that the
interplay of conformal symmetry with the on-shell limit is rather subtle. We
carefully studied how conformal symmetry acts on LSZ–reduced amplitudes. We
found that, in the case where the scattered particles have an anomalous
dimension, the conformal symmetry generator does not commute with the on-shell
condition $p^{2}=0$. As a consequence of this non-commutativity, an anomalous
term appears in the conformal Ward identity.

One of the main results of this paper is the anomaly formula eq. (65) for
general $n$-particle amplitudes, generalising to the $d$-dimensional case a
previous result due to ref. Chicherin:2017bxc , which is valid for finite
conformal loop integrals. We showed that the anomaly in eq. (66) is comprised
of two terms that have different physical origin: while the first term is
proportional to the elementary field anomalous dimension, the second term can
be traced to certain collinear regions of the loop momenta. Importantly, for
both terms, at $L$-loops, only $(L-1)$-loop information is required to compute
the anomaly. This makes our anomaly formula useful for practical computations.

Furthermore, we showed that for $n$-particle amplitudes the conformal anomaly
is a sum of local contributions. In particular, leveraging the method of
regions, we found that the collinear anomaly term can be written as a
convolution of lower-loop amplitudes over a certain collinear kernel,
$\Omega$, cf. eq. (85), which we computed to two loops in perturbation theory.
This provides an important piece of universal information for using our
conformal Ward identities in practice.

There are a number of interesting future directions:

  * •

Test our $n$-point anomaly formula for one-loop $\phi^{3}$ amplitudes, and use
it to bootstrap two-loop amplitudes.

  * •

Understand the collinear kernel $\Omega$ more generally, and explore its
relationship to splitting functions considered in the QCD literature.

  * •

Extend the methods developed in this work to gauge theory scattering
amplitudes (which are famously conformally invariant at tree level
Witten:2003nn ). A natural starting point may be the finite one-loop all-plus
and single-minus amplitudes. It is known that the one-loop all-plus helicity
amplitudes are conformally invariant Henn:2019mvc , but the single-minus
helicity amplitudes are not. Understanding the origin of its non-invariance
would be very interesting, and may shed light on the conformal properties of
loop amplitudes in gauge theories.

## Acknowledgements

We are grateful to Vladimir Braun, Einan Gardi, Grisha Korchemsky, Emery
Sokatchev and Alexander Manashov for many insightful discussions. We thank
Bláithín Power for collaboration at the early stage of this project blaithin .
JMH thanks Erik Panzer and Yang Zhang for collaboration on canonical
differential equations for Feynman integrals with non-integer powers. SZ
wishes to thank Robin Brüser for helping with Qgraf. We have used JaxoDraw
Binosi:2008ig  to draw Feynman diagrams. This project has received funding
from the European Union’s Horizon 2020 research and innovation programmes
Novel structures in scattering amplitudes (grant agreement No 725110), and
High precision multi-jet dynamics at the LHC (grant agreement No 772099). DC
is supported by the French National Research Agency in the framework of the
Investissements d’avenir program (ANR-15-IDEX-02). DC and SZ gratefully
acknowledge the computing resources provided by the Max Planck Institute for
Physics. This research was supported by the Munich Institute for Astro-,
Particle and BioPhysics (MIAPbP) which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy – EXC-2094 – 390783311.

##  Appendix A Conformal Ward identity for renormalised correlation functions

In this appendix we review the CWIs in the renormalised theory adopting the
presentation from Derkachov:1993uw ; Vasilev:2004yr ; Braun:2003rp . We rely
on the path-integral representation for the renormalised correlators,

$\displaystyle\langle{\phi(x_{1})\ldots\phi(x_{n})}\rangle=\frac{1}{N}\int\left[{\cal
D}\phi\right]e^{\textup{i}S}\phi(x_{1})\ldots\phi(x_{n})\,,$ (115)

where the normalisation $N=\int\left[{\cal D}\phi\right]e^{\textup{i}S}$ is
irrelevant in what follows. The path integration is weighted with the
renormalised action of the theory (1),

$\displaystyle S=\int\mathrm{d}^{d}x\,{\cal L}(x)\,,\qquad{\cal
L}:=\frac{Z_{1}}{2}(\partial\phi^{a})^{2}+\frac{1}{6}g\mu^{\epsilon}Z_{g}Z_{1}^{\frac{3}{2}}d^{abc}\phi^{a}\phi^{b}\phi^{c}\,,$
(116)

with renormalisation factors $Z_{1}=Z_{1}(u,\epsilon)$ and
$Z_{g}=Z_{g}(u,\epsilon)$. The renormalisation factors $Z_{1}$ and
$Z_{3}=Z_{g}Z_{1}^{3/2}$ were calculated in the minimal subtraction scheme up
to order $u^{2}$ in ref. Braun:2013tva . We spell them out here for the
convenience of the readers:

$\displaystyle
Z_{1}=1-\frac{\textsc{n}^{2}-4}{2\textsc{n}}\left[\frac{u}{6\epsilon}-\frac{u^{2}}{36}\left(\frac{1}{\epsilon^{2}}\frac{\textsc{n}^{2}-16}{\textsc{n}}-\frac{1}{\epsilon}\frac{\textsc{n}^{2}-100}{12\textsc{n}}\right)\right]+\mathcal{O}(u^{3})\,,$
(117) $\displaystyle
Z_{3}=1-\frac{u}{4\epsilon}\frac{\textsc{n}^{2}-12}{\textsc{n}}+\frac{u^{2}}{16}\left(\frac{1}{\epsilon^{2}}\frac{\textsc{n}^{2}-16}{\textsc{n}}\frac{\textsc{n}^{2}-12}{\textsc{n}}-\frac{1}{\epsilon}\frac{\textsc{n}^{4}-100\textsc{n}^{2}+960}{6\textsc{n}^{2}}\right)+\mathcal{O}(u^{3})\,.$

The path integral is invariant under the infinitesimal variations of the
fields given in eq. (2), which result in the conformal Ward
identities121212The measure of the path integration is not invariant under
conformal variations, but the resulting contributions vanish in the
dimensional regularisation.

$\displaystyle
D_{\Delta}\langle{\phi(x_{1})\ldots\phi(x_{n})}\rangle=-\textup{i}\langle{\left(D_{\Delta}S\right)\phi(x_{1})\ldots\phi(x_{n})}\rangle\,,$
(118) $\displaystyle
K^{\mu}_{\Delta}\langle{\phi(x_{1})\ldots\phi(x_{n})}\rangle=-\textup{i}\langle{\left(K^{\mu}_{\Delta}S\right)\phi(x_{1})\ldots\phi(x_{n})}\rangle\,.$
(119)

It remains to choose $\Delta$ such that the right-hand sides of these Ward
identities evaluate as a UV-finite correlation function.

The bare action (1) in six dimensions is invariant under conformal variations
(2) with $\Delta=2$, which is the classical dimension of the scalar field.
Thus, a nonzero variation of the renormalised action in $d$-dimensions (116)
may result only from the $\Delta$-dependent terms of the conformal generators
in eqs. (3) and (5), which count the dimension. In the following the choose
$\Delta=\Delta_{\gamma}$ (21). Then the dimension of the kinematic term of the
action equals to $2\Delta_{\gamma}+2-d=2\gamma$ and the dimension of the cubic
interaction $3\Delta_{\gamma}-d=3\gamma-\epsilon$, namely

$\displaystyle\begin{aligned} \textup{i}D^{\mu}_{\Delta_{\gamma}}S&=2\gamma
Z_{1}\int\mathrm{d}^{d}x\frac{1}{2}(\partial\phi^{a})^{2}+(3\gamma-\epsilon)gZ_{g}Z_{1}^{\frac{3}{2}}\frac{\mu^{\epsilon}}{6}\int\mathrm{d}^{d}x\,d^{abc}\phi^{a}\phi^{b}\phi^{c}\,,\\\
-\textup{i}K^{\mu}_{\Delta_{\gamma}}S&=2\gamma
Z_{1}\int\mathrm{d}^{d}x(-2x^{\mu})\frac{1}{2}(\partial\phi^{a})^{2}+(3\gamma-\epsilon)gZ_{g}Z_{1}^{\frac{3}{2}}\frac{\mu^{\epsilon}}{6}\int\mathrm{d}^{d}x(-2x^{\mu})d^{abc}\phi^{a}\phi^{b}\phi^{c}\,.\end{aligned}$
(120)

Further, we would like to establish that the given above conformal variations
of the action are proportional to the beta-function. We will need the
following expressions for derivatives of the renormalisation constants.
According to the definition of the anomalous dimension $\gamma$ (20) and of
the $\beta$-function (19), we have that

$\displaystyle\gamma(u)=\frac{1}{2}\frac{du}{d\log\mu}\,\partial_{u}\log
Z_{1}=\frac{\beta(u)}{2Z_{1}}\partial_{u}Z_{1}\,.$ (121)

Moreover, the bare coupling does not depend on the renormalisation scale,
namely $dg_{0}/d\mu=0$. Hence, through to eq. (17), it follows that

$\displaystyle\epsilon gZ_{g}=-\beta(u)\partial_{u}(gZ_{g})\,.$ (122)

Equations (121) and (122) enable us to simplify the coefficients in eqs.
(120),

$\displaystyle 2\gamma Z_{1}=\beta(u)\partial_{u}Z_{1}\,,$ (123)
$\displaystyle(3\gamma(u)-\epsilon)gZ_{g}Z_{1}^{\frac{3}{2}}=\beta(u)gZ_{g}\,\partial_{u}Z_{1}^{\frac{3}{2}}+\beta(u)Z_{1}^{\frac{3}{2}}\partial_{u}(gZ_{g})=\beta(u)\,\partial_{u}(gZ_{g}Z_{1}^{\frac{3}{2}})\,.$
(124)

Thus we find that the conformal variations of the action are proportional to
the $\beta$-function,

$\displaystyle\textup{i}D_{\Delta_{\gamma}}S$
$\displaystyle=\beta(u)\int\mathrm{d}^{d}x\,\partial_{u}{\cal
L}(x)=\beta(u)\,\partial_{u}S\,,$ (125)
$\displaystyle-\textup{i}K^{\mu}_{\Delta_{\gamma}}S$
$\displaystyle=\beta(u)\int\mathrm{d}^{d}x(-2x^{\mu})\,\partial_{u}{\cal
L}(x)\,.$ (126)

It remains to be proven that $\partial_{u}{\cal L}(x)$ is a UV-finite local
operator.

The renormalised correlator is UV-finite, and so is its derivative in the
coupling. The latter results in the insertion of the action,

$\displaystyle\partial_{u}\langle{\phi(x_{1})\ldots\phi(x_{n})}\rangle=\textup{i}\langle{\phi(x_{1})\ldots\phi(x_{n})\partial_{u}S}\rangle\,.$
(127)

Consequently, the operator
$\partial_{u}S=\int\mathrm{d}^{d}x\,\partial_{u}{\cal L}$ is UV-finite. In
perturbation theory, the renormalised action (116) is split into the tree-
level part, $S_{\rm tree}:=S\bigl{|}_{Z_{g},Z_{1}\to 1}$, and the counter
terms. The role of the counter-terms is to cancel out UV-divergences in the
loop corrections. Thus, by adding counter-terms to the operator
$\partial_{u}S_{\rm tree}$ we a obtain UV-finite operator $\partial_{u}S$,
namely

$\displaystyle\int\mathrm{d}^{d}x\,\left[\partial_{u}{\cal L}_{\rm
tree}\right]_{R}(x)=\int\mathrm{d}^{d}x\,\partial_{u}{\cal L}(x)\,,$ (128)

where explicitly $\partial_{u}{\cal L}_{\rm
tree}=g\mu^{\epsilon}d^{abc}\phi_{a}\phi_{b}\phi_{c}/(12u)$. Combining
together eqs. (118), (125) and (128) we obtain the Ward identity for the
dilatation, eq. (22).

Lifting up integration over $x$ in (128), we conclude that the renormalised
local operator $\left[\partial_{u}{\cal L}_{\rm tree}\right]_{R}(x)$ can
differ from $\partial_{u}{\cal L}(x)$ by a total derivative. There is no local
operator with a single total derivative in the theory which could mix with the
Lagrangian upon renormalisation, e.g.
$\partial_{\mu}\left(\phi^{a}\partial^{\mu}\phi^{a}\right)=\partial^{2}(\phi^{a})^{2}/2$,
therefore

$\displaystyle\left[\partial_{u}{\cal L}_{\rm
tree}\right]_{R}(x)=\partial_{u}{\cal L}(x)+\partial^{2}F(x)\,,$ (129)

with a polynomial $F(x)$ in the fields $\phi^{a}$ and their space-time
derivatives. Hence, we can insert a factor of $x^{\mu}$ into the integration
in eq. (128) without introducing a contribution from the unknown $F$,

$\displaystyle\int\mathrm{d}^{d}x\,x^{\mu}\partial_{u}{\cal
L}(x)=\int\mathrm{d}^{d}x\,x^{\mu}\left[\partial_{u}{\cal L}_{\rm
tree}\right]_{R}(x)\,.$ (130)

Combining together eqs. (119), (126) and (130) we obtain the Ward identity for
the conformal boost, eq. (23).

Finally, let us note that the dilatation Ward identity is equivalent to the
renormalisation group equation. Indeed, according to eqs. (118), (125) and
(127), we have that

$\displaystyle\textup{i}D_{\Delta_{\gamma}}\langle{\phi(x_{1})\ldots\phi(x_{n})}\rangle=-\textup{i}\beta(u)\langle{\left(\partial_{u}S\right)\phi(x_{1})\ldots\phi(x_{n})}\rangle=-\beta(u)\partial_{u}\langle{\phi(x_{1})\ldots\phi(x_{n})}\rangle\,.$
(131)

Furthermore, the renormalised correlator carries the canonical dimension,
namely

$\displaystyle\mu\partial_{\mu}\langle{\phi(x_{1})\ldots\phi(x_{n})}\rangle=\sum_{j=1}^{n}\left(x_{j}^{\mu}\partial_{x_{j}^{\mu}}+\Delta_{\phi}\right)\langle{\phi(x_{1})\ldots\phi(x_{n})}\rangle\,.$
(132)

Putting eqs. (131) and (132) together gives the renormalisation group
equation,

$\displaystyle\left[\mu\partial_{\mu}+\beta(u)\partial_{u}+n\gamma\right]\langle{\phi(x_{1})\ldots\phi(x_{n})}\rangle=0\,.$
(133)

##  Appendix B Implications of conformal symmetry for amplitudes away from the
conformal fixed point

In this appendix we show how the knowledge of a (renormalised) $n$-point
correlator at the conformal fixed point $u^{*}$ constrains strongly its form
away from the conformal fixed point. Completely analogous observations hold
for the scattering amplitudes. Let us assume we know the correlator $C$ at the
conformal fixed point,

$\displaystyle C\bigl{|}_{u=u^{*}}=\sum_{k\geq 0}\epsilon^{k}C^{*}_{k}\,.$
(134)

The dependence on the kinematics is understood. Away from the conformal fixed
point, the renormalised correlator has the generic form

$\displaystyle C=C_{0}+\sum_{\ell\geq 1}u^{\ell}\sum_{m\geq
0}C_{\ell}^{[m]}\epsilon^{m}\,.$ (135)

By equating eq. (134) with eq. (135) evaluated at $u=u^{*}$,

$\displaystyle u^{*}=\sum_{m\geq 1}u_{m}\epsilon^{m}\,,$ (136)

and solving for the $C_{\ell}^{[0]}$’s we obtain

$\displaystyle C_{0}=C^{*}_{0}\,,$ (137) $\displaystyle
C_{1}^{[0]}=\frac{1}{u_{1}}C^{*}_{1}\,,$ $\displaystyle
C_{2}^{[0]}=\frac{1}{u_{1}^{2}}C^{*}_{2}-\frac{u_{2}}{u_{1}^{3}}C^{*}_{1}-\frac{1}{u_{1}}C_{1}^{[1]}\,,$

and so on. Up to one loop conformal symmetry fixes entirely the finite part of
the renormalised correlator. This follows from the fact that the tree-level
correlator does not depend on $\epsilon$. The finite part of the two-loop
correlator is instead determined by conformal symmetry together with the
order-$\epsilon$ part of the one-loop correlator. In general, the finite part
of a renormalised correlator at a given loop order is determined by the
correlator at the conformal fixed point together with higher-$\epsilon$ but
lower-loop information.

##  Appendix C Conformal symmetry and momentum conservation

Momentum-space correlations functions and scattering amplitudes are always
accompanied by an overall $\delta$ function imposing momentum conservation. In
this appendix we motivate why, when studying the conformal properties of these
objects, it is possible to effectively neglect such $\delta$ function.

Let us consider a $n$-point correlation function $\mathcal{C}^{(n)}$ in
momentum space. We define the reduced correlator $C^{(n)}$ by stripping off
the overall momentum-conservation $\delta$ function,

$\displaystyle\mathcal{C}^{(n)}(p_{1},\ldots,p_{n})=\delta^{(d)}(P)\,C^{(n)}(p_{1},\ldots,p_{n})\,,$
(138)

where $P=p_{1}+\ldots+p_{n}$. The commutator between the conformal boost
generator $K^{\mu}_{\Delta}$ given in eq. (13) and the momentum-conservation
$\delta$ function is

$\displaystyle\left[K^{\mu}_{\Delta}\,,\delta^{(d)}(P)\right]C^{(n)}(p_{1},\ldots,p_{n})=2\left([C^{(n)}]+(n-1)d-n\Delta\right)(\partial_{P^{\mu}}\delta)C^{(n)}(p_{1},\ldots,p_{n})\,,$
(139)

where $[C^{(n)}]$ is the dimension of $C^{(n)}$ in units of energy. The
dimension of the _full_ reduced correlator $[C^{(n)}]$ (i.e. not truncated to
some perturbative order) is exactly such that the right-hand side of eq. (139)
vanishes. Therefore we can focus on the action of the conformal generator on
the reduced correlator,

$\displaystyle
K^{\mu}_{\Delta}\delta^{(d)}\left(P\right)C^{(n)}(p_{1},\ldots,p_{n})=\delta^{(d)}\left(P\right)K^{\mu}_{\Delta}C^{(n)}(p_{1},\ldots,p_{n})\,.$
(140)

Effectively this means we can neglect the momentum-conservation $\delta$
function and, for this reason, we refer to the reduced correlator as
correlator throughout this paper.

It is important to stress that $K^{\mu}_{\Delta}$ and
$\delta^{(d)}\left(P\right)$ do _not_ commute if applied only to certain terms
of the perturbative expansion of $C^{(n)}$. For instance, $C^{(3)}$ has
dimension $-6+\epsilon$, whereas its tree-level term,

$\displaystyle C^{(3)}_{0}=\frac{1}{p_{1}^{2}s_{2}s_{3}}\,,$ (141)

has dimension $-6$. For this reason,

$\displaystyle\left[K^{\mu}_{\Delta}\,,\delta^{(d)}(P)\right]C^{(3)}_{0}=\mathcal{O}(\epsilon)\,.$
(142)

This is the case whenever we truncate the perturbative expansion to a certain
order: differences due to different ways of implementing momentum conservation
kick in at higher orders in $\epsilon$, as expressed by eq. (139).

##  Appendix D Exact results for two- and three-point correlators

|  |  |  |
---|---|---|---|---
Figure 6: Representative Feynman diagrams contributing to the two-point
correlator $C^{(2)}$ up to two-loop order.

In this section we discuss how conformal symmetry constrains the renormalised
two- and three-point correlators at the conformal fixed point. We start off in
position space, where the correlators have particularly simple expressions,
and then Fourier-transform them to momentum space. While the Fourier transform
of the two-point correlator is trivial, that of the three-point one requires
the computation of a one-loop triangle integral with $\epsilon$-dependent
powers of the propagators. We then compute perturbatively the correlators up
to two loop orders using Feynman diagrams, and show that the perturbative
result matches the conformal prediction at the conformal fixed point.

We begin by defining the notation. The main characters of this section are the
two- and three-point correlators at the conformal fixed point (25),

$\displaystyle\begin{aligned}
&\mathcal{C}^{(2)}_{ab}(x_{1},x_{2})=\langle\phi^{a}(x_{1})\phi^{b}(x_{2})\rangle\bigl{|}_{u=u^{*}}\,,\\\
&\mathcal{C}^{(3)}_{abc}(x_{1},x_{2},x_{3})=\langle\phi^{a}(x_{1})\phi^{b}(x_{2})\phi^{c}(x_{3})\rangle\bigl{|}_{u=u^{*}}\,.\end{aligned}$
(143)

Both correlators have a single and overall $su(\textsc{n})$ factor, which it
is convenient to strip off as

$\displaystyle\begin{aligned}
&\mathcal{C}^{(2)}_{ab}(x_{1},x_{2})=\delta^{ab}C^{(2)}(x_{1},x_{2})\,,\\\
&\mathcal{C}^{(3)}_{abc}(x_{1},x_{2},x_{3})=d^{abc}C^{(3)}(x_{1},x_{2},x_{3})\,.\end{aligned}$
(144)

We define the momentum-space correlators by Fourier transforming the position-
space ones as in eq. (31).131313To avoid the proliferation of symbols we
slightly abuse the notation, and use $C^{(n)}$ for the correlators in both
momentum and position space. The arguments make the distinction. It is long
known that conformal symmetry fixes them up to the overall normalisation (see
e.g. ref. Polyakov:1970xd ). In position space their expressions are extremely
simple,

$\displaystyle
C^{(2)}(x_{1},x_{2})=\frac{c_{12}(u^{*})}{\left(x_{12}^{2}-\textup{i}0\right)^{\Delta_{\gamma}}}\,,$
(145) $\displaystyle
C^{(3)}(x_{1},x_{2},x_{3})=\frac{c_{123}(u^{*})}{\left(x_{12}^{2}-\textup{i}0\right)^{\frac{\Delta_{\gamma}}{2}}\left(x_{23}^{2}-\textup{i}0\right)^{\frac{\Delta_{\gamma}}{2}}\left(x_{31}^{2}-\textup{i}0\right)^{\frac{\Delta_{\gamma}}{2}}}\,,$
(146)

where $x_{ij}=x_{i}-x_{j}$. We recall that we are working at the conformal
fixed point $u=u^{*}$, hence $\Delta_{\gamma}$ and the normalisation factors
$c_{12}$ and $c_{123}$ are functions of $\epsilon$. The anomalous dimension
$\gamma$ at the conformal fixed point is given by

$\displaystyle\begin{aligned} \gamma\left(u^{*}\right)&=\sum_{k\geq
1}\gamma_{k}\,\epsilon^{k}\\\
&=-\epsilon\,\frac{\textsc{n}^{2}-4}{3(\textsc{n}^{2}-20)}-4\epsilon^{2}\frac{(\textsc{n}^{2}-4)(\textsc{n}^{4}-94\textsc{n}^{2}+840)}{27(\textsc{n}^{2}-20)^{3}}+\mathcal{O}\left(\epsilon^{3}\right)\,.\end{aligned}$
(147)

In the next subsections we will consider these correlators in momentum space,
and compare them against the result of a perturbative computation up to two-
loop order.

###  D.1 Fourier transform of the conformal correlators

To warm up, we begin with the two-point correlator given by eq. (145). Since
it is a function of a single scalar invariant only, the Fourier transform has
no mystery. It is given by

$\displaystyle
C^{(2)}\left(p^{2}\right)=\tilde{c}_{12}(u^{*})\,\left(-p^{2}-\textup{i}0\right)^{-1+\gamma}\,,$
(148)

where we absorbed the factors coming from the Fourier transform into the
normalisation,

$\displaystyle\tilde{c}_{12}(u^{*})=-\textup{i}\,e^{\textup{i}\pi(\gamma-\epsilon)}\frac{4^{1-\gamma}\pi^{\frac{d}{2}}\Gamma\left(1-\gamma\right)}{\Gamma(2-\epsilon+\gamma)}\,c_{12}(u^{*})\,.$
(149)

We choose the minus sign for $p^{2}$ in eq. (148) so that the series expansion
around $\epsilon=0$ can be directly compared against the perturbative results
containing Feynman integrals, which we compute in the Euclidean region, namely