aainstitutetext: Department of Physics, Brown University, Providence, RI
02912, USAbbinstitutetext: Brown Theoretical Physics Center, Brown University,
Providence, RI 02912, USA

# Loop-level gluon OPEs in celestial holography

Rishabh Bhardwaj a Luke Lippstreu a Lecheng Ren a,b Marcus Spradlin a Akshay
Yelleshpur Srikant a and Anastasia Volovich<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute one-loop corrections to the OPE of gluons in the celestial
conformal field theory corresponding to Yang-Mills coupled to arbitrary
matter. We exploit universal hard/soft factorization to derive an IR finite
OPE for the hard gluon operators. This OPE involves logarithms and operators
that resemble logarithmic partners of primary operators. We derive an exact
all-loop OPE in a limit of the Higgs-regulated planar $\mathcal{N}=4$ super
Yang-Mills theory.

##  1 Introduction

The celestial holography framework Pasterski:2016qvg ; Pasterski:2017kqt ;
Pasterski:2017ylz  (see Raclariu:2021zjz ; Pasterski:2021rjz ;
Pasterski:2021raf ; McLoughlin:2022ljp  for recent reviews) suggests
interpreting four-dimensional scattering amplitudes as correlation functions
(also called “celestial amplitudes”) of primary operators in a celestial
conformal field theory (CCFT). It seems clear that any putative CCFT would
have some quite mysterious features, but this framework has revealed
interesting previously hidden features of scattering amplitudes, such as the
fact that the infinite tower of tree-level soft graviton symmetries generate a
symmetry algebra with $w_{1+\infty}$ structure, under which soft gluon
symmetries also transform irreducibly Guevara:2021abz ; Strominger:2021lvk .

A central tool in the study of CFT is the operator product expansion (OPE); in
particular the $w_{1+\infty}$ structure is imprinted in celestial OPE
coefficients. Celestial amplitudes for gluons at tree level were considered
first in Pasterski:2017ylz , and the OPE of celestial gluon primaries was
worked out in Fan:2019emx ; Pate:2019lpp . Aspects of loop corrections to
gluon amplitudes have been studied in several papers in Banerjee:2017jeg ;
Albayrak:2020saa ; Gonzalez:2020tpi ; Gonzalez:2021dxw ; Nastase:2021izh . In
particular, in Gonzalez:2020tpi  it was noted that in dimensionally-regulated
planar $\mathcal{N}=4$ super-Yang-Mills (SYM) theory, loop corrected celestial
gluon amplitudes can be expressed as differential operators acting on the
corresponding tree-level celestial amplitude.

This paper has two parts. In the first part we use one-loop corrections to
splitting functions to compute one-loop corrections to the OPE of two positive
helicity gluons in dimensionally regulated Yang-Mills theory coupled to
arbitrary (fundamental) matter. The result, shown in (47), has a piece that
can be expressed as a differential operator acting on the tree-level OPE, but
for general matter content also has a second term involving the negative
helicity gluon. Using the hard/soft factorization theorem for massless gauge
theories, whose implication for celestial amplitudes was analyzed in
Gonzalez:2021dxw , we can subtract the universal IR divergent terms. The
resulting finite OPE of the hard operators is shown in (64).

In the second part we study the amplitudes of massless gluons in a particular
corner of the Coulomb branch of planar $\mathcal{N}=4$ super-Yang-Mills (SYM)
theory that has received attention for example in Alday:2009zm ; Henn:2010bk ;
Henn:2010ir . The motivation is two-fold. First, these Higgs-regulated
amplitudes are infrared finite, with a physical mass parameter $m$ replacing
the dimensional regularization parameter $\epsilon$ whose interpretation in
the context of celestial holography is rather unclear. Second, the four- and
five-gluon amplitudes are believed to be known exactly (to all loop order) in
the limit when $m^{2}$ is very small compared to the Mandelstam
invariants—they are given by exponential formulas reminiscent of the BDS
ansatz Bern:2005iz  in dimensional regularization, whose celestial incarnation
was computed in Gonzalez:2020tpi . Indeed, the computation of the Higgs-
regulated celestial four-gluon amplitude is essentially identical to that of
Gonzalez:2020tpi ; see appendix A. We derive the all-loop OPE for two positive
helicity gluon operators in the Higgs-regulated theory in (88).

One reason to be interested in loop-corrected or exact OPEs is to see whether
and how they exhibit $w_{1+\infty}$ structure. In the case of self-dual
gravity, for example, it is known that the $w_{1+\infty}$ symmetry is
perturbatively exact Ball:2021tmb . Unfortunately the loop-corrected gluon
OPEs computed in this paper contain logarithms that obstruct the usual CFT
mode expansion and therefore complicate the analysis of the symmetry algebra;
this remains an interesting problem for future work.

This paper is organized as follows. In section 2 we briefly review the well-
known one-loop splitting functions in gauge theory with various matter
content. In section 3 we use these splitting functions to compute one-loop
corrections to the OPE of celestial gluon operators and utilize the hard/soft
factorization to define a finite OPE for hard gluon operators. In section 4 we
compute the all-loop OPE in the Higgs-regulated planar $\mathcal{N}=4$ SYM
theory. We find that the OPE of hard operators computed in dimensional
regularization agrees precisely with the OPE in the Higgs-regulated theory.
Finally in section 5 we make some preliminary comments and speculation
regarding the potential need for using the machinery of logarithmic CFT for
properly understanding these celestial OPEs.

##  2 Collinear singularities of one-loop amplitudes in gauge theory

Scattering amplitudes in Yang-Mills theory with gauge group $SU(N_{c})$ have a
perturbative expansion

$\displaystyle\mathcal{A}_{n}^{a_{1}\dots a_{n}}\left(p_{1},\sigma_{1};\dots
p_{n},\sigma_{n}\right)=g^{n-2}\sum_{L=0}^{\infty}\alpha^{L}\mathcal{A}_{n}^{(L)a_{1}\dots
a_{n}}\left(p_{1},\sigma_{1};\dots p_{n},\sigma_{n}\right),$ (1)

where $g$ is the Yang-Mills coupling constant, $\alpha=a\left(4\pi
e^{-\gamma_{E}}\right)^{\epsilon}$ with $a=\frac{g^{2}N_{c}}{8\pi^{2}}$ the ’t
Hooft coupling, $\epsilon$ the dimensional regularization parameter,
$\left\\{p_{1},\dots p_{n}\right\\}$ the momenta of the external gluons,
$\sigma_{i}=\pm 1$ their helicities, and $a_{i}$ the color indices. They
develop singularities when two particles become collinear. If $p_{1},p_{2}$
are becoming collinear, we can parameterize them111We will only consider the
collinear limit of two outgoing gluons. by

$\displaystyle p_{1}=t\,P\,,\qquad\qquad p_{2}=\left(1-t\right)P\,.$ (2)

In this limit the tree-level amplitude behaves as

$\mathcal{A}_{n}^{(0)a_{1}a_{2}\dots
a_{n}}\left(p_{1},\sigma_{1};p_{2},\sigma_{2};\dots p_{n},\sigma_{n}\right)\\\
\overset{1\parallel
2}{\longrightarrow}\,if^{a_{1}a_{2}a_{p}}\,\sum_{\sigma=\pm}\,\text{Split}^{(0)}_{-\sigma}(t;1^{\sigma_{1}},2^{\sigma_{2}})\,\mathcal{A}_{n-1}^{(0)a_{p}a_{3}\dots
a_{n}}(P,\sigma;p_{3},\sigma_{3};\dots)\,.$ (3)

The singular behavior is entirely captured by the splitting functions
Mangano:1987xk ; Berends:1987me

$\displaystyle\text{Split}^{(0)}_{-}(t;1^{-},2^{-})=0\,,\qquad$
$\displaystyle\text{Split}^{(0)}_{-}(t;1^{+},2^{+})=\frac{1}{\sqrt{t(1-t)}\braket{12}}\,,$
(4)
$\displaystyle\text{Split}^{(0)}_{-}(t;1^{+},2^{-})=\frac{t^{2}}{\sqrt{t(1-t)}[12]}\,,\qquad$
$\displaystyle\text{Split}^{(0)}_{-}(t;1^{-},2^{+})=\frac{(1-t)^{2}}{\sqrt{t(1-t)}[12]}\,,$
(5)

where $\left\langle 12\right\rangle$ and $\left[12\right]$ are the usual
spinor brackets. Loop amplitudes exhibit a similar behavior Bern:1994zx ;
Bern:1996je ; Bern:1995ix ; Mangano:1990by ; Dixon:1996wi ; Bern:1999ry ;
Kosower:1999rx ; Kosower:1999xi ; Bern:2004cz , captured by

$\mathcal{A}^{(1)a_{1}a_{2}a_{3}...a_{n}}_{n}\overset{1\parallel
2}{\longrightarrow}if^{a_{1}a_{2}a_{p}}\sum_{\sigma=\pm}\Big{[}\text{Split}^{(0)}_{-\sigma}(t;1^{\sigma_{1}},2^{\sigma_{2}})\mathcal{A}_{n-1}^{(1)a_{P}a_{3}...a_{n}}\\\
+\,\text{Split}^{(1)}_{-\sigma}(t;1^{\sigma_{1}},2^{\sigma_{2}})\mathcal{A}_{n-1}^{(0)a_{P}a_{3}...a_{n}}\Big{]}.$
(6)

Here and in what follows we will suppress the momentum and spin dependence of
amplitudes where possible. The one-loop splitting functions used in this paper
are Bern:1994zx ; Bern:1994fz ; Bern:1995ix

$\displaystyle\text{Split}^{(1)}_{-}(t;a^{+},b^{+})$
$\displaystyle=(G^{f}+G^{n})\text{Split}^{(0)}_{-}(t;a^{+},b^{+})\,,$ (7)
$\displaystyle\text{Split}^{(1)}_{-}(t;a^{\pm},b^{\mp})$
$\displaystyle=G^{n}\,\text{Split}^{(0)}_{-}(t;a^{\pm},b^{\mp})\,,$ (8)
$\displaystyle\text{Split}^{(1)}_{+}(t;a^{+},b^{+})$
$\displaystyle=G^{f}\frac{\left[ab\right]}{\sqrt{t(1-t)}\left\langle
ab\right\rangle^{2}}\,,$ (9)

where the factorizing $G^{f}$ and non-factorizing $G^{n}$ parts read222It is
sufficient to retain terms up to order $\epsilon^{0}$ in the splitting
functions at one loop. We will need to retain terms to higher order in
$\epsilon$ while performing higher loop computations. All-order expressions
for the one-loop splitting functions can be found in Bern:2005iz .,333For
outgoing particles $s_{ab}>0$ and the prescription for analytic continuation
of the logarithm to negative values is $\ln(-|s_{ab}|)=\ln|s_{ab}|-i\pi$
Bern:2004cz .

$\displaystyle G^{f}$
$\displaystyle=\frac{1}{6}\Big{(}1+\frac{n_{s}}{N_{c}}-\frac{n_{f}}{N_{c}}\Big{)}t(1-t)+\mathcal{O}(\epsilon)\,,$
(10) $\displaystyle G^{n}$
$\displaystyle=\hat{c}_{\Gamma}\Bigg{[}-\frac{1}{\epsilon^{2}}\Bigg{(}\frac{\mu^{2}}{-s_{ab}t(1-t))}\Bigg{)}^{\epsilon}+2\ln(t)\ln(1-t)-\frac{\pi^{2}}{6}\Bigg{]}+\mathcal{O}(\epsilon)\,,$
(11)

where $n_{s}$ and $n_{f}$ are the numbers of massless scalars and fermions,
respectively, presumed to be in the fundamental representation, and

$\hat{c}_{\Gamma}=\frac{e^{\epsilon\gamma}}{2}\frac{\Gamma(1+\epsilon)\Gamma^{2}(1-\epsilon)}{\Gamma(1-2\epsilon)}\,.$
(12)

A complete listing of one-loop splitting functions, including ones for
fermions, can be found in Bern:1994zx ; Bern:1994fz .

##  3 Celestial one-loop OPEs from collinear limits

Now consider the celestial amplitude $\tilde{\mathcal{A}}_{n}$ (involving $n$
gluons), which is related to the usual scattering amplitude via a Mellin
transform Pasterski:2016qvg

$\tilde{\mathcal{A}}_{n}^{a_{1}\dots
a_{n}}\left(\Delta_{1},\sigma_{1};\dots;\Delta_{n},\sigma_{n}\right)=\int_{0}^{\infty}\prod_{i=1}^{n}\,d\omega_{i}\,\,\omega_{i}^{\Delta_{i}-1}\,\mathcal{A}_{n}^{a_{1}\dots
a_{n}}\left(p_{1},\sigma_{1};\dots;p_{n},\sigma_{n}\right),$ (13)

where we have parameterized the massless four-momenta of the external gluons
as

$p_{i}^{\mu}=\epsilon_{i}\frac{\Lambda\,\omega}{2}\left(1+z_{i}\bar{z}_{i},z_{i}+\bar{z}_{i},-i(z_{i}-\bar{z}_{i}),1-z_{i}\bar{z}_{i}\right)$
(14)

where $\epsilon=-/+$ for incoming/outgoing particles respectively and
$\Lambda$ is a new parameter that has dimensions of energy and $\omega$ is
kept dimensionless for the Mellin integral to make sense dimensionally. We
work in a $(+,-,-,-)$ signature bulk spacetime throughout, and assume all
particles to be outgoing. A choice of spinor helicity variables corresponding
to this parameterization, and the resulting spinor brackets, are

$\displaystyle\lambda_{i}=\sqrt{\Lambda\omega_{i}}\begin{pmatrix}1\\\
z_{i}\end{pmatrix},\qquad\tilde{\lambda}_{i}=\sqrt{\Lambda\omega_{i}}\epsilon_{i}\begin{pmatrix}1\\\
\bar{z}_{i}\end{pmatrix},$ (15) $\displaystyle\left\langle
ij\right\rangle=\Lambda\sqrt{\omega_{i}\,\omega_{j}}z_{ij}\,,\qquad\left[ij\right]=\Lambda\epsilon_{i}\epsilon_{j}\sqrt{\omega_{i}\,\omega_{j}}\bar{z}_{ij}\,,$

where $z_{ij}=z_{i}-z_{j}$. It has been conjectured Pasterski:2016qvg  that
celestial amplitudes can be interpreted as correlation functions in a 2D CFT
living on the celestial sphere at null infinity, referred to as celestial
conformal field theory (CCFT); choosing a convenient normalization, we
have444Henceforth we omit color indices, where possible, in order to avoid
notational clutter.

$\displaystyle\tilde{\mathcal{A}}_{n}=g^{n-2}\left\langle\mathcal{O}_{\Delta_{1},\sigma_{1}}\left(z_{1},\bar{z}_{1}\right)\dots\mathcal{O}_{\Delta_{n},\sigma_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle.$
(16)

Carrying over the decomposition in (1) to celestial amplitudes gives

$\displaystyle\tilde{\mathcal{A}}_{n}\left(\Delta_{1},\sigma_{1};\dots;\Delta_{n},\sigma_{n}\right)=g^{n-2}\sum_{L=0}^{\infty}\alpha^{L}\tilde{\mathcal{A}}_{n}^{(L)}\left(\Delta_{1},\sigma_{1};\dots;\Delta_{n},\sigma_{n}\right)$
(17)

where

$\displaystyle\tilde{\mathcal{A}}_{n}^{(L)}\left(\Delta_{1},\sigma_{1};\dots;\Delta_{n},\sigma_{n}\right)=\int\prod_{i=1}^{n}d\omega_{i}\,\omega_{i}^{\Delta_{i}-1}\mathcal{A}_{n}^{(L)}\left(p_{1},\sigma_{1};\dots;p_{n},\sigma_{n}\right)$
(18)

and we identify

$\displaystyle\left\langle\mathcal{O}_{\Delta_{1},\sigma_{1}}\left(z_{1},\bar{z}_{1}\right)\dots\mathcal{O}_{\Delta_{n},\sigma_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle^{(L)}:=\tilde{\mathcal{A}}_{n}^{(L)}\,.$
(19)

###  3.1 OPEs from splitting functions

One consequence of the dictionary in (16) is that the OPE is determined by
collinear limits of amplitudes. Here we work out the general form of this
relation. In a nutshell, we will characterize the splitting functions in terms
of certain functions $f(z_{12},\bar{z}_{12},\sigma,t)$, and the corresponding
OPE coefficients are then determined by integrals of these functions against
the measure
$\mathrm{d}\mathcal{T}=\mathrm{d}t\,t^{\Delta_{1}-1}\,\left(1-t\right)^{\Delta_{2}-1}$.
Examples are worked out explicitly in the next section.

Using the parameterization (2) for the collinear limit $1||2$, with

$\displaystyle
P=\frac{\Lambda\omega}{2}\left(1+z_{1}\bar{z}_{1},z_{1}+\bar{z}_{1},-i\left(z_{1}-\bar{z}_{1}\right),1-z_{1}\bar{z}_{1}\right)$
(20)

and taking $z_{12},\bar{z}_{12}\rightarrow 0$, it follows from the singular
behavior of the amplitudes in (3) that the corresponding celestial amplitude
behaves as

$\displaystyle\tilde{\mathcal{A}}_{n}^{(0)}\left(\Delta_{1},\sigma_{1};\Delta_{2},\sigma_{2};\dots;\Delta_{n},\sigma_{n}\right)$
(21) $\displaystyle\qquad\qquad\xrightarrow[]{z_{12},\bar{z}_{12}\to
0}\int_{0}^{1}\mathrm{d}\mathcal{T}\int_{0}^{\infty}\mathrm{d}\omega\,\omega^{\Delta_{1}+\Delta_{2}-1}\sum_{\sigma=\pm}\,\Big{[}\text{Split}^{(0)}_{-\sigma}(t;1^{\sigma_{1}},2^{\sigma_{2}})$
$\displaystyle\hskip 170.71652pt\times\prod_{j\neq
1,2}\int_{0}^{\infty}\mathrm{d}\omega_{j}\,\omega_{j}^{\Delta_{j}-1}\mathcal{A}_{n-1}^{(0)}\left(P,\sigma;p_{3},\sigma_{3};\dots;p_{n},\sigma_{n}\right)\Big{]}$

where we use the measure defined above and we have suppressed all color
dependence in the above equation because it is identical to (3). Noting that
all of the tree-level splitting functions (3) take the form

$\displaystyle\text{Split}^{(0)}_{-\sigma}(t;1^{\sigma_{1}},2^{\sigma_{2}})=\frac{1}{\omega}f^{(0)}\left(z_{12},\bar{z}_{12},\sigma,t\right)$
(22)

(this equation is meant to define $f^{(0)}$, whose dependence on
$\sigma_{1},\sigma_{2}$ we suppress) we see that

$\tilde{\mathcal{A}}_{n}^{(0)}\left(\Delta_{1},\sigma_{1};\Delta_{2},\sigma_{2};\dots;\Delta_{n},\sigma_{n}\right)\\\
\xrightarrow[]{z_{12},\bar{z}_{12}\to
0}\sum_{\sigma=\pm}\tilde{\mathcal{A}}_{n-1}^{(0)}\left(\Delta_{1}+\Delta_{2}-1,\sigma;\Delta_{3},\sigma_{3};\dots;\Delta_{n},\sigma_{n}\right)\int_{0}^{1}\mathrm{d}\mathcal{T}\,f^{(0)}\left(z_{12},\bar{z}_{12},\sigma,t\right).$
(23)

This behavior of celestial amplitudes implies the same for the corresponding
CCFT correlation functions:

$\displaystyle\left\langle\mathcal{O}_{\Delta_{1},\sigma_{1}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{\Delta_{2},\sigma_{2}}\left(z_{2},\bar{z}_{2}\right)\dots\mathcal{O}_{\Delta_{n},\sigma_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle^{(0)}\xrightarrow[]{z_{12},\bar{z}_{12}\to
0}$ (24)
$\displaystyle\qquad\sum_{\sigma=\pm}\left\langle\mathcal{O}_{\Delta_{1}+\Delta_{2}-1,\sigma}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{\Delta_{3},\sigma_{3}}\left(z_{3},\bar{z}_{3}\right)\dots\mathcal{O}_{\Delta_{n},\sigma_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle^{(0)}\int_{0}^{1}\mathrm{d}\mathcal{T}\,f^{(0)}\left(z_{12},\bar{z}_{12},\sigma,t\right).$

The tree-level OPE coefficients were determined in this manner in Pate:2019lpp
; Fan:2019emx  for gluons and gravitons.

This formalism readily extends to loop level where the one-loop splitting
functions have the generic form (cf. (7))

$\displaystyle\alpha\,\text{Split}^{(1)}_{-\sigma}(t,\omega;1^{\sigma_{1}},2^{\sigma_{2}})=\frac{1}{\omega}\sum_{r=0}^{2}af_{r}^{(1)}\left(z_{12},\bar{z}_{12},\sigma,t\right)\ln^{r}\left(-\omega^{2}t^{2}(1-t)^{2}z_{12}\bar{z}_{12}\frac{\Lambda^{2}}{\mu^{2}}\right)$
(25)

for certain functions $f_{r}^{(1)}$ (see section 3.2 for specific examples).
The combination $\omega^{2}t^{2}(1-t)^{2}z_{12}\bar{z}_{12}$ that occurs in
the argument of the logarithms is Lorentz invariant in (1,3) signature
spacetime. We can rewrite this in a convenient way by noting that

$\int_{0}^{\infty}d\omega\,\omega^{\Delta_{1}+\Delta_{2}-1}\,t^{\Delta_{1}-1}\left(1-t\right)^{\Delta_{2}-1}\ln\left(-\omega^{2}t^{2}(1-t)^{2}z_{12}\bar{z}_{12}\frac{\Lambda^{2}}{\mu^{2}}\right)(\dots)\\\
:=2\mathcal{\hat{D}}_{12}\int_{0}^{\infty}\mathrm{d}\omega\,\omega^{\Delta}\,t^{\Delta_{1}-1}\left(1-t\right)^{\Delta_{2}-1}(\dots)\Big{|}_{\Delta=\Delta_{1}+\Delta_{2}-1}$
(26)

where $(\dots)$ is an arbitrary function of $\omega$ and $t$ and in the second
line we have defined a new “covariant derivative” operator555This operator
transforms appropriately only in the OPE limit $z_{ij},\bar{z}_{ij}\rightarrow
0$.,

$\mathcal{\hat{D}}_{ij}=\partial_{\Delta_{i}}+\partial_{\Delta_{j}}+\partial_{\Delta}+\frac{1}{2}\ln\left(-z_{ij}\bar{z}_{ij}\frac{\Lambda^{2}}{\mu^{2}}\right).$
(27)

Importantly, $\Delta$ must be assumed to be independent of
$\Delta_{1},\Delta_{2}$ before taking the derivative. Also, it is to be
understood that we always take $\Delta=\Delta_{1}+\Delta_{2}-1$ after
differentiating. Using (6), we can express the collinear behavior of the loop-
level celestial amplitude as

$\displaystyle\tilde{\mathcal{A}}_{n}^{(0)}(\Delta_{1},\sigma_{1};\Delta_{2},\sigma_{2};\dots\Delta_{n},\sigma_{n})+\tilde{\mathcal{A}}_{n}^{(1)}(\Delta_{1},\sigma_{1};\Delta_{2},\sigma_{2};\dots\Delta_{n},\sigma_{n})\xrightarrow[]{z_{12},\bar{z}_{12}\to
0}(\ref{eq:collinearlimitcelestial})\,+$
$\displaystyle\qquad\qquad\sum_{\sigma=\pm}\tilde{\mathcal{A}}_{n-1}^{(1)}\left(\Delta_{1}+\Delta_{2}-1,\sigma;\Delta_{3},\sigma_{3};\dots\Delta_{n},\sigma_{n}\right)\int_{0}^{1}\mathrm{d}\mathcal{T}\,f^{(0)}\left(z_{12},\bar{z}_{12},\sigma,t\right)$
(28)
$\displaystyle\qquad\qquad+\sum_{r=0,\sigma=\pm}^{r=2}2^{r}\mathcal{\hat{D}}_{12}^{r}\tilde{\mathcal{A}}_{n-1}^{(0)}\left(\Delta,\sigma;\Delta_{3},\sigma_{3};\dots\Delta_{n},\sigma_{n}\right)\int_{0}^{1}\mathrm{d}\mathcal{T}\,f^{(1)}_{r}\left(z_{12},\bar{z}_{12},\sigma,t\right).$

The operator $\partial_{\Delta}$ in $\mathcal{\hat{D}}_{12}$ acts only on the
celestial amplitude while $\partial_{\Delta_{1}}$ and $\partial_{\Delta_{2}}$
act only on the integral (recall that the measure $\mathrm{d}\mathcal{T}$
contains $\Delta_{1}$ and $\Delta_{2}$). Rewritten in terms of correlation
functions, this is

$\displaystyle\left\langle\mathcal{O}_{\Delta_{1},\sigma_{1}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{\Delta_{2},\sigma_{2}}\left(z_{2},\bar{z}_{2}\right)\dots\mathcal{O}_{\Delta_{n},\sigma_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle\xrightarrow[]{z_{12},\bar{z}_{12}\to
0}$ (29)
$\displaystyle\quad\sum_{\sigma=\pm}\int_{0}^{1}\mathrm{d}\mathcal{T}\,f^{(0)}\left(z_{12},\bar{z}_{12},\sigma,t\right)\left\langle\mathcal{O}_{\Delta_{1}+\Delta_{2}-1,\sigma}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{\Delta_{3},\sigma_{3}}\left(z_{3},\bar{z}_{3}\right)\dots\mathcal{O}_{\Delta_{n},\sigma_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle$
$\displaystyle\quad+a\sum_{r,\sigma=\pm}2^{r}\mathcal{\hat{D}}_{12}^{r}\int_{0}^{1}\mathrm{d}\mathcal{T}\,f^{(1)}_{r}\left(z_{12},\bar{z}_{12},\sigma,t\right)\left\langle\mathcal{O}_{\Delta,\sigma}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{\Delta_{3},\sigma_{3}}\left(z_{3},\bar{z}_{3}\right)\dots\mathcal{O}_{\Delta_{n},\sigma_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle$
$\displaystyle\quad+\mathcal{O}\left(a^{2}\right).$

We emphasize that in the second line, we’ve replaced the tree-level correlator
by the full correlator since we’re only working to $\mathcal{O}(a)$. This
approximation is crucial for us to be able to infer the OPE

$\mathcal{O}_{\Delta_{1},\sigma_{1}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{\Delta_{2},\sigma_{2}}\left(z_{2},\bar{z}_{2}\right)\sim\sum_{\sigma=\pm}\int_{0}^{1}\mathrm{d}\mathcal{T}\,f^{(0)}\left(z_{12},\bar{z}_{12},\sigma,t\right)\mathcal{O}_{\Delta_{1}+\Delta_{2}-1,\sigma}\left(z_{2},\bar{z}_{2}\right)\\\
+a\sum_{r,\sigma=\pm}2^{r}\mathcal{\hat{D}}_{12}^{r}\int_{0}^{1}\mathrm{d}\mathcal{T}\,f^{(1)}_{r}\left(z_{12},\bar{z}_{12},\sigma,t\right)\mathcal{O}_{\Delta,\sigma}(z_{2},\bar{z}_{2}).$
(30)

In the computation of all-loop OPEs in section 4.1, we will also encounter a
splitting function of the form

$\text{Split}_{-\sigma}(t,\omega;1^{\sigma_{1}},2^{\sigma_{2}})=\frac{f^{(0)}\left(z_{12},\bar{z}_{12},\sigma,t\right)}{\omega}\\\
\times\exp\left[a\sum_{r=0}^{2}f_{r}^{(1)}\left(z_{12},\bar{z}_{12},\sigma,t\right)\ln^{r}\left(-\omega^{2}t^{2}(1-t)^{2}z_{12}\bar{z}_{12}\frac{\Lambda^{2}}{\mu^{2}}\right)\right],$
(31)

which gives the all-loop OPE

$\mathcal{O}_{\Delta_{1},\sigma_{1}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{\Delta_{2},\sigma_{2}}\left(z_{2},\bar{z}_{2}\right)\\\
\sim\sum_{\sigma=\pm}\int_{0}^{1}\exp\left[a\sum_{r=0}^{2}f_{r}^{(1)}\left(z_{12},\bar{z}_{12},\sigma,t\right)2^{r}\hat{\mathcal{D}}_{12}^{r}\right]f^{(0)}\left(z_{12},\bar{z}_{12},\sigma,t\right)\mathcal{O}_{\Delta,\sigma}\left(z_{2},\bar{z}_{2}\right)\mathrm{d}\mathcal{T}.$
(32)

###  3.2 One-loop gluon OPEs

In this section we will employ the formalism developed above to compute the
one-loop correction to the OPE of two positive helicity gluons. The relevant
splitting functions expressed in the parameterization of (14), (15) are

$\displaystyle\text{Split}^{(0)}_{-}(t;1^{+},2^{+})=\frac{1}{t(1-t)}\frac{1}{\omega}\frac{1}{z_{12}}\,,$
(33)
$\displaystyle\text{Split}^{(1)}_{-}(t;1^{+},2^{+})=\,\frac{1}{6}\frac{1}{\omega}\frac{1}{z_{12}}\left(1+\frac{n_{s}}{N_{c}}-\frac{n_{f}}{N_{c}}\right)+\frac{1}{\omega}\frac{1}{t\left(1-t\right)z_{12}}G^{n}\,,$
(34)
$\displaystyle\text{Split}^{(1)}_{+}(t;1^{+},2^{+})=\,\frac{1}{6}\frac{1}{\omega}\frac{\bar{z}_{12}}{z^{2}_{12}}\left(1+\frac{n_{s}}{N_{c}}-\frac{n_{f}}{N_{c}}\right),$
(35)

where

$\displaystyle\alpha\,G^{n}$
$\displaystyle=\alpha\,\hat{c}_{\Gamma}\left[-\frac{1}{\epsilon^{2}}\left(\frac{-1}{t^{2}(1-t)^{2}z_{12}\bar{z}_{12}\omega^{2}\frac{\Lambda^{2}}{\mu^{2}}}\right)^{\epsilon}+2\ln(t)\ln(1-t)-\frac{\pi^{2}}{6}\right]$
(36)
$\displaystyle=-\frac{a}{2\epsilon^{2}}+\frac{a}{2\epsilon}\left[c_{E}+\ln\left(-t^{2}(1-t)^{2}\omega^{2}z_{12}\bar{z}_{12}\frac{\Lambda^{2}}{\mu^{2}}\right)\right]-\frac{a}{4}\ln^{2}\left(-t^{2}(1-t)^{2}\omega^{2}z_{12}\bar{z}_{12}\frac{\Lambda^{2}}{\mu^{2}}\right)$
(37)
$\displaystyle\quad-\frac{a}{24}\left[\pi^{2}+6c_{E}^{2}-24\ln(t)\ln(1-t)+12c_{E}\ln\left(-t^{2}(1-t)^{2}\omega^{2}z_{12}\bar{z}_{12}\frac{\Lambda^{2}}{\mu^{2}}\right)\right].$
(38)

We have introduced $c_{E}=\gamma_{E}-\ln(4\pi)$, where $\gamma_{E}$ is the
Euler-Mascheroni constant, and we expanded everything to
$\mathcal{O}\left(\epsilon\right)$. From this, we can read off that the
functions $f^{(1)}_{r}$ appearing in (25) are

$\displaystyle
f^{(0)}\left(z_{12},\bar{z}_{12},+,t\right)=\frac{1}{z_{12}}\frac{1}{t\left(1-t\right)}\,,$
(39) $\displaystyle
f^{(1)}_{0}\left(z_{12},\bar{z}_{12},+,t\right)=\frac{1}{2}\frac{1}{z_{12}}\left[\frac{1}{3}\left(1+\frac{n_{s}}{N_{c}}-\frac{n_{f}}{N_{c}}\right)+\frac{1}{t\left(1-t\right)}\left(-\frac{1}{\epsilon^{2}}+\frac{c_{E}}{\epsilon}\right.\right.$
(40) $\displaystyle\left.\left.\hskip
179.25244pt-\frac{1}{12}\left(\pi^{2}+6c_{E}^{2}-24\ln
t\ln\left(1-t\right)\right)\right)\right],$ $\displaystyle
f^{(1)}_{1}\left(z_{12},\bar{z}_{12},+,t\right)=\frac{1}{2}\frac{1}{t\left(1-t\right)z_{12}}\left[\frac{1}{\epsilon}-c_{E}\right],$
(41) $\displaystyle
f^{(1)}_{2}\left(z_{12},\bar{z}_{12},+,t\right)=-\frac{1}{4}\frac{1}{t\left(1-t\right)z_{12}}\,,$
(42) $\displaystyle
f^{(1)}_{0}\left(z_{12},\bar{z}_{12},-,t\right)=\frac{1}{6}\frac{\bar{z}_{12}}{z^{2}_{12}}\left(1+\frac{n_{s}}{N_{c}}-\frac{n_{f}}{N_{c}}\right).$
(43)

According to (30), the OPE coefficients are obtained by integrating these
functions with the measure $\mathrm{d}\mathcal{T}$. Using the formulas

$\displaystyle\int_{0}^{1}\mathrm{d}\mathcal{T}=\frac{\left(\Delta_{1}-1\right)\left(\Delta_{2}-1\right)\text{B}\left(\Delta_{1}{-}1,\Delta_{2}{-}1\right)}{\left(\Delta_{1}+\Delta_{2}-1\right)\left(\Delta_{1}+\Delta_{2}-2\right)}\,,$
(44)
$\displaystyle\int_{0}^{1}\mathrm{d}\mathcal{T}\frac{1}{t\left(1-t\right)}=\text{B}\left(\Delta_{1}{-}1,\Delta_{2}{-}1\right),$
(45)
$\displaystyle\int_{0}^{1}\mathrm{d}\mathcal{T}\frac{1}{t\left(1-t\right)}\ln
t\ln\left(1-t\right)=\partial_{\Delta_{1}}\partial_{\Delta_{2}}\text{B}\left(\Delta_{1}{-}1,\Delta_{2}{-}1\right),$
(46)

where $\text{B}(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$, we obtain the OPE

$\mathcal{O}^{a}_{\Delta_{1},+}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{b}_{\Delta_{2},+}\left(z_{2},\bar{z}_{2}\right)\sim\frac{if^{abc}}{z_{12}}\Big{[}1+\frac{a}{2}\Big{(}C^{(1)}_{0,+}+C^{(1)}_{1,+}\mathcal{\hat{D}}_{12}\\\
+C^{(1)}_{2,+}\mathcal{\hat{D}}_{12}^{2}\Big{)}\Big{]}C^{(0)}_{+}\mathcal{O}^{c}_{\Delta,+}\left(z_{2},\bar{z}_{2}\right)+if^{abc}\,\frac{a}{2}\,\frac{\bar{z}_{12}}{z^{2}_{12}}C^{(1)}_{0,-}\,\mathcal{O}^{c}_{\Delta_{1}+\Delta_{2}-1,-}\left(z_{2},\bar{z}_{2}\right)$
(47)

where $C^{(0)}_{+}=\text{B}\left(\Delta_{1}-1,\Delta_{2}-1\right)$ is the
tree-level OPE coefficient Pate:2019lpp ; Fan:2019emx  and

$\displaystyle C^{(1)}_{0,+}=H_{0,+}^{(1)}+S_{0,+}^{(1)}\,,\qquad
S_{0,+}^{(1)}=-\frac{1}{\epsilon^{2}}+\frac{c_{E}}{\epsilon}-\frac{c_{E}^{2}}{2}\,,$
$\displaystyle
H_{0,+}^{(1)}=\frac{1}{3}\left(1+\frac{n_{s}}{N_{c}}-\frac{n_{f}}{N_{c}}\right)\frac{\left(\Delta_{1}-1\right)\left(\Delta_{2}-1\right)}{\left(\Delta_{1}+\Delta_{2}-1\right)\left(\Delta_{1}+\Delta_{2}-2\right)}-\frac{\pi^{2}}{12}+2\partial_{\Delta_{1}}\partial_{\Delta_{2}}\,,$
(48) $\displaystyle
C^{(1)}_{1,+}=2\left(\frac{1}{\epsilon}-c_{E}\right),\qquad
C^{(1)}_{2,+}=-2\,,\qquad
C^{(1)}_{0,-}=\frac{1}{3}\Big{(}1+\frac{n_{s}}{N_{c}}-\frac{n_{f}}{N_{c}}\Big{)}\text{B}\left(\Delta_{1},\Delta_{2}\right).$

We pause here to point out a few salient features of this OPE. First, it is
interesting to note that it was possible to write the loop corrections as
operators acting on the tree-level OPEs. The one exception is the term
proportional to $\mathcal{O}_{\Delta_{1}+\Delta_{2}-1,-}$ in (47). This has no
counterpart at tree-level in Yang-Mills theory. Secondly, the OPE now has
second order poles in the $\Delta_{i}$ in contrast to the simple poles at
tree-level.

The splitting behavior considered so far was for bare amplitudes. Upon
renormalization, the splitting functions are modified as (following the
renormalization scheme in section 2 of Bern:1995ix )

$\text{Split}^{(1)}_{\text{ren.}}(t;a^{\sigma_{a}},b^{\sigma_{b}})=\text{Split}^{(1)}(t;a^{\sigma_{a}},b^{\sigma_{b}})-\frac{\hat{c}_{\Gamma}}{\epsilon}\Big{(}\frac{11}{6}-\frac{1}{3}\frac{n_{f}}{N_{c}}-\frac{1}{6}\frac{n_{s}}{N_{c}}\Big{)}\text{Split}^{(0)}(t;a^{\sigma_{a}},b^{\sigma_{b}})\,.$
(49)

This modifies the OPE to

$\displaystyle\mathcal{O}^{a}_{\Delta_{1},+}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{b}_{\Delta_{2},+}\left(z_{2},\bar{z}_{2}\right)\sim(\ref{eq:++OPEfinal})+\frac{if^{abc}}{z_{12}}\frac{a}{4}C_{1,+}^{(1)}C_{+}^{(0)}\Big{(}\frac{11}{6}-\frac{1}{3}\frac{n_{f}}{N_{c}}-\frac{1}{6}\frac{n_{s}}{N_{c}}\Big{)}\mathcal{O}_{\Delta_{1}+\Delta_{2}-1,+}^{c}\,.$
(50)

###  3.3 Hard/soft factorization and the hard OPE

The OPE derived in section 3.2 is IR divergent. A natural way to extract a
finite OPE is by appealing to the hard/soft factorization theorem for massless
gauge theories Sen:1982bt ; Dixon:2008gr ; Gardi:2009qi ; Gardi:2009zv ;
Becher:2009qa ; Becher:2009cu ; Feige:2014wja ; Ma:2019hjq  which takes the
form

$\displaystyle\bigg{[}\mathcal{A}^{\text{ren}}_{n}\Big{(}\frac{p_{i}}{\mu},\alpha_{s}(\mu)\Big{)}\bigg{]}^{a_{1}a_{2}...a_{n}}=\bigg{[}\textbf{Z}_{n}\Big{(}\frac{p_{i}}{\mu},\alpha_{s}(\mu)\Big{)}\bigg{]}^{a_{1}a_{2}...a_{n}}_{a_{1}^{\prime}a_{2}^{\prime}...a_{n}^{\prime}}\bigg{[}\mathcal{H}_{n}\Big{(}\frac{p_{i}}{\mu},\alpha_{s}(\mu)\Big{)}\bigg{]}^{a_{1}^{\prime}a_{2}^{\prime}...a_{n}^{\prime}}\,.$
(51)

Here $\mathcal{A}^{\text{ren}}_{n}$ is the color-dressed, UV renormalized,
$n$-gluon amplitude in dimensional regularization and $a_{i}$ are color
indices. All IR divergences are contained in the color operator
$\textbf{Z}_{n}$. The hard operators are defined by

$\displaystyle\braket{H^{a_{1}}_{\Delta_{1},\sigma_{1}}\left(z_{1},\bar{z}_{1}\right)\dots
H^{a_{n}}_{\Delta_{n},\sigma_{n}}\left(z_{n},\bar{z}_{n}\right)}:=\int\prod_{i=1}^{n}d\omega_{i}\,\omega_{i}^{\Delta_{i}-1}\mathcal{H}_{n}^{a_{1}\dots
a_{n}}\,.$ (52)

Note that $\textbf{Z}_{n}=1+\mathcal{O}\left(a\right)$ which implies that

$\displaystyle\braket{H^{a_{1}}_{\Delta_{1},\sigma_{1}}\left(z_{1},\bar{z}_{1}\right)\dots
H^{a_{n}}_{\Delta_{n},\sigma_{n}}\left(z_{n},\bar{z}_{n}\right)}^{(0)}=\tilde{\mathcal{A}}_{n}^{(0)}\,.$
(53)

This lets us infer the OPE

$H^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1})H^{b}_{\Delta_{2},+}(z_{2},\bar{z}_{2})\sim\frac{if^{abc}}{z_{12}}\left[\text{B}(\Delta_{1}-1,\Delta_{2}-1)H^{c}_{\Delta_{1}+\Delta_{2}-1,+}(z_{2},\bar{z}_{2})\right.\\\
\left.\quad+a\,C_{H}(z_{12},\bar{z}_{12},\Delta_{1},\Delta_{2},\Delta,\sigma)\,H^{c}_{\Delta,\sigma}(z_{2},\bar{z}_{2})\right]+\mathcal{O}(a^{2})\,,$
(54)

where $C_{H}(z_{12},\bar{z}_{12},\Delta_{1},\Delta_{2},\Delta,\sigma)$ is the
one-loop correction that we must determine666For simplicity we have already
presumed the color structure of $C_{H}$ to be $f^{abc}$, which a more general
analysis demonstrates to be correct.. We will typically suppress the arguments
of $C_{H}$ in the rest of the paper. In the parameterization (14), the
universal soft factor can be written as777We will work in the dipole
approximation in this paper since deviations are expect only start at three
loops and beyond.

$\displaystyle\textbf{Z}_{n}\Big{(}\frac{p_{i}}{\mu},\alpha(\mu)\Big{)}=e^{n\,J}\textbf{Z}_{n}\Big{(}\frac{\Lambda^{2}}{\mu^{2}}\left|z_{ij}\right|^{2},\alpha(\mu)\Big{)}\prod_{k=1}^{n}\left(\frac{\omega_{k}}{\mu}\right)^{-\frac{K}{2}}$
(55)

where

$\displaystyle K$
$\displaystyle=\frac{1}{2}\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\gamma\left(\alpha(\lambda^{2})\right)=-a\left(\frac{1}{\epsilon}-c_{E}\right)+\mathcal{O}\left(a^{2}\right)=-\frac{a}{2}C_{1,+}^{(1)}$
(56)

and

$\displaystyle J$
$\displaystyle=\frac{1}{8}\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\ln\Big{(}\frac{\lambda^{2}}{\mu^{2}}\Big{)}\gamma(\alpha)-\frac{1}{2}\gamma_{J}(\alpha)$
(57)
$\displaystyle=\frac{a}{2}\left(-\frac{1}{\epsilon^{2}}+\frac{c_{E}}{\epsilon}-\frac{c_{E}^{2}}{2}\right)+\frac{a}{2}\left(-\frac{1}{\epsilon}+c_{E}\right)\left(\frac{11}{6}-\frac{1}{3}\frac{n_{f}}{N_{c}}-\frac{1}{6}\frac{n_{s}}{N_{c}}\right)$
$\displaystyle=\frac{a}{2}S_{0,+}^{(1)}+\frac{a}{4}\left(\frac{11}{6}-\frac{1}{3}\frac{n_{f}}{N_{c}}-\frac{1}{6}\frac{n_{s}}{N_{c}}\right)C_{1,+}^{(1)}\,.$

In arriving at the above results, we have used the one-loop values of the cusp
anomalous dimension $\gamma\left(\alpha\right)$ and collinear anomalous
dimension of gluons $\gamma_{J_{i}}$ DelDuca:2014cya  appropriate for this
theory, performed the integral after using the one-loop RG equation to compute
the running coupling. Finally, we have expressed them in terms of the OPE
coefficients defined in (48). The soft factor can be written as the correlator
of $2$D vertex operators. Building upon observations made in Arkani-
Hamed:2020gyp ; Kalyanapuram:2020epb ; Nande:2017dba , it was shown in
Magnea:2021fvy ; Nastase:2021izh ; Gonzalez:2021dxw  that the soft factor in
non-abelian gauge theories can be expressed as a correlator of vertex
operators, i.e.

$\displaystyle
e^{n\,J}\bigg{[}\textbf{Z}_{n}\Big{(}\frac{p_{i}}{\mu},\alpha(\mu)\Big{)}\bigg{]}^{a_{1}a_{2}...a_{n}}_{a_{1}^{\prime}a_{2}^{\prime}...a_{n}^{\prime}}=\Big{\langle}[\textbf{V}_{\kappa_{1}}(z_{1},\bar{z}_{1})]^{a_{1}}_{a_{1}^{\prime}}...[\textbf{V}_{\kappa_{n}}(z_{n},\bar{z}_{n})]^{a_{n}}_{a_{n}^{\prime}}\Big{\rangle}\,.$
(58)

Consequently, the celestial amplitude factorizes as

$\displaystyle\tilde{\mathcal{A}}^{a_{1}...a_{n}}=\Big{\langle}[\textbf{V}_{\kappa_{1}}(z_{1},\bar{z}_{1})]^{a_{1}}_{a_{1}^{\prime}}...[\textbf{V}_{\kappa_{n}}(z_{n},\bar{z}_{n})]^{a_{n}}_{a_{n}^{\prime}}\Big{\rangle}\Big{\langle}H^{a_{1}^{\prime}}_{\sigma_{1},\Delta_{1}^{\prime}}(z_{1},\bar{z}_{1})...H^{a_{n}^{\prime}}_{\sigma_{n},\Delta_{n}^{\prime}}(z_{n},\bar{z}_{n})\Big{\rangle}\,.$
(59)

Note that the conformal dimensions of the hard operators are
$\Delta_{i}^{\prime}=\Delta_{i}-K$. This shift is due to the factor
$\prod_{i=1}^{n}\left(\frac{\omega_{i}}{\mu}\right)^{-K}$ in (55). The OPE of
the vertex operators was derived in Magnea:2021fvy 888Our vertex operators
differ by a factor of $e^{J_{i}}$ from those in Magnea:2021fvy .. For the
purposes of this paper, it is convenient to express the OPE in the form

$\displaystyle
f^{a^{\prime}b^{\prime}c^{\prime}}[\textbf{V}_{\kappa_{1}}(z_{1},\bar{z}_{1})]^{a}_{a^{\prime}}\,\,[\textbf{V}_{\kappa_{2}}(z_{2},\bar{z}_{2})]^{b}_{b^{\prime}}\,\sim\,e^{J}\left(z_{12}\bar{z}_{12}\right)^{-\frac{K}{2}}f^{abc}[\textbf{V}_{\kappa}(z_{1},\bar{z}_{1})]_{c^{\prime}}^{c}\,.$
(60)

This form of the OPE can be derived from the OPE for
$[\textbf{V}_{\kappa_{1}}(z_{1},\bar{z}_{1})]^{a}_{a^{\prime}}\,\,[\textbf{V}_{\kappa_{2}}(z_{2},\bar{z}_{2})]^{b}_{b^{\prime}}$
given in Magnea:2021fvy  by contracting it with
$f^{a^{\prime}b^{\prime}c^{\prime}}$ and using the Jacobi identity. Expanding
to first order in $a$, we arrive at

$\displaystyle
f^{a^{\prime}b^{\prime}c^{\prime}}[\textbf{V}_{\kappa_{1}}(z_{1},\bar{z}_{1})]^{a}_{a^{\prime}}\,\,[\textbf{V}_{\kappa_{2}}(z_{2},\bar{z}_{2})]^{b}_{b^{\prime}}\sim$
(61) $\displaystyle\qquad
f^{abc}\left(1+\frac{a}{2}S_{0,+}^{(1)}+\frac{a}{4}\Bigg{(}\frac{11}{6}-\frac{1}{3}\frac{n_{f}}{N_{c}}-\frac{1}{6}\frac{n_{s}}{N_{c}}\Bigg{)}C_{1,+}^{(1)}+\frac{a}{4}C_{1,+}^{(1)}\ln\left(z_{12}\bar{z}_{12}\frac{\Lambda^{2}}{\mu^{2}}\right)\right)[\textbf{V}_{\kappa}(z_{1},\bar{z}_{1})]_{c^{\prime}}^{c}\,.$

Introducing the hard/soft decomposition of the operators

$\displaystyle\mathcal{O}^{a}_{\Delta,\sigma}(z,\bar{z})=[\textbf{V}_{\kappa}(z,\bar{z})]^{a}_{a^{\prime}}H^{a^{\prime}}_{\Delta^{\prime},\sigma}(z,\bar{z})\,,$
(62)

we can use (54), (61) to express the OPE of two gluons as

$\displaystyle\mathcal{O}^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1})\mathcal{O}^{b}_{\Delta_{2},+}(z_{2},\bar{z}_{2})\sim\frac{if^{abc}}{z_{12}}\Bigg{[}C_{+}^{(0)}\mathcal{O}^{c}_{\Delta_{1}+\Delta_{2}-1,+}+\frac{a}{2}\left(S_{0,+}^{(1)}+C_{1,+}^{(1)}\hat{\mathcal{D}}_{12}\right)C_{+}^{(0)}\mathcal{O}^{c}_{\Delta_{1}+\Delta_{2}-1,+}$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{a}{4}\left(\frac{11}{6}-\frac{1}{3}\frac{n_{f}}{N_{c}}-\frac{1}{6}\frac{n_{s}}{N_{c}}\right)C_{1,+}^{(1)}C_{+}^{(0)}\mathcal{O}^{c}_{\Delta_{1}+\Delta_{2}-1,+}+a\,C_{H}\,\mathcal{O}^{c}_{\Delta,\sigma}\Bigg{]}.$
(63)

Comparing this with (47), we infer the value of $C_{H}$ and deduce the hard
OPE to be

$H^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1})H^{b}_{\Delta_{2},+}(z_{2},\bar{z}_{2})\sim\frac{if^{abc}}{z_{12}}\left[1+aH_{0,+}^{(1)}+aC_{2,+}^{(1)}\hat{\mathcal{D}}_{12}^{2}\right]C_{+}^{(0)}H_{\Delta_{1}+\Delta_{2}-1,+}^{c}\\\
+if^{abc}\frac{a}{2}\frac{\bar{z}_{12}}{z^{2}_{12}}C_{0,-}^{(1)}H_{\Delta_{1}+\Delta_{2}-1,-}^{c}\,.$
(64)

Similarly, for $g^{+}g^{-}\rightarrow g^{+}$ the hard OPE reads

$H^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1})H^{b}_{\Delta_{2},-}(z_{2},\bar{z}_{2})\sim\\\
\frac{if^{abc}}{\bar{z}_{12}}\left[1-a\left(\frac{\pi^{2}}{24}-\partial_{\Delta_{1}}\partial_{\Delta_{2}}+\mathcal{\hat{D}}^{2}_{12}\right)\right]\text{B}(\Delta_{1}{+}1,\Delta_{2}{-}1)H^{c}_{\Delta_{1}+\Delta_{2}-1,+}\,,$
(65)

and for $g^{+}g^{-}\rightarrow g^{-}$ the hard OPE reads

$H^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1})H^{b}_{\Delta_{2},-}(z_{2},\bar{z}_{2})\sim\\\
\frac{if^{abc}}{z_{12}}\left[1-a\left(\frac{\pi^{2}}{24}-\partial_{\Delta_{1}}\partial_{\Delta_{2}}+\mathcal{\hat{D}}^{2}_{12}\right)\right]\text{B}(\Delta_{1}{-}1,\Delta_{2}{+}1)H^{c}_{\Delta_{1}+\Delta_{2}-1,-}\,.$
(66)

In the case of planar $\mathcal{N}=4$ SYM theory one simply omits from (64)
the non-factorizing term to obtain

$H^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1})H^{b}_{\Delta_{2},+}(z_{2},\bar{z}_{2})\sim\\\
\frac{if^{abc}}{z_{12}}\Bigg{[}1-a\Bigg{(}\frac{\pi^{2}}{24}-\partial_{\Delta_{1}}\partial_{\Delta_{2}}+\hat{\mathcal{D}}^{2}_{12}\Bigg{)}\Bigg{]}\text{B}(\Delta_{1}{-}1,\Delta_{2}{-}1)H^{c}_{\Delta_{1}+\Delta_{2}-1,+}\,.$
(67)

##  4 All-loop OPE in planar $\mathcal{N}=4$ SYM

###  4.1 Amplitudes in Higgs-regulated planar $\mathcal{N}=4$ SYM

In planar $\mathcal{N}=4$ super Yang-Mills (SYM) theory, the exponential
structure of IR divergences is captured to all orders by the BDS ansatz
Bern:2005iz

$\begin{split}M_{n}^{\text{BDS}}(\epsilon)=\exp\left[\sum_{L=1}^{\infty}\alpha^{L}\left(\frac{1}{4}\gamma^{(L)}+\frac{\epsilon
L}{2}\mathcal{G}^{(L)}\right)M_{n}^{(1)}\left(L\epsilon\right)+R_{n}\left(a,s_{ij}\right)+\mathcal{O}\left(\epsilon\right)\right],\end{split}$
(68)

where
$M_{n}^{\text{BDS}}(\epsilon)=\frac{\mathcal{A}_{n}(\epsilon)}{\mathcal{A}_{n}^{(0)}}$
(cf. (1)), $s_{ij}=\left(p_{i}+p_{j}\right)^{2}$, $\gamma^{(L)}\left(a\right)$
is the $L$-loop cusp anomalous dimension Korchemskaya:1992je  and
$\mathcal{G}^{(L)}$ is the collinear anomalous dimension Magnea:1990zb .
$R_{n}$ is the remainder function which is nonvanishing only for $n>5$
Drummond:2007cf ; Drummond:2007au ; Bern:2008ap ; Cachazo:2008hp . The
consistency of this expression with the hard/soft factorization of section 3.3
can be checked by noting that the IR divergent part of the one-loop amplitude
is

$\displaystyle
M_{n}^{(1)}=-\frac{1}{2\epsilon^{2}}\sum_{i=1}^{n}\left(-\frac{\mu^{2}}{s_{i,i+1}}\right)^{\epsilon}+\mathcal{O}\left(\epsilon^{0}\right).$
(69)

The all-loop celestial amplitude at four points has been computed in
Gonzalez:2020tpi  via the BDS Ansatz. The avatar of the hard/soft
factorization at the level of celestial amplitude has been analyzed in
Gonzalez:2021dxw .

One feature of the BDS ansatz is that, due to cross terms present in the power
series expansion of the exponential, the argument inside the exponential must
be computed to increasingly higher orders in $\epsilon$ at higher loop order
in order to determine $\mathcal{O}(\epsilon^{0})$ contributions to $M_{n}^{\rm
BDS}$. Dimensional regularization also breaks the dual superconformal symmetry
of the amplitude. A way to circumvent both of these difficulties was developed
in Alday:2009zm  where the authors proposed a Higgs mechanism based regulator,
which preserves an appropriately generalized version of dual conformal
symmetry. Spontaneously breaking $U(N+M)\longrightarrow U(N)\times U(M)$ in
$\mathcal{N}=4$ SYM makes some of the gauge bosons massive. In the limit $N\gg
M\gg 1$, the dominant contribution to the scattering of massless $U(M)$ gauge
bosons comes from planar graphs with a single external loop of massive gauge
bosons, and only massless $U(N)$ bosons propagating inside the loop. Based on
their computation at one and two loops and taking inspiration from the BDS
ansatz, the authors postulated the following formula for the Higgs-regulated
four-point amplitude:

$\displaystyle\ln M_{4}(s,t)=$
$\displaystyle\frac{\gamma\left(a\right)}{4}M_{4}^{(1)}-\tilde{\mathcal{G}}(a)\left[\ln\left(\frac{s}{m^{2}}\right)+\ln\left(\frac{t}{m^{2}}\right)\right]+\tilde{c}_{4}(a)+\mathcal{O}(m^{2})\,,$
(70)

where

$M^{(1)}_{4}(p_{1},p_{2},p_{3},p_{4})=-\frac{1}{2}\left[\ln^{2}\left(\frac{s_{12}}{m^{2}}\right)+\ln^{2}\left(\frac{s_{13}}{m^{2}}\right)-\ln^{2}\left(\frac{s_{12}}{s_{13}}\right)\right]+\frac{\pi^{2}}{2}+\mathcal{O}(m^{2})\,,$
(71)

$\gamma\left(a\right)=\sum_{L=0}^{\infty}\gamma^{(L)}\left(a\right)$ while
$\tilde{\mathcal{G}}$ is the analog of the collinear anomalous dimension
appearing in (68), $\tilde{c}_{4}(a)$ is a finite additive constant and the
coupling in this theory is just the t‘Hooft coupling $a$. Further support for
this conjecture was provided by three- and four-loop computations performed in
Henn:2010bk ; Henn:2010ir . Their calculations also supported the expected
exponentiation of the five-point amplitude

$\begin{split}\ln{M_{5}}&=\frac{\gamma(a)}{4}M_{5}^{(1)}-\frac{\tilde{\mathcal{G}}_{0}(a)}{2}\sum_{i=1}^{5}\ln\left(\frac{s_{i}}{m^{2}}\right)+\mathcal{O}(m^{2})\end{split}$
(72)

where $s_{i}=s_{i,i+1}$ and

$\displaystyle M_{5}^{(1)}=$
$\displaystyle-\frac{1}{4}\sum_{i=1}^{5}\left[\ln^{2}\left(\frac{s_{i}}{m^{2}}\right)-2\ln\left(\frac{s_{i}}{s_{i+1}}\right)\ln\left(\frac{s_{i+2}}{s_{i-2}}\right)\right]+\frac{5\pi^{2}}{12}+\mathcal{O}(m^{2})\,.$
(73)

is the five-point, one-loop amplitude.

The $m\to 0$ IR divergences of Higgs-regulated amplitudes take the form of
powers of $\ln m$, and since $m^{a}\ln({s}/{m^{2}})^{b}\xrightarrow[]{m\to
0}0$ for any positive $a,b$, there are no cross terms present in the power
series expansion of the exponential, unlike in dimensional regularization. At
non-zero values of $m$, this provides an example of an all-loop, IR finite
amplitude999Of course, an important order-of-limits issue is hiding here: when
integrating over all values of energy $s$, regions of exponentially large and
small values $s\sim m^{2}e^{\pm\frac{1}{m^{a}}}$ do contribute to a term like
$m^{a}\ln\left(\frac{s}{m^{2}}\right)$. Hence it is necessary to take
$m\rightarrow 0$ before taking a Mellin transform to the celestial amplitude..

It is instructive to compare the Higgs-regulated amplitude to the one we
obtain via the dimensionally regularization BDS formula (68). Working with
$n=4$ and dropping the $\frac{1}{\epsilon}$ and $\frac{1}{\epsilon^{2}}$ terms
we have

$\ln
M_{4}^{\text{BDS}}(s,t)|_{\text{finite}}=-\frac{1}{4}\gamma(a)\left[\ln^{2}\left(-\frac{\mu^{2}}{s}\right)+\ln^{2}\left(-\frac{\mu^{2}}{t}\right)\right]\\\
-\frac{\mathcal{G}(a)}{2}\left[\ln\left(-\frac{\mu^{2}}{s}\right)+\ln\left(-\frac{\mu^{2}}{t}\right)\right]+\frac{\gamma(a)}{8}\ln^{2}\left(\frac{s}{t}\right)+C(a)+\mathcal{O}(\epsilon)\,.$
(74)

The amplitude is equal (up to an additive constant and redefinition of
collinear and cusp anomalous dimension) to the amplitude in (70) obtained via
Higgs regularization.

It is natural to expect that this relationship continues to hold at the level
of celestial amplitudes. Testing this expectation requires us to first
identify an IR finite celestial amplitude in the dimensionally regulated
theory. To this end, we can take advantage of the fact that celestial
amplitudes in $\mathcal{N}=4$ SYM factorize as

$\tilde{\mathcal{A}}(\\{z_{i},\bar{z}_{i},\Delta_{i}\\})=\left<\mathcal{V}_{\tau_{1}}(z_{1},\bar{z}_{1})\cdots\mathcal{V}_{\tau_{n}}(z_{n},\bar{z}_{n})\right>\big{<}\hat{\mathcal{O}}_{\Delta^{\prime}_{1},l_{1}}(z_{1},\bar{z}_{1})\cdots\hat{\mathcal{O}}_{\Delta^{\prime}_{n},l_{n}}(z_{n},\bar{z}_{n})\big{>}\,,$
(75)

where $\Delta_{i}^{\prime}=\Delta_{i}+\frac{\gamma\left(a\right)}{8\epsilon}$
and $\mathcal{V}_{i}$ are vertex operators whose dimensions are all equal to
$\frac{\gamma\left(a\right)}{8\epsilon}$101010Where
$\sigma_{i}=-\frac{\gamma_{1}(a)}{8\epsilon}$, $\\{\tau_{i}\\}$ are analogs of
the $SU(N_{c})$ generators $T^{a}$ given in section 3.3; they may also be seen
as a set of vectors in $\mathbb{R}^{n}$ with the following property
$\mathbf{\tau}_{i}\cdot\mathbf{\tau}_{j}=\delta_{i,j}-\delta_{i,j+1}-\delta_{i+1,j}+\delta_{i+1,j+1},\hskip
28.45274pt\sum_{i}\tau^{a}_{i}=0.$ (76) the vertex operators are dressed as
$\mathcal{V}_{\mathbf{\tau_{k}}}(z,\bar{z})=\mathcal{J}(\epsilon,a):\exp{\left[i\mathbb{\tau}_{k}\cdot\Phi(z,\bar{z})\right]}:$
(77) where $\Phi^{a}(z,\bar{z})$ are colored scalar fields.. This was shown in
Gonzalez:2021dxw , where it allowed the authors to define and compute an IR
finite celestial amplitude. Mellin transforming (70) we find

$\tilde{\mathcal{A}}_{4}(\\{\Delta_{i},\sigma_{i},z_{i},\bar{z}_{i}\\})=\int_{0}^{\infty}\prod_{i=1}^{4}d\omega_{i}\,\omega_{i}^{\Delta_{i}-1}M_{4}=f(z,\bar{z})\prod_{i<j}^{4}z_{ij}^{\frac{h}{3}-h_{i}-h_{j}}\bar{z}_{ij}^{\frac{\bar{h}}{3}-\bar{h}_{i}-\bar{h}_{j}}$
(78)

where

$f(z,\bar{z})=4\left(\frac{m}{\Lambda}\right)^{\beta}g^{2}\mathcal{C}(a)\delta(z-\bar{z})\Theta(z-1)z^{\frac{\beta+10}{6}-\tilde{\mathcal{G}}_{0}(a)-\frac{i\pi}{4}\gamma(a)}(z-1)^{\frac{\beta+4}{6}}\\\
\times\sqrt{\frac{\pi}{\gamma(a)}}\exp{\left[\frac{1}{\gamma(a)}\left(\frac{\beta}{2}-\frac{\gamma(a)}{4}\ln(-z)-2\tilde{\mathcal{G}}_{0}(a)\right)^{2}+\mathcal{O}(m^{2})\right]},$
(79)

and

$\displaystyle\beta=\sum_{i=1}^{4}\Delta_{i}-4\,,\quad
z=\frac{z_{12}z_{34}}{z_{23}z_{41}}\quad\text{and }\quad\mathcal{C}(a)$
$\displaystyle=\exp{\left(\frac{\pi^{2}}{8}\gamma(a)+i\pi\tilde{\mathcal{G}}_{0}(a)+\tilde{c}_{4}(a)\right).}$
(80)

We refer the reader to appendix A for the complete computation. This is in
agreement with the four-point correlator of the hard operators found in (2.35)
of Gonzalez:2021dxw 111111We are only comparing the dependence on kinematics
and ignore overall numerical factors and a redefinition of the collinear
anomalous dimension.. We will demonstrate below that this correspondence
continues to hold at the level of OPEs. The upshot is that the Higgs-regulated
theory seems to offer an efficient way to perform computations involving the
hard operators.

Since $z$ is always real and positive due the constraints coming from delta
and theta functions in (79), the amplitude has a nontrivial imaginary part
arising from $\ln(-z)$. Recently the authors of Garcia-Sepulveda:2022lga
discussed unitarity and factorization on the celestial sphere in the context
of the three-dimensional $O(N)$ sigma model. They concluded that the imaginary
part of the celestial amplitude can be expressed as a sum over residues of the
bulk S-matrix at appropriate singularities in $\omega$. It might be
interesting to study (79) along those lines.

###  4.2 All-loop OPE in the Higgs-regulated $\mathcal{N}=4$ SYM

In order to compute the OPE we first compute the corresponding splitting
function for this theory. Starting with (73), parameterizing $p_{1}$ and
$p_{2}$ as in (2), we find121212The subscript $H$ stands for Higgs-regulated
and not hard as in section 3.3.

$M^{(1)}_{5}\overset{1\parallel
2}{\longrightarrow}M^{(1)}_{4}(P,p_{3},p_{4},p_{5})+r_{H}^{(1)}(s,m^{2};t)+\mathcal{O}(m^{2})$
(81)

where

$r_{H}^{(1)}(s,m^{2};t)=-\frac{1}{4}\ln^{2}\left(\frac{m^{2}}{\Lambda^{2}st(1-t)}\right)+\ln{t}\ln(1-t)-\frac{\pi^{2}}{12}\,.$
(82)

Recall that $M_{n}^{(1)}={\mathcal{A}^{(1)}_{n}}/{\mathcal{A}_{n}^{(0)}}$ and
that131313Here we assume particles 1 and 2 have positive helicity. (cf. (3))

$\displaystyle\mathcal{A}_{5}^{(0)}\left(p_{1},+;p_{2},+,\dots\right)\overset{1\parallel
2}{\longrightarrow}\text{Split}^{(0)}_{-}\left(t;1^{+},2^{+}\right)\mathcal{A}_{4}^{(0)}\left(P,+,\dots\right).$
(83)

We can rewrite (81) as

$\displaystyle\mathcal{A}^{(1)}_{5}\overset{1\parallel
2}{\longrightarrow}\text{Split}^{(0)}_{-}\left(t;1^{+},2^{+}\right)\mathcal{A}_{4}^{(1)}\left(P,+,\dots\right)+\text{Split}^{(0)}_{-}\left(t;1^{+},2^{+}\right)r_{H}^{(1)}(s,m^{2};t)\mathcal{A}_{4}^{(0)}\left(P,+,\dots\right)$
(84)

which is in agreement with (6) and (7). The factorizing part of the splitting
function doesn’t contribute here. Following the procedure outlined in section
3 we get

${\mathcal{O}}^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1}){\mathcal{O}}^{b}_{\Delta_{2},+}(z_{2},\bar{z}_{2})\sim\\\
\frac{if^{abc}}{z_{12}}\left[1-a\left(\hat{\mathcal{D}}_{12}^{2}-\partial_{\Delta_{1}}\partial_{\Delta_{2}}+\frac{\pi^{2}}{12}\right)\right]\text{B}(\Delta_{1}{-}1,\Delta_{2}{-}1){\mathcal{O}}^{c}_{\Delta-1,+}(z_{2},\bar{z}_{2})$
(85)

with ${\mathcal{D}}_{12}$ defined as in (27) with the replacement $\mu\to m$.
We emphasize once again that we treat $\Delta$ as independent prior to the
action of the differential operator. We can now obtain the OPE for the all-
loop amplitude in (72). Inserting the collinear behavior of the one-loop
amplitude yields

$\begin{split}M_{5}(p_{1},+;p_{2},+;\dots)\overset{1\parallel
2}{\longrightarrow}r_{H}(s,m^{2};t)M_{4}(P,+;\dots)\,,\end{split}$ (86)

where the all-loop splitting function is

$r_{H}(s,m^{2};t)=\exp\left[\frac{\gamma(a)}{4}r_{H}^{(1)}(s,m^{2};t)+\frac{\tilde{\mathcal{G}}_{0}(a)}{2}\ln\left(\frac{m^{2}}{\Lambda^{2}st(1-t)}\right)\right].$
(87)

This implies the all-loop OPE (cf. (31), (32))

${\mathcal{O}}^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1}){\mathcal{O}}^{b}_{\Delta_{2},+}(z_{2},\bar{z}_{2})\sim\frac{if^{abc}}{z_{12}}e^{-\frac{\gamma\left(a\right)\pi^{2}}{48}}\\\
\times\exp\left[-\frac{\gamma(a)}{4}\left(\hat{\mathcal{D}}^{2}_{ij}-\partial_{\Delta_{i}}\partial_{\Delta_{j}}\right)-\tilde{\mathcal{G}}_{0}(a)\hat{\mathcal{D}}_{ij}\right]\text{B}(\Delta_{1}{-}1,\Delta_{2}{-}1){\mathcal{O}}^{c}_{\Delta,+}(z_{2},\bar{z}_{2})\,.$
(88)

###  4.3 All-loop OPE in dimensionally regulated $\mathcal{N}=4$ SYM

In this section, we will derive the all-loop OPE in the dimensionally
regulated theory. Based on the discussion presented towards the end of section
4.1, we expect the finite part of this OPE to match the one in (88). Starting
with (68), and using the collinear behavior of the one-loop
amplitude141414Since the collinear behavior is independent of the number of
legs, we can work with $n=5$ where the remainder function vanishes

$M^{(1)}_{n}(\epsilon)\longrightarrow
M^{(1)}_{n-1}(\epsilon)\,+r_{D}^{(1)}(s,\epsilon;t)\,,$ (89)

where $r^{(1)}_{D}(s,\epsilon;t)$ is the analog of
$r_{H}^{(1)}\left(s,m^{2};t\right)$ (cf. (82)) and is given by

$\begin{split}r^{(1)}_{D}(L\epsilon;t,s)&=\hat{c}_{\Gamma}\left[-\frac{1}{(L\epsilon)^{2}}\left(\frac{-\mu^{2}}{\Lambda^{2}t(1-t)s}\right)^{L\epsilon}+2\ln{t}\ln{(1-t)}-\frac{\pi^{2}}{6}\right]+\mathcal{O}\left({\epsilon}\right)\end{split}$
(90)

where $\hat{c}_{\Gamma}$ was defined in (12). Using this in the full
amplitude, we extract the all-loop splitting function

$r_{D}(s,\epsilon;t)=\exp{\left[\sum_{L=1}^{\infty}\alpha^{L}\left(f^{(L)}(L\epsilon)r^{(1)}_{D}(L\epsilon;t,s)+\mathcal{O}(\epsilon)\right)\right]}$
(91)

where
$f^{(L)}(\epsilon)=\gamma^{(L)}/4+\epsilon\,\mathcal{G}^{(L)}/2+\mathcal{O}(\epsilon^{2})$.
Expanding in $\epsilon$ and dropping terms of
$\mathcal{O}\left(\epsilon\right)$, and using (31), (32), we get

$\displaystyle
O^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1})O^{b}_{\Delta_{2},+}(z_{2},\bar{z}_{2})$
(92)
$\displaystyle\sim\frac{if^{abc}}{z_{12}}e^{-\frac{\gamma(\alpha)\pi^{2}}{96}+\frac{i\pi}{8\epsilon}\gamma_{1}(\alpha)}\mathcal{J}(\epsilon,\alpha)\exp\left[\hat{\mathcal{S}}_{12}+\frac{\gamma_{1}\left(\alpha\right)}{4\epsilon}\hat{\mathcal{D}}_{12}+\mathcal{O}(\epsilon)\right]\text{B}\left(\Delta_{1}{-}1,\Delta_{2}{-}1\right)O^{c}_{\Delta,+}(z_{2},\bar{z}_{2})$

with

$\hat{\mathcal{S}}_{ij}(\epsilon,\alpha;z_{ij})=-\frac{\gamma(\alpha)}{4}\left(\hat{\mathcal{D}}^{2}_{ij}-\partial_{\Delta_{i}}\partial_{\Delta_{j}}\right)+\frac{\mathcal{G}_{0}(\alpha)}{2}\hat{\mathcal{D}}_{ij}\,,$
(93)

and the jet function

$\mathcal{J}(\epsilon,\alpha)=\exp\left(-\frac{i\pi}{8\epsilon}\gamma_{1}(\alpha)-\frac{\gamma_{2}(\alpha)}{8\epsilon^{2}}-\frac{\mathcal{G}_{1}(\alpha)}{4\epsilon}\right)$
(94)

is a non-dynamical factor depending on $\alpha$ only via the modified
anomalous dimensions as defined in Gonzalez:2021dxw

$\displaystyle\gamma_{m}(\alpha)=\sum_{L=1}^{\infty}L^{-m}\alpha^{L}\gamma^{(L)}\,,\qquad\mathcal{G}_{m}(\alpha)=\sum_{L=1}^{\infty}L^{-m}\alpha^{L}\mathcal{G}^{(L)}\,.$
(95)

With the goal of bringing (92) to a form similar to (88), we note that we can
use

$\exp\left(\frac{i\pi}{8\epsilon}\gamma_{1}(\alpha)+\frac{\gamma_{1}\left(\alpha\right)}{4\epsilon}\hat{\mathcal{D}}_{12}\right)\text{B}\left(\Delta_{1}{-}1,\Delta_{2}{-}1\right)O^{c,+}_{\Delta}(z_{2},\bar{z}_{2})\\\
=\left(\frac{\mu^{2}}{\Lambda^{2}z_{12}\bar{z}_{12}}\right)^{-\frac{\gamma_{1}\left(a\right)}{8\epsilon}}\text{B}\left(\Delta_{1}^{\prime}{-}1,\Delta_{2}^{\prime}{-}1\right)O_{\Delta+\frac{\gamma}{4\epsilon}}^{c,+}\left(z_{2},\bar{z}_{2}\right)$
(96)

to rewrite (92) as

$O^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1})O^{b}_{\Delta_{2},+}(z_{2},\bar{z}_{2})\sim\frac{if^{abc}}{z_{12}}\left[\mathcal{J}(\epsilon,\alpha)\left(\frac{\mu^{2}}{\Lambda^{2}z_{12}\bar{z}_{12}}\right)^{-\frac{\gamma_{1}\left(\alpha\right)}{8\epsilon}}\right]e^{-\frac{\gamma(\alpha)\pi^{2}}{96}}\\\
\times\exp\left[\hat{\mathcal{S}}_{12}+\mathcal{O}(\epsilon)\right]\text{B}\left(\Delta^{\prime}_{1}{-}1,\Delta^{\prime}_{2}{-}1\right)O^{c,+}_{\Delta+\frac{\gamma}{4\epsilon}}(z_{2},\bar{z}_{2})\,.$
(97)

Finally, we recall from (75) that the celestial amplitude factorizes and we
can decompose

$\mathcal{O}_{\Delta,+}^{a}(z_{i},\bar{z}_{i})=\,\,:\\!\mathcal{V}_{\sigma_{i}}(z_{i},\bar{z}_{i})\\!:\,\,:\\!\hat{\mathcal{O}}_{\Delta^{\prime},+}^{a}(z_{i},\bar{z}_{i})\\!:$
(98)

and that the OPE of the vertex operators is Gonzalez:2021dxw ,

$\mathcal{V}_{\tau_{1}}(z_{1},\bar{z}_{1})\mathcal{V}_{\tau_{2}}(z_{2},\bar{z}_{2})=\frac{\mathcal{J}(\epsilon,\alpha)\mathcal{V}_{\tau_{1}+\tau_{2}}(z_{2},\bar{z}_{2})}{\left(z_{12}\bar{z}_{12}\right)^{-\frac{\gamma_{1}\left(\alpha\right)}{8\epsilon}}}\,.$
(99)

This lets us write

$O^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1})O^{b}_{\Delta_{2},+}(z_{2},\bar{z}_{2})\sim\frac{if^{abc}}{z_{12}}\left(\mathcal{V}_{\tau_{1}}(z_{1},\bar{z}_{1})\mathcal{V}_{\tau_{2}}(z_{2},\bar{z}_{2})\right)\left(\frac{\mu^{2}}{\Lambda^{2}}\right)^{-\frac{\gamma_{1}\left(\alpha\right)}{8\epsilon}}e^{-\frac{\gamma(\alpha)\pi^{2}}{96}}\\\
\times\exp\left[\hat{\mathcal{S}}_{12}+\mathcal{O}(\epsilon)\right]\text{B}\left(\Delta^{\prime}_{1}{-}1,\Delta^{\prime}_{2}{-}1\right)\hat{O}^{c}_{\Delta^{\prime},+}(z_{2},\bar{z}_{2})\,,$
(100)

where we note that
$\Delta^{\prime}=\Delta^{\prime}_{1}+\Delta^{\prime}_{2}-1$. Stripping off the
factor
$\mathcal{V}_{\tau_{1}}(z_{1},\bar{z}_{1})\mathcal{V}_{\tau_{2}}(z_{2},\bar{z}_{2})$
from the LHS, we see that the OPE of the hard operators coincides with the OPE
derived from the Higgs-regulated amplitude.

##  5 Discussion

###  5.1 Role of logarithmic CFTs and operators in CCFT

The IR finite OPEs obtained in (64), (88) after soft factorization contain
logarithms of $z$ and operators of the form
$\partial_{\Delta}\mathcal{O}_{\Delta,\sigma}$. The most familiar OPE which
contains logarithms is the free boson OPE. In that case, it is a consequence
of working with a weight zero primary whose conformal properties are not well-
defined and may be fixed by working with derivatives or exponentials of the
operator. A second instance in which logarithms appear is when one performs
perturbation theory in a CFT. Logs then appear as a consequence of anomalous
dimensions and operator mixing. Upon diagonalizing the dilation operator and
summing the perturbative series, one should then find that the logarithms sum
to a power law if the conformal symmetry is preserved. It would be of interest
to determine whether the all-loop result (88) is consistent with this
scenario.

If not, it raises the more exotic possibility that celestial CFT may be a type
of logarithmic CFT (LCFT) (see for example Gurarie:1993xq ). In an LCFT states
are organized in logarithmic multiplets of rank $r\geq 1$ Hogervorst:2016itc .
These are built on the top of $r$ primary operators (of the same conformal
dimension) which mix under the action of the dilatation operator. The
dilatation operator in an LCFT is non diagonalizable rendering the theory non-
unitary. Under a conformal transformation parameterized by $f(z)$, a rank-$r$
primary $O_{a}(z)$ transforms as

$O_{a}(z)=\left(\frac{\partial f(z)}{\partial
z}\right)^{\Delta}\sum_{b=a}^{r}\frac{1}{(b-a)!}\ln^{b-a}\left(\partial_{z}f(z)\right)O_{b}(f(z))\,.$
(101)

The logarithmic partners of a primary are defined by Hogervorst:2016itc ;
Flohr:2001zs

$O_{a}\equiv\frac{1}{(r-a)!}\partial^{r-a}_{\Delta}O_{\Delta}\,.$ (102)

The multiplet is

$\hat{O}_{\Delta}=\left(\frac{1}{r!}O^{(r)}_{\Delta},\frac{1}{(r-1)!}O^{(r-1)}_{\Delta},\cdots,O^{{}^{\prime}}_{\Delta},O_{\Delta}\right)^{T}.$
(103)

For some applications of LCFTs, we refer the reader to Hogervorst:2016itc ;
Flohr:2001zs ; Nivesvivat:2020gdj . The all-loop OPE (88) seems to suggest
that at loop order $L$, the corresponding CCFT is described by a rank-$(2L+1)$
multiplet. It would be interesting to analyze this further and to consider the
effect this has on the conformal block decomposition in light of
Hogervorst:2016itc . As $L\to\infty$ we have an rank-$\infty$ family of log
primaries. Not much is known of infinite rank LCFTs; see Rasmussen:2004jc  for
a discussion.

###  5.2 Associativity

The presence of logarithms prevents us from performing a straightforward check
of OPE associativity along the lines of Costello:2022upu ; Ren:2022sws .
However, in this section we propose a conjectural identity to test OPE
associativity when it contains logarithms. The statement resembles the “double
residue formula” used in Ren:2022sws  to check the associativity of the tree-
level OPE. Consider a double-OPE limit of three holomorphic operators,
$\mathcal{O}_{1}(z_{1})$, $\mathcal{O}_{2}(z_{2})$ and
$\mathcal{O}_{3}(z_{3})$. The expansion of
$(\mathcal{O}_{1}\mathcal{O}_{2})\mathcal{O}_{3}$ can be brought to the form:

$\displaystyle(\mathcal{O}_{1}\mathcal{O}_{2})\mathcal{O}_{3}\sim\frac{1}{z_{12}}\frac{1}{z_{23}}\sum_{r,s}\ln^{r}(z_{12})\ln^{s}(z_{23})K(1,2,3;r,s;z_{3})\,,$
(104)

where we suppress the color factors for the moment. We conjecture that if the
OPE is associative, the following should be satisfied:

$\sum_{\begin{subarray}{c}r,s\geq 0\\\
r+s=m\end{subarray}}\left(K(1,2,3;r;s;z_{3})+K(2,3,1;r;s;z_{3})+K(3,1,2;r;s;z_{3})\right)=0\,,$
(105)

where $K\left(2,3,1;r,;s;z_{3}\right)$ and $K\left(3,1,2;r;s;z_{3}\right)$ are
obtained by taking the OPE limits in different orders.

In order to check if this identity is satisfied by the one-loop OPE of hard
operators (67), we first note that if we restore the color factors then each
$K$ has the same color structure, specifically
$K\left(i,j,k;r;s;z_{3}\right)f^{a_{i}a_{j}b}f^{ba_{k}c}$. The identity (105)
is satisfied if $K\left(i,j,k;r;s;z_{3}\right)$ is symmetric under cyclic
permutations of $i,j,k$ by virtue of the Jacobi identity satisfied by the
color factors. Evaluating the $K$ coefficients for planar $\mathcal{N}=4$ SYM
(67) we find151515Here the $K^{(n)}_{m}(\Delta_{i},\Delta_{j},\Delta_{k})$
stands for the coefficient in front of the $n$th logarithmic partner with
logarithmic power $m$.

$\displaystyle K^{(2)}_{0}$
$\displaystyle=-2\frac{\Gamma(\Delta_{i}-1)\Gamma(\Delta_{j}-1)\Gamma(\Delta_{k}-1)}{\Gamma\left(\Delta_{i}+\Delta_{j}+\Delta_{k}-3\right)}\,,$
(106) $\displaystyle K^{(1)}_{0}$
$\displaystyle=-2(\partial_{\Delta_{i}}+\partial_{\Delta_{j}}+\partial_{\Delta_{k}})\frac{\Gamma(\Delta_{i}-1)\Gamma(\Delta_{j}-1)\Gamma(\Delta_{k}-1)}{\Gamma\left(\Delta_{i}+\Delta_{j}+\Delta_{k}-3\right)}\,,$
(107) $\displaystyle K^{(1)}_{1}$ $\displaystyle=K^{(2)}_{0}\,,\hskip
14.22636ptK^{(0)}_{1}=K^{(1)}_{0}\,,\hskip
14.22636ptK^{(0)}_{2}=-K^{(2)}_{0}\,.$ (108)

Interestingly, all of the terms are fully symmetric in the exchange of
conformal dimensions and therefore pass the check (105), this is not true for
the single term

$\begin{split}K^{(0)}_{0}&=\left(\partial^{2}_{\Delta_{i}}+\partial^{2}_{\Delta_{j}}+\partial^{2}_{\Delta_{k}}\right)\frac{\Gamma(\Delta_{i}-1)\Gamma(\Delta_{j}-1)\Gamma(\Delta_{k}-1)}{\Gamma\left(\Delta_{i}+\Delta_{j}+\Delta_{k}-3\right)}\\\
&\qquad+\text{B}(\Delta_{i}{+}\Delta_{j}{-}2,\Delta_{k}{-}1)\partial_{\Delta_{i}}\partial_{\Delta_{j}}\text{B}(\Delta_{i}{-}1,\Delta_{j}{-}1)\\\
&\qquad+\text{B}(\Delta_{i}{-}1,\Delta_{k}{-}1)\partial_{\Delta_{i}}\partial_{\Delta_{k}}\text{B}(\Delta_{i}{+}\Delta_{j}{-}2,\Delta_{k}{-}1)\end{split}$
(109)

where the exchange symmetry is ruined by the latter double derivative terms of
the beta function. In fact one sees that the only two parts that lead to a
failure of associativity are the one above and the $F^{3}$ self-dual term
(already considered in Costello:2022upu ) regardless of the matter content
$\\{N_{c},n_{f},n_{s}\\}$.

###### Acknowledgements.

We are grateful to Shamik Banerjee, Shounak De, Yangrui Hu, Andrew McLeod and
Francisco Rojas for useful comments and discussions. This work was supported
in part by the US Department of Energy under contract DE-SC0010010 (Task F)
and by Simons Investigator Award #376208.

##  Appendix A Calculation of the celestial four-point Higgs-regulated
amplitude

For the Higgs-regulated version of $\mathcal{N}=4$ SYM the general MHV
(maximum helicity violating) four-point gluon amplitude is simply the tree-
level Parke-Taylor amplitude $\mathcal{A}^{(0)}_{4}$ times the exponential of
the function Henn:2010ir

$\ln
M_{4}(s,t)=-\frac{1}{8}\gamma(a)\left[\ln^{2}\left(\frac{s}{m^{2}}\right)+\ln^{2}\left(\frac{t}{m^{2}}\right)\right]-\tilde{\mathcal{G}}_{0}(a)\left[\ln\left(\frac{s}{m^{2}}\right)+\ln\left(\frac{t}{m^{2}}\right)\right]\\\
+\frac{\gamma(a)}{8}\left[\ln^{2}\left(\frac{s}{t}\right)+\pi^{2}\right]+\tilde{c}_{4}(a)+\mathcal{O}(m^{2})\,.$
(110)

From Nastase:2021izh  we have that any general color-stripped amplitude can be
brought into the form

$\mathcal{A}(1^{\ell_{1}},2^{\ell_{2}},3^{\ell_{3}},4^{\ell_{4}})=z^{\alpha_{1}}(z-1)^{\alpha_{2}}\mathcal{B}(s,t)\left(\frac{z_{12}}{\bar{z}_{12}}\right)^{-\frac{1}{2}(\ell_{1}+\ell_{2}-\ell_{3}-\ell_{4})}\left(\frac{z_{13}}{\bar{z}_{13}}\right)^{-\frac{1}{2}(\ell_{1}+\ell_{3}-\ell_{2}+\ell_{4})}\\\
\times\left(\frac{z_{23}}{\bar{z}_{23}}\right)^{-\frac{1}{2}(\ell_{2}+\ell_{3}-\ell_{1}-\ell_{4})}\left(\frac{z_{24}}{\bar{z}_{24}}\right)^{-\ell_{4}}\,.$
(111)

The Mellin transform of this formula gives the general four-point celestial
gluon amplitude

$\tilde{\mathcal{A}}_{4}(\\{\Delta_{i},\sigma_{i},z_{i},\bar{z}_{i}\\})=f(z,\bar{z})\prod_{i<j}^{4}z_{ij}^{\frac{h}{3}-h_{i}-h_{j}}\bar{z}_{ij}^{\frac{\bar{h}}{3}-\bar{h}_{i}-\bar{h}_{j}}$
(112)

where the nontrivial function of conformal cross ratios is given as

$f(z,\bar{z})=2g^{2}\delta(z-\bar{z})\Theta(z-1)z^{\frac{\beta+4}{6}+\alpha_{1}}(z-1)^{\frac{\beta+4}{6}+\alpha_{2}}\int_{0}^{\infty}dw\,w^{\frac{\beta-2}{2}}\mathcal{B}(zw,-w)\,.$
(113)

For the MHV four-point gluon amplitude we have $\alpha_{1}=0$ and
$\alpha_{2}=1$, and we have defined $\beta=\sum_{i=1}^{4}\Delta_{i}-4$. For
the amplitude (110) the kinematic factor $\mathcal{B}(s,t)$ is given as

$\mathcal{B}(s,t)=g^{2}\left(\frac{s}{m^{2}}\right)^{-\frac{\gamma(a)}{8}\ln\left(\frac{s}{m^{2}}\right)}\left(\frac{t}{m^{2}}\right)^{-\frac{\gamma(a)}{8}\ln\left(\frac{t}{m^{2}}\right)}\left(\frac{s}{m^{2}}\right)^{-\tilde{\mathcal{G}}_{0}(a)}\\\
\times\left(\frac{t}{m^{2}}\right)^{-\tilde{\mathcal{G}}_{0}(a)}\left(\frac{s}{t}\right)^{\frac{\gamma(a)}{8}\ln\left(\frac{s}{t}\right)}e^{\frac{\pi^{2}}{8}\gamma(a)+\tilde{c}_{4}(a)}\,.$
(114)

Under the change of variables $s=\Lambda zw$, $t=-\Lambda w$ and after some
straightforward algebra we get

$\mathcal{B}(zw,-w)=\exp{\left(\frac{\pi^{2}}{8}\gamma(a)+i\pi\tilde{\mathcal{G}}_{0}(a)+\tilde{c}_{4}(a)\right)}z^{-\tilde{\mathcal{G}}_{0}(a)-\frac{i\pi}{4}\gamma(a)}\\\
\times\left(w\frac{\Lambda^{2}}{m^{2}}\right)^{-\left(\frac{\gamma(a)}{4}\ln(-z)+2\tilde{\mathcal{G}}_{0}(a)+\frac{\gamma(a)}{4}\ln\left(w\frac{\Lambda^{2}}{m^{2}}\right)\right)}\,.$
(115)

After the substitution $x=w\Lambda^{2}/m^{2}$ this then implies that the
contribution to the Mellin transform is

$\displaystyle f(z,\bar{z})$
$\displaystyle\propto\int_{0}^{\infty}dw\,w^{\frac{\beta-2}{2}}\mathcal{B}(zw,-w)=\left(\frac{m}{\Lambda}\right)^{\beta}\int_{0}^{\infty}dx\,x^{\frac{\beta-2}{2}}\mathcal{B}(z,x)$
(116)

where

$\mathcal{B}(z,x)=\mathcal{C}(a)z^{-\tilde{\mathcal{G}}_{0}(a)+\frac{i\pi}{4}\gamma(a)}x^{c_{1}(a,z)+c_{2}(a)\ln
x}$ (117)

for the constants given as

$\displaystyle\mathcal{C}(a)$
$\displaystyle=\exp{\left(\frac{\pi^{2}}{8}\gamma(a)+i\pi\tilde{\mathcal{G}}_{0}(a)+\tilde{c}_{4}(a)\right)},$
(118) $\displaystyle c_{1}(a,z)$
$\displaystyle=-\left(\frac{\gamma(a)}{4}\ln(-z)+2\tilde{\mathcal{G}}_{0}(a)\right),$
(119) $\displaystyle c_{2}(a)$ $\displaystyle=-\frac{\gamma(a)}{4}\,.$ (120)

Integrating the above function for Re$(c_{2})<0$ (this is always true as the
cusp anomalous dimension satisfies $\gamma(a)>0$161616The importance of this
fact for the existence of the Mellin transform was emphasized in
Gonzalez:2021dxw .) we get a Gaussian integral which is elementary to evaluate
and gives

$f(z,\bar{z})\propto
2\left(\frac{m}{\Lambda}\right)^{\beta}\mathcal{C}(a)\sqrt{\frac{\pi}{\gamma(a)}}z^{-\tilde{\mathcal{G}}_{0}(a)-\frac{i\pi}{4}\gamma(a)}\exp{\left[-\frac{1}{4c_{2}(a)}\left(\frac{\beta}{2}+c_{1}(a,z)\right)^{2}\right]}.$
(121)

Substituting in for the constants from (119) and (120) we have our final
result

$f(z,\bar{z})=4\left(\frac{m}{\Lambda}\right)^{\beta}g^{2}\mathcal{C}(a)\delta(z-\bar{z})\Theta(z-1)z^{\frac{\beta+10}{6}-\tilde{\mathcal{G}}_{0}(a)-\frac{i\pi}{4}\gamma(a)}(z-1)^{\frac{\beta+4}{6}}\\\
\times\sqrt{\frac{\pi}{\gamma(a)}}\exp{\left[\frac{1}{\gamma(a)}\left(\frac{\beta}{2}-\frac{\gamma(a)}{4}\ln(-z)-2\tilde{\mathcal{G}}_{0}(a)\right)^{2}+\mathcal{O}(m^{2})\right]}$
(122)

with the constant $\mathcal{C}(a)$ is given by (118).

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