text:
\section{Introduction}

It is a simple kinematical fact that massless 4-momenta can be parametrized by a point $(z,\bar{z})$ on the celestial sphere, along with a frequency $\omega$. In this parametrization, collinear limits of massless particles are described by collision of the corresponding points on the celestial sphere. In gauge theory or gravity, the amplitudes factorize in terms of well-known collinear splitting functions, whose leading behaviour in the collinear limit is proportional to a simple pole in the inner product between the collinear momenta (cf., \cite{Altarelli:1977zs,Mangano:1990by,Bern:1998sv}). All subleading corrections to the splitting function are regular in the inner product of collinear momenta, and the factor multiplying the splitting function is itself a lower-point scattering amplitude.

On the celestial sphere, the structure of the collinear limit resembles, at least heuristically, an operator product expansion (OPE). It turns out that this can be made precise: replacing the frequency $\omega$ of each external particle with a conformal dimension $\Delta$ by means of a Mellin transform turns the scattering amplitude into an object which transforms like a conformal correlation function on the celestial sphere~\cite{Pasterski:2016qvg,Pasterski:2017kqt}. The resulting Mellin-transformed object is often called a celestial amplitude, and the external states are conformal primary wavefunctions. The Mellin transform precisely relates the collinear limit in momentum space with the OPE limit of colliding operator insertions on the celestial sphere~\cite{Fan:2019emx,Pate:2019lpp}. So with a slight abuse of terminology, one can refer to both the collinear limit in momentum space and the OPE limit in the conformal primary basis as \emph{celestial OPEs} defined on the sphere.

Celestial OPEs play a key role in celestial holography, which aims to describe massless S-matrix elements for `bulk' theories in flat spacetime in terms of a two-dimensional conformal field theory (CFT) (see~\cite{Pasterski:2021raf,McLoughlin:2022ljp} and references therein). Indeed, celestial OPEs encode the data (the OPE coefficients and spectrum) of such a celestial CFT (CCFT), so determining them is important for any bottom-up construction of celestial holography. The singular terms contributing to tree-level celestial OPEs have been determined by direct Mellin transform or symmetry arguments for a broad array of massless field theories~\cite{Fan:2019emx,Pate:2019lpp,Himwich:2021dau,Jiang:2021ovh}, and this in turn led to the discovery of various infinite-dimensional asymptotic symmetry algebras underlying gluon and graviton scattering~\cite{Guevara:2021abz,Strominger:2021lvk,Himwich:2021dau}. There have now been an extensive number of studies investigating the structural properties (e.g., associativity) of the singular contributions to tree- and loop-level celestial OPEs~\cite{Fan:2021isc,Guevara:2021tvr,Atanasov:2021cje,Fan:2021pbp,Costello:2022upu,Ren:2022sws,Monteiro:2022lwm,Bhardwaj:2022anh,Ball:2022bgg,Bittleston:2022jeq}.

This singular OPE data would be enough to uniquely determine a local, unitary CFT, but it is clear that any CCFT will have exotic properties (e.g., distributional terms in 2- and 3-point correlation functions~\cite{Pasterski:2017ylz} and complicated analytic dependence on conformal dimensions~\cite{Schreiber:2017jsr}), and the singular celestial OPEs are not sufficient. To constrain CCFTs, one requires not just the singular terms, but the regular, subleading terms of the celestial OPE as well. Even in momentum space (where such terms represent subleading corrections to the collinear splitting function) these terms probe the degree to which four-dimensional scattering amplitudes are governed by an associative OPE of a two-dimensional CFT (celestial or otherwise).

Starting with the work of~\cite{Banerjee:2020kaa}, there has been an effort to determine regular contributions to the celestial OPE using symmetry constraints or brute force calculations in the collinear expansion of low-multiplicity gluon or graviton amplitudes~\cite{Banerjee:2020zlg,Ebert:2020nqf,Banerjee:2020vnt,Banerjee:2021cly,Banerjee:2021dlm}. These strategies get complicated very quickly, and it is not at all clear how to proceed to arbitrary order in the OPE, even for specific scattering amplitudes at tree-level. One approach to tackling this problem is to find a \emph{dynamical} principle for computing celestial OPEs directly, rather than relying on kinematical constraints or Mellin transforming explicit momentum space calculations.

\medskip

In this paper, we compute \emph{all} of the regular terms in the tree-level celestial OPE between gluons for the maximal helicity violating (MHV) sector using \emph{twistor string theory}~\cite{Witten:2003nn,Berkovits:2004hg}. This builds on the deep and long-standing connections between twistor theory and the asymptotic geometry of spacetimes~\cite{Newman:1976gc,Penrose:1976js,Hansen:1978jz,Eastwood:1982,Adamo:2014yya,Geyer:2014lca,Adamo:2015fwa}. The twistor string provides a dynamical principle for direct calculation of the celestial OPE, and the connection between twistor theory and celestial holography has now been established in a variety of ways~\cite{Adamo:2021lrv,Costello:2021bah,Adamo:2021zpw,Bu:2021avc,Costello:2022wso,Costello:2022jpg}. In a prior paper~\cite{Adamo:2021zpw}, we showed that the worldsheet OPE of (ambi)twistor string theory localizes to the singular OPE limit on the celestial sphere and hence dynamically generates all singular terms in the celestial OPE without any truncation or approximations. However, it was not clear how to obtain regular terms in the OPE with this framework, which localized on the region of the string moduli space corresponding to the singular terms only.

By restricting our attention to the MHV sector, where the twistor string worldsheet is holomorphically identified with the celestial sphere, we obtain an effective description of tree-level MHV gluon scattering in terms of a two-dimensional CFT on the celestial sphere itself. This means that all terms in the celestial OPE (singular and regular) can be read off from the OPEs between vertex operators in this effective CFT. It is crucial that the resulting OPEs are organized in terms of soft gluon descendants to obtain expressions which close on the gluon vertex operators at each order in the OPE expansion. This provides an interesting contrast with the work of~\cite{Banerjee:2020zlg,Ebert:2020nqf,Banerjee:2020vnt,Banerjee:2021cly,Banerjee:2021dlm}, where the expansion is organized in terms of Kac-Moody and Virasoro descendants.

Our results depend only on the fact that the two gluons involved in the celestial OPE are external states in a tree-level MHV amplitude; the multiplicity of this amplitude or the specific structure of the Parke-Taylor formula~\cite{Parke:1986gb} or its Mellin transform is irrelevant. This provides both an all-multiplicity proof of the subleading celestial OPE and an explicit realization of the OPE by a two-dimensional CFT on the sphere. The latter guarantees that the resulting MHV celestial OPE is associative as well as invariant under holographic symmetry algebras.

The all-order MHV gluon celestial OPEs are surprisingly compact. For instance, the OPE for positive helicity gluons with on-shell 4-momenta $k_{i}^{\alpha\dot\alpha}=\kappa_{i}^{\alpha}\,\tilde{\kappa}_i^{\dot\alpha}$ represented on the celestial sphere by vertex operators $\cU_{+}^{\msf{a}}(\kappa_i,\tilde{\kappa}_j)$, the celestial OPE for the MHV sector is:
\be
\cU^\msf{a}_{+}(\kappa_i,\tilde{\kappa}_i)\,\cU^\msf{b}_{+}(\kappa_j,\tilde{\kappa}_j) = \sum_{p=0}^{\infty}\la i\,j\ra^{p-1}\sum_{k=0}^p\sum_{\ell=0}^{p-k}\frac{(-[i\,\p_j])^{\ell}}{\ell!}\,J^\msf{a}_{-p}[k](\tilde\kappa_i)\,\cU_+^\msf{b}(\kappa_j,\tilde\kappa_i+\tilde\kappa_j)\,.
\ee
Here, $\msf{a},\,\msf{b}$ are the colour indices of the two gluons, $\la i\,j\ra=\kappa_i^{\alpha}\,\kappa_{j\,\alpha}$, $[i\,\partial_j]=\tilde{\kappa}_i^{\dot\alpha}\frac{\p}{\p\tilde{\kappa}_j^{\dot\alpha}}$, and $J^{\msf{a}}_{-p}[k]$ are the soft gluon descendants. This expression can easily be Mellin transformed to yield the all-order, tree-level celestial OPE in CCFT between positive helicity gluons in the MHV sector. For example, the result when both gluons are outgoing is
\begin{multline}
\cU_{+,\Delta_i}^{\msf{a}}(z_i,\bar{z}_i)\,\cU_{+,\Delta_j}^{\msf{b}}(z_j,\bar{z}_j)= \sum_{p=0}^{\infty}z_{ij}^{p-1}\sum_{k=0}^p\sum_{\ell=0}^{p-k}\sum_{m=0}^{\infty}\frac{\bar z_{ij}^m}{m!}\,\frac{\bar{D}_k^{\ell}}{\ell !}\, J^{\msf{a}}_{-p}[k](\bar z_i)\\
    \times\,B(\Delta_i+k+\ell+m-1,\,\Delta_j-1)\,\bar\partial^m_j\,\cU_{+,\Delta_i+\Delta_j+k-1}^{\msf{b}}(z_j,\bar z_j)\,,
\end{multline}
where $\Delta_i,\Delta_j$ are the conformal dimensions of the conformal primary wavefunctions, $z_{ij}=z_i-z_j$, $\bar{z}_{ij}=\bar{z}_i-\bar{z}_j$ denote the holomorphic and anti-holomorphic separations on the celestial sphere, $\bar{D}_k=-\bar{z}_{ij}\dbar_j+\Delta_i+\Delta_j+k-3$ is a differential operator on the celestial sphere, and $B(x,y)$ is the Euler Beta function.

More generally, we obtain expressions for the tree-level celestial OPE between two gluons of positive or mixed positive/negative helicity in any incoming/outgoing configuration in the MHV sector. The intrinsic chirality of the MHV sector and twistor string theory means that there are no
nontrivial OPE expressions for two negative helicity gluons that can be obtained in this framework.

\medskip

The paper is structured as follows: Section~\ref{Sec:TS} reviews the salient aspects of twistor string theory and the induced description of the tree-level MHV sector in terms of a 2d CFT on the celestial sphere. Section~\ref{Sec:OPE} then sets out the details of the celestial OPE computation, defining the soft gluon descendants used to organize the expansion and performing the calculation for both momentum eigenstates and the conformal primary basis. Section~\ref{Sec:Conc} concludes with a discussion of open problems and future directions. Appendix~\ref{app:exp} proves some identities used during the OPE computation.

\section{Twistor strings and the MHV sector on the celestial sphere}\label{Sec:TS}

The maximal helicity violating (MHV) sector of tree-level gluon scattering amplitudes -- with two negative helicity and arbitrarily many positive helicity external gluons -- is remarkably simple: the amplitudes are captured at all multiplicities by the famous Parke-Taylor formula~\cite{Parke:1986gb}. Nair observed long ago that the Parke-Taylor formula can be understood as a worldsheet correlator in twistor space~\cite{Nair:1988bq}, and Witten refined and generalized this to formulate tree-level Yang-Mills theory as a twistor string~\cite{Witten:2003nn}.

In twistor string theory, the MHV sector corresponds to holomorphic maps from a Riemann sphere to twistor space which are \emph{linear}. When vertex operators for the external gluons correspond to momentum eigenstates or conformal primary wavefunctions -- parametrized by points on the celestial sphere and a frequency or scaling dimension, respectively -- this means that the worldsheet is holomorphically identified with the celestial sphere.

\medskip

Broadly speaking, twistor strings are 2d chiral worldsheet CFTs governing holomorphic maps from a closed Riemann surface to \emph{twistor space}, a complex projective space which is related non-locally to spacetime (see Section 3 of~\cite{Geyer:2022cey} for a recent review). Twistor strings have been formulated to describe several massless field theories, including supergravity~\cite{Skinner:2013xp} and ABJM theory~\cite{Engelund:2014sqa}, but we will be interested in the Berkovits-Witten twistor string describing four-dimensional Yang-Mills theory~\cite{Witten:2003nn,Berkovits:2004hg,Berkovits:2004jj,Reid-Edwards:2012vwx}.

A fact about this twistor string theory is that genus zero correlation functions of vertex operators representing positive and negative helicity gluons are equal to tree-level gluon scattering amplitudes. The worldsheet path integral imposes a relationship between the degree of the holomorphic map from the worldsheet to twistor space and the number of negative helicity gluon vertex operators: maps of degree $d$ correspond to correlators involving $d+1$ negative helicity gluons. In other words, the helicity grading of the worldsheet correlator localizes the degree of the holomorphic map. After this localization, the resulting expression, often referred to as the Roiban-Spradlin-Volovich-Witten (RSVW) formula~\cite{Witten:2003nn,Roiban:2004yf}, gives the tree-level N$^{d-1}$MHV gluon scattering amplitude in Yang-Mills theory. The formula can be proved to correctly capture these scattering amplitudes using on-shell recursion relations~\cite{Skinner:2010cz,Dolan:2011za} or worldsheet factorization arguments~\cite{Adamo:2013tca}.

For our purposes, it suffices to simply state the gluon vertex operators and the OPEs between the worldsheet fields which result from quantizing the model at genus zero. Let $Z^{A}=(\mu^{\dot\alpha},\lambda_{\alpha})$ be homogeneous coordinates on projective twistor space, $\PT$, which is the open subset of $\CP^3$ corresponding to $\lambda_{\alpha}\neq 0$. This means that the $Z^A$ are considered only up to overall projective rescalings, $Z^A\sim r\,Z^A$ for any non-vanishing complex number $r\in\C^*$. As worldsheet fields on the Riemann sphere $\CP^1$, these are bosons with zero conformal weight, taking values in degree $d$ holomorphic maps $\CP^1\to\PT$.

Positive and negative helicity gluons are represented by the (integrated) vertex operators
\be\label{VOs}
\cU_+^{\msf{a}} = \int_{\Sigma} j^{\msf{a}}\, a(Z)\,, \qquad \cU_-^{\msf{a}} = \int_{\Sigma} \,j^{\msf{a}}\, \msf{O}\, b(Z)\,.
\ee
Here, $a$ and $b$ are twistor representatives for the positive and negative helicity gluons, respectively; they are valued in
\be\label{cohreps}
a\in H^{0,1}(\PT,\cO)\,, \qquad b\in H^{0,1}(\PT,\cO(-4))\,,
\ee
where $H^{0,1}(\PT,\cO(k))$ denotes the cohomology group on $\PT$ of $(0,1)$-forms which are homogeneous of degree $k$ and $\dbar$-closed but not exact. The Penrose transform, relating cohomology on $\PT$ to massless free fields in (complexified) Minkowski spacetime, ensures that these representatives correspond to on-shell positive and negative helicity gluons~\cite{Penrose:1969ae,Ferber:1977qx,Eastwood:1981jy}. In the negative helicity vertex operator, $\msf{O}$ is constructed from fermionic fields $\chi^{a}$, $a=1,\ldots,4$:
\be\label{fermO}
\msf{O}:=\frac{\epsilon_{abcd}}{4!}\,\chi^{a}\,\chi^{b}\,\chi^{c}\,\chi^{d}=\frac{\chi^{4}}{4!}\,,
\ee
where $\chi^a$ have vanishing conformal weight, are valued in degree $d$ holomorphic polynomials on the worldsheet and have the same projective scaling as the twistor coordinates $Z^A$. This ensures that the integrand of the negative helicity vertex operator is homogeneous of weight zero, and thus well-defined.

The conformal weight $(1,0)$ currents $j^{\msf{a}}$ appearing in both vertex operators arise from a worldsheet current algebra, with the index $\msf{a}$ in the associated Lie algebra. The current algebra contributes all of the non-trivial Wick contractions in any correlation function of the vertex operators \eqref{VOs} through the OPE between the currents:
\begin{equation}
    j^{\msf{a}}(\sigma)\,j^{\msf{b}}(\sigma')\sim \frac{k\,\delta^{\msf{a}\msf{b}}}{(\sigma-\sigma')^2}+ \frac{f^{\msf{a}\msf{b}\msf{c}}\,j^{\msf{c}}(\sigma')}{\sigma-\sigma'}\,,
\end{equation}
where $\sigma,\sigma'$ are affine coordinates on the genus zero worldsheet, $k$ is the level of the current algebra and $f^{\msf{abc}}$ are the structure constants of the associated gauge group. The double pole in this OPE leads to gravitationally-mediated double-trace terms in correlation functions, so to decouple these gravitational degrees of freedom (which correspond to four-derivative conformal gravity, so are also non-unitary) we set the level of the worldsheet current algebra to zero: $k\to0$~\cite{Berkovits:2004jj,Adamo:2018hzd}\footnote{Ordinarily the level is required to be a positive integer for the worldsheet current algebra to be globally well-defined, but for calculations based only on local worldsheet OPEs, having $k=0$ is not problematic.}. Hence, the current OPE on the worldsheet will be taken to have only a simple pole
\begin{equation}\label{jj_OPE}
    j^{\msf{a}}(\sigma)\,j^{\msf{b}}(\sigma')\sim  \frac{f^{\msf{a}\msf{b}\msf{c}}\,j^{\msf{c}}(\sigma')}{\sigma-\sigma'}\,,
\end{equation}
from now on.

To obtain answers in momentum space, one chooses momentum eigenstate representatives for the twistor representatives in the vertex operators \eqref{VOs}. For an on-shell 4-momentum decomposed into spinors as $k^{\alpha\dot\alpha}=\kappa^{\alpha}\tilde{\kappa}^{\dot\alpha}$, the corresponding positive and negative helicity gluon representatives in twistor space are (cf., \cite{Adamo:2011pv}):
\be\label{momeig0}
    a(Z)=\int_{\C^*}\frac{\d s}{s}\,\bar{\delta}^{2}(\kappa-s\,\lambda)\,\e^{\im\,s[\mu\,\tilde{\kappa}]}\,, \qquad b(Z)=\int_{\C^*}\d s\,s^{3}\,\bar{\delta}^{2}(\kappa-s\,\lambda)\,\e^{\im\,s[\mu\,\tilde{\kappa}]}\,,
\ee
with the scale parameter $s\in\C^{*}$ ensuring that each representative has the appropriate homogeneity on twistor space and
\be\label{holdelt}
\bar{\delta}^{2}(\kappa-s\,\lambda):=\frac{1}{(2\pi\im)^2}\bigwedge_{\alpha=0,1}\dbar\left(\frac{1}{\kappa_{\alpha}-s\,\lambda_{\alpha}}\right)\,,
\ee
is the two-dimensional holomorphic delta function.

\medskip

Now, given the relation between the degree $d$ of the holomorphic map from the worldsheet to twistor space, and the helicity grading of the resulting scattering amplitude (namely, N$^{d-1}$MHV), the MHV sector of twistor string theory corresponds to linear, $d=1$ holomorphic maps. In this case, the zero modes of the $Z^{A}$ worldsheet fields are
\be\label{wszm1}
Z^{A}(\sigma)=U^{A}_{\alpha}\,\sigma^{\alpha}\,,
\ee
where $\sigma_{\alpha}=(\sigma_0,\sigma_1)$ are homogeneous coordinates on the $\CP^1$ worldsheet and $\{U^{A}_{\alpha}\}$ are the map moduli. The remaining GL$(2,\C)$ redundancy in this description can be fixed by picking (compatible) values for four of the moduli; a particularly convenient choice is to set $U^{\beta}_{\alpha}=\delta^{\alpha}_{\beta}$. This holomorphically identifies the homogeneous coordinates $\sigma_{\alpha}$ on the worldsheet $\Sigma\cong\CP^1$ with the $\lambda_{\alpha}$ components of the twistor field. With this identification, the remaining components of the degree one map to twistor space are simply
\begin{equation}\label{worldsheet_incidence}
\mu^{\dot\alpha}(\lambda)=x^{\alpha\dot\alpha}\,\lambda_{\alpha}\,, \qquad \chi^{a}(\lambda)=\theta^{a\alpha}\,\lambda_{\alpha}\,,
\end{equation}
where $(x^{\alpha\dot\alpha},\theta^{a\alpha})$ are the remaining four bosonic and eight fermionic moduli.

The relations \eqref{worldsheet_incidence} are easily recognizable as the standard twistor incidence relations for chiral Minkowski superspace, relating a point $(x,\theta)\in\M$ to a holomorphic, linearly embedded sphere in $\PT$; for the MHV sector, this Riemann sphere in twistor space is identified (holomorphically) with the string worldsheet itself. Furthermore, the sphere in twistor space with homogeneous coordinates $\lambda_{\alpha}$ is precisely the celestial sphere (cf., \cite{Adamo:2021lrv}).

This holomorphic identification between the twistor string worldsheet and the celestial sphere for the MHV sector can be implemented directly at the level of the individual vertex operators. This allows us to effectively describe the MHV sector in terms of a 2d CFT living on the celestial sphere itself. To do this, we parametrize the on-shell 4-momentum of the $i^{\mathrm{th}}$ external gluon as $k_i^{\alpha\dot\alpha}=\kappa_i^{\alpha}\,\tilde{\kappa}_{i}^{\dot\alpha}$ with
\be\label{momparam}
\kappa_{i\,\alpha}=(1,\,z_i)\,, \qquad \tilde{\kappa}_{i\,\dot\alpha}=\varepsilon_i\,\omega_i\,\bar{z}_{i\,\dot\alpha}=\varepsilon_i\,\omega_i\,(1,\,\bar{z}_i)\,,
\ee
where $\omega_i$ is the frequency, $(z_i,\bar{z}_i)$ is a point on the celestial sphere (in an affine coordinate patch), and $\varepsilon_i=\pm1$ denotes whether the momentum in incoming ($\varepsilon_i=-1$) or outgoing ($\varepsilon_i=1$). Note that this is a slightly non-standard parametrization, associating the frequency with the anti-holomorphic coordinates on the celestial sphere. This simplifies subsequent formulae by identifying holomorphic momentum spinor contractions with holomorphic displacements on the celestial sphere, $\la i\,j\ra=z_i-z_j$, the only price is that holomorphic coordinates on the celestial sphere now carry little group weight.

Consider a positive helicity gluon vertex operator $\cU_{+,i}^{\msf{a}}$ in the MHV sector of the twistor string. With the GL$(2,\C)$-fixing of \eqref{worldsheet_incidence}, for which $\sigma_{i\,\alpha}=\lambda_{i\,\alpha}$, we choose an affine patch of the Riemann sphere so that $\lambda_{i\,\alpha}=(1,\lambda_i)$. The holomorphic delta function in the vertex operator can be decomposed using a normalized spinor dyad $a^{\alpha}=(1,0)$, $b^{\alpha}=(0,-1)$:
\be\label{EFTVO1}
\begin{split}
\cU^{\msf{a}}_{+,i}&=\int\limits_{\CP^1\times\C^*}j^{\msf{a}}(\lambda_i)\,\frac{\d s_i}{s_i}\,\bar{\delta}(\la a\,i\ra-s_i\,\la a\,\lambda_i\ra)\,\bar{\delta}(\la b\,i\ra-s_i\,\la b\,\lambda_i\ra)\,\e^{\im\,s_i\,[\mu(\lambda_i)\,i]} \\
 &=\int\limits_{\CP^1\times\C^*}j^{\msf{a}}(\lambda_i)\,\frac{\d s_i}{s_i}\,\bar{\delta}(1-s_i)\,\bar{\delta}(s_i\,\lambda_i-\,z_i)\,\e^{\im\,s_i\,[\mu(\lambda_i)\,i]}\,,
\end{split}
\ee
with $\la a\,b\ra=1$. The scale integral in $s_i$ can now be performed against the first delta function, leaving
\be\label{EFTVO2}
\cU^{\msf{a}}_{+,i}=\int_{\CP^1}j^{\msf{a}}(\lambda_i)\,\bar{\delta}(\lambda_i-z_i)\,\e^{\im\,[\mu(\lambda_i)\,i]}\,,
\ee
with the remaining delta function explicitly implementing the holomorphic identification between the vertex operator insertion on the worldsheet ($\lambda_i$) and the point on the celestial sphere corresponding to the gluon momentum ($z_i$).

The worldsheet integral in \eqref{EFTVO2} can now be done against this remaining delta function to leave a vertex operator defined entirely on the celestial sphere:
\be\label{effective_vertex_operator_+}
\cU^{\msf{a}}_{+,i}(z_i,\tilde{\kappa}_i) = j^\msf{a}(z_i)\,\exp\left(\im\,[\mu(z_i)\,i]\right)\,.
\ee
Here, we have abused notation by denoting the current $j^{\msf{a}}$ with the same symbol that previously denoted the worldsheet current of conformal weight $(1,0)$; in particular $j^{\msf{a}}_{\text{old}}(\sigma)=j^{\msf{a}}_{\text{new}}(\sigma)\,\d\sigma$ in terms of affine coordinates. From now on, $j^{\msf{a}}$ denotes the Kac-Moody current with zero conformal weight. An identical procedure can be used to reduce the negative helicity gluon vertex operator to a vertex operator on the celestial sphere:
\be\label{effective_vertex_operator_-}
\cU^{\msf{a}}_{-,i}(z_i,\tilde{\kappa}_i) = j^\msf{a}(z_i)\,\msf{O}(z_i)\,\exp\left(\im\,[\mu(z_i)\,i]\right)\,,
\ee
with $j^{\msf{a}}$ again denoting the pure conformal weight zero Kac-Moody current.

It is straightforward to Mellin transform these expressions to obtain the corresponding celestial sphere vertex operators in the conformal primary basis~\cite{Pasterski:2016qvg,Pasterski:2017kqt}. This Mellin transform exchanges the frequency $\omega_i$ with a scaling dimension $\Delta_i$:
\be\label{cVOs}
\cU^{\msf{a},\varepsilon_i}_{\pm,\Delta_i}(z_i,\bar{z}_i)=\int_{0}^{\infty}\d\omega_{i}\,\omega_i^{\Delta_i-1\mp1}\,\cU^{\msf{a}}_{\pm,i}(z_i,\tilde{\kappa}_i)\,,
\ee
where the additional helicity dependent factors of $\omega_i$ arise due to the non-standard little group scaling associated with the parametrization \eqref{momparam}. For these conformal primary vertex operators, the incoming/outgoing parameter is explicit, as the Mellin transform strips off the prefactors $\varepsilon_i\,\omega_i$ of the momentum spinor $\tilde{\kappa}_i$.

\medskip

Regardless of whether one considers momentum eigenstates or conformal primary wavefunctions, it is clear that the positive and negative helicity gluon vertex operators can be reduced to operator insertions on the celestial sphere for the MHV sector. Interactions between these vertex operators can now be captured by an effective 2d CFT defined on the celestial sphere itself:
\be\label{2dEFT}
S^{\text{MHV}}=\frac{1}{2\pi}\int_{\CP^1}\tilde{\lambda}_{\dot\alpha}(z)\,\dbar\mu^{\dot\alpha}(z)+S_{\text{current}}\Big|_{\CP^1}\,,
\ee
where $S_{\text{current}}|_{\CP^1}$ denotes the worldsheet current algebra restricted to the celestial sphere. This CFT induces OPEs between the fields on the celestial sphere:
\be\label{CSOPE}
\mu^{\dot\alpha}(z_i)\,\tilde{\lambda}_{\dot\beta}(z_j)\sim\frac{\delta^{\dot\alpha}_{\dot\beta}}{z_{ij}}\,, \qquad j^{\msf{a}}(z_i)\,j^{\msf{b}}(z_j)\sim\frac{f^{\msf{abc}}\,j^{\msf{c}}(z_j)}{z_{ij}}\,,
\ee
where $z_{ij}:=z_i-z_j$.

This 2d CFT is `effective' in the sense that it only describes correlation functions of the vertex operators in the MHV configuration. Indeed, it is easy to see that these correlators reproduce the Parke-Taylor formula for the MHV amplitude. Taking only one colour-ordered term contributing to the correlation function of Kac-Moody currents, one finds
\begin{multline}\label{EFTcorr}
\left\la \cU_{-,i}\,\cU_{-,j}\,\prod_{l\neq i,j}\cU_{+,l}\right\ra=\frac{1}{z_{12}\,z_{23}\cdots z_{n1}}\,\int\d^{4}x\,\d^{4}\theta\,\msf{O}(z_i)\,\msf{O}(z_j)\,\exp\left(\im\sum_{l=1}^{n}k_l\cdot x\right) \\
 =(2\pi)^{4}\,\delta^{4}\!\left(\sum_{l=1}^{n}k_l\right)\,\frac{z_{ij}^4}{z_{12}\,z_{23}\cdots z_{n1}} = (2\pi)^{4}\,\delta^{4}\!\left(\sum_{l=1}^{n}k_l\right)\,\frac{\la i\,j\ra^{4}}{\la1\,2\ra\,\la2\,3\ra\cdots\la n\,1\ra}\,,
\end{multline}
as desired, having used
\be\label{Grassint}
\int \d^{8}\theta\,\msf{O}(z_i)\,\msf{O}(z_j)=z_{ij}^{4}\,,
\ee
when evaluating the fermionic moduli integral.

More generally, OPEs between fields or the vertex operators \eqref{effective_vertex_operator_+} \eqref{effective_vertex_operator_-}, or \eqref{cVOs} defined by the CFT \eqref{2dEFT} only hold in the MHV sector. Within this sector, such OPEs in the 2d CFT are, by definition, OPEs on the celestial sphere itself.

\section{OPE computation}\label{Sec:OPE}

Armed with the description of the MHV sector on the celestial sphere in terms of the vertex operators \eqref{effective_vertex_operator_+}, \eqref{effective_vertex_operator_-} and the 2d CFT \eqref{2dEFT}, we can now consider the OPE between gluon insertions. For momentum eigenstates, the resulting OPE expansion corresponds to an expansion in the holomorphic collinear limit, and by Mellin transforming the result we obtain the descendant expansion of the celestial OPE for the MHV sector of CCFT.

Since the effective theory lives on the celestial sphere itself, we are able to perform the descendant expansions in this OPE to \emph{all-orders} in both the holomorphic and anti-holomorphic degrees of freedom: in momentum space, this means that we obtain all-order collinear expansions, while in CCFT this results in all-order expressions for the celestial OPE with all regular terms included. The resulting expressions are organized in terms of soft gluon descendants.

\subsection{Soft descendants}

In this section, we identify the soft gluon descendants used to organize the terms in the exact OPE between two vertex operators. The only non-trivial Wick contractions between gluon vertex operators (positive or negative helicity) are through the Kac-Moody current. While the singular part of the OPE is given by \eqref{CSOPE}, to obtain all-order expressions we require the \emph{exact} OPE between two Kac-Moody currents. This is obtained in the usual way, by expanding the current $j^{\msf{a}}(z)$ in modes (cf., \cite{Ketov:1995yd,DiFrancesco:1997nk,Blumenhagen:2009zz}):
\begin{equation}\label{jj}
    j^{\msf{a}}(z_i)\,j^{\msf{b}}(z_j) = \frac{f^{\msf{abc}}\,j^{\msf{c}}(z_j)}{z_{ij}} + \sum_{n=0}^{\infty}\frac{z_{ij}^n}{n!}:\!\p^nj^{\msf{a}}\,j^{\msf{b}}\!:(z_j)\,,
\end{equation}
where $\partial$ denotes holomorphic differentiation and normal ordering $:\,:$ is defined by
\be\label{normalorder}
:\!\p^nj^{\msf{a}}\,j^{\msf{b}}\!:(z_j)=\frac{1}{2\pi\im}\oint_{C(z_j)}\d z_i\,\frac{\p^n j^{\msf{a}}(z_i)\, j^{\msf{b}}(z_j)}{z_{ij}}\,,
\ee
for $C(z_j)$ a small contour enclosing $z_j$. From now on, normal ordering with this prescription will be implicitly assumed.

To define soft gluon descendants of the vertex operators, we first identify the Kac-Moody descendants through the OPE between the Kac-Moody current and a gluon vertex operator (of either helicity) on the celestial sphere:
\be\label{KMexpand}
j^{\msf{a}}(z_i)\,\cU^{\msf{b}}_{\pm,j}(z_j,\tilde{\kappa}_j) = \frac{f^{\msf{a}\msf{b}\msf{c}}\,\cU^{\msf{c}}(z_j,\tilde\kappa_j)}{z_{ij}} + \sum_{n=1}^\infty z_{ij}^{n-1}\,j^{\msf{a}}_{-n}\,\cU^{\msf{b}}_{\pm,j}(z_j,\tilde\kappa_j)\,,
\ee
where $\{j^{\msf{a}}_{-n}\}$ are the modes of the Kac-Moody current. The action of these modes on vertex operators is easily identified by comparing \eqref{KMexpand} and \eqref{jj}.
Now, conformally soft gluon currents on the celestial sphere are encoded in twistor space by polynomials in the two components of $\mu^{\dal}$, along with the Kac-Moody current~\cite{Adamo:2021lrv,Adamo:2021zpw}:
\be\label{softcurrents}
J^{\msf{a}}[k,l](z) := \im^{k+l}\,j^{\msf{a}}(z)\,\big(\mu^{\dot1}\big)^k(z)\,\big(\mu^{\dot2}\big)^l(z)\,.
\ee
Note that $j^{\msf{a}} \equiv J^{\msf{a}}[0,0]$ in this decomposition. Soft gluon vertex operators are now defined by
\be\label{softgluons}
J^{\msf{a}}[p](z,\tilde\kappa):= \frac{\im^{p}}{p!}\,j^{\msf{a}}(z)\,[\mu(z)\,\tilde\kappa]^p = \sum_{k+l=p}\frac{\tilde\kappa_{\dot1}{}^k\,\tilde\kappa_{\dot2}{}^l}{k!\,l!}\,J^{\msf{a}}[k,l]\,,\qquad p\in\Z_{\geq0}\,,
\ee
with the second equality following from the expansion of $[\mu\,\tilde\kappa]^p=(\mu^{\dot1}\tilde\kappa_{\dot1}+\mu^{\dot2}\tilde\kappa_{\dot2})^p$ using the binomial theorem. Acting with these soft gluons on a momentum eigenstate gives OPEs of the form
\begin{equation}\label{soft_descendants}
    J^{\msf{a}}[p](z_i,\tilde\kappa_i)\,\cU^{\msf{b}}_{\pm,j}(z_j,\tilde\kappa_j) = \frac{[i\,\p_j]^p}{p!}\,\frac{f^{{\msf{a}}{\msf{b}}\msf{c}}\,\cU^{{\msf{c}}}_{\pm,j}(z_j,\tilde\kappa_j)}{z_{ij}} +  \sum_{n=1}^\infty z_{ij}^{n-1}\,J_{-n}^{\msf{a}}[p](\tilde\kappa_i)\,\cU^{\msf{b}}_{\pm,j}(z_j,\tilde\kappa_j)\,,
\end{equation}
where $[i\,\p_j]\equiv\tilde\kappa_i^{\dal}\,\p/\p\tilde\kappa_j^{\dal}$. The coefficients $J_{-n}^{\msf{a}}[p](\tilde\kappa_i)$ are the \emph{soft gluon descendants} in terms of which all OPEs between the gluon vertex operators can be organized.

Using \eqref{softgluons} and \eqref{soft_descendants}, along with the basic OPEs of the Kac-Moody currents, it is straightforward to identify the action of the soft gluon descendants at each order. For instance, the action of the first soft descendant $J^{\msf{a}}_{-1}[1](\tilde{\kappa}_i)$ is given by
\be\label{JV1}
J^{\msf{a}}_{-1}[1](\tilde{\kappa}_i)\,\cU^{\msf{b}}_{\pm,j}=\im\,f^{\msf{abc}}\,j^{\msf{c}}(z_j)\,[\partial\mu(z_j)\,i]\,\e^{\im\,[\mu(z_j)\,j]}+\im\,j^{\msf{a}}\,j^{\msf{b}}(z_j)\,[\mu(z_j)\,i]\,\e^{\im\,[\mu(z_j)\,j]}\,,
\ee
More generally, by writing any exact OPE as a series in $z_{ij}$ and comparing against the soft descendant expansion one can identify the action of the soft descendants at each order in the OPE expansion.

Now that we have the definition of soft descendants in hand, we are ready to write the OPEs between vertex operators in terms of them.

\subsection{The OPE in momentum space}\label{sec_momentum_space}

Armed with the soft descendant expansion, we can now consider the OPE between two positive helicity gluon vertex operators, in a momentum eigenstate basis, on the celestial sphere. The strategy is to perform the computation by writing all terms in the OPE as a series in $z_{ij}$, then recasting terms at each order in terms of combinations of soft descendants.

We start by computing the OPE of two positive helicity vertex operators \eqref{effective_vertex_operator_+}; using the Kac-Moody OPE \eqref{jj} and the fact that the $\mu^{\dot\alpha}(z)$ fields have no singular OPEs between themselves, it follows that
\begin{align}\label{VVope}
\cU_{+,i}^{\msf{a}}(z_i,\tilde\kappa_i)\,\cU_{+,j}^{\msf{b}}(z_j,\tilde\kappa_j) &= j^\msf{a}(z_i)\,j^\msf{b}(z_j)\,\e^{\im\,[\mu(z_i)\,i]}\,\e^{\im\,[\mu(z_j)\,j]}\nonumber \\
&=\left(\sum_{m=0}^{\infty} z_{ij}^{m-1}\,j^{\msf{a}}_{-m}\,j^{\msf{b}}\right)\left(\sum_{n=0}^{\infty}\frac{z_{ij}^n}{n!}\,\partial^n\!\left(\e^{\im\,[\mu\,i]}\right)\,\e^{\im\,[\mu\,j]}\right)\,,
\end{align}
where
\be\label{expmodes}
j^{\msf{a}}_{-m}\,j^{\msf{b}}\vcentcolon=
\begin{cases}
f^{\msf{abc}}\,j^{\msf{c}}(z_j) & \mbox{when } m=0 \\
\vspace{-1em}\\
\displaystyle\frac{\partial^{m-1}j^{\msf{a}}\,j^{\msf{b}}(z_j)}{(m-1)!} & \mbox{when } m\geq 1
\end{cases}
\ee
$\partial\equiv\p/\p z_j$ and all fields on the second line of \eqref{VVope} are now evaluated at $z_j$ and implicitly normal-ordered. In writing this expression, we have only assumed that the fields $\mu^{\dal}(z)$ are analytic away from $z=0$ when Taylor expanding $\e^{\im\,[\mu(z_i)\,i]}$ around $z_j$. Schematically, \eqref{VVope} takes the form of an expansion in the separation $z_{ij}$:
\be
\cU_{+,i}^{\msf{a}}(z_i,\tilde\kappa_i)\,\cU_{+,j}^{\msf{b}}(z_j,\tilde\kappa_j)  = \sum_{p=0}^\infty z_{ij}^{p-1}\,U^{\msf{a}\msf{b}}_p(z_j,\tilde\kappa_i,\tilde\kappa_j)\,,
\ee
where the coefficients $U^{\msf{a}\msf{b}}_p$ are what we want to determine. For instance, the coefficient at $O(z_{ij}^{-1})$ (i.e., the leading, singular part of the OPE) is
\be\label{U-1}
U_{0}^{\msf{a}\msf{b}} = f^{\msf{a}\msf{b}\msf{c}}\,j^\msf{c}\,\e^{\im\,[\mu\,i]+\im\,[\mu\,j]} = f^{\msf{abc}}\,\cU_+^\msf{c}(z_j,\tilde\kappa_i+\tilde\kappa_j)\,
\ee
which corresponds to the well-known collinear splitting function for positive helicity gluons~\cite{Altarelli:1977zs,Birthwright:2005ak}.

\medskip

From \eqref{VVope}, one identifies
\be
U_p^\msf{ab}(z_j,\tilde{\kappa}_i,\tilde{\kappa}_j) = \sum_{n=0}^p\frac{1}{n!}\,j^\msf{a}_{n-p}\,j^b\,\e^{\im\,[\mu\,j]}\,\p^n\!\left(\e^{\im\,[\mu\,i]}\right)\,.
\ee
In order to evaluate $\p^n(\e^{\im\,[\mu\,i]})$, one needs the following main identity: for any smooth $f(z)$, the $n^\text{th}$ derivative $\p^n\e^{f(z)}$ of its exponential is given by:
\be\label{dnef0}
\p^n\e^{f(z)} = e^{f(z)}\,\sum_{\ell=0}^n\sum_{k=0}^\ell\frac{(-1)^{\ell-k}}{k!\,(\ell-k)!}\,f^{\ell-k}(z)\,\p^nf^{k}(z)\,.
\ee
We review a brief proof of this identity in appendix~\ref{app:exp}. Applying it to $f(z)=\im\,[\mu(z)\,i]$ gives
\be\label{Upab}
U_p^\msf{ab} = \sum_{n=0}^p\frac{1}{n!}\,j^\msf{a}_{n-p}\,j^{\msf{b}}\,\e^{\im\,[\mu\,i]+\im\,[\mu\,j]}\,\sum_{\ell=0}^n\sum_{k=0}^\ell\frac{(-1)^{\ell-k}\,\im^\ell}{k!\,(\ell-k)!}\,[\mu\,i]^{\ell-k}\,\p^n\left([\mu\,i]^{k}\right)\,.
\ee
A second useful identity, valid for any smooth $f(z)$, states that:
\be
\sum_{k=0}^\ell\frac{(-1)^{\ell-k}}{k!(\ell-k)!}\,f^{\ell-k}\,\p^nf^k = 0\,,\qquad \forall\,\ell>n\,,
\ee
see equation \eqref{goodboi} of appendix \ref{app:exp}. Using this for $f(z)=\im\,[\mu(z)\,i]$, the second sum in \eqref{Upab} can be trivially extended from $0\leq\ell\leq n$ to $0\leq\ell\leq p$, since the summands for $n+1\leq\ell\leq p$ vanish identically:
\be\label{Upab1}
U_p^\msf{ab} = \sum_{n=0}^p\frac{1}{n!}\,j^\msf{a}_{n-p}\,j^{\msf{b}}\,\e^{\im\,[\mu\,i]+\im\,[\mu\,j]}\sum_{\ell=0}^p\sum_{k=0}^\ell\frac{(-1)^{\ell-k}\,\im^\ell}{k!\,(\ell-k)!}\,[\mu\,i]^{\ell-k}\,\p^n\!\left([\mu\,i]^{k}\right)\,.
\ee
As a result, the sum over $n$ can be commuted through the other sums. This expression is now ready to be recast in terms of soft descendants of the gluon operator $\cU_+^\msf{c}(z,\tilde\kappa_1+\tilde\kappa_2)$.

\medskip

The OPE of a hard gluon $\cU^\msf{b}_{+,j}$ with a soft gluon $J^\msf{a}[k](z_i,\tilde\kappa_i) = j^\msf{a}(z_i)\,[\mu(z_i)\,i]^k$ can be written out along the sames lines as \eqref{VVope}, giving:
\begin{align}
J^\msf{a}[k](z_i,\tilde\kappa_i)\,\cU_{+,j}^\msf{b}(z_j,\tilde\kappa_j) &= \frac{\im^k}{k!}\,j^\msf{a}(z_i)\, j^\msf{b}(z_j)\,[\mu(z_i)\,i]^k\,\e^{\im\,[\mu(z_j)\,j]}\nonumber\\
&= \frac{\im^k}{k!}\left(\sum_{m=0}^{\infty}z_{ij}^{m-1}\,j^\msf{a}_{-m}\,j^\msf{b}\right)\left(\sum_{n=0}^{\infty}\frac{z_{ij}^n}{n!}\,\e^{\im\,[\mu\,j]}\,\p^n\!\left([\mu\,i]^k\right)\right)\,.
\end{align}
Comparing with \eqref{soft_descendants}, we see that a typical soft descendant has the explicit expression
\be
J^\msf{a}_{-p}[k](\tilde\kappa_i)\,\cU_{+,j}^\msf{b}(z_j,\tilde\kappa_j) = \frac{\im^k}{k!}\,\sum_{n=0}^p\frac{1}{n!}\,j^\msf{a}_{n-p}\,j^\msf{b}\,\p^n\!\left([\mu\,i]^k\right)\,\e^{\im\,[\mu\,j]}\,.
\ee
By shifting $\tilde\kappa_j\mapsto\tilde\kappa_j+\tilde\kappa_i$ in this expression, one also obtains
\be\label{softdesc}
J^\msf{a}_{-p}[k](\tilde\kappa_i)\,\cU_+^\msf{b}(z_j,\tilde\kappa_i+\tilde\kappa_j) = \frac{\im^k}{k!}\,\sum_{n=0}^p\frac{1}{n!}\,j^\msf{a}_{n-p}\,j^\msf{b}\,\p^n\!\left([\mu\,i]^k\right)\,\e^{\im\,[\mu\,i]+\im\,[\mu\,j]}\,,
\ee
which contains precisely the kind of terms entering \eqref{Upab1}.

Performing the sum over $n$ in \eqref{Upab1} using \eqref{softdesc} gives all of the expansion coefficients in the OPE between two positive helicity gluons in a basis of soft gluon descendants:
\be\label{Upab2}
U_p^\msf{ab} = \sum_{\ell=0}^p\sum_{k=0}^\ell\frac{(-\im\,[\mu\,i])^{\ell-k}}{(\ell-k)!}\,J^\msf{a}_{-p}[k](\tilde\kappa_i)\,\cU_+^\msf{b}(z_j,\tilde\kappa_i+\tilde\kappa_j)\,.
\ee
Written in terms of these coefficients, we now obtain an all-order expression for the celestial OPE
\be
\cU^\msf{a}_{+,i}\,\cU^\msf{b}_{+,j} = \sum_{p=0}^{\infty}z_{ij}^{p-1}\sum_{\ell=0}^p\sum_{k=0}^\ell\frac{(-[i\,\p_j])^{\ell-k}}{(\ell-k)!}\,J^\msf{a}_{-p}[k](\tilde\kappa_i)\,\cU_+^\msf{b}(z_j,\tilde\kappa_i+\tilde\kappa_j)\,,
\ee
where factors of $\im\,[\mu\,i]$ have been replaced with the differential operators
\be\label{opreplace}
\im\,[\mu\,i]\longleftrightarrow[i\,\p_j]\equiv\tilde\kappa_i^{\dal}\frac{\p}{\p\tilde\kappa_j^{\dal}}\,,
\ee
when acting on $\e^{\im\,[\mu\,i]+\im\,[\mu\,j]}$.

Exchanging the sums over $\ell$ and $k$, then shifting $\ell\mapsto\ell+k$ simplifies the result further to
\be\label{++mom}
\boxed{\cU^\msf{a}_{+,i}\,\cU^\msf{b}_{+,j} = \sum_{p=0}^{\infty}z_{ij}^{p-1}\sum_{k=0}^p\sum_{\ell=0}^{p-k}\frac{(-[i\,\p_j])^{\ell}}{\ell!}\,J^\msf{a}_{-p}[k](\tilde\kappa_i)\,\cU_+^\msf{b}(z_j,\tilde\kappa_i+\tilde\kappa_j)\,.}
\ee
This is the exact tree-level celestial OPE, to all orders in $z_{ij}$, for two positive helicity gluons in the MHV sector in a momentum eigenstate basis. Note that this expression also encodes all orders in the anti-holomorphic collinear limit as well: for each $p\geq0$, the soft descendant $J^\msf{a}_{-p}[k](\tilde\kappa_i)\,\cU_+^\msf{b}(z_j,\tilde\kappa_i+\tilde\kappa_j)$ can be Taylor expanded around $\tilde{\kappa}_i=\tilde{\kappa}_j$. Since this anti-holomorphic dependence is exponential, this expansion generates all-order terms in $[i\,j]$ as well.

\medskip

The calculation of the celestial OPE proceeds along identical lines for the mixed-helicity configuration, involving one positive and one negative helicity gluon. In this case, the only difference is the insertion of $\msf{O}(z)$ -- built from the twistor fermions -- which accompanies the negative helicity vertex operator \eqref{effective_vertex_operator_-}. The end result is a celestial OPE
\be\label{+-mom}
\boxed{\cU^\msf{a}_{+,i}\,\cU^\msf{b}_{-,j} = \sum_{p=0}^{\infty}z_{ij}^{p-1}\sum_{k=0}^p\sum_{\ell=0}^{p-k}\frac{(-[i\,\p_j])^{\ell}}{\ell!}\,J^\msf{a}_{-p}[k](\tilde\kappa_i)\,\cU_-^\msf{b}(z_j,\tilde\kappa_i+\tilde\kappa_j)\,,}
\ee
with all the resulting terms proportional to soft gluon descendants of a \emph{negative} helicity gluon.

At first, it may seem that the result is incomplete: the leading collinear splitting function for mixed-helicity gluons includes two terms, one for each helicity, whereas the $p=0$ term in \eqref{+-mom} contains only the negative helicity contribution to this splitting function. The key is to remember that \eqref{+-mom} holds only for the MHV sector, where positive helicity terms in the mixed-helicity OPE are identically zero. Phrased differently, the MHV amplitude is homogeneous with respect to collinear limits (i.e., all collinear limits of a MHV amplitude give lower-point MHV amplitudes), and positive helicity terms on the right-hand-side of \eqref{+-mom} would violate this homogeneity. Indeed, such terms would correspond to generating a tree-level gluon scattering amplitude with only one negative helicity particle, which are identically zero.

\medskip

Finally, one can consider the OPE between two negative helicity vertex operators. This OPE is on a different footing to the others due to the inherent chirality of the MHV sector; this is captured in the fact that the OPE between $\cU^{\msf{a}}_{-,i}$ and $\cU^{\msf{b}}_{-,j}$ is proportional to
\be\label{--ope1}
\msf{O}(z_i)\,\msf{O}(z_j)=\msf{O}(z_j)\,\sum_{m=0}^{\infty}\frac{z_{ij}^m}{m!}\,\partial^{m}\msf{O}(z_j)\,,
\ee
where $\msf{O}(z)\equiv \chi^{4}(z)$. Since
\be
\chi^{a}(z)=\theta^{a0}+z\,\theta^{a1}\,,
\ee
and there are only 8 degrees of freedom in the fermionic $\theta^{a\alpha}$, only the $m=4$ term from the expansion \eqref{--ope1} survives, consistent with \eqref{Grassint}. The rest of the OPE calculation proceeds as before, leaving:
\be\label{--ope2}
\cU^{\msf{a}}_{-,i}\,\cU^{\msf{b}}_{-,j}=\frac{\theta^{4}}{4!}\,\sum_{p=0}^{\infty}z_{ij}^{p+3}\sum_{k=0}^p\sum_{\ell=0}^{p-k}\frac{(-[i\,\p_j])^{\ell}}{\ell!}\,J^\msf{a}_{-p}[k](\tilde\kappa_i)\,\cU_-^\msf{b}(z_j,\tilde\kappa_i+\tilde\kappa_j)\,,
\ee
where
\be\label{thetaab}
\theta^{4}:=\frac{\epsilon_{abcd}}{4!}\theta^{a1}\,\theta^{b1}\,\theta^{c1}\,\theta^{d1}\,,
\ee
denotes the combination of fermionic moduli produced by the expansion.

Note that this OPE has no singular term, as expected for the MHV sector. If this OPE were singular, it would correspond to generating a collinear limit of the tree-level MHV amplitude which had one negative helicity gluon, and such tree amplitudes are identically zero. This is tied to the presence of $\theta^4$ in \eqref{--ope2}, which indicates that the OPE only makes sense in the context of the complete correlator \eqref{EFTcorr}, where the $\d^8\theta$ integral must be saturated. Regularity of this expression is also consistent with generating non-trivial form factors for two anti-self-dual fields in self-dual Yang-Mills theory~\cite{Costello:2022wso}.

Hence, the expression \eqref{--ope2} is not a self-contained celestial OPE, since it cannot be written purely in terms of locations on the celestial sphere and descendants of the vertex operators. This is due to the underlying chirality of the twistor string, for which the OPE between negative helicity vertex operators is not really well defined. As such, \eqref{--ope2} will not provide good OPE data in the context of CCFT, and we will not consider this helicity configuration in the MHV sector any further.

\subsection{The OPE in Celestial CFT}

By Mellin transforming the results \eqref{++mom} and \eqref{+-mom} to the conformal primary basis, we can now obtain all-order celestial OPEs in the context of CCFT, which include all regular contributions in the MHV sector. To do this, we follow the rule \eqref{cVOs} to implement the Mellin transform while making all dependence on the frequencies explicit using \eqref{momparam} and \eqref{softdesc}. In doing this, it is easier to replace $[i\,\partial_j]$ with $\im\,[\mu\,i]$ according to \eqref{opreplace} at the beginning of the calculation.

Starting with the OPE between positive helicity gluons, the Mellin transformed OPE is
\begin{multline}\label{MT1}
\cU_{+,\Delta_i}^{\msf{a},\varepsilon_i}(z_i,\bar z_i)\,\cU_{+,\Delta_j}^{\msf{b},\varepsilon_j}(z_j,\bar z_j)=\int_{\R_+^2}\d\omega_i\,\omega_i^{\Delta_i-2}\,\d\omega_j\,\omega_j^{\Delta_j-2}\sum_{p=0}^{\infty}z_{ij}^{p-1} \\
\times\sum_{k=0}^{p}\sum_{\ell=0}^{p-k}(\varepsilon_i\,\omega_i)^{\ell}\,\frac{(-\im\,[\mu\,\bar{z}_i])^\ell}{\ell!}\,(\varepsilon_i\,\omega_i)^{k}
\, J^{\msf{a}}_{-p}[k](\bar{z}_i)\,j^{\msf{b}}\,\e^{\im\varepsilon_i\omega_i[\mu\bar z_i]+\im\varepsilon_j\omega_j[\mu\bar z_j]}\,,
\end{multline}
where all fields are evaluated at $z_j$ with implicit normal-ordering on the right-hand-side of the OPE relation, and we have abbreviated $\bar z_{ij\,\dal}\equiv\bar z_{i\,\dal}-\bar z_{j\,\dal} = (0,\bar z_{ij})$ (with $\bar z_{ij}=\bar z_i-\bar z_j$ as usual). Now, the argument of the exponential in this expression can be conveniently rewritten as
\begin{equation}
    \im\,\varepsilon_i\,\omega_i\,[\mu\,\bar z_i]+\im\,\varepsilon_j\,\omega_j\,[\mu\,\bar z_j] = \im\,(\varepsilon_i\omega_i+\varepsilon_j\omega_j)\left([\mu\,\bar z_j] + \frac{[\mu\,\bar z_{ij}]}{1+\frac{\varepsilon_j\omega_j}{\varepsilon_i\omega_i}}\right)\,,
\end{equation}
and combined with rescaling the first Mellin integral by $\omega_i\rightarrow\omega_i\omega_j$ this leaves
\begin{multline}
\cU_{+,\Delta_i}^{\msf{a},\varepsilon_i}\,\cU_{+,\Delta_j}^{\msf{b},\varepsilon_j} = \sum_{p=0}^{\infty}z_{ij}^{p-1}\sum_{k=0}^p\sum_{\ell=0}^{p-k} \int_{\mathbb{R}_+^2} \d\omega_i\,\omega_i^{\ell+k+\Delta_i-2}\, \d\omega_j\,\omega_j^{k+\ell+\Delta_i+\Delta_j-3}\\
   \times\, \varepsilon_i^{\ell+k}\,\frac{(-\im\,[\mu\,\bar{z}_i])^\ell}{\ell!}\;J_{-p}^\msf{a}[k](\bar z_i)\, j^{\msf{b}}\,\text{exp}\bigg[{\im\,(\varepsilon_i\omega_i+\varepsilon_j)\,\omega_j\left([\mu\,\bar z_j] + \frac{[\mu\,\bar z_{ij}]}{1+\frac{\varepsilon_j}{\varepsilon_i\omega_i}}\right)}\bigg]\,.
\end{multline}
After rescaling the second Mellin integral by $\omega_j \rightarrow \frac{\omega_j}{|\varepsilon_j+\varepsilon_i\omega_i|}$, it is possible to Taylor expand the exponential in $\bar{z}_{ij}$:
\begin{multline}\label{inter1}
\cU_{+,\Delta_i}^{\msf{a},\varepsilon_i}\,\cU_{+,\Delta_j}^{\msf{b},\varepsilon_j}= \sum_{p=0}^{\infty}z_{ij}^{p-1}\sum_{k=0}^p\sum_{\ell=0}^{p-k}\sum_{m=0}^{\infty} \int_{\mathbb{R}_+^2} \frac{\d\omega_i\,\omega_i^{\ell+k+m+\Delta_i-2}}{|\varepsilon_j+\varepsilon_i\omega_i|^{\Delta_i+\Delta_j+k+\ell+m-2}}\,\d\omega_j\,\omega_j^{\Delta_i+\Delta_j+k+\ell-3} \\
    \times\,\varepsilon_i^{\ell+k+m}\,[\text{sgn}(\varepsilon_i\omega_i+\varepsilon_j)]^{m}\,J_{-p}^\msf{a}[k](\bar z_i)\,j^{\msf{b}}\,\frac{\bar z_{ij}^{m}}{m!}\,\frac{(-\im\,[\mu\,\bar{z}_i])^{\ell}}{\ell!} \\
    \times\,\bar\partial^m_j \,\text{exp}\bigg[{\im\,\text{sgn}(\varepsilon_i\omega_i+\varepsilon_j)\,\omega_j\,[\mu\,\bar{z}_j] }\bigg]\,,
\end{multline}
where $\dbar_j\equiv\frac{\p}{\p\bar{z}_j}$. Already, we see that the expression explicitly contains all orders in both $z_{ij}$ \emph{and} $\bar{z}_{ij}$.

Using the fact that $[\mu\,\bar{z}_i]=\mu^{\dot{0}}+\bar{z}_i\,\mu^{\dot{1}}$, a short calculation demonstrates that
\be\label{opreplace2}
-\im\,[\mu\,\bar{z}_i]\longleftrightarrow \frac{1}{\omega_j\,\text{sgn}(\varepsilon_i\,\omega_i+\varepsilon_j)}\left(-\bar{z}_{ij}\,\dbar_j-\omega_j\,\frac{\partial}{\partial\omega_j}\right)\,,
\ee
when acting on the exponential in \eqref{inter1}. The Euler vector $\omega_j\,\frac{\partial}{\partial\omega_j}$ can then be integrated-by-parts to extract the homogeneity in $\omega_j$ of all non-exponential factors in each term of the expression. This results in:
\begin{multline}
\cU_{+,\Delta_i}^{\msf{a},\varepsilon_i}\,\cU_{+,\Delta_j}^{\msf{b},\varepsilon_j}= \sum_{p=0}^{\infty}z_{ij}^{p-1}\sum_{k=0}^{p}\sum_{\ell=0}^{p-k}\sum_{m=0}^{\infty}\frac{\bar{z}^m_{ij}}{m!}\,\varepsilon_i^{m+\ell+k}\,\frac{\bar{D}_k^{\ell}}{\ell!}\,J^{\msf{a}}_{-p}[k](\bar{z}_i)\,j^{\msf{b}}\,\bar\partial^m_j \\
\times \int_{\mathbb{R}_+^2}\frac{\d\omega_i\,\omega_i^{\ell+k+m+\Delta_i-2}\,\d\omega_j\,\omega_j^{\Delta_i+\Delta_j+k-3}}{\left|1+\frac{\varepsilon_i\omega_i}{\varepsilon_j}\right|^{\Delta_i+\Delta_j+k+\ell+m-2}}\,[\text{sgn}(\varepsilon_i\,\omega_i+\varepsilon_j)]^{m+\ell}\,\e^{\im\,\text{sgn}(\varepsilon_i\omega_i+\varepsilon_j)\,\omega_j\,[\mu\,\bar{z}_j] }\,,
\end{multline}
where
\be
\bar{D}_{k}:=-\bar{z}_{ij}\,\dbar_j+\Delta_i+\Delta_j+k-3\,,
\ee
is a differential operator on the celestial sphere, dependent on the conformal dimensions as well as the summation index $k$.

At this stage, we observe that
\begin{equation}
\int_{0}^{\infty}\!\d\omega_j\,\omega_j^{\Delta_i+\Delta_j+k-3}\,j^{b}(z_j)\,\text{exp}\bigg[{\im\,\text{sgn}(\varepsilon_i\omega_i+\varepsilon_j)\,\omega_j\,[\mu\,\bar{z}_j] }\bigg]=\cU^{\msf{b},\mathrm{sgn}(\varepsilon_i\omega_i+\varepsilon_j)}_{+,\Delta_i+\Delta_j+k-1}(z_j,\bar{z}_j)\,,
\end{equation}
by virtue of \eqref{cVOs}. Combined with some trivial algebraic rearrangements, we are left with the all-orders celestial OPE in the conformal primary basis
\begin{equation}\label{++_final}
\boxed{
\begin{aligned}
 \cU&_{+,\Delta_i}^{\msf{a},\varepsilon_i}(z_i,\bar z_i)\,\cU_{+,\Delta_j}^{\msf{b},\varepsilon_j}(z_j,\bar z_j)= \sum_{p=0}^{\infty}z_{ij}^{p-1}\sum_{k=0}^p\sum_{\ell=0}^{p-k}\sum_{m=0}^{\infty}\frac{\bar z_{ij}^m}{m!} \,\varepsilon_i^{m+\ell+k}\,\frac{\bar{D}_k^{\ell}}{\ell !}\, J^{\msf{a}}_{-p}[k](\bar z_i)\\
    &\times\,\int\limits_{0}^{\infty} \frac{\d\omega_i\,\omega_i^{\ell+k+m+\Delta_i-2}}{\left|\frac{\varepsilon_i\omega_i}{\varepsilon_j}+1\right|^{\Delta_i+\Delta_j+k+\ell+m-2}}\,[\text{sgn}(\varepsilon_i\,\omega_i+\varepsilon_j)]^{m+\ell}
    \,\bar\partial^m_j\,\cU_{+,\Delta_i+\Delta_j+k-1}^{\msf{b}, \text{sgn}(\varepsilon_i\omega+\varepsilon_j)}(z_j,\bar z_j)\,.
\end{aligned}}
\end{equation}
This provides a `master' formula for the celestial OPE, valid for any configuration of incoming/outgoing positive helicity gluons in the MHV sector.

It is easy to see that the $p=0$ terms in \eqref{++_final} correctly reproduce the master formula for the singular contribution to the OPE~\cite{Adamo:2021zpw}. It is illustrative to evaluate the OPE for explicit incoming/outgoing configurations, where the remaining Mellin frequency integral can be performed explicitly. For instance, when both gluons are in the same configuration ($\varepsilon_i=\varepsilon_j=\varepsilon$), one obtains
\begin{multline}
\cU_{+,\Delta_i}^{\msf{a},\varepsilon}\,\cU_{+,\Delta_j}^{\msf{b},\varepsilon}= \sum_{p=0}^{\infty}z_{ij}^{p-1}\sum_{k=0}^p\sum_{\ell=0}^{p-k}\sum_{m=0}^{\infty}\frac{\bar z_{ij}^m}{m!}\,\frac{\bar{D}_k^{\ell}}{\ell !}\, J^{\msf{a}}_{-p}[k](\bar z_i)\\
    \times\,B(\Delta_i+k+\ell+m-1,\,\Delta_j-1)\,\bar\partial^m_j\,\cU_{+,\Delta_i+\Delta_j+k-1}^{\msf{b},\varepsilon}(z_j,\bar z_j)\,,
\end{multline}
where $B(x,y)$ denotes the Euler Beta function. The $p=0$ terms in this expression contain the single tower of SL$(2,\R)$ descendants that come with the pole in $z_{ij}$ which were previously determined in the literature~\cite{Fan:2019emx,Pate:2019lpp,Adamo:2021zpw}; while these singular terms are valid in general, all terms for $p>0$ hold only for the celestial OPE within the MHV sector of gluon scattering.

\medskip

The Mellin transform of the celestial OPE between a positive and a negative helicity gluon \eqref{+-mom} follows similar lines; the only difference is in the initial definition of the transform with respect to the frequency of the negative helicity particle via \eqref{cVOs}. The resulting master formula is:
\begin{equation}\label{+-_final}
\boxed{
\begin{aligned}
\cU&_{+,\Delta_i}^{\msf{a},\varepsilon_i}(z_i,\bar z_i)\,\cU_{-,\Delta_j}^{\msf{b},\varepsilon_j}(z_j,\bar z_j)= \sum_{p=0}^{\infty}z_{ij}^{p-1}\sum_{k=0}^p\sum_{\ell=0}^{p-k}\sum_{m=0}^{\infty}\frac{\bar z_{ij}^m}{m!} \,\varepsilon_i^{m+\ell+k}\,\frac{\bar{D}_k^{\prime\,\ell}}{\ell !}\, J^{\msf{a}}_{-p}[k](\bar z_i)\\
    &\times\,\int\limits_{0}^{\infty} \frac{\d\omega_i\,\omega_i^{\ell+k+m+\Delta_i-2}}{\left|\frac{\varepsilon_i\omega_i}{\varepsilon_j}+1\right|^{\Delta_i+\Delta_j+k+\ell+m}}\,[\text{sgn}(\varepsilon_i\,\omega_i+\varepsilon_j)]^{m+\ell}
    \,\bar\partial^m_j\,\cU_{-,\Delta_i+\Delta_j+k-1}^{\msf{b}, \text{sgn}(\varepsilon_i\omega+\varepsilon_j)}(z_j,\bar z_j)\,,
\end{aligned}}
\end{equation}
where
\be\label{primeop}
\bar{D}_k^{\prime}:=-\bar{z}_{ij}\,\dbar_j+\Delta_i+\Delta_j+k-1\,,.
\ee
As in momentum space, this OPE features only contributions from soft gluon descendants of the negative helicity gluon on the celestial sphere. Indeed, the $p=0$ terms in \eqref{+-_final} only contain one (chiral) half of those appearing in the master formula for the singular, mixed helicity celestial OPE~\cite{Adamo:2021zpw}. This is, once again, an intrinsic feature of the mixed helicity OPE in the MHV sector (as discussed above). Whereas the singular terms in the same helicity OPE \eqref{++_final} are valid beyond the MHV sector, even the singular terms in the mixed helicity OPE are sensitive to the helicity configuration on the celestial sphere.

For the sake of concreteness, one can consider the mixed helicity OPE for gluons in the same incoming/outgoing configuration ($\varepsilon_i=\varepsilon_j=\varepsilon$). This leads to an explicit all-orders tower of OPE coefficients
\begin{multline}
\cU_{+,\Delta_i}^{\msf{a},\varepsilon}\,\cU_{-,\Delta_j}^{\msf{b},\varepsilon}= \sum_{p=0}^{\infty}z_{ij}^{p-1}\sum_{k=0}^p\sum_{\ell=0}^{p-k}\sum_{m=0}^{\infty}\frac{\bar z_{ij}^m}{m!}\,\frac{\bar{D}_k^{\prime\,\ell}}{\ell !}\, J^{\msf{a}}_{-p}[k](\bar z_i)\\
    \times\,B(\Delta_i+k+\ell+m-1,\,\Delta_j+1)\,\bar\partial^m_j\,\cU_{-,\Delta_i+\Delta_j+k-1}^{\msf{b},\varepsilon}(z_j,\bar z_j)\,,
\end{multline}
defined in terms of the Euler Beta function. It is easy to see that the $p=0$ terms in this expression coincide with one chiral half of the singular terms for one SL$(2,\R)$ descendant tower in the OPE of mixed helicity celestial gluons which have been appeared previously~\cite{Pate:2019lpp,Adamo:2021zpw}.

\subsection{Null states}

Comparing our results with previous expressions at subleading level in the literature~\cite{Ebert:2020nqf} is an involved process. This is due to the fact that we use soft-descendants to organize the OPE expansion, while the expression in~\cite{Ebert:2020nqf} is written in terms of Kac-Moody and Virasoro descendants. The way to bridge this gap is to consider the existence of null states in CCFT. On these states, we expect that the action of the soft currents can be recast in terms of the Kac-Moody and Virasoro currents. That is, soft descendants of these states can be rewritten in a basis of Kac-Moody and Virasoro descendants.

While these null relations are not \emph{a priori} known, we can leverage our free field realization of the target space algebra at MHV level to calculate them explicitly. This is rather trivial at first order. In this case the leading soft current is given by the residue around $\Delta=1$ of the hard particle $\mathcal{U}_{\Delta}$ (cf., \cite{Donnay:2018neh,Adamo:2019ipt,Adamo:2021zpw}). In our model, this is the residue of the vertex operator \eqref{cVOs} which is the Kac-Moody current $j^\msf{a}$ itself; this is just a restatement of \eqref{softcurrents}, where $J^\msf{a}[0,0]=j^\msf{a}$. The next soft current is more interesting. From \eqref{softcurrents}, it is a combination of the Kac-Moody current and the $\mu$ operators. In a conformal basis it can be written as
\begin{equation}
J^\msf{a}[1]=J^\msf{a}[1,0]+\bar{z}\,J^\msf{a}[0,1]=j^\msf{a}\,[\mu(z)\,\bar{z}]\,,
\end{equation}
and its descendant as
\begin{equation}\label{nulldesc}
    J^\msf{a}_{-1}[1]=j^\msf{a}_0\,[\mu_{-1}\,\bar z]+j^\msf{a}_{-1}\,[\mu_{0}\,\bar z].
\end{equation}
The descendants of $\mu^{\dot{\alpha}}$ acting on a gluon operator are given by a Taylor expansion, since there is no short distance singularity:
\begin{equation}
    \mu^{\dot{\alpha}}(z_i)\,\cU^\msf{a}_\Delta(z_j)=\sum_{n=0}^{\infty}\frac{z^n_{ij}}{n!}\partial^n\mu^{\dot{\alpha}}\,\cU^\msf{a}_\Delta(z_j)\equiv\sum_{p=1}^{\infty}z^{p-1}_{ij}\,\mu_{-p}^{\dot{\alpha}}\,\cU^\msf{a}_\Delta(z_j)\,,
\end{equation}
suppressing irrelevant helicity and incoming/outgoing labels. This means that
\begin{equation}
[\mu_0\,\bar z]\,\cU^\msf{a}_\Delta(z)=[\mu\,\bar z]\,\cU^\msf{a}_\Delta(z)=(\Delta-2)\,\cU^\msf{a}_{\Delta-1}(z)\,,
\end{equation}
and
\begin{equation}
    [\mu_{-1}\,\bar z]\,\cU^\msf{a}_\Delta(z)=[\partial\mu\,\bar z]\,\cU^\msf{a}_\Delta(z)\,,
\end{equation}
in terms of the modes appearing in the soft gluon descendant \eqref{nulldesc}.

Using these relations, the action of $J^\msf{a}_{-1}[1]$ on a hard gluon in the conformal primary basis can be written as
\begin{equation}\label{null_state}
    J^\msf{a}_{-1}[1]\,\cU^{\msf{b}}_{\Delta}=(\Delta-1)\,j^\msf{a}_{-1}\,\cU^b_{\Delta-1}-j_{-1}^\msf{b}\,\cU^\msf{a}_{\Delta-1}-f^{\msf{a}\msf{b}\msf{c}}\,L_{-1}\,\cU^\msf{c}_{\Delta-1}\,,
\end{equation}
to lowest order in $\bar{z}$, where $L_{-1}=\partial$ is the SL$(2,\C)$/Virasoro generator. Equation \eqref{null_state} is the same as the null state condition of~\cite{Banerjee:2020zlg}, and can be used to rewrite the OPE at subleading order in a basis of Kac-Moody and Virasoro descendants matching the subleading terms found in~\cite{Ebert:2020nqf}. We expect that similar null relations hold for other soft current descendants. These can, in principle, be derived from the representation \eqref{softcurrents} for the soft currents and their action on hard particles using the effective OPEs, though the process quickly becomes rather cumbersome.

\section{Discussion}\label{Sec:Conc}

In this paper, we used twistor string theory to provide all-order expressions for tree-level celestial OPEs of gluons in the MHV sector. The results \eqref{++mom}, \eqref{+-mom}, \eqref{++_final} and \eqref{+-_final} are organized in terms of soft gluon descendants, when enables the OPE to close on the gluon vertex operators. These provide the first instances where all regular contributions to the celestial OPE, needed to constrain any putative CCFT, are captured in their entirety.

There are many interesting directions which can be explored following on from this work. While the twistor string provides a dynamical principle for generating the celestial OPEs, it is natural to ask if there are alternative ways to obtain our results using more standard amplitudes technology; we will return to this in future work \cite{renta}. It should also be possible to obtain all-order celestial OPEs for the MHV sector of \emph{graviton} scattering; we attempted to do this using the twistor string of~\cite{Skinner:2013xp} for $\cN=8$ supergravity, but ran into technical obstructions related to the worldsheet supersymmetry of that model. A more straightforward route is probably to use chiral twistor sigma models~\cite{Adamo:2021bej,Adamo:2021lrv,Sharma:2022arl}, which describe the MHV sector in terms of a classical 2d CFT on the celestial sphere directly.

In addition, one can ask to what extent it is possible to push these all-order results beyond the MHV sector. From the perspective of twistor string theory, it is unclear how to achieve this: N$^{k}$MHV scattering corresponds to degree $k+1$ holomorphic maps from the Riemann sphere to twistor space, so the worldsheet is no longer identified with the celestial sphere at generic points in the moduli space. While this identification does emerge in the strict OPE limit on the worldsheet~\cite{Adamo:2021zpw}, it only captures the singular contributions to the celestial OPE.

\acknowledgments

We thank Lecheng Ren, Anders Schreiber and Diandian Wang for useful discussions. TA is supported by a Royal Society University Research Fellowship and by the Leverhulme Trust (RPG-2020-386). WB is supported by a Royal Society PhD studentship. EC thanks the support of the Frankel-Goldfield-Valani Research Fund. AS is supported by a Black Hole Initiative fellowship, which is funded by the Gordon and Betty Moore Foundation and the John Templeton Foundation.

redpajama_set_name: RedPajamaArXiv


text:
CNBC Business News and Finance
MarketsBusinessInvestingTechPoliticsMake It
Tillerson reportedly described Trump as 'a moron' and was set to resign in July
Published Wed, Oct 4 2017 6:33 AM EDT
Updated Wed, Oct 4 2017 12:34 PM EDT
David Reid
@davyreid73
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Secretary of State Rex Tillerson openly referred to President Donald Trump as "a moron" after a July 20 meeting at the Pentagon, according to NBC News.
Tillerson made the comment in front of members of Trump's national security team and Cabinet members, three senior officials told NBC.
NBC also said Tillerson was on the verge of resigning this summer and had to be talked out of it by Vice President Mike Pence.
Secretary of State Rex Tillerson openly referred to President Donald Trump as a "moron" after a July 20 meeting at the Pentagon, according to NBC News, citing multiple senior administration officials.
According to three officials, Tillerson made the comment in front of members of Trump's national security team and Cabinet members. The report also said Tillerson was on the verge of resigning this summer and had to be talked out of it by Vice President Mike Pence.
Tillerson later denied that he ever considered leaving the Trump administration.
Officials said the tensions reached a high point around the time Trump gave a politicized speech to the Boy Scouts of America in late July. Tillerson was once leader of the organization.
The White House was not immediately available for comment when contacted by CNBC and declined to comment on the record to NBC News for the story.
President Donald Trump and Secretary of State Rex Tillerson confer during a working lunch with African leaders during the U.N. General Assembly in New York, Sept. 20, 2017.
Kevin Lamarque | Reuters
Relations between Trump and the former Exxon CEO appear to still be strained. Just last weekend, Trump publicly denied Tillerson's announcement that the U.S. was attempting diplomatic talks with North Korea.
"I told Rex Tillerson, our wonderful Secretary of State, that he is wasting his time trying to negotiate with Little Rocket Man...," Trump said in a tweet.
In its report, NBC said Tillerson's top State Department spokesman, R.C. Hammond, denied that the secretary had considered quitting this summer and denied that Tillerson called Trump a "moron."
Click here to read more on the NBC News website.
WATCH: Tillerson lacks support on the left and right: Axios' Jonathan Swan
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Priced at Rs. 175, this is quite an economical offer considering the other ON Net Free offers by other operators in the state. Earlier it used to come with a talk time of Rs. 50 as well for other networks but that has been done away with.
The validity of the STV has been reduced to 28 days would remain the same as 30 days. This new offer has been extended till 04th October, 2017 till further notice.
Effective 08th July 2017 – BSNL HIM Power 175 is now offering BSNL to BSNL Unlimited Local & STD Voice Calls, 50 Local SMS to any Local network, 90 mins for BSNL to other networks for Local & STD calls & 2GB data for 28 days.
Also BSNL has launched a new Pan India Unlimited BSNL to BSNL Local & STD Calls Special Tariff Voucher. Priced at Rs. 148 in Himachal Pradesh, BSNL offers BSNL to BSNL Unlimited Local & STD calls with 1 GB Data for 26 days. You can read more about it at this page.
You can also check other Unlimited Calls STV in Himachal Pradesh at this page.


Manufacturers operate using a network of systems including equipment, machines, and processes that are run in a centralized manner and are often backed up on a company owned server. The increased level of cyber attacks occurring within the industry are forcing manufacturers to strengthen security for company data and systems through security software, firewalls, and other security devices.
Manufacturers are well aware of the cyber risk of data loss and systems control. The threat of these cyber attacks comes from individual hackers or groups, organized crime rings, or under the auspices of foreign governments, with the purpose to steal trade secrets and intellectual property.
Despite all the security efforts to protect your company network and its proprietary information, manufacturers often overlook a critical security loophole — your company website.
Having a company server doesn't automatically qualify you to host your own website. Whereas the confidential company data on your server is password protected and encryption-enabled, and you continually update internal software vulnerabilities, the challenge is maintaining the same level of security with your website.
Hosting your own website leaves you open to hackers who are looking to get into your server for blueprints, your banking system, and your customer's bank accounts. Manufacturers make attractive victims for cyber criminals due to your unique assets and the fact that you are often easy targets. In fact, believing you aren't a target is one of the most common contributing factors to a manufacturer's vulnerability.
A client-server model of networks has its own set of computer security vulnerabilities. Remember, your network is only as secure as your administrators and designers make it. There are many ways your server is vulnerable to cyber attacks, including the choice of server and how it's configured and encrypted.
Data theft due to improper configuration of servers.
Denial of Service (DoS) attacks on a server causing it to crash and lose data.
There is no way to prevent cyber attacks, just as there's no way you can prevent someone from attempting to rob you. However, although no one is immune, you can reduce the risks by choosing strong cyber security programs and a website hosting service who's dedicated to maintaining a secure server and continually managing it for optimum performance.
Your web hosting company will help you keep your site up and running, giving you reliable and advanced security, uptime, and disaster recovery. The greater the cyber defense of your website, the less likely you are to attract the interest of cybercriminals, and the safer your business will be.
Keep an eye on your website analytics. If there's something amiss, keen attention to your site's analytics can reveal unusual activity. For example, if you see a sharp increase in unexplained visits from a foreign country, it could be a sign that you're being targeted.
Update everything. This means your antivirus software, operating systems, and all your website software, including WordPress software, its plugins and themes. The majority of these updates are security based. Hackers will take advantage of any security hole you've left wide open by not updating.
Manufacturers should always plan for the worst outcome. By assuming cyber criminals will target you, you will be prepared by giving yourself the best protection to your data and systems. The best safeguard for your manufacturing company isn't a one-time effort or expense. It's an ongoing process that involves utilizing security at all ports of entry, including your company website.


text:
class QuestionsController < ApplicationController

  def index
    @questions = Question.all
    render json: @questions
  end

  def show
    @question = Question.find(params[:id])
    render json: @question
  end

end

redpajama_set_name: RedPajamaGithub


Is the EPA actually working to protect the public from the potential dangers of Roundup®?
Whitewash is a detailed examination of the tactics Monsanto uses to keep control of the glyphosate narrative despite growing data and activism against the weed killer. Gillam contends, and backs up with ample, detailed evidence obtained through the Freedom of Information Act and the discovery process from active lawsuits, that Monsanto has undue influence among regulatory agencies, primarily the Environmental Protection Agency (EPA), and is using all its substantial resources to suppress any question of glyphosate's safety for humans or the environment. This book is very timely and thorough, but it is not for the beginner as it reads more like a research paper than a narrative and could have benefited from some graphics to bring some clarity to the dense information.
Although the book does delve into some (but not all) of the potential and documented side effects of glyphosate use (cancer, super weeds, monarch decline, soil microbe disruption, etc.), there is no smoking gun pointing to any definite conclusions on the side effects. What it does point out is that there are many serious and valid questions about the use of glyphosate and its formulations which are not being addressed.
At the same time, private testing is showing that glyphosate is being found in unexpected areas such as in honey and in human urine and at much higher levels than expected. Unfortunately, no government agency is tracking this information, so how much actual exposure the public is subjected to is unknown.
The questions this book raises could not be timelier as there is currently an effort by some U.S. Congressmen to pressure the World Health Organization cancer research program into recanting the designation of glyphosate as a "potential human carcinogen" by pulling their funding.
Carey Gillam has provided a detailed assessment of how our government is failing to protect the public from a multifaceted threat. Her book may be too detailed for a general audience, but her message warrants attention and action.
Kerry O'Kelly is a Certified Landscape Designer, an Accredited Organic Land Care Professional and Owner of Garden Dance Landscape Design in Andover, MA. She has been involved in the ELA community for many years and is currently serving as an ELA Board Member and participates in several committees. She can be reached by email at Kerry@GardenDanceLandscape.com.


Bath time can be a pretty crazy time in our home with all three girls in the tub together. So much for a nice relaxing soak before bed!
Mathilde is the trouble maker of the trio. She splashes, climbs over her sisters, walks around the tub, pulls hair... She just doesn't stay still. I think the photo below says it all! I always have to take Mathilde out early so Ava and Lola can bathe without being terrorized by their little sister.
Out of the tub the girls get dry and snuggly warm in their Puj Hug and Puj Big Hug towels.
Once the craziness of bath time is over the chaos of bedtime begins. But that's a whole other story for another day!
Love the photos. Anton is the same will not sit down! P.s. I am waiting for you to announce first steps.
Mathilde drives me crazy in the bath, especially when I'm showering as she keeps turning the water off!
I'm sure we'll both have littler walkers in not too long. Now that's going to be scary!


.Water gilding, also called gilding on bole, consists of applying loose sheets of gold on a layer of red clay (bole). The bole supplies a flexible surface that allows for polishing the gold with an agate burnisher thereby obtaining a brilliant and very smooth effect. The bole gives the gold red reflections due to transparency. This technique of gilding requires a significant investment in materials and long practice to obtain good results.
Very close attention must be paid to the quality of the levkas that will receive water gilding. The surface must be perfectly flat, smooth and carefully sanded as the slightest imperfection risks causing scratches in the gold during burnishing.
The Armenian clay is a mixture of very fine ground and iron oxide of yellow, red or brown color. It can be found in specialized art supply stores. The clay used in iconography is always of red color.
- Soak 10 grams of rabbit skin glue in 250 ml of water. It will turn into a gelatinous substance overnight.
- Crush the Armenian clay until it becomes a very fine dust.
- The following day, warm the mixture of water and glue in a double boiler. Add a teaspoon of clay, and stir until completely dissolved.
- Filter the bole through an old sheer stocking.
- Allow the mixture to settle overnight. The bole is ready to use.
- Soak 10 grams of rabbit skin glue in 200 ml of water overnight.
- Heat the mixture in a double boiler.
- Gradually add the bole into the glue mixture in a ratio of 1 volume of paste to 1 volume of glue mixture.
- Stir it well and let settle overnight.
- Moisten the entire surface to be gilded with clear water to soften the levkas. This will allow the bole to penetrate the levkas and guarantee better adhesion.
- Heat the bole in a double boiler and apply a thin and regular layer to the surface. Let it dry at least six hours.
- If the layer cracks or peels off, the glue is too strong: carefully remove the bole with extra-fine sandpaper, add more water to the mixture and apply again.
- Apply four layers of bole in the same manner. Allow six hours of drying time between each layer.
- Scrape very delicately the surface of the bole with a razor blade.
- Then rub the surface using a rough rag, the rough side of a sheet of Kraft paper or horsehair cloth.
- The smoothing of the bole must be perfect as the slightest grain will hit the agate and cause a stripe or hole in the gilding.
Fill a large glass with 200 ml. of water. Add 50 ml. of 90 proof alcohol (or Vodka!) and some grains of rabbit skin glue. Heat slightly in a double boiler and let it cool. Gilding water size can be used cold because of its very low glue content.
Other precautions should be taken: nothing that comes into contact with the gold leaves should be touched with the fingers. The gilder's knife must be degreased with alcohol before use. Don't touch the cushion directly with the hand as it is covered with a degreased calfskin on which gold does not stick. Finally, do not try to hold or to pick up the gold sheet with the fingers as it would be destroyed at once. Only pick it up with the gilder's tip.
- Hold the booklet of gold leaves above the cushion and slip onto the cushion a gold leaf using the gilder's knife.
- Lay out the leaf on the cushion using the knife while blowing on the gold softly from above in the direction you want the gold to lay.
- Découper la feuille d'or en petit carrés proportionnés à la grandeur de la dorure à effectuer.
Cut the gold leaf into small squares according to the size of the gilding area with a gilding knife, a very sharp stainless knife. Place the blade on the gold leaf and, pressing slightly, make a small back and forth movement. Then raise the knife carefully. Be careful not to incise the leather of the cushion!
- Moisten de surface to be gilded abundantly with the water size, using a squirrel brush.
- Carry a piece of gold using the gilder's tip and lay it out on the water size.
To make the piece of gold stick to the tip, slightly rub it on your cheek or on the back of your hand to create a small charge of static electricity which will attract the gold.
Once the gold is laid on the puddle, it will extend itself out onto the bole. If air bubbles appear under the gold, drive them out by tapping gently with a soft brush.
- Repeat these steps until the entire surface to be gilded is covered. Overlap the pieces of gold to avoid holes in the area being gilded.
- If a hole does appear, spread a little of the water size on the bare place and apply another piece of gold.
- After two hours of drying time, polish the gilded area with an agate burnisher tool.
- Press the agate slightly on to the gold and make circular movements.
The area that is being polished will become mirror-brilliant in contrast to the rest of the gold, which will retain a matte finish.
- Finish burnishing, clean the surface by delicately removing excess gold with a gilding mop.
- If holes appear in the gilding after the cleaning process, they may be repaired by spreading a little of the water size over the bare places, applying pieces of gold that are larger than the holes, waiting an hour and burnishing once again.
- Protect the gilding with a coat of bleached shellac diluted in 50% alcohol.
Last update : Tue, Dec 9, 2008, P. Grall © ASA 2000 - 2009 All rights reserved.


Today went a lot swifter than yesterday and we reached our target of Bywith Lock service block. We have heard a lot of stuff about Stalybridge with suggestions that there is a penchant for the local youth to stone passing boats but our passage through was without incident and, if the reports aren't correct it seems a shame to avoid the place because there is a profusion of moorings that look pretty good. Rather than take the risk and later regret it however we have pressed on to the service block which is over a mile out of Stalybridge.
This puts us comfortable on our target to get to Standedge Tunnel for the passage through on Friday, there is even the suggestion that tomorrow we go on to Upper Mill and stay there for a couple of days, travelling the last few miles up to the tunnel entrance on Thursday. We'll see what we feel like doing.


text:
India Raises Curtain on Historic World Environment Day
World Environment Day will be celebrated on June 5th, with events organized in the lead-up to the day.
India is the host of this year's World Environment Day
Events are being held across the country, with additional celebrations around the world.
New Delhi, 25 May 2018 – As global hosts to World Environment Day 2018, India today launched its official countdown to June 5 with a historic slate of activities ranging from nationwide clean-ups, to single-use plastic bans across States, Universities and National Parks.
At a press conference here, government officials outlined plans from communities large and small aimed at beating plastic pollution through civic engagement and celebration. With support from an inspiring cross section of Indian society, ranging from cricket pitches to board rooms, the scope of efforts in India represent an unprecedented national commitment to this global cause, with the promise to make this the largest and most substantial World Environment Day ever.
"India has demonstrated the magnitude of what is possible when leaders, individuals and businesses come together to tackle a challenge – even one as great as plastic pollution," said Erik Solheim, Head of UN Environment. "The momentum for World Environment Day on June 5 is picking up, and all across India we are witnessing exactly the kind of global leadership we need to save our planet from the rising tide of plastic pollution."
The press conference marked the launch of a 10-day countdown to World Environment Day on June 5.
The countdown culminates with the opening of the World Environment Day Exhibition in New Delhi on June 2,showcasing green innovations and programs from every corner of the country. Plenaries on plastic pollution, marine litter and, new plastic economy, extended producer responsibility and many others are also being organized. Environment experts, thinkers, corporates, industries and multinational companies are participating for a truly engaging global dialogue on environmental challenges. State environment ministers will also meet to discuss the way forward on India's green future and the global push to beat plastic pollution.
And finally, on June 5, Prime Minister Modi will visit the exhibition and make a global call to action, outlining India's commitments in the battle to beat plastic pollution.
The events highlight India's leadership of a movement for a simple yet ambitious goal: Inspire the world to beat plastic pollution. Complimenting this nationwide effort, are global initiatives taking place in in hundreds of countries, mobilizing dozens of partners and connecting millions of supporters worldwide.
It's all part of an unprecedented effort that invites us to consider how we can make changes in our everyday lives to reduce the heavy burden of plastic pollution on our natural places, our wildlife and our own health.
Ranging from record-breaking beach cleanups to inspiring art installations the sprint to World Environment Day, launched here today will bring together interests from all parts of the globe and all sectors of society for this common cause.


Guests are welcome to take a bottle of wine and a picnic lunch and relax under the shade trees by the lake or to take in the view on the upper terrace overlooking the lake.
Bridlewood Estate Winery features an eclectic selection of Central Coast AVA blends, Estate Syrah and Zinfandel and Monterey County Pinot Noir and Chardonnay. Of particular note is the 2012 Reserve Viognier from Santa Barbara AVA. Wafts of honeysuckle, orange blossoms and vanilla meet the nose in this Viognier from Volgelzang Ranch in nearby Happy Canyon AVA.
The 2012 Santa Lucia Highlands Reserve Chardonnay from Monterey County Chardonnay is fruit forward with flavors of pear, honey and lemony citrus tones with a creamy, mouth-filling texture ~ an excellent pairing for shellfish, pate or fresh fruits.
On the darker side, the 2012 Santa Rita Hills Reserve Pinot Noir is spicy, yet floral with notes of cranberry, mushroom and dark cola flavors. And for the Syrah lover, the 2009 Estate Syrah has layers and layers of flavor. Aromas of licorice and spice are followed by flavors of melted chocolate and dried cherry, forming a nice, round palate.


This guide is not personal advice but will explain how pension tax relief currently works and how some high earners could receive £58,500 in tax relief. Download the guide below.
This simple, no-nonsense guide will explain how pension tax relief currently works and how some high earners could receive £58,500 in tax relief. Download the guide below.
Every year the government gives billions back to investors (£41 billion at the last count). Every UK resident under 75 can claim a share. Find out how with our free guide to pension tax relief.
This guide is not personal advice, consider taking advice if you're not sure. The value of investments can go down as well as up and you could get back less than you invested. Tax reliefs will depend on your personal circumstances and the pension and tax rules are subject to change by the government. Money in a pension is not usually accessible until age 55 (57 from 2028).
This guide is not personal advice. The value of investments can go down as well as up and you could get back less than you invested. Tax reliefs will depend on your personal circumstances and the pension and tax rules are subject to change by the government. Money in a pension is not usually accessible until age 55 (57 from 2028).


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These are the smaller communities Canadians are moving to
New population data from Statistics Canada shows that Canadians are continuing to move away from large, urban centers amid the COVID-19 pandemic, rather than living in smaller communities located near larger cities.
New reports released Thursday found that Carleton Place, Ont., which is located about 46 kilometers outside Ottawa, saw its population increase by about four percent last year, making it the fastest-growing community in Canada. Has been made.
according to statcanCowansville, Quebec, came in second with a growth rate of 3.7 percent, followed by Wasaga Beach in Ontario at 3.7 percent, and then Woodstock, Ont., which saw a population increase of 3.1 percent.
The report considers the combined areas to be a Census Group (CAs) – those with a core population of at least 10,000 – and Census Metropolitan Areas (CMAs), which have a total population of at least 100,000, of whom 50,000 or more reside. Corner.
Overall, the number of Canadians living in rural areas increased by 0.7 percent between July 2020 and July 2021, with Ontario and Quebec having moved to rural areas within their respective provinces since at least 2001. Has reported the highest number of residents.
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The report found that except for the Ottawa-Gatineau region and the city of Lethbridge, Ontario and Alberta recorded the greatest number of residents moving to other provinces.
"These projections, to first give us a look at a sub-provincial level for a full year of the COVID-19 pandemic, tell us about a trend towards accelerating migration from large urban centres, implying their continued growth. is largely supported by international migration," the StatCan report said.
StatCan says population growth in urban areas – both large and small – has slowed for the second year in a row.
According to the report, a record number of people have opted to relocate outside some of Canada's largest CMAs, with Toronto and Montreal being the largest net to expatriate exchanges along with other regions in their respective provinces since at least 2001. damage has been done.
StatCan says personal health, the ability to work remotely, and high housing costs are among the "most important factors" contributing to many Canadians' decision not to live in large urban centers.
The report said permanent immigrants in Canada accounted for most of the population growth seen between July 2020 and July 2021, while the loss of non-permanent residents due to the COVID-19 pandemic has "significantly reduced growth". has contributed.
According to Statistics CanadaThe communities with the highest population growth rate in the last year are:
Carlton Place, Ont. (3.8 percent)
Cowansville, Q. (3.7 percent)
Wasaga Beach, Ont. (3.5 percent)
Woodstock, Ont. (3.1 percent)
Lachut, K. (2.9 percent)
Collingwood, Ont. (2.9 percent)
Kelowna, BC (2.6 percent)
Squamish, BC (2.5 percent)
Whitehorse, Yukon (2.4 percent)
Arnprior, Ont. (2.3 percent)
Oshawa, Ontario (2.3 percent)
Wellington Center, Ont. (2.2 percent)
Salaberry-de-Valleyfield, Ky. (2.1 percent)
Halifax, NS (2.1 percent)
Summerside, PEI (2 percent)
Moncton, NB (2 per… ,
Airline executives implore feds to keep AT&T, Verizon's 5G away from airports
Airlines and Pearson ask government to drop mandatory arrival testing requirement
Canada's biggest airlines and the nation's...


The WELLNET team as part of the Royal Free Hospital and University College London Cancer Institute have a large programme researching the cause of NETs, the genomic changes, as well as studies aimed at earlier diagnosis (e.g. circulating tumour cells and DNA) and development of new imaging modalities and therapies.
The team are involved in many clinical trials into NETs. If we thought it was appropriate to involve you in research studies we would happily discuss this with you. Research is core to our philosophy as this is how we can better personalise treatment as well as improve the longer term future of our patients.


See other types of restaurants & cuisines in Belmont Cragin, including area Thai and Mexican.
In nearby Chicago, take a look at a number of other restaurants, including avec and Trivoli.
Don't forget to check out the many restaurants located in and around North Center, Horner Park & Roscoe Village, IL and Museum campus & Printer's Row, IL.


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Q: routerLinkActive is not working when queryParams is binded to a property defined as getter In my component controller I have a property -
 public get navigationParams() {
    return {
      fromDate: sessionStorage.getItem('fromDate'),
      toDate: sessionStorage.getItem('toDate')
    };
  }

Note: I need it in session storage so that I can redirect from other pages where it is not part url state.

 Revenue

It should apply active class which it is applying for the first time but as date changes through datepicker(ng2-bootstrap) it updates fromDate and toDate in sessionStorage reloads nested state-
this.router.navigate([], {relativeTo: this.activatedRoute, queryParams: this.navigationParams) and the now the style is gone.
However it works if I use a property like
public queryParams: any; instead of navigationParams

 Revenue

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Q: Convergent algorithm for minimizing nonconvex smooth function Let $\Phi$ be the Gaussian CDF and for $\gamma\ge 0$ and $h>0$, define a loss function $\ell_h:\{\pm 1\} \times \mathbb R$ by
$$
\ell_{\gamma,h}(y,y') := \phi_{\gamma,h}(yy') := \Phi((yy'-\gamma)/h).
$$
Note that

*

*$\phi_{\gamma,h}$ is nonconvex.

*$\phi_{\gamma,h}$ is nondecreasing

*$\phi_{\gamma,h}$ is $O(1/h)$-Lipschitz continuous and its derivative is $O(1/h^2)$-Lipschitz.

Let $(x_1,y_1),\ldots,(x_n,y_n)$ be $n$ points in $\mathbb R^{d} \times \{\pm 1\}$, and for any $(w,b) \in \mathbb R^{d+1}$, define
$$
F(w,b) := \frac{1}{n}\sum_{i=1}^n\phi_{\gamma,h}(y_i(x_i^\top w - b)).
$$
Fix $p \in [1,\infty]$ and let $B_p$ be the unit-ball in $\mathbb R^d$ w.r.t the $\ell_p$-norm. I'm particularly interested in the case $p \in \{1,2\}$.

Question. Is there a convergent gradient-based algorithm which minimizes $F$ over $B_p \times \mathbb R$, or even $B_p \times \{0\}$ ?

Ideally, an explicit rate of convergence would be provided too.


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FIGO President discusses how to make labour safer for women worldwide
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Labour, birth and the immediate postnatal period is the most dangerous time for mothers and their newborn babies, and there is more that medical professionals need to do to help reduce mortality rates around the world, says FIGO President Professor Sir Sabaratnam Arulkumaran in his plenary lecture today at the 2014 RCOG World Congress in Hyderabad, India.
Professor Sir Arul's presentation, 'Making labour safe: the achievable dream', focuses on maternal and neonatal mortality during childbirth. The deaths of mothers and babies during labour and childbirth are an unnecessary and often avoidable tragedy, and with the 2015 deadline for the UN's millennium development goals fast-approaching, Professor Sir Arul says that while significant progress has been made we are still falling short of the mark.
According to UN statistics there has been a decline in maternal mortality rates of 47 per cent since 1990 with eastern and southern Asia recording a decline of almost two-thirds. Professor Sir Arul says that while maternal mortality ratios vary greatly between the developed and developing world, the main causes of maternal mortality remain the same, just on different scales.
While cardiac complications and sepsis are the biggest causes for maternal mortality in the UK, in India haemorrhage and hypertensive diseases are the biggest contributors. Prof Sir Arul acknowledges that there are major problems that require urgent redress for safer maternity services globally, particularly in under-resourced settings where the lack of skilled birth attendants during labour and standardised care throughout pregnancy and birth cause major health disparity among women.
However, in addition to this Professor Sir Arul stresses the importance of data recording. The accurate recording of data, particularly serious incidents and adverse health outcomes, is a simple task that can be overlooked in busy birth settings but is hugely important to prevent maternal mortality in all countries. For example, the establishment of confidential enquiries into maternal death in the UK have been very successful in improving outcomes and reducing deaths by highlighting deficiencies in care.
The same applies to neonatal mortality, which has also decreased over the past two decades and which Professor Sir Arul defines as not only the survival rate of newborns during labour but the longevity of their health as children.
Professor Sir Arul says that in countries like India, there is an increased need for better monitoring and management of labour to prevent birth problems such as shoulder dystocia among newborns. Whereas in regions like the UK, which is generally a safe place to give birth, he says that stillbirth is still a big problem.
These problems arise due to the growing birth rate along with the high number of complex pregnancies, making labour more hazardous in developed countries, says Professor Sir Arul. He emphasizes that this only puts further importance on the fact that these health problems need to be addressed not only from a clinical but an organisational standpoint.
Professor Sir Arul said:
"For better health outcomes for mothers and their babies, there needs to be a two-pronged approach to making labour safer. As medical professionals we need to address the deficiencies within our organisational structures as well as improving our clinical standards through research and training. One will not work without the other.
"The availability of statistics and maternity dashboards, particularly in regard to serious incident reporting, is vital in all birth settings. This provides us with valuable information on where we need to improve as units and where our time and money can be spent more effectively.
"There are many promising trials for new devices and techniques that show great promise for the future of safer childbirth, but research outcomes can only be assessed properly if we have fundamental baseline data to compare it with.
"Everyone who is attending this Congress is doing so because they are devoted to improving women's health worldwide. And while we may not be able to do much in terms of providing the hundreds of skilled health professionals that are needed in maternity services across the world, we can play a part in ensuring the best standards and clinical practice are kept up-to-date. Through training and auditing our work as medical professionals, we will ensure that our profession keeps progressing to reduce mortality among women and provide newborns with the best possible start to life."
For further information, please contact the RCOG Media and PR team on +44 20 7772 6300 or email pressoffice@rcog.org.uk.
Professor Sir Sabaratnam Arulkumaran is a Professor Emeritus of Obstetrics and Gynaecology at St George's University, London. He is the current President of the International Federation of Obstetrics and Gynaecology (FIGO) and the President of the British Medical Association. He also served as President of the Royal College of Obstetricians and Gynaecologists (RCOG) from 2007–2010.
Read the RCOG report Safer Childbirth: Minimum standards for the organisation and delivery of care in labour.
Royal College of Obstetricians and Gynaecologists
10 –18 Union Street
SE1 1SZ


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Hans Viking Axelson, född 8 juni 1934 i Kungsbacka, död 26 maj 2020 i Sollentuna distrikt, var en svensk entreprenör inom friskvård, grundare av utbildningsföretaget Axelsons Gymnastiska Institut. Axelson introducerade flera olika i dag välkända metoder och tekniker inom alternativ behandling i Sverige. 2006 tog han emot The Ashley Montague Award och 2007 utsågs han till Årets Företagare av organisationen Företagarna.

Yrkesliv
Hans Axelson växte upp i Kungsbacka och var 13 år gammal när bestämde sig för att leva sunt med vegetarisk kost och yoga, utan sprit och tobak. Inspirationskällor var Johan Lindström Saxon och Are Waerland. 1957–58 utbildade han sig till sjukgymnast vid Göteborgs Gymnastiska Institut.

1961 grundade Hans Axelson Yogaskolan i Malmö, en av Sveriges första yogaskolor, som snart fick filialer i Halmstad och Göteborg. 1962 slog han upp portarna till Axelsons Gymnastiska Institut – först i Malmö – och började med massageutbildning. Efter att ha utbildat sig till fotvårdsspecialist i Tyskland utökade han verksamheten med fotvårdsutbildning 1963. 1964 öppnade han Wiking Träningsstudio i Malmö, ett av Sveriges allra första gym. 1965 fick Axelsons Gymnastiska Institut egna lokaler i Stockholm, med massageutbildning och gymträning. De alternativa terapier som existerade vid denna tid var enbart klassisk massage, kiropraktik, homeopati och örtbehandling. 1970 startade Hans Axelson tillsammans med Naprapatic School en yrkesutbildning i naprapati. Följande år höll Axelsons även Sveriges första kurser i zonterapi och shiatsu, och till landets första kurs i bindvävsmassage anlitades enbart läkare som lärare. I slutet av 70-talet började Hans Axelson vidareutveckla teknikerna inom massageutbildningen med muskulering och longering, med syftet att genom töjning och förlängning av muskler förbättra möjligheterna att arbeta bort muskelspänningar. 1982 lanserade Hans Axelson Rosenmetoden i Sverige och inledde ett långvarigt samarbete med grundaren, Marion Rosen, som blev hans mentor under flera decennier. Tillsammans reste de ut i Europa för att introducera metoden även i andra länder. 1983 startades en utbildning i kiropraktik, som avknoppades och senare blev till Skandinaviska Kiropraktorhögskolan. I mitten av 80-talet etablerades Axelsons även i Köpenhamn, Helsingfors, Oslo och Göteborg.

Andra projekt
1996 arrangerade Axelsons den första beröringskonferensen i Sverige med forskare och föreläsare från hela världen. Samma år inledde Hans Axelson sitt stora ideella projekt "Fredlig beröring" med massage som verktyg för att minska våldet i samhället. En av vägarna var att etablera massage inom skola och förskola, med flera syften: trygga och empatiska barn, minskad mobbning och ökad samhörighet. Hittills har Axelsons utbildat mer än 15 000 anställda inom skola och förskola. Konceptet har introducerats även internationellt.

1998 grundades Branschrådet Svensk Massage med Hans Axelson som en av initiativtagarna. Syftet var kvalitetssäkring och certifiering av massageutbildningar.

Under 2000-talet grundades Insamlingsstiftelsen för forskning om manuella terapier, med Axelson som en av initiativtagarna och sponsorerna.

Referenser
 Eva Sanner: En man som berör (2002, 1:a upplagan). Stockholm: HÄLSAböcker.
 Friskvårdsmagasinet september 2012, sid 7-12

Noter

Friskvård
Svenska entreprenörer
Födda 1934
Avlidna 2020
Män
Personer inom alternativmedicin
Personer från Kungsbacka
Personer inom Sveriges näringsliv under 1900-talet

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