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Now consider g(p1) + g(p2) → H(−p3) + g(−p4) + g(−p5) +g(-p6) where p6 gluon is soft istead of gluon with p5....and do the whole calculation as done BELOW

\section{Soft and next-to-soft corrections}
\label{sec:shifts}
In this section, we briefly review the soft and next-to-soft theorems in terms of colour ordered scattering amplitudes. Any colour ordered scattering amplitude involving $n$ particles (quarks and gluons) with specific helicities can be represented as,
\begin{align}
\mathcal{A},=, \mathcal{A}{n}\left(\left{ |1\rangle,|1]\right} ,\ldots,\left{ |n\rangle,|n]\right} \right),,
\end{align}
where $ \angspinor{i} $ and $ \sqspinor{i} $ denote the holomorphic and anti-holomorphic spinors associated with the particle $i$ carrying momentum $p_i$. Let us now consider that a gluon $s$ with momenta $p_s$ and helicity \lq $+$\rq, is being emitted from this scattering process. Scaling the momentum of the radiated gluon $p_s\to\lambda,p_s$, the scattering amplitude for $ n+1 $ particle can be expressed in powers of $ \lambda$ as written here under~\cite{Luo:2014wea,Casali:2014xpa},
\begin{equation}
\mathcal{A}{n+1}\bigg(\left{ \lambda\angspinor{s},\sqspinor{s}\right} ,\left{ \angspinor{1},\sqspinor{1}\right} ,\ldots,\left{ \angspinor{n},\sqspinor{n}\right} \bigg)=\left(S^{(0)}+S^{(1)}\right)\mathcal{A}{n}\bigg(\left{ \angspinor{1},\sqspinor{1}\right} ,\ldots,\left{ \angspinor{n},\sqspinor{n}\right} \bigg),.
\label{eq:factorization}
\end{equation}
Here $ S^{(0)} $ and $ S^{(1)} $ denote LP and NLP terms that are of $\mathcal{O}(1/\lambda^2)$ and $\mathcal{O}(1/\lambda)$ respectively, and are given by,
\begin{align}
S^{(0)} &, =,\frac{\braket{n1}}{\braket{s1}\braket{ns}} ,, \nn
S^{(1)}&, =,\frac{1}{\braket{s1}}\sqspinor{s}\frac{\partial}{\partial\sqspinor{1}}-\frac{1}{\braket{sn}}\sqspinor{s}\frac{\partial}{\partial\sqspinor{n}},.
\label{eq:leading}
\end{align}
In order to obtain the above formulae, holomorphic soft limit\cite{Casali:2014xpa,Luo:2014wea} is being used {\em i.e.,}
\begin{align}
\angspinor{s}\rightarrow \lambda, \angspinor{s},,\quad \sqspinor{s}\rightarrow \sqspinor{s} ,,
\end{align}
under the BCFW\cite{Britto:2004ap,Britto:2005fq} deformation of $s$ and $n$ pair, while particles $1$ and $s$ always form a three particle amplitude involving the on-shell cut propagator that carries complex momentum. With the help of eq.\eqref{eq:leading}, the colour ordered amplitude of eq.\eqref{eq:factorization} can be rewritten as,
\begin{align}
\mathcal{A}{n+1}^{\text{LP+NLP}}\bigg(\left{ \lambda\angspinor{s},\sqspinor{s}\right} ,\left{ \angspinor{1},\sqspinor{1}\right} ,\ldots,\left{ \angspinor{n},\sqspinor{n}\right} \bigg)& , = , \frac{1}{\lambda^2} \frac{\braket{1n}}{\braket{1s}\braket{ns}} \times \nn
& \quad \mathcal{A}n\bigg(\left {\angspinor{1}, \sqspinor{1^\prime} \right }, \ldots,\left { \angspinor{n}, \sqspinor{n^\prime} \right }\bigg) ,,
\label{eq:LPplusNLP}
\end{align}
where
\begin{align}
\sqspinor{1^\prime} & ,=, \sqspinor{1}+\Delta_s^{(1,n)}\sqspinor{s}\nonumber ,,\
\sqspinor{n^\prime} & ,=,\sqspinor{n}+\Delta_s^{(n,1)}\sqspinor{s} ,,
\label{eq:gen-shifts}
\end{align}
and,
\begin{equation}
\Delta_s^{(i,j)}=\lambda \frac{\braket{js}}{\braket{ji}},.
\label{eq:mom-ratio}
\end{equation}
This form of eq.~\eqref{eq:LPplusNLP} signifies that the leading and subleading behaviour of the amplitude can be obtained in terms of simple shifts in the spinors of tree amplitudes. Note that the emitted soft gluon is placed in between the $1$ and $n$ particles in the colour ordered amplitudes and forms a colour dipole $\mathcal{D}{1n}$. Such colour dipole structures play an important role in understanding the IR singularities of scattering amplitudes~\cite{Catani:1996vz,Gardi:2009qi,Becher:2009cu}.

Emission of soft gluon with \lq $-$\rq , helicity can be treated analogously by taking anti-holomorphic soft gluon limit and interchanging angle and square spinors. Equipped with these formulae, we now move on to calculate the LP and NLP amplitudes for Higgs plus one jet production in the gluon fusion channel.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{LP and NLP amplitudes for $ gg \rightarrow Hg$}
\label{sec:LPplusNLP}
The most dominant mechanism for Higgs boson production at the LHC is via the gluon fusion channel. In this section we first reproduce all independent helicity amplitudes for Higgs plus one jet production via gluon fusion with(out) one extra gluon emission. Then we obtain NLP amplitudes by -- ({\em i}) taking soft gluon limit on $gg\to Hgg$ amplitudes, ({\em ii}) shifting spinors in $gg\to Hg$ amplitudes. Both ways lead to the exactly same results. Finally we discuss that for Higgs plus one jet production NMHV amplitudes do not contribute to the NLP threshold corrections.

\subsection{Higgs-gluon amplitudes}
The Standard Model of particle physics forbids gluons to interact with Higgs at the tree level, however they can interact via a massive quark loop. As the top quark is the heaviest among massive quarks, the coupling of Higgs with gluons is dominated via a top quark loop. In the large top mass limit $ m_t\rightarrow\infty $, we can integrate out the heavy top quark effect to obtain an effective Lagrangian as follows~\cite{Wilczek:1977zn, Shifman:1978zn},
\begin{align}
\mathcal{L}{,\text{eff}},=, -\frac{1}{4}, G, ,H ,\text{Tr} (F{\mu \nu }^a F^{\mu \nu,a}),,
\end{align}
where $ F_{\mu \nu}^a $ is the QCD field strength tensor.
The effective coupling is given at lowest order by $ G=\alpha_{s}/3\pi v $, where $v$ is the vacuum expectation value of the Higgs field and $ \alpha_s $ is the strong coupling constant.
%Our topic of interest are processes with Higgs plus two gluons (LO) and Higgs plus three gluons (NLO) production.
The general form of an amplitude consisting of one Higgs boson and $ n $-gluons can be represented as,
\begin{equation}
\mathcal{A}n (p_i,h_i,c_i),=, i,\left (\frac{\alpha{s}}{6\pi v}\right )g_s^{n-2} \sum_{\sigma \in \mathcal{S}_{n'}} \text{Tr} \left( \tbf{c_1}\tbf{c_2} \ldots \tbf{c_n} \right )\mathcal{A}n^{{c_i}} \left (h_1,h_2,h_3 \ldots ,{h_n };H \right),.
\label{eq:gen-ng}
\end{equation}
Here $ \mathcal{S}{n'} $ represents the set of all $ (n-1)! $ non-cycling permutations of $ 1,2,\ldots,n $. $ \tbf{c_i} $ denote the SU(3) colour matrix in the fundamental representation and the are normalized as, $ \text{Tr} (\tbf{c_1},\tbf{c_2})= \delta^{c_1 c_2} $. For brevity, we avoid writing $H$ explicitly in $\mathcal{A}_n^{{c_i}}$ in the rest of this paper.

The leading order process for Higgs plus one gluon production can be written as,
\begin{align}
g (p_1)+ g(p_2) \rightarrow H(-p_3) +g (-p_4),.
\label{eq:momassign}
\end{align}
There are two independent colour ordered helicity amplitudes for this process as given below,
\begin{align}
\mathcal{A}^{124}{+++} & =\frac{m{H}^{4}}{\braket{12}\braket{24}\braket{41}},, \qquad
\mathcal{A}^{124}_{-++} =\frac{[24]^{3}}{[12][14]},,
\label{eq:LO}
\end{align}
and amplitudes for all other helicity configurations can be constructed using these two.

Now, we consider that a gluon with momenta $ p_5 $ is being emitted from the leading order process, {\em i.e.,}
\begin{align}
g (p_1)+g(p_2) \rightarrow H(-p_3)+g(-p_4)+g(-p_5) , .
\end{align}
For this process, there are only three independent helicity amplitudes and remaining helicity configurations can be obtained by switching external momenta and spinors. These three independent helicity amplitudes containing Higgs plus four gluons are given by,
\begin{align}
\mathcal{A}^{1245}{++++},& =\frac{m{H}^4} {\braket{1 2} \braket{2 4}
\braket{ 45} \braket{5 1}} ,,\nn
%
\mathcal{A}^{1245}{-+++},& =\frac{\langle 1|4+5|2 ]^3}{\langle4|1|2] \braket{15} \braket{45} s{145}}+\frac{[25] [45] \langle 1|4+5|2 ]^2}{\langle4|1|2] s_{15} s_{145}} +\frac{[24] \langle 1|2+4|5 ]^2}{\braket{24} s_{12} s_{124}}
\nn& \quad +\frac{[25] \langle 1|2+4|5 ]^2}{ \braket{14} \braket{24} [15] s_{12}} -\frac{[25]^2 \langle 1|2+5|4 ]^2}{s_{12} s_{15} s_{125}} ,,\nn
%
\mathcal{A}^{1245}{--++},&=-{\langle 1 2\rangle^4\over \langle 12 \rangle
\langle 24 \rangle \langle 45\rangle \langle 51\rangle}
- {[45]^4 \over [1 2] [24] [45] [51]},.
\end{align}
Here $s{ij}=(p_i+p_j)^2$ and $s_{ijk}=(p_i+p_j+p_k)^2$. These amplitudes were calculated for the first time in\cite{Kauffman:1996ix}.
% In the upcoming section we discuss the forms of Higgs plus four gluon amplitudes at NLP.
Following eq.\eqref{eq:gen-ng}, we can write the full amplitude for a given helicity configuration as,
\begin{align}
& \mathcal{A}({p_i,h_i,c_i}),\nn& =, i , \left (\frac{\alpha_{s}}{6\pi v}\right ), g_s^2 \Bigg[ \left {\text{Tr} \left (\tbf{c_1}\tbf{c_2}\tbf{c_4}\tbf{c_5}\right )+ \left (\tbf{c_1}\tbf{c_5}\tbf{c_4}\tbf{c_2}\right )\right } \mathcal{A}^{1245}{h_1h_2h_4h_5} \nn &
+\left {\text{Tr} \left (\tbf{c_1}\tbf{c_4}\tbf{c_5}\tbf{c_2}\right )+ \left (\tbf{c_1}\tbf{c_2}\tbf{c_5}\tbf{c_4}\right )\right } \mathcal{A}^{1452}{h_1h_2h_4h_5} \nn &
+\left {\text{Tr} \left (\tbf{c_1}\tbf{c_5}\tbf{c_2}\tbf{c_4}\right )+ \left (\tbf{c_1}\tbf{c_4}\tbf{c_2}\tbf{c_5}\right )\right } \mathcal{A}^{1524}{h_1h_2h_4h_5} \Bigg] ,.
\label{eq:gen4gamp}
\end{align}
Squaring the above equation and summing over colours, we obtain the expression of squared amplitude as,
\begin{align}
\sum{\text{colours}}|\mathcal{A}({p_i,h_i,c_i})|^2,& =, \bigg[\left (\frac{\alpha_{s}}{6\pi v}\right ), g_s^2\bigg]^2 (N^2-1) \bigg { 2, N^2 \left ( |\mathcal{A}^{1245}|^2+|\mathcal{A}^{1452}|^2+|\mathcal{A}^{1524}|^2\right) \nn
& -4 \frac{(N^2-3)}{N^2} \big|\mathcal{A}^{1245}+\mathcal{A}^{1452}+\mathcal{A}^{1524}\big|^2 \bigg},.
\label{eq:genAsq}
\end{align}
Here, for simplicity, we have suppressed the labels that represent helicity configurations. Due to the dual Ward identity~\cite{Dixon:2004za,Kauffman:1996ix}, the term in the second line of the above equation vanishes and we are left with only the first term.
%

\subsection{Spinor shifts and colour dipoles}
In order to obtain NLP amplitudes for the Higgs plus two gluon production process, one needs to expand the $gg\to Hgg$ helicity amplitudes in the powers of the soft momentum keeping the sub-leading contributions. In parallel, following the arguments presented in section~\ref{sec:shifts}, we can get NLP amplitudes using the shifts in the spinors of $gg\to Hg$ amplitudes. We start our calculation by noting the fact that the gluon with momentum $ p_5 $ be emitted from any of the three gluons present at the leading order and as discussed in the previous section, the emission of a soft gluon always engenders shifts in two adjacent spinors present in the colour ordered non-radiative Born amplitudes.

In case of emission of a next-to-soft gluon from Higgs plus $ n $ gluon amplitudes, a total $ ^nC_2= n(n-1)/2 $ number of colour dipoles can be formed. Therefore, for amplitudes consisting of Higgs plus three gluons, three dipoles are generated due to the emission of a next-to-soft gluon and NLP amplitudes can be realised by shifting appropriate spinors depending on the helicity of the emitted gluon. For a \lq $+$\rq , gluon emisison from the dipole $\mathcal{D}{14}$ made up of momenta $p_1$ and $p_4$, the LP+NLP amplitude can be expressed as,
\begin{align}
\mathcal{A}^{1245}{h_1 h_2 h_4 +} , =, \frac{\braket{14}}{\braket{15}\braket{45}}\mathcal{A}^{1^\prime2,4^\prime}{h_1 h_2 h_4} ,,
\label{eq:gen-shift14}
\end{align}
where $\mathcal{A}^{1^\prime2,4^\prime}{h_1 h_2 h_4}$ denotes that the \sqspinor{1} and \sqspinor{4} spinors are shifted in the colour ordered leading amplitude obeying eq.(\ref{eq:gen-shifts}). Similar contributions coming from the dipoles $\mathcal{D}{24}$ and $\mathcal{D}{12}$ can be written as,
\begin{align}
\mathcal{A}^{1452} {h_1h_2h_4+} , =, \frac{\braket{24}}{\braket{25}\braket{54}}
\mathcal{A}^{1,2'4'} {h_1h_2h_4},,
\label{eq:gen-shift24}
\end{align}
and
\begin{align}
\mathcal{A}^{1524} {h_1h_2h_4+} , =, \frac{\braket{12}}{\braket{15}\braket{52}}\mathcal{A}^{1'2',4}{h_1h_2h_4} ,.
\label{eq:gen-shift12}
\end{align}
So the full amplitude of eq.\eqref{eq:gen4gamp} can now be rewritten using eqs.\eqref{eq:gen-shift14}--\eqref{eq:gen-shift12} as,
\begin{align}
& \mathcal{A}{h_1h_2h_4+}|{,\text{LP+NLP}},=, i , \left (\frac{\alpha_{s}}{6\pi v}\right ), g^2 \nn \times &\Bigg[ \left {\text{Tr} \left (\tbf{C_1}\tbf{C_2}\tbf{C_4}\tbf{C_5}\right )+ \left (\tbf{C_1}\tbf{C_5}\tbf{C_4}\tbf{C_2}\right )\right } \frac{\braket{14}}{\braket{15}\braket{45}}\mathcal{A}^{1^\prime2,4^\prime}{h_1 h_2 h_4} \nn &
-\left {\text{Tr} \left (\tbf{C_1}\tbf{C_4}\tbf{C_5}\tbf{C_2}\right )+ \left (\tbf{C_1}\tbf{C_2}\tbf{C_5}\tbf{C_4}\right )\right } \frac{\braket{24}}{\braket{25}\braket{45}} \mathcal{A}^{1,2'4'} {h_1h_2h_4} \nn &
-\left {\text{Tr} \left (\tbf{C_1}\tbf{C_5}\tbf{C_2}\tbf{C_4}\right )+ \left (\tbf{C_1}\tbf{C_4}\tbf{C_2}\tbf{C_5}\right )\right } \frac{\braket{12}}{\braket{15}\braket{25}}\mathcal{A}^{1'2',4}_{h_1h_2h_4} \Bigg],.
\label{eq:gen4gLPNLP}
\end{align}
To derive the above equation, we have used the reflection identity\cite{Dixon:2004za} that applies for Higgs plus $n$-gluon amplitudes.
%Note that the shift in the spinors corresponds to the dipole factors precede them.
This equation is one of the central results of this paper which identifies the direct correspondence of colour ordered amplitudes in the next-to-soft limit to the non-radiative colour ordered Born amplitudes with shifted spinors. Shift in each non-radiative spinor pair represents one colour ordered radiative amplitude. The validity of this formula relies only on the cyclic and antisymmetric properties of Higgs plus gluon amplitudes. Thus, this formula is applicable to any process that satisfy such properties, namely pure gluon amplitudes in Yang-Mills theories or gluons with a quark-antiquark pair in QCD.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{NLP Amplitudes: Absence of NMHV contribution}
As evident from the discussion in the previous section, colour ordered LP amplitudes always appear as a product of Born amplitudes and the corresponding Eikonal factors such as,
\begin{align}
\left.\mathcal{A}^{1245} {h_1h_2h_4+}\right|{,\rm LP},& =, \frac{\braket{14}}{\braket{15}\braket{45}} \mathcal{A}^{124} {h_1h_2h_4} ,,\nn
\left.\mathcal{A}^{1245} {h_1h_2h_4-}\right|{,\rm LP},& =, \frac{[14]}{[15][45]} \mathcal{A}^{124} {h_1h_2h_4} ,.
\end{align}
In this section we provide the details of NLP amplitudes for different helicity configurations. For Higgs plus four gluon amplitudes, there are altogether $ 2^4=16 $ helicity configurations possible. Out of these sixteen helicity amplitudes, one needs to calculate only eight, as the remaining conjugate configurations can easily be obtained by flipping the helicity of all the external gluons. As discussed earlier, NLP amplitudes can be calculated considering emission of both \lq $+$\rq , and \lq $-$\rq , helicity gluons from all possible Born amplitudes. In doing so, we find that the NMHV amplitudes do not add to the NLP contribution. We illustrate this by a simple example. Let us consider emission of a \lq $+$\rq , helicity gluon out of the $\mathcal{A}^{124}{+--}$ amplitude, which following eq.~\eqref{eq:LO} can be presented as,
\begin{align}
\mathcal{A}^{124}{+--} =-\frac{\braket{24}^{3}}{\braket{12}\braket{14}},.
\end{align}