In these notes we describe how to formulate the Lagrangian insertion technique in a way that mimics generalized unitarity. We introduce a notion of cuts in position space and show that the cuts of the correlators in the super-correlators/super-amplitudes duality correspond to generalized unitarity cuts of the equivalent amplitudes. The cuts consist of correlation functions of operators in the chiral part of the stress-tensor multiplet as well as other half-BPS operators. We will also discuss the application of the method to other correlators as well as non-planar contributions.
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2$ MHV form factors will have a fermionic content in addition to the super-momentum conserving delta function. If we define the quantity $\widetilde{\mathcal{F}}_{\mathcal{T}_d}$ as the form factor excluding the super-momentum conserving delta function: \begin{align} \mathcal{F}_{\mathcal{T}_d}(\gamma_{i+a}^\alpha,1,\cdots,n)&=\widetilde{\mathcal{F}}_{\mathcal{T}_d}(1,\cdots,n)\delta^8\left((i)^{+a}_{A}\gamma_{i+a}^\alpha-\sum_{r=1}^n\eta_{rA}\lambda_r^\alpha\right), \end{align} then $\widetilde{\mathcal{F}}^{\rm MHV}_{\mathcal{T}_d}$ will be a polynomial of degree $2(d-2)$ in $\eta_{-a}=(\bar{\imath})_{-a}^A\eta_A$. Some interesting relations between the form factors for an operator $\mathcal{T}_d$ and form factors for an operator $\mathcal{T}_{d-1}$ were found in~\cite{Penante-ml-2014sza} using BCFW recursion. However we are not interested in the explicit expressions for $\widetilde{\mathcal{F}}$. We only need to know its degree, and that it contains non-zero terms with $d-2$ factors of $\eta_{i-a}\epsilon^{ab}\eta_{i-b}$ for any set of $i$'s. The second fact follow from simple Feynman diagrams as there is always a non-zero form factor for $\mathrm{Tr}((\phi^{++})^d)$ with $d$ external scalars and any number of positive helicity gluons regardless of the ordering of the external states. Conservation of super momentum can then be used to make $\widetilde{\mathcal{F}}$ independent of two of the $\eta_-$'s. The $\overline{\rm MHV}$ form factor can be written as follows: \begin{align} \mathcal{F}^{\overline{\mathrm{MHV}}}_{\mathcal{T}_2}(\gamma_{i+a}^\alpha,1,\cdots,n)&=\frac{\delta^4\left(\gamma_{i+a}^\alpha-(\bar{\imath})^A_{-a'}\sum_{r=1}^n\eta_{rA}\lambda_r^\alpha\right)}{[12][23]\cdots [n1]}\\ &\quad \int \left(\prod_{j=1}^nd^4\tilde{\eta}_j\right)e^{i\sum_{j=1}^n\eta_{jA}\tilde{\eta}^A_j}\delta^4\left((\bar{\imath})^A_{+a}\sum_{r=1}^n\eta_{rA}\lambda_r^\alpha\right).\nonumber \end{align} After performing the $\tilde{\eta}$-integrations, this formula becomes a Grassmann polynomial of degree $4n$. For the case $n=2$, it is equivalent to the MHV formula while for $n>2$ it is a Grassmann polynomial of a higher degree than the MHV formula. Unlike for scattering amplitudes where the three-point $\overline{\rm MHV}$ amplitude is a Grassmann polynomial of only degree 4, there are no special form factors for the operators in~\eqref{T_2} with a lower degree than the MHV formula. \begin{figure} \centering \begin{tabular}{cc} \includegraphics[trim = 0 0 20 15, clip, scale=0.8]{Eksempel1-eps-converted-to.pdf}&\includegraphics[trim = 0 0 20 15, clip, scale=0.8]{Eksempel2-eps-converted-to.pdf}\\ (a)&(b)\\[3pt] \includegraphics[trim = 0 0 20 15, clip, scale=0.8]{Eksempel3-eps-converted-to.pdf}&\includegraphics[trim = 0 0 20 15, clip, scale=0.8]{Eksempel4-eps-converted-to.pdf}\\ (c)&(d) \end{tabular} ]]>
0$. By introducing Feynman parameters and performing the momentum integrals this expression can be rewritten as: \begin{align} L_{\mu_1\cdots\mu_n}(x_1,x_2,p_1\cdots p_n)&=\frac{(-i)^{n+1}}{(2\pi)^D}\prod_{j=1}^n\left(-2i\frac{\partial}{\partial x_1^{\mu_j}}+2\sum_{r=1}^{j-1}p_{r\mu_j}+p_{j\mu_j}\right)\nonumber\\ &\quad \times\left(\prod_{j=1}^n\int_{t_{j-1}}^1dt_j\right)e^{-i\sum_{j=1}^np_j\cdot (x_2t_j+x_1(1-t_j))} \label{gammelt2}\\ &\quad \int_0^\infty d\zeta\zeta^n\frac{\pi^{D/2}}{(-i\zeta)^{D/2}}e^{i\zeta f(t_j,p_j)-i\frac{(x_1-x_2)^2}{4\zeta}-\epsilon\zeta-\epsilon/\zeta},\nonumber \end{align} with $f(t_j,p_j)$ being a function whose specific expression is irrelevant. For this to be as divergent as a single scalar propagator in the light-like, the factor $\zeta^n$ in the $\zeta$-integral will have to be removed. This means that only the term involving all $n$ derivatives will survive in the light-like limit. The derivatives will become proportional to $(x_1-x_2)^\mu$, and the expression will act like a single Wilson line. In general, it was concluded that the number of derivatives minus the number of propagators decided how divergent a side was. As the operator in equation~\eqref{T_2} includes both fermions and field strenghts, one might expect the correlation functions in the duality to contain something more divergent than simple scalar propagators. However due to the chirality of the operator this is not the case. Only the scalars in the operators $T_2$ can connect through free propagators. The other fields have to connect through interaction vertices that will lower the divergences to that of simple scalar propagators. Because of this, the Born-level correlator $G_n^{(0)}$ in~\eqref{duality eq} is just a collection of scalar propagators while the full correlation function $G_n$ do not become more divergent than $G_n^{(0)}$. Lagrangian insertions can be included in this analysis as the chiral on-shell Lagrangian is simply the highest component of the operator~\eqref{T_2}. So we should not get anything more divergent than scalar propagators. This also matches the conclusions referenced in section~\ref{LI afsnit} coming from the operation product expansion. We will use the approach of~\cite{Engelund-ml-2011fg} in the case of a purely scalar polygon where it will provide some clear insight. We will not use it for the supersymmetric case because it becomes rather cumbersome, especially finding the correct fields that sit at the corners of the polygon. The sides of the polygon do seem to act like the supersymmetric Wilson loops of~\cite{Mason-ml-2010yk,CaronHuot-ml-2010ek} but the appearance of ghosts at higher loop orders complicates matters. ]]>
2$ there will some additional Grassmann variables in $\widetilde{\mathcal{F}}^{MHV}_{\mathcal{T}_d}$. We can use conservation of super-momentum to write this function without any explicit dependence on either $\eta_{P_{j-1}}$ or $\eta_{P_{j}}$. Subsequently, we find the term proportional to $\eta_{P_{\tilde{1}}-a}\epsilon^{ab}\eta_{P_{\tilde{1}}-b}$ as well as similar factors for all other directions that has been made light-like as part of the cut. From a Feynman diagram perspective we know that such a term should always be present. The integration over the variables $\gamma_a$ can be used to remove this factor: \begin{align} &\int d^4\eta_{P_{\tilde{1}}}\int d^2\gamma\delta^8\left(\gamma_{\tilde{1}+a}^\alpha(\tilde{1})^{+a}_A-\sum_r\eta_{rA}\lambda_r^\alpha-\eta_{P_{\tilde{1}}A}\lambda_{P_{\tilde{1}}}^\alpha\right)\eta_{P_{\tilde{1}}-a}\epsilon^{ab}\eta_{P_{\tilde{1}}-b}\label{fermi3}\\ &\qquad \qquad \qquad \qquad\qquad \qquad \qquad =2{\bf (j\tilde{1})}\langle P_jP_{j-1}\rangle^2\delta^4\left(\langle P_{\tilde{1}}\hat{\gamma}_{\tilde{1}+a}\rangle-\sum_r\eta_{rA}\langle P_{\tilde{1}}r\rangle\right).\nonumber \end{align} By imposing shift symmetries for all point made light-like separated from $x_j$ (apart from $x_{j-1}$ and $x_{j+1}$), $\widetilde{\mathcal{F}}^{MHV}_{\mathcal{T}_d}$ is reduced to purely bosonic factors. This can therefore be written as $\widetilde{\mathcal{F}}^{MHV}_{\mathcal{T}_2}$ multiplied by some spinor products. Potentially, it should be possible to use generalized unitarity to find the correct spinor factor in a systematic way by exploiting relations like the ones found in~\cite{Penante-ml-2014sza}. However, we will instead use that we already found this factor for the scalar polygon in section~\ref{afsnit skalar}. Combining the information gained from the two approaches, we get: \begin{align} &\lim_{(x_i-y_{\tilde{1}})^2=0}\frac{(x_i-y_{\tilde{1}})^2}{\bf (i\tilde{1})}\int d^4\theta_{\tilde{1}}\begin{minipage}{2.5cm} \includegraphics[trim = 5 0 19 15, clip, scale=0.8]{fler-punkt1-eps-converted-to.pdf} \end{minipage}\label{vertex relation susy}\\ &\qquad \qquad \qquad =\frac{\langle i-1i\rangle}{\langle i-1\tilde{1}\rangle\langle \tilde{1}i\rangle}\frac{1}{{\bf (i\tilde{1})}}\int d^4\theta_{\tilde{1}}\delta^2(\langle \tilde{1}\theta_{\tilde{1}}^{+a}\rangle-\langle \tilde{1}\theta_i^A\rangle(\tilde{1})_A^{+a})\begin{minipage}{2.5cm} \includegraphics[trim = 5 0 19 15, clip, scale=0.8]{fler-punkt2-eps-converted-to.pdf}\nonumber \end{minipage}\qquad . \end{align} Again full lines represent distances made light-like after dividing out scalar propagators, and vertices where $d$ lines meet correspond to an operator $\mathcal{T}_d$. As we used that all of the Grassmann variables from $\widetilde{\mathcal{F}}_{\mathcal{T}_d}^{MHV}$ in~\eqref{fermi2} were removed by imposing shift symmetries, it is assumed that there are delta functions similar to~\eqref{deltafunktion} for all but two of the lines meeting at $x_i$. We will now apply all this to a full cut. A string of $m$ Lagrangian insertions will be made light-like separated from each other and the polygon such that the inside of the polygon is split into two with $x_i$ being light-like separated from $y_{\tilde{1}}$ and $x_j$ from $y_{\tilde{m}}$. In terms of the diagram in figure~\ref{cut full susy}(a), the cut can be defined as: \begin{align} \mathbf{Cut}&=\lim_{(x_i-y_{\tilde{1}})^2=0}\frac{(x_i\!-\!y_{\tilde{1}})^2}{\bf (i\tilde{1})}\lim_{(y_{\tilde{1}}-y_{\tilde{2}})^2=0}\cdots\lim_{(y_{\tilde{m}}-x_j)^2=0}\frac{(y_{\tilde{m}}\!-\!x_j)^2}{\bf (\tilde{m}j)}\int\prod_{r=1}^md^4\theta_{y_{\tilde{r}}}\,{\rm figure~\ref{cut full susy}(a)}. \end{align} \begin{figure} \centering \begin{tabular}{cc} \includegraphics[trim = 0 0 15 13, clip, scale=1]{Susy-cut1-eps-converted-to.pdf}&\includegraphics[trim = 0 0 15 13, clip, scale=1]{Susy-cut2-eps-converted-to.pdf}\\ (a)&(b) \end{tabular} ]]>
2$ as shown in figure~\ref{ny cf}. This sort of diagram will appear as part of the cuts used in section~\ref{Duality afsnit} but one could also use this as the starting point. Since equations~\eqref{fermi1} and~\eqref{fermi2} do not rely on integration over the super-space variables, it should be dual to three different four-point amplitudes and a single six-point amplitude provided we introduce some additional fermionic delta functions like the ones in~\eqref{deltafunktion}. \begin{figure} \centering \includegraphics[trim = 0 3 0 3, clip, scale=0.5]{Flower-eps-converted-to.pdf} ]]>