<figure><img src="image_13.png" /><figcaption>Figure 11. Flux-tube representation of the 3 OPE components of the $n = 4$ super form factor of $\mathcal{T}_3$ at leading twist. Left panel: a scalar $\phi$ is produced by the bottom pentagon; it propagates freely through the next pentagon before being absorbed at the top. Central panel: a fermion $\bar{\psi}$ is produced at the bottom, converted into a scalar $\phi$ in the second pentagon and absorbed at the top. The right panel corresponds to the strict collinear limit to the $n = 3$ form factor, with the scalar $\phi$ produced in the second pentagon.</figcaption></figure>

also introduced a suitable power of $q^2$ to cancel the scaling dimension of the R-invariants and make the ratio function dimensionless. In what follows, we will drop the $|q|$ dependence and work with the normalization $|q| = 1$.

The second important ingredient for the comparison is the mapping between the $\eta$ variables on the edges of the super Wilson loop and the $\chi$ variables carried by the superpentagons,

$$ 
\begin{aligned}
W_{3,4}(\eta) &= \langle \mathcal{F}_3 | \mathbb{P}_t \circ \mathbb{P}_b \rangle \\
&= \langle \mathcal{F}_3 | P \circ P_{12} \rangle \chi_b \cdot \chi_b + 2 \langle \mathcal{F}_3 | P_1 \circ P_2 \rangle \chi_t \cdot \chi_b + \langle \mathcal{F}_3 | P_{12} \circ P \rangle \chi_t \cdot \chi_t,
\end{aligned} 
\quad (4.36) $$

with $t, b$ referring to the top and bottom pentagon in the sequence and with $\chi \cdot \chi' = \frac{1}{2}\epsilon_{ab}\chi^a\chi^b$. As there are more edges than pentagons, the mapping between $\eta$'s and $\chi$'s is not one-to-one. It only becomes one-to-one after setting the $\eta$'s to zero on two edges, using the supertranslation symmetry of the super loop. In the OPE frame where $\eta_1^a = \eta_4^a = 0$, with $a = 1, 2$, the 3 components in the right-hand side of eq. (4.36) can be put in correspondence with the 3 invariants $\eta_i \cdot \eta_j$, with $1 < i \le j < 4$, standing in the left-hand side.

The precise dictionary reads, see section 4.2,

$$ 
\begin{aligned}
\frac{\partial}{\partial \chi_b} &= \alpha_2 \frac{\partial}{\partial \eta_2}, \\
\frac{\partial}{\partial \chi_t} &= \alpha_3 \left[ \frac{\partial}{\partial \eta_3} + \frac{\langle 4^{-124} \rangle}{\langle 4^{-134} \rangle} \frac{\partial}{\partial \eta_2} \right],
\end{aligned} 
\quad (4.37) $$

where $Z_i$ is the momentum twistor of edge $i$ and $Z_{i-}$ the twistor of the same edge in the preceding period. Going to the OPE edge labelling, $\{4^-, 1, 2, 3, 4\} \to \{\mathbf{3}, \mathbf{1}, -\mathbf{1}, \mathbf{0}, \mathbf{2}\}$, and using the OPE parametrization of the momentum twistors in appendix B.3, we find

$$ \alpha_2^4 = \frac{\langle -\mathbf{1012} \rangle^4}{(\mathbf{0})_b^4 (\mathbf{1})_b^4 (\mathbf{2})_b^4} = -1, \quad \alpha_3^4 = \frac{\langle \mathbf{0123} \rangle^4}{(\mathbf{1})_t^4 (\mathbf{2})_t^4 (\mathbf{3})_t^4} = -e^{4\tau_b + 2i\phi_b}, \quad (4.38) $$

with $\phi_b$ being the OPE angle variable in the bottom channel. Additionally, in this parametrization, $\langle 4^{-124} \rangle = \langle 4^{-134} \rangle$ and $q^2 = -1$.
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Flux-tube representation of the 3 OPE components of the $n = 4$ super form factor of $\mathcal{T}_3$ at leading twist. Left panel: a scalar $\phi$ is produced by the bottom pentagon; it propagates freely through the next pentagon before being absorbed at the top. Central panel: a fermion $\bar{\psi}$ is produced at the bottom, converted into a scalar $\phi$ in the second pentagon and absorbed at the top. The right panel corresponds to the strict collinear limit to the $n = 3$ form factor, with the scalar $\phi$ produced in the second pentagon.</figcaption></figure>

also introduced a suitable power of $q^2$ to cancel the scaling dimension of the R-invariants and make the ratio function dimensionless. In what follows, we will drop the $|q|$ dependence and work with the normalization $|q| = 1$.

The second important ingredient for the comparison is the mapping between the $\eta$ variables on the edges of the super Wilson loop and the $\chi$ variables carried by the superpentagons,

$$ W_{3,4}(\eta) = \langle \mathcal{F}_3 | \mathbb{P}_t \circ \mathbb{P}_b \rangle \\ = \langle \mathcal{F}_3 | P \circ P_{12} \rangle \chi_b \cdot \chi_b + 2 \langle \mathcal{F}_3 | P_1 \circ P_2 \rangle \chi_t \cdot \chi_b + \langle \mathcal{F}_3 | P_{12} \circ P \rangle \chi_t \cdot \chi_t, \quad (4.36) $$

with $t, b$ referring to the top and bottom pentagon in the sequence and with $\chi \cdot \chi' = \frac{1}{2}\epsilon_{ab}\chi^a\chi^b$. As there are more edges than pentagons, the mapping between $\eta$'s and $\chi$'s is not one-to-one. It only becomes one-to-one after setting the $\eta$'s to zero on two edges, using the supertranslation symmetry of the super loop. In the OPE frame where $\eta_1^a = \eta_4^a = 0$, with $a = 1, 2$, the 3 components in the right-hand side of eq. (4.36) can be put in correspondence with the 3 invariants $\eta_i \cdot \eta_j$, with $1 < i \le j < 4$, standing in the left-hand side.

The precise dictionary reads, see section 4.2,

$$ \begin{aligned} \frac{\partial}{\partial \chi_b} &= \alpha_2 \frac{\partial}{\partial \eta_2}, \\ \frac{\partial}{\partial \chi_t} &= \alpha_3 \left[ \frac{\partial}{\partial \eta_3} + \frac{\langle 4^{-124} \rangle}{\langle 4^{-134} \rangle} \frac{\partial}{\partial \eta_2} \right], \end{aligned} \quad (4.37) $$

where $Z_i$ is the momentum twistor of edge $i$ and $Z_i^-$ the twistor of the same edge in the preceding period. Going to the OPE edge labelling, $\{4^-, 1, 2, 3, 4\} \to \{\mathbf{3}, \mathbf{1}, -\mathbf{1}, \mathbf{0}, \mathbf{2}\}$, and using the OPE parametrization of the momentum twistors in appendix B.3, we find

$$ \alpha_2^4 = \frac{\langle -\mathbf{1012} \rangle^4}{(\mathbf{0})_b^4 (\mathbf{1})_b^4 (\mathbf{2})_b^4} = -1, \quad \alpha_3^4 = \frac{\langle \mathbf{0123} \rangle^4}{(\mathbf{1})_t^4 (\mathbf{2})_t^4 (\mathbf{3})_t^4} = -e^{4\tau_b + 2i\phi_b}, \quad (4.38) $$

with $\phi_b$ being the OPE angle variable in the bottom channel. Additionally, in this parametrization, $\langle 4^{-124} \rangle = \langle 4^{-134} \rangle$ and $q^2 = -1$.
Applying these derivatives to the left-hand side of eq. (4.36) and taking the collinear limit $\tau, \tau_b \to \infty$, we arrive at

$$ \begin{aligned} \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} &= \frac{e^{-\tau-\tau_b}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} (R_1 + R_4) + \dots, \\ \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} &= \frac{e^{-\tau-\tau_b} - \frac{i\phi_b}{2}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} \left[ \frac{e^{\sigma-\sigma_b}}{e^{\sigma} + e^{-\sigma}} (R_3 + R_4) + e^{\sigma_b} (R_3 - R_1) \right] + \dots, \end{aligned} \quad (4.39) $$

with $R_i$ defined earlier, see eq. (4.34), and with dots standing for subleading terms. Lastly, we have

$$ \frac{\partial}{\partial \chi_t^2} \frac{\partial}{\partial \chi_t^1} W_{3,4} = \frac{1}{(e^{\tau} + e^{-\tau})(e^{\sigma} + e^{-\sigma} + e^{-2\tau-\sigma})} (R_2 + R_3) + \mathcal{O}(e^{-\tau_b}), \quad (4.40) $$

which holds for any $\tau$ at large $\tau_b$. This contribution corresponds to the strict collinear limit $3||2$, onto the 3pt form factor of $\mathcal{T}_3$. In this limit, (123), (234) $\to 0$ and (341) $= (412)$, meaning that the bottom pentagon is empty, as depicted in figure 11. This collinear limit is, of course, built into the perturbative data, so the comparison with the flux-tube formula for this component reduces to the analysis carried out previously.

The other two components are more interesting. They correspond, respectively, to the OPE processes

$$ \text{vacuum} \to \phi \to \phi \to \mathcal{F}_3, \quad \text{vacuum} \to \bar{\psi} \to \phi \to \mathcal{F}_3. \quad (4.41) $$

In both cases, the sequences end with a single scalar $\phi$ at the top, absorbed by the half-BPS operator $\mathcal{T}_3$. However, whereas in the first case the scalar is produced from the onset, using the scalar component $P_{12}$ of the superpentagon, in the second case it is created through a fermion $\bar{\psi}$ propagating in the bottom channel. See figure 11.

The flux-tube representations of these two processes in (4.41) are expressed in terms of the (charged) pentagon transitions $P_{\phi\phi}$ and $P_{\bar{\psi}\phi}$ worked out in refs. [50, 51, 53, 55, 56]. In the normalization used in ref. [53], they read

$$ \begin{aligned} \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} &= \langle \mathcal{F}_3 | P \circ P_{12} \rangle = g^{-1} \int \frac{dudv}{(2\pi)^2} \hat{\mu}_{\phi}(u) P_{\phi\phi}(u|v) \tilde{\mu}_{\phi}(v) + \dots, \\ \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} &= \langle \mathcal{F}_3 | P_2 \circ P_1 \rangle = g^{-\frac{5}{4}} e^{-i\phi_b/2} \int \frac{dudv}{(2\pi)^2} \hat{\mu}_{\bar{\psi}}(u) P_{\bar{\psi}\phi}(u|v) \tilde{\mu}_{\phi}(v) + \dots, \end{aligned} \quad (4.42) $$

with the overall coupling factors determined by the requirement that the processes start at order $\mathcal{O}(g^0)$ at weak coupling. Here, in order to avoid cluttering the formulae, we introduced notations for the integration measures,

$$ \hat{\mu}_X(u) = \mu_X(u) e^{-E_X(u)\tau_b - ip_X(u)\sigma_b}, \quad \tilde{\mu}_\phi(v) = \sqrt{\mu_\phi(v) \nu_\phi(v)} e^{-E_\phi(v)\tau + ip_\phi(v)\sigma}, \quad (4.43) $$

with $X = \phi, \bar{\psi}$ and $\tau, \sigma, \tau_b, \sigma_b$ denoting the OPE spacetime variables in the top and bottom channels.<fnref n="10" /> The rapidity $u$ is integrated along the real axis with a small negative imaginary

<footnote n="10">¹⁰Note that in order to match the conventions in ref. [53] we must rescale the scalar measure and transition, using $\mu_{\phi}^{[\alpha]} \to \mu_{\phi}^{[\alpha]}/g$ and $P_{\phi\phi}^{[\alpha]} \to gP_{\phi\phi}^{[\alpha]}$.</footnote>
Applying these derivatives to the left-hand side of eq. (4.36) and taking the collinear limit $\tau, \tau_b \to \infty$, we arrive at

$$ \begin{aligned} \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} &= \frac{e^{-\tau-\tau_b}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} (R_1 + R_4) + \dots, \\ \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} &= \frac{e^{-\tau-\tau_b} - \frac{i\phi_b}{2}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} \left[ \frac{e^{\sigma-\sigma_b}}{e^{\sigma} + e^{-\sigma}} (R_3 + R_4) + e^{\sigma_b} (R_3 - R_1) \right] + \dots, \end{aligned} \quad (4.39) $$

with $R_i$ defined earlier, see eq. (4.34), and with dots standing for subleading terms. Lastly, we have

$$ \frac{\partial}{\partial \chi_t^2} \frac{\partial}{\partial \chi_t^1} W_{3,4} = \frac{1}{(e^{\tau} + e^{-\tau})(e^{\sigma} + e^{-\sigma} + e^{-2\tau-\sigma})} (R_2 + R_3) + \mathcal{O}(e^{-\tau_b}), \quad (4.40) $$

which holds for any $\tau$ at large $\tau_b$. This contribution corresponds to the strict collinear limit $3||2$, onto the 3pt form factor of $\mathcal{T}_3$. In this limit, (123), (234) $\to 0$ and (341) $= (412)$, meaning that the bottom pentagon is empty, as depicted in figure 11. This collinear limit is, of course, built into the perturbative data, so the comparison with the flux-tube formula for this component reduces to the analysis carried out previously.

The other two components are more interesting. They correspond, respectively, to the OPE processes

$$ \text{vacuum} \to \phi \to \phi \to \mathcal{F}_3, \quad \text{vacuum} \to \bar{\psi} \to \phi \to \mathcal{F}_3. \quad (4.41) $$

In both cases, the sequences end with a single scalar $\phi$ at the top, absorbed by the half-BPS operator $\mathcal{T}_3$. However, whereas in the first case the scalar is produced from the onset, using the scalar component $P_{12}$ of the superpentagon, in the second case it is created through a fermion $\bar{\psi}$ propagating in the bottom channel. See figure 11.

The flux-tube representations of these two processes in (4.41) are expressed in terms of the (charged) pentagon transitions $P_{\phi\phi}$ and $P_{\bar{\psi}\phi}$ worked out in refs. [50, 51, 53, 55, 56]. In the normalization used in ref. [53], they read

$$ \begin{aligned} \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} &= \langle \mathcal{F}_3 | P \circ P_{12} \rangle = g^{-1} \int \frac{dudv}{(2\pi)^2} \hat{\mu}_{\phi}(u) P_{\phi\phi}(u|v) \tilde{\mu}_{\phi}(v) + \dots, \\ \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} &= \langle \mathcal{F}_3 | P_2 \circ P_1 \rangle = g^{-\frac{5}{4}} e^{-i\phi_b/2} \int \frac{dudv}{(2\pi)^2} \hat{\mu}_{\bar{\psi}}(u) P_{\bar{\psi}\phi}(u|v) \tilde{\mu}_{\phi}(v) + \dots, \end{aligned} \quad (4.42) $$

with the overall coupling factors determined by the requirement that the processes start at order $\mathcal{O}(g^0)$ at weak coupling. Here, in order to avoid cluttering the formulae, we introduced notations for the integration measures,

$$ \hat{\mu}_X(u) = \mu_X(u) e^{-E_X(u)\tau_b - ip_X(u)\sigma_b}, \quad \tilde{\mu}_\phi(v) = \sqrt{\mu_\phi(v) \nu_\phi(v)} e^{-E_\phi(v)\tau + ip_\phi(v)\sigma}, \quad (4.43) $$

with $X = \phi, \bar{\psi}$ and $\tau, \sigma, \tau_b, \sigma_b$ denoting the OPE spacetime variables in the top and bottom channels.<fnref n="10" /> The rapidity $u$ is integrated along the real axis with a small negative imaginary

<footnote n="10">¹⁰Note that in order to match the conventions in ref. [53] we must rescale the scalar measure and transition, using $\mu_{\phi}^{[\alpha]} \to \mu_{\phi}^{[\alpha]}/g$ and $P_{\phi\phi}^{[\alpha]} \to gP_{\phi\phi}^{[\alpha]}$.</footnote>
part whereas the rapidity $v$ is given a small positive imaginary part. This prescription is needed to avoid the pole at $u = v$ in the first integral, coming from the Gamma function in eq. (3.14), and to deal with the pole at $v = 0$ in the second integral, coming from the fermion measure, see refs. [51, 53].

To check our conjecture for $\nu_\phi$, we only need to compare the outcome of integrations in eqs. (4.42) with the corresponding coefficients $R_i$. The analysis presents no major difficulty. We illustrate it here for the first integral.

As in the three-point case, one may simplify the analysis by exploiting a connection to NMHV amplitudes. The main observation is that the first integral in eq. (4.42) is almost identical to the one introduced in ref. [50] to study the double collinear limit of the scalar component of the NMHV heptagon Wilson loop “$W_{\text{hept}}^{(7145)}$”. The only difference is that for the heptagon one should replace $\nu_\phi$ with $\mu_\phi$ in the first integral of eq. (4.42). To be precise, one has

$$ \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} = W_{\text{hept}}^{(7145)} + \int \frac{dudv}{(2\pi)^2} \hat{\mu}_\phi(u) P_{\phi\phi}(u|v) \Delta \tilde{\mu}_\phi(v) + \mathcal{O}(e^{-3\tau}), \quad (4.44) $$

with

$$ \Delta \tilde{\mu}_{\phi}(v) = \left[ \sqrt{\mu_{\phi}(v) \nu_{\phi}(v)} - \mu_{\phi}(v) \right] e^{-E_{\phi}(v)\tau + ip_{\phi}(v)\sigma}. \quad (4.45) $$

As stressed earlier, at weak coupling, $\nu_\phi/\mu_\phi = 1 + \mathcal{O}(g^2)$. Hence, the $\Delta$ term vanishes at tree level, giving [50]

$$ \begin{aligned} \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} &= e^{-\tau-\tau_b} \int \frac{dudv}{(2\pi)^2} \Gamma(\frac{1}{2}-iu) \Gamma(iu-iv) \Gamma(\frac{1}{2}+iv) e^{-2iu\sigma_b+2iv\sigma} + \mathcal{O}(g^2) \\ &= \frac{e^{-\tau-\tau_b}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} + \mathcal{O}(g^2). \end{aligned} \quad (4.46) $$

It agrees precisely with the first line of eq. (4.39), using $R_i = 1/2 + \mathcal{O}(g^2)$. At one loop, one may refer to formulae in ref. [50] to evaluate the heptagon integral. The extra bit is easily computed,

$$ \begin{aligned} & \int \frac{dudv}{(2\pi)^2} \hat{\mu}_{\phi}(u) P_{\phi\phi}(u|v) \Delta \tilde{\mu}_{\phi}(v) \\ &= \frac{g^2 e^{-\tau-\tau_b}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} \\ & \times \left[ \zeta_2 + \log \left[ \frac{e^{\sigma_b-\sigma}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} \right] \log \left[ \frac{e^{\sigma-\sigma_b} + e^{\sigma+\sigma_b}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} \right] \right] + \mathcal{O}(g^4). \end{aligned} \quad (4.47) $$

Combining all the pieces together, one easily verifies agreement with the perturbative result in eqs. (4.30) and (4.35). A similar analysis applies to the second integral in eq. (4.42), with a different component of the NMHV heptagon Wilson loop.

One could proceed similarly at higher loops using the all-order OPE conjectures. The connection to scattering amplitudes indicates that the contributions to the form factor should live in the same function space as the NMHV heptagon in the double collinear limit, facilitating their exploration at higher loops. It would be interesting to perform this analysis explicitly at two loops, using known result for the heptagon OPE integral [50]. It would also be fascinating to see if one could combine insights from the leading OPE integrals and the structure of the NMHV heptagon amplitude [11, 75–77] to bootstrap the $n = 4$ super form factor of $\mathcal{T}_3$ in general kinematics at higher loops, as done recently through two loops for $\mathcal{T}_2$ in refs. [27, 29].
part whereas the rapidity $v$ is given a small positive imaginary part. This prescription is needed to avoid the pole at $u = v$ in the first integral, coming from the Gamma function in eq. (3.14), and to deal with the pole at $v = 0$ in the second integral, coming from the fermion measure, see refs. [51, 53].

To check our conjecture for $\nu_\phi$, we only need to compare the outcome of integrations in eqs. (4.42) with the corresponding coefficients $R_i$. The analysis presents no major difficulty. We illustrate it here for the first integral.

As in the three-point case, one may simplify the analysis by exploiting a connection to NMHV amplitudes. The main observation is that the first integral in eq. (4.42) is almost identical to the one introduced in ref. [50] to study the double collinear limit of the scalar component of the NMHV heptagon Wilson loop “$W_{\text{hept}}^{(7145)}$”. The only difference is that for the heptagon one should replace $\nu_\phi$ with $\mu_\phi$ in the first integral of eq. (4.42). To be precise, one has

$$ \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} = W_{\text{hept}}^{(7145)} + \int \frac{dudv}{(2\pi)^2} \hat{\mu}_\phi(u) P_{\phi\phi}(u|v) \Delta \tilde{\mu}_\phi(v) + \mathcal{O}(e^{-3\tau}), \quad (4.44) $$

with

$$ \Delta \tilde{\mu}_{\phi}(v) = \left[ \sqrt{\mu_{\phi}(v) \nu_{\phi}(v)} - \mu_{\phi}(v) \right] e^{-E_{\phi}(v)\tau + ip_{\phi}(v)\sigma}. \quad (4.45) $$

As stressed earlier, at weak coupling, $\nu_\phi/\mu_\phi = 1 + \mathcal{O}(g^2)$. Hence, the $\Delta$ term vanishes at tree level, giving [50]

$$ \begin{aligned} \frac{\partial}{\partial \chi_b^2} \frac{\partial}{\partial \chi_b^1} W_{3,4} &= e^{-\tau-\tau_b} \int \frac{dudv}{(2\pi)^2} \Gamma(\frac{1}{2}-iu) \Gamma(iu-iv) \Gamma(\frac{1}{2}+iv) e^{-2iu\sigma_b+2iv\sigma} + \mathcal{O}(g^2) \\ &= \frac{e^{-\tau-\tau_b}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} + \mathcal{O}(g^2). \end{aligned} \quad (4.46) $$

It agrees precisely with the first line of eq. (4.39), using $R_i = 1/2 + \mathcal{O}(g^2)$. At one loop, one may refer to formulae in ref. [50] to evaluate the heptagon integral. The extra bit is easily computed,

$$ \begin{aligned} & \int \frac{dudv}{(2\pi)^2} \hat{\mu}_{\phi}(u) P_{\phi\phi}(u|v) \Delta \tilde{\mu}_{\phi}(v) \\ &= \frac{g^2 e^{-\tau-\tau_b}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} \\ & \times \left[ \zeta_2 + \log \left[ \frac{e^{\sigma_b-\sigma}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} \right] \log \left[ \frac{e^{\sigma-\sigma_b} + e^{\sigma+\sigma_b}}{e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b}} \right] \right] + \mathcal{O}(g^4). \end{aligned} \quad (4.47) $$

Combining all the pieces together, one easily verifies agreement with the perturbative result in eqs. (4.30) and (4.35). A similar analysis applies to the second integral in eq. (4.42), with a different component of the NMHV heptagon Wilson loop.

One could proceed similarly at higher loops using the all-order OPE conjectures. The connection to scattering amplitudes indicates that the contributions to the form factor should live in the same function space as the NMHV heptagon in the double collinear limit, facilitating their exploration at higher loops. It would be interesting to perform this analysis explicitly at two loops, using known result for the heptagon OPE integral [50]. It would also be fascinating to see if one could combine insights from the leading OPE integrals and the structure of the NMHV heptagon amplitude [11, 75–77] to bootstrap the $n = 4$ super form factor of $\mathcal{T}_3$ in general kinematics at higher loops, as done recently through two loops for $\mathcal{T}_2$ in refs. [27, 29].
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Leading flux-tube contribution to the form factor of $\mathcal{T}_4$. A scalar is produced by each pentagon. The two scalars are then absorbed by the operator. Only one scalar can be created by a single pentagon, due to the fact that the pentagons can only be charged under a chiral half of the R-symmetry group.</figcaption></figure>

## 4.5 Minimal four-point form factor

We conclude our perturbative checks with a brief discussion of the four-point form factor of $\mathcal{T}_4$. This is a minimal form factor with a single $\eta$ structure, namely, (123) (341). Stripping off the overall factor $\chi_b^1\chi_b^2\chi_t^1\chi_t^2$, one gets for the ratio function,

$$ \mathcal{R}_{4,4} = \Omega_4 \times [1 - g^2 V'_0 + \mathcal{O}(g^4)], \quad (4.48) $$

where $V'_0$ denotes the finite part of $V_0$ in eq. (4.35) and with

$$ \Omega_4 = \sqrt{u_1 u_2 u_3 u_4}. \quad (4.49) $$

In the collinear limit, $\tau, \tau_b \to \infty$, one finds

$$ \Omega_4 = \frac{e^{-2\tau-\tau_b}}{(e^{\sigma} + e^{-\sigma})(e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b})} + \mathcal{O}(e^{-2\tau-2\tau_b}, e^{-4\tau-\tau_b}), \quad (4.50) $$

using the OPE parametrization given in appendix B.3.

The leading OPE process on the flux-tube side is depicted in figure 12. It is described by a sequence of two $P_{12}$ pentagon transitions, producing a two-scalar state in the top OPE channel. The corresponding integral formula reads

$$ \mathcal{R}_{4,4} = g^{-6} \int \frac{du \, dv_1 \, dv_2}{2(2\pi)^3} \hat{\mu}_{\phi}(u) P_{\phi|\phi\phi}(u|v_1, v_2) F_{\phi\phi}(v_1, v_2) \tilde{\mu}_{\phi}(v_1) \tilde{\mu}_{\phi}(v_2) + \dots, \quad (4.51) $$

with the overall power of $g$ adjusted such that $\mathcal{R}_{4,4} = \mathcal{O}(g^0)$ at weak coupling. The measures are defined in eq. (4.43) and the $u$ integral is taken along $\mathbb{R} - i0$. The multi-particle pentagon transition can be written concisely in terms of fundamental pentagon transitions [50, 78]

$$ P_{\phi|\phi\phi}(u|v_1, v_2) = \frac{P_{\phi\phi}(u|v_1) P_{\phi\phi}(u|v_2)}{P_{\phi\phi}(v_1|v_2)}, \quad (4.52) $$

and our conjecture (3.30) for the 2-scalar form factor reads

$$ F_{\phi\phi}(v_1, v_2) = \frac{1}{Q_{\phi\phi}(v_2|v_1)}, \quad (4.53) $$
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Leading flux-tube contribution to the form factor of $\mathcal{T}_4$. A scalar is produced by each pentagon. The two scalars are then absorbed by the operator. Only one scalar can be created by a single pentagon, due to the fact that the pentagons can only be charged under a chiral half of the R-symmetry group.</figcaption></figure>

## 4.5 Minimal four-point form factor

We conclude our perturbative checks with a brief discussion of the four-point form factor of $\mathcal{T}_4$. This is a minimal form factor with a single $\eta$ structure, namely, (123) (341). Stripping off the overall factor $\chi_b^1 \chi_b^2 \chi_t^1 \chi_t^2$, one gets for the ratio function,

$$ \mathcal{R}_{4,4} = \Omega_4 \times [1 - g^2 V'_0 + \mathcal{O}(g^4)], \quad (4.48) $$

where $V'_0$ denotes the finite part of $V_0$ in eq. (4.35) and with

$$ \Omega_4 = \sqrt{u_1 u_2 u_3 u_4}. \quad (4.49) $$

In the collinear limit, $\tau, \tau_b \to \infty$, one finds

$$ \Omega_4 = \frac{e^{-2\tau-\tau_b}}{(e^{\sigma} + e^{-\sigma})(e^{\sigma-\sigma_b} + e^{\sigma_b-\sigma} + e^{\sigma+\sigma_b})} + \mathcal{O}(e^{-2\tau-2\tau_b}, e^{-4\tau-\tau_b}), \quad (4.50) $$

using the OPE parametrization given in appendix B.3.

The leading OPE process on the flux-tube side is depicted in figure 12. It is described by a sequence of two $P_{12}$ pentagon transitions, producing a two-scalar state in the top OPE channel. The corresponding integral formula reads

$$ \mathcal{R}_{4,4} = g^{-6} \int \frac{du \, dv_1 \, dv_2}{2(2\pi)^3} \hat{\mu}_{\phi}(u) P_{\phi|\phi\phi}(u|v_1, v_2) F_{\phi\phi}(v_1, v_2) \tilde{\mu}_{\phi}(v_1) \tilde{\mu}_{\phi}(v_2) + \dots, \quad (4.51) $$

with the overall power of $g$ adjusted such that $\mathcal{R}_{4,4} = \mathcal{O}(g^0)$ at weak coupling. The measures are defined in eq. (4.43) and the $u$ integral is taken along $\mathbb{R} - i0$. The multi-particle pentagon transition can be written concisely in terms of fundamental pentagon transitions [50, 78]

$$ P_{\phi|\phi\phi}(u|v_1, v_2) = \frac{P_{\phi\phi}(u|v_1) P_{\phi\phi}(u|v_2)}{P_{\phi\phi}(v_1|v_2)}, \quad (4.52) $$

and our conjecture (3.30) for the 2-scalar form factor reads

$$ F_{\phi\phi}(v_1, v_2) = \frac{1}{Q_{\phi\phi}(v_2|v_1)}, \quad (4.53) $$
with $Q_{\phi\phi}$ the tilted transition at $\alpha = 0$, see eqs. (3.14) and (3.21). It yields

$$ \mathcal{R}_{4,4} = g^{-6} \int \frac{du \, dv_1 \, dv_2}{2(2\pi)^3} \hat{\mu}_{\phi}(u) \frac{P_{\phi\phi}(u|v_1) P_{\phi\phi}(u|v_2)}{P_{\phi\phi}(v_1|v_2) Q_{\phi\phi}(v_2|v_1)} \tilde{\mu}_{\phi}(v_1) \tilde{\mu}_{\phi}(v_2) + \dots, \quad (4.54) $$

with all ingredients determined to all loops using formulae in section 3.3.

This integral is significantly harder to evaluate than the previous ones. In order to simplify the analysis, we will consider it in the regime $\sigma_b, \sigma \gg 1$, corresponding to the double soft limit $p_2, p_3 \to 0$.<fnref n="11" /> In this regime, the measures in eq. (4.54) oscillate wildly and the integral is dominated by the singularities in the complex planes of $u$ and $v_{1,2}$. At weak coupling, through any loop order $L$, these singularities are poles located at $u = -i/2 - in$ and $v_{1,2} = i/2 + im_{1,2}$, with $n, m_{1,2} = 0, 1, 2, \dots$. They generate contributions that can be easily compared with the perturbative data (4.48); they are of the type

$$ \sum_{m_1+m_2=m} \mathrm{res}_{n,m_{1,2}} = e^{-2\tau-\tau_b-2(m+1)\sigma-(2n+1)\sigma_b} \times P_{m,n}^{(2L)}(\sigma, \sigma_b, \tau, \tau_b), \quad (4.55) $$

where $P_{n,m}^{(2L)}$ is a polynomial of degree $2L$ in the OPE spacetime variables. For illustration, after plugging inside eq. (4.54) the one-loop expressions for $P, Q, \mu, \nu$, together with the energy and momentum of a scalar, one finds

$$ \mathcal{R}_{4,4} = e^{-2\tau-\tau_b-2\sigma-\sigma_b} \left[ 1 + 2g^2(\zeta_2 + 2\sigma\sigma_b - 2\sigma\tau - 2\sigma_b\tau_b) + \mathcal{O}(g^4) \right] + \dots, \quad (4.56) $$

from the leading singularity, $n = m_{1,2} = 0$. It agrees perfectly with the Feynman diagrammatic formula (4.48) in the limit of interest. We proceeded similarly for a few higher residues, finding agreement in every cases. Finally, we performed checks at two loops using the symbol in refs. [31, 79].<fnref n="12" />

## 5 Conclusion

In this paper, we proposed a dual description for MHV form factors of half-BPS operators in terms of matrix elements of infinite periodic super Wilson loops with states composed of zero-momentum scalars. Based on this description, we generalized the OPE program, recently developed for the form factors of the stress-tensor multiplet, to the super form factors of half-BPS operators. The form factor transitions, which describe the absorption of the flux-tube state by the operator, were found to be controlled by the same building blocks that appeared previously in the case of the stress-tensor multiplet. We performed a series of checks of our construction at weak coupling, finding agreement for the three- and four-point form factors of $\mathcal{T}_3$ and the four-point form factor of $\mathcal{T}_4$. Higher-loop data may be easily produced, allowing one to set up a function bootstrap for the three-point form factor of $\mathcal{T}_3$ [72], along the lines of what has been done for $\mathcal{T}_2$ [25, 26, 28]. It would also be interesting to use the same methods for studying the four-point form factors of $\mathcal{T}_3$ and $\mathcal{T}_4$ at higher

<footnote n="11">¹¹In terms of the kinematic invariants (4.31), $u_4, u_1, u_2, v_4, v_1 \to 0$ and $u_3, v_2, v_3 \to 1$.</footnote>

<footnote n="12">¹²Note that the matching requires flipping the overall sign of the two-loop remainder function in eq. (5.9) of ref. [31], as was done in ref. [79].</footnote>
with $Q_{\phi\phi}$ the tilted transition at $\alpha = 0$, see eqs. (3.14) and (3.21). It yields

$$ \mathcal{R}_{4,4} = g^{-6} \int \frac{du \, dv_1 \, dv_2}{2(2\pi)^3} \hat{\mu}_{\phi}(u) \frac{P_{\phi\phi}(u|v_1) P_{\phi\phi}(u|v_2)}{P_{\phi\phi}(v_1|v_2) Q_{\phi\phi}(v_2|v_1)} \tilde{\mu}_{\phi}(v_1) \tilde{\mu}_{\phi}(v_2) + \dots, \quad (4.54) $$

with all ingredients determined to all loops using formulae in section 3.3.

This integral is significantly harder to evaluate than the previous ones. In order to simplify the analysis, we will consider it in the regime $\sigma_b, \sigma \gg 1$, corresponding to the double soft limit $p_2, p_3 \to 0$.<fnref n="11" /> In this regime, the measures in eq. (4.54) oscillate wildly and the integral is dominated by the singularities in the complex planes of $u$ and $v_{1,2}$. At weak coupling, through any loop order $L$, these singularities are poles located at $u = -i/2 - in$ and $v_{1,2} = i/2 + im_{1,2}$, with $n, m_{1,2} = 0, 1, 2, \dots$. They generate contributions that can be easily compared with the perturbative data (4.48); they are of the type

$$ \sum_{m_1+m_2=m} \mathrm{res}_{n,m_{1,2}} = e^{-2\tau-\tau_b-2(m+1)\sigma-(2n+1)\sigma_b} \times P_{m,n}^{(2L)}(\sigma, \sigma_b, \tau, \tau_b), \quad (4.55) $$

where $P_{n,m}^{(2L)}$ is a polynomial of degree $2L$ in the OPE spacetime variables. For illustration, after plugging inside eq. (4.54) the one-loop expressions for $P, Q, \mu, \nu$, together with the energy and momentum of a scalar, one finds

$$ \mathcal{R}_{4,4} = e^{-2\tau-\tau_b-2\sigma-\sigma_b} \left[ 1 + 2g^2(\zeta_2 + 2\sigma\sigma_b - 2\sigma\tau - 2\sigma_b\tau_b) + \mathcal{O}(g^4) \right] + \dots, \quad (4.56) $$

from the leading singularity, $n = m_{1,2} = 0$. It agrees perfectly with the Feynman diagrammatic formula (4.48) in the limit of interest. We proceeded similarly for a few higher residues, finding agreement in every cases. Finally, we performed checks at two loops using the symbol in refs. [31, 79].<fnref n="12" />

## 5 Conclusion

In this paper, we proposed a dual description for MHV form factors of half-BPS operators in terms of matrix elements of infinite periodic super Wilson loops with states composed of zero-momentum scalars. Based on this description, we generalized the OPE program, recently developed for the form factors of the stress-tensor multiplet, to the super form factors of half-BPS operators. The form factor transitions, which describe the absorption of the flux-tube state by the operator, were found to be controlled by the same building blocks that appeared previously in the case of the stress-tensor multiplet. We performed a series of checks of our construction at weak coupling, finding agreement for the three- and four-point form factors of $\mathcal{T}_3$ and the four-point form factor of $\mathcal{T}_4$. Higher-loop data may be easily produced, allowing one to set up a function bootstrap for the three-point form factor of $\mathcal{T}_3$ [72], along the lines of what has been done for $\mathcal{T}_2$ [25, 26, 28]. It would also be interesting to use the same methods for studying the four-point form factors of $\mathcal{T}_3$ and $\mathcal{T}_4$ at higher

<footnote n="11">¹¹In terms of the kinematic invariants (4.31), $u_4, u_1, u_2, v_4, v_1 \to 0$ and $u_3, v_2, v_3 \to 1$.</footnote>

<footnote n="12">¹²Note that the matching requires flipping the overall sign of the two-loop remainder function in eq. (5.9) of ref. [31], as was done in ref. [79].</footnote>
loops. This analysis could shed light on the origins of the recently discovered antipodal (self) duality [26, 27], currently only observed for the form factors of $T_2$.

While we have only considered the MHV case, it would be interesting to generalize our results to the $N^{k'}$MHV components. There is a natural proposal for how this generalization could work. The MHV super Wilson loop only depends on the $\eta^-$ components of the on-shell supermultiplets. At higher values of $k'$, $\eta^+$ components will enter in the construction. This implies that the super Wilson loop edge and vertex insertions, eqs. (2.27) and (2.29), must be promoted to their unrestricted values defined in refs. [9, 44, 45]. Recall, however, that the $\eta^+$ components are not periodic, but instead get shifted by $\gamma^+$, the supersymmetric analog of $q$: $\eta_{i+n}^+ = \eta_i^+ + \lambda_i \gamma^+$. While this feature may be swiftly accommodated in the periodic super Wilson loop set-up, see ref. [19], it is not immediately clear how to incorporate it into the OPE framework. At tree level, we expect the $m=2$ amplituhedron picture to be broken by the $N^{k'}$MHV corrections, since these ones depend on all components of the supertwistors, including the non-periodic ones. It would be interesting to study the tree-level form factors for all values of $k$ and $k'$ using techniques like the BCFW recursion relations [80, 81] or the MHV diagrams in momentum twistor space [82, 83], and compare the results with other twistorial approaches [84–86].

Lastly, it would be interesting to extend the Wilson loop and OPE descriptions to form factors of unprotected operators, studied, for instance, in refs. [79, 87–89]. Although the problem looks daunting at the level of the Wilson loop, one may hope that progress can be made on the OPE side, by combining the integrable description of the local operators with the Form Factor OPE. To do so, one could draw inspiration from the integral representations obtained for the correlation functions of local operators using the method of the Separation of Variables (SoV), see e.g. refs. [90–92], which share many similarities with the OPE formulae. They suggest that a general solution to the form factor axioms [22, 24] might be found in terms of the Baxter Q-functions describing the SoV wave functions of the operators. It would be fascinating to see if this analogy can be made precise and turned into an OPE-SoV framework for calculating form factors of generic operators. Reciprocally, it would be interesting to see if the tilted transitions, and notably the $\alpha=0$ one, play a role in the definition of the SoV measures at higher loops [92]. It would also be natural to seek connections with the correlation functions of large-spin operators, studied recently in refs. [93, 94].

## Acknowledgments

It is a pleasure to thank Lance Dixon, Lionel Mason, Amit Sever and Matthias Wilhelm for inspiring discussions and comments on the manuscript. We are also grateful to Amit Sever for several clarifications on periodic Wilson loops and to Lance Dixon for collaboration on a related project. AT was supported by the Institut Philippe Meyer at the Ecole Normale Supérieure in Paris.

## A IR divergences

To complete the equivalence between form factors and Wilson loops at one loop, we should address the few extra diagrams describing the interactions between the outgoing scalars. Examples of such diagrams are shown in figure 13. Their specificity is that they contain
loops. This analysis could shed light on the origins of the recently discovered antipodal (self) duality [26, 27], currently only observed for the form factors of $T_2$.

While we have only considered the MHV case, it would be interesting to generalize our results to the $N^{k'}$MHV components. There is a natural proposal for how this generalization could work. The MHV super Wilson loop only depends on the $\eta^-$ components of the on-shell supermultiplets. At higher values of $k'$, $\eta^+$ components will enter in the construction. This implies that the super Wilson loop edge and vertex insertions, eqs. (2.27) and (2.29), must be promoted to their unrestricted values defined in refs. [9, 44, 45]. Recall, however, that the $\eta^+$ components are not periodic, but instead get shifted by $\gamma^+$, the supersymmetric analog of $q$: $\eta_{i+n}^+ = \eta_i^+ + \lambda_i \gamma^+$. While this feature may be swiftly accommodated in the periodic super Wilson loop set-up, see ref. [19], it is not immediately clear how to incorporate it into the OPE framework. At tree level, we expect the $m=2$ amplituhedron picture to be broken by the $N^{k'}$MHV corrections, since these ones depend on all components of the supertwistors, including the non-periodic ones. It would be interesting to study the tree-level form factors for all values of $k$ and $k'$ using techniques like the BCFW recursion relations [80, 81] or the MHV diagrams in momentum twistor space [82, 83], and compare the results with other twistorial approaches [84–86].

Lastly, it would be interesting to extend the Wilson loop and OPE descriptions to form factors of unprotected operators, studied, for instance, in refs. [79, 87–89]. Although the problem looks daunting at the level of the Wilson loop, one may hope that progress can be made on the OPE side, by combining the integrable description of the local operators with the Form Factor OPE. To do so, one could draw inspiration from the integral representations obtained for the correlation functions of local operators using the method of the Separation of Variables (SoV), see e.g. refs. [90–92], which share many similarities with the OPE formulae. They suggest that a general solution to the form factor axioms [22, 24] might be found in terms of the Baxter Q-functions describing the SoV wave functions of the operators. It would be fascinating to see if this analogy can be made precise and turned into an OPE-SoV framework for calculating form factors of generic operators. Reciprocally, it would be interesting to see if the tilted transitions, and notably the $\alpha=0$ one, play a role in the definition of the SoV measures at higher loops [92]. It would also be natural to seek connections with the correlation functions of large-spin operators, studied recently in refs. [93, 94].

## Acknowledgments

It is a pleasure to thank Lance Dixon, Lionel Mason, Amit Sever and Matthias Wilhelm for inspiring discussions and comments on the manuscript. We are also grateful to Amit Sever for several clarifications on periodic Wilson loops and to Lance Dixon for collaboration on a related project. AT was supported by the Institut Philippe Meyer at the Ecole Normale Supérieure in Paris.

## A IR divergences

To complete the equivalence between form factors and Wilson loops at one loop, we should address the few extra diagrams describing the interactions between the outgoing scalars. Examples of such diagrams are shown in figure 13. Their specificity is that they contain
<figure><img src="image_15.png" /><figcaption>Figure 13. Diagrams renormalizing the outgoing scalars at one loop.</figcaption></figure>

IR divergences, which are dual to the UV divergences of the local operators. Since we are considering states dual to half-BPS operators, these extra diagrams should cancel out. We explain below how this is happening.

To handle the IR divergences, we will use dimensional regularization, $D = 4 - 2\epsilon$, and give small non-zero momenta, $P_1, \dots, P_{k-2}$, to all the outgoing scalars. These momenta should not be confused with the form-factor momenta, $p_1, \dots, p_n$. The latter are encoded in the contour of the Wilson loop, whereas the former are just regulators for the sate. A proper treatment of the self-energy diagrams also requires working with off-shell particles. Hence, we will take $P_i^2 \neq 0$ and consider the limit

$$|P_i^2| \ll |P_i \cdot P_j| \to 0. \qquad (A.1)$$

We also assume $Pq = 0$, where $P = \sum_{i=1}^{k-2} P_i$, for consistency with the periodicity condition.

Essentially, all the diagrams in figure 13 can be accounted for by considering a single simple building block, which consists of two scalars inserted at points $X_1$ and $X_2$, with the segment of the Wilson loop between them. This object can be calculated as a Sudakov form factor of a bilocal operator,

$$\langle \phi(P_1)\phi(P_2)|\bar{\phi}(X_1)[X_1, X_2]\bar{\phi}(X_2)|0\rangle = e^{i\sum_{j=1,2} P_j X_j} \times F, \qquad (A.2)$$

with $\phi = \phi^{12}$, $\bar{\phi} = \bar{\phi}_{12}$, and with $[X_1, X_2]$ denoting the Wilson line between $\phi(X_1)$ and $\phi(X_2)$. Our normalization is such that $F = 1$ at tree level. The scalar fields can be inserted anywhere along the loop, at the cusps or on the edges, but they are assumed to be next to each other as far as the color ordering is concerned.<fnref n="13" />

To avoid double counting, we mod out by the diagrams connecting the Wilson line to itself. Hence, the only diagrams in which the Wilson line plays a role are the ones in which a gluon connects it to an outgoing particle, as shown in the left panel of figure 13. We get two such diagrams, for each particle in (A.2). Importantly, in the limit, $P_{1,2} \to 0$, these diagrams do not depend on the contour. They only depend on the distance $\Delta X = X_1 - X_2$ and read

$$F_{\text{W-line}} = I(\Delta X) - I(0) + \mathcal{O}(P_i), \qquad (A.3)$$

where $I$ are the so-called bubble integrals, with two propagators,

$$I(\Delta X) = \frac{i\lambda}{2} \int \frac{d^D\ell}{(2\pi)^D} \frac{e^{i\ell\Delta X}}{(P_1 + \ell)^2 \ell^2} + \frac{i\lambda}{2} \int \frac{d^D\ell}{(2\pi)^D} \frac{e^{i\ell\Delta X}}{\ell^2 (P_2 - \ell)^2}. \qquad (A.4)$$

<footnote n="13">¹³When $\phi(X_i)$ is on an edge, one should take a derivative $\partial_{X_i}$ of the form factor. However, this distinction proves immaterial, since, as we will see, the one-loop correction to form factor (A.2) vanishes for any $X_1, X_2$ when $P_{1,2} \to 0$.</footnote>
<figure><img src="image_15.png" /><figcaption>Figure 13. Diagrams renormalizing the outgoing scalars at one loop.</figcaption></figure>

IR divergences, which are dual to the UV divergences of the local operators. Since we are considering states dual to half-BPS operators, these extra diagrams should cancel out. We explain below how this is happening.

To handle the IR divergences, we will use dimensional regularization, $D = 4 - 2\epsilon$, and give small non-zero momenta, $P_1, \dots, P_{k-2}$, to all the outgoing scalars. These momenta should not be confused with the form-factor momenta, $p_1, \dots, p_n$. The latter are encoded in the contour of the Wilson loop, whereas the former are just regulators for the sate. A proper treatment of the self-energy diagrams also requires working with off-shell particles. Hence, we will take $P_i^2 \neq 0$ and consider the limit

$$|P_i^2| \ll |P_i \cdot P_j| \to 0. \qquad (A.1)$$

We also assume $Pq = 0$, where $P = \sum_{i=1}^{k-2} P_i$, for consistency with the periodicity condition.

Essentially, all the diagrams in figure 13 can be accounted for by considering a single simple building block, which consists of two scalars inserted at points $X_1$ and $X_2$, with the segment of the Wilson loop between them. This object can be calculated as a Sudakov form factor of a bilocal operator,

$$\langle \phi(P_1)\phi(P_2)|\bar{\phi}(X_1)[X_1, X_2]\bar{\phi}(X_2)|0\rangle = e^{i\sum_{j=1,2} P_j X_j} \times F, \qquad (A.2)$$

with $\phi = \phi^{12}$, $\bar{\phi} = \bar{\phi}_{12}$, and with $[X_1, X_2]$ denoting the Wilson line between $\phi(X_1)$ and $\phi(X_2)$. Our normalization is such that $F = 1$ at tree level. The scalar fields can be inserted anywhere along the loop, at the cusps or on the edges, but they are assumed to be next to each other as far as the color ordering is concerned.<fnref n="13" />

To avoid double counting, we mod out by the diagrams connecting the Wilson line to itself. Hence, the only diagrams in which the Wilson line plays a role are the ones in which a gluon connects it to an outgoing particle, as shown in the left panel of figure 13. We get two such diagrams, for each particle in (A.2). Importantly, in the limit, $P_{1,2} \to 0$, these diagrams do not depend on the contour. They only depend on the distance $\Delta X = X_1 - X_2$ and read

$$F_{\text{W-line}} = I(\Delta X) - I(0) + \mathcal{O}(P_i), \qquad (A.3)$$

where $I$ are the so-called bubble integrals, with two propagators,

$$I(\Delta X) = \frac{i\lambda}{2} \int \frac{d^D\ell}{(2\pi)^D} \frac{e^{i\ell\Delta X}}{(P_1 + \ell)^2 \ell^2} + \frac{i\lambda}{2} \int \frac{d^D\ell}{(2\pi)^D} \frac{e^{i\ell\Delta X}}{\ell^2 (P_2 - \ell)^2}. \qquad (A.4)$$

<footnote n="13">¹³When $\phi(X_i)$ is on an edge, one should take a derivative $\partial_{X_i}$ of the form factor. However, this distinction proves immaterial, since, as we will see, the one-loop correction to form factor (A.2) vanishes for any $X_1, X_2$ when $P_{1,2} \to 0$.</footnote>
The two other diagrams in figure 13 stand for the scalar quartic interactions,

$$F_{\text{quartic}} = -\frac{i\lambda}{2} \int \frac{d^D\ell}{(2\pi)^D} \frac{e^{i\ell\Delta X}}{(P_1+\ell)^2 (P_2-\ell)^2}, \qquad (A.5)$$

and the gluon exchange,

$$F_{\text{gluon}} = \frac{i\lambda}{2} \int \frac{d^D\ell}{(2\pi)^D} \frac{(2P_1+\ell)(2P_2-\ell)}{(P_1+\ell)^2 \ell^2 (P_2-\ell)^2} e^{i\ell\Delta X}. \qquad (A.6)$$

Lastly, we have the self-energy diagrams shown in the top row of figure 14. We should include half the self energy for each leg, giving<fnref n="14" />

$$F_{\text{self}} = I(0). \qquad (A.7)$$

A great simplification occurs when we put all these integrals together. Namely, the bubble integrals cancel out. For instance, the integral $I(0)$ in eq. (A.3) is cancelled by the self energy. To eliminate the rest, one should massage the numerator in eq. (A.6), using

$$(2P_1 + \ell)(2P_2 - \ell) = 4P_1P_2 + P_1^2 + P_2^2 - (P_1 + \ell)^2 - (P_2 - \ell)^2 + \ell^2, \qquad (A.8)$$

and observe that the last three terms in this equation remove $I(\Delta X)$ and $F_{\text{quartic}}$. As a consequence, the form factor is proportional to a triangle integral, with three propagators,

$$F^{1-\text{loop}} = \frac{i\lambda}{2} \int \frac{d^D\ell}{(4\pi)^D} \frac{4P_1P_2 + P_1^2 + P_2^2}{(P_1+\ell)^2 \ell^2 (P_2-\ell)^2} e^{i\ell\Delta X}. \qquad (A.9)$$

It is then straightforward to take the zero-momentum limit. Continuing the integral to $D = 4 - 2\epsilon$, we bring the limit $P_i \to 0$ below the integral sign and conclude that

$$\lim_{P_{1,2} \to 0} F^{1-\text{loop}} = 0, \qquad (A.10)$$

no matter the distance $\Delta X$.

Let us stress that this cancellation heavily depends on the vanishing of the bubble integrals. Unlike the integral (A.9), these integrals are logarithmically divergent and stay non-trivial when $P_i \to 0$ and $(\Delta X)^2 \neq 0$. E.g.,

$$\lim_{P_i \to 0} I(\Delta X) = i\lambda \int \frac{d^D\ell}{(2\pi)^D} \frac{e^{i\ell\Delta X}}{(\ell^2)^2} = g^2 \frac{[-(\Delta X)^2]^\epsilon}{\epsilon} \pi^\epsilon \Gamma(1-\epsilon), \qquad (A.11)$$

with $g^2 = \lambda/(4\pi)^2$, and similarly for $F_{\text{scalar}}$. These integrals drop out in our case because we are considering symmetric scalar states, dual to half-BPS local operators. For scalar states in other representations, extra bubbles will contribute to $F_{\text{quartic}}$ and the cancellation will be incomplete. This pattern is reminiscent of the one found for the UV divergences of local scalar operators [95] and it would be very interesting to perform a careful comparison of the two analyses for generic scalar states/operators. It may give important clues for extending the state-operator dictionary to non-BPS operators.

<footnote n="14">¹⁴In fact, only a quarter of the self energy is being used here, since the self-energy diagrams are shared between two planar form factors.</footnote>
The two other diagrams in figure 13 stand for the scalar quartic interactions,

$$F_{\text{quartic}} = -\frac{i\lambda}{2} \int \frac{d^D\ell}{(2\pi)^D} \frac{e^{i\ell\Delta X}}{(P_1+\ell)^2 (P_2-\ell)^2}, \qquad (A.5)$$

and the gluon exchange,

$$F_{\text{gluon}} = \frac{i\lambda}{2} \int \frac{d^D\ell}{(2\pi)^D} \frac{(2P_1+\ell)(2P_2-\ell)}{(P_1+\ell)^2 \ell^2 (P_2-\ell)^2} e^{i\ell\Delta X}. \qquad (A.6)$$

Lastly, we have the self-energy diagrams shown in the top row of figure 14. We should include half the self energy for each leg, giving<fnref n="14" />

$$F_{\text{self}} = I(0). \qquad (A.7)$$

A great simplification occurs when we put all these integrals together. Namely, the bubble integrals cancel out. For instance, the integral $I(0)$ in eq. (A.3) is cancelled by the self energy. To eliminate the rest, one should massage the numerator in eq. (A.6), using

$$(2P_1 + \ell)(2P_2 - \ell) = 4P_1P_2 + P_1^2 + P_2^2 - (P_1 + \ell)^2 - (P_2 - \ell)^2 + \ell^2, \qquad (A.8)$$

and observe that the last three terms in this equation remove $I(\Delta X)$ and $F_{\text{quartic}}$. As a consequence, the form factor is proportional to a triangle integral, with three propagators,

$$F^{1-\text{loop}} = \frac{i\lambda}{2} \int \frac{d^D\ell}{(4\pi)^D} \frac{4P_1P_2 + P_1^2 + P_2^2}{(P_1+\ell)^2 \ell^2 (P_2-\ell)^2} e^{i\ell\Delta X}. \qquad (A.9)$$

It is then straightforward to take the zero-momentum limit. Continuing the integral to $D = 4 - 2\epsilon$, we bring the limit $P_i \to 0$ below the integral sign and conclude that

$$\lim_{P_{1,2} \to 0} F^{1-\text{loop}} = 0, \qquad (A.10)$$

no matter the distance $\Delta X$.

Let us stress that this cancellation heavily depends on the vanishing of the bubble integrals. Unlike the integral (A.9), these integrals are logarithmically divergent and stay non-trivial when $P_i \to 0$ and $(\Delta X)^2 \neq 0$. E.g.,

$$\lim_{P_i \to 0} I(\Delta X) = i\lambda \int \frac{d^D\ell}{(2\pi)^D} \frac{e^{i\ell\Delta X}}{(\ell^2)^2} = g^2 \frac{[-(\Delta X)^2]^\epsilon}{\epsilon} \pi^\epsilon \Gamma(1-\epsilon), \qquad (A.11)$$

with $g^2 = \lambda/(4\pi)^2$, and similarly for $F_{\text{scalar}}$. These integrals drop out in our case because we are considering symmetric scalar states, dual to half-BPS local operators. For scalar states in other representations, extra bubbles will contribute to $F_{\text{quartic}}$ and the cancellation will be incomplete. This pattern is reminiscent of the one found for the UV divergences of local scalar operators [95] and it would be very interesting to perform a careful comparison of the two analyses for generic scalar states/operators. It may give important clues for extending the state-operator dictionary to non-BPS operators.

<footnote n="14">¹⁴In fact, only a quarter of the self energy is being used here, since the self-energy diagrams are shared between two planar form factors.</footnote>
<figure><img src="image_16.png" /><figcaption>**Figure 14.** Gluon, fermion and scalar contributions to the self energy of a scalar. (The gluon contribution also contains a quartic diagram not represented here.) Because the diagrams in the bottom row wrap around the cylinder, they are exclusive to the $k = 3$ case.</figcaption></figure>

Our analysis above may appear a bit fast given that the integral in eq. (A.9) also contains infrared divergences in $D = 4$ when the momenta are small. In fact, when $P_i^2 = 0$ and $\Delta X = 0$, our integral coincides with the traditional Sudakov form factor, with well-known double pole,<fnref n="15" />

$$F^{1-\text{loop}} = -g^2 \frac{(-2P_1 P_2)^{-\epsilon}}{\epsilon^2}. \qquad (A.12)$$

Consistency with (A.10) requires that the limit $P_i \to 0$ is taken for $\epsilon < 0$, that is, $D > 4$. With this choice, the interactions die off at large distances and with them all the dangerous terms at small $P$.

There may still be a pitfall in analyzing divergences this way at higher loops. The problem is that the condition $\epsilon < 0$ is *opposite* to the one used to regularize the cusp divergences of the Wilson loop. The latter give rise to contributions $\sim 0^\epsilon$ which are hard, if not impossible, to interpret if $\epsilon < 0$. This dual use of the dimensional regulator may be an issue for diagrams containing both UV and IR divergences, for which it may not be obvious which condition to use and how. In such cases, one may need to separate the problematic regions more carefully. However, at one loop, we see nothing wrong in using different $\epsilon$'s for different diagrams, depending on the nature of their divergences.

The above analysis explains the vanishing of the extra diagrams for generic $k$. However, it does not extend to $k = 3$, where there is only a single scalar in the state. In this case, the relevant form factor is

$$\langle \phi(P) | \text{Tr} \bar{\phi}(X) [X, X+q] | 0 \rangle = e^{iPX} F(P, q), \qquad (A.13)$$

<footnote n="15">¹⁵Up to an irrelevant factor $\tilde{c} = (4\pi)^\epsilon \Gamma(1+\epsilon) \Gamma(1-\epsilon)^2 / \Gamma(1-2\epsilon) = 1 + \mathcal{O}(\epsilon)$.</footnote>
<figure><img src="image_16.png" /><figcaption>**Figure 14.** Gluon, fermion and scalar contributions to the self energy of a scalar. (The gluon contribution also contains a quartic diagram not represented here.) Because the diagrams in the bottom row wrap around the cylinder, they are exclusive to the $k = 3$ case.</figcaption></figure>

Our analysis above may appear a bit fast given that the integral in eq. (A.9) also contains infrared divergences in $D = 4$ when the momenta are small. In fact, when $P_i^2 = 0$ and $\Delta X = 0$, our integral coincides with the traditional Sudakov form factor, with well-known double pole,<fnref n="15" />

$$F^{1-\text{loop}} = -g^2 \frac{(-2P_1 P_2)^{-\epsilon}}{\epsilon^2}. \qquad (\text{A.12})$$

Consistency with (A.10) requires that the limit $P_i \to 0$ is taken for $\epsilon < 0$, that is, $D > 4$. With this choice, the interactions die off at large distances and with them all the dangerous terms at small $P$.

There may still be a pitfall in analyzing divergences this way at higher loops. The problem is that the condition $\epsilon < 0$ is *opposite* to the one used to regularize the cusp divergences of the Wilson loop. The latter give rise to contributions $\sim 0^\epsilon$ which are hard, if not impossible, to interpret if $\epsilon < 0$. This dual use of the dimensional regulator may be an issue for diagrams containing both UV and IR divergences, for which it may not be obvious which condition to use and how. In such cases, one may need to separate the problematic regions more carefully. However, at one loop, we see nothing wrong in using different $\epsilon$'s for different diagrams, depending on the nature of their divergences.

The above analysis explains the vanishing of the extra diagrams for generic $k$. However, it does not extend to $k = 3$, where there is only a single scalar in the state. In this case, the relevant form factor is

$$\langle \phi(P) | \text{Tr} \bar{\phi}(X) [X, X+q] | 0 \rangle = e^{iPX} F(P, q), \qquad (\text{A.13})$$

<footnote n="15">¹⁵Up to an irrelevant factor $\tilde{c} = (4\pi)^\epsilon \Gamma(1+\epsilon) \Gamma(1-\epsilon)^2 / \Gamma(1-2\epsilon) = 1 + \mathcal{O}(\epsilon)$.</footnote>
<figure><img src="image_17.png" /><figcaption>Figure 15. OPE parametrizations of two-, three-, and four-point form factors. The blue square is the reference square. The red square is deformed with an arbitrary group element that leaves the blue square intact, while the green square is deformed by both the symmetries of the blue and the red squares.</figcaption></figure>

with $\bar{\phi}(X)$ connected to its image at $X + q$ through a Wilson line 'going around the world'. As before, this line generates the interaction

$$F_{\text{W-line}}(P, q) = I(q) - I(0), \qquad (A.14)$$

at low momentum, with $I$ given by eq. (A.4) with $P_1 = P_2 = P$. The self energy still takes care of the $I(0)$ part but, in the lack of additional diagrams, it may not be clear how to cancel $I(q)$. The resolution is that for a single particle we must also include *wrapping diagrams* with virtual particles traveling around the period $q$. These diagrams produce double-trace corrections for $k > 3$, which are subleading at large $N$. However, the very same diagrams become planar for a single outgoing scalar. Their structure mirrors the one for the self energy, and for each contribution to the usual self energy we should include a diagram in which the virtual particle is wrapping around the period, as shown in figure 14. There is also a minus sign coming from the different color ordering. Adding up all diagrams, one finds

$$F_{\text{self}} \rightarrow I(0) - I(q), \qquad (A.15)$$

which perfectly cancels the Wilson loop contribution (A.14).

## B Twistor parametrization

In this appendix we give explicit parametrizations of the momentum twistors in terms of the OPE variables in two-, three- and four-point cases. The geometry of the corresponding Wilson loop configurations is presented in figure 15.

### B.1 Two-point form factor

The centerpiece of the construction is the reference square, formed by four twistors that can be arbitrarily chosen to have the following form,

$$ \begin{aligned} Z_{\text{left}} &= (0, 0, 0, 1), & Z_{\text{right}} &= (0, 0, 1, 0), \\ Z_{\text{top}} &= (1, 0, 0, 0), & Z_{\text{bot}} &= (0, 1, 0, 0). \end{aligned} \qquad (B.1) $$
<figure><img src="image_17.png" /><figcaption>Figure 15. OPE parametrizations of two-, three-, and four-point form factors. The blue square is the reference square. The red square is deformed with an arbitrary group element that leaves the blue square intact, while the green square is deformed by both the symmetries of the blue and the red squares.</figcaption></figure>

with $\bar{\phi}(X)$ connected to its image at $X + q$ through a Wilson line 'going around the world'. As before, this line generates the interaction

$$F_{\text{W-line}}(P, q) = I(q) - I(0), \qquad (A.14)$$

at low momentum, with $I$ given by eq. (A.4) with $P_1 = P_2 = P$. The self energy still takes care of the $I(0)$ part but, in the lack of additional diagrams, it may not be clear how to cancel $I(q)$. The resolution is that for a single particle we must also include *wrapping diagrams* with virtual particles traveling around the period $q$. These diagrams produce double-trace corrections for $k > 3$, which are subleading at large $N$. However, the very same diagrams become planar for a single outgoing scalar. Their structure mirrors the one for the self energy, and for each contribution to the usual self energy we should include a diagram in which the virtual particle is wrapping around the period, as shown in figure 14. There is also a minus sign coming from the different color ordering. Adding up all diagrams, one finds

$$F_{\text{self}} \rightarrow I(0) - I(q), \qquad (A.15)$$

which perfectly cancels the Wilson loop contribution (A.14).

## B Twistor parametrization

In this appendix we give explicit parametrizations of the momentum twistors in terms of the OPE variables in two-, three- and four-point cases. The geometry of the corresponding Wilson loop configurations is presented in figure 15.

### B.1 Two-point form factor

The centerpiece of the construction is the reference square, formed by four twistors that can be arbitrarily chosen to have the following form,

$$ \begin{aligned} Z_{\text{left}} &= (0, 0, 0, 1), & Z_{\text{right}} &= (0, 0, 1, 0), \\ Z_{\text{top}} &= (1, 0, 0, 0), & Z_{\text{bot}} &= (0, 1, 0, 0). \end{aligned} \qquad (B.1) $$
One segment of the Wilson loop is formed by the $Z_{\text{bot}}$ and $Z_{\text{right}}$ twistors, while twistors in the subsequent periods can be obtained by acting with the periodic shift operator $P$: $Z_i^{[n]} = P^n \cdot Z_i$. This, in particular, implies that

$$Z_{\text{right}} = P \cdot Z_{\text{left}}, \quad Z_{\text{top}} = P \cdot Z_{\text{bot}}. \qquad (B.2)$$

One particular choice of $P$ that is consistent with this constraint is

$$P = \begin{pmatrix} 2 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & -1 & 0 \end{pmatrix}. \qquad (B.3)$$

The two-dimensional space of twistors invariant under the action of $P$ represents the point at infinity and, therefore, enters the definition of the two-bracket,

$$P \cdot \mathcal{I}_i = \mathcal{I}_i, \quad \langle ij \rangle = \langle \mathcal{I}_1 \mathcal{I}_2 ij \rangle. \qquad (B.4)$$

Explicitly,

$$\mathcal{I}_1 = (1, -1, 0, 0), \quad \mathcal{I}_2 = (0, 0, 1, -1). \qquad (B.5)$$

Under this definition, the two-bracket is manifestly shift invariant: $\langle i^{[n]}j^{[m]} \rangle = \langle ij \rangle$.

## B.2 Three-point form factor

The three conformal symmetries that leave the reference square (B.1) invariant can be encoded into the following matrix

$$M = \begin{pmatrix} e^{-\tau + \frac{i\phi}{2}} & 0 & 0 & 0 \\ 0 & e^{\tau + \frac{i\phi}{2}} & 0 & 0 \\ 0 & 0 & e^{\sigma - \frac{i\phi}{2}} & 0 \\ 0 & 0 & 0 & e^{-\sigma - \frac{i\phi}{2}} \end{pmatrix}. \qquad (B.6)$$

In order to obtain the most general three-point configuration, one has to deform the twistors of the red square in figure 15 with this matrix. We arrive at the following expressions for the twistors that form the Wilson loop,

$$Z_1 = M \cdot (0, 1, 0, 1), \quad Z_2 = M \cdot (1, 1, -1, 0), \quad Z_3 = (0, 0, 1, 0), \qquad (B.7)$$

while the internal twistor $Z_{\text{mid}}$ is simply equal to $Z_{\text{bot}}$.

The three OPE variables that parametrize $M$ can be recast in the form that is common in the amplitudes literature,

$$\begin{align} u_1 &= \frac{\langle 1231^+ \rangle \langle 31 \rangle}{\langle 3^- 131^+ \rangle \langle 12 \rangle} = \frac{1}{1 + e^{2\sigma} + e^{-2\tau}}, \notag \\ u_2 &= \frac{\langle 231^+ 2^+ \rangle \langle 31 \rangle^2}{\langle 3^- 131^+ \rangle \langle 12 \rangle \langle 23 \rangle} = \frac{e^{2\sigma}}{(1 + e^{-2\tau})(1 + e^{2\sigma} + e^{-2\tau})}, \tag{B.8} \\ u_3 &= \frac{\langle 31^+ 2^+ 3^+ \rangle \langle 31 \rangle}{\langle 3^- 131^+ \rangle \langle 23 \rangle} = \frac{1}{1 + e^{2\tau}}. \notag \end{align}$$

As expected, the dependence on the angular variable $\phi$ does not enter any of the cross-ratios.
One segment of the Wilson loop is formed by the $Z_{\text{bot}}$ and $Z_{\text{right}}$ twistors, while twistors in the subsequent periods can be obtained by acting with the periodic shift operator $P$: $Z_i^{[n]} = P^n \cdot Z_i$. This, in particular, implies that

$$Z_{\text{right}} = P \cdot Z_{\text{left}}, \quad Z_{\text{top}} = P \cdot Z_{\text{bot}}. \qquad (B.2)$$

One particular choice of $P$ that is consistent with this constraint is

$$P = \begin{pmatrix} 2 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & -1 & 0 \end{pmatrix}. \qquad (B.3)$$

The two-dimensional space of twistors invariant under the action of $P$ represents the point at infinity and, therefore, enters the definition of the two-bracket,

$$P \cdot \mathcal{I}_i = \mathcal{I}_i, \quad \langle ij \rangle = \langle \mathcal{I}_1 \mathcal{I}_2 ij \rangle. \qquad (B.4)$$

Explicitly,

$$\mathcal{I}_1 = (1, -1, 0, 0), \quad \mathcal{I}_2 = (0, 0, 1, -1). \qquad (B.5)$$

Under this definition, the two-bracket is manifestly shift invariant: $\langle i^{[n]}j^{[m]} \rangle = \langle ij \rangle$.

## B.2 Three-point form factor

The three conformal symmetries that leave the reference square (B.1) invariant can be encoded into the following matrix

$$M = \begin{pmatrix} e^{-\tau + \frac{i\phi}{2}} & 0 & 0 & 0 \\ 0 & e^{\tau + \frac{i\phi}{2}} & 0 & 0 \\ 0 & 0 & e^{\sigma - \frac{i\phi}{2}} & 0 \\ 0 & 0 & 0 & e^{-\sigma - \frac{i\phi}{2}} \end{pmatrix}. \qquad (B.6)$$

In order to obtain the most general three-point configuration, one has to deform the twistors of the red square in figure 15 with this matrix. We arrive at the following expressions for the twistors that form the Wilson loop,

$$Z_1 = M \cdot (0, 1, 0, 1), \quad Z_2 = M \cdot (1, 1, -1, 0), \quad Z_3 = (0, 0, 1, 0), \qquad (B.7)$$

while the internal twistor $Z_{\text{mid}}$ is simply equal to $Z_{\text{bot}}$.

The three OPE variables that parametrize $M$ can be recast in the form that is common in the amplitudes literature,

$$\begin{align}
u_1 &= \frac{\langle 1231^+ \rangle \langle 31 \rangle}{\langle 3^- 131^+ \rangle \langle 12 \rangle} = \frac{1}{1 + e^{2\sigma} + e^{-2\tau}}, \nonumber \\
u_2 &= \frac{\langle 231^+ 2^+ \rangle \langle 31 \rangle^2}{\langle 3^- 131^+ \rangle \langle 12 \rangle \langle 23 \rangle} = \frac{e^{2\sigma}}{(1 + e^{-2\tau})(1 + e^{2\sigma} + e^{-2\tau})}, \nonumber \\
u_3 &= \frac{\langle 31^+ 2^+ 3^+ \rangle \langle 31 \rangle}{\langle 3^- 131^+ \rangle \langle 23 \rangle} = \frac{1}{1 + e^{2\tau}}. \tag{B.8}
\end{align}$$

As expected, the dependence on the angular variable $\phi$ does not enter any of the cross-ratios.
## B.3 Four-point form factor

In the four-point case, we require an additional matrix that encodes the symmetries of the red square,

$$ M_b = \begin{pmatrix} e^{\tau_b + \frac{i\phi_b}{2}} & 0 & 0 & 0 \\ e^{\tau_b + \frac{i\phi_b}{2}} - e^{-\tau_b + \frac{i\phi_b}{2}} & e^{-\tau_b + \frac{i\phi_b}{2}} & 0 & e^{\sigma_b - \frac{i\phi_b}{2}} - e^{-\tau_b + \frac{i\phi_b}{2}} \\ e^{-\sigma_b - \frac{i\phi_b}{2}} - e^{\tau_b + \frac{i\phi_b}{2}} & 0 & e^{-\sigma_b - \frac{i\phi_b}{2}} & 0 \\ 0 & 0 & 0 & e^{\sigma_b - \frac{i\phi_b}{2}} \end{pmatrix}. \quad (B.9) $$

The Wilson line is formed by the following four twistors and their periodic images,

$$ \begin{aligned} Z_1 &= M \cdot (0, 1, 0, 1), & Z_2 &= M \cdot M_b \cdot (1, 3, -1, 1), \\ Z_3 &= M \cdot M_b \cdot (1, 1, -2, 0), & Z_4 &= (0, 0, 1, 0), \end{aligned} \quad (B.10) $$

while the twistors in the middle, which separate the squares, are given by

$$ Z_{\text{upper mid}} = (0, 1, 0, 0), \quad Z_{\text{lower mid}} = M \cdot (1, 1, -1, 0). \quad (B.11) $$

Given this parametrization we can introduce a set of kinematical variables $u_i, v_i$,

$$ \begin{aligned} u_1 &= \frac{\langle 1234 \rangle \langle 41 \rangle^2}{\langle 4^{-}141^{+} \rangle \langle 12 \rangle \langle 34 \rangle} = \frac{e^{-2\tau_b}}{(1 + e^{2\tau})(1 + e^{2\sigma} + e^{-2\tau} + e^{-2\tau_b})}, \\ u_2 &= \frac{\langle 2341^{+} \rangle \langle 41 \rangle}{\langle 4^{-}141^{+} \rangle \langle 23 \rangle} \\ &= [1 + e^{-2\tau} + e^{2(\sigma-\sigma_b)} (1 + e^{2\sigma_b} + e^{-2\tau} + e^{-2\tau_b} + 2e^{\sigma_b-\tau_b} \cos \phi_b)]^{-1}, \quad (B.12) \end{aligned} $$

$$ u_3 = \frac{\langle 341^{+}2^{+} \rangle \langle 41 \rangle^2}{\langle 4^{-}141^{+} \rangle \langle 12 \rangle \langle 34 \rangle} = e^{2(\sigma+\tau+\tau_b)} u_1, $$

$$ u_4 = \frac{\langle 41^{+}2^{+}3^{+} \rangle \langle 41 \rangle}{\langle 4^{-}141^{+} \rangle \langle 23 \rangle} = e^{2(\sigma-\sigma_b-\tau)} u_2, $$

as well as

$$ \begin{aligned} v_1 &= \frac{\langle 1241^{+} \rangle \langle 41 \rangle}{\langle 4^{-}141^{+} \rangle \langle 12 \rangle} = 1 + u_1 - u_3 - v_4, \\ v_2 &= \frac{\langle 231^{+}2^{+} \rangle \langle 41 \rangle}{\langle 4^{-}141^{+} \rangle \langle 23 \rangle} = 1 + u_2 - u_4 - v_1, \\ v_3 &= \frac{\langle 342^{+}3^{+} \rangle \langle 41 \rangle^2}{\langle 4^{-}141^{+} \rangle \langle 23 \rangle \langle 34 \rangle} = 1 + u_3 - u_1 - v_2, \end{aligned} \quad (B.13) $$

$$ v_4 = \frac{\langle 41^{+}3^{+}4^{+} \rangle \langle 41 \rangle}{\langle 4^{-}141^{+} \rangle \langle 34 \rangle} = \frac{1}{1 + e^{2\tau}}. $$

## C Mirror and crossing

In this appendix, we summarize the properties of the tilted functions $f_{a,s}^{[\alpha]}$, defined in (3.12), under the mirror transformations and illustrate how they translate into the crossing properties of the tilted transitions. Here, we closely follow similar analyses performed in refs. [24, 68].
## B.3 Four-point form factor

In the four-point case, we require an additional matrix that encodes the symmetries of the red square,

$$ M_b = \begin{pmatrix} e^{\tau_b + \frac{i\phi_b}{2}} & 0 & 0 & 0 \\ e^{\tau_b + \frac{i\phi_b}{2}} - e^{-\tau_b + \frac{i\phi_b}{2}} & e^{-\tau_b + \frac{i\phi_b}{2}} & 0 & e^{\sigma_b - \frac{i\phi_b}{2}} - e^{-\tau_b + \frac{i\phi_b}{2}} \\ e^{-\sigma_b - \frac{i\phi_b}{2}} - e^{\tau_b + \frac{i\phi_b}{2}} & 0 & e^{-\sigma_b - \frac{i\phi_b}{2}} & 0 \\ 0 & 0 & 0 & e^{\sigma_b - \frac{i\phi_b}{2}} \end{pmatrix}. \quad (B.9) $$

The Wilson line is formed by the following four twistors and their periodic images,

$$ \begin{aligned} Z_1 &= M \cdot (0, 1, 0, 1), & Z_2 &= M \cdot M_b \cdot (1, 3, -1, 1), \\ Z_3 &= M \cdot M_b \cdot (1, 1, -2, 0), & Z_4 &= (0, 0, 1, 0), \end{aligned} \quad (B.10) $$

while the twistors in the middle, which separate the squares, are given by

$$ Z_{\text{upper mid}} = (0, 1, 0, 0), \quad Z_{\text{lower mid}} = M \cdot (1, 1, -1, 0). \quad (B.11) $$

Given this parametization we can introduce a set of kinematical variables $u_i, v_i$,

$$ \begin{aligned} u_1 &= \frac{\langle 1234 \rangle \langle 41 \rangle^2}{\langle 4^{-}141^{+} \rangle \langle 12 \rangle \langle 34 \rangle} = \frac{e^{-2\tau_b}}{(1 + e^{2\tau})(1 + e^{2\sigma} + e^{-2\tau} + e^{-2\tau_b})}, \\ u_2 &= \frac{\langle 2341^{+} \rangle \langle 41 \rangle}{\langle 4^{-}141^{+} \rangle \langle 23 \rangle} \\ &= [1 + e^{-2\tau} + e^{2(\sigma-\sigma_b)} (1 + e^{2\sigma_b} + e^{-2\tau} + e^{-2\tau_b} + 2e^{\sigma_b-\tau_b} \cos \phi_b)]^{-1}, \quad (B.12) \end{aligned} $$

$$ u_3 = \frac{\langle 341^{+}2^{+} \rangle \langle 41 \rangle^2}{\langle 4^{-}141^{+} \rangle \langle 12 \rangle \langle 34 \rangle} = e^{2(\sigma+\tau+\tau_b)} u_1, $$

$$ u_4 = \frac{\langle 41^{+}2^{+}3^{+} \rangle \langle 41 \rangle}{\langle 4^{-}141^{+} \rangle \langle 23 \rangle} = e^{2(\sigma-\sigma_b-\tau)} u_2, $$

as well as

$$ \begin{aligned} v_1 &= \frac{\langle 1241^{+} \rangle \langle 41 \rangle}{\langle 4^{-}141^{+} \rangle \langle 12 \rangle} = 1 + u_1 - u_3 - v_4, \\ v_2 &= \frac{\langle 231^{+}2^{+} \rangle \langle 41 \rangle}{\langle 4^{-}141^{+} \rangle \langle 23 \rangle} = 1 + u_2 - u_4 - v_1, \\ v_3 &= \frac{\langle 342^{+}3^{+} \rangle \langle 41 \rangle^2}{\langle 4^{-}141^{+} \rangle \langle 23 \rangle \langle 34 \rangle} = 1 + u_3 - u_1 - v_2, \end{aligned} \quad (B.13) $$

$$ v_4 = \frac{\langle 41^{+}3^{+}4^{+} \rangle \langle 41 \rangle}{\langle 4^{-}141^{+} \rangle \langle 34 \rangle} = \frac{1}{1 + e^{2\tau}}. $$

## C Mirror and crossing

In this appendix, we summarize the properties of the tilted functions $f_{a,s}^{[\alpha]}$, defined in (3.12), under the mirror transformations and illustrate how they translate into the crossing properties of the tilted transitions. Here, we closely follow similar analyses performed in refs. [24, 68].
The mirror transformation shifts the rapidity $u$ of a scalar excitation to $u^{\pm\gamma} = u \pm i$. Performing this deformation involves crossing one of the cuts that stretch between $-2g \pm \frac{i}{2}$ and $2g \pm \frac{i}{2}$, as a result of which we find ourselves on a different sheet of the Riemann surface after the transformation.

In order to perform this transformation, it is convenient to introduce an integral representation for $f_{a,s}^{[\alpha]}$ in terms of auxiliary functions $\gamma_{\alpha}^v$ and $\tilde{\gamma}_{\alpha}^v$,

$$ \gamma_{\alpha}^{v} = \frac{1}{1 + \mathbb{K}(\alpha)} \kappa_{\alpha}^{v}, \qquad \tilde{\gamma}_{\alpha}^{v} = \frac{1}{1 + \mathbb{K}(\alpha)} \tilde{\kappa}_{\alpha}^{v}. \qquad (C.1) $$

These equations naturally define these objects as infinite-dimensional vectors in Bessel function space, but we can conveniently repackage them into functions,

$$ \gamma_{\alpha}^{v}(2gt) = \sum_{n=1}^{\infty} 2n \gamma_{\alpha,n}^{v} J_{n}(2gt), \qquad \tilde{\gamma}_{\alpha}^{v}(2gt) = \sum_{n=1}^{\infty} 2n \tilde{\gamma}_{\alpha,n}^{v} J_{n}(2gt). \qquad (C.2) $$

Due to the kernel acting differently on even and odd components, we are splitting these functions into the corresponding parts,

$$ \gamma_{\alpha}^{v} = \gamma_{\alpha, \bullet}^{v} + \gamma_{\alpha, \circ}^{v}, \qquad \tilde{\gamma}_{\alpha}^{v} = \tilde{\gamma}_{\alpha, \bullet}^{v} + \tilde{\gamma}_{\alpha, \circ}^{v}. \qquad (C.3) $$

Equations (C.1) can be rewritten as the following set of integral equations,

$$ \gamma_{\alpha, 2n+1}^{v} + \int \frac{dt}{t} J_{2n+1}(2gt) \frac{2 \sin \alpha \cos \alpha \gamma_{\alpha, \bullet}^{v}(2gt) + 2 \cos^2 \alpha \gamma_{\alpha, \circ}^{v}(2gt)}{e^t - 1} = \kappa_{\alpha, 2n+1}^{v}, \qquad (C.4) \\ \gamma_{\alpha, 2n}^{v} + \int \frac{dt}{t} J_{2n}(2gt) \frac{2 \cos^2 \alpha \gamma_{\alpha, \bullet}^{v}(2gt) - 2 \sin \alpha \cos \alpha \gamma_{\alpha, \circ}^{v}(2gt)}{e^t - 1} = \kappa_{\alpha, 2n}^{v}, $$

and an identical one for $\tilde{\gamma}_{\alpha}^v$. These equations can be repackaged into the following set of cut-crossing identities,

$$ \begin{aligned} & \int \frac{dt}{t} \sin(ut) \left[ \gamma_{\alpha, \circ}^{v}(2gt) \frac{e^t + \cos 2\alpha}{e^t - 1} + \gamma_{\alpha, \bullet}^{v}(2gt) \frac{\sin 2\alpha}{e^t - 1} + \sin 2\alpha \frac{\cos(vt)e^{t/2} - J_0(2gt)}{e^t - 1} \right] = 0, \\ & \int \frac{dt}{t} \cos_J(ut) \left[ \gamma_{\alpha, \circ}^{v}(2gt) \frac{e^t + \cos 2\alpha}{e^t - 1} - \gamma_{\alpha, \circ}^{v}(2gt) \frac{\sin 2\alpha}{e^t - 1} + 2\cos^2\alpha \frac{\cos(vt)e^{t/2} - J_0(2gt)}{e^t - 1} \right] = 0, \\ & \int \frac{dt}{t} \sin(ut) \left[ \tilde{\gamma}_{\alpha, \circ}^{v}(2gt) \frac{e^t + \cos 2\alpha}{e^t - 1} + \tilde{\gamma}_{\alpha, \circ}^{v}(2gt) \frac{\sin 2\alpha}{e^t - 1} - 2\cos^2\alpha \frac{\sin(vt)e^{t/2}}{e^t - 1} \right] = 0, \\ & \int \frac{dt}{t} \cos_J(ut) \left[ \tilde{\gamma}_{\alpha, \circ}^{v}(2gt) \frac{e^t + \cos 2\alpha}{e^t - 1} - \tilde{\gamma}_{\alpha, \circ}^{v}(2gt) \frac{\sin 2\alpha}{e^t - 1} + \sin 2\alpha \frac{\sin(vt)e^{t/2}}{e^t - 1} \right] = 0. \end{aligned} \qquad (C.5) $$

which are valid inside the interval $u^2 \le 4g^2$. Here, $\cos_J(ut)$ is a shorthand notation for $\cos(ut) - J_0(2gt)$.

Instead of using the symmetric and antisymmetric $f$-functions in eq. (3.12), for the purposes of the finite coupling analysis of this section we will instead use,

$$ \begin{aligned} f_1^{[\alpha]}(u, v) &= \frac{1}{\cos^2 \alpha} \tilde{\kappa}_{\alpha}^{u} \mathbb{Q} \frac{1}{1 + \mathbb{K}(\alpha)} \kappa_{\alpha}^{v}, & f_2^{[\alpha]}(u, v) &= \frac{1}{\cos^2 \alpha} \kappa_{\alpha}^{u} \mathbb{Q} \frac{1}{1 + \mathbb{K}(\alpha)} \tilde{\kappa}_{\alpha}^{v}, \\ f_3^{[\alpha]}(u, v) &= \frac{1}{\cos^2 \alpha} \tilde{\kappa}_{\alpha}^{u} \mathbb{Q} \frac{1}{1 + \mathbb{K}(\alpha)} \tilde{\kappa}_{\alpha}^{v}, & f_4^{[\alpha]}(u, v) &= \frac{1}{\cos^2 \alpha} \kappa_{\alpha}^{u} \mathbb{Q} \frac{1}{1 + \mathbb{K}(\alpha)} \kappa_{\alpha}^{v}, \end{aligned} \qquad (C.6) $$
The mirror transformation shifts the rapidity $u$ of a scalar excitation to $u^{\pm\gamma} = u \pm i$. Performing this deformation involves crossing one of the cuts that stretch between $-2g \pm \frac{i}{2}$ and $2g \pm \frac{i}{2}$, as a result of which we find ourselves on a different sheet of the Riemann surface after the transformation.

In order to perform this transformation, it is convenient to introduce an integral representation for $f_{a,s}^{[\alpha]}$ in terms of auxiliary functions $\gamma_{\alpha}^v$ and $\tilde{\gamma}_{\alpha}^v$,

$$ \gamma_{\alpha}^{v} = \frac{1}{1 + \mathbb{K}(\alpha)} \kappa_{\alpha}^{v}, \qquad \tilde{\gamma}_{\alpha}^{v} = \frac{1}{1 + \mathbb{K}(\alpha)} \tilde{\kappa}_{\alpha}^{v}. \qquad (C.1) $$

These equations naturally define these objects as infinite-dimensional vectors in Bessel function space, but we can conveniently repackage them into functions,

$$ \gamma_{\alpha}^{v}(2gt) = \sum_{n=1}^{\infty} 2n \gamma_{\alpha,n}^{v} J_{n}(2gt), \qquad \tilde{\gamma}_{\alpha}^{v}(2gt) = \sum_{n=1}^{\infty} 2n \tilde{\gamma}_{\alpha,n}^{v} J_{n}(2gt). \qquad (C.2) $$

Due to the kernel acting differently on even and odd components, we are splitting these functions into the corresponding parts,

$$ \gamma_{\alpha}^{v} = \gamma_{\alpha, \bullet}^{v} + \gamma_{\alpha, \circ}^{v}, \qquad \tilde{\gamma}_{\alpha}^{v} = \tilde{\gamma}_{\alpha, \bullet}^{v} + \tilde{\gamma}_{\alpha, \circ}^{v}. \qquad (C.3) $$

Equations (C.1) can be rewritten as the following set of integral equations,

$$ \gamma_{\alpha,2n+1}^{v} + \int \frac{dt}{t} J_{2n+1}(2gt) \frac{2 \sin \alpha \cos \alpha \gamma_{\alpha, \bullet}^{v}(2gt) + 2 \cos^2 \alpha \gamma_{\alpha, \circ}^{v}(2gt)}{e^t - 1} = \kappa_{\alpha,2n+1}^{v}, \qquad (C.4) \\ \gamma_{\alpha,2n}^{v} + \int \frac{dt}{t} J_{2n}(2gt) \frac{2 \cos^2 \alpha \gamma_{\alpha, \bullet}^{v}(2gt) - 2 \sin \alpha \cos \alpha \gamma_{\alpha, \circ}^{v}(2gt)}{e^t - 1} = \kappa_{\alpha,2n}^{v}, $$

and an identical one for $\tilde{\gamma}_{\alpha}^v$. These equations can be repackaged into the following set of cut-crossing identities,

$$ \begin{aligned} & \int \frac{dt}{t} \sin(ut) \left[ \gamma_{\alpha, \circ}^{v}(2gt) \frac{e^t + \cos 2\alpha}{e^t - 1} + \gamma_{\alpha, \bullet}^{v}(2gt) \frac{\sin 2\alpha}{e^t - 1} + \sin 2\alpha \frac{\cos(vt)e^{t/2} - J_0(2gt)}{e^t - 1} \right] = 0, \\ & \int \frac{dt}{t} \cos_J(ut) \left[ \gamma_{\alpha, \circ}^{v}(2gt) \frac{e^t + \cos 2\alpha}{e^t - 1} - \gamma_{\alpha, \circ}^{v}(2gt) \frac{\sin 2\alpha}{e^t - 1} + 2\cos^2\alpha \frac{\cos(vt)e^{t/2} - J_0(2gt)}{e^t - 1} \right] = 0, \\ & \int \frac{dt}{t} \sin(ut) \left[ \tilde{\gamma}_{\alpha, \circ}^{v}(2gt) \frac{e^t + \cos 2\alpha}{e^t - 1} + \tilde{\gamma}_{\alpha, \circ}^{v}(2gt) \frac{\sin 2\alpha}{e^t - 1} - 2\cos^2\alpha \frac{\sin(vt)e^{t/2}}{e^t - 1} \right] = 0, \\ & \int \frac{dt}{t} \cos_J(ut) \left[ \tilde{\gamma}_{\alpha, \circ}^{v}(2gt) \frac{e^t + \cos 2\alpha}{e^t - 1} - \tilde{\gamma}_{\alpha, \circ}^{v}(2gt) \frac{\sin 2\alpha}{e^t - 1} + \sin 2\alpha \frac{\sin(vt)e^{t/2}}{e^t - 1} \right] = 0, \end{aligned} \qquad (C.5) $$

which are valid inside the interval $u^2 \le 4g^2$. Here, $\cos_J(ut)$ is a shorthand notation for $\cos(ut) - J_0(2gt)$.

Instead of using the symmetric and antisymmetric $f$-functions in eq. (3.12), for the purposes of the finite coupling analysis of this section we will instead use,

$$ \begin{aligned} f_1^{[\alpha]}(u, v) &= \frac{1}{\cos^2 \alpha} \tilde{\kappa}_{\alpha}^{u} \mathbb{Q} \frac{1}{1 + \mathbb{K}(\alpha)} \kappa_{\alpha}^{v}, & f_2^{[\alpha]}(u, v) &= \frac{1}{\cos^2 \alpha} \kappa_{\alpha}^{u} \mathbb{Q} \frac{1}{1 + \mathbb{K}(\alpha)} \tilde{\kappa}_{\alpha}^{v}, \\ f_3^{[\alpha]}(u, v) &= \frac{1}{\cos^2 \alpha} \tilde{\kappa}_{\alpha}^{u} \mathbb{Q} \frac{1}{1 + \mathbb{K}(\alpha)} \tilde{\kappa}_{\alpha}^{v}, & f_4^{[\alpha]}(u, v) &= \frac{1}{\cos^2 \alpha} \kappa_{\alpha}^{u} \mathbb{Q} \frac{1}{1 + \mathbb{K}(\alpha)} \kappa_{\alpha}^{v}, \end{aligned} \qquad (C.6) $$
with

$$f_a^{[\alpha]}(u, v) = f_2^{[\alpha]}(u, v) - f_1^{[\alpha]}(u, v), \quad f_s^{[\alpha]}(u, v) = f_4^{[\alpha]}(u, v) - f_3^{[\alpha]}(u, v). \quad (C.7)$$

Under the flip of the arguments, $f_{1,2,3,4}(v, u) = f_{2,1,3,4}(u, v)$. These functions admit the following representations in terms of $\gamma_\alpha^v$ and $\tilde{\gamma}_\alpha^v$,

$$ \begin{aligned} f_1^{[\alpha]}(u, v) &= \frac{1}{\cos \alpha} \int \frac{dt}{t} \frac{\sin(ut) e^{t/2}}{e^t - 1} \left( \cos \alpha \gamma_{\alpha, \circ}^v (2gt) + \sin \alpha \gamma_{\alpha, \bullet}^v (2gt) \right), \\ f_2^{[\alpha]}(u, v) &= \frac{1}{\cos \alpha} \int \frac{dt}{t} \frac{\cos(ut) e^{t/2} - J_0(2gt)}{e^t - 1} \left( \cos \alpha \tilde{\gamma}_{\alpha, \circ}^v (2gt) - \sin \alpha \tilde{\gamma}_{\alpha, \circ}^v (2gt) \right), \\ f_3^{[\alpha]}(u, v) &= \frac{1}{\cos \alpha} \int \frac{dt}{t} \frac{\sin(ut) e^{t/2}}{e^t - 1} \left( \cos \alpha \tilde{\gamma}_{\alpha, \circ}^v (2gt) + \sin \alpha \tilde{\gamma}_{\alpha, \circ}^v (2gt) \right), \\ f_4^{[\alpha]}(u, v) &= \frac{1}{\cos \alpha} \int \frac{dt}{t} \frac{\cos(ut) e^{t/2} - J_0(2gt)}{e^t - 1} \left( \cos \alpha \gamma_{\alpha, \circ}^v (2gt) - \sin \alpha \gamma_{\alpha, \circ}^v (2gt) \right). \end{aligned} \quad (C.8) $$

Now, to perform the mirror transformation, we follow the standard procedure of shifting the rapidity $u$ all the way to the cut, crossing the cut using the identities (C.5), and finally shifting $u$ the rest of the way to $u^{\pm\gamma}$. As a result of this procedure, we find the following transformation properties for the functions $f_{1,2,3,4}^{[\alpha]}$,

$$ \begin{aligned} f_1^{[\alpha]}(u^{\pm\gamma}, v) &= -\cos 2\alpha f_1^{[\alpha]}(u, v) \mp i \sin 2\alpha f_4^{[\alpha]}(u, v) \\ & \mp i \sin 2\alpha \int \frac{dt}{t} \frac{\cos(vt) e^{t/2} - J_0(2gt)}{e^t - 1} (e^{\mp i u^{\pm} t} - J_0(2gt)), \\ f_2^{[\alpha]}(u^{\pm\gamma}, v) &= -\cos 2\alpha f_2^{[\alpha]}(u, v) \mp i \sin 2\alpha f_3^{[\alpha]}(u, v) \\ & \mp \sin 2\alpha \int \frac{dt}{t} \frac{\sin(vt) e^{t/2}}{e^t - 1} (e^{\mp i u^{\pm} t} - J_0(2gt)), \end{aligned} \quad (C.9) $$

$$ \begin{aligned} f_3^{[\alpha]}(u^{\pm\gamma}, v) &= -\cos 2\alpha f_3^{[\alpha]}(u, v) \mp i \sin 2\alpha f_2^{[\alpha]}(u, v) \\ & \mp 2i \int \frac{dt}{t} \frac{\sin(vt) e^{t/2}}{e^t - 1} \left[ (\cos(u^{\pm}t) - J_0(2gt)) \sin^2 \alpha \pm i \sin(u^{\pm}t) \cos^2 \alpha \right], \\ f_4^{[\alpha]}(u^{\pm\gamma}, v) &= -\cos 2\alpha f_4^{[\alpha]}(u, v) \mp i \sin 2\alpha f_1^{[\alpha]}(u, v) \\ & -2 \int \frac{dt}{t} \frac{\cos(vt) e^{t/2} - J_0(2gt)}{e^t - 1} \left[ (\cos(u^{\pm}t) - J_0(2gt)) \cos^2 \alpha \pm i \sin(u^{\pm}t) \sin^2 \alpha \right], \end{aligned} $$

where in the right-hand side $u^{\pm} = u \pm i/2$. In terms of the antisymmetric and symmetric scattering phases,

$$f_a^{[\alpha]}(u^{\pm\gamma}, v) = -\cos 2\alpha f_a^{[\alpha]}(u, v) \pm i \sin 2\alpha f_s^{[\alpha]}(u, v) \pm i \sin 2\alpha \mathcal{J}(\pm u, \pm v), \quad (C.10)$$

$$f_s^{[\alpha]}(u^{\pm\gamma}, v) = -\cos 2\alpha f_s^{[\alpha]}(u, v) \pm i \sin 2\alpha f_a^{[\alpha]}(u, v) - \mathcal{J}(\mp u^{\pm 2}, \mp v) - \cos 2\alpha \mathcal{J}(\pm u, \pm v).$$

with $u^{[\pm 2]} = u \pm i$ and

$$ \begin{aligned} \mathcal{J}(u, v) &= \int \frac{dt}{t} \frac{(e^{t/2-iu t} - J_0(2gt))(e^{t/2+iv t} - J_0(2gt))}{e^t - 1} \\ &= \log \left[ \frac{\Gamma(iu - iv)}{\Gamma(\frac{1}{2} + iu) \Gamma(\frac{1}{2} - iv)} \right] + J_{\phi}(u) + J_{\phi}(-v), \end{aligned} \quad (C.11) $$
with

$$f_a^{[\alpha]}(u, v) = f_2^{[\alpha]}(u, v) - f_1^{[\alpha]}(u, v), \quad f_s^{[\alpha]}(u, v) = f_4^{[\alpha]}(u, v) - f_3^{[\alpha]}(u, v). \quad (C.7)$$

Under the flip of the arguments, $f_{1,2,3,4}(v, u) = f_{2,1,3,4}(u, v)$. These functions admit the following representations in terms of $\gamma_\alpha^v$ and $\tilde{\gamma}_\alpha^v$,

$$ 
\begin{aligned}
f_1^{[\alpha]}(u, v) &= \frac{1}{\cos \alpha} \int \frac{dt}{t} \frac{\sin(ut) e^{t/2}}{e^t - 1} \left( \cos \alpha \gamma_{\alpha, \circ}^v (2gt) + \sin \alpha \gamma_{\alpha, \bullet}^v (2gt) \right), \\
f_2^{[\alpha]}(u, v) &= \frac{1}{\cos \alpha} \int \frac{dt}{t} \frac{\cos(ut) e^{t/2} - J_0(2gt)}{e^t - 1} \left( \cos \alpha \tilde{\gamma}_{\alpha, \circ}^v (2gt) - \sin \alpha \tilde{\gamma}_{\alpha, \circ}^v (2gt) \right), \\
f_3^{[\alpha]}(u, v) &= \frac{1}{\cos \alpha} \int \frac{dt}{t} \frac{\sin(ut) e^{t/2}}{e^t - 1} \left( \cos \alpha \tilde{\gamma}_{\alpha, \circ}^v (2gt) + \sin \alpha \tilde{\gamma}_{\alpha, \circ}^v (2gt) \right), \\
f_4^{[\alpha]}(u, v) &= \frac{1}{\cos \alpha} \int \frac{dt}{t} \frac{\cos(ut) e^{t/2} - J_0(2gt)}{e^t - 1} \left( \cos \alpha \gamma_{\alpha, \circ}^v (2gt) - \sin \alpha \gamma_{\alpha, \circ}^v (2gt) \right).
\end{aligned} 
\quad (C.8) $$

Now, to perform the mirror transformation, we follow the standard procedure of shifting the rapidity $u$ all the way to the cut, crossing the cut using the identities (C.5), and finally shifting $u$ the rest of the way to $u^{\pm\gamma}$. As a result of this procedure, we find the following transformation properties for the functions $f_{1,2,3,4}^{[\alpha]}$,

$$ 
\begin{aligned}
f_1^{[\alpha]}(u^{\pm\gamma}, v) &= -\cos 2\alpha f_1^{[\alpha]}(u, v) \mp i \sin 2\alpha f_4^{[\alpha]}(u, v) \\
&\quad \mp i \sin 2\alpha \int \frac{dt}{t} \frac{\cos(vt) e^{t/2} - J_0(2gt)}{e^t - 1} (e^{\mp iu^{\pm}t} - J_0(2gt)), \\
f_2^{[\alpha]}(u^{\pm\gamma}, v) &= -\cos 2\alpha f_2^{[\alpha]}(u, v) \mp i \sin 2\alpha f_3^{[\alpha]}(u, v) \\
&\quad -\sin 2\alpha \int \frac{dt}{t} \frac{\sin(vt) e^{t/2}}{e^t - 1} (e^{\mp iu^{\pm}t} - J_0(2gt)), \\
f_3^{[\alpha]}(u^{\pm\gamma}, v) &= -\cos 2\alpha f_3^{[\alpha]}(u, v) \mp i \sin 2\alpha f_2^{[\alpha]}(u, v) \\
&\quad \mp 2i \int \frac{dt}{t} \frac{\sin(vt) e^{t/2}}{e^t - 1} [(\cos(u^{\pm}t) - J_0(2gt)) \sin^2 \alpha \pm i \sin(u^{\pm}t) \cos^2 \alpha], \\
f_4^{[\alpha]}(u^{\pm\gamma}, v) &= -\cos 2\alpha f_4^{[\alpha]}(u, v) \mp i \sin 2\alpha f_1^{[\alpha]}(u, v) \\
&\quad -2 \int \frac{dt}{t} \frac{\cos(vt) e^{t/2} - J_0(2gt)}{e^t - 1} [(\cos(u^{\pm}t) - J_0(2gt)) \cos^2 \alpha \pm i \sin(u^{\pm}t) \sin^2 \alpha],
\end{aligned} 
\quad (C.9) $$

where in the right-hand side $u^{\pm} = u \pm i/2$. In terms of the antisymmetric and symmetric scattering phases,

$$ 
\begin{aligned}
f_a^{[\alpha]}(u^{\pm\gamma}, v) &= -\cos 2\alpha f_a^{[\alpha]}(u, v) \pm i \sin 2\alpha f_s^{[\alpha]}(u, v) \pm i \sin 2\alpha \mathcal{J}(\pm u, \pm v), \quad (C.10) \\
f_s^{[\alpha]}(u^{\pm\gamma}, v) &= -\cos 2\alpha f_s^{[\alpha]}(u, v) \pm i \sin 2\alpha f_a^{[\alpha]}(u, v) - \mathcal{J}(\mp u^{\pm 2}, \mp v) - \cos 2\alpha \mathcal{J}(\pm u, \pm v).
\end{aligned} $$

with $u^{[\pm 2]} = u \pm i$ and

$$ 
\begin{aligned}
\mathcal{J}(u, v) &= \int \frac{dt}{t} \frac{(e^{t/2-iu t} - J_0(2gt))(e^{t/2+iv t} - J_0(2gt))}{e^t - 1} \\
&= \log \left[ \frac{\Gamma(iu - iv)}{\Gamma(\frac{1}{2} + iu) \Gamma(\frac{1}{2} - iv)} \right] + J_{\phi}(u) + J_{\phi}(-v),
\end{aligned} 
\quad (C.11) $$
which satisfies $\mathcal{J}(-u, -v) = \mathcal{J}(v, u)$. It also happens to coincide with the logarithm of the simple part of the transition (3.14), not captured by $f_a$ and $f_s$. One can see that up to inhomogeneous terms $\propto \mathcal{J}$, the mirror transformation is equivalent to the combination of a rotation by $2\alpha$ and a parity transformation in the plane $(f_s, if_a)$,

$$ \begin{pmatrix} f_s \\ if_a \end{pmatrix} \xrightarrow{\pm\gamma} - \begin{pmatrix} \cos 2\alpha & \mp \sin 2\alpha \\ \pm \sin 2\alpha & \cos 2\alpha \end{pmatrix} \begin{pmatrix} f_s \\ if_a \end{pmatrix} + \dots \quad (C.12) $$

When performing mirror transformations on the tilted transition (3.14), the simple part of it, given by the exponent of $\mathcal{J}(u, v)$, also needs to be transformed. The simplest move to consider is the inverse mirror rotation $u \to u^{-\gamma}$. As one can see from its integral representation, $\mathcal{J}(u, v)$ is an analytic function of $u$ for $\mathrm{Im}(u) < 1/2$. Hence, there is no cut to cross and we simply get<fnref n="16" />

$$ \mathcal{J}(u^{-\gamma}, v) = \mathcal{J}(u^{[-2]}, v). \quad (C.13) $$

This term neatly cancels when combined with $f_s^{[\alpha]}(u^{-\gamma}, v)$. Putting all pieces together, we arrive at the following compact representation for the mirror transformation of the tilted transition (3.14)

$$ \begin{aligned} \log[g^2 P^{[\alpha]}(u^{-\gamma}|v)] &= if_a^{[\pi/4]}(u^{-\gamma}, v) + f_s^{[\alpha]}(u^{-\gamma}, v) + \mathcal{J}(u^{-\gamma}, v) \\ &= f_s^{[\pi/4]}(u, v) - \cos 2\alpha f_s^{[\alpha]}(u, v) - i \sin 2\alpha f_a^{[\alpha]}(u, v) + 2 \sin^2 \alpha \mathcal{J}(v, u), \end{aligned} \quad (C.14) $$

using $\mathcal{J}(-u, -v) = \mathcal{J}(v, u)$. We may also consider the transformation under $v \to v^{\gamma}$. This transformation follows from the above equations after recalling the permutation properties of $f_a$ and $f_s$,

$$ f_s^{[\alpha]}(u, v^{\gamma}) = f_s^{[\alpha]}(v^{\gamma}, u), \quad f_a^{[\alpha]}(u, v^{\gamma}) = -f_a^{[\alpha]}(v^{\gamma}, u). \quad (C.15) $$

Using these, one can verify that the $v^{+\gamma}$ move is equivalent to the $u^{-\gamma}$ one,

$$ P_{\phi\phi}^{[\alpha]}(u^{-\gamma}|v) = P_{\phi\phi}^{[\alpha]}(u|v^{\gamma}). \quad (C.16) $$

This equation holds true for any rapidities $u, v$ and may be equivalently written as

$$ P_{\phi\phi}^{[\alpha]}(u|v) = P_{\phi\phi}^{[\alpha]}(u^{\gamma}|v^{\gamma}), \quad (C.17) $$

after replacing $u \to u^{\gamma}$ on both sides. This is the mirror symmetry axiom. Notice that it holds for any $\alpha$.

In the particular case of $\alpha = \pi/4$, mirror transformation (C.14) takes the transition back to its original form, up to a permutation of the rapidities. That is, we verify that

$$ P_{\phi\phi}(u^{-\gamma}|v) = P_{\phi\phi}(v|u), \quad (C.18) $$

$$ \mathcal{J}(u^{\gamma}, v) = \mathcal{J}(u, v) - \mathcal{J}(v, u) + \mathcal{J}(v, u^{[+2]}) + \log \left[ \frac{g^2}{(u-v)(u-v+i)} \right]. $$

<footnote n="16">¹⁶Moving the rapidity in the opposite direction is more complicated, due to the presence of a cut at $\mathrm{Im}u = 1/2$. It results in the more involved relation</footnote>
which satisfies $\mathcal{J}(-u, -v) = \mathcal{J}(v, u)$. It also happens to coincide with the logarithm of the simple part of the transition (3.14), not captured by $f_a$ and $f_s$. One can see that up to inhomogeneous terms $\propto \mathcal{J}$, the mirror transformation is equivalent to the combination of a rotation by $2\alpha$ and a parity transformation in the plane $(f_s, if_a)$,

$$ \begin{pmatrix} f_s \\ if_a \end{pmatrix} \xrightarrow{\pm\gamma} - \begin{pmatrix} \cos 2\alpha & \mp \sin 2\alpha \\ \pm \sin 2\alpha & \cos 2\alpha \end{pmatrix} \begin{pmatrix} f_s \\ if_a \end{pmatrix} + \dots \qquad (C.12) $$

When performing mirror transformations on the tilted transition (3.14), the simple part of it, given by the exponent of $\mathcal{J}(u, v)$, also needs to be transformed. The simplest move to consider is the inverse mirror rotation $u \to u^{-\gamma}$. As one can see from its integral representation, $\mathcal{J}(u, v)$ is an analytic function of $u$ for $\mathrm{Im}(u) < 1/2$. Hence, there is no cut to cross and we simply get<fnref n="16" />

$$ \mathcal{J}(u^{-\gamma}, v) = \mathcal{J}(u^{[-2]}, v). \qquad (C.13) $$

This term neatly cancels when combined with $f_s^{[\alpha]}(u^{-\gamma}, v)$. Putting all pieces together, we arrive at the following compact representation for the mirror transformation of the tilted transition (3.14)

$$ \begin{aligned} \log[g^2 P^{[\alpha]}(u^{-\gamma}|v)] &= if_a^{[\pi/4]}(u^{-\gamma}, v) + f_s^{[\alpha]}(u^{-\gamma}, v) + \mathcal{J}(u^{-\gamma}, v) \\ &= f_s^{[\pi/4]}(u, v) - \cos 2\alpha f_s^{[\alpha]}(u, v) - i \sin 2\alpha f_a^{[\alpha]}(u, v) + 2 \sin^2 \alpha \mathcal{J}(v, u), \end{aligned} \qquad (C.14) $$

using $\mathcal{J}(-u, -v) = \mathcal{J}(v, u)$. We may also consider the transformation under $v \to v^{\gamma}$. This transformation follows from the above equations after recalling the permutation properties of $f_a$ and $f_s$,

$$ f_s^{[\alpha]}(u, v^{\gamma}) = f_s^{[\alpha]}(v^{\gamma}, u), \qquad f_a^{[\alpha]}(u, v^{\gamma}) = -f_a^{[\alpha]}(v^{\gamma}, u). \qquad (C.15) $$

Using these, one can verify that the $v^{+\gamma}$ move is equivalent to the $u^{-\gamma}$ one,

$$ P_{\phi\phi}^{[\alpha]}(u^{-\gamma}|v) = P_{\phi\phi}^{[\alpha]}(u|v^{\gamma}). \qquad (C.16) $$

This equation holds true for any rapidities $u, v$ and may be equivalently written as

$$ P_{\phi\phi}^{[\alpha]}(u|v) = P_{\phi\phi}^{[\alpha]}(u^{\gamma}|v^{\gamma}), \qquad (C.17) $$

after replacing $u \to u^{\gamma}$ on both sides. This is the mirror symmetry axiom. Notice that it holds for any $\alpha$.

In the particular case of $\alpha = \pi/4$, mirror transformation (C.14) takes the transition back to its original form, up to a permutation of the rapidities. That is, we verify that

$$ P_{\phi\phi}(u^{-\gamma}|v) = P_{\phi\phi}(v|u), \qquad (C.18) $$

$$ \mathcal{J}(u^{\gamma}, v) = \mathcal{J}(u, v) - \mathcal{J}(v, u) + \mathcal{J}(v, u^{[+2]}) + \log \left[ \frac{g^2}{(u-v)(u-v+i)} \right]. $$

<footnote n="16">¹⁶Moving the rapidity in the opposite direction is more complicated, due to the presence of a cut at $\mathrm{Im}u = 1/2$. It results in the more involved relation</footnote>
for $P_{\phi\phi} = P_{\phi\phi}^{[\pi/4]}$, as it should be for a pentagon. The other case of interest corresponds to $\alpha = 0$, for which the transition does not map back into itself. Yet, its mirror image takes a very simple form

$$Q_{\phi\phi}(u^{-\gamma}|v) = g^{-2} \exp \left[ f_s^{[\pi/4]}(u, v) - f_s^{[0]}(u, v) \right]. \qquad (C.19)$$

Importantly, this function is symmetric under permutation of $u$ and $v$,

$$Q_{\phi\phi}(u^{-\gamma}|v) = Q_{\phi\phi}(v^{-\gamma}|u). \qquad (C.20)$$

Combined with eq. (C.17) it leads to the crossing relation

$$Q_{\phi\phi}(u^{-2\gamma}|v) = Q_{\phi\phi}(v^{-\gamma}|u^{-\gamma}) = Q_{\phi\phi}(v|u). \qquad (C.21)$$

A slightly more complicated relation may be written for the crossing transformation in the opposite direction, $Q_{\phi\phi}(u^{2\gamma}|v)$. To derive it, one may first apply Watson equation,

$$Q_{\phi\phi}(u^{2\gamma}|v) = S_{\phi\phi}(u^{2\gamma}, v) Q_{\phi\phi}(v|u^{2\gamma}). \qquad (C.22)$$

Taking then into account mirror symmetry, $Q_{\phi\phi}(v|u^{2\gamma}) = Q_{\phi\phi}(v^{-2\gamma}|u)$, eq. (C.21), and the crossing-unitary relations of the flux-tube S-matrix,

$$S_{\phi\phi}(u, v) S_{\phi\phi}(v, u) = 1, \qquad S_{\phi\phi}(u^{2\gamma}, v) S(u, v) = \frac{u-v}{u-v+2i}, \qquad (C.23)$$

one finds

$$Q_{\phi\phi}(u^{2\gamma}|v) = \frac{u-v}{u-v+2i} Q_{\phi\phi}(v|u). \qquad (C.24)$$

This identity is the key formula derived in ref. [24] for proving the crossing symmetry of the singlet form factor transition in eq. (3.23).

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[5] J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [INSPIRE].
for $P_{\phi\phi} = P_{\phi\phi}^{[\pi/4]}$, as it should be for a pentagon. The other case of interest corresponds to $\alpha = 0$, for which the transition does not map back into itself. Yet, its mirror image takes a very simple form

$$Q_{\phi\phi}(u^{-\gamma}|v) = g^{-2} \exp \left[ f_s^{[\pi/4]}(u, v) - f_s^{[0]}(u, v) \right]. \qquad (C.19)$$

Importantly, this function is symmetric under permutation of $u$ and $v$,

$$Q_{\phi\phi}(u^{-\gamma}|v) = Q_{\phi\phi}(v^{-\gamma}|u). \qquad (C.20)$$

Combined with eq. (C.17) it leads to the crossing relation

$$Q_{\phi\phi}(u^{-2\gamma}|v) = Q_{\phi\phi}(v^{-\gamma}|u^{-\gamma}) = Q_{\phi\phi}(v|u). \qquad (C.21)$$

A slightly more complicated relation may be written for the crossing transformation in the opposite direction, $Q_{\phi\phi}(u^{2\gamma}|v)$. To derive it, one may first apply Watson equation,

$$Q_{\phi\phi}(u^{2\gamma}|v) = S_{\phi\phi}(u^{2\gamma}, v) Q_{\phi\phi}(v|u^{2\gamma}). \qquad (C.22)$$

Taking then into account mirror symmetry, $Q_{\phi\phi}(v|u^{2\gamma}) = Q_{\phi\phi}(v^{-2\gamma}|u)$, eq. (C.21), and the crossing-unitary relations of the flux-tube S-matrix,

$$S_{\phi\phi}(u, v) S_{\phi\phi}(v, u) = 1, \qquad S_{\phi\phi}(u^{2\gamma}, v) S(u, v) = \frac{u-v}{u-v+2i}, \qquad (C.23)$$

one finds

$$Q_{\phi\phi}(u^{2\gamma}|v) = \frac{u-v}{u-v+2i} Q_{\phi\phi}(v|u). \qquad (C.24)$$

This identity is the key formula derived in ref. [24] for proving the crossing symmetry of the singlet form factor transition in eq. (3.23).

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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[78] A.V. Belitsky, S.E. Derkachov and A.N. Manashov, Quantum mechanics of null polygonal Wilson loops, Nucl. Phys. B 882 (2014) 303 [arXiv:1401.7307] [INSPIRE].

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[91] Y. Jiang, S. Komatsu, I. Kostov and D. Serban, *Clustering and the Three-Point Function*, J. Phys. A 49 (2016) 454003 [arXiv:1604.03575] [INSPIRE].

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[89] A. Brandhuber, M. Kostacinska, B. Penante and G. Travaglini, *Tr$(F^3)$ supersymmetric form factors and maximal transcendentality Part I: $N = 4$ super Yang-Mills*, JHEP 12 (2018) 076 [arXiv:1804.05703] [INSPIRE].

[90] S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, *Separation of variables for the quantum SL(2, $\mathcal{R}$) spin chain*, JHEP 07 (2003) 047 [hep-th/0210216] [INSPIRE].

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[93] C. Bercini, V. Gonçalves, A. Homrich and P. Vieira, *The Wilson loop — large spin OPE dictionary*, JHEP 07 (2022) 079 [arXiv:2110.04364] [INSPIRE].

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