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Scattering of open strings $\chi$ attached to distinct probe D3-branes at positions $z_i = 1/m_i$. The strings stretched between the probe branes and the stack of $N$ coincident D3-branes correspond to the massive W-bosons.

A characteristic shape of the leading Regge trajectory at finite coupling. We also depict the relevant limits of the Wilson line with a cusp. The Regge trajectory is curved, it has a logarithmic asymptotic at large $s$, and it exhibits an accumulation point at $s=4m^2$. For $s>4 m^2$ it develops a non-zero imaginary part.

Planar diagrams with external $\chi$ fields (wavy lines) attached to a frame made of W-bosons (thick lines), filled with massless fields from the unbroken gauge group $U(N)$ (dashed lines).

One-loop box diagram in double-line notation.

Scattering of open strings on a probe D3-brane at fixed radial position $z_{D3}$ in $AdS$.

Analytic structure of $f(s,t)$ at finite coupling $g$ for fixed $t$. There is an infinite set of bound states for $s < 4m^2$ and a two-particle cut from $s = 4m^2$.Spectrum constraints in the bootstrap. For $z < 4m^2$, the spectral density $c_J(z)$ from the $SO(4)$ decomposition \eqref{eq:cJDecomposition} is used, while for $z > 4m^2$, the $SO(1,3)$ decomposition with the spectral density $c_\nu(z)$ applies.

Summary of the different limits captured by $j_0(s) = -1-\Gamma_{\text{cusp}}(g,\phi)$, with $s = 4m^2 \sin^2(\phi/2)$. Close to $s = 0$ (i.e. $\phi = 0$), $j_0(s)$ is related to the Bremsstrahlung function $B(g)$, which is exactly known from supersymmetric localization \cite{Correa:2012at} and integrability \cite{Correa:2012hh}. When $s \to 4m^2$ (i.e. $\phi \to \pi$), the Wilson lines become anti-parallel and the Regge trajectory diverges. This limit captures the quark-antiquark potential \eqref{eq:Gamma_cusp_pi}. The Euclidean angle $\phi$ can be analytically continued to its Lorentzian version $\varphi = i \phi$: the behavior as $\varphi \to \infty$ of $\Gamma_{\text{cusp}}$ captures the limit $|s| \to \infty$ of $j_0(s)$ \eqref{eq:light_like_cusp}. In this regime, the trajectory scales as a logarithm and is controlled by $\Gamma_{\text{cusp}}^{\infty}(g)$, which is the cusp anomalous dimension of a light-like Wilson line \cite{Korchemsky:1987wg}. The region $s>4m^2$ corresponds to $\varphi = \pi + i \phi$.

Summary of the different limits captured by $j_1(s) = -1-\Gamma_{\text{cusp},\Phi}(g,\phi)$, with $s = 4m^2 \sin^2(\phi/2)$. Close to $s = 0$ (i.e. $\phi = 0$), $j_1(s)$ is related to the scaling dimension of the \textit{parallel} scalar inserted in the cusp $\Delta_{\Phi}(g)$, which can be computed using QSC \cite{Grabner:2020nis} and it is known at weak and strong coupling \eqref{eq:Delta_Phi_g}. When $s \to 4m^2$ (i.e. $\phi \to \pi$), the Wilson lines become anti-parallel and the subleading Regge trajectory diverges with residue $\Omega_{\Phi}$ \eqref{eq:Gamma_cusp_pi_Phi}. In \cite{Klebanov:2006jj} it was argued that $\Omega_{\Phi}(g) = 0$ for $g \leq g_c$, with $g_c = 1/4$ from the ladder approximation. Our integrability computation, reported in \secref{sec:qsc_analysis_j1}, confirms this behavior and the critical coupling that we obtain is $g_c \approx 0.28$.

A Maldacena-Wilson line with the cusp angle $\phi$ and an insertion of operator $O_i$. In the more general case the coupling of the lines with the scalars is parametrized by two unit vectors $\vec{n}, \vec{n}_{\theta}$ in the R-symmetry space, such that $\vec{n} \cdot \vec{n}_{\theta} = \cos \theta$. In the case analyzed in this paper $\theta = 0$ \eqref{eq:n_all_aligned}.

QSC data point for $\Gamma_{\text{cusp}}(g,\phi)$ as a function of $(g,\phi)$. Here we summarize our workflow: in the left corner, at small $g$, the QSC is initialized using the perturbative solution. Then we move in $\phi$ at fixed $g$ or in $g$ at fixed $\phi$.

Comparison between $\mathrm{j}_0^{\text{ansatz}}(\phi)$ and QSC data for $g = \{5/32, 2/5, 4/5\}$ as a function of the cusp angle $\phi$.

Plot of $\mathrm{j}_0(\phi)$ for higher values of $g$. The results are obtained from the $\mathrm{j}_0^{\text{ansatz}}(g,\phi)$.

Values of $\Omega_{\Phi}^{(N)}(g)$ for $g = 1/2$ as a function of the order $N$ of the fit \eqref{eq:fit_j1}. In order to extrapolate the value of $\Omega_{\Phi}(g)$ at $N \to \infty$ we perform a fit with the function $\Omega_{\Phi}(g) + \frac{b}{N^c}$, with $\Omega_{\Phi}(g), b, c$ fitting parameters. The results of $\Omega_{\Phi}(g)$ for different values of $g$ are reported in \tabref{tab:tab_residue_j1}. For the specific fit in the plot we have used all the points with $N \geq 5$.

Plot of the residue of $\mathrm{j}_1(\phi) \simeq {\Omega_{\Phi}(g) \over \pi - \phi}$ for different values of $g$. For $g < g_{c} \approx 0.28$ the residue is effectively zero, while it increases for $g \geq g_{c}$. The orange dots were obtained using extrapolation of the QSC results at $\phi = \pi$.

Comparison between $\mathrm{j}_1^{\text{ansatz}}(\phi)$ and QSC data for $g = \{5/32, 2/5, 4/5, 6/5\}$ as a function of the cusp angle $\phi$. As computed above, the $g = 5/32$ trajectory does not diverge at $\phi = \pi$.

Summary of the argument for the Regge/cusp correspondence. Thanks to dual conformal invariance, the Regge limit and the soft limit are equivalent. By matching the behavior of the amplitude in the two limits it is possible to get the equivalence \eqref{eq:reggecusp}.

Plot of $m_{\text{gap}}$ as a function of $g$. Data from QSC interpolate the weak-coupling and strong-coupling expansions \eqref{eq:expansion_mgap}.

Contribution of different states to the dispersion relation \eqref{eq:dispagain}. Blue: close to $s = 0$ flat-space states contribute. There are two other regions that we have control over: close to the two-particle threshold (yellow), where the non-relativistic EFT expansion applies, and at $s \to \infty$ (green), where the behavior of $f_s(s,0)$ in $g$ is fixed by the Bremsstrahlung function $B(g)$, see \eqref{eq:crosssection}.

Asymptotic regions in the parameter space for both the $SO(4)$ and $SO(1,3)$ sectors. Regions where additional constraints are applied are highlighted in red.

Exclusion plots for $(\bar{a}_{1,0}, \bar{a}_{2,0})$ and $(\bar{a}_{1,0}, \bar{a}_{2,1})$ at the extremal couplings $g \to 0$ and $g \to \infty$. In the $g\to0$ plots (left panels) we highlight the difference between results obtained with (light brown) and without (black dashed) dual conformal invariance. The 1-loop box amplitude \eqref{eq:weak1loop} is located at the bottom-left kink. In the $g \to \infty$ plots (right panels) we show the bounds with (orange) and without (black) the pole constraints. Both sets of bounds are obtained using $SO(4)$ partial waves. The super-Veneziano amplitude lies close to the upper-right kink.

Spectra at weak and strong coupling. Left: $g \to 0$, with no bound states and only a two-particle cut. Right: $g \to \infty$, exhibiting $SO(4)$ bound states on linear Regge trajectories. The right panel also illustrates the spectral assumptions used to generate the bounds in this limit, shown in \figref{fig:a10_a20_g=0}.

Comparison of the bounds at $g \to 0$ (light brown) and $g \to \infty$ (dark brown), together with the weak-coupling (red) and strong-coupling (black) expansions for $(\bar{a}_{1,0},\bar{a}_{2,0})$ and $(\bar{a}_{1,0},\bar{a}_{2,1})$. A Pad\'{e} approximation (blue, with $\mu = 2$, see \eqref{eq:pade_band_def}) provides a possible prediction for the Wilson coefficients, smoothly interpolating between the two extremal regimes. The weak-coupling expansion (dashed red) is shown for $g\leq 0.2$, while the strong-coupling expansion (dashed black) corresponds to $g \geq 0.4$.

Exclusion plot for $\bar a_{1,0}$ as a function of the coupling $g$. Black: bounds derived without imposing the pole constraints. Orange: bounds with pole constraints included. Blue: Pad\'{e} band of $\bar a_{1,0}(g)$ from \secref{sec:Pade model for Wilson coefficients} with $\mu = 1$, see \eqref{eq:pade_band_def}. The asymptotic $g \to \infty$ value is indicated by the dashed blue line.

Exclusion plot for $(\bar a_{1,0},\bar a_{2,0})$ at various values of the coupling $g$. The black contour shows the allowed region obtained without the pole constraints, while the orange contour includes them. The red dot marks the one-loop box amplitude, corresponding to the $g \to 0$ limit. The blue ellipse indicates the Pad\'{e}-model prediction with $\mu = 2$, see \eqref{eq:pade_band_def}.

Exclusion plot for $(\bar{a}_{1,0},\bar{a}_{2,1})$ at various values of the coupling $g$. The black contour shows the allowed region obtained without the pole constraints, while the orange contour includes them. The red dot marks the one-loop box amplitude, corresponding to the $g \to 0$ limit. The blue ellipse indicates the Pad\'{e}-model prediction with $\mu=2$, see \eqref{eq:pade_band_def}.

Exclusion plot for $\bar\lambda_{0,0}^2$ as a function of the Wilson coefficient $\bar a_{1,0}$ at fixed intermediate coupling $g=1$. Black: bounds obtained without pole constraints. Orange: bounds with pole constraints included. Blue: Pad\'{e}-model prediction with $\mu = 1$, see \eqref{eq:pade_band_def}. The size in the $\bar \lambda_{0,0}^2$ direction has no meaning as we did not create an error estimate for this observable.

Consistency of the Pad\'{e} bands of \secref{sec:Pade model for Wilson coefficients} with the bootstrap bounds derived by imposing that all the Wilson coefficients $\bar{a}_{n,l}$ with $n = 1,\dots,5$ except the one that we are bounding have to be inside the Pad\'{e} bands. Notice that the bootstrap bounds are perfectly consistent with the bands and reproduce the expected shape. Here, we do not rescale the Pad\'{e} bands and take $\mu=1$, see \eqref{eq:pade_band_def}.

Bound on the residue $\bar \lambda_{0,0}^2$ at finite coupling. In the black and orange curves, we show the upper bound without $\bar a_{n,\ell}$ Pad\'{e} constraints. In red/green, we show the bands obtained the $\bar a_{n,\ell}$ Pad\'{e} bands. In blue, we compare with the conformal-Pad\'{e}-[1,1] \eqref{eq:lambda_pade} which asymptotes to super-Veneziano.

Two-sided bounds on the amplitude and its derivatives at $g=1$, we set $\mgap=1$. In all cases we fixed $s\text{ or } t=-1$ and and varied the other Mandelstam. In black, we present the two-sided bootstrap bounds obtained without using the Pad\'{e} bands and in red, the two-sided bootstrap bounds, where the Wilson coefficients were forced to lie within their Pad\'{e} bands (with scale $\mu=1$). In the top row, we used the black dashed curve to show the contribution of $1/st$ to the observables.

Two-sided bounds on the dimensionless amplitude $\bar f_{0,0}(s,t)$, see \eqref{eq:funcitself00}, at $g=1$, setting $\mgap = 1$. The resulting region is shown as the red solid volume, obtained using Padé bands for the Wilson coefficients (with scale $\mu = 1$). The orange surface corresponds to the one-loop box \eqref{eq:weak1loop}, and the purple surface to the super-Veneziano amplitude \eqref{eq:super_Veneziano}. Recall that $\bar f_{0,0}(0,0) = 1$ by definition \eqref{eq:funcitself}.

Examples of null constraints that cannot be satisfied on the allowed spectrum. The black dots represent the leading trajectory $j_0$, and the allowed spectrum is on the right of $j_0$. The filled colours represent when the sum rule for the null constraint $\chi_{n,0}^{SO(4)}(z_{5d}, J)$ is positive.

Definition of the orthogonal unit vectors $n_{\pm}$ w.r.t. the Maldacena-Wilson line with a cusp. Reflections around these two vectors are used to classify the discrete symmetry properties of local operators.

Regge trajectories and the spectrum of bound states in flat space from the decomposition of irreducible representations in $10$d. The numbers on top of the states denote degeneracy.

Error of the Chebyshev approximation to \eqref{eq:Omegainapp} at $t=-1$ and $\nu=0,10$.

Maximum value of $\bar{\lambda}_{0,0}^2$ at $g = 1$ for different numbers of null constraints $n_{\text{max}}$.