002947035 001__ 2947035
002947035 005__ 20251030042643.0
002947035 0248_ $$aoai:cds.cern.ch:2947035$$pcerncds:FULLTEXT$$pcerncds:CERN:FULLTEXT$$pcerncds:CERN
002947035 037__ $$9arXiv:reportnumber$$aCERN-TH-2025-202
002947035 037__ $$9arXiv$$aarXiv:2510.19909$$chep-th
002947035 035__ $$9arXiv$$aoai:arXiv.org:2510.19909
002947035 035__ $$9Inspire$$aoai:inspirehep.net:3072588$$d2025-10-29T17:51:56Z$$h2025-10-30T03:00:06Z$$mmarcxml$$ttrue$$uhttps://inspirehep.net/api/oai2d
002947035 035__ $$9Inspire$$a3072588
002947035 041__ $$aeng
002947035 100__ $$aAlday, Luis F.$$uOxford U., Inst. Math.$$vMathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quar-ter, Woodstock Road, Oxford, OX2 6GG, U.K.
002947035 245__ $$9arXiv$$aFrom Partons to Strings: Scattering on the Coulomb Branch of $\mathcal{N}=4$ SYM
002947035 269__ $$c2025-10-22
002947035 300__ $$a109 p
002947035 500__ $$9arXiv$$a109 pages, 37 figures, 1 ancillary Mathematica notebook
002947035 520__ $$9arXiv$$aWe study scattering on the Coulomb branch of planar ${\mathcal{N}}=4$ SYM at finite 't Hooft coupling. This setup defines a family of classical open-string S-matrices that smoothly interpolates between perturbative parton scattering at weak coupling and flat-space string scattering at strong coupling. We focus on the four-point amplitude, which exhibits a remarkably rich structure: nonlinear Regge trajectories, dual conformal invariance, an intricate spectrum of bound states with an accumulation point, and a two-particle cut. Dual conformal invariance relates the spectrum of Regge trajectories to the energy spectrum of the Maldacena-Wilson cusp Hamiltonian. This connection allows us to use integrability to compute the leading and subleading Regge trajectories at finite coupling, which we then input in the bootstrap analysis. At strong coupling, we use the worldsheet bootstrap to construct the first $AdS$-curvature correction to the Veneziano amplitude. We apply dispersion relations and S-matrix bootstrap techniques to derive bounds on Wilson coefficients, couplings to bound states, and the overall shape of the amplitude. We find that the $\mathcal{N}=4$ amplitude saturates the bootstrap bounds at weak coupling and nearly saturates them at strong coupling. At intermediate coupling, the amplitude traces a nontrivial path through the allowed space of observables. To characterize this path, we combine the weak- and strong-coupling information about the amplitude to construct a finite-coupling model for Wilson coefficients using a Padé approximation. The resulting model satisfies bootstrap constraints and yields sharp predictions for the finite-coupling behavior of the amplitude. We provide evidence that complete monotonicity of the scattering amplitude, previously observed perturbatively, persists at finite coupling.
002947035 541__ $$aarXiv$$chepcrawl$$d2025-10-24T04:01:24.049731$$e10852048
002947035 540__ $$3preprint$$aCC BY 4.0$$uhttp://creativecommons.org/licenses/by/4.0/
002947035 595__ $$aCERN-TH
002947035 65017 $$2arXiv$$ahep-th
002947035 65017 $$2SzGeCERN$$aParticle Physics - Theory
002947035 690C_ $$aCERN
002947035 690C_ $$aPREPRINT
002947035 700__ $$aArmanini, Elisabetta$$uCERN$$uEPFL, Lausanne, FSL$$vCERN, Theoretical Physics Department, Geneva, Switzerland$$vFields and Strings Laboratory, Institute of Physics, École Polytechnique Fédéral de Lausanne (EPFL), Route de la Sorge, CH-1015 Lausanne, Switzerland
002947035 700__ $$aHäring, Kelian$$uDelta ITP, Amsterdam$$vInstitute for Theoretical Physics, University of Amsterdam, 1090 GL Amsterdam, The Nether-lands
002947035 700__ $$aZhiboedov, Alexander$$uCERN$$vCERN, Theoretical Physics Department, Geneva, Switzerland
002947035 8564_ $$82791022$$s39119$$uhttps://cds.cern.ch/record/2947035/files/plotj0_high_g.png$$y00011 Plot of $\mathrm{j}_0(\phi)$ for higher values of $g$. The results are obtained from the $\mathrm{j}_0^{\text{ansatz}}(g,\phi)$.
002947035 8564_ $$82791023$$s94548$$uhttps://cds.cern.ch/record/2947035/files/behaviours_j1_no_yellow.png$$y00007 Summary of the different limits captured by $j_1(s) = -1-\Gamma_{\text{cusp},\Phi}(g,\phi)$, with $s = 4m^2 \sin^2(\phi/2)$. Close to $s = 0$ (i.e. $\phi = 0$), $j_1(s)$ is related to the scaling dimension of the \textit{parallel} scalar inserted in the cusp $\Delta_{\Phi}(g)$, which can be computed using QSC \cite{Grabner:2020nis} and it is known at weak and strong coupling \eqref{eq:Delta_Phi_g}. When $s \to 4m^2$ (i.e. $\phi \to \pi$), the Wilson lines become anti-parallel and the subleading Regge trajectory diverges with residue $\Omega_{\Phi}$ \eqref{eq:Gamma_cusp_pi_Phi}. In \cite{Klebanov:2006jj} it was argued that $\Omega_{\Phi}(g) = 0$ for $g \leq g_c$, with $g_c = 1/4$ from the ladder approximation. Our integrability computation, reported in \secref{sec:qsc_analysis_j1}, confirms this behavior and the critical coupling that we obtain is $g_c \approx 0.28$.
002947035 8564_ $$82791024$$s23121$$uhttps://cds.cern.ch/record/2947035/files/plota10vsg1_arrow.png$$y00022 Exclusion plot for $\bar a_{1,0}$ as a function of the coupling $g$. Black: bounds derived without imposing the pole constraints. Orange: bounds with pole constraints included. Blue: Pad\'{e} band of $\bar a_{1,0}(g)$ from \secref{sec:Pade model for Wilson coefficients} with $\mu = 1$, see \eqref{eq:pade_band_def}. The asymptotic $g \to \infty$ value is indicated by the dashed blue line.
002947035 8564_ $$82791025$$s17252$$uhttps://cds.cern.ch/record/2947035/files/plotresvsa10.png$$y00025 Exclusion plot for $\bar\lambda_{0,0}^2$ as a function of the Wilson coefficient $\bar a_{1,0}$ at fixed intermediate coupling $g=1$. Black: bounds obtained without pole constraints. Orange: bounds with pole constraints included. Blue: Pad\'{e}-model prediction with $\mu = 1$, see \eqref{eq:pade_band_def}. The size in the $\bar \lambda_{0,0}^2$ direction has no meaning as we did not create an error estimate for this observable.
002947035 8564_ $$82791026$$s238934$$uhttps://cds.cern.ch/record/2947035/files/workflow_j0_3.png$$y00009 QSC data point for $\Gamma_{\text{cusp}}(g,\phi)$ as a function of $(g,\phi)$. Here we summarize our workflow: in the left corner, at small $g$, the QSC is initialized using the perturbative solution. Then we move in $\phi$ at fixed $g$ or in $g$ at fixed $\phi$.
002947035 8564_ $$82791027$$s23686$$uhttps://cds.cern.ch/record/2947035/files/spectrum.png$$y00005 Analytic structure of $f(s,t)$ at finite coupling $g$ for fixed $t$. There is an infinite set of bound states for $s < 4m^2$ and a two-particle cut from $s = 4m^2$.Spectrum constraints in the bootstrap. For $z < 4m^2$, the spectral density $c_J(z)$ from the $SO(4)$ decomposition \eqref{eq:cJDecomposition} is used, while for $z > 4m^2$, the $SO(1,3)$ decomposition with the spectral density $c_\nu(z)$ applies.
002947035 8564_ $$82791028$$s151285$$uhttps://cds.cern.ch/record/2947035/files/behaviours_j0_corr_grigio.png$$y00001 A characteristic shape of the leading Regge trajectory at finite coupling. We also depict the relevant limits of the Wilson line with a cusp. The Regge trajectory is curved, it has a logarithmic asymptotic at large $s$, and it exhibits an accumulation point at $s=4m^2$. For $s>4 m^2$ it develops a non-zero imaginary part.
002947035 8564_ $$82791029$$s65963$$uhttps://cds.cern.ch/record/2947035/files/extrapolate_residue.png$$y00012 Values of $\Omega_{\Phi}^{(N)}(g)$ for $g = 1/2$ as a function of the order $N$ of the fit \eqref{eq:fit_j1}. In order to extrapolate the value of $\Omega_{\Phi}(g)$ at $N \to \infty$ we perform a fit with the function $\Omega_{\Phi}(g) + \frac{b}{N^c}$, with $\Omega_{\Phi}(g), b, c$ fitting parameters. The results of $\Omega_{\Phi}(g)$ for different values of $g$ are reported in \tabref{tab:tab_residue_j1}. For the specific fit in the plot we have used all the points with $N \geq 5$.
002947035 8564_ $$82791030$$s98019$$uhttps://cds.cern.ch/record/2947035/files/behaviours_j0_corr.png$$y00006 Summary of the different limits captured by $j_0(s) = -1-\Gamma_{\text{cusp}}(g,\phi)$, with $s = 4m^2 \sin^2(\phi/2)$. Close to $s = 0$ (i.e. $\phi = 0$), $j_0(s)$ is related to the Bremsstrahlung function $B(g)$, which is exactly known from supersymmetric localization \cite{Correa:2012at} and integrability \cite{Correa:2012hh}. When $s \to 4m^2$ (i.e. $\phi \to \pi$), the Wilson lines become anti-parallel and the Regge trajectory diverges. This limit captures the quark-antiquark potential \eqref{eq:Gamma_cusp_pi}. The Euclidean angle $\phi$ can be analytically continued to its Lorentzian version $\varphi = i \phi$: the behavior as $\varphi \to \infty$ of $\Gamma_{\text{cusp}}$ captures the limit $|s| \to \infty$ of $j_0(s)$ \eqref{eq:light_like_cusp}. In this regime, the trajectory scales as a logarithm and is controlled by $\Gamma_{\text{cusp}}^{\infty}(g)$, which is the cusp anomalous dimension of a light-like Wilson line \cite{Korchemsky:1987wg}. The region $s>4m^2$ corresponds to $\varphi = \pi + i \phi$.
002947035 8564_ $$82791031$$s46433$$uhttps://cds.cern.ch/record/2947035/files/f_vs_st.png$$y00029 Two-sided bounds on the dimensionless amplitude $\bar f_{0,0}(s,t)$, see \eqref{eq:funcitself00}, at $g=1$, setting $\mgap = 1$. The resulting region is shown as the red solid volume, obtained using Padé bands for the Wilson coefficients (with scale $\mu = 1$). The orange surface corresponds to the one-loop box \eqref{eq:weak1loop}, and the purple surface to the super-Veneziano amplitude \eqref{eq:super_Veneziano}. Recall that $\bar f_{0,0}(0,0) = 1$ by definition \eqref{eq:funcitself}.
002947035 8564_ $$82791032$$s114714$$uhttps://cds.cern.ch/record/2947035/files/regge_cusp_corr_u_v_2.png$$y00015 Summary of the argument for the Regge/cusp correspondence. Thanks to dual conformal invariance, the Regge limit and the soft limit are equivalent. By matching the behavior of the amplitude in the two limits it is possible to get the equivalence \eqref{eq:reggecusp}.
002947035 8564_ $$82791033$$s46174$$uhttps://cds.cern.ch/record/2947035/files/plota21vsa10_fin_blue.png$$y00024 Exclusion plot for $(\bar{a}_{1,0},\bar{a}_{2,1})$ at various values of the coupling $g$. The black contour shows the allowed region obtained without the pole constraints, while the orange contour includes them. The red dot marks the one-loop box amplitude, corresponding to the $g \to 0$ limit. The blue ellipse indicates the Pad\'{e}-model prediction with $\mu=2$, see \eqref{eq:pade_band_def}.
002947035 8564_ $$82791034$$s87327$$uhttps://cds.cern.ch/record/2947035/files/regimes_bs.png$$y00017 Contribution of different states to the dispersion relation \eqref{eq:dispagain}. Blue: close to $s = 0$ flat-space states contribute. There are two other regions that we have control over: close to the two-particle threshold (yellow), where the non-relativistic EFT expansion applies, and at $s \to \infty$ (green), where the behavior of $f_s(s,0)$ in $g$ is fixed by the Bremsstrahlung function $B(g)$, see \eqref{eq:crosssection}.
002947035 8564_ $$82791035$$s29948$$uhttps://cds.cern.ch/record/2947035/files/plotconsistency_pade.png$$y00026 Consistency of the Pad\'{e} bands of \secref{sec:Pade model for Wilson coefficients} with the bootstrap bounds derived by imposing that all the Wilson coefficients $\bar{a}_{n,l}$ with $n = 1,\dots,5$ except the one that we are bounding have to be inside the Pad\'{e} bands. Notice that the bootstrap bounds are perfectly consistent with the bands and reproduce the expected shape. Here, we do not rescale the Pad\'{e} bands and take $\mu=1$, see \eqref{eq:pade_band_def}.
002947035 8564_ $$82791036$$s55171$$uhttps://cds.cern.ch/record/2947035/files/Feynman_diagram_2.png$$y00002 Planar diagrams with external $\chi$ fields (wavy lines) attached to a frame made of W-bosons (thick lines), filled with massless fields from the unbroken gauge group $U(N)$ (dashed lines).
002947035 8564_ $$82791037$$s43446$$uhttps://cds.cern.ch/record/2947035/files/plotj1.png$$y00014 Comparison between $\mathrm{j}_1^{\text{ansatz}}(\phi)$ and QSC data for $g = \{5/32, 2/5, 4/5, 6/5\}$ as a function of the cusp angle $\phi$. As computed above, the $g = 5/32$ trajectory does not diverge at $\phi = \pi$.
002947035 8564_ $$82791038$$s38287$$uhttps://cds.cern.ch/record/2947035/files/plotj0.png$$y00010 Comparison between $\mathrm{j}_0^{\text{ansatz}}(\phi)$ and QSC data for $g = \{5/32, 2/5, 4/5\}$ as a function of the cusp angle $\phi$.
002947035 8564_ $$82791039$$s25992$$uhttps://cds.cern.ch/record/2947035/files/string_spectrum_flat.png$$y00032 Regge trajectories and the spectrum of bound states in flat space from the decomposition of irreducible representations in $10$d. The numbers on top of the states denote degeneracy.
002947035 8564_ $$82791040$$s111103$$uhttps://cds.cern.ch/record/2947035/files/D3_brane_2.png$$y00004 Scattering of open strings on a probe D3-brane at fixed radial position $z_{D3}$ in $AdS$.
002947035 8564_ $$82791041$$s28413$$uhttps://cds.cern.ch/record/2947035/files/plotpred2_with_arrows.png$$y00021 Comparison of the bounds at $g \to 0$ (light brown) and $g \to \infty$ (dark brown), together with the weak-coupling (red) and strong-coupling (black) expansions for $(\bar{a}_{1,0},\bar{a}_{2,0})$ and $(\bar{a}_{1,0},\bar{a}_{2,1})$. A Pad\'{e} approximation (blue, with $\mu = 2$, see \eqref{eq:pade_band_def}) provides a possible prediction for the Wilson coefficients, smoothly interpolating between the two extremal regimes. The weak-coupling expansion (dashed red) is shown for $g\leq 0.2$, while the strong-coupling expansion (dashed black) corresponds to $g \geq 0.4$.
002947035 8564_ $$82791042$$s28371$$uhttps://cds.cern.ch/record/2947035/files/j_x_nu_x_regions.png$$y00018 Asymptotic regions in the parameter space for both the $SO(4)$ and $SO(1,3)$ sectors. Regions where additional constraints are applied are highlighted in red.
002947035 8564_ $$82791043$$s13270$$uhttps://cds.cern.ch/record/2947035/files/conv_nmax.png$$y00034 Maximum value of $\bar{\lambda}_{0,0}^2$ at $g = 1$ for different numbers of null constraints $n_{\text{max}}$.
002947035 8564_ $$82791044$$s61324$$uhttps://cds.cern.ch/record/2947035/files/1-loop_double.png$$y00003 One-loop box diagram in double-line notation.
002947035 8564_ $$82791045$$s100049$$uhttps://cds.cern.ch/record/2947035/files/mgap_new.png$$y00016 Plot of $m_{\text{gap}}$ as a function of $g$. Data from QSC interpolate the weak-coupling and strong-coupling expansions \eqref{eq:expansion_mgap}.
002947035 8564_ $$82791046$$s64617$$uhttps://cds.cern.ch/record/2947035/files/fit_Omega_Phi.png$$y00013 Plot of the residue of $\mathrm{j}_1(\phi) \simeq {\Omega_{\Phi}(g) \over \pi - \phi}$ for different values of $g$. For $g < g_{c} \approx 0.28$ the residue is effectively zero, while it increases for $g \geq g_{c}$. The orange dots were obtained using extrapolation of the QSC results at $\phi = \pi$.
002947035 8564_ $$82791047$$s87395$$uhttps://cds.cern.ch/record/2947035/files/stack_Dbranes_fin.png$$y00000 Scattering of open strings $\chi$ attached to distinct probe D3-branes at positions $z_i = 1/m_i$. The strings stretched between the probe branes and the stack of $N$ coincident D3-branes correspond to the massive W-bosons.
002947035 8564_ $$82791048$$s38083$$uhttps://cds.cern.ch/record/2947035/files/cuspedWL.png$$y00008 A Maldacena-Wilson line with the cusp angle $\phi$ and an insertion of operator $O_i$. In the more general case the coupling of the lines with the scalars is parametrized by two unit vectors $\vec{n}, \vec{n}_{\theta}$ in the R-symmetry space, such that $\vec{n} \cdot \vec{n}_{\theta} = \cos \theta$. In the case analyzed in this paper $\theta = 0$ \eqref{eq:n_all_aligned}.
002947035 8564_ $$82791049$$s53575$$uhttps://cds.cern.ch/record/2947035/files/chebychev_nu_example.png$$y00033 Error of the Chebyshev approximation to \eqref{eq:Omegainapp} at $t=-1$ and $\nu=0,10$.
002947035 8564_ $$82791050$$s61730$$uhttps://cds.cern.ch/record/2947035/files/res_vs_g_with_Pade.png$$y00027 Bound on the residue $\bar \lambda_{0,0}^2$ at finite coupling. In the black and orange curves, we show the upper bound without $\bar a_{n,\ell}$ Pad\'{e} constraints. In red/green, we show the bands obtained the $\bar a_{n,\ell}$ Pad\'{e} bands. In blue, we compare with the conformal-Pad\'{e}-[1,1] \eqref{eq:lambda_pade} which asymptotes to super-Veneziano.
002947035 8564_ $$82791051$$s52367$$uhttps://cds.cern.ch/record/2947035/files/plota20vsa10_fin_blue.png$$y00023 Exclusion plot for $(\bar a_{1,0},\bar a_{2,0})$ at various values of the coupling $g$. The black contour shows the allowed region obtained without the pole constraints, while the orange contour includes them. The red dot marks the one-loop box amplitude, corresponding to the $g \to 0$ limit. The blue ellipse indicates the Pad\'{e}-model prediction with $\mu = 2$, see \eqref{eq:pade_band_def}.
002947035 8564_ $$82791052$$s150384$$uhttps://cds.cern.ch/record/2947035/files/example_no_doubleDisc.png$$y00030 Examples of null constraints that cannot be satisfied on the allowed spectrum. The black dots represent the leading trajectory $j_0$, and the allowed spectrum is on the right of $j_0$. The filled colours represent when the sum rule for the null constraint $\chi_{n,0}^{SO(4)}(z_{5d}, J)$ is positive.
002947035 8564_ $$82791053$$s13650139$$uhttps://cds.cern.ch/record/2947035/files/2510.19909.pdf$$yFulltext
002947035 8564_ $$82791054$$s46388$$uhttps://cds.cern.ch/record/2947035/files/plotg0andginf_1.png$$y00019 Exclusion plots for $(\bar{a}_{1,0}, \bar{a}_{2,0})$ and $(\bar{a}_{1,0}, \bar{a}_{2,1})$ at the extremal couplings $g \to 0$ and $g \to \infty$. In the $g\to0$ plots (left panels) we highlight the difference between results obtained with (light brown) and without (black dashed) dual conformal invariance. The 1-loop box amplitude \eqref{eq:weak1loop} is located at the bottom-left kink. In the $g \to \infty$ plots (right panels) we show the bounds with (orange) and without (black) the pole constraints. Both sets of bounds are obtained using $SO(4)$ partial waves. The super-Veneziano amplitude lies close to the upper-right kink.
002947035 8564_ $$82791055$$s50438$$uhttps://cds.cern.ch/record/2947035/files/parity_WL_2.png$$y00031 Definition of the orthogonal unit vectors $n_{\pm}$ w.r.t. the Maldacena-Wilson line with a cusp. Reflections around these two vectors are used to classify the discrete symmetry properties of local operators.
002947035 8564_ $$82791056$$s91400$$uhttps://cds.cern.ch/record/2947035/files/spectrum_plot.png$$y00020 Spectra at weak and strong coupling. Left: $g \to 0$, with no bound states and only a two-particle cut. Right: $g \to \infty$, exhibiting $SO(4)$ bound states on linear Regge trajectories. The right panel also illustrates the spectral assumptions used to generate the bounds in this limit, shown in \figref{fig:a10_a20_g=0}.
002947035 8564_ $$82791057$$s77130$$uhttps://cds.cern.ch/record/2947035/files/amplitude_vs_s.png$$y00028 Two-sided bounds on the amplitude and its derivatives at $g=1$, we set $\mgap=1$. In all cases we fixed $s\text{ or } t=-1$ and and varied the other Mandelstam. In black, we present the two-sided bootstrap bounds obtained without using the Pad\'{e} bands and in red, the two-sided bootstrap bounds, where the Wilson coefficients were forced to lie within their Pad\'{e} bands (with scale $\mu=1$). In the top row, we used the black dashed curve to show the contribution of $1/st$ to the observables.
002947035 960__ $$a11
002947035 980__ $$aPREPRINT