In this work, we present a recurrence relation for the instanton partition function of the $$ \mathcal{N} $$ = 2 SYM $\textit{SU}$ ($\textit{N}$) gauge theory with 2$\textit{N}$ fundamental multiplets. The main difficulty lies in determining the asymptotic behaviour of the partition function in the regime of large vacuum expectation values of the Higgs field. Using the saddle point method and the $\textit{qq}$-characters technique, we demonstrate that, in this limit, the partition function is governed by the Quantum Seiberg-Witten curves, as in the Nekrasov-Shatashvili limit, up to a normalisation constant. With the asymptotic behaviour found, we are able to write the recurrence relation for the partition function and to find the effective infrared coupling constant. The resulting effective constant is an inverse of a modular function with respect to a certain triangle group, and the asymptotic itself is a product of modular functions and forms with respect to triangle groups.

Italy

Aleksei Bykov

Ekaterina Sysoeva

Zamolodchikov recurrence relation and modular properties of effective coupling in $$ \mathcal{N} $$ = 2 SQCD

We present a new method to compute the soft function for the $\textit{N}$ -Jettiness variable for arbitrary $\textit{N}$ at high perturbative orders in QCD. It is based on the observation that the most singular part of the soft function, the dipole contribution, can be represented by a sum of an analytically calculable inclusive soft function and a remainder. The latter is absent at NLO, is immediately finite at NNLO, and we expect that can be made finite with the help of simple NLO-like infrared subtractions at N LO. As a byproduct of this approach, we derive a very simple formula for the tripole contribution to the $\textit{N}$ -Jettiness NNLO soft function, which results in a fast numerical evaluation. We apply this method to compute the $\textit{N}$ -Jettiness soft function at NNLO, and report numerical results for up to five jets for the hadron-collider soft function. We finally outline the prospects for applications at N LO.

Switzerland

Luca Buonocore

Maximilian Delto

Germany

Kirill Melnikov

Pier Monni

United Kingdom

Andrey Pikelner

Gherardo Vita

$\textit{N}$-Jettiness soft functions made simple

We study the ODE/IM correspondence between the linear problem associated with the supersymmetric affine Toda field equation for the twisted affine Lie superalgebra $\textit{C}$(2) = $$ \mathfrak{osp}{\left(2|2\right)}^{(2)} $$ and two-dimensional $$ \mathcal{N} $$ = 1 superconformal field theories (SCFTs). On the ODE side, we introduce a boundary condition more suitable for the conformal limit and the subsequent WKB analysis and diagonalize the resulting Lax operator. This leads to a WKB expansion from which we extract the WKB periods and semi-local conserved quantities up to tenth order. On the IM side, we compute the eigenvalues of the local integrals of motion on the cylinder in both the Neveu-Schwarz and Ramond sectors of 2d $$ \mathcal{N} $$ = 1 SCFTs. We then compare the two sides and verify, up to sixth order, that the WKB periods coincide with the eigenvalues of the local integrals of motion for highest-weight states in the Neveu-Schwarz sector.

Japan

Naozumi Tanabe

The ODE/IM correspondence between $\textit{C}$(2)(2)-type linear problems and 2d $$ \mathcal{N} $$ = 1 SCFT

We investigate the elliptic-flow splitting between prompt <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>D</mi><mn>0</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></math> mesons within a heavy-quark transport framework with sequential hadronization, in which <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></math> forms at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1.2</mn><mspace width="0.16em"></mspace><msub><mi>T</mi><mi>c</mi></msub></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>D</mi><mn>0</mn></msup></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>c</mi></msub></math>. We present model calculations for the <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>T</mi></msub></math>-differential <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>2</mn></msub></math> and predictions for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>/</mo><msup><mi>D</mi><mn>0</mn></msup></mrow></math> yield ratio in O-O <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0</mn><mo>–</mo><mspace width="0.16em"></mspace><mn>20</mn><mo>%</mo></mrow></math> collisions at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>=</mo><mn>5.36</mn></mrow></math> TeV, where preliminary ALICE data are available. The sequential scenario reproduces the observed <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>v</mi><mn>2</mn></msub><mrow><mo>(</mo><msup><mi>D</mi><mn>0</mn></msup><mo>)</mo></mrow><mo>&gt;</mo><msub><mi>v</mi><mn>2</mn></msub><mrow><mo>(</mo><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>)</mo></mrow></mrow></math> ordering and predicts an enhanced <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>/</mo><msup><mi>D</mi><mn>0</mn></msup></mrow></math> ratio at low <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mi>T</mi></msub></math>, whereas a simultaneous baseline yields the opposite ordering. Decomposing the hadronic splitting into its partonic components, we show that the <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>2</mn></msub></math> ordering is driven by the late-stage flow accumulated by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>D</mi><mn>0</mn></msup></math>-parent charm quarks during the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1.2</mn><mspace width="0.16em"></mspace><msub><mi>T</mi><mi>c</mi></msub><mo>→</mo><msub><mi>T</mi><mi>c</mi></msub></mrow></math> interval, with hadronic rescattering essential to preserve the signal in small systems. A systematic scan across nine collision configurations spanning O-O and Pb-Pb centralities reveals a universal linear scaling between the hadronic splitting and the partonic flow increment accumulated during this window. This establishes the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>–</mo><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></mrow></math> flow splitting as a hadronization chronometer of the QGP at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>=</mo><mn>5.36</mn></mrow></math> TeV.

China

Hui Du

Xiao-Wei Hao

Wei Dai

Jiaxing Zhao

Ben-Wei Zhang

Enke Wang

System-size dependence of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>0</mn></msup><mtext>−</mtext><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></mrow></math> flow splitting from early <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></math> formation at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>=</mo><mn>5.36</mn></mrow></math> TeV

We review the structure of superconformal anomalies in 4d $$ \mathcal{N} $$ = 4 conformal supergravity (CSG) coupled to a number N of $$ \mathcal{N} $$ = 4 vector multiplets and 6d (2,0) CSG coupled to N of (2,0) tensor multiplets. Anomalies cancel if N = 4 and N = 26 respectively. If the CSG part of the action is dropped and N = 6 + n , the first theory is classically equivalent to the 4d $$ \mathcal{N} $$ = 4 Poincaré supergravity (PSG) coupled to n vector multiplets, while the second one with N = 5 + n is classically equivalent to the 6d (2,0) PSG coupled to n tensor multiplets. We suggest that these facts imply that divergences in the 4d PSG with n vectors should be proportional to n + 2 and similarly in the 6d PSG with n tensors to n – 21. This conjecture appears to be consistent with most of the known results of explicit scattering amplitude computations in these 4d and 6d PSG theories, apart from one coefficient in the 3-loop mixed vector scattering amplitude computed in arXiv:1305.4876.

United States

R. Kallosh

Russia

A. Tseytlin

On anomaly-free 4d $$ \mathcal{N} $$ = 4 and 6d (2,0) conformal supergravities and UV finiteness of Poincaré supergravities

Quantum Field Theory (QFT) introduced us to the notion that a causal diamond in space-time corresponded to a subsystem of a quantum mechanical system defined on the global space-time. Work by Jacobson [1], Fischler and Susskind [2] and particularly Bousso [3–5] suggested that in the quantum theory of gravity this subsystem should have a density matrix of finite entropy. These authors formalized older intuitive arguments based on black hole physics. Although mathematically, Type II von Neumann algebras admit finite entropy density matrices, the black hole arguments suggest that the number of physical states in these subsystems is finite. The conjecture that de Sitter (dS) space has a finite number of physical states was first made in [6, 7]. Leutheusser and Liu [8] showed that, in the $\textit{N}$ = $\textit{∞}$ limit, causal diamonds with finite area in AdS radius units had Type $\textit{III}$ von Neumann sub-algebras of the full operator algebra. They claimed that this was true for finite values of the UV cutoff, and that the algebra was the algebra of bulk local fields in the diamond. We will argue that the second part of this conjecture is incorrect and that the bulk field algebra emerges only in a double scaled limit, where the boundary UV cutoff is taken to infinity as $\textit{N}$ is taken to infinity. There is never a bulk field theory description that resolves distances smaller than the AdS radius.

A Sidan

Tom Banks

Diamonds in the bulk and large-$\textit{N}$ scaling in AdS/CFT

We propose a numerical method based on the master field for large-$\textit{N}$ reduced matrix models. While the master field is originally an infinite-dimensional matrix, in this method it is regularized to a finite dimension, with the requirement that it satisfies the loop equations as much as possible. This formulation can be directly implemented for numerical computation, and since there is no sign problem at the fundamental level, the method can be applied regardless of whether the model is of Euclidean or Minkowski type. In numerical calculations for one- and two-matrix models, the exact solution is well reproduced in the Euclidean case, while perturbative results are well reproduced in the Minkowski case. This demonstrates the effectiveness of the method and supports the idea that the matrix models studied in this paper admit a regularized master-field description.

Canada

Reishi Maeta

Regularized master-field approximation for large-$\textit{N}$ reduced matrix models

We derive an exact, scheme-independent generating function <math altimg='si51.svg'><mrow><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> for the massive <italic>O</italic>( <italic>N</italic>)-symmetric <italic>&#981;</italic> 4 two-point observable <math altimg='si52.svg'><mrow><msub><mi>R</mi><mstyle mathvariant='normal'><mi>&#915;</mi></mstyle></msub><mrow><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>,</mo><mi>a</mi><mo>,</mo><msup><mi>m</mi><mn>2</mn></msup><mo>)</mo></mrow><mo>&#8801;</mo><msup><mstyle mathvariant='normal'><mi>&#915;</mi></mstyle><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup><mrow><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>,</mo><mi>a</mi><mo>,</mo><msup><mi>m</mi><mn>2</mn></msup><mo>)</mo></mrow><mo linebreak='goodbreak'>/</mo><msup><mi>Q</mi><mn>2</mn></msup></mrow></math> within Scale-Factorized Quantum Field Theory (SF-QFT). Path-integral factorization at the physical matching scale <math altimg='si5.svg'><mrow><msup><mi>Q</mi><mo>*</mo></msup><mo linebreak='goodbreak'>=</mo><mi>m</mi></mrow></math>, followed by Seeley&#8211;DeWitt heat-kernel integration of the infrared modes, extracts five universal scheme-independent inputs: the two-loop <italic>&#946;</italic>-function coefficients <italic>&#946;</italic> <inf>0</inf>, <italic>&#946;</italic> <inf>1</inf>, the mass anomalous dimension coefficients <italic>d</italic> <inf>1</inf>, <italic>d</italic> <inf>2</inf>, and the two-loop field anomalous dimension coefficient <italic>e</italic> <inf>2</inf>, in the normalization of Refs. [1,2] with <math altimg='si6.svg'><mrow><mi>a</mi><mo>&#8801;</mo><mi>&#955;</mi><mo linebreak='goodbreak'>=</mo><mi>g</mi><mo linebreak='goodbreak'>/</mo><mo>(</mo><mn>2</mn><msup><mi>&#960;</mi><mn>2</mn></msup><mo>)</mo></mrow></math>. The observable <math altimg='si46.svg'><msub><mi>R</mi><mstyle mathvariant='normal'><mi>&#915;</mi></mstyle></msub></math> carries its own renormalization-group anomalous dimension <math altimg='si53.svg'><mrow><msub><mi>&#947;</mi><msub><mi>R</mi><mstyle mathvariant='normal'><mi>&#915;</mi></mstyle></msub></msub><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo linebreak='goodbreak'>=</mo><msub><mi>&#945;</mi><mn>1</mn></msub><mi>a</mi><mo linebreak='goodbreak'>+</mo><msub><mi>&#945;</mi><mn>2</mn></msub><msup><mi>a</mi><mn>2</mn></msup><mo linebreak='goodbreak'>+</mo><mo>&#8943;</mo></mrow></math>, with <math altimg='si54.svg'><mrow><msub><mi>&#945;</mi><mn>1</mn></msub><mo linebreak='goodbreak'>=</mo><msub><mi>d</mi><mn>1</mn></msub></mrow></math> and <math altimg='si55.svg'><mrow><msub><mi>&#945;</mi><mn>2</mn></msub><mo linebreak='goodbreak'>=</mo><msub><mi>d</mi><mn>2</mn></msub><mo linebreak='goodbreak'>+</mo><mn>2</mn><msub><mi>e</mi><mn>2</mn></msub></mrow></math>; the factor of two on <italic>e</italic> <inf>2</inf> is the two-leg field renormalization of the two-point function. The Principle of Observable Effective Matching (POEM) at <math altimg='si5.svg'><mrow><msup><mi>Q</mi><mo>*</mo></msup><mo linebreak='goodbreak'>=</mo><mi>m</mi></mrow></math> fixes the boundary data <math altimg='si56.svg'><mrow><msub><mi>f</mi><mn>1</mn></msub><mo linebreak='goodbreak'>=</mo><msub><mi>&#945;</mi><mn>1</mn></msub></mrow></math> and <math altimg='si57.svg'><mrow><msub><mi>f</mi><mn>2</mn></msub><mo linebreak='goodbreak'>=</mo><msub><mi>&#945;</mi><mn>2</mn></msub><mo linebreak='goodbreak'>/</mo><mn>2</mn></mrow></math> and yields a universal three-sector Effective Dynamical Renormalization (EDR) recursion whose coefficients <italic>f</italic> <inf><italic>n</italic>,0</inf> define <math altimg='si31.svg'><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub></math> exactly and converge geometrically with ratio &#8239;&#8764;&#8239;0.003 to 0.01. Summing the recursion gives the master ordinary differential equation (ODE) <math altimg='si58.svg'><mrow><msubsup><mi mathvariant='script'>G</mi><mn>0</mn><mo>&#8242;</mo></msubsup><mrow><mo>(</mo><mn>1</mn><mo linebreak='badbreak'>&#8722;</mo><mi>k</mi><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub><mo>)</mo></mrow><mo linebreak='goodbreak'>=</mo><msub><mi>&#945;</mi><mn>1</mn></msub><mo linebreak='goodbreak'>+</mo><mrow><mo>[</mo><msub><mi>&#945;</mi><mn>2</mn></msub><mo linebreak='badbreak'>&#8722;</mo><mrow><mo>(</mo><mn>1</mn><mo linebreak='badbreak'>+</mo><mi>k</mi><mo>)</mo></mrow><msubsup><mi>&#945;</mi><mrow><mn>1</mn></mrow><mn>2</mn></msubsup><mo>]</mo></mrow><mi>z</mi><mo linebreak='goodbreak'>+</mo><mrow><mo>(</mo><msub><mi>&#945;</mi><mn>1</mn></msub><mo linebreak='badbreak'>+</mo><msub><mi>&#945;</mi><mn>2</mn></msub><mi>z</mi><mo>)</mo></mrow><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub></mrow></math>, with <math altimg='si18.svg'><mrow><mi>k</mi><mo>&#8801;</mo><msub><mi>&#946;</mi><mn>1</mn></msub><mo linebreak='goodbreak'>&#8722;</mo><msub><mi>&#946;</mi><mn>0</mn></msub></mrow></math>, whose two branches are distinguished by whether the anomalous-dimension feedback is active: the massless branch ( <math altimg='si30.svg'><mrow><msub><mi>&#945;</mi><mi>i</mi></msub><mo linebreak='goodbreak'>=</mo><mn>0</mn></mrow></math>) integrates to the square-root <italic>F</italic> <inf>0</inf> of Ref. [3], while the massive branch is an Abel equation of the second kind that admits no elementary closed form, the obstruction being controlled by <italic>&#945;</italic> <inf>1</inf> through <math altimg='si59.svg'><mrow><mi>d</mi><mi mathvariant='script'>S</mi><mo linebreak='goodbreak'>/</mo><mi>d</mi><mi>z</mi><mo linebreak='goodbreak'>=</mo><msub><mi>&#945;</mi><mn>1</mn></msub><mi>c</mi><mo linebreak='goodbreak'>/</mo><msup><mi>&#981;</mi><mrow><mo>&#8242;</mo><mn>2</mn></mrow></msup></mrow></math> with <math altimg='si60.svg'><mrow><mi>c</mi><mo linebreak='goodbreak'>=</mo><mi>k</mi><mrow><mo>(</mo><mi>k</mi><mo linebreak='badbreak'>+</mo><mn>1</mn><mo>)</mo></mrow><msubsup><mi>&#945;</mi><mrow><mn>1</mn></mrow><mn>2</mn></msubsup></mrow></math> and a non-constant Chini invariant. The Lambert- <italic>W</italic> structure nonetheless survives in renormalized form, <math altimg='si61.svg'><mrow><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub><mo linebreak='goodbreak'>=</mo><mo>&#8722;</mo><mn>1</mn><mo linebreak='goodbreak'>+</mo><msup><mi>&#956;</mi><mrow><mo>&#8722;</mo><mn>1</mn></mrow></msup><msub><mi>W</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>&#956;</mi><mspace width='0.16em' /><msup><mi>e</mi><mrow><mstyle mathvariant='normal'><mi>&#934;</mi></mstyle><mo>(</mo><mi>z</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow></mrow></math> with a self-consistently dressed argument <math altimg='si62.svg'><mrow><msup><mstyle mathvariant='normal'><mi>&#934;</mi></mstyle><mo>&#8242;</mo></msup><mo linebreak='goodbreak'>=</mo><msup><mi>&#981;</mi><mo>&#8242;</mo></msup><mo linebreak='goodbreak'>/</mo><mrow><mo>(</mo><mi>k</mi><mo linebreak='badbreak'>+</mo><mn>1</mn><mo>)</mo></mrow><mo linebreak='goodbreak'>&#8722;</mo><msubsup><mi>&#945;</mi><mrow><mn>1</mn></mrow><mn>2</mn></msubsup><mi>z</mi><mo linebreak='goodbreak'>/</mo><mrow><mo>(</mo><mn>1</mn><mo linebreak='badbreak'>+</mo><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow></math>, reducing to the explicit closed form in the massless limit. Predictions for all <italic>O</italic>( <italic>N</italic>) universality classes follow from the recursion, and a direct comparison with conventional fixed-order perturbation theory shows the renormalization-scale band of the latter contracting toward the single scheme-free SF-QFT value at every loop order.

Farrukh A. Chishtie

Exact scheme-independent generating function for the massive <italic>O</italic>( <italic>N</italic>) <italic>&#981;</italic> 4 two-point function in scale-factorized quantum field theory

We compute the complete nonplanar integrand for the correlation function of four lightest scalar operators in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></math> super Yang-Mills theory at five-loop order. This is equivalent to the supercorrelator of nine stress-tensor multiplets in the self-dual theory. Starting with an ansatz of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi></math> graphs, we impose constraints from light-cone limits, and fix the remaining freedom by using the reformulation of the theory in twistor space. We develop an efficient GPU-based algorithm for the numerical evaluation of the twistor rules. Our result for the integrand suggests a sharpened <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>f</mi></math>-graph genus conjecture. As an application, we extract the five-loop nonplanar anomalous dimension of the Konishi operator, whose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mo>/</mo><msubsup><mi>N</mi><mi mathvariant="normal">c</mi><mn>2</mn></msubsup></math> term is entirely given in terms of odd zeta values of transcendentality five and higher. Our code and result are provided in ancillary files.

Till Bargheer

Albert Bekov

Nonplanar four-point integrand and Konishi dimension in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></mrow></math> super-Yang-Mills theory at five loops

In this paper I continue the program of studying the strong coupling expansion of certain observables in $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory, which are given by a determinant with a matrix Bessel kernel. I show that, by reorganizing the transseries of the determinant at large values of the ’t Hooft coupling, a simple underlying structure emerges, in which each exponentially suppressed correction is related to the perturbative series in a simple way. This new approach provides an efficient method to generate the full transseries for $$ \mathcal{N} $$ = 4 SYM observables, such as the cusp anomalous dimension, multi-gluon scattering amplitudes, and the octagon form factor. Using high-precision numerical analysis, I verify the results and provide a complete description of the resurgence structure of the strong coupling expansion.

Hungary

Bercel Boldis

Strong coupling structure of $$ \mathcal{N} $$ = 4 SYM observables with matrix Bessel kernel