scholarly journals A conformal dispersion relation: correlations from absorption

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Dean Carmi ◽  
Simon Caron-Huot
Keyword(s):  
Dispersion Relation ◽  
Perfect Matching ◽  
Elliptic Integral ◽  
Absorptive Part ◽  
Integral Relation ◽  
Integral Function ◽  
Conformal Block ◽  
Tree Level ◽  
Conformal Field ◽  
Free Fields

Abstract We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its “absorptive part”, defined as a double discontinuity, times a theory-independent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the “inverted” conformal block with the ordinary conformal block.

scholarly journals Towards Feynman rules for conformal blocks

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sarah Hoback ◽  
Sarthak Parikh
Keyword(s):  
Field Theory ◽  
Effective Field Theory ◽  
Hypergeometric Functions ◽  
Power Series Expansion ◽  
Effective Field ◽  
Conformal Block ◽  
Tree Level ◽  
Conformal Blocks ◽  
Simple Set ◽  
Feynman Rules

Abstract We conjecture a simple set of “Feynman rules” for constructing n-point global conformal blocks in any channel in d spacetime dimensions, for external and exchanged scalar operators for arbitrary n and d. The vertex factors are given in terms of Lauricella hypergeometric functions of one, two or three variables, and the Feynman rules furnish an explicit power-series expansion in powers of cross-ratios. These rules are conjectured based on previously known results in the literature, which include four-, five- and six-point examples as well as the n-point comb channel blocks. We prove these rules for all previously known cases, as well as two new ones: the seven-point block in a new topology, and all even-point blocks in the “OPE channel.” The proof relies on holographic methods, notably the Feynman rules for Mellin amplitudes of tree-level AdS diagrams in a scalar effective field theory, and is easily applicable to any particular choice of a conformal block beyond those considered in this paper.

scholarly journals Bootstrap bounds on closed Einstein manifolds

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
James Bonifacio ◽  
Kurt Hinterbichler
Keyword(s):  
Compact Riemannian Manifold ◽  
Tree Level ◽  
Conformal Field Theories ◽  
Einstein Manifolds ◽  
Consistency Conditions ◽  
Conformal Field ◽  
Geometric Data ◽  
Conformal Bootstrap ◽  
Laplacian Operators ◽  
Kaluza Klein

Abstract A compact Riemannian manifold is associated with geometric data given by the eigenvalues of various Laplacian operators on the manifold and the triple overlap integrals of the corresponding eigenmodes. This geometric data must satisfy certain consistency conditions that follow from associativity and the completeness of eigenmodes. We show that it is possible to obtain nontrivial bounds on the geometric data of closed Einstein manifolds by using semidefinite programming to study these consistency conditions, in analogy to the conformal bootstrap bounds on conformal field theories. These bootstrap bounds translate to constraints on the tree-level masses and cubic couplings of Kaluza-Klein modes in theories with compact extra dimensions. We show that in some cases the bounds are saturated by known manifolds.

scholarly journals CFT unitarity and the AdS Cutkosky rules

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
David Meltzer ◽  
Allic Sivaramakrishnan
Keyword(s):  
Flat Space ◽  
Tree Level ◽  
Optical Theorem ◽  
Conformal Field Theories ◽  
Strongly Coupled ◽  
Conformal Field ◽  
Weakly Coupled ◽  
S Matrix ◽  
Space Limit ◽  
Diagrammatic Method

Abstract We derive the Cutkosky rules for conformal field theories (CFTs) at weak and strong coupling. These rules give a simple, diagrammatic method to compute the double-commutator that appears in the Lorentzian inversion formula. We first revisit weakly-coupled CFTs in flat space, where the cuts are performed on Feynman diagrams. We then generalize these rules to strongly-coupled holographic CFTs, where the cuts are performed on the Witten diagrams of the dual theory. In both cases, Cutkosky rules factorize loop diagrams into on-shell sub-diagrams and generalize the standard S-matrix cutting rules. These rules are naturally formulated and derived in Lorentzian momentum space, where the double-commutator is manifestly related to the CFT optical theorem. Finally, we study the AdS cutting rules in explicit examples at tree level and one loop. In these examples, we confirm that the rules are consistent with the OPE limit and that we recover the S-matrix optical theorem in the flat space limit. The AdS cutting rules and the CFT dispersion formula together form a holographic unitarity method to reconstruct Witten diagrams from their cuts.

scholarly journals Gauge-fixed Lattice QCD and the dispersion relation of Wilson fermions

2022 ◽  
Vol 258 ◽  
pp. 02003
Author(s):  
Giuseppe Burgio ◽  
Hannes Vogt
Keyword(s):  
Dispersion Relation ◽  
Lattice Qcd ◽  
Correlation Functions ◽  
Landau Gauge ◽  
Coulomb Gauge ◽  
Tree Level ◽  
Wilson Fermions

We show that, when investigating Wilson-fermions correlation functions on the lattice, one is bound to encounter major difficulties in defining their dispersion relation, even at tree level. The problem is indeed quite general and, although we stumbled upon it while studying Coulomb-gauge applications, it also affects gauge fixed studies in covariant gauges, including their most popular version, Landau gauge. In this paper we will discuss a solution to this problems based on a redefinition of the kinematic momentum of the fermion.

scholarly journals Model-dependence of minimal-twist OPEs in d > 2 holographic CFTs

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
A. Liam Fitzpatrick ◽  
Kuo-Wei Huang ◽  
David Meltzer ◽  
Eric Perlmutter ◽  
David Simmons-Duffin
Keyword(s):  
Stress Tensor ◽  
Central Charge ◽  
Effective Field ◽  
Tree Level ◽  
Probe Field ◽  
Conformal Field ◽  
Primary Operator ◽  
Higher Dimensional ◽  
Higher Spin ◽  
Light Probe

Abstract Following recent work on heavy-light correlators in higher-dimensional conformal field theories (CFTs) with a large central charge CT, we clarify the properties of stress tensor composite primary operators of minimal twist, [Tm], using arguments in both CFT and gravity. We provide an efficient proof that the three-point coupling $$ \left\langle {\mathcal{O}}_L{\mathcal{O}}_L\left[{T}^m\right]\right\rangle $$ O L O L T m , where $$ {\mathcal{O}}_L $$ O L is any light primary operator, is independent of the purely gravitational action. Next, we consider corrections to this coupling due to additional interactions in AdS effective field theory and the corresponding dual CFT. When the CFT contains a non-zero three-point coupling $$ \left\langle TT{\mathcal{O}}_L\right\rangle $$ TT O L , the three-point coupling $$ \left\langle {\mathcal{O}}_L{\mathcal{O}}_L\left[{T}^2\right]\right\rangle $$ O L O L T 2 is modified at large CT if $$ \left\langle TT{\mathcal{O}}_L\right\rangle \sim \sqrt{C_T} $$ TT O L ∼ C T . This scaling is obeyed by the dilaton, by Kaluza-Klein modes of prototypical supergravity compactifications, and by scalars in stress tensor multiplets of supersymmetric CFTs. Quartic derivative interactions involving the graviton and the light probe field dual to $$ {\mathcal{O}}_L $$ O L can also modify the minimal-twist couplings; these local interactions may be generated by integrating out a spin-ℓ ≥ 2 bulk field at tree level, or any spin ℓ at loop level. These results show how the minimal-twist OPE coefficients can depend on the higher-spin gap scale, even perturbatively.

Integral relation for the t-dependence of the It = 1 ππ absorptive part

1973 ◽  
Vol 45 (1) ◽  
pp. 48-52 ◽  
Author(s):  
J.L. Basdevant ◽  
C. Schomblond
Keyword(s):  
Absorptive Part ◽  
Integral Relation

scholarly journals An application of the new function method to the Zhiber–Shabat equation

2017 ◽  
Vol 7 (3) ◽  
pp. 271-274
Author(s):  
Tolga Aktürk ◽  
Yusuf Gürefe ◽  
Yusuf Pandır
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Elliptic Integral ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Integral Function ◽  
Jacobi Elliptic Functions ◽  
New Approach ◽  
Function Solution ◽  
Wave Solutions

This paper applies a new approach including the trial equation based on the exponential function in order to find new traveling wave solutions to Zhiber-Shabat equation. By the using of this method, we obtain a new elliptic integral function solution. Also, this solution can be converted into Jacobi elliptic functions solution by a simple transformation.

scholarly journals Helicity basis for three-dimensional conformal field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Simon Caron-Huot ◽  
Yue-Zhou Li
Keyword(s):  
Conformal Symmetry ◽  
Mean Field ◽  
Three Dimensional ◽  
Flat Space ◽  
Three Dimensions ◽  
Tree Level ◽  
Conformal Field ◽  
Anomalous Dimensions ◽  
Conserved Currents ◽  
Level Order

Abstract Three-point correlators of spinning operators admit multiple tensor structures compatible with conformal symmetry. For conserved currents in three dimensions, we point out that helicity commutes with conformal transformations and we use this to construct three-point structures which diagonalize helicity. In this helicity basis, OPE data is found to be diagonal for mean-field correlators of conserved currents and stress tensor. Furthermore, we use Lorentzian inversion formula to obtain anomalous dimensions for conserved currents at bulk tree-level order in holographic theories, which we compare with corresponding flat-space gluon scattering amplitudes.

scholarly journals Two-photon processes in conformal QCD: resummation of the descendants of leading-twist operators

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
V.M. Braun ◽  
Yao Ji ◽  
A.N. Manashov
Keyword(s):  
Deeply Virtual Compton Scattering ◽  
Virtual Compton Scattering ◽  
Field Theories ◽  
Tree Level ◽  
Operator Product ◽  
Conformal Field Theories ◽  
Closed Expression ◽  
Conformal Field ◽  
Two Photon ◽  
Twist Operators

Abstract Using some techniques of conformal field theories, we find a closed expression for the contribution of leading twist operators and their descendants, obtained by adding total derivatives, to the operator product expansion (OPE) of two electromagnetic currents in QCD. Our expression resums contributions of all twists and to all orders in perturbation theory up to corrections proportional to the QCD β-function. At tree level and to twist-four accuracy, our result agrees with the expression derived earlier by a different method. The results are directly applicable to deeply-virtual Compton scattering and, e.g., γγ∗ annihilation in two mesons. As a byproduct, we derive a simple representation for the OPE of two scalar currents that is convenient for applications.

scholarly journals Theoretical analysis of stress intensity factor for two unequal collinear cracks subjected to internal pressure and compressive stress

2021 ◽  
Vol 37 ◽  
pp. 584-596
Author(s):  
Xianshang Zhang ◽  
Qingming Long ◽  
Tao Zheng ◽  
Zheming Zhu ◽  
Meng Wang ◽  
...  
Keyword(s):  
Stress Intensity ◽  
Internal Pressure ◽  
Compressive Stress ◽  
Elliptic Integral ◽  
Complex Stress ◽  
Integral Function ◽  
Collinear Cracks ◽  
Mode Ii Crack ◽  
The Difference ◽  
Good Agreement

Abstract In this paper, a new solution of the stress intensity factors (SIFs) for two unequal collinear cracks is developed considering internal pressure, friction and compressive stress. The complex stress function and elliptic integral function are used to obtain the analytical solution of the SIFs, and it shows a good agreement with the previous researchers’ solutions and numerical results. The theoretical results show that the difference and interaction of the SIFs at cracks’ tips are caused by crack geometry parameters, and they also indicate that the internal pressure leads to the SIFs of a mode I crack and affects the SIFs of a mode II crack because of friction.

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