Alternative to the embedding formalism, we provide a group theoretic approach to the conformal primary basis for the massless field with arbitrary helicity. To this end, we first point out that <math><mi>s</mi><mi>l</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math> isometry gets enhanced to <math><mi>s</mi><mi>l</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo><mo>×</mo><mi>D</mi></math> symmetry for the solution space of the massless field with <math><mi>D</mi></math> the bulk dilatation. Then, associated with <math><mi>s</mi><mi>l</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo><mo>×</mo><mi>D</mi></math> symmetry, we introduce the novel quadratic Casimirs and relevant tensor/spinor fields to derive two explicit constraints on the bulk dilatation and <math><mi>s</mi><mi>l</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math> Casimirs. With this, we further argue that the candidate conformal primary basis can be constructed out of the infinite tower of the descendants of the left and right highest (lowest) conformal primary wave function of <math><mi>s</mi><mi>l</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math> Lie algebra, and the corresponding celestial conformal weights are determined by the bulk scaling dimension through solving out the exact on-shell conformal primary wave functions, where on top of the two kinds of familiar-looking on-shell conformal primary wave functions, we also obtain another set of independent on-shell conformal primary wave functions for the massless field with helicity <math><mo>|</mo><mi>s</mi><mo>|</mo><mo>≥</mo><mn>1</mn></math>. In passing, we also develop the relationship between the four-dimensional Lorentz Lie algebra and two-dimensional conformal Lie algebra from scratch, and present an explicit derivation for the two important properties associated with the conformal primary wave functions.

China

Yuan Chen

Mingfeng Li

Kai Shi

Hongbao Zhang

United States

Jingchao Zhang

<math><mi>s</mi><mi>l</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo><mo>×</mo><mi>D</mi></math> symmetry and conformal primary basis for massless fields

The algebraic approach to the spectrum of quasinormal modes has been made as simple as possible for the BTZ black hole by the strategy developed in [16]. By working with the self-dual warped AdS black hole, we demonstrate in an explicit way that such a strategy can be well adapted to those warped AdS balck holes with the SL(2, R) × U(1) isometry. To this end, we first introduce two associated tensor fields with the quadratic Casimir of SL(2, R) × U(1) Lie algebra in the self-dual warped AdS black hole and show that they correspond essentially to the metric and volume element up to a constant prefactor, respectively. Then without appealing to any concrete coordinate system, we can further show that the solutions to the equations of motion for the scalar, vector, spinor fields all fall into the representations of the SL(2, R) × U(1) Lie algebra by a purely abstract tensor and spinor analysis. Accordingly, the corresponding spectrum of quasinormal modes for each fixed azimuthal quantum number can be derived algebraically as the infinite tower of descendants of the highest weight mode of the SL(2, R) Lie subalgebra.

Wei Guo

SL(2, R) × U(1) symmetry and quasinormal modes in the self-dual warped AdS black hole

Abstract The open N = 2 string theory is defined on the 4D space-time with the split signature (+, +, −, −). The string field theory action of the open N = 2 string theory is described by the 4D Wess–Zumino–Witten (WZW$_{4}$) model. The equation of motion of the WZW$_{4}$ model is the Yang equation, which is equivalent to the anti-self-dual Yang–Mills equation. In this paper, we study soliton-type classical solutions of the WZW$_{4}$ model in the split signature by calculating the action density of the WZW$_{4}$ model. We find that the action density of the one-soliton solutions is localized on a 3D hyperplane. This shows that there would be codimension-one-solitonic objects, or equivalently some kind of three-branes in the open N = 2 string theory. We also prove that, in the asymptotic region of the space-time, the action density of the n-soliton solutions is a “non-linear superposition” of n one-solitons. This suggests the existence of n intersecting three-branes in the N = 2 strings. Finally we make a reduction to a (1 + 2)D real space-time to calculate the energy densities of the soliton solutions. We can successfully evaluate the energy distribution for the two-soliton solutions and find that there is no singularity in the interacting region. This implies the existence of smooth intersecting codimension-one branes in the whole region. Soliton solutions in the Euclidean signature are also discussed.

Japan

Masashi Hamanaka

Shan-Chi Huang

Hiroaki Kanno

Solitons in open = 2 String Theory Kahler-Chen-Simons theory A tour through N=2 strings On soliton solutions of the anti-self-dual Yang-Mills equations from the perspective of integrable systems Soliton solutions in an integrable chiral model in 2+1 dimensions On the space-time monopole equation A review on tachyon condensation in open string field theories Soliton equations as dynamical systems on infinite dimensional Grassmann manifold

Stimulated by the newly observed charged hidden-charm state <math><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo>(</mo><mn>3985</mn><msup><mo>)</mo><mo>−</mo></msup></math> by BESIII Collaboration, <math><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo>(</mo><mn>4000</mn><msup><mo>)</mo><mo>+</mo></msup></math>, <math><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo>(</mo><mn>4220</mn><msup><mo>)</mo><mo>+</mo></msup></math> and the excited <math><msubsup><mi>B</mi><mi>s</mi><mn>0</mn></msubsup></math> states by LHCb Collaboration, a full calculation including masses, decay widths, and the inner structures of the states has emerged in the chiral quark model. For <math><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub></math> states, we assign quantum numbers <math><mi>I</mi><mo>(</mo><msup><mi>J</mi><mi>P</mi></msup><mo>)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>(</mo><msup><mn>1</mn><mo>+</mo></msup><mo>)</mo></math> and quark composition <math><mi>c</mi><mover><mi>c</mi><mo>¯</mo></mover><mi>s</mi><mover><mi>u</mi><mo>¯</mo></mover></math> according to the experiment. For <math><msubsup><mi>B</mi><mi>s</mi><mn>0</mn></msubsup></math> states, systematically, investigations are performed with <math><mi>I</mi><mo>(</mo><msup><mi>J</mi><mi>P</mi></msup><mo>)</mo><mo>=</mo><mn>0</mn><mo>(</mo><msup><mn>0</mn><mo>+</mo></msup><mo>)</mo><mo>,</mo><mn>0</mn><mo>(</mo><msup><mn>1</mn><mo>+</mo></msup><mo>)</mo><mo>,</mo><mn>0</mn><mo>(</mo><msup><mn>2</mn><mo>+</mo></msup><mo>)</mo></math> in both two-body <math><mi>b</mi><mover><mi>s</mi><mo>¯</mo></mover></math> and four-body <math><mrow><mi>b</mi><mover><mrow><mi>s</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>q</mi><mover><mrow><mi>q</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></math> (<math><mrow><mi>q</mi><mo>=</mo><mi>u</mi></mrow></math> or <math><mrow><mi>d</mi></mrow></math>) systems. Each tetraquark calculation takes all structures including meson-meson, diquark-antidiquark, and all possible color configurations into account. Among the numerical techniques to solve the two-body and four-body Schrödinger equation, the spatial wave functions are expanded in series of Gaussian basis functions for high precision, which is the way Gaussian expansion method (GEM) so called. Our results indicate that the low-lying states of the four-quark system are all higher than the corresponding thresholds either for <math><mi>c</mi><mover><mi>c</mi><mo>¯</mo></mover><mi>s</mi><mover><mi>u</mi><mo>¯</mo></mover></math> or for <math><mi>b</mi><mover><mi>s</mi><mo>¯</mo></mover><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover></math> systems. With the help of the real scaling method, we found two molecular resonance states with masses of 4023 and 4042 MeV for the <math><mi>c</mi><mover><mi>c</mi><mo>¯</mo></mover><mi>s</mi><mover><mi>u</mi><mo>¯</mo></mover></math> system. The state <math><mi>c</mi><mover><mi>c</mi><mo>¯</mo></mover><mi>s</mi><mover><mi>u</mi><mo>¯</mo></mover><mo>(</mo><mn>4042</mn><mo>)</mo></math> has a close mass and decay width with the recent observed state <math><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo>(</mo><mn>3985</mn><msup><mo>)</mo><mo>−</mo></msup></math>. For the <math><mi>b</mi><mover><mi>s</mi><mo>¯</mo></mover><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover></math> system with <math><mi>J</mi><mo>=</mo><mn>0</mn></math>, some resonance states are also found. The newly observed excited <math><msubsup><mi>B</mi><mi>s</mi><mn>0</mn></msubsup></math> states can be accommodated in the chiral quark model as <math><mn>2</mn><mi>S</mi></math> or <math><mn>1</mn><mi>D</mi></math> states, and the mixing with four-quark states also needs to be considered.

Xiaoyun Chen

Yue Tan

Study on <math><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub></math> and excited <math><msubsup><mi>B</mi><mi>s</mi><mn>0</mn></msubsup></math> states in the chiral quark model

We extend on-shell bootstrap methods from spacetime amplitudes to the world sheet objects of the Cachazo-He-Yuan formalism. We observe that the integrands corresponding to tree-level nonlinear sigma model, Yang-Mills and <math><mo>(</mo><mi>D</mi><mi>F</mi><msup><mo>)</mo><mn>2</mn></msup></math> theory are determined by demanding enhanced UV scaling under Britto-Cachazo-Feng-Witten shifts. Back in spacetime, we also find that <math><mo>(</mo><mi>D</mi><mi>F</mi><msup><mo>)</mo><mn>2</mn></msup></math> theory is fixed by gauge invariance/UV scaling and simple locality assumptions.

France

Laurentiu Rodina

UV consistency conditions for Cachazo-He-Yuan integrands

The Cachazo-He-Yuan formula was originally proposed to describe on-shell scattering of particles from a single massless field. We present a method to modify it to include several interacting scalar fields, all possessing different masses and possibly off-shell momenta. The method is applied to Yukawa interactions between a number of scalar nucleons and pions, and to the <math><msub><mi>ϕ</mi><mn>1</mn></msub><msub><mi>ϕ</mi><mn>2</mn></msub><msub><mi>ϕ</mi><mn>3</mn></msub></math> coupling of three different scalar fields. Composite models constructed from existing theories can be used to broaden the scope of the method. The modification is applied to describe Compton scattering from a massive particle and to photon bremsstrahlung. It is also employed to generalize the disk function <math><mi>Z</mi></math> and the sphere function <math><mi>J</mi></math>.

Canada

C. S. Lam

Cachazo-He-Yuan theory for several fields

Möbius invariance is used to construct gluon tree amplitudes in the Cachazo, He, and Yuan (CHY) formalism. If it is equally effective in steering the construction of off-shell tree amplitudes, then the S-matrix CHY theory can be used to replace the Lagrangian Yang-Mills theory. Unfortunately that is not possible. We find that the CHY formula can indeed be modified to obtain a Möbius-invariant off-shell amplitude <math><msub><mi>M</mi><mi>P</mi></msub></math>, but this modified amplitude lacks local gauge invariance, which can be restored to give the correct Yang-Mills amplitude only by the addition of a complementary amplitude <math><msub><mi>M</mi><mi>Q</mi></msub></math>. Although neither <math><msub><mi>M</mi><mi>P</mi></msub></math> nor <math><msub><mi>M</mi><mi>Q</mi></msub></math> is fully gauge invariant, both are partially gauge invariant in a sense to be explained.

Off-shell Yang-Mills amplitude in the Cachazo-He-Yuan formalism

We show that the use of Sudakov variables greatly simplifies the study of the solutions to the scattering equations in the Cachazo-He-Yuan formalism. We work in the center-of-mass frame for the two incoming particles, which partially fixes the SL(2, ℂ) redundancy in the integrand defining the scattering amplitudes, the remaining freedom translates into a global shift in the azimuthal angle of the outgoing particles. Studying four- and five-particle amplitudes, we see how an appropriate choice of this phase allows for algebraic simplifications when finding solutions to the scattering equations, as well as in the expression of the scattering amplitudes in terms of the locations of the punctures in the sphere. These punctures themselves are remarkably simple functions of the Sudakov parameters.

Spain

Grigorios Chachamis

Diego Jiménez

Agustín Vera

Miguel Vázquez-Mozo

Sudakov representation of the Cachazo-He-Yuan scattering equations formalism

The last few years have been witness to a proliferation of new results concerning heavy exotic hadrons. Experimentally, many new signals have been discovered that could be pointing towards the existence of tetraquarks, pentaquarks, and other exotic configurations of quarks and gluons. Theoretically, advances in lattice field theory techniques place us at the cusp of understanding complex coupled-channel phenomena, modelling grows more sophisticated, and effective field theories are being applied to an ever greater range of situations. It is thus an opportune time to evaluate the status of the field. In the following, a series of high priority experimental and theoretical issues concerning heavy exotic hadrons is presented.

R.A. Briceño

T.D. Cohen

S. Coito

J.J. Dudek

E. Eichten

Germany

C.S. Fischer

M. Fritsch

W. Gradl

A. Jackura

M. Kornicer

Brazil

G. Krein

R. F. Lebed

F. A. Machado

R. E. Mitchell

C.J. Morningstar

Ireland

M. Peardon

M. R. Pennington

K. Peters

J. M. Richard

C. P. Shen

M.R. Shepherd

T. Skwarnicki

E. S. Swanson

A. P. Szczepaniak

C. Z. Yuan

Issues and Opportunities in Exotic HadronsSupported by U.S. Department of Energy (Cohen); the Institute of Modern Physics and Chinese Academy of Sciences under contract Y104160YQ0 and agreement No. 2015-BH-02 (Coito); the U.S. Department of Energy, for grant DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, manages and operates Jefferson Laboratory and DE-SC0006765, Early Career award (Dudek); Fermilab, operated by the Fermi Research Alliance under contract number DEAC02-07CH11359 with the U.S. Department of Energy (Eichten); BMBF, under contract No. 06GI7121, and the DAAD under contract No. 56889822 and by the Helmholtz International Center for FAIR within the LOEWE program of the State of Hesse (Fischer); the German Research Foundation DFG under contract number Collaborative Research Centre CRC-1044 (Gradl); the Conselho Nacional de Desenvolvimento CientÃ­fico e TecnolÃ³gico - CNPq, Grant No. 305894/2009-9 and FundaÃ§Ã£o de Amparo Ã Pesquisa do Estado de SÃ£o Paulo - FAPESP, Grant No. 2013/01907-0 (Krein); U.S. National Science Foundation, under grants PHY-1068286 and PHY-1403891 (Lebed); the Brazilian National Council for Scientific and Technological Development under grant CNPq/CAPES-208188/2014-2 (Machado); U.S. Department of Energy under grant DE-FG02-05ER41374 (Mitchell); U.S. National Science Foundation under grant PHY-1306805 (Morningstar); U.S. Department of Energy, supported by Jefferson Science Associates, LLC under contract No. DE-AC05-06OR23177 (Pennington); the National Natural Science Foundation of China (NSFC) under contract No. 11575017, the Ministry of Science and Technology of China under Contract No. 2015CB856701 (Shen); U.S. Department of Energy, under grant DE-FG02-05ER41374 (Shepherd); U.S. National Science Foundation under grant PHY-1507572 (Skwarnicki); U.S. Department of Energy, under contract DE-AC05-06OR23177 and grant DE-FG0287ER40365 (Szczepaniak); the National Natural Science Foundation of China (NSFC) under contract numbers 11235011 and 11475187 (Yuan).

In the present paper, the exact solutions of spherically symmetrical Einstein field equations describing wormholes supported by phantom energy that violates the null energy condition from Shan–Chen background fluid are obtained. We have considered the important case of the model parameter ψ≈1 , which corresponds to the “saturation effect”, and this regime corresponds to an effective form of “asymptotic freedom” for the fluids, but occurring at cosmological rather than subnuclear scales. Then we investigate the allowed range for the values of the model parameters g and ω when the spacetime metrics describe wormholes and discuss the possible singularities of the solutions, finding that the obtained spacetimes are geodesically complete. Furthermore, we construct two traversable wormholes through matching our obtained interior solutions to the exterior Schwarzschild solutions and analyze the traversablities of the wormholes. Finally, we consider the case of anisotropic pressure and discover that the transverse pressure also crosses the phantom divide -1 with the growth of the wormhole dimension, and it tends to be the same as the radial pressure with the growth of the wormhole radius.

Deng Wang

Xin-He Meng

Wormholes supported by phantom energy from Shan–Chen cosmological fluids