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

Recently, Guo and Sun derived an identity for factorial Grothendieck polynomials which is a generalization of the one for Schur polynomials by Fehér, Némethi and Rimányi. We analyze the identity from the point of view of quantum integrability, based on the correspondence between the wavefunctions of a five-vertex model and the Grothendieck polynomials. We give another proof using the quantum inverse scattering method. We also apply the same idea and technique to derive an identity for factorial Grothendieck polynomials for rectangular Young diagrams. Combining with the Guo-Sun identity, we get a duality formula. We also discuss a q-deformation of the Guo-Sun identity.

Japan

Kohei Motegi

Integrability approach to Fehér-Némethi-Rimányi-Guo-Sun type identities for factorial Grothendieck polynomials

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 transverse momentum distributions of final-state particles are very important for high energy collision physics. In this work, we investigate <math id="M3"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> and <math id="M4"><mrow><mi>Υ</mi></mrow></math> meson distributions in the framework of a particle-production source, where Tsallis statistics are consistently incorporated. The results are in good agreement with the experimental data in <math id="M5"><mi>p</mi><mtext>-</mtext><mi>p</mi></math> and <math id="M6"><mrow><mi>p</mi></mrow></math>-Pb collisions at LHC energies. The temperature of the emission source and the nonequilibrium degree of the collision system are extracted.

China

Bao-Chun Li

Ting Bai

Yuan-Yuan Guo

Fu-Hu Liu

On And Transverse Momentum Distributions in High Energy Collisions

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.

United States

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).

The dihadron azimuthal correlations in p-p collisions at <math id="M3"><msqrt><msub><mrow><mi>s</mi></mrow><mrow><mtext>N</mtext><mtext>N</mtext></mrow></msub></msqrt><mo>=</mo><mn>7</mn></math> TeV and p-Pb collisions at <math id="M4"><msqrt><msub><mrow><mi>s</mi></mrow><mrow><mtext>N</mtext><mtext>N</mtext></mrow></msub></msqrt><mo>=</mo><mn>5.02</mn></math> TeV are investigated in the framework of a multisource thermal model. The model can approximately describe the experimental results measured in the Large Hadron Collider. We find the <math id="M5"><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></math> amplitude of the source is magnified and the source translates along the direction.

Dihadron Azimuthal Correlations in p-p Collisions at TeV and p-Pb Collisions at TeV

We analyze transverse momentum spectra of <math id="M6"><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msubsup></mrow></math> and <math id="M7"><mrow><msup><mrow><mi>K</mi></mrow><mrow><mi>*</mi><mn>0</mn></mrow></msup></mrow></math> at midrapidity in <math id="M8"><mrow><mi>d</mi></mrow></math> + Au, Cu + Cu, and <math id="M9"><mi>p</mi><mo>+</mo><mi>p</mi></math> collisions at <math id="M10"><msqrt><msub><mrow><mi>s</mi></mrow><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>=</mo><mn>200</mn></math> GeV in the formworks of Tsallis statistics and Boltzmann statistics, respectively. Both of them can describe the transverse momentum spectra and extract the thermodynamics parameters of matter evolution in the collisions. The parameters are helpful for us to understand the thermodynamics factors of the particle production.

Guo-Xing Zhang

Transverse Momentum Spectra of And at Midrapidity in + Au, Cu + Cu, and Collisions at GeV

A proof is given of the formula, recently proposed by Cachazo, He and Yuan (CHY) for gluon tree amplitudes in pure Yang-Mills theory in arbitrary dimension. The approach is to first establish the corresponding result for massless ϕ 3 theory using the BCFW recurrence relation and then to extend this to the gauge theory case. Additionally, it is shown that the scattering equations introduced by CHY can be generalized to massive particles, enabling the description of tree amplitudes for massive ϕ 3 theory.

Louise Dolan

Peter Goddard

Proof of the formula of Cachazo, He and Yuan for Yang-Mills tree amplitudes in arbitrary dimension