<p>We consider evolution of observables which depend on a small but fixed value of longitudinal momentum fraction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>, to high rapidity, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>η</mi><mo>&gt;</mo><mi>ln</mi><mtext> </mtext><mn>1</mn><mo>/</mo><mi>x</mi></mrow></math>. We show that this evolution is not given by the Jalilian-Marian–Iancu–McLerran–Weigert–Leonidov–Kovner (JIMWLK) (or Balitsky-Kovchegov) equation. We derive the evolution Hamiltonian—<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>H</mi><mrow><mi>CSS</mi><mo>−</mo><mi>JIMWLK</mi></mrow></msub></math> which generates this evolution in the cases of dilute and dense projectile wave function. The two limits yield identical results for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>H</mi><mrow><mi>CSS</mi><mo>−</mo><mi>JIMWLK</mi></mrow></msub></math>. We show that the resulting evolution for the gluon Transverse-Momentum-Dependent is identical to the (double logarithmic) perturbative Collins-Soper-Sterman evolution equation in the longitudinal resolution parameter at a fixed and very large transverse resolution.</p>

United States

Haowu Duan

Alex Kovner

Israel

Michael Lublinsky

Collins-Soper-Sterman Hamiltonian: High energy evolution of rapidity dependent observables

We continue exploring the Unitary Toy Model (UTM) as a playground for high energy collisions in QCD. Our new approach is based on the diagonalization of the evolution Hamiltonian. Part of the spectrum can be identified with intercepts of dressed Pomerons. Analogously to QCD, a multi-Pomeron expansion of the S-matrix is badly divergent asymptotic series. Yet we succeeded to establish resummation procedures resulting in a well behaved S-matrix. In addition the Hamiltonian possesses negative eigenvalues, which dominate the approach of the S-matrix to saturation. We are hopeful that important lessons about BFKL-based Pomeron calculus could be taken from the toy world to real QCD.

Eugene Levin

High energy scattering in the Unitary Toy Model

We reanalyze the origin of the large transverse logarithms associated with the QCD one loop β function coefficient in the NLO JIMWLK Hamiltonian. We show that some of these terms are not associated with the running of the QCD coupling constant but rather with the DGLAP evolution. The DGLAP-like resummation of these logarithms is mandatory within the JIMWLK Hamiltonian, as long as the color correlation length in the projectile is larger than that in the target. This regime in fact covers the whole range of rapidities at which JIMWLK evolution is supposed to be applicable. We derive the RG equation that resums these logarithms to all orders in α  s  in the JIMWLK Hamiltonian. This is a nonlinear equation for the eikonal scattering matrix S(x). We solve this equation, and perform the DGLAP resummation in two simple cases: the dilute limit, where both the projectile and the target are far from saturation, and the saturated regime, where the target correlation length also determines its saturation momentum.

Vladimir Skokov

Zichen Zhao

Not all that is β 0 is β-function: the DGLAP resummation and the running coupling in NLO JIMWLK

Various conventional running coupling prescriptions reproducing β 0-dependent terms of NLO JIMWLK are reviewed and found to be theoretically inconsistent: the JIMWLK evolution Hamiltonian with running coupling violates the requirement of positive semidefiniteness. This requirement appears to be tightly related to the possibility of having a Langevin formulation for the evolution.

Poland

Tolga Altinoluk

Guillaume Beuf

On running coupling in the JIMWLK evolution and its Langevin formulation

We revisit the calculation of the cross section for forward inclusive single hadron production in <math><mi>p</mi><mi>A</mi></math> collisions within the hybrid approach. We show that the proper framework to perform this calculation beyond leading order is not the collinear factorization, as has been assumed so far, but the transverse momentum dependent (TMD) factorized framework. Within the TMD factorized approach we show that all the large transverse logarithms appearing in the fixed order calculation are resummed into the evolution of the TMD parton distribution functions and TMD fragmentation functions with factorization scale. The resulting expressions, when written in terms of TMDs evolved to the appropriate, physically well understood factorization scale, contain no additional large logarithms. The absence of any large logarithms in the resummed result should ensure positivity of the cross section and eradicate the persistent problem that has plagued the previous attempts at calculating this observable in the hybrid approach.

Spain

Néstor Armesto

Alexander Kovner

Single inclusive particle production at next-to-leading order in proton-nucleus collisions at forward rapidities: Hybrid approach meets TMD factorization

NLO evolution of the Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) equation with massless quarks was derived a few years ago. We make a step further to compute the evolution kernels focusing on the effects due to finite quark masses. To this goal, the light-cone wave function of a fast moving dilute hadronic projectile is computed up to   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (g 3) in QCD coupling constant. Compared with the massless case, a new IR divergence emerges, which is eventually canceled by a mass dependent counter term. Our results extend the theoretical tools used in physics of gluon saturation and aim at improving precision in future phenomenological applications.

Germany

Lin Dai

NLO JIMWLK evolution with massive quarks

We explore possible extensions of the t-channel and s-channel unitary model of high energy evolution in zero transverse dimensions appropriate to very high energy/atomic number where the dipole density in a toy hadron is parametrically high. We suggest that the appropriate generalization is to allow emission of more than one dipole in a single step of energy evolution. We construct explicitly such a model that preserves the t-channel and s-channel unitarity and have the correct low density limit, and study the particle multiplicity distribution resulting from this evolution. We consider initial conditions of a single dipole and many dipoles at initial rapidity. We observe that the saturation regime in this model is preceded by a parametric range of rapidities   <math> <mfrac> <mn>1</mn> <msub> <mi>α</mi> <mi>s</mi> </msub> </mfrac> <mo>ln</mo> <mfrac> <mn>1</mn> <msub> <mi>α</mi> <mi>s</mi> </msub> </mfrac> <mo>&lt;</mo> <mi>Y</mi> <mo>&lt;</mo> <mfrac> <mn>1</mn> <msub> <mi>α</mi> <mi>s</mi> </msub> </mfrac> <mo>ln</mo> <mfrac> <mn>1</mn> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mfrac> </math>  $$ \frac{1}{\alpha_s}\ln \frac{1}{\alpha_s}&lt;Y&lt;\frac{1}{\alpha_s}\ln \frac{1}{\alpha_s^2} $$ , where the saturation effects are still unimportant, but multiple emissions determine the properties of the evolution. We also discuss the influence of the saturation on the parton cascade and, in particular, find that in the saturation regime the entropy of partons becomes S ≈   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{1}{2} $$  ln N where N is the mean multiplicity.

Chile

Nuclei in the toy world: beyond the Pomeron in zero transverse dimensions

Within the dense-dilute Color Glass Condensate approach, and using the Golec–Biernat–Wuesthoff model for the dipole scattering amplitude, we calculate  $$v_2^2$$  <math> <msubsup> <mi>v</mi> <mn>2</mn> <mn>2</mn> </msubsup> </math>   as well as the correlations between  $$v_2^2$$  <math> <msubsup> <mi>v</mi> <mn>2</mn> <mn>2</mn> </msubsup> </math>   and both the total multiplicity and the mean transverse momentum of produced particles. We find that the correlations are generally very small consistent with the observations. We note an interesting sharp change in the value of  $$v^2_2$$  <math> <msubsup> <mi>v</mi> <mn>2</mn> <mn>2</mn> </msubsup> </math>   as well as of its correlations as a function of the width of the transverse momentum bin. This crossover is associated with the change from Bose enhancement dominance of the correlation for narrow bin to HBT dominated correlations for larger bin width.

Angular correlations in pA collisions from CGC: multiplicity and mean transverse momentum dependence of  $$v_2$$  <math> <msub> <mi>v</mi> <mn>2</mn> </msub> </math>

The placement of a magnetic monopole into an electrically neutral chiral plasma with a nonzero axial density results in an electric polarization of the matter. The electric current produced by the chiral magnetic effect is balanced by charge diffusion and Ohmic dissipation, which generates a nontrivial charge distribution. In turn, the latter induces a separation of chiralities along the magnetic field of the monopole due to the chiral separation effect. We find the stationary states of such a system, with vanishing total electric current and stationary axial current balanced by the chiral anomaly. In this solution, the monopole becomes “dressed” with an electric charge that is proportional to the averaged chiral density of the matter—forming a chiral dyon. The interplay between the chiral effects on the one hand, and presence of magnetic field of the monopole on the other, may affect the evolution of the monopole density in the early Universe, contribute to the process of baryogenesis, and can also be instrumental for detection of relic monopoles using chiral materials.

Jared Reiten

Russia

Andrey V. Sadofyev

Magnetic monopole in a chiral plasma: Chiral dyon

An effective action for diffusion of a conserved U(1) charge is derived to all orders in the derivative expansion within a holographic model dual to the Schwinger-Keldysh closed time path. A systematic approach to solution of the 5D Maxwell equations in a doubled Schwarzschild-AdS5 black brane geometry is developed. Constitutive relation for the stochastic charge current is shown to have a term induced by thermal fluctuations (coloured noise). All transport coefficient functions parameterising the effective action and constitutive relations are computed analytically in the hydrodynamic expansion, and then numerically for finite momenta.