We develop an approach to QCD evolution based on the sequential Born-Oppenheimer approximations that include higher and higher frequency modes as the evolution parameter is increased. This Born-Oppenheimer renormalization group is a general approach which is valid for the high energy evolution as well as the evolution in transverse resolution scale $\textit{Q}$ . In the former case it yields the frequency ordered formulation of high energy evolution, which includes both the eikonal splittings which produce gluons with low longitudinal momentum, and the DGLAP-like splittings which produce partons with high transverse momentum. In this, first paper of the series we lay out the formulation of the approach, and derive the expression for the evolved wave function of a hadronic state. We also discuss the form of the $\textit{S}$-matrix which is consistent with the frequency ordeing.

United States

Haowu Duan

Germany

Alex Kovner

Israel

Michael Lublinsky

Born-Oppenheimer renormalization group for high energy scattering: the setup and the wave function

In [1], we have introduced the Born-Oppenheimer (BO) renormalization group approach to high energy hadronic collisions and derived the BO approximation for the light cone wave function of a fast moving projectile hadron. In this second paper, we utilize this wave function to derive the BO evolution of partonic distributions in the hadron — the gluon transverse momentum and integrated parton distributions (TMD and PDF respectively). The evolution equation for the TMD contains a linear and a nonlinear term. The linear term reproduces the Collins-Soper-Sterman (CSS) equation with a physical relation between the transverse and longitudinal resolution scales. We explain how this equivalence arises, even though the BO and CSS cascades are somewhat different in structures. The nonlinear term in the evolution has a very appealing physical meaning: it is a correction due to stimulated emission, which enhances emission of gluons (bosons) into states with a nonzero occupation. For the evolution of the PDF we again find a linear and nonlinear term. At not very small Bjorken $\textit{x}$, the linear term recovers the DGLAP equation in the leading logarithmic approximation. At small $\textit{x}$ however there are contributions from gluon splittings which are in the BFKL kinematics leading to a modification of the DGLAP equation. The nonlinear terms have the same physical origin as in the equation for the TMD — the stimulated emission corrections. Interestingly the nonlinear corrections are the most important for the virtual terms, so that the net correction to the DGLAP is negative and mimics shadowing, although the physical origin of the nonlinearity is very different.

Born-Oppenheimer renormalization group for high energy scattering: CSS, DGLAP and all that

We explore the recently derived equation that resums DGLAP corrections to the JIMWLK Hamiltonian in the simplified setting of the SU(2) gauge theory. We solve the equation numerically for the scattering matrix of a dressed gluon for a particular initial condition, that corresponds to a dipole initial state. We evolve the $\textit{S}$-matrix of a single dressed gluon from the scale $\textit{Q}$ , which is the inverse color correlation length in the projectile to $\textit{Q}$ ≫ $\textit{Q}$ which corresponds to the hard resolution scale provided by the target. As expected, $\textit{S}$ ceases to be unitary if evolved to significant ln $$ {Q}^2/{Q}_P^2 $$ . Our numerical results indicate an interesting universal (independent of the coupling constant) pattern for this deviation from unitarity.

Spain

Néstor Armesto

Switzerland

Víctor López-Pardo

Exploring the DGLAP resummation in the JIMWLK Hamiltonian

<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>

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

We calculate the entanglement entropy of a model proton wave function in coordinate space by integrating out degrees of freedom outside a small circular region <math><mover><mi>A</mi><mo>¯</mo></mover></math> of radius <math><mi>L</mi></math>, where <math><mi>L</mi></math> is much smaller than the size of the proton. The wave function provides a nonperturbative distribution of three valence quarks. In addition, we include the perturbative emission of a single gluon and calculate the entanglement entropy of gluons in <math><mover><mi>A</mi><mo>¯</mo></mover></math>. For both quarks and gluons, we obtain the same simple result: <math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>=</mo><mo>−</mo><mo>∫</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>Δ</mi><mi>x</mi></mrow></mfrac><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mi>log</mi><mo>[</mo><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>]</mo></mrow></math>, where <math><mi>a</mi></math> is the UV cutoff in coordinate space and <math><mi>Δ</mi><mi>x</mi></math> is the longitudinal resolution scale. Here <math><msub><mi>N</mi><mi>S</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> is the number of partons (of the appropriate species) with longitudinal momentum fraction <math><mi>x</mi></math> inside an area <math><mi>S</mi></math>. It is related to the standard parton distribution function by <math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></mfrac><mi>Δ</mi><mi>x</mi><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math>, where <math><msub><mi>A</mi><mi>p</mi></msub></math> denotes the transverse area of the proton.

Adrian Dumitru

Vladimir V. Skokov

Entanglement entropy of the proton in coordinate space

We extend the results of A. Kovner, M. Li, and V. V. Skokov [Phys. Rev. Lett. 128, 182003 (2022)], where we argued that in the controlled environment of the deep inelastic scattering experiments, Bose-Einstein correlation between gluons in a hadronic wave function can be accessed through the production of the diffractive dijet plus a third jet. In this observable, Bose-Einstein correlation causes the enhancement of the production cross sections at the zero relative angle between the transverse momentum imbalance of the photon-going dijet and the transverse momentum of the gluon jet, when the magnitude of the momentum imbalance is about the same as the magnitude of the produced gluon. In the present paper, we account for multiple scattering and nonlinear effect in the target wave function. Although our equations can be applied to any high-energy DIS kinematics, to make them tractable numerically, we consider the high-momentum limit (momentum larger than <math><msub><mi>Q</mi><mi>s</mi></msub></math>) for the total momentum of the dijet, momentum imbalance, and the momentum of the produced gluon. By performing explicit numerical calculations, we confirm that the signal is present after accounting for multiple scattering.

Ming Li

Probing gluon Bose correlations in nuclear wave function in deep inelastic scattering

We study correlations originating from the quantum nature of gluons in a hadronic wave function. Bose-Einstein correlation between identical particles lead to the enhancement in the number of pairs of gluons with the same quantum numbers and small relative momentum. We show that these preexisting correlations can be probed in deep inelastic scattering experiments at high energy. Specifically, we consider diffractive dijet plus a third jet production. The azimuthal dependence displays a peak at the zero relative angle between the transverse momentum imbalance of the photon-going dijet and the transverse momentum of the hadron-going jet. Our calculations explicitly show that the peak originates from Bose enhancement. Comparing electron-proton to electron-nucleus collisions, we demonstrate that the nuclear target enhances the relative strength of the peak. With the future high luminosity electron-ion collider the proposed measurements of gluon Bose enhancement become experimentally feasible.

Probing Gluon Bose Correlations in Deep Inelastic Scattering

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.