arXiv subjects

Heribert Weigert

Publications and source records attributed to Heribert Weigert.

At least 19 recordsLinked to original sources

arXiv2019

Compact construction algorithms for the singlets of SU(N) over mixed tensor product spaces

The irreducible representations of SU(N) over a mixed quark-antiquark Fock space component have been studied for many years. In analogy to the case for the quark-only Fock space component, there exist efficient tools to classify the irreducible representations of SU(N) over a mixed tensor product space using tableaux. Unlike the quark-only case, the only general algorithm known to us for constructing the associated projection operators onto irreducible multiplets involves translating N-1 fundamental factors into an antifundamental factor using the Leibniz rule, which turns out to be computationally extremely inefficient. If one is interested only in singlets, this problem can be entirely avoided as is demonstrated below where we provide an efficient algorithm to construct the projection operatators onto the irreducible representations of dimension 1 of the special unitary group SU(N) over a mixed Fock space component that transparently gives access to N dependence, and discuss the relative merits in comparison to an alternative algorithm briefly discussed in a different context by Keppeler and Sjoedahl.

math-ph↗

arXiv2018

A simple counting argument of the irreducible representations of SU(N) on mixed product spaces

It is a well known result that the number of irreducible representations of SU(N) on a tensor product containing k factors of a vector space V is given by the number of involutions in the symmetric group on k letters. In this paper, we present an alternative proof for this fact using a basis of projection and transition operators of the algebra of invariants of SU(N). This proof easily generalizes to the irreducible representations of SU(N) on mixed tensor product spaces (consisting of factors of V as well as its dual space). This implies that the number of irreducible representation of SU(N) on such a space remains unchanged if one exchanges factors V for its dual and vice versa, as long as the total number of factors remains unchanged.

math.RT↗

arXiv2016

Simplification Rules for Birdtrack Operators

This paper derives a set of easy-to-use tools designed to simplify calculations with birdtrack op- erators comprised of symmetrizers and antisymmetrizers. In particular, we present cancellation rules allowing one to shorten the birdtrack expressions of operators, and propagation rules identifying the circumstances under which it is possible to propagate symmetrizers past antisymmetrizers and vice versa. We exhibit the power of these simplification rules by means of a short example in which we apply the tools derived in this paper on a typical operator that can be encountered in the representation theory of SU(N) over the product space $V^{\otimes m}$ . These rules form the basis for the construction of compact Hermitian Young projection operators and their transition operators addressed in companion papers.

math-ph↗

arXiv2016

Compact Hermitian Young Projection Operators

In this paper, we describe a compact and practical algorithm to construct Hermitian Young projection operators for irreducible representations of the special unitary group SU(N), and discuss why ordinary Young projection operators are unsuitable for physics applications. The proof of this construction algorithm uses the iterative method described by Keppeler and Sjödahl. We further show that Hermitian Young projection operators share desirable properties with Young tableaux, namely a nested hierarchy when "adding a particle". We end by exhibiting the enormous advantage of the Hermitian Young projection operators constructed in this paper over those given by Keppeler and Sjödahl.

math-ph↗

arXiv2016

Transition Operators

In this paper, we give a generic algorithm of the transition operators between Hermitian Young projection operators corresponding to equivalent irreducible representations of SU(N), using the compact expressions of Hermitian Young projection operators derived in a companion paper. We show that the Hermitian Young projection operators together with their transition operators constitute a fully orthogonal basis for the algebra of invariants of $V^{\otimes m}$ that exhibits a systematically simplified multiplication table. We discuss the full algebra of invariants over $V^{\otimes 3}$ and $V^{\otimes 4}$ as explicit examples. In our presentation we make use of various standard concepts such as Young projection operators, Clebsch-Gordan operators, and invariants (in birdtrack notation). We tie these perspectives together and use them to shed light on each other.

math-ph↗

arXiv2011

HERA-data in the light of small x evolution with state of the art NLO input

Both total and diffractive cross sections from HERA are successfully confronted with JIMWLK evolution equations in the asymptotic pseudo-scaling region. We present a consistent, simultaneous description of both types of cross sections that includes NLO corrections in the form of running coupling and energy conservation corrections. The inclusion of energy conservation corrections allows to match all available data with x below .02 i.e. up to Q^2 of 1200 GeV^2. We discuss the effects of quark masses including charm, contrast asymptotic and pre-asymptotic fit strategies, and survey non-perturbative uncertainties related to impact parameter dependence.

hep-ph↗

arXiv2010

New observables to test the Color Glass Condensate beyond the large-N_c limit

The JIMWLK framework offers a powerful tool to calculate the energy dependence of QCD observables at high energies. Despite a growing number of observables considered for phenomenological analysis, few features of JIMWLK evolution beyond its evolution speed are yet well constrained by experiment. We argue that meson production cross-sections have the potential to provide qualitatively new insights and allow to address issues both beyond the large-N_c limit and at higher twist. These cross-sections generically contain four point functions whose evolution is shown to follow from the JIMWLK framework. The Gaussian truncation is used to provide an efficient and practical means of calculating the evolution of four point correlators beyond the large-N_c limit.

hep-ph↗

arXiv2009

Subleading-N_c corrections in non-linear small-x evolution

We explore the subleading-N_c corrections to the large-N_c Balitsky-Kovchegov (BK) evolution equation by comparing its solution to that of the all-N_c Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) equation. In earlier simulations it was observed that the difference between the solutions of JIMWLK and BK is unusually small for a quark dipole scattering amplitude, of the order of 0.1%, which is two orders of magnitude smaller than the naively expected 1/N_c^2 or 11%. In this paper we argue that this smallness is not accidental. We show that saturation effects and correlator coincidence limits fixed by group theory constraints conspire with the particular structure of the dipole kernel to suppress subleading-N_c corrections reducing the difference between the solutions of JIMWLK and BK to 0.1%. We solve the JIMWLK equation with improved numerical accuracy and verify that the remaining 1/N_c corrections, while small, still manage to slow down the rapidity-dependence of JIMWLK evolution compared to that of BK. We demonstrate that a truncation of JIMWLK evolution in the form of a minimal Gaussian generalization of the BK equation captures some of the remaining 1/N_c contributions leading to an even better agreement with JIMWLK evolution. As the 1/N_c corrections to BK include multi-reggeon exchanges one may conclude that the net effect of multi-reggeon exchanges on the dipole amplitude is rather small.

hep-ph↗

arXiv2008

Collinear Singularities and Running Coupling Corrections to Gluon Production in CGC

We analyze the structure of running coupling corrections to the gluon production cross section in the projectile-nucleus collisions calculated in the Color Glass Condensate (CGC) framework. We argue that for the gluon production cross section (and for gluon transverse momentum spectra and multiplicity) the inclusion of running coupling corrections brings in collinear singularities due to final state splittings completely unaffected by CGC resummations. Hence, despite the saturation/CGC dynamics, the gluon production cross section is not infrared-safe. As usual, regularizing the singularities requires an infrared cutoff Lambda_coll that defines a resolution scale for gluons. We specifically show that the cutoff enters the gluon production cross section in the argument of the strong coupling constant alpha_s(Lambda_coll^2). We argue that for hadron production calculations one should be able to absorb the collinear divergence into a fragmentation function. The singular collinear terms in the gluon production cross section are shown not to contribute to the energy density of the produced matter, which is indeed an infrared-finite quantity.

hep-ph↗

arXiv2007

Quark loop contribution to BFKL evolution: Running coupling and leading-N_f NLO intercept

We study the sea quark contribution to the BFKL kernel in the framework of Mueller's dipole model using the results of our earlier calculation. We first obtain the BFKL equation with the running coupling constant. We observe that the ``triumvirate'' structure of the running coupling found previously for non-linear evolution equations is preserved for the BFKL equation. In fact, we rederive the equation conjectured by Levin and by Braun, albeit for the unintegrated gluon distribution with a slightly unconventional normalization. We obtain the leading-N_f contribution to the NLO BFKL kernel in transverse momentum space and use it to calculate the leading-N_f contribution to the NLO BFKL pomeron intercept for the unintegrated gluon distribution. Our result agrees with the well-known results of Camici and Ciafaloni and of Fadin and Lipatov. We show how to translate this intercept to the case of the quark dipole scattering amplitude and find that it maps onto the expression found by Balitsky.

hep-ph↗

arXiv2006

Triumvirate of Running Couplings in Small-x Evolution

We study the inclusion of running coupling corrections into the non-linear small-x JIMWLK and BK evolution equations by resumming all powers of alpha_s N_f in the evolution kernels. We demonstrate that the running coupling corrections are included in the JIMWLK/BK evolution kernel by replacing the fixed coupling constant alpha_s in it with alpha_s (1/r_1^2) alpha_s (1/r_2^2) / alpha_s (1/R^2), where r_1 and r_2 are transverse distances between the emitted gluon and the harder gluon (or quark) off of which it was emitted to the left and to the right of the interaction with the target. In the formalism of Mueller's dipole model r_1 and r_2 are the transverse sizes of ``daughter'' dipoles produced in one step of the dipole evolution. The scale R is a function of two-dimensional vectors r_1 and r_2, the exact form of which is scheme-dependent. We propose using a particular scheme which gives us R as an explicit function of r_1 and r_2.

hep-ph↗

arXiv2006

Running coupling and power corrections in nonlinear evolution at the high--energy limit

A main feature of high-energy scattering in QCD is saturation in the number density of gluons. This phenomenon is described by non-linear evolution equations, JIMWLK and BK, which have been derived at leading logarithmic accuracy. In this paper we generalize this framework to include running coupling corrections to the evolution kernel. We develop a dispersive representation of the dressed gluon propagator in the background of Weiszacker Williams fields and use it to compute O(beta_0^{n-1} alpha_s^n) corrections to the kernel to all orders in perturbation theory. The resummed kernels present infrared-renormalon ambiguities, which are indicative of the form and importance of non-perturbative power corrections. We investigate numerically the effect of the newly computed perturbative corrections as well as the power corrections on the evolution and find that at present energies they are both significant.

hep-ph↗

arXiv2005

Extension of the Color Glass Condensate Approach to Diffractive Reactions

We present an evolution equation for the Bjorken x dependence of diffractive dissociation on hadrons and nuclei at high energies. We extend the formulation of Kovchegov and Levin by relaxing the factorization assumption used there. The formulation is based on a technique used by Weigert to describe interjet energy flow. The method can be naturally extended to other exclusive observables.

hep-ph↗

arXiv2005

Evolution at small x_bj: The Color Glass Condensate

When probed at very high energies or small Bjorken x_bj, QCD degrees of freedom manifest themselves as a medium of dense gluon matter called the Color Glass Condensate. Its key property is the presence of a density induced correlation length or inverse saturation scale R_s=1/Q_s. Energy dependence of observables in this regime is calculable through evolution equations, the JIMWLK equations, and characterized by scaling behavior in terms of Q_s. These evolution equations share strong parallels with specific counterparts in jet physics. Experimental relevance ranges from lepton proton and lepton nucleus collisions to heavy ion collisions and cross correlates physics at virtually all modern collider experiments.

hep-ph↗

arXiv2003

Non-global jet evolution at finite N_c

Resummations of soft gluon emissions play an important role in many applications of QCD, among them jet observables and small x saturation effects. Banfi, Marchesini, and Smye have derived an evolution equation for non-global jet observables that exhibits a remarkable analogy with the BK equation used in the small x context. Here, this analogy is used to generalize the former beyond the leading N_c approximation. The result shows striking analogy with the JIMWLK equation describing the small x evolution of the color glass condensate. A Langevin description allows numerical implementation and provides clues for the formulation of closed forms for amplitudes at finite N_c. The proof of the new equation is based on these amplitudes with ordered soft emission. It is fully independent of the derivation of the JIMWLK equation and thus sheds new light also on this topic.

hep-ph↗

arXiv2003

Universal features of JIMWLK and BK evolution at small x

In this paper we present the results of numerical studies of the JIMWLK and BK equations with a particular emphasis on the universal scaling properties and phase space structure involved. The results are valid for near zero impact parameter in DIS. We demonstrate IR safety due to the occurrence of a rapidity dependent saturation scale Q_s(τ). Within the set of initial conditions chosen both JIMWLK and BK equations show remarkable agreement. We point out the crucial importance of running coupling corrections to obtain consistency in the UV. Despite the scale breaking induced by the running coupling we find that evolution drives correlators towards an asymptotic form with near scaling properties. We discuss asymptotic features of the evolution, such as the τ- and A-dependence of Q_s away from the initial condition.

hep-ph↗

arXiv2001

Unitarity at small Bjorken x

This paper presents a solution to the nonlinear small x ``projectile side'' evolution equations as derived by Balitskii in 1996. The solution is based on functional Fokker-Planck methods. The fixed point at small x is explicitly calculated and all correlation functions in this limit are determined. They show clear saturation and unitarization properties. Scaling laws that hold during the saturation phase and throughout the whole course of the evolution are established. The corresponding Langevin equations are given as a basis for numerical simulations opening the field for future studies of dynamical issues of the evolution not analytically accessible. The methods used may be extended to the ``target side'' equations of Jalilian-Marian, Kovner, Leonidov and Weigert.

hep-ph↗

arXiv2000

Relating different approaches to nonlinear QCD evolution at finite gluon density

We analyze the relation between evolution equations at low x that have been derived in different approaches in the last several years. We show that the equation derived by Balitsky and Kovchegov is obtained from the Jalilian-Marian-Kovner-Leonidov-Weigert (JKLW) equation in the limit of small induced charge density. We argue that the higher nonlinearities resummed by the JKLW equation correspond, in physical terms, to the breakdown of the eikonal approximation when the gluon fields in the target are large.

hep-ph↗