ELEMENTARY PARTICLE PHYSICS

Introduction

Quantum Chromodynamics (QCD) is without any reasonable doubt the correct fundamental theory of the strong interactions between quarks and gluons. Thus, it describes exactly all properties of particles like the proton, which is a bound state of quarks and gluons, so-called hadrons. However, while it is rather straight­forward to calculate all properties of atoms, i.e. bound states of electrons and nuclei, from Quantum Electrodynamics, the theory which describes the electromagnetic force, this is not the case for QCD. In fact, the QCD­equations are so difficult to solve that many or even most properties of hadrons are only poorly understood at the QCD level. Simulations on powerful computers like SuperMUC­NG, using Lattice QCD provide some, but only very limited insight, because they require physical time to be analytically continued to imaginary time which precludes, e.g., the possibility to describe any real time dynamics. Quantum computers might improve this situation some day but, at present, this day still seems to be far away. Another very severe technical problem is the so-called operator mixing which further reduces the number of calculable quantities dramatically. Thus, people explore already for many years alternative formulations, which might allow to broaden the range of calculable quantities, so far, unfortunately without decisive success. In our research projects we explore two such ideas. In this report we will focus on just one of them which goes under the name of LaMET (Large Momentum Effective Theory).

Research in this field proceeds in two steps. First, one develops complete parameterizations of those hadron properties which can be experimentally observed in terms of a hierarchy of functions (typically with rather exotic names, e.g. transversity, worm-gear functions, distribution amplitudes, or Boer-Mulders function.) This hierarchy is artfully constructed such that it groups these functions according to the difficulty of calculating them. It turns out that at each level of this hierarchy there are angular momentum dependent and independent functions and proceeding from the simple to the more difficult ones one typically calculates all of them before continuing to the next, more difficult level. This is what we did by calculating first the quark transversity and, after this was successful, the quark Boer-Mulders function in the proton. The Boer-Mulders function is a so-called Transverse Momentum Dependent distribution (TMD) which in addition depends on the transverse momentum (relative to the total proton momentum) of the hit quark. TMDs have very complicated properties closely related to the properties auf gauge links, which are the most basic, gluonic building blocks of QCD. The extraction of TMDs is, e.g., the primary goal of the new accelerator center EIC (Electron Ion Collider) to be built in the US, as well as its Chinese competition, the EicC (Electron ion collider in China). Without going into any detail let us add that in all probability experiments alone will not allow to reach this goal. Instead, to do so will require complementary information from the lattice of the type we are providing. Let us finally stress that we per -form only pioneering work aiming primarily at clarifying the many open questions of TMD physics. LaMET requires taking several non-trivial limits (most importantly the proton momentum must be larger and nobody knows as of today how large to reach a certain accuracy) and our simulationstest whether these can be controlled with real world resources. The upshot of this report is that they can and that we do. This success paves the way for future high-precision simulations, which, unfortunately, will require far more computer time.

Results and Methods

Work on QCD is dominated by two techniques namely, analytic, perturbative calculations (pQCD) and numerical lattice QCD (LQCD). LaMET combines both. The quantities one has to calculate are non-local in time and space and, as explained above, time looses its original meaning when analytically continued. (In fact, it turns into an inverse temperature.) Therefore, the calculation of functions which are non-local in time is not feasible in standard LQCD. In LaMET one calculated instead correlators which are only non-local in space and matches them to the required objects by continuum perturbation theory. More precisely, one calculates the perturbative matching function with which the LaMET lattice result, called e.g. quasi parton distribution function must be convoluted. This strategy was pioneered by X. Ji (University of Maryland) and is meanwhile employed by several large collaborations, primarily in the US and China. (We are members of the Lattice Parton Collaboration based in China.) While all of this might sound rather natural and straight forward, the technical problems are hellishly difficult. Still, the cooperative effort of all these groups has succeeded to overcome most problems such that TMDs and the similarly complicated Generalized Parton Distributions (GPDs) can meanwhile be calculated.

Ongoing Research / Outlook

Our future work has two main objectives namely to ex-tend the range of applications of LaMET and to improve the precision reached. The latter has two aspects. The purely statistical precision is rather straight forward to improve as so far, we have used only rather limited computer resources. In contrast, identifying and solving the remaining conceptual problems of LaMET is a very demanding task. The most difficult remaining such problem is to develop methods allowing to reach substantially larger hadron momenta at acceptable cost beyond what was suggested in [2]. Though all groups in this field try to do so, progress so far was rather incremental than decisive.

References and Links

[1] F. Yao et al., PRL 131 (2023) 261901; doi:10.1103/PhysRevLett.131.261901.

[2] G.S. Bali et al., Phys. Rev. D 93 (2016) 094515; doi:10.1103/PhysRevD.93.094515.