Quantum chromodynamics through the geometry of M¨obiusstructures John Mashford School of Mathematics and Statistics University of Melbourne, Victoria 3010, Australia E-mail: [email protected] September 13, 2021 Abstract This paper describes a rigorous mathematical formulation providing a diver- gence free framework for QCD and the electroweak theory in curved space-time. The starting point of the theory is the notion of covariance which is interpreted as (4D) conformal covariance rather than the general (diffeomorphism) covari- ance of general relativity. Fock space which is a graded algebra composed of tensor valued function spaces associated with spaces of multiparticle bosonic or fermionic states is defined concretely. Algebra bundles whose typical fibers are algebras such as the Fock algebra are defined. Scattering processes are associ- ated with linear maps between fibers in such bundles. Such linear maps can be generated by linear operators between the multiparticle function spaces which intertwine with the actions of the group SU(3) associated with the strong in- teraction and a group locally isomorphic to SL(2; C)×U(1) which is associated arXiv:1809.04457v5 [physics.gen-ph] 10 Sep 2021 with the electroweak interaction and such intertwining, or covariant, operators may be generated by covariant integral kernels. It is shown how quark-quark scattering and gluon-gluon scattering are associated with such covariant ker- nels. The rest of the paper focuses on QCD vacuum polarization in order to compute and display the QCD running coupling. Through an application of the technique called spectral regularization the densities associated with the quark, gluon and ghost bubbles are computed without requiring renormalization and 1 hence the QCD vacuum polarization tensor is determined. It is found that the spectral QCD running coupling does not manifest a Landau pole and has the property known as \freezing of αs" in the infrared. Keywords: axiomatic QCD, conformal covariance, Fock space algebra, vacuum polarization, divergence-free, running coupling Contents 1 Introduction 3 2 General objects transforming covariantly at a point x 2 X where X denotes space-time 7 3 Covariant infinitesimal objects 10 3.1 Covariant infinitesimal objects for M¨obiusstructures . 10 3.2 Covariant infinitesimal objects for su(3) and M¨obiusstructures . 12 4 The Fock space F = L1 F of test functions for mul- k1;k2;k3;k4=0 (k1;k2;k3;k4) tiparticle states 13 5 Particle scattering processes and intertwining operators between F and P 0 0 0 0 15 (k1;k2;k3;k4) (k1;k2;k3;k4) 6 Quark-quark scattering 18 7 Gluon-gluon scattering and the extended Fock space 20 8 QCD vacuum polarization 23 8.1 The quark fermion bubble . 23 8.2 The gluon bubble . 26 8.3 The ghost bubble . 32 9 The QCD running coupling on the basis of 1-loop vacuum polariza- tion 34 9.1 The contribution of vacuum polarization to the QCD running coupling 34 9.2 Computation of the QCD running coupling . 37 9.3 Determination of the bare QCD coupling constant gs ......... 40 2 10 Higher order calculations 42 11 Discussion 44 12 Conclusion 46 List of Figures 1 Feynman diagram for gluon-gluon scattering . 20 2 1-loop spectral QCD running coupling versus log energy (GeV) between 1 and 10000 GeV . 41 3 1-loop spectral QCD running coupling versus log energy (eV) between 1eV and 10 GeV . 42 4 2-loop spectral QCD running coupling versus log energy (GeV) between 1 and 10000 GeV . 45 5 2-loop spectral QCD running coupling versus log energy (eV) between 1eV and 10 GeV . 45 1 Introduction Quantum chromodynamics (QCD) is the highly successful theory which organizes and explains the \particle zoo" made up of the baryons and the mesons, collectively known as hadrons, which experience the strong force. In QCD baryons and mesons are made up of collections of quarks, which are fermions with non-zero masses analogous to electrons in QED, and the strong force is mediated by gluons which are massless bosons analogous to photons in QED. One of the main problems met with in perturbative quantum chromodynamics (pQCD) is the difficulty with setting the renormalization scale µr of the QCD running coupling such that fixed order predictions for physical observables can be obtained. In the usual method a single µr as the argument of the QCD running coupling is simply guessed at, and its value is then varied over an arbitrary range [1]. This procedure is ad hoc and contradicts renormalization scheme (RS) invariance. Two approaches to getting around this problem are the principle of maximum conformality (PMC) [1] and the Brodsky-Lepage-Mackenzie (BLM) approach [2]. The running of the coupling in QCD (and in QED) is an important phenomenon 3 [3, 4, 5] which needs to be taken into account in carrying out calculations in QCD (e.g. [6]). The running fine structure constant in QCD αs can be determined ex- perimentally in a number of ways including deep inelastic scattering, τ decay and hadron-hadron scattering [4]. In addition it can be determined from lattice gauge theory computations. In all cases its determination involves theoretical QCD calcu- lations involving renormalization. In QCD the running coupling is not an observable but observables are calculated from αs in the high energy domain where, due to asymptotic freedom [7, 8], pQCD is applicable, by expanding to nth order a series in αs [9]. The methods for setting the renormalization scale such as the PMC and the BLM approach are all formulated in the context of the renormalization group (RG) ap- proach involving the RG equations [10, 11, 12]. Such calculations are neccessary when renormalization is carried out in order to circumvent divergences in non-tree level pQCD calculations. We have developed a method of regularization for quantum field theory (QFT) which we call spectral regularization and which is described in Refs. [13, 14, 15]. When using spectral regularization renormalization is not required and we may obtain an expression for the QCD running coupling αs(τ) as a function of energy τ which does not require the setting of a renormalization scale µr. QCD is based on the non-abelian \gauge" group SU(3). Nobody really has ex- plained why the group SU(3) should be of such fundamental importance in describing the physical world at the subatomic level, though some not entirely convincing rea- sons have been suggested. The group SU(3) is described as an internal symmetry group because it is believed that it does not relate to space-time symmetry. Space- time symmetry in QFT is encapsulated in the Lorentz group and more generally in the Poincar´egroup. QED and QCD are formulated in Minkowski space which has the Poincar´egroup as the maximal symmetry group. More general space-time symmetries are conformal covariance, which is covariance with respect to conformal diffeomorphisms, and the \general covariance" of general relativity (GR), which is covariance with respect to arbitrary C1 diffeomorphisms. GR is the currently accepted theory of gravity. QCD has certain general properties related to conformal symmetry [16, 17, 18, 19]. Gluodynamics has been considered in the context of curved conformally flat space- time [20]. In a previous paper [21] we have presented a theory in which space-time is modeled 4 as a (causal) locally conformally flat Lorentzian manifold so we are considering a subset of the set of space-times considered in GR. Locally conformally flat space- times form a wide class of space-times, including the well studied AdS space-time, and, in general, the class has a wide variety of geometric and topological properties. Actually we do not specify a metric, we only specify the causal, or interaction, structure. This results in any such structure being associated with an equivalence class of conformally equivalent metrics. Such objects are closely related to what are known as M¨obiusstructures. A M¨obiusstructure is a set X together with an atlas A = f(Ui; φi)gi2I where I is some index set and for all i 2 I, Ui ⊂ X and φi : Ui ! Vi 4 is a bijection of Ui onto some open subset Vi ⊂ R , such that 8i; j 2 I for which −1 1 Ui \ Uj 6= ;; φi ◦ φj is a C conformal diffeomorphism. Throughout this paper we let X = (X; A) be a M¨obiusstructure where A is a maximal atlas. The category of M¨obiusstructures has the same logical status as the category of smooth manifolds which forms the starting point for GR except that the pseu- dogroup of transformations (Kobayashi and Nomizu, 1963 [22], p. 1) is the conformal pseudogroup Γ(1; 3) [21] rather than the pseudogroup of general coordinate trans- formations. Γ(1; 3) is made up of all C1 conformal diffeomorphisms between open subsets of Minkowski space. In Ref. [21] we described how there is a natural principal bundle with structure group C(1; 3), the group of maximal conformal transformations in Minkowski space, associated with any M¨obiusstructure which is obtained by taking the unique element of C(1; 3) associated with each overlap conformal diffeomorphism as a (constant) C(1; 3) valued transition function. We will consider, in this paper, M¨obiusstruc- tures whose transition functions are generated by constant U(2; 2) valued transition functions. Consider G = U(2; 2) in its matrix representation defined with respect to the Hermitian form ! 0 1 g = 2 : (1) 12 0 We define the subgroup H ⊂ G by ( ! ) a b H = 2 U(2; 2) : b = 0 ; (2) c d 5 and the subgroup K ⊂ H by ( ! ) a 0 K = : a 2 GL(2; C); jdet(a)j = 1 : (3) 0 a†−1 K is a homomorphic image of H and is locally isomorphic to SL(2; C) × U(1).