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Wavelet Regularization of Gauge Theories

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Wavelet Regularization of Gauge Theories PHYSICAL REVIEW D 101, 105004 (2020) Wavelet regularization of gauge theories M. V. Altaisky * Space Research Institute RAS, Profsoyuznaya 84/32, Moscow, 117997, Russia (Received 7 December 2019; accepted 23 March 2020; published 6 May 2020) P ψ → ωA A ψ ∈ Rd A Extending the principle of local gauge invariance ðxÞ exp ð{ A ðxÞT Þ ðxÞ;x , with T being the generators of the gauge group A, to the fields ψðgÞ ≡ hχjΩÃðgÞjψi, defined on a locally compact Lie group G, g ∈ G, where ΩðgÞ is a suitable square-integrable representation of G, it is shown that taking the coordinates (g) on the affine group, we get a gauge theory that is finite by construction. The renormalization group in the constructed theory relates to each other the charges measured at different scales. The case of the A ¼ SUðNÞ gauge group is considered. DOI: 10.1103/PhysRevD.101.105004 I. INTRODUCTION principle of gauge invariance, explicitly manifested in the existence of gauge bosons—the carriers of gauge inter- Gauge theories form the basis of modern high-energy action. The role of the renormalization group (RG) is to physics. Quantum electrodynamics (QED)—a quantum field theory model based on the invariance of the view the physics changing with scale in an invariant way Lagrangian under local phase transformations of the matter depending on charges and parameters related to the given scale, absorbing all divergences in renormalization factors. fields ψðxÞ → e{wðxÞψðxÞ—was the first theory to succeed According to the author’s point of view [5], the cause of in describing the effect of the vacuum energy fluctuations divergences in quantum field theory is an inadequate choice on atomic phenomena, such as the Lamb shift, with an 2 Rd extremely high accuracy of several decimal digits [1]. The of the functional space L ð Þ. Due to the Heisenberg crux of QED is that in representing the matter fields by uncertainty principle, nothing can be measured at a sharp Δ square-integrable functions in Minkowski space, it yields point: it would require an infinite momentum p to keep Δ → 0 Δ Δ ≥ ℏ formally infinite Green functions, unless a special pro- x with p x 2. Instead, the values of physical cedure, called renormalization, is applied to the action fields are meaningful on a finite domain of size Δx, and functional [2,3]. Much later, it was discovered that all other hence the physical fields should be described by scale- ψ known interactions of elementary particles, viz. weak dependent functions ΔxðxÞ. As was shown in previous – ψ interaction and strong interaction, are also described by papers [5 7], having defined the fields aðxÞ as the wavelet gauge theories. The difference from QED consists in the transform of square-integrable fields, we yield a quantum fact that the multiplets of matter fields are transformed by field theory of scale-dependent fields—a theory finite by unitary matrices ψðxÞ → UðxÞψðxÞ, making the theory construction with no renormalization required to get rid non-Abelian. Due to ’t Hooft, we know such theories to of divergences. be renormalizable, and thus physically meaningful [4]. The present paper makes an endeavor to construct a Now they form the standard model (SM) of elementary gauge theory based on local unitary transformations of the ψ → ψ particles—an A ¼ SUð2Þ × Uð1Þ × SUcð3Þ gauge theory scale-dependent fields: aðxÞ UaðxÞ aðxÞ. The physi- supplied with the Higgs mechanism of spontaneous sym- cal fields in such a theory are defined on a region of finite- metry breaking. size Δx centered at x as a sum of all scale components from A glimpse at the stream of theoretical papers in high- Δx to infinity by means of the inverse wavelet transform. energy physics, from Ref. [2] till the present time, shows The Green functions are finite by construction. The RG that renormalization takes a bulk of technical work, symmetry represents the relations between the charges although the role of it is subjunctive to the main physical measured at different scales. This is essentially important for quantum chromody- *[email protected] namics, the theory of strong interactions, where the ultimate way of analyzing the hadronic interactions at both Published by the American Physical Society under the terms of short and long distances remains the study of the depend- the Creative Commons Attribution 4.0 International license. ence of the coupling constant αS on only one parameter— Further distribution of this work must maintain attribution to 2 the author(s) and the published article’s title, journal citation, the squared transferred momentum Q . Naturally, one can and DOI. Funded by SCOAP3. suggest that two parameters may be better than one. As has 2470-0010=2020=101(10)=105004(16) 105004-1 Published by the American Physical Society M. V. ALTAISKY PHYS. REV. D 101, 105004 (2020) been realized in the classical physics of strongly coupled by changing the partial derivative ∂μ into the covariant nonlinear systems—first in geophysics [8]—the use of two derivative parameters (scale and frequency) may solve a problem that appears hopeless for spectral analysis. Attempts of a similar Dμ ¼ ∂μ þ {AμðxÞ: ð3Þ kind have also been made in QCD [9], although they have not received further development. The modified action I must admit, in this respect, that one of the challenges of modern QCD is to separate the short-distance behavior of Z quarks, where the perturbative calculations are feasible, 0 4 ψγ¯ ψ ψψ¯ SE ¼ d x½ μDμ þ {m ð4Þ from the long-distance dynamics of quark confinement— and then somehow to relate the parameter Λ, which describes the short-range dynamics, to the mass scale of remains invariant under the phase transformations of hadrons [10]. This may not be the ultimate solution of the Eq. (2) if the gauge field AμðxÞ is transformed accordingly: problem: sometimes it is easier to scan the whole range of U † † scales with some continuous parameter than to separate the Aμ ¼ UðxÞAμðxÞU ðxÞþ{ð∂μUðxÞÞU ðxÞ: ð5Þ small- and the large-scale modes [11]. The remainder of this paper is organized as follows: In Equation (2) represents gauge rotations of the matter- Sec. II, I briefly review the formalism of local gauge field multiplets. The matrices wðxÞ can be expressed in the theories as it is applied to the Standard Model and QCD. basis of appropriate generators Section III summarizes the wavelet approach to X quantum field theory, developed by the author in previous A A papers [5–7], which yields finite Green functions wðxÞ¼ w ðxÞT ; ϕ …ϕ A h a1 ðx1Þ an ðxnÞi for scale-dependent fields. Its applica- tion to gauge theories, however, remains cumbersome. where TA are the generators of the gauge group A, acting Section IV is the main part of the paper. It presents the on matter fields in the fundamental representation. For the formulation of gauge invariance in scale-dependent for- Lie group, they satisfy the commutation relations malism, sets up the Feynman diagram technique, and gives the one-loop contribution in a pure gauge theory to the A B ABC C three-gluon vertex in scale-dependent Yang-Mills theory. ½T ;T ¼{f T The developed formalism aims to catch the effect of A B AB 1 asymptotic freedom in a non-Abelian gauge theory that and are normalized as Tr½T ;T ¼TFδ ; where TF ¼ 2 is finite by construction, and hopefully, with fermions being is a common choice. For the Yang-Mills theory, I included, to describe the color confinement and enable assume the symmetry group to be SUðNÞ. The trivial case analytical calculations in QCD. The problems and pro- of N ¼ 1 corresponds to the Abelian theory—quantum spectives of the developed methods are summarized in the electrodynamics. Conclusion. The Yang-Mills action, which describes the action of the gauge field AμðxÞ itself, should be added to the action in II. LOCAL GAUGE THEORIES Eq. (4). It is expressed in terms of the field-strength tensor The theory of gauge fields stems from the invariance Z 1 4 of the action functional under the local phase transforma- SYM½A¼2 2 TrðFμνFμνÞd x; ð6Þ tions of the matter fields. Historically, it originated in g quantum electrodynamics, where the matter fields ψ—the 1 where spin-2 fermions with mass m—are described by the action functional − ∂ − ∂ Z Fμν ¼ {½Dμ;Dν¼ μAν νAμ þ {½Aμ;Aνð7Þ 4 S ¼ d x½ψγ¯ μ∂μψ þ {mψψ¯ ; ð1Þ E is the strength tensor of the gauge field, and g is a formal coupling constant obtained by redefinition of the gauge γ written in Euclidean notation, with matrices satisfying the fields Aμ → gAμ. anticommutation relations fγμ; γνg¼−2δμν. It should be noted that the free-field action [Eq. (1)] that The action functional [Eq. (1)] can be made invariant has given rise to gauge theory was written in a Hilbert 2 under the local phase transformations space of square-integrableR functions L ðRdÞ, with the scalar product hψjϕi¼ ψ¯ ðxÞϕðxÞddx. In what follows, ψ → ψ ≡ {wðxÞ ðxÞ UðxÞ ðxÞ;UðxÞ e ð2Þ the same will be done for more general Hilbert spaces. 105004-2 WAVELET REGULARIZATION OF GAUGE THEORIES PHYS. REV. D 101, 105004 (2020) III. SCALE-DEPENDENT QUANTUM both the position and the resolution—the scale of FIELD THEORY observation. The Green functions for such fields hϕ ðx1Þ…ϕ ðx Þi can be made finite by construction A. The observation scale a1 an n under certain causality conditions [7,14]. The dependence of physical interactions on the scale The introduction of resolution into the definition of the of observation is of paramount importance.
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