sign in sign up
<<
Home , Regularization (physics)

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 ≡ hχjΩðgÞjψi, 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 that is finite by construction. The 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 (RG) is to . (QED)—a 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 , 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 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 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 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. In classical field function has a clear physical interpretation. If the physics, when the position and the momentum can be particle, described by the field ϕ, has been initially measured simultaneously, one can average the measured prepared in the interval ðx − Δx ;xþ ΔxÞ, the probability Δx 2 2 quantities over a region of a given size centered at point of registering this particle in this interval is generally less x . For instance, the Eulerian velocity of a fluid, measured at than unity, because the probability of registration depends x Δx point within a cubic volume of size ,isgivenby on the strength of interaction and on the ratio of typical Z 1 scales of the measured particle and the measuring equip- d vΔxðxÞ ≔ vðxÞd x: ment. The maximum probability of registering an object of Δ d d ð xÞ ðΔxÞ typical scale Δx by equipment with typical resolution a is achieved when these two parameters are comparable. In quantum physics, it is impossible to measure any field For this reason, the probability of registering an electron ϕ sharply at a point x. This would require an infinite Δ ∼ ℏ Δ Δ → 0 by visual-range photon scattering is much higher than momentum transfer p = x, with x , making that by long radio-frequency waves. As a mathematical ϕðxÞ meaningless. That is why any such field should be ϕ generalization, we should say that if a set of measuring designated by the resolution of observation: ΔxðxÞ.In equipment with a given spatial resolution a fails to high-energy physics experiments, the initial and final states register an object, prepared on a spatial interval of width of particles are usually determined in momentum basis Δx, with certainty, then tuning the equipment to all jpi—the plane wave basis. For this reason, the results of possibleR resolutions a0 would lead to the registration measurements—i.e., the correlations between different 2 jϕ ðxÞj dμða; xÞ¼1, where dμða; xÞ is some measure events—are considered as functions of squared momentum a that depends on resolution a. This certifies the fact of the transfer Q2, which play the role of the observation existence of the measured object. scale [10,12]. A straightforward way to construct a space of scale- In theoretical models, the straightforward introduction of dependent functions is to use a projection of local fields a cutoff momentum Λ as the scale of observation is not ϕðxÞ ∈ L2ðRdÞ onto some basic function χðxÞ with good always successful. A physical theory should be Lorentz localization properties, in both the position and momentum invariant, should provide energy and momentum conser- spaces, and scaled to a typical window width of size a. vation, and may have gauge and other symmetries. The use This can be achieved by using a continuous wavelet of the truncated fields transform [15]. Z ddk ϕ< x ≔ e−{kxϕ˜ k ð Þ ð Þ d B. Continuous wavelet transform jkj<Λ ð2πÞ Let H be a Hilbert space of states for a quantum field jϕi. may destroy the symmetries. In the limiting case of Let G be a locally compact Lie group acting transitively Λ → ∞, this returns to the standard Fourier transform, on H, with dμðνÞ; ν ∈ G being a left-invariant measure ϕ …ϕ making some of the Green functions h ðx1Þ ðxnÞi on G. Then, any jϕi ∈ H can be decomposed with respect infinite and the theory meaningless. A practical solution to a representation ΩðνÞ of G in H [16,17]: of this problem was found in the renormalization group Z (RG) method [13], first discovered in quantum electrody- 1 jϕi¼ ΩðνÞjχidμðνÞhχjΩ†ðνÞjϕi; ð8Þ namics [2]. The bare charges and the bare fields of the Cχ G theory are then renormalized to some “physical” charges and fields, the Green functions for which become finite. where jχi ∈ H is referred to as a mother wavelet, satisfying The price to be paid for it is the appearance in the theory of the admissibility condition 2 some new normalization scale μ . The comparison of the Z 1 model prediction to the experimental observations now χ Ω ν χ 2 μ ν ∞ 2 2 Cχ ¼ 2 jh j ð Þj ij d ð Þ < : requires the use of two scale parameters, (Q , μ ) [13]. kχk G A significant disadvantage of the RG method is that in renormalized theories, we are often doomed to ignore the The coefficients hχjΩ†ðνÞjϕi are referred to as wavelet finite parts of the Feynman graphs. The solution of the coefficients. Wavelet coefficients can be used in quantum divergences problem may be the change of the functional mechanics in the same spirit as the coherent states are space to the space of functions that explicitly depend on used [18,19].

105004-3 M. V. ALTAISKY PHYS. REV. D 101, 105004 (2020)

If the group G is Abelian, the wavelet transform [Eq. (8)] system and the state vector corresponding to the localiza- with G∶x0 ¼ x þ b0 is the Fourier transform. Next to the tion at the point x: ϕðxÞ ≡ hxjϕi, the modified theory [5,21] Abelian group is the group of the affine transformations of should respect the resolution of the measuring equipment. the Euclidean space Rd: Namely, we define the resolution-dependent fields

G∶ x0 ¼ aRðθÞxþb; x; b ∈ Rd;a∈ R ; θ ∈ SOðdÞ; þ ϕaðxÞ ≡ hx; a; χjϕi; ð14Þ ð9Þ also referred to as the scale components of ϕ, where where RðθÞ is the SOðdÞ rotation matrix. Here we define hx; a; χj is the bra-vector corresponding to localization of the representation of the affine transform [Eq. (9)] with the measuring device around the point x with the spatial respect to the mother wavelet χðxÞ as follows: resolution a; in optics χ labels the apparatus function of the equipment, an aperture function [20]. 1 − −1 x b In QED, the common calculation techniques are based Uða; b; θÞχðxÞ¼ χ R ðθÞ : ð10Þ ad a on the basis of plane waves. However, the basis of plane waves is not obligatory. For instance, if the inverse size of Thus, the wavelet coefficients of the function ϕðxÞ ∈ a QED microcavity is compared to the energy of an L2ðRdÞ with respect to the mother wavelet χðxÞ in interlevel transition of an atom, or a quantum dot inside Euclidean space Rd can be written as the cavity, we can (at least in principle) avoid the use of Z plane waves and use some other functions to estimate the 1 x − b dependence of vacuum energy effects on the size and ϕ χ −1 θ ϕ d a;θðbÞ¼ d R ð Þ ðxÞd x: ð11Þ shape of the cavity. In this sense, the mother wavelet can Rd a a be referred to as an aperture function. In QCD, all The wavelet coefficients (11) represent the result of the measuring equipment is far removed from the collision measurement of function ϕðxÞ at the point b at the scale a domain, and the approximation of plane waves may be with an aperture function χ rotated by the angle(s) θ [20]. most simple technically, but it is not justified unambig- The function ϕðxÞ can be reconstructed from its wavelet uously: some other sets of functions may be used as well. coefficients [Eq. (11)] using the formula Eq. (8): Discrete wavelet basis, for instance, has been already used Z for common QCD models in Ref. [9]. The field theory of 1 1 − d extended objects with the basis χ defined on the spin ϕ χ −1 θ x b ϕ dad b μ θ ðxÞ¼ d R ð Þ aθðbÞ d ð Þ; Cχ a a a variables was considered in Refs. [22,23]. The goal of the present paper is to study the scale ð12Þ dependence of the running coupling constant in non- Abelian gauge theory constructed directly on scale- μ θ where d ð Þ is the left-invariant measure on the SOðdÞ dependent fields. Assuming the mother wavelet χ to be rotation group, usually written in terms of the Euler angles: isotropic, we drop the angle argument θ in Eq. (12) and Z perform all calculations in Euclidean space. Yd−2 π k The interpretation of the real experimental results in dμðθÞ¼2π sin θkdθk: 0 terms of the wave packet χ is a nontrivial problem to be of k¼1 special concern in future. It can be addressed by construct- The normalization constant Cχ is readily evaluated using ing wavelets in the Minkowski space and by analytic the Fourier transform. In what follows, I assume isotropic continuation from the Euclidean space to the Minkowski wavelets and omit the angle variable θ. This means that the space [7,24]. mother wavelet χ is assumed to be invariant under SOðdÞ For the same reason, I also do not consider here rotations. This is quite common for the problems with no the of scale-dependent fields, which was preferable directions. For isotropic wavelets, addressed elsewhere [7,25,26]. A prospective way to do Z Z this, as suggested in Refs. [7,27], is the use of light-cone ∞ d coordinates [28,29]. With these remarks, we can under- χ˜ 2 da χ˜ 2 d k ∞ Cχ ¼ j ðakÞj ¼ j ðkÞj d < ; ð13Þ stand the physically measured fields, at least in local 0 a Sdjkj theories like QED and the φ4 model, as the integrals over 2πd=2 d all scale components from the measurement scale (A)to where Sd ¼ Γ 2 is the area of the unit sphere in R , with ðd= Þ infinity: ΓðxÞ being Euler’s gammaR function. A tilde denotes the χ˜ {kxχ d Z Fourier transform: ðkÞ¼ Rd e ðxÞd x. 1 If the standard quantum field theory defines the field ϕðAÞðxÞ¼ hxjχ; a; bidμða; bÞhχ; a; bjϕi: function ϕðxÞ as a scalar product of the state vector of the Cχ a≥A

105004-4 WAVELET REGULARIZATION OF GAUGE THEORIES PHYS. REV. D 101, 105004 (2020) Z R The limit of an infinite resolution (A → 0) certainly drives − ϕ ϕ d Z½J¼N e SE½ þ JðxÞ ðxÞd xDϕ ð17Þ us back to the known divergent theories.

C. An example of scalar field theory [where JðxÞ is a formal source, used to calculate the Green To illustrate the wavelet method, following the previous functions, and N is a formal normalization constant of papers [5,30], I start with the phenomenological model of a the Feynman integral] by perturbation expansion; see, e.g., scalar field with nonlinear self-interaction ϕ4ðxÞ, described Ref. [33]. by the Euclidean action functional The parameter λ in the action functional [Eq. (15)]isa phenomenological coupling constant, which knows noth- Z 2 1 m λ ing about the scale of observation, and becomes the running S ½ϕ¼ ddx ð∂ϕÞ2 þ ϕ2 þ ϕ4 : ð15Þ E 2 2 4! coupling constant only because of renormalization or cutoff introduction. The straightforward way to introduce the This model is an extrapolation of a classical interacting spin scale dependence into the model [Eq. (15)] is to express the model to the continual limit [31]. Known as the Ginzburg- local field ϕðxÞ in terms of its scale components ϕaðbÞ Landau model [32], it describes phase transitions in super- using the inverse wavelet transform [Eq. (12)]: conductors and other magnetic systems fairly well, but it Z produces divergences when the correlation functions 1 1 − d ϕ χ x b ϕ dad b ðxÞ¼ d aðbÞ : ð18Þ δn ln Z J Cχ a a a ðnÞ … ½ G ðx1; ;xnÞ¼δ …δ ð16Þ Jðx1Þ JðxnÞ J¼0 This leads to the generating functional for scale-dependent are evaluated from the generating functional fields:

Z Z 1 d d N Dϕ − ϕ − ϕ da1d x1 da2d x2 ZW½Ja¼ aðxÞ exp a1 ðx1ÞDða1;a2;x1 x2Þ a2 ðx2Þ 2 Cχa1 Cχa2 Z Z d d d d d λ … da1d x1 da2d x2 da3d x3 da4d x4 dad x − a1; ;a4 ϕ ϕ ϕ Vx1;…;x4 a1 ðx1Þ a4 ðx4Þ þ JaðxÞ aðxÞ ; ð19Þ 4! Cχa1 Cχa2 Cχa3 Cχa4 Cχa

pffiffiffi where Dða1;a2;x1 − x2Þ is the wavelet transform of the where d is the space dimension, and 1= ζ is a kind of ordinary , and N is a formal normalization weighted scale. constant. For Feynman diagram expansion, the substitution of — The functional (19) R if integrated over all scale argu- the fields by Eq. (18) is naturally performed in the Fourier ∞ dai ments in infinite limits 0 —will certainly drive us back representation ai to the known divergent theory. All scale-dependent fields Z Z ∞ d 1 da d k − [ϕaðxÞ] in Eq. (19) still interact with each other with the ϕ {kxχ˜ ϕ˜ ðxÞ¼ d e ðakÞ aðkÞ; same coupling constant λ, but their interaction is now Cχ 0 a ð2πÞ modulated by the wavelet factor Va1a2a3a4, which is the ˜ ˜ Q x1x2x3x4 ϕ ðkÞ¼χ˜ðakÞ ϕðkÞ: 4 χ˜ a Fourier transform of i¼1 ðaikiÞ. In coordinate form, for λ ϕN the N! interaction, these coefficients, calculated with the Doing so, we have the following modification of the above mentioned first derivative of the Gaussian taken as Feynman diagram technique [6]: the mother wavelet, have the form (1) Each field ϕ˜ ðkÞ is substituted by the scale compo- ˜ ˜ ˜ ˜ nent ϕðkÞ → ϕaðkÞ¼χðakÞ ϕðkÞ. d XN 2 (2) Each integration in the momentum variable is … 2π 2 1 a1; ;aN N bk V … ¼ð−1Þ exp − accompanied by the corresponding scale integration b1; ;bN ζ 2 a k¼1 k d d 1 ξ2 YN 1 ξ d k → d k da d d : × exp − b ; ð2πÞ ð2πÞ a Cχ 2ζ dþ1 ζ i i¼1 ai XN 1 XN (3) Each interaction vertex is substituted by its wavelet ζ ≡ ξ ≡ bk 2 ; 2 ; transform; for the Nth-power interactionQ vertex, this k¼1 ak k¼1 ak N χ˜ gives multiplication by the factor i¼1 ðaikiÞ.

105004-5 M. V. ALTAISKY PHYS. REV. D 101, 105004 (2020)

4 According to these rules, the bare Green function in Eq. (21) [30]. The Γð Þ contribution to the effective action wavelet representation takes the form ðAÞ is shown in diagram (22):

ð2Þ χ˜ða1pÞχ˜ð−a2pÞ G0 ða1;a2;pÞ¼ : p2 þ m2

The finiteness of the loop integrals is provided by the following rule: There should be no scales ai in internal lines smaller than the minimal scale of all external lines [5,6]. Therefore, the integration in ai variables is performed from the minimal scale of all external lines up to infinity. ð22Þ For a theory with local ϕNðxÞ interaction, the presence of two conjugated factors χ˜ðakÞ and χ˜ðakÞ on each diagram Each vertex of the Feynman diagram corresponds to −λ, line, connected to the interaction vertex, simply means and each external line of the 1PI diagram contains the that each internal line of the Feynman diagram carrying wavelet factor χ˜ða k Þ, hence momentum k is supplied by the cutoff factor f2ðAkÞ, where i i Z 4 1 ∞ Γð Þ 2 da A 3 fðxÞ ≔ jχ˜ðaÞj ;fð0Þ¼1; ð20Þ ð Þ ¼ λ − λ2XdðAÞ: ð23Þ Cχ x a χ˜ða1p1Þχ˜ða2p2Þχ˜ða3p3Þχ˜ða4p4Þ 2 where A is the minimal scale of all external lines of this The value of the one-loop integral diagram. This factor automatically suppresses all UV divergences. Z 2 2 ddq f ðqAÞf ððq − sÞAÞ The admissibility condition [Eq. (13)] for the mother XdðAÞ¼ ; ð24Þ 2π d 2 2 − 2 2 wavelet χ is rather loose. At best, χðxÞ would be the ð Þ ½q þ m ½ðq sÞ þ m aperture function of the measuring device [20]. In practice, any well-localized function with χ˜ð0Þ¼0 will suit. For where s ¼ p1 þ p2 and A ¼ minða1;a2;a3;a4Þ, depends analytical calculations, the mother wavelet should be easy on the mother wavelet χ by means of the cutoff function to integrate, and for this reason, as in previous papers fðxÞ. The integral in Eq. (24) with the Gaussian cutoff [5,7,30], we choose the derivative of the Gaussian function function [Eq. (20)] can be easily evaluated. In physical 2 2 as a mother wavelet: dimension d ¼ 4 in the limit s ≫ 4m , this gives [5]

2 −k −2α2 χ˜ − 2 2 2 ðkÞ¼ {ke : ð21Þ 4 α2 e α − 1 − α2 2α α2 lim X ð Þ¼ 2 2 ½e e Ei1ð Þ s2≫4m2 16π α 1 2 This gives Cχ ¼ 2 and provides the exponential cutoff 2α2 2α 2α2 2 þ e Ei1ð Þ; ð25Þ factor [Eq. (20)]: fðxÞ¼e−x . As usual in functional renormalization group technique where α ¼ As is a dimensionless scale, and [34], we can introduce the effective action functional: Z ∞ e−xt Γ½ϕ¼−W½JþhJϕi; Ei1ðxÞ ≡ dt 1 t the functional derivatives of which are the vertex functions: is the exponential integral of the first type. All integrals are finite now, and the coupling constant becomes running, Z X∞ 1 λ ¼ λðα2Þ, only because of its dependence on the dimen- Γ ϕ Γð0Þ ΓðnÞ … ðAÞ½ a¼ ðAÞ þ n ðAÞða1;b1; ;an;bnÞ sionless observation scale α: Cχ n¼1 d d 2 da1d b1 dand bn ∂λ ∂ 4 3λ2 2α2 1 − α × ϕ ðb1Þ…ϕ ðb Þ … : 2 2 X þ e −2α2 a1 an n ¼ 3λ α ¼ e ; ð26Þ a1 an ∂μ ∂α2 16π2 α2 The subscript (A) indicates the presence in the theory of where μ ¼ − ln A þ const. The dimensionless scale varia- minimal scale—the observation scale. ble α is the product of the observation scale A and the total Γð4Þ Let us consider the one-loop vertex function ðAÞ in the momentum s. The analogue of Eq. (25) in standard field scale-dependent ϕ4 model with the mother wavelet theory subjected to cutoff at momentum Λ is

105004-6 WAVELET REGULARIZATION OF GAUGE THEORIES PHYS. REV. D 101, 105004 (2020) Z d 1 ψ → −{eΛðxÞψ d d q ðxÞ e ðxÞ: ð28Þ XΛ ¼ d 2 2 2 2 : jqj≤Λ ð2πÞ ðq þ m Þððq − sÞ þ m Þ The transformation of the gauge field—the electromagnetic Symmetrizing the latter equation in the loop momenta field—is the gradient transformation: q → q þ s=2, we get, in the same limit of s2 ≫ 4m2 and the dimension d ¼ 4: AμðxÞ → AμðxÞþ∂μΛðxÞ: ð29Þ 2 4 1 Λ XΛ ¼ ln 4 þ 1 : ð27Þ In view of the linearity of the wavelet transform, Eq. (29) 16π2 s keeps the same form for all scale components of the gauge Λ2 field Aμ;aðxÞ—in contrast to the matter field transformation We can compare Eq. (27) to Eq. (25) by setting 2 ¼ 1=r in s [Eq. (28)], which is nonlinear—and thus, the gauge trans- α2 momentum space, and ¼ r in wavelet space. Graphs form of the matter fields in a local gauge theory is not the showing the dependence of the one-loop contribution to the change of all scale components ψ ðxÞ by the same phase. ϕ4 a vertex as a function of scale for both the standard The Euclidean QED Langangian is [Eq. (27)] and the wavelet-based formalism [Eq. (25)] are presented in Fig. 1 below. The slopes of both curves in the 1 1 ∂λ 3λ2 ψ¯ ψ ∂ 2 UV limit (A → 0) are the same: ¼ 2. L ¼ ðxÞð=D þ {mÞ ðxÞþ FμνFμν þ ð μAμÞ ; ∂μ 16π 4 2|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}α λ α2 The running coupling constant ð Þ can be understood as the coupling that folds into its running all quantum ≡∂ ∂ − ∂ effects characterized by a scale larger than the observation Dμ μ þ {eAμðxÞ; with Fμν ¼ μAν νAμ; ð30Þ scale A. For small α, Eq. (26) tends to the known result. This is which is the field-strength tensor of the electromagnetic because we have started with the local Ginzburg-Landau field AμðxÞ. (The slashed vectors denote the convolution theory, where the fluctuations of all scales interact with the Dirac γ matrices: =D ≡ γμDμ.) with each other, with the interaction of neighboring scales The wavelet regularization technique works for QED in being most important; see, e.g., Ref. [35] for an excellent the same way as it does for the above considered scalar field discussion of the underlying physics. theory. This means that each line of the Feynman diagram carrying momentum p acquires a cutoff factor f2ðApÞ. D. QED: Wavelet regularization of a local gauge theory In this way, in one-loop approximation, we get the Quantum electrodynamics is the simplest gauge theory electron self-energy, shown in Fig. 2: of the type given in Eq. (4), with the gauge group being the Z ΣðAÞ 4 γ p − − γ 1 ðpÞ 2 d q FAðp; qÞ μ½2 =q m μ Abelian group Uð Þ: −{e ; χ˜ χ˜ − 0 ¼ 4 p 2 2 p 2 ðapÞ ð a pÞ ð2πÞ ½ð2 − qÞ þ m ½2 þ q 0.1 wavelet scale cutoff ð31Þ momentum cutoff where 0.01 ≡ 2 p 2 p − FAðp; qÞ f A 2 þ q f A 2 q

2 2 2 2 X4(r) ¼ e−A p −4A q

0.001 is the product of the wavelet cutoff factors, and A ¼ minða; a0Þ is the minimal scale of two external lines of the diagram in Fig. 2. Similarly, for the vacuum polarization diagram of QED, 0.0001 0.001 0.01 0.1 1 shown in Fig. 3, we get [21] squared dimensionless scale (r)

FIG. 1. Plot of the one-loop contribution to the ϕ4 vertex, calculated with the first derivative of the Gaussian as the mother wavelet as a function of the squared observation scale r ¼ A2s2, compared to that calculated with the standard cutoff at the cutoff momentum Λ ¼ A−1 in Euclidean d ¼ 4 dimensions. FIG. 2. Electron self-energy diagram in scale-dependent QED.

105004-7 M. V. ALTAISKY PHYS. REV. D 101, 105004 (2020)

use wavelets for gauge theories, QED and QCD, was undertaken for the first time by P. Federbush [9] in a form of discrete wavelet transform. Later, it was extended by using the wavelet transform in lattice simulations and theoretical studies of related problems [37–39]. The con- FIG. 3. Vacuum polarization diagram in scale-dependent QED. sideration was restricted to axial gauge and a special type of divergency-free wavelets in four dimensions. The con- Z text of that application was the localization of the wavelet ΠðAÞ 4 μν ðpÞ 2 d q basis, which may be beneficial for numerical simulation, ¼ −e F ðp; qÞ χ˜ðapÞχ˜ð−a0pÞ ð2πÞ4 A but is not tailored for analytical studies, and does not link the gauge invariance to the dependence on scale. The Trðγμð=q þ p== 2 − mÞγνð=q − p== 2 − mÞÞ × 2 2 2 − 2 2 2 : discrete wavelet transform approach to different quantum ½ðq þ p= Þ þ m ½ðq p= Þ þ m field theory problems has been further developed in ð32Þ Hamiltonian formalism, but for scalar theories with local interaction [39,40]. The electron-photon interaction vertex, in one-loop ap- Now is a point to think of how we can build a gauge- proximation, with the photon propagator taken in the invariant theory of fields that depend on both the position Feynman gauge, gives the equation (x) and the resolution (a). To do this, we recall that the free fermion action [Eq. (1)] can be considered as a matrix ðAÞ Z 4 Γμ;r 2 d f p= − f= − m element of the Dirac operator: ¼ e γα χ˜ð−pa0Þχ˜ð−qrÞχ˜ðkaÞ ð2πÞ4 ðp − fÞ2 þ m2 S ¼hψjγμ∂μ þ {mjψi: ð34Þ − = − E γ =k f m γ × μ 2 2 α ðk − fÞ þ m Assuming a scalar product h·j·i in a general Hilbert space 1 H, in accordance with the original Dirac’s formulation of × f2ðAðp − fÞÞ f2 quantum fieldP theory [41], we can insert arbitrary partitions ˆ 2 2 of unity 1 ¼ jcihcj into Eq. (34), so that × f ðAðk − fÞÞf ðAfÞ: ð33Þ c X ψ γ ∂ 0 0 ψ The vertex function [Eq. (33)] and the inverse propagator SE ¼ h jcihcj μ μ þ {mjc ihc j i: 0 are related by the Ward-Takahashi identities, which are c;c wavelet transforms of corresponding identities of the An important type of the unity partition in Hilbert space ordinary local gauge theory [7,36]. The detailed one-loop H is a unity partition related to the generalized wavelet calculations, except for the contribution to the vertex, transform [Eq. (8)]: can be found in Ref. [7]. As for the vertex contribution Z [Eq. (33)], shown in Fig. 4, the calculation is rather μ ν ˆ d Lð Þ † cumbersome, but it can be done numerically. 1 ¼ ΩðνÞjχi hχjΩ ðνÞ: ð35Þ G Cχ

IV. GAUGE INVARIANCE FOR Our main criterion for this choice is to find a group G SCALE-DEPENDENT FIELDS which pertains to the physics of quantum measurement and For a non-Abelian gauge theory, both terms in the gauge provides the fields defined on finite domains rather than field transformation [Eq. (5)] are nonlinear. The wavelet points. The group that can leverage this task is a group of transform [Eq. (18)] can hardly be applied to the theory affine transformations: without violation of local gauge invariance. An attempt to ∶ 0 ∈ R 0 ∈ Rd G x ¼ ax þ b; a þ;x;x;b : ð36Þ

Following Refs. [5,7], we consider an isotropic theory. The representation of the affine group [Eq. (36)]inL2ðRdÞ is chosen as 1 x − b ½Ωða; bÞχðxÞ ≔ χ ; ð37Þ ad a

FIG. 4. One-loop vertex function in scale-dependent QED. and the left-invariant Haar measure is

105004-8 WAVELET REGULARIZATION OF GAUGE THEORIES PHYS. REV. D 101, 105004 (2020)

daddb Assuming the formal coupling constant of the gauge field dμLða; bÞ¼ : ð38Þ a Aμ;aðbÞ to be dependent on scale only, we can rewrite the covariant derivative by changing Aμ;aðbÞ to gðaÞAμ;aðbÞ: In view of the linearity of the wavelet transform

Z Dμ ¼ ∂μ þ {gAμ ðbÞ: ð44Þ 1 x − b ;a ;a ψ → ψ χ¯ ψ d ðxÞ aðbÞ¼ d ðxÞd x; ð39Þ Rd a a This means we have a collection of identical gauge theories for the fields ψ ðbÞ;Aμ ðbÞ, labeled by the scale variable a, the action on the affine group [Eq. (36)] keeps the same a ;a which differ from each other only by the value of the scale- form as the action of the genuine theory [Eq. (34)]. Thus, dependent coupling constant g ¼ gðaÞ. It is a matter of we get the action functional for the fields ψ ðbÞ defined on a choice whether to keep the scale dependence in gðaÞ,or the affine group: solely in Aμ ðbÞ. The Euclidean action of the multiscale Z ;a 1 theory takes the form SE ¼ ½ψ¯ aðbÞγμ∂μψ aðbÞ Cχ R ⊗Rd Z þ 1 d dad b ψ¯ γ ψ daddb SE ¼ aðbÞð μDμ;a þ {mÞ aðbÞ ψ¯ ψ Cχ a þ {m aðbÞ aðbÞ ; ð40Þ a 1 A A þ Fμν;aFμν;a þ gauge fixing terms; ð45Þ where the derivatives ∂μ are now taken with respect to 4 spatial variables bμ. The meaning of the representation Eq. (40) is that the action functionalR is now a sum of where da independent scale components SE ¼ SðaÞ , with no a A ∂ A − ∂ A − ABC B C interaction between the scales. Fμν;a ¼ μAν;a νAμ;a gf Aμ;aAν;a: StartingR from the locally gauge invariant action 4 The difference between the standard quantum field theory SE ¼ d xψ¯ ð=D þ {mÞψ, we destroy such independence by the cubic term ψ¯ =Aψ, which yields cross-scale terms. formalism and the field theory with action (45), defined However, knowing nothing about the point-dependent on the affine group, consists in changing the integration d gauge fields AμðxÞ at this stage, we should certainly ask measure from d x to the left-invariant measure on the affine the question of how one can make the theory of Eq. (40) group [Eq. (38)]. So, the generating functional can be invariant with respect to a phase transformation defined written in the form locally on the affine group: Z R 4 −S ½Φþ dad bΦ ðbÞJ ðbÞ DΦ E Cχ a a a X Z½JaðbÞ ¼ aðbÞe ; ð46Þ A A UaðbÞ¼exp { wa ðbÞT ? ð41Þ A where ΦaðbÞ¼ðAa;μðbÞ; ψ aðbÞ; …Þ is the full set of all Since the action [Eq. (40)], for each fixed value of the scale-dependent fields present in the theory. Since the “ ” scale a, has exactly the same form as the standard action Lagrangian in action (45), for each fixed value of a, [Eq. (1)], we can introduce the invariance with respect has exactly the same form as that in standard theory, the to local phase transformation separately at each scale Faddeev-Popov gauge-fixing procedure [42] can be intro- ∂ duced to the scale-dependent theory in a straightfor- by changing the derivative ∂μ ≡ into the covariant ∂bμ ward way. derivative A. Feynman diagrams Dμ;a ¼ ∂μ þ {Aμ;aðbÞð42Þ The same as in wavelet regularization of a local theory, with the gauge transformationP law for the scale-dependent described in Sec. III, here we understand the physically A A gauge field Aμ;aðbÞ¼ A Aμ;aðbÞT identical to Eq. (5): observed fields as the sums of scale components from the observation scale A to infinity [5]: 0 † ∂ † Aμ;aðbÞ¼UaðbÞAμ;aðbÞUaðbÞþ{ð μUaðbÞÞUaðbÞ: Z Z 1 ∞ da 1 x − b ψ A d χ ψ ðxÞ¼ d b d aðbÞ: Similarly, for the field strength tensor and for the Yang- Cχ A a a a Mills Lagrangian: The free-field Green functions at a given scale a are 1 − YM projections of the ordinary Green function to the scale a Fμν;a ¼ {½Dμ;a;Dν;a;La ¼ 2 TrðFμν;aFμν;aÞ: ð43Þ 2g performed by the χ wavelet filters:

105004-9 M. V. ALTAISKY PHYS. REV. D 101, 105004 (2020)

… χ¯˜ …χ¯˜ … Ga1;…;an ðk1; ;knÞ¼ ða1k1Þ ðanknÞGðk1; ;knÞ:

The interacting-field Green functions, according to the action [Eq. (45)], can be constructed if we provide the equalityQ of all scale arguments by ascribing the multiplier δ − δ − The gluon-to-fermion coupling: gðaÞ i ðln ai ln aÞ to each vertex, and ðln ai ln ajÞ to each line of the Feynman diagram. This is different from the local theory, described in Sec. III, where all scale components do interact with each other. Now, we do not yield the cutoff factor f2ð·Þ on each internal line, with fðxÞ given by the scale integration [Eq. (20)]. Instead, we have to put the wavelet filter modulus squared on each internal line. This suppresses not only the UV divergences, but also the IR divergences. As a result, we arrive at the following The three-gluon vertex: diagram technique, which is (up to the above mentioned cutoff factors) identical to standard Feynman rules for Yang-Mills theory; see, e.g., Ref. [43]. The propagator for the spin-half fermions:

where c, d are the indices of the fermion representation of 47 the gauge group. ð Þ The propagator of the gauge field (taken in the Feynman gauge): All momenta are incident to the vertex: p þ q þ r ¼ 0.

Similarly, for the four-gluon vertex:

The propagator:

The gluon-to-ghost interaction vertex:

with r þ p þ q ¼ 0.

105004-10 WAVELET REGULARIZATION OF GAUGE THEORIES PHYS. REV. D 101, 105004 (2020)

For simplicity in the following calculations, I use the first for each line of the diagram, calculated for the scale a of the derivative of Gaussian as a mother wavelet [Eq. (21)], considered model. which provides the cancellation of both the UV and the IR divergences by virtue of jχ˜ð·Þj2 on each propagator line. B. Scale dependence of the gauge coupling constant For the chosen wavelet [Eq. (21)], the wavelet cutoff To study the scale dependence of the gauge coupling factor is constant, we can start with a pure gauge field theory without fermions, along the lines of Ref. [44]. The total

2 2 one-loop contribution to three-gluon interaction is given by 2 −a p FaðpÞ¼ðapÞ e ð48Þ the diagram in Eq. (49):

ð49Þ

In standard QCD theory, the one-loop contribution to the ABC 3 CA ABC one-loop Γμ μ μ ¼ −{g ðaÞ f Vμ μ μ ðp1;p2;p3Þ; ð50Þ three-gluon vertex is calculated in the Feynman gauge [45]. 1 2 3 2 1; 2; 3 This was later generalized to an arbitrary covariant gauge [46]. These known results, being general in kinematic where the common color factor is CA ¼ 2TFNC, NC is structure, are based on dimensional regularization, and thus 1 are determined by the divergent parts of integrals. Different the number of colors, and TF ¼ 2 is the usual normalization corrections to the perturbation expansion based on analy- of generators in fundamental representation; see, e.g., ticity have been proposed [47,48], but they are still based Ref. [51]. on divergent graphs. In this context, QCD is often con- sidered as an effective theory, which describes the low- 1. Gluon loop contribution energy limit for a set of asymptotically observed fields, We calculate the one-loop tensor structure obtained by integrating out all heavy particles [49]. The one-loop Vμ1;μ2;μ3 ðp1;p2;p3Þ in the Feynman gauge. After symmet- effective theory is believed to be derivable from a future rization of the loop momenta in diagram (50), unified theory, which includes gravity. The essential artifact of renormalized QCD is the logarithmic decay of the running coupling constant p3 −p2 p1 −p3 p2 −p1 2 2 l1 ¼ f þ ;l2 ¼ f þ ;l3 ¼ f þ ; αsðQ Þ at infinite momentum transfer Q → ∞, known 3 3 3 as asymptotic freedom. With the help of MS, the calcu- lations are available up to the five-loop approximation [50]. and the tensor structure of the diagram takes the form In the present paper, I do not pretend to derive the logarithmic law. Instead, I have shown that if our under- standing of gauge invariance is true in an arbitrary func- tional basis, based on a Lie group representation we use to measure physical fields, the resulting theory is finite by construction. The restriction of calculations to the Feynman gauge and the specific form of the mother wavelet are technical simplifications, with which we proceed to make the results viewable. The first term on the rhs of Eq. (49) is the unrenormal- ized three-gluon vertex [Eq. (47)]. The second graph is the FIG. 5. Gluon loop contribution to three-gluon vertex; gluon loop shown in Fig. 5: Its value is p1 þ p2 þ p3 ¼ 0.

105004-11 M. V. ALTAISKY PHYS. REV. D 101, 105004 (2020)

one-loop − is the tensor structure of the three-gluon interaction Vμ1;μ2;μ3 ðp1;p2;p3;fÞ¼Vμ1;α;βðp1;l3; l2Þ vertex [Eq. (47)]. × Vα μ δ −l3;p2;l1 ; 2; ð Þ The tensor structure of Eq. (50) can be represented as a − × Vδ;μ3;βð l1;p3;l2Þ; ð51Þ sum of two terms: the first term is free of loop momentum f, and the second term is quadratic in it: where

≔ − δ one-loop 0 1 Vμ1;μ2;μ3 ðp1;p2;p3Þ ðp3;μ1 p2;μ1 Þ μ2;μ3 Vμ1;μ2;μ3 ðp1;p2;p3;fÞ¼V ðp1;p2;p3ÞþV ðp1;p2;p3;fÞ; − δ þðp1;μ2 p3;μ2 Þ μ3;μ1 − δ þðp2;μ3 p1;μ3 Þ μ1;μ2 ð52Þ with

1 3 − − − Vμ1;μ2;μ3 ðp1;p2;p3;fÞ¼ ½fμ1 fμ3 ðp1;μ2 p3;μ2 Þþfμ1 fμ2 ðp2;μ3 p1;μ3 Þþfμ2 fμ3 ðp3;μ1 p2;μ1 Þ

7 2 þ f ½ðp3 μ − p2 μ Þδμ μ þðp1 μ − p3 μ Þδμ μ þðp2 μ − p1 μ Þδμ μ 3 ; 1 ; 1 2; 3 ; 2 ; 2 3; 1 ; 3 ; 3 1; 2 2 þ ½δμ μ fμ fαðp2 α − p1 αÞþδμ μ fμ fαðp1 α − p3 αÞþδμ μ fμ fαðp3 α − p2 αÞ: 3 1 2 3 ; ; 1 3 2 ; ; 2 3 1 ; ;

1 δμν 2 Integrating the equation Vμ μ μ p1;p2;p3;f with the Gaussian weight, we substitute fμfν → f into the Gaussian R 1; 2; 3 ð Þ d 2 d d −ζf 2 d d 2 −2−1 2 integral e f d f ¼ 2 π ζ . With ζ ¼ 3a , d ¼ 4, this gives the tensor structure

13 2 2 2 2 2 1 − a ½p þp þp −p1p2−p1p3−p2p3 V ðp1;p2;p3Þ¼ e 9 1 2 3 × Vμ μ μ ðp1;p2;p3Þ: 864π2 1; 2; 3

RThe part of the tensor structure that does not contain f contributes a term proportional to the Gaussian integral 2 π d −ζf d 2 e d f ¼ðζÞ . This gives

2 a 2 2 2 2 − a ½p þp þp1p2 0 e 3 1 2 Vμ μ μ ðp1;p2;p3 ¼ −p1 − p2Þ; 144π2 1; 2; 3 where

0 4 5 Vμ μ μ ðp1;p2Þ¼ ðp2 μ p2 μ p2 μ − p1 μ p1 μ p1 μ Þþ ðp2 μ p2 μ p1 μ − p1 μ p1 μ p2 μ Þ 1; 2; 3 3 ; 1 ; 2 ; 3 ; 1 ; 2 ; 3 3 ; 1 ; 2 ; 3 ; 1 ; 2 ; 3 2 1 þ ðp2 μ p2 μ p1 μ − p1 μ p1 μ p2 μ Þþ ðp1 μ p1 μ p2 μ − p2 μ p2 μ p1 μ Þ 3 ; 2 ; 3 ; 1 ; 1 ; 3 ; 2 3 ; 2 ; 3 ; 1 ; 1 ; 3 ; 2

37 58 2 2 þ δμ μ p1p2ðp2 μ − p1 μ Þþ δμ μ ðp p2 μ − p p1 μ Þ 27 1 2 ; 3 ; 3 27 1 2 1 ; 3 2 ; 3

5 2 2 2 þ ðδμ μ p p1 μ Þ − δμ μ p p2 μ − δμ μ p p2 μ Þ 27 2 3 1 ; 1 1 3 2 ; 2 1 2 2 ; 3

32 2 2 þ ðδμ μ p1 μ ðp þ p1p2Þ − δμ μ p2 μ ðp þ p1p2ÞÞ 27 1 3 ; 2 1 2 3 ; 1 2

16 2 2 53 2 2 47 þ ðδμ μ p p2 μ − δμ μ p p1 μ Þþ ðδμ μ p p1 μ − δμ μ p p2 μ Þþ p1p2ðδμ μ p1 μ − δμ μ p2 μ Þ: 27 1 3 1 ; 2 2 3 2 ; 1 27 1 3 2 ; 2 2 3 1 ; 1 27 2 3 ; 1 1 3 ; 2

Summing these two terms, we get ABC C f 2 2 2 2 13 ABC 3 A − a ðp þp þp1p2Þ 2 0 Γμ μ μ ðp1;p2Þ¼−{g ðaÞ × e 3 1 2 a Vμ μ μ ðp1;p2Þþ Vμ μ μ ðp1;p2; −p1 − p2Þ ; ð53Þ 1 2 3 2 144π2 1 2 3 6 1 2 3

where Vμ1μ2μ3 , given by Eq. (52), is the tensor structure of the unrenormalized three-gluon vertex.

105004-12 WAVELET REGULARIZATION OF GAUGE THEORIES PHYS. REV. D 101, 105004 (2020)

2. Contribution of four-gluon vertex The next one-loop contribution to the three-gluon vertex comes from the diagrams with four-gluon interaction, of the type shown in Fig. 6. In the case of four-gluon contribution, the common color factor cannot be factorized: instead, there are three similar diagrams with gluon loops inserted in each gluon leg: p3, p2, and p1, respectively. The case of p3 is shown in Fig. 6. The one-loop contribution to a three-gluon vertex shown in Fig. 6 can be easily calculated, taking into account that the squared momenta in gluon are canceled by FIG. 6. One-loop contribution to the three-gluon vertex pro- wavelet factors [Eq. (48)]. This gives vided by four-gluon interaction.

Z 4 − DEC 2 − δ − − δ 2 − δ − 2 AEX BDX δ δ − δ δ a ð {gðaÞÞf ½ð p3 fÞδ μ3ϵ þð f p3Þϵ μ3δ þð f p3Þμ3 ϵδ × ð gðaÞÞ ½f f ð μ1μ2 ϵδ ϵμ2 μ1δÞ

4 BEX ADX DEX BAX 2 2 2 2 d f þ f f ðδμ μ δϵδ − δϵμ δμ δÞþf f ðδδμ δϵμ − δϵμ δμ δÞ × expð−a f − a ðf þ p1 þ p2Þ Þ : 1 2 1 2 2 1 2 1 ð2πÞ4

9 The presence of four-gluon interaction does not allow for ABC C f ½δμ μ s2 − δμ μ sμ ; the factorization of the common color factor. Instead, there 2 A 1 3 2 3 1 are three different terms in color space: and thus the whole integral C fDECfAEXfBDX ¼ − A fABC; 2 3 2 2 9 {g ðaÞ −a s CA C ABC 2 ABC δ − δ DEC BEX ADX A ABC Vμ1μ2μ3 ðsÞ¼ 2 e f ½ μ1μ3 sμ2 μ2μ3 sμ1 : f f f ¼þ 2 f ; 64π 2 DEC DEX BAX ABC ð56Þ f f f ¼ −CAf ; ð54Þ with the normalization condition Two more contributing diagrams, symmetric to Fig. 6, are different from the above calculated VðA; μ1;p1; B; μ2;p2; fACDfBCD ¼ C δ : A AB C; μ3;p3Þ only by changing B; μ2;p2 ↔ C; μ3;p3 and μ ↔ μ There are two Gaussian integrals contributing to the A; 1;p1 C; 3;p3, respectively. This gives us two more diagram shown in Fig. 6: terms,

Z 4 1 2 2 3 d f −2 2 2−2 2 a s − 2 2 9 a f a sf 2 ABC {g ðaÞ a t CA ACB IðsÞ¼ 4 e ¼ 2 4 e ; Vμ μ μ ðtÞ¼ e 2 f ½δμ μ tμ − δμ μ tμ ; ð2πÞ 64π a 1 2 3 64π2 2 1 2 3 2 3 1 Z 4 3 2 2 2 2 9 d f −2 2 2−2 2 sμ a s {g ðaÞ −a u CA a f a sf − 2 ABC 2 CBA δ − δ IμðsÞ¼ fμe ¼ e ; Vμ1μ2μ3 ðuÞ¼ 2 e f ½ μ1μ3 uμ2 μ2μ1 uμ3 ; ð2πÞ4 128π2a4 64π 2 where s ¼ p1 þ p2. where t ¼ p1 þp3 ¼ −p2 and u ¼ p2 þ p3 ¼ −p1. Taking Thus, we can express the tensor coefficients at the three 1 into account the common topological factor 2 standing terms in Eq. (54) as before all these diagrams in Eq. (49), finally we get 3 3 3 T1 ¼ − δμ μ sμ þ δμ μ sμ ; 2 2 2 1 3 2 2 2 3 1 ABC g ðaÞ ABC −a s Γμ μ μ ðp1;p2Þ¼{ 9C f ½e 2 ðδμ μ sμ −δμ μ sμ Þ 3 3 1 2 3 256π2 A 1 3 2 2 3 1 2 2 T2 ¼þ δμ μ sμ − δμ μ sμ ; −a p 2 1 3 2 2 2 3 1 2 2 δ −δ þe ð μ1μ2 p2;μ3 μ2μ3 p2;μ1 Þ 3 δ − δ T3 ¼ ð μ2μ3 sμ1 μ1μ3 sμ2 Þ; ð55Þ − 2 2 a p1 2 δ −δ þe ð μ1μ3 p1;μ2 μ1μ2 p1;μ3 Þ; ð57Þ CA ABC respectively. The sum of all three terms − 2 f T1 þ CA ABC ABC 2 f T2 − CAf T3 gives where t ¼ −p2, u ¼ −p1, p3 ¼ −p1 − p2.

105004-13 M. V. ALTAISKY PHYS. REV. D 101, 105004 (2020)

The contributions containing four-gluon vertexes (without fermions) give

3 2 2 ABC;4 g ðaÞ −a p Γμ μ μ p −{ 9C fABCe 2 V p; −p; 0 : 60 1 2 3 ð Þ¼ 256π2 A ð Þ ð Þ

The contributions of two ghost loops give

ABC FIG. 7. Ghost loop contribution to three-gluon vertex; ghost 3 CA f −2 2 2 Γμ μ μ p; −p; 0 −{g a e 3a p p1 þ p2 þ p3 ¼ 0. 1 2 3 ð Þ¼ ð Þ 2 144π2 1 2 4 1 3. Ghost loop contribution × a pμ pμ pμ þ Vðp; −p; 0Þ : 9 3 1 2 3 2 The last one-loop contribution, not shown in Eq. (49),is the ghost loop diagram (Fig. 7), and one more diagram ð61Þ symmetric to it. The color factor of the diagram Fig. 7 is C Therefore, due to the use of a localized wavelet basis fDEAfFDBfEFC ¼ − A fABC. The tensor structure for dia- 2 with a window width of size a, we obtain an exponential gram Fig. 7 is l2 μ l3 μ l1 μ , and it is l3 μ l1 μ l2 μ for the ; 1 ; 2 ; 3 ; 1 ; 2 ; 3 decay of the vertex function proportional to p2. symmetric diagram [51]. Ghost propagators multiplied by 3 6 −3 2 2−2 2 2 2 The gauge interaction in the action functional [Eq. (45)] − a f 3a ðp1þp2þp1p2Þ wavelet factors give ð {Þ a e , and one is not identical to that of local gauge theory [Eq. (4)]. more (−1) accounts for the fermion loop. Finally, this gives At this point, I cannot definitely claimR that physical ∞ da ϕ −2 2 2 2 observables are integrals of the form A a F½ aðbÞ.If C e 3a ðp1þp2þp1p2Þ Γghost − 3 A ABC the parameter A of a wavelet-regularized local theory ¼ {g ðaÞ 2 f 144π2 [Eq. (19)] were a counterpart of a 1=μ normalization scale, 2 1 in our theory, with scale-dependent gauge invariance, the × a V0 þ Vμ μ μ ðp1;p2;p3 ¼ −p1 − p2Þ ; 18 1 2 3 scale parameter a should be treated as an independent 1 coordinate on a (d þ 1)-dimensional group manifold (a, x), V0 ¼ ðp1 μ − p2 μ Þðp1 μ p2 μ − 2p1 μ p2 μ Þ 27 ; 3 ; 3 ; 2 ; 1 ; 1 ; 2 with the scale transformations given by the generator ∂ 4 D ¼ a a. – þ ðp2 μ p2 μ p2 μ − p1 μ p1 μ p1 μ Þ Using the simplified vertex contributions [Eqs. (59) 27 ; 1 ; 2 ; 3 ; 1 ; 2 ; 3 (61)] of the one-loop scale-dependent Yang-Mills theory, 5 − we can estimate the renormalization of the coupling þ ðp2;μ1 p2;μ2 p1;μ3 p1;μ1 p1;μ2 p2;μ3 Þ: ð58Þ 27 constant g in the considered theory with scale-dependent gauge invariance [Eq. (41)]. Since the scale a in such a theory plays the role of the normalization scale 1=μ of C. Study of simplified three-gluon vertex ðp; − p;0Þ common models, we can calculate the β function: To study the scale dependence of the coupling constant, let us start with a trivial situation: p1 ¼ p, p2 ¼ −p, ∂g β −a2 p3 ¼ 0. The unrenormalized vertex takes the form ¼ ∂ 2 a g0¼const: ΓABC − ABC − 0 μ1μ2μ3 ðpÞ¼ {gðaÞf Vðp; p; Þ; from the equality g0 ¼ Z1g, with where 2 g CA 10 −2 2 2 9 −1 2 2 1 3a p 2a p − 0 ≡ δ δ − 2 δ Z1 ¼ þ 2 e þ e ð62Þ Vðp; p; Þ pμ1 μ2μ3 þ pμ2 μ1μ3 pμ3 μ1μ2 : 16π 81 16

The triangle gluon loop contribution, shown in Fig. 5,is calculated from the one-loop expansion [Eq. (49)] with the – 2 2 2 vertex contributions [Eqs. (59) (61)]. This gives −3a p ABC;3 3 CA e Γμ μ μ p −{g a fABC 1 2 3 ð Þ¼ ð Þ 2 144π2 ∂ 1 β − 2 ¼ ga 2 : ð63Þ 13 ∂a Z1 2 − 0 × a V0 þ 6 Vðp; p; Þ 2 The equation (63) differs from standard renormalization 4 3 p 2 V0 ¼ pμ pμ pμ − ð5δμ μ pμ schemes by the absence of the factor Z for field renorm- 3 1 2 3 27 2 3 1 3 alization. This is because each of the scale-dependent fields þ 5δμ μ pμ þ 32δμ μ pμ Þ: ð59Þ 1 3 2 1 2 3 Aμ;aðbÞ dwells on its own scale a, and is not subjected to

105004-14 WAVELET REGULARIZATION OF GAUGE THEORIES PHYS. REV. D 101, 105004 (2020)

0 have symmetries more complex than the Abelian group of translations. The simplest generalization is the affine group -0.05 G∶x0 ¼ ax þ b, considered in this paper—a tool for study- )

A -0.1 ing scaling properties of physical systems. In this paper, I C

3 have considered the possibility to extend quantum field

/(g Rd 2 -0.15 theory models of gauge fields, usually defined in or π Minkowski space, to more general space—the group of

*16 -0.2 β affine transformations, which includes not only translations and rotations, but also scale transformation. The peculiarity -0.25 of the scale parameter (a) is that the scale transformation generator D ¼ a∂ , in contrast to coordinate or momentum -0.3 a 0 1 2 3 4 5 operators, is not a Hermitian operator. Hence, the scale a is x=ap not a physical observable—it is a parameter of measure- ment—say, a scale we use in our measurements. FIG. 8. Dependence of the [Eq. (64)]onthe Following the previous papers [5,7,30], introducing dimensionless scale x ¼ ap. The dependence is shown in units 3 explicitly a basis χða; ·Þ to describe quantum fields, the g CA of 16π2 . current paper presents a gauge theory with a gauge transformation defined separately on each scale, Ω ψ → { aðxÞψ renormalization [30]. Taking the scale derivative in Eq. (63) aðxÞ e aðxÞ. The transformation from the usual ψ explicitly, we get local fields ðxÞ to the scale-dependent fields, which may be referred to as the scale components of the field ψ with 3 2 respect to the basic function χ at a given scale a,is g CAðapÞ 20 −2 2 2 9 −1 2 2 β − e 3a p e 2a p < 0: — ¼ 16π2 243 þ 32 ð64Þ performed by means of continuous wavelet transform a versatile tool of group representation theory. This repre- The dependence of this function on dimensionless scale sentation is physically similar to coherent state representa- x ¼ ap is shown in Fig. 8 below. tion [18]. The Green functions in the scale-dependent theory become finite, because both the UV and the IR The factors similar to field renormalization may be 2 required depending on the type of observation—if we divergences are suppressed by the wavelet factor jχ˜ðapÞj hAAAi on each internal line of the Feynman diagrams. assume the observable quantity to be dependent on 3 , hAi As a practical example of calculations, the paper presents where the averaging h·i involves integration over a certain one-loop correction to the three-gluon vertex in a pure range of scales. A more detailed study of the subject is Yang-Mills theory. The calculations are done with the planned for future research. Since the action in Eq. (45) mother wavelet χ being the first derivative of the Gaussian. comprises the fields of different scales which do not The Green functions vanish at high momenta, which is interact to each other, to derive a phenomenological usual for the theories with asymptotic freedom. interpretation of the proposed model, we need to study it The existence of such a theory is merely an exciting within a wider framework of the Standard Model with the 2 1 3 mathematical possibility. The author does not know which SUð Þ × Uð Þ × SUcð Þ gauge group to calculate the type of interaction takes place in real processes: standard observable quantities. local gauge theory, where all scales talk to each other due to locally defined gauge invariance, or the same-scale V. CONCLUSION interaction proposed in this paper. This subject needs The basis of Fourier harmonics, an omnipresent tool of further investigation—at least, it seems not less elegant quantum field theory, is just a particular case of the than the existing finite-length and noncommutative decomposition of the observed field ϕ with respect to geometry models [52,53]. representations ΩðgÞ of the symmetry group G responsible for observations. It is commonly assumed that the sym- ACKNOWLEDGMENTS metry group of measurement is a translation group (or, more generally, the Poincar´e group), the representations of The author is thankful to Dr. A. V. Bednyakov, Dr. S. V. which are used. We can imagine, however, that the Mikhailov, and Dr. O. V. Tarasov for useful discussions, measurement process itself is more complex, and may and to the anonymous referee for useful comments.

105004-15 M. V. ALTAISKY PHYS. REV. D 101, 105004 (2020)

[1] F. Dyson, Advanced (World Scientific, [28] S. J. Brodsky and G. F. de T´eramond, Phys. Rev. D 77, Singapore, 2007). 056007 (2008). [2] E. Stueckelberg and A. Petermann, Helv. Phys. Acta 26, 499 [29] W. Polyzou, arXiv:2002.02311. (1953). [30] M. V. Altaisky, Phys. Rev. D 93, 105043 (2016). [3] N. Bogoliuobov and D. Shirkov, Nuovo Cimento 3, 845 [31] J. Glimm and A. Jaffe, Quantum Physics: A Functional (1956). Integral Point of View (Springer-Verlag, Berlin, 1981). [4] G. ’t Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972). [32] V. Ginzburg and L. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 [5] M. V. Altaisky, Phys. Rev. D 81, 125003 (2010). (1950). [6] M. V. Altaisky, in Group 24: Physical and Mathematical [33] J. Zinn-Justin, Quantum Field Theory and Critical Aspects of Symmetries (IoP, 2002), pp. 893–897. Phenomena (Oxford University Press, New York, NY, [7] M. V. Altaisky and N. E. Kaputkina, Phys. Rev. D 88, 1999). 025015 (2013). [34] C. Wetterich, Phys. Lett. B 301, 90 (1993). [8] P. Goupillaud, A. Grossmann, and J. Morlet, Geoexplora- [35] K. G. Wilson and J. Kogut, Phys. Rep. 12, 75 (1974). tion; international Journal of mining and technical geophys- [36] S. Albeverio and M. V. Altaisky, New Adv. Phys. 5,1 ics and related subjects 23, 85 (1984). (2011). [9] P. Federbush, Prog. Theor. Phys. 94, 1135 (1995). [37] I. G. Halliday and P. Suranyi, Nucl. Phys. B436, 414 [10] A. Deur, S. Brodsky, and G. de T´eramond, Prog. Part. Nucl. (1995). Phys. 90, 1 (2016). [38] G. Battle, Wavelets and Renormalization Group (World [11] D. Sonechkin and N. Datsenko, Pure Appl. Geophys. 157, Scientific, Singapore, 1999). 653 (2000). [39] T. Michlin, W. N. Polyzou, and F. Bulut, Phys. Rev. D 95, [12] G. Altarelli, Proc. Sci., CORFU2012 (2013) 002. 094501 (2017). [13] J. C. Collins, Renormalization (Cambridge University [40] G. K. Brennen, P. Rohde, B. C. Sanders, and S. Singh, Phys. Press, Cambridge, England, 1984). Rev. A 92, 032315 (2015). [14] J. D. Christensen and L. Crane, J. Math. Phys. (N.Y.) 46, [41] P. Dirac, The Principles of Quantum Mechanics 4th ed. 122502 (2005). (Oxford University Press, New York, 1958). [15] I. Daubechies, Ten Lectures on Wavelets (S.I.A.M., [42] L. Faddeev and V. Popov, Phys. Lett. 25B, 29 (1967). Philadelphia, 1992). [43] P. Ramond, Field Theory: A Modern Primer 2nd ed. [16] A. L. Carey, Bulletin of the Australian Mathematical Soci- (Addison-Wesley, Reading, MA, 1989). ety 15, 1 (1976). [44] D. J. Gross and F. Wilczek, Phys. Rev. D 8, 3633 (1973). [17] M. Duflo and C. C. Moore, J. Funct. Anal. 21, 209 (1976). [45] J. Ball and T.-W. Chiu, Phys. Rev. D 22, 2550 (1980). [18] I. Daubechies, A. Grossmann, and Y. Meyer, J. Math. Phys. [46] A. Davydychev, P. Osland, and O. V. Tarasov, Phys. Rev. D (N.Y.) 27, 1271 (1986). 54, 4087 (1996). [19] J. R. Klauder and R. F. Streater, J. Math. Phys. (N.Y.) 32, [47] D. Shirkov and I. Solovtsov, Theor. Math. Phys. 150, 132 1609 (1991). (2007). [20] E. Freysz, B. Pouligny, F. Argoul, and A. Arneodo, Phys. [48] A. Bakulev, S. Mikhailov, and N. Stefanis, J. High Energy Rev. Lett. 64, 745 (1990). Phys. 06 (2010) 85. [21] M. V. Altaisky, Symmetry Integr. Geom. Methods Appl. 3, [49] H. Georgi, Annu. Rev. Nucl. Part. Sci. 43, 209 (1993). 105 (2007). [50] P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn, Phys. Rev. [22] D. Gitman and A. Shelepin, Eur. Phys. J. C 61, 111 (2009). Lett. 118, 082002 (2017). [23] V. Varlamov, Int. J. Theor. Phys. 51, 1453 (2012). [51] A. Grozin, Lectures on QED and QCD: Practical Calcu- [24] E. Gorodnitskiy and M. Perel, J. Math. Phys. (N.Y.) 45, lation and Renormalization of One- and Multi-Loop Feyn- 385203 (2012). man Diagrams (World Scientific, Singapore, 2007). [25] F. Bulut and W. Polyzou, Phys. Rev. D 87, 116011 (2013). [52] L. Freidel and E. R. Livine, Phys. Rev. Lett. 96, 221301 [26] M. Altaisky and N. Kaputkina, Int. J. Theor. Phys. 55, 2805 (2006). (2016). [53] D. N. Blaschke, Eur. Phys. Lett. 91, 11001 (2010). [27] M. Altaisky and N. Kaputkina, Russ. Phys. J. 55, 1177 (2013).

105004-16