Non-local Non-Abelian Gauge Theory: Conformal Invariance and $\beta$-function

This paper focuses on extending our previous discussion of an Abelian U(1) gauge theory involving infinite derivatives to a non-Abelian SU(N) case. The renormalization group equation (RGEs) of the SU(N) gauge coupling is calculated and shown to reproduce the local theory $\beta$-function in the limit of the non-local scale M $\rightarrow \infty$. Interestingly, the gauge coupling stops its running beyond the scale $M$, approaching an asymptotically conformal theory.


INTRODUCTION
With no clinching evidence of new particles in physics Beyond the Standard Model (BSM) by any of the current searches at experiments world-wide, such as the Large Hadron Collider (LHC), an alternative philosophy for BSM could be the modification of the standard canonical kinetic terms through the introduction of infinite derivatives, instead of introducing new particles (new states). Motivated by string field theory [1][2][3][4][5][6][7][8][9][10], infinite derivative formulation is expressed in the form of an entire function [11] and the higher order derivatives are accompanied by a suppression by a scale M , which we call the "non-local scale." The choice of this derivative function does not appear to be unique, as long as it acts to suppress terms in the high energy regime.
We have previously considered a non-local Abelian U(1) gauge theory within this framework and have shown that the evolution of the gauge coupling becomes fixed or "UV-insensitive" in the energy regime well above the non-local scale M [12]. It was also shown that Higgs vacuum instability problem [13] is cured, leading to a stable Abelian Higgs theory. The theory is ghost-free [14] and predicts a unique scattering phenomenology leading to transmutation of energy scales which has its own cosmological and astrophysical implications [15]. The phenomenology of dark matter in this theory is also investigated in Ref. [16] and shown DM experiments can be a novel probe for the scale on non-locality. Strongly coupled regime of the theory was considered in Refs. [17,18] in Higgs and Yang-Mills versions and it was found that the mass gap generated gets diluted due to non-local effects. On aspects of gravity, Ref. [19] showed that the most general quadratic curvature gravitational action (parity-invariant and torsion-free) with infinite covariant derivatives makes the gravitational sector free from the Weyl ghost and is devoid of any classical singularities, such as black hole [19][20][21][22][23][26][27][28][29][30]  In this paper, we extend the same idea involving infinite series of higher-order derivatives to a non-Abelian SU(N) gauge theory, investigate the gauge invariance, and compute the running of the SU(N) gauge coupling. Here too we expect that within this framework, the standard renormalization group equations (RGEs) should be reproduced in the local limit (M → ∞).

NON-ABELIAN EXTENSION
For local non-Abelian SU(N) gauge theory, the Lagrangian includes the gauge boson kinetic term, The trace is over the SU(N) group indices and the fieldstrength tensor is given by where the f abc represents the group structure constant. For implementation of the non-local modification, we follow our approach in Ref. [12]. The gauge boson kinetic term is then described as where the covariant derivative is given by The fermionic part of the Lagrangian is given by the standard form as in Refs. [11,12]: where D 2 = η µν D µ D ν (µ, ν = 0, 1, 2, 3), assuming all gauge and fermionic particles being massless.
Refs. [8,10]. 2 For supersymmetric versions on non-locality in the matter section, see for instance, Refs. [24,25] arXiv:2010.15919v2 [hep-ph] 3 Jul 2021 The exponential term is introduced by the non-local modification and the Lagrangian includes an infinite series of higher dimensional operators that are all suppressed by the non-local scale M . As a result, their contribution can be largely ignored at energies lower than M . In other words, the conventional Lagrangian is reproduced in the limit of M → ∞. We take the metric with η = diag(+1, −1, −1, −1) to implement our procedure for UV completion upon the Wick rotation.

Non-Abelian Gauge Field Propagator
To find the massless non-Abelian gauge boson propagator, we follow the same gauge-fixing prescription as in Ref. [11], and the non-Abelian gauge and ghost Lagrangians are given by and where ξ is the gauge fixing parameter and D ab µ = ∂ µ δ ab − igA c µ (T c ) ab is the covariant derivative in the adjoint representation. In order to have consistency with the standard gauge fixing procedure, we choose the entire function f (2) = e − 2 2M 2 . The non-local Faddeev-Popov procedure is given in Appendix. In the Euclidean space, the gauge boson and ghost propagators have the following forms: in the Feynman-'t Hooft gauge ξ = 1. And the massless fermion propagator is given by [11]:

GAUGE COUPLING RUNNING
In Ref. [12], we have obtained the RGE for the gauge coupling in the Abelian U(1) gauge theory with one Dirac fermion having a unit U(1) charge: The standard result for the beta function is obtained in the local limit of M → ∞. Interestingly, for the nonlocal U(1) theory is "UV-complete," as the beta function is vanishing beyond the non-local scale. This behavior is opposed to the local U(1) theory where the running is asymptotically non-free. In the case of a non-Abelian theory, we expect a similar behavior, namely, the gauge coupling stops running beyond the non-local scale. The beta function for the non-Abelian gauge coupling incorporates the gauge and ghost field contributions; we break up the derivation of each contribution below for clarity.

Gauge Wave Function Renormalization
First we consider the wave function renormalization for the gauge fields. We use the standard group theory parameterizations: f acd f bcd = T (A)δ ab and Tr[T a T b ] = T (R)δ ab with the Dynkin indices for the adjoint (T (A) = N ) and the fundanmental (T (R) = 1/2) representations, respectively. The quadratic Casimir for the adjoint representatio is C 2 (A) = N . All the following calculations are performed with the Feynman gauge (ξ = 1).
Our methodology for calculating the wave function renormalization follows from the same procedure in Ref. [12]. In our calculation, we employ the cutoff regularization scheme. Although one should use a gauge invariant regularization scheme such as the dimensional regularization, the cutoff regularization scheme can be practically used, since there is a one-to-one correspondence between the two schemes (at the one-loop level), log Λ 2 = 1 , where Λ is the cutoff parameter and is the dimensional regularization parameter. For the gaugegauge-gauge contribution: where β ≡ 1/M 2 , and N µν is given by Using the Schwinger parameters, α 1 and α 2 , the integral is recast as It is convenient to introduce new parameters, s : {0, ∞} and α : {0, 1}, which are defined as α 1 + α 2 = s and α = α 1 s (which gives α 2 = s(1 − α)). In addition, we take the momentum shift p → p − αs+β s+2β q and then whereN µν is given bȳ and A = 1+ αs+β s+2β , B = 1− 2αs+2β s+2β , and C = 2− αs+β s+2β . To find the corrections ∆Z Gauge , we focus on the coefficient of terms only proportional to q µ q ν , which gives Here, O(q 2 ) contains higher order momentum terms which are sub-leading and ignored: Employing the same procedure for the ghost contribution, we find where β = β/2 due to the ghost propagator's exponential factor in (29). Again, picking up the q µ q ν terms and ignoring the further sub-leading terms, Contributions from the fermion loops are the same as in Ref. [12] with the addition of the SU(N) group factor T (R), for N F fermions Extracting all the relevant terms from (15) and (17),

Fermion Wavefunction Renormalization
Proceeding in a similar manner, lastly, the non-Abelian gauge contribution to the fermion wave function renormalization is

Non-Abelian Vertex Correction
Next we consider the corrections to the non-Abelian fermion-fermion-gauge vertex. They are evaluated in a similar fashion, with the main difference being external momentum is set to zero. We work in the Feynman-'t Hooft gauge, and find ∆g= g 3 16π 2 Having derived wave-function renormalization and vertex correction we proceed to investigate the β−function studies 3 .

Non-Local Non-Abelian Beta Function
Bringing these contributions together in the standard way we culminate with the beta function 4 for the gauge couplings in the infinite derivative framework.
According to the standard QFT procedure, we extract the β-function from Γ 2 by taking ∂/∂logΛ and replacing the cutoff Λ with the re-normalization scale µ 5 .
In the limit M → ∞ (β → 0) we recover the standard SU(N) RGE, which is what we expect in the infrared (IR) limit of the non-local theory: In Fig. 1 we show the gauge coupling running of the non-local (local) non-Abelian SU(N) gauge theory represented by the solid (dashed) line, where we have set N = 3 and N F = 6 & N F = 0. Beyond the non-local scale that we set M = 10 5 GeV, the running becomes "conformally complete", in the sense that the running becomes frozen. On the other hand, the standard gauge coupling running exhibits the usual asymptotic free behavior.

CONCLUSIONS AND DISCUSSIONS
The central attractive feature of our non-local extension of QFT is that the theory becomes scale free (scale invariant) at energies beyond the non-local scale M . In other words, the theory becomes conformal, and M signifies the UV-fixed point 6 . The UV behaviour of the non-Abelian Higgs model is expected to be very similar to what we have discussed in the Abelian case [12]. The RG evolution of the Higgs self-coupling freezes beyond M and the Higgs potential never develops any instability.
Classically scale-invariant models are of immense interest in QFT. Usually the conformal symmetries are 3 In deriving the RGE, one may consider 1-loop corrections to the gauge boson self-interactions or 1-loop corrections to the gauge coupling of the ghost. Thanks to the gauge invariance, the RGE devised in this way is coincide with the one presented in this paper. 4 The RGE is described as δg = ∆g − g(∆Z f ermion + 1 2 ∆Zgauge) 5 Also see Ref. [12] for details of the procedure. 6 See Refs. [70,71] for UV-fixed points in supersymmetric context. anomalous in QFT in four-dimensions except for a specific system such as N=4 supersymmetric Yang-Mills theory. We may classify scale-invariant theories as: (i) Exact scale invariance is always maintained; (ii) Scale Invariance breaks only at the quantum level.
In infinite derivative non-Abelian gauge theory, we find a unique scenario where the theory does not possess the scale invariance in the IR, but theory becomes scale invarinat in the UV whose scale is set by the non-local scale M . Thus, the symmetry breaking (to generate a scale) maybe considered as an artefact of only the low energy behaviour of the theory leading to the concept of scale-dependence, or in other words, the running of the coupling constants. This is very similar to the classical scale invariant theory with "soft" symmetry-breaking, which would not suffer from the naturalness issue arising from the UV sensitivity of scalar mass squared corrections. However, in non-local theory, the beta function is exponentially suppressed for µ > M , so that energy M practically works as the conformal fixed point. This leads to scale invariance in the UV in spite of "quantum" interactions and the presence of "scale" in the IR, thereby denoting a "scale-insensitivity" of the tree-level action 7 .
In the infinite derivative theory, the Higgs mass squared corrections are exponentially suppressed at the scale beyond M , and the non-local scale works as an effective cutoff for the corrections, ∆m 2 H ∼ M 2 [11]. This means that the Higgs mass fine-tuning is reduced to , with M P lanck being the Planck mass. Beyond the scale of non-locality to infinite energy, all the beta-functions are vanishing and all the couplings are approaching a fixed point, determined by the nonlocal scale (see Fig. 1). This means that no Landau-pole exists in the theory.
We end our discussion with the speculation that the non-local extension of gauge field theories may allow us to provide a unified framework of Conformal Invariance without having to encounter with Landau poles, and thus paves a theoretical pathway for theories being perturbatively stable and valid to infinite energy. However, the details are beyond the scope of the current investigation, and we will take this issue up explicitly in a future publication.

Appendix A: NL Non-Abelian Faddeev-Popov Procedure
Focusing solely on the kinetic term of the non-abelian gauge field, the partition function in the Euclidean space is given by where G a (x) = f (2)∂ µ A a µ (x) − w a (x) and G a (x) transforms as G a (x) → f (2)∂ µ (A a µ (x) + D ab µ α b (x)) − w a (x) under a SU(N) gauge transformation. Because the function Z(J) is independent of w a (x) we introduce the arbitrary functional dependent on w a (x) and employ the usual gauge fixing procedure to arrive at Using the familiar relation between a functional determinant and a path integral over complex Grassmann variables, we obtain the lagrangians given in Eqs. (5) and (6). In order to provide consistency with the gauge fixing procedure in the local limit, we have chosen f (2) = e −2/2M 2 .

Power Counting
In conventional quantum field theories, the kinetic terms of gauge fields contain up to two derivatives. In momentum space this means that the propagators behave as k −2 . In four dimensions each momentum loop provides a k 4 factor in any quantum loop integral. The superficial degree of divergence of a Feynman diagram in the local theory is therefore given by D = no. of factors of internal momentum in the numerator −no. of factors in the denominator = 4L − 2I + 2V, where L is the number of loops, V is the number of vertices, and I is the number of internal propagators.
When one has exponential factors in the loops, vertices and propagators, an exponential suppression will always be dominant over a polynomial growth. Therefore, we see that as long as these exponential factors come with a negative power, the integrals should be manifestly convergent. In the counting of the superficial degree of divergence in the infinite derivative theory, we note that each propagator comes with an exponential suppression, (see Eq. (7) and Ref. [12]) and each vertex also comes with an exponential suppression. Thus, the power of exponential suppression factor should be: By using the topological relation: we have Since V is an integer, E < 0, for all loops and the corresponding scattering amplitudes are superficially convergent.

BRST Invariance
Next we discuss the convergence in the complete BRST-invariant infinite-derivative gauge theory action. The conclusion is exactly the same. The quantized action action is of the form: where ξ is the gauge fixing parameter, B is the auxiliary field, and c andc are the ghost and anti-ghost fields, respectively. The BRST transformations for non-Ableian gauge theories express a residual symmetry of the effective action which remains after the original gauge invariance has been broken by the addition of the gauge-fixing and ghost action terms. In our theory, we introduce the following BRST transformations: δ BRSTc where δλ is an infinitesimal anti-commuting constant parameter. We show the BRST-invariance of S total,quantized (following Refs. [37,39]): the BRST transformation of the gauge field is just a gauge transformation of A µ generated by c a δλ. Therefore, any gauge-invariant functionals of F µν , like the first term in Eq. (29) gives M 2 (F µν ) a )) = 0. The second term in Eq. (29) gives δ BRST ( ξ 2 (B a ) 2 ) = 0 from Eq. (33). For the third term in Eq. (29), the transformation of A a µ cancels the transformation ofc in the last term, due to Eqs. (30, 31 & 32) leaving us with which is is equal to 0, using the Jacobi identity (see Ref. [38]). The transformation of c σ is nilpotent, while the transformation of A µ is also nilpotent, Hence, the action in Eq. (29) is BRST-invariant. Noting the fact that the only part of the ghost action which varies under the BRST transformations is that of the anti-ghost (c a ), the central idea behind our proof of BRST-invariance is that we have chosen the BRST variation of the anti-ghost (c a ) (see Eqs. (32)) to cancel the variation of the gauge-fixing term.
Next we proceed in the same manner as in Ref. [39] and introduce BRST-invariant couplings of the ghosts and gauge bosons to external fields K µ (anti-commuting) and L σ (commuting), so that the effective action S is which is also BRST-invariant. Let us now compute the superficial degree of divergence for the BRST-invariant action. To proceed, we introduce the following notations: • n A is the number of gauge boson vertices, • n G is the number of anti-ghost-gauge boson-ghost vertices, • n K is the number of K-gauge boson-ghost vertices, • n L is the number of L-ghost-ghost vertices, • I A is the number of internal gauge boson propagators, • I G is the number of internal ghost propagators, • E c is the number of external ghosts, • Ec is the number of external anti-ghosts.
By counting the exponential contributions of the propagators and the vertex factors, as discussed earlier, we can now obtain the superficial degree of divergence, which is given by By using the following topological relation, we obtain Employing the momentum conservation law for ghost and anti-ghost lines, we obtain It is clear that the degree of divergence reduces as n K , E c and Ec increase. Thus, one may conclude the most divergent diagrams are those for which n K = E c = Ec = 0, i.e., the diagrams whose external lines are all gauge bosons. In this case, the degree of divergence is given by E = 1 − 2n A − L. Since n A is an integer, E < 0, namely, the corresponding loop amplitudes (L ≥ 1) are superficially convergent. Here, we have utilized powercounting-in-exponentials to understand the degree of divergence. In the local QFT limit of M → ∞, all the exponential suppression factors disappear, and we will obtain divergences according to power-counting-in-polynomials as usual in the standard QFT. We justify our procedure in the way that by taking M → ∞, our results should reproduce the local theory results.