2023 Re-normalisable Non-local Quark–Gluon Interaction: Mass Gap, Chiral Symmetry Breaking & Scale Invariance

We derive a Nambu–Jona-Lasinio (NJL) model from a non-local gauge theory and show that it has conﬁning properties at low energies. In particular, we present an extended approach to non-local QCD and a complete revision of the technique of Bender, Milton and Savage applied to non-local theories providing a set of Dyson–Schwinger equations in diﬀerential form. In the local case, we obtain closed form solutions in the simplest case of the scalar ﬁeld and extended it to the Yang–Mills ﬁeld. In general, for non-local theories, we use a perturbative technique and a Fourier series and show how higher-order harmonics are heavily damped due to the presence of the non-local factor. The spectrum of the theory is analysed for the non-local Yang–Mills sector and found to be in agreement with the local results on the lattice in the limit of the non-locality mass parameter running to inﬁnity. In the non-local case, we contend ourselves to have the non-locality mass suﬃciently large compared to the mass scale arising from the integration of the Dyson–Schwinger equations. Such a choice grants good agreement, in the proper limit, with the spectrum of the local theory. We derive the gap equation for the fermions in the theory that gives some indication of quark conﬁnement also in the non-local NJL case. This result is really important, as it seems to point to the fact the conﬁnement (and breaking of scale invariance) could be an ubiquitous eﬀect in nature that removes some degrees of freedom in a theory in order to favour others.

free of ghosts [33] 1 and also cure the Higgs vacuum instability problem of the SM Higgs [43], as analysed by one of the authors. It was shown that the β function reaches a conformal limit resolving the Landau poles issue in QFT [44]. Therefore, by capturing the infinite derivatives by an exponential of an entire function we obtain a softened ultraviolate (UV) behaviour in the most suitable manner, without the cost of introducing any new degrees of freedom that contribute to the particle mass spectrum, since there are no new poles in the propagators of infinite-derivative extensions. 2 A bound on the scale of non-locality from observations from LHC and dark matter physics is M ≥ O(10) TeV [37,46]. Moreover, the non-local theory leads to interesting implications for the proton decay and Grand Unified Theories [48], as well as in braneworld models [49]. In addition, non-perturbative strongly coupled regimes, exact β functions and conditions of confinement in higher-derivative nonlocal theories are being actively investigated [50][51][52][53][54][55]. The results obtained so far shows that the effect of the non-locality in the strong coupling limits is to dilute any mass gap (that may be present in the theory) in the UV regime the system generates. 3 In order to render charge renormalisation finite, infinite-derivative terms should be introduced for the fermion fields as well [44]. An analogous mechanism is the well-known Pauli-Villars regularisation scheme where a mass dependence Λ in introduced as the cut-off.
In the infinite-derivative case, the non-local energy scale M plays the role of the ultraviolet cut-off. The infinite-derivative approach is to promote the Pauli-Villars cut-off Λ to the non-local energy scale M of the theory. Recently, the infinite-derivative model for QED and Yang-Mills has been reconsidered in view of its generalisation to the SM [43,44]. 4 Indeed, this model leads to a theory which is naturally free of quadratic divergencies, thus providing an alternative way to the solution of the hierarchy problem [37]. Higher dimensional operators, containing new interactions, naturally appear in higher-derivative theories with non-abelian gauge structure. These operators soften quantum corrections in the UV regime and extinguish divergencies in the radiative corrections. Nevertheless, as can be easily understood by power counting arguments, the new higher-dimensional operators do not break renormalisability [44]. This is due to the improved ultraviolet behaviour of the bosonic 1 The unitarity issues are well addressed due the certain prescription [38][39][40][41][42]. 2 For astrophysical implications, dimensional transmutation and dark matter and dark energy phenomenology in these theories see Refs. [45][46][47]. 3 In the context to gravity theories one can get rid of classical singularities, such as black hole singularities [19,24,[56][57][58][59][60][61][62][63][64][65] and cosmological singularities [66][67][68][69][70][71][72]. Recently, false vacuum tunneling studies were carried out in Ref. [73]. 4 See also Refs. [37,46] for a few phenomenological applications to LHC physics and dark matter physics.
Note that the presence of the non-local energy scale M, associated to the infinitederivative term, manifestly breaks (at classical level) the conformal symmetry of the unbroken gauge sector. Therefore, one may wonder whether this term can also trigger (dynamically) chiral symmetry breaking at low energy, or in other words, whether the fermion field with mass m could dynamically obtain a mass m satisfying the condition m < M.
The aim of the present paper is to investigate this issue by analysing a general class of renormalisable models containing infinite-derivative terms. In this paper, we will show that a non-vanishing mass term for the fermion field can indeed be generated, depending on the kind of interaction at hand, as a solution of the mass gap equation. The fermion mass can be predicted, and it turns out to be a function of the energy scale M.
The paper is organised as follows: In Sec. II we introduce the infinite-derivative SU(N).
In Sec. III we derive the set of Dyson-Schwinger equations for non-local QCD. We solve these equations in a perturbative manner by noting that higher harmonics are heavily damped by the non-local factor. In Sec. IV the spectrum of the theory is analysed for the non-local Yang-Mills sector and found to be in agreement with the local results on lattice in the limit of the non-local scale running to infinity. In the non-local case, we contend ourselves to have the non-local scale sufficiently large with respect the mass scale arising from the integration of the Dyson-Schwinger equations. In Sec. V we derive the gap equation for the fermion in the theory and show that an identical argument as given in Refs. [54,55] can be applied here, giving some indication of quark confinement also in the non-local case. Sec. VI contains our conclusions. In Appendix A we show how to derive the Dyson-Schwinger equations for a φ 4 theory in differential form. In Appendix B we derive the scalar two-point function.

II. INFINITE-DERIVATIVE SU(N)
In infinite-derivative SU(N) with massless fermions we start with the Lagrangian [44]: Both the Yukawa term and the gauge fixing term are delocalised by the infinite-derivative exponential. A source term j µ a A a µ is added in order to 4 use the Lagrangian for the generating functional. As an example we can use f (D 2 ) = D 2 /M 2 where D ab µ = δ ab ∂ µ − igA c µ (T c ) ab is the covariant derivative in adjoint representation. For a large non-local scale M 2 it has been shown in Refs. [50,51] that e f (D 2 ) can be approximated by e f ( ) , where = ∂ 2 . Therefore, one can start with Dealing with the non-locality, we use an approach without redefinition of fields originally devised in Ref. [47]. Inserting F a µν and performing integration by parts, one obtains In the following we use Feynman gauge ξ = 1. The Euler-Lagrange equation obtained from the Lagrangian in Eq. (3) are given by Applying the formalism of Bender, Milton and Savage [74] to the Lagrangian (3) will lead to the tower of Dyson-Schwinger equations which is formulated in terms of multiple-point Green functions. This approach is displayed in Appendices A and B for a local φ 4 theory.

III. SOLUTION OF THE DYSON-SCHWINGER EQUATIONS
The creation of the tower of Dyson-Schwinger equations is explained in detail in Appendix B of Ref. [50]. We do not repeat it here to avoid to reprint already published material. For the one-and two-point functions one obtains and by variation with respect to j λ h (y) The system of equations of motion for the complete set of components of the Green functions for the Yang-Mills field cannot be treated exactly. Instead, we use a mapping to the scalar case. The mapping theorem introduced in Refs. [75,76] is based on Andrei Smilga's solution for the problem that "the Yang-Mills system is not exactly solvable, . . . in contrast to . . . some early hopes" (cf. Sec. 1.2 in Ref. [77]). Indeed, for more than one independent component, in solving the system of equations one observes chaotic behaviour (cf. Refs. [78][79][80]).
Even though the importance of such observations for the macroscopic behaviour is questionable, the safe path is to use a mapping to a single scalar function. This is done here by using where η a µ are the components of the polarisation vector and η µν are the components of the Minkowski metric.
The two-point Green functions from x to x become constants with vanishing derivative while n-point Green functions with n > 2 can be set to zero. One obtains and Contracting with η µ a and δ ah η µλ and using the orthogonality and completeness relations and One has f abc f abd = N c δ cd for two and, therefore, f abc f abc = N c (N 2 c − 1) for three summed indices. The common factor cancels, and one obtains where λ = N c g 2 s and ∆m 2 G = (D − 1)λG 2 (0). The solution for the corresponding local one-point function, given by φ 0 (x) =φ 0 (kx) = µ sn(kx + θ|κ) [75], that obeys the dispersion relation k 2 = ∆m 2 G + λµ 2 /2 with κ = (∆m 2 G − k 2 )/k 2 and θ = (1 + 4N)K(κ), can be expanded in a Fourier series, For non-negative values of n one obviously has However, this can be extended also to negative values, as for Writing e −f ( )φ 1 (kx) + ∆m 2 Gφ 0 (kx) + λφ 0 (kx) 3 = 0 as the first iteration towards the non-local solution, one can use the property sn ′′ (ζ|κ) = −(1 + κ) sn(ζ|κ) + 2κ sn 3 (ζ|κ) (17) to replace the cube by a second derivative. One haŝ and inserting this one obtains which is quite obvious, as the right hand side is nothing else thanφ 0 (kx). Inserting the explicit form ofφ 0 (kx) results in We can assume thatφ 1 (kx) = −iµη m odd a ′ m e imηkx/2 has the same shape with coefficients a ′ m to be determined. With this ansatz one obtains and the comparison gives a ′ m = e f (−m 2 η 2 k 2 /4) a m . Therefore, the first order solution readŝ The dispersion relation k 2 = ∆m 2 G + λµ 2 /2 tells us that for large values of λ the higher harmonics are suppressed in case that f ( ) = /M 2 . Using φ(x) =φ 1 (kx) as one-point function, the equation of motion for the two-point function leads to Again, the Gaussian factors can be used to restrict to the lowest harmonics m, n = ±1, (m 2 G = −6λµ 2 η 2 a 2 1 e 2f (−η 2 k 2 /4) = 12κη 2 k 2 a 2 1 e 2f (−η 2 k 2 /4) ) or with∆ Eq. (25) can be solved iteratively up to arbitrary orders of m 2 G , as it is shown in Ref. [52]. For simplicity, here we take only the leading order approximationG 2 (p) ≈∆(p).
Dimensional regularisation is not applicable in this case, as the integral is not possible to calculate analytically, different from the local limit M → ∞. One might think about ρ 2 = M 2 as an upper cutoff. In this case the integral will diverge as M 2 , and a truncation of this highest power is necessary in order to obtain a finite local limit. Instead, our proposal is to use a fixed upper cutoff Λ 2 which is determined by matching the result to values obtained on the lattice for different values of N c = 2, 3, 4, 6, 8 [81], in the local case analysed in Refs. [82,83] with excellent agreement. In order to compare with the lattice, instead of λ as input one has to use λ = N c g 2 s = 2N 2 c /β with β = 2N c /g 2 s . The values for β are taken from Ref. [81] and listed in the second line of Tab. I. 5 The agreement is excellent. For the fit of the mass values in the local limit we use the method of least squares by minimising with M 2 G = m 2 G + ∆m 2 G . Out of the N = 5 lattice values we have excluded from this method the less reliable value for N c = 6. The result is given by Λ 2 = 73.5015(5)k 2 . As this result suggests, the agreement between the lattice values and our estimates is very good, as these are found in the remaining lines of Tab. I. In Fig. 1 we show the solution of the mass gap equation (27) for M 2 G for the upper limit Λ 2 = 73.5015k 2 in dependence on the non-local 5 A recent paper about the Casimir effect in non-Abelian gauge theories on the lattice [84] has shown that the ground state of the local Yang-Mills theory is not the same as that found in lattice computations [81] (cf. Tab. I). The authors of Ref. [84]obtain M lat G = 1.0(1) that corresponds for us to the choice n = 0 in the glueball spectrum displayed in Appendix B and Refs. [82,83]. This is also in agreement with the picture of the Casimir effect in local Yang-Mills theory presented in Ref. [85].
scale M 2 , together with the dependence of the parameter −κ. It is interesting to point out that, in this way, we can obtain a physical understanding of the non-local scale and its proper order of magnitude as compared to local physics.

V. MASS GAP EQUATION OF THE QUARK
The solution obtained in the previous section is input for the determination of the dynamical quark mass. Namely, the Green function G (0)ab 2µν (x, y) = η µν δ ab G 2 (x − y) with G 2 (x − y) satisfying the differential equation (13) have to be convoluted with the left hand side of the equation of motion (4) in order to obtain This result is inserted into the Euler-Lagrange equation for the quark field, to obtain This equation of motion can be understood as the equation of motion of a non-local Nambu-

Jona-Lasinio (NJL) model. After integration by parts, we derive the NJL Lagrangian
The standard procedure to reach up to the mass gap equation is taken from Ref. [86] and consists of serveral steps. First of all, a Fierz rearrangement leads to the action integral This action integral is ingredient for the functional integral In the following we use iG 2 (z) = GC 2 (z)/2 with C 2 (z)d 4 z = 1. The quartic interaction term can be removed by introducing the meson field (φ α ) = (σ, π) via the factor to the functional integral, interchanging the integrations over w and z, and performing the functional "shift" In doing so and returning in part to x and y, one ends up with the action functional The functional integral reads Performing a Fourier transform and integrating out the fermionic degrees of freedom, one ends up with the functional determinant and the bosonised action where the logarithm of the determinant is understood between momentum states, p ′ | · · · |p .
While in the mean field approximation the first term givesσ 2 /2G, the representation of the determinant is diagonal, leading to with M q (p) = g sC2 (p)σ. The variation of the action integral with respect to the mean valuē σ leads toσ and the insertion into the definition of M q (p) finally leads to the mass gap equation where we insertedC 2 (p) = −2iG 2 (p)/G and used to obtain In order to solve the gap equation for the quark, we assume that the gap mass M q (p) does not depend explicitly on the momentum. In this case one has to solve the equation In principle, M q can be cancelled on both sides, leading to a gap equation similar to Eq. (3.7) in the original NJL publication [87]. However, in order to solve this equation iteratively, it is more appropriate to multiply it instead with M q in order to determine M 2 q from 13 The upper limit is taken to be the same as in the previous section, and the solution M 2 G of the mass gap equation for the glueball is used. The fixed point problem converges, and one obtains a dependence on the non-local scale M 2 which is displayed in Fig. 2. We can see that chiral symmetry breaking appears in non-local QCD as well. It is shown in Refs. [54,55] that an educated guess for confinement is given by the threshold M q /M G > 0.39, above which two real pole solutions for p 2 in the equation change to imaginary pole solutions. According to Fig. 3, this is the case for the non-local scale below 115 ÷ 125k 2 .

VI. DISCUSSION AND CONCLUSIONS
The spectrum of the Yang-Mills field is analysed for the non-local Yang-Mills sector and found in agreement with the local results on lattice in the limit of the non-local scale running to infinity (cf. Tab. I). The agreement with lattice is astonishingly good, and also with respect to previous computations [82,83]. Because of this, this spectrum becomes our In this appendix we show how to derive the Dyson-Schwinger equations for a φ 4 theory in differential form, as exemplified in Refs. [88,89] using the technique devised in Ref. [74].
Starting point is the (massless) Lagrangian leading to the generating functional First of all, after integration by parts the variation of this generating functional with respect to φ leads to the Dyson-Schwinger master equation where (A4) Following a notation used e.g. by Abbott [90], the application of functional derivatives with respect to the current j to the effective action W [j] = ln Z[j] on the one and to the generating functional Z[j] on the other hand leads to the tower of Green functions Inverting this system step by step one obtains the tower In obtaining equations for the Green functions by setting j = 0, at the same time we use translation invariance to write the Green functions in the form to obtain As G 2 (0) and G 3 (0, 0) are constants, this equation is an ordinary (though, nonlinear) differential equation for G 1 (x). For 3λG 2 (0) = ∆m 2 G and G 3 (0, 0) = 0 one has ∂ 2 G 1 + ∆m 2 G G 1 + λG 3 1 = 0. As shown in Ref. [89], this nonlinear differential equation is solved by where we have used cn 2 (ζ|κ) + sn 2 (ζ|κ) = 1 and dn 2 (ζ|κ) + κ sn 2 (ζ|κ) = 1. One can start with sn(θ|κ) = 1 which is satisfied if θ = (1 + 4N)K(κ), and as a consequence of this Cω(1 − κ) = i which is solved by C = i/ω(1 − κ). Therefore, we end up with As Jacobi's elliptic functions sn(ζ|κ), cn(ζ|κ) and dn(ζ|κ) are periodic functions, it is possible to expand them in a Fourier series. Using the nome q = exp(−πK * /K) where K * (κ) = K(1 − κ) and one has Inserting this Fourier series into our solution forḠ 2 (t) leads tō with θ = (1 + 4N)K(κ). Inserting these values for θ, one has cos (2n + 1) πωt 2K(κ) + π 2 + 2πN = −(−1) n sin (2n + 1) πωt 2K(κ) .