# Strongly Coupled String-inspired Infinite Derivative Non-local Yang-Mills:
Diluted Mass Gap 111This paper is the original version of the article accepted
for publication JHEP (2021) with the title ”Diluted Mass Gap in Strongly
Coupled Non-local Yang-Mills.

Marco Frasca ID<EMAIL_ADDRESS>Rome, Italy  Anish Ghoshal ID
<EMAIL_ADDRESS>INFN - Sezione Roma “Tor Vergata”, Via della
Ricerca Scientifica 1, 00133, Roma, Italy

###### Abstract

We investigate the non-perturbative regimes in the class of non-Abelian
theories that have been proposed as an ultraviolet completion of 4-D Quantum
Field Theory (QFT) generalizing the kinetic energy operators to an infinite
series of higher-order derivatives inspired by string field theory. We prove
that, at the non-perturbative level, the physical spectrum of the theory is
actually corrected by the ”infinite number of derivatives” present in the
action. We derive a set of Dyson-Schwinger equations in differential form, for
correlation functions till two-points, the solution for which are known in the
local theory. We obtain that just like in the local theory, the non-local
counterpart displays a mass gap, depending also on the mass scale of non-
locality, and show that it is damped in the deep UV asymptotically. We point
out some possible implications of our result in particle physics and cosmology
and discuss aspects of non-local QCD-like scenarios.

##  I Introduction

Adopting novel perspectives, our understanding of the fundamental nature of
reality may reside in the putative finiteness of superstring loop amplitudes
via the Polyakov representation Polchinski:1998rq ; Polchinski:1998rr . This
novel formalism invokes the fact that the fundamental constituent to be
quantized is an extended non-local object.

In this context, an infinite derivative non-local approach was motivated
starting from string field theory Moffat:1990jj ; Evens:1990wf ;
Tomboulis:1997gg ; Moffat:2011an ; Tomboulis:2015gfa ; Kleppe:1991rv ; sft1 ;
sft2 ; sft3 ; padic1 ; padic2 ; padic3 ; Frampton-padic ; Tseytlin:1995uq ;
marc ; Siegel:2003vt ; Calcagni:2013eua ; Modesto:2011kw ; Modesto:2012ga ;
Modesto:2015foa ; Modesto:2017hzl  where attempts where made to address the
divergence problem in QFT by generalizing the kinetic energy operators of the
Standard Model (SM) to an infinite series of higher order derivatives
suppressed by the scale of non-locality ($M$) at which the higher order
derivatives come into the picture Biswas:2014yia . Such a theoretical
construction naturally solves the SM vacuum instability problem as the
$\beta$-functions at the scale of non-locality vanish beyond M, without
introducing any new degrees of freedom in the particle spectrum Olive:2016xmw
; Ghoshal:2017egr . They have been explicitly shown to be ghost-free
Buoninfante:2018mre , predicting conformal invariance in the UV, trans-
planckian scale transmutation and dark matter phenomenology, leading to a new
directions of UV-completion of 4D Quantum Field Theories, valid and
perturbative up to infinite energy scales Ghoshal:2018gpq ;
Buoninfante:2018gce  222Theories of gravity with infinite derivatives were
studied in Ref.Biswas:2011ar ; Giacchini:2018wlf ; Burzilla:2020utr ,
particularly for singularities, such as black hole singularities Biswas:2011ar
; Biswas:2013cha ; Frolov:2015bia ; Frolov:2015usa ; Koshelev:2018hpt ;
Koshelev:2017bxd ; Buoninfante:2018xiw ; Cornell:2017irh ; Buoninfante:2018rlq
; Buoninfante:2018stt ; Abel:2019zou ; Buoninfante:2020ctr  and cosmological
singularities Biswas:2005qr ; Biswas:2006bs ; Biswas:2010zk ; Biswas:2012bp ;
Koshelev:2012qn ; Koshelev:2018rau  and in context to inflationary cosmology &
predictions in the CMB, Koshelev:2016vhi ; SravanKumar:2018dlo ;
Koshelev:2020fok ; Koshelev:2020xby ; Koshelev:2020foq . For SUSY see Refs.
Gama:2017ets ; Gama:2020pte .

Strongly coupled sectors in such a scenario was studied in a scalar field
theory case and was found to predict diluted mass gap in the UV, with M
determining the corrections to the mass spectrum arising from the infinite
sets of higher-derivatives present in the theory. Alongside with detailed
investigations of N-point correlation functions were performed in
Ref.Frasca:2020jbe . The purpose of this paper is to extend our analysis to
Yang-Mills case and understand the Green’s functions in the otherwise non-
perturbative regions of the interactions. We will see that the mass gap
developed in the local theory gets asymptotically diluted and scale invariance
is asymptotically reached in the deep UV.

This paper is organized as follows: In Sec.II, we introduce the non-local
Yang-Mills theory and fix the formalism. In Sec.III, we obtain the Dyson-
Schwinger equations for n-point correlation functions (dubbed nP-functions in
the following) and obtain the structure of the 2P-function and the mass gap.
In Sec.IV, we put this work in perspective to future possible applications
framing better its understanding. Finally, in Sec.V, we yield the conclusions.

##  II Infinite Derivative SU(N)

For local non-Abelian SU(N) gauge theory, the Lagrangian includes the gauge
boson kinetic term,

$\mathcal{L}_{g}=-\frac{1}{4}F^{a\mu\nu}F_{a\mu\nu}.$ (1)

The trace is over the SU(N) group indices and the field-strength tensor is
given by

$F_{\mu\nu}^{a}=\partial_{[\mu}A_{\nu]}^{a}-gf^{abc}A^{b}_{\mu}A^{c}_{\nu}\ ,$
(2)

where the $f^{abc}$ represents the group structure constant and $g$ the
coupling that is dimensionless. For implementation of the non-local
modification, we follow the approach in Refs. Ghoshal:2017egr ;
Ghoshal:2020lfd .

It should be pointed out that a common definition in literature is the
following Tomboulis:1997gg ; Ghoshal:2020lfd

$L_{f}=-\frac{1}{4}F^{a}_{\mu\nu}e^{-f(D^{2})}F^{a\mu\nu}.$ (3)

This definition is somewhat misleading for the following reason. Let us take

$f(D^{2})=\frac{D^{2}}{M^{2}}$ (4)

where $D_{\mu}^{ab}=\partial_{\mu}\delta^{ab}-igA^{c}_{\mu}(T^{c})^{ab}$ is
the covariant derivative in the adjoint representation and, as usual, $M$ is a
very large mass that sets the scale of non-locality. It should be very large
as non-local effects have not been uncovered so far. So, generally speaking,
the variation in momenta scale of the $D^{2}$ will be at most comparable but
lesser to $M^{2}$. Now,

$D^{2}=(\partial_{\mu}-igT^{a}A^{a}_{\mu})^{2}=\partial^{2}-ig\partial^{\mu}\left(T^{a}A^{a}_{\mu}\right)-igT^{a}A^{a}_{\mu}\partial^{\mu}-g^{2}T^{a}T^{b}A^{a}_{\mu}A^{b\mu}.$
(5)

By a direct application of the Backer-Campbell-Hausdorff formula (BCH), this
yields

$\displaystyle e^{-\frac{D^{2}}{M^{2}}}$ $\displaystyle=$ $\displaystyle
e^{-\frac{\Box}{M^{2}}}e^{-\frac{1}{M^{2}}\left(-ig\partial^{\mu}\left(T^{a}A^{a}_{\mu}\right)-igT^{a}A^{a}_{\mu}\partial^{\mu}-g^{2}T^{a}T^{b}A^{a}_{\mu}A^{b\mu}\right)}\times$
$\displaystyle
e^{-\frac{1}{2M^{4}}\left[\Box,-ig\partial^{\mu}\left(T^{a}A^{a}_{\mu}\right)-igT^{a}A^{a}_{\mu}\partial^{\mu}-g^{2}T^{a}T^{b}A^{a}_{\mu}A^{b\mu}\right]}\times\ldots,$

that implies that the commutators are all higher orders with respect to first
two exponentials entering as powers of the field. Indeed, there are two kind
of approximations that we introduce at this stage. Firstly, we are assuming
that the gauge field $A^{a}_{\mu}$ is very small with respect to the non-local
mass scale. Secondly, we assume that also its space-time variations happen on
scale smaller than that of the non-local mass scale. This will imply that a
gauge change will not affect the action in a sensible way keeping invariance
in a proper limit.

In order to explicitly evaluate this in the strong coupling limit, let us
consider

$e^{-\frac{D^{2}}{M^{2}}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\left(\frac{D^{2}}{M^{2}}\right)^{n}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!M^{2n}}\left(\partial^{2}-ig\partial^{\mu}\left(T^{a}A^{a}_{\mu}\right)-igT^{a}A^{a}_{\mu}\partial^{\mu}-g^{2}T^{a}T^{b}A^{a}_{\mu}A^{b\mu}\right)^{n}.$
(7)

In the strong coupling limit, the theory acquires a scale mass $\mu$ and the
background solution we start from for SU(N) has the form
$A_{\mu}^{a}\propto\mu(2/Ng^{2})^{\frac{1}{4}}f(x,\mu,Ng^{2})$, where
$f(x,\mu,Ng^{2})$ is a bounded function (see Frasca:2015yva  for the local
case). This implies that we get a power series in $\mu/M$ and terms like
$M^{-1}\partial_{\mu}$. We are granted that gauge invariance is preserved
wherever $\mu/M\ll 1$ and gradient terms can be retained. At the same time,
the strong coupling limit $Ng^{2}\gg 1$ holds consistently and helps to
enforce our approximation. Therefore, our techniques can be safely applied.
Such an ordering is essential for what follows where we assume that fields are
large locally but not with respect to the non-local mass scale $M$. So, adding
further fields to preserve gauge invariance, at this stage, would imply to
keep higher order terms in our approximation as the scale $\mu/M\ll 1$ will
determine their importance.

This argument is consistent with Ref. Tomboulis:1997gg  where all the non-
local factors in the gauge field propagator are properly evaluated with $\Box$
rather than $D^{2}$ but in that case, the author works in the weak coupling
limit. Our approximation is just dual to it, for the local limit, and
consistent as well.

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 $M\to\infty$. We take
the metric convention with ${\rm diag}(+1,-1,-1,-1)$ to implement our
procedure for UV completion upon the Wick rotation.

###  II.1 Gauge Field Re-definition

Now, we show that we can treat eq.(3) as for the scalar field. Let us define

${\hat{A}}_{\mu}^{a}=e^{-\frac{1}{2}f(\Box)}A_{\mu}^{a}.$ (8)

The idea with this change of variables is to eliminate the exponential non-
local factor from the kinetic term just as happens for scalar field theory.
Indeed, our work will parallel the one already presented in Frasca:2020jbe .

We will arrive at the Lagrangian

$\displaystyle\mathcal{L}_{f}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\hat{A}}_{\mu}^{a}(\Box\eta^{\mu\nu}-\partial^{\mu}\partial^{\nu}){\hat{A}}_{\nu}^{a}$
(10)
$\displaystyle-\frac{g}{4}f^{abc}e^{-f(\Box)}\left[e^{\frac{1}{2}f(\Box)}\left(\partial_{\mu}{\hat{A}}^{a}_{\nu}-\partial_{\nu}{\hat{A}}_{\mu}^{a}\right)\left(e^{\frac{1}{2}f(\Box)}{\hat{A}}^{b\mu}e^{\frac{1}{2}f(\Box)}{\hat{A}}^{c\nu}\right)\right]$
$\displaystyle-\frac{g}{4}f^{abc}e^{-f(\Box)}\left[e^{\frac{1}{2}f(\Box)}{\hat{A}}^{b\mu}e^{\frac{1}{2}f(\Box)}{\hat{A}}^{c\nu}e^{\frac{1}{2}f(\Box)}\left(\partial_{\mu}{\hat{A}}^{a}_{\nu}-\partial_{\nu}{\hat{A}}_{\mu}^{a}\right)\right]$
$\displaystyle-\frac{g^{2}}{4}f^{abc}f^{cde}e^{-f(\Box)}\left[e^{\frac{1}{2}f(\Box)}{\hat{A}}^{b\mu}e^{\frac{1}{2}f(\Box)}{\hat{A}}^{c\nu}e^{\frac{1}{2}f(\Box)}{\hat{A}}^{d}_{\mu}e^{\frac{1}{2}f(\Box)}{\hat{A}}^{e}_{\nu}\right]$
$\displaystyle+j_{\mu}^{a}e^{\frac{1}{2}f(\Box)}{\hat{A}}^{a\mu},$

where we added an arbitrary source term $j_{\mu}^{a}$ that will be useful in
the following. This is similar in the way the non-local scalar field theory is
formulated. The main difference is the multiplication of the interaction part
by the non-local factor $e^{-f(\Box)}$.

The non-Abelian ghost and gauge-fixing Lagrangians are given by

$\mathcal{L}_{ghost}=-\bar{c}^{a}e^{-f(\Box)}(\partial^{\mu}D_{\mu}^{ab})c^{b},$
(11)

and

$\mathcal{L}_{g-f}=\frac{1}{2\xi}{\hat{A}}_{\mu}^{a}e^{-f(\Box)}\partial^{\mu}\partial^{\nu}{\hat{A}}_{\nu}^{a},$
(12)

where $\xi$ is the gauge fixing parameter. In order to have consistency with
the standard gauge fixing procedure, we choose the entire function
$e^{-f(\Box)}$. The standard result is obtained in the local limit of
$M\to\infty$.

Similarly as we have done for the gauge field, also the ghost field can be
redefined as

$c^{a}=e^{-\frac{1}{2}f(\Box)}{\hat{c}}^{a}.$ (13)

This will yield

$\mathcal{L}_{ghost}=-{\bar{\hat{c}}^{a}}\partial^{\mu}\left(\partial_{\mu}\delta^{ab}-ige^{\frac{1}{2}f(\Box)}{\hat{A}}^{c}_{\mu}(T^{c})^{ab}\right){\hat{c}}^{b}+{\bar{\eta}}^{a}e^{\frac{1}{2}f(\Box)}{\hat{c}}^{a}+e^{\frac{1}{2}f(\Box)}{\bar{\hat{c}}^{a}}\eta.$
(14)

Also in this case, we added arbitrary source terms $\eta^{a}$ and
${\bar{\eta}}^{a}$.

Finally, our Lagrangian will be given by

$\mathcal{L}=\mathcal{L}_{f}+\mathcal{L}_{g-f}+\mathcal{L}_{ghost}.$ (15)

##  III N-point Correlation Functions

The aim of this section is to obtain the Dyson-Schwinger equations for the
correlation functions of the non-local Yang-Mills theory. Then, we solve them
using a generalized form of the mapping theorem Frasca:2007uz ; Frasca:2009yp
. This theorem permits to match the classical solutions of a local scalar
field theory on the classical solutions of a local Yang-Mills theory. We will
show how this works also for the non-local case. This will provide the
spectrum of the non-Abelian theory for the latter case.

###  III.1 Classical infinite derivative Yang-Mills theory

The aim of this section is to show that there is a way to solve the classical
Yang-Mills theory that successfully parallels the Dyson-Schwinger approach.
This is a current expansion, assuming the gauge potentials as functions of the
currents themselves. This was initially presented in Frasca:2013tma  for the
scalar field but it can be applied straightforwardly to any field theory.

We are working fixing the gauge to Landau-Lorenz. The main reason to do this
is that, when we will quantize the theory, this choice will make our
computations simpler. By no means, this imply that the same computations
cannot be done in another or more convenient gauge. Our main result will be
the spectrum of the theory that is a gauge-invariant observable and can be
obtained in lattice computations at least for the local case.

We can write down the equations of motion for the Yang-Mills field in the form

$\displaystyle\Box
A_{\mu}^{a}+gf^{abc}e^{-\frac{1}{2}f(\Box)}\partial_{\nu}\left[e^{\frac{1}{2}f(\Box)}{\bar{A}}^{b}_{\mu}e^{\frac{1}{2}f(\Box)}A^{c\nu}\right]+$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}A^{b\nu}e^{\frac{1}{2}f(\Box)}(\partial_{\mu}A^{c}_{\nu}-\partial_{\nu}A^{c}_{\mu})\right]+$
$\displaystyle
g^{2}f^{abc}f^{cde}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}A^{b\nu}e^{\frac{1}{2}f(\Box)}A^{d}_{\nu}e^{\frac{1}{2}f(\Box)}A^{e}_{\mu}\right]$
$\displaystyle=$ $\displaystyle e^{\frac{1}{2}f(\Box)}j^{a}_{\mu},$ (16)

where we have no ghost field, this will enter when we will quantize the
theory. We have removed the hat from the fields being the context clear. Then,
assuming we are in a strong coupling limit, we can evaluate a strong coupling
series by a functional Taylor series in the currents as

$\displaystyle A_{\mu}^{a}[j]$ $\displaystyle=$ $\displaystyle
A_{\mu}^{a}[0]+\int d^{4}x_{1}\left.\frac{\delta A_{\mu}^{a}}{\delta
j_{\nu}^{b}(x_{1})}\right|_{j=0}j_{\nu}^{b}(x_{1})$ (17)
$\displaystyle+\frac{1}{2!}\int
d^{4}x_{1}d^{4}x_{2}\left.\frac{\delta^{2}A_{\mu}^{a}}{\delta
j_{\nu}^{b}(x_{1})j_{\lambda}^{c}(x_{2})}\right|_{j=0}j_{\nu}^{b}(x_{1})j_{\lambda}^{c}(x_{2})+O(j^{3}).$

The coefficients, that play a similar role to the correlation functions of the
quantum theory that we will see in a moment, can obtained from eq.(III.1) by
deriving it successively with respect to the currents and solving the
corresponding PDEs one obtains. We do not give here this set of equations
because are not of interest for what follows.

Anyway, we point out that, In order to solve this hierarchy of equations,
there is a class of solutions for the local Yang-Mills theory that maps on a
$\phi^{4}$ theory. In principle, this permits to solve all the hierarchy of
the Dyson-Schwinger equations Frasca:2015yva . Besides, it gives the spectrum
of the theory in very close agreement with lattice computations Frasca:2017slg
. Then, it is legit to guess that we can have such solutions at the leading
order that are properly corrected by non-local effects, if we refer to the
mass scale $M$ being very high. This mapping was firstly proposed in Refs.
Frasca:2007uz ; Frasca:2009yp .

The technique is the following Frasca:2019ysi . Let us introduce a set of
symbols: $\eta-$symbols for SU(N) (this can be possibly extend to the SO(N)
case in a straightforward way. It should be seen for other groups.). For the
sake of simplicity, we work out the case for SU(2) where the proof of their
existence is straightforward. Indeed, in this case, they can be defined as
Smilga:2001ck ; Frasca:2019ysi

$\eta_{\mu}^{a}=((0,1,0,0),(0,0,1,0),(0,0,0,1)),$ (18)

that yields

$\eta_{\mu}^{1}=(0,1,0,0),\ \eta_{\mu}^{2}=(0,0,1,0),\
\eta_{\mu}^{3}=(0,0,0,1),$ (19)

that implies $\eta_{\mu}^{a}\eta^{a\mu}=3$. This easily generalizes to SU(N)
as

$\eta_{\mu}^{a}\eta^{a\mu}=N^{2}-1.$ (20)

Similarly, by generalizing the SU(2) case,

$\eta_{\mu}^{a}\eta^{b\mu}=\delta_{ab},$ (21)

and

$\eta_{\mu}^{a}\eta_{\nu}^{a}=\frac{1}{2}\left(g_{\mu\nu}-\delta_{\mu\nu}\right),$
(22)

being $g_{\mu\nu}$ the Minkowski metric and $\delta_{\mu\nu}$ the identity
tensor. Then, we choose the solutions as for the local case in the form

$A_{\mu}^{a}(x)=\eta_{\mu}^{a}\phi(x)$ (23)

and put this into the action. This will work exactly in the Lorenz gauge. that
we fix to carry on the quantum theory otherwise we will get asymptotic
corrections Frasca:2007uz ; Frasca:2009yp  in the limit of a strong coupling.

###  III.2 Partition function

Give the Lagrangian in Eqn.(15), the partition function is given by

$Z[j,{\bar{\eta}},\eta]=\int D[A]D[c]D[{\bar{c}}]e^{-\int
d^{4}x\mathcal{L}_{f}-\int d^{4}x\mathcal{L}_{g-f}-\int
d^{4}x\mathcal{L}_{ghost}}.$ (24)

We will refer to it when we will do averaging in quantum field theory. We are
working in Euclidean metric.

###  III.3 Dyson-Schwinger equations

We use the Bender-Milton-Savage method Bender:1999ek , described in Appendix
A, to obtain the set of Dyson-Schwinger equations for our case. Then, we will
approach these equations choosing a set of solutions mapped from a scalar
field. This will yield in the end the main result of the paper that is, the
spectrum of the theory.

We start from the Lagrangian given in Eqn.(10). After we have chosen the
Landau-Lorenz gauge aiming at quantization, as already said, we obtain the
motion equations for the gauge fields given by

$\displaystyle\Box
A_{\mu}^{a}+gf^{abc}e^{-\frac{1}{2}f(\Box)}\partial_{\nu}\left[e^{\frac{1}{2}f(\Box)}{\bar{A}}^{b}_{\mu}e^{\frac{1}{2}f(\Box)}A^{c\nu}\right]+$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}A^{b\nu}e^{\frac{1}{2}f(\Box)}(\partial_{\mu}A^{c}_{\nu}-\partial_{\nu}A^{c}_{\mu})\right]+$
$\displaystyle
g^{2}f^{abc}f^{cde}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}A^{b\nu}e^{\frac{1}{2}f(\Box)}A^{d}_{\nu}e^{\frac{1}{2}f(\Box)}A^{e}_{\mu}\right]+$
$\displaystyle gf^{abc}e^{\frac{1}{2}f(\Box)}\bar{c}^{b}\partial_{\mu}c^{c}$
$\displaystyle=$ $\displaystyle e^{\frac{1}{2}f(\Box)}j^{a}_{\mu},$ (25)

and for the ghost field is

$-\Box
c^{a}+gf^{abc}\left(e^{\frac{1}{2}f(\Box)}A_{\mu}^{b}\right)\partial^{\mu}c^{c}=e^{\frac{1}{2}f(\Box)}\eta^{a}.$
(26)

For our computations to follow, we point out that the propagator in the Landau
gauge can be written as Peskin:1995ev

$G_{\mu\nu}^{ab}(x,y)=\delta_{ab}\left(\eta_{\mu\nu}-\frac{\partial_{\mu}\partial_{\nu}}{\Box}\right)G_{2}(x,y).$
(27)

Our aim will be to get the structure of the 2P-function for the non-local
Yang-Mills theory and the mass gap, and study its nature. These are presented
in Appendix B.

###  III.4 Solution of the Dyson-Schwinger equations

We can try to get a solution to the Dyson-Schwinger set of equations for the
infinite-derivative Yang-Mill theory in a similar way this is done for the
local theory Frasca:2015yva . The idea is to select a set of solutions and
work out all the results (background field method). Anyway, it is possible to
start with a different set of solutions, e.g. a Yang-Wu monopole Wu:1967vp ,
but in this way the results obtained for the local theory are quite different
and closed form solutions could not possibly be found. The Dyson-Schwinger set
of equations, till 2-P functions, is derived in Appendix B. In order to solve
them, starting with the 1P-functions, we look for a solution mapped on a
scalar field by taking Frasca:2015yva

$G_{1\mu}^{a}(x)=\eta_{\mu}^{a}\phi(x).$ (28)

Using the properties of the $\eta$-symbols and the fact that one can take the
1P-functions, $P_{1}^{a}(x)=0$ and $K_{2}^{ab}(0)=0$, we get for eq.(Appendix
B: Derivation of the Dyson-Schwinger equations for 1P- and 2P-functions)

$\Box\phi(x)+Ng^{2}e^{-\frac{1}{2}f(\Box)}\left[2G_{2}(0)e^{f(\Box)}\phi(x)+\left(e^{\frac{1}{2}f(\Box)}\phi(x)\right)^{3}\right]=0.$
(29)

This is very similar to the result we obtained for the scalar field in
Frasca:2020jbe . The most important change arises from the mass shift due to
quantum effects that, in this case, is given by

$\delta m^{2}=2Ng^{2}G_{2}(0).$ (30)

The factor 2 arises just by the algebraic properties of the gauge group and
can change depending on it.

Next, we can consider the 2P-function by applying the mapping with the scalar
field in eq.(28) and the 2P-function in the Landau gauge as in eq.(27). For
reasons of completeness, we need to report the mapped equation. One gets

$\displaystyle\delta_{ah}\left(\eta_{\mu\lambda}-\frac{\partial_{\mu}\partial_{\lambda}}{\Box}\right)\Box
G_{2}(x,y)$
$\displaystyle+gf^{abc}e^{-\frac{1}{2}f(\Box)}\partial^{\nu}\left[e^{\frac{1}{2}f(\Box)}G_{3\mu\nu\lambda}^{bch}(0,y)+e^{\frac{1}{2}f(\Box)}\delta_{bh}\left(\eta_{\mu\lambda}-\frac{\partial_{\mu}\partial_{\lambda}}{\Box}\right)G_{2}(x,y)\times\right.$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}\eta_{\nu}^{c}\phi(x)+e^{\frac{1}{2}f(\Box)}\eta_{\mu}^{b}\phi(x)e^{\frac{1}{2}f(\Box)}\delta_{ch}\left(\eta_{\nu\lambda}-\frac{\partial_{\nu}\partial_{\lambda}}{\Box}\right)G_{2}(x,y)\right]-$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{3\mu\nu\lambda}^{bch}(0,y)+e^{\frac{1}{2}f(\Box)}\delta_{bh}\left(\eta_{\mu\lambda}-\frac{\partial_{\mu}\partial_{\lambda}}{\Box}\right)\partial^{\nu}e^{\frac{1}{2}f(\Box)}G_{2}(x,y)\eta_{\nu}^{c}e^{\frac{1}{2}f(\Box)}\phi(x)\right.$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}\eta_{\mu}^{b}\partial^{\nu}\phi(x)\delta_{ch}\left(\eta_{\nu\lambda}-\frac{\partial_{\nu}\partial_{\lambda}}{\Box}\right)e^{\frac{1}{2}f(\Box)}G_{2}(x,y)\right]-$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}\partial_{\mu}G_{3\nu\lambda}^{bch\nu}(0,y)+e^{\frac{1}{2}f(\Box)}\delta_{bh}\left(\eta_{\nu\lambda}-\frac{\partial_{\nu}\partial_{\lambda}}{\Box}\right)\partial_{\mu}G_{2}(x,y)\eta^{c\nu}e^{\frac{1}{2}f(\Box)}\phi(x)+\right.$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}\delta_{ch}\left(\eta_{\lambda}^{\nu}-\frac{\partial^{\nu}\partial_{\lambda}}{\Box}\right)\partial_{\mu}G_{2}(x,y)\eta_{\nu}^{b}e^{\frac{1}{2}f(\Box)}\partial_{\mu}\phi(x)\right]+$
$\displaystyle
g^{2}f^{abc}f^{cde}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}G_{3\mu\nu\lambda}^{(j)bdh}(0,y)e^{\frac{1}{2}f(\Box)}\eta^{\nu
e}e^{\frac{1}{2}f(\Box)}\phi(x)\right.$ $\displaystyle
e^{\frac{1}{2}f(\Box)}\delta_{bd}\left(\eta_{\mu\nu}-\frac{\partial_{\mu}\partial_{\nu}}{\Box}\right)G_{2}(0)e^{\frac{1}{2}f(\Box)}\delta_{eh}\left(\eta_{\lambda}^{\nu}-\frac{\partial_{\lambda}\partial^{\nu}}{\Box}\right)G_{2}(x,y)+e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{4\mu\nu\lambda}^{bdeh\nu}(0,0,y)+$
$\displaystyle
e^{\frac{1}{2}f(\Box)}\delta_{bh}\left(\eta_{\mu\lambda}-\frac{\partial_{\mu}\partial_{\lambda}}{\Box}\right)G_{2}(x,y)\eta_{\nu}^{d}\eta^{e\nu}\left(e^{\frac{1}{2}f(\Box)}\phi(x)\right)^{2}+$
$\displaystyle
e^{\frac{1}{2}f(\Box)}\delta_{dh}\left(\eta_{\nu\lambda}-\frac{\partial_{\nu}\partial_{\lambda}}{\Box}\right)G_{2}(x,y)\eta_{\mu}^{b}\eta^{e\nu}\left(e^{\frac{1}{2}f(\Box)}\phi(x)\right)^{2}+$
$\displaystyle
e^{\frac{1}{2}f(\Box)}\delta_{eh}\left(\eta_{\lambda}^{\nu}-\frac{\partial^{\nu}\partial_{\lambda}}{\Box}\right)G_{2}(x,y)\eta_{\mu}^{b}\eta^{d}_{\nu}\left(e^{\frac{1}{2}f(\Box)}\phi(x)\right)^{2}+$
$\displaystyle
e^{\frac{1}{2}f(\Box)}G_{3\mu\lambda}^{beh\nu}(0,y)e^{\frac{1}{2}f(\Box)}\eta_{\nu}^{d}\phi(x)+$
$\displaystyle\delta_{be}\left(\eta_{\mu}^{\nu}-\frac{\partial_{\mu}\partial^{\nu}}{\Box}\right)e^{\frac{1}{2}f(\Box)}G_{2}(0)\delta_{dh}\left(\eta_{\nu\lambda}-\frac{\partial_{\nu}\partial_{\lambda}}{\Box}\right)e^{\frac{1}{2}f(\Box)}G_{2}(x,y)$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}G_{3\nu\lambda}^{deh\nu}(0,y)e^{\frac{1}{2}f(\Box)}\eta_{\mu}^{b}\phi(x)+\delta_{de}\left(\eta_{\nu}^{\nu}-\frac{\partial_{\nu}\partial^{\nu}}{\Box}\right)e^{\frac{1}{2}f(\Box)}G_{2}(0)\delta_{bh}\left(\eta_{\mu\lambda}-\frac{\partial_{\mu}\partial_{\lambda}}{\Box}\right)e^{\frac{1}{2}f(\Box)}G_{2}(x,y)\right]-$
$\displaystyle
gf^{abc}e^{\frac{1}{2}f(\Box)}\left\\{{\bar{J}}_{2\lambda}^{bh}(x,y)e^{\frac{1}{2}f(\Box)}\left[\partial_{\mu}P_{1}^{c}(x)\right]\right.+$
$\displaystyle\left.{\bar{P}}_{1}^{b}(x)e^{\frac{1}{2}f(\Box)}\left[\partial_{\mu}J_{2\lambda}^{ch}(x,y)\right]+\partial_{\mu}\left[W_{3\lambda}^{bch}(0,y)\right]\right\\}=$
$\displaystyle
e^{\frac{1}{2}f(\Box)}\delta^{ah}\eta_{\mu\lambda}\delta^{4}(x-y).$ (31)

This equation can be simplified by noticing that the the terms linear
$f^{abc}$ goes to zero, due to anti-symmetry of $f^{abc}$, granting the
correct symmetry arising from the Krönecker symbol in the first term. The
reason for this is quite simply. One has terms like
$f^{abc}\eta_{b}\delta_{ch}$ but, for the equation to have a non-trivial
solution, one must have only the terms with $a=h$ due to the fact that we have
a factor $\delta_{ah}$ on other terms. This sets those contributions to zero.
We also assume that the contributions coming from $G_{3}$ and $G_{4}$ are
zero. Finally, we use the solution $P_{1}^{a}(x)=0$ introduced in the
computations of the 1P-function. This will give in the end

$\Box G_{2}(x,y)+\delta
m^{2}e^{\frac{1}{2}f(\Box)}G_{2}(x,y)+3Ng^{2}\left(e^{\frac{1}{2}f(\Box)}\phi(x)\right)^{2}e^{\frac{1}{2}f(\Box)}G_{2}(x,y)=e^{\frac{1}{2}f(\Box)}\delta^{4}(x-y).$
(32)

Similarly, for the ghost 2P-functions one gets

$-\Box K_{2}^{ch}(x,y)=e^{\frac{1}{2}f(\Box)}\delta^{ch}\delta^{4}(x-y),$ (33)

and

$-\Box J_{2}^{ch\nu}(x,y)=0.$ (34)

We see that, also for non-local Yang-Mills field theory the ghost decouples in
the Landau gauge and has the propagator of a free massless particle. This
means that we can also assume $J_{2}^{ch\nu}(x,y)=0$ consistently.

###  III.5 Mass gap

In this section, we aim to compute the spectrum of the theory. This is a
gauge-independent observable that is obtained by the 2P-function that,
otherwise, are gauge dependent. It seen in the local case that, indeed, the
correct spectrum is obtained by comparison with the lattice data
Frasca:2017slg . Particularly, in the non-local case, the main result is just
the mass gap.

As already seen for the scalar field Frasca:2020jbe , due to the non-local
effects being small corrections to the local case $\phi_{c}(x)$, we do a
negligible error if we put $\phi(x)\approx\phi_{c}(x)$ into the equation for
the 2P-function being Frasca:2015yva

$\phi_{c}(x)=\mu\left(\frac{2}{Ng^{2}}\right)^{\frac{1}{4}}{\rm
sn}\left(p\cdot x+\theta\right)$ (35)

being $\mu$ and $\theta$ arbitrary integration constants and provided that

$p^{2}=\mu^{2}\sqrt{\frac{Ng^{2}}{2}}.$ (36)

We can use the equation Frasca:2020jbe

$\displaystyle\left(e^{\frac{1}{2}f(\Box)}\phi_{c}\right)^{2}$
$\displaystyle=$
$\displaystyle\left[\mu\left(\frac{2}{Ng^{2}}\right)^{\frac{1}{4}}\frac{2\pi}{K(i)}\sum_{n=0}^{\infty}(-1)^{n}\frac{e^{-\left(n+\frac{1}{2}\right)\pi}}{1+e^{-(2n+1)\pi}}e^{\frac{1}{2}f\left(-(2n+1)\frac{\pi}{2K(i)}p^{2}\right)}\right.\times$
$\displaystyle\left.\sin\left((2n+1)\frac{\pi}{2K(i)}(p\cdot
x+\theta)\right)\right]^{2}$ (37)

that, for our aims, takes the form

$\displaystyle\left(e^{\frac{1}{2}f(\Box)}\phi_{c}\right)^{2}$
$\displaystyle=$
$\displaystyle\mu^{2}\left(\frac{2}{Ng^{2}}\right)^{\frac{1}{2}}\frac{4\pi^{2}}{K^{2}(i)}\times$
(38)
$\displaystyle\left[\frac{e^{-\pi}}{(1+e^{-\pi})^{2}}e^{f\left(-\frac{\pi^{2}}{4K^{2}(i)}p^{2}\right)}\sin^{2}\left(\frac{\pi}{2K(i)}(p\cdot
x+\theta)\right)+\right.$
$\displaystyle\left.\sum_{n=1}^{\infty}D_{n}(x)\right],$

being $D_{n}$ all the contributions arising from the square of the series with
$n>0$. It is an effect of the non-locality to flatten the spectrum of the
theory retaining a mass gap that is diluted in the ultraviolet, as already
seen for the scalar field. This happens also in this case as we get the
equation for the 2P-function

$\displaystyle-\Box
G_{2}(x,y)+3Ng^{2}\mu^{2}\left(\frac{2}{Ng^{2}}\right)^{\frac{1}{2}}\frac{4\pi^{2}}{K^{2}(i)}\times$
$\displaystyle\left\\{\frac{e^{-\pi}}{(1+e^{-\pi})^{2}}e^{f\left(-\frac{\pi^{2}}{4K^{2}(i)}p^{2}\right)}\left[1-\cos\left(\frac{\pi}{K(i)}(p\cdot
x+\theta)\right)\right]+\sum_{n=1}^{\infty}D_{n}(x)\right\\}e^{\frac{1}{2}f(\Box)}G_{2}(x,y)$
$\displaystyle+\delta
m^{2}e^{\frac{1}{2}f(\Box)}G_{2}(x,y)=e^{\frac{1}{2}f(\Box)}\delta^{4}(x-y).$
(39)

This equation is linear and we can apply a Fourier transform. Anyway, already
at this stage, we can note a mass term given by

$\Delta
m^{2}=\mu^{2}\left(18Ng^{2}\right)^{\frac{1}{2}}\frac{4\pi^{2}}{K^{2}(i)}\frac{e^{-\pi}}{(1+e^{-\pi})^{2}}e^{f\left(-\frac{\pi^{2}}{4K^{2}(i)}p^{2}\right)}+\delta
m^{2}.$ (40)

This must be completed by the gap equation

$\delta m^{2}=2Ng^{2}G_{2}(0)=2Ng^{2}\int\frac{d^{4}p}{(2\pi)^{4}}G_{2}(p).$
(41)

This result agrees quite well with the mass gap presented in Frasca:2020jbe
for the scalar field, aside from the factor 2 in the gap equation that is 3
for the scalar field and here arises from algebraic properties of the gauge
group.

Working as already done for the scalar field, we can write down the structure
of the 2P-function as Frasca:2020jbe

$G_{2}(k)=\frac{e^{\frac{1}{2}f(-k^{2})}}{k^{2}+\Delta
m^{2}e^{\frac{1}{2}f(-k^{2})}}\frac{1}{1-\Pi(k)}$ (42)

where the function $\Pi(k)$ and the corresponding sign can be properly
obtained by an iteration procedure already described in Frasca:2020jbe .

By comparison with Frasca:2020jbe , we see that the structure of the two-point
function in Eqn.(42) is very similar to the one obtained for the scalar field.
This does not come out as a surprise as we used a mapped solution but it is
indeed remarkable that all the techniques devised for the local field theory
apply here as well yielding absolutely non-trivial results.

##  IV Non-local QCD-like Theory: A Heuristic Discussion

###  IV.1 Non-local QCD-like Model

Adding Fermions to the theory we discussed in this paper implies that it would
be interesting to study its infrared limit. Indeed, we can use the 2P-function
we obtained in Eqn.(42) to devise a non-local Nambu-Jona-Lasinio
approximation. We can possibly write

$\mathcal{L}_{nlNJL}={\bar{\psi}}(x)e^{\frac{1}{2}f(\Box)}(i{\not{\partial}}-m)\psi(x)+g^{2}\int
d^{4}y{\bar{\psi}}(x)\psi(x)G_{2}(x,y){\bar{\psi}}(y)\psi(y)$ (43)

where we used the result given in Ghoshal:2017egr  for the non-local Dirac
equation and omitted internal degrees of freedom. Anyway, this kind of model
can imply breaking of chiral symmetry and confinement as recently shown
Frasca:2019ysi . This kind of model can also describe bounded states and all
the properties of QCD at very low-energy can be translated to the non-local
case. The advantage to have an entire function multiplying the propagator can
assure convergence of the integrals, notably for the chiral condensate. A
natural cut-off is also granted by the non-local mass scale $M$.

###  IV.2 Non-local Confinement Criterion

In a recent paper Ghoshal:2020lfd , it was shown that BRST invariance does not
change too much with respect to the local case. This is a rather interesting
result in view of the Kugo-Ojima criterion for confinement in a Yang-Mills
theory without fermions Kugo:1977zq ; Kugo:1979gm . It was shown recently that
this criterion, obtained by BRST invariance, provides the exact beta function
for the local theory that is shown to confine Chaichian:2018cyv . Therefore,
we are in a very similar situation in the non-local case and we could get a
beta function, even if approximately, describing the behaviour of the theory
on all the energy range providing a confinement proof.

We extended these ideas to the non-local case and we have shown that, in the
infrared limit, there is the coupling running to infinity without a Landau
pole. This is a different way to obtain confinement to a trivial fixed point.
Also, we see that in the integral that determines the Kugo-Ojima confinement
criterion, easily extended to the non-local case, the very existence of a non-
local mass scale yields UV contributions that, when integrated, cannot be
neglected lowering the energy scale. This is expected as such an integral
ranges on all the enrgy scale to infinity fratp .

###  IV.3 Asymptotically Diluted Mass Gap

Considering Eqn.(40) and (41) in the UV-limit, the mass gap of the non-local
Yang-Mills theory becomes diluted in the UV. So, the mass gap becomes even
more negligible as the energy increases. This property of the theory can be
helpful to understand several phenomena as dark energy understood as
quintessence.

###  IV.4 Toy Model for Dark Matter

As we have seen, the non-local Yang-Mills theory appears to have a single
stable glueball in the spectrum while higher order excitations are displaced
in the very far UV. This could be a possible candidate for dark matter.
Possible paths to pursue are comprehensively described in Boddy:2014yra ;
Boddy:2014qxa .

##  V Conclusion

We investigated non-local Yang-Mills theory in 4-D with non-Abelian SU(N)
gauge interactions and predicted the mass gap and the mass spectrum of the
states in the strong coupled regime. Alongside we compared the results with
that of the local theory and discussed implications regarding realistic QCD-
like scenarios. We summarize the main findings of our paper below:

  * •

We particularly studied the mass gap generated in non-local infinite-
derivative Yang-Mills field theory and showed, as an example, for the case of
non-local Gaussian operator, that the non-local scale M is responsible for no
extra poles in the propagator even in the non-perturbative regime (see
Eqn.(42)). This feature was already speculated while investigating the scalar
non-local field theory in Ref. Frasca:2020jbe .

  * •

As in the case of the scalar field, the generated mass gap gets diluted and
asymptotically vanishes in the UV in the limit $M\rightarrow\infty$ or becomes
very small near the non-local mass scale..

  * •

We have shown that, also for non-local Yang-Mills theory, the mass spectrum is
yielded by a single mass gap with higher excitations displaced far away in the
energy scale with respect to the local theory (see Eqn.(42)).

The approach of evaluation mass gap and N-point functions is general enough
that can be extended, in principle, to the case for gravity, as gauge theory
in the form of non-local gravity as a direction of re-normalizable, ghost-free
quantum gravity. Several possibilities involving particle physics and
cosmology open up with our obtained results as already shown in Sectns.IV.
which will pursue the avenues of research incorporating proper treatment of
quantum gravity and cosmology in future publications.

##  VI Acknowledgements

Authors acknowledge Luca Buoninfante for very useful comments and suggestions
on the manuscript.

## Appendix A: Dyson-Schwinger Equations & Bender-Milton-Savage Technique

In this appendix, we describe the Bender-Milton-Savage technique Bender:1999ek
to obtain the hierarchy of equations for the nP-functions of a quantum field
theory. The main idea is to retain the PDE form for them without ever
introducing the vertex functions.

In order to fix the ideas, we work with a single component scalar field having
the following generic partition function

$Z[j]=\int[D\phi]e^{iS(\phi)+i\int d^{4}xj(x)\phi(x)}.$ (44)

The first equation for the 1P-function is obtained by

$\left\langle\frac{\delta S}{\delta\phi(x)}\right\rangle=j(x)$ (45)

where

$\left\langle\ldots\right\rangle=\frac{\int[D\phi]\ldots e^{iS(\phi)+i\int
d^{4}xj(x)\phi(x)}}{\int[D\phi]e^{iS(\phi)+i\int d^{4}xj(x)\phi(x)}}$ (46)

We complete the procedure by setting $j=0$. Then, we derive the above equation
dependent on $j$ to obtain the equation for the 2P-function. We point out that
the nP-functions are defined by

$\langle\phi(x_{1})\phi(x_{2})\ldots\phi(x_{n})\rangle=\frac{\delta^{n}\ln(Z[j])}{\delta
j(x_{1})\delta j(x_{2})\ldots\delta j(x_{n})}.$ (47)

Therefore,

$\frac{\delta G_{k}(\ldots)}{\delta j(x)}=G_{k+1}(\ldots,x).$ (48)

So, for a $\phi^{4}$ theory one will have

$S=\int
d^{4}x\left[\frac{1}{2}(\partial\phi)^{2}-\frac{\lambda}{4}\phi^{4}\right],$
(49)

and we have to evaluate

$\partial^{2}\langle\phi\rangle+\lambda\langle\phi^{3}(x)\rangle=j(x).$ (50)

This yields

$Z[j]\partial^{2}G_{1}^{(j)}(x)+\lambda\langle\phi^{3}(x)\rangle=j(x).$ (51)

Using the definition of the 1P-function it is

$Z[j]G_{1}^{(j)}(x)=\langle\phi(x)\rangle.$ (52)

Deriving this with respect to $j(x)$ we obtain

$Z[j][G_{1}^{(j)}(x)]^{2}+Z[j]G_{2}^{(j)}(x,x)=\langle\phi^{2}(x)\rangle.$
(53)

Another derivation step gives

$Z[j][G_{1}^{(j)}(x)]^{3}+3Z[j]G_{1}^{(j)}(x)G_{2}(x,x)+Z[j]G_{3}^{(j)}(x,x,x)=\langle\phi^{3}(x)\rangle.$
(54)

This is substituted into eq.(50) giving

$\partial^{2}G_{1}^{(j)}(x)+\lambda[G_{1}^{(j)}(x)]^{3}+3\lambda
G_{2}^{(j)}(0)G_{1}^{(j)}(x)+G_{3}^{(j)}(0,0)=Z^{-1}[j]j(x)$ (55)

We realize that, by the effect of renormalization, a mass term appeared. This,
by setting $j=0$, yields the first Dyson-Schwinger equation into differential
form

$\partial^{2}G_{1}(x)+\lambda[G_{1}(x)]^{3}+3\lambda
G_{2}(0)G_{1}(x)+G_{3}(0,0)=0.$ (56)

We derive again eq.(55) with respect to $j(y)$ obtaining

$\begin{split}&\partial^{2}G_{2}^{(j)}(x,y)+3\lambda[G_{1}^{(j)}(x)]^{2}G_{2}^{(j)}(x,y)+\\\
&3\lambda G_{3}^{(j)}(x,x,y)G_{1}^{(j)}(x)+3\lambda
G_{2}^{(j)}(x,x)G_{2}^{(j)}(x,y)+G_{4}^{(j)}(x,x,x,y)=\\\
&Z^{-1}[j]\delta^{4}(x-y)+j(x)\frac{\delta}{\delta
j(y)}(Z^{-1}[j]).\end{split}$

Setting $j=0$, one gets the equation for the 2P-function as

$\partial^{2}G_{2}(x,y)+3\lambda[G_{1}(x)]^{2}G_{2}(x,y)+3\lambda
G_{3}(0,y)G_{1}(x)+3\lambda G_{2}(0)G_{2}(x,y)+G_{4}(0,0,y)=\delta^{4}(x-y).$
(57)

Such a procedure can be iterated to whatever order giving, in principle, all
the set of the Dyson-Schwinger hierarchy’s equations in PDE form.

## Appendix B: Derivation of the Dyson-Schwinger equations for 1P- and
2P-functions

In this appendix we show all the step needed to obtain the Dyson-Schwinger
equations for the 1P- and 2P-functions for the non-local Yang-Mills theory
with the Bender-Milton-Savage method. After averaging the equations of motion
we get

$\displaystyle\Box
G_{1\mu}^{(j)a}+gf^{abc}e^{-\frac{1}{2}f(\Box)}\left\langle\partial_{\nu}\left[e^{\frac{1}{2}f(\Box)}{\bar{A}}^{b}_{\mu}e^{\frac{1}{2}f(\Box)}A^{c\nu}\right]\right\rangle+$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left\langle\left[e^{\frac{1}{2}f(\Box)}A^{b\nu}e^{\frac{1}{2}f(\Box)}(\partial_{\mu}A^{c}_{\nu}-\partial_{\nu}A^{c}_{\mu})\right]\right\rangle$
$\displaystyle
g^{2}f^{abc}f^{cde}e^{-\frac{1}{2}f(\Box)}\left\langle\left[e^{\frac{1}{2}f(\Box)}A^{b\nu}e^{\frac{1}{2}f(\Box)}A^{d}_{\nu}e^{\frac{1}{2}f(\Box)}A^{e}_{\mu}\right]\right\rangle+$
$\displaystyle+gf^{abc}e^{\frac{1}{2}f(\Box)}\left\langle\bar{c}^{b}\partial_{\mu}c^{c}\right\rangle$
$\displaystyle=$ $\displaystyle e^{\frac{1}{2}f(\Box)}j^{a}_{\mu},$ (58)

and similarly for the ghost

$-\Box
P_{1}^{(\eta)a}+gf^{abc}\left\langle\left(e^{\frac{1}{2}f(\Box)}A_{\mu}^{c}\right)\partial^{\mu}c^{b}\right\rangle=e^{\frac{1}{2}f(\Box)}\eta^{a}.$
(59)

At this stage, we have introduced the 1P-functions

$\displaystyle G_{1\mu}^{(j)a}(x)$ $\displaystyle=$ $\displaystyle
Z^{-1}\langle A_{\mu}^{a}(x)\rangle$ $\displaystyle P_{1}^{(\eta)a}(x)=$
$\displaystyle=$ $\displaystyle Z^{-1}\langle c^{a}(x)\rangle.$ (60)

A similar equation holds also for ${\bar{c}}^{a}$ that gives
${\bar{P}}_{1}^{(\eta)a}(x)$. Then, we evaluate the averages appearing in
these equations as follows. Let us consider the definitions given above
written as

$\displaystyle Z[j,\eta,{\bar{\eta}}]e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)a}(x)$
$\displaystyle=$ $\displaystyle\langle
e^{\frac{1}{2}f(\Box)}A_{\mu}^{a}(x)\rangle$ $\displaystyle
Z[j,\eta,{\bar{\eta}}]P_{1}^{(\eta)a}(x)$ $\displaystyle=$
$\displaystyle\langle c^{a}(x)\rangle.$ (61)

We have introduced the apexes $(j)$ and $(\eta)$ to remember the explicit
dependence on the currents that will be set to zero to the end of computation.
Then, we derive one time with respect to $j(x)$ on the first equation to get

$Ze^{\frac{1}{2}f(\Box)}G_{2\mu\nu}^{(j)ab}(x,x)+Ze^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)a}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)b}(x)=\langle
e^{\frac{1}{2}f(\Box)}A_{\mu}^{a}(x)e^{\frac{1}{2}f(\Box)}A_{\nu}^{b}(x)\rangle.$
(62)

Therefore, by applying the space derivative $\partial^{\nu}$, we also have

$Ze^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{2\mu\nu}^{(j)ab}(x,x)+Ze^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{1\mu}^{(j)a}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)b}(x)=\langle
e^{\frac{1}{2}f(\Box)}\partial^{\nu}A_{\mu}^{a}(x)e^{\frac{1}{2}f(\Box)}A_{\nu}^{b}(x)\rangle.$
(63)

This step is important as such averages enter into the equation for the
1P-function. A further derivation of eq.(62) with respect to $j^{c\nu}$ will
yield

$\displaystyle
Ze^{\frac{1}{2}f(\Box)}G_{2\mu\nu}^{(j)ab}(x,x)e^{\frac{1}{2}f(\Box)}G_{1}^{(j)\nu
c}(x)+Ze^{\frac{1}{2}f(\Box)}G_{3\mu\nu}^{(j)abc\nu}(x,x,x)+$ $\displaystyle
Ze^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)a}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)b}(x)e^{\frac{1}{2}f(\Box)}G_{1}^{(j)\nu
c}(x)+Ze^{\frac{1}{2}f(\Box)}G_{2\mu}^{(j)ac\nu}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)b}+$
$\displaystyle
Ze^{\frac{1}{2}f(\Box)}G_{2\nu}^{(j)bc\nu}(x)e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)a}(x)=\langle
e^{\frac{1}{2}f(\Box)}A_{\mu}^{a}(x)e^{\frac{1}{2}f(\Box)}A_{\nu}^{b}(x)e^{\frac{1}{2}f(\Box)}A^{c\nu}(x)\rangle.$
(64)

In order to complete the computation, we need also to evaluate the averages
for the ghost field. From eq.(Appendix B: Derivation of the Dyson-Schwinger
equations for 1P- and 2P-functions) we write

$Z[j,\eta,\bar{\eta}]P_{1}^{(\eta)a}(x)=\langle c^{a}(x)\rangle.$ (65)

Then, we derive this equation with respect to $\partial_{\mu}$ and then with
respect to $\bar{\eta}$, the first derivative just enters into the equation of
motion of the ghost field while the latter is needed to obtain the correlation
function. One gets

$Z{\bar{P}}_{1}^{(\eta)b}(x)e^{\frac{1}{2}f(\Box)}\partial^{\mu}P_{1}^{(\eta)a}(x)+Z\partial^{\mu}K_{2}^{(\eta)ab}(x,x)=\langle{\bar{c}}^{b}\partial^{\mu}c^{a}(x)\rangle.$
(66)

In this equation, we have introduced the new 2P-function

$K_{2}^{(\eta)ab}(x,y)=\frac{1}{Z}\frac{\delta
P_{1}^{(\eta)a}(x)}{\delta\eta^{b}(y)}.$ (67)

Also, we need the following 2P-function

$J_{2\mu}^{(\eta,j)ab}(x,y)=\frac{1}{Z}\frac{\delta P_{1}^{(\eta)a}(x)}{\delta
j^{b\mu}(y)}.$ (68)

Therefore, by deriving eq.(65) with respect to $j^{b\mu}(x)$, we get

$Ze^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)b}(x)\partial^{\mu}P_{1}^{(\eta)a}(x)+Z\partial^{\mu}J_{2\mu}^{(\eta,j)ab}(x,x)=\langle
A^{b}_{\mu}(x)\partial^{\mu}c^{a}(x)\rangle.$ (69)

Finally, we can collect all these computations to get

$\displaystyle\Box
G_{1\mu}^{(j)a}+gf^{abc}e^{-\frac{1}{2}f(\Box)}\partial^{\nu}\left[e^{\frac{1}{2}f(\Box)}G_{2\mu\nu}^{(j)bc}(x,x)+e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)b}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)c}(x)\right]-$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{2\mu\nu}^{(j)bc}(x,x)+e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{1\mu}^{(j)b}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)c}(x)\right]-$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}\partial_{\mu}G_{2\nu}^{(j)bc\nu}(x,x)+e^{\frac{1}{2}f(\Box)}\partial_{\mu}G_{1\nu}^{(j)b}(x)e^{\frac{1}{2}f(\Box)}G_{1}^{(j)c\nu}(x)\right]+$
$\displaystyle
g^{2}f^{abc}f^{cde}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}G_{2\mu\nu}^{(j)bd}(x,x)e^{\frac{1}{2}f(\Box)}G_{1}^{(j)\nu
e}(x)+e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{3\mu\nu}^{(j)bde\nu}(x,x,x)+\right.$
$\displaystyle
e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)b}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)d}(x)e^{\frac{1}{2}f(\Box)}G_{1}^{(j)\nu
e}(x)+e^{\frac{1}{2}f(\Box)}G_{2\mu}^{(j)be\nu}(x,x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)d}(x)+$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}G_{2\nu}^{(j)de\nu}(x,x)e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)b}(x)\right]-$
$\displaystyle
gf^{abc}e^{\frac{1}{2}f(\Box)}\left\\{{\bar{P}}_{1}^{(\eta)b}(x)e^{\frac{1}{2}f(\Box)}\left[\partial_{\mu}P_{1}^{(\eta)c}(x)\right]+\partial_{\mu}\left[K_{2}^{(\eta)bc}(x,x)\right]\right\\}$
$\displaystyle=$ $\displaystyle e^{\frac{1}{2}f(\Box)}j^{a}_{\mu},$ (70)

It is interesting to point out that, if the exponential due to the non-
locality would be set to 1 (i.e. we are taking the local limit
$M\rightarrow\infty$), we have just recovered the same equation computed for
this case in Frasca:2015yva . The same is true for the ghost field that yield
instead

$-\Box
P_{1}^{(\eta)c}-gf^{abc}e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)a}(x)\partial^{\mu}P_{1}^{(\eta)b}(x)-gf^{abc}\partial^{\mu}J_{2\mu}^{(\eta,j)ab}(x,x)=e^{\frac{1}{2}f(\Box)}\eta^{c}.$
(71)

Following the computational step reported here, we get the first equation for
the 1P-function by setting all the currents to zero to obtain

$\displaystyle\Box
G_{1\mu}^{a}+gf^{abc}e^{-\frac{1}{2}f(\Box)}\partial^{\nu}\left[e^{\frac{1}{2}f(\Box)}G_{2\mu\nu}^{bc}(x,x)+e^{\frac{1}{2}f(\Box)}G_{1\mu}^{b}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{c}(x)\right]-$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{2\mu\nu}^{bc}(x,x)+e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{1\mu}^{b}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{c}(x)\right]-$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}\partial_{\mu}G_{2\nu}^{bc\nu}(x,x)+e^{\frac{1}{2}f(\Box)}\partial_{\mu}G_{1\nu}^{b}(x)e^{\frac{1}{2}f(\Box)}G_{1}^{c\nu}(x)\right]+$
$\displaystyle
g^{2}f^{abc}f^{cde}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}G_{2\mu\nu}^{bd}(x,x)e^{\frac{1}{2}f(\Box)}G_{1}^{\nu
e}(x)+e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{3\mu\nu}^{bde\nu}(x,x,x)+\right.$
$\displaystyle
e^{\frac{1}{2}f(\Box)}G_{1\mu}^{b}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{d}(x)e^{\frac{1}{2}f(\Box)}G_{1}^{(j)\nu
e}(x)+e^{\frac{1}{2}f(\Box)}G_{2\mu}^{be\nu}(x,x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{d}(x)+$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}G_{2\nu}^{de\nu}(x,x)e^{\frac{1}{2}f(\Box)}G_{1\mu}^{b}(x)\right]-$
$\displaystyle
gf^{abc}e^{\frac{1}{2}f(\Box)}\left\\{{\bar{P}}_{1}^{b}(x)e^{\frac{1}{2}f(\Box)}\left[\partial_{\mu}P_{1}^{c}(x)\right]+\partial_{\mu}\left[K_{2}^{bc}(x,x)\right]\right\\}=0,$
(72)

We see that, in the local limit, we have recovered the Dyson-Schwinger
equation for the 1P-function of the Yang-Mills theory Frasca:2015yva .
Similarly, for the ghost field we get

$-\Box
P_{1}^{c}-gf^{abc}e^{\frac{1}{2}f(\Box)}G_{1\mu}^{a}(x)\partial^{\mu}P_{1}^{b}(x)-gf^{abc}\partial^{\mu}J_{2\mu}^{ab}(x,x)=0.$
(73)

and similarly for ${\bar{P}}_{1}^{c}(x)$. Equations for 1P-functions are
fundamental to apply our background field method. Then, given their solutions,
we are able to evaluate the 2P-functions and obtain the observables of the
theory.

Moving from eq.(Appendix B: Derivation of the Dyson-Schwinger equations for
1P- and 2P-functions), we can get the equations for the 2P-functions in the
following way. We derive it with respect to $j^{\lambda h}(y)$ obtaining

$\displaystyle\Box
G_{2\mu\lambda}^{(j)ah}(x,y)+gf^{abc}e^{-\frac{1}{2}f(\Box)}\partial^{\nu}\left[e^{\frac{1}{2}f(\Box)}G_{3\mu\nu\lambda}^{(j)bch}(x,x,y)+e^{\frac{1}{2}f(\Box)}G_{2\mu\lambda}^{(j)bh}(x,y)\times\right.$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)c}(x)++e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)b}(x)e^{\frac{1}{2}f(\Box)}G_{2\nu\lambda}^{(j)ch}(x)\right]-$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{2\mu\nu\lambda}^{(j)bch}(x,x,y)+e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{2\mu\lambda}^{(j)bh}(x,y)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)c}(x)+\right.$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{1\mu}^{(j)b}(x)e^{\frac{1}{2}f(\Box)}G_{2\nu\lambda}^{(j)ch}(x,y)\right]-$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}\partial_{\mu}G_{3\nu\lambda}^{(j)bch\nu}(x,x,y)+e^{\frac{1}{2}f(\Box)}\partial_{\mu}G_{2\nu\lambda}^{(j)bh}(x,y)e^{\frac{1}{2}f(\Box)}G_{1}^{(j)c\nu}(x)+\right.$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}\partial_{\mu}G_{1\nu}^{(j)b}(x)e^{\frac{1}{2}f(\Box)}G_{2\lambda}^{(j)ch\nu}(x,y)\right]+$
$\displaystyle
g^{2}f^{abc}f^{cde}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}G_{3\mu\nu\lambda}^{(j)bdh}(x,x,y)e^{\frac{1}{2}f(\Box)}G_{1}^{(j)\nu
e}(x)+\right.$ $\displaystyle
e^{\frac{1}{2}f(\Box)}G_{2\mu\nu}^{(j)bd}(x,x)e^{\frac{1}{2}f(\Box)}G_{2\lambda}^{(j)\nu
eh}(x,y)+e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{4\mu\nu\lambda}^{(j)bdeh\nu}(x,x,x,y)+$
$\displaystyle
e^{\frac{1}{2}f(\Box)}G_{2\mu\lambda}^{(j)bh}(x,y)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)d}(x)e^{\frac{1}{2}f(\Box)}G_{1}^{(j)\nu
e}(x)+$ $\displaystyle
e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)b}(x)e^{\frac{1}{2}f(\Box)}G_{2\nu\lambda}^{(j)dh}(x,y)e^{\frac{1}{2}f(\Box)}G_{1}^{(j)\nu
e}(x)+$ $\displaystyle
e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)b}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)d}(x)e^{\frac{1}{2}f(\Box)}G_{2\lambda}^{(j)\nu
eh}(x,y)+$ $\displaystyle
e^{\frac{1}{2}f(\Box)}G_{3\mu\lambda}^{(j)beh\nu}(x,x,y)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{(j)d}(x)+$
$\displaystyle
e^{\frac{1}{2}f(\Box)}G_{2\mu}^{(j)be\nu}(x,x)e^{\frac{1}{2}f(\Box)}G_{2\nu\lambda}^{(j)dh}(x,y)+$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}G_{3\nu\lambda}^{(j)deh\nu}(x,x,y)e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)b}(x)+e^{\frac{1}{2}f(\Box)}G_{2\nu}^{(j)de\nu}(x,x)e^{\frac{1}{2}f(\Box)}G_{2\mu\lambda}^{(j)bh}(x,y)\right]-$
$\displaystyle
gf^{abc}e^{\frac{1}{2}f(\Box)}\left\\{{\bar{J}}_{2\lambda}^{(\eta,j)bh}(x,y)e^{\frac{1}{2}f(\Box)}\left[\partial_{\mu}P_{1}^{(\eta)c}(x)\right]\right.+$
$\displaystyle\left.{\bar{P}}_{1}^{(\eta)b}(x)e^{\frac{1}{2}f(\Box)}\left[\partial_{\mu}J_{2\lambda}^{(\eta)ch}(x,y)\right]+\partial_{\mu}\left[W_{3\lambda}^{(\eta,j)bch}(x,x,y)\right]\right\\}=$
$\displaystyle
e^{\frac{1}{2}f(\Box)}\delta^{ah}\eta_{\mu\lambda}\delta^{4}(x-y),$ (74)

where we have introduced the 3P-function

$W_{3\lambda}^{(\eta,j)abc}(x,y,z)=Z^{-1}\frac{\delta
K_{2}^{(\eta)ab}(x,y)}{\delta j^{\lambda c}(z)}.$ (75)

Turning to the 1P-function for the ghost, We derive it with respect to
$\eta^{h}(y)$ to obtain

$\displaystyle-\Box
K_{2}^{(\eta)ch}(x,y)-ige^{\frac{1}{2}f^{abc}f(\Box)}L_{2\mu}^{(\eta,j)ah}(x,y)\partial^{\mu}P_{1}^{(\eta)b}(x)$
$\displaystyle-
igf^{abc}e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)a}(x)\partial^{\mu}K_{2}^{(\eta)bh}(x,y)-igf^{abc}\partial^{\mu}W_{3\mu}^{(\eta,j)abh}(x,x,y)$
$\displaystyle=e^{\frac{1}{2}f(\Box)}\delta^{ch}\delta^{4}(x-y).$ (76)

Here we have introduced the 2P-function

$L_{2\mu}^{(\eta,j)ab}(x,y)=\frac{\delta G_{1}^{(j)a}(x)}{\delta\eta^{b}(y)}.$
(77)

Then, we derive with respect to $j^{h\nu}(y)$. This yields the equation for
$J_{2}$ as

$\displaystyle-\Box
J_{2}^{(\eta)ch\nu}(x,y)-igf^{abc}e^{\frac{1}{2}f(\Box)}G_{2\mu\nu}^{(j)ah}(x,y)\partial^{\mu}P_{1}^{(\eta)b}(x)$
$\displaystyle-
igf^{abc}e^{\frac{1}{2}f(\Box)}G_{1\mu}^{(j)a}(x)\partial^{\mu}J_{2}^{(\eta,j)bh\nu}(x,y)$
$\displaystyle-igf^{abc}\partial^{\mu}J_{3\mu}^{(\eta,j)abh}(x,x,y)=0,$ (78)

where we have introduced the 3P-function

$J_{3\mu}^{(\eta,j)abc}(x,y,z)=\frac{\delta J_{2\mu}^{(\eta,j)ab}(x,y)}{\delta
j^{c\mu}(z)}.$ (79)

Again, setting all the currents to 0, gives the equations for the 2P-functions
as

$\displaystyle\Box
G_{2\mu\lambda}^{ah}(x,y)+gf^{abc}e^{-\frac{1}{2}f(\Box)}\partial^{\nu}\left[e^{\frac{1}{2}f(\Box)}G_{3\mu\nu\lambda}^{bch}(x,x,y)+e^{\frac{1}{2}f(\Box)}G_{2\mu\lambda}^{bh}(x,y)\times\right.$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}G_{1\nu}^{c}(x)++e^{\frac{1}{2}f(\Box)}G_{1\mu}^{b}(x)e^{\frac{1}{2}f(\Box)}G_{2\nu\lambda}^{ch}(x)\right]-$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{2\mu\nu\lambda}^{bch}(x,x,y)+e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{2\mu\lambda}^{bh}(x,y)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{c}(x)+\right.$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{1\mu}^{b}(x)e^{\frac{1}{2}f(\Box)}G_{2\nu\lambda}^{ch}(x,y)\right]-$
$\displaystyle
gf^{abc}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}\partial_{\mu}G_{3\nu\lambda}^{bch\nu}(x,x,y)+e^{\frac{1}{2}f(\Box)}\partial_{\mu}G_{2\nu\lambda}^{bh}(x,y)e^{\frac{1}{2}f(\Box)}G_{1}^{c\nu}(x)+\right.$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}\partial_{\mu}G_{1\nu}^{b}(x)e^{\frac{1}{2}f(\Box)}G_{2\lambda}^{ch\nu}(x,y)\right]+$
$\displaystyle
g^{2}f^{abc}f^{cde}e^{-\frac{1}{2}f(\Box)}\left[e^{\frac{1}{2}f(\Box)}G_{3\mu\nu\lambda}^{bdh}(x,x,y)e^{\frac{1}{2}f(\Box)}G_{1}^{\nu
e}(x)+\right.$ $\displaystyle
e^{\frac{1}{2}f(\Box)}G_{2\mu\nu}^{bd}(x,x)e^{\frac{1}{2}f(\Box)}G_{2\lambda}^{\nu
eh}(x,y)+e^{\frac{1}{2}f(\Box)}\partial^{\nu}G_{4\mu\nu\lambda}^{bdeh\nu}(x,x,x,y)+$
$\displaystyle
e^{\frac{1}{2}f(\Box)}G_{2\mu\lambda}^{bh}(x,y)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{d}(x)e^{\frac{1}{2}f(\Box)}G_{1}^{\nu
e}(x)+$ $\displaystyle
e^{\frac{1}{2}f(\Box)}G_{1\mu}^{b}(x)e^{\frac{1}{2}f(\Box)}G_{2\nu\lambda}^{dh}(x,y)e^{\frac{1}{2}f(\Box)}G_{1}^{\nu
e}(x)+$ $\displaystyle
e^{\frac{1}{2}f(\Box)}G_{1\mu}^{b}(x)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{d}(x)e^{\frac{1}{2}f(\Box)}G_{2\lambda}^{\nu
eh}(x,y)+$ $\displaystyle
e^{\frac{1}{2}f(\Box)}G_{3\mu\lambda}^{beh\nu}(x,x,y)e^{\frac{1}{2}f(\Box)}G_{1\nu}^{d}(x)+$
$\displaystyle
e^{\frac{1}{2}f(\Box)}G_{2\mu}^{be\nu}(x,x)e^{\frac{1}{2}f(\Box)}G_{2\nu\lambda}^{dh}(x,y)+$
$\displaystyle\left.e^{\frac{1}{2}f(\Box)}G_{3\nu\lambda}^{deh\nu}(x,x,y)e^{\frac{1}{2}f(\Box)}G_{1\mu}^{b}(x)+e^{\frac{1}{2}f(\Box)}G_{2\nu}^{de\nu}(x,x)e^{\frac{1}{2}f(\Box)}G_{2\mu\lambda}^{bh}(x,y)\right]-$
$\displaystyle
gf^{abc}e^{\frac{1}{2}f(\Box)}\left\\{{\bar{J}}_{2\lambda}^{bh}(x,y)e^{\frac{1}{2}f(\Box)}\left[\partial_{\mu}P_{1}^{c}(x)\right]\right.+$
$\displaystyle\left.{\bar{P}}_{1}^{b}(x)e^{\frac{1}{2}f(\Box)}\left[\partial_{\mu}J_{2\lambda}^{ch}(x,y)\right]+\partial_{\mu}\left[W_{3\lambda}^{bch}(x,x,y)\right]\right\\}=$
$\displaystyle
e^{\frac{1}{2}f(\Box)}\delta^{ah}\eta_{\mu\lambda}\delta^{4}(x-y).$ (80)

For the ghost, one has

$\displaystyle-\Box
K_{2}^{ch}(x,y)-ige^{\frac{1}{2}f^{abc}f(\Box)}L_{2\mu}^{ah}(x,y)\partial^{\mu}P_{1}^{(\eta)b}(x)$
$\displaystyle-
igf^{abc}e^{\frac{1}{2}f(\Box)}G_{1\mu}^{a}(x)\partial^{\mu}K_{2}^{bh}(x,y)-igf^{abc}\partial^{\mu}W_{3\mu}^{abh}(x,x,y)$
$\displaystyle=e^{\frac{1}{2}f(\Box)}\delta^{ch}\delta^{4}(x-y),$ (81)

and

$\displaystyle-\Box
J_{2}^{ch\nu}(x,y)-igf^{abc}e^{\frac{1}{2}f(\Box)}G_{2\mu\nu}^{ah}(x,y)\partial^{\mu}P_{1}^{b}(x)$
$\displaystyle-
igf^{abc}e^{\frac{1}{2}f(\Box)}G_{1\mu}^{a}(x)\partial^{\mu}J_{2}^{bh\nu}(x,y)$
$\displaystyle-igf^{abc}\partial^{\mu}J_{3\mu}^{abh}(x,x,y)=0.$ (82)

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