aainstitutetext: Centro de Astropartículas y Física de Altas Energías,
Departamento de Física TeóricaUniversidad de Zaragoza, C/ Pedro Cerbuna 12,
E-50009 Zaragoza, Spainbbinstitutetext: Departamento de Física – ICE,
Universidade Federal de Juiz de Fora, Rua José Lourenço Kelmer, Campus
Universitário, Juiz de Fora, 36036-900, MG, Brazil

# Asymptotic freedom and higher derivative gauge theories

M. Asorey a F. Falceto b L. Rachwał<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The ultraviolet completion of gauge theories by higher derivative terms can
dramatically change their behavior at high energies. The requirement of
asymptotic freedom imposes very stringent constraints that are only satisfied
by a small family of higher derivative theories. If the number of derivatives
is large enough ($n>4$) the theory is strongly interacting both at extreme
infrared and ultraviolet regimes whereas it remains asymptotically free for a
low number of extra derivatives ($n\leqslant 4$). In all cases the theory
improves its ultraviolet behavior leading in some cases to ultraviolet finite
theories with vanishing $\beta$-function. The usual consistency problems
associated to the presence of extra ghosts in higher derivative theories may
not harm asymptotically free theories because in that case the effective
masses of such ghosts are running to infinity in the ultraviolet limit.

###### Keywords:

gauge theories, renormalization group, higher derivatives regularization,
asymptotic freedom

##  1 Introduction

Field theories with higher derivatives were first considered as covariant
ultraviolet regularizations of gauge theories FS -LZ . However, in the last
years there is a renewed interest in these theories mainly due to the
rediscovery that they provide a renormalizable field theoretical framework for
the quantization of gravity Stelle ; Stelle2 . Field theories with higher
derivatives can be also considered as very efficient effective theories in
strongly correlated regimes of standard gauge field theories and extended
gravity theories in inflationary scenarios Starobinsky ; FT . From a
fundamental viewpoint higher derivatives theories were disregarded in the past
due to the fact that they face pathological behaviors concerning causality and
unitarity principles alsh ; arsh . However, such a behavior is not a problem
in some higher derivative theories such as Lee-Wick theories leewick –AP2  and
theories with ghost condensation phenomena. We shall show that the small
family of consistent gauge theories may be extended to asymptotically free
gauge theories.

These theories might provide ultraviolet completions of the Standard Model.
Higher derivative theories of spin-0 and spin-1/2 fields with local
interactions are finite. However, in general relativity and non-abelian gauge
theories one cannot get rid of one-loop ultraviolet (UV) divergences that
require renormalization FS -LZ . We shall show that there are some very
special higher derivative theories that do not require renormalization because
all UV divergences cancel out. These finite theories do not have UV
singularities even classically, which is particularly interesting in gravity
theories because they lead to black holes without hidden singularities and
evolutions of the Universe without Big Bang singularity BGKM ; BCKM . However,
this does not exclude that some of those theories can present infrared (IR)
divergences.

In this paper, we analyze from this perspective the ultraviolet behavior of
higher derivative gauge theories and, in particular, the asymptotic freedom
behavior at extremely high energies. We also analyze the emergence of IR
divergences and how the renormalization group (RG) can smoothly interpolate
between these two asymptotic regimes and solve the consistency problems
associated to the appearance of extra ghost fields.

##  2 Higher derivative gauge theories

In the Euclidean formalism a pure gauge theory with higher derivatives can be
defined by the following action

$S=\frac{1}{4g^{2}}\int
d^{4}x\,F_{\mu\nu}^{a}F^{\mu\nu}{}^{a}+\frac{1}{4g^{2}{\Lambda^{2n}}}\int
d^{4}x\,F_{\mu\nu}^{a}\,{\Delta^{n}}\,F^{\mu\nu}{}^{a}\,,$ (1)

where the operator $\Delta$ defined by

$\Delta_{\mu\,a}^{\nu\,b}=-D^{2}\delta^{\nu}_{\mu}\delta^{b}_{a}+2f^{b}{}_{ca}\,F_{\mu}{}^{\nu
c}$ (2)

is the Hodge-covariant Laplacian operator
$\Delta=d_{A}^{\ast}d_{A}+d_{A}d_{A}^{\ast}$ acting on the Lie algebra
$\mathfrak{g}{\rm-valued}$ 2-forms $F$ of the gauge field strength
$F_{\mu\nu}$ and $D_{\mu}$ is the gauge-covariant derivative. The gauge group
is denoted by $G$ here and its corresponding algebra by $\mathfrak{g}$. The
gauge-covariant exterior differential on forms we denote by $d_{A}$ and
$d_{A}^{\ast}$ is its Hodge-dual.

These new theories are not conformally invariant, in contrast with standard
Yang-Mills theory in $d=4$ spacetime dimensions. But they preserve all
solutions of the Yang-Mills vacuum equation of motion $d_{A}^{\ast}F=0$,
including standard instantons ($F=\pm\ast F$) for Euclidean theory, as exact
solutions of their equations of motion.

Although one-loop corrections around such solutions are different than that of
ordinary Yang-Mills theories and might have a softer UV behavior, they share
the same pathological IR behavior, which usually spoils the instanton physical
implications.

For $n\geqslant 2$ the new theories (1) are superrenormalizable. They only
have one-loop UV divergences FS ; LZ . Higher order contributions become
finite once the one-loop divergences are renormalized af1 ; af2 .

The calculation of UV divergences can be performed in $\alpha$-gauge by adding
the following gauge-fixing functional to the action (1)

$S_{\alpha}=\frac{\alpha}{2g^{2}\Lambda^{2n}}\int
d^{4}x\,\partial^{\mu}A^{a}_{\mu}(-\partial^{\sigma}\partial_{\sigma})^{n}\partial^{\nu}A^{a}_{\nu}$
(3)

and by using dimensional regularization with $\epsilon=4-d$, where $d$ is the
regularized dimension of Euclidean space. Returning to Minkowski spacetime
formalism, the result for the UV-divergent contribution of the two-point
gluonic function coming from one-loop vacuum polarization diagrams is (see
Appendix)

$\Gamma_{\mu\nu}^{ab}(p)=-c_{n}{{\frac{C_{2}(G)}{16\pi^{2}\epsilon}}}i\delta^{ab}\left(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu}\right)$
(4)

with

$c_{0}={\alpha}{}-\frac{13}{3},\ \ \ c_{1}=-\frac{43}{3}\ \ \ \hbox{and}\ \ \
c_{n}=5n^{2}-23n+\frac{29}{3},\,\ \ \hbox{for}\ \ \,n\geqslant 2,\ $ (5)

which agrees for $n\geqslant 2$ with the results of Asorey-Falceto af1 ; af2
and for $n=1$ with those of Babelon-Namazie babelon . In the radiative
correction to the two-point function (4), $p$ is the momentum of the incoming
gluon, $\eta_{\mu\nu}$ denotes the Minkowski metric of flat spacetime and
$C_{2}(G)$ is the quadratic Casimir operator of the gauge group $G$. Notice
the independence of $c_{n}$ on the gauge fixing parameter $\alpha$ for
$n\geqslant 1$, because in that case all $\alpha$-dependent terms are finite
at any loop order (see Appendix for a detailed explanation). It can be shown
that in such a case the divergent contributions to 3-point and 4-point gluonic
functions preserve gauge invariance, and thus, all UV one-loop divergences can
be removed by a simple counterterm

$S_{\rm
count}=c_{n}\frac{C_{2}(G)}{128\pi^{2}}\left(\frac{2}{\epsilon}+\log{\displaystyle\frac{\Lambda^{2}_{\rm
QCD}}{\Lambda^{2}}}\right)\,F_{\mu\nu}^{a}F^{\mu\nu}{}^{a},$ (6)

where the $\Lambda_{\rm QCD}$ scale has been introduced by a renormalization
prescription, which adds a finite counterterm to the minimal renormalization
scheme to recover in the IR the renormalized two-point function of standard
QCD towards our higher derivative theory (1) tends to.

In the case $n=1$, there are still some two-loop divergent contributions which
require additional renormalization. In the case $n=0$, besides the
corresponding one-loop counterterm a most careful analysis is required,
because the 3-point and 4-point contributions cannot be simply absorbed by a
renormalization of the YM coupling constant $g$, but also need a field
renormalization

$A_{r,\mu}=\left(1+c_{0}\frac{C_{2}(G)}{32\pi^{2}\epsilon}\right)A_{\mu},$ (7)

which absorbs all $\alpha$-dependence of $c_{0}$. On the other hand the
$\alpha$-independence of one-loop radiative corrections to the coupling
constant $g$ follows from the BRST invariance of the theory. Moreover, as it
is well known in the last case the renormalization process has to be extended
to any loop order.

The effect of higher derivative terms leads to a modified $\beta$-function. In
the case $n\geqslant 1$, the result is

$\beta_{n}=c_{n}{{\frac{g^{3}C_{2}(G)}{32\pi^{2}}}},$ (8)

whereas in the case $n=0$, once the wave-functions of gluons and ghosts are
renormalized, one gets the standard Yang-Mills $\beta$-function of the
coupling constant af2 .

Summing up all orders of perturbation theory one gets the renormalization
group flow of the bare coupling constant

$g_{{}_{\mathrm{bare}}}^{2}(\mu)=\frac{g^{2}}{1-\frac{g^{2}C_{2}(G)}{(4\pi)^{2}}c_{n}\log\mu/\Lambda{}},$
(9)

which is running to $0$ as the renormalization scale $\mu$ goes to infinity.
Since the higher derivative terms of the action (1) do not get any divergent
radiative correction, the coefficient
$g^{2}\Lambda^{2n}=g_{{}_{\mathrm{bare}}}^{2}\Lambda_{{}_{\mathrm{bare}}}^{2n}$
remains unrenormalized. Thus, the ultraviolet behavior of the bare mass
$\Lambda_{{}_{\mathrm{bare}}}$ parameter is just the oposite of
$g_{{}_{\mathrm{bare}}}$, i.e. $\beta_{\Lambda}=-\frac{\Lambda}{ng}\beta_{n}$,
which implies that $\Lambda_{{}_{\mathrm{bare}}}$ is running to infinity as
the renormalization scale $\mu$ goes to infinity.

This observation allows to extend the family of consistent higher derivative
theories beyond the expected tree-level bounds, because the masses of all
pathological ghosts breaking causality and unitarity are proportional to
$\Lambda_{{}_{\rm{bare}}}$, and for asymptotically free theories these ghosts
become infinitely heavy and decouple from the physical spectrum in the UV
regime.

##  3 Asymptotic Freedom

The behavior of the renormalized two-point functions is more involved. The
one-loop result at leading order in $\Lambda$ is

$\Gamma_{\mu\nu}^{ab}(p)=-\frac{C_{2}(G)}{32\pi^{2}}i\delta^{ab}\left(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu}\right)\Pi(p^{2}),$
(10)

with

$\Pi(p^{2})=\left(b_{n}\log\frac{p^{2}+\Lambda^{2}}{\Lambda^{2}}+c_{0}\log\frac{p^{2}}{{\Lambda^{2}_{\rm
QCD}}}\right),$ (11)

where we identified an arbitrary renormalization scale $\mu$ with the higher
derivative scale $\Lambda$ and

$b_{0}=0,\quad b_{1}=-10-\alpha,\quad\hbox{and}\quad
b_{n}=14-\alpha-23n+5n^{2}\ \quad\hbox{for}\ n\geqslant 2$ (12)

or generally $b_{n}=c_{n}-c_{0}$ for any $n$.

There are two different asymptotic regimes, the $p\gg\Lambda$ UV regime, where

$\Gamma_{\mu\nu}^{ab}(p)=-c_{n}\frac{C_{2}(G)}{32\pi^{2}}i\delta^{ab}\log\frac{p^{2}}{\Lambda^{2}}\left(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu}\right)$
(13)

and the intermediate IR regime $\Lambda_{\rm QCD}<p\ll\Lambda$, where we
recover the renormalized two-point function of standard QCD

$\Gamma_{\mu\nu}^{ab}(p)=-c_{0}\frac{C_{2}(G)}{32\pi^{2}}i\delta^{ab}\log\frac{p^{2}}{\Lambda^{2}_{\rm
QCD}}\left(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu}\right).$ (14)

The renormalization group (RG) flow of the higher derivative Yang-Mills theory
interpolates for $n\geqslant 1$ between these two asymptotic regimes with two
different beta functions

$\beta_{\rm UV}=c_{n}{{\frac{g^{3}C_{2}(G)}{32\pi^{2}}}}\quad{\rm
and}\quad\beta_{\rm IR}=-\frac{22}{3}{\frac{g^{3}C_{2}(G)}{32\pi^{2}}}.$ (15)

In the infrared regime one recovers the standard values of the
$\beta$-function for asymptotically free Yang-Mills scenario. However, in the
ultraviolet regime there is a variety of scenarios involving either asymptotic
freedom or asymptotic slavery, except in the case $n=0$ where the
$\beta$-function remains frozen in the QCD asymptotically free regime for all
$p>{\Lambda_{\rm QCD}}$ (see Figure 1).

Figure 1:  Momentum dependence of one-loop radiative corrections to the
2-point function form factor $\Pi(p^{2})$ for different higher derivative
theories. The cases $n=1$ and $n=3$ give rise to the same form factors. The
results for large momenta are independent of the gauge fixing parameter
$\alpha$, whereas in the IR we used a Lorentz gauge with $\alpha=0$. In the
infrared regime $\Lambda_{\rm QCD}<p\ll\Lambda$, the behavior is independent
on the number of extra derivatives and agrees with that of standard gauge
theories. In the UV regime $p\gg\Lambda$, the behavior depends on the number
of extra derivatives, but in all five cases the theory is asymptotically free.
We assumed that the higher derivative scale $\Lambda$ is of the order of $100$
TeV, i.e. $\Lambda\approx 10^{6}\Lambda_{\rm QCD}\,.$

Assuming that $n$ has to be an integer in order to preserve locality, the
number of UV asymptotically free theories is very limited. It reduces to
theories with negative coefficients of the $\beta$-function, i.e., only to
five cases $n=0,1,2,3,4$:

$\widetilde{c}_{0}=-\frac{22}{3}\,,\quad c_{1}=-\frac{43}{3}\,,\quad
c_{2}=-\frac{49}{3}\,,\quad c_{3}=-\frac{43}{3}\,,\quad c_{4}=-\frac{7}{3}\,.$
(16)

For all other integer values of $n$, $c_{n}>0$. This shows how the limitation
to UV asymptotically free theories imposes severe constraints on higher
derivative gauge theories.

In order to enlarge the family of theories with UV asymptotically free regimes
one can introduce an extra dimensionless coupling $\lambda$ in the theory, and
replace the Hodge Laplacian operator (2) in (1) by the operator
${}^{\lambda}\Delta$ defined by

${}^{\lambda}\Delta{}_{\mu\,a}^{\nu\,b}=-\delta^{b}_{a}\delta^{\nu}_{\mu}D^{2}+2\lambda
f^{b}{}_{ca}\,F_{\mu}{}^{\nu c}\,.$ (17)

For $\lambda\neq 1$ instantons ($d_{A}^{\ast}F=0$) are not anymore solutions
of the Euclidean equations of motion. The Euclidean extension of operator
${}^{\lambda}\Delta$ is positive for $\lambda=1$ and $\lambda=0$, but not for
$n$ odd and larger than $1$. However, the action of the theory remains
positive for any even $n$, which is a physically relevant requirement. In this
case, the structure of one-loop divergences gets modified because now

$\displaystyle
c_{n}=\begin{cases}\displaystyle-9\lambda^{2}-18\lambda{}+\frac{38}{3},&\hbox{for}\
n=1\\\\[12.91663pt]
\displaystyle\left(5n^{2}-18n+16\right)\lambda^{2}{{-}}\left(4n^{2}+10n+4\right)\lambda+4n^{2}+5n-\frac{7}{3},&\hbox{for}\
n\geqslant 2.\hfill\end{cases}$

Although these coefficients reduce to (5) in a continuous way when $\lambda\to
1$.

The coupling $\lambda$ does not get any divergent radiative contribution and,
thus, is kept fixed under RG flow as an extra parameter of the theory. On the
other hand, for higher values of $\lambda\gg 1$ the constraint of having a
window with asymptotic freedom is even more stringent. In fact, in the limit
$\lambda\to+\infty$, only the theory with $n=2$ is marginally asymptotically
free. Actually, for $n=2$ and $\lambda\to+\infty$ the theory becomes UV-
finite. Notice that the minimal choice of scalar covariant Laplacian
${}^{0}\Delta=-D^{2}$ never gives rise to asymptotically free theories. In
fact, the above results imply that the Hodge-covariant Laplacian $\Delta$ is
the optimal choice to get asymptotic freedom in a larger number of higher
derivative gauge theories.

Figure 2:  Domain of asymptotically free gauge theories (shadow blue area).
Horizontal lines correspond to local theories and $\lambda$ is the extra
parameter of the generalized Laplacian operator (17). Red points mark theories
with Hodge Laplacians ($\lambda=1$) and blue points UV-finite theories. In the
limit $\lambda\to\infty$ the upper and lower curves converge at $n=2$.

It is also interesting to remark that the theories are UV-finite for special
values of the parameter $\lambda$:

$n=1$ | $\lambda_{1}=-2.55$ | $\lambda_{2}=0.55$
---|---|---
$n=2$ | $\lambda_{\phantom{1}}=\phantom{-}0.59$ |
$n=3$ | $\lambda_{1}=\phantom{-}0.75$ | $\lambda_{2}=9.25$
$n=4$ | $\lambda_{1}=\phantom{-}0.96$ | $\lambda_{2}=3.54$

without any need of supersymmetry. Notice that the introduction of operator
${}^{\lambda}\Delta$ recalls the addition of terms cubic or quartic in field
strengths to the action of a higher derivative theory, which is crucial for
achieving UV-finiteness MR ; MR2 ; fingauge .

For $G=SU(N)$ the interaction with fundamental quarks generates a lower
(absolute) value of the $\beta$-function

$\beta_{n}=\left(c_{n}N+{\textstyle\frac{4}{3}N_{q}}\right){\frac{g^{3}}{32\pi^{2}}}\,$
(18)

for $n\geqslant 1$, and thus, slightly stronger constraints that shrink the
window of UV asymptotically free regime. In the case $\lambda=1$, the window
is preserved for small number of quarks $N_{q}<\frac{43}{4}N$. For a larger
number of quarks the window shrinks till its disappearance in the large
$N_{q}$ limit.

##  4 Discussion

In summary, gauge theories with higher derivatives can be asymptotically free
in the UV regime, if the total number of higher derivatives is not bigger than
8. Otherwise, the theories become UV-slave and exhibit a strong interacting
regime both in the UV and in the deep IR. There is an intermediate regime
$\Lambda^{2}_{\rm QCD}<p^{2}\ll\Lambda^{2}$, where the theory is always
asymptotically free with the same scaling properties as in the standard QCD at
high energies.

This means that there are UV-completions of gauge theories, where the behavior
of the theory for energy scales larger than $\Lambda$ can dramatically change
from asymptotic freedom to asymptotic slavery.

One remarkable feature of higher derivative theories is the possibility of
having an UV scale-invariant fixed point with $\beta=0$, where the theory
becomes finite, without any UV divergence fingauge . The behavior of these
theories in the presence of a $\theta$-term deserves further analysis. In
particular, it will be interesting to find the appearance of non-standard
effective potentials, which might have implications for axion physics
phenomenology.

An interesting remaining open problem is the analysis of the dynamics of the
hidden ghost sector and its implications for unitarity, causality and
stability of these theories. Ghost fields appear in conjugate pairs of poles
of the propagator as in the case of Lee-Wick theories leewick –AP2 . However,
in the present case of asymptotically free theories the running of the bare
mass $\Lambda_{{}_{\mathrm{bare}}}$ towards $+\infty$ in the UV regime can
make the pathological effects of ghosts harmless. It is remarkable that this
property only holds for asymptotically free theories, which enhances the
relevance of such a UV behavior for the consistency of higher derivative
theories.

These results are compatible with the behavior of lattice gauge theories. The
Wilson formulation of lattice gauge theories satisfies in the Euclidean
formalism the reflection positivity property which guarantees unitarity and
stability of the corresponding quantum theory. The first two leading terms in
the continuum limit of Wilson’s action of lattice gauge theories do coincide
with those of (1) for $n=1$. On the other hand, non-abelian lattice gauge
theories turn out to be also asymptotically free. These two relevant
properties can provide an ultimate argument for the consistency of
asymptotically free higher derivative gauge theories.

In spite of this fact some physical effects of the pairs of complex poles can
still be traced down in the behavior of the trace anomaly on curved
backgrounds. Under certain consistency conditions $a$-theorem establishes that
the coefficient of the Gauss-Bonnet term of the anomaly $a$ must evolve in a
monotonically decreasing way under the renormalization group flow atheorem .
However, in the case of the UV asymptotically free theories analyzed in this
paper $a$ is positive in standard infrared regime whereas in the UV regime it
can reach negative values PS ; GH ; ARSh . This breaking of $a$-theorem raises
some questions that require a deeper analysis.

The above results open a new interesting perspective for the analysis of
higher derivative theories of quantum gravity, which deserves further study.

## Acknowledgements

The work of M.A. and F.F. is partially supported by Spanish MINECO/FEDER grant
PGC2018-095328-B-I00 and DGA-FSE grant 2020-E21-17R, and COST action programs
MP1405-37241 and QGMM-CA18108. The work of L.R. was partially supported by a
Short Term Scientific Mission (STSM), within the COST action MP1405-37241 for
the group Effective Theories of Quantum Gravity and Quantum Structure of
Spacetime WG3, in QSpace program. L.R. would like to thank Department of
Theoretical Physics, of University of Zaragoza for the kind hospitality during
the initial stage of this project.

##  Appendix A Two-point functions: One-loop gluonic UV divergences.

One-loop pure gluonic contribution to the two-point gluon function is given by
Feynman diagrams of Figure 3.

For theories with $n>1$ the UV-divergent contribution of diagram (1) of Figure
3 is given in dimensional regularization by

$\displaystyle{}^{(1)}\Gamma^{ab}_{\mu\nu}(p)$
$\displaystyle=\frac{iC_{2}(G)\delta^{ab}}{64\pi^{2}\epsilon}p^{2}\eta_{\mu\nu}\Bigl{[}\left(4n^{2}+16n+16\right)\lambda^{2}+\left(8n^{2}+32n+32\right)\lambda$
(19) $\displaystyle\hskip
42.67912pt+\frac{2n^{4}}{3}+\frac{8n^{3}}{3}-8n^{2}+\frac{8n}{3}+\frac{50}{3}+\left(\frac{10n^{2}}{3}+\frac{32n}{3}+8\right)\alpha\Bigr{]}$
$\displaystyle-\frac{iC_{2}(G)\delta^{ab}}{64\pi^{2}\epsilon}p_{\mu}p_{\nu}\Bigl{[}\left(4n^{2}+16n+16\right)\lambda^{2}+\left(8n^{2}+32n+32\right)\lambda$
$\displaystyle\hskip
36.98866pt-\frac{4n^{4}}{3}-\frac{16n^{3}}{3}-16n^{2}+\frac{8n}{3}+\frac{56}{3}+\left(\frac{4n^{2}}{3}+\frac{20n}{3}+8\right)\alpha\Bigr{]},$

where $\epsilon=4-d$.

Figure 3:  Gluonic one-loop Feynman diagrams contributing to the 2-point
functions of higher derivative theory.

The corresponding UV-divergent contribution of diagram (2) of Figure 3 is

$\displaystyle{}^{(2)}\Gamma^{ab}_{\mu\nu}(p)$
$\displaystyle=\frac{iC_{2}(G)\delta^{ab}}{64\pi^{2}\epsilon}p^{2}\eta_{\mu\nu}\Bigl{[}\left(-24n^{2}+56n-80\right)\lambda^{2}+\left(8n^{2}+8n-16\right)\lambda$
(20) $\displaystyle\hskip
56.9055pt-\frac{2n^{4}}{3}-\frac{8n^{3}}{3}-8n^{2}-\frac{68n}{3}-8-\left(\frac{10n^{2}}{3}+\frac{32n}{3}+8\right)\alpha\Bigr{]}$
$\displaystyle-\frac{iC_{2}(G)\delta^{ab}}{64\pi^{2}\epsilon}p_{\mu}p_{\nu}\Bigl{[}\left(-24n^{2}+56n-80\right)\lambda^{2}+\left(8n^{2}+8n-16\right)\lambda$
$\displaystyle\hskip
85.35826pt+\frac{4n^{4}}{3}+\frac{16n^{3}}{3}-\frac{68n}{3}-8-\left(\frac{4n^{2}}{3}+\frac{20n}{3}+8\right)\alpha\Bigr{]}.$

The sum of these diagrams

$\displaystyle\\!\\!\\!\\!\\!\\!{}^{(1)}\Gamma^{ab}_{\mu\nu}(p)+\,^{(2)}\Gamma^{ab}_{\mu\nu}(p)$
$\displaystyle=-\frac{iC_{2}(G)\delta^{ab}}{16\pi^{2}\epsilon}\Bigl{[}\left(5n^{2}-18n+16\right)\lambda^{2}-\left(4n^{2}+10n+4\right)\lambda$
(21) $\displaystyle\hskip
14.22636pt+\,4n^{2}+5n-\frac{13}{6}\Bigr{]}\left(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu}\right)-\frac{iC_{2}(G)\delta^{ab}}{32\pi^{2}\epsilon}\
p_{\mu}p_{\nu}$

cancels out the $n^{4}$ and $n^{3}$ terms and the $\alpha$-dependence. The
cancelation of $\alpha$-dependence can be understood in background gauge
formalism alsh ; BV  as a consequence of the fact that the infinitesimal
variation of the effective action under changes of $\alpha$ is given by a
functional equation where one of the factors is proportional to the equations
of motion of the theory alsh . Since these equations involve higher derivative
terms which do no get any divergent contribution from radiative corrections we
can conclude that there are no $\alpha$-dependent UV divergences due to one-
loop radiative corrections to any gluonic $n$-point function, which is in
agreement with the above explicit calculation (21).

The contribution of Faddeev-Popov ghosts to $\Gamma^{ab}_{\mu\nu}$ is given by
diagram (3) of Figure 4

$^{\mathrm{FP}}\Gamma^{ab}_{\mu\nu}(p)=\frac{iC_{2}(G)\delta^{ab}}{32\pi^{2}\epsilon}\left[\frac{1}{3}(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu})+p_{\mu}p_{\nu}\right].$
(22)

Notice that it is the same as that of the standard Faddeev-Popov ghosts in
ordinary Yang-Mills theory. This is a consequence of the factorization of the
higher derivative Faddeev-Popov determinant

$\det[(-\partial^{\sigma}\partial_{\sigma})^{n}(-\partial^{\mu}D_{\mu})]=\det(-\partial^{\sigma}\partial_{\sigma})^{n}\det(-\partial^{\mu}D_{\mu})$
(23)

and the fact that $-(\partial^{\sigma}\partial_{\sigma})^{n}$ does not give
any contribution to the gluonic $n$-point functions.

Figure 4:  One-ghost-loop Feynman diagram contributing to the 2-point
functions.

Finally, the sum of all diagrams contributing to the two-point function reads

$\displaystyle\Gamma^{ab}_{\mu\nu}(p)$
$\displaystyle=-\frac{iC_{2}(G)\delta^{ab}}{16\pi^{2}\epsilon}\Bigl{[}\left(5n^{2}-18n+16\right)\lambda^{2}-\left(4n^{2}+10n+4\right)\lambda$
(24) $\displaystyle\hskip
85.35826pt+\,4n^{2}+5n-\frac{7}{3}\Bigr{]}\left(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu}\right).$

The explicit transverse structure of the two-point function
$\Gamma^{ab}_{\mu\nu}(p)$ is a consequence of BRST invariance of the theory.

Diagrams involving two external ghosts and any number of external gluons are
finite because in the corresponding Feynman graphs there is always one more
gluon propagator with higher derivatives than vertices with higher
derivatives. This implies that the gluonic wave function does not require
renormalization. Thus all divergences can be absorbed by gauge coupling
renormalization that by BRST symmetry cannot depend on the gauge fixing
parameters. This fact explains why in higher derivatives theories the gluonic
2-point function does not depend on the gauge fixing parameter $\alpha$ as
shown by the explicit calculation (24). Notice that for this reason the result
holds even for gauge fixing conditions with less derivatives whenever the
number of extra derivatives of the gauge fixing terms is larger than one. This
fact guarantees that both the transverse and longitudinal parts of the wave
functions do not acquire any divergent contribution from radiative
corrections.

Similar results hold for the UV-divergent contributions in $n=1$ higher
derivative theories. The gluonic one-loop contribution to
$\Gamma^{ab}_{\mu\nu}(p)$ is given by the two diagrams of Figure 3. The
contribution of diagram (1) of Figure 3 is

$\displaystyle{}^{(1)}\Gamma^{ab}_{\mu\nu}(p)$
$\displaystyle=\frac{iC_{2}(G)\delta^{ab}}{32\pi^{2}\epsilon}\left[\left(18\lambda^{2}+36\lambda+\frac{22}{3}+11\alpha\right)(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu})+\left(8+3\alpha\right)p_{\mu}p_{\nu}\right]$

and that of diagram (2) is

$\displaystyle{}^{(2)}\Gamma^{ab}_{\mu\nu}(p)$
$\displaystyle=-\frac{iC_{2}(G)\delta^{ab}}{32\pi^{2}\epsilon}\left[\left(33+11\alpha\right)(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu})+\left(9+3\alpha\right)p_{\mu}p_{\nu}\right].$
(25)

And in the sum of these two diagrams,

$\displaystyle\Gamma^{ab}_{\mu\nu}(p)$
$\displaystyle=\frac{iC_{2}(G)\delta^{ab}}{32\pi^{2}\epsilon}\left[\left(18\lambda^{2}+36\lambda-\frac{77}{3}\right)(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu})-p_{\mu}p_{\nu}\right]$
(26)

the $\alpha$-dependence cancels out by the same reasons that in the
$n\geqslant 2$ case.

When we add the contribution of diagram (3) of Figure 4 with one FP ghost loop
given in (22) we finally obtain

$\displaystyle\Gamma^{ab}_{\mu\nu}(p)$
$\displaystyle=\frac{iC_{2}(G)\delta^{ab}}{16\pi^{2}\epsilon}\left(9\lambda^{2}+18\lambda-\frac{38}{3}\right)\left(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu}\right),$
(27)

which again has a transverse structure in agreement with BRST symmetry.

## References

  * (1) A. A. Slavnov, Invariant regularization of nonlinear chiral theories, Nucl. Phys. B31 (I971) 301-315.
  * (2) A. A. Slavnov, Invariant regularization of gauge theories, Theor. Math. Phys. 13 (1972) 174-177.
  * (3) B. W. Lee and J. Zinn-Justin, Spontaneously broken gauge symmetries, Phys. Rev. D5 (1972) 3121-3137.
  * (4) K. S. Stelle, Classical gravity with higher derivatives, Gen. Rel. Grav. 9 (1978) 353-371
  * (5) K. S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D 16 (1977) 953-969
  * (6) A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B91 (1980) 99-102.
  * (7) E. S. Fradkin, A. A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B201 (1982) 469-491.
  * (8) M. Asorey, J. L. Lopez and I. L. Shapiro, Some remarks on high derivative quantum gravity, Int. J. Mod. Phys. A 12 (1997) 5711-5734.
  * (9) M. Asorey, L. Rachwal, I. Shapiro, Unitary issues in some higher derivative field theories, Galaxies 6 (2018) 23:1-7.
  * (10) T. D. Lee and G. C. Wick, Negative metric and the unitarity of the S-matrix, Nucl. Phys. B 9 (1969) 209-243.
  * (11) T. D. Lee and G. C. Wick, Finite theory of quantum electrodynamics, Phys. Rev. D2 (1970) 1033-1048.
  * (12) D. Anselmi and M. Piva, Perturbative unitarity of Lee-Wick quantum field theory, Phys. Rev. D 96 (2017) 045009:1-15.
  * (13) D. Anselmi and M. Piva, A new formulation of Lee-Wick quantum field theory, JHEP 06 (2017) 066:1-23.
  * (14) T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Towards singularity and ghost free theories of gravity, Phys. Rev. Lett. 108 (2012) 031101:1-4.
  * (15) T. Biswas, A. Conroy, A. S. Koshelev and A. Mazumdar, Generalized ghost-free quadratic curvature gravity, Class. Quant. Grav. 31 (2013) 015022:1-20.
  * (16) M. Asorey and F. Falceto, Geometric regularization of gauge theories, Nucl. Phys. B 327 (1989) 427-460.
  * (17) M. Asorey and F. Falceto, On the consistency of the regularization of gauge theories by high covariant derivatives, Phys. Rev. D 54 (1996) 5290-5301.
  * (18) O. Babelon and M. A. Namazie, Comment on the ghost problem in a higher derivative Yang-Mills theory, J. Phys. A 13 (1980) L27-L30.
  * (19) L. Modesto and L. Rachwal, Super-renormalizable and finite gravitational theories, Nucl. Phys. B889 (2014) 228-248.
  * (20) L. Modesto and L. Rachwal, Universally finite gravitational and gauge theories, Nucl. Phys. B900 (2015) 147-169.
  * (21) L. Modesto, M. Piva and L. Rachwal, Finite quantum gauge theories, Phys. Rev. D 94 (2016) 025021:1-13.
  * (22) Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099:1-19.
  * (23) P. I. Pronin and K. V. Stepanyantz, One-loop divergences in theories with an arbitrary nonminimal operator in curved space, Theor. Math. Phys. 110:3 (1997) 277-294.
  * (24) D. M. Ghilencea, Higher dimensional operators and their effects in (non)super-symmetric models, Mod. Phys. Lett. A23 (2008) 711-720
  * (25) M. Asorey, L. Rachwal and I. Shapiro, a-theorem in higher derivative gauge theories, in preparation.
  * (26) A. O. Barvinsky and G. A. Vilkovisky, The generalized Schwinger-Dewitt technique in gauge theories and quantum gravity, Phys. Rep. 119 (1985) 1.