aainstitutetext: Instituto de Física, Pontificia Universidad Católica de
Valparaíso, Casilla 4950, Valparaísobbinstitutetext: Departamento de Física,
Universidad Técnica Federico Santa María, y Centro Científico-Tecnológico de
Valparaíso, Casilla 110-V, Valparaíso, Chile

# Non-perturbative SQED beta function using functional renormalization group
approach and NSVZ exact beta function.

Jeremy Echeverria b Iván Schmidt<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The renormalization group equations of $\mathcal{N}=1$ SQED with a cubic
superpotential are studied using the functional renormalization group
approach. Applying a supercovariant derivative expansion of the fields, we
compute the non-perturbative beta function. In the local potential
approximation, we verify the exact NSVZ $\beta$ function for the pure SQED
case, and we find an NSVZ relation for an effective fine-structure constant
when the superpotential is considered. Furthermore, we improve the calculation
of the beta function by incorporating the influence of momentum modes on the
propagation of the chiral and vector superfields in the non-perturbative
running of the electric charge, applying a second-order truncation for the
derivative expansion, which we use to verify that the momentum contributions
are canceled and the exact NSVZ $\beta$ function is again obtained for the
pure SQED case, and the effective constant is modified due to the momentum
modes when the superpotential is considered.

##  1 Introduction

Supersymmetry is one of the leading candidates to be the fundamental theory
behind the fundamental interactions of nature because on the one hand, it
relates bosons and fermions through transformations that make forces and
matter the same entity gol1989extension , and on the other hand there are
supersymmetric theories that unify the gauge interactions aulakh2004minimal .
Despite its astonishing theoretical achievements, no evidence for
supersymmetry has been found at low energies, but there is still hope that
high-energy experiments such as the Large Hadron Collider will provide clues.
Some of the reasons for this hope are the elegance with which it solves
problems such as the hierarchy problem martin2010supersymmetry , the smallness
of the cosmological constant froggatt2006smallness ;
aitchison2007supersymmetry  or the renormalizability of supergravity
grisaru1976one , as well as its essential role in string theory and the
minimal supersymmetric standard model martin2010supersymmetry ;
aitchison2007supersymmetry ; bagger1996weak ; on the other hand, when
supersymmetric models are implemented with a $U(1)$ $R-$symmetry, the lightest
supersymmetric particle (LSP) become heavy, neutral and stable, making it
excellent candidate for dark matter jungman1996supersymmetric .

Special cases of study are the supersymmetric extensions of the gauge theories
because they are the building blocks with which supersymmetric models for the
fundamental interactions beyond the standard model are built. Their
supersymmetric properties give them important advantages over their non-
supersymmetric counterparts, one of which is that they have a better UV
behavior due to the cancellations between bosons and fermions, which gives way
to non-renormalization theorems; for example, in the $\mathcal{N}=1$
supersymmetric gauge theories the superpotential is not renormalized
grisaru1979improved . Another important aspect of supersymmetric gauge
theories is that they allow explicit and non-divergent condensate
calculations, which are key to the non-perturbative study of vacuum states,
for example when we perturb around non-trivial backgrounds such as instantons
novikov1983instanton . One feature of supersymmetric $\mathcal{N}=1$ gauge
theories that follow from this is the existence of an exact beta function that
relates the beta function to the anomalous dimensions of the chiral
superfields, e.g. for $\mathcal{N}=1$ supersymmetric quantum electrodynamics
with one flavor one has that (following the nomenclature of goriachuk2018class
; goriachuk2020exact )

$\displaystyle\beta(\alpha)=\frac{\alpha^{2}}{\pi}\left(\frac{1}{2}-\gamma(\alpha)\right),$
(1)

where $\gamma(\alpha)$ is the anomalous dimension of the chiral superfield and
$\alpha$ the fine-structure constant. This is known as the Novikov-Shifman-
Vainshtein-Zakharov or NSVZ exact beta function, named after those who first
obtained it using the exact results for the instanton measure in conjunction
with renormalizability novikov1983exact ; novikov1986beta . Subsequently,
other researchers obtained the same exact beta function by different methods,
such as the structure of anomalies jones1983more ; jones1984beta ;
jones1984chiral ; vainshtein1984axial ; novikov1985supersymmetric ;
shifman1986solution  or the non-renormalization theorems for topological terms
kraus2003supersymmetric .

This relation has also been tested using perturbation theory in a class of
renormalization schemes known as NSVZ-type schemes goriachuk2018class , and
its validity is also verified to all orders in perturbation theory, using
regularization by higher derivatives stepanyantz2011derivation ;
kataev2013nsvz ; stepanyantz2014nsvz .

In this article we study the beta function of $\mathcal{N}=1$ supersymmetric
quantum electrodynamics (SQED), using the functional (or non-perturbative)
renormalization group method dupuis2021nonperturbative , which describes the
flow of the renormalization group of the effective average action (EAA)
wetterich1991average ; wetterich1993average ; wetterich1993improvement ;
wetterich1993exact , a functional that has as the infrared limit the effective
action. The advantage of this approach is that it allows us to calculate the
electric charge flow in the non-perturbative regime and that it preserves
supersymmetry at all the steps of the implementation.

The application of the functional renormalization group in supersymmetric
theories has been studied in supersymmetric quantum mechanics by considering
the non-perturbative flow of the superpotential and the convergence of the
derivative expansion of the effective average action synatschke2009flow .
Following that line, several Wess-Zumino models at zero temperature and finite
temperature have been studied, in which it has been shown that the non-
renormalization theorem for the superpotential is respected by the flow of the
effective average action synatschke2010phase ; synatschke2010two ;
synatschke2010n ; mastaler2012supersymmetric ; feldmann2016functional ;
feldmann2018critical . The convergence of the derivative expansion has also
been investigated for these models heilmann2015convergence . In a previous
work, this method was applied to the case of a single chiral superfield
coupled with a $U(1)$ vector superfield, where the behavior of the
renormalization group equations was analyzed and it was found that the
coupling with vector superfields does not modify the non-renormalization
theorem for the superpotential echeverria2023functional .

The main result of this study is the demonstration that the non-perturbative
flow of the fine structure constant is consistent with the exact NSVZ beta
function in the pure SQED case, both in the leading order and in the next-to-
leading order

$\displaystyle\beta(\alpha)=\frac{\alpha^{2}}{2\pi}.$ (2)

If the self-interactions of the chiral superfield degrees of freedom are taken
into account, i.e. if a superpotential is added, then the anomalous dimensions
of the chiral superfields appear as a contribution to the beta function, which
corresponds to a NSVZ exact beta function for an effective fine structure
constant to which the mass of the chiral superfields contributes.

##  2 $\mathcal{N}=1$ supersymmetric quantum electrodynamics

The classical action of $\mathcal{N}=1$ supersymmetric quantum electrodynamics
or SQED, including a superpotential with quadratic and cubic term is

$\displaystyle S=$ $\displaystyle\int
d^{4}xD_{\mu}{\phi_{-}}^{\dagger}D^{\mu}\phi_{-}+{\left(D_{\mu}{\phi_{+}}^{\dagger}\right)}^{\dagger}D^{\mu}{\phi_{+}}^{\dagger}+i\bar{\psi}_{-}\bar{\sigma}^{\mu}D_{\mu}\psi_{-}+i\bar{\psi}_{+}\bar{\sigma}^{\mu}{D_{\mu}}^{\dagger}\psi_{+}+{F_{-}}^{\dagger}F_{-}+{F_{+}}^{\dagger}F_{+}$
$\displaystyle-\frac{1}{4e^{2}}{F}_{\mu\nu}{F}^{\mu\nu}+\frac{1}{2\chi
e^{2}}{A}_{\mu}{A}^{\mu}+\frac{i}{e^{2}}\bar{\lambda}\bar{\sigma}^{\mu}\partial_{\mu}\lambda+\frac{1}{2e^{2}}D^{2}+i\sqrt{2}\left[{\phi_{-}}^{\dagger}\psi_{-}\lambda-{\phi}^{\dagger}_{+}\psi_{+}\lambda\right]$
$\displaystyle+i\sqrt{2}\left[\phi_{+}\bar{\psi}_{+}\bar{\lambda}-\phi_{-}\bar{\psi}_{-}\bar{\lambda}\right]+\left({\phi_{-}}^{\dagger}\phi_{-}D-{\phi_{+}}^{\dagger}\phi_{+}D\right)+m\left(\phi_{-}F_{+}+\phi_{+}F_{-}-\psi_{-}\cdot\psi_{+}+\text{h.c}.\right)$
$\displaystyle+\frac{y}{2\sqrt{2}}\left(-\phi_{-}^{2}F_{-}+\phi_{-}^{2}F_{+}+\phi_{+}^{2}F_{-}-\phi_{+}^{2}F_{+}+2\phi_{-}\phi_{+}F_{-}+2\phi_{-}\phi_{+}F_{+}+\phi_{-}\psi_{-}\cdot\psi_{-}-\phi_{-}\psi_{+}\cdot\psi_{+}\right.$
$\displaystyle\left.+\phi_{+}\psi_{+}\cdot\psi_{+}-\phi_{+}\psi_{-}\cdot\psi_{-}-\phi_{-}\psi_{-}\cdot\psi_{+}-\phi_{-}\psi_{+}\cdot\psi_{-}-\phi_{+}\psi_{-}\cdot\psi_{+}-\phi_{+}\psi_{+}\cdot\psi_{-}+\text{h.c}.\right),$
(3)

where the subscript “$-$” denotes the electron and “$+$” the positron, and
$D_{\mu}=\partial_{\mu}+i{A}_{\mu}$ is the $U(1)$ covariant derivative. This
action can also be formulated in the superspace

$\displaystyle S=$ $\displaystyle\int d^{4}xd^{2}\theta
d^{2}\bar{\theta}\left[{\Phi_{-}}^{\dagger}e^{-2V}\Phi_{-}+{\Phi_{+}}^{\dagger}e^{2V}\Phi_{+}\right]+\frac{1}{32e^{2}}\int
d^{4}xd^{2}\theta W^{\alpha}W_{\alpha}+\frac{1}{\chi e^{2}}\int
d^{4}xd^{2}\theta d^{2}\bar{\theta}V^{2}$ $\displaystyle+m\left\\{\int
d^{4}xd^{2}\theta\Phi_{-}\Phi_{+}+\text{h.c}\right\\}-\frac{1}{2\sqrt{2}}y\left\\{\int
d^{4}xd^{2}\theta\left[\frac{1}{3}\Phi_{-}^{3}-\frac{1}{3}\Phi_{+}^{3}+\Phi_{-}^{2}\Phi_{+}+\Phi_{+}^{2}\Phi_{-}\right]+\text{h.c}\right\\},$
(4)

where $\Phi_{\mp}$ are chiral superfields containing the matter fields and $V$
is a $U(1)$ vector superfield containing the gauge boson. More details in the
Appendix A.

##  3 The functional renormalization group

In order to obtain the non-perturbative running of the electric charge, we
will compute the beta function using the functional renormalization group
approach. This is constructed using the effective average action (EAA)
$\Gamma_{k}$, a scale-dependent functional that interpolates between the IR
and UV sectors of the theory. The main idea is that in the IR sector, the
theory contains all quantum fluctuations, so for $k\to 0$ the EAA tends to the
ordinary effective action (EA) $\Gamma$, the generator of 1PI Green functions,
whereas in the UV sector, characterized by a momentum cutoff $\Lambda$, all
fluctuations are neglected, so for $k\to\Lambda$ the EAA will be the bare
action in eq.(3). The EAA will be a path in the space of differential
operators with bounded conditions at $k\to 0,\Lambda$. To respect the bound
conditions, the EAA is constructed with a term,

$\displaystyle\Delta S_{k}[\Psi]=\int
d^{4}q\Psi^{\dagger}(q)R_{k}(q^{2})\Psi(q),$ (5)

which acts as a regulator for the functional. The cutoff kernel $R_{k}=\Delta
S^{(2)}_{k}$ is an operator containing functions that regulate the propagators
of the theory and is required to be zero when $k\to 0$ and infinite when
$k\to\Lambda$. At intermediate momentum scales, $R_{k}$ modulates the momentum
of the propagators and defines a path in the operator space for the EAA.

Starting with a bare action $S[\varphi]$ and adding the regulator term, we can
define the partition function as

$\displaystyle e^{W_{k}[J]}=\int\mathcal{D}\varphi e^{-S[\varphi]-\Delta
S_{k}[\varphi]-\int d^{4}xJ(x)\varphi(x)}.$ (6)

To construct the EAA, we perform a Legendre transformation (for more details,
see section 2.1 of gies2012introduction ),

$\displaystyle\Gamma_{k}[\Psi]=\int d^{4}xJ(x)\Psi(x)-W_{k}[J]-\Delta
S_{k}[\Psi],$ (7)

where the fields $\Psi=\langle\varphi\rangle_{J}$ are the expectation values
of the fields in the presence of a current $J$ (and the reason why we can
extract non-perturbative information from the theory).

The flow on the energy scale of the EAA is described by the Wetterich equation
wetterich1993exact

$\displaystyle\partial_{t}\Gamma_{k}=\frac{1}{2}\textrm{STr}\left\\{\left[\Gamma^{(2)}_{k}+R_{k}\right]^{-1}\partial_{t}R_{k}\right\\},$
(8)

where $t=\ln{\frac{k}{\Lambda}}$ and the supertrace denotes sum over the field
arrangement (see Appendix B) and momentum integration. $\Gamma^{(2)}_{k}$ is
the second functional derivative of the EAA with respect to the fields

$\displaystyle\left[\Gamma^{(2)}_{k}\right]_{ij}(p,q)=\frac{\vec{\delta}}{\delta\Psi_{i}(p)}\Gamma_{k}\frac{\reflectbox{$\vec{\reflectbox{$\delta$}}$}}{\delta\Psi_{j}(q)}.$
(9)

To ensure that EAA preserves supersymmetry in all steps of its implementation,
we need to add a regulating term that respects supersymmetry. According to
echeverria2023functional , for a supersymmetric gauge theory, we can use the
regulating term

$\displaystyle\Delta S_{k}=$ $\displaystyle\int d^{4}xd^{2}\theta
d^{2}\bar{\theta}\left[{\Phi_{-}}^{\dagger}r(\square)e^{-2V}\Phi_{-}+{\Phi_{+}}^{\dagger}r(\square)e^{2V}\Phi_{+}\right]+\frac{1}{32}\int
d^{2}\theta{W}^{\alpha}t(\square){W}_{\alpha},$ (10)

where the functions $r$ and $t$ are the optimized Litim-type regulator
functions litim2000optimisation  for fermions and bosons. The explicit forms
of the dimensionless and renormalized regulators (more details in the Appendix
B) in momentum space are

$\displaystyle r=$ $\displaystyle\left(\dfrac{1}{q}-1\right)\theta(1-q^{2})$
$\displaystyle t=$
$\displaystyle\left(\dfrac{1}{q^{2}}-1\right)\theta(1-q^{2}).$ (11)

##  4 The local potential approximation

In order to extract useful information from the EAA flow, we will use a
truncation scheme. In our first approach, we will use a truncation in which
only the superpotential is scale-dependent. This framework is known as "local
potential approximation" (LPA). In this case, this truncation is insufficient
because, as we will see later, the beta function depends on the anomalous
dimension of the vector superfield, so following the procedure of perturbative
renormalization, the fields, and electric charge have been re-escalated using
wave function renormalization functions that are independent of both the
fields and the energy scale,

$\displaystyle\Phi\to{Z_{\Phi}}^{\frac{1}{2}}\Phi,\quad
V\to{Z_{V}}^{\frac{1}{2}}V,\quad e\to Z_{e}e.$

This improvement of LPA is called LPA’ truncation and is our leading order in
the supercovariant derivative expansion.

To avoid violating gauge symmetry, the re-scaling function of the electric
charge cannot be independent of the other functions, in particular,
$Z_{e}={Z_{V}}^{-\frac{1}{2}}$, which gives us the following relation

$\displaystyle e=Z_{e}e_{R}={Z_{V}}^{-\frac{1}{2}}e_{R}.$ (12)

As a consequence, the beta function in $d-$dimensions is defined as

$\displaystyle\beta(e_{R})=\dfrac{de_{R}}{d\ln{k}}=\left(-\frac{1}{2}\eta_{V}+\frac{d-4}{2}\right)e_{R},$
(13)

and

$\displaystyle\beta(e_{R})=\dfrac{de_{R}}{d\ln{k}}=-\frac{1}{2}\eta_{V}e_{R}$
(14)

in $d=4$. The term $\eta_{\mathcal{O}}=-\partial_{t}\ln{Z_{\mathcal{O}}}$ is
the anomalous dimension of the operator $\mathcal{O}$. Similarly, the
anomalous dimension of the chiral superfield is given by

$\displaystyle\gamma=\eta_{\Phi}=-\dfrac{d\ln Z_{\Phi}}{d\ln{k}}.$ (15)

The LPA’ truncation for the effective action has the form

$\displaystyle\Gamma_{k}=$ $\displaystyle Z_{\Phi}\int d^{4}xd^{2}\theta
d^{2}\bar{\theta}\left[{\Phi_{-}}^{\dagger}e^{2V}\Phi_{-}+{\Phi_{+}}^{\dagger}e^{-2V}\Phi_{+}\right]+\frac{{Z_{V}}^{2}}{32e^{2}}\int
d^{4}xd^{2}\theta W^{\alpha}W_{\alpha}+\frac{{Z_{V}}^{2}}{\chi e^{2}}\int
d^{4}xd^{2}\theta d^{2}\bar{\theta}V^{2}$ $\displaystyle+m\left\\{\int
d^{4}xd^{2}\theta\Phi_{-}\Phi_{+}+\text{h.c}\right\\}-\frac{1}{2\sqrt{2}}y\left\\{\int
d^{4}xd^{2}\theta\left[\frac{1}{3}\Phi_{-}^{3}-\frac{1}{3}\Phi_{+}^{3}+\Phi_{-}^{2}\Phi_{+}+\Phi_{+}^{2}\Phi_{-}\right]+\text{h.c}\right\\},$
(16)

or, in components of superfields

$\displaystyle\Gamma_{k}=$ $\displaystyle\int
d^{4}xZ_{\Phi}\left[D_{\mu}{\phi_{-}}^{\dagger}D^{\mu}\phi_{-}+{\left(D_{\mu}{\phi_{+}}^{\dagger}\right)}^{\dagger}D^{\mu}{\phi_{+}}^{\dagger}+i\bar{\psi}_{-}\bar{\sigma}^{\mu}D_{\mu}\psi_{-}+i\bar{\psi}_{+}\bar{\sigma}^{\mu}{D_{\mu}}^{\dagger}\psi_{+}+{F_{-}}^{\dagger}F_{-}+{F_{+}}^{\dagger}F_{+}\right]$
$\displaystyle+\frac{{Z_{V}}^{2}}{e^{2}}\left[-\frac{1}{4}{F}_{\mu\nu}{F}^{\mu\nu}+\frac{1}{2\chi}{A}_{\mu}{A}^{\mu}+i\bar{\lambda}\bar{\sigma}^{\mu}\partial_{\mu}\lambda+\frac{1}{2}D^{2}\right]+i\sqrt{2}\left[{\phi_{-}}^{\dagger}\psi_{-}\lambda-{\phi}^{\dagger}_{+}\psi_{+}\lambda\right]$
$\displaystyle+i\sqrt{2}\left[\phi_{+}\bar{\psi}_{+}\bar{\lambda}-\phi_{-}\bar{\psi}_{-}\bar{\lambda}\right]+\left({\phi_{-}}^{\dagger}\phi_{-}D-{\phi_{+}}^{\dagger}\phi_{+}D\right)+m\left(\phi_{-}F_{+}+\phi_{+}F_{-}-\psi_{-}\cdot\psi_{+}+\text{h.c}.\right)$
$\displaystyle+\frac{y}{2\sqrt{2}}\left(-\phi_{-}^{2}F_{-}+\phi_{-}^{2}F_{+}+\phi_{+}^{2}F_{-}-\phi_{+}^{2}F_{+}+2\phi_{-}\phi_{+}F_{-}+2\phi_{-}\phi_{+}F_{+}+\phi_{-}\psi_{-}\cdot\psi_{-}-\phi_{-}\psi_{+}\cdot\psi_{+}\right.$
$\displaystyle\left.+\phi_{+}\psi_{+}\cdot\psi_{+}-\phi_{+}\psi_{-}\cdot\psi_{-}-\phi_{-}\psi_{-}\cdot\psi_{+}-\phi_{-}\psi_{+}\cdot\psi_{-}-\phi_{+}\psi_{-}\cdot\psi_{+}-\phi_{+}\psi_{+}\cdot\psi_{-}+\text{h.c}.\right).$
(17)

The LPA’ truncation of the coupling between chiral and vector superfields
respects the non-renormalization theorem for the superpotential
echeverria2023functional , and because of this we can simply put the
truncation of the superpotential inside the LPA’ truncation; on the other
hand, the wave function renormalization function for the electron and positron
are the same, because both are components of a Dirac fermion.

Since the renormalization functions of the wave function have no dependence on
the energy scale, we can restrict all fields to real constants in momentum
space $\Psi_{i}(q)=(\sqrt{2\pi})^{4}\Psi_{i}\delta(q)$ without affecting the
flow of the renormalization functions. To extract the corresponding
information, we have to project eq.(8) in a suitable configuration of the
fields in such a way that it is possible to isolate the term containing the
information. For example, projecting on
$\phi_{-}=\phi_{+}=\psi_{-}=\psi_{+}=F_{+}=F_{-}=A=\lambda=0$ we get

$\displaystyle\partial_{t}\Gamma_{k}=(\sqrt{2\pi})^{4}\delta^{(4)}\left(0\right)\partial_{t}\left[\frac{{Z_{V}}^{2}}{2e^{2}}D^{2}\right].$
(18)

From this point, we can see that

$\displaystyle-\frac{1}{2}\eta_{V}=\frac{e^{2}}{4{Z_{V}}^{2}}\left.\left\\{\left.\dfrac{d^{2}}{dD^{2}}\partial_{t}\Gamma_{k}\right|_{\phi_{-}=\phi_{+}=\psi_{-}=\psi_{+}=F_{+}=F_{-}=A=\lambda=0}\right\\}\right|_{D=0}.$
(19)

In the same way we can project eq.(17) onto
$\phi_{-}=\phi_{+}=\psi_{-}=\psi_{+}=F_{+}=D=A=\lambda=0$ in order to get,

$\displaystyle\partial_{t}\Gamma_{k}=(\sqrt{2\pi})^{4}\delta^{(4)}\left(0\right)\partial_{t}\left[Z_{\Phi}{F_{-}}^{\dagger}F_{-}\right],$
(20)

therefore111Note that in eq.(19) and eq.(21) we are also taking into
consideration the conjugate form of the fields.,

$\displaystyle\gamma=-\frac{1}{{Z_{\Phi}}}\left.\left\\{\left.\frac{d^{2}}{dF_{-}d{F_{-}}^{\dagger}}\partial_{t}\Gamma_{k}\right|_{\phi_{-}=\phi_{+}=\psi_{-}=\psi_{+}=F_{+}=D=A=\lambda=0}\right\\}\right|_{F_{-}=0}.$
(21)

The right-hand sides of eq.(19) and eq.(21) have been obtained from eq.(8) in
momentum space

$\displaystyle\partial_{t}\Gamma_{k}=\frac{1}{2}\int
d^{4}q\textrm{Tr}\left\\{\left(\left[\Gamma^{(2)}_{k}\right](p,q)+R_{k}(p,q)\right)^{-1}\partial_{t}R_{k}(p,q)\delta^{(4)}\left(p-q\right)\right\\}.$
(22)

Since all the fields are constant, we can simply invert the matrix in Appendix
C and take the matrix trace. For the momentum integration, both the regulator
functions as well as the energy scale, and the coupling constants have been
expressed in a dimensionless and renormalized way (see Appendix B). This leads
to the anomalous dimension of the chiral superfield

$\displaystyle\gamma=\frac{3y^{2}(m^{2}+1)}{24\pi^{2}(m^{2}-1)^{3}+y^{2}(m^{2}+1)}$
(23)

and the flow equation for the electric charge

$\displaystyle\beta(\alpha)=$
$\displaystyle-\frac{12\pi\alpha^{2}(m^{2}+1)}{24\pi^{2}(m^{2}-1)^{3}+2y^{2}(m^{2}+1)}$
(24)

where $\alpha$ is the fine-structure constant $\alpha=e_{R}^{2}/4\pi$. Eq.(24)
can be writing in the form

$\displaystyle\beta(\alpha)=\frac{\alpha^{2}}{\pi}\left(\frac{1+m^{2}}{(1-m^{2})^{3}}\right)\left(\frac{1}{2}-\frac{\gamma}{6}\right)$
(25)

###  4.1 Pure SQED case

The pure SQED limit corresponds to switching off the superpotential, or in
other words to setting $m=y=0$ in eq.(23), eq.(24) and eq.(25) which leaves us
with the following result

$\displaystyle\gamma=$ $\displaystyle 0,$ $\displaystyle\beta(\alpha)=$
$\displaystyle\frac{\alpha^{2}}{2\pi}$ (26)

satisfying the NSVZ relation when the anomalous dimension vanishes.

###  4.2 The non-vanishing anomalous dimension case

Keeping the superpotential, we get contributions from the mass and the
anomalous dimension to the beta function, as in eq.(25). However, it is
possible to modify it to obtain the functional form of a NSVZ beta function by
defining the following effective constant

$\displaystyle\alpha^{2}=\frac{{\alpha_{0}}^{2}(1+m^{2})^{2}}{(1-m^{2})^{6}}$
(27)

where $\alpha_{0}$ is the usual fine-structure constant. This relation allows
the removal of the dependence on the mass at the cost of the addition of a
dependence on the effective constant in the anomalous dimension. The anomalous
dimension and the beta function for this effective fine-structure constant
will be

$\displaystyle\gamma(\alpha)=-\frac{3y^{2}\alpha}{24\pi^{2}\alpha_{0}-y^{2}\alpha}$
$\displaystyle\beta(\alpha)=\frac{12\pi\alpha_{0}\alpha}{24\pi^{2}\alpha_{0}-y^{2}\alpha},$
(28)

which can be rewrite as

$\displaystyle\beta(\alpha)=\frac{\alpha^{2}}{\pi}\left(\frac{1}{2}-\frac{\gamma(\alpha)}{6}\right).$
(29)

##  5 Second order of derivative expansion

Now, we will improve the calculation of the beta function by using the second
order in the derivative expansion, using arbitrary functions $Z_{k}(\square)$
modulating the kinetic part of the Lagrangian, this allows us to add the
influence of momentum modes in the propagation of both superfields in the
running of the electric charge.

The second-order truncation for the effective action will be

$\displaystyle\Gamma_{k}=$ $\displaystyle\int d^{4}xd^{2}\theta
d^{2}\bar{\theta}Z_{\Phi,k}(\square)\left[{\Phi_{-}}^{\dagger}e^{2V}\Phi_{-}+{\Phi_{+}}^{\dagger}e^{-2V}\Phi_{+}\right]+\frac{1}{32e^{2}}\int
d^{4}xd^{2}\theta{Z_{V,k}(\square)}^{2}W^{\alpha}W_{\alpha}$
$\displaystyle+\frac{1}{\alpha e^{2}}\int d^{4}xd^{2}\theta
d^{2}\bar{\theta}{Z_{V,k}(\square)}^{2}V^{2}+m\left\\{\int
d^{4}xd^{2}\theta\Phi_{-}\Phi_{+}+\text{h.c}\right\\}$
$\displaystyle-\frac{1}{2\sqrt{2}}y\left\\{\int
d^{4}xd^{2}\theta\left[\frac{1}{3}\Phi_{-}^{3}-\frac{1}{3}\Phi_{+}^{3}+\Phi_{-}^{2}\Phi_{+}+\Phi_{+}^{2}\Phi_{-}\right]+\text{h.c}\right\\}.$
(30)

To respect the results of the local potential approximation, we require that
the arbitrary functions are the ordinary wave function renormalization
functions in the limit $q\to 0$ (in momentum space),

$\displaystyle Z_{\Phi/V,k}(q\to 0)\to Z_{\Phi/V}$ (31)

which has been implemented by parameterizing the momentum information in a
function $\zeta_{\Phi/V,k}(q)$ with momentum dependence feldmann2018critical
such that

$\displaystyle Z_{\Phi/V,k}(q)=Z_{\Phi/V}\zeta_{\Phi/V,k}(q)$ (32)

with $\zeta_{\Phi/V,k}(0)=1$.

Since this time the renormalization functions of the wave function have a
dependence on the momentum, we cannot restrict the totality of the field to
real constants, for example, if we project the EAA in the term proportional to
$D^{2}$ and we make $D(q)$ a real constant, this will happen

$\displaystyle\frac{1}{(\sqrt{2\pi})^{4}}\int
d^{4}q\frac{{Z_{V,k}(q)}^{2}}{2e^{2}}D(q)D(-q)=\delta^{(4)}\left(0\right)\frac{{Z_{V}}^{2}}{2e^{2}}D^{2}$
(33)

and we have lost all information about the momentum modes, so we will restrict
all fields except $D(q)$. Projecting eq.(8) on
$\phi_{-}=\phi_{+}=\psi_{-}=\psi_{+}=F_{+}=F_{-}=A=\lambda=0$ we obtain

$\displaystyle\partial_{t}\Gamma_{k}=\int
d^{4}p\partial_{t}\left[\frac{{Z_{V,k}(p)}^{2}}{2e^{2}}D(p)D(-p)\right].$ (34)

From this point, we can see that

$\displaystyle\frac{1}{2Z_{V}}\partial_{t}Z_{V,k}(p)=$
$\displaystyle-\frac{1}{2}\eta_{V}\zeta_{V,k}(p)+\frac{1}{2}\partial_{t}\zeta_{V,k}(p)$
$\displaystyle=$
$\displaystyle\frac{e^{2}}{4Z_{V}Z_{V,k}(p)}\left.\left\\{\left.\frac{\delta^{2}}{\delta
D(p)\delta
D(-p)}\partial_{t}\Gamma_{k}\right|_{\phi_{-}=\phi_{+}=\psi_{-}=\psi_{+}=F_{+}=F_{-}=A=\lambda=0}\right\\}\right|_{D=0}.$
(35)

In the same way we can project onto
$\phi_{-}=\phi_{+}=\psi_{-}=\psi_{+}=F_{+}=D=A=\lambda=0$, keeping $F_{-}(q)$
and ${F_{-}}^{\dagger}(q)$ as momentum-dependent and we get

$\displaystyle\partial_{t}\Gamma_{k}=\int
d^{4}p\partial_{t}\left[{Z_{\Phi,k}(p)}F_{-}(p){F_{-}}^{\dagger}(p)\right],$
(36)

therefore,

$\displaystyle\frac{1}{Z_{\Phi}}\partial_{t}Z_{\Phi,k}(p)=$
$\displaystyle-\gamma\zeta_{\Phi,k}(p)+\partial_{t}\zeta_{\Phi,k}(p)$
$\displaystyle=$
$\displaystyle\frac{1}{Z_{\Phi}}\left.\left\\{\left.\frac{\delta^{2}}{\delta
F_{-}(p)\delta{F_{-}}^{\dagger}(p)}\partial_{t}\Gamma_{k}\right|_{\phi_{-}=\phi_{+}=\psi_{-}=\psi_{+}=F_{+}=D=A=\lambda=0}\right\\}\right|_{F_{-}=0}.$
(37)

The right-hand sides of eq.(35) and eq.(37) have been obtained from eq.(8) in
momentum space

$\displaystyle\partial_{t}\Gamma_{k}=\frac{1}{2}\int
d^{4}q\textrm{Tr}\left\\{\left(\left[\Gamma^{(2)}_{k}\right](p,q)+R_{k}(p,q)\right)^{-1}\partial_{t}R_{k}(p,q)\right\\}$
(38)

with the important difference that in this case, the operators have non-
trivial dependence on momentum, therefore we can not compute eq.(40) taking
the inverse matrix of the operator $G_{k}=\Gamma_{k}^{(2)}+R_{k}$. The
calculation strategy is decompose the operator $G_{k}$ in a field-independent
part $G_{0,k}$ and in a field-dependent part $\Delta G_{k}$ as is shown in
gies2002renormalization

$\displaystyle G_{k}=G_{0,k}+\Delta G_{k},$ (39)

whose specific forms are given in the Appendix D. Substituting this into
eq.(8) allows us to perform an expansion of the Wetterich equation in the
field dependent part of the operator

$\displaystyle\partial_{t}\Gamma_{k}=$
$\displaystyle\frac{1}{2}\text{STr}\tilde{\partial}_{t}\ln{\left[G_{0,k}+\Delta
G_{k}\right]}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\text{STr}\tilde{\partial}_{t}\left({G_{0,k}}^{-1}\Delta
G_{k}\right)-\frac{1}{4}\text{STr}\tilde{\partial}_{t}\left({G_{0,k}}^{-1}\Delta
G_{k}\right)^{2}+\cdots$ (40)

where $\tilde{\partial}_{t}$ only affects the operator $R_{k}$. The inverse of
the field-independent part can be computed by simply inverse matrix. Because
eq.(35) and eq.(37) are quadratic in fields we only need to expand up to the
second order in the field-dependent part. After taking the matrix trace and
performing the functional derivatives, we obtain the flow of the momentum-
dependent wave function renormalization functions before the momentum
integration (more details in Appendix D).

$\displaystyle\frac{1}{2Z_{V}}\partial_{t}Z_{V,k}(p)=-\frac{\Omega_{d}}{(2\pi)^{d}}\int^{1}_{0}dqq^{d-1}\frac{e_{R}^{2}(r(q-p)+\zeta_{\Phi,k}(q-p))(m^{2}+q^{2}(r(q)+\zeta_{\Phi,k}(q))^{2}\partial_{t}r(q)}{(m^{2}-q^{2}r(q)+\zeta_{\Phi,k}(q)^{2})^{2}(m^{2}-(q-p)^{2}(r(q-p)+\zeta_{\Phi,k}(q-p))^{2})}$
$\displaystyle\frac{1}{Z_{\Phi}}\partial_{t}Z_{\Phi,k}(p)=\frac{\Omega_{d}}{(2\pi)^{d}}\int^{1}_{0}dqq^{d-1}\frac{2y^{2}(r(q-p)+\zeta_{\Phi,k}(q-p))(m^{2}+q^{2}(r(q)+\zeta_{\Phi,k}(q))^{2}\partial_{t}r(q)}{(m^{2}-q^{2}r(q)+\zeta_{\Phi,k}(q)^{2})^{2}(m^{2}-(q-p)^{2}(r(q-p)+\zeta_{\Phi,k}(q-p))^{2})}.$
(41)

At this point the functions $\zeta_{\Phi/V,k}(p)$ remain arbitrary, to extract
explicit information about the momentum modes we will polynomially truncate
$\zeta_{\Phi/V,k}(p)$ feldmann2016functional

$\displaystyle\zeta_{\Phi/V,k}(p)=\sum^{\infty}_{n=0}\zeta_{\Phi/V,n}p^{n}$
(42)

with $\zeta_{\Phi/V,0}=1$. The beta function and the anomalous dimension are
obtained replacing $p=0$

$\displaystyle\frac{1}{2Z_{V}}\partial_{t}Z_{V,k}(0)=-\frac{1}{2}\eta_{V}\zeta_{V,k}(0)+\frac{1}{2}\partial_{t}\zeta_{V,k}(0)=-\frac{1}{2}\eta_{V}=\frac{1}{e_{R}}\beta(e_{R})$
$\displaystyle\frac{1}{Z_{\Phi}}\partial_{t}Z_{\Phi,k}(0)=-\gamma\zeta_{\Phi,k}(0)+\partial_{t}\zeta_{\Phi,k}(0)=-\gamma.$
(43)

Performing $p=0$ in eq.(41) we get

$\displaystyle-\frac{1}{2}\eta_{V}=-\frac{\Omega_{d}}{(2\pi)^{d}}\int^{1}_{0}dqq^{d-1}\frac{e_{R}^{2}(r(q)+\zeta_{\Phi,n}q^{n})(m^{2}+q^{2}(r(q)+\zeta_{\Phi,n}q^{n})^{2})\partial_{t}r(q)}{(m^{2}-q^{2}(r(q)+\zeta_{\Phi,n}q^{n})^{2})^{3}}$
$\displaystyle-\gamma=\frac{\Omega_{d}}{(2\pi)^{d}}\int^{1}_{0}dqq^{d-1}\frac{2y^{2}(r(q)+\zeta_{\Phi,n}(q-p)^{n})(m^{2}+q^{2}(r(q)+\zeta_{\Phi,n}q^{n})^{2})\partial_{t}r(q)}{(m^{2}-q^{2}r(q)+\zeta_{\Phi,n}q^{n})^{2}(m^{2}-(q-p)^{2}(r(q-p)+\zeta_{\Phi,n}(q-p)^{n})^{2})},$
(44)

where the terms $\zeta_{\Phi,n}q^{n}$ denotes a sum over $n$. Using the
regulator functions in Appendix B, the integrals have been solved for $d=4$
giving the anomalous dimension of the chiral superfield

$\displaystyle\gamma=$
$\displaystyle-\frac{3y^{2}(1+n)(m^{2}+\zeta_{\Phi,n}^{2})\zeta_{\Phi,n}^{2}}{y^{2}(1+n)\zeta_{\Phi,n}^{2}(m^{2}+\zeta_{\Phi,n}^{2})+12\pi^{2}m^{2}(3+n)\zeta_{\Phi,n}^{2}(m^{2}-\zeta_{\Phi,n}^{2})-8\pi^{2}(3+n)(m^{6}-\zeta_{\Phi,n}^{6})}$
(45)

and the beta function for the fine-structure constant

$\displaystyle\beta(\alpha)=$
$\displaystyle\frac{12\pi\alpha^{2}(1+n)(m^{2}+\zeta_{\Phi,n}^{2})\zeta_{\Phi,n}^{2}}{y^{2}(1+n)\zeta_{\Phi,n}^{2}(m^{2}+\zeta_{\Phi,n}^{2})+12\pi^{2}m^{2}(3+n)\zeta_{\Phi,n}^{2}(m^{2}-\zeta_{\Phi,n}^{2})-8\pi^{2}(3+n)(m^{6}-\zeta_{\Phi,n}^{6})}$
(46)

which can be write in the form

$\displaystyle\beta(\alpha)=$
$\displaystyle-\frac{\alpha^{2}}{\pi}\left(\frac{3(1+n)(m^{2}+\zeta_{\Phi,n}^{2})\zeta_{\Phi,n}^{2}}{(3+n)(m^{2}-\zeta_{\Phi,n}^{2})^{3}}\right)\left(\frac{1}{2}-\frac{\gamma}{6}\right).$
(47)

###  5.1 Pure SQED case

Again we consider the pure SQED case, at first glance, it seems that the next-
to-leading order gives us corrections to the fine-structure constant coming
from the momentum modes encoded in the functions $\zeta_{\Phi,V}$, so again we
will have an effective constant, however, this is not the case because, from
eq.(41), we can see that

$\displaystyle\frac{1}{Z_{\Phi}}\partial_{t}Z_{\Phi,k}(p)=$
$\displaystyle-\gamma\zeta_{\Phi,k}(p)+\partial_{t}\zeta_{\Phi,k}(p)$
$\displaystyle=$
$\displaystyle-\gamma-\sum_{n=1}^{\infty}(\gamma+\zeta_{\Phi,n})p^{n}=0,$ (48)

from $p^{0}$ coefficient we recover the result of the local potential
approximation

$\displaystyle\gamma=0,$ (49)

which is consistent with eq.(45). For all orders of the expansion eq.(42) we
obtain

$\displaystyle\zeta_{\Phi,n}=-\gamma=0,$ (50)

then it can be seen that the result obtained in the LPA is also valid in the
second order of the derivative expansion, as follows

$\displaystyle\gamma=$ $\displaystyle 0,$ $\displaystyle\beta(\alpha)=$
$\displaystyle\frac{\alpha^{2}}{2\pi}.$ (51)

###  5.2 The non-vanishing anomalous dimension case

From eq.(47) we see that if we consider the superpotential we will have, in
addition to the mass and the anomalous dimension, contributions from the
momentum modes, but we can follow the same procedure as in the LPA case and
define an effective fine structure constant for each truncation of eq.(42).
For example, in the case $n=1$

$\displaystyle\zeta_{\Phi,k}=1+\zeta_{\Phi,1}p,$ (52)

calculation of the integrals in eq.(41) shows that

$\displaystyle\gamma=\frac{3y^{2}(1+\zeta_{\Phi,1})(2+3\zeta_{\Phi,1})\left(m^{2}+(1+\zeta_{\Phi,1})^{2}\right)}{48\pi^{2}(m^{2}-(1+\zeta_{\Phi,1})^{2})^{3}-y^{2}(1+\zeta_{\Phi,1})(2+3\zeta_{\Phi,1})(m^{2}+(1+\zeta_{\Phi,1})^{2})}$
$\displaystyle\beta(\alpha)=-\frac{24\pi\alpha^{2}(1+\zeta_{\Phi,1})(2+3\zeta_{\Phi,1})\left(m^{2}+(1+\zeta_{\Phi,1})^{2}\right)}{96\pi^{2}(m^{2}-(1+\zeta_{\Phi,1})^{2})^{3}-2y^{2}(1+\zeta_{\Phi,1})(2+3\zeta_{\Phi,1})(m^{2}+(1+\zeta_{\Phi,1})^{2})}$
(53)

wich can be write in the form

$\displaystyle\beta(\alpha)=-\frac{\alpha^{2}}{\pi}\left(\frac{(1+\zeta_{\Phi,1})(2+3\zeta_{\Phi,1})\left(m^{2}+(1+\zeta_{\Phi,1})^{2}\right)}{2(m^{2}-(1+\zeta_{\Phi,1})^{2})^{3}}\right)\left(\frac{1}{2}-\frac{\gamma}{6}\right).$
(54)

This tells us that we can define an effective constant

$\displaystyle\alpha^{2}=\frac{{\alpha_{0}}^{2}(1+\zeta_{\Phi,1})^{2}(2+3\zeta_{\Phi,1})^{2}\left(m^{2}+(1+\zeta_{\Phi,1})^{2}\right)^{2}}{2((1+\zeta_{\Phi,1})^{2}-m^{2})^{6}}$
(55)

where $\alpha_{0}$ is the usual fine-structure constant. The anomalous
dimension and the beta function depending on this effective constant will be

$\displaystyle\gamma(\alpha)=-\frac{3y^{2}\alpha}{24\pi^{2}\alpha_{0}-y^{2}\alpha}$
$\displaystyle\beta(\alpha)=\frac{12\pi\alpha_{0}\alpha}{24\pi^{2}\alpha_{0}-y^{2}\alpha},$
(56)

which can be rewrite as

$\displaystyle\beta(\alpha)=\frac{\alpha^{2}}{\pi}\left(\frac{1}{2}-\frac{\gamma(\alpha)}{6}\right).$
(57)

This process can be made for each truncation of eq.(42).

##  6 Conclusions

In this work, the beta function of $\mathcal{N}=1$ supersymmetric quantum
electrodynamics has been calculated in the leading and next-to-leading order
of the supercovariant derivative expansion of the effective average action in
order to study its non-perturbative behavior.

In the leading order, the local potential approximation, we truncated the EAA
by assuming that only the superpotential has an energy dependence, and we also
considered the wave function renormalization functions in order to access the
anomalous dimensions of the fields. Here we obtained a relation between the
beta function and the anomalous dimension as in the NSVZ relation, with the
difference of a mass-dependent factor, which motivated us to define an
effective fine-structure constant whose beta function respects a similar
relation. Furthermore, due to such a definition, the anomalous dimension
obtained a dependence on the fine-structure constant through the mass; this
NSVZ relation is compatible with the exact beta function for the pure SQED
case in eq.(2).

In the next-to-leading order, we added the contributions of the momentum modes
in the propagation of the chiral superfields and the vector superfield to
improve the calculation of the beta function by promoting the wave function
renormalization functions to momentum-dependent functions through arbitrary
functions, which were then polynomially truncated to obtain concrete results.
In this case, a relation between the beta function and the anomalous dimension
was again obtained, but the factor now depends on both the mass and the
coefficients of the polynomial truncation, and it was shown that, given a
fixed order of truncation, it is possible to define an effective fine-
structure constant which has as its limit the result obtained in the local
potential approximation when the coefficients are zero. In the pure SQED case
it has been shown that the coefficients of the polynomial truncation are all
zero, so that the beta function receives no higher order corrections, again in
agreement with eq.(2). When considering the superpotential something curious
happens, in the local potential approximation we saw that the effective fine-
structure constant was a combination of both the ordinary constant and the
mass of the chiral superfields; however, at second order the coefficients of
the polynomial truncation will obtain a dependence on the cubic coupling
constant via eq.(41), therefore the flow of the effective average action shows
that although the electric charge does not exactly comply with a kind of NSVZ
relation, a combination of all the parameters of the theory does.

The main contribution of this work is to show that the results of the
application of the functional renormalization group in supersymmetric gauge
theories agree with the non-perturbative results obtained by geometric
methods, and to provide a non-perturbative verification of the exact NSVZ beta
function in the case of an abelian supersymmetric gauge theory; moreover, such
results have been improved with the second order of the derivative expansion.
Further investigations can explore the behavior of the effective fine-
structure constant and the non-perturbative effects contributed by the
momentum modes. These results can also be generalized to non-abelian
supersymmetric gauge theories and more general superpotentials.

##  Appendix A Superspace formulation

Supersymmetric theories can be constructed from an action formulated in terms
of its field content, as in eq.(3), or in a more compacted form in terms of
superfields over a superspace. Superfields are functions of both the space-
time coordinates ${x}^{\mu}$, $\mu=(t,x,y,z)$ and the non-conmutative
Grassmann variables
$\left\\{{\theta}_{\alpha},{\bar{\theta}}_{\dot{\alpha}}\right\\}$. For
$\mathcal{N}=1$ SQED we need only one pair of Grassmann variables.

A general superfield $\Sigma(x,\theta,\bar{\theta})$ must be constrained to
define chiral and vector superfields, which will be the supersymmetric version
of matter and gauge superfields.

A chiral superfield $\Phi(x,\theta,{\theta}^{\dagger})$ is defined as such if
it satisfies the conditions

$\displaystyle{\bar{D}}_{\dot{\alpha}}\Phi=0$
$\displaystyle{D}_{\alpha}{\Phi}^{\dagger}=0$ (58)

where

$\displaystyle{D}_{\alpha}=\dfrac{\partial}{\partial\theta^{\alpha}}+i(\sigma^{\mu})_{\alpha\dot{\alpha}}\partial_{\mu}$
$\displaystyle{\bar{D}}_{\dot{\alpha}}=-\dfrac{\partial}{\partial{\bar{\theta}}^{\dot{\alpha}}}-i{\theta}^{\alpha}(\sigma^{\mu})_{\alpha\dot{\alpha}}\partial_{\mu}.$
(59)

are the covariant chiral/anti chiral derivatives.

The expansion of the chiral superfield in terms of Grassmann variables is
given by

$\displaystyle\Phi(x,\theta,\bar{\theta})=$
$\displaystyle\phi(x)+\sqrt{2}\theta\psi(x)+\theta\theta
F(x)+i\partial_{\mu}\phi(x)\theta\sigma^{\mu}\bar{\theta}$
$\displaystyle-\frac{i}{\sqrt{2}}\theta\theta\partial_{\mu}\psi(x)\sigma^{\mu}\bar{\theta}-\frac{1}{4}\partial_{\mu}\partial^{\mu}\phi(x)\theta\theta\bar{\theta}\bar{\theta}$
$\displaystyle\Phi(x,\theta,\bar{\theta})=$
$\displaystyle{\phi}^{\dagger}(x)+\sqrt{2}\bar{\theta}\bar{\psi}(x)+\bar{\theta}\bar{\theta}{F}^{\dagger}(x)-i\partial_{\mu}{\phi}^{\dagger}(x)\theta\sigma^{\mu}\bar{\theta}$
$\displaystyle+\frac{i}{\sqrt{2}}\bar{\theta}\bar{\theta}\theta\sigma^{\mu}\partial_{\mu}\bar{\psi}(x)-\frac{1}{4}\partial_{\mu}\partial^{\mu}{\phi}^{\dagger}(x)\theta\theta\bar{\theta}\bar{\theta}.$
(60)

To get a Lagrangian density in terms of the above superfield components, it is
necessary to integrate the Grassmann directions. A renormalizable
supersymmetric action for chiral superfields is

$\displaystyle S=$ $\displaystyle\int d^{4}xd^{2}\theta
d^{2}\bar{\theta}\Phi^{\dagger}\Phi=\int
d^{4}x\partial_{\mu}{\phi}^{\dagger}\partial^{\mu}\phi+i\bar{\psi}\bar{\sigma}^{\mu}\partial_{\mu}\psi+{F}^{\dagger}F$
(61)

This is known as the free Wess-Zumino model wess1974lagrangian . The most
general interaction for this model has the form aitchison2007supersymmetry

$\displaystyle\mathcal{L}_{\textrm{int}}=\dfrac{dW(\phi)}{d\phi_{i}}F_{i}-\frac{1}{2}\frac{d^{2}W(\phi)}{d\phi_{i}d\phi_{j}}\psi_{i}\cdot\psi_{j}+\textrm{h.c},$
(62)

where the function $W(\phi)$ is the superpotential, an holomorphic function of
the scalar fields. In superspace, the interaction action is

$\displaystyle S_{\textrm{int}}=\int d^{4}xd^{2}\theta W(\Phi)+\int
d^{4}xd^{2}\bar{\theta}\bar{W}({\Phi}^{\dagger}).$ (63)

On the other hand, $V(x,\theta,{\theta}^{\dagger})$ is defined as a vector
superfield if it satisfies the condition

$\displaystyle V=V^{\dagger}.$ (64)

This condition causes non-physical degrees of freedom, which can be gauge away
using the transformation

$\displaystyle V\to V+i\left(\Omega-{\Omega}^{\dagger}\right),$ (65)

where $\Omega$ is a chiral superfield. The transformation (65) is called a
$U(1)$ supergauge transformation and we can semi-fix it so that it only holds
for the usual gauge transformations. This is known as the Wess-Zumino gauge
ferrara1974supergauge  and its expansion in Grassmann variables is

$\displaystyle V_{\textrm{W-Z
gauge}}=\theta\sigma^{\mu}\bar{\theta}{A}_{\mu}+i\theta\theta\bar{\theta}\bar{\lambda}-i\bar{\theta}\bar{\theta}\theta\lambda+\frac{1}{2}\theta\theta\bar{\theta}\bar{\theta}D.$
(66)

Finally, $W_{\alpha}$ is the gauge invariant abelian superfield.

$\displaystyle{W}_{\alpha}\equiv\bar{D}^{2}{D}_{\alpha}V$ (67)

which allows us to define a renormalizable supersymmetric and gauge invariant
action for vector superfields.

$\displaystyle\frac{1}{32}\int d^{4}xd^{2}\theta{W}^{\alpha}{W}_{\alpha}=\int
d^{4}x-\frac{1}{4}{F}_{\mu\nu}{F}^{\mu\nu}+i\bar{\lambda}\bar{\sigma}^{\mu}\partial_{\mu}\lambda+\frac{1}{2}D^{2}.$
(68)

In essence, supersymmetric gauge theories are the coupling between chiral and
vector superfields. To describe charged fields, it is necessary to consider
the left and right chiral components separately. Starting with two chiral
superfields $\Phi_{1}$ and $\Phi_{2}$, we can define complex chiral
superfields

$\displaystyle\Phi_{-}=$ $\displaystyle\frac{1}{\sqrt{2}}(\Phi_{1}-i\Phi_{2})$
$\displaystyle\Phi_{+}=$ $\displaystyle\frac{1}{\sqrt{2}}(\Phi_{1}+i\Phi_{2})$

which, under the $U(1)$ gauge transformation, it transform as

$\displaystyle\Phi_{\pm}^{\prime}\to e^{\pm 2ie\Omega}\Phi_{\pm}$ (70)

then the products

$\displaystyle{\Phi_{-}}^{\dagger}e^{-2eV}\Phi_{-},$
$\displaystyle{\Phi_{+}}^{\dagger}e^{2eV}\Phi_{+}$ (71)

are gauge invariant. Finally, the action for a supersymmetric $U(1)$ gauge
theory will be

$\displaystyle S=\int d^{4}xd^{2}\theta
d^{2}\bar{\theta}\left[{\Phi_{-}}^{\dagger}e^{-2eV}\Phi_{-}+{\Phi_{+}}^{\dagger}e^{2eV}\Phi_{+}\right]+\frac{1}{32}\int
d^{4}xd^{2}\theta W^{\alpha}W_{\alpha}.$ (72)

The action defined in eq.(4) has the vector superfield redefined as
$V\to\dfrac{1}{e}V$ and a gauge-fixing term defined by the fact that in the
Wess-Zumino gauge eq.(66)

$\displaystyle{V_{\textrm{W-Z
gauge}}}^{2}(x,\theta,\bar{\theta})=\frac{1}{2}\theta\theta\bar{\theta}\bar{\theta}{A}_{\mu}{A}^{\mu}.$
(73)

Finally, for the interaction part, we consider a superpotential with quadratic
and cubic interactions in order to obtain a non-vanishing anomalous dimension
for chiral superfields

$\displaystyle W(\Phi_{1},\Phi_{2})=$
$\displaystyle\frac{1}{2}m\Phi_{1}^{2}+\frac{1}{2}m\Phi_{2}^{2}+m\Phi_{1}\Phi_{2}$
$\displaystyle+\frac{1}{6}y\Phi_{1}^{3}+\frac{1}{6}y\Phi_{2}^{3}+\frac{1}{2}y\Phi_{1}^{2}\Phi_{2}+\frac{1}{2}y\Phi_{2}^{2}\Phi_{1}+\text{h.c}.$
(74)

Using the definitions in eq.(LABEL:chiralfields) the superpotential in terms
of the electron and positron superfields is

$\displaystyle W(\Phi_{-},\Phi_{+})=$ $\displaystyle
m\Phi_{-}\Phi_{+}-\frac{1}{6\sqrt{2}}y\Phi_{-}^{3}-\frac{1}{6\sqrt{2}}y\Phi_{+}^{3}+\frac{1}{2\sqrt{2}}y\Phi_{-}^{2}\Phi_{+}+\frac{1}{2\sqrt{2}}y\Phi_{+}^{2}\Phi_{-}+\text{h.c}$
(75)

from which we obtain the action in terms of the field content in eq.(3). For
more details of the superspace formalism see aitchison2007supersymmetry ;
martin2010supersymmetry ; bailin1994supersymmetric .

##  Appendix B Conventions, regulators and renormalized quantities

The matrix representation of all operators used to calculate the Wetterich
equation is given with respect to the following arrangement,

$\displaystyle\Psi=$
$\displaystyle\left\\{\phi_{-},{\phi_{-}}^{\dagger},F_{-},{F_{-}}^{\dagger},\phi_{+},{\phi_{+}}^{\dagger},F_{+},{F_{+}}^{\dagger},D,A,\psi_{-},{\bar{\psi}_{-}}^{\dagger},\psi_{+},{\bar{\psi}_{+}}^{\dagger},\lambda,{\bar{\lambda}}^{\dagger}\right\\},$
$\displaystyle{\Psi}^{\dagger}=$
$\displaystyle\left\\{{\phi_{-}}^{\dagger},\phi_{-},{F_{-}}^{\dagger},F_{-},{\phi_{+}}^{\dagger},\phi_{+},{F_{+}}^{\dagger},F_{+},D,A,\bar{\psi}_{-},{\psi_{-}}^{\dagger},\bar{\psi}_{+},{\psi_{+}}^{\dagger},\bar{\lambda},{\lambda}^{\dagger}\right\\}.$
(76)

The convention for momentum space is

$\displaystyle F(q)=\dfrac{1}{\left(\sqrt{2\pi}\right)^{d}}\displaystyle\int
d^{d}xF(x)e^{i{q}_{\mu}{x}^{\mu}}.$ (77)

Dimensionless and renormalized electric charge is defined by

$\displaystyle e=$ $\displaystyle Z_{e}k^{\frac{d-4}{2}}e_{R},$ (78)

similarly the renormalized mass and cubic coupling constant are

$\displaystyle m=$ $\displaystyle Z_{\Phi}km_{R}$ $\displaystyle y=$
$\displaystyle\sqrt{(Z_{\Phi})^{3}}k^{\frac{d-4}{2}}y_{R}.$ (79)

The form of dimensionless and renormalized regulators in eq.(11) are the
optimized Litim regulator functions litim2000optimisation

$\displaystyle R_{\textrm{opt}}(p)=$ $\displaystyle(k-p)\theta(k^{2}-p^{2})$
$\displaystyle T_{\textrm{opt}}(p^{2})=$
$\displaystyle(k^{2}-p^{2})\theta(k^{2}-p^{2})$ (80)

defining dimensionless momentum

$\displaystyle q=\frac{p}{k}$ (81)

we can define eq.(80) in terms of a dimensionless regulator functions $r$, $t$

$\displaystyle R_{\textrm{opt}}(q)=$ $\displaystyle qr^{\prime}(q),$
$\displaystyle T_{\textrm{opt}}(q^{2})=$ $\displaystyle q^{2}t^{\prime}(q)$
(82)

and finally we choose $r^{\prime}(q)$ and $t^{\prime}(q^{2})$ proportional to
$Z_{\Phi/V}$

$\displaystyle r^{\prime}(q)=Z_{\Phi}r(q),$ $\displaystyle
t^{\prime}(q)=Z_{V}t(q^{2}).$ (83)

In the calculation involving polynomial truncation of the momentum modes, the
regulating functions that have been used are

$\displaystyle r=$
$\displaystyle\zeta_{\Phi,k}(q)\left(\frac{1}{q}\frac{\zeta_{\Phi,k}(1)}{\zeta_{\Phi,k}(q)}-1\right)\theta\left(1-q^{2}\right)$
$\displaystyle t=$
$\displaystyle\zeta_{\Phi,k}(q)\left(\frac{1}{q^{2}}\frac{\zeta_{\Phi,k}(1)}{\zeta_{\Phi,k}(q)}-1\right)\theta\left(1-q^{2}\right),$
(84)

named as “Litim-Type Regulator II” in feldmann2016functional . The advantage
of these functions is that they eliminate dependencies on q from the
denominator for all orders of truncation.

##  Appendix C LPA’ Flow

In eq.(22) the flow of the LPA’ truncation was computed using the operators

$\displaystyle G_{k}=\Gamma_{k}^{(2)}+R_{k}$ (85)

and

$\displaystyle\partial_{t}R_{k}.$ (86)

The explicit forms of these operators are

$\displaystyle G_{k}=\begin{pmatrix}G_{BB}&G_{BF}\\\
G_{FB}&G_{FF}\end{pmatrix}$ (87)
$\displaystyle\partial_{t}R_{k}=\begin{pmatrix}{\partial_{t}R_{k}}_{BB}&0\\\
0&{\partial_{t}R_{k}}_{FF}\end{pmatrix}$ (88) $\displaystyle G_{BB}=$
$\displaystyle\begin{pmatrix}K_{-}&\mathcal{Y}_{-}^{{F}^{\dagger}}&0&\mathcal{Y}_{-}^{{\phi}^{\dagger}}&0&\mathcal{Y}_{+}^{{F}^{\dagger}}&0&m+\mathcal{Y}_{+}^{{\phi}^{\dagger}}&Z_{\Phi}\phi_{-}&-2Z_{\Phi}A\phi_{-}\\\
\mathcal{Y}_{-}^{F}&K_{-}&\mathcal{Y}_{-}^{\phi}&0&\mathcal{Y}_{+}^{F}&0&m+\mathcal{Y}_{+}^{\phi}&0&Z_{\Phi}{\phi_{-}}^{\dagger}&-2Z_{\Phi}A{\phi_{-}}^{\dagger}\\\
0&\mathcal{Y}_{-}^{{\phi}^{\dagger}}&\mathcal{R}&0&0&m+\mathcal{Y}_{+}^{{\phi}^{\dagger}}&0&0&0&0\\\
\mathcal{Y}_{-}^{\phi}&0&0&\mathcal{R}&m+\mathcal{Y}_{+}^{\phi}&0&0&0&0&0\\\
0&\mathcal{Y}_{+}^{{F}^{\dagger}}&0&m+\mathcal{Y}_{+}^{{\phi}^{\dagger}}&K_{+}&-\mathcal{Y}_{-}^{{F}^{\dagger}}&0&-\mathcal{Y}_{-}^{{\phi}^{\dagger}}&-Z_{\Phi}\phi_{+}&2AZ_{\Phi}\phi_{+}\\\
\mathcal{Y}_{+}^{F}&0&m+\mathcal{Y}_{+}^{\phi}&0&-\mathcal{Y}_{-}^{F}&K_{+}&-\mathcal{Y}_{-}^{\phi}&0&-Z_{\Phi}{\phi_{+}}^{\dagger}&2AZ_{\Phi}{\phi_{+}}^{\dagger}\\\
0&m+\mathcal{Y}_{+}^{{\phi}^{\dagger}}&0&0&0&-\mathcal{Y}_{-}^{{\phi}^{\dagger}}&\mathcal{R}&0&0&0\\\
m+\mathcal{Y}_{+}^{\phi}&0&0&0&-\mathcal{Y}_{-}^{\phi}&0&0&\mathcal{R}&0&0\\\
Z_{\Phi}{\phi_{-}}^{\dagger}&Z_{\Phi}\phi_{-}&0&0&-Z_{\Phi}{\phi_{+}}^{\dagger}&-Z_{\Phi}\phi_{+}&0&0&\mathcal{T}&0\\\
-2Z_{\Phi}A{\phi_{-}}^{\dagger}&-2Z_{\Phi}A\phi_{-}&0&0&2AZ_{\Phi}{\phi_{+}}^{\dagger}&2AZ_{\Phi}\phi_{+}&0&0&0&J\end{pmatrix}$
(89) $\displaystyle
G_{BF}=\begin{pmatrix}0&\mathcal{Y}_{-}^{\bar{\psi}}&0&-\mathcal{Y}_{+}^{\bar{\psi}}&\sqrt{2}Z_{\Phi}\sigma_{2}\psi_{-}^{T}&0\\\
-\mathcal{Y}_{-}^{\psi^{T}}&\sqrt{2}Z_{\Phi}\sigma_{2}\bar{\lambda}&\mathcal{Y}_{+}^{\psi^{T}}&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&\mathcal{Y}_{+}^{\bar{\psi}}&0&-\mathcal{Y}_{-}^{\bar{\psi}}&-\sqrt{2}Z_{\Phi}\sigma_{2}\psi_{+}^{T}&0\\\
\mathcal{Y}_{+}^{\psi^{T}}&0&\mathcal{Y}_{+}^{\psi^{T}}&-\sqrt{2}Z_{\Phi}\sigma_{2}\bar{\lambda}&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
Z_{\Phi}\bar{\sigma}\bar{\psi}_{-}&0&Z_{\Phi}\bar{\sigma}\bar{\psi}_{+}&0&0&0\end{pmatrix}$
(90) $\displaystyle
G_{FB}=\begin{pmatrix}0&\mathcal{Y}_{-}^{\bar{\psi}^{T}}&0&0&0&-\mathcal{Y}_{+}^{\bar{\psi}^{T}}&0&0&0&Z_{\Phi}\bar{\sigma}\psi_{-}\\\
-\mathcal{Y}_{-}^{\psi}&\sqrt{2}Z_{\Phi}\sigma_{2}\lambda&0&0&\mathcal{Y}_{+}^{\psi}&0&0&0&0&0\\\
0&-\mathcal{Y}_{+}^{\bar{\psi}^{T}}&0&0&0&-\mathcal{Y}_{-}^{\bar{\psi}^{T}}&0&0&0&-Z_{\Phi}\bar{\sigma}\psi_{+}\\\
\mathcal{Y}_{+}^{\psi}&0&0&0&\mathcal{Y}_{-}^{\psi}&-\sqrt{2}Z_{\Phi}\sigma_{2}\lambda&0&0&0&0\\\
\sqrt{2}Z_{\Phi}\sigma_{2}\bar{\psi}_{-}^{T}&0&0&0&-\sqrt{2}Z_{\Phi}\sigma_{2}\bar{\psi}_{+}^{T}&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\end{pmatrix}$ (91) $\displaystyle
G_{FF}=\begin{pmatrix}\hat{K}_{+}&\mathcal{Y}_{-}^{{\phi}^{\dagger}}&0&-\mathcal{Y}_{+}^{{\phi}^{\dagger}}&0&0\\\
-\mathcal{Y}_{-}^{\phi}&\mathcal{\hat{R}}&i\sigma_{2}m+\mathcal{Y}_{+}^{\phi}&0&\sqrt{2}Z_{\Phi}\sigma_{2}{\phi_{-}}^{\dagger}&0\\\
0&-i\sigma_{2}m-\mathcal{Y}_{+}^{{\phi}^{\dagger}}&\hat{K}_{-}&-\mathcal{Y}_{-}^{{\phi}^{\dagger}}&0&0\\\
\mathcal{Y}_{+}^{\phi}&0&\mathcal{Y}_{-}^{\phi}&\mathcal{\hat{R}}&-\sqrt{2}Z_{\Phi}\sigma_{2}{\phi_{+}}^{\dagger}&0\\\
0&\sqrt{2}Z_{\Phi}\sigma_{2}\phi_{-}&0&-\sqrt{2}Z_{\Phi}\sigma_{2}\phi_{+}&\mathcal{\hat{T}}&0\\\
0&0&0&0&0&\mathcal{\hat{T}}\end{pmatrix}$ (92)

$\displaystyle{\partial_{t}R_{k}}_{BB}=\begin{pmatrix}q^{2}\partial_{t}r&0&0&0&0&0&0&0&0&0\\\
0&q^{2}\partial_{t}r&0&0&0&0&0&0&0&0\\\ 0&0&\partial_{t}r&0&0&0&0&0&0&0\\\
0&0&0&\partial_{t}r&0&0&0&0&0&0\\\ 0&0&0&0&q^{2}\partial_{t}r&0&0&0&0&0\\\
0&0&0&0&0&q^{2}\partial_{t}r&0&0&0&0\\\ 0&0&0&0&0&0&\partial_{t}r&0&0&0\\\
0&0&0&0&0&0&0&\partial_{t}r&0&0\\\ 0&0&0&0&0&0&0&0&\frac{1}{e^{2}}t&0\\\
0&0&0&0&0&0&0&0&0&\frac{q^{2}}{e^{2}\chi}t\end{pmatrix}$ (93)
$\displaystyle{\partial_{t}R_{k}}_{FF}=\begin{pmatrix}\bar{\sigma}\cdot
q\partial_{t}r&0&0&0&0&0\\\ 0&\bar{\sigma}\cdot q\partial_{t}r&0&0&0&0\\\
0&0&\bar{\sigma}\cdot q\partial_{t}r&0&0&0\\\ 0&0&0&\bar{\sigma}\cdot
q\partial_{t}r&0&0\\\ 0&0&0&0&\frac{\bar{\sigma}\cdot
q}{e^{2}}\partial_{t}t&0\\\ 0&0&0&0&0&\frac{\bar{\sigma}\cdot
q}{e^{2}}\partial_{t}t\end{pmatrix}$ (94)

where

$\displaystyle\mathcal{R}=r+Z_{\Phi}$
$\displaystyle\mathcal{T}=\frac{1}{e^{2}}\left(t+{Z_{V}}^{2}\right)$
$\displaystyle\mathcal{\hat{R}}=\bar{\sigma}\cdot q\mathcal{R}$
$\displaystyle\mathcal{\hat{T}}=\bar{\sigma}\cdot q\mathcal{R}$ $\displaystyle
K_{\mp}=q^{2}\mathcal{R}\mp Z_{\Phi}D\pm Z_{\Phi}A^{2}$ (95) $\displaystyle
J=\frac{q^{2}}{\chi}\mathcal{T}-2Z_{\Phi}\phi_{-}{\phi_{-}}^{\dagger}+2Z_{\Phi}\phi_{+}{\phi_{+}}^{\dagger}$
$\displaystyle\hat{K}_{\pm}=\mathcal{\hat{R}}\pm Z_{\Phi}\bar{\sigma}\cdot A$
$\displaystyle\mathcal{Y}_{\pm}^{F}=\frac{y}{\sqrt{2}}(F_{+}\pm F_{-})$
$\displaystyle\mathcal{Y}_{\pm}^{\phi}=\frac{y}{\sqrt{2}}(\phi_{+}\pm\phi_{-})$
$\displaystyle\mathcal{Y}_{\pm}^{\psi}=i\frac{y}{\sqrt{2}}(\psi_{-}\pm\psi_{+}).$
(96)

Making the product between (85) and (86) and taking the trace, we are able to
perform eq.(19) and eq.(21). Using the dimensionless and renormalized
quantities of Appendix B and performing eq.(21) we obtain the following
integral in $d-$dimensions for the anomalous dimension of the chiral
superfield

$\displaystyle\gamma=$
$\displaystyle\frac{\Omega_{d}}{(2\pi)^{d}}\int^{1}_{0}dqq^{d-1}\frac{2y^{2}(r(q)+1)(m^{2}+q^{2}(r(q)+1)^{2})\partial_{t}r(q)}{(m^{2}-q^{2}(r(q)+1)^{2})^{3}}$
$\displaystyle=$
$\displaystyle\frac{\Omega_{d}}{(2\pi)^{d}}\int^{1}_{0}dqq^{d-1}\frac{2y^{2}(r(q)+1)(m^{2}+q^{2}(r(q)+1)^{2})(-\gamma-q\partial_{q})r(q)}{(m^{2}-q^{2}(r(q)+1)^{2})^{3}}$
(97)

where
$\Omega_{d}=\dfrac{2(\pi)^{\frac{d}{2}}}{\Gamma\left[\frac{d}{2}\right]}$ is
the surface of a $d-1$ dimensional sphere. Replacing eq.(11) we can solve the
integral (97)

$\displaystyle\gamma=\frac{4y^{2}(d-1)(m^{2}+1)}{4y^{2}(m^{2}+1)+2^{d}\pi^{\frac{d}{2}}(d-2)(d-1)(m^{2}-1)^{3}\Gamma\left[\frac{d}{2}\right]}.$
(98)

On the other hand, from eq.(19), we obtain the anomalous dimension of the
vector superfield

$\displaystyle-\frac{1}{2}\eta_{V}=$
$\displaystyle\frac{\Omega_{d}}{(2\pi)^{d}}\int^{1}_{0}dqq^{d-1}\frac{e_{R}^{2}\partial_{t}r(q)(r(q)+1)(m^{2}+q^{2}(r(q)+1)^{2})}{(m^{2}-q^{2}(r(q)+1)^{2})^{3}}$
$\displaystyle=$
$\displaystyle\frac{\Omega_{d}}{(2\pi)^{d}}\int^{1}_{0}dqq^{d-1}\frac{e_{R}^{2}(r(q)+1)(m^{2}+q^{2}(r(q)+1)^{2})(-\gamma-q\partial_{q})r(q)}{(m^{2}-q^{2}(r(q)+1)^{2})^{3}}$
$\displaystyle=$
$\displaystyle-\frac{2e_{R}^{2}(d-1)(m^{2}+1)}{4y^{2}(m^{2}+1)+2^{d}\pi^{\frac{d}{2}}(d-2)(d-1)(m^{2}-1)^{3}\Gamma\left[\frac{d}{2}\right]}$
(99)

and, from the eq.(13) the beta function in $d-$ dimensions

$\displaystyle\beta(e_{R})=-\frac{2e_{R}^{3}(d-1)(m^{2}+1)}{4y^{2}(m^{2}+1)+2^{d}\pi^{\frac{d}{2}}(d-2)(d-1)(m^{2}-1)^{3}\Gamma\left[\frac{d}{2}\right]}+\frac{d-4}{2}e_{R},$
(100)

which are reduced to eq.(23) and eq.(24) in $d=4$.

##  Appendix D NLO Flow

In eq.(39) we decompose the operator $G_{k}$ into $G_{0,k}$ and $\Delta
G_{k}$. The explicit forms of the operators are

$\displaystyle G_{0,k}=\begin{pmatrix}{G_{0,k}}_{BB}&0\\\
0&{G_{0,k}}_{FF}\end{pmatrix}$ (101)
$\displaystyle{G_{0,k}}_{BB}=\begin{pmatrix}q^{2}\mathcal{R}&0&0&0&0&0&0&m&0&0\\\
0&q^{2}\mathcal{R}&0&0&0&0&m&0&0&0\\\ 0&0&\mathcal{R}&0&0&m&0&0&0&0\\\
0&0&0&\mathcal{R}&m&0&0&0&0&0\\\ 0&0&0&m&q^{2}\mathcal{R}&0&0&0&0&0\\\
0&0&m&0&0&q^{2}\mathcal{R}&0&0&0&0\\\ 0&m&0&0&0&0&\mathcal{R}&0&0&0\\\
m&0&0&0&0&0&0&\mathcal{R}&0&0\\\ 0&0&0&0&0&0&0&0&\mathcal{T}&0\\\
0&0&0&0&0&0&0&0&0&\frac{1}{\chi}\mathcal{T}\end{pmatrix}$ (102)
$\displaystyle{G_{0,k}}_{FF}=\begin{pmatrix}\mathcal{\hat{R}}&0&0&0&0&0&0&0&0&0\\\
0&\mathcal{\hat{R}}&i\frac{m}{2}\sigma_{2}&0&0&0&0&0&0&0\\\
0&-i\frac{m}{2}\sigma_{2}&\mathcal{\hat{R}}&0&0&0&0&0&0&0\\\
0&0&0&\mathcal{\hat{R}}&0&0&0&0&0&0\\\ 0&0&0&0&\mathcal{\hat{R}}&0&0&0&0&0\\\
0&0&0&0&0&\mathcal{\hat{R}}&0&0&0&0\\\ 0&0&0&0&0&0&\mathcal{\hat{R}}&0&0&0\\\
0&0&0&0&0&0&0&\mathcal{\hat{R}}&0&0\\\ 0&0&0&0&0&0&0&0&\mathcal{\hat{T}}&0\\\
0&0&0&0&0&0&0&0&0&\mathcal{\hat{R}}\end{pmatrix}$ (103) $\displaystyle\Delta
G_{k}=\begin{pmatrix}{\Delta G_{k}}_{BB}&{\Delta G_{k}}_{BF}\\\ {\Delta
G_{k}}_{FB}&{\Delta G_{k}}_{FF}\end{pmatrix}$ (104) $\displaystyle{\Delta
G_{k}}_{BB}=$
$\displaystyle\begin{pmatrix}K_{-}&\mathcal{Y}_{-}^{{F}^{\dagger}}&0&\mathcal{Y}_{-}^{{\phi}^{\dagger}}&0&\mathcal{Y}_{+}^{{F}^{\dagger}}&0&\mathcal{Y}_{+}^{{\phi}^{\dagger}}&Z_{\Phi}\phi_{-}&-2Z_{\Phi}A\phi_{-}\\\
\mathcal{Y}_{-}^{F}&K_{-}&\mathcal{Y}_{-}^{\phi}&0&\mathcal{Y}_{+}^{F}&0&\mathcal{Y}_{+}^{\phi}&0&Z_{\Phi}{\phi_{-}}^{\dagger}&-2Z_{\Phi}A{\phi_{-}}^{\dagger}\\\
0&\mathcal{Y}_{-}^{{\phi}^{\dagger}}&\mathcal{R}&0&0&\mathcal{Y}_{+}^{{\phi}^{\dagger}}&0&0&0&0\\\
\mathcal{Y}_{-}^{\phi}&0&0&\mathcal{R}&\mathcal{Y}_{+}^{\phi}&0&0&0&0&0\\\
0&\mathcal{Y}_{+}^{{F}^{\dagger}}&0&\mathcal{Y}_{+}^{{\phi}^{\dagger}}&K_{+}&-\mathcal{Y}_{-}^{{F}^{\dagger}}&0&-\mathcal{Y}_{-}^{{\phi}^{\dagger}}&-Z_{\Phi}\phi_{+}&2AZ_{\Phi}\phi_{+}\\\
\mathcal{Y}_{+}^{F}&0&\mathcal{Y}_{+}^{\phi}&0&-\mathcal{Y}_{-}^{F}&K_{+}&-\mathcal{Y}_{-}^{\phi}&0&-Z_{\Phi}{\phi_{+}}^{\dagger}&2AZ_{\Phi}{\phi_{+}}^{\dagger}\\\
0&\mathcal{Y}_{+}^{{\phi}^{\dagger}}&0&0&0&-\mathcal{Y}_{-}^{{\phi}^{\dagger}}&\mathcal{R}&0&0&0\\\
\mathcal{Y}_{+}^{\phi}&0&0&0&-\mathcal{Y}_{-}^{\phi}&0&0&\mathcal{R}&0&0\\\
Z_{\Phi}{\phi_{-}}^{\dagger}&Z_{\Phi}\phi_{-}&0&0&-Z_{\Phi}{\phi_{+}}^{\dagger}&-Z_{\Phi}\phi_{+}&0&0&\mathcal{T}&0\\\
-2Z_{\Phi}A{\phi_{-}}^{\dagger}&-2Z_{\Phi}A\phi_{-}&0&0&2AZ_{\Phi}{\phi_{+}}^{\dagger}&2AZ_{\Phi}\phi_{+}&0&0&0&J\end{pmatrix}$
(105) $\displaystyle{\Delta
G_{k}}_{BF}=\begin{pmatrix}0&\mathcal{Y}_{-}^{\bar{\psi}}&0&-\mathcal{Y}_{+}^{\bar{\psi}}&\sqrt{2}Z_{\Phi}\sigma_{2}\psi_{-}^{T}&0\\\
-\mathcal{Y}_{-}^{\psi^{T}}&\sqrt{2}Z_{\Phi}\sigma_{2}\bar{\lambda}&\mathcal{Y}_{+}^{\psi^{T}}&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&\mathcal{Y}_{+}^{\bar{\psi}}&0&-\mathcal{Y}_{-}^{\bar{\psi}}&-\sqrt{2}Z_{\Phi}\sigma_{2}\psi_{+}^{T}&0\\\
\mathcal{Y}_{+}^{\psi^{T}}&0&\mathcal{Y}_{+}^{\psi^{T}}&-\sqrt{2}Z_{\Phi}\sigma_{2}\bar{\lambda}&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
Z_{\Phi}\bar{\sigma}\bar{\psi}_{-}&0&Z_{\Phi}\bar{\sigma}\bar{\psi}_{+}&0&0&0\end{pmatrix}$
(106) $\displaystyle{\Delta
G_{k}}_{FB}=\begin{pmatrix}0&\mathcal{Y}_{-}^{\bar{\psi}^{T}}&0&0&0&-\mathcal{Y}_{+}^{\bar{\psi}^{T}}&0&0&0&Z_{\Phi}\bar{\sigma}\psi_{-}\\\
-\mathcal{Y}_{-}^{\psi}&\sqrt{2}Z_{\Phi}\sigma_{2}\lambda&0&0&\mathcal{Y}_{+}^{\psi}&0&0&0&0&0\\\
0&-\mathcal{Y}_{+}^{\bar{\psi}^{T}}&0&0&0&-\mathcal{Y}_{-}^{\bar{\psi}^{T}}&0&0&0&-Z_{\Phi}\bar{\sigma}\psi_{+}\\\
\mathcal{Y}_{+}^{\psi}&0&0&0&\mathcal{Y}_{-}^{\psi}&-\sqrt{2}Z_{\Phi}\sigma_{2}\lambda&0&0&0&0\\\
\sqrt{2}Z_{\Phi}\sigma_{2}\bar{\psi}_{-}^{T}&0&0&0&-\sqrt{2}Z_{\Phi}\sigma_{2}\bar{\psi}_{+}^{T}&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\end{pmatrix}$ (107) $\displaystyle{\Delta
G_{k}}_{FF}=\begin{pmatrix}\hat{K}_{+}&\mathcal{Y}_{-}^{{\phi}^{\dagger}}&0&-\mathcal{Y}_{+}^{{\phi}^{\dagger}}&0&0\\\
-\mathcal{Y}_{-}^{\phi}&\mathcal{\hat{R}}&\mathcal{Y}_{+}^{\phi}&0&\sqrt{2}Z_{\Phi}\sigma_{2}{\phi_{-}}^{\dagger}&0\\\
0&-\mathcal{Y}_{+}^{{\phi}^{\dagger}}&\hat{K}_{-}&-\mathcal{Y}_{-}^{{\phi}^{\dagger}}&0&0\\\
\mathcal{Y}_{+}^{\phi}&0&\mathcal{Y}_{-}^{\phi}&\mathcal{\hat{R}}&-\sqrt{2}Z_{\Phi}\sigma_{2}{\phi_{+}}^{\dagger}&0\\\
0&\sqrt{2}Z_{\Phi}\sigma_{2}\phi_{-}&0&-\sqrt{2}Z_{\Phi}\sigma_{2}\phi_{+}&\mathcal{\hat{T}}&0\\\
0&0&0&0&0&\mathcal{\hat{T}}\end{pmatrix}$ (108)
$\displaystyle\partial_{t}R_{k}=\begin{pmatrix}{\partial_{t}R_{k}}_{BB}&0\\\
0&{\partial_{t}R_{k}}_{FF}\end{pmatrix}$ (109)
$\displaystyle{\partial_{t}R_{k}}_{BB}=\begin{pmatrix}q^{2}\partial_{t}r&0&0&0&0&0&0&0&0&0\\\
0&q^{2}\partial_{t}r&0&0&0&0&0&0&0&0\\\ 0&0&\partial_{t}r&0&0&0&0&0&0&0\\\
0&0&0&\partial_{t}r&0&0&0&0&0&0\\\ 0&0&0&0&q^{2}\partial_{t}r&0&0&0&0&0\\\
0&0&0&0&0&q^{2}\partial_{t}r&0&0&0&0\\\ 0&0&0&0&0&0&\partial_{t}r&0&0&0\\\
0&0&0&0&0&0&0&\partial_{t}r&0&0\\\ 0&0&0&0&0&0&0&0&\frac{1}{e^{2}}t&0\\\
0&0&0&0&0&0&0&0&0&\frac{q^{2}}{e^{2}\chi}t\end{pmatrix}$ (110)
$\displaystyle{\partial_{t}R_{k}}_{FF}=\begin{pmatrix}\bar{\sigma}\cdot
q\partial_{t}r&0&0&0&0&0\\\ 0&\bar{\sigma}\cdot q\partial_{t}r&0&0&0&0\\\
0&0&\bar{\sigma}\cdot q\partial_{t}r&0&0&0\\\ 0&0&0&\bar{\sigma}\cdot
q\partial_{t}r&0&0\\\ 0&0&0&0&\frac{\bar{\sigma}\cdot
q}{e^{2}}\partial_{t}t&0\\\ 0&0&0&0&0&\frac{\bar{\sigma}\cdot
q}{e^{2}}\partial_{t}t\end{pmatrix}$ (111)

where

$\displaystyle\mathcal{R}=r+Z_{\Phi,k}$
$\displaystyle\mathcal{T}=\frac{1}{e^{2}}\left(t+{Z_{V,k}}^{2}\right)$
$\displaystyle\mathcal{\hat{R}}=\bar{\sigma}\cdot q\mathcal{R}$
$\displaystyle\mathcal{\hat{T}}=\bar{\sigma}\cdot q\mathcal{R}$ $\displaystyle
K_{\mp}=\mp Z_{\Phi,k}D\pm Z_{\Phi,k}A^{2}$ (112) $\displaystyle
J=-2Z_{\Phi,k}\phi_{-}{\phi_{-}}^{\dagger}+2Z_{\Phi,k}\phi_{+}{\phi_{+}}^{\dagger}$
$\displaystyle\hat{K}_{\pm}=\pm Z_{\Phi,k}\bar{\sigma}\cdot A$
$\displaystyle\mathcal{Y}_{\pm}^{F}=\frac{y}{\sqrt{2}}(F_{+}\pm F_{-})$
$\displaystyle\mathcal{Y}_{\pm}^{\phi}=\frac{y}{\sqrt{2}}(\phi_{+}\pm\phi_{-})$
$\displaystyle\mathcal{Y}_{\pm}^{\psi}=i\frac{y}{\sqrt{2}}(\psi_{-}\pm\psi_{+}).$
(113)

We need the quadratic term in the field-dependent part of eq.(40). Defining

$\displaystyle W(q,q^{\prime})={G_{0,k}}^{-1}(q)\Delta G_{k}(q+q^{\prime})$
(114)

we can see that the quadratic terms have the form

$\displaystyle W^{2}(q,q^{\prime})=\int
d^{4}q^{\prime\prime}W(q,q^{\prime\prime})W(q^{\prime\prime},q^{\prime})=\int
d^{4}q^{\prime\prime}{G_{0,k}}^{-1}(q)\Delta
G_{k}(q^{\prime\prime}+q){G_{0,k}}^{-1}(q^{\prime\prime})\Delta
G_{k}(q^{\prime\prime}+q^{\prime}),$ (115)

whose trace corresponds to integrating in $q$ by doing $q^{\prime}\to q$. Thus

$\displaystyle-\frac{1}{4}\text{STr}\tilde{\partial}_{t}$
$\displaystyle\left({G_{0,k}}^{-1}\Delta G_{k}\right)^{2}=$
$\displaystyle-\frac{1}{4}\int
d^{4}qd^{4}q^{\prime}{G_{0,k}}^{-1}(q)\partial_{t}R_{k}(q){G_{0,k}(q)}^{-1}\Delta
G_{k}(q^{\prime}+q){G_{0,k}(q^{\prime})}^{-1}\Delta G_{k}(q^{\prime}+q)$
$\displaystyle+\frac{1}{4}\int d^{4}qd^{4}q^{\prime}{G_{0,k}(q)}^{-1}\Delta
G_{k}(q^{\prime}+q){G_{0,k}}^{-1}(q^{\prime})\partial_{t}R_{k}(q^{\prime}){G_{0,k}(q^{\prime})}^{-1}\Delta
G_{k}(q^{\prime}+q).$ (116)

By performing the matrix products, taking the matrix trace, and computing the
$q^{\prime}-$integral, we obtain the expressions in eq.(41) after performing
the corresponding functional derivatives.

###### Acknowledgements.

We would like to thank D. Salinas-Arizmendi for his valuable advice on some
calculations. This research was partially supported by ANID PIA/APOYO
AFB180002 and AFB220004 (Chile), by FONDECYT (Chile) under Grant No. 1230391.
J.E thanks to “Beca doctorado nacional”, by ANID (Chile) No. 21212381 and
“Programa de Incentivo a la Iniciación Científica” (PIIC), by UTFSM.

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