# Dimensional Regularization in Quantum Field Theory with Ultraviolet Cutoff

Durmuş Demir1, Canan Karahan2, Ozan Sargın1 1 Sabancı University, Faculty of
Engineering and Natural Sciences, 34956 Tuzla, İstanbul, Turkey
2 National Defence University, Turkish Naval Academy, Department of Basic
Sciences, 34942 Tuzla, İstanbul, Turkey

###### Abstract

In view of various field-theoretic reasons, in the present work, we study the
question of if the usual dimensional regularization can be extended to quantum
field theories with an ultraviolet cutoff (Poincare-breaking scale) in a way
preserving all the properties of the dimensional regularization. And we find
that it can indeed be. The resulting extension gives a framework in which the
power-law and logarithmic divergences get detached to involve different
scales. This new regularization scheme, the detached regularization as we call
it, enables one to treat the power-law and logarithmic divergences differently
and independently. We apply the detached regularization to the computation of
the vacuum energy and to two well-known QFTs namely the scalar and spinor
electrodynamics. As a case study, we consider Fujikawa’s subtractive
renormalization in the framework of the detached regularization, and show its
effectiveness up to two loops by specializing to scalar self energy. We
discuss various application areas of the detached regularization.

##  I Introduction

Quantum field theories (QFTs) develop divergences at the loop level. The
ultraviolet (UV) divergences refer to infinities arising at large loop
momenta. The methodology to deal with the divergences – renormalization –
requires the divergent and finite terms to be separated appropriately –
regularization. Various regularization methods have been proposed since the
early days of the QFTs [1]. The simplest and physically most intuitive
regularization scheme is the cutoff regularization in which QFTs are ascribed
a Poincare-breaking UV edge. In this scheme, QFTs are endowed with a hard
momentum cutoff so that loop integrals run up to the cutoff not to the
infinity [2]. The effective QFT at the end can have quartic, quadratic and
logarithmic dependencies on the UV cutoff [3]. The most conspicuous drawback
of this scheme is that it breaks explicitly all the gauge symmetries in the
QFT because each gauge boson acquires a mass proportional to the UV cutoff [4,
5, 6, 7]. (For clarity, one can consider the symmetric phase of the QFT in
which none of the gauge symmetries is broken spontaneously so that the entire
gauge breaking comes from the UV cutoff.)

The cutoff regularization has a Poincare-conserving alternative: the Pauli-
Villars regularization [8]. In this scheme, one introduces a soft momentum
cutoff $M$ (which is the mass of an auxiliary field) instead of the hard
momentum cutoff $\Lambda$ (which is not a particle mass though it can be
numerically equal to a particle mass). Practically, in Pauli-Villars
regularization one makes the replacement
$(p^{2}-m^{2})^{-1}\rightarrow(p^{2}-m^{2})^{-1}-(p^{2}-M^{2})^{-1}$ in the
loop integration so that the propagator decreases faster as the loop momenta
$p$ tends to infinity. The auxiliary particle provides a Poincare-conserving
cutoff (with opposite statistics with respect to the actual particle of mass
$m$). The Pauli-Villars regularization has an advantage over the cutoff
regularization in that it leaves the theory gauge invariant at each order of
perturbation theory thanks to the precise prescription of the use of the
regulators [6]. The gauge invariance here is restricted to the Abelian case,
which is what we are interested in the present work. The difficulty of finding
a physical interpretation for the auxiliary fields is the main drawback of the
Pauli-Villars regularization.

The cutoff-based regularizations above have two alternatives: Analytic
regularization (changing the power of the loop propagator) and dimensional
regularization (changing the dimension of the momentum space). The analytic
regularization is based on the concept of analytic continuation [9, 10, 11].
The parameter which is exploited for this purpose is the one that is obtained
by carrying the power of the denominator of the propagator to the complex
plane. For instance, suppose we have the propagator for a particle of mass $m$
and four-momentum $p_{\alpha}$ which contains the factor
$(p^{2}-m^{2}+i0)^{-1}$, what analytic regularization does is that it replaces
the propagator with $(p^{2}-m^{2}+i0)^{-n}$, where the regulating parameter
$n$ is a complex number in general. The crucial point is that $\mathrm{Re}(n)$
must be large enough so that the integrals are made to converge. After the
result is obtained, which normally depends on the parameter $n$, it is
analytically continued to the domain which contains the integral value of
$n=1$. The divergences do all manifest themselves as simple poles in the limit
$n\to 1$. This lets one to renormalize the QFT simply by subtracting the
divergent pole terms. The finite parts which remain constitute the desired
renormalized QFT. The analytic regularization respects the Poincare symmetry,
however, as it is the case in the cut-off regularization, breakdown of the
gauge symmetries is unavoidable.

The dimensional regularization is the most popular regularization method due
to its ability to preserve the local symmetries of the QFT such as the gauge
symmetries [12, 13]. It has been successful all along in making precise
predictions (higher loop calculations), which have been tested at the LEP, LHC
and other colliders. The idea behind this method is to change the integration
measure of the loop integrals by carrying the dimensionality $D$ of the
measure to the complex domain. Let us illustrate this with a simple example
and see how it helps. Consider a loop integral of the form

$I=\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{(p^{2}+i0)[(p-q)^{2}-m^{2}+i0]}\,$
(1)

which is a four dimensional integral over the loop momentum $p$. In this form
and in the limit of large loop momenta, namely $p\to\pm\infty$, this integral
diverges. However, if we consider the same integral evaluated in three space-
time dimensions

$I=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{(p^{2}+i0)[(p-q)^{2}-m^{2}+i0]}\,$
(2)

we see that, in the limit $p\to\pm\infty$, it readily converges to a finite
value. What this example teaches us is that unbounded loop integrals over four
dimensional spacetime can be rendered convergent if it is possible to reduce
the dimensionality of the momentum space [12, 13, 14, 6]. The dimensional
regularization involves only logarithmic divergences since at the end of
calculations one takes $D\rightarrow 4$. In essence, dimensional
regularization isolates divergences via the poles of $1/(D-4)$ such that their
subtraction via MS or $\overline{MS}$ schemes leads to renormalization of the
underlying QFT. Independence of the bare parameters from $\mu$ gives rise to
renormalization group equations and their solutions determine how various
interactions vary with $\mu$ (the scale of the experiments). This can be done
up to any loop order. For instance, the fine structure constant varies with
$\mu$ such that its value at $\mu=M_{Z}$ agrees with the LEP measurements. The
same thing happens with the LHC measurements and, is expected to be the same
also at the FCC and ILC if there is no new physics at their scales.

Having briefly discussed the four main regularization schemes in the
literature, let us now go back to the cutoff regularization. Let us consider a
QFT with a Poincare-breaking UV cutoff $\Lambda$ [2]. The matter loops lead to
three types of UV-sensitivites:

  1. 1.

Quartic sensitivities going like $\Lambda^{4}$,

  2. 2.

Quadratic sensitivities going like $\Lambda^{2}$,

  3. 3.

Logarithmic sensitivities involving $\log\Lambda$.

These UV sensitivities can be exemplified by a typical loop integral

$\displaystyle
I_{n}=\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{(p^{2}-m^{2}+i0)^{n}}$ (3)

which is seen to diverge at large loop momenta for $n\leq 2$. It can be
rendered finite (regularized) by cutting off all loop momenta above $\Lambda$
so that one gets the $\Lambda$-dependent finite results

$\displaystyle
I_{n}(\Lambda^{2})=\left\\{\begin{array}[]{ll}\frac{i}{32\pi^{2}}\Lambda^{4}&\quad
n=0\\\ \\\
\frac{-i}{16\pi^{2}}\left(\Lambda^{2}-m^{2}\log\frac{\Lambda^{2}+m^{2}}{m^{2}}\right)&\quad
n=1\\\ \\\
\frac{i}{16\pi^{2}}\left(-1+\log\frac{\Lambda^{2}+m^{2}}{m^{2}}\right)&\quad
n=2\end{array}\right.$ (9)

corresponding, respectively, to the three types of the UV-sensitivities listed
above. (As we deal with a renormalizable QFT, we do not consider non-
renormalizable corrections of the form $1/\Lambda^{n}$ ($n>2$) or we assume
that such corrections are absorbed in redefinitions of the QFT parameters.)
The power-law (quadratic and quartic) UV-sensitivities are local in that they
live at the scale $\Lambda$. The logarithmic UV-sensitivities, on the other
hand, extend in the entire range since the correction
$M^{2}\log\Lambda^{2}/M^{2}$ varies with the particle mass $M$. Besides, the
logarithmic UV-sensitivities set the beta functions and conformal anomalies in
the QFT while the power-law divergences do not play such roles. In view of
these differences, certain QFT studies have attempted to treat the power-law
and logarithmic divergences differently and independently. One example is
subtraction of the power-law divergences as in [20, 21, 22]. Another example
is induced gravity models in which $\Lambda^{2}$ leads to Newton’s constant
[23, 24]. Yet another example is emergent gravity models in which
$\Lambda^{2}$ is promoted to curvature (in reminiscence to promotion of vector
boson masses to Higgs field) [25, 26, 27]. In all such attempts, the hampering
problem is that both the logarithmic and power-law divergences involve one and
the same scale – the UV cutoff $\Lambda$. They are thus not eligible for
treating differently and independently. In view of the examples above,
however, we ask the crucial question: Is it possible to find a new
regularization scheme in which the power-law and logarithmic divergences are
detached to involve independent scales in place of a single scale like
$\Lambda$? This question brings up a whole new approach to the idea of
regularization. Indeed, if it can be answered positively then it will be
possible to analyze power-law and logarithmic divergences with independent
scales. In an attempt to find an answer, one comes to realize that an
efficient method is to start from a known regularization scheme and deform it
judiciously to arrive at the “detached regularization” implied in the
question. In this regard, we repose the question above in a more specialized
form

$\displaystyle How\ to\ extend\ the\ dimensional\ regularization\ scheme\ to\
QFTs\ with\ an\ \ UV\ cutoff$ (10) $\displaystyle such\ that\ all\ the\
features\ of\ the\ dimensional\ regularization\ are\ preserved?$

In this form, we construct the detached regularization as an extension or
deformation of the dimensional regularization. This new question is actually
highly nontrivial because dimensional regularization scheme is specific to
QFTs without an UV scale [12, 13, 14, 6]. Indeed, the question of how to
inject a cutoff scale in a dimensionally-regularized amplitude has no clear
answer. To this end, following the nascent ideas in [28, 29], in Sec. II
below, we study the question (10) by extending the dimensional regularization
to $D=0$ and $D=2$ momentum space dimensions [15, 16, 17, 18, 30] at a scale
involving not only the renormalization scale $\mu$ but also the UV cutoff
$\Lambda_{\wp}$ of the QFT. By a judicious structuring, the end result will
have $\Lambda_{\wp}$ setting the power-law UV-sensitivities (replacing
$\Lambda$ in $\Lambda^{2}$ and $\Lambda^{4}$ terms of cutoff regularization)
and $\mu$ parametrizing the logarithmic UV-sensitivities (replacing $\Lambda$
in $\log\Lambda$ terms in cutoff regularization). This two-scale
regularization scheme will lead to the sought-for detached regularization if
the poles at $D=0$, $D=2$ and $D=4$ are all included. Resting on the
dimensional regularization, the detached regularization generates no
evanescent contributions different from the ones expected in dimensional
regularization at higher loops.

Having established the detached regularization in Sec. II, we apply it to
computation of the vacuum energy in a general QFT in Sec. III.

In Sec. IV, we give applications of the detached regularization to two well-
known QFTs: the scalar electrodynamics in Sec. IV.1 and the spinor
electrodynamics in Sec. IV.2. Their calculational details are given in Sec.
VII (Appendices A and B).

In Sec. V, we illustrate how subtractive renormalization can be realized in
the framework of detached regularization. We in particular show that what
remains after the subtraction of power-law corrections is just the
dimensionally-regularized ${\overline{MS}}$-renormalized QFT.

In Sec. VI we conclude.

##  II Detached Regularization

In this section our goal is to answer the question (10). We want to construct
a regularization scheme in which these two properties hold:

  1. (a)

Power-law and logarithmic divergences are both contained in the
regularization, and

  2. (b)

Power-law and logarithmic divergences involve independent scales.

To achieve the property (a) we adopt the dimensional regularization scheme
[12, 13, 14] and exploit the fact that dimensional regularization starts
involving quartic and quadratic divergences when the momentum space dimension
is set to $D=0$ and $D=2$, respectively [16, 17, 18, 30]. This dimensional
change is a highly useful property but it is far from sufficient for achieving
the property (b). To that end, we introduce a generalization of the
dimensional regularization by introducing a new scale $\Lambda_{\wp}$ besides
the usual renormalization scale $\mu$. In explicit terms, we consider an
extension of the form

$\displaystyle\mu^{4-D}\int\frac{d^{D}p}{(2\pi)^{D}}\frac{1}{(p^{2}-m^{2}+i0)^{n}}\longrightarrow
f(\Lambda_{\wp},\mu,D)\int\frac{d^{D}p}{(2\pi)^{D}}\frac{1}{(p^{2}-m^{2}+i0)^{n}}$
(11)

in which the new function $f(\Lambda_{\wp},\mu,D)$ is to be structured
judiciously. To this end, we impose the following conditions:

  1. (i)

It should suffice to take $f(\Lambda_{\wp},\mu,D)$ as a polynomial of the form
$f(\Lambda_{\wp},\mu,D)=\Lambda_{\wp}^{a}\mu^{b}$, where $a$ and $b$ are
functions of $n$ and $D$.

  2. (ii)

It should be possible to split $f(\Lambda_{\wp},\mu,D)$ as
$f_{0,2}(\Lambda_{\wp},\mu,D)+f_{4}(\Lambda_{\wp},\mu,D)$ such that
$f_{0,2}(\Lambda_{\wp},\mu,D)\rightarrow 0$ as $D\rightarrow 4$, and
$f_{4}(\Lambda_{\wp},\mu,D)\rightarrow 0$ as $D\rightarrow 0,2$.

  3. (iii)

The function $f_{4}(\Lambda_{\wp},\mu,D)$ should give the usual dimensional
regularization amplitude in the left-hand side of (11).

  4. (iv)

The function $f_{0,2}(\Lambda_{\wp},\mu,D)$ should lead to the same powers of
$\Lambda_{\wp}$ compared to the cutoff regularization results in (9).

These requirements and limit values put the regularization function
$f(\Lambda_{\wp},\mu,D)$ in this compact form

$\displaystyle
f(\Lambda_{\wp},\mu,D)=\frac{1}{(8\pi)^{2-n}}\left(\delta_{[D]0}+\delta_{[D]2}\right)\Lambda_{\wp}^{4-2n}\mu^{2n-D}+\delta_{[D]4}\;\mu^{4-D}$
(12)

in which $[D]$ designates the integer part of $D$ so that
$[0-\epsilon]=0,[2-\epsilon]=2$ and $[4-\epsilon]=4$ for an infinitesimal
$\epsilon$. Needless to say, $\delta_{ij}$ is Kronecker delta, which is equal
to 1 (0) if $i=j$ ($i\neq j$). The normalization factor $1/(8\pi)^{2-n}$ is
attached to make coefficients of the $\Lambda_{\wp}^{4}$ and
$\Lambda_{\wp}^{2}$ to remain parallel, respectively, to those of the
$\Lambda^{4}$ and $\Lambda^{2}$ in the cutoff regularization integrals in (9).
In fact, after using $f(\Lambda_{\wp},\mu,D)$, the original loop integral in
(3) takes the form

$\displaystyle I_{n,D}(\Lambda_{\wp},\mu)$ $\displaystyle=$
$\displaystyle\Bigg{[}\frac{1}{(8\pi)^{2-n}}\left(\delta_{[D]0}+\delta_{[D]2}\right)\Lambda_{\wp}^{4-2n}\mu^{2n-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\int\frac{d^{D}p}{(2\pi)^{D}}\frac{1}{(p^{2}-m^{2}+i0)^{n}}$
(13) $\displaystyle=$
$\displaystyle\frac{i(-1)^{n}}{(4\pi)^{D/2}}\frac{1}{(8\pi)^{2-n}}\frac{\Gamma(n-D/2)}{\Gamma(n)}\left(\delta_{[D]0}+\delta_{[D]2}\right)\Lambda_{\wp}^{4-2n}\left(\frac{\mu}{m}\right)^{2n-D}$
(14) $\displaystyle+$
$\displaystyle\frac{i(-1)^{n}}{(4\pi)^{D/2}}\frac{\Gamma(n-D/2)}{\Gamma(n)}\delta_{[D]4}\;\mu^{4-2n}\left(\frac{\mu}{m}\right)^{2n-D}$
(15)

which is seen to be power-law in $\Lambda_{\wp}$ and yet logarithmic in $\mu$.
It is so because power of $\Lambda_{\wp}$ is independent of $D$ but that of
$\mu$ depends on $D$ and gives rise to $\log(\mu/m)$ terms when the gamma
functions are expanded about momentum space dimensions $D=0,2,4$. (In view of
the earlier literature [16, 17, 18] on power-law divergences in the
dimensional regularization scheme, we do not consider odd dimensions $D=1,3$.)
It is clear that the sought-for detachment is achieved: While $\Lambda_{\wp}$
arises only in power-law terms $\mu$ appears only in logarithmic terms, and
hence, the power-law and logarithmic UV-sensitivities get completely detached.
This is the sought-for detached regularization. The detachment can be
explicitly seen by evaluating $I_{n,D}(\Lambda_{\wp},\mu)$ for the relevant
values of $D$ and $n\leq D/2$

$\displaystyle
I_{n,D}(\Lambda_{\wp},\mu)=\left\\{\begin{array}[]{ll}\frac{i}{32\pi^{2}}\Lambda_{\wp}^{4}&\quad
n=0,D=0\\\ \\\
-\frac{i}{32\pi^{2}}\Lambda_{\wp}^{2}\log\frac{\mu^{2}}{m^{2}}&\quad
n=1,D=2\\\ \\\
\frac{im^{2}}{16\pi^{2}}\left(1+\log\frac{\mu^{2}}{m^{2}}\right)&\quad
n=1,D=4\\\ \\\ \frac{i}{16\pi^{2}}\log\frac{\mu^{2}}{m^{2}}&\quad
n=2,D=4\end{array}\right.$ (23)

after employing the $\overline{MS}$ subtraction scheme [6, 14]. These
individual loop integrals shed enough light on the roles of the scales
$\Lambda_{\wp}$ and $\mu$. The role of $\mu$ is as usual in that the QFT under
consideration runs from scale to scale via the renormalization group equations
in terms of $\mu$ [6, 14]. The role of $\Lambda_{\wp}$, on the other hand, is
also as usual in that it acts as the UV cutoff, as can be seen by comparing
(23) with the cutoff regularization integrals in (9). This is seen also from
the fact that $\Lambda_{\wp}$ terms arise only in $D=0$ and $D=2$ limits,
which correspond, respectively, to the quartic and quadratic UV divergences
[16, 17, 18]. The $\Lambda_{\wp}^{4}$ term from $D=0$ integral and
$\Lambda_{\wp}^{2}$ term from the $D=2$ integral both vanish identically in
dimensional regularization. Incorporation of these terms by the new
regularization method in (13) enables us to take into account the power-law
and logarithmic divergences all at once in a way detached from each other. It
is clear that the $\Lambda^{4-2n}_{\wp}$ factor in (13) is much more than a
simple multiplicative factor in that it reveals the UV sensitivity of the QFT
as a function of the propagator order $n$. The detachment of the power-law and
logarithmic divergences, which was attempted also by the loop regularization
method [32, 33, 34] and by other methods based on implicit regularization
[35], enables us to analyze the two types of divergences separately and
independently.

It is clear that for a proper analysis of the UV behavior of the QFT it is
necessary to include each and every pole in (23) [3, 31, 30]. Thus, we gather
residues of the poles at $D=0,2,4$ to construct the actual loop amplitude in
(3)

$\displaystyle I_{n}\xrightarrow{\rm detached\
regularization}I_{n}(\Lambda_{\wp},\mu)=\sum_{\begin{subarray}{c}D=0,2,4\\\
(n\leq D/2)\end{subarray}}I_{n,D}(\Lambda_{\wp},\mu)\ \equiv\ {\rm the\
answer\ to\ the\ question\ in\ eq.\,(\ref{question-new})}$ (24)

where one keeps in mind that in actual calculations the mass parameter $m^{2}$
is a combination of the masses, external momenta, and appropriate Feynman
parameters [6, 14].

Having revealed its effects by the loop integrals in (23), it is timely to
examine the question of if the detached regularization is unique or not. It
actually is unique. In exact terms, if $\Lambda_{\wp}$ is to remain polynomial
(no logarithm of $\Lambda_{\wp}$) and $\mu$ logarithmic (no polynomial in
$\mu$) then the detached regularization is unique to the extent the
dimensional regularization itelf is unique. It is so because the power-law
divergences in the detached regularization result from the fact that the
dimensional regularization generate power-law terms in dimensions $D=0$ and
$2$ [16, 17]. In this sense, what is novel in detached regularization is the
existence of two separate scales $\Lambda_{\wp}$ and $\mu$, with distinct
roles. To sum up, according to all four requirements below equation (11), the
detached regularization prescription in (12) stands out as a unique
prescription.

The cutoff regularization results are given in (9) above (in $D=4$ momentum
space). It is seen that cutoff regularization method leads to power-law
divergences $\Lambda^{4}$ and $\Lambda^{2}$ as well as the logarithmic
divergences $\log(\Lambda^{2}/M^{2})$. Both divergences involve one and the
same scale $\Lambda$. Now, let us compare these divergences with the detached
regularization results in (23). The power-law divergences $\Lambda_{\wp}^{4}$
and $\Lambda_{\wp}^{2}$ still arise. They are parallel to those in the cutoff
regularization results (excepting the appearance of $\log\mu$ in front of
$\Lambda_{\wp}^{2}$). The main difference is that logarithmic divergences
involve the subtraction scale $\mu$ not the hard momentum cutoff
$\Lambda_{\wp}$ (corresponding to the cutoff $\Lambda$ in equation (9)). In
the last two lines of (23) there is no involvement of $\Lambda_{\wp}$ simply
because these integrals are for $D=4$. (This is consistent with the structure
of dimensional regularization in which power-law divergences are probed by
going to $D=0$ and $D=2$ dimensions, as discussed in Refs. [16] and [17]).

To conclude, the detached regularization prescription in equation (24)
constitutes an affirmative answer to the question in (10). In our derivations
we have focused on a typical loop integral like (3) but the detached
regularization is general enough to apply all loop amplitudes. In fact, we
will illustrate this generality when we apply the detached regularization to
the vacuum energy in Sec. III, the scalar electrodynamics in Sec. IV.1, and
spinor electrodynamics in Sec. IV.2. These two QFTs will suffice for
illustrating all the important aspects of the detached regularization method.
(We will defer the realistic case of the standard model of elementary
particles to future work since it involves multi-faceted calculations and
analyses in the gauge, Higgs and the fermion sectors.)

##  III One-Loop Corrections to Vacuum Energy

In general, one-loop corrections can be cast as a change $\delta S$ in the QFT
action $S$ [14, 6]. In fact, $\delta S$ is a sum over the individual
corrections

$\displaystyle\delta S_{\psi}=\frac{i}{2}(-1)^{s_{\psi}}\int
d^{4}x\int\frac{d^{4}p}{(2\pi)^{4}}\log\frac{\left(p^{2}-m_{\psi}^{2}+T_{int}(\psi,\psi_{other})\right)}{M_{0}^{2}}$
(25)

for each field $\psi$ of spin $s_{\psi}$ and mass $m_{\psi}$. In this formula,
$M_{0}$ is a mass scale and $T_{int}(\psi,\psi_{other})$ collects couplings of
$\psi$ to the self and other fields $\psi_{other}$ in the QFT. In the
perturbative regime, the $\psi$ action above can be expanded as

$\displaystyle\delta S_{\psi}=\frac{i}{2}(-1)^{s_{\psi}}\int
d^{4}x\int\frac{d^{4}p}{(2\pi)^{4}}\left[\underbrace{\log\frac{(p^{2}-m_{\psi}^{2})}{M_{0}^{2}}}_{\rm
non-
diagrammatic}+\underbrace{\frac{T_{int}(\psi,\psi_{other})}{p^{2}-m_{\psi}^{2}}-\frac{1}{2}\left(\frac{T_{int}(\psi,\psi_{other})}{p^{2}-m_{\psi}^{2}}\right)^{2}+\dots}_{\rm
diagrammatic\ (see\ Sec.\ \ref{sec:apps})}\right]$ (26)

which splits into a non-diagrammatic part plus a series expansion that can be
graphed via Feynman diagrams. The latter will be studied in detail in Sec. IV
by considering the illustrative cases of the scalar and spinor
electrodynamics. The former (non-diagrammatic part) contributes to the vacuum
energy, and in the large loop momentum regime it can be recast as

$\displaystyle\int\frac{d^{4}{\tilde{p}}}{(2\pi)^{4}}-\int\frac{d^{4}p}{(2\pi)^{4}}\frac{m_{\psi}^{2}}{p^{2}-\mu_{IR}^{2}}-\frac{1}{2}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{m_{\psi}^{4}}{(p^{2}-\mu_{IR}^{2})^{2}}$
(27)

after defining $d^{4}{\tilde{p}}\equiv d^{4}p\log\frac{p^{2}}{M_{0}^{2}}$,
introducing an infrared regulator $\mu_{IR}$ which we identify with $m_{\psi}$
($\mu_{IR}=m_{\psi}$), and discarding the terms finite in the UV. Now, using
the detached regularization integrals in (23) we get from (27) the following
$\psi$-contribution to vacuum action

$\displaystyle(\delta S_{\psi})^{\rm(vac)}=\int
d^{4}x\left\\{-\frac{(-1)^{s_{\psi}}}{64\pi^{2}}\Lambda_{\wp}^{4}+\frac{(-1)^{s_{\psi}}}{64\pi^{2}}\Lambda_{\wp}^{2}m_{\psi}^{2}\log\frac{m_{\psi}^{2}}{\mu^{2}}+\frac{(-1)^{s_{\psi}}}{32\pi^{2}}m_{\psi}^{4}\left(1-\frac{3}{2}\log\frac{m_{\psi}^{2}}{\mu^{2}}\right)\right\\}$
(28)

such that inclusion of the contributions of all the QFT fields leads to the
total vacuum action

$\displaystyle(\delta S)^{\rm(vac)}=\int
d^{4}x\left\\{-\frac{(n_{b}-n_{f})}{64\pi^{2}}\Lambda_{\wp}^{4}+\frac{1}{64\pi^{2}}\Lambda_{\wp}^{2}{\rm
str}\left[M^{2}\log\frac{M^{2}}{\mu^{2}}\right]+\frac{1}{32\pi^{2}}{\rm
str}\left[M^{4}\left(1-\frac{3}{2}\log\frac{M^{2}}{\mu^{2}}\right)\right]\right\\}$
(29)

in which $n_{b}(n_{f})$ is the total number of bosons (fermions) in the QFT,
$M^{2}$ is the mass-squared matrix of fields, and ${\rm
str}[\dots]=\sum_{s}(-1)^{s}{\rm tr}[\dots]$ is the supertrace over spins.
This vacuum action will be added to the results of the diagrammatic
calculations in Sec. IV below (power-law in $\Lambda_{\wp}$ and logarithmic in
$\mu$).

In the symmetric phase of the QFT where all gauge symmetries are exact (to be
spontaneously broken after the loop corrections are included), it turns out
that the mass matrix $M$ in (29) can pertain only to the scalar fields,
singlet fermions and vector-like fermions. It is worth noting that while the
quartic term involves all the fields in the QFT via $n_{b}-n_{f}$, the
quadratic term (in the symmetric phase of the QFT) involves only the scalars
and singlet/vector-like fermions. In the SM, only the Higgs field contributes
(as four massive physical scalar fields in the symmetric phase). Inclusion of
the neutrino masses brings in the right-handed neutrinos as the singlet-
fermion sector. Inclusions of dark matter, inflaton, axion and others bring in
scalars or singlet/vector-like fermions so that the quadratic correction in
(29) proves to be a sensitive probe of the new particles beyond the SM. These
new particles do not have to couple to the SM particles unless required by
empirical facts (like inflaton decay and neutrino Majorana masses) or by
symmetry reasons (like gauge symmetry and broken supersymmetry).

##  IV Applications of the Detached Regularization

In this section, we shall give applications of the detached regularization to
specific loop amplitudes. To this end, we shall perform a detailed analysis of
the scalar electrodynamics and spinor electrodynamics as two comprehensive
applications of the detached regularization.

###  IV.1 Scalar Electrodynamics

In order to illustrate the use of detached regularization, we consider in this
subsection a simple QFT composed of a charged spin-zero particle ($\phi$) and
the gauge field ($A_{\mu}$) – the so-called scalar electrodynamics. The
Lagrangian defining this theory is

${\cal
L}=(D^{\mu}\phi)^{\dagger}(D_{\mu}\phi)-m^{2}\phi^{{\dagger}}\phi-\frac{1}{4}\lambda(\phi^{{\dagger}}\phi)^{2}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$
(30)

where $F_{\mu\nu}$ is the field strength tensor and $D_{\mu}$ is the gauge
covariant derivative defined as

$D_{\mu}=\partial_{\mu}-ieA_{\mu}\,.$ (31)

The Lagrangian in (30) leads to the three interaction vertices depicted in
FIG. 1.

Figure 1: The three basic vertices of scalar electrodynamics and their vertex
factors.

Now, we will illustrate the usage of the newly-developed detached
regularization in the context of loop corrections in scalar electrodynamics.
The details of the calculation can be found in Appendix A at the end of the
paper. The relevant diagrams fall into two main categories. The first category
concerns the one-loop corrections to the photon propagator due to the charged
scalar which is given in FIG. 2.

|
---|---
(a) | (b)
Figure 2: One-loop corrections to the photon propagator in scalar
electrodynamics.

It is more convenient to combine the self-energy and tadpole diagrams given in
FIG. 2 (a) and 2 (b) into a single amplitude ($k_{\mu}$ ($p_{\mu}$) is loop
(external) momentum)

$i\Pi^{\mu\nu}_{\ref{fig:sqed_photon}a+\ref{fig:sqed_photon}b}\,(p)=e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{(2k+p)^{\mu}(2k+p)^{\nu}-2g^{\mu\nu}[(k+p)^{2}-m^{2}]}{[(k+p)^{2}-m^{2}][k^{2}-m^{2}]}\,,$
(32)

from which it will be easier to see the eventual transversality of the
logarithmic part. Indeed, applying the detached regularization scheme in (13)
to this amplitude, we get in the $\overline{MS}$ subtraction scheme (see
Appendix A)

$i\Pi^{\mu\nu}_{\ref{fig:sqed_photon}a+\ref{fig:sqed_photon}b}\,(p)=-\>\frac{ie^{2}}{16\pi^{2}}\,g^{\mu\nu}\;\Lambda_{\wp}^{2}\>-\frac{ie^{2}}{4\pi^{2}}\>[p^{2}g^{\mu\nu}-p^{\mu}p^{\nu}]\;\int\limits_{-1/2}^{1/2}dy\Bigg{(}y^{2}\>\log\Bigg{[}\frac{\mu^{2}}{p^{2}(y^{2}-\frac{1}{4})+m^{2}}\Bigg{]}\Bigg{)}\;,$
(33)

in which the first term is the power-law part (similar to what one would find
by employing the cutoff regularization results in (9)), and the second term
which is the finite-transverse part we are familiar from the dimensional
regularization [6, 14].

|  |
---|---|---
(a) | (b) | (c)
Figure 3: One-loop corrections to the scalar propagator in scalar
electrodynamics.

The second class of loop corrections that is going to be examined in scalar
electrodynamics is the one-loop corrections to the scalar propagator itself.
The relevant diagrams pertaining to this type of correction is given in FIG.
3. The self-energy correction in FIG. 3 (a) is given by the amplitude (see
Appendix A)

$i\Pi_{\ref{fig:sqed_scalar}a}(p)=-e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{P_{\mu\nu}(k)(k+2p)^{\mu}(k+2p)^{\nu}}{k^{2}\,[(k+p)^{2}-m^{2}]}$
(34)

in which

$P_{\mu\nu}(k)=g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}$ (35)

is the projector of the photon propagator in Lorenz gauge. The detached
regularization scheme leads to

$i\Pi_{\ref{fig:sqed_scalar}a}(p)=\frac{-3\,i\,e^{2}\,p^{2}}{16\pi^{2}}\,\Bigg{\\{}\log\;\Bigg{(}\frac{\mu^{2}}{m^{2}-p^{2}}\Bigg{)}-\frac{m^{2}}{p^{2}}\,\log\Bigg{(}\frac{m^{2}}{m^{2}-p^{2}}\Bigg{)}+\frac{4}{3}\Bigg{\\}}\;,$
(36)

where $\overline{MS}$ subtraction scheme is applied again. This result is
essentially the same as the one you would obtain in the dimensional
regularization framework [6, 14].

The scalar-scalar tadpole correction in FIG. 3 (b) is given by the amplitude

$i\Pi_{\ref{fig:sqed_scalar}b}=\lambda\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{1}{k^{2}-m^{2}}$
(37)

where $m$ is the mass of the charged scalar. The detached regularization (13)
with the $\overline{MS}$ subtraction results in the amplitude (see Appendix A)

$i\Pi_{\ref{fig:sqed_scalar}b}=\frac{-i\,\lambda}{32\pi^{2}}\,\Lambda_{\wp}^{2}\,\log\;\Bigg{(}\frac{\mu^{2}}{m^{2}}\Bigg{)}\;+\frac{i\,\lambda}{16\pi^{2}}\,m^{2}\,\Bigg{\\{}\log\;\Bigg{(}\frac{\mu^{2}}{m^{2}}\Bigg{)}+1\Bigg{\\}}\;.$
(38)

It is interesting to note that the second term is the same as the one you
would get in dimensional regularization scheme [6, 14]. The first term, on the
other hand, is rather specific to the detached regularization method (13).

The third and last diagram to consider in scalar electrodynamics is the one in
FIG. 3 (c), which is the scalar-photon tadpole diagram. The amplitude for this
is

$i\Pi_{\ref{fig:sqed_scalar}c}=2\;e^{2}\;\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{g^{\mu\nu}P_{\mu\nu}(k)}{k^{2}-m_{\gamma}^{2}}$
(39)

where $P_{\mu\nu}(k)$ is again given by (35). Here, we added a fictitious
photon mass $m_{\gamma}$ to the photon propagator as an infrared regulator.
The detached regularization scheme (13) applied to this diagram results in the
following amplitude after the $\overline{MS}$ subtraction (see Appendix A):

$i\Pi_{\ref{fig:sqed_scalar}c}=\frac{i\,e^{2}}{8\pi^{2}}\,\Lambda_{\wp}^{2}\;-\frac{i\,e^{2}}{16\pi^{2}}\,\Lambda_{\wp}^{2}\log\Bigg{(}\frac{\mu^{2}}{m_{\gamma}^{2}}\Bigg{)}$
(40)

which would vanish in the dimensional regularization scheme (more precisely,
the first term would disappear and $\Lambda_{\wp}^{2}$ in the second term
would be replaced by $m_{\gamma}^{2}$). This result, which is quadratic in the
scale $\Lambda_{\wp}$ and logarithmic in the renormalization scale $\mu$, is
specific to the detached regularization scheme (13).

Before closing, it proves useful to dwell on the loop-induced photon mass in
equation (33). In this regard, one notes that different regularization schemes
can be contrasted by typical scattering processes such as the Drell- Yan
scattering $e^{+}e^{-}\rightarrow\gamma^{\star}\rightarrow f{\bar{f}}$ ($f=$
leptons, quarks) (see [1] for a detailed study). The detached regularization
has a different take compared to those in [1] in that it aims at revealing
first the effects of the UV cutoff. Indeed, it is clear that photon acquires a
mass (as in (33) above) in the presence of the cutoff $\Lambda_{\wp}$, and
consequently, the Drell-Yan cross section exhibits resonance behavior at the
loop-induced photon mass. But it is also clear that a finite photon mass is
unphysical despite the fact that the UV cutoff exists as a concrete scale.
This means that one has to do something about the UV cutoff. That “something”
could be Sakharov’s induced gravity [23, 24] ($\Lambda_{\wp}$ leads to
Newton’s constant) or Demir’s emergent gravity [25, 26, 27] ($\Lambda_{\wp}$
is promoted to affine curvature as a spurion in resemblance to the Higgs
mechanism) or the subtractive renormalization scheme ($\Lambda_{\wp}^{2}$
terms are subtracted away) or some other mechanism. Our goal in the present
work is to prepare a framework (like the photon polarization as in (33) above)
in which one can do “something” about the UV cutoff such that what is left
after that “something” is the usual dimensionally-regularized QFT. In other
words, after the UV cutoff $\Lambda_{\wp}$ is dealt with the Drell-Yann
scattering $e^{+}e^{-}\rightarrow\gamma^{\star}\rightarrow f{\bar{f}}$
proceeds as in the dimensional regularization at the subtraction scale $\mu$
(as discussed in (33), for instance).

###  IV.2 Spinor Electrodynamics

We continue applications of the detached regularization with the spinor
electrodynamics, which is a simple QFT composed of a charged Dirac fermion
($\psi$) and the gauge field ($A_{\mu}$). The calculational details of this
section are all given in Appendix B. The theory is governed by the Lagrangian

${\cal
L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\overline{\psi}\left(i\not{\partial}-e\not{A}-m_{f}\right)\psi$
(41)

where $F_{\mu\nu}$ is the field strength tensor for $A_{\mu}$.

|  |
---|---|---
(a) | (b) | (c)
Figure 4: One-loop corrections in spinor electrodynamics.

In regard to the applications of the detached regularization method in (13),
we will calculate the amplitudes for the one-loop Feynman diagrams depicted in
FIG. 4.

We begin with the photon vacuum polarization diagram in FIG. 4 (a). Its is
given by

$i\Pi^{\mu\nu}_{\ref{fig:qed}a}(p)=-e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{{\rm
Tr}\left[\gamma^{\mu}(\not{k}+\not{p}+m_{f})\gamma^{\nu}(\not{k}+m_{f})\right]}{\left[k^{2}-m_{f}^{2}\right]\left[(k+p)^{2}-m_{f}^{2}\right]}\;.$
(42)

Regularization of this amplitude by the detached regularization in (13) leads
to (see Appendix B)

$i\Pi^{\mu\nu}_{\ref{fig:qed}a}(p)=\frac{ie^{2}}{16\pi^{2}}\,g^{\mu\nu}\;\Lambda_{\wp}^{2}\>-\frac{ie^{2}}{2\pi^{2}}\>[p^{2}g^{\mu\nu}-p^{\mu}p^{\nu}]\;\int\limits_{0}^{1}dx\left\\{\>(x-x^{2})\>\left(-\>\,\frac{1}{2}+\log\left[\frac{\mu^{2}}{p^{2}(x^{2}-x)+m_{f}^{2}}\right]\right)\right\\}\;,$
(43)

whose first term is a quadratic correction (similar to what one would get by
cutoff regularization). Its second term is finite and transverse just like the
corresponding vacuum polarization diagram in (33) in the scalar
electrodynamics. It is what one would get from the dimensional regularization
[6, 14].

The fermion self-energy diagram in FIG. 4 (b) obtains the amplitude

$-i\Sigma_{\ref{fig:qed}b}(p)=-e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}(\not{k}+m_{f})\gamma_{\mu}}{\Big{[}k^{2}-m_{f}^{2}\Big{]}\Big{[}(p-k)^{2}-m_{\gamma}^{2}\Big{]}}\;$
(44)

and its regularization by the detached regularization in (13) results in (see
Appendix B)

$-i\Sigma_{\ref{fig:qed}b}(p)=\frac{ie^{2}}{16\pi^{2}}\>\int\limits_{0}^{1}dx\left[2\,m_{f}-2x\,\not{p}\right]\;-\;\frac{ie^{2}}{16\pi^{2}}\>\int\limits_{0}^{1}dx\left\\{\left[4\,m_{f}-2x\,\not{p}\right]\;\log\left[\frac{\mu^{2}}{p^{2}(x^{2}-x)+x\,m_{\gamma}^{2}+(1-x)\,m_{f}^{2}}\right]\right\\}$
(45)

after applying the $\overline{MS}$ subtraction scheme.

Lastly, the fermion-photon vertex of diagram in FIG. 4 (c) has the amplitude

$-ie\Gamma^{\mu}_{\ref{fig:qed}c}(p)=-e^{3}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma_{\nu}(\not{p}^{{}^{\prime}}-\not{k}+m_{f})\gamma^{\mu}(\not{p}-\not{k}+m_{f})\gamma^{\nu}}{k^{2}\,\Big{[}(p-k)^{2}-m_{f}^{2}\Big{]}\Big{[}(p^{{}^{\prime}}-k)^{2}-m_{f}^{2}\Big{]}}\;$
(46)

whose detached regularization (13) gives rise to (see Appendix B)

$-ie\Gamma^{\mu}_{\ref{fig:qed}c}(p)=-2\,e^{3}\int\limits_{0}^{1}dx\,\int\limits_{0}^{1-x}dy\left\\{\frac{i\gamma^{\mu}}{16\pi^{2}}\left(-\frac{5}{2}\,+\,\log\frac{\mu^{2}}{\Delta^{2}}\right)\,-\,\frac{i\tilde{N}^{\mu}}{32\,\pi^{2}\,\Delta^{2}}\right\\}$
(47)

in which

$\Delta^{2}=\,m_{f}^{2}\,(x+y)+p^{2}\,(x^{2}-x)+{p^{{}^{\prime}}}^{2}\,(y^{2}-y)\,+2\,p\cdot
p^{{}^{\prime}}(xy)$ (48)

and

$\tilde{N}^{\mu}=\gamma_{\nu}\left[\not{p}^{{}^{\prime}}(1-y)-\not{p}x+m_{f}\right]\gamma^{\mu}\left[\not{p}(1-x)-\not{p}^{{}^{\prime}}y+m_{f}\right]\gamma^{\nu}\;.$
(49)

Needless to say, the fermion-photon vertex is what one would find in
dimensional regularization. It is an important property of the detached
regularization that all the results and properties of the dimensional
regularization are maintained.

In parallel with the discussion at the end of Sec. IV A, here it should be
emphasized that the goal in the present work is to construct a framework
(detached regularization scheme) in which one can deactivate the photon mass
in equation (43) in a way leaving behind only a dimensionally-regularized QFT.
It is after the deactivation of the photon mass that the Drell- Yan scattering
$e^{+}e^{-}\rightarrow\gamma^{\star}\rightarrow f{\bar{f}}$ ($f=$ leptons,
quarks) proceeds as in the dimensional regularization [1]. In the next
section, we discuss how the subtractive renormalization can be naturally
realized in the framework of detached regularization.

##  V Subtractive renormalization in the framework of detached regularization

As was mentioned in Introduction, subtractive renormalization [21, 20] is one
instance in which distinction between power-law and logarithmic divergences is
a must. The detached regularization constructed in Sec. II is one such
distinctive regularization framework. Its applications in Sec. III and Sec. IV
(see also Appendix A and Appendix B for details) have shown that the power-law
and logarithmic divergences get manifestly detached from each other under the
regularization function $f(\Lambda_{\wp},\mu,D)$ in (12) (supplemented with
the ${\overline{MS}}$ subtraction).

One immediate field theoretic setup in which one can benefit from the merits
of the detached regularization is the subtractive renormalization of the
$\lambda\phi^{4}$ theory. To this end, we analyze the scalar two-point
function, and show how subtractive renormalization works in the detached
regularization scheme. One recalls that in subtractive renormalization [21,
20], one subtracts out the quadratically divergent ($\Lambda_{\wp}^{2}$) terms
by adding relevant counter terms to the theory. However, since the commonly
employed regularization methods do not detach power-law and logarithmic
dependencies, mass renormalization factor $Z_{m}$ still contains a term
proportional to the logarithm of $\Lambda_{\wp}^{2}$ such as [21]

$\displaystyle\frac{\lambda}{32\pi^{2}}\log\left(\frac{\Lambda_{\wp}^{2}}{m^{2}}\right)$
(50)

for a real scalar with mass $m$. This means that even though the quadratic
terms ($\Lambda_{\wp}^{2}$) are subtracted out the remnant logarithmically
divergent terms ($\log\Lambda_{\wp}$) continue to involve the UV cutoff
$\Lambda_{\wp}^{2}$.

This remnant $\Lambda_{\wp}$ problem does not arise in the detached
regularization. Indeed, in detached regularization two-point function of a
real scalar field $\phi$ follows from (38) after scaling by $1/2$, and takes
the compact form

$\displaystyle\Gamma^{(2)}(p^{2})=p^{2}-m^{2}-\frac{\lambda}{64\pi^{2}}\Lambda_{\wp}^{2}\log\left(\frac{\mu^{2}}{m^{2}}\right)+\frac{\lambda}{32\pi^{2}}m^{2}+\frac{\lambda}{32\pi^{2}}m^{2}\log\left(\frac{\mu^{2}}{m^{2}}\right)$
(51)

where $p$ is the momentum of the particle. Now, application of the subtractive
renormalization [21] to this two-point function removes the
$\Lambda_{\wp}^{2}$ term to leave behind $\log\mu$–involving terms as in the
dimensional regularization [6, 14]. It is in this sense that it becomes
possible to subtract away power-law UV-sensitivities such that what remains
after the subtraction is precisely what one would find in the dimensional
regularization (with no dependence on $\Lambda_{\wp}$). In the language of
[20], the local power-law terms are cleaned of the regularized theory in a way
causing no physical effects.

Figure 5: Two-loop sunset diagram for a real scalar field $\phi$ with
$\lambda\phi^{4}$ coupling.

Having done with the subtractive renormalization of the one-loop self energy
in detached regularization, it is now time to discuss workings of the detached
regularization at two-loop order. This can be done by analyzing the sunset
diagram in Fig. 5, for instance. The goal is to determine if the
regularization function $f(\Lambda_{\wp},\mu,D)$ in (12) properly works at two
loops. The superficial degree of divergence of the sunset diagram is two. It
therefore is expected to be quadratically divergent (proportional to
$\Lambda_{\wp}^{2}$). In fact, direct calculation gives

$-i\Sigma=\frac{i\lambda^{2}}{96\pi^{2}}\int\int\int
dx\,dy\,dz\,\delta(x+y+z-1)\frac{1}{\sqrt{\alpha\beta}}\Bigg{\\{}\frac{\Lambda_{\wp}^{2}}{32\pi^{2}}\log\frac{\mu^{2}}{{\overline{m}}^{2}}-\frac{{\overline{m}}^{2}}{16\pi^{2}}\Big{(}\log\frac{\mu^{2}}{{\overline{m}}^{2}}+1\Big{)}\Bigg{\\}}$
(52)

in which

$\alpha=x+z\,,\quad\beta=\frac{xz+y(x+z)}{x+z}\,,\quad\gamma=\frac{xyz}{xz+y(x+z)}\,,\quad{\overline{m}}^{2}=-\gamma
p^{2}+(x+y+z)m^{2}\,,$ (53)

and $m$ is the mass of the scalar, $p$ is the external momentum. The first
term inside the curly bracket in (52) is the aforementioned quadratic
divergence of the sunset diagram in Fig. 5. The second term in (52), on the
other hand, is the expression one would find in dimensional regularization.
Clearly, the two-loop self energy piece (52) ensures that, at higher loops,
the detached regularization continues to comprise both the power-law
($\Lambda_{\wp}^{2}$) and logarithmic ($\log\mu$) UV-sensitivities. As
mentioned above while discussing the one-loop self-energy, subtractive
renormalization weeds out $\Lambda_{\wp}^{2}$ terms to leave behind a
$\log\mu$-involving expression. All this implies that the detached
regularization provides a natural setting for subtractive renormalization at
and beyond one loop.

##  VI Conclusion

The dimensional regularization sets the common ground for regularizing QFTs
(see Ref. [1] for various variants) thanks mainly to its ability to preserve
the gauge invariance. It gives a gauge-invariant description of how the QFT
changes with the renormalization scale $\mu$ (essentially the scale of the
experiment). But the dimensional regularization holds good provided that the
QFT under concern is devoid of any cutoff scale. (The cutoff $\Lambda_{\wp}$
is not a particle mass. It is the UV boundary of the QFT in the Wilsonian
sense). The cutoff does necessarily break the gauge symmetries (violation of
the Ward identities). In the present work, the goal has been to extend the
dimensional regularization to QFTs with a UV cutoff such that power-law
divergences (involving $\Lambda_{\wp}$ and breaking gauge symmetry) and
logarithmic divergences (involving $\mu$ and respecting gauge symmetry) are
combined in one single regularization scheme. This detachment of the two
divergences is motivated by their natures. Indeed, power-law divergences are
local in that they live at the scale $\Lambda_{\wp}$. The logarithmic
divergences, on the other hand, extend in the entire range since the
correction $M^{2}\log\Lambda_{\wp}^{2}/M^{2}$ varies with the particle mass
$M$. Besides, the logarithmic UV-sensitivities set the beta functions and
conformal anomalies in the QFT while the power-law divergences do not play
such roles.

In addition to the above, there is the fact that the conventional cut-off
regularization does not assign different mass scales to the power-law and
logarithmic divergences. In the case of the induced gravity the UV cutoff sets
Newton’s constant. In the case of emergent gravity, on the other hand, the UV
cutoff gets promoted to affine curvature as a spurion field (similar in
philosophy to the Higgs mechanism). Finally, in the subtractive
renormalization procedure, power-law divergences are cancelled away by the
introduction of appropriate counter terms. In all these mechanisms, the
problem boils down to the fact that while one operates on the terms power-law
in $\Lambda_{\wp}$ (whether it be identification or promotion or cancellation)
one has to overlook the dependence of the logarithmic terms on the same scale
$\Lambda_{\wp}$. We illustrated this problem by a discussion of the
subtractive renormalization in Sec. V. In that case, even though one subtracts
away the power-law dependencies on $\Lambda_{\wp}$, one is still left with the
$\log\Lambda_{\wp}$ dependencies which survive in gauge-invariant corrections
(including the fermion masses). It is clear that if one had a regularization
method that separates out the scales on which the power-law and logarithmic
divergences depend, all these predicaments would evaporate. This is the main
motivation of the present manuscript and it is the main supremacy of detached
regularization over the cut-off regularization.

In this work, for the first time in the literature, we have extended the usual
dimensional regularization to involve the logarithmic ($D=4$) and power-law UV
sensitivities ($D=0$ and $D=2$) in a way detached to involve different scales.
We have demonstrated benefits of the detached regularization by giving its
basic uses in different circumstances. Firstly, we have applied it to the
computation of the vacuum energy in Sec. III, finding the usual UV structure.
Secondly, we have applied it to regularization of the scalar and spinor
electrodynamics in Sec. IV. Thirdly, we have applied it to subtractive
renormalization to show how eligible the detached regularization is for
treating the power-law and logarithmic divergences independently. In general,
the detached regularization is unique in that it separates out the scales of
the power-law and logarithmic corrections, where the latter sets the running
of the parameters of the theory by virtue of the beta functions.

The detachment of the power-law and logarithmic divergences can be of broad
interest for the renormalization of QFTs. The renormalization schemes in which
both logarithmic and power-law divergences are kept can prove useful (as did
in nuclear physics applications [30]) for revealing the UV sensitivity of the
QFTs. If one wants to renormalize away the quadratic UV dependencies, one way
to go by is to employ the subtractive renormalization [20, 21]. In
conventional subtractive renormalization quadratic and logarithmic divergences
formally appear as separate terms but the logarithmic terms embedded in the
bare mass still inhabit the same scale as the power-law ones which are
subtracted out via counter terms. This may seem like a simple issue. However
it is more involved than that, since the renormalized mass term depends on the
scale $\Lambda_{\wp}$. In contrast to this conventional structure, the newly-
introduced detached regularization enables a complete detachment of the power-
law and logarithmic divergences, and this detachment in return enables the
subtractive renormalization to subtract away all quadratic divergences,
leaving behind exactly the logarithmic terms one would find in the dimensional
regularization.

The detached regularization can have potential applications in various
problems. It can be utilized in a broad class of renormalization methods or
field-theoretic mechanisms. One example is subtraction of the power-law
divergences as in [20, 21, 22]. Another example is induced gravity models in
which UV cutoff leads to Newton’s constant [23, 24]. Yet another example is
emergent gravity models in which UV cutoff is promoted to curvature (in
reminiscence to promotion of vector boson masses to Higgs field) [25, 26, 27].
These examples can be furthered with other possible applications.

## Acknowledgements

The work of O. S. is supported by the TÜBİTAK BİDEB-2218 national postdoctoral
fellowship grant 118C522. D. D. acknowledges the contribution of the COST
Action CA21106 - COSMIC WISPers in the Dark Universe: Theory, astrophysics and
experiments (CosmicWISPers). The authors are grateful to conscientious
reviewers for their comments, criticisms and suggestions.

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##  VII Appendices

### Appendix A: Scalar Electrodynamics

#### Vacuum polarization in Scalar Electrodynamics

Combining the self-energy and tadpole diagrams given in FIG. 2 (a) and 2 (b)
we obtain

$i\Pi^{\mu\nu}_{\ref{fig:sqed_photon}a+\ref{fig:sqed_photon}b}\,(p)=e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{(2k+p)^{\mu}(2k+p)^{\nu}-2g^{\mu\nu}[(k+p)^{2}-m^{2}]}{[(k+p)^{2}-m^{2}][k^{2}-m^{2}]}\,,$
(54)

to which we apply Feynman parametrization in the denominator and shift the
loop momenta in the numerator accordingly. Getting rid of the terms linear in
the shifted loop momenta $q$, we get to the point

$\displaystyle
i\Pi^{\mu\nu}_{\ref{fig:sqed_photon}a+\ref{fig:sqed_photon}b}\,(p)$
$\displaystyle=$ $\displaystyle
e^{2}\int\limits_{0}^{1}dx\;\left\\{g^{\mu\nu}\left(\frac{4}{D}-2\right)\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{q^{2}}{\left[q^{2}-\Delta^{2}\right]^{2}}\;\right\\}$
$\displaystyle+$
$\displaystyle\;e^{2}\int\limits_{0}^{1}dx\;\left\\{\left[p^{\mu}p^{\nu}(2x-1)^{2}-2g^{\mu\nu}[p^{2}(1-x)^{2}-m^{2}]\right]\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{1}{\left[q^{2}-\Delta^{2}\right]^{2}}\right\\}$

where

$\Delta^{2}=p^{2}(x^{2}-x)+m^{2}\>,$ (56)

and the first integral in (VII) can be reduced to two irreducible integrals.
The result of this reduction is to take the amplitude (VII) into

$i\Pi^{\mu\nu}_{\ref{fig:sqed_photon}a+\ref{fig:sqed_photon}b}\,(p)=e^{2}\int\limits_{0}^{1}dx\;\Big{\\{}I_{1}+I_{2}\Big{\\}}$
(57)

where

$I_{1}=g^{\mu\nu}\left(\frac{4}{D}-2\right)\int\frac{d^{4}q}{(2\pi)^{4}}\>\frac{1}{q^{2}-\Delta^{2}}$
(58)

and

$I_{2}=g^{\mu\nu}\left(\frac{4}{D}-2\right)\,\Delta^{2}\;\int\frac{d^{4}q}{(2\pi)^{4}}\>\frac{1}{\left[q^{2}-\Delta^{2}\right]^{2}}\,+\,\left[p^{\mu}p^{\nu}(2x-1)^{2}-2g^{\mu\nu}[p^{2}(1-x)^{2}-m^{2}]\right]\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{1}{\left[q^{2}-\Delta^{2}\right]^{2}}\>.$
(59)

Now, we should first apply the new regularization (13) to $I_{1}$ in (58). The
power of the denominator of $I_{1}$ is $n=1$. Therefore (13) applied to (58)
reads

$I_{1}=g^{\mu\nu}\left(\frac{4}{D}-2\right)\;\Bigg{[}\frac{1}{8\pi}\left(\delta_{[D]0}+\delta_{[D]2}\right)\Lambda_{\wp}^{2}\>\>\mu^{2-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\;\int\frac{d^{D}q}{(2\pi)^{D}}\>\frac{1}{q^{2}-\Delta^{2}}$
(60)

where the integral on the RHS amounts to

$\int\frac{d^{D}q}{(2\pi)^{D}}\>\frac{1}{q^{2}-\Delta^{2}}=\frac{-i}{(4\pi)^{D/2}}\frac{\Gamma(1-D/2)}{\Gamma(1)}(\Delta^{2})^{D/2-1}\>.$
(61)

Replacing (61) into (60) we obtain the following expression which is a
function of the dimensionality $D$

$I_{1}=-i\;g^{\mu\nu}\left(\frac{4}{D}-2\right)\;\Bigg{[}\frac{1}{8\pi}\left(\delta_{[D]0}+\delta_{[D]2}\right)\Lambda_{\wp}^{2}\>\>\mu^{2-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\;\frac{(\Delta^{2})^{D/2-1}}{(4\pi)^{D/2}}\frac{\Gamma(1-D/2)}{\Gamma(1)}\>.$
(62)

Now, the crucial point to keep in mind is that in evaluating (62), not only do
we have to consider the $D\to 4$ limit but we should also take the $D\to 2$
limit so that we don’t leave out the quadratic corrections in $\Lambda_{\wp}$.
In the $D\to 2$ limit (62) yields

$(I_{1})_{D\to 2}=-i\;g^{\mu\nu}\>\frac{\Lambda_{\wp}^{2}}{16\pi^{2}}$ (63)

making use of the $\overline{MS}$ subtraction scheme.

Before calculating the $D\to 4$ limit of $I_{1}$, first let us evaluate
$I_{2}$ in (59). The power of the denominator of the divergent integrals in
$I_{2}$ is $n=2$, therefore the new regularization scheme (13) applied to it
reads

$\displaystyle I_{2}$
$\displaystyle=g^{\mu\nu}\left(\frac{4}{D}-2\right)\,\Delta^{2}\Bigg{[}\left(\delta_{[D]0}+\delta_{[D]2}\right)\mu^{4-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\>\int\frac{d^{D}q}{(2\pi)^{D}}\,\frac{1}{\left[q^{2}-\Delta^{2}\right]^{2}}$
$\displaystyle+$
$\displaystyle\left[p^{\mu}p^{\nu}(2x-1)^{2}-2g^{\mu\nu}[p^{2}(1-x)^{2}-m^{2}]\right]\Bigg{[}\left(\delta_{[D]0}+\delta_{[D]2}\right)\mu^{4-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{\left[q^{2}-\Delta^{2}\right]^{2}}$

where the divergent integral evaluates to

$\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{\left[q^{2}-\Delta^{2}\right]^{2}}=\frac{i}{(4\pi)^{D/2}}\frac{\Gamma(2-D/2)}{\Gamma(2)}(\Delta^{2})^{D/2-2}\>.$
(65)

Making use of (65) in (VII) before summing it up with (62) and then taking the
($D\to 4$) limit for $I_{1}+I_{2}$ leads to

$(I_{1}+I_{2})_{D\to
4}=\frac{i}{16\pi^{2}}\>\left[p^{\mu}p^{\nu}(2x-1)^{2}-p^{2}\,g^{\mu\nu}(2x-1)^{2}+p^{2}\,g^{\mu\nu}(2x-1)\right]\log\frac{\mu^{2}}{\Delta^{2}}$
(66)

where $\Delta^{2}$ is as given in (56). The next step is to plug (66) and (63)
into

$i\Pi^{\mu\nu}_{\ref{fig:sqed_photon}a+\ref{fig:sqed_photon}b}\,(p)=e^{2}\int\limits_{0}^{1}dx\;\Big{\\{}(I_{1})_{D\to
2}+(I_{1}+I_{2})_{D\to 4}\Big{\\}}$ (67)

which can be put into the following final form via the change of variable
$y=x-1/2$

$i\Pi^{\mu\nu}_{\ref{fig:sqed_photon}a+\ref{fig:sqed_photon}b}\,(p)=-\>\frac{ie^{2}}{16\pi^{2}}\,g^{\mu\nu}\;\Lambda_{\wp}^{2}\>-\frac{ie^{2}}{4\pi^{2}}\>[p^{2}g^{\mu\nu}-p^{\mu}p^{\nu}]\;\int\limits_{-1/2}^{1/2}dy\Bigg{(}y^{2}\>\log\Bigg{[}\frac{\mu^{2}}{p^{2}(y^{2}-\frac{1}{4})+m^{2}}\Bigg{]}\Bigg{)}\;.$
(68)

#### Scalar propagator in Scalar Electrodynamics

One-loop corrections to the scalar propagator in SQED is depicted in FIG. 3.
The amplitude of the self energy correction in FIG. 3 (a) reads

$i\Pi_{\ref{fig:sqed_scalar}a}(p)=-e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{P_{\mu\nu}(k)(k+2p)^{\mu}(k+2p)^{\nu}}{k^{2}\,[(k+p)^{2}-m^{2}]}$
(69)

where

$P_{\mu\nu}(k)=g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}}$ (70)

is the projector of the photon propagator in Lorenz gauge which satisfies
$k^{\mu}P_{\mu\nu}=0$ and $k^{\nu}P_{\mu\nu}=0$. Using these two relations we
obtain

$i\Pi_{\ref{fig:sqed_scalar}a}(p)=-4e^{2}p^{2}\left(1-\frac{1}{D}\right)\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{1}{k^{2}\,[(k+p)^{2}-m^{2}]}\,.$
(71)

First, Feynman-parameterizing the denominator and then applying the
regularization (13), one obtains

$i\Pi_{\ref{fig:sqed_scalar}a}(p)=-4e^{2}p^{2}\int\limits_{0}^{1}dx\left(1-\frac{1}{D}\right)\,\Bigg{[}\left(\delta_{[D]0}+\delta_{[D]2}\right)\mu^{4-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\int\frac{d^{D}q}{(2\pi)^{D}}\,\frac{1}{[q^{2}-\Delta^{2}]^{2}}\,,$
(72)

where the momentum integral is again given by (65) and
$\Delta^{2}=p^{2}(x^{2}-x)+m^{2}x$. Since the power of the divergent integral
is $n=2$ we need only consider the ($D\to 4$) limit for (72) while applying
the new regularization. This results in

$i\Pi_{\ref{fig:sqed_scalar}a}(p)=\frac{-3\,i\,e^{2}\,p^{2}}{16\pi^{2}}\,\Bigg{\\{}\log\;\Bigg{(}\frac{\mu^{2}}{m^{2}-p^{2}}\Bigg{)}-\frac{m^{2}}{p^{2}}\,\log\Bigg{(}\frac{m^{2}}{m^{2}-p^{2}}\Bigg{)}+\frac{4}{3}\Bigg{\\}}\;,$
(73)

where $\overline{MS}$ subtraction scheme is applied again.

The amplitude for the scalar-scalar tadpole correction in FIG. 3 (b) is given
by

$i\Pi_{\ref{fig:sqed_scalar}b}=\lambda\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{1}{k^{2}-m^{2}}$
(74)

where $m$ is the mass of the charged scalar. Taking note of the fact that the
power of the denominator of the propagator is $n=1$, the regularization (13)
yields

$i\Pi_{\ref{fig:sqed_scalar}b}=\lambda\;\Bigg{[}\frac{1}{8\pi}\left(\delta_{[D]0}+\delta_{[D]2}\right)\Lambda_{\wp}^{2}\>\>\mu^{2-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\int\frac{d^{D}k}{(2\pi)^{D}}\,\frac{1}{k^{2}-m^{2}}$
(75)

where again the integral is given by

$\int\frac{d^{D}k}{(2\pi)^{D}}\>\frac{1}{k^{2}-m^{2}}=\frac{-i}{(4\pi)^{D/2}}\frac{\Gamma(1-D/2)}{\Gamma(1)}(m^{2})^{D/2-1}\>.$
(76)

Since $n=1$, we need to take both ($D\to 2$) and ($D\to 4$) limits in
performing the detached regularization on $\Pi_{\ref{fig:sqed_scalar}b}$. The
analytical continuation to $D=2$ results in

$(i\Pi_{\ref{fig:sqed_scalar}b})_{D\to
2}=\frac{-i\,\lambda}{32\pi^{2}}\,\Lambda_{\wp}^{2}\,\log\;\Bigg{(}\frac{\mu^{2}}{m^{2}}\Bigg{)}\;,$
(77)

while analytical continuation to $D=4$ gives

$(i\Pi_{\ref{fig:sqed_scalar}b})_{D\to
4}=\frac{i\,\lambda}{16\pi^{2}}\,m^{2}\,\Bigg{\\{}\log\;\Bigg{(}\frac{\mu^{2}}{m^{2}}\Bigg{)}+1\Bigg{\\}}\;$
(78)

where $\overline{MS}$ subtraction scheme is applied in both $D=2$ and $D=4$
cases. Putting it all together, the new regularization (13) and
$\overline{MS}$ subtraction scheme give the final result

$i\Pi_{\ref{fig:sqed_scalar}b}=\frac{-i\,\lambda}{32\pi^{2}}\,\Lambda_{\wp}^{2}\,\log\;\Bigg{(}\frac{\mu^{2}}{m^{2}}\Bigg{)}\;+\frac{i\,\lambda}{16\pi^{2}}\,m^{2}\,\Bigg{\\{}\log\;\Bigg{(}\frac{\mu^{2}}{m^{2}}\Bigg{)}+1\Bigg{\\}}\;.$
(79)

The last diagram that we will examine which contributes to the scalar
propagator at one-loop level is the scalar-photon tadpole given in FIG. 3 (c).
The amplitude for this is

$i\Pi_{\ref{fig:sqed_scalar}c}=2\;e^{2}\;\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{g^{\mu\nu}P_{\mu\nu}(k)}{k^{2}-m_{\gamma}^{2}}$
(80)

where $P_{\mu\nu}(k)$ is again given by (35) and whose contraction with the
metric is

$g^{\mu\nu}P_{\mu\nu}(k)=D-1\>.$ (81)

This makes (80)

$i\Pi_{\ref{fig:sqed_scalar}c}=2\;e^{2}\;(D-1)\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{1}{k^{2}-m_{\gamma}^{2}}\>.$
(82)

The detached regularization applied to (82) yields

$i\Pi_{\ref{fig:sqed_scalar}c}=2\;e^{2}\;(D-1)\Bigg{[}\frac{1}{8\pi}\left(\delta_{[D]0}+\delta_{[D]2}\right)\Lambda_{\wp}^{2}\>\>\mu^{2-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\int\frac{d^{D}k}{(2\pi)^{D}}\,\frac{1}{k^{2}-m_{\gamma}^{2}}$
(83)

in which the momentum integral amounts to

$\int\frac{d^{D}k}{(2\pi)^{D}}\>\frac{1}{k^{2}-m_{\gamma}^{2}}=\frac{-i}{(4\pi)^{D/2}}\frac{\Gamma(1-D/2)}{\Gamma(1)}\;(m_{\gamma}^{2})^{D/2-1}\>.$
(84)

Since the power of the denominator is $n=1$, both ($D\to 2$) and ($D\to 4$)
limits should be considered in performing the detached regularization. The
($D\to 2$) limit yields

$(i\Pi_{\ref{fig:sqed_scalar}c})_{D\to
2}=\frac{i\,e^{2}}{8\pi^{2}}\,\Lambda_{\wp}^{2}\;-\frac{i\,e^{2}}{16\pi^{2}}\,\Lambda_{\wp}^{2}\log\Bigg{(}\frac{\mu^{2}}{m_{\gamma}^{2}}\Bigg{)}\;$
(85)

whereas the ($D\to 4$) limit yields no contribution since it basically amounts
to an amplitude which is proportional to the square of the photon mass.
Therefore the resultant amplitude in the $\overline{MS}$ subtraction scheme
becomes

$i\Pi_{\ref{fig:sqed_scalar}c}=\frac{i\,e^{2}}{8\pi^{2}}\,\Lambda_{\wp}^{2}\;-\frac{i\,e^{2}}{16\pi^{2}}\,\Lambda_{\wp}^{2}\log\Bigg{(}\frac{\mu^{2}}{m_{\gamma}^{2}}\Bigg{)}\;.$
(86)

### Appendix B: Spinor Electrodynamics

#### Vacuum polarization in Spinor Electrodynamics

According to FIG. 4 (a), the amplitude of vacuum polarization is given by

$i\Pi^{\mu\nu}_{\ref{fig:qed}a}(p)=-e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{Tr\left[\gamma^{\mu}(\not{k}+\not{p}+m_{f})\gamma^{\nu}(\not{k}+m_{f})\right]}{\left[k^{2}-m_{f}^{2}\right]\left[(k+p)^{2}-m_{f}^{2}\right]}\;.$
(87)

The trace in the numerator amounts to

$Tr\left[\gamma^{\mu}(\not{k}+\not{p}+m_{f})\gamma^{\nu}(\not{k}+m_{f})\right]=D\left[(k^{\mu}+p^{\mu})k^{\nu}+(k^{\nu}+p^{\nu})k^{\mu}+g^{\mu\nu}(m_{f}^{2}-[k+p]\cdot
k)\right]\>.$ (88)

Applying the Feynman parametrization one arrives at

$\displaystyle
i\Pi^{\mu\nu}_{\ref{fig:qed}a}(p)=-e^{2}\>D\int\limits_{0}^{1}dx\;\int\frac{d^{4}q}{(2\pi)^{4}}\,\left\\{\left[\frac{2}{D}g^{\mu\nu}\frac{q^{2}}{\left[q^{2}-\Delta^{2}\right]^{2}}\;-\frac{g^{\mu\nu}}{q^{2}-\Delta^{2}}\right]+\frac{2(x-x^{2})[p^{2}g^{\mu\nu}-p^{\mu}p^{\nu}]}{\left[q^{2}-\Delta^{2}\right]^{2}}\right\\}$
(89)

where $\Delta^{2}=p^{2}(x^{2}-x)+m_{f}^{2}$. The first term inside the square
brackets in (89) can be decomposed into two irreducible integrals the result
of which is

$i\Pi^{\mu\nu}_{\ref{fig:qed}a}\,(p)=-e^{2}\int\limits_{0}^{1}dx\;\Big{\\{}D\>I_{1}+D\>I_{2}\Big{\\}}$
(90)

where

$I_{1}=g^{\mu\nu}\left(\frac{2}{D}-1\right)\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{1}{q^{2}-\Delta^{2}}$
(91)

and

$I_{2}=\frac{2}{D}\;g^{\mu\nu}\Delta^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{1}{\left[q^{2}-\Delta^{2}\right]^{2}}+2(x-x^{2})[p^{2}g^{\mu\nu}-p^{\mu}p^{\nu}]\int\frac{d^{4}q}{(2\pi)^{4}}\,\frac{1}{\left[q^{2}-\Delta^{2}\right]^{2}}\>.$
(92)

First, we apply the regularization (13) to $I_{1}$, paying attention to the
fact that the power of the denominator is $n=1$. This means that we need to
consider not only the analytical continuation to $D=4$ but also to $D=2$. The
regularization applied to $I_{1}$ reads

$I_{1}=g^{\mu\nu}\left(\frac{2}{D}-1\right)\;\Bigg{[}\frac{1}{8\pi}\left(\delta_{[D]0}+\delta_{[D]2}\right)\Lambda_{\wp}^{2}\>\>\mu^{2-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\;\int\frac{d^{D}q}{(2\pi)^{D}}\>\frac{1}{q^{2}-\Delta^{2}}$
(93)

where the momentum integral is again given by (61). In the $D\to 2$ limit (93)
yields

$(I_{1})_{D\to 2}=-ig^{\mu\nu}\frac{\Lambda_{\wp}^{2}}{32\pi^{2}}$ (94)

where we used the $\overline{MS}$ subtraction scheme.

Now, let’s evaluate $I_{2}$. The power of the denominator of the divergent
integrals in $I_{2}$ is $n=2$, therefore the new regularization scheme (13)
applied to it reads

$\displaystyle I_{2}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}\frac{2}{D}\,\Delta^{2}\Bigg{[}\left(\delta_{[D]0}+\delta_{[D]2}\right)\mu^{4-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\>\int\frac{d^{D}q}{(2\pi)^{D}}\,\frac{1}{\left[q^{2}-\Delta^{2}\right]^{2}}$
$\displaystyle+$ $\displaystyle
2(x-x^{2})[p^{2}g^{\mu\nu}-p^{\mu}p^{\nu}]\Bigg{[}\left(\delta_{[D]0}+\delta_{[D]2}\right)\mu^{4-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\int\frac{d^{D}q}{(2\pi)^{D}}\frac{1}{\left[q^{2}-\Delta^{2}\right]^{2}}$

where the momentum integral is given by (65). Next we sum (VII) with (93) and
then take the ($D\to 4$) limit for $I_{1}+I_{2}$. This leads to

$(I_{1}+I_{2})_{D\to
4}=\frac{i}{8\pi^{2}}\>\left[p^{\mu}p^{\nu}-p^{2}\,g^{\mu\nu}\right](x^{2}-x)\log\frac{\mu^{2}}{\Delta^{2}}$
(96)

where $\Delta^{2}=p^{2}(x^{2}-x)+m_{f}^{2}$. The next step is to plug (96) and
(94) into

$i\Pi^{\mu\nu}_{\ref{fig:qed}a}\,(p)=-e^{2}\int\limits_{0}^{1}dx\;\Big{\\{}[D\>I_{1}]_{D\to
2}+[D\>(I_{1}+I_{2})]_{D\to 4}\Big{\\}}$ (97)

which results in the regularized amplitude

$i\Pi^{\mu\nu}_{\ref{fig:qed}a}(p)=\frac{ie^{2}}{16\pi^{2}}g^{\mu\nu}\Lambda_{\wp}^{2}-\frac{ie^{2}}{2\pi^{2}}\>[p^{2}g^{\mu\nu}-p^{\mu}p^{\nu}]\;\int\limits_{0}^{1}dx\left\\{\>(x-x^{2})\>\left(-\>\,\frac{1}{2}+\log\left[\frac{\mu^{2}}{p^{2}(x^{2}-x)+m_{f}^{2}}\right]\right)\right\\}\;,$
(98)

where the second term is the finite transverse term.

#### Fermion propagator in Spinor Electrodynamics

The correction to the fermion propagator is given by the diagram in FIG. 4 (b)
which corresponds to

$-i\Sigma_{\ref{fig:qed}b}(p)=-e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma^{\mu}(\not{k}+m_{f})\gamma_{\mu}}{\Big{[}k^{2}-m_{f}^{2}\Big{]}\Big{[}(p-k)^{2}-m_{\gamma}^{2}\Big{]}}\;$
(99)

wherein the numerator equals

$\gamma^{\mu}(\not{k}+m_{f})\gamma_{\mu}=(2-D)\not{k}+m_{f}D\;.$ (100)

The next step is to Feynman parameterize the denominator and shift the loop
momenta accordingly. The outcome of this step is

$-i\Sigma_{\ref{fig:qed}b}(p)=e^{2}\int\limits_{0}^{1}dx\left\\{\left[(D-2)\not{p}x-m_{f}D\right]\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{\Big{[}q^{2}-\Delta^{2}\Big{]}^{2}}\right\\}\;$
(101)

where the terms linear in the shifted loop momenta are dropped since they
evaluate to zero. Next, the regularization (13) is applied

$-i\Sigma_{\ref{fig:qed}b}(p)=e^{2}\int\limits_{0}^{1}dx\left\\{\left[(D-2)\not{p}x-m_{f}D\right]\Bigg{[}\left(\delta_{[D]0}+\delta_{[D]2}\right)\mu^{4-D}+\delta_{[D]4}\;\mu^{4-D}\Bigg{]}\int\frac{d^{D}q}{(2\pi)^{D}}\,\frac{1}{[q^{2}-\Delta^{2}]^{2}}\right\\}\;$
(102)

in which the momentum integral is again given by (65) and
$\Delta^{2}=p^{2}(x^{2}-x)+xm_{\gamma}^{2}+(1-x)m_{f}^{2}$. Since the power of
the divergent integral is $n=2$ we need only consider the ($D\to 4$) limit.
This results in

$-i\Sigma_{\ref{fig:qed}b}(p)=\frac{ie^{2}}{16\pi^{2}}\>\int\limits_{0}^{1}dx\left[2\,m_{f}-2x\,\not{p}\right]\;-\;\frac{ie^{2}}{16\pi^{2}}\>\int\limits_{0}^{1}dx\left\\{\left[4\,m_{f}-2x\,\not{p}\right]\;\log\left[\frac{\mu^{2}}{p^{2}(x^{2}-x)+x\,m_{\gamma}^{2}+(1-x)\,m_{f}^{2}}\right]\right\\}$
(103)

via the $\overline{MS}$ subtraction scheme.

#### One-loop vertex correction in Spinor Electrodynamics

The correction to the fermion-photon vertex in spinor QED is given by the
diagram FIG. 4 (c) which is

$-ie\Gamma^{\mu}_{\ref{fig:qed}c}(p)=-e^{3}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\gamma_{\nu}(\not{p}^{{}^{\prime}}-\not{k}+m_{f})\gamma^{\mu}(\not{p}-\not{k}+m_{f})\gamma^{\nu}}{k^{2}\,\Big{[}(p-k)^{2}-m_{f}^{2}\Big{]}\Big{[}(p^{{}^{\prime}}-k)^{2}-m_{f}^{2}\Big{]}}\;.$
(104)

The first step is Feynman parameterizing the denominator. This leads to

$-ie\Gamma^{\mu}_{\ref{fig:qed}c}(p)=-2\,e^{3}\int\limits_{0}^{1}dx\,\int\limits_{0}^{1-x}dy\int\frac{d^{4}q}{(2\pi)^{4}}\frac{\gamma_{\nu}[\not{p}^{{}^{\prime}}(1-y)-\not{p}x-\not{q}+m_{f}]\gamma^{\mu}[\not{p}(1-x)-\not{p}^{{}^{\prime}}y-\not{q}+m_{f})\gamma^{\nu}}{\left[q^{2}-\Delta^{2}\right]^{3}}\;$
(105)

where

$\Delta^{2}=\,m_{f}^{2}\,(x+y)+p^{2}\,(x^{2}-x)+{p^{{}^{\prime}}}^{2}\,(y^{2}-y)\,+2\,p\cdot
p^{{}^{\prime}}(xy)\>$ (106)

and $q$ is the shifted loop momentum. The numerator of (105) can be written as

$\gamma_{\nu}[\not{p}^{{}^{\prime}}(1-y)-\not{p}x-\not{q}+m_{f}]\gamma^{\mu}[\not{p}(1-x)-\not{p}^{{}^{\prime}}y-\not{q}+m_{f})\gamma^{\nu}=\gamma_{\nu}\not{q}\gamma^{\mu}\not{q}\gamma^{\nu}+\tilde{N}^{\mu}+{\rm\
(terms\ linear\ in\ }q)$ (107)

where

$\tilde{N}^{\mu}=\gamma_{\nu}\left[\not{p}^{{}^{\prime}}(1-y)-\not{p}x+m_{f}\right]\gamma^{\mu}\left[\not{p}(1-x)-\not{p}^{{}^{\prime}}y+m_{f}\right]\gamma^{\nu}\>.$
(108)

The gamma matrix algebra in $D$ dimensions applied to (105) leads to

$-ie\Gamma^{\mu}_{\ref{fig:qed}c}(p)=-2\,e^{3}\int\limits_{0}^{1}dx\,\int\limits_{0}^{1-x}dy\left\\{\frac{(2-D)^{2}}{D}\gamma^{\mu}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{q^{2}}{\left[q^{2}-\Delta^{2}\right]^{3}}+\tilde{N}^{\mu}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{\left[q^{2}-\Delta^{2}\right]^{3}}\right\\}\;$
(109)

wherein the first integral can be reduced to two irreducible ones which are

$I_{1}=\frac{(2-D)^{2}}{D}\gamma^{\mu}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{\left[q^{2}-\Delta^{2}\right]^{2}}$
(110)

and

$I_{2}=\frac{(2-D)^{2}}{D}\gamma^{\mu}\Delta^{2}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{\left[q^{2}-\Delta^{2}\right]^{3}}$
(111)

and the last integral in (109)

$I_{3}=\tilde{N}^{\mu}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{\left[q^{2}-\Delta^{2}\right]^{3}}$
(112)

is kept intact. The first integral is regularized via (13) taking note of the
fact that $n=2$ therefore we only make the analytical continuation $D\to 4$
which leads to

$I_{1}=\frac{i\gamma^{\mu}}{16\pi^{2}}\left(-2\,+\,\log\frac{\mu^{2}}{\Delta^{2}}\right)\>.$
(113)

The other two integrals $I_{2}$ and $I_{3}$ are already convergent. Therefore
there is no need to regularize them. They amount to

$I_{2}=\frac{-i\gamma^{\mu}}{32\pi^{2}}$ (114)

and

$I_{3}=\frac{-i\tilde{N}^{\mu}}{32\pi^{2}\Delta^{2}}\>.$ (115)

Putting it all together as

$-ie\Gamma^{\mu}_{\ref{fig:qed}c}(p)=-2\,e^{3}\int\limits_{0}^{1}dx\,\int\limits_{0}^{1-x}dy\left\\{I_{1}+I_{2}+I_{3}\right\\}\;$
(116)

the regularized form of the vertex correction becomes

$-ie\Gamma^{\mu}_{\ref{fig:qed}c}(p)=-2\,e^{3}\int\limits_{0}^{1}dx\,\int\limits_{0}^{1-x}dy\left\\{\frac{i\gamma^{\mu}}{16\pi^{2}}\left(-\frac{5}{2}\,+\,\log\frac{\mu^{2}}{\Delta^{2}}\right)\,-\,\frac{i\tilde{N}^{\mu}}{32\,\pi^{2}\,\Delta^{2}}\right\\}\>,$
(117)

where $\Delta^{2}$ is given by (106) and $\tilde{N}^{\mu}$ by (108).