# Denominator Regularization in Quantum Field Theory

W. A. Horowitz<EMAIL_ADDRESS>Department of Physics, University of
Cape Town, Rondebosch 7701, South Africa

###### Abstract

We propose a novel regularization scheme in quantum field theory, denominator
regularization (den reg). As simple to apply as dimensional regularization,
and similarly compatible with a minimal subtraction renormalization scheme,
den reg manifestly 1) maintains Lorentz invariance, 2) maintains gauge
invariance, 3) maintains supersymmetry, 4) correctly predicts the axial
anomaly, and 5) yields Green functions that satisfy the Callan-Symanzik
equation. Den reg also naturally enables regularization in asymmetric
spacetimes, finite spacetimes, curved spacetimes, and in thermal field theory.

##  I Introduction

In the usual calculation of quantities in quantum field theory, in which an
expansion of a time- or path-ordered exponential is evaluated order by order,
contributions that correspond to loops are often formally infinite [1]. These
infinities must be tamed by a regularization procedure such that final,
physical quantities are finite as the regulator is smoothly removed. Examples
of such regularization procedures include: momentum cutoff, Pauli-Villars [2],
dimensional regularization [3] and dimensional reduction [4], zeta and
operator regularization [5, 6], and analytic regularization [7, 8, 9]. These
procedures all have serious practical and, sometimes, conceptual shortcomings.
We propose a novel regularization procedure, “denominator regularization,”
similar to dimensional reduction and analytic regularization, which overcomes
these shortcomings, with the added advantage that the procedure also permits
the calculation of higher order corrections to quantum field theoretic
quantities in asymmetric spacetimes, finite spacetimes, curved spacetimes, and
thermal field theory.

The origin of the ultraviolet (UV) divergences in quantum field theories (QFT)
is integration up to infinitely large values of unconstrained momenta in loop
diagrams. One may render these divergences finite trivially by imposing an
upper limit momentum cutoff. Such a regularization procedure provides a
natural way to understand the renormalization group [10]. However, a finite
momentum cutoff explicitly breaks Lorentz invariance and, e.g., violates the
Ward identity in quantum electrodynamics (QED) [1]. In Pauli-Villars
regularization, massive fictitious particles are introduced with statistics
such that their associated propagators make the loop integrands go to zero
fast enough that the loop integrals converge. This procedure naturally
preserves Lorentz invariance, but requires multiple particles and is
cumbersome when one wishes to preserve gauge invariance [11, 12].

Dimensional regularization, “dim reg,” is by far the most common
regularization procedure in QFT, almost always used in conjunction with the
modified minimal subtraction ($\overline{\mathrm{MS}}$) renormalization scheme
[13, 14, 1]. In dim reg, the number of spacetime dimensions in the problem is
analytically continued from $d$ to $d-\epsilon$, where $d$ is usually 4. By
reducing the number of dimensions, the convergence properties of the integrand
are improved. The result is expanded in powers of $\epsilon$, where
$1/\epsilon$ divergences are either cancelled naturally or absorbed in the
renormalization procedure. Crucially, one performs a replacement
$\ell^{\mu}\ell^{\nu}\rightarrow(1/d)\eta_{\mu\nu}\ell^{2}$, where
$\eta_{\mu\nu}$ is the usual Minkowski metric, in the integrands of loop
momentum integrals where all other dependence on $\ell$ is through the
invariant $\ell^{2}$. Dim reg has a number of advantages: it is often,
relatively speaking, simple to implement; gauge invariance is manifestly
satisfied at all orders; and minimal subtraction, which is a straightforward
and transparent renormalization scheme, is trivial to implement with dim reg.
However, by changing the number of spacetime dimensions, dim reg breaks
supersymmetry (SUSY) [4] and unitarity [15]. Worse, conceptually, the analytic
continuation in the number of dimensions is not consistently applied: the
continuation is only applied to the spacetime index $\mu$; however, the fields
still use the 4 dimensional representations of SO(1,3)111While there are
generalizations to representations of SO(1,$n$), these are very complicated
and technically challenging [16, 17], thus spoiling the simplicity of the
usual dimensional regularization.. Because dim reg relies so heavily on the
“rotational” symmetry of spacetime, one is limited to computing finite volume
effects in only highly symmetric spacetimes, for example in the calculation
[18] of the finite size corrections to critical exponents, and it’s unclear
how to generalize dim reg to spacetimes with curvature [5]. By analytically
continuing the number of spacetime dimensions, one has difficulty defining the
$\gamma^{5}$ Dirac matrix [19, 20]. The usual BMHV choice for $\gamma^{5}$ in
dim reg, in which an infinite set of $\gamma^{\mu}$ are introduced, with
$\gamma^{5}$ anticommuting for $\mu=1,\,2,\,3,\,4$ and commuting otherwise,
explicitly breaks Lorentz invariance [19, 21]. Perhaps worst of all, one
simply cannot define the completely anti-symmetric Levi-Civita symbol in dim
reg. This latter catastrophe makes the determination of the axial anomaly in
dim reg at best extremely awkward, and generally requires abandoning manifest
Lorentz invariance [19, 20, 1].

Other regularization schemes only partially address dim reg’s shortcomings, or
have other shortcomings of their own. Dimensional reduction [4] modifies dim
reg such that only momenta are treated in $d$ dimensions, whereas $\gamma$
matrices and gauge fields remain ordinary 4-vectors. As a result,
supersymmetry is not broken in dimensional reduction; however, there are still
fundamental issues in defining and using $\gamma^{5}$ and, as a result,
dimensional reduction either does not yield the correct axial anomaly or
breaks supersymmetry and gauge invariance already at one loop order [22]. Zeta
regularization was invented to allow for an unambiguous way to renormalize in
curved spacetimes [5]. Operator regularization is the extension of zeta
regularization to higher loops [6]. However, zeta and operator regularization
in general violate BRS symmetry [23]. Analytic regularization analytically
continues the power of the denominator in momentum space propagators [7, 8,
9]. The procedure is more complicated than dim reg because the continuation of
the power introduces extra terms when combining denominators with Feynman
parameters; these additional terms also lead to a violation of gauge
invariance [23].

In denominator regularization, or den reg, denominators are first combined
using Feynman parameters. Then the overall power of the single denominator is
analytically continued from $n$ to $n+\epsilon$, where $\epsilon$ is taken
sufficiently large to ensure that the integral converges in the UV. All
subsequent manipulations—i.e. interchanging Feynman parameter integrations
with momentum integrations, Wick rotating, and ultimately integral
evaluation—are rigorously well defined and valid mathematical operations. Like
in dim reg, a fictitious scale $\mu$ is introduced to preserve the dimensions
of the amplitude. Thus the Green functions derived using den reg will
manifestly and straightforwardly satisfy Callan-Symanzik equations. The final
ingredient in den reg is the slight generalization of the analytic
continuation of the integrand to include in addition to the scale
$\mu^{2\epsilon}$ an overall coefficient function $f_{(n,p)}$ that smoothly
goes to 1 as $\epsilon\rightarrow 0$. This function depends only on
$\epsilon$; the original power of the denominator, $n$; and the superficial
degree of divergence of the integral, $p$. $f_{(n,p)}$ is _uniquely_ specified
by minimally requiring that the Laurent expansion of the amplitude in
$\epsilon$ has only a simple pole at $\epsilon=0$, including in the massless
limit. Thus the physics is fixed by the analytic properties of the amplitude
or, equivalently, by the requirement that the amplitude is finite for any
value of the convergence factor $\epsilon>0$. With the given scheme, gauge
invariance is maintained and the axial anomaly is correctly predicted
manifestly in den reg. At all times in the calculation the number of spacetime
dimensions is fixed. As a result, SUSY is preserved [4], there is no ambiguity
in the definition of $\gamma^{5}$, and one need only consider fields in
representations of SO(1,$i$), $i\in\mathbb{N}$ fixed. Since no where does den
reg rely on the symmetries of the spacetime, den reg is applicable to
asymmetric spacetimes, finite spacetimes, curved spacetimes, and thermal field
theory. We will show that the manipulations and results are as simple to apply
and arrive at as in dim reg.

##  II Den Reg in Scalar Theories

$p_{A}$$p_{B}$$p_{1}$$p_{2}$$p+k$$k$ Figure 1:  The $s$ channel diagram
contributing to the NLO correction to $2\rightarrow 2$ scattering in
$\phi^{4}$ theory, which we define as
$(-i\lambda)^{2}iV\big{(}(p_{1}+p_{2})^{2};\mu\big{)}$.

A precursor to den reg was first introduced in [24] in the context of
computing finite system size corrections to $2\rightarrow 2$ scattering in
massive $\phi^{4}$ theory at next-to-leading order (NLO). A relevant diagram,
Fig. (1), which is logarithmically divergent, is regulated as222There is a
subleading $\mathcal{O}(\epsilon)$ contribution from shifting the integration
variable in the logarithmically divergent integrand, which we may safely
ignore.

$\displaystyle V\rightarrow
i\frac{({-}\mu^{2})^{\epsilon}f_{(2,0)}(\epsilon)}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\int_{0}^{1}dx\frac{1}{(k^{2}-\Delta^{2})^{2+\epsilon}},$
$\displaystyle\Delta^{2}\equiv m^{2}-x(1-x)p^{2}-i\varepsilon,\qquad p\equiv
p_{1}+p_{2}$ (1)

The above integral is well defined for $\epsilon>0$. We introduced, as is done
in dim reg, a dimensionful scale $\mu$ that keeps the dimensions of $V$ a
constant. One should interpret the introduction of this $\mu$ as a part of the
overall analytic continuation of the integrand such that the original
integrand is reproduced as $\epsilon\rightarrow 0$. The minus sign multiplying
$\mu$ cancels the $({-}1)^{-\epsilon}$ introduced by den reg after Wick
rotation (that is absent in dim reg). The $f_{(2,0)}$ function indicates that
the original power of the denominator is 2 and the power of the numerator is
such that the integral is only logarithmically divergent. After exchanging
integration orders and Wick rotating, we need the following integral:

$\displaystyle\int
k_{E}^{3}dk_{E}\frac{\mu^{2\epsilon}}{(k_{E}^{2}+\Delta^{2})^{2+\epsilon}}=\frac{1}{2}\frac{1}{\epsilon(1+\epsilon)}\Big{(}\frac{\mu^{2}}{\Delta^{2}}\Big{)}^{\epsilon},$
(2)

which holds for general $m^{2},\,p^{2}\in\mathbb{R}$. Similar to dim reg, the
$1/\epsilon$ pole from den reg captures the logarithmic UV divergence of the
integral.

Taking $m\rightarrow 0$ and integrating over the Feynman $x$ yields the beta
function $B(1-\epsilon,1-\epsilon)$. In order to minimally cancel the IR poles
at $\epsilon\in\mathbb{N}^{+}$, we are uniquely led to take

$\displaystyle f_{(2,0)}(\epsilon)\equiv\Gamma^{-2}(1-\epsilon).$ (3)

Expanding in powers of $\epsilon$ about $\epsilon=0$ we find

$V(p^{2};\,\mu)={-}\frac{1}{2(4\pi)^{2}}\int_{0}^{1}dx\left[\frac{1}{\epsilon}-1-2\gamma_{E}+\ln\big{(}\frac{\mu^{2}}{\Delta^{2}}\big{)}\right]\\\
+\mathcal{O}(\epsilon),$ (4)

where $\gamma_{E}$ is the usual Euler constant. After slightly modifying
modified minimal subtraction to subtract the constant ${-}1-2\gamma_{E}$, the
result is identical to that from dim reg taking the usual $d=4-2\epsilon$.
Thus den reg explicitly respects unitarity to one loop in $\phi^{4}$ theory.

After determining a valuable analytic continuation of the Epstein zeta
function, one may compute the above NLO contribution [24] in a system confined
in a box of side lengths $L_{i}$, $i=1,\,2,\,3$. After a modified minimal
subtraction, the result is

$\overline{V}(p^{2},\\{L_{i}\\};\mu)=-\frac{1}{2}\frac{1}{(4\pi)^{2}}\int_{0}^{1}dx\left\\{\ln\frac{\mu^{2}}{\Delta^{2}}\right.\\\
\left.+2\sideset{}{{}^{\prime}}{\sum}_{\vec{n}\in\mathbb{Z}^{3}}e^{-2\pi\hskip
0.70004pti\hskip 0.70004ptx\sum
n_{i}p^{i}L_{i}}K_{0}\Big{(}2\pi\sqrt{\Delta^{2}\sum
n_{i}^{2}L_{i}^{2}}\Big{)}\right\\},$ (5)

where $K_{0}$ is the usual modified Bessel function. The above can be shown to
satisfy unitarity and reproduces the infinite volume result Eq. (4) in the
limit $L_{i}\rightarrow\infty$ [24]. One may then directly compute the Callan-
Symanzik equation, extract the beta function, and compute the finite system
size corrections to the running coupling [25].

##  III Den Reg in Gauge Theories:
Boson Self Energy

The QED and QCD gauge boson two point functions are interesting physically
because gauge invariance requires that they be transverse; i.e. the two point
functions must satisfy the Ward identity. Mathematically, the QED and QCD
gauge boson two point functions are interesting because they involve
quadratically divergent integrals.

$\mu$$\nu$$q$$q+k$$k$ Figure 2:  The NLO contribution to the photon two point
function in QED.

_QED_. Let us now apply den reg to the photon two point function in QED to one
loop, Fig. (2). One finds after combining the fermion propagator denominators,
shifting the momentum integral333There is a subleading $\mathcal{O}(\epsilon)$
contribution from shifting the integration variable in the quadratically
divergent integrand, which we may safely ignore., and discarding terms that
integrate to zero by symmetry that for a photon of incoming momentum
$q^{\mu}$,

$\displaystyle
i\Pi_{2}^{\mu\nu}=-4e^{2}\int\frac{d^{4}\ell}{(2\pi)^{4}}\int_{0}^{1}dx$
$\displaystyle\frac{2\ell^{\mu}\ell^{\nu}-\eta^{\mu\nu}\ell^{2}-2x(1-x)q^{\mu}q^{\nu}+\eta^{\mu\nu}\big{(}x(1-x)q^{2}+m^{2}\big{)}}{(\ell^{2}-\Delta^{2})^{2}}$
$\displaystyle\rightarrow-4e^{2}\int\frac{d^{4}\ell}{(2\pi)^{4}}\int_{0}^{1}dx({-}\mu^{2})^{\epsilon}\Big{[}-\frac{1}{2}\eta^{\mu\nu}f_{(2,2)}(\epsilon)\ell^{2}$
$\displaystyle-
f_{(2,0)}(\epsilon)[2x(1-x)q^{\mu}q^{\nu}-\eta^{\mu\nu}\big{(}x(1-x)q^{2}+m^{2}\big{)}]\Big{]}$
$\displaystyle\hskip 126.47249pt\times(\ell^{2}-\Delta^{2})^{-(2+\epsilon)},$
(6) $\displaystyle\Delta^{2}\equiv m^{2}-x(1-x)q^{2}-i\varepsilon.$

In the last line we used the symmetry of the integrand to replace
$\ell^{\mu}\ell^{\nu}$ by $\eta^{\mu\nu}\ell^{2}/4$. (In dim reg, the
replacement is $\ell^{\mu}\ell^{\nu}\rightarrow\eta^{\mu\nu}\ell^{2}/d$; it is
precisely this replacement that makes dim reg manifestly gauge invariant.)
Note that due to the quadratic divergence, we have a new analytic continuation
function $f_{(2,2)}(\epsilon)$ and the right hand side of Eq. (III) is only
well defined for $\epsilon>1$.

After exchanging integration orders and Wick rotating, we need the following
integral

$\int\ell_{E}^{3}d\ell_{E}\frac{\mu^{2\epsilon}\ell_{E}^{2}}{(\ell_{E}^{2}+\Delta^{2})^{2+\epsilon}}={-}\frac{\Delta^{2}}{(1-\epsilon)\epsilon(1+\epsilon)}\Big{(}\frac{\mu^{2}}{\Delta^{2}}\Big{)}^{\epsilon},$
(7)

which holds for general $m^{2},\,p^{2}\in\mathbb{R}$. Notice how the quadratic
divergence is captured by the $1/(1-\epsilon)$ pole, which diverges as
$\epsilon\rightarrow 1$, exactly at the value of $\epsilon$ at which the
original integral fails to converge. In order to cancel this quadratic UV
divergence we must take $f_{(2,2)}(\epsilon)\sim 1-\epsilon$.444Note that this
UV pole cancellation is equivalent to what happens manifestly in dim reg. In
dim reg, the quadratically divergent integral produces a
$\Gamma\big{(}1-\frac{2}{d}\big{)}$, which diverges logarithmically at $d=2$.
But the replacement
$2\ell^{\mu}\ell^{\nu}-\eta^{\mu\nu}\ell^{2}\rightarrow\big{(}1-\frac{d}{2}\big{)}\eta^{\mu\nu}\ell^{2}/d$
gives an overall prefactor such that
$\big{(}1-\frac{2}{d}\big{)}\Gamma\big{(}1-\frac{2}{d}\big{)}=\Gamma\big{(}2-\frac{2}{d}\big{)}$,
which cancels the logarithmic divergence at $d=2$ and effectively softens the
quadratic divergence at $d=4$ to only a logarithmic divergence. When combined
with the logarithmically divergent contributions, integration over the Feynman
$x$ in the $m\rightarrow 0$ limit again yields $B(1-\epsilon,1-\epsilon)$,
which uniquely fixes

$\displaystyle f_{(2,2)}(\epsilon)\equiv(1-\epsilon)\Gamma^{-2}(1-\epsilon).$
(8)

For completeness, one finds in den reg that

$\displaystyle\Pi_{2}^{\mu\nu}=-\frac{2\alpha}{\pi}\frac{\eta^{\mu\nu}q^{2}-q^{\mu}q^{\nu}}{\epsilon(1+\epsilon)\Gamma^{2}(1-\epsilon)}\int_{0}^{1}dx\,x(1-x)\Big{(}\frac{\mu^{2}}{\Delta^{2}}\Big{)}^{\epsilon}.$
(9)

Note crucially the relative factor of ${-}1/2$ between the logarithmically
divergent integral Eq. (2) and the quadratically divergent integral Eq. (7).
This relative factor is identical to the dim reg case. As a result, all orders
proofs of gauge invariant regularization in the den reg case will go through
identically to the dim reg case.

$a,\,\mu$$b,\,\nu$$q$$q+P$$P$$P$

$q+P$$P$

Figure 3:  The NLO contributions to the gluon two point function in Yang-Mills
theory: the three gluon vertex, four gluon vertex, and ghost loop.

_Yang-Mills_. The three diagrams contributing to the gluon two point function
in Yang-Mills theory are given in Fig. (3). In Feynman gauge the den reg’d
three gluon vertex bubble diagram is

$-\frac{g^{2}}{2}C_{2}(G)\delta^{ab}\int\frac{d^{4}P}{(2\pi)^{4}}\int_{0}^{1}dx({-}\mu^{2})^{\epsilon}\Big{[}-\frac{9}{2}f_{(2,2)}(\epsilon)\eta^{\mu\nu}P^{2}\\\
+f_{(2,0)}(\epsilon)\big{[}2q^{\mu}q^{\nu}(1+5x-5x^{2})-\eta^{\mu\nu}q^{2}(5-2x+2x^{2})\big{]}\Big{]}\\\
(P^{2}-\Delta^{2})^{-(2+\epsilon)},$ (10)

the four gluon vertex bubble diagram can be manipulated to

$-g^{2}C_{2}(G)\delta^{ab}\int\frac{d^{4}P}{(2\pi)^{4}}\int_{0}^{1}dx({-}\mu^{2})^{\epsilon}\\\
\frac{3\eta^{\mu\nu}\big{(}f_{(2,2)}(\epsilon)P^{2}+f_{(2,0)}(\epsilon)(1-x)^{2}q^{2}\big{)}}{(P^{2}-\Delta^{2})^{2+\epsilon}},$
(11)

and the ghost loop diagram is

$-g^{2}C_{2}(G)\delta^{ab}\int\frac{d^{4}P}{(2\pi)^{4}}\int_{0}^{1}dx({-}\mu^{2})^{\epsilon}\\\
\frac{\frac{1}{4}\eta^{\mu\nu}f_{(2,2)}(\epsilon)P^{2}-f_{(2,0)}(\epsilon)x(1-x)q^{2}}{(P^{2}-\Delta^{2})^{2+\epsilon}},$
(12)

where $\Delta^{2}\equiv-x(1-x)q^{2}-i\varepsilon$. We have already computed
the relevant integrals, Eq. (2) and Eq. (7). If we use the same minimal
$f_{(2,0)}$ and $f_{(2,2)}$, then the full one loop Yang-Mills self energy is

$\Pi_{2}^{\mu\nu\,ab}=C_{2}(G)\delta^{ab}\frac{\alpha_{s}}{4\pi}\frac{\eta^{\mu\nu}q^{2}-q^{\mu}q^{\nu}}{\epsilon(1+\epsilon)\Gamma^{2}(1-\epsilon)}\\\
\times\int_{0}^{1}dx(3-4x^{2})\Big{(}\frac{\mu^{2}}{\Delta^{2}}\Big{)}^{\epsilon},$
(13)

where, by seeing that $\Delta^{2}$ is symmetric under $x\leftrightarrow 1-x$,
we have replaced $x\rightarrow 1/2$ in the numerator. Note that the $3-4x^{2}$
makes the integral over the Feynman $x$ in the $m\rightarrow 0$ limit more
complicated than in the QED case, but does not introduce any new poles in
$\epsilon$.

We thus see that the exact same choices for the analytic continuation of the
divergent integrals in the photon two point function, which were dictated by
the analytic structure of the two point function in $\epsilon$, automatically
yield a gauge invariant non-Abelian two point function (at one loop order).

Expanding in $\epsilon$ and performing a modified minimal subtraction, one
sees that the NLO Yang-Mills self-energy in den reg is identical to that given
by dim reg.

##  IV Axial Anomaly

Physically, the axial anomaly is interesting because no previous
regularization scheme naturally, manifestly yields the correct value for the
anomaly. (As an example, in dim reg, the definition of $\gamma^{5}$ in $d\neq
4$ dimensions is ambiguous. The usual choice made breaks manifest Lorentz
invariance [1, 19]. In any case, computing the axial anomaly in dim reg is at
best extremely subtle [20].) Mathematically, the anomaly is interesting
because the amplitude involves (superficially) linear divergences.

$q_{\mu}$$p_{\nu}$$k_{\lambda}$$\ell+p$$\ell$$\ell-k$$q_{\mu}$$k_{\lambda}$$p_{\nu}$$\ell+k$$\ell-p$$\ell$

Figure 4:  The leading order diagrams that contribute to the axial anomaly.
The incoming photon momentum $q_{\mu}$ is treated as an off-shell, internal
line. The outgoing photon momenta $p_{\nu}$ and $k_{\lambda}$ are on-shell.

One of the two triangle diagrams with incoming, off-shell photon momentum
$q^{\mu}$ and outgoing, on-shell photon momenta $p^{\nu}$ and $k^{\lambda}$,
shown in Fig. (4), gives

$\displaystyle i\mathcal{M}^{\mu\nu\lambda}(p,k)$
$\displaystyle=e^{2}\int\frac{d^{4}\ell}{(2\pi)^{4}}\mbox{tr}\Big{[}\gamma^{\mu}\gamma^{5}\frac{i}{\not{\ell}-\not{k}-m+i\varepsilon}$
$\displaystyle\qquad\qquad\qquad\qquad\gamma^{\lambda}\frac{i}{\not{\ell}-m+i\varepsilon}\gamma^{\nu}\frac{i}{\not{\ell}+\not{p}-m+i\varepsilon}\Big{]}$
$\displaystyle={-}ie^{2}\int\frac{d^{4}\ell}{(2\pi)^{4}}\Big{\\{}{-}\mbox{tr}[\mu
5\lambda\nu\alpha](\ell^{2}-m^{2})(\ell_{\alpha}+p_{\alpha}-k_{\alpha})$
$\displaystyle\qquad\qquad+2\mbox{tr}[\mu
5\beta\nu\alpha]p_{\alpha}\ell^{\lambda}\ell_{\beta}-2\mbox{tr}[\mu
5\alpha\lambda\beta]k_{\alpha}\ell^{\nu}\ell_{\beta}$
$\displaystyle\qquad\qquad\qquad\qquad-\mbox{tr}[\mu
5\alpha\lambda\beta\nu\gamma]k_{\alpha}\ell_{\beta}p_{\gamma}\Big{\\}}$
$\displaystyle\times\frac{1}{[(\ell-k)^{2}-m^{2}+i\varepsilon][\ell^{2}-m^{2}+i\varepsilon][(\ell+p)^{2}-m^{2}+i\varepsilon]},$
(14)

where for simplicity we show only the contravariant indices of the Dirac
$\gamma$ matrices in the traces. The other triangle diagram contribution can
be found from $i\mathcal{M}^{\mu\lambda\nu}(k,p)$, known as Bose
symmetrization. We wish to compute the axial Ward anomaly
$iq_{\mu}i\mathcal{M}^{\mu\nu\lambda}$ and the vector Ward identities
$p_{\nu}\mathcal{M}^{\mu\nu\lambda}$ and
$k_{\lambda}\mathcal{M}^{\mu\nu\lambda}$. One can easily show that
$p_{\mu},\,p_{\nu},\,k_{\mu},$ or $k_{\lambda}$ dotted into the last term in
the curly bracket of Eq. (14) is zero. Thus for our purposes we may safely
ignore this term. To evaluate the other terms, we require a number of new
integrals. After combining denominators and analytically continuing, we have
for the most important integrals555We have suppressed all contributions
$\mathcal{O}(\epsilon)$ in the following

$\displaystyle I_{0}$
$\displaystyle\equiv-i\int\frac{d^{4}\ell}{(2\pi)^{4}}\int_{0}^{1}dx\int_{0}^{1-x}dy$
$\displaystyle\qquad\qquad\frac{2f_{(3,0)}(\epsilon)(-\mu^{2})^{\epsilon}\ell^{2}}{[\ell^{2}+2xp\cdot\ell-2yk\cdot\ell-m^{2}+i\varepsilon]^{3+\epsilon}},$
(15)
$\displaystyle=\frac{2\pi^{2}}{(2\pi)^{4}}\int_{0}^{1}dx\int_{0}^{1-x}dy\tilde{I}_{0},$
$\displaystyle\tilde{I}_{0}$
$\displaystyle\equiv\frac{f_{(3,0)}(\epsilon)}{(1+\epsilon)(2+\epsilon)}\left[\frac{2}{\epsilon}+\frac{2xyp\cdot
k}{\Delta^{2}}\right]\left(\frac{\mu^{2}}{\Delta^{2}}\right)^{\epsilon}$ (16)
$\displaystyle I_{0,\alpha\beta}$
$\displaystyle\equiv-i\int\frac{d^{4}\ell}{(2\pi)^{4}}\int_{0}^{1}dx\int_{0}^{1-x}dy$
$\displaystyle\qquad\quad\frac{2f_{(3,0)}(\epsilon)(-\mu^{2})^{\epsilon}\ell_{\alpha}\ell_{\beta}}{[\ell^{2}+2xp\cdot\ell-2yk\cdot\ell-m^{2}+i\varepsilon]^{3+\epsilon}},$
(17)
$\displaystyle=\frac{2\pi^{2}}{(2\pi)^{4}}\int_{0}^{1}dx\int_{0}^{1-x}dy\tilde{I}_{\eta}\eta_{\alpha\beta}+\tilde{I}_{xy}(p_{\alpha}k_{\beta}+p_{\beta}l_{\alpha})$
$\displaystyle\hskip
90.3375pt-\tilde{I}_{x^{2}}p_{\alpha}p_{\beta}+\tilde{I}_{y^{2}}k_{\alpha}k_{\beta},$
(18) $\displaystyle\tilde{I}_{\eta}$
$\displaystyle\equiv\frac{f_{(3,0)}(\epsilon)}{(1+\epsilon)(2+\epsilon)}\left[\frac{1}{2\epsilon}\right]\left(\frac{\mu^{2}}{\Delta^{2}}\right)^{\epsilon},$
(19) $\displaystyle\tilde{I}_{x^{i}y^{j}}$
$\displaystyle\equiv\frac{f_{(3,0)}(\epsilon)}{(1+\epsilon)(2+\epsilon)}\left[\frac{x^{i}y^{j}}{2\epsilon}\right]\left(\frac{\mu^{2}}{\Delta^{2}}\right)^{\epsilon},$
(20) $\displaystyle I_{1,\alpha}$
$\displaystyle\equiv-i\int\frac{d^{4}\ell}{(2\pi)^{4}}\int_{0}^{1}dx\int_{0}^{1-x}dy$
$\displaystyle\qquad\qquad\frac{2f_{(3,1)}(\epsilon)(-\mu^{2})^{\epsilon}\ell^{2}\ell_{\alpha}}{[\ell^{2}+2xp\cdot\ell-2yk\cdot\ell-m^{2}+i\varepsilon]^{3+\epsilon}},$
(21)
$\displaystyle=\frac{2\pi^{2}}{(2\pi)^{4}}\int_{0}^{1}dx\int_{0}^{1-x}dy\tilde{I}_{1}(p_{\alpha}-k_{\alpha}),$
$\displaystyle\tilde{I}_{1}$ $\displaystyle\equiv
f_{(3,1)}(\epsilon)\Big{\\{}\frac{x}{2}$
$\displaystyle\quad\quad\quad-\frac{x}{(1+\epsilon)(2+\epsilon)}\Big{[}\frac{3}{\epsilon}+\frac{2xyp\cdot
k}{\Delta^{2}}\Big{]}\Big{(}\frac{\mu^{2}}{\Delta^{2}}\Big{)}^{\epsilon}\Big{\\}},$
(22) $\displaystyle\Delta^{2}$ $\displaystyle\equiv m^{2}-2xyp\cdot
k-i\varepsilon.$ (23)

Of the above integrals, the only non-trivial one is the (superficially)
linearly divergent one, Eq. (IV). Integrals that diverge like odd powers in
the UV are subtle because shifting the dummy integration variable can leave a
finite remainder [26, 27]. Following [27], one may derive the exact result

$S^{\mu}(a)\equiv{-}i\int\frac{d^{4}\ell}{(2\pi)^{4}}\left(\frac{\ell^{2}(\ell^{\mu}+a^{\mu})}{[\ell+a)^{2}-M^{2}+i\varepsilon]^{3}}\right.\\\
\left.-\frac{\ell^{2}\ell^{\mu}}{[\ell^{2}-M^{2}+i\varepsilon]^{3}}\right)=\frac{2\pi^{2}}{(2\pi)^{4}}\frac{1}{4}a^{\mu}.$
(24)

If one evaluates $S^{\mu}$ in den reg, one can show that
$S^{\mu}_{den}(a;\,\epsilon)=\frac{2\pi^{2}}{(2\pi)^{4}}\frac{1}{4}a^{\mu}+\mathcal{O}(\epsilon)$.
As an aside, shockingly, if one computes $S^{\mu}$ in dim reg, one finds that
the result is _identically zero_ for any $d<4$; i.e. even though the
integrands in the dimensionally regularized result smoothly go to the
integrands of Eq. (24) as $\epsilon\rightarrow 0$, it is _impossible_ to
correctly derive Eq. (24) by smoothly taking the $\epsilon\rightarrow 0$ limit
of the integral, $S^{\mu}_{dim}$. The origin of this discontinuity is the
replacement $\ell^{\mu}\ell^{\nu}\rightarrow(1/d)\eta^{\mu\nu}\ell^{2}$. For
$d=4$ there is an exact cancellation that reduces the highest power in the
numerator by 1. For $d<4$ there is no highest power cancellation, and the
extra contribution to the integral _exactly_ cancels all the other
contributions.

Eq. (IV) contains only a $1/\epsilon$ pole, indicating that Eq. (IV) is
actually only logarithmically divergent. Thus none of the $f_{(n,p)}$ have
contributions from UV poles and are fixed entirely by the $m\rightarrow 0$ IR
physics. In order to minimally cancel the IR poles from the beta functions
that result from integrating over the Feynman $x$ and $y$ parameters when
$m=0$, we are uniquely led to

$\displaystyle f_{(3,0)}$ $\displaystyle\equiv\Gamma^{-2}(1-\epsilon)$ (25)
$\displaystyle f_{(3,1)}$
$\displaystyle\equiv\Gamma^{-1}(1-\epsilon)\Gamma^{-1}(2-\epsilon).$ (26)

Then, automatically and straightforwardly, one finds that

$\displaystyle
p_{\nu}i\mathcal{M}^{\mu\nu\lambda}=k_{\lambda}i\mathcal{M}^{\mu\nu\lambda}=\mathcal{O}(\epsilon),\qquad
m\geq 0$ (27) $\displaystyle
iq_{\mu}i\mathcal{M}^{\mu\nu\lambda}=\frac{e^{2}}{4\pi^{2}}\epsilon^{\alpha\lambda\beta\nu}k_{\alpha}p_{\beta}+\mathcal{O}(\epsilon),\qquad
m=0.$ (28)

Thus den reg manifestly maintains gauge invariance for all $m$ and gives the
correct $m=0$ axial anomaly [1, 26, 27].

##  V Conclusions and Outlook

We introduced the denominator regularization scheme in quantum field theory.
In den reg, the power of the denominator in an amplitude, after combined by
Feynman parameters, is analytically continued; the number of spacetime
dimensions is always fixed. The main ingredient to den reg is the further
analytic continuation of the integrand by coefficient functions
$f_{(n,p)}(\epsilon)$ that depend on the original power $n$ of the denominator
and of the superficial degree of divergence of the integral, $p$. The
$f_{(n,p)}$ are uniquely fixed by minimally requiring that the Laurent
expansion of the amplitude is free of poles other than a single $1/\epsilon$;
i.e. we choose $f_{(n,p)}$ such that the amplitude converges for all
$\epsilon>0$. Crucially, we require the $f_{(n,p)}$ to cancel UV poles of the
form $(\frac{p}{2}-\epsilon)^{-1}$ _and_ IR poles that emerge for
$\epsilon\in\mathbb{N}^{+}$ when the theory is massless. With this scheme in
place, we showed that den reg manifestly maintains Lorentz and gauge
invariance, preserves SUSY, and manifestly predicts the axial anomaly. As in
dim reg, a fictitious scale $\mu$ is introduced to keep the dimensions of
amplitudes fixed. Thus Green functions computed in den reg satisfy the Callan-
Symanzik equation. No other regularization scheme satisfies all these
criteria.

The integrals that emerge in den reg are of a type and difficulty similar to
dim reg. As a result, den reg is thus well suited, as done here, to the
minimal subtraction renormalization scheme, and the determination of the
$f_{(n,p)}$ is straightforward and easy. So far, the $f_{(n,p)}$ appear to be
universal; once fixed for an integral in one amplitude, the same $f_{(n,p)}$
emerges from other amplitudes. We are thus led to speculate that the
$f_{(n,p)}$ are, in fact, universal.

It will be interesting to apply den reg to other divergent amplitudes. Den reg
and dim reg will give manifestly different results for linearly and higher
divergent amplitudes in scalar theories: in dim reg, these divergences will
keep their same superficial degree of divergence, while den reg will soften
all UV divergences to logarithmic. We speculate that den reg will manifestly
conserve the energy momentum tensor and simultaneously manifestly predict the
correct Weyl anomaly. We have checked that den reg correctly reproduces the
leading order results in thermal field theory [28, 29]. It will also be
interesting to apply den reg to Chern-Simons theory [30], quantities in curved
spacetime [5], and to compute the finite size corrections to running couplings
[25] and critical exponents in asymmetric systems, especially in the
universality class of $\phi^{4}$ theory through the resummed two point
function [31]. The latter may provide valuable insight, e.g., in detecting the
critical endpoint of the QCD phase diagram from measurements of particle
fluctuations in hadronic collisions [32, 33].

## Acknowledgments

The author wishes to thank the South African National Research Foundation and
the SA-CERN Collaboration for support. The author wishes to thank Jean Du
Plessis, Matthew Sievert, Alexander Rothkopf, Robert de Mello Koch, Kurt
Hinterbichler, Bowen Xiao, Jonathan Shock, and Kevin Bassler for valuable
discussions.

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