††institutetext: School of Physics and Astronomy, University of Nottingham,
University Park, Nottingham NG7 2RD, United Kingdom††institutetext: The
Nottingham Centre of Gravity, University of Nottingham, Nottingham NG7 2RD, UK

# Smoothed asymptotics: from number theory to QFT

Antonio Padilla  and Robert G. C. Smith<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Inspired by the method of smoothed asymptotics developed by Terence Tao, we
introduce a new ultra-violet regularisation scheme for loop integrals in
quantum field theory which we call $\eta$ regularisation. This allows us to
reveal a surprising connection between the elimination of divergences in
divergent series of powers and the preservation of gauge invariance in the
regularisation of loop integrals in quantum field theory. In particular, we
note that a method for regularising the series of natural numbers so that it
converges to minus one twelfth inspires a regularisation scheme for non-
abelian gauge theories coupled to Dirac fermions that preserves the Ward
identity for the vacuum polarisation tensor. We also comment on a possible
connection to Schwinger proper time integrals.

##  1 Introduction

The problem of infinity dates back to the sixth century BCE when Anaximander,
a tutor of Pythagoras, conjured up apeiron, an indefinite and limitless source
from which everything is born and to which everything will return
moore2018infinite . However, the ancient Greeks did not embrace the infinite
with Aristotle accepting only the potential for infinity, rejecting it in
actuality. Centuries later, Gauss, perhaps the greatest mathematician of the
modern era, warned that “the use of an infinite quantity as a completed one …
is never permissible in mathematics. The infinite is only a façon de parler,
where one is really speaking of limits where certain ratios come as close as
one likes, while others are allowed to grow without restriction”
waterhouse1979gauss .

As physicists, how can we heed these warnings and at the same time make sense
of divergent series and integrals that emerge unapologetically in the
mathematics we use to describe fundamental physics? Perturbative quantum field
theory (QFT) is well established as a microscopic theory of the fundamental
interactions, enjoying predictive power and stunning experimental success
ParticleDataGroup:2022pth , and yet the presence of divergences is well
documented. Indeed, as Dyson noted (Dyson52, ), the perturbative expansion
used in quantum electrodynamics does not converge, even after renormalisation.
Although this does not hinder its predictive power when the expansion
parameter is small, it does raise concerns around the concepts upon which the
theory is built. Many notable advances have been made in this regard using
Borel summation, particularly under the heading of resurgence theory
(Dorigoni19, ; Dunne15qft, ; Marino14, ). Here the usual perturbative
expansion is replaced with a trans-series expansion, including non-
perturbative instanton contributions.

Even at finite order in perturbative QFT, we encounter divergences from the
integration over momenta running through loops. These problems were first
identified for quantum electrodynamics by Oppenheimer Oppenheimer:1930zz  and
later solved using the method of renormalisation developed by Tomonoga
Tomonaga:1946zz , Schwinger Schwinger:1948iu  and Feynman Feynman:1948fi , and
extended to non-abelian gauge theories by ’t Hooft and Veltman tHooft:1972tcz
. As for General Relativity, this works well as a perturbative quantum field
theory at scales below the Planck scale, but cannot be extended to arbitrarily
high energies as the theory is known to be perturbatively non-renormalisable.
Said another way, for General Relativity, the number of counter-terms required
to renormalise the divergences arising from loops is itself divergent!

Adopting Aristotle’s philosophy in rejecting actual infinity, at least in a
natural context, one might hope that the correct theory of quantum gravity
coupled to matter will be free of all such divergences. If this is the case,
it may point towards a preferred way of regularising the divergences in the
low energy QFT. String theory has certainly provided multiple insights in this
regard, particularly in terms of a natural exponential damping of ultra-violet
(UV) divergences, which can be seen, for instance, in the modular invariance
of the worldsheet theory (dhoker22, ; Polchinski98, ). Further, in the Gross-
Mende regime (GrossMende87, ; Gross:1987ar, ) where the high energy, fixed
angle, behaviour of string scattering amplitudes is considered, the sum over
all Riemann surfaces is dominated by a saddle point, and we recover the
exponential damping from a stringy tower of states. With a view to
understanding this from a simple particle perspective, Abel and others
(Abel19, ; Abel20, ) have constructed UV complete particle theories based on
the Schwinger representation and worldline formalism. This includes particle
theories with a tower of worldline internal degrees of freedom that mimic
stringy behaviour in the UV, from Gross-Mende saddle points to modular
invariance (Abel20, ). Finally, string field theory has revealed how such
exponential damping comes partly from adding stubs to loop diagrams (Erbin21,
), which can also be translated to the point particles of QFT (Chiaffrino22,
).

Mathematically rigorous and axiomatic formulations of perturbative QFT have
progressed significantly in recent decades, aiding our understanding on the
nature of UV divergences, with the view that all regularisation in QFT is
really just an effort in obtaining sensible results for products of
distributions (Latorre94, ). But, with this progress and insight, not much has
been said fundamentally about regularisation besides it being a useful, if not
altogether ad hoc, tool. This philosophy does not align with the hopes
outlined in the previous paragraph and the quest to identify a preferred
method of regularisation inherited from string theory, or some other
consistent quantum theory of gravity. Where else can we take our inspiration?
One possibility is through analytic number theory and the study of divergent
series. In particular, we ask if there are clever ways to regularise these
series that eliminate the divergences altogether and if so, can we connect
them, via QFT, to the softening of scattering amplitudes at high energies in a
fundamental microscopic theory of nature?

A particularly interesting approach to divergent series is the method of
smoothed asymptotics, elegantly discussed by Tao (Tao11, ). For monomial
series, this makes use of the Euler-Maclaurin summation formula and as such is
closely related to Ramanujan’s methods (Berndt85, ; Hardy73, ). The importance
of smoothed asymptotics in understanding divergent series is best illustrated
with an example. Consider the infinite series of natural numbers,
$\sum_{n=1}^{\infty}n$, which was famously assigned the seemingly absurd value
of $-1/12$ by Ramanujan. The high school technique for evaluating infinite
series is to take the limit of partial sums, which for the sum of natural
numbers yields, $\sum_{n=1}^{N}n=\frac{1}{2}N(N+1)$. There is no sign of
$-1/12$ in this expression, leading many to scoff at Ramanujan’s claim.
However, this is simply an artefact of the discontinuities that occur as we
increase the cut-off $N$. To remedy this, we note that partial sums are really
just infinite series where each term is weighted by a step function
$\theta(n/N)$ that equates to unity for $n\leq N$ and vanishes for $n>N$.
However, suppose we replace the step function with another regulator function,
$\eta(n/N)$, where $\eta(x)$ is a bounded smooth function with compact support
on the non-negative real line and with $\eta(0)=1$ and $\eta(x)\to 0$ at large
$x$. Tao shows that
$\sum_{n=1}^{\infty}n\eta(n/N)=C_{1}[\eta]N^{2}-1/12+\mathcal{O}(1/N)$, where
$C_{1}[\eta]=\int_{0}^{\infty}dx\ x\eta(x)$ is the Mellin transform of the
regulator function. As we will explicitly show, the same method can also be
extended to regulator functions that are Schwartz class.

Tao’s results have several interesting features. The first concerns the
divergence as $N\to\infty$. Unlike the partial sum, there is no linear
divergence, while the quadratic divergence is dependent on the choice of
regulator. This is in stark contrast to the finite term which is universal,
returning Ramanujan’s famous result. These features extend to infinite series
of polynomials and are reminiscent of a well known result regarding UV
divergences in QFT. In particular, when we compute the one loop effective
action for a theory cut-off at some scale, $\Lambda$, the precise couplings
associated with power law divergences are meaningless. This is in contrast to
the logarithmic divergences where the couplings are universal, much like the
finite terms in regularising divergent series of polynomials using smoothed
asymptotics. This overlap suggests that one loop divergences in QFT may be
connected to divergent series.

The power law divergences that emerge from regularising a series with smoothed
asymptotics can also be eliminated with a suitable choice of regulator. We
call these enhanced regulators. For the series of natural numbers, an enhanced
regulator is one for which the corresponding Mellin transform is vanishing
$C_{1}[\eta]=0$. As we will show, such enhanced regulators are relatively easy
to find and we present several algorithms for finding them. A particularly
elegant choice is the enhanced regulator
$\eta(n/N)=e^{-\frac{n}{N}}\cos(n/N)$. When this regulator is used for the
series of natural numbers the result converges to $-1/12$ and there is no
divergence whatsoever!

To build these ideas into QFT, we introduce the concept of $\eta$
regularisation. Here the integrand in (Euclidean) loop integrals is weighted
by a regulator function $\eta(|k|/\Lambda)$, where $|k|$ is the norm of the
loop momentum and $\Lambda$ is the cut-off. It is convenient to work directly
with one fold irreducible loop integrals (ILIs), introduced by Wu Wu03  as the
basic building block of all one particle irreducible graphs. When we implement
$\eta$ regularisation we see, as expected, that power law divergences are
regulator dependent while logarithmic divergences are universal. As with
divergent series and smoothed asymptotics, the power law divergences are
multiplied by Mellin transforms of the regulator. By choosing enhanced
regulators, we can eliminate the power law divergences at will.

However, the real question is whether or not the elimination of power law
divergences by enhanced regulators has any deeper meaning from a QFT
perspective. Remarkably, we find that the answer is yes. In a series of papers
Wu03 ; Wu04 ; Wu12 ; Wu13 ; Wu14 ; Wu17 , Wu and others have derived a set of
consistency relations that are necessary in order for the regulator to
preserve gauge invariance. Requiring that these hold for $\eta$
regularisation, we find that certain regulators must be enhanced. In other
words, regulators that allow the infinite series of natural numbers to
converge towards $-1/12$ are intimately connected to the preservation of gauge
invariance at one loop in a wide class of non-abelian gauge theories coupled
to an arbitrary number of Dirac fermions. This is the main result of our work.

The rest of the paper is organised as follows: in section 2, we review several
ideas from analytic number theory and the study of divergent series. We
introduce a number of important concepts, including a brief review of the
Euler-Maclaurin summation formula and Ramanujan summation in section 2.1. In
section 2.2, we review Tao’s work on smoothed asymptotics, extending the
analysis to include regulators that are Schwartz and to series of general
polynomials. We also introduce the concept of enhanced regulators that
eliminate divergences altogether and present several algorithms for finding
them, including one that is inspired by the Schwinger proper time formalism.
In section 3, we switch gears to divergent integrals in QFT, drawing analogies
between these and the results that emerged from regularising divergent series
with smoothed asymptotics. We formally introduce $\eta$ regularisation in
section 3.1 and study the implications for irreducible loop integrals in
section 3.1.1. In section 3.2 we implement Wu’s consistency conditions,
demonstrating the role of enhanced regulators in preserving gauge invariance.
We comment explicitly on the connection to Schwinger proper time in section
3.2.1. In section 4, we conclude.

##  2 Divergent series: a physicist’s review

Let us begin with the following well known expressions for the finite sums of
powers:

$\displaystyle\sum_{n=1}^{N}n$ $\displaystyle=$
$\displaystyle\frac{1}{2}N+\frac{1}{2}N^{2},$ (1)
$\displaystyle\sum_{n=1}^{N}n^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{6}N+\frac{1}{2}N^{2}+\frac{1}{3}N^{3},$ (2)
$\displaystyle\sum_{n=1}^{N}n^{3}$ $\displaystyle=$
$\displaystyle=\frac{1}{4}N^{2}+\frac{1}{2}N^{3}+\frac{1}{4}N^{4}.$ (3)

These relationships date back more than two millennia. The first of them,
corresponding to a sum of natural numbers, can be traced to the Pythagorean
school in the 6th century BCE (Peng02a, ; Sant17, ). Archimedes of Syracuse
(circa 287-212 BCE), considered the greatest mathematician of antiquity,
discovered the second relationship for a sum of the squares. The sum of cubes
can be found in the work of Nicomachus of Gerasa (circa 60 -120 CE), along
with the remarkable theorem that bears his name

$\sum\limits_{n=1}^{N}n^{3}=\left(\sum\limits_{n=1}^{N}n\right)^{2}.$

The expressions for the three finite sums (1) to (3) are special cases of
Faulhaber’s formula Knuth_1993

$\sum_{n=1}^{N}n^{z}=\frac{1}{z+1}\sum_{n=0}^{z}\binom{z+1}{n}B_{n}N^{z-n+1},$
(4)

where $z$ is a positive integer and $B_{n}$ are the Bernoulli numbers (with
$B_{1}=1/2$), which are defined by the following recursion relation

$\sum_{n=0}^{z}\binom{z+1}{n}B_{n}=z+1,\qquad B_{0}=1$ (5)

or, equivalently, from the exponential generating function

$\frac{t}{1-e^{-t}}=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!}.$ (6)

In the limit where we take $N\to\infty$, Faulhaber’s formula will obviously
break down. This is because the infinite series
$S(z)=\sum_{n=1}^{\infty}n^{z}$ is known to be divergent in the Cauchy sense
whenever $\Re(z)\geq-1$. According to Cauchy’s definition, an infinite series
is said to be convergent if the sequence of partial sums converges to some
finite limit; otherwise, the series is divergent. If the series is not
convergent in the Cauchy sense, then it will generally be considered as one of
two types of divergent series (Hardy73, ; Graham91ConcreteMaths, ): (i) a
series that grows in absolute value without limit, or (ii) a series that is
bounded but whose sequence of partial sums does not approximate any specific
value.

At high school we are taught to think of an infinite series,
$\sum_{n=1}^{\infty}a_{n}$ as the limit of its partial sums,
$\lim_{N\to\infty}\sum_{n=1}^{N}a_{n}$. Partial sums are favoured because they
allow us to perform standard arithmetic without issue. Of course, the method
works well for convergent series, less so for divergent series. With the
latter the task is to develop a summation method that shares the most
important properties of partial sums but allows the result to be generalised
to infinity without giving a divergent answer. Alternative summation methods
include Cesaro summation, where we compute the sequence of partial sums
$s_{N}=\sum_{n=1}^{N}a_{n}$ and find the limiting value of their average
$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}s_{n}$; and Abel summation, where
we compute $\lim_{t\to 1^{-}}\sum_{n=1}^{\infty}a_{n}t^{n}$. Hardy has argued
that any new summation method should satisfy three properties: regularity,
linearity and stability (Hardy73, ). Regularity states that a summation method
yields the known results for convergent series obtained using partial sums.
For linearity, we require that $\sum_{n=1}^{\infty}{\lambda a_{n}+\mu
b_{n}=}\lambda\sum_{n=1}^{\infty}a_{n}+\mu\sum_{n=1}^{\infty}b_{n}$. The third
property of stability, namely
$\sum_{n=1}^{\infty}a_{n}=a_{1}+\sum_{n=1}^{\infty}a_{n+1}$, is considered
less crucial and known to fail in some important cases, including Ramanujan
summation (Candelpergher17, ).

The real question is whether any meaning can be extracted from the finite
results obtained by these alternative methods. This is more than just a
mathematical curiosity. In Physics, the mathematical description of physical
phenomena is often given in terms of divergent series and/or integrals.
Although the formal treatment of divergent series remains an open question, a
commitment to truncated partial sums ignores the fact that divergences can
lead to inconsistent results in physical situations. Of course, such
divergences may just be a reflection of a breakdown in the relevant
mathematical description. However, in some cases, physically consistent
results can be obtained by using alternative summation methods that yield
finite as opposed to infinite answers. Indeed, the infinite series of natural
numbers appears in computations of the Casimir force Schwartz14  and in the
critical dimension of bosonic string theory Polchinski98 , where it is
necessarily interpreted as the seemingly absurd formula
$\sum_{n=1}^{\infty}n=-1/12$ (Hardy73, ; Berndt85, ).

Although $\sum_{n=1}^{\infty}n=-1/12$ is often associated with Ramanujan
Berndt85 , the result is best derived using zeta function regularisation. When
$\Re(z)>1$, the Riemann zeta function is defined by the convergent series
$\zeta(z)=\sum_{n=1}^{\infty}n^{-z}$. Of course, when $\Re(z)\leq 1$ the
series is no longer convergent. However, we can analytically continue
$\zeta(z)$ into $\Re(z)<1$ using the following formula

$\zeta(z)=2^{z}\pi^{z-1}\sin\left(\frac{\pi
z}{2}\right)\Gamma(1-z)\zeta(1-z).$

We may now identify the series given above, $S(z)=\sum_{n=1}^{\infty}n^{z}$
with $\zeta(-z)$, even when it is divergent, $\Re(z)\geq-1$. In particular, we
obtain $S(1)=\sum_{n=1}^{\infty}n\overset{\zeta}{=}\zeta(-1)=-1/12$. More
generally, to sum a divergent series $\sum_{n=1}^{\infty}a_{n}$ using zeta
function regularisation, we identify a convergent series
$\sum_{n=1}^{\infty}a_{n}^{-z}$ for $z$ in some complex domain, then
analytically continue the result to $z=-1$.

The majority of issues with divergent series come with the transition from
partial sums to infinity, as notably exposed by Ramanujan (Berndt85, ;
Hardy73, ). Although it does not appear that he fully understood the validity
of his asymptotic expansions, he was able to extract some remarkable results.
As we will see in a moment, Ramanujan developed a study of divergent series
based upon the Euler-Maclaurin summation formula, often employing it in
creative ways. The Euler-Maclaurin formula also plays an important role in
understanding the results of smoothed asymptotic expansions, an intuitive
approach to the regularisation of divergent series beautifully elucidated by
Tao (Tao11, ). Indeed, using Tao’s methodology we will begin to understand how
a divergent series of positive numbers could ever be identified with a finite
negative number in a meaningful way.

###  2.1 Ramanujan and the Euler-Maclaurin summation formula

In the early 1730s, Euler solved one of the most intriguing mathematical
puzzles of the time: the Basel problem (Euler08Reciprocals, ). First
formulated by Leibniz and the Bernoulli brothers (Peng02a, ), the Basel
problem concerns the infinite series whose terms are the reciprocal squares of
the natural numbers. Using calculus to relate the discrete sum of an arbitrary
function $\sum_{n}\limits f(n)$ to an integral of the form $\int dxf(x)$,
Euler was able to prove that
$\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6}$. For a historical
review of the development of some of Euler’s key ideas, see (Peng07, ;
Ferraro98, ).

From elementary calculus, we know that a sum and an integral provide first
approximations to one other. In particular, the sum can be interpreted as the
total area of rectangles forming a step graph along some curve, while the
integral, evaluated between end points, can be interpreted as the area under
the same curve. More formally, the Euler-Maclaurin summation formula provides
an estimation of the sum $\sum\limits_{n=a}^{b}f(n)$ in terms of the integral
$\int_{a}^{b}\ dx\ f(x)$ and the derivatives of the function $f(x)$. As an
estimate of how much the trapezoid rule fails, an error term is also included
given by an integral involving Bernoulli polynomials. The formula can be
written as (Candelpergher17, )

$\sum_{n=a}^{b}f(n)=\int^{b}_{a}dxf(x)+\frac{f(b)+f(a)}{2}+\sum\limits_{k=2}^{m}\frac{B_{k}}{k!}(f^{(k-1)}(b)-f^{(k-1)}(a))+R_{m},$
(7)

where $a,b$ are integers such that $a\leq b$ and the integer $m\geq 2$. We
denote the $k$th derivative $f^{(k)}(x)=\frac{d^{k}}{dx^{k}}f$. The sum on the
right-hand side contains the non-vanishing Bernoulli numbers $B_{2n}$ (recall
that $B_{2n+1}=0$ for $n\geq 1$). As these numbers grow asymptotically for
large $k$ as $B_{2n}\sim(-1)^{n+1}4n^{2n}(\pi e)^{-2n}\sqrt{\pi n}$, this sum
often diverges as $m\to\infty$, and is best treated as an asymptotic series.
The corresponding error or remainder is given by

$R_{m}=(-1)^{m+1}\int^{b}_{a}dx\ \frac{b_{m}(x)}{m!}f^{(m)}(x),$ (8)

where $b_{m}(x)$ is the periodic function $B_{m}(x-\lfloor x\rfloor)$ given in
terms of the Bernoulli polynomials. The latter are defined by the generating
function

$\frac{te^{xt}}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}(x)\frac{t^{n}}{n!}.$ (9)

As emphasized in Tao11 , the level of approximation offered by (7) is heavily
dependent on the asymptotic behaviour of $R_{m}$. It is often the case that
the expansion remains valid even after taking the limits $a\rightarrow-\infty$
and/or $b\rightarrow\infty$. Generally, the integral on the right-hand side
can be evaluated in closed form in terms of elementary functions, even though
the sum on the left-hand side cannot. In the best case scenario, all the terms
in the asymptotic series can be expressed in terms of elementary functions.

The idea behind Ramanujan’s summation method is to use the Euler-Maclaurin
formula in the following asymptotic form

$\sum_{n=1}^{N}f(n)\sim\int^{N}_{a}dxf(x)+\frac{f(N)}{2}+\sum\limits_{k=2}^{\infty}\frac{B_{k}}{k!}f^{(k-1)}(N)+C(f,a),$
(10)

where $C(f,a)$ is the so-called “constant of the series”. This will depend on
the choice of lower limit on the integral, first noted by Hardy Hardy73 . We
will return to this ambiguity in a moment. This constant is identified with
the corresponding infinite series in the limit where $N\to\infty$ such that

$\sum_{n=1}^{\infty}f(n)\overset{(\mathcal{R},a)}{=}C(f,a).$ (11)

The $\mathcal{R}$ corresponds to Ramanujan, whereas the $a$ reflects the
ambiguity in choosing the limits on the integral. To recover $C(f,a)$, we
first define a sequence of constants $C_{m}(f,a)$ via the Euler-Maclaurin
formula. In particular, we write

$\sum_{n=1}^{N}f(n)=\int^{N}_{a}dxf(x)+\frac{f(N)}{2}+\sum\limits_{k=2}^{m}\frac{B_{k}}{k!}f^{(2k-1)}(N)+(-1)^{m}\int^{\infty}_{N}dx\
\frac{b_{m}(x)}{(2m)!}f^{(m)}(x)+C_{m}(f,a),$ (12)

so that

$C_{m}(f,a)=-\int^{1}_{a}dxf(x)+\frac{f(1)}{2}-\sum\limits_{k=2}^{m}\frac{B_{k}}{k!}f^{(k-1)}(1)+(-1)^{m+1}\int^{\infty}_{1}dx\
\frac{b_{m}(x)}{m!}f^{(m)}(x).$ (13)

Assuming that $f\in\mathcal{C}^{\infty}$ and the integral above is convergent
for sufficiently large values of $m$, one is able to show that $C_{m}(f,a)$
becomes independent of $m$ at large $m$, settling down to $C(f,a)$. For
$f(n)=n^{z}$, where $z$ is a positive integer, it is sufficient to take any
$m\geq z+1$ and evaluate the constant of the series, giving

$\sum_{n=1}^{\infty}n^{z}\overset{(\mathcal{R},a)}{=}\frac{1}{z+1}(a^{z+1}-1)+\frac{1}{2}-\frac{1}{z+1}\sum\limits_{k=2}^{z+1}\binom{z+1}{k}B_{k}=\frac{a^{z+1}}{z+1}-\frac{B_{z+1}}{z+1}$
(14)

where we have used the recursion relation (5) for the Bernoulli numbers. Given
that $\zeta(-z)=-\frac{B_{z+1}}{z+1}$ for positive integers, $z$, we see the
connection of this result to zeta function regularisation in the limit where
$a=0$.

To better understand the role played by the arbitrary real number $a$ in
general, we first note that

$C(f,b)-C(f,a)=\int_{a}^{b}dxf(x).$ (15)

Furthermore, when the series $\sum_{n=1}^{\infty}f(n)$ is convergent, it is
easy to see from the asymptotic formula (10) that Candelpergher17

$C(f,1)=\sum_{n=1}^{\infty}f(n)-\int^{\infty}_{1}dxf(x).$ (16)

It immediately follows that $\sum_{n=1}^{\infty}f(n)=C(f,\infty)$,
demonstrating the fact we should choose $a=\infty$ for the case of convergent
series. For a divergent series with $f(n)=\sum_{z=0}^{s}c_{z}n^{z}$ a
polynomial of degree $s$, it is clear that we must take $a=0$ in order to
match the result obtained via analytic continuation. When we develop the
method of smooth asymptotics advocated by Tao Tao11  in the next section, we
will see how $C(f,0)$ also picks out the finite term that is independent of
the choice of cut-off.

###  2.2 Smoothed asymptotics

When we use partial sums to sum an infinite series $\sum_{n=1}^{\infty}a_{n}$,
we truncate the series at some finite value $N$ and then compute the sum
$\sum_{n=1}^{N}a_{n}$ before taking $N\to\infty$. We can think of the partial
sum as modifying the infinite series with a step function

$\theta(x)=\begin{cases}1&x\leq 1\\\ 0&x>1\end{cases}$ (17)

such that

$\sum_{n=1}^{\infty}a_{n}\to\sum_{n=1}^{\infty}a_{n}\theta\left(\frac{n}{N}\right)=\sum_{n=1}^{N}a_{n}.$
(18)

In the case of a convergent series, the partial sum tends towards a unique
finite value as we take larger and larger values of $N$. For a divergent
series, we have seen how there exist alternative summation that yield finite
answers. Unfortunately, there is no obvious trace of those finite values
anywhere in the partial sum. This is readily seen in Faulhaber’s formula (4)
where we see no evidence for the finite value of
$S(z)=\sum_{n=1}^{\infty}n^{z}$ obtained via analytic continuation. In Tao11 ,
Tao argues that this is an artefact of the jump discontinuity in $\theta(x)$
and can be remedied by generalising $\theta(x)$ to a smooth regulator
function, as opposed to a sharp cut-off.

To this end, we regularise the infinite series with a smooth function,
$\eta(x)$, such that

$\sum_{n=1}^{\infty}a_{n}\to\sum_{n=1}^{\infty}a_{n}\eta\left(\frac{n}{N}\right).$
(19)

We assume that $\eta(x)$ is a smooth bounded function, defined for the non-
negative real numbers, with $\eta(0)=1$ and $\eta(x)\to 0$ at large $x$.
Indeed, Tao assumes that $\eta(x)$ is a bump function - a smooth function with
compact support, vanishing whenever $x\notin[0,1]$. Here we will relax this
condition a little, and allow $\eta(x)$ to be a Schwartz function. That means
that $\eta(x)$ and all its derivatives are rapidly decreasing at large $x$,
going to zero at infinity faster than $x^{\alpha}$ for all $\alpha<0$.

For series that were already absolutely convergent, the smoothing by $\eta(x)$
does not affect the asymptotic value Tao11 . However, for divergent series,
smoothing can significantly improve the convergence properties. For example,
we can easily sum Grandi’s series $\sum_{n=1}^{\infty}(-1)^{n-1}$ using a
regulator function $\eta(x)=e^{-x}$, so that the corresponding series
converges towards one half.

Let us consider the impact of smoothing on a divergent series
$\sum_{n=1}^{\infty}f(n)$ where $f(n)=\sum_{z=0}^{s}c_{z}n^{z}$ is a
polynomial of degree $s$. From the Euler-Maclaurin formula (7) applied to the
function $g_{N}(x)=f(x)\eta(x/N)$ with $a=0$ and $b=\infty$, we readily obtain
the following expression

$\sum_{n=1}^{\infty}g_{N}(n)=\int^{\infty}_{0}dx\
g_{N}(x)-\frac{g_{N}(0)}{2}-\sum\limits_{k=2}^{m}\frac{B_{k}}{k!}g_{N}^{(k-1)}(0)+R_{m},$
(20)

where we have used the fact that $g_{N}(x)$ and all of its derivatives vanish
at infinity and the remainder term is given by

$R_{m}=(-1)^{m+1}\int^{\infty}_{0}dx\ \frac{b_{m}(x)}{m!}g_{N}^{(m)}(x).$ (21)

As $g_{N}^{(k)}(x)=f^{(k)}(x)\eta(x/N)+\mathcal{O}(1/N)$, and given the
polynomial form for $f(n)=\sum_{z=0}^{s}c_{z}n^{z}$, we choose $m\geq s+2$ and
find that

$\sum_{n=1}^{\infty}f(n)\eta\left(\frac{n}{N}\right)=\sum_{z=0}^{s}c_{z}\left[C_{z}[\eta]N^{z+1}+\zeta(-z)\right]+\mathcal{O}(1/N),$
(22)

where $C_{z}[\eta]=\int_{0}^{\infty}dx\ x^{z}\eta(x)$ is the Mellin transform
of the regulator function. Here we have used the fact that
$\zeta(-z)=-\frac{B_{z+1}}{z+1}$ for natural numbers $z$. The appearance of
the zeta function follows in a very similar way to its appearance in equation
(14). In choosing $m\geq s+2$, we guarantee that the remainder term
$R_{m}=\mathcal{O}(1/N)$. This is not immediately obvious. In particular we
find that

$\displaystyle R_{m}$ $\displaystyle=$
$\displaystyle(-1)^{m+1}\sum_{z=0}^{s}c_{z}\int^{\infty}_{0}dx\
\frac{b_{m}(x)}{m!}\frac{d^{m}}{dx^{m}}\left[x^{z}\eta\left(\frac{x}{N}\right)\right]$
(23) $\displaystyle=$
$\displaystyle(-1)^{m+1}\sum_{z=0}^{s}c_{z}N^{z+1-m}\int^{\infty}_{0}dy\
\frac{b_{m}(Ny)}{m!}\frac{d^{m}}{dy^{m}}(y^{z}\eta\left(y\right)).$ (24)

Now since $\eta$ is Schwartz and $b_{m}$ is bounded, it follows that the
integrand $\frac{b_{m}(Ny)}{m!}\frac{d^{m}}{dy^{m}}(y^{z}\eta\left(y\right))$
is also Schwartz, and so the corresponding integral is bounded111If $F(x)$ is
Schwartz then for each $n\geq 0$, $|x^{n}F(x)|<Q_{n}$ for finite constants
$Q_{n}$. Consider the corresponding integral
$\int_{0}^{\infty}dxF(x)=\int_{0}^{1}dxF(x)+\int_{1}^{\infty}dxF(x)$. $F(x)$
is smooth so the first integral running over a finite range,
$\int_{0}^{1}dxF(x)$, is bounded. As for the second integral, note that
$\left|\int_{1}^{\infty}dxF(x)\right|\leq\int_{1}^{\infty}dx|F(x)|<\int_{1}^{\infty}dxQ_{n}/x^{n}=Q_{n}/(n-1)$.
Taking $n\geq 2$ we immediately see that $\int_{1}^{\infty}dxF(x)$ is bounded.
It follows that $\int_{0}^{\infty}dxF(x)$ is bounded.. Since $m\geq s+2$, and
$s$ is finite, we immediately see that $R_{m}=\mathcal{O}(1/N)$.

For monomials $f(n)=n^{z}$, there is a single divergence as $N\to\infty$ in
the corresponding expression for the regularised series (22), and there is a
unique finite piece given by $\zeta(-z)$. This coincides with the result
obtained using Ramanujan or analytic continuation. The connection to Ramanujan
summation (with $a=0$) is understood via the following identity derived
directly from (22)

$\sum_{n=1}^{\infty}n^{z}\overset{(\mathcal{R},0)}{=}C(f,0)=\lim_{N\to\infty}\left[\sum_{n=1}^{\infty}n^{z}\eta\left(\frac{n}{N}\right)-\int_{0}^{\infty}dxx^{z}\eta\left(\frac{x}{N}\right)\right]=\zeta(-z).$
(25)

Using the method of smoothed asymptotics, we see how Ramanujan performed a
delicate cancellation between infinities that can be rigorously understood.
The procedure is reminiscent of renormalsation in perturbative QFT.

As we will explore in more detail in the next section, another feature of
equation (22) reminiscent of QFT is the regulator dependence in the power law
divergences. In QFT, this regulator dependence suggests the details of these
divergences are unphysical. At best they indicate a sensitivity to new high
energy physics in an effective field theory framework, if indeed such new
physics exists. The regulator dependence in the divergences of (22) raises the
possibility that there are families of enhanced regulators for which the
divergences vanish altogether!

This is indeed the case: an enhanced regulator is one for which the Mellin
transform $C_{z,\eta}=\int_{0}^{\infty}dx\ x^{z}\eta(x)$ vanishes for integer
values of $z\geq 0$. More precisely, for all natural numbers $z$, we define an
enhanced regulator of order $z$ to be a regulator function $\eta_{[z]}(x)$ for
which $\int_{0}^{\infty}dx\ x^{z}\eta_{[z]}(x)=0$. We immediately see from
(22) that if we use an enhanced regulator of order $z$ to regularise the
monomial series $S(z)=\sum_{n=0}^{\infty}n^{z}$, there are no divergences
whatsoever. Indeed, the regularised series converges to the finite value
$\zeta(-z)$ as $N\to\infty$, without any need to renormalise.

An elegant example of an enhanced regulator of order $z$ is given by the
function

$\eta_{[z]}(x)=e^{-x\cot\left(\frac{\frac{\pi}{2}-\theta}{z+1}\right)}\frac{\cos(x+\theta)}{\cos\theta},$
(26)

where $0<\theta<\frac{\pi}{2}$ and $z$ is any natural number. One can readily
check that we have $\int_{0}^{\infty}dx\ x^{z}\eta_{[z]}(x)=0$ and so the
smoothed series $\sum_{n=1}^{\infty}n^{z}\eta_{[z]}\left(\frac{n}{N}\right)$
converges to $\zeta(-z)$. We can actually take $\theta$ to be zero, except in
the case of $z=0$ where non-vanishing $\theta$ is required in order to ensure
that the regulator remains Schwartz. Note that for $\theta=0$ and $z=1$ we
recover the rather beautiful enhanced regulator of order one,
$\eta_{[1]}(x)=e^{-x}\cos x$, from which we infer that222To our best knowledge
this remarkable regulator does not formally appear in the literature. However,
in the process of this research, we found this formula appears on Wiki without
citation or context. It is also the topic of discussion on this
mathematics.stackexchange post, in which the author notices the nature of the
function being Schwartz and also relates it to Tao’s method of smooth
summation.

$\lim_{N\to\infty}\sum_{n}ne^{-\frac{n}{N}}\cos\left(\frac{n}{N}\right)=-\frac{1}{12}.$

Since $\int_{0}^{\infty}dx\ x^{z}e^{-x}\cos
x=2^{-\frac{1}{2}(1+z)}z!\cos\left(\frac{\pi}{4}(z+1)\right)$ it just so
happens that $e^{-x}\cos x$ is also an enhanced regulator of order $z=4m+1$
for any natural number $m$. Likewise, it turns out that
$\eta_{[z]}(x)=e^{-x\cot\left(\frac{\pi}{2(z+1)}\right)}\cos x$ is also
enhanced regulator of order $z^{\prime}=z+2(z+1)m$ for any natural number $m$
and $z\geq 1$.

More generally, we can employ a few simple algorithms for finding enhanced
regulators of any given order. The first assumes that we already know an
enhanced regulator of order zero, $\eta_{[0]}(x)$, with vanishing integral
$\int_{0}^{\infty}dx\eta_{[0]}(x)=0$. By changing variables from $x\to
x^{z+1}$, we see that $\eta_{[z]}(x)=\eta_{[0]}(x^{z+1})$ is an enhanced
regulator of order $z$. The second algorithm requires no previous knowledge of
enhanced regulators: take any Schwartz function $\chi(x)$ defined on the
positive real axis with the property that $\chi(0)=0$ and $\chi^{(z+1)}(0)=1$,
then $\eta_{[z]}(x)=\chi^{(z+1)}(x)$ is an enhanced regulator of order $z$. To
see this, we first note that since $\chi(x)$ is a Schwartz function, the same
is trivially true of $\chi^{(z+1)}(x)$. Furthermore, since
$\chi^{(z+1)}(0)=1$, it is clear that $\chi^{(z+1)}(x)$ has all the properties
of a regulator as defined at the beginning of this section. To see that it is
an enhanced regulator, we compute the relevant Mellin transform

$\int_{0}^{\infty}dx\
x^{z}\chi^{(z+1)}(x)=\left[\sum_{k=0}^{z}(-1)^{k}\frac{z!}{(z-k)!}x^{z-k}\chi^{(z-k)}(x)\right]^{\infty}_{0}=0$
(27)

verifying the defining condition for an enhanced regulator of order $z$.

A third and final algorithm connects enhanced regulators to Schwinger’s proper
time formalism SchwingerPT . Indeed, for any natural number $n$, consider a
regulator of the form

$\lambda_{n}(x)=\frac{1}{(n-1)!}\int_{0}^{\infty}\frac{du}{u}\rho(u)(ux^{2})^{n}e^{-ux^{2}},$
(28)

where the bounded function $\rho(u)\to 0$ faster than any power as $u\to 0$
and $\rho(u)\to 1$ as $u\to\infty$. The properties of $\rho(u)$ guarantee that
this function is Schwartz with $\lambda_{n}(x)\to 1$ as $x\to 0^{+}$, as
required for a regulator. After changing variables from $x$ to $ux^{2}$, it is
not difficult to show that the Mellin transform is given by

$\int_{0}^{\infty}dx\
x^{z}\lambda_{n}(x)=\frac{\Gamma\left(n+\frac{z+1}{2}\right)}{2(n-1)!}\int_{0}^{\infty}du\
\rho(u)u^{-\left(\frac{z+3}{2}\right)}.$ (29)

For any natural numbers $n\neq m$, we can now construct an enhanced regulator
of order $z$ as follows

$\eta_{[z]}(x)=\frac{\frac{(n-1)!}{\Gamma\left(n+\frac{z+1}{2}\right)}\lambda_{n}(x)-\frac{(m-1)!}{\Gamma\left(m+\frac{z+1}{2}\right)}\lambda_{m}(x)}{\frac{(n-1)!}{\Gamma\left(n+\frac{z+1}{2}\right)}-\frac{(m-1)!}{\Gamma\left(m+\frac{z+1}{2}\right)}}.$
(30)

This is clearly normalised so that it is classed as a regulator. To see that
it is enhanced we simply check that the corresponding Mellin transform
vanishes $\int dx\ x^{z}\eta_{[z]}(x)=0$ using (29).

Although the enhanced regulators described above work well for summing
specific monomial series, they are less suited to eliminating divergences for
generic polynomial series. For example, the regulators given in equation (26)
will yield convergent series for $\sum_{n}n^{z}\eta_{[z]}(n/N)$ but not for,
say, $\sum_{n}(n^{z-1}+n^{z})\eta_{[z]}(n/N)$. To remedy these problems for
all natural numbers $r\leq s$, we introduce the notion of an enhanced
regulator of order $[r,s]$ defined to be a regulator function
$\eta_{[r,s]}(x)$ for which $\int_{0}^{\infty}dx\ x^{z}\eta_{[r,s]}(x)=0$ for
all natural numbers $z\in[r,s]$. It now follows from (22) that if we use an
enhanced regulator of order $[0,s]$ to regularise the series over polynomials
of degree $s$, there are no divergences whatsoever. The regularised series for
the polynomial $f(n)=\sum_{z=0}^{s}c_{z}n^{z}$ converges to the finite value
$\sum_{z=0}^{s}c_{z}\zeta(-z)$ as $N\to\infty$ without any need to
renormalise. For the example given above, we find that
$\lim_{N\to\infty}\sum_{n}(n^{z-1}+n^{z})\eta_{[0,z]}(n/N)=\zeta(1-z)+\zeta(-z)$
for $z\geq 1$.

We can generalise the second algorithm described above to find enhanced
regulators of order $[r,s]$: take any Schwartz function $\chi(x)$ defined on
the positive real axis with the property that $\chi^{(k)}(0)=0$ for all
natural numbers $k\in[0,s-r]$ and with $\chi^{(s+1)}(0)=1$. It follows that
$\eta_{[r,s]}(x)=\chi^{(s+1)}(x)$ is an enhanced regulators of order $[r,s]$.
The argument for $\eta_{[r,s]}(x)$ being a regulator is the same as for
$\eta_{[z]}(x)$. To see that it is enhanced at order $[r,s]$, we compute the
Mellin transforms for all natural numbers $z\in[r,s]$ and see that they all
vanish

$\int_{0}^{\infty}dx\
x^{z}\chi^{(s+1)}(x)=\left[\sum_{k=0}^{z}(-1)^{k}\frac{z!}{(z-k)!}x^{z-k}\chi^{(s-k)}(x)\right]^{\infty}_{0}=0.$
(31)

As an example, we consider the case where
$\chi(x)=e^{-x}\frac{x^{s+1}}{(s+1)!}$ which generates an enhanced regulator
of order $[0,s]$ given by

$\eta_{[0,s]}(x)=e^{-x}\sum_{k=0}^{s+1}\binom{s+1}{k}\frac{(-x)^{k}}{k!}.$
(32)

The existence of enhanced regulators challenges the idea that divergent series
of polynomials must be renormalised in order to recover their corresponding
finite values. Indeed, they explicitly demonstrate a rigorous method of taking
the series towards infinity without encountering any divergence. Of course,
there is a magic of sort: the enhanced regulators must always change sign and
in just the right way in order for large positive contributions to be
cancelled by large negative ones. It is here that we are reminded of the words
of Euler who said “the quantities greater than infinity are also smaller than
nothing and the quantities smaller than infinity also correspond to the
quantities greater than nothing”(Euler18DeSeriebusDivergentibus, ).

##  3 Divergent integrals in quantum field theory

Divergent integrals are encountered with unsettling regularity in interacting
QFTs. As revealed by Tomonoga Tomonaga:1946zz , Schwinger Schwinger:1948iu
and Feynman Feynman:1948fi  in the context of quantum electrodynamics and by
’t Hooft and Veltman for non-abelian gauge theories tHooft:1972tcz , for
renormalisable theories, these divergences can be carefully regularised and
absorbed into a renormalisation of a finite number of couplings. The origin of
these divergences is understood to be a consequence of pushing the
corresponding theory too hard, and assuming it to apply even at the shortest
distance scales, or equivalently, at the highest energies. Thanks to the
Applequist-Carazzone decoupling theorem, the effects of very heavy particles
on low energy scattering processes are hidden inside a renormalisaton of the
low energy couplings Appelquist:1974tg . Said another way, even though our
QFTs encounter divergences that betray limitations on their validity at all
scales, they can still be made to work well as effective field theories at low
energies.

It is interesting to note parallels between divergences in effective field
theory and the characteristic behaviour we see whenever we regularise a
divergent series with a smooth cut-off. Indeed, it is well known that power
law divergences in QFT are regulator dependent and the corresponding couplings
carry no physical meaning. This is in stark contrast to the universal
prediction obtained from the logarithmic divergences. In order to develop the
analogy with our divergent series, it is instructive to examine power law and
logarithmic divergences in QFT in a little more detail. Closely following the
discussion presented in deRham:2014wfa , we consider the Wilson action
$S_{\Lambda}[\varphi]$ for modes lighter than some scale $\Lambda$ defined
according to the path integral Burgess:2020tbq

$e^{iS_{\Lambda}[\varphi]}=\int_{k>\Lambda}D[\varphi]e^{iS[\varphi]}.$ (33)

This generically contains power law and logarithmic divergences, and can be
written as

$S_{\Lambda}[\varphi]=\int
d^{4}x\left[\sum_{i}\Lambda^{4-d_{i}}g_{i}(\Lambda)\mathcal{O}_{i}(x)+\ln(\Lambda/\mu)\sum_{a}g_{a}(\Lambda)\mathcal{O}_{a}(x)+\ldots\right],$
(34)

where $\mathcal{O}_{i}$ are relevant operators of dimension $d_{i}=0,2$ and
$\mathcal{O}_{a}$ are marginal operators of dimension $4$. The ellipsis denote
finite terms which may also depend on $\mu$, an arbitrarily chosen mass scale.
Now consider the Wilson action, $S_{\Lambda}^{\prime}[\varphi]$, defined at
some lower scale, $\Lambda^{\prime}<\Lambda$, and given by

$e^{iS_{\Lambda}^{\prime}[\varphi]}=\int_{\Lambda^{\prime}<k<\Lambda}D[\varphi]e^{iS_{\Lambda}[\varphi]}.$
(35)

This is obviously independent of the choice of $\Lambda$, but highly dependent
on $\Lambda^{\prime}$ and correspondingly takes the form

$S_{\Lambda^{\prime}}[\varphi]=\int d^{4}x\left[\sum_{i}\Lambda^{\prime
4-d_{i}}g_{i}(\Lambda^{\prime})\mathcal{O}_{i}(x)+\ln(\Lambda^{\prime}/\mu)\sum_{a}g_{a}(\Lambda^{\prime})\mathcal{O}_{a}(x)+\ldots\right].$
(36)

It follows that $S_{\Lambda}^{\prime}=S_{\Lambda}+\Delta\Gamma$ where
$\Delta\Gamma$ is the contribution from integrating out intermediate modes
$\Lambda^{\prime}\leq k\leq\Lambda$. Focussing on the relevant and marginal
operators, we must have

$\Delta\Gamma=\int
d^{4}x\left[\sum_{i}c_{i}(\Lambda^{\prime},\Lambda)\mathcal{O}_{i}(x)+\sum_{a}c_{a}(\Lambda^{\prime},\Lambda)\mathcal{O}_{a}(x)+\ldots\right],$
(37)

where the contribution from the intermediate modes
$c_{i}(\Lambda^{\prime},\Lambda)=\Lambda^{\prime
4-d_{i}}g_{i}(\Lambda^{\prime})-\Lambda^{4-d_{i}}g_{i}(\Lambda)$ completely
cancels the power law divergences in passing from one Wilson action to
another. This is just Wilsonian renormalisation in action, and illustrates the
folly in attaching too much significance to the specific couplings associated
with power law divergences. In contrast, the cancellation of the $\Lambda$
dependence for the logarithmic couplings yields
$c_{a}(\Lambda^{\prime},\Lambda)=\ln(\Lambda^{\prime}/\mu)\sum_{a}g_{a}(\Lambda^{\prime})-\ln(\Lambda/\mu)\sum_{a}g_{a}(\Lambda^{\prime})$.
In order for the $\mu$ to consistently drop out to leading order, we see that
there must be a universal property $g_{a}(\Lambda)\sim
g_{a}(\Lambda^{\prime})\sim g_{a}$.

In a similar vein, if we compute the one loop effective action

$e^{i\Gamma_{\text{eff}}[\varphi]}=\int_{k<\Lambda}D[\delta\varphi]e^{iS_{\Lambda}[\varphi+\delta\varphi]}$
(38)

so that
$\Gamma_{\text{eff}}[\varphi]=S_{\Lambda}[\varphi]+\Delta\Gamma_{\text{eff}}[\varphi]$,
we find that any power law divergences generated in $\Delta\Gamma$ are
cancelled by the corresponding terms in the Wilson action, $S_{\Lambda}$. In
contrast, the couplings associated with logarithmic divergences can survive as
a universal feature of the effective action, necessarily independent of
$\Lambda$.

These universal features of logarithmic divergences, in contrast to power law
divergences, are strongly reminiscent of the results we saw in the previous
section. When we regularised a divergent series with a smoothed cut-off, we
saw how it was the finite terms that were universal, yielding the results
obtained via analytic continuation, in contrast to the power law divergences
that were regulator dependent. How far can we push the similarities? For
example, having introduced the enhanced regulators to eliminate divergences
appearing in number theory, it is tempting to ask if they can also play an
interesting role in the loop divergences arising in QFT and whether this can
be linked to the absence of divergences in finite theories. Indeed, the
reluctance of string theory to travel deep into the UV is a direct
manifestation of the smoothing or smearing effect of the string length scale
as seen in a study of modular invariance of the worldsheet. This was first
observed in string theory by Shapiro (Shapiro72, ), but a more modern
treatment can be found in (Polchinski98, ).

###  3.1 $\eta$ regularisation in QFT

To control the ultra-violet divergences that plague perturbative QFT, several
regularisation schemes are included in the literature. A sharp momentum cut-
off connects intuitively with Wilson’s understanding of the renormalisation
group Kadanoff:1966wm ; Wilson:1971bg ; Wilson:1971dh ; Wilson:1974mb  but is
known to break translational invariance and, perhaps more importantly, the
gauge invariance of the underlying theory. The standard way to regularise loop
integrals whilst preserving gauge invariance is dimensional regularisation
tHooft:1972tcz . However, in analytically continuing the dimensionality of the
spacetime, we run into difficulties when defining quantities unique to four
dimensions (such as the $\gamma_{5}$ matrix).

For infinite series, the method of smoothed asymptotics amounts to replacing
an infinite sum with a weighted infinite sum

$\sum_{n=0}^{\infty}\\#\rightarrow\sum_{n=0}^{\infty}\eta\left(\frac{n}{N}\right)\\#.$
(39)

The generalisation of this to divergent loops is straightforward: working in
Euclidean signature, we simply replace the loop integral with a weighted loop
integral

$\int\ d^{4}k\\#\rightarrow\int\ d^{4}k\
\eta\left(\frac{|k|}{\Lambda}\right)\\#,$ (40)

where $|k|$ is the norm of the Euclidean four-momentum and $\Lambda$ is a cut-
off scale. Of course, if $\eta$ were just a step function, this would be
nothing more than a sharp momentum cut-off. However, inspired by the method of
smoothed asymptotics discussed in the previous section, we shall assume that
$\eta$ is a smooth regulator, corresponding to a Schwartz function that
equates to unity at the origin. We dub this $\eta$ regularisation.

The idea of smoothing the cut-off is, of course, nothing new although in most
applications, this is implemented at the level of the propagator. For example,
in his derivation of the exact renormalisation group equation, Polchinski
makes use of a modified propagator that is exponentially damped at high
momenta (Polchinski84, ). As a modification of the integral measure, $\eta$
regularisation is similar to dimensional regularisation albeit with the
advantage that it does not require us to analytically continue the dimension
of spacetime. However, the proposed scheme has most in common with the smooth
operator regularisation method inspired by Ball88  and developed in
Oleszczuk94 ; Liao96  using Schwinger proper time SchwingerPT . Gauge
invariance is also preserved in this case, as it is with dimensional
regularisation. This is because the regularisation is transferred to the
proper time integral, leaving the (gauge invariant) momentum integral
unchanged. For a recent application of the smooth operator regularisation and
further discussion on symmetry preservation, see Xing:2022jtt .

Does $\eta$ regularisation also preserve gauge invariance? Generically, it
seems obvious that it will not. Just as we saw for sharp momentum cut-offs,
gauge invariance is typically broken as soon as we regularise the momentum
integral directly. However, if $\eta$ regularisation has any chance of opening
up a better understanding of how divergences are absent in string theory, it
must, at the very least, admit a preferred set of regulators that preserve
gauge invariance. This may seem unlikely at first glance, although as we will
see in a moment, it will be possible to preserve gauge invariance and key to
that are the enhanced regulators that eliminate divergences. We will also see
how this connects to regularisation of Schwinger proper time integrals.

####  3.1.1 Irreducible loop integrals and eliminating divergences

To develop $\eta$ regularisation in more detail, it is convenient to introduce
the concept of irreducible loop integrals (ILIs) (Wu03, ; Wu04, ; Wu14, ). In
general, $n$-fold ILIs are defined as the $n$-loop integrals for which there
are no longer the overlapping factors $(k_{i}-k_{j}+p)$ in the denominator of
the integrand and no factors of the scalar momentum $k^{2}$ in the numerator
Wu14 . In this work we focus on regularising ultra-violet divergences at one-
loop, postponing a detailed discussion of higher loops to future work
(PadillaSmith23, ). It was shown in Wu03  that upon use of the Feynman
parameter method, all one-loop perturbative Feynman integrals of the one-
particle irreducible graphs can be evaluated as the following one-fold ILIs in
Minkowski spacetime:

$\displaystyle I_{-2\alpha}(\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}+\mathcal{M}^{2})^{2+\alpha}},$
(41) $\displaystyle I_{-2\alpha}^{\mu\nu}(\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{\mu}k^{\nu}}{(k^{2}+\mathcal{M}^{2})^{3+\alpha}},$
(42) $\displaystyle I_{-2\alpha}^{\mu\nu\rho\sigma}(\mathcal{M}^{2})$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{\mu}k^{\nu}k^{\rho}k^{\sigma}}{(k^{2}+\mathcal{M}^{2})^{4+\alpha}},$
(43)

where the subscript $(-2\alpha)$ labels the power counting dimension (of
energy-momentum) with $\alpha=-1$ and $\alpha=0$ corresponding to quadratic
and logarithmically divergent integrals. The mass term
$\mathcal{M}^{2}=\mathcal{M}^{2}(m^{2}_{1},p_{1}^{2},\dots)$ is a function of
Feynman parameters, external momenta, $p_{i}$ and corresponding mass scales,
$m_{i}$. Note that $k^{2}=g_{\mu\nu}k^{\mu}k^{\nu}$ where the metric
$g_{\mu\nu}$ is written with mostly positive signature.

The ILIs of tensor type are related to scalar integrals in the usual way:

$\displaystyle I_{-2\alpha}^{\mu\nu}(\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{4}g^{\mu\nu}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{2}}{(k^{2}+\mathcal{M}^{2})^{3+\alpha}},$
(44) $\displaystyle I_{-2\alpha}^{\mu\nu\rho\sigma}(\mathcal{M}^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{4!}S^{\mu\nu\rho\sigma}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{4}}{(k^{2}+\mathcal{M}^{2})^{4+\alpha}},$
(45)

where
$S^{\mu\nu\rho\sigma}=g^{\mu\nu}g^{\rho\sigma}+g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho}$
is a totally symmetric tensor of rank four. Notice that these relations are
merely a consequence of the rotational symmetries of the four dimensional
space and not to be confused with the gauge consistency relations derived in
(Wu03, ; Wu04, ; Wu14, ). As we will see later, the latter can be accommodated
by using different $\eta$s for regularising different ILIs.

When implementing the $\eta$ regularisation, we Wick rotate the integrals to
Euclidean signature $k_{0}\to ik_{4}$ and insert a factor of
$\eta(|k|/\Lambda)$ in the integrand. This yields $I_{-2\alpha}^{\cdots}\to
iJ_{-2\alpha}^{\cdots}[\eta]$, where the form of $\eta$ need not be universal
for all one-fold ILIs, at least in principle. In particular, we now have that

$\displaystyle J_{-2\alpha}[\eta](\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{(k^{2}+\mathcal{M}^{2})^{2+\alpha}}\eta\left(\frac{|k|}{\Lambda}\right),$
(46) $\displaystyle J_{-2\alpha}^{\mu\nu}[\eta](\mathcal{M}^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{4}g^{\mu\nu}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{2}}{(k^{2}+\mathcal{M}^{2})^{3+\alpha}}\eta\left(\frac{|k|}{\Lambda}\right),$
(47) $\displaystyle J_{-2\alpha}^{\mu\nu\rho\sigma}[\eta](\mathcal{M}^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{4!}S^{\mu\nu\rho\sigma}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{4}}{(k^{2}+\mathcal{M}^{2})^{4+\alpha}}\eta\left(\frac{|k|}{\Lambda}\right),$
(48)

where the integration is over four dimensional Euclidean space in each case.
Making use of partial fractions, the regularised tensor ILIs can be written
explictly in terms of scalar counterparts

$\displaystyle J_{-2\alpha}^{\mu\nu}[\eta](\mathcal{M}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{4}g^{\mu\nu}\left[J_{-2\alpha}[\eta](\mathcal{M}^{2})-\mathcal{M}^{2}J_{-2(\alpha+1)}[\eta](\mathcal{M}^{2})\right],$
(49) $\displaystyle J_{-2\alpha}^{\mu\nu\rho\sigma}[\eta](\mathcal{M}^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{4!}S^{\mu\nu\rho\sigma}\left[J_{-2\alpha}[\eta](\mathcal{M}^{2})-2\mathcal{M}^{2}J_{-2(\alpha+1)}[\eta](\mathcal{M}^{2})+\mathcal{M}^{4}J_{-2(\alpha+2)}[\eta](\mathcal{M}^{2})\right]\qquad\quad.$
(50)

To understand the role of divergences, it is now sufficient to study the
scalar integrals of the form (46). After integrating out the three-sphere,
these can be written as

$J_{-2\alpha}[\eta](\mathcal{M}^{2})=\frac{1}{8\pi^{2}}\int_{0}^{\infty}dk\
\frac{k^{3}}{(k^{2}+\mathcal{M}^{2})^{2+\alpha}}\eta\left(\frac{|k|}{\Lambda}\right).$
(51)

For $\alpha>0$ the integrals are convergent as $\Lambda\to\infty$ and one
readily obtains

$J_{-2\alpha}[\eta](\mathcal{M}^{2})\sim\frac{1}{16\pi^{2}\alpha(1+\alpha)\mathcal{M}^{2\alpha}}.$
(52)

For $\alpha\leq 0$, the integrals diverge as $\Lambda\to\infty$, where they
take the following asymptotic form

$\displaystyle J_{0}[\eta](\mathcal{M}^{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{8\pi^{2}}\left[\ln(\Lambda/|\mathcal{M}|)+\gamma[\eta]-\frac{1}{2}\right],$
(53) $\displaystyle J_{2}[\eta](\mathcal{M}^{2})$ $\displaystyle\sim$
$\displaystyle\frac{1}{8\pi^{2}}\left[\Lambda^{2}C_{1}[\eta]-\mathcal{M}^{2}\left(\ln(\Lambda/|\mathcal{M}|)+\gamma[\eta]\right)\right],$
(54)

and

$J_{2s+4}[\eta](\mathcal{M}^{2})\sim\frac{1}{8\pi^{2}}\left[\sum_{z=0}^{s}\binom{s}{z}C_{2z+3}[\eta]\mathcal{M}^{2(s-z)}\Lambda^{2z+4}\right]$
(55)

for any natural number $s$. These expressions are valid provided $\eta$ is a
regulator: a smooth Schwartz function with $\eta(0)=1$. Here we have also
defined the finite integral

$\gamma[\eta]=-\int_{0}^{\infty}dx\ \eta^{\prime}(x)\ln x,$ (56)

and recall that $C_{z}[\eta]=\int_{0}^{\infty}x^{z}\eta(x)$ is the
corresponding Mellin transform for any natural number $z$. We immediately see
the parallels with the divergent series in the previous section. Power law
divergences are regulator dependent and weighted by the corresponding Mellin
transform. These can always be eliminated with a judicious use of enhanced
regulators. Logarithmic divergences are independent of the regulator, just as
the finite terms were in the divergent series. This universal form for the log
divergences agrees with the corresponding terms obtained by other
regularisation methods, such as dimensional regularisation. Of course, for our
QFT integrals, there are also finite terms, although unlike those that appear
in the divergent series, they are regulator dependent. This is a direct
consequence of the logarithm and the correction to the finite term that arises
when we rescale the cut-off $\Lambda$. Indeed, for any regulator $\eta(x)$ and
real number $\lambda>0$, we can define another regulator
$\eta_{\lambda}(x)=\eta(\lambda x)$ which is equivalent to rescaling the cut-
off. It then follows that $\gamma[\eta_{\lambda}]=\gamma[\eta]-\ln\lambda$
while $C_{z}[\eta_{\lambda}]=\lambda^{-1-z}C_{z}[\eta]$. These relations will
be useful when we consider the role of gauge invariance in the next section.

###  3.2 Gauge invariant $\eta$ regularisation

To investigate how gauge invariance is affected by $\eta$ regularisation, we
follow Wu03 ; Wu04 ; Wu14  and consider a general gauge theory where the gauge
group has dimension $d_{G}$ and where $N_{f}$ Dirac spinors $\Psi_{n}$
($n=1,\ldots,N_{f}$) are interacting with the Yang Mills field $A^{a}_{\mu}$
($a=1,\ldots,d_{G}$). Such a theory is described by a Lagrangian

$\mathcal{L}=\bar{\psi}_{n}(i\gamma^{\mu}D_{\mu}-m)\psi_{n}-\frac{1}{4}F_{\mu\nu}^{a}F^{\mu\nu}_{a},$
(57)

where

$F^{a}_{\mu\nu}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}-gf_{abc}A^{b}_{\mu}A^{c}_{\nu},\qquad
D_{\mu}\psi_{n}=(\partial_{\mu}+igT^{a}A^{a}_{\mu})\psi_{n},$ (58)

and $T^{a}$ are the generators of the gauge group whose commutator
$[T^{a},T^{b}]=if^{abc}T^{c}$ defines the structure constants $f^{abc}$. A
careful computation of the vacuum polarisation for the gauge field at one-loop
(see Wu03 ; Wu04 ; Wu14  for details) yields an expression of the form

$\Pi^{ab}_{\mu\nu}(p)=\Pi^{(g)ab}_{\mu\nu}(p)+\Pi^{(f)ab}_{\mu\nu}(p),$ (59)

where $p^{\mu}$ is the external momentum. Here $\Pi^{(g)ab}_{\mu\nu}(p)$ are
the pure Yang Mills contributions coming from gauge field loops and ghost
loops. $\Pi^{(f)ab}_{\mu\nu}(p)$ are the contributions from fermion loops,
arising from the interaction of the fermions with the gauge field. Gauge
invariance is understood in terms of the Ward identities
$p^{\mu}\Pi^{ab}_{\mu\nu}=\Pi^{ab}_{\mu\nu}p^{\nu}=0$. Requiring this to hold
for any gauge theory and with any number of fermions means that Ward
identities should hold separately for the gauge field and fermionic
contributions

$p^{\mu}\Pi^{{(g)}ab}_{\mu\nu}=\Pi^{(g)ab}_{\mu\nu}p^{\nu}=0,\qquad
p^{\mu}\Pi^{{(f)}ab}_{\mu\nu}=\Pi^{(f)ab}_{\mu\nu}p^{\nu}=0.$ (60)

When the vacuum polarisation is computed in terms of the regularised ILIs,
$I^{\cdots}_{-2\alpha}|_{\text{regularised}}$, these generalized Ward
identities impose strict consistency conditions on the regularisation scheme.
In particular, we find that for $\alpha=-1,0,1$ we must have Wu03 ;
Xing:2022jtt

$\displaystyle I^{\mu\nu}_{-2\alpha}|_{\text{regularised}}$
$\displaystyle\sim$
$\displaystyle\frac{1}{2(\alpha+2)}g^{\mu\nu}I_{-2\alpha}|_{\text{regularised}},$
(61) $\displaystyle I^{\mu\nu\rho\sigma}_{-2\alpha}|_{\text{regularised}}$
$\displaystyle\sim$
$\displaystyle\frac{1}{4(\alpha+2)(\alpha+3)}S^{\mu\nu\rho\sigma}I_{-2\alpha}|_{\text{regularised}},$
(62)

in the asymptotic limit. We now wish to apply these conditions in the context
of $\eta$ regularisation. As mentioned earlier, it is important to note that,
in principle, different ILIs use different regulators. At this stage, we do
this in order to keep things general. As we will see shortly, the Ward
identities impose relations between different $\eta$s for different loop
topologies, which may point to a deeper underlying structure. We will
investigate the meaning of these relations in more detail in our forthcoming
work PadillaSmith23 .

In the following we denote the regulators for scalar ILIs $I_{-2\alpha}$ by
$\eta_{-2\alpha}$ and the regulators for the corresponding tensor ILIs of rank
two and four by $\theta_{-2\alpha}$ and $\kappa_{-2\alpha}$ respectively. It
follows that

$\displaystyle I_{-2\alpha}|_{\text{regularised}}$ $\displaystyle=$
$\displaystyle iJ_{-2\alpha}[\eta_{-2\alpha}],$ (63) $\displaystyle
I^{\mu\nu}_{-2\alpha}|_{\text{regularised}}$ $\displaystyle=$ $\displaystyle
iJ^{\mu\nu}_{-2\alpha}[\theta_{-2\alpha}],$ (64) $\displaystyle
I^{\mu\nu\rho\sigma}_{-2\alpha}|_{\text{regularised}}$ $\displaystyle=$
$\displaystyle iJ^{\mu\nu\rho\sigma}_{-2\alpha}[\kappa_{-2\alpha}],$ (65)

We now impose the conditions (61) and (62) making use of the decomposition
formulae (49) and (50) giving

$\displaystyle\frac{\alpha}{\alpha+2}J_{-2\alpha}\left[\tilde{\theta}_{-2\alpha}\right]$
$\displaystyle\sim$
$\displaystyle\mathcal{M}^{2}J_{-2(\alpha+1)}[\theta_{-2\alpha}],$ (66)
$\displaystyle\frac{\alpha(\alpha+5)}{(\alpha+2)(\alpha+3)}J_{-2\alpha}\left[\tilde{\kappa}_{-2\alpha}\right]$
$\displaystyle\sim$ $\displaystyle
2\mathcal{M}^{2}J_{-2(\alpha+1)}[\kappa_{-2\alpha}]-\mathcal{M}^{4}J_{-2(\alpha+2)}[\kappa_{-2\alpha}],$
(67)

as $\Lambda\to\infty$ and where

$\tilde{\theta}_{-2\alpha}=\frac{(\alpha+2)\theta_{-2\alpha}-2\eta_{-2\alpha}}{\alpha},\qquad\tilde{\kappa}_{-2\alpha}=\frac{(\alpha+2)(\alpha+3)\kappa_{-2\alpha}-6\eta_{-2\alpha}}{\alpha(\alpha+5)}.$
(68)

Note that for $\alpha\neq 0$ the combination of regulators appearing in (68)
are themselves regulators, being Schwartz and equating to unity at the origin,
$\tilde{\theta}_{-2\alpha}(0)=\tilde{\kappa}_{-2\alpha}(0)=1$. It remains to
plug in the asymptotic expressions for the scalar ILIs given by (52), (53),
(54) and (55). For the convergent ILIs, with $\alpha\geq 1$, we can use the
asymptotic formula (52) to show that the consistency relations hold
automatically in the limit as $\Lambda\to\infty$, as, of course, they should.
For $\alpha=0$, the conditions (66) and $\eqref{Jcond2}$ can be understood via
the $\alpha\to 0$ limit, giving

$\displaystyle J_{0}[\theta_{0}]-J_{0}[\eta_{0}]$ $\displaystyle\sim$
$\displaystyle\mathcal{M}^{2}J_{-2}[\theta_{0}],$ (69) $\displaystyle
J_{0}[\kappa_{0}]-J_{0}[\eta_{0}]$ $\displaystyle\sim$ $\displaystyle
2\mathcal{M}^{2}J_{-2}[\kappa_{0}]-\mathcal{M}^{4}J_{-4}[\kappa_{0}].$ (70)

Plugging in the asymptotic formulae, we see that there are logarithmic
divergences, although they cancel and the leftover finite parts yield the
following constraints

$\gamma[\theta_{0}]-\gamma[\eta_{0}]=\frac{1}{4},\qquad\gamma[\kappa_{0}]-\gamma[\eta_{0}]=\frac{5}{12}.$
(71)

For $\alpha=-1$, we get both quadratic and logarithmic divergences. The latter
cancel when we impose the two consistency conditions (66) and (67), just as
they did for $\alpha=0$, and we are left with

$\displaystyle\Lambda^{2}C_{1}[\tilde{\theta}_{2}]+2\mathcal{M}^{2}\left[\gamma[\theta_{2}]-\gamma[\eta_{2}]-\frac{1}{4}\right]$
$\displaystyle\sim$ $\displaystyle 0,$ (72)
$\displaystyle\Lambda^{2}C_{1}[\tilde{\kappa}_{2}]+\frac{3}{2}\mathcal{M}^{2}\left[\gamma[\kappa_{2}]-\gamma[\eta_{2}]-\frac{5}{12}\right]$
$\displaystyle\sim$ $\displaystyle 0,$ (73)

where we recall that $\tilde{\theta}_{2}=2\eta_{2}-\theta_{2}$ and
$\tilde{\kappa}_{2}=\frac{3}{2}\eta_{2}-\frac{1}{2}\kappa_{2}$. Cancellation
of the quadratic divergences requires that $\tilde{\theta}_{2}$ and
$\tilde{\kappa}_{2}$ are enhanced regulators of order one

$C_{1}[\tilde{\theta}_{2}]=C_{1}[\tilde{\kappa}_{2}]=0.$ (74)

Here we see for the first time the connection between gauge invariance and the
elimination of quadratic divergences in both divergent series and ILIs. The
remaining finite parts in (72) and (73) yield constraints that are very
similar to (71)

$\gamma[\theta_{2}]-\gamma[\eta_{2}]=\frac{1}{4},\qquad\gamma[\kappa_{2}]-\gamma[\eta_{2}]=\frac{5}{12}.$
(75)

At this time, we have not been able to develop any further insight into the
consistency conditions arising for the finite parts, given by (71) and (75).
However, using the fact that $\gamma[\eta_{\lambda}]=\gamma[\eta]-\ln\lambda$
and $C_{z}[\eta_{\lambda}]=\lambda^{-1-z}C_{z}[\eta]$ for any regulator
$\eta_{\lambda}(x)=\eta(\lambda x)$ with a rescaled cut-off, we can arrive at
a rather elegant solution to the full set of consistency conditions. For
$\alpha\geq-1$, this is given by

$\eta_{-2\alpha}(x)=\eta_{[1]}(x),\quad\theta_{-2\alpha}(x)=\eta_{[1]}(\lambda
x),\quad\kappa_{-2\alpha}(x)=\eta_{[1]}(\mu x),$ (76)

where $\lambda=e^{-1/4}$, $\mu=e^{-5/12}$, and $\eta_{[1]}(x)$ is any enhanced
regulator of order one.

####  3.2.1 Connecting to Schwinger proper time

It was noted in Xing:2022jtt  that regularised Schwinger proper time integrals
provide an elegant framework for satisfying Wu’s consistency relations (61)
and (62). In particular, we write the regularised ILIs as follows

$\displaystyle I_{-2\alpha}(\mathcal{M}^{2})|_{\text{regularised}}$
$\displaystyle=$
$\displaystyle\frac{i}{(\alpha+1)!}\int_{0}^{\infty}\frac{d\tau}{\tau}\
\rho(\Lambda^{2}\tau)\tau^{2+\alpha}\int\frac{d^{4}k}{(2\pi)^{4}}e^{-\tau(k^{2}+\mathcal{M}^{2})},$
(77) $\displaystyle
I_{-2\alpha}^{\mu\nu}(\mathcal{M}^{2})|_{\text{regularised}}$ $\displaystyle=$
$\displaystyle\frac{i}{(\alpha+2)!}\int_{0}^{\infty}\frac{d\tau}{\tau}\
\rho(\Lambda^{2}\tau)\tau^{3+\alpha}\int\frac{d^{4}k}{(2\pi)^{4}}k^{\mu}k^{\nu}e^{-\tau(k^{2}+\mathcal{M}^{2})},$
(78) $\displaystyle
I_{-2\alpha}^{\mu\nu\rho\sigma}(\mathcal{M}^{2})|_{\text{regularised}}$
$\displaystyle=$
$\displaystyle\frac{i}{(\alpha+3)!}\int_{0}^{\infty}\frac{d\tau}{\tau}\
\rho(\Lambda^{2}\tau)\tau^{4+\alpha}\int\frac{d^{4}k}{(2\pi)^{4}}k^{\mu}k^{\nu}k^{\rho}k^{\sigma}e^{-\tau(k^{2}+\mathcal{M}^{2})},\quad$
(79)

where $\rho(u)\to 0$ faster than any power as $u\to 0$ and $\rho(u)\to 1$ as
$u\to\infty$. To make contact with the current work, we note that we can write
these formulae in the form of equations (63) to (65), provided we generalise
the regulators as follows:

$\displaystyle\eta_{-2\alpha}(x)\to\eta_{-2\alpha}(x,\mathcal{M}_{\Lambda})$
$\displaystyle=$
$\displaystyle\frac{\int_{0}^{\infty}\frac{du}{u}\rho(u)[u(x^{2}+\mathcal{M}_{\Lambda}^{2})]^{\alpha+2}e^{-u(x^{2}+M_{\Lambda}^{2})}}{\int_{0}^{\infty}\frac{du}{u}\rho(u)[u\mathcal{M}_{\Lambda}^{2}]^{\alpha+2}e^{-uM_{\Lambda}^{2}}},$
(80)
$\displaystyle\theta_{-2\alpha}(x)\to\theta_{-2\alpha}(x,\mathcal{M}_{\Lambda})$
$\displaystyle=$
$\displaystyle\frac{\int_{0}^{\infty}\frac{du}{u}\rho(u)[u(x^{2}+\mathcal{M}_{\Lambda}^{2})]^{\alpha+3}e^{-u(x^{2}+M_{\Lambda}^{2})}}{\int_{0}^{\infty}\frac{du}{u}\rho(u)[u\mathcal{M}_{\Lambda}^{2}]^{\alpha+3}e^{-uM_{\Lambda}^{2}}},$
(81)
$\displaystyle\kappa_{-2\alpha}(x)\to\kappa_{-2\alpha}(x,\mathcal{M}_{\Lambda})$
$\displaystyle=$
$\displaystyle\frac{\int_{0}^{\infty}\frac{du}{u}\rho(u)[u(x^{2}+\mathcal{M}_{\Lambda}^{2})]^{\alpha+4}e^{-u(x^{2}+M_{\Lambda}^{2})}}{\int_{0}^{\infty}\frac{du}{u}\rho(u)[u\mathcal{M}_{\Lambda}^{2}]^{\alpha+4}e^{-uM_{\Lambda}^{2}}},$
(82)

where $u=\Lambda^{2}\tau$, $x=|k|/\Lambda$, and
$\mathcal{M}_{\Lambda}=|\mathcal{M}|/\Lambda$. These do not have the precise
form of the $\eta$ regulators described in this paper as the $\Lambda$
dependence is spread between $x=|k|/\Lambda$ and
$\mathcal{M}_{\Lambda}=|\mathcal{M}|/\Lambda$. We will consider much more
general regulators in our forthcoming work PadillaSmith23 , but for now let us
examine the form of these regulators in the limit where
$\mathcal{M}_{\Lambda}\to 0$. In particular, note that in this limit, the
denominators can generically be written as

$\displaystyle\int_{0}^{\infty}\frac{du}{u}\rho(u)[u\mathcal{M}_{\Lambda}^{2}]^{n}e^{-uM_{\Lambda}^{2}}$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{dy}{y}\rho\left(\frac{y}{\mathcal{M}_{\Lambda}^{2}}\right)y^{n}e^{-y}$
(83) $\displaystyle\sim$
$\displaystyle\int_{0^{+}}^{\infty}\frac{dy}{y}y^{n}e^{-y}=(n-1)!$ (84)

It follows that as $\mathcal{M}_{\Lambda}\to 0$, the regulators fall into the
class described by equation (28), with

$\eta_{-2\alpha}(x,0)=\lambda_{2+\alpha}(x),\qquad\theta_{-2\alpha}(x,0)=\lambda_{3+\alpha}(x),\qquad\kappa_{-2\alpha}(x,0)=\lambda_{4+\alpha}(x).$
(85)

Using the expression (30), it is easy to verify that
$\tilde{\theta}_{2}=2\eta_{2}(x,0)-\theta_{2}(x,0)$ and
$\tilde{\kappa}_{2}=\frac{3}{2}\eta_{2}(x,0)-\frac{1}{2}\kappa_{2}(x,0)$ are
enhanced regulators of order one, consistent with the gauge consistency
condition (74). Of course, this had to be true given the claim in Xing:2022jtt
that regularised Schwinger proper time integrals automatically satisfy Wu’s
consistency relations but it serves as a nice check of our formalism.

We can also compute the finite integral (56) for each of the regulators in
(85). Indeed, from (28), we change the integration variable to $y=ux^{2}$ and
write

$\lambda_{n}(x)=\frac{1}{(n-1)!}\int_{0}^{\infty}\frac{dy}{y}\rho(y/x^{2})y^{n}e^{-y},$
(86)

to then find that

$\displaystyle\gamma[\lambda_{n}]$ $\displaystyle=$
$\displaystyle\frac{1}{(n-1)!}\int_{0}^{\infty}dx\int_{0}^{\infty}dy\rho^{\prime}(y/x^{2})\frac{y^{n}}{x^{3}}e^{-y}(2\ln
x)$ (87) $\displaystyle=$
$\displaystyle\frac{1}{2(n-1)!}\int_{0}^{\infty}dy\int_{0}^{\infty}du\rho^{\prime}(u)y^{n-1}e^{-y}(\ln
y-\ln u)$ (88) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\Psi(n)-\int_{0}^{\infty}du\rho^{\prime}(u)\ln
u\right],$ (89)

where $\Psi(z)=\Gamma^{\prime}(z)/\Gamma(z)$ is the digamma function and we
have used the fact that $\rho(u)\to 0$ as $u\to 0$ and $\rho(u)\to 1$ as
$u\to\infty$. It immediately follows that

$\displaystyle\gamma[\theta_{-2\alpha}(x,0)]-\gamma[\eta_{-2\alpha}(x,0)]$
$\displaystyle=$ $\displaystyle\frac{1}{2\alpha+4},$ (90)
$\displaystyle\gamma[\kappa_{-2\alpha}(x,0)]-\gamma[\eta_{-2\alpha}(x,0)]$
$\displaystyle=$
$\displaystyle\frac{2\alpha+5}{2\left(2+\alpha\right)\left(\alpha+3\right)}.$
(91)

For the logarithmically divergent ILIs corresponding to $\alpha=0$, the
expressions (90) and (91) satisfy the finite parts of the consistency
condition, given by (71). However, for the quadratically divergent ILIs
corresponding to $\alpha=-1$, they do not agree with the corresponding
consistency conditions (75). The difference is easily understood. Although
Xing:2022jtt  has shown that ILIs regularised using Schwinger proper time
automatically satisfy Wu’s consistency relations, recall that this is not
exactly equivalent to $\eta$ regularisation with regulators of the form
$\eta(|k|/\Lambda)$ for which we derived the consistency conditions (74), (71)
and (75). The equivalence only emerges in the limit $\mathcal{M}_{\Lambda}\to
0$, suggesting that in the limit of large $\Lambda$, only the leading order
constraints should align. This is indeed the case. For quadratically divergent
ILIs, the leading order constraint is (74) and this holds for the derived
regulators, $\eta_{2}(x,0)$, $\theta_{2}(x,0)$ and $\kappa_{2}(x,0)$, whereas
the subleading constraint (75) does not, exactly as expected. For
logarithmically divergent ILIs, the leading order constraint is (71), which we
have just seen to hold for the derived regulators, $\eta_{0}(x,0)$,
$\theta_{0}(x,0)$ and $\kappa_{0}(x,0)$, just as it should.

##  4 Conclusions

We have explored possible connections between analytic number theory and the
study of divergent series and the ultra-violet regularisation of loop
integrals in perturbative quantum field theory. On the number theory side, we
have extended Tao’s work on smoothed asymptotics Tao11  which offers a
tantalising taste of QFT. Both exhibit regulator dependence of power law
divergences, while the universal features of finite terms in Tao’s study of
divergent series mirror the universal features of logarithms in QFT. However,
our analysis runs much deeper than this elegant analogy. Inspired by Tao’s
work, we have developed a new and general method for regularising divergent
integrals in QFT which we dub $\eta$ regularisation. As a result, we have
discovered a surprising connection between the regularisation of divergent
series and the elimination of divergences in analytic number theory and the
preservation of gauge invariance at one loop in a regularised quantum field
theory. Is this just a coincidence, or does it signal something deeper and a
possible window into how divergences are eliminated at high energies in a
complete microscopic theory?

There is certainly much more to be learnt before we can answer this question.
Indeed, it is notable that the consistency relations for preserving gauge
invariance, at least in the context of $\eta$ regularisation, also include a
constraint on the finite terms, as well as the divergences. We have not been
able to glean a deeper understanding of the meaning of these additional
constraints, beyond noting that they are satisfied in the appopriate limit by
$\eta$ regulators inspired by Schwinger proper time integrals.

In a forthcoming publication, we will extend our study of $\eta$
regularisation to a much broader class of regulators, going beyond the
structure inspired by Tao PadillaSmith23 . This will allow us to capture all
known methods of regularisation and more beyond. Implementing Wu’s consistency
relations Wu03 ; Wu04 ; Wu14  in this general setting will also be shown to
yield a master equation for the regularisation scheme in the form of a
differential equation. Finding solutions will be the same as finding gauge
invariant regulators, which will include dimensional regularisation, as well
as the enhanced $\eta$ regulators discussed here. We will also describe how
$\eta$ regularisation can be extended to arbitrary loop order, consistent with
unitarity, locality, and causality.

There are several ways in which we may continue to build the bridge towards
string theory and the softening of amplitudes at high energies. Given that
critical string theory is a fundamentally higher dimensional theory, it would
be interesting to explore the dimensional dependence of Wu’s consistency
relations and the implications on $\eta$ regularisation. It is also important
to better understand the role played by gravity. Is there a gravitational
analogue of Wu’s consistency relations that guarantees diffeomorphism
invariance is preserved by the regulator? Further, how do we implement $\eta$
regularisation in curved space?

The work of (Abel19, ; Abel20, ) could open up a path connecting $\eta$
regularisation (in the Schwinger representation) to string theory via non-
local particle theories that preserve the higher dimensional properties of
strings. In this context, we are particularly interested in those regulators
that exhibit some remnant of modular invariance, such as those presented in
(Abel21, ). We would also like to deepen this analysis and say something about
the nature of QFT and the effect of smoothing in position space.

Although our main focus is on developing QFT and building a bridge towards
string theory, there is much to be explored at the level of analytic number
theory. For example, it would be interesting to see if any other gauge
invariant regulators identified in PadillaSmith23  have a role to play in the
suppression of divergences in divergent series. Further, how do these enhanced
regulators connect to analytic continuation? Tao has already provided some
insight in this regard Tao11 . To develop more understanding it would be
useful to extend Tao’s results to series of non-polynomial functions such as
logarithms and polylogarithms, which are also expected to appear in
applications to QFT. What relation, if any, can then be made to resurgence
theory and trans-series? In particular, it would be interesting to see if any
connection can be made between $\eta$ regularisation and some general
resummation procedure that might restore some, or all, non-perturbative
information.

##### Acknowledgements

We are grateful to John Barrett, Pete Millington, Lubos Motl, Benjamin Muntz,
Terence Tao and Paul Saffin for useful comments and discussions. RGCS was
supported by a Bell Burnell Studentship and AP by STFC consolidated grant
number ST/T000732/1. For the purpose of open access, the authors have applied
a CC BY public copyright licence to any Author Accepted Manuscript version
arising. No new data were created during this study.

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