# Discrete heat kernel, UV modified Green’s function, and higher derivative
theories

Nahomi Kan<EMAIL_ADDRESS>National Institute of Technology, Gifu College,
Motosu-shi, Gifu 501-0495, Japan  Masashi Kuniyasu<EMAIL_ADDRESS>Kiyoshi Shiraishi<EMAIL_ADDRESS>Zhenyuan Wu
<EMAIL_ADDRESS>Graduate School of Sciences and Technology for
Innovation, Yamaguchi University, Yamaguchi-shi, Yamaguchi 753–8512, Japan

###### Abstract

We perform the UV deformation of the Green’s function in free scalar field
theory using a discrete heat kernel method. It is found that the simplest UV
deformation based on the discretized diffusion equation leads to the well-
known Pauli–Villars effective Lagrangian. Furthermore, by extending
assumptions on the discretized equation, we find that the general higher
derivative theory is derived from the present UV deformation. In some specific
cases, we also calculate the vacuum expectation values of the scalar field
squared in nontrivial background spaces and examine their dependence on the UV
cutoff constant.

###### pacs:

03.70.+k, 11.10.Lm, 11.10.Kk .

##  I Introduction

Taking the effects of quantum gravity into account, we can expect the
emergence of fundamental lengths in effective field theories. In recent years,
various types of UV deformed Green’s functions incorporating a fundamental
length have been proposed based on the expression of the Green’s function
obtained by the heat kernel method Pad1 ; Pad2 ; Pad3 ; Pad4 ; Pad5 ; Pad6 ;
SSP ; KSSP ; AD ; ABM ; AL ; Siegel .111The authors of Refs. Pad1 ; Pad2 ;
Pad3 ; Pad4 ; Pad5 ; Pad6 ; SSP ; KSSP ; AD ; ABM ; AL  explored the
construction of them mainly assuming the duality in the path integral
amplitude. Putting a lower limit of integral on the Schwinger’s proper-time
parameter in the standard expression is known as the easiest way to regularize
the Green’s function in the heat kernel method. Siegel Siegel  pointed out
that this treatment is equivalent to considering infinite derivative effective
field theory instead of original canonical theory.

When we consider UV deformation from the viewpoint of regularization, we can
imagine that there can be too many variations in the way of deformation.
However, it is desirable to maintain properties related to the symmetry of the
original field theory as much as possible.

In our previous paper KKSW , we calculated the vacuum expectation values of
the stress tensor in nontrivial background spaces using the UV deformed
Green’s function Pad1 ; Pad2 ; Pad3 ; Pad4 ; Pad5 ; Pad6 ; SSP ; KSSP ; AD ;
ABM ; AL ; Siegel . The dependence on the cutoff constant (which may be
identified with the Planck length) was clarified by using numerical
calculations. However, a primitive approach, in which we applied only the raw
deformation of scalar Green’s functions to the original theory, resulted in
small violations of the conservation law, which is negligible in the limit of
a small cutoff length and large physical scales.

In the present paper, we start with modifying the original heat kernel method.
We consider a heat kernel that satisfies a discretized heat diffusion
equation. Of course, we first define the basic scheme which leads to the
standard Green’s function without change. Then, by changing the lower limit of
the sum, instead of integral, UV deformation of the Green’s function is
realized. In this case, the Green’s function containing higher order terms in
the momentum space is obtained. It coincides with Pauli–Villars’ Green
function in its simplest form IZ ; Collins ; PS .222The Pauli–Villars-type
scalar field model is also considered SS  to study the possible effects of
high energy corrections to the inflationary perturbation spectrum. We identify
the higher derivative theory as the effective field theory that yields the UV
modified Green’s function.

In the second half of the paper, based on the formulation used in the first
half, we consider more general UV deformation and evaluate the vacuum
expectation values of the scalar field squared in nontrivial background spaces
as specific cases. In each case, attention is paid to the cutoff dependence of
the physical quantity. The dimensionful cutoff constant can be taken as a
parameter in a mere regularization, but similarly to the introduction of the
aforementioned UV deformed Green’s function Pad1 ; Pad2 ; Pad3 ; Pad4 ; Pad5 ;
Pad6 ; SSP ; KSSP ; AD ; ABM ; AL ; Siegel , it may be resulted from yet-
unknown quantum gravity theory (possibly directly related to the Planck
length). Thus, the cutoff constant can be considered as a fundamental constant
in nature. The discussion in this paper is the first stage of
‘phenomenological’ demonstration which is aimed to future theories with
absolute UV completion.

The structure of this paper is as follows. In Sec. II, we describe the Green’s
function constructed from the heat kernel which satisfies the discretized
equations. In this paper, we consider a locally flat Euclidean space as a
background space, and here we consider a uniformly isotropic $D$-dimensional
space $\mathbf{R}^{D}$. Then, UV deformation is introduced based on a simple
assumption. We discuss more generalized UV modification in Sec. III. In the
effective theory due to the UV modification obtained in the previous sections,
we calculate the vacuum expectation values of the square of the scalar field
in two kinds of nontrivial background spaces, a Kaluza–Klein-type space and a
conical space in Sec. IV. In the last section, we discuss the results and
future prospects.

##  II Discrete heat kernel and the simplest UV modified Green’s function

First of all, we review some basic facts of the standard Green’s functions and
heat kernels Vassilevich ; Camporesi . The standard Green’s function
$G(x,x^{\prime})$ of a scalar field with mass $m$ in a metric space with the
metric $g_{\mu\nu}$ satisfies

$(-\Box_{x}+m^{2}){G}(x,x^{\prime})=\frac{1}{\sqrt{g}}\delta(x,x^{\prime})\,,$
(1)

where $\Box=\frac{1}{\sqrt{g}}\partial_{\mu}\sqrt{g}g^{\mu\nu}\partial_{\nu}$
is the d’Alembert operator on scalar fields, and
$\frac{1}{\sqrt{g}}\delta(x,x^{\prime})$ is the $D$-dimensional covariant
delta function.

The main tool that we select here is the heat kernel of the Laplacian. The
heat kernel $K(x,x^{\prime};s)$ is introduced as a representation of the
Green’s function constructed by using it:

${G}(x,x^{\prime})=\int_{0}^{\infty}ds\,K(x,x^{\prime};s)\,.$ (2)

Here, the heat kernel follows the equation

$\left[\frac{\partial}{\partial
s}+(-\Box_{x}+m^{2})\right]K(x,x^{\prime};s)=0\,,$ (3)

with the initial condition $\lim_{s\rightarrow
0}K(x,x^{\prime};s)=\frac{1}{\sqrt{g}}\delta(x,x^{\prime})$.

Next, we consider the Fourier transform $\tilde{K}(p;s)$ of the heat kernel in
an Euclidean space $\mathbf{R}^{D}$. This is defined through the relation

$K(x,x^{\prime};s)=K(x-x^{\prime};s)=\int\frac{d^{D}p}{(2\pi)^{D}}\tilde{K}(p;s)e^{ip\cdot(x-x^{\prime})}\,.$
(4)

Then, the heat equation becomes

$\left[\frac{\partial}{\partial s}+(p^{2}+m^{2})\right]\tilde{K}(p;s)=0\,,$
(5)

with the ‘initial’ condition $\tilde{K}(p;0)=1$. One can easily find the
solution

$\tilde{K}(p;s)=e^{-s(p^{2}+m^{2})}\,,$ (6)

and immediately obtain the Fourier transform of the Green’s function
$\tilde{G}(p)$ in $\mathbf{R}^{D}$ as follows:

$\tilde{G}(p)=\int_{0}^{\infty}\exp\left[-s(p^{2}+m^{2})\right]ds=\frac{1}{p^{2}+m^{2}}\,.$
(7)

Then, the real space Green’s function is found to be

$G(x,x^{\prime})=G(x-x^{\prime})=\int\frac{d^{D}p}{(2\pi)^{D}}e^{ip\cdot(x-x^{\prime})}\tilde{G}(p)=\int\frac{d^{D}p}{(2\pi)^{D}}\frac{e^{ip\cdot(x-x^{\prime})}}{p^{2}+m^{2}}\,.$
(8)

We can exchange the order of integrations. That is, the real space heat kernel
is

$K(x-x^{\prime};s)=\frac{1}{(4\pi
s)^{D/2}}\exp\left({-\frac{|x-x^{\prime}|^{2}}{4s}-m^{2}s}\right)\,,$ (9)

and the Green’s function is recovered as

$G(x-x^{\prime})=\int_{0}^{\infty}ds\,K(x-x^{\prime};s)=\int_{0}^{\infty}ds\frac{1}{(4\pi
s)^{D/2}}\exp\left({-\frac{|x-x^{\prime}|^{2}}{4s}-m^{2}s}\right)\,.$ (10)

Anyway, the closed form for $G(x-x^{\prime})$ turns out to be

$G(x-x^{\prime})=\frac{m^{D-2}}{(2\pi)^{D/2}(m|x-x^{\prime}|)^{\frac{D-2}{2}}}K_{\frac{D-2}{2}}({m|x-x^{\prime}|})\,,$
(11)

where $K_{\nu}(z)$ is the modified Bessel function of the second kind.
Incidentally, for the massless case ($m=0$), we find

$G(x-x^{\prime})=\frac{\Gamma\left({\textstyle\frac{D-2}{2}}\right)}{4\pi^{D/2}|x-x^{\prime}|^{D-2}}\,,$
(12)

where $\Gamma(z)$ is the gamma function.

Now, we turn to considering a discretized heat kernel formulation, whose
continuous limit reproduces the standard method constructing the Green’s
function. To this end, we propose the discrete proper-time heat equation in
$\mathbf{R}^{D}$ as333Here we used the backward difference, but using the
forward difference does not give essentially different results.

$\frac{\tilde{K}_{k}-\tilde{K}_{k-1}}{\epsilon}=-(p^{2}+m^{2})\tilde{K}_{k}\,,$
(13)

where $k=1,\dots,N$ and $\epsilon$ is a positive constant. We adopt the
initial value $\tilde{K}_{0}=1$. Then, the solution of the recurrence equation
(13) is given by

$\tilde{K}_{k}=\left[\frac{1}{1+\epsilon(p^{2}+m^{2})}\right]^{k}\,.$ (14)

On the other hand, the following sum is directly concluded from (13):

$(p^{2}+m^{2})\left[\sum_{k=1}^{N}\epsilon\tilde{K}_{k}\right]=\tilde{K}_{0}-\tilde{K}_{N}\,.$
(15)

Observing that
$\lim_{N\rightarrow\infty}\left[{1+\epsilon(p^{2}+m^{2})}\right]^{-N}=0$, we
find

$\sum_{k=1}^{\infty}\epsilon\tilde{K}_{k}=\frac{1}{p^{2}+m^{2}}\,,$ (16)

which is just the discretized version of the integral (7).

The continuum limit is realized by taking $k\epsilon\rightarrow s$,
$\tilde{K}_{k}\rightarrow\tilde{K}(p;s)$, and $\epsilon\rightarrow 0$. Then,
we find that

$\left[\frac{1}{1+\epsilon(p^{2}+m^{2})}\right]^{k}\rightarrow\exp[-s(p^{2}+m^{2})]\,,\quad\sum_{k=1}^{\infty}\epsilon\tilde{K}_{k}\rightarrow\int_{0}^{\infty}\tilde{K}(p;s)ds=\frac{1}{p^{2}+m^{2}}\,.$
(17)

After making the above preparation, we consider the UV modification of the
Green’s function. For this purpose, we define the modified Green’s function in
momentum space

$\tilde{G}_{n}(p)\equiv\sum_{k=n}^{\infty}\epsilon\tilde{K}_{k}=\frac{1}{(p^{2}+m^{2})[1+\epsilon(p^{2}+m^{2})]^{n-1}}\,.$
(18)

To see the effect of the modification, we investigate a special case, in which
we assume that $\epsilon=\epsilon^{\prime}/n$ and $n\rightarrow\infty$. In
this case, $\tilde{G}$ reduces to

$\tilde{G}_{\infty}(p)=\frac{e^{-\epsilon^{\prime}(p^{2}+m^{2})}}{p^{2}+m^{2}}=\int_{\epsilon^{\prime}}^{\infty}\exp[-(p^{2}+m^{2})s]ds\,,$
(19)

which was studied by Siegel Siegel . It is also known that the present manner
is used in the UV regularization in the standard heat kernel formalism.

Here, we study the simplest modification, the case with $n=2$. Namely, we use

$\tilde{G}_{2}(p)=\frac{1}{(p^{2}+m^{2})[1+\epsilon(p^{2}+m^{2})]}=\frac{1}{p^{2}+m^{2}}-\frac{1}{p^{2}+m^{2}+1/\epsilon}\,.$
(20)

This is the simplest Pauli–Villars-type propagator (Green’s function) IZ ;
Collins ; PS . The effective scalar field theory that leads to this Green’s
function is

$S_{2}=\int\frac{d^{D}p}{(2\pi)^{D}}\,\tilde{\cal
L}_{2}=-\frac{1}{2}\int\frac{d^{D}p}{(2\pi)^{D}}\tilde{\phi}(-p)(p^{2}+m^{2})[1+\epsilon(p^{2}+m^{2})]\tilde{\phi}(p)\,,$
(21)

in the Fourier transformed form, and in the real coordinate space, it turns
out to be

$S_{2}=\int d^{D}x\,{\cal L}_{2}=-\frac{1}{2}\int
d^{D}x\,{\phi}(x)(-\Box+m^{2})[1+\epsilon(-\Box+m^{2})]{\phi}(x)\,,$ (22)

where $\phi$ is the real scalar field and $\tilde{\phi}$ is its Fourier
transform.

We can introduce an auxiliary field to continue study, referring to works on
higher derivative theories GOW ; CL  motivated by the seminal Lee–Wick model
LW . We consider the alternative Lagrangian

${\cal
L}^{\prime}_{2}=-\frac{1}{2}\phi(-\Box+m^{2})\phi-\psi(-\Box+m^{2})\phi+\frac{1}{2\epsilon}\psi^{2}\,.$
(23)

The equation of motion of the initially auxiliary field $\psi$ from the
Lagrangian is

$\psi=\epsilon(-\Box+m^{2})\phi\,.$ (24)

Thus, substituting this equation to ${\cal L}^{\prime}_{2}$ yields the
original Lagrangian ${\cal L}_{2}$. On the other hand, setting
$\chi\equiv\phi+\psi$, we find

${\cal
L}^{\prime}_{2}=-\frac{1}{2}\chi(-\Box+m^{2})\chi+\frac{1}{2}\psi(-\Box+m^{2}+1/\epsilon)\psi\,.$
(25)

This effective Lagrangian describes two free scalar fields, whose masses are
$m$ and $\sqrt{m^{2}+1/\epsilon}$. The scalar field $\psi$ with mass
$\sqrt{m^{2}+1/\epsilon}$ governed by the part of the Lagrangian with a
‘wrong’ sign. This field decouples from the physical spectrum if
$\epsilon\rightarrow 0$, since its mass becomes infinitely large.

The real space Green’s function

$G_{2}(x-x^{\prime})=\frac{m^{\frac{D-2}{2}}K_{\frac{D-2}{2}}({m|x-x^{\prime}|})-(m^{2}+1/\epsilon)^{\frac{D-2}{4}}K_{\frac{D-2}{2}}({\sqrt{m^{2}+1/\epsilon}|x-x^{\prime}|})}{(2\pi)^{D/2}|x-x^{\prime}|^{\frac{D-2}{2}}}\,,$
(26)

thus involves two mass scales $m$ and $\sqrt{m^{2}+1/\epsilon}$. For the
massless case ($m=0$), we find

$G_{2}(x-x^{\prime})=\frac{\Gamma\left({\textstyle\frac{D-2}{2}}\right)}{4\pi^{D/2}|x-x^{\prime}|^{D-2}}-\frac{\epsilon^{-\frac{D-2}{4}}K_{\frac{D-2}{2}}({\epsilon^{-\frac{1}{2}}|x-x^{\prime}|})]}{(2\pi)^{D/2}|x-x^{\prime}|^{\frac{D-2}{2}}}\,.$
(27)

At small $|x-x^{\prime}|$, the Green’s function $G_{2}(x-x^{\prime})$ behaves
as $\sim\epsilon^{-1}|x-x^{\prime}|^{4-D}$ for $D\neq 4$ and
$\sim\epsilon^{-1}\ln(\epsilon^{-1/2}|x-x^{\prime}|)$ for $D=4$.

In the next section, we will consider possible generalization of the UV
modification along with the present line of thought.

##  III generalization of discretization and interesting cases

The generalization of (13) is not difficult. The parameter $\epsilon$ can have
dependence on the integer $k$, i.e.,

$\frac{\tilde{K}_{k}-\tilde{K}_{k-1}}{\epsilon_{k}}=-(p^{2}+m^{2})\tilde{K}_{k}\,,$
(28)

where positive parameters $\epsilon_{k}$ are restricted by

$\epsilon_{k}=\epsilon_{n}\quad\mbox{for~{}}k>n\,.$ (29)

Then, the summation yields

$\tilde{G}_{n}(p)\equiv\sum_{k=n}^{\infty}\epsilon_{k}\tilde{K}_{k}=\frac{1}{(p^{2}+m^{2})\prod_{k=1}^{n-1}[1+\epsilon_{k}(p^{2}+m^{2})]}\,.$
(30)

Note that $\tilde{G}_{1}(p)=\frac{1}{p^{2}+m^{2}}$ and
$\tilde{G}_{2}(p)=\frac{1}{(p^{2}+m^{2})[1+\epsilon_{1}(p^{2}+m^{2})]}$. They
are the same as those discussed in Sec. II. Now, $\tilde{G}_{n}(p)$ involves
$n-1$ parameters in general.

The effective action that leads to this modified propagator in an expression
with the momentum argument is

$S=-\frac{1}{2}\int\frac{d^{D}p}{(2\pi)^{D}}\tilde{\phi}(-p)[\tilde{G}_{n}(p)]^{-1}\tilde{\phi}(p)\,,$
(31)

since the free propagator is obtained by the Gaussian integration in path
integral formalism.

Provided that values of $\epsilon_{k}$ are all different, the Green’s function
in the momentum space can be expanded by partial fractions as follows:

$\frac{1}{(p^{2}+m^{2})\prod_{k=1}^{n-1}[1+\epsilon_{k}(p^{2}+m^{2})]}=\frac{1}{p^{2}+m^{2}}+\sum_{k=1}^{n-1}\frac{c_{k}\epsilon_{k}}{1+\epsilon_{k}(p^{2}+m^{2})}\,,$
(32)

where the coefficient $c_{k}$ reads

$c_{k}=\frac{-\epsilon_{k}^{n-2}}{\prod_{i\neq
k}^{n-1}(\epsilon_{k}-\epsilon_{i})}\,.$ (33)

Now, we consider concrete examples. We here consider two particular cases for
$\\{\epsilon_{k}\\}$. We shall call them model A and model B.

$\bullet$ Model A

First, we consider
$\\{\epsilon_{k}\\}=\\{\epsilon,\frac{\epsilon}{2},\frac{\epsilon}{3},\ldots\frac{\epsilon}{n-1}\\}$.
We know the mathematical formula

$\tilde{G}_{A}(p)\equiv\frac{1}{(p^{2}+m^{2})\prod_{k=1}^{n-1}[1+\frac{\epsilon}{k}(p^{2}+m^{2})]}=\sum_{k=0}^{n-1}{\textstyle{n-1}\choose{k}}\frac{(-1)^{k}\epsilon/k}{1+\frac{\epsilon}{k}(p^{2}+m^{2})}\,,$
(34)

thus,

$c_{k}=(-1)^{k}{\textstyle{n-1}\choose{k}}\qquad(k=1,\ldots,n-1)\,.$ (35)

Note that the case $n=2$ trivially reduces to the simplest case treated in
Sec. II. Thus, this first example is a straightforward extension of the
simplest case.

$\bullet$ Model B

The second example enjoys the infinite derivatives, i.e., defined with
$n\rightarrow\infty$. We set
$\\{\epsilon_{k}\\}=\\{\epsilon,\frac{\epsilon}{2^{2}},\frac{\epsilon}{3^{2}},\ldots\\}$.
Now, the Green’s function in the momentum space becomes

$\displaystyle\tilde{G}_{B}(p)\equiv\frac{1}{(p^{2}+m^{2})\prod_{k=1}^{\infty}[1+\frac{\epsilon}{k^{2}}(p^{2}+m^{2})]}$
$\displaystyle=$
$\displaystyle\frac{\pi\sqrt{\epsilon}}{\sqrt{p^{2}+m^{2}}\sinh[\pi\sqrt{\epsilon(p^{2}+m^{2})}]}$
(36) $\displaystyle=$
$\displaystyle\frac{1}{p^{2}+m^{2}}+\sum_{k=1}^{\infty}\frac{2(-1)^{k}\epsilon/k^{2}}{1+\frac{\epsilon}{k^{2}}(p^{2}+m^{2})}\,,$

and we find

$c_{k}=(-1)^{k}\cdot 2\qquad(k\geq 1)\,.$ (37)

This type of the Green’s function has appeared in Refs. FS1 ; FS2 ; Fujikawa
as generalizations of the Pauli–Villars regularization. Incidentally, the
Green’s function for Model B with $m=0$ in configuration space can be written
as

$G_{B}(x-x^{\prime})=\frac{\sqrt{\epsilon}}{\pi^{\frac{D-1}{2}}}\sum_{k=0}^{\infty}\frac{\Gamma\left(\frac{D-1}{2}\right)}{\left[|x-x^{\prime}|^{2}+4\pi^{2}\left(k+\frac{1}{2}\right)^{2}\epsilon\right]^{\frac{D-1}{2}}}\qquad(\mbox{for~{}}m=0)\,.$
(38)

The limit $\epsilon\rightarrow 0$ reduces this expression to the normal
massless Green’s function (12). Furthermore, especially for $D=5$, one can
obtain a closed form of $G_{B}(w)$ as

$G_{B}(w)=\frac{1}{8\pi^{2}}\left[\frac{1}{w^{3}}\tanh\frac{w}{2l}-\frac{1}{2lw^{2}}\mbox{sech}^{2}\frac{w}{2l}\right]\qquad(\mbox{for~{}}m=0,D=5)\,,$
(39)

where $w\equiv|x-x^{\prime}|$.

In the next section, we evaluate vacuum expectation values of the scalar field
squared in the effective theory of Models A and B in nontrivial spaces.

##  IV vacuum expectation value of the scalar field squared for the UV
modified cases in nonsimply connected spaces

Now, we can express the one-loop vacuum expectation value of the square of the
scalar field $\langle\phi^{2}\rangle$, or also called as the quantum
fluctuation of the scalar field, in the effective higher derivative theory. We
find, with the aid of (32),

$\langle\phi^{2}\rangle=\lim_{x^{\prime}\rightarrow
x}G_{n}(x,x^{\prime})=\lim_{x^{\prime}\rightarrow
x}\left[G(x,x^{\prime};m^{2})+\sum_{k=1}^{n-1}c_{k}G(x,x^{\prime};M^{2}_{k})\right]\,,$
(40)

where $G(x,x^{\prime};M^{2})$ is the Green’s function of a canonical scalar
field with mass $M$ and $M^{2}_{k}=m^{2}+1/\epsilon_{k}$.

We should note that the value of $\langle\phi^{2}\rangle$ is still a ‘bare’
quantity. Even if the highest divergence (e.g. quadratic divergences in four
dimensions) has been canceled, divergences remain for a sufficient high
dimensional case (see the calculation in this section below). Moreover, even
if divergences have ceased, the bare value of the stress tensor should be of
order of $\epsilon^{-\alpha}$ $(\alpha>0)$. If one would like to combine the
cutoff constant $\epsilon$ with some fundamental theory, the finite but huge
quantities that appears in quantum expectation values should be subtracted by
some method. In the present analysis, similar to many past works including our
work KKSW , subtraction is performed by hand, but done on the basis of
physical meaning.

The calculation of the vacuum expectation values of the free scalar field
squared in nonsimply connected space has been studied by many authors in
recent decades. We consider two specific cases. We pick up a Kaluza–Klein-type
space and a conical space.

###  IV.1 The Kaluza–Klein-type space

First, we consider a $D$-dimensional space whose metric is given by

$ds^{2}=dx^{\lambda}dx_{\lambda}=\sum_{i=1}^{D-1}dz_{i}^{2}+dy^{2}\,,$ (41)

where the Greek index runs over $1,\dots,D$, while the Roman index runs over
$1,\dots,D-1$. The coordinate $y$ is the coordinate on a circle ($S^{1}$) and
we assume that $0\leq y<L$, where $L$ is the circumference of the circle. The
periodic boundary condition in the direction of $y$ is supposed to be applied
on the massless scalar, i.e., $\phi(y+L)=\phi(y)$. In this space,
$\mathbf{R}^{D-1}\otimes S^{1}$, the standard massless Green’s function for a
scalar field takes the form

$G(x,x^{\prime};M^{2})=\int_{0}^{\infty}\frac{ds}{(4\pi
s)^{D/2}}\sum_{q=-\infty}^{\infty}\exp\left[\textstyle-\frac{\zeta^{2}+(\eta-
qL)^{2}}{4s}-M^{2}s\right]\,,$ (42)

where $\zeta=|z-z^{\prime}|$ and $\eta=y-y^{\prime}$. Then, we define the
quantities with subtraction

$\langle\bar{\phi^{2}}\rangle[M^{2}]\equiv\lim_{x^{\prime}\rightarrow
x}\bar{G}(x,x^{\prime};M^{2})\,,$ (43)

where
$\bar{G}(x,x^{\prime};M^{2})\equiv{G}(x,x^{\prime};M^{2})-\lim_{L\rightarrow\infty}{G}(x,x^{\prime};M^{2})$.
This subtraction is aimed to take the difference from the quantity in the
infinitely extending space $\mathbf{R}^{D}$. After straightforward
manipulations, we get

$\langle\bar{\phi^{2}}\rangle[M^{2}]=\frac{2}{(4\pi)^{D/2}}\int_{0}^{\infty}\frac{ds}{s^{D/2}}\sum_{q=1}^{\infty}\exp\left[{\textstyle-\frac{q^{2}L^{2}}{4s}-M^{2}s}\right]=\frac{1}{\pi}\sum_{q=1}^{\infty}\left(\textstyle\frac{M}{2\pi
qL}\right)^{\frac{D-2}{2}}K_{\frac{D-2}{2}}(qML)\,,$ (44)

and thus, we can evaluate the vacuum expectation value in the effective theory
as

$\langle\bar{\phi^{2}}\rangle=\langle\bar{\phi^{2}}\rangle[m^{2}]+\sum_{k=1}^{n-1}c_{k}\langle\bar{\phi^{2}}\rangle[M_{k}^{2}]\,.$
(45)

In general, in the higher derivative scalar field theory (31) which leads to
the Green’s function (32), the vacuum expectation value for the scalar field
squared $\langle\bar{\phi^{2}}\rangle$ is written by

$\langle\bar{\phi^{2}}\rangle=\frac{2}{(4\pi)^{D/2}}\int_{0}^{\infty}\frac{ds}{s^{D/2}}\sum_{q=1}^{\infty}\exp\left[{\textstyle-\frac{q^{2}L^{2}}{4s}}-m^{2}s\right]\varrho(s)\,,$
(46)

where $\varrho(s)\equiv 1+\sum_{k=1}^{n-1}c_{k}\exp[-s/\epsilon_{k}]$.

For Model A, we find

$\varrho_{A}(s)=\sum_{k=0}^{n-1}(-1)^{k}{\textstyle{n-1}\choose{k}}e^{-ks/\epsilon}=(1-e^{-s/\epsilon})^{n-1}\,.$
(47)

For Model B, we find

$\varrho_{B}(s)=\sum_{k=-\infty}^{\infty}(-1)^{k}e^{-k^{2}s/\epsilon}=\vartheta_{4}(0,e^{-s/\epsilon})=\sqrt{\frac{\pi\epsilon}{s}}\vartheta_{2}(0,e^{-\pi^{2}\epsilon/s})=\sqrt{\frac{\pi\epsilon}{s}}\sum_{k=-\infty}^{\infty}e^{-\pi^{2}(k+1/2)^{2}\epsilon/s}\,,$
(48)

where $\vartheta_{j}(v,q)$ is the Jacobi theta function.

We now numerically evaluate $\langle\bar{\phi^{2}}\rangle$ for the case with
$m=0$ and $D=5$. We define a cutoff length scale, $l\equiv\sqrt{\epsilon}$ and
use it hereafter.

Figure 1: The vacuum expectation values of the square of the scalar field
regulated as $L^{3}\langle\bar{\phi^{2}}\rangle$ for $m=0$ and $D=5$ are
plotted against the cutoff length $l$ divided by the circumference $L$ of
$S^{1}$ in the cases with $n=2$, $n=4$, and $n=6$ in Model A. Figure 2: The
vacuum expectation values of the square of the scalar field
$l^{2}L\langle\bar{\phi^{2}}\rangle$ for $m=0$ and $D=5$ are plotted against
the circumference $L$ of $S^{1}$ divided by the cutoff length $l$ in the cases
with $n=1$, $n=2$, $n=4$, and $n=6$ in Model A. The black curve for the $n=1$
case indicates the usual case, i.e., the case that the cutoff length is set to
zero.

Fig. 1 shows $L^{3}\langle\bar{\phi^{2}}\rangle$ in Model A as a function of
$l/L$ for $m=0$ and $D=5$. Note that $\lim_{l\rightarrow
0}L^{3}\langle\bar{\phi^{2}}\rangle=\frac{\zeta_{R}(D-2)}{2\pi^{D/2}}$, where
$\zeta_{R}(z)$ is Riemann’s zeta function, and it is $\approx 0.0343574$ for
$D=5$. One can see that $L^{3}\langle\bar{\phi^{2}}\rangle$ monotonically
decreases as $l$ increases. Fig. 2 shows $l^{2}L\langle\bar{\phi^{2}}\rangle$
in Model A as a function of $L/l$ for $m=0$ and $D=5$. The value of
$l^{D-3}L\langle\bar{\phi^{2}}\rangle$ in $D$ dimensions at $L=0$ becomes
finite if $n>\frac{D-1}{2}$, and is expressed by the following integral as for
Model A:

$\lim_{L/l\rightarrow
0}l^{D-3}L\langle\bar{\phi^{2}}\rangle=\frac{1}{(4\pi)^{D-1}}\int_{0}^{\infty}\frac{(1-e^{-t})^{n-1}}{t^{\frac{D-1}{2}}}dt\,.$
(49)

For $D=5$ and $n=4$, $\lim_{L/l\rightarrow
0}l^{2}L\langle\bar{\phi^{2}}\rangle_{4}\approx 0.0054653$, while for $D=5$
and $n=6$, $\lim_{L/l\rightarrow
0}l^{2}L\langle\bar{\phi^{2}}\rangle_{6}\approx 0.00369362$.

For Model B, we find the expression of the vacuum expectation value
$\langle\bar{\phi^{2}}\rangle=\bar{G}_{B}(0)$ in $\mathbf{R}^{D-1}\otimes
S^{1}$ for $m=0$:

$\langle\bar{\phi^{2}}\rangle=\frac{2l}{\pi^{\frac{D-1}{2}}}\sum_{q=1}^{\infty}\sum_{k=0}^{\infty}\frac{\Gamma\left(\frac{D-1}{2}\right)}{\left[q^{2}L^{2}+4\pi^{2}\left(k+\frac{1}{2}\right)^{2}l^{2}\right]^{\frac{D-1}{2}}}\,.$
(50)

In Fig. 3, the value of $L^{3}\langle\bar{\phi^{2}}\rangle$ in Model B with
$D=5$ and $m=0$ is plotted against $l/L$. Fig. 4 shows
$l^{2}L\langle\bar{\phi^{2}}\rangle$ in Model B with $D=5$ and $m=0$ against
$L/l$. The limit $L\rightarrow 0$ gives a finite value for
$l^{D-3}L\langle\bar{\phi^{2}}\rangle$ in Model B, in arbitrary dimensions. It
turns out to be

$\lim_{L/l\rightarrow
0}l^{D-3}L\langle\bar{\phi^{2}}\rangle=\frac{1-2^{-(D-2)}}{\pi^{D-2}}\frac{\Gamma(\frac{D-2}{2})\zeta_{R}(D-2)}{\pi^{\frac{D-2}{2}}}\,.$
(51)

In particular, we find $\lim_{L/l\rightarrow
0}l^{2}L\langle\bar{\phi^{2}}\rangle=\frac{7\zeta_{R}(3)}{16\pi^{4}}\approx
0.00539888$.

Figure 3: The vacuum expectation value regulated as
$L^{2}\langle\bar{\phi^{2}}\rangle$ for $m=0$ and $D=5$ are plotted against
the cutoff length $l$ divided by the circumference $L$ of $S^{1}$ in Model B.
Figure 4: The vacuum expectation value regulated as
$L^{2}\langle\bar{\phi^{2}}\rangle$ for $m=0$ and $D=5$ are plotted against
the circumference $L$ of $S^{1}$ divided by the cutoff length $l$ in Model B.
The black curve indicates the canonical case, i.e., the case that the cutoff
length is set to zero and the other curve indicate the present UV modified
case of Model B.

###  IV.2 The conical space

Next, we consider the vacuum expectation values in a conical space. A conical
space, or a space with a conical singularity at the coordinate origin, is
described by the metric

$ds^{2}=\sum_{i=1}^{D-2}(dz^{i})^{2}+dr^{2}+\frac{r^{2}}{\nu^{2}}d\theta^{2}\,,$
(52)

where $\nu$ is a constant greater than unity. This metric is equivalent to

$ds^{2}=\sum_{i=1}^{D-2}(dz^{i})^{2}+dr^{2}+r^{2}d\tilde{\theta}^{2}\,,$ (53)

where the range of $\tilde{\theta}$ is $0<\tilde{\theta}\leq 2\pi/\nu$. This
metric adequately describes a locally flat Euclidean space except for the
coordinate origin if $\nu\neq 1$.

The standard Green’s function in a conical space (without a fundamental
length) is known Smith ; SH ; CKV ; Moreira . Because it is natural to take
$\bar{G}(x,x^{\prime};M^{2})=G(x,x^{\prime};M^{2})-G(x,x^{\prime};M^{2})|_{\nu=1}$,
we find KKSW

$\displaystyle\bar{G}(x,x^{\prime};M^{2})=\bar{G}(r,r^{\prime},\varphi,\zeta;M^{2})$
(54) $\displaystyle\equiv$
$\displaystyle\int_{0}^{\infty}ds\,{\textstyle\frac{e^{-\frac{r^{2}+{r^{\prime}}^{2}+\zeta^{2}}{4s}-M^{2}s}}{2\pi(4\pi
s)^{D/2}}}\int_{0}^{\infty}e^{-\frac{rr^{\prime}}{2s}\cosh
v}\left[{\textstyle\frac{\nu\sin\nu(\tilde{\varphi}-\pi)}{\cosh\nu
v-\cos\nu(\tilde{\varphi}-\pi)}-\frac{\nu\sin\nu(\tilde{\varphi}+\pi)}{\cosh\nu
v-\cos\nu(\tilde{\varphi}+\pi)}}\right]dv\,,$

where $x=(r,\theta,z^{i})$, $x^{\prime}=(r^{\prime},\theta^{\prime},z^{\prime
i})$, and
$\tilde{\varphi}=\tilde{\theta}-\tilde{\theta}^{\prime}=\varphi/\nu=(\theta-\theta^{\prime})/\nu$.
Moreover, the vacuum expectation value of a single, canonical free scalar
field squared can be rewritten as
$\langle\bar{\phi^{2}}\rangle[M^{2}]=\bar{G}(x,x;M^{2})$. Using this, the
vacuum expectation value is reduced to be in a similar form to (45) and
equivalently

$\langle\bar{\phi^{2}}\rangle=\sum_{k=0}^{n-1}c_{k}\bar{G}(x,x;M_{k}^{2})\,,$
(55)

where we define $c_{0}\equiv 1$ and $M_{0}\equiv m$, in addition.

In the present case of the conical space, we thus find

$\langle\bar{\phi^{2}}\rangle=\int_{0}^{\infty}ds\,\varrho(s){\frac{e^{-\frac{r^{2}}{2s}-m^{2}s}}{2\pi(4\pi
s)^{D/2}}}\int_{0}^{\infty}e^{-\frac{r^{2}}{2s}\cosh
v}\left[{\frac{-2\nu\sin\nu\pi}{\cosh\nu v-\cos\nu\pi}}\right]dv\,,$ (56)

where we again used
$\varrho(s)\equiv\sum_{k=0}^{n-1}c_{k}\exp[-s/\epsilon_{k}]$, which is already
defined in the previous subsection.

Now, we can evaluate the vacuum expectation values of
$\langle\bar{\phi^{2}}\rangle$ in Models A and B, very efficiently using these
expressions (56). We assume that the original, unmodified theory is a free
massless scalar theory ($m=0$).

(a) (b) (c)

Figure 5: The vacuum expectation values regulated as
$r^{2}\langle\bar{\phi^{2}}\rangle$ for $m=0$ and $D=4$ are plotted against
the cutoff length $l$ divided by the distance from the origin $r$ in Model A,
where (a) for $n=2$, (b) for $n=4$, and (c) for $n=6$.

(a) (b) (c)

Figure 6: The vacuum expectation values regulated as
$l^{2}\langle\bar{\phi^{2}}\rangle$ for $m=0$ and $D=4$ are plotted against
the distance from the origin $r$ divided by the cutoff length $l$ in Model A,
where (a) for $n=2$, (b) for $n=4$, and (c) for $n=6$. Note that the values
diverge near $r/l\sim 0$ for $n=2$, the numerical values near $r/l$ are
expressed up to a finite cut in the figure (a).

In Fig. 5, we illustrate the $l/r$ dependence of the vacuum expectation values
of the square of the scalar field in Model A. The correction due to the scale
$l$ has a little dependence on $n$ for a small $l/r$. Note that
$\lim_{l/r\rightarrow
0}r^{2}\langle\bar{\phi^{2}}\rangle=\frac{\nu^{2}-1}{48\pi^{2}}$ Smith . The
vacuum expectation values become small faster for larger $n$, though the
changes are qualitatively the same.

In Figs. 6, we illustrate the $r/l$ dependence of the vacuum expectation
values in Model A. In general, when $n>D/2$, the vacuum expectation values are
finite at $r=0$.444Note that, in Fig. 6 (a), the values near $r=0$ is cut by a
certain finite value. The value at the origin is finite as

$\lim_{r/l\rightarrow
0}l^{D-2}\langle\bar{\phi^{2}}\rangle=\frac{(\nu-1)l^{D-2}}{(4\pi)^{D/2}}\int_{0}^{\infty}\frac{ds}{s^{D/2}}\varrho_{A}(s)=\frac{\nu-1}{(4\pi)^{D/2}}\int_{0}^{\infty}\frac{ds}{t^{D/2}}(1-e^{-t})^{n-1}\,.$
(57)

For $D=4$ and $n=4$, this becomes $\approx 0.0054653\times(\nu-1)$, and for
$D=4$ and $n=6$, this becomes $\approx 0.00369362\times(\nu-1)$.

(I) (II)

Figure 7: (I) The vacuum expectation values for $m=0$ and $D=4$ regulated as
$r^{2}\langle\bar{\phi^{2}}\rangle$ are plotted against the cutoff length $l$
divided by the distance from the origin $r$. (II) The vacuum expectation
values regulated as $l^{2}\langle\bar{\phi^{2}}\rangle$ for $m=0$ and $D=4$
are plotted against the distance from the origin $r$ divided by the cutoff
length $l$ in Model B.

In Fig. 7, we illustrate the $l/r$ dependence and the $r/l$ dependence of the
vacuum expectation values in Model B. The values at $r=0$ are finite. In both
cases, qualitative features are almost the same as in the case of Model A. The
value at the origin is finite as

$\lim_{r/l\rightarrow
0}l^{D-2}\langle\bar{\phi^{2}}\rangle=\frac{\nu-1}{(4\pi)^{D/2}}\int_{0}^{\infty}\frac{ds}{s^{D/2}}\varrho_{B}(s)=(\nu-1)\frac{1-2^{-(D-1)}}{\pi^{D-1}}\frac{\Gamma(\frac{D-1}{2})\zeta_{R}(D-1)}{\pi^{\frac{D-1}{2}}}\,.$
(58)

For $D=4$, this becomes $\approx 0.00539888\times(\nu-1)$.

##  V Conclusion

In this paper, we first introduced a discrete heat kernel formulation of the
Green’s function for a free scalar field, and considered the UV deformation
that can be naturally obtained from it. The effective Lagrangian that derives
this UV modified Green’s function has been found to be higher derivative
theory. In the simplest case, it corresponds to a simple Pauli–Villars model,
or, a Lee–Wick scalar field theory. In a further generalization, it was found
that the present UV deformation yields a theory involving arbitrarily higher
order derivatives, and the vacuum expectation values of the scalar field
squared were evaluated in nontrivial spaces. We demonstrated that the cutoff
length has an influence on the physical quantities at a short distance in the
configuration space.

The advantage of using the heat kernel method is that we can keep the symmetry
of the space (or break it at will). It is also easy to incorporate the effect
of a constant external field (including spatial metric). On the other hand,
one of the disadvantages is the handling of general interactions of dynamical
fields. In addition, although the form of UV deformation is limited in our
approach, it is also a problem that the sequence $\epsilon_{k}$ introduced in
this paper has a wide range of arbitrariness other than the case of the two
models.

To eliminate UV divergences completely, higher order derivatives that satisfy
the conditions depending on the number of dimensions are required. Even if
there is no divergence, a constant depending on the cutoff constant comes out.
Therefore, some mechanism for canceling the finite contribution in the UV
limit is still required. However, the present demonstration shows that the
finite cutoff dependence of physical quantities occurs when the typical scale
in the considered system is very close to the cutoff scale.

In the effective higher derivative theory, a scalar field with a negative norm
appears, so the addition of interaction may bring about instability and
violate unitarity in the Lorentzian spacetime. This is a subject that has been
frequently addressed and has recently been actively discussed AP ; Anselmi ,
and in particular, infinite derivative theory is being studied in various
situations Modesto ; BT ; BLM ; BLMTY . One of the models we have proposed
(Model B) involves infinite derivatives, and the unitarity of the theory and
its generalization should be studied further. Unfortunately, our current
understanding has not led us to discuss the unitarity of the models presented
in this paper in more detail. We consider that detailed discussions about
nonlinear terms, i.e., interaction terms of fields are needed in the
(infinitely) higher derivative theory, because it seems important to properly
allocate higher order derivatives to the part of propagation and the part of
interaction vertices.

It is interesting to study whether the theory with a basic cutoff scale can be
derived naturally, for example from a model of quantum gravity, or not.555Some
specific models were discussed in the last decade WZ1 ; WZ2 ; RS . That is a
topic for the future, but one way might be to proceed with the tentative goal
of deriving behavior like the simple models we examined here.

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