# The effect of non-local derivative on Bose-Einstein condensation

F.E. Bouzenna${}^{\textsf{\footnotesize{\mbox{?}}}}$ 111e-mail:
<EMAIL_ADDRESS>M.T. Meftah${}^{\textsf{\footnotesize{\mbox{?}}}}$
222e-mail<EMAIL_ADDRESS>M.
Difallah${}^{\textsf{\footnotesize{\mbox{?}}}}$ 333e-mail<EMAIL_ADDRESS>
(Received June 20, 2020)

###### Abstract

In this paper, we study the effect of non-local derivative on Bose-Einstein
condensation. Firstly, we consider the Caputo-Fabrizio derivative of
fractional order $\alpha$ to derive the eigenvalues of non-local Schrödinger
equation for a free particle in a 3D box. Afterwards, we consider 3D Bose-
Einstein condensation of an ideal gas with the obtained energy spectrum.
Interestingly, in this approach the critical temperatures $T_{c}$ of
condensation for $1<\alpha<2$ are greater than the standard one. Furthermore,
the condensation in 2D is shown to be possible. Second and for comparison, we
presented, on the basis of a spectrum established by N. Laskin, the critical
transition temperature as a function of the fractional parameter $\alpha$ for
a system of free bosons governed by an Hamiltonian with power law on the
moment ($H\sim p^{\alpha}$). In this case, we have demonstrated that the
transition temperature is greater than the standard one. By comparing the two
transition temperatures (relative to Caputo-Fabrizio and to Laskin), we have
found for fixed $\alpha$ and the density $\rho$ that the transition
temperature relative to Caputo-fabrizio is greater than relative to Laskin.

Key words: phase transition, critical phenomena

###### Abstract

Ó öé ðîáîò ìè âèâàìî âïëèâ íåëîêàëüíî¿ ïîõäíî¿ íà êîíäåíñàöþ Áîçå-Åéíøòåéíà.
Ñïåðøó, ìè ðîçãëÿäàìî ïîõäíó Êàïóòî-Ôàáðöî äðîáîâîãî ïîðÿäêó $\alpha$ äëÿ
âèâåäåííÿ âëàñíèõ çíàåíü íåëîêàëüíîãî ðâíÿííÿ Øðåäíãåðà äëÿ âëüíî¿ àñòèíêè â
3D áîêñ. Ïîòì ìè ðîçãëÿíåìî 3D-êîíäåíñàöþ Áîçå-Åéíøòåéíà äåàëüíîãî ãàçó ç
îòðèìàíèì åíåðãåòèíèì ñïåêòðîì. Öêàâî, ùî ïðè òàêîìó ïäõîä êðèòèí òåìïåðàòóðè
$T_{c}$ êîíäåíñàö¿ äëÿ $1<\alpha<2$ ïåðåâèùóþòü ñòàíäàðòíó. Êðì òîãî,
ïîêàçàíî, ùî êîíäåíñàöÿ â 2D ìîæëèâîþ. Ïî-äðóãå äëÿ ïîðâíÿííÿ, ìè ïðåäñòàâèëè
íà îñíîâ ñïåêòðó, âñòàíîâëåíîãî Í. Ëàñêíèì, êðèòèíó òåìïåðàòóðó ïåðåõîäó ÿê
ôóíêöþ äðîáîâîãî ïàðàìåòðà $\alpha$ äëÿ ñèñòåìè âëüíèõ áîçîíâ, êåðîâàíèõ
ãàìëüòîíàíîì ç ñòåïåíåâèì çàêîíîì ùîäî ìîìåíòó ($H\sim p^{\alpha}$). Ó öüîìó
âèïàäêó ìè ïðîäåìîíñòðóâàëè, ùî òåìïåðàòóðà ïåðåõîäó ïåðåâèùó ñòàíäàðòíó.
Ïîðâíþþè äâ òåìïåðàòóðè ïåðåõîäó (ïî âäíîøåííþ äî Êàïóòî-Ôàáðöî òà Ëàñêíà), ìè
âèÿâèëè äëÿ ôêñîâàíèõ $\alpha$ òà ãóñòèíè $\rho$, ùî òåìïåðàòóðà ïåðåõîäó ïî
âäíîøåííþ äî Êàïóòî-Ôàáðèöî âèùà, íæ ïî âäíîøåííþ äî Ëàñêíà.

Ключов слова: ôàçîâèé ïåðåõä, êðèòèí ÿâèùà

##  1 Introduction

The fractional quantum mechanics is a new branch in physics introduced
recently by Laskin [1] through generalizing the Feynman path integration to
Lévy formulation. In this approach, the order of the space derivative in the
Schrödinger equation is the non-integer number $\alpha$ instead of 2. In this
case, the modified equation is the so-called fractional Schrödinger equation
which is a non-local equation. On the one hand, Laskin used the Riesz
fractional derivative (FD) to find the associated energy spectrum and the
wavefunctions of the studied systems. On the other hand, the authors [2] and
[3] used the Caputo derivative that provided different solutions.

Afterwards, the fractional quantum mechanics was applied by Alisultanov and
Meilanov to study ideal quantum statistical systems with Laskin energy
spectrum [4, 5]. The latter has a fractional (non-quadratic) power of the
momentum. Moreover, using the Green’s function approach, they showed that
these systems can emulate real systems with physical inter-particle
interactions. In this context, [6] studied thermodynamic properties of 3D
ideal Fermi and Bose gases, 3D independent harmonic oscillators, and black-
body radiation. Interestingly, [5, 6] found that the critical condensation
temperatures of bosons with this spectrum are larger than in the standard
ideal case. More recently, [7, 8] extended this approach to explore
d-dimensional statistical systems. To the best of our knowledge, no large-
scale study has revealed the different effects of fractional derivative
definitions on Bose-Einstein condensation.

In this work, we investigate Bose-Einstein condensation through the Caputo-
Fabrizio derivative [9] and compare our findings with those presented in [6].
Unlike Riesz and Caputo FD, the Caputo-Fabrizio derivative is improved, so
that the kernel becomes non-singular. Therefore, this newly proposed
derivative was applied to various fields of physics, for example, classical
mechanics [10], heat transfer [11], fluid mechanics [12], and quantum
mechanics [13].

Our paper is organized as follows: section 2 studies 3D ideal Bose gas with
single-particle energy spectrum derived from a fractional Schrödinger equation
for a free particle in a hard-wall box using Caputo-Fabrizio FD. In section 3,
by using Laskin spectrum for the ideal bose gas, we presented the transition
temperature as function of the fractionnal order derivative $\alpha$. Section
4 discusses the obtained critical temperatures compared with those found
according to Laskin’s and to Fabrizio-Caputo energy spectra. Conclusion is
then presented in section 5.

##  2 The Bose-Einstein condensation based on Caputo-Fabrizio derivative

We consider the Caputo-Fabrizio derivative in resolving the non-local
Schrödinger equation because; (i) the Caputo-Fabrizio derivative of a constant
function is exactly zero, (ii) the Caputo-Fabrizio kernel has no singularity.
Physically, these two properties are very important and desirable.

The Caputo-Fabrizio derivative of the order $\gamma$ where $0<\gamma<1$ is
given as follows [9]

${}^{CF}\mathfrak{D}^{\gamma}f(x)=\frac{(2-\gamma)M(\gamma)}{2(1-\gamma)}\int_{0}^{x}f^{\prime}(t)\mathrm{d}t\exp\left[\frac{-\gamma}{1-\gamma}(x-t)\right],\quad
x\geqslant 0,$ (2.1)

where $M(\gamma)=2/(2-\gamma)$ is a normalization constant that depends on
$\gamma$. By applying the definition (2.1), the solution of the following non-
local differential equation

${}^{CF}\mathfrak{D}^{\gamma}f(x)=u(x),~{}~{}~{}~{}0<\gamma<1$ (2.2)

is given by [10]

$f(x)=(1-\gamma)u(x)+\gamma\int_{0}^{x}u(t)\mathrm{d}t+f(0).$ (2.3)

Consider now the non-local Schrödinger equation for a free particle moving in
1D box with length $L$ [14, 15]

${}^{CF}\mathfrak{D}^{2\gamma}\psi(x)~{}=-\left(\frac{a_{0}^{2\gamma}}{\hbar^{2\gamma}D_{2\gamma}}\right)E_{2\gamma}\psi(x),~{}~{}~{}~{}x\geqslant
0~{}~{},~{}~{}\frac{1}{2}<\gamma<1,$

or equivalently

${}^{CF}\mathfrak{D}^{\alpha}\psi(x)~{}=-\varepsilon_{\alpha}\psi(x),~{}~{}~{}1<\alpha=2\gamma<2,$
(2.4)

where the operator ${}^{CF}\mathfrak{D}^{\alpha}$ has the same meaning as
(2.1) and $x=X/a_{0}$ being the reduced space coordinate, $a_{0}$ is the
characteristic length, $D_{2\gamma}$ is a generalized coefficient and
$\varepsilon_{\alpha}=\varepsilon_{2\gamma}=a_{0}^{2\gamma}E_{2\gamma}/\hbar^{2\gamma}D_{2\gamma}$
is the reduced energy. To seek for a solution of Schrödinger equation (2.4),
we take: ${}^{CF}\mathfrak{D}^{\gamma}f(x)=u(x)~{}$ and
${}^{~{}CF}\mathfrak{D}^{2\gamma}f(x)=^{CF}\mathfrak{D}^{\gamma}u(x)=g(x)$.
Hence, according to the solution (2.3) we obtain

$u(x)=(1-\gamma)g(x)+\gamma\int_{0}^{x}g(t)\mathrm{d}t+u(0),$ (2.5)

and

$f(x)=(1-\gamma)u(x)+\gamma\int_{0}^{x}u(t)\mathrm{d}t+f(0).$ (2.6)

Plugging (2.5) into (2.6), the solution of non-local differential equation
${}^{~{}CF}\mathfrak{D}^{2\gamma}f(x)=g(x)$ reads

$\displaystyle
f(x)=(1-\gamma)^{2}g(x)+2\gamma(1-\gamma)\int_{0}^{x}g(t)\mathrm{d}t+\gamma^{2}\int_{0}^{x}\mathrm{d}t\int_{0}^{t}g(t)\mathrm{d}t+(\gamma
x-\gamma+1)u(0)+f(0).$ (2.7)

Let us now consider $f(x)=\psi(x)$ and $g(x)=-\varepsilon_{\alpha}\psi(x)$,
then we derive twice from the above equation, and we find:

$\displaystyle\psi^{\prime\prime}(x)+\frac{2\alpha(2-\alpha)\varepsilon_{\alpha}}{4+(2-\alpha)^{2}\varepsilon_{\alpha}}\psi^{\prime}(x)+\frac{\alpha^{2}\varepsilon_{\alpha}}{4+(2-\alpha)^{2}\varepsilon_{\alpha}}\psi(x)=0,~{}~{}~{}~{}\alpha=2\gamma,$
(2.8)

or

$\psi^{\prime\prime}(x)+2\Lambda\psi^{\prime}(x)+\omega_{0}^{2}\psi(x)=0,$
(2.9)

where
$\Lambda=\alpha(2-\alpha)\varepsilon_{\alpha}/\left(4+(2-\alpha)^{2}\varepsilon_{\alpha}\right)$
and
$\omega_{0}=\alpha\left[\varepsilon_{\alpha}/\left(4+(2-\alpha)^{2}\varepsilon_{\alpha}\right)\right]^{1/2}$.
The solution of equation (2.9) has the following form

$\psi(x)=\phi(x)\mathrm{e}^{-\Lambda x},$ (2.10)

where $\phi(x)$ fulfills the equation

$\phi(x)+k^{2}\phi(x)=0,$ (2.11)

with $k=\left(\omega_{0}^{2}-\Lambda^{2}\right)^{1/2}$ being the reduced
wavenumber ($\omega_{0}>\Lambda$). Then, the solution of (2.9) is given as
follows:

$\psi(x)=\mathrm{e}^{-\Lambda x}[A\sin\left(kx\right)+B\cos\left(kx\right)].$
(2.12)

By using the hard-wall boundary conditions

$\left\\{\begin{array}[c]{c}\psi(0)=0\\\ \psi(a)=0{}\end{array}\right.$ (2.13)

where $a=L/a_{0}$, we find

$\psi(x)=A\mathrm{e}^{-\Lambda x}\sin(kx),$ (2.14)

and

$k=\frac{\piup}{a}n,~{}~{}~{}~{}n\in Z^{+},$ (2.15)

here, the magnitude $A$ is determined by the normalization condition
$\int_{0}^{a}|\psi(x)|^{2}\mathrm{d}x=1$. In addition, we have

$k=\frac{2\alpha\varepsilon_{\alpha}^{1/2}}{4+(2-\alpha)^{2}\varepsilon_{\alpha}},$
(2.16)

therefore,

$k(2-\alpha)^{2}\varepsilon_{\alpha}-2\alpha\sqrt{\varepsilon_{\alpha}}+4k=0.$
(2.17)

This equation has two real solutions, $\epsilon_{\alpha}^{1}$ and
$\epsilon_{\alpha}^{2}$, only if

$0<k<\frac{\alpha}{2(2-\alpha)}=B(\alpha).$ (2.18)

The first one is

$\epsilon_{\alpha}^{1}=\frac{\alpha-\left[\alpha^{2}-4k^{2}(2-\alpha)^{2}\right]^{1/2}}{k(2-\alpha)^{2}}=\alpha\frac{1-\left[1-4k^{2}\left(\frac{2-\alpha}{\alpha}\right)^{2}\right]^{1/2}}{k(2-\alpha)^{2}}.$
(2.19)

In the limit $\alpha\rightarrow 2$, the solution (2.19) tends to $k$, and we
recover the standard energy spectrum

$E=\frac{\hbar^{2}D_{2}}{a_{0}^{2}}k^{2}=\frac{\piup^{2}\hbar^{2}}{2mL^{2}}n^{2},~{}~{}~{}D_{2}=\left(2m\right)^{-1}.$
(2.20)

The second solution is not valid because in the limit $\alpha\rightarrow 2$ it
does not yield $\sqrt{\varepsilon}=k$. The energy spectrum of the equation
(2.4) can be written as

$\varepsilon_{\alpha}=\alpha^{2}\frac{\left[1-\sqrt{1-4k^{2}\left(\frac{2-\alpha}{\alpha}\right)^{2}}\right]^{2}}{(2-\alpha)^{4}k^{2}},$
(2.21)

or

$E_{\alpha}=\frac{\hbar^{\alpha}D_{\alpha}}{a_{0}^{\alpha}}\frac{\alpha^{2}}{(2-\alpha)^{4}}\frac{\left[1-\sqrt{1-4k^{2}\left(\frac{2-\alpha}{\alpha}\right)^{2}}\right]^{2}}{k^{2}}.$
(2.22)

From the above results we note: (i) the number of energy levels is limited by
the condition (2.18), and reaches infinity at $\alpha=2$ [$B(2)=\infty$, see
equation (2.19)]. (ii) For $k=0$, the energies $E_{\alpha}=0$ for all $\alpha$
values. This means that, regardless of the form of the energy spectrum, the
effect of Caputo-Fabrizio definition seems to limit the number of energy
levels and to relate it to the derivative order $\alpha$. Therefore, according
to Caputo-Fabrizio approach, we expect that the critical temperature of Bose-
Einstein condensation will be higher (see figure 1). To check this result, we
consider an ideal Bose gas in a hard-wall box with the volume $V=[0,L]^{3}$,
so that the single-particle energy spectrum is defined by (2.22) as

$E_{\alpha}=E_{\alpha}^{1}+E_{\alpha}^{2}+E_{\alpha}^{3}=\frac{\hbar^{\alpha}D_{\alpha}}{a_{0}^{\alpha}}\frac{\alpha^{2}}{(2-\alpha)^{4}}\sum_{i=1}^{3}\frac{\left[1-\sqrt{1-4k_{i}^{2}\left(\frac{2-\alpha}{\alpha}\right)^{2}}\right]^{2}}{k_{i}^{2}}.$
(2.23)

Figure 1: The critical temperature curves at different $\alpha$ values within
the Caputo-Fabrizio approach.

In the thermodynamic limit, the total mean density $\rho$ of this gas is given
by

$\displaystyle\rho=\lim_{V\rightarrow\infty}\frac{N}{V}=\rho_{0}+\frac{1}{\piup^{3}a_{0}^{3}}\int_{0}^{+\infty}\mathrm{d}^{3}k\left[\mathrm{e}^{\beta(E_{\alpha}-\mu)}-1\right]^{-1}=\rho_{0}+\rho_{e}\,,$
(2.24)

where $\rho_{0}$ is the occupation mean density in the ground state and
$\rho_{e}$ is the occupation mean density in the excited levels $\left(k\neq
0\right)$ given by

$\rho_{e}=\frac{1}{\piup^{3}a_{0}^{3}}\int_{0}^{+\infty}\mathrm{d}^{3}k\left[\mathrm{e}^{\beta(E_{\alpha}-\mu)}-1\right]^{-1}.$
(2.25)

For $T=T_{c}$ ($T_{c}$ is the critical temperature of condensation at which
the chemical potential $\mu=0$), the total mean density of these bosons in the
box ($V=L^{3}$) becomes

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{\piup^{3}a_{0}^{3}}\int_{0}^{B}\int_{0}^{B}\int_{0}^{B}\frac{\mathrm{d}k_{1}\mathrm{d}k_{2}\mathrm{d}k_{3}}{\exp\left\\{\beta_{c}\frac{\hbar^{\alpha}D_{\alpha}}{a_{0}^{\alpha}}\frac{\alpha^{2}}{(2-\alpha)^{4}}\sum_{i=1}^{3}\frac{\left[1-\sqrt{1-4k_{i}^{2}\left(\frac{2-\alpha}{\alpha}\right)^{2}}\right]^{2}}{k_{i}^{2}}\right\\}-1}$
(2.26) $\displaystyle=$ $\displaystyle\left[\frac{\alpha}{2\piup
a_{0}(2-\alpha)}\right]^{3}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{\mathrm{d}K_{1}\mathrm{d}K_{2}\mathrm{d}K_{3}}{\exp\left[\frac{4}{(2-\alpha)^{2}}\frac{\hbar^{\alpha}D_{\alpha}}{a_{0}^{\alpha}k_{\text{B}}}\frac{1}{T_{c}}\sum_{i=1}^{3}\left(\frac{1-\sqrt{1-K_{i}^{2}}}{K_{i}}\right)^{2}\right]-1}$
$\displaystyle=$ $\displaystyle\left[\frac{\alpha}{2\piup
a_{0}(2-\alpha)}\right]^{3}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{\mathrm{d}K_{1}\mathrm{d}K_{2}\mathrm{d}K_{3}}{\exp\left[\frac{4\piup^{-\alpha}}{(2-\alpha)^{2}}\frac{E_{0}}{k_{\text{B}}}\frac{1}{T_{c}}\sum_{i=1}^{3}\left(\frac{1-\sqrt{1-K_{i}^{2}}}{K_{i}}\right)^{2}\right]-1},$

where $K_{i}=2\left(2-\alpha\right)k_{i}/\alpha$ and $D_{\alpha}$ is given as
[5]

$D_{\alpha}=\frac{E_{0}}{\hbar^{\alpha}k_{0}^{\alpha}},$ (2.27)

where $E_{0}=\hbar^{2}k_{0}^{2}/2m$, $k_{0}=\piup/a_{0}$, and $a_{0}$ are the
characteristic energy, wavenumber and the length of the system, respectively,
($a_{0}$ can be shown as the diameter of atoms). For $\alpha=2$, the equation
(2.27) yields $D_{2}=1/2m$. From the above formula (2.26), we plot the curves
of $T_{c}$ against the density $\rho$ for various $\alpha$ values as presented
in figure 1. We choose $a_{0}=2.56\cdot 10^{-10}~{}\mathrm{m}$ and
$m=6.64\cdot 10^{-27}~{}\mathrm{kg}$ being the diameter and the mass of the
helium atoms, respectively. In this case, $k_{0}\approx 1.23\cdot
10^{10}~{}\mathrm{m}^{-1}$ and $E_{0}/k_{\text{B}}\approx 9.12$ K.

##  3 Bose-Einstein condensation based on Laskin approach

In his paper [1], Laskin showed that bosons have the single-particle energy
spectrum defined by

$\displaystyle
E_{\alpha}=D_{\alpha}\left(\piup\hbar\right)^{\alpha}\left[\frac{n_{1}^{2}}{L^{2}}+\frac{n_{2}^{2}}{L^{2}}+\frac{n_{3}^{2}}{L^{2}}\right]^{\alpha/2}=D_{\alpha}\hbar^{{}^{\alpha}}k^{{}^{\alpha}},\quad(n_{1},n_{2},n_{3})\in\mathbb{Z}$
(3.1)

Subsequently, [6] derived the critical temperature for an ideal Bose gas
confined in a hard-wall box with the volume $V=[0,L]^{3}$

$\displaystyle
T_{c}=\frac{\hbar^{\alpha}D_{\alpha}}{k_{\text{B}}}\left[\frac{2\piup^{2}\alpha\rho}{\Gamma\left(\frac{3}{\alpha}\right)\zeta\left(\frac{3}{\alpha}\right)}\right]^{\alpha/3}=\frac{E_{0}}{k_{\text{B}}k_{0}^{\alpha}}\left[\frac{2\piup^{2}\alpha\rho}{\Gamma\left(\frac{3}{\alpha}\right)\zeta\left(\frac{3}{\alpha}\right)}\right]^{\alpha/3}.$
(3.2)

If we put $\alpha=2$, we obtain

$T_{c}=\frac{h^{2}}{2\piup
mk_{\text{B}}}\left[\frac{\rho}{\zeta\left(\frac{3}{2}\right)}\right]^{2/3},$
(3.3)

which is the well-known standard critical temperature since, in this case, the
Laskin spectrum coincides with the standard one [16].

We plotted $T_{c}$ given by equation (3.2) with respect to the density $\rho$
[see figure 2] for some specific helium parameters ($a_{0}$ and $m$).

Figure 2: The critical temperature curves at different $\alpha$ values
according to Laskin approach.

##  4 Discussion

From figures 1 and 2, we can deduce that: (i) The critical temperature $T_{c}$
grows with the density $\rho$ both in non-local and in standard approaches as
expected. (ii) In the non-local approach, $T_{c}$ is increased for $\alpha<2$.
It tends to the standard critical temperature $T_{c}$ for the case $\alpha=2$
where the Caputo-Fabrizio derivative is in complete agreement with the results
known from the standard derivative. It turns out that the transition
temperature obtained via the Caputo-Fabrizio approach is larger than the
standard transition temperature and than the one obtained by using Laskin
approach. For example when $\alpha=1.3$ and $\rho\approx 10^{28}~{}{\rm
m^{-3}}$, $T_{c}$ exceeds the standard one 10 times. In our opinion, the
difference of critical condensation temperatures is due to the allowed number
of energy levels in non-local case. In other words, the finite number of
energy levels increases $T_{c}$. From the condition (2.18), this number
decreases when $\alpha$ decreases. Therefore, $T_{c}$ increases as well. On
the contrary, the number of energy levels is infinite in the standard
approach.

##  5 Conclusion

The study of Bose-Einstein condensation from Caputo-Fabrizio point of view on
the momentum power law has revealed various effects of the non-local
derivative on the critical condensation temperature. This goal is reached by
considering an ideal bosonic gas enclosed in a hard-wall box for a different
energy spectrum. This spectrum is derived based on a non-local Schrödinger
equation defined in terms of Caputo-Fabrizio derivative. The main findings can
be summarized as follows: (i) The number of energy levels is bounded and
related to the derivative order $\alpha$. (ii) As a result of the bounded
number of energy levels, the critical temperature for $\alpha<2$ is greater
than the standard one. (iv) When $\alpha=2$, all the results become standard
because, in this case, the Caputo-Fabrizio derivative collapses to the
ordinary one. (v) Additionally, the condensation for $\alpha<2$ is possible in
two dimensions. The last finding will be investigated more in detail in our
next research paper where the thermodynamic properties of such systems will be
reviewed. Furthermore, new effects might be highlighted by extending this
study to other quantum statistical systems such as Fermi gas.

To compare, we have presented the transition temperature based on the Laskin
spectrum of the ideal Bose gas. It turns out that this temperature is grater
than the standard temperature transition but smaller than those obtained from
the Caputo-fabrizio spectrum.

## Acknowledgements

All authors would like to thank the head of the LABTHOP laboratory, Professor
Mansour Abdelouahab, Mathematics Department at University of El-Oued, Algeria,
for the material support.

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Ukrainian 0

Âïëèâ íåëîêàëüíî¿ ïîõäíî¿ íà êîíäåíñàöþ Áîçå-Åéíøòåéíà Ô.Å.
Áóçåííà${}^{\textsf{\footnotesize{\mbox{?}}}}$, M.T.
Ìåôòàõ${}^{\textsf{\footnotesize{\mbox{?}}}}$, M.
Äôàëàõ${}^{\textsf{\footnotesize{\mbox{?}}}}$

${}^{\textsf{\footnotesize 1}}$ Ôçèíå âääëåííÿ ëàáîðàòîðÿ LEVRES, ôàêóëüòåò
òîíèõ íàóê, óíâåðñèòåò Åëü Óåäà, 39000, Àëæèð
${}^{\textsf{\footnotesize 2}}$ Ôçèíå âääëåííÿ ëàáîðàòîðÿ LEVRES, ôàêóëüòåò
ìàòåìàòèêè ìàòåðàëîçíàâñòâà, óíâåðñèòåò Êàñä Ìåðáàõ, Óàðãëà 30000, Àëæèð
${}^{\textsf{\footnotesize 3}}$ Ôçèíå âääëåííÿ ëàáîðàòîðÿ LEVRES, ôàêóëüòåò
òîíèõ íàóê, óíâåðñèòåò Åëü Óåäà, 39000, Àëæèð


# Discrete time heat kernel and UV modified propagators with Dimensional
Deconstruction

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

###### Abstract

We revisit the dimensionally deconstructed scalar quantum electrodynamics and
consider the (Euclidean) propagator of the scalar field in the model. Although
we have previously investigated the one-loop effect in this model by obtaining
the usual heat kernel trace, we adopt discrete proper-time heat kernels in
this paper and aim to construct the modified propagator, which has improved
behaviors in the ultraviolet region, by changing the range of sum of the
discrete heat kernels.

###### pacs:

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

##  I Introduction

In recent years, various ideas have been discussed to consider Pad1 ; Pad2 ;
SSP ; SP ; KSSP ; Pad3 ; Pad4 ; Pad5 ; Pad6 ; Pad7 ; Pad8 ; Pad9 ; Nicolini ;
AD ; ABM ; AL ; Siegel ; KKSW1 ; KKSW2 ; Mondal  how to take changes in the
Schwinger’s proper-time parameter of the heat kernel Vassilevich ; Camporesi
for a certain Laplace operator as a way to improve ultraviolet (UV) behavior
in quantum field theory. In our previous paper KKSW2 , we proposed a method to
derive propagators (Green’s functions) with moderate behavior at an
infinitesimally short distance by discretizing the proper-time parameter and
adjusting the range of summation. The modified propagator obtained with this
prescription is corresponding to the one in the theory of the original
canonical field action modified by adding higher-order derivative terms KKSW2
.

For the simplest example of the self-interacting canonical scalar field,
consider the $\lambda\varphi^{4}$ theory. The scalar one-loop effect gives the
radiative correction $\delta m^{2}\sim\lambda\langle\varphi^{2}\rangle$ (with
zero momentum transfer). This suffers from a $(D-2)$-th order divergence in
$D$-dimensional spacetime, since $\langle\varphi^{2}\rangle\sim G(x,x)\sim\int
d^{D}p/p^{2}$, where $G(x,x^{\prime})$ is the propagator of the scalar field,
and the Fourier-transformed propagator $\tilde{G}(p^{2})$ then behaves as
$\sim 1/p^{2}$ at high energy. Phenomenologically, this fact is known as an
origin of the hierarchy problem in the standard model of particle physics with
$D=4$.111Needless to say, loop effects of fermions such as heavy quarks are
also important in the hierarchy problem. We will discuss them in the last
section. The way to obtain softer behavior of $\tilde{G}(p^{2})\sim 1/p^{4}$
or more at high energy has been studied for some time until now LW ; GOW ; CL
.

On the other hand, many authors have also considered the direction of assuming
the discretized background spacetime (motivated by a way of thinking about
quantum gravitational consideration tHooft ). By the way, in a limited sense,
discretization is reminiscent of dimensional deconstruction (DD) ACG ; HPW ;
HL . The idea of DD has even been incorporated into phenomenological models
and variously explored. Suppose a number of copies of a four-dimensional
theory and linking pairs of these individual sites in the theory space. The
resulting whole theory mimics a higher-dimensional theory. This is an attempt
to introduce the discrete extra space into the theory, and so interesting
characteristics of higher dimensional theory can be inherited by four
dimensional theory. We should note, however, that high energy behavior of the
theory becomes even worse because the extra dimensional contributions are
summed up to obtain a four dimensional effective theory.

In the present paper, we study the UV modified propagators in the
deconstructed model by using the discrete time heat kernel. Concretely, we
find the Euclidean propagator $G(x,x^{\prime})$ of the complex scalar fields
$\phi_{\nu}$ and its coincidence limit $G(x,x)$, which is proportional to the
one-loop vacuum polarization $\langle\phi^{\dagger}_{\nu}\phi_{\nu}\rangle$,
of the dimensionally deconstructed $(D+1)$-dimensional scalar quantum
electrodynamics (QED). Here we assume the presence of the background of
pseudo-Nambu–Goldstone boson (PNGB) field HPW ; HL  (which corresponds to the
constant $U(1)$ gauge field in a higher-dimensional theory) in one discrete
extra dimensional direction.

The model treated here is the same as the one studied in Ref. KSS  using the
usual heat kernel trace method. In the study, we have a heat kernel trace that
employs the graph Laplacian associated to a cycle graph as a part of the
Laplace operator, known from spectral graph theory,222For the spectral graph
theory, see Refs. Mohar1 ; Mohar2 ; Mohar3 ; Merris . which coincides with
that obtained from the heat kernel discussed more generally later in Refs.
CJK1 ; CJK2 ; Dowker . It is known that the DD model using the cycle graph has
a continuous limit, which yields the Kaluza–Klein model with an extra
dimension $S^{1}$. It should be noted here that, as we have already pointed
out in Ref. KKSW2 , even if the Laplace operator is in the form of a direct
sum, the heat kernel cannot be expressed in a form of a direct product when
the proper time is discrete. Therefore, it is significant to study the
mathematical properties of the discrete time heat kernel, albeit for a simple
model. Fortunately, discrete time heat kernel for a certain class of graph
Laplacians has recently been discussed in Ref. CHJSV , so we can manage to
apply it to our calculations.

Although the model considered in this paper is the simplest one, there is a
future goal to develop this toy model into various field theories with higher
symmetries to explore continuous and discrete versions of the Hosotani
mechanism Hosotani  with UV modification. At the same time, extension from DD
to models using various graph Laplacians will also come into future view. As a
natural extension of DD, we can study the discretization of our real spacetime
in a similar method, which we will link to future research and work by other
authors tHooft .

The structure of this paper is as follows. In Section II, we review the model
setup, the derivation of the usual heat kernel, and the calculations leading
to the propagator. They are necessary for the comparison with those obtained
from the discrete time heat kernel later. Section III introduces two types of
discrete time heat kernels and discusses how to soften or moderate the UV
divergence in the short range behavior of the (Euclidean) propagator by
changing the sum of kernels. The final section is devoted to summary and
future prospects.

##  II (continuous time) heat kernel for DD with a cycle graph

We revisit the deconstructed massless scalar QED model introduced in Ref. KSS
. Its action of the scalar field sector is expressed as follows:

$S=-\sum_{\nu=1}^{N}\sum_{\nu^{\prime}=1}^{N}\int
d^{D}x\,\phi_{\nu}^{\dagger}(x)\Bigl{[}-I_{\nu\nu^{\prime}}\Box+f^{2}\Delta_{\nu\nu^{\prime}}(C_{N},\chi)\Bigr{]}\phi_{\nu^{\prime}}(x)\,,$
(1)

where $I_{\nu\nu^{\prime}}$ denotes the $N\times N$ identity matrix and the
Hermitian matrix $\Delta(C_{N},\chi)$ is given by

$\Delta(C_{N},\chi)\equiv\left(\begin{array}[]{cccccc}2&-e^{i\chi}&0&\cdots&&-e^{-i\chi}\\\
-e^{-i\chi}&2&-e^{i\chi}&\cdots&&0\\\ 0&-e^{-i\chi}&2&\cdots&&0\\\
\vdots&\vdots&\vdots&\ddots&&\vdots\\\ &&&&2&-e^{i\chi}\\\
-e^{i\chi}&0&0&\cdots&-e^{-i\chi}&2\end{array}\right)\,,$ (2)

and $f$ is a constant with dimension of mass. We consider the $D$-dimensional
Euclidean space. The d’Alembert operator
$\Box=\sum_{j=0}^{D-1}\partial^{j}\partial_{j}$ in (1) acts on scalar fields.
The label of the fields are considered as periodic modulo $N$, e.g.,
$\phi_{N+1}\equiv\phi_{1}$, $\phi_{0}\equiv\phi_{N}$, and so on. The matrix
$\Delta(C_{N},0)$ is the graph Laplacian for a cycle graph $C_{N}$ Mohar1 ;
Mohar2 ; Mohar3 ; Merris , whose $N$ vertices form a discrete circle. The
constant $\chi$ stands for the ‘twist’ factor, which comes from the constant
background PNGB field corresponding to the background $U(1)$ gauge field in
the extra dimensions if the continuous limit is taken. We omit the background
gauge field in the flat large dimensions in the present analysis.

Here, we introduce a heat kernel $K_{\nu}(x,x^{\prime};s)$ that satisfies the
equation

$\sum_{\nu^{\prime}=1}^{N}\left[I_{\nu\nu^{\prime}}\frac{\partial}{\partial
s}+\Bigl{(}-I_{\nu\nu^{\prime}}\Box_{x}+f^{2}\Delta_{\nu\nu^{\prime}}(C_{N},\chi)\Bigr{)}\right]K_{\nu^{\prime}}(x,x^{\prime};s)=0\,,$
(3)

subject to the initial condition $\lim_{s\rightarrow
0}K_{0}(x,x^{\prime};s)=\delta(x,x^{\prime})$ and $\lim_{s\rightarrow
0}K_{\nu}(x,x^{\prime};s)=0$ for $\nu\neq 0$. The d’Alembert operator
$\Box_{x}$ acts on the coordinate $x$. In our present model, the heat kernel
can be written in the form333Obviously, we use $\nu-\nu^{\prime}$ instead of
$\nu$ for an arbitrary pair of the scalar fields on the sites labeled by $\nu$
and $\nu^{\prime}$.

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

where we notice that $\tilde{K}_{\nu}(p^{2};s)$ is a function of
$p^{2}=\sum_{j=1}^{D-1}p_{j}p^{j}$, for the homogeneity and isotropy of the
Euclidean space $\mathbf{R}^{D}$. Then, the heat equation (4) reduces to

${\partial_{s}}\tilde{K}_{\nu}(p^{2};s)=f^{2}\Bigl{[}e^{i\chi}\tilde{K}_{\nu+1}(p^{2};s)-(2+p^{2}/f^{2})\tilde{K}_{\nu}(p^{2};s)+e^{-i\chi}\tilde{K}_{\nu-1}(p^{2};s)\Bigr{]}\,,$
(5)

with the condition $\lim_{s\rightarrow 0}\tilde{K}_{0}(p^{2};s)=1$ and
$\lim_{s\rightarrow 0}\tilde{K}_{\nu}(p^{2};s)=0$ for $\nu\neq 0$. Here, the
abbreviation $\partial_{s}\equiv\frac{\partial}{\partial s}$ has been used.

The solution for (5) can be expressed by using the modified Bessel function
$I_{\nu}(z)$ GR

$I_{\nu}(z)=\left(\frac{z}{2}\right)^{\nu}\sum_{n=0}^{\infty}\frac{(z/2)^{2n}}{n!\Gamma(\nu+n+1)}\,.$
(6)

Note that $I_{0}(0)=1$, $I_{\nu}(0)=0$ for $\nu\neq 0$, and
$I_{-\nu}(z)=I_{\nu}(z)$ if
$\nu\in\mathbf{Z}\equiv\\{\dots,-2,-1,0,1,2,\dots\\}$. We should also notice
that

$\partial_{z}I_{\nu}(z)=\frac{1}{2}[I_{\nu+1}(z)+I_{\nu-1}(z)]\,.$ (7)

A solution for the heat equation is found to be CJK1 ; CJK2 ; Dowker

$e^{-i\nu\chi}e^{-(p^{2}+2f^{2})s}I_{\nu}(2f^{2}s)\,,$ (8)

which becomes unity if $\nu=s=0$ and vanishes for the case with $s=0$ and
$\nu\neq 0$. For our present model, the kernel should be periodic such as
$K_{\nu+N}(p^{2};s)=K_{\nu}(p^{2};s)$. Therefore, the solution of the heat
equation subject to the boundary conditions turns out to be

$\tilde{K}_{\nu}(p^{2};s)=e^{-(p^{2}+2f^{2})s}\sum_{q=-\infty}^{\infty}e^{-i(\nu+qN)\chi}I_{\nu+qN}(2f^{2}s)\,.$
(9)

The ‘genuine’ trace of the heat kernel is written as

$\displaystyle N\int d^{D}x\,K_{0}(x,x;s)$ $\displaystyle=$ $\displaystyle
NV\int\frac{d^{D}p}{(2\pi)^{D}}\tilde{K}_{0}(p^{2};s)$ (10) $\displaystyle=$
$\displaystyle\frac{NV}{(4\pi)^{D/2}s^{D/2}}e^{-2f^{2}s}\sum_{q=-\infty}^{\infty}\cos(qN\chi)I_{qN}(2f^{2}s)\,.$

Here, $V$ denotes the volume of the Euclidean spacetime $\int d^{D}x$, which
is often omitted in the works on the heat kernel trace in quantum field
theory. Similarly, the factor $N$ is regarded as the ‘volume’ of the discrete
circle, or simply recognized as the ‘symmetry factor’ for $N$ scalar fields.

In the paper Ref. KSS , we sought the heat kernel trace from the beginning.
Here we briefly describe it. The eigenvalues of the matrix (2) are
$4\sin^{2}\left(\frac{\pi k}{N}+\frac{\chi}{2}\right)$ ($k=0,1,\dots,N-1$).
Then, utilizing the mathematical formula GR , we find

$\displaystyle\sum_{k=0}^{N-1}\exp\left[-4f^{2}\sin^{2}\left(\frac{\pi
k}{N}+\frac{\chi}{2}\right)s\right]$ $\displaystyle=$ $\displaystyle
e^{-2f^{2}s}\sum_{p=0}^{N-1}\sum_{\ell=-\infty}^{\infty}\cos\left[\ell\left(\frac{2\pi
p}{N}+{\chi}\right)\right]I_{\ell}(2f^{2}s)$ (11) $\displaystyle=$
$\displaystyle
Ne^{-2f^{2}s}\sum_{q=-\infty}^{\infty}\cos\left(qN{\chi}\right)I_{qN}(2f^{2}s)\,,$

so, the heat kernel trace has been certainly reproduced.

Now, we consider the ‘partial trace’, i.e., the trace only on the label of
scalar fields (or equivalently, vertices). This operation corresponds to
‘integrating out the extra discrete dimensions’444Consequently, the one-loop
effect which comes from the extra discrete circle is included. and creating a
physical $D$-dimensional perspective. Accordingly, the partially-traced
Fourier transform of the propagator is

$\displaystyle\tilde{G}(p^{2})\equiv
N\int_{0}^{\infty}\tilde{K}_{0}(p^{2};s)\,ds=N\int_{0}^{\infty}e^{-(p^{2}+2f^{2})s}\sum_{q=-\infty}^{\infty}\cos(qN\chi)I_{qN}(2f^{2}s)ds$
(12) $\displaystyle=$
$\displaystyle\frac{N}{\sqrt{p^{2}(4f^{2}+p^{2})}}\sum_{q=-\infty}^{\infty}\cos(qN\chi)\left(\frac{2f^{2}}{2f^{2}+p^{2}+\sqrt{p^{2}(4f^{2}+p^{2})}}\right)^{|q|N}$
$\displaystyle=$
$\displaystyle\frac{N/f^{2}}{2\sinh\beta}\frac{\sinh(N\beta)}{\cosh(N\beta)-\cos(N\chi)}\,,$

where, in the last line, the new parameter

$\frac{p^{2}}{f^{2}}\equiv
4\sinh^{2}\frac{\beta}{2}\sim\left\\{\begin{array}[]{cc}\beta^{2}&(p^{2}/f^{2}\ll
1)\\\ e^{\beta}&(p^{2}/f^{2}\gg 1)\end{array}\right.\,,$ (13)

has been used.

As a special but familiar case, for $\chi=0$, we find

$\tilde{G}(p^{2})\Bigl{|}_{\chi=0}=\frac{N/f^{2}}{2\sinh\beta}\frac{\cosh(N\beta/2)}{\sinh(N\beta/2)}\,,$
(14)

and further if $\beta$ is small,555Of course, as a trivial check, we also
observe $\tilde{G}(p^{2})|_{\chi=0}=1/p^{2}$ if $N=1$.
$\tilde{G}(p^{2})|_{\chi=0}\approx 1/(f^{2}\beta^{2})\approx{1}/{p^{2}}$, that
is the usual Fourier-transformed propagator, also known as the propagator in
the momentum space.666Note that the addition of a common mass $m$ to scalar
fields leads to replacing $p^{2}\rightarrow p^{2}+m^{2}$ in the propagator in
momentum space. Then, the propagator in the $D$-dimensional spacetime
coordinates,

$G(x,x^{\prime})=\int\frac{d^{D}p}{(2\pi)^{D}}\,\tilde{G}(p^{2})\,e^{ip\cdot(x-x^{\prime})}\,,$
(15)

is proportional to $1/r^{D-2}$, where $r=|x-x^{\prime}|$. On the other hand,
if $p^{2}/f^{2}\gg 1$, $\tilde{G}(p^{2})|_{\chi=0}\approx N/p^{2}$, as
expected for finite $N$. What we have to be careful about is when $N/f\equiv
L$ is fixed and $N$ and $f$ approaches infinity, then
$\tilde{G}(p^{2})|_{\chi=0}\approx L/(2p)$ for $pL\gg 1$. Therefore, in this
case, $G(x,x^{\prime})\propto L/r^{D-1}$ for a small $r$, whose behavior
corresponds to the $(D+1)$ dimensional one.

Next, let us see the divergence behavior of
$G(x,x)\sim\langle\phi^{\dagger}_{\nu}\phi_{\nu}\rangle$ for $D=4$. If we use
the cutoff momentum $\Lambda$, we find

$\displaystyle G(x,x)$ $\displaystyle=$
$\displaystyle\frac{2\pi^{2}N}{(2\pi)^{4}}\int_{0}^{\Lambda}\tilde{G}(p^{2})\,p^{3}dp$
(16) $\displaystyle=$
$\displaystyle\frac{N}{16\pi^{2}}\left[\sqrt{\Lambda^{2}(\Lambda^{2}+4f^{2})}-4f^{2}\ln\left(\frac{\Lambda}{2f}+\sqrt{1+\frac{\Lambda^{2}}{4f^{2}}}\right)\right]$
$\displaystyle-\sum_{q=1}^{\infty}\frac{2}{8\pi^{3}q(q^{2}N^{2}-1)}\cos(qN\chi)\Biggl{[}\left(q^{2}N^{2}\Lambda^{2}+qN\sqrt{\Lambda^{2}(\Lambda^{2}+4f^{2})}+2f^{2}\right)$
$\displaystyle\times\left(\frac{2f^{2}}{\Lambda^{2}+2f^{2}+\sqrt{\Lambda^{2}(\Lambda^{2}+4f^{2})}}\right)^{Nq}-2f^{2}\Biggr{]}\,.$

The first term in the last line, which corresponds to the $q=0$ term,
apparently diverges quadratically when $\Lambda\rightarrow\infty$. This
divergent term is independent of $\chi$, the background PNGB field. For $N\geq
2$, the remaining terms converge when $\Lambda\rightarrow\infty$, and become

$\sum_{q=1}^{\infty}\frac{f^{2}}{2\pi^{3}q(q^{2}N^{2}-1)}\cos(qN\chi)\,.$ (17)

Consequently, we find that $G(x,x)$ has the quadratic divergence
($\sim\Lambda^{2}$) in four dimensions, and the divergent term is independent
of the background field $\chi$ in the present model.

Finally, in the remainder of this section we see the further correspondence
with already known results. This will also serve as a check on our
calculations. Because $\tilde{G}(p^{2})$ can be regarded as the trace of an
inverse matrix $A(p^{2})^{-1}$, where

$A(p^{2})=f^{2}\left[(p^{2}/f^{2})I+\Delta(C_{N},\chi)\right]\,,$ (18)

the integral connects the propagator $\tilde{G}(p^{2})$ and the determinant of
$A(p^{2})$ as follows.

$\displaystyle\ln\Biggl{[}\frac{\det A(p^{2})}{\det A(\Lambda^{2})}\Biggr{]}$
$\displaystyle=$
$\displaystyle\int_{\Lambda^{2}}^{p^{2}}\tilde{G}(\mu^{2})\,d\mu^{2}=N\int_{\lambda}^{\beta}\frac{\sinh(Nz)}{\cosh(Nz)-\cos(N\chi)}dz$
(19) $\displaystyle=$
$\displaystyle\ln\Biggl{[}\frac{\cosh(N\beta)-\cos(N\chi)}{\cosh(N\lambda)-\cos(N\chi)}\Biggr{]}=\ln\Biggl{[}\frac{\sinh^{2}(N\beta/2)+\sin^{2}(N\chi/2)}{\sinh^{2}(N\lambda/2)+\sin^{2}(N\chi/2)}\Biggr{]}\,,$

where $\Lambda$ is a constant and
$4\sinh^{2}\frac{\lambda}{2}=\frac{\Lambda^{2}}{f^{2}}$. This result is
consistent with the calculation of the determinant found in Ref. KS .
Incidentally, this determinant can be used to calculate the one-loop vacuum
energy.777The method to obtain the one-loop effective action from the
propagator has been well-known, for example, see Refs. CR ; DC .888The one-
loop contribution of the vector field is neglected in this time. Of course,
for $D=4$, there is a quartic divergence, but since the part depending on
$\chi$ is finite, the four dimensional case gives the well-known form of the
effective potential for $\chi$:ACG ; HPW ; HL ; KSS

$V(\chi)=-\frac{3f^{4}}{2\pi^{2}}\sum_{q=1}^{\infty}\frac{\cos(qN\chi)}{q(q^{2}N^{2}-1)(q^{2}N^{2}-4)}\,.\quad(N\geq
3)$ (20)

In the next section, we will consider the discrete time heat kernel.

##  III discrete time heat kernel for DD with a cycle graph

The discrete time heat kernels for the $D$-dimensional canonical scalar model
was discussed in Ref. KKSW2 . The discrete time heat kernels for the graph
Laplacian are recently dealt with in Ref. CHJSV . Here we extend the technique
to the case of the DD model investigated so far.

There are two types of the difference operator: The forward difference
operator $\Delta$ is defined by

$\Delta f(t)\equiv f(t+1)-f(t)\,,$ (21)

while the backward difference operator $\nabla$ is defined by

$\nabla f(t)\equiv f(t)-f(t-1)\,.$ (22)

In the two subsections below, we will obtain the heat kernel as a solution of
the equation using each type of difference, and show the construction of the
momentum-space propagator. After that, the UV modification is studied in the
third subsection.

###  III.1 The solution of the forward difference equation

In this subsection, we consider the discrete time heat kernel defined as the
unique solution of the forward difference equation

$\Delta\tilde{K}_{\nu}(p^{2};t)=\epsilon
f^{2}\Bigl{[}e^{i\chi}\tilde{K}_{\nu+1}(p^{2};t)-(2+p^{2}/f^{2})\tilde{K}_{\nu}(p^{2};t)+e^{-i\chi}\tilde{K}_{\nu-1}(p^{2};t)\Bigr{]}\,,$
(23)

with the condition $\tilde{K}_{0}(p^{2};0)=1$ and $\tilde{K}_{\nu}(p^{2};0)=0$
for $\nu\neq 0$. Compared with the differential equation (5), we find that the
parameter $s$ corresponds to $\epsilon t$
($t\in\mathbf{N}_{0}\equiv\\{0,1,2,\dots\\}$) and that the differential
equation (5) is recovered by the continuum limit, since
$\frac{1}{\epsilon}\Delta\rightarrow\frac{\partial}{\partial s}$, if
$\epsilon\rightarrow 0$.

The solution of (23) can be expressed by using the discrete modified Bessel
function $I^{c}_{\nu}(t)$ CHJSV ; BC ; Slavik ,

$\displaystyle I^{c}_{\nu}(t)$ $\displaystyle\equiv$
$\displaystyle\frac{(-c/2)^{\nu}\Gamma(-t+\nu)}{\nu!\,\Gamma(-t)}F\Bigl{(}\frac{\nu-t}{2},\frac{\nu-t+1}{2};\nu+1;c^{2}\Bigr{)}$
(24) $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{\Gamma(t+1)(c/2)^{2n+\nu}}{n!\,\Gamma(t-2n-\nu+1)\Gamma(\nu+n+1)}\,,$

where $F(\alpha,\beta;\gamma;z)$ is the Gauss’ hypergeometric function. Note
that $I^{c}_{-\nu}(z)=I^{c}_{\nu}(z)$ for $\nu\in\mathbf{Z}$.

Note also that we can express $I^{c}_{\nu}(t)$ as the form

$I^{c}_{\nu}(t)=\frac{(c/2)^{\nu}t^{\underline{\nu}}}{\nu!}F\Bigl{(}\frac{\nu-t}{2},\frac{\nu-t+1}{2};\nu+1;c^{2}\Bigr{)}\,.$
(25)

Here the falling power $t^{\underline{\nu}}$ is defined as

$t^{\underline{\nu}}\equiv(-1)^{\nu}(-t)_{\nu}\,,$ (26)

where the Pochhammer symbol means

$(x)_{k}\equiv x(x+1)\cdots(x+k-1)=\Gamma(x+k)/\Gamma(x)\,.$ (27)

One can see that $\Delta\,t^{\underline{n}}=n\,t^{\underline{n-1}}$. It has
been known that $I^{1}_{\nu}(t)$ is obtained by replacing $t^{k}$ in the
Maclaurin series of $I_{\nu}(t)$ by $t^{\underline{k}}$ BC .

The key property of $I^{c}_{\nu}(t)$ is that

$\Delta
I^{c}_{\nu}(t)=I^{c}_{\nu}(t+1)-I^{c}_{\nu}(t)=\frac{c}{2}[I^{c}_{\nu+1}(t)+I^{c}_{\nu-1}(t)]\,.$
(28)

Therefore, the solution of the discrete heat equation (23) can be written as

$\tilde{K}_{\nu}(p^{2};t)=\sum_{q=-\infty}^{\infty}e^{-i(\nu+qN)\chi}a^{t}I^{b}_{\nu+qN}(t)\,,$
(29)

where

$a\equiv 1-\epsilon f^{2}(2+p^{2}/f^{2})\quad\mbox{and}\quad b\equiv 2\epsilon
f^{2}/a\,.$ (30)

The partially-traced propagator $\tilde{G}(p^{2})$ in the momentum space is
considered to be rederived if we take a continuum limit $\epsilon\rightarrow
0$ of

$\epsilon\sum_{t=0}^{\infty}N\tilde{K}_{0}(p^{2};t)\,,$ (31)

which is just the discretized version of the integral (12), in which $s$
corresponds to $\epsilon t$.

Fortunately, the authors of Ref. CHJSV  have even examined the following
function:

$f^{c}_{\nu}(z)\equiv\sum_{t=0}^{\infty}z^{t}I^{c}_{\nu}(t)\,,$ (32)

and they found the closed form of it: CHJSV

$\displaystyle f^{c}_{\nu}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(1-z)^{2}-c^{2}z^{2}}}\left(\frac{1-z}{cz}-\sqrt{\frac{(1-z)^{2}-c^{2}z^{2}}{c^{2}z^{2}}}\right)^{|\nu|}\,.$
(33)

Using their result, we find

$\sum_{q=-\infty}^{\infty}\cos(qN\chi)f^{c}_{qN}(z)=\frac{1}{\sqrt{(1-z)^{2}-c^{2}z^{2}}}\frac{1-B^{2N}}{1+B^{2N}-2\cos(N\chi)B^{N}}\,,$
(34)

where

$B=B_{c}(z)\equiv\frac{1-z}{cz}-\sqrt{\frac{(1-z)^{2}-c^{2}z^{2}}{c^{2}z^{2}}}\,.$
(35)

A lengthy but straightforward computation gives

$\epsilon\sum_{t=0}^{\infty}N\tilde{K}_{0}(p^{2};t)=\epsilon
N\sum_{q=-\infty}^{\infty}\cos(qN\chi)f^{b}_{qN}(a)=\frac{N/f^{2}}{2\sinh\beta}\frac{\sinh(N\beta)}{\cosh(N\beta)-\cos(N\chi)}\,,$
(36)

where $\frac{p^{2}}{f^{2}}=4\sinh^{2}\frac{\beta}{2}$. Note that
$B_{b}(a)=e^{-\beta}$. As the previous analysis of Ref. KKSW2  for the flat
spacetime, this result for the propagator from the discrete time heat kernel
is the same as that from the usual continuous one, and it is even not
necessary to take the limit of $\epsilon\rightarrow 0$.

###  III.2 The solution of the backward difference equation

In this subsection, we start with the backward difference equation

$\nabla\tilde{K}_{\nu}(p^{2};t)=\epsilon
f^{2}\Bigl{[}e^{i\chi}\tilde{K}_{\nu+1}(p^{2};t)-(2+p^{2}/f^{2})\tilde{K}_{\nu}(p^{2};t)+e^{-i\chi}\tilde{K}_{\nu-1}(p^{2};t)\Bigr{]}\,.$
(37)

Obviously, the continuum limit ($\epsilon\rightarrow 0$) of this equation is
the same differential equation (5) we considered.

First, following the previous subsection, we define another discrete modified
Bessel function $\bar{I}^{c}_{\nu}(t)$:

$\bar{I}^{c}_{\nu}(t)\equiv\frac{(c/2)^{\nu}t^{\overline{\nu}}}{\nu!}F\Bigl{(}\frac{\nu+t}{2},\frac{\nu+t+1}{2};\nu+1;c^{2}\Bigr{)}\,,$
(38)

where the rising power $t^{\overline{\nu}}$ is defined by the Pochhammer
symbol,

$t^{\overline{\nu}}\equiv(t)_{\nu}\,,$ (39)

which satisfies $\nabla\,t^{\overline{n}}=n\,t^{\overline{n-1}}$. The function
$\bar{I}^{1}_{\nu}(t)$ is obtained by replacing $t^{k}$ in $I_{\nu}(t)$ by
$t^{\overline{k}}$. The other expression of $\bar{I}^{c}_{\nu}(t)$ is

$\bar{I}^{c}_{\nu}(t)=\sum_{n=0}^{\infty}\frac{\Gamma(t+2n+\nu)(c/2)^{2n+\nu}}{n!\,\Gamma(t)\Gamma(\nu+n+1)}\,,$
(40)

and then, we see that $I^{c}_{-\nu}(z)=I^{c}_{\nu}(z)$ for $\nu\in\mathbf{Z}$.
The identity one can find is

$\nabla\bar{I}^{c}_{\nu}(t)=\bar{I}^{c}_{\nu}(t)-\bar{I}^{c}_{\nu}(t-1)=\frac{c}{2}[\bar{I}^{c}_{\nu+1}(t)+\bar{I}^{c}_{\nu-1}(t)]\,.$
(41)

Note also that $\bar{I}^{c}_{\nu}(0)=0$ for $\nu\neq 0$ and
$\bar{I}^{c}_{0}(0)=1$.

Now, the solution for the backward difference equation (37) turns out to be

$\tilde{K}_{\nu}(p^{2};t)=\sum_{q=-\infty}^{\infty}e^{-i(\nu+qN)\chi}\bar{a}^{t}\bar{I}^{\bar{b}}_{\nu+qN}(t)\,,$
(42)

where

$\bar{a}\equiv\Bigl{[}1+\epsilon
f^{2}(2+p^{2}/f^{2})\Bigr{]}^{-1}\quad\mbox{and}\quad\bar{b}\equiv 2\epsilon
f^{2}\bar{a}\,.$ (43)

As previously, we first consider the generating function

$\bar{f}^{c}_{\nu}(z)\equiv\sum_{t=1}^{\infty}z^{t}\bar{I}^{c}_{\nu}(t)\,,$
(44)

where we should notice that the sum starts from $t=1$. We follow the similar
path as the derivation of $f^{c}_{\nu}(z)$ in Ref. CHJSV . We find that the
series sum $\bar{f}^{c}_{\nu}(z)$ satisfies

$(1-z)\bar{f}^{c}_{\nu}(z)=z\delta_{\nu
0}+\frac{c}{2}(\bar{f}^{c}_{\nu+1}(z)+\bar{f}^{c}_{\nu-1}(z))\,.$ (45)

We assume that the function takes the form
$\bar{f}^{c}_{\nu}(z)=\bar{A}_{c}(z)(\bar{B}_{c}(z))^{\nu}$. Then, the
recursion relation (45) shows

$(\bar{B}_{c}(z))^{2}-\frac{2(1-z)}{c}\bar{B}_{c}(z)+1=0\quad\mbox{and}\quad\bar{A}_{c}(z)=\frac{z}{1-z-c\bar{B}_{c}(z)}\,.$
(46)

The solution for $\bar{B}_{c}(z)$ is

$\bar{B}_{c}(z)=\frac{1-z}{c}-\sqrt{\frac{(1-z)^{2}-c^{2}}{c^{2}}}\,,$ (47)

and accordingly, we obtain

$\displaystyle\bar{f}^{c}_{\nu}(z)$ $\displaystyle=$
$\displaystyle\frac{z}{\sqrt{(1-z)^{2}-c^{2}}}\left(\frac{1-z}{c}-\sqrt{\frac{(1-z)^{2}-c^{2}}{c^{2}}}\right)^{|\nu|}\,.$
(48)

This result can be verified by comparing the coefficient of the first order
term of the Maclaurin series of $\bar{f}^{c}_{\nu}(z)$ with
$\bar{I}^{c}_{\nu}(1)=\frac{1}{\sqrt{1-c^{2}}}\left(\frac{c}{1+\sqrt{1-c^{2}}}\right)^{|\nu|}$,
which is derived from the formula in Ref. Lebedev  for example.

Using the result above, we find

$\sum_{q=-\infty}^{\infty}\cos(qN\chi)\bar{f}^{c}_{qN}(z)=\frac{z}{\sqrt{(1-z)^{2}-c^{2}}}\frac{1-\bar{B}^{2N}}{1+\bar{B}^{2N}-2\cos(N\chi)\bar{B}^{N}}\,,$
(49)

where $\bar{B}=\bar{B}_{c}(z)$, and

$\epsilon\sum_{t=1}^{\infty}N\tilde{K}_{0}(p^{2};t)=\epsilon
N\sum_{q=-\infty}^{\infty}\cos(qN\chi)\bar{f}^{\bar{b}}_{qN}(\bar{a})=\frac{N/f^{2}}{2\sinh\beta}\frac{\sinh(N\beta)}{\cosh(N\beta)-\cos(N\chi)}\,,$
(50)

where $\frac{p^{2}}{f^{2}}=4\sinh^{2}\frac{\beta}{2}$. Note that
$\bar{B}_{\bar{b}}(\bar{a})=e^{-\beta}$. This result from the discrete time
heat kernel from the backward difference equation is also the same as that
from the usual continuous one, and it is also not necessary to take the limit
of $\epsilon\rightarrow 0$.

###  III.3 UV modification of the propagator

After making the above preparation, we consider the UV modification of the
propagator. In Ref. KKSW2 , we introduced the modified propagator (Green’s
function) of the free massive scalar field in momentum space by omitting a
finite number of discrete heat kernels $\tilde{K}(p^{2};t)$,
$t=0,1,2,\dots,n-1$ (in the notation of the present paper), in the infinite
sum. Here, we only state the results ((2.18) in Ref. KKSW2 ),

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

where $m$ is the mass of the scalar field. This method of modification is the
discrete counterpart of the Siegel’s modification Siegel , which converts the
integration range from $[0,\infty]$ to $[\varepsilon,\infty]$, where
$\varepsilon$ is a small constant. It is also known that such a manner is
often used in the UV regularization in the standard heat kernel formalism.

Now, let us return to our present DD model. For the heat kernel from the
forward difference equation, the simplest modification method described above
turns out not to be effective, because $I^{c}_{0}(0)=1$, $I^{c}_{n}(t)=0$ for
$t<n$ CHJSV  and $a=1-\epsilon f^{2}(2+p^{2}/f^{2})\rightarrow-\infty$ when
$p^{2}/f^{2}\rightarrow\infty$, it is not immediately clear if the elimination
of finite terms for small numbers $t$ makes the propagator behaves better at
high energy ($p^{2}/f^{2}\rightarrow\infty$). Another idea for the heat kernel
from the forward difference equation, we consider eliminating the infinite
terms of the even powers of $a$ in the sum over $t$. Indeed, the behavior of
$\tilde{G}(p^{2})$ at large $p^{2}$ becomes better, but the sign change of $a$
at large $p^{2}$ results in a ‘cut’ in the complex plane of $p^{2}$ that are
difficult to interpret.

If we adopt the heat kernel from the backward difference equation, the
prospects of omission of finite terms seems good. Since
$\bar{a}\equiv\Bigl{[}1+\epsilon f^{2}(2+p^{2}/f^{2})\Bigr{]}^{-1}>0$ for
$p^{2}/f^{2}\geq 0$, it can be easily inferred that
$\tilde{K}_{\nu}(p^{2};t)\sim(p^{2})^{-t}$ at large $p^{2}$.

Therefore, we define the modified propagator in momentum space

$\tilde{G}_{n}(p^{2})\equiv\epsilon\sum_{t=n}^{\infty}N\tilde{K}_{0}(p^{2};t)\,,$
(51)

where $\tilde{K}_{0}(p^{2};t)$ is the solution of the backward difference
equation. Note, of course, that $\tilde{G}_{1}(p^{2})=\tilde{G}(p^{2})$.

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

$\displaystyle\tilde{G}_{2}(p^{2})=\epsilon\sum_{t=2}^{\infty}N\tilde{K}_{0}(p^{2};t)=\tilde{G}(p^{2})-\epsilon
N\tilde{K}_{0}(p^{2};1)$ (52) $\displaystyle=$
$\displaystyle\frac{N}{\sqrt{p^{2}(4f^{2}+p^{2})}}\sum_{q=-\infty}^{\infty}\cos(qN\chi)\left(\frac{2f^{2}}{2f^{2}+p^{2}+\sqrt{p^{2}(4f^{2}+p^{2})}}\right)^{|q|N}$
$\displaystyle-\frac{N}{\sqrt{(\epsilon^{-1}+p^{2})(\epsilon^{-1}+4f^{2}+p^{2})}}$
$\displaystyle\times\sum_{q=-\infty}^{\infty}\cos(qN\chi)\left(\frac{2f^{2}}{\epsilon^{-1}+2f^{2}+p^{2}+\sqrt{(\epsilon^{-1}+p^{2})(\epsilon^{-1}+4f^{2}+p^{2})}}\right)^{|q|N}\,.$

Before analyzing this modified propagator, we propose another subtraction
scheme. As the other way, we eliminate the terms of the odd order in $t$ in
the propagator. Namely, we introduce

$\displaystyle\tilde{G}_{e}(p^{2})\equiv
2\epsilon\sum_{t=2,4,6\dots}^{\infty}N\tilde{K}_{0}(p^{2};t)$ (53)
$\displaystyle=$
$\displaystyle\frac{1}{2}(2\epsilon)\left[N\sum_{q=-\infty}^{\infty}\cos(qN\chi)\bar{f}^{\bar{b}}_{qN}(\bar{a})+N\sum_{q=-\infty}^{\infty}\cos(qN\chi)\bar{f}^{\bar{b}}_{qN}(-\bar{a})\right]$
$\displaystyle=$
$\displaystyle\frac{N}{\sqrt{p^{2}(4f^{2}+p^{2})}}\sum_{q=-\infty}^{\infty}\cos(qN\chi)\left(\frac{2f^{2}}{2f^{2}+p^{2}+\sqrt{p^{2}(4f^{2}+p^{2})}}\right)^{N|q|}$
$\displaystyle-\frac{N}{\sqrt{(2\epsilon^{-1}+p^{2})(2\epsilon^{-1}+4f^{2}+p^{2})}}$
$\displaystyle\times\sum_{q=-\infty}^{\infty}\cos(qN\chi)\left(\frac{2f^{2}}{2\epsilon^{-1}+2f^{2}+p^{2}+\sqrt{(2\epsilon^{-1}+p^{2})(2\epsilon^{-1}+4f^{2}+p^{2})}}\right)^{N|q|}\,.$

Interestingly, above two cases result in similar deformations of the
propagators:

$\tilde{G}_{2}(p^{2})=\tilde{G}(p^{2})-\tilde{G}(p^{2}+\epsilon^{-1})\,,\quad\tilde{G}_{e}(p^{2})=\tilde{G}(p^{2})-\tilde{G}(p^{2}+2\epsilon^{-1})\,.$
(54)

That is, we find the same form as the propagator in the simplest Lee–Wick
theory, or almost equivalent to the one with the simplest Pauli–Villars
subtraction PV ; PU ; IZ ; Collins ; PS  as a result: For example, the
propagator in momentum space for a canonical scalar field with mass $m$ will
be modified as

$\frac{1}{p^{2}+m^{2}}\rightarrow\frac{1}{p^{2}+m^{2}}-\frac{1}{p^{2}+M^{2}}\,,$
(55)

where $M$ is the mass which would be taken as infinitely large.

The two types of modified propagators behave like $p^{-4}$ at high energies,
and the UV behavior is improved as in the Lee–Wick theory. The important part
of the results here is that both of the two subtraction methods lead to the
Lie–Wick type (albeit with two different mass parameters). Especially in the
subtraction of the odd-number terms, it is a nontrivial result to be
represented by only one parameter ($2\epsilon^{-1}$). It is also interesting
to note that, for $\tilde{G}_{n}~{}(n\geq 3)$, complicated functional forms
different from the original $\tilde{G}$ inevitably appear, and that
$\tilde{G}_{e}$ is represented by being combined into a remarkable simple
form.

Now, we turn to examine the behavior of divergence in $G(x,x)$ in our case for
$D=4$. If we use the cutoff scale $\Lambda$, it is written as

$G(x,x)=\frac{2\pi^{2}}{(2\pi)^{4}}\int_{0}^{\Lambda}\tilde{G}(p^{2})\,p^{3}dp\,.$
(56)

If we use the modified propagators proposed above, we can write $G(x,x)$ using
the results we have obtained so far, since

$\int_{0}^{\Lambda}\tilde{G}(p^{2})\,p^{3}dp\rightarrow\int_{0}^{\Lambda}\Bigl{[}\tilde{G}(p^{2})-\tilde{G}(p^{2}+M^{2})\Bigr{]}\,p^{3}dp\,,$
(57)

where we should read $M^{2}=\epsilon^{-2}$ for $\tilde{G}_{2}$, while
$M^{2}=2\epsilon^{-2}$ for $\tilde{G}_{e}$. In the limit of
$\Lambda\rightarrow\infty$, the only divergent term is the term with $q=0$ in
the sum-form representation of $\tilde{G}(p^{2})$ (12). Indeed, the
contribution is found to be

$\displaystyle\frac{N}{16\pi^{2}}\Biggl{[}M^{2}\ln\frac{\Lambda^{2}+M^{2}+2f^{2}+\sqrt{(\Lambda^{2}+M^{2})(\Lambda^{2}+M^{2}+4f^{2})}}{M^{2}+2f^{2}+\sqrt{M^{2}(M^{2}+4f^{2})}}+\sqrt{M^{2}(M^{2}+4f^{2})}$
$\displaystyle-2f^{2}\ln\frac{M^{2}+2f^{2}+\sqrt{M^{2}(M^{2}+4f^{2})}}{2f^{2}}-\frac{M^{2}(2\Lambda^{2}+M^{2}+4f^{2})}{\sqrt{(\Lambda^{2}+M^{2})(\Lambda^{2}+M^{2}+4f^{2})}+\sqrt{\Lambda^{2}(\Lambda^{2}+4f^{2})}}$
$\displaystyle+2f^{2}\ln\frac{\Lambda^{2}+M^{2}+2f^{2}+\sqrt{(\Lambda^{2}+M^{2})(\Lambda^{2}+M^{2}+4f^{2})}}{\Lambda^{2}+2f^{2}+\sqrt{\Lambda^{2}(\Lambda^{2}+4f^{2})}}\Biggr{]}\,,$
(58)

and this gives the logarithmic divergence $\sim M^{2}\ln\Lambda$ when
$\Lambda\rightarrow\infty$, instead of the quadratic divergence
$\sim\Lambda^{2}$ known in the ordinary scalar one-loop effect.

Now we consider $\tilde{G}_{3}(p^{2})$ and the diverging behavior of
$G_{3}(x,x)$. First we note that

$\tilde{G}_{3}(p^{2})=\tilde{G}_{2}(p^{2})-\epsilon
N\tilde{K}_{0}(p^{2};2)\,,$ (59)

and the calculations on the divergent part above can be used. Since
$\bar{I}_{0}^{c}(2)=\frac{1}{(1-c^{2})^{3/2}}$, the divergent contribution
comes from the $q=0$ term in the sum form of $\epsilon
N\tilde{K}_{0}(p^{2};2)$ (42) and it turns out to be

$\frac{N\epsilon^{-1}(p^{2}+2f^{2}+\epsilon^{-1})}{[(p^{2}+\epsilon^{-1})(p^{2}+4f^{2}+\epsilon^{-1})]^{3/2}}\,.$
(60)

Note that this behaves $\sim 1/p^{4}$ for large $p$. One can find that the
momentum integration of $\epsilon N\tilde{K}_{0}(p^{2};2)$ (60) with the
cutoff $\Lambda$ is

$\displaystyle\frac{N}{16\pi^{2}}\Biggl{[}M^{2}\ln\frac{\Lambda^{2}+M^{2}+2f^{2}+\sqrt{(\Lambda^{2}+M^{2})(\Lambda^{2}+M^{2}+4f^{2})}}{M^{2}+2f^{2}+\sqrt{M^{2}(M^{2}+4f^{2})}}$
$\displaystyle-\frac{M^{2}\Lambda^{2}}{\sqrt{(\Lambda^{2}+M^{2})(\Lambda^{2}+M^{2}+4f^{2})}}\Biggr{]}\,,$
(61)

where $M^{2}=\epsilon^{-1}$. Then, subtraction of (61) from (58) gives

$\displaystyle G_{3}(x,x)$ $\displaystyle=$
$\displaystyle\frac{N}{16\pi^{2}}\Biggl{[}\sqrt{M^{2}(M^{2}+4f^{2})}-2f^{2}\ln\frac{M^{2}+2f^{2}+\sqrt{M^{2}(M^{2}+4f^{2})}}{2f^{2}}\Biggr{]}$
(62) $\displaystyle+(\mbox{finite, $\chi$-dependent terms})\,,$

in the lmit of $\Lambda\rightarrow\infty$. The result that $G_{3}(x,x)\sim
M^{2}$ is an expected one, but is involving the scale $f$ in a nontrivial
form.

##  IV Summary and outlook

In the present paper, we consider the discrete time heat kernel in the simple
model of the dimensionally deconstructed scalar QED. The Euclidean propagator
of the scalar field is obtained by the sum of the discretized kernels, which
are not individually expressed by the direct products of the kernel on the
flat space and that on the discrete circle. The propagator thus obtained by an
infinite sum of kernels is found to be in perfect agreement with the usual one
obtained from the integral of the continuous time heat kernel. A nontrivial
result here is that it holds in both forward and backward discretization
cases, and not relying on the scale of the discretization unit $\epsilon$,
that is, there is no need to take the limit of $\epsilon\rightarrow 0$.

Furthermore, we considered the behavior of the two-point coincidence limit of
the propagator $G(x,x)$, which is a measure of the UV divergence of the one-
loop mass correction to the interacting scalar field. For a usual propagator,
it gives a quadratic divergence in the four-dimensional spacetime, namely,
$G(x,x)\sim\Lambda^{2}$, where $\Lambda$ is the cutoff momentum. In the
previous paper KKSW2 , we proposed the method to modify the propagator by
subtracting contributions of a finite number of discretized heat kernels. This
idea is a discretized version of Siegel’s method Siegel  of moving the lower
bound of the integration range of the kernel. We apply similar method to the
present model with DD. We found that both cases with the subtraction of the
first kernel of $t=1$ and with the subtraction of the odd-numbered kernels of
$t=1,3,5,\dots$ gives the same form of the resulting propagator,
$G(p^{2})-G(p^{2}+M^{2})$; $M^{2}=\epsilon^{-1}$ in the former and
$M^{2}=2\epsilon^{-1}$. This is a nontrivial result, and this modified
propagator has exactly the same structure as that of the Lee–Wick theory, or
the regularized propagator of Pauli–Villars. Consequently, this modified
propagator gives $G(x,x)\sim\ln\Lambda$ in four dimensions. Finally, we showed
that the modified propagator by subtracting the kernels of $t=1$ and $t=2$
gives $G(x,x)\sim M^{2}=\epsilon^{-1}$.

It turns out that the exact propagator can be obtained without taking the
continuum limit with the solutions of two difference equations. From this
fact, it seems that we may be able to develop more free ideas regarding the
handling of the proper time method, even though it looks like an ad hoc
assumption at the current primitive stage of study. Of course, it is necessary
to pursue the principle underlying the physical inevitability of use of
discrete proper-time.

Anyway, the discretization of the heat kernel for the Dirac operator on
continuous and discrete real spaces is interesting mathematically and may be
important for physical applications such as an approach to the hierarchy
problem. Since the Dirac operators on graphs have been studied so far (see for
example, Ref. KS2 ), the study on heat kernel for them can be tackled in near
future.

Although it may seem trivial in the present treatment, it is interesting that
the finite part involving the PNGB field is also subject to change when the
propagator is modified under our prescription. This effect would similarly
affect the Hosotani-like mechanism in DD theory, though the treatment of the
divergence in the potential requires further consideration.

The UV modified propagator obtained in our method includes additional mass
poles in the complex plane of $-p^{2}$. Such poles appear in general higher
derivative theories and have been studied by many authors. In the higher
derivative theory, the addition of interaction may bring about specific
instability and may violate unitarity in the Lorentzian spacetime, since
analysis with auxiliary fields indicates the appearance of negative norms.
This is a subject that has been frequently addressed and has recently been
actively discussed AP ; Anselmi1 ; Anselmi2 . 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
unitarity and other physics on self-interacting models should be studied
further. At the same time, we have to reconsider the heat kernel derived from
the forward difference equation, which yields the momentum-space propagator
involving an apparently bizarre cut when discrete subtraction is carried out,
in similar sense.999It is likely to be related to the studies on field
theories with not only higher derivatives but also fractional powers of
derivative operators CCLR .

Finally, we would like to write about the discretization of real whole
spacetime, which is a radical extension of the DD model, and the study of its
heat kernel as well. Mathematically, it would be straightforward to replace
the continuum space with a certain graph structure. Many difficulties can be
expected from physical requirements, but it will be great if we could use the
mathematics of heat kernel theory to uncover non-obvious features of the
general discretized models.

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