55

沼津工業高等専門学校研究報告
第 58 号，2024 年 2 月

Lattice Green Function for a Free Massive Boson
KOMA Yoshiaki ∗Abstract: The Green functions characterize interaction properties between particles in quantum
ﬁeld theories. We evaluate the Green functions for a massive scalar boson deﬁned on a threedimensional Euclidean lattice numerically, and investigate the lattice eﬀect by comparison withthe continuum Green function. We again ﬁnd a characteristic diﬀerence at very short distances
quantiﬁed approximately by an exponential function as in the massless case.
Key Words: Green function, Propagator, Lattice formulation
1.

Introduction

Formulating a quantum ﬁeld theory on a dis-results with the continuum one will help to expose thelattice artifacts.
Previously, we investigated the lattice Green

cretized spacetime, which we call the lattice formulation in short, regularizes the continuum theory thathas an inﬁnite number of degrees of freedom with aﬁnite number of them. In this formulation, ﬁeld variables of the theory are treated as a ﬁnite number ofsite or link variables, and then, it is possible to solve
the ﬁeld equations by using a kind of ﬁnite diﬀerence
methods or to compute the partition function of thetheory by using the Monte-Carlo method. A weak
point, however, may be that the numerical results always suﬀer from systematic errors due to ﬁnite cutoﬀ

function for the massless boson deﬁned on a threedimensional Euclidean lattice numerically and compared the results with the continuum one [1]. We
then found a characteristic diﬀerence at very shortdistances, which is quantiﬁed approximately by an exponential function. As we mentioned, the remnant ofsuch an exponential behavior may cause a problem
when a theory that can generate a mass dynamicallyis analyzed, since the eﬀect of the mass usually appears as an exponential function.
In the present report, we extend our previous

and ﬁnite volume even if one obtains precise results.
This is because one cannot perform computation practically in inﬁnite volume with inﬁnitesimally small lattice spacing. Therefore, it is important to control thelattice artifacts in the numerical results.
The evaluation of the Green functions deﬁned
on the lattice, which we call the lattice Green functions, may provide us with some hints for controlling
the lattice eﬀect. The Green functions play the roleof propagators in quantum ﬁeld theories, which contain information on the interaction properties between

work to the case that a propagating particle has aﬁnite mass m. More speciﬁcally, we investigate thelattice Green functions for a free massive scalar bosonin three-dimensional Euclidean space. In the continuum theory, it is known that the particle obeys the
Klein-Gordon equation, so that the propagator withthe momentum pµ is given by 1/(p2 + m2 ) and theFourier transformation leads to the Yukawa potential∝ e−mr /r. We shall compare the lattice results withthis continuum behavior.

particles. For instance, when a massless boson, suchas the photon in quantum electrodynamics, carries acontinuum momentum pµ , its propagator in the mo-

2.

The lattice Green functions

mentum representation is given by 1/p2 , which corre-

In the continuum theory, the Green function

sponds to the well-known Coulombic potential ∝ 1/r

G(x; m) for a massive scalar boson is deﬁned by thein the coordinate representation. On the lattice, how-relation

ever, the momentum is discretized and its range is also
restricted due to ﬁniteness of the lattice spacing and(−∆ + m2 )G(x; m) = δ(x) ,(1)the volume, which should aﬀect the behavior of theGreen function. Thus, the comparison of the latticewhere ∆ denotes the Laplacian and δ(x) the Diracdelta function.

The solution of Eq. (1) in the n-dimensional space can be obtained by performing the∗ Division of Liberal Arts

Fourier transformation of the Green function in the

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沼津工業高等専門学校研究報告

momentum representation 1/(p2 + m2 ) as
G(x; m) =

∫ ∞

dn p
eipx
,
n 2
2
−∞ (2π) p + m

The Fourier transformation is carried out similarly as
∫ π n
d p
eipx
G∞ (x; m) =n 2
2
−π (2π) p̂ + m
∫ π n
d p
eipx
=
∑
n
2
n
2
−π (2π) 4µ=1 sin (pµ /2) + m∫ ∞2=dα e−(2n+m )α(2)∑nµ=1 pµ pµ , ipx = iµ=1 pµ xµ . By using∫∞
the integral formula F −1 = 0 dαe−αF , the Fourier

where p2 =∑ntransformation is carried out as [2]∫ ∞ ∫G(x; m) =dα

第 58 号0×nd p ipx −α(p2 +m2 )
e e
n
(2π)
0
∫ ∞
n ∫ ∞
∏
dpµ −α(pµ − ixµ )2 − x2µ
−αm2
2α
4α
e
=
dα e
2π
0
µ=1 −∞( )n ( ) n∫ ∞1
π 2 − x2
−αm2
=
dα e
e 4α
2π
α
0
1 ( m ) n2 −1=
K n2 −1 (mr) ,(3)n(2π) 2 r

=

n ∫ π
∏
dpµ

2π
µ=1 −πe2α cos pµ +ipµ xµ

∫ ∞
n
∏
2
dα e−(2n+m )α
Ixµ (2α) ,
0

(8)

µ=1

where the ﬁnal expression is based on the integral formula of the modiﬁed Bessel function of the ﬁrst kind∫
1 πIν (z) =dθez cos θ cos νθ (ν ∈ Z) .
(9)
π 0

where the ﬁnal expression is based on the integral for-Since the integration over α cannot be carried out

mula of the modiﬁed Bessel function of the second

analytically, one may resort to numerical integra-kindtion. The convergence property of this integration,Kν (z) =1 ( z )ν
2 2

∫ ∞

z2

dt e−t− 4t t−ν−10

however, is expected to be quite slow for large α,(ν ∈ R) . (4)due to the behavior of the modiﬁed Bessel function√Iν (2α) ∼ 1/(4πα)e2α for large α. In fact, the inte-

Therefore, for the n = 3 case the Green function is

grand will be of O(α−n/2 ) and this asymptotic behav-

reduced to the well-known Yukawa potential
G(x; m) =

1
(2π)

3
2

( m ) 12r

K 12 (mr) =

e−mr
,
4πr

ior indicates that it may be diﬃcult to obtain precise
numerical values for lower dimensions n ≤ 3 except(5)

for the large mass cases.

For the massless case with n = 4, 3, 2, this prob-√
where K1/2 (z) = π/(2z) e−z is used.

lem is solved [3–5]. If one applies to a similar method,
the procedure is modiﬁed as in Ref. [6] with a relation∫ π n
d p ipx(∇∗µ +∇µ )G∞ (x; m) = xµe ln(p̂2+m2 ) . (10)n−π (2π)On the lattice in inﬁnite volume, the Green func-tion G∞ (x; m) should satisfy(−∆ + m2 )G∞ (x; m) = δx0 ,where ∆ =(6)

Inserting Eq. (10) into Eq. (6) for x ̸= 0, one obtains∑na recursion relation∗µ=1 ∇µ ∇µ denotes the lattice Laplacian,which is a combination of the forward and backward∑ndiﬀerences, so that ∆f (x) = µ=1 [f (x + µ̂) + f (x −G∞ (x + µ̂; m) = G∞ (x − µ̂; m)
n
2xµ ∑
m2
+
)G∞ (x; m) − G∞ (x − ν̂; m)] (11)
[(1 +
ρ ν=1
2n

µ̂) − 2f (x)], and δx0 is the Kronecker delta, satisfying
δx0 = 1 for x = 0 and δx0 = 0 for x ̸= 0. The∑nµ=1 xµfor ρ ̸= 0. We do not proceed

momentum carried by a particle is deﬁned by p̂µ =with ρ =2 sin(pµ /2), where pµ ∈ [−π, π]. The lattice spacing isfurther this method in this report, but examine if thenumerical result satisﬁes the relation in Eq. (6).

set to be one throughout this report. Note that due
to hypercubic symmetry, the lattice Green functions∏n

On the lattice in ﬁnite volume of the size V =µ=1 Lµ , where periodic boundary conditions are im-near the origin satisfy a relation

posed in all µ directions, the Green function GL (x; m)2(2n + m )G∞ (0; m) − 2nG∞ (µ̂; m) = 1 .

should satisfy the same relation as in Eq. (6), but the

(7)
2

57

Lattice Green Function for a Free Massive Boson

Table 1: The lattice Green functions for m = 0.50 in inﬁnite volume and in ﬁnite volume of V = 243 .
r G∞ (r; m)
G∞ (r; m) − G∞ (0; m)
0 2.111547903170267×10−1 0
1 5.328623991356951×10−2 −1.578685504034572×10−1
2 1.708696890778575×10−2 −1.940678214092410×10−1
3 6.642001737189745×10−3 −2.045127885798370×10−1
4 2.934166508491701×10−3 −2.082206238085350×10−1
5 1.402515601993884×10−3 −2.097522747150329×10−1
6 7.042904660635528×10−4 −2.104504998509632×10−1
7 3.654127104022456×10−4 −2.107893776066245×10−1
8 1.940082167762876×10−4 −2.109607821002504×10−1
9 1.047841774237187×10−4 −2.110500061396030×10−1
10 5.735009830886571×10−5 −2.110974402187179×10−1
11 3.172329212786405×10−5 −2.111230670248989×10−1
12 1.770079724664004×10−5 −2.111370895197801×10−1

GL (r; m)
GL (r; m) − GL (0; m)
2.111549304663567×10−1 05.328638590245488×10−2 −1.578685445639018×10−1
1.708713461306398×10−2 −1.940677958532927×10−1
6.642208288145088×10−3 −2.045127221782116×10−1
2.934449547507940×10−3 −2.082204809188487×10−1
1.402937062890787×10−3 −2.097519934034659×10−1
7.049587589070812×10−4 −2.104499717074496×10−1
3.665181733324653×10−4 −2.107884122930242×10−1
1.958861987254369×10−4 −2.109590442676312×10−1
1.080283405645461×10−4 −2.110469021257921×10−1
6.301773058981373×10−5 −2.110919127357669×10−1
4.171101085926890×10−5 −2.111132194554974×10−1
3.544003416786985×10−5 −2.111194904321888×10−1

momentum is modiﬁed to pµ (k) = 2πlµ (k)/Lµ with

0.05

integers lµ (k) = 0, 1, 2, ..., Lµ − 1, reﬂecting the pe-

0.00

Ĝ∞ (r; m)transformation is then given by summation instead of−0.05

ĜL=32 (r; m)integration as−0.10

ĜL=64 (r; m)

riodicity GL (x; m) = GL (x + Lµ µ̂; m). The Fourieri∑n2πlµ (k)xe µ=1 Lµ µ
1 ∑GL (x; m) =. (12)∑n2 πlµ (k)2Vk 4µ=1 sin ( Lµ ) + m

−0.15
−0.20

In contrast to the massless case, there is no need to−0.25

remove the zero momentum mode. The summation

0

10

20

30

40

50

60

r

over all momenta is easily achieved by using a FFT
10algorithm. When the lattice volume is isotropic such

0

10−3

as Lµ = L for all µ directions, the relation like in
Eq. (7) holds for GL due to hypercubic symmetry.
3.

ĜL=16 (r; m)

10−6
10−9

Numerical results

ĜL=16 (r; m) − Ĝ∞ (r; m)

10−12

We shall present numerical results in three di-ĜL=32 (r; m) − Ĝ∞ (r; m)

10−15

mensions. Since our aim is to reveal the lattice eﬀects
in the lattice Green function, we ﬁrst focus on the dif-

10−18

ference of the lattice Green functions in inﬁnite andĜL=64 (r; m) − Ĝ∞ (r; m)
0

10

20

30

40

50

60

r

ﬁnite volumes, where the mass is set to be m = 0.50

Fig. 1: The normalized lattice Green function in inﬁnite
volume Ĝ∞ (r; m) and in ﬁnite volume ĜL (r; m)(upper),and the diﬀerence ĜL (r; m) − Ĝ∞ (r; m) (lower).

as an example.
In Table 1, we show the numerical values ofEq. (8) evaluated by using Wolfram Mathematica,
and of Eq. (12) obtained by using a FFT algorithmof SciPy on the V = 243 lattice, both along the on-

discrepancy, which seems to increase gradually withaxis x = (r, 0, 0). An interesting observation is thatthe distance r. The reason will be obvious by plotting

G∞ (r; m) and GL (r; m) are quite similar with eachthe numerical values of various lattice volumes for all

carefully, however, one may notice that there is a small

increase of the diﬀerence just reﬂects the periodicity

range of r ∈ [0, L] as shown in Fig. 1. The systematicother. If one looks at the numerical values in Table 13

58

沼津工業高等専門学校研究報告

0.10

第 58 号

tinuum case, the Green function at the origin is diverG∞ (r; m)

0.08

gent due to the 1/r behavior, so that we pay attentionG(r; m)0.06

to the range for r ≥ 1. We ﬁnd that both lattice and0.04

with an exponential suppression for r, and only at

0.02

very short distances for r ≤ 5 there appears a characteristic diﬀerence by about ten percent. A chi-square

0.00

ﬁtting analysis for the diﬀerence G∞ (r; m) − G(r; m)−0.02

continuum Green functions show a similar behavior

indicates that this approximately obeys an exponen0

5

10

15

20

25

30

tial functionrf (r) = 0.0119(3)e−0.85(2)r .

0.010
G∞ (r; m) − G(r; m)

0.008

f (r) = 0.0119(3)e−0.85(2)r(13)

What is remarkable is that the functional form and0.006

the ﬁtting parameters are quite similar to that in the

0.004

massless case [1]. We note that if we assume the func-

0.002

tional form of the G∞ (r; m) as of the continuum one,0.000

parameter corresponding to the mass, p2 , can be ap-−0.002p1 e−p2 r /r, and perform chi-square ﬁtting analyses, the

0

5

10

15

20

25

propriate only when we neglect the data at short dis-

30

tances, such that p2 = 0.5009(4) for 6 ≤ r ≤ 20, whilerp2 = 0.535(2) for 2 ≤ r ≤ 20.

Fig. 2: The lattice Green function in inﬁnite volume
G∞ (r; m) and the continuum Green function G(r; m) =e−mr /(4πr) (upper), and the diﬀerence G∞ (r; m) −G(r; m) (lower).

4.

Summary

We have investigated the lattice Green functionof the lattice in ﬁnite volume. This means that it canfor a scalar massive boson numerically both in ﬁnite

easily be controlled by taking a reasonably large lat-and inﬁnite volumes in three-dimensional Euclidean

tice volume. Thus, in order to investigate the lattice

space. Our results again exhibit a characteristic diﬀer-

eﬀect based on comparison with the continuum Greenence from the continuum Green function at very shortfunction, either type of the lattice Green functions candistances for r ≤ 5, which is quantiﬁed by using an exponential function as in the massless case. Thus, one

be used (of course, the lattice volume should appropriately be large when the ﬁnite volume one is used).

should be careful to use the continuum Green function

Note for the massless case that the coincidence of theto extract physical parameters such as a mass fromlattice Green function in inﬁnite and ﬁnite volumes

the lattice result when only the short distance data

can be observed only for the normalized one [1].

is available. The origin of the exponential diﬀerence
need to be investigated further.

We then compare the lattice Green function in

The author is grateful to Miho Koma for fruitful

inﬁnite volume G∞ (r; m) and the continuum Greenfunction G(r; m) = e−mr /(4πr) in Fig. 2. In the con-collaboration.

[1] Y. Koma, National Institute of Technology, Numazu
College Research Annual No.56 (2022) 59-62.
[2] B. Paladini, J. Sexton, Phys. Lett. B448 (1999) 76-84.
[3] M. Lüscher, P. Weisz, Nucl. Phys. B445 (1995) 429-450.

[4] S. Necco, R. Sommer, Nucl. Phys. B622 (2002) 328-346.
[5] D.S. Shin, Nucl. Phys. B525 (1998) 457-482.
[6] B. Borasoy, H. Krebs, Phys.Rev. D 72 (2005) 056003.

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