#### Derivation of the cut-off length from the quantum quadratic enhancement of a mass in vacuum energy constant Lambda

Eur. Phys. J. C
Derivation of the cut-off length from the quantum quadratic enhancement of a mass in vacuum energy constant Lambda
Kimichika Fukushima 0 2
Hikaru Sato 1
0 Theoretical Division, South Konandai Science Research , 9-32-2-701, Konandai, Konan-ku, Yokohama 234-0054 , Japan
1 Emeritus, Department of Physics, Hyogo University of Education , Yashiro-cho, Kato-shi, Hyogo 673-1494 , Japan
2 Advanced Reactor System Engineering Department, Toshiba Nuclear Engineering Services Corporation , 8, Shinsugita-cho, Isogo-ku, Yokohama 235-8523 , Japan
Ultraviolet self-interaction energies in field theory sometimes contain meaningful physical quantities. The self-energies in such as classical electrodynamics are usually subtracted from the rest mass. For the consistent treatment of energies as sources of curvature in the Einstein field equations, this study includes these subtracted selfenergies into vacuum energy expressed by the constant Lambda (used in such as Lambda-CDM). In this study, the self-energies in electrodynamics and macroscopic classical Einstein field equations are examined, using the formalisms with the ultraviolet cut-off scheme. One of the cut-off formalisms is the field theory in terms of the step-functiontype basis functions, developed by the present authors. The other is a continuum theory of a fundamental particle with the same cut-off length. Based on the effectiveness of the continuum theory with the cut-off length shown in the examination, the dominant self-energy is the quadratic term of the Higgs field at a quantum level (classical selfenergies are reduced to logarithmic forms by quantum corrections). The cut-off length is then determined to reproduce today's tiny value of Lambda for vacuum energy. Additionally, a field with nonperiodic vanishing boundary conditions is treated, showing that the field has no zero-point energy.
1 Introduction
Self-interaction energies in field theory, which contain
ultraviolet divergences in continuum theory, sometimes reveal
meaningful properties in physics [1–8]. In our previous paper
[9–12], we formulated a field theory in terms of the
stepfunction-type basis functions (SFT field theory), which is
based on the finite element theory [9,12–14] (the
formulation is rather different from that by Bender et al.), and cuts
off high-frequency oscillations of wave functions at short
distances. Owing to the space-time continuum and
differentiable step-function-type basis functions, this formalism
is Poincaré covariant and removes ultraviolet divergences at
short distances. The advantage of our formalism is the
availability to perform self-energy evaluation. (We note that the
conventional finite element method is widely used [13]. The
validity of theories is of course justified solely by the
correctness of the logical deduction. The support based only on
the fact, where an article was published, is insufficient for
the true justification of the theory. The assessment of the
theory is beyond the range of the work by the authors.) The
meaningful self-energy appears in the Lamb shift [1], which
is caused by finite parts of the self-energy in higher-order
terms, and the divergent parts are subtracted from the rest
mass. In contrast, the self-energy also appears in the φ3 model
(the mass is sometimes not renormalized when the mass is a
value in vacuum without containing additional interactions).
In our previous paper [12], we derived excited states such as
meta-stable states at stationary states, which are not always
orthogonal to the ground state.
In the Einstein field equations [15,16], the rest energy
works as a source of the curvature. The mass renormalization
in such as electrodynamics subtracts self-energies, which can
be finite using the cut-off scheme. It is then expected that the
self-energies are involved in the Einstein field equations.
In our formalism, four-dimensional space-time is divided
into many hyper-octahedrons, whose shape are arbitrary and
have the size Δ (cut-off length) in four-dimensional
spacetime. For simplicity, we consider three-dimensional space
and divide the region into many cubes. The classical wave
function φ (x , y, z) is expressed in terms of the
step-functiontype basis functions Ω˜ 3p(x , y, z) in three-dimensional space
[the step-function-type basis function in one dimension is
defined by Eq. (
19
)]
φ (x , y, z) =
φ pΩ˜ 3p(x , y, z),
p
where the basis function takes a value of 1 in a cube (each
cubic region is identified by index p) and vanishes outside
the cube. The coefficient φ p is a constant within the cubic
region identified by the index p.
Motivated by the above expectation, this paper is aimed
at presenting a formulation to include the subtracted
selfenergies into vacuum energy with the constant Λ
(cosmological constant) [16–32] of the macroscopic classical Einstein
field equations. The self-energy in classical electrodynamics
is calculated by the continuum theory with a finite cut-off
length. The self-energy is also derived using the field
theory in terms of the step-function-type basis functions, which
was developed by the present authors, and the result is
compared with that calculated by the continuum theory. We also
examine the curvature (gravitational) self-energy of the
fundamental particle with the energy of a rest mass. Considering
the examinations that the self-energies derived in terms of
the step-function-type basis functions and that by the
continuum theory with the same cut-off length are not so
different, the classical self-energies are reduced to the
logarithmic forms. However, the self-energy of the Higgs boson has
the larger quadratic form. The derived self-energy is
subtracted and involved in the repulsive vacuum energy with
the constant Λ. Under a classical gravitational field, whose
strength is small for scales larger than the Planck scale, we
consider the contribution from the self-energy of a Higgs
boson to vacuum energy. The cut-off length is then
determined to reproduce the observed vacuum energy constant Λ.
This theoretical vacuum energy constant Λ has today’s tiny
value.
This paper is organized as follows: Section 2 presents
the formalism and analysis procedure. We exhibit a
formalism for the subtraction of the self-energy by including the
energy into vacuum energy constant Λ (cosmological
constant). Subsequently, the field theory in terms of the
stepfunction-type basis functions is described to derive finite
self-energies. Section 3 examines the self-energy in
classical electrodynamics and from the macroscopic classical
Einstein field equations. The self-energies are calculated
by the continuum theory and the field theory in terms of
the step-function-type basis functions. Section 4 describes
the relationship between the subtracted self-energies and
the vacuum energy constant Λ, and we derive the
cutoff length to reproduce vacuum energy with the constant
Λ, followed by Sect. 5, which summarizes the
conclusions.
2 Formalism for self-energies and the field theory in
terms of the step-function-type basis functions
(
1
)
2.1 Formalism for the subtraction of the self-energy by
involving the energy into vacuum energy constant Λ
−c3
Lg = 16π G R,
R = gμν Rμν ,
Rμν = Rμρρν ,
with Rμν being defined by
where G is the gravitational constant and R is the scalar
curvature written by
In this subsection, we present the formalism for the
inclusion of subtracted self-energies produced by interactions (in
such as electrodynamics) into vacuum energy constant Λ.
Throughout this paper, the notation x 0 = ct (c is the velocity
of light) is the time coordinate, and the xi are space
coordinates, where x 1 = x , x 2 = y and x 3 = z. The infinitesimal
squared distance (according to the notation by Bjorken and
Drell [33]) is denoted
(ds)2 = gμν dx μdx ν ,
where gμν is the metric tensor and the indices run over 0, 1,
2 and 3. We use the summation conventions such as
gμν dx ν = gμ0dx 0 + gμ1dx 1 + gμ2dx 2 + gμ3dx 3,
for Greek indices and
gμi dxi = gμ1dx 1 + gμ2dx 2 + gμ3dx 3,
for Latin indices. The metric tensor of gμν in a flat Minkowski
space is given by
⎡ 1
0
gμν = ⎢⎢ 0
⎣
0
Lg√−gdV 4,
where g is the determinant of gμν , g = det(gμν ), and dV 4 =
dx 0dx 1dx 2dx 3. For the gravity,
(
2
)
(
3
)
(
4
)
(
5
)
(
6
)
(
7
)
(
8
)
(
9
)
using the Riemann curvature tensor Rμρρν . The tensor Rμρν
ρ
is expressed in terms of the Christoffel symbol Γμλν as
ρ
∂Γνσ
Rμρρν = ∂ x μ −
ρ
∂Γμσ
∂ x ν + ΓμρλΓνλσ − ΓνρλΓμλσ ,
(
10
)
where
1 gλρ
Γμλν = 2
∂gρμ ∂gρν ∂gμν
∂ x ν + ∂ x μ + ∂ x ρ
.
Meanwhile, the action functional of the matter is denoted
1
Sm = c
Lm
√−gdV 4,
where Lm is the Lagrangian density of the matter, and the
energy-momentum tensor of the matter is obtained from the
relation
1 √
2
−gTμν = −
∂ xρ
∂√−gLm
∂ ∂∂gxμρν
−
∂√−gLm
∂gμν
.
(
13
)
The variational calculus with respect to δgμν of the total
action functional,
(
11
)
(
12
)
(
14
)
(
15
)
(
16
)
(
17
)
(
18
)
δSg + δSm
−c3
= 16π G
1
Rμν − 2 gμν R −
8π G
c4 Tμν δgμν √−gdV 4,
yields the Einstein field equations,
1
Rμν − 2 gμν R =
The renormalization of the mass by interactions in such as
electrodynamics subtracts self-energies from the rest mass.
Because the energy of the rest mass produces curvature
(gravity), the subtracted energies are included in vacuum energy
with the constant Λ. In the above equation, we then add the
following tensor for the removal of self-energies produced
by interactions (such as in electrodynamics):
c4
Tμ(Sν) = 8π G gμν Λ(S),
where Λ(S) is regarded as the vacuum energy constant Λ
(cosmological constant). The Einstein field equations given
by Eq. (
15
) are rewritten as follows:
1
Rμν − 2 gμν R =
which corresponds to the Einstein field equations with
vacuum energy constant Λ. Consequently, subtracted
selfenergies in the interactions are involved in a vacuum energy
with constant Λ.
2.2 Field theory in terms of the step-function-type basis
functions
In describing physical quantities at short distances, theories
are required to remove ultraviolet divergences. We
formulated the field theory [9–12], which is expressed in terms of
the step-function-type basis functions to realize the removal
of the ultraviolet divergences. In this subsection, the
formalism is described so as to express the fields in terms of the
step-function-type basis functions in the form used by this
paper. Our described formalism divides four-dimensional
real space-time into hyper-octahedrons with arbitrary shapes
of the boundaries. The hyper-octahedron in real space-time
is mapped from a hypercube with flat boundary surfaces in
a parameter space-time. A basis function defined around a
center of a hypercube takes a value of unity and vanishes
outside the hypercube.
In this paper, the cubic region in three-dimensional space
is approximated by the spherical region for simplicity and
convenience. We calculate fields in spherical coordinates
and divide the spherical symmetric region into shells. The
results can be generalized to the case in which the region is
divided into many hyper-octahedrons with arbitrary shapes.
Grid (lattice) points along the radial r -axis (r = (x 2 +
y2 + z2)1/2) are denoted r1, r2, . . . , rk , . . . , rNr +1, with
k = 1, 2, c, . . . , Nr + 1, where Nr is the number of
lattice points and k = Nr + 1 is the lattice index for a boundary.
We here set the radial cut-off length Δh (corresponding to
the cut-off length Δ with Δ = 2Δh) to the lattice spacing by
Δh = rk −rk−1 and define the notations rk−1/2 = rk − Δh/2
and rk+1/2 = rk + Δh/2. The step-function-type basis
function used is defined by
where δ(r ) is the Dirac delta function.
The field φ (r ) in spherical coordinates is transformed to
(
19
)
(
20
)
(
21
)
(
22
)
(
23
)
u(r ) = r φ (r ),
u(r ) =
uk Ω˜ kE (r ).
k
and this wave function u(r ) is expressed in terms of basis
functions defined by Eq. (
19
):
Thus, we have prepared the formalism to the analysis of the
self-energies one finds in the next section.
3 Analysis of self-energies from interactions by classical
fields
3.1 Self-energy and mass renormalization in classical
electrodynamics by the continuum theory
This subsection examines and summarizes the self-energy
in classical electrodynamic interactions using the continuum
theory [15,34–37]. The mass density μm of a fundamental
particle with a mass mE and size RE is denoted
μm = (4π/m3)E(RE)3 .
We divide three-dimensional (3D) space into identical cubic
elements, which were considered in Sect. 2.2. The cubic
region is approximated by a spherical region with radius R0.
The charge Q and mass M of the spherical region occupied
by the fundamental particle are expressed by
M = (4π/3)(R0)3μm ,
Q = (4π/3)(R0)3ρ(e),
(
25
)
respectively, where ρ(e) is the charge density. A radial
cutoff length Δh = Δ/2 = R0 in spherical coordinates,
corresponding to the cut-off length Δ, is introduced for simplicity
and convenience. From the conventional energy-momentum
tensor of electrodynamics, the self-energy of the static
electric field has the form
E C(e) = dV 3 21 |E|2, (
26
)
where dV 3 = dx dydz. The classical electric field E is
produced as div(E) = ρ(e) from the electric charge density ρ(e)
and is written by E = −∇φ(e), where φ(e) is the electric
potential and satisfies
− ∇2φ(e) = ρ(e).
The above self-energy,
E (e)
C = −
dV 3
E · ∇φ(e),
1
2
We consider the case, in which the charges exist in the
region r ≤ R0 and ρ(e) = 0 for r > R0 using Q in Eq.
(
25
). Gauss’ theorem for Eq. (
27
) then gives the following
potential:
Q
φ(e)(r ) = 4πr
4πr 2
for r > R0.
Similarly, for r ≤ R0, we have
(
35
)
(
37
)
(
38
)
(
24
)
(
27
)
(
28
)
(
29
)
(
30
)
(
31
)
yielding
φ(e)(r )
−
dr
followed by
φ(e)(r ) =
dr
−
φ(e)(r )
dr
= fe − 34c2 E C(e) ddvtC ,
dvC
dt
= fe.
(The relativistic version was given by Dirac and Rohrlich,
where the factor 1 appears corresponding to the above factor
4/3 [34–37].) Due to the requirement from the continuum
relativistic theory, the fundamental particle is considered to
be pointlike. Then the above self-energy diverges, which is
why mass renormalization is required in electrodynamics. In
mass renormalization, the self-energy is subtracted from the
term with the mass.
To connect φ(e)(r ) for r ≤ R0 in Eq. (
33
) continuously with
that in Eq. (
30
) for r > R0 at r = R0, we shift φ(e)(r ) in Eq.
(
33
) to
φ(e)(r ) = 16 r 2ρ(e) − 61 R02ρ(e)
Q
+ 4π R0
Using Eqs. (
29
), (
34
) and Q in Eq. (
25
), we obtain the
selfenergy by the classical electric interaction in the continuum
theory:
Under an external force fe, the classical Newtonian
equation of motion for the above charged object, with a small
velocity vC compared to the speed of light c, is expressed by
(small magnetic contributions are dropped) [34–37]
M ddvtC = fe + dV 3(ρ(e)E), (
36
)
where M is the mass of the charged object in Eq. (
24
). Using
the self-energy in Eq. (
29
), the lower-order terms expanded
with respect to 1/c amounts to
3.2 Self-energy derivation for classical electrodynamics
using the field theory in terms of the step-function-type
basis functions
In contrast to the analysis of Sect. 3.1, this subsection studies
the self-energy of the same object in Sect. 3.1 in classical
electrodynamic equations, using the step-function-type basis
functions. As mentioned in Sect. 2.2 and by Eqs. (
24
)–(
25
),
we divide three-dimensional space into cubic elements with
the cut-off length Δ, and each cubic region is approximated
by a sphere. The action functional for the electric field φ(e)(r )
can be written in the form [considering the form −∇2φ(e) −
ρ(e) = 0 on the left in Eq. (
27
)]
S(e)
f
As in Sect. 2.2 and by Eqs. (
24
) and (
25
), we divide
threedimensional space, containing the above sphere with radius
R0 centered at the origin in spherical coordinates, into shells
(the number of cells enclosing the central sphere is Nr − 1).
The radial width (lattice spacing implying the radial cut-off
length) of each shell is Δh, which is equal to the radius R0
of the enclosed central sphere. As Eq. (
23
), the above wave
function u(e)(r ) is expressed by
u(e)(r ) =
u(ke)Ω˜ kE (r ),
in terms of the step-function-type basis functions Ω˜ kE (r )
in Eq. (
19
). From Eq. (
41
), it follows that (k, K =
1, 2, . . . , Nr+1)
where δk,K is the Kronecker delta. By similar calculations
for the elements of Sf(e) given by Eqs. (46)–(48), the total
Sf(e) in Eq. (44) amounts to
S(e)
f
=
4π 1
2 Δh k,K
−u(ke)u(e)
K −1δk,K −1
+ 2u(ke)u(e) K +1δk,K +1 .
K δk,K − u(ke)u(e)
On the other hand, the action functional of the electric
charge of the matter for spherical coordinates is expressed
using u(e)(r ) in Eq. (
40
) by [considering also the form
−∇2φ(e) − ρ(e) = 0 on the left in Eq. (
27
)]
S(e)
f
With the help of the lattice spacing Δh mentioned above
Eq. (
19
), an element such as Sf(e)−− in Eq. (45) is reduced to
Sf(e)−− = 4π 21
k,K
u(ke)u(Ke)[δ(rk−1/2 − rK −1/2)]
1
= 4π 2 k,K ΔΔhh u(ke)u(Ke)[δ(rk−1/2 − rK −1/2)]
Using Eqs. (
20
) and (
21
), Sf(e) above is decomposed into
Sf(e) = Sf(e)−− + Sf(e)−+ + Sf(e)+− + Sf(e)++,
(44)
(45)
(46)
(47)
(49)
(50)
(51)
Subsequently, by the expansion of u(e)(r ) given by Eq. (
40
)
in terms of the basis functions in Eq. (
19
), the above action
becomes
Because R0 = Δh as mentioned below Eq. (
24
),
which implies ρ(e) = 0 for k > 1 (the index 1 is one) in Eq.
(52), and rk−1/2 = 0 (or rk−1/2 = with → 0 after the
calculation). Using Q in Eq. (
25
) and R0 = Δh, we have
S(e)
m = −ρ(e)
= −
k
u(e) 3 4π (Δh)3
k 2 3 Δh
δk,1
k
u(e) 3 1
k 2 Δh Qδk,1,
where δk,1 is the Kronecker delta (the index 1 is one).
From Eqs. (50) and (54), the variation with respect to u(ke),
+[2(β − 1)Q + Q − [(β − 1)Q + Q ]
= 3(β − 1)Q + Q .
δ Sf(e) + δ Sm(e) = 0,
leads to
1
− Δh
Q
with Q = 4π .
u(ke−)1 − 2u(ke) + u(ke+)1 = 23 ΔQh δk,1
This equation is equivalent to
u(ke−)1 − 2u(ke) + u(ke+)1
(Δh)2
3 Q
= − 2 (Δh)2 δk,1,
corresponding to Eq. (
27
) for φ(e)(r ) = r u(e)(r ).
We then have
u(e)
k−1 − 2u(ke) + u(ke+)1 = 0
which is rewritten by
u(e)
k−1 − u(ke) = u(ke) − u(ke+)1
for k > 1,
Additionally, for the boundary rk−1/2 with k = 1 (the index
is 1)
r1−1/2 =
> 0
(we set → 0 after the calculation), the basis function is not
given in the region for r < 0. Considering this boundary for
Eq. (57), we obtain
−2u(ke) + u(ke+)1
(Δh)2
3 Q
= − 2 (Δh)2
In contrast, using Q in Eq. (
25
) and R0 = Δh for the charge,
the outer boundary condition imposed is
u(e)
N +1 = Q ,
which implies φ(e)(rN +1) = Q /rN +1 in Eq. (
40
). Then Eq.
(59) becomes
u(e) N − Q .
N −1 − u(Ne) = u(e)
We consider a solution that takes
u(e)
N = (β − 1)Q + Q
for k = N ,
where β is a constant to be determined below. Equations (59)
and (62)–(64) lead to
u(e)
N −1 = [(β − 1)Q + Q ]
+[(β − 1)Q + Q − Q
= 2(β − 1)Q + Q ,
u(e)
N −2 = [2(β − 1)Q + Q ]
Using Eqs. (58), the sequential manipulations result in
u(e)
k = [(N − k + 1)(β − 1)]Q + Q
for k > 1. (67)
Because the above solution diverges unless β = 1 for k = 2,
we derive the following solution, by setting β = 1 and using
Q = Q/(4π ) in Eq. (56):
u(e) Q
k = Q = 4π
Furthermore, from Eqs. (61) and (68) as well as Q =
Q/(4π ) in Eq. (56), we have the solution (at the remaining
point) for k = 1:
3.3 Self-energy in macroscopic classical Einstein field
equations
This subsection presents the analysis of the curvature
selfenergy in the Einstein field equations. Although the
gravitational field is different from the charged particle fields, we
treat the Newtonian approximation case, which is similar to
the charged particle case. When the renormalization is
difficult in this case, it is possible to use the cut-off length. The
self-energy is first evaluated by the continuum field theory.
Subsequently, the self-energy is evaluated using the
formalism in terms of the step-function-type basis functions. As
described by Landau and Lifshitz [15] (owing to the
negligible contributions of higher-order terms with respect to 1/c in
the Lagrangian with c being the velocity of light), the
Newtonian approximation, within the scheme of the Einstein field
equations for the matter with the slow velocities compared
to c, has the infinitesimal squared distance expressed by
(ds)2 = gμν dx μdx ν = (ημν + hμν )dx μdx ν
and φ is the Newtonian potential. We note that the Newtonian
potential (field) φ is distinguished from electric field φ(e).
Letting μm be the density of the mass, we have the
energymomentum tensor
T ν
μ =
μmc2
0
for μ = 0 and ν = 0 .
for μ = 0 or ν = 0
It is well known that the field equation Eq. (
15
) can be
rewritten
Rμν =
where δμν is the unit tensor and
T = gμν Tμν .
Furthermore, using the well-known relations for Eq. (74)
R00 = c12 ∂∂x2φi2 ,
and from Eqs. (72)–(75), we obtain the Newtonian equation
(78)
(79)
From Eq. (72), the term δg00√−g in Eq. (
14
) is
approximated by (higher-order terms with respect to 1/c in √−g
are neglected)
We then approximate the action functional for directly
leading to the Newtonian equation as follows. Because the action
functional for the matter is linear with respect to φ, this action
is approximated by
S(N)
m
−c3
= − 16π G
Meanwhile, we approximate the following action functional
of the gravity, which is consistent with the above equation
(the factor 1/2 appears considering the variational of both
∂2φ/∂ xi2 and φ with respect to φ), with integration by parts:
By variation with respect to φ, the above action functionals
Sg(N) and Sm(N) lead to the Newtonian equation given by Eq.
(78).
In the Newtonian approximation within the Einstein
scheme, the energy-momentum tensor has a similar form to
that in Eq. (
29
) for the static electric field. Using the notation
∇ = (∂ x 1, ∂ x 2, ∂ x 3), the static energy is written
From Eqs. (90) and (92), the variational calculus with
respect to uk
1
= 2 0
1
= − 2
R0
In contrast to the above analysis, we next study the
self-energy in the Einstein field equations, using the
stepfunction-type basis functions. We also use the above 3D
sphere with the radial cut-off length R0 = Δ/2 related to
the cut-off length Δ. The mass density μm in Eq. (
24
) of the
fundamental particle leads to the mass M in Eq. (
25
). As in
Sect. 2.2, we divide three-dimensional space, containing the
above sphere (with the radius R0 = Δh) centered at the
origin in spherical coordinates, into shells (the number of cells
enclosing the central sphere is Nr − 1). The radial width of
each shell is Δh, which is equal to the radius of the sphere
R0.
To use the basis functions in Sect. 2.2, the action functional
for the gravity in Eq. (81) with the factor
(Δx 0 is the time interval and can be dropped for the
present static case) is rewritten (considering the form ∇2φ −
4π Gμm = 0 of Eq. (78))
The above wave function u(r ) is then expressed in terms of
the step-function-type basis functions in Eq. (
19
). From Eqs.
(
23
) and (88), we have (k, K = 1, 2, . . . , Nr+1)
Meanwhile, using u(r ) in Eq. (
22
), the action functional
of the matter in Eq. (80) for spherical coordinates becomes
(91)
(92)
(93)
(94)
(95)
(96)
(98)
[considering the form ∇2φ − 4π Gμm = 0 of Eq. (78)]
Sm(N) = −4π Gγg
dr (4π )r 2μmφ (r )
= −4π Gγg
dr (4π )r μmu(r ).
By the expression of u(r ) given by Eq. (
23
) in terms of basis
functions denoted in Eq. (
19
), the above action is written by
Sm(N) = −4π Gγg
dr (4π )r μm
uk Ω˜ k (r )
(88)
Consequently, from Eqs. (
22
), (82) and (97) with r1 =
Δh/2 = (Δ/2)/2, we derive the following classical
curvature self-energy in the region with the radial cut-off length
Δh = R0 (related to the cut-off length Δ) and the mass
M = (4π/3)(Δh)3(μm) [in Eqs. (
24
)–(
25
)]:
E (N)
Ω
= 21 43π (Δh)3μm
(−5) G M
4
4 Relationship between the subtracted self-energy and
vacuum energy constant Λ
The continuum relativistic theory requires that a fundamental
particle be considered pointlike, and the radius of a pointlike
particle leads to ultraviolet divergences. However, our
formalism can obtain finite self-energies by expressing fields
in terms of the step-function-type basis functions. As in
Sect. 2.1, the self-energy subtracted from the energy of the
rest mass is included in vacuum energy expressed in terms
of the constant Λ (cosmological constant). The self-energy
calculated using the step-function-type basis function with
the cut-off length Δ is not so different from that calculated
by the continuum theory with the same cut-off length as was
shown in Sect. 3. For a fundamental particle, the self-energy
caused by the classical electrodynamics was proportional to
1/Δ. This self-energy is reduced to the following
logarithmic form by quantum corrections [33,38] (h¯=h/(2π ) with h
being the Planck constant):
E (e) 3
Q = 4π
e2
4π h¯c
mEc2 ln
(h¯c/Δ)2
(mEc2)2
(Δ is the cut-off length and mE is the rest mass of an
electrodynamically interacting fundamental particle.) This
reduction of the Coulomb-type self-energy also occurs in
chromodynamics with the asymptotic freedom at short distances
[39–41]. However, a stronger divergence of the self-energy
at a quantum level appears in the Higgs boson case.
(Fundamental particles (quarks) receive a mass through the coupling
to the Higgs field.) The gravitational field is treated at a
classical level, because the cut-off length below in this section is
longer than the Planck scale and the gravitational strength is
small. Under such a small gravitational field, our treatment
in this section relates vacuum energy to the Higgs boson
self-energy, which is dominant among other interactions at
a quantum level. This relation (between vacuum energy and
the Higgs self-energy) determines the cut-off length,
reproducing today’s tiny value of the cosmological constant Λ as
described below.
It is well known that the matter is mainly composed of
protons. The averaged energy of the rest mass of the
fundamental particles is mEc2 ≈ 3.23 [MeV]. Considering that
the contribution from the mass of the fundamental particles
to that of a proton is very small, we set
energy of proton
γE = energy of fundamental particles
energy density of matter
= energy density of fundamental particle ≈ 96.7. (100)
To derive the cut-off length, the ratio γΛ is defined by
γΛ =
energy density of vacuum energy with Λ
energy density of matter
self-energy density 0.73
.
= energy density of matter ≈ 0.04
(101)
The fundamental particle (quark) mass mE is due to the
coupling to the Higgs field with the coupling constant λ f
written by
√2
λ f = v mE, (102)
where v is the vacuum expectation value of the
symmetrybroken Higgs field. The Higgs self-energy EH (included in
vacuum energy) for the cut-off length Δ and λ f above is
written by
√2 hc 2
|EH| = 4π λ f ¯Δ c
√2 √2 mEc2 h¯c 1 1 mEc2 h¯c . (103)
= 4π v Δ = 2π v Δ
Meanwhile, from Eqs. (100) and (101) it follows that
|EH|
mEc2 = γE γΛ.
1 1 1 h¯c .
Δ = γE γΛ 2π v
Combining Eqs. (103) and (104), we have
Because v ≈ 246 [GeV], we derive the cut-off length
Δ ≈ 7.2 × 10−8 [fm], which corresponds to ≈ 2.7 × 106
[GeV], that is, Δh ≈ 3.6 × 10−8 [fm] corresponding to
≈ 5.5 × 106 [GeV].
Even if modifications of values or definitions are required
for the above calculations, we obtain a similar value of
Δ. Owing to the above cut-off length Δ derived, the field
theory may be advanced without ultraviolet divergences.
When the renormalization is difficult, the analysis is
possible by using the cut-off length. In general, gμν is written
gμν (x ) = ημν (x ) + hμν (x ) as given in Eq. (71), and the
tensor field hμν (x ) is expressed by
hμν (x ) =
hμνpΩ˜ 4p(x ),
p
where the coefficients hμνp are tensor elements, and Ω˜ 4p(x )
is the four-dimensional basis function, which takes the
value of unity in a hyper-octahedron with the index p
in four-dimensional space-time and vanishes out of the
hyper-octahedron. Namely, the present theory divides the
space-time continuum of classical general relativity into
pieces (hyper-octahedrons). The expression of wave
functions in terms of step-function-type basis functions restricts
the degrees of freedom of the wave functions in a
hyperoctahedron (cuts off high-frequency contributions), meaning
the quantization of space-time in classical general relativity.
In Sect. 3, it was shown that the quantities calculated using
the step-function-type basis functions are similar to the
corresponding quantities calculated by using the continuum theory
with the cut-off. The formalism and calculated quantities in
the continuum theory are mapped to the corresponding
formalism and quantities using step-function-type basis
functions. Then, from Eqs. (103)–(105), the cut-off length used
for the step-function-type basis functions is related to Higgs
self-energy, which amounts to the vacuum energy expressed
by the constant Λ (of such as Λ-CDM).
In the Einstein field equations, the energy of the rest mass
is the source of the curvature, and the renormalization (by
(104)
(105)
(106)
such as the electrodynamic interaction) subtracts the
selfenergy from the rest mass. The subtracted self-energies can
be involved in vacuum energy constant Λ in the Einstein
field equations, as described in Sect. 2. (Concerning the
curvature self-energy by the gravitational coupling between the
mass and the produced field, it is well known that the general
curvature self-energy is not always within the
renormalization scheme.) The relatively large cut-off length (compared
to the Planck length) of the present theoretical formalism has
an advantage with naturalness that the Higgs self-energy is
suppressed, and this cut-off is related to today’s tiny
vacuum energy expressed by Λ, without fine tuning. In contrast,
by the relatively small cut-off at the Planck scale in other
models arises the huge Higgs self-energy, which needs the
following fine tuning. In a highly precise fine tuning, the huge
Higgs self-energy for the cut-off at the Planck scale is
canceled by another physical quantity to adjust the Higgs mass.
Furthermore, the Planck energy composing Λ
(cosmological constant) in other models needs fine tuning to obtain the
present tiny value of Λ by the cancellation from such as a
huge Higgs self-energy for the cut-off at the Planck scale.
Therefore, the present formalism provides an answer to the
fine-tuning problem. The present formalism also has the
possibility to offer a fundamental physical theory, predicting a
cut-off length that may play the role of a fundamental
physical constant, if experimentally observed. The present model
has another merit: that the initial universe has no possibility to
form a black hole because of the relatively large cut-off length
unlike the cut-off at the Planck scale near the black hole
size of the whole universe. Furthermore, the gravity in the
present formalism is weak compared to the other
fundamental interactions, and space-time coordinates do not largely
deviate from classical numbers. Moreover, the present
theoretical vacuum energy constant Λ decreases to today’s order
of magnitude expressing the vacuum energy density, telling
the ratio of vacuum energy density to the energy density of
the matter.
If the expansion of the universe is matter dominated (in the
present case the vacuum energy caused by the self-energy has
the same property of the matter), the vacuum energy constant
Λ seems to be proportional to tU−2 (at least) at present, where
tU is the age of the universe at each point in time. This is due
to the well-known fact that the solution of the Friedmann
equation indicates the scale of the universe a(tU) as a
function of tU to be a(tU) ∝ tU2/3, that is, [a(tU)]−3 ∝ tU−2. The
matter density ρm (we can include the dark matter and
vacuum energy into the matter) is written by ρm ∝ [a(tU)]−3,
which leads to ρm ∝ t −2 and Λ ∝ tU−2, because the vacuum
U
energy expressed by the vacuum energy constant Λ obeys the
relation Λ ∝ ρm in our scheme. This is the reason why the
vacuum energy constant Λ seems to be proportional to tU−2. In
contrast, at the Planck scale the wave packet size for the mass
of the Planck energy (≈ 1019 [GeV]), which seems to be the
The variation with respect to Φk ,
δ SB = 0,
yields
1
Δ (Φk−1 − 2Φk + Φk+1) = 0,
which leads to
1
Δ2 (Φk−1 − 2Φk + Φk+1) = 0,
where k, K = 1, 2, ..., Nx . For the above equation, the
following boundary conditions on the wave function are
imposed (Nx +2 is the number of lattice points, and the lattice
indices of the boundary points are denoted by k = 0, Nx +1):
Φ0 = 0,
ΦNx +1 = 0.
Similar to the classical vibrational case [42], the
eigenvector for a diagonalization of the action is expressed as
1
ΦK = CN sin
k K π
whole energy of the universe, is equal to the
gravitationalbased radius, and the conditions between the above two cases
are quite different. When the present model is generalized to
the early universe, the initial size of the universe is the
cutoff length of the present model. Because Λ ∝ t −2 mentioned
U
above diverges in the limit as tU → 0 (tU is larger than the
cut-off length), the early universe expands rapidly, although
the expansion rate is different from the exponential expansion
of inflation models.
Finally, we add that, owing to the nonperiodic boundary
condition, the zero-point energy for the candidate of vacuum
energy constant Λ is not seen in the present formalism. Let
us consider a simple action for the wave function Φ(x ),
1
SB = 2
dx
dΦ(x ) dΦ(x )
dx dx
.
The wave function in terms of the basis functions in Eq. (
19
)
with the lattice spacing Δ is given by
Φ(x ) =
Φk Ω˜ kE (x ).
k
Similar to Eq. (50), we write the action functional
1
SB = 2
k,K
1 1
= 2 Δ
k,K
dx
Φk ΦK
dΩ˜ kE (x ) dΩ˜ KE (x )
dx dx
(Φk ΦK )(−δk−1,K + 2δk,K − δk+1,K ). (109)
(107)
(108)
(110)
(111)
(112)
(113)
(114)
1 1
SB,k = 2 Δ
= 2Δ CN
where CN is a normalization constant. Then the element SB,k
of the action in Eq. (109) is diagonalized giving
(−δk−1,K + 2δk,K − δk+1,K )ΦK
×
− sin
− sin
k(k − 1)π
with k = 1, 2, · · ·, Nx . Consequently, the zero-point energy
for the candidate vacuum energy is not seen in the present
system because of the boundary condition in Eq. (113).
(Similarly, eigenvalues in higher dimensions are obtained [10, 12].)
We note that when the system is considered using a box
normalization, in which wave functions are defined in a
box with periodic boundary conditions at the box surfaces,
eigenvalues may have zero-point energies. However,
physical quantities such as the transition amplitude are calculated
without using the zero-point energies by expressing plane
waves in the form of complex exponential functions. The
zero-point energies dropped in this case may not be included
in vacuum energy, because the zero-point energies appearing
here are due to the non-vanishing periodicity (which seems
to lack in the real expanding universe) in approximate
calculational manipulations.
As the dark matter, we considered the classical solution
with quantum field fluctuations in chromodynamics in Ref.
[12]. Although the Big Bang is out of the scope of this paper,
an expansion may arise making the Big Bang like the
vaporization of water in vacuum by absorbing heat.
5 Conclusions
For the renormalization of the mass, we have considered the
subtracted self-energies, which act as sources of the curvature
in the Einstein field equations. It was shown that this
consistency is satisfied by including these self-energies into the
vacuum energy expressed by the constant Λ. The self-energies
in electrodynamics and Einstein field equations were
investigated by using the ultraviolet cut-off length. The field theory,
which was developed by the present authors, expresses wave
functions in terms of the step-function-type basis functions to
cut off oscillations at short distances. In the other continuum
theory, we used the same cut-off length as that used for the
former theory. From the examination, the continuum theory
with the cut-off length is effective. Classical self-energies are
reduced to logarithmic forms by quantum corrections, and the
quadratic Higgs self-energy is dominant at a quantum level.
The cut-off length was determined so as to reproduce the
observed vacuum energy constant Λ, using the self-energy
derived from the above cut-off theories. The derived vacuum
energy expressed by the constant Λ is of the order of the
matter (composed of the conventional matter such as atoms
and dark matter), showing that the vacuum energy constant
Λ has today’s tiny value.
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