#### Pauli–Zeldovich cancellation of the vacuum energy divergences, auxiliary fields and supersymmetry

Eur. Phys. J. C
Pauli-Zeldovich cancellation of the vacuum energy divergences, auxiliary fields and supersymmetry
Alexander Yu. Kamenshchik 1 2
Alexei A. Starobinsky 0 1
Alessandro Tronconi 2
Tereza Vardanyan 2
Giovanni Venturi 2
0 National Research University Higher School of Economics , Moscow 101000 , Russia
1 L. D. Landau Institute for Theoretical Physics , Moscow 119334 , Russia
2 Dipartimento di Fisica e Astronomia, Università di Bologna and INFN , via Irnerio 46, 40126 Bologna , Italy
We have considered the Pauli-Zeldovich mechanism for the cancellation of the ultraviolet divergences in vacuum energy. This mechanism arises because bosons and fermions give contributions of the opposite signs. In contrast with the preceding papers devoted to this topic wherein mainly free fields were studied, here we have taken their interactions into account to the lowest order of perturbation theory. We have constructed some simple toy models having particles with spin 0 and spin 1/2, where masses of the particles are equal while the interactions can be quite non-trivial.
1 Introduction
Many years ago Pauli [1] suggested that the vacuum
(zeropoint) energies of all existing fermions and bosons
compensate each other. This possibility is based on the fact that
vacuum energy of fermions has a negative sign whereas that of
bosons has a positive one. As is well known, such a
cancellation indeed takes place in supersymmetric models (see
e.g. [2]). Subsequently in a series of papers Zeldovich [3,4]
related vacuum energy to the cosmological constant.
However rather than eliminating divergences through the
bosonfermion cancellation, he suggested the Pauli-Villars
regularization of all divergences by introducing a number of massive
regulator fields. Covariant regularization of all contributions
then leads to finite values for both the energy density ε and
(negative) pressure p corresponding to a cosmological
constant, i.e. connected by the equation of state p = −ε.
In our preceding paper [5] we examined the conditions for
the cancellation of the ultraviolet divergences of the vacuum
energy to the leading order in h¯, i.e. by considering free
theories and neglecting interactions. Such conditions are reduced
to some sum rules involving the masses of particles present
in the model. We formulated these conditions not only for the
Minkowski spacetime, but also for the de Sitter one. In the
latter case, the radius of the de Sitter universe also enters into
the mass sum rules. In paper [6] we applied such
considerations to observed particles of the Standard Model (SM) and
also studied the finite part of vacuum energy. This last
contribution should be very small, so as to obtain a result
compatible with the observed value of the cosmological constant
(almost zero with respect to SM particle masses). We showed
[6] that it was impossible to construct a minimal extension
of the SM by finding a set of boson fields which, besides
canceling ultraviolet divergences, could compensate
residual huge contribution of known fermion and boson fields of
the Standard Model to the finite part of the vacuum energy
density.
On the other hand, we found that addition of at least one
massive fermion field was sufficient for the existence of a
suitable set of boson fields which would permit such
cancellations and obtained their allowed mass intervals. On
examining one of the simplest SM extensions satisfying the
constraints, we found that the mass range of the lightest massive
boson was compatible with the Higgs mass bounds which
were known at the time of the publication of the paper [6].
As is well known, later the Higgs boson was discovered at
the LHC [7,8]. For some time it appeared that there might
exist an the observed diphoton excess at 750 GeV [9]. This
excess, had it been confirmed, could be interpreted as an
indication for the existence of a new heavy elementary or
composite particle with a mass of the order of 750 GeV. Later
this phenomenon disappeared, nonetheless inspiring in the
meanwhile quite a few theoretical works. In particular, we
also studied in our preprint how the presence of such a new
particle could be included into our scheme of the cancellation
of ultraviolet divergences of vacuum energy [10].
In our preceding papers [5,6,10] we studied only free
theories without interactions. It is also interesting to take
interactions into account, at least to the lowest order of
perturbation theory. This is not easy, and in the present paper we shall
concentrate on the construction of relatively simple toy
models where the “Pauli–Zeldovich cancellation” of ultraviolet
divergences still takes place.
We wish to emphasize that the approach employed in the
present paper represents a whole direction in quantum field
theory which goes well beyond effective low energy field
theory and, although based on some hypothesis, has not been
proven to be wrong. We also wish to mention the paper by
Ossola and Sirlin [11], where contributions of fundamental
particles to the vacuum energy density were discussed with
a special attention to relations between different
regularization schemes and to the appearance of power divergences in
different contexts. Other related approaches are presented in
Refs. [12–14].
In the recent paper [15], it was noticed that under certain
circumstances (in particular, but not limited to finite QFTs),
the Pauli cancellation mechanism would survive the
introduction of particle interactions. It was pointed out there that
for the mass sum rules to be valid at different mass scales, it
is necessary to impose some relations on mass runnings with
energy. Thus, the corresponding relations between
anomalous mass dimensions were formulated [15]. However,
concrete examples were not constructed.
In the present paper we discuss some relatively simple
examples of models where the Pauli–Zeldovich
cancellation takes place to the first order of perturbation theory.
Being inspired by the famous supersymmetric Wess–Zumino
model [16], we consider models with spinor, scalar and
pseudoscalar fields only. We hope to treat vector (gauge) fields in
future works. The models which we discuss are not
supersymmetric, but they have one important feature which makes
them akin to supersymmetric models: the number of the
fermion and boson degrees of freedom in them is the same.
That implies an unexpected feature: the necessity to take so
called auxiliary fields into account. Such fields are
necessary in the supersymmetric models, because they allow one
to formulate supersymmetry transformations in a coherent
way.
But their role is even more ubiquitous. To conserve
supersymmetry, it is necessary to have the balance between
fermion and boson degrees of freedom not only on shell, but
also off shell. However, the number of degrees of freedom
of a spinor field doubles when it is off shell. For example,
a Majorana spinor has two complex components, i.e. four
degrees of freedom off shell. When we require the
satisfaction of the first-order Dirac equation, the number of degrees
of freedom becomes equal to two. Thus, for example, in the
Wess–Zumino [16] model one has two fermion degrees of
freedom of the Majorana spinor and two boson degrees of
freedom associated with the scalar and pseudoscalar fields.
Off shell the number of fermion degrees of freedom becomes
equal to four, while the role of two additional boson fields
is played by two auxiliary fields which become in a sense
independent off shell. If we consider non-supersymmetric
models with the Pauli–Zeldovich mechanism of cancellation
of ultraviolet divergences for vacuum energy in the presence
of interactions, then the number of the boson and fermion
degrees of freedom should be equal not only on shell, but
also off shell. This means that we should introduce auxiliary
fields. Further, when we consider a model with interactions,
we should not only take into account running of masses of
the fields, but also consider cancellations of contributions
coming from the potential terms in the Lagrangians. It is
there that the introduction of the auxiliary fields becomes
very convenient. Fortunately, we shall see that, at least in the
considered class of spinor-scalar models, the introduction of
auxiliary fields is equivalent to a simple rule for the
calculation of some contribution to the scalar fields self-interaction.
Here we can add that, in principle, one can perform all
calculations and show that in the formalism where auxiliary fields
are excluded, vacuum energy in the supersymmetric models
is equal to zero. However, in this case there are no
separate cancellations of the potential energy and of the kinetic
energy between bosons and fermions. Thus, verification of
the analogous cancellation in non-supersymmetric models
becomes more complicated. Hence, it is better to implement
rather simple rules, equivalent to the explicit introduction of
auxiliary fields, which will be used in the present paper.
Here we present a model consisting of a Majorana fermion
and two scalar fields with the same mass and with different
kinds of interactions, and we show that for such a model,
one can find a family of coupling constants such that the
Pauli–Zeldovich mechanism for the cancellation still works.
Then we find an analogous family of models with a Majorana
fermion, a scalar field and a pseudoscalar field. Obviously,
the Wess–Zumino model belongs to this family. We also
discuss briefly models where particles with different masses are
present.
The structure of the paper is as follows: in the second
section we briefly discuss the mass sum rules for theories
without interactions; in the third section we formulate rules
for the conservation of the mass sum rules when interactions
are switched on. In Sect. 4 we discuss the vacuum expectation
values of the potential terms and the role of auxiliary fields. In
Sect. 5 we present a model with one Majorana field and two
scalar fields. In the sixth section we consider a model with one
Majorana field, one scalar field and one pseudoscalar field.
Section 7 is devoted to the discussion of models with
nondegenerate masses, the last section contains some concluding
remarks.
2 Vacuum energy and the balance between the fermion
and boson fields
One knows that vacuum energy of the harmonic oscillator
is equal to h¯2ω . If one has a massive field with mass m, then
ω = √k2c2 + m2c4, where k is the wave number. In the
following we shall set h¯ = 1 and c = 1. The energy density
of vacuum energy of a scalar field treated as free oscillators
with all possible momenta is given by the divergent integral
[3]:
1
ε = 2
= 2π
d3k k2 + m2
dkk2 k2 + m2.
We can regularize this integral by introducing a cutoff Λ. In
this case
ε = 2π
dkk2 k2 + m2
= 2π m4
⎡ Λ
⎣ 8m
2Λ2 + 1
m2
Λ2
m2 + 1
1 ⎛ Λ
− 8 ln ⎝ m +
⎞ ⎤
Λ2
m2 + 1⎠ ⎦ .
0
0
∞
Λ
On expanding this expression with respect to the small
parameter Λm , one obtains
ε = π2 Λ4 + π2 Λ2m2 + 1π6 m4(1 − 4 ln 2) − π4 m4 ln Λm
m
+o Λ . (
3
)
The contribution of one fermion degree of freedom coincides
with that of Eq. (
1
) with the opposite sign. It now follows
from Eq. (
3
) that to cancel the quartic ultraviolet divergences
proportional to Λ4, one has to have equal numbers of boson
and fermion degrees of freedom:
NB = NF .
The conditions for the cancellation of quadratic and
logarithmic divergences are
and
m2S + 3
m4S + 3
m2V = 2
m4V = 2
m2F
m4F ,
respectively. Here the subscripts S, V and F denote scalar,
massive vector and massive spinor Majorana fields
respec−2
m4F ln m F .
tively (for Dirac fields it is sufficient to put a 4 instead of 2
on the right-hand sides of Eqs. (
5
) and (
6
)). For the case in
which the conditions (
4
), (
5
) and (
6
) are satisfied, the
remaining finite part of the vacuum energy density is equal to
εfinite =
m4S ln ms + 3
m4V ln mV
Let us now calculate the vacuum pressure. This pressure is
given by the formula [3]
(
7
)
(
8
)
(
9
)
(
10
)
(
11
)
(
1
)
(
2
)
(
4
)
(
5
)
(
6
)
On expanding this expression with respect to the small
parameter Λm , we obtain
p = π6 Λ4 − π6 Λ2m2
7π
− 48
+ π4 m4 ln Λm + o Λm
m4
.
π
+ 4 ln 2
On then comparing the expressions (
3
) and (
10
), we see that
the quartic divergence satisfies the equation of state for
radiation p = 31 ε, the quadratic divergence satisfies the equation
of state p = − 13 ε, which sometimes is identified with the
so called string gas (see e.g. [17,18]), while the logarithmic
divergence behaves as a cosmological constant with p = −ε.
If all these divergences cancel, then the finite part of the
pressure is
pfinite = −
−2
ms4 ln ms + 3
m4F ln m F ,
m4V ln mV
which also behaves as a cosmological constant. We can
emphasize that the real (renormalized) vacuum energy does
not have radiation-like (or, p = −ρ/3) equation of state.
Instead, following Pauli and Zeldovich we investigate
conditions under which it (and the whole average value of the
vacuum energy-momentum tensor) can be finite without any
renormalization in the absence of an exact supersymmetry
(which is not realized in nature). If it is finite indeed, then
any method of calculation – covariant or non-covariant – will
show it.
Let us first calculate the term Σ2:
dd k 1
(2π )d (( p − k)2 − m2)(k2 − M 2) .
On making a Wick rotation, we obtain the integral on the
Euclidean momenta:
dd kE
(2π )d (( p − k)2E + m2)(k2E + M 2)
1
,
3 Running masses and anomalous mass dimensions
Here d is the dimensionality of the spacetime such that
In what follows we shall consider models having only
particles with spin zero and spin 1/2. If we include the
interactions, the masses begin their running and the conservation
of the relations (
5
) and (
6
) implies some new restrictions
on the masses and on the coupling constants. Namely, the
conservation of the relation (
5
) gives
γmS = 2
γmF,
where γm is the mass anomalous dimension defined as
γm ≡ μ
∂m2
∂μ
,
where as usual μ is the renormalization mass parameter. The
conservation of the relation (
6
) gives
m2SγmS = 2
m2F γmF.
These relations coincide with those presented in paper [15].
We shall here derive the expressions for these anomalous
mass dimensions. Generally the technique of such
calculations was developed many years ago [19–25]. However,
for convenience and completeness we shall perform all the
derivations from the start. On considering our toy models
with degenerate masses, we shall not really use them
explicitly. It will be enough to study shifts of masses induced by
radiative corrections for different fields present in the models
under consideration. However, when one considers models
where particles with different masses are present, the
formulas given in this section become necessary.
Our treatment of the anomalous mass dimensions in the
presence of quadratic divergences is based on the approach
presented in paper [26], which in turn uses the version of
renormalization group formalism connected with
dimensional regularization [27].
Let us consider the model, including a Dirac spinor with
a mass M and a scalar field with a mass m.
L = 21 ∂μφ∂μφ −
2 − 4!
+ i ψ¯ γ μ∂μψ − M ψ¯ ψ − gψ¯ ψ φ.
m2φ2
λφ4
The full propagator of the fermion field is given by
i
S( p) = pˆ − M − i Σ ,
where Aˆ ≡ γμ Aμ and where Σ is the self-energy operator
of the fermion field. This operator in the one-loop
approximation is given by the formula
(
12
)
(
13
)
(
14
)
(
15
)
(
16
)
(
17
)
Σ = g2
= pˆΣ1 + M Σ2.
dd k 1 kˆ + M
(2π )d ( p − k)2 − m2 k2 − M 2
(
18
)
(
19
)
(
21
)
(
22
)
(
23
)
(
24
)
(
25
)
(
26
)
(
27
)
(
28
)
(
29
)
To find Σ1, we shall take the 41 Tr( pˆΣ ). Then
g2
Σ1 = p2
Using the identity
dd k kp
(2π )d (( p − k)2 − m2)(k2 − M 2) .
pk = 21 ( p2 + k2 − ( p − k)2),
we transform the expression (
25
) as
g
Σ1 = 2 p2
dd k 1 1
(2π )d k2 − M 2 − ( p − k)2 − m2
m2 − M 2 − p2 .
− (( p − k)2 − m2)(k2 − M 2)
This expression only contains a logarithmic divergence. On
making a Wick rotation, integrating in the Euclidean
momentum space and keeping only the poles in ε, we obtain
i g2
Σ1 = 16π 2ε .
Thus,
i g2 i g2
Σ = pˆ 16π 2ε + M 8π 2ε .
On substituting the formula (
29
) into Eq. (
16
) we see that
the fermion propagator in the one-loop approximation is
In the same approximation, this propagator can be rewritten
as
To compensate this shift, we should introduce a
counterterm into the Lagrangian, or in other terms, we should
introduce a bare mass MB which is connected with the
renormalized mass M through the relation
Further, to have a canonically normalized fermion field, or
in other words, to compensate a non-trivial divergent factor
in the numerator of the formula (31), we should introduce a
bare fermion field
On now, following the scheme elaborated in paper [27],
we introduce and calculate the anomalous mass dimension
for the fermion mass M . Let us remember that when we
use the dimensional regularization, the renormalized
quantities depend on the renormalization mass parameter μ. At the
same time the bare quantities depend on the regularization
parameter ε, but do not depend on the renormalization mass
parameter μ. Thus, we can write down a general equation
Generally, the renormalization constant Z M has the
following structure:
MB = μ
Z M
M + Z M μ
M = 0.
(37)
On introducing
∂ M
γM ≡ μ ∂μ
,
we can rewrite Eq. (37) as follows:
γM
1 +
γM
1 +
∞ an
εn
n=1
∞ an
εn
n=1
+ M
+ M
∞
n=1
∞
n=1
∂an 1
μ ∂μ εn = 0.
∂g2 dan 1
μ ∂μ dg2 εn = 0.
For the case wherein the residues an depend only on the
Yukawa coupling constant g, Eq. (40) becomes
We now introduce the β – function for the Yukawa constant
g:
∂g2
βg ≡ μ ∂μ + εg2.
Then Eq. (41) reads:
γM
1 +
∞ an
εn
n=1
2 da1
γM = g dg2
.
3g2 M
γM = 16π 2 .
+ M (βg − εg2)
n=1
∞ dan 1
dg2 εn = 0.
The above equation should be correct in any order in ε. To
the zeroth order it gives:
From Eq. (34) we immediately obtain
The calculation of the analogous quantity for the scalar
field is more complicated because the mass renormalization
in this case includes quadratic divergences. To treat them, we
shall follow the approach developed in paper [26]. The full
propagator of the scalar field is
i
D( p) = p2 − m2 − i Π
,
where Π is the self-energy operator. The contribution of the
scalar field self-interaction to the one-loop order in the
operator Π is
quadratic divergences the factor με becomes crucial. A direct
calculation gives
Π =
−i λμεmd−2
d
2(4π ) 2
Γ
d
1 − 2 .
One can see that this expression has the pole at d = 4 and
also the pole at d = 2, corresponding to quadratic
divergence [26]. Indeed, it is well known that in the theory with
the Lagrangian (
15
) the index of divergence of a diagram
G, ω(G) is
3
ω(G) = 4 − 2 E F − E B ,
where E F is a number of the external fermion lines and E B is
a number of the external boson lines. Thus, the diagrams with
E F = 0, E B = 2 are quadratically divergent. Let us now
consider d-dimensional spacetime. In this case, the formula
(49) is replaced by
3
ω(G) = (d − 4)L + 4 − 2 E F − E B ,
where L is the number of loops. Let us again consider a
diagram with E F = 0, E B = 2. This diagram, which is
quadratically divergent at d = 4 becomes logarithmically
divergent (ω(G) = 0) at d = 4 − L2 . That means that the
quadratic divergence is represented as a pole of the quantity
Calculation of this integral gives
− i Π =
ε(L) = 4 − d − L2 ,
ε(
1
) = 2 − d.
λm2
i Πd→4 = − 16π 2ε ,
λμ2
i Πd→2 = 4π ε(
1
) .
and in the case of the one-loop approximation
Thus, expanding the expression (48) around d = 4, we have
while expansion of the same expression at d → 2 gives
Thus, the infinite shift of the mass squared in the full
propagator of the scalar field due to its self-interaction is
δm2 = −m2 16πλ 2ε + μ2 4πλε(
1
) .
We can analogously calculate the contribution of the
fermion loop to the self-energy of the scalar field.
2 ε
Π = −g μ
dd k
(2π )d [(Tkr[+(kˆ p+)2pˆ−+MM2])[(kkˆ2+−MM)2]] .
(48)
(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
Γ d2 − 1
Γ (d − 2)
2
To compensate this shift, we introduce a bare scalar field
mass following the procedure elaborated in the paper [26]:
m2B = Zm m2 + Zμμ2,
where the renormalization constants in the one-loop
approximation are
λ 3g2 M 2 g2
Zm = 1 + 16π 2ε − 2π 2m2ε + 4π 2ε
and
λ 2g2
Zμ = − 4π ε(
1
) + π ε(
1
) .
Further, to have a canonical normalization of the scalar
field, we introduce a bare field as follows:
φB = Zφ1/2φ,
where
g2
Zφ = 1 + 4π 2ε .
and requiring the independence of the bare mass (63) on
the renormalization mass parameter μ and using the explicit
expressions (64) and (65), we obtain in the one-loop
approximation
λm2
γm = 16π 2 −
.
(69)
L = −hψ¯ γ 5ψ χ .
Here
(γ 5)2 = −1, γ 5kˆγ 5 = kˆ.
On repeating the preceding calculations and using the
formulae (71), we see that the contribution of the interaction (70)
to the anomalous mass dimension of the fermion field is
h2 M
γM = 16π 2 .
The contribution of the fermion loop to the anomalous mass
dimension of the pseudoscalar field χ is
h2 M 2 h2m2
γmχ = − 2π 2 + 4π 2 +
2h2μ2
π
.
4 Contribution of potential terms into the vacuum
energy and the auxiliary fields
(70)
(71)
(72)
(73)
(74)
(75)
When we switch on the interactions and require the
cancellation of ultraviolet divergences in the expression for vacuum
energy, we should consider not only the mass sum rules,
but also the potential terms. The contributions of the
potential terms to the vacuum energy density have the following
structure
Epot =
0|T (V exp(i d4x Lint))|0 .
0|T exp(i d4x Lint)|0
Here, Lint is the interaction Lagrangian and the exponent
should be expanded up to necessary order in the perturbation
theory while V represents potential terms. For the term
On now introducing the anomalous mass dimension
where the integral I is defined as
(68)
I =
.
,
∂m2
V = λφ4,
E1 = −3λI 2,
to obtain the result in the two-loop approximation which we
study in the present paper, it is enough to take only the zeroth
order of the expansion of the exponent of the action in the
formula (74). The corresponding contribution is equal to
The contribution of the Yukawa interaction term is given
by the structure
E2 = 0|T (gψ¯ ψ φ × (−i g)ψ¯ ψ φ|0 ,
where the second factor (−i g)ψ¯ ψ φ comes from the
firstorder term in the expansion of the T -exponent. This
contribution (for the case of a Majorana spinor) is equal to
where M is the fermion mass and m is the scalar mass. A
simple calculation shows that for the case of the Wess–Zumino
model, when m = M and there are well-known relations
between the coupling constants [16], the quartic divergences
present in the contributions (76) and (79) do not cancel each
other (we shall present detailed calculations in the next
section). Namely, the contribution of the spinors is twice that
of the scalars. The reason for this mismatch was already
discussed in the Introduction. The point is that the number of
fermion degrees of freedom is doubled off shell. To
compensate this effect, we should introduce the auxiliary scalar
fields as is done in supersymmetric models. A simple
example shows that this exactly gives the doubling of the
leading contribution to vacuum energy. Indeed, let us consider a
model with the Lagrangian
2
On shell this theory is equivalent to the theory where the
auxiliary field is excluded F by means to the equation of
motion
(77)
(78)
(80)
(81)
(82)
(83)
(84)
(85)
and which has a Lagrangian
L = 21 (∂μφ)2 − 21 h2φ4.
3 2 2
Evacuum = − 2 h I .
(76)
0|T (F F )|0 = i.
It follows from Eq. (76) that vacuum energy in the theory
with the Lagrangian (82) is equal to
Let us calculate an analogous (two-loop) contribution to
vacuum energy in the model with the Lagrangian (80). It is
equal to
Evacuum = 0|T (−h F φ2 × (i h)F φ2|0 = −3h2 I 2.
Here we have used the fact that the propagator of the auxiliary
field in the massless theory is given [28] by
Thus, the requirement of the explicit account of auxiliary
fields arises only in the diagrams possessing quartic
ultraviolet divergences and including only boson propagators,
because their contribution is proportional to the number of
degrees of freedom present off shell in the model under
consideration. This fact gives us a practical recipe: when one
calculates vacuum energy contribution of the scalar field
diagrams, having the shape of “eight”, one should multiply it by
the factor 2.
Concluding this section, we wish to make one more
comment. In the action (80) the term F2
2 is also present. One
can consider this term as a part of the kinetic energy. The
contribution of this term into vacuum energy is given by the
formula
0 T
3 2 2
= + 2 h I .
Thus, on summing (87) and (84) we reproduce the result (83).
This means that, in the end, the results for vacuum energy
in the model (80) with an auxiliary field and in the model
(82), where the auxiliary field is eliminated, coincide.
However, the expressions for the contributions of the potential
energy and of the kinetic energy do not coincide separately.
As we have already mentioned in the Introduction, a similar
effect can be observed in the supersymmetric Wess–Zumino
model. If we consider the formalism in the absence of
auxiliary fields, vacuum energy is still equal to zero, but the
potential and kinetic energy are not equal to zero separately.
5 A model with one Majorana and two scalar fields
Let us consider a model with a Majorana field ψ and two
scalar fields A and B. All the fields have the same mass m
and the interaction is given by
Hint = λ1 A4 + λ2 B4 + λ3 A2 B2
+g1ψ¯ ψ A + g2ψ¯ ψ B
+mh1 A3 + mh2 B3 + mh3 A2 B + mh4 A B2.
We see that in this case the result is doubled because of
the effective doubling of the number of degrees of freedom.
One can check also that the contributions to the self-energy
operator of the scalar field to the order of h2 coincide in the
models with the Lagrangians (80) and (82):
0 T
φφ ×
i h2φ4
The two tadpole diagrams for fields A and B should be
cancelled to avoid the necessity of introducing linear in fields
terms into the Lagrangian. The tadpole for the field A arises
due to the contraction of this field with the vertices A3, A B2
and ψ¯ ψ A. All these contributions are proportional to the
integral (77). The corresponding combinatorial factors are
3mh1, for A3, mh4 for A B2 and −4mg1 for the vertex ψ¯ ψ A.
The last contribution arises due to the trace of the fermion
propagator which is proportional to the mass m. Thus, the
cancellation of the tadpole diagram for A requires
(89)
(90)
(91)
dk
(92)
(93)
(94)
(95)
(96)
Now the self-energy operator for the propagator of the
field A obtains the contributions from the vertex A4, from the
vertex A2 B2 and from the pair of vertexes ψ¯ ψ A, A3, A2 B
and A B2. The contributions of two quartic vertexes are both
proportional to the integral I . The corresponding coefficients
are 12λ1 and 2λ3. The contribution of the fermion loop is
Tr(( pˆ + kˆ + m)(kˆ + m)
[( p + k)2 − m2][k2 − m2] dk,
where the factor 2 arises due to the Majorana nature of the
fermion. Then
k2 + kp + m2
[( p + k)2 − m2][k2 − m2] dk
(k2 − m2) + ((k + p)2 − m2) − p2 + 4m2
[( p + k)2 − m2][k2 − m2]
= −8g12 I + (4 p2 − 16m2)g12 K ,
dk
[( p + k)2 − m2][k2 − m2] .
C1 = −8g12
= −4g12
where
K =
Quadratic divergences present in the integral I should be
canceled because such divergences do not arise in the
selfenergy correction to the fermion propagator. Thus, we have
Thus, the propagator of the scalar field A in the one-loop
approximation has the form
The contribution coming from the two scalar-fermion
vertices is given by the integral
Normalizing as usual the wave function, i.e. making the
coefficient at p2 equal to 1, we obtain
.
Thus, this effective shift of the mass squared given by
i m2(−12g12 + 18h21 + 4h23 + 2h24)K
defines the running of the mass for the scalar field A. The
analogous shift for the second scalar field is
i m2(−12g22 + 18h22 + 4h24 + 2h23)K .
The self-energy contribution to the fermion propagator is
= (g12 + g22)(2 pˆ + 4m)K ,
C3 = 4(g12 + g2 )
2
kˆ + m
[( p − k)2 − m2][k2 − m2]
where the factor 4 arises due to the Majorana nature of the
fermion. The fermion propagator is now
G F = pˆ(1 − 2i (g12 + g22)K ) − m(1 + 4i (g12 + g22)K
.
i
i
(97)
(98)
(99)
(100)
(101)
(102)
(103)
(104)
(105)
(106)
(107)
On normalizing the term at pˆ, we obtain
i
G F = pˆ − m(1 + 6i (g12 + g22)K
.
The shift of the mass squared is
12i m2(g12 + g22)K .
The running of the masses and, hence, the shifts (99), (100)
and (104) should be equal and we obtain two equations:
18h21 + 4h23 + 2h24 − 12g12 = 12(g12 + g2 )
2
and
18h22 + 4h24 + 2h23 − 12g22 = 12(g12 + g22).
Let us now consider the contribution of the potential term
(88) to vacuum energy. The contribution of the quartic terms
is
E1 = (3λ1 + 3λ2 + λ3)I 2.
where
L =
provided Eqs. (94), (95), (105) and (106) are satisfied. The
coefficient 2 in front of the term E1 is introduced to take into
account the fact that the number of boson and fermion degrees
of freedom should be equal also off shell. It is equivalent to
the introduction of two auxiliary fields in supersymmetric
models, as was explained in the preceding section.
On now substituting the expressions for g1 and g2 from
Eqs. (89) and (90) into Eqs. (105) and (106), we obtain the
following pair of the consistency conditions on the constants
h1, h2, h3 and h4:
These equations are homogeneous in h1, h2, h3 and h4.
Thus, we can fix h1 = 1 and we can then change the value of
h2. Then we shall have a system of two quadratic equations
for h3 and h4. This system, which is equivalent to one
quartic equation for one variable, is solvable analytically but the
solutions are very cumbersome. Thus, we shall just present
some numerical solutions. In any case we have four
solutions, two of them are complex and two of them are real. We
shall only take real solutions into account.
Then
Let us note that the negative values of the coupling constants
are not essential because they are in front of the odd (third)
power of fields A and B. The lower bound of the scalar field
potential exists and is determined by the quartic terms with
positive constants λ1, λ2 and λ3.
One can meanwhile introduce the quartic interactions
using auxiliary fields in a manner similar to that used in the
Wess–Zumino model. It is enough to introduce the
following terms into the Lagrangian instead of terms with quartic
interactions:
6 Model with a Majorana field, a scalar field and a
pseudoscalar field
Let us consider another toy model where the field B is a
pseudoscalar. In this case
In this case we have only one condition for the tadpole
cancellation for the scalar field A which coincides with that given
by Eq. (89). The conditions for the cancellation of quadratic
divergences in the propagators of the scalar and pseudoscalar
fields are also the same (94) and (95). However, the shifts of
the mass squared for the fields A, B and ψ are different. They
are proportional to
and
and
which implies a negative value for g22 or for g12 as follows from
the couple of equations (116) and (117). Thus, the choice
(119) should be discarded.
We have seen that for the case with one Majorana field,
one scalar and one pseudoscalar we have less freedom in the
choice of the coupling constants than in the case of two scalar
fields and one Majorana field, but this choice is still broader
than that in the Wess–Zumino model.
In principle, one can also consider a model with one
Majorana field and two pseudoscalar fields. In this case the triple
scalar interactions do not exist and the constants h1, h2, h3
and h4 all are equal to zero. The requirement of the absence
of quadratic divergences in the shifts of the mass squared of
two pseudoscalar propagators is again given by Eqs. (94) and
(95). However, the shifts of the mass squared for this
propagators are equal to zero, while the shift of the mass squared
of the fermion propagator is given by
−4g12 − 4g22.
Thus, the Yukawa interactions should vanish as well. Now,
the conditions (94) and (95) can be satisfied if
λ3 = −6λ1 = −6λ2,
but the corresponding quartic potential of these two
pseudoscalar fields is unbounded from below and is hardly
interesting.
7 Models with non-degenerate masses
It is interesting to find toy models with masses which are
not degenerate. In this case it is necessary to consider at least
four boson and four fermion degrees of freedom [5]. The
simplest models of this kind are those which include a certain
number of “triplets” of the types described in two
preceding sections, i.e. with degenerate masses inside any triplet
and with coupling constants (again, describing interactions
within a triplet) which satisfy the relations obtained in the
Sects. 5 and 6. Naturally, in this case, if there are no
interactions between the fields belonging to different triplets, then
the Pauli–Zeldovich mechanism does work. If we introduce
interactions between different triplets with different masses,
then the coupling constants should satisfy some constraints.
We shall illustrate this by considering a model, where there
are two triplets. Both triplets contain a Majorana fermion
and two scalar fields. The mass of all the particles in the
first triplet is equal to m1, in the second triplet – m2 and the
coupling constants describing interactions within the triplets
are chosen in such a way that vacuum energy is equal to zero.
Let us then introduce the following interaction Hamiltonian
between fields belonging to different triplets:
H = λAC A2C 2 + λAD A2 D2 + λBC A2C 2 + λB D A2 D2
+gAχ χ¯ χ A + gBχ χ¯ χ B + gCψ ψ¯ ψ C + gDψ ψ¯ ψ D
+h AC1 A2C + h AC2 AC 2 + h AD1 A2 D + h AD2 A D2
+h BC1 B2C + h BC2 BC 2
+h B D1 B2 D + h B D2 B D2,
(120)
where the scalar fields A and B and the Majorana spinor ψ
belong to the first triplet, while the scalar fields C and D and
the Majorana spinor χ belong to the second triplet.
It is now possible to find relations which constrain the
choice of the coupling constants given in the interaction
Hamiltonian (120). However, this task is rather cumbersome
and we shall postpone it and the construction of other models
for future work [29].
8 Concluding remarks
In this paper we have studied the Pauli–Zeldovich
mechanism for the cancellation of ultraviolet divergences in
vacuum energy which is associated with the fact that bosons and
fermions produce contributions to it having opposite signs.
In contrast with the preceding papers devoted to this topic
where only free fields were considered, here we have taken
interactions up to the lowest order of perturbation theory into
account. We have constructed a number simple toy models
having particles with spin 0 and spin 1/2, wherein masses
of the particles are equal while interactions can be quite
non-trivial. To make calculations simpler and more
transparent, it was found useful to introduce some auxiliary fields. It
appears that the presence of these fields is equivalent to the
modification of some contributions of the physical fields to
the vacuum expectation of the potential energy. We hope to
construct more complicated models including particles with
different masses and in the presence of vector fields in future
work [29].
Acknowledgements We are grateful to A. O. Barvinsky, F. Bastianelli
and O. V. Teryaev for useful discussions.
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