Effective Potential in Curved Space and Cut-Off Regularizations

Abstract

We consider derivation of the effective potential for a scalar field in
curved space-time within the physical regularization scheme, using two sorts of
covariant cut-off regularizations. The first one is based on the local momentum
representation and Riemann normal coordinates and the second is operatorial
regularization, based on the Fock-Schwinger-DeWitt proper-time representation.
We show, on the example of a self-interacting scalar field, that these two
methods produce equal results for divergences, but the first one gives more
detailed information about the finite part. Furthermore, we calculate the
contribution from a massive fermion loop and discuss renormalization group
equations and their interpretation for the multi-mass theories.

Full text

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arXiv:1107.2262v1 [gr-qc] 12 Jul 2011
Effective Potential in Curved Space and Cut-Off Regularizations
Fl´ avia Sobreira(a), Baltazar J. Ribeiro and Ilya L. Shapiro(b)
(a) IFT-UNESP, S˜ ao Paulo, SP, Brazil
(b) Departamento de Fisica, UFJF, Juiz de Fora, MG, Brazil
Abstract.
space-time within the physical regularization scheme, using two sorts of covariant cut-
off regularizations. The first one is based on the local momentum representation and
Riemann normal coordinates and the second is operatorial regularization, based on the
Fock-Schwinger-DeWitt proper-time representation. We show, on the example of a self-
interacting scalar field, that these two methods produce equal results for divergences,
but the first one gives more detailed information about the finite part. Furthermore, we
calculate the contribution from a massive fermion loop and discuss renormalization group
equations and their interpretation for the multi-mass theories.
We consider derivation of the effective potential for a scalar field in curved
1 Introduction
Recently there was a growing interest in the more physical regularization and renor-
malization schemes in curved space-time. In particular, one can mention the papers on
deriving the energy-momentum tensor of vacuum in momentum cut-off regularization
[1, 2, 3, 4, 5, 6, 7] from one side and intensive discussions of physical interpretation of
renormalization group from another one [8, 9, 10, 11, 12, 13, 14, 15, 16, 17].
One of the outputs of the works on the cut-off approach is that this regularization may
produce an explicit breaking of the local Lorentz invariance and also of general covariance.
Therefore, it would be interesting to have an example of the cut-off-based calculations
which preserve both symmetries explicitly.
The effective potential of a scalar field in curved space-time have been studied in
a number of papers starting from [18, 19, 20, 21] (see [22] for further references). In
particular, a very general expression for such an effective potential has been obtained in
[21] via the renormalization group method. Indeed, this means the Minimal Subtraction
scheme of renormalization, when the effect of masses of the quantum fields are either
ignored or taken into account through the heuristic method. An additional motivation
for a more physical renormalization and regularization schemes comes from inflationary
side. In the recent paper [23] it was shown (see also previous works [24] in this direction)

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that the Higgs-based inflation, originally invented by A. Guth [25], can be consistent with
known observational tests if assuming that the Higgs field H couples non-minimally to
scalar curvature. Let us remark that the corresponding term ξRH∗H is requested in
order to make Standard Model of elementary particles multiplicatively renormalizable in
curved space-time [22]. The value of ξ should be of the order of 104− 105, but this does
not pose a problem, because the dimensional quantity |ξR| does not exceed the square
of the Higgs mass. The great difference between the Higgs-based and inflaton-based
inflationary models is that the Higgs field probably does exist. Therefore, the model of
[25, 23] should be considered as the first candidate to describe an inflationary paradigm
[26]. According to the further works on Higgs inflation [27] and [28] (see also [29] and
references therein), the renormalization group - based quantum correction to the Higgs
potential play an essential role in this inflationary model, such that taking them into
account leads to important restrictions for the Higgs mass. This result was essentially
based on the well-known renormalization group derivation of effective potential in curved
space-time, completely equivalent to the one which was first performed in [21, 22]), and
concerns both one- and two-loop contributions. However, as far as this derivation is based
on the Minimal Subtraction scheme of renormalization, it would be interesting to verify
what is the effects of the masses of quantum fields by direct calculation.
In order to address the issues of covariant cut-off and of the effect of masses on the
Higgs potential in curved space, we perform direct calculation of the one-loop effective
potential of a scalar field in curved space-time. We consider two sorts of covariant cut-off
regularizations. The first one is based on the local momentum representation, which is
due to the use of Riemann normal coordinates, and the second is the so-called operatorial
regularization, based on the Fock-Schwinger-DeWitt proper-time representation. It was
demonstrated recently in [30] that these two types of regularizations give equivalent results
in flat space-time. In view of this, our calculations can be seen as an extension of the
same statement to a curved space.
The paper is organized as follows. In the next section we perform calculation for
a self-interacting scalar field through the cut-off regularization in the local momentum
representation. In Sect. 3 we consider a technically simpler scheme of operatorial regu-
larization cut-off. In Sect. 4 we extend the previous results to the fermion contributions.
In Sect. 5 the µ-dependence and renormalization group equations for the parameters of
the theory are discussed. Finally, in Sect. 6 we draw our conclusions.
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2 Covariant momentum cut-off calculation
The effective potential is defined as the zero-order term in the derivative expansion of the
effective action of a mean scalar field1,
Γ[ϕ, gµν] =
?
d4x√g
?
− Veff(ϕ) +1
2Z(ϕ)gµν∂µϕ∂νϕ + ...
?
. (1)
The calculation of Veff(ϕ) can be performed for constant ϕ, in different theories with
different content of quantum fields. In this paper we consider two examples, namely self-
interacting scalar field and also fermion field with Yukawa coupling to the background
scalar, both in curved space-time.
Our starting point will be the action of a real scalar field
S0=
?
d4x√g
?1
2gµν∂µϕ∂νϕ −1
2(m2− ξ R)ϕ2− V (ϕ)
?
, (2)
where V (ϕ)+m2ϕ2is the minimal potential term and ξRϕ2is the non-minimal addition,
which is necessary for formulating renormalizable theory in curved space-time. In flat
space R = 0 and hence the non-minimal term vanish. Our purpose is to derive one-loop
correction to Eq. (2) in the constant scalar case. We perform calculations in four space-
time dimensions. Hence we are mainly interested in the renormalizable case V = fϕ4/4.
However in this section we shall use general notation V (ϕ), as being more compact and
general.
In what follows we briefly consider the flat case first (one can see, e.g., [32] for a
very pedagogical exposition with full details, despite there is some difference with our
method) and then take care about linear in curvature correction. We stop at the first order
because it is sufficient for our purposes and because calculations become too cumbersome
in the next order approximation. However, the normal coordinate method enables one, in
principle, to perform calculations to any given order in curvature tensor and its derivatives
and also can be helpful to evaluate higher loops contributions.
2.1 Flat space calculation
The result for the flat space is pretty well known [31]. The derivation for the massive case
can be found, e.g., in the text-book [32], where it was obtained via Feynman diagrams.
We can also arrive at the same result via the path integral functional methods. The
starting point is the following expression:
Veff(ϕ) = m2ϕ2+ V (ϕ) +¯V0(ϕ), (3)
1We use Euclidean signature.
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where
¯V0(ϕ) =1
2Tr lnS2(ϕ) −1
2Tr lnS2(ϕ = 0), (4)
where S2(ϕ) is the bilinear form of the classical action in the background-field formalism
[33]. The last term in (4) can be seen as normalization of a functional integral. This
term arises naturally through the diagrammatic representation of effective potential (see,
e.g., [32]). In curved space-time the second term gets dependent on the metric and hence
become relevant. Here and below we omit an infinite volume factor. Let us note that
the one-loop contribution (4) represents a quantum correction to the complete expression
V (ϕ)+m2ϕ2/2 and not just for V (ϕ). The same notations will be used in what follows.
By introducing four-dimensional momentum cut-off Ω, we arrive at the result
¯V0(ϕ,ηµν) =
1
32π2
Ω
?
0
k2dk2ln
?k2+ m2+ V′′
k2+ m2
?
. (5)
After taking this intergal we obtain
¯V0(ϕ,ηµν) =
¯V0 =¯Vdiv
1
32π2
1
32π2
0
+¯Vfin
0
, (6)
¯Vdiv
0
=
?
?1
Ω2V′′−1
?m2+ V′′?2ln?1 +V′′
2
?m2+ V′′?2lnΩ2
m2
?−1
?
, (7)
¯Vfin
0
=
2m2
4
?m2+ V′′?2?
. (8)
In the last expressions we have included the ϕ-independent m4-type terms, which are
indeed part of the second term in (4). The naive quantum contribution (6) must be
supplemented by an appropriate local counterterm, which we choose in the form2
∆V0 =
1
32π2
?
− Ω2V′′+1
2
?m2+ V′′?2lnΩ2
µ2+1
4
?m2+ V′′?2?
.(9)
As a result we eliminate both quadratic and logarithmic divergences and arrive at the
simple form of renormalized effective potential
Vren
eff,0(ηµν, ϕ) =m2ϕ2+ V +¯V0+ ∆V0
= m2ϕ2+ V +
1
64π2
?m2+ V′′?2ln
?m2+ V′′
µ2
?
.(10)
Looking at the counterterms (9) it is easy to see that the renormalizable theory is the
one which has V (ϕ) = const×ϕ4. The reason is that for this potential the counterterms
have the same form as the classical potential with an additional cosmological constant. At
the next stage we will see that the same feature holds in curved space if the non-minimal
term ξRϕ2is introduced.
2For the sake of convenience we have included into ∆V the finite term, this can be easily compensated
by changing µ.
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2.2 Riemann normal coordinates
Riemann normal coordinates represent a useful tool for deriving local quantities, such as
divergences or effective potential. These coordinates are based on the geodesic lines which
link some fixed point P′(xµ′) with other points. We can always assume that gµν(P′) = ηµν.
One can fix the initial conditions for the geodesic lines in such a way that the metric in
the point P(xµ) becomes a Taylor series in the deviation yµ= xµ′− xµ. The coefficients
of such an expansion are curvature tensor, its contractions and covariant derivatives at
the point P′. In the present work we will be interested only in the first order in curvature
terms, and therefore all expansions will be taken in linear approximation.
For instance, for the metric tensor we meet [34]
gαβ(x) = gαβ(x′) −1
3Rαµβν(x′)yµyν. (11)
The bilinear operator of the action (2) is
−ˆH= −1
= ηµν∂µ∂ν+1
√g
δ2S0
δϕ(x)δϕ(x′)= ? + m2− ξR + V′′
βyαyβ∂µ∂ν−2
3Rµαν
3Rα
βyβ∂α+ m2− ξR + V′′. (12)
Of course, the term −ξR must be also expanded, but as far as we keep only first order
in curvature, this is not relevant.
The main advantage of the local momentum representation is that all calculations
can be performed in flat space-time (but with modified elements of Feynman technique)
and the result for some local quantity can be always presented in a covariant way. For
instance, the equation for the propagator of the scalar field has the form
ˆH G(x,x′) = −g1/4(x′)δ(x,x′)g1/4(x). (13)
It proves better to work with the modified propagator [35]¯G(x,x′), where
ˆH¯G(x,x′) = −δ(x,x′). (14)
It is important for us that the r.h.s. of the last relation does not depend on the met-
ric, because we are going to use the relation Tr lnˆH = −Tr lnG(x,x′) to obtain the
dependence on curvature.
The explicit form of¯G(x,x′) is known for a long time [35] for the free V′′= 0 case. As
far as V′′= const, we can simply replace m2by ˜ m2= m2+ V′′and obtain, in the linear
in curvature approximation,
¯G(y) =
?
d4k
(2π)4eiky?
1
k2+ ˜ m2−(ξ − 1/6)R
(k2+ ˜ m2)2
?
.(15)
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