# Effective Actions for Spin in Curved Spacetimes

###### Abstract

We calculate the effective potentials for scalar, Dirac and Yang-Mills
fields in curved backgrounds using a new method for the determination of the
heat kernel involving a partial resummation of the Schwinger-DeWitt series.
Self-interactions are treated both to one loop order as usual
and slightly beyond one-loop order by means of a mean-field approximation.
The new
approach gives the familiar result for scalar fields, the Coleman-Weinberg
potential plus corrections such as the leading-log terms, but the
actual calculation is much faster.
We furthermore show how to go systematically to higher loop
order. The Schwarzschild spacetime is used to exemplify the procedure.

Next we consider phase transitions and we show that for a classical critical
point to be a critical point of the effective potential too, certain
restrictions
must be imposed on as well its value as on the form of the classical potential
and the background geometry.
We derive this extra condition for scalar fields with arbitrary self couplings
and comment on the case of fermions and gauge bosons. Critical points of
the effective action which are not there classically are also discussed.

The renormalised energy-momentum tensor for a
scalar field with arbitrary self-interaction and non-minimal coupling to
the gravitational background is calculated to this improved one-loop order as
is the resulting conformal anomaly. Conditions for the violation of energy
conditions are given.

All calculations are performed in the case of
dimensions.

## 1 Introduction

The Coleman-Weinberg formula for the one-loop order
effective potential of a theory has many applications,
e.g. to inflation and to the study of the
standard model of particle physics. But for calculations in curved spacetime,
such as more realistic inflationary scenarios and the study of the early
universe, one should take curvature into account.

In this paper we will use the
heat kernel method for finding the effective potential as described in e.g.
the textbook by Ramond,
[1], but this time in curved spacetime using methods developed by
the authors [2]. We should emphasise that even
the simple approach of the first part of this
paper can be used to go beyond one-loop order, as we can easily include
higher order derivatives of the curvature and of the classical fields corresponding
to quantum corrections to the kinetic part of the effective action. In fact, we
will argue that some of the terms in our final expression corresponds to a
summation of leading log-terms.
Furthermore, the mean-field approach, to which we resort in order to extend
the results to the case of self-interacting fields, is essentially
a non-perturbative approximation to the full effective action, as it allow us to
perform the functional integral. This mean-field approach can be iterated
to reach, in principle, any accuracy desired. We finally compare our method
to that of other authors. For generality, we let the classical potential of
the scalar field be completely arbitrary.

The next step is to treat Dirac fermions, and it is shown how to relate the
effective actions
for such spin- fields to that of the non-minimally coupled
scalar field. Subsequently the
renormalised mass is found. Comments on Weyl fermions are also
made, as are
non-minimally coupled fermions, i.e., fermions coupling to the background
torsion, and spinors coupling to Yang-Mills fields.

Thirdly we consider Yang-Mills fields in curved spacetime, and we present
the effective
action for this case too, both to one loop (or higher) order and in the
mean field approach. In both
cases can the ghost contribution be related to the effective action for a free
scalar field.

Next we show how, using a prescription for propagators,
[3], we can in principle go to any loop order. The relationship between
the Green’s function and the heat kernel is used to write down an explicit
formula for the second-loop order contribution to the effective
action in a certain approximation.

We also discuss phase transitions and critical points, both of the original
classical potential and the resulting effective one. In particular we see when
a classical critical point is also a critical point of the effective
potential. It turns out that this leads to restrictions not only on the
couplings but also on the background geometry.

From the effective action we can then also find the
renormalised energy-momentum tensor. It turns out that this
in general violates the weak energy condition, and a
measure for this violation is found.

At the end, we provide a discussion and an outlook.

### Contents

Part I (Theory)

1. Introduction

2. Determining the curved space Coleman-Weinberg Potential

3. The Mean-field Approach

3.1. Example: Schwarzschild Spacetime.

4. Fermions

4.1. Free Dirac Fermions

4.2. On Weyl Fermions

4.3. On Coupling to Yang-Mills Fields and Torsion

4.4. Example: Schwarzschild Spacetime

5. Yang-Mills Fields

5.1. Mean-field Approximation and Symmetry Restoration

5.2. Example: Schwarzschild Spacetime

6. Beyond One Loop Order

Part II (Applications)

7. Quantum Modification of Classical Critical Points

7.1 Spinor and Vector Fields

8. Quantum Critical Points

9. The Energy-Momentum Tensor

9.1. Conformal Anomaly

9.2. Violation of Energy Conditions

10. Discussion and Conclusion

## 2 Determining the Curved Space Coleman-Weinberg Potential

As shown in e.g. Ramond [1], the effective potential to one-loop order can be found rather quickly by the heat kernel method:

(1) |

where the zeta function is calculated assuming a constant configuration and where denotes the classical field with . In curved space this must be generalised by determining the heat kernel of the curved space scalar field operator (from which one can calculate the zeta-function), , which thus obeys

(2) |

where denotes the curved spacetime d’Alembertian

(3) |

with

(4) |

and where we have introduced vierbeins (i.e. and , ).

The computation of the coefficients of the heat kernel is
rather straightforward.
^{1}^{1}1One should notice that most of the calculations to follow
don’t really assume , only in the final result for the
renormalised mass and coupling constant is this assumption needed.

One should furthermore
notice that although the result quoted here is for we can actually find
the heat kernel to arbitrary order of accuracy even for . The heat
kernel would then depend on , the geodesic distance squared,
as well as on integrals of powers of the curvature scalar and its derivatives
along the geodesic from to , assuming such a curve exists
[3]. The equation to be solved is (where is found from (2))

(5) |

subject to the boundary condition

(6) |

We solve this equation by writing

(7) |

with the heat kernel of , which in dimensions is

(8) |

with the square of the geodesic distance (the so-called Synge world function) and

being the Van Vleck-Morette determinant.^{2}^{2}2Fortunately, we only
need these
quantities evaluated at the diagonal for the present purposes, which
simplifies matters a lot. Doing this leads to the following
equation for

(9) |

subject to the condition

(10) |

Taylor expanding ,

(11) |

with because of the initial value condition, we get the following recursion relation when

(12) |

From the heat kernel equation it furthermore follows that
. Finding the higher coefficients is then trivial,
[2].

We must emphasise that these coefficients differ from the usual
Schwinger-DeWitt
ones, [8], a difference due to the different nature of the
two expansions used.

One
should also note that, contrary to the Schwinger-DeWitt case, the coefficients
given by our expansion are relatively straightforward to obtain explicitly.
The expression (7) can actually be viewed as a partial resummation of the
Schwinger-DeWitt series. In fact, the series is the so-called
cumulant of the (divergent/asymptotic) series of Schwinger and DeWitt,
which is strictly speaking only valid for small.^{3}^{3}3If is a divergent/asymptotic series the cumulant is given by a
series . Our series also
has better convergence properties due the pressence of an -term, which is in itself an indication of an implicit summation
of leading log-terms, [22].
Besides the different expansions used, the major cause for simplification
lies in
the usage of vierbeins and hence co-moving coordinates. This being so,
it turns out that one can actually rather easily write down expressions for
the coefficients in our series, whereas one can in general only compute the
first few in the Schwinger-DeWitt case.

Thus, we arrive at

(13) | |||||

where

(14) | |||||

(15) |

with containing vierbeins and their derivatives only

(16) |

In formula (13) higher order terms have been
omitted, i.e., the result is valid for and strong but
sufficiently slowly
varying in order for us to discard third and higher derivatives. It should be
noted that that is the only approximation we have made. Furthermore, one could
in principle include higher derivatives – it is not difficult to find the
corresponding coefficients – but the integral over which we will
ultimately perform becomes difficult to make analytically.

With this expression for the heat kernel it is thus possible to
calculate the zeta function (see e.g.
[1, 2, 7])

and we finally arrive (from equations (11,14,15,18)) at

The quantity appears because we are using vierbeins and
hence do not
have a coordinate basis, , and
because is
not the true d’Alembertian.

Notice that this result, (LABEL:eq:veff), holds even for not
constant. This means
that even though (1) is only valid to one-loop order we can get some of the
quantum corrections to the kinetic part, which otherwise belong to higher loop
order. Of course, we cannot expect to get all the corrections this way, but
we may get some indication of the nature of the leading terms.

From (LABEL:eq:veff) we can obtain the renormalised values of the mass and of
the coupling
constant, putting and assuming, for sake of
argument,
:

(20) | |||||

(21) | |||||

(22) | |||||

Similarly, by writing and considering this an action for the gravitational degrees of freedom, we get the following values for the renormalised cosmological and Newtonian constants respectively

(23) | |||||

(24) | |||||

this shows how the masses of the fundamental fields modify the Newtonian
constant, whether this effect is testable or not is difficult to say at
present. The quantum modifications of the coupling constants are due to (1)
the mass of the scalar field, (2) the value of and , and (3)
the variations thereoff. Consequently, also massless scalar fields will lead
to . Such effects may have implications similar to
dark matter.

Let us at this stage pause and compare with the results found by other
authors.
Hu and O’Connor, [12], have calculated the effective potential for a
theory in a static, homogeneous spacetime in which the interaction
term can be assumed constant. They then use
dimensional regularisation and the Schwinger-DeWitt expansion to find the
following expressions for the change in mass and
couplings constants to one-loop order:

Comparing this with our result (we always take ) we notice that the
two set of expressions agree provided we take in
Hu and O’Connor’s result, remove the special curvature terms coming from the
noncoordinate basis (i.e., the -terms), assuming and
ignoring all higher order terms. The -term is easy to understand as
it is a constant in their approach,
similarly one would expect to be absent in a Riemann normal coordinate
patch to
the lowest order. As mentioned earlier is a result of not using a
coordinate frame in our case, .

There is something odd about the result of Hu and O’Connor, though,
the non-minimal coupling, , clearly acts like a mass term in the
Lagrangian and one would thus expect the
renormalised mass to depend on . In fact, even if the field was massless
classically we would expect a curvature induced mass to turn up, but their
result is that a massless field does not acquire any mass to one-loop order.
This is in stark contrast to what one would expect in view of Shore’s work,
[10], where precisely such a curvature induced mass is responsible for
symmetry restoration in scalar QED. Shore’s result, which we
will also comment on later, is based on an Euclidean version of de
Sitter space, namely , and he then explicitly calculates the
zeta function for scalar QED. In fact, he shows that a minimal coupling
would lead to an anomalous mass term.
The discrepancy between the work of Hu and O’Connor on the one hand and Shore
and this paper on the other we think
rests on the approximations made by the authors of [12]: a static,
homogeneous spacetime using the Schwinger-DeWitt expansion in a Riemann normal
coordinate patch. As
mentioned earlier, the method put forward here is a partial resummation of the
Schwinger-DeWitt series and will thus inevitably give different results.
What all these methods agree on, of course, is the result in the flat space
limit, where
we must recover the Coleman-Weinberg potential at least as the leading
contribution. Only keeping
the very lowest order terms in (LABEL:eq:veff) we recover (except for the
quantity ), the result by Hu and O’Connor.

The effective potential for a -theory in some model spacetimes
have been calculated by
a number of authors. For instance, Hu and O’Connor has also found the effective
potential in a mix-master universe.
Futamase, [15], has used the -function technique in the
Bianchi I spacetime for a -theory with and without a coupling to an el
ectromagnetic field. The formula he finds is valid for
(the inhomogeneous parameter, showing the discrepency between the
homogenous Friedmann-Robertson-Walker spacetime and the inhomogeneous Bianchi
I) small. Similarly, Berkin [17], has found a formula by Taylor
expanding the heat kernel in powers of . His result too, is valid
only for small .
These two authors agree with each other. Contrary to this, Huang [16],
has used an adiabatic approximation
and get a different result. He finds, for instance, that symmetry
restoration (a point we will be returning to later) is possible even for
large, whereas the other authors only found this to be possible for
small. We will not comment further on these results.

Ishikawa, [18], has calculated the effective potential for a
massless
, scalar QED
theory in a more general background, but using Riemann normal coordinates and
only going to
assuming throughout that . He uses a
renormalisation-group improved technique and gets

This differs from our result due to the approximations made (linearity in ,
no derivatives of the curvature, no -term).

In general then, the previous research have mostly been limited to the
case , in which case one gets results linear in or
the (slightly improved) Coleman-Weinberg
potential from flat spacetime with .
Contrary to this, we do not require small. For purely practical purposes
we have, though, limited ourselves to the regime in which are small
(compared to ). However, as pointed out in [2], we could in
principle remove this limitation.

## 3 The Mean-Field Approach

Only Gaussian functional integrals can be calculated in any reliable
manner. We must consequently find a way of transforming the original
functional integral
defining the partition function, , into a Gaussian one.
An improvement over the previous method can be
obtained by writing^{4}^{4}4i.e., instead of making use of the prescription
(perhaps keeping
constant, and in any case only keeping terms
when functionally integrating out the fluctuating part )
for part of the interaction term we make the prescription
where
is the actual mean
value of the the field due to the propagation of virtual particles in
curved space..

(25) |

where is a mean field. A similar approach for Yang-Mills fields have been developed in [2], and we will briefly outline it for the simpler case of a scalar field. Supposing that we have calculated , the heat kernel is (cf. equation (13))

(26) |

with

With a recursion relation identical in form to the one given in (12).
In the previous calculation we have ignored the term in
as this is a fourth order derivative of the curvature essentially.

From this one easily obtains the effective action.

Also, the mean field is obtained to the lowest order
from

(27) |

where is the action of with , i.e., only the
kinetic and gravitational
terms are included; the self-interaction is ignored. To the next order one
could define an
with and given by
(27), this can then be
iterated to any order of accuracy wanted – the functional integrals are
easily solved, as one
just needs to calculate the new coefficients in the expansion above
of the
heat kernel. For completeness, the first iterated coefficients are listed at
the end of this section.

Now, the inverse of the second derivative of the action is the Green’s function
. This too can be found from the heat kernel (see [3]) as

(28) |

which follows directly from the spectral decompositions of the heat kernel and
the Green’s function respectively.

Inserting the above expansion (26) with we get a
divergent result
for as one would expect (it is proven
in [3] that the propagator one gets from our expression for the
heat kernel satisfies the
Hadamard condition and thus has the correct singularity structure as
), but the infinities can be
removed quite simply by using principal values instead (the singularity is of
the form ) [4]. The final result thus becomes

(29) |

with , and where
we
did not iterate the mean field.^{5}^{5}5Actually, one can think of this
calculation as determining the contribution to the
mean field due to the propagation
of virtual particles in curved space whereas the next iteration includes
the self-interaction and thus, essentially, is a one or higher loop-order
calculation (in the matter fields, this time). The superscript refers
to the lack of iteration. Explicitly:

and so on. Discarding third and higher derivatives of (i.e., going only to one loop order in gravity) the remaining coefficients all vanish. Thus

is the formula we find for the curvature induced mean field.

By the definition of the Lagrangian, (25), the mean field can
be seen as a
redefinition of the mass and the non-minimal coupling

(30) | |||||

(31) |

which shows the non-perturbative nature of this approximation very clearly (it
is non-polynomial in the coupling to the curvature, ).

The coefficients which enter the full heat kernel are then

As this result is based on a mean field approximation it is non-perturbative, and iterating the process as described above, we could further increase the accuracy of the calculation, it is clear, however, that this would be somewhat cumbersome, though not at all difficult in principle. To give an example of the procedure, we just list the first iterated mean-field

(32) |

with

The procedure is now transparent. One should note, as follows from
(29), that our normalisation is such that the
mean field vanishes whenever .

Using (32) in the the general expression for , eq
(LABEL:eq:veff), i.e. putting
we can find the new corrections to the mass and non-minimal coupling

(33) | |||||

(34) |

One can then repeat this procedure ad infinitum. Apparently, by going to infinite order in we would get the effective non-minimal coupling to be

corresponding to a finite, multiplicative renormalisation of , when
, and infinite, multiplicative renormalisation otherwise.

One should note that this method is also of use in flat spacetime for strongly
self-interacting scalar fields.

Bunch and Davies, [28], have found a formula for
in de Sitter space valid for arbitrary:

with , and is a small
number due to the regularisation procedure chosen.
The finite part of this differs from ours, by the addition of a constant term,
due to a different choice of renormalisation. Also, of course, as we go to
higher order, we have non-linear terms in and furthermore include the
derivatives of the curvature scalar.

For a constant (or for the derivatives of negligible) we can
find the mean-field to infinite order in , and get , and where

leading to

which can be seen as a finite renormalisation of and .

### 3.1 An Example, the Schwarzschild Spacetime

In this case and the only quantity we need to evaluate is

(35) |

where the first term is ignored as an artifact, arising from the
approximations made; the final result should of course be independent of the
angles.^{6}^{6}6Of course the full heat kernel must be covariant, the apparent
non-covariance – the -dependence – comes from splitting-up the d’Alembertian
in a non-covariant way. If we were able to carry out the full re-summation this problem
would dissappear by covariance of the procedure, but as we have only a truncated
expression for the heat kernel, we must expect some unphysical effects like this. The
full re-summation would remove the angular-dependence by means of summation formulas
for trigonometric functions. This is likely to give rise to finite term, which we
cannot find by our method at this stage, we will therefore ignore this angular term
altogether. Improved re-summation techniques will then show how accurate this
approximation is, and whether this finite term leads to any testable effect or not.

Since Schwarzschild spacetime is a solution to the vacuum Einstein equations
we get a very simple result for the mean field, namely

(36) |

which is independent of .

The effective potential then becomes (again taking to be a
potential) from (19)

(37) | |||||

from which we get the renormalised mass and couplings to be from (21, 22)

(38) | |||||

(39) | |||||

Far away from the gravitational source, these take on their Minkowski spacetime values as one would expect – there is no gravitational effect far away. We do, however, notice that the arguments of the logarithms can become negative for certain values of , this then leads to an imaginary part of the effective action, i.e. to particle production. We have plotted the renormalised mass as a function of the distance to the black hole in figure 1 with . Note that the quantum effects are very much confined to a narrow region around the horizon, and very quickly fall off to their non-gravitational values. Note also that