Vacuum energy for a massive Dirac neutrino propagating in a neutron medium
Brian Woodahl and Ephraim Fischbach
Physics Department, Purdue University, West Lafayette, Indiana 47907
Received 23 November 1999; published 23 May 2000
We calculate the energy acquired by the vacuum due to both a massless and massive Dirac neutrino
propagating in an external neutron field, utilizing the quantum field theoretic approach of Schwinger. This
method computes the vacuum energy to all orders in the coupling, using the interaction Hamiltonian for a
neutrino in the presence of an external background of neutrons. To verify our results, we develop heuristic
arguments to compute the energy of the neutrino condensate arising from a neutron-induced chiral potential
well. The results for the massive case are new, and are discussed in some detail.
PACS numbers: 11.10.Ef
I. INTRODUCTION
Neutrino-exchange forces, and the behavior of neutrinos
in a dense medium such as a neutron star, have lately been
the subject of renewed interest 1–19. It is known that the
exchange of virtual neutrinos between neutrons gives rise to
long-range forces through which the neutrons can interact
with one another 1–10. However, the focus of this paper
will not be on the neutron-neutron interaction i.e. the self-
energy of the neutron field, but rather on the energy arising
from the interaction between a virtual neutrino and an exter-
nal neutron field in which the neutrino is propagating. To this
end we assume in the present calculation that the neutron star
medium is an external fixed background which is unaffected
by the neutrino ‘‘back reaction.’’ This vacuum energy can be
viewed as equivalent to the energy of the neutrinos which
condense out of the vacuum condensate energy due to the
presence of the external fixed neutron medium. In this paper
we analytically evaluate the vacuum energy for both a mass-
less and massive Dirac neutrino using the formalism of
Schwinger 4 and Hartle 5. Our results for the massive
case are new, and serve as a basis for distinguishing between
the vacuum energy computed here and the energy of inter-
action between the neutrons themselves arising from
neutrino-exchange self-energy of the neutron field20.
The results for the massless case are equivalent to those ob-
tained previously by other authors using different methods
11–17.
The outline of this paper is as follows: In Sec. II we
evaluate the vacuum energy for a massless Dirac neutrino
propagating in the presence of a neutron background. This
result is in agreement with that obtained by others 11–17.
In Sec. III we derive the same result using a different method
which evaluates the energy of the neutrino condensate
formed in the external neutron background. In Sec. IV we
introduce the formalism for dealing with massive neutrinos,
which we then use for repeating the two previous calcula-
tions. Section V presents our conclusions, and in the Appen-
dix we evaluate an integral which arises in our formalism.
II. ENERGY OF THE VACUUM
Our starting point is the low-energy Lagrangian density
describing the interaction between neutrons and massless
neutrinos 21 which, in the notation and conventions of
Sakurai 22, is given by
L=-
¯
x
” +
4
1 +
5
x . 1
In Eq. 1, ( x ) is the 4-component Dirac neutrino field for
any flavor, and is the electroweak neutrino-neutron poten-
tial arising from neutral current interactions induced by a
classical neutron number density ,
G
F
a
n
2
=
-G
F
8
. 2
Here G
F
is the Fermi constant, and a
n
=-
1
2
is the neutrino-
neutron coupling constant 21. To calculate the vacuum en-
ergy we use Schwinger’s formula for the vacuum expecta-
tion value of the interaction Hamiltonian 4, where the
Hamiltonian is obtained from the Lagrangian in Eq. 1. Fol-
lowing Schwinger 4 and Hartle 5, we subtract the bare
Hamiltonian ( H
0
) to renormalize the theory, and therefore
include only the energy due to the interaction:
W 0 | H| 0 - 0 | H
0
| 0 =i
dE
2
Tr log 1 +D
F
(0)
r
, E ,
3
where
D
F
(0)
r
, E i
4
1 +
5
S
F
(0)
r
, E , 4
and S
F
(0)
( r
, E ) is the neutrino vacuum propagator in the
coordinate-energy representation 5,7. The trace Tr in Eq.
3 is over both configuration and spinor space.
When is taken to be a constant over all space i.e. an
external field with no spatial variation, completeness of
states can be used to collapse all but one spatial integration.
The generalized trace Tr reduces to a spinor trace tr, and
the vacuum energy W
vac
arising from Eq. 3 becomes
W
vac
=i
d
3
x
d
3
p
2
3
dE
2
tr log 1 +D
F
(0)
p , 5
where
PHYSICAL REVIEW D, VOLUME 61, 125010
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