Dark energy and particle masses
Corrado Massa
Via Fratelli Manfredi 55
42124 Reggio Emilia -- Italy
In the following natural units c = h = 1 are employed
(c = the speed of light, h = the reduced Planck constant).
In the absence of any pressure, the mass M 0 of a body
with mass density μ and volume V is M 0 = μ V; in the
presence of a scalar pressure P the mass of the body is,
according to the general relativity theory [ 1 ]
M = M 0 + 3 P V ( 1 )
If the only pressure acting on the body is the negative
pressure P = – D due to the dark energy field with
positive mass density D then the total observable mass
of the body is
M = M0 - 3 D V ( 2 )
where M0 is the mass the body would have in the
absence of the dark energy field. I call it the “ bare mass “
of the body.
Now I make the bold assumption that eq ( 2 ) applies not
only to macroscopic bodies but also to elementary particles;
V should be interpreted as the natural volume related to
the particle
V = η / M 3 ( 3 )
where η is a dimensionless factor of the order of unity.
This is the ” natural volume” because it is impossible to
localize a particle of energy E with a better accuracy than
about 1 / E , that is 1 / M in a frame at rest with respect
to the particle [ 2 ].
Insert eq ( 3 ) into eq ( 2 ), rearrange terms, and get:
M 0 = M + 3 η D / M 3 ( 4 )
The bare mass M0 , function of the total, observable mass
M, has its minimal value M0 (min) = m (say) at
M = ( 9 η D ) 1 / 4 ( 5 )
which for eq ( 4 ) gives
m = ( 4 / 31 / 2 ) ( η D ) 1 / 4 ~ M ( 6 )
Displaying c and h, this reads:
m = ( 4 / 31 / 2 ) ( η D ) 1 / 4 ( h / c ) 3 / 4 ~ M ( 6 a )
What is m ? According to our procedure, m must be
interpreted as the minimal bare mass in the universe.
If the non gravitational self – interactions are the
dynamical source of the particles bare masses, as
assumed e.g. in [ 3, 4, 5, 6 ], then the bare mass of a
particle is related to the dimensionless coupling
constant of the main interaction the particle is subject
to, and the main interaction related to a particle with
bare mass m 0 should be the weakest interaction in
the world, since m 0 is the minimal bare mass in the
world. Valev [ 6 ] suggests the following linear relation
between bare masses and dimensionless coupling constants:
m 1 / m 2 = K 1 / K 2 ( 7 )
m 1 , m 2 are the bare masses of two whatever particles;
K 1 , K 2 are the dimensionless coupling constants of the
main interaction such particles are subject to; for instance,
if m 1 is the pion bare mass, then K1 is the strong coupling
constant [5 ], if m 2 is the electron mass, then K 2 is the
electromagnetic coupling constant, and so on. Accordingly,
the smallest dimensionless coupling constant related to the
weakest interaction (of non gravitational nature ) in the world
is expected to be ( with α = the electromagnetic coupling and
M e = the electron mass; of course other choices are possible):
K (min) = m ( α / M e ) ~
( α / M e )( η D ) 1 / 4 ( h / c ) 3 / 4 ( 8 )
Observations on the accelerated expansion of the universe
give D ~ 10 – 29 g / cm 3 hence from eq ( 6 ) and eq ( 8 ) one
finds
M ~ 10 – 3 eV ~ 10 – 36 g , K (min) ~ 10 – 12
that are respectively the neutrino mass (according to
most GUTs ) and the dimensionless coupling constant
of the weak interaction [ 6 ]. The weak interaction is
very likely the weakest non gravitational interaction in
the world.
REFERENCES
[ 1 ] Tai L. Chow: Gravity, Black Holes, and the Very early
Universe (Springer, 2008) Sec 9, p. 185.
[ 2 ] L. J. Garay: Int. J. Mod. Phys. A 10, 145, ( 1995 )
[ 3 ] K.W. Ford: The World of Elementary Particles, Chap. 6,
Blaisdell (New York) 1963.
[ 4 ] R. P. Feynman, R. B. Leighton, M. Sands:
The Feynman Lectures on Physics, Vol II,
Secs 28 – 6 and 8 – 5,
Addison - Wesley, Reading, MA ( 1964 ).
[ 5 ] C. Massa: American J. Phys. 53, 908 ( 1985 )
[ 6 ] D. Valev: arXiv : hep – ph / 05 07 25 5 v 5 (12 May 2006)