Ref: Indian Science Congress 68th
Session,
Mathematics Section, Banaras
Hindu University, 3-8th January 1981.
GRAVITON-ANTI GRAVITON
ANNIHILATION AND CREATION OF
FERMI QUARK - F-MESON PAIR MULTIPLET
K. L.
Narayana,
Shivaji
University, Kolhapur – 416004.
Present
Address: Prof. Dr. Kotcherlakota Lakshmi Narayana,
17-11-10,
Narasimha Ashram, Official Colony, Maharanipet.P.O., Visakhapatnam-530002.
ABSTRACT
Within
the context of an unified Quasi U(2,10) supersymmetric Gravity theory the
process Graviton + Anti-Graviton à Fermi Quark + Anti-Fermi Quark + f-meson +
Anti–f-meson has been analyzed as a four body decay process and the life time
(Half-life) has been obtained as τ f i = C - 2 . 10 - 59 where the new coupling constant is
shown to be as fundamental and the Newtonian coupling constant. Also is the
suggested “connectivity” between the two space-time domains of interactions one
of the elementary particles and the other of Graviton-like particles. The
relationship between the mass ratios of f-meson and Graviton-like Spin 2 (and
Spin 5/2) and the new coupling constant C, of the universe has been explicitly
given adopting the COSMOD transformations (Il Nuovo Cimento, Vol.33A, p.641,
1976) and the PERANGON
transformations(Int. Symp. on Appl. Math. Tech. to Phys. Sciences,
December 31, 1976). The decay cross-section expressions are reported.
Ref: Indian Science Congress 68th
Session, Paper No.114, Mathematics Section, Banaras Hindu University, Varanasi,
3-8th January 1981.
GRAVITON-ANTIGRAVITON
ANNIHILATION AND CREATION OF
FERMI QUARK - f-MESON PAIR MULTIPLET
K. L.
Narayana,
Shivaji
University, Kolhapur – 416004.
Present
Address: Prof. Dr. Kotcherlakota Lakshmi Narayana,
17-11-10,
Narasimha Ashram, Official Colony, Maharanipet.P.O., Visakhapatnam-530002.
INTRODUCTION
Dirac
[1] form of gravitation equation, of motion has been earlier formulated by the
author. The underlying group has been suggested as one of a direct product of
Kemmer-Duffin ring structure with the Spin 5/2 angular momentum operators. This
set of generators analogous with SU(2,2) ≃ O(4,2) isomorphic group have
been utilized to construct a unified Lie Algebraic structure i.e. isomorphic
with the sub-algebra of a quasi U(2,10) group. The formulation also leads to
other kinds of Gravity possibilities of gravitation such as the non-compact
Graviton spectrum existence of a Graviton vital space and as well that the
Charm Graviton to be able to be classified as a singlet of unified description
of Graviton-Lepton-Hadron super-symmetric structures [2].
The fact that the Graviton has spin 5/2
leads to either possibility that it may obey Einstein-Bose statistics or
Fermi-Dirac statistics. The Bosonic Graviton of half-integral spin 5/2 was
commented upon by the author extensively [2] previously. The object of the
present paper is to examine more closely the feasibility of its obeying a
Fermi-Dirac statistics in the context of four body decay processes that would
consequently become allowed. Obviously, such a formulation has the immediate
physical consequence of existence of an Anti-Graviton. The Anti-Graviton
(though the authors approach is distinct) concept is not new one and is already
discussed within the frame work of super-symmetry models, that are being
investigated in literature by Ferrara,[3], Wess and Zumino[4] and recently by
Davis [5] and Narayana [6] certain features of the Graviton- Anti-Graviton
processes have been considered by Zeldovich[7].
For our investigation presented herein
firstly it may be recognized that the Fermionic Graviton of Spin 5/2 would have
a corresponding Anti-Graviton of opposite momentum. The Graviton and
Anti-Graviton system would lead to a compound system of spin angular momentum
total of five units. The particular possibility that this system can decay into
a Quark - Anti-Quark and two f-mesons is one of a great interest, as it would
lead to events that are verifiable experimentally. The author earlier model
that the hadrons and as well the Leptons are the bound structures of a
fundamental super-quark system of Quasi U(2,10) group certainly implies a decay
process of the kind above mentioned. It is interesting that a number of
features of this decay mode are amenable for an analysis without explicitly
adopting either the covariant-perturbation theory or the dispersion technique
for the evaluation of the cross-section. Analogous approaches of analysis were
previously adopted in the early stages of formulation of theories of β-decay
(Low Energy) and in recent times, in the studies of Vector Boson mediated
electro-weak forces (High Energy).
In section-1 the details of a four body
decay of final states are given. In section-2 the expressions of the decay mode
in the center-of-mass frame of reference have been detailed. Section-3 presents
the results and conclusions drawn by the present study of the Graviton – Anti-Graviton
annihilation process.
SECTION-
1: Decay to four particles final state:
The four body reaction is written as,
a + b à d + e + f + g
with
the decay cross-section given by,
Fig.1
where
σ is the cross-section of the
process from the initial to the final state of momenta Pi to Pf
respectively, V the volume of spatial extension and Mfi the matrix
amplitude of the process. From the later one can construct the scattering
matrix
Fig.2
with
Mfi = τ f i / N
where N is the appropriate normalization factor. The transition probability per
unit time ω = (1/(VT)) |Rfi|2
with the standard relation of perturbation theory connecting Rfi with the Mfi.
Also we can write
Fig.3
SECTION
– 2: Center-of-mass frame of
reference
In
the center-of-mass system, we have,
Fig.4
The
differential cross-section,
Fig.5
An
important step that has been adopted in deriving this, the expressions are
valid in the limit of zero mass as well just as in β-decay
Fermi-coupling constant derivation. But, the quarks and as well the f-mesons we
note are massive. The mass consideration however, does arise since the Graviton
and Anti-Graviton rest mass energies are of finite magnitude. This is a rather
radical departure from the conventional understanding of gravitation and its
propagation by the Graviton mediator. In our theory, being presented here, the
Graviton and rest-mass and as well has the Anti-Graviton has a negative energy
and negative momentum.
Fig.6
where
mf, Ef are the mass and energy of the F-meson.
Fig.7
Where
C2 is proportional to the product of strong interaction strength g
2 / 4π = 14, and that of weak
interaction strength
(g
2ω m2 π
/4π)
≃
10-14.
SECTION
– 3: PERANGON TRANSFORMATIONS AND RESULTS
It
is necessary for practical calculations then to take the derived formula for
the half-life (τ f i ) of the decay process.
Graviton + Anti-Graviton à Quark Pair + (Pair) F-meson
or
g + ̅g
à
ν + ̅ν + f
+ ̅f
to be given as follows,
C
2 mf7 /( 64x64 π5) *(Δ
/ mf)6 ≃ 1/ τ f i
when
Δ =
2 mf it simplifies to
1/
τ
f i = C
2 mf7 / 64 π5
when
mf is about 1250 MeV we get,
1/
τ
f i = C2 . 1059
as
the order of magnitude.
Thus we arrive at an incredulously low
half-life of the decay
process i.e.
about 10 - 45 sec. What may be the physically reasonable life-time
for the process that is envisaged in the present work? It can be said that it
is many fold move quicker than any known strong interaction order of time
periods! What is the optimum half life-time order of magnitude for the
creation process following the annihilation of Graviton - Anti-Graviton pair?
PERANGON Transformation has been
defined earlier which connect the domains of two types of spin 5/2 particles
while discussing about the “Cosmod Transformations”. The author has stressed that the ratio of masses between the two
different kinds of Spin 2 particles, should be taken as fundamental as the
Newtonian coupling constant. In relevance to these investigations, the
present work reveals that there, exists a new type of coupling constant which
is actually to be taken as fundamental as the Newtonian coupling constant.
There exists an intimate relationship of this new coupling constant with the
mass ratios of the two domains of particles, and also with the half life-time
of the decay process outlined above. The interaction coupling strengths and the
mass ratios both play a significant role in the determination of the half
life-times of the particle creation and annihilation processes.
What’s the new “connectivity” that
exists between the two domains of interactions one that of particle would as
observed to possess the Charm, Hyper-charge, Strangeness etc. quantum numbers
and the other domain of Graviton-like particles, such as spin 5/2 and spin 2
particles. This topic would be elaborated in a forthcoming publication.
CONCLUSION
To conclude the present work we state that
the value of τfi = 10-45 sec is almost double the usual
interaction life-time of the order of 10-23 sec., for the strong interaction
processes. Since the gravitational interaction are much faster relative to the
strong interactions. The value of τfi = 10-45 sec is not physically
unreasonable.
ACKNOWLEDGEMENT
The
author is grateful to Late Prof. K. R. Rao, D. Sc. (Madras) D.Sc. (London) for
his stimulating research endeavor at his laboratories in Andhra University,
Waltair.
REFERENCES
1. Dirac
2. Bosonic
quanta Narayana
3. Ferrar
4. Wess
and Zumino
5. Davis
6. Narayana
7. Zeldovich
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