PROTON DECAY
The Holy Grail in particle physics is the unification of all of the
fundamental forces into a single fundamental force. However, a slightly less ambition
goal, and a goal perhaps more easily attainable, is the Unification of the
three vector forces, forces for which a successful quantum theory exists. This
effort goes by the name of Grand Unification Theory. (GUT)
The most straight forward attempt utilizing SU(5) symmetry , has not proved
successful because it predicts a decay constant for the proton which is too
small based on the ongoing attempts to observe proton decay. In fact, it may
well be, that in order to derive a successful Grand Unification theory we must
incorporate a new symmetry of nature, called Super Symmetry, a symmetry which
relates to any future quantum theory of gravity based on String Theory. Or
alternately, we may need a larger symmetry based on the group structure SO
(10).
Nevertheless, the basic picture given to us by SU (5) may well contain an
essentially correct description of the fundamental particles of nature in broad
terms,, as SU (5) can be derived from the larger symmetry of SO(10).
Therefore it will be instructive to look as
certain aspects of SU (5) , especially as to how it relates to Proton Decay.
SU (5) symmetry is based on unifying the color interaction, Quantum
chromodynamics (QCD) described by
(SU (3) _c) symmetry
And the electroweak interaction described by
(SU2)_w X U (1) _em) symmetry
These interactions have a three and two “color” charge structure
respectively. (Here color having nothing to do with its normal meaning of course. There
is some reason to view these color charge structures as topological and
being dual to the corresponding Noether charges of QCD and the electroweak
model.) The more universal electric and hyper charges are derived from the unified
structure of the QCD and electroweak
charge structure. Therefore we can see that:
SU (5) = SU (3) _c X SU2) _L X U (1) _Y
This structure forms a five dimensional vector and a ten dimensional anti
symmetric tensor structure for the fermion particle states of the standard
model.
Psi_5 = {d_c, ebar, nubar_e} _R
And
Psi_10= {d_c, U_c, Ubar_c, ebar} _L
And
CP [Y_5> & CP [Y_10>
Notice that there is no right handed neutrino or left handed anti Neutrino.
These singlet states do exist in the larger SO (10) Symmetry which is
postulated to be involved in the unique mass generation scheme proposed for the
neutrinos under the assumption they are Majorana Spinors , the so called See Saw
model.
In SU (5) we can define each fermion particle state in terms of its five
color charges.
[Psi>= [r,w,b,g,y,>
So that
[U_L> _r = [10010> [Ubar_L> = 01100>
d_L>_r = [10001> [dbar_L>_r = [-1 0000>
[ e_L > = [ 000 -1 0> [ebar_L>= [ 00011>
[nu_L> = [0000-1 >
[U_R>_r = [0-1-1 00> [Ubar_R>_r= [-1 00-1 0>
[d_R>_r> = [100000> [dbar_R>_r = [-1 000-1>
[ e_R>= [ 000-1-1> [ nubar_R>= [ 00001>
This gives us
Q_weak= (1/2)*{g-y, y-g}
Q_color= (1/2)* {r-b, w-r, b-w}
Q_Y= (-2/3)*(r+w+b) + (g+y)
Q_em= (-1/3)*(r+w+b) +g
I_3= (1/2) *(g-y)
Giving us the QCD charge structure
{U_r,d_r}_L = {100} { Ubar_r,dbar_R}= {-100}
U_R= {0-1-1} d_r= {100}
Ubar_L= {011} dbar_L= {-100}
And the WEAK charge structure
{u,d}_L= {10, 01} {ubar,dbar}_R= { -1 0 ,0-1}
{e,nu}_L= {-10, 0-1} {ebar,nu}_R = {10, 01}
U_R= {00} Ubar_L= {00}
d_R= {0, 0} dbar_L= {0, 0}
e_R= {-1 -1} ebar_L= {1 1}
In the full SU (5) symmetry there are 24 generators corresponding to the 24
interchange bosons. Therefore SU (5) postulates twelve additional new bosons
beyond the standard model commonly called the X and Y bosons. These bosons
carry color and fractional electric charge and the X boson carries weak charge.
We can write (color indices in caps are anti colors)
X (4/3) _R= U_b + U_w = ebar+dbar_R
Y (1/3) _R = U_b +d_w= nubar+dbar_R
This provides a unique symmetry which unites the quark and lepton sector,
predicting the decay of the proton.
So we have
(X channel)
d_r= X (-4/3) _r + ebar & X (-4/3) _r+ U_b = Ubar_W
P= pion(0)+ ebar = 2Xphoton +ebar
(Y Channel)
d_r= nubar + Y (-1/3) _r & Y (-1/3) _r + U_b= dbar_W
P=pion(+) =nu_e = nu_e + antimuon + nu_muon
The mean time for decay is a function of the Compton time of the proton and
the ratio of the mass of the Grand Unification boson to the proton mass.
Obviously the greater the mass of the GUT boson the lower the probability of the
quantum fluctuation needed to affect proton decay.
Taking these factors under consideration we get
Lambda (decay constant) = m_prot^5*c^2/ (h*m_X(Y) ^4
So that the probability is
P (t) = 1-exp [-(m_prot^5*c^2/ (h*m_X(Y) ^4)*t]
The simplest version of SU (5) predicts a GUT boson mass no larger than 1E16
proton masses. This gives a mean life time of just over 1E33 years. This
value has been ruled out by observation. Since the actual proton mean decay time
must be greater, some additional mechanism to drive up the GUT boson mass is
needed. Some Physicists think the existence of super partners may do this.
Alternatively the larger SO (10) symmetry also predicts a longer mean decay
time for the proton.
Therefore attempts are being made to observe the possibility of proton decay
using even more sensitive detection arrangements. So far no confirmed proton
decays have been observed. The search continues.
Bob Zannelli
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