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[Phys-l] CP Violation and the Arrow of Time.




CP Violation and the Arrow of Time.


The weak interactions are not invariant under the parity transformation.
We can see this via the decay

Pion(+)= Anti Muon(+) + muon neutrino

Where in every case the anti muon comes out left handed. Nor are the weak
interactions invariant under the C transformation as revealed by:


Pion(-) = Muon(-) + anti muon neutrino


Where the Muon always comes out as right handed. However if we combine
the two operations, the CP operator we would seem to have a conserving
operation. But in fact this is not the case. In nature we do observe
violations of CP invariance as will be related below. Given that almost certainly
the operation of CPT is strictly conserved, the discovery of CP violation is
a discovery of an intrinsic breakdown of time reversal invariance.

In this post we will take a look at CP violation for the Kaon particle,
though the exact same mechanism of CP violation is seen for the D^0 and B^0
pseudo scalars.

The K mesons are known to known to violate strangeness charge conservation
because the physical manifestation of K mesons are linear combinations of
the Kaon and anti Kaon.

We have the following Eigenstates;


[ K(0) > = [ d sbar> = [-1/2 +1/2> [ K(0)bar > = [ dbar s>
= [+1/2 -1/2>


Being pseudoscalars we have


P[K(0)> = - [K(0)> P[K(0)bar>= -[K(0bar>


Therefore under the CP transform we have;


CP[ k(0)>= - [K(0)bar)> CP[K(0)bar> = - [ K(0)>



However due to the internal exchange of the W boson in the physical
pseudoscalars we get the following linear combination as the physical
manifestation of the Kaon particles.


[K(1)> = (1/srt[2])*{ [ K(0)- K(0)bar>}


[K(2)> = (1/srt[2])*{ [ K(0)+ K(0)bar>}


These two physical particle states, as can very easily be seen, have
opposite eigenvalues in terms of their CP Eigenstate.


CP[K(1)> = [ K(1) > CP[K(2)> = - [ K(2)>


Assuming CP is strictly conserved the K(1) can only decay into a state with
CP= +1 and the K(2) can only decay into a state with CP=-1. Since a
prominent decay channel for Kaon decay is into the pseudoscalars pions we should
always observe the K(1) decay into two pions (- 1 *-1 =+1) and the K(2)
decay into three pions. ( -1 *-1 *-1 = -1 )



This would give the K(1) a much shorter mean decay time than the K(2) due
to energy considerations. And if fact we observe

t_K(1) = 8.9 E -11 seconds and t_K(2)= 5.2 E-8 seconds


Note that K(1) and K(2) are not antiparticles of each other. Both particles
are their own anti particles, a kind of pseudoscalars Majorana State. Also
these particles have a mass split.


m_1-m_2= 3.5 E-6 ev


It is important to understand that real observable physical states for
the Kaon system are K(1) and K(2) because these states carry the unique
lifetimes.


Based on this is if we prepare a beam of Kaons and CP is conserved we
should observe no two pions decays after a time significantly greater than
8.9 E-11 seconds. But this is not what is observed. Given any time of
flight for the Kaons we always observe a tiny fraction of two pion decays.
Therefore CP is violated in the Kaon System.

This means that physical Kaons are not pure K(1) and K(2) states but rather
mixtures of this states we call K_L and K_S. We can write the equation for
this as;


{ K_s K_L } = U*{K(1) K(2) }


U_11 = 1/sqrt[1+a^2] U_12= - a/sqrt[1+a^2]



U_21= a/sqrt[1+a^2] U_22= 1/sqrt[1+a^2]


Where a= 2.3 E-3 for the Kaon system.


We can write this equation in terms of the CP phase angle


phi_CP= cos^-1[ 1/sqrt[1+a^2]] = sin^-1[ a/sqrt[1+a^2] ]



So that we have



{ K_s K_L } = exp[-i*phi_CP*sigma_y* {K(1) K(2) }


For the Kaon system we find that phi_CP= 0.13178 degrees.


This angle is far too small to explain the photon to Baryon ratio we
observe in the Universe today but the CP violation process does reveal how
our Universe came to be dominated by matter.


It's very important to see that in this system CP violation, hence
time reversal violation is not the result of any thermodynamic like process,
but rather is inherent in the superposition state of observable Kaon
particle states. We have'


K_s= K(1)*cos[phi_CP] - K(2)*sin[phi_CP]



K_L= K(1)*sin[phi_CP] + K(2)*cos[phi_CP]


The degree of time reversal invariance is determined by the value of
phi_CP. If phi_CP= 0 there would no CP based arrow of time.


Bob Zannelli



















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