SYMMETRIES
of B s and K s
A Pedagogical Consideration:
Simplification to a homework problem
This Month‘s Special
RUF‘s Theorem
Unitarity of D
H.-J. Gerber, ETHZ
[email protected]
MITP Workshop on T violation and CPT tests in neutral-meson systems 15.- 16 April 2013, Uni Mainz.
CPLEAR
<σk> = <ψ(t)|σk|ψ(t)>
CP violation
k = 1, 2, 3.
T violation
ψ(t) = e-iΛt ψ(0)
CONTRADICTS
Time reversal symmetry of Hweak
ψ(t) → <σk>
Homomorphism SL(2,C) onto sLT
CP is violated, whenever ...
KL
x3
Re(ε-δ)
x1
x2
Im(ε-δ)
Cannot have CP conserved, T and CPT violated.
Reward: Conservation Law for CP.
CPT (δ) and/or T (Re(ε)) is violated
The Parameters of Measurement
Time evolution governed by „Grand Schödinger Equation“
iħ(∂/∂t) |ψALL> = H |ψALL>
with H = Hstr+elm + Hweak
Reduce to 2 dimensions: |ψALL>  |ψ> .
Amplitude for evolution and decay of B: ABf = <f |T e-iΛt |B>
Measure T , Λ
Reward for CP: (for not having its own parameter)
If CP-1 T CP = T and CP+ = CP-1 (unitary)
then <f |T| B> = < CP f |T| CP B > ( = < f | CP+ T CP | B > )
Measure the matrix T for B s
Assumptions
Λ is Time reversal- and CPT symmetric, ΔΓ = 0 ,
Ti≠j = 0 „Δb = ΔQ rule“ ,
No FSI.
Then
U=
e-iΛt
= U0
cos(Δm t/2)
-i sin(Δm t/2)
-i sin(Δm t/2)
cos(Δm t/2)
, |U0|2 = e-Γt .
Matrix representation of |ABf |2 = | <f |T e-iΛt_ |B> |_2
|ABf |2 = fi*fj T*iiTjj U*ikUjm bk* bm . Basis K0K0, B0B0.
Strategies
significance and choice
Greatest sensitivity of |ABf |2 to T.
|ABf |2 /e-Γt =
(1/4) (|T11|2+ |T22|2)[(f12+f22) (b12+b22) + (f12-f22) (b12-b22) cos(Δm t)]+
(1/4) (|T11|2 - |T22|2)[(f12- f22) (b12+b22) + (f12+f22) (b12-b22) cos(Δm t)]+
2 Re(T11* T22)
[ f1f2b1b2 ]+
Im(T11* T22)
[ f1f2 (b12-b22) sin(Δm t)] .
(1)
Examples:
B0
Shorthands:
K0 ~
~
0
0~
B
_ ~K
_
1
0
0
1
(Phase convention)
J/ψKS ~
1
J/ψKL ~
1
1
-1
(|T11|2 - |T22|2) ≡ TCPT
Im(T11* T22) ≡ TT
CP violation
TT ≠ 0 or/and TCPT ≠ 0 .
(- TCPT cos(mΔ t) – 2 TT sin(mΔ t) ) / ( |T11|2 + |T22|2).
From (1).
Data show
Sine ! No sign of cosine ?
Is the question
sin(mΔ t)
TT found
Fourier analysis by eye
Aubert et. al, (BABAR Collaboration)
Observation of CP violation in the B0 meson system
Phys. Rev. Lett. 87, 091801 (2001) .
CPT violation
Calculate
_
Fwd = < B_ B0 >
Bwd = < B_ B0 >
Find
( |Fwd|2 - |Bwd|2 ) /e-Γt = - TCPT cos(Δm t).
Compare
- TCPT cos(Δm t)
Experiments
- TCPT cos(Δm t) – 2 TT sin(Δm t)
CPT
CP
T violation ! DIRECT ?
T violation and Motion Reversal
Motion Reversal: Compare B0  B+ vs B+  B0.
Identify the B+ (~ KL) by observing the decay to KS („first decay“)
of its orthogonal entangled partner (B-).
Let |h> = state of first decay’s products. The surviving state
is |surv> = iσ2K T-1 |h>. σ2: Pauli matrix, K : Complex conjugation.
The backward amplitude Bwd is then
Bwd = <B |U iσ2K T-1| h> , to compare with
Fwd = <f |TU| B> .
Need a matrix representation of | Bwd |2 = | <B | UD | h> |2
“Disentanglement Operator” D ≡ iσ2K T-1 .
MR violation,
is it T
violation ?
| Bwd |2 |T11 T22|2 /e-Γt = | <B | U D | h> |2 |T11 T22|2 /e-Γt =
(1/4) (|T11|2+ |T22|2)[(h12+h22) (b12+b22) + (h12- h22) (b12-b22) cos(Δm t)]
+ (1/4) TCPT
[(h12- h22) (b12+b22) + (h12+h22) (b12-b22) cos(Δm t)]
+ 2 Re(T*11T22) [ h1h2b1b2 ]
+ TT
[ h1h2 (b12 -b22) sin(Δm t)] .
(2)
NO ?
Assumption
|T11
T22|2
= 1 (preliminary).
( sometimes YES )
Apply (1) and (2) to B0  B+ vs B+  B0. |f > = |KL > , |h> = |KS >.
MRV(B0 B+) = ( |Fwd|2 - |Bwd|2 ) /e-Γt =
TCPT cos(Δm t) – 2 TT sin(Δm t) . (3)
Compare CP violation
- TCPT cos(mΔ t) – 2 TT sin(mΔ t) .
RUF‘s Theorem
Background
Bernabeu, Martinez-Vidal, and
Villanueva-Perez, JHEP 08 (2012) 064
F. Martinez-Vidal (CERN EP Seminar 2012)
~
~
„B+ and B+ , and B_ and B_ have to be the same states.“
T. Ruf
~
„If the surviving state B_ needs to be the same as the state B_ ,
then
|T11| = |T22| .”
This implies TCPT = 0 .  MRV, eq.(3), is insensitive to CPTV.
Another Proof (unitarity)
Ruf‘s Theorem follows also as a special case from the physical
requirement, that the Disentanglement Operator D be unitary.
This entails unitarity of T.
With the „Δb = ΔQ rule“, T12 = T21 = 0, the result follows.
Let D + = D -1.
D = iσ2 T-1*.
Note: iσ2 = real, orthogonal, anti-symmetric, non-singular.
Thus, T is also unitary:
E. g.
T+ = T-1.
|T11|2 + |T21|2 = |T22|2 + |T12|2 =1
T11 T12* + T21 T22* = 0.
Corollary (RUF)
Apply “Δb = Q rule“ T12 = T21 = 0 and find
|T11| = |T22| ( = 1 ) .
Apply Unitarity of D
CPT violation:
_
MRV(B0 B_ vs B_  B0) =
( |Fwd|2 - |Bwd|2 ) /e-Γt = --TCPT cos(Δm t) ≡ 0.
Motion Reversal together with Disentanglement
excludes testing of CPT violation, but fine for TT .
Link to Basic Physics, Summary
The symmetry properties of Hweak in the
„Grand Schrödinger Equation“ (Significance for Basic Physics)
require for the parameters in the 2 dimensional representation:
This means, in the model of the „homework problem“
Compatible with symmetry
Incompatible with symmetry
Experiment
MRV „T“
MRV „CPT“
in Hweak
in Hweak
CPT
T, CP
CPT chosen
CPT chosen
T, (CP)
none
GREAT THANKS TO
MARIA FIDECARO
CERN
THOMAS RUF
CERN
and YOU !
FAQ s
1 Two Amplitudes in One Channel
2 Selected References
FAQ 1
FAQ 2
Selected References
• Kaons on the globe (and 121 references)
The fundamental symmetries in the neutral kaon system – a pedagogical choice
Maria Fidecaro and Hans-Jürg Gerber, Rep. Prog. Phys.69 (2006) 1713-1770.
• Significance of the parameters for Hweak (and much on fundamentals)
On the phenomenological description of CP violation for K-mesons and its
consequences
C. P. Enz and R. R. Lewis, Helv. Phys. Acta 38 (1965) 860-876.
Reprinted in L. Wolfenstein (ed), CP Violation, North-Holland (1989) .