Introduction
Lowest Order
∆
I
=
3 / 2 Decays
∆
I
=
3 / 2 Quenched
K
ππ
Chris Sachrajda
School of Physics and Astronomy
University of Southampton
Southampton
UK
UKQCD Collaboration Meeting
January 23 – 24 2006
K
→
ππ
Decay Amplitudes
UKQCD Collaboration Meeting, January 23/24 2006

Introduction
Lowest Order
∆
I
=
3 / 2 Decays
∆
I
=
3 / 2 Quenched
• During recent Grid Conferences of the DWF collaborators we have been discussing possible K →
ππ projects.
(Jun Noaki and I prepared a discussion document which has been replied to by Norman Christ and RBC collaborators.)
• Status: In 2001, the RBC and CP-PACS collaborations published some very interesting results on the
∆
I = 1 / 2 rule and of
ε
0 /
ε
:
Collaboration(s)
RBC
Re A
0
/ Re A
2
ε
0 / ε
25 .
3 ± 1 .
8 − ( 4 .
0 ± 2 .
3 ) × 10 − 4
CP-PACS
Experiments
9 ÷ 12
22.2
(-7 ÷ -2) × 10 − 4
( 17 .
2 ± 1 .
8 ) × 10 − 4
• The two collaborations evaluated (quenched) matrix elements of the form h M | O | M i
(where M is a pseudoscalar meson) and used LO
χ
PT to determine the corresponding K →
ππ matrix elements.
K
→
ππ
Decay Amplitudes
UKQCD Collaboration Meeting, January 23/24 2006

Introduction
Lowest Order
∆
I
=
3 / 2 Decays
∆
I
=
3 / 2 Quenched
20
15
25
10 experiment
ω
=Re A
0
/Re A
2
16
24
3
3 x32 x32 chiral log.
quadratic
CP-PACS
15
10
5
0
25
20
5
0
0 0.1
0.2
0.3
m
M
0.4
2
[GeV
2
]
0.5
Re A
0
/Re A
2
0.6
0.7
0.8
KTeV
NA48
ε
’/
ε
[10 ]
16
3
24
3 x32 x32
0 0.1
0.2
0.3
m
0.4
M
2
[GeV
2
]
0.5
ε
0 /
ε
0.6
0.7
0.8
K
→
ππ
Decay Amplitudes
UKQCD Collaboration Meeting, January 23/24 2006

∆
I
=
3 / 2 Decays
Introduction
Lowest Order
∆
I
=
3 / 2 Quenched
1.
Repetition of the RBC/CP-PACS Calculations with the QCDOC
Configurations.
2.
A state-of-the-art Study of
∆
I = 3 / 2 K →
ππ
Decays.
3.
∆
I = 3 / 2 K →
ππ
Decays at Physically Small Masses in the Quenched
Approximation.
4.
∆
I = 1 / 2 K →
ππ
Decays. (Nothing to say, other than to note that new ideas and R&D are necessary.)
5.
Chiral Perturbation Theory and NPR. (I will not discuss this today.)
K
→
ππ
Decay Amplitudes
UKQCD Collaboration Meeting, January 23/24 2006

Introduction
Lowest Order
∆
I
=
3 / 2 Decays
∆
I
=
3 / 2 Quenched
The 3 major sources of error in the RBC/CP-PACS calculations were:
I quenching;
I
The chiral extrapolation (the calculations were performed at lowest-order in
χ
PT and with mesons in the range 400 – 800 MeV);
I
The subtraction of power divergences, particularly in the matrix elements of O
6
.
Repeating the calculation with the QCDOC configurations would
I eliminate quenching;
I
Reduce the quark masses substantially ⇒ we will be able to check that the octet enhancement increases as the quark mass decreases.
The
χ
PT remains at LO however ⇒ significant errors remain (at least of
O ( m
2
K
/ Λ )) ;
I
Increase the error due to the subtraction of power divergences (smaller statistics).
K
→
ππ
Decay Amplitudes
UKQCD Collaboration Meeting, January 23/24 2006

Introduction
Lowest Order
∆
I
=
3 / 2 Decays
∆
I
=
3 / 2 Quenched
• Status: This project is high priority for the RBC collaboration and it will be carried out. At the moment there is no UK commitment.
K
→
ππ
Decay Amplitudes
UKQCD Collaboration Meeting, January 23/24 2006

Introduction
Lowest Order
∆
I
=
3 / 2 Decays
∆
I
=
3 / 2 Quenched
∆
I
=
3 / 2
• The collaboration is in a position to carry out a major state-of-the-art project on
∆
I = 3 / 2 decays and I = 2
ππ phase-shifts.
This would be a significant improvement on the SPQR project
( hep-lat/0412029 ):
I it would be an unquenched calculation;
I the finite-volume effects in the matrix elements are now understood and so can be accounted for;
I smaller masses ⇒ χ
PT may be applicable;
I better chiral behaviour for DWF than improved-Wilson fermions;
I
I for the EWP operators we would be able to compare the direct calculations of K →
ππ matrix elements with a NLO determination using
K →
π and K → 0 (see Jun’s recent work);
I = 2
ππ phase-shifts will be calculated (with twisted boundary conditions to increase the momentum range).
K
→
ππ
Decay Amplitudes
UKQCD Collaboration Meeting, January 23/24 2006

Introduction
Lowest Order
∆
I
=
3 / 2
∆
I
=
3 / 2 Decays
∆
I
=
3 / 2 Quenched
• Status: This project is high priority for the Southampton Group and it will be carried out.
I have volunteered to write a detailed manifesto of what should be calculated.
K
→
ππ
Decay Amplitudes
UKQCD Collaboration Meeting, January 23/24 2006

Introduction
Lowest Order
∆
I
=
3 / 2 Decays
∆
I
=
3 / 2
∆
I
=
3 / 2 Quenched
RBC (with some interest from Southampton) are keen to perform a quenched calculation of
∆
I = 3 / 2 decay amplitudes at approximately the physical quark masses.
• Status: This is still open for discussion.
K
→
ππ
Decay Amplitudes
UKQCD Collaboration Meeting, January 23/24 2006