Strategies Flavour-Symmetry to Extract
Robert Fleischer DESY Hamburg, Theory Group WIN 02, Christchurch, New Zealand, 2126 January 2002
Setting the Stage Isospin + SU (3) + Dynamics: from B K, U -Spin Strategies:
Focus on the following systems:
Bd + , Bs K + K B(s) K . Conclusions
Setting the Stage
Preliminaries
Central target of CP-B studies:
Im (,)
UT
Rb
Rt
0
1 2 /2 ,
1
1 2 /2
Re
Particularly interesting element for tests of KM picture:
Direct determination of
Comparison between dierent approaches. Comparison with UT ts, yielding 60.
Key Problem in Determination of
Wolfenstein (LO): CKM elements real, apart from
Vtd = |Vtd |ei
and Vub = |Vub |ei .
Sensitivity to due to interference eects between dierent CKM amplitudes in non-leptonic B decays:
CKM unitarity
A(B f ) = |A1 |ei1 + ei |A2 |ei2 A(B f ) = |A1 |ei1 + e+i |A2 |ei2 ,
where enters through Vub and |A1,2 |ei1,2 CP-conserving strong amplitudes ( hadron dynamics!?):
direct CP asymmetry:
ACP = |A(B f )|2 |A(B f )|2 |A(B f )|2 + |A(B f )|2
=
2|A1 ||A2 | sin(1 2) sin . |A1 |2 + 2|A1 ||A2 | cos(1 2) cos + |A2 |2
Goal: extraction of from ACP! Problem: hadronic uncertainties due to |A1,2 |ei1,2 :
|A|e
i
Ck ()
k pert. QCD
f |Qk ()|B .
unknown
Major Approaches to Extract
Try to calculate f |Qk ()|B : interesting progress
QCD factorization [Beneke et al.] PQCD [Li et al.].
B K,
Use decays of neutral Bd- or Bs-mesons:
0 0 Interference eects due to Bq Bq mixing!
Fortunate cases, where hadronic matrix elements cancel:
Bd D()K
[Dunietz & Sachs ...]
() Bs Ds K ()
2
Bd KS
+
0 Bq
f
s
Bs
+
[Aleksan, Dunietz & Kayser ...]
0 Bq
Amplitude relations to eliminate the hadronic uncertainties:
Exact relations: [Gronau & Wyler; Dunietz; R.F. & Wyler ...] Tree decays B KD or Bc DsD. Flavour symmetries, i.e. isospin, SU (3) or U -spin:
B(s) , K, KK
Focus of this talk!
Isospin + SU (3) + Dynamics:
from B K,
Basic Features of B K Decays
B K decays are governed by QCD penguins:
Example:
W
s
0 Bd K +
111 000 111 000 111 000 B000 111 d 111 000 111 000
b u, c, t
11 00 11 00 + 11 00 K 11 00 11 00 11 00
u
G
d d
W
u
111 000 111 000 + 111 000 111 000K 111 u 000 111 000
s u
11 00 11 00 11 00 11 00 11 00 11 00
Bd
11 00 11 00 11 00 11 00 11 00
b
d
111 000 111 000 111 000 111 000 111 000
penguins
|Vus Vub/(Vts Vtb )| 0.02
tree
penguins dominate!
Rle of EW penguins (large top-quark mass!): o
0 Bd K + , B + + K 0 :
contribute in colour-suppressed form and are expected to play a minor rle: factorization O(1%). o
0 B + 0 K + , Bd 0 K 0 :
contribute also in colour-allowed form and may compete with tree-diagram-like topologies O(20%)!
SU (2) isospin relations:
2A(B + 0 K + ) + A(B + + K 0 ) 2A(Bd K ) + A(Bd K )
0 0 0 0 +
=
C = |T + C|eiT +C ei + (Pew + Pew ) ei + qew . Trees EW Penguins
Amplitude relation with analogous phase structure also 0 for the mixed B + + K 0, Bd K + system.
Combinations of B K decays to probe :
B K , Bd K (mixed)
[R.F. (95); R.F. & Mannel (97); Gronau & Rosner (98)]
B K , B 0 K (charged)
[Gronau, Rosner, London (94); Neubert, Rosner; Buras, R.F. (98)]
Bd 0 K , Bd K (neutral)
[Buras & R.F. (98 00)]
Interestingly, already CP-averaged branching ratios may lead to highly non-trivial constraints on .
[R.F. & Mannel (97); Neubert & Rosner (98)]
Extracting from B K Decays
Key observables:
R A0
0 0 BR(Bd K + ) BR(Bd + K ) B + BR(B + + K 0 ) + BR(B K 0 ) B 0 d
Rc Ac 0
2
BR(B + 0 K + ) BR(B 0 K ) BR(B + + K 0 ) + BR(B K 0 )
Rn An 0
0 + + 0 1 BR(Bd K ) BR(Bd K ) . 0 0 2 BR(Bd 0 K 0 ) + BR(Bd 0 K 0 )
Employing the SU (2) avour symmetry and dynamical assumptions, concerning mainly the smallness of FSI:
(c,n)
R(c,n) , A0
= functions q(c,n) , r(c,n) , (c,n) , .
Here the following variables are involved:
q(c,n): ratio of EW penguins to trees. r(c,n): ratio of trees to QCD penguins. (c,n): strong phase between trees and QCD penguins.
[Buras & R.F. (98); alternative parametrization: Neubert (98)]
The q(c,n) can be xed through theoretical arguments:
B K , Bd K : q 0, as EW penguins contribute only in colour-suppressed form.
[R.F. (95); R.F. & Mannel (97); Gronau & Rosner (98)]
B K , B 0 K : qc can be xed through the SU (3) avour symmetry (no dynamics !).
[Neubert & Rosner (1998)]
Bd 0 K , Bd K : qn can also be xed through the SU (3) avour symmetry (no dynamics !).
[Buras & R.F. (1998)]
The r(c,n) can be xed as follows:
B K , Bd K : r can be xed using factorization or Bs K modes.
[R.F. (95); Gronau & Rosner (98,00); Beneke et al. (01)]
B K , B 0 K : rc can be xed from the B + + 0 branching ratio by using the SU (3) avour symmetry (no dynamics !).
[Gronau, Rosner & London (1994)]
Bd 0 K , Bd K : rn can also be xed through SU (3) from B + + 0 (no dynamics !).
[Buras & R.F. (1998)]
Uncertainties can be reduced through QCD factorization.
[Beneke, Buchalla, Neubert & Sachrajda (2001)]
Comments on Rescattering Eects
Whereas the determination of q and r as sketched above may be aected by rescattering eects, this is not the case for the qc,n and rc,n, since here SU (3) suces. Nevertheless, we have to assume that B + + K 0 or Bd 0 K do not involve a CP-violating weak phase:
i A(B + + K 0 ) = |P |e P = A(B K 0 ).
This relation may be aected by rescattering processes:
A(B
+ + 0 i i i K ) = |P |e P 1 + c e c e . 2 Rb
Example:
W
+
B
11 00 11 00 11 00 11 00 11 00 11 00
b
u
u
111 11 000 00 111 11 000 + 00 111 11 000 00 111 11 000K 00 111 11 u 000 00u 111 11 000 00 111 11 000 00 u 111 11 000 00 111 11 000 0 00 111 11 000 00 111 11 000 00 111 11 000 00 111 11 000 00
s
s
d d
u
11 00 11 00 11 00 0 11 00 K 11 00 11 00 11 00 11 00 11 00 + 11 00 11 00 11 00 11 00
Can be taken into account through additional input, i.e. SU (3) and data on B K K . In the case of the neutral strategy, rescattering processes can be included in an exact manner with the help of Amix (Bd 0 KS). CP QCD factorization is in favour of small eects!
Back to the Determination of ...
Observables:
R(c,n) q(c,n) , r(c,n) , (c,n) ,
(c,n) A0
q(c,n) , r(c,n) , (c,n) ,
(c,n) = (c,n) q(c,n) , r(c,n) ,
= q(c,n) , r(c,n) .
Interesting constraints on already from R(c,n):
(c,n) suers from large hadronic uncertainties! However, we can get rid of (c,n) by keeping it as a free variable, yielding minimal and maximal values for R(c,n):
R(c,n)
ext (c,n)
= function
q(c,n) , r(c,n) , .
Keeping, in addition, r(c,n) as a free variable, we obtain another less restrictive minimal value for R(c,n):
min R(c,n) (c,n) ,r(c,n)
= function
q(c,n) , sin2 .
These extremal values of R(c,n) imply constraints on , as the following cases are excluded:
min R(c,n) < R(c,n) , exp max R(c,n) > R(c,n) . exp
Dependence of extremal values of Rc on (qc = 0.68):
2.2 2 1.8 1.6 1.4 1.2 Rc 1 0.8 0.6 0.4 0.2 0 0 15 30 45 60 75 90 105 [deg] 120 135 150 165 180 rc=0.21 rc=0.27 rc=0.15 Rmin
Dependence of extremal values of Rn on (qn = 0.68):
2.2 2 1.8 1.6 1.4 1.2 Rn 1 0.8 0.6 0.4 0.2 0 0 15 30 45 60 75 90 105 [deg] 120 135 150 165 180 rn=0.17 rn=0.23 rn=0.11 Rmin
Observable R Rc Rn
CLEO (00) 1.00 0.30 1.27 0.47 0.59 0.27
BaBar (01) 0.97 0.23 1.19 0.35 1.02 0.40
Belle (01) 1.50 0.66 2.38 1.12 0.60 0.29
May Arrive at Puzzling Situation!
Constraints in the plane: note that qc,n 1/Rb !
Example: Rn = 0.6, rn = 0.17
0.5
0.4
0.3 .bar 0.2 Rb=0.38 0.1 0 0.5
0.4
0.3
0.2
0.1
0 .bar
0.1
0.2
0.3
0.4
0.5
Impact of lower bound on Ms : < 90!
0.5
0.4 Ms=15/ps Ms=20/ps 0.2 Ms=25/ps Ms=30/ps 0.1
0.3 .bar 0 0.5
0.4
0.3
0.2
0.1
0 .bar
0.1
0.2
0.3
0.4
0.5
In addition > to 90, as indicated by Rc and Rn , CLEO & Belle may point towards another puzzle: cos c > 0 and cos n < 0!
[Buras & R.F. (2000)]
Towards Calculations of B K,
Interesting theoretical progress:
QCD factorization [Beneke et al.] PQCD [Li et al.].
QCD factorization formula of the following structure:
A(B M1 M2 ) = M2 |j2 |0 M1|j1 |B [1 + O(s ) + O(/mb)]
O(s) can be calculated in a systematic way. O(/mb ) represent major limitation!
[Beneke, Buchalla, Neubert & Sachrajda (19992001)]
Detailed recent analysis: [Beneke et al., hep-ph/0104110]
QCD factorization allows a reduction of the theoretical uncertainties of rc,n and qc,n to the level of
O 1 ms md NC mb =O 1 ms md NC mb .
Complementary approaches to probe , making more extensive use of QCD factorization:
Rle of /mb corrections: hot topic: o
Charming penguins ... [Ciuchini et al., hep-ph/0104126]
U -Spin Strategies
... employ U -spin-related B decays: ds
[Prehistory: Dunietz, Snowmass 93 proceedings; Lipkin (1997); Buras, R.F. & Mannel (1997); Falk, Kagan, Nir and Petrov (1997); ...]
+ Bs(d) KS, Bd(s) Dd(s) Dd(s) or K 0K 0:
[R.F. (1999)]
Bd + and Bs K +K : and
[R.F. (19992000)]
Strategies employing angular distributions: , ,
[R.F. (1999)]
Bd K and Bs K + B K :
[Gronau & Rosner (2000); Chiang & Wolfenstein (2000)]
Bs(d) J/ :
[Skands (2000)]
Extracting and from Bd and Bs K K
+ +
[R.F., PLB 459 (1999) 306; EPJC 16 (2000) 87]
The Bd + , Bs K +K System
111 000 111 000 111 000 111 000 111 u 000 111 000
d(s) u
W
b u, c, t
d(s)
W
11 00 11 00 11 Bd(s) 00 11 00 11 00
b
d(s)
111 000 111 000 111 000 111 000 111 000
111 000 111 000 Bd(s) 111 000 111 000 111 000
11 00 11 00 11 00 11 00 11 00 11 00
u u
G
d(s) d(s)
11 00 11 00 11 00 11 00 11 00 11 00
(j {u, c, t})
d(s) u Vud(s) Vub
d(s) Vjd(s) Vjb j
Structure of decay amplitudes:
A(Bd ) = u Atree + Apen + c Apen + t Apen A(Bs K K ) = u Atree + Apen
0 + s u u 0 + d u u d c d t
+ c Apen + t Apen .
s
c
s
t
Unitarity of CKM matrix:
t = q q u c
q
0 A(Bd + ) = C ei dei 0 A(Bs K + K ) = C
e
i
+
1 2 2
d ei
dei = Pen , d ei = Pen . Tree Bd + Tree Bs K + K
[d, d : real hadronic numbers; , : CP-conserving strong phases]
CP asymmetries:
aCP (Bq (t) f ) = cosh(q t/2) A sinh(q t/2) Adir cos(Mq t) + Amix sin(Mq t) CP CP
CP-violating observables:
ACP (Bd ) ACP (Bd ) ACP (Bs K K ) ACP (Bs K K )
mix + dir + mix + dir +
= =
function(d, , ) function(d, , , d = 2) function(d , , ) function(d , , , s 0 ).
Bs
= =
Bd + and Bs K +K are related to each other by interchanging all down and strange quarks:
U -spin symmetry
d=d,
=.
4 observables, depending on 4 unknowns: d, , d = 2 , ,
i.e. these quantities can be determined!
No dynamical assumptions required, only U spin!
Minimal Use of the U -Spin Symmetry
The use of the U -spin-symmetry arguments can be minimized, if we employ also d = 2 as an input:
Adir (Bd + ) and Amix (Bd + ) allow us CP CP then to eliminate the strong phase :
d = d()
Adir (Bs K +K ) and Amix (Bs K +K ) allow CP CP us to eliminate the strong phase in an analogous way:
d = d ()
The corresponding contours in the d and d planes can be determined in a theoretically clean way!
and d, , can now be extracted with the help of
d =d
Example:
Input parameters:
0 0 negligible Bs Bs mixing phase, i.e. s = 0
2 = 44, = 60, d = d = 0.3, = = 210
Output for the observables:
Bd + : Adir = +19%, Amix = +62% CP CP Bs K +K : Adir = 17%, Amix = 27%. CP CP
Contours in the d and d planes:
0.6
0.5 Bd> 0.4 Bs>KK
d
0.3
()
0.2
0.1
0
0
15
30
45
60
75
90 105 [deg]
120
135
150
165
180
Experimental accuracy of O(10 ) and O(1) for at Tevatron-II and BTeV/LHC, respectively
very promising!
U -spin-breaking Eects
Interestingly, d ei = dei does not depend on decay constants and form factors, and is not aected by U -spinbreaking corrections within the BSS mechanism:
Strengthens condence into d ei = dei !
Moreover, experimental insights:
In addition to , d = d , also , can be determined: First consistency check is provided by = . Moreover, normalization factors |C| and |C | can be determined from the CP-averaged branching ratios:
C C =
fact ?
fK f
decay constants
2 FBs K (MK ; 0+ ) 2 FB (M ; 0+ ) d
.
form factors
Another interesting implication of d ei = dei :
Adir (Bs K + K ) CP Adir (Bd + ) CP C = C
2
BR(Bd + ) BR(Bs K + K )
Bs B
d
.
Similar relations between other U -spin-related B decays and further experimental tests ...
[R.F. (1999); Gronau (2000)]
Some Interesting Constraints
Useful quantity:
K 1 C C
2
BR(Bd + ) Bs BR(Bs K + K ) Bd
Parametrizations given above, and d ei = dei :
1 2d cos cos + d2 K= 2 , + 2 d cos cos + d2
with
2 . 1 2
Allows us to determine C cos cos as function of d:
1 C +1 constraints on d and Adir ! CP
Bs K +K not accessible at (4S) Bd K :
SU (3) avour symmetry & dynamical assumptions:
ACP (Bs K K ) ACP (Bd K )
BR(Bs K K ) BR(Bd K )
+ Bs dir + dir
B
.
d
Determination of K [data reported in spring 2001]:
K 1 fK f
2
BR(Bd ) BR(Bd K )
+
7.3 2.9 7.2 2.3 = 8.5 3.7
(CLEO) (BaBar) (Belle).
[Details: R.F., Eur. Phys. J. C16 (2000) 87]
C = cos cos as a function of d:
1 0.8 0.6 0.4 C=coscos 0.2 0 0.2 0.4 0.6 0.8 1 K=4.0 K=7.5 K=11
0
0.1
0.2
0.3
0.4
0.5
0.6 d
0.7
0.8
0.9
1
1.1
1.2
The maximal direct CP asymmetries for Bd + (upper curves) and Bs K +K Bd K :
1 0.9 0.8 0.7 0.6 |A |max 0.5 0.4 0.3 0.2 0.1 0 K=4.0 K=7.5 K=11
dir
0
15
30
45
60
75
90 105 [deg]
120
135
150
165
180
Shaded regions: d d /d [0.8, 1.2] for K = 7.5.
What about Amix(Bd + )? CP
In the following, we assume that d has been measured through the gold-plated mode Bd J/ KS. Using cos = C/ cos to eliminate , extremal values of Amix (Bd + ) can be obtained as a function of : CP
1 0.8 0.6 0.4 0.2
mix
=1.0 =0.8 =1.2 no pen.
K = 7.5, d = 50
A
0 0.2 0.4 0.6 0.8 1
0
15
30
45
60
75
90 105 [deg]
120
135
150
165
180
For given , the allowed range for Amix (Bd + ) is CP usually very large. On the other hand, a measurement of Amix (Bd + ) CP would imply a rather restricted range for !
If in addition to K and Amix (Bd + ) also direct CP CP + violation in Bd or Bd K is measured, and d, can be determined.
Extraction of from B(s) K Decays
[M. Gronau and J. Rosner, Phys. Lett. B482 (2000) 71]
The B(s) K System
Another interesting U -spin pair:
0 0 Bd K + and Bs + K .
Amplitudes in the strict U -spin limit:
A(Bd K ) = P
0 A(Bs + K ) = P 0 +
= 2 /(1 2 )
i i
1 re e 1+ 1
rei ei
.
At rst sight: 3 observables, depending on , r , . However, only 2 of them independent! Consequently, further information required:
Assuming both negligible rescattering eects and coloursuppressed EW penguins, we obtain
P = A(B + + K 0 ) 3 independent observables: A0 = As = 2r sin sin R = 1 2r cos cos + r 2 Rs = 2r cos cos +
r2
, r, !
Complements mixed B K approach (see above).
Conclusions
There are many approaches to extract . Particularly promising strategies for B experiments:
B K, strategies:
e+e B -factories
Make use of avour-symmetry relations and plausible dynamical assumptions. Constraints on from CP-averaged branching ratios. Data may point towards > 90 in contrast to UT ts and a puzzling situation for strong phases!?
U -Spin strategies: hadron machines
Several approaches! Particularly promising systems: Bd + , Bs K +K & B(s) K . As a by-product, also insights into hadron dynamics:
Strong phases & penguin parameters!
QCD factorization & PQCD:
reduction of theoretical uncertainties and complementary approaches to probe through B K, decays!