Merging Flavour Symmetries with QCD Factorisation for B-->KK Decays

0.15 MB·English

by  Joaquim Matias

7 Merging Flavour Symmetries with QCD Factorisation for
0 B → KK Decays.∗
0
2
Joaquim Matias
n
a
J IFAE, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

5
1

1
6v withThtheeinrteecrepnlatlyybmetewaeseunrefldavBodursymKm0Ke¯tr0iedseccoanyneacntdingQBCsD→FaKctKoridseactaioyns
1 opensnewstrategiestodescribet→hedecaysBs K0K¯0 andBs K+K−
→ →
1 in the SM andin supersymmetry. A new relation,emerging fromthe sum-
1 rule for the Bs K0K¯0 decay mode, is presented offering a new way to
0 determine the w→eak mixing angle φs of the Bs system.
7
0
PACS numbers: 13.25.Hw,11.30.Er,11.30.Hv,12.39.St
/
h
p
- 1. Introduction
p
e The huge effort on the experimental side at present B facilities (Babar,
h
Belle and CDF) to increase the precision on data measurements force us to
:
v revisethestrategiesonthetheorysidetoproducemoreaccuratepredictions.
i
X Non leptonic B decays offer different strategies to determine the Unitarity
r Triangle, to search for New Physics (NP)[1] but also to rule out models [2].
a
While alotof attention has beendevoted to theB πK[3,4,5,6,7]decay
→
modes, here we will focus on B KK decays that has been observed at
CDF [8] (B K+K−) and at B→abar [9] and Belle [10] (B K0K¯0).
s d
→ →
There are two main approaches in the literature to describe B KK
→
decays: flavour symmetries and 1/m expansion methods (QCD Factori-
b
−
sation [11, 12]/ soft collinear effective theories [13] or PQCD [14]). Each
of those methods has pros and cons, that we will discuss in turn. Flavour
symmetries, like U-spin symmetry that relates B K+K− with B
s d
π+π− [15, 16, 17, 18], provide a model independen→t analysis and extra→ct
most of the needed hadronic parameters from data. However, this method
has the disadvantage that it relies strongly on the accuracy of data, and, at
present, there is still not full agreement between Babar and Belle data on
the CP asymmetries of the B π+π− mode. As a consequence, error bars
d
→
∗ Presented at Final Euridice Meeting, Kazimierz, August 2006
(1)
2 BtoKK printed on February 2, 2008
are still quite large, see for instance, the prediction BR(B K+K−) =
s
→
35+73 10−6 [7] or, more recently, 4.2 10−6 BR(B K+K−)
6(cid:16)0.7−20(cid:17)10×−6 [18]. Also when relating B ×π+π− w≤ith B s →K+K− som≤e
d s
× → →
of the needed U-spin parameters can only be roughly estimated and they
are usually taken to be of the order of 20%.
Concerning 1/m -expansion methods, here we will focus on QCD Fac-
b
torisation (QCDF) [11, 12, 19]. The main idea is to exploit the existence of
a large scale m Λ together with colour transparency, that applies
b QCD
≫
when the outgoing meson that does not contain the spectator quark is very
energetic. At leading power in Λ/m all long distance contributions can
b
be parametrized in terms of form factors and light cone distribution ampli-
tudes, while the contribution from energetic gluons comes in a perturbative
series in α and it is incorporated into the hard scattering kernels. QCDF
s
predicts some of the hadronic parameters reducing the error bars, however
in the computation one has to face chirally enhanced IR divergences. They
are formally suppressed by a power of 1/m , but can be numerically sig-
b
nificant. They are modelled and induce an important uncertainty in the
predictions.
However, there is a third possibility and it is the proposal presented
in [20] that combines QCDF and Flavour symmetries giving rise to rather
accurate predictions for the branching ratios of the above mentioned B
s
→
KK decays in SM and in supersymmetry. Moreover, the method predicts
someoftheSU(3)breakingparameterswhichcan beusefulforotherflavour
approaches and, at the same time, deals with the problem of the chirally
enhanced IR divergences coming at order Λ/m (see also [21]).
b
Since IR divergences play a central role in this discussion, it is worth to
mention the two sources of IR divergences in QCDF:
Hard spectator-scattering: Hard gluons exchange between spectator
•
quark and the outgoing energetic meson gives rise to integrals of the
following type (see [12] for definitions):
1 1 Φ (x)Φ (y) Φ (x)Φ (y)
H (M M )= C dx dy M2 M1 +rM1 M2 m1 ,
i 1 2 Z Z (cid:20) x¯y¯ χ xy¯ (cid:21)
0 0
wherethe second term (formally of order Λ/m )diverges when y 1.
b
→
Weak annihilation: These type of diagrams also exhibit endpoint IR
•
divergences as it is explicit in the corresponding integrals:
1 1
Ai = πα dxdy Φ (x)Φ (y)
1 s Z (cid:26) M2 M1 (cid:20)y(1 xy¯)
0
−
1 2
+ +rM1rM2Φ (x)Φ (y) .
x¯2y(cid:21) χ χ m2 m1 x¯y(cid:27)
BtoKK printed on February 2, 2008 3
Both divergences are modelled in the same way [12]:
1 dy 1 dy 1 dy
Φ (y)=Φ (1) + Φ (y) Φ (1) Φ (1)XM1 +r,
Z0 y¯ m1 m1 Z0 y¯ Z0 y¯ h m1 − m1 i≡ m1 H,A
where r is a finite piece and the divergent piece is cut-off by a physical scale
of order Λ with an arbitrary complex coefficient to take into account
QCD
possible multiple soft scattering: X = (1+ρ )ln(m /Λ).
H,A H,A b
2. Sum-rules: α and φs
The SM amplitude for a B-decay into two mesons can be split into tree
and penguin contributions [19]1 :A¯ A(B¯ MM¯) = λ(q)TqC +λ(q)PqC,
≡ q → u M c M
withC denotingthechargeofthedecayproducts,andtheproductsofCKM
factors λ(q) = V V∗. Tree and penguin contributions in B¯ K0K¯0 in
p pb pq d
→
QCDF are:
1 1
Tˆd0 = αu αu +βu +2βu βu βu
4 − 2 4EW 3 4 − 2 3EW − 4EW
1 1
Pˆd0 = αc αc +βc+2βc βc βc , (1)
4− 2 4EW 3 4 − 2 3EW − 4EW
where Pˆd0 = Pd0/Ad , Tˆd0 = Td0/Ad , the super-scripts identify the
KK KK
channel, the normalisation is Aq = M2 FB¯q→K(0)f G /√2 (see [12, 20]
KK Bq 0 K F
for the corresponding expression of the B KK channels). Following
s
→
the observation in [20] that the structure of the IR divergences is the same,
independently of the charm or up quark running in the loop, we identified
an IR-safe quantity at NLO in QCDF that we called ∆ Td0 Pd0.
d
≡ −
All chirally enhanced IR divergences cancel exactly in this quantity at this
order. Its explicit expression in terms of the coefficients in Eq.(1) is:
∆ = Ad [αu αc +βu βc+2βu 2βc],
d kk 4 − 4 3 − 3 4 − 4
where electroweak contributions are neglected. This quantity can be safely
evaluated in QCDF and the result found in [20] was: ∆ = (1.09 0.43)
d
10−7+i( 3.02 0.97) 10−7GeV. Thelargestuncertaintyentering∆± comes·
d
− ± ·
from theratio m /m andthe scale dependence. Interestingly, this quantity
c b
can be expressed in terms of observables, providing a relation between the
direct induced CP-asymmetry (Ad0 ), the mixing induced CP-asymmetry
dir
(Ad0 ) and the branching ratio (BRd0) of B¯ K0K¯0:
mix d
→
BRd0
∆ 2= x +[x sinφ x cosφ ]Ad0 [x cosφ +x sinφ ]Ad0 ,
| d| L { 1 2 d− 3 d mix − 2 d 3 d ∆}
d
1 Conventionally, we will call “tree” the piece proportional to λ(q) and “penguin” the
u
piece proportional toλ(q), even if applied to decayswith no actual tree diagram.
c
4 BtoKK printed on February 2, 2008
where Ad0 2 + Ad0 2 + Ad0 2 = 1, φ is the weak mixing angle for the
| ∆| | dir| | mix| d
B system, L = τ M2 4M2/(32πM2 ) and x are functions of γ and
d d dq Bd − K Bd i
CKM elements. All SM inputs are taken as in [20] following [22].
Moreover, it was found in [23] that this sum-rule encodes also a very
interesting information. It provides a new way of measuring sinα:
BRd0
sin2α = 1 1 Ad0 2 Ad0 2 .
4L λ(d) 2 ∆ 2 (cid:18) −q − dir − mix (cid:19)
d u d
| | | |
(See [24] for a recent review on the extraction of α). In a similar way, it was
foundin [20] a correspondingIR safe quantity ∆ Ts0 Ps0 and sum-rule
s
for the decay B¯ K0K¯0: ≡ −
s
→
BRs0
∆ 2= y +[y sinφ y cosφ ]As0 [y cosφ +y sinφ ]As0 ,
| s| L { 1 2 s− 3 s mix − 2 s 3 s ∆}
s
which provides a completely new way to determine the weak mixing angle
φ that we present here (see [20] for definitions):
s
φ BRs0
sin2 s = 1 1 As0 2 As0 2 .
2 4L λ(s) 2 ∆ 2 (cid:18) −q − dir − mix (cid:19)
s c s
| | | |
This implies that a measurement of the branching ratio, direct CP asym-
metry and mixing induced CP asymmetry of the decay B K0K¯0 auto-
s
→
matically translates into a value for sin2φ /2. Finally, given the relation
s
∆ = f∆ , wheref = As /Ad a new relation between sinα and sinφ /2
s d KK KK s
immediately emerges.
3. Description of the Method: Flavour Symmetries & QCDF
3.1. B K0K¯0
s
→
The SM amplitude of this b s penguin decay is given by:
→
A¯ A(B¯ K0K¯0) = λ(s)Ts0+λ(s)Ps0.
s u c
≡ →
Its dynamics is described in terms of three parameters: Ts0 , Ps0 and
the relative strong phase arg(Ps0/Ts0) (remember that Ts|0 sta|nd|s fo|r the
(s)
piece proportional to λ but it is not due to an actual tree diagram in this
u
case). Its hadronic parameters can be related via U-spin with the hadronic
parameters (Td0 , Pd0 and arg(Pd0/Td0)) of the also penguin governed
mode B |K0|K¯0|. T|his has several advantages: first, we can expect
d
→
similar final state interactions (although not equal), second, the sources of
U-spin breaking can be better controlled using QCDF. These sources are:
BtoKK printed on February 2, 2008 5
i) the factorisable ratio f = As /Ad (extrapolated from the lattice) ii)
KK KK
U-spin breaking 1/m suppressed terms δα and δβ : sensitive to the differ-
b i i
ence of B and B distribution amplitudes and spectator quark dependent
d s
contributions coming from a gluon emitted from the d or s quark. This
leads to the relations Ps0/(fPd0) 1 3% and Ts0/(fTd0) 1 3%.
The next step is to|determine th−e h|a≤dronic para|meters (Td0−, P|d0≤) of the
decay B¯ K0K¯0. This is done using as inputs the BR(B¯ K0K¯0) and
d d
∆ (fromQ→CDF).ThedirectCPasymmetryA (B¯ K0K¯→0)(denotedby
d dir d
Ad0 )willbetakenasafreeparameter. Thecombinatio→nofthoseconstraints
dir
gives rise to a set of non-linear equations (see definitions in [20]):
x +iy = ∆ (1 cosγ/R)/a,
C C d
− −
r2 = ρ2/[aλ(d) 2] [sinγ ∆ /(aR)]2,
0 | u | − | d|
y x = y x ρ2Ad0 /(2λ(d)λ(d) sinγ), (2)
P ∆ ∆ P − 0 dir | u c |
that determines Pd0 = x +iy , then using ∆ one gets Td0. Two remarks
P P d
are important here: first, there is a twofold ambiguity in the sign of ImPd0
(solved in the next subsection). Second, current data together with our
knowledge on ∆ limits the Ad0 asymmetry (by means of Eqs.(2)) within
d dir
a restricted range between 0.2 Ad0 0.2.
− ≤ dir ≤
Finally, our SMpredictions for the branchingratio and CP asymmetries
of B¯ K0K¯0 are obtained using the relations between Ps0 Pd0 and
s
Ts0 →Td0 mentioned above and including all U-spin breaking↔sources to-
↔
gether with the QCDF uncertainties in ∆ . The resulting predictions are
d
BR(B K0K¯0) = (18 7 4 (2)) 10−6, A (B K0K¯0)
s dir s
1.1 10−→2 and A (B ±K0±K¯0)± 1.5×10−2. T| he sign o→f these asy|m≤-
mix s
met×ries can be fi|xed once→Ad0 will b|≤e meas×ured with enough accuracy.
dir
3.2. B K+K−
s
→
The analysis of the decay mode B K+K− follows similar steps, but
s
→
withsomeimportantdifferences: i)wewilluseU-spinandisospintoconnect
B K0K¯0 with B K+K−, ii) B K+K− contains a tree (denoted
d s s
byα¯→inQCDF)withn→ocounterpartinB→ K0K¯0,iii)BR(B K+K−)
1 d s
→ →
has been measured with excellent precision at CDF[8] and iv) we will use
the information only on the sign predictions (not the absolute value) for
the CP-asymmetries of B K+K− from the strategy that uses U-spin to
s
relate B K+K− with B→ π+π−.
s d
→ →
The hadronic parameters describing the amplitude:
A(B¯ K+K−) = λ(s)Ts±+λ(s)Ps±
s u c
→
are obtained from the relations containing the sub-leading 1/m U-spin
b
breaking: Ps±/(fPd0) 1 2%, Ts±/(As α¯ ) 1 Td0/(Ad α¯ ) 4%.
| − | ≤ | kk 1 − − kk 1 | ≤
6 BtoKK printed on February 2, 2008
Those errors, estimated within QCDF, are stretched roughly by a factor
two to be conservative.
The two-fold ambiguity on the sign of ImPd0 is lifted here using U-spin
argumentsbasedontheB π+π− strategy. WhilethesignsofImPd0 and
d
As± are correlated (being b→oth positive or negative), the prediction based
dir
on the B π+π− strategy points towards a positive sign for As±[18],
d → dir
discarding then the solution with ImPd0 <0.
Our result in the SM for the branching ratio of B K+K− averaging
s
over all values of Ad0 is BR(B K+K−) = (20 8→ 4 (2)) 10−6,
dir s → ± ± ± ×
where the last error in parenthesis stands for a rough estimate of finite,
non-enhanced Λ/m corrections. Finally, confronting our predictions for
b
B K+K− withthedataonB π+π− [25]: Td± = (5.48 0.42) 10−6
s d ππ
→ → | | ± ×
and Pd±/Td± = 0.13 0.05, arg Pd±/Td± = (131 18)◦, provides a
ππ ππ ππ ππ
(cid:12) (cid:12) ± (cid:16) (cid:17) ±
doub(cid:12)le informa(cid:12)tion. First, we can give predictions for the U-spin breaking
param(cid:12) eters: C(cid:12) = Ts±/Tπdπ± = 2.0 0.6 and ξ = Ps±/Ts± /Pπdπ±/Tπdπ± =
0.8 0.3 conRnectin|g B K| +K−±with B π|+π−. The|s|e paramet|ers
s d
± → →
can be compared with the QCD sum rules predictions in ref. [26]. Notice
that while QCD sum rules gives only the factorizable part, our predictions
include, in principle, the full contribution. Second, a comparison between
the two relative strong phases arg(Ps±/Ts±) and arg Pd±/Td± selects
ππ ππ
(cid:16) (cid:17)
Ad0 0. Then, if we restricts only to positive values of Ad0 according to
dir ≥ dir
the previous arguments, our SM predictions turn out to be [20, 27]:
BR(B K+K−) = (17 6 3 (2)) 10−6,
s
→ ± ± ± ×
0.22 As± 0.49 and 0.55 As± 0.02. (3)
− ≤ dir ≤ − ≤ mix ≤
AnotherargumentinfavourofAd0 0comesfromthepreferenceofAs± <
0 of the U-spin based B π+diπr−≥strategy [17] (see the anti-correlmaitxion
d
between Ad0 and As± in T→able 1 of [20]).
dir mix
The accuracy on these CP-asymmetries will be substantially improved
onceaprecisemeasurementonAd0 willbeavailableortheerrorofBR(B
K0K¯0) and the QCD uncertaintdieirs on ∆ (mainly m /m and scale depden→-
d c b
dence) will be reduced.
3.3. Supersymmetry
The leading gluino-squark box and penguin contributions [3] to B
s
K0K¯0 andB K+K− wereevaluated firstin[18]usingtheU-spinflavo→ur
s
strategy with →B π+π− and, afterwards, in [27] using the new method
d
→
combining flavour and QCDF [20]. The relative size of this contribution
compared to the SM penguin is (α /M2 )/(α/M2 ) 1. The amplitude
s susy W ∼
of these decays in presence of NP contains an extra contribution: (B0
s
A →
BtoKK printed on February 2, 2008 7
K+K−) = s± + ueiΦu, (B0 K0K¯0) = s0 + deiΦd. These NP
amplitudes AuSMeΦu aAnd deΦAd inste→rms of WilsonAcSoMefficiAents are:
A A
G 1 1 2α χ
qeiΦq= F χ( c¯q +c¯q) (c¯q c¯q) (c¯q c¯q) λ sC¯eff(1+ ) A
A √2h− 3 1 2 − 3 3− 6 − 4− 5 − t 3π 8g 3 i
with q = u,d, A = i(m2 m2 )f FBs→K and χ = 1.18 (see [27] for defi-
B − K K
nitions). These Wilson coefficients are sensitive to the s˜ ˜b mass splitting.
−
After including the constraints coming from BR(B X γ), B πK and
s
→ →
∆M wefoundthatalarge isospinviolation controlled by themass splitting
s
uR-dR is possible between the NP amplitudes ueiΦu and deiΦd. For the
region of parameters considered in this superAsymmetric sAcenario, ueiΦu
ceanebe up to a factor three larger than deiΦd. The specific resAults in
A
supersymmetry for each decay mode are [27]:
B0 K+K−: Thebranchingratio is very little affected by SUSY.At
s
• mos→t, the SM prediction can be increased by 15% for Ad0 = 0.1, in-
dir
creasing a bitthe already good agreement with the new CDF data [8].
The direct CP asymmetry within SUSY falls inside the range 0.1 <
A (B0 K+K−)SUSY < 0.7 for 0.1 Ad0 0.1. The de−viatio∼n
dir s → − ≤ dir ≤
depends on the relative si∼ze of the competing SM tree versus the NP
amplitude. A (B0 K+K−)SUSY can take any value from [-1,1].
mix s
→
B0 K0K¯0: The impact of SUSY on BR(B0 K0K¯0) is even
s s
• smal→ler, reflecting the reduced allowed region for →deiΦd as compared
to ueiΦu. The situation is very different for thAe CP asymmetries,
A
that are particularly promising, due to the very small size of their SM
prediction [20]. The direct CP asymmetry in SUSY can be 10 times
larger than the SM one. A (B0 K0K¯0)SUSY covers the entire
mix s
→
range, and so this asymmetry can be large in the presence of SUSY,
contrary to the SM prediction.
The method discussed here is beingapplied to other non-leptonic B-decays.
Acknowledgements: Research partly supported by EU contract EURIDICE
(HPRN-CT2002-00311), PNL2005-41, FPA2002-00748 and the Ramon y
Cajal Program.
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