WG3: Theory SummaryUlrich Haisch

University of Zürich

4th International Workshop on the CKM Unitarity Triangle,December 12−16, 2006, Nagoya, Japan

Many interesting talks ...*

P. Ball, E. Baracchini, T. Becher, F. Bedeschi, B. Casey, S. Cohen, T. Ewerth, P. Gambino, E. Gardi, B. Grinstein, J. van Hunen, R. Itoh, M. Iwasaki, H. Lee, Z. Ligeti, N. Magini, F. Mescia, R. Mumford, M. Nakao, M. Neubert, A. Oyanguren, M. Papucci, M. Pierini, B. Quinn, G. Ruggiero, A. Schwartz, G. Sciolla, C. Smith, S. Stone, N. Tantalo, C. Tarantino, V. Vagnoni, R. van de Water, G. Weber, and L. Widhalm

*

• my apologies• due to lack of time, have to omit interesting results

• discussed topics reflect interest(s) of speaker

• ...

*C. Smith

Status of rare K-decays in SM*

LD28%

CKM(mt, αs)

37%

mc 22%

μ13%

• errors in K+→π+νν • error in KL→π0l+l−

∆aS [%] ∼ B(KS → π0l+l−)

[%]

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

!

"#

CKM fit

$

%C K M

f i t t e r

Future (?)

1 – CL

*C. Smith

Future (?) CKM fit from K→πνν*

B(K+ → π+νν̄) = (8.0± 0.8)× 10−11

B(KL → π0νν̄) = (3.0± 0.3)× 10−11

Future (?)σ(|Vtd|)|Vtd|

= ±4.0%

σ(sin 2β) = ±0.024

σ(γ) = ±4.7◦

σ(|Vtd|)|Vtd|

= ±1.0%

σ(sin 2β) = ±0.006

σ(γ) = ±1.2◦

NLO

NNLO • nice CKM fit from K→πνν, but golden modes more interesting to look for NP in particular non-MFV

K+ → π+νν̄

KL → π0νν̄

• large PT corrections associated with• running coupling effects (renormalons)

• Sudakov logarithms (endpoint region)

• both effects resummed by DGE• for Γ77 NNLO, NLO for other Γij• access to NPT power corrections

B→Xsγ by dressed gluon exponentiation (DGE)*

*E. Gardi

B Xs

m2X → 0nµ

Q

k

∼ (αsβ0 ln k2/Q2)n ∼ (CF αs ln2 m2X/Q2)n

B→Xsγ: total rate and moments using DGE*

*E. Gardi

• important findings• cut-dependence uncertainty smaller than error on

total decay rate • Eγ moments more sensitive to mb and NPT power

corrections than BF

• sophisticated calculations• NNLO for dB/dq2 and AFB• model-independent NLO with mX cut

• SM predictions with 10−15% errors

• NPT effects• consistent to cut out ψ and ψ´ and

compare data with SD calculation?

• like in B→Xsγ non-local power corrections αsΛQCD/mb ~ 5%

• difficult to quantify error/size given present command of NPT QCD

B→Xsl+l−: solved problems, open issues*

Z. Ligeti, M. Neubert

*

Q7

Q8b bs

q q

gγ

γ

...

5 10 150 200

1

2

3

4

q2 [GeV2]

low-q2 high-q2ψ ψ′

NNLO

dB/d

q2[1

0−7

GeV

2]

b→sγ, b→sl+l−: signs of penguins*

M. Iwasaki,M. Pierini

*

• inclusive modes• well-known way to avoid b→sγ is

to reverse sign of C7 • possibility disfavored as it leads to a

b→sl+l− rate ~ 3σ above measurement

• exclusive modes• measurement of FB asymmetry in

B→K*l+l− excludes C9C10 > 0 at 95% CL

• hints towards exclusion of large destructive Z-penguin

∆C7

0.05

0.25

0.5

0.75

1

-0.5 0 0.5 1.0 1.5

0

-1

-2

-3

1

∆C

1-CL

SM

C7 > 0

C < 0

8 12 164 200

q2 [GeV2]

-1

-0.5

0.5

1

0

ĀFB(B̄→

K∗ l

+l−

)SM

C7 < 0

C10 > 0

C9C10 > 0C9 < 0, C10 > 0

b→sγ, b→sl+l−: signs of penguins*

M. Iwasaki,M. Pierini

*

∆C7

0.05

0.25

0.5

0.75

1

-0.5 0 0.5 1.0 1.5

0

-1

-2

-3

1

∆C

1-CL

SM

C7 > 0

C < 0

8 12 164 200

q2 [GeV2]

-1

-0.5

0.5

1

0

ĀFB(B̄→

K∗ l

+l−

)SM

C7 < 0

C10 > 0

C9C10 > 0C9 < 0, C10 > 0

• inclusive modes• well-known way to avoid b→sγ is

to reverse sign of C7 • possibility disfavored as it leads to a

b→sl+l− rate ~ 3σ above measurement

• exclusive modes• measurement of FB asymmetry in

B→K*l+l− excludes C9C10 > 0 at 95% CL

• hints towards exclusion of large destructive Z-penguin

HT,L,A(q21 , q22) ≡

∫ q22

q21

dq2HT,L,A(q2)

b→sl+l−: learning effectively from 1ab-1 data*

*Z. Ligeti

(s = q2/m2b , z = cos θ, θ : ! b, l+)

d2Γdsdz

∼{

(1 + z2)

[(C9 +

2sC7

)2+ C210

]

+(1− z2)[(C9 + 2C7)2 + C210

]

−4zsC10(

C9 +2sC7

) }

≡ HT + HL + HA{ {

∼ AFB∼ Γ

• angular decomposition

4

-6 -4 -2 0 2 4 6 8 10C9

-8

-6

-4

-2

0

2

4

6

8

C10

!!1,3.5"

!!3.5,6"

HA!1,3.5"HA!3.5,6"

-6 -4 -2 0 2 4 6 8 10C9

-8

-6

-4

-2

0

2

4

6

8

C10

-6 -4 -2 0 2 4 6 8 10C9

-8

-6

-4

-2

0

2

4

6

8

C10

!!1,6"

HL!1,6"HT!1,6"

HA!1,6"

-6 -4 -2 0 2 4 6 8 10C9

-8

-6

-4

-2

0

2

4

6

8

C10

FIG. 1: Constraints in the C9 − C10 plane. Left: Γ(1, 3.5) (light gray), Γ(3.5, 6) (dark gray), HA(1, 3.5) (light green), andHA(3.5, 6) (dark green). Right: HT (1, 6) (blue), HL(1, 6) (orange), HA(1, 6) (green), and for comparison Γ(1, 6) (gray). Theblack dot denotes the SM values used as central values. The yellow regions show the combined constraints. The error estimatesare discussed in the text.

HA(1 , 3.5)/Γ0 = −C10 (70.19 C9 + 1401. C7 − 121.5) ,HA(3.5 , 6)/Γ0 = −C10 (111.8 C9 + 1051. C7 − 80.91) ,

HL(1 , 6)/Γ0 = 315.2 (C29 + C210) + 1377. C27 + 1299. C7C9 + 33.41 C9 + 86.86 C7 − 72.87 . (14)

The major uncertainties in Eqs. (14) arise from higherorder perturbative corrections, mb and mc. Varying therenormalization scale between mb/2 and 2mb, we get lessthan 5% uncertainty in the coefficients of the dominantterms in Eq. (14). Since the difference mb −mc is knownprecisely [27] we vary mb and mc in a correlated man-ner, which gives a 1–5% uncertainty. The uncertaintiesfrom other input parameters and higher order correc-tions in 1/mb are much smaller. The uncertainties fromthe electroweak matching scale, µ0, and the top-quarkmass, mt, in Eq. (14) are negligible, because they pri-marily enter via the SM values of the Ci. Using the SMvalues of Ci from Table I and Eq. (14) we obtain theSM prediction for the B → Xs!+!− branching ratio for1 GeV2 < q2 < 6 GeV2,

τBΓ(1, 6) =(

1.575 ± 0.067[µ] ± 0.051[mb,mc]± 0.041[mt] ± 0.019[µ0]

)

× 10−6 . (15)

This agrees well with Refs. [16, 20].To obtain a reasonable estimate of the future uncer-

tainties, we scale the current measurements [2, 3] to1 ab−1 luminosity, which gives about 10% statistical un-certainty for Γ(1, 6). We assume that the Hi are mea-sured with the central values given by the SM. The sta-tistical error of HT and HL is obtained by scaling by

the number of events compared to Γ(1, 6). In the case ofHA we take the same absolute statistical error for 3/4HAas for the total rate integrated over the same q2-region.The reason is that 3/4HA corresponds to the differencebetween the rates for positive and negative cos θ, whichhas the same absolute statistical error as the sum. Tothis we add in quadrature a 20% systematic uncertaintyfor all Hi, to account for experimental systematics andtheoretical uncertainties. From each observable’s totalerror we build χ2 for the individual and combined con-straints, and Figs. 1 and 2 show the ∆χ2 = 1 regionsin the C9 − C10 plane. Since B → Xsγ will always bemeasured with higher precision than B → Xs!+!−, weconsider the value of C27 to be known from B → Xsγ andassume its sign is negative as in the SM (since there isan overall sign ambiguity).

On the left-hand side in Fig. 1 we show the con-straints from Γ(1, 3.5) (light gray), Γ(3.5, 6) (dark gray),HA(1, 3.5) (light green), and HA(3.5, 6) (dark green).This plot shows that splitting Γ(1, 6) into two regionsis not really useful because a very similar linear combi-nation of Wilson coefficients is constrained. As is well-known, splitting HA(1, 6) into two regions is very use-ful, since different combinations of coefficients are con-strained by each region. (This is also the reason why the

SM

*Z. Ligeti

b→sl+l−: learning effectively from 1ab-1 data*

• angular decomposition

d2Γdsdz

∼{

(1 + z2)

[(C9 +

2sC7

)2+ C210

]

+(1− z2)[(C9 + 2C7)2 + C210

]

−4zsC10(

C9 +2sC7

) }

≡ HT + HL + HA{ {

∼ AFB∼ Γ

(s = q2/m2b , z = cos θ, θ : ! b, l+) HT,L,A(q21 , q22) ≡∫ q22

q21

dq2HT,L,A(q2)

5

-6 -4 -2 0 2 4 6 8 10C9

-8

-6

-4

-2

0

2

4

6

8

C10

HL!1,6"HT!1,6"

HA!1,3.5"HA!3.5,6"

-6 -4 -2 0 2 4 6 8 10C9

-8

-6

-4

-2

0

2

4

6

8

C10

-6 -4 -2 0 2 4 6 8 10C9

-8

-6

-4

-2

0

2

4

6

8

C10

HT!3.5,6"

HL!1,6"

HT!1,3.5"

HA!1,3.5"HA!3.5,6"

-6 -4 -2 0 2 4 6 8 10C9

-8

-6

-4

-2

0

2

4

6

8

C10

FIG. 2: Constraints in the C9 − C10 plane. Left: HT (1, 6) (blue), HL(1, 6) (orange), HA(1, 3.5) (light green), and HA(3.5, 6)(dark green). Right: as on the left, but HT split into HT (1, 3.5) (blue), and HT (3.5, 6) (light blue). The dark blue regions inthe right plot show the constraint from the two HT observables alone. All other notations are as in Fig 1.

zero of the forward-backward asymmetry is interesting tostudy.) The plot on the right in Fig. 1 shows that split-ting Γ(1, 6) into HT (1, 6) (blue) and HL(1, 6) (orange)gives a very powerful constraint. This shows the powerof separately measuring HT and HL as advocated in theintroduction. The observables shown in this figure aresufficient to extract the absolute values |Ci| and the signof C9 relative to C7. Also displayed is the constraint fromΓ(1, 6) (gray), which shows that separating Γ into HTand HL significantly improves the constraints. However,because of its large relative error, HA(1, 6) (green) doesnot provide a good constraint.

The left plot in Fig. 2 shows that HA still gives sensitiv-ity to the sign of C10. Using the two integrals HA(1, 3.5)and HA(3.5, 6) allows one to distinguish between the pos-itive and negative solutions for C10. Combined with thetighter constraints from splitting Γ into HT and HL, thesplitting of HA into these two regions can substitute forthe information from the zero of the AFB. The ploton the right in Fig. 2 shows that splitting HT (1, 6) intoHT (1, 3.5) and HT (3.5, 6) can further overconstrain thedetermination of the Wilson coefficients. The dark blueregion in the right plot in Fig. 2 shows the combined con-straint from only the two HT integrals. The requirementthat the HL constraint has to overlap with it effectivelyprovides a consistency test on the value of C7 extractedfrom B → Xsγ. It can also play an important role inthe search for physics beyond the SM. If the new physicsintroduces low energy operators with a helicity structuredifferent from the SM, it will affect HL and HT differ-ently, because of their different polarizations.

It would also be interesting to explore experimentally

whether the influence of the J/ψ resonance turns on atsimilar q2 values in HT , HA, and HL. Since we haveno information on the J/ψ polarization in inclusive B →J/ψXs decay, it is possible that the upper cut on q2 canbe extended past 6 GeV2 in some (but maybe not all)of these observables, which may improve the statisticalaccuracy of the measurement.

IV. EXCLUSIVE B → K∗!+!−

We now turn to the exclusive decay B → K∗#+#−.While the theoretical uncertainties are larger than inthe inclusive analysis, measuring the exclusive mode issimpler and it may be the only possibility at LHCb.Compared to inclusive decays, the exclusive measure-ments go closer to q2 = m2ψ; 8.1 GeV

2 at Belle [4] and

8.4 GeV2 at Babar [5]. In this region of phase spaceEK∗ varies only between 1.9 GeV < EK∗ < 2.7 GeV,which helps controlling some theoretical uncertainties.In our general discussion we will consider for simplic-ity 0.1 GeV2 < q2 < 8 GeV2, where the precise valuesof neither limits are important (the lower limit can bereplaced by any experimentally appropriate value above4m2"). However, for numerical comparisons of our resultswith the data, we use the limits used in the experimentalanalysis.

In this section we explain that, similar to inclusivedecays, all the information obtainable can be extractedfrom a few integrated rates. To obtain the most infor-mation on the ratios of Wilson coefficients, Belle [4] per-formed a maximum-likelihood fit to the double differen-

SM

B→Vγ: warm-up*

P. Ball,F. Mescia

*

• QCD factorization

: PT QCD quantities

: NPT QCD quantities

+ O(ΛQCD/mb)

• higher theoretical accuracy on form factor ratios than on T1B→V(0) itself

QCD sum rules on light cone (LCSR): results*

*P. Ball

• form factors/ratios

• CP-averaged branching ratiosBranching Ratios (CP-averaged)

B × 106 QCDF + WA + soft gluons th. error

B− → ρ−γ 1.05 1.11 1.16 ±0.26

B0 → ρ0γ 0.49 0.53 0.55 ±0.13

B0 → ωγ 0.40 0.42 0.44 ±0.10

B− → K∗−γ 39.7 38.3 39.4 only exp. error

B0 → K∗0γ 37.1 39.9 41.0 only exp. error

B0s → K̄∗0γ 1.12 1.23 1.26 ±0.31

B0s → φγ 34.6 38.3 39.4 ±11.9

BaBar: HFAG:

B− → ρ−γ B0 → ρ0γ B0 → ωγ B− → K∗−γ B0 → K∗0γ

1.10 ± 0.38 0.79 ± 0.23 < 0.78 40.3 ± 2.6 40.1 ± 2.0

– p.4

B

q

B

q

VV

(a) (b)

b

q

Db D

Figure 1: (a): WA diagram. The square denotes insertion of the operator Qi. Photon emissionfrom lines other than the B spectator is power-suppressed, except for emission from the final-

state quark lines for the operators Q5,6, denoted by crosses. (b): soft-gluon emission from a

quark loop. Again the square dot denotes the insertion of the operator Qi. There is also a

second diagram where the soft gluon is picked up by the B meson.

• CKM suppression;

• size of hadronic matrix elements;

have to be taken into account. This is a consequence of the fact that in radiative transitionsthe “naively” leading term in Q7 is loop suppressed, which is qualitatively different fromother applications of QCDF, for instance in B− → π−π0, where the leading hadronicmatrix element describes a tree-level process.

The exclusive B → V γ process is actually described by two physical amplitudes, onefor each polarisation of the photon:

ĀL(R) = A(B̄ → V γL(R)) , AL(R) = A(B → V̄ γL(R)) , (5)

where B̄ denotes a (bq̄) and V a (Dq̄) bound state.1 In the notation introduced in Ref. [11]

in the context of QCDF, the decay amplitudes can be written as

ĀL(R) =GF√

2

(λDu a

u7(V γL(R)) + λ

Dc a

c7(V γL(R))

)〈V γL(R)|QL(R)7 |B̄〉

≡GF√

2

(λDu a

u7L(R)(V ) + λ

Dc a

c7L(R)(V )

)〈V γL(R)|Q

L(R)7 |B̄〉 ,

AL(R) =GF√

2

((λDu )

∗au7R(L)(V ) + (λDc )

∗ac7R(L)(V ))〈V̄ γL(R)|(QR(L)7 )†|B〉 . (6)

1Note that in this paper K∗ is a (sq̄) bound state, in contrast to the standard labelling, accordingto which K∗0 = (ds̄) and K̄∗0 = (sd̄). This is because the calculation of form factors and other matrixelements involves light-cone distribution amplitudes of the vector meson V and that in the standardnotation used in that context, K∗ always contains an s quark, and K̄∗ an s̄ quark. This distinction isrelevant because of a sign change of G-odd matrix elements under (sq̄) ↔ (qs̄), see Tabs. 3, 5, 6.

5

weak annihilation

B

q

B

q

VV

(a) (b)

b

q

Db D

Figure 1: (a): WA diagram. The square denotes insertion of the operator Qi. Photon emissionfrom lines other than the B spectator is power-suppressed, except for emission from the final-

state quark lines for the operators Q5,6, denoted by crosses. (b): soft-gluon emission from a

quark loop. Again the square dot denotes the insertion of the operator Qi. There is also a

second diagram where the soft gluon is picked up by the B meson.

• CKM suppression;

• size of hadronic matrix elements;

have to be taken into account. This is a consequence of the fact that in radiative transitionsthe “naively” leading term in Q7 is loop suppressed, which is qualitatively different fromother applications of QCDF, for instance in B− → π−π0, where the leading hadronicmatrix element describes a tree-level process.

The exclusive B → V γ process is actually described by two physical amplitudes, onefor each polarisation of the photon:

ĀL(R) = A(B̄ → V γL(R)) , AL(R) = A(B → V̄ γL(R)) , (5)

where B̄ denotes a (bq̄) and V a (Dq̄) bound state.1 In the notation introduced in Ref. [11]

in the context of QCDF, the decay amplitudes can be written as

ĀL(R) =GF√

2

(λDu a

u7(V γL(R)) + λ

Dc a

c7(V γL(R))

)〈V γL(R)|QL(R)7 |B̄〉

≡GF√

2

(λDu a

u7L(R)(V ) + λ

Dc a

c7L(R)(V )

)〈V γL(R)|Q

L(R)7 |B̄〉 ,

AL(R) =GF√

2

((λDu )

∗au7R(L)(V ) + (λDc )

∗ac7R(L)(V ))〈V̄ γL(R)|(QR(L)7 )†|B〉 . (6)

1Note that in this paper K∗ is a (sq̄) bound state, in contrast to the standard labelling, accordingto which K∗0 = (ds̄) and K̄∗0 = (sd̄). This is because the calculation of form factors and other matrixelements involves light-cone distribution amplitudes of the vector meson V and that in the standardnotation used in that context, K∗ always contains an s quark, and K̄∗ an s̄ quark. This distinction isrelevant because of a sign change of G-odd matrix elements under (sq̄) ↔ (qs̄), see Tabs. 3, 5, 6.

5

soft gluon emission

|Vtd/Vts| and γ from B→Vγ using LCSR*

*P. Ball

• |Vtd/Vts| depends on cos(γ)• discrete ambiguity γ ↔ -γ

∣∣∣∣VtdVts

∣∣∣∣ = 0.199+0.022−0.025(exp)± 0.014(th)

γ = (61.0+13.5−16.0(exp)+13.5−16.0(th))

◦

• CP asymmetry in B→D(*)K(*) depends on sin2(γ)

• discrete ambiguity γ ↔ γ+π

• remove discrete ambiguity! γ < 180º is favored!

B→K*γ decay form factor on lattice*

*F. Mescia

TB→K∗

1 (0) = 0.25(4)

TB→K∗

1 (0)TB→ρ1 (0)

= 1.2(1)

T (0,MH)M32H = a0 +

a1MH

+a2

M2H

• rough control on discretisation errors: 2 lattice spacings, a-1 = 3, 4 GeV

• quadratic and linear extrapolation in heavy quark mass

• systematic errors due to unquenched calculation

T1B→K*(0): lattice QCD (LQCD) vs. LCSR*

P. Ball,F. Mescia

*

*C. Tarantino

B-hadron lifetime ratios*

• basic features• lifetime diversity due to

spectator effects ~ (ΛQCD/mb)3

• hadronic matrix elements from LQCD

τ(B+)τ(Bd)

τ(Bs)τ(Bd)

τ(Λb)τ(Bd)

LO

LO

LO

NLO

NLO

NLO

LO 1.01(3) 1.00(1) 0.93(4)

NLO 1.06(3) 1.00(1) 0.90(5)

NLO +(ΛQCD/mb)4 1.06(2) 1.00(1) 0.88(5)

HFAG ‘06 1.076(8) 0.957(27) 0.84(5)

τ(B+)τ(Bd)

τ(Bs)τ(Bd)

τ(Λb)τ(Bd)

• theoretical value of Λb ratio used to be above experimental average ...

*C. Tarantino

B-hadron lifetime ratios*

• basic features• lifetime diversity due to

spectator effects ~ (ΛQCD/mb)3

• hadronic matrix elements from LQCD

τ(B+)τ(Bd)

τ(Bs)τ(Bd)

τ(Λb)τ(Bd)

LO

LO

LO

NLO

NLO

NLO

• ... new CDF value higher than experimental WA by ~ 2.5σ

• is there a reversed Λb ”puzzle”? time will tell ...

NLO +(ΛQCD/mb)4 0.88(5)

HFAG ‘06 0.84(5)

CDF ‘06 1.041(57)

D0 ‘06 0.87(11)

τ(Λb)τ(Bd)

*C. Tarantino

B-meson width differences*

• important fact• possible BSM effects may

enter through EW box diagram (M12 ⇔ CB, φB)

• in contrast to lifetimes, ΔΓ very sensitive to NP

NLO +(ΛQCD/mb)4 2.3(8) 7(3)

HFAG ‘06 9(37) 14(6)

∆ΓdΓd

× 103 ∆ΓsΓs

× 102

NLOLO

LONLO

∆ΓsΓs

× 102

∆ΓdΓd

× 103

• quite large uncertainties for NLO + (ΛQCD/mb)4 predictions

• are observed strong cancellation accidental?

*C. Tarantino

A. Lenz and U. Nierste: “Let’s change basis!”*

• observation • large corrections in ΔΓ could

be artifact of poor choice of operator basis

• different basis might lead to better convergence of series in αs and ΛQCD/mb

• shift of central value could also indicate large αs2 and αsΛQCD/mb terms!? ...

HFAG ‘06old NLO +(ΛQCD/mb)4

new NLO +(ΛQCD/mb)4

14(6) 7(3) 12.7(2.4)∆ΓsΓs

× 102

old NLO

∆ΓsΓs

× 102

WA

new NLO

∆ΓsΓs

× 102

5 10 15 20 25 30

!!"s""s#102

0

0.05

0.1

0.15

0.2

B̂K = 0.81(3),∆B̂KB̂K

= 4%

S. Cohen, N. Tantalo

*

Lattice calculations of BK: summary*

• unquenched average

• provocative average

• accounting for spread of results

B̂K = 0.78(2)(9)

• a calculation that controls all systematics is missing

B̂K = 0.78(2),∆B̂KB̂K

= 3%

fDs = 248(5) MeV,∆fDsfDs

= 2%

fDs = 252(7) MeV,∆fDsfDs

= 3%

Lattice calculations of fD: summary*

N. Tantalo,R. van de Water

*

fDs = 252(7)(14) MeVfDsfD

= 1.23(2)(4)

fDsfD

= 1.19(3),∆RDRD

= 2%

fDsfD

= 1.23(2),∆RDRD

= 2%

Lattice calculations of fB: summary*

*N. Tantalo

fBs = 245(15) MeV,∆fBsfBs

= 6%

fBs = 269(19) MeV,∆fBsfBs

= 7%

fBs = 269(19)(20) MeVfBsfB

= 1.19(2)(5)

fBsfB

= 1.24(5),∆RBRB

= 4%

fBsfB

= 1.19(2),∆RBRB

= 1.5%

Lattice calculations of BB: summary*

*N. Tantalo

BB(mb) = 0.83(1)(6)

BBs(mb) = 0.85(2),∆BBsBBs

= 2%

BBs(mb) = 0.84(3),∆BBsBBs

= 3%

BBs(mb) = 0.84(3)(5)

BB(mb) = 0.85(2),∆BBBB

= 2%

BB(mb) = 0.83(1),∆BBBB

= 2%

Thank you!*

*V. Lubicz for help in prepairing slides