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The width difference of B

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					Carmel

September 2000

The width difference of

Bs-mesons
Ulrich Nierste Fermilab

Work done in collaboration with Martin Beneke, Gerhard Buchalla, Christoph Greub and Alexander Lenz.

Outline
1. Heavy Quark Expansion 2. Lifetime differences 3. The width difference of Bs–mesons 4. Summary

1

1. Heavy Quark Expansion
Voloshin, Shifman; Bigi, Uraltsev, Vainshtein

Optical theorem for total decay rate ;:

;

/ /

Im

hBj i d4x T Heff (x)Heff (0)jBi
=mb) hBjOj{z ) jB} ( i |

Z

HQE = Operator product expansion:

Heff is the effective j Bj =1 hamiltonian.
;
2 X m8;dj c G

F

j

b

j(

O Om
d

dj ;3
QCD

cj :

O

Wilson coefficients containing physics from scales = ( b) j : local operators with dimension j 3.
QCD

Effect: Expansion of ; in

=mb and s(mb).

2

; (B ! all) = ; (b ! all) + O
First term: QCD corrected parton model. First corrections are

m2 b

2

!

QCD

O

motion of the b-quark and the chromomagnetic interaction with the light degrees of freedom.

m2 b

2

QCD

and come from Fermi

Validity of HQE

, Quark-hadron duality

3

2. Lifetime differences
Dominant source of lifetime differences between d, s and mesons: Participation of the spectator quark in the weak

B
in

B B

decay. Effect of order (Exception:

O

m2 b

2

(Bs) ;

O

16 m3 b (Bd) stems from SU (3)F
QCD

2

3

breaking

QCD

matrix elements.)

Bd–B
d

lifetime difference:
d u u d u

b

c

WA

b

b

c

PI

b

Bigi, Shifman, Uraltsev, Vainshtein Neubert, Sachrajda

4

Lifetime difference of Bs

Bs mesons: bs Bs
1 h = p j Bs i 2

bs

Standard Model: Negligible CP-violation in Bs–Bs–mixing:

j BL H i
Width difference

j Bs i

i

;Bs =

;L ; ;H 1

;M
c

Bs

Im

hBsj i d4x T Heff (x)Heff (0)jBsi
s b

Z

from final states common to Bs and Bs
b s c c s c b s b

Measurement at Tevatron Run-II: Compare average s ; + with ( s L) = lifetime ( s) measured in s s 1 ;L measured in s (CP-even component).

=

B

B !

B !D

B

B

5

Compare:

insensitive to new physics

(B +)= (Bd):

mildly sensitive to new physics in penguin coefficients
Keum, U.N.

(Bs)= (Bd) ' 1 (Bs L)= (Bs H ):

O(1%) (in Standard Model):

) tests HQE

New CP-violating physics in Bs–Bs–mixing can suppress ;Bs below its SM value.
Grossman

6

Why calculate lifetime differences to O (
to reduce the sizable -dependence consistent use of
MS

s )?

meaningful use of lattice results for hadronic matrix elements like

hBsjOjBsi, hBsjOS jBsi

QCD corrections are of order 30%. verify infrared safety of the j ’s. Test of quark-hadron duality: Need to go beyond leading logarithmic approximation.

c

7

3. The width difference of Bs–mesons
b s b s b s b s

s b

E1 b
s c b

s

E2 b
c s b

s

E3 b
c s b

s

E4 b
c s

s b

c

D1

b s

s b

c

D2

b s

s b

c

D3

b s

s b

c

D4

b s

c

c

c

c c s b

D5
c

b s

s b

c

D6

b s

s b

c

D7

b s

s b

c

D8

b s

c

c

c s c

c

Q8 D11 b
s

D9

b

s

c

D10 b

s

D12 b

Result: Im

hBsj i d4x T Heff (x)Heff (0)jBsi
2

Z

IR-singularities cancel via two mechanisms:

F and FS are IR-safe functions of z = m2=m2. c b
1 Bloch-Nordsiek cancellations among different cuts of the same diagram 2 factorization of IR-singularities, which end up in s s , s S s

G2 m2 ;V V = ; F b cb cs 12

F (z)hBsjQjBsi + FS (z)hBsjQS jBsi

hB jOjB i hB jO jB i

8

Nonperturbative QCD in

hBsjQjBsi hBsjQS jBsi

8 2 2 f M B 3 Bs Bs 2 MBs B 5 2 2 = ; fBs MBs 3 (mb + ms)2 S =
QCD

Include corrections of order

=mb:

Beneke, Buchalla, Dunietz 1996

; = ; Bs

245 MeV

fBs

2

0:008 B + 0:204 BS ; 0:086]

for the MS-scheme at Quenched lattice QCD:

= mb.
Hashimoto (Lattice ’99) Yamada et al. (Hiroshima)

B( = mb) = 0:80 0:15 BS ( = mb) = 1:19 0:20
; fBs = ; Bs 245 MeV
2

(0:162

0:041 ??? (latt. syst.))

9

Summary
1. Need ( s) corrections to test the HQE predictions for the lifetime differences of mesons.

O

B

2. New CP-violation in

Bs-mixing affects ;Bs .
;Bs
are infrared

3. Next-to-leading QCD-corrections to safe and reduce ;Bs by 30%. ;Bs ;Bs = (16 7)%.

=

10


				
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