Effective wea Lagrangians in the Standard Model and B decays Andrey Grozin A.G.Grozin@inp.ns.su Buder Institute of Nuclear Physics
b s
Dipole operator b q s
Dipole operator b q s s L G a µνt a σ µν b
Dipole operator b q s s L G a µνt a σ µν b R
Dipole operator b q s O g = m b s L G a µνt a σ µν b R
Dipole operator b q s O g = m b s L G a µνt a σ µν b R g 2 2g s M 2 W q=u,c,t x q = m2 q M 2 W V qb V qsex q )
Dipole operator b q s O g = m b s L G a µνt a σ µν b R g 2 2g s M 2 W q=u,c,t x q = m2 q MW 2 V qb Vqs = 0 q=u,c,t V qb V qs Ex q ) E0))
Dipole operator b q s O g = m b s L G a µνt a σ µν b R g2g 2 s V MW 2 tb Vts Ex t ) E0)) x q = m2 q M 2 W
Penguin operator b q s s L D ν G a µνt a γ µ b L
Penguin operator b q s s L D ν G a µνt a γ µ b L EOM D ν G a µν = g s qt a γ µ q Up to an EOM-vanishing operator q s L t a γ α b L ) q qt a γ α q)
GIM mechanism Penguin operator b q s s L D ν G a µνt a γ µ b L EOM D ν G a µν = g s qt a γ µ q Up to an EOM-vanishing operator q s L t a γ α b L ) q qt a γ α q)
Full set of operators s b c c O 1c = c Li γ α b i L) s Lj γ α c j L ) O 2c = c Li γ α b j L ) s Ljγ α c i L)
Full set of operators s b c c O 1c = c Li γ α b i L) s Lj γ α c j L ) O 2c = c Li γ α b j L ) s Ljγ α c i L) O 1u = ū Li γ α b i L) s Lj γ α u j L ) O 2u = ū Li γ α b j L ) s Ljγ α u i L)
O 1 = c Li γ α b i L) s Lj γ α c j L ) O 2 = c Li γ α b j L ) s Ljγ α c i L)
Penguin operators O 1 = c Li γ α b i L) s Lj γ α c j L ) O 2 = c Li γ α b j L ) s Ljγ α c i L) O 3 = s Li γ α b i L) q O 4 = s Li γ α b j L ) q q j γ α q j ) q j γ α q i ) O 5 = s Li γ α γ β γ γ b i L) q O 6 = s Li γ α γ β γ γ b j L ) q q j γ γ γ β γ α q j ) q j γ γ γ β γ α q i )
Penguin operators O 1 = c Li γ α b i L) s Lj γ α c j L ) O 2 = c Li γ α b j L ) s Ljγ α c i L) O 3 = s Li γ α b i L) q O 4 = s Li γ α b j L ) q q j γ α q j ) q j γ α q i ) O 5 = s Li γ α γ β γ γ b i L) q O 6 = s Li γ α γ β γ γ b j L ) q q j γ γ γ β γ α q j ) q j γ γ γ β γ α q i ) Dipole operator O g = m b s L G a µνt a σ µν b R
Lagrangian [ L = g2 2 V 2M usv W 2 ub c 1u O 1u + c 2u O 2u ) + VcsV cb c 1c O 1c + c 2c O 2c ) 6 ] + VtsV tb c i O i i=3
<O 0 1> [ α ] s 1 2C F <O 1 > 4πε
<O 0 1> [ α ] s 1 2C F <O 1 > 4πε α s 1 C F T 1 4πε 4 γµ γ λ γ α γ λ γ µ γ α
<O 0 1> T F T 2 T 1 N c ) α s 1 4πε 4 γα γ λ γ µ γ µ γ λ γ α
<O 0 1> T F T 2 T 1 N c ) α s 1 4πε 4 γα γ λ γ µ γ µ γ λ γ α T F T 2 T 1 N c ) α s 1 4πε 4 γα γ λ γ µ γ α γ λ γ µ γ α γ λ γ µ γ α γ λ γ µ = γ α γ λ γ µ γ µ γ λ γ α + 20γ α γ α
<O 0 1> b s q I ρσ = 1 iπ d/2 = G 11 q 2 ) ε 4d 1) d d ρ q) σ 2 q) 2 [ q 2 g ρσ + d 2)q ρ q σ] T F T 2 T ) 1 αs q 2 g ρσ + 2q ρ q σ γ N c 4πε 12q 2 α γ ρ γ µ γ σ γ α γ µ = T F T 2 T ) 1 αs 2 N c 4πε 3 γµ γ µ
<O 0 1> result <O1> 0 = <O 1 > α s 6T F <O 2 > <O ) 1> 4πε N c 2 3 T α s F <O 4 > <O ) 3> 4πε N c
<O 0 2> Penguin diagram has 0 color factor <O 0 2> = <O 2 > 6T F α s <O 1 > <O ) 2> 4πε N c
<O 0 3> T F T 2 T 1 N c ) α s 1 4πε 4 γα γ λ γ µ γ µ γ λ γ α
<O 0 3> T F T 2 T 1 N c ) α s 1 4πε 4 γα γ λ γ µ γ µ γ λ γ α T F T 2 T 1 N c ) α s 1 4πε 4 γα γ λ γ µ γ α γ λ γ µ γ α γ λ γ µ γ α γ λ γ µ = γ α γ λ γ µ γ µ γ λ γ α + 20γ α γ α
<O 0 3> b s q T F T 2 T ) 1 αs q 2 g ρσ + 2q ρ q σ γ N c 4πε 12q 2 α γ ρ γ µ γ σ γ α γ µ = T F T 2 T ) 1 αs 2 N c 4πε 3 γµ γ µ
<O 0 3> result <O3> 0 = <O 3 > α s + T F <O 6 > <O ) 5> 4πε N c 32 3 T α s F <O 4 > <O ) 3> 4πε N c
<O 0 4> [ α ] s 1 2C F <O 4 > 4πε
<O 0 4> [ α ] s 1 2C F <O 4 > 4πε T F T 1 T ) 2 αs 1 N c 4πε 4 γµ γ λ γ α γ λ γ µ γ α
<O 0 4> α s 1 C F T 2 4πε 4 γα γ λ γ µ γ µ γ λ γ α
<O 0 4> α s 1 C F T 2 4πε 4 γα γ λ γ µ γ µ γ λ γ α T F T 1 T 2 N c ) α s 1 4πε 4 γα γ λ γ µ γ α γ λ γ µ γ α γ λ γ µ γ α γ λ γ µ = γ α γ λ γ µ γ µ γ λ γ α + 20γ α γ α
<O 0 4> b s q T F n f T 2 T ) 1 αs q 2 g ρσ + 2q ρ q σ γ N c 4πε 12q 2 α γ ρ γ µ γ σ γ α γ µ = T F n f T 2 T ) 1 αs 2 N c 4πε 3 γµ γ µ
<O 0 4> result <O 0 4> = [ α s 1 2C F 4πε α s + 8T F 4πε α s ] <O 4 > <O 3 > <O 4> + 1 2 C F 4πε <O 6> + 1 2 T F + 2 3 T F n f α s 4πε N c α s 4πε ) <O 4 > <O 3> N c <O 5 > <O ) 6> N ) c
<O 0 5> [ α ] s 1 2C F <O 5 > 4πε
<O 0 5> [ α ] s 1 2C F <O 5 > 4πε α s 1 C F T 1 4πε 4 γµ γ λ γ α γ β γ γ γ λ γ µ γ γ γ β γ α cancels
T F T 2 T 1 N c ) α s 1 4πε 4 γα γ β γ γ γ λ γ µ γ µ γ λ γ γ γ β γ α γ α γ β γ γ γ λ γ µ γ µ γ λ γ γ γ β γ α = 20γ α γ β γ γ γ γ γ β γ α 64γ α γ α
T F T 2 T 1 N c ) α s 1 4πε 4 γα γ β γ γ γ λ γ µ γ µ γ λ γ γ γ β γ α γ α γ β γ γ γ λ γ µ γ µ γ λ γ γ γ β γ α = 20γ α γ β γ γ γ γ γ β γ α 64γ α γ α T F T 2 T 1 N c ) α s 1 4πε 4 γα γ β γ γ γ λ γ µ γ γ γ β γ α γ λ γ µ γ α γ β γ γ γ λ γ µ γ γ γ β γ α γ λ γ µ = 64γ α γ α
<O 0 5> b s q T F T 2 T ) 1 αs q 2 g ρσ + 2q ρ q σ γ N c 4πε 12q 2 α γ β γ γ γ ρ γ µ γ σ γ γ γ β γ α γ µ = T F T 2 T ) 1 αs 8 N c 4πε 3 γµ γ µ
<O 0 5> result <O5> 0 = <O 5 > α s + 10T F <O 6 > <O ) 5> 4πε N c 200 3 T α s F <O 4 > <O ) 3> 4πε N c
<O 0 6> [ α ] s 1 2C F <O 6 > 4πε
<O 0 6> [ α ] s 1 2C F <O 6 > 4πε T F T 1 T 2 N c ) αs 4πε 1 4 γµ γ λ γ α γ β γ γ γ λ γ µ γ γ γ β γ α
α s 1 C F T 2 4πε 4 γα γ β γ γ γ λ γ µ γ µ γ λ γ γ γ β γ α γ α γ β γ γ γ λ γ µ γ µ γ λ γ γ γ β γ α = 20γ α γ β γ γ γ γ γ β γ α 64γ α γ α
α s 1 C F T 2 4πε 4 γα γ β γ γ γ λ γ µ γ µ γ λ γ γ γ β γ α γ α γ β γ γ γ λ γ µ γ µ γ λ γ γ γ β γ α = 20γ α γ β γ γ γ γ γ β γ α 64γ α γ α T F T 1 T 2 N c ) α s 1 4πε 4 γα γ β γ γ γ λ γ µ γ γ γ β γ α γ λ γ µ γ α γ β γ γ γ λ γ µ γ γ γ β γ α γ λ γ µ = 64γ α γ α
<O 0 6> b s q T F n f T 2 T ) 1 αs q 2 g ρσ + 2q ρ q σ γ N c 4πε 12q 2 α γ β γ γ γ ρ γ µ γ σ γ γ γ β γ α γ µ = T F n f T 2 T ) 1 αs 8 N c 4πε 3 γµ γ µ
<O 0 6> result <O 0 6> = [ α s 1 2C F 4πε α s + 2T F 4πε α s ] <O 6 > <O 5 > <O 6> N c + 1 2 C F 4πε 20<O 6> 64<O 4 >) α s 32T F <O 3 > <O ) 4> 4πε N c + 8 3 T α s F n f <O 4 > <O ) 3> 4πε N c )
1-loop anomalous dimensions
B 0 B 0
b q d b d d q b d q q b O = d L γ α b L ) d L γ α b L ) L = g 4 2 CO 512π 2 MW 2
b q d b d d q b d q q b O = d L γ α b L ) d L γ α b L ) L = g 4 2 CO 512π 2 MW 2 C = q,q =u,c,t x q = m2 q M 2 W V qbv qd V q bv q dsx q, x q )
b q d b d d q b d q q b O = d L γ α b L ) d L γ α b L ) L = g 4 2 CO 512π 2 MW 2 C = q =u,c,t q,q =u,c,t V qbv qd V q bv q d [Sx q, x q ) Sx q, 0)] V q bv q d = 0
b q d b d d q b d q q b O = d L γ α b L ) d L γ α b L ) L = g 4 2 CO 512π 2 MW 2 C = V tbv td q=u,c,t V qbv qd [Sx q, x t ) Sx q, 0)]
b q d b d d q b d q q b O = d L γ α b L ) d L γ α b L ) L = g 4 2 CO 512π 2 MW 2 C = V tbv td q=u,c,t q=u,c,t V qbv qd = 0 V qbv qd [Sx q, x t ) Sx q, 0) S0, x t ) + S0, 0)]
b q d b d d q b d q q b O = d L γ α b L ) d L γ α b L ) L = g 4 2 CO 512π 2 MW 2 C = V tbv td ) 2 [Sx t, x t ) Sx t, 0) S0, x t ) + S0, 0)]
b q d b d d q b d q q b O = d L γ α b L ) d L γ α b L ) L = g 4 2 CO 512π 2 MW 2 C = V tbv td ) 2 [Sx t, x t ) 2Sx t, 0) + S0, 0)]
1-loop anomalous dimension γ 0 = λ + = 12T F 1 1 N c )