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Transcript of Chiral symmetry and B decays in the SCET › talks › WorkShops › int_05_1 › People › ... ·...
Dan Pirjol (MIT)
Chiral symmetry and B decays in the SCET
• Effective field theories: HQET, heavy hadron chiral perturbation theory
• The Soft-Collinear Effective Theory
• Combining the SCET and chiral perturbation theory
• Applications
• Conclusions and outlook
Outline
Energy scales in B physicsHeavy quark (b) interacting with soft quarks and gluons
Two relevant scales: mb ∼ 4.6 GeVΛ ∼ 500 MeV ,
1. Systematic expansion in Λ/mb ∼ 0.1
2. Heavy-quark spin and flavor symmetry at leading order in Λ/mb
HQET - the appropriate effective theory
π, ρ
HQET - symmetries
Superfield Ha =1 + v/
2[B∗
µγµ− Bγ5]
Constructing operators using only H automatically incorporates heavy quark symmetry
Heavy meson fields B, B* form a spin doublet j! =
1
2
−
→ JP= 0
−, 1−
Example: semileptonic B -> D(*) decay
〈D(∗)|cΓb|B〉 = ξ(v.v′)Tr[H(c)v′ ΓHv] + O(Λ/mc)
Isgur-Wise function
Chiral symmetryMassless QCD mq = 0 has a chiral symmetry
broken down spontaneously to the flavor group
Goldstone bosons M =
1√2π0 +
1√6η π+ K+
π−−
1√2π0 +
1√6η K0
K− K0−
2√6η
SU(3)L × SU(3)R → SU(3)V
ξ = eiM/fΣ = ξ2
Transformations:
Σ → LΣR†
ξ → LξU†= UξR†
Can be extended to include also e.m. interactions
Write down the most general Lagrangian invariant under the chiral transformations
L =f2
8Tr∂µΣ∂µΣ +
f2B0
4Tr(mqΣ + mqΣ
†) + · · ·
(pπ, mq)Organize the Lagrangian in an expansion in
π
π π
π π
Gasser, Leutwyler
Heavy hadron chiral perturbation theory
Transformation under the heavy quark spin rotations
and chiral symmetry
H → SH
H → HU†
L = −TrHaiv · DbaHb + gTrHaHbγµγ5Aµba + · · ·
V µab =
1
2(ξ†∂µξ + ξ∂µξ†)
Aµab =
i
2(ξ†∂µξ − ξ∂µξ†)
Dµab = δab∂
µ− V
µab
πaB
(∗)
B B∗
Wise, Burdman+Donoghue
Application
Jµa = qaγνPLb transforms as (3L ⊗ 1R)
matches in the chiral perturbation theory as
Jµa →
1
2ifBmBTr[γµPLHbξ
†ba] + · · ·
Semileptonic B -> pi decays with a soft pion
〈π(p′)|uγµPLb|B(p)〉 = f+(q2)(p + p′)µ + f−(q2)(p − p′)µ
π
B π B∗
B
f+(Eπ) = gfBmB
2fπ
1
Eπ + ∆
Result at LO in the chiral expansion
∆ = mB∗ − mB
∼ 50 MeV
Valid for not too energetic pions Eπ < 4πfπ ∼ 1 GeV
0 5 10 15 20 25 30
q2 (GeV
2)
0.0
1.0
2.0
f0,+(q
2)
UKQCD
APE
Fermilab
JLQCD
NRQCD
LCSR
LCSR
Lattice QCD
q2
> 17 GeV2
5 10 15 20 25
0.5
1
1.5
2
2.5
3
g=0.5
g=0.3g=0.1
Energetic hadronsConstruct a heavy-quark expansion for B processes
involving both soft and energetic light particles
Example: decays
BW
Energy scales:- Soft
- Hard
- Hard-collinear
mb ∼ 4.6 GeV
Λ ∼ 500 MeV ,
√Λmb ∼ 1.4 GeV
B → π"ν (Eπ ∼ mB/2 ∼ 2.2 GeV)
π
The soft-collinear effective theory (SCET)
• Systematic power counting in implemented at the level of momenta, fields, operators
• Construct effective Lagrangians for strong and weak interactions expanded in powers of
• New symmetries (gauge, reparameterization invariance)
• Nonlocal operators and Lagrangians
Λ/mb
Λ/mb
Degrees of freedomIntroduce distinct fields for each relevant degree of freedom, with well defined momentum scaling
modes field pµ ∼ (+,−,⊥)
hardhard-collinear
collinear
soft/ultrasoft
An,q, ξn,p
An,q, ξn,p
As, q, bv
(Q, Q, Q)
(Λ, Q,√
QΛ)
(Λ2/Q, Q,Λ)
(Λ,Λ,Λ)
Integrate out successively the hard scales Q2→ QΛ
QCD -> SCETI -> SCETIIIntegrate out hard collinear modes
SCETII = theory containing only soft and collinear
modes p2
s, p
2
c∼ Λ
2
p2
hc ∼ QΛ
SCETI SCETII
Soft-collinear decoupling at LO
LSCETII= L
(0)S
+ L(0)C
+ · · ·
O = OS ×OC 〈O〉 = 〈OS〉 × 〈OC〉
factorization
Combining SCET with xPTObjective: construct an effective theory describing
energetic hadrons and soft Goldstone bosons
Relevant scales: hard scale Q
hard-collinear scale√
ΛQ
soft scale
|!pπ | ∼ mπ
Λ
pion momenta
•
•
•
•
Hierarchy: Q !
√QΛ ! Λ ∼ pπ ∼ mπ
Use SCET for soft-collinear factorization and apply xPT to the soft part
B decays without collinear hadrons in the final state
Factorization in semileptonic radiative decays
Cirigliano, DPB → πSγ#ν!
Factorization in Simplest exclusive B decay, proceeding through weak annihilation of the b quark with the light spectator
B → γ"ν!
W
B
γ
B ⊗
QCD SCETIIeiq.xT{jem
µ (x) , Vν(0)} → H(Eγ)
∫dλeiλn·xJ(k+)[q(λn)Yn(λ, 0)Γb(0)]
hard jet soft⊗ ⊗
Lunghi,DP,Wyler;Becher,Hill,Lange,Neubert
Factorization relation
A(B → γeν) ∼ C(v)(2Eγ , µ)
∫dk+J(k+)φ+
B(k+)
J(k+) =παs(
√QΛ)CF
k+
(1 + O(αs(
√QΛ))
)Jet function - hard-collinear scales
Wilson coeff - hard momenta C1(ω) = 1 + O(αs(Q))
B light-cone wave function
〈0|q(λn)Yn(λ, 0)Γb(0)|B〉 =
∫dk+e−iλk+Tr [
1 + v/
2(n/φ+
B(k+) + n/φ−
B(k
−))γ5Γ]
•
•
•
B
W
γ
B ⊗
QCD SCETII
eiq.xT{jemµ (x) , Vν(0)} → H(Eγ)
∫dλeiλn·xJ(k+)[q(λn)Yn(λ, 0)Γb(0)]
Operator matching
Independent on the soft state (B -> vacuum)
Can describe equally well B → π B → ππ, · · ·,
Find the chiral representation of the soft light cone operator
OaL(k+) =
∫dλe−iλk+ qa
L(λn)Yn(λ, 0)Γb(0)
(3L,0R)Transforms like under chiral SU(3)LxSU(3)R
Most general chiral LO operators
OaL(k+) → Tr [α(k+)ΓHbξ
†ba
] + · · ·
α(k+) = a1 + n/a2 + v/a3 + [n/, v/]a4
Heavy quark symmetry fixes a1-a4 in terms of the B light-cone wave functions
All B -> Goldstone boson matrix elements of <OL>
are fixed by heavy quark and chiral symmetry
π
No need for additional low-energy constants!
B(∗)
B(∗)
Previous work on chiral representation of nonlocal operators:
Twist-2 DIS and DVCS structure functionsJ.W.Chen,M.Savage
SU(3) breaking in , wave functions π K
J.W.Chen,I.Stewart
Application: B → πγ#ν
In the kinematical region with an energetic photon and one soft pion, SCET+xPT give a factorization theorem
1 2
1
2
3
E
E
g
p q2
= 4 GeV2
Validity region
Eγ ! Λ
Eπ < ΛχSB
Important for a precise measurement of the
B → π"ν branching fractions
from SCET + xPT
π
B∗
BB π
Leading order diagrams
B → πγ#ν
Factorization relation (schematic)
A(B → πγ#ν) ∼ C(2Eγ)
∫dk+J(k+)φ+
B(k+)S(Pπ)
Results
Form-factor parameterization compatible with the Ward identity
T Jµν = i
∫d4xeiq.x〈π(p′)|jem
µ (x) , Jν(0)|B(p)〉
Jν = Vν , Aν
The amplitude is given byB(p) → π(p′)γ(q, ε)$ν
W = p − p′ − q
ε∗µVµν = V1(ε∗
ν −
ε∗ · W
q · Wqν)
+(p′ · ε∗ −(p′ · q)(ε∗ · W )
q · W)(V2qν + V3Wν + V4p
′
ν)
+ε∗ · W
q · WFν(p, p′)
qµVµν = Fν ≡ 〈π|Vν |B〉
V1−4 are functions of (Eπ, Eγ , W 2)
Analogous results for the axial current
Factorization relations for
SL, SR = calculable functions of the pion momentum
V1 = C(v)1 (2Eγ)SL(pπ)I(Eγ , µ)
V2 =1
2Eγ(C(v)
2 (2Eγ) + 2C(v)3 (2Eγ))SR(pπ)I(Eγ , µ)
V3 =1
n · WC
(v)2 (2Eγ)SR(pπ)I(Eγ , µ)
V4 = O(Λ
Q)
I(Eγ , µ) =
∫dk+J(k+, µ)φ+
B(k+)
B → πγ#ν
Decays with one collinear hadron in the final state
B → KnπS"+"−Forward-backward asymmetry in
Grinstein, DP
SCETII matchingMatching a current onto SCETII produces both
factorizable and nonfactorizable operators
qΓ⊥µ b → c1(ω)qn,ωγ⊥
µ PLbv
+b1L ⊗ J⊥ ⊗ (qn/γ⊥µ γ⊥
λ PRbv) ⊗ (qn,ω1n/γ⊥
λ qn,ω2)
+b1R ⊗ J‖ ⊗ (qn/γ⊥µ PRbv) ⊗ (qn,ω1
n/PLqn,ω2)
+
b q
Factorization relations for form factors
Bauer, DP, StewartBeneke, FeldmannLange,NeubertHill, Becher, Lee,
Neubert
Form factors can be written as combinations of soft and hard
scattering terms
fB→Mi (E) = ciζ
BM + biJ ⊗ φ+
B⊗ φM
ci, bi are Wilson coefficients in SCET-1
Predictions for transverse helicity amplitudes at LO in 1/mb for B -> M decays
H+(B → V ) = c10 + b1R0
H−(B → V ) = c1ζ⊥ + b1L ⊗ J⊥ ⊗ φ+B⊗ φV
⊥
1. The right-handed amplitude vanishes exactly at LO in 1/mb
Symmetry relations
mB
mB + mV
V (E) =mB + mV
2EA1(E)
T1(E) =mB
2ET2(E)
Burdman, Hiller
2. Prediction independent on the hadron spin
Multibody decaysLO predictions for B -> M
H−(B → Mnπ) = c1ζBMπ⊥ + b1LJ⊥ ⊗ 〈π|OS |B〉 ⊗ φ⊥
M
H+(B → Mnπ) = c10 + b1RJ‖ ⊗ 〈π|OS |B〉 ⊗ φ‖M
π
1. The right-handed amplitude is non-vanishing!
Completely factorizable -> calculable if the soft matrix element is known
2. New soft functions contributing to the left-handed amplitude
w/ soft pion
Factorization in B → Kπ"+"−
π
B
KB*
π
B
KB* K
B K*π
O(1) O(Λ/mb)
Use chiral perturbation theory to compute the soft matrix element 〈π|OS |B〉
ζBKπ⊥New soft function
Factorization relations
H+(B → Kπ) =1
2m2
B
( g
fπ
ε∗+ · pπ
v · pπ + ∆
)∫dzb1R(z)ζBK
J (z)
Right-handed amplitude
Same convolution as that appearing in B->K form factors
ζBKJ (z) =
fBfK
mB
∫dxdk+J(x, z, k+)φ+
B(k+)φK(x)
FB asymmetryNew contribution to the FBA of the lepton momentum distribution
K∗
!+
!−
AFB(q2) =Γ(θ+ > 0) − Γ(θ+ < 0)
Γ(θ+ > 0) + Γ(θ+ < 0)
The FB asymmetry measures the interference of the 2 transverse helicity amplitudes
∼
(C9 + 2mb
MB
q2C7
)ζ(EK∗)+∆ζfact
AFB(q2) ∼ Re [H(A)∗−
H(V )−
− H(A)∗+ H
(V )+ ]
Results for the zero
0
1
2
3
4
5
6
7
0 1 2 3 4 5
q2
M2Kp
µ=4.8 GeVµ=9.6 GeVµ=2.4 GeV
Scale dependence Soft pion corrections
0
1
2
3
4
5
6
7
0 1 2 3 4 5
q2
M2Kp
Ep=300 MeVEp=150 MeV
µ=4.8 GeV
For given M(K+pi), predict a zero at q0[M(K+pi)]
The effect depends on the size of the factorizable m.e. ζBK
J
q20 |MKπ=MK+Mπ
= 3.69+0.42−0.53GeV
2
Other applications
Semileptonic and radiative B decays into multibody states (one collinear + multiple soft pions)
B → πnπS"ν B → KnπS"+"−e.g.
Larger branching ratios, more observables
Tests of the Standard Model with nonresonant modes: the photon helicity in
Ligeti et al.
B → KSπ0γ
• 20-odd pages in the PDG on nonleptonic and semileptonic B decays
• SCET helps reduce the number of independent hadronic parameters
• Model-independent analyses of exclusive semileptonic and nonleptonic B decays are possible
• The SCET+xPT combination extends these methods to multibody decays
Rich phenomenology