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