Sudakov and heavy-to-light form factors in SCET

11

Sudakov and heavy-to-light Sudakov and heavy-to-light form factors in SCETform factors in SCET

Zheng-Tao Wei Zheng-Tao Wei

Nankai UniversityNankai University

2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing

2009.9.9, KITPC, Beijing2009.9.9, KITPC, Beijing 22

Wei, PLB586 (2004) 282,Wei, hep-ph/0403069,Lu, et al., PRD (2007)

Introduction to SCET

Sudakov form factor

Heavy-to-light transition form factors

Summary

33

I. Soft-Collinear Effective TheoryI. Soft-Collinear Effective Theory

The soft-collinear effective theory is a low energy effective theory for collinear and soft particles. (Bauer, Stewart , et al.; Beneke, Neubert….)

(1) It simplifies the proof of factorization theorem at the Lagrangian and operator level.

(2) The summation of large-logs can be performed

in a new way.


44

. regluon whe

soft a with interactsquark energetican :processelmentary an Consider

QCDQ

• Diagrammatic analysis and effective Lagrangian

)0,,0( Qpc

eikonal approximation


55

Transforming the diagrammatic analysis into an effective Lagrangian

0.WD where :onredefiniti Field ss0 inWs

}.)( exp{)( line Wilson The x

tnAigndtigPxW

LEET


66

• Power counting

),,(~ :ultrasoft

),,(~ :soft

),1,(~ :collinear

222

2

us

s

c

p

p

p

MomentumField

23

2/3

2

~ ,~

~ ,~

),1,(~ ,~

usus

ss

c

Aq

Aq

A

• Degrees of freedom

SCET(I)for /

SCET(II)for /

QCD

QCD

Q

Q

Reproduce the full IR physics


77

),,Q(~

) ,1 ,(~

,~ :SCET(I)

222

2

QCD2

us

c

c

p

Qp

Qp

The effective Lagrangian:

• The effective interaction is non-local in position space.

• Two different formulae: hybrid momentum-position space and position space representation.

/QCD Q


88

Gauge invariance

• The ultrasoft field acts as backgroud field compared to collinear field.

The collinear and ultrasoft gauge transformation are constrained in corresponding regions,


99

Wilson lines

ccc

us

Win

WDin

W

11

on redefiniti Field0

Gauge invariant operators: (basic building blocks)

No interaction with usoft gluons

sv , , qWhWW ususcc

. , :ionfactorizatusoft -collinear 00 uscuscsu WAWAW


1010

),,Q(~

),1,(~

,~ :SCET(II)2

22

s

c

c

p

Qp

p /QCD Q

Matching: mismatch? New mode, such as soft-collinear mode proposed by Neubert et al.?

Endpoint singularity?


1111

Q

Q

SCET(I)

SCET(II)

Two step matching:

1. Integrate out the high momentum fluctuations of order Q,

2. Integrate out the intermediate scale (hard-collinear field)

SCET(II)SCET(I)QCD 21



Form factor

The matrix elements of current operator between initial and final states are represented by different form factors.

Form factors are important dynamical quantity for describing the inner properties of a fundamental or composite particle.

III. Sudakov form factorIII. Sudakov form factor

1313

The form factor (only the first term F1(Q2)) in the asymptotic limit q2→∞ is called Sudakov form factor. (in 1956)

).()](2

)()['( 22

21 puqF

m

qiqFpu

The interaction of a fermion with EM current is represented by

At q2=0 , the g-factor is given by and the anomalous magnetic moment is

sgm

e

qFg

2

)0(2

2 22


1414

The large double-logarithm spoils the convergence of pertubative expansion. The summation to all orders is an exponential function. The form factor is strongly suppressed when Q is large. In phenomenology, it relates to most high energy process in certain momentum regions, DIS, Drell-Yan, pion form factor, etc.

The naïve power counting is strongly modified (at tree level F=1).


1515

• Methods of momentum regions (by Beneke, Smirnov, etc)

The basic idea is to expand the Feynman diagram integrand in the momentum regions which give contributions in dimensional regularization. Each region is involved by one scale.

). , ,(~ :regionsoft the)4(

); ,1 ,(~ :region B-collinear the)3(

); , ,1(~ :regionA -collinear the(2)

);1 ,1 ,1(~gluon virtual the:region hard the(1)

:from come onscontributi thefactor, formquark In the

222

2

2

Qk

Qk

Qk

Qk



Regularization method

Introduce a cutoff scale δ in both k+ and k-.

DOF

),,Q(~

),1,(~

,~ :SCET(I)

222

2

2

us

c

c

p

Qp

Qp

Bauer (2003)


1818

Factorization

Step 1: the separation of hard from collinear contributions.

Step 2: the separation of soft from collinear functions.

. , :onredefiniti Field 00 scscs WAWAW

Q,at SCET ofoperator theonto QCDin current theMatching AB


1919

Evolution

The anomalous dimension depends on the renormalization scale. The exponentiation is due to the RGE. The suppression is caused by the positive anomalous dimension.

Two-step running:

]lnexp[]lnexp[)()( 2

Q S

Q

Q C ddQCQF


2020

Exponentiation and scaling

.dg')(g'

) ,(g'Exp)C()C(

0,(g) and ) (g, :III Case

ion.approximat LLin ,)(

)()()C(

0,(g) and (g) :II Case

law.-power )()C(

0,(g) and zero-nonbut constant, :I Case

.ln

)(ln :RGE

)(

)g(0

)2/(

00

00

0

00

g

s

sC

C

d

Cd

Exponential of logs can be considered as a generalized scaling.


2121

Comparisons with other works

1. The leading logarithmic approximation method sums the leading contributions from ladder graphs to all orders. The ladder graphs constitutes a cascade chain: qq->qq->…->qq. There are orderings for Sudakov parameters.

2. Korchemsky et al. used the RGE for a soft function whose evolution is determined by cusp dimension. The cusp dimension contains a geometrical meaning.

).( space

n Minkowskiain anglewith

)( ),(

gg cuspcusp


2222

3. CSS use a diagrammatic analysis to prove the factorization. The RGEs are derived from gauge-dependence of the jet and hard function. The choice

of gauge is analogous to the renormalization scheme.

).(2)(lnln

);,/(),/(ln

ln

ggd

dK

d

dG

gQGgmKQ

F

cuspK


2323

IV. Heavy-to-light transition form factorIV. Heavy-to-light transition form factor

The importance of heavy-to-light form factors:

CKM parameter Vub

QCD, perturbative, non-perturbative basic parameters for exclusive decays in QCDF or SCET new physics…

At large recoil region q2<<mb2, the light meson moves

close to the light cone.

QCD2

QCD ~ ,~ ,/~ PmPmP bb

Light cone dominance


2424

Hard gluon exchange: soft spectator quark → collinear quark Perturbative QCD is applicable.

2QCDQCD

2 ~)(

momentumgluon exchanged The

bcs mpp

Hard scattering


2525

Endpoint singularity

1

0

1

0 2...

)()()1()()()12(

x

xx

x

xddxF BBB

0

,0

x

endpoint singularity

Factorization of pertubative contributions from the non-perturbative part is invalid. There are soft contributions coming from the endpoint region.


2626

Hard mechanism -- PQCD approach

The transverse momentum are retained, so no endpoint singularity. Sudakov double logarithm corrections are included.


Momentum of one parton in the light meson is small (x->0). Soft interactions between spectator quark in B and soft quark in light meson.

Methods: light cone sum rules, light cone quark model… (lattice QCD is not applicable.)

Soft mechanism

2727

Spin symmetry for soft form factor

In the large energy limit (in leading order of 1/mb),

The total 10 form factors are reduced to 3 independent factors. 3→1 impossible!

J. Charles, et al., PRD60 (1999) 014001.


2828

Definition

)*)((2| |

)v*)((2| |

)(2| |

v

||v5

Pv

EEBhV

EmiBhV

EEBhP

V


2929

QCDF and SCET

In the heavy quark limit, to all orders of αs and leading order in 1/mb,

)()(),( )()( 2 yxyxTdxdECqF BMiiii

Sudakovcorrections

Soft form factors,with singularity andspin symmetry Perturbative, no singularity

The factorization proof is rigorous. The hard contribution ~ (Λ/mb)3/2, soft form factor ~ (Λ/m b)2/3 (?) About the soft form factors, study continues, such as zero-bin method…


3030

Zero-bin method by Stewart and Manohar

A collinear quark have non-zero energy. The zero-bin contributions should be subtracted out. After subtracting the zero-bin contributions, the remained is finite and can be factorizable.

For example,

)/(ln)0(')0(')0()()( 1

0 2

1

0 2

pdx

x

xxdx

x

x


3131

Soft overlap mechanism

The soft part form factor is represented by the convolution of initial and final hadron wave functions.

Fock

jjLiiBjjiiji

kxkxdkdxdkdxq ),(),()(,

2


3232

Dirac’s three forms of Hamiltonian dynamics( S. Brodsky et al., Phys.Rep.301(1998) 299 )


3333

Advantage of LC framework

LC Fock space expansion provides a convenient description of a hadron in terms of the fundamental quark and gluon degrees of freedom. The LC wave functions is Lorentz invariant. ψ(xi, k┴i ) is independent of the bound state momentum. The vacuum state is simple, and trivial if no zero-modes. Only dynamical degrees of freedom are remained. for quark: two-component ξ, for gluon: only transverse components A┴.

Disadvantage

In perturbation theory, LCQCD provides the equivalent results as the covariant form but in a complicated way.

It’s difficult to solve the LC wave function from the first principle.


3434

LC Hamiltonian Kinetic Vertex

Instantaneous interaction

LCQCD is the full theory compared to SCET.

Physical gauge is used A+=0.


3535

LC time-ordered perturbation theory

Diagrams are LC time x+-ordered. (old-fashioned) Particles are on-shell. The three-momentum rather than four- is conserved in each vertex. For each internal particle, there are dynamic and instantaneous lines.


3636

Perturbative contributions:

Only instantaneous interaction in the quark propagator.

The exchanged gluons are transverse polarized.

Instantaneous, no singularitybreak spin symmetry

have singularity,conserve spin symmetry


3737

Basic assumptions of LC quark model

Valence quark contribution dominates.

The quark mass is constitute mass which absorbs some dynamic effects.

LC wave functions are Gaussian.


3838

LC wave functions

2MPPH LC

In principle, wave functions can be solved if we know the Hamiltonian (T+V).

Choose Gaussian-type

Power law: )/exp(-~)( QCD2

bmq

The scaling of soft form factor depends on the light meson wave function at the endpoint.


3939

Melosh rotation 22

0

'0

'

)(

)(2

),()',()(

pMxm

npiMxmm

spuspuR

ii

ssiii

iiDssM

meson. for vector

),,()(

),(~2

1

meson;ar pseudoscalfor

),,(),(~2

1

:) ,( seigenstatehelicity LC

ofout ),(spin of of state a constructs

2221

11

0

1

22511

0

00

21

21

21

21

kvw

kkku

MR

kvkuM

R

SSR

V

S

zSS

z

z


4040

Numerical results

The values of the three form factors are very close, but they are quite different in formulations.


4141

Comparisons with other approaches


4242

SummarySummary

SCET provides a model-independent analysis of processes with energetic hadrons: proof of factorization theorem, Sudakov resummation, power corrections.

SCET analysis of Sudakov form factor emphasizes the scale point of view.

LC quark model is an appropriate non-perturbative method to study the soft part heavy-to-light form factors at large recoil.

How to treat the endpoint singularity is still a challenge.


ThanksThanks


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Sudakov and heavy-to-light form factors in SCET