Three partons in kT factorization

Hsiang-nan LiAcademia Sinica

May 16, 2012Ref: Chen and Li, 1104.5398;

1112.5059

Outlines

• Introduction• Gauge invariance• 3-parton contributions• B -> pi form factors• Summary

Introduction • kT factorization has been pushed to

subleading level• NLO for pion transiton, EM form factors, B->pi

form factors• Next-to-leading power in 1/Q needs to be

examined too• Have examined 2-parton twist-3• Consider 3-parton contributions, which should

not be separated from 2-parton twist-3

Power expansion in kT

• kT is kept in propagator denominators• Can this be extended to higher power

consistently?• Will there be double counting?• Is there gauge invariance at higher power?

Form factors• Pion EM form factor is symmetric under flip of

initial and final states• 3 partons on both sides, power of 1/Q^2• B->pi form factor is not symmetric• 3 partons on one side only, power of 1/mB• 3-parton contribution vanishes as mB->0• Need to confirm gauge invariance first• 3-parton contributions negligible, few

percents

Gauge invariance

Gauge dependence

• Two sources of gauge dependence:• Transverse momenta of 2-parton state

• 3-parton state

• The two sources cancel as combined into

kinematics• LO diagrams for pion EM form factor

• kinematics

Fermion and color flows• Fierz transformation

• Color identity

j i

kl

j i

kl

focus onthis one

2-parton 3- parton

2-to-2 gauge dependence • Spin projectors for initial and final state in LO

diagrams

• Gluon propagator in covariant gauge

gauge parameter

Amplitude from Fig. 1(a)• Gauge dependent piece

• Extract term proportional to k1 and k2, ie., partial derivative of quark fields Ward identity

valence quark

valence anti-quark

Amplitude from Fig.1(b)

valence quark

valence anti-quark

3-to-2 gauge dependence• Diagrams

• A, B,…, and H represent attachments of additional valence gluon from initial state

• Attachments to initial valence lines should be included for U(1) gauge invariance, which lead to 2-parton twist-3 DAs

Attachment A as an example• Color factorization

• Initial-state spin projectorb

a

Extraction of gauge dependence• Amplitude from Attachment A

• Extract term proportional to k2

Other 3-to-2

Gauge invariance• Sum over all attachments

• A and B added into with color factor• Second term of G and H added into• Sum is independent of l1, which can be

integrated out,• Equation of motion for

2-to-3 and 3-to-3• 2-to-3 gauge dependence• 3-to-3

• Use equation of motion again

3-parton contributions

Three-parton contributions• Consider the matrix element

• Insert does not change power behavior • Employ . Just need to

consider 3-parton state

• gives 3-parton twist-4• does not contribute

Parton momenta and structures

• Initial quark, anti-quark, gluon carry

• Structures for initial- and final-states

Dominant diagram

• With 4-gluon vertex

Factorization formula• For the dominant diagram

obey equation of motion with 2-parton DAs

Other diagrams

More diagrams

Numerical results

B -> pi form factors

Gauge dependence from 2 partons• LO diagrams for B->pi form factor

• kinematics

Amplitude from Fig.1(a)• Spin projectors for initial and final states

• Gauge dependence

• Extract term proportional to k2

Amplitude from Fig.1(b)• Gauge dependent piece

• Extract term proportional to k2

• Gauge dependence from Figs.1(a) and 1(b) cancel

Gauge dependence from 3 partons• 2-to-3 diagrams with one additional valence

gluon from the pion side

• Spin projector for the pion replaced by• Color factorization for Attachment A

Amplitudes from all attachments• Other attachments vanish

• They cancel each other. No need of equation of motion

2-to-3 contribution• B -> pi form factors

• Hard kernels proportional to mB

3-parton B wave function• 3-parton matrix elements

• Sum rules by Grozin, Neubert

• Nishikawa, Tanaka

3-to-2 contribution• Adopt 3-parton B meson wave function

• 3-to-2 hard kernel, also proportional to mb

Wave functions

Numerical results• Cancellation between 2-to-3 and 3-to-2

contributions same order of magnitude as fromGegenbauer terms in 2-parton pion DAs

Figures

• Contributions from GN parameters larger than NT parameters

LO

Summary on various contributions • B meson spin projector for 2 partons

• 1st , leading power; 2nd, 30%, 3rd, few percents• 3-parton contributions are also few percents• 3-parton contributions are of the same order

of magnitude as higher Gegenbauer terms of 2-parton DAs

integration of