Three partons in kT factorization Hsiang-nan Li Academia Sinica May 16, 2012 Ref: Chen and Li, ;

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 k T k T 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 ji kl ji kl focus on this one 2-parton3- 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 projector b 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 from Gegenbauer 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 1 st, leading power; 2 nd, 30%, 3 rd, 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