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Three partons in kT factorization. Hsiang- nan Li Academia Sinica May 16, 2012 Ref: Chen and Li, 1104.5398; 1112.5059. Outlines . Introduction Gauge invariance 3-parton contributions B -> pi form factors Summary. Introduction . - PowerPoint PPT Presentation
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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