Three partons in kT factorization
description
Transcript of Three partons in kT factorization
![Page 1: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/1.jpg)
Three partons in kT factorization
Hsiang-nan LiAcademia Sinica
May 16, 2012Ref: Chen and Li, 1104.5398;
1112.5059
![Page 2: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/2.jpg)
Outlines
• Introduction• Gauge invariance• 3-parton contributions• B -> pi form factors• Summary
![Page 3: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/3.jpg)
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
![Page 4: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/4.jpg)
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?
![Page 5: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/5.jpg)
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
![Page 6: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/6.jpg)
Gauge invariance
![Page 7: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/7.jpg)
Gauge dependence
• Two sources of gauge dependence:• Transverse momenta of 2-parton state
• 3-parton state
• The two sources cancel as combined into
![Page 8: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/8.jpg)
kinematics• LO diagrams for pion EM form factor
• kinematics
![Page 9: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/9.jpg)
Fermion and color flows• Fierz transformation
• Color identity
j i
kl
j i
kl
focus onthis one
2-parton 3- parton
![Page 10: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/10.jpg)
2-to-2 gauge dependence • Spin projectors for initial and final state in LO
diagrams
• Gluon propagator in covariant gauge
gauge parameter
![Page 11: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/11.jpg)
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
![Page 12: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/12.jpg)
Amplitude from Fig.1(b)
valence quark
valence anti-quark
![Page 13: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/13.jpg)
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
![Page 14: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/14.jpg)
Attachment A as an example• Color factorization
• Initial-state spin projectorb
a
![Page 15: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/15.jpg)
Extraction of gauge dependence• Amplitude from Attachment A
• Extract term proportional to k2
![Page 16: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/16.jpg)
Other 3-to-2
![Page 17: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/17.jpg)
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
![Page 18: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/18.jpg)
2-to-3 and 3-to-3• 2-to-3 gauge dependence• 3-to-3
• Use equation of motion again
![Page 19: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/19.jpg)
3-parton contributions
![Page 20: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/20.jpg)
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
![Page 21: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/21.jpg)
Parton momenta and structures
• Initial quark, anti-quark, gluon carry
• Structures for initial- and final-states
![Page 22: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/22.jpg)
Dominant diagram
• With 4-gluon vertex
![Page 23: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/23.jpg)
Factorization formula• For the dominant diagram
obey equation of motion with 2-parton DAs
![Page 24: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/24.jpg)
Other diagrams
![Page 25: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/25.jpg)
More diagrams
![Page 26: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/26.jpg)
Numerical results
![Page 27: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/27.jpg)
B -> pi form factors
![Page 28: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/28.jpg)
Gauge dependence from 2 partons• LO diagrams for B->pi form factor
• kinematics
![Page 29: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/29.jpg)
Amplitude from Fig.1(a)• Spin projectors for initial and final states
• Gauge dependence
• Extract term proportional to k2
![Page 30: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/30.jpg)
Amplitude from Fig.1(b)• Gauge dependent piece
• Extract term proportional to k2
• Gauge dependence from Figs.1(a) and 1(b) cancel
![Page 31: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/31.jpg)
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
![Page 32: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/32.jpg)
Amplitudes from all attachments• Other attachments vanish
• They cancel each other. No need of equation of motion
![Page 33: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/33.jpg)
2-to-3 contribution• B -> pi form factors
• Hard kernels proportional to mB
![Page 34: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/34.jpg)
3-parton B wave function• 3-parton matrix elements
• Sum rules by Grozin, Neubert
• Nishikawa, Tanaka
![Page 35: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/35.jpg)
3-to-2 contribution• Adopt 3-parton B meson wave function
• 3-to-2 hard kernel, also proportional to mb
![Page 36: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/36.jpg)
Wave functions
![Page 37: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/37.jpg)
Numerical results• Cancellation between 2-to-3 and 3-to-2
contributions same order of magnitude as fromGegenbauer terms in 2-parton pion DAs
![Page 38: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/38.jpg)
Figures
• Contributions from GN parameters larger than NT parameters
LO
![Page 39: Three partons in kT factorization](https://reader035.fdocuments.us/reader035/viewer/2022062411/568166e5550346895ddb1f72/html5/thumbnails/39.jpg)
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