# Presentation: 449846

## Noncollinear parton structure in hard processes

- Piet Mulders

- ‘The partonic structure of hadrons’
- Trento, May 9-14, 2005

- mulders@few.vu.nl

## Content

- Intrinsic transverse momenta
- Partonic structure of hadrons:
- 	correlators (distribution/fragmentation)
- Color gauge invariance
- Gauge link structure
- Transverse moments
- gluonic poles

## Intrinsic transverse momenta

- Hard processes: Sudakov decomposition for momenta:
- 		p = xPH + pT + n
- zero: pT.PH = n2 = pT.n
- 	large: PH.n ~ s
- 	hadronic: pT2 ~ PH2 = MH2
- 	small:  ~ (p.PH,p2,MH2)/s
- Parton virtuality enters in  and is integrated out  Hq(x,pT)
- Lightlike vector n enters in the treatment of distributions (x,pT), but is irrelevant in cross sections

-     Similarly for
-     fragmentation:
-         k = zKh + kT + ’n’
-     correlator qh(z,kT)

## Intrinsic transverse momenta

- Aim in a hard process is to get acces to the partons (quarks and gluons).
- Determine momenta from observed momenta,
- 	e.g.	DIS	x = xB = Q2/2P.q
- 		SIDIS	z = zh = P.Kh/P.q
- Also possible for transverse momenta
- 	e.g. SIDIS	qT = kT – pT
- 			    = q + xBP – Ph/zh = - Kh/zh

## Partonic structure of hadrons

- PH

- H

- Ph

- p

- k

- h

- distribution
- correlator

- fragmentation
- correlator

- hard process

- PH

- Ph

- Ph

- PH

- Need dp... for ‘separation’

## (calculation of) cross section in DIS

- Full calculation

- + ...

- +

- +

- +

- LEADING (in 1/Q)

## (calculation of) cross section in SIDIS

- Full calculation

- +

- + ...

- +

- +

- LEADING (in 1/Q)

## Correlators for hadrons

- Describe the partonic structure of hadrons
- Hadrons enter as QM states: |PH>
- Partons enter as field operators: i,r() or Aa()
- Correlators are matrix elements of nonlocal combinations of quark and gluon operators:
- nonlocality  momentum dependence
- 	lightfront correlators (x,pT): nonlocality +.n = 0
- 	lightcone correlators (x):  nonlocality .n = T = 0
- The fact that T  0 leads to nontrivial gauge link structure
- Related to antiquark-hadron scattering amplitude after integration over p (T-ordering irrelevant)
- 	(Jaffe, Diehl, Gousset)

## Lightfront correlators

- Collins & Soper
- NP B 194 (1982) 445

- Jaffe & Ji,
- PRL 71 (1993) 2547;
- PRD 57 (1998) 3057

## Gauge link in DIS

- In limit of large Q2 the result
- 	of ‘handbag diagram’ survives
- ... + contributions from A+ gluons
- 	ensuring color gauge invariance

## Color gauge invariance

- Nonlocal combinations of colored fields must be joined by a gauge link:
- Gauge link structure is determined
- 	by interactions of gluons between
- 	soft and hard part
- Link structure for pT-dependent functions in SIDIS involves T

## Parametrization and interpretation

-  Parametrization of  (x,pT) and (x) in terms of
- 	 distribution functions f..(x,pT2) and explicit linear pT
-  Parametrization of  (z,kT) and (z) in terms of
- 	 fragmentation functions D..(z,kT2) and explicit linear kT
-  Constraints come from P, C and sometimes T
-  Time reversal:
- Fields have a particular behavior: T-even or T-odd
- Behavior of states: T|P0,**P**,...>in = |P0,**P**,...>out
- For plane waves: T|P0,**P**> = |P0,**P>   ** (in = out)
- Behavior of gauge link: T U[0,] T† = U[0,-]
-  (x): no T-odd functions
- 	 (z): T-odd functions (but only subleading: H, DT and EL)
- 	 (x,pT): T-odd functions f1T and h1
- 	 (z,kT): T-odd functions D1T and H1
-  True for any link, but links may lead to certain functions

## Parametrization of lightcone correlator

- Jaffe & Ji    NP B 375 (1992) 527
- Jaffe & Ji    PRL 71 (1993) 2547

## Basis of partons

-  ‘Good part’ of Dirac
-    space is 2-dimensional
-  Interpretation of DF’s

- unpolarized quark
- distribution

- helicity or chirality
- distribution

- transverse spin distr.
- or transversity

## Matrix representationfor M = [(x)+]T

-  Off-diagonal elements (RL or LR) are chiral-odd functions
-  Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY

- Matrix representationfor M = [(x)+]T

- Quark production matrix, directly related to the
- helicity formalism

- Anselmino et al.

- Bacchetta, Boglione, Henneman & Mulders
- PRL 85 (2000) 712

## Parametrization of (x,pT) 

- Link dependence allows also T-odd distribution functions since T U[0,] T† = U[0,-]
- Functions h1  and f1T (Sivers) nonzero!
- Similar functions (of course) exist as fragmentation functions (no T-constraints) H1 (Collins) and D1T

## Interpretation

- unpolarized quark
- distribution

- helicity or chirality
- distribution

- transverse spin distr.
- or transversity

- need pT

- need pT

- need pT

- need pT

- need pT

- T-odd

- T-odd

## Matrix representationfor M = [±](x,pT)+]T

-  pT-dependent
-    functions

- T-odd: g1T  g1T – i f1T and h1L  h1L + i h1   (imaginary parts)

- Matrix representationfor M = [±](x,pT)+]T

- Bacchetta, Boglione, Henneman & Mulders
- PRL 85 (2000) 712

## T-odd  single spin asymmetry

- with time reversal constraint only even-spin asymmetries
- the time reversal constraint cannot be applied in DY or in  1-particle inclusive DIS or ee
- In those cases single spin asymmetries can be used to measure T-odd quantities (such as T-odd distribution or fragmentation functions)

- *

- *

- time
- reversal

- symmetry
- structure

- parity

- hermiticity

- *

- *

## Lepto-production of pions

- H1 is T-odd and chiral-odd

## Color gauge invariance

- Nonlocal combinations of colored fields must be joined by a gauge link:
- Gauge link structure is determined by interactions of gluons between soft and hard part
- Link structure for pT dependent functions depends on the hard process!

## Other hard processes

- e.g.

- C. Bomhof, P.J. Mulders and F. Pijlman
- PLB 596 (2004) 277

- qq-scattering as hard subprocess
- insertions of gluons collinear with parton 1 are possible at many places
- this leads for ‘external’ parton fields to a gauge link to lightcone infinity

## Other hard processes

- qq-scattering as hard subprocess
- insertions of gluons collinear with parton 1 are possible at many places
- this leads for ‘external’ parton fields to a gauge link to lightcone infinity
- The correlator (x,pT) enters for each contributing term in squared amplitude with specific link

******[U****+****Tr(U****□****)]**(x,pT)

**U****□**** = U****+****U********†**

******[U****+****U****□****]**(x,pT)

## Transverse moments

- The correlators [U](x,pT) and [U](z,kT) depend on gauge link, which is process dependent.
- This will very likely lead to severe problems for factorization including pT,
- 		 = f(x,pT2)  ...   D(z,kT2) ...
- Look at transverse moments obtained by integration over transverse momentum after weighing
- 		[U](x) =  d2pT pT [U](x,pT)
- Beware of QCD corrections. Things only works if
- 		pT [U](x,pT)   s(pT2)/pT2

****

- pT2 

## Difference between [+] and  [-] upon integration 

****

- integrated quark
- distributions

- transverse
- moments

- measured in azimuthal asymmetries

- ±

- Back to the lightcone (theoretically clean)

- twist 2

- twist 2 & 3

## Difference between [+] and  [-] upon integration

- gluonic pole m.e. (T-odd)

- In momentum space:

- Note: T-odd parts are gluon-driven (QCD interactions)

## Time reversal constraints for distribution functions

- Time reversal(x,pT)  (x,pT)

- G

- 

- 

- 

- T-even
- (real)

- T-odd
- (imaginary)

- Conclusion:
- T-odd effects in SIDIS and DY have opposite signs

## Time reversal constraints for fragmentation functions

- Time reversalout(z,pT)  in(z,pT)

- G

- 

- 

- 

- T-even
- (real)

- T-odd
- (imaginary)

## Time reversal constraints for fragmentation functions

- G out

- out

- out

- out

- T-even
- (real)

- T-odd
- (imaginary)

- Time reversalout(z,pT)  in(z,pT)

- Conclusion:
- T-odd effects in SIDIS and ee are not related
- (except when G= 0)

## Gluonic poles

- In general one has
- [U](x) = (x) + CG[U] G(x,x)
- Examples:	CG[U±] = ±1
- 			CG[U+U□] = 3
- But uniquely defined functions in gluonic pole m.e. (T-odd for distributions)
- 	E.g. there is only one Sivers function f1T(1)(x) contained in G
- One needs to consider specific hard cross sections for gluonic poles (in order to absorb the factors CG[U])
- 	l_qg_lq = +lqlq  for SIDIS
- 	_qg_q*ll* = qq*ll*  for DY
- 	_qg_qqq =  CG[D] qqqq  for pp

- [D]

- [D]

- In preparation: Bacchetta, Bomhof, Pijlman, Mulders

## Conclusions

- Intrinsic transverse momentum dependence is of interest for the partonic structure of hadrons
- Intrinsic transverse momentum accessible in experiment via azimuthal asymmetries, often in combination with (transverse) polarization (among others the simplest access to transverse quark polarization)
- Going beyond collinearity gives access to gluon dynamics in hadrons, which can be done in a controlled way via weighted asymmetries (twist limited, t  3), use of chirality, and the specific time-reversal behavior of gluonic pole matrix element
- Identification of gluonic pole matrix elements may be possible from more processes than only SIDIS and DY (sign flip) and can be made transparant by calculating the appropriate ‘gluonic pole hard cross sections’.