Theoretical developments

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Theoretical developments piet mulders Transversity worksho Trento, 17/06/2004 [email protected]

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Transversity workshop Trento, 17/06/2004. Theoretical developments. piet mulders. [email protected]. Content. Spin structure & transversity Transverse momenta & azimuthal asymmetries T-odd phenomena & single spin asymmetries. Collinear hard processes, e.g. DIS. - PowerPoint PPT Presentation

Transcript of Theoretical developments

Page 1: Theoretical developments

Theoretical developments

piet mulders

Transversity workshopTrento, 17/06/2004

[email protected]

Page 2: Theoretical developments

Content

• Spin structure & transversity

• Transverse momenta & azimuthal asymmetries

• T-odd phenomena & single spin asymmetries

Page 3: Theoretical developments

Collinear hard processes, e.g. DIS

• Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone)

• Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv.

• There are three ‘leading’ DF’s f1

q(x) = q(x), g1q(x) = q(x), h1

q(x) = q(x) • Transverse gluons appear at 1/Q and are

contained in (higher twist) qqG-correlators• DF’s are quark densities that are directly

linked to lightcone wave functions squared• Perturbative QCD evolution (reflecting

the large pT-behavior of the correlator proportional to s/pT

2 in the pT-integration)

Page 4: Theoretical developments

Leading order DIS

• In limit of large Q2 the resultof ‘handbag diagram’ survives

• … + contributions from A+ gluonsensuring color gauge invariance

A+ gluons gauge link

Ellis, Furmanski, PetronzioEfremov, Radyushkin

A+

Page 5: Theoretical developments

Collinear hard processes, e.g. DIS

• Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone)

• Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv.

• There are three ‘leading’ DF’s f1

q(x) = q(x), g1q(x) = q(x), h1

q(x) = q(x) • Transverse gluons appear at 1/Q and are

contained in (higher twist) qqG-correlators• DF’s are quark densities that are directly

linked to lightcone wave functions squared• Perturbative QCD evolution (reflecting

the large pT-behavior of the correlator proportional to s/pT

2 in the pT-integration)

Page 6: Theoretical developments

Parametrization of lightcone correlator

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

leading part

• M/P+ parts appear as M/Q terms in • T-odd part vanishes for distributions but is important for fragmentation

Page 7: Theoretical developments

Collinear hard processes, e.g. DIS

• Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone)

• Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv.

• There are three ‘leading’ DF’s f1

q(x) = q(x), g1q(x) = q(x), h1

q(x) = q(x) • Transverse gluons appear at 1/Q and are

contained in (higher twist) qqG-correlators• DF’s are quark densities that are directly

linked to lightcone wave functions squared• Perturbative QCD evolution (reflecting

the large pT-behavior of the correlator proportional to s/pT

2 in the pT-integration)

Page 8: Theoretical developments

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

Related to thehelicity formalism

Anselmino et al.

Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712

Page 9: Theoretical developments

Collinear hard processes, e.g. DIS

• Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone)

• Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv.

• There are three ‘leading’ DF’s f1

q(x) = q(x), g1q(x) = q(x), h1

q(x) = q(x) • Transverse gluons appear at 1/Q and are

contained in (higher twist) qqG-correlators• DF’s are quark densities that are directly

linked to lightcone wave functions squared• Perturbative QCD evolution (reflecting

the large pT-behavior of the correlator proportional to s/pT

2 in the pT-integration)

Page 10: Theoretical developments

Non-collinear processes, e.g. SIDIS

• Relevant in electroweak processes with two hadrons (SIDIS, DY)

• Beyond just extending DIS by tagging quarks …

• Transverse momenta of partons become relevant, appearing in azimuthal asymmetries

• DF’s and FF’s depend on two variables, (x,pT) and (z,kT)

• Gauge link structure is process dependent ( [])

• pT-dependent distribution functions and (in general) fragmentation functions are not constrained by time-reversal invariance

• This allows T-odd functions h1 and f1T

(H1 and D1T

) appearing in single spin asymmetries

Page 11: Theoretical developments

Leading order SIDIS

• In limit of large Q2 only resultof ‘handbag diagram’ survives

• Isolating parts encoding soft physics

? ?

Page 12: Theoretical developments

Lightfront correlators

no T-constraintT|Ph,X>out = |Ph,X>in

Collins & SoperNP B 194 (1982) 445

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

Page 13: Theoretical developments

Non-collinear processes, e.g. SIDIS

• Relevant in electroweak processes with two hadrons (SIDIS, DY)

• Beyond just extending DIS by tagging quarks …

• Transverse momenta of partons become relevant, appearing in azimuthal asymmetries

• DF’s and FF’s depend on two variables, [](x,pT) and [](z,kT)

• Gauge link structure is process dependent ( [])

• pT-dependent distribution functions and (in general) fragmentation functions are not constrained by time-reversal invariance

• This allows T-odd functions h1 and f1T

(H1 and D1T

) appearing in single spin asymmetries

Page 14: Theoretical developments

Distribution

From AT() m.e.

including the gauge link (in SIDIS)

A+

One needs also AT

G+ = +AT

AT()= AT

(∞) + d G+

Belitsky, Ji, Yuan, hep-ph/0208038Boer, M, Pijlman, hep-ph/0303034

Page 15: Theoretical developments

Distribution

A+

A+

including the gauge link (in SIDIS or DY)

SIDIS

SIDIS [-]DY DY [+]

Page 16: Theoretical developments

Non-collinear processes, e.g. SIDIS

• Relevant in electroweak processes with two hadrons (SIDIS, DY)

• Beyond just extending DIS by tagging quarks …

• Transverse momenta of partons become relevant, appearing in azimuthal asymmetries

• DF’s and FF’s depend on two variables, [](x,pT) and [](z,kT)

• Gauge link structure is process dependent ( [])

• pT-dependent distribution functions and (in general) fragmentation functions are not constrained by time-reversal invariance

• This allows T-odd functions h1 and f1T

(H1 and D1T

) appearing in single spin asymmetries

Page 17: Theoretical developments

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!

• These functions (of course) exist as fragmentation functions (no T-symmetry) H1

(Collins) and D1T

Page 18: Theoretical developments

Interpretation

unpolarized quarkdistribution

helicity or chiralitydistribution

transverse spin distr.or transversity

need pT

need pT

need pT

need pT

need pT

T-odd

T-odd

Page 19: Theoretical developments

pT-dependent functions

T-odd: g1T g1T – i f1T and h1L

h1L + i h1

Matrix representationfor M = [±](x,pT)+]T

Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712

Page 20: Theoretical developments

pT-dependent DF’stwist structure

• For integrated correlator (x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE

• For unintegrated correlators [](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist t

• Transverse moments (x,pT) d2pT pT

(x,pT) project out the parts in [](x,pT) proportional to pT

. They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon)

• Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum

• The difference between [](x) and

[](x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd

• The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link

• Factorization of explicit pT-dependent functions requires ‘soft factors’• Universality of subleading functions (appearing at M/P+ level) in

[](x) seems ok, but this seems not the case for subleading functions

in [](x) [so f1T

(1) ok but f(1) problematic: in cos h asymmetry pT f

at order 1/Q gets pQCD correction s f1/|pT|, which shows up as s f1 at order 1]

Page 21: Theoretical developments

Difference between [+] and [-] upon integration

integrated quarkdistributions

transverse moments

measured in azimuthal asymmetries

±

Back to the lightcone

Page 22: Theoretical developments

pT-dependent DF’stwist structure

• For integrated correlator (x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE

• For unintegrated correlators [](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist t

• Transverse moments (x,pT) d2pT pT

(x,pT) project out the parts in [](x,pT) proportional to pT

. They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon)

• Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum

• The difference between [](x) and

[](x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd

• The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link

• Factorization of explicit pT-dependent functions requires ‘soft factors’• Universality of subleading functions (appearing at M/P+ level) in

[](x) seems ok, but this seems not the case for subleading functions

in [](x) [so f1T

(1) ok but f(1) problematic: in cos h asymmetry pT f

at order 1/Q gets pQCD correction s f1/|pT|, which shows up as s f1 at order 1]

Page 23: Theoretical developments

Difference between [+] and [-] upon integration

gluonic pole m.e.(T-odd)

In momentum space:

Page 24: Theoretical developments

pT-dependent DF’stwist structure

• For integrated correlator (x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE

• For unintegrated correlators [](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist t

• Transverse moments (x,pT) d2pT pT

(x,pT) project out the parts in [](x,pT) proportional to pT

. They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon)

• Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum

• The difference between [](x) and

[](x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd

• The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link

• Factorization of explicit pT-dependent functions requires ‘soft factors’• Universality of subleading functions (appearing at M/P+ level) in

[](x) seems ok, but this seems not the case for subleading functions

in [](x) [so f1T

(1) ok but f(1) problematic: in cos h asymmetry pT f

at order 1/Q gets pQCD correction s f1/|pT|, which shows up as s f1 at order 1]

Page 25: Theoretical developments

T-odd phenomena

• T-odd phenomena appear in single spin asymmetries• T-odd parts for distribution functions are in the gluonic pole part,

hence in [](x) and

[](x) they have opposite signs

• T-odd parts for fragmentation functions in [](x) and

[](x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links

• Contributions in other hard processes, such as pp X involving three hadrons require a careful analysis

Page 26: Theoretical developments

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 ee

• In those cases single spin asymmetries can be used to select T-odd quantities

*

*

W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)

W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)

W(q;P,S;Ph,Sh) = W(q;P, S;Ph, Sh)

W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)

_

___

_ ____

__ _time

reversal

symmetrystructure

parity

hermiticity

Page 27: Theoretical developments

T-odd phenomena

• T-odd phenomena appear in single spin asymmetries• T-odd parts for distribution functions are in the gluonic pole part,

hence in [](x) and

[](x) they have opposite signs

• T-odd parts for fragmentation functions in [](x) and

[](x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links

• Contributions in other hard processes, such as pp X involving three hadrons require a careful analysis

Page 28: Theoretical developments

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

Page 29: Theoretical developments

T-odd phenomena

• T-odd phenomena appear in single spin asymmetries• T-odd parts for distribution functions are in the gluonic pole part,

hence in [](x) and

[](x) they have opposite signs

• T-odd parts for fragmentation functions in [](x) and

[](x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links

• Contributions in other hard processes, such as pp X involving three hadrons require a careful analysis

Page 30: Theoretical developments

Time reversal constraints for fragmentation functions

Time reversalout(z,pT)

in(z,pT)

G

T-even(real)

T-odd(imaginary)

Page 31: Theoretical developments

Time reversal constraints for fragmentation functions

G out

out

out

out

T-even(real)

T-odd(imaginary)

Time reversalout(z,pT)

in(z,pT)

Conclusion:T-odd effects in SIDIS and ee are not related

Page 32: Theoretical developments

T-odd phenomena

• T-odd phenomena appear in single spin asymmetries• T-odd parts for distribution functions are in the gluonic pole part,

hence in [](x) and

[](x) they have opposite signs

• T-odd parts for fragmentation functions in [](x) and

[](x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links

• Contributions in other hard processes, such as pp X involving three hadrons require a careful analysis

Page 33: Theoretical developments

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 gauge link to lightcone infinity

Page 34: Theoretical developments

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 gauge link to lightcone infinity

• The correlator (x,pT) enters for each contributing term in squared amplitude with specific link

• The link may enhance the effect of the gluonic pole contribution involving also specific color factors