Transverse spin physics

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Transverse spin physics
Transverse spin physics RIKEN Spinfest, June 28
29, 2007

• Piet Mulders

mulders_at_few.vu.nl
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Abstract
QCD is the theory underlying the strong
interactions and the structure of hadrons. The
properties of hadrons and their response in
scattering processes provide in principle a large
number of observables. For comparison with theory
(lattice calculations or models), it is
convenient if these observables can be identified
with well-defined correlators, hadronic matrix
elements that involve only one hadron and known
local or nonlocal combinations of quark and gluon
operators. Well-known examples are static
properties, such as mass or charge, form factors
and parton distribution and fragmentation
functions. For the partonic structure, accessible
in high-energy (hard) scattering processes, a lot
of information can be obtained, in particular if
one finds ways to probe the transverse
structure (momenta and spins) of partons.
Relevant scattering experiments to extract such
correlations usually require polarized beams and
targets and measurements of azimuthal
asymmetries. Among these, single spin asymmetries
are special because of their particular
time-reversal behavior. The strength of single
spin asymmetries depends on the flow of color in
the hard scattering process, which affects the
nonlocal structure of quark and gluon field
operators in the correlators.
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Content

• Lecture 1
• Partonic structure of hadrons
• correlators distribution/fragmentation
• Lecture 2
• Correlators parameterization, interpretation,
  sum rules
• Orbital angular momentum?
• Lecture 3
• Including transverse momentum dependence
• Single spin asymmetries
• Lecture 4
• Hadronic scattering processes
• Theoretical issues on universality and
  factorization

4
Valence structure of hadrons global properties
of nucleons

• mass
• charge
• spin
• magnetic moment
• isospin, strangeness
• baryon number

• Mp ? Mn ? 940 MeV
• Qp 1, Qn 0
• s ½
• gp ? 5.59, gn ? -3.83
• I ½ (p,n) S 0
• B 1

Quarks as constituents
5
A real look at the proton

• g N ? .

Nucleon excitation spectrum E 1/R 200 MeV R
1 fm
6
A (weak) look at the nucleon
n ? p e- n

• 900 s
• ? Axial charge
• GA(0) 1.26

7
A virtual look at the proton
_

• g ? N N

g N ? N
8
Local forward and off-forward m.e.
Local operators (coordinate space densities)
Form factors
Static properties
9
Nucleon densities from virtual look
neutron
proton

• charge density ? 0
• u more central than d?
• role of antiquarks?
• n n0 pp- ?

10
Quark and gluon operators
Given the QCD framework, the operators are known
quarkic or gluonic currents such as
probed by gravitons
11
Towards the quarks themselves

• The current provides the densities but only in
  specific combinations, e.g. quarks minus
  antiquarks and only flavor weighted
• No information about their correlations,
  (effectively) pions, or
• Can we go beyond these global observables (which
  correspond to local operators)?
• Yes, in high energy (semi-)inclusive measurements
  we will have access to non-local operators!
• LQCD (quarks, gluons) known!

12
Deep inelasticexperiments
fragmenting quark
proton remnants
xB
Results directly reflect quark, antiquark and
gluon distributions in the proton
scattered electron
13
QCD Standard Model

• QCD framework (including electroweak theory)
  provides the machinery to calculate cross
  sections, e.g. gq ? q, qq ? g, g ? qq, qq ?
  qq, qg ? qg, etc.
• E.g.
• qg ? qg
• Calculations work for plane waves

_
_
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Confinement in QCD

• Confinement limits us to hadrons as quark
  sources or targets
• These involve nucleon states
• At high energies interference terms between
  different hadrons disappear as 1/P1.P2
• Thus, the theoretical description/calculation
  involves for hard processes, a forward matrix
  element of the form

quark momentum
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Partonic structure of hadrons

• Hard (high energy) processes
• Inclusive leptoproduction
• 1-particle inclusive leptoproduction
• Drell-Yan
• 1-particle inclusive hadroproduction
• Elementary hard processes
• Universal (?) soft parts
• - distribution functions f
• - fragmentation functions D

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Partonic structure of hadrons
Need PH.Ph s (large) to get separation of soft
and hard parts Allows ? ds ? d(p.P)
hard process
p
k
H
Ph
h
PH
fragmentation correlator
distribution correlator
D(z, kT)
F(x, pT)
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Intrinsic transverse momenta

• Hard processes Sudakov decomposition for
  momenta
• p xPH pT s n
• zero pT.PH n2 pT.n
• large PH.n ?s
• hadronic pT2 PH2 MH2
• small s (p.PH,p2,MH2)/?s
• Parton virtuality enters in s and is integrated
  out ? FH?q(x,pT) describing quark distributions
• Integrating pT ? collinear FH?q(x)
• Lightlike vector n enters in F(x,pT), but is
  irrelevant in cross sections

• Similarly for
• quark fragmentation
• k z-1Kh kT s n
• correlator Dq?h(z,kT)

18
(calculation of) cross section in DIS
Full calculation




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Lightcone dominance in DIS
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Parametrization of lightcone correlator
Jaffe Ji NP B 375 (1992) 527 Jaffe Ji
PRL 71 (1993) 2547
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Basis of partons

• Good part of Dirac
• space is 2-dimensional
• Interpretation of DFs

unpolarized quark distribution
helicity or chirality distribution
transverse spin distr. or transversity
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Matrix representationfor M F(x)gT
Bacchetta, Boglione, Henneman Mulders PRL 85
(2000) 712
Quark production matrix, directly related to
the helicity formalism
Anselmino et al.

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

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Results for deep inelastic processes
24

• End lecture 1

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• Lecture 2

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Local forward and off-forward
Local operators (coordinate space densities)
Form factors
Static properties
Examples (axial) charge mass spin magnetic
moment angular momentum
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Nonlocal - forward
Nonlocal forward operators (correlators)
Specifically useful squares
Momentum space densities of f-ons
Sum rules ? form factors
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Quark number

• Quark distribution and quark number
• Sum rule
• Next higher moment gives momentum sum rule

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Quark axial charge/spin sum rule

• Quark chirality distribution and quark spin/axial
  charge
• Sum rule
• This is one part of the spin sum rule

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Full spin sum rule

• The angular momentum operators in this spin sum
  rule
• The off-forward matrix elements of the
  (symmetric) energy momentum tensor give access to
  JQ and JG

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Local forward and off-forward
Local operators (coordinate space densities)
Form factors
Static properties
reminder
32
Nonlocal - forward
Nonlocal forward operators (correlators)
Specifically useful squares
Momentum space densities of f-ons
reminder
Sum rules ? form factors
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Nonlocal off-forward
Nonlocal off-forward operators (correlators AND
densities)
Sum rules ? form factors
GPDs
b
Forward limit ? correlators
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Quark tensor charge

• Quark chirality distribution and quark spin/axial
  charge
• Sum rule
• Note that this is not a spin measure, even if
  h1(x) is the distribution of transversely
  polarized quarks in a transversely polarized
  nucleon!

Transverse spin
(no decent operator!)
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A transverse spin rule

• One can write down a transverse spin sum rule
• It was first discussed by Teryaev and Ratcliffe,
  but it involves the twist-3 funtion gTg1g2
• (Burkhardt-Cottingham sumrule)
• and a similar gluon sumrule
• It does not involve the transverse spin. This
  appears in the Bakker-Leader-Trueman sumrule
  (which involves the assumption of having free
  quarks).

(my version of Trieste meeting)
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• End lecture 2

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• Lecture 3

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Issues

• Knowledge of partonic structure can be extended
  by looking at the transverse structure
• Time reversal invariance provides a nice
  discriminator for special effects
• Example is the color flow in hard processes,
  which is reflected in the nonlocal structure of
  matrix elements and shows up in single spin
  asymmetries
• Single spin asymmetries are being measured
  (HERMES_at_DESY, JLAB, COMPASS_at_CERN, KEK,
  RHIC_at_Brookhaven)

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The partonic structure of hadrons

• The cross section can be expressed in hard
  squared QCD-amplitudes and distribution and
  fragmentation functions entering in forward
  matrix elements of nonlocal combinations of quark
  and gluon field operators (f ? y or G)

lightfront x 0
TMD
lightcone
FF
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Partonic structure of hadrons
Need PH.Ph s (large) to get separation of soft
and hard parts Allows ? ds ? d(p.P)
hard process
reminder
p
k
H
Ph
h
PH
fragmentation correlator
distribution correlator
D(z, kT)
F(x, pT)
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(calculation of) cross section in SIDIS
Full calculation


LEADING (in 1/Q)


42
Lightfront dominance in SIDIS
Three external momenta P Ph q transverse
directions relevant qT q xB P Ph/zh or qT
-Ph/zh
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Gauge link in DIS

• In limit of large Q2 the result
• of handbag diagram survives
• contributions from A gluons
• ensuring color gauge invariance

44
Distribution

including the gauge link (in SIDIS)
A
One needs also AT Ga ? ATa ATa(x) ATa(8)
? dh Ga
Belitsky, Ji, Yuan, hep-ph/0208038 Boer, M,
Pijlman, hep-ph/0303034
From lty(0)AT(?)y(x)gt m.e.
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Parametrization of F(x,pT)

• Link dependence allows also T-odd distribution
  functions since T U0,? T U0,-?
• Functions h1 and f1T (Sivers) nonzero!
• Similar functions (of course) exist as
  fragmentation functions (no T-constraints) H1
  (Collins) and D1T

46
Interpretation
unpolarized quark distribution
need pT
T-odd
helicity or chirality distribution
need pT
T-odd
need pT
transverse spin distr. or transversity
need pT
need pT
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Matrix representationfor M F(x,pT)gT

• pT-dependent
• functions

T-odd g1T ? g1T i f1T and h1L ? h1L i h1
(imaginary parts)
Bacchetta, Boglione, Henneman Mulders PRL 85
(2000) 712
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T-odd ? single spin asymmetry
symmetry structure
hermiticity


parity
time reversal

• 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 measure T-odd quantities (such as T-odd
  distribution or fragmentation functions)

49
Lepto-production of pions
H1? is T-odd and chiral-odd
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• End lecture 3

51

• Lecture 4

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Quarks

• Integration over x- x.P allows twist
  expansion
• Gauge link essential for color gauge invariance
• Arises from all leading matrix elements
  containing y A...A y

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Sensitivity to intrinsic transverse momenta

• In a hard process one probes partons (quarks and
  gluons)
• Momenta fixed by kinematics (external momenta)
• DIS x xB Q2/2P.q
• SIDIS z zh P.Kh/P.q
• Also possible for transverse momenta
• SIDIS qT q xBP Kh/zh ? -Kh?/zh
• kT pT
• 2-particle inclusive hadron-hadron scattering
• qT K1/z1 K2/z2- x1P1- x2P2 ? K1?/z1
  K2?/z2
• p1T p2T k1T k2T
• Sensitivity for transverse momenta requires ? 3
  momenta
• SIDIS g H ? h X
• DY H1 H2 ? g X
• ee- g ? h1 h2 X
• hadronproduction H1 H2 ? h1 h2 X
• ? h X (?)

p ? x P pT k ? z-1 K kT
Knowledge of hard process(es)!
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Generic hard processes

• E.g. qq-scattering as hard subprocess
• Matrix elements involving parton 1 and additional
  gluon(s) A A.n appear at same (leading) order
  in twist expansion
• insertions of gluons collinear with parton 1 are
  possible at many places
• this leads for correlator involving parton 1 to
  gauge links to lightcone infinity

Link structure for fields in correlator 1
C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596
(2004) 277 hep-ph/0406099 EPJ C 47 (2006) 147
hep-ph/0601171
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SIDIS
SIDIS ? FU F
DY ? FU- F-
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A 2 ? 2 hard processes qq ? qq

• E.g. qq-scattering as hard subprocess
• The correlator F(x,pT) enters for each
  contributing term in squared amplitude with
  specific link

U? UU-
FTr(U?)U(x,pT)
FU?U(x,pT)
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Gluons

• Using 3x3 matrix representation for U, one finds
  in gluon correlator appearance of two links,
  possibly with different paths.
• Note that standard field displacement involves C
  C

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Integrating F(x,pT) ? F(x)
?
collinear correlator
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Integrating F(x,pT) ? F?a(x)
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Gluonic poles

• Thus FU(x) F(x)
• F?Ua(x) F?a(x) CGU pFGa(x,x)
• Universal gluonic pole m.e. (T-odd for
  distributions)
• pFG(x) contains the weighted T-odd functions
  h1?(1)(x) Boer-Mulders and (for transversely
  polarized hadrons) the function f1T?(1)(x)
  Sivers
• F?(x) contains the T-even functions h1L?(1)(x)
  and g1T?(1)(x)
• For SIDIS/DY links CGU 1
• In other hard processes one encounters different
  factors
• CGU? U 3, CGTr(U?)U Nc

Efremov and Teryaev 1982 Qiu and Sterman
1991 Boer, Mulders, Pijlman, NPB 667 (2003) 201
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A 2 ? 2 hard processes qq ? qq

• E.g. qq-scattering as hard subprocess
• The correlator F(x,pT) enters for each
  contributing term in squared amplitude with
  specific link

U? UU-
FTr(U?)U(x,pT)
FU?U(x,pT)
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examples qq?qq in pp
Bacchetta, Bomhof, Pijlman, Mulders, PRD 72
(2005) 034030 hep-ph/0505268
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examples qq?qq in pp
Bacchetta, Bomhof, DAlesio,Bomhof, Mulders,
Murgia, hep-ph/0703153
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Gluonic pole cross sections

• In order to absorb the factors CGU, one can
  define specific hard cross sections for gluonic
  poles (which will appear with the functions in
  transverse moments)
• for pp
• etc.
• for SIDIS
• for DY

Bomhof, Mulders, JHEP 0702 (2007) 029
hep-ph/0609206
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examples qg?qg in pp
collinear
Transverse momentum dependent
D1
Only one factor, but more DY-like than SIDIS
D2
D3
D4
Note also etc.
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examples qg?qg
e.g. relevant in Bomhof, Mulders, Vogelsang,
Yuan, PRD 75 (2007) 074019
collinear
Transverse momentum dependent
D1
D2
D3
D4
D5
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examples qg?qg
collinear
Transverse momentum dependent
68
examples qg?qg
collinear
Transverse momentum dependent
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examples qg?qg
collinear
Transverse momentum dependent
It is also possible to group the TMD functions in
a smart way into two! (nontrivial for nine
diagrams/four color-flow possibilities)
But still no factorization!
70
Residual TMDs

• We find that we can work with basic TMD functions
  F(x,pT) junk
• The junk constitutes process-dependent residual
  TMDs
• The residuals satisfies Fint ?(x) 0 and pFint
  G(x,x) 0, i.e. cancelling kT contributions
  moreover they most likely disappear for large kT

no definite T-behavior
definite T-behavior
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Conclusions

• Appearance of single spin asymmetries in hard
  processes is calculable
• For integrated and weighted functions
  factorization is possible
• For TMDs one cannot factorize cross sections,
  introducing besides the normal partonic cross
  sections some gluonic pole cross sections
• Opportunities the breaking of universality can
  be made explicit and be attributed to specific
  matrix elements

Related Qiu, Vogelsang, Yuan, hep-ph/0704.1153 Co
llins, Qiu, hep-ph/0705.2141 Qiu, Vogelsang, Yan,
hep-ph/0706.1196 Meissner, Metz, Goeke,
hep-ph/0703176