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Single spin asymmetries in pp scattering

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Single Spin Asymmetries (SSA) in pp scattering

- Introduction: what are we after?
- SSA and time reversal invariance
- Transverse momentum dependence (TMD)
Through TMD distribution and fragmentation functions totransverse momentsandgluonic poles

- Electroweak processes (SIDIS, Drell-Yan and annihilation)
- Hadron-hadron scattering processes
- Gluonic pole cross sections
- What can pp add?
- Conclusions

_

_

For (semi-)inclusive measurements, cross sections in hard scattering processes factorize into a hard squared amplitude and distribution and fragmentation functions entering in forward matrix elements of nonlocal combinations of quark and gluon field operators (f y or G)

lightcone

TMD

lightfront

FF

pictures?

appendix

- Quark distribution functions (DF) and fragmentation functions (FF)
- unpolarized
q(x) = f1q(x) and D(z) = D1(z)

- Polarization/polarimetry
Dq(x) = g1q(x) and dq(x) = h1q(x)

- Azimuthal asymmetries
g1T(x,pT) and h1L(x,pT)

- Single spin asymmetries
h1(x,pT) and f1T(x,pT); H1(z,kT) and D1T(z,kT)

- unpolarized
- Form factors
- Generalized parton distributions

FORWARD

matrix elements

x section

one hadron in inclusive or semi-inclusive scattering

NONLOCAL

lightcone

NONLOCAL

lightfront

OFF-FORWARD

Amplitude

Exclusive

LOCAL

NONLOCAL

lightcone

- QCD is invariant under time reversal (T)
- Single spin asymmetries (SSA) are T-odd observables, but they are not forbidden!
- For distribution functions a simple distinction between T-even and T-odd DF’s can be made
- Plane wave states (DF) are T-invariant
- Operator combinations can be classified according to their T-behavior (T-even or T-odd)

- Single spin asymmetries involve an odd number (i.e. at least one) of T-odd function(s)
- The hard process at tree-level is T-even; higher order as is required to get T-odd contributions
- Leading T-odd distribution functions are TMD functions

f2 - f1

K1

df

K2

pp-scattering

- In a hard process one probes partons (quarks and gluons)
- Momenta fixed by kinematics (external momenta)
DISx = xB = Q2/2P.q

SIDIS z = zh = P.Kh/P.q

- Also possible for transverse momenta
SIDIS qT = kT – pT

= q + xBP – Kh/zh-Kh/zh

2-particle inclusive hadron-hadron scattering

qT = p1T + p2T – k1T – k2T

= K1/z1+ K2/z2- x1P1- x2P2 K1/z1+ K2/z2

- Sensitivity for transverse momenta requires 3 momenta
SIDIS: g* + H h + X

DY: H1 + H2 g* + X

e+e-: g* h1 + h2 + X

hadronproduction: H1 + H2 h + X

h1 + h2 + X

p x P + pT

k z-1 K + kT

In collinear cross section

In azimuthal asymmetries

Transverse moment

pictures?

quark correlator

F(x, pT)

- T-odd
- Transversely
- polarized quarks

- Nonlocal combinations of colored fields must be joined by a gauge link:
- Gauge link structure is calculated from collinear A.n gluons exchanged between soft and hard part
- Link structure for TMD functions
depends on the hard process!

DIS F[U]

SIDIS F[U+] =F[+]

DY F[U-] = F[-]

collinear correlator

transverse moment

FG(p,p-p1)

T-even

T-odd

_

What about other hard processes (e.g. pp and pp scattering)?

- Thus
F[±]a(x) = Fa(x) + CG[±]pFGa(x,x)

- CG[±] = ±1
- with universal functions in gluonic pole m.e. (T-odd for distributions)
- There is only one function h1(1)(x) [Boer-Mulders] and (for transversely polarized hadrons) only onefunction f1T(1)(x) [Sivers] contained in pFG
- These functions appear with a process-dependent sign
- Situation for FF is (maybe) more complicated because there are no T-constraints

Efremov and Teryaev 1982; Qiu and Sterman 1991

Boer, Mulders, Pijlman, NPB 667 (2003) 201

Metz and Collins 2005

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

Link structure for fields in correlator 1

- 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

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

U□ = U+U-†

F[Tr(U□)U+](x,pT)

F[U□U+](x,pT)

- Thus
F[U]a(x) = Fa(x) + CG[U]pFGa(x,x)

- CG[U±] = ±1
CG[U□U+] = 3, CG[Tr(U□)U+] = Nc

- with the same uniquely defined functions in gluonic pole matrix elements (T-odd for distributions)

Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268

D1

CG [D1]

= CG [D2]

D2

D3

CG [D3]

= CG [D4]

D4

(gluonic pole cross section)

y

- In order to absorb the factors CG[U], 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:

- Similarly for gluon processes

Bomhof, Mulders, Pijlman, EPJ; hep-ph/0601171

D1

For Nc:

CG [D1] -1

(color flow as DY)

end

- Single spin asymmetries in hard processes can exist
- They are T-odd observables, which can be described in terms of T-odd distribution and fragmentation functions
- For distribution functions the T-odd functions appear in gluonic pole matrix elements
- Gluonic pole matrix elements are part of the transverse moments appearing in azimuthal asymmetries
- Their strength is related to path of color gauge link in TMD DFs which may differ per term contributing to the hard process
- The gluonic pole contributions can be written as a folding of universal (soft) DF/FF and gluonic pole cross sections

Belitsky, Ji, Yuan, NPB 656 (2003) 165

Boer, Mulders, Pijlman, NPB 667 (2003) 201

Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030

Bomhof, Mulders, Pijlman, EPJ; hep-ph/0601171

Eguchi, Koike, Tanaka, hep-ph/0604003

Ji, Qiu, Vogelsang, Yuan, hep-ph/0604023

D

P

P’

Local operators (coordinate space densities):

Form factors

Static properties:

Examples:

(axial) charge

mass

spin

magnetic moment

angular momentum

Nonlocal forward operators (correlators):

Specifically useful: ‘squares’

Selectivity at high energies: q = p

Momentum space densities of f-ons:

Sum rules form factors

Nonlocal off-forward operators (correlators AND densities):

Selectivity q = p

Sum rules form factors

GPD’s

b

Forward limit correlators

back

- We study forward matrix elements, including transverse momentum dependence (TMD), i.e. f(p||,pT) with enhanced nonlocal sensitivity!
- This is not a measurement of orbital angular momentum (OAM). Direct measurement of OAM requires off-forward matrix elements, i.e. GPD’s.
- One may at best make statements like:
linear pT dependence nonzero OAM

no linear pT dependence no OAM

unpolarized

hadrons

back

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