remarks on angular momentum
DESCRIPTION
Trieste, November 2006. Remarks on angular momentum. Piet Mulders. [email protected]. Comments. Parton model is not frame dependent (IMF)! Angular momentum is space integral (but space ambiguous!) In QM wave packets are allowed (Gallilean invariance, c infinity). - PowerPoint PPT PresentationTRANSCRIPT
Comments
• Parton model is not frame dependent (IMF)!• Angular momentum is space integral (but space ambiguous!)• In QM wave packets are allowed (Gallilean invariance, c infinity).• In relativistic QM (Lorentz/Poincare invariance) there is a problem.
• Can one not avoid problem with spin vector (parameterisation of density matrix) by using explicit spin basis, e.g. helicity states?
• These are projections of the fermion fields. Make sure you use a ‘good basis’.
• Expansion of nucleon state in terms of partons ‘dangerous’. Do it in front form Lightcone wave functions, etc.
• Transverse spin sumrule can be written down, but use ‘operator expressions’.
| , ( ) | ,p s dq q q s
(Angular) momentum operators in QCD
Q GT T T
M x T x T
M x x
Q G
Kinematic operators
Front form quantization Instant form quantization
Local – forward and off-forward
.1 2' | ( ) | ( ) ( )i xP O x P e G t i G t
2t
1(0) | ( ) |G P O x P
2 (0) | ( ) |G P x O x P
Local operators (coordinate space densities):
P P’
Static properties:
Examples:(axial) chargemassspinmagnetic momentangular momentum
Form factors
Nonlocal - forward
†0, 0 ...O x x
| , | | 0, |P O y y x P P O x P
2. †| 0 | | 0 | ( )ip xdy e P x P P p P f p
Nonlocal forward operators (correlators):
Specifically useful: ‘squares’
Momentum space densities of -ons:
1( ) (0)dp f p GSum rules form factors
Selectivity at high energies: q = p
Nonlocal – off-forward
. † .1 2' | | ( , ) ( , )ip x i ydx e P y y x P e f t p i f t p
. †1| | (0, )ip xdx e P y y x P f p
. †2| | (0, )ip xdx e P y y y x P f p
Nonlocal off-forward operators (correlators AND densities):
1 1( , ) ( )dp f t p G t2 2( , ) ( )dp f t p G t
Sum rules form factors
Forward limit correlators
GPD’sb
2t
Selectivity q = p
Caveat
• 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
Aspects of high energy processes
• Ability to access matrix elements of specific operators (‘incoherence’) in inclusive processes and
• 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
Densities and (spacelike) formfactors
20 01.' | ( ) | ( , ,...) ( , ,...)i x GP GO x P t n i t ne
1 23 3 ' | (0) |
' | (0) | (2 ) ( ' )P O P
i P OG P PG PV
3 . ' | ( ) |i x P P
Vd x
O xe
( )O x
'P P
10 2 ( ' )n P P
10 2P En
10 2'P En
2 1
4E M t 2 2t
Forward limits of (spacelike) form factors
13
2. ' | ( ) |i x P O x
dP
x G iV
Ge
1 22 2 2
Gi
t t
GG
23
1. ' | ( ) |i xd x i i
P xO x P
VG Ge
' 1' | ( ) |i ( )m 0l P P GP O x P
' 2 (0' | )m |l ( )i P P P x O x GP
Caveat
• 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
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