Spin structure calculations from ChPT
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Transcript of Spin structure calculations from ChPT
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Spin Spin structure structure
calculations calculations from ChPTfrom ChPT
Marc VanderhaeghenJohannes Gutenberg Universität, Mainz
Workshop on “Spin Structure at Long Distance”
JLab, March 12-13, 2009
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Forward double virtual Compton scattering : brief review
Inclusion of in chiral EFT framework :
masses, NΔ transition form factors chiral EFT used in dual role : as a framework to predict e p -> e p π observables (Q2 dependence of structure functions) and to perform chiral extrapolation of lattice data
Sum rule / analyticity constraints on EFT calculations
evaluation of sum rules in perturbation theory
OutlineOutline
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Forward double virtual Compton scattering (VVCS)
optical theorem
nucleon parton distribution
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Dispersion relation for fT
subtraction
elastic contribution singularities at : = § B ( $ xB=1)
with B = Q2/2MN
low energy expansion
for the non-pole part
0: threshold
subtracted dispersion relation
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“DIS”
(W < 2 GeV)
(F2)
resonance estimate (W < 2 GeV, MAID + + ) Drechsel,
Pasquini, Vdh (2003)
CLAS data: Osipenko et al. (2003)
DIS (MRST 01)
+ +
+ +
Q2 >>
Generalized Baldin sum ruleGeneralized Baldin sum rule
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Spin dependent forward double VCS
link between S1, S2 & gTT, gLT
optical theorem
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UNsubtracted dispersion relation
Dispersion relations for gDispersion relations for gTTTT and and SS11
low energy expansion for inelastic part
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Generalized GDH sum rule for Generalized GDH sum rule for protonproton
resonance estimate W < 2 GeV: + + Drechsel, Pasquini, Vdh (2003)
resonance estimate + DIS (VMD)Anselmino, Ioffe, Leader / Burkert, Ioffe
DIS: Bluemlein, Boettcher (2003)
“DIS”: W < 2 GeV
rel. BChPt O(p4) : Bernard et al. (2002)
HBChPt O(p4): Ji, Kao, Osborne (2000)
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HBChPT O(p4): Ji, Kao, Osborne (2000)
rel. BChPT O(p4): Bernard, Meissner, Hemmert (2002)
resonance estimate (W < 2 GeV): N (MAID)resonance + DIS estimate (VMD)
Drechsel, Pasquini, Vdh (2003)
DIS:
Bluemlein, Boettcher (2003)
“DIS”: W < 2 GeV
GDH sum rule for GDH sum rule for p - np - n
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resonance estimate (W < 2 GeV) : (MAID)
Forward spin polarizability Forward spin polarizability ofofprotonproton
JLab/CLAS data : Prok et al. (2008)
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UNsubtracted dispersion relation
Dispersion relation for gDispersion relation for gLTLT
low energy expansion for inelastic part
Longitudinal – Transverse spin polarizability
first moment of g2 :
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JLab/Hall A data : E94010 (2004)
HBChPT: Kao, Spitzenberg, Vdh (2003)
resonance estimate (MAID) : Drechsel, Pasquini, Vdh (2003)
0
HBChPT: p4
HBChPT: p3
MAID
E94010
LT
E94010
MAID
HBChPT: p3
HBChPT: p4
Forward spin polarizability of Forward spin polarizability of neutronneutron
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Burkhardt – Cottingham sum rule
resonance estimate (W < 2 GeV): (MAID)HBChPT O(p4) : Kao, Spitzenberg, Vdh (2002)
E155 :
0.02 · x · 0.8
Burkhardt-Cottingham sum rule for Burkhardt-Cottingham sum rule for protonproton
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Burkhardt-Cottingham sum rule for Burkhardt-Cottingham sum rule for neutronneutron
inelastic part
elastic part
Burhardt-Cottingham
sum rule is satisfied
for the neutron
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scale scale dependencedependencescaling limit, Q2 -> ∞ :
arbitrary Q2 :
enter in low energy expansion of VVCS
low Q2 : ChPT
HBChPT : p3 p4
Lz =1
L T0 ±1
Twist-3 : quark-gluon correlations
in nucleon
Twist-2 : Wandzura-Wilczek
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Scale dependence of Scale dependence of dd22
HBChPT O(p4)
HBChPT O(p3)
HBChPT O(p3) incl. Δ
ELASTIC
resonance estimate : π (MAID)
E155 (2002)
JLab/Hall C
RSS Coll. (2008)
Kao, Drechsel, Kamalov, Vdh
(2003)
JLab/Hall A (2003)
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N and Δ masses : chiral EFT calculation chiral extrapolation of lattice data NΔ transition form factors / e p -> e p π observables chiral EFT used in dual role : as a framework to predict e p -> e p π observables
(extraction of NΔ form factors) and to perform chiral extrapolation of lattice data
work done in coll. with V. Pascalutsa
-> N and Δ masses : PLB 636 (2006) 31
-> NΔ transition : PRL 95 (2005) 232001 and PRD 73 (2006) 034003
-> Δ MDM : PRL 94 (2005) 102003 and PRD 77 (2008) 014027
-> Pascalutsa, Vdh, Yang : Phys. Rept. 437 (2007) 125
(1232)-resonance(1232)-resonance in in chiral EFTchiral EFT
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N N and and ΔΔ chiral chiral LagrangiansLagrangians
ππNNΔΔ
Δ part is such that # spin d.o.f. is constrained to physical number
Couplings involving Δ are consistent : respect spin-3/2 gauge symmetry
ππΔΔΔΔ
ππNNNN
Pascalutsa (1998)
Pascalutsa, Timmermans (1999)
Chiral behavior of
masses
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Include the as an explicit d.o.f. , described by a spin-3/2 (Rarita-Schwinger) isospin-3/2 (isoquartet) fieldPower counting:Jenkins & Manohar (1991), Hemmert et al. (1998) … (2006) N and propagators:
Pascalutsa & Phillips (2003)
= + … = O(p3) = O(3 )
Chiral Lagrangians Chiral Lagrangians withwith andand power countingpower counting
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N N andand ΔΔ masses masses
Chiral loops :
depend on 2 light scales
LEC (fit parameters)
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N N andand ΔΔ masses masses : covariant chiral : covariant chiral loopsloops
renormalizes MB
(0)
c1B
2 light scales : μ and δ
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N N andand ΔΔ masses masses : covariant chiral : covariant chiral loops (contd.)loops (contd.)
in an expansion to third power in ( mπ , Δ = MΔ – MN )
mπ < Δ
mπ > Δ
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ππN and N and ππΔΔ chiral loops : chiral loops : leadingleading non-analytic non-analytic
behaviorbehavior
Banerjee, Milana (1995)
Young, Leinweber, Thomas, Wright (2002)
Bernard, Hemmert, Meissner (2003)
LNA terms agree with :
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N massN mass : : mmππ dependence dependence
c1N = - 1.26 (9) GeV-1
MN(0) = 0.883 (29) GeV
full QCD
lattice calculations :
ETMC
Alexandrou et al. (2008)
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ΔΔ mass mass : : mmππ dependence dependence
lattice : MILC (2001)
πΔ : mπ3 term
(HBChPT)
πN + πΔ : covariant, c2Δ = 0
MΔ(0) = 1.26 (54) GeV
c1Δ = - 1.16 (17) GeV-1
lattice : ETMC (2008)
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Pion-nucleon scattering Pion-nucleon scattering in thein the ΔΔ regionregion
RenormalizedNLO propagator
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chiralchiral EEffectiveffective FFieldield Theory Theory calculation ofcalculation of
e p -> e p e p -> e p ππ00 inin ΔΔ(1232)(1232) regionregion
calculation to NLO in
δ expansion for e p -> e p π0
Power counting scheme Power counting scheme ::
in threshold region : momentum p ~ mπ
in Δ region : p ~ MΔ - MN
LO
vertex corrections : unitarity & gauge invariance exactly preserved to NLO
Pascalutsa, Vdh (2005)
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magnetic (M1) magnetic (M1) && electric electric (E2) (E2)
N -> N -> ΔΔ transitiontransition2 free parameters !
G*M = 2.97 G*
E = 0.07 (E2/M1 = -2.3 %)
Δ pole
Δ pole + Born
Δ pole + Born
+ vertex corr.
MAID (2003)
SAID (2003)
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N -> N -> ππ N N inin ΔΔ(1232)(1232) region : region : observablesobservables
NLO χEFT
DMT01
Sato-Lee
DUO (Utrecht-Ohio)
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QQ22 dependence ofdependence of E2/M1 E2/M1 andand C2/M1 C2/M1 ratiosratios
EFT calculation predicts the Q2 dependence
data points :
MIT-Bates
(Sparveris et al., 2005)
MAMI :
Q2 = 0 (Beck et al., 2000)
Q2 = 0.06 (Stave et al., 2006)
Q2 = 0.2 (Elsner et al., 2005, Sparveris et al.,
2006)
no pion loopspion loops included
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Compute both sides of the sum rule in perturbation theory. Is the GDH sum rule verified?
GDH sum rule in GDH sum rule in QEDQED
electron anomalous magnetic moment (loop effect)
to 1-loop accuracy :
for Dirac particle :
experiment electron : g = 2.002319304374 ± 8 . 10-12 !
Schwinger
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Altarelli, Cabibbo, Maiani (1972)
GDH sum rule in GDH sum rule in QEDQED : : O(eO(e44))
+
2
at O(e4) O(e4) ::
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Dicus, Vega (2001)
+
GDH sum rule in GDH sum rule in QED QED :: O(e O(e66))
+ ...
+ ...
2
+
GDH sum rule is satisfied in QED to order O(e6)
tree level 1-loop diagrams
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verification of GDH sum rule in Chiral Effective Field Theory
(ChEFT)
O(g2):
to lowest order in g NN
0 =
2
=
κ = κ0 + δκ
loop contributi
on
trial value
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Verification in ChEFT (cont’d)
Yes!
However, only in the fully covariant calculation. Any “heavy-baryon” type of expansion does not do it.Reason: violation of analyticity…
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agreeswith direct calculation!
Nucleon anomalous magnetic moments
equivalent to a sideways dispersion relation
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Chiral behavior of nucleon anomalous magnetic moments
HB LO Rel. corr.
goes as 1/mq
Exactly as expected from a naïve quark model. Coincidence?
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analyticity constraints on magnetic moments
covariant chiral loops (SR) compared with heavy-baryon expansion (HB) or Infrared-Regularized ChPT (IR)
Red curve is the single-parameter fit to lattice data The parametrization is based on SR result
Pascalutsa, Holstein, Vdh (2004)
lattice : Zanotti et al. (2004)
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forward spin polarizability
Our sum rule (SR) evaluation, using the lowest order (Born) total cross-section, upon expansion in pion mass agrees with NLO heavy-baryon result of [Ji, Kao, Osborne (2000); Kumar, McGovern, Birse (2000)], not [Gellas, Hemmert, Meissner (2000)]The covariant ChPT calculation (using Infrared Regularization) [Bernard, Hemmert, Meissner, PRD (2003)] agrees with sum rule calculation up to terms analytic in quark mass :
IR destroys analyticity (IR-regulated integrals do not satisfy DRs)
BHM
SR
Chiral behavior of spin polarizabilities
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Covariant chiral loops (RLO) should describe chiral behavior better than the heavy-baryon expansion (HBLO). Green dashed : HBLO + 1st rel. corr.“Exp.” is a fit to lattice QCD calculations :
[Leinweber et al. (1999)]
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Conclusions & Outlook (1232)-resonance(1232)-resonance in in chiral EFT chiral EFT
quantitative framework for pion electroproduction
observables at low Q2 (moments of) structure functions
Sum rules of forward double VCS
improve/extend chiral expansion : calculate/resum higher order terms (guidance by analyticity)
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The End…