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

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

Forward double virtual Compton scattering (VVCS)

optical theorem

nucleon parton distribution

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

“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

Spin dependent forward double VCS

link between S1, S2 & gTT, gLT

optical theorem

UNsubtracted dispersion relation

Dispersion relations for gDispersion relations for gTTTT and and SS11

low energy expansion for inelastic part

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)

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

resonance estimate (W < 2 GeV) : (MAID)

Forward spin polarizability Forward spin polarizability ofofprotonproton

JLab/CLAS data : Prok et al. (2008)

UNsubtracted dispersion relation

Dispersion relation for gDispersion relation for gLTLT

low energy expansion for inelastic part

Longitudinal – Transverse spin polarizability

first moment of g2 :

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

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

Burkhardt-Cottingham sum rule for Burkhardt-Cottingham sum rule for neutronneutron

inelastic part

elastic part

Burhardt-Cottingham

sum rule is satisfied

for the neutron

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

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)

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

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

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

N N andand ΔΔ masses masses

Chiral loops :

depend on 2 light scales

LEC (fit parameters)

N N andand ΔΔ masses masses : covariant chiral : covariant chiral loopsloops

renormalizes MB

(0)

c1B

2 light scales : μ and δ

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

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 :

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)

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)

Pion-nucleon scattering Pion-nucleon scattering in thein the ΔΔ regionregion

RenormalizedNLO propagator

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)

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)

N -> N -> ππ N N inin ΔΔ(1232)(1232) region : region : observablesobservables

NLO χEFT

DMT01

Sato-Lee

DUO (Utrecht-Ohio)

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

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

Altarelli, Cabibbo, Maiani (1972)

GDH sum rule in GDH sum rule in QEDQED : : O(eO(e44))

+

2

at O(e4) O(e4) ::

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

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

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…

agreeswith direct calculation!

Nucleon anomalous magnetic moments

equivalent to a sideways dispersion relation

Chiral behavior of nucleon anomalous magnetic moments

HB LO Rel. corr.

goes as 1/mq

Exactly as expected from a naïve quark model. Coincidence?

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)

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

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)]

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)

The End…