Effective Field Theory for N* AnalysisEffective Field Theory for N* Analysis
Vladimir Pascalutsa
Physics Department, The College of William & MaryTheory Center, Jefferson Laboratory
andECT*, Trento, Italy
Supported by
@ N* Analysis 06, @ N* Analysis 06, JLabJLab, Newport News, VA, Newport News, VA
Nov 5, 2006Nov 5, 2006
Low‐energy QCD… in the presence of resonances
[ Weinberg (1979), Gasser & Leutwyler (1984), …]Low‐energy QCD ~ ChPT
Lagrangian:
S‐matrix:
However, near a resonance (or a bound state):
Low‐energy QCD in the presence of Δ(1232)
Δ(1232) – first nucleon resonance,
For example, Compton scattering on the nucleon
Generic features:(i) below π production threshold (ω < mπ), the Δ is a high‐energy degree of freedom – can be integrated out (because ω/Δ << 1 ) – ChPT with no Δ’s(ii) above, rapid change with energy, at ω ∼ Δ ‐‐ PT break‐down How to obtain this behavior in ChEFT ?
Effective Lagrangians with Δ: higher‐spin aspects
Include the Δ as an explicit d.o.f. , [Jenkins & Manohar (1991), …]described by a spin‐3/2 (Rarita‐Schwinger) isospin‐3/2 (isoquartet) field
Details in:V.P., PRD (1998); PLB (2001)V.P. & Timmermans, PRC (1999)Deser, V.P. & Waldron PRD (2000)
The couplings are also required to be gauge symmetric, to ensure the decoupling of the spin‐1/2 components –a higher‐spin consistency condition.
Decoupling of lower spin…
Covariance, locality, correct spin DOF content
δ expansion
OΔR propagator
∼
p∼ mπ , 1/Δ = O(1/δ) [c.f., SN∼ 1/p = O(1/δ2)]
p∼ Δ , 1/(p-Δ-Σ )= O(1/δ3 )
Σ = + … = O(p3) = O(δ3 )
[ V.P. & Phillips, PRC (2003)]
Power counting:
Pion‐nucleon scattering in the resonance region
1.17 1.19 1.21 1.2321.25 1.27 1.29W�GeV�
40
60
80
100
120
Δ�degrees
�
P33
RenormalizedNLO propagator
non-zero values for REM and RSM : measure of non-spherical distribution of charges
J P=3/2+ (P33), MΔ ' 1232 MeV, ΓΔ ' 115 MeVN → Δ transition: π N → Δ (99%), γ N → Δ (<1%)
N ‐> Δ electromagnetic transition
γγ**
N N ∆∆
3 electromagnetic transitions :M1 -> GM
*
E2 -> GE*
C2 -> GC*
q : photon momentum in Δ rest frame
Ratios:
(a) (b)
(c)
ρ
(d) (e) (f)
LO
Calculation to NLO in the δ expansion:
chiral loop corrections: unitarity & e.m. gauge‐invariance exact to NLO
[V.P. & Vanderhaeghen, PRL 95 (2005); PRD 73 (2006)]
Pion Electroproduction (e N ‐> e N π ) in Δ(1232) region
4 free parameters –to GM
LECs corresponding, GE, GC at Q2=0, and GM radius
γ N Δ vertex to NLO
W = 1.232 GeV , Q2 = 0.127 GeV2
NLO ChEFT (4 LECs)
theory error bands due to NNLO
data points : MIT-Bates
(Sparveris et al., 2005)
e N ‐> e N π in Δ(1232) region: observables
data points :
MAMI : [Beck et al.(2000), Pospischil et al. (2001),
Elsner et al. (2005), Stave et al (2006), Sparveris et al (2006) ]
NLO (LECs fixed from observables)
“Bare” N->Delta (no chiral corr.’s)
MAID
SAID
MIT-Bates [Sparveris et al. (2005) ]
Prediction of the Q2 dependence of REM=E2/M1 and RSM=C2/M1
curves :
W=1.232 GeV, Q2 = 0.1 GeV2
linear extrapolation
in mq ~ mπ2
discrepancy with lattice due to the chiral dynamics
data points : MAMI, MIT-Bates
quenched lattice QCD results :
mπ = 0.37, 0.45, 0.51 GeV
Nicosia – MIT group [Alexandrou et al., PRL 94 (2005)]
ChEFT prediction
Prediction of the mq dependence of REM=E2/M1 and RSM=C2/M1
mπ = Δ
• First observations of the magnetic moment of a strongly unstable particle.
• High-precision experiment at MAMI (Mainz) using Crystal Ball and TAPS detectors. completed 2005, analysis in progress.
• Theory input needed
Δ(1232) magnetic moment – μΔ from Radiative Pion Photoproduction (γN ‐> Nπ γ’ )
V.P. & Vanderhaeghen, PRL 94 (2005)Calculation to NLO in the δ expansion
2 free LECs – μ(s) and μ(v)
Data points from an older exptTAPS@MAMI 2002
≤≤ ≤
≤
μΔ from γ p ‐> p π0 γ’ observables
Lattice data points from [1] D.B. Leinweber, Phys. Rev. D (1992); I.C. Cloet, D.B. Leinweber and A.W.Thomas, Phys. Lett. B563 (2003).[2] F.X. Lee et.al., hep-lat/0410037
Real parts
Imag. parts
Δ++
Δ+
p
Chiral behavior of the Δ++ and Δ+
magnetic moments
mπ = Δ
Summary
1. Strict low‐energy expansion is limited by the excitation energy of a resonance. In the N‐sector the limit is set by
2. δ−expansion, EFT with 2 light scales, designed to provide an adequate power‐counting for the (rapidly changing) resonant contributions.
3. Successful ChEFT description of processes (π N scattering, electroproduction, Compton, …) in the Δ‐resonance region.
4. Allows to extract low energy hadronic properties from experiment: γ*N ‐> Δ transition from e p ‐> e p π0 process
ΔMDM from γ p ‐> p π0 γ’ processsynergy with lattice QCD : describes the chiral behavior of hadronic properties
γ*N ‐> Δ transition may explain the discrepancy between lattice and exptΔMDM : may complement (euclidean) calculations done for stable particles
Reviewhep‐ph/0609004
Higher N*s
Include the Ν∗ as an explicit d.o.f. ,
N* Δ
MN*‐MN MN*‐MΔ MΔ ‐MN
“hard” “soft”
N N --> > ΔΔ DVCS and DVCS and GPDsGPDs*
x + ξ x - ξ
Q2 >> t = Δ2
N Δ
low –t process: -t << Q2
HM, HE, HC, H4 (x, ξ ,t)
Jones-Scadron
N -> Δ form factors
N N --> > ΔΔ magnetic dipolemagnetic dipole form factorform factor
modified Regge model
Regge model
large Nc limit
N N --> > ΔΔ magnetic dipole magnetic dipole GPDGPD
b⊥ (fm)
x
HM
N N --> > ΔΔ E2 E2 andand C2 C2 form factorsform factors
Buchmann, Hester, Lebed (2002)large Nc limit of QCD :
Nc = 3 EXP : rn2 = - 0.113 (3) fm2
large Nc : Qp -> Δ+ = - 0.080 fm2
EXP : Qp -> Δ+ = - 0.085 (3) fm2
finite (low) Q2 :
VP, Vanderhaeghen (2006)
N N --> > ΔΔ E2 E2 andand C2C2 form factorsform factors
VP, Vanderhaeghen
hep-ph/0611050
modified Regge model
data :MAMI, BATES, JLab/CLAS, JLab/Hall A
large Nc limit
