Transcript Slide 1

Resonances in chiral EFT
and lattice QCD
Vladimir Pascalutsa
European Centre for Theoretical Studies (ECT*) , Trento, Italy
Supported by
Presented @ Erice School “Quarks and Hadrons in Nuclei” ( Erice, Italy, 17-24 Sep, 2007)
Unstable particles (resonances)
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Muon:
Neutron:
0(1193):
(1116):
(1232):
many more…
+p scattering cross-section
Bubble chamber events
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Low-energy QED (an example of
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EFT)
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Towards low-energy QCD
The massless quark Lagrangian,
is invariant under “chiral rotations”:
Chiral symmetry is an SU(nf )L SU(nf )R symmetry of mq=0 QCD
For
, SU(nf ) isospin symmetry
u
u
q     exp ( iθ  τ )  
d 
d 
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Chiral Symmetry Breaking
L
good SU(2)L x SU(2)R chiral symmetry,
:  SU(3)L  SU(3)R
:
u
u
d
d
R
The symmetry NOT visible in hadron spectrum (no parity doublets),
- spontaneously broken down to isospin symmetry, giving rise to
(massless) Goldstone bosons:
• Spontaneous chiral SB characterized by the
non-zero quark condensate:
• Broken explicitly by the quark masses, Goldstone bosons acquire mass:
(Gell-Mann – Oakes – Renner)
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Chiral Perturbation Theory and Resonances
Low-energy QCD ~ ChPT [ Weinberg (1979), Gasser & Leutwyler (1984), …]
Lagrangian:
S-matrix:
However, near a resonance (or a bound state):
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Chiral Lagrangians with 
(1232) – first nucleon resonance,
Include the  as an explicit d.o.f. ,
[Jenkins & Manohar (1991), …]
described by a spin-3/2 (Rarita-Schwinger) isospin-3/2 (isoquartet) field
Power counting:
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Example: Nucleon mass
Leading order pion-nucleon interaction:
Power counting index:
tells us that a graph is of O(pn)
LO nucleon self-energy =
renormalization
LECs
prediction
On-mass-shell renormalization [Gegelia et al. (1999), (2003)],
not MS-bar [Gasser et al. (1989)].
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N and Δ masses: pion-mass dependence
Lattice :
MILC
ChEFT:
Covariant p3
V.P. & Vanderhaeghen,
PLB 636 (2006)
physical
world
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Low-energy QCD in the presence of (1232)
Compton scattering on the nucleon
Generic features:
(i) below  production threshold ( < m), the  is a high-energy degree of
freedom – can be integrated out (because / << 1 ) – ChPT with no ’s
(ii) above, rapid change with energy, at    -- PT break-down
How to obtain this behavior in ChEFT ?
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Power countings
[Jenkins & Manohar (1991); Hemmert et al. (1998) … (2006)]
N and  propagators:
[ V.P. & Phillips, PRC (2003) ]
OR propagator
»
=
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+ … = O(p3) = O(3 )
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Pion-nucleon scattering in the resonance region
Renormalized
NLO propagator
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Electromagnetic excitation of the (1232) resonance
VP, Vanderhaeghen & S.N. Yang, Phys Rept 427 (2007);
“Shape of Hadrons” ed. Bernstein & Papanicolas, AIP 904 (2007)
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N
Δ
3 e.m. transitions :
M1, E2, C2
non-zero values of REM =E2/M1, RSM =C2/M1:
measure of non-spherical shape
G.A. Miller, arXiv:0708.2297; Kvinikhidze & Miller, PRC 76:025203,2007
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Pion Electroproduction in Chiral EFT
[V.P. & Vanderhaeghen, PRL 95 (2005); PRD 73 (2006)]
Calculation of e N -> e N π to NLO in the δ expansion:
LO
4 free parameters – LECs corresponding
to GM, GE, GC at Q2=0, and GM radius
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chiral loop corrections:
unitarity & e.m. gaugeinvariance exact to NLO
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e N -> e N π in Δ(1232) region: observables
data points :
MIT-Bates
W = 1.232 GeV , Q2 = 0.127 GeV2
(Sparveris et al., 2005)
NLO ChEFT (4 LECs)
theory error bands
due to NNLO
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Prediction of the Q2 dependence of E2/M1 and C2/M1
data points :
MIT-Bates [Sparveris et al. (2005) ]
MAMI :
[Beck et al.(2000), Pospischil et al. (2001),
Elsner et al. (2005), Stave et al (2006),
Sparveris et al (2006) ]
curves :
NLO (LECs fixed from observables)
“Bare” N->Delta (no chiral corr.’s)
MAID
SAID
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Prediction of the mq dependence of E2/M1 and C2/M1
W=1.232 GeV, Q2 = 0.1 GeV2
Nicosia – MIT group
[Alexandrou et al.,
PRL 94 (2005)]
quenched lattice QCD
results :
mπ = 0.37, 0.45, 0.51 GeV
linear extrapolation
in mq ~ mπ2
ChEFT prediction
m = 
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data points : MAMI, MIT-Bates
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Radiative Pion Photoproduction (N -> Nπ ’ )
E / MeV
´
1232
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
Chiang, Vanderhaeghen, Yang & Drechsel,
PRC (2005).
proton
938
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
V.P. & Vanderhaeghen, PRL 94 (2005)
Machavariani, Faessler & Buchmann,
Calculation
to NLO in(2001).
the δ expansion
NPA
(1999), Erratum-ibid
2 free LECs – (s) and (v)
Drechsel et al, PLB (2000)
Drechsel & Vanderhaeghen, PRC (2001)
First observations of the magnetic
moment of a very unstable particle.
High-precision experiment at MAMI
(Mainz) using Crystal Ball and TAPS
detectors, see M. Kotulla’s talk.
Theory input needed
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Circular beam asymmetry
VP & Vanderhaeghen, in prep.
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Chiral behavior of the  magnetic moment
quenched lattice points :
Leinweber (1992)
Cloet,Leinweber,Thomas (2003)
Lee et.al. (2004)
– revised (2006)
p
m = 
Real parts
Imag. parts
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Curves - chiral EFT calculations
V.P & Vanderhaeghen PRL (2005)
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New quenched lattice results
Leinweber et al. (to be publ.)
Wrong trend
[Young, Leinweber,
Thomas (2006)]
Quenching pathology
– full QCD needed
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Resonance properties from lattice QCD
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Lattice QCD, an Euclidean field
theory (real numbers)
A method to compute a resonant
phase-shift on the lattice [ Luscher,
NPB364 (1991) ] – study the
volume dependence of the energy
spectrum
NPLQCD:  scattering length;
QCDSF: -meson
Rusetsky, Bernard, Lage, Meissner:
-resonance volume dependence
in ChPT, hep-lat/0702012
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Conclusion and challenges
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-resonance in ChEFT
1. Effective fields, require resummations (strict low-energy
expansion fails), which must make sense within the power-counting
scheme; acquire width via the absorptive part of the self-energy…
2. Successful description of both the momentum dependence of
scattering amplitudes (connection to experiment) and the pionmass dependence of static quantities (connection to lattice)
Higher resonances in ChEFT ?
Resonances in LQCD
impossible in quenched
tough in full
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