chiral extrapolations for nn, ndelta form factors --- update thomas r. hemmert theoretische physik...

Chiral Extrapolations for NN, Chiral Extrapolations for NN, NDelta Form Factors --- Update NDelta Form Factors --- Update

Thomas R. Hemmert

Theoretische Physik T39

Physik Department, TU München

Workshop on Computational Hadron PhysicsUniversity of Cyprus, Nicosia

Sep 14 -17, 2005

T.R. Hemmert, "Chiral Extrapolation for NN, NDelta Form Factors" Sep 16, 2005

OutlineOutlineOutlineOutline

• Basics regarding covariant BChPT versus non-relativistic BChPT (HBChPT)

• Covariant analysis of the isovector magnetic moment of the nucleon

• Momentum-dependence of the NDelta-transition form factors revisited

• On the road to a quantitative chiral extrapolation of NDelta-form factors

• Outlook


Baryon ChPTBaryon ChPTBaryon ChPTBaryon ChPT

• Low energy effective theory of QCD– Pions as (Quasi-) Goldstone Bosons of the

spontaneously broken chiral symmetry of QCD– In addition explicit breaking of chiral symmetry due

to finite quark masses– Baryons added as matter fields in a chirally invariant

procedure (CCWZ)

• Perturbation Theory organised in powers pn

• Careful: Baryon ChPT has 2 scales:

MN, Λχ ~ 1 GeV (at physical point)


NR-BChPTNR-BChPTNR-BChPTNR-BChPT

• Non-relativistic framework (HBChPT) (Jenkins, Manohar 1991)

→ only terms ~ 1/MN appear in calculation

• Organise perturbative calculation as a simultaneous expansion in 1/MN and 1/ Λχ (Bernard et al. 1992)

1/Mn-1


Baryon ChPTBaryon ChPTBaryon ChPTBaryon ChPT

• NR-BChPT/HBChPT very successful for near-threshold scattering experiments (mπ=140 MeV)

• Chiral extrapolation rather tedious in HBChPT: Low order polynomial in mπ compared to smoothly varying lattice results

• Idea: Utilize covariant BChPT– At each order pn the result to that order is given in

terms of smoothly-varying analytic functions f(μ) with μ=mπ /MN

– Different organization of the perturbative expansion


Regularization in BChPTRegularization in BChPTRegularization in BChPTRegularization in BChPT

• Ultraviolet divergences can be absorbed via counterterms in the effective theory; UV is not the (main) issue

• Troublesome are terms ~ MN/Λχ ~O(1),

(which for example appear in MS-scheme)

→ uncontrolled finite renormalization of coupling constants

1/Mn-1


Infrared RegularizationInfrared RegularizationInfrared RegularizationInfrared Regularization

• Use a regularization scheme that avoids terms

~ (mq)a (MN/Λχ)b

– e.g. Infrared Regularization (IR) (Becher, Leutwyler 1999)

• Idea: Add an extra integral in Feynman-parameter space that contains an infinite string of quark-mass insertions which cancel these terms

02

12

1

02

1

02

1111

MBdx

MBdx

MBdx

MBdx

MS IRRegulator Integral


Covariant BChPT with IRCovariant BChPT with IRCovariant BChPT with IRCovariant BChPT with IR• Controlled coupling constant renormalization

• Reorganization of perturbative expansion with exact HBChPT limit• Successful scheme at physical point (e.g. nucleon spin structure,

neutron form factor, …) • Promising results for chiral extrapolation of nucleon mass in finite

volume (QCDSF collaboration, Nucl. Phys. B689, 175 (2004))

• However: Can be problematic in large mπ , large Q2 behaviour

1/Mn-1







• Outlook


Magnetic Moments on the Lattice Magnetic Moments on the Lattice Magnetic Moments on the Lattice Magnetic Moments on the Lattice

• Isovector anomalous magnetic moment of the nucleon: κv

– extrapolated to q2=0 via Dipole-fits

• Slope ? Caldi-Pagels ↓ !

Quenched (improved) Wilson Data: QCDSF collaboration;Phys. Rev. D71, 034508 (2005)

Turning-points in quark-mass dependence ?Breakdown of ChEFT ?

κproton-κneutron = 3.71 n.m.


Magnetic Moments ?Magnetic Moments ? Magnetic Moments ?Magnetic Moments ?• Nucleon isovector electromagnetic current:

• Dirac FF: F1v(0)=1;

• Pauli FF: F2v(0)=κv=κp-κn=3.71 [n.m.]

with [n.m.]=e/2M=3.15 10-14 MeV T-1

at mπ=140 MeV• We are not interested in the quark-mass dependence of

the nuclear magneton unit ! → express lattice results in units of the physical [n.m]

spuqiqFM

eqFespuNqqeN duq ,

2´´, 2

22

1

Quark mass dependent magneton Quark mass

dependent Pauli formfactor


Normalized Magnetic MomentsNormalized Magnetic MomentsNormalized Magnetic MomentsNormalized Magnetic Moments

• Measure Pauli form factor in physical [n.m.]

• Not required, but elucidates quark mass dependence of the magnetic moment more clearly !

• Note: Similar complications occur in the NΔ-transition form factors!

qlatticev

qlatticeN

physN

qnormv m

mM

Mm

• Use lattice data for the normalization• Set lattice scale via MN → normalization factor becomes 1 at physical point


Magnetic Moments IIMagnetic Moments IIMagnetic Moments IIMagnetic Moments II

• Anomalous isovector magnetic moment of nucleon κv

– Caldi-Pagels prediction ~ - mπ (= HBChPT O(p3) )

κv(mπ) measured in physical nuclear magnetons [n.m.] → the remaining quark mass dependence is flat! (compare <x>, gA)



Magnetic Moments IIIaMagnetic Moments IIIaMagnetic Moments IIIaMagnetic Moments IIIa

• Go beyond Caldi-Pagels !

• e.g. include explicit Delta(1232) degrees of freedom (NR SSE)

(TRH, Weise, EPJA 15, 487 (2002)) • Physical point and

lattice data in good agreement within assumptions for Delta parameters

→ Covariant BChPT ?Quenched (improved) Wilson Data: QCDSF collaboration;Phys. Rev. D71, 034508 (2005)


Magnetic Moments IIIbMagnetic Moments IIIbMagnetic Moments IIIbMagnetic Moments IIIb

• O(p4) covariant BChPT with (modified) IR-regularization

(T. Gail, TRH, forthcoming)

• ci couplings fixed from πN and NN scattering

• 2 unknown LECs κ0v,

E1(λ) fit to QCDSF data → one free parameter less than in SSE calculation

ci


Comparison Magnetic MomentsComparison Magnetic MomentsComparison Magnetic MomentsComparison Magnetic Moments

• Comparable results BChPT - NR-SSE, need better data to study differences (T. Gail and TRH, forthcoming)• However: Results at finite Q2 in covariant BChPT still require work/thought (Q2 dependence in SSE very successful)


O(p4) BChPT

NR-SSE Leading-one-loop

O(p3) BChPT


NN Form factorsNN Form factorsNN Form factorsNN Form factors• Isovector NN form factors can be analyzed up to Q2 < 0.4 GeV2 and

mπ< 600 MeV in SSE to O(ε3) (M. Göckeler et al., Phys. Rev. D71, 034508 (2005))

• Note: Direct comparison with simulation data at finite Q2 possible !

→ We can even avoid extra uncertainties from dipole fits in this window

Physics beyond the radii !!







• Outlook


NDelta Form FactorsNDelta Form FactorsNDelta Form FactorsNDelta Form Factors

• 3 complex (isovector) transition form factors:

G1(Q2), G2(Q2), G3(Q2) (real for mπ > MΔ-MN)

• Known to O(ε3) in SSE (G.C. Gellas et al., Phys. Rev. D60, 054022 (1999))

mπ=140 MeV


Multipole BasisMultipole BasisMultipole BasisMultipole Basis• 3 complex NDelta transition form factors:

G1(Q2), G2(Q2), G3(Q2) → GM1*(Q2), GE2*(Q2), GC2*(Q2)

• 2 free parameters at O(ε3) in SSE: G1(0), G2(0)

→ Fix at GM1*(Q2=0)

and at

EMR(Q2=0) =

Re[GM1*(Q2)GE2(Q2)]/|GM1(Q2)|2

G.C. Gellas et al., PRD 60, 054022 (1999),T. Gail and TRH, forthcoming

Abs!

mπ=140 MeV


EMR(QEMR(Q22) ?) ?EMR(QEMR(Q22) ?) ?

• O(ε3) SSE: EMR(Q2) in multipole basis ? (% effects !)

• Problem results from G2(Q2): Rising with Q2 !??!– , no c.t. at this order! → check effect of extra c.t. in radius

Abs!

mπ=140 MeV


Prediction for CMR(QPrediction for CMR(Q22) ) Prediction for CMR(QPrediction for CMR(Q22) ) • GM1*(Q2) still okay, small radius correction in G2 shows large effects

• CMR(Q2) = Re[GM1*(Q2)GC2(Q2)]/|GM1(Q2)|2 is a prediction

→ Comparison to new Mainz data ?

Abs!

mπ=140 MeV

Abs!







• Outlook


Chiral Extrapolation IChiral Extrapolation IChiral Extrapolation IChiral Extrapolation I

• (Quenched) Lattice Data for GM1*(Q2) at low Q2 (C. Alexandrou et al., [hep-lat/0307018])

Note:GM1*(Q2) increases with increasing quark mass → similar to situation in GM(Q2)

!


Chiral Extrapolation IIChiral Extrapolation IIChiral Extrapolation IIChiral Extrapolation II• First step: Extrapolate all dynamical factors in NDelta transition current

to the physical point mπ=140 MeV (not just the magneton!)– strong rise in mπ is gone !– lattice data are now lower than the mπ=140 MeV curve

Here: Data at Q2=0.135 GeV2 from C. Alexandrou et al., [hep-lat/0307018]→ Need MN(mπ) and MΔ(mπ) with correct extrapolation to the physical point to do this !T. Gail and TRH, forthcoming


Nucleon and Delta MassNucleon and Delta MassNucleon and Delta MassNucleon and Delta Mass

• We are utilizing the covariant O(ε4) SSE result:

V. Bernard, TRH, U.-G. Meißner,[hep-lat/0503022]

3-flavour data !MILC[hep-lat/0104002]

Rising NDelta mass-splitting near the chiral limit ?


Chiral Extrapolation IIChiral Extrapolation IIChiral Extrapolation IIChiral Extrapolation II

• Study quark mass dependence of the form factors with physical point kinematics – directly at low Q2 and low mπ region without dipole

extrapolationsNote:Before one can address the tiny EMR(mπ), CMR(mπ) ratios, one needs to get GM1*(mπ,Q2) right !

T. Gail and TRH, forthcoming


Chiral Extrapolation IIIChiral Extrapolation IIIChiral Extrapolation IIIChiral Extrapolation III• Essential NDelta intermediate state needs to be added for chiral

extrapolation (see TRH, Weise, EPJA 15, 487 (2002))– formally NLO, but essential at intermediate mπ (no effect on Q2-dependence) – similar to κv(mπ), but no steep slope due to Caldi Pagels !

T. Gail and TRH, in preparation!


Outlook/SummaryOutlook/SummaryOutlook/SummaryOutlook/Summary• κv(mπ) to O(p4) in covariant BChPT now under control

– smooth mπ dependence in modified IR-regularization– comparable to NR-SSE result– new data? (Cyprus-Athens?)

• Q2 dependence of nucleon form factors in covariant BChPT still needs more work

– Q2 dependence of NR-SSE compares well with phenomenology– direct comparison with lattice data in low Q2 window without dipole extrapolations

• Q2 dependence of all 3 NDelta form factors now well under control– New compared to Gellas et al. calculation: Radius correction in G2(Q2) – comparison to new Mainz data ?

• Chiral extrapolation of NDelta form factors– evaluate kinematical factors at physical point → most headaches seem to be gone– focus first on GM1*(mπ,Q2), direct comparison with small GE2*(Q2), GC2*(Q2) is step 2– at the moment only qualitative results for EMR(mπ), CMR(mπ), comparable to

Pascalutsa, Vanderhaghen (same diagrams as in Gellas et al.)– Guess: Quantitative chiral extrapolation of quadrupole form factors will require a lot

more effort than O(ε3) SSE


Magnetic Moment in naive IRMagnetic Moment in naive IRMagnetic Moment in naive IRMagnetic Moment in naive IR

• O(p3) BChPT in naive IR-regularization– Kubis, Meißner NPA

679, 698 (2001)– TRH, Weise EPJA 15,

487 (2002)• Problems:

– turning points in mπ

– artefacts of naive IR• Alternative: Correct for

the quark-mass dependence of the analytic structures in f(μ) „by hand“ to soften the curve

(see e.g. MIT-meeting, Hemmert 2004)


chiral extrapolations for nn, ndelta form factors --- update thomas r. hemmert theoretische physik...

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