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Howard Budd, Univ. of Rochester1 Vector and Axial Form Factors Applied to Neutrino Quasi-Elastic Scattering Howard Budd University of Rochester (in collaboration with A. Bodek and J. Arrington) http://www.pas.rochester.edu/~bodek/nuint04-howard.ppt Talk Given in NUINT04, ITALY March 2004

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Howard Budd, Univ. of Rochester2 Review of BBA-form factors Reanalyze the previous deuterium quasi- elastic data by calculating M A with their assumptions and with BBA-form factor to extract a new value of M A Use the previous deuterium quasi-elastic data to extract F A Look at what MINER A can do See what information anti-neutrinos can give Outline

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Howard Budd, Univ. of Rochester3

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5 Our Constants BBA-Form Factors and our constants

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Howard Budd, Univ. of Rochester6 Neutron G M N is negative Neutron (G M N / G M N dipole ) At low Q2 Our Ratio to Dipole similar to that nucl-ex/0107016 G. Kubon, et al Phys.Lett. B524 (2002) 26-32 Neutron (G M N / G M N dipole )

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Howard Budd, Univ. of Rochester7 Neutron G E N is positive New Polarization data gives Precise non zero G E N hep-ph/0202183(2002) Neutron, G E N is positive - Imagine N=P+pion cloud Neutron (G E N / G E P dipole ) Krutov (G E N ) 2 show_gen_new.pict Galster fit Gen

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Howard Budd, Univ. of Rochester8 Functional form and Values of BBA Form Factors G E P.N (Q 2 ) = {e i q. r (r) d 3 r } = Electric form factor is the Fourier transform of the charge distribution for Proton And Neutron (therefore, odd powers of Q should not be there at low Q)

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Howard Budd, Univ. of Rochester9 Type in their d /dQ2 histogram. Fit with our best Knowledge of their parameters : Get M A =1.118+-0.05 (A different central value, but they do event likelihood fit And we do not have their the event, just the histogram. If we put is best knowledge of form factors, then we get M A =1.090+-0.05 or M A = -0.028. So all their Values for M A. should be reduced by 0.028

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Howard Budd, Univ. of Rochester10 Using these data we get M A to update to for latest ga+form factors. (note different experiments have different neutrino energy Spectra, different fit region, different targets, so each experiment requires its own study). A Pure Dipole analysis, with ga=1.23 (Shape analysis) - if redone with best know form factors --> M A = -0.047 (I.e. results need to be reduced by 0.047) for different experiments can get M A from -0.025 to -0.060 Miller did not use pure dipole (but did use Gen=0)

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Howard Budd, Univ. of Rochester11 The dotted curve shows their calculation using their fit value of 1.07 GeV They do unbinned likelyhood to get M A No shape fit Their data and their curve is taken from the paper of Baker et al. The dashed curve shows our calculation using M A = 1.07 GeV using their assumptions The 2 calculations agree. If we do shape fit to get M A With their assumptions -- M A =1.079 GeV We agree with their value of M A If we fit with BBA Form Factors and our constants - M A =1.055 GeV. Therefore, we must shift their value of M A down by -0.024 GeV. Baker does not use a pure dipole The difference between BBA-form factors and dipole form factors is -0.049 GeV Determining m A, Baker et al. – BNL deuterium

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Howard Budd, Univ. of Rochester12 The dotted curve shows their calculation using their fit value of M A =1.05 GeV They do unbinned likelyhood, no shape fit. The dashed curve shows our calculation using M A =1.05 GeV and their assumptions The solid curve is our calculation using their fit value M A =1.05 GeV The dash curve is our calculation using our fit value of M A =1.19 GeV with their assumption However, we disagree with their fit value. Our fit value seem to be in better agreement with the data than their fit value. We get M A =1.175 GeV when we fit with our assumptions Hence, -0.019 GeV should be subtracted from their M A. Kitagaki et al. FNAL deuterium

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Howard Budd, Univ. of Rochester13 Dotted curve – their calculation M A =0.95 GeV is their unbinned likelyhood fit The dashed curve – our calculation using their assumption We agree with their calculation. The solid curve – our calculation using theirs shape fit value of 1.01 GeV. We are getting the best fit value from their shape fit. The dashed curve is our calculation using our fit value M A =1.075 GeV. We slightly disagree with their fit value. We get M A =1.049 GeV when we fit with BBA – Form Factors and our constants. Hence, -0.026 GeV must be subtracted from their value of M A Barish et al. ANL deuterium

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Howard Budd, Univ. of Rochester14 Miller is an updated version of Barish with 3 times the data The dotted curve – their calculation taken from their Q 2 distribution figure, M A =1 GeV is their unbinned likely hood fit. Dashed curve is our calculation using their assumptions We don't quite agree with their calculation. Their best shape fit for M A is 1.05 Dotted is their calculation using their best shape M A Our M A fit of using their assumptions is 1.119 GeV Our best shapes agree. Our fit value using our assumptions is 1.09 GeV Hence, -0.027 GeV must be subtracted from their fit value. Miller – ANL deuterium

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Howard Budd, Univ. of Rochester15 Summary of Results

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Howard Budd, Univ. of Rochester16 Hep-ph/0107088 (2001) Difference in Ma between Electroproduction And neutrinos is understood For M A from QE neutrino expt. On free nucleons No theory corrections needed 1.11=M A -0.026 -0.028 Neutrinos 1.026+-0.021-=M A average From Neutrino quasielasti c From charged Pion Electroproduction Average value of 1.069->1.014 when corrected for theory hadronic effects to compare to neutrino reactions =1.014 when corrected for hadronic effect to compare to neutrino reactions Ma=1.06+-0.14 (using dipole FF) from K2K goes down to 1.01 with BBA form factors For updated M A expt. need to be reanalyzed with new g A, and G E N More correct to use 1.00+-0.021=M A

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Howard Budd, Univ. of Rochester17 Nuclear correction uses NUANCE calculation Fermi gas model for carbon. Include Pauli Blocking, Fermi motion and 25 MeV binding energy Nuclear binding on nucleon form factors as modeled by Tsushima et al. Model valid for Q 2 < 1 Binding effects on form factors expected to be small at high Q 2.

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Howard Budd, Univ. of Rochester18 Neutrino quasi-elastic cross section Most of the cross section for nuclear targets low

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Howard Budd, Univ. of Rochester19 Anti-neutrino quasi-elastic cross section Mostly on nuclear targets Even with the most update form factors and nuclear correction, the data is low

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Howard Budd, Univ. of Rochester20 A comparison of the Q 2 distribution using 2 different sets of form factors. The data are from Baker The dotted curve uses Dipole Form Factors with m A =1.10 GeV. The dashed curve uses BBA-2003 Form Factors with m A =1.05 GeV. The Q 2 shapes are the same However the cross sections differ by 7-8% Shift in m A – roughly 4% Nonzero GEN - roughly 3% due Other vector form factor – roughly 2% at low Q 2 Effects of form factors on Cross Section

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Howard Budd, Univ. of Rochester21 K2K using dipole form factor and set m A =1.11 instead of nominal value of 1.026 This plot is the ratio of BBA with m A =1 vs dipole with m A =1.11 GeV This gets the cross section wrong by 12% Need to use the best set of form factors and constants Effect of Form Factors on Cross Section

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Howard Budd, Univ. of Rochester22 These plots show the contributions of the form factors to the cross section. This is d(d /dq)/dff % change in the cross section vs % change in the form factors The form factor contribution neutrino is determined by setting the form factors = 0 The plots show that F A is a major component of the cross section. Also shows that the difference in G E P between the cross section data and polarization data will have no effect on the cross section. I Extracting the axial form factor

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Howard Budd, Univ. of Rochester23 We solve for F A by writing the cross section as a(q 2,E) F A (q 2 ) 2 + b(q 2,E)F A (q 2 ) + c(q 2,E) if (d /dq 2 )(q 2 ) is the measured cross section we have : a(q 2,E)F A (q 2 ) 2 + b(q 2,E)F A (q 2 ) + c(q 2,E) – (d /dq 2 )(q 2 ) = 0 For a bin q 1 2 to q 2 2 we integrate this equation over the q 2 bin and the flux We bin center the quadratic term and linear term separately and we can pull F A (q 2 ) 2 and F A (q 2 ) out of the integral. We can then solve for F A (q 2 ) Shows calculated value of F A for the previous experiments. Show result of 4 year Miner a run Efficiencies and Purity of sample is included. Measure F A (q 2 )

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Howard Budd, Univ. of Rochester24 For Miner a - show G E P for polarization/dipole, F A errors, F A data from other experiments. For Miner a – show G E P cross section/dipole, F A errors. Including efficiencies and purities. Showing our extraction of F A from the deuterium experiments. Shows that we can determine if F A deviates from a dipole as much as G E P deviates from a dipole. However, our errors, nuclear corrections, flux etc., will get put into F A. Is there a check on this? F A /dipole

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Howard Budd, Univ. of Rochester25 d(d /dq 2 )/dff is the % change in the cross section vs % change in the form factors Shows the form factor contributions by setting ff=0 At Q 2 above 2 GeV 2 the cross section become insensitive to F A Therefore at high Q 2, the cross section is determined by the electron scattering data and nuclear corrections. Anti-neutrino data serve as a check on F A. Do we get new information from anti-neutrinos?

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Howard Budd, Univ. of Rochester26 Errors on F A for antineutrinos The overall errors scale is arbitrary The errors on F A become large at Q 2 around 3 GeV 2 when the derivative of the cross section wrt to F A goes to 0 Bottom plot shows the % reduction in the cross section if F A is reduced by 10% At Q 2 =3 GeV 2 the cross section is independent of F A

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Howard Budd, Univ. of Rochester27 Conclusion Using BBA-form factors we derive a new value of m A = 1.00 GeV ~7-8% effect on the neutrino cross section from the new value m A and with the updated vector form factors Extract F A from the d /dq 2 MINER A can measure F A and determine deviations from the dipole form The anti-neutrinos at high Q 2 serves as a check on F A

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