Neutron SSFs from RSS proton and deuteron data

• Technote on extracting neutron SSFs by Oscar
• Slides on smeared g1n and g2n
• Deuteron RCs by Mark
  
• Table of preliminary results for g1,g2 for p(smear),d(data),n(smear) vs Xbj (see Apr23,07)
• Report on deuteron and neutron analysis
• Outline of the archival paper

• Deuteron results were updated on jun03, 2007 by Mark. Here are the updated plots of g1n, g2n, and a1n&a2n . For g1n and g2n, We used 180Mev bin for x<0.576 and 60Mev bin otherwise so that RSS error bars are compatible with the world data errors. Updated data points are basically on top of the old points. Here is the table, of g1n, g2n etc with no-rebinning. NOTE: Starting on Apr16, We do not use deuteron 'fits' to obtain neutron results. deuteron data and data error have been used to get them, however. Added on Sep 04: Here are the 'nicer' plots of RSS g1n and g2n compared with the world data.
  

• Preliminary results of g1, g2 for smeared proton, deuteron (data), and smeared neutron versus Xbj are available here . The corresponding W bin size is 30 MeV. Full DState Correction (using the results by W.Melnitchouk et al) has been applied.

• Results of smeared f1n and f2n were obtained from peter's routine, F1F2IN06.f.  The original routine outputs the unsmeared F1n, F2n so I modified it slightly with the help from Peter.
  

• Smeared A1n and A2n were extracted from our measured g1n and g2n along with F1n (shown above). This procedure is much simpler than our old procedure in which the neutron extraction is done at the level of cross section (explained in detail on sep.11, 2007 below). I assumed the relative uncertainty of 10% in the meantime. (Update: Peter thinks that the systematic error on F1n is about 5-6% because of the stat errors on deuteron he used to make a fit is about 5% and the proton data error is much smaller than that.) The DState correction has already been applied to A1n,A2n because that was already done to get g1n and g2n.  Comparison of different DState corrections (no correction, const factor, and FULL correction) is here . One comment about the FULL correction: the D-state correction factor is about ~1/0.925 for   x<0.7 and it then starts icreasing for x beyond 0.7 (see the reference given on Apr.10). At our lowest W, the D-state factor becomes close to 1.0, so we don't see much difference between "no Dstate" and "full correction" results in the low W region.
  

• As requested at the meeting last week, g1n and g2n were obtained using wider bin sizes. I used 180Mev bin for x<0.576,  and 60Mev bin otherwise. Results are still consistent with world data.
  

• Effect of D-state correction on g1n,g2n using the results from W. Melnitchouk et al (see Apr10,2007 below) is here: g1n , g2n . Difference between this result and result from last week (using constant D-state correction factor of 1/0.925) is not large. (The largest difference is 13% for g1n at xbj=0.82, and 7% for g2n at xbj=0.79.)
  
  

• Updated plots of neutron g1,g2 are posted here g1n_extraction , g1n_plot , g2n_extraction , g2n_plot , and smearing_effect for the proton. Our preliminary results basically agree with world data. The same binning as of last week was used for the neutron.

• Using our extracted g1n and g2n and F1n from Peter's code (F1F2IN06.f), neutron A1,A2 can be extracted: A1n = ( g1n - gamma^2 g2n ) / F1n  ;  A2n = gamma * ( g1n + g2n ) / F1n . This plot shows F1,F2 for deuteron, proton, neutron from Peter's code. Note that his code returns structure function per "nucleus". Note that these are 'unsmeared' results. Neutron results are obtained by setting Z=0, A=1. In his code, F1,F2 is multiplied by the EMC factor but that is 1.0 for A<2.5, meaning no EMC correction for deuteron.
  

• Thanks to Karl and Mark, I extracted neutron g1,g2 using the smeared proton (using v6c fit) and the latest deuteron fit (for A1d) with linear DIS form obtained by Mark. I used the constant fit for deuteron A2 (A2d = 0.081 +/- 0.017) as posted at Mark's webpage.  The D-state correction has been applied when neutron g1,g2 are extracted from the deuteron fit and smeared proton: g1n=g1d/(1-1.5*W_D) - g1p, where W_D=0.05, and similarly for g2n. To calculate the error on neutron, the error on smeared proton was calculated numerically using the procedure explained on Apr02,2007 below. For the deuteron, we assumed the same relative error as the RSS deuteron data. Results of g1n and g2n . Each file has three pages: (P1-- No Rebin result [30MeV], p2--Neutron results rebinned (using 90MeV bin for x<0.576 or W>1.37: otherwise 30MeV bin), P3--With or without D-state correction ) Because the deuteron fit shows clear resonance structures, the extracted neutron also shows peaks although the error bars are large. Regarding the D-state correction, the correction factor at high x (x>0.7) deviates from ~0.925 and goes up to ~1.06 at our largest x value of x=0.82. That needs to be corrected (see W. Melnitchouk, G. Piller and A. W. Thomas Phys. Lett. B 346, 165 (1995) ).
  

•  Smeared g1p, g2p fit errors are obtained using the A1,A2 fit error. Fit Version V6c is used. Error on g1p due to A1 only is estimated by calculating g1p with A1+da1. Similarly calculated g1p with A2+da2. The total error is given by the sqrt of sum of the errors squared. Similarly for g2p. Results of g1p fit error and  g2p fit error   .  d(ga1)=g1(A1+dA1, A2)-g1(A1,A2), and d(ga2)=g1(A1, A2+dA2)-g1(A1,A2). The total fit error is therefore given bt d(g1) = Sqrt( d(ga1)^2 + d(ga2)^2 ) because A1 and A2 are independent.

• Results of A1,A2 fit errors by Karl

•  I applied the same procedure used on Feb12 below. Here are the preliminary results of fit error on smeared g1p and g2p but I think these are still overestimate.
  

• In the smearing code, off-shell W could lie outside of the resonance region. But our A1 and A2 fit routines return zero for w value outside of the resonance region. The contribution from DIS (W>2) and Elastic (W=Mp) regions can be included in the calculation of smeared g1p and g2p. For DIS g1p, we use Regge fit obtained by Oscar and we also assume g2p = g2ww which can be calculated using Regge g1p. Regge fit result at RSS kinematics is here.  Results of g1p and g2p at elastic kinematics are given in Karl's thesis p210.I choose the elastic region to be Mp +/- 5MeV.
  

• The results of smeared g1p and g2p from outside of the resonance region seems to be very small... (I need to double check my calculation). This plot shows the contribution from DIS and elastic regions only to g1p and g2p in the resonance W region.
  

• We use proton A1, A2 "fits" to calculate proton cross sections (dsigma_para, dsigma_perp) which are smeared when neutron quantities are extracted. We've been assuming that the relative stat errors before and after the smearing of proton do not change. To better estimate the stat errors after smearing, the A1p and A2p errors from our data at discrete W values are fitted. The error fit results are here .

• In the smearing code, the stat 'error' obtained from the fit (dA1 and dA2) is added to or subtracted from A1p and A2p from the fit function, and smeared g1p, g2p are calculated. The difference between the original and the new g1p,g2p could be interpreted as the stat error on smeared g1p,g2p. Results of g1p and g2p . To get g1p upper error, I used A1+dA1 and A2+dA2. Similarly, A1-dA1,A2-dA2 for g1p lower limit. For g2, A1-dA1 and A2+dA2 are used for g2p upper error, and A1+dA1 and A2-dA2 for lower error. Note: g1=C*(A1+gamma*A2), g2=C*(-A1+A2/gamma), C=F1/(1+gamma^2).
  

• I used Mark's results of fully-radiated (Born) deuteron g1d and g2d (as of Nov.14) to update the smeared neutron results (g1ns and g2ns) . Note that g1n_smear = g1d - g1p_smear, and similarly for g2n_smear.
  

• I also updated smeared A1n and A2n (with A2n Soffer limit) with 60mev bins using the updated deuteron results. R1998 and hallc2h.f were used. For A1n, the RC effect is large especially in low W region. For A2n, the RC effect is basically small. A2_n at W=1300MeV is not as prominent as the A2_d spike that we saw probably because I'm using a wider bin (60MeV). The procedure to get A1n, A2n from the deuteron and proton results is summarized on Sep 11, 2006 below. In order to be consistent with the deuteron DF calculation, I should update the results of A1n and A2n using Peter's models instead of hallc2h model.

• [Update on Feb16, 2007] We added A2n Soffer limit to the plot. The Soffer inequality for A2 is given by A2 .le. sqrt(R*(1+A1)/2). To get the A2n limit at each W, we used our A1n data and R from Eric's routine (Q^2 value obtained from the Summary table).
  

• A plot of offshell W (w_off) calculated at constant W values of 1220, 1520, 1820 MeV, weighted by the fermi momentum probability distribution and jacobian factor. The fermi momentum ranges from 0 to 497.5MeV.
  

• R1998 instead of R1990 is now used to get A1n and A2n. This plot (Note: updated plot with error bars) shows the comparison between R1990 and R1998. The error bars are obtained from the subroutine output. The relative difference between the two becomes as large as ~15% at around x=0.7. However, the extracted A1n and A2n using R1998 (plot) shows that the asymmetry results are almost unchanged (difference is 1% or less).
  

• I've been using the proton kinematics values (such as thetae, phie, E', Q2, etc from the proton data file) to calculate A1n and A2n from Apara_n and Aperp_n. But these kin values are not identical to those from the deuteron data file. I recalculated A1n and A2n using the deuteron  values: For A1n, the results are almost unchanged. But for A2n in which electron 'phi' is used, the difference for low W (W<1.25) is as large as 5% and less than 1% for higher W (W>1.25).
  

• Planning to check and study the details of smearing effects in our data.
  

• Extracted smeared neutron xsec(plot), which is equal to "sigma_d" minus "sigma_p_smear",  together with a plot of Q2. For W>1.8, the xsecs rise while Q2 drops. To understand this more, I made a plot showing the proton and deuteron xsecs at RSS kinematics as well as those xsecs at the min and max Q2 (Q2_min=1.04, Q2_max=1.44 in RSS data). At high (low) W edge, the RSS xsec agrees with the xsec at Q2=Q2_min (Q2=Q2_max) as expected.

• A plot of smeared Apara_n and Aperp_n from the RSS deuteron data and smeared proton results. To obtain A_para_n (= dsigma_para_n / (2*sigma_n)), dsigma_para_n was obtained by subtracting the smeared dsigma_para_p from dsigma_d (data). Similarly for Aperp_n. The errors are basically dominated by the Deuteron Apara and Aperp data from which deuteron xsec differences are calculated. Here is a reference plot of the proton and deuteron asymmetries (made by Karl?), but note that Apara_n is not equal to "Apara_d - Apara_p".
  

• Extracted smeared A1n and A2n (30MeV and 60MeV bins). R1990 has been used to get A1n and A2n. Two resonance peaks for A1_n at higher W are seen on the 60MeV plot. (The arrows indicate where the resonances should show up.) At low W, the QE contribution is so significant that the delta resonance peak (at W~1.2 GeV) is not yet seen.
  

• I received from oscar the proton and deuteron xsec data and plots at RSS central kinematics. When I was comparing Oscar's xsec results with mine, I found a couple of errors in my program which caused a strange behavior at high W as posted on Aug17. After correcting the errors, my results agree with oscar's. Difference is in the order of 0.1% but become 1% at the W edges.
  

• Following Karl's suggestion, I changed the way kinematic quantities are calculated in my program. I extracted thetae,hsna1_h, and hsca1pa_h from the summary file and calculated E'. I also used the average beam energy of 5752.515MeV as found by Karl.

• Results of smeared neutron asymmetries (Apara, Aperp, A1, A2). The error bars are not assigned yet. The resonance structure is not clear. To get A1 and A2, R1990 was used. The electron phi angle was from the perp data.

• Need to check the difference between proton and deuteron kinematics.
  

• Last week, the deuteron cross section I calculated from F1d and F2d were obviously incorrect (because I then got the negative neutron cross section!). This week I used the hallc2h.f routine, which has been used in our MC, to calculate the deuteron cross section. Eric's routines were used to obtain the proton xsec, which has been smeared.
  

• This plot shows the cross sections for deuteron, proton, smeared-proton, and smeared-neutron . Smeared neutron sigma is given by "deuteron sigma minus smeared proton sigma". I also obtained some results for Apara, Aperp, A1, A2 for the neutron, but I need to rebin the results and also calculate the error bars.
  

• I've been working on extracting A1n(w) and A2n(w). But My code has some errors and I've been debugging the code.
  

• Results of smeared w1p and w2p . Using these, smeared proton sigma is calculated. Smeared neutron sigma is given by "deuteron sigma minus smeared proton sigma".

• Results of g1n and g2n using bin sizes of 180MeV for low x (x<0.576 or W>1.37) and 30 MeV otherwise. The error bars at low x becomes comaprable to those at high x for g1n. For g2n, the error bars at low x are still large.
  

• Working on extracting 'smeared' neutron asymmetries A1n(W) and A2n(W) from the smeared neutron cross section and smeared dsigma_para and perp. The detailed procedure is explained in Oscar's emails on 6/30 and 7/17
  

• Note that SLAC reports the 'pre nucleon' structure functions. They used, for ex, the relation g1d ~ (g1p+g1n)/2 (omitting a correction factor) to extract g1n from their g1p and g1d. See for ex, Phys. Lett. B 493 (2000) 19-28, Phys. Lett. B 553 (2003) 18-24
  

• It turnd out that we've been also using the 'per nucleon' F1 to get g1d and g2d from A1d and A2d. So we would need to multiply our g1d and g2d by a factor 2.
  

• Updated results of g1n and g2n using the corrected g1d and g2d data (multiplied by '2' as mentioned above) given to me by Oscar. See his email dated Jul 03 for details. 30MeV bins are used for x>0.45 (W<1.565). Otherwise, bin size is 60MeV. Also plotted are two sets of Hall A data and SLAC E155 data. Dotted lines are our INCORRECT results as of last month. Our updated results of g1n and g2n are consistent with the world data. But note that the RCs are not applied to our data yet.
  

• [Comments received] Both our g1d and g2d diviate from 0 at high x (x>0.8). That may be due to the quasi-elastic tail which has not been subtracted. For low x, better to use much wider bin (say 90 MeV bin or more) to reduce our error bar.
  

•  Extraction of A1d(W) and A2d(W) in progress.
  

• extracted g1n and g2n  from the deuteron and smeared proton data.
  

• Neutron SSFs are quite insensitive to the deuteron models(Paris, Bonn, Hulthen-Yamaguchi). Plot of the three deuteron models versus the fermi momentum shows that they are basically on top of each other. Extracted g1n and g2n using those deuteron models agree very well.
  

• Extraction of g1n and g2n from the deuteron and smeared proton data. We assumed that the relative uncertainty for the proton g1p and g2p remains the same after the smearing.
  

• Distribution of 'off-shell W' (y-axis) versus 'W'(x-axis) is plotted (BOX plot, COLOR plot). Two red lines indicate the W value for pion threshold (lower line) and the highest W from our data (upper line). I'm not sure how to calculate the STATISTICAL ERROR for the smeared  dsigma_para and perp.
  

• Plot of smeared dsigma_para and perp (para:black, perp: red)calculated from our proton data and the Paris WF for the deuteron.