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A New Parameterization of the Nucleon Elastic Form Factors
Nuclear Physics B (Proc. Suppl.) 159 (2006) 127–132 www.elsevierphysics.com
A New Parameterization of the Nu leon Elasti Form Fa tors
R. Bradford,aA. Bodek,a H. Budd,a and J. Arringtonb a Department of Physi s and Astronomy, University of Ro hester, Ro hester NY 14627-0171, USA b Argonne National Laboratory Argonne, IL 60439, USA The nu leon elasti form fa tors are generally interpreted as a mapping of the harge and magneti urrent distributions of the proton and neutron. about
Gep
New high
Q2
measurements have opened up fundamental questions
that remain to be answered. This talk will summarize urrent developments surrounding the nu leon
form fa tors and explain why they are important to neutrino physi ists.
New parameterizations of the nu leon
form fa tors, suitable for use by neutrino physi ists, will be introdu ed and dis ussed.
1. Introdu tion
While the nu leon elasti form fa tors have been measured for 50 years in e N s attering, re ent measurements from Je erson Lab have shown unexpe ted stru ture in the ratio of Gp Gmpep . Understanding the new measurements has been a major fo us of the Jlab ommunity. As the elasti nu leon form fa tors are input for neutrino simulations pa kages, it is important for neutrino physi ists to understand the nu leon form fa tors and the urrent ontroversy. This talk will begin by presenting an overview of the nu leon form fa tors in Se tion 2. Se tion 3 will dis uss two te hniques for measuring the form fa tors and brie y dis uss the new Je erson Lab measurements. Se tion 4 motivates the role of the nu leon form fa tors in neutrino physi s. The talk ends by presenting a new parametrization of the form fa tors in Se tions 5 and 6. 2. Overview
In the single-photon ex hange approximation, the nu leon elasti form fa tors arise in the elasti ele tron-nu leon s attering ross se tion a
ording to: d 2 Ee0 os 2e h 2 2 i 1 = G G + eN " mN d 4Ee3 sin4 2e 1 + b(1) where E is the in ident ele tron energy, E 0 is the s attered ele tron energy, e is the ele tron s at0920-5632/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2006.08.028
tering angle, and = 4QM 2 (with M being the nu leon mass). " = 1 + 2 (1 + ) tan2 2e 1 is the polarization of the ex hange photon mediating the intera tion. GeN is the nu leon ele tri form fa tor, and GmN is the magneti form fa tor. While unique form fa tors exist for both nu leons, this paper will use GeN and GmN when making statements that may be applied to both the proton and neutron. The form fa tors a
ount for the e e ts of the spatial size of the nu leons on the elasti ross se tion. In Equation (1), the overall oeÆ ient e 2 of 4EE3 esin os4 ((2e )) is known as the Mott ross se e 2 tion, whi h is the standard ross se tion for elasti s attering of point-like parti les. The Mott
ross se tion was originally thought to explain elasti ele tron-nu leon s attering. Rosenbluth developed a ross se tion formula in 1950 that introdu ed a \form fa tor" to a
ount for the ase that the nu leon may not be point-like parti le. Rosenbluth's formula ame into wide use after the Mott ross se tion formula failed to explain early measurements from Hofstadter and M Allister [2℄. While there are various parameterizations of the elasti nu leon di erential ross se tion, Equation (1) employs the Sa hs form fa tors. In the non-relativisti limit, these may be interpreted as the Fourier transform of the nu leon
harge and urrent distributions. 2
0
128
R. Bradford et al. / Nuclear Physics B (Proc. Suppl.) 159 (2006) 127–132
3. Experimental Measurements
Over the years, two di erent te hniques for measuring the elasti form fa tors have been developed. The rst, and oldest te hnique, is to perform a Rosenbluth separation, while the \newer" te hnique involves using re oil polarization measurements to extra t the form fa tor ratio. The next two se tions will dis uss ea h type of measurement. 3.1. Early
measurements:
Rosenbluth
Separation
The earliest form fa tor measurements were made in the 1950's using a te hnique known as Rosenbluth separation. Rosenbluth separation takes advantage of Equation (1)'s linear dependen e on ". The idea was fairly straightforward: The elasti ross se tion was measured at various values of " by holding Q2 onstant while varying e . A line was t to the resulting ross se tion, and the t parameters yielded the various form fa tors - the inter ept gave a measurement of G2mN , while the slope yielded G2eN . Early measurements of the form fa tors appeared to be t well by a parameterization known as the dipole form fa tor:
Gd = 1 +
Q2
2
(2) 2 where 2 =0.71 GeV2 . Three of the four form fa tors, Gep , Gmp , and Gmn were well modeled p n by this parameterization. Despite the early su
ess of the Rosenbluth separation, the method did have some weaknesses. Be ause the method involved rst measuring the elasti ross se tion, it was sus eptible to the systemati errors that are inherent in ross se tion measurements. In addition, the method ould only produ e pre ise measurement of GeN below Q2 = 1GeV 2 . At higher Q2 , the GeN term is damped by a fa tor of 1 , as seen in Equation (1). The ross se tion, then, be omes dominated by GmN above Q2 = 1GeV 2 . Finally, the whole formalism rests on the assumption of single photon ex hange. While the measured ross se tions are
orre t for terms beyond one photon ex hange, the two photon ex hange orre tions to the form
Figure 1. Gp Gmpep measured in Hall A at Je erson Lab [3℄. Based on previous measurements, we expe ted Gp Gmpep to be at and 1. However, the new results here based on re oil polarization measurements drop linearly in Q2. The urves are for various model al ulations, referen es to whi h an be found in [3℄. The shaded bar towards the bottom of the plot represents expe ted systemati errors. fa tors are not well understood. Signi ant two photon ontributions to the ross se tion ould undermine the validity of the formalism. 3.2. Polarization Measurements
A se ond method of measuring the elasti form fa tors involves s attering polarized ele trons from the nu leon, and then measuring orthogonal omponents of the nu leon's re oil polarization. The ratio of the polarization omponents is related to the ratio of the ele tri and magneti form fa tors by GeN Pt (Ee + Ee0 ) e = tan (3) GmN Pl 2M 2 ; where Pl and Pt are longitudinal and transverse (with respe t to the nu leon momentum) omponents of the nu leon's polarization. Be ause the measurement involves a ratio of polarization omponents, many of the possible systemati errors traditionally asso iated with
R. Bradford et al. / Nuclear Physics B (Proc. Suppl.) 159 (2006) 127–132
the use of a polarimeter divide out of the measurement. This method is viewed as being systemati ally more robust than measurements made with a Rosenbluth separation. 3.3. Re ent Jlab Measurements
While the re oil polarization measurements have been made sin e the 1970's, it was not until re ently that the two methods were shown to disagree. Figure 1 shows data published in 2000 that was taken in Hall A at Je erson Lab [3℄. The gure shows the ratio Gp Gmpep . Based on experien e from earlier measurements employing the Rosenbluth Separation te hnique, we expe ted this ratio to be approximately one as a fun tion of Q2 . However, the newer data drop o linearly with Q2 . This was a great surprise to the nu lear physi s
ommunity. The result has spawned mu h dialog, s rutiny of older datasets, and a number of additional experiments attempting to verify, refute, or explain the dis repan y. Further re oil polarization results from Jlab using the same experimental setup show that the dis repan y persists at higher values of Q2 [4,5℄, while re-analysis of older datasets show that the Rosenbluth measurements are self- onsistent [6℄. Results from a \SuperRosenbluth" experiment, whi h attempted a pre ise Rosenbluth separation having systemati errors omparable to the re oil polarization experiments, are also onsistent with older measurements based on the Rosenbluth separation method [7℄. While the sour e of the dis repan y remains an open question, urrent e orts fo us on possible two-photon ontributions to the elasti ep
ross se tion [8{11℄. The extent of this e e t is
urrently under investigation. If this me hanism proves to explain the dis repan y, then the resulting errors will be more pronoun ed in the ase of the Rosenbluth form fa tors, making form fa tors based on the polarization transfer te hnique preferred. 4. Neutrino Physi s
The elasti nu leon form fa tors have dire t bearing on neutrino physi s. The ve tor part
129
of the neutrino ross se tion an be expressed in terms of GVE and GVM , the ve tor ele tri and magneti form fa tors. Through the onserved ve tor
urrent hypothesis, these form fa tors may then be related to the elasti nu leon form fa tors measured in elasti eN s attering as shown by: GVE Q2 = Gep Q2 Gen Q2 (4) and GVM Q2 = Gmp Q2 Gmn Q2 : (5) Current neutrino simulation programs use the elasti nu leon form fa tors to parameterize the ve tor part of the elasti A ross se tion. In light of the re ent ontroversy, it is important for neutrino physi ists to understand the state of nu leon form fa tor measurements and realize that there are open questions that are urrently being investigated. Attention must be paid to the eld as it develops over the next fews years, and parameterizations sele ted for used in simulations must be hosen arefully. Ideally, one should employ a parameterization with reasonable onstraints at both low- and high-Q2 . 5. New Parameterizations
The re ent ontroversy has lead physi ists to question the validity of the Gd parameterization. Many new parameterizations have been developed [12,14℄ based on ts to experimental data. We have developed a new parameterization whi h builds on earlier work from our group [12℄ and e orts by Kelly [14℄. The parameterization was developed by tting a single fun tional form for all four elasti form fa tors. Datasets used in the tting were similar to those used by Kelly [14℄, although we did not in lude measurements of mean nu leon radius in our ts. The data emphasized measurements based on the polarization transfer te hnique and ex luded Rosenbluth measurements of Gep above Q2 > 1GeV 2 . For this analysis, we have ex luded the data requiring large two photon ex hange orre tions in the form fa tor extra tion. A more omplete analysis,
urrently underway, would orre t both the ross se tion and polarization results for two photon ex hange, but is beyond the s ope of this work.
R. Bradford et al. / Nuclear Physics B (Proc. Suppl.) 159 (2006) 127–132
Table 1: Fit parameters, given Gen > 0 and ud =0.2. The a0 parameter was used to ensure the orre t low Q2 limit and was not varied during the ts. Observable a0 a1 a2 b1 b2 b3 b4 GEp 1 0:0578 :166 11:1 0:217 13:6 1:39 33:0 8:95 GMp 1 0:150 0:312E 1 11:1 0:103 19:6 0:281 7:54 0:967 GEn 0 1:25 0:368 1:30 1:99 9:86 6:46 305 28:6 758 77:5 802 156 GMn 1 1:81 0:402 14:1 0:597 20:7 2:55 68:7 14:1
130
The fun tional form is given by
G Q2 =
Pn
ak k k: k=1 bk
k=0 P
(6) 1+ While this form has been used by other parameterizations in the past [14℄, this is the rst time that this parti ular form has been employed for all four form fa tors. We will refer to our parameterizations as the \BBBA05" form fa tors throughout the rest of this talk. An additional feature of our new parameterizations is the implementation of two onstraints applied in the tting. The rst onstraint omes from lo al duality. R is de ned as the ratio of form fa tors. In the elasti limit, R takes the form 4 M 2 G2eN 2 R x = 1; Q = (7) N
Q2
G2mN
As Q2 ! 1, Rn = Rp , so our rst onstraint takes the form
Gen Gmn
2
= GGep
2
mp
(8)
A se ond onstraint is based on QCD sum rules and a further appli ation of duality. In the elasti limit, we an express the FF2n ratio as 2p
F 2n F 2p
2
mn = G Gmp
2
(9)
In the ! 1 and Q2 ! 1 and xed x limits, the F2 form fa tor be omes a simple quark- ounting exer ise X F2 = x e2i fi (x) : (10) i
Inserting (10) into (9), we arrive at our se ond
onstraint: Gmn 2 1 + 4 ud = (11) Gmp 4 + ud The value of ud is somewhat subje tive. We a tually ran three sets of ts, ea h with a di erent value of ud =0, 0.2, or 0.5. Our preferred value was ud =0.2.
R. Bradford et al. / Nuclear Physics B (Proc. Suppl.) 159 (2006) 127–132
131
Figure 2. The solid bla k line shows the ratio of the BBBA05 form fa tors to Gd , and the dashed blue line is the ratio of the Kelly form fa tors to Gd . The di eren es in the two parameterizations for GGepd and Gen are due to the onstraints applied to the BBBA05 form fa tors. All gures have a y-axis ranging Gd from Q2 = 0GeV 2 to Q2 = 30GeV 2 . In the lower limit (Q2 = 0GeV 2 ), all ratios approa h unity, ex ept for Gen , whi h approa hes zero. The above onstraints were implemented by s aling the high Q2 data-points of Gmp and then adding these s aled points to the datasets for Gen and Gmn during the ts. The \ onstraint data" are not shown in the gures of this paper. While we initially tried to apply the onstraints expli itly to the t parameters that determined the high Q2 behavior of the less well-measured form fa tors, the onvergen e of the onstraints was very slow. We were looking for something to onverge around Q2 = 30GeV 2 . Using these additional data-points satis ed this riteria. Errors on the \ onstraint data" were in ated to keep these additional points from wielding too strong an in uen e on the ts. Be ause the above onstraints are all on squares of form fa tors, one may argue about different sign onventions that one ould use in the appli ation of the onstraints, parti ularly at high Q2, where many of the form fa tors are poorly measured. At large Q2 values, where Gep and Gen might hange sign, their ontributions to the
neutrino ross se tion are extremely small [13℄. We ran ts with both signs for Gen and preferred the positive Gen in the end. Gen < 0 yielded odd os illatory behavior in the onstraint at the sign
hange. The plots that will be shown in this paper are based on Gen > 0, and with ud = 0:2. The t parameters are shown in Table 1. 6. Plots and dis ussion of new parameterization
Plots of the new parameterizations are shown in Figure 2. As shown in the gure, the BBBA05 parameterizations of GMp and GMn are lose to the Kelly form fa tors. However, the new fun tional form and added high Q2 onstraints ause the BBBA05 parameterization of GEn to die o mu h more qui kly at high Q2 than does the Kelly parameterization. Plots demonstrating the behavior of the onstraints are shown in Figures 3 and 4. The ratios here both appear to satisfy the onstraint at one
132
R. Bradford et al. / Nuclear Physics B (Proc. Suppl.) 159 (2006) 127–132
2
Figure 3. The e e t of onstraining GGmn mp at high Q2 is demonstrated here. GGmn for the mp BBBA05 parameterization (solid bla k line) interse ts the asymptoti value (blue dashed line) at a single point around Q2 = 20GeV 2 . Data points are average ratios of available data a ross bins 250 MeV 2 -wide in Q2 where data exist for the appropriate form fa tors. point around Q2 = 20GeV 2 . This behavior is a related to our implementation of the onstraints as additional data-points (data from Gmp s aled a
ording to the onstraints and added to the t datasets for Gen and Gmn ). Hen e, we laim that our parameterization is valid up to approximately Q2 = 18GeV 2 . 7. A knowledgments
Work supported in part by the US Department of Energy, OÆ e of Nu lear Physi s, under
ontra t W-31-109-ENG-38, and the DOE OÆ e of High Energy Physi s under grant DE-FG0291ER40685. REFERENCES
1. M.N. Rosenbluth, Phys Rev 79, 615 (1950). 2. R.W. M Allister and R. Hofstadter, Phys Rev 102, 851 (1956). 3. M.K. Jones et al, PRL 84, 1398 (2000).
2
ep Figure 4. The two urves represent GGmp (solid en bla k) and GGmn (dashed blue) at high Q2 . Data points are average ratios of available data a ross bins 250 MeV 2 -wide in Q2 where data exist for the appropriate form fa tors.
4. O. Gayou et al, PRC 64, 038202 (2001). 5. O. Gayou et al, PRL 88, 092301 (2002). 6. J. Arrington, PRC 68, 034325 (2003). 7. I.A. Quattan et al, PRL 94, 142301 (2005). 8. J. Arrington, PRC 69, 022201 (2004). 9. J. Arrington, PRC 71, 015202 (2005). 10. P.G. Blunden, W. Melnit houk, and J.A. Tjon, PRL 91, 142304 (2003). 11. P.A.M Gui hon and M. Vanderhaeghen, PRL 91, 142303 (2003). 12. H. Budd, A. Bodek, and J. Arrington, arXiv:hep-ex/0308005 (2003). 13. H. Budd, A. Bodek, and J. Arrington, arXiv:hep-ex/0410055 (2004). 14. J.J. Kelly, PRC 70, 068202 (2004).
A New Parameterization of the Nu leon Elasti Form Fa tors
R. Bradford,aA. Bodek,a H. Budd,a and J. Arringtonb a Department of Physi s and Astronomy, University of Ro hester, Ro hester NY 14627-0171, USA b Argonne National Laboratory Argonne, IL 60439, USA The nu leon elasti form fa tors are generally interpreted as a mapping of the harge and magneti urrent distributions of the proton and neutron. about
Gep
New high
Q2
measurements have opened up fundamental questions
that remain to be answered. This talk will summarize urrent developments surrounding the nu leon
form fa tors and explain why they are important to neutrino physi ists.
New parameterizations of the nu leon
form fa tors, suitable for use by neutrino physi ists, will be introdu ed and dis ussed.
1. Introdu tion
While the nu leon elasti form fa tors have been measured for 50 years in e N s attering, re ent measurements from Je erson Lab have shown unexpe ted stru ture in the ratio of Gp Gmpep . Understanding the new measurements has been a major fo us of the Jlab ommunity. As the elasti nu leon form fa tors are input for neutrino simulations pa kages, it is important for neutrino physi ists to understand the nu leon form fa tors and the urrent ontroversy. This talk will begin by presenting an overview of the nu leon form fa tors in Se tion 2. Se tion 3 will dis uss two te hniques for measuring the form fa tors and brie y dis uss the new Je erson Lab measurements. Se tion 4 motivates the role of the nu leon form fa tors in neutrino physi s. The talk ends by presenting a new parametrization of the form fa tors in Se tions 5 and 6. 2. Overview
In the single-photon ex hange approximation, the nu leon elasti form fa tors arise in the elasti ele tron-nu leon s attering ross se tion a
ording to: d 2 Ee0 os 2e h 2 2 i 1 = G G + eN " mN d 4Ee3 sin4 2e 1 + b(1) where E is the in ident ele tron energy, E 0 is the s attered ele tron energy, e is the ele tron s at0920-5632/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2006.08.028
tering angle, and = 4QM 2 (with M being the nu leon mass). " = 1 + 2 (1 + ) tan2 2e 1 is the polarization of the ex hange photon mediating the intera tion. GeN is the nu leon ele tri form fa tor, and GmN is the magneti form fa tor. While unique form fa tors exist for both nu leons, this paper will use GeN and GmN when making statements that may be applied to both the proton and neutron. The form fa tors a
ount for the e e ts of the spatial size of the nu leons on the elasti ross se tion. In Equation (1), the overall oeÆ ient e 2 of 4EE3 esin os4 ((2e )) is known as the Mott ross se e 2 tion, whi h is the standard ross se tion for elasti s attering of point-like parti les. The Mott
ross se tion was originally thought to explain elasti ele tron-nu leon s attering. Rosenbluth developed a ross se tion formula in 1950 that introdu ed a \form fa tor" to a
ount for the ase that the nu leon may not be point-like parti le. Rosenbluth's formula ame into wide use after the Mott ross se tion formula failed to explain early measurements from Hofstadter and M Allister [2℄. While there are various parameterizations of the elasti nu leon di erential ross se tion, Equation (1) employs the Sa hs form fa tors. In the non-relativisti limit, these may be interpreted as the Fourier transform of the nu leon
harge and urrent distributions. 2
0
128
R. Bradford et al. / Nuclear Physics B (Proc. Suppl.) 159 (2006) 127–132
3. Experimental Measurements
Over the years, two di erent te hniques for measuring the elasti form fa tors have been developed. The rst, and oldest te hnique, is to perform a Rosenbluth separation, while the \newer" te hnique involves using re oil polarization measurements to extra t the form fa tor ratio. The next two se tions will dis uss ea h type of measurement. 3.1. Early
measurements:
Rosenbluth
Separation
The earliest form fa tor measurements were made in the 1950's using a te hnique known as Rosenbluth separation. Rosenbluth separation takes advantage of Equation (1)'s linear dependen e on ". The idea was fairly straightforward: The elasti ross se tion was measured at various values of " by holding Q2 onstant while varying e . A line was t to the resulting ross se tion, and the t parameters yielded the various form fa tors - the inter ept gave a measurement of G2mN , while the slope yielded G2eN . Early measurements of the form fa tors appeared to be t well by a parameterization known as the dipole form fa tor:
Gd = 1 +
Q2
2
(2) 2 where 2 =0.71 GeV2 . Three of the four form fa tors, Gep , Gmp , and Gmn were well modeled p n by this parameterization. Despite the early su
ess of the Rosenbluth separation, the method did have some weaknesses. Be ause the method involved rst measuring the elasti ross se tion, it was sus eptible to the systemati errors that are inherent in ross se tion measurements. In addition, the method ould only produ e pre ise measurement of GeN below Q2 = 1GeV 2 . At higher Q2 , the GeN term is damped by a fa tor of 1 , as seen in Equation (1). The ross se tion, then, be omes dominated by GmN above Q2 = 1GeV 2 . Finally, the whole formalism rests on the assumption of single photon ex hange. While the measured ross se tions are
orre t for terms beyond one photon ex hange, the two photon ex hange orre tions to the form
Figure 1. Gp Gmpep measured in Hall A at Je erson Lab [3℄. Based on previous measurements, we expe ted Gp Gmpep to be at and 1. However, the new results here based on re oil polarization measurements drop linearly in Q2. The urves are for various model al ulations, referen es to whi h an be found in [3℄. The shaded bar towards the bottom of the plot represents expe ted systemati errors. fa tors are not well understood. Signi ant two photon ontributions to the ross se tion ould undermine the validity of the formalism. 3.2. Polarization Measurements
A se ond method of measuring the elasti form fa tors involves s attering polarized ele trons from the nu leon, and then measuring orthogonal omponents of the nu leon's re oil polarization. The ratio of the polarization omponents is related to the ratio of the ele tri and magneti form fa tors by GeN Pt (Ee + Ee0 ) e = tan (3) GmN Pl 2M 2 ; where Pl and Pt are longitudinal and transverse (with respe t to the nu leon momentum) omponents of the nu leon's polarization. Be ause the measurement involves a ratio of polarization omponents, many of the possible systemati errors traditionally asso iated with
R. Bradford et al. / Nuclear Physics B (Proc. Suppl.) 159 (2006) 127–132
the use of a polarimeter divide out of the measurement. This method is viewed as being systemati ally more robust than measurements made with a Rosenbluth separation. 3.3. Re ent Jlab Measurements
While the re oil polarization measurements have been made sin e the 1970's, it was not until re ently that the two methods were shown to disagree. Figure 1 shows data published in 2000 that was taken in Hall A at Je erson Lab [3℄. The gure shows the ratio Gp Gmpep . Based on experien e from earlier measurements employing the Rosenbluth Separation te hnique, we expe ted this ratio to be approximately one as a fun tion of Q2 . However, the newer data drop o linearly with Q2 . This was a great surprise to the nu lear physi s
ommunity. The result has spawned mu h dialog, s rutiny of older datasets, and a number of additional experiments attempting to verify, refute, or explain the dis repan y. Further re oil polarization results from Jlab using the same experimental setup show that the dis repan y persists at higher values of Q2 [4,5℄, while re-analysis of older datasets show that the Rosenbluth measurements are self- onsistent [6℄. Results from a \SuperRosenbluth" experiment, whi h attempted a pre ise Rosenbluth separation having systemati errors omparable to the re oil polarization experiments, are also onsistent with older measurements based on the Rosenbluth separation method [7℄. While the sour e of the dis repan y remains an open question, urrent e orts fo us on possible two-photon ontributions to the elasti ep
ross se tion [8{11℄. The extent of this e e t is
urrently under investigation. If this me hanism proves to explain the dis repan y, then the resulting errors will be more pronoun ed in the ase of the Rosenbluth form fa tors, making form fa tors based on the polarization transfer te hnique preferred. 4. Neutrino Physi s
The elasti nu leon form fa tors have dire t bearing on neutrino physi s. The ve tor part
129
of the neutrino ross se tion an be expressed in terms of GVE and GVM , the ve tor ele tri and magneti form fa tors. Through the onserved ve tor
urrent hypothesis, these form fa tors may then be related to the elasti nu leon form fa tors measured in elasti eN s attering as shown by: GVE Q2 = Gep Q2 Gen Q2 (4) and GVM Q2 = Gmp Q2 Gmn Q2 : (5) Current neutrino simulation programs use the elasti nu leon form fa tors to parameterize the ve tor part of the elasti A ross se tion. In light of the re ent ontroversy, it is important for neutrino physi ists to understand the state of nu leon form fa tor measurements and realize that there are open questions that are urrently being investigated. Attention must be paid to the eld as it develops over the next fews years, and parameterizations sele ted for used in simulations must be hosen arefully. Ideally, one should employ a parameterization with reasonable onstraints at both low- and high-Q2 . 5. New Parameterizations
The re ent ontroversy has lead physi ists to question the validity of the Gd parameterization. Many new parameterizations have been developed [12,14℄ based on ts to experimental data. We have developed a new parameterization whi h builds on earlier work from our group [12℄ and e orts by Kelly [14℄. The parameterization was developed by tting a single fun tional form for all four elasti form fa tors. Datasets used in the tting were similar to those used by Kelly [14℄, although we did not in lude measurements of mean nu leon radius in our ts. The data emphasized measurements based on the polarization transfer te hnique and ex luded Rosenbluth measurements of Gep above Q2 > 1GeV 2 . For this analysis, we have ex luded the data requiring large two photon ex hange orre tions in the form fa tor extra tion. A more omplete analysis,
urrently underway, would orre t both the ross se tion and polarization results for two photon ex hange, but is beyond the s ope of this work.
R. Bradford et al. / Nuclear Physics B (Proc. Suppl.) 159 (2006) 127–132
Table 1: Fit parameters, given Gen > 0 and ud =0.2. The a0 parameter was used to ensure the orre t low Q2 limit and was not varied during the ts. Observable a0 a1 a2 b1 b2 b3 b4 GEp 1 0:0578 :166 11:1 0:217 13:6 1:39 33:0 8:95 GMp 1 0:150 0:312E 1 11:1 0:103 19:6 0:281 7:54 0:967 GEn 0 1:25 0:368 1:30 1:99 9:86 6:46 305 28:6 758 77:5 802 156 GMn 1 1:81 0:402 14:1 0:597 20:7 2:55 68:7 14:1
130
The fun tional form is given by
G Q2 =
Pn
ak k k: k=1 bk
k=0 P
(6) 1+ While this form has been used by other parameterizations in the past [14℄, this is the rst time that this parti ular form has been employed for all four form fa tors. We will refer to our parameterizations as the \BBBA05" form fa tors throughout the rest of this talk. An additional feature of our new parameterizations is the implementation of two onstraints applied in the tting. The rst onstraint omes from lo al duality. R is de ned as the ratio of form fa tors. In the elasti limit, R takes the form 4 M 2 G2eN 2 R x = 1; Q = (7) N
Q2
G2mN
As Q2 ! 1, Rn = Rp , so our rst onstraint takes the form
Gen Gmn
2
= GGep
2
mp
(8)
A se ond onstraint is based on QCD sum rules and a further appli ation of duality. In the elasti limit, we an express the FF2n ratio as 2p
F 2n F 2p
2
mn = G Gmp
2
(9)
In the ! 1 and Q2 ! 1 and xed x limits, the F2 form fa tor be omes a simple quark- ounting exer ise X F2 = x e2i fi (x) : (10) i
Inserting (10) into (9), we arrive at our se ond
onstraint: Gmn 2 1 + 4 ud = (11) Gmp 4 + ud The value of ud is somewhat subje tive. We a tually ran three sets of ts, ea h with a di erent value of ud =0, 0.2, or 0.5. Our preferred value was ud =0.2.
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Figure 2. The solid bla k line shows the ratio of the BBBA05 form fa tors to Gd , and the dashed blue line is the ratio of the Kelly form fa tors to Gd . The di eren es in the two parameterizations for GGepd and Gen are due to the onstraints applied to the BBBA05 form fa tors. All gures have a y-axis ranging Gd from Q2 = 0GeV 2 to Q2 = 30GeV 2 . In the lower limit (Q2 = 0GeV 2 ), all ratios approa h unity, ex ept for Gen , whi h approa hes zero. The above onstraints were implemented by s aling the high Q2 data-points of Gmp and then adding these s aled points to the datasets for Gen and Gmn during the ts. The \ onstraint data" are not shown in the gures of this paper. While we initially tried to apply the onstraints expli itly to the t parameters that determined the high Q2 behavior of the less well-measured form fa tors, the onvergen e of the onstraints was very slow. We were looking for something to onverge around Q2 = 30GeV 2 . Using these additional data-points satis ed this riteria. Errors on the \ onstraint data" were in ated to keep these additional points from wielding too strong an in uen e on the ts. Be ause the above onstraints are all on squares of form fa tors, one may argue about different sign onventions that one ould use in the appli ation of the onstraints, parti ularly at high Q2, where many of the form fa tors are poorly measured. At large Q2 values, where Gep and Gen might hange sign, their ontributions to the
neutrino ross se tion are extremely small [13℄. We ran ts with both signs for Gen and preferred the positive Gen in the end. Gen < 0 yielded odd os illatory behavior in the onstraint at the sign
hange. The plots that will be shown in this paper are based on Gen > 0, and with ud = 0:2. The t parameters are shown in Table 1. 6. Plots and dis ussion of new parameterization
Plots of the new parameterizations are shown in Figure 2. As shown in the gure, the BBBA05 parameterizations of GMp and GMn are lose to the Kelly form fa tors. However, the new fun tional form and added high Q2 onstraints ause the BBBA05 parameterization of GEn to die o mu h more qui kly at high Q2 than does the Kelly parameterization. Plots demonstrating the behavior of the onstraints are shown in Figures 3 and 4. The ratios here both appear to satisfy the onstraint at one
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Figure 3. The e e t of onstraining GGmn mp at high Q2 is demonstrated here. GGmn for the mp BBBA05 parameterization (solid bla k line) interse ts the asymptoti value (blue dashed line) at a single point around Q2 = 20GeV 2 . Data points are average ratios of available data a ross bins 250 MeV 2 -wide in Q2 where data exist for the appropriate form fa tors. point around Q2 = 20GeV 2 . This behavior is a related to our implementation of the onstraints as additional data-points (data from Gmp s aled a
ording to the onstraints and added to the t datasets for Gen and Gmn ). Hen e, we laim that our parameterization is valid up to approximately Q2 = 18GeV 2 . 7. A knowledgments
Work supported in part by the US Department of Energy, OÆ e of Nu lear Physi s, under
ontra t W-31-109-ENG-38, and the DOE OÆ e of High Energy Physi s under grant DE-FG0291ER40685. REFERENCES
1. M.N. Rosenbluth, Phys Rev 79, 615 (1950). 2. R.W. M Allister and R. Hofstadter, Phys Rev 102, 851 (1956). 3. M.K. Jones et al, PRL 84, 1398 (2000).
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ep Figure 4. The two urves represent GGmp (solid en bla k) and GGmn (dashed blue) at high Q2 . Data points are average ratios of available data a ross bins 250 MeV 2 -wide in Q2 where data exist for the appropriate form fa tors.
4. O. Gayou et al, PRC 64, 038202 (2001). 5. O. Gayou et al, PRL 88, 092301 (2002). 6. J. Arrington, PRC 68, 034325 (2003). 7. I.A. Quattan et al, PRL 94, 142301 (2005). 8. J. Arrington, PRC 69, 022201 (2004). 9. J. Arrington, PRC 71, 015202 (2005). 10. P.G. Blunden, W. Melnit houk, and J.A. Tjon, PRL 91, 142304 (2003). 11. P.A.M Gui hon and M. Vanderhaeghen, PRL 91, 142303 (2003). 12. H. Budd, A. Bodek, and J. Arrington, arXiv:hep-ex/0308005 (2003). 13. H. Budd, A. Bodek, and J. Arrington, arXiv:hep-ex/0410055 (2004). 14. J.J. Kelly, PRC 70, 068202 (2004).