Electromagnetic Form Factors of Nucleons with QCD Constraints Sytematic Study of the Space and Time-like Regions
aa r X i v : . [ h e p - ph ] S e p Electromagnetic Form Factors of Nucleons with QCDConstraints
Sytematic Study of the Space and Time-like Regions
Susumu FURUICHI ∗ Department of Physics, Rikkyo University, Toshima Tokyo 171-8501, JapanHirohisa ISHIKAWA † Department of EconomyCMeikai University, Urayasu Chiba, 279-8550, JapanKeiji WATANABE ‡ Department of Physics, Meisei University, Hino Tokyo 191-8506, Japan
Abstract
Elastic electromagnetic form factors of nucleons are investigated both for the time-like and the space-like momentums under the condition that the QCD constraints aresatisfied asymptotically. The unsubtracted dispersion relation with the superconver-gence conditions are used as a realization of the QCD conditions. The experimentaldata are analyzed by using the dispersion formula and it is shown that the calculatedform factors reproduce the experimental data reasonably well.
We have investigated the nucleon electromagnetic form factors by using the dispersionrelation, which worked in understanding the low energy data very well. We were ableto realize the low energy experimental data for the space-like momentum transfer [1]-[3].For the low momentum, the vector dominance model is qualitatively valid in explainingthe electromagnetic form factors of nucleons except for the ρ meson mass, which shouldbe taken much smaller than the experimental value. The problem of the ρ meson masswas solved by taking into account the uncorrelated two pion contribution. The dispersionrelation turned out to be very effective for this purpose. ∗ Present address: Sengencho 3-2-6, Higashikurume Tokyo 203-0012 † e-mail address: [email protected] ‡ e-mail address [email protected]; Present address: Akazutumi, 5-36-2, Setagaya Tokyo156-0044 G NM ( t ) and G NE ( t ) respectively, are proportional to the dipole formula G D ( t ) = 1 / (1 + | t | / . ,where t is the squared space-like momentum transfer expressed in the unit of (GeV /c ) . The dipole formula represents the experimental data of form factors fairly well for largerange of momentum.To be precise, the experimental data, however, decrease more rapidly than the dipoleformula. This is compatible with the prediction of perturbative QCD (PQCD), where theelastic form factors of hardons decrease asymptotically for large squared momentum ascompared with the dipole formula [6]: For the boson form factors, F ( t ) → const / ln | t | for | t | → ∞ and for the nucleon form factors, F N and F N , the charge and magneticmoment form factors, respectively, decrease as F N ( t ) → const t − (ln | t | ) − γ and F N ( t ) → const t − (ln | t | ) − γ with γ ( ≥
2) being a constant. Consequently, G NM and G NE decrease as t − (ln | t | ) − γ .We showed that the QCD conditions are incorporated by assuming superconvergentdispersion relation (SCDR) for the form factors and investigated the pion and kaon elec-tromagnetic form factors. It was shown that the experimental data are reproduced bothfor the space-like and the time-like momentum [12, 13, 14].Different from the boson form factors, for the nucleons the unphysical regions of ab-sorptive parts are not observed for s < m , with m being the nucleon mass. This makesthe problem of the nucleon form factors difficult, as was observed by R. Wilson in his re-view article [4], in which he emphasized the importance of systematic study of the time-likeand space-like regions for the nucleon form factors.It is the purpose of this paper to examine the nucleon electromagnetic form factors tosee if it is possible to realize the experimental data with the QCD constraints satisfied,where the space-like and time-like regions are treated on equal footing in the chi squareanalysis.Organization of this paper is given as follows: In Sec.2 we summarize on the dispersionrelation for the nucleon electromagnetic form factors with the QCD condition imposed. InSec.3, imaginary parts of the form factors are given for the low, intermediate and asymp-totic regions. In Sec.4 remarks conserning numerical analysis are given and the numericalresults are summarized in Sec.5. The final section is devoted to general discussions. As is mentioned in Sec.1, the electromagnetic form factors approach zero asymptoticallyfor t → ∞ . Therefore, we may assume the unsubtracted dispersion relations for the chargeand magnetic moment form factors F I and F I , respectively, with I denoting the isospinstate I = 0 ,
1. That is, F Ii = 1 π Z ∞ t dt ′ Im F Ii ( t ′ ) t ′ − t , (1) We adopt the convention t < t > t = − Q for the space-like momentum and t = s > t = 4 µ . Here µ is the pion mass being taken as the average ofthe neutral and charged pions.We briefly summarize the asymptotic theorems which are used to incorporate theconstraints of PQCD [6], where the proof is given in Ref.[12]. Let F ( t ) satisfy the dispersionrelation (1), and Im F be given asIm F ( t ′ ) = c [ln( t ′ /Q )] γ + O (cid:18) t ′ /Q )] γ +1 (cid:19) (2)for t → ∞ with γ >
1. Then F ( t ) approaches F ( t ) = 1 π Z ∞ t dt ′ c ( t ′ − t )[ln( t ′ /Q )] γ → cπ ( γ −
1) ln( | t | /Q )] γ − (3)for t → ±∞ .Generally, if Im F ( t ) tends to t ′ n +1 Im F ( t ′ ) → c [ln( t ′ /Q )] γ + O (cid:18) t ′ /Q )] γ +1 (cid:19) (4)for t → ±∞ and the superconvergence conditions Z ∞ t dt ′ t ′ k Im F ( t ′ ) = 0 , k = 0 , , · · · , n, (5)are satisfied, F ( t ) given by (1) approaches for t → ±∞ F ( t ) = 1 π Z ∞ t dt ′ Im F ( t ′ ) t ′ − t → t n +1 cπ ( γ − | t | /Q ) , (6)which can be proved by using (3), (4) and (5) together with the identity1 t ′ − t = − t n t ′ t + · · · + (cid:16) t ′ t (cid:17) n o + 1 t n +1 t ′ n +1 t ′ − t . The QCD constraints for the nucleon form factors are, therefore, attained by assumingthe unsubtracted dispersion relation and the superconvergence conditions for Im F Ii π Z ∞ t dt ′ Im F I ( t ′ ) = 1 π Z ∞ t dt ′ t ′ Im F I ( t ′ ) = 0 , (7)1 π Z ∞ t dt ′ Im F I ( t ′ ) = 1 π Z ∞ t dt ′ t ′ Im F I ( t ′ )= 1 π Z ∞ t dt ′ t ′ Im F I ( t ′ ) = 0 , (8)where Im F Ii ( t ′ ) satisfies the asymptotic conditions for t ′ → ∞ t ′ i Im F Ii ( t ′ ) → const / [ln( t ′ /Q )] γ ( i = 1 , . (9)3 and γ ( ≥
2) are constants, the latter of which is written in terms of the anomalousdimension of the renormalization group in QCD.In addition to the conditions (7) and (8) we impose the normalization conditions at t = 0: 12 = 1 π Z ∞ t dt ′ Im F I ( t ′ ) /t ′ , (10) g I = 1 π Z ∞ t dt ′ Im F I ( t ′ ) /t ′ , (11)where g I are the anomalous moments of nucleons with the isospin I . Let us discuss the imaginary parts of nucleon form factors, which are broken up into threeparts: The low momentum, the intermediate, and the asymptotic regions.
The imaginary parts of the charge and magnetic moment form factors, Im F Vi , are givenin terms two pion contributionsIm[ F V ( t ) /e ] = m t − µ )4 m − t (cid:18) t − µ t (cid:19) / × Re h M ∗ ( t ) n f ( − )1+ ( t ) − t m m √ f ( − )1 − ( t ) oi , (12)Im[2 mF V ( t ) /e ] = m t − µ )(4 m − t ) (cid:18) t − µ t (cid:19) / × Re h M ∗ n m √ f ( − )1 − ( t ) − f ( − )1+ ( t ) oi , (13)where f ( − )1 ± ( t ) are helicty amplitudes for ππ ↔ N ¯ N , M ( t ) is the pion form factor and µ is the pion mass. For the helicity amplitudes we use the numerical values given by H¨ohlerand Schopper [5] and parameterize M ( t ) according to them. M ( t ) = t ρ { ρ /m ρ d ) } [ t ρ − t − im ρ Γ ρ ( q t /q ρ ) √ t ] − , (14)where m ρ and Γ ρ are the ρ meson mass and width respectively and t ρ = m ρ , q ρ = q t ρ − µ , d = 3 µ πt ρ ln m ρ + 2 q ρ µ + m ρ πq ρ (cid:16) − µ t ρ (cid:17) . (15)The imaginary parts thus obtained are denoted as Im F Hi ( i = 1 ,
2) hereafter. It must beremarked that the ρ meson contribution is included in the helicity amplitudes of Ref.[5].4 .2 Intermediate region The intermediate states 4 µ ≤ t ≤ Λ are approximated by the addition of the Breit-Wigner terms. with the imaginary part parameterized as follow:Im f BWR ( t ) = g ( t − M R ) + g , (16)where g = Γ M R ( M R + t res ) t res ( M R − t ) / r ( t − t ) t t ( t + t res ) , (17)where M R and Γ are the mass and width of resonance, respectively. t is the threshold t = 4 µ and t res , being treated as an adjustable parameter, is introduced to cut-off theBreit-Wigner formula.We write the intermediate part as the summation of resonancesIm F BW,Ii = X n a I,in f InR , (18)where I is the isospin and n is the labeling of resonances (see Table I). Here the suffix i denotes i = 1 ,
2, corresponding to the charge and magnetic moment form factors F N and F N . The same formulas for f InR are used for i = 1 and i = 2. To calculate the absorptive part of form factors for the asymptotic region, we need therunning coupling constant α for the time-like momentum. We perform the analytic contin-uation to the time-like momentum by assuming the dispersion relation for α ; applicationof the so called analytic regularization [7]-[9].Let α S ( Q ) be the running coupling constant calculated by the perturbative QCD asa function of the squared momentum Q for the space-like momentum expressed in thePade form. α S ( Q ) = 4 πβ h ln( Q / Λ ) + a ln { ln( Q / Λ ) } + a ln { ln( Q / Λ ) } ln( Q / Λ ) + a ln( Q / Λ ) + · · · i − . (19)Here Λ is the QCD scale parameter, and a i are given in terms of the β function of QCD, a = 2 β /β , a = 4 β β , a = 4 β β (cid:18) − β β β (cid:19) , (20)where β = 11 − n f , β = 51 − n f , β = 2357 − n f + 32527 n f (21)5ith n f being the number of flavor. We perform the analytic continuation of the squaredmomentum Q to the time-like region, s , by the replacement in (19) Q → e − iπ s. (22)Then the effective coupling constant becomes complex; α S ( q ) = Re[ α S ( s )] + i Im[ α S ( s )]with Re[ α S ( s )] = 4 πuβ D ( s ) , (23)Im[ α S ( s )] = 4 πvβ D ( s ) . (24)We write α S ( s ) = 1 / ( u − iv ) = u + ivD ,D = u + v , where u = ln( s/ Λ ) + a { ln ( s/ Λ ) + π } + a ln ( s/ Λ ) + π h
12 ln( s/ Λ ) ln { ln ( s/ Λ ) + πθ } i + a ln( s/ Λ )ln ( s/ Λ ) + π , (25) v = π + a θ − a ln ( s/ Λ ) + π h π { ln ( s/ Λ ) + π } − θ ln( s/ Λ ) i − πa ln ( s/ Λ ) + π , (26)and θ = tan − { π/ ln( s/ Λ ) } . (27)The running coupling constant is given by the dispersion integral both for the space-likeand the time-like momentum α R ( t ) = Z ∞ Q dt ′ σ ( t ′ ) t ′ − t (28)with σ ( t ′ ) = 4 πv/β D. (29) α R ( t ) represented by (28) is called analytically regularized running coupling constant asit has no singular point for t <
0. The regularization eliminates the ghost pole of α S ( Q )appearing at the point Q = Q ∗ = Λ e u ∗ , (30)6here u ∗ = 0 . · · · for the number of flavor n f = 3. Calculating (28), we find that α R ( t ) is approximately given by the simple formula with the ghost pole subtracted α R ( t ) ≈ α S ( Q ) − A ∗ / ( Q − Q ∗ ) , (31)where the residue A ∗ is A ∗ = 4 π Λ e u ∗ / n β (cid:16) a u ∗ − a ln u ∗ u ∗ + a − a u ∗ (cid:17)o . (32)We use (31) as the regularized coupling constant; for the time-like momentum we replace Q = e − iπ s as was mentioned before.The QCD parts, F QCD, Ii ( i = 1 , I = 0,1) are written as follows: F QCD, Ii ( t ) = ˆ F QCD, Ii ( t ) h i ( t ) , (33)where ˆ F QCD, Ii ’s are given as expansion in terms of the running coupling constantˆ F QCD, Ii ( t ) = X j ≥ c QCD, Ij { α R ( t ) } j (34)for the space-like momentum ( t < h i ( t ) in (33) to assurethe convergence of the superconvergence conditions (7) and (8). The following formula isassumed for h i ( t ): h i ( t ) = (cid:18) t − t Q t + t (cid:19) / (cid:18) t t + t (cid:19) i +1 ( i = 1 , , (35)which may be interpreted as the form factor for γ → q ¯ q with t Q being the threshold of thequark antiquark pair. The parameters t Q , t and t are taken as adjustable parametersand will be determined by the analysis of experimental data.For the time-like momentum ( t > α R ( t ) to α R ( s ) through the equation α R ( s ) = α R ( Q e − iπ ) = Re[ α R ( s )] + i Im[ α ( s )] . (36)We take three loop approximation for the effective coupling constant and express the QCDpart as follows: ˆ F QCD,Ii ( s ) = X ≤ j ≤ c QCD, I i,j { α R ( t ) } j . (37)The summation in (37) begins in the second order in the effective coupling constant so asto realize the logarithmic decrease of the nucleon form factors.Imaginary part of (37) is obtained to beIm ˆ F QCD,Ii = 2 c QCD, I i, Re α R Im α R + c QCD, I i, [3(Re α R ) Im α R − (Im α R ) ]+ c QCD, Ii, [4(Re α R ) Im α R − α R (Im α R ) ] + · · · , (38)7nd Im F QCD, Ii ( s ) = Im ˆ F QCD Ii ( s ) h i ( s ).We write the low energy part, intermediate resonance part and asymptotic QCD partsof form factors as F H i , F BW,Ii and F QCD, Ii , which are given by the dispersion integral withthe imaginary parts (13), (18) and (38), respectively. The form factors F Ii are definedby adding them up. We impose the conditions (7) and (8) on Im F Ii so that the QCDconditions are satisfied. We analyzed the experimental data of nucleon electro-magnetic form factors for the space-like momentum G pM /µ p G D , G pE /G D , G nM /µ n G D , G nE and the ratio µ p G pE /G pM and for thetime-like momentum | G p | and | G n | . The parameters in the form factors are determined sothat the calculated results realize the experimental data. In addition to the data used inour previous analysis [3], [10] we used the data in Refs.[15]- [38]. In our analysis we treatboth of space-like and time-like regions on equal footing in the chi square analysis.Space-like region:For G NM /µ N and G NE we used the ratio to the dipole formula G D as we have done inRefs.[2], [3], [10]. In addition to them we take into account in the chi square analysis thedata of the ratio µ p G pE /G pM obtained by the polarization experiments.Time-like region:Experimentally, the proton and neutron form factors | G p | and | G n | for the time-likemomentum are obtained by using the formula for the cross section σ for the processes e + ¯ e → N + ¯ N or N + ¯ N → e + ¯ e , which is given as σ = 4 πα ν s (cid:18) m N s (cid:19) | G ( s ) | , (39)where α is the fine structure constant, m N and ν are the mass and velocity of the nucleon N , respectively. | G NM | are estimated from | G | under the assumption G M = G E or G E = 0[37]. σ is now expressed in terms of G NM and G NE σ = 4 πα ν s (cid:18) | G NM | + 2 m N s | G NE | (cid:19) . (40)Equating (39) and (40), we have | G | = | G NM | + 2 m N | G NE | /s m N /s . (41)Substituting our calculated result of form factors to the right hand side of (41), we get thetheoretical value for | G | , which is compared with the experimental data for the magneticform factor obtained under the assumption G M = G E .8he parameters appearing in our analysis are the following:Residues at resonances, coefficients appearing in the expansion by the QCD effective cou-pling constants, cut-offs for the low, intermediate and asymptotic regions Λ , Λ and Λ ,respectively. In addition to them we have parameters in the Breit-Wigner formula andthe convergence factor h of QCD contribution, t , t res , t , t , t .We have taken the masses and the widths of resonances as adjustable parameters. Asthe superconvergence constraints impose very stringent conditions on the form factors, itwas necessary to take the masses and widths as parameters. We give in Table I and II the results for the parameters; in Table I the masses and widthsof resonances and residues at resonance poles obtained by our analysis and in Table IIthe coefficients c QCD, Ii,j ( i = 1 , j = 2 , , I = 0 ,
1) in the expansion in terms of theeffective coupling constant α R of QCD defined by (37).The value of χ is obtained to be χ tot = 267 .
0, which includes both the data of space-likeand time-like regions. The χ value for the space-like data is χ space = 233 .
3. The totalnumber of data used in the chi square analysis is 209 and the number of parameters is 36.Therefore, χ tot /Df = 1 .
54. For the time-like momentum, the data of Ablikim et al. [37](2005) is systematically smaller than that of Antonelli et al. [38] (1998). In the presentanalysis we restricted ourselves to the (2005) data. We were able to get much better resultboth for the time-like and for the space-like part than the result obtained by the restrictionto (1998) data.We summarize the parameters obtained by the chi square analyis in Table I and II.isospin
I n mass (GeV) width (GeV) a I,n (GeV ) a I,n (GeV )1 1.367 0.324 -6.7 1.062 1.376 0.220 9.562613 -17.81168 I = 1 3 1.6096 0.26 -8.323391 11.180874 1.832 0.381 5.746830 -5.9410325 2.320 0.430 -0.40 0.401 0.78256 0.844 × − × − -3.302706 0.5371316 I = 0 3 1.227 0.1609 0.6303140 × -1.9816664 1.472 0.2123 -1.734902 -2.5896605 1.530 0.1416 -2.460598 4.147711Table I Parameters obtained by the analysis. Residues at resonances.9 −2 −1 G M p / µ p G D −2 −1 G E p / G D Fig.1 Proton form factor for the space-like momentum: Magnetic and electric form factors.isospin
I i c
QCD, Ii, c QCD, Ii, c QCD, Ii, I = 1 1 -2.403217 0.220 × -4.402 5.072373 -0.510 × × I = 0 1 2.964186 -0.195 × -0.687 × × -0.1982 × Table II Coefficients of expansion in terms of the effectiveof coupling constant QCD defined in (37).The parameters t , t , t res , t Q and Λ are determined as follows: t = 0 . × (GeV / c) , t = 0 . × (GeV / c) , t res = 0 . × (GeV / c) and BW cut = Λ = 0 . × GeV/c. QCD threshold = p t Q = 0 . × GeV / c.We take the number of flavor as n f = 3 and the QCD scale parameter Λ QCD = 0 .
213 GeV.The calculated results are illustrated in Figs.1-4; In Figs.1, 2 we give the results for thespace-like momentum for G pM /µ p , G pE , G nM /µ n , G nE and in Fig.3 the ratio of electric andmagnetic form factors of proton µ p G pE /G pM . In Fig.4 the results for the time-like momen-tum are illustrated for the proton and neutron form factors | G p | and | G n | , respectively.Experimental data are taken from [15]-[38].10 −2 −1 G M n / µ n G D G E n Fig.2 Neutron form factors for the space-like momentum: Magnetic and electric formfactors. G E p / µ p G M p Fig.3 Ratio of the electric and magnetic form factors of proton for the space-like momen-tum. 11 ) | G p | Bardin et al. (1994)Antonelli et al. (1998)Ablikim et al. (2005) | G n | Fig.4 Nucleon form factors for the time-like momentum: the proton and neutron formfactors.
We have demonstrated that our superconvergent dispersion relation works in synthesizingthe low and the high momentum parts of nucleon electromagnetic form factors for thespace-like and time-like momentums as we did for the bosons.For the space-like momentum we were able to reproduce the experimental data, but forthe time-like momentum we did not have very good results. If we restrict ourselves only tothe data of space-like momentum, leaving out the time-like data in the chi square analysis,the result for the space-like momentum is improved a little; we have χ = 217. By usingthe parameters thus determined, we calculated the time-like part | G | , which turned out tobe very large; the value of chi square became as large as χ = 1 . × . Incorporationof the data for the time-like momentum seems to be necessary in the systematic study ofspace-like and time-like momentum, although the number of data is limited.We used the experimental data for the helicity amplitudes obtained by H¨oher andSchopper in which the contribution from the ρ meson is included. As their data are limitedto low t ( ≤ . ), we do not have sufficient data for the region s ≤ m N . Wesupplemented the unphysical region for I = 1 state by introducing vector bosons with thesmall mass, m V < ∼ . . For the isoscalar state we also introduced a vector bosonwith small mass.In our calculation we treated all of the vector boson masses and widths as parame-ters. If they are kept at experimental values, we get poor results. The superconvergenceconditions are so strong that the value of χ is very sensitive to the mass and width. Themasses are obtained to be smaller than the experimental value and the existence of vectorbosons with the masses around 1.2 ∼ are implied.12o conclude the paper we remark on the mass about 1.2 GeV/c . Both for I = 0 and I = 1 states there are indications of resonances observed by the processes e + e − → ηπ + π − , γp → ωπ p and B → D ∗ ωπ − [39]. Incorporation of further resonances may improve re-sults for the time-like momentum.The authors wish to express gratitude to Professor M. Ishida for the valuable discus-sions and comments. We also would like to thank Professor T. Komada for the informationon the vector bosons with the mass around 1.2 GeV/c . References [1] S. Furuichi, H. Kanada and K. Watanabe, Prog. Theor. Phys. (1969), 861.[2] S. Furuichi and K. Watanabe, Prog. Theor. Phsy. (1990), 565.[3] S. Furuichi and K. Watanabe, Prog. Theor. Phys. (1990), 1188.[4] R. Wilson, Phys. Today (1967), 47.[5] G. H¨ohler and H.H. Schopper, Landolt-B¨ornstein, I/9b 2, (1983).[6] S.J.Brodsky and G.R.Farrar, Phys. Rev. Lett. (1973) 1153; Phys. Rev. D (1975)1309; G.P.Lepage and S.J.Brodsky, Phys. Rev. D (1980) 2157.[7] H. F. Jones and I. L. Solovstov, Phys. Lett. B (1995), 519.[8] Yu. L. Dokshitzer and B. R. Webber, Phys. Lett. B (1995), 451.[9] Yu. L. Dokshitzer, G. Marchesini and B. R. Webber, Nucl. Phys. B (1996), 93.[10] S. Furuichi and K. Watanabe, Prog. Theor. Phys. (1994), 339.[11] S. Furuichi and K. Watanabe, Nuovo Cimento A 577 (1997).[12] M. Nakagawa and K. Watababe, Nouvo Cimento
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