Nucleon form factors and a nonpointlike diquark

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Bloch, J. C. R.; Roberts, C. D.; Schmidt, S. M.; Bender, Axel; Frank, M. R.

© 1999 American Physical Society

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March 2013

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Nucleon form factors and a nonpointlike diquark

J. C. R. Bloch,¹C. D. Roberts,¹ S. M. Schmidt,¹ A. Bender,²and M. R. Frank³

1Physics Division, Building 203, Argonne National Laboratory, Argonne, Illinois 60439-4843

2Centre for the Subatomic Structure of Matter, University of Adelaide, Adelaide SA 5005, Australia

3Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195 共Received 30 July 1999; published 2 November 1999兲

Nucleon form factors are calculated on q²苸关0,3兴GeV²using an ansatz for the nucleon’s Faddeev amplitude motivated by quark-diquark solutions of the relativistic Faddeev equation. Only the scalar diquark is retained, and it and the quark are confined. A good description of the data requires a,, nonpointlike diquark correlation with an electromagnetic radius of 0.8 r_␲. The composite, nonpointlike natu,re of the diquark is crucial. It provides for diquark-breakup terms that are of greater importance than the diquark photon absorption contribution.关S0556-2813共99兲51711-1兴

PACS number共s兲: 24.85.⫹p, 13.40.Gp, 14.20.Dh, 12.38.Lg Mesons present a two-body problem, and the Dyson- Schwinger equations 共DSEs兲 have been widely used in the calculation of their properties and interactions 关1,2兴. Many studies have focused on electromagnetic processes, such as the form factors of light pseudoscalar 关3,4兴 and vector mesons 关5兴, and the ␥*␲⁰→␥ 关6–8兴, ␥*␲→␳ 关7兴, and ␥␲*

→␲␲关9兴transition form factors, all of which are accessible at the Thomas Jefferson National Accelerator Facility. These studies provide a foundation for the exploration of nucleons, which is fundamentally a three-body problem.

The nucleon’s bound state amplitude can be obtained from a relativistic Faddeev equation关10兴. Its analysis may be simplified by using the feature that ladderlike dressed-gluon exchange between quarks is attractive in the color antitriplet channel. Then, in what is an analogue of the rainbow-ladder truncation for mesons, the Faddeev equation can be reduced to a sum of three coupled equations, in which the primary dynamical content is dressed-gluon exchange generating a correlation between two quarks and the iterated exchange of roles between the dormant and diquark-participant quarks.

Following this approach, the diquark correlation is repre- sented by the solution of a homogeneous Bethe-Salpeter equation in the dressed-ladder truncation and hence its contribution to the quark-quark scattering matrix, Mqq, is that of an asymptotic bound state; i.e., it contributes a simple pole. That is an artifact of the ladder truncation 关11兴 and complicates solving the Faddeev equation 关12兴by introduc- ing spurious free-particle singularities in the kernel.

Studies of DSE models关1,2兴suggest that confinement can be realized via the absence of a Lehmann representation for colored Green’s functions, and have led to a phenomenologi- cally efficacious parametrization of the dressed-quark Schwinger function关3兴. A similar parametrization of the diquark contribution toMqq, advocated in Ref.关13兴, has been used to good effect in solving the Faddeev equation关14兴. We use such representations herein.

The nucleon-photon current is¹

J_␮共P⬘^{, P}兲⫽ie u¯共P⬘兲⌳_␮共q, P兲u共P兲, 共1兲 where the spinors satisfy ␥•Pu( P)⫽i M u( P), u¯ (P)␥•P

⫽i M u¯ (P), with M⫽0.94 GeV the nucleon mass, and q

⫽( P⬘^⫺P). The complete specification of a fermion-vector- boson vertex requires 12 independent scalar functions:

i⌳_␮共q, P兲⫽i␥_␮^f1⫹i␴_␮␯^q_␯^f2⫹R_␮f₃⫹i␥•RR_␮f₄

⫹i␴␯␳R_␮q_␯R_␳f₅⫹i␥5␥␯␧_␮␯␳␴q_␳R_␴f₆⫹•••, 共2兲 where f_i⫽f_i(q²,q•P, P²), R⫽( P⬘⫹P), and q•R⫽0 for elastic scattering. However, using the definition of the nucleon spinors, Eq.共1兲can be written

J_␮共P⬘^{, P}兲⫽ieu¯共P⬘兲

冉

^␥^␮^F¹^共^q²^兲⫹^{2 M}¹ ^␴^␮␯^q^␯^F²^共^q²^兲

冊

^u^共^P^兲^,

共3兲 where the Dirac and Pauli form factors are

F₁⫽f₁⫹2 M f₃⫺4 M²f₄⫺2 M q²f₅⫺q²f₆, 共4兲 F₂⫽2 M f₂⫺2 M f₃⫹4 M²f₄⫹2 M f₅⫺4 M²f₆, 共5兲 in terms of which one has the electric and magnetic form factors:

G_E共q²兲⫽F₁共q²兲⫺ q²

4 M²F₂共q²兲, 共6兲 G_M共q²兲⫽F₁共q²兲⫹F₂共q²兲. 共7兲 To calculate these form factors we represent the nucleon as a three-quark bound state involving a diquark correlation, and require the photon to probe the diquark’s internal structure. Antisymmetrization ensures there is an exchange of roles between the dormant and diquark-participant quarks and this gives rise to diquark ‘‘breakup’’ contributions. We describe the propagation of the dressed-quarks and diquark correlation by confining parametrizations and hence pinch singularities associated with quark production thresholds are absent. Our calculation is related to many studies of meson properties关3–7,9兴.

1In our Euclidean formulation, p•q⫽兺_i⫽1⁴ piqi,兵^␥␮,␥␯其

⫽2␦␮␯,␥_␮^†⫽␥_␮, ␴_␮␯⫽i/2关␥_␮,␥␯兴, and tr_D关␥5␥_␮␥␯␥␳␥␴兴

⫽⫺4⑀␮␯␳␴,⑀1234⫽1.

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We write the Faddeev amplitude of the nucleon as关15兴

⌿_␣^␶共p₁,␣1,␶¹^{; p}2,␣2,␶²^{; p}3,␣3,␶³兲

⫽␧c₁c₂c₃␦^␶␶³␦␣␣₃␺共p₁⫹p₂, p₃兲⌬共p₁⫹p₂兲

⫻⌫_␣

1␣₂

␶¹␶²共p₁, p₂兲, 共8兲

where␧c₁c₂c₃ effects a singlet coupling of the quarks’ color indices, ( p_i,␣i,␶ⁱ) denote the momentum and the Dirac and isospin indices for the ith quark constituent, ␣ ^and ␶ ^are these indices for the nucleon itself, ␺^(l1,l₂) is a Bethe- Salpeter-like amplitude characterizing the relative- momentum dependence of the correlation between diquark and quark,⌬(K) describes the propagation characteristics of the diquark, and

⌫_␣

1␣₂

␶¹␶²共p₁, p₂兲⫽共Ci␥5兲_␣₁_␣₂共i␶2兲^␶¹^␶²⌫共p₁, p₂兲共9兲 represents the momentum-dependence, and spin and isospin character of the diquark correlation; i.e., it corresponds to a diquark Bethe-Salpeter amplitude.

With this form of⌿, we retain inMqq only the contribution of the scalar diquark, which has the largest correlation length 关13兴: ␭0⫹ª1/m₀⫹⫽0.27 fm. For all (ud) correlations with J^P⫽1^⫹,␭ud⬍0.5␭0^⫹. The axial-vector correlation is different: ␭1⫹⫽0.78␭0⫹, and it is quantitatively important in the calculation of baryon masses (ⱗ30%) 关14兴. Hence we anticipate that neglecting the 1^⫹ correlation will prove the primary defect of our ansatz. However, it is a help- ful expedient in this exploratory calculation, which is made complicated by our desire to elucidate the effect of the di- quarks’ internal structure.

Our impulse approximation to the nucleon form factor is depicted in Fig. 1. Enumerating from top to bottom, the diagrams represent

⌳_␮¹共q, P兲⫽3

冕

_共2^d␲⁴^l兲⁴␺共K, p₃⫹q兲⌬共K兲

⫻␺共K, p₃兲Q_F⌳_␮^q共p₃⫹q, p₃兲, 共10兲 with² K⫽␩^P⫹l, p₃⫽(1⫺␩^{) P}⫺l, p₂⫽K/2⫺k, Q_F

⫽diag(2/3,⫺1/3), ⌳_␮^q(k₁,k₂)⫽S(k₁)⌫_␮(k₁,k₂)S(k₂),

⌳_␮²共q, P兲⫽6

冕

_共2^d␲⁴^k兲⁴ d⁴l

共2␲兲⁴⍀共p₁⫹q, p₂, p₃兲

⫻⍀共p₁, p₂, p₃兲tr_D关⌳_␮^q共p₁⫹q, p₁兲S共p₂兲兴

⫻S共p₃兲¹3I_F, 共11兲 which contributes equally to the proton and neutron and con- tains the diquark electromagnetic form factor, with 6

⫽␧c₁c₂c₃␧c₁c₂c₃ and

⍀共p₁, p₂, p₃兲⫽␺共p₁⫹p₂, p₃兲⌬共p₁⫹p₂兲⌫共p₁, p₂兲, 共12兲

⌳_␮³共q, P兲⫽6

冕

共2␲兲⁴⍀共p₁⫹q, p₃, p₂兲

⫻⍀共p₁, p₂, p₃兲S共p₂兲共i␶2兲^TQ_F共i␶2兲

⫻⌳_␮^q共p₁, p₁⫹q兲S共p₃兲, 共13兲

⌳_␮⁴共q, P兲⫽6

冕

共2␲兲⁴⍀共p₁, p₃, p₂⫹q兲

⫻⍀共p₁, p₂, p₃兲Q_F⌳_␮^q共p₂⫹q, p₂兲S共p₁兲S共p₃兲, 共14兲

⌳_␮⁵共q, P兲⫽6

冕

共2␲兲⁴⍀共p₁, p₃⫹q, p₂兲

⫻⍀共p₁, p₂, p₃兲S共p₂兲S共p₁兲Q_F⌳_␮共p₃⫹q, p₃兲. 共15兲 The nucleon-photon vertex is

⌳_␮^q共q, P兲⫽⌳_␮¹共q, P兲⫹2i

兺

⫽2 5

⌳_␮ⁱ共q, P兲. 共16兲

Equation共16兲is fully defined once⌿⬃␺⌫⌬, S, and⌫_␮ are specified. S and ⌫_␮ are primary elements in studies of meson properties and are already well constrained. For the dressed-quark propagator,

2␩ describes the partitioning of the nucleon’s total momentum, P⫽p₁⫹p₂⫹p₃, between the diquark and quark, a necessary feature of a covariant treatment.

FIG. 1. Our impulse approximation to the electromagnetic current requires the calculation of five contributions, Eqs.共10兲–共15兲.␺^:␺^(l1,l2) in Eq. 共8兲; ⌫: Bethe-Salpeter-like diquark amplitude in Eq. 共9兲; solid line:

S(q), quark propagator in Eq.共17兲; dotted line:⌬(K), diquark propagator in Eq. 共28兲. The lowest three diagrams, which describe the interchange between the dormant quark and the diquark participants, effect the antisymmetrization of the nucleon’s Faddeev amplitude. Current conservation fol- lows because the photon-quark vertex is dressed, given in Eq.共23兲.

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S共p兲⫽⫺i␥•p␴V共p²兲⫹␴S共p²兲共17兲

⫽关i␥•pA共p²兲⫹B共p²兲兴^⫺¹, 共18兲 we use the algebraic parametrizations 关3兴:

␴

¯S共x兲⫽2m¯F„2共x⫹m¯²兲…⫹F共b₁x兲F共b₃x兲关b₀⫹b₂F共⑀^x兲兴, 共19兲

␴

¯V共x兲⫽ 1

x⫹m¯²关1⫺F„2共x⫹m¯²兲…兴, 共20兲 with F(y )⫽(1⫺e^⫺^y)/ y , x⫽p²/␭², m¯⫽m/␭, ␴^¯S(x)

⫽␭␴S( p²), and ␴^¯V(x)⫽␭²␴V( p²). The mass scale, ␭

⫽0.566 GeV, and parameter values

m¯ b₀ b₁ b₂ b₃

0.00897 0.131 2.90 0.603 0.185

共21兲 were fixed in a least-squares fit to light-meson observables.

关⑀⫽10^⫺⁴ in Eq. 共19兲 acts only to decouple the large- and intermediate-p² domains.兴 This algebraic parametrization combines the effects of confinement and DCSB with free- particle behavior at large spacelike p² 关2兴.

In Eqs.共10兲–共15兲,⌫_␮ is the dressed-quark-photon vertex.

It satisfies the vector Ward-Takahashi identity,

共l₁⫺l₂兲_␮i⌫_␮共l₁,l₂兲⫽S^⫺¹共l₁兲⫺S^⫺¹共l₂兲, 共22兲 which ensures current conservation 关3兴.⌫_␮ has been much studied 关16兴and, although its exact form remains unknown, its qualitative features have been elucidated so that a phe- nomenologically efficacious ansatz has emerged 关17兴:

i⌫_␮共l₁,l₂兲⫽i⌺A共l₁²,l₂²兲␥_␮⫹共l₁⫹l₂兲_␮

冋

¹²ⁱ^␥^•共^l¹^⫹^l²^兲

⫻⌬A共l₁²,l₂²兲⫹⌬B共l₁²,l₂²兲

册

^, ^共²³^兲

⌺F共l₁²,l₂²兲⫽¹2关F共l₁²兲⫹F共l₂²兲兴, 共24兲

⌬F共l₁²,l₂²兲⫽F共l₁²兲⫺F共l₂²兲

l₁²⫺l₂² , 共25兲 where F⫽A,B; i.e., the scalar functions in Eq.共18兲. A feature of Eq. 共23兲is that⌫_␮ is completely determined by the dressed-quark propagator. Further, we estimate that calcu- lable improvements would modify our results by ⱗ15 % 关18兴.

The new element herein is the model of the nucleon’s Faddeev amplitude, Eq.共8兲. For the Bethe-Salpeter-like amplitudes we use the one-parameter model forms

⌫共q₁,q₂兲⫽ 1

N_⌫F共q²/␻_⌫²兲, qª¹2共q₁⫺q₂兲共26兲

␺共l₁,l₂兲⫽ 1

N_⌿F共l²/␻_␺²兲, lª共1⫺␩兲l₁⫺␩^l2. 共27兲 Our impulse approximation is founded on a dressed-ladder kernel in the Faddeev equation and ⌫_␮ satisfies Eq. 共22兲. Hence, the canonical normalization conditions for the diquark and nucleon amplitudes translate to the constraints that the (ud) diquark must have charge 1/3 and the proton unit charge, which fix N_⌫ and N_⌿. For the diquark propagator we use the one-parameter form

⌬共K²兲⫽ 1

m_⌬²F共K²/␻_⌫²兲, 共28兲 and interpret 1/m_⌬ as the diquark correlation length.

We fix the model’s three parameters by optimizing a fit to G_E^p(q²) and ensuring G_Eⁿ(0)⫽0, which yields³

␻_␺ ␻_⌫ ^m_⌬

␩⫽2/3 0.20 1.0 0.63

共29兲 all in GeV (1/m_⌬⫽0.31 fm). Using Monte-Carlo methods to evaluate the multidimensional integrals, these values give

Emp. Calc.

r²_p(fm)² (0.87)² (0.79)² r_n²(fm)² ⫺(0.34)² ⫺(0.43)²

␮p(␮N) 2.79 2.88

␮n(␮N) ⫺1.91 ⫺1.58

␮n/␮p ⫺0.68 ⫺0.55 共30兲 where the statistical error is ⱗ1 %. The sensitivity of our results to the model’s parameters is illustrated in Table I. It is clear that the fit is stable but does not bracket the experimen-

3Our results are sensitive to ␩ because Eqs. 共26兲 and 共27兲 are equivalent to retaining only the leading Dirac amplitude in the ex- pression for these functions and neglecting their q•K, l•P dependence when solving the Bethe-Salpeter and Faddeev equations.␩

⫽2/3 is required for this ansatz to transform correctly under charge conjugation. Accounting for the q•K, l•P dependence would eliminate this artifact关14,19兴.

TABLE I. A variation of the model parameters ␻_␺, ␻⌫, and m_⌬ 共in GeV兲illustrates the sensitivity and stability of our results.

The column labeled ‘‘rn’’ listssign(rn 2)兩rn

2兩^1/2.共Radii in fm, magnetic moments in units of␮N. The statistical errors are⭐1%.兲

␻_␺ ␻⌫ m_⌬ r_p r_n ␮p ␮n ␮n/␮p

0.20 1.0 0.63 0.79 ⫺0.43 2.88 ⫺1.58 ⫺0.55 0.16 1.0 0.63 0.84 ⫺0.46 2.83 ⫺1.55 ⫺0.55 0.24 1.0 0.62 0.75 ⫺0.41 2.89 ⫺1.59 ⫺0.55 0.20 0.8 0.62 0.80 ⫺0.40 2.93 ⫺1.64 ⫺0.56 0.20 1.2 0.63 0.78 ⫺0.45 2.84 ⫺1.54 ⫺0.54

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tal domain; i.e., the model lacks a relevant degree of free- dom, a defect we expect the inclusion of an axial-vector diquark to ameliorate.

The charge radii are obtained via

r_p,n² ⫽⫺6 d

dq²F₁^p,n共q²兲兩q²⫽0⫹ 3

2 M²F₂^p,n共0兲, 共31兲 ª共r_p,n^I 兲²⫹共r^F_p,n兲² 共32兲 and in this calculation共in fm²)

共r_p^I兲²⫽共0.70兲², 共r_p^F兲²⫽共0.35兲²,

共r_n^I兲²⫽⫺共0.29兲², 共r_n^F兲²⫽⫺共0.32兲². 共33兲 A 20% reduction in␻⌫ 共Table I, row 4兲reduces兩r_n兩by 7%.

However, that results from a 21% reduction in兩r_n^I兩 and 2%

increase in 兩r_n^F兩. We attribute our overestimate of 兩r_n²兩 to a poor description of F₁ⁿ(q²), which involves many cancella- tions between terms because of the (u,d,d) electric charge combinations and must vanish at q²⫽0.

Five diagrams contribute to our impulse approximation and diagram 2 involves the diquark form factor. The calculated value of the associated elastic charge radius provides a measure of the size of the ‘‘constituent’’ diquark:

r₀²_⫹⫽共0.45 fm兲²⫽共0.80r_␲兲², 共34兲 with r_␲calculated in the same model关3兴, and in quantitative agreement with another estimate 关22兴. This is important because, with ␻⌫ allowed to vary, r₀⫹ is a qualitative predic- tion of the model. Thus an optimal description of the data requires a nonpointlike diquark.

Table II provides a guide to each diagram’s relative importance. In all cases the first diagram, describing scattering from the dormant quark, is the most significant. For the charge radii the breakup contributions are comparable in magnitude to the second diagram, photon-diquark scattering.

The magnetic moments are of particular interest. A scalar diquark does not have a magnetic moment, and that is ex- pressed in our calculation by the very small contribution from diagram 2. It is not identically zero because of the confinement of the spectator quark; i.e., the absence of a mass shell. Diagrams 3 –5 only appear because the diquark is a nonpointlike composite and they provide ⬃50% of

␮p,␮n. Discarding these contributions one obtains ␮n/␮p

⭓⫺0.5, and in pointlike diquark models the axial-vector has

alone been forced to remedy that defect 关23兴. Our results indicate that approach to be erroneous, attributing too much importance to the axial-vector correlation.

The calculated form factors are depicted in Figs. 2 and 3 and it is obvious in Fig. 2 that we used G_E^p(q²) to constrain our fit. The 0^⫹ (ud) diquark correlation in ⌿ ensures that G_{E fit}ⁿ (q²)⬅”0, and the presence of diquark correlations can also explain the N-⌬ mass difference. Our result for G_Eⁿ(q²) is well described by关20兴

FIG. 2. Upper panel: Calculated proton electric form factor: ⫹, compared with the empirical dipole fit: Femp(q²)⫽1/(1⫹q²/memp

2 )², memp

⫽0.84 GeV. Lower panel: Calculated neutron electric form factor: ⫹, compared with the experimental data关20兴as extracted using the Argonne V18 potential关21兴. In both calculations the Monte-Carlo errors are smaller than the symbols.

TABLE II. Relative contribution to the charge radii and magnetic moments of each of the five diagrams in our impulse approximation: Fig. 1, Eqs.共10兲–共15兲.

Diagram 1 2 3 4 5

(r_p²)ⁱ/r_p² 0.68 0.11 ⫺0.02 0.12 0.12 (r_n²)ⁱ/r_n² 1.14 ⫺0.37 ⫺0.15 0.19 0.19

␮p

i/␮p 0.60 0.01 0.04 0.17 0.17

␮n

i/␮n 0.55 ⫺0.02 0.15 0.16 0.16

FIG. 3. Calculated proton and neutron magnetic form factors, normal- ized by兩␮p,n兩in Eq.共30兲. The curves are dipole fits with masses共in GeV兲: m_p⫽0.95, m_n⫽1.0, 13% and 19% larger than m_emp in Fig. 2.

关␮p,n

empFemp(q²) describes the data very well.兴

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G_{E fit}ⁿ 共q²兲⫽⫺␮n

empF_emp共q²兲 a²␶

1⫹b²␶^, ^共³⁵^兲 with ␶⫽q²/(2 M )², F_emp(q²) given in Fig. 2, and a

⫽1.33, b⫽1.00, and the discrepancy between our calculation and experiment can be discussed in terms of these parameters. a characterizes the charge radius and it is ⱗ30%

too large, as can be anticipated from Eq.共30兲. b describes the magnitude at intermediate momenta and it is only

⬃23–35 % of the empirical value. That is a systematic defect shared by other studies 关24兴 that only retain the scalar diquark correlation. Unlike those studies, however, our calculated magnetic form factors, Fig 3, agree well with the data and, as we have seen, that is because we include the diquark breakup diagrams. It must be borne in mind that in our calculation a and b are not independent. Modifying the parameters in Eq. 共29兲 so as to reduce a automatically and substantially increases b. However, notwithstanding our ob- servation that its importance has previously been overesti- mated, without an axial-vector diquark correlation it is not possible to accurately describe all observables simulta- neously.

We have employed a three-parameter model of the nucleon’s Faddeev amplitude,⌿, to calculate an impulse approximation to the electromagnetic form factors.⌿ represents the nucleon as a bound state of a confined quark and confined, nonpointlike scalar diquark, and the exchange of roles between the dormant and diquark-participant quarks is an inte- gral feature. Five processes contribute: direct quark-photon scattering with a spectator diquark; photon-diquark scatter-

ing with a spectator quark; and three distinct diquark breakup diagrams. We obtain a good description of all form factors except G_Eⁿ, which is too large in magnitude. That defect is shared by all models that do not include more than a scalar diquark correlation. The nonpointlike nature of the diquark correlation is important, especially via the breakup contributions which provide large contributions to the magnetic moments and ensure ␮n/␮p⬍⫺0.5.

Including a nonpointlike axial-vector diquark is an obvious improvement of the model. That must be done in anal- ogy with the scalar diquark because an accurate interpreta- tion of the model parameters is impossible if the breakup diagrams are discarded. Another avenue for improvement is a direct solution of the Faddeev equation, retaining the axial- vector correlation and the breakup contributions to the form factor. That would provide a model for correlating meson and baryon observables in terms of very few parameters.

Models of the nucleon such as ours have hitherto been applied only at small and intermediate q². Based on the ob- servation 关4兴 that a description of the large-q² behavior of F_␲(q²) is only possible if the subleading pseudovector components of the pion’s Bethe-Salpeter amplitude are retained, we anticipate that a successful description of the nucleon form factors on that domain will require a parametrization of the Faddeev amplitude that includes the analogous subleading Dirac components.

This work was supported by the U.S. Department of En- ergy, Nuclear Physics Division, under Contract No. W-31- 109-ENG-38. S.M.S. thanks the A.v. Humboldt foundation.

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