Current quark mass dependence of nucleon magnetic moments and radii
I.C. Cloet, G. Eichmann, V.V. Flambaum, C.D. Roberts, M.S. Bhagwat, A. Holl
aa r X i v : . [ nu c l - t h ] A p r Few-Body Systems 0, 1–23 (2018)
Few-
Body
Systems c (cid:13) by Springer-Verlag 2018Printed in Austria Current quark mass dependence of nucleonmagnetic moments and radii
I. C. Clo¨et, G. Eichmann, , V. V. Flambaum, , C. D. Roberts, M. S. Bhagwat † and A. H¨oll. ‡ Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA Institut f¨ur Physik, Karl-Franzens-Universit¨at Graz, A-8010 Graz, Austria Argonne Fellow, Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA School of Physics, University of New South Wales, Sydney 2052, Australia Institut f¨ur Physik, Universit¨at Rostock, D-18051 Rostock, Germany
Abstract.
A calculation of the current-quark-mass-dependence of nucleonstatic electromagnetic properties is necessary in order to use observationaldata as a means to place constraints on the variation of Nature’s fundamentalparameters. A Poincar´e covariant Faddeev equation, which describes baryonsas composites of confined-quarks and -nonpointlike-diquarks, is used to calcu-late this dependence. The results indicate that, like observables dependent onthe nucleons’ magnetic moments, quantities sensitive to their magnetic andcharge radii, such as the energy levels and transition frequencies in Hydrogenand Deuterium, might also provide a tool with which to place limits on theallowed variation in Nature’s constants.
It is a feature anticipated of models for the unification of all interactions thatthe so-called fundamental “constants” actually exhibit spatial and temporal vari-ation. In consequence there is an expanding search for this variation via as-tronomical, geochemical and laboratory measurements [1, 2]. An interpretationof these measurements can materially benefit from calculations of the current-quark-mass-dependence of observables characterising hadronic and nuclear sys-tems.One example is found in the behaviour of hadron masses [3, 4]. A variationin light-meson masses modifies the internucleon potential, and a variation inthe nucleon’s mass affects the kinetic energy term in the nuclear Hamiltonian. † Current address: Department of Radiation Oncology, DF/BWH Cancer Center, Harvard Med-ical School, Boston, MA 02115 ‡ Current address: BMWi, Villemombler Str. 76, D-53123 Bonn, Germany Variation of nucleon magnetic moments and radii
Such changes modify nuclear binding energies and can thereby have a substantialimpact on Big Bang Nucleosynthesis [5]. These results enable the use of obser-vational data to place constraints on the variation of Nature’s constants; e.g.,Ref. [6].Other effects of a variation in hadron masses are an alteration in the locationof energy levels and in the positions of compound resonances in heavy nuclei.For example, Refs. [7, 8] explore the sensitivity to changes in the light-quarkmasses, m , of the nuclear clock transition between the ground- and first-excitedstates in Th and the position of the 0 . Sm.It is noteworthy that the shift of the Sm resonance, as determined from theOklo natural nuclear reactor, currently provides the best terrestrial limit onthe temporal variation of Nature’s fundamental parameters; namely, | ˙ X q /X q | < . × − y − , X q := m/Λ QCD [9, 10, 11].Hadron magnetic moments can also depend upon current-quark mass. Calcu-lations of this dependence are necessary for the interpretation of measurementsof quasar absorption spectra and superprecise atomic clocks in terms of thevariation of Nature’s fundamental parameters [12, 13, 14]. It is notable that inrudimentary constituent-quark models for hadron bound-states composed of de-generate light-quarks, in which the constituents are weakly bound and hence thebound-state’s mass is accurately approximated as the sum of constituent-quarkmasses, the bound-states’ magnetic moments are independent of quark mass.Any quark-mass variation in hadron magnetic moments is therefore a gauge ofhadron structure.Herein we report a calculation of the mass-dependence of neutron and protonmagnetic moments based on a Poincar´e covariant Faddeev equation model forthe nucleon. Since the magnetic and electric form factors are obtained simul-taneously, we also describe the variation of the nucleons’ magnetic and chargeradii. The background material for our calculation is provided in Sects. 2 and 3,and in appendices. Our results are presented and discussed in Sect. 4. Section 5is a brief summation.
In quantum field theory a nucleon appears as a pole in a six-point quark Greenfunction. The pole’s residue is proportional to the nucleon’s Faddeev amplitude,which is obtained from a Poincar´e covariant Faddeev equation that adds-up allpossible quantum field theoretical exchanges and interactions that can take placebetween three dressed-quarks.A tractable truncation of the Faddeev equation is based [15] on the obser-vation that an interaction which describes mesons also generates diquark corre-lations in the colour-¯3 channel [16]. The dominant correlations for ground stateoctet and decuplet baryons are scalar (0 + ) and axial-vector (1 + ) diquarks be-cause, for example: the associated mass-scales are smaller than the baryons’masses [17, 18], namely (in GeV) m [ ud ] = 0 . − . , m ( uu ) = m ( ud ) = m ( dd ) = 0 . − . . V. Flambaum, et al. 3 = a Ψ Pp q p d Γ b Γ − a p d p q b Ψ P q Figure 1.
Poincar´e covariant Faddeev equation, Eq. (A.11), employed herein to calculate nu-cleon properties. Ψ in Eq. (A.1) is the Faddeev amplitude for a nucleon of total momentum P = p q + p d . It expresses the relative momentum correlation between the dressed-quark and-diquarks within the nucleon. The shaded region demarcates the kernel of the Faddeev equa-tion, Sect. A.2, in which: the single line denotes the dressed-quark propagator, Sect. A.2.1; Γ is the diquark Bethe-Salpeter-like amplitude, Sect. A.2.2; and the double line is the diquarkpropagator, Sect. A.2.3. and the electromagnetic size of these correlations [19] r [ ud ] ≈ . , r ( ud ) ∼ . , (2)is less than that of the proton. (The last result is an estimate based on the ρ -meson/ π -meson radius-ratio [20, 21].)The kernel of the Faddeev equation is completed by specifying that the quarksare dressed, with two of the three dressed-quarks correlated always as a colour-¯3diquark. As illustrated in Fig 1, binding is then effected by the iterated exchangeof roles between the bystander and diquark-participant quarks.The Faddeev equation is detailed in Appendix A: Faddeev Equation . With allits elements specified, as described therein, the equation can be solved to obtainthe nucleon’s mass and amplitude. Owing to Eq. (A.36), in this calculation themasses of the scalar and axial-vector diquarks are the only variable parameters.The axial-vector mass is chosen so as to obtain a desired mass for the ∆ , andthe scalar mass is subsequently set by requiring a particular nucleon mass.We have written here of desired rather than experimental mass values becauseit is known that the masses of the nucleon and ∆ are materially reduced bypseudoscalar meson cloud effects. This is discussed in detail in Refs. [22, 23].Hence, a baryon represented by the Faddeev equation described above mustpossess a mass that is inflated with respect to experiment so as to allow for anadditional attractive contribution from the pseudoscalar cloud. As in previouswork [3, 24] and reported in Table 1, we require M N = 1 .
18 GeV and m ∆ =1 .
33 GeV. The results and conclusions of our study are essentially unchangedshould even larger masses and a smaller splitting M ∆ − M N be more realistic, apossibility suggested by Ref. [25]. This is natural because the spin- and isospin-3 / ∆ contains only an axial-vector diquark. Therelevant Faddeev equation is not different in principle to that for the nucleon. It is describede.g. in Ref. [3]. Variation of nucleon magnetic moments and radii Table 1.
Mass-scale parameters (in GeV) for the scalar and axial-vector diquark correlations,fixed by fitting nucleon and ∆ masses offset to allow for “pion cloud” contributions [22]. Wealso list ω J P = 1 √ m J P , which is the width-parameter in the ( qq ) J P Bethe-Salpeter amplitude,Eqs. (A.30) & (A.31): its inverse is an indication of the diquark’s matter radius. Row 3 illustrateseffects of omitting the axial-vector diquark correlation: the ∆ cannot be formed and M N issignificantly increased. Evidently, the axial-vector diquark provides significant attraction in theFaddeev equation’s kernel. M N M ∆ m + m + ω + ω + The nucleon’s electromagnetic current is J µ ( P ′ , P ) = ie ¯ u ( P ′ ) Λ µ ( q, P ) u ( P ) , (3)= ie ¯ u ( P ′ ) (cid:18) γ µ F ( Q ) + 12 M σ µν Q ν F ( Q ) (cid:19) u ( P ) , (4)where P ( P ′ ) is the momentum of the incoming (outgoing) nucleon, Q = P ′ − P ,and F and F are, respectively, the Dirac and Pauli form factors. They are theprimary calculated quantities, from which one obtains the nucleon’s electric andmagnetic (Sachs) form factors G E ( Q ) = F ( Q ) − Q M F ( Q ) , G M ( Q ) = F ( Q ) + F ( Q ) . (5)The nucleons’ magnetic moments are defined through µ n = κ n = G nM (0) , µ p = 1 + κ p = G pM (0) , (6)where κ N , N = n, p , are referred to as the anomalous magnetic moments. Ofcourse, the nucleon’s electric charges, G NE (0), are conserved and cannot dependon the current-quark mass. That is not true of their electric and magnetic radii: r p := − dds G pE ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 , r n := − dds G nE ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 , (7)( r µN ) := − dds ln G NM ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 . (8)In order to calculate the magnetic moments and charge radii, and their depen-dence on current-quark mass, one must know the manner in which the nucleondescribed in Sect. 2 couples to a photon. That is derived in Ref. [26], illustratedin Fig. 2 and detailed in Appendix C: Nucleon-Photon Vertex . Naturally, as ap-parent in that Appendix, the current depends on the electromagnetic propertiesof the diquark correlations. . V. Flambaum, et al. 5 ii Ψ Ψ P f f P Q ii Ψ Ψ P f f P Q ii Ψ Ψ PP f f Q Γ − Γ scalaraxial vector ii Ψ Ψ P f f P Q µ ii X Ψ Ψ P f f QP Γ − µ ii X −Ψ Ψ P f f P Q Γ Figure 2.
Vertex which ensures a conserved current for on-shell nucleons described by theFaddeev amplitudes, Ψ i,f , described in Sect. 2 and Appendix A: Faddeev Equation . The singleline represents S ( p ), the dressed-quark propagator, Sec. A.2.1, and the double line, the diquarkpropagator, Sec. A.2.3; Γ is the diquark Bethe-Salpeter amplitude, Sec. A.2.2; and the remainingvertices are described in Appendix C: the top-left image is Diagram 1; the top-right, Diagram 2;and so on, with the bottom-right image, Diagram 6. The framework we have described hitherto provides what might be called thequark-core contribution to the nucleons’ electromagnetic form factors. As withthe mass, the nucleons’ magnetic moments, and charge and magnetic radii receivematerial contributions from the pseudoscalar meson cloud [27, 28]. There are twotypes of contribution: regularisation-scheme-dependent terms, which are analyticfunctions of m in the neighbourhood of vanishing current-quark mass, m = 0;and nonanalytic scheme-independent terms.For the magnetic moments and radii the leading-order scheme-independentcontributions are [29]( µ n/p ) − loop NA m π ≃ = ± g A M N π f π m π , (9) h r n/p i − loopNA m π ≃ = ± g A π f π ln( m π M N ) , (10) h ( r µN ) i − loopNA m π ≃ = − g A π f π ln( m π M N ) + g A M N πf π µ v m π , (11)where, experimentally, g A = 1 . f π = 0 . / (2 .
13 fm) and µ V = Variation of nucleon magnetic moments and radii µ p − µ n = 4 .
7. These terms reduce the magnitude of both neutron and protonmagnetic moments, and increase the magnitudes of the radii.While these scheme-independent terms are important, at physical valuesof the pseudoscalar meson masses they do not usually provide the dominantcontribution to observables. That is provided by the regularisation-parameter-dependent terms, as is apparent for baryon masses in Ref. [22] and for the pioncharge radius in Ref. [30]. This is particularly significant for the magnetic mo-ments, in which connection the regularisation-scheme-dependent terms providea nonzero contribution in the chiral limit and have the net effect of increasing | µ N | .This last fact was overlooked in Ref. [24] so that Eq. (82) therein is a poormodel for the net contribution to µ n,p from pseudoscalar meson loops. A minimalimprovement is( µ n/p ) − loop R = (cid:18) µ π n/p ± g A M N π f π m π (cid:19) π arctan( λ m π ) , (12)where µ π n/p are the chiral limit values of the meson loop contributions. Equa-tion (12) reproduces Eq. (9) but also enables us to express reasonable estimatesof the regularisation-parameter-dependent parts of the chiral loops. The param-eter λ is a mass-scale. Its presence echoes a physical effect; namely, that finitemeson size guarantees an intrinsic regularisation of loop integrals. EmulatingRef. [24], we employ λ = 0 . / [0 .
66 fm].A recent estimate from numerical simulations of lattice-regularised QCD [31]gives the following chiral-loop contributions to the nucleons’ magnetic momentsat the physical pion mass: µ πn = − . , µ πp = 0 . . (13)They are obtained with µ π n = − . , µ π p = 0 . , (14)in Eq. (12). Equation (13) suggests that the dressed-quark core we have describedabove should yield the follow magnetic moments for the nucleons at the physicalcurrent-quark mass, Eq. (A.23): µ q ( qq ) n = − . , µ q ( qq ) p = 2 . . (15)These values provide the comparison column in Table 2.It was argued in Ref. [24] that the dressed-quark core described herein iscompatible with augmentation by a sensibly regulated pseudoscalar meson cloud.Indeed, it was argued strongly that agreement with experimental data should notbe achieved in the absence of such contributions. Given Eq. (12), we reconsiderthis below. NB. The expression in Eq. (12) vanishes when the pion mass is much larger than the regu-larisation scale. This is required because very massive states must decouple from low-energyphenomena. Variations in the regularisation mass-scale have no impact; e.g., using λ = 0 . Table 2.
Values of the magnetic moments defined in Eq. (6) calculated with the diquark massvalues in Table. 1 and χ + = 1. Experimental values are [32] µ n = − . µ p = 2 . Eq. (15) – quark-core µ + = 0 µ + = 0 .
37 =: µ f1 + target value κ T = 0 κ T = 0 .
12 =: κ f T µ q ( qq ) n -1.51 -1.44 -1.51 µ q ( qq ) p As explained in Appendix C:
Nucleon-Photon Vertex , the nucleons’ electromag-netic current involves three parameters, which characterise the axial-vector di-quarks’ electromagnetic properties; viz., the diquarks’ magnetic moment – µ + ,their quadrupole moment – χ + , and the scalar ↔ axial-vector transition strength– κ T . Calculations have shown [24] that the quadrupole moment has no mate-rial impact. Its greatest effect is on the neutron’s charge radius, Eq. (7). How-ever, that is notoriously sensitive to details of the neutron’s Faddeev amplitude,which determine the magnitude and sign of the Dirac form factor’s slope, and tomeson-related contributions, so that, within the accuracy of the model describedherein, the net result for all quantities considered is essentially independent of χ + ∈ [0 , . µ + and κ T that effect thisare much smaller than those for an on-shell axial-vector, which were the focusof Ref. [24]. This marked suppression is physically reasonable because the di-quarks are far from being on-shell within the nucleon, a fact observed explicitlyin Ref. [33]. Hence, the dominant contributions to the form factors are providedby Diagrams 1 and 3.It follows that the dressed-quark core defined herein is compatible with dress-ing by a sensibly regulated pseudoscalar meson cloud so long as the magneticinteractions of axial-vector diquark correlations within the nucleon are commen-surate in magnitude with those of the meson cloud. (See also Table 3.) We can now describe the method by which we computed the m -dependence ofthe nucleons’ magnetic moments and radii.We first solved the Faddeev equation at each of a number of current-quark Variation of nucleon magnetic moments and radii masses within a domain that includes the physical current-quark mass, m phys given in Eq. (A.23). In our Faddeev equation the dependence on current-quarkmass appears explicitly in the dressed-quark propagator described in Sect. A.2.1and implicitly in the diquark masses. At each value of m the dressed-quarkpropagator is obtained in an obvious manner, Eqs. (A.19) and (A.20), while thediquark masses are given by m J P ( m ) = m phys J P + σ qq ( m/m phys − , (16)with m phys J P given in Row 2 of Table 1 and σ qq ≈
26 MeV being the diquarks’ σ -term [3, 4]. These observations prescribe the m -dependence of the Faddeevkernel. Solving the Faddeev equation then provides a range of nucleon Faddeevamplitudes, one for each selected current-quark mass. We worked with m = 3 MeV + j δm , δm = 1 MeV , j = 0 , , . . . , . (17)The next step was to calculate, at each value of m in Eq. (17), the nucleons’electric and magnetic form factors on Q ∈ (0 ,
1] GeV . In doing this we alsoallowed the diquarks’ magnetic properties to evolve with current-quark mass.Owing to the fact that Γ J P C := Γ J P C † , where Γ J P is a diquark’s Bethe-Salpeteramplitude, satisfies approximately the same Bethe-Salpeter equation as a J − P colour-singlet meson [16], we referred to Ref. [21] for guidance in constrainingthe variation of the diquarks’ magnetic parameters. For the ρ -meson we inferredthat in the neighbourhood of the physical u/d -quark mass µ ρ ( m ) = µ phys ρ + 0 .
002 ( m/m phys − , µ phys ρ = 2 . , (18)from which one extracts the renormalisation group invariant ratio δµ ρ µ ρ / δmm = 0 . . (19)Plainly, the response is weak and (perhaps surprisingly, given Table 4 herein) the ρ -meson magnetic moment increases with increasing current-quark mass. Basedon this analysis, in our calculations we employed µ + ( m ) = µ f1 + + ς µ .
002 ( m/m phys − , (20) κ T ( m ) = κ f T + ς κ .
002 ( m/m phys − , (21)where µ f1 + and κ f T are given in Table 2. Since the response in Eq. (18) is weakand the sign unexpected, we included the parameters ς µ = ± ς κ = ± δr ρ r ρ / δmm ≈ δ ( r µρ ) ( r µρ ) / δmm ≈ − . . (22) . V. Flambaum, et al. 9 Table 3.
Quark-core and pseudoscalar loop [Eqs. (12), (26), (27)] contributions to the momentsand radii, calculated at the physical current-quark mass, Eq. (A.23). The radii are listed in fm .Experimental values are quoted from Ref. [32], where available, and otherwise from Ref. [37]. µ n µ p h r n i h r p i h ( r µn ) i h ( r µp ) i experiment -1.91 2.79 − (0 . (0 . (0 . (0 . q ( qq ) -1.51 2.55 − (0 . (0 . (0 . (0 . π -loop -0.40 0.24 − (0 . (0 . (0 . (0 . total -1.91 2.79 − (0 . (0 . (0 . (0 . This dressed-quark core value possesses the same sign and a similar magnitudeto the response of the proton’s dressed-quark core radii, described subsequentlyin connection with Table 4. We have also analysed the response of the pion’sdressed-quark core. A quick estimate can be obtained via the Nambu-Jona–Lasinio model, in which [34] r π = 34 π f π . (23)It is straightforward to solve this model’s gap equation, and therefrom calculate f π and its dependence on current-quark mass. We find thereby q ¯ qqr π := δr π r π / δmm = − . . (24)The same result is found by evaluating Eq. (23) with the m -dependence of f π depicted in Fig. 3 of Ref. [35], which is obtained from a rainbow truncation ofQCD’s gap equation, Finally, from Fig. 6 of Ref. [36], which reports an internallyconsistent Dyson-Schwinger equation calculation, we infer q ¯ qqr π = − .
06. A naturalscale associated with the response of dressed-quark core radii is now apparent.Given Eq. (18) it was not surprising for us to discover that the nucleons’magnetic moments respond slowly to changing m . Therefore very precise resultsfor the form factors are required in order to compute the slope. In order toachieve the numerical accuracy necessary we made significant modifications tothe computer code employed in Ref. [24]. These changes reduced execution timesby an order of magnitude, and made practical and reasonable the use of 10 adaptive Monte-Carlo sample points in the evaluation of Diagrams 3, 5 and 6,which are two-loop integrals.Proceeding as described above we arrived at a set of form factors, one at eachvalue of the current-quark mass in Eq. (17), and thereby the magnetic momentsand radii as a function of m . Each quantity, designated generically by Q , can beexpressed in the form Q = Q q ( qq ) + Q π (25)where Q q ( qq ) is provided by our Faddeev equation results and Q π is an estimate ofthe contribution from pseudoscalar meson loops. At the physical current-quark NB. By construction the Faddeev equation is intended to describe a nucleon’s dressed-quarkcore. It explicitly excludes all diagrams that can appear in the calculation of Q π .0 Variation of nucleon magnetic moments and radii Table 4.
Calculated variation of the nucleons’ magnetic moments and radii. We list q s Q := − δ Q s Q s / δmm , where s = q ( qq ), π , t = total= q ( qq ) + π , and f Q = − δ Q q ( qq ) Q t / δmm . q q ( qq ) µ n q πµ n q t µ n f µ n q q ( qq ) µ p q πµ p q t µ p f µ p q q ( qq ) r n q πr n q t r n f r n q q ( qq ) r p q πr p q t r p f r p q q ( qq )( r µn ) q π ( r µn ) q t( r µn ) f ( r µn ) q q ( qq )( r µp ) q π ( r µp ) q t( r µp ) f ( r µp ) h r n/p i − loop R = ± g A π f π ln( m π m π + λ ) , (26) h ( r µN ) i − loop R = − g A π f π ln( m π m π + λ )+ g A M N πf π µ v m π π arctan( λm π ) . (27)These values, too, are reported in the Table, from which it is apparent thatpseudoscalar meson loops contribute materially to the nucleons’ static properties.As already remarked, the neutron’s electric charge radius is particularly sensitiveto details of the neutron’s Faddeev amplitude and, we see here, to the size of themeson cloud contribution. In the computation of form factors, that contributionremains significant for Q . . (See, e.g., Ref. [39].)The dependence of the magnetic moments and radii on current-quark massis expressed through δ Q = d Q dm q ( qq ) + d Q dm π ! δm . (28)We have described in detail the manner in which we calculated the first term. Thesecond is computed from Eqs. (12), (26), (27) using Eq. (A.27). Our results arelisted in Table 4, from which it is immediately apparent that the moments andradii all decrease with increasing current-quark mass. Moreover, for nucleons,which are composed of u/d valence quarks, the response of the contributionsfrom the dressed-quark core to a change in current-quark mass is 8 ±
3% of thatarising from the pseudoscalar meson cloud.In connection with the magnetic moments it is natural to make a comparisonwith Ref. [12], which reports in Eq. (A7): µ n = − . µ p = 3 .
48. These valuescan be compared with our results µ q ( qq ) n + µ π n = − . , µ q ( qq ) p + µ π p = 3 . . (29) NB. The ( ς µ , ς κ )-variation discussed in connection with Eqs. (20), (21) affects no result bymore than 2%.. V. Flambaum, et al. 11 In addition, Eqs. (25), (27) list q t µ n = 0 . q t µ p = 0 .
087 and Eqs. (A1), (A5),(A6) enable one to calculate f µ n = 0 . f µ p = 0 . ≈ .
5. Our dressed-quark core contribution to thevariation of µ n is almost the same, whereas for the proton it is larger by a factorof two. We emphasise that our results were obtained through an internallyconsistent calculation performed directly at the physical light-quark current-mass. In contrast, Ref. [12] extrapolated lattice-regularised QCD results obtainedat approximately the strange-quark mass.These comments notwithstanding, our calculations plainly provide an in-dependent confirmation of the magnitude and sign of the effects discussed inRef. [12]. This agreement supports a view that all the results reported in Table 4are reliable estimates.
We used a Poincar´e covariant Faddeev equation model for the dressed-quarkcore of the nucleon, augmented by a nucleon-photon vertex which automaticallyfulfills the Ward-Takahashi identity for on-shell nucleons and a rudimentary es-timate of the contribution from pseudoscalar meson loops, to obtain insight intothe response of the nucleons’ static electromagnetic properties to changes incurrent-quark mass.Our key results are discussed in connection with Table 4 and summarised bythe renormalisation group invariant ratios presented here: Q µ n µ p r n r p ( r µn ) ( r µp ) − δ QQ / δmm .
186 0 .
139 0 .
483 0 .
202 0 .
350 0 . . (30)Those for the magnetic moments can assist in constraining the allowed tempo-ral variation of the current-quark mass via, e.g., experiments with atomic clocks[2, 14] and various astrophysical measurements [2, 13, 40, 41]. Our results alsosuggest that observables dependent on the nucleons’ magnetic and charge radiimight provide a useful means by which to place limits on the allowed variation inNature’s fundamental parameters. It is notable, for example, that the calculatedenergy levels and transition frequencies in Hydrogen and Deuterium, which aresome of the most precise theoretical predictions in physics, are quite sensitive tothe value of the proton’s charge radius [42]. Moreover, all hyperfine transition fre-quencies in atomic clocks [2, 14] and astrophysics [2, 13, 40, 41] react to a changein nucleonic charge and magnetisation distributions: the former alters electronicwave functions and the latter changes the hyperfine interaction Hamiltonian.As a byproduct of this study, we arrived at an improved understanding ofdiquark correlations. Within the nucleon they are usually far from being on-shell and hence it is a poor approximation to represent, for example, the activemagnetic properties of the axial-vector correlations by on-shell values. Acknowledgement.
We are grateful to B. El-Bennich, T. Kl¨ahn and R. D. Young for profitablediscussions. This work was supported by: the Department of Energy, Office of Nuclear Physics, Appendix D:
Charge Symmetry explains the relative magnitudes of q q ( qq ) µ n and q q ( qq ) µ p .2 Variation of nucleon magnetic moments and radiicontract no. DE-AC02-06CH11357; the Austrian Science Fund FWF under grant no. W1203;and benefited from the facilities of ANL’s Computing Resource Center.
Appendix A: Faddeev Equation
A.1 General structure
The nucleon is represented by a Faddeev amplitude Ψ = Ψ + Ψ + Ψ , (A.1)where the subscript identifies the bystander quark and, e.g., Ψ , are obtained from Ψ by acorrelated, cyclic permutation of all the quark labels. We employ the simplest realistic repre-sentation of Ψ . The spin- and isospin-1 / Ψ ( p i , α i , τ i ) = N + + N + , (A.2)with ( p i , α i , τ i ) the momentum, spin and isospin labels of the quarks constituting the boundstate, and P = p + p + p the system’s total momentum. The scalar diquark piece in Eq. (A.2) is N + ( p i , α i , τ i ) = [ Γ + ( 12 p [12] ; K )] τ τ α α ∆ + ( K ) [ S ( ℓ ; P ) u ( P )] τ α , (A.3)where: the spinor satisfies (Appendix B: Euclidean Conventions )( iγ · P + M ) u ( P ) = 0 = ¯ u ( P ) ( iγ · P + M ) , (A.4)with M the mass obtained by solving the Faddeev equation, and it is also a spinor in isospinspace with ϕ + = col(1 ,
0) for the proton and ϕ − = col(0 ,
1) for the neutron; K = p + p =: p { } , p [12] = p − p , ℓ := ( − p { } + 2 p ) / ∆ + is a pseudoparticle propagator for the scalardiquark formed from quarks 1 and 2, and Γ + is a Bethe-Salpeter-like amplitude describing theirrelative momentum correlation; and S , a 4 × S , Γ + and ∆ + are discussed in Sect. A.2.) The colour antisymmetryof Ψ is implicit in Γ J P , with the Levi-Civita tensor, ǫ c c c , expressed via the antisymmetricGell-Mann matrices; viz., defining { H = iλ , H = − iλ , H = iλ } , (A.5)then ǫ c c c = ( H c ) c c . [See Eqs. (A.30), (A.31).]The axial-vector component in Eq. (A.2) is N + ( p i , α i , τ i ) = [ t i Γ + µ ( 12 p [12] ; K )] τ τ α α ∆ + µν ( K ) [ A iν ( ℓ ; P ) u ( P )] τ α , (A.6)where the symmetric isospin-triplet matrices are t + = 1 √ τ + τ ) , t = τ , t − = 1 √ τ − τ ) , (A.7)and the other elements in Eq. (A.6) are straightforward generalisations of those in Eq. (A.3).The general forms of the matrices S ( ℓ ; P ), A iν ( ℓ ; P ) and D νρ ( ℓ ; P ), which describe themomentum space correlation between the quark and diquark in the nucleon are described inRefs. [43, 44]. The requirement that S ( ℓ ; P ) represent a positive energy nucleon; namely, thatit be an eigenfunction of Λ + ( P ), Eq. (B.7), entails S ( ℓ ; P ) = s ( ℓ ; P ) I D + “ iγ · ˆ ℓ − ˆ ℓ · ˆ P I D ” s ( ℓ ; P ) , (A.8) NB. Herein we assume isospin symmetry of the strong interaction; i.e., the u - and d -quarksare indistinguishable but for their electric charge. This simplifies the form of the Faddeevamplitudes.. V. Flambaum, et al. 13where ( I D ) rs = δ rs , ˆ ℓ = 1, ˆ P = −
1. In the nucleon rest frame, s , describe, respectively, theupper, lower component of the bound-state nucleon’s spinor. Placing the same constraint onthe axial-vector component, one has A iν ( ℓ ; P ) = X n =1 p in ( ℓ ; P ) γ A nν ( ℓ ; P ) , i = + , , − , (A.9)where (ˆ ℓ ⊥ ν = ˆ ℓ ν + ˆ ℓ · ˆ P ˆ P ν , γ ⊥ ν = γ ν + γ · ˆ P ˆ P ν ) A ν = γ · ˆ ℓ ⊥ ˆ P ν , A ν = − i ˆ P ν , A ν = γ · ˆ ℓ ⊥ ˆ ℓ ⊥ ,A ν = i ˆ ℓ ⊥ µ , A ν = γ ⊥ ν − A ν , A ν = iγ ⊥ ν γ · ˆ ℓ ⊥ − A ν . (A.10)One can now write the Faddeev equation satisfied by Ψ as » S ( k ; P ) u ( P ) A iµ ( k ; P ) u ( P ) – = − Z d ℓ (2 π ) M ( k, ℓ ; P ) » S ( ℓ ; P ) u ( P ) A jν ( ℓ ; P ) u ( P ) – . (A.11)The kernel in Eq. (A.11) is M ( k, ℓ ; P ) = " M ( M ) jν ( M ) iµ ( M ) ijµν (A.12)with M = Γ + ( k q − ℓ qq / ℓ qq ) S T ( ℓ qq − k q ) ¯ Γ + ( ℓ q − k qq / − k qq ) S ( ℓ q ) ∆ + ( ℓ qq ) , (A.13)where: ℓ q = ℓ + P/ k q = k + P/ ℓ qq = − ℓ + 2 P/ k qq = − k + 2 P/ M ) jν = t j Γ + µ ( k q − ℓ qq / ℓ qq ) × S T ( ℓ qq − k q ) ¯ Γ + ( ℓ q − k qq / − k qq ) S ( ℓ q ) ∆ + µν ( ℓ qq ) , (A.14)( M ) iµ = Γ + ( k q − ℓ qq / ℓ qq ) × S T ( ℓ qq − k q ) t i ¯ Γ + µ ( ℓ q − k qq / − k qq ) S ( ℓ q ) ∆ + ( ℓ qq ) , (A.15)( M ) ijµν = t j Γ + ρ ( k q − ℓ qq / ℓ qq ) × S T ( ℓ qq − k q ) t i ¯ Γ + µ ( ℓ q − k qq / − k qq ) S ( ℓ q ) ∆ + ρν ( ℓ qq ) . (A.16) A.2 Kernel of the Faddeev equation
To complete the Faddeev equations, Eq. (A.11), one must specify the dressed-quark propagator,the diquark Bethe-Salpeter amplitudes and the diquark propagators.
A.2.1 Dressed-quark propagator
The dressed-quark propagator has the general form S ( p ) = − iγ · p σ V ( p ) + σ S ( p ) = 1 / [ iγ · p A ( p ) + B ( p )] (A.17)and can be obtained from QCD’s gap equation; namely, the Dyson-Schwinger equation for thedressed-quark self-energy [45]. The gap equation has been much studied and features of itssolution elucidated. Indeed, it is a longstanding prediction of DSE studies in QCD that forlight-quarks the wave function renormalisation and dressed-quark mass: Z ( p ) = 1 /A ( p ) , M ( p ) = B ( p ) /A ( p ) , (A.18)respectively, receive strong momentum-dependent corrections at infrared momenta [45, 46, 47]: Z ( p ) is suppressed and M ( p ) enhanced. These features are an expression of dynamical chiralsymmetry breaking (DCSB) and, plausibly, of confinement [48]. The enhancement of M ( p ) iscentral to the appearance of a constituent-quark mass-scale and an existential prerequisite for4 Variation of nucleon magnetic moments and radiiGoldstone modes. The mass function evolves with increasing p to reproduce the asymptoticbehaviour familiar from perturbative analyses, and that behaviour is unambiguously evidentfor p &
10 GeV [49]. These DSE predictions are confirmed in numerical simulations of lattice-regularised QCD [50], and the conditions have been explored under which pointwise agreementbetween DSE results and lattice simulations may be obtained [51, 52, 53].The impact of this infrared dressing on hadron phenomena has long been emphasised [54]and, while numerical solutions of the quark DSE are now readily obtained, the utility of analgebraic form for S ( p ) when calculations require the evaluation of numerous multidimensionalintegrals is self-evident. An efficacious parametrisation of S ( p ), which exhibits the featuresdescribed above, has been used extensively in hadron studies [55]. It is expressed via¯ σ S ( x ) = 2 ¯ m F (2( x + ¯ m )) + F ( b x ) F ( b x ) [ b + b F ( ǫx )] , (A.19)¯ σ V ( x ) = 1 x + ¯ m ˆ − F (2( x + ¯ m )) ˜ , (A.20)with x = p /λ , ¯ m = m/λ , F ( x ) = 1 − e − x x , (A.21)¯ σ S ( x ) = λ σ S ( p ) and ¯ σ V ( x ) = λ σ V ( p ). The mass-scale, λ = 0 .
566 GeV, and parametervalues ¯ m b b b b . .
131 2 .
90 0 .
603 0 . , (A.22)were fixed in a least-squares fit to light-meson observables [56, 57]. The dimensionless u = d current-quark mass in Eq. (A.22) corresponds to m = 5 .
08 MeV =: m phys . (A.23)The parametrisation yields a Euclidean constituent-quark mass M Eu,d = 0 .
33 GeV , (A.24)defined as the solution of p = M ( p ).The ratio M E /m = 65 is one expression of DCSB in the parametrisation of S ( p ). Itemphasises the dramatic enhancement of the dressed-quark mass function at infrared momenta.Another is the chiral-limit vacuum quark condensate [54] − h ¯ qq i ζ = λ π b b b ln ζ Λ (A.25)which assumes the value ( Λ QCD = 0 . − h ¯ qq i ζ =1 GeV = (0 .
221 GeV) . (A.26)A detailed discussion of the vacuum quark condensate in QCD can be found in Ref. [58, 59]An exact formula for pseudoscalar meson masses was derived in Ref. [60] and in the presentcontext it can be expressed for the pion as f π m π = − m h ¯ qq i π , (A.27)where h ¯ qq i π is an in-pion condensate [61]. In order to calculate this quantity and f π oneneeds the pion’s Bethe-Salpeter amplitude. An Ansatz compatible with the parametrisation ofthe dressed-quark propagator described above is explained in Ref. [57], and together they yield − h ¯ qq i π = (0 .
250 GeV) , f π = 0 .
090 GeV , m π = 0 .
140 GeV . (A.28) The parameters b , , , are assumed to be m -independent. In the current application, that ispossibly a weakness of the parametrisation. For example, it leads to a constituent-quark σ -termthat is 30% smaller than that obtained from the solution of a well-constrained gap equation[3]. ǫ = 10 − in Eq. (A.19) acts only to decouple the large- and intermediate- p domains.. V. Flambaum, et al. 15 A.2.2 Diquark Bethe-Salpeter amplitudes
The rainbow-ladder DSE truncation yields asymptotic diquark states in the strong interactionspectrum. Such states are not observed and their appearance is an artefact of the truncation.Higher-order terms in the quark-quark scattering kernel, whose analogue in the quark-antiquarkchannel do not much affect the properties of vector and flavour non-singlet pseudoscalar mesons,ensure that QCD’s quark-quark scattering matrix does not exhibit singularities which corre-spond to asymptotic diquark states [62]. Nevertheless, studies with kernels that do not producediquark bound states, do support a physical interpretation of the masses, m ( qq ) JP , obtainedusing the rainbow-ladder truncation: the quantity l ( qq ) JP = 1 /m ( qq ) JP may be interpreted as arange over which the diquark correlation can persist inside a baryon. These observations moti-vate an Ansatz for the quark-quark scattering matrix that is employed in deriving the Faddeevequation: [ M qq ( k, q ; K )] turs = X J P =0 + , + ,... ¯ Γ J P ( k ; − K ) ∆ J P ( K ) Γ J P ( q ; K ) . (A.29)One means of specifying the Γ J P in Eq. (A.29) is to employ the solutions of a rainbow-ladderquark-quark Bethe-Salpeter equation (BSE), as e.g. in Refs. [18, 19]. Using the properties of theGell-Mann matrices one finds easily that Γ J P C := Γ J P C † satisfies exactly the same equation asthe J − P colour-singlet meson but for a halving of the coupling strength [16]. This makes clearthat the interaction in the ¯3 c ( qq ) channel is strong and attractive.A solution of the BSE equation requires a simultaneous solution of the quark-DSE. However,since we have already chosen to simplify the calculations by parametrising S ( p ), we also employthat expedient with Γ J P , using the following one-parameter forms: Γ + ( k ; K ) = 1 N + H a Ciγ iτ F ( k /ω + ) , (A.30) t i Γ + µ ( k ; K ) = 1 N + H a iγ µ C t i F ( k /ω + ) , (A.31)with the normalisation, N J P , fixed by requiring2 K µ = » ∂∂Q µ Π ( K, Q ) – K = − m JP Q = K , (A.32) Π ( K, Q ) = tr Z d q (2 π ) ¯ Γ ( q ; − K ) S ( q + Q/ Γ ( q ; K ) S T ( − q + Q/ . (A.33)The Ans¨atze of Eqs. (A.30), (A.31) retain only that single Dirac-amplitude which wouldrepresent a point particle with the given quantum numbers in a local Lagrangian density. Theyare usually the dominant amplitudes in a solution of the rainbow-ladder BSE for the lowestmass J P diquarks [17, 18] and mesons [61, 63, 64]. A.2.3 Diquark propagators
Solving for the quark-quark scattering matrix using the rainbow-ladder truncation yields freeparticle propagators for ∆ J P in Eq. (A.29). As already noted, however, higher-order contri-butions remedy that defect, eliminating asymptotic diquark states from the spectrum. Theattendant modification of ∆ J P can be modelled efficiently by simple functions that are free-particle-like at spacelike momenta but pole-free on the timelike axis [62]; namely, ∆ + ( K ) = 1 m + F ( K /ω + ) , (A.34) ∆ + µν ( K ) = „ δ µν + K µ K ν m + « m + F ( K /ω + ) , (A.35) These forms satisfy a sufficient condition for confinement because of the associated violationof reflection positivity. See Sect. 2 of Ref. [48] for a brief discussion.6 Variation of nucleon magnetic moments and radiiwhere the two parameters m J P are diquark pseudoparticle masses and ω J P are widths charac-terising Γ J P . Herein we require additionally that ddK „ m J P F ( K /ω J P ) « − ˛˛˛˛˛ K =0 = 1 ⇒ ω J P = 12 m J P , (A.36)which is a normalisation that accentuates the free-particle-like propagation characteristics ofthe diquarks within the hadron. Appendix B: Euclidean Conventions
In our Euclidean formulation: p · q = X i =1 p i q i ; (B.1) { γ µ , γ ν } = 2 δ µν ; γ † µ = γ µ ; σ µν = i γ µ , γ ν ] ; tr [ γ γ µ γ ν γ ρ γ σ ] = − ǫ µνρσ , ǫ = 1 . (B.2)A positive energy spinor satisfies¯ u ( P, s ) ( iγ · P + M ) = 0 = ( iγ · P + M ) u ( P, s ) , (B.3)where s = ± is the spin label. It is normalised:¯ u ( P, s ) u ( P, s ) = 2 M (B.4)and may be expressed explicitly: u ( P, s ) = √ M − i E χ s σ · P M − i E χ s ! , (B.5)with E = i √ P + M , χ + = „ « , χ − = „ « . (B.6)For the free-particle spinor, ¯ u ( P, s ) = u ( P, s ) † γ .The spinor can be used to construct a positive energy projection operator: Λ + ( P ) := 12 M X s = ± u ( P, s ) ¯ u ( P, s ) = 12 M ( − iγ · P + M ) . (B.7)A negative energy spinor satisfies¯ v ( P, s ) ( iγ · P − M ) = 0 = ( iγ · P − M ) v ( P, s ) , (B.8)and possesses properties and satisfies constraints obtained via obvious analogy with u ( P, s ).A charge-conjugated Bethe-Salpeter amplitude is obtained via¯ Γ ( k ; P ) = C † Γ ( − k ; P ) T C , (B.9)where “T” denotes a transposing of all matrix indices and C = γ γ is the charge conjugationmatrix, C † = − C . Appendix C: Nucleon-Photon Vertex
In order to explicate the vertex depicted in Fig. 2 we write the scalar and axial-vector compo-nents of the nucleons’ Faddeev amplitudes in the form [cf. Eq. (A.11)] Ψ ( k ; P ) = » Ψ ( k ; P ) Ψ iµ ( k ; P ) – = » S ( k ; P ) u ( P ) A iµ ( k ; P ) u ( P ) – , i = 1 , . . . , . (C.1). V. Flambaum, et al. 17For explicit calculations, we work in the Breit frame: P µ = P BFµ − Q µ / P ′ µ = P BFµ + Q µ / P BFµ = (0 , , , i p M n + Q / D P ′ | ˆ J emµ | P E = Λ + ( P ′ ) " γ µ G E + M n P BFµ P BF ( G E − G M ) Λ + ( P ) (C.2)= Z d p (2 π ) d k (2 π ) ¯ Ψ ( − p, P ′ ) J emµ ( p, P ′ ; k, P ) Ψ ( k, P ) . (C.3)In Fig. 2 we have broken the current, J emµ ( p, P ′ ; k, P ), into a sum of six terms, each of which wesubsequently make precise. NB. Diagrams 1, 2 and 4 are one-loop integrals, which we evaluateby Gaußian quadrature. The remainder, Diagrams 3, 5 and 6, are two-loop integrals, for whoseevaluation Monte-Carlo methods are employed. C.1 Diagram 1
This represents the photon coupling directly to the bystander quark. It is expressed as J quµ = S ( p q ) ˆ Γ quµ ( p q ; k q ) S ( k q ) “ ∆ + ( k s ) + ∆ + ( k s ) ” (2 π ) δ ( p − k − ˆ ηQ ) , (C.4)where ˆ Γ quµ ( p q ; k q ) = Q q Γ µ ( p q ; k q ), with Q q = diag[2 / , − /
3] being the quark electric chargematrix, and Γ µ ( p q ; k q ) is the dressed-quark-photon vertex. In Eq. (C.4) the momenta are k q = ηP + k , p q = ηP ′ + p ,k d = ˆ ηP − k , p d = ˆ ηP ′ − p , (C.5)with η + ˆ η = 1. The results reported herein were obtained with η = 1 /
3, which provides a singlequark with one-third of the baryon’s total momentum and is thus a natural choice. Notably,as our approach is manifestly Poincar´e covariant, the precise value is immaterial so long asthe numerical methods preserve that covariance. Calculations converge most quickly with thenatural choice.It is a necessary condition for current conservation that the quark-photon vertex satisfythe Ward-Takahashi identity: Q µ iΓ µ ( ℓ , ℓ ) = S − ( ℓ ) − S − ( ℓ ) , (C.6)where Q = ℓ − ℓ is the photon momentum flowing into the vertex. Since the quark is dressed,Sec. A.2.1, the vertex is not bare; i.e., Γ µ ( ℓ , ℓ ) = γ µ . It can be obtained by solving an inho-mogeneous Bethe-Salpeter equation, which was the procedure adopted in the DSE calculationthat successfully predicted the electromagnetic pion form factor [20, 64]. However, since wehave parametrised S ( p ), we follow Ref. [54] and write [65] iΓ µ ( ℓ , ℓ ) = iΣ A ( ℓ , ℓ ) γ µ + 2 k µ ˆ iγ · k µ ∆ A ( ℓ , ℓ ) + ∆ B ( ℓ , ℓ ) ˜ ; (C.7)with k = ( ℓ + ℓ ) / Q = ( ℓ − ℓ ) and Σ F ( ℓ , ℓ ) = 12 [ F ( ℓ ) + F ( ℓ )] , ∆ F ( ℓ , ℓ ) = F ( ℓ ) − F ( ℓ ) ℓ − ℓ , (C.8)where F = A, B ; viz., the scalar functions in Eq. (A.17). It is critical that Γ µ in Eq. (C.7) satisfiesEq. (C.6) and very useful that it is completely determined by the dressed-quark propagator. C.2 Diagram 2
This figure depicts the photon coupling directly to a diquark correlation. It is expressed as J dqµ = ∆ i ( p d ) h ˆ Γ dqµ ( p d ; k d ) i ij ∆ j ( k d ) S ( k q )(2 π ) δ ( p − k + ηQ ) (C.9)with [ ˆ Γ dqµ ( p d ; k d )] ij = diag[ Q + Γ + µ , Q + Γ + µ ], where Q + = 1 / Γ + µ is given in Eq. (C.12),and Q + = diag[4 / , / , − /
3] with Γ + µ given in Eq. (C.14). Naturally, the diquark propaga-tors match the line to which they are attached.8 Variation of nucleon magnetic moments and radiiIn the case of a scalar correlation, the general form of the diquark-photon vertex is Γ + µ ( ℓ , ℓ ) = 2 k µ f + ( k , k · Q, Q ) + Q µ f − ( k , k · Q, Q ) , (C.10)and it must satisfy the Ward-Takahashi identity: Q µ Γ + µ ( ℓ , ℓ ) = Π + ( ℓ ) − Π + ( ℓ ) , Π J P ( ℓ ) = { ∆ J P ( ℓ ) } − . (C.11)The evaluation of scalar diquark elastic electromagnetic form factors in Ref. [19] is a first step to-ward calculating this vertex. However, in providing only an on-shell component, it is insufficientfor our requirements. We therefore adapt Eq. (C.7) to this case and write Γ + µ ( ℓ , ℓ ) = k µ ∆ Π ( ℓ , ℓ ) , (C.12)with the definition of ∆ Π ( ℓ , ℓ ) apparent from Eq. (C.8). Equation (C.12) is the minimal Ansatz that: satisfies Eq. (C.11); is completely determined by quantities introduced already;and is free of kinematic singularities. It implements f − ≡
0, which is a requirement for elasticform factors, and guarantees a valid normalisation of electric charge; viz.,lim ℓ ′ → ℓ Γ + µ ( ℓ ′ , ℓ ) = 2 ℓ µ ddℓ Π + ( ℓ ) ℓ ∼ = 2 ℓ µ , (C.13)owing to Eq. (A.36). NB. We have factored the fractional diquark charge, which therefore ap-pears subsequently in our calculations as a simple multiplicative factor.For the case in which the struck diquark correlation is axial-vector and the scattering iselastic, the vertex assumes the form [66]: Γ + µαβ ( ℓ , ℓ ) = − X i =1 Γ [i] µαβ ( ℓ , ℓ ) , (C.14)with ( T αβ ( ℓ ) = δ αβ − ℓ α ℓ β /ℓ ) Γ [1] µαβ ( ℓ , ℓ ) = ( ℓ + ℓ ) µ T αλ ( ℓ ) T λβ ( ℓ ) F ( ℓ , ℓ ) , (C.15) Γ [2] µαβ ( ℓ , ℓ ) = [ T µα ( ℓ ) T βρ ( ℓ ) ℓ ρ + T µβ ( ℓ ) T αρ ( ℓ ) ℓ ρ ] F ( ℓ , ℓ ) , (C.16) Γ [3] µαβ ( ℓ , ℓ ) = − m + ( ℓ + ℓ ) µ T αρ ( ℓ ) ℓ ρ T βλ ( ℓ ) ℓ λ F ( ℓ , ℓ ) . (C.17)This vertex satisfies: ℓ α Γ + µαβ ( ℓ , ℓ ) = 0 = Γ + µαβ ( ℓ , ℓ ) ℓ β , (C.18)which is a general requirement of the elastic electromagnetic vertex of axial-vector bound statesand guarantees that the interaction does not induce a pseudoscalar component in the axial-vector correlation. We note that the electric, magnetic and quadrupole form factors of anaxial-vector bound state are expressed [66] G + E ( Q ) = F + 23 τ + G + Q ( Q ) , τ + = Q m + (C.19) G + M ( Q ) = − F ( Q ) , (C.20) G + Q ( Q ) = F ( Q ) + F ( Q ) + (1 + τ + ) F ( Q ) . (C.21)Owing to the fact that Γ J P C := Γ J P C † satisfies exactly the same Bethe-Salpeter equation asthe J − P colour-singlet meson but for a halving of the coupling strength, the vector meson formfactor calculation in Ref. [21] might become useful as a guide in understanding the form factorsin Eqs. (C.14)–(C.17). However, in providing only an on-shell component, that information isinsufficient for our requirements. Hence, we employ the following Ans¨atze : F ( ℓ , ℓ ) = ∆ Π ( ℓ , ℓ ) , (C.22) F ( ℓ , ℓ ) = − F + (1 − τ + ) ( τ + F + 1 − µ + ) d ( τ + ) (C.23) F ( ℓ , ℓ ) = − ( χ + (1 − τ + ) d ( τ + ) + F + F ) d ( τ + ) , (C.24) If the scattering is inelastic the general form of the vertex involves eight scalar functions [67].For simplicity, we ignore the additional structure in this
Ansatz .. V. Flambaum, et al. 19with d ( x ) = 1 / (1 + x ) . This construction ensures a valid electric charge normalisation for theaxial-vector correlation; viz.,lim ℓ ′ → ℓ Γ + µαβ ( ℓ ′ , ℓ ) = T αβ ( ℓ ) ddℓ Π + ( ℓ ) ℓ ∼ = T αβ ( ℓ ) 2 ℓ µ , (C.25)owing to Eq. (A.36), and current conservationlim ℓ → ℓ Q µ Γ + µαβ ( ℓ , ℓ ) = 0 . (C.26)The diquark’s static electromagnetic properties follow: G + E (0) = 1 , G + M (0) = µ + , G + Q (0) = − χ + . (C.27)For an on-shell or pointlike axial-vector: µ + = 2; and χ + = 1, which corresponds to anoblate charge distribution. In addition, Eqs. (C.14)–(C.17) with Eqs. (C.22)–(C.24) realise theconstraints of Ref. [68]; namely, independent of the values of µ + & χ + , the form factors assumethe ratios G + E ( Q ) : G + M ( Q ) : G + Q ( Q ) Q →∞ = (1 − τ + ) : 2 : − . (C.28)We note that within a nucleon the diquark correlation is not on-shell. Hence, in con-trast with Ref. [24], herein we do not assume that point-particle values for the magnetic andquadrupole moments in Eqs. (C.27) serve as a good reference point. For the processes describedby Fig. 2, the values can be much smaller in magnitude [33], as we find in Table 2. C.3 Diagram 3
This image depicts a photon coupling to the quark that is exchanged as one diquark breaks upand another is formed. It is expressed as J exµ = − S ( k q ) ∆ i ( k d ) Γ i ( p , k d ) S T ( q ) ˆ Γ quTµ ( q ′ , q ) S T ( q ′ ) ¯ Γ jT ( p ′ , p d ) ∆ j ( p d ) S ( p q ) , (C.29)wherein the vertex ˆ Γ quµ appeared in Eq. (C.4). While this is the first two-loop diagram we havedescribed, no new elements appear in its specification: the dressed-quark-photon vertex wasdiscussed in Sec. C.1. In Eq. (C.29) the momenta are q = ˆ ηP − ηP ′ − p − k , q ′ = ˆ ηP ′ − ηP − p − k ,p = ( p q − q ) / , p ′ = ( − k q + q ′ ) / ,p ′ = ( p q − q ′ ) / , p = ( − k q + q ) / . (C.30)It is noteworthy that the process of quark exchange provides the attraction necessary inthe Faddeev equation to bind the nucleon. It also guarantees that the Faddeev amplitude hasthe correct antisymmetry under the exchange of any two dressed-quarks. This key feature isabsent in models with elementary (noncomposite) diquarks. The full contribution is obtainedby summing over the superscripts i, j , which can each take the values 0 + , 1 + . C.4 Diagram 4
This differs from Diagram 2 in expressing the contribution to the nucleons’ form factors owingto an electromagnetically induced transition between scalar and axial-vector diquarks. Theexplicit expression is given by Eq. (C.9) with [ ˆ Γ dqµ ( p d ; k d )] i = j = 0, and [ ˆ Γ dqµ ( p d ; k d )] , = Γ SA and [ ˆ Γ dqµ ( p d ; k d )] , = Γ AS . This transition vertex is a rank-2 pseudotensor, kindred to thematrix element describing the ρ γ ∗ π transition [69], and can therefore be expressed Γ γαSA ( ℓ , ℓ ) = − Γ γαAS ( ℓ , ℓ ) = iM N T ( ℓ , ℓ ) ε γαρλ ℓ ρ ℓ λ , (C.31)where γ , α are, respectively, the vector indices of the photon and axial-vector diquark. Forsimplicity we proceed under the assumption that T ( ℓ , ℓ ) = κ T ; (C.32)0 Variation of nucleon magnetic moments and radiiviz., a constant, for which a typical on-shell value is κ T ∼ µ + and χ + , we recognise herein that this value is not a useful reference point because, for the processesdescribed by Fig. 2, κ T can be much smaller in magnitude.In the nucleons’ rest frame, a conspicuous piece of the Faddeev amplitude that describesan axial-vector diquark inside the bound state can be characterised as containing a bystanderquark whose spin is antiparallel to that of the nucleon, with the axial-vector diquark’s spinparallel. The interaction pictured in this diagram does not affect the bystander quark but thetransformation of an axial-vector diquark into a scalar effects a flip of the quark spin within thecorrelation. After this transformation, the spin of the nucleon must be formed by summing thespin of the bystander quark, which is still aligned antiparallel to that of the nucleon, and theorbital angular momentum between that quark and the scalar diquark. This argument, whilenot sophisticated, does motivate an expectation that Diagram 4 will impact upon the nucleons’magnetic form factors. C.5 Diagrams 5 & 6
These two-loop diagrams are the so-called “seagull” terms, which appear as partners to Dia-gram 3 and arise because binding in the nucleons’ Faddeev equations is effected by the exchangeof nonpointlike diquark correlations [26]. The explicit expression for their contribution to thenucleons’ form factors is J sgµ = 12 S ( k q ) ∆ i ( k d ) “ X iµ ( p q , q ′ , k d ) S T ( q ′ ) ¯ Γ jT ( p ′ , p d ) − Γ i ( p , k d ) S T ( q ) ¯ X jµ ( − k q , − q, p d ) ” ∆ j ( p d ) S ( p q ) , (C.33)where, again, the superscripts are summed.The new elements in these diagrams are the couplings of a photon to two dressed-quarks asthey either separate from (Diagram 5) or combine to form (Diagram 6) a diquark correlation. Assuch they are components of the five point Schwinger function which describes the coupling of aphoton to the quark-quark scattering kernel. This Schwinger function could be calculated, as isevident from the computation of analogous Schwinger functions relevant to meson observables[71]. However, such a calculation provides valid input only when a uniform truncation of theDSEs has been employed to calculate each of the elements described hitherto. We must insteademploy an algebraic parametrisation [26], which for Diagram 5 reads X J P µ ( k, Q ) = e by k µ − Q µ k · Q − Q h Γ J P ( k − Q/ − Γ J P ( k ) i + e ex k µ + Q µ k · Q + Q h Γ J P ( k + Q/ − Γ J P ( k ) i , (C.34)with k the relative momentum between the quarks in the initial diquark, e by the electric chargeof the quark which becomes the bystander, and e ex the charge of the quark that is reabsorbedinto the final diquark. Diagram 6 has¯ X J P µ ( k, Q ) = e by k µ + Q µ k · Q + Q h ¯ Γ J P ( k + Q/ − ¯ Γ J P ( k ) i + e ex k µ − Q µ k · Q − Q h ¯ Γ J P ( k − Q/ − ¯ Γ J P ( k ) i , (C.35)where ¯ Γ J P ( ℓ ) is the charge-conjugated amplitude, Eq. (B.9). Plainly, these terms vanish if thediquark correlation is represented by a momentum-independent Bethe-Salpeter-like amplitude;i.e., the diquark is pointlike. Another component of the amplitude has the bystander quark’s spin parallel to that of thenucleon while the axial-vector diquark’s is antiparallel: this q ↑ ⊕ ( qq ) ↓ + system has one unit ofangular momentum. That momentum is absent in the q ↑ ⊕ ( qq ) + system. Other combinationsalso contribute via Diagram 3 but all mediated processes inevitably require a modification ofspin and/or angular momentum. An analysis of the contribution from quark orbital angularmomentum to a nucleon’s spin is presented in Ref. [44].. V. Flambaum, et al. 21It is naturally possible to use more complicated Ans¨atze . However, like Eq. (C.12),Eqs. (C.34) & (C.35) are simple forms, free of kinematic singularities and sufficient to ensurethe nucleon-photon vertex satisfies the Ward-Takahashi identity when the composite nucleonis obtained from the Faddeev equation.
Appendix D: Charge Symmetry
Our analysis assumes m u = m d . Hence the only difference between the u - and d -quarks istheir electric charge. Our equations and computer codes therefore exhibit the following chargesymmetry relations: µ un = − µ dp , µ dn = − µ up , (D.1)where µ un means the contribution from the u -quark to the magnetic moment of the neutron,etc.; and furthermore δµ pn = − δµ dp , δµ dn = − δµ up , (D.2)where, in this section, δµ pn means the variation in µ pn owing to a small change in current-quarkmass. Using these equations, one obtains δµ p µ p = δµ up + δµ dp µ up + µ dp , δµ n µ n δµ up + 4 δµ dp µ up + 4 µ dp . (D.3)Our Faddeev equation model yields µ up = 2 . , δµ up = − . ,µ dp = 0 . , δµ dp = 0 . , (D.4)from which we obtain the results in Tables 3 and 4.It is interesting to provide a context for the results in Eq. (D.4). Suppose one were requiredto reproduce µ q ( qq ) p in Eq. (15) with nonrelativistic pointlike constituent-quarks. Such quarkshave the magnetic moments: µ U = 2¯ µ Q , µ D = − µ Q , (D.5)in terms of which µ Q ( QQ ) p = 43 µ U − µ D =: µ Up + µ Dp . (D.6)With ¯ µ Q = 0 .
85 one reproduces Eq. (15) and finds µ Up = 2 . , µ Dp = 0 . . (D.7)A comparison between Eqs. (D.4) and (D.7) indicates the presence of correlations in our Faddeevamplitude for the nucleon. Relative to a generic constituent-quark model, they increase theprobability for a u -quark to have its spin aligned with that of the proton, and markedly decreasethat probability for the d -quark. References
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