Peripheral transverse densities of the baryon octet
from chiral effective field theory and dispersion analysis
The baryon electromagnetic form factors are expressed in terms of two-dimensional densities describing the distribution of charge and magnetization in transverse space at fixed light-front time. We calculate the transverse densities of the spin- flavor-octet baryons at peripheral distances using methods of relativistic chiral effective field theory (EFT) and dispersion analysis. The densities are represented as dispersive integrals over the imaginary parts of the form factors in the timelike region (spectral functions). The isovector spectral functions on the two-pion cut are calculated using relativistic EFT including octet and decuplet baryons. The EFT calculations are extended into the meson mass region using an method that incorporates the pion electromagnetic form factor data. The isoscalar spectral functions are modeled by vector meson poles. We compute the peripheral charge and magnetization densities in the octet baryon states, estimate the uncertainties, and determine the quark flavor decomposition. The approach can be extended to baryon form factors of other operators and the moments of generalized parton distributions.
keywords:Electromagnetic form factors, chiral lagrangians, dispersion relations, hyperons, charge distribution
Pacs:12.40.Vv, 13.40.Gp, 11.55.Fv, 13.60.Hb
Exploring the spatial structure of hadrons and their interactions is a central objective of modern strong interaction physics. The description of hadrons as extended objects in relativistic space-time enables an intuitive understanding of their structure and permits the formulation of novel approximation methods based on physical distance scales. The most basic information about spatial structure comes from the transition matrix elements (or form factors) of electroweak current operators, which reveal the distributions of charge and current in the system. In non-relativistic systems, such as nuclei in nuclear many-body theory, the form factors are represented as the Fourier transform of 3-dimensional spatial distributions of charge and current at a fixed instant of time const. These distributions can be related to probability densities of the -particle Schrödinger wave function. In relativistic systems such as hadrons the formulation of a density requires separating the structure of the “system” from that of the “probe” in the presence of vacuum fluctuations that change the number of constituents. This is accomplished in a frame-independent manner by viewing the system at fixed light-front time (light-front quantization) Dirac:1949cp (); Leutwyler:1977vy (); Lepage:1980fj (); Brodsky:1997de (). In this framework the form factors are represented in terms of 2-dimensional spatial distributions of charge and current at fixed light-front time const. Soper:1976jc (); Burkardt:2000za (); Burkardt:2002hr (); Miller:2007uy (). The transverse densities are frame-independent (they remain invariant under longitudinal boosts and transform kinematically under transverse boosts) and provide an objective spatial representation of the hadron as a relativistic system. The properties of the transverse densities of mesons and baryons, their extraction from experimental data, and their explanation in dynamical models have been discussed extensively in the literature Miller:2007uy (); Carlson:2007xd (); Vanderhaeghen:2010nd (); Venkat:2010by (); Miller:2009sg (); see Ref. Miller:2010nz () for a review. An important aspect is that the transverse densities are naturally related to the partonic structure of hadrons probed in high-energy short-distance processes (generalized parton distributions, or GPDs). The densities describe the cumulative charge and current carried by the quarks/antiquarks in the hadron at a given transverse position and represent the integral of the GPDs over the quark/antiquark light-front momentum fraction Burkardt:2000za (); Burkardt:2002hr (); Diehl:2002he (); see Refs. Belitsky:2005qn (); Boffi:2007yc () for a review. This relation places the study of elastic form factors in the context of exploring the 3-dimensional quark-gluon structure of hadrons in QCD.
Of particular interest are the transverse densities at distances of the order , where the pion mass is regarded as parametrically small on the hadronic mass scale (as represented e.g. by the vector meson mass ) Strikman:2010pu (); Granados:2013moa (). At such “peripheral” distances the densities are governed by the effective dynamics resulting from the spontaneous breaking of chiral symmetry and can be computed model-independently using methods of chiral effective field theory (EFT) Gasser:1983yg (); Gasser:1984gg (); see Refs. Bernard:1995dp (); Scherer:2002tk (); Scherer:2012zzd () for a review. The peripheral densities are generated by chiral processes in which the hadron couples to the current operator through soft-pion exchange in the -channel. The nucleon’s peripheral charge and magnetization densities (related to the electromagnetic form factors and ) were studied in Ref. Granados:2013moa () using relativistic EFT in the leading-order approximation. The densities decay exponentially at large with a range of the order , and exhibit a rich structure due to the coupling of the two-pion exchange to the nucleon. It was found that the peripheral charge and current densities are similar in magnitude and related by an approximate inequality. These properties are a consequence of the essentially relativistic nature of chiral dynamics [the typical pion momenta are ] and can be understood in a simple quantum-mechanical picture of peripheral nucleon structure Granados:2015rra (); Granados:2015lxa (); Granados:2016jjl (). It was also shown that the EFT densities exhibit the correct -scaling behavior if contributions from isobar intermediate states are included Granados:2013moa (); Granados:2016jjl ().
The study of peripheral transverse densities is closely connected with the dispersive representation of the nucleon form factors. The form factors are analytic functions of the invariant momentum transfer and can be represented as dispersive integrals of their imaginary parts (spectral functions) on the two-pion cut at timelike . The integration extends over the unphysical region below the nucleon-antinucleon threshold, where the spectral functions cannot be measured directly, but can be calculated theoretically using EFT Gasser:1987rb (); Bernard:1996cc (); Becher:1999he (); Kubis:2000zd (); Kaiser:2003qp () and dispersion theory Frazer:1960zza (); Frazer:1960zzb (); Hohler:1974ht (); Belushkin:2005ds (); Hoferichter:2016duk (), or determined by fits to the spacelike form factor data Hohler:1976ax (); Belushkin:2006qa (); Lorenz:2012tm (). The transverse densities are given by similar dispersive integrals, in which the spectral functions are integrated with an exponential weight factor , suppressing the contribution from large masses Strikman:2010pu (); Miller:2011du (). At distances the integration is limited to the near-threshold region , where the spectral functions are governed by chiral dynamics. The peripheral densities thus represent “clean” chiral observables and are ideally suited for EFT calculations. Quantitative studies in Ref. Miller:2011du () found that at distances the dispersive integrals for the densities are dominated by the region , where the isovector spectral functions can be computed directly using fixed-order EFT. To obtain the densities at smaller distances one needs to construct the spectral functions at larger values of , where they are affected by strong rescattering in the -channel and the meson resonance at . This can be accomplished in an approach that combines EFT with a dispersive technique based on elastic unitarity in the -channel.
In this article we study the peripheral transverse densities of the spin-1/2 flavor-octet baryons using methods of EFT and dispersive analysis. We calculate the spectral functions of the isovector electromagnetic form factors on the two-pion cut at , using relativistic EFT with flavor group and explicit spin-3/2 decuplet degrees of freedom at in the small-scale expansion. The calculations are extended from the near-threshold region to masses and higher, using a dispersive technique that incorporates rescattering through elastic unitarity and the pion electromagnetic form factor data. The method allows us to compute the isovector transverse densities at distances with controlled uncertainties. For the isoscalar densities we construct an empirical parametrization of the spectral function in terms of vector meson poles () using symmetry. We present the isovector and isoscalar peripheral transverse densities of the individual octet baryons and discuss their properties. We also calculate the flavor decomposition of the peripheral densities and compare it to the baryons’ valence quark content.
The treatment of spectral functions and transverse densities presented here goes beyond that of previous studies in several aspects. First, the EFT calculations are performed with the flavor group, which allows us to study for the first time the peripheral densities of all octet baryons resulting from two-pion exchange in the -channel (contributions from two-kaon exchange are included as well but turn out to be negligible at peripheral distances). Second, the spin-3/2 decuplet baryons are included in the EFT calculations using a consistent Lagrangian, which eliminates off-shell effects in physical observables. This approach, together with a fully relativistic formulation of EFT with baryons via the extended-on-mass-shell (or EOMS) scheme Fuchs:2003qc (), overcomes certain specific difficulties that arise in EFT calculations of fundamental processes Becher:2001hv (); Alarcon:2011kh (); Alarcon:2012kn (); Lensky:2014dda (); Blin:2016itn () and has been applied successfully to the form factors of the nucleon and the isobar Ledwig:2011cx (), the magnetic moments of the octet and decuplet baryons Geng:2008mf (), the axial-vector charge of the octet Ledwig:2014rfa (), and to the calculation of quantities like the nucleon -terms and two photon exchange corrections to the muonic hydrogen Lamb shift Alarcon:2011zs (); Alarcon:2012nr (); Alarcon:2013cba (), which are used in searches of physics beyond the standard model. Earlier EFT calculations of peripheral densities Granados:2013moa (); Granados:2016jjl () included the isobar through a “minimal” coupling with off-shell dependence; this scheme was sufficient to achieve proper -scaling of the spectral functions and densities, but could not fix the contact terms associated with the isobar contribution; see Ref. Granados:2013moa (); Granados:2016jjl () for details.
Third, we extend the range of the EFT calculation of the spectral function beyond the near-threshold region by combining it with a dispersive method based on elastic unitarity. It uses the representation of the spectral functions on the two-pion cut as the product of the -channel partial-wave amplitudes in the channel, , and the complex-conjugate pion form factor, . Chiral EFT is used to calculate the real function , in which the common phase due to rescattering cancels by virtue of the Watson theorem Frazer:1960zza (); Frazer:1960zzb (); Hohler:1974ht (); Watson:1954uc (); the result is then multiplied by the empirical measured in annihilation experiments, which contains the rescattering effects and the resonance. The “improved” EFT spectral functions thus obtained agree well with the empirical spectral functions (obtained by analytic continuation of the partial wave amplitudes) Hohler:1976ax (); Belushkin:2005ds () up to and have qualitatively correct behavior even in the meson mass region. The agreement is achieved without adding free parameters and represents a genuine prediction of EFT at ; the inclusion of higher-order chiral corrections might further improve the accuracy of the results in the meson mass region. The method allows us to compute the isovector transverse densities down to distances with controlled accuracy and results in a major overall improvement in the description of peripheral nucleon structure. The method could be extended to the spectral functions of other form factors with two-pion cut and their transverse densities InPreparation (). The technique described here is a variant of the method of amplitude analysis Chew:1960iv (), which was employed in applications of EFT to meson-meson, meson-baryon, and baryon-baryon scattering Alarcon:2011kh (); Oller:1998zr (); Oller:2000ma (); Meissner:1999vr (); Oller:2000fj (); Albaladejo:2011bu (); Albaladejo:2012sa (). A similar approach was used in a recent study of the - transition form factors Granados:2017cib ().
The article is structured as follows. In Sec. 2 we summarize the basic properties of the transverse densities and discuss their dispersive representation. In particular, we analyze the distribution of strength in the dispersion integral for the densities at different distances and identify the regions of where we need to calculate the spectral functions. In Sec. 3 we describe the EFT calculation of the isovector spectral functions, the improvement using dispersion theory, and the modeling of isoscalar spectral functions. In Sec. 4 we present the results for the charge and magnetization densities of the octet baryons and discuss their properties. We also perform the flavor separation of the transverse densities and discuss the connection with partonic structure. A summary and outlook on further studies are given in Sec. 5. Technical material is collected in the appendices. A describes the validation of the form factor calculations. B discusses the role of the off-shell behavior of the EFT with isobars and compares our results with previous calculations. C describes our model for the coupling of isoscalar vector mesons to the octet baryons. Preliminary results of the present study were reported in Ref. Alarcon:2017lkk ().
2 Transverse densities
2.1 Form factors and transverse densities
The transition matrix element of the electromagnetic current between states of a spin-1/2 baryon with 4-momenta and is of the general form
where and are the Dirac spinors of the baryon states, is the baryon mass, is the 4-momentum transfer, and . The form factors and (Dirac and Pauli form factor) are functions of the Lorentz-invariant momentum transfer , with in the physical region of elastic scattering. They are invariant functions and can be measured and interpreted without specifying the form of relativistic dynamics or choosing a reference frame. The isospin decomposition of Eq. (1) in the case of octet baryons () will be described below.
In the light-front form of relativistic dynamics one considers the evolution of strong interactions in light-front time . The baryon states are parametrized by their light-front momentum and transverse momentum , whereas plays the role of the energy, with on the baryon mass shell . In this context one naturally considers the current matrix element in a class of reference frames where the momentum transfer has only transverse components, . The form factors are then represented as Fourier transforms of two-dimensional spatial densities (transverse densities)
where is a transverse coordinate variable and ; we follow the conventions of Ref. Granados:2013moa (). The spatial integral of the densities reproduces the total charge and anomalous magnetic moment of the baryons,
The functions thus describe the transverse spatial distribution of charge and magnetization in the baryon. The interpretation of and as spatial densities has been discussed extensively in the literature Burkardt:2000za (); Burkardt:2002hr (); Miller:2010nz () and is summarized in Ref. Granados:2013moa (). In a state where the baryon is localized in transverse space at the origin, and its spin (or light-front helicity) quantized in the -direction, the expectation value of the current at and transverse position is given by
Here represents a factor resulting from the normalization of states (see Ref. Granados:2013moa () for details), is the angle of the vector relative to the -axis, and is the spin projection in the -direction in the baryon rest frame (see Fig. 1). describes the spin-independent part of the plus current in a localized baryon state, while describes the spin-dependent part of the current in a transversely polarized baryon. Other aspects of the transverse densities, such as their connection with the partonic structure and GPDs, are discussed in the literature and reviewed in Ref. Miller:2010nz ().
In the present work the concepts of light-front quantization are used only for the interpretation of the transverse densities but will not be employed in the actual calculations. Dynamical calculations (chiral EFT, dispersion theory) will be performed at the level of the invariant form factors, from which the densities can be obtained by inverting the Fourier transform of Eq. (2). The advantage of transverse densities is precisely that they connect light-front structure with the invariant form factors, for which well-tested theoretical methods are available. An alternative approach, in which dynamical calculations in chiral EFT are performed directly in light-front quantization, is described in Refs. Granados:2015rra (); Granados:2015lxa (); Granados:2016jjl ().
2.2 Dispersive representation
The baryon form factors are analytic functions of , with singularities (branch points, poles) on the positive real axis. Assuming power-like asymptotic behavior , as expected in perturbative QCD and supported by present experimental data for the nucleon, the form factors satisfy unsubtracted dispersion relations of the form
Here the form factors are expressed as integrals of their imaginary parts (or spectral functions) on the principal cut on the physical sheet. The singularities at correspond to processes in which the current produces a hadronic state that couples to the baryon-antibaryon system, current hadronic state (see Fig. 2). The hadronic state with the lowest mass is the two-pion state with threshold at , produced by the isovector current (anomalous thresholds with in the strange octet baryon form factors will be discussed in Sec. 3.2). At larger values the dominant states are the vector mesons appearing as resonances in the two-pion channel (, isovector) and the three-pion channel (, isoscalar). At even higher the spectral functions receive contributions from states and the resonance, and from other multi-hadron states whose composition is not known in detail. Most of the relevant states lie below the threshold ( GeV for the nucleon), in the unphysical region of the current process, where the spectral functions cannot be measured directly in conversion experiments, but can only be calculated theoretically. Methods for constructing the spectral functions in different regions of (EFT, dispersion analysis) will be described below.
A similar dispersive representation can be derived for the transverse densities associated with the form factors, Eq. (2). Calculating the transverse Fourier transform of the form factors as given by Eq. (7), and using Eq. (6), one obtains Strikman:2010pu (); Miller:2011du ()
where denotes the modified Bessel functions of the second kind. This representation has several interesting properties. (a) The modified Bessel functions decay exponentially at large arguments,
The dispersive integrals for the densities therefore converge exponentially at large , while the original integrals for the form factors converge only like a power. This greatly reduces the contribution from the high-mass region where the spectral functions are poorly known. (b) The density at a certain is given by the integral of the spectral function with the “exponential filter” . The distance therefore represents an external parameter that allows one to effectively select (emphasize or de-emphasize) certain mass regions in the spectral function. The large-distance behavior of the densities is governed by the lowest-mass states in a given channel. We shall use this property to identify the region of in which the densities are dominated by values of close to the two-pion threshold, where the spectral functions can be computed using EFT and dispersion theory. (c) The representations Eqs. (8) and (9) incorporate the correct analytic behavior of the form factors and can be used to study the large-distance asymptotics of the densities.111Form factor representations with incorrect analytic properties in (e.g., with artificial singularities at complex ) are principally not adequate for studying the large-distance asymptotics of the densities, even if they provide good fits to the spacelike form factor data at small ; see Ref. Miller:2011du () for a discussion.
It is instructive to study the distribution of strength in in the dispersive integrals for the baryon densities, Eqs. (8) and (9), at different Miller:2011du (). For this purpose we consider the nucleon form factors, whose isovector spectral functions at have been calculated using amplitude analysis techniques (analytic continuation of the partial-wave amplitudes Belushkin:2005ds (); Hohler:1976ax (), Roy-Steiner equations Hoferichter:2016duk ()). Figures 3a and b show the spectral functions of the nucleon’s isovector form factors and obtained in Ref. Hoferichter:2016duk (). One observes the rise of the spectral functions in the region above the two-pion threshold , and the prominent role of the resonance at (the rapid variation on the right shoulder of the is due to - mixing caused by isospin breaking). Figures 3c and d show the distribution of strength in the dispersive integrals Eqs. (8) and (9), i.e., the spectral functions multiplied by and , for and 3 fm. (The plots show the distributions in normalized to unit area under the curves, to facilitate comparison of the relative distribution of strength at different .) One sees that for fm the dominant contribution to the integrals arises from the near-threshold region GeV, while the vector meson region is very strongly suppressed. For fm the near-threshold region still gives the main contribution to the integrals, but the vector meson region becomes noticeable, especially in . For fm, finally, the vector meson mass region gives the main contribution to the integrals. These estimates illustrate in what regions of we need to calculate (or model) the spectral functions in order to compute the densities at a given , and how the uncertainties of the spectral function in the different regions affect the overall uncertainty of the densities. Similar estimates can be performed for the isoscalar densities of the nucleon and the densities of the other flavor-octet baryons, using the spectral functions calculated below.
In this work we use relativistic EFT to calculate the isovector spectral functions of the flavor-octet baryons on the two-pion cut. The relativistic EFT framework correctly implements the subthreshold singularities of the form factors on the unphysical sheet, which result from the baryon Born terms and account for the steep rise of the isovector spectral function above threshold Gasser:1987rb (); Bernard:1996cc (); Becher:1999he (); Kubis:2000zd (); Kaiser:2003qp (). The inclusion of the decuplet baryons as dynamical degrees of freedom implements also the Born term singularities further removed from the physical region, which become important at larger . To extend the EFT calculations of the spectral functions into the vector meson mass region we combine them with a dispersive technique that incorporates the pion form factor data from annihilation experiments Frazer:1960zza (); Frazer:1960zzb (); Hohler:1976ax (). In this way we construct the isovector spectral functions of the octet baryons up to GeV with an estimated uncertainty , and up to GeV with somewhat larger uncertainty. The isoscalar spectral functions we model by phenomenological vector meson exchange (). Non-resonant isoscalar contributions from kaon loops in the EFT turn out to be negligible. Altogether, these methods allow us to compute the peripheral densities of the octet baryons at distances 1 fm with controlled accuracy.
2.3 Isospin structure
In the present study we consider the form factors and densities of baryons in the octet representation of the flavor group. The electromagnetic current operator in QCD with 3 flavors is given by , where is the quark field and the quark charge matrix,
where are the Gell-Mann matrices. The term in the current transforms as an isovector under isospin rotations (); the term transforms as an isoscalar (), and the current can be represented as the sum of an isovector and an isoscalar current, . The matrix elements of the isovector and isoscalar currents can be specified for each baryon state.
The baryons in the octet representation of form four isospin multiplets
Within each multiplet we write the form factors as the sum/difference of an isoscalar and isovector component,
For the nucleon this corresponds to the standard definition of the isoscalar and isovector form factors as . The form factors are pure isoscalar. In the form factors the isovector component is absent because the transition is forbidden by isospin symmetry. The - transition form factors are pure isovector. Equation (13) embodies the constraints imposed on the octet baryon form factors by isospin symmetry. The same decomposition applies to the spectral functions and the transverse densities.
3 Spectral functions
3.1 Chiral effective field theory
For the calculation of the baryon isovector spectral functions we use a manifestly Lorentz-covariant version of EFT with flavor group including spin-1/2 flavor-octet and spin-3/2 flavor-decuplet baryons. The pseudoscalar mesons and spin-1/2 baryons are described by fields in the octet representation (rank-2 tensors)
The spin-3/2 baryons are described by fields in the decuplet representation (rank-3 totally symmetric tensor)
The spin-1/2 fields are introduced as relativistic bispinor fields (Dirac fields). The spin-3/2 fields are introduced as 4-vector-bispinor fields , which have to be subjected to relativistically covariant constraints to eliminate spurious spin-1/2 degrees of freedom. In the present formulation of the EFT the projection on spin-3/2 is implemented through the use of consistent interaction vertices (see below); i.e., the spin-1/2 degrees of freedom are allowed to propagate but are filtered out in the interaction Pascalutsa:1998pw (); Pascalutsa:1999zz (); Pascalutsa:2000kd (); Krebs:2008zb (). Electromagnetic interactions are introduced by the coupling to a 4-vector background field in the octet representation,
The construction of the chiral Lagrangian with spin-3/2 fields and the formulation of a small-scale expansion have been described in Refs. Hemmert:1996xg (); Hemmert:1997ye (). In the -expansion the octet-decuplet baryon mass splitting is counted at the same order as the pseudoscalar octet meson mass and momentum,
where denotes the generic expansion parameter. The chiral order of a diagram is given by
where is the number of loops, the number of propagators of hadrons of the type and the number of vertices of a Lagrangian of order . The standard power counting of EFT is recovered with the extended-on-mass-shell (EOMS) scheme Fuchs:2003qc (), whereby the divergent parts of loop diagrams, along with the power-counting-breaking terms, are absorbed into the low-energy constants following the prescription.
The set of EFT diagrams contributing to the spectral functions of the form factors on the two-pion cut at is shown in Fig. 4.222To clarify the parametric counting, we note that refers to the order of the baryon electromagnetic vertex function, in which the background field is counted as . The actual order of the spectral function extracted in this way is therefore . Following standard usage we refer to our calculation as , keeping in mind this distinction. These are the loop diagrams in which the electromagnetic field couples to the baryon through two-pion exchange in the -channel: the “triangle” diagrams (a) and (c) with octet and decuplet intermediate states, and the “tadpole” diagram (b). Within the dispersive representation Eq. (7) these diagrams account for the densities at peripheral distances . The two-kaon cut of the form factor at , which results from the corresponding two-kaon exchange diagrams in Fig. 4, contributes only to the densities at much shorter distances and can be neglected in the periphery (see below). The EFT diagrams in which the electromagnetic field couples to the baryon or the meson-baryon vertices (not shown in Fig. 4) have cuts only at , or no cuts at all (polynomials); their contributions modify the densities only at distances or through delta functions at and can be neglected in the periphery.
For reference we list here the Lagrangians generating the propagators and vertices in the diagrams of Fig. 4. The lowest-order Lagrangian for the interaction between octet mesons and the electromagnetic field is
where the vielbein is given by , , ; and the mass term is given by with , where are the quark masses; and is the meson decay constant in the chiral limit. For the couplings with the octet baryons we need the leading-order meson-baryon Lagrangian
where the covariant derivative acts as , with the chiral connection given by . The axial-vector fields are set to zero in our case, and the two axial couplings are fixed as and Alarcon:2012nr (). Finally, when introducing the decuplet baryons, the Lagrangians that are needed are
where is the octet-decuplet axial coupling, , , and . The covariant derivative acts on the decuplet as , where . The vertices in Eqs. (22) and (23) are consistent vertices (they transform covariantly under point transformations), which effectively implement the projection on spin-3/2 degrees of freedom.
The spectral functions (imaginary parts) resulting from the -channel cut of the diagrams of Fig. 4 can be obtained by applying cutting rules Granados:2013moa () and are given by finite phase-space integrals, which do not require renormalization. We have also calculated the form factors themselves from the full set of diagrams for the baryon electromagnetic vertex function (not shown in Fig. 4), including loop diagrams and their renormalization, and verified electromagnetic gauge invariance of the result. We have confirmed that our expressions reproduce the ones of Ref. Ledwig:2011cx () when reduced to the flavor group. Further tests validating our calculation are described in A.
For the numerical evaluation we consider a range of values for the couplings and baryon masses that goes from the values ( MeV, MeV and MeV, ; see Ref. Ledwig:2011cx ()) to the average values ( MeV, MeV, , ; see Refs. Ledwig:2014rfa (); Alarcon:2012nr ()). This provides an uncertainty band associated with the systematic errors and gives an estimate of the expected higher-order corrections. The meson masses are taken at their physical values, MeV, MeV. According to the chiral expansion the meson mass differences are of the same order as the meson masses themselves, whereas the baryons have a common mass in the chiral limit and the mass splittings are suppressed. This circumstance has important consequences for the description of peripheral baryon structure, as it ensures that the pion and kaon exchange contributions to the densities are evaluated with the physical masses and possess the correct ranges and already in the lowest order.
The EFT results for the nucleon isovector spectral functions are shown in Fig. 5. The steep rise above the threshold at is caused by the subthreshold singularity at (on the unphysical sheet of the nucleon form factor), which results from the triangle diagram Fig. 4 (a) with an intermediate nucleon and is a general feature of the analytic structure. In the diagram (c) with intermediate isobar the subthreshold singularity is further removed from threshold, resulting in a smaller contribution. The intermediate contribution reduces the intermediate result in the case of , and enhances it in the case of . We note that this pattern ensures the proper scaling behavior of the form factors in the large- limit of QCD, where the and become degenerate Granados:2013moa (); Granados:2016jjl ().
In B we compare the results of the present calculation with those of Ref. Granados:2013moa (); Granados:2016jjl (), which used an “inconsistent” form of the vertex. The results for the contribution agree at but differ by higher-order terms, which cause numerical differences of 20% in (50% in ) at (see Fig. 12).
3.2 Improvement through unitarity
The EFT expressions by themselves describe the baryon isovector spectral functions only in the near-threshold region . This can be seen clearly in the case of the nucleon, where the EFT expressions can be compared with the spectral functions obtained from dispersion theory (see below) and was noticed in earlier EFT calculations Gasser:1987rb (); Bernard:1996cc (); Becher:1999he (); Kubis:2000zd (); Kaiser:2003qp (). The reason for the discrepancy is the strong rescattering within the system in the -channel, which manifests itself in the meson resonance at . In order to construct the baryon densities down to distances fm we need to extend the calculation of isovector spectral functions into the meson mass region (see Sec. 2.2 and Fig. 3). This can be accomplished in an approach that combines the EFT calculations with dispersion theory. We describe this approach first for the baryon form factors in which the two-pion cut starts at the normal threshold (such as the nucleon ); the role of anomalous thresholds with in the strange baryon form factors will be considered subsequently.
where is the center-of-mass momentum of the system in the -channel, is the complex partial wave amplitude, and is the complex pion form factor in the timelike region (see Fig. 6). Equation (24) follows from the unitarity condition in the -channel and is valid strictly in the region up to the four-pion threshold, ; if contributions from four-pion states are neglected it can effectively be applied up to . The expression on the right-hand side of Eq. (24) is real because the complex functions and have the same phase on the two-pion cut (Watson theorem Watson:1954uc ()). It is convenient to rewrite Eq. (24) in the form
This representation has two advantages: (a) The function is real at and therefore has no two-pion cut; (b) the squared modulus can be extracted directly from the exclusive annihilation cross section, without determining the phase of the complex form factor. Equation (25) is a variant of the method of amplitude analysis Chew:1960iv (). The -channel partial-wave amplitude is represented in the form , such that the right-hand cut (related to the -channel exchanges) appears only in the factor and the left-hand cut (related to the -channel intermediate states) in the factor . In the case at hand the function is naturally chosen as the inverse pion form factor, Hohler:1974ht ().
The representation Eq. (25) suggests a new approach to calculating the spectral function on the two-pion cut. We use EFT to calculate the real function at to a fixed order. We then multiply the result with the empirical , which contains the effects of rescattering and the meson resonance. The major advantage of this approach is that the EFT calculation is not affected by the strong rescattering, which would require higher-order unitarity corrections when treated within the EFT. We therefore expect this approach to show much better convergence than direct EFT calculations of the spectral function. From a general perspective, the organization according to Eq. (25) is consistent with the idea of separation of scales basic to EFT. The function is dominated by the singularities of the baryon Born diagrams (or diagrams with inelastic intermediate states in higher orders), which are governed by the scales and (the octet-decuplet mass difference). The -dependence of the pion form factor, in contrast, is governed by the chiral-symmetry-breaking scale , which is of the order of the vector meson mass.
In our EFT calculation of the spectral function at
in Sec. 3.1 the function is effectively
evaluated at (cf. Footnote 2).
At this level the pion form factor in the denominator enters at leading order,
, so that the EFT result for
is the same as that for itself. The prescription of Eq. (25)
therefore simply amounts to multiplying our results
for the spectral functions by ,
This formula permits an extremely simple implementation of unitarity at . For the pion form factor we use the Gounaris-Sakurai parametrization including effects of - mixing Gounaris:1968mw (); Barkov:1985ac (), with the parameters determined in Ref. Lorenz:2012tm (). The prescription Eq. (26) results in a remarkable improvement of the EFT predictions for the spectral functions. The improved EFT results for the nucleon () reproduce the spectral functions obtained from amplitude analysis (analytic continuation of the partial-wave amplitudes Belushkin:2005ds (); Hohler:1976ax (), Roy-Steiner equations Hoferichter:2016duk ()) up to within errors (see Fig. 7a and c). Note that this is achieved without adding any free parameters and represents a genuine prediction of EFT. The improved results have qualitatively correct behavior even in the meson mass region. We expect that higher-order EFT corrections to the ratio would further improve the agreement with the dispersion-theoretical results at GeV InPreparation ().
The improvement according to Eq. (26) has a dramatic effect on the peripheral transverse densities of the nucleon (see Fig. 7b and d). At distances fm the improved predictions are up to an order of magnitude larger than what would be obtained with the original EFT results. At distances fm the change is still by a factor . At asymptotically large distances (much larger than those shown in the figure) the improvement would change the densities by the constant factor .
In order to extend the dispersive improvement to the other octet baryons it is necessary to discuss the role of anomalous thresholds in the strange baryon form factors Karplus:1958zz (); Landau:1959fi (). In the normal situation considered so far, the Born terms in the amplitudes give rise to singularities on the unphysical sheet of the baryon form factor, and the principal cut starts at the normal threshold . Such is the case with the and Born terms in the amplitudes. In certain special situations the Born term singularities in the amplitudes move onto the physical sheet of the form factor and give rise to cuts with anomalous thresholds at . This occurs if Karplus:1958zz ()
where is the mass of the baryon pole in the Born term, and the anomalous threshold is located at
Such is the case for the pole in the amplitudes, which produces an anomalous threshold at . In a general dispersion-theoretical treatment with exact masses the contribution from the anomalous cut would need to be considered explicitly in the spectral representation of the form factors and the densities. In the present treatment based on leading-order EFT anomalous thresholds do not occur, as the octet baryon masses are taken at a common value in the chiral limit (this is required by the small-scale expansion) and therefore in Eq. (27). We can thus use Eq. (26) at leading order for the entire baryon octet. This feature is a consequence of our particular combination of EFT and dispersion theory and permits a major simplification of the description of the strange baryon form factors. The anomalous threshold in the form factors appears only in higher-order EFT calculations and affects the behavior of the densities at very large distances; its contribution at realistic distances fm is expected to be small (cf. the discussion in Sec. 2.2 and Fig. 3).
An anomalous threshold would also appear in the amplitudes through the Born term, at , cf. Eq. (28), if the exact baryon masses were used. Again this feature is absent in our leading-order EFT calculation, where the and are taken at a common mass. In any case this anomalous threshold is only marginally lower than the normal threshold, so that its cut corresponds to a high-mass contribution that can be neglected in the peripheral densities on the same grounds as the normal cut.
Based on the preceding considerations we now use the prescription Eq. (26) to calculate the isovector spectral functions of the strange octet baryons on the two-pion cut. While we cannot compare with a dispersion-theoretical result in this case, we expect the approximation to be of similar quality as in the case of the nucleon. The results for and are shown in Fig. 8 [see Eq. (13) for the definition of the isovector component in the octet baryon states]. They exhibit several interesting features. (a) In the near-threshold region (see Fig. 8a, c) the dominant contribution to the spectral functions comes from the triangle diagram with intermediate octet baryons, Fig. 4a. The ratios of the spectral functions for the different baryons are therefore determined approximately by the ratios of the products of the couplings for the relevant intermediate states, which follow directly from the chiral Lagrangian Eq. (21):
(b) At larger values of (see Fig. 8b, d) the spectral functions receive sizable contributions from the triangle diagram with intermediate decuplet baryons and the contact terms, Fig. 4c and b. These contributions change the relative order of the different baryons compared to the near-threshold region dominated by intermediate octet baryons. The signs of the intermediate octet and decuplet contributions are summarized in Table 1. In the Dirac form factor the intermediate decuplet contributes to Im with opposite sign to the intermediate octet, and to Im and Im with the same sign. As a result, Im is now larger than Im , and Im is comparable to Im . In the Pauli form factor the intermediate decuplet contributes to Im with the same sign as the intermediate octet, and to Im and Im with different sign. As a result the relative order of Im and Im remains the same, while Im becomes negative at larger values of . These changes between the near-threshold region and larger values of are a non-trivial consequence of the inclusion of decuplet degrees of freedom in the EFT.
To complete our assessment we want to quantify also the contribution of the two-kaon cut to the peripheral isovector densities in EFT. Numerical evaluation shows that in the nucleon at the isovector density from the two-kaon cut is smaller than that from the two-pion cut by an order of magnitude; at it is already smaller by two orders of magnitude. Similar ratios are obtained for the strange octet baryons. We can therefore neglect the contributions of the two-kaon cut in the peripheral isovector densities. The quoted kaon/pion ratios refer to the EFT densities before the dispersive improvement. An interesting theoretical question is how the dispersive improvement could be extended to the kaon sector through a coupled-channel formalism.
3.3 Isoscalar spectral functions
The isoscalar component of the baryon spectral functions behaves very differently from the isovector one and requires separate treatment. In the limit of exact isospin symmetry the isoscalar spectral functions arise from the exchange of an odd number of pions in the -channel (G-parity). The lowest-mass exchange with vector quantum numbers is the three-pion exchange (). In EFT this contribution appears only at and is very small Bernard:1996cc (). Inclusion of the resonance through unitarity in the three-pion channel, in analogy to our treatment of the in the two-pion channel, would be possible in principle but requires three-body unitarity techniques that are not readily available.
The present -flavor EFT generates an isoscalar term in the spectral function through two-kaon exchange. Its contribution to the peripheral densities is extremely small compared to the isovector densities, or the isoscalar densities resulting from and exchange (see below). Treatment of the as a resonance in the two-kaon channel, along the lines of the in the two-pion channel, is impractical because of the mixing with the three-pion and other hadronic channels; see Refs. Hammer:1998rz (); Hammer:1999uf () for a discussion.
In the present study we therefore model the isoscalar spectral functions of the octet baryons at through phenomenological vector meson exchange (),
The vector meson couplings of the octet baryons are obtained from symmetry, certain assumptions about the ratio, and the empirical vector meson couplings to the nucleon; see C. The contribution from states with is strongly suppressed in the peripheral densities, so that the two-pole parametrization Eq. (30) is sufficient for our purposes. We do not aim for a precise description of the isoscalar sector here, as the peripheral densities are dominated by the isovector component.
4 Octet baryon densities
4.1 Charge and magnetization densities
Using the spectral functions calculated in Sec. 3 we now calculate the peripheral transverse densities of the octet baryons and study their properties. The studies of Sec. 2.2 have shown that at distances the dispersion integrals Eqs. (8) and (9) are dominated by masses , where our approximations in the spectral functions are justified.
In order to quantify the uncertainties of the peripheral densities we propagate the estimated theoretical uncertainties of the spectral functions through the dispersion integrals Eqs. (8) and (9). In the isovector spectral functions we use the estimated EFT errors in the region (as shown by the error bands in Fig. 7a and c) and assign an additional error of in the meson mass region (these error bands are not shown in the insets of Fig. 7a and c but inferred from the comparison of the improved EFT results with the amplitude analysis results Belushkin:2005ds (); Hoferichter:2016duk () in this region). In the isoscalar spectral functions we generate an error band from the uncertainties of the empirical couplings extracted from fits to the isoscalar nucleon form factor data Belushkin:2006qa () (see C and the parameters in Table 3). Note that the error estimates performed here are specific to the peripheral densities and do not take into account constraints resulting from the total isovector/isoscalar charges of the baryons, which would involve the densities at central as well as peripheral distances; these constraints would significantly reduce the error of the overall densities at