A dispersive approach to two-photon exchange in elastic electron–proton scattering

A dispersive approach to two-photon exchange in elastic electron–proton scattering


We examine the two-photon exchange corrections to elastic electron–proton scattering within a dispersive approach, including contributions from both nucleon and intermediate states. The dispersive analysis avoids off-shell uncertainties inherent in traditional approaches based on direct evaluation of loop diagrams, and guarantees the correct unitary behavior in the high energy limit. Using empirical information on the electromagnetic nucleon elastic and transition form factors, we compute the two-photon exchange corrections both algebraically and numerically. Results are compared with recent measurements of to cross section ratios from the CLAS, VEPP-3 and OLYMPUS experiments, as well as with polarization transfer observables.


I Introduction

The nucleon’s electroweak form factors are some of the cornerstone observables that characterize its extended spatial structure. Since the original observation Hofstadter and McAllister (1955) some 60 years ago that elastic scattering from the proton deviates from point-like behavior at large scattering angles, considerable information has been accumulated on the detailed structure of the proton’s and neutron’s electric and magnetic responses. Almost universally the underlying scattering reaction has been assumed to proceed through the exchange of a single photon between the lepton (typically electron) beam and nucleon target.

A major paradigm shift occurred around the turn of the last century with the observation of a significant discrepancy between the ratio of electric to magnetic form factors of the proton measured using the relatively new polarization transfer technique Jones et al. (2000); Gayou et al. (2002) and previous extractions of the same quantity from cross section measurements via Rosenbluth separation. It was soon realized Blunden et al. (2003); Guichon and Vanderhaeghen (2003) that a large part of the discrepancy could be understood in terms of additional, hadron structure-dependent two-photon exchange (TPE) contributions, which had not been included in the standard treatments of electromagnetic radiative corrections Tsai (1961); Mo and Tsai (1969).

A number of approaches have been adopted to computing the TPE corrections to elastic scattering, including direct calculation of the loop contributions in terms of hadronic degrees of freedom Blunden et al. (2003); Kondratyuk et al. (2005); Blunden et al. (2005); Kondratyuk and Blunden (2007); Nagata et al. (2009); Graczyk (2013); Lorenz et al. (2015); Zhou and Yang (2015); Zhou (2017), modeling the high energy behavior of box diagrams at the quark level through generalized parton distributions Chen et al. (2004); Afanasev et al. (2005), or more recently dispersion relations Gorchtein (2007); Borisyuk and Kobushkin (2008, 2015); Tomalak and Vanderhaeghen (2015). Each of these methods has its own advantages as well as limitations (for reviews, see Refs. Carlson and Vanderhaeghen (2007); Arrington et al. (2011); Afanasev et al. (2017)), and to date no single approach has been able to provide a universal description valid at all kinematics.

Most of the attention on the TPE corrections in recent experiments has been focussed on the region of small and intermediate values of the four-momentum transfer squared,  few GeV, where the expectation is that hadrons retain their identity sufficiently well that calculations in terms of physical degrees of freedom give reliable estimates. Traditionally, this approach has required direct evaluation of the real parts of the two-photon box and crossed-box diagrams, with nucleons or other excited state hadrons in the intermediate state parametrized through half off-shell form factors (with one nucleon on-shell and one off-shell). Because the off-shell dependence of these form factors is not known, usually one approximates the half off-shell form factors by their on-shell limits.

For nucleon intermediate states, the off-shell uncertainties are not expected to be severe. On the other hand, for transitions to excited state baryons described by effective interactions involving derivative couplings, such as for the resonance, the off-shell dependence leads to divergences in the forward angle (or high energy) limit, and signals a violation of unitarity. Furthermore, from a more technical perspective, in order to evaluate the TPE corrections analytically in terms of Passarino-Veltman functions, the loop integration method requires the transition form factors to be parametrized as sums or products of monopole functions. This can prove cumbersome in some applications, since such parametrizations are usually only valid in a limited region of spacelike , and may be prone to roundoff errors in numerical evaluation. It would naturally be highly desirable to be able to compute the loop integrations with a more robust numerical method that is valid for form factor parametrizations based on more general classes of functional forms.

The limitations of the previous loop calculations are especially problematic in view of new measurements of ratios of to elastic scattering cross sections Rimal et al. (2016); Rachek et al. (2015); Nikolenko et al. (2014); Henderson et al. (1969), which have provided high precision data that are directly sensitive to TPE effects. Some of these data are in the small-angle region, where the off-shell ambiguities in the loop calculations make the calculations unreliable. To enable meaningful comparison between the data and TPE calculations over the full range of kinematics currently accessible, clearly a different approach to the problem is needed.

In this paper, we revisit the calculation of TPE corrections within the hadronic approach, but using dispersion relations to construct the real part of the TPE amplitude from its imaginary part. The dispersive method involves the exclusive use of on-shell transition form factors, thereby avoiding the problem of unphysical violation of unitarity in the high energy limit. The dispersive approach to TPE was developed at forward angles by Gorchtein Gorchtein (2007), and at non-forward angles by Borisyuk and Kobushkin Borisyuk and Kobushkin (2008, 2012, 2014, 2015), and more recently by Tomalak and Vanderhaeghen Tomalak and Vanderhaeghen (2015). A feature of the latter two analyses has been the use of monopole form factor parametrizations, which allowed the computations to be performed semi-analytically. In this work we extend the dispersion relation approach to allow for more general classes of transition form factors.

In Sec. II of this paper we review the formalism for elastic electron–nucleon scattering for both one-photon and two-photon exchange processes, and introduce the main elements of the dispersive approach. We describe analytical calculations of the imaginary part of the TPE corrections using the more restrictive monopole form factors, for which one can obtain analytic expressions in terms of elementary logarithms. We also describe the more general numerical method that allows standalone calculation of the imaginary part using a general class of transition form factors.

The results of the calculations are presented in Sec. III, where we critically examine the differences between the new dispersive method and the previous loop calculations with off-shell intermediate states. While the differences are relatively small for the nucleon elastic contributions, the effects for intermediate states are dramatic at high energies and forward scattering angles. We also compare in Sec. III the new results with the recent data on to cross section ratios from the CLAS Rimal et al. (2016), VEPP-3 Rachek et al. (2015); Nikolenko et al. (2014) and OLYMPUS Henderson et al. (1969) experiments, as well as with polarization data sensitive to TPE contributions Meziane et al. (2011). Finally, in Sec. IV we summarize our results, and discuss possible future developments in theory and experiment. For completeness, in the appendices we give the full expressions for the generalized form factors in Appendix A, and analytic expressions for the imaginary parts of Passarino-Veltman functions in Appendix B. We also provide convenient reparametrizations of the nucleon and vertex form factors in Appendix C that can be used in the analytic calculations.

Ii Formalism

In this section we present the formalism on which the electron–nucleon scattering analysis in this paper will be based. After summarizing the kinematics and main formulas for the elastic scattering amplitudes and cross sections at the Born and TPE level, we proceed to describe the new elements of the analysis that make use of dispersive methods, including both analytic and numerical evaluation of integrals.

ii.1 Elastic scattering

For the elastic scattering process the four-momenta of the initial and final electrons (taken to be massless) are labeled by and , and of the initial and final protons (mass ) by and , respectively, as depicted in Fig. 1. The four-momentum transfer from the electron to the proton is given by , with . One can express the elastic cross section in terms of any two of the Mandelstam variables (total electron–proton invariant mass squared), , and , where


with the constraint .

Figure 1: Contributions to elastic electron–nucleon scattering from (a) one-photon exchange, and (b) two-photon exchange amplitudes, with particle momenta as indicated. For TPE we have . Only the -channel “box” diagram is drawn. The “crossed-box” contribution, which can be obtained by applying crossing symmetry , is implied.

The elastic scattering cross section can be defined in terms of any two of the dimensionless quantities


The inverse relationships are also useful,


In the target rest frame the variables are given by


where is the energy of the incident (scattered) electron, is the electron scattering angle, and () is identified with the relative flux of longitudinal virtual photons.

One-photon exchange

In the Born (OPE) approximation the electron–nucleon scattering invariant amplitude can be written as


where is the electric charge, and the matrix elements of the electromagnetic leptonic and hadronic currents are given in terms of the lepton () and nucleon () spinors by


The electromagnetic hadron current operator is parametrized by the Dirac () and Pauli () form factors as


where the Born form factors are functions of a single variable, . In our convention, the reduced Born cross section is given by


where the Sachs electric and magnetic form factors are defined in terms of the Dirac and Pauli form factors as


Two-photon exchange

Using the kinematics illustrated in the box diagram in Fig. 1(b), the contribution to the TPE box amplitude from an intermediate hadronic state of invariant mass can be written in the general form Blunden et al. (2003); Kondratyuk et al. (2005)


with , and an infinitesimal photon mass is introduced to regulate any infrared divergences. (In general the mass can have a distribution which can be integrated over, but here we specialize to the case of a narrow state .) The leptonic and hadronic tensors here are given by


with , , and the electron propagator is


The hadronic transition current operator is written in a general form that allows for a possible dependence on the incoming momentum of the photon and the outgoing momentum of the hadron, while and are Lorentz indices.

The hadronic state propagator in this work will describe the propagation of a baryon with either spin-\sfrac12 or spin-\sfrac32. For spin-\sfrac12 intermediate states, such as the nucleon, this reduces to


and the transition operator involves one free Lorentz index. For spin-\sfrac32 intermediate states, such as the baryon, the propagator can be written


where the projection operator


ensures the presence of only spin-\sfrac32 components. Unphysical spin-\sfrac12 contributions are suppressed by the condition on the vertex .

One can obtain the crossed-box (“xbox”) contribution directly from the box term (10) by applying crossing symmetry. For example, in the unpolarized case, we have


In general, has both real and imaginary parts, whereas is purely real. The total squared amplitude for the sum of the one- and two-photon exchange processes shown in Fig. 1 is then


where the relative correction to the cross section due to the interference of the one- and two-photon exchange amplitudes is defined as


Within the framework of the simplest hadronic models, analytic evaluation of is made possible by writing the transition form factors at the -hadron vertices as a sum and/or product of monopole form factors Blunden et al. (2003, 2005), which are typically fit to empirical transition form factors over a suitable range in spacelike four-momentum transfer. Four-dimensional integrals over the momentum in the one-loop box diagram can then be expressed in terms of the Passarino-Veltman (PV) scalar functions , , and  ’t Hooft and Veltman (1979); Passarino and Veltman (1979). This reduction to scalar integrals is automated by programs such as FeynCalc Mertig et al. (1991); Shtabovenko et al. (2016). The PV functions can then be evaluated numerically using packages such as LoopTools Hahn and Perez-Victoria (1999). In this paper we are interested only in the imaginary parts of these PV functions, which are considerably simpler than the full expressions. This will be discussed in detail in the next section.

Note that the expressions (10) and (19) contain infrared (IR) divergences arising from the elastic intermediate state when the momentum of either photon vanishes. In analyzing the TPE corrections for scattering, it is convenient to subtract off these conventional IR-divergent parts, which are independent of hadronic structure, and which are usually already included in experimental analyses using a specific prescription (e.g. Mo & Tsai Mo and Tsai (1969), Grammer & Yennie Grammer and Yennie (1973), or Maximon & Tjon Maximon and Tjon (2000)).

In general, the TPE amplitude at the IR poles ( or ) has the form


where is a function containing all the IR divergences that is independent of hadronic structure. Its form depends on the particular IR prescription being used. This is discussed extensively in the TPE review by Arrington, Blunden, and Melnitchouk Arrington et al. (2011), and we defer to that paper for details. The hard-TPE correction of interest is then


In this paper we follow the prescription used by Maximon and Tjon Maximon and Tjon (2000), which is to evaluate the contribution to the numerator of Eq. (10) arising from the poles , while keeping the propagators in the denominator intact. In this prescription,


This expression has both real and imaginary parts. In our convention, for , so explicitly the real and imaginary parts are


After accounting for conventional radiative corrections, the measured reduced cross section is related to the Born cross section by


In practice, most experimental cross section analyses use the IR-divergent expression of Mo and Tsai Mo and Tsai (1969), so that if one uses the Maximon and Tjon prescription Maximon and Tjon (2000) (as we do in this paper) then the difference should be accounted for when comparing to experimental data (see Ref. Arrington et al. (2011) for further discussion).

The total TPE amplitude can be rewritten in terms of “generalized form factors”, generalizing the expressions of Eqs. (5)–(7), as described by Guichon and Vanderhaeghen Guichon and Vanderhaeghen (2003). Although the decomposition is not unique, and different generalized form factor conventions have been used in the literature, in this paper we use the basis of form factors denoted by , and , defined via


where the vector and generalized form factors are the TPE analogs of the Dirac and Pauli form factors, while the axial vector generalized form factor has no Born level analog.

Rather than construct and subtract the IR-divergent terms, as in Eq. (21), it is convenient to incorporate the IR subtractions directly into the generalized form factors and ( is not IR-divergent),


where refer to the unregulated expressions. In terms of these regulated generalized form factors, the relative TPE correction is given by


ii.2 Dispersive approach

As noted earlier, the TPE amplitude has both real and imaginary parts. The real and imaginary parts can be related through dispersion relations Gorchtein (2007); Borisyuk and Kobushkin (2008), which forms the basis of the dispersive method discussed in this section. Our discussion in this section follows the formalism of Tomalak and Vanderhaeghen Tomalak and Vanderhaeghen (2014, 2015). An alternative treatment by Borisyuk and Kobushkin Borisyuk and Kobushkin (2008) starts from the annihilation channel, .

Using the parametrization of the TPE amplitude in terms of the generalized form factors , and , we note that these TPE amplitudes have the symmetry properties Gorchtein (2007); Borisyuk and Kobushkin (2008)


and satisfy the fixed- dispersion relations


Here denotes the Cauchy principal value integral, and is the threshold for the elastic cut, corresponding to an electron of energy . The physical threshold for electron scattering is at (or ), which requires , with . This integral therefore extends into an unphysical region of parameter space, which requires knowledge of the transition form factors in the timelike region of four-momentum transfer. The crossed-box terms in the real part of the TPE amplitudes are generated by incorporating the symmetry properties into the dispersive integrals, which is equivalent to the use of Eq. (17) in the loop calculation.

For the interaction of point particles, such as in elastic scattering, the real parts generated in this way agree completely with those obtained directly from the four-dimensional loop integrals of Eq. (10Tomalak and Vanderhaeghen (2014). In general, however, there may be momentum dependence in the -hadron interaction, such as for the vertex (see Sec. III.2 below). In fact, the momentum dependence in a transition vertex function allows one to construct different parametrizations of that vertex function, for example, by using the Dirac equation, that are equivalent on-shell but differ off-shell. The additional momentum-dependence associated with this freedom will affect one-loop integrals because the intermediate hadronic states are not on-shell. This ambiguity is not present in the dispersive method, for which all the intermediate states are on-shell. In the context of TPE, this means that for any momentum-dependent interactions one should not expect agreement between the real parts of the generalized form factors calculated from Eqs. (29) and those calculated using the loop integration method. We will quantify these differences for the cases of the nucleon and intermediate states in Sec. III.

Analytic method

The analytic approach used in previous work Blunden et al. (2003); Kondratyuk et al. (2005); Blunden et al. (2005); Kondratyuk and Blunden (2007); Tjon and Melnitchouk (2008); Nagata et al. (2009); Tjon et al. (2009); Graczyk (2013); Lorenz et al. (2015); Zhou and Yang (2015) relies on a parameterization of the transition form factors as a sum and/or product of monopole form factors. The most basic relation is


which is to be applied at each photon–hadron vertex (). More complicated constructions are straightforward to generate by repeatedly applying the feature that the product of any two monopoles is proportional to their difference. The general expression for an amplitude with form factors will thus involve a sum of “primitive” integrals with different photon mass parameters and for each of the two photon propagators, modified according to Eq. (30). The primitive integrals may yield spurious ultraviolet or infrared divergences, but these divergences will cancel when taking the sum. We give details on these constructions in Appendix B.

By means of the PV reduction scheme, a one-loop integral for the box-diagram amplitude can be written in terms of a set of scalar PV functions , , and , corresponding to one-, two-, three-, and four-point functions. This can be visualized as a “pinching” of the four various propagators in the box diagram due to cancellations of the terms in the numerator with the propagator terms in the denominator. The PV functions can be evaluated numerically, and there are various computer programs to do this Hahn and Perez-Victoria (1999); van Hameren (2011); Carrazza et al. (2016); Denner et al. (2017).

In general the scalar PV functions are complex-valued. The imaginary parts of the TPE amplitudes are contained entirely in these functions. For the box (and crossed-box) diagrams in elastic scattering there are only four of the PV functions that have imaginary parts. These four functions are the ones that arise in the -channel box diagram with the electron and intermediate hadronic states on-shell. This is illustrated in Fig. 2.

Figure 2: Visualization of the Passarino-Veltman functions from the TPE amplitudes that have an imaginary part, for (a) ; (b) ; (c) , where is the total invariant mass of the system. Here external legs have been amputated. The double line indicates a hadronic state of invariant mass , and the dotted line indicates that the electron and hadronic states are on-shell. For elastic scattering, is the same whichever photon propagator is pinched, and .

Recall that an amplitude becomes imaginary when the intermediate state particles become real, or on their mass shells. This is formalized by the well-known Cutkosky cutting rules Cutkosky (1958). Namely, as a consequence of unitarity, the imaginary part of a scattering amplitude can be obtained by summing all possible cuttings of the corresponding Feynman diagram, where a cut is across any two internal propagators separating the external states from the rest of the diagram. Cut propagators are then put on-shell according to the rule .

For elastic scattering, the two functions arising when either photon propagator is pinched are identical, so there are only three distinct PV functions. Other PV functions where the electron or hadronic intermediate state (or both) are pinched have no imaginary parts for scattering, and the -channel crossed-box diagram also has no imaginary part. (Recall that the crossed-box amplitude can be obtained by replacing , with an appropriate overall changed in sign given in Eq. (28).) We will denote these three functions as , , and . The full expression for these functions is

In addition to the explicit dependence on and , there is also an implied dependence on , , and that is suppressed for clarity of notation (see Appendix B for details).

We define the imaginary parts of the PV functions by . According to the Cutkosky rules, the imaginary parts correspond to putting the electron and intermediate hadronic states on-shell, and . Working in the center-of-mass (CM) frame, we define the electron variables


In this frame we have


with the shorthand notation


In the physical region, the CM scattering angle satisfies the constraint , requiring . However, the dispersive integral of Eq. (29) only requires , meaning there is an unphysical region of parameter space where , and is purely imaginary. Therefore, in the dispersive approach expressions for the imaginary parts of the TPE amplitudes need to be analytically continued into this unphysical region.

Recall that in terms of the electron energy in the laboratory frame, , the -invariant mass squared is . After changing the integration variable from to , and using the on-shell conditions, we find after some algebra the expressions


where are the squared four-momenta of the virtual photons (), with


and .

The integral in Eq. (35) is trivial. Through the use of Eq. (36), the other integrals can be brought into the form


The integrand here has poles when , which can arise in the unphysical region when becomes timelike. A simpler version of this integral was considered by Mandelstam Mandelstam (1958) for the case where the target and scattering particles have equal masses. The general expression has been given by Beenakker and Denner Beenakker and Denner (1990),


For the function, we set , , , , and . For , we set , , , and . In the unphysical region, , so that is purely imaginary. However, we note that the combination , and therefore , independent of the value of . Thus Eq. (38) for is the proper analytic continuation of the integral for into the unphysical region. Explicit expressions for , and , including the IR limits , are given in Appendix B.

In previous work Blunden et al. (2003, 2005) the TPE amplitudes were obtained by numerical evaluation of the PV functions using the program LoopTools Hahn and Perez-Victoria (1999). The real parts were used directly, and the imaginary parts were not needed. Here, we have constructed analytic expressions for the imaginary parts in terms of elementary logarithms, thus allowing a completely analytic evaluation of the imaginary parts of the TPE amplitudes. The real parts are then constructed from a numerical evaluation of the dispersion integrals of Eq. (29). The imaginary parts obtained here are, of course, identical with those obtained numerically in the earlier work Blunden et al. (2003, 2005). The real parts are numerically identical for the elastic and TPE amplitudes, while the amplitude differs, but in a numerically insignificant way. For the inelastic states there are significant differences, especially as . These differences will be discussed further in Sec. III.

As an alternative to using the PV reduction method implemented in FeynCalc Mertig et al. (1991); Shtabovenko et al. (2016), one can work entirely with on-shell quantities. Using the on-shell conditions, we find that the TPE amplitudes are sums of integrals of the general form


where is a polynomial function of combined degree in and ,


The coefficients are functions of , , and , and satisfy for elastic scattering due to the symmetry under . Thus we can write


with the “primitive” integrals defined as


For nucleon intermediate states we find , indicating that monopole form factors are sufficient to eliminate the UV divergences (there is one power of at each photon–nucleon vertex in the term of ). For intermediate states, however, we find , which implies that dipole form factors (or a product of monopoles) are needed to eliminate the UV divergences, as there are up to two powers of at each vertex in (see Secs. III.1 and III.2 below). The integrals of Eq. (42) up to are given in Table 1 of Appendix B, and can easily be extended to more complicated form factor constructions.

Numerical method

In analogy with Eq. (39), the TPE amplitudes of interest have the general form


where is a polynomial function of combined degree 2 (3) in for intermediate states, and are form factors that are real-valued and finite for all spacelike values of . For elastic scattering, the total integral is symmetric under the interchange . For the nucleon intermediate state, it is convenient to bring the IR subtractions of Eq. (26) into this integral. This is consistent with the Maximon and Tjon IR regularization scheme whereby the numerator of Eq. (43) vanishes whenever . It also vanishes for excited states under these conditions. Therefore we could actually set without encountering any singularities in the integrals. This is unlike the analytic expressions of the previous section, where only the sum of individual IR-divergent expressions is independent of .

In the physical region there are no singularities in the integrand of Eq. (43), so evaluation of the integral is a straightforward 2-dimensional numerical quadrature over the domain and , following Eq. (36). However, this approach fails in the unphysical region. To get around this, Tomalak and Vanderhaeghen used a contour integration method Tomalak and Vanderhaeghen (2015), and applied it to the calculation of TPE amplitudes with monopole form factors. By summing the residue at the poles enclosed by the contour they were able to obtain algebraic expressions for the TPE amplitudes. These expressions are equivalent to the ones we obtained in the previous section using algebraic expressions for the PV functions. In this section we will follow this method, with modifications, to implement a numerical contour integration of Eq. (43) that allows for a more general parametrization of the transition form factors than a sum and/or product of monopoles.

In the complex plane, we define a timelike half with , and spacelike half with . In general, the allowed form factors can have poles in anywhere in the timelike half of the complex plane. With certain restrictions, which we will state explicitly, the form factors can have poles in the spacelike half as well.

Without providing a rigorous mathematical proof, we can nonetheless specify certain restrictions on the type of allowed form factors. Namely, they should have a simple functional form in , such as exponentials, polynomials, or inverse polynomials, that can be analytically continued to the complex plane. There should be no branch cuts, and any poles should either lie along the negative, real axis (at timelike ), or occur in complex conjugate pairs. This is the case for a commonly used form factor parametrization [see Eq. (51)] in terms of a ratio of polynomials Kelly (2004); Arrington et al. (2007); Venkat et al. (2011).

The area of integration in Eq. (43) can be visualized as an integral over the photon virtual momenta and of Eq. (36), which form a symmetric ellipse in vs. , centered at  Gorchtein and Horowitz (2008); Tomalak and Vanderhaeghen (2015). The boundary of the ellipse is defined by . Following Ref. Tomalak and Vanderhaeghen (2015), we make a change of variables to elliptic coordinates,


The contours of constant represent concentric ellipses with radial parameter . From Eq. (36), in elliptic coordinates we have


In the physical region the integral over can be rewritten as a contour integral on the unit circle , with regarded as functions of (and ) using




In anticipation of extending this formalism to the unphysical region, Eq. (45) can be simplified into the form




Recall that is given in Eq. (33) in terms of and . Expressed in this way, it is clear that . Because the integrands of the full TPE amplitudes are symmetric under the interchange , this means that for every value associated with the poles in in the complex plane, there is a corresponding pole at . In addition, for every pole in , both and are poles in the complex plane, where lies inside the unit circle, and lies outside the unit circle. The values are the IR-divergent poles, where for . These could be regulated by introducing the photon mass parameter, but as the integrand vanishes in our regularization scheme when , there is no IR divergence, and one can set . Analogously, for , both and are poles inside and outside a circle of radius , respectively. The values are the IR-divergent poles, where for . These results are valid in either the physical region, where , or the unphysical region, where .

As an illustration of these points, consider the monopole form factors as given in Eq. (39), for which there is a pole in along the negative real axis at . This yields poles and along the positive real axis in the complex plane, with the interior pole lying between 0 and 1. For the corresponding poles associated with , in the physical region, with , these lie along a line at angle . In the unphysical region, with , they lie along the negative real axis, with lying between and 0. A graphical representation of these results in the complex plane is shown in Fig. 3, with the dots representing the “inside” points and , and the crosses representing the “outside” points and .

Figure 3: (a): The contour around the unit circle in the physical region, . For every pole inside the contour (denoted by a dot), there is a corresponding pole at lying outside the contour (denoted by a cross). For monopole form factors, interior poles in