Analyticity and Regge asymptotics in virtual Compton scattering on the nucleon

The virtual Compton scattering on the nucleon, in particular the doubly-virtual forward scattering $\gamma^{*}(q)+N(p)\to\gamma^{*}(q)+N(p)$, is of much interest for the calculation of the electromagnetic mass difference between the proton and the neutron by Cottingham formula Cottingham:1963zz and for the study of nucleon polarizabilities. This process has been intensively studied a long time ago Harari:1966mu ; Gross:1968zz ; Creutz:1968ds ; Damashek:1969xj ; Dominguez:1970wu ; Elitzur:1970yy ; Brodsky:1971zh ; Zee:1971df ; Brodsky:1972vv , especially in the frame of Regge theory. In particular, a question that received much attention was the possible existence of a fixed pole at $J=0$ in the angular momentum plane for the Compton amplitude. This problem was discussed also more recently, see for instance Brodsky:2008qu ; Muller:2015vha and references therein.

Dispersion theory is an important tool for investigating the virtual Compton amplitude.¹¹1Lattice calculations started to produce competitive results in recent years. For an updated comparison of lattice and dispersive results, see Fig. 8 of Gasser:2020hzn .. Early dispersive evaluations of Cottingham formula have ben performed in Harari:1966mu ; Gross:1968zz ; Damashek:1969xj ; Dominguez:1970wu ; Elitzur:1970yy ; Gasser:1974wd . There has been a recent renewed interest in this problem: dispersive treatments of the Compton amplitude, using modern data on the structure functions, have been reported in WalkerLoud:2012bg ; WalkerLoud:2012en ; Erben:2014hza ; Thomas:2014dxa ; Gasser:2015dwa ; Caprini:2016wvy ; Walker-Loud2018 ; Tomalak:2018dho ; Gasser:2020hzn ; Gasser:2020mzy .

The forward virtual Compton scattering is described by the time-ordered product of two electromagnetic currents between on-shell proton states, summed over the spin orientations:

By Lorentz covariance, this tensor is decomposed as

in terms of two invariant amplitudes, $T_{1}(\nu,Q^{2})$ and $T_{2}(\nu,Q^{2})$, free of kinematical singularities and zeros. We recall the notations: $\nu=p\cdot q/m$ and $Q^{2}=-q^{2}$, where $p$ and $q$ are the nucleon and photon momenta and $m$ is the nucleon mass.

The invariant amplitudes are even functions of $\nu$ due to crossing symmetry. Moreover, they are real analytic functions in the $\nu^{2}$ complex plane, with singularities along the real axis imposed by unitarity: poles due to the elastic contributions, and cuts produced by the inelastic states. Schwarz reflection principle implies that the discontinuities across the cuts are related to the imaginary parts on the upper edge of the cut. By unitarity, the contribution of the inelastic states is:

where $\nu_{th}=m_{\pi}+(m_{\pi}^{2}-Q^{2})/2m$ is the lowest threshold due to the $\pi N$ intermediate state, and for $Q^{2}\geq 0$ the structure functions $V_{i}$ are obtained from the cross sections of lepton-nucleon scattering.

Besides analyticity, the asymptotic behaviour of the amplitudes for large $\nu$ at fixed $Q^{2}$ plays an important role in the dispersion theory. It is generally accepted that the amplitude $T_{2}(\nu,Q^{2})$ can be expressed by an unsubtracted dispersion relation, while the dispersion relation for $T_{1}(\nu,Q^{2})$ requires a subtraction. This means that for recovering the amplitude, the structure function is not sufficient, one needs also the subtraction function, i.e. the value of the amplitude at a certain point, for instance $S_{1}(Q^{2})\equiv T_{1}(0,Q^{2})$. It turns out that the subtraction function plays a crucial role in the dispersion theory, in particular for the proton-neutron electromagnetic mass difference. As we will discuss below, there are several different approaches to the determination of the subtraction function, and they depend on the specific assumptions about the behaviour of the amplitude at high energies.

The asymptotic behaviour of the Compton amplitude is assumed in general to be governed by Reggeon exchanges, which manifest as moving poles in the angular momentum plane. For the proton and neutron amplitudes, the dominant contributions are due to the exchange of the Pomeron. A Reggeon with the quantum numbers $I^{C}=1^{+}$ and a trajectory with intercept around 0.5, identified with $a_{2}$, is the most important non-leading contribution.

In Ref. Gasser:1974wd it was assumed that the Reggeons dominate the asymptotic behaviour. This assumption was implemented by the requirement

where $T_{1}^{R}(\nu,Q^{2})$ is the leading Regge contribution. This Reggeon dominance hypothesis implies that the amplitude $T_{1}(\nu,Q^{2})$ does not contain a fixed pole at $J=0$, which would contribute with a real constant term at large $\nu$.

As shown in Gasser:1974wd , by exploiting (4) one can derive a sum rule which uniquely determines the subtraction function $S_{1}(Q^{2})$ in terms of the cross section of lepton-nucleon scattering and the residues of the Regge poles. A variant of this sum rule was proposed by Elitzur and Harari Elitzur:1970yy , on the basis of duality and finite energy sum rules. The recent analyses with modern data on the structure functions, performed in Gasser:2015dwa ; Gasser:2020hzn ; Gasser:2020mzy , show that this sum rule leads to precise dispersive determinations of proton-neutron electromagnetic mass difference and nucleon polarizabilities.

We mention also a more general frame of exploiting Regge asymptotics, investigated in Caprini:2016wvy , where it was assumed that the modulus of the amplitude is given above a certain energy by the modulus of Reggeon exchanges. Using this assumption, upper and lower bounds on the subtraction function have been derived, which turned out to be consistent with the sum-rule predictions made in Gasser:2015dwa .

On the other hand, in the analyses performed in WalkerLoud:2012bg ; WalkerLoud:2012en ; Erben:2014hza ; Thomas:2014dxa ; Walker-Loud2018 , the knowledge of the Regge behaviour of the amplitude at high energies was not taken into account in an explicit way. As a consequence, the subtraction function was assumed to be an independent quantity, which cannot be calculated in terms of the lepton-nucleon cross sections. Instead, the function $S_{1}(Q^{2})$ was parametrized by simple algebraic expressions, which interpolate between the low and high $Q^{2}$ regimes, known from effective field theory and perturbative QCD, respectively. A detailed comparison of the predictions of these models with the subtraction function derived from Reggeon dominance is presented in Sect. 24.2 of Gasser:2020hzn .

From the above discussion, it follows that the assumptions about the validity of Regge asymptotics in virtual Compton scattering play an important role in the dispersive predictions. In the present paper, we propose a test of Regge asymptotics, which exploits analyticity and the phenomenological input on the nucleon structure functions, available up to a certain energy. The question that we consider is whether this information can say something about the difference between the exact amplitude and the Reggeon contributions above that energy.

It turns out that the standard dispersion relations are not useful for providing an answer to this question. Fortunately, there exist also other methods of exploiting analyticity, based on functional analysis and optimization theory (for a review, see Caprini:2019osi ). By solving an extremal problem on a suitable class of analytic function, we obtain a rigorous lower bound on the maximum difference between the true amplitude and the leading Reggeon contributions above a certain energy. As we will show in the paper, the value of this lower bound and its variation when the energy is increased offer insights on the onset of the Regge regime. By solving also a more general extremal problem, we investigate the dependence of the lower bound on the subtraction function and argue that a reasonable prescription for finding this quantity assuming Regge asymptotics is to look for the value that achieves the minimum of the lower bound.

The paper is organized as follows: in the next section we specify the frame and formulate the extremal problems of interest for our study. In Sect. III we briefly discuss the phenomenological input used in the calculations. The results are presented in Sect. IV, while Sect. V contains a summary and our conclusions. Finally, in appendix A we present the solution of the extremal problems formulated in Sect. II.

To illustrate our approach we consider the difference of the invariant amplitudes $T_{1}$ for the proton and neutron. Also, in order to facilitate the comparison with the previous works Gasser:2015dwa ; Caprini:2016wvy , we consider only the “inelastic” amplitude:

defined by subtracting from the total amplitude the elastic contribution due to the nucleon pole. The explicit expression of $T_{1}^{\text{el}}(\nu,Q^{2})$ is given in Gasser:2015dwa .

As discussed above, the amplitude $T_{1}^{\text{inel}}(\nu,Q^{2})$ obeys at fixed $Q^{2}$ a subtracted dispersion relation in the $\nu$ variable. Choosing the subtraction point at $\nu=0$, the dispersion relation has the form

where $\nu_{th}$ is defined below Eq. (3) and

As already mentioned, the dispersion relation, which is based on the Cauchy integral relation, is not the only way to implement the analyticity of the amplitude in the $\nu$ complex plane. In our approach, analyticity will be exploited by functional-analysis techniques. We first specify the ingredients of this approach.

Let $\nu_{h}(Q^{2})$ be a certain high energy, which will be treated as a free parameter in the analysis. Below this energy, we assume that the imaginary part is known on the unitarity cut, i.e. we have

At high energies, the amplitude is described by the the exchange of Reggeons. Since we consider the difference of the proton and neutron amplitudes, the Pomeron contributions cancel out and the asymptotic behaviour is given by the leading Reggeon. We write this contribution as²²2A sligthly different representation, given in Eq. (18) of Gasser:2020hzn , coincides with (9) at high $\nu$.

where $\alpha>0$ is the $a_{2}$ trajectory at zero momentum transfer. Subleading contributions are expected to be present at finite energies.

It is of interest to define a quantity which measures how much the exact amplitude deviates from the prediction of the Regge model for $\nu\geq\nu_{h}(Q^{2})$. Taking into account the fact that the amplitude exhibits asymptotically a growth like $\nu^{\alpha}$ with $\alpha>0$, a reasonable candidate is

which involves the difference of functions bounded in the complex plane.

Since the physical amplitude is not known for $\nu\geq\nu_{h}(Q^{2})$, the exact value of this supremum defined in (10) is not known. However, it turns out that it must exceed a certain value, which can be calculated. Namely, let us consider the functional extremal problem

where the minimization is taken upon the class of functions $\{T_{1}^{\text{inel}}\}$ which are real-analytic in the $\nu^{2}$ plane cut for $\nu^{2}\geq\nu_{\text{th}}^{2}$, obey the unitarity condition (8) and exhibit an asymptotic increase bounded by $\nu^{\alpha}$. Since the true amplitude belongs to this class, the difference (10) corresponding to it must be larger than $\delta_{0}^{R}$. Therefore, $\delta_{0}^{R}$ is a lower bound on the exact difference (10). Moreover, unlike the exact difference which is not known, the lower bound can be calculated in terms of the phenomenological input.

The above extremal problem can be generalized to the case when more informations about the class of analytic functions $T_{1}^{\text{inel}}$ are available. Suppose that the value of $T^{\text{inel}}(\nu,Q^{2})$ at a point $\nu_{1}$ inside the analyticity domain is given at each $Q^{2}$. We take in particular $\nu_{1}=0$, because this was the choice of the subtraction point in the dispersion relation (6), but the more general case of an arbitrary $\nu_{1}<\nu_{\text{th}}$ can be treated in a similar way. Then, we consider the modified extremal problem

where the minimization is performed upon the class of functions $\{T_{1}^{\text{inel}}\}$ with the properties specified below (11), which satisfy in addition the constraint (7). Therefore, $\widehat{\delta}^{R}_{0}(S_{1}^{\text{inel}})$ is a lower bound on the difference between the physical amplitude $T_{1}^{\text{inel}}$ and the Regge expression $T_{1}^{R}$ for $\nu>\nu_{h}$, when the value of the physical amplitude at $\nu=0$ is supposed to be given. The notation in (12) emphasizes the fact that the lower bound depends on the input value $S_{1}^{\text{inel}}(Q^{2})$.

The solution of the extremal problems (11) and (12) is presented in appendix A), where it is shown that the lower bounds $\delta_{0}^{R}$ and $\widehat{\delta}_{0}^{R}$ are obtained by a relatively simple numerical algorithm. Before discussing the significance and the applications of these quantities, we briefly review in the next section the phenomenological input used in the calculations.

We used as input modern data on the nucleon structure functions.³³3I am grateful to J. Gasser and H. Leutwyler for detailed information on the experimental input. Experimental measurements are available from various groups, depending on the c.m.s. energy $W$, defined as

At low energies, $W\leq 1.3\,\text{GeV}$, we used the data on the cross sections measured in Drechsel:2007if ; Kamalov:2000en ; Hilt:2013fda ; MAID . In the range $1.3<W\leq 3\,\text{GeV}$ we used the parametrization of Bosted and Christy Bosted:2007xd ; Christy:2007ve . For $W>3\,\text{GeV}$ and $Q^{2}\leq 1\,\text{GeV}^{2}$, the input has been taken from the vector-meson dominance model of Alwall and Ingelman Alwall:2004wk . For $W>3\,\text{GeV}$ and $1<Q^{2}\leq 3\,\text{GeV}^{2}$, we used the solution of the DGLAP equations constructed by Alekhin, Blümlein and Moch, who obtained numerical values for the structure functions over a wide range: $1\leq Q^{2}\leq 2\cdot 10^{5}$ and $10^{-7}\leq x\equiv Q^{2}/(2m\nu)\leq 0.99$. The underlying analysis is described in Alekhin:2013nda ; Alekhin:2017kpj ; Alekhin:2019ntu . As discussed in Gasser:2015dwa ; Gasser:2020hzn , a conservative attitude is to attach large overall uncertainties, of about $30\%$, to the values of the structure function in this range.

The parametrization of the structure function for $W>3\,\text{GeV}$ and $Q^{2}\leq 1\,\text{GeV}^{2}$ proposed in Alwall:2004wk tends in the limit of high $\nu$ to the imaginary part of the Regge expression (9) for $\alpha=0.55$, allowing us to extract the $Q^{2}$-dependent residue $\beta(Q^{2})$. For $Q^{2}>1\,\text{GeV}^{2}$, the residue is obtained as in Gasser:2020hzn from a Regge fit of the numerical table given in Alekhin:2013nda .

For completeness, we show in Fig. 1 the function $\beta(Q^{2})$ for $Q^{2}$ in the range $(0,3)\,\text{GeV}^{2}$, with an overall error of about 30%. The figure shows that the two representations obtained from different sources for $Q^{2}<1\,\text{GeV}^{2}$ and $Q^{2}>1\,\text{GeV}^{2}$ match remarkably well. Note that the slightly different Regge expression given in Eq. (18) of Gasser:2020hzn coincides at large $\nu$ with (9) for $\beta(Q^{2})=(2m)^{\alpha}\beta_{\alpha}(Q^{2})$, where $\beta_{\alpha}(Q^{2})$ is the residue represented in Fig. 4 of Gasser:2020hzn .

The purpose of this work was to investigate the validity of Regge asymptotics for the amplitude of virtual Compton scattering on the nucleon. Specifically, the aim was to check whether the data on the nucleon structure function, used as input up to a certain energy $\nu_{h}$, are consistent with a behaviour given by Reggeon exchanges for $\nu\geq\nu_{h}$.

The work was motivated by the present applications of dispersion theory in Compton scattering. Since one of the invariant amplitudes, $T_{1}(\nu,Q^{2})$, increases at high $\nu$, the dispersion relation at fixed $Q^{2}$ requires a subtraction. Therefore, the amplitude cannot be recovered completely from the knowledge of the structure function known phenomenologically on the unitarity cut, but depends also on the subtraction function $S_{1}(Q^{2})$, i.e. the value of the amplitude at $\nu=0$.

In general, if the asymptotic behaviour of the amplitude is not known in detail, the subtraction constants cannot be calculated and are viewed as independent ingredients in the dispersion relation. For Compton scattering, this view was adopted in WalkerLoud:2012bg ; WalkerLoud:2012en ; Erben:2014hza ; Thomas:2014dxa ; Walker-Loud2018 .

However, if one assumes that Regge asymptotics is valid, the subtraction function can be determined in terms of the structure function by exploiting in a suitable way the known behaviour. Using the Reggeon dominance hypothesis (4), a sum rule for the subtraction function in terms of the structure function was derived independenly a long time ago in Elitzur:1970yy and Gasser:1974wd , and was evaluated with modern input in the recent analyses Gasser:2015dwa ; Gasser:2020hzn ; Gasser:2020mzy .

The validity of Regge behaviour at high energies is therefore important for improving the precision of dispersion theory for the Compton amplitude. In the present paper, we proposed a test of Regge asymptotics using methods of functional analysis. For illustration, we considered the difference of the inelastic amplitudes $T_{1}^{\text{inel}}(\nu,Q^{2})$ relevant for proton and neutron.

The central role in our analysis is played by the quantities $\delta^{R}_{0}$ and $\widehat{\delta}^{R}_{0}$, defined as the solutions of the functional minimization problems (11) and (12). They represent lower bounds on the maximum difference between the exact amplitude $T_{1}^{\text{inel}}$ and the Reggeon contribution $T_{1}^{R}$ above a sufficiently high energy $\nu_{h}$, imposed by analyticity and the structure function $V_{1}$ known for $\nu<\nu_{h}$. As shown in appendix A, the lower bounds can be calculated by a numerical algorithm.

The results given in Tables 1 and 2 show that, for each $Q^{2}$ in the range $(0,3)\,\text{GeV}^{2}$, $\delta^{R}_{0}$ exhibits an impressive decrease when the energy $W_{h}$ is increased, and becomes only a tiny fraction of the leading Reggeon contribution.

As discussed in Sect. IV, large values of the lower bound at high energies would put doubts on the validity of the Regge asymptotics. However, the small values of the lower bound given in Table 1 show that this is not the case. If the true amplitude $T_{1}^{\text{inel}}$ would saturate the lower bound, there would be little room for subasymptotic corrections in the asymptotic behaviour of this amplitude. We note in particular that a fixed Regge pole at $J=0$ would contribute to the difference (10) as a term $C/\nu_{h}^{\alpha}$, with real $C$. If the lower bound would be saturated by the exact amplitude $T_{1}^{\text{inel}}$, neglecting other subleading contributions to this amplitude, one would have $\delta_{0}^{R}\sim C/\nu_{h}^{\alpha}$, allowing the extraction of the residue $C$. But, since the residue is not dimensionless, its values do not have a direct meaning. The magnitude of the fixed-pole contribution can be controlled by comparing it with the corresponding contribution of the dominant Reggeon. This amounts to consider the ratio of the quantities $C/\nu_{h}^{\alpha}$ and $|T^{R}_{1}|/\nu_{h}^{\alpha}$. Assuming that the lower bound is saturated, i.e. $C/\nu_{h}^{\alpha}\sim\delta_{0}^{R}$, it follows that an appropriate dimensionless parameter that controls the magnitude of the fixed-pole contribution is the ratio $r$ defined in (14). The values of this quantity, presented in Table 2, show that the fixed pole brings a negligible contribution compared to the leading Regge term in the range of $Q^{2}$ considered.

Of course, the existence of a lower bound does not limit from above the difference between the exact amplitude and the Reggeon contribution. So, a strong statment about the contribution of a fixed pole cannot be made. On the other hand, the small value of the lower bound mean that a small value of the exact difference is not forbidden. Therefore, the results given in Tables 1 and 2 allow us to make the conservative statement that the data on the structure function are consistent with the standard Regge asymptotics.

Finally, in Section IV.2 we proposed a prescription for obtaining the subtraction function $S_{1}^{\text{inel}}(Q^{2})$ in a frame consistent with Regge asymptotics. Specifically, our conjecture is to take the value which minimizes the lower bound $\widehat{\delta}_{0}^{R}$ defined in (12). The behaviour of $\widehat{\delta}_{0}^{R}$ as a function of $S_{1}^{\text{inel}}(Q^{2})$ and the stability of the minimum when $W_{h}$ is increased, shown in Fig. 2, support this conjecture. Remarkably, for $0.5\leq Q^{2}\leq 1\,\text{GeV}^{2}$ the results presented in Fig. 3 are very close to the central values of the uncertainty band obtained in Gasser:2015dwa from the Reggeon dominance hypothesis (4).

The significance of this result deserves attention, having in view that in our formalism Regge asymptotics is implemented a way similar, but not identical to the Reggeon dominance hypothesis. Indeed, both approaches use as input the imaginary part of the amplitude along the cut up to a certain $\nu_{h}$, and the behaviour of the full amplitude above $\nu_{h}$. However, the Reggeon dominance hypothesis refers to the asymptotic limit $\nu_{h}\to\infty$, while in the present formalism $\nu_{h}$ is kept finite. Moreover, the Reggeon dominance assumes that the difference between the true amplitude and the Reggeon contribution is strictly zero at $\nu\to\infty$, while in the present approach one finds a lower bound on this difference for $\nu\geq\nu_{h}$. One might think that the Reggeon dominance hypothesis is somehow obtained as a limit of the present formalism for large $\nu_{h}$. However, the limit $\nu_{h}\to\infty$ cannot be defined in a strict mathematical sense in the present formalism, which assumes explicitely that $\nu_{h}$ is finite.

As mentioned above, the position of the minimum of $\widehat{\delta}_{0}^{R}$ as a function of $S_{1}^{\text{inel}}(Q^{2})$ is stable, while the corresponding value of $\widehat{\delta}_{0}^{R}$ decreases when $W_{h}$ is increased. This trend continuous in a spectacular way when $W_{h}$ is further increased up to high values of about 10000 GeV: the minimum occurs at the same place, whereas the corresponding values of $\widehat{\delta}_{0}^{R}$ and $|T_{1}^{\text{inel}}-T_{1}^{R}|$ become smaller and smaller. As mentioned above, in our formulation of the problem, the limit $\nu_{h}\to\infty$ cannot be defined in a formal way. However, the numerical trend⁵⁵5Numerical instabilities are expected to occur at very large $\nu_{h}$, since the exact limit $\nu_{h}\to\infty$ does not exist. suggests that the difference $|T_{1}^{\text{inel}}-T_{1}^{R}|$ approaches extremely small values, consistent with the Reggeon dominance hypothesis (4). This explains why the sum rule derived from Reggeon dominance in Gasser:2015dwa ; Gasser:2020hzn and the prescription proposed in this work, although mathematically quite different, lead to close predictions for the subtraction function.

We emphasize that the present work has an exploratory character: the aim was to propose a new approach for testing Regge asymptotics and its implications. For illustration we considered the amplitude $T_{1}^{\text{inel}}$ and used only central values of the phenomenological input. The investigation of the amplitude $\bar{T}(\nu,Q^{2})$, used recently in Gasser:2020hzn ; Gasser:2020mzy for the evaluation of Cottingham formula, and a detailed analysis of the uncertainties, will be considered in a future work.