Chiral constraints on the isoscalar electromagnetic spectral functions of the nucleon from leading order vector meson couplings

Using baryon chiral perturbation theory including vector mesons, we analyse various continuum contributions to the isoscalar electromagnetic spectral functions of the nucleon induced by the leading order couplings.


Introduction
A premier tool to analyse the electromagnetic structure of the nucleon are dispersion relations. The physics is encoded in the socalled spectral functions, which feature vector meson resonances and continuum contributions. Much is known about the continuum contribution to the isovector spectral function of the electric and magnetic nucleon form factors, see e.g. Ref. [1] for the most recent and most precise investigation. The situation for the corresponding continuum contributions to the isoscalar spectral functions is different, as discussed in detail in Ref. [2]. Chiral perturbation theory has been used to analyze the three-pion continuum [3,4] within the threshold region starting at t 0 = 9M 2 π . These two-loop contributions, however, turn out to be rather small, because the potential enhancement from the anomalous threshold at t 8.9M 2 π is efficiently masked by phase space factors. On the other hand, earlier modelling suggests an important contribution from the πρ continuum [5] and dispersion relations have been used to calculate the KK contribution [6,7]. In the sector with strange quarks, there have been suggestions of cancellation between contributions from KK , K K * and K * K * loops, although these have very different thresholds [8]. It might be worth to put our work in proper context: We work with a chiral Lagrangian with explicit vector mesons, whereas in Ref. [5] a meson-exchange model was used. In contrast to that, the approach used here can be systematically improved. As concerns the dispersive approach used in Refs. [6,7], it in principle contains the vector meson contributions. However, as noted in these papers, the required data for the analytic continuation of the kaon-nucleon scattering amplitudes are not precise enough to uniquely fix the isoscalar spectral functions. Thus, the approach used here gives complementary information.
In this paper, we want to re-analyze some of these continua as well as some vector meson contributions to the isoscalar electromagnetic spectral functions based on chiral perturbation theory with explicit vector mesons. This extends the earlier tree-level analyses of vector meson contributions to the nucleon electromagnetic (em) form factors as given e.g. in Refs. [9,10]. Here, we work at leading one-loop order and consider the vector coupling induced contributions. The paper is organized as follows: In Sec. 2, we briefly recall the pertinent definitions for the electromagnetic form factors and their spectral functions. The πρ contribution is worked out and discussed in Sec. 3. The contributions involving K and K * mesons are given in Sec. 4. We end with a summary and outlook. Some technicalities are given in the Appendix.

Isoscalar electromagnetic spectral functions
The electromagnetic form factors of the nucleon are defined via the matrix element of the electromagnetic current,  (1) with t = (p f − p i ) 2 the invariant momentum transfer squared, m N the nucleon mass and Q the quark charge matrix. Throughout, we work in the isospin limit m N = (m p + m n )/2, with m p (m n ) the proton (neutron) mass. The functions F 1 (t) and F 2 (t) are the Dirac and Pauli form factors of the nucleon, respectively. In the isospin basis, they can be decomposed into isoscalar and isovector components, following the notation of Ref. [11] An unsubstracted dispersion relation can be written down for each form factor defined by where t 0 is the threshold energy for hadronic intermediate states.
Here, we focus entirely on the isoscalar spectral function with t 0 = 9M 2 π , as the three-pion state is the lowest mass intermediate state possible. For a more general discussion of the spectral functions, see e.g. Refs. [2,12,13].

The πρ contribution
In this section we focus on the calculation of the imaginary parts of the isoscalar nucleon electromagnetic form factors generated from the πρ continuum based on relativistic twoflavor baryon chiral perturbation theory. At lowest order in the quark mass and momentum expansion, the relevant interaction Lagrangians are given by [14][15][16] Here, denotes the nucleon doublet, ρ μ the isovector-vector ρ-meson, ω μ the isoscalar-vector ω-meson, A μ the photon field, . . . denotes the trace in flavor space and μναβ is the totally antisymmetric Levi-Civita tensor with 0123 = 1. Further, g A is the nucleon axial-vector coupling, g A 1.27, and the various vector meson couplings are taken as g ρN = 5.92, g ωρπ = 11.6 [17] and f ω = 0.1 [18]. As f ω is not well determined in the analysis of Ref. [18], we also use f ω = 0.4 corresponding to the larger coupling g ω from Ref. [2], keeping the product g ω f ω fixed. This defines a reasonable range for this coupling constant. In the SU (2) sector, due to the universality of the ρ-meson coupling the equality of the ρ-meson self-coupling and the coupling to the nucleons g = g ρN , holds [19]. As can be seen from the Lagrangian in Eq. (4), we work in the vector meson dominance approximation, i.e. the photon couples only via vector mesons to the hadrons. This is e.g. realized in the hidden symmetry approach of Ref. [20] for the parameter a = 2. However, in general, the gauged Wess-Zumino-Witten term also leads to a direct γρπ coupling e.g. in the massive Yang-Mills approach. It can, however, be shown that all these different approaches are equivalent, see e.g. the detailed discussion in Ref. [21]. Our choice is driven by simplicity. For more thorough discussion of the Wess-Zumino-Witten term in the presence of vector mesons and photons, see the reviews [20,21]. Note also that we neglect the small OZI-violating π -ρ contribution due to the φ-meson coupling.
The following power counting rules for the loop diagrams are used: vertices from L (n) count as O(q n ), so we count the vector meson, nucleon and pion propagators as O(q 0 ), O(q −1 ) and O(q −2 ), in order. Thus the order of the diagrams in Fig. 1 at low energies, i.e. for small t.
We work in the centre of momentum frame of the nucleon pair with q = − 2E p , 0 . The initial and the final momentum of the nucleons are, respectively, To calculate the imaginary part of the amplitude for the diagrams which are shown in Fig. 1 with q 2 = t = (p f − p i ) and + stands for → 0 + . The corresponding imaginary parts can be readily determined From these, the expressions for the imaginary part of the isoscalar electric and magnetic form factor can be given as follows Im G S Im G S The resulting electric and magnetic spectral functions Im G S E,M (t) and the weighted spectral functions Im G S E,M (t)/t 2 are shown in Fig. 2. The magnetic one shows a peak at t 1.1 GeV 2 , which is consistent with the modelling in Ref. [5], where the calculated πρ distribution was approximated by a sharp vector meson poles with a mass of about 1.1 GeV. For a more detailed comparison, we would also have to include the ρ-meson tensor couplings that appear at next-to-leading order.

The contribution from strange loops
Next, we consider the effect of loops with KK , K K * and K * K * loops. We note, however, that the latter two only start to have an imaginary part at about 2 GeV 2 and 3 GeV 2 , which is already far inside the resonance dominated region. The relevant effective Lagrangians in the SU(3) basis to obtain the contributions from KK , K K * and K * K * loops read [20][21][22][23][24][25] where V μ and V μ represent the vector meson octet and the vector meson nonet particles, respectively. We take the couplings as G F = g and G D = 0 which results from the constraint analysis and the perturbative renormalizability condition on the vector meson octet and baryon octet interaction of Ref. [26]. The low-energy constants D and F can be determined by fitting to the semi-leptonic baryon decays at tree level. We use the values F = 0.5 and D = 0.8 from Ref. [27]. Throughout, we use the pion decay constant F π for the leading order meson decay constant. In addition to the scalar loop integrals in Eq. (6), we also need the following functions corresponding to the K K , K * K * , K K , K K , K * K * and K * K * loops with the equal masses The corresponding imaginary parts are determined as where M i and m j denote the masses of the K , K * mesons and of the , baryons, respectively. We work in the particle basis using the SU(3) symmetric Lagrangian densities given above. An example of such a calculation can be found in the appendix. The imaginary parts of the Sachs form factors for the K K loop are obtained as Im G S (12) and Im G S The definitions of J (t), J (t) and P K (t) can be found in Eq. (11). In Fig. 4, we show the KK loop contribution including the one from the φ-meson. To account for its finite width, we have MeV is the width of   the φ. We find similar spectral functions for the electric and magnetic form factors except for the sign. Similar to Ref. [6], there is no enhancement on the left wing of the φ which sits directly at the KK threshold. Our results with vector mesons at one loop can, however, not directly be compared to dispersion-theoretical result of Ref. [7] as their calculation does not include all of the φ-meson strength and ours would also require the inclusion of the NLO tensor coupling. Still, it is comforting to see that in the region of the φ there is little continuum strength. As noted earlier, the spectral functions generated by K * K and K * K * start at much larger t and turn out to be rather small, see Fig. 5 and Fig. 6, respectively.

Summary
In this paper, we have used baryon chiral perturbation theory including vector mesons to derive new chiral constraints on the isoscalar electromagnetic spectral functions based on the leading order vector coupling. As noted earlier, the π -ρ loop contribution is the most substantial non-resonant effect and needs to be included. The non-resonant KK contribution that sits under the φ-meson is less pronounced, which is consistent with earlier findings, used e.g. in Ref. [2]. The inclusion of the tensor coupling effects should be considered next.
Here, k 1 and k 2 denote the momenta for the K + and the K − , in order. The φ vector meson has momentum q and Lorentz index μ.
The amplitude for the diagram in Fig. 7 reads with n the number of dimensions and μ the scale of dimensional regularization.