The K-meson form factor and charge radius: linking low-energy data to future Jefferson Laboratory measurements

The European Physical Journal C, Jul 2017

Starting from a successful model of the \(\pi \)-meson electromagnetic form factor, we calculate a similar form factor, \(F_{K}(Q^{2})\), of the charged K meson for a wide range of the momentum transfer squared, \(Q^{2}\). The only remaining free parameter is to be determined from the measurements of the K-meson charge radius, \(r_{K}\). We fit this single parameter to the published data of the NA-7 experiment which measured \(F_{K}(Q^{2})\) at \(Q^{2}\rightarrow 0\) and determine our preferred range of \(r_{K}\), which happens to be close to recent lattice results. Still, the accuracy in the determination of \(r_{K}\) is poor. However, future measurements of the K-meson electromagnetic form factor at \(Q^{2}\lesssim 5.5\) GeV\(^{2}\), scheduled in Jefferson Laboratory for 2017, will test our approach and will reduce the uncertainty in \(r_{K}\) significantly.

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The K-meson form factor and charge radius: linking low-energy data to future Jefferson Laboratory measurements

Eur. Phys. J. C The K -meson form factor and charge radius: linking low-energy data to future Jefferson Laboratory measurements A. F. Krutov 2 S. V. Troitsky 1 V. E. Troitsky 0 0 D.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University , Moscow 119991 , Russia 1 Institute for Nuclear Research of the Russian Academy of Sciences , 60th 2 Samara University , 443086 Samara , Russia Starting from a successful model of the π -meson electromagnetic form factor, we calculate a similar form factor, FK (Q2), of the charged K meson for a wide range of the momentum transfer squared, Q2. The only remaining free parameter is to be determined from the measurements of the K -meson charge radius, rK . We fit this single parameter to the published data of the NA-7 experiment which measured FK (Q2) at Q2 → 0 and determine our preferred range of rK , which happens to be close to recent lattice results. Still, the accuracy in the determination of rK is poor. However, future measurements of the K -meson electromagnetic form factor at Q2 5.5 GeV2, scheduled in Jefferson Laboratory for 2017, will test our approach and will reduce the uncertainty in rK significantly. 1 Introduction and outline Quantitative description of particle systems in the strongcoupling regime remains one of the most challenging problems of contemporary particle theory. The electromagnetic structure of light mesons represents an ideal testbed for various approaches to practical calculations at strong coupling, including low-energy effective theories of strong interactions and their connection to Quantum Chromodynamics (QCD). Not surprisingly, experimental understanding of various aspects of the meson structure comprises an important part of the scientific program of the upgraded Jefferson Laboratory (JLab) [ 1 ], presently ready for its start. In particular, the E12-09-011 experiment, scheduled in JLab Hall C for 2017, will attempt to measure the K -meson electromagnetic form factor, FK (Q2), at the momentum transfer squared up to Q2 ∼ 5.5 GeV2 [ 2 ]. In this work, we address some important implications of this expected result and revisit previous scarce data on FK (Q2) in the frameworks of a successful theoretical model. Some time ago, a model for the electromagnetic form factor of the charged π meson, Fπ (Q2), has been developed (see Refs. [ 3–5 ] for a detailed description), possessing a few free parameters, fixed in 1998 [ 6 ] from the experimental data available at that time. The model predicted subsequent JLab experimental results on Fπ (Q2) surprisingly well [ 7 ] without further tuning of parameters, despite the fact that the experimentally accessible range of Q2 was extended by an order of magnitude [ 8 ]. Moreover, the model with the very same parameters predicts the correct QCD asymptotics of Fπ (Q2) at large Q2 [ 9,10 ]. Subsequently, the model was also applied to the calculation of electroweak parameters of the ρ meson [11], for which particular interesting relations have been obtained. The theoretical grounds for the model include a relativistic-invariant approximation [ 3 ] to the instant form of the relativistic Hamiltonian dynamics (see e.g. Ref. [ 12 ]), while the model’s quantitative parameters in the light-quark sector are fixed from the successful π -meson study [ 6 ]. The two principal benefits of our model have been demonstrated in the π -meson study. The first one is its predictivity: provided the experimental value of the decay constant is fixed, only one parameter remains to fit the mean square radius. Any further dependence on the model details, e.g. in particular the choice of the phenomenological wave function, is negligible [ 6 ]. The second advantage is matching with QCD predictions in the ultraviolet limit: when constituentquark masses are switched off, as expected at high energies, the model reproduces correctly not only the functional form of the QCD asymptotics, but also the numerical coefficient; see Refs. [ 9,10 ] for details. To the best of our knowledge, this is the only available low-energy model reproducing the QCD limit without introducing additional parameters. Analytical properties of the pion form factor, as a function on the complex Q2 plane, obtained in our model, are the same as follows from the general quantum field theory principles [28]. Since the model predicted successfully the values of Fπ (Q2) up to Q2 ∼ 2.5 GeV2, we expect that it can be used for FK (Q2) at least in the same domain of momentum transfers. Our model shares its limitations with other constituent-quark models of mesons: at present, their parameters cannot be consistently derived from QCD without additional experimental input. In this paper, the model is applied to the K meson. This brings two additional parameters into the game, one being the mass of the strange constituent quark, Ms , and another describing the interaction in the light-heavy quark system. In the case of the π meson, the two corresponding parameters were uniquely determined from two measured observables, the π -meson decay constant, fπ , and charge radius, rπ , which determines the form-factor behavior at low Q2. We will demonstrate below that, for the K meson, the decay constant, f K , fixes one combination of parameters, while the charge radius, rK , is known with large uncertainties, which makes it difficult to use it for fixing the remaining parameter. We therefore keep it free and obtain a range of the model parameters consistent with the present data. We address old measurements of FK (Q2) at Q2 → 0 obtained in the NA7 experiment at CERN SPS [ 13 ], which represents the most precise source of experimental input for determination of rK . The value of rK obtained in Ref. [ 13 ], since then extensively used in numerous experimental and theoretical works, was estimated from fitting the data points in the pole approximation. We demonstrate that the range of Q2 studied in Ref. [ 13 ] was sufficiently large for deviations from this approximation to become important. We fit the NA-7 data points with our exact functions for FK (Q2), which results in a slightly shifted value of rK . We note that the obtained allowed range of rK , bounded by the 68% CL agreement with the data and by the consistency of the model, is in a better agreement with the recent lattice results [ 14 ] than the original result of Ref. [ 13 ]. Turning to higher energies, we use the constraints on rK to fix the allowed range of FK (Q2) functions at modestly large Q2 6 GeV2. We observe that the spread of the theoretical curves exceeds considerably the expected precision of the E12-09-011 experiment in JLab. Therefore, within our approach, the E12-09-011 results might be used not only to study FK (Q2) at moderate Q2 but also to constrain its behavior at Q2 → 0 and to further narrow the experimentally allowed range of rK . This improvement in the value of a very soft parameter represents an unexpected application of the coming experiment, aimed presumably at the studies at much higher momentum transfers, Q2 ∼ (0.5–5.5) GeV2. The rest of the paper is organized as follows. In Sect. 2, we describe briefly the model for FK (Q2), with the emphasis on the differences between the π - and K -meson models and on the two new parameters we have to introduce. Sect. 3 addresses the experimental constraints on rK . The NA-7 data are reanalyzed here in the frameworks of our model. In Sect. 4, we present the expected FK (Q2) behavior at larger Q2 and demonstrate how the E12-09-011 JLab experiment may improve the precision of the rK measurement. We briefly conclude in Sect. 5 and present a more detailed description of the model in the Appendix. 2 The K -meson electromagnetic form factor The approach we used is based on the instant-form Dirac relativistic Hamiltonian dynamics (see e.g. Ref. [ 12 ]) supplemented by the relativistic-invariant modified impulse approximation [ 3 ]. The form factor of a system of two quarks with different masses1 has been calculated, within this approach, in Ref. [ 15 ]. For completeness, all necessary formulas are collected in the Appendix. In general, we need to know five parameters to proceed with the real numerical calculation. They include the masses of the two constituent quarks, Mu and Ms ; the parameter of the two-quark phenomenological wave function, b, with the physical meaning of the confinement scale; and two anomalous magnetic moments of quarks, κu and κs . The values of the latter are fixed in the same way as it was done for the π meson, that is through the Gerasimov sum rules; see Ref. [ 16 ] for details and the appendix for explicit expressions. The value of κu is, clearly, the same as it was used for the π meson. We also take advantage of the working model for Fπ (Q2) where Mu = 0.22 GeV was fixed. The phenomenological confinement scale, b, may in principle be different for different systems, and, for the moment, we keep it as a free parameter, together with Ms . At this point, we note that we do not vary the shape of the phenomenological wave function u(k) used in the calculation, but fix it instead from the π -meson study and leave only the scale b as a free parameter. In early theoretical studies of our model [ 15,18 ], various wave functions have been used, which, in general, resulted in different predictions. However, once the model was applied to phenomenology, the dependence on the wave-function choice was found negligible, provided the value of the meson decay constant was fixed [6]. The theoretical systematic uncertainty related to the choice of the wave-function shape is small compared to the experimental uncertainties. Note that the weak dependence of the results of calculation of electromagnetic form factors of pseudoscalar mesons on the shape of the wave function of constituent quarks has been pointed out also in Ref. [ 29 ]. 1 The π meson is well described with Mu = Md . The choice of the light constituent-quark mass Mu was determined from a fit to experimental values of Fπ (Q2) at small Q2 in Ref. [ 6 ] and confirmed by subsequent experimental data. It is interesting to note that, with the same value of Mu , the mass spectrum of light mesons had been successfully described in Ref. [ 30 ], within a different framework. We leave the study of the meson masses in our model for future work. Within our approach, we build up a consistent phenomenologically successful global fit of electroweak properties of light mesons. The same parameters have been used first for the best-studied π meson [ 6,10 ], then for certain properties of the ρ meson [11]; now we proceed with the K meson. To fix the two free parameters, b and Mu , for the π meson, two experimental observables were used. One was the meson decay constant, fπ , and another was the meson charge radius, rπ , determined from experimental measurements of Fπ (Q2) at Q2 → 0. For the K meson, we may equally well use the decay constant, fK = (0.1562 ± 0.0010) GeV [ 17 ], to eliminate one of the parameters. The expression relating f K to the model parameters was derived in Ref. [ 18 ] and is presented in the Appendix. It would be natural to use the experimental information on the K -meson charge radius, rK , to fix the single remaining free parameter and to predict the behavior of FK (Q2) in the yet unexplored domain of large Q2. However, as we will see in the next section, this approach is limited by the poor experimental knowledge of rK . 3 Experimental constraints on the K -meson charge radius The form factor FK (Q2) was measured by the NA-7 experiment at CERN SPS, Ref. [ 13 ]. This measurement of 1986 remains the most recent and the most precise one, and we will concentrate on its results in what follows.2 Figure 1 presents the experimental data points. The authors of Ref. [ 13 ] used these data to extract the K -meson charge radius by fitting their data with the pole approximation, FK (Q2) = 1/ 1 + Q2 r K2 /6 . We note in passing that a better fit to data points was obtained in Ref. [ 13 ] when the condition FK (0) = 1 was not used, though a departure from this condition is unphysical. This resulted in the value of r K2 = 0.34 ± 0.05 fm2 (50% CL), widely used in subsequent studies. However, one may note that the actual FK (Q2) function may deviate from the pole approximation already at 2 Inclusion of an earlier measurement of Ref. [ 31 ] cannot change our result because of larger error bars and smaller number of data points, all of which lay farther away from Q2 = 0. Q2 ∼ 0.1 GeV2, so that corrections to the pole approximation are already important for the NA-7 data range. This means that, to determine the derivative of FK (Q2) at Q2 = 0, and hence rK , one should use either a shorter range of Q2 or a more complicated approximation. The first option fails because of the insufficient number of data points. Fortunately, as described above, we can calculate the function FK (Q2) within our approach. The model has one free parameter which we use to fit, by means of the usual χ 2 method, the NA-7 data points with their experimental error bars. To do that, we consider the two-dimensional parameter space (Ms , b) of the model; see Fig. 2. Fixing the value of f K implies a constraint on (Ms , b), so that a one-parametric space remains (one can see from Fig. 2 that the precision of the experimental value of f K is so good that its uncertainty may be neglected). The remaining freedom is therefore parametrized by a line on the (Ms , b) plane; different points on the line correspond to different values of rK . Changing Ms and always keeping b(Ms ) to satisfy the f K constraint, one may fit the experimental data points for FK (Q2). One may note, however, from Fig. 2, that not all values of rK may be achieved, provided the f K constraint is satisfied. Indeed, at r 2 (Ms , b) < 0.39 fm2, the two curves K determined by f K (Ms , b) and rK (Ms , b), Fig. 2, have no intersection points. Since all other parameters beyond Ms and b are fixed from the π -meson studies and their values are confirmed experimentally, we have no freedom to change this picture. Therefore, the limitation in simultaneous description of fK and rK is a consequence of our requirement of a joint description of π and K mesons, and not of the construction Fig. 2 The two parameters for the K -meson form factor. The experimental value of fK is reproduced on the red dashed line, with the pink shade representing the experimental uncertainty (this condition leaves essentially one-dimensional parameter space). Thin black lines correspond to different values of r K2 , indicated by numbers (in fm2) of our relativistic model, whose predictivity is manifested in this way. The value of Ms ≈ 0.27 GeV corresponds to the lowest achievable rK , which splits the b(Ms ) line into two branches, so that a larger value of rK may be obtained for two distinct values of Ms . In Fig. 3, we illustrate the results of the fit by presenting the χ 2(rK ) function determined by this method. We note that the best fit (χ 2 = 13.5 for 14 degrees of freedom) corresponds to the lowest value of rK , allowed in our approach, and that 0.39 fm2 ≤ r K2 ≤ 0.42 fm2(68% CL, this work). (1) Our best-fit curve is also presented in Fig. 1. It is interesting to note that our best-fit value of r K2 = 0.39 fm2 is in a better agreement with the recent lattice calculations [ 14 ], r K2 ,lattice = 0.380 ± 0.033 fm2, than the best-fit value derived with the pole approximation. However, we note that, being derived from the same data, our value for rK shares similar large statistical uncertainties with the original result, and both are compatible at 68% CL. It is interesting to compare our result also with the values obtained from the data analysis in the frameworks of the chiral perturbation theory [ 32 ], 2 which vary between rK ,ChPT,min = 0.354 ± 0.071 fm2 and 2 rK ,ChPT,max = 0.431 ± 0.071 fm2: our 68% CL interval for r K2 is contained in that of Ref. [ 32 ] for all their assumptions. 4 Large Q2 and the future JLab experiment The use of the low-Q2 data constrains, by Eq. (1), the remaining free parameter of the model through the fitting procedure described in the previous section. In principle, this allows us to calculate the FK (Q2) function for a large range of Q2 and to make predictions for the future JLab measurements. This prediction is presented in Fig. 4, where the Q2 range probed by the E12-09-011 JLab experiment and estimated uncertainties of the measurement [ 27 ] are shown. For comparison, we present also predictions obtained within other approaches. Considering Fig. 4, one immediately notes that the uncertainty in our predictions, determined by the uncertainty in the rK measurements, exceeds the expected precision of the experiment and is of order the typical difference between model predictions. The E12-09-011 experiment would be able to distinguish between our model and several other ones. Since these approaches differ in their original assumptions as well as in approximations being used, future JLab experiments will be able, in principle, to contribute to the choice between various models of the description of nonperturbative dynamics of strong interactions at large and intermediate distances. In this context, it is instructive to compare our results with those of Ref. [ 24 ]. Their approach differs from ours by the choice of the form of relativistic dynamics: they use the light-front dynamics. Another difference is in the approximations: while we use the so-called modified impulse approximation, see e.g. Ref. [ 5 ], the conventional impulse approximation was used in Ref. [ 24 ]. Our modified impulse approximation, unlike the conventional one, does not violate covariance conditions, nor the current conservation. Another difference with Ref. [ 24 ], which may be important at large Q2, is in the Q2-dependence of quark form factors: we use a renormalization-group inspired logarithmic function, while in Ref. [ 24 ], a dipole form is used. Note that the value of Mu and the expression for the quark radius coincide in the two works. In addition, this consideration opens a surprising new possibility to constrain the low-Q2 behavior of FK (Q2) and to reduce the experimental uncertainty in rK . Indeed, the expected error bars of the experiment are smaller than the spread of FK (Q2) curves predicted in our model. This spread is determined by the 68% C.L. allowed variations of the single not firmly fixed parameter of the model, rK , which determines the behavior of FK (Q2) at Q2 → 0. Hence, limiting the spread at large Q2 would narrow the allowed range of rK . Figure 5 illustrates how the measurement of FK (5.5 GeV2) in JLab would constrain rK in the case that the uncertainty of the measurement agrees with the estimate of Ref. [ 27 ]. We note that the reduced uncertainty in rK transforms, within our approach, into a more precise knowledge of the model Fig. 5 The effect of future JLab measurements on the precision of the rK value. Suppose that JLab finds the value of FK (Q2 = 5.5 GeV2) shown in the horizontal axis, with the uncertainty given in Ref. [ 27 ] (conservative value). Then, within our approach, one would be able to constrain rK to lay between the lower and upper thick red curves for this value of FK (the strip with red vertical hatching). For comparison, rK of Ref. [ 13 ] is shown by the dashed line (best fit), dark blue shade (68% CL) and light blue shade (95% CL). The horizontal strip with orange diagonal hatching represents the lattice result of Ref. [ 14 ] parameter; for instance, Ms , which may be interesting from a theoretical point of view (cf. Fig. 2). 5 Conclusions In this work, we discussed present and future experimental data on the K -meson electromagnetic form factor, FK (Q2), in the context of the model for the electroweak structure of light mesons based on the relativistic-invariant modified impulse approximation in the frameworks of the instant-form relativistic Hamiltonian dynamics. All but two parameters of the model are fixed from its successful application to the π -meson form factor in previous studies, where, experimentally, the model predictions had been confirmed precisely by a number of subsequent measurements, and theoretically, the correct QCD asymptotics was reproduced. Of the two parameters specific for the K -meson case, one combination is fixed from the K -meson decay constant, f K , while the remaining one is related to the K -meson charge radius, rK . The latter is known experimentally with large uncertainties. We revisited the NA-7 data on FK (Q2) at Q2 0.1 GeV2 and found the rK range allowed by the data within our model. This range agrees well with the recent lattice results. Still, the error bars in rK remain large, which makes the predictions of the model uncertain in the large-Q2 range to be probed by the coming E12-09-011 experiment in JLab. However, this suggested an unexpected application of the coming JLab measurements of FK (Q2), which, despite being performed at Q2 ∼ (0.5–5) GeV2, would improve the accuracy of the rK measurements. Acknowledgements We thank A. Gasparian and G. Huber for interesting information on the coming JLab K -meson experiments. The work of AK was supported in part by the Ministry of Education and Science of the Russian Federation (Grant No. 1394, state task). The work of ST on the nonperturbative description of strongly coupled QCD bound states is supported by the Russian Science Foundation, Grant 14-22-00161. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. Appendix A: Formulas for the form-factor calculation Appendix A.1: The form factor of a system with two quarks of different masses This form factor was calculated, in the present approach, in Ref. [ 15 ]. The free form factor is given by √ss [s2 − 2s(Ms2 + Mu2) + η2][s 2 − 2s (Ms2 + Mu2) + η2] Bu (s, Q2, s ) + Bs¯(s, Q2, s ) , g0(s, Q2, s ) = Q2(s + s + Q2) × 2[λ(s, −Q2, s )]3/2 where Mq is the mass of the constituent quark q, η = Mu2 − M 2, s Bs¯(s, Q2, s ) = f1(s¯)(s + s + Q2 − 2η) cos(ω1 + ω2) − f2(s¯) Ms ξ(s, Q2, s ) sin(ω1 + ω2) θ (s, Q2, s ), 2 ξ(s, Q2, s ) = −λ(s, −Q2, s )Ms2 + ss Q2 − ηQ2(s + s + Q2) + Q2η2, λ(a, b, c) = a2 + b2 + c2 − 2(ab + ac + bc), f (s¯) = 1 2Ms G(Es¯)(Q2) 4Ms2 + Q2 ; f2(s¯) = − Ms 4Ms2 + Q2 , the Wigner rotation parameters are ω1 = arctan ω2 = arctan where eq are quark charges, fq (Q2) = 1/(1 + log(rq2 Q2/6)) and rq2 = 0.3/Mq2. The anomalous magnetic moments κq of quarks q are calculated following Ref. [ 16 ]. The values of κu and κd¯ should satisfy [ 16 ] eu + κu ed + κd = −1.77. The π -meson form factor depends [ 6 ] on the sum κu + κd¯, which has been fixed in Ref. [ 6 ] from the condition that the constituent-quark parameters providing a good description of the data do not depend on the choice of the shape of the phenomenological wave function: in this way, the value κu + κd¯ = 0.0268 has been found for Mu = 0.22 GeV, and we use this value in the present study. Together with Eq. (A.1), this condition determines κu and κd unambiguously. The value ¯ of κs is determined [ 16 ] from (A.1) eedu++κκud + ed +κd es +κs es+κs 1 + ed +κd 2 = 0.42. This relation, for fixed κu and κ ¯, determines the value of d κs up to the choice of the sign at the square root from the ¯ right-hand side. We choose the negative sign because in the opposite case, the solution gives an unphysically large value of κs¯. We arrive at the values of κu ≈ −0.01055 and κs¯ ≈ −0.08099, which are used in this study. The form factor FK (Q2) is given by the double integral, FK (Q2) = √ √ d s d s ϕ(k) g0(s, Q2, s ) ϕ(k ), where ϕ(k) = √s(1 − η2/s2) u(k) k, k = (s2 − 2s(Ms2 + Mu2) + η2)/4s, is the phenomenological wave function of the two-quark system. Following previous studies of the π meson, we choose the power-law function, u(k) = N (k2/b2 + 1)−3, where the normalization N = 16 by the condition 0 ∞ dk k2u(k)2 = 1. Appendix A.2: The K -meson decay constant The expression for the decay constant f K was calculated, in the present approach, in Ref. [ 18 ]. It reads 2/(7π b3) is determined f K = where G0(s) = ∞ Ms +Mu √ 3 2√2π √s × pq0 = Mq2 + k2. √ d s G0(s)ϕ(s), ( ps0 + Ms )( pu0 + Mu ) k2 1 − ( ps0 + Ms )( pu0 + Mu ) , 1. J. Dudek et al., Physics opportunities with the 12 GeV upgrade at Jefferson Lab . Eur. Phys. J. A 48 , 187 ( 2012 ). arXiv: 1208 .1244 [hep-ex] 2. P. Bosted et al., Studies of the L-T separated kaon electroproduction cross section from 5-11 GeV . 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A. F. Krutov, S. V. Troitsky, V. E. Troitsky. The K-meson form factor and charge radius: linking low-energy data to future Jefferson Laboratory measurements, The European Physical Journal C, 2017, 1-8, DOI: 10.1140/epjc/s10052-017-5038-8