Low-Energy Quark Mass Test for U and D Via Elastic Form Factors

Introduction

The up (u) and down (d) quarks are the lightest generation of quarks. As major constituents of matter, they form the proton (uud) and neutron (udd). Their respective masses, m u=2.2+^0,5- 0.4MeV and md=4.7^+0,5-0.3MeV, are estimates in a Mass-independent Subtraction (MS) scheme [19]. They were first observed by experiments at the Stanford Linear Accelerator Center (SLAC) in 1968 [3,5] through deep inelastic scattering (DIS) experiments which indicated the protons to be made-up of three of these fundamental substructures [8]. Despite being common, the bare masses of u and d are not well determined. However, Lattice QCD calculations have a more precise value of 2.01±0.14MeV/c² and 4.79±0.16MeV/c², respectively [7].

Masses of quarks are fundamental parameters of the Standard Model. Quarks are confined inside the hadrons and are not observed as physical particles, therefore, quark masses cannot be measured directly but must be determined through their influence on hadronic properties.

Any quantitative statement about the value of a quark mass must refer to the particular theoretical framework that is used to define it. The quark masses for light quarks are often referred to as the current (bare) quark masses. Non-relativistic quark models use constituent (effective) quark masses, which are in the order of ~350MeV for the u and d quarks. Constituent quark masses model the effects of dynamical chiral symmetry breaking and are not directly related to the quark mass parameters of the QCD Lagrangian. Constituent masses are only defined in the context of a particular hadronic model. For mass measurements in lattice gauge theory one computes a convenient and appropriate set of physical quantities, frequently chosen to be a set of hadronic masses, for a variety of input values of the quark masses; precise measurements are determined by the lattice spacing a, that is the distance between neighboring points of the lattice and quark masses. The true physical values of the quark masses are those which correctly reproduce the set of physical quantities being used for the calibration. In the particle data listings, quark masses have been obtained by using a wide variety of methods. Each method involves its own set of approximations and uncertainties. In most cases, the errors are an estimate of the size of neglected higher-order corrections or other uncertainties. It is also important to note that the quark mass values can be significantly different for different schemes. At low energy Quantum Chromodynamics (QCD) where both perturbation theory and asymptotic freedom are not possible, the collective interactions between valence and sea quarks become significant. The effects of virtual quarks and gluons in the sea of quarks are assigned to some particular quarks, which get surrounded by the dense cloud of virtual quarks and gluons. This cloud is a high energy barrier concealing the current quark at the core. This system is called constituent quark with an effective mass. The bare masses of u and d are so light that they cannot be straightforwardly calculated because relativistic effects have to be taken into account.

Form factors used to generate differential cross section of elastic electron-proton scattering (dcsep) were measured through various methods. One of which is by Rosenbluth Extraction Method which obtains them from the plot of the reduced cross section versus the square of the transfer momentum at several angles by determining the plot’s slope and intercept, and performing linear regression with it [14]. With the world data [1,2,13,17,20], the form factor ratio is consistent to ~1.0 at very low energies. Another is by Polarization Transfer Method, wherein the form factor ratio is measured through polarization transfer where longitudinally polarized electron beam is scattered from an unpolarized proton target. For ep-scattering in the single-photon exchange, it was shown that the normal component of polarization vanishes and the transverse and longitudinal components satisfy certain conditions [15]. Without the need to measure the cross-sections, this gives the form factor ratio but which does not agree well Rosenbluth measurements. Form factor ratio from polarization transfer are well fit by 1-0.13( Q2-0.04) [9]. The Super-Rosenbluth method of form factor extraction is associated with smaller angular-dependent corrections. And here, it is the protons that are scattered rather than electrons. Experimental data and results of elastic simulation from [14] used the Bosted global fit of previous Rosenbluth data from [4] due to the slight variation of Q² across the finite momentum and angle acceptance. P.E. Bosted ensured that form factors from the elastic simulation gave the closest corresponding final cross section. However, it is only valid at 0<Q²<7GeV². Another fitting procedure was presented in [22] and explained with details in [16]. For the proton, simultaneous fitting on form factors to the data were performed and the fit is a bounded polynomial z-expansion [12]. This global data fitting procedure is valid up to Q²~30GeV².

Since the masses of u and d are not directly calculated, a device test can be formulated to estimate their masses via the measurement of the dcs generated from the form factors of elastic ep-scattering. The main objective of this study is to develop a technique in the mass estimation of u- and d-quarks with the proton mass as a parameter (and vice-versa) using the dcsep generated from theory and experiments. The investigation was mainly to determine the energy and dcs intersections of dcsep, dcseu and dcsed where the quarks assume the minimum/ maximum bare and effective masses by modifying the relativistic recoil factor using the proton mass at the low energy regime.

Methods

The generated dcsep data sets are compared to the modified dcseq, where q is either u and d, and their intersections were determined within the low energy regime. In the generation of dcseq, the quarks were assigned bare and effective (low energy) masses [10,18,19,21], separately. The relativistic recoil factor of dcseq of the spin-averaged eq-scattering is modified by using the mass of the proton in order to shift the distribution of dcseq closer to dcsep and thereby putting a possibility of finding an energy intersection with it. This energy intersection actually means the square of momentum transfer at which the two dcs meet. This materializes the proton mass a parameter in the determination of the quark mass estimates. The dipole proton form factors (dff) [11] and form factors derived from the fitting of experimental data, such as polarization transfer fitting (ptf) [14], Bosted global fitting (bgf) [4] and global data fit (gdf) [22]; and the average of all form factors (aff) were used to generate the data for dcsep. Using these form factors, the dcsep, dcseu, and dcsed were generated simultaneously in ROOT Data Analysis Framework [6]. The value for the anomalous magnetic moment of the proton used is ^=2.793 and the mass of proton is set at m p = 0.938272081±0.000000023 GeV. A total of 3000 events each for dcsep, dcseu, and dcsed were gathered at various scattering angles randomly selected within 0^o to 180^o. Then, they were averaged for each data point.

Analysis

The dcsep and dcseq are curves in the dcs versus Q² plot but there is no assurance of their intersection in at least one point unless alterations have to be implemented. So, the proton mass was required as a parameter in the quarks mass test. This can be done through modifying the relativistic recoil factor of dcseq using the mass of proton, instead. For a fixed scattering angle or considering all angles, the relativistic recoil factor is just a constant at a particular transfer momentum. Doing so does not alter the distribution, however, it shifts the dcs vertically. Considering this modification could make the possible attainment of an energy intersection. With the proton mass in the recoil factor, the existence of an energy intersection at low-energy is highly probable. The proton mass parameter becomes the link between the two dcs curves. It should be noted that polarization transfer fitting (ptf) diverges at Q²~7.7GeV² and Bosted global fit (bgf) is valid only at Q²~7GeV². Hence, it is important that the intersections should be below these valid transfer momentum limit. To give a general applicability of the technique, the values of u- and d-quark bare masses (minimum and maximum) and the effective masses were used. Effective quark masses are important since they dominate at low energy. Table 1 summarizes the bare and effective masses used as inputs to determine the energy and cross section intersections.

Table 1. Mass range for u and d in MeV.

Estimation of the masses of u- and d-quarks using the generated dcsep from form factors coming from different fitting models has to establish the energy at which dcsep and modified dcseq intersect with the globally accepted proton mass as a parameter. Once the energy intersections and their corresponding form factors are determined and established, then the dcs can be generated and from it the range of masses will be obtained. Oppositely, with an established energy intersection, a range of values of u and d quark masses can be plugged-in to the modified dcseq to obtain the accepted experimentally derived proton mass. Hence, a technique is developed for the mass estimation of u- and d-quarks using the generated dcsep from form factors derived from experiments. Different experiments give different dcs corresponding to different form factors, whichever come first at a given energy, vesting a range of values for the quark masses as inputs to obtain the mass of the proton which is also determined in a multitude of ways.

Figure 1. (a) The differential cross section (dcs) of the eu-scattering (red) and ed- scattering (blue) carrying effective masses are compared to that of ep-scattering (black) generated from the averaged form factors (aff). (b) The dcseu (red) and dcsed (blue) carrying effective masses are compared to dcsep (black) generated from aff with scattering angles ranging from 0^o-180^o. The intersections are pronounced here.

Results

The global data fit (gdf), one of the four form factor data fittings considered, does not agree well with the other three. This is due to the coefficient parameters of the curve. The limits of coefficient parameters are chosen only when actual data are available for analysis. For this study, it is assumed that the dcs data points generated from this fitting procedure are the deviants to the expected outcomes and they constitute 25% to the dcs generated from the average form factors (aff). The choice of coefficient parameters for gdf, to be within -0.214 to 0.214, is a compromised one and was based on the criteria that dcsep generated from gdf should produce an intersection with the modified dcseq not beyond 3GeV for both the modified dcseu and dcsed. It should be noted that gdf has a validity of up to ~30GeV², that is way beyond the limits of polarization transfer fitting (ptf) and Bosted global fit (bgf) which are valid only up to ~7GeV².

Comparisons of the generated dcsep between dcseu and dcsed, where quarks assume the bare masses, have large disparities and no intersections were observed. However, in Figure 1(a) where quarks have effective masses, intersection can exist between the curves, see Tables 2 and 3, for the summary of energy and dcs intersections. The extrapolated average energy intersection of dcsep from aff with dcseu is 346.00008MeV². It has no intersection with dcsed but they were closest at ~1954.54339MeV². Their corresponding extrapolated dcs are 4.58726x10-4 and ~1.1597x10^-5 in the units of barn, respectively.

Table 2. Accessible low energy intersections of dcsep and dcseq in MeV² with bin size of 1 MeV² and where the quarks assume effective masses.

*there is no intersection at this energy but the dcs are closest here.

Table 3. Corresponding dcs (barns) intersections of ep and eq-scatterings in Table 2.

Figure 2. The dcseu (black) with minimum bare mass are compared to dcsep generated from (a) dff, (b) ptf, (c) bgf and (d) gdf form factor data sets.

Modifying the relativistic recoil factors caused the dcseq distributions to shift nearer to the dcsep, see Figure 1(b) for quarks carrying effective masses and Figure 2 for quarks carrying minimum bare masses. Due to the shift, intersections were observed. The energy intersections between dcseu and the generated dcsep from different form factor fitting were around the extrapolated value of 407MeV² to 450MeV² except for gdf which is beyond 7GeV². For dcsed, the energy intersections with dcsep occur beyond 1GeV² except for the ones generated by gdf which register around 183MeV2. In order to provide a range for the bare mass estimate of the quarks, their minimum and maximum values from MS scheme were used as inputs for this test.

Table 4. Low energy intersections of dcsep and dcseq in MeV² with bin size of 1 MeV².

Table 5. Corresponding dcs (barns) intersections of ep and eq-scatterings in Table 4.

The extrapolated energy intersection of dcsep generated from aff with the modified dcseu carrying the minimum bare mass is 349.00016MeV² and this is greater than that of the modified dcseu carrying the maximum bare mass at 349.00015MeV²; their respective extrapolated dcs intersections were at 4.51368x10^-4 and 4.51363x10^-4. The extrapolated energy intersection of dcsep generated from aff with the modified dcsed carrying the minimum bare mass is 1272.00005MeV² and this is greater than that of the modified dcsed carrying the maximum bare mass at 1272.00004MeV²; their respective extrapolated dcs intersections were both at 0.26820x10^-4. For quarks carrying effective masses, the extrapolated energy intersection of dcsep generated from aff with the modified dcseu is 226.00013MeV² and this is less than that of the modified dcsed at 1093.00004MeV² with respective dcs intersections at 9.83012x10^-4 and 0.36743x10^-4. For quark masses calculated from Lattice QCD, the extrapolated energy intersection of dcsep generated from aff with the modified dcseu carrying the minimum and maximum bare masses were both at 349.00016MeV²; their respective extrapolated dcs intersections were at 4.51367x10^-4 and 4.51366x10^-4. The extrapolated energy intersection of dcsep generated from aff with the modified dcsed carrying the minimum bare mass is 1272.00005MeV² and this is greater than that of the modified dcsed carrying the maximum bare mass at 1272.00004MeV²; their respective extrapolated dcs intersections were both at 0.26820x10^-4. All intersections are summarized in Tables 4 and 5.

Conclusions and Recommendations

Indeed, a device to estimate the range of quark masses for u and d via the elastic ep-scattering at low momentum transfer can be possible by generating dcsep data sets from form factors and comparing them to the modified dcseq, where q is either u and d, wherein the proton mass is a parameter. The theoretical dipole form factor and some form factor fitting models used in experiments such as the polarization transfer fitting, Bosted global fitting, global data fit and their average were used to generate the dcsep. The dcsep, dcseu and dcsed would have been independent of each other without the modification of the relativistic recoil factor since they do not possess a critical intersection that could be exploited. By using the mass of the proton into the relativistic recoil factor of the spin-averaged eq-scattering, the dcseq distribution was shifted closer dcsep and thereby increasing the possibility of intersection. Materializing the proton mass as a parameter, intersections were observed within the low energy regime by using the experimental quark masses as inputs. These intersections were summarized in Tables 2 through 5. Once established, these intersections can be used on experimental elastic ep-scattering data in order to estimate the masses of u and d, as well. Using most, if not all, data from actual experiments, a global range of u and d can be estimated. For the estimation of effective quark masses, both the intersections of the raw and modified dcseq have to be used simultaneously. One of the things to be done in making the intersections formidable would be incorporating additional form factor fitting models and to integrate the results of the actual experiments as they come. Generating the dcs with much smaller bin sizes is recommended to give a more precise measurement of the intersections. Also, choosing the optimum coefficient parameters for gdf could lead to a better result. Although the protons are measured in a multitude of ways, but more precise mass ranges of u and d are led by the precise measurement of its mass being the parameter of the test. The accuracy of the results can also be improved by generating even more events and considering more scattering angles. All of these recommendations, however, would entail much more computing power.

Acknowledgment

The Mindanao State University - Iligan Institute of Technology (MSU-IIT) and its Department of Physics and the Premier Research Institute for Science and Mathematics (PRISM) of Iligan City, Philippines; Research Center for Theoretical Physics (RCTP) of Jagna, Philippines; and Centro de Investigacion en Computacion - Instituto Politecnico Nacional (CIC-IPN) of CDMX, Mexico are acknowledged for their conducive venues in making this research possible. Gratitude is extended to the Department of Science and Technology (DOST) of the Philippines and MSU-IIT for their financial support. The inspiration and encouragements from Prof. Christopher Bernido, Prof. Maria Victoria Bernido, Prof. Ludwig Streit and Prof. Wolfgang Bock are highly appreciated.

References