Project sponsor: The Foundation for Polish Science,

Project number: 2014/15/N/ST2/03451 (in Polish only)

Principle investigator: Dr. Arkadiusz P. Trawiński, University of Warsaw

Scientific advisor: Prof. Stanisław D. Głazek, University of Warsaw

Time of realization: from Jun 20, 2015 to Jan 19, 2017

Examination of the proton light-front wave function

The model of proton built from two up quarks, u₁ and u₂, and one down quark, d₃, was studied as part of the PRELUDIUM project. Following the Ref.[1] the state of the proton of momentum P and a spin σ is defined as,

where Ioffe currents [2] are denoted by I_k. The scalar function f for quarks of the same mass m reads,

Using this triquark model, the quark-diquark model of proton has been derived [3]. In the first step, momenta p_i^T and p_i⁺ had been expressed by relative momenta, p^T and k^T, as well as by ratios of longitudinal momenta x and y. In our notation k^T is inner momentum of a diquark and p^T is momentum between an active quark and the diquark. In the second step, one had to integrate over all inner degrees of freedom of diquark. Finally it leads to the quark-diquark wave function characterized by the width $\varkappa$.

For m = 308 MeV, ϰ = 310 MeV and a₁ = 1.944, a₂ = -0.279, a₃= -4.112, it has been found that: proton magnetic moment is μ = 2.59, where experimental data form PDG gives <μ> = 2.79; proton electric radius is r_E² = 20.94 GeV^-2, where <r_E²> = 19.15 GeV^-2; proton magnetic radius is r_M² =15.30 GeV^-2, where <r_M²> = 15.05 GeV^-2. For these parameters an agreement between calculated electric and magnetic form factor and corresponding experimental data has been found, see Fig.1. Moreover, we have calculated the wave function square, see Fig.2 and the unpolarized distribution of up and down quarks, see Fig.3.

Figure 1: The figures shows the electric and magnetic proton form factor as a function of x. Data denoted by red squres are from Ref. [4], and by blue triangles from Ref. [5]. Figure 2: The figure shows the quark-diquark wave function square as a function of x. On the left panel for parallel spins of active quark and proton, on the right panel for anti-parallel. The magnitude of the momentum p=|p^T| is denoted by different colors: black p=0 GeV, blue p=0.1 GeV, green p=0.2 GeV, yellow p=0.3 GeV, orange p=0.4 GeV, red p=0.5 GeV, violet p=0.6 GeV. Figure 3: The figure shows the unpolarized distribution of quarks in proton as a function of x. On the left panel for up quark, on the right panel for down quark. The magnitude of the momentum transfer Q² is denoted by different colors: black Q=0 GeV², blue Q²=0.1 GeV², green Q²=0.2 GeV², yellow Q²=0.3 GeV², orange Q²=0.4 GeV², red Q²=0.5 GeV², violet Q²=0.6 GeV².

References:

[1] Stanisław D. Głazek and J.M. Namyslowski, Act.Phys.Pol. B19, 569 (1988)

[2] B.L. Ioffe, Nucl.Phys. B188, 317 (1981)

[3] Stanisław D. Głazek and Arkadiusz P. Trawiński, Phys.Rev. D88, 105025 (2013)

[4] B. Dudelzak, G. Sauvage, and P. Lehmann, Nuovo Cim. 28, 18 (1963)

[5] L. E. Price, J. R. Dunning, M. Goitein, K. Hanson, T. Kirk, and R. Wilson, Phys. Rev. D4, 45 (1971)