The puzzle of the proton radius A K F Val Baker Torus Tech LLC. San Clemente, California, 92673, USA Hawaii Institute for Unified Physics, Kailua Kona, Hawaii, 96740, USA Email: [email protected] Abstract. The radius of the proton is a topic of debate and has yet to be confirmed. Measurements from different techniques yield conflicting values, with some experimental methods yielding a value in disagreement with the standard model. This review article outlines the current status of the experimental results and summarizes the possible solutions to this puzzle. Keywords: nucleonic parameters: radius – particle physics: proton – fundamental physics: standard model 1. Introduction describe the spatial distributions of electric charge and current inside the nucleon, To understand the workings of the which for the non-relativistic case is given Universe, from the very big to the very in terms of the volume integral small, we need to observe and measure the F()() q2= r eiq . r d 3 r [2]. fundamental characteristics of the material world. Advances in technology enables measurements of greater accuracy and thus The radius of an oscillating electrostatic either confirms theories or leads us in new, field, such as a proton, defines an effective and sometimes unknown, directions. charge boundary in that region of space. This is typically referred to as a ‘charge One of the most important characteristics radius’ and is described by its electric form of the objects we study is their spatial factor. The standard approach to extent. However, when dealing with determining this radius thus relies on subatomic particles, our ability to directly indirect measurements of the energy measure the size is limited by the interaction at the charge surface boundary. instruments we use. For example, Measurements of this energy interaction techniques such as scanning probe are typically done utilizing electron-proton microscopy, although achieving subatomic scattering and/or hydrogen spectroscopy resolutions on the order of 0.1 nm [1], methods. Both these methods have are unable to reveal the spatial extent of consistently yielded similar results, where subatomic particles. We therefore rely on the latest 2014 CODATA value, more indirect methods which instead of rp = 0.8751( 61) fm , is based on a least- measuring the object in co-ordinate space squares approximation between both measures the distribution of changes of methods [3]. However, spectroscopic momentum. This distribution is called the measurements of the Lamb shift were also form factor which, based on theoretical made utilizing muonic hydrogen as oppose assumptions, compensates for irregularity to electronic hydrogen and yielded much in the shape of an object and is essentially smaller values than that of the combined a Fourier transform. For example, the electronic methods as given by CODATA. nucleon electromagnetic form factors The first measurement of muonic hydrogen was made in 2010 and yielded a proton radius, rp = 0.8356(20) fm , similar value of rp = 0.84184(67) fm [4], differing to that found from muonic hydrogen [10]. from the 2010 CODATA value, rp = 0.8775(51) fm , by 7 [5]. These These results suggest that spectroscopic measurements were repeated in 2013 and methods yield smaller values than those found a more precise value of found from scattering techniques, with r= 0.84087(39) fm [6], differing from muonic spectroscopy measurements p yielding the smallest values. the 2010 CODATA value by 7.2 . However, new experiments measuring the The question is, why are these different 22SP− Lamb shift in electronic methods (electronic and muonic) yielding hydrogen found a significantly smaller different results? This is a problem that the value, r= 0.833(10) fm , more in physics community have been trying to p reconcile, and the failure to understand it is agreement with that found by muonic known as the proton radius puzzle. hydrogen and muonic deuterium [12]. Their approach claims to utilize special Due to the high correlation between the techniques to reduce quantum interference, proton radius and the Rydberg constant, from which they conclude yields a more the same spectroscopic experiments are precise and accurate result. If their result used to determine both values, thus any and the previous muonic measurements are change in the charge radius of the proton correct, then what does this tell us about also implies a change in the Rydberg previous electronic measurements and constant. The value for the Rydberg quantum interference correction constant is known with a relative techniques. −12 uncertainty of 1.9 10 and is one of the Furthermore, in another recent study most accurate fundamental constants to carried out at the Jefferson Laboratory, date. Therefore, the electrical size of the Virginia, USA, the measurement of the proton, its charge radius, is now a proton radius was made utilizing high limitation (4− 7 ) to the accuracy of the precision electron-proton scattering Rydberg constant found from this techniques, yielding a value of comparison of theory [7] [8] [9]. r= 0.831(19) fm [13]. This value is 3.2 p In 2016 this discrepancy became even smaller than the most recent high-precision more significant when a measurement of electron-proton scattering measurement of r= 0.879(15) fm [14] and 2.7 smaller the Lamb shift was made utilizing muonic p deuterium, as oppose to electronic than the average of all electron-proton hydrogen, and yielded a value for the scattering measurements [3]. Remarkably, charge radius of deuteron of it is in a significant agreement with both the muonic hydrogen spectroscopy rd = 2.12562(78) fm [10]. This value found from muonic deuterium is smaller measurements and the latest electronic than the 2010 CODATA value of hydrogen measurements. So, it could be that the proton radius puzzle is not a r= 2.1424(21) fm by 7.5 [5] and d puzzle anymore. smaller than the value of rd = 2.1415(45) fm found by electronic In this paper we will outline the two deuterium spectroscopy techniques by methods for measuring and determining 3.5 [11]. This smaller radius for the proton radius and discuss the possible deuteron also yields a smaller value for the sources of error or resolutions to the proton radius puzzle. 2. Measuring the radius of the proton the charge distribution is not static due to recoil effects, and the magnetic and 2.1 Electron scattering experiments electric distributions will be independently effected by the spin, the form factors are 2 Electron scattering was first applied by separated for electric GQE () and Robert McAllister and Robert Hofstadter magnetic GQ()2 distributions. in the 1950’s as a technique for measuring M the charge distribution of the proton, and to this day provides one of the most The multiplicative factor is therefore a powerful tools to investigate nucleonic function of both the magnetic and electric structure [15] [16]. form factors and the solution is given as first order from perturbation theory, known Measurements of the proton radius are as the Rosenbluth formula [17], typically done using elastic electron-proton scattering, in which a beam of electrons is dd 1 2 2 2 2 = GQGQEM()() + fired at a proton source (i.e. liquid dd Mott (1 + ) hydrogen) and the scattering angles are measured. The scattering probability can 2 Q be calculated as a function of proton where = 2 , radius, and thus the proton radius can be 4M 2 −1 found by measuring the scattering angle. =1 + 2(1 + ) tan ( / 2) is the Q2 In the case of electron-proton scattering, kinematic factor and GFFEDP=− 2 the differential scattering cross section 4M dd of the scattered electrons can be and GFFMDP=+ are the electric and described by the Mott scattering formula. magnetic form factors in terms of the This is an extension of low-energy Dirac and Pauli form factors FD and FP , Coulomb scattering (i.e. Rutherford respectively. scattering), which includes the effects of relativistic energies and spin-coupling Similarly, the form factors can be inelastic Coulomb scattering, and is given determined by measuring the differential as, cross section of a variety of Q2 and scattering angles . The electric and 2 2 3 dE4 cos ( / 2) magnetic radius can thus be found as the = 4 d Mott Q E slope of the proton form factor at zero momentum transfer, where is the interaction cross section, is the solid angle, is the fine 6 2 dG() Q2 r 2 =− EM/ structure constant, is the scattering EM/ 2 GEM/ (0) dQ Q2 =0 angle, E is the incoming electron lab energy, E is the outgoing electron lab The precision to which the radius is energy and Q2 is the four-momentum determined is therefore dependent on the transfer. data available at low and the However, for an extended target, which is associated uncertainties. Recent work [14] the assumed case for a proton, the utilising the Mainz Microtron (MAMI) differential scattering cross section has to located at the Johannes Gutenberg be modified by multiplying by the square University in Mainz, Germany, successfully performed high-precision of the form factor FQ()2 . As well, since measurements and significantly expanded The first term defines the fundamental the data set at low . These results were structure of hydrogen as a function of its included in the global examination of principle energy level whereas the second elastic electron–proton scattering data as and third terms account for quantum given by Arrington and Sick [18] and corrections (i.e. relativistic recoil, vacuum Bernauer and Distler [3]. The latest results polarization, nuclear finite-size and recommended by CODATA 2014 are nuclear motion). Specifically, the second found as the weighted mean between these term accounts for relativistic corrections as two values, r= 0.880(11) fm as given by well as contributions coming from the p interactions of the bound-state system with Bernauer and Distler [3] and the quantum vacuum and is given as a rp = 0.879(11) fm as given by Arrington function in terms of the fine structure and Sick [18], yielding a value of constant, and the electron-to-proton rp = 0.879(13) fm .