On proton charge radius definition

We find a relation for the proton charge radius in closed form. The result 0.841235641(10) fm agrees with determinations from muonic hydrogen. Our relation is based on a mapping from an intrinsic configuration space for the proton to the observed size in laboratory experiments. The torus of the configuration space leads to periodic potentials in dynamical toroidal angles with the proton charge radius as a scale parameter in the mapping to laboratory space. The periodic potentials allow the introduction of Bloch phase factors which open for period doublings in the wave function. The opening of Bloch degrees of freedom is mediated by the Higgs mechanism. Thus, we find that pi times the proton charge radius equals twice the proton Compton wavelength.

The scattering method is sensitive to the proton form factors which modify the differential scattering cross-section for point particle scattering, cf. p. 194 in [17]. The form factors are expanded in powers of the charge radius defined as moments of the proton charge density. Relativistically invariant moments for radius determination have recently been introduced from two-dimensional charge densities [18]. A variation of the scattering method is the MAMI initial state radiation (photon bremsstrahlung) experiment [19] which is sensitive to smaller momentum exchange and, therefore, potentially more accurate in form factor extraction for r e p determination. Presently this method has not reached a level of accuracy compatible with muonic spectroscopy. The MAMI value is (0.870 ± 0.014 stat ± 0.024 sys ± 0.003 mod ) fm and thus covers the muonic value within one standard deviation. For a future perpective on simultaneous electron and muon scattering on protons, cf. [20]. In the present work we find the following relation, between the proton charge radius r p and the Compton wavelength λ p ≡ h/(m p c) of the proton. The relation (1) derives from our description of the neutron decay and 21001-p1 yields r p = 0.841235641(10) fm based on the experimental proton mass m p [8] inserted via λ p in (1) and it is in striking agreement with the result using muons. In particular we note the result r p = 0.8412(15) fm using a higher-order determination of the muonic hydrogen Lamb shift [22]. Further experimental establishment of (1) would confirm the working of the Higgs mechanism in the neutron transformation to the proton [23]. As we shall see, the factor π in (1) originates in a toroidal structure from the intrinsic configuration space on which we describe the proton state. The factor two on the Compton wavelength originates in the period doubling involved in parametrizing the protonic state from the exterior laboratory space, cf. fig. 1. The present work is edited and updated from [21].
Intrinsic states. -As described in [24], we consider protons, neutrons and other baryons as stationary states on the intrinsic configuration space U (3), (2) In lattice gauge theory [25][26][27][28][29], u is a gluon plaquette variable and a is the lattice spacing. In the new interpretation used here, u = e iχ ∈ U (3) is an intrinsic configuration variable for the entire baryonic state Ψ, c/a ≡ Λ is an energy scale, Δ is the Laplacian and 1 2 Tr χ 2 is a representation independent and gauge invariant potential [28] depending only on the three eigenvalues e iθj of u [30]. We call the dynamical variables θ j , j = 1, 2, 3 eigenangles. The potential is independent on left translations in configuration space because such translations correspond to conjugations u → vuv −1 to diagonalize u and leave the trace unchanged (Tr χ 2 ≡ d 2 (e, u) = d 2 (v, vu) = d 2 (e, vuv −1 ), for details cf. footnote 4 in [31]). This means that the potential is independent on the choice of origo e in configuration space. It can be shown that a left translation in configuration space corresponds to a (local) gauge transformation in laboratory space [23,32]. For a compact proof cf. [33]. The potential splits into a sum of three independent periodic terms in the eigenangles θ j [30], 1 2 Tr where (cf. fig. 1) (4) It may be interpreted as the Euclidean measure from the laboratory space folded onto the group manifold [34].
The configuration variable u can be excited from the laboratory space by nine kinematic generators. Thus, with intrinsic momentum generators and spin and Laplace-Runge-Lenz generators which in a coordinate representation (pp. 210 in [35]) spell out as, e.g., The emergence of quarks from (2) is used in [24] to derive u and d parton distribution functions for the proton and described in more detail together with gluons in [23]. In [31] the generation of quark fields is also mentioned to yield the proton spin structure function and a relation between flavour and colour degrees of freedom is used to suggest an explanation for the value of the Cabibbo angle. The potential (3) is a Manton analogue [28] of the Wilson trace here used on the configuration variable u [25,27], The two expressions (3) and (8) where e iθj are the three eigenvalues of u in a 3 × 3 matrix representation and θ j are dynamical eigenangles with action-angle quantization conditions The polar decomposition is analogous to the Euclidean Laplacian in polar coordinates for instance used in solving the hydrogen atom. The "Jacobian" of the parametrization in (9) is (cf. p. 197 in [39]) and S k and M k are off-toroidal generators equivalent to off-diagonal Gell-Mann matrices, cf. p. 210 in [35], carrying spin and flavour degrees of freedom, respectively. The Laplacian (9) in its polar form matches the potential (3) since the trace expression only depends on the eigenvalues e iθj of the configuration variable. Due to the match in the parametrization of the Laplacian and the potential, the wave function factorizes into To solve (2), we first differentiate through J 2 in (9), then multiply by J and finally integrate out the six offtoroidal degrees of freedom α 1 , α 2 , α 3 , β 1 , β 2 , β 3 to obtain a dimensionless equation for the toroidal part R ≡ Jτ of the measure-scaled wave function Φ ≡ JΨ [24], ⎡ Here E = E/Λ and The nominator 4 in (16) is the minimum value of (S 2 + M 2 )/ 2 [23,24]. The eigenvalues of (15) with the potential (16) yields the N and Δ baryon resonance spectrum with no serious missing resonance problem, cf. [23,32].
To solve (15) by a Rayleigh-Ritz method [40] for period doubled states, we may expand R on Slater determinants, We have not restricted the set of integers p, q, r to make {b pqr } a complete set and, therefore, we know only approximate solutions with p = 0, 1, . . . , P − 1; q = 1, 2, . . . , P ; r = p + 1, p + 2, . . . , P . Work is in progress on this 2 . However, the point of interest here for the proton radius is the fact that for all terms in the expansion of R, the parametrizations via the θ j 's in (17) introduce a period doubling.
What is the origin of the period doublings? They come from exploiting the Higgs mechanism in the neutron decay to lower the eigenvalue of the intrinsic baryonic state from that of the neutron to that of the proton. The mechanism is summed up in fig. 2.

Period doubling and proton radius. -
The periodic potential W in (16) calls for a description with concepts from solid-state physics where periodic potentials offer Bloch degrees of freedom. In our case the wave function needs to have single-valued square on the configuration space. This selects Bloch phase factors e iκθ with κ = 0, ± 1 2 . Choosing κ = 0 gives neutronic states, whereas κ = ± 1 2 introduces a topological winding and lowers the energy as seen to the right in fig. 2. We interpret the topological change as the creation of electric charge in the decay n → p + e + ν e .
To allow the period doubling we invoked the Higgs mechanism [23]. The Higgs mechanism is normally presented in the lepton sector as the agent behind the spontaneous breakdown of the U (2) symmetry of a leptonic doublet, into a chirally broken SU (2) symmetry times a remaining U (1) symmetry providing three massive gauge bosons W ± , Z 0 and a massless photon. But the same symmetry breakdown happens for the electroweak degrees of freedom of the quarks in hadrons, cf. pp. 378-379 in [42] and p. 240 in [43]. For instance the up and down flavour quarks are also thought to break up into right-handed SU (2)-singlets and a left-handed doublet, Quarks have both colour characteristics for strong interactions and flavour for electroweak interactions and we take the similarity in (20) to support a relation among strong and electroweak degrees of freedom. We make the following trailing Ansatz to correlate the sectors: Here Λ and α scale the trailing among the colour angle field θ and the scalar Higgs field ϕ. We found the Higgs field vacuum expectation value ϕ 0 by the 2π shift from one trough to a neighbouring one in the periodic potential (4), cf. fig. 2, Λ2π = αϕ 0 .
The shift corresponds to the exchange of one quantum of action h between the two sectors 3 .
Higgs potential and electroweak scale. -We fitted the Higgs potential to the periodic potential (4) as seen in fig. 2

and found [23]
Note that the standard model value v SM is related to our v by the u and d quark mixing matrix element V ud , thus v SM = v |V ud | [23,44]. From the Higgs potential and the scale (24) we got an accurate expression for the Higgs mass [23,38] the value of which we update here:  We use the classical electron radius re [46,47] to set the length scale a for the projection of the intrinsic dynamics. Figure  taken from [23]. α e = 1/137.035999139 (31) at the electron scale [8], respectively. The electron mass enters from our use of the classical electron radius, cf. [46] and p. 97 in [47], r e = e 2 4π 0 m e c 2 (26) to set the length scale a in (2) by the projection relation, cf. fig. 3 [24], πa = r e .
This projection rests on the existence of a map from the intrinsic torus to space coordinates, thus The choice of r e in (27) to set the exterior scale in mapping the intrinsic scale is heuristically motivated by the fact that the electron with its rest mass m e is created simultaneously with the topological change giving p its charge in the transformation from the neutral neutron n to the charged p. We have called the electon a "peel-off" [24] and we have described the proton as "charge scarred". The scale introduced in (26) led to an accurate value for the electron-to-neutron mass ratio [24] m e m n = α(m n ) π where the fine structure coupling is to be taken at nucleonic energies. The same scale is to be used for the baryon spectra. Cf. [23,32] for approximate solutions for charged states. The result in (29) is calculated with α −1 (m n ) = 133.61 obtained by sliding with radiative corrections from α −1 (m τ ) = 133.476(7) [8] and with E n = E n /Λ = 4.382 (2) 21001-p4 On proton charge radius definition where E n is the neutron ground-state eigenvalue found in a Rayleigh-Ritz solution [38] of (15) with 3078 base functions [23]. With r e = e 2 /(4π 0 m e c 2 ) eq. (27) gives a = 0.92 . . . fm for the length scale used in the space projection (28) of the toroidal angles in the neutronic state.
Proton charge as topological. -After the neutron decay the emerging proton lives in laboratory space as an extended charge-scarred object. Being a quantum object it will have "fuzzy edges", with an extension conventionally described by its Compton wavelength This wavelength is an experimentally observable quantity, e.g., in Compton scattering. For a lucid description of Compton scattering of photons off electrons cf. p. 6 in [48]. The photon, in its role of being the U (1) gauge field of electromagnetism, interacts with the electric charge. We therefore take the Compton wavelength λ p to be the projected extension in laboratory space relating to the charge of the proton. We are aware that the Compton wavelength is defined also for neutral particles and is determined by the mass of the particle. We come to this more general conception in the next section. The length scale to be used in the mapping of the intrinsic charge scar we will identify as the proton charge radius r p . We do not want to imply that it would mean that the proton as a space object has a sharp, well-defined radius. Rather as stated, the radius is a scale parameter for a mapping analogous to the one in (28). Now, as the mapping of the intrinsic protonic state has to take into account the period doubling from the Bloch phase factors e iθ/2 in (17), the projectional relation analogous to (28) reads or, for a full extension projection, as shown in fig. 1 and equivalent to eq. (1), The result for r p is r p = 0.841235641(10) fm (33) in agreement with the root-mean-square value of the proton charge distribution [8] r μ p = r 2 1 2 = 0.84087(39) fm (34) as observed by spectroscopy on muonic hydrogen [1,3,4]. The uncertainty in (33) is from the experimental uncertainty on the proton mass m p and Planck's constant h, both known with a relative uncertainty of 6 · 10 −9 .
That is, we use m p c 2 = 938.272081 (6) MeV and c = 197.3269788 (12) MeV fm [8]. Allowing for a possible systematic correlation among the two uncertainties, the uncertainty we cite in (33) is a conservative estimate from summing the relative uncertainties instead of combining them independently in quadrature. The relation (32) for the proton charge radius does not depend on any approximation. It solely depends on the topological structure of the wave function, i.e., on the fractional Bloch phase factors from eq. (17) manifested in the fraction 1 2 in (32) -and signifying the creation of the proton charge. Note that the same kind of period doublings lie behind the approximate protonic state from which parton distribution functions were generated in ref. [24].
Mass as intrinsic energy-momentum. -We here connect our relation (32) to semi-classical considerations. We start out by the diffential geometry underlying the treatment of (2). From the compactness of the intrinsic configuration space U (3) and the exchange of energymomentum through the exterior derivatives (momentum forms) dR and dΦ operating on the intrinsic wave function we get an intuitive conception of baryonic massiveness as introtangled energy-momentum. The exterior derivatives act on the momentum generators and rotation generators in laboratory space and thereby map the intrinsic wave function to laboratory space. In particular, by the projection of the toroidal degrees of freedom [24] on the spatial degrees of freedom (28), we imagine the extension of a baryon as a compromise on the impossibility of pointlike localization and mutual, definite "curled up" momentum [49]. We take the general relation among energy E, momentum p and mass m as defined in spacetime where c is the velocity of pure energy configurations and assume formally that intrinsic dynamics is that of pure energy (zero mass and no gravity in intrinsic space). Then from an intrinsic point of view In the complementary real-space configuration, the state is a particle at rest (zero momentum), so we have The right-hand sides of (36) and (37) are squares of corresponding eigenvalues for the same baryonic entity in (2) only seen from complementary spaces. Thus, With the quantization inherent in the concept of the de Broglie wavelength and the Bohr-Sommerfeld quantization condition for the action integral p dq = nh, n ∈ Z + (cf. p. 36 in [50]) we infer for n = 1 The fourth expression is the Compton wavelength of the particle as conceived in real space. The last expression opens for relating the proton charge radius in (1) to the classical electron radius by using πa = r e from (27) in (39) to yield Here the main uncertainties lie in E = 4.382(2) determined from (15) and in the necessary sliding from the fine-structure coupling α −1 (m τ ) = 133.476(7) at tauonic energies [8] to nucleonic energies in the neutron-to-proton transformation where α −1 (m n ) = 133.61(1). This has only been done for radiative corrections and not for fermionic corrections since here we are outside the pertubative regime of QCD. We may insert the classical electron radius (26) in (40) where the last expression involves the proton charge in the fine-structure coupling at nucleonic energies.
Conclusion. -We have defined a proton charge radius from a conception of the proton structure based on a Hamiltonian on the intrinsic configuration space, the Lie group U (3). The compactness of the intrinsic space manifests itself in parameter space as periodic potentials.
The periodic potentials open for Bloch phase factors allowing period doublings with topological consequences in the wave function of the proton relative to that of the neutron thereby lowering the nucleon ground state. Our result relates the proton charge radius to its Compton wavelength and the result is in excellent agreement with experimental results using spectroscopy on muonic hydrogen. Further confirmation of our result would stress the connection between the period doublings and the Higgs mechanism which opens the necessary degrees of freedom. We look forward to results from simultaneous electron and muon scattering on protons. * * * We thank The Technical University of Denmark for the institutional framework. We thank Thomas Lyspil for an observation on the Rydberg constant. In particular we thank our colleague Mogens Stibius Jensen for co-work on the Higgs mass.