How big is a proton? Experiments during the past decade have called well-established measurements of the proton’s radius into question – even prompting somewhat outlandish suggestions that new physics might be at play. Soon-to-be-published results promise to settle the proton-radius puzzle once and for all.
Contrary to popular depictions, the proton does not have a hard physical boundary like a snooker ball. Its radius was traditionally deduced from its charge distribution via electron-scattering experiments. Scattering from a charge distribution is different from scattering from a point-like charge: the extended charge distribution modifies the differential cross section by a form factor (the Fourier transform of the charge distribution). For a proton this takes the form of a dipole with respect to the scale of the interaction, and an exponentially decaying charge distribution as a function of the distance from the centre of the proton. Scattering experiments found the root mean square (RMS) radius to be about 0.88 fm.
Since the turn of the millennium, a modest increase in precision on the proton radius was made possible by comparing measurements of transitions in hydrogen with quantum electrodynamics (QED) calculations. Since atomic energy levels need to be corrected due to overlapping electron clouds in the extended charge distribution of the proton, precise measurements of the transition frequencies provide a handle on the proton’s radius. A combination of these measurements yielded the most recent CODATA value of 0.8751(61) fm.
The surprise came in 2010, when the CREMA collaboration at the Paul Scherrer Institute (PSI) in Switzerland achieved a 10-fold improvement in precision via the Lamb shift (the 2S–2P transition) in muonic hydrogen, the bound state of a muon orbiting a proton. As the muon is 200 times heavier than the electron, its Bohr radius is 200 times smaller, and the QED correction due to overlapping electron clouds is more substantial. CREMA observed an RMS proton radius of 0.8418(7) fm, which was five sigma below the world average, giving rise to the so-called “proton radius puzzle”. The team confirmed the measurement in 2013, reporting a radius of 0.8409(4) fm. These observations appeared to call into question the cherished principle of lepton universality.
More recent measurements have reinforced the proton’s slimmed-down nature. In 2016 CREMA reported a radius of 0.8356(20) fm by measuring the Lamb shift in muonic deuterium (the bound state of a muon orbiting a proton and a neutron). Most interestingly, in 2017 Axel Beyer of the Max Planck Institute of Quantum Optics in Garching and collaborators reported a similarly lithe radius of 0.8335(95) fm from observations of the 2S–4P transition in ordinary hydrogen. This low value is confirmed by soon-to-be-published measurements of the 1S–3S transition by the same group, and of the 2S–2P transition by Eric Hessels of York University, Canada, and colleagues. “We can no longer speak about a discrepancy between measurements of the proton radius in muonic and electronic spectroscopy,” says Krzysztof Pachucki of CODATA TGFC and the University of Warsaw.
But what of the discrepancy between spectroscopic and scattering experiments? The calculation of the RMS proton radius using scattering data is tricky due to the proton’s recoil, and analyses must extrapolate the form factor to a scale of Q2 = 0. Model uncertainties can therefore be reduced by performing scattering experiments at increasingly low scales. Measurements may now be aligning with a lower value consistent with the latest results in electronic and muonic spectroscopy. In 2017 Miha Mihovilovic of the University of Mainz and colleagues reported an interestingly low value of 0.810(82) fm using the Mainz Microtron, and results due from the Proton Radius Experiment (pRad) at Jefferson Lab will access a similarly low scale with even smaller uncertainties. Preliminary pRad results presented in October 2018 at the 5th Joint Meeting of the APS Division of Nuclear Physics and the Physical Society of Japan in Hawaii indicate a proton radius of 0.830(20) fm. These electron-scattering results will be complemented by muon-scattering results from the COMPASS experiment at CERN, and the MUSE experiment at PSI.
For now, says Pachucki, the latest CODATA recommendations published in 2016 list the higher value obtained from electron scattering and pre-2015 hydrogen-spectroscopy experiments. If the latest experiments continue to line up with the slimmed-down radius of CREMA’s 2010 result, however, the proton radius puzzle may soon be solved, and the world average revised downwards.
101 count = 0
105 for i = 0 to 20
106 print
107 next i
110 Dim x2(7),sign(7), dx(7)
120 Rem Init some vars
125 aminerr=1000
130 rem res = 8.5e-17
132 res = 1e-4
200 xpi=3.14159265358979323846
250 rem 8 coefs here
255 Rem elementary charge (e)-0
260 x2(0)=1.60217662e-19
265 Rem Planck's constant (h)-1
300 x2(1)=6.62607004e-34
350 rem eletron mass (Me)-2
400 x2(2)=9.109383560899034e-31
450 Rem Proton radius (Rp)-3
500 x2(3)=8.41235640479985e-16
550 Rem Rydberg Constant (R_H or R_{\infty})-4
600 x2(4)=10973731.5685083
610 Rem Permittivity of free space (e0)-5
620 x2(5)=8.854187817e-12
701 Rem Speed of Light (c)-6
702 x2(6)=299792458.0
703 Rem Proton mass (Mp)-7
704 x2(7)=1.672621898209999e-27
710 For i = 0 to 6
720 sign(i)=0
730 dx(i) = 0.01*res*(x2(i))
740 next i
750 Rem define bit to be twiddled here
755 sign(0)=1
760 sign(1)=1
770 sign(2)=1
780 sign(3)=1
790 sign(4)=1
795 sign(5)=1
797 sign(6)=1
1000 rem starting error
1010 xerr1=x2(2)*x2(0)^4/(8*x2(6)*x2(5)^2*x2(1)^3*x2(4))
1015 yerr2=-xpi*x2(3)*x2(6)*x2(2)/(2*x2(1))
1017 err = xerr1 + yerr2 - 1.0
1020 digits=int(-log(abs(yerr+0.00001))/log(10)+0.5)
1030 Print "Starting err is: ";err
1032 Print "Starting xerr1 is: ";xerr1
1034 Print "Starting yerr2 is: ";yerr2
1040 print "Digits resolution ";digits
1050 print "aminerr error is: ";aminerr
1060 for i = 0 to 6
1061 print "i= ";i, x2(i)
1062 next i
1100 print "pi= ";xpi
1111 rem end
2000 Rem Main Loop
2010 For i = 0 to 6
2020 x2(i)=x2(i)+sign(i)*dx(i)
2030 xerr1=x2(2)*x2(0)^4/(8*x2(6)*x2(5)^2*x2(1)^3*x2(4))
2032 yerr2=-xpi*x2(3)*x2(6)*x2(2)/(2*x2(1))-1.0
2034 err=abs(xerr1+yerr2)
2035 if err>aminerr then sign(i)=-1*sign(i)
2037 if err<aminerr then aminerr = err
2040 next i
2050 if err < 1.5*res then goto 5000
2055 print "Working ";err
2058 count = count+1
2060 goto 2000
5000 Rem
5001 xresstop=2e-15
5002 if res > xresstop then res = res/10.0
5010 For i = 0 to 6
5030 dx(i) = 0.01*res*(x2(i))
5040 next i
5100 if res > xresstop then goto 2000
5106 Print "Done."
5107 print "Coef "
5109 for i = 0 to 7
5110 print x2(i);" "
5120 next i
6000 rem Final error
6010 xerr1=(x2(2)*x2(0)^4)/(8*x2(6)*x2(5)^2*x2(1)^3*x2(4))
6012 yerr2=-xpi*x2(3)*x2(6)*x2(2)/(2*x2(1)) - 1.0
6014 err=abs(xerr1+yerr2)
6020 digits=int(-log(abs(err+1e-16))/log(10)+0.5)
6030 Print "Starting err is: ";err
6040 print "Digits resolution ";digits
6050 print "Final error is: ";aminerr
6060 print "Iterations= ";count
6070 print xpi;" <-ideal"
6080 print xpi+err;" <-calc'd
6085 print "Calc'd proton mass= ";2*x2(1)/(xpi*x2(3)*x2(6))
6087 print "Input proton mass= ";x2(7)
6090 print "Proton/electron mass ratio=";x2(7)/x2(2)
6100 end
If you look at precisely how they solved the second order differential SchrΓΆdinger equation for the hydrogen atom (the only one with an analytical solution), and see how they derived the equation that precisely links the QUANTUM energy steps to frequency/energy (the photon frequency/energy for when the electron jumps shells; Rydberg equation - THE FULL ONE), AND solve it precisely without making their reduced mass approximation, you get the solution to the proton radius puzzle and much more. Requires iteration on a computer to solve AND it is precise.
Abstract: The status of the proton radius puzzle (as of the date of the Confer- ence) is reviewed. The most likely potential theoretical and experimental explanations are discussed. Either the electronic hydrogen experiments were not sufficiently accurate to measure the proton radius, the two- photon exchange effect was not properly accounted for, or there is some kind of new physics.
Abstract: We extract the proton charge radius from the elastic form factor data using a theoretical framework combining chiral effective field theory and dispersion analysis. Complex analyticity in the momentum transfer correlates the behavior of the spacelike form factor in different $Q^2$ regions and permits the use of data up to $Q^2 \sim$ 0.5 GeV$^2$ in constraining the radius.
The work done by the 4th Order Polynomial paper solves all of this and it is a refinement of Haramein’s work:
