Sizing up the proton

An international team of scientists has now found that the tiny proton is even smaller than previously thought.

PROTON is a fundamental constituent of all matter in the universe. It carries a unit positive charge. Along with chargeless neutrons, protons make up the nuclei of atoms. Electrons, equal to the number of protons in the nuclei, orbit around the nuclei to yield electrically neutral atoms of different elements. Hydrogen is the simplest of all atoms, having a single electron moving around a solitary proton. Unlike the electron, the proton is not a point-like particle and has a finite size of the order of about a millionth of a billionth of a metre (10-15, or a femtometre, fm) only. The proton, in turn, also has a substructure. It is a composite of (fractionally charged) quarks and chargeless gluons that are permanently confined and, therefore, not observable, but will have observable effects on the proton's structure and interactions.

In a hydrogen atom, the positively charged proton attracts the electron, which, according to quantum theory, can move in different shells corresponding to different discrete energy levels (designated as 1S, 2S, 2P, 3S, 3P, 3D, and so on). The energy of an electron increases when it jumps to the next higher shell by absorbing energy. When it moves to a lower level, it loses energy by emitting photons whose frequency determines the energy difference between the levels (the transition energy). Quantum theory actually tells us that electron orbits are not fixed paths in space but fuzzy cloud-like distributions (called orbitals, Figure 1) around the nuclei. The proton, likewise, is not a hard sphere with a sharp edge but a fuzzy ball determined by the charge distribution within, which falls off to zero smoothly near the edge. So, when we say the proton radius what is meant is the radius of the charge distribution, the charge radius.

An international team of scientists has now found that the tiny proton is even smaller than previously thought. At the Paul Scherrer Institute (PSI) in Switzerland, the team, which includes scientists from the Max Planck Institute for Quantum Optics, Garching, Germany; the Swiss Federal Institute for Technology, Zurich; the University of Stuttgart; the University of Coimbra, Portugal; and the Kastler Brossel Laboratory, Paris, has been engaged over the past 10 years in determining the proton radius with much greater accuracy than before using high-precision laser spectroscopic techniques. The newly determined value is significantly less than the hitherto accepted value.

Until recently, the best value for the proton radius, as estimated by the Committee on Data for Science and Technology (CODATA), was 0.8768 fm with an uncertainty of 0.0069 fm (about 1 per cent). It has been measured to be only 0.84184 fm a difference of 0.0350 fm with an uncertainty of 0.00067 fm. Though 10 times more precise, the measured value is smaller by 4 per cent than the expected value. The results were published recently in Nature and they have caused quite a bit of stir in the world of physics. A 4 per cent difference may not seem to be a great deal for our day-to-day concerns, but the results challenge known physics in a fundamental way. As of now, the origin of this large discrepancy is not known, say the authors in the paper.

Beginning of Precision Spectography

Because of its relatively simple atomic structure, the hydrogen atom has been central to the theories of atomic structure, quantum mechanics and quantum electrodynamics (QED). QED describes how matter and light interact in a quantum theoretical framework, and is perhaps the most successful theory of modern physics so far. For example, the Dirac equation for the electron, which incorporates relativistic effects into the quantum theory, predicts that the energies of the electron levels in the hydrogen atom designated as 2S and 2P should be the same.

But in 1947, W.E. Lamb and R.C. Retherford observed a small deviation in their energies. This 2S-2P energy splitting from the prediction of the Dirac equation is called the Lamb shift. The detection of the Lamb shift, in fact, triggered the development of QED as a full-fledged theory. It also marked the beginning of precision spectroscopy. Nobel Laureate Theodor W. Hnsch of Max Planck Institute for Quantum Optics, who developed the frequency-comb technique that led to major advances in high-precision laser spectroscopy (see interview), has said: The spectrum of hydrogen atom has proved to be the Rosetta Stone of modern physics.

In addition to the Lamb shift, electron energies are also affected by the nuclear spin that causes hyperfine splitting of the energy levels and the finite size of the proton. The finite size affects the ground state (1S) of hydrogen, whose orbital is the closest to the proton, more. In the orbital picture, the effect of the finite size can be understood as follows: In the P-orbital, the effect is not so much because the electron's dumbbell-shaped orbital hardly overlaps the proton cloud. But in the S-orbital the electron overlaps the proton significantly, and therefore, feels the effect of the proton much more strongly. The charges of the proton and electron cancel whenever the electron flits through the proton, effectively reducing attractive force on the electron, and hence its energy. QED calculations quantify this and the QED equations for the energy levels explicitly display their dependence on the proton radius.

However, before the present experiment, comparison of such precision spectroscopy measurements with theory was limited by the uncertainties in the value of the proton radius. In the conventional hydrogen atom, the effect is so small that even the most accurate measurements have a great degree of uncertainty. The main uncertainties in the comparison of the Lamb shift measurements with theory are: from experiment (2 ppm), QED calculations (1 ppm) and proton size (6 ppm). Hitherto CODATA estimates were based on QED theoretical calculations for the hydrogen energy levels (with the corrections for finite proton size included). Using the observed transition frequency data, the radius is extracted by treating it as an adjustable parameter. The radius is also determined (less accurately) from electron-proton scattering data. A recent analysis of this data gave a less precise value of 0.895 fm with an uncertainty of 0.018 fm, an uncertainty as large as 2 per cent.

Working with Muonic Hydrogen Atom

The present experiment has reduced the uncertainty by an order of magnitude. This was achieved by not working with the conventional hydrogen atoms but muonic hydrogen atom (mu-p) in which the electron is replaced by a muon, which moves around the proton. The muon is actually a heavier cousin of the electron with all its properties except that it is 207 times heavier. The total diameter of the atom, therefore, shrinks by that factor (as the orbit size is inversely proportional to the mass) and, on the average, the muon orbital has greater overlap with the proton. Therefore, its orbitals, particularly the S-orbitals, will be much more sensitive to the proton size. Accordingly, the S-energy levels are shifted down more than in the case of normal hydrogen atom.

The current experiment was basically designed to measure the Lamb shift (the 2S-2P energy difference) in muonic hydrogen. For nearly 40 years, it has been known that a measurement of the mu-p Lamb shift would yield more accurate results for the proton radius and better tests of QED. But only recent progress in muon beams, laser technology and detectors has made this possible. While PSI has developed the requisite technology of muon beams, the Garching institute and the University of Stuttgart have developed the laser technology. The aim of this experiment was to measure the mu-p Lamb shift with 30 ppm precision and to deduce the proton charge radius with 1,000 ppm, or one-in-thousand (1x10-3) accuracy, and, in fact, the experiment has succeeded in doing marginally better.

The basic principle is to irradiate mu-p atoms in the 2S state by a short pulse of infrared radiation whose wavelength (of about 6 micrometre) corresponds to the muonic Lamb shift energy difference so that muons are excited from the 2S to the higher 2P level. What is detected is the number of muon de-excitations from the 2P state to the ground state by observing the 1.9 kilo electronvolt (keV) x-ray that is emitted (Figure 2). The lifetime of muons in the 2P state is extremely short, about 8.5 picoseconds. So the measurement involved is the number of 2P-1S transitions, which occur almost simultaneously with the laser pulse tuned to the correct resonance frequency (around 6 micrometre) that causes the transition between the two muon orbitals.

But these are extremely tricky steps requiring very precise techniques to succeed. The first step is to create mu-p atoms. Only the PSI has a cyclotron producing low-energy (3-6 keV) muons with sufficient intensity (about 500 per second). From the accelerator, muons are extracted and transported to a hydrogen target, which is basically hydrogen gas at low pressure of 1 mbar. The idea is to capture negative muons whereby mu-p in highly excited states can be formed. Muons get slowed down by ionisation when an electron in the hydrogen atom is replaced by a muon to form a mu-p atom. But it turns out that nearly 99 per cent of these de-excite very quickly (within 100 nanoseconds) to the ground state. The issue is to exploit the 1 per cent of muons that populate the metastable 2S state. The lifetime of the muon in the metastable state is crucial for the experiment. In free space, the lifetime of a muon is 2.2 microseconds. But in the hydrogen gas, because of collisions, it gets reduced and at a low pressure of 1 mbar it has a lifetime of 1 microsecond (Figure 2). What is required to excite the 2S muons to the 2P state is a laser that can be tuned with extreme precision so that its energy can be changed in very small equal steps.

When the laser frequency is resonant, the 2S-2P transition is induced. The laser must also inject the pulse within a microsecond after it receives the signal for having registered a muonic hydrogen atom in the set-up. The excitation is immediately followed by de-excitation to the ground state with the emission 2P-1S x-ray of 1.9 keV. The detection of the x-ray coincident with the laser pulse is used as a signature for the 2S-2P transition. A resonance curve is obtained (Figure 3) by plotting the number of detected 2 keV x-rays coincident with the laser pulse as a function of the laser frequency. From the centroid position of this peak, a proton radius of 0.84184 fm has been deduced, which, however, is in disagreement by a large 5 standard deviations with the accepted CODATA value. Standard deviation (s.d., also termed as sigma) is a statistical measure of the extent of variation from the average or expected value. Five s.d. marks a significant departure and demands explanation (Figure 4).

On the face of it, however, a consequence of this finding is either the atomic calculations, in particular those for muonic hydrogen atom, made using QED are incorrect or the value of the Rydberg Constant, which sets the scale for energy levels in atoms and is one of the most precisely determined fundamental constants in physics, has to be changed. If the problem lies in the mu-p energy level calculations, a very large term (larger than one-thousandth of the Lamb shift energy) would be missing. Theorists argue that it is improbable that a term of such a magnitude could be missed. If the theory used to extract the proton radius from muonic hydrogen is correct, then the problem arises from hydrogen spectroscopic measurements (of the frequencies of 1S-2S, 2S-8S and 2S-12D transitions) used to calculate the Rydberg Constant.

The authors have argued that if the Lamb shift calculations for mu-p are correct, using their highly precise hydrogen 1S-2S transition measurement, the Rydberg Constant extracted using the present radius value yields a value of the Constant that is 4.6 times more precise but 4.6 s.d. away from the CODATA value. We chose to use the best value for the charge radius, and the most precisely measured frequency in simple atomic system (hydrogen 1S-2S transition) to deduce the Rydberg Constant, says Randolf Pohl of the Max Planck Institute and the leading author of the paper. This of course discards all other measurements in hydrogen, but they are two orders of magnitude less precise, he adds.

At this point of time, it is not clear if the discrepancy is due to some significant errors in the complicated QED calculations of atomic energy levels or whether it would require an important change in the fundamentals of the theory itself. It is true that QED rests on techniques whose mathematical foundations are yet to be firmly established. But given the enormous success of the theory, in terms of highly accurate predictions so far, physicists would tend to bet on the former or even argue that there could be problems with the experiment itself. But the experimenters are extremely sure of their result.

Telltale signal

According to Pohl, the team had observed the telltale signal on July 5, 2009, itself (Figure 3) and the experiment was continued until August. From October we analysed the data, Pohl pointed out in an e-mail exchange. We had to make very sure that we had not overlooked something. After all this is a 5 sigma discrepancy. Of course, we had studied all systematics before we started to take data, but our result was so puzzling that we questioned again every detail of the experiment. We also performed more auxiliary measurements with the laser system, to re-check again the laser frequency calibration to verify the accuracy that we are claiming now and possible systematic effects. In parallel to all of this, we rechecked the theory calculations. This meant writing more software and careful calculation of a lot of QED terms. After we had become convinced that we had not overlooked any systematic effect, we knew the result only in February this year and we submitted the paper in March, he says.

Personally, he says, I would not be much surprised if one of the QED calculations turned out to be incorrect. These calculations are terribly difficult, and until now the theorists had no benchmark, because the proton radius was not well known. Now with more accurate value, theorists have a way to check their calculations.

The result may also be a pointer to new physics, Krzysztof Pachucki, a physicist involved in the Lamb shift calculations for long, believes. Muon may lead to quantum fluctuations hitherto not considered, like the creation of lots of electron-positron pairs. This, he argues, could increase polarisation within the muonic atom, which could, in principle, resolve the Lamb shift discrepancy without having to alter the proton radius.

Could the not-well-understood composite nature of the proton be the cause for errors in QED calculations? The inner complexity, Rudolf Faustov of the Russian Academy of Sciences in Moscow, has been quoted as saying, makes it difficult for physicists to handle precisely the electromagnetic force between the proton and the muon in their calculations. According to him, it is not clear how these interactions must be separated and, in particular, physicists may have to consider how the muon affects the proton. The finding may also be a pointer to a more precise understanding of the compositeness of the proton within the framework of the theory of quarks and gluons known as quantum chromodynamics (QCD). But QCD is nowhere near being able to predict something like the size of the proton. The present proton radius measurement has clearly set the benchmark and opened up avenues for all such theoretical calculations.