## Friday, 3 December 2010

### Update on Muonic Hydrogen

5 months ago an experimental group at PSI announced the measurement of the Lamb shift in muonic hydrogen. Since muon is 200 times heavier than electron, the muonic hydrogen atom is 200 times smaller than the ordinary hydrogen. Therefore, finite proton size effects are far more pronounced in the former, and end up contributing as much as 2 percent to the Lamb shift. Assuming that the system is adequately described by QED, the PSI result can be interpreted as a new measurement of the size (charge radius) of the proton. The surprise was the deduced value of the charged radius turned out to be inconsistent at the 5 sigma level with the previous determinations based on the spectroscopy of hydrogen and electron-proton scattering data. Something is wrong. Either there is an experimental error, or there is an error in the theoretical computations of the Lamb shift, or maybe some new forces are in the game.

Of course, it is the last of the above possibilities that makes the anomaly attractive to hoards of hungry-eyed particle theorists. In fact, it's not the first mysterious result related to the muon: a 3-point-something-sigma anomaly in the muon anomalous magnetic moment has been nudging us for years. It is tempting to speculate that both these muon anomalies have a common explanation in terms of yet unknown fundamental forces. Furthermore, as I explained here, new hidden forces have recently become very popular in the particle circles for other, completely unrelated reasons. Yet ArXiv has not been flooded with theory papers on muonic hydrogen, so far. The reason is that it's difficult to write down a new physics model that explains the measured Lamb shift without violating constraints from atomic precision physics. The most painful constraints come from
• Ordinary hydrogen spectroscopy,
• Anomalous magnetic moment of the electron,
• Low energy neutron scattering experiments,
• Interactions of neutrinos with matter.
These constraint exclude popular models, such as the hidden photon or a new Higgs-like scalar, as the explanation of the anomaly. One could thus conclude that there is nothing interesting here and move on. Or one could promote the positive attitude. Like in this paper last week by David Tucker-Smith and Itay Yavin who, apart from mounting difficulties, also proposes a solution.

The paper proposes how to shift the energy levels of muonic hydrogen without violating other experimental constraints. The first part is easy: a scalar or vector particle could provide for the new attractive force that does the job. One possibility is to take the mass of the new particle to be of order MeV, and the coupling to muons and protons of order $10^{-4}$ (the contribution to the Lamb shift scales as $g_\mu g_p/m^2$ for m above 1 MeV and $g_\mu g_p m^2/m_\mu^4$ for m less than 1 MeV; thus other choices of the parameters are possible, for example, for a larger mass one would need correspondingly larger couplings). With the couplings and the mass in the same ballpark one could also obtain a new contribution to the muon anomalous magnetic moment that resolves the tension with experiment, see the blue band in the plot.

Now comes the tricky part, that is addressing other experimental constraints. There are some older muonic-atom experiments, for example the one with Mg and Si, who constrain the couplings of new force carriers to muons and protons. However, they are not inconsistent with the coupling strength needed to explain the muonic hydrogen anomaly. But it seems the new force carrier has to couple only to muons and protons and virtually nothing else. For example, the coupling to electrons has to be at least an order of magnitude smaller than that to muons in order to avoid excessive contributions to the anomalous magnetic moment of the electron. The coupling to neutrons is even more strongly constrained by some prehistoric experiments (from 1966!!! back when England last won the world cup!!! ;-) involving low energy neutrons scattering on lead atoms. Furthermore, B-factories strongly constrain the couplings to b-quarks, neutrino experiments strongly constrain the couplings to neutrinos, and so on.

It is simple to cook up a model where the coupling of the new force carrier to electrons is suppressed (a particle coupled to mass), or when the coupling to neutrons is suppressed (a particle coupled to charge), but to achieve both at the same time is a model-building challenge. However this possibility cannot be excluded in a model independent manner, so it open to experimental verification. If a new force carrier is the reason for the muonic anomalies, there should be shifts in the spectrum of other muon systems, such as muonic helium or the true muonium (a bound state of muon and antimuon). Those systems have not been investigated yet, but with the present technology they seem to be within reach. So, if you have some free time this weekend you could try to make the true muonium and measure its energy levels. Depending on the result, life could get very interesting, or it could get as usual...

See also here and here to better appreciate the problems with model building. For a fresh review and reevaluation of the standard QED contributions to the muonic hydrogen energy levels, see here.

Anonymous said...

interesting post, thank you! what's really missing here is an ab initio determination of the proton radius. but that's a really tough one.

Luboš Motl said...

I bet it's a mistake in the theoretical calculation, it's a damn confusing one.

It's very easy to propose a new force carrier at a few MeVs - it's actually much harder to calculate the Lamb shift with all the right terms needed for the accuracy.

This problem is complex because the usual Lamb shift calculation is tough even for the electron-based atoms: one kinds of needs all the major loop corrections from QFT but use them in a problem that resembles the problems in non-relativistic QM of static systems.

The frequencies are sensitive to the width of the muon - which decays - and much more sensitive to the shape of the proton/nucleus because the atom is smaller and the lepton spends much more time in the vicinity of the nucleus/proton than in the case of the electron.

In this sense, I think that e.g. Tucker-Smith and Yavin are choosing the "easier solution" (although involving a more extraordinary claim!) which is not necessarily the right one.

In my opinion, it would be much healthier to see papers with alternative, independent calculations of the Lamb shift than lots of catchy far-reaching speculations what a discrepancy between 1+1 papers (only) could mean.

Anonymous said...

Lubos, I think what Itay and David have done is worthwhile -- they've shown precisely how ridiculous a result would be needed to explain this with new physics. A particle that couples to muons and protons but not electrons and neutrons is absurd; by establishing that this is what is needed to explain the data with new physics, they've provided further incentive for someone to do the hard work of revisiting the SM calculation. A lot of the early press, including things the experimenters were saying, tried to sell this as possible new physics. It clearly isn't, and they've helped to establish that.

Luboš Motl said...

Dear Anonymous, do I agree with you that the features of the new force make it ludicrous? Absolutely.

Do I agree that it was their goal to show that ludicrous new physics would be needed? No. Just look how Jester described it. They actually proposed a solution that will save us, finally.

It's not just Jester. While they say it's "theoretically challenging" to obtain the bizarre flavor dependence of the coupling, they do a lot to argue that the model actually has been supported by evidence because they can "solve" (affect, in roughly the right direction) both the muon Lamb shift problem and the error in the muon anomalous magnetic moment.

This may or may not be true or accurate but of course this positive argument is negligible relatively to the negative evidence - the need for the weird flavor-dependence of the force.

So while the two of us interpret the weird new physics equally, the authors of the paper and most readers of the paper just don't. They should but they don't. I think it means to deny the obvious (sociological) observations if we would think that our interpretation of the situation is universally - or even commonly - shared. It's not. So I think that this paper, while OK, won't stimulate anyone to redo the calculations and look at the old-physics problems again. It will reinforce the belief of people that it's hot to propose arbitrarily unreasonable and sensational "solutions" to any minor anomaly in the data.

Best wishes
Lubos

Anonymous said...

I'm surprised to see no mention here of the work of de Rujula on this.

Luboš Motl said...

I only heard about the De Rujula paper from a commenter on my blog. The author proposed a theory fix based on Zemach moments etc.

His proposal was violently attacked by Michael Distler et al.. The only other citation so far is a paper by Barger et al. that proposes new particles again.

Can't someone do a comprehensive calculation where he guarantees all the pieces instead of tearing individual lines and terms out of the context and arguing about them? Can't it be calculate just right?

Jester said...

The point of De Rujula is that the Lamb shift depends not only on the proton charge radius r^2, but also on the so-called 3rd Zemach radius r^3 . The theoretical formula for the Lamb shift in muonic hydrogen is
\Delta E = [209.9779 - 5.2262 r^2 /fm^2 + 0.0347 r^3/fm^3] meV
Thus, rather than interpreting the PSI experiment in terms of a smaller-than-expected one could also argue that it points to a larger-than-expected . However the 3rd Zemach radius needed explain the experiment would have to be humongous, r^3 \sim 30 fm^3. Meanwhile, several determinations of the 3rd Zemach radius exist in the literature, some depending on a particular model of the proton structure, some claiming to be model independent. They all seem to converge to r^3 \sim 3 fm^3 with a 5 percent error, that is 10 times too small to explain the PSI measurement. For this reason de Rujula's explanation is considered very unlikely by experts. See Section 3.4 of 1011.5275 for a recent discussion.

Luboš Motl said...

Dear Jester,

could you please explain me something that you implicitly seem to assume - and you're not the only one?

Why do you think that the "proton charge radius" is the same for all probes - such as the electron and the muon that fly around?

They "see" different physics inside the proton, don't they? The muon is more capable to resolve the individual quarks, isn't it? Why does everyone assume that the "proton charge radius" is independent, so to say, on the renormalization mass scale, or the energy and type of the probes that try to "feel it"?

Cheers
LM

Jester said...

For two reasons.
1) The Bohr radius of the muonic hydrogen, being of order 1/MeV, is still fairly large compared to the QCD length scale, so no dramatic effects are expected.
2) The proton radius has also been measured in electron-proton scattering experiments at a momentum transfer of order 100 MeV, and the result agrees very well with the proton radius extracted from hydrogen spectroscopy where the typical momentum transfer is of order 5 keV.

It thus seems unlikely that something happens to the proton radius at the intermediate MeVish momentum transfer relevant for muonic hydrogen. There is no theoretical reasons that anything dramatic should happen. Although, strictly speaking, the present experimental data cannot exclude such a possibility.

Luboš Motl said...

Dear Jester, what you say in 2) sounds very surprising to me - I can't even imagine what the statement could operationally mean, given the fact that you're essentially entering the deep inelastic scattering mode. Can you give me reference, please?

Concerning, 1) - and 2) - 1 MeV is 0.7% of 150 MeV, the QCD scale, so I wouldn't find it surprising to see linear - of order O(1%) - corrections to various things. And indeed, the observed change of the proton radius is in O(1%), too.

Well, changes in the perception of a proton at an MeV are not "dramatic", but the observed change of the proton radius was not "dramatic", either.

I think you're using different standards for the word "dramatic" in different contexts. It really does sound like some non-quantitative disciplines that enjoy claims about "dramatic" phenomena.

Luboš Motl said...

By the way, even if the influence on the proton radius from the probes (muon) with total energy 100 MeV and momentum 1 MeV - velocity is always the fine-structure constant or whatever - were just 1% and you needed 4%, it is a very relevant effect because it may reduce a 5-sigma discrepancy to 4 sigma and then similarly 3 sigma by one more thing, and you're done.

So you just couldn't "strictly" neglect all things just because they are 5 times smaller. There can be many of them and even if there are not many, they may substantially reduce the error.

Anonymous said...

Just to state clearly what is being discussed: the charge form factor is just the matrix element of the EM current, schematically (i.e., I'm being lazy about order-one numbers and dropping the dipole term):

< p(q)| J_μ(q) | p(0)> = F(q^2) ubar γ_μ u.

So it's true that F(q^2) depends on the momentum scale of the scattering, but the charge radius is by definition a fixed coefficient in the small q^2 expansion:

F(q^2) = 1 + 1/6 <r_p^2> q^2 + O(q^4)

At least in pure QCD, this is a good nonperturbative definition, in principle measurable on the lattice. It's just a fixed number, not anything with scale dependence.

Now there are two issues:

In e-p scattering, the scale at which q^2 is probed is large enough that higher-order terms in the series matter. The experimenters have to do some sort of fit or extrapolation to extract just the leading coefficient. The question is whether they estimated their errors correctly or have more model-dependence.

Note though that issue does not arise in the comparison of ordinary hydrogen and muonic hydrogen, because as Jester said in both cases the q^2 scale is such that higher-order terms in the form factor are small unless their coefficients are abnormally large.

The second issue is that once you turn on QED, there can be IR divergences from the massless photons. Naively I thought that this adds new scale-dependence to the definition of the charge form factor and leads to ambiguities in what the charge radius means. I would have thought this could give a ~ alpha log(m_μ/m_e) correction of order a percent. But this doesn't seem consistent with what the experts who have studied this say, and I haven't tried to understand the issue carefully. Maybe someone can clarify it here.

Luboš Motl said...

Oh I see, it's a coefficient in a q-expansion. That was helpful, Anonymous.

The logarithmic threat may be relevant, too.

Jester said...

The latest measurement of r^2 in e-p collisions is from an experiment at Mainz Microtron, http://arxiv.org/abs/1007.5076v2.

Luboš Motl said...

If I return to the expansion a bit, how does one prove that there are no terms e.g. with fractional powers of "q" in the formula? And, less radically, odd powers of "q"? And if there's the "q^3" term, how can one see that it's not high enough?

Chris Oakley said...

Hi Jester,
Interesting post. Do you know anything about Lamb shift in e+e-, μ+e- (or μ-e+) or μ+μ-? I would have thought that these would be the first places to look for problems in QED.

Anonymous said...

Lubos: you have written a post and commented on a topic which you did not even know the definition of?
WOW

Jester said...

Chris, mu+mu- has not been created in a laboratory yet. As for ee and emu, I don't know if Lamb shift was measured in those; I'd guess that yes, and that it agrees with the QED predictions.

Luboš Motl said...

Dear Anonymous,

I don't need to know the definition to know that it is bullshit that new science can be found in this way.

It doesn't mean how one defines or parameterized all these concepts: it's clear that different processes will see different internal processes of the proton. Knowing the name of something is something very different than knowing something.

By the way, obviously, no one else on this blog knows the answer to my last question, otherwise it would have already been answered.

I won't be bullied by anonymous know-nothings.

Cheers, LM

Itay said...

Jester, thank you for featuring our article and sorry for the late posting. There is much to be said about the issue, but let me concentrate on a few things that came up in the comments.

First, regarding the reason for writing the article. To me, there were three interesting issues. 1) such a new force is, as far as we can tell, not excluded in a model-independent way by any other experiment. 2) it also explains the muon g-2 if indeed you believe the discrepancy. 3) It provides for fairly concrete predictions for systems that will soon be measured as well (more below). Together, these three points make for an interesting story, and that's all I take it to be at this point, an interesting story.

At the moment the model building looks pretty tough and the possibilities rather ugly. But, that might just be a lack of our imagination. Ultimately,
I don't quite see the point of arguing too much about matters of taste, as soon we'll have results from muonic deuterium and muonic helium that will test this possibility directly (true muonium will probably take a few more years and the Lamb shift measurement in this system even longer).

Regarding the QED calculation, the part that does not involve the nucleus structure is actually easier in the muonic system compared with normal hydrogen, so I doubt this is the issue. The nucleus structure part is more complicated because the second order contribution (the Zemach effect) is not negligible like in normal hydrogen. There has been much discussion about this as well, and I have nothing intelligent to add. But, again, it seems to me like this is not where the resolution is.

Itay said...

One point to keep in mind regarding the proton radius measurement using normal hydrogen is that none of the measurements in the CODATA (about 14 I think) is more than about 2\sigma from the muonic hydrogen result. They are all consistent with each other and consistently higher than the new result. It's their average which is 5\sigma away. I will not be surprised if we see a new measurement of the proton radius in normal hydrogen again soon (with blind analysis this time around). See also the nice paper by Hill and Paz for the corresponding issues with scattering data.

All in all, I think it is fair to say that the situation is pretty confusing at the moment. Of course, it is most likely not new physics, but that's a cheap statement that is true for anything but the most blatant discrepancy from the SM (like a di-muon resonance at 800GeV, Amen, :-)). As it stands right now, it is pretty clear that the coupling necessary does not jive with anything aesthetically appealing. That might be enough to convince most theorists to abandon any possible new physics explanation of this result. I sympathize and agree, but the empiricist in me leans towards allowing the coming measurements (mu-D, mu-He) to inform us some more before making a final judgement. If nothing else, we now have one more discrepancy in a muonic system.

Happy holidays everyone!