Possible explanation for muon g-2 anomaly: Gravity?

[vanhees71]

I don’t see their calculation make use of, or state, that noninertial frame is important, and if it were, there would be some term proportional to the christoffel symbol for a stationary world line, i.e. 1/r². Their words mention curvature, but I can’t see it anywhere in their calculation. Everything is in terms of metric coefficients, and for these, to PN accuracy, the sun’s contribution would be larger.

Of course, what they calculate are corrections to the magnetic moment of the muon, i.e., they evaluate the Dirac-particle Hamiltonian (in 1st-quantization notation, see e.g. Eq. (16) in paper (1)). Of course, there the "potential occur" (both the em. four-potential as well as the metric components of the Schwarzschild spacetime, which is the GR analogue of potentials for the em. field), as usual in classical as well as quantum Hamiltonian theory.

 

[vanhees71]

I don't think that is a good approach. There are well-known effects where the Sun has a stronger influence, e.g. gravitational deflection of light (which depends on the potential and the angle).

Well, the deflection of light of course involves also not the metric components directly but is calculated from solving the geodesic equation for light rays, i.e., finding the null geodesics of the Schwarzschild metric.

Based on various tweets and blog posts: Several experts say the calculations are incorrect.

My favorite one:

This is a more serious criticism, indeed. Unfortunately, I'm not an expert in field quantization in curved spacetime. So I can't say whether this is a valid critcism or not without further study.

On the other hand, there is this very basic experiment with neutrons in the gravitational field on Earth, which is in full agreement with the standard-QM 1 problem to evaluate the motion of a neutron in the gravitational field. It's of course completely non-relativistic, but also there you write down simply the naive non-relativistic Hamiltonian with the Newtonian gravitational potential ##V=-m g z## for the gravitational field (usually assuming a reflecting ground; so that the neutrons are altogether in a binding potential, and what you get are Airy functions in momentum representation for the energy eigenstates). Also there the potential occurs in the Hamiltonian and not the force ##\vec{F}=-m g \vec{e}_z##. So the argument you quoted is a bit too naive to buy it without careful further investigation.

It's as with the Aharonov-Bohm effect: At the first glance it seems as if the potentials play an observable role all of a sudden, which cannot be, because gauge dependent quantities cannot be observable by construction, and indeed what goes into the observable relative phase shift in the AB effect is the line integral over the vector potential, which can be cast into a surface integral of the magnetic field (which indeed is observable and gauge invariant!) over a surface encircled by the paths around the magnetic fiber, along which one has to integrate to get the relative phase shift for particles going around it the one or the other way.

I'm not sure, that such arguments apply to the calculations in question, but I think, it's not so easy to find a fundamental error in the argument. Let's see what the reviewers of the article(s) think, i.e., whether it appears in the journal. I'd not believe too much in blog posts ;-))).

 

[George Jones]

I'd not believe too much in blog posts ;-))).

I agree, but I still find the comments to be interesting, including the set of comments at "Not Even Wrong",

http://www.math.columbia.edu/~woit/wordpress/?p=9986#comments

 

[PAllen]

I agree, but I still find the comments to be interesting, including the set of comments at "Not Even Wrong",

http://www.math.columbia.edu/~woit/wordpress/?p=9986#comments

I especially like the comment by Matt Visser, who is certainly a top expert on GR, who is saying the same things I’ve been saying, only better.

 

[Arman777]

As I said before, just take everyday experience. How much do you feel from the gravitational field of the Sun and how much do you feel from the gravitational field of the Earth? The reason is that the Earth including us is freely falling in the gravitational field of the Sun, and all that acts from the Sun from our point of view are her effects on the tides, but this is not relevant for the very local interactions we feel in everyday life, and the same holds for the very accurate measurement of the muon's (g-2), which is a local experiment.

I kind of said it it my previous post (#19)

http://www.math.columbia.edu/~woit/wordpress/?p=9986#comments

From the last, you should definately read the fifth post (by vmarko). I am not expert of course but it seems informative.

 

[PAllen]

I kind of said it it my previous post (#19)
From the last, you should definately read the fifth post (by vmarko). I am not expert of course but it seems informative.

"Note that the apparatus does *not* freely fall in the gravitational field of the Earth (as opposed to the Sun and the galaxy), but is being “pushed” off its geodesic trajectory upwards by the floor of the lab. This is an electromagnetic effect (spiced up by the Pauli exclusion principle), despite being described by the gravitational potential ϕ" role="presentation">

, in Newtonian language. This force is real, we all feel it when we stand up, and it has nothing whatsoever to do with any violation of the equivalence principle."

If the effect is due to this force it is purely an SR non-inertial frame effect not a GR effect, unless it vioates the principle of equivalence between a local accelerated frame near a massive body versus empty space. Thus it either is not due to GR or violates the POE, exactly as Visser said. I would say Vmarko completely missed Visser's point.

"Third, one should distinguish the contribution coming from the potential ϕ" role="presentation">

and the contribution coming from the gradient of the potential, ∇ϕ" role="presentation">

. These are different, and the latter is much smaller than the former, as explained in Appendix C of the paper."

Except that the potential per se should not be able to influence anything, only changes in potential, i.e. gradient.

It seems to me that Vmarko has neither understood nor answered Visser's points.

 

[Arman777]

So can we say papers are incomplete, insufficient or etc (but definitely not accurate) ?

 

[Vanadium 50]

I've thought about this some more. I'm not qualified to discuss the calculation itself, but there are some things that still trouble me.

(1) The fact that there is a "naked phi" in the equation of the shift is troublesome. This is the root source of several other complaints, such as not being gauge-invariant. I believe that the only sensible way to treat this is to really mean ΔΦ, the difference between local Φ and one "sufficiently far away" where spacetime is "sufficiently flat:"

(2) If you do that, you have to deal with the fact that the definition of meters and seconds "far away" is different than that at the experiment. The comparison that should be made is the calculation and measurement of g locally. "You'd get a different magnetic moment if you use these rulers and clocks instead" is maybe of academic interest, but it doesn't help with the anomaly. My earlier comment on comparison NMR is along these lines, although now I have come to the conclusion that things are more general than I first thought.

(3) I don't think putting the experiment in orbit helps either theoretically or experimentally (ignoring practicality). If what matters is potential, the potential is not that much different in LEO. If what matters is acceleration, you're never going to get the muons in free fall, because they are undergoing terrific acceleration just keeping them circulating in the storage ring.

 

[websterling]

I especially like the comment by Matt Visser

Visser placed a paper on the arXiv, probably the first reply to the 3 papers. It's basically an expansion of his comments on Woit's blog.

Post-Newtonian particle physics in curved spacetime

From his introduction-

Unfortunately these articles do not correctly implement the Einstein equivalence principle, and so their methods and conclusions are in direct conflict with general relativity.

 

[vanhees71]

I kind of said it it my previous post (#19)
From the last, you should definately read the fifth post (by vmarko). I am not expert of course but it seems informative.

Well, vmarko indeed argues exactly as I did, but as I said, I've not done the calculation myself, and I think one should confirm it by a proper QFT calculation too. On the other hand, I don't see something that looks obviously wrong.

 

[vanhees71]

Visser placed a paper on the arXiv, probably the first reply to the 3 papers. It's basically an expansion of his comments on Woit's blog.

Post-Newtonian particle physics in curved spacetime

From his introduction-

Hm, if you argue like this, you can also claim that the standard textbook treatment of the hydrogen atom is wrong, because it is done in a specific gauge (namely Coulomb gauge) in the approximation to take into account only the electrostatic Coulomb potential, which is of course also unjustified. I think this paper doesn't really get the point that in QT you always deal with potentials in the Hamiltonian, not directly with forces. Of course, the gauge-invariance issue is a serious argument, but it has to be clarified by a real calculation and not some four-page hand-waving argument which looks a bit too naive. The original authors at least were more careful with their arguments (at least in paper I, including the appendices). That their calculation is not manifestly generally covariant is, however, clear due to the PN (PPN) approximations made, but for solar-system gravitational fields these approximations are very well justified.

I guess the real resolution would be to formulate everything systematically in QFT in a given background spacetime. In my opinion, as a first approximation the Schwarzschild metric due to the Earth's gravitational field should suffice for this. Then you can calculate the effect of gravity on the magnetic moment (including radiative corrections), but that's a huge effort.

 

[mfb]

You have potentials in the Hamiltonian, but the absolute value does not matter for observables. The situation is analog to the hydrogen atom: You might choose a particular gauge because it makes calculations easier, but the result does not - cannot - depend on the gauge.
Sure, have potentials in the calculation, but if the result explicitly depends on them something went wrong. And even if not, then you have to consider all potentials, not cherry-pick one.

 

[vanhees71]

Yes, indeed. That's my point. So you cannot simply claim the calculation is wrong, only because the potentials occur. Also at this discussion linked by somebody above,

http://www.math.columbia.edu/~woit/wordpress/?p=9986#comments

they have the two camps of opinion. It's for sure not an easy thing to disentangle, and I hope that the reviewer(s) do a careful job for PTEP ;-)).

 

[mfb]

So you cannot simply claim the calculation is wrong, only because the potentials occur.

I questioned it because the potentials occur in the final result. Experts like Matt Visser claimed it is wrong for the same reason.

 

[vanhees71]

Ok, then something has gone wrong with the P(P)N approximation. It's very puzzling indeed. There are also some doubts about the correctness of the entire calculation itself, not even related to the gravity part:

http://www.science20.com/comments/206921

So, we'll have to wait until the debate in the community has settled or to do a calculation ourselves (the latter being a very time-consuming effort, at least for me, unfamiliar with the details of QFT in a curved background spacetime).

 

[George Jones]

they have the two camps of opinion

This just collapsed to the single camp that the effect is negligible.

 

[Arman777]

This just collapsed to the single camp that the effect is negligible.

Well, vmarko indeed argues exactly as I did, but as I said, I've not done the calculation myself, and I think one should confirm it by a proper QFT calculation too. On the other hand, I don't see something that looks obviously wrong.

Vmarko added in his recent post on the site,

"It appears that I was wrong regarding my comment to the response of Chris Polly. Namely, in experiment, the a.m.m. is not being determined from equation (44) but from equation (8). The authors correctly point out that this is calculated using the skewed value of ##γ## which appears to be a valid remark given the GR correction term in (40). So I decided to calculate the variation of (8) with respect to ##γ## taking into account (40), to see what happens when the value of ##γ## is slightly shifted. And indeed, the variation turns out to be proportional to ##β## x ##E##
as well, as Chris wrote. So Chris is right that this effect is weighted with the magnitude of the electric field, which is apparently small enough to suppress the GR correction beyond the experimental resolution.

In the end, it appears that the correction term in (45) and in Table 3 is really just a numerical coincidence."

 

[ohwilleke]

So sad. Refutations are coming in from all directions to these papers. It appears that they have at least two distinct flaws. One related to the relative strength of the B and E fields in the experiment and another more general objection related to GR.

 

[mfb]

It means the muon g-2 discrepancy is still there. That is interesting as well.

 

[vanhees71]

Ok, if there are calculational and interpretational shortcomings in these papers, then it's likely to be wrong :-(.

 

[Vanadium 50]

Ok, if there are calculational and interpretational shortcomings in these papers, then it's likely to be wrong

Yes, but apart from that...

 

[Urs Schreiber]

So sad. Refutations are coming in from all directions to these papers. It appears that they have at least two distinct flaws. One related to the relative strength of the B and E fields in the experiment and another more general objection related to GR.

I imagine a proper discussion of QFT corrections due to a background gravitational field ought to proceed in direct analogy to the well-known discussion of QFT corrections due to a background electromagnetic field as in the Lamb shift.

This should require evaluating vertex corrections to lepton-graviton Feynman amplitudes (a concept that is curiously missing from the discussion of the three articles cited in #19). But this must have been considered before. (?)

Digging around, I find

• F. del Aguila, A. Culatti, R. Munoz-Tapia, M. Perez-Victoria,
  "Supergravity corrections to ##(g-2)_{\mathrm{lepton}}## in differential renormalization",
  Nuclear Physics B 504 (1997) 532-550
  hep-ph/9702342

which (on the first page of its introduction) points to

• [13]
  F.A. Berends, R. Gastmans,
  Phys. Lett. B55 (1975) 311

as:

This seems relevant.

Now, I haven't seen that Berends-Gastmans article yet. Maybe the reference [13] is garbled, or my spire-search foo is lacking. Might anyone have a copy?

 

[Urs Schreiber]

Now,I haven't seen that Berends-Gastmans article yet.

Sorry, got it now:

• F. A. Berends, R. Gastmans
  "Quantum gravity and the electron and muon anomalous magnetic moment"
  Phys. Lett. B55 Issue 3
  Feb 1975 311-312
  https://doi.org/10.1016/0370-2693(75)90608-5

Okay, I see, these authors do not consider corrections due to a background field, just the 1-loop gravity corrections due to these diagrams:

So it's not directly relevant to the claim cited in #19.

But nevertheless, it seems to me that if one wanted to study that effect in #19, it's this kind of QFT computation that should be used.

 

[ohwilleke]

A short new pre-print from a Wellington, New Zealand based physicist posted three days after the original papers concludes that these papers are flawed:

In three very recent papers, (an initial paper by Morishima and Futamase, and two subsequent papers by Morishima, Futamase, and Shimizu), it has been argued that the observed experimental anomaly in the anomalous magnetic moment of the muon might be explained using general relativity. It is my melancholy duty to report that these articles are fundamentally flawed in that they fail to correctly implement the Einstein equivalence principle of general relativity. Insofar as one accepts the underlying logic behind these calculations (and so rejects general relativity) the claimed effect due to the Earth's gravity will be swamped by the effect due to Sun (by a factor of fifteen), and by the effect due to the Galaxy (by a factor of two thousand). In contrast, insofar as one accepts general relativity, then the claimed effect will be suppressed by an extra factor of [(size of laboratory)/(radius of Earth)]^2. Either way, the claimed effect is not compatible with explaining the observed experimental anomaly in the anomalous magnetic moment of the muon.

Matt Visser, "Post-Newtonian particle physics in curved spacetime" (February 2, 2018).

There is also an official statement from the g-2 collaboration:

The response from the g-2 collaboration (from the spokesperson Chris Polly):

Our spokes already replied to the authors since they made a mistake in the final conclusion. While the additional effect in the bxE term they calculate is 2ppm, they then attribute this full term to be the change in g-2. However, they forgot that that additional contribution needs to be weighted by the relative strength of the bxE term which is 1330ppm of the B field. So even if their calculation was correct, the actual contribution is 2ppm*1330ppm=2ppb. That’s negligible for the ongoing experiments measuring to ~100ppb precision. And this argument does not even involve any judgement on the validity of the additional term they calculate.

and from the same source http://www.science20.com/comments/206921:

Re: Gravitational Effects Explain Muon Magnetic Moment Anomaly A

Regardless of whether or not the GR is correct, the authors make an error at the end of their paper by failing to take the relative strengths of the E and B fields used by the experiment into account. The vast majority of the muon precession is driven by the B-field while the E-field is only a small perturbation. The maximum E-field experienced by a muon in the g-2 storage ring is 30kV/5cm while the B-field is 1.5T. That means betaXE is very small compared to B...to be precise betaXE is 1300 parts per million (ppm) compared to B. So, in their treatment they find an additional modification to the coefficient in front of the betaXE term that shifts the value of the coefficient by 2ppm. Therefore, the overall impact on the anomalous magnetic moment extracted by the experiment would change by 2ppm x 1300 ppm = 2.7 parts per billion (ppb), which is well below the 500ppb error on the BNL experiment and the 140ppb error targeted at Fermilab. This is actually an overestimate since we used the maximum E-field a muon can experience in the g-2 ring in the calculation. If you cannot find anywhere in the paper where they state the average magnitudes of E and B observed by muons in the experiment, then you know there is a problem. For instance, they would find the same correction arising from the betaXE term would apply to the experiment proposed at J-PARC even though the novel design of that experiment has E=0 by construction.

There also also many comments in the blog posts that have covered this development and some have updates to their body texts or follow up posts.

 

[mfb]

I moved six posts from this thread into this thread as they have the same topic and don't fit well in the other thread.

 

[Demystifier]

From Aharonov-Bohm effect, we know that sometimes in QM the potential itself (not its derivative) has a physical role. The gravitational redshift in classical GR is also formulated in terms of the potential (not its derivative). Could it be that something similar is happening here?

 

[mfb]

From Aharonov-Bohm effect, we know that sometimes in QM the potential itself (not its derivative) has a physical role.

Even there it is only the difference. Gauge symmetry stays, and adding a constant term to the potential is a trivial gauge symmetry everywhere.

The gravitational redshift in classical GR is also formulated in terms of the potential (not its derivative)

Yes, there something moves from A to B.

 

[Demystifier]

Even there it is only the difference. Gauge symmetry stays, and adding a constant term to the potential is a trivial gauge symmetry everywhere.Yes, there something moves from A to B.

Well, maybe the correction to g-2 from ##g_{\mu\nu}\neq\eta_{\mu\nu}## really depends on ##\phi(R)-\phi(\infty)##, where ##r=\infty## corresponds to the point where ##g_{\mu\nu}=\eta_{\mu\nu}##, and the authors work in the gauge in which ##\phi(\infty)=0##.

 

[Demystifier]

I am trying to understand what could possibly be wrong with their calculation. Eq. (2) in paper I is general covariant. From that they derive Eq. (4). The second line of Eq. (4) contains only the derivative of the potential, so it should not be problematic. The potential itself appears only in the first line, which vanishes when the EM fields ##{\bf E}## and ##{\bf B}## vanish. This suggests that there could be something wrong with their calculation of ##{\bf E}## and ##{\bf B}##. Indeed, they do not state how ##{\bf E}## and ##{\bf B}## are defined. I suspect that they define ##{\bf E}## and ##{\bf B}## as the corresponding components of ##F^{\mu\nu}## in (2), but if they do, that's wrong. In general, electric and magnetic field are defined covariantly as (see https://arxiv.org/abs/1302.5338 )
$$E^{\mu}=F^{\mu\nu}o_{\nu}$$
$$B^{\mu}=-\tilde{F}^{\mu\nu}o_{\nu}$$
where ##o_{\nu}## is the 4-velocity of the observer. In a gravitational background, the velocity ##o_{\nu}## also depends on the potential, which might cancel the dependence on the potential in the first line of (4). Someone should do a detailed calculation to check it!

 

[mfb]

Well, maybe the correction to g-2 from ##g_{\mu\nu}\neq\eta_{\mu\nu}## really depends on ##\phi(R)-\phi(\infty)##, where ##r=\infty## corresponds to the point where ##g_{\mu\nu}=\eta_{\mu\nu}##, and the authors work in the gauge in which ##\phi(\infty)=0##.

If that would be true, we would be back at the Sun/galaxy question.

 

[Demystifier]

If that would be true, we would be back at the Sun/galaxy question.

You are right. Now I suspect that their calculation could be wrong due to the reason explained in #64 above.

 

[Demystifier]

I have found a rather trivial error in their paper, so I have written a short paper on it:
http://lanl.arxiv.org/abs/1802.04025

 

[vanhees71]

• [13]
  F.A. Berends, R. Gastmans,
  Phys. Lett. B55 (1975) 311

as:

View attachment 219802

This seems relevant.

Now, I haven't seen that Berends-Gastmans article yet. Maybe the reference [13] is garbled, or my spire-search foo is lacking. Might anyone have a copy?

https://doi.org/10.1016/0370-2693(75)90608-5

 

[Urs Schreiber]

https://doi.org/10.1016/0370-2693(75)90608-5

Yeah, I had found in #58, right after my first message. Sorry for the trouble.

 

[Urs Schreiber]

I have found a rather trivial error in their paper, so I have written a short paper on it:
http://lanl.arxiv.org/abs/1802.04025

Great. Trivial or not, you seem to be the first to actually identify the error, instead of just making broad comments about the plausibility of the result.