Possible explanation for muon g-2 anomaly: Gravity?

[mfb]

For many years, the measurements of the Landé g-factor of the muon have been puzzling, as the experimental value and the theoretical predictions showed some disagreement - 3.6 standard deviations for the last years. Experimental and theoretical uncertainties have a similar size, so work on both sides helps.
Muon g-2 at Fermilab is currently taking more data to improve the experimental result, while CERN is studying a muon-electron scattering experiment to improve some experimental values that go into the theory calculations. But a few days a completely new idea was made public.

The difference might come from the gravitational field of Earth. This sounds absurd - something you learn early as particle physicist is that gravity is completely negligible (unless you design detectors!). But we are talking about extremely precise measurements - parts per billion. Three authors investigated if gravity can have an effect - and found a possible contribution that is of the size of the observed discrepancy. Adding this to the theoretical prediction reduces the discrepancy from 3.6 standard deviations to 0.1 standard deviations. If the calculations are correct, theory and experiment are actually in excellent agreement.

The authors uploaded three papers to arXiv:
Theoretical framework
Effect on electrons
Effect on muons

The effect on electrons is very important: Our measurements there are a factor 1000 more precise, and they agree nicely. The authors calculate that gravity doesn't influence the electron measurements due to the different measurement principles - the measurements are done with slow electrons, while the muons are relativistic.

Other theorists are checking these calculations now. If they can confirm the results, the muon g-2 anomaly is gone. On the positive side, we get a direct influence of curved spacetime on particle physics measurements - an interesting effect to study in more detail.

Other links:
Blog article covering the topic

Something I don't understand: If the effect scales with $1-\frac{3GM}{rc^2}$ for g as given in the abstract and the conclusion, then the Sun should have an effect a factor 14 larger. Yet it is not mentioned at all.

 

[Jimster41]

Something I don't understand: If the effect scales with 1−3GMrc21−3GMrc21-\frac{3GM}{rc^2} for g as given in the abstract and the conclusion, then the Sun should have an effect a factor 14 larger. Yet it is not mentioned at all.

Do you have any ideas as to why they didn't mention it or consider it? Is there an inertial frame argument based on the idea the sun would be affecting the experimental apparatus and the muon in a way that would almost completely obscure the sun's effect on the muon?

I have no idea, but you seem to be saying they didn't consider something totally obvious.

 

[websterling]

It's the gravitational potential, not the gravitational field. Since the Earth and the apparatus are in orbit (free fall) around the Sun, the equivalence principle says the Sun's field has no effect on the local measurement. Unless, of course, I'm wrong

 

[ohwilleke]

Mentor note: This post was originally in this thread and was moved to keep the discussion together.

The question in the original post is no longer hypothetical. With this latest theoretical value adjustment, the experimental value and the theoretical value are within one sigma of each other.

The magnetic moment of free fermions in the curved spacetime has been studied based on the general relativity. Adopting the Schwarzschild metric for the background spacetime, the effective value of the magnetic moment has been calculated up to the post-Newtonian order O(1/c2) for three cases (A) Dirac particles with g=2, (B) neutral fermions with g≠2 and e=0 and (C) charged fermions with g≠2 and e≠0. The result shows their gravity dependence is given as μeffm=(1+3ϕ/c2)μm for all of these cases in which the coupling between fermions and the electromagnetic field is essentially different. It implies that the magnetic moment is influenced by the spacetime curvature on the basis of the general relativity commonly for point-like fermions, composite fermions and spread fermions dressed with the vacuum fluctuation. The gravitational effect affects the gyro-magnetic ratio and the anomalous magnetic moment as geff≃(1+3ϕ/c2)g, aeff≃a+3(1+a)ϕ/c2. Consequently, the anomalous magnetic moment of fermions with g≃2 measured on the Earth's surface contains the gravitational effect as |aeff|≃3|ϕ|/c2≃2.1×10−9, which implies that the gravitational anomaly of 2.1×10−9 is induced by the curvature of the spacetime on the basis of the general relativity in addition to the quantum radiative corrections for all fermions including electrons and muons.

Post-Newtonian effects of Dirac particle in curved spacetime - I : magnetic moment in curved spacetime" (January 30, 2018).

The general relativistic effects to the anomalous magnetic moment of the electron ge-2 in the Earth's gravitational field have been examined. The magnetic moment of electrons to be measured on the Earth's surface is evaluated as μeffm≃(1+3ϕ/c2)μm on the basis of the Dirac equation containing the post-Newtonian effects of the general relativity for fermions moving in the Earth's gravitational field. This implies that the anomalous magnetic moment of 10−9 appears in addition to the radiative corrections in the quantum field theory. This may seem contradictory with the fact of the 12th digit agreement between the experimental value measured on the ground level ge(EXP) and the theoretical value calculated in the flat spacetime ge(SM). In this paper, we show that the apparent contradiction can be explained consistently with the framework of the general relativity.

Takahiro Morishima, Toshifumi Futamase, Hirohiko M. Shimizu, "Post-Newtonian effects of Dirac particle in curved spacetime - II : the electron g-2 in the Earth's gravity" (January 30, 2018).

The general relativistic effects to the anomalous magnetic moment of muons moving in the Earth's gravitational field have been examined. The Dirac equation generalized to include the general relativity suggests the magnetic moment of fermions measured on the ground level is influenced by the Earth's gravitational field as μeffm≃(1+3ϕ/c2)μm, where μm is the magnetic moment in the flat spacetime and ϕ=−GM/r is the Earth's gravitational potential. It implies that the muon anomalous magnetic moment measured on the Earth aμ≡gμ/2−1 contains the gravitational correction of |aμ|≃2.1×10−9 in addition to the quantum radiative corrections. The gravitationally induced anomaly may affect the comparison between the theoretical and experimental values recently reported: aμ(EXP)−aμ(SM)=28.8(8.0)×10−10(3.6σ). In this paper, the comparison between the theory and the experiment is examined by considering the influence of the spacetime curvature to the measurement on the muon gμ−2 experiment using the storage ring on the basis of the general relativity up to the post-Newtonian order of O(1/c2).

Takahiro Morishima, Toshifumi Futamase, Hirohiko M. Shimizu, "Post-Newtonian effects of Dirac particle in curved spacetime - III : the muon g-2 in the Earth's gravity" (January 30, 2018).

 

[mfb]

It's the gravitational potential, not the gravitational field. Since the Earth and the apparatus are in orbit (free fall) around the Sun, the equivalence principle says the Sun's field has no effect on the local measurement. Unless, of course, I'm wrong

Why would the gravitational potential of Earth (instead of the local acceleration) matter then?

 

[Haelfix]

I haven't had the time to look at this much, but just glancing at the paper, they seem to be working in some sort of post Newtonian (weak field) approximation using the Schwarzschild background. If you included the sun, you wouldn't have the Schwarzschild metric as a valid solution, and you would need to work numerically (not unlike what they do for binary star mergers). To even get some sort of to first order solution for the combined system would be an absolute mess, and I don't know what you could expand around to even attempt that calculation.

In any event, the suns contribution wouldn't scale like ~ 1 - Mg/r.

 

[mfb]

What is wrong with using the Sun with Schwarzschild metric?

In any event, the suns contribution wouldn't scale like ~ 1 - Mg/r.

MG/r is literally the expression they have in the papers. Multiplied by 3 it is the 2*10^-9 discussed.

 

[PAllen]

I've looked at the paper just a little, and I am also confused. If the effect were related to the christoffel symbols, that would make clear why the sun was irrelevant, but as mfb says, all expressions they end up with are based on GM/r (which, of course, is the deviation of metric from Minkowski). The role of the post Newtonian approximation just seems to be to express the isotropic SC metric in an algebraically convenient form up to the precision they care about.

 

[Haelfix]

The Schwarzschild metric is a model for one large spherically symmetric system. You can't use it to model the composite effects of another large mass some distance away and then superpose solutions like it was a central force problem. The MG/r in the papers is referring to the Earths mass. I don't see how it would make sense to refer to the sun at all in this context. Maybe I am mistaken, do the papers talk about the effect of the sun at all explicitly?

 

[PAllen]

The Schwarzschild metric is a model for one large spherically symmetric system. You can't use it to model the composite effects of another large mass some distance away and then superpose solutions like it was a central force problem. The MG/r in the papers is referring to the Earths mass. I don't see how it would make sense to refer to the sun at all in this context. Maybe I am mistaken, do the papers talk about the effect of the sun at all explicitly?

No, that is the question. It appears the effect from the sun should be larger, but it is not mentioned. That is, if you ignored the Earth and used the same method for the sun, you would get a larger anomaly.

 

[mfb]

The Schwarzschild metric is a model for one large spherically symmetric system. You can't use it to model the composite effects of another large mass some distance away and then superpose solutions like it was a central force problem. The MG/r in the papers is referring to the Earths mass. I don't see how it would make sense to refer to the sun at all in this context. Maybe I am mistaken, do the papers talk about the effect of the sun at all explicitly?

If the potential would be the important quantity, then they should take the Sun and neglect the Earth - to get an effect a factor 14 larger. They do not mention the Sun at all (I searched for the word).
If the local gravitational acceleration is important, then the Sun is negligible - but why doesn't it appear in the final expression (GM/r² instead of GM/r) then?

 

[Arman777]

this might be explain why,

'However those effective values in curved spacetime are respectively
different from those of values in the flat spacetime, the gravitational contribution is canceled
in the ratio Eq. (43) and the anomalous magnetic moment in the curved spacetime coincides
with the case in the flat spacetime."

(Page 11, third article)

They are assuming the magnetic moment of muon in the flat space-time when they considering the Earth gravity (or solution based on Earth's gravitational potential)

If we try to pass the sun we cannot use the magnetic moment of the muon in flat space-time since in the presence of the sun we should use the magnetic moment in the curved space-time value.

But as in the qoute says there will be cancellation. So they will be the same.

 

[Keith_McClary]

My (dim) understanding is that the effect is not due to the potential or the acceleration/force, but to the curvature (or rate of change of the acceleration with distance). The latter would be small for the Sun's gravity.

 

[PAllen]

this might be explain why,

'However those effective values in curved spacetime are respectively
different from those of values in the flat spacetime, the gravitational contribution is canceled
in the ratio Eq. (43) and the anomalous magnetic moment in the curved spacetime coincides
with the case in the flat spacetime."

(Page 11, third article)

They are assuming the magnetic moment of muon in the flat space-time when they considering the Earth gravity (or solution based on Earth's gravitational potential)

If we try to pass the sun we cannot use the magnetic moment of the muon in flat space-time since in the presence of the sun we should use the magnetic moment in the curved space-time value.

But as in the qoute says there will be cancellation. So they will be the same.

But that doesn’t jive with the math they show. If they were talking about curvature in r t components, it would be 1/r² effect, while in tangent components the curvature is 1/r.

The issue remains they really need to mention the sun and justify ignoring it.

 

[PAllen]

I should also add that at the level of post Newtonian approximation you can easily consider the combination of sun and Earth - JPL publishes PN equations combining the effects of all the major solar system bodies, for use in high precision ephemeris calculation.

 

[vanhees71]

Why would the gravitational potential of Earth (instead of the local acceleration) matter then?

Because we are not freely falling around the Earth but sit on it, hold by the electromagnetic force and the Pauli principle of the electrons in the molecules of the matter around us and ourselves.

 

[mfb]

Where exactly is that considered? They just use the gravitational potential. Not the acceleration of the experiment or anything similar. These two do not have a unique relation.

 

[vanhees71]

I have to read the paper for the details about this idea on (g-2).

But just think about our daily experience: The effects of the gravity of the sun are completely negligible for nearly anything, the only exception are of course the tides of the ocean. To calculate the free fall of the apple hitting Newton's head inspiring him to discover his universal law of gravitation, you indeed don't need to consider the gravity of the sun.

 

[Arman777]

I have to read the paper for the details about this idea on (g-2).

But just think about our daily experience: The effects of the gravity of the sun are completely negligible for nearly anything, the only exception are of course the tides of the ocean. To calculate the free fall of the apple hitting Newton's head inspiring him to discover his universal law of gravitation, you indeed don't need to consider the gravity of the sun.

I considered that too but again it comes to the point where PAllen said,

That is, if you ignored the Earth and used the same method for the sun, you would get a larger anomaly.

Or am I mistaken ?

 

[Arman777]

I think we should consider the physical interpretation. We all know that muon is a subatomic particle. In this sense, considering the "size" of the muon and Earth's gravtiational energy we can possibly define such relationship as described in the article ( magnetic moment of muon affected by Earth's potential energy)

The Sun has larger mass yes but in this case, how can we define for a gravitational potential energy for such a small object in so large distance ? Mathematically we can put the distance between eart and the sun, but does that makes sense ?

 

[Vanadium 50]

Mathematically we can put the distance between eart and the sun, but does that makes sense ?

Of course it does.

The authors uploaded three papers to arXiv:

Unfortunately, that doesn't quite close the loop. The way the magnetic fields are measured, at least for the muon experiments, are with NMR. One needs to understand how this transforms as well.

 

[Haelfix]

So yea, I just went through some of this and there is definitely something weird about having a potential and not an acceleration term. I thought they might have suppressed a 1/r factor somewhere in their calculation, but I can't seem to find that in their notation.

It still wouldn't be self consistent to include the sun in the calculation.. You would need to change the background to a perturbed Schwarzschild solution with the relevant corrections at 1PN and 2PN orders, and then do your qft in curved space on top of that. However as far as I'm aware what people do when they do Ephemerides calculations is that they treat the sun/earth/moon as effective point masses and treat the N body problem with Lagrangian techniques, so the modeling techniques wouldn't quite match up.

(Estabrook,F.B. (1971a). “Derivation of Relativistic Lagrangian for n-Body Equations containing Relativity Parameters β and γ” JPL IOM, Section 328, 14 June 1971)

But yes the results don't seem to be gauge invariant, and that looks like a real problem.

 

[MTd2]

The experimental apparatus stay functioning at random hours. The effect of the sun should cancel due Earth's rotation.

 

[mfb]

The effect of the sun should cancel due Earth's rotation.

Why exactly do you expect this?
Where in the papers is the orientation of the apparatus considered?

 

[MTd2]

There are minimal deviations from free fall due Earth's rotation. These should cancel by means of day - night cycle and by means of being on different sides of the orbit, given that it's pretty much a circle.

If you look for correction due Sun's gravity on GPS, you will see that they are more due slow vertical deviation's due tidal effects: https://goo.gl/V4ACUG But I think the apparatus is too small for this type of thing. (And even so, they are cyclical effects).

 

[mfb]

The muons are not in free fall, and the gravitational acceleration does not appear in the calculations. The second point is exactly what causes my confusion. If the gravitational acceleration would appear, it would be obvious that the Earth has a stronger influence.

If you look for correction due Sun's gravity on GPS

It would be large if our second would be based on interstellar space. It is not, because our reference frame is the surface of Earth.

Actually: If it is the gravitational potential with respect to "far away from any masses", the galaxy has an even deeper potential. Something odd is going on. I think it is not the potential itself, but maybe there is some aspect that leads to the potential of the mass with the largest acceleration being dominant.

 

[MTd2]

The galaxy has a very shallow potential, where we are located, the distances are indeed so great that you can consider that the curvature is null, where we are located. The stars are indeed too far. Even in this case, the same cycles cancels out the influence.

Indeed, the muons are not in free fall, in respect to Earth. This is what matters.

 

[mfb]

Sorry to be so direct, but you are missing the point of this discussion, and I don't see how to help further - I would just repeat myself.

 

[MTd2]

Sorry to be so direct, but you are missing the point of this discussion, and I don't see how to help further - I would just repeat myself.

I think the same about you, so, whatever.

 

[vanhees71]

The muons are not in free fall, and the gravitational acceleration does not appear in the calculations. The second point is exactly what causes my confusion. If the gravitational acceleration would appear, it would be obvious that the Earth has a stronger influence.It would be large if our second would be based on interstellar space. It is not, because our reference frame is the surface of Earth.

Actually: If it is the gravitational potential with respect to "far away from any masses", the galaxy has an even deeper potential. Something odd is going on. I think it is not the potential itself, but maybe there is some aspect that leads to the potential of the mass with the largest acceleration being dominant.

The Earth with all equipment measuring the muons and the muons themselves are in free fall around the Sun. It's also sufficiently local, i.e., the tidal forces from the Sun's gravitational field can be neglected. On the other hand the muons and the equipment measuring their magnetic moment are at rest relative to Earth due to other (mostly electromagnetic) forces and thus not in free fall relative to the Earth. That's why things fall down when made free due to the gravitational interaction with the Earth's gravitational field (for questions like this, it's more intuitive to think in terms of fields rather than in the overemphasized geometric meaning of GR) and that's why you can expect an effect due to gravity on the value of the magnetic moment of the muons. The prediction by the authors should be easily to test: Just put the measurement of the (g-2) to the international space station. There you should not measure the deviation from the Minkowski-space result, if the gravitational field of the Earth is really the cause of the measured deviations. As far as I can see, the order of magnitude calculated by the authors fit, and it's also interesting to see, whether the papers get published in PTEP. From just reading superficially, I don't see any obvious problems with their approach.

 

[mfb]

If the authors would use g=GM/R² in their calculation, there wouldn't be anything to discuss. But they do not. So why exactly do we discuss GM/R² which doesn't seem to be relevant for the calculations?

The prediction by the authors should be easily to test: Just put the measurement of the (g-2) to the international space station.

"Easy"... the muon g-2 magnet alone has 150% the mass of the whole ISS, and the muon source is a big accelerator facility. You wouldn't add this to the ISS, you would add the ISS to this experiment.

 

[PAllen]

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.

 

[PAllen]

What I would really hope to see in a calculation like this, if it were curvature related, would be in terms of local Fermi Normal coordinates, i.e. translated Rindler coordinates with the first order PN curvature correction. This would clearly make the solar or galactic contributions irrelevant. Further, it would distinguish curvature effects from SR non inertial effects. For example, it took a while for people to realize that the Pound Rebka experiment was over 7 orders of magnitude short of detecting curvature.

 

[vanhees71]

I considered that too but again it comes to the point where PAllen said,
Or am I mistaken ?

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.

What the authors do in their 1st paper is to derive the effect of the gravitational field of the Earth, modeled by the PPN (parametrized post-Newtonian) approximation of the Schwarzschild metric, on the radiation corrections to the magnetic moment of Dirac particles. Of course, I've not checked their calculation in detail, but I don't see any obvious flaw in their argument. We'll see, whether they get the papers published.

 

[mfb]

As I said before, just take everyday experience.

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

I've been speaking with some of the scientists working on the Fermilab g-2 experiment, and they are confident that this paper is substantially incorrect.

As its been explained to me, the authors of this paper calculated a ppm effect on the motional term, but this term is ~100 times smaller than the B-field term, making the correction ~100 times smaller than claimed.
https://twitter.com/DanHooperAstro/status/959475926273871872

The Muon g-2 expt team have responded to authors whose correction would get rid of the #gminus2 anomaly. A subtle mistake in interpreting the general relativity correction, they say, mean the fix is actually ~1000 times smaller than existing uncertainties, and so negligible.
https://twitter.com/LizzieGibney/status/959493675352055808

I've heard from several experts saying this fix for the #gminus2 anomaly is incorrect or needs more investigation. So, simmer down, folks.
https://twitter.com/emcconover/status/959439155683045382

My favorite one:

This is true, because the result depends only on the gravitational potential, whose absolute value has no physical meanining.
[...]
Now, in their calculation, in the result nothing depends on the derivative of ϕ, and so it must reduce to the flat space result.
http://realselfenergy.blogspot.com/2018/02/update-on-muon-g-2-story-of-debacle.html