Muon g-2 Result

The long awaited FNAL muon g-2 result was announced today, you can watch a video of the seminar here, look at the paper and a discussion of it at Physical Review Letters, or read stories from Natalie Wolchover at Quanta and Dennis Overbye at the New York Times. Tommaso Dorigo has an extensive discussion at his blog. In terms of the actual new result, it’s not very surprising: quite similar to the previous Brookhaven result (see here), with similar size uncertainties. It’s in some sense a confirmation of the Brookhaven result. If you combine the two you get a new, somewhat smaller uncertainty and ($a_\mu=\frac{1}{2}(g-2)$)
$$a_\mu(Exp)=116592061(41)×10^{−11}$$

The measurement uncertainties are largely statistical, and this is just using data from Run 1 of the experiment. They have accumulated a lot more data since Run 1, and once that is analyzed the FNAL experiment should be able to provide an experimental value with much lower uncertainty.

The big excitement over the g-2 experimental number has to do with it being in conflict (by 4.2 sigma now) with the Standard Model theoretical calculation, described here, which gives
$$a_\mu(Theory)=116591810(43)×10^{−11}$$
An actual discrepancy between the SM theory and experimental value would be quite exciting, indicating that something was missing from our understanding of fundamental particle physics.

The problem is that while the situation with the experimental value is pretty clear (and uncertainties should drop further in coming years as new data is analyzed), the theoretical calculation is a different story. It involves hard to calculate strong-interaction contributions, and the muon g-2 Theory Initiative number quoted above is not the full story. The issues involved are quite technical and I certainly lack the expertise to evaluate the competing claims. To find out more, I’d suggest watching the first talk from the FNAL seminar today, by Aida El-Khadra, who lays out the justification for the muon g-2 Theory Initiative number, but then looking at a new paper out today in Nature from the BMW collaboration. They have a competing calculation, which gives a number quite consistent with the experimental result:
$$a_\mu(BMW)=116591954(55)×10^{−11}$$

So, the situation today is that unfortunately we still don’t have a completely clear conflict between the SM and experiment. In future years the experimental result will get better, but the crucial question will be whether the theoretical situation can be clarified, resolving the current issue of two quite different competing theory values.

Update: Also recommended, as always: Jester’s take.

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41 Responses to Muon g-2 Result

  1. Mark Weitzman says:

    I wish the New York Times would have better science reporting – typical error:

    “That leads the factor g for the muon to be less than 2, hence the name of the experiment: Muon g-2.”

    Of course (g_mu-2)/2 from PDG website is about (10^-10) * 11659208.9± 5.4±3.3 and |g_mu|>2.

  2. Roger says:

    So BMW showed up to crash the party with the news that their paper is now published in Nature. Its not often we get that degree of excitement in particle physics.

    A lot of people worked on that result. They didn’t deserve their result day being hijacked the way it was.

    That said, I’d be interested in knowing the back story of what went on and why the BMW paper wasn’t included in the calculation that led to 4.2 sigma. Was there a prescribed procedure for deciding whether or not a calculation would be included or was this decided on the fly in response to the BMW work ?

    To be clear, I certainly don’t think that it was the intention of the g-2 collaboration and the Theory Initiative to put across a misleading estimate of the size of the data-theory discrepancy (and they not have done this at all, even inadvertently). However, given the events of today, there is an unfortunate impression of a stitch-up.

    For an outsider like me (I’m a collider physicist), today’s events were interesting from both a scientific and sociological perspective.

  3. Alessandro Strumia says:

    4.2σ anomaly with respect to one theoretical prediction, omitting the other theoretical prediction close to the measured g-2.

  4. Amitabh Lath says:

    Muons are having a moment.

    So what physics could be causing this 4.2 sigma (assuming it remains)? After the LHCb results there was a lot of chatter about leptoquarks, but I don’t see much here?

  5. Henry McFly says:

    Assuming that the muon g-2 Theory Initiative’s calculation is correct, what kind of BSM explanations are there to explain the discrepancy between the SM prediction and the experimental measurement?

  6. e says:

    Roger,

    The value for the leading-order HVP recommended by the g-2 Theory Initiative is that of the data-driven approach. The white paper notes that

    “For HVP, the current uncertainties in lattice calculations are too large to perform a similar average and the future confrontation of phenomenology and lattice QCD crucially depends on the outcome of forthcoming lattice studies. For this reason, we adopt [the data-driven evaluations of HVP] as our final estimate [..]”
    [Section 8, above (8.12)]

    From v1 of 2002.12347 to the paper published today, the quoted result for the LO HVP in units of 10^{-10} has shifted from 712.4(4.5) [v1, Feb ’20] to 708.7(5.3) [v2, Aug ’20 — after the white paper] to 707.5(5.5) [published], so the assessment that all lattice results need to be further scrutinized seems to have been warranted.

  7. Rollo Burgess says:

    The media are saying that these results can be taken in conjunction with the LHCb data also recently discussed to give an indication of something going on beyond the SM.

    I was a bit unsure about this.

    Is it correct that these 2 anomalies could be related and the the probability of BSM physics being present is higher than just the sum of either being real? Or is this not right?

  8. Mark says:

    I’m an experimentalist, but am very curious about the BMW 20 result. The theory seminar speaker implied, I recall, one should not take it seriously because it was not “data-driven”, unlike the 3 calculations that show the 4.2 sigma discrepancy. Is this a conclusion that is widely shared in the theory community?

    When I read the abstract of the BMW20 paper last night they imply their calculation is independent because it does not rely on the e+e- experimental results that the others use and position their calculation as a new independent result which resolves the discrepancy in that it agrees with the experimental measurement, but not the other theory estimates quoted. Reading the full paper today on the archive I don’t get the same impression that they claim that though and instead they talk about further calculations being needed to confirm or refute their result, thus implying its unresolved as to which calculations are the most correct.

  9. Roger says:

    “e”

    The argument for not including the lattice number is plausible but that’s not really the point.

    When it comes to stats, as an experimentalist, I’m taught to decide on the analysis strategy and to stick to it (unless this becomes impossible in practice) so that any conclusion I draw from the data will have a certain statistical integrity.

    The key question is whether the lattice number would have been omitted if it had agreed with the dispersion-based estimates. If it would have been omitted then fair enough. If it wouldn’t have been omitted then this further weakens any attempt to claim a 4.2 sigma discrepancy.

  10. Peter Woit says:

    Something that strikes me about the BMW vs. Theory Initiative story. At least one of the two is wrong, having made a significant mistake in their calculation/analysis. One would hope that both groups will look very carefully at wherever they are getting inconsistent numbers and focus on finding the error(s), whether their own or errors of the other group.

  11. André W-L says:

    Some comments on the theory discrepancy and lattice QCD (from a lattice QCD practitioner who does not work on g-2):

    1 – Even if the theoretical determination of the hadronic vacuum polarization (HVP) moves the theory prediction to match the experimentally measured muon g-2, this just shuffles the tension to other parts of the electroweak constraints (with similarly sized tension)
    https://arxiv.org/abs/2003.04886

    The only way to eliminate this global tension without invoking new physics, presently, is if the experimental value moves towards the current standard model theory prediction that is in 4.2-sigma tension with the experimental result.

    2 – The Theory White Paper (WP – arXiv:2006.04822) considered both lattice QCD determinations of the HVP as well as data-drive (dispersive) ones. Besides the BMW result, no other lattice QCD results have a precision comparable to the dispersive approaches. Therefore, they decided to not use a lattice QCD determination in the final estimate, until the lattice results have an uncertainty small enough to compete with the data-driven determination (as noted above).

    3 – Regarding BMW (arXiv:2002.12347) not being included in the the WP average, there was a well known deadline of 15th Oct. 2019 for papers to be included, if and only if they were published (not just arXiv’d). This deadline was relaxed to 31 March 2020, but is still too soon for the BMW paper to be included in the averaging procedure. See 3rd bullet on the 10th page of the arXiv version.

    4 – Regarding lattice QCD results as theory predictions, as Aida El-Khadra nicely summarized in her presentation, we do not base our lattice theory predictions on any single lattice QCD result, as they all contain systematic uncertainties which must be controlled and eliminated. For example, different choices of discretizing QCD lead to finite differences that are not universal and so the continuum limit must be obtained through an extrapolation (for many quantities, all discretization effects arise from higher dimensional “irrelevant” operators). But, there is no proof this works, it is just a very well-motivated expectation based on asymptotic freedom. Therefore, we rely upon multiple independent calculations to be performed as a cross check that these systematic effects are under control. This is very synonymous to the situation that we do not accept new physics results until multiple experiments can confirm the finding (as best we can).

    5 – Regarding the precision of the BMW result, as Aida also nicely summarized, the BMW result is so stochastically precise, the final result is dominated by the systematic uncertainties, which for HVP calculations, involve more complex systematics than in more typical calculations. This increases the importance of having multiple groups independently perform the computations since systematic uncertainties do not automatically improve with more statistics and computing power. It is not statistically very significant, but as a measure of the challenge with these systematics, the BMW value shifted 1-sigma over time (from first arXiv release to publication): 712(4) [v1] to 709(5) [v2] 708(5), based upon community “criticism” of how they handled some of the systematic uncertainties, not through additional computations etc. (as noted above).

    It is therefore premature to rely upon lattice QCD results to constrain the hadronic vacuum polarization contribution to g-2 with sufficient precision to make a statement about new physics or not. It is anticipated that by the time the higher-precision experimental results are available, the lattice results will be sufficiently precise and cross-checked to provide a constraining determination of the HVP and hadronic-light-by-light contributions.

  12. clayton says:

    André W-L — thanks for chiming in here, I for one consider your perspective to be informative 🙂

    You say in closing that “the lattice results will be sufficiently precise and cross-checked” by the time the experiment becomes more precise, which is in the next year or two as a rough timescale. That indicates that other lattice groups are already working on this ab initio calculation. Do you know which groups are working on this?

  13. WTW says:

    In re. “criticisms” causing updates of their original paper, the BMW group has actively solicited such evaluations and critiques from other experts, including from the g-2 Theory Initiative team. While there may inevitably be some level of competitiveness between groups, in general they seem to have tried to use any available resources to improve the results. Hopefully such collaboration will continue, to help reach a consensus from lattice QCD calculations.

    As to many people noting the LHCb report of possible EW lepton flavour violation in B-hadron decays as somehow further validation of these g-2 results, Tommaso Dorigo’s comments seem extremely timely.

    It should also be noted that a recent ATLAS report “sets a new constraint on lepton-flavour-violating effects in weak interactions, searching for Z-boson decays… ” — https://arxiv.org/abs/2010.02566 And as to some new ultra-light particle being the culprit affecting the anomaly in the muon’s anomalous magnetic moment, there is a recent paper showing no detection of any Axion-like particle having any EM interaction within a network of synchronized ultra-precision optical clocks, down to a level of ~ 8 x 10^-18 second. (https://www.nature.com/articles/s41586-021-03253-4.epdf) Likewise, CERN’s Baryon/Anti-Baryon Symmetry Experiment (BASE) has recently published new limits on (non) detection of coupling of Axion-like Particles and photons, using a cryogenic Penning trap: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.126.041301#fulltext

    Throw in the ANAIS experiment failing to replicate DAMA’s problematic detection of dark matter particles after 3 years of data, and it seems that the tactic of just adding some more particles to the mix might be getting more constrained of late.

  14. André W-L says:

    It was pointed out to me that in their latest version, BMW addresses the electroweak tension pointed out by Crivellin et al – page 82 of the BMW arXiv is section 26, addressing this point. They present a detailed argument that the global electroweak tension (if g-2 theory moves to be consistent with experiment) is more like 2.4 sigma (than ~3-4).

  15. Roger says:

    André W-L
    Thanks a lot for your comments. I’m pleased that a rule wasn’t made up to exclude BMW.

    Beyond this, while I accept and respect your experience with lattice QCD, I found your last remark a bit baffling “It is therefore premature to rely upon lattice QCD results to constrain the hadronic vacuum polarization contribution to g-2 with sufficient precision to make a statement about new physics or not.”

    Nobody is suggesting that lattice QCD be used to make a statement about new physics. There is simply no statement to be made with the current precision of the data and theory calculations. Instead we are in a situation in which a calculation has been made which has been peer-reviewed and has now appeared in Nature and this should surely now merit being shown next to the data even if the experiments would also feel the need for a caveat. Perhaps two plots could have been prepared. I saw the CERN seminar of the g-2 results (theory talk + experiment talk) and felt the experiment was making a mistake by not showing the new calculation in its standard publicity plot since it gave the impression of being in bed with its own theorists , which is very unfortunate. Perhaps that was ok – doing the right thing isn’t always easy, if it was the right thing. However, whether the experiment and Theory Initative like it or not, people are making their own plots. It will be difficult for the experiment to keep up its standard publicity plot with its “approved” prediction given that Nature has just published a plot that is “unapproved” for comparison. Politically they need to think carefully about how to proceed.

    Regarding the tension moving elsewhere, this is certainly interesting but should have nothing to do with a decision on whether to show a theory prediction for a specific observable in comparison to a measurement of that observable. Indeed,the whole business of much of particle physics these days is looking for tensions and trying to interpret them.

  16. Martin H. says:

    Dear Andre,

    thanks for your comments! If I may add to your point about the electroweak fit: in our paper we presented several scenarios how big the tension would be depending on the energy range in which the changes in the e+e- cross section occur. In response, BMW provided arguments that these changes will be concentrated at low energies, in which case the tension in the electroweak fit only increases moderately.

    However, this then implies that the low-energy cross sections would have to be changed by a lot, see
    https://arxiv.org/abs/2006.12666
    https://arxiv.org/abs/2010.07943
    so unless the Fermilab number changed, the tensions would be moved somewhere else.

  17. Joseph Conlon says:

    Whether it’s in Nature (the journal) is neither here nor there. Glossy covers and fancy editorials do not correct science make.

  18. André W-L says:

    Martin H. – thank you for the update! (those who don’t know, unless Martin H has conspicuously the same initials – he is one of the leading experts on the data-driven dispersive methods for g-2 and also an expert on the electroweak fits).

    clayton – if you look at Table 8 (page 80) of the theory white paper (https://arxiv.org/abs/2006.04822), the top 3 panels are results from various groups working on the problem. It is one of the most intensely studied problems with lattice QCD these days, so there are a large number of groups working on it. Figure 44, on the next page, gives you a visual representation of the same results.

    WTW – I did not mean to imply BMW were not looking for feedback. My point is that the systematic uncertainties are sufficiently challenging that even one of the worlds leading lattice QCD groups did not “get it right” the first time.

    Roger – BMW, in their paper, claim there is no need for new physics, the “or not” part of my statement about it being premature to rely upon lattice QCD results.

    If I were one of the co-authors, I would make a similar claim as I’m certain they put a tremendous amount of effort in trying to ensure everything was controlled, and are therefore confident that within the uncertainties they quote, they have the correct Standard Model prediction. As someone without “skin in the game”, I see the value in “community consensus”, meaning more LQCD results need to be performed with uncertainties similar to BMW before we (the lattice and broader community) can have higher confidence.

    Take it another way – if the LQCD results do not agree with the data-driven dispersive results for the HVP, at a similar level of tension that is currently quoted for g-2, that would be evidence for new physics (or a larger than expected systematic uncertainty) since LQCD is pure Standard Model and the data-driven analysis is nature.

    I also agree that it is unfortunate, and a mistake not to show the BMW results on the same, or at least a companion plot. Their result is among the highest quality lattice QCD results available, if not the highest quality.

  19. SG says:

    @Martin H.: If all the changes are indeed below 1 GeV what is the significance of the discrepancy between the e+e- cross-section measurements and BMW ? From the lower panel of Fig 30 of https://arxiv.org/pdf/2002.12347.pdf (BMW) it seems that the first bin has a discrepancy that is more than 3 sigma. Am I reading this correctly?

  20. Martin H. says:

    SG: not quite. This plot shows bins in the space-like region, which you cannot directly compare to (time-like) e+e- cross sections. But the emerging picture, see Fig. 8 in
    https://arxiv.org/abs/2010.07943
    is similar: the e+e- data would have to be wrong way outside their quoted uncertainties. As for the significance, the global tension between e+e- and BMW (v3) is now 2.1sigma, but since the changes need to be concentrated at low energies, the local tension is bigger than that. The exact significance depends on energy range and data compilation, but to give you some idea: for a partial (“window”) result the BMW paper quotes a tension of 3.7sigma, see caption of Fig. 4 in
    https://www.nature.com/articles/s41586-021-03418-1
    Such detailed comparisons, especially with forthcoming new data and results from other lattice collaborations, should allow for some clarification in the near future.

  21. Pingback: Muon g−2 del Fermilab incrementa a 4.2 sigmas la desviación en el momento magnético anómalo del muón - La Ciencia de la Mula Francis

  22. SG says:

    @Martin H– Thanks a lot for your answer. So is it correct to say that BMW either has a 3 sigma(-ish) tension with electroweak precision tests or if the changes to e+e- cross section are concentrated at low energies, it has a 3 sigma (-ish) tension with e+e- data ? So whichever way we see it there is a significant tension between theory and experiment.

  23. cssm says:

    In the theoretical prediction, I noticed, components from individual gauge sector are mentioned. I was just wondering what happened to the cross terms from gauge mixing. Also if there is some new physics, shouldn’t that be already present in the lepton-hadron cross-section data? In that case iterative approach and convergence test should also be incorporated.

  24. Anonyrat says:

    Experimentalists, please explain how this is disentangled: Bhabha scattering is used to measure the integrated luminosity in electron-positron colliders, which is a key to measurement of the hadron vacuum polarization. But hadron vacuum polarization is apparently necessary to calculate Bhabha scattering to the required precision.

    Recently (2020), a newly computed Bhabha scattering cross-section, reducing it by 0.048% resolved a 2-decade-old 2-sigma discrepancy of LEP data with the Standard Model.
    https://www.sciencedirect.com/science/article/pii/S0370269320301234

    If BMW is correct, there is a correction needed to the Bhabha scattering cross-section, and to the measured e+e- data. How big is it, and will its direction go toward resolving the tension mentioned in the comments above?

  25. someone says:

    @SG: No, BMW is in 3sigma tension with with chiPT predictions from e+-e-. The direct lattice prediction effort of e+-e- is just starting.

  26. Pingback: Wobble in the Standard Model | Gödel's Lost Letter and P=NP

  27. SG says:

    @someone: I did not mean comparison with lattice but experimental data on e+e-. Fig. 8 in https://arxiv.org/abs/2010.07943 , pointed out by Martin H., is a comparison between real data and the shift in e+e- cross-section implied by BMW. This is, of course, just a restatement of the tension between the BMW and data-driven value for the HVP contribution but now focussing on the <1 GeV region for the e+e- data.

  28. Luca says:

    Honestly I find the work on the non-perturbative behavior of the standard model even more exciting than potential new particles. From this situation is clear that we do not understand the model well enough to make some predictions, or at least that a substantial amount of work needs to be done here. New physics is where physicists work and it is perhaps time to give more glamour to the hard work of connecting our best theory to the reality we have instead of always looking for jet another reality. High energy physics is alive and well.

  29. Will Sawin says:

    Exciting!

    Are there other interesting situations where the discrepancies between different attempts to calculate the standard model prediction differ by multiple sigma, so that we aren’t necessarily able to tell whether an experiment is evidence for a failure of the standard model?

    I am tempted to ask “Could we have already observed, in some experiment, the effects of beyond-the-standard-model physics, but not have noticed it because our calculation of the standard model prediction is different from the correct standard model prediction but not from the experimental data?” but I realize this is probably not a very helpful question, in part do the the fact that “the correct standard model prediction” might not be a mathematically well-defined concept outside of the existing calculational techniques.

  30. Peter Woit says:

    Will Sawin,
    The calculational problems with the Standard Model are pretty much just with the strong interactions (there are problems of principle with perturbation expansions for weak and electromagnetic interactions being only asymptotic expansions, but these issues only show up at too high orders of the expansion to matter).

    The problems with strong interaction calculations are very serious: typically the best you can do is a lattice gauge theory calculation (Monte-Carlo simulation of a path integral), and there you’re lucky to get a calculation to 1%. The only reason the g-2 calculation is so accurate is that the strongly interacting contributions to the calculation are a very small part.

    So, for most things involving strongly interacting particles, the theoretical calculation has much larger errors than the experimental measurement (e.g. the masses of such particles). In general the much more precise measurements are consistent with the low precision calculations. An example where there’s a possible problem involves spin and the proton, see eg. “Proton spin crisis” on Wikipedia or someplace better. But there, the theoretical understanding of how to calculate things is so bad that it’s likely that the problems are just with the calculations, not problems with the underlying theory.

  31. Luca says:

    Hi Peter,

    “But there, the theoretical understanding of how to calculate things is so bad that it’s likely that the problems are just with the calculations, not problems with the underlying theory.”

    Can we actually say to have a theory then? We have an asymptotic perturbative theory for sure, but does this fully constraints the non-perturbative behavior, especially concerning bound states? To my naïve eye, it is more a rough set of ideas than a full theory.
    I have the impression that the “beyond the standard model” things we should look for are protons, atoms and tables and chairs, not superstrings. It puzzles me why we have a fundamental theory that is utterly incapable of predicting the behavior of something as elementary as a hydrogen atom and jet we think that all that physics is boring and already understood. Where does all this confidence comes from? Who decided that the perturbative regime of a theory is its only fundamental aspect? Am I missing something? I mean I know that the people who actually work in these fields takes these problems very seriously, but jet the general consensus seems to be that there isn’t new physics to be understood here. Why? Because all the physics is implicitly in the Lagrangian? But as far as I know we do not even a proper way to make sense of the Lagrangian theory in the non-perturbative regime. Do we have strong evidence to suggest that the perturbative behavior can be continued in a unique way to the non-perturbative regime without introducing anything new?

  32. Peter Woit says:

    Luca,
    What we have for QCD is not a “rough set of ideas”, but a well-defined cut-off theory, the lattice theory, so precisely defined that you can write code and do computer calculations. What we don’t have is rigorous or calculational control of the continuum limit, there all we have is conjectures that there’s a well-defined limit with certain properties, together with all the computational evidence being consistent with those conjectures. To convince yourself this is more than a “rough set of ideas”, look for instance at
    https://arxiv.org/abs/1203.1204
    in particular the hadron spectrum in figure 2.

    The one place where there might be an actual hole in our ability to write down a precise non-perturbative theory is not for the strong interactions, but for the weak interactions, where the gauge interactions are chiral. That’s a complicated story, and there since the interactions are weak, perturbation theory gives all we need to compare to experiment.

  33. Luca says:

    Peter,

    Thank you for the clarification. I am aware that lattice QCD is a well defined theory. However, it’s value is as a proxy for an underlying continuous limit which is truly the theory we would like to have. I understand that the conjecture of a unique and well defined limit is supported by evidence, but is as far as I understand this is limit to specific settings. Your discussion about the proton spin suggests that in many, perhaps most, situations we do not have idea how to approach this limit. Is it just a problem of limited computational resources? Are you confident that a correct MC estimate of the proton spin composition could be obtained if we just had bigger computers? Or it will require novel ideas? If the latter is true then we cannot say to have a complete theory.

  34. Mark Weitzman says:

    Hi Peter:

    Have all the fundamental difficulties of lattice gauge theory, such as fermion doubling problem been resolved, or is it simply the case that workarounds have been found for the current calculations. I have always felt that a physical theory is not really well defined unless it can be simulated on a computer in principle.

  35. Peter Woit says:

    Luca/Mark Weitzman,
    I’m not that well-informed about the latest state of the technology of QCD calculations with fermions, but not aware of any problems of principle, i.e. that can’t be solved with more computational power. This is different for the chiral weak interactions, where there may very well be remaining problems of principle.

    These issues are actually related to the work I’ve been doing with twistors, going way back. Early in my career I worked on pure lattice gauge theory calculations, trying to take into account the role of topology, so needed to understand well the geometry of what was going on. I was very interested in the fact that the geometry was straightforward for pure gauge theory, but that when you introduced spinor fields, some sort of new geometry was coming into play and it was unclear both what it might be and whether lattice difficulties were due to not taking spinor geometry into account.

    One way of thinking about what’s going on is that there’s a natural lattice version of fermions if they’re differential form fields, with 0-forms given by fields on the sites, 1-forms on the links, 2-forms on the plaquettes, etc. These are “Kahler-Dirac” fermions. The problem is that they’re not spinor-valued fields, but anti-symmetric tensor valued fields. Spinors are in some sense “square-roots” of the anti-symmetric tensors, but to construct such a square root, you need to choose a complex structure on R^4. Then spinors are (up to a phase factor) either the holomorphic or anti-holomorphic differential forms.

    There is an S^2 of appropriate choices for the complex structure on the tangent space at each point in R^4, and a point in the projective twistor space is just a point in R^4 together with a choice of the complex structure (a point in the fiber S^2). So, on projective twistor space, the spinor fields have a straightforward geometrical interpretation (although in terms of holomorphic geometry).

    I haven’t thought seriously about this in a long time, but back when I was thinking about it, I didn’t see any obvious way to use this to solve any of the problems of lattice fermion fields. Some of the things I’ve started thinking about now (how to relate chiral Yang-Mills theory on R^4 to holomorphic theories on projective twistor space) might though give some new ideas on the lattice stuff, we’ll see…

  36. Andre W-L says:

    Luca, Mark, Peter, Will,

    A few more comments related to your comments and questions.

    1. The fermion doubling problem is practically solved in several ways now, and each method has been used to make post-dictions of well known quantities that agree with the experimental numbers and each other (after the continuum limit extrapolation and physical quark mass extrapolation/interpolation).

    2. It is now common for quantities of mesons (pions, kaons, charmed and bottom mesons) that lattice QCD results are obtained at the sub-percent level of precision with a fully quantified uncertainty budget. This level of precision requires the inclusion of QED and Weak interactions to reduce the uncertainty from the ~1% level down to the 0.2% level (the target precision for many quantities). This is starting to become the case for simple properties of nucleons, which have now started to reach the 1% level of precision.

    3. The “Proton spin crisis” is a red herring. In units of h-bar, we know the spin is exactly 1/2. What is complicated is to determine how the various constituents of the nucleon contribute to the spin. Now, there are several complete calculations including the quark contribution and gluon contribution that obtains the total spin. For example, this result obtains the total spin with a 15% uncertainty:
    https://arxiv.org/abs/2003.08486

    The “spin crisis” came because, based upon the quark model (which is an extremely naive and in many cases surprisingly successful description of protons, neutrons etc), there was an expectation that the spin of the proton would come almost entirely from the “valence” quarks of the proton (which has up, up, down valence quarks). It was found experimentally that the total contribution from these valence quarks fell far short of adding up to 1/2. But, we know these valence quarks are confined (very strongly) by gluons, and so it is not surprising at all that there is substantial contributions to the spin from the gluons and the orbital angular momentum contributions from the proton’s constituents.

    4. There are several interesting puzzles related to strong interactions. There is the more famous “proton radius” puzzle in which the size of the proton when measured from electrons or muons disagreed by as much as 7-sigma. This puzzle is now expected by most in the community to have come from systematic uncertainties in the experimental determinations which are now beginning to be resolved.

    There is the “neutron lifetime puzzle” in which the lifetime of the neutron when measured in a “beam” and “bottle” experiment disagree at the 4-sigma level. The beam experiment only measures Standard Model decay modes and the bottle measurement simply counts the number of neutrons vs time and so measures all effects. This generated a lot of interest that there might be new, relatively light, new physics decay modes. Many expect this discrepancy is driven by a systematic uncertainty.

    In both cases, lattice QCD can be used in principle to help resolve these puzzles, but the challenge of obtaining a sufficient precision to be useful has prevented the calculations from contributing so far. Calculations of nucleons generically require exponentially more stochastic samples to obtain the same relative precision as for mesonic quantities (this is a well understood Monte-Carlo sampling problem related to sign-problems and stochastic sampling of fermions).

    5. There are other exciting examples in “nuclear physics” where lattice QCD can be used to compute quantities with a 20% uncertainty which will be useful. For example, the search for neutrinoless double beta decay of large nuclei requires an understanding of short distance contributions to how two-neutrons can simultaneously beta-decay to two protons, assuming the neutrino is Majorana in nature (it is at least in part is its own anti-particle – the only fermion we know of for which this is possible, as it does not cary any Standard Model “charge”). While we can never use lattice QCD to calculate the decay of a large nucleus, we can compute the basic neutron-neutron to proton-proton amplitude for various models of Majorana neutrino exchange between two nucleons. Knowing the QCD contribution to this amplitude with a 20% uncertainty is more than sufficient to help the field. Unlike many other processes, there is no experiment that can measure this neutron-neutron to proton-proton process because if it occurs, it is only in a big nucleus, so isolating the underlying amplitude is not possible.

    Lattice QCD can also be used to constrain the interactions of strange-matter (hyperons) with neutrons in principle much better than experiment. This is because the hyperons rapidly decays through Weak interactions and so the data set is very limited (there are now several international experiments that will significantly improve the data set). In lattice QCD, we are free to “turn off” the Weak interactions so the hyperons are stable. What is interesting is that an improved understanding of hyperon-neutron forces, and triple-neutron forces, may improve our ability to predict the nuclear equation of state which is relevant for understanding neutron stars and how they might merge in binary systems, and if supernovae will result in a neutron star or black hole.

    We (the community) are optimistic that as the exa-scale supercomputers come online (starting next year), we may get enough computing power to reliably compute these processes. I could go on, but I’ve already written a “bit” more than I intended.

  37. Gianni says:

    Andre W-L,
    I am very grateful for the extensive and stimulating, though short, list of issues that the lattice community is able to investigate either with great precision, good precision, or an acceptable one.
    I would like to ask your opinion on the challenge presented by lattice calculations of the parton distribution functions, which are so relevant in the analysis of HEP experiments. I know that there efforts along this directions.

  38. André W-L says:

    Gianni,

    That is a very exciting topic these days also. Until a few years ago, the lattice community had come to the conclusion that lattice QCD could be used to only compute moments of the parton distribution functions. A few years ago now, the field was re-vitalized by a proposal of Xiangdong Ji, that, instead of computing these structure functions directly, one could approach the problem by computing a related quantity (quasi-parton distribution functions) that in the limit of large nucleon momentum, become the parton distribution functions. It took a few years for the community to sort out the most essential details and now great progress is being made (the critical difficulty was understanding how to renormalize these objects). There are now at least three variants of the idea being used in the community, each with their own advantages and disadvantages, which is also great as it provides multiple ways to get at the same quantities (perhaps in different kinematic regimes). There are at least as many lattice groups working on this problem as those who are working on g-2 calculations. The community is now pushing the idea to compute the transverse momentum dependent distribution functions with some promising ideas being pursued.

    One of the biggest challenges in these computations is related to the same signal-to-noise problem mentioned above. As you look at a boosted nucleon, the noise issue becomes more severe because, the boosted nucleon has more energy, and so the correlation functions decays more rapidly in Euclidean time, while the noise of the correlation function has an energy scale that does not increase as the momentum is increased, and thus the mass gap which dictates the degredation of the signal-to-noise becomes larger, causing a more rapid loss of the signal. People are actively working on clever ideas to circumvent this problem (in addition to using bigger/faster computers). These calculations are well on track to help with HEP and NP problems (the planned electron-ion-collider).

    There are already collaborations between lattice groups and those who work on the structure functions to begin figuring out how to use lattice QCD results as inputs to help determine and constrain them, so I anticipate the next several years will be very fruitful for this type of research.

  39. Gianni says:

    André W-L ,
    thank you very much for the very nice overview. Let us hope to see soon reliable calculations also on parton distribution distributions and TMDs! I heard that the issue of the renormalization within the Ji framework is highly non trivial, but as you pointed out the sinergy between different approaches should lead us to the goal.

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