The Triumphant Standard Model

The Standard Model Is Triumphant

The most notable apparently observational deviation from the Standard Model, apparent lepton universality violations in B meson decays, have disappeared with better data and more rigorous analysis. 

This is a capstone to the complete triumph of the Standard Model in modern high energy physics despite the myriad beyond the Standard Model (BSM) physics proposals made in papers filed on arXiv every business day.

If there are no lepton universality violation, as the new LHCb results would tend to show, 

AND there is no muon g-2 anomaly, as new lattice computations of the SM prediction for muon g-2 are increasingly showing, 

AND the Higgs boson is just the SM Higgs boson (which is confirmed ever more tightly every few months by new LHC data),

AND there are no non-standard neutrino interactions or sterile neutrinos (which experiments are tending to show and which I won't discuss again in this post although I have blogged about this many times), 

THEN, there is less and less room for BSM physics at energy scales that can be tested experimentally at current colliders or next generation colliders (i.e. up to the hundreds of TeV energy scale).

The State of the Muon g-2 "Anomaly"

The g-2 discrepancy could simply be a problem with the theory prediction (and I think that's the most likely explanation). There are two different methods for the calculation, one is in agreement with the experimental values.

Muon g-2 (i.e. the magnetic moment of the muon, minus two, and divided by two) is the sum of an electromagnetic contribution (QED) (the lion's share with negligible uncertainty), a weak force contribution (a modest contribution with a very small uncertainty), and a strong force (QCD) contribution which is small but not insignificant given current experimental precision in measuring muon g-2 that has a comparatively huge uncertainty.

The consensus value of the QED plus weak force contribution to SM prediction for muon g-2 in units of 10^-11 is:

116,584,872.53 ± 1.01 (with most of the uncertainty coming from the weak force contribution)

In the Theory Initiative analysis the QCD amount is 6937(44) which is broken out into two parts: hadronic vacuum polarization (HVP) = 6845(40), which is a 0.6% relative error, and hadronic light by light (HLbL) = 98(18) which is about a 20% relative error.

The calculation from the Theory Initiative of the SM prediction (which mixed experimental data for parts of the HVP calculation and lattice computations for other parts of the HVP calculations) is in tension with the experimental measurement.

But, the first principles lattice calculation of the HVP part of the SM prediction for muon g-2 by the BMW group is consistent with the experimental results (see below for the actual values times 10^-11):

Fermilab (2021): 116,592,040(54)

Brookhaven's E821 (2006): 116,592,089(63)

Combined measurement: 116,592,061(41)

Theory Initiative calculation: 116,591,810(43)

BMW calculation: 116,591,954(55)

Combined measurement - Theory Initiative: 251(59)

Combined measurement - BMW: 107(69)

Essentially all of the subsequent work has confirmed the BMW calculation in parts that have been replicated, especially the HVP "window", which is a key subcomponent of the overall HVP contribution that is somewhat easier to calculate. Some of these papers have even narrowed down to some extent where the discrepancy between the two SM predictions is coming from. See, e.g., https://arxiv.org/abs/2212.09340 and https://arxiv.org/abs/2212.10490

In addition, on the day that the new muon g-2 experimental results was released a new calculation of the hadronic light by light contribution to the muon g-2 calculation was also released on arXiv which doesn't seem to be part of the BMW calculation. This increases the contribution from that component from 92(18) x 10^-11 to 106.8(14.7) x 10^-11. This boost of 14.8 in the overall QCD component isn't as big as the BMW HVP calculation's impact on it, but the two combined narrow the gap even more.

With these two SM calculation refinements the discrepancy between the combined measurement and the BMW plus new HLbL prediction is about 85(68) x 10^-11, so barely more than a 1 sigma difference.

Why care?

Because muon g-2 is an indirect global measure of the consistency of the SM with experiment that is sensitive to new or different particles and/or forces at scales into the TeV to tens of TeVs (or more if the deviation from the SM is really strong) scale, because all three SM forces and all SM particle parameters contribute to it to some extent.

If the SM in consistent with experiment at the parts per billion or ten billion level, then there is basically no room for BSM physics that don't cancel out in the muon g-2 calculation at the energy scales of the next generation collider.

For example, it is basically impossible to have SUSY with 1-10 TeV scale sparticles without tweaking muon g-2. Likewise, adding leptoquarks to the SM, which have been a popular BSM physics explanation for hints of lepton universality violations (which now seem to be basically ruled out) should also tweak muon g-2.

What do we know about how strong the fit of what we observe to the SM Higgs is?

Scientists haven't established beyond all doubt that the Higgs boson we have seen is the SM Higgs boson yet, but every few months since it has been discovered the constraints on differences from the SM Higgs have gotten smaller and more restricted. The data has also ruled out many hypotheses for additional Higgs bosons.

There is basically no data that is contrary to the predictions of the SM Higgs hypothesis made about 50 years ago (subject to determining its mass), and for a given Higgs boson mass the properties of the SM Higgs boson are completely predetermined with no wiggle room at all down to parts per ten million or better.

The global average value for the mass of the Higgs boson is currently 125.25±0.17 GeV, a relative accuracy of about 1.4 parts per thousand.

There is also basically no data strongly suggesting one or more additional BSM Higgs bosons (although there is a bit of an anomaly at 96 GeV), even though BSM Higgs bosons aren't directly ruled out yet above the hundreds of GeVs. BSM Higgs bosons are also allowed in pockets of allowed parameter spaces at lower masses if the properties of the hypothetical particles are just right. For example, new Higgs bosons with a charge of ± 2 are ruled out at masses up to about 900 GeV, and so are many other heavy Higgs boson hypotheses. Indirect constraints also greatly limit the parameter space of BSM Higgs bosons unless they have precisely the right properties (which turn out to be not intuitively plausible or well-motivated theoretically).

The data strongly favor the characterization of the observed Higgs boson as a spin-0 particle, just like the SM Higgs boson, and strongly disfavors any other value of spin for it.

The data is fully consistent at the 0.6 sigma level with an even parity SM Higgs boson, see here, while the pure CP-odd Higgs boson hypothesis is disfavored at a level of 3.4 standard deviations. In other words, the likelihood that the Higgs boson is not pure CP-odd is about 99.9663%.

A mix of a CP-odd Higgs boson and a CP-even Higgs boson of the same mass is (of course) harder to rule out as strongly, particularly if the mix is not equal somehow and the actual mix is more CP-even than CP-odd. There isn't a lot of precedent for those kinds of uneven mixings, however, in hadron physics (i.e., the physics of composite QCD bound particles), for example.

Eight of the nine Higgs boson decay channels theoretically predicted to be most common in a SM Higgs of about 125 GeV have been detected. Those channels, ranked by branching fraction are:

b-quark pairs, 57.7% (observed)

W boson pairs, 21.5% (observed)

gluon pairs, 8.57%

tau-lepton pairs, 6.27% (observed)

c-quark pairs, 2.89% (observed May 2022)

Z boson pairs, 2.62% (observed)

photon pairs, 0.227% (observed)

Z boson and a photon, 0.153% (observed April 2022)

muon pairs, 0.021 8% (observed)

electron-positron pairs, 0.000 000 5%

All predicted Higgs boson decay channels, except gluon pairs, with a branching fraction of one part per 5000 or more have been detected.

Decays to gluon pairs are much harder to discern because the hadrons they form as they "decay" are hard to distinguish from other background processes that give rise to similar hadrons to those from gluon pairs at high frequencies. Even figuring out what the gluon pair decays should look like theoretically due to QCD physics, so that the observations from colliders can be compared to this prediction, is very challenging.

The total adds 99.9518005% rather than to 100% due to rounding errors, and due to omitted low probability decays including strange quark pairs (a bit less likely than muon pairs), down quark pairs (slightly more likely than electron-positron pairs), up quark pairs (slightly more likely than electron positron pairs), and asymmetric boson pairs other than Z-photon decays (also more rare than muon pairs).

The Higgs boson doesn't decay to top quarks, but the measured top quark coupling is within 10% of the SM predicted value in a measurement with an 18% uncertainty at one sigma in one kind of measurement, and within 1.5 sigma of the predicted value using another less precise kind of measurement.

The Particle Data Group summarizes the strength of some of the measured Higgs boson couplings relative to the predicted values for the measured Higgs boson mass, and each of these channels is a reasonably good fit relative to the measured uncertainty in its branching fraction.

Combined Final States = 1.13±0.06

W W∗= 1.19±0.12

Z Z∗= 1.01±0.07

γγ= 1.10±0.07

bb= 0.98±0.12

μ+μ−= 1.19±0.34

τ+τ−= 1.15+0.16−0.15

ttH0Production = 1.10±0.18

tH0production = 6±4

The PDG data cited above predates the cc decay and Zγ channel discovery made this past spring, so I've omitted those from the list above in favor of the data from the papers discovering the new channels.

One of these papers shows that the branching fraction in the Zγ channel relative to the SM expectation is μ=2.4±0.9. The ratio of branching fractions B(H→Zγ)/B(H→γγ) is measured to be 1.5+0.7−0.6, which agrees with the standard model prediction of 0.69 ± 0.04 at the 1.5 standard deviation level. 

The branching fraction of the cc channel isn't very precisely known yet, but isn't more than 14 times the SM prediction at the 95% confidence level.

The Higgs boson self-coupling is observationally constrained to be not more than about ten times stronger than the SM expected value, although it could be weaker than the SM predicted value. But the crude observations of its self-coupling are entirely consistent with the SM expected value so far. This isn't a very tight constraint, but it does rule out wild deviations from the SM paradigm.

The width of the Higgs boson (equivalently, its mean lifetime) is consistent to the best possible measurements with the theoretical SM prediction for the measured mass. The full width Higgs boson width Γ is 3.2+2.8−2.2MeV, assuming equal on-shell and off-shell effective couplings (which is a quite weak assumption). The predicted value for a 125 GeV Higgs boson is about 4 MeV.

There are really no well motivated hypotheses for a Higgs boson with properties different from the SM Higgs boson that could fit the observations to date this well.

For a particle that has only been confirmed to exist for ten and a half years, that's a pretty good set of fits. And, the constraints on deviations from the SM Higgs boson's properties have grown at least a little tighter every year since its discovery announced on July 4, 2012.

Higgs, W, and Z boson properties as constraints on BSM physics

This reasonably good fit of the observed properties of the Higgs boson to the properties it is predicted to have in SM at its measured mass is especially notable because the decay properties and couplings of the Higgs boson, like muon g-2, are good global tests of the SM, although not as comprehensive muon g-2, and not extending to BSM phenomena in excess of about 62.5 GeV (half the Higgs boson mass), which is a much lower threshold than the muon g-2 indirect exclusion which is in the TeVs.

Any BSM particle that couples to the Higgs boson in proportion to its rest mass, as the SM Higgs boson is predicted to do, with a mass between about 1 GeV and 62.5 GeV would have thrown off the branching fractions of the Higgs boson that have been observed to date dramatically. On the other hand, a new BSM massive fundamental particle that coupled to the Higgs boson in proportion to its rest mass with a mass of less than 20 MeV would not discernibly change the properties of the Higgs boson observed to date at all.

All quarks, charged leptons, and massive fundamental bosons in the Standard Model get their mass from the Higgs mechanism and couple to the Higgs boson (the source of the neutrino masses is unknown at this time), so it would be surprising to see some new massive fundamental particle that got its mass in some other manner.

In the same way, W and Z boson decays are sufficiently close to the SM predicted values that we can be confident that there are no particles that couple to the weak force with the strength that SM particle that do so, at any rest mass whatsoever from 0 to 45 GeV.

Incidentally, all known massive fundamental particles in the SM (quarks, charged leptons, neutrinos, W bosons, Z bosons, and Higgs bosons) couple to the weak force with the same "weak force charge" strength, and none of the zero rest mass fundamental particles in the SM (i.e. photons and gluons) couple directly to the weak force in the SM.

The number of SM "left handed" neutrinos that exist, for example, must be exactly three in the mass range from 0 to 45,000,000,000 eV. We know that none of the SM neutrinos can have an absolute mass of more than about 1 eV from direct measurements of lightest neutrino mass together with neutrino oscillation data. Indirect cosmology limits combined with neutrino oscillation based mass differences putting the upper limit on the mass of the most massive neutrino eigenstate closer to 0.07 eV at 95% confidence.

There are no good theoretical motivations for a hypothetical fourth generation Standard Model neutrino to be so profoundly more massive than neutrinos in the three known generations of Standard Model fermions. This is why searches for BSM neutrinos almost entirely focuses on new "sterile" a.k.a. "right handed" neutrinos.

And, since mathematical consistency in the SM calls for generations of new fermions to always include an up-type quark, a down-type quark, a charged lepton, and a neutrino, the non-existence of a SM left-handed neutrino at masses up to 45 GeV pretty much rules out the possibility that any fourth generation SM fermions exist.