*Some technical details on particle physics today…*

Papers are pouring out of particle theorists’ offices regarding the latest significant challenge to the Standard Model, namely the W boson mass coming in about 0.1% higher than expected in a measurement carried out by the Tevatron experiment CDF. *(See here and here for earlier posts on the topic.)* Let’s assume today that the measurement is correct, though possibly a little over-stated. Is there any reasonable extension to the Standard Model that could lead to such a shift without coming into conflict with previous experiments? Or does explaining the experiment require convoluted ideas in which various effects have to cancel in order to be acceptable with existing experiments?

The most common way (judging from the papers that have appeared this week) for theoretical physicists to try to explain a shift in the W boson mass, and many other details of the Standard Model, is to add new fields and particles which affect the W boson through “quantum fluctuations” — disturbances in the fields which aren’t particles (though they are often called “virtual particles” anyway.) Famous theories which can do this, in principle at least, include models with extra sterile and/or extra Standard Model-like Higgs bosons, minimal supersymmetry, and certain “compositeness models” in which the Higgs boson is understood not to be an elementary particle, but instead to be composite, somewhat like a proton. The problem with this approach is that it generally predicts new particles whose masses are not much bigger than the Higgs boson itself, so small that one would expect they would have been seen at the Large Hadron Collider (LHC). There are plenty of ways around this problem, though they require some finesse.

However, there are a couple of easy, well-known, and very reasonable ways to shift the W boson’s mass without requiring any quantum fluctuations, without leading to low mass particles, and without impacting much else in the Standard Model. Perhaps the simplest is to add another Higgs field, of a type called a “triplet”: let’s call it “T”, to distinguish it from the Standard Model’s own Higgs field “H”. The T field leads to a triplet of new Higgs bosons, one (T^{0}) which is electrically neutral, one (T^{+}) which has electric charge +1 (like a proton), and one (T^{–}) which has electric charge -1 (like a electron). The T^{+} and T^{–} have the same mass, which typically differs only slightly from the mass of the T^{0}.

Triplet Higgs fields have been discussed for decades, going back to the late 1960s or early 1970s. As for combining the Standard Model with a T field, that too must go back to the 1970s; I know of papers from the mid-1980s whose authors knew everything I’m about to tell you. *(A few of the theory papers [1,2,3,4] that appeared on Monday night, and perhaps others that I missed, pointed out parts of what I’m going to say here; I’m sure there will be more of them as the week goes by.)*

How does this idea work? First, a quick review. The H field, uniquely among the Standard Model’s fields, has a non-zero, constant and uniform value in empty space (a so-called “vacuum expectation value”). Most of the known elementary particles are affected by the H field’s value, and obtain their mass from it.

But other, as-yet-unknown particles need not follow this rule, and the T field’s particles — the triplet Higgs bosons — are examples. For this reason, it can easily happen that the T bosons can all have mass much larger than the H boson, perhaps by ten times or so. At the LHC, that increase in mass has a huge cost; production rates for the T bosons are tremendously smaller than the production rate for the H boson. As a result, they won’t have been discovered yet, and aren’t discoverable in the near term.

Meanwhile, when the H field switches on, it can cause the T field to switch on too, as a knock-on effect. It can easily happen that the T^{0} field (but **not** the T^{+} or T^{–} fields) develops a non-zero, constant and uniform value — though one that is more than fifty times smaller than that of the H field. The only Standard Model’s particles that can interact directly with the T field are the W boson, the Z boson and the Higgs boson. But here’s the key point: it turns out that the W^{+} and W^{–} interact with T^{0} but the Z boson does not! So **an appropriate small but non-zero vacuum expectation value for T ^{0} shifts the W boson’s mass upward by about 1/1000, and leaves the Z boson’s mass alone**.

Meanwhile the H has its mass shifted too, but its mass is not precisely predicted by the Standard Model, so that effect isn’t measurable. What is measurable is **mixing between the T ^{0} and the H**, such that the 125 GeV/c

^{2}Higgs boson that we already know and love is mostly H but a little bit T. This alteration, making the known Higgs boson differ slightly from what is predicted in the Standard Model, is probably too small to observe with current LHC data. But it might someday be observable at the LHC, and certainly would be seen at a “Higgs factory” machine that may be built sometime in the future.

*[The math behind these statements is not particularly challenging, so maybe I’ll write a post later showing how it works, if there is interest among readers.]*

It seems that this very simple addition to the Standard Model can cause a shift in the W mass of a sufficient size to match the CDF result without messing up anything else (or did I overlook a subtle effect?) So it would seem impossible to argue that the CDF measurement’s discrepancy from the Standard Model is too large to be plausible. Even a very simple model from decades ago can potentially generate it.

That said, this model is really **too** simple — too simple to be interesting, even if it is correct. With just the T field and nothing else new, it would represent merely a small wrinkle on the Standard Model. However, it might be embedded in a larger theoretical structure; for instance, there are composite Higgs models and unusual supersymmetry models (and surely others) that **require** this kind of triplet Higgs field. Maybe one of these models, or some other like it, might be right, and might help explain one of the various other discrepancies with the Standard Model, such as those in B meson decays and in the magnetic properties of the muon? That would be a lot more satisfying and informative.

But of course, it doesn’t matter what physicists would prefer. And of course, all this assumes the discrepancy observed by CDF is real.

## 19 Responses

Would love to see the math that you mention, and many thanks for the clear post! I’ve just sent to Bj Bjorken who is of course interested in the results…

I really love this page. I can get as much knowledge as my limited skills can allow me to acquire. I think that’s because the explanations are straightforward albeit detailed. I owe my thanks tk prof. Strassler and to those who contribute with smart comments herein.

Please do write about the math of the triplet Higgs.

Thanks for your clear writing Matt. As a full time composer, your blog enables me to follow what you consider interesting [and so do I,

unerringly] I have shared this piece on my FB Page.

I’m glad you find it fits your needs… I can’t always direct every post at this level, but I do realize there are readers who are seeking this type of technical detail.

Your text (“almost none”) indicates perhaps another SM particle would interact with the T. My only guess would be t. Is that it?

No, that’s just bad writing. Somewhere along the way I rewrote the sentence and left it ambiguous. It should say “Almost none…; the only ones are…” or it should say “None of the… except…”. I’ll fix it.

Hey Matt, I would love to see the math. Please post it.

Any comment on what value superstring theory would place on the W boson mass and how it compares with the CDF results?

String theory, like quantum field theory, seems to be a setting for universes, and does not predict a specific one. Quantum field theory doesn’t predict the W mass; only a *specific example* of quantum field theory — the Standard Model — makes a precise prediction, and only after you’ve first measured the Z boson mass, the top quark mass, and the Higgs boson mass. String theory, similarly, is unlikely to ever make such a detailed prediction; at best, you’d have to see if you could find a specific universe (or “vacuum of string theory”) that contains the Standard Model and see what it predicts. The problem is that there may be many such universes, all with slightly different predictions; we simply don’t know. And furthermore, the mathematical techniques for correctly identifying and studying realistic vacua of string theory do not yet exist. So… your question is at best too general and too early, and the jury is out as to whether the answer might be that there are thousands or trillions of possible predictions within string theory (just as there are endless numbers of predictions for the W mass within quantum field theory, once you start adding new fields and particles to the Standard Model in all ways you can think of.)

The triplet is here too https://arxiv.org/abs/2204.04191 (and of course in many old papers).

Not in version one of your paper, Alessandro. I read it carefully, and you say: “For example a Higgs triplet T with vacuum expectation vT would contribute as Tb = −g^2 v^2_T /M^2_W , while the M_W anomaly favours a positive correction to Tb. In the next section we focus on a specific plausible tree-level source of a positive Tb: an extra heavy Z0 vector boson.” Apparently you recognized your mistake/omission and fixed it in version 2; but that doesn’t count. Such is the price of rushing to be first; the first shall be last.

v1 (monday) had the triplet of little Higgs. v2 (tuesday) includes all 3plets and 4plets. That’s the spirit of the gold rush.

To try to explain a shift in the W boson mass, and many other details of the Standard Model, is to add new fields and particles which affect the W boson through “quantum fluctuations” — disturbances in the fields which aren’t particles (though they are often called “virtual particles” anyway.)

Higgs boson that we already know and love is mostly H but a little bit T.

The only way for this is the gauge invariance created by Diffeomorphism?

When the H field switches on, it can cause the T field to switch on too.

Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown.

Why “spontaneous breakdown”?

It is colder than CMB?

T field is is like HOLE in LED need more ENERGY (like blue light) to dig out to create a particle.

Some disturbance (like virtual particle in condensed matter) in High-energy CERN allowed the W (Goldstone) to eat the disturbance?

Meanwhile the H has its mass shifted too, but its mass is not precisely predicted by the Standard Model, so that effect isn’t measurable. What is measurable is mixing between the T0 and the H, such that the 125 GeV/c2 Higgs boson that we already know and love is mostly H but a little bit T.

Thank you, very interesting and instructive.

Yes please to « The math behind these statements is not particularly challenging, so maybe I’ll write a post later showing how it works, if there is interest among readers »

Nice piece, Matt, but IMO the key sentence in it is the following:

“And of course, all this assumes the discrepancy observed by CDF is real.”

A claim of a 9 MeV uncertainty is unprecedented, and it will take a while to verify (or falsify) it, especially as the CDF paper does not go into sufficient derails to allow a proper scrutiny of the result. I hope our CDF colleagues will provide further material to support this amazing (if true) uncertainty of their measurement.

Agreed, of course. I would hope that CDF is preparing a paper with more details on their measurement, since indeed there’s not nearly enough information in the current paper. I have not heard anyone confirm this is planned.

The point of today’s post, though, is that it’s not hard to accommodate a shift of this size theoretically, possibly with minimal impact to LHC other than on properties of the 125 GeV Higgs. This is in contrast to various other discrepancies that we’ve seen, some of which are harder to produce. So the challenges we face here seem mainly to be experimental, not theoretical.