© *Matt Strassler [May 8, 2022]*

*Please feel free to point out errors or suggest improvements; this is a dynamic document.*

This page is the third of several (first one is here, second is here) that together will explain the “Triplet Model”, a very simple method for shifting the W boson mass upward relative to the predictions of the Standard Model.

After our first two posts on the triplet model, we have all the basic ingredients:

- a potential for fields,
- a strategy to find the vacuum of the field theory from the potential,
- an understanding of how to read off particle masses from the potential,
- and a mechanism whereby the vacuum expectation value (vev) of one field can change the mass of another.

Last time we saw the mechanism in question in a cartoon of the Standard Model. Next we’ll work out a more elaborate cartoon, showing how this last mechanism is changed in the presence of a triplet-like field T.

### The Potential and the W Mass

In the first article in this series, we considered the potential

- V = – ½ μ
_{H}^{2}H^{2}+ ¼ λ H^{4}+ ½ m_{T}^{2}T^{2}– r H^{2}T

to see how a vev for H could trigger a small vev for T, if r is small. In the second article, we considered the potential

- V = – ½ μ
_{H}^{2}H^{2}+ ¼ λ H^{4}+ ½ m_{W}^{2}W^{2}+ ⅛ g^{2}H^{2}W^{2}

which gave us a cartoon of how the Standard Model’s mass-generating mechanism works: the field H, representing the Higgs field, develops an expectation value, and that in turn shifts the mass of a W particle (by shifting the terms proportional to W^{2} in the potential.)

All we need to get a cartoon for the triplet model is to combine these two potentials, and add one term proportional to T^{2} W^{2} :

- V = – ½ μ
_{H}^{2}H^{2}+ ¼ λ H^{4}+ ½ m_{T}^{2}T^{2}– r H^{2}T + ⅛ g^{2}H^{2}W^{2}+ ½ g^{2}T^{2}W^{2}

(The ⅛ and ½ in the last two terms are correct, but where they come from won’t be clear until a later post.) We know from earlier that if r is small enough, the vacuum is approximately given by **< H > ** **= μ _{H} /λ^{1/2}** and

**<T> = r<H>**. But then, by looking at the potential above and examining the W

^{2}/m_{T}^{2}^{2}terms, we find the W mass is

**W mass = g (¼<H>**^{2}+ <T>^{2})^{1/2}= ½**g****<H>**(1 + 4 r^{2}**<H>**^{2}/m_{T}^{4})^{1/2}

** which is shifted upward compared to the r=0 case**. You’ll often see this written in notation where we replace < H > with “v”, in which case

**W mass = ½****g****v**(1 + 4 r^{2}**v**^{2}/m_{T}^{4})^{1/2}

In short, because of the presence of T and its interaction with the H field via the interaction parameterized by r, the W’s mass is larger than it would have been had we only had the fields H and W.

### Lessons and Questions

Notice there are three ways that this mass shift could be avoided:

- The T field could be removed from the theory altogether
- The coefficient r in front of the H
^{2}T piece of the potential could be set to zero. - The mass of the T particle,
**m**, could be taken to be very large; as it becomes infinite, the shift disappears_{T}

These three ways to avoid the shift are exactly the same as those that apply to the W boson’s mass shift when you extend the Standard Model by adding a triplet T. This makes it very easy for the shift to be very small, as the CDF experiment claims it is.

So now you now know the basic math, and a significant conceptual point: *if the T boson has a large mass m _{T}, and r is not too large, then the shift in the W mass is naturally going to be very small.*

But there are all sorts of details I haven’t explained. In particular, where does this potential come from? How did I know the plus and minus signs for the most important terms? Why aren’t there all sorts of other terms in the potential that would mess up this structure? Why, for example, isn’t there a “T” term in the potential? What about T^{4} or W^{4} or H^{4} T terms?

Taking an a larger perspective, the questions I still have to address include

- what does the Standard Model’s potential look like and why?
- why do the W boson and Z boson have different masses in the Standard Model?
- why does the addition of the triplet field give the potential like the one above?
- why does the triplet vev shift the W boson mass but not the Z boson mass?
- what does it mean that the H boson and the T boson “mix”, and why is it significant?

This will take us a few more pages.

Absolutely great and interesting. Thank you so much Pr Strassler