Questions about the Higgs Field

[MadRocketSci]

I've seen the same popular news articles everyone else has about the discovery of the Higgs boson. I have always had some questions about it though that I haven't seen addressed, and I'm still beating my head against introductory QFT, so I'm not to the level where I'd pick up the answers from understanding QFT.

1. Prior to actually building the LHC and finding the interaction signature, what made the Higgs field any more elegant than just adding the field rest-masses manually? Are there any fewer degrees of freedom in the theory with a Higgs field and a bunch of (arbitrary?) interaction strengths than there are in a theory with just the rest masses added in manually?

2. What gave physicists any idea of what the rest field-value of the Higgs field should be? It seems to me that there should be freedom to dial down the resting value of the higgs field and dial up the interaction strengths (or vice versa) arbitrarily.

3. Is giving the Higgs field a rest mass just moving the ball for what gives particles rest mass? Can the existence of rest mass be explained without being inserted "by hand" in at least one field?

 

[mfb]

You cannot add masses manually. It would break the whole idea of gauge interactions and Lorentz invariance.

2. What gave physicists any idea of what the rest field-value of the Higgs field should be? It seems to me that there should be freedom to dial down the resting value of the higgs field and dial up the interaction strengths (or vice versa) arbitrarily.

The bosons of the weak interactions fix that parameter. They don't fix the Higgs mass, that was still unknown.

3. Is giving the Higgs field a rest mass just moving the ball for what gives particles rest mass? Can the existence of rest mass be explained without being inserted "by hand" in at least one field?

If neutrinos are majorana particles, they can have their own mass term. Somewhat ironically, the Higgs boson has its own mass term. Apart from neutrinos and Higgs, all particles that participate in the weak interaction (=everything apart from gluons and photons) cannot have an explicit mass term.

 

[MadRocketSci]

You cannot add masses manually. It would break the whole idea of gauge interactions and Lorentz invariance.

What isn't Lorenz invariant about adding a rest-mass term to the dispersion relation for your particle propagators? That seems to be what everyone does in the intro chapters/lectures in QFT courses. The Klein-Gordon equation and Dirac equation with a nonzero rest-mass term transforms into itself when you do Lorenz transforms?

PS - thank you for your reply. I figure asking a bunch of questions on a forum might end up being more productive than continuing to beat my head against textbooks. (I'll need to ask more basic questions about what I'm stuck on currently later...)

PS - there seems to be something special about the weak interaction?: I've heard it stated that it is the source of much of the lack of symmetry in the standard model (time reversal, parity, matter/antimatter asymmetry etc.)

Since I'm still messing around with non-interacting fields, it isn't obvious to me why any given field or interaction has any special properties. Don't they all start with some spin-object (or geometric object/relation to geometry) (scalar, spinor, vector, 3/2-spinor, ...), a rest mass, and some sort of Lorenz invariant dispersion relation for propagation?

They don't fix the Higgs mass, that was still unknown.

I had heard that there were predictions being made for the Higgs rest-mass prior to the construction of the LHC. What would these prediction be based on? (And I suppose, alternatively, what would make any given unknown particle signature at a given rest mass Higgs-like? Vs being treated as a signature for some other unknown particle/interaction?)

 

[PeterDonis]

@MadRocketSci please note that I have used magic moderator powers to edit your post #3 to properly use PF quotes; they aren't quite the same as HTML quote tags. Using the "Quote" or "Reply" buttons at the lower right of a post, or highlighting a portion of a post and using the "Quote" or "Reply" buttons that pop up, will help you to use the PF quote feature.

 

[mfb]

The weak interaction only interacts with left-handed particles. But what does that mean if the particles have a direct mass term? You can’t separate left and right handed if you add such a term. I’m not a theorist, if you want all the mathematical details we’ll have to wait for them. I moved the thread to the particle physics section.

Without the weak interaction mass terms for the fermions would be fine.

The mass of the Higgs boson has an impact on the properties of other particles (via loop processes mainly) - precision measurements gave some rough idea what the mass should be.

 

[Orodruin]

In order to describe weak interactions properly, the Standard Model treats left- and right-handed particles differently as @mfb has already pointed out. The way that this manifests itself is that you generally have a right-handed singlet Weyl fermion (say a right-handed electron $e_R$) and a left-handed Weyl fermion that is a doublet under the SM SU(2) gauge group. The components of the doublet are the left-handed electron neutrino and the left-handed electron and we can call the doublet
$$
L = \begin{pmatrix}\nu_{eL} \\ e_L \end{pmatrix}.
$$
Now, a mass term for the electron would be on the form $\overline{e_L}e_R + {\rm h.c.}$, but this would break gauge invariance as $e_L$ would transform to part neutrino under general SU(2) transformations.

So how does the Higgs field help with this? The Higgs field is a scalar SU(2) doublet $\Phi$, which means that you can write down a Yukawa interaction term on the form
$$
y \overline L \Phi e_R + {\rm h.c.},
$$
where $y$ is some constant called a Yukawa coupling. At spontaneous symmetry breaking, the Higgs field takes a vacuum expectation value $v$ (or $v/\sqrt{2}$ depending on convention) in the second component. This leads to the Yukawa interaction (apart from leaving interactions with the Higgs and Goldstone bosons) being the source of mass terms
$$
m_e \overline{e_L} e_R + {\rm h.c.} = yv \overline{e_L} e_R + {\rm h.c.}.
$$

 

[CAF123]

@Orodruin at the beginning of textbooks etc, it is common to see a treatment in which scalar fields and the Dirac field have an explicit mass term wrote into their respective lagrangians by hand, as OP points out. Various symmetries are then discussed. Is it okay to write a mass term here because we are dealing with only a sector of the SM which does not include/ignores weak interactions or higgs physics? So the presence of only left handed couplings is blind to the non weak sector and so e.g $m_e \bar \psi \psi$ can be written in the Dirac lagrangian? The correct mass term relevant for SM physics is then generated properly in a gauge invariant manner via Higgs mechanism.

 

[Orodruin]

Various symmetries are then discussed. Is it okay to write a mass term here because we are dealing with only a sector of the SM which does not include/ignores weak interactions or higgs physics?

The point in the SM is that the SU(2) of the electroweak interactions couples differently to left- and right-handed fields. This means that a mass term directly breaks this gauge invariance. If you take any gauge field that couples in the same way to left- and right-handed fields, then you will not have the same problem and could write down the mass term directly and it would be gauge invariant. However, in the SM this is not the case. If it was not for the SU(2), then you could write down the mass terms directly. With the SU(2), mass terms would directly break the gauge invariance and it should therefore not come as any surprise that such mass terms are induced once you break the SU(2) gauge invariance spontaneously through the vev of the Higgs field.