Understanding the Running of Mass in Particle Interactions

[metroplex021]

I have a question on quantities that 'run' under the renormalization group flow. What's clear is that the properties through which particles couple -- such as charge, isospin, color etc -- change in their strengths as the energy increases. But this running is also true of the mass, which is a property that particles have even when they're not interacting. However, the other properties that we can define in a free field theory -- spin and parity, say -- don't change under the RG flow. So what is it about the kinematical property of mass in particular that allows us to treat it as a coupling?

(A weird question I know, but I'd be interested to hear any thoughts!)

 

[Einj]

The quantities that are scale dependent under renormalization flow are those quantities that appear as parameters of the Lagrangian. The mass and charge of the particles explicitely appear in the Lagrangian while properties such as spin don't.

 

[metroplex021]

Thanks very much: of course that's the answer. But may I ask one more thing.

All the parameters appearing in the Lagrangian are interaction couplings, with the exception of from mass. Is there a reason that mass is singled out in this way? Is there perhaps some reason that we can think of it as a coupling in some sense, even though at present we await a model of quantum gravity?

 

[Einj]

I guess it dependends on how you define "coupling". Consider for example a scalar field theory. The mass appears in front of the quadratic term $m^2\phi^2$ which is included in the Lagrangian of the free theory:
$$
\mathcal{L}_{free}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2.
$$
This is the free Lagrangian since it gives the equation of motion of a free particle. When you add other terms to this Lagrangian they will appear with coupling "contants" in front of them. Therefore, one usually defines "coupling" those parameter that appear in the interaction Lagrangian.
The mass is a parameter of the free Lagrangian and hence it's not a coupling in the usual sense.

 

[haushofer]

I would say that mass can be treated as a coupling; it's just the coupling of a field to itself. See e.g. Susskind,



What you do is to write down all the powers in the field with a constant in front of it, and consider renormalizability. All the parameters you use in this process are basically "couplings".

 

[metroplex021]

That's great people: I think I know the Susskind lecture and I'll watch it again. Final question. Does anyone know of a place where a conjectured Lagrangian for a QFT of gravity is written down? I would like to see how the mass 'couplings' that occur in, say, the Standard Model relate to the terms that couple masses in a gravitational theory. But thanks again folks!

 

[The_Duck]

To couple a theory to gravity you take the Lagrangian density $\mathcal{L}$ of your theory, replace all instances of the Minkowski metric $\eta_{\mu\nu}$ with the GR metric $g_{\mu\nu}$, and write down the action
$$S = \int d^4x \sqrt{-g} \mathcal{L}$$
where $g$ is the determinant of the metric tensor. This turns out to give a coupling between the metric (i.e., the gravitational field) and the stress-energy tensor of the theory original theory. The terms in $\mathcal{L}$ that give mass to particles, for example the $m^2 \phi^2/2$ term in a scalar field theory, contribute to the stress-energy tensor and so produce couplings between massive particles and the gravitational field. But gravity also couples to all other sources of stress-energy, including the energy of massless particles.

 

[metroplex021]

Thanks, The_Duck. Fascinating!

 

[Avodyne]

You also have to add to $\cal L$ the Einstein-Hilbert term $R/16\pi G$, where $R$ is the Ricci scalar and $G$ is Newton's constant: http://en.wikipedia.org/wiki/Einstein–Hilbert_action