Why do radiative corrections means changing particle masses?

[jeebs]

I'm currently trying to find out about radiative corrections to Feynman diagrams, where you have a particle traveling along (just represented by a line or whatever) then you can have it momentarily changing into a particle/antiparticle pair or whatever even if energy is not conserved - as long as conservation of energy is recovered within the confines of the energy-time uncertainty principle. Also I am aware that there can be infinitely many higher order corrections, ie. you can draw as many different Feynman diagrams as you can think of as long as you recover your original particle at the end. That's fair enough, I'm familiar with that much.

What I'm not understanding is, what has this got to do with the mass of a particle? For instance, apparently the simplest first order radiative correction to the higgs particle (1 loop) causes a mass increase of 10^19 GeV or something ridiculous. Why do more possible loops mean the mass has to be modified?

I ask this because I'm learning the basics of SUSY and the like, and apparently (correct me if I'm wrong) by introducing supersymmetric partners it enables you to draw more diagrams that give you a very delicate cancellation of the radiative corrections and you end up with sensible particle masses?

 

[tom.stoer]

... even if energy is not conserved - as long as conservation of energy is recovered within the confines of the energy-time uncertainty principle.

The internal lines are called "virtual particles" b/c they do NOT obey p² = m²; nevertheless at each vertex 4-momentum p is strictly conserved.

 

[jeebs]

The internal lines are called "virtual particles" b/c they do NOT obey p² = m²; nevertheless at each vertex 4-momentum p is strictly conserved.

I'm not sure what you mean here - I thought that the virtual particles were allowed to have as high a mass or momentum or anything as they liked, as long as they only existed for as brief a time span as the uncertainty principle allows?

 

[henry_m]

I'd be very wary of trying to interpret the existence of virtual particles too literally. They serve as a nice interpretation and mnemonic for the Feynman rules, but in my opinion there's no sense in which they are 'real' objects. To talk about particles with 'off-shell momentum' and wave your hands with time-energy uncertainty relations and so on can be quite confusing and not terribly clarifying (though I can perhaps see some merit in these idea for a 'popular science' sort of audience).

As Tom points out though, four momentum must always be conserved at interaction vertices, though it may be unphysical. For example, in a loop when you integrate over the possible energies and momenta running through the loop you include negative energies.

The only way to properly understand what's going on with radiative corrections to masses, interactions and so on is to do it for yourself. Any good books will have full explanations and examples. From a conceptual point of view, I think Zee is very good, as long as you make the effort to fill in all the gaps in the calculations.

Good luck and enjoy!

 

[jeebs]

I'd be very wary of trying to interpret the existence of virtual particles too literally. They serve as a nice interpretation and mnemonic for the Feynman rules, but in my opinion there's no sense in which they are 'real' objects. To talk about particles with 'off-shell momentum' and wave your hands with time-energy uncertainty relations and so on can be quite confusing and not terribly clarifying (though I can perhaps see some merit in these idea for a 'popular science' sort of audience).

As Tom points out though, four momentum must always be conserved at interaction vertices, though it may be unphysical. For example, in a loop when you integrate over the possible energies and momenta running through the loop you include negative energies.

The only way to properly understand what's going on with radiative corrections to masses, interactions and so on is to do it for yourself. Any good books will have full explanations and examples. From a conceptual point of view, I think Zee is very good, as long as you make the effort to fill in all the gaps in the calculations.

Good luck and enjoy!

lol oh well, I wish I had the luxury of time to sit and figure this stuff out properly, but I'm 4 days away from an exam in a module that has discussed this stuff in the most hand-wavy non-mathematical way you could imagine.

 

[tom.stoer]

First of all I agree to what henry m is saying: don't take these virtual particles too literally; they are just math.

Regarding Faynman diagrams and 4-momentum conservation:

Think about a real photon with with m=0 and 4-momentum (p,p) with p = |p|. Of course p²-p² = p²-p²=0.

Now think about a virtual electron-positron-loop. 4-momentum conservation for the photon-electron-positron vertex requires (E,q) and (p-E,p-q) for electron and positron; 4-momentum, but E²-q² is now arbitrary and no longer fixed to m². Therefore E and q become free variables which are integrated over.

So the virtual electron-positron pair is nothing else but a calculational rule how to introduce two vertices with 4-momentum conservation, new variables E and q, and a certain rule how to integrate over E and q.

 

[henry_m]

lol oh well, I wish I had the luxury of time to sit and figure this stuff out properly, but I'm 4 days away from an exam in a module that has discussed this stuff in the most hand-wavy non-mathematical way you could imagine.

Haha fair enough. I'll try to be a bit more helpful then!

The mass of a particle is synonymous with its energy when it is at rest. If we have, say, a charged particle, which interacts with light, then its energy will change by virtue of the fact that it is interacting.

Imagine that the particle, while very tiny, still has finite size and is made of some sort of fluid. We can imagine building the particle by starting with some of this fluid very widely separated, and then bringing it all together. If the fluid is not charged, the mass of the particle will just be the mass of the fluid. But if it is charged, assembling the particle will require lots of energy to work against electric repulsion of the fluid, and the resulting particle will be more massive because it has lots of electrostatic potential energy. This is a 'radiative correction' of the mass due to its charge.

In fact, we could imagine that the fluid had no intrinsic mass at all, and that the mass of the electron comes entirely from the fluid's self-interaction. The smaller the space we pack the fluid into, the higher the energy. (The size of the ball of fluid needed to give the mass we measure is sometimes called the classical electron radius). If we assume the electron is actually a point we get the horrible result that it must have infinite mass! This behaviour of infinities also occurs in the quantum treatment, and is perhaps the large mass increase referred to in the OP. It is ultimately dealt with by the process of regularisation and renormalisation.

In this classical context, we regularise by assuming the electron has an unknown small but finite radius, and we renormalise by assuming the 'bare' mass of the electron (what it would be without interactions) is just right to cancel the very large self-interaction energy to give the total mass we see.

 

[jeebs]

ahh ok cheers for that. henry_m, you've hit on another thing I've not been to sure about - renormalization.

You say it gets rid of infinities in mass, but I was under the impression that each level of complexity you add to a diagram comes with a certain probability - and sometimes the probabilities increase with increasing energy to values greater than 1 - so renormalization gets rid of that bad behaviour.

So does this renormalization technique apply to both mass and probability or am I mistaken?
Are you able to outline a brief summary of what it involves, if it's not too much trouble?

Thanks.

 

[henry_m]

Okay, I'll give a brief summary of the ideas of renormalisation. This is from the modern point of view of effective field theories, from which any quantum field theory we come up with comes implicitly with some domain of validity. Beyond some energy scale, it stops working and some new theory takes over. The essence of a renormalisable field theory is that the results we get at low enough energies are independent of the exact form of this high energy theory.

Let's take quantum electrodynamics as an example, with just the electromagnetic field (photons) and the electron field (electrons and positrons). We know that in high energy experiments, we're going to get all sorts of other exotic things happening, with hadrons and so on. But we don't want the existence of these complicated particles and interactions to mess up what happens at the low energies we're interested in.

We can do scattering calculations with just the tree diagrams and we get good agreement with experiment at low energies. But when we try to make more accurate calculations with loop diagrams we get infinities, coming from contributions to the scattering amplitude from very high energies running through the loops. The naïve assumption that the theory is good at arbitrarily high energies has come back to bite us.

So we can start by regularising, and simply throwing away the high energy contributions and integrating momenta and energies only up to a scale $\Lambda$. This gives us a finite number, but it depends on $\Lambda$, and since we don't know what this might be, and we were very simplistic in the way the cutoff is imposed, we don't want our predictions to rely on this parameter. But we also notice that the loop diagram affects the physical masses of the particles, and the masses we started with as parameters in the theory (bare masses) no longer have a measurable role. So we eliminate the bare mass in favour of the the physical mass everywhere. If we do the same for the charge of the electron, and the 'scaling' of the field, what we ultimately find is that $\Lambda$ disappears entirely from measurable quantities (or at least appears only in corrections like $E/\Lambda$, where E is a typical energy of the process we are measuring). This is the process of renormalisation, and the fact that we only needed to fix finitely many physical parameters to achieve this cutoff independence is what we mean by renormalisability.

The clever thing is that whatever field theory we have at high energies, it turns out that at sufficiently low energies it will look like a renormalisable theory. But that's a story for another time...