# I The weak force as an attractive force

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1. May 16, 2018

### cmcraes

Hi all, how much merit does this answer have on Quora? There also an identical answer on Physics Stack Exchange so it'd be nice to confirm or deny the validity of these.

Thanks!

2. May 16, 2018

### Orodruin

Staff Emeritus
None. Weak interactions are based on a non-Abelian gauge symmetry and furthermore the up/down charges are not interaction eigenstates (ie, there are interactions that change these charges). You cannot talk about a classical force when it comes to weak interactions and for strong interactions you can do so only for the residual strong nuclear force responsible for keeping colourless nucleons together by exchange of virtual pions.

3. May 16, 2018

### ohwilleke

This doesn't sound right to me. Although I'm not certain of the answer and not very confident of the one posted in Physics.SE.

Thinking about W boson interactions isn't a very clean thought experiment since W bosons have electromagnetic couplings as well as weak interaction couplings. The clean analogy to photon exchanges in electromagnetism that we know and love and understand is with Z boson exchanges.

You can certainly have an interaction in which a left parity particle emits a Z boson and another left parity particle (or right parity anti-particle) absorbs that Z boson before it decays. This interaction follows the same propagator as a photon (since Z bosons have no weak isospin and no weak hypercharge themselves), except for mass and a replacement of electromagnetic charge (Q) with the weak hypercharge and weak isospin of the interacting particles. If the particles emitting and absorbing the Z bosons are neutrinos and/or anti-neutrinos, you have no electromagnetic charge or strong force confounds either in this very clean thought experiment (although the neutrinos have to be extremely energetic to have enough energy to emit a real on shell Z boson).

This neutral current interaction (or an equivalent virtual Z boson interaction) is what gives rise to the elastic scattering of neutrinos which is a force interaction much like a classical force.

The logic of the answers cited in the question would imply that if both particles are neutrinos in that scenario, then the Z boson exchange would have a repulsive effect (albeit very small and short range), and if one was a neutrino and the other an antineutrino in that scenario, that the Z boson exchange would be attractive (albeit very small and short range).

This is the opposite of my intuition (which isn't to say that it is wrong), because when a Z boson decays to a neutrino and an antineutrino of the same generation, one commonly imagines them flying apart from each other in opposite directions in a very similar (maybe even identical excluding a rotation of the respective time and space axis) Feynman diagram to one neutrino emitting a Z boson and another anti-neutrino absorbing one.

Last edited: May 16, 2018
4. May 16, 2018

### Orodruin

Staff Emeritus
Then how can you authoritatively say that it does not sound right?

First of all, the question was about weak interactions. That is not just Z exchange but also includes W exchange. You cannot change the question to try to squeeze in your answer. Even if that were the case, the answer would be wrong. The Z has different couplings to L/R due to its admixture of hypercharge.

I strongly suggest against trying to make sense of those answers.

5. May 17, 2018

### Staff: Mentor

Closed for moderation.
[Edit: for those who are wondering why: Several problematic posts are hidden while we decide what to with them]
[Edit: Opened for corrections. Please avoid ad hominem argumentations. Two posts still under consideration.]
[Edit: Two posts in moderation were deleted.]

Last edited by a moderator: May 17, 2018
6. May 17, 2018

Staff Emeritus

First, since the weak interaction is short range, discussing it in terms of the Newtonian concept of "force" is doomed to failure. It's intrinsically quantum mechanical.

However, that doesn't completely close the door. Since we're only concerned about attraction and repulsion, one could ask what happens if I took the equations for the weak force and adjusted them to create an analogous theory where it's just like the weak force, only longer-range: long enough so that the Newtonian picture was valid enough to determine if the force were attractive or repulsive. In this case, one can only talk about the electroweak force, because what remains when you subtract off the electric force depends on the details of just how you modeled the force as long range. With that caveat:
• neutrinos repel each other
• neutrinos and antineutrinos attract
• charged leptons repel
• charged leptons and antileptons attract
• up-type quarks repel
• up-type quarks and antiquarks attract
• up-type quarks attract charged leptons and repel antileptons
The other combinations depend on the details of just how you modeled the force as long range, so there truly is no answer to this question.

It doesn't matter that the mass eignestates are not weak eigenstates, because what matters is that a quark is up-type, not up.

7. May 17, 2018

### Orodruin

Staff Emeritus
These caveats are essentially what I consider missing from the quoted posts. It is mentioned that what is discussed is weak isospin interactions, bit it is glossed over how those relate to what many students would call ”weak interactions”, ie, W and Z exchange. Weak isospin only affects left-handed fermions and as long as you do not break electroweak symmetry those are massless and do not couple to their right-handed counterparts.

8. May 17, 2018

Staff Emeritus
There's a little more information than that, though, and that's what I was trying to bring out. For example, a u_L repels another u_L because they have the same isospin and hypercharge, same for the u_R and the u_R. The mixed combination, u_L and u_R, have no force due to isospin (because the u_R is a singlet), but are repelled by hypercharge. So two physical up quarks, irrespective of their admixture of u_L and u_R, will repel.

Not all the combinations are this clean: down-type quarks are not.

9. May 17, 2018

### Staff: Mentor

I think part of the issue with the original questions, though, is that they assume that anything that falls under the heading "weak interaction" must be an attraction or a repulsion, which is not correct.

10. May 18, 2018

### DrDu

I just want to point out that it was always a little bit unclear to me how you actually see from staring at some Feynman diagrams whether a force is attractive or repulsive. How to see this, I first learned from Zee's book "QFT in a nutshell". Basically you look at the propagator of the field carrying the interaction between two sources of (electroweak) charge. This propagator is proportional to $\exp(-iEt)$ where the attraction is attractive if E is negative and repulsive if it is positive.

11. May 18, 2018

### DrDu

That's an astonishing statement. Wasn't Newton the guy who played, together with his buddy Hooke with springs and all the like to explain the nature of forces? I don't see that the force excerted by a spring is especially long range. Maybe a more modern and relevant example are Yukawa forces. I also don't see why you can't consider them already in classical Newtonian mechanics and many models of the nucleus do so.

12. May 18, 2018

### Orodruin

Staff Emeritus
In this case, those forces are very very long range in comparison. "Short range" in terms of weak interactions are of the order of $10^{-18}$ meters (essentially the inverse of the W boson mass). At that scale it is rather irrelevant to talk about "forces" in the typical sense of having a classical path that a particle follows being affected by forces that determine its acceleration. In the case of models of the nucleus, you have a potential which is related to the concept of forces but not exactly the same. You have the potential in quantum mechanical descriptions as well without having to refer to forces.

Also, you could of course consider Yukawa forces also in classical mechanics, the point is that at the scales relevant for weak interactions you are really not in that limit.

13. May 18, 2018

### DrDu

I agree, but this is all rather a question of scale than of principle. In principle, even a bound state of a neutrino and an anti-neutrino is possible, although the binding energy would be fantastically small.

14. May 18, 2018

### Orodruin

Staff Emeritus
Are you sure that the neutrino-antineutrino bound state would be possible? As far as I understand, the existence of a bound state is not guaranteed for a Yukawa potential (there is a critical value of the screening parameter, i.e., the mass of the mediator in this case). A quick search revealed this: