What are the different decay processes shown in this Feynman diagram?

[spaghetti3451]

Consider the following momentum-space Feynman diagram

https://upload.wikimedia.org/wikipedia/commons/b/b5/Lepton-interaction-vertex-eeg.svg

This Feynman diagram is the leading-order contribution to the any of the following processes:

1. $e^{-} \rightarrow e^{+} + \gamma$
2. $e^{+} \rightarrow e^{-} + \gamma$
3. $\gamma \rightarrow e^{+} + e^{-}$

The process in 3 is clearly the decay of a photon to an electron-positron pair.

Why can't the processes in 1 and 2 be considered to be decay processes of the electron and positron respectively?

 

[stoomart]

The process in 3 is clearly the decay of a photon to an electron-positron pair.

Why can't the processes in 1 and 2 be considered to be decay processes of the electron and positron respectively?

My understanding is a single photon cannot decay into a positron/electron pair per this recent discussion:

https://www.physicsforums.com/threads/electron-positron-production-by-photon.895704/

 

[spaghetti3451]

What if I replace the photon by a massive scalar boson?

 

[stoomart]

What if I replace the photon by a massive scalar boson?

https://en.m.wikipedia.org/wiki/Higgs_boson#Decay

For a Higgs boson with a mass of 126 GeV/c²the SM predicts a mean life time of about 1.6×10^-22 s.

Also see:
https://en.m.wikipedia.org/wiki/Par...e_elementary_and_composite_particle_lifetimes

 

[spaghetti3451]

Ok, so why is the process $h \rightarrow h + h$ allowed?

In the rest frame, the total energy is the rest energy of $h$, but the outgoing particles have twice the rest energy, so energy is not conserved.

 

[stoomart]

Ok, so why is the process $h \rightarrow h + h$ allowed?

Sorry, this doesn't make any sense to me, I better let the pros take this one.

 

[Orodruin]

Ok, so why is the process $h \rightarrow h + h$ allowed?

In the rest frame, the total energy is the rest energy of $h$, but the outgoing particles have twice the rest energy, so energy is not conserved.

It is not allowed for free Higgses.

Neither is any of your reactions listed in the OP.

 

[spaghetti3451]

Ah! I see!

But consider the following Lagrangian

$$\mathcal{L} = \bar{\psi}_{e}(i\gamma^{\mu}{\partial_{\mu}}-y_{e}\nu)\psi_{e}-y_{e}\bar{\psi}_{e}h\psi_{e}+\frac{1}{2}(\partial_{\mu}h)(\partial^{\mu}h)-\frac{1}{2}\left(2|\kappa^{2}|\right)h^{2}-\frac{\lambda}{6}\nu h^{3}-\frac{\lambda}{24}h^{4}$$

which has the intreaction term $-y_{e}\bar{\psi}_{e}h\psi_{e}$. This term leads to an interaction vertex of the form given in the diagram:

https://i.imgsafe.org/64394a297d.jpg

This diagram describes the leading-order Feynman diagram of the process $h \rightarrow e^{-} + e^{+}$.

Can we have a process like $h \rightarrow h$, where the electron and anti-electron are in a loop? Can we call this process the decay of $h$?

 

[Orodruin]

Ah! I see!

But consider the following Lagrangian

$$\mathcal{L} = \bar{\psi}_{e}(i\gamma^{\mu}{\partial_{\mu}}-y_{e}\nu)\psi_{e}-y_{e}\bar{\psi}_{e}h\psi_{e}+\frac{1}{2}(\partial_{\mu}h)(\partial^{\mu}h)-\frac{1}{2}\left(2|\kappa^{2}|\right)h^{2}-\frac{\lambda}{6}\nu h^{3}-\frac{\lambda}{24}h^{4}$$

which has the intreaction term $-y_{e}\bar{\psi}_{e}h\psi_{e}$. This term leads to an interaction vertex of the form given in the diagram:

https://i.imgsafe.org/64394a297d.jpg

This diagram describes the leading-order Feynman diagram of the process $h \rightarrow e^{-} + e^{+}$.

Can we have a process like $h \rightarrow h$, where the electron and anti-electron are in a loop? Can we call this process the decay of $h$?

No, the $h$ can decay to the electron positron pair if its mass is large enough. The $h \to h$ diagram is not a decay, it is a contribution to the self energy of the $h$.

 

[spaghetti3451]

No, the $h$ can decay to the electron positron pair if its mass is large enough. The $h \to h$ diagram is not a decay, it is a contribution to the self energy of the $h$.

Ok, so (assuming that the mass of $h$ is greater than the mass of $e^{-}$) apart from $h \rightarrow e^{-} + e^{+}$, what other decays of $h$ are possible?

 

[Orodruin]

Ok, so (assuming that the mass of $h$ is greater than the mass of $e^{-}$) apart from $h \rightarrow e^{-} + e^{+}$, what other decays of $h$ are possible?

The mass needs to be greater than two electron masses for the decay to occur.

What other decays that will occur depends on what else you stuff into your theory.

As given, none (apart from possible higher order decays to several electron-positron pairs, but those are suppressed).

 

[spaghetti3451]

Why are higher-order decays suppressed?

 

[Orodruin]

Because they contain more vertices and are therefore only available at higher order in perturbation theory. If the total mass of the decay products is close to that of the decaying particle, you will also get additional phase space suppression.

 

[spaghetti3451]

But you can't deny that these higher-order decay processes still exist!

It's just that they are suppressed since they contain more vertices, right?

 

[Orodruin]

But you can't deny that these higher-order decay processes still exist!

Where did I do that?

Also, they will not exist unless the sum of the masses of the outgoing particles is smaller than the $h$ mass.

It's just that they are suppressed since they contain more vertices, right?

This is what I said.

 

[spaghetti3451]

Is the image in the following link the leading order Feynman diagram for the (higher-order) decay process $h \rightarrow e^{-} + e^{+} + e^{-} + e^{+}$

https://i.imgsafe.org/650622580e.jpg?

 

[Orodruin]

No. It has three electrons and three positrons in the out state. Also, there are several different diagrams contributing.

 

[spaghetti3451]

How do I understand that there are several different diagrams contributing?

By several different diagrams, do you mean several interaction vertices?

 

[Orodruin]

How do I understand that there are several different diagrams contributing?

Because you can draw them while respecting the Feynman rules.

By several different diagrams, do you mean several interaction vertices?

No.

 

[spaghetti3451]

Okay, so if instead, we had the Lagrangian

$$\mathcal{L} = \bar{\psi}_{e}(i\gamma^{\mu}{\partial_{\mu}}-y_{e}\nu)\psi_{e}-y_{e}\bar{\psi}_{e}h\psi_{e}+\bar{\psi}_{\mu}(i\gamma^{\nu}{\partial_{\nu}}-y_{\mu}\nu)\psi_{\mu}-y_{\mu}\bar{\psi}_{\mu}h\psi_{\mu}+\frac{1}{2}(\partial_{\mu}h)(\partial^{\mu}h)-\frac{1}{2}\left(2|\kappa^{2}|\right)h^{2}-\frac{\lambda}{6}\nu h^{3}-\frac{\lambda}{24}h^{4}$$

we can write down processes like

$$h \rightarrow e^{-} + e^{+}, \qquad\qquad h \rightarrow e^{-} + e^{+} + e^{-} + e^{+}, \dots$$

$$h \rightarrow \mu^{-} + \mu^{+}, \qquad\qquad h \rightarrow \mu^{-} + \mu^{+} + \mu^{-} + \mu^{+}, \dots$$

$$h \rightarrow e^{-} + e^{+} + \mu^{-} + \mu^{+}, \dots $$

I was wondering if a particle (e.g. muon) must always be produced with its corresponding antiparticle (e.g. antimuon).

Also, I was wondering if there are only a finite number of processes for this reaction.

 

[ChrisVer]

I was wondering if a particle (e.g. muon) must always be produced with its corresponding antiparticle (e.g. antimuon).

if Lepton number and charge is conserved, the muon will be produced with an antimuon...

 

[spaghetti3451]

if Lepton number and charge is conserved, the muon will be produced with an antimuon...

1. Can we have the Higgs decay into an electron-positron pair and a muon-anti-muon pair?

2 .Can we have the Higgs decay into a muon and a positron?

 

[ChrisVer]

1. Can we have the Higgs decay into an electron-positron pair and a muon-anti-muon pair?

yes... However the Yukawa couplings to electrons or muons are small... From dilepton channels, the Higgs has only been observed in the ditau ones (by combination of CMS and ATLAS results).
The dimuon as far as I know is the next promising one.

2 .Can we have the Higgs decay into a muon and a positron?

Not in the Standard Model, because this violates the lepton flavor... There are models which predict such decays (not only for Higgs but for Z as well) and they are studied in experiment.

 

[spaghetti3451]

yes... However the Yukawa couplings to electrons or muons are small... From dilepton channels, the Higgs has only been observed in the ditau ones (by combination of CMS and ATLAS results).
The dimuon as far as I know is the next promising one.Not in the Standard Model, because this violates the lepton flavor... There are models which predict such decays (not only for Higgs but for Z as well) and they are studied in experiment.

The questions were asked relative to this Lagrangian

$$\mathcal{L} = \bar{\psi}_{e}(i\gamma^{\mu}{\partial_{\mu}}-y_{e}\nu)\psi_{e}-y_{e}\bar{\psi}_{e}h\psi_{e}+\bar{\psi}_{\mu}(i\gamma^{\nu}{\partial_{\nu}}-y_{\mu}\nu)\psi_{\mu}-y_{\mu}\bar{\psi}_{\mu}h\psi_{\mu}+\frac{1}{2}(\partial_{\mu}h)(\partial^{\mu}h)-\frac{1}{2}\left(2|\kappa^{2}|\right)h^{2}-\frac{\lambda}{6}\nu h^{3}-\frac{\lambda}{24}h^{4}$$

 

[ChrisVer]

The questions were asked relative to this Lagrangian

do you see any $\bar{\psi}_\mu h \psi_e$ in tat lagrangian?
What would the Feynman diagram for the production of emu look like, and would the vertex correspond to any of those terms' couplings?

Also your Lagrangian doesn't look too different to the SM one (to suggest that you don't have some symmetries)

 

[spaghetti3451]

So, Higgs decay into electron and anti-muon not possible.

Is Higgs decay into four particles: an electron-positron pair and a muon-anti-muon pair possible?

 

[ChrisVer]

as far as I can see, yes...? what is your question?

 

[spaghetti3451]

I was wondering if tree-level diagrams are not possible with the process

$$h \rightarrow e^{-} + e^{+} + e^{-} + e^{+}$$

Consider the following attempt to draw a tree-level Feynman diagram for this process:

I end up with a Higgs in the final decay product, which is not allowed:

https://i.imgsafe.org/77134d5878.jpg

 

[Orodruin]

Why are you adding that line then? There is no reason for it to be there.

 

[spaghetti3451]

I am adding that line, because each interaction vertex must have three lines popping out of it, with the Higgs for one line and the fermion and anti-fermion on the other two lines

 

[Orodruin]

I am adding that line, because each interaction vertex must have three lines popping out of it, with the Higgs for one line and the fermion and anti-fermion on the other two lines

But without having that extra higgs you still satisfy that! You just have one vertex less.

 

[spaghetti3451]

Got it!

Is this the only tree-level Feynman diagram for this process?

 

[Orodruin]

No. You have two diagrams. One where the electron of the initial vertex is external and one where the positron is.

 

[spaghetti3451]

I see! So, for a process like

$$h \rightarrow e^{+} + e^{-} + \mu^{+} + \mu^{-}$$

you have only the following two tree-level Feynman diagrams, right?

https://i.imgsafe.org/79327227a1.jpg

https://i.imgsafe.org/7932eec9b7.jpg

 

[Orodruin]

No. You also have the diagrams where the initial vertex is with the electron line.