Basic question on the pertubative Standard Model

[Breo]

Dear all,

how to 100% know if a process is allowed in standard model?

And when a process is allowed, how to know what diagrams contribute, and what of those are the dominant ones?

Thank you in advance.

 

[ChrisVer]

how to 100% know if a process is allowed in standard model?

You check whether your process violates any symmetry/law that it is not supposed to.

And when a process is allowed, how to know what diagrams contribute, and what of those are the dominant ones?

I don't think there is any straightforward answer to that. Any diagram can contribute depending on the process. In general all diagrams contribute- some contribute less some contribute more, it depends on the order and the couplings/propagators...

 

[Breo]

In general all diagrams contribute- some contribute less some contribute more, it depends on the order and the couplings...

But if for example you only have the process like: xx -> yy (x and y as unknown variables :P) how to know which diagrams contribute more at the process just by looking at xx -> yy?

 

[ChrisVer]

Quite oftenly in such cases, you take only the 1st order interaction, and you can take just 1 propagator between the particles.
However that's not a must.

 

[Breo]

And that/those propagator/s is the one/s allowed by Feynman rules, right?

 

[ChrisVer]

as an example, it's know that in the SM, the FCNC first order diagrams are forbidden, so you are looking at second order diagrams...

 

[ChrisVer]

And that/those propagator/s is the one/s allowed by Feynman rules, right?

It depends on how you choose x and y's to interact... you could as well write a Z0 boson or a photon or an X scalar or whatever... how does x couples to itself? or how does x couples to y?

 

[Breo]

It depends on how you choose x and y's to interact... you could as well write a Z0 boson or a photon or an X scalar or whatever... how does x couples to itself? or how does x couples to y?

Are you trying to tell me to take care about loops, etc?

 

[ChrisVer]

what precision do you want? and what is your x and y?

 

[Breo]

Let us say One-Loop Level.

If I understood well, with a process , let's say: H -> gg.

We treat H like a scalar field boson so we with the Feynman Rules in front, draw the possible diagrams at tree level, which should be "easy". And then... well, for one-loop level could be many ones so how to know which contribute more? This is a more generic and basic question. Because I can draw loops here and there. I suppose that in our case we just have to draw one diagram with a loop on the gluon external line, another diagram on the other gluon external line, other diagram with a loop on the H incoming line and a last diagram with a loop on the vertex?

 

[ChrisVer]

How could you join a Higgs with two gluons except for using quarks?

 

[ChrisVer]

 

[Breo]

One Loop-order?

But how could I computed a virtual top quark loop is produced, and not any other quark?

 

[ChrisVer]

But how could I computed a virtual top quark loop is produced, and not any other quark?

There could be more (eg bottoms), but the Higgs dominantly couples to top quarks (they are the heaviest or should I say they have the largest Yukawa coupling).

Also I don't think there is any lower order for this transition... there is no tree coupling between Higgs and gluons since the Higgs is an SU(3) singlet (color-chargeless).

 

[Breo]

There could be more (eg bottoms), but the Higgs dominantly couples to top quarks (they are the heaviest or should I say they have the largest Yukawa coupling).

Also I don't think there is any lower order for this transition... there is no tree coupling between Higgs and gluons since the Higgs is an SU(3) singlet (color-chargeless).

Where can I read about that?

 

[Breo]

I can see if a process is allowed just by checking Feynman rules... but I want to learn to justify and compute the process allowed not just by looking to the "solutions"

 

[ChrisVer]

Where can I read about that?

About what? I said two things...
The last is a common knowledge, I mean there can only be effective couplings between the higgs and gluons... or you can't write in your lagrangian a term such as $h GG$ without breaking your SM symmetry ($h$ is a doublet of SU(2) )... this terms appears effectively via the triangle diagram.

The first thing, is just by looking at the Yukawa couplings? (couplings of your fermions[quarks] to scalar fields[higgs this time] )

 

[ChrisVer]

I can see if a process is allowed just by checking Feynman rules... but I want to learn to justify and compute the process allowed not just by looking to the "solutions"

What do you mean by checking the Feynman rules?
The Feynman rules gives you some relation (simply put the relation between a feynman diagram and the mathematical formulae)

 

[Breo]

What do you mean by checking the Feynman rules?

If I have for example H > gg, I look to the posible vertex which have gluons, those are quarks, ghosts or more gluons. Then I check to H possible vertex and there is one with 2 fermions so the only possible way is to make a triangle of 3 quarks in between as the diagram you posted.

I do not like this way of work. That is why I am asking here how to compute these kind of things more "seriously".

The Feynman rules gives you some relation (simply put the relation between a feynman diagram and the mathematical formulae)

I am not sure I do understand what you mean by relation. I can see the diagram and the mathematical formulae for each one.

 

[mfb]

There is no general way to look at diagrams and say "this is more important". There are many cases where it is possible, but sometimes you just have to calculate it with QFT (which takes a lot of time).

"Higgs couples to mass" is one of those rules that help in many diagrams - every quark, every charged lepton and the W boson can run in this loop for the diphoton decay, but top and W are by far the heaviest particles in this group so their contribution will be dominant.

 

[Breo]

There is no general way to look at diagrams and say "this is more important". There are many cases where it is possible, but sometimes you just have to calculate it with QFT (which takes a lot of time).

I want to learn how to computate it. Any reference?

 

[mfb]

Books about quantum field theory? And books about tons of other stuff you need to understand the concepts used there. And books about stuff needed to understand those books. And ... this webcomic explains it quite well (click on the image to continue).

 

[Breo]

Books about quantum field theory? And books about tons of other stuff you need to understand the concepts used there. And books about stuff needed to understand those books. And ... this webcomic explains it quite well (click on the image to continue).

xD

Just let me know if the more important are those with bigger amplitude or so

 

[mfb]

Sure, larger amplitudes are more important. All amplitudes are added to calculate the probability of a process. If you look at the result of 100+1=101, the first summand is more important.

 

[ChrisVer]

I am not sure I do understand what you mean by relation. I can see the diagram and the mathematical formulae for each one.

The Feynman rules tells you what mathematical relationship corresponds to each vertex, each propagator, and each external lines... so that you can go on further and calculate the amplitudes and the cross sections later on...

If I have for example H > gg, I look to the posible vertex which have gluons, those are quarks, ghosts or more gluons. Then I check to H possible vertex and there is one with 2 fermions so the only possible way is to make a triangle of 3 quarks in between as the diagram you posted.

This sounds tiring... but in general it's a way... I don't know whether (unconsciously) I am doing this process when I try to think of a diagram...
I would just remember from the Lagrangian that the scalars couple to fermions (so the Higgs->fermion+fermion would be the thing that would come in my mind). Then I would think that you need two external gluons, and so what else could it be but the fermions to be quarks (no other fermions couple to gluons)... then I already think you have been told enough about which quark would be the most dominant one. It's all in the Lagrangian. So I guess I am moving with the symmetries. I wouldn't follow your way, which gives me the impression that you follow the algorithm:
Solve 4+5:
1- write all numbers from 1 to 100
2- choose the number 4 from them
3- write all numbers from 1 to 100
4- choose the number 5 from them
6- combine to the number that corresponds to 4+5.
(I'd drop the 1 and 3 line)

Now a general interaction cannot be answered in 1 way, because there may be more interactions possible... take for example the $e^-e^+ \rightarrow f^- f^+$ (with f some fermion). This interaction can also be done through photons as well as through Z-bosons.

 

[ChrisVer]

Sure, larger amplitudes are more important. All amplitudes are added to calculate the probability of a process. If you look at the result of 100+1=101, the first summand is more important.

And that's the idea of perturbation- you need to set up your desired precision...

 

[Breo]

I would just remember from the Lagrangian that the scalars couple to fermions (so the Higgs->fermion+fermion would be the thing that would come in my mind). Then I would think that you need two external gluons, and so what else could it be but the fermions to be quarks (no other fermions couple to gluons)

And how do you take care of bosons (with his positive or negative charges in the W case) and leptons possibilities just by looking to the lagrangian?

 

[ChrisVer]

And how do you take care of bosons (with his positive or negative charges in the W case) and leptons possibilities just by looking to the lagrangian?

How would the positive/negative charge of a W boson stop you from writing a diagram? (sorry right now I can't think of any problem arising from that).
lepton possibilities? if you have more than one possible leptons in a process, in general you have to take it into account. For example I wouldn't even have to write the $t$ quark in the triangle propagator I posted before. There are all fermions going around. Of course (as I mentioned) the dominant one was the top... if you don't know what is the dominant, you have to write all of them, because all these diagrams would in general contribute...
That is something happening quite often when you get quark loops... each loop of these quarks is appearing with a multiplicity of 3 (because of the colors). So you could write three diagrams with the same quarks but with different colors, or you could write 3 times the one diagram (exploiting the color symmetry).

I editted the above message btw...

 

[Breo]

So, in order to know the more dominant you should to compute the amplitudes for all the possibilities, right? It is the only way.

 

[ChrisVer]

In general yes. Sometimes the diagrams can have almost identical contributions...
Sometimes the diagrams don't... the couplings in this case can play some role (eg. know that the higgs coupling to fermions depends on their masses),because you are taking their squares.
But again don't think of this so lightly...for example, even though the Higgs couples to heavy fermions, there will be searches for the Higgs channel of $H \rightarrow \tau \tau$ in the next ATLAS run , so they will try to measure its CP... (the interaction exists)...The taus are relatively heavy, but compared to tops they are light (of course there can't be a $H \rightarrow t t$ with asymptotic top states due to the mass) and still lighter that bottoms (but bottoms can be more diluted by QCD background)...in the last you are looking at VH (vector boson+higgs) decays, which help you distinguish the bottom products...