Supersymmetry: Theory and Experiments

[itssilva]

Supersimmetry, by itself, is a neat, elegant concept; I've read somewhere else (think Griffiths' Introduction to Elementary Particles, if memory serves me right) that it allows the various running couplings of the Standard Model to converge to a single value at high enough energies; however, besides that, what are the theoretical/experimental motivations to study this complication of standard QT? I'm under the impression that this theory haven't been garnering much love from the non-string physics community, of late.

 

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[Orodruin]

Supersymmetry is popular in large parts of the particle physics community, not only with string theorists. The main reason for this is that it would provide possible solutions to some problems with the Standard Model. For example, it would solve the hierarchy problem (assuming the SUSY scale is low enough) and it provides viable dark matter candidates.

 

[Ilja]

One point of supersymmetry seems that the additional particles it introduces somehow "solve" the "problem" of a unification of couplings. If this is really a problem is a first question - it is, of course, a problem for GUTs, which unify all forces into some simple gauge group, which requires that at some fundamental scale of unbroken symmetry the interaction coupling has to be the same for all parts.

One can argue that it could be also a problem for an effective theory, which assumes a common critical length scale where the field theory breaks down into some sub-theory, say some "atomic ether" or so. In this case, one would reasonably expect that all the effects should have a similar order at this scale.

My question is a different one. Suppose one simply has some usual field theory, containing all what the SM contains but in different numbers - say, more fermions, more scalar fields, above with masses. Computing the corresponding functions for the couplings should be, I would guess, something already done in a quite general form, so that it could be sufficient to put in the numbers of the additional particles, their spin, their masses, I would guess also the representations of the SM gauge groups acting on them, to find out if in this theory the couplings unify or not.

Any suggestions where such things can be found?

 

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[Orodruin]

I would guess, something already done in a quite general form, so that it could be sufficient to put in the numbers of the additional particles, their spin, their masses, I would guess also the representations of the SM gauge groups acting on them, to find out if in this theory the couplings unify or not.

You cannot simply add an arbitrary number of fermions to the SM without caring for anomaly cancellations. In the SM, the gauge couplings are such that the anomalies cancel generation by generation.

What you can do is to add extra Higgses. This does change the running behaviour of the gauge couplings and using an appropriate number of extra Higgs doublets (if I remember correctly, the required number was seven additional doublets, but do not quote me on that) you can get the couplings to unify.

 

[Ilja]

You cannot simply add an arbitrary number of fermions to the SM without caring for anomaly cancellations.

Thanks, I know.

But I don't care much about this anomaly stuff, for a reason which is possibly completely wrong:

My understanding is that the problem they cause is non-renormalizability. But in this case, this would be a non-problem for an effective field theory with some explicit cutoff. There, the non-renormalizable terms would be those which descrease faster than the renormlizable ones in the large distance limit, so they would disappear automatically if the cutoff is sufficiently small, even if they would have comparable order at the critical distance. Roughly speaking, one could leave problems with non-renormalizability to the long distance limit. Or is this completely off, and there are other problems with them?

What you can do is to add extra Higgses. This does change the running behaviour of the gauge couplings and using an appropriate number of extra Higgs doublets (if I remember correctly, the required number was seven additional doublets, but do not quote me on that) you can get the couplings to unify.

What I would like to add (for completely different reasons) is a single scalar field for each electroweak doublet. The triplets for quarks would be colored, those for leptons would not interact with gauge fields at all.

Now, naive counting gives equal numbers for bosons and fermions: 8+3+1 gauge bosons, 3 x (1+3) new scalars against the 3 x (1+3) electroweak doublets. So, it seems to make sense to look if some of the advantages of supersymmetry - whatever they are - would be present - by accident - in this theory too.

 

[Orodruin]

What I would like to add (for completely different reasons) is a single scalar field for each electroweak doublet. The triplets for quarks would be colored, those for leptons would not interact with gauge fields at all.

This is inconsistent. You cannot have an electroweak doublet that does not interact with gauge fields.

 

[Ilja]

This is inconsistent. You cannot have an electroweak doublet that does not interact with gauge fields.

But I can have scalar fields which do not interact with gauge fields, and they can be associated with electroweak doublets.

This association would be a correspondence for numbers (there will be one scalar field for every electroweak doublet). Moreover, each scalar field borrows its gauge charges from one preferred component of "its" electroweak doublet. In the case of leptons, this preferred component will be the right-handed neutrino, for quarks the right-handed anti-down-type quark (EM charge 1/3).

 

[Orodruin]

Why don't you write down your intended Lagrangian? It will be more effective (haha) than trying to describe your idea in words.

 

[Ilja]

I don't think this would simplify something. The Lagrangian of scalar fields interacting with gauge fields would be nothing, new, thus, only a misdirection of interest (one would think there is something interesting in this formula) and what matters would be anyway the description of the number of fields and their charges.

 

[Orodruin]

what matters would be anyway the description of the number of fields and their charges

Exactly, and I am not getting a clear picture of exactly what you want to do, which fields you want to couple to the scalars and how. This would be much simpler if you just wrote down the (interaction) Lagrangian you had in mind.

 

[Ilja]

If $$\mathcal{L}= \frac12 D_\mu \phi^*_{ga} D^\mu \phi_{ga} - \frac12 m^2_{ga} \phi^*\phi$$ with $$D_\mu = \partial_\mu + i A^b_\mu T^b$$ makes you happy. Here g is the generation index, a is from 0 (for leptons) to 3 (for the three quark colors), b goes over the SM gauge fields, and the $T^b$ describe the representation I have to describe in words anyway: It acts trivially on the $\phi_{g0}$ and by the standard three-dimensional representation of U(3) (obtained by factorizing out weak interactions $SU(2)_L$ and $\mathbb{Z}_3$ from the SM gauge group) on the three-dimensional space of the $\phi_{gi}$ with $i=1,2,3$, g fixed.

 

[ChrisVer]

I'm under the impression that this theory haven't been garnering much love from the non-string physics community, of late.

That's true from my case since both Dark Matter scenarios (supersymmetric WIMPs) as well as the Hierarchy problem solutions of SUSY become less favorable as you go to larger energies. As already mentioned the hierarchy problem could be solved by SUSY if the scale of it was around the TeV scale. Obviously we haven't yet found anything in the LHC and the parameter space for the theory to exists gets tighter. On the other hand they always try to save SUSY models like MSSM, either by extending it or I don't know exactly , say that it might still be invisible at the LHC.

I still like SUSY but the normal things the theory could solve, have been cornered a lot.

The unification of the coupling constants is (for me) an aesthetic need of some people and I don't really like it (as a reason)
Check eg. this figure :
http://scienceblogs.com/startswithabang/files/2013/05/running_coupling.gif
The left is the SM and the right is the MSSM.
I don't personally find any reason to have the lines changing at only 1 given energy (at around 10^3 GeV ~ 1 TeV scale).

If you don't care about these problems and don't care about experiments (like string theorists) then you can allow for SUSY to exist at any energy.

 

[Orodruin]

So what are your Yukawa couplings?

If your fields transform non-trivially under any of the gauge groups, they are going to have gauge couplings so what is the point of having the covariant derivative there at all?

 

[Ilja]

The coupling constants are the same as those of the quarks.

 

[Orodruin]

The coupling constants are the same as those of the quarks.

Which coupling constants? The Yukawa couplings are coupling constants between fermions and scalars. If you have gauge coupling constants your scalar fields interact with the gauge fields. Or are you saying that you want to have a scalar field that does not have any quantum number? What is going to stop this field from coupling to every fermion in the SM?

 

[Ilja]

Yes, these scalar fields are not intended as Higgs fields which have to give any fermions any mass.

 

[Orodruin]

It is not a matter of having to give fermions mass, it is a matter of which terms are allowed in your Lagrangian. If they are allowed, they will be there or you need to explain why they are not. If you have a singlet scalar $\phi$, what stops me from writing down the interaction term $\phi \bar q_R q_R$?

 

[Ilja]

So ok, you are free to add whatever you want or think it is somehow unpreventable. I have seen no justification or necessity to write down such terms, so I do not write them down by Ockham's razor, that's all.

edit: Thinking about this a little bit more, such a Yukawa term, which would connect the scalar field with the corresponding electroweak doublet, would be quite natural.

 

[ChrisVer]

whatever you want

Whatever you are allowed to... If you are allowed to write something, then you have to write it, otherwise explain how this thing went on missing.
If you could discard terms like this in the Lagrangian, then there would be no strong-CP-problem and there would be no "bad" proton decays in several other theories.

 

[Orodruin]

So ok, you are free to add whatever you want or think it is somehow unpreventable. I have seen no justification or necessity to write down such terms, so I do not write them down by Ockham's razor, that's all.

edit: Thinking about this a little bit more, such a Yukawa term, which would connect the scalar field with the corresponding electroweak doublet, would be quite natural.

Ockham's razor would actually go in the other direction. If you do not prohibit the coupling using some symmetry, it is generally going to be generated by running at one scale even if it is zero at another scale.

 

[Orodruin]

Thinking about this a little bit more, such a Yukawa term, which would connect the scalar field with the corresponding electroweak doublet, would be quite natural.

And I almost forgot: Regardless of how "natural" this would seem to you, it would induce flavour changing neutral currents.

 

[Ilja]

And I almost forgot: Regardless of how "natural" this would seem to you, it would induce flavour changing neutral currents.

What would be the problem with this?

 

[Orodruin]

What would be the problem with this?

That flavour changing neutral currents are not observed in Nature?

 

[Ilja]

That flavour changing neutral currents are not observed in Nature?

In this case, please explain in more detail how all this happens.

I start with a scalar particle which does exactly nothing, except of having a mass. You tell me, that it is somehow unavoidable that there appear some terms of some Yukawa-like interaction $\psi* \psi \phi$. Ok, maybe, so what, and I have not a big problem with accepting such terms, they look not that unproblematic if that scalar is connected with the fermions of its own associated electroweak doublet. Anyway, except for the right-handed neutrino they are all connected with each other by gauge fields. Now you tell me that this leads somehow to some process you have named "flavour changing neutral current" which does not exist in the SM/is not observed in Nature.

Sorry, but this is already too much mystery, so please details. In particular, why (AFAIU, correct me if not) everything is fine if the Higgs particle somehow interacts in some terms with all fermions, but everything is fine with this, but if my completely innocent scalar field is doing the same thing all is unavoidably wrong.

The second point I don't understand the terminology. How would you name neutrino oscillations - AFAIU they change the flavour of the neutrino, and whatever happens there is neutral, not?

 

[Orodruin]

"flavour changing neutral current" which does not exist in the SM/is not observed in Nature.

A flavour changing neutral current gives rise to processes such as $e + X \to \mu + X$ or $e^- + e^- \to \mu^+ + \mu^-$. There are strong constraints on this type of processes. Even if you introduce one scalar per type of particle and per generation, you will likely run into problems with lepton/quark universality (which is essentially saying that all leptons/quarks interact with the same strengths.

In particular, why (AFAIU, correct me if not) everything is fine if the Higgs particle somehow interacts in some terms with all fermions

Yes, this is the point. The Higgs interactions define what flavour is. Anything else you do in the scalar sector is likely to screw this up unless you engineer your model in a way that ensures that it does not, the general idea here is that what gives rise to the Higgs Yukawa couplings also gives rise to other interactions which could potentially be flavour violating.

but if my completely innocent scalar field is doing the same thing all is unavoidably wrong.

Adding scalars is not something that is easy and essentially no scalar field is "innocent" as far as interactions go. The point is that your scalar fields would have a structure which is different from the flavour structure imposed by the Higgs Yukawas and therefore induce flavour changing neutral currents.

How would you name neutrino oscillations - AFAIU they change the flavour of the neutrino, and whatever happens there is neutral, not?

No, neutrino oscillations are due to charged current interactions (this is how you create and detect neutrinos). It is no stranger than flavour changing charged currents in the quark sector and is precisely based on the Higgs Yukawas not being diagonal in the interaction basis. The SM is fine with flavour changing charged currents and they are observed in Nature. The reason you get oscillations is that the neutrino mass eigenstates, unlike the quark mass eigenstates, have very small mass differences, leading to interference terms when you compute certain processes.

 

[ChrisVer]

Yes, this is the point. The Higgs interactions

define what flavour is. Anything else you do in the scalar sector is likely to screw this up unless you engineer your model in a way that ensures that it does not, the general idea here is that what gives rise to the Higgs Yukawa couplings also gives rise to other interactions which could potentially be flavour violating.

I thought that the Higgs also introduced FCNCs with its coupling to top quark, but GIM mechanism +CKM matrix strong suppresses it?

Adding scalars is not something that is easy and essentially no scalar field is "innocent" as far as interactions go. The point is that your scalar fields would have a structure which is different from the flavour structure imposed by the Higgs Yukawas and therefore induce flavour changing neutral currents.

Well for that case one could look for how H2DMs (two higgs doublet models) deal with the problem of FCNCs at low energies. I think you have to add an additional U(1)-type symmetry for yukawa couplings, and either allow only the one higgs to give mass to the fermions (type I models) or allow for the one Higgs to give the mass to up-type quarks and another to down-type quarks (type II models).

Or at least that's what I understood from this:
http://arxiv.org/pdf/hep-ph/9806282v3.pdf
(2nd paragraph of introduction)

 

[arivero]

theoretical/experimental motivations

Well, the point that it somehow generates the momentum operator is awesome for mathematics, or should be. Of course you are asking for the motivation in physics...

 

[Ilja]

I will come back to this, because there are clearly some points I have not yet understood and I have to ask about. But part of this is, clearly, a point where I am possibly completely off for quite similar reasons.

The point is that I do not understand the reasons for introducing Higgs bosons at all. So what I read is that simply defining massive gauge fields would give a non-renormalizable theory, and using a gauge-invariant theory together with a Higgs somehow circumvents this. But, again, from point of view of effective field theory this looks like a solution of a non-problem: Simply define massive gauge fields from the start, at the critical distance, and let the long distance limit sort out the non-renormalizable parts of it.

Ok, if you really want to compute something, using a renormalizable theory may be technically simpler, even if one has to introduce additional fields, because overwise one would have to care about non-renormalizable terms and to show how they are suppressed in the long distance limit, which may be technically more difficult. But these would be, so to say, human arguments, Nature does not have to care about, so if I want to make guesses about theories at the critical distance, I should not care about such simplicity too. And some approximate gauge symmetry at the critical distance would be, from point of view of Occams razor, simpler. What is wrong with this, and better in the Higgs model?

One point could be that gauge theories with exact gauge symmetry can be made manifestly Lorentz-invariant. Gupta-Bleuler vs. the old Dirac-Fermi quantization. But this is also not decisive for me, nothing prevents a not-manifestly-Lorentz-covariant theory from being fine as an effective theory, anyway general relativity is fine only as an effective fields theory.

Another point could be unification with the massless EM field. If one wants to unify photons with weak fields, one would have to use the same number for fundamental degrees of freedom for them, so, once weak fields are massive, and the photon is not massive, one would have, nonetheless, to introduce its gauge degree of freedom, which would become simply an independent scalar field, not connected with the other, observable degrees of freedom of the EM field. What would prevent this field from playing the same role as the Higgs?

 

[haushofer]

But, again, from point of view of effective field theory this looks like a solution of a non-problem:

Actually, I also never really understood this; it is often given as the explanation for the Higgs, but in retrospect and in the view of effective field theory this also seems to me not a good reason. I'm sure someone here can elaborate on that :)

It's related to the question "why should the SM be renormalizable?" which, I think, has as answer "it doesn't need to be".

 

[haushofer]

I asked this question here,

https://www.physicsforums.com/threads/understanding-the-need-for-the-higgs-field.724415/