# Spontaneusly broken R-symmetry

• diracologia
In summary, R-symmetry is a symmetry in supersymmetric theories that treats the supersymmetric partners differently. If this symmetry is spontaneously broken, it can also lead to the breaking of supersymmetry. This is important in constructing models of supersymmetry. However, this statement is not always true and requires additional conditions, such as a generic superpotential. A generic superpotential includes all allowed operators and is necessary for the breaking of R-symmetry to also break supersymmetry.
diracologia
Spontaneously broken R-symmetry is a sufficient condition for spontaneously supersymmetry breaking.

My question is: What is a spontaneously R-symmetry breaking?

This is technical. Since I have no idea what your level is, I'll give you the full answer. But if you don't understand any of this, then the short answer is, "this is a technical thing that theoretical physicists consider when constructing mathematical models of supersymmetry."

An R symmetry in SUSY is a symmetry that treats the supersymmetric partners differently. The most famous example is "R-parity" where the supersymmetric particles are "odd" and the ordinary particles are "even".

When you have a SUSY theory, you often (perhaps ALWAYS - I can't think of a counterexample) have an "R symmetry" as well. For so called "N=1 SUSY" the R symmetry is U(1). For "N=2 SUSY" the R symmetry is SU(2), etc. The statement you are making is that if you spontaneously break the R symmetry, you also tend to break SUSY as well. This is very important if you are trying to build a model of SUSY, for example, and you are trying to figure out how SUSY is broken - if you can find a way to break the R symmetry, then you also break SUSY while you're at it!

The minimal supersymmetric standard model (MSSM) is an example of N=1 SUSY whose U(1) R symmetry was broken down to a Z_2 parity. I have written papers that consider N=2 SUSY that does NOT break the R symmetry (we called it Minimal-R-symmetric-Supersymmetric-Standard-Model, or MRSSM).

The above statement you made is also not true in general - you need something called "generic conditions". But that's another technical point...

Hope that helps!

blechman said:
This is technical. Since I have no idea what your level is, I'll give you the full answer. But if you don't understand any of this, then the short answer is, "this is a technical thing that theoretical physicists consider when constructing mathematical models of supersymmetry."

An R symmetry in SUSY is a symmetry that treats the supersymmetric partners differently. The most famous example is "R-parity" where the supersymmetric particles are "odd" and the ordinary particles are "even".

When you have a SUSY theory, you often (perhaps ALWAYS - I can't think of a counterexample) have an "R symmetry" as well. For so called "N=1 SUSY" the R symmetry is U(1). For "N=2 SUSY" the R symmetry is SU(2), etc. The statement you are making is that if you spontaneously break the R symmetry, you also tend to break SUSY as well. This is very important if you are trying to build a model of SUSY, for example, and you are trying to figure out how SUSY is broken - if you can find a way to break the R symmetry, then you also break SUSY while you're at it!

The minimal supersymmetric standard model (MSSM) is an example of N=1 SUSY whose U(1) R symmetry was broken down to a Z_2 parity. I have written papers that consider N=2 SUSY that does NOT break the R symmetry (we called it Minimal-R-symmetric-Supersymmetric-Standard-Model, or MRSSM).

The above statement you made is also not true in general - you need something called "generic conditions". But that's another technical point...

Hope that helps!

Hi Blechman,

First of all, I was studying and thinking about my initial question and maybe I got the answer.

May I say that R-symmetry is spontaneously broken if and only if any R-charged superfield get a non-vanishing VEV?

Another question:

In fact spontaneously broken R-sym is a sufficient cond. for susy breaking only if the superpotential is generic. I gave a look at the paper of Nelson, Seiberg, but I didn't understand exactly what is a generic superpotential. Would you give me some hint?

Tank you.

diracologia said:
Hi Blechman,
May I say that R-symmetry is spontaneously broken if and only if any R-charged superfield get a non-vanishing VEV?

That is correct.

In fact spontaneously broken R-sym is a sufficient cond. for susy breaking only if the superpotential is generic. I gave a look at the paper of Nelson, Seiberg, but I didn't understand exactly what is a generic superpotential. Would you give me some hint?

That is right. When you build models (or do any kind of "effective field theory", or EFT), the idea is that you have a list of fields and symmetries, and you write down every operator you can that is made of products of the fields and is consistent with those symmetries. In ordinary EFT, you are REQUIRED to do this, since if you miss an operator, quantum corrections will generate those operators upon renormalization. This is why, for example, you can never set the Higgs mass to zero in the Standard Model.

But SUSY is different: thanks to non-renormalization theorems, even if you don't include all the allowed operators, you will NOT generate them from quantum corrections. This is very special to supersymmetric theories and is not true in general.

So when building SUSY models, you can write down "incomplete" superpotentials that are consistent even at the quantum level. These superpotentials are called "non-generic". A "generic" superpotential would be one that includes every operator allowed by all the symmetries. If you had such a superpotential, and it is arranged so that there is a spontaneously broken R symmetry, then Nelson-Seiberg point out that (at least in N=1) SUSY is also broken.

Note that broken R symmetry is sufficient but **NOT** necessary.

## 1. What is spontaneously broken R-symmetry?

Spontaneously broken R-symmetry refers to a phenomenon in theoretical physics where the symmetry of a system is broken at a low energy scale, even though the underlying laws of the system are symmetric. This can occur in systems that possess a global R-symmetry, which is a type of symmetry that does not depend on the position or orientation of the system.

## 2. How does spontaneously broken R-symmetry occur?

Spontaneously broken R-symmetry occurs when the ground state of a system does not possess the same symmetry as the underlying laws that govern the system. This can happen due to the presence of a Higgs field, which is a type of field that gives mass to particles and can also cause symmetry breaking.

## 3. What are the implications of spontaneously broken R-symmetry?

The implications of spontaneously broken R-symmetry are significant in theoretical physics, as it can help explain the origin of mass in the universe and can also have implications for the search for a unified theory of all fundamental forces. It also has applications in cosmology, as it can play a role in the evolution of the early universe.

## 4. Can spontaneously broken R-symmetry be observed in experiments?

Yes, the existence of spontaneously broken R-symmetry has been confirmed through various experiments, such as the discovery of the Higgs boson at the Large Hadron Collider. Additionally, the predictions of this phenomenon have been successfully tested in other experiments, providing further evidence for its existence.

## 5. How is spontaneously broken R-symmetry related to other types of symmetry breaking?

Spontaneously broken R-symmetry is related to other types of symmetry breaking, such as spontaneous symmetry breaking in particle physics and spontaneous gauge symmetry breaking in the Standard Model. However, it differs in that it involves the breaking of a global R-symmetry rather than a local symmetry. It also has connections to other areas of physics, such as condensed matter physics and cosmology.

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