Why Complex Scalars in 4D Supersymmetric Theories?

[Emilie.Jung]

The scalar fields of supersymmetric theories in 4 spacetime dimensions are a set of complex fields (usually denoted by $z^{\alpha}$). How can this be physically translated?

More precisely, we know that in 5D, those scalars are real, so what is that makes them real here but complex there?

 

[haushofer]

They carry a U(1) charge.

 

[Emilie.Jung]

Sorry, can you elaborate more? @haushofer

 

[haushofer]

Well, write down the Lagrangian for a real scalar field. Does it have a U(1)-invariance? I'm sure you'll find it has not. Now write down the Lagrangian for a complex scalar field. You can check now that this Lagrangian does have U(1)-invariance, and you can calculate the conserved current. An interpretation is that if you can couple this current to a vector field A. We say that the scalar is charged under U(1).

Do you have this U(1) charge in the five dimensional case also?

 

[Emilie.Jung]

It shouldn't have U(1) charge in 5 D, given what you said, right? @haushofer

 

[haushofer]

Exactly.

 

[Emilie.Jung]

1)scalar being charged under U(1) is said to be complex? If so, why? @haushofer

 

[samuelphysics]

@haushofer But you start by setting as a given that the lagrangian holds scalar fields or complex fields and you build on that, meanwhile the question was why is this the case? Why in 4D you have complex scalars and in 5D you have real scalars? It seems you answered the question by setting the supposed answer as a given.

 

[haushofer]

I'm sorry, I've misread the question. I suppose you have to do a counting of the amount of dof's for d=4 and d=5. I'll see if I can find something explicit.

 

[samuelphysics]

Maybe @fzero can give us an insight on how to tackle this and if there a different approach to the answer?

 

[Emilie.Jung]

Thanks @haushofer for your input anyway, will be glad if you could add also a more explicit answer if you could.

 

[haushofer]

I've looked for some explanation and found this paper on N=2, D=5 SUGRA (I'm more familiar with SUGRA, but the idea is the same):

http://arxiv.org/abs/hep-th/0004111

Have you done a counting of the (on-shell) degrees of freedom? Also, have you tried to consult Van Proeyen his Paris lectures (page 8),

http://itf.fys.kuleuven.be/~toine/LectParis.pdf

I haven't done the counting by myself before and I'm not that familiar with the D=5 case, but maybe an important difference with D=4 is that one cannot choose Majora or Weyl-fermions in D=5 (only so-called symplectic Majorana spinors). This should affect the counting.

Here,

http://bolvan.ph.utexas.edu/~vadim/Classes/01f/396T/table.pdf

it's said that for N=2, D=5 "Each vector multiplet contains one vector field, five real scalars and two Dirac spinors.". Let's do the counting, first of-shell:

A dirac spinor has 2^{[5/2]} = 4 complex components = 8 real components, so two Dirac spinors have 16 real components
A vector field has due to gauge symmetry 4 real degrees of freedom
5 real scalars have 5 real degrees of freedom

On-shell we get

Two Dirac spinors have 16/2=8 real components
A vector field has D-2=3 real components
5 real scalars have 5 real degrees of freedom.

So on-shell I get bosonic dof = 3+5=8 = fermionic dof, which seems to be right; complex scalars would add 5 more real degrees of freedom (on-shell and off-shell).

I hope this helps :)

 

[haushofer]

I haven't looked at the action, but you should also find that the five scalars are not charged under the U(1) of the vector.

 

[Emilie.Jung]

Thanks a lot for your reply and you giving all the effort to search about this. I will have to think about it and then comment on it. @haushofer

 

[Emilie.Jung]

Thank you again for your answer, it is until today that I had time to sit and research about few things you included in your answer, @haushofer

Have you done a counting of the (on-shell) degrees of freedom?

Why would I? What exactly are on- and off-shell d.o.f? Sorry for this question which I understand you consider it as very basic thing in your nNswer, but I don't know what they are or what they can tell us. I understood though how did you cont them later on, so no need to point out the counting procedure.

So on-shell I get bosonic dof = 3+5=8 = fermionic dof, which seems to be right

What seems to be right here?

 

[haushofer]

Well, that on-shell i.e. with using the equations of motion, the amount of fermionic degrees of freedom equals the bosonic ones. If the scalars would be complex this would not be the case; you would have 5 more bosonic dof's, and supersymmetry would not be realized.

Maybe it helps to say that 1 complex dof equals 2 real dofs. It could be a nice exercise for you to check this counting for some (to you) familiar cases. For this you need to know the spinor-representations in different dimensions (when can i choose Weyl, Majorana,...). Van Proeyen's Tools for Susy is an excellent review for this. E.g., in four dimensions you can choose Majorana spinors, which have in four dimensions four real components. The Dirac equations makes half of them dependent on the other half, leaving 2 components (dofs) on-shell.

Let me know if this clearifies anything ;)

 

[haushofer]

So on-shell =using the EOM
Of-shell = not using the EOM

For scalars the counting is the same (why?). For vectors the EOM make one dof dependent on the other ones, which removes one dof. For spinors the Dirac eqn. makes half of the components dependent on the other half, cutting the dof's to half.

 

[Emilie.Jung]

Thanks loads @haushofer for your answers, I have few questions to clarify our entire conversation:

A) So, let me get this straight. If we referred only to my question above, we would say that in order to know why is it that in 4D we have complex scalars and in 5D we have scalar ones, we have to go check the on-shell degrees of freedom because if the fermionic on-shell degrees of freedom equal the bosonic on-shell degrees of freedom then the scalars are real scalars, is that correct?

B) Then you said if scalars were complex instead, then we would have more bosonic on-shell degrees of freedom than the fermionic ones, is this also correct?

C) Finally you said, supersymmetry would not be realized if scalars were complex, right? But how come? We have complex scalars in 4D theories and although this is the case, we have supersymmetric solutions in N=2, D=4 supergravity (supersymmetric) theories, which means that supersymmetry is realized when we have complex scalars, do you disagree with this?

 

[haushofer]

A) yes B) yes C) I said this in the D=5, N=2 case for the vectormultiplet. For e.g. D=4, N=1 the Wess-zumino model consists of a Majorana (i.e. Real ) fermion and a complex scalar. You should now be able to argue why this scalar must be complex in order to realize on-shell susy, and why you need a complex auxiliary field to realize off-shell susy.

So whether scalars in multiplets are real or complex depends on your spacetime dimension. So no, I don't disagree with the fact that D=4 has complex scalars (half the amount of real scalars). Notice also that the five real scalars in D=5 cannot be written in terms of complex scalars (5/2= 2 1/2) while the complex scalar for the D=4 Wess Zumino theory are just two real scalars.

 

[Emilie.Jung]

A)Ok @haushofer ... this is a little bit confusing... because we agreed above that on-shell dof are there to tell us whether our scalars must be complex or real, but now you write

You should now be able to argue why this scalar must be complex in order to realize on-shell susy, and why you need a complex auxiliary field to realize off-shell susy.

the confusing part was the off-shell, what do we have to do with off-shell? I thought we shouldn't care about it... :(
B)Finally, you say in your second paragraph something that I understood as follows:
If complex scalars have even number, then susy holds? If not then no? If yes, does this rule hold in any dimension like N=2, D=4 one?

 

[haushofer]

No, susy holds when your amount of fermionic dofs equals the amount of bosonic dofs. This can be on-shell or of-shell. In the first case you need to use the EOM in order to close the susy-algebra on the fields, in the second you don't.

You shouldn't care about whether your scalars must be real or complex, but just how much of them you need. For the wess zumino model in d=4 n=1, you have one fermion. In four dimensions a dirac fermion has 4 complex components. But in four dimensions you are allowed to choose majorana representations, so your fermion can be majorana, having 4 real components. On shell these 4 real components are reduced to 2 real ones. So on-shell you need two real scalars to realize susy. But two real scalars can be stacked into one complex scalar. Off-shell you need 4 real scalars to match the four real fermionic dofs. These can be stacked into two complex scalars. But then one complex scalar should't be dynamical but auxiliary, otherwise the on-shell counting is messed up.

Now review my counting for the d=5, n=2 vector multiplet. There you need 5 real scalars. But you cannot stack 5 real scalars into a complex one, that doesn't make sense. I think it should be possible, on basis of counting, to stack them into two complex plus one real, but I'm not fully sure about that because I haven't considered the action.

 

[haushofer]

If you check the closure of the susy-algebra of your theory, do you need the EOM for it or not?