Constraints on Chiral superfield

In summary, according to the expert, a left chiral superfield can be obtained by imposing the constraint that the supercovariant derivative on it will have to vanish. This will bring some constraints about the component fields in the expansion of the superfield.
  • #1
ChrisVer
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Suppose we have a superfield [itex]\Phi(x,\theta,\bar{\theta})[/itex] this can be expanded in component fields in the standard way as:
[itex] \Phi(x,\theta,\bar{\theta})= c(x) + \theta \psi(x) + \bar{\theta} \bar{ζ}(x) + \theta^{2} F(x) + \bar{\theta}^{2} Z(x) + \theta \sigma^{\mu} \bar{\theta} u_{\mu}(x) + \theta^{2} \bar{\theta} \bar{λ}(x) + \bar{\theta}^{2} \theta β(x) + \bar{\theta}^{2} \theta^{2} D(x) [/itex]

In case we want to say that this superfield is chiral, in general we have to impose the constraint that the supercovariant derivative on it will have to vanish (depends on if it's right or left which we choose)... Nevertheless. This will bring some contraints about the component fields in [itex]\Phi[/itex]'s expansion above, which you can work by asking for the cov. derivative of it to vanish. However I'd like to ask if there's a faster way to do that, and avoid the tedious calculations... For example I'd take a left chiral superfield:
[itex] \Phi_{L}( y, \theta) = \phi(y) + \theta \psi(y) + \theta^{2} F(y) [/itex]
and bring it from [itex]S_{L}[/itex] repr back to [itex]S[/itex], by doing a translation in the usual way:
[itex] \Phi_{L} (x+ i \theta \sigma \bar{\theta}, \theta)[/itex]
[itex]= \phi(x) + i (\theta \sigma^{\mu} \bar{\theta}) \partial_{\mu} \phi(x) - \frac{1}{2} (\theta \sigma^{\mu} \bar{\theta})(\theta \sigma^{\nu} \bar{\theta}) \partial_{\mu} \partial_{\nu} \phi(x) + \theta \psi(x) + i (\theta \sigma^{\mu} \bar{\theta}) (\theta \partial_{\mu} \psi)+ \theta^{2} F(x) [/itex]

If I now try to compare the first superfield's components with the last one, shouldn't I get the constraints needed for it to be a chiral superfield?
eg fastly:
[itex]\bar{ζ}(x)=0[/itex]
[itex] Z(x)=0 [/itex]
[itex]D(x)= -\frac{1}{4} \partial_{\mu} \partial^{\mu} \phi[/itex] ( because [itex] (\theta \sigma^{\mu} \bar{\theta})(\theta \sigma^{\nu} \bar{\theta})=\frac{1}{2} n^{\mu \nu} \theta^{2} \bar{\theta}^{2}[/itex])
[itex] u_{\mu}= i \partial_{\mu} \phi(x) [/itex]
[itex]β(x)=0 [/itex]

etc...
 
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  • #2
##x →x+ i \theta \sigma \bar{\theta}## is precisely the shift which makes covariant derivative a simple derivative against ##\bar{\theta}## for left chiral superfield and hence make sure that there is no explicit dependence on ##\bar{\theta}## of the superfield. So it is not in any sense arbitrary. Now if you want to use it to put constraint on Superfield, good for you. (Have to run now)
 
  • #3
I don't really get the answer...
A shift [itex]x\rightarrow x + i \theta \sigma \bar{\theta}[/itex] is going to change the cov.derivative of the Left repr to the Cov derivative of the S-repr...

[itex] \bar{D}_{\dot{A}}= - \frac{\partial}{\partial \bar{\theta}^{\dot{A}}} [/itex]

[itex]= - \frac{\partial \bar{\theta}'^{\dot{B}}}{\partial \bar{\theta}^{\dot{A}}} \frac{\partial }{\partial \bar{\theta}'^{\dot{B}}}- \frac{\partial y^{\mu}}{\partial \bar{\theta}^{\dot{A}}} \partial_{\mu}^{(y)}[/itex]

[itex]= - \frac{\partial}{\partial \bar{\theta}^{\dot{A}}} + i (θσ^{\mu})_{\dot{A}} \partial_{\mu}^{(y)}[/itex]

which isn't the same as the cov derivative in the general representation... :confused: any idea?
However I am sure after using it so many times that the coord change above is the one that's supposed to move me from [itex]S_{L} \rightarrow S[/itex] repr...
 

1. What is a chiral superfield?

A chiral superfield is a type of field in supersymmetric theories that has both bosonic and fermionic components. It is a fundamental building block for constructing supersymmetric Lagrangians.

2. What are the constraints on chiral superfields?

The constraints on chiral superfields are mathematical conditions that restrict the allowed values of their components. These constraints are necessary for the consistency of supersymmetric theories and include the chirality condition and the F-term condition.

3. How do the constraints on chiral superfields affect supersymmetric theories?

The constraints on chiral superfields play a crucial role in determining the properties and interactions of particles in supersymmetric theories. They ensure the stability and predictability of the theory, and also help to simplify calculations.

4. Can the constraints on chiral superfields be broken?

While the constraints on chiral superfields are fundamental to supersymmetric theories, there are scenarios where they can be broken. This can lead to interesting phenomena such as the generation of mass for particles that were previously massless.

5. What are the implications of the constraints on chiral superfields for experimental searches?

The constraints on chiral superfields provide important predictions for the properties and behavior of supersymmetric particles, which can guide experimental searches for these particles. They also help to rule out certain supersymmetric models that do not satisfy the constraints.

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