SUSY multiplets - simple question

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In summary, the SUSY (N=1) massless supermultiplet consists of a scalar and a Weyl fermion in the chiral multiplet, and a Weyl fermion and a vector field in the vector multiplet. The Weyl fermions in the two multiplets cannot be identified with each other due to their different transformation properties under the gauge group.
  • #1
Orion2321
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Hi,
I have a conceptual problem in understanding the SUSY (N=1) massless supermultiplet.
Using appropriately normalized creation and annihilation operators Q, Q+ (only one component survives in this representation) we have for the quark state:

Q+|p,-1/2>=0 (quark) where the 1/2 labels the eigenvalue of J3 spin operator.

For the gluino we can write
Q+|p,-1/2>=Q+Q|p,-1>=(2E-QQ+)|p,-1> ~ |p,-1> (gluon)

I don't understand why can we go from the gluino to the gluon but for the quark (also spin 1/2 like the gluino) the superpartner is a scalar.
 
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  • #2
Orion2321 said:
Hi,
I have a conceptual problem in understanding the SUSY (N=1) massless supermultiplet.
Using appropriately normalized creation and annihilation operators Q, Q+ (only one component survives in this representation)

I'm not sure why you think that only one component survives. In 4d, you need both spinor components.

we have for the quark state:

Q+|p,-1/2>=0 (quark) where the 1/2 labels the eigenvalue of J3 spin operator.

For the gluino we can write
Q+|p,-1/2>=Q+Q|p,-1>=(2E-QQ+)|p,-1> ~ |p,-1> (gluon)

I don't understand why can we go from the gluino to the gluon but for the quark (also spin 1/2 like the gluino) the superpartner is a scalar.

These are two different representations. To describe a supermultiplet, you also need to describe its representation under the Poincare algebra. So it's best to start with a Poincare multiplet, which is described by a mass [tex]M[/tex] and a spin [tex]s[/tex]. The states are labeled as [tex]|M,s,m_s\rangle[/tex]. Since you want massless multiplets, we'll take [tex]M=0[/tex] and just write the states as [tex]|s,m_s\rangle[/tex].

We view [tex]Q_\alpha[/tex] as lowering operators and [tex]Q^\dagger_{\dot{\alpha}}[/tex] as raising operators. The lowest weight state is

[tex]|\Omega_s \rangle = Q_1 Q_2 | s, m_s \rangle,[/tex]

since

[tex]Q_1 | \Omega_s\rangle =Q_2 | \Omega_s\rangle =0.[/tex]

The rest of the states in the supermultiplet are generated by action with [tex]Q^\dagger_{\dot{\alpha}}[/tex]. So the supermultiplet is

[tex] |\Omega_s \rangle , ~~ Q^\dagger_{\dot{1}} | \Omega_s\rangle,~ Q^\dagger_{\dot{2}} | \Omega_s\rangle , ~~Q^\dagger_{\dot{1}} Q^\dagger_{\dot{2}} | \Omega_s\rangle . [/tex]

For the scalar, or chiral, supermultiplet, we start with the state

[tex]|\Omega_0 \rangle = Q_1 Q_2 | 0, 0 \rangle . [/tex]

Acting with [tex]Q^\dagger_{\dot{\alpha}}[/tex] gives us a Weyl fermion

[tex]
Q^\dagger_{\dot{1}} | \Omega_s\rangle,~ Q^\dagger_{\dot{2}} | \Omega_s\rangle \sim | \tfrac{1}{2},\pm \tfrac{1}{2} \rangle ,
[/tex]

while

[tex]
Q^\dagger_{\dot{1}} Q^\dagger_{\dot{2}} | \Omega_s\rangle \sim | 0,0\rangle
[/tex]

pairs with the lowest weight state to form a complex scalar.

For the vector multiplet, we start with a Weyl fermion

[tex]|\Omega_{1/2} \rangle = Q_1 Q_2 | \tfrac{1}{2}, \pm \tfrac{1}{2} \rangle . [/tex]

Acting with [tex]Q^\dagger_{\dot{\alpha}}[/tex] gives us states that fill out a Lorentz vector

[tex]Q^\dagger_{\dot{\alpha}}|\Omega_{1/2} \rangle \rightarrow | 1 , m_1 \rangle[/tex],

while acting with [tex]Q^\dagger_{\dot{1}} Q^\dagger_{\dot{2}} [/tex] gives us a Weyl fermion of the opposite chirality.
 
  • #3
Thank you very much for your reply.
I'm not sure why you think that only one component survives. In 4d, you need both spinor components.
Here I mean that for the massless case [tex]p_{\mu}=(E,0,0,E)[/tex] the algebra yields [tex]\{Q_{a},\bar{Q}_{\dot{b}}\}[/tex]=[tex]2(\sigma^{\mu})_{a\dot{b}}P_{\mu}[/tex]=[tex]2E(\sigma^0+\sigma^3)_{a\dot{b}}[/tex]=[tex]{\left(\begin{array}{cc} 1 &0 \\ 0 &0 \end{array}\right)_{a\dot{b}}}[/tex] implying that [tex]Q_{2}[/tex] is zero in this representation.

Apart from that I agree with everything you said in your post. My final question is
For the vector multiplet, we start with a Weyl fermion
why can't we identify this Weyl fermion with the quark? In that case could we raise the spin eigenvalue [tex]m_{s}[/tex] by 1/2 and find a massless spin 1 state pairing with the quark?
 
  • #4
Orion2321 said:
Thank you very much for your reply.

Here I mean that for the massless case [tex]p_{\mu}=(E,0,0,E)[/tex] the algebra yields [tex]\{Q_{a},\bar{Q}_{\dot{b}}\}[/tex]=[tex]2(\sigma^{\mu})_{a\dot{b}}P_{\mu}[/tex]=[tex]2E(\sigma^0+\sigma^3)_{a\dot{b}}[/tex]=[tex]{\left(\begin{array}{cc} 1 &0 \\ 0 &0 \end{array}\right)_{a\dot{b}}}[/tex] implying that [tex]Q_{2}[/tex] is zero in this representation.

Apart from that I agree with everything you said in your post.

OK. I've given massive supermultiplets then. I've also missed a scalar in the massive vector multiplet.

My final question is

why can't we identify this Weyl fermion with the quark? In that case could we raise the spin eigenvalue [tex]m_{s}[/tex] by 1/2 and find a massless spin 1 state pairing with the quark?

The Weyl fermion in the vector multiplet must transform in the same representation of the gauge group as the vector field, so the adjoint. This explains why the gauginos are in adjoint representations, but it also means that you can't identify them with fundamental quarks. We can put chiral multiplets in arbitrary representations of the gauge group, so they're necessary.
 

1. What is a SUSY multiplet?

A SUSY multiplet is a group of particles that are related to each other through supersymmetry, a theoretical framework that proposes a symmetry between particles with integer and half-integer spin.

2. How are SUSY multiplets classified?

SUSY multiplets are classified based on their spin and other quantum numbers, such as electric charge and color charge. The most common types are chiral multiplets, gauge multiplets, and gravity multiplets.

3. What is the significance of SUSY multiplets in particle physics?

SUSY multiplets are important in particle physics because they provide a solution to the hierarchy problem, which is the large discrepancy between the strength of the electroweak and gravitational forces. They also offer a potential explanation for dark matter and can help unify the fundamental forces of nature.

4. Can SUSY multiplets be observed in experiments?

While SUSY multiplets have not yet been directly observed in experiments, their existence is supported by theoretical predictions and indirect evidence. Physicists are currently searching for evidence of SUSY multiplets through experiments at particle accelerators.

5. Are SUSY multiplets a proven concept?

No, SUSY multiplets are still a theoretical concept and have not been conclusively proven. However, they are a well-studied and widely accepted idea in particle physics, and their potential implications continue to be investigated through both theoretical and experimental research.

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