What is a Quark Field? Answers & Explanation

[dextercioby]

What is a quark field??

According to the SM of particles and interractions,a quark (any of 'em) is the quanta of a quark field.So what is a quark field??
Is it:
a)an irreductible representation of the SU(3)-color symmetry group,
b)an irreductible representation of the SU(6)-flavor symmetry group,
c)a $$(\frac{1}{2},0) + (0,\frac{1}{2})$$ representation of the full Poincaré group,
d) something else??

Please,give me a correct answer...

 

[humanino]

Quarks are a weak doublet : $${\mathbf Q}_i=\left( \begin{array}{c}{\mathbf u}_i\\{\mathbf d}_i\end{array}\right)_L\sim({\mathbf 2},{\mathbf 3}^c)_{+\frac{1}{3}}$$ with notation $$(SU(2)_W,SU(3)_c)_Y$$
In fact, all fermions are represented by a two-component Weyl left-handed field.

That means : $${\mathcal L}=\sum_i {\mathbf Q}_i^\dagger\sigma^\mu{\mathcal D}_\mu{\mathbf Q}_i$$ with covariant derivative $$\mathcal D}_\mu{\mathbf Q}_i=(\partial_\mu+\imath{\mathbf A}_\mu+\imath{\mathbf W}_\mu+\frac{\imath}{2}yB_\mu){\mathbf Q}_i$$ (the factor $$\frac{1}{2}$$ in front of the hypercharge $$y$$ is conventional) and $$\mathbf W}_\mu=\frac{1}{2}W_\mu^a(x)\tau^a$$ and $$\mathbf A}_\mu=\frac{1}{2}A_\mu^A(x)\lambda^A$$ where $$\tau$$ and $$\lambda$$ are the Pauli and Gell-Mann matrices for $$SU(2)_W$$ and $$SU(3)_c$$

So I would say :
a) yes
b) no
c) somehow... they do have mass though, so eventually the chiralities mix
d) yes, at least a weak doublet

See for instance :
www.df.unipi.it/~astrumia/SM.pdf

 

[Haelfix]

In regards to b)

Its actually a good approximation if you consider there to be an SU(2) flavor symmetry, assuming you just consider u and d quarks.. Sometimes people even include s with su(3). They then model it as having the quark masses generating the symmetry breaking.

However in general the full Su(6) flavor symmetry is badly broken in reality, and just not applicable to QCD.

 

[dextercioby]

Quarks are a weak doublet : $${\mathbf Q}_i=\left( \begin{array}{c}{\mathbf u}_i\\{\mathbf d}_i\end{array}\right)_L\sim({\mathbf 2},{\mathbf 3}^c)_{+\frac{1}{3}}$$ with notation $$(SU(2)_W,SU(3)_c)_Y$$
In fact, all fermions are represented by a two-component Weyl left-handed field.

That means : $${\mathcal L}=\sum_i {\mathbf Q}_i^\dagger\sigma^\mu{\mathcal D}_\mu{\mathbf Q}_i$$ with covariant derivative $$\mathcal D}_\mu{\mathbf Q}_i=(\partial_\mu+\imath{\mathbf A}_\mu+\imath{\mathbf W}_\mu+\frac{\imath}{2}yB_\mu){\mathbf Q}_i$$ (the factor $$\frac{1}{2}$$ in front of the hypercharge $$y$$ is conventional) and $$\mathbf W}_\mu=\frac{1}{2}W_\mu^a(x)\tau^a$$ and $$\mathbf A}_\mu=\frac{1}{2}A_\mu^A(x)\lambda^A$$ where $$\tau$$ and $$\lambda$$ are the Pauli and Gell-Mann matrices for $$SU(2)_W$$ and $$SU(3)_c$$

So I would say :
a) yes
b) no
c) somehow... they do have mass though, so eventually the chiralities mix
d) yes, at least a weak doublet

See for instance :
www.df.unipi.it/~astrumia/SM.pdf

Thenx...You gave me a definition wrt to their interractions with the particles from the electroweak theory.I meant a definition wrt to the QCD.It doesn't matter,anyway.Wait a minute...Wouldn't it be fair to include interraction of electroweak particles with the quarks in a unifying theory,e.g.SU(5) Georgi-Glashow??

 

[humanino]

I did put emphasis on the eletroweak part, yet I included the color part. The reason for me to put emphasis on the electroweak part is that we understand it better and what we know is more elaborated on this part. On the color part, we know that quark belong to the fundamental representation of SU(3), antiquarks to the conjugate fundamental, and gluons are in the adjoint representation (for which one has to remove the symmetric "white" combination which otherwise would not be confined). Then, we do not understand much more. As Healfix said, the SU(6) flavor is badly broken (it is not fundamental). On the other hand, the electroweak part is more elaborated. There are several steps related to the construction of the massive vector bosons through the Higgs mechanism, involving the Weinberg angle as well as the Cabibbo mixing one. Then the consistency has to be checked in the cancellation of anomalies (necessary to ensure Ward-Takahashi identities and proofs of renormalizability). Of course, to be honnest I should mention the corresponding construction on the color part, with the Slavnov-Taylor identities. But as far as I understand, those do not put further restrictions on the representations for quarks. On the contrary, they tell us to add ghosts in the Feynman rules for QCD.

To present day, SU(5) unification is not necessary.

 

[dextercioby]

So it suffices to include quarks and gluons in the electroweak theory,because the theory itself is renormalizable and gives physically acceptable results...