All Questions
Tagged with quantum-chromodynamics representation-theory
43
questions
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How can I calculate action of $\mathfrak{su}(3)$ or other simple algebra ladder operators on "states" from the algebra commutators?
I wanted a way to "derive" Gell-Mann matrices for $\mathfrak{su}(3)$ and generalise this to other semi-simple algebras $\mathfrak{g}$. The way I wanted to approach this is start from the ...
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1
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449
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Why, in QCD, are quarks in the fundamental representation of $SU(3)$?
QCD is built from the notion that Dirac's Lagrangian should be invariant under gauge colour transformations.
Here, quarks are elements of $\psi_{\alpha,f,c}(x)$, where $\alpha$, $f$ and $c$ stand for ...
- 343
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0
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244
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One-loop renormalization of the gauge coupling
Quoting Yuji Tachikawa, chapter 3 of "${\cal N}=2$ Supersymmetric Dynamics for Pedestrians":
Recall the one-loop renormalization of the gauge coupling in a general Lagrangian field theory $$...
user242231
4
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3
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Is the concept of bicolored gluons mathematically precise/meaningful? Please explain
Each flavour of quark carries a colour quantum number: red, green or blue. I know what it means mathematically. But elementary textbooks (e.g, particle physics by Griffiths) also say that gluons are ...
- 11.8k
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2
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Quantum chromodynamics - why are there no $rrb$ or $ggr$ terms?
$$\Psi_{colour}^{qqq} = \frac{1}{\sqrt{6}}(rgb + gbr + brg -grb - rbg - bgr)$$
My lecturer stated that there cannot be any $rrb$ or $ggr$ terms in the expression above. I would like to understand what ...
- 13
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2
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How do we understand the ${\bf 3}$ of $Q_L({\bf 3}, {\bf 2})_{1/3}$?
A left-handed quark doublet of the Standard Model is specified as $Q_L({\bf 3}, {\bf 2})_{1/3}=(u,d)^T$. I have a problem understanding this quark doublet as a triplet of ${\rm SU}(3)$. Any help? I ...
- 11.8k
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How many colors really are there in QCD?
In abelian gauge theory (electrodynamics), the matter fields transform like (please correct me if I am wrong)
$$
|\psi\rangle\rightarrow e^{in\theta(x)}|\psi\rangle\tag{1}
$$
under a gauge ...
- 133
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How to decompose tensor products of $SU(3)$ representations? [duplicate]
Formally, one can arrange the quark flavors in a $SU(n)$ fundamental representation. One can then do tensor products for flavor and spin to construct other representations like baryons and mesons. An ...
- 11
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Do the “$SU(3)$ colors” live in a 3-dimensional vector space?
Previously I asked a question about the visualized colors:
Do the "colors" live in a 3-dimensional vector space?
(My earlier question is unfortunately closed)
Now I like to ask the “$SU(3)$ ...
- 4,336
3
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2
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313
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Normalisation of QCD Lagrangian
In QCD, and more generally in representations of $\mathfrak{su}(N)$, there is a freedom to choose the normalisation of the generators,
$$
\mathrm{Tr} \, \left[R(T^a) R(T^b)\right] = T_R \delta^{ab}.\...
- 264
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1
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Physical significance of the reality of an ${\bf N}$ representation: how the nature of interactions is affected?
Background The fundamental representation of ${\rm SU(N)}$ is denoted by ${\bf N}$ and the conjugate of the fundamental is denoted by ${\bar{\bf N}}$. If the representation ${\bf N}$ is related to ${\...
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0
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What is the application of dimension $6$ representation of $SU(3)$ in particle physics?
As we know, the $uds$ transforms in fundamental representations of $SU(3)$. It has the antifundamental partner. According to representation theory,
$$
\mathbf{3} \otimes \mathbf{\bar{3}}= \mathbf{8} \...
3
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2
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2k
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Confusions with gluons. How many of them are there?
Gluons are bicolored objects. They are made out of one color and one anticolor. Therefore, there seems to be nine possible states $r\bar{r},r\bar{b},r\bar{g},b\bar{r},b\bar{b},b\bar{g},g\bar{r},g\bar{...
- 11.8k
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Clarification about confinement of colour charged objects
In lecture today we were reviewing the QCD lagrangian, and discussing hadronic wavefunctions. My lecturer said that QCD alone allows for states of colored hadrons, however because we do not see ...
- 1,117
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$\mathfrak{su}(3)$ structure constants
The $\mathfrak{su}(3)$ structure constants $f^{abc}$ are defined by $$[T^a,T^b] = i f^{abc} T^c,$$ with $T^a$ being the generators of the group $\mathrm{SU}(3)$. They are usually written out in a very ...
- 109
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What prohibits fundamental fermions transforming like the $6$ and $6^*$ IRR's of $SU(3)$?
The lowest IRRs of SU(3) are 3,3* (the fundamental reps), 6,6*, and 8 (the adjoint rep). The quark fields are chosen to transform as 3, 3*, and the gluons as 8 under SU(3), but there is no ...
- 417
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134
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What is the physical meaning of Lie congruence classes?
Any weight $\lambda$ characterising a representation of $\mathfrak{su}(N)$ is an element of one of the $N$ congruence classes defined by (ref.1)
$$
\lambda_1+2\lambda_2+\cdots+(N-1)\lambda_{N-1}\quad\...
2
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1
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263
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axial anomaly for adjoint fermion v.s. fundamental fermion
It is known that the axial anomaly (chiral anomaly, the left L- right R) shows that $U(1)_A$-axial symmetry is not a global symmetry at quantum level.
In particular, one can consider the (1) ...
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630
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"Color charge" of the adjoint fermion?
What kind of "color charge" does the adjoint fermion carry?
Let us consider the SU(N) gauge theory. The gauge field is in the adjoint representation (rep).
Well-Konwn: If the fermion is in SU(N) ...
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2
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Breaking of $SU(3)$ symmetry by bi-fundamental representation
Are there any general theorems which fix the possible symmetry breaking patterns of Lie groups (such as $SU(3)$) by vacuum expectation values of fields in specific representations (such as the quark ...
- 1,783
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$SU(3)$ Color Symmetry
I have following (maybe a bit general) question about the $SU(3)$-symmetry of color by quarks:
If I consider the analogy to the $SU(2)$-symmetry of isospin $I$ crucially it concers the conservation ...
- 1,395
12
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2
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$SU(3)$ vs $SO(3)$ color gauge
I have kind of a dumb question: what would happen if the color gauge group is $SO(3)$ instead of $SU(3)$, assuming there are still colors and physical states are still color singlets? Will we e.g. get ...
- 175
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297
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Formal definition of gauge field and spinors in QFT
I am trying to pin down what spaces a spinor and gluon gauge field exactly occupy. I know that the spinor is a quantity $\psi_{i\alpha f}(\vec x, t)$ where
$i$ is a color index in the fundamental ...
- 8,439
5
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2
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664
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Quark model extension to all six flavors
Gell-Mann's $SU(3)$ quark model is extremely successful at describing the bound states of the three light quarks $u,d,s$. The bound states fall neatly into the irreducible representations of $\...
3
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1
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195
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Why can gluino (superpartner of gluon) have a Majorana mass?
I read in a paper by Scott Willenbrock that gluinos can have a Majorana mass although they have SU(3) color symmetry. The explanation was that gluinos transform under the adjoint representation which ...
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Why does the $U(2n)$ flavor symmetry break down to a $U(1)$ group and an $SU(2n)$ group?
I am studying quantum field theory using Srednicki's textbook. Problem 83.1 is:
Suppose that the color group is $G_C=SO(3)$ rather than $SU(3)$, and that each quark flavor is represented by a Dirac ...
- 1,653
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How are the generators of $\mathrm{SU}(3)$ represented on the gluon space?
I was watching some new lectures on QCD from Colorado and I have a few questions about what I heard:
The $\lambda^a_{ij}$ are generators of $\mathrm{SU}(3)$ in the fundamental representation so are $...
- 3,569
2
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1
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688
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What is the effect of the Gell-Mann matrices on color antiparticles?
I'm studying QCD and I can't understand how exactly are defined the color antiparticles. Indeed, we have the particle color triplet $(r,g,b)$. With the usual SU(3) algebra, we define the 8 Gell-Mann ...
- 2,296
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2
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1k
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Why are 3 colors used in QCD?
The mapping of strong charge to RGB left me believing that there are only 3 conserved quantities in QCD. I recently came to the understanding that there are in fact 8 conserved quantities, as ...
- 403
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SU(3) adjoint representation's invariant tensors
Considering a complex scalar field $\varphi^a$ that transform in the adjoint representation (8) of SU(3).
A quartic interaction term SU(3) invariant is
$$\lambda C^{abcd}\varphi^{\dagger a} \varphi^...
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