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31

The problem In case you were not aware of this, finding a proof for confinement is one of the Millenium Problems by the Clay Mathematics Institute. You can find the (detailed) answer to your question in the official problem description by Arthur Jaffe and Edward Witten. In short: proving confinement is essentially equivalent to showing that a quantum ...


22

In QED there are 4 kinds of divergences: Ultraviolet divergences. Naive calculations depend on the cut-off in such a way that they go to infinity as the cut-off do. However, QED is a perturbatively renormalizable theory so that non-naive, well-done computations (see regularization and renormalization) give sensible results. Landau pole. The coupling ...


21

in the late 1960s, the strongly interacting particles were a jungle. Protons, neutrons, pions, kaons, lambda hyperons, other hyperons, additional resonances, and so on. It seemed like dozens of elementary particles that strongly interacted. There was no order. People thought that quantum field theory had to die. However, they noticed regularities such as ...


21

Color charge is the representation of the SU(3) gauge group. The representation theory of SU(3) is described below: The basic representation is called the "3" or the fundamental, or defining, representation. It is a triplet of complex numbers $V^i$, which transform under a 3 by 3 SU(3) matrix by getting multiplied by the matrix. The value of "i" is ...


20

From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force $V(r) = - \frac{4}{3} \frac{\alpha_s(r) \hbar c}{r} + kr$ where the constant $k$ determines the field energy per unit length and is called string tension. For short distances this resembles the ...


15

A mass-gap means that aside from the vacuum (totally empty space), the next higher energy state has an energy which is bigger than zero by a finite amount, not by an arbitrarily small amount. This usually means no massless particles, since massless particles can have arbitrarily low energy. Another way of saying mass-gap which is somewhat more mathematical ...


14

No, there is none such equation. Reason is that these equations are highly classical and invalid in both relativistic (there is an action at a distance, incompatible with finite speed of light) and quantum mechanical regime (distances strong force is important at are quite microscopic). Also, strong force is confining, meaning you can't ever observe ...


12

Dear qftme, I agree that your question deserves a more expansive answer. The answer, "pions" or "gluons", depends on the accuracy with which you want to describe the strong force. Historically, people didn't know about quarks and gluons in the 1930s when they began to study the forces in the nuclei for the first time. In 1935, Hideki Yukawa made the most ...


11

The utility of using branes to realize gauge theories in string theory, compared to using heterotic, lies in the ease with which we can decouple bulk gravity. Basically you can zoom in to the branes to isolate the degrees of freedom on them, forgetting the gravity. In contrast, in heterotic compactificarions, both gauge fields and gravity live in the same ...


10

At the level of quantum hadron dynamics (i.e. the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form $$ V(r) = - \frac{g^2}{4 \pi c^2} \frac{e^{-mr}}{r} $$ where $m$ is roughly the pion mass and $g$ is an effective coupling constant. To get the force related to ...


9

As you say, "$G(x,Q^2)$ is the probability of finding a gluon with momentum fraction $x$ inside the hadron if the transmitted four-momentum is $Q^2$." In other words, $G(x,Q^2)$ is a probability density function. As you can see from the article, in this case the expectation value of the variable is $E[X] = \int_{0} ^{1} x\cdot G(x,Q^2) dx$ The plot of ...


9

Suppose that $\text{U}(3)$ was the gauge group. We can decompose this as $$\text{U}(3)=\text{U}(1)\times\text{SU}(3),$$ which implies that in addition to the $\text{SU}(3)$ that has eight generators corresponding to eight gluons, there would be an additional generator for $\text{U}(1)$. The latter in principle corresponds to an additional gauge boson, but ...


8

From the beginning of the wikipedia page on Yang-Mills theory (have you read it?): "Yang–Mills theory is a gauge theory based on the SU(N) group ... ... In early 1954, Chen Ning Yang and Robert Mills extended the concept of gauge theory for abelian groups, e.g. quantum electrodynamics, to nonabelian groups to provide ... ... This prompted a significant ...


8

One of the candidate explanations of the QCD color confinement involves the distinction between the Yang-Mills field electric and magnetic components. This model of confinement was qualitatively proposed in the 70s, and according to which, the quark confinement is explained by assuming the QCD vacuum to be composed of a magnetic monopole condensate in a ...


8

If by large density you mean large baryon density, then I believe one of the fundamental large $N_c$ results is that at densities of order nuclear densities, but below the density where the baryons have dissolved into quarks, baryonic matter forms a crystalline structure. This has been analyzed in the Skyrme model. I think this paper by Klebanov was one of ...


8

There is no known reason that you can't have bound states like $qq\bar{q}\bar{q}$ or $qqqq\bar{q}$ or higher number excitations, but none have been observed to date. You do have to make a color-neutral state, of course. In the mid-2000 some folks thought that they had of pentaquark states (that the $qqqq\bar{q}$) for a while, but it was eventually ...


8

Once upon a time, I asked an experienced phenomenologist who worked on particle physics in the 60s why even and odd signatured trajectories lie on top of each other. He said the phenomenon was called 'exchange degeneracy' and that so far no one has an explanation. I'm looking back at my notes on Dual Resonance Models, and it looks like by introducing ...


8

It is known that for an element $U$ of the group, in matrix sence: $$ Ad_Ux=UxU^{-1}.\,\,(1) $$ Now, we note that the target space of the adjoint rep is spanned by $N^2-1$ traceless matrices $t_a$. So, if we add the unity matrix, we get a full basis in $\mathrm{Mat}_N(\mathbb{C})$. We now note that that the adjoint action is trivially extended to this space, ...


8

Texts on QCD don't divide the generators of $SU(3)$ – and therefore "bicolors of gluons" – into two groups because this separation is completely unphysical and mathematically artificial (basis-dependent). Moreover, the number of "bicolors of gluons" i.e. generators of $SU(3)$, the gauge group of QCD, isn't nine as you seem to think but only eight. The group ...


7

Global invariance under $SU(N)$ is equivalent to the conservation of $N^2-1$ charges – these charges are nothing else than the generators of the Lie algebra ${\mathfrak su}(N)$ that mix some components of $SU(N)$ multiplets with other components of the same multiplets. These charges don't commute with each other in general. Instead, their commutators are ...


7

No, quarks couldn't turn out to be non-existent anymore. The evidence that quarks exist involves more than just some playful games in which we add electric charges to construct hadrons. Quarks show up in many processes. Historically, the important experimental observation was that of the deep inelastic scattering. Much like Rutherford observed the nucleus ...


7

As usual with these things, the presence of QFT and a non-Abelian gauge theory makes life hard, so let's take the prototypical theory that we can actually calculate with easily: Maxwell's equations. The question is then something like "what happens if I put a single electron in a (classical) lattice simulation?". Immediately, one realises that this question ...


7

Let me first make a general remark about internal symmetry groups, unrelated to our problem of the correct symmetry group for QCD. The symmetry must act on Hilbert space as a unitary operator for the conservation of probability. Now let us turn to the strong interaction. The most important experimental facts were that Observed hadron spectrum was ...


7

They do, just as all quantum objects do. They have momenta, and since they are massless, their frequency/wavelength/energy/momentum relations are the same as for photons. But since you will never detect a free gluon, as they are color-charged and thus confined, this is not a sensible thing to say. Quantum objects are not waves (just as they are not ...


6

It's not a sufficient explanation. There are asymptotically free theories which are not strongly coupled in the IR. The rate at which the coupling gets strong is important. In QCD, it seems to get strong very quickly near the confinement scale, so that beyond a certain scale, you only see hadrons. It is not really understood how this works. The ...


6

The situation is well represented in the following very pictorial picture but this is a very active field of study. It is interesting to note that a real proof of existence for the critical endpoint (CEP, indicated as a critical point in the figure), both from a theoretical and numerical point of view, does not exist yet. The reason, at least for the ...


6

Pure QED, unlike Pure Yang Mills ('pure' in the sense that there is only an $F^2$ term in the lagrangian, and it doesn't couple to matter) is a free theory. That means that it's boring, there's no need for renormalization or perturbation theory or anything. So the coupling constant (in this case the wave function renormalization of the photon) doesn't run ...


6

That really depends on what you call necessary. If you completely forget all about $SU(2)_L$ (say, in an alternate universe with no Weak Interactions). Then mass terms in the Lagrangian for quarks and leptons are not forbidden by any symmetry and you would not need the Higgs field to generate the mass of the quarks or of the electron. Now, in OUR ...


6

The particles that communicate the Weak interaction, i.e W Bosons and Z bosons are massive. So unlike Electromagnetism which is communicated by massless particles(Photons), the weak interaction has a very short range. For Massive particles the Potential of interaction falls as $V(x) = -K \frac{1}{r} e^{-m r} $ The range of this force is approximately ...


6

Dear dbrane, $\Lambda_{\rm QCD}$ is the only dimensionful parameter of pure QCD (pure means without extra matter). It is dimensionful and replaces the dimensionless parameter $g_{\rm QCD}$, the QCD coupling constant. The process in which a dimensionless constant such as $g$ is replaced by a dimensionful one such as $\Lambda$ is called the dimensional ...



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