# I Does Lattice QCD use virtual gluons?

1. Jan 8, 2019

### friend

I'm given to understand that perturbation methods don't work in QCD because the coupling constant is too large. So they use supercomputers to calculate equations at various points on a lattice. Does this lattice method still take into account the virtual gluons that we might see in perturbation methods? Thanks.

2. Jan 8, 2019

### king vitamin

The whole concept of individual "virtual particles" largely only makes sense when describing particular Feynman diagrams. With that said, people will often colloquially refer to virtual "gluons/quarks/fluctuations" in the abstract when referring non-perturbative effects of quantum fluctuations even when they're using a calculation method which doesn't incorporate them. Ostensibly, the idea is that the whole non-perturbative theory can be thought of in terms of exactly summing the entire diagrammatic series exactly, so whatever method you use to do calculations you can also think of your results as in principle obtainable that way.*

*Although to be precise, diagrammatic expansions are often asymptotic so they do not actually converge.

3. Jan 8, 2019

### friend

So in practice the lattice QCD is solving the path integral exactly without reference to the virtual particles of perturbation methods?

4. Jan 8, 2019

### king vitamin

Yes, when people discuss "lattice QCD" they largely are referring to numerically computing the path integral directly.

Though for completedness, I'll add that there are some other methods in lattice field theory which have some use in studying quantum field theory, like duality transformations and strong/weak coupling lattice expansions. For an intro to these, I highly recommend John Kogut's review article, https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.51.659, as well as Polyakov's fantastic textbook Gauge Fields and Strings.

5. Jan 8, 2019

Thanks ever so much. I hope this is not too far aside. But I read that there is a $11 million prize for anyone figuring out why the color charge in hadrons has to add up to white. So I take it that this is a property put into the QCD wave function(s) by hand and is not derived, is this right? I know it is put there to comply with the Pauli Exclusion Principle. Could this color neutral property be considered an entanglement property (which is no explanation)? 6. Jan 8, 2019 ### king vitamin I assume you're referring to the$1 million dollar Millennium prize? One of the big stumbling points for that particular prize is that you must construct Yang-Mills theory (QCD without quarks/hadrons) in the continuum rather than on a lattice, and then prove that the continuum theory you've written down does or doesn't have an energy gap between its ground and first excited states (the latter property is related to the fact that the only particles in the theory are colorless).

So working with the lattice theory doesn't necessarily get you closer to the prize, but in the lattice theory it is rather uncontroversial that all particles are colorless and there is an energy gap. In fact the Kogut article I linked in my last post shows this using a strong-coupling expansion.

The fact that all hadrons are colorless does not have much to do with either the Pauli Exclusion Principle or entanglement.

7. Jan 9, 2019

Staff Emeritus

I'll go s atep further - I believe this doesn't exist. Oh sure, there is probably some prize somewhere for proving something about QCD, but not what you wrote. I really wish you would stop posting incorrect statements as facts. It starts the discussion from a very bad place. If I am wrong, I'll apologize.

8. Jan 9, 2019

### Reggid

Perturbative methods don't work for QCD at low energies. At high energies (e.g. hard interactions in a collider environment) of course we are using perturbative methods also for QCD.

No, because they have a finite lattice spacing, which is another kind of approximation.

9. Jan 9, 2019

### friend

Sorry, that was a typo. Thankfully, it was not germane to any issue.

10. Jan 9, 2019

### friend

I'm getting confused. All quarks have a color charge, right? And gluons have a color and anti-color charge, right? Quarks change their color due to interactions with gluons, right? Are gluons modeled in lattice theory as particles that travel at the speed of light between quarks and other gluons? If we take out the perturbation method away from lattice theory, do we still have gluons interacting with other gluons? Or are those only virtual processes associated with perturbation methods? Thank you.

11. Jan 9, 2019

Staff Emeritus
The whole sentence was a typo? Wow...that's quite a typo!

Anyway, I think the reason your questions are not being answered to your satisfaction is that there are many built-in assumptions in them that are simply not true. They are syntactically valid questions, but because of the lack of foundation, are only syntactically valid. There is no answer because these aren't really questions. It's like asking "how many sides does a square circle have?"

12. Jan 9, 2019

### friend

Well of course, you, or anyone else is certainly welcome to compose a summary of the QCD efforts. I'm certainly open to improve my language. The responses I've gotten so far make it seem that the popular things I've been reading are not even close. So perhaps a summary is in order or a link to it or to a video that describes it. At this point, I'm not even sure gluons even interact with each other, at least in lattice methods. You point out that I lack a foundation. OK, fair enough. Can someone please give an acceptable summary of that foundation? What do these sophisticated methods assume? What is the approach to the problems?

PS. your rebuke of your last post is not even an attempt to help. Surely the questions of post 10 should be easy. Are you afraid to commit to an answer?

13. Jan 9, 2019

### Staff: Mentor

Popular science is not trying to teach a topic in a proper way. It is trying to make things sound cool.
You can't fix problems from popular science with more popular science. Get a book. Start with the basics.
That's a book.

14. Jan 9, 2019

### friend

Oh please! Are you telling me these questions are not actually answerable without resorting to popularizations?

15. Jan 9, 2019

### king vitamin

So one of the big problems with the way popular science discusses QCD is the way it describes color charges. In particular, it likes to say that the "white" meson can be made up of either red*antired, green*antigreen, or blue*antiblue in quark-antiquark pairs. This is even how I learned it in an advanced undergraduate quantum mechanics class. But a little thought already shows this is weird - shouldn't every meson then be tripled, since we have three different ways of combining the quark and antiquark to get the "white" meson?

Later, when I took quantum field theory and actually learned QCD mathematically, I found that a lot of what was said even in that advanced undergraduate course was just an inaccurate representation of what is actually mathematically going on. In particular, the nature of how non-abelian charges add up to become neutral ("white") is much more complicated than what is said using the "color" language.

So let's say each quark and antiquark comes in three possible colors and anticolors respectively - this accurately represents the math. (As the more advanced readers know, this is because quarks and antiquarks are in the fundamental and antifundamental representations of SU(3) respectively.) Now, how can we combine a quark and an antiquark to get a "color-neutral" object? If you learn the math behind QCD, you learn that this is a problem in group theory: the technical term is that you need to decompose tensor product of the representations of the quark and antiquark into direct sums, and then the allowed "white" states are any parts of that direct sum which are singlets (are trivial under SU(3)).

(If you have taken an undergraduate quantum mechanics course, you were actually doing part of this process with a different group when you learned the addition of angular momentum and Clebsch-Gordan coefficients.)

Now, if I call the three indices of the quark red, blue, and green, and the three indices of the antiquark antired, antiblue, and antigreen, then the mathematical procedure I mentioned above would actually tell me that there is only one white state, and it is given by the superposition (red*antired + blue*antiblue + green*antigreen).

This is before getting to baryons. The popsci version says you combine red*green*blue=white. But the actual decomposition is some massively ugly group theory computation which I would need to brush up on my math skills to do. The problem is that the color "white" is actually referring to something called an "irreducible representation of a Lie algebra" (specifically white= "the trivial representation"), and the way in which representations combine to give you "white" cannot be described using some nice simple rule about mixing colors.

So I guess this whole post is to caution you against taking what you hear about color charge in pop sci (or even at some more advanced levels) at face value. It can lead you astray.

Last edited: Jan 9, 2019
16. Jan 10, 2019

### friend

Very good. I love it when people take the time to give me something meaningful to think about. I knew superposition would come into it.

Presently, I'm watching this video about QCD, both perturbation and lattice theory. It was the only one I could find in American English. The speaker seems to be constructing the summary I'm looking for. Some of it even sounds familiar. Do you have any comment about its reliability.

You've mentioned that the white state (for mesons) consisted of a superposition of various color/anti-color states. Does a superposition of red, green, blue states of various quarks and gluons also make up the white state for baryons? You seemed to have stopped short of directly saying that.

One of the conceptual problems I'm having is that the reason given for the color charge is due to some symmetry principle. This makes the symmetry more fundamental than the things that are symmetric, so to speak.

17. Jan 11, 2019 at 8:36 PM

### king vitamin

I don't have time to watch this, but Iain Stewart is a tenured professor at MIT specializing in particle physics so I'm sure it is reliable. This looks like a lecture on perturbative QCD though, which does not give you much info about low-energy hadronic physics.

Essentially yes, there is some complicated superposition of different combinations of colors in a baryon. But QCD is a strongly coupled theory when expressed in terms of quarks and gluons. In my opinion, it is not very useful (or even correct) to think of hadrons as a bound state of some definite number of quarks and gluons. Hadronic physics is really an emergent low-energy phenomenon, and trying to reconstruct what the physics looks like in terms of the quarks/gluons isn't really possible.

I don't understand this statement.

18. Jan 11, 2019 at 8:52 PM

### friend

I don't know. I think it is just that academia has accepted the SM symmetries and is now teaching it as one of the axioms of the field. You should not feel any obligation to answer this.

Thank you, king vitamin, for your responses. You're a friend. One last question if you don't mind. You might like this one.

What happens to the SM if we have to add new particles to the mix, like super-symmetric partners or axions, etc? Is it just a matter of shifting the coupling constants around to accommodate the new interactions? Do we destroy the present U(1)SU(2)SU(3) symmetry and start over? Or must we add new symmetries on top of the existing ones in order to preserve what we have? Thanks.

19. Jan 11, 2019 at 10:02 PM

### king vitamin

Most of academia considers the SM to only be approximately true, but since it is so well-verified experimentally, the approximation appears to be very good.

Once again, it seems that the Standard Model holds to a good approximation, so we expect that the gauge "symmetries" hold at higher energies, and they possibly are a part of an even larger group, say SU(5) or something. But I should say that this is special to the gauge "symmetries." But I don't personally like to call gauge invariance a symmetry because it is more like a redundancy of description. This is why we expect them to continue to hold, they exist partially because of our chosen method of description.

In contrast, actual symmetries like baryon conservation may be broken at higher energies and only approximately conserved in the SM. In fact, arguments from quantum gravity imply that they must eventually break. (Just throw a proton into a black hole and watch the number of baryons in the universe change!)

For a more fleshed out version of everything in this post, I recommend reading this fantastic article by Ed Witten: https://arxiv.org/abs/1710.01791. There is a lot of very nice discussion about how gauge symmetries are fundamental but global ones are not.

20. Jan 14, 2019 at 12:29 PM

Staff Emeritus
A few points:

A. Friend, by continuing to post at the A-level, you seem to think you understand physics at the graduate level (and many of your messages suggest that you think you understand it better than practitioners). You don't. You need to back up and fill in the blanks if you want to progress.

B. It's not realistic to post an hour and a half video and expect people to comment on it.

C. Your swipe at academic and symmetries shows a profound lack of understanding. What the Lie algebras are telling us is that the same equations have the same solutions. You have the same equations with spin and isospin, so of course they have the same solutions. The k-l degeneracy in the hydrogen atom is a direct result of the overall SO(4) symmetry of the system decomposing to two SU(2)'s.

It is simply not possible for the same equations to have different solutions. The best you could possibly hope for is that these equations are approximations or subsets of a more complicated theory. There is nobody in academia who will tell you that this is not the case - but they will point out that if these equations are approximate, they are very close. In some cases, fifteen or twenty digits of accuracy close.

D. As king vitamin has pointed out, the discussion of QCD color in popularizations is really bad. It's worse than oversimplification, in that it picks the wrong group to oversimplify - insofar as it is a group at all, it's U(1) x U(1) x U(1) and not the SU(3) of nature. Thinking about "redness" and "blueness" as physical entities is just plain wrong. This is why you can't trust popularizations: simple and explanations of the wrong thing abound.

E. The book advice was good. You should take it.