(You may download this Lab Note in a PDF file at: qcd-confinement-handout-10.pdf)

This is part 1 of a Lab Note dealing with the origin of baryons and confinement in Yang-Mills theory, and attempting to lay the foundation for a solution to the so-called “Mass Gap” problem. I have organized this into eight brief, bite-sized sections.

1. __What Makes Yang-Mills Gauge Theory Different from an Abelian Gauge Theory like QED?__

In an Abelian Gauge Theory such as QED, a field strength two-form is expressed in terms of a potential one-form for a field of vector bosons, in this case photons, using the compact language of differential forms, as:

, (1.1)

where .

In Yang Mills theory, also known as non-Abelian gauge theory, there is an extra term in the field strength, and in particular, if the vector potential one-form is now , then:

, (1.2)

where , and *g* is the group “running charge” strength.

**The only difference is the existence of this extra term !**

***__Mathematical Review Notes for Section 1__:

If you need a brief review of Yang Mills, remember that and are NxN matrices for any simple Yang-Mills group SU(N). The group structure is specified by and the Latin internal symmetry index is raised and lowered with the unit matrix . Equation (1.2) can thus be expanded to the NxN equation , and, with all components explicit, to the commonly-written . The are often referred to as the group generators. In subsequent discussion, to diminish cluttering, we shall omit explicit rendering of the wedge products.

Regarding differential forms, recall also, that defines the differential operator *d* as applied to any p-form *H*.

Finally, we will often use the commutator notation , and on occasion, the anticommutator .

2. __What Do Source Current Source Densities Look Like in Yang-Mills Theory, versus QED?__

In QED, one has both an electric and a magnetic current source (probability and flux) density. In forms language, the electric current source density is:

, (2.1)

which expands to the familiar with the gauge condition , while the magnetic source current density is:

, (2.2)

the latter vanishing because for any two successive exterior derivatives. This expands to the familiar : there are *no magnetic charges* in QED.

In Yang-Mills theory, the source densities are related to the field strengths in the same manner, i.e., and , but, because of the extra term, we find in contrast, using (1.2), that:

, (2.3)

which expands to using the gauge condition . We also find that:

, (2.4)

which expands to .

While part of the magnetic source density still vanishes in the usual way because , *there is also a non-vanishing term in the magnetic source density*: . Put differently: in Yang-Mills theory, magnetic sources do not vanish, as has been pointed out in the past by t’Hooft & Polyakov and others. The “Yang-Mills electric” current density three-form in (2.3) also acquires an extra term .

**Might these non-vanishing magnetic three-forms (2.4) represent anything observed in the physical world?**

***__Mathematical Review Notes for Section 2__:

Recall that and , which makes use of the duality formalism and first developed by Reinich and Wheeler and later applied to differential forms by Hodge. Also, note that in Yang-Mills theory, , .

3. __Boundary Integration Properties of Yang-Mills Magnetic Sources__

Differential forms are tailor-made for examining surface and volume integrals over a closed boundary. So, to try to understand the magnetic three-form *P* of (2.4), we first examine the volume and surface integrals over *P*.

Taking the 3-volume integral of the *P* in (2.4), using , and applying Gauss’ law, enables us to rewrite (2.4) in integral form (in the below, represents integration over a closed d-dimensional surface):

(3.1)

In part, the above employs Gauss’ law, in the form . In further part, the above contains the expression . Combining these two parts of (3.1), enables us to deduce that:

. (3.2)

Now, setting (3.2) into (3.1) yields a simplified version of (3.1):

. (3.3)

Further, using (1.2) in (3.2), in the form , and again using Gauss’ law, now in the form , yields an expanded version of (3.2):

. (3.4)

Equations (3.3)and (3.4) tell us, mathematically, how these Yang-Mills magnetic sources behave at their boundaries. These two equations will be a primary focus of the discussion to follow.

**How do we physically interpret (3.3) and (3.4)?**

***__Mathematical Review Notes for section 3__:

Recall that Gauss’ law For a given p-form *H* states that , where *d* is the dimensionality of the closed surface over which the integration takes place.

4. __An Important Gauge Symmetry over Closed Surfaces of Yang-Mills Magnetic Sources__

Although perhaps not immediately apparent, equation (3.2), , tells us that there is no net flux of non-Abelian vector fields across any closed surface over the magnetic three-form source density *P*. To see this, subject the field strength two-form *F* to the transformation:

, (4.1)

which expands to . Now we ask: what effect does the transformation (4.1) have *over a closed 2-dimensional surface* surrounding the magnetic three-form *P*, as well as *on the magnetic charge within the enclosed 3-dimensional volume*?

Substituting (4.1) into (4.3), we obtain:

. (4.2)

That is, under the transformation , we find that and .

The above reveals a very important, apparently unknown, gauge symmetry of Yang-Mills field theory. Consider, by way of contrast, that QED and related theories are invariant under the transformation . This means that the scalar “phase” is not observable. Consider also by way of contrast, the that the field equations of gravitation are invariant under the (similar to (4.1)) gauge transformation . This means that is not a gravitational observable.

So, when under the transformation , this means that the non-Abelian vector fields are not observable over any closed 2-D surface defined around the magnetic three-form *P*. More to the point: these is *no net flux of non-Abelian vector fields* across any closed surface containing *P*. In addition, tells us that under the same transformation , the total magnetic charge within the specified 3-volume also does not change. More to the point: *this transformations does not remove any net magnetic charge out of the specified 3-volume.* All of these consequences emerge from .

Finally, let’s return to (3.3), which we expand to the form (see following (1.2)):

(4.3)

This is *non-zero*, which means that there *is* a net flux across the 2-D surface in the above, of whatever physical entities are represented by !

**How might all of this relate to QCD confinement?**

5. __Possible Parallels with Four Main Features of QCD Confinement__

There are four main features of QCD confinement, which appear to parallel the development of the previous section. These parallels are best specified with reference to baryons, as follows: Establish any *closed* surface over a baryon source density *P*. Then:

1) While gluons may flow ** within** the closed surface across various

*surfaces,*

**open***there can be no*

**net**flux of gluons in to or out of any**closed**surface.This may possibly be represented by , and the invariance of under the transformation .

2) While quarks may flow ** within** the closed surface across various

**surfaces,**

*open**there can be no*.

**net**flux of**individual**quarks in to or out of any**closed**surfaceThis may possibly be represented by the invariance of under the transformation .

3) While there can be no * net* flux of individual quarks in to or out of any closed surface, there can indeed be a net flux of

*in to or out of any closed surface. The antiquark cancels the quark, thereby averting a*

**quark-antiquark pairs***flux, and in this way, quarks do flow in to or out of the closed surface,*

**net***.*

**but only paired with antiquarks, as mesons**This may possibly be represented as .

4) It does not matter how hard or in what manner one “smashes” a baryon, one can still never extract a net flux of quarks or a net flux of gluons, but only a large number of meson jets.

This may be possibly represented by the fact that in all of the foregoing, the volume and surface integrals apply to *any and all closed surfaces*. One can choose a small closed surface, a large closed surface, a spherical closed surface, an oblong closed surface, and indeed, a closed surface of *any shape and size*. *The choice of closed surface does not matter.* These mathematical rules for what does and does not flow across any closed surface, in fact, thereby impose very stringent dynamical constraints on the behaviors of these non-Abelian magnetic sources: No matter what flows across various *open* surfaces, they may never be a *net* flux of anything across *any closed* surface. The only exceptions, which may flow across a closed surface, are physical entities represented by .

**So, w****here are we going with this?**

6. __What the Author Believes can be Proven to be True__

1. The magnetic three-form *P*, and its associated third-rank antisymmetric tensor , has all the characteristics of a baryon current density.

2. These , among their other properties, are naturally occurring sources *containing exactly three fermions*. These constituent fermions are most-sensibly interpreted as quarks.

3. , or the surface symmetry under the transformation , tells us that there is no *net* flow of gluons across any *closed* surface over the baryon density.

4. The volume symmetry under , tells us that there is no *net* flow of quarks across any *closed* surface over the baryon density.

5. The physical entities represented by , when examined in further detail, have the characteristics of mesons.

6. tells us that mesons are the only entities which may flow across any *closed* surface of the baryon density. But, there is one remaining question of paramount importance:

**What about the “Mass Gap”?**

7. __How do we Fill the Mass Gap?__

The one question remaining is how the mesons may become *massive*, given the seemingly-massless nature of the gluons. This is necessary to explain why the nuclear force is strong but short-ranged, thereby filling the so-called “mass gap.”

This problem may be solved starting with equation (2.3), , which we have not yet explored in depth here. Expanded, and with , (2.3) is:

, (7.1)

which we abbreviate, functionally, as .

The trick to solving the mass gap, is to make use of the above-noted symmetries under , and also, to exactly obtain the *inverse* relationship . In QED, where the term does not exist, and , , and are all simple four-vectors, this is trivial, because in summary, one starts with , uses to turn this into , and then “inverts,” to obtain , which is also connected in a known way to the photon propagator . The term is easily put into the denominator, because is a scalar.

Non-Abelian gauge theory is trickier, because a) there is the extra term , b) the and are *matrices* of four-vectors, and not the simple non-matrix four-vectors and of QED, c) the four-momentum vectors which are analogs of , are also *matrices* of four-vectors, and d) because of these matrices, one must be very careful to employ commutators when performing the analog to the substitution.

But most importantly, the aforementioned matrix character of , , and means that the of QED will migrate over to the form for SU(N) in general, and that one must then obtain the *matrix inverse* of this to obtain . For the special case of SU(2), this is a diagonal matrix with each diagonal element identical, so inversion is simple. This is why it has proven possible to do accurate calculations of vector boson masses in weak and electroweak theory. But for SU(3) and larger, this matrix is non-diagonal and non-trivial. What one normally thinks of as the propagator, is now an , specifically related to the inverse matrix . So, when one finally gets to , one has an equation of the form . When used in amplitudes , together with suitable SU(N) scalar multiplets which break symmetry similarly to how this is done in electroweak theory, the result is that the mesons become massive, they also obtain imaginary mass components which give them a short lifetime, the interactions become short range, and the mass gap can be filled.

**What About Some Pictures?**

8. __Some Feynman-Type Diagrams that Result from all of this__

While we will not show the detailed development here, the above can be developed into the following three Feynman-type diagrams:

**Thank you for listening!**

Hi Jay,

There are a bunch of resonances going on between this and the stuff I’m working on. Of course I think that these sorts of diagrams are also useful for preons where the quarks and leptons are composed of three preons each. And in addition, the use of matrices here somehow reminds me of Margaret Hawton’s photon density wave function, but I can’t put my finger on it. In her case, it was spin-1/2 case of SU(2) that was simple (in that angular momentum commutes with density, which allows density to be treated as a scalar), while the spin-1 and larger cases of SU(2) for massless particles end up without the ability to define a complete(!) set of angular momentum operators (as spin-1 massless has 3 degrees of freedom, but only 2 are populated, and which two depends on orientation). But in your case the simplicity of SU(2) is trumped by the complexity of SU(N>2).

The other thing I was wondering about were the Feynman diagrams. As you know, I think bound states of quarks (as well as preons) should be represented by things that look very similar to what you’ve got here. However, I always want to keep one end of my Feynman diagrams in the past and the other end in the future, so that the propation is from old to new.

In your Figure 1 Feynman diagrams, it seems that this sort of thing could only be done by breaking the dashed lines. Are you working in Euclidean or Wick-rotated spacetime so that time is no longer the usual? Then I guess I understand these, at least superficially.

Basically, when you un-Wick rotate Figure 1, you end up with three fermions propagating from the past into the future, with each exchanging gauge bosons by pairs. After cutting the dashed lines, you can choose either end to be the initial or final state, and you get various permutations of the order in which the gauge bosons are created or annihilated.

Finally, Figure 3 kind of bothers me. If you are using Wick rotated Feynman diagrams, what happens when I un rotate it? I don’t know what to do when a fermion loop has two dashed lines in it. I can only cut one of them I think. But to get two baryons, I really must make 6 cuts, so that means I have to cut that one fermion loop twice. Then it seems like what is going on is some sort of vertex exchange…

I guess I’d prefer to see a meson exchange between two baryons that leaves the structure of the two baryons almost unchanged. So the baryon would interact with itself almost entirely, except for every now and then having a much rarer interaction with a fermion loop that is the meson. I.e., putting the bound quark on the left and the fermion loop / meson on the right, (I’ll add periods to prevent WordPress from killing spaces):

+.+~+………………..

+.+.+………………..

+~~~+………………..

+.+.+………………..

+.+~+………………..

+~+.+…..++++++>+++…..

+.+.+~~~~+……….+~~~~

+.+.+…..++++++>+++…..

+.+~+………………..

+~+.+………………..

Carl

Comment by carlbrannen — January 29, 2008 @ 4:44 am |

Oh well, I guess that has to be written out in “Courier” to be properly spaced.

Comment by carlbrannen — January 29, 2008 @ 4:45 am |

Hi Carl:

I guess your spacing didn’t come out right; feel free to try again.

Euclidean versus Wick space depends on a factor of “i” and therefore a rotation of 90 degrees through a complex plane. I really hadn’t thought about it in those terms, so let me tell you what I do have in mind and maybe you can make some sense of it.

Insofar as Figure 1, I think of the baryon as a “finite state machine.” (In fact, just this past week, I watched some of Feynman’s QED old lectures, and if you look at him go through Feynman diagrams, he also takes the approach that nature is in one state, then a photon is exchanged, then it is in the next state, then another photon exchange to the next state, etc. And that everying we observe macroscopically (and classically) is built up out of trillions of these individual, finite state, photon exchanges.) So, you can think of Figure 1 as iterating perhaps 10^20 times per second or more from subscripted states 1 to 2 to 3 and then, via the dashed lines, back to 1. The temporal component is in the iteration over and over. Three quarks. Then gluon exchanges. Back to three quarks. Then gluon exchanges. And so on. The Dashed lines are meant to schematically indicate that state 3 is really just recycled to state 1 once again. That is, 1 –> 2 –> 3/1 –>2 –> 3/1, etc. THE DASHED LINES CAN BE REMOVED IF YOU WISH.

Another way to think about Figure 1, which lays the foundation for Figure 3, is to take away all dashed lines, since they are meant to be iterative anyway. From Figure 2, right side, you can draw a meson as either two quark lines going in “opposite” directions (upper right), or as one quark and one antiquark line going in the same direction (lower right). In either case, in these figures, time is an independent line flowing from right to left.

So, if you take the dashed lines out of Figure 1, you see that the only thing that flows “out,” or “in,” are quarks and antiquarks in pairs. Equivalently and alternatively, for every quark that flows in, another quark flows out. Think about it: that is exactly what we know about baryons — their only way of INTERacting strongly (versus INTRAacting strongly), is by emitting quark-anti-quark pairs, or equivalently and alternatively, absorbing one quark for every emitted one quark. The baryon can emit a meson, and then reabsorb it immediately. Or, the meson can vanish entirely with a quark/antiquark anihillation — these are short-lived, after all. When the dashed lines are retained in Figure 1, that is showing the INTRAactions of the baryon.

Now go to Figure 3. Remove, it you wish, all four of the dashed “recycling” lines, and turn the lines connecting the two baryons from dashed to solid. Now, we see a quark sent from the left baryon “simultaneously” with a quark being sent from the right baryon to the left baryon. Equivalently and alternatively, we see a meson emitted from the left baryon and absorbed by the right baryon. Or, if you move the time arrow the other way, there is a meson emitted from the right baryon and being absorbed by the left baryon. This is the most interesting of phenomenon, because this is the nuclear force in action.

This final thing to keep in mind regarding the flow of time, which is perplexing you, is this: When I put arrows on the diagrams, it is so as to think of this in terms of s,t,u channel processes, bringing in all the rules and techniques for doing scattering calculations in this context. Thus, the diagram for each baryon is really a closed, three-node Mandelstam diagram, and one can use that formalism in place of having to think explicitly about time. Thus, in each baryon, with an s or a t or a u possible at each of three nodes, one gets 3^3=27 possible combinations, though some are redundant. On top of this, one can play with the vertices by doing various “crossings,” because the quarks and gluons can take various INTRAbaryon paths. For example, look at all the various crossing diagrams I have posted on my web site http://home.nycap.rr.com/jry/FermionMass.htm , about 3/4 of the way down the page. I believe this is an exhaustive first order group of diagrams. Once the gluons start to beget more gluons, or quark loops, then you are into second order and higher.

I hope that fills in a bit more.

Best as always,

Jay.

Comment by Jay R. Yablon — January 29, 2008 @ 11:36 pm |