Video Transcript
The Feynman diagram shows two up
quarks exchanging a gluon. What is the color charge of the
gluon?
Before we go anywhere with
answering this question, since it’s a question about color charge, let’s add some
color to our diagram. We’ve now colored the up quark
symbols and their corresponding lines on the diagram so that their color matches the
color in the parentheses. This isn’t actually a necessary
step for solving the problem, but it will help us visualize what we’re working
with. Since we’re dealing with color
charge in this question, let’s recall some key facts about color charge.
Color charge is a property of
quarks and gluons, and it comes in three pairs of values. We abstractly call them red and
antired, green and antigreen, and blue and antiblue. Just like a positive electric
charge is the opposite value of a negative electric charge, an anticolor charge is
the opposite value of the corresponding color charge. For example, antiblue is the
opposite value of blue. Also like electric charge, color
charge is a conserved quantity. That is, the total color of all the
particles entering an interaction must be the same as the total color of all the
particles leaving an interaction.
These few facts about color charge
are actually all that we need to answer this question. There is, however, one useful fact
about the color charge of gluons that will help us make sure that our answer is on
the right track. When gluons participate in
interactions, they always do so with two color charges, one color — red, green, or
blue — and one anticolor — antired, antigreen, or antiblue. With all this in hand, let’s
conserve color charge in our Feynman diagram, starting with the top vertex.
Using our notation for representing
color charge, we can write that green color charge becomes blue color charge plus
the unknown color charges of the gluon. Since blue appears in the final
configuration but not in the initial configuration, it must be balanced by antiblue
in the color of the gluon. Furthermore, since green is the
only color appearing in the initial configuration, it must also appear in the final
configuration.
Since the final color of the up
quark is blue, not green, this green must be accounted for in the color of the gluon
as well. So, a gluon with color charge
green-antiblue will conserve color charge at this vertex. Let’s double-check that this gluon
also conserves color charge at the second vertex where an up quark with blue color
charge becomes an up quark with green color charge.
Going into the vertex, we have blue
from the quark and green-antiblue from the gluon. Going out of the vertex, we only
have green from the quark. Adding up the total color of the
particles entering the interaction, blue from the up quark is exactly cancelled by
antiblue from the gluon, which leaves a net color of green, also from the gluon. The total color leaving the
interaction is just the single green charge of the up quark. Because the total color both before
and after this interaction is green, we see that a green-antiblue gluon does indeed
ensure that color charge is conserved at this vertex. So, by conserving color charge at
each vertex, we’ve both determined and confirmed that the color charge of this gluon
is green-antiblue.
