**3.3. Running Couplings**

Grand Unified Theories
point to such an explanation - a unified model from which one can
derive the values and ratios of
the coupling constants. In these unification schemes,
the three Standard Model coupling constants derive from
one unified coupling (which is still arbitrary at this level).
The logarithmic running of coupling strength
with energy, derived from renormalization theory, leads to the large ratio
between unification scale and the proton mass. Although
gravity is not included in these theories, the inferred
unification scale (10^{16} GeV) is close to
the Planck mass; the running couplings thus account for most
of the ``largeness'' of the astrophysical Large Numbers.

Phenomenological
coupling constants such as those we have been using
(e.g., ) are not really constant but
``run'' or change with energy scale
(Wilczek 1999).
The vacuum is full of virtual particles which are polarized
by the presence of a charge. An electrical charge
(or a weak isospin charge) attracts
like charges, which tend to screen its charge as measured from far away.
At small distances there is less screening, so the charge
appears bigger, so the effective coupling grows with energy.
On the other hand a strong color charge attracts mostly virtual like-color
charged gluons, so it is antiscreened and the coupling changes
with the opposite sign - it gets weaker at high energy,
and is said to display ``asymptotic freedom''. The freedom
comes about from the antiscreening by gluons.
^{(7)}

The bookkeeping of how the constants change with the energy scale *M*
of interactions is done by renormalization group calculations. These show that
the running coupling constant of U(1),
_{1} =
*g*_{1}^{2}, obeys

where the sum is over the charges *Q*_{i} of
all fermions of mass less than *M*. The amount of charge screening
by virtual particles increases if the vacuum contains more degrees of
freedom that can be excited at a given energy. If all
fermions in the Standard Model are
included (and no more), the total sum on the right side is 14/3,
yielding a slope of -14/9.

For SU(3), there is again a screening term depending on the number of color-charged fermions, but there is also an antiscreening term from the (known number of) gluons,

where *n*_{f} is the number of quark flavors of mass less
than *M*.
The factor of 11 from gluons dominates if the number of
quark flavors is not too large, giving asymptotic freedom.
In the Standard Model, *n*_{f} = 6, yielding a slope of +
7/4.

The running of couplings depends on the particle degrees of freedom at each energy scale, that is, counting virtual particles with rest mass below that energy. Thus in reality the slopes change with energy scale and with the addition of new species, if there are any.

It has been known for over 20 years that the gauge groups
of the Standard Model fit nicely into larger groups of certain Grand Unified
Theories (GUTs), the simplest
ones being SU(5) and SO(10). The coupling constants of *SU*(3),
*SU*(2),
*U*(1) all approach each other logarithmically, merging at the GUT
scale, about 10^{16} GeV. In recent years measurements
of the couplings near *m*_{Z} have steadily improved and
for some GUTs (such as minimal SU(5)) the three couplings no
longer meet at a point; however, the agreement survives impressively
well in supersymmetric models
(Langacker and
Polonsky 1994),
or in models such as SO(10).
There is thus some reason to believe that these models work
up to the large scale of unification,
which is already close to the Planck mass.

^{7} The reason for the
difference is related to the zero point energies being opposite for
fermion and boson modes, which also enters into considerations
about their cancelling contributions to the
cosmological constant in supersymmetric vacua. Back.