SOFT-GLUON DYNAMICS FOR HEAVY QUARK-ANTIQUARK SYSTEMS Page: 4 of 8
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and has a finite lifetime2 - 1/As r/ > r. Single-gluon emission
from (or absorption by) a color-singlet QQ state inevitably induces a
color-singlet to color-octet (1 + 8) transition of the Q9 state, no
matter how soft the emitted gluon is. This reminds us of a close analogy
between the QQ system interacting with soft gluons and a Dirac particle
interacting with soft photons (as studied by means of the Foldy-
Wouthuysen transformation3. In fact, the one-body fermion, the fermion
mass and soft photons in the latter are replaced by the two-body QQ
system, the binding energy As and soft gluons in the former, respectively.
This analogy is the starting point of the present investigation.
Because of the basic difference between color-singlet and color-
octet Q states, it is natural to label soft-gluon interactions accord-
ing to the associated color change of the Q state [i.e. 1 + 1
(color-singlet to color-singlet) or 1 -+ 8 or _ -++a 8 transitions.
These interactions are further divided according to their ranges.
We adopt the Coulomb gauge. The simplest way to construct the
color-Coulomb potential is to eliminate the temporal gluon field A (x)
(which is a Lagrange multiplier) in the Lagrangian. Then the Coulomb
potential is given by ab(x, y; A) =<x,al (Pk[A]Dk[A]) -Ily, b > ,
ab ab acb c
where [ (A] = d 2 a g f Ak . The emission of transverse gluons
from Coulomb gluon lines is a feature specific to the color-Coulomb
potential. The Coulomb gluons fall into three groups: Those exchanged
between the Q pair, those coupled to either O or 9, and those exchanged
between trasnverse gluons. The last group belongs to the pure gluon
sector. Let us denote the Hamiltonians corresponding to the other two
groups by H and HA, and project them onto the Q two-body subspace
(assuming no pair creation of heavy quarks)ab ab _ _ 2 a'a * b'b ce- -a -
< L (7) (Te) (x, y'; A) + aa b ab
HA
where Tc 2 c
indices of the Qa4
quark (antiquark).
tional of transversesgy ,
rAO(x) (TC) a'a b'b c- aa *bb {
0 c O A(y)6a (T)* 1 *
c 2 c ' and (a,b)[(a',b')] denotes the color
a 9b) system. Here x() is the position of the
In HA and in what follows. AO stands for a func-
0 Ak '
se gluons Ak,A0(x,t) -f d3z ab(x z; k A]Ak(z, t) b , (2)
which is the Coulomb-gauge temporal gluon field in the pure gluon
sector. H and HA are (spatially) nonlocal functionals of Coulomb
0
gluons and Ak. The Q system is surrounded by these gluons. The
gluons which closely surround the Q system, i.e. gluons whose momenta
are as hard as the Q size r, will predominantly build up the Q
binding. The gluons which are distributed over the size 1/As will
mainly cause the color fluctuation (i.e. 1 +-a 8 transitions) of the
Q system. On the other hand, softer gluons (i.e. symbolically,
gA < As) will tend to connect this fluctuating 09 system with external
perturbations.
Our next task is to construct effective soft-gluon interactions
out of H and HA ; this is achieved by selective summation of the
0
contribution of hard gluons (gA > As) in these Hamiltonians. A
natural expansion scheme that emerges is a multipole expansion of the
gluon field around the Q system, developed in powers of p = (99 size)/
(1/As) = rAe. This expansion classifies into multipole moments the
gluons responsible for the color fluctuation of the 99 system (i.e.
1/r > gA > Ae). A useful prescription for the separation of hard-gluon
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Shizuya, Ken-ichi. SOFT-GLUON DYNAMICS FOR HEAVY QUARK-ANTIQUARK SYSTEMS, article, March 1, 1980; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc845859/m1/4/: accessed March 28, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.