7 Relativistic Nucleus-Nucleus Collisions and the QCD Matter Phase Diagram
335
QCD. Due to the intrinsic non-linearity of QCD [70, 71], gluon showers generate
more gluon showers, producing an avalanche toward small x. As a consequence
of this exponential growth the spatial density of gluons (per unit transverse area
per unit rapidity) of any hadron or nucleus must increase as x decreases [65].
This follows because the transverse size, as seen via the total cross section, rises
more slowly toward higher energy than the number of gluons. This is illustrated
in Fig. 7.14 (right side). In a head-on view of a hadronic projectile more and more
partons (mostly gluons) appear as x decreases. This picture reflects a representation
of the hadron in the “infinite momentum frame” where it has a large light-cone
longitudinal momentum P + M. In this frame one can describe the hadron wave
function as a collection of constituents carrying a fraction p + = xP + , 0 ≤ x < 1,
of the total longitudinal momentum [73] (“light cone quantization” method [74]). In
DIS at large sqrts and Q 2 one measures the quark distributions dN q /dx at small x,
deriving from this the gluon distributions xG(x, Q 2 ) of Fig. 7.14.
It is useful [75] to consider the rapidity distribution implied by the parton
distributions, in this picture. Defining y = y hadron − ln(1/x) as the rapidity of the
potentially struck parton, the invariant rapidity distribution results as
dN/dy = x dN/dx = xG(x, Q
2 ).
(7.8)
At high Q 2 the measured quark and gluon structure functions are thus simply related
to the number of partons per unit rapidity, resolved in the hadronic wave function.
The above textbook level [74, 75] recapitulation leads, however, to an important
application: the dN/dy distribution of constituent partons of a hadron (or nucleus),
determined by the DIS experiments, is similar to the rapidity distribution of
produced particles in hadron-hadron or A+A collisions as we expect the initial
gluon rapidity density to be represented in the finally observed, produced hadrons,
at high
√
s. Due to the longitudinal boost invariance of the rapidity distribution, we
can apply the above conclusions to hadron-hadron or A+A collisions at high
√
s,
by replacing the infinite momentum frame hadron rapidity by the center of mass
frame projectile rapidity, y proj , while retaining the result that the rapidity density of
potentially interacting partons grows with increasing distance from y proj like
≡ y proj − y = ln(1/x).
(7.9)
At RHIC energy,
√
s = 200 GeV, y at mid-rapidity thus corresponds to x < 10 −2
(well into the domain of growing structure function gluon density, Fig. 7.14), and
the two intersecting partonic transverse density distributions thus attempt to resolve
each other given the densely packed situation that is depicted in the lower circle
of Fig. 7.14 (right panel). At given Q 2 (which is modest, Q 2 ≤ 5 GeV 2 , for bulk
hadron production at mid-rapidity) the packing density at mid-rapidity will increase
toward higher
√
s as
y
midrap
≈ ln(
√
s/M), i.e. 1/x ≈
√
s/M
(7.10)
335
QCD. Due to the intrinsic non-linearity of QCD [70, 71], gluon showers generate
more gluon showers, producing an avalanche toward small x. As a consequence
of this exponential growth the spatial density of gluons (per unit transverse area
per unit rapidity) of any hadron or nucleus must increase as x decreases [65].
This follows because the transverse size, as seen via the total cross section, rises
more slowly toward higher energy than the number of gluons. This is illustrated
in Fig. 7.14 (right side). In a head-on view of a hadronic projectile more and more
partons (mostly gluons) appear as x decreases. This picture reflects a representation
of the hadron in the “infinite momentum frame” where it has a large light-cone
longitudinal momentum P + M. In this frame one can describe the hadron wave
function as a collection of constituents carrying a fraction p + = xP + , 0 ≤ x < 1,
of the total longitudinal momentum [73] (“light cone quantization” method [74]). In
DIS at large sqrts and Q 2 one measures the quark distributions dN q /dx at small x,
deriving from this the gluon distributions xG(x, Q 2 ) of Fig. 7.14.
It is useful [75] to consider the rapidity distribution implied by the parton
distributions, in this picture. Defining y = y hadron − ln(1/x) as the rapidity of the
potentially struck parton, the invariant rapidity distribution results as
dN/dy = x dN/dx = xG(x, Q
2 ).
(7.8)
At high Q 2 the measured quark and gluon structure functions are thus simply related
to the number of partons per unit rapidity, resolved in the hadronic wave function.
The above textbook level [74, 75] recapitulation leads, however, to an important
application: the dN/dy distribution of constituent partons of a hadron (or nucleus),
determined by the DIS experiments, is similar to the rapidity distribution of
produced particles in hadron-hadron or A+A collisions as we expect the initial
gluon rapidity density to be represented in the finally observed, produced hadrons,
at high
√
s. Due to the longitudinal boost invariance of the rapidity distribution, we
can apply the above conclusions to hadron-hadron or A+A collisions at high
√
s,
by replacing the infinite momentum frame hadron rapidity by the center of mass
frame projectile rapidity, y proj , while retaining the result that the rapidity density of
potentially interacting partons grows with increasing distance from y proj like
≡ y proj − y = ln(1/x).
(7.9)
At RHIC energy,
√
s = 200 GeV, y at mid-rapidity thus corresponds to x < 10 −2
(well into the domain of growing structure function gluon density, Fig. 7.14), and
the two intersecting partonic transverse density distributions thus attempt to resolve
each other given the densely packed situation that is depicted in the lower circle
of Fig. 7.14 (right panel). At given Q 2 (which is modest, Q 2 ≤ 5 GeV 2 , for bulk
hadron production at mid-rapidity) the packing density at mid-rapidity will increase
toward higher
√
s as
y
midrap
≈ ln(
√
s/M), i.e. 1/x ≈
√
s/M
(7.10)