7 Relativistic Nucleus-Nucleus Collisions and the QCD Matter Phase Diagram
415
7.7.3 Critical Fluctuation of the Sigma-Field, and Related
Pionic Observables
Earlier investigations of critical QCD phenomena that might occur in high energy
nuclear collisions were based on QCD chiral field theory [276]. The QCD critical
point is associated with the chiral phase transition in so far as it appears as a remnant
of a tri-critical point [147] corresponding to the “ideal” chiral limit that would occur
if m u = m d = 0. Therefore the existence of a second-order critical point, at μ B > 0,
is a fundamental property of QCD with small but non-zero quark masses [277]. The
magnitude of the quark condensate, which plays the role of an order parameter of
the spontaneously broken symmetry (generating hadronic mass), has the thermal
expectation value
qq =
1
Z
n
n | qq | n exp(−E n /T )
(7.77)
with the partition function of hadronic states E n
Z =
n
exp(−E n /T ).
(7.78)
The low energy behavior of the matrix elements n | qq | n can be worked out
in chiral perturbation theory [277]. At the QCD critical point the order parameter
fluctuates strongly. Its magnitude qq is identified with an isoscalar quantity, the
so-called σ -field. The critical point communicates to the hadronic population via
the σ ↔ ππ reaction, generating fluctuating fractions of the direct pion yield
present near T = T c , which thus gets imprinted with a fluctuation of transverse
momentum (in the low p T domain) stemming from σ mass fluctuation, downward
toward the critical point. At it the isoscalar field ideally approaches zero mass, in
order to provide for the long wavelength mode required by the divergence of the
correlation length [147].
Note the relatively fragile structure of the argument. In an ideal, stationary
infinite volume situation the sigma field would really become massless, or at least
fall below the π + π − threshold; thus its coupling to π + π − becomes weak, and
restricted to very small p T . Furthermore, such primary soft pions, already small
in number, are subject to intense subsequent re-absorption and re-scattering in the
final hadronic cascade evolution [114]. In fact, experimental investigations of event
by event p T fluctuations in central A+A collisions, covering the entire
√
s domain
from low SPS to top RHIC energy have not found significant dynamic effects [278–
281]. Figure 7.65 illustrates the first such measurement by NA49 [278] in central
Pb+Pb collisions at
√
s = 17.3 GeV, at forward rapidity 4 < y < 5.5, showing the
distribution of event-wise charged particle average transverse momentum, a perfect
Gaussian. It is very closely approximated by the mixed event distribution, ruling
out a significant value of σ dyn from Eq. (7.76). More sensitive measures of changes,
415
7.7.3 Critical Fluctuation of the Sigma-Field, and Related
Pionic Observables
Earlier investigations of critical QCD phenomena that might occur in high energy
nuclear collisions were based on QCD chiral field theory [276]. The QCD critical
point is associated with the chiral phase transition in so far as it appears as a remnant
of a tri-critical point [147] corresponding to the “ideal” chiral limit that would occur
if m u = m d = 0. Therefore the existence of a second-order critical point, at μ B > 0,
is a fundamental property of QCD with small but non-zero quark masses [277]. The
magnitude of the quark condensate, which plays the role of an order parameter of
the spontaneously broken symmetry (generating hadronic mass), has the thermal
expectation value
qq =
1
Z
n
n | qq | n exp(−E n /T )
(7.77)
with the partition function of hadronic states E n
Z =
n
exp(−E n /T ).
(7.78)
The low energy behavior of the matrix elements n | qq | n can be worked out
in chiral perturbation theory [277]. At the QCD critical point the order parameter
fluctuates strongly. Its magnitude qq is identified with an isoscalar quantity, the
so-called σ -field. The critical point communicates to the hadronic population via
the σ ↔ ππ reaction, generating fluctuating fractions of the direct pion yield
present near T = T c , which thus gets imprinted with a fluctuation of transverse
momentum (in the low p T domain) stemming from σ mass fluctuation, downward
toward the critical point. At it the isoscalar field ideally approaches zero mass, in
order to provide for the long wavelength mode required by the divergence of the
correlation length [147].
Note the relatively fragile structure of the argument. In an ideal, stationary
infinite volume situation the sigma field would really become massless, or at least
fall below the π + π − threshold; thus its coupling to π + π − becomes weak, and
restricted to very small p T . Furthermore, such primary soft pions, already small
in number, are subject to intense subsequent re-absorption and re-scattering in the
final hadronic cascade evolution [114]. In fact, experimental investigations of event
by event p T fluctuations in central A+A collisions, covering the entire
√
s domain
from low SPS to top RHIC energy have not found significant dynamic effects [278–
281]. Figure 7.65 illustrates the first such measurement by NA49 [278] in central
Pb+Pb collisions at
√
s = 17.3 GeV, at forward rapidity 4 < y < 5.5, showing the
distribution of event-wise charged particle average transverse momentum, a perfect
Gaussian. It is very closely approximated by the mixed event distribution, ruling
out a significant value of σ dyn from Eq. (7.76). More sensitive measures of changes,