Asymmetry of in-medium rho-mesons as a signature of Cherenkov effects
Asymmetry of in-medium ρ-mesons
as a signature of Cherenkov effects
I.M. Dremin∗, V.A. Nechitailo†
P.N.Lebedev Physics Institute RAS, 119991 Moscow, Russia
December 3, 2018
Abstract
Cherenkov gluons may be responsible for the asymmetry of dilep-
ton mass spectra near ρ-meson observed in experiment. They can be
produced only in the low-mass wing of the resonance. Therefore the
dilepton mass spectra are flattened there and their peak is slightly
shifted to lower masses compared with the in-vacuum ρ-meson mass.
This feature must be common for all resonances.
PACS: 12.38Bx, 13.87.-a
There exist numerous experimental data [1, 2, 3, 4, 5, 6, 7, 8, 9] about the
in-medium modification of widths and positions of prominent vector-meson
resonances. They are mostly obtained from the shapes of dilepton mass and
transverse momentum spectra in nucleus-nucleus collisions. Such in-medium
effects were tied theoretically to chiral symmetry restoration a long time ago
[10].
The dilepton mass spectra decrease approximately exponentially with in-
crease of masses albeit with substantial declines from the average approx-
imation of the general trend by the exponent in the low-mass region. A
significant excess of low-mass dilepton pairs yield over expectations from
hadronic decays is observed in experiment. The shape of the excess mass
spectra shown in [1, 2, 3] is dominated by ρ-mesons. Their ratio to other
vector meson resonances can be estimated as ρ : ω : φ=10:1:2.
∗email: dremin@lpi.ru
†email: nechit@lpi.ru
http://arxiv.org/abs/0704.1081v2
Several approaches have been advocated for explanation of the excess.
Strong dependence of the parameters of the effective Lagrangian on the tem-
perature and the chemical potential was assumed in [11, 12]. The hydro-
dynamical evolution was incorporated in [13] to describe the spectra. The
QCD sum rules and dispersion relations have been used [14, 15] to show that
condensates decrease in the medium leads to both broadening and slight
downward mass shift of resonances. The similar conclusions have been ob-
tained from more traditional attempts using either the empirical scattering
amplitudes with parton-hadron duality [16, 17] or the hadronic many-body
theory [18, 19, 20].
In the latest approach, which pretends on the best description of experi-
mental plots, the in-medium V-meson spectral functions are evaluated. The
excess of dilepton pairs below ρ-mass is ascribed to anti-/baryonic effects.
This conclusion is the alternative to more common ideas about the chiral
restoration at high energies. It asks for some empirical constraints to fit the
observed excess.
In this paper we propose another possible source of low-mass lepton pairs.
Namely, the emission of Cherenkov gluons may provide a substantial contri-
bution to the low mass region.
Considered first for processes at very high energies [21], the idea about
Cherenkov gluons was extended to resonance production [22, 23]. For Che-
renkov effects to be pronounced in ordinary or nuclear matter, the (either
electromagnetic or nuclear) index of refraction of the medium n should be
larger than 1. Qualitatively, the observed low mass excess of lepton pairs is
easy to ascribe to the gluonic Cherenkov effect if one reminds that the index
of refraction of any medium exceeds 1 within the lower wing of any resonance
(the ρ-meson, in particular).
This feature is well known in electrodynamics (see, e.g., Fig. 31-5 in [24])
where the atoms behaving as oscillators emit as Breit-Wigner resonances
when get excited. This results in the indices of refraction larger than 1
within their low-energy wings. In QCD, one can imagine that the nuclear
index of refraction for gluons in the hadronic medium behaves in a similar
way in the resonance regions. This statement is more general and can be
valid also at other energies if the relation (see, e.g., [25]) between the index
of refraction and the forward scattering amplitude F (E, 0o) is fulfilled not
only for photons but for gluons as well:
∆n = Ren− 1 ∝ ReF (E, 0o)/E. (1)
Here E is the photon (gluon) energy. In classical electrodynamics, it is the
dipole excitation of atoms in the medium by light which results in the Breit-
Wigner shape of the amplitude F (E, 0o). In hadronic medium, there should
be some modes (quarks, gluons or their preconfined bound states, conden-
sates, blobs of hot matter...?) which can get excited by the impinging parton,
radiate coherently if n > 1 and hadronize at the final stage as hadronic res-
onances [22, 23]. The hadronic Cherenkov effect can provide insight into
the substructure of the medium formed in nucleus-nucleus collisions. The
resonance amplitudes are chosen for F (E, 0o) at comparatively low energies.
The scenario, we have in mind, is as follows. The initial parton, belonging
to a colliding nucleus, emits a gluon which traverses the nuclear medium. On
its way, it collides with some internal modes. Therefore it affects the medium
as an ”effective” wave which accounts also for the waves emitted by other
scattering centers (see, e.g., [25]). Beside incoherent scattering, there are
processes which can be described as the refraction of the initial wave along
the path of the coherent wave. The Cherenkov effect is the induced coherent
radiation by a set of scattering centers placed on the way of propagation of
a gluon. That is why the forward scattering amplitude plays such a crucial
role in formation of the index of refraction. At low energies its excess over 1
is related to the resonance peaks as dictated by the Breit-Wigner shapes of
the amplitudes. In experiment, usual resonances are formed during the color
neutralization process. However, only those gluons whose energies are within
the left-wing resonance region of n > 1 give rise also to collective Cherenkov
effect proportional to ∆n.
Thus, apart from the ordinary Breit-Wigner shape of the cross section
for resonance production, the dilepton mass spectrum would acquire the
additional term proportional to ∆n at masses below the resonance peak.
Therefore its excess near the ρ-meson can be described by the following
formula1
(m2ρ −M
2)2 +M2Γ2
1 + w
m2ρ −M
θ(mρ −M)
HereM is the total c.m.s. energy of two colliding objects (the dilepton mass),
mρ=775 MeV is the in-vacuum ρ-meson mass. The first term corresponds
to the Breit-Wigner cross section. According to the optical theorem it is
proportional to the imaginary part of the forward scattering amplitude. The
second term is proportional to ∆n where it is taken into account that the
ratio of real to imaginary parts of Breit-Wigner amplitudes is
ReF (M, 0o)
ImF (M, 0o)
m2ρ −M
. (3)
1We consider only ρ-mesons here. To include other mesons, one should evaluate the
corresponding sum of similar expressions.
This term vanishes for M > mρ because only positive ∆n lead to the Che-
renkov effect. Namely it describes the distribution of masses of Cherenkov
states. In these formulas, one should take into account the in-medium mod-
ification of the height of the peak and its width. In principle, one could
consider mρ as a free in-medium parameter as well. We rely on experimental
findings that its shift in the medium is small. All this asks for some dynamics
to be known. In our approach, it is not determined. Therefore, first of all,
we just fit the parameters A and Γ by describing the shape of the mass spec-
trum at 0.75 < M < 0.9 GeV measured in [3] and shown in Fig. 1. In this
way we avoid any strong influence of the φ-meson. Let us note that w is not
used in this procedure. The values A=104 GeV3 and Γ = 0.354 GeV were
obtained. The width of the in-medium peak is larger than the in-vacuum
ρ-meson width equal to 150 MeV.
Thus the low mass spectrum at M < mρ depends only on a single pa-
rameter w which is determined by the relative role of Cherenkov effects and
ordinary mechanism of resonance production. It is clearly seen from Eq. (2)
that the role of the second term in the brackets increases for smaller masses
M . The excess spectrum in the mass region from 0.4 GeV to 0.75 GeV has
been fitted by w = 0.19. The slight downward shift about 40 MeV of the
peak of the distribution compared with mρ may be estimated from Eq. (2) at
these values of the parameters. This agrees with the above statement about
small shift compared to mρ. The total mass spectrum (the dashed line) and
its widened Breit-Wigner component (the solid line) according to Eq. (2)
with the chosen parameters are shown in Fig. 1. The overall description of
experimental points seems quite satisfactory. The contribution of Cherenkov
gluons (the excess of the dashed line over the solid one) constitutes the no-
ticeable part at low masses. The formula (2) must be valid in the vicinity of
the resonance peak. Thus we use it for masses larger than 0.4 GeV only.
The experimental data plotted in Fig. 1 have not been corrected for the
acceptance of the experiment, which strongly depends on mass and transverse
momentum of the muon pairs. However, due to an approximate cancellation
between the variations of the thermal radiation mediated by the rho and
those of the acceptance, the data as shown can roughly be interpreted as
spectral function of the rho, averaged over momenta and the complete space-
time evolution of nuclear collision [3]. To use these data without further
corrections is therefore justified as long as the pT spectra of the radiation and
those of the Cherenkov process are not dramatically different. From general
principles one would expect slightly lower pT for low-mass dilepton pairs from
coherent Cherenkov processes than for incoherent scattering. Qualitatively,
this conclusion is supported by experiment [3]. The Cherenkov dominance
region of masses from 400 MeV to 600 MeV below the ρ-resonance has softer
Figure 1: Excess dilepton mass spectrum in semi-central In(158 AGeV)-In
of NA60 (dots) compared to the in-medium ρ-meson peak with additional
Cherenkov effect (dashed line).
pT -distribution compared to the resonance region from 600 MeV to 900 MeV
filled in by usual incoherent scattering. More accurate statements can be
obtained after the microscopic theory of Cherenkov gluons developed.
We should mention that the expression (2) may be applied for ∆n ≪ 1.
The RHIC experiments revealed rather large ∆n ≈ 2. If the same values are
typical at lower energies of SPS then the more general formulas (see [23])
should be used. The qualitative conclusions stay valid.
Whether the in-medium Cherenkov gluonic effect is as strong as shown
in Fig. 1 can be verified by measuring the angular distribution of the lepton
pairs with different masses. The trigger-jet experiments similar to that at
RHIC are necessary to check this prediction. One should measure the angles
between the companion jet axis and the total momentum of the lepton pair.
The Cherenkov pairs with masses between 0.4 GeV and 0.7 GeV should tend
to fill in the rings around the jet axis. The angular radius θ of the ring is
determined by the usual condition
cos θ =
as discussed in more detail in [22].
Another way to demonstrate it is to measure the average mass of lepton
pairs as a function of their polar emission angle (pseudorapidity) with the
companion jet direction chosen as z-axis. Some excess of low-mass pairs
may be observed at the angle (4). Baryon-antibaryon effects can not possess
signatures similar to these ones.
In practice, these procedures can be quite complicated at comparatively
low energies if the momenta of decay products are comparable to the trans-
verse momentum of the resonance. It can be a hard task to pair leptons in
reliable combinations. The Monte Carlo models could be of some help.
In non-trigger experiments like that of NA60 there is another obstacle.
Everything is averaged over directions of initial partons. Different partons
are moving in different directions. The angle θ, measured from the direction
of their initial momenta, is the same but the total angles are different, corre-
spondingly. The averaging procedure would shift the maxima and give rise to
more smooth distribution. Nevertheless, some indications on the substruc-
ture with maxima at definite angles have been found at the same energies by
CERES collaboration [27]. It is not clear yet if it can be ascribed to Che-
renkov gluons. To recover a definite maximum, it would be better to detect
a single parton jet, i.e. to have a trigger.
The prediction of asymmetrical in-medium widening of any resonance at
its low-mass side due to Cherenkov gluons is universal. This universality is
definitely supported by experiment. Very clear signals of the excess on the
low-mass sides of ρ, ω and φ mesons have been seen in [5, 6]. This effect for
ω-meson is also studied in [8]. Slight asymmetry of φ-meson near 0.9 - 1 GeV
is noticeable in the Fig. 1 shown above but the error bars are large there.
We did not try to fit it just to deal with as small number of parameters as
possible. There are some indications at RHIC (see Fig. 6 in [7]) on this effect
for J/ψ-meson.
At much higher energies one can expect better alignement of the mo-
menta of initial partons. This would favour the direct observation of emitted
by them rings in non-trigger experiments. The first cosmic ray event [26]
with ring structure gives some hope that at LHC energies the initial partons
are really more aligned and this effect can be found. The possible addi-
tional signature at high energies could be the enlarged transverse momenta
of particles within the ring.
To conclude, the new mechanism is proposed for explanation of the low-
mass excess of dilepton pairs observed in experiment. It is the Cherenkov
gluon radiation which adds to the ordinary processes at the left wing of any
resonance.
Acknowledgments
We thank S. Damjanovic for providing us with experimental data. I.D.
is grateful to S. Damjanovic and H. Specht for very illuminating and fruitful
discussions, in particular, on the role of the experimental acceptance.
This work has been supported in part by the RFBR grants 06-02-16864,
06-02-17051.
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