Implications of local parity breaking in heavy ion collisions
Recent data on dilepton production in heavy ion collisions revealed an abnormal excess in the region of invariant masses below 1 GeV. Our proposal is the creation of a slowly varying time-dependent pseudoscalar condensate within the hot nuclear fireball that comes from the very collision. The local parity breaking effect that immediately arises substantially modifies the vector meson properties leading to an excess of lepton pairs that could be a part of the explanation for the observed abnormal dilepton yield.
excess from LPB
\FullConferenceThe XXth International Workshop High Energy Physics and Quantum Field Theory
September 24-October 1, 2011
1 Motivation of local parity breaking
It is well known that parity is a well established global symmetry of strong interactions. However, some time ago, it was proposed that the QCD vacuum can possess metastable domains leading to violation. Accordingly, there is no reason to think that this symmetry cannot be broken in a finite volume. It is conjectured that the presence of non-trivial angular momentum (or magnetic field) in heavy ion collisions (HIC) then leads to the so-called Chiral Magnetic Effect (CME) .
where parity is locally broken. Our goal, with all the machinery developed for the study of local parity breaking (LPB) in HIC, consists in giving a qualitative answer to the PHENIX anomaly.
2 Axial baryon charge and chiral chemical potential
The strong interaction shows a highly non-trivial vacuum energy due to its non-abelian character. Such a behaviour allows different topological sectors to be present in the vacuum state, separated by high energy barriers. Nevertheless, these non-equivalent configurations may be connected due to large quantum fluctuations of the vacuum state in the presence of a hot medium via sphaleron transitions. The topological charge in a finite volume associated with a fireball reads
which is not a gauge invariant object under large gauge transformations. However, its jump can be associated to the integral of a gauge-invariant quantity. Then, in order to examine LPB in nuclear matter, we can integrate the local PCAC and connect and the quark axial charge :
In the chiral limit, thus, the exact conservation of the axial current states that the presence of a non-trivial topological charge in a finite volume is directly related to the emergence of a quark axial charge, a clear indication LPB.
The incorporation of QED into this framework entails the addition of an extra electromagnetic axial charge . At this point, the sum of both contributions is the one directly related to the change of in Eq. (2). In this sense, the description of the total axial charge may be described by a chiral chemical potential .
After bosonization of QCD in the light meson sector following the standard Vector Meson Dominance (VMD) prescription, the quark axial charge is substituted by its equivalent operator incorporating the lightest vector mesons fields. Accordingly, the QED Lagrangian may be extended with an extra piece given by the anomalous term where has to be understood as the derivative of some time dependent but approximately spatially homogeneous background pseudoscalar field induced within the fireball resulting from a HIC. Therefore, we are just left with one component of , where the hat notation stands for the isospin content that carried by this object. In addition, the absolute magnitude of is directly associated to the chiral chemical potential and one may expect it to be of order MeV, taking the condensate time of formation fm and the value of condensate of order of .
In summary, the existence of a non-trivial thermal average of the topological charge is translated into a prediction of . However, we propose the inverse procedure: if one is able to measure the average value of in a HIC, then the topological charge average can be automatically found using known techniques on the lattice. In this work, we provide some reasonable values for in our attempt to unravel the anomalous dilepton production in the PHENIX experiment.
3 Vector Meson Dominance approach to LPB
The appropriate framework to describe electromagnetic interactions of hadrons at low energies is the VMD containing the lightest vector mesons , and in the flavor sector. Quark-meson interactions are described by
where , , and . These values are extracted from vector meson decays. The Maxwell and mass terms are
where the mixing among vector mesons and photons appears. In addition, as stressed before, a parity-odd term has to be taken into account that produces an extra mixing:
In a HIC, where a fireball of a lifetime fm is created, the chiral charge that arises due to a jump in the topological charge has a characteristic oscillation time governed by inverse quark masses. Thus, for , quarks MeV fm, so the oscillation can be neglected and during the fireball lifetime we may consider the chiral charge to be approximately constant. However, for the quark the situation is different as MeV fm and even that a topological charge persists during fireball lifetime, the mean value of strange quark chiral charge is around zero due to essential left-right oscillations. Furthermore, the meson lifetime is much larger than the fireball; so finally one expects the strange sector to decouple from the rest of the system, thus being a good approximation to reduce from 3 to 2 flavors. In this sense, from now on we shall just consider a general to be a linear combination of isosinglet and isotriplet cases, in the same way as in .
In the case of an experiment that creates a very hot nuclear fireball rather than dense, i.e. ; an isosinglet pseudoscalar background is expected to be dominant, as it is the case for RHIC and LHC. In such environment , with a mixing matrix . Unlikely, when the opposite condition holds, like for the future experiments FAIR and NICA, the expected dominant background is the isotriplet one, where , with a CS matrix :
In this work, we shall not consider about the second case and we will just focus on the first one that could be a source of dilepton enhancement to give a qualitative answer to the PHENIX anomaly. Following these lines, one may solve the equations of motion after simultaneous diagonalization of the matrices and . Then, it is possible to show that photons remain massless and undistorted while mesons split into three polarizations () with masses
This splitting unambiguously signifies local parity breaking as well as breaking of Lorentz invariance due to the time-dependent background. The enlargement of the resonant region potentially leads to a substantial enhancement of their contribution to dilepton production away from their nominal vacuum resonance position.
On the other side, the averaged topological charge should be expected to vanish due to an equal probability to find positive and negative jumps. However, the change in the sign of is equivalent to the change of the polarization to the one, producing the same result. As a consequence, we shall just focus in the averaged modulus of .
4 Manifestation of LPB in heavy ion collisions
The main contributions in the cocktail of hadron decays presented by the PHENIX collaboration in the intermediate mass region MeV comes from the direct processes , the Dalitz decays and ; in addition to the background of . In this work we will focus on the first two decays as Dalitz decays result to be much more complicated to simulate. However, in the left panel of Fig. 2, we show how the Dalitz decay is affected by acceptance correction, described below. Work on the incorporation of anomalous propagators in Dalitz decays is in progress.
From the technical point of view, the acceptance correction constitutes an important issue in order to compare with experimental results. Following these lines, we draw our attention to the PHENIX experiment where a (single lepton) transverse momentum cut is set to MeV and a dilepton rapidity one to so that the outgoing lepton pairs are sufficiently separated to reconstruct such events. Moreover, the lepton pair invariant mass cannot be perfectly resolved as events are collected in bins of 10 MeV. To overcome this limitation, simulations need to incorporate a gaussian smearing of width of the same size as the bins used experimentally. In Fig. 2 we show how different channels are affected by acceptance correction.
All in all, acceptance correction breaks Lorentz invariance and the final phase space integration cannot be analytically computed but needs of numerical simulation since the easiest process in the hadron cocktail unavoidably requires at least 5 non-trivial integrals. For Dalitz decays, calculations are even more tricky with 7-8 integrals to solve. Therefore, we have used the VEGAS routine to numerically compute the phase space integrals.
Finally, the vector mesons contribution to the dilepton productions after acceptance corrections is given by the following expression:
where for and cases respectively; and . The standard definition for transverse mass is used separating transverse components from parallel ones and the chiral projectors are detailed in ref. . The coefficient absorbs combinatorial factors different for both channels, meson chemical potential and finite volume suppression so as they are not known with precision, the relative weights are used as free parameters in the cocktail by PHENIX collaboration and we shall follow the same procedure here.
5 Numerical results for dilepton excess
Introducing now the characteristic inputs of the PHENIX experiment, namely, the experimental acceptance cuts and the average temperature, that is MeV, as stated in ref. , we present in Fig. 3 (left), the spectral function together with the separate contributions for each different polarization for MeV. Similar results may be obtained modifying this parameter, so when increases, the chiral resonances appear to be more separated from the vacuum one.
Yet, this realistic value allows us to qualitatively show and stress the polarization asymmetry that is perfectly visible aside the peak. In the invariant mass region above the vacuum peak, it is clear that a large percentage of lepton pairs are chiral, unlike expected in vacuum or in medium parity-even calculations. Of course, in the mass region below the peak, the asymmetry is also present but does not show such a spectacular behaviour as above the peak. Therefore, the precise experimental measure of event-by-event dilepton polarization may reveal in an unambiguous way the existence of LPB, confirming the hypothesis of pseudoscalar condensate formation in the fireball of a HIC.
As discussed, LPB is not forbidden by any physical principle in QCD at finite temperature and density so the possibility of a pseudoscalar condensate to appear as a result from a HIC inside the hot nuclear fireball has been discussed. The effect leads to unexpected isospin-dependent modifications of vector mesons in-medium properties. After a theoretical analysis, one may to compute the dilepton production in the PHENIX experiment, thus realizing that LPB seems capable of explaining in a natural way the abnormal dilepton production. Moreover, we also want to stress that event-by-event measurements of the lepton polarization asymmetry aside the peak may reveal in an ambiguous way the existence of LPB.
Finally, the Dalitz and decays are thought to be mainly responsible for the enhancement at MeV, while the isotriplet condensate could also play an important role and should be analysed and incorporated to the calculation. This work is in progress and we expect it to be published soon.
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