Collisional Energy Loss
of a Fast Parton in a QGP
The suppression of hadron spectra in high energy central heavy-ion collisions compared to proton-proton collisions, referred to as ’jet-quenching’, is currently attributed to partonic energy loss in the hot medium created in the collision. The RHIC experiments show that at large enough , hadron quenching is strong, and of similar magnitude for light and heavy flavours. This point is difficult to understand in a parton energy loss scenario, where the energy loss is believed to be dominantly radiative, and quantitatively different for light partons and heavy quarks: gluon radiation off a heavy quark is suppressed at small angles – within the so-called dead cone – and a heavy quark suffers less radiative energy loss than a light parton. In this context it is important to reconsider the collisional contribution to partonic energy loss. Although it seems difficult to see how collisional losses could substantially increase heavy flavour quenching without simultaneously increasing light hadron quenching, it is theoretically important to establish correct results for the heavy quark collisional energy loss.
address=SUBATECH, UMR 6457, Université de Nantes
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Collisional Energy Loss
of a Fast Parton in a QGP
Keywords: QCD, quark-gluon plasma
At the Relativistic Heavy Ion Collider (RHIC), the spectrum of light hadrons has been observed phenix (); star () to be strongly suppressed in central ultrarelativistic heavy-ion collisions, as compared to proton-proton collisions. This is illustrated by the factor ,
appropriately normalized by the number of binary nucleon-nucleon collisions in a nucleus-nucleus collision, to be much smaller than unity, as can be seen on Fig. 1. For pions for GeV.
This nuclear attenuation of hadron production, , has been quoted as jet-quenching, and is one of the most important signals for the new state of matter – referred to below as a quark-gluon plasma or QGP – produced in a (central) heavy-ion collision.
The effect of jet-quenching was anticipated by Bjorken bj (), as a consequence of medium-induced partonic energy loss in the QGP. Although Bjorken illustrated his idea by performing an estimate of the collisional loss of light partons, parton energy loss generally arises from both collisional and radiative processes. In fact, for a parton energy , where is the parton mass and the plasma temperature, the radiative component is believed to be dominant. For a medium of large size we have Baier:1994bd ()
The critical length reads Baier:1994bd ()
but still dominates over the collisional loss as long as . Thus it seems legitimate to neglect collisional energy loss in phenomenological studies of light hadron quenching, and to attribute hadron quenching solely to the radiative parton energy loss111For a light quark and for fm, a careful comparison of collisional and radiative contributions within the same consistent model Zakharov:2007pj () gives the estimate at GeV. Using fm, this is in agreement with our estimate when . In Ref. Wicks:2005gt () a surprisingly large value is used at GeV, obviously changing the qualitative understanding of quenching..
The data on heavy flavour quenching Adler:2005xv (); Abelev:2006db (), obtained indirectly by observing the electrons and positrons arising from D and B meson semi-leptonic decays, shows that the nuclear attenuation of heavy and quarks is as strong as that of light partons, (see Fig. 2).
This is difficult to explain when assuming that heavy quarks, similarly to light partons, lose energy dominantly through radiation. Indeed, for heavy quarks (), gluon radiation at an angle is suppressed Dokshitzer:2001zm (). This is the so-called ‘dead cone effect’, leading to a strong suppression (at not too large ), of the average radiative loss of a heavy quark compared to a light parton,
If the electrons and positrons from heavy flavour decays observed in the experiment would arise dominantly from D mesons, then the dead cone effect would not be too drastic, due the moderate value of the charm quark mass. In this case explaining the heavy flavour quenching with purely radiative quark energy loss might still be possible Zakharov:2007pj (). However, the relative contributions from D and B decays are likely to be of the same order, as in collisions bcratio (). The dead cone effect being stronger for than for quarks, this leads to a reduction of the effective radiative loss. As a consequence, with similar contributions from and quarks, the current phenomenological models typically underestimate heavy flavour quenching.
Thus for heavy quarks, purely radiative energy loss seems insufficient to explain the observed attenuation. This has renewed the interest in the collisional part of the parton energy loss. Could the collisional loss be much larger for a heavy quark than for a light parton, which would increase heavy flavour quenching without spoiling the reasonably good description of light hadron quenching with purely radiative loss?
We have reconsidered the average collisional loss of a heavy quark, and of a light parton. In the next section we summarize our calculations, which include the effect of the running coupling. We will see that the very definition of parton energy loss is different in the tagged heavy quark case and in the untagged light parton case. As a result, in the high energy limit , the heavy quark collisional loss contains an additional term – a collinear logarithm arising from -channel exchange – which had been previously overlooked. This term makes the heavy quark collisional loss (25) slightly larger than the light quark loss (26), but is however small compared to the well-known soft logarithm ( being the Debye mass of the QCD plasma) arising from -channel scattering. This is mainly due to a different associated color factor. We can thus approximate (see also (27))
We stress that the latter estimate holds in the ultra-high energy limit . However, taking GeV for the heavy quark energy, and MeV, we see that the data represented in Fig. 2 correspond to the region for the charm quark, but not for the bottom quark for which . Since is an increasing function of we must have
Thus in the kinematical region of Fig. 2, we expect both collisional and radiative contributions to the quark energy loss to be suppressed compared to the or light quark case. In this respect the observed strong heavy flavour quenching, with an important contribution from quarks, remains a puzzle.
This conclusion should however be considered as preliminary. Indeed, an average loss smaller for than for light quarks is a priori not incompatible with heavy and light flavour quenching being of similar magnitudes, due to the different energy loss probability distributions in both cases.
Parton collisional energy loss
The collisional energy loss rate of an incoming parton of 4-momentum due to scattering off thermal particles of type () is given by
where we denote . The expression (8) represents the thermal average – with appropriate Bose-Einstein or Fermi-Dirac thermal distributions – of the elastic scattering amplitude squared , weighted by the energy loss in a single elastic scattering (to be defined precisely below). We will consider a fast parton of velocity . The appropriate sum and average of over spin and color is implicit in (8) and in the following.
Tagged heavy quark
In the case the incoming particle is a heavy quark of mass , we can neglect the presence of quarks and antiquarks of the same flavour in the plasma, and the scattering amplitude is given by the diagrams shown in Fig. 3. In the following we will denote , , , and use the Mandelstam invariants , , and , where . We will focus on the limit , implying .
When the heavy quark is tagged in the final state, the energy loss appearing in (8) is simply the difference between the heavy quark initial and final energies, . We stress that for a tagged particle, events corresponding to full stopping, i.e., to or equivalently to maximal energy transfer , contribute to .
where is the degeneracy factor of target particles of type , and
is the corresponding differential cross section. In Eq. (9) the bounds on are set by kinematics, , and .
The result (9) is exact in the limit and can actually be simply understood. Noting that , the factor to be inserted in (8) reads . From the result (9), we see that effectively we can write , the term vanishing after the angular integrals performed to obtain (9) starting from (8). Without the last factor in the r.h.s. of (9) we recognize , with the spatial density of thermal scatterers, the total elastic scattering cross section, and the incoming particle’s mean free path. Thus we can read the r.h.s. of (9) as .
soft logarithm from -channel exchange
We used , which is justified to logarithmic accuracy, and as an effective infrared cut-off, where is the Debye screening mass in the QGP BI (). The logarithm in (11) thus arises from the broad logarithmic domain .
The inequality implies that the energy transfer in a single collision satisfies . Thus the logarithm in (11) arises from kinematical configurations where the heavy quark is the leading parton in the 2-parton final state of the elastic scattering represented in Fig. 3.
collinear logarithm from -channel exchange
The contribution from the square of the -channel amplitude shown in Fig. 3c brings another logarithm in the heavy quark energy loss, arising from the domain . Indeed, in this region we can approximate , and (recall that ). Changing variables from to in (9), we thus obtain the contribution
We emphasize that the logarithmic contribution (12) arises from , or . Hence, in contrast to the soft logarithm of (11), the collinear logarithm of (12) arises from full stopping of the incoming heavy quark, where the leading particle in the final state of Fig. 3 is the struck thermal parton, to which the incoming heavy quark has tranferred all its energy. This effect is the QCD analogue of QED Compton backscattering of laser beams, used to produce energetic photons from energetic electrons. Full stopping must be taken into account in the case of a tagged particle, and this was overlooked in previous calculations of heavy quark energy loss TG (); BTqcd ().
Let us remark that the above energy loss process via (-channel) Compton scattering is rare but very efficient. Neglecting logarithms and writing heuristically
we see that despite a large Compton mean free path , the typical energy loss in Compton scattering is large, , and the related rate can thus compete with the -channel contribution.
beyond leading logarithms
In order to determine the constant beyond the leading logarithm in (11) and in (12), the following procedure can be used PPqed (); PPqcd (). Subtracting the leading logarithms and from the complete expression of , the remaining integrals are dominated by either or . An accurate calculation of the former integrals requires using the resummed hard thermal loop BI (); HTL () gluon propagator in the -channel, as pictured in Figs. 3a and 3d. The integrals dominated by can be evaluated with bare exchanged propagators, but require using exact kinematics and exact expressions of the Born elastic scattering amplitude. Some previous attempts to calculate the heavy quark collisional loss beyond leading logarithm TG (); BTqcd () were misleading, for the simple reason that the collinear leading logarithm had been overlooked.
Untagged light parton
Here we briefly discuss the case of a light incoming parton. When there is no heavy-quark tagged jet, it is impossible to know whether the detected jet arises, in the elastic scattering, from the scattered incoming parton of from the struck thermal parton. Moreover, a light incoming parton can be annihilated, due to the presence of partons of the same flavour in the plasma. Thus, in the untagged case, the loss in elastic scattering must be defined with respect to the leading parton in the final state. Defining momenta as in Fig. 3 we have (neglecting compared to )
Since , there is no full stopping with this definition.
Thus when , the apparent energy loss, defined with respect to the leading parton, is not as in the heavy-quark tagged case, but . Let us express the condition using Mandelstam invariants. We have
where we used and in the second line. When , it is easy to check from energy-momentum conservation that and are quasi-collinear, implying . Taking the ratio of the two equations in (16) we thus obtain
To get the last equality we used and assumed that , which will be justified in the following. Reexpressing (15) in terms of Mandelstam invariants we get
We can readily see that the first term of (19) will yield the same -channel leading logarithm as in the tagged case, but that the second term does not produce any -channel logarithm, due to the energy loss weight instead of . The term of which is indeed contributes to the second term of (19) as
where the integral over is dominated by . The contribution (20) is subleading compared to the -channel logarithmic contribution.
As a result, the energy loss of an untagged parton, to logarithmic accuracy, is given by the first term of (14) only,
where for a quark and for a gluon. The expression (21) is Bjorken’s result bj () and is specific to the case of an untagged parton. We stress the importance of defining correctly the observable (tagged or untagged) parton energy loss in order to obtain meaningful results, even to leading logarithmic accuracy.
The results (14) and (21) obtained in the fixed coupling approximation are not truly predictive, since the scale at which to evaluate is not specified. In order to obtain predictive results, we must repeat our calculations including the effect of running coupling.
In the case of -channel scattering (Figs. 3a and 3d), we must set the scale of to , as dictated by renormalization in PQCD at zero temperature APrunning (). Setting this scale in the factor of the cross section amounts to perform a specific all-order resummation, corresponding physically to the final partons being accompanied by collinear gluon radiation. The true exclusive cross section involving charged (coloured) particles vanishes Dokshitzer:1991wu ().
With running coupling, the -channel leading logarithm in (11) becomes
Using , where , we find that using a running coupling amounts to the replacement
We have already mentioned that the constant appearing next to the leading logarithms (in (14) or (21)) is given by integrals where the typical value of is determined by either or . Since there is no logarithmic spread there, the terms in (14) and (21) which are not logarithmically enhanced can be evaluated at a scale chosen arbitrarily between and .
and the collisional loss of a light energetic () parton as
We refer to PPqcd () for a discussion of the theoretical uncertainty.
Due to the additional logarithm in (25), when the collisional loss is (slightly) larger for a heavy than for a light (untagged) quark. Taking and ,
We discussed the average collisional loss of a heavy quark crossing a QGP, in the limit , and briefly reviewed the case of an energetic light parton. For fixed , at leading order , with a coefficient which is logarithmically enhanced. A soft logarithm is present for both heavy and light partons. A collinear logarithm contributes additionally to the heavy quark loss, but not to the light parton loss, due the difference between the definition of energy loss for a tagged heavy quark and for an untagged light parton. Implementing the running of we obtained the corresponding results (25) and (26).
At very high energy the heavy quark collisional loss is slightly enhanced compared to the light quark loss, see (27). However, as mentioned in section Jet-quenching phenomenology, for , the quark collisional (as well as radiative) loss should be reduced compared to the light quark case, challenging our understanding of heavy flavour quenching.
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