Production of two pairs in gluongluon scattering
in high energy protonproton collisions
Abstract
We calculate cross sections for in the highenergy approximation in the mixed (longitudinal momentum fraction, impact parameter) and momentum space representations. Besides the total cross section as a function of subsystem energy also differential distributions (in quark rapidity, transverse momentum, , invariant mass) are presented. The elementary cross section is used to calculate production of in singleparton scattering (SPS) in protonproton collisions. We present integrated cross section as a function of protonproton center of mass energy as well as differential distribution in . The results are compared with corresponding results for doubleparton scattering (DPS) discussed recently in the literature. We find that the considered SPS contribution to production is at high energy ( 5 TeV) much smaller than that for DPS contribution.
pacs:
13.87.a, 11.80La,12.38.Bx, 13.85.tI Introduction
The cross section for production in protonproton or protonantiproton collisions at high energy is quite large book (); LMSccbar2011 () because the gluongluon luminosity grows quickly with energy. We have shown recently that the cross section for production of two pairs of in doubleparton scattering (DPS) grows even faster with incident centerofmass energy LMSDPS2011 () and becomes very large at large energies. In order to verify the DPS contribution a singleparton scattering (SPS) contribution has to be evaluated. This was not done so far in the literature as it requires calculation of subprocesses. At LHC a measurement of the twopairs of production should be possible. This could be a good test of methods of calculating higherorder QCD corrections.
It is the aim of the present paper to calculate contribution of singleparton scattering to the inclusive cross section. In the present paper we shall use highenergy approximation in calculating elementary cross section. At low incident energy and/or low invariant mass production a carefull treatment of the threshold effects is required. The elementary cross section is convoluted next with gluon distribution functions. The result is compared with that for DPS presented recently LMSDPS2011 (). A prospects how to disentangle SPS and DPS contributions will be discussed in the Result section.
Ii Theoretical framework
ii.1
In the high energy limit of the gluongluon subprocess, the amplitude will be dominated by the channel gluon exchange. Furthermore, if the pairs are produced in the respective fragmentation regions of the incoming gluons an intuitively appealing approach based on lightcone perturbation theory is possible. Namely we are looking for the cross section of excitation of the Fock states of the colliding gluons.
We base our calculations on previous works NPZ94 (); NPZ96 (), where the total cross section for the process and has been obtained, as well as on SingleJet (), where more detailed kinematical distributions can be found.
Let us briefly recapitulate the derivation of the key formulas. We start by writing out the lightcone Fockstate expansion of the incoming, physical, gluon, taking into account only the heavyquarkantiquark fluctuation in the twobody sector (see, NPZ96 ()):
(1) 
Here, quark and antiquark in the gluon carry fractions of the gluon’s large lightcone plusmomentum and are seperated by a distance in the impact parameter plane. Notice that the quarkantiquark pair before the interaction is not a color dipole, but carries a coloroctet charge. Infrared safety of the relevant cross section is a consequence of the fact, that arbitrarily longwavelength gluons cannot induce the transition and decouple.
The normalized colorstates of the quarkantiquark system in the coloroctet and colorsinglet states are, respectively:
(2) 
A similar Fockspace expansion as (1) can be written for the second gluon participating in the interaction, which will have a large momentum in the lightcone minusdirection.
The interaction between the right and left moving parton systems is then mediated by the gluon exchange which acts like a helicity conserving potential Gunion_Soper () between partons and .
(3) 
Here is the impact parameter between the two initial gluons, and are the impact parameter distances of and relative to their respective parent gluon which are conserved during the interaction. The matrices are the generators of color acting on parton in the relevant representation. Notice, that the gluon mass parameter is not needed to make the cross section finite, as mentioned above there are no singularities associated with the small behaviour of the gluon propagator. As a physical parameter it enforces a finite propagation radius of gluons in the transverse plane, as is in fact enforced by confinement. In practice, for the problem at hand its precise value is unimportant: as long as our results are practically independent of . The relevant Feynman diagrams for our process are shown in Figs.(1) and (2), and the corresponding scattering amplitude for in the highenergy limit of interest takes the form:
Let us now turn to one of the factors in the square brackets (the socalled “impactfactors”). The quark in the gluon is located at the distance , and the antiquark at , where . In fact we are interested in the respective twobody Fockstate components orthogonal to the physical gluon, and the relevant piece of the amplitude for the excitation is then given by (see also NPZ96 ()):
(5)  
Here is the color index of the channel gluon which carries the transverse momentum . Clearly, the impact factor vanishes for . A completely analogous expression can be written for the other factor in square brackets, and the full amplitude is obtained as:
(6) 
The two impact factors correspond to the upper and lower gluons, respectively. The total cross section is then obtained after integrating the squared amplitude over the impact parameter and averaging over initial gluon colors
(7) 
Similar impact factor representations for related QED problems are known for a long time Cheng_Wu (). Introducing the shorthand notation
(8) 
after some calculation, we obtain the impact factors relevant for the total cross section:
(9)  
The lightcone wave function for the transition can be obtained from the wellknown case for the photon as NZ90 (); NPZ96 ():
(10) 
where are generalized Bessel functions, and in the spirit of collinear factorization, we took the gluon to be onshell.
ii.2 Dipoledipole cross section
It is now convenient to introduce the total cross section for two color dipoles of sizes , , in the twogluon exchange Bornapproximation NZZ93 ():
(11)  
where
(12) 
The Born level dipoledipole cross section now reads
Notice, that it is finite for .
The total cross section for the partonlevel process can now be written in terms of the dipoledipole cross section and the lightcone wavefunctions for the transitions as
(14) 
where
(15)  
ii.3 Momentum distributions
The mixed representation given above could in principle be used to obtain distributions in longitudinal momenta, simply by stripping off the and/or integrations. For the more interesting transverse momentum distributions it is better to start all over in momentum space, where the impact factor reads
(16)  
After squaring, we obtain naturally the same structure as in SingleJet ():
(17)  
The explicit form for the squares of the lightcone wave functions can been found in SingleJet () and reads:
(18)  
With all of this given, we can put together the differential cross section
A brief comment on the kinematics of quarks is in order. The fourmomenta of quarks are fully specified by the variables . Incoming gluons carry longitudinal momentum fractions of the momenta of the incoming protons. The latter are
(20) 
so that . We parametrize fourmomenta in lightcone coordinates
(21) 
Then, the four momentum, say of the quark/antiquark belonging to parent gluon , read
(22) 
We still need to give the relation between the transverse momenta of the (anti)quark and the momenta used above. These relations read:
(23) 
or, alternatively
(24) 
Notice, that is the total transverse momentum of the pair, while is the lightcone relative transverse momentum which was conjugate to the dipole size. Now having the four momenta, we can calculate all sorts of kinematical variables, e.g. the rapidity will be given by
(25) 
The kinematics of the “lower” pair is treated analogously. Obviously the transverse momentum of the second pair is just , because incoming gluons are collinear. When constructing fourmomenta, we only need to be careful that the large component is along the lightconeminus direction
(26) 
And here
(27) 
ii.4 On normalization and phasespace
It is important to remember, that the expressions we derived are valid in a high energy limit, in which the invariant masses of pairs are much smaller than the centerofmass energy squared of the gluongluon collision: .
In the practical calculation of the hadronlevel cross section, we integrate over all momentum fractions carried by initial state gluons, and hence over the cmsenergy of the gluongluon subprocess. In practice, we want that our cross section behaves smoothly also at low energies.
Firstly notice, that our states are normalized in such a way, that the formulas are simple in the high energy limit. In particular, the constrained twobody phase space is just .
Effectively, the fourbody phase space in the highenergy limit is just
(28) 
Obviously it doesn’t vanish if approaches the threshold. To improve upon this, let us first rescale the amplitude, and introduce the Feynmanamplitude
which is normalized such, that the fully differential cross section takes the form:
(30) 
Here, the constrained nbody phase space, with is
(31) 
We now introduce invariant masses , and write the fourbody phasespace as
(32) 
Now, going over to lightcone coordinates, we have
(33)  
Here we integrated out the components from the onshell conditions. Let us now write , then one of the overall deltafunctions gives us, that . Furthermore, we can use the onshell condition to write . then we obtain finally
(34)  
Which agrees, modulo the factors now absorbed in the Feynman amplitude with the result obtained in the highenergy limit. The major simplification in the highenergy limit occurs in the first factor of the fourbody phase space (32). Namely we can write the phase space for two clusters . which are seperated by a large rapidity distance as
(35) 
For brevity, we wrote Now we can neglect the pluscomponent of and the minuscomponent of in the overall fourmomentum conservation, so that
(36) 
Hence all the integrations except for the transverse momentum ones can be done immediately. From the integrals over components, we only get a factor of
(37) 
so that, finally:
(38) 
If we approach the threshold, the exact phasespace goes to zero and our approximation is very bad. Still, we know, that the integrated twocluster phasespace will be just
(39) 
In order to improve our calculation, we therefore introduce the correction factor
(40) 
which within the approximations of the highenergy limit of course is exactly unity. A deviation from unity, or as a matter of fact any energydependence of the subprocess cross section thus indicates for us how far we are from the highenergy domain.
ii.5 inclusive cross section
The cross section for protonproton (protonantiproton) can be calculated as usually in the parton model as
(41) 
where . The last factor is the elementary cross section discussed in the previous sections. In calculating (41) we take into account kinematical constraints and the threshold correction factor (40). The elementary cross section is calculated first on a subsystem energy grid and a simple interpolation is done then when using it in formula (41). We shall use different parton (gluon) distributions from the literature. The factorization scale of the gluon distribution in principle depends on the kinematics of the final state quarks. We use when calculating the integral (41). The parton formula (41) is very useful to make differential distribution in invariant mass of the system. In principle it will be desirable to obtain results for opencharm mesons, but the inclusion of hadronization goes beyond the scope of the present paper where we wish to present only a first estimation of the cross section for SPS production of .
Iii Results
iii.1
To make final calculations we have to fix in formulae (18) and (LABEL:sigma_momentum_space). We shall use leadingorder running strong coupling constant in impact factors (see Eq.(18)) and with for gluon exchange (see Eq.(LABEL:sigma_momentum_space)).
In Fig.3 we show total cross section for the as a function of gluongluon energy. We show cross section with extra cut and with extra correction factor (see Eq.(40)). The latter cross section will be used then to calculate corresponding cross section for protonproton collisions. In the following calculations we have fixed the regularization parameter = 0.5 GeV. There is only a marginal dependence on the value of the nonperturbative parameter.
Before we go to real observables let us show an auxiliary distributions in and (see Fig.(4)).
The transverse momentum distributions of quark (antiquark) is shown in Fig.5 for a few selected subsystem energies. The higher subsystem energy the bigger transverse momenta are available kinematically.
Rapidity distributions of quark (antiquark) for different energies are shown in Fig.6. We show distributions for quarks emitted from the upper line (solid line) and from the lower line (dashed line). The higher the energy the better separation of the two contributions can be seen.
The rapidity separation can be better seen in the distributions in rapidity distance between (anti)quark(anti)quark. The distance between quarkantiquark from the same pair is very small compared to the distance bewtween (anti)quark(anti)quark from different pairs. In Fig.7 we show an example for subsystem energy = 200 GeV. The distance in rapidity between quarks and antiquarks emitted from different pairs reminds a bit situation in doubleparton scattering. There the distance between quarks and antiquarks emitted from two different hard processes can also be large LMSDPS2011 ().
Finally we close presentation of our results for
by showing distributions
in quarkantiquark invariant masses.
Here there are two distinct classes of subsystems. We shall introduce
the notation:
(category I)
for quarkaniquark emitted in the same pair and
(category II)
for quarkantiquark emitted from different pairs.
One can see that the average invariant mass of quarkantiquark
from the same pair is smaller than the average invariant mass from the
different pairs. At large invariant masses the emission from different
pairs dominates over the emission from the same pair.
The situation reminds that for doubleparton scattering
LMSDPS2011 ().
iii.2
Let us come now to protonproton scattering. In Fig.9 we show distribution in () for 7 TeV (LHC). We see that typical and are not too small, of the order of 10 – 10. This is the region where the gluon distributions are relatively well known. There is no strong dependence of the cross section on the choice of gluon distribution function (GDF).
The distribution in invariant mass of the system (equal to subsystem energy) is shown in Fig.10. We show distribution for CTEQ6 GDFs CTEQ6 (). For comparison we show distribution obtained for doubleparton scattering (see LMSDPS2011 ()). While the double parton scattering contribution dominates in the region of small invariant masses, the single parton scattering contribution takes over above 500 GeV. This is the region where large rapidity gaps between quarks and antiquarks occur.
The energy dependence of the inclusive cross section is shown in Fig.11. One can observe that the inclusive cross section for the final state is much smaller than that for the final state but grows somewhat faster at low energies. At higher energies the ratio is almost constant of the order of 1%. This is in contrast to double parton scattering contribution for the production which grows much faster than the cross section for single production LMSDPS2011 ().
Iv Conclusions
We have presented for the first time formulae for the production of two pairs of in singleparton scattering. The elementary cross section was given in two different representations: socalled mixed one (longitudinal momentum fraction, impact parameter) called also dipole representation, and momentum space one within a high (subsystem) energy approximation. While the dipole representation is easy to include energy dependence of the dipoledipole interaction the momentum representation seems better suited to include threshold effects. We have discussed how to correct the highenergy formulae close to threshold where the phase space is rather limited by energymomentum conservation.
We have presented energy dependence as well as different differential distributions for the elementary cross section for . We have shown that the elementary cross section varies quickly from the kinematical threshold () up to 100 GeV where almost a plateau can be observed. The invariant mass distributions are much steeper than that for . where 1 and 2 are from the first pair and 3 and 4 are from the other pair.
Our highenergy approach does not include processes when a second pair is produced via gluon emitted from quark/antiquark of a first pair and its “subsequent” splitting into . This processes may be important for small rapidity distance between quarks and antiquarks. A good example can be LHCb kinematics when both quarks (or both antiquarks) are emitted within two units of rapidity. This mechanism should not be, however, important for the case of large rapidity interval between or discussed recently in the context of double parton scattering LMSDPS2011 ().
The elementary cross section has been convoluted next with gluon distributions in the proton. We have presented inclusive cross section for as a function of incident energy as well as invariant mass distribution of the system. The results have been compared with corresponding contribution of doubleparton scattering. We have found that the singleparton scattering contribution is significantly smaller than that for the doubleparton scattering. We conclude therefore that a measurement of two pairs of at LHC would be very useful in testing models of double parton scattering.
An evalution of distributions for charmed mesons would be very useful in planning and interpreting current measurements at the LHC. The LHCb collaboration started already such an analysis Belaev2012 ().
Acknowledgment
We are indebted to Rafał Maciuła for a discussion of double parton scattering contribution and help in preparation of some figures and Vanja Belaev for the discussion of LHCb measurement. This work was partially supported by the polish MNiSW grant DEC2011/01/B/ST2/04535.
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