If multiple independent minijet production indeed takes place in an inelastic proton proton scattering, one could be concerned with the conservation of momentum.
Breaking of this would be an obvious sign of violating the assumed independence of partonparton collisions. Clearly, the sum of the momenta of the individual partons cannot exceed the initial momenta of the protons they are from. To take this eect into account for example in (5.4), one must limit the phase space of each subsequent calculation ofσjet. In calculatingGn, we should then calculate the σjetn in (5.4) as
where θ is the Heaviside step function, and x1 and x2 depend on the integral variables via (2.7) and (2.8). The integration limits in (5.5) are now mutually dependent, so the integrals cannot be calculated separately, but as a one, 3n -dimensional integral. As the -dimensionality of the integrals increases so rapidly, Monte Carlo methods are needed to solve Equation (5.5) numerically. We also note that the factor
e−σjet(s,k0)A(b)= 1
1 +σjet(s, k0)A(b) + 2!1(σjet(s, k0)A(b))2+. . .,
i.e. the probability for no minijet production, should also be modied accordingly, with σnjet(s, k0) from Equation (5.5).
Chapter 6 Results
First, we wanted to estimate the error of using the single eective subprocess approximation described in Chapter 2 in calculating σjet. Figure 6.1 shows σjet calculated using both the full partonic bookkeeping and the SES approximation with a constant momentum cuto k0 = 2GeV. Next the same comparison was made by using the momentum cutosk0that were found by tting theσtotto data,
Figure 6.1: Minijet cross-sectionσjet(s, k0)calculated as a function of CMS energy
√s in leading order pQCD, with a constant momentum cuto k0 = 2GeV, and computed with the full bookkeeping (2.17) and the SES approximation (2.26).
Figure 6.2: Jet cross-sectionσjet(s, k0(√
s))calculated as a function of CMS energy
√s in leading order pQCD, with a momentum cuto k0(√
s) such thatσtot ts to data (see Figure 6.5).
as explained in Section 5.1. The results of the latter calculation are represented in Figure 6.2. Both Figures 6.1 and 6.2 show a minuscule relative error in using SES approximation, it undershoots the value of σjet by at most 1.8% in the √
s-range studied. The relative error tolerance of the integration routine was 10−4, so the dierence cannot be explained just due to numerical inaccuracy.
The SES approximation is based on the assumption that processes gg → gg and gq → gq dominate the minijet production. These subprocess cross-sections have largest values, as can be seen from Figure 2.4. The subprocess cross-sections are, in Equation (2.17), weighted by partonic luminosities x1fi(x1, Q2)x2fj(x2, Q2). The gluonic PDFs have such high values at high energies (see Figure 5.1) that the gluonic processes clearly dominate over other processes. To demonstrate this eect, we calculated the luminosities x1fi(x1, Q2)x2fj(x2, Q2) with i =g,u and j =g,u for values √
s = 1TeV and kT = 3GeV, as a function of rapidities y1 and y2 using Equations (2.7) and (2.8). The results are shown in Figure 6.3. The shape of the plots in this gure can be understood with the help of Figure 5.1:
when y1 = y2 = 0, x1 = x2 = 2k√Ts = 6 ·10−3, where the gluonic PDFs have their global maximum, and up quark PDFs have a local minimum. Moving in any direction from this point decreases gluonic PDFs' values, hence the lone peak
Figure 6.3: Parton luminosities x1fg(x1, Q2)x2fg(x2, Q2) (top left picture), x1fg(x1, Q2)x2fu(x2, Q2) (top right picture), and x1fu(x1, Q2)x2fu(x2, Q2) (bot-tom picture) calculated with CT14LO [38] PDFs, at√
s= 1TeV andkT = 3GeV, as a function of rapidities y1 and y2 using Equations (2.7) and (2.8).
in x1fg(x1, Q2)x2fg(x2, Q2). The Equations (2.7) and (2.8) are symmetric with respect to exchanging y1 and y2, and x1fu(x1, Q2)x2fu(x2, Q2) is symmetric with respect to exchange ofx1 andx2. In the positive quadrant, both quark PDF terms grow to form a peak, as x1 grows from the PDF's local minimum and x2 shrinks.
In the quadrant where y1 grows and y2 shrinks (or other way around), both x1 and x2 grow and form a peak at the quark PDF's local maximum. The plot of the mixed term x1fg(x1, Q2)x2fu(x2, Q2)follows the gluonic PDF's form as it has much higher values than the up quark PDF. Overall, from the Figure 6.3 we can clearly see that gluonic subprocesses' cross-sections get a very dominant weight in the whole integral domain in Equation (2.17). These two eects combined lead to the dominance of the processes gg → gg and gq → gq over all the others in the minijet production. To further demonstrate this eect, we calculated the fractional contribution from each subprocess of Table 2.1 to σjet. These results are shown in Figure 6.4, from which we can read that the gluon dominance is very clear at high energies.
After these calculations, we tted the total cross-section σtot to the best t of experimental data [41]:
σtot, tpp,pp¯ (s) = 42.6 (s)−0.46±33.4 (s)−0.545+ 0.307 log2 s 29.1
+ 35.5
Figure 6.4: The contributions of individual subprocesses (see Table 2.1) to the minijet cross-section σjet calculated as a function of CMS energy √
s in leading order pQCD, with a momentum cuto k0 such that σtot ts to data (see Figure 6.5). Each σsub is calculated otherwise exactly like σjet, but with setting all other subprocess cross-sections to zero than the subprocess in question.
using the momentum cutok0inσjet(√
s, k0)as a tting parameter. The procedure is explained in Section 5.2. The tted k0 as a function of the CMS energy √
s is represented in Figure 6.5. The data points fell somewhat on a straight line on a log-log plot, suggesting that the momentum cuto is ruled by a power law k0 ∝ (√
s)0.19. Interestingly, this behaviour is qualitatively similar to what is predicted in the pQCD saturation model [42].
Next, we proceeded to calculate the cross-sections σin and σel using the obtained momentum cutos (Figure 6.5) as explained in Chapter 5.1. The results are shown in Figure 6.6. From these, we can see that, when the Gaussian widthσ = 0.43 fm and when the model parameterk0 is tuned so thatσtot ts the experimental data, the eikonal minijet model slightly undershoots the inelastic cross-section σin. In the results calculated with σ = 0.43fm in Figure 6.6, the problem is in the proportionality of σinand σel with respect toσtot. Therefore, this problem cannot be corrected by simply taking higher order pQCD corrections into σjet, as this would generally just raise the value of σjet. As can be seen from Equations (4.1), (4.2), and (4.3), raising the value of σjet (thus raising the value of χ) leads into
Figure 6.5: The tted momentum cutok0 as a function of CMS energy√
s, from tting the total cross-section σtot to the experimental data with width parameters σ = 0.43(crosses and the green curve) and σ= 0.53(stars and the orange curve).
The tting procedure is explained in detail in Section 5.2. Notice the power-law -like behaviour.
raising values of all three cross-sections,σin,σel, andσtot, not into loweringσelwhile raising σin and keepingσtot still, which would be needed. As a matter of fact, the contributions of higher order pQCD corrections have already been eectively taken into account by the tting of k0, because it forces theσjet to take the appropriate value, no matter what terms lie in it.
Another modication one could make to the eikonal minijet model would be to allow for a real part in χ, i.e. considering also purely elastic events. As can be seen from Equation (3.39), this would keep the value ofσinunchanged, but, as can be seen from Equations (3.36) and (3.38), raise bothσtot and σel, thus making the proportionality issue even worse.
To understand the eect of the width of the proton thickness function Tn on the ratios σσtotin and σσtotel , let us consider a toy model, where Tn is such that the partonic overlap function A in Equation (4.6) becomes the Heaviside θ in the
Figure 6.6: The calculated cross-sectionsσtot (pink curve),σin (black curves), and σel (blue curves) as functions of CMS energy √
s. The solid lines are calculated using width parameter σ = 0.53fm and the dotted lines using σ = 0.43fm. The experimental data is from [30, 31, 32,33, 34, 35].
radial direction:
A(b) = 1
πσ2θ(σ−b) Z
A(b)d2b = 1. (6.1) Clearly, as can be seen from the Gaussian A, the width parameter σ here is in direct correlation with that in Tn. Inserting this overlap function into the eikonal minijet model, Equations (4.1), (4.2), and (4.3) yield straightforwardly ratios
σin σtot = 1
2(1 +e−σjet2πσ(s,k20)) and (6.2) σel
σtot = 1
2(1−e−σjet2πσ(s,k20)). (6.3) From these, we directly see that in order to raise σσtotin and lower σσtotel , one has to raise the width parameterσ relative toσjet. Then, sinceσtot = σjetx (1−e−x), where x= 2πσσjet2, we can see that the lowering ofxmust be compensated with the lowering of σjet to keepσtot unchanged.
We found by the method of trial and error that raising the width parameter σ of the proton thickness functions Tn to a value of σ = 0.53fm ts our model very well with the experimental data. Very interestingly, this value also nearly ts the
Figure 6.7: Probabilities Gn of the production of n jets for CMS energies √ s = 200GeV,550GeV,1800GeV,8000GeV,13000GeV, and50000GeV with width pa-rameters σ = 0.43fm (purple curves) and σ= 0.53fm (green curves).
error margins of the value σ = (0.43±0.09)fm obtained from Ref. [28]. The cross-sections calculated with σ = 0.53fm are also shown in Figures 6.2 and 6.6, and the re-tted k0 is represented in Figure 6.5.
Next we calculated the probabilities Gn of the production of n pairs of minijets for some selected values of √
s for both values σ = 0.43fm and σ = 0.53fm of the width parameter. The procedure used is explained in Section 5.3. The results are shown in Figure 6.7. As expected, at lower energies most of the inelastic collisions produce only one pair of minijets, and then as the energy grows, the
Figure 6.8: ProbabilitiesGn of the production ofn jets, multiplied by n, for CMS energies √
s = 200GeV,550GeV,1800GeV,8000GeV,13000GeV, and 50000GeV with width parametersσ= 0.43fm (purple curves) andσ = 0.53fm (green curves).
In each plot there is also the quantitieshni=P
n
n·Gn given.
production of multiple pairs of minijets becomes more signicant, while the one-pair production stays as the most probable one in all cases studied. There is, however, a signicant dierence in the results for dierent width parameter values.
This is also reasonable, as can be seen from Figure 6.5, for narrower protons the tting parameter k0 is lower, leading to higher σjet, and higher probabilities for multiple minijet pairs produced. Looking at Figure 6.7 might be a bit misleading for intuition, as even though multiple minijet productions have low probability,
they produce indeed many pairs of minijets. This shifts our attention to the quantity n·Gn, as the expected value of minijetpairs produced in a single inelastic protonproton collision can be calculated ashni=
∞
P
n=1
n·Gn = σσjet
in. The quantities n·Gn as well as hni for some selected values of √
s for both values σ = 0.43fm andσ = 0.53fm of the width parameter are shown in Figure 6.8. The observations made from Figure 6.7 are also valid for this gure. One should also note that changing the width parameter changes the expected value hni very dramatically.
We still have one more experimentally obtainable quantity we can validate the eikonal minijet model against, the slope parameter B of ddσtel ∝ e−B|t|. Approxi-mating Equation (3.40) to the rst order in the Mandelstam variable t yields
dσel
where we have dened the slope parameter
B ≡ 1
We calculatedB in the eikonal minijet model at various CMS energies√
swith the width parameters σ= 0.43fm andσ = 0.53fm. These results are shown in Figure 6.9, along with experimental data from Refs. [43, 44, 45, 46, 47, 48, 49, 50]. The results calculated with the larger width parameterσ = 0.53fm t the experimental data, in fact, surprisingly well in the studied range, although the slope ofB in√
s seems to be slightly steeper in the experimental data than in the calculated results.
Interestingly, this slope ofB seems to be approximately same withσ = 0.43fm and σ = 0.53fm. The message of Figure 6.9 is rather obvious: the width parameter σ should apparently slightly still grow as a function of √
s. Such tting is, however, beyond the scope of this Master's thesis.
Finally, we proceeded to calculate the probabilities Gn with σ = 0.43 fm taking also momentum conservation into account (Equation (5.5)) with Monte Carlo in-tegration as described in Section 5.4. Because of the limited calculational capacity,
Figure 6.9: The slope parameterB as a function of the CMS energy√
s, calculated from the eikonal minijet model with width parameterσ = 0.43fm (red plus signs) and σ = 0.53fm (blue crosses). The experimental data (black stars) is from [43, 44, 45,46, 47, 48,49, 50].
we only calculated the eect in the region √
s = 100..1000GeV and up to 8-fold minijet production, i.e. 24-dimensional integrals. In this region, we found the ef-fect negligible, changing values ofGn by at most3%. This result is not surprising, as from Figure 6.7 one can read that single dijet production dominates the σin in this region.
Chapter 7 Conclusion
In this thesis, we have derived the eikonal approximation for a quantum scattering problem. In this framework, we then studied an analytically simplistic model, the eikonal minijet model, for describing the CMS-energy dependence of the cross-sections in high energy protonproton collisions. We then tested this model against experimental data gathered so far. As a part of our numerical analysis, we also studied the validity of the single eective subprocess approximation for minijet production at high energies. We found the approximation to be highly accurate at the energy scales studied.
The eikonal minijet model depends, in the leading order QCD perturbation theory, on two parameters, the momentum cutok0 and the Gaussian thickness parameter σ. The cuto k0 sets the perturbatively calculable hard subprocesses apart from the nonperturbative soft subprocesses which we left untouched in this thesis as we loweredk0 to as small values as possible. Using numerical analysis and varying the thicknessσfrom0.43 fmto0.53 fm, we foundk0's so that our calculated total cross-sections for protonproton collisions t the experimental data in the CMS energy range√
s = 0.1..100TeV. Interestingly,k0 ΛQCD in all cases studied. With the model parameters xed, we then proceeded to calculate predictions for inelastic scattering cross-sections in the mentioned range. Results of these calculations are shown in Figures 6.5 and 6.6. We found that despite its apparent simplicity, the model t the experimental data surprisingly well.
The eikonal minijet model is based on the notion that inelastic protonproton scattering on high energies can be treated perturbatively in the collision one or more distinct pairs of partons, minijets, scatter from each other. Based on this we calculated within the framework of the model the probabilities of inelastic events to be understood as productions of n pairs of minijets. These probabilities are shown in Figure 6.7. As expected, the greater the energy of the colliding protons,
the more probable it is to have a multiple minijet production. In doing this, we updated and conrmed the results of Ref. [25].
As another test of the eikonal minijet model, we calculated the slope parameterB of the elastic dierential cross section in the CMS energy range√
s= 0.1..10TeV, and compared it to experimental data. These results are shown in Figure 6.9. Even in this test, the eikonal minijet model represented experimental data surprisingly well, in spite of its simplicity.
A noteworthy detail of the model is that one does not need to impose strong ad hoc√
s-scaling in the partonic overlap function A, even though protons appear to eectively widen at larger CMS energies. This eect is instead inherently taken care of via the tting of the parameterk0 andσjetwhich follows fromk0. According to our results in Figure 6.6, once a tting width parameterσis found at some CMS energy, the same σ works well at all the CMS energies studied. Even so, as can be seen from Figure 6.9, the eikonal minijet model could perhaps represent the experimental data even better if some√
s-dependence would be introduced in the overlap functionA, which leads one also to reconsider the factorization ofσjet from A(b) assumed in Equation (4.5).
As the centre of momentum energy rises, the contribution of the multiple mini-jet production events in inelastic scattering cross-section increases. This raises a question of how does the conservation of momenta t into the picture. The sum of the momenta of the partons participating in jet productions cannot be more than the initial momenta of the protons in the collision. To see if this plays a role in the probabilities of having multiple minijet production, one has to perform high-dimensional Monte Carlo integration. We calculated the eect of the mo-mentum conservation on the probabilities of having multiple minijet productions in the region√
s = 100..1000GeV, up to 8-fold jet production, and found it neg-ligible. This was somewhat expected, as in this region the production of one pair of minijets is very dominant. Studying the eect on higher energies was out of the scope of our work, as much more calculational capacity would be needed to perform the increasingly high-dimensional Monte Carlo Integrals.
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