In this article, a model based on Glauber Monte Carlo is built to simulate the procedure of jet quenching in Quark-Gluon Plasma (QGP). In this model, energy loss of jets in QGP is parametrized by two quantities: path length of jets in QGP, L, and the initial transverse momentum of the jet P T. The path length of each branch of the jet are labeled L 1 and L 2. As input data, original jet energy data of p + p collisions were obtained from CMS measurement. After being processed by our model, simulated Pb + Pb jet energy data could be given and were compared to the data of ATLAS’s measurement in Ref. [1]. Distributions of (1/N)dN/dx <sub>J</sub> where x<sub>J</sub>=P<sub>T2</sub>/P<sub>T1</sub>, also noted as “frequency”, are presented as a function of P T1 and collision centrality. As the final result, two different forms of energy loss formula were found, both of which have good adaptation to certain centrality and P T ranges.
Relativistic heavy-ion collisions can produce a medium made up of free quarks and gluons, known as the Quark-Gluon Plasma (QGP). Hard scattering processes occurring in these collisions produce high transverse momentum ( p T ) partons called jets which can be one or multiple. Due to the reason that jets carry color charges, they strongly couple with the QGP, thus lose energy as they propagate through the medium, resulting in the phenomenon of “jet quenching”, which is widely concerned and studied by various scholars [2 - 4].
A number of research groups have already published their measurement results related to energy of jets produced in these heavy-ion collisions at various collision energy levels. Some experimental data were compared to theoretical results [5 , 6] while others remained unexplained by theory. In this article, a model based on Glauber Monte Carlo is built to simulate the measurement results of dijet events p T correlations in Ref. [
In the Glauber Monte Carlo model section of this article, frequently used terms include impact parameter, b defined as the distance between the center of two colliding heavy-ions and number of nucleons that are hit by at least one nucleon in heavy-ion collisions, N p a r t . These nucleons are also called participants.
In the real circumstance of a collider, two beams of heavy nuclei move in opposite direction. They randomly distribute in a cross-section area (of mm in dimension) far greater than the cross-section area of one nucleon (of fm in dimension).
To determine the probability density of a collision taking part at impact parameter range b ∼ b + d b , first switch to a reference frame where one nucleon in one of the two beams is located at the origin. The positive direction of z axis is selected as the same as the velocity of this nucleon. In this reference frame, incident nuclei are evenly distributed in the whole xy plane.
The number of incident nuclei that fall on the thin ring corresponding to impact parameter range b ∼ b + d b is
d N = σ 2 π b d b
where σ is the area density of incident nucleon. Then the probability density function
f ( b ) = d N d b = 2 π σ b ∝ b (1)
The specific procedure of generating a Monte Carlo Glauber Model is well explained in Ref. [
ρ ( r ) ∝ 1 + w ( r / R ) 2 1 + exp ( r − R a ) (2)
In the computer program, parameter r should be randomly generated according to the probability density function
f ( r ) ∝ 4 π r 2 1 + w ( r / R ) 2 1 + exp ( r − R a ) (3)
where the specific ration index does not make a difference. For Pb + Pb collision, the Fermi parameters are listed in
Implementing the Monte Carlo Glauber, we can plot a 2-D graph showing the distribution of all nucleons of two Pb nuclei on the transverse plane, as shown in
After that, we can get the number of participants—impact parameter and number of collisions— number of participants relationship, as shown in
In the procedure of Monte Carlo Glauber model, impact parameter b, the distance between the two centers of Pb nucleon, is an independent variable. In the ATLAS measurement Ref. [
The definition of centrality cannot be written explicitly in formula. It is based on the division of the histogram of N p a r t as shown in
The curve of N p a r t histogram and x-axis form a certain shape which corresponds to a certain area. In this graph, we can find a series of critical values x 0 , x 1 , x 2 , ⋯ , x 9 , x 10 such that
| Nucleus | R [fm] | a [fm] | w [fm] |
|---|---|---|---|
| 208Pb | 6.62 | 0.546 | 0 |
x 0 = ( N p a r t ) max = 416
∫ x 1 x 0 f ( x ) d x = 10 % ∫ 0 x 0 f ( x ) d x
∫ x 2 x 1 f ( x ) d x = 10 % ∫ 0 x 0 f ( x ) d x
⋯
∫ x 9 x 10 f ( x ) d x = 10 % ∫ 0 x 0 f ( x ) d x
x 10 = ( N p a r t ) min = 2
Because N p a r t is a discrete variable. The integrals above degenerate into summation. The specific values of these critical points are listed in
To simulate the energy loss of jets produced in QGP, the path length of each branch of the jet first needs to be calculated. Note that the QGP forms as a result of a number of nucleons colliding together, the shape of QGP is likely to be irregular and its boundary does not have a clear definition. It is thus more practical to define the path length based on position of participants, the origin of the jet and the equation of the jet line.
However, there still exists various ways to define jet length. The definition we adopt is the number of nucleon that each branch of the jet passes through. Under such definition, the path length is a discrete function. Another possible definition of the path length is the geometric distance from the origin of the jet and the point where the jet leaves the QGP. This definition was finally abandoned because of the following reasons. For one thing, as mentioned above, the boundary of the QGP is not clear. More importantly, it
| Centrality | N p a r t |
|---|---|
| 10% | 304 |
| 20% | 203 |
| 30% | 128 |
| 40% | 81 |
| 50% | 52 |
| 60% | 30 |
| 70% | 18 |
| 80% | 8 |
| 90% | 3 |
has already been proved that the density in each part of QGP is not homogeneous [
The specific criterion to judge whether a participant contribute to the jet and if it does, which branch of the jet it contribute to can be explained by
In this graph, all participants in the orange shadow area make one-unit contribution to upper branch of the jet and all participants in the blue shadow area make one-unit contribution to lower branch of the jet. The width of the shadow area is D.
In each centrality range, the histogram of path length of each branch of the jet, noted as L 1 and L 2 , is shown in
As the final step of our computer model, the form of the energy loss formula needs to be determined. As a jet propagates through QGP, the amount of energy it loses may be correlated to the path length and the initial energy of the jet. It is worth mentioning that the jets studied in our investigation have momenta of 100 GeV order of magnitude and thus are all ultra-relativistic. Their energy is approximately proportional to their momenta:
E = p c .
Thus energy loss is directly proportional to momentum loss.
Before the energy loss formula is finally determined, we can only come up with possible forms of it.
Δ p T = C f ( L ) g ( p T ) R (4)
where C is a constant parameter to adjust the overall magnitude of energy loss, L is path length and R is a randomly generated parameter to simulate fluctuations. Each term of the formula may have the following forms:
f ( L ) = L α , α = 1 , 2 , 3 , ⋯ (5)
g ( p T ) = { p T M [ ln ( p T p 0 ) ] ⋯ (6)
R { = const follows uniform distribution follows Gaussian distribution ⋯ (7)
where the sign M [ f ( x ) ] is defined as M [ f ( x ) ] = f ( x ) η ( x ) = f ( x ) if f ( x ) ≥ 0 , M [ f ( x ) ] = 0 if
f ( x ) < 0 . Note that if g ( p T ) = ln ( p T / p 0 ) , g ( p T ) < 0 when p T < p 0 , this sign is introduced to prevent
“negative” energy loss. A jet can never gain energy from QGP as it propagates through the medium. As shown in Ref. [
form g ( p T ) = M [ ln ( p T / p 0 ) ] . On the other hand, the form f ( L ) = L 2 is derived in Ref. [
investigation, f ( L ) = L α is adopted, leaving the power of path length as a free variable. This reduces the total number of parameters to 3, C, α and p 0 . The minima of this 3-dimensional parameter space is relatively easy to look for, and we search for it by traversing the parameter space, evenly selecting points first in larger intervals and then in smaller intervals.
As input data to our model, original data of jet events in pp collisions at s N N = 2.76 TeV are obtained from CMS collaboration’s experimental results. Jets produced in pp collisions are considered to be the initial, unquenched state of jets produced in Pb + Pb collisions. These data are first selected based on the same selection rule of ATLAS’s measurement [
A pair of path length L 1 and L 2 corresponding to a certain centrality range is randomly generated by the Glauber Monte Carlo model for each jet-jet event. Given the energy loss formula, the amount of energy loss and the final energy of each jet can be calculated. Then the jets are selected based on final p T ranges and classified into different x J = p T 2 / p T 1 where p T 2 < p T 1 ranges. The total number of events in each x J range is counted and normalized by dividing the total number of events which leads to the eventual simulated result.
Even though the energy loss formula consists of several independent parameters, and they act on energy of unquenched jets as a whole. It is still discovered that different parameters have different modulation effect on the shape of simulated curve.
Firstly, it is found that the greater the variance of the random term R, the less obvious the peak at centrality range 0 - 10% and p T range 100 - 126 GeV is. Let R follow Gaussian distribution with different variances and R = 1 as average value, the result of the curve is shown in
Secondly, we found that g ( p T ) = ln ( p T p 0 ) can usually make the predicted formula fit to Pb + Pb
curves in both different centrality ranges and different p T 1 ranges, even though g ( p T ) = p T can
sometimes outperforms g ( p T ) = ln ( p T p 0 ) in a fixed p T 1 range.
Thirdly, it is found that for the form g ( p T ) = ln ( p T p 0 ) , the position of the curve is mostly dominated
by the parameter p 0 when C is fixed, as shown in
In the procedure of adjusting the energy loss formula, f ( L ) is usually decided at first. After that, two parameters C and p 0 are adjusted to match the data in Ref. [
D = ∑ i = 1 10 | y i Measure − y i Simulate |
which is the sum of the distances between simulated data points and experimental data points in 10 x J ranges. Also, the average deviation, D ¯ , is defined as:
D ¯ = average value of D in six centrality ranges
The simulated data is considered to better resemble ATLAS’s experimental results when average deviation D ¯ is smaller.
In our investigation, there are basically two ways to select the parameters. One is to minimize average deviation of centrality ranges 0 - 10% and 10% - 20% at p T range 100 - 126 GeV, recorded as D ¯ 2 . The conspicuous peak of ATLAS’s measurement is located in this region.
The other is to minimize average deviation of all 9 different combinations of centrality ranges and p T ranges, recorded as D ¯ 9 .
The reason why these two ways of parameter selection were developed is that when D ¯ 9 reaches its minumum, tendency of the simulated curve of centrality ranges 0 - 10% and 10% - 20% deviates significantly from the measured result, failing to match the most significant feature of ATLAS’s measurement. As shown in the Result section, these two ways of parameter selection will lead to two different forms of energy loss formula with better fitness at different centrality and p T ranges.
By the end of our investigation, the two ways of parameter selection did not give similar forms energy loss formula, leading to two two different forms of energy loss formula with better fitness at different centrality and p T ranges instead. And the formula that has the best fitness under two criteria, minimum D ¯ 2 and D ¯ 9 are shown separately.
For the first way of parameter selection, the relationship of minimum D ¯ 2 to α is shown and the dependence of D ¯ 2 on C and p 0 at α = 5.35 is shown in
The energy loss formula with best fitness is:
Δ p T = 205 ( L / 40 ) 5.35 M [ ln ( p T / 62 ) ] (8)
corresponding to deviation D ¯ 2 = 1.87 and D ¯ 9 = 2.45 . Predicted data is shown in
For the second way of parameter selection, the relationship of minimum D ¯ 9 to α and the dependence of D ¯ 9 on C and p 0 at α = 0.8 is shown in
The energy loss formula with best fitness is:
Δ p T = 11 ( L / 40 ) 0.8 M [ ln ( p T / 1.3 ) ] (9)
corresponding to deviation D ¯ 9 = 1.84 and D ¯ 2 = 3.56 . Predicted data is shown in
For the first way of optimization, the simulated curves deviate from experimental data at centrality ranges 20% - 30% and 30% - 40% most significantly and for the second way of optimization, simulated curves fail to resemble the conspicuous peak at centrality range 0 - 10% and when 100 < p T 1 < 126 GeV .
It is not satisfying enough that an energy loss formula that can simulate all centrality and p T ranges very well has not been found. So possibility remains that 1) the procedure of jet quenching in QGP cannot be described by a simple form of energy loss formula in Equation (4), and 2) the unfolding procedure done by the ATLAS collaboration cannot fully represent physical reality as the peculiar peak at low centrality ranges was not reported in measurements done by other experimental groups [
In conclusion, our investigation has simulated the procedure of jet quenching in Quark-Gluon Plasma using Glauber Monte Carlo Model, using energy data of jets produced in pp collision obtained from the CMS collaboration. The research has characterised the energy loss of jets in QGP with path length of jets, L, and their initial transverse momentum, p T . Simulated data were compared with ATLAS’s measurement in Ref. [
gives the most possible form of interaction between QGP and hadron jets. With our findings, the method of using jets to probe the features of QGP can be more effective as how they interact with each other is determined. We expect our findings can find their application in future studies of the features of Quark- Gluon Plasma.
We gratefully acknowledge Professor Gunther Roland for his instruction on this paper, and Yan Lu for his early stage contribution to this investigation.
The authors declare no conflicts of interest regarding the publication of this paper.