Z boson production in Pb-p collisions at
Introducing a model to predict the Z boson production differential cross section for a proton lead collision.
19-02-2017
Report Bachelor Project Physics and Astronomy – Paul Gorris
[ABSTRACT]Parton Distribution Functions (PDF) are essential for hadron collider experiments, without them barely any prediction can be made. The production of electro weak Z bosons is one of the ways to probe the hadronic structure of either a free proton or a proton inside a nucleus. Inclusion of the nuclear Z boson data measured at hadron colliders can provide important information of the nuclear modification present in nuclear PDF. The nuclear PDF of a lead atom is probed by colliding it with a proton and measuring, amongst other quantities, the Z boson production. This thesis presents a calculation of a proton-lead collision using different PDF set predictions and comparing this model to available data from ATLAS. From the comparison, it can be concluded that for most rapidity regions the model predicts the Z boson production satisfactory.
STUDENTNUMBER UVA 10675205 SUPERVISOR DR. JUAN ROJO
2nd EXAMINATOR PROF. DR. PIET MULDERS
CONDUCTED BETWEEN SEPTEMBER 2017 AND MARCH 2018
SIZE 12 EC
FACULTY PHYSICS AND ASTRONOMY
SPECIALIZATION THEORETICAL PHYSICS
Samenvatting
Alle materie is opgebouwd uit deeltjes uit het standaard model. Deze deeltjes worden opgedeeld in twee groepen, de leptonen en de quarks met hun respectievelijke anti-deeltjes. Er zijn twee soorten quarks die allebei drie generaties hebben; de up-like (u, c en t) en de down-like quarks (d, s en b). Doordat quarks, naast normale lading, een kleur lading hebben kunnen quarks nooit individueel worden waargenomen maar alleen in gebonden toestanden met andere quarks. In deze gebonden toestanden heffen de kleur ladingen van de verschilldende quarks elkaar op zodat de toestand kleur neutraal is. Deze gebonden toestanden heten hadronen en de bekendste voorbeelden zijn de proton (uud) en het neutron (ddu). In de hadronen reageren de quarks constant met elkaar door middel van gluonen, de sterke kernkracht ijkboson.
Deze gluonen die dus aanwezig zijn in protonen, kunnen vervallen in een quark anti-quark paar in het proton. De kleur en normale lading van een quark en anti-quark vallen tegen elkaar weg waardoor alle meetbare constanten van de proton hetzelfde blijven, ookal heeft het proton meer quarks in zich. De drie orginele quarks worden de valence-quarks genoemd en de overige gecreërde quarks heten de sea-quarks. Doordat er interactie plaatsvindt tussen quarks, is er een kans dat een bepaalde quarksoort een deel van het momentum van de proton draagt. De kansverdeling voor momentum van alle partons in een hadron heet de Parton Distribution Function (PDF), waar, naast de quarks, ook de gluonen een deel van het momentum dragen. De PDF van een hadron is een universele grootheid, waardoor deze gemeten kan worden en daarna kan worden gebruikt voor voorspellingen van nieuwe experimenten.
Protonen zijn met neutronen aanwezig in atoomkernen. Uit onderzoek is gebleken dat de PDF van een hadron in een atoomkern niet gelijk is aan de PDF van een vrije hadron en dat er nucleare aanpassingen op de PDF gemaakt moeten worden. Hierdoor moet voordat er voorspellingen gedaan kunnen worden voor nucleaire hoge energie botsingen, zoals die in Genève bij de LHC, eerst de nucleare PDF (nPDF) worden bepaald.
De nPDFs worden bepaald door middel van gemeten werkzame doorsnedes bij hoge energie botsingen waaronder de Z-boson productie bij een botsing. Een Z-boson wordt geproduceerd wanneer er tijdens een botsing een quark en zijn anti-quark annihileren. Het Z-boson vervalt vervolgens weer heel snel waardoor deze niet direct waargenomen kan worden. Een Z-boson heeft meerdere verval mogelijkheden, één daarvan is het verval in een lepton anti-lepton paar. Deze leptonen hebben een langere vervaltijd waardoor deze wel kunnen worden waargenomen tijdens botsingen in bijvoorbeeld de ATLAS detector bij CERN. Wanneer er dus een lepton anti-lepton paar waargenomen wordt in een detector kan dat worden teruggeleid naar een Z-boson. De Z-boson productie differentiële werkzame doorsnede van een proton-lood botsing kan voorspeld worden door middel van een beschikbare proton PDF set (CT14) en een beschikbare
nPDF set (CT14+EPPS16) van lood. In deze scriptie is ervoor gekozen om een model te hanteren voor voor het vinden van de differentiële doorsnede voor een proton-lood botsing. Eerst worden de differentiële doorsnedes van een proton-proton botsing gevonden en die van een lood-lood botsing bij een center of mass energie van 5.02TeV met behulp van het programma MCFM. Daarna wordt hier het geometrische gemiddelde van genomen om zo een proton-lood botsing te modelleren. Dit model wordt vervolgens vergeleken met gemeten ATLAS data van een proton-lood botsing met ook een center of mass energie van 5.0.2TeV. Uit de vergelijking blijkt dat het model en de gemeten ATLAS data goed overeenkomen, alleen is er één rapidity gebied waarin het model een te lage waarde aangeeft in vergelijking met de data.
Contents
1 Introduction 2 2 Theoretic Background 4 2.1 Standard Model . . . 4 2.1.1 Fundamental Particles . . . 4 2.2 Cross Sections . . . 5 2.2.1 Transition Matrix . . . 52.2.2 Scattering Cross Section . . . 7
2.2.3 Differential Cross Section . . . 7
2.3 Electron-Proton Scattering . . . 8
2.3.1 Elastic Electron-Proton Scattering . . . 8
2.3.1.1 Kinematics . . . 8
2.3.1.2 Rosenbluth Formula . . . 10
2.3.2 Deep Inelastic Electron-Proton Scattering . . . 10
2.3.2.1 Kinematics . . . 10
2.3.2.2 Elastic Quark-Electron Scattering . . . 11
2.3.3 Hadron-Hadron Collisions . . . 12 2.4 PDFs . . . 13 2.4.1 Proton PDF . . . 13 2.4.1.1 Proton PDF Parameterization . . . 16 2.4.1.2 PDFs At The LHC . . . 16 2.4.2 Nuclear PDFs . . . 17 2.4.2.1 Nuclear PDF Parameterization . . . 18 2.4.2.2 Structure Functions . . . 19 2.4.3 PDF Determination . . . 20
2.5 Z Boson Production In Hadron Collisions . . . 20
3 Method 23 3.1 PDF Selection . . . 23 3.2 MCFM Parameters . . . 23 3.3 Geometric Mean . . . 23 3.4 Average Ratio . . . 24 4 Results 25
5 Discussion 26
6 Conclusion 28
1
Introduction
Hadron colliders have been playing an important role in the understanding of fundamental laws of physics and the verification of theories, such as the particles in the standard model. The most recent example is the discovery of the Higgs boson at the Large Hadron Collider (LHC) in Geneva which was predicted over 50 years ago (ATLAS, 2012). In the LHC, two hadrons are accelerated and then collide at relativistic velocities. Quarks are the building blocks of hadrons but because quarks interact amongst each other using the strong nuclear force, with a big coupling constant, perturbative QCD theory can usually not be applied to the partons. Only during high energy collisions, such as the ones at the LHC, permit the use of perturbative QCD (Forte, 2013). Hadrons are described with parton distribution functions (PDFs) which are universal and encode the structure of the hadron. PDFs can not be computed from first principles, and therefore have to be extracted from hard experiments. Without PDFs to rely on in particle physics experiments, doing a prediction of the outcome of a collision would be next to impossible (Forte, 2013).
With the lead-lead (Pb-Pb) run at the LHC in 2015, twice the energy was reached compared to any other accelerator experiment before that time. During the Pb-Pb runs, two bundles of lead atoms are accelerated to relavistic velocities after which they will collide, the same as the hadrons. One of the important findings was that the proton PDF is not the same for a free proton versus a proton inside a nucleus (Arneodo, 1994) and so the nuclear PDFs (nPDFs) have to be extracted from collisions as well. Because the free proton PDF is measured with a good accuracy, the nPDF of lead is extracted from proton-lead (p-Pb) collisions to form a baseline for heavy ion collisions (Armesto, 2016).
A method for extracting the nDPF of lead in a p-Pb collision is with the measurement of the Z boson differential cross section, which is sensitive to the PDF with parton momentum fraction, x, between 4 ∗ 10−4 and 0.43 (Chatrchyan, 2012). The Z boson differential cross section is predicted by PDFs and nPDFs of the colliding particles. In this thesis the geometric mean of the Z boson differential cross sections of a p-p collision and a Pb-Pb collision is used as a model for the prediction of the differential cross section of a p-Pb collision. For the free proton PDF CT14 is used and for a lead proton CT14 with nuclear correction EPPS16 is used. The model for the Z boson differential cross section will be compared to p-Pb collision data from ATLAS at the LHC.
To what extend can the geometric mean of the Z boson production differential cross section of a p-p and a Pb-Pb collision model a p-Pb collision?
To answer this question, first a p-p collision differential cross section will be computed using the CT14 free proton PDF. Secondly, the differential cross section of Z boson production in
a Pb-Pb collision will be computed using the CT14+EPPS16 PDF. The computing process is done using the MCFM program, in which specific parameters of the process can be put in. Lastly the geometrical mean of the p-p and Pb-Pb collisions is calculated and compared to the measured data.
In the second chapter an introduction into the theory will be given, starting at the standard model and working up to the Z boson production in a hadron collision. After the theory, the method will be discussed in the third chapter with which parameters were used in MCFM. In the fourth chapter the results will be illustrated in a graph and elucidated. In the fifth chapter a discussion on the results and the theory will be held after which a conclusion will be presented in the last chapter.
2
Theoretic Background
2.1 Standard Model
During the last century, particle physicists have attempted to gain an understanding of all particles and the forces by which they interact. Experimental discoveries lead to the introduc-tion of the Standard Model (SM) in which all particles and forces are represented and has not been discredited since its introduction. There are three fundamental forces: electromagnetic, weak, and strong force. Despite that the SM has never been discredited, it is not complete. SM can not explain gravitational force nor dark matter, for which additions to the SM are required.
2.1.1 Fundamental Particles
As aforementioned, the SM is a representation of all the fundamental matter particles and forces. The particles are divided into two subgroups: the leptons and quarks. Both these subgroups have three generations. The first generation lepton is the electron (e−) with its corresponding neutrino (electron neutrino), the second is the muon (µ), and the third is the tau (τ ) lepton with each a corresponding neutrino. The quarks are also divided into three generations. The first generation quarks are the up and down quark (u and d), the second generation is represented by the charm and strange quarks (c and s), and the last generation is given by the top and bottom quarks (t and b). The most stable particles are the particles from generation one and as the generation increases so too does the mass and the instability of the particles. The u, c, and t quarks all have a charge of 2/3e and the d, s, and b quarks all have a charge of -1/3e. All particles are susceptible to the electromagnetic and weak force. But because quarks additionally to a regular charge also carry a colour charge, they are also susceptible to the strong force. Colour charges can not be observed, so an isolated quark does not exist. Quarks exist in bound stages where the colour charges of the different quarks cancel. These bound stages are called hadrons. The two most well known examples of hadrons are the proton (uud) and the neutron (ddu). In these hadrons, the quarks are being kept together predominantly by the strong force (Johnson, 1979).
The fundamental forces are being carried by the gauge bosons, the second group of the SM. Bosons are spin-1 particles and can be massless. The photon is the mediator for the electromagnetic force and has zero mass. The electromagnetic force is observable both on the subatomic scale as well as on the macroscopic scale. The W±and the Z bosons are the atomic force carriers, which do poses a mass, mW= 80.385 ± 0.015 GeV and mZ= 91.1876 ± 0.0021
of importance in the atomic range or smaller. The last force is the strong force. This force is carried by the gluon which, like the photon, is massless. The strong force is a lot stronger than the other two forces, but decreases rapidly with distance. Therefore the strong force is only of interest in the subatomic regions. The last boson is the Higgs boson which is a scalar boson. The Higgs boson is a spin-0 scalar boson, introduced by the Higgs mechanism, to be able to explain masses for particles and gauge bosons (Higgs, 1964). The SM is visualised in Figure 1 with the spin, mass, and charge of all the particles.
Figure 1: Schematic display of the standard model with the three generations of up- and down-type quarks (purple), three generations of the charged and neutral leptons (green). The gauge bosons are represented in the fourth column (red), and in the last column (yellow) the scalar Higgs boson is represented.
2.2 Cross Sections
2.2.1 Transition Matrix
During an interaction, there is a shift in the states of the interacting particles. Defined as the initial state (i) and the final state (f ). The interaction rate from state i to f (Γf i) is defined
Γf i= 2π|Tf i|2ρ(Ei) (1)
with Tf i as the transition matrix element and ρ(Ei) as the density of states at the initial
energy. The transition matrix element is not Lorentz invariant, and therefore has to be revisited. The matrix element Tf i is defined as
Tf i= hψi|H0|ψfi (2)
with H0 as the Hamiltonian, on the left side of H’ all the initial wave functions, and on the right all the final wave functions. This definition was found using that one particle was present in a volume V. But due to special relativity this volume is not the same in every inertial frame. The volume will be reduced with a factor of 1γ. In the boosted frame of reference, this results inmE particles in the box. Therefore, the wave functions need to be proportional to E in order to be Lorentz invariant, the conventional choice is 2E particles per unit volume.
The wave functions as specified in (2) are not Lorentz invariant and are normalised by Z
V
ψ∗ψ d3x = 1. (3)
The Lorentz invariant wave functions represented by ψ0 are normalised by Z
V
ψ0∗ψ0d3x = 2E. (4)
So the two wave functions are related by ψ0 =√2E ψ. The Lorentz invariant matrix element is defined as Mf i= hψ 0 i|H 0|ψ0 fi =p2Ei· 2EfTf i, (5)
in which Ei represents the energy of all the initial state particles and Ef represents the energy
of all the final state particles. With (5) as the expression for Mf i, the transition rate can be
made Lorentz invariant. With the density of states dn dE Ei = Z dn dEδ(Ei− E)dE, (6)
and energy conservation Fermi’s golden rule can be written in an alternative form Γf i= 2π
Z
|Tf i|2δ(Ei− E)dn. (7)
dn = (2π)3 d 3p 1 2π3 d3p2 2π3 δ 3(p a+ pb− p1− p2) . (8)
In (8) pa and pb represent the initial state particles and p1 and p2 represent the final state
particles. Substituting (8) into (7) yield the transition rate for a scattering process
Γf i= (2π)4 Z |Tf i|2δ(Ei− E) δ3(pa+ pb− p1− p2) d3p1 2π3 d3p2 2π3. (9)
2.2.2 Scattering Cross Section
The cross section of an interaction is defined asnumber of interactions per unit time per target particleincident f lux in which the flux is defined by the relative motion between the target particles and the incident particles. This gives an expression for the cross section of an scattering process, in which a + b → 1 + 2.
σ = Γf i va+ vb
(10) With σ as the cross section and va and vb the speed of the incident and target particles
respectively. Substituting (9) and (5) into (10) gives the expression for the cross section of a scattering process. σ = 1 16EaEb(va+ vb)π2 Z |Mf i|2δ(E a+ Eb− E1− E2)δ3(pa+ pb− p1− p2) d3p1 2E1 d3p2 2E2 . (11) Because the cross section is Lorentz invariant, it can be calculated in every reference frame. For a scattering interaction the most convenient choice is the center of mass frame. In this frame pa= −pb = p∗i, p1 = −p2 = pf and
√
s = (Ea∗+ Eb∗). With s = (p1+ p2)2 = (Ea+ Eb)2,
so the total energy available is the center of mass frame squared. With these constrictions due to the choice of reference frame a cross section for any two-body → two-body process is given by σ = 1 64π2 p∗f p∗i Z |Mf i|2dΩ∗. (12)
2.2.3 Differential Cross Section
In many cases, it is not only the total cross section that is of interest but also the distribution of a kinematic variable. For example during an inelastic scattering process between an electron and a proton. After the interaction has taken place in an inelastic scattering process the proton breaks up into different hadrons. During a inelastic scattering, the angular distribution of the
scattered electron is of importance. For this, the differential cross section is used for the rate of an electron going through an area element (dΩ) dΩdσ = number of particles trough dΩ per target particleincident f lux . When the differential cross section is integrated over all the possible areas (or in some other cases all the possible energies) it gives the cross section.
To find a cross section, where the laboratory frame corresponds to the center of mass frame, (12) can be written in differential form to give
dσ dΩ∗ = 1 64π2s p∗f p∗i|Mf i| 2. (13)
This is the case when two particles with the same mass are fired upon each other. When two different particles are fired upon each other, or one of the particles is stationary while the other is moving, the lab frame does not correspond with the center of mass frame. In this case (13) has to be made Lorentz invariant. This can be done by finding that
dt = 2p∗1p∗3d(cosθ) (14) so dσ dt = 1 64πsp2i|Mf i| 2. (15)
(15) is valid in all reference frames since it is Lorentz invariant.
2.3 Electron-Proton Scattering
There are two important distinguishable scattering processes between electrons and protons; elastic and inelastic scattering.
2.3.1 Elastic Electron-Proton Scattering
In elastic scattering, the electrons interact with the proton as a whole. When the energy of the electron is relatively small, so that the wavelength of the gauge photon is much bigger than the radius of the proton, the scattering can be described only by the potential of the proton. When the energy of the electron gets bigger so that λ ∼ rp, apart from the electrostatic
potential of the proton, there also has to be accounted for the magnetic moment distribution of the proton.
2.3.1.1 Kinematics
Because (15) is valid in all rest frames, it can be implemented into the electron-proton elastic scattering in the laboratory frame, visualised in Figure 2.
In the laboratory frame the four momenta of the initial- and final state particles can be ex-pressed as
Figure 2: Schematic display of an electron-proton elastic collision in the proton rest frame. The electron hits the proton with momentum p1. After the collision the
electron has a momentum p2 with an angle θ relative to the original direction of its
velocity. The proton, after being struck, has a momentum of p4.
p1= (E1, 0, 0, E1)
p2= (E3, 0, E3sinθ, E3cosθ)
p3= (mp, 0, 0, 0)
p4= (E4, p4).
The expectation value of the matrix element squared is, for an elastic electron proton scat-tering process, defined as
h|M |2i = 8e4 (p1− p2)4 (p1· p3) ∗ (p2· p4) + (p1· p4)(p3· p2) − m2p.(p1· p2) (16)
With the expressions for the momenta, and working out the products the average matrix element squared, h|M |2i can be expressed as
h|M |2i = m 2 pe4 E1E3sin4(θ/2) cos2θ 2− q2 2m2 p sin2θ 2 (17) with q2 represents the momentum loss of the electron (p1− p2). Now that the matrix element
is known, the expression for the differential cross section of the scattered electron can be computed using (15), resulting in
dσ dΩ = α2 4E12sin4(θ 2) E3 E1 cos2θ 2− q2 2m2 p sin2θ 2 . (18)
This equation was derived by assuming that the proton is a point like particle and neglecting the electron mass.
2.3.1.2 Rosenbluth Formula
When accounting for the finite size of the proton the differential cross section of an e−p → e−p scattering is given by introducing two form factors. One is related to the charge distribution of the proton (GE,) the other is related to the magnetic moment distribution within the
proton (GM).The differential cross section is given by the Rosenbluth formula (Rosenbluth,
1950), dσ dΩ = α2 4E12sin4(θ 2) E3 E1 G2 E+ τ G2M 1 + τ cos 2θ 2 + 2τ G 2 Msin2 θ 2 . (19) Where τ = mQ22
p and GE and GM represent the charge and magnetic moment distribution of the proton respectively.
2.3.2 Deep Inelastic Electron-Proton Scattering 2.3.2.1 Kinematics
In deep inelastic scattering, the proton breaks up due to one of its quark’s interaction with the photon emitted by the electron. This results in multiple hadrons being produced because of the collapse of the proton. The invariant mass of the hadronic system, W , depends on the four momenta of the photon: W2 = p24 = (p2+ q)2, with p2 the proton four momentum.
Because W has a range of possible values in inelastic scattering (in elastic scattering it was always the four momentum of the proton) the kinematics have to be described using two variables instead of one. There are four Lorentz invariant variables of which two can be used. The first variable, Q2, is defined by the four momentum of the virtual photon
Q2 = −q2. (20)
When expressed in the four momenta of the initial and final electron, and neglecting the electron mass, Q2 can be expressed as
Q2 = 4E1E3sin2
θ 2
. (21)
The Lorentz invariant variable x is defined as
x = Q
2
2p2· q
in terms of W: x = Q2+WQ22−m2
p. Because mp is the mass of the lowest hadronic state (the proton) W ≥ mp. Therefore the value of x must be between 0 and 1. When W is a proton,
so the scattering is elastic, x equals 1. Q2 and x are the most common used variables in deep inelastic scattering. The other two, y and υ, are variables used to describe the energy loss of the electron y = p2· q p2· p1 (23) and υ = p2· q mp . (24)
The Rosenbluth formula can be made Lorentz invariant using the expressions of Q and y. GM, GE and τ are all Q2 dependent and can be absorbed into other functions f1 and f2
dσ dQ2 = 4πα2 Q4 1 − y − m 2 py2 Q2 ! f2(Q2) + 1 2y 2f 1(Q2) . (25)
In elastic scattering, where x equals 1, the functions f1 and f2 are independent of x. To
extend the Rosenbluth formula to the inelastic realm, a few adjustments have to be made to (25). Together with Q2 m2
py2 in an inelastic collision gives
dσ2 dxdQ2 = 4πα2 Q4 (1 − y)F2(Q 2, x) x + y 2F 1(Q2, x) . (26)
During experiments it was first shown that both F1 and F2 are (fairly) independent of Q2.
Secondly, in the deep inelastic realm it was shown that F2 = 2xF1 (Bodek, 1979). With these
experimental data, it was concluded that the photon did not interact with the proton as a whole but had an elastic scattering with a spin 1/2 particle (a quark).
2.3.2.2 Elastic Quark-Electron Scattering
The spin-averaged expectation value of the matrix element squared of the elastic QED scat-tering is given by h|Mf i|2i = 2Q2 qe4 (p1· p2)2+ (p1· p4)2 (p1· p3)2 (27) The most convenient frame to choose is the center of mass frame and to express the matrix element in terms of θ∗ as shown in Figure 3.
The four momenta can easily be obtained from the angle and the starting momenta. From which the inner products can be formulated leading to expressing the matrix element of a QED scattering between a quark and an electron as
Figure 3: Schematic display of an electron-quark scattering in the center of mass frame with the beam axis as the z axis and with the x axis perpendicular to the beam axis. p1 and p2 represents the electron and quark four momentum before the
interaction and p3 and p4 after the interaction.
h|Mf i|2i = 2Q2 qe4
4E4+ E4(1 + cosθ∗)2
E4(1 − cosθ∗)2 , (28)
with the expression for the matrix element it is straight forward to give the differential cross section using (13). For a Lorentz invariant differential cross section express cosθ∗ in terms of s and q2 and changing the variables using the spin-averaged expectation value of the matrix element squared of the elastic QED scattering is given by
dθ dq2 = dθ dΩ∗ dΩ∗ dq2 . (29)
Which leads to the final expression for the cross section in terms of s and q2
dσ dq2 = 2πα2Q2q q4 1 + 1 +q 2 s 2 . (30) 2.3.3 Hadron-Hadron Collisions
In a hadron-hadron collision the momentum fraction of the reacting partons (x1 and x2)
are unknown and the kinematics of the collision has to be described using three variables. The momentum fractions of the reacting partons can be extracted from other well measured quantities. One of these quantities is the rapidity (Y ). The rapidity is defined as
Y = 1 2ln E + pz E − Pz (31)
with E the energy, and pz the momentum in the is along the beam axis. This is a useful
quantity because differences in rapidity are Lorentz invariant. In Hadron-Hadron collisions the center of mass frame is not the same in the hadron frame as in the lab frame, so specifically for these collisions the rapidity is convenient. In collider experiments the mass component of the energy is negligible and so a new quantity can be introduced, the pseudorapidity, defined as η = − ln tanθ 2 . (32)
Where θ stands for the angle between the beam direction and the produced particles. The pseudorapidity of several angels is given in Figure 4.
Figure 4: The pseudorapidity for several angles with the z axis as the beam axis so the rapidity perpendicular to the beam axis is equivalent to a pseudorapidity of 0.
2.4 PDFs
2.4.1 Proton PDF
A proton is made up from partons (quarks). Each quark carries a part of the momentum of the proton, pq= p2=(E2,0,0,E2). In the infinite momentum frame the assumption can be
made that the mass of the proton can be neglected as well as any motion of a struck parton perpendicular to the motion of the proton. The four momentum of the quark squared, after the interaction with a photon, is the mass of the quark so
(p2+ q)2 = 2p22+ 2p2· q + q2 = m2q (33)
= Q
2
2p1· q
= x, (34)
so in the parton model Bjorken x can be identified as the fraction of the proton momentum carried by the struck quark.
The differential cross section can be deducted from (30) with the substitutions xq=1 and
yq=y which yields dσ dQ2 = 4πα2Q2q Q4 (1 − y) + y 2 2 . (35)
This equation is the cross section for electron quark elastic scattering, where the quark has a fraction x of the total momentum of the proton.
Quarks group together into hadrons, like a proton. Inside the proton the quarks interact with each other using the strong force carrier, the gluon. This results into a momenta distribution of the quarks inside a hadron. The distributions of the quarks form the Parton Distribution Functions (PDF) of the hadron which is a universal property of each hadron. The up quark PDF inside a proton is defined as
up(x)δx, (36)
which represent the number of up quarks inside the proton with momentum fraction between x and δx.
The differential cross section is for electron-proton inelastic scattering is obtained by (35) and the definition of the PDF.
d2σep dxdQ2 = 4πα2 Q4 (1 − y) +y 2 2 X i Q2iqpi(x), (37)
with the sum being carried out over all the quark flavours with charge Q. When compared to (19), predictions for F2ep and F1ep for deep inelastic scattering between a proton and an electron can be made,
F2ep(x) = 2xF1ep(x) = xX
i
Q2iqip(x). (38)
Finding the structure functions F1 and F2 has to be done experimentally because the strong
force has a too big coupling constant, αs, so that perturbation theory can not be applied.
Experiments have indicated that about half of the proton’s momentum is being carried by all of the proton’s quarks. In the proton there are constantly quark anti-quark pairs being created by gluons, which are responsible for the other half of the momentum. Assuming that the quarks created are only first generation quarks (up, down, anti-up, and anti-down) an expression for F2ep can be derived with (38)
F2ep= x 4 9u p(x) + 1 9d p(x) + 4 9u¯ p(x) + 1 9 ¯ dp(x) . (39)
With the functions in the equation as the PDFs for the corresponding quark or anti-quark. In a neutron, with two down quarks and one up quark, the PDF of the up quark in a proton can be used for the PDF of the down quark in a neutron and the same goes for the down quark therefore F2en can be found by substituting dn for up and un for dp in (39). This assumption is called isosymetry.
The quarks inside a hadron can be categorised into two groups, the valence quarks and the sea quarks produced by the gluons. Because these sea quarks and valence quarks interact with each other, there is a smooth distribution of the momentum, x, carried by different quark species. The PDFs for the up and down quarks can be written as
up(x) = upv(x) + ups(x) (40)
for the up quark in the proton and similarly
dp(x) = dpv(x) + dps(x) (41)
for the down quark. Because the up and the down quarks are the only valence quarks inside a proton, the rest of the quark and anti-quark distribution functions are solely based on the sea quarks. Because the valence quarks of the proton are two up quarks and one down quark, the valence quark functions are normalised:
Z 1 0 upvdx = 2, (42) Z 1 0 dpvdx = 1. (43)
For the sea quark contributions some assumptions can be made. The first assumption is that because the quarks are produced by quark anti-quark production the sea PDFs of the corresponding quark and anti-quark are the same. Secondly, because the masses of the up and down quark are almost equal, it is expected that the production of an up anti-up pair is just as favourable as an down anti-down pair and therefore the difference of the PDFs will be negligible. With these assumptions a function, S(x), can represent all the first generation sea quark PDFs us(x) = ¯us(x) = ds(x) = ¯ds(x). With these assumptions a prediction can be
made for the ratio between the two structure functions of on one side the proton and on the other side the neutron:
F2en Fep =
4dv+ uv+ 10S
4uv+ dv+ 10S
2.4.1.1 Proton PDF Parameterization
Because the PDFs have to be found empirically, the functions to connect the data points have parameters. A common Parameterization used for the proton PDF is given by (Charchua, 1992)
fip = Axai(x − 1)bi, (45)
so there wont be any parton with either all the momentum nor none of the momentum. The parameterization given by (45) is a basic version. There are several research groups that use different parameterizations with more parameters, or use neural networks to fit the data. An example of a proton PDF for all partons is visualised in Figure 5.
Figure 5: Parton distribution functions, CT14NNLO, of a proton with x on the bottom axis. All valence quarks and their respective anti-particles are shown as well as the anti-strange quark distribution and the gluon distribution. It can be seen that the valence quark distributions are the highest. This graph was adapted from Dulat et al, 2016
2.4.1.2 PDFs At The LHC
Finding the gluon and quark structure of the proton is an important part of collider physics. Due to the fact that the internal structure of the proton is driven by non-perturbative dy-namics, PDFs can not be computed by first principles and have to be extracted from data measured during hard-scattering lepton-proton and proton-proton collisions. The PDFs of
the proton and their uncertainties play an important role in High Energy Collisions at the LHC. For example, the PDF and their respective uncertainties represent the dominant theo-retical uncertainty for the determination of Higgs boson coupling (Gao, 2017). With a better understanding of the internal structure of the proton, which leads to smaller uncertainties, the Higgs boson coupling can be found with smaller errors. Due to tightly fixed predictions by the standard model and a with smaller uncertainty for the coupling, evidence for new physics can be found when these do not comply. Next to this example, PDFs and their uncertainties are also of importance for Beyond the Standard Model (BSM) scenarios during high-mass resonance production which probe the proton structure at high x for which little data is available. Lastly, for measurements of SM parameters such as the W and Z boson masses, and the strong coupling constant the PDF with corresponding uncertainties are important. For these parameters the PDF uncertainties represent a limiting factor for its measurements, which could be reduced by furthering the understanding of the proton and thus decrease the PDF uncertainty (Gao, 2017).
2.4.2 Nuclear PDFs
The nucleus of an atom consist of protons and neutrons. What could be expected is that there will be no nuclear effects on the PDFs of the neutrons and protons so that the PDF of a proton inside a nucleus of atomic number A is the same as the PDF of a free proton
fip/A(x, Q2) = fip(x, Q2). (46)
In the case of a lead nucleus Z = 82 and A = 208 with Z the number of protons and A the number of total nucleons. The up quark PDF inside a nucleus can be written as
uP b=
Zup+ (A − Z)un
A , (47)
or with isospin symmetry,
uP b =
Zup+ (A − Z)dp
A . (48)
If this is correct the measured up quark PDF of a lead nucleus (uexpP b) and (48) have a ratio,
R = uP b
uexpP b = 1. (49)
However data from the LHC has shown that the ratio is not equal to one and therefore there is a nuclear effect on the PDFs of quarks inside a nucleus.
Just like the regular PDFs for free nucleons, the nPDFs have to be extracted via experiments for instance at the LHC. In these experiments a proton beam and, as an example, a lead
nucleus beam collide. One of the quarks inside the proton and one of the quarks inside the nucleus interact with each other, consequently both the lead nucleus and the proton break up and will form new hadrons. Subsequently, there is also a possibility of leptons being created due to decaying weak nuclear force gauge bosons created by annihilation of partons.
2.4.2.1 Nuclear PDF Parameterization
The parameterization of the nuclear PDFs is done by the hand of the proton PDF multiplied by the ratio function R from (49),
fip/A= RAi (x, Q2)fip(x, Q2). (50)
The ratio is a function of both x and Q2. R can be parameterized in several ways but its usually represented with several formulas for specific x ranges (Eskola, 2017),
RAi (x, Q2) = a0+ a1(x − xa)2 x ≤ xa b0+ b1xα+ b2x2α+ b3x3α xa≤ x ≤ xe c0+ (c1− c2x)(1 − x)−β xe≤ x ≤ 1 ,
where all parameters on the right hand side are dependent of the quark species as well as A. An example of this parameterization is shown in Figure 6
Figure 6: (Eskola, 2017) Illustration of the nuclear ratio function with the different x regions, from the parameterization, shown on the axis.
For this thesis the Z boson production differential cross section will be computed for a p-p and a Pb-Pb collision to model a p-Pb differential cross section. For comparison p-Pb collision data from ATLAS will be used. Several other research groups already include the ATLAS data which are being modelled in this thesis.
In this thesis, the Z boson production model will be tested to available data from ATLAS. The first p-Pb run at the LHC was preformed in 2013 and several research groups already include the measured data in their nuclear PDF set. Together with Z boson production, other quantities such as the W± production and jet cross sections are measured during collisions at ATLAS. To include new data the Bayesian reweighting method is applied, which uses PDF replicas to fit the new data. The effect the inclusion of the new p-Pb data had on the nuclear ratio of two nPDF sets can be seen in Figure 7.
Figure 7: Impact of the first ATLAS p-Pb run on the nPDFs of EPS (top 3 graphs) and DSSZ (bottom 3 graphs). Shown are both the ratio function before the reweight-ing (black) and after (red) with correspondreweight-ing errors. On the left, the nuclear ratio function for the valence quarks is shown, in the middle the nuclear ratio function for sea quarks is shown, and on the right the gluon distribution ratio function is shown. This graph was adapted from Armesto et al, 2016
2.4.2.2 Structure Functions
The structure function, F2A , for a nucleus of A nucleons Z protons and N neutrons, can be written as F2A= Z AF p,A 2 + N AF n,A 2 (51)
There is a nucleus which has an equal number of protons and neutrons, a deuterium nucleus (A = 2). It is assumed that in these nuclei there is no nuclear effect on the PDFs. Therefore the ratio of the structure functions F2A
FD 2
can be interpreted as the nuclear effect of the PDFs of nucleus with A nucleons. The structure function of a deuterium nucleus is given by
F2A=2= 1 2F p,A=2 2 + 1 2F n,A=2 2 . (52) Substituting (51) gives F2A=2= βF2A (53) with β given by β = A 2 1 +F n,A 2 F2p,A Z + NF n,A 2 F2p,A . (54)
The assumption can be made that the ratio of the two structure functions is free from nuclear effects
F2n,A F2p,A =
F2n
F2p. (55)
Therefore when the structure functions of a free proton and a free neutron are known β can be found for a nucleus when A, Z and N are known. From β and the structure function of deuterium, which can be extracted from experiments, the structure function of any nucleus can be found.
2.4.3 PDF Determination
PDFs are universal functions, so that they can be found imperially during an experiment and used to predict different experiments (Forte, 2013). The PDFs are parameterized at a fixed reference value of Q2, Q20. Using evolution equations, such as the integro-differential equations, tables of PDFs are produced as a function of x and Q2 from which the PDF can be applied in a different setting. These evolution equations are perturbative series in the strong coupling constant αs. The strong coupling constant is a relative to the coupling
constant of other forces big constant so that the leading order (LO) of the series alone won’t be extremely dominant over the next-to leading order (NLO) and the next-to next-to leading order (NNLO). Up until now the proton PDF sets are evolved until NNLO, as the perturbative series are known up to this accuracy. However the nPDF sets are evolved up to NLO and so to be consistent with the order of evolution the Z boson production cross section will for both the p-p and Pb-Pb collisions be computed with NLO accuracy.
2.5 Z Boson Production In Hadron Collisions
In p-p and p-Pb collisions Z bosons are produced by quark anti-quark annihilation in a Drell-Yan process,
q + q → Z, (56) the leading order of Z boson production is illustrated in Figure 8.
Figure 8: Z boson production in a hadron collision. A quark from hadron one with momentum fraction x1 annihilates with an anti-quark from hadron two with
momentum fraction x2, which form a Z boson which in turn decays into a lepton
anti-lepton pair.
There are two next-to leading order classes that can contribute to the Drell-Yan process of Z boson production; the virtual loop corrections and the real emission of a gluon or quark. One quark can emit a gluon and later on absorb it again, or it can be absorbed by the quark which later on interacts with the quark that did the emitting, these three possibilities are the virtual loop corrections. For the real gluon or quark emission there are four possibilities: A quark emits a gluon after which it annihilates with an other quark to form a Z boson, a gluon is absorbed by a quark after which the quark emits a Z boson, a quark emits a Z boson after which the quark annihilates with a different quark into a gluon or a quark emits a Z boson after which it absorbs a gluon. These seven possibilities are visualised in figure 9
For a quark and anti-quark to annihilate and create a Z boson, the sum of the charge and colour charge of the quarks must be equal to zero so the charge is conserved.
Z bosons have a number of possible decay options. One of these options is for the Z boson to decay into a lepton and its respective anti-particle. Because leptons do not carry a colour
Figure 9: NLO corrections to Z boson production in Drell-Yan process. The top three visualise the virtual loop corrections, the bottom four represent the real gluon or quark emissions.
charge, these particles are not susceptible to the strong force and, after the Z boson has decayed, can not react with other particles by the use of the strong force. Because of this, this decay channel of the Z boson has a very clean measurement. Next to this, because a fermion and its anti-particle is produced it is easy to trace it back to a Z boson decay (Nadolski, 2005). When on opposing sides a fermion and an anti-fermion is detected during an experiment and these can be traced back to the same origin point these has to have been a Z boson there.
3
Method
To verify if the geometric mean of a p-p and a Pb-Pb collision would result in the same Z boson production differential cross section as a Pb-p collision, different PDFs will be used. These PDFs will be used in a program called MCFM, in which the specifics of the process can be put in, such as the center of mass energy.
3.1 PDF Selection
There are several groups that update their PDF set on a regular basis. For the verification, a set has to be chosen for both the proton PDF and the lead PDF for MCFM to use. In the first place a MCFM run was tried with the NNPDF proton PDF set, but due to a lack of computing power this was rendered unsuccessful. Due to this the choice was made for the CT14 proton PDF as the PDF set. This choice was made because this PDF set was already installed in the MCFM program. For the lead atom PDF the same proton PDF was used, CT14. But because this concerns a nuclear PDF a nuclear correction factor has to be implemented, this was done with EPPS16.
For the Z boson production cross section for both the p-p an Pb-Pb collision, only the central value was computed. The errors of the PDF sets, which result in an error in the differential cross section bins, have been omitted for the purposes of this thesis.
3.2 MCFM Parameters
MCFM has several input parameters in which you can specify the specific process. First of all the process is Z boson production, with a decay to a lepton anti-lepton pair. The center of mass energy is set to be 5.02 TeV, the p-Pb run in CERN was done with this center of mass energy and therefore there is data, measured at ATLAS, available for comparison. Some cuts were also implemented into the program. First of all the mass cut that was used for the leptonic pair is 60GeV < m34 < 120GeV . Secondly there was a rapidity cut implemented,
|Yz|< 3.5. The experimental data was measured with a rapidity bin width of 0.5, and so MCFM was also set to have a rapidity bin width of 0.5.
The Z boson production was computed with next-to-leading order (NLO) precision for both the p-p and the Pb-Pb collision.
3.3 Geometric Mean
Due to technical complications with the MCFM program, a direct p-Pb collision could not be computed. What could be computed with MCFM were a p-p and a Pb-Pb collision. Because
these collisions were available, a model had to be chosen for the p-Pb collision consisting of the available collisions. This thesis uses the geometric mean of a p-p and Pb-Pb collision as a model for the Z boson production differential cross section of a p-Pb collision.
To model a p-Pb collision the geometric mean will be taken from the p-p collision and the Pb-Pb collision for all bins. The geometric mean of a certain bin of the p-Pb model is defined as δσ δY M od = s δσ δY p−pδσ δY P b−P b . (57)
The outcome of this model will be compared to the data generated in the p-Pb run at CERN with a center of mass energy of 5.02 TeV.
The measured data the computed values will be compared to were extracted using the EasyN-Data tool. The graph that was used in the EasyNdata tool can be found in ’Vector boson production in pPb and PbPb collisions at the LHC and its impact on nCTEQ15 PDFs’ by Kusina et al (Kusina, 2017, Fig.3). The EasyNData tool is used to obtain data from graphs, calibrating the picture after which all the data points can be extracted by the program.
3.4 Average Ratio
The computed Z boson differential cross section histograms of CT14 and CT14+EPPS16 generated by MCFM were too small compared to the measured data. The original values of the histograms were all a factor of between five and ten too small compared with the measured ATLAS data. So for all the steps in the histogram the ratio was calculated between its value and the measured data point. From this the average ratio (Rav) was calculated and all of the
original values were multiplied with this average δσ δY N ew = δσ δY Old ∗ Rav. (58)
The average ratio was found to be 7.43. In computing the average the last data point (YZ=3.25) was omitted because of its big uncertainty and low computed values, which lead
to a high rise in the average ratio ascending the rest of the histogram above the measured data points.
4
Results
The geometric mean of the computed p-p and a Pb-Pb collision predicts the Z boson pro-duction differential cross section for a p-Pb collision reasonably well. The only area where the geometric mean is too low is in the area between YZ=-2.25 and YZ=0. For All the other
measured data points the Geometric mean histogram goes through the error bars of the mea-sured ATLAS data. It was beyond the scope of this thesis to compute the Z boson production errors in the histogram which originate from the PDF uncertainty. These errors are of impor-tance in the quantitative interpretation of the results. The outcome of the computed values with the geometric mean and the ATLAS data points for comparison have been visualised in Figure 10.
Figure 10: Z production differential cross section computed with MCFM for a proton-proton collision (CT14, red) and a Pb-Pb collision (CT14+EPPS16, blue) with calculated Geometric mean of the CT14 and CT14+EPPS16 (Green). The data measured at ATLAS for a proton lead collision with center of mass energy √
5
Discussion
In this thesis, a model was used to predict the Z boson production differential cross section in a p-Pb collision at a center of mass energy of 5.02TeV. The used model was the geometric mean of the differential cross section of a p-p and a Pb-Pb collision both at a center of mass energy of 5.02TeV. When this model is compared to the data, the computed differential cross section predicts most of the measured data, as can be seen in Figure 10. At the data points between a rapidity of -2.25 and 0 the Geometric mean undervalues compared to the measured data of ATLAS. For all other data points the computed histogram values are inside the corresponding data error bars.
The chosen model does have shortcomings, which could be the explanation for the difference in the histogram between a rapidity of -2.25 and 0 and the measured data. First of all, the used PDF sets all are measured in experiments and therefore have an error, this will lead to an error in the predicted differential cross section. It was beyond the scope of this thesis to compute the errors for both the p-p collision cross section and the Pb-Pb collision cross section. The computation of the errors and implementing them into the results will lead to a bigger chance of finding the computed value in the measured data errors.
Secondly, due to practical constraints this model had to be chosen. This model does not perfectly reflect the reality of a p-Pb collision, so it is plausible that if a p-Pb collision is computed in the MCFM program the computed cross section values are closer to the measured ATLAS data especially in the rapidity region between -2.25 and 0.
For this thesis the p-p and Pb-Pb collisions have been computed using the NLO corrections. Of course there are further order corrections during hadron collisions which, when taken into account, could result into a higher differential cross section so then the computed values will be better in occurrence with the measured data.
As stated in the method section, the differential cross section values computed by MCFM were all too low, by a factor of between 5 and 10, compared to the measured data. That MCFM computed too low values could be explained by that MCFM only computed the Z bosons that would decay into a lepton anti-lepton pair whilst the ATLAS data measures all the Z bosons produced. Z bosons can also decay into other final states and the probability through which channel a Z boson decays is given by the branching factor. The branching factor was too high to multiply the computed values with, as it would have risen all the computed values above the measured data. So it was decided to compute the average ratio, as the exact origin of the ratio was unknown and could theoretically not be verified.
To verify if the geometric mean of a p-p and a (different from lead) nucleus-nucleus collision does predict a collision of a proton and said nucleus further research can use the presented
setup for lead which can be used for other nuclei as well, with a different nuclear PDF. The computed value for the differential cross section from these other nuclei can be compared to experimental data as to verify if the geometrical mean predicts the Z boson production for other nuclei as well.
6
Conclusion
In this thesis an answer has been sought to the question: ’To what extend can the geometric mean of the Z boson production differential cross section of a p-p and a Pb-Pb collision model a p-Pb collision?’ To answer this question both a p-p and a Pb-Pb collision Z boson cross section were computed, from which the geometric mean was calculated.
From the results, it can be concluded that the geometric mean of a p-p and a Pb-Pb collision predicts the Z boson production fairly well. Except for the region between Y =-2.25 and Y =0, where the measured ATLAS data is higher than the prediction provided by the model as can be seen in Figure 10.
When for an analysis only the rough numbers of a Z boson production cross section of a proton-nucleus collision are needed the geometric mean can be applied for finding a cross section for p-Pb collision but when rough numbers are not sufficient it is advisable to compute a p-Pb collision directly so the results will be more in occurrence with the data found at ATLAS.
7
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