Higgs at the LHeC Electron-Proton Collisions at the Energy Frontier

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National Institute for

Subatomic Physics University of Amsterdam

Higgs at the LHeC

Electron-Proton Collisions at the Energy

Frontier

Author:

Tal van Daalen Student nr: 10247513

Supervisor: Prof. Stan Bentvelsen Second Assessor: Dr. Ivo van Vulpen

Verslag van Bachelorproject Natuur en Sterrenkunde, omvang 15 EC, uitgevoerd tussen 31-03-2014 en 09-07-2014

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Abstract

In this preliminary study, the prospects of precision Higgs measurements at the Large Hadron Electron Collider (LHeC) are investigated. In the LHeC project, 60 GeV electrons collide with 7 TeV protons from the LHC ring, corresponding to√s = 1.3 TeV and expanding the deep inelastic scattering (DIS) experimental range to x ranging from 10−6 to nearly 1 and Q2 extending to over 1 TeV2. Higgs production in DIS occurs through radiation from W± or Z boson probes and was analysed with Monte Carlo simulations performed by MadGraph. This resulted in promising prospects for the LHeC to become a precision Higgs facility. Also, a novel signal reconstruction method was created, using the final-state electron and parton to reconstruct the Higgs mass, which has the potential to further boost the accuracy of Higgs measurements at the LHeC.

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1 Introduction

With the recent discovery of the Higgs boson at the Large Hadron Collider (LHC), the Standard Model has passed its toughest test yet. The particle content of the Standard Model has been confirmed and all seems to be going smoothly. Is the field of particle physics running out of things to discover?

That is what one might naively think. Luckily it is far from true. The horizons are only just opening up to the exotic physics that many believe underlie the Standard Model. The High-Luminosity stage of the LHC will most likely go online in 2022, offering the best opportunity so far of revealing these exotic physics. When such major steps in any field of study are taken, it is important to take into account any unexpected turnouts and keep all options open. This is where the Large Hadron Electron Collider (LHeC) comes into play.

In this Bachelor thesis I research the physics and prospects of precision Higgs meas-urements at the LHeC. Firstly, an outline of the LHeC project is given. Secondly, a theoretical description of electron-proton scattering is given to provide a basis for the main goal: understanding Higgs production as it will occur in the LHeC. Thirdly, this process is analysed using Monte Carlo simulations performed by MadGraph to invest-igate the LHeC potential of becoming a precision Higgs facility.

2 The Large Hadron Electron Collider

The Large Hadron Electron Collider is an electron beam upgrade to the existing proton-proton beam configuration of the LHC. Its physics programme and design studies are being performed by the LHeC Study Group, with contributors from 79 institutes from over twenty countries [5]. The LHeC programme was first proposed in 2008 at the inaugural meeting of the LHeC workshop on electron-proton (ep) and electron-ion (eA) collisions at the LHC. The physics of electron-ion collisions is not of further concern in this thesis.

2.1 LHeC Design

In order to maximally benefit the implementation of an electron beam in the LHC, many different design configurations have been considered. The baseline design is briefly discussed here to give a general idea of the scale of the LHeC operation.

2.1.1 Detector

The LHeC requires the installation of a new detector at the LHeC collision site. This detector has to be tailor-made for ep/eA collision events, for which high level precision is of great importance. The beam energy configuration is highly asymmetric, see Tab.

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1. Therefore, it is crucial that the detector acceptance extends as close as possible to the beam pipe.

The detailed detector physics and techniques will be left out here. A schematic cross section of a candidate for the LHeC detector is given in Fig. 1. This detector allows for a polar angle coverage for final-state particle detection between 1◦ and 179◦.

Figure 1: Schematic cross section in yz-plane of the baseline design for the LHeC detector

2.1.2 Accelerator

Two main options for injecting the electron beam into the LHC ring have been con-sidered, consisting of a ring-ring and a linac-ring option. The ring-ring option would consist of an electron accelerator ring to be placed on top of the existing LHC ring. This design is schematically sketched in Fig. 2. The linac-ring option would include a linear accelerator with energy recovery trough a 9 km racetrack configuration, allowing for multiple passes. This design is schematically sketched in Fig. 3.

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Figure 2: Schematic sketch of possible ring-ring configuration for electron beam

injector

Figure 3: Schematic sketch of possible linac configuration for electron beam

injector

The LHeC complements the ongoing research of the LHC. Therefore, it is important that no interference occurs. The existing designs allow for the ongoing proton-proton collisions to continue, but simultaneously run electron-proton collisions. Currently, due to considerations of physical parameters and installation issues, the preferred design is the linac-ring configuration.

2.1.3 Beam Parameters

The current LHeC design projects a 60 GeV electron beam colliding with a 7 TeV proton beam or a 2.76 TeV ion beam. Therefore, an electron-proton collision at the LHeC has a centre-of-mass energy of 1.3 TeV. The projected luminosity during electron-proton running is 10−33 cm−2 s−1. The accessed deep inelastic scattering regime extends up to Q2 _{of over 1 TeV}2 _{and x ranging from 10}−6 _{to nearly 1, [4]. An explanation of these}

terms can be found throughout Sec. 3.

A preliminary summary of the LHeC run parameters, adapted from [2], is given in Tab. 1. Furthermore, an extremely detailed paper on the design and physics programme of the LHeC from 2012 can be found in [4].

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Parameter [unit] LHeC

species e p, 208_{P b}82+

beam energy (/nucleon) [GeV] 60 7000, 2760

bunch spacing [ns] 25, 100 25, 100

bunch intensity (nucleon) [1010_] _{0.1 (0.2), 0.4} _{17 (22), 2.5}

beam current [mA] 6.4 (12.8) 860 (1110), 6

rms bunch length [mm] 0.6 75.5

polarization [%] 90 none, none

normalized rms emittance [µm] 50 3.75 (2.0), 1.5 geometric rms emittance [nm] 0.43 0.50 (0.31) IP beta function β_x,y∗ [m] 0.12 (0.032) 0.1 (0.05)

IP spot size [µm] 7.2 (3.7) 7.2 (3.7)

synchrotron tune Qs 1.9 · 10−3

hadron beam-beam parameter 0.0001 (0.0002)

lepton disruption parameter D 6 (30)

crossing angle 0 (detector-integrated dipole) hourglass reduction factor Hhg 0.91 (0.67)

pinch enhancement factor HD 1.35

CM energy [GeV] 1300, 810

luminosity / nucleon [10−33 cm−2 s−1 ] 1 (10), 0.2

Table 1: Projected LHeC run parameters for both ep and eA collisions. The quantities in between parentheses would optimize the ep luminosity.

3 Theory

Electron-proton scattering experiments are a powerful tool for researching the internal structure of the proton. However, in light of the recent discovery of the Higgs boson, electron-proton experiments may provide the keystone to fully understand this particle. Before venturing into the regime of high energy electron-proton collisions and possible Higgs phenomena, an introduction to elastic and inelastic scattering is given. Then, the specification to deep inelastic scattering will be made, before discussing Higgs production and the intricacies involved. As mentioned previously, the physics of eA collisions will be left out.

3.1 Electron-Proton Scattering

The nature of an electron-proton collision is determined by the wavelength of the virtual photon that is exchanged between the electron and the proton. This wavelength relates to the transferred momentum, Q2_{, according to}

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λ ∼ 1

pQ2, (1)

where Q2 is related to the four-momentum of the virtual proton as

Q2 ≡ −q2_. ₍₂₎

Note that, for now, this is the definition that will be used throughout the theoretical description of electron-proton scattering. During the process of Higgs production this might give rise to some ambiguity, since in that case the four-momenta of the probes radiated by the electron and the parton are different. This issue will be further elucidated in Sec. 3.3.

3.1.1 Different Processes

Depending on the ratio of the wavelength to the radius of the proton, which can be imagined as a measure for the resolution with which the proton is being examined, different processes occur during electron-proton collisions:

1. If the energy involved is low, and the electron is non-relativistic, the wavelength of the photon is very large compared to the proton radius, λ rp. Here, the

electron-proton scattering is effectively the electron scattering of the 1/r electro-static potential of a point-like proton.

2. When the energy is somewhat higher, so that λ ∼ rp, relativistic effects of the

electron have to be taken into account.

3. At higher energies, when the photon wavelength becomes smaller than the proton radius, λ < rp, the cross section of elastic scattering becomes small and inelastic

scattering becomes the dominant process. Here, the photon interacts with one of the constituent quarks inside the proton and makes the proton break up. For the elastic case, the complicated structure of the proton has to be included.

4. When the energy available to the photon is so high that its wavelength becomes much shorter than the proton radius, λ rp, the elaborate internal structure of

the proton can be probed. This is called the deep inelastic scattering (DIS) regime. Again, the photon interacts with a single quark and the proton always breaks up in the process. At these energies, the proton manifests itself as a conglomeration of (sea) quarks and gluons constantly interacting with each other to create a coherent entity. It is also at these energies that the exchange of W and Z bosons can occur instead of photons. This will turn out to be the key to Higgs production in DIS.

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3.1.2 Kinematics and Cross Sections

When one calculates the cross section of the above described processes, different assump-tions and limits have to be taken for each case. These cases will be discussed briefly, but since the primary interest of this research lies with Higgs production, the main focus will be towards understanding DIS.

The calculations of the cross sections of the above described processes can be made with the consideration of a form factor. This is the three-dimensional Fourier transform of the charge distribution of the proton, and is a function of q2. Qualitatively, the form factor accounts for the effect that the extended charge distribution of the proton has on the scattered photon wave. Different parts of the wave, scattering from different points on the proton potential, can interfere constructively or destructively with the entire scattered photon wave. Therefore, it is very intuitive that the form factor depends on q2_{, which in turn determines the wavelength of the probing photon. That’s why, for each}

case of different probing photon wavelength, a corresponding form factor must be used. For case 1), the form factor is of only trivial importance. Since the electrostatic potential used to calculate the cross section is that of a point-like proton, the charge distribution function is simply a delta-peak. Therefore there are no interference effects of any sort contributing to the scattered photon and the corresponding form factor is flat. The differential cross section of case 1) is called the Rutherford scattering cross section and is given by

dσ dΩ Rutherf ord = α 2 16E2 esin 4_(θ/2), (3)

where Ee is the energy of the electron and θ its scattering angle. No derivation will be

given here, but this result can be derived by assuming a non-relativistic electron and keeping the proton as a fixed source and neglecting its magnetic moment.

For case 2), the corresponding cross section is the Mott scattering cross section. Here, the electron is considered to be relativistic, but still the charge distribution of the proton is taken as that of a point charge. So again, the form factor is trivial. The Mott scattering cross section is given by

dσ dΩ M ott = α 2 4E2 esin4(θ/2) cos2 θ 2 , (4)

and it is valid when approximately me Ee mp. The Mott scattering cross section

can be derived by taking relativistic effects into account, but still neglecting the recoil of the proton along with the magnetic spin-spin interaction.

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Case 3) approaches the realm of high energy collisions and so the scattering physics become more complicated. Firstly, the recoil of the proton cannot be ignored anymore and also the effects of the magnetic spin-spin interaction start to play an important role. Therefore, regarding cases 3) and 4), it is convenient to work according to the Feynman rules.

Figure 4: Electron-proton scattering with four-momenta labeled, adapted from [7]

In the rest-frame of the proton, the four-momenta of the particles involved, labeled as seen in Fig. 4, can be expressed as

p1 = (E1, 0, 0, E1)

p2 = (mp, 0, 0, 0)

p3 = (E3, 0, E3sin θ, E3cos θ)

p4 = (E4, p4) .

(5) Here, the electron mass is taken to be negligibly small compared to its energy. Applying the Feynman rules for a t-channel diagram to acquire the matrix element gives

h|Mf i|2i = 8e4 (p1− p3) 4 (p1· p2) (p3· p4) + (p1· p4) (p2 · p3) − m2p(p1· p3) . (6) Using energy and momentum conservation to eliminate p4 and expressing the inner

products in terms of the initial and final-state electron energies, E1 and E3 respectively,

and the scattering angle θ, (6) becomes h|Mf i|2i = 8e4 (p1− p3)4 mpE1E3 (E1− E3) sin2 θ 2 + mpcos2 θ 2 . (7)

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The term (p1− p3) 4

, which originates from the photon propagator term as q4_{, can also be}

written in terms of E1, E3 and θ. Moreover, it is convenient to work with the kinematic

variable Q2_{, so that} Q2 ≡ −q2 _{= − (p} 1− p3)2 = 4E1E3sin2 θ 2 . (8)

Finally, the energy lost by the electron, E1− E3, can also be expressed in terms of Q2,

as

E1− E3 =

Q2 2mp

(9) Using these steps to refine the matrix element in (7), the differential cross section be-comes dσ dΩ = 1 64π2 E3 mpE1 2 h|Mf i|2i = α 2 4E2 1sin4(θ/2) E3 E1 cos2 θ 2 + Q 2 2m2 p sin2 θ 2 . (10)

This is a very important result concerning the physics of elastic scattering. Equation (10) contains E3, Q2 and θ, which are all variables that are initially unkwown.

How-ever, E3 as well as Q2can be expressed in terms of θ and initially known values, as follows

E3 = E1mp mp+ E1(1 − cos θ) , (11) Q2 = 2mpE 2 1(1 − cos θ) mp+ E1(1 − cos θ) . (12) Thus, the complete kinematics of an elastic collision are solely determined by one in-dependent variable. Measuring the scattering angle θ provides all acquirable information about the interaction.

The above derivation of the differential cross section does not yet account for the magnetic moment and extended charge distributions of the proton. Including these into the calculation of the differential cross section, one eventually obtains the Rosenbluth formula, given by dσ dΩ = α 2 4E2 1sin4(θ/2) E3 E1 G2 E + τ G2M 1 + τ cos 2 θ 2 + 2τ G2_Msin2 θ 2 , (13) with τ defined as τ = Q 2 4m2 p . (14)

The Rosenbluth formula describes an elastic scattering interaction in its most general form. The factors GE and GM represent the form factors and thus are functions of

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Q2_{. They respectively account for the charge and magnetic moment distribution of the}

proton. Their exact shapes will not be discussed here, but their proportionality to Q2 is of great importance. Measurements have shown that both GE and GM scale with Q2

with the shape of a dipole function. In the limit of high Q2_{, the dipole function reduces}

to a Q−4 shape and thus

dσ dΩ elastic ∝ 1 Q6 dσ dΩ M ott . (15)

Hence, the cross section for elastic scattering decreases rapidly as Q2 _increases.

Con-sequently, as mentioned before, electron-proton scattering at high energy scales predom-inantly consists of inelastic processes. The Q2_{dependence that causes this can be traced}

back to the effect of the extended charge distribution of the proton. One can therefore expect that any cross section expressions with high Q2 that involve an interaction with a proton would have an analogous Q2 _{suppression. However, the cross section for case}

4) does not have this Q2 _{dependence, which suggests that this process does not involve}

the entire proton, with its extended charge distribution, but rather (one of) the con-stituent quarks inside it. This observation lead to the discovery of the compositeness of the proton and is exactly what characterizes DIS.

3.2 DIS

The regular electron-proton DIS scattering process can be described as an e−p → e−X process, shown in Fig. 5. X specifies the hadronic final-state from the proton that was broken up. The invariant mass of the hadronic final-state X, usually denoted W , is given by

W2 = p2₄ = (p2+ q)2. (16)

Because W2 _{depends on q, the four-momentum of the probing photon, the value}

of W2 is not predetermined by the known collision properties. Thus, immediately, a major difference between the kinematics of elastic and inelastic scattering arises. The kinematics of an elastic scattering collision are solely determined by one variable, whereas an inelastic scattering collision has an additional degree of freedom that results from the breaking up of the proton. Hence, to fully understand an inelastic scattering event, two independent variables have to be measured.

3.2.1 Kinematic variables

There are several kinematic variables that are used to characterize an inelastic electron-proton scattering event. One can use two out of the four most commonly used, Lorentz-invariant quantities x, Q2_{, y and ν. These will be discussed shortly. However, x and Q}2

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Figure 5: Electron-proton inelastic scattering, adapted from [7]

The dimensionless quantity x is called the Bjorken x, after American physicist James Bjorken, and is defined as

x ≡ Q

2

2p2· q

. (17)

Effectively, x is a measure for the “elasticity” of a collision. x can take any value between 0 and 1, where the case of x = 1 corresponds to elastic scattering where the hadronic final-state is simply the proton, W2 _{= m}2

p. Also, it appears that x expresses

the fraction of the momentum of the initial-state proton that the struck quark carries. The measurement of x therefore provides information about the way the momentum of the entire proton is distributed between the quarks it consists of. For more on this, see Sec. 3.2.2.

The definition of Q2 _{in DIS is the same as for elastic scattering, according to (2).}

The kinematic range of a scattering experiment can be clearly displayed in a (x,Q2₎

diagram. This will be discussed more thoroughly in Sec. 4.

The quantity y can be seen as the “inelasticity” of a collision and is defined as y ≡ p2· q

p2· p1

. (18)

In the initial-state proton rest frame, y expresses the fractional energy lost by the elec-tron, as

y = 1 −E3 E1

, (19)

which directly shows that y too can take a value between 0 and 1.

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ν ≡ p2· q mp

. (20)

In the initial-state proton rest frame, ν is simply a measure for the energy lost by the electron:

ν = E1− E3. (21)

Using these definitions, it is possible to construct an expression for the cross section of inelastic electron-proton scattering. This cross section will be related to the cross section for electron-quark elastic scattering, which is the underlying process. This is an important step for eventually understanding Higgs production.

The full derivation of the inelastic scattering cross section will not be included here. It is important to notice that, because an inelastic scattering event is determined by two independent variables, the cross section will be expressed as a double differential cross section. The quantities x and Q2 are the most suitable for expressing this cross section. Starting from the Rosenbluth formula, (13), and making several assumptions that are valid for DIS, one obtains the following expression for the cross section of electron-proton inelastic scattering mediated by a virtual photon:

dσ2 dx dQ2 ≈ 4πα 2 Q4 (1 − y)F2(x, Q 2₎ x + y 2_F 1(x, Q2) . (22)

Although y appears in this expression, it is related to x and Q2 according to

y = Q 2 s − m2 p x ≈ Q 2 sx (23)

where s is the centre-of-mass energy of the process. This can be derived from the definitions of x, Q2_{, y and ν. The factors F}

1 and F2 are called structure functions and

are directly related to the form factors as they appear in (13). See also Sec. 3.2.2. 3.2.2 Elastic Electron-Quark Scattering

Continuing to make the specification to elastic electron-quark scattering provides a basis for understanding Higgs production through exhange of a virtual W± or Z boson. The cross section of the e−q → e−q process has to be derived according to the quark-parton model. This is the viewpoint from which the electron scatters off a single quark inside the proton. The quark-parton model description of this process is given in Fig. 6, in contrast to the hadronic viewpoint given in Fig. 5.

The cross section can be derived by first calculating the spin-averaged matrix element of the process, by applying the Feynman rules, and then expressing it in terms of the kinematic variables discussed in Sec. 3.2.1. Implementing this into the general formula for the cross section results in (24). Again, a full derivation is left out. Note that Q2

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Figure 6: Electron-quark scattering in the quark-parton model, adapted from [7]

defined with respect to the electron-quark system, instead of the entire electron-proton system. dσ dΩ = 4πα 2_Q2 q Q4 h (1 − y) + y 2 2 i . (24)

This equation resembles (22), where the double differential cross section of inelastic electron-proton scattering was described in terms of x and the structure functions. Even though x does not appear in (24), the implicit dependence on x follows from (23).

The study of the quark-parton model mainly revolves around the determination of Parton Distribution Functions (PDFs). These describe the internal momentum distri-butions between the quarks inside a proton. As can be expected, PDFs are directly related to the structure functions as they appear in (22).

3.2.3 Charged Current and Neutral Current

At sufficiently high energy scales, electron-proton collisions can occur with the exchange of weak gauge bosons, the force carriers of the weak interaction. The weak interaction differs very much from the QED and QCD interaction, which both follow roughly the same set of Feynman rules. QED and QCD are both mediated by massless spin-1 bosons. The weak interaction is mediated by massive, neutral Z bosons and massive, charged W± bosons. Furthermore, the weak interaction violates parity conservation, also unlike QED and QCD interactions. This introduces intricacies in calculations of electroweak processes, but is not of great concern here.

With the possibility of DIS events being mediated by charged W± bosons, two dis-ctinct possibilities for the final-state particles arise. In a DIS event mediated by a photon or a Z boson, none of the initial-state particles changes flavour and so the initial- and final-state are identical. This process is called neutral current (NC) DIS and is displayed in Fig. 8. In a DIS event mediated by a W±boson, however, both the initial-state lepton

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and quark change flavour and so the final-state contains a neutrino and an opposite fla-vour quark compared to the initial-state. This process is called charged current (CC) DIS and is displayed in Fig. 7.

Figure 7: Feynman diagram of charged current DIS

Figure 8: Feynman diagram of neutral current DIS

For the calculation of the cross section of electroweak CC and NC DIS events, consid-eration of all properties of the particles involved is of great importance. A large difference between photon mediated DIS and Z or W± boson mediated DIS is due to the masses of the electroweak probes. These introduce a factor of (Q2+ M_Z2)−1 or (Q2+ M_W2 )−1 to the cross sections. The electroweak probes also have an additional degree of freedom due to their masses, in contrast to massles bosons. This introduces some complexity in the summation over the possible polarization states, which is needed to derive the propagator term in the Feynman rules. A full derivation of electroweak CC and NC DIS events is beyond the scope of this paper.

3.3 Higgs Production

A Higgs boson can be produced in electroweak NC or CC DIS by coupling to either the Z boson propagator (through a ZZH coupling) or the W boson propagator (through a W W H coupling). The leading-order Feynman diagrams of NC and CC Higgs production are displayed in Fig. 9 and Fig. 10 respectively.

A fully detailed desctription of Higgs production is not very insightful. However, without immersing in highly complex electroweak physics, it is still possible to describe Higgs production in a phenomenological way. The W W H and ZZH couplings contribute to the likelihood of a Higgs boson being produced. These appear in the Lagrangian of the Higgs field and are essentially just parameters of the Standard Model. Also the couplings of the W± and Z bosons to fermions affect the cross section of Higgs production. The Standard Model coupling parameters differ significantly for W± and Z bosons, which

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Figure 9: LO Feynman diagrams of NC Higgs production in DIS (generated by MadGraph)

Figure 10: LO Feynman diagrams of CC Higgs production in DIS (generated by MadGraph)

causes the values for CC and NC Higgs production cross sections to differ further. As mentioned earlier, another small difference in the cross section arises from the mass difference of the W± boson and the, slightly heavier, Z boson.

An analytic description of the Higgs production cross section would not be insightful. Instead, the cross section has been numerically calculated using MadGraph for several run parameters. See also Sec. 4 for more on MadGraph. The polarization of the electron beam also influences the cross section significantly, which has to do with the left-handedness of the interaction. This will not be discussed further here. In Tab. 2 the numerically calculated cross sections are displayed. As can be seen, the CC cross section is generally higher.

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Beam energy 50 GeV + 7 TeV 100 GeV + 7 TeV 150 GeV + 7 TeV Event type CC NC CC NC CC NC Cross section [fb] P = 0 72.89 13.11 150.9 28.62 218.5 41.27 Cross section [fb] P = 0.8 14.59 10.68 30.21 23.43 43.71 33.97 Cross section [fb] P = -0.8 131.2 15.51 271.2 33.74 393.6 48.63

Table 2: CC and NC Higgs production cross sections [fb] for several beam energy configurations and electron beam polarizations, P

The current baseline beam configuration of the LHeC consists of a 90% left handed polarized 60 GeV electron beam and a 7 TeV proton beam, see Tab. 1. The correspond-ing cross sections for CC and NC Higgs production are

σH,CC = 196 fb (25) σH,N C = 25 fb (26)

3.4 Higgs Decay and Background

The detection of a Higgs signal has to be done by detecting its decay products, since the Higgs boson itself has a lab-frame lifetime in the order of 10−22 s. The possible Higgs decay channels are presented in Fig. 11. The dominant decay mode is H → b¯b, with a branching ratio value of approximately 60 % [3]. Therefore, only the decay into a b¯b pair will be considered for the simulation analysis from here on.

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Figure 11: Higgs branching ratio, with a dotted line indication at mH = 125 GeV

The observability of a e−p → e−p H (→ b¯b) event is determined by its distinguishab-ility from the background. Thus, in this case, the background consists of all events of the form e−p → X b¯b. The presence of an electron in the final state helps to determine whether the background has to be accounted for NC or CC Higgs production. See also Sec. 4. A few examples of NC and CC background processes are displayed in Fig. 12 and Fig. 13.

Figure 12: Two examples of e−p → e−X b¯b background Feynman diagrams, generated by MadGraph

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Figure 13: Two examples of e−p → νeX b¯b background Feynman diagrams, generated by

MadGraph

To get an idea for the amount of produced Higgs bosons in a certain running period of electron-proton collisions, one must compare the cross sections for signal events with those for all background events. The CC and NC background cross sections are roughly 1 pb and 22 pb respectively. This may appear to be a very large difference compared to the fb-scale cross sections given in (25) and (26). However, the kinematic behaviour of the produced Higgs boson and its decay products and how much they differ from the background eventually determines the observability of a signal. This will be discussed in the next section.

4 Simulations with MadGraph

To analyse Higgs production in electron-proton collisions MadGraph was used to gen-erate the signal and background events. To an attentive reader it must be noted that signal and background events were created separately, thus eliminating the possibility of any interference that might affect the results in reality. However, this is of no concern here since the amount of interference in this case is expected to be negligible.

4.1 About MadGraph

MadGraph is a computational framework for simulating any Standard Model (SM) and Beyond Standard Model (BSM) phenomena. MadGraph has an extraordinary flexibility towards the user, as any property of the entered process can be adjusted, such as coupling strengths and masses. QCD SM processes can be simulated to next-to-leading-order (NLO) accuracy by MadGraph, whereas the accuracy of other SM and BSM processes is limited to leading-order (LO).

MadGraph is also capable of combining its produced data with third-party software, such as Pythia and PGS. Thus, for example, the user has the option to simulate hadronization and showering of the final state with Pythia and also run these through

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a simulated detector with PGS, fully named “Pretty Good Simulation”. More on this can be found in Sec. 4.4. In such a simulated event chain, several .root files are created, each containing the process’ Tree level information. These files can be interpreted by Root, the widely used data analysis framework produced at CERN.

For this research, version 2.1.1 of MadGraph5 aMC@NLO was used. See [1] for a detailed description of MadGraph by its developers.

4.2 NC Parton Level Results

First, the relevant kinematic distributions of signal and background events are discussed. These were created with a custom-built Root analysis to interpret the MadGraph generated .root files. At parton level, the properties of all particles that take part in the interaction are known. These results therefore are the most suitable for understanding the kinematics of signal and background events. It has to be noted that the following kinematic distributions were all created using an equal number of signal and background events to clearly bring out the differences. Also, no polarization of the electron beam was implemented.

To keep the length of this thesis within limits, the discussion of parton level results is restricted to NC events. These have a lower Higgs production cross section than CC events, but are easier to reconstruct because of the presence of an electron in the final-state, instead of an undetectable neutrino. Moreover, in Sec. 4.3, a reconstruction method is presented which particularly depends on this electron to reconstruct the Higgs boson mass.

4.2.1 (x, Q2_{) Reconstruction}

With the Root analysis, the value of Q2 _{was reconstructed from the outgoing electron,}

according to (8). From Fig. 14, it can be seen that background entries of Q2 peak at a much lower value than signal entries.

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Figure 14: Q2 values for both signal and background events

Figure 15: (x, Q2) kinematic range scatter plot for both signal and

background events

Furthermore, a reconstruction of (x, Q2_{) for DIS provides information about which}

physical properties are accessible. For both signal and background events the (x, Q2) diagram is displayed in Fig. 15. It can be seen that Higgs events occur practically only at events with both high x and Q2_.

4.2.2 Higgs Kinematics

The detection of a Higgs event will eventually depend on the detection of its decay products, which in turn have their kinematics determined by the kinematics of the mother Higgs boson. The Higgs transverse momentum and pseudorapidity are displayed in Fig. 16 and Fig. 17 respectively.

Figure 16: Transverse momentum of produced Higgs boson

Figure 17: Pseudorapidity of produced Higgs boson

The transverse momentum distribution shows that the pT values peak at around

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Figure 18: Energy of produced Higgs boson

shows in the distribution of the pseudorapidity, with a peak at η around 2. Note that the projected detector acceptance allows for particle detection up to a polar angle of 1◦ from the beam axis, corresponding to |η| < 4.74. The Higgs energy distribution evidently starts at its mass, but extends to very high values.

4.2.3 b¯b Kinematics

In order to get an idea for the observability of a Higgs signal, the kinematics of the produced b and ¯b quarks have to be compared to see how easy signal and background can be separated. The kinematics of individual b and ¯b quarks are displayed in Fig. 19, Fig. 20 and 21. Note that these distributions therefore contain two entries per event.

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Figure 19: Transverse momentum of produced b/¯b quarks

Figure 20: Pseudorapidity of produced b/¯b quarks

Figure 21: Energy of produced b/¯b quarks

It can be seen that the transverse momentum of the b and ¯b quarks that originated from a Higgs boson peaks at around 50 GeV/c, compared to only a few GeV/c for the background b and ¯b quarks. The difference in pseudorapidity is not very striking, but this will change when the combined system of b and ¯b quarks is analysed. The energy of b and ¯b quarks also peaks at a higher value for signal events than for background events. This is to be expected, since energy and transverse momentum are closely related.

Next, the Root analysis is used to combine the two separate b and ¯b quarks to a single b¯b system. This will highlight the differences between signal and background events further. The kinematics of produced b¯b pair are displayed in Fig. 22, Fig. 23 and

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Fig. 24.

Figure 22: Transverse momentum of produced b¯b pair

Figure 23: Pseudorapidity of produced b¯b pair

Figure 24: Energy of produced b¯b pair

An observant reader may notice that, as expected, the signal distributions of the b¯b pair are exactly the same as those of the produced Higgs boson, displayed in Fig. 16, Fig. 17 and Fig. 18. The difference with background b¯b pairs is what is to be highlighted here.

Again, the transverse momentum of the b¯b pair originating from a Higgs boson peaks at a higher value than background b¯b pairs, and also extends to very high values in the tail of the distribution. The background pseudorapidity has a higher spread than

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the signal distribution, which has a more distinct peak value. Similar to the transverse momentum, the energy of signal b¯b pairs is generally higher and also extends to higher values for a significant number of events.

Another quantity one might analyse to separate signal and background events is the invariant mass of a system of particles. For a single particle, this simply corresponds to the mass of the particle. So for a Higgs boson, and also a b¯b pair that originates from a Higgs boson, the invariant mass has to be 125 GeV. The invariant mass distribution of the produced b¯b pairs is displayed in Fig. 25.

Figure 25: Invariant mass of the produced b¯b pair

Figure 26: ∆R between the produced b and ¯b quarks

Furthermore, the angle between the b and ¯b quarks is another quantity that can be analysed. A measure for the angle between two particles is usually given by ∆R, for which the distribution is given in Fig. 26. As can be seen, the signal distribution starts at a ∆R value of nearly 1 and falls of rapidly after ∆R exceeds 3. This further helps to distinguish the signal from the background.

4.3 Alternate Signal Reconstruction

So far, only the Higgs signal reconstruction through the b and ¯b quark kinematics has been discussed. However, in this research it is noted the mass of a Higgs boson can also be reconstructed solely from the kinematics of the final-state electron and outgoing parton. In this section, a derivation of the corresponding formula is provided and shown to yield the correct result. Evidently, this method cannot be used in the case of a CC event because of the absence of an electron in the final-state. See also Sec. 3.2 for more on the measurements of DIS events.

An event reconstruction method from the outgoing electron and parton could offer an increase in the accuracy of DIS measurements. Moreover, it could provide a better means for the separation of signal from background by imposing extra conditions on the final-state kinematics. An analysis based on Higgs boson mass reconstruction only from

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b¯b pair detection comes with high systematic uncertainty caused by the final-state jets. An electron, however, can be measured with a significantly higher accuracy. The above mentioned reconstruction method utilizes this benefit in reconstructing the mass of the produced Higgs boson from the kinematics of the outgoing electron and quark.

4.3.1 Higgs Mass Reconstruction from Outgoing Electron and Parton The initial- and final-state kinematics of the electron and struck parton are described in the centre-of-mass frame, in order to impose energy and momentum conservation. Starting with the initial-state, the total energy and momentum are, respectively

E = Ee+ Eq, (27) p = 0 = pe+ pq. (28)

Whereas the total energy and momentum of the final-state can be written as E = E_e0 + E_q0 + EH, (29) p = 0 = p0e+ p

0

q+ pH. (30)

The final-state is a three-particle system, and therefore there is also a dependence on the polar angle φ. The masses of the electron and quark can be neglected compared to their total energies. Written in terms of the measurable energies and angles, the four-momenta of the outgoing electron and quark are

p0_e= (E_e0, p0_e) = (E_e0 , E_e0 sin θecos φe, Ee0 sin θesin φe, Ee0cos θe) , (31)

and

p0_q= E_q0 , p0_q = E_q0 , E_q0 sin θqcos φq, Eq0 sin θqsin φq, Eq0 cos θq . (32)

The invariant mass of the produced Higgs boson is dependent on the energy and momentum difference between the initial- and final-state of the electron-quark system. Combining (27) and (28) with (29) and (30) and using

m2_H = E_H2 − | p2_H|, (33)

the invariant mass of the produced Higgs boson becomes m2_H = Ee+ Eq− Ee0 − E

0 q

2

− |(pe0− pq0)|2. (34)

The expression for the Higgs boson mass in terms of the measurable quantities E_{e, q}0 , θe, q and φe, q can now be acquired by filling in (31) and (32), which gives

m2_H = Ee+ Eq− Ee0 − E 0 q

2

− E_e0sin θecos φe+ Eq0 sin θqcos φq

2 − E_e0 sin θesin φe+ Eq0sin θqsin φq

2

− E_e0 cos θe+ Eq0 cos θq

2

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This formula can be easily tested by implementing it into the Root analysis. To convert the system to its centre-of-mass frame, a suitable Lorentz-boost has to be ap-plied. Since the initial-state is known, the Lorentz-boost vector can be simply created by assessing the momentum asymmetry.

As expected, the result is a clean Breit-Wigner peak shaped plot, centered around 125 GeV. See Fig. 27.

Figure 27: Higgs mass reconstructed from the final-state electron and struck quark using (35)

4.3.2 Electron and Struck Parton Kinematics

In order to correctly apply this alternate reconstruction method, the two required particles have to be distinguished from the other final-state particles. The electron is generally measurable with high accuracy. Its kinematic distributions of transverse momentum and pseudorapidity are displayed in Fig. 28 and Fig. 29.

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Figure 28: Transverse momentum of outgoing electron

Figure 29: Pseudorapidity of outgoing electron

The transverse momentum distribution shows that, in signal events, the scattered electron obtains momenta extending to much higher values than in background events. This also shows in the pseudorapidity distribution, where it can be seen that the electron is usually deflected approximately perpendicular to the beam axis in signal events. In background events, on the other hand, the electron deviates significantly less from its original direction.

The outgoing parton may prove to be more difficult to distinguish from other quarks present in the final-state, because these all become jets with intrinsically high systematic uncertainty. This problem can possibly be avoided, however, because when dealing with a suspected signal event, the reconstruction method could calculate the Higgs mass by iteratively using all electron-jet-pairs and checking which pair has the best fit. Moreover, the b¯b pair jets can be B-tagged1. Thus, there will be a better discrimination between jets originating from the initial-state proton and jets that formed in the scattering event itself.

The outgoing parton distributions of transverse momentum and pseudorapidity are displayed in Fig. 30 and Fig. 31.

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Figure 30: Transverse momentum of struck quark

Figure 31: Pseudorapidity of struck quark

Similarly to the electron transverse momentum, the outgoing parton transverse mo-mentum extends to high values for signal events, whereas the background distribution starts to drop off from a low value. The pseudorapidity distribution shows a wider spread in background events, whereas signal events have a more distinct peak with η around -3.

4.4 CC Study from LHeC Study Group

A full simulation of Higgs production and H → b¯b decay was performed by the LHeC Study Group. However, only CC Higgs events were counted as signal and all NC events were rejected. The study was performed with a chain of MadGraph, Pythia and PGS. See 4.1.

The most recent results from the LHeC CC study can be found in [6], along with the used cuts. For a luminosity of 100 fb−1, the achieved signal to noise ratio, N_√signal

Nbkg

, is 27.5. A distribution of the invariant mass of the b¯b pair jets is displayed in Fig. 32.

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Figure 32: Dijet invariant mass distribution, adapted from [6]

5 Discussion

All in all, the wide scope of this thesis prevented any of its subjects to really be fully investigated. For example, the alternate reconstruction method using the outgoing elec-tron and parton, Sec. 4.3, could have been inquired more thoroughly. As mentioned previously, it has to be performed in the centre-of-mass frame. To acquire a legitimate centre-of-mass boost in a real collision is no simple task. It might be possible to assess the collision asymmetry by calculating x from the outgoing electron, which in turn provides the energy of the initial-state struck parton and allows the construction of a suitable boost vector. Or perhaps another, more accessable frame of reference would be suitable for a similar reconstruction method. To determine the feasibility of accomplishing such a reconstruction method in reality would require an in-depth investigation.

Also, to include the LHeC design, electron-proton scattering theory and Monte Carlo simulations into the research frequently required a switch of perspective, which did not benefit the overall efficiency.

Furthermore, I did not have any experience with the programming frameworks used for the simulations before starting this research. Learning how to use MadGraph, Root and writing analysis scripts in C++ took a lot of time. In the end, I even wrote a script for analysing the NC output from a full MadGraph-Pythia-PGS chain to obtain similar results as the LHeC Study Group did for CC events, see Sec. 4.4, but ended up not using it due to time restrictions. However, I did learn a lot from writing the parton level analysis that successfully generated my main results.

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6 Conclusion

This has been a preliminary study, inquiring the prospect of Higgs precision measure-ments at the LHeC. The LHeC would require a new linear accelerator and detector to be built and installed adjacent to the LHC ring. 60 GeV linac electrons would collide with 7 TeV LHC protons, expanding the DIS experimental range to a centre-of-mass energy of 1.3 TeV and an (x, Q2_{) coverage of x ranging from 10}−6 _{to nearly 1 and Q}2

extending to over 1 TeV2.

The production of a Higgs boson in DIS is a process of the weak interaction and can occur with the mediation of either a W± boson in a CC event or Z boson in a NC event. The corresponding cross sections are approximately 196 fb and 25 fb. The distinction of NC Higgs production and decay into a b¯b pair compared to background events was assessed at parton level. The difference in kinematics indicates promising separation between signal and background events. With the implementation of a novel reconstruction method, utilizing the final-state electron and outgoing parton, a signi-ficant increase in accuracy could be achieved. This would require further investigation into the construction of a compatible frame of reference, though.

Overall, the prospect of Higgs precision measurements at the LHeC is only a part of its entire physics programme. The decision of whether or not LHeC project is to be carried through is up to CERN’s management, funding opportunities and, above all, the physics community.

7 Acknowledgments

First and foremost, I would like to thank Stan Bentvelsen, my supervisor. Despite his frequent absence, he maintained a structured organization throughout the project, preventing me from immersing too deep in highly complex physics. Also, I appreciate the lively atmosphere created by my fellow students occupying the same office during the past three months, namely Bram, David, Jennifer, Martijn and Danne.

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A

Appendix: the Higgs Mechanism

The Higgs mechanism is an essential part of the Standard Model, accounting for the masses of its constituents - the fundamental particles. Specifically the W± and Z bo-sons, the carriers of the weak force, are observed to be very heavy but would be massless without the Higgs mechanism ‘giving’ them their mass. The working of this mechanism is very complicated and it is important to note that only a brief outline is given here. For further details, see for example ‘Modern Particle Physics’ by Mark Thomson [7].

W± and Z bosons have observed masses of approximately 80 GeV and 91 GeV re-spectively. A Higgsless standard model predicts these particles to be massless and thus is inconsistent with nature. The working of the Higgs mechanism can be understood by introducing a real scalar field, spanning all of space-time - the Higgs field. Massive particles acquire their mass by interacting with this field.

To keep it simple, we’ll start by assuming the Higgs field to be real. The idea of symmetry breaking, which will lead to mass terms associated with this scalar field, will be introduced around real scalar fields. Afterwards we will make the generalization to symmetry breaking in complex scalar fields. This is needed to create a continuous degeneracy in the potential minimum and eventually allows the mass terms that we are looking for. Let’s define a scalar field φ with a corresponding potential.

V (φ) = 1 2µ

2_φ2₊1

4λφ

4_, ₍₃₆₎

where µ and λ are free parameters. The minimum of this potential corresponds to the vacuum state of the field. In other words, the state in which the least amount of energy is contained within the field. For this value to be finite, λ has to be positive. If both λ and µ2 _{are positive, the vacuum state of the field is zero and occurs at φ = 0 (see}

Fig. 33, left). The parameter µ can be chosen freely however, and the possibility that µ2 < 0 also has to be taken into account. If this is the case, the potential V(φ) has two minima at φ = ±p−µ2_{/λ (see Fig. 33, right), and thus there is a degeneracy in}

the vacuum state. In this situation, choosing between the two vacuum states results in breaking the symmetry of the Lagrangian of the potential — a process known as spontaneous symmetry breaking.

Now to make the generalization to a complex Higgs field φ, consisting of a real and an imaginary part:

φ = √1

2(φ1+ iφ2), (37)

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Figure 33: left: Scalar field potential with both λ and µ2 positive. right: The same potential, except now with µ2 negative.

Figure 34: left: Complex scalar field potential with both λ and µ2 positive. right: “Mexican hat” potential, resulting from µ2 being negative.

V (φ) = 1 2µ 2_φ∗ φ + 1 4λ(φ ∗ φ)2. (38)

As can be seen quickly, the same choice of the parameters λ as positive and µ2 _{as either}

positive or negative gives a similar shape of the previous potential, but instead three-dimensional (see Fig. 34). With µ2 _{positive, there is a unique minimum when both φ}

1

and φ2 are zero. With µ2 negative, on the other hand, the potential has its minima on

a circle, given by:

φ2₁+ φ2₂ = −µ

2

λ = v

2_. ₍₃₉₎

Any point on this circle corresponds to the nonzero vacuum state of the field. The actual choice of this state is arbitrary though, and for convenience we can choose it to

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be along the real axis, at:

φ1 =

r −µ2

2λ = v, φ2 = 0 (40)

Clearly, after making this choice, the system is not rotationally invariant anymore, and thus its symmetry is spontaneously broken. Any correct field theory is assumed to be invariant under local gauge transformations of the form:

φ(r, t) → φ0(r, t) = exp − iqf (r, t)φ(r, t), (41) but the case where µ2 is negative, this is not the case. By using covariant derivatives in the Lagrangian, instead of normal derivatives, this problem can be avoided (see [7]). The covariant derivative comes with the introduction of a new field, which we shall call Aµ.

Resulting from interactions with the Higgs field with a nonzero vacuum state, the gauge boson associated with this new field Aµ becomes massive. Also, the new Lagrangian

corresponding to the field after transformation (41) now contains terms associated with interactions with other particles as well, along with a term accounting for excitations of the Higgs field associated with the Higgs boson itself. The strength of these interactions is given by coupling constants which are dependent of the mass of the particles of interest.

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B

Appendix: Populair wetenschappelijke

samenvatting

Higgs bij de LHeC

In de “Large Hadron Collider” (LHC) in Gen`eve worden protonen, een van de bouwstenen van de materie, tot enorme snelheden versneld om vervolgens op elkaar te botsen. Bij zulke botsingen komt ongelooflijk veel energie vrij en kan het voorkomen dat heel zware deeltjes gemaakt worden. Op deze manier is in 2012 het higgsboson ontdekt, wat door velen als de belangrijkste ontdekking in de deeltjesfysica wordt gezien. Het higgsbo-son was namelijk het enige missende deeltje uit het standaardmodel, de theorie die de krachten in de natuur beschrijft op het niveau van de elementaire deeltjes. De rol die het higgsboson hierin speelt is niet onbelangrijk: het is namelijk verantwoordelijk voor de massa van ieder deeltje. Echter, nog niet alles is bekend over dit zogenaamde “God-deeltje”. We weten zelfs niet eens of het deeltje dat twee jaar geleden werd ontdekt wel het higgsboson uit het standaardmodel is, en niet een of ander exotisch deeltje dat we nog niet begrijpen.

Onder andere om deze reden heeft het bestuur van CERN, het samenwerkingsverband achter dit grootschalige experiment, voorgesteld om ook elektronen te laten botsen op protonen, in plaats van alleen protonen op protonen zoals dat nu gaande is. Dit geeft de mogelijkheid om nog beter te kijken welke eigenschappen het higgsboson precies heeft. Dit project heeft de passende naam “Large Hadron Electron Collider” (LHeC) gekregen. In dit verslag wordt onderzocht hoe een higgsboson eigenlijk gemaakt wordt bij een elektron-proton botsing en welke natuurkunde hierbij komt kijken. Om precies te be-grijpen hoe higgsboson productie werkt, wordt eerst de natuurkunde omtrent elektron-proton botsingen besproken. Het blijkt dat er hierbij veel verschillende processen kunnen gebeuren. Door middel van simulaties met het programma MadGraph zijn elektron-proton botsingen gesimuleerd zoals ze bij de LHeC zouden plaatsvinden. Hiermee kun je kijken welke moeilijkheden je tegen kunt komen wanneer de LHeC eenmaal online zou gaan, en dit soort botsingen ge¨ınterpreteerd moeten worden. Bijvoorbeeld, hoe kun je een higgsboson het beste detecteren? En hoe onderscheid je een dergelijk signaal van andere signalen die erop lijken? Dit onderzoek richt zich specifiek op de meting van de massa van het higgsboson. Bovendien wordt een methode gegeven waarmee deze bepaald kan worden zonder direct te kijken naar de deeltjes waarin het higgsboson uiteenvalt.

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References

[1] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, et al. The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations. 2014.

[2] O Bruening and M Klein. The Large Hadron Electron Collider. May 2013. A brief review on the LHeC to appear in Modern Physics Letters A.

[3] S. Dittmaier et al. Handbook of LHC Higgs Cross Sections: 1. Inclusive Observables. 2011.

[4] J L Abelleira Fernandez et al (LHeC Study Group). A large hadron electron collider at cern report on the physics and design concepts for machine and detector. Journal of Physics G: Nuclear and Particle Physics, 39(7):075001, 2012.

[5] LHeC Study Group. A large hadron electron collider at cern - website. [6] B. Mellado. Status of higgs studies. Presentation, June 2014.

Higgs at the LHeC Electron-Proton Collisions at the Energy Frontier

Higgs at the LHeC

Electron-Proton Collisions at the Energy

Frontier

Contents

1

Introduction

2

The Large Hadron Electron Collider

2.1

LHeC Design

3

Theory

3.1

Electron-Proton Scattering

3.2

DIS

3.3

Higgs Production

3.4

Higgs Decay and Background

4

Simulations with MadGraph

4.1

About MadGraph

4.2

NC Parton Level Results

4.3

Alternate Signal Reconstruction

4.4

CC Study from LHeC Study Group

5

Discussion

6

Conclusion

7

Acknowledgments

A

Appendix: the Higgs Mechanism

B

Appendix: Populair wetenschappelijke

samenvatting

Higgs bij de LHeC

References