Beyond the Standard Model
So far, the Standard Model is our most successful theory of particle physics. Most of its predictions have been confirmed by many experiments, and almost all the observed phenomena in the particle colliders all over the world are well explained. However, the standard Model is not a perfect, or complete theory. The SM fails to provide dark matter candidates or convincing explanations to the dark energy, massive neutrino or matter-antimatter asymmetry in our universe. Physicists are also expecting mechanism that breaks the CP symmetry in the strong interaction, however such violation has not been observed experimentally yet. Moreover, some problems of the SM always exist within its mathematical framework, e.g. the SM is not compatible with the general relativity, and therefore cannot explain the forth fundamental interaction, gravity. The hierarchy problem is also unsolved: some quantum corrections (e.g. on the Higgs mass) are so much larger than the effective value itself, and the fine tuning on this seems unnatural. In addition, there are a few experimental results that deviate a lot from the SM expectation, such as the famous anomalous magnetic dipole moment of muon.
Answering to this kind of problem requires more precise and careful experiments as well.
Fortunately, solving the existing problems of the SM does not mean that we need to reject the whole theory. A lot of excellent ideas are raised by physicists known as “Beyond the Standard Model” (BSM), which are modifications of the SM in a subtle way so that the new models would still be consistent with the current data and observations. Two models are briefly introduced in this section as examples, and both of them predict new resonances in the diphoton final state. They can be seen as the physical motivation of the analysis part of this thesis.
The Two-Higgs-Doublet Models
As discussed in Sec. 1.1.3, the Standard Model assumes a simple scalar structure with only one SU(2) doublet, while experimentally the existence of extended scalar sectors is still allowed. The Two-Higgs-Doublet Models (2HDM) are some of the simplest extension of the SM, which extend the SM Higgs sector into two scalar doublets. An additional Higgs doublet might be an elegant solution to many problems.
For example, the 2HDMs are able to generate baryon asymmetry of the universe while the SM cannot; an additional Higgs doublet is needed for cancellation of anomalies in supersymmetry; with two Higgs doublets, it is also possible to imposing a global U(1) symmetry, which is needed to deal with a CP-violating term in the QCD Lagrangian in the Peccei-Quinn model.
The 2HDMs are categorized according to the way the Higgs doublets couple to the quarks and leptons. There are four types of 2HDMs: Type-I, Type-II, lepton−specific and flipped models. A serious potential problem of general 2HDMs is the existence of tree-level flavour-changing neutral currents (FCNC), which are excluded by the data. A solution to circumvent this problem is to impose discrete symmetries. The four types of 2HDMs mentioned above are all free from the flavour-changing neutral current, although models with tree-level FCNCs also exist, such as the Type-III model listed in Tab. 1.2, together with the coupling of the two doublets 1 and 2 with the fermions.397 One can rewrite the Higgs potential in Eq. 1.42 for two complex scalar doublets under some necessary assumptions (e.g. CP conservation in the Higgs sector). After symmetry breaking, minimization of this potential ends up in eight fields, among which three are used to generate mass for the W± and Z bosons; the five remaining fields are physical states. There is one neutral CP-odd pseudoscalar A, two charged Higgs H±, and two neutral CP-even Higgs H and h with different masses. The free parameters of 2HDM are: the four Higgs masses mh, mH, mA and mH±403 , the ratio between the two vacuum expectation values (tan = v2 v1 404 ), and the mixing angle of the neutral CP-even 2HDM Higgs bosons. With these parameters, we can express the 2HDM couplings in terms of the SM couplings.For example, the light CP-even Higgs
Beyond the Standard Model
boson h coupling to WW407 or ZZ is given by the SM coupling multiplied by a factor of sin( −), and the coupling of the heavier Higgs H is given by the SM coupling multiplied by cos(−).5 Assuming the SM Higgs discovered in 2012 with mass of 125 GeV being the neutral Higgs boson H or h, we might be able to discover the other one as well in the lower- or higher-mass region.
The Randall-Sundrum model
A new spin-2 resonance is predicted by the Randall-Sundrum (RS) model. This mechanism was proposed for solving the hierarchy problem, where the electroweak scale (MEW 1010 GeV) is much lower than the Planck mass scale (Mpl 1019 GeV). To illustrate the RS model, we need to start with the central idea of the brane cosmology, brane and bulk. Our visible, three-dimensional universe is restricted to a “brane” inside a higher-dimensional space, called the “bulk” (or “hyperspace”). At least some of the extra dimensions of the bulk are extensive, so that other branes may be moving through this bulk. Assuming the simplest case: the higher dimensional
spacetime is approximately a product of a 4-dimensional spacetime with a n-dimensional compact space. Then, the effective four-dimensional (reduced) Planck scale M¯pl (M¯pl =Mpl/ p423 8) can be determined by the fundamental (4+n)-dimensional Planck scale M424 , and the geometry of the extra dimensions: ¯ M2 pl =Mn+2 Vn (1.50) where Vn 425 is the n-dimensional volume of the compact space. By taking the compact space to be very large, the hierarchy between the weak scale and Planck scale may be eliminated. Particularly, the RS models describe our universe as a 5-dimensional warped429 geometry universe. There were two models with one extra dimension proposed in 1999 by Lisa Randall and Raman Sundrum: one is called RS1 model, which has a finite size of extra dimensions with two branes, one as each end; the other is called RS2 model, which has only one brane left since the other brane is placed infinitely far away. The following discussion is based on RS1 model. As illustrated in Fig. 1.10, it involves a finite 5-dimensional bulk that is extremely warped and contains two branes: the Planck brane (also called « gravity brane » where gravity is a relatively strong) and the TeV brane (also called « weak brane »). The trick is that all the SM particles and forces are confined to a 4-dimensional subspace (TeV brane), while gravity is free to propagate in the full spacetime (bulk). The exponential drop of the probability439 that the gravity would be much weaker on the TeV brane than on the Planck brane.
The resulting 5-dimensional metric is non-factorizable, given by: ds2 = e−2krcμdxμdx +r2 cd2 (1.51) where k and rc 442 are the curvature and compactification radius of the extra dimension; is the Minkowski metric; xμ are the traditional coordinates for the four dimensions; is the coordinate for the extra dimension, in the range 0 < < . With reasonable krc 445 (e.g. krc 12), the hierarchy problem can be eliminated.
With the spacetime configured above, the TeV scale is related to the Planck scale, given by: =M¯pl exp(−krc) (1.52) When the graviton travels freely in the bulk, a series of massive graviton excitations come out as a consequence. This set of possible graviton mass values are called a Kaluza-Klein (KK) tower. They are visible on the TeV brane, meaning that we could observe the KK gravitons just like other SM particles. The KK gravitons have spin 2, and a universal dimensionless coupling to the SM fields of k/M¯pl452 . Its mass mG is splitted between the different KK levels on the TeV scale.The Large Hadron Collider and the ATLAS detector
The world’s largest and most powerful particle accelerator, the Large Hadron Collider (LHC) , is located beneath the France-Switzerland border near Geneva.
It lies in the former Large Electron-Positron collider (LEP)  tunnel, which is 27 km in circumference, around 100 m underground.
The LHC is a two-ring-superconducting-hadron accelerator, designed to collide proton and heavy ion beams with a centre-of-mass energy up to 14 TeV. In December 1994, the approval of the LHC project was given by the European Organization for Nuclear Research (CERN). The construction of the LHC started in 1998. After the LEP was closed to liberate its tunnel in 2000, the LHC was finished in 2008 under the cooperation of many scientists, universities and laboratories across the world. Seven detectors, each designed for different purposes, are positioned at the four crossing points of the collider. There are four main experiments: ATLAS, CMS, LHCb and ALICE. The two high luminosity experiments, ATLAS (A Toroidal LHC ApparatuS) and CMS (Compact Muon Solenoid) are general-purpose detectors, both designed to operate at a peak luminosity of L = 1034 cm−2s−1 for proton operation.
The low luminosity experiment LHCb (Large Hadron Collider beauty) is designed for B-physics, capable of data-taking at a peak luminosity of L = 1032 cm−2s−1. The dedicated heavy ion experiment ALICE (A Large Ion Collider Experiment) is designed to study of the physics of strongly interacting matter at extreme energy densities, aiming at a peak luminosity of L = 1027 cm−2s−1 for nominal lead-lead ion operation.
On 10 September 2008, the first beam was circulated through the LHC. Nine days later, however, a quench occurred and the collider had to be stopped. After one year of repairs and reviews from the consequential damages, the first operation run (Run 1) started on 20 November 2009. The proton beam energy was 3.5 TeV (corresponding to centre-of-mass energy of 7 TeV) in 2010, and increased to 4 TeV s = 8 TeV) in 483 2012. On 13 February 2013, the LHC was shut down for a two-year upgrade, enabling collisions at its designed energy and enhancing the detectors and pre-accelerators. After the Long Shutdown 1 (LS1), the second operation run (Run 2) started on 5 April 2015 with collision energy of 13 TeV. On 10 December 2018, the Long Shutdown 2 (LS2) started for the purposes of maintaining and upgrading of the LHC and ATLAS complex. After which, Run 3 is planned to start in February 2022. The implementation of the High Luminosity Large Hadron Collider (HL-LHC) project has been preparing since LS2, aiming to be used in Run 4 in the future. The beam parameters and hardware configuration are designed for the HL-LHC to reach a peak luminosity of 5×1034 cm−2s−1, allowing an integrated luminosity of 250 fb−1 492 494 2.1.1 The LHC injection chain
In order to accelerate protons and heavy ions to the required energy, a chain of accelerators is used as shown in Fig. 2.1. The LHC injection chain for protons is Linac 2 — Proton Synchrotron Booster (PSB) — Proton Synchrotron (PS) — Super Protos Synchrotron (SPS). The protons are first stripped of the hydrogen gas by an electric field.
Table of contents :
1.1 The Standard Model of particle physics
1.1.1 The gauge theory
1.1.2 The Standard Model Lagrangian
1.1.3 Spontaneous symmetry breaking and the Higgs mechanism
1.1.4 The production and decay of Higgs boson
1.1.5 Non-resonant diphoton production
1.2 Beyond the Standard Model
1.2.1 The Two-Higgs-Doublet Models
1.2.2 The Randall-Sundrum model
2 The Large Hadron Collider and the ATLAS detector
2.1 The Large Hadron Collider
2.1.1 The LHC injection chain
2.1.2 Luminosity and performance
2.2 The ATLAS detector
2.2.1 Inner detector
2.2.3 Muon spectrometer
2.2.4 Magnet system
2.2.5 Forward detectors
2.2.6 Trigger system
3 Photon reconstruction and performance
3.1 Photon reconstruction
3.1.1 Energy reconstruction
3.1.2 Track matching
3.2 Energy calibration
3.3 Photon identification
3.4 Photon isolation
4 Photon energy calibration uncertainties from shower leakage mismodeling
4.1.1 Definition of leakage variables
4.1.2 Data and simulated samples
4.1.3 Background subtraction in the diphoton sample
4.2 Measurement of the lateral leakage and double difference
4.2.1 Measurement of the lateral leakage
4.2.2 Measurement of the double difference
4.3 Studies on the double difference
4.3.1 pT and dependence
4.3.2 Leakage along and directions
4.3.3 Pile-up dependence
4.3.4 Impact of additional material
4.3.5 Other effects
4.4 Refined double difference measurement and final results
4.4.1 Corrections on the double difference
4.4.2 Systematic uncertainty of background subtraction method for diphoton sample
4.4.3 Final results
5 Search for diphoton resonances
5.1 Data and Monte Carlo samples
5.1.1 Low-mass samples
5.1.2 High-mass samples
5.2 Event selection
5.3 Signal modeling
5.3.1 Narrow-width signal modeling
5.3.2 Large-width signal modeling
5.4 Background modeling
5.4.1 Non-resonant background
5.4.2 Resonant background
5.4.3 Background modeling results
5.5 Fiducial and total acceptance corrections
5.5.1 Fiducial volume and correction factor
5.5.2 Acceptance factor
5.6 Systematic uncertainties
5.6.1 Signal modeling uncertainties
5.6.2 Signal yield uncertainties
5.6.3 Background modeling
5.6.4 Migration between categories
5.6.5 Systematics uncertainties summary
5.7 Statistical method
5.7.1 Profile log-likelihood ratio method
5.7.2 Discovery p-value
5.7.3 Look-elsewhere effect
5.7.4 Upper limits
5.7.5 Statistical models
5.8.1 Low-mass search results
5.8.2 High-mass search results
5.9.1 Low-mass analysis
5.9.2 High-mass analysis
A Stitching of the sliced MC background samples
B Functional Decomposition smoothing