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Term Paper for the course 8.811: Particle Physics II Fall 2004 Standard Model Higgs Detection and Measurements at a Linear Collider Georgios Choudalakis ABSTRACT The Higgs boson is believed to be one of the fundamental constituents of the Standard Model. In this work I am firstly presenting the significance of the Higgs Mechanism and what has been achieved so far in the experimental search for it. Then, I will investigate how a Linear Collider should be to designed to improve our knowledge of its properties. Finally I am going to present an outline of the method that would allow a compelling measurement of its properties, focussing on the measurement of its mass
Contents 1 Motivation 1.1 Outline of the Higgs Mechanism 2 History 2.1H 2.1.2 Knowledige of the sin 7789 ints on mg······ 32t 3.3 The det 33.1 The magnet.................... 33 The Flect 3.34 3.3.5 The Muons System .. 29 4 42 Mass analysis ee+Xor4+X,,., 5 Conclusions 44 6Appendices tum 49
Contents 1 Motivation 3 1.1 Outline of the Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 History 7 2.1 How the SM gives hints about mH . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Knowledge of the mW . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 Knowledge of the sin χeff W . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 The parameters �υ5 and υs(MZ) had, mt . . . . . . . . . . . . . . . . . 9 2.2 Constraints on mH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 The Linear Collider 13 3.1 The e+e− source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 The accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.1 Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 The detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3.1 The magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3.2 The tracker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3.3 The Electromagnetic Calorimeter . . . . . . . . . . . . . . . . . . . . 26 3.3.4 The Hadronic Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.5 The Muons System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4 Analysis plan 31 4.1 Higgs production and decays . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Mass analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 The channel ZH ≈ e+e− + X or µ+µ− + X . . . . . . . . . . . . . . . . . . 33 4.3.1 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3.2 Higgs Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.4 Optimization and statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.5 LC vs LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5 Conclusions 44 6 Appendices 49 A Magnetic field, tracker radius, tracker spacial resolution and transverse momentum resolution 49 1
B Estimation of thero of momentum using the ideal tracker 50 CAngular separation of the decay products of a particle in motion 53 D Calculation of the invariant mass of Z and its error 的 D.4 Combining the Tracker and the ECal 65 2
B Estimation of the error of momentum using the ideal tracker 50 C Angular separation of the decay products of a particle in motion 53 D Calculation of the invariant mass of Z and its error 55 D.1 1st approach: Using the ECal . . . . . . . . . . . . . . . . . . . . . . . . . . 58 D.2 2nd approach: Using the Tracker . . . . . . . . . . . . . . . . . . . . . . . . . 59 D.3 A comment about multiple scattering and β�χ . . . . . . . . . . . . . . . . . 60 D.4 Combining the Tracker and the ECal . . . . . . . . . . . . . . . . . . . . . . 65 2
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1 Motivation In accelerator experiments physicists probe the way elementary particles interact. Excluding experiments with cosmic radiation (CR), we have investigated how nature works at collisions of energy up to 1.96 TeV. This is the center-of-momentum energy at Tevatron, currently the most powerful accelerator. LHC is expected to operate by 2008, stretching this frontier to 14 TeV. The Standard Model (SM) has been extremely successful at describing almost all phenomena we have observed so far in this energy range [1, 2, 3]. However, we still ignore the mechanism that causes some of the phenomena we see, first being the fact that W ±, Z, quarks and leptons (including neutrinos [4]) are massive particles. For the particles to be given masses, it is necessary to break in some way the symmetry of the electroweak interaction. If the SM were just a mathematical conception instead of a model intending to describe the real world, the demand for local gauge invariance of electroweak interaction would be satisfied even without any mass terms in the Lagrangian [5]. But masses must somehow be introduced in the Model. One may think that it would be a good idea to manually plug mass terms in the Lagrangian of the SM, but it can be proved that this naive method would make the theory unrenormalizable. The Higgs scalar field is only one mechanism of electroweak symmetry breaking; other approaches, based on very different dynamics, also exist. For example, one can introduce new fermions and new dynamics (i.e., new forces[8, 9]), or implement Higgs in the context of Supersymmetry (SUSY)[10]. Though there is no direct observation of the Higgs Boson, there are indications that can be interpreted as favoring its existence, as we will see in the next section. The significance of the question of the existence of Higgs is obvious. If it exists, as the majority of physicists expects today, then its discovery would advance our confidence and trust in the SM to an even higher level. Then, the SM would become a renormalizable theory which succeeds to explain the fact that particles have mass. The involvement of Higgs in important phenomena, such as CP violation and renormalizability, makes the determination of its properties a requirement to understand better the behaviour of quarks and leptons. Its observation, and the check whether it is a SM Higgs or one of the particles that SUSY predicts for example is important because if SUSY is a correct theory then Particle Physics goals will turn to the direction of determining the parameters of SUSY and checking if this SM extension can answer issues that are not yet understood, such as the apparent plentiness of dark matter, the hierarchy puzzle, the nature of gravity, Grand Unification etc. Also, the mass of the Higgs is correlated with the scale of energy � at which the SM is expected to fail describing things right. We know that at some energy lower than the Planck energy MP L ≡ 1019GeV , this failure has to happen, since gravity is not included in the SM (fig. 1). Even more funtamentally and within the energies we have already studied, the SM needs the Higgs boson to exist so as the cross-sections of several SM processes (such as uu¯ ≈ W+W−) not to diverge. 3
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� � Figure 1: The dependence of the Higgs mass MH and the energy scale � at which the SM is expected to break down. If MH � 120GeV , then the SM appears to extend up to the Planck scale. Actually, Supersymmetry is expected to take over much before MP L, even if MH is in that small window. On the other hand, if Higgs is not observed, then alternative theories will need to be investigated more carefully, possibly giving raise to new experiments. New theories might be motivated, or theories like Strings Theory may gain ground, proposing alternative approaches to the puzzles of particles. Before even being observed, Higgs has inspired hundreds of thousands of publications, involving almost all areas of Particle Physics, from renormalizability [12] to extra-dimensions theories and Superstrings[13]. This is a characteristic indication of its great importance which justifies its popularity. 1.1 Outline of the Higgs Mechanism The Higgs Mechanism is one possible answer to the puzzle of mass acquisition. The rigorous description of the Mechanism has been done by several authors ([7, 5, 6]), so here we are going to describe only its outline with two examples. In essence, the assumption is made that the universe is filled with a spin-zero field, called a Higgs field, that is a doublet in the SU(2) space and carries non-zero U(1) hypercharge and also is a color singlet. In the presence of this field gauge bosons and fermions appear to have non-zero mass. The simplest example of spontaneous symmetry breaking is to consider a Lagrangian 1 1 1 L = T − V = τµδτµδ − µ2 δ2 + ηδ4 . (1) 2 2 4 This potential V is symmetric under δ ≈ −δ. For η > 0, if µ2 = 0 then the vacuum, which 4
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� � � is the δ that minimizes V , corresponds to δ = 0. But there is no physical reason for µ2 to be positive. If µ2 < 0 then the potential V has two minima, corresponding to δ = −µ2 ± ∝ γ. (2) η So, the (Higgs) field has a non-zero value in the vacuum. To determine the particle spectrum, we must study the theory in the region of the minimum, so we put δ(x) = γ + σ(x) (3) so that we are expanding around σ = 0. We could have chosen to expand around the negative minimum (−γ), which would give the same physics. This arbitrarity is the reason we talk about spontaneous symmetry breaking. After substitution of (3) in the L, we get an expression for the Lagrangian which is 1 1 L = (τµστµσ) − ηγ2 σ2 + ηγσ3 + ησ4 + constant. (4) 2 4 Now, instead of a L of the field δ we have a L of the perturbation σ. This Lagrangian represents the description of a (Higgs) particle with mass m2 2 = 2ηγ2 = −2µ , (5) θ deriving from its self-interaction. Similarly we need to work on a U(1) or SU(2) symmetric Lagrangian to attribute mass to the corresponding vector bosons, Z0 and W±. For the U(1) case, we can start from a complex scalar field δ = (δ1 + iδ2)/ �2, and L = (τµδ) � (τµδ) − µ2 δ� δ − η(δ� δ) 2 . (6) As before, µ2 < 0. To make L invariant under local U(1) gauge transformations δ(x) ≈ δ√ (x) = ei�(x) δ(x), we must rewrite it in terms of the covariant derivative Dµ = τµ − igAµ, simultaneously introducing the gauge field, transforming as Aµ ≈ Aµ √ = A 1 µ − τµ�(x). The g Lagrangian is then 1 L = (Dµδ) � (Dµδ) − µ2 δ� δ − η(δ� δ) 2 − 4 FµυFµυ. (7) The potential energy has a minimum along circle of radius δ2 1 + δ2 = −µ τ 2 = γ2. The typical 2 perturbation method is now to expand δ around a point along this circle: (γ + σ(x) + iπ(x)) δ = � (8) 2 5
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δ(x) = Because of the local gauge invariance, we have the freedom to rewrite the field in the form (γ+h(x)) � . Then, 2 1 1 (τµh)(τµh) + g2 γ2 AµAµ − ηγ2 h2 − ηγh3 L = 2 2 η 1 1 −4 h4 + g2 γhAµAµ + 2 g2 h2 AµAµ − 4 FµυFµυ. (9) A Remarkably, we see that the gauge boson A has mass MA = gγ. If we had substituted δ as it is parametrized in eq. (8), we would have ended up with one massless boson term in the Lagrangian. This is the renouned Goldstone boson. Using this alternative parametrization, this unnatural boson has given its place to a third (longitudinal) polarization that the massive µ can have. The process to apply the Higgs Mechanism for an SU(2) local gauge invariant field is analogous, just more mathematically tedious. 6
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2 History As a result of the very high interest in the Higgs boson, a lot of intense research has already been conducted in the direction of • directly detecting the Higgs, and • constraining its mass from electroweak precision measurements The most important experiments that have contributed to this research are those that ran at LEP (Large Electron Positron collider), SLC (Stanford Linear Collider) and Tevatron, located at CERN, SLAC and Fermilab respectively. Efforts for direct observation of the Higgs at those experiments has been fruitless [11, 50]. What has been achieved however is the combination of several other SM measurements, which are sensitive to the mass of the Higgs, in order to indirectly set constrains for its mass. 2.1 How the SM gives hints about mH Physicists have been trying to put constrains to mH by measuring very precisely other parameters of the SM which are related to mH. The first thing that should be pointed out is that whenever such an attempt is made, one has to assume the correctness of some model, and based on that assumption he can proceed. As mentioned, the MSM (Minimal Standard Model) can not be a complete theory, since it is expected to break down at some energy scale �. But the mass of the Higgs is expected to be lower than 1 TeV, so probably the assumption of a MSM is numerically accurate enough. A SM Higgs would affect all currently measured electroweak observables, primarily through correction in the polarization of vacuum to W and Z bosons, through diagrams like those in fig. 2. Figure 2: Vacuum polarization to W and Z vector bosons, denoted by V . More specifically, the parameters that are mostly used to constrain the MSM Higgs mass are mW and sin2χw. Those parameters are related to the mH by the following two expressions [15]: 7
mw=80.3505 .007 -0.51802-1+0537(tr）°- -0.085 -1 in28形= 0.231540+5.2310-41mm +4410-[-1] 1) omWs10 o tand血，e的 agm 2.1.1 Knowledge of themw of the for the fig. 8
� � � � Figure 3: Invariant mass distribution of two different final states of W decays at ALEPH. m mH mH W = 80.3805 − 0.0581 ln 100 GeV − 0.0078 ln2 100 GeV �η5 ⎦ �2 − mt 0.518 had + 0.537 0.0280 − 1 175 GeV − 1 ⎤ ⎞ −0.085 ηs(MZ ) (10) 0.118 − 1 sin2 χeff W = 0.231540 + 5.23 · 10−4 ln mH 100 GeV � � � � +0.00986 �η5 had 0.0280 − 1 − 0.00268 ⎦ mt 175 GeV�2 − 1 ⎤ ⎞ +4.4 · 10−4 ηs(MZ ) 0.118 − 1 (11) From equations 10 and 11 it is obvious that, except for mW and sin χeff W , we also need to know the parameters �υ5 had, mt and υs(MZ). Another remark is that mH is met as an argument of the log function, which means that its errors are multiplicative, instead of additive. 2.1.1 Knowledge of the mW The mass of the W boson has been measured at experiments at Tevatron (CDF, D0) [16] and at CERN (LEP II) [17]. Precise determination has been achieved using W ≈ qqqq ¯ ¯ and qqλ¯ ∂¯λ processes (fig. 3). The world average of the mW presently is 80.426 ± 0.034 [18], but there are several different assumptions which can be made, resulting into slightly different estimations. An overview of the experimental results for the W mass is in fig. 4. 8
W-Boson Mass IGeV] 十80,454±0.050 Averag 80420±0.034 +00a LEP1/SLDim, 80.380±0.02 80 02 ,[GeV] 下igure4:Different mea 2.1.2 Knowledge of the si ke领e名-nd sin=(1-va/er)/4. andsorelatedtothelktf-igtcouplingasynmmetryglg (13) 公w 2.1.3 The parameters Aom and (M) The strong couping n)is currently measured to be87() The ()defined as 2=此的- 14) 9
Figure 4: Different measurements of the W mass. 2.1.2 Knowledge of the sin χeff W The effective weak mixing angle for the Z − λ+λ− coupling is defined in terms of the vector and axial vector effective couplings, sin2 χeff W = (1 − γλ/υλ)/4, (12) and is also related to the left-right coupling asymmetry [19]: λ λ (gL)2 − (gR)2 2γλυλ 2(1 − 4 sin2 χeff ) Aλ = λ = = W λ (gL)2 + (gR)2 γλ 2 + υλ 2 1 + (1 − 4 sin2 χeff )2 (13) W The measurement of sin2 χeff W has mostly been done at SLD and LEP, with analyses which use severel asymmetries where the value of sin2 χeff W is involved. The world average today is 0.23149(15) [20] (fig. 5). 2.1.3 The parameters �υ5 and υs(MZ) had, mt The top quark mass measurement is still ongoing at experiments mostly at Tevatron [21] where t was first observed. The current value for mt is 174.3±5.1 GeV from direct observation and 178.1+10.4 − GeV from SM electroweak fit [22]. 8.3 The strong coupling constant υ(MZ) is currently measured to be 0.1187(20) [20]. The �υhad(M2 Z), defined as υ(q2) − υ0 χχ (q2 ) − �√ �υhad = = �√ χχ (0) (14) υ(q2) 9