The Higgs boson. Marina Cobal University of Udine
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1 The Higgs boson Marina Cobal University of Udine
2 Suggested books F.Halzen, A.D.Martin, Quarks & Leptons: An Introductory Course in Modern Particle Physics, Wiley 1984 Cap.14,15 W.E.Burcham,M.Jobes, Nuclear and Particle Physics, Longman 1995 Cap.13 R.K.Ellis, W.J.Stirling, B.R.Webber QCD and Collider Physics, Cambridge U.P Cap. 8, 10, 11 Other useful text (more advanced level) L.B.Okun, Leptoni e Quarks, Ed. Riuniti 1986 Cap.19,20 F.Mandl, G.Shaw, Quantum Field Theory, Wiley 1984 Cap. 11,12,13 J.F.Donoghue,E.Golowich,B.R.Holstein Dynamics of the Standard Model, Cambridge U.P Cap.15
3 Introduction The starting point for the construction of the Standard Model is a Lagrangian of free or auto-interacting fields, which is invariant under a certain group of global symmetries. The Lagrangian is invariant for transformations like: U is a symmetric matrix, T α (hermitians) are the generators of the group G of global simmetry. If G is a group of the SU(N) type, then we will have N 2-1 hermitian generators with hermitians generators (a traccia nulla).
4 The interaction terms and the fields of the gauge bosons are introduced by making the G group symmetry local: U(θ) U( θ(x) ) = exp( ig T α θ(x) ) The symmetry of the Lagrangian is saved by introducing the covariant derivative, which means applying the substitution: It is possible to introduce a kinetic term for the gauge fields which turns out to be of the type: The expression of the Lagrangian which contains the matter and the gauge boson fields will be therefore:
5 Such a Lagrangian cannot contain mass terms in the gauge fileds, which would violate the gauge symmetry! F µν F µν, includes cubic and quartic terms of gauge fields autointeraction. In particular, in the SM construction we can consider the following symmetry groups: U(1) 1 generator (QED, γ) SU(2) 3 generators (electroweak sector, W,Z) SU(3) 8 generators (QCD, gluons) If we consider the electroweak sector of the Standard Model, the symmetry group G is given by;
6 It is possible to give mass to the gauge boson through the Higgs mechanism. Lets consider the field doublet: With the Lagrangian given by: The V potential is; The µ 2 value is important, as can be seen from the shape of the V potential. All the states which give Are a minimum for the V potential
7 A phase transformation( U(1) ) connects all the status of mimimum for the V potential. If we choose the status of minimum, then such a symmetry is broken The global lagrangian saves its gauge symmetry! In particular, it is possible to choose a gauge, in which the vacuum status is: : The Higgs fied can be written as: H(x) is a real field The second degree of freedom has been absorbed in the choice of the gauge. It shows up again in the transformations of the gauge fields.
8 Terms for the Potential Substituting the expression of the Higgs field in V(φ) we get: There are: 1) Higgs mass term: 2λv 2 2) Auto-interaction terms (H 3 e H 4 )
9 Kinetic terms Starting from this expression: Doing the calculations, one arrives at the conclusions The W and Z get a mass equal to From the measurement of the G F constant one obtains the value of v:
10 Kinetic terms The coupling between Higgs and the gauge bosons is: And turns out to be proportional to the gauge bosons masses!
11 Mass terms for fermions In the Standard Model Lagrangian, the mass terms for fermions would violate the gauge symmetry. Also in this case the Higgs mechanism is used by introducing a Yukawa coupling of the type g f [ (ψ L H) ψ R + h.c.] One gets that the fermion masses are: m f = g f v/ 2 and so even in this case the coupling is proportional to the fermion masses.
12 Feynmann rules In the Standard Model all the Higgs couplings are fixed. The only free parameter is:
13 Limits on the boson Higgs mass Solving the equation of the group renormalization for the coupling constant λ which appears in the Higgs Lagrangian, one obtains
14 Limits on the boson Higgs mass A lower limit can be obtained if λ(µ) 0 in the energy range where we think the theory is reliable.
15 Limits on the boson Higgs mass An upper limit is obtained by requiring that the theory is a perturbative one and therefore λ(µ) 1 for all the values of µ<λ (Λ theory scale)
16 Limits on the boson Higgs mass Let s consider the scattering of longitudinally polarized Z bosons: Requiring that the unitarity limit of the perturbative development is valid, it turns out that (amplitude in S-wave): This limit becomes stronger if we conider also the scattering of other bosons (800 GeV).
17 Indirect measurements Some electroweak observables are sensitive to parameters like m t and m H. High precision measureemnts of these observables, allow to gain some information on m t and m H even without a direct measurement!! For example, M W is dependent from and from m H through the existence of higher order diagrams.
18 Using the precision measurement (LEP & Tevatron) is interesting to consider the following plot: These results seem to indicate the presence of a light Higgs boson, compatible with what is predicted by the Standard Model.
19 The indication of a light Higgs becomes even stronger if we consider the results of a fit on all the observable parameters of the Standard Model in the electroweak sector.
20 Higgs Production at e+e- Colliders
21 Higgs Production at Hadron Colliders gg Fusion 10x > W/Z Fusion > (~= at low mass) H-radiation
22 Higgs Decay Modes H mass Γ(H) Dom. proc <130 GeV ~10MeV H->bb (ff) >130 GeV ~10MeV H->V*V # >180 GeV <GeV H->VV tt-channel not very relevant m(h)>500 -> Γ ~ m H->γγ: rare decays, but clean signature #: V*V: one real and one virtual Vector Boson: W*W or Z*Z
23 Higgs at LEP BR* LEP1 sensitive to: 17% 6% e + e - -> (H 0 -> bb) (Z 0 ->νν) e + e - -> (H 0 -> bb) (Z 0 ->l + l - ) (l: e or µ) ET miss! leptonic! 60% 10% LEP2 additionally involved decay modes e + e - -> (H 0 -> bb) (Z 0 ->qq) e + e - -> (H 0 -> ττ) (Z 0 ->qq) Background! e + e - -> (H 0 -> qq) (Z 0 ->ττ) m(h) = 115 GeV
24 LEP -> GeV! 1-sigma (BG) 2-sigma (BG) BG (simulated) S (simulated) Q = L (s+b) /L (b) BUT: Combined data only 1.7 sigma 95% CL of Higgs mass lower bound of 114 GeV
25 Tevatron
26 Constrains on Higgs mass m(h)>114 GeV (LEP II) m(h)<166 GeV (LEP II) Results from precision electroweak measurements: M(H) = 85 (+39) (-28) GeV
27 Bunch-crossing frequency = 40 MHz Interaction frequency~10 9 L = cm -2 s -1 Collected events 100 Hz (Rejection factor: 10 7 ) Higgs Hz σ tot = 80 mb 10 9 interactions/ sec Top W 10 Hz 2 Khz
28 LHC Cross-sections 10 to 100 x larger at the LHC (depending on Higgs mass)
29 Gluon Fusion Vector Boson Fusion Higgs Strahlung tth
30 LHC M(H)<120 GeV Dominant decay channel: H-bb (but only usable in associated production mode) Good b-tagging needed t-tagging to reduce bck Signal x-secion*br 0.36pb BG x-secion (ttbb) 60pb fb -1 M(H)<140 GeV 2 forward jets Higgs decay products in central region -> ~ 40 fb -1 needed for discovery
31 LHC M(H)<150 GeV X-section * BR 50 fb, But very clean signature EM Calorimetry efficiency Crucial (ATLAS vs CMS performance) M(H)>130 GeV X-section * BR 5.7 fb, very clean signature
32 LHC: Discovery potential
33 Irreducible background coming from the processes: gg,qq γγ. The signal can be observed on the continuum background only if the experimental resolution on M γγ is very good (1%)
34 How can one claim a discovery? Suppose a new narrow particle X γγ is produced: peak width due to detector resolution S Signal significance : = N S N N B B m γγ N S = number of signal events N B = number of background events error on number of background events S > 5 : signal is larger than 5 times error on background. Probability that background fluctuates up by more than 5σ : 10-7 discovery in peak region
35 Two critical parameters to maximise S: detector resolution: if σ m increases by e.g. two, then need to enlarge peak region by two. N B increases by ~ 2 (assuming background flat) N S unchanged S 1 / σ m S =N S / N B decreases by 2 detector with better resolution has larger probability to find a signal Note: only valid if Γ H << σ m. If Higgs is broad detector resolution is not relevant. Γ H ~ m 3 H Γ H ~ MeV (~100 GeV) m H =100 (600) GeV integrated luminosity : N S ~ L N B ~ L S ~ L
36 Irreducible background coming from the process: qq ZZ* 4leptons Irreducible background: tt bbww (semileptonic decay of the b) Zbb
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