The Higgs boson Marina Cobal University of Udine
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The Higgs boson
Marina Cobal University of Udine
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. 1996 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. 1992 Cap.15
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, Tα (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 N2 -1 hermitian generators with hermitians generators (a traccia nulla).
Introduction
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 Tα 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:
Such a Lagrangian cannot contain mass terms in the gauge fileds, which would violate the gauge symmetry!
FµνFµν, includes cubic and quartic terms of gauge fields auto-interaction.
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;
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 µ2 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
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.
Substituting the expression of the Higgs field in V(φ) we get:
There are:
1) Higgs mass term: 2λv2
2) Auto-interaction terms (H3 e H4)
Terms for the Potential
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 GF constant one obtains the value of v:
Kinetic terms
The coupling between Higgs and the gauge bosons is:
And turns out to be proportional to the gauge bosons’ masses!
Kinetic terms
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
gf [ (ψLH) ψ R + h.c.]
One gets that the fermion masses are: mf= gf v/√2 and so even in this case the coupling is proportional to the fermion masses.
Mass terms for fermions
In the Standard Model all the Higgs couplings are fixed. The only free parameter is:
Feynmann rules
Solving the equation of the group renormalization for the coupling constant λ which appears in the Higgs Lagrangian,
one obtains
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.
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)
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).
Limits on the boson Higgs mass
Some electroweak observables are sensitive to parameters like mt and mH. High precision measureemnts of these observables, allow to gain
some information on mt and mH even without a direct measurement!!
For example, MW is dependent from and from mH through the existence of higher order diagrams.
Indirect measurements
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.
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.
Higgs Production at e+e- Colliders
Higgs Production at Hadron Colliders
gg Fusion
W/Z Fusion
H-radiation
10x >
> (~= at low mass)
Higgs Decay Modes H mass Dom. proc <130 GeV H->bb (ff) >130 GeV H->V*V #
Γ(H) ~10MeV
<GeV
#: V*V: one real and one virtual Vector Boson: W*W or Z*Z
>180 GeV H->VV tt-channel not very relevant
m(H)>500 -> Γ ~ m
~10MeV
H->γγ: rare decays, but clean signature
Higgs at LEP
LEP1 sensitive to:
e+e- -> (H0 -> bb) (Z0->νν)
e+e- -> (H0 -> bb) (Z0->l+l-) (l: e or µ)
LEP2 additionally involved decay modes e+e- -> (H0 -> bb) (Z0->qq)
e+e- -> (H0 -> ττ) (Z0->qq)
e+e- -> (H0 -> qq) (Z0->ττ)
ETmiss ! leptonic !
BR*
* @ m(H) = 115 GeV
17%
6%
60%
10% Background!
Higgs @ LEP -> 3 sigma @ 115 GeV!
Q = L(s+b)/L(b)
BG (simulated) S (simulated)
1-sigma (BG) 2-sigma (BG)
BUT:
Combined data only 1.7 sigma 95% CL of Higgs mass lower bound of 114 GeV
Higgs @ Tevatron
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
• Bunch-crossing frequency = 40 MHz • Interaction frequency~109 Hz @ L = 1034 cm-2 s-1
• Collected events≈100 Hz (Rejection factor: 107)
σtot = 80 mb
109 interactions/sec
Higgs 10-2 - 10-1 Hz
Top 10 Hz
W 2 Khz
Higgs @ LHC
Cross-sections 10 to 100 x larger at the LHC (depending on Higgs mass)
Gluon Fusion
Vector Boson Fusion
Higgs Strahlung
ttH
Higgs @ LHC M(H)<120 GeV Signal x-secion*BR
0.36pb BG x-secion (ttbb)
60pb
Dominant decay channel: H-bb (but only usable in
associated production mode) Good b-tagging needed t-tagging to reduce bck
M(H)<140 GeV
2 forward jets Higgs decay products in
central region
3.4 sigma @ 30 fb-1
-> ~ 40 fb-1 needed for discovery
Higgs @ 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
Higgs @ LHC: Discovery potential
Irreducible background coming from the processes: gg,qq →γγ.
The signal can be observed on the continuum background only if the experimental resolution
on Mγγ is very good (1%)
How can one claim a discovery ? Suppose a new narrow particle X → γγ is produced:
Signal significance :
B
S
NN =S NS= number of signal events
NB= number of background events in peak region
√NB ≡ 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
peak width due to detector resolution
mγγ
Two critical parameters to maximise S: • detector resolution:
if σm increases by e.g. two, then need to enlarge peak region by two.
→ NB increases by ~ 2 (assuming background flat)
NS unchanged
⇒ S =NS/√NB decreases by √2
⇒ S ≈1 /√σm detector with better resolution has larger probability to find
a signal
Note: only valid if ΓH << σm. If Higgs is broad detector resolution is not relevant.
ΓH ~ mH3 ΓH ~ MeV (~100 GeV) mH =100 (600) GeV
• integrated luminosity : NS ~ L NB ~ L ⇒ S ~ √L
Irreducible background coming from the process:
qq → ZZ*→4leptons Irreducible background:
tt →bbWW (semileptonic decay of the b)
Zbb
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