1 Why Does the Standard Model Need the Higgs Boson ? II Augusto Barroso Sesimbra 2007

2 Scattering  e-e- e+e+ W-W- W+W+ Incoming e- e+e+ Outgoing W-W- W+W+

3 Scattering  and Z s-channel Both terms grow as s. However, notice the cancelation between gamma and Z.

4 t-channel diagram It cancels with the similar contribution from the s-channel, i. e. Conclusion: All contributions that grow with s or cancel

5 Notice that: The previous results were obtained with m=0. Otherwise there would be an additional contribution to the t-channel diagram, namely:

6 Again the scalar exchange rescues the theory !

7 The Standard Model SU(3)xSU(2)xU(1) x Gravity Number of Parameters: 27 parameters. 23 are generated by the Higgs Sector. Another one, the Cosmological Constant, “does not like the Higgs sector”.

8 The Higgs Sector and The Higgs gives: The Higgs gives: Cosmology gives: Cosmology gives: =2x m -2

9 The Electroweak Sector

10 Example: only SU(2) But QFT is more than L g

11 Ex: Calculate the Self Energy is absorbed in the Wave Function renormalization. …=finite terms + +

12 Now, introduce fermions, F We don’t want to decide if F is left, right or whatever. Hence we use the coupling: There is another contribution to the gauge field self-energy. The new singular term requires a mass renormalization.

13 Renormalization of massless non- Abelian Gauge Theory Without Fermions Without Fermions –Simply With massive fermions axial coupled With massive fermions axial coupled –It requires This implies a mass term for the gauge bosons.

14 Loop Corrections Because the W and Z self energies are different we have a correction to . Because the W and Z self energies are different we have a correction to .

15 Summary: Parity conservation in QED Electron in Dirac Rep. Mass terms are Left-right Parity violation in Weak Int. The weak currents are left Bad behaviour of WW scattering Scalar boson couples to W Bad behaviour of e + e - WW The scalar couples to fermions Scalar is a doublet of SU(2)

16 Solution with Spontaneous symmetry breaking. Three degrees of freedom become the longitudinal polarizations of W and Z. The remaining is the Higgs Boson.

17 Higgs Boson From the value of G F one obtains: The parameter can be traded off with the Higgs mass. Inserting the LEP upper bound: M H < 144 GeV (95%CL) gives One can safely use perturbation theory.

18 FAQ The Higgs mechanism is a “clever trick”. Can we do it without ending up with an The Higgs mechanism is a “clever trick”. Can we do it without ending up with an Higgs Boson? Higgs Boson? NO. Cif. the example on WW scattering. NO. Cif. the example on WW scattering. Unless …

19 FAQ If we are not careful can we also break the electromagnetic gauge group and give the photon a mass? If we are not careful can we also break the electromagnetic gauge group and give the photon a mass? No, in the minimal model with a single doublet. No, in the minimal model with a single doublet.

20 Conclusion LEP results tested the SM in such a way that it is hard to believe that one is not checking the loop expansion of a QFT. LEP results tested the SM in such a way that it is hard to believe that one is not checking the loop expansion of a QFT. Then, the standard model is a QFT valid at the same level as, for instance, QED. Then, the standard model is a QFT valid at the same level as, for instance, QED. Hence, the Higgs boson ought to exist. Hence, the Higgs boson ought to exist.