The minimal B-L model naturally realized at TeV scale Yuta Orikasa(SOKENDAI) Satoshi Iso(KEK,SOKENDAI) Nobuchika Okada(University of Alabama) Phys.Lett.B676(2009)81 Phys.Rev.D80(2009)115007

The Standard Model is the best theory of describing the nature of particle physics, which is in excellent agreement with almost of all current experiments. However SM has hierarchy problem. It is the problem that the quadratic divergence in quantum corrections to the Higgs self energy, which should be canceled by the Higgs mass parameter with extremely high precision when the cutoff scale is much higher than the electroweak scale. 2 Λ:cutoff scale

Conformal symmetry and hierarchy problem 3 SM is classically conformal invariant except for the Higgs mass term. Possibility The chiral symmetry protects fermion masses, even in quantum level no fermion mass. The classical conformal symmetry protects mass scale. Even in quantum level this symmetry may protect the quadratic divergences. Therefore once this symmetry is imposed on SM, it can be free from hierarchy problem. W.A. Bardeen, FERMILAB-CONF T We know one similar example.

Classically conformal SM 4 If theory has the classical conformal invariance, the Higgs mass term is forbidden. Therefore there is no electroweak symmetry breaking at the classical level. We need to consider origin of the symmetry breaking. Coleman-Weinberg Mechanism (radiative symmetry breaking) Calculate quantum correction

CW potential in SM 5 The stability condition The extremum condition The CW mechanism occurs under the balance between the tree- level quartic coupling and the terms generated by quantum correction. ?

In the classically conformal SM, due to the large top mass the effective potential is rendered unstable, and CW mechanism does not work. 6 However, top quark is heavy, so the stability condition does not satisfy. The effective potential is not stabilized. We need to extend SM. We propose classically conformal minimal B-L extended model.

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Classically conformal B-L extended Model Gauge symmetry New particles right-handed neutrino Three generations of right-handed neutrinos are necessarily introduced to make the model free from all the gauge and gravitational anomalies. SM singlet scalar The SM singlet scalar works to break the U(1)B-L gauge symmetry by its VEV. gauge field 8

Lagrangian We assume classical conformal invariance Yukawa sector Potential 9 Dirac YukawaMajorana Yukawa See-Saw mechanism associates with B-L symmetry breaking. The mass terms are forbidden by classical conformal invariance.

B-L symmetry breaking If the mixing term of SM doublet Higgs and singlet Higgs is very small, we consider SM sector and singlet Higgs sector separately. 10 small First, we consider singlet Higgs sector.

11 The potential minimal is realized by the balance between the tree-level quartic coupling and the 1-loop correction. The extremum condition The stability condition 1-loop CW potential This coupling relation generates the mass hierarchy between singlet scalar and Z’ boson.

12 In our model, if majorana Yukawa coupling is small, the stability condition satisfies. The potential has non-trivial minimum. B-L symmetry is broken by CW mechanism.

Electroweak symmetry breaking 13 Effective tree-level mass squared is induced, and if λ’ is negative, EW symmetry breaking occurs as usual in the SM. Once the B-L symmetry is broken, the SM Higgs doublet mass is generated through the mixing term between H and Φ in the scalar potential. Φ has VEV M.

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LEP bound 15 LEP experiments provided a severe constraint. LEP bound

Theoretical bound 16 The bound of B-L gauge coupling We impose the condition that B-L gauge coupling does not blow up to Planck scale. For TeV scale B-L symmetry breaking, we find αB-L scale Planck scale

Naturalness constraint We should take care of the loop effects of the heavy states, since there is a small hierarchy between the electroweak scale and the B-L breaking scale. Here we estimate the loop corrections of heavy states on the Higgs boson mass. We have imposed the classical conformal invariance to solve the gauge hierarchy problem. 17 Once B-L symmetry is broken, heavy states associated with this breaking contribute to effective Higgs boson mass.

The dominant contribution comes from 2-loop effect involving the top-quarks and the Z’ boson, because of the large top Yukawa coupling. This contribution should be smaller than the EW scale. Naturalness constraint 18

Summary of phenomenological bound 19 U(1)Y The figure indicates that if the B-L gauge coupling in not much smaller than the SM gauge couplings, Z’ boson mass is around a few TeV. Coupling blow up Disfavored by naturalness LEP excluded

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Z’ boson at LHC 21 We calculate the dilepton production cross section through the Z’ boson exchange together with the SM processes mediated by Z boson and photon. SM background Z’ exchange A clear peak of Z’ resonance

Z’ boson at ILC (International Linear Collider) 22 We calculate the cross section of the process → at the ILC with a collider energy =1 TeV. The deviation of the cross section in our model from the SM one is shown as a function of Z’ boson mass. Assuming the ILC is accessible to 1% deviation, the TeV scale Z’ boson can be discovered at ILC.

Allowed parameter region together with search reach at future colliders The figure indicates that if the B-L gauge coupling in not much smaller than the SM gauge couplings, Z’ gauge boson can be discovered by near future collider experiments. 23

Conclusions The classical conformal theory may be free from the hierarchy problem. CW mechanism does not work in classically conformal SM since the large top Yukawa coupling destabilizes the effective Higgs potential. SM needs to be extended. We propose the classically conformal minimal B- L model. 24

conclusion B-L symmetry and EW symmetry are broken by CW mechanism. Our model naturally predicts B-L breaking scale at TeV. Z’ boson can be discovered in the near future. Because of CW type symmetry breaking, the singlet Higgs boson mass is smaller than the Z’ gauge boson mass. 25