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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.

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the minimal b l model naturally realized at tev scale

The minimal B-L model naturally realized at TeV scale



Nobuchika Okada(University of Alabama)




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.

Λ:cutoff scale

conformal symmetry and hierarchy problem
Conformal symmetry and hierarchy problem

SM is classically conformal invariant except for the Higgs mass term.

W.A. Bardeen,



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.

We know one similar example.

The chiral symmetry protects fermion masses,

even in quantum level no fermion mass.

classically conformal sm

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.

Classically conformal SM

Coleman-Weinberg Mechanism

(radiative symmetry breaking)

Calculate quantum correction

cw potential in sm
CW potential in SM

The extremum condition

The CW mechanism occurs under the balance between the tree-level quartic coupling and the terms generated by quantum correction.

The stability condition



However, top quark is heavy, so the stability condition does not satisfy.

The effective potential

is not stabilized.

In the classically conformal SM, due to the large top mass the effective potential is rendered unstable, and CW mechanism does not work.

We need to extend SM.

We propose classically conformal minimal B-L extended model.

classically conformal b l extended model
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



We assume classical conformal invariance

  • Yukawa sector
  • Potential

Dirac Yukawa

Majorana Yukawa

See-Saw mechanism associates with B-L symmetry breaking.

The mass terms are forbidden by classical conformal invariance.

b l symmetry breaking
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.


First, we consider singlet Higgs sector.


1-loop CW potential

The extremum condition

The potential minimal is realized by the balance between the tree-level quartic coupling and the 1-loop correction.

The stability condition

This coupling relation generates the mass hierarchy between singlet scalar and Z’ boson.


In our model, if majoranaYukawa coupling is small, the stability condition satisfies.

The potential has non-trivial minimum.

B-L symmetry is broken by CW mechanism.

electroweak symmetry breaking
Electroweak symmetry breaking

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.

Φhas VEV M.

Effective tree-level mass squared is induced, and if λ’ is negative, EW symmetry breaking occurs as usual in the SM.

lep bound
LEP bound

LEP experiments provided a severe constraint.

LEP bound

theoretical bound

Planck scale

Theoretical bound

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

naturalness constraint
Naturalness constraint

We have imposed the classical conformal invariance to solve the gauge hierarchy problem.

Once B-L symmetry is broken, heavy states associated with this breaking contribute to effective Higgs boson mass.

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.


Naturalness constraint

The dominant contribution comes from 2-loop effect involving the top-quarks and the Z’ boson, because of the large top Yukawacoupling.

This contribution should be smaller than the EW scale.

summary of phenomenological bound
Summary of phenomenological bound

Coupling blow up

LEP excluded


Disfavored by naturalness

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.

z boson at lhc
Z’ boson at LHC

We calculate the dilepton production cross section through the Z’ boson exchange together with the SM processes mediated by Z boson and photon.

Z’ exchange

A clear peak of Z’ resonance

SM background

z boson at ilc international linear collider
Z’ boson at ILC(International Linear Collider)

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
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.

  • 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.
  • 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.