A new mechanism of Higgs-radion mixing

Igor Volobuev (SINP MSU)

QFTHEP'2011 Sochi, 29 September 2011

Introduction

It is a common knowledge in QFT that fields with the

same quantum numbers can mix, if there is an interaction between them.

A canonical example is the mixing of the weak hyper-charge U(1) gauge filed and of the neutral component of the SU(2) gauge field, which gives rise to the electromag- netic field and the field of the Z boson.

In extensions of the SM additional fields can mix with the fields of the SM, if they have the same quantum numbers.

QFTHEP'2011 Sochi, 29 September 2011

1 The Randall-Sundrum model L. Randall and R.

Sundrum,

“A large mass hierarchy from

a small extra dimension”,

Phys. Rev. Lett. 83 (1999) 3370

Two branes with tension at the fixed points of the orbifold S1/Z2:

QFTHEP'2011 Sochi, 29 September 2011 The solution for the background metric:

The parameters k, Λ и λ1,2 satisfy the fine tuning conditions:

The linearized gravity is obtained by the substitution

QFTHEP'2011 Sochi, 29 September 2011 It is a gauge theory with the gauge transformations

The functions ξM(x,y) satisfy the orbifold symmetry conditions

The field hMN can be transformed to the gauge

QFTHEP'2011 Sochi, 29 September 2011 The distance between the branes along the geodesic

x = const

The equations for the fields hμν(x,y) и Φ(x) can be decoupled by the substitution

QFTHEP'2011 Sochi, 29 September 2011

QFTHEP'2011 Sochi, 29 September 2011

The hiearrchy problem is solved, if M ~ k ~ 1 TeV и kL~ 35.

There appears a tower of tensor fields on the brane with the lowest mass of the order of M and the coupling to the SM fields of the order of 1/M.

The branes in the Randall-Sundrum model can oscillate with respect to each other, which manifests itself as a massless four-dimensional scalar field, -- the radion field.

The coupling of the radion to matter on the brane at y=L is too strong and contradicts the experimental constraints even at the level of classical gravity!

QFTHEP'2011 Sochi, 29 September 2011

The radion field can mix with the Higgs field, if they are coupled.

Such a coupling was put forward in paper

G.F. Giudice, R. Rattazzi and J.D. Wells,

“Graviscalars from higher-dimensional metrics and curvature-Higgs mixing”, Nucl. Phys. B 595 (2001) 250

[arXiv:hep-ph/0002178]

and it looks like

QFTHEP'2011 Sochi, 29 September 2011 2 Stabilized Randall-Sundrum model

Stabilization mechanisms:

W. D. Goldberger and M.B. Wise,

“Modulus stabilization with bulk fields”, Phys. Rev. Lett. 83 (1999) 4922

O. DeWolfe, D.Z. Freedman, S.S. Gubser and A. Karch, “Modeling the fifth dimension with scalars and gravity”, Phys. Rev. D 62 (2000)

046008

QFTHEP'2011 Sochi, 29 September 2011 The second model is more consistent. Its action is taken

to be

If the potential V is of the form

one can find exact solutions for this model.

QFTHEP'2011 Sochi, 29 September 2011 For

and the finetuned brane potentials

the solution for the scalar field and the metric looks like

For uL<<1 we have

QFTHEP'2011 Sochi, 29 September 2011 The physical degrees of freedom of the model in the linear

approximation were isolated in the paper E.E. Boos, Y.S. Mikhailov, M.N. Smolyakov and I.P. Volobuev, “Physical degrees of freedom in stabilized brane world models”, Mod. Phys. Lett. A 21 (2006) 1431

They are: tensor fields bμν

n(x), n=0,1, … with masses mn (m_0 = 0) and wave functions in the space of extra

dimension ψn(y), scalar fields φn(x), n=1,2, … with masses μn and wave

functions in the space of extra dimension gn(y).

QFTHEP'2011 Sochi, 29 September 2011 The Higgs-radion mixing due to the curvature term in this

model was discussed in paper

C. Csaki, M.L. Graesser and G.D. Kribs, “Radion dynamics and electroweak physics”, Phys. Rev. D 63 (2001) 065002

[arXiv:hep-th/0008151]

but only the lowest mode φ0(x) of the scalar KK tower was taken into account.

It turns out that the cumulative effect of the higher scalar modes can be comparable with that of the first mode and in this case should be taken into account!

QFTHEP'2011 Sochi, 29 September 2011 3 New mechanism of Higgs-radion mixing

We modify the action of the brane at y=L as follows:

Then the equations of motion give

We pass to the linearised theory by the substitution

QFTHEP'2011 Sochi, 29 September 2011 The terms responsible for the Higgs-radion mixing are

Expanding the field f in terms of four-dimensional modes, we get

One can put here

QFTHEP'2011 Sochi, 29 September 2011 Thus, we get the interaction Lagrangian

Next we denote

and observe that for M ~ 2TeV and μ1 ~ 200 Gev

QFTHEP'2011 Sochi, 29 September 2011 Thus, we get a mass matrix

QFTHEP'2011 Sochi, 29 September 2011 It is not difficult to find that

One can approximately find the eigenvalues of this matrix.

We introduce the following notations:

QFTHEP'2011 Sochi, 29 September 2011 Then for the Higgs and the radion masses we obtain

We also introduce normalization constants:

QFTHEP'2011 Sochi, 29 September 2011 The mass eigenstate fields are

QFTHEP'2011 Sochi, 29 September 2011 And their couplings to fermions and to the trace of the

energy-momentum tensor look like

QFTHEP'2011 Sochi, 29 September 2011 where the constants are given by

QFTHEP'2011 Sochi, 29 September 2011 Thus we see that though the interaction of the individual

excited scalar states with the Higgs field may be weak, there cumulative effect on the Higgs-radion mixing may be observable due to sizable values of the parameters

Δ and ρ.

A similar contribution of the directly unobservable KK modes to scattering processes was discussed in

E.E. Boos, V.E. Bunichev, M.N. Smolyakov and I.P. Volobuev, “Testing extra dimensions below the production threshold of Kaluza-Klein excitations” Phys.Rev.D79:104013,2009 arXiv:0710.3100v4 [hep-ph].