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High Energy Physics and Quantum Field Theory - 2015

Higgs field as the Goldberger-Wise field
Vadim Egorov, Igor Volobuev (SINP MSU)

Samara, 2015


QFTHEP'2015

Higgs field as the Goldberger-Wise field

The question of the stabilization of the size of the extra dimension with help of the Higgs field was raised earlier in the papers: · L. Vecchi, "A Natural Hierarchy and a low New Physics scale from a Bulk Higgs,"(2011); · M. Geller, S. Bar-Shalom and A. Soni, "Higgsradion unification: Radius stabilization by an SU(2) bulk doublet and the 126 GeV scalar," (2014). There was considered a perturbative solution. We attempt to find an exact one.
Egorov Vadim 1/14


QFTHEP'2015

Higgs field as the Goldberger-Wise field

The Randall-Sundrum model
We consider two branes with tension interacting with gravity in a fivedimensional space-time E M 4 S 1 / Z

2

In this report the interbrane separation is assumed to be stabilized by a twocomponent complex scalar field. On "our" brane it will implement the Higgs mechanism of spontaneous symmetry breaking.
Egorov Vadim 2/14


QFTHEP'2015

Higgs field as the Goldberger-Wise field

The background solution
1 Let us consider a scalar field . 2
The action of the model can be written as

S Sg S
S g 2M
and S SM model:


SM

,

where the gravitational action S g is given by
3



d 4 x dyR g ,
L

L

is an action of the scalar field, branes and the Standard
L

S

SM

M d 4 x dy M M V ( ) g
L









yL

~ 1 ( ) g d 4 x


yL

2 ( ) LSM

HP

~ ( , ) g d 4 x.
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Egorov Vadim


QFTHEP'2015

Higgs field as the Goldberger-Wise field

A solution for the metric, which preserves the Poincare invariance in any fourdimensional subspace y const, is sought in the form:

ds
2

MN

dx dx e
M N

2 A( y )

dx dx dy ,
2





and for the multidimensional Higgs field in the form:

( x, y ) ( y ).
Egorov Vadim 4/14


QFTHEP'2015

Higgs field as the Goldberger-Wise field

By the variation of the action we get the equations of motion:
1 2 V ( y) ( M M 1 2 2 12 M A V 2 dV 1 d1 1 d2 ( y) d M d M d dV 1 d1 1 d2 ( y) d M d M d
1 2
Egorov Vadim

2 y L) 2 M 2 3 A 6 A , 0, ( y L) 4 A , ( y L) 4 A .





5/14


QFTHEP'2015

Higgs field as the Goldberger-Wise field

An ansatz looks like:
1 dW dW 1 2 W ( ) , V 2 4 d d 24 M 1 dW 1 dW ( y ) sign ( y ) , ( y ) sign ( y ) , 2 d 2 d 1 A( y ) sign ( y ) W ( ). 2 24 M

The equations of motion are valid everywhere, except for the branes.
Egorov Vadim 6/14


QFTHEP'2015

Higgs field as the Goldberger-Wise field

We choose the function W ( ) in the form:

W 24M k 2u .
2

Then the brane potentials should be defined as follows:
1 ( ) MW ( ) 1 , 2
2 1 2

2 ( ) MW ( ) 2 . 2
2 2
Egorov Vadim

2

A Higgs-like potential

7/14


QFTHEP'2015

Higgs field as the Goldberger-Wise field

We finally get:

0 ( y ) e u 2

y L

, 246 GeV ,

A( y ) k y L



2 2

96M

e

2u y L



1 .



The interbrane distance is defined by the boundary conditions for the scalar field and is expressed in terms of the parameters of the model by the relation:

1 1 L ln , u
so we have the size of the extra dimension stabilized.
Egorov Vadim 8/14


QFTHEP'2015

Higgs field as the Goldberger-Wise field

The equation for the fluctuations of the scalar field
In order to build the linearized theory we represent the metric and the five-dimensional Higgs field in the unitary gauge as: 1 g MN ( x, y ) MN ( y ) hMN ( x, y ), 2M 3 0 ( x, y ) ( y ) 1 f ( x, y ) . 2 2M
Egorov Vadim 9/14


QFTHEP'2015
2 A( y )

Higgs field as the Goldberger-Wise field

Let us define a new function g e h44 ( x, y) . After the mode decomposition of g we get the equation in the Sturm-Liouville form:

e2 A 2 g n 2 y (0, L) , and the d dy

e2 A e4 2 g n n g n 6M 2 2

A



2

,

boundary conditions on the branes:

1 d 2 1 2 2 g n n e2 A g 4 M d 2 2 2

n y 0

0, 0.
y L 0
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1 d 2 2 2 2 g n n e2 A g 4 M d 2 2 2
Egorov Vadim

n


QFTHEP'2015

Higgs field as the Goldberger-Wise field

Using the results of the paper: Edward E. Boos, Yuri S. Mikhailova, Mikhail N. Smolyakov, Igor P. Volobuev, "Physical degrees of freedom in stabilized brane world models", in the case uL 1 we get the following mass of the lowest excitation of the scalar field identified as the Higgs boson:

u 2 uM m . 2 2 3M 2 uk
2 H 2 2 2
Egorov Vadim 11/14


QFTHEP'2015

Higgs field as the Goldberger-Wise field

If we choose M 2 TeV and 2 we get the model parameters as follows:

u 1.76 TeV , k 186 TeV ,

1 345 TeV ,
1

L 0.2 TeV

2 10

18

cm.

The Higgs boson can now interact with the energy-momentum tensor: H hT , where h g1.

k 1 The coupling: H ~ 1TeV . 3 24M
Egorov Vadim 12/14


QFTHEP'2015

Higgs field as the Goldberger-Wise field

Conclusion
· The stabilization of the size of the extra dimension in the Randall-Sundrum model and the spontaneous symmetry breaking on "our" brane are explained simultaneously with help of the fivedimensional Higgs field. · The equation of motion for this field is found and a solution is obtained.
Egorov Vadim 13/14


QFTHEP'2015

Higgs field as the Goldberger-Wise field

· In this case the Higgs boson is the radion at the same time, and it now has an interaction with the energy-momentum tensor that can affect its properties significantly. · The possible values of the model parameters are estimated, which give the correct value of the Higgs boson mass.

Egorov Vadim

14/14