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arXiv:hep-ph/0510399v1 30 Oct 2005

UCI-TR-2005-37

Branon radiative corrections to collider

physics and precision observables

J. A. R. Cembranos1, A. Dobado2and A. L. Maroto2

1Department of Physics and Astronomy,

University of California, Irvine, CA 92697, USA

2Departamento de F´ ısica Te´ orica,

Universidad Complutense de Madrid, 28040 Madrid, Spain

ABSTRACT

In the context of brane-world scenarios, we study the effects produced by

the exchange of virtual massive branons. A one-loop calculation is performed

which generates higher-dimensional operators involving SM fields suppressed

by powers of the brane tension scale. We discuss constraints on this scenario

from colliders such as HERA, LEP and Tevatron and prospects for future

detections at LHC or ILC. The most interesting phenomenology comes from

new four-particles vertices induced by branon radiative corrections, mainly

from four fermion interactions. The presence of flexible branes modifies also

the muon anomalous magnetic moment and the electroweak precision ob-

servables.

PACS: 11.25Mj, 11.10Lm, 11.15Ex

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1 Introduction

In recent years, it has been shown that a generic property of brane-world

models [1] with low tension (τ ≡ f4≪ Λ4, where τ is the brane tension

and Λ is the scale below which the description given by the brane-world

scenario is appropriate) is the presence of new modes πα(x) called branons

which roughly correspond to excitations of the brane position along the extra

compactified dimensions. The relevant tree-level phenomenology of branons

has been studied for colliders and also for astrophysics and cosmology in

terms of their mass M and brane tension parameter f, and it has been

suggested that massive branons could be natural candidates for dark matter

in this kind of models [2].

In this work we study the phenomenology of branon radiative corrections.

Branon loops are interesting mainly for two reasons. First because preci-

sion tests of the Standard Model (SM) usually enforce strong constraints on

physics beyond it and thus make possible to reject many new models, or at

least to set bounds on their parameters. The second reason is that branon

loops provide new physical effects, such as four fermion interactions, which

can be searched for in present and next generation colliders.

As it is the case for branon tree-level effects, the loop corrections can be

obtained from the effective action for branons described in detail in [3]. This

effective action can be expanded in powers of ∂π/f2and M2π2/f4[4, 5]:

Seff[π] = S(0)

eff[π] + S(2)

eff[π] + S(4)

eff[π] + ...(1)

The zeroth order term is just a constant, the O((π/f)2) contribution contains

the branon free action:

S(2)

eff[π] =1

2

?

M4d4x(δαβ∂µπα∂µπβ− M2

αβπαπβ). (2)

where M2

branon excitations πα, with α running from one to the number of effective

extra dimensions N. The couplings to the SM fields Φ living on the brane

(or any suitable extension of it) in the presence of a gravitational background

which for simplicity we will assume to be flat (gµν= ηµν), can be described

at low energies by the action:

αβis the squared branon mass matrix corresponding to the different

SSM[Φ,π] =

?

M4d4x

?

LSM(Φ) +1

2δαβ∂µπα∂µπβ−1

2M2

αβπαπβ

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+

1

8f4(4δαβ∂µπα∂νπβ− M2

αβπαπβηµν)Tµν

SM

?

+ O(π4). (3)

where Tµν

ground metric:

SMis the conserved SM energy-momentum evaluated in the back-

Tµν

SM= −

?

gµνLSM+ 2δLSM

δgµν

??????gµν=ηµν

(4)

It is interesting to note that there is no single branon interactions due to

the parity conservation on the brane by the gravitational action. Thus bra-

nons are absolutely stable. This fact is crucial for the branon phenomenology

and in particular for cosmology since it makes them natural WIMP candi-

dates for dark matter [2]. In addition, the quadratic expression in (3) is valid

for any internal extra-dimension space KN, regardless the particular form of

its metric γαβ. Indeed the form of the couplings only depends on the num-

ber N of branon fields and the brane tension. Dependence on the geometry

of the extra dimensions will appear only at higher orders. Here we are as-

suming that the bulk D dimensional space-time (D = 4 + N) can be split

as MD= M4× KN, where M4is the standard four-dimensional Minkowski

space and KNis some compact and homogeneous space of dimension N with

gaussian coordinates yα. Then the N branon fields (α = 1,...,N) can be

chosen so that πα(x) = f2yα(x) where yα= yα(x) represents the position of

the excited brane in the extra-dimension space KN. The brane ground state

corresponds to πα= 0.

From the action above it is clear that branons always interact by pairs

with the SM matter fields. In addition, due to their geometric origin, those

interactions are very similar to the gravitational ones since the παfields

couple to all the matter fields through the energy-momentum tensor and

with the same strength suppressed by a f4factor. The interaction between

bulk gravitons and SM fields is given by:

Sh =

1

¯ MP

?

p

?

M4d4xh(p)

µν(x)Tµν

SM(x) (5)

where ¯ M2

1019GeV) and h(p)

hµν(x,y) corresponding to the 4 × 4 part of the bulk metric:

gµν(x,y) = ηµν+2hµν(x,y)

P≡ M2

P/4π is the squared reduced Planck mass (MP = 1.2 ×

µνare the Kaluza-Klein (KK) modes of the bulk graviton

¯ M1+N/2

D

(6)

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where¯ MD−2

damental scale of gravity) related with the usual four-dimensional Planck

scale by M2

D

with V (KN) being the volume of the internal

space KN (notice that M4≡ MP). By introducing a complete orthonormal

set of functions fp= fp(y) on KNwith normalization:

D

= MD−2

D

/(4π) and MDis the D dimensional Planck scale (fun-

P= V (KN)MD−2

?

KndV(KN)f∗

p(y)fq(y) = V (Kn)δpq

(7)

and fp(0) = 1, the KK mode decomposition for the graviton field becomes

hµν(x,y) =

1

?

V (KN)

?

p

h(p)

µν(x)fp(y). (8)

When computing radiative corrections, divergent integrals appear. As

our effective actions are not renormalizable all our results will be given in

terms of some energy cut-off Λ, which could be taken as the value where the

whole brane-world picture breaks down and a more fundamental approach is

needed. Then our results will be given in terms of four parameters, namely

the number of branons or extra dimensions N, the branon mass M (for

simplicity we will assume at the end that all of them are degenerate Mαβ=

δαβMβ= δαβM), the brane tension scale f (τ = f4) and the cut-off Λ.

The plan of the paper goes as follows: In section 2 we reobtain the result

of [7] concerning the suppression of the coupling between SM fields on the

brane and bulk fields, by integrating out the branon fields instead of using

arguments based on normal ordering. In section 3 we study the effects of

branon loops on the SM particle parameters and find the effective action

describing the new induced interactions. The corresponding phenomenolog-

ical consequences are considered in section 4, where we also set the bounds

coming from the branon loops on the parameters f, M, N and the scale Λ.

Further constraints can be obtained from two loops effects and their impact

on the electroweak precision observables and the muon anomalous magnetic

momentum which can be found in section 5. In section 6 we summarize and

comment our results and in Appendix A, B and C we define the divergent

integrals appearing in our computations, the Feynman rules corresponding

to the effective Lagrangian describing the branon loops effects, and the as-

sociated cross-sections.

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2Graviton coupling suppression

Probably the most immediate effect of virtual branons is the suppression of

the coupling of SM particles and the KK modes bulk fields like the graviton.

When branon fluctuations are taken into account this effective coupling is

described by the action:

Sh =

1

¯ MP

?

p

?

M4d4xh(p)

µν(x)Tµν

SM(x)fp(π) (9)

due to the fact that the brane is no more sitting at π = 0 but is moving

around this point. Now the branons fields can be integrated out in the usual

way to find:

Sh =

1

¯ MP

?

p

?

M4d4xh(p)

µν(x)Tµν

SM(x)?fp(π)? (10)

where the fpexpectation value is given by

?fp(π)? =

?

[dπ]eiS(2)

eff[π]fp(π) (11)

In the limit of massless branons, the branon effective action is just a non

linear sigma model (NLSM) based on a coset which is isomorphic to KN.

Therefore the invariant path integral measure should include an additional

factor proportional to the square root determinant of the coset metric to

ensure that quantum corrections do not spoil the Ward identities of the

NLSM. The extra term in the measure amounts to an extra term in the

effective action proportional to Λ4. This term is important when dealing

with branon loop corrections to the branon self-interactions (for instance

branon-branon elastic scattering) [6]. However in this work we are mainly

interested in interactions between a couple of branons and SM particles and

hence we can safely neglect this measure term.

To compute the path integral above, we need to know the precise form of

the f(π) functions which depends on the KNgeometry. For example for the

case of the torus KN= TN:

f? n(y) = exp

?

i? n? y

R

?

(12)

where ? n = (n1,n2,...nN) is a N dimensional vector with integer and positive

or zero components and R is the torus radius (common for all coordinates).

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