1. Extra dimensions hypothesis in high energy physics
Edward Boos, Vyacheslav Bunichev, Maxim Perfilov,
Mikhail Smolyakov, Igor Volobuev (SINP MSU)

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Introduction
The idea that the microscopic structure of space-time can be different from the macroscopic one was first formulated by B. Riemann.
Bernhard Riemann, "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (1854)
(On the Hypotheses which lie at the Bases of Geometry)

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The first physical theory, in which an attempt was made to use five-dimensional space-time for a unification of electromagnetism and relativistic scalar gravity, was put forward 60 years later. Gunnar Nordström, ''Über die Möglichkeit, das elektromagnetische Feld und das Gravitationsfeld zu vereinigen''‚ (On the possibility of unifying the electromagnetic and the gravitational fields) (1914) The appearance of this theory was intimately connected with the development of general theory of relativity.

4. Later this idea was developed by T. Kaluza and O. Klein:
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Later this idea was developed by T. Kaluza and O. Klein:
Kaluza T. ''Zum Unitätsproblem in der Physik''‚
(On the problem of unity in physics)
Sitzungsber. Preuss. Akad. Wiss.
Berlin. (Math. Phys.) P ;
Klein O. ''Quantentheorie und fünfdimensionale
Relativitätstheorie'', (Quantum theory
and five-dimensional theory of relativity)
Zeit. Physik V. 37. P .

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In the paper A. Einstein and P. Bergmann, “On a Generalization of Kaluza's Theory of Electricity,” Annals Math. V. 39 (1938) 683, it was show that the unobservability of the extra dimension could be explained by its compactness and extremely small size, - of the order of the Planck length LPl = 1/MPl

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The original Kaluza-Klein model: gravity in five-dimensional space-time E=M4 x S1 with action being the five dimensional gravitational constant, being the five-dimensional scalar curvature and the signature of the metric being sign gMN = (-,+,+,+, +), M,N = 0,1,2,3,4.

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To isolate the physical degrees of freedom, it is useful to consider the linearized theory This theory is a gauge theory of tensor field with gauge transformations which allows one to impose the gauge condition

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Then the linearized equations of motion look like If the energy-momentum tensor is equal to zero, the equations take the form The corresponding eigenvalue problem for the mass spectrum looks as follows

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Thus, the mass spectrum can be found to be It includes the massless zero modes, for which n = 0 and whose wave functions do not depend on the coordinate of the extra dimension, and the higher Kaluza-Klein modes that are very heavy, because the size of the extra dimension should be of the order of the Planck length to provide for the unobservability of the extra dimension.

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The zero mode five-dimensional metric gMN can be decomposed as Since it does not depend on the extra dimension coordinate y, we get R(4) being the scalar curvature of the four-dimensional space-time M4 with metric Ɣμν.

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In the papers by Kaluza and Klein the field φ was assumed to be constant, and the field Aμ was identified with the electromagnetic field. The theory gives a relation between the five-dimensional (M) and the four-dimensional (MPl) Planck masses: Any field in space-time E=M4 x S1 can be expanded in a Fourier series in the coordinate y. For every four-dimensional field there exists a tower of fields with the same quantum numbers and the masses of the order of MPl, which cannot be observed at the energies available nowadays.

12. 1 Large extra dimensions
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1 Large extra dimensions
Localization of fields:
V.A.Rubakov and M.E. Shaposhnikov,
“Do We Live Inside A Domain Wall?”
Phys. Lett. 125 (1983) 136.
“Extra Space-Time Dimensions: Towards A
Solution Of The Cosmological Constant Problem”,
Phys. Lett. 125 (1983) 139.

13. N. Arkani-Hamed, S.Dimopoulos and G.R. Dvali,
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The fields of the SM can be localized on a domain wall in a multidimensional space. If the thickness of the domain wall goes to zero, then it turns into a membrane, or just a “brane”.
ADD scenario
N. Arkani-Hamed, S.Dimopoulos and G.R. Dvali,
“The hierarchy problem and new dimensions at a
millimeter”, Phys. Lett. B 429 (1998) 263

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A single brane without tension (i.e. energy density) in a space-time with an arbitrary number of compact extra dimensions.
The scenario provides a solution to the hierarchy problem: it gives a strong gravity in the multidimensional space-time and a weak gravity on the brane
The approximation of the zero brane tension turns out to be rather too rough, and the proper gravitational field of the brane cannot be taken into account perturbatively.

15. “A large mass hierarchy from a small extra dimension”,
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For the equations of Einstein gravity in space–time with compact extra dimensions to be consistent, there should exist at least two branes with tension, and the number of extra dimensions can be either one or two.
L. Randall and R. Sundrum,
“A large mass hierarchy from
a small extra dimension”,
Phys. Rev. Lett. 83 (1999) 3370

16. Sean M. Carroll, Monica M. Guica,
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Sean M. Carroll, Monica M. Guica,
“Sidestepping the cosmological constant
with football shaped extra dimensions e-Print: hep-th/
In Einstein gravity it is impossible to find braneworld solutions with more than two extra dimensions.
A possible way out is to pass to Lovelock gravity
D. Lovelock,
"The Einstein tensor and its generalizations",
J. Math. Phys. 12 (1971)

17. 2 The Randall-Sundrum model
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2 The Randall-Sundrum model
Two branes with tension at the fixed points of the orbifold S1/Z2:

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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

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The field hMN can be transformed to the gauge The field φ(x) is called the radion field. The distance between the branes along the geodesic x = const

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The metric in the zero mode approximation looks like Substituting this metric into the action and integrating over the coordinate of the extra dimension one gets an effective action Galilean coordinates: gμν = diag(-1,1,1,1).

21. Coordinates {x} are Galilean for c=0,
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Coordinates {x} are Galilean for c=0,

22. Coordinates {x} are Galilean for c = - kL,
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Coordinates {x} are Galilean for c = - kL,

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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 coupling of the radion to matter on the brane is too strong and contradicts the experimental restrictions even at the level of classical gravity. The Randall-Sundrum model must be stabilized!

24. 3 Stabilized Randall-Sundrum model
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3 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)

25. M.N. Smolyakov and I.P. Volobuev, “Physical degrees of freedom
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The second model is more consistent. We consider such values of the model parameters that the background metric of the stabilized model is close to that of the unstabilized model.
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

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They are:
tensor fields bμνn(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).
In the Standard Model extension based on the stabilized Randall-Sundrum model there appear new interactions of the SM fields described by the Lagrangian
Tμν being the energy-momentum tensor of the SM.

27. 4 Scalar sector: radion interactions and Higgs-radion mixing
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4 Scalar sector: radion interactions and Higgs-radion mixing
The radion field is a combination of the metric fluctuations in the extra dimension and of the fluctuations of the stabilizing scalar field. Due to its origin, it couples to the trace of the energy-momentum tensor of the Standard Model:

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In the paper E. Boos, S. Keizerov, E. Rahmetov and K. Svirina, “Higgs boson-radion similarity in production processes involving off-shell fermions” Phys. Rev. D 90 (2014) 9, it was shown that this interaction Lagrangian leads to the same production and decay properties of a single radion as those of the Higgs boson. The radion and the Higgs boson having the same quantum numbers, the radion field and its excitations can mix with the Higgs field, if they are coupled.

29. G. F. Giudice, R. Rattazzi and J. D.Wells
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Higgs-radion coupling in the unstabilized Randall-Sundrum model arising due to a Higgs-curvature term on the brane was introduced in the 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
It looks like , H(x) denoting the Higgs field.
In the linearized theory

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The Higgs-curvature term gives rise to the interaction term ,,,, which leads to a Higgs-radion mixing and to a term with the Fierz-Pauli Lagrangian that affects the boundary conditions for the tensor fields.

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In stabilized brane world models, instead of the Higgs-curvature term, one can introduce an interaction between the stabilizing scalar field and the Higgs field of the form
that replaces the Higgs potential and leads to spontaneous symmetry breaking on our brane.
This term gives rise to an interaction of the physical Higgs boson field σ(x) with the radion field φ1(x) and its Kaluza-Klein tower.

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At low energies the heavy Kaluza-Klein scalar modes can be integrated out, the interaction between the Higgs field σ(x) and the radion field φ1(x) can be diagonalized by a rotation , which gives an effective Lagrangian for the interaction of the Higgs-dominated field h(x) and the radion-dominated field r(x) with the fields of the Standard Model.

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The parameter c takes into account the contributions of the integrated out heavy scalar modes and the conformal anomaly is explicitly given by

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We have studied two a priori possible scenarios that the scalar state discovered at 125 GeV is either a Higgs-dominated state or a radion-dominated state. In our analysis we used only two signal strengths corresponding to the channels gg → ɣɣ and gg → ZZ*. E. Boos, V. Bunichev, M. Perfilov, M. Smolyakov and I. Volobuev, “Higgs-radion mixing in stabilized brane world models,” Phys. Rev. D 92 (2015) no.9,

35. e Exclusion contours for the combined χ2 fit in
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Exclusion contours for the combined χ2 fit in
the (mr , sinθ ) plane for Λr = 3 TeV e

36. Exclusion contours for the combined χ2 fit in
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Exclusion contours for the combined χ2 fit in
the (mr , sinθ ) plane Λr = 5 TeV

37. Exclusion contours for the combined χ2 fit in
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Exclusion contours for the combined χ2 fit in
the (mh , sinθ ) plane Λr = 3 TeV

38. Exclusion contours for the combined χ2 fit in
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Exclusion contours for the combined χ2 fit in
the (mh , sinθ ) plane Λr = 5 TeV

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Next we have studied the impact of the constraints from the measurements of the parameters of the observed 125 GeV Higgs boson and from the unconfirmed 750 GeV diphoton excess in the LHC experiments on the properties of a possible extra scalar boson predicted in various Standard Model extensions. E.E. Boos, V.E. Bunichev and I.P. Volobuev, “Heavy scalar boson in view of the unconfirmed 750 GeV LHC diphoton excess,” J. Exp. Theor. Phys. 124 (2017) no.5,

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The production cross section of the radion-dominated state with mass 750 GeV as a function of the mixing angle sinθ.

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Exclusion contours for the global χ2 fit in the (sinθ, 1/ Λr ) plane for the LHC at 8 TeV and mh=125 GeV, mr=750 GeV, c = cmax.

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Exclusion contours for the global χ2 fit in the (sinθ, 1/ Λr ) plane for the LHC at 13 TeV and mh=125 GeV, mr=750 GeV, c = cmax.

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The region in the (mr , sinθ ) plain still allowed by the common fit of the CMS and ATLAS exclusion limits for heavy Higgs searches.

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Mr = 30 GeV
LEP 210 GeV LHC 8TeV

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Mr = 30 GeV
LEP 210 GeV LEP + LHC

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Mr = 60 GeV
LEP 210 GeV LHC 8 TeV

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Mr = 60 GeV
LEP 210 GeV LEP + LHC

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Conclusion
The Standard Model extension based on the stabilized Randall-Sundrum model is phenomenologically acceptable. If the values of the fundamental parameters lie in the TeV energy range, then the effects due to the radion and its massive KK modes can be observed in the Higgs boson decay.
The LHC data on BSM neutral bosons give an estimate for the parameter Λπ:

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Thank you!