ADD Large Extra Dimensions

• Gluino pair production with ˜g → g+ ˜χ⁰₁: Assumed 100% branching ratio for this decay, with the rest of SUSY particles decoupled, but the neutralino.

• Gluino pair production with ˜g→b¯b+ ˜χ⁰₁: Same prescription as gluino pair production with ˜g→g+ ˜χ⁰₁.

˜ g

˜ g

˜ q

q q

˜ q

q q

p p

˜ χ⁰₁ g

˜ χ⁰₁

g

Figure 3.7: Feynman diagrams for the inclusive squark/gluino production processes studied. Left: inclusive squark pair production, with the squarks decaying each to a quark and a neutralino. Center: gluino pair production with the gluino decaying each to a gluon and a neutralino. Right: gluino pair production, each decaying to a bottom and antibottom quarks and a neutralino.

3.2.5.3 Processes involving direct production of charginos or neu-tralinos

SUSY electroweak particles are usually produced in the cascade decays.

However, they can also be produced directly in electroweak-driven processes, with much lower cross sections in comparison to SUSY strong production [41, 42]. In the analysis presented, two simplified model processes involving electroweakinos (whose Feynman diagrams are shown in Figure 3.8) are studied:

• pp → q˜+ ˜χ⁰₁ production: First and second squark generations are degenerated in mass, while the third generation squarks and the gluino are decoupled from the theory.

• pp→ χ˜⁰₂+ ˜χ^±₁ and pp →χ˜^±₁ + ˜χ^∓₁ production: All squarks, sleptons and gluino masses are set to more than 5 TeV.

3.3 ADD Large Extra Dimensions

The Arkani-Hamed-Dimopoulos-Dvali [43] (ADD) model of large extra di-mensions is a framework which aims to solve the hierarchy problem without

˜ q

p p

˜ χ⁰₁ q

˜ χ⁰₁

Figure 3.8: Feynman diagrams for the electroweak SUSY production pro-cesses studied. Left: squark-neutralino production, with the squark decay-ing to a quark and a neutralino. Center: chargino pair production. Right:

Chargino-neutralino production.

relying in supersymmetry. In particular, this model provides an explanation to the 16 orders of magnitude difference between the electroweak and the Planck scales. With this objective,nextra spatial compactified dimensions with radius R are added to the 3 + 1 space-time dimensions. Under this assumption, two test massesm1 and m2 placed at a distance r R would feel a gravitational potential,

V(r)∼ m₁m₂ M_Dⁿ⁺²

1

rⁿ⁺¹, (rR), (3.55)

where M_D is the Planck scale assuming n extra dimensions. However, if rR,

V(r)∼ m1m2

M_Dⁿ⁺² 1

Rⁿr, (rR), (3.56)

which can be related to the Newtonian gravitational potential:

V(r)∼ m1m2

M_P² 1

r, (3.57)

if the “effective” 4-dimensional Planck scale is equivalent to

M_P² ∼M_D²⁺ⁿRⁿ. (3.58)

In the ADD model, the electroweak scalem_EW is the only fundamental short scale in nature. Therefore, ifMD ∼mEW, Equation 3.58 leads to:

R∼10³⁰ⁿ⁻¹⁷cm×

1 TeV m_EW

. (3.59)

This result points to the fact thatMD, the truth strength of the gravita-tional interaction can be as low as the electroweak scale providing values of R as large as a millimeter. For n = 1,R ∼10¹³cm, which should produce

3.3. ADD LARGE EXTRA DIMENSIONS 49 deviations from Newtonian gravity over solar system distances, and there-fore is empirically excluded. Forn= 2,R ∼100µm−1 mm, thus leading to deviations in the gravitational predictions that could be proved in the up-coming years¹. Highern values would lead to lower compactification radii R.

In the ADD model, the SM particles can only propagate in a 4-dimensional submanifold, while gravitons, understood as excitations of then-dimensional metric, are the only particles allowed to 4 +ndimensional bulk.

In terms of 4-dimensional indices, the metric tensor contains 2, spin-1 and spin-0 particles, which can be expressed as a tower of Kaluza-Klein modes [44, 45]. The mass of each Kaluza-Klein mode corresponds to the modulus of its momentum in the direction transverse to the 4-dimensional brane. In fact, the picture of a massless graviton propagating in a n-dimensional space or a massive Kaluza-Klein tower of massive gravitons propagating in a 4-dimensional space is completely equivalent. At low en-ergy and small curvature, the equations of motion of the effective theory reduce to Einstein equation in n= 4 +δ dimensions:

G_AB ≡ R_AB−1

2gABR=− TAB

M¯_D^2+δ A, B= 1, . . . , n, (3.60) where ¯M_D is the reduced Planck scale of the n-dimensional theory, M_D = (2π)^δ/(2+δ)M¯D. If the metric gAB is expanded around its Minkowski value η_AB,

g_AB =η_AB+ 2 ¯M_D^−1−δ/2h_AB, (3.61) and therefore, Equation 3.60 can be rewritten as:

M¯_D^1+δ/2G =2h_AB −∂_A∂^Ch_CB−∂_B∂^Ch_CA+∂_A∂_Bh^C_C

−ηAB2h^C_C +ηAB∂^C∂^DhCD =−M¯_D^−1−δ/2TAB,

(3.62)

keeping only the first power of h. The previous expression can be derived from the followingn-dimensional graviton lagrangian:

L_grav=−1

2h^AB2h_AB+1

2h^A_A2h^B_B

−h^AB∂_A∂_Bh^C_C+h^AB∂_A∂_Ch^C_B− 1

M¯_D^−1−δ/2h^ABT_AB. (3.63)

1At present, gravity has been proven at the level of a centimeter

This lagrangian becomes the sum over the Kaluza-Klein modes of:

L_grav=X

all~n

−1

2G^(−~n)µν(2+m²)G^(−~_µνⁿ⁾+1

2G^(−~_µ ^n)µ(2+m²)G^(−~_ν ^n)ν

−G^(~n)µν∂µ∂νG^(~_λ^n)λ+G^(~n)µν∂µ∂_λG^(~_ν^n)λ− 1

M_PG^(~n)µνTµν, +· · ·

(3.64)

under the unitary gauge and the parametrization from Reference [46]. In this equation, the ellipses refer to spin-2, spin-1 and spin-0 particles that are not coupled (or their coupling is very suppressed) to the SM energy-momentum tensorT_µν, and therefore play no role in a collider experiment.

The latest term is the graviton interaction lagrangian. IfTµν is expanded, the Feynman rules for the interactions between gravitons and SM fields can be retrieved.

For not very highδ (i.e. δ.6) the mass difference between the graviton modes is small and the contributions of the different modes can be integrated over the mass. Under this approximation, the differential cross section for inclusive graviton production is expressed as:

d²σ

dt dm = 2π^δ/2 Γ(δ/2)

M_P²

M_D^2+δm^δ−1d σ_m

dt , (3.65)

wheredσm/dtis the differential cross section for producing a single Kaluza-Klein graviton of massm, found to be:

dσm

dt (qq¯→gG) = αs

36 1

M_P²sF1(t/s, m²/s) dσm

dt (qg→qG) = αs

96 1

M_P²sF2(t/s, m²/s) dσm

dt (gg→gG) = 3αs

16 1

M_P²sF3(t/s, m²/s),

(3.66)

with the expressions F₁, F₂ and F₃ reported in Reference [46]. Figure 3.9 shows the Feynman diagrams for the graviton production at colliders at LO.

The ADD model is an effective theory and therefore, it is only valid up to a given scale, which is assumed to be of the order ofm_EW. For this reason, the effects of the hypothetical underlying theory are expected to emerge at energy scales close toMD.