2.4 Models for physics beyond the Standard Model
2.4.2 Dark Matter and WIMP pair production at LHC
There is a certain consensus among physicists about the existence in the universe of a non-luminous matter, called “dark matter” (DM), that interacts gravitationally with SM particles (see reference [21] for a recent review). One of the most striking evidence of DM, comes from the measurement of the rotational velocity of stars in spiral galaxies.
These galaxies are composed of a disc of matter that rotates around an axis, and the measurement of the rotational velocity as a function of the radius gives information about the amount of matter and its distribution inside the galaxy. Observations made until now require an amount of matter much larger than expected (see figure2.7), and are compatible with the presence of a large DM halo, with a mass three to ten times larger than that corresponding to the visible matter.
Figure 2.7: Rotational velocity of stars as a function of the radius in the spiral galaxy NGC6503 [22]. The contributions of the disk (dashed curve), gas (dotted curve), and dark matter halo (dot-dashed curve) are also shown separately.
Other evidencies of the existence of DM come from observations of the motion of
galaxies within galaxy clusters [23, 24], and from measurements of the “cosmic mi-crowave background” (CMB). The CMB is the thermal radiation background that is measured in the universe, corresponding to roughly 2.7 K of temperature. It is almost homogeneous in all directions (in a part over 105), and is stronger in the microwave sector of the spectrum. This radiation is not associated with a specific object (star, galaxy...) and is interpreted as the relic radiation emitted when the universe became transparent to photons (3·105 years after the Big Bang). Detailed study of the an-gular correlation in the CMB fluctuations gives information about the geometry of the universe (confirming that is flat), about its evolution in early stages, as well as its energy-matter content. At present, the most precise measurements of the CMB is made with the PLANCK experiment [25] (see figure 2.8). As measured from the PLANCK data, in the present universe the density of the ordinary matter, DM, and dark energy are 5%, 27%, and 68%, respectively.
Figure 2.8: Cosmic Microwave Background measured by the PLANCK experiment [25].
The colors represent the variation of measured temperature. The blue and red areas differ from each other by a temperature of 0.6 mK.
From the CMB measurement, one can also infer that DM was present in much larger quantity in the early universe. A striking coincidence in cosmology is that if DM would annihilate to SM particles with an interaction strength close that of the weak force, that would result exactly in the decrease of DM density observed between early
and present universe (see reference [26] for a concise review). This coincidence leads to the idea that DM could be composed of Weak Interacting Massive Particles (WIMPs).
These are expected to have a mass between few GeV and a TeV and couple to SM particles through a generic weak interaction. Many new models for physics beyond the SM designed to solve the hierarchy problem (Supersymmetry for instance) also predict WIMP candidates.
WIMPs have being searched for with a variety of detection strategies. In “direct detection” experiments the aim is to observe WIMP-nucleon elastic scattering , by measuring the nuclear recoil. Instead, “indirect detection” experiments search for the SM products from WIMP annihilation. In the last decade, the field of DM detection has attracted a lot of interest because there have been several published results that can be interpreted as detection of WIMP particles. Possible hints of detection of a light WIMP (∼10 GeV) have emerged from data obtained by the DAMA/LIBRA [27], CDMS II [28], CRESST-II [29], CoGeNT [30, 31], but the interpretation of these events as due to scattering of a WIMP has been challenged by several other experiments such as XENON100 [32,33] (see figure2.9). Indirect detection experiments, such as AMS [34], FERMI [35] and PAMELA [36], have shown an anomalous excess of high-momentum positrons in the galaxy. Such an excess is consistent with the hypothesis of WIMP annihilation, but not yet sufficiently conclusive to rule out other explanations, like for example pulsars. The next generation of direct and indirect detection experiments, characterized by very low background, larger volumes and improved energy resolution, is awaited to shine a light on these ambiguous hints of detection.
A third strategy to search for WIMPs is based on the direct production at colliders.
In collider experiments, the WIMPs are undetected and if produced in association of an initial state parton (or photon), result in a mono-jet (or mono-photon) final state. As it will be shown in this thesis, the sensibility of collider searches is competitive with (and in some cases higher than) those of direct and indirect detection experiments. In case of discovery, the complementarity of these three strategies (see sketch in figure2.10) will be essential to measure the properties of the DM candidate(s), with less assumptions, and with reduced ambiguities.
Figure 2.9: Cross sections for spin-independent coupling versus WIMP mass. The figure is taken from reference [37], that details the experimental results. Shaded 68% and 95%
CL regions are from Supersymmetry predictions on WIMP candidates, that include recent LHC constraints, and are taken from references [38,39].
WIMP production at hadron colliders
Production of WIMPs at colliders can happen in many ways depending on the physics beyond the SM that one is considering. In the following, an effective field theory is used to describe possible interactions between WIMPs and partons (see reference [40]).
Here, WIMPs are assumed to be a Dirac-like fermion, and odd under the Z2 symmetry (R-parity in SUSY, or KK-parity in extra dimensions), so that each coupling involves an even number of WIMPs. Different effective operators (listed in table4.10) are taken into account to mimic the different nature of the mediators in a Fermi-like point interaction.
For each operator, a parameterM∗ caracterizes the strength of the interaction. In table 4.10, the operators D1 and D11 correspond to a scalar mediator that couples to quarks and gluons, respectively. The operators D5 and D8 would correspond instead to a vector and axial vector mediator, respectively.
The effective field theory used here is a good approximation of the real theory if the mass of the mediator (or mediators) is too heavy to be produced directly. Supposing
Figure 2.10: Sketch of the different strategies for Dark Matter search.
Name Operator Name Operator
D1 (Mm!q)3χχ¯ qq¯ D2 (Mm!q)3χγ¯ 5χ¯qq D3 (Mm!q)3χχ¯¯ qγ5q D4 (Mm!q)3χγ¯ 5χqγ¯ 5q D5 (M1!)2χγ¯ µχ¯qγµq D6 (M1!)2χγ¯ µγ5χqγ¯ µq D7 (M1!)2χγ¯ µχqγ¯ µγ5q D8 (M1!)2χγ¯ µγ5χ¯qγµγ5q D9 (M1!)2χσ¯ µνχqσ¯ µνq D10 (M1!)2,µναβχσ¯ µνγ5χqσ¯ αβq D11 (4M1!)3χχα¯ s(Gaµν)2 D12 (4M1!)3χγ¯ 5χαs(Gaµν)2 D13 (4M1!)3χχα¯ sGaµνG˜a,µν D14 (4M1!)3χγ¯ 5χαsGaµνG˜a,µν
Table 2.2: Operators coupling Dirac fermion WIMPs to Standard Model quarks or gluons.
a mediator of mass M, the suppression scale M∗ ∼ M(gSMgDM)−12 where gSM and gDM are the coupling of the mediator with SM and DM particles. So even for mod-erateM∗ the theory can be still valide if the couplings are sufficiently large. Since an effective theory requiresM >2mχand couplings aregSMgDM ≤(4π)2 to be treated in perturbation theory, the interaction with strongest coupling satisfymχ= 2πM∗. This means that for eachmχthere is aM∗ lower bound of validity below which the effective theory is not reliable anymore. In case the effective approach does not strictly apply, it is hard to know whether the predictions under-estimate or over-estimate the cross sections due to the lack of knowledge on the underlying ultraviolet theory.
Collider results compared with direct and indirect detection experiments Results from WIMP pair production at colliders can be compared with those from direct and indirect detection experiments. Exclusion limits on M∗ are translated into bounds on the WIMP-nucleon cross section [40,41]:
σD10 = 1.60·10−37cm2( µχ
1GeV )2$
20 GeV M∗
&6
, (2.18)
σD50 = 1.38·10−37cm2( µχ 1 GeV
)2$
300GeV M∗
&4
, (2.19)
σ0D8 =σ0D9 = 9.18·10−40cm2( µχ
1 GeV )2$
300 GeV M∗
&4
, (2.20)
σD110 = 3.83·10−41cm2( µχ 1GeV
)2$
100GeV M∗
&6
. (2.21)
whereµχ the reduced mass of the WIMP-nucleon system,µχ=mχ∗mN/(mχ+mN).
For this comparison, one needs to keep in mind the different kinematic regime in which the interaction is happening. In direct detection experiments, the typical transferred momentum is of the order of a keV. In this regime the propagator of a particle with massm >>1 KeV that mediate the interaction cannot be resolved, making a Fermi-like point interaction suitable. On the contrary, at LHC the center of mass scale of the hard scattering √
ˆ
s can be up to the TeV scale.
Finally, results at colliders can be translated into WIMP annihilation cross section, relevant for indirect detection experiments. DM annihilation rate is proportional to the quantity*σv+, whereσ is the annihilation cross section,vis the relative velocity of the annihilating particles, and the average is over the DM velocity distribution. Using the effective field theory approach already introduced, we find aσv for operator D5 (vector interaction), and for operator D8 (axial-vector interaction) are [42]:
σD5v= 1 16πM∗4
!
q
*
1− m2q m2χ
+
24(2m2χ+m2q) +8m4χ−4m2χm2q+ 5m4q m2χ−m2q v2
, , (2.22)
σD8v = 1 16πM∗4
!
q
*
1− m2q m2χ
+
24m2q+8m4χ−22m2χm2q+ 17m4q m2χ−m2q v2
,
, (2.23)
where the sum
-q
runs over all kinematically accessible quarks. Note that for other effective operators (for instance the scalar operator D1), the bounds from colliders can be much stronger, especially for low *v2+. It has to be noticed that the annihilation rate *σv+ is calculated in the assumption of equal coupling for all quarks, and only annihilation to quarks is considered. In the case of DM coupled with leptons, the exclusion limits are weakened by a factor 1/BR(χχ→qq).¯