Supersymmetry. Theory, Experiment, and Cosmology

The LSP and dark matter 111

The couplings air and bir describe scalar and pseudoscalar LSP–squark–quark interactions:

i r=1

where the index r refers to the two mass eigenstates in the case of sfermion mixing (see (5.58)). They can be obtained from (5.81) and read explicitly

ai1 = − 12 cos θqi (Xi + Ziq) + sin θqi (Yi + Ziq ) ,

with cos θqi replaced by − sin θqi and sin θqi replaced by cos θqi for r = 2.

Realizing that |air | = |bir | in the chiral limit (Ziq goes to zero with the fermion mass), we see that fi vanishes in this limit. This is true also for the contribution coming from H, h exchange since their couplings to quarks are proportional to the mass, see Table 5.3. The scalar contribution is thus naturally disfavored for the light quarks that are abundant in nucleons. Heavy quarks couple to nucleons only through a loop [123, 124]. The combined e ects lead to a dynamical suppression of the scalar interaction. We may note that, going to large values of tan β may significantly increase the cross-section because, as can be seen from Table 5.3, it enhances the couplings of H, h to down-type quarks [123].

Regarding the spin-dependent cross-section, we infer from (5.77) that a large contribution from Z exchange requires a large Higgsino component, which is disfavored if we want a substantial relic abundance.

To give an order of magnitude, in models where the LSP is mostly a bino, one typically obtains spin-dependent cross-sections σχSD−p in the 10−7 to 10−5 pb and scalar cross-sections σχSI−p in the range 10−10 to 10−7 pb range.

5.5.5Direct detection

Goodman and Witten (1985) were the first to propose to use bolometers to detect dark matter candidates. Typically, the event rate R is given by the number NN = NA/A of nuclei in the target (NA is Avogadro’s number and A the atomic number) times the flux F = ρχvχ/mχ (ρχ is the local WIMP density and vχ the average WIMP velocity) times the cross-section σ. Putting typical numbers, this gives

where µ is the reduced mass (µ−1 = MA−1 + m−χ 1). For the processes discussed in the preceding section, M is typically of the form g2/M 2, where g is a gauge coupling and

The LSP and dark matter 113

where S(|q|)/S(0) is the nuclear spin form factor, J is the total spin of the nucleus, and λ is a nucleonic matrix element that basically describes the fraction of the total spin that is due to the spin of the nucleons:

where Sp,n is the expectation value of the proton (neutron) group spin content of the nucleus.

To recapitulate, the di erential detection rate is calculated to be

The total rate R is, in this case, easily computed. Assuming an experimental cut-o ET for the recoil energy measured,

where we have made explicit in the last term the dependence in the neutralino density and mass. If an experiment puts an upper limit on R, this translates into a upper limit on ρχ of the order of

This behaves linearly with mχ for large mχ and grows exponentially for small mχ. Thus, exclusion plots typically look like Fig. 5.3: assuming a given value of ρχ leads to the exclusion of neutralino masses in a range [mminχ , mmaxχ ].

Since the rate depends on the velocity of the incident WIMP, one expects an annual modulation due to the motion of the Earth in the Galactic frame. More precisely, the

114 The minimal supersymmetric model

ρχ

Local galactic density of halo

Excluded

Fig. 5.3 Typical form of the exclusion plot for a WIMP in the plane local density versus mass.

Earth motion is a superposition of its rotation around the Sun and of the rotation of the Sun around the Galactic Center (with velocity v = 220 km/s):

where t is measured in days and tp 153 days (June 2) is the day when the direction of the Earth’s motion around the Sun matches the direction of the Sun’s motion around the Galactic Center [164]. The numerical coe cients reflect the 60◦ angle between the two planes of rotation. The yearly modulation comes from the fact that, in the lab (Earth) frame, one must shift the velocity distribution: f (v + v ).

5.5.6Indirect detection

WIMPs annihilate and their annihilation products can be detected. The variety of these particles (gamma rays, neutrinos, positrons, antiprotons, antinuclei) leads to expect to extract rich information from correlated observations. Because the annihilation rate is proportional to the square of dark matter density, one increases the expected flux by looking at places where the dark matter accumulates. Typical places are the center of the Earth or the Sun: WIMPs loose energy by scattering with the nucleons of matter and sink to the center. Or the Galactic Center: the density profile of the dark matter halo is expected to grow as a power law (ρ(r) r−γ ); this e ect may even be enhanced by adiabatic accretion on the central massive black hole leading to a central “spike” [201].

WIMPs which have accumulated in the Earth or the Sun annihilate into ordinary particles. Most of the decay products of these particles will be absorbed, except for high energy neutrinos (with typical energy 13 to 12 the WIMP mass) which pass through the Earth or the Sun. Thus energetic neutrinos are a signature for dark matter. Interactions of such neutrinos in the Earth give energetic muons which are detected in large underwater (or ice) detectors such as AMANDA, ANTARES, or NESTOR.

The capture rate of neutralinos is proportional to the nucleus–neutralino elastic

SI,SD

scattering cross-section, and thus depends on σχ−p,n. In the case of the Earth, scalar

The LSP and dark matter 115

interactions are favored because of the zero spin of Fe56. For the Sun, the spin of hydrogen allows spin-dependent interactions and capture is more e cient.

On general grounds, the annihilation cross-section of neutralinos at rest (a good approximation in the case of neutralinos trapped in the center of a celestial body) into massless fermions vanishes [195]. Indeed, Fermi statistics imposes that the spin of the two neutralinos in the initial state (S-wave) are opposite. In the final state, the required spin flip is ensured by a mass insertion; hence the cross-section is proportional to the

mass of the fermions. Direct annihilation into neutrinos is thus strongly disfavored and

→ ¯ ¯ + − + − → neutrinos are produced as secondary particles: χχ tt, bb, cc,¯ τ τ , W W , ZZ

Xν. The more energetic the neutrinos are, the better their conversion into muons and the longer the distance they cover. Hence energetic neutrinos are more easy to detect.

Annihilation of neutralinos in the halo also gives monoenergetic photons (through the one-loop processes χχ → γγ, γZ) but also a continuous spectrum of photons through the decay of annihilation products (mostly from the decay of π0 produced in hadronization). Generally speaking, the annihilation flux is proportional to (i) the number of annihilations per second in a unit volume σannv ρ2(r)/M 2 for dark matter particles of mass M and annihilation cross-section σann, (ii) the spectrum dNi/dE of secondary particles of type i. Performing an integration over the line of sight, one obtains the observed flux of secondary particles of type i in a direction making an angle ψ with the direction of the galactic center:

where the galactocentric coordinate is expressed in terms of the coordinate s along the line of sight and the angle ψ as: r2 = s2 + R02 − 2sR0 cos ψ (R0 8 kpc is the solar distance from the galactic center).

There is thus a strong dependence on the density profile of dark matter ρ(r) which is poorly known. It is usually parametrized as

where R is a characteristic length and α, β, and γ are parameters. Table 5.5 gives their values obtained in typical models based on N -body simulations. Because of the strong model dependence of the astrophysics input in the computation of the

¯

flux, it proves useful to factorize it and to introduce J(∆Ω), the average of the function

Nonminimal models 117

5.6Nonminimal models

We end this chapter with a discussion of the simplest extension of the MSSM, obtained by adding a gauge singlet S and all its renormalizable interaction [129,295]. This model is called the next-to-minimal supersymmetric model (NMSSM, sometimes also referred to as the (M+1)SSM). We will see in Chapters 9 and 10 that singlets naturally appear in the context of grand unified models or string models.

The most general superpotential compatible with the symmetries is:

where W (2) and W (3) are the standard MSSM quadratic and cubic terms given in equations (5.1) and (5.2).

One of the motivations for the NMSSM is that it provides a framework to account for a dynamical origin of the µ term. Indeed, let us assume the existence of a discrete symmetry that prohibits any quadratic term in the superpotential: µ = µS = 0. A rationale behind this assumption is that the typical mass scale of the fundamental theory underlying this e ective low energy theory is very large: any dimensionful parameter should be fixed by this superheavy scale or vanishing. The low energy theory then has only dimensionless parameters such as κS , λS and the Yukawa couplings, and its mass scale appears only through symmetry breaking. Indeed, if S acquires a vacuum expectation value, then we obtain an e ective µ-term with

which generates an e ective Bµ term: Bµ = AλλS S .

The spectrum of the NMSSM is the one of the MSSM plus one scalar, one pseudoscalar, and one Weyl fermion (often referred to as the singlino). The scalar mixes with h0 and H0, the pseudoscalar with A0. The singlino mixes with the other neutralinos: the neutralino mass matrix Mn is now five-dimensional. In most of the parameter space, these mixings are small and the fields associated with the S superfield basically decouple. Otherwise, one may encounter large di erences with the phenomenology of the MSSM [135].

Further reading

•H. Haber and G. Kane, The search for supersymmetry: probing physics beyond the Standard Model, Physics Reports 117 (1985) 75.

•G. Jungman, M. Kamionkowski and K. Griest, Supersymmetric dark matter, Physics Reports 267 (1996) 195–373.

118 The minimal supersymmetric model

Exercises

Exercise 1 Explain in detail why the MSSM has 105 real parameters besides the 19 of the Standard Model, i.e. 124 real parameters in total.

Hints: See Chapter 12, Section 12.1.

Exercise 2 Show that the potential V ≡ VF + VD + VSB given by (5.10)-(5.12) has only charge-conserving minima: H1± = 0 = H2± .

Exercise 3 Compute the mass matrices corresponding to the scalar potential of the MSSM (Section 5.3.1) for the neutral scalars (Re Hi0) and pseudoscalars (Im Hi0). Deduce (5.28)–(5.30) and (5.32)–(5.33).

Exercise 4 Using the Lagrangians presented in Chapter 3, prove (5.43)–(5.44) where

Hints: Use equations (3.30) and (3.52) of Chapter 3. Note that, because λ1 and λ2 are

Majorana spinors and because charge conjugation implies hermitian conjugation, one

√

has λc± = λ . Also λ± = (λ1 ± iλ2)/ 2.

Exercise 5 Compute the spectrum of charginos and neutralinos in the limit case where

MW , MZ M1 < M2 < |µ|.

Exercise 6 Check the terms in the d-type squark mass matrix (5.54) or slepton mass matrix (5.57).

Exercise 7 We wish to study the evolution with time of the number N of neutralinos in the center of a celestial body under the influence of capture and annihilation.