Supersymmetry. Theory, Experiment, and Cosmology

Cosmological constant 359

A model [11] has been proposed which goes one step further: the dynamical component, a scalar field, is called k-essence and the model is based on the property observed in string models that scalar kinetic terms may have a nontrivial structure. Tracking occurs only in the radiation-dominated era; a new attractor solution where quintessence acts as a cosmological constant is activated by the onset of matter domination.

Models of dynamical supersymmetry breaking easily provide a model of the tracker field type just discussed [46]. Let us consider supersymmetric QCD with gauge group

¯g

SU (Nc) and Nf < Nc flavors, i.e. Nf quarks Qg (resp. antiquarks Q ), g = 1, . . . , Nf ,

¯

in the fundamental Nc (resp. antifundamental Nc) of SU (Nc) (as studied in Section 8.4.1 of Chapter 8). At the scale of dynamical symmetry breaking Λ where the

The dynamics is described by a superpotential which can be computed nonperturbatively using standard methods, see equation (8.53) of Chapter 8:

Such a superpotential has been used in the past but with the addition of a mass or interaction term (i.e. a positive power of M ) in order to stabilize the condensate. One does not wish to do that here if M is to be interpreted as a runaway quintessence component. For illustration purpose, let us consider a condensate diagonal in flavor space: Mf g ≡ φ2δfg . Then the potential for φ has the form (12.94), with α = 2(Nc + Nf )/(Nc − Nf ). Thus,

which clearly indicates that the meson condensate is a potential candidate for a quintessence component.

One may note that, in the tracker model, when φ reaches values of order mP , it satisfies the slow-roll conditions of an inflation model. The last possibility that I will discuss goes in this direction one step further. It is known under several names: deflation [344], kination [240], quintessential inflation [305]. It is based on the remark that, if a field φ is to provide a dynamical cosmological constant under the form of quintessence, it is a good candidate to account for an inflationary era where the evolution is dominated by the vacuum energy. In other words, are the quintessence component and the inflaton the same unique field?

In this kind of scenario, inflation (where the energy density of the Universe is dominated by the φ field potential energy) is followed by reheating where matterradiation is created by gravitational coupling during an era where the evolution is driven by the φ field kinetic energy (which decreases as a−6). Since matter-radiation energy density is decreasing more slowly, this turns into a radiation-dominated era until the φ energy density eventually emerges as in the quintessence scenarios described above.

360 The challenges of supersymmetry

Quintessential problems

However appealing, the quintessence idea is di cult to implement in the context of realistic models [68,256]. The main problem lies in the fact that the quintessence field must be extremely weakly coupled to ordinary matter. This problem can take several forms:

•We have assumed until now that the quintessence potential monotonically decreases to zero at infinity. In realistic cases, this is di cult to achieve because the couplings of the field to ordinary matter generate higher order corrections that are increasing with larger field values, unless forbidden by a symmetry argument.

For example, in the case of the potential (12.94), the generation of a correction

term λd m4P−dφd puts in jeopardy the slow-roll constraints on the quintessence field, unless very stringent constraints are imposed on the coupling λd. But one typically expects from supersymmetry breaking λd MSB4 /m4P where MSB is the supersymmetry breaking scale.

Similarly, because the vev of φ is of order mP , one must take into account the full supergravity corrections. One may then argue [49] that this could put in jeopardy the positive definiteness of the scalar potential, a key property of the quintessence potential. This may point towards models where W = 0 (but not its

derivatives, see (6.36) of Chapter 6) or to no-scale type models: in the latter case,

Chapter 6).

• The quintessence field must be very light. If we return to our example of supersymmetric QCD in (12.94), V (mP ) provides an order of magnitude for the mass-squared of the quintessence component:

using (12.98). This might argue for a pseudo-Goldstone boson nature of the scalar field that plays the rˆole of quintessence. This field must in any case be very weakly coupled to matter; otherwise its exchange would generate observable long range forces. E¨otv¨os-type experiments put very severe constraints on such couplings.

Again, for the case of supersymmetric QCD, higher order corrections to the K¨ahler potential of the type

will generate couplings of order 1 to the standard matter fields φi, φ†j since Q is of order mP .

•It is di cult to find a symmetry that would prevent any coupling of the form β(φ/mP )nF µν Fµν to the gauge field kinetic term. Since the quintessence behavior is associated with time-dependent values of the field of order mP , this would generate, in the absence of fine tuning, corrections of order one to the gauge coupling. But we have seen in Chapter 11 that the time dependence of the fine structure

Cosmological constant 361

constant for example is very strongly constrained: |α/α˙ | < 5 × 10−17yr−1. This

Pseudo-Goldstone boson

There exists a class of models [166] very close in spirit to the case of runaway quintessence: they correspond to a situation where a scalar field has not yet reached its stable groundstate and is still evolving in its potential.

More specifically, let us consider a potential of the form:

where M is the overall scale, f is the vacuum expectation value φ and the function v is expected to have coe cients of order one. If we want the potential energy of the field (assumed to be close to its vev f ) to give a substantial fraction of the energy density at present time, we must set

However, requiring that the evolution of the field φ around its minimum has been overdamped by the expansion of the Universe until recently imposes

Let us note that this is again one of the slow-roll conditions familiar to the inflation scenarios.

From (12.105) and (12.106), we conclude that f is of order mP (as the value of the field φ in runaway quintessence) and that M 10−3 eV (not surprisingly, this is the scale Λ typical of the cosmological constant, see (12.62)). As we have seen, the field φ must be very light: mφ h0 × 10−60mP h0 × 10−33 eV. Such a small value is only natural in the context of an approximate symmetry: the field φ is then a pseudoGoldstone boson. A typical example of such a field is provided by the string axion field. In this case, the potential simply reads:

All the preceding shows that there is extreme fine tuning in the couplings of the quintessence field to matter, unless they are forbidden by some symmetry. This is somewhat reminiscent of the fine tuning associated with the cosmological constant. In fact, the quintessence solution does not claim to solve the cosmological constant (vacuum energy) problem described above. If we take the example of a supersymmetric theory, the dynamical cosmological constant provided by the quintessence component clearly does not provide enough amount of supersymmetry breaking to account for

362 The challenges of supersymmetry

the mass di erence between scalars (sfermions) and fermions (quarks and leptons): at least 100 GeV. There must be other sources of supersymmetry breaking and one must fine tune the parameters of the theory in order not to generate a vacuum energy that would completely drown ρφ.

However, the quintessence solution shows that, once this fundamental problem is solved, one can find explicit fundamental models that e ectively provide the small amount of cosmological constant that seems required by experimental data.

Further reading

•Y. Nir, B Physics and CP violation, Proceedings of the 2005 Les Houches summer school, ed. by D. Kazakov and S. Lavignac.

•S. Weinberg, The cosmological constant, Rev. Mod. Phys. 61 (1989) 1.

Exercises

Exercise 1 Consider a 3 × 3 matrix Λ where the entries Λij are of order λnij , λ being a small parameter and nij ≥ 0. Defining

show that we have the following eigenvalue patterns:

p≥ 2q : O(1), ±O(λp)

p≤ 2q : O(1), O(λp), O(λq−p).

Hints: Consider the characteristic equation of the hermitian combination Λ†Λ ([43, 44]).

Exercise 2 Introducing the mixed U (1)X gravitational anomaly Cg = Tr x, express the charges a0, b0, c0, d0 and e0 defined in Table 12.2, in terms of the anomaly coe cients C1, C2, C3 in (12.23), Cg and a0 + b0. Show that one can redefine the x charge by combining it to the y charge in order to obtain a0 + b0 = a0 + c0 = 0 and 3(d0 + e0) = −(h1 + h2) in the absence of anomalies.

Hints: Cg = 3(6a0 + 3b0 + 3c0 + 2d0 + e0)

a0 = + 13 (a0 + c0) + 13 CD

Exercises 363

with CD = −Cg /3 + C1/6 + C2/2 + 5C3/9.

Exercise 3 Tracker field solution.

In the case where the evolution of the Universe is driven by the background (matter or radiation) energy, we search a scaling solution to the set of equations (12.83), (12.84) and (12.75), for the potential (12.94), i.e. V (φ) = λΛ4+α/φα, with α > 0. In other words, we are looking for a solution such that

1.One assumes that the background has equation of state pB = wB ρB . If this background energy determines the evolution of the Universe, how does a(t) evolves with time?

2.By plugging the scaling solution (12.109) into (12.75) and (12.83), express x and y in terms of α and wB .

3.Show that ρφ/ρB increases in time until it reaches the value aQ where it becomes of order 1 (and where our starting assumptions are no longer valid).

4.Compute φ/mP and ρφ/(λΛ4+α/mαP ) in terms of a/aQ, wB and α.

5.Compute the pressure pφ and deduce (12.97).

6.Show that, as long as φ φ(aQ), one reaches the slow-roll regime ε, η 1 (cf. equations (D.109) and (D.110) of Appendix D). Solve the equations of motion in this context (note that ρφ ρB then).

Hints:

1.a(t) t2/[3(1+wB )] (see the beginning of Section D.3 of Appendix D)

2.All terms in a given equation should have the same time dependence:

Appendix A

A review of the Standard Model and of various notions of quantum field theory

We review in this appendix the basics of the Standard Model of electroweak interactions. We will take this opportunity to recall some basic notions of quantum field theory, especially in connection with the concepts of symmetry, spontaneous symmetry breaking, and quantum anomalies.

which depends on a generic field Φ(x) and its spacetime derivatives. The equations of motion for the field Φ(x) are expressed by the Euler–Lagrange equation:

δL

∂µ = 0. (A.2) δ(∂µΦ)

We suppose that the action is invariant under a transformation Φ(x) → Φ (x ) which depends on a continuous parameter α (or a collection of such parameters). If this parameter has an infinitesimal value δα, one may write the transformation as:

We see that in the case of spacetime transformations, δΦ(x) depends on the spacetime derivatives of the field: δΦ(x) = δaµ∂µΦ(x).

and δΦ(x) = −iδθΦ(x): for an internal symmetry, δΦ(x) only depends on the field Φ(x) itself.

Whenever the action has a continuous symmetry, there is a procedure, known as the Niether procedure (see for example [318]), which allows us to construct a conserved

Note that, in the case where the transformation is internal, i.e. leaves spacetime

In this case, if the transformation is not a symmetry of the Lagrangian, the current is no longer conserved but one may still write a simple equation:

where we have assumed that the fields vanish at infinity (and thus the integral of a total spatial divergence vanishes). Since in the quantum theory dQ/dt = i[H, Q], we have

The transformation is said to be a symmetry of the Hamiltonian. We recall the standard equal-time commutation relations (assuming from now on that the field Φ(x) is a scalar field φ(x)):

366 A review of the Standard Model and of various notions of quantum field theory

where π is the canonical conjugate momentum

We note therefore that the time component of the Noether current (A.9) is simply expressed in terms of the conjugate momentum as J0 = πδφ/δα. In the case of an internal symmetry, where δφ is a function of φ only (and not of its spacetime derivatives), one may deduce an interesting commutation relation. Indeed, from (A.14), (A.15)

δφ

[J0(y), φ(x)]x0=y0 = [π(y), φ(x)]x0=y0 δα (y) = −iδ3(x − y) δαδφ (x).

Hence, using the fact that the charge is a constant of motion (i.e. time independent)

In other words, once we have the charge operator, we can reconstruct the infinitesimal transformation, and even the finite transformation. Indeed the unitary transformation U ≡ e−iαQ acts infinitesimally on φ as follows:

It thus corresponds to the finite transformation.

We now illustrate these notions on several examples.

A.1.2 Isospin symmetry

The Lagrangian describing a Dirac spinor field

has the well-known global phase invariance

These transformations form an abelian group: Tθ1 Tθ2 = Tθ1+θ2 = Tθ2 Tθ1 . The associated charge just counts the number of ψ fields (if ψ is the quark, it can be interpreted as the baryon number).

Such abelian global transformations have a nonabelian generalization. The simplest case is provided by the formalism of isospin which we now describe. From the point of

Symmetries 367

view of strong interactions, protons and neutrons have identical properties. This can be formalized by saying that physics should remain the same if one makes a unitary

rotation on these states. Let us thus write a doublet

ψp

ψN = (A.21)

ψn

which we call the nucleon. Physics should be invariant under the SU (2) transformation

where U is a unitary 2 × 2 matrix of determinant unity.

This is obviously reminiscent of the spin formalism. It is why such a transformation is called an isospin transformation. The two components of the doublet (A.21) have respectively I3 = +1/2 for the proton and I3 = −1/2 for the neutron. We may write the SU (2) matrix U as:

"

where |α| ≡ (α1)2 + (α2)2 + (α3)2. The Pauli matrices σa, a = 1, 2, 3, are explicitly

because the transformation ψN → U ψN is global and the unitary matrix commutes with the spacetime derivatives ∂µ. One may develop (A.27)

where we have written explicitly spinor indices (s, t) to distinguish them from the indices of the internal symmetry (p, n).

368 A review of the Standard Model and of various notions of quantum field theory

Thus, proton and neutron must have the same mass if isospin is to be a symmetry (see Exercise 1): this is the standard way of identifying a symmetry (such as isospin) in the spectrum of a theory.

A.1.3 Local abelian gauge transformation. Quantum electrodynamics

It is well known that, in the case of quantum electrodynamics, the phase invariance (A.20) is promoted to the level of a local transformation, i.e. the phase θ depends on the spacetime point considered

However, introducing a vector field Aµ(x) (the photon field) which transforms simultaneously as

It is called a covariant derivative to express the fact that it transforms as the field itself. Then obviously the generalized Lagrangian

is invariant under the local gauge transformation, i.e. (A.29) and (A.31).

In the case of quantum electrodynamics (QED), one sets g = e and q = −1 (the electron charge) and one introduces a dynamics for the photon field

where Fµν = ∂µAν − ∂ν Aµ is the field strength invariant under the gauge transformation (A.31). The complete Lagrangian is the QED Lagrangian

The electron–photon coupling is called minimal coupling since it has been obtained by a purely geometric argument: completing the spacetime derivative into the covariant derivative to ensure invariance under local gauge transformations. We note that this coupling is of the form Jµ(x)Aµ(x) where Jµ is the Noether current (cf. (A.9)).