Supersymmetry. Theory, Experiment, and Cosmology

Exercises 309

Further reading

•J. Polchinski, String theory, volume 1, Cambridge University Press.

•M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, volume 2, Cambridge monographs in mathematical physics, Cambridge University Press.

Exercises

Exercise 1 Compute, in a D-dimensional spacetime with d − 1 infinite spatial dimensions and D − d spatial dimensions of finite size L, the gravitational force between two masses m1 and m2 placed at a distance r much larger than L (cf. (10.28)).

Exercise 2 We generalize the analysis of the Kaluza–Klein modes of a scalar field performed in Section 10.2.1 to the case of several compact dimensions. We analyze the problem using this time the action (instead of the equation of motion).

Let us consider a massive real scalar field Φ in D = 4 + N dimensions. Its action in flat (4 + N )-dimensional spacetime reads

identification:

yk = yk + 2πR.

(a) Show that the field Φ can be decomposed as:

where {nk} is a set of N integers (one for each compact coordinate).

(b)How does the reality condition on Φ translate on the components φ{nk }?

(c)Using the decomposition (10.176), show that the action (10.175) can be written as:

S = d4x 12 ∂µφ{nk }∂µφ{nk } − 12 m2{nk }φ{nk }φ{nk } nk

and determine m2{nk }.

(d) Show that all mass levels except the lowest one are degenerate.

310 An overview of string theory and string models

Hints:

(b)Φ{−nk } = Φ{nk }.

(c)Use dyei(n+n)y/R = (2πR)1/2 δn+n,0, to prove

(d)There is only one state of mass m0({nk} = {0, . . . , 0}); all other mass levels are degenerate (for example {nk} = {1, 0, . . . , 0}, {0, 1, 0, . . . , 0}, {0, 0, 1, . . . , 0},...

correspond to the first such level, with degeneracy N ).

Exercise 3 Let us consider the torus T2 represented in Fig. 10.8b. We construct in this exercise the orbifold T2/Z3 which is obtained by identifying points which are transformed into one another by a rotation of 2π/3 around the origin.

(a)Draw the fixed points of the transformation.

(b)It is more convenient to use complex notations and to represent the torus of

where x0 is the coordinate of the fixed point considered. The commutation relations are of the form:

&'

11

Supersymmetry and the early Universe

11.1The ultimate laboratory

The introduction of supersymmetry allows us to introduce new physics with a fundamental mass scale which is very large. A certain number of observations plead for such a possibility, among which the value of gauge couplings compatible with unification and the scale of neutrino masses. More generally the fact that fundamental physics is described by gauge theories, which are known to dominate in the infrared (i.e. low energy) limit, tends to accommodate the idea that the scale of underlying physics is much higher.

There are indirect ways to test at low energy a theory with a large fundamental scale, through e ective interactions which scale like inverse powers of the fundamental scale. The best-known example is proton decay: an excessively rare event because the corresponding amplitude scales like the inverse of the square of the grand unification scale.

On the other hand, there is no hope to construct accelerators that would reach the desired mass scales to test these theories directly. One has to resort to natural cosmic accelerators or to the study of the early Universe , in which, at least in the hot BigBang scenario, temperatures have reached high enough values to excite superheavy degrees of freedom.

From this point of view, the recent successes of observational cosmology, which have turned it into a quantitative science, are a strong encouragement. It is fair however to say that we have, until now, tested quantitatively the evolution of our Universe up to nucleosynthesis (see Appendix D and Table D.1), which corresponds to an energy scale which is a fraction of MeV. The physics at higher scales thus merely provides boundary conditions for the observationally testable cosmological evolution. But this is su cient to test indirectly the evolution in the very early Universe. The most famous example is inflation: inflationary expansion takes place at a very early stage of the expansion of the Universe but provides fluctuations that develop in later stages of the evolution to show up in the cosmic microwave background.

Supersymmetric models provide a wealth of new fields and new mechanisms that may play a significant rˆole in cosmology. In particular, fundamental scalar fields, which are one of the building blocks of supersymmetry, have played an increasing rˆole in cosmology. They are nowadays called upon to solve all kinds of problems from the time variation of constants to dark energy or inflation. If the associated energy scale is

Cosmological relevance of moduli fields 315

For the case of a single scalar field, this gives limits on |α(φ0)|2 which depend on the range λ = 1/m of the interaction, from 10−3 in the (10 m, 10 km) range to 10−8 in the (104, 105) km range. As we have stressed in Chapter 10, the law of gravity is poorly determined below the mm region.

In the case of massless fields, the expression (11.7) does not modify the 1/r2 law; it simply leads to a rescaling of Newton’s constant. One has then to resort to the postNewtonian limit (terms smaller by a factor v2/c2 or Gm/(rc2)) to put constraints on such theories). The limit obtained on |α|2 is typically 10−3.

It is also possible to appeal to the cosmological evolution to account for the smallness of such coe cients in scalar-tensor theories. For example, [95] have found an attractor mechanism towards general relativity.

This mechanism exploits the stabilization of the dilaton-type scalar through its conformal coupling to matter. Indeed, assuming that this scalar field φ couples to matter with equation of state parameter wB through the action (11.3)–(11.4) (with γ(φ) = 1), then its equation of motion takes the form:

where φ = dφ/d ln a. This equation can be interpreted as the motion of a particle of velocity-dependent mass 2/(3 − φ 2) subject to a damping force (1 − wB )φ in an external force deriving from a potential ve (φ) = (1 − 3wB ) ln A2(φ). If this e ective potential has a minimum, the field quickly settles there.

An e cient mechanism of this type has been devised for the string dilaton by Damour and Polyakov [96] under the assumption that the dilaton dependent coupling functions terms entering the e ective string action have a common extremum (cf. the tree level action (10.93) of Chapter 10 where these functions are a universal e−2φ).

11.2.2Time variation of fundamental constants

Since couplings and scales are often given in terms of moduli fields, it is tempting to consider that, since some moduli may not have been stabilized, some of these quantities are still presently varying with time or have been doing so in the course of the cosmological evolution. This leads to the fascinating possibility that some of the fundamental constants of nature are time-dependent.

Such an idea was put forward by Dirac [119, 120]. According to him, a fundamental theory should not involve fundamental dimensionless parameters (i.e. dimensionless ratios of fundamental parameters) which are very large numbers. Such numbers should instead be considered as resulting from the evolution of the Universe and the corresponding dimensionless parameters be variables characterizing the evolving state of the Universe. Obviously this leads to some time-dependent fundamental parameters.

To illustrate Dirac’s approach, one may consider, besides dimensionless parameters such as the fine structure constant α = e2/(4π c) or the strong gauge coupling α3:

•the ratio of the electromagnetic to gravitational force between a proton and an electron e2/(GN mpme) 1039;

316 Supersymmetry and the early Universe

•the age of the Universe (of the order of H0−1) in microscopic time units (a typical atomic time scale is 3/(mee4), as can be checked on Bohr’s model of the hydrogen

atom) me4/( 3H0) 1034h−0 1.

The second ratio obviously evolves linearly with time (just replace the Hubble constant H0 by the Hubble parameter H t−1). The dependence of the first one with time would involve for example the time dependence of Newton’s constant.

In the context of supersymmetric theories where many of these dimensionless ratios are fixed by the values of moduli fields, one may expect some time dependence. For example, in heterotic string theory we have seen that the Planck scale (hence Newton’s constant) is given in terms of the string scale by the vacuum expectation value of the string dilaton. Similarly for the four-dimensional coupling, evaluated at the string scale (close to unification scale). If the dilaton is not properly stabilized at low energy, that is if the flat direction is not lifted or if its minimum remains too shallow, one thus expects a possible time dependence of the dimensionless ratio MP /MS or of the fine structure constant.

There are, however, some stringent bounds on the possible time evolution of funda-

region whereas the presence in the Oklo uranium in Gabon of a natural fission reactor which operated some 109 yr ago puts a limit [94] on |α/α˙ | in the 10−17 yr−1 region.

11.2.3 Moduli and gravitino problems

Because moduli are light and have gravitational interactions, they are long lived. There are then two potential dangers. If their lifetime is smaller than the age of our Universe, their decay might have released a very large amount of entropy in the Universe and diluted its content. If their lifetime is larger than the age of our Universe, they might presently still be oscillating around their minimum and the energy stored in these oscillations may overclose the Universe. One refers to these problems as the moduli problem or sometimes as the Polonyi problem since they were first discussed in the context of the Polonyi model described in Section 6.3.2 of Chapter 6 [90,133,199]. Taken at their face values, such constraints forbid any modulus field which is not superlight or very heavy. We now proceed to make these statements quantitative.

We first define two quantities which play a central rˆole in this discussion. A modulus field φ has typically gravitational interactions and thus its decay constant Γφ scales like κ2 m−P 2. Since the only available scale is the scalar field mass mφ, one infers

One thus deduces that the modulus will decay at present times if Γφ H0, that is if

its mass mφ is of order H0m2P 1/3 20 MeV.

The other relevant quantity is the initial value fφ of the scalar field with respect to its ground state value φ0. Presumably at very high energy (that is above the phase

Cosmological relevance of moduli fields 317

transition associated with dynamical supersymmetry breaking) where the flat direction is restored, one expects generically that fφ mP since this is the only scale available.

Let us first consider the case where mφ < 20 MeV, that is a field which has not yet decayed at present time. The equation of evolution for the field φ reads (compare with (D.103) of Appendix D)

where the last term accounts from particle creation due to the time variation of φ. If we assume that the Universe is initially radiation dominated, then H T 2/mP

˙

(see (D.62) of Appendix D). As long as H > mφ, the friction term 3Hφ dominates in the equation of motion and the field φ remains frozen at its initial value fφ. When H mφ, i.e. for TI (mφmP )1/2, the field φ starts oscillating around the minimum φ0 of its potential which we approximate as:

hand side term in equation (11.10) which simply reads, within our approximations,

We recover the standard result that coherent oscillations behave like nonrelativistic matter i.e.

Since the radiation energy density ρR(T ) behaves as T 4, ρφ/ρR increases as the temperature of the Universe decreases and one reaches a time where the energy of the scalar field oscillations dominates the energy density of the Universe. One should then make sure that ρφ(T0) < ρc. Using (11.14) and TI = (mφmP )1/2, one may write this condition as

where we have set fφ mP . Thus, if 10−26 eV < mφ < 20 MeV, there is too much energy stored in the φ field (which has not yet decayed in present times).

318 Supersymmetry and the early Universe

We next consider the case where mφ > 20 MeV, that is the scalar field has already decayed at present times. Decay occurs at a temperature TD when H(TD) Γφ, i.e.

where we have used (11.14), assuming that, at TD, the scalar field energy density dominates over radiation; ρφ(TI) m2φfφ2. At decay, all energy density is transferred into radiation. Thus, the reheating temperature TRH , that is the temperature of radiation issued from the decay, is given by the condition

where we have used (D.61) of Appendix D. Using (11.16) to express ρφ(TD), we obtain

This gives, using TI m1φ/2m1P/2 and ρφ(TI) m2φfφ2, σ fφ2/(mφmP ). With fφ mP , this gives a very large entropy release as long as the modulus mass remains much

smaller than the Planck scale.

This entropy release must necessarily precede nucleosynthesis since otherwise it would dilute away its e ects. This condition, namely TRH > 1 MeV, gives mφ > 10 TeV. Thus for 20 MeV < mφ < 10 TeV, the entropy release following the decay of the modulus field is too large to be consistent with present observations.

In the absence of other e ects, we are left with only superlight moduli fields (mφ < 10−26 eV) or heavy ones (mφ > 10 TeV).

This moduli problem may be discussed in parallel with a similar problem associated with gravitinos [360]. Indeed gravitinos are also fields with gravitational interactions and thus a decay constant given by (11.9) with mφ replaced by m3/2 . The formula (11.19) for entropy release remains valid with ρφ(TI) replaced by ρ3/2 (TI) = m3/2 n3/2 (TI). Using (D.74) and (D.77) of Appendix D which give the number density of

Again, if 20 MeV < m3/2 < 10 TeV, this gives too large an entropy release which dilutes away the products of nucleosynthesis.

The preferred value of the order of a TeV thus seems ruled out by such a constraint. This was also a theoretically preferred value for the moduli mass since m3/2 is the characteristic mass scale describing the e ects of supersymmetry breaking at low