Supersymmetry. Theory, Experiment, and Cosmology

290 An overview of string theory and string models

This shows that the string scale is of the order of the Planck scale (taking for αU the value at unification: 1/24). Moreover, if the compact manifold is isotropic, MC represents, to a first approximation, the scale where the theory becomes truly unified and is thus interpreted as the gauge coupling unification scale MU . It is clear in this context that (10.95) implies a large string scale MH .

We next turn to the strongly coupled SO(32) heterotic string which is equivalent to the weakly coupled type I open string. We now note the string scale MI = (α )1/2 and the string coupling λI . The e ective supergravity action reads

(10.96) where the gauge symmetry arises from p-branes (p = 3, 5, 7 or 9) [312]: the dilaton dependence is e−φ because the gauge term corresponds to an open string tree level amplitude (disk of Fig. 10.10c). One obtains after compactification

Thus, at least for p = 9 or 7, the string scale can be as low as the type I string coupling λI can be taken small. In the case of 9-branes, whose world-volume fills the 10-dimensional spacetime, we simply have MI = mP (2παU λI )1/2.

Finally, we may write the e ective 11-dimensional Hoˇrava–Witten [227, 228] supergravity action for M-theory compactified on the orbifold S1/Z2 or line segment

(10.100)

292 An overview of string theory and string models

Before we proceed, let us be more explicit on the way the string dilaton and the T modulus appear in four dimensions. We work here in the string frame and the corresponding fundamental mass scale is the string scale MS . In order to introduce the degree of freedom associated with the overall size of the compact manifold, we introduce the “breathing mode” eσ through the compact space part of the metric:

If we consider specifically the weakly coupled heterotic string, then the terms

The couplings of the s field are reminiscent of the Wess–Zumino terms which restore scale invariance through a dilaton field (see (A.268) of Appendix Appendix A).

Phenomenological aspects of superstring models 293

The sign of the kinetic term for s is not a problem because s is coupled to the spacetime curvature in the string frame metric gµν(4). If we go to the Einstein frame by performing a Weyl transformation on the four-dimensional metric:

One readily checks that the corresponding kinetic terms (see (6.19) of Chapter 6) yield the kinetic terms for s and t just found.

The Lagrangian (10.109) is invariant under the group SL(2, Z) of modular transformations

Chiral supermultiplets describing matter may be added: as we have seen earlier, they arise from the components of the 10-dimensional gauge supermultiplet with compact space indices. In the simplest compactification scheme [375], the K¨ahler potential is then generalized into

!

K(S, T ) = − ln S + S† − 3 ln T + T † − ΦiΦi† , (10.113)

i

where Φi are the matter fields. The modular transformations (10.111) must thus be complemented by

[We recognize in the structure thus unravelled the no-scale supergravity discussed in Section 6.12 of Chapter 6.]

14[The S field does not transform under T -duality because of the transformation law of the string dilaton (cf. (10.45)).]

Phenomenological aspects of superstring models 295

This equivalence is generalized to the corresponding supermultiplets: a linear supermultiplet is equivalent on-shell to a chiral supermultiplet. For example the linear supermultiplet with bosonic components s and bµν is equivalent to a chiral supermultiplet with scalar field S (see Section C.4 of Appendix C).

In the case of the heterotic string, it turns out that the antisymmetric tensor transforms nontrivially under Yang–Mills gauge transformations:

This extra term plays a central rˆole in the cancellation of gauge anomalies in 10 dimensions. Indeed, the hexagonal diagram of Fig. 10.12a, which is responsible of the anomaly in 10 dimensions (just as a triangular diagram is responsible of the anomaly in four dimensions, see Section A.6 of Appendix Appendix A), is cancelled by the diagram of Fig. 10.12b which represents the tree level exchange of a bMN field. The vertex on the left-hand side originates from the kinetic

the right-hand side is associated with the counterterm introduced by [205]

SGS = d10x M1···M10 bM1M2 Tr (FM3M4 FM5M6 FM7M8 FM9M10 ) . (10.125)

We note that the gauge completion of the 3-index field strength in (10.124) induces, after a Hodge duality transformation, a coupling of the pseudoscalar

aFor N indices between brackets, we average (factor 1/N !) the sum of the terms obtained by permutations of the N indices, with ±1 for even or odd perturbations.

296 An overview of string theory and string models

This is a Wess–Zumino term of the type encountered in anomaly cancellation

mechanisms. Since this is reminiscent of the way the axion field couples to

"

the gauge fields, the field a ≡ 2/3θ is called the string axion. The S field introduced earlier is S = s + ia and the action term (10.126) is simply a supersymmetric completion of the last term in (10.109).

bµν

Fig. 10.12 Hexagonal anomaly diagram (a) and antisymmetric tensor field exchange tree diagram (b).

In realistic compactifications, the situation is of course more involved than the one that we have presented. Let us illustrate this on the example of Calabi–Yau compactification. Matter fields appear in representations 27 and 27 of gauge group E6. As can be checked from Tables 10.3 and 10.4, there are as many T fields, i.e. radius or K¨ahler moduli, as there are 27 representations (their number is given by the Hodge number

h1,1).

There exists another class of moduli (noted U (i) in Table 10.4) which describes the complex structure of the compact manifold. Let us illustrate the di erence on the example of a torus. We may represent a torus in the complex plane by the square of unit length (0, 1, i, 1 + i) with opposite sides identified. Changing the radius of this torus amounts to multiply all its dimensions by a factor λ. On the other hand, the case where only the imaginary direction is dilated cannot be described in the complex plane by a holomorphic transformation z → λz: it corresponds to a change in the complex structure of the torus. In Calabi–Yau compactification, there are as many complex structure moduli as there are 27 (h2,1).

[One may show in this context that the same holomorphic function determines the superpotential of the matter fields and the K¨ahler potential of the moduli15. More

15Such a relation between superpotential and K¨ahler potential is reminiscent of a N = 2 supersymmetry which lies in the background of this type of compactification.

298 An overview of string theory and string models

therefore very similar to the case of a Higgs field φ in the adjoint representation of

the gauge group (U eiφ) with a nonzero vacuum expectation value18. Indeed, in

=

much the same way, E6 is broken to the gauge group that commutes with U . This is sometimes called the Hosotani mechanism [226].

Gauge symmetry is broken to the gauge group that commutes with U . Since we want the residual symmetry to contain at least SU (3)c ×SU (2)L ×U (1)Y , this imposes constraints on the form of U .

To derive them, it is easier to consider the SU (3)c × SU (3)L × SU (3)R subgroup of E6 considered in Section 9.5 of Chapter 9 and to write U as

where Uc,L,R are 3 × 3 matrices, elements of the gauge groups SU (3)c,L,R. To fully fix our notation, we have to say which SU (2) subgroup of SU (3) is SU (2)L. We will therefore suppose that the t3L element of the algebra SU (2)L is

Now, from the fact that all the generators of SU (3)c × SU (2)L commute with U , one easily infers that

where det Uc = η3 = 1. For simplicity we will take in the following η = 1.

But any matrix that commutes with [Y ] will commute with the two matrices (notation follows from Section 9.5 of Chapter 9):

Therefore the gauge symmetry is broken by topological breaking at most to SU (3)c × SU (2)L × U (1)YL × U (1)QR . One combination of the two U (1) charges yields the weak hypercharge y = yL + 2qR; the orthogonal combination is yη = qR − 2yL.

18Except that, since [U (γ)]n = 1, the eigenvalues of U are quantized.