Supersymmetry. Theory, Experiment, and Cosmology

Phenomenological aspects of superstring models 299

Thus, an extra U (1)η appears in the gauge symmetry pattern at the scale of compactification. It remains to be seen whether this extra U (1) remains unbroken down to the low energies where it could be observed.

Since U must commute with [YL] and [YR], we find

γ

UR = (10.133)

V

where V is 2 × 2 matrix and γ det V = 1. If Π1(K) = G is commutative, then all matrices Ug , g G must commute and V is diagonal. In this case

For example if G Z , βn = γn = δn = n = 1. = n

Let us turn to a specific example where G Z . To build U , it is easy to convince

= 3 R

oneself that only two cases arise: (I) all its entries are equal, (II) all its entries are di erent.

To see which gauge symmetry remains unbroken, we consider the generators in the adjoint representation of E6. Their decomposition under SU (3)c × SU (3)L × SU (3)R was given in Chapter 9, equation (9.114):

Clearly, all the generators in the adjoint representation of SU (3)c × SU (3)L × SU (3)R (the first three terms) are invariant ([T, U ] = 0). For the remaining ones, we have

300 An overview of string theory and string models

Among the generators of SU (3)c ×SU (3)L ×SU (3)R only (8, 1, 1), (1, 8, 1) and the two diagonal generators λ3, λ8 in (1, 1, 8) are invariant. Among the rest, only (3c, 3L, t3R =

−1/2) and (3c, 3L, t3R = +1/2) are invariant; for example U (3c, 3L, t3R = −1/2) = 1 × α × α−1(3c, 3L, t3R = −1/2). One can check that these 8 + 8 + 2 + 9 + 9 = 36

generators are the generators of SU (6)×U (1). This last example is interesting because it shows that although we chose to represent U under the maximal subgroup SU (3)3 of E6, it does not mean that the residual symmetry has to be a subgroup of SU (3)3.

We have seen that in the case of a compactification on a Calabi–Yau manifold matter fields fall into full representations 27 and 27 of E6, the net number of families being fixed by the Euler characteristics of the manifold. This is modified when E6 is broken by topological gauge symmetry breaking: matter fields fall in representations of the residual gauge symmetry group – the subgroup of E6 that commutes with U . Indeed we saw that matter fields survive compactification if they obey the twisted boundary conditions (10.62).

10.4.4Gauge coupling unification

There is no guaranteed unification of the gauge couplings in string theory. For example in type I models the couplings of gauge groups which correspond to di erent stacks of D-branes have no a priori reason to be equal. Because of the apparent success of the gauge coupling unification one might therefore be inclined to search for string models with gauge coupling unification.

One should stress at this point that string theories yield more possibilities of gauge coupling unification than standard grand unified theories. In fact the relation (10.107) may be generalized to

where ki is known as a Kaˇc–Moody level. Crudely speaking, in the case of a nonabelian symmetry, it represents the relative strength between the gravitational coupling and the trilinear self-coupling of the gauge bosons19; it is an integer. In the case of an abelian symmetry, ki is simply a real number necessary to account for the normalization fixed by (10.138). Thus the unification condition reads

The successes of coupling unification do impose to take k2 = k3.

19[This may be expressed more quantitatively through the operator product expansion between two world-sheet currents associated with the gauge symmetry:

where αi is the longest root. The double pole term corresponds to a gravitational coupling whereas the single pole term is associated with the trilinear self-coupling.]

Phenomenological aspects of superstring models 301

There are reasons to go beyond k = 1. Indeed, the choice k = 1 puts some restrictions on the possible representations. Such restrictions are welcome in the case of the Standard Model since they yield as only possible representations 1 and 2 for SU (2) and 1, 3 and 3 for SU (3). But, in the case of grand unified groups, they limit the representations (e.g. 1, 5 and 5, 10 and 10 for SU (5), 1, 27 and 27 for E6) in such a way as to prevent further breaking. In this case one should go to higher Kaˇc–Moody level or resort to partial unification.

Regarding the running of couplings at scales below M , it is important to note the following: since string theory amplitudes are finite, there are no ultraviolet divergences and thus no renormalization group type of running. The couplings that we have defined above are couplings of the e ective field theory which describes the interactions of the massless string modes; this e ective theory has ultraviolet divergences and thus running couplings. Thus, in a sense, the running of these couplings is associated with the infrared properties of the underlying string theory.

The e ective theory is obtained by integrating out the massive string modes. One thus expects some corresponding threshold contributions at the string scale:

where the second (resp. third) term on the right-hand side is due to massless (resp. massive) string modes running in loops. The massive string threshold corrections ∆i depend on the precise distribution of the massive string spectrum; they are generally moduli-dependent.

They have been computed in some explicit cases. We will take the example of fourdimensional string models which have N = 2 spacetime supersymmetry [121,247]: they are obtained through the toroidal compactification of six-dimensional string models with N = 1 spacetime supersymmetry. One obtains

where X is a numerical constant, T, U are the complex moduli that parametrize the two-dimensional compactification from six dimensions, and the Dedekind η function

2∞

is defined as: η(τ ) = eiπτ /12 n=1 1 − e2iπnτ . Under the modular transformation (10.111), η(iT ) transforms into (icT + d)1/2 η(iT ); threshold corrections are thus in-

which thus behaves like the radius squared. Hence, in the decompactification limit, such threshold corrections may become large (in the N = 2 case considered here, they

Phenomenological aspects of superstring models 303

Couplings of the pseudo-anomalous U (1)

We use the superfield formalism described in Appendix C to obtain the bosonic couplings of a pseudo-anomalous U (1)X gauge supermultiplet [116]. We note VX the gauge vector superfield and S = s + ia the dilaton chiral superfield.

We note the presence of a term which is a remnant of the Green–Schwarz counterterm (10.125) in four dimensions (by integration by parts µνρσbµν FXρσµνρσhµνρAσX ∂σaAσX ) as well as a mass term for the gauge fields: the pseudoanomalous gauge symmetry is broken by the gauge anomalies. We note that

Once we solve for DX , we obtain a scalar potential in the form of a D-term:

304 An overview of string theory and string models

where we have have restored the Planck scale and used (10.107) and (10.138): 1/s = kX gX2 . The presence of the field-dependent Fayet–Iliopoulos term usually

induces a nonzero vacuum expectation value for one or more field of x charge

"

opposite to δGS. The scale |ξX | determines the energy scale at which the U (1)X symmetry is broken. A string computation gives in the context of the heterotic string [13]

which is of the form Cg /kg as in (10.147), Cg being the mixed gravitational anomaly proportional to TrX. In this case, the scale

"

|ξX | is at most one order

of magnitude smaller than MS .

We finally note that the dilaton supermultiplet consists of the dilaton, dilatino, and antisymmetric tensor and, strictly speaking, fits into a linear supermultiplet (dual to the chiral supermultiplet S used above). The corresponding real linear superfield L is introduced in Section C.4 of Appendix Ca. The Green–Schwarz counterterm then takes in this formulation the following form

The set of relations (10.147) allowed L. [234] to relate the value of the Weinberg angle to the mixed anomaly coe cients of the anomalous U (1). Indeed, using (10.140), one finds:

We finally note that, in the context of type I or type IIB strings, pseudo-anomalous U (1) symmetries also appear (the corresponding axion field is not the string axion but may be provided by the imaginary part of moduli fields). The scale is no longer restricted to be of the order of the string scale and may be much smaller. Moreover, there may be several such symmetries in a given string model.

10.4.6Supersymmetry breaking

As for any supersymmetric scenario, supersymmetry breaking is a key issue in string models. A favoured mechanism is gaugino condensation in a hidden sector which we have discussed in some details in subsection 7.4.2. But gaugino condensation is formulated in the context of the e ective field theory and is not strictly speaking of a stringy nature (although it is largely motivated by string models).

Phenomenological aspects of superstring models 305

Other mechanisms are more directly relevant to the string context, in the sense that identifying them within the experimental spectrum of supersymmetric particles would be a clear sign of some of the more typical aspects of string models. We will discuss briefly here the Scherk–Schwarz mechanism, the role of the pseudo-anomalous U (1) and fluxes.

Scherk–Schwarz mechanism

This mechanism, proposed by Scherk and Schwarz [330], makes use of the symmetries of the higher dimensional theory to break supersymmetry through the boundary conditions imposed on fields in the compact dimensions. These symmetries are symmetries that do not commute with supersymmetry, e.g. R-symmetries or fermion number (−1)F .

Let us illustrate this on the example of a single compact dimension of radius R: 0 ≤ y ≤ 2πR. We consider bosonic and fermionic fields Φi(xµ, y) which obey the following boundary conditions:

where the matrix U is associated with a symmetry that does not commute with supersymmetry (and is thus di erent for bosons and fermions). The corresponding expansion is thus (compare with (10.38))

n Z

Scherk and Schwarz take U = exp (ωM y/R) where M is an anti-Hermitian matrix. Kinetic terms in the compact direction generate fermion-boson splittings of order ω/R.

Let us apply this to the case of one massless hypermultiplet, introduced in Section 4.4.2 of Chapter 4 [151, 152]. As can be seen from the Lagrangian (4.43)

there is a R-symmetry that leaves the fermion field Ψ invariant and rotates the two complex scalar fields. We thus choose M = iσ2. The decomposition (10.158) then gives the following four-dimensional Lagrangian

mechanism generates soft masses of order ω/R.

Supergravity versions of this mechanism may be worked out using for example the string dilaton and K¨ahler moduli (see for example [126]). They show that, although in the strict sense supersymmetry is broken explicitly, Scherk–Schwarz breaking is in many ways similar to an F -type spontaneous breaking.

306 An overview of string theory and string models

Pseudo-anomalous U(1)

Pseudo-anomalous U (1)X symmetries may play a significant rˆole in supersymmetry breaking. Since they have mixed anomalies with the other gauge symmetries –those of the Standard Model as well as of the hidden sector–, it is not surprising that the whole issue of supersymmetry breaking through gaugino condensation is modifed in such models. Moreover, because in the Green–Schwarz mechanism all the mixed anomalies are non-vanishing and proportional to one another, there must exist fields charged under U (1)X in the observable as well as in the hidden sector. The U (1)X gauge symmetry thus serves as a messenger interaction competitive with the gravitational interaction.

We will give an explicit example to stress the modifications that the presence of such an anomalous U (1)X symmetry is bringing to the scenario of a dynamical supersymmetry breaking through gaugino condensation.

The model that we consider [33] is an extension of the model of Section 8.4.1: supersymmetric SU (Nc) with Nf < Nc flavors. Quarks Qi in the fundamental of

¯i

SU (Nc) have U (1)X charge q and antiquarks Q in the antifundamental of SU (Nc) have charge q˜.

Since we want to avoid SU (Nc) breaking in the U (1)X flat direction, we require that the charges q and q˜ are positive. We then need at least one field of negative charge in order to cancel the D-term. For simplicity we introduce a single field φ of U (1)X charge normalized to −1.

We write the classical Lagrangian compatible with the symmetries in the superfield language of Appendix C. The reader interested only in the phenomenological results may go directly to equation (10.171).

[We write L = Lkin + Lcouplings, where we assume a flat K¨ahler potential for the

where kX and kN are the respective Kaˇc–Moody levels of U (1)X and SU (Nc).

The mixed anomaly U (1)X [SU (Nc)]2 which fixes, through (10.147), all the mixed

We thus require q + q˜ > 0, which in turn justifies the presence of the superpotential term (10.162).

The two scales present in the problem are:

• the scale at which the anomalous U (1)X symmetry is broken which is set by