Higher derivatives and brane-localised kinetic terms in gauge theories on orbifolds

We perform a detailed analysis of one-loop corrections to the self-energy of the (off-shell) gauge bosons in six-dimensional N=1 supersymmetric gauge theories on orbifolds. After discussing the Abelian case in the standard Feynman diagram approach, we extend the analysis to the non-Abelian case by employing the method of an orbifold-compatible one-loop effective action for a classical background gauge field. We find that bulk higher derivative and brane-localised gauge kinetic terms are required to cancel one-loop divergences of the gauge boson self energy. After their renormalisation we study the momentum dependence of both the higher derivative coupling h(k^2) and the {\it effective} gauge coupling g_eff(k^2). For momenta smaller than the compactification scales, we obtain the 4D logarithmic running of g_eff(k^2), with suppressed power-like corrections, while the higher derivative coupling is constant. We present in detail the threshold corrections to the low energy gauge coupling, due to the massive bulk modes. At momentum scales above the compactification scales, the higher derivative operator becomes important and leads to a power-like running of g_eff(k^2) with respect to the momentum scale. The coefficient of this running is at all scales equal to the renormalised coupling of the higher derivative operator which ensures the quantum consistency of the model. We discuss the relation to the similar one-loop correction in the heterotic string, to show that the higher derivative operators are relevant in that case too, since the field theory limit of the one-loop string correction does not commute with the infrared regularisation of the (on-shell) string result.Comment: 1+45 pages, 2 figures, JHEP style file, version to be published in JHE

Higher Derivative Operators as Counterterms in Orbifold Compactifications

In the context of 5D N=1 supersymmetric models compactified on S_1/Z_2 or S_1/(Z_2 x Z_2') orbifolds and with brane-localised superpotential, higher derivative operators are generated radiatively as one-loop counterterms to the mass of the (brane or zero mode of the bulk) scalar field. It is shown that the presence of such operators which are brane-localised is not related to the mechanism of supersymmetry breaking considered (F-term, discrete or continuous Scherk-Schwarz breaking) and initial supersymmetry does not protect against the dynamical generation of such operators. Since in many realistic models the scalar field is commonly regarded as the Higgs field, and the higher derivative operators seem a generic presence in orbifold compactifications, we stress the importance of these operators for solving the hierarchy problem.Comment: Contribution to the Conference "Supersymmetry 2005", Durham; 13 pages, LaTe

On gauge unification in Type I/I' models

We discuss whether the (MSSM) unification of gauge couplings can be accommodated in string theories with a low (TeV) string scale. This requires either power law running of the couplings or logarithmic running extremely far above the string scale. In both cases it is difficult to arrange for the multiplet structure to give the MSSM result. For the case of power law running there is also enhanced sensitivity to the spectrum at the unification scale. For the case of logarithmic running there is a fine tuning problem associated with the light closed string Kaluza Klein spectrum which requires gauge mediated supersymmetry breaking on the ``visible'' brane with a dangerously low scale of supersymmetry breaking. Evading these problems in low string scale models requires a departure from the MSSM structure, which would imply that the success of gauge unification in the MSSM is just an accident.Comment: 10 pages, LaTeX, 2 figures; minor change

Quantum scale invariance in gauge theories and applications to muon production

We discuss quantum scale invariance in (scale invariant) gauge theories with both ultraviolet (UV) and infrared (IR) divergences. Firstly, their BRST invariance is checked in two apparently unrelated approaches using a scale invariant regularisation (SIR). These approaches are then shown to be equivalent. Secondly, for the Abelian case we discuss both UV and IR quantum corrections present in such theories. We present the Feynman rules in a form suitable for offshell Green functions calculations, together with their one-loop renormalisation. This information is then used for the muon production cross section at one-loop in a quantum scale invariant theory. Such a theory contains not only new UV poles but also IR poles. While the UV poles bring new quantum corrections (in the form of counterterms), finite or divergent, that we compute, it is shown that the IR poles do not bring new physics. The IR quantum corrections, both finite and divergent, cancel out similarly to the way the IR poles themselves cancel in the traditional approach to IR divergences (in the cross section, after summing over virtual and real corrections). Hence, the evanescent interactions induced by the scale-invariant analytical continuation of the SIR scheme do not affect IR physics, as illustrated at one-loop for the muon production ($e^+ e^- \to \mu^+\mu^-$) cross section.Comment: 20 page

Quadratic Divergences in Kaluza-Klein Theories

We investigate the so-called ``Kaluza-Klein regularisation'' procedure in supersymmetric extensions of the standard model with additional compact dimensions and Scherk-Schwarz mechanism for supersymmetry breaking. This procedure uses a specific mathematical manipulation to obtain a finite result for the scalar potential. By performing the full calculation, we show that the finiteness of this result is not only a consequence of the underlying supersymmetry, but also the result of an implicit fine-tuning of the coefficients of the terms that control the ultraviolet behaviour. The finiteness of the Higgs mass at one-loop level seems therefore to be an artefact of the regularisation scheme, and quadratic divergences are expected to reappear in higher orders of perturbation theory.Comment: 10 pages, LaTe

Higher Derivative Operators from Scherk-Schwarz Supersymmetry Breaking on T^2/Z_2

In orbifold compactifications on T^2/Z_2 with Scherk-Schwarz supersymmetry breaking, it is shown that (brane-localised) superpotential interactions and (bulk) gauge interactions generate at one-loop higher derivative counterterms to the mass of the brane (or zero-mode of the bulk) scalar field. These brane-localised operators are generated by integrating out the bulk modes of the initial theory which, although supersymmetric, is nevertheless non-renormalisable. It is argued that such operators, of non-perturbative origin and not protected by non-renormalisation theorems, are generic in orbifold compactifications and play a crucial role in the UV behaviour of the two-point Green function of the scalar field self-energy. Their presence in the action with unknown coefficients prevents one from making predictions about physics at (momentum) scales close to/above the compactification scale(s). Our results extend to the case of two dimensional orbifolds, previous findings for S^1/Z_2 and S^1/(Z_2 x Z_2') compactifications where brane-localised higher derivative operators are also dynamically generated at loop level, regardless of the details of the supersymmetry breaking mechanism. We stress the importance of these operators for the hierarchy and the cosmological constant problems in compactified theories.Comment: 23 pages, LaTeX, one figure, published version in JHE

Higher Derivative Operators from Transmission of Supersymmetry Breaking on S_1/Z_2

We discuss the role that higher derivative operators play in field theory orbifold compactifications on S_1/Z_2 with local and non-local (Scherk-Schwarz) breaking of supersymmetry. Integrating out the bulk fields generates brane-localised higher derivative counterterms to the mass of the brane (or zero-mode of the bulk) scalar field, identified with the Higgs field in many realistic models. Both Yukawa and gauge interactions are considered and the one-loop results found can be used to study the ``running'' of the scalar field mass with respect to the momentum scale in 5D orbifolds. In particular this allows the study of the behaviour of the mass under UV scaling of the momentum. The relation between supersymmetry breaking and the presence of higher derivative counterterms to the mass of the scalar field is investigated. This shows that, regardless of the breaking mechanism, (initial) supersymmetry cannot, in general, prevent the emergence of such operators. Some implications for phenomenology of the higher derivative operators are also presented.Comment: 29 pages, LaTeX. Added Section 4 ("Phenomenological implications: living with ghosts?") and Appendix

Unification through extra dimensions at two loops

The presence of an extra dimension of size R\equiv M_c^{-1} introduces corrections of order (\mu/M_c)\alpha to the gauge and Yukawa couplings and accelerates their running at scales \mu larger than M_c. This could result in a grand unification scale M_X\approx 20 M_c. We study the corrections at the two-loop level. We find corrections of order (\mu/M_c)\alpha^2 for the gauge couplings and of order (\mu/M_c)^2\alpha^2 for the Yukawa couplings. Therefore, in the Yukawa sector one and two-loop contributions can be of the same order below M_X. We show that in the usual scenarios the dominant gauge and Yukawa couplings are decreasing functions of the scale, in such a way that (\mu/M_c)\alpha becomes approximately constant and two-loop contributions introduce just a 30% correction which does not increase with the scale.Comment: 14 pages, added references, corrected typo

TeV-Scale Z' Bosons from D-branes

Generic D-brane string models of particle physics predict the existence of extra U(1) gauge symmetries beyond hypercharge. These symmetries are not of the E_6 class but rather include the gauging of Baryon and Lepton numbers as well as certain Peccei-Quinn-like symmetries. Some of the U(1)'s have triangle anomalies, but they are cancelled by a Green-Schwarz mechanism. The corresponding gauge bosons typically acquire a mass of order the string scale M_S by combining with two-index antisymmetric fields coming from the closed string sector of the theory. We argue that in string models with a low string scale M_S proportional to 1-10 TeV, the presence of these generic U(1)'s may be amenable to experimental test. Present constraints from electroweak precision data already set important bounds on the mass of these extra gauge bosons. In particular, for large classes of models, rho-parameter constraints imply M_S >= 1.5 TeV. In the present scheme some fraction of the experimentally measured Z^0 mass would be due not to the Higgs mechanism, but rather to the mixing with these closed string fields. We give explicit formulae for recently constructed classes of intersecting D6- and D5-brane models yielding the Standard Model (SM) fermion spectrum.Comment: 46 pages, LaTeX, JHEP.cls, 21 Figures. minor correction

Adjoint bulk scalars and supersymmetric unification in the presence of extra dimensions

There are several advantages of introducing adjoint superfields at intermediate energies around $M=10^{13}$ GeV. Such as (i) gauge couplings still unify (ii) neutrino masses and mixings are produced (iii) primordial lepton asymmetry can be produced. We point out that if adjoint scalars have bulk excitations along with gauge bosons whereas fermions and the doublet scalar live on boundary then N=2 supersymmetric beta functions $\tilde{b_i}$ vanish. Thus even if extra dimensions open up at an intermediate scale $\mu_0$ and all N=2 Yang-Mills fields as well as N=2 matter fields in the adjoint representation propagate in the bulk, still gauge couplings renormalize beyond $\mu_0$ just like they do in 4-dimensions with adjoint scalars. Consequently unification is achieved in the presence to extra dimensions, mass scales are determined uniquely via Renormalization Group Equations(RGE) and unification scale remains high enough to suppress proton decay. This scenario can be falsified if we get signatures of extra dimensions at low energy.Comment: New references added. This version will appear in Phys. Rev.