Generalized Green-Schwarz Mechanism Research Articles

The Green-Schwarz mechanism and geometric anomaly relations in 2d (0,2) F-theory vacua

We study the structure of gauge and gravitational anomalies in 2d N = (0, 2) theories obtained by compactification of F-theory on elliptically fibered Calabi-Yau 5-folds. Abelian gauge anomalies, induced at 1-loop in perturbation theory, are cancelled by a generalized Green-Schwarz mechanism operating at the level of chiral scalar fields in the 2d supergravity theory. We derive closed expressions for the gravitational and the non-abelian and abelian gauge anomalies including the Green-Schwarz counterterms. These expressions involve topological invariants of the underlying elliptic fibration and the gauge background thereon. Cancellation of anomalies in the effective theory predicts intricate topological identities which must hold on every elliptically fibered Calabi-Yau 5-fold. We verify these relations in a non-trivial example, but their proof from a purely mathematical perspective remains as an interesting open problem. Some of the identities we find on elliptic 5-folds are related in an intriguing way to previously studied topological identities governing the structure of anomalies in 6d N = (1, 0) and 4d N = 1 theories obtained from F-theory.

Retrofitted natural supersymmetry from a U(1)

We propose that a single, spontaneously broken, U(1) gauge symmetry may be responsible for suppressing both the first two generation Yukawa couplings, and also, in a correlated manner, parameters in the dynamical supersymmetry (SUSY) breaking sector by the mechanism of retrofitting. In the dynamical SUSY breaking sector, these small parameters are typically required in order to introduce R-symmetry breaking in a controlled manner and obtain phenomenologically viable meta-stable vacua. The heavy U(1) multiplet mediates a dominant contribution to the first two generation MSSM sfermion soft masses, while gauge mediation provides a parametrically suppressed soft term contribution to the stop and most other states, so realising a natural SUSY spectrum in a fashion consistent with SUSY unification. In explicit models the spectra obtained can be such that current LHC limits are evaded, and predictions of flavour changing processes are consistent with observation. We examine both implementations with low scale mediation, and string-motivated examples where the U(1) is anomalous before the inclusion of a generalised Green-Schwarz mechanism.

Anomaly cancellation and abelian gauge symmetries in F-theory

We study 4D F-theory compactifications on singular Calabi-Yau fourfolds with fluxes. The resulting $\mathcal{N}=1$ effective theories can admit non-Abelian and U(1) gauge groups as well as charged chiral matter. In these setups we analyze anomaly cancellation and the generalized Green-Schwarz mechanism. This requires the study of 3D $\mathcal{N}=2$ theories obtained by a circle compactification and their M-theory duals. Reducing M- theory on resolved Calabi-Yau fourfolds corresponds to considering the effective theory on the 3D Coulomb branch in which certain massive states are integrated out. Both 4D gaugings and 3D one-loop corrections of these massive states induce Chern-Simons terms. All 4D anomalies are captured by the one-loop terms. The ones corresponding to the mixed gauge-gravitational anomalies depend on the Kaluza-Klein vector and are induced by integrating out Kaluza-Klein modes of the U(1) charged matter. In M-theory all Chern-Simons terms classically arise from G 4-flux. We find that F-theory fluxes implement automatically the 4D Green-Schwarz mechanism if non-trivial geometric relations for the resolved Calabi-Yau fourfold are satisfied. We confirm these relations in various explicit examples and elucidate the general construction of U(1) symmetries in F-theory. We also compare anomaly cancellation in F-theory with its analog in Type IIB orientifold setups.

The physics of a new gauge boson in a Stueckelberg extension of the two-Higgs-doublet model

String theory constructions using D-brane physics offer a framework where ingredients like extra abelian factors in the gauge group, more than one Higgs doublet and a generalized Green-Schwarz mechanism appear at the same time. Motivated by works towards the direction of obtaining the Standard Model in orientifold constructions, we study in the present work a Stueckelberg extension of the two-Higgs-doublet model. The distinctive features of our model are i) a sharp decay width for the heavy gauge boson, and ii) a charged Higgs boson having two main decay channels at tree level with equal branching ratios.

The generalized Green-Schwarz mechanism for type IIB orientifolds with D3- and D7-branes

In this paper, we work out in detail the tadpole cancellation conditions as well as the generalized Green-Schwarz mechanism for type IIB orientifold compactifications with D3- and D7-branes. We find that not only the well-known D3- and D7-tadpole conditions have to be satisfied, but in general also the vanishing of the induced D5-brane charges leads to a non-trivial constraint. In fact, for the case $h^{1,1}_{-} \neq 0$ the latter condition is important for the cancellation of chiral anomalies. We also extend our analysis by including D9- as well as D5-branes and determine the rules for computing the chiral spectrum of the combined system.

MSSM via Pati–Salam from intersecting branes on T6/(Z2×Z2′)

We construct an MSSM-like model via Pati–Salam from intersecting D-branes in type IIA theory on the Z 2 × Z 2 ′ orientifold where the D-branes wrap rigid 3-cycles. Because the 3-cycles are rigid, there are no extra massless fields in the adjoint representation, arising as open-string moduli. The presence of these unobserved fields would create difficulties with asymptotic freedom as well as the prediction of gauge unification. The model constructed has four generations of MSSM matter plus right-handed neutrinos, as well as additional vector-like representations. In addition, we find that all of the required Yukawa couplings are allowed by global symmetries which arise from U ( 1 ) 's which become massive via a generalized Green–Schwarz mechanism. Furthermore, we find that the tree-level gauge couplings are unified at the string scale.

Flipped [formula omitted] from D-branes with type IIB fluxes

We construct flipped SU(5) GUT models as type IIB flux vacua on Z2×Z2 orientifolds. Turning on supergravity self-dual NSNS and RR three-form fluxes fixes the toroidal complex structure moduli and the dilaton. We give a specific example of a three-generation flipped SU(5) model with a complete Higgs sector where supersymmetry is softly broken by the supergravity fluxes in the closed string sector. All of the required Yukawa couplings are present if global U(1) factors resulting from a generalized Green–Schwarz mechanism are broken spontaneously or by world-sheet instantons. In addition, the model contains extra chiral and vector-like matter, potentially of mass O(Mstring) via trilinear superpotential couplings.

Standard model compactifications of IIA [formula omitted] orientifolds from intersecting D6-branes

We discuss the construction of chiral four-dimensional T 6 / ( Z 3 × Z 3 ) orientifold compactifications of IIA theory, using D6-branes intersecting at angles and not aligned with the orientifold O6 planes. Cancellation of mixed U ( 1 ) anomalies requires the presence of a generalized Green–Schwarz mechanism mediated by RR partners of closed string untwisted moduli. In this respect we describe the appearance of three quark and lepton family SU ( 3 ) C × SU ( 2 ) L × U ( 1 ) Y non-supersymmetric orientifold models with only the massless spectrum of the SM at low energy that can have either no exotics present and three families of ν R 's ( A ′ -model class) or the massless fermion spectrum of the N = 1 SM with a small number of massive non-chiral colour exotics and in one case with extra families of ν R 's ( B ′ -model class). Moreover we discuss the construction of SU ( 5 ) , flipped SU ( 5 ) and Pati–Salam SU ( 4 ) c × SU ( 2 ) L × SU ( 2 ) R GUTS—the latter also derived from adjoint breaking—with only the SM at low energy. Some phenomenological features of these models are also briefly discussed. All models are constructed with the Weinberg angle to be 3 / 8 at the string scale.

A K-theory anomaly free supersymmetric flipped [formula omitted] model from intersecting branes

We construct an N = 1 supersymmetric three-family flipped SU ( 5 ) model from type IIA orientifolds on T 6 / ( Z 2 × Z 2 ) with D6-branes intersecting at general angles. The model is constrained by the requirement that Ramond–Ramond tadpoles cancel, the supersymmetry conditions, and that the gauge boson coupled to the U ( 1 ) X factor does not get a string-scale mass via a generalised Green–Schwarz mechanism. The model is further constrained by requiring cancellation of K-theory charges. The spectrum contains a complete grand unified and electroweak Higgs sector, however the latter in a non-minimal number of copies. In addition, it contains extra matter both in bi-fundamental and vector-like representations as well as two copies of matter in the symmetric representation of SU ( 5 ) .

Supersymmetric (non-)abelian bundles in the type-I and SO(32) heterotic string

We discuss perturbative four-dimensional compactifications of both the SO(32) heterotic and the Type I string on smooth Calabi-Yau manifolds endowed with general non-abelian and abelian bundles. We analyse the generalized Green-Schwarz mechanism for multiple anomalous U(1) factors and derive the generically non-universal one-loop threshold corrections to the gauge kinetic function as well as the one-loop corrected Fayet-Iliopoulos terms. The latter can be interpreted as a stringy one-loop correction to the Donaldson-Uhlenbeck-Yau condition. Applying S-duality, for the Type I string we obtain the perturbative Pi-stability condition for non-abelian bundles on curved spaces. Some simple examples are given, and we qualitatively discuss some generic phenomenological aspects of this kind of string vacua. In particular, we point out that in principle an intermediate string scale scenario with TeV scale large extra dimensions might be possible for the heterotic string.

Loop-corrected compactifications of the heterotic string with line bundles

We consider the E8 x E8 heterotic string theory compactified on Calabi-Yau manifolds with bundles containing abelian factors in their structure group. Generic low energy consequences such as the generalised Green-Schwarz mechanism for the multiple anomalous abelian gauge groups are studied. We also compute the holomorphic gauge couplings and induced Fayet-Iliopoulos terms up to one-loop order, where the latter are interpreted as stringy one-loop corrections to the Donaldson-Uhlenbeck-Yau condition. Such models generically have frozen combinations of Kaehler and dilaton moduli. We study concrete bundles with structure group SU(N) x U(1)^M yielding quasi-realistic gauge groups with chiral matter given by certain bundle cohomology classes. We also provide a number of explicit tadpole free examples of bundles defined by exact sequences of sums of line bundles over complete intersection Calabi-Yau spaces. This includes one example with precisely the Standard Model gauge symmetry.

Witten-Veneziano from Green-Schwarz

We consider the U(1) problem within the AdS/CFT framework. We explain how the Witten-Veneziano formula for the eta' mass is related to a generalized Green-Schwarz mechanism. The closed string mode, that cancels the anomaly of the gauged U(1) axial symmetry, is identified with the eta' meson. In a particular set-up of D3-branes on a C3/(Z3xZ3) orbifold singularity, the eta' meson is a twisted-sector R-R field.

Intersecting D5-brane models with massive vector-like leptons

We construct eight-stack intersecting D5-brane models, with an orbifold transverse space, that yield the (non-supersymmetric) standard model up to vector-like leptons. The matter includes right-chiral neutrinos and the models have the renormalisable Yukawa couplings to tachyonic Higgs doublets needed to generate mass terms for {\it all} matter, including the vector-like leptons. The models are constrained by the requirement that twisted tadpoles cancel, that the gauge boson coupled to the weak hypercharge $U(1)_Y$ does not get a string-scale mass via a generalised Green-Schwarz mechanism, and that there are no surviving, unwanted gauged U(1) symmetries coupled to matter. Gauge coupling constant ratios close to those measured are easily obtained for reasonable values of the parameters, consistently with having the string scale close to the electroweak scale, as required to avoid the hierarchy problem. Unwanted (colour-triplet, charged-singlet, and neutral-singlet) scalar tachyons can be removed by a suitable choice of the parameters.

Standard-like models from intersecting D5-branes

We construct intersecting D5-brane orbifold models that yield the (non-supersymmetric) standard model up to vector-like matter and charged-singlet scalars. The models are constrained by the requirement that twisted tadpoles cancel, and that the gauge boson coupled to the weak hypercharge U(1)Y does not get a string-scale mass via a generalised Green–Schwarz mechanism. Gauge coupling constant ratios close to those measured are easily obtained for reasonable values of the parameters, consistently with having the string scale close to the electroweak scale, as required to avoid the hierarchy problem.

Deformed Intersecting D6-Brane GUTS I

By employing D6-branes intersecting at angles in $D = 4$ type IIA strings, we construct {\em four stack string GUT models} (PS-I class), that contain at low energy {\em exactly the three generation Standard model} with no extra matter and/or extra gauge group factors. These classes of models are based on the Pati-Salam (PS) gauge group $SU(4)_C \times SU(2)_L \times SU(2)_R$. They represent deformations around the quark and lepton basic intersection number structure. The models possess the same phenomenological characteristics of some recently discussed examples (PS-A class) of four stack PS GUTS. Namely, there are no colour triplet couplings to mediate proton decay and proton is stable as baryon number is a gauged symmetry. Neutrinos get masses of the correct sizes. Also the mass relation $m_e = m_d$ at the GUT scale is recovered. Moreover, we clarify the novel role of {\em extra} branes, the latter having non-trivial intersection numbers with quarks and leptons and creating scalar singlets, needed for the satisfaction of RR tadpole cancellation conditions. The presence of N=1 supersymmetry in sectors involving the {\em extra} branes is equivalent to the, model dependent, orthogonality conditions of the U(1)'s surviving massless the generalized Green-Schwarz mechanism. The use of {\em extra} branes creates mass couplings that predict the appearance of light fermion doublets up to the scale of electroweak scale symmetry breaking.

New standard-like models from intersecting D4-branes

We construct a new set of intersecting D4-brane models that yield the (non-supersymmetric) standard model up to vector-like matter and, in some cases, extra U(1) factors in the gauge group. The models are constrained by the requirement that twisted tadpoles cancel, and that the gauge boson coupled to the weak hypercharge U(1)Y does not get a string-scale mass via a generalised Green–Schwarz mechanism. We find six-stack models that contain all of the Yukawa couplings to the tachyonic Higgs doublets that are needed to generate mass terms for the fermions at renormalisable level, but which have charged-singlet scalar tachyons and an unwanted extra U(1) gauge symmetry after spontaneous symmetry breaking. There is also a six-stack model without any unwanted gauged U(1) symmetries, but which only has the Yukawa couplings to generate masses for the u quarks and charged leptons. A particular eight-stack model is free of charged-singlet tachyons and has gauge coupling strengths whose ratios at the string scale are close to those measured at the electroweak scale, consistent with the string scale being at most a few TeV.

New Standard Model Vacua from Intersecting Branes

We construct new D6-brane model vacua (non-supersymmetric) that have at low energy exactly the standard model spectrum (with right handed neutrinos). The minimal version of these models requires five stacks of branes. and the construction is based on D6-branes intersecting at angles in $D = 4$ type toroidal orientifolds of type I strings. Three U(1)'s become massive through their couplings to RR couplings and from the two surviving anomaly free U(1)'s, one is the standard model hypercharge generator while the extra anomaly free U(1) could be broken from its non-zero couplings to RR fields and also by triggering a vev to previously massive particles. We suggest that extra massless U(1)'s should be broken by requiring some intersection to respect N=1 supersymmetry thus supporting the appearance of massless charged singlets at the supersymmetric intersection. Proton is stable as baryon number is gauged and its anomalies are cancelled through a generalized Green-Schwarz mechanism. Neutrinos are of Dirac type with small masses, as in the four stack standard models of hep-th/0105155, as a result of the existence of a similar PQ like-symmetry. The models are unique in the sense that they predict the existence of only one supersymmetric particle, the superpartner of $\nu_R$.

More modular invariant anomalous U(1) breaking

We consider the case of several scalar fields, charged under a number of U(1) factors, acquiring vacuum expectation values due to an anomalous U(1). We demonstrate how to make redefinitions at the superfield level in order to account for tree-level exchange of vector supermultiplets in the effective supergravity theory of the light fields in the supersymmetric vacuum phase. Our approach builds upon previous results that we obtained in a more elementary case. We find that the modular weights of light fields are typically shifted from their original values, allowing an interpretation in terms of the preservation of modular invariance in the effective theory. We address various subtleties in defining unitary gauge that are associated with the noncanonical Kähler potential of modular invariant supergravity, the vacuum degeneracy, and the role of the dilaton field. We discuss the effective superpotential for the light fields and note how proton decay operators may be obtained when the heavy fields are integrated out of the theory at the tree-level. We also address how our formalism may be extended to describe the generalized Green–Schwarz mechanism for multiple anomalous U(1)'s that occur in four-dimensional Type I and Type IIB string constructions.

Exact Standard Model Compactifications from Intersecting Branes

We construct six stack D6-brane vacua (non-supersymmetric) that have at low energy exactly the standard model (with right handed neutrinos). The construction is based on D6-branes intersecting at angles in $D = 4$ type toroidal orientifolds of type I strings. Three U(1)'s become massive through their couplings to RR fields and from the three surviving massless U(1)'s at low energies, one is the standard model hypercharge generator. The two extra massless U(1)'s get broken, as suggested recently (hep-th/0205147), by requiring some intersections to respect N=1 supersymmetry thus supporting the appearance of massless charged singlets. Proton and lepton number are gauged symmetries and their anomalies are cancelled through a generalized Green-Schwarz mechanism that gives masses to the corresponding gauge bosons through couplings to RR fields. Thus proton is stable and neutrinos are of Dirac type with small masses as a result of a PQ like-symmetry. The models predict the existence of only two supersymmetric particles, superpartners of $\nu_R$'s.

GUT Model Hierarchies from Intersecting Branes

By employing D6-branes intersecting at angles in $D = 4$ type I strings, we construct the first examples of three generation string GUT models (PS-A class), that contain at low energy exactly the standard model spectrum with no extra matter and/or extra gauge group factors. They are based on the group $SU(4)_C \times SU(2)_L \times SU(2)_R$. The models are non-supersymmetric, even though SUSY is unbroken in the bulk. Baryon number is gauged and its anomalies are cancelled through a generalized Green-Schwarz mechanism. We also discuss models (PS-B class) which at low energy have the standard model augmented by an anomaly free U(1) symmetry and show that multibrane wrappings correspond to a trivial redefinition of the surviving global U(1) at low energies. There are no colour triplet couplings to mediate proton decay and proton is stable. The models are compatible with a low string scale of energy less that 650 GeV and are directly testable at present or future accelerators as they predict the existence of light left handed weak fermion doublets at energies between 90 and 246 GeV. The neutrinos get a mass through an unconventional see-saw mechanism. The mass relation $m_e = m_d$ at the GUT scale is recovered. Imposing supersymmetry at particular intersections generates non-zero Majorana masses for right handed neutrinos as well providing the necessary singlets needed to break the surviving anomaly free U(1), thus suggesting a gauge symmetry breaking method that can be applied in general left-right symmetric models.