Higgs versus Matter in the Heterotic Landscape
In supersymmetric extensions of the standard model there is no basic difference between Higgs and matter fields, which leads to the well known problem of potentially large baryon and lepton number violating interactions. Although these unwanted couplings can be forbidden by continuous or discrete global symmetries, a theoretical guiding principle for their choice is missing. We examine this problem for a class of vacua of the heterotic string compactified on an orbifold. As expected, in general there is no difference between Higgs and matter. However, certain vacua happen to possess unbroken matter parity and discrete -symmetries which single out Higgs fields in the low energy effective field theory. We present a method how to identify maximal vacua in which the perturbative contribution to the -term and the expectation value of the superpotential vanish. Two vacua are studied in detail, one with two pairs of Higgs doublets and one with partial gauge-Higgs unification.
In the standard model there is a clear distinction between Higgs and matter: Quarks and leptons are chiral fermions whereas a scalar field describes the Higgs boson. The most general renormalizable lagrangian consistent with gauge and Lorentz invariance yields a very successful description of strong and electroweak interactions . Furthermore, with appropriate coefficients, the unique dimension-5 operator can account for Majorana neutrino masses, and the baryon number violating dimension-6 operators are consistent with the experimental bounds on proton decay.
In supersymmetric extensions of the standard model the distinction between Higgs and matter is generically lost. Since the lepton doublets and one of the Higgs doublets have the same gauge quantum numbers the most general supersymmetric gauge invariant lagrangian contains unsuppressed -parity violating terms which lead to rapid proton decay. In grand unified models (GUTs)  colour triplet exchange can also generate dangerous baryon number violating dimension-5 operators. These problems can be overcome by introducing continuous or discrete symmetries which distinguish between Higgs and matter fields, such as -symmetry, Peccei-Quinn type symmetries or matter parity. However, in the context of four-dimensional (4D) field theories the origin and theoretical justification of these symmetries remain unclear.
Higher-dimensional theories provide a promising framework for unified extensions of the supersymmetric standard model . In particular the heterotic string  with gauge group is the natural candidate for a unified theory including gravity. Its compactifications on orbifolds [4, 5] yield chiral gauge theories in four dimensions including the standard model as well as GUT gauge groups. During the past years some progress has been made in deriving unified field theories from the heterotic string [6, 7, 8], separating the GUT scale from the string scale on anisotropic orbifolds , and a class of compactifications yielding supersymmetric standard models in four dimensions have been successfully constructed [10, 11, 12].
The heterotic string model  has a 6D orbifold GUT limit, where two compact dimensions are much larger than the other four, with 6D bulk gauge group and unbroken symmetry at two fixed points. The corresponding supergravity model has been explicitly constructed in , and it has been shown that all bulk and brane anomalies are canceled by the Green-Schwarz mechanism. Furthermore, a class of vacua has been found which have a pair of bulk Higgs fields and two bulk families in addition to the two brane families. At the fixed points these fields form an GUT model. In 4D one obtains one quark-lepton ‘family’ and a pair of Higgs doublets from split bulk multiplets together with the two brane families.
What distinguishes Higgs from matter fields with the same quantum numbers in an orbifold GUT? In the vacuum studied in  there is no distinction, which leads to unacceptable -parity violating Yukawa couplings. In  interesting 4D vacua with unbroken matter parity were found, which allow to forbid the dangerous -parity violating couplings. Some of these vacua also have gauge-Higgs unification for which an intriguing relationship exists between -term and gravitino mass. Indeed, several vacua with semi-realistic Yukawa couplings could be identified where to order six in powers of standard model singlets -term and gravitino mass both vanish.
In this paper we further analyse the vacua of the 6D orbifold GUT . Since , we consider vacua with expectation values (VEVs) of all 6D zero modes. One then obtains further vacua with unbroken matter parity. The localized Fayet-Iliopoulos terms of anomalous symmetries may indeed stabilize two compact dimensions at the GUT scale [13, 14] but the study of stabilization and profiles of bulk fields  is beyond the scope of this paper. In the following we concentrate on local properties of the model at the GUT fixed points, in particular the decoupling of exotics and the generation of superpotential terms.
The existence of a matter parity is not sufficient to distinguish Higgs from matter. One also needs that the -term is much smaller than the decoupling mass of exotic states. In principle, there are two obvious solutions: Either a non-zero -term is generated at very high powers in standard model singlets, or the perturbative part of the -term vanishes exactly and a non-perturbative contribution, possibly related to supersymmetry breaking, yields a correction of the order of the electroweak scale. In Section 4, we shall discuss how to identify ‘maximal’ vacua with vanishing -term, as well as extended vacua with -terms generated at high orders. This is the main point of our paper.
The maximal vacua with vanishing -term do not include the case of gauge-Higgs unification. Instead, we find a vacuum with two pairs of massless Higgs doublets and one with partial gauge-Higgs unification only for which gives mass to up-type quarks. This is perfectly consistent with the fact that a large top-quark mass is singled out. The original symmetry between - and -plets is violated by selecting vacua where matter belongs to - and -plets.
There are also other promising approaches which use elements of unification to find realistic string vacua. This includes compactifications on Calabi-Yau manifolds with vector bundles [16, 17, 18, 19, 20, 21], which are related to orbifold constructions whose singularities are blown up [22, 23]. Very recently, also interesting GUT models based on F-theory have been discussed [24, 25, 26].
The paper is organized as follows. In Section 2 we recall some symmetry properties of effective field theories, which are relevant for the -term and baryon number violating interactions. The relevant features of the 6D orbifold GUT model  are briefly reviewed in Section 3. New vacua of this model with vanishing -term and gravitino mass are analyzed in Section 4, and the corresponding unbroken discrete -symmetries are determined. Yukawa couplings for these vacua are calculated in Section 5.
2 Effective low energy field theory
The heterotic 6D GUT model  has local SU(5) invariance corresponding to Georgi-Glashow unification. Hence, the superpotential of the corresponding low energy 4D field theory has the general form,
where we have included dimension-5 operators. Here denote generation indices, and for simplicity we have kept the notation. Note that the colour triplets contained in the Higgs fields and are projected out. and yield the well known renormalizable baryon (B) and lepton (L) number violating interactions, and the coefficients and of the dimension-5 operators are usually obtained by integrating out states with masses . In supergravity theories also the expectation value of the superpotential is important since it determines the gravitino mass. One expects
if the scale of electroweak symmetry breaking is related to supersymmetry breaking.
Experimental bounds on the proton lifetime and lepton number violating processes imply , and . Furthermore, one has to accommodate the hierarchy between the electroweak scale and the GUT scale, . On the other hand, lepton number violation should not be too much suppressed, since yields the right order of magnitude for neutrino masses.
These phenomenological requirements can be implemented by means of continuous or discrete symmetries. Imposing an additional factor with
where denotes the standard model hypercharge, one has , since these operators contain only violating terms. On the other hand, conserves and is therefore not affected. The canonical charges read
The wanted result, , , can be obtained with a subgroup of , which contains the ‘matter parity’ ,
Matter parity, however, does not solve the problem , and also the hierarchy remains unexplained.
In supersymmetric extensions of the standard model, electroweak symmetry breaking is usually tied to supersymmetry breaking. It is then natural to have for unbroken supersymmetry. One easily verifies that in this case, for , the superpotential aquires a unique Peccei-Quinn type symmetry with charges
together with an additional symmetry with -charges
Note that the -symmetry implies the wanted relations , with unconstrained. On the other hand, the Peccei-Quinn symmetry only yields .
The latter relations can also be obtained by imposing only a discrete subgroup with -parities
On the contrary, the familiar -parity, which is preserved by non-zero gaugino masses,
implies , whereas , and are all allowed.
In summary, the unwanted terms in the lagrangian (2.1) can be forbidden by a continuous global -symmetry. Supersymmetry breaking will also break to -parity, which may lead to an -axion. The dangerous terms and will then be proportional to the soft supersymmetry breaking terms and therefore strongly suppressed. Alternatively, the unwanted terms in (2.1) can be forbidden by discrete symmetries, such as matter parity, PQ-parity or -parity.
In ordinary 4D GUT models continuous or discrete symmetries can be introduced by hand. It is interesting to see how protecting global symmetries arise in higher-dimensional theories. The global symmetry (2.8) indeed occurs naturally , and it has been used in 5D and 6D orbifold GUTs . However, as we shall see in the following sections, orbifold compactifications of the heterotic string single out discrete symmetries, which may or may not commute with supersymmetry.
3 Heterotic model in six dimensions
Let us now briefly describe the main ingredients of the 6D orbifold GUT model derived in . The starting point is the heterotic string propagating in the space-time background . Here , and represents four-dimensional Minkowski space; and are the tori associated with the root lattices of the Lie groups and , respectively. By construction the twist yielding the orbifold has and subtwists which act trivially on the and the plane, respectively. As a consequence, the model has bulk fields living in ten dimensions and fields from twisted sectors, which are confined to six or four dimensions.
The model has twelve fixed points
The 6D orbifold GUT has supersymmetry and unbroken gauge group
with the corresponding massless vector multiplets
In addition one finds the bulk hypermultiplets
where we have dropped the charges. It is convenient to decompose all 6D multiplets in terms of 4D multiplets. The 6D vector multiplet splits into a pair of 4D vector and chiral multiplets, , whereas a hypermultiplet contains of a pair of chiral multiplets, ; here and are left-handed, is right-handed. It is often convenient to use the charge conjugate field instead of so that all degrees of freedom are contained in left-handed chiral multiplets. In the following we use the same symbol for a hypermultiplet and its left-handed chiral multiplet; the superscript ’’ indicates that the field is the charge conjugate of a right-handed chiral multiplet contained in a hypermultiplet. As an example, the chiral multiplets and are both -plets of , but they belong to different hypermultiplets which transform as and , respectively.
As we shall see, the four non-Abelian singlets, denoted as , play a crucial role in vacua with unbroken matter parity; the -plet contains part of one quark-lepton generation. At the fixed points one has
In addition to the vector and hypermultiplets from the untwisted sector of the string, there are 6D bulk fields which originate from the twisted sectors and of the orbifold model. They are localized at the fixed points of the subtwist in the plane, but bulk fields in the plane which is left invariant by this subtwist. In contrast, fields of the twisted sectors and are localized at fixed points in the plane. For simplicity, we shall list in the following only the states of the ‘visible’ sector, the complete set of fields can be found in . For each of the three fixed points in the plane, one finds
where the omitted charges depend on . The multiplicity factor 3 is related to three different localizations in the plane; and denote singlets under the non-Abelian part of . At the fixed points , Eq. (3.5) reads
where denote singlets. Note that each hypermultiplet contains two chiral multiplets and with opposite gauge quantum numbers.
At the two inequivalent fixed points in the plane the bulk gauge group is broken to the subgroups and , respectively,
At these fixed points chiral multiplets from the twisted sectors and are localized. At each fixed point one has
This provides two quark-lepton families and additional singlets whose vacuum expectation values, together with those of and can break unwanted symmetries. Note that , and form together a -plet of which is unbroken at two equivalent fixed points of the 6D orbifold . Hence is one of the ‘right-handed’ neutrinos in the theory.
According to Eqs. (3.4) and (3.6), the 6D theory dimensionally reduced to 4D is vectorlike. In terms of chiral multiplets there are two ’s, two ’s, 19 ’s and 19 ’s. The chiral spectrum in 4D is a consequence of the further orbifold compactification. At the fixed points of the plane two chiral families, , occur. Furthermore, the boundary conditions for the 6D bulk fields at the fixed points lead to a chiral massless spectrum. Zero modes require positive ‘parities’ for bulk fields at all fixed points. As shown in , positive parities at the fixed points reduce the 18 ’s and 18 ’s in Eq. (3.6) to 10 ’s and 8 ’s, i.e., to a chiral spectrum.
The model clearly has a huge vacuum degeneracy. In most vacua the standard model gauge group will be broken. This can be avoided by allowing only VEVs of the SM singlet fields,
but most vacua will have a massless spectrum different from the MSSM. An interesting subset of vacua can be identified by observing that the products and are total gauge singlets for which one can easily generate masses at the fixed points. This allows the decoupling of 6 pairs of ’s and ’s ,
after which one is left with three -plets, five -plets and two -plets,
The decoupling scale will be discussed in more detail later on. We are now getting rather close to the standard model. The bulk fields, together with the localized fields (3.9), can account for four quark-lepton families, and the additional three pairs of - and -plets may contain a pair of Higgs fields.
How can one distinguish between Higgs and matter fields and which fields should be decoupled? The discussion in Section 2 suggests to search for the symmetry among the six factors at the fixed points, so that the extended gauge symmetry contains ,
Here are generators of the six local
We can now demand the canonical charges (2.4) for the localized fields and the bulk - and -plets. This fixes four coefficients: . Two - and two -plets then have the charges of the Higgs multiplets and , respectively,
This leaves , and as candiates for matter fields. The requirement to identify two -plets which, together with and , form two generations, uniquely determines the last two coefficents, and , so that
The remaining charge assignments read
One can also embed the symmetry (2.7) in the product . One finds
However, in the vacua considered in the next section, this symmetry is completely broken.
To proceed further we now consider the zero modes of the - and -plets listed in Table 3.1: and yield exotic colour triplets and therefore have to be decoupled,
where the decoupling scale will be discussed in more detail later on. and contain a canonical colour-triplet and lepton doublet, respectively. Finally, and are candidates for , whereas and are candidates for .
For the matter fields we now have a clear picture. There are two localized
and two further families of bulk fields,
At the fixed points , these chiral multiplets form a local GUT theory. The corresponding Yukawa couplings are matrices which are generated locally ,
according to the string selection rules. Projecting the bulk fields to their zero modes,
yields one quark-lepton generation in the effective 4D theory. From (3.21) one deduces the corresponding Yukawa matrices,
which avoid the unsuccessful prediction of 4D GUTs.
Like all factors at the fixed points, the symmetry has to be spontaneously broken at low energies. As we saw in Section 2, it is then crucial to maintain a subgroup, which includes matter parity, to distinguish between Higgs and matter fields. In order to see whether this is possible in the present model one has to examine the charges of the singlet fields (3.10), which are listed in Table 3.3. In the vacuum selected in  fields with obtained a VEV breaking completely. This led to phenomenologically unacceptable -parity violating couplings.
Varying the discrete Wilson line in the plane, in  4D models with conserved matter parity were found. In these models only SM singlets with even charge aquire VEVs. These fields are zero modes of the 4D theory. In a 6D orbifold GUT model, in principle all 6D zero modes can aquire VEVs, even if they do not contain 4D zero modes, since the negative mass squared induced by the local Fayet-Iliopoulos terms can compensate the positive Kaluza-Klein GUT mass term. Hence, one can include the fields and , which have (see Table 3.3), in the set of vacuum fields. Not allowing VEVs of singlets with then preserves matter parity. Note that not all vacua of the 6D orbifold GUT can be obtained from the 4D zero modes.
The pairwise decoupling (3.11), the decoupling of the exotic - and -plets, and the matter parity preserving breaking of can be achieved with the minimal vacuum
As we shall discuss in detail in the following section, the couplings needed to decouple the -pairs satisfy all string selection rules. Note that no exotic matter is located at the fixed points . Most of the exotic matter at can be decoupled by VEVs of just a few singlet fields (cf. ). This decoupling takes place locally at one of the fixed points, which is a crucial difference compared to previous discussions of decoupling in four dimensions [10, 11]. The unification of gauge couplings yields important constraints on the decoupling masses and the GUT scale . This question goes beyond the scope of our paper. Detailed studies have recently been carried out for the 6D model  in  and for a heterotic 6D model similar to the one described here in .