#
Cern-Ph-Th/2012-079

Dimensional Reduction of , SYM over
and its four-dimensional effective
action

###### Abstract

We present an extension of the Standard Model inspired by the Heterotic String. In order that a reasonable effective Lagrangian is presented we neglect everything else other than the ten-dimensional supersymmetric Yang-Mills sector associated with one of the gauge factors and certain couplings necessary for anomaly cancellation. We consider a compactified space-time , where is the nearly-Kähler manifold and is a freely acting discrete group on . Then we reduce dimensionally the on this manifold and we employ the Wilson flux mechanism leading in four dimensions to an gauge theory with the spectrum of a supersymmetric theory. We compute the effective four-dimensional Lagrangian and demonstrate that an extension of the Standard Model is obtained with interesting features including a conserved baryon number and fixed tree level Yukawa couplings and scalar potential. The spectrum contains new states such as right handed neutrinos and heavy vector-like quarks.

CERN-PH-TH/2012-079

Dimensional Reduction of , SYM over and its four-dimensional effective action

George Zoupanos ^{†}^{†}footnotemark:

Theory Group, Physics Department CERN, Geneva, Switzerland

E-mail: George.Zoupanos@cern.ch

\abstract@cs

## 1 Introduction

Superstring Theory is often regarded as the best candidate for a quantum theory of gravitation, or more generally as a unified theory of all fundamental interactions. On the other hand the main goal expected from a unified description of interactions by the Particle Physics community is to understand the present day large number of free parameters of the Standard Model (SM) in terms of a few fundamental ones. Indeed the celebrated SM had so far outstanding successes in all its confrontations with experimental results. However its apparent success is spoiled by the presence of a plethora of free parameters mostly related to the ad-hoc introduction of the Higgs and Yukawa sectors in the theory.

It is worth recalling that various dimensional reduction schemes, with the Coset Space Dimensional Reduction (CSDR) [1, 2, 3] and the Scherk-Schwarz reduction [4] being pioneers, suggest that a unification of the gauge and Higgs sectors can be achieved in higher than four dimensions. The four-dimensional gauge and Higgs fields are simply the surviving components of the gauge fields of a pure gauge theory defined in higher dimensions, while the addition of fermions in the higher-dimensional gauge theory leads naturally after CSDR to Yukawa couplings in four dimensions. The last step in this unified description in high dimensions is to relate the gauge and fermion fields, which can be achieved by demanding that the higher-dimensional gauge theory is supersymmetric, i.e. the gauge and fermion fields are members of the same vector supermultiplet. Furthermore a very welcome additional input coming from Superstring Theory (for instance the heterotic string [5]) is the suggestion of the space-time dimensions and the gauge group of the higher-dimensional supersymmetric theory [6].

Superstring Theory is consistent only in ten dimensions and therefore the following crucial issues have to be addressed, (a) distinguish the extra dimensions from the four observable ones which are experimentally approachable, i.e. determine a suitable compactification which is a solution of the theory (b) reduce the higher-dimensional theory to four dimensions and determine the corresponding four-dimensional theory, which may subsequently be compared to observations. Among superstring theories the heterotic string [5] has always been considered as the most promising version in the prospect to find contact with low-energy physics studied in accelerators, mainly due to the presence of the ten-dimensional gauge sector. Upon compactification of the ten-dimensional space-time and subsequent dimensional reduction the initial gauge theory can break to phenomenologically interesting Grand Unified Theories (GUTs), where the SM could in principle be accommodated [7]. Moreover, the presence of chiral fermions in the higher-dimensional theory serves as an advantage in view of the possibility to obtain chiral fermions also in the four-dimensional theory. Finally, the original supersymmetry can survive and not get enhanced in four dimensions, provided that appropriate compactification manifolds are used. In order to find contact with the minimal supersymmetric standard model (MSSM), the non-trivial part of this scenario was to invent mechanisms of supersymmetry breaking within the string framework.

The task of providing a suitable compactification and reduction scheme which would lead to a realistic four-dimensional theory has been pursued in many diverse ways for more than twenty years. The realization that Calabi-Yau (CY) threefolds serve as suitable compact internal spaces in order to maintain an supersymmetry after dimensional reduction from ten dimensions to four [8] led from the beginning to pioneering studies in the dimensional reduction of superstring models [9, 10]. However, in CY compactifications the resulting low-energy field theory in four dimensions contains a number of massless chiral fields, known as moduli, which correspond to flat directions of the effective potential and therefore their values are left undetermined. The attempts to resolve the moduli stabilization problem led to the study of compactifications with fluxes (for reviews see e.g. [11]). In the context of flux compactifications the recent developments suggested the use of a wider class of internal spaces, called manifolds with -structure, which contains CYs. Admittance of an -structure is a milder condition as compared to -holonomy, which is the case for CY manifolds, in the sense that a nowhere-vanishing, globally-defined spinor can be defined such that it is covariantly constant with respect to a connection with torsion and not with respect to the Levi-Civita connection as in the CY case. An interesting class of manifolds admitting an -structure is that of nearly-Kähler manifolds. The homogeneous nearly-Kähler manifolds in six dimensions have been classified in [12] and they are the three non-symmetric six-dimensional coset spaces (see table 1 Appendix B) and the group manifold . In the studies of heterotic compactifications the use of non-symmetric coset spaces was introduced in [13, 15, 14] and recently developed further in [16, 17]. Particularly, in [17] it was shown that supersymmetric compactifications of the heterotic string theory of the form exist when background fluxes and general condensates are present. Moreover, the effective theories resulting from dimensional reduction of the heterotic string over nearly-Kähler manifolds were studied in [18].

Last but not least it is worth noting that the dimensional reduction of ten-dimensional supersymmetric gauge theories over non-symmetric coset spaces led in four dimensions to softly broken theories [19, 20].

Here we would like to present the significant progress that has been made
recently concerning the dimensional reduction of the N=1 supersymmetric
gauge
theory resulting in the field theory limit of the heterotic string over the
nearly-Kähler manifold . Specifically an extension of
the Standard Model (SM) inspired by the heterotic string was
derived [21]^{1}^{1}1For earlier attempts to obtain realistic
models by CSDR see ref
[2, 31, 38, 50].. In addition in
order to make the presentation
self-contained we
present first a short review of the CSDR.

## 2 The Coset Space Dimensional Reduction.

Given a gauge theory defined in higher dimensions the obvious way to dimensionally reduce it is to demand that the field dependence on the extra coordinates is such that the Lagrangian is independent of them. A crude way to fulfill this requirement is to discard the field dependence on the extra coordinates, while an elegant one is to allow for a non-trivial dependence on them, but impose the condition that a symmetry transformation by an element of the isometry group of the space formed by the extra dimensions corresponds to a gauge transformation. Then the Lagrangian will be independent of the extra coordinates just because it is gauge invariant. This is the basis of the CSDR scheme [1, 2, 3], which assumes that is a compact coset space, .

In the CSDR scheme one starts with a Yang-Mills-Dirac Lagrangian, with gauge group , defined on a -dimensional spacetime , with metric , which is compactified to with a coset space. The metric is assumed to have the form

(2.0) |

where and is the coset space metric. The requirement that transformations of the fields under the action of the symmetry group of are compensated by gauge transformations lead to certain constraints on the fields. The solution of these constraints provides us with the four-dimensional unconstrained fields as well as with the gauge invariance that remains in the theory after dimensional reduction. Therefore a potential unification of all low energy interactions, gauge, Yukawa and Higgs is achieved, which was the first motivation of this framework.

It is interesting to note that the fields obtained using the CSDR approach are the first terms in the expansion of the -dimensional fields in harmonics of the internal space [2, 49]. The effective field theories resulting from compactification of higher dimensional theories might contain also towers of massive higher harmonics (Kaluza-Klein) excitations, whose contributions at the quantum level alter the behaviour of the running couplings from logarithmic to power [22]. As a result the traditional picture of unification of couplings may change drastically [23]. Higher dimensional theories have also been studied at the quantum level using the continuous Wilson renormalization group [24] which can be formulated in any number of space-time dimensions with results in agreement with the treatment involving massive Kaluza-Klein excitations. However we should stress that in ref [25] the CSDR has been shown to be a consistent scheme.

Before we proceed with the description of the CSDR scheme we need to recall some facts about coset space geometry needed for subsequent discussions. Complete reviews can be found in [2, 26].

### 2.1 Coset Space Geometry.

Assuming a -dimensional spacetime with metric given in (2) it is instructive to explore further the geometry of all coset spaces .

We can divide the generators of , in two sets : the generators of , , and the generators of , ( , and . Then the commutation relations for the generators of are the following:

(2.0) |

So is assumed to be a reductive but in general non-symmetric coset space. When is symmetric, the in (2.1) vanish. Let us call the coordinates of space , where is a curved index of the coset, is a tangent space index and defines an element of which is a coset representative, . The vielbein and the -connection are defined through the Maurer-Cartan form which takes values in the Lie algebra of :

(2.0) |

Using (2.0) we can compute that at the origin , and . A connection on which is described by a connection-form , has in general torsion and curvature. In the general case where torsion may be non-zero, we calculate first the torsionless part by setting the torsion form equal to zero,

(2.0) |

while using the Maurer-Cartan equation,

(2.0) |

we see that the condition of having vanishing torsion is solved by

(2.0) |

where is symmetric in the indices , therefore . The can be found from the antisymmetry of , , leading to

(2.0) |

In turn becomes

(2.0) |

where

The ’s can be related to ’s by a rescaling [2]:

where the ’s depend on the coset radii. Note that in general the rescalings change the antisymmetry properties of ’s, while in the case of equal radii . Note also that the connection-form is -invariant. This means that parallel transport commutes with the action [26]. Then the most general form of an -invariant connection on would be

(2.0) |

with an -invariant tensor, i.e.

This condition is satisfied by the ’s as can be proven using the Jacobi identity.

In the case of non-vanishing torsion we have

(2.0) |

where

with

(2.0) |

while the contorsion is given by

(2.0) |

in terms of the torsion components . Therefore in general the connection-form is

(2.0) |

The natural choice of torsion which would generalize the case of equal radii [15, 27, 28], would be except that the ’s do not have the required symmetry properties. Therefore we must define as a combination of ’s which makes completely antisymmetric and -invariant according to the definition given above. Thus we are led to the definition

(2.0) |

In this general case the Riemann curvature two-form is given by [2], [28]:

(2.0) |

whereas the Ricci tensor is

(2.0) |

By choosing the parameter to be equal to zero we can obtain the Riemannian connection . We can also define the canonical connection by adjusting the radii and so that the connection form is , i.e. an -gauge field [15]. The adjustments should be such that . In the case of where the metric is , we have and in turn . In the case of , where the metric is , we have to set and then to obtain the canonical connection. Similarly in the case of , where the metric is , we should set and take . By analogous adjustments we can set the Ricci tensor equal to zero [15], thus defining a Ricci flattening connection.

### 2.2 Reduction of a -dimensional Yang-Mills-Dirac Lagrangian.

The group acts as a symmetry group on the extra coordinates. The CSDR scheme demands that an -transformation of the extra coordinates is a gauge transformation of the fields that are defined on , thus a gauge invariant Lagrangian written on this space is independent of the extra coordinates.

To see this in detail we consider a -dimensional Yang-Mills-Dirac theory with gauge group defined on a manifold which as stated will be compactified to , , :

(2.0) |

where

(2.0) |

with

(2.0) |

the spin connection of , and

(2.0) |

where , run over the -dimensional space. The fields and are, as explained, symmetric in the sense that any transformation under symmetries of is compensated by gauge transformations. The fermion fields can be in any representation of unless a further symmetry such as supersymmetry is required. So let , , be the Killing vectors which generate the symmetries of and the compensating gauge transformation associated with . Define next the infinitesimal coordinate transformation as , the Lie derivative with respect to , then we have for the scalar,vector and spinor fields,

(2.0) | |||||

depend only on internal coordinates and represents a gauge transformation in the appropriate representation of the fields. represents a tangent space rotation of the spinor fields. The variations satisfy, and lead to the following consistency relation for ’s,

(2.0) |

Furthermore the W’s themselves transform under a gauge transformation [2] as,

(2.0) |

Using (2.0) and the fact that the Lagrangian is independent of we can do all calculations at and choose a gauge where .

The detailed analysis of the constraints (2.0) given in refs.[1, 2] provides us with the four-dimensional unconstrained fields as well as with the gauge invariance that remains in the theory after dimensional reduction. Here we give the results. The components of the initial gauge field become, after dimensional reduction, the four-dimensional gauge fields and furthermore they are independent of . In addition one can find that they have to commute with the elements of the subgroup of . Thus the four-dimensional gauge group is the centralizer of in , . Similarly, the components of denoted by from now on, become scalars at four dimensions. These fields transform under as a vector , i.e.

(2.0) |

Moreover act as an intertwining operator connecting induced representations of acting on and . This implies, exploiting Schur’s lemma, that the transformation properties of the fields under can be found if we express the adjoint representation of in terms of :

(2.0) |

Then if , where each is an irreducible representation of , there survives an multiplet for every pair , where and are identical irreducible representations of .

Turning next to the fermion fields [2, 30, 31, 29] similarly to scalars, they act as intertwining operators between induced representations acting on and the tangent space of , . Proceeding along similar lines as in the case of scalars to obtain the representation of under which the four-dimensional fermions transform, we have to decompose the representation of the initial gauge group in which the fermions are assigned under , i.e.

(2.0) |

and the spinor of under

(2.0) |

Then for each pair and , where and are identical irreducible representations there is an multiplet of spinor fields in the four-dimensional theory. In order however to obtain chiral fermions in the effective theory we have to impose further requirements. We first impose the Weyl condition in dimensions. In dimensions which is the case at hand, the decomposition of the left handed, say spinor under is

(2.0) |

So we have in this case the decompositions

(2.0) |

Let us start from a vector-like representation for the fermions. In this case each term in (2.0) will be either self-conjugate or it will have a partner . According to the rule described in eqs.(2.0), (2.0) and considering we will have in four dimensions left-handed fermions transforming as . It is important to notice that since is non self-conjugate, is non self-conjugate too. Similarly from we will obtain the right handed representation but as we have assumed that is vector-like, . Therefore there will appear two sets of Weyl fermions with the same quantum numbers under . This is already a chiral theory, but still one can go further and try to impose the Majorana condition in order to eliminate the doubling of the fermionic spectrum. We should remark now that if we had started with complex, we should have again a chiral theory since in this case is different from non self-conjugate). Nevertheless starting with vector-like is much more appealing and will be used in the following along with the Majorana condition. Majorana and Weyl conditions are compatible in dimensions. Then in our case if we start with Weyl-Majorana spinors in dimensions we force to be the charge conjugate to , thus arriving in a theory with fermions only in .

An important requirement is that the resulting four-dimensional theories should be anomaly free. Starting with an anomaly free theory in higher dimensions, Witten [32] has given the condition to be fulfilled in order to obtain anomaly free four-dimensional theories. The condition restricts the allowed embeddings of into by relating them with the embedding of into , the tangent space of the six-dimensional cosets we consider [2, 33]. To be more specific if are the generators of into and are the generators of into the condition reads

(2.0) |

According to ref. [33] the anomaly cancellation condition (2.0) is automatically satisfied for the choice of embedding

(2.0) |

which we adopt here. Furthermore concerning the abelian group factors of the four-dimensional gauge theory, we note that the corresponding gauge bosons surviving in four dimensions become massive at the compactification scale [32, 34] and therefore, they do not contribute in the anomalies; they correspond only to global symmetries.

### 2.3 The Four-Dimensional Theory.

Next let us obtain the four-dimensional effective action. Assuming that the metric is block diagonal, taking into account all the constraints and integrating out the extra coordinates we obtain in four dimensions the following Lagrangian :

(2.0) |

where and with the connection of the coset space, while is the volume of the coset space. The potential is given by:

(2.0) |

where, and ’ s are the structure constants appearing in the commutators of the generators of the Lie algebra of S. The expression (2.0) for is only formal because must satisfy the constraints coming from eq.(2.0),

(2.0) |

where the generate . These constraints imply that some components ’s are zero, some are constants and the rest can be identified with the genuine Higgs fields. When is expressed in terms of the unconstrained independent Higgs fields, it remains a quartic polynomial which is invariant under gauge transformations of the final gauge group , and its minimum determines the vacuum expectation values of the Higgs fields [35, 36, 37]. The minimization of the potential is in general a difficult problem. If however has an isomorphic image in which contains in a consistent way then it is possible to allow the to become generators of . That is with suitable combinations of generators, a generator of and is also a coset-space index. Then

because of the commutation relations of . Thus we have proven that which furthermore is the minimum, because is positive definite. Furthermore, the four-dimensional gauge group breaks further by these non-zero vacuum expectation values of the Higgs fields to the centralizer of the image of in , i.e. [2, 35, 36, 37]. This can been seen if we examine a gauge transformation of by an element of . Then we have

We note that the v.e.v. of the Higgs fields is gauge invariant for the set of ’s that commute with . That is belongs to a subgroup of which is the centralizer of in .

In the fermion part of the Lagrangian the first term is just the kinetic term of fermions, while the second is the Yukawa term [38]. Note that since is a Majorana-Weyl spinor in ten dimensions the representation in which the fermions are assigned under the gauge group must be real. The last term in (2.0) can be written as

where

(2.0) | |||||

(2.0) |

and we have used the full connection with torsion [28] given by

(2.0) |

with

(2.0) |

and

(2.0) |

We have already noticed that the CSDR constraints tell us that . Furthermore we can consider the Lagrangian at the point , due to its invariance under -transformations, and as we mentioned at that point. Therefore (2.0) becomes just and the term in (2.3) is exactly the Yukawa term.

Let us examine now the last term appearing in (2.3). One can show easily that the operator anticommutes with the six-dimensional helicity operator [2]. Furthermore one can show that commutes with the ( close the -subalgebra of ). In turn we can draw the conclusion, exploiting Schur’s lemma, that the non-vanishing elements of are only those which appear in the decomposition of both irreps and , e.g. the singlets. Since this term is of pure geometric nature, we reach the conclusion that the singlets in and will acquire large geometrical masses, a fact that has serious phenomenological implications. In supersymmetric theories defined in higher dimensions, it means that the gauginos obtained in four dimensions after dimensional reduction receive masses comparable to the compactification scale. However as we shall see in the next section this result changes in presence of torsion. We note that for symmetric coset spaces the operator is absent because are vanishing by definition in that case.

## 3 Dimensional Reduction of over and soft supersymmetry breaking

In this model we consider the coset space on which we reduce the ten-dimensional theory. To determine the four-dimensional gauge group, the embedding of in is suggested by the decomposition

(3.0) |

Then, the surviving gauge group in four dimensions is

The of decomposes under in the following way:

(3.0) |

The decomposition of the vector and spinor representations of (see table 1, Appendix B) is

and

respectively. Thus applying the CSDR rules we find that the surviving fields in four dimensions are three vector multiplets , (where is an , index and the other two refer to the two ) containing the gauge fields of . The matter content consists of three chiral multiplets (, , ) with an , index and three chiral multiplets (, , ) which are singlets and carry only charges.

To determine the potential we examine further the decomposition of the adjoint of the specific under , i.e.

(3.0) |

Then according to the decomposition (3.0) the generators of can be grouped as

(3.0) |

The non trivial commutator relations of generators (3.0) are given in the table 2 given in the Appendix B. The decomposition (3.0) suggests the following change in the notation of the scalar fields,

(3.0) |

The potential of any theory reduced over is given in terms of the redefined fields in (3.0) by

(3.0) | |||||