Lepton flavour violation in realistic non-minimal supergravity models

J. G. Hayes, S. F. King, I. N. R. Peddie ^{1}^{1}1Address after
1st September, 2005: Physics Division, School of Technology,
Aristotle University of Thessaloniki, Thessaloniki 54124, Greece.

School of Physics and Astronomy, University of Southampton,

Southampton, SO17 1BJ, U.K.

Abstract

Realistic effective supergravity models have a variety of sources of lepton flavour violation (LFV) which can drastically affect the predictions relative to the scenarios usually considered in the literature based on minimal supergravity and the supersymmetric see-saw mechanism. We catalogue the additional sources of LFV which occur in realistic supergravity models including the effect of D-terms arising from an Abelian family symmetry, non-aligned trilinear contributions from scalar F-terms, as well as non-minimal supergravity contributions and the effect of different Yukawa textures. In order to quantify these effects, we investigate a string inspired effective supergravity model arising from intersecting D-branes supplemented by an additional family symmetry. In such theories the magnitude of the D-terms is predicted, and we calculate the branching ratios for and for different benchmark points designed to isolate the different non-minimal contributions. We find that the D-term contributions are generally dangerously large, but in certain cases such contributions can lead to a dramatic suppression of LFV rates, for example by cancelling the effect of the see-saw induced LFV in models with lop-sided textures. In the class of string models considered here we find the surprising result that the D-terms can sometimes serve to restore universality in the effective non-minimal supergravity theory.

January 21, 2021

## 1 Introduction

Lepton flavour violation (LFV) is a sensitive probe of new physics in supersymmetric (SUSY) models [1, 2]. In SUSY models, LFV arises due to off-diagonal elements in the slepton mass matrices in the ‘super-CKM’ (SCKM) basis, in which the quark and lepton mass matrices are diagonal [3]. In supergravity (SUGRA) mediated SUSY breaking the soft SUSY breaking masses are generated at the Planck scale, and the low energy soft masses relevant for physical proceses such as LFV are therefore subject to radiative corrections in running from the Planck scale to the weak scale. Off-diagonal slepton masses in the SCKM basis can arise both directly at the high energy scale (due to the effective SUGRA theory which is responsible for them), or can be radiatively generated by renormalisation group running from the Planck scale to the weak scale, for example due to Higgs triplets in GUTs or right-handed neutrinos in see-saw models which have masses intermediate between these two scales.

Neutrino experiments confirming the Large Mixing Angle (LMA) MSW solution to the solar neutrino problem [4] taken together with the atmospheric neutrino data [5] shows that neutrino masses are inevitable [6]. The presence of right-handed neutrinos, as required by the see-saw mechanism for generating neutrino masses, will lead inevitably to LFV, due to running effects, even in minimal SUGRA (mSUGRA) which has no LFV at the high energy GUT or Planck scale [7, 8]. Therefore, merely assuming SUSY and the see-saw mechanism, one expects LFV to be present. This has been studied, for example, in mSUGRA models with a natural neutrino mass hierarchy [9]. There is a large literature on the case of minimal LFV arising from mSUGRA and the see-saw mechanism [10].

Despite the fact that most realistic string models lead to a low energy effective non-minimal SUGRA theory, such theories have not been extensively studied in the literature, although a general analysis of flavour changing effects in the mass insertion approximation has recently been performed [11]. Such effective non-minimal SUGRA models predict non-universality of the soft masses at the high energy scale, dependent on the structure of the Yukawa matrices. Moreover there can be additional sources of LFV which also enter the analysis. For example, realistic effective SUGRA theories arising from string-inspired models will also typically involve some gauged family symmetry which can give an additional direct (as opposed to renormalisation group induced) source of LFV. This is because the D-term contribution to the scalar masses generated when the family symmetry spontaneously breaks adds different diagonal elements to each generation in the ‘theory’ basis, and generates non-zero off-diagonal elements in the SCKM basis, leading to LFV. This effect depends on the strength of the D-term contribution, which is expected to be close in size to the size of the uncorrected scalar masses. There can also be a significant contribution to non-universal trilinear soft masses leading to flavour violation [12, 13, 14] arising from the F-terms of scalars associated with the Yukawa couplings (for example the flavons of Froggatt-Nielsen theories).

The purpose of the present paper is to catalogue and quantitatively study the importance of all the different sources of LFV present in a general non-minimal SUGRA framework, including the effects of gauged family symmetry. Although the different effects have all been identified in the literature, there has not so far been a coherent and quantitative dedicated study of LFV processes, beyond the mass insertion approximation, which includes all these effects within a single framework. In order to quantify the importance of the different effects it is necessary to investigate these disparate souces of LFV numerically, both in isolation and in association with one another, within some particular SUGRA model. To be concrete we shall study the effective SUGRA models of the kind considered in [14] which have a sufficiently rich structure to enable all of the effects to be studied within a single framework. Within this class of models we shall consider specific benchmark points in order to illustrate the different effects. Some of these benchmark points were already previously considered in [14]. However in the previous study the important effect of D-terms arising from the Abelian family symmetry was not considered. Here we shall show that such D-terms are in fact calculable within the framework of the particular model considered, and can lead to significant enhancement (or suppression) of LFV rates, depending on the particular model considered.

The outline of this paper is as follows. In Section 2 we discuss soft supersymmetry breaking masses in supergravity. In Section 3 we summarise the flavour problem and catalogue the distinct sources of lepton flavour violation. In Section 4, we outline the aspects of the specific models that we shall study. In Section 5 we discuss the soft SUSY-breaking sector within this class of models, parameterise the F-terms, and write down the soft terms, including the D-term contribution to the scalar masses. We also discuss the D-terms associated with a family symmetry that are expected to lead to large lepton flavour violation. In Section 6 we specify two models in this class, define benchmark points and present the results of our numerical analysis of LFV for these benchmark points. Section 7 concludes the paper.

## 2 Soft terms from supergravity

We summarise here the standard way of getting soft SUSY breaking terms from supergravity. Supergravity is defined in terms of a Kähler function, , of chiral superfields (). Taking the view that supergravity arises as the low energy effective field theory limit of a string theory, the hidden sector fields are taken to correspond to closed string moduli states (), and the matter states are taken to correspond to open string states. In string theory, the ends of the open string states are believed to be constrained to lie on extended solitonic objects called .

Using natural units,

(1) |

is the Kähler potential, a real function of chiral superfields. This may be expaned in powers of :

(2) |

is the Kähler metric. is the superpotential, a holomorphic function of chiral superfields.

(3) |

We expect the supersymmetry to be broken; if it is broken, then the
auxilliary fields for some . As we lack a model of
SUSY breaking, we introduce Goldstino angles as parameters that will
enable us to explore different methods of breaking supersymmetry. We
introduce a matrix, that canonically normalises the Kähler metric,
^{2}^{2}2 The subscripts on the Kähler potential means
. However, the subscripts on the F-terms are just
labels. [15]. We also introduce a column
vector which satisfies . We are
completely free to parameterise in any way which satisfies
this constraint.

Then the un-normalised soft terms and trilinears appear in the soft SUGRA breaking potential [16]

(4) |

The non-canonically normalised soft trilinears are then

(5) | |||||

In this equation, it should be noted that the index runs over . However, by definition, the hidden sector part of the Kähler potential and the Kähler metrics is independent of the matter fields.

Assuming that the terms , the canonically normalised equation for the trilinear is

(6) |

If the Yukawa hierarchy is taken to be generated by a Froggatt-Nielsen (FN) field, , such that , then we expect , and then , and so even though these fields are expected to have heavily sub-dominant F-terms, they contribute to the trilinears on an equal footing as the moduli.

If the Kähler metric is diagonal and non-canonical, then the canonically normalised scalar mass-squareds are given by

(7) |

and the gaugino masses are given by

(8) |

where is the ‘gauge kinetic function’. enumerates -branes in the model. In type I string models without twisted moduli these have the form .

Specifically, we use a Kähler potential that doesn’t have any twisted-moduli [17]:

(9) | |||||

The notation is that the field theory scalars, the dilaton and the untwisted moduli originate from closed strings. Open string states are required to have their ends localised onto . The upper index then specifies which brane(s) their ends are located on, and if both ends are on the same brane, the lower index specifies which pair of compacitified extra dimensions the string is free to vibrate in.

## 3 Sources of lepton flavour violation

There are two parts to the flavour problem. The first is understanding the origin of the Yukawa couplings (and heavy Majorana masses for the see-saw mechanism), which lead to low energy quark and lepton mixing angles. In low energy SUSY, we also need to understand why flavour changing and/or CP violating processes induced by SUSY loops are so small. A theory of flavour must address both problems simultaneously. For a full discussion of this see the review [3].

There are two contributions that can lead to large amounts of flavour violation. The first is the non-alignment of the trilinear soft coupling matrices to the corresponding Yukawa matrices, due to the contribution , . The reasons why this can lead to large flavour violation are have been given before [14], where a numerical investigation of a model very similar to those considered herein finds that there is a large amount of flavour violation. The second contribution can come from scalar mass matrices which are not proportional to the identity in the theory basis, and lead to off-diagonal entries in the SCKM basis, resulting in flavour violation.

In this section we begin by defining the SCKM basis, and
in the following subsections we systematically
discuss a number of distinct sources of Lepton Flavour Violation (LFV) in
SUGRA models. As well as considering generic SUGRA models, we also allow for a
family symmetry, which easily lead to non-universal scalar mass matrices, and
non-aligned trilinear matrices^{3}^{3}3By non-aligned trilinears, we mean that
.

### 3.1 The SCKM Basis

The most convenient basis to work in for considering flavour violating decays, such as is the super-CKM (SCKM) basis, which is the basis where the Yukawa matrices are diagonal. If we define the unitary rotation matrices by

(10) |

such that has positive
eigenvalues. To convert the physical mass matrices to the SCKM basis,
we rotate by the relevant matrix; for the left-handed scalar
matrices and for the right-handed. Then flavour violation is
proportional to the off-diagonal elements in the SCKM basis, and is
suppressed by the diagonal values. The selectron mass matrix is 6 by
6, and the sneutrino mass matrix is 3 by 3 ^{4}^{4}4The heavy right
handed neutrinos cause the right-handed part of the sneutrino mass
matrix to decouple by the electroweak scale.. The selectron mass
matrix is

(11) |

where . The sneutrino mass matrix is then

(12) |

Off diagonal elements in any of the 3 by 3 submatrices in the SCKM basis will lead to flavour violation. We will now consider the block of . The arguments follow for any other block of or . The transformation to the SCKM basis is carried out by

(13) |

is unitary, and is diagonal, so the first two
terms will be diagonal. Any off-diagonality must come from the third
term. If this is proportional to the identity at the GUT scale, it
will be approximately equal to the identity at the electroweak scale,
which is the scale we should be working at. The fact that this is only
approximate is due to the presence of the right handed neutrino fields
in the running of the soft scalar mass squared matrices. If, however,
the soft mass squared matrices are not proportional to the identity at
the GUT scale, then large off-diagonal values will be generated when
rotating to the SCKM basis, unless the rotation happens to be
small. Generally this won’t be the case. Since the family D-term
contribution is not proportional to the identity^{5}^{5}5This
statement assumes that the generational charges are not the same for
both left- and right-handed fields. This would remove the point of
the family symmetry generating the fermion mass hierarchy. this will
usually be the case^{6}^{6}6One can, however, imagine some model with
aberrant points in its parameter space where a non-universal non-zero D-term corrects a
non-universal base mass matrix to give a universal net mass matrix.
and so we expect large flavour violation in models with Abelian family
symmetries when the D-terms correct the scalar mass matrices.

### 3.2 The relevance of the Yukawa textures

There is one subtlety concerning the size of the off-diagonal elements of the
scalar mass matrices in the SCKM basis. This comes back to the
definition of the SCKM basis as the basis in which the Yukawa matrices
are diagonal. The larger the SCKM transformation between any ‘theory’
basis and the mass eigenstate basis for the Yukawa matrices, the
larger the SCKM transformation that must be performed on the
scalar mass matrices in going to the SCKM basis,
hence the larger the off-diagonal elements of the
scalar mass matrices in the SCKM basis generated from non-equal
diagonal elements in the ‘theory’ basis.
^{7}^{7}7
Note that the D-terms make us sensitive to right-handed mixings in the Yukawa
matrices, so the non-universal family charge structure for the right-handed
scalar masses may lead to a non-universal generational hierarchy in
the right-handed scalar mass matrices.
The larger the off-diagonal entries in the SCKM basis
compared to the diagonal ones, the greater will be
the flavour violation. Also, the greater the mass difference between the
diagonal elements in the ‘theory’ basis,
the greater the size of the off-diagonal entries
produced when rotating from the ’theory’ basis, hence the larger the
flavour violating effect. Clearly these effects are sensitive to the
size of the transformation required to go to the SCKM basis,
which in turn is sensitive to the particular choice of Yukawa
textures in the ‘theory’ basis. In this way, the choice of Yukawa
texture can play an important part on controlling the magnitude
of flavour violation, and we shall see examples of this later.

### 3.3 Running effects

Consider the case where, at the high-energy scale, the scalar mass matrices are proportional to the identity matrix and each soft trilinear coupling matrix is aligned to the corresponding Yukawa matrix:

(14) |

This is often referred to as mSUGRA. In the quark sector, due to the quark flavour violation responsible for CKM mixing, when the scalar squark mass matrices are run down to the electroweak scale, they will run to non-universal scalar mass matrices and non-aligned trilinear coupling matrices. If this is the case, then in the SCKM basis, which is the basis where the Yukawa matrices are diagonal, off-diagonal elements in the scalar squark mass matrices or the trilinear squark mass matrices lead to flavour violation.

In the lepton sector, in the absence of neutrino masses the separate lepton flavour numbers are conserved and mSUGRA will not lead to any LFV induced by running the matrices down to low energy. However, in the presence of neutrino masses, with right-handed neutrino fields included to allow a see-saw explanation of neutrino masses and mixing angles, the separate lepton flavour numbers will be violated and, even in the mSUGRA type scenario, running effects will generate off-diagonal elements in the scalar mass matrices in the SCKM basis, resulting in low energy LFV.

### 3.4 Diagonal scalar mass matrices not propotional to the unit matrix

#### 3.4.1 Non-minimal SUGRA

In non-minimal SUGRA the scalar mass matrices may be diagonal at the high-energy scale, but not proportional to the identity. In this case, there will be non-zero off-diagonal elements in the SCKM basis even with no contribution to running effects, or contribution from the trilinear coupling matrices.

One way of getting diagonal mass matrices not proportional to the unit matrix is from a SUGRA model corresponding to the low energy limit of a string model with D-branes. If each generation from the field theory viewpoint corresponds to a string attaching to different branes, then the masses predicted in the SUGRA can be different. This leads to diagonal but non-universal scalar mass matrices.

#### 3.4.2 D-term contributions from broken family gauge groups

Another way of getting diagonal mass matrices not proportional to the unit matrix is by having a model with a gauge family symmetry, which is broken spontaneously. When the Higgs which breaks the family group, the flavon, gets a vev, it contributes a squark (slepton) mass contribution through the four point scalar gauge interaction which has two flavons and two squarks (sleptons).

To make the point more explicitly, consider a family group. Then the mass contribution is proportional to the charge under the family symmetry. As the point of a family symmetry is to explain the hierarchy of fermion masses, small quark mixing angles and large neutrino mixing angles, the charges are usually different.

Then, even if the mass matrix starts off as a universal matrix, it will be driven non-universal by the D-term contribution:

(15) |

### 3.5 Non-aligned trilinears

#### 3.5.1 Non-minimal SUGRA

One way of getting non-aligned trilinear matrices is by having the same sort of non-minimal SUGRA setup that leads to diagonal but non-universal mass matrices, as described in Section 3.4.1. From the supegravity equations from Section 2, the trilinears that appear in the soft Lagrangian, will be non-aligned if the trilinears predicted by the SUGRA model, are not democratic, i.e. if . From a string-inspired/SUGRA standpoint, if each generation is assigned to a different brane and extra-dimensional vibrational direction, then in general we expect to be different, due to the differing values of the Kähler metrics for the different brane assignents . When is transformed to the SCKM basis at the electroweak scale, there will then be large off-diagonal elements which contribute to flavour violating processes.

#### 3.5.2 Flavon contributions from the Yukawa couplings

In general when one considers a family symmetry in order to understand the origin of the Yukawa couplings, the new fields arising from this can develop F-term vevs, and contribute to the supersymmetry breaking F-terms in a non-universal way. This leads to a dangerous source of flavour violating non-aligned trilinears: [12, 13, 14, 18],

(16) |

where the Yukawa coupling in Eq. (16) arises from the an effective FN operator and is a polynomial of the FN field , , leading to

(17) |

However the auxiliary field is proportional to the scalar component,

(18) |

An example of this with an arbritary family symmetry is

(19) |

The are arbiratry couplings, all of which should be for the symmetry to be considered natural. The are integers appearing as a power for the -th element of the above Yukawa, and it comprises the sum of the family charges for the th-generation left-handed field and th-generation right-handed field. In principle, if the Yukawa texture is set up so that each power is different, then each element in will be different from each other, and the physical trilinear matrix, will be non-aligned to the corresponding Yukawa. Due to the dependence on the charges of the different fields, this contribution to the trilinears is not diagonalised when we transform to the SCKM basis.

## 4 Intersecting D-brane models with an Abelian family symmetry

### 4.1 Symmetries and symmetry breaking

In order to study the effects of LFV elucidated in the previous section, it is necessary to specialize to a particular effective non-minimal SUGRA model which addresses the question of flavour (i.e. provides a theory of the Yukawa couplings). The specific model we shall discuss is defined in Table 1. This model is an extension of the Supersymmetric Pati-Salam model discussed in ref.[19], based on two branes which intersect at degrees and preserve SUSY down to the TeV energy scale. The generic D-brane set-up that we use is illustrated in Fig.1, where the string assignment notation is defined. The gauge group of the sector is , and the gauge group of the sector is (e.g., we assume the of the sector are broken). The symmetry breaking pattern of this model takes place in two stages, which we assume occur at very similar scales . In the first stage, the groups are broken to the diagonal subgroup via diagonal VEV’s of bifundamentals; the resulting theory is an effective Pati-Salam model (with additional ’s) which then breaks to the MSSM (and a number of additional ) via the usual Higgs pair of bifundamentals. The string scale is taken to be equal to the GUT scale, about GeV.

The symmetry breaking pattern leads to the following relations among the gauge couplings of the SM gauge groups in terms of the gauge couplings and associated with the gauge groups of the and sectors:

(20) | |||||

(21) | |||||

(22) |

The extension is to include an additional family symmetry and the FN operators responsible for the Yukawa couplings as in [20] (see also [21]). The charges under the Abelian symmetry are left arbritary for now. The present ‘42241’ Model is the same as the model considered in [14], with the following modifications considered; firstly, we allow the Froggatt-Nielsen field to be either an intersection state or attached to the brane. The location of dramatically changes the value of the D-term contribution to the scalar masses coming from the FN sector.

The quark and lepton fields are contained in the representations which are assigned charges under . In Table 1 we list the charges, string assignments and representations under the string gauge group .

The field represents both Electroweak Higgs doublets that
we are familiar with from the MSSM. The fields and
are the Pati-Salam Higgs scalars;^{8}^{8}8We will also refer to these as
“Heavy Higgs”; this has nothing to do with the MSSM heavy neutral higgs
state . the bar on the second is used to note that it is in the conjugate
representation compared to the unbarred field.

The extra Abelian gauge group is a family symmetry, and is broken at the high energy scale by the vevs of the FN fields [22] , which have charges and respectively under . We assume that the singlet field arises as an intersection state between the two , transforming under the remnant s in the 4224 gauge structure. In general the FN fields are expected to have non-zero F-term vevs.

The two gauge groups are broken to their diagonal subgroup at a high scale due to the assumed vevs of the bifundamental Higgs fields , [19]. The symmetry breaking at the scale

(23) |

is achieved by the heavy Higgs fields , which are assumed to gain vevs [21]

(24) |

This symmetry breaking splits the Higgs field into two Higgs doublets, , . Their neutral components then gain weak-scale vevs.

(25) |

The low energy limit of this model contains the MSSM with right-handed neutrinos. We will return to the right handed neutrinos when we consider operators including the heavy Higgs fields , which lead to effective Yukawa contributions and effective Majorana mass matrices when the heavy Higgs fields gain vevs.

### 4.2 Yukawa operators

The (effective) Yukawa couplings are generated by operators
involving the FN field with
the following structure:^{9}^{9}9The field will not
enter the Yukawa operators because will be positive for
any .
[21]:

(26) |

where the integer is the total charge of and has a charge of zero. The tensor structure of the operators in Eq. (26) is

(27) |

One constructs invariant tensors that combine and representations of into , , , and representations [21]. Similarly we construct tensors that combine the representations of into singlet and triplet representations. These tensors are contracted together and into to create singlets of , and . Depending on which operators are used, different Clebsch-Gordan coefficients (CGCs) will emerge.

We will return to these in section 6, when we define the two models that we will be using for the numerical analysis.

### 4.3 Majorana operators

We are interested in Majorana fermions because they can contribute neutrino masses of the correct order of magnitude via the see-saw effect. The operators for Majorana fermions are of the form

(28) |

There do not exist renormalisable elements of this infinite series of operators, so Majorana operators are not defined, except in the element. We assume that a neturino mass term is allowed at leading (but non-renormalisable) order. A similar analysis goes through as for the Dirac fermions; however the structures only ever give masses to the neutrinos, not to the electrons or to the quarks.

## 5 Soft supersymmetry breaking masses

### 5.1 Supersymmetry breaking F-terms

In [19] it was assumed that the Yukawas were field-independent, and hence the only -vevs of importance were that of the dilaton (), and the untwisted moduli (). Here we set out the parameterisation for the F-term vevs, including the contributions from the FN field and the heavy Higgs fields . Note that the field dependent part follows from the assumption that the family symmetry field, is an intersection state.

(29) | |||||

(30) | |||||

(31) | |||||

(32) | |||||

(33) |

We introduce a shorthand notation:

(34) |

The F-terms above use values of and which are given in terms of the gauge couplings as:

(35) |

The gauge couplings are given from Eq. (20)-(22) [19] as

(36) |

where we shall assume that at the scale we have,

(37) |

The values of are assumed to be equal and are obtained from the string relation

(38) |

as

(39) |

where we have taken

(40) |

These rather small gauge couplings imply

(41) |

In [14] the string relation was not used and it was assumed incorrectly that which resulted in .

### 5.2 Soft scalar masses

There are two contributions to scalar mass squared matrices, coming from SUGRA and from D-terms. In this subsection we calculate the SUGRA predictions for the matrices at the GUT scale, and in the next subsection we add on the D-term contributions.

The SUGRA contributions to soft masses are detailed in Section 2.

From Eq. (7) we can get the family independent form for all scalars:

(42) | |||||

(43) | |||||

(44) | |||||

(45) | |||||

(46) | |||||

(47) |

where

(48) | |||||

(49) | |||||

(50) |

Here represents the left handed scalar mass squared matricies and . represents the right handed scalar mass squared matricies , , and . A discussion of the equations for can be found in Section 5.4.

### 5.3 D-term contributions

There are two D-term contributions to the scalar masses. The first is the well known [23, 14] contribution from the breaking of the Pati-Salam group to the MSSM group. Note that these D-terms are different to those quoted in the references above as we now consider the D-terms generated by breaking a family symmetry. See Appendix B for a full derivation. The second D-term comes solely from the breaking of the family symmetry. The corrections lead to the following mass matrices:

(51) | |||||

(52) | |||||

(53) | |||||

(54) | |||||

(55) | |||||

(56) | |||||

(57) | |||||

(58) |

The charges are the charges under of
and respectively, as shown in
Table 2. The correction factors
are calculated explicitly in Appendix
B in terms of the gauge couplings and soft
masses as^{10}^{10}10 is defined to be , thus
for Model 1 and for Model 2. See Appendix B for
more details.

(59) | |||||

(60) |

We note that the factors of appearing in the mass matrices are cancelled by the in the definition of .

The D-terms associated with the family symmetry depend on the charges
of the left-handed and right-handed matter representations
under the family symmetry. It is well
known^{11}^{11}11For an explanation, see for example [24]. that
for Pati-Salam, one can choose any set of charges, and there will be
an equivalent, shifted set of charges that are anomaly free due to
the Green-Schwartz anomaly cancellation mechanism. The charges used for
the D-term calculation should be the anomaly free charges.

The gauge couplings and mass parameters in Eqs. (59 , 60) are predicted from the model, in terms of the parameters and as shown in Eqs. (45 – 47) and Eqs. (62 – 64). Note that the D-terms will be zero if , or if the brane assignment is the same as . Choosing the second of these conditions is useful since it gives a comparison case where there are no D-terms; this comparison will make the D-term contribution to flavour violation immediately apparent.

### 5.4 Magnitude of -terms for different assignments

The main point worth emphasising is that in the string model the magnitudes of the -terms are calculable. We have assumed throughout that the FN field is an intersection string state , but have not specified the string assignment of . Thus takes various values depending on the string assignment for .

From Eq. (60), we see that we have calculable D-terms,

(61) |

so the value of depends on the choice of where the field lives. We use Table 1 and Eqs. (42 - 50) to quantify the -term for each possible string assignment. As always lives at the intersection, on , our first choice of on is trivial: it gives . For on , is equivalent to , as this is also on . So using Eq. (44) for and Eq. (47) for in Eq. (61), we have

(62) |

Similarly, the other two choices yield

(63) | |||||

(64) |

### 5.5 Soft gaugino masses

The soft gaugino masses are the same as in [19], which we quote here for completeness. The results follow from Eq. (8) applied to the gauginos, which then mix into the gauginos whose masses are given by

(65) | |||||

(66) | |||||

(67) |

The values of and are proportional to the brane gauge couplings and , which are related in a simple way to the MSSM couplings at the unification scale. This is discussed in [19].

When we run the MSSM gauge couplings up and solve for and we find that approximate gauge coupling unification is achieved by