#
Resurrecting the minimal renormalizable

supersymmetric SU(5) model

Abstract

It is a well-known fact that the minimal renormalizable supersymmetric SU(5) model is ruled out assuming superpartner masses of the order of a few TeV. Giving up this constraint and assuming only SU(5) boundary conditions for the soft terms, we find that the model is still alive. The viable region of the parameter space typically features superpartner masses of order to , with values between and , but lighter spectra with single states around TeV are also possible. The main constraints come from proton decay, the Higgs mass, the requirement of the SU(5) spectrum being reasonably below the Planck scale, and the lifetime of the universe. A generic feature of the model is metastability of the electroweak vacuum. In the absence of a suitable dark matter particle in the neutralino sector, a light (order GeV or smaller) gravitino is a natural candidate.

###### Contents

- 1 Introduction
- 2 The minimal renormalizable supersymmetric SU(5) model
- 3 Renormalization group equations
- 4 Solutions of the RGEs for soft terms
- 5 Proton decay
- 6 Vacuum (meta)stability
- 7 Results and discussion
- 8 Conclusions
- A Procedure for solving the soft term RGEs
- B Proton lifetime computation
- C UFB constraints
- D Decays into CCB vacua

## 1 Introduction

The minimal renormalizable supersymmetric SU(5) model [1],
with just 3 pairs of fermion representations and an adjoint
plus a pair in the Higgs sector,
is the simplest supersymmetric
Grand Unified extension of the Standard Model.
It is therefore particularly important
to test this model in detail, and possibly to rule it out.
Although the choice of the gauge group, supersymmetry and minimality do not need a special
motivation, it is more difficult to justify the absence of non-renormalizable terms in the superpotential.
Experimental evidence tells us that some of these terms must be strongly suppressed.
For instance, the superpotential operators ()
induce proton decay at an unacceptable rate unless they come with coefficients smaller
than about .
Their smallness will lead us to assume in this paper that for some (to us unknown) reason
all non-renormalizable operators can be neglected, and to adopt the minimal renormalizable
supersymmetric SU(5) model as a benchmark^{5}^{5}5Another option is to give up renormalizability
and to assume that the non-renormalizable operators giving rise to fast proton decay are
suppressed, while harmless higher-dimensional couplings can be sizable. Under this assumption,
it is possible to avoid fast proton decay from heavy colour triplet
exchange [2, 3, 4, 5]
and to correct the (phenomenologically inaccurate) SU(5) relations between
charged lepton and down-type quark masses [6]
(for a review on these issues, see Ref. [4])..

It has been shown long ago [7] that the region of the parameter
space of the minimal renormalizable supersymmetric SU(5) model corresponding
to TeV-scale soft terms is excluded^{6}^{6}6In the case of decoupling
scenario of heavy first two generations of sfermions considered in [7]
proton decay can still be consistent with the experimental bounds providing the flavor sfermion
sector gets a very specific form [3].. The reason put forward was the incompatibility
between the colour triplet mass constraints associated with gauge coupling unification
on the one hand, and with proton decay on the other hand.
This conclusion relies however on the assumption of a relatively light superpartner spectrum
(although masses as large as for the first two generations of sfermions
were considered in Ref. [7]).
The purpose of this paper is to ascertain whether it can be extended to the region of
larger superpartner masses.

In order to answer this question, several constraints have to be imposed on the model:
precise gauge coupling unification, correct predictions for charged fermion masses,
the Higgs mass constraint, the experimental bound on the proton lifetime, and finally
the experimental bounds from flavour physics and from direct searches for supersymmetric particles.
We also require perturbativity of the model.
The observed down-type quark and charged lepton masses are accounted for by
generation-dependent supersymmetric threshold corrections with large
A-terms [8, 9, 10, 11, 12, 13, 14],
which in turn are bounded by considerations related to the stability of the electroweak
vacuum [15, 16, 17].
The soft supersymmetry breaking terms are required to respect SU(5) invariance
but are not assumed to be flavour universal as in the so-called mSUGRA model
(as a matter of fact, generation-dependent A-terms are needed to correct the SU(5)
fermion mass relations). To avoid potentially dangerous flavour effects,
we will therefore take the GUT-scale sfermion soft terms to be
aligned with fermion masses^{7}^{7}7Flavour violating soft terms
are going to be generated from renormalization group running (due to the CKM matrix,
and possibly to right-handed neutrino couplings and R-parity violating couplings),
but this will lead only to small effects in flavour-changing processes, especially in view
of the large superpartner masses. We shall therefore neglect these small RG effects. [18].
Neutrino masses can be generated either
through bilinear R-parity violation^{8}^{8}8The SU(5)-invariant
bilinear R-parity violating operators also contain baryon number
violating terms, which however are harmless by virtue of the doublet-triplet splitting
if [19, 20].
Contrary to Ref. [21], we assume negligible trilinear R-parity violating
couplings. [19, 22, 23, 24] or
by adding right-handed
neutrinos to the model in order to implement the seesaw mechanism [25].
Finally, if the neutralino sector does not contain a suitable candidate for dark matter
(either because the lightest neutralino is not the lightest supersymmetric particle,
or because it is too heavy), this role may be played by the gravitino.

Not surprisingly, the main constraint on the superpartner mass scale comes from proton decay, which pushes the spectrum above the scale (the conflict between the proton decay and unification constraints pointed out in Ref. [7] is resolved by relaxing the upper bound on the superpartner masses). Perturbativity imposes an upper bound on the coloured triplet mass, which translates into an upper limit on superpartner masses once gauge coupling unification is imposed. The observed Higgs mass, together with vacuum metastability constraints associated with the stop A-term, also excludes large portions of the parameter space. A priori, there is no guarantee that the minimal renormalizable supersymmetric SU(5) model will survive all these constraints, even if very large values of the soft terms are allowed.

To give an idea of how the parameter space is restricted by the various phenomenological requirements, we show in Fig. 1 the approximate constraints in the (, ) plane obtained by making several simplifying assumptions. Namely, all sfermion masses are taken to be equal to at the low scale, as well as the parameter and ( is assumed, and and the parameter are computed from the electroweak symmetry breaking conditions), while the gaugino masses assume a common value at the GUT scale and are split by renormalization group running. Obviously, these inputs are not consistent with SU(5) symmetry of the soft terms at the GUT scale, but they make it possible to show several constraints in a single plot. Imposing SU(5) boundary conditions will significantly affect quantities such as the heavy colour triplet mass and more crucially the proton lifetime, making the investigation of the parameter space of the minimal renormalizable supersymmetric SU(5) model more involved than suggested by Fig. 1. As we are going to see, phenomenologically viable points typically feature superpartners in the range, with values of between and , but lighter spectra with supersymmetric particles as light as a few can also be found.

In the process we have generalized the procedure of Refs. [26, 27] for deriving approximate semi-analytic solutions to the one-loop renormalization group equations (RGEs) for the MSSM soft terms. In this way we are able to write the low-energy soft terms as linear or quadratic functions of the initial (GUT-scale) parameters, making it possible to explore the parameter space without having to solve the RGEs for each point. In practice, one just needs to solve numerically the RGEs for gauge and Yukawa couplings for each choice of and , the matching scale between the SM and the MSSM. The low-energy soft terms and their dependence on the other model parameters are then simply given by linear and quadratic algebraic equations.

The paper is organized as follows. Section 2 introduces the minimal renormalizable supersymmetric SU(5) model. In Section 3, the running of the model parameters is discussed, and the semi-analytical procedure used to solve the renormalization group equations (RGEs) is presented in Section 4 (more details can be found in Appendix A). Proton decay and the constraints associated with the metastability of the electroweak vacuum are addressed in Sections 5 and 6, respectively, with technical details relegated to Appendices B, C and D. Finally, we present our results in Section 7.

## 2 The minimal renormalizable supersymmetric SU(5) model

In this section, we briefly describe the minimal renormalizable supersymmetric SU(5) model [1] and present our notations. The Higgs sector includes the adjoint , which spontaneously breaks the SU(5) gauge group to SU(3)SU(2)U(1), and a fundamental and anti-fundamental representations and containing the two light Higgs doublets responsible for electroweak symmetry breaking. The Higgs fields also includes a heavy pair of colour triplet and antitriplet that mediate proton decay through operators. All matter fields belong to and representations (leaving aside right-handed neutrinos in the singlet representation that may also be present), where is the generation index.

In order to connect the minimal renormalizable supersymmetric SU(5) model with experimental data, one has to deal with three different theories: SU(5) above the unification scale , the Minimal Supersymmetric Standard Model (MSSM) between and the supersymmetry scale , and the Standard Model (SM) between and the weak scale . Since the heavy GUT states (resp. the superpartners) are not degenerate in mass, the matching between the SU(5) theory and the MSSM at (resp. between the MSSM and the SM at ) will involve threshold corrections.

In the next subsections we give the relevant parts of the corresponding Lagrangians (i.e. the Higgs and Yukawa sectors and the soft supersymmetry breaking terms, which determine the superpartner spectrum) and we specify our notations and assumptions.

### 2.1 The SU(5) model

The superpotential of the minimal renormalizable supersymmetric SU(5) model is determined by its field content, gauge invariance and renormalizability. It can be divided into two parts describing the Higgs and Yukawa sectors, respectively:

(2.1) | ||||

(2.2) |

in which we have omitted terms involving right-handed neutrinos as well as R-parity violating couplings that may be present, depending on how neutrino masses are generated. After having solved the equations of motion for the in the SM singlet direction:

(2.3) |

and performed the fine-tuning needed to achieve doublet-triplet splitting in the Higgs sector, one can write down the masses of the heavy states in terms of the SU(5) superpotential parameters:

(2.4) |

where is the mass of the SU(5) gauge bosons in the representations of the SM gauge group; is the mass of the colour triplet and antitriplet pair contained in ; and , and are the masses of the SM singlet, SU(2) triplet and SU(3) octet components of , respectively. Demanding that the superpotential couplings and be in the perturbative regime and taking into account the fact that the unified gauge coupling is of order , one obtains the following constraint:

(2.5) |

We also assume that supersymmetry breaking is coming from above the GUT scale, as for example in supergravity. In practice this means that the soft terms should be SU(5) symmetric at the GUT scale:

(2.6) | |||||

We will consider the possibility of generation-dependent soft terms (as explained in the introduction, generation-dependent A-terms are needed to correct the SU(5) fermion mass relations), but in order to comply with the strong constraints coming from flavour physics we must ensure that they do not induce large flavour-violating effects. To this end, we assume that the soft sfermion mass matrices are diagonal in the basis in which the Yukawa couplings are diagonal, and that the A-term matrices are diagonal in the corresponding fermion mass eigenstate basis:

(2.7) | |||||

(2.8) |

so that all flavour violation at the GUT scale is concentrated in the up squark sector and controlled by the CKM angles, yielding an effective alignment of sfermion soft terms with fermion masses [18]. In addition, we assume that the soft masses of the first two generations of sfermions are degenerate:

(2.9) |

Finally, we will take and the A-terms (as well as the parameter ) to be real. This may be more than what we need to evade flavour and CP constraints from low-energy experiments, especially in view of the fact that the superpartner spectrum is heavy, but this choice also helps reducing the number of parameters. In our subsequent exploration of the parameter space of the minimal renormalizable supersymmetric SU(5) model we shall completely neglect flavour violation in the sfermion sector, including the small amount of flavour violation that is generated from the running of the soft terms.

### 2.2 Mssm

Below the scale , the relevant theory is the MSSM, with superpotential

(2.10) |

and soft supersymmetry breaking terms

(2.11) | |||||

where the contraction of indices is understood (for instance, stands for , where are indices and is the totally antisymmetric tensor with ). Due to the boundary conditions (2.7) and (2.8), and to the fact that we are neglecting the effects of the CKM matrix in the running, the sfermion soft terms keep a diagonal form all the way down to low energies.

### 2.3 Standard Model

Below the matching scale

(2.12) |

(where the last approximation is valid as long as the mixing in the stop sector is small, i.e. ), the relevant theory is the Standard Model. The Higgs potential is given by (with the SM Higgs doublet given by in the decoupling limit):

(2.13) |

while the Yukawa Lagrangian is

(2.14) |

where .

## 3 Renormalization group equations

In this section, we collect the SM and MSSM renormalization group equations (RGEs) and various expressions used in our analysis (from boundary to matching conditions). We use the SM RGEs [28] between and , and the MSSM RGEs [29] between and . The gauge and Yukawa couplings, as well as the Higgs quartic coupling are evolved with the 2-loop RGEs, with the 1-loop threshold corrections accounting for the splitting of superpartner masses added at the scale . All soft parameters (A-terms, gaugino masses and soft scalar masses) are run at 1 loop.

### 3.1 Gauge couplings

The 2-loop RGEs for the gauge couplings ( for the gauge groups , and , respectively, with ) read:

(3.1) |

where , being the renormalization group scale, and the -function coefficients below and above are given by:

(3.2) | |||||

(3.3) | |||||

(3.4) |

In the last term of Eq. (3.1), the MSSM Yukawa couplings should be replaced with the SM ones () below .

At , the running gauge couplings should be converted from the scheme to the scheme, in which the MSSM RGEs are written:

(3.5) | ||||

(3.6) | ||||

(3.7) |

where and .

#### 3.1.1 Threshold corrections to gauge couplings

Imposing gauge coupling unification at the GUT scale:

(3.8) |

implies certain relations among the masses of the various thresholds (supersymmetric partners of the SM fields and heavy GUT fields). Adding 1-loop threshold corrections [30, 31, 32, 33, 34] to the running gauge couplings evolved with the 2-loop MSSM RGEs between the scales and yields the following relations ():

(3.9) |

where the denote the values of the gauge couplings obtained by solving numerically the 2-loop MSSM RGEs with all superpartner masses at the scale , and runs over the superpartners. Their contributions to the -function coefficients are given by:

(3.10) |

while the contributions of the heavy GUT fields are:

(3.11) |

Taking appropriate combinations of the three equations (3.9), one obtains:

(3.12) | ||||

(3.13) | ||||

(3.14) |

At the 1-loop level, the matching scales and drop out from Eqs. (3.1.1) and (3.1.1), while only drops out from Eq. (3.1.1). Since , one can neglect the mixing between the higgsinos and the electroweak gauginos and identify:

(3.15) | ||||

(3.16) | ||||

(3.17) |

where is the ratio of the two MSSM Higgs doublet vevs, and satisfies the electroweak symmetry breaking (EWSB) condition (again neglecting ):

(3.18) |

In spite of the initial historical success [35, 36, 37, 38], it is well known that gauge couplings do not unify accurately at the 2-loop level in the MSSM with TeV-scale superpartners. High-energy threshold corrections thus play a crucial role in achieving precise unification [33]. Using Eqs. (3.1.1) and (3.1.1), one can express the combinations of GUT state masses needed for exact 2-loop unification in terms of the superpartner masses [33]. Assuming that all superpartners have masses equal to , one obtains for the colour triplet mass and for the combination of heavy gauge boson and adjoint Higgs masses :

(3.19) | ||||

(3.20) |

where we have used the fact that in the minimal renormalizable supersymmetric SU(5) model. For superpartner masses in the TeV range, the colour triplet is far too light and makes the proton decay too fast, which led Ref. [7] to conclude that the minimal renormalizable supersymmetric SU(5) model is excluded (this conclusion has been found to be mitigated at the three-loop level [39] though, and can be avoided for a specific choice of the soft terms [3]).

### 3.2 Yukawa couplings

For the Yukawa couplings, we use the 2-loop Standard Model RGEs [28] below the scale , and the 2-loop MSSM RGEs [29] above it. We neglect all CKM contributions and work with diagonal Yukawa matrices. In the absence of threshold corrections, the matching conditions at are:

(3.21) | ||||

(3.22) | ||||

(3.23) |

where the couplings (resp. the couplings) are the diagonal entries of the SM Yukawa matrices (resp. of the MSSM Yukawa matrices ).

#### 3.2.1 Threshold corrections to Yukawa couplings

At the GUT scale, the SU(5)-invariant boundary conditions apply:

(3.24) | ||||

(3.25) |

Threshold corrections due to the splitting of the heavy GUT state masses slightly modify these relations (see later). After running down to low energy, Eqs. (3.25) lead to predictions for down-type quark and charged lepton masses that are in gross contradiction with the data. Supersymmetric threshold corrections at the scale may cure this problem.

##### Supersymmetric threshold corrections to light fermion masses

In the following, we will neglect supersymmetric threshold corrections to the leptonic
Yukawa couplings, and consider only the corrections to the down-type quark
Yukawa couplings^{9}^{9}9Supersymmetric threshold corrections to up-type quark
masses remain under control, as no large A-terms are needed to correct the SU(5)
prediction. As for the top quark mass, even the large stop mixing that may be needed
to reproduce the measured Higgs boson mass does not induce sizable threshold corrections.,
whose dominant contributions are proportional to and
(see however Ref. [40]).
This will allow us to derive the SU(5) Yukawa couplings by simply running
the charged lepton couplings up to the GUT scale.
The leading supersymmetric threshold corrections to down-type quark masses are given
by the gluino and higgsino contributions
[8, 9, 10, 12, 13, 14]
(the latter can safely be neglected for the first two generations):

(3.26) |

where

(3.27) | ||||

(3.28) | ||||

(3.29) |

and the loop function is defined by:

(3.30) |

with the limits

(3.31) | ||||

(3.32) |

The matching is done at the scale :

(3.33) |

where

(3.34) |

in which .
As a first approximation, – Yukawa unification is a relatively successful
prediction of SU(5), while the discrepancy between the prediction and the data
is much more important for the first two generations.
As can be seen from Eq. (3.26), the non-holomorphic
() contributions to and
are the same for equal first two generation squark masses, while the ratios
and are widely different. This implies that large A-terms and
are needed to bring these ratios into agreement with experimental data^{10}^{10}10Incidentally,
it turns out that the correct ratio cannot be obtained
from the corrections proportional to and to alone,
and that a large is also needed., which in turn
makes the electroweak vacuum metastable [12, 13].
This issue will be discussed in Section 6.

##### High-scale thresholds corrections to and

In addition to supersymmetric corrections at the superpartner mass scale, Yukawa couplings are also subject to high-scale threshold corrections due to the heavy GUT states. These may affect in particular bottom-tau Yukawa unification, which as explained before is an important constraint on the model, hence one must take them into account. In practice, all one needs is the difference between and induced by the GUT-scale threshold corrections. One can check that it is given by:

(3.35) |

GUT threshold corrections also affect strange quark-mu and down quark-electron Yukawa unification, but the numerical effect is negligible compared with the size of the low-scale (supersymmetric) threshold corrections that are needed to account for the observed masses.

### 3.3 Higgs quartic coupling

For heavy stop masses, the proper way to compute the lightest supersymmetric Higgs boson mass (for the standard computation, see Refs. [41, 42]) is to consider the effective theory below the scale , in which all superpartners and heavy Higgs bosons have been integrated out and the Higgs boson mass is determined from the Higgs quartic coupling , with the value of determined by the supersymmetric theory valid above . At tree level, the matching condition is , while at the 1-loop level it is given by:

(3.36) |

where the first term on the right-hand side of Eq. (3.36) is the tree-level contribution, accounts for the conversion of the gauge couplings from the scheme to the scheme, and and are the one-loop threshold corrections due to scalars and electroweak gauginos/higgsinos, respectively, whose expressions can be found in Ref. [43]. The dominant contributions are the ones proportional to in , which in the case where all sparticle masses lie close to (and in particular ) reduce to the leading stop mixing term:

(3.37) |

For large values of and/or large values of , one should also include on the RHS of Eq. (3.37) the leading sbottom and stau contributions, which in the limit read: