Resurrecting the minimal renormalizable supersymmetric SU(5) model

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 $10^2$ to $10^4$ TeV, with tan beta values between 2 and 5, but lighter spectra with single states around 10 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.


Introduction
The minimal renormalizable supersymmetric SU(5) model [1], with just 3 pairs of 10 ⊕5 fermion representations and an adjoint 24 plus a 5 ⊕5 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 Q 1 Q 1 Q 2 L k /M P lanck (k = 1, 2, 3) induce proton decay at an unacceptable rate unless they come with coefficients smaller than about 10 −7 . 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 1 .
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 2 . 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 10 TeV 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 3 [18]. Neutrino masses can be generated either through bilinear R-parity violation 4 [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 10 TeV 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 (tan β, m susy ) plane obtained by making several simplifying assumptions. Namely, all sfermion masses are taken to be equal to m susy at the low scale, as well as the µ parameter and m Hu (m 2 Hu > 0 is assumed, and m 2 H d and the B parameter are computed from the electroweak symmetry breaking conditions), while the gaugino masses assume a common value M 1/2 = m susy 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 O(10 2 − 10 4 ) TeV range, with values of tan β between 2 and 5, but lighter spectra with supersymmetric particles as light as a few 10 TeV 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 per turb ativ ity pr ot on de ca y vacuum metastability H ig g s m a s s tanβ m susy Figure 1: Approximate constraints on the minimal renormalizable supersymmetric SU(5) model in the (tan β, m susy ) plane. The region to the left of the red curve is excluded by the measured Higgs boson mass, while the area to the right of the purple curve is excluded by vacuum stability considerations (namely, the value of the stop A-term needed to reproduce the observed Higgs boson mass would render the electroweak vacuum unstable with a lifetime shorter than the age of the universe). The experimental lower bound on the proton lifetime is satisfied above the green line, and the perturbativity constraint m T < M P lanck /10, where m T is the colour triplet mass, is satisfied below the black line. The green area is consistent with all four constraints. In this plot, the sfermion masses, the µ parameter and m Hu are all equal to m susy , as well as the common gaugino mass at the GUT scale. Since these values are not consistent with SU(5) boundary conditions, the figure should be considered as illustrative only.
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 tan β and m susy , 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 24 H , which spontaneously breaks the SU(5) gauge group to SU(3)×SU(2)×U(1), and a fundamental and anti-fundamental representations 5 H and5 H containing the two light Higgs doublets responsible for electroweak symmetry breaking. The 5 H ⊕5 H Higgs fields also includes a heavy pair of colour triplet and antitriplet that mediate proton decay through d = 5 operators. All matter fields belong to 10 i and5 i representations (leaving aside right-handed neutrinos in the singlet representation that may also be present), where i = 1, 2, 3 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 M GU T , the Minimal Supersymmetric Standard Model (MSSM) between M GU T and the supersymmetry scale m susy , and the Standard Model (SM) between m susy and the weak scale m Z . Since the heavy GUT states (resp. the superpartners) are not degenerate in mass, the matching between the SU(5) theory and the MSSM at M GU T (resp. between the MSSM and the SM at m susy ) 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.

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: 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 24 H in the SM singlet direction: 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: where m V is the mass of the SU(5) gauge bosons in the representations (3, 2) −5/3 ⊕(3,2) +5/3 of the SM gauge group; m T is the mass of the colour triplet and antitriplet pair (T,T ) contained in 5 H ⊕5 H ; and m 1 , m 3 and m 8 are the masses of the SM singlet, SU(2) triplet and SU(3) octet components of 24 H , respectively. Demanding that the superpotential couplings λ 5 and η H be in the perturbative regime and taking into account the fact that the unified gauge coupling g GU T is of order 1, one obtains the following constraint: 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: 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 Λ5 are diagonal, and that the A-term matrices are diagonal in the corresponding fermion mass eigenstate basis:

7)
A 10 ij = A 10 i δ ij in the basis Λ 10 ij = Λ 10 i δ ij , (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: Finally, we will take M 1/2 and the A-terms (as well as the µ parameter µ ≡ µ H − 3η H σ 0 ) 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.

MSSM
Below the scale M GU T , the relevant theory is the MSSM, with superpotential and soft supersymmetry breaking terms where the contraction of SU (2) L indices is understood (for instance, H u H d stands for are SU (2) L indices and ij is the totally antisymmetric tensor with 12 = +1). 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.

Standard Model
Below the matching scale (where the last approximation is valid as long as the mixing in the stop sector is small, , the relevant theory is the Standard Model. The Higgs potential is given by (with the SM Higgs doublet given by h = √ 2 (cos β h d + sin β iσ 2 h * u ) in the decoupling limit): while the Yukawa Lagrangian is whereh ≡ iσ 2 h * .

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 m Z and m susy , and the MSSM RGEs [29] between m susy and M GU T . 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 m susy . All soft parameters (A-terms, gaugino masses and soft scalar masses) are run at 1 loop.

Threshold corrections to gauge couplings
Imposing gauge coupling unification at the GUT scale: 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 m susy and M GU T yields the following relations (i = 1, 2, 3): while the contributions of the heavy GUT fields are: Taking appropriate combinations of the three equations (3.9), one obtains: . (3.14) At the 1-loop level, the matching scales M GU T and m susy drop out from Eqs. (3.12) and (3.13), while only m susy drops out from Eq. (3.14). Since m susy m Z , one can neglect the mixing between the higgsinos and the electroweak gauginos and identify: Hu (m susy ))(tan 2 β + 1) / (tan 2 β − 1) , (3.16) mw/mg α 2 (mg)/α 3 (mg) , (3.17) where tan β ≡ h 0 u / h 0 d is the ratio of the two MSSM Higgs doublet vevs, and µ satisfies the electroweak symmetry breaking (EWSB) condition (again neglecting m 2 Z ): 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. (3.12) and (3.13), 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 m susy , one obtains for the colour triplet mass and for the combination of heavy gauge boson and adjoint Higgs masses m   where we have used the fact that m 3 = m 8 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]).

Yukawa couplings
For the Yukawa couplings, we use the 2-loop Standard Model RGEs [28] below the scale m susy , 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 m susy are: where the h couplings (resp. the λ couplings) are the diagonal entries of the SM Yukawa matrices H U,D,E (resp. of the MSSM Yukawa matrices Λ U,D,E ).

Threshold corrections to Yukawa couplings
At the GUT scale, the SU(5)-invariant boundary conditions apply: 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 m susy 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 5 , whose dominant contributions are proportional to α 3 and λ 2 t (see however Ref. [40]). This will allow us to derive the SU(5) Yukawa couplings Λ5 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): (3.29) and the loop function I 3 is defined by: , (3.30) with the limits The matching is done at the scale m susy : (3.34) in which v = h 0 . As a first approximation, b -τ 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. This implies that large A-terms A d and A s are needed to bring these ratios into agreement with experimental data 6 , which in turn makes the electroweak vacuum metastable [12,13]. This issue will be discussed in Section 6.
High-scale thresholds corrections to λ b and λ τ 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 λ b (M GU T ) and λ τ (M GU T ) induced by the GUT-scale threshold corrections. One can check that it is given by: 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.

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 m susy , in which all superpartners and heavy Higgs bosons have been integrated out and the Higgs boson mass is determined from the Higgs quartic coupling λ, with the value of λ(m susy ) determined by the supersymmetric theory valid above m susy . At tree level, the matching condition is λ = (3g 2 1 /5 + g 2 2 ) cos 2 2β/4, while at the 1-loop level it is given by: where the first term on the right-hand side of Eq. (3.36) is the tree-level contribution, ∆λ reg accounts for the conversion of the gauge couplings from the DR scheme to the MS scheme, and ∆λ φ and ∆λ χ 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 (m t /v) 4 in ∆λ φ , which in the case where all sparticle masses lie close to m susy (and in particular mQ 3 mũ 3 m susy ) reduce to the leading stop mixing term: For large values of tan β and/or large values of X b,τ , one should also include on the RHS of Eq. (3.37) the leading sbottom and stau contributions, which in the limit mQ 3 md 3 mL 3 mẽ 3 m susy read: Considering only the leading term (3.37), one can see that for each value of m susy and tan β there exist either 4, 2 or 0 different values of X t satisfying the matching condition (3.36). There also exists a lower bound on m susy for each value of tan β, reached when X t becomes where the stop mixing contribution reaches its maximum [44] (when higher order corrections are included the Higgs mass also depends on the sign of X t ). This is illustrated in Fig. 2, in which the lower bound on m susy is represented by the green curve, assuming a superpartner spectrum as in Fig

A-terms
As discussed in the introduction and in Subsection 3.2, we assume the A-term matrices to be diagonal 7 in the corresponding fermion mass eigenstate basis, with generation-dependent

entries:
(3.42) These matrices are run with the 1-loop MSSM RGEs, with the SU(5) boundary conditions imposed at the GUT scale:

Gaugino masses
For the running of gaugino masses we use the 1-loop RGEs: and impose the SU(5) boundary condition at the GUT scale: by the small CKM angles (the same statement holds for the soft sfermion mass matrices). We shall therefore neglect this effect in the following.

Soft scalar masses
We also assume the soft sfermion mass matrices m 2X (X = Q, u c , e c , L, d c ) to be diagonal with generation-dependent entries: however with m 2X 1 = m 2X 2 imposed at the GUT scale. Furthermore, SU(5) invariance requires the following relations to hold at M GU T (i = 1, 2, 3): Hence the splitting of the soft sfermion masses within SU (5) representations and between the first two generations is only due to the running, performed at the 1-loop level as for the other soft terms. As for the soft Higgs masses, we allow the possibility of different boundary conditions for the two Higgs doublets of the MSSM, namely m 2 . Note that our boundary conditions are less restrictive than the so-called minimal supergravity ansatz (mSUGRA), which assumes universal scalar and gaugino masses as well as A-terms proportional to the Yukawa couplings. By contrast we allow for some generation dependence in the sfermion soft terms, but we require them to be aligned with fermion masses (exactly at the GUT scale, approximately at low energy due to the running) in order to minimize supersymmetric contributions to flavour-violating processes.

µ and Bµ terms
Contrary to the other MSSM parameters, the µ and Bµ terms are not fixed at the GUT scale and renormalized down to low energy, but rather determined from the tree-level electroweak symmetry breaking conditions at the scale m susy , as is usually done (by inserting the running soft terms in the tree-level Higgs potential one ensures that the most relevant 1-loop radiative corrections are taken into account). This procedure yields: while the sign of µ remains undetermined.

Solutions of the RGEs for soft terms
Here we present (approximate) semi-analytic solutions to the 1-loop renormalization group equations for the MSSM soft terms. More details about the procedure used to derive them can be found in Appendix A.

Gauge and Yukawa couplings
As a first step, for each choice of m susy and tan β, one runs the gauge couplings g i (t) (i = 1, 2, 3) and the Yukawa couplings

Gaugino masses
The solutions to the 1-loop RGEs for gaugino masses read:

A-terms
As explained in Appendix A, the hierarchy of Yukawa couplings makes it possible to solve the 1-loop RGEs for A-terms in a sequential way, from A t to A e . With the convention that the index n runs over the ordered values {t, b, τ, c, s, µ, u, d, e} (such that e.g. p < c means p = t, b or τ ), one can write the (approximate) solutions as: where β A n is a function of the gauge and Yukawa couplings, and γ A n is the product of λ n times a linear combination of g 2 i M i and λ m A m with m < n, all of which are already known quantities, since the RGEs for the A m 's are solved in order of increasing m. One can therefore rewrite Eq. (4.2) in the form: in which the coefficients a nm (t) and b n (t) are integrals that can be evaluated numerically after having solved the MSSM RGEs for the gauge and Yukawa couplings. Imposing the In practice it may be more convenient to express the A n (t) in terms of other input parameters, for example M 3 (m susy ) and the A d i (m susy ) (i = 1, 2, 3), since these quantities enter the supersymmetric threshold corrections to the down-type quark masses needed to fit the experimental values. One may also want to trade A 10 3 for A t (m susy ), which is directly constrained by the measured value of the Higgs mass. To this end, it suffices to invert the linear relations (4.3) so that the A n (t) can be rewritten as a function of the desired input parameters. In the following, we shall choose:

Soft scalar masses
Following Appendix A, we first introduce the combinations of masses (which appear in the RGEs for soft scalar masses): and The 1-loop RGE for S is easily integrated to give: due to the SU(5) boundary conditions on soft scalar masses, while the RGEs for the Σ n 's can be solved sequentially in a similar way to the Aterm RGEs, taking advantage of the hierarchy of Yukawa couplings to neglect subdominant terms. We then define: With these ingredients one can express the solutions to the 1-loop RGEs for soft scalar masses as: where the index α runs over {H u , H d ,Q 3,2,1 ,ũ c 3,2,1 ,ẽ c 3,2,1 ,L 3,2,1 ,d c 3,2,1 }, and the numerical coefficients T α n , U α n and V α i (resp. R α ) are given in Table 4 (resp. Table 3) of Appendix A. Plugging the previously derived semi-analytic expressions for A n (t), Eq. (4.3), and for Σ n (t) into Eq. (4.13), one then arrives at in which the coefficients c αβ (t), d αnm (t), e αn (t) and f α (t) are integrals that can be evaluated numerically just from the knowledge of the solutions to the 2-loop MSSM RGEs for gauge and Yukawa couplings. Finally, imposing the SU(5) boundary conditions on soft terms, one obtains the running soft scalar masses m 2 α (t) at an arbitrary scale as quadratic functions of the SU(5) soft parameters , with numerical coefficients depending on the choice of m susy and tan β (and very mildly on M GU T , m T and m V ).
As was done for A-terms, one can easily rewrite the running soft scalar masses in terms of other input variables by inverting the system of equations (4.3) and/or (4.14). As explained before, a convenient choice for exploring the parameter space of the minimal renormalizable supersymmetric SU(5) model is to trade the 7 SU(5) parameters M 1/2 , A 10 i and A5 i for M 3 (m susy ), A d i (m susy ) and A u i (m susy ). One may also invert 8 of the 17 equations (4.14) in order to replace the GUT-scale masses m 10 i , m5 i , m Hu (M GU T ) and m H d (M GU T ) by 8 low-energy soft scalar masses, so that (for given values of m susy and tan β) the whole supersymmetric spectrum is parametrized by 15 low-energy input variables.

Proton decay
Proton decay is one of the main prediction of Grand Unified Theories, and since it has not been observed yet, it sets strong constraints on the parameter space of the minimal renormalizable supersymmetric SU(5) model. Qualitatively, the proton lifetime behaves as: implying a tan β-dependent lower bound on the superpartner mass scale.
To compute precisely the proton lifetime we will need the following input parameters: here written in the Wolfenstein's parametrization, with [45]: Details about the computation of the proton lifetime can be found in Appendix B. The predicted proton lifetime is compared with the experimental constraint τ (p → K +ν ) > 2.3 × 10 33 yrs (90% C.L.) [46].

Vacuum (meta)stability
In the general MSSM, some regions of the parameter space lead to instabilities of the electroweak vacuum. One may encounter two kinds of dangerous situations.
The first one is the possible existence of directions in field space along which the potential is unbounded from below (UFB). To remain on the safe side, we will allow only the points in parameter space that do not possess any such direction. The associated constraints on the model parameters will be summarized in Subsection 6.1.
The second one is the metastability of the electroweak vacuum, which due to the large trilinear soft terms needed to correct the SU(5) predictions for fermion masses cannot be avoided. This means that there are minima in field space, lower than the electroweak vacuum, into which it will eventually decay. These new minima are the so-called charge and colour breaking (CCB) vacua. Although one cannot forbid the decay of the electroweak vacuum into a lower minimum, one can check whether its lifetime is long enough on cosmological time scales. The procedure followed in our analysis is summarized in Subsection 6.2.
All computations in this section are done at the tree level. The generalization to higher orders is conceptually straightforward, although technically much more involved, so we will leave it for future work. Below we summarize the discussion presented in Appendices C and D, where technical details and relevant references can be found.

Unbounded from below directions
As explained in Appendix C, the tree-level constraints associated with the absence of UFB directions can be written as (neglecting m Z ): which must be satisfied for any of the three slepton generations.

Charge and colour breaking vacua
We shall discuss the constraint applying to A t separately from the ones applying to all other A-terms. The reason to treat them differently is that in the second case the D-terms can be considered to be vanishing to a good approximation, thus providing constraints on the fields, while in the first case this assumption is not justified due to the large top Yukawa coupling 8 . We will nevertheless set the colour D-terms to zero for simplicity, but allow for non-vanishing hypercharge and SU(2) D-terms.

Constraints on A n (n = t)
Let us first define: and write the SU(2) D-term constraint: where ψ,ψ are sfermion fields and H,H Higgs fields that parametrize a specific direction in field space along which a CCB minimum is present. The different possibilities for the constant fields ψ,ψ, H,H, their mass parameters and the coefficients c and c z are summarized in Table 5 of Appendix D.
To evaluate the lifetime of the electroweak vacuum, we consider the (normalized) bounce action, which for A n = A t can be approximated by: ψ, H,H) . (6.6) This action is then minimized by varying the direction in field space (ψ,ψ, H,H), subject to the constraint (6.5). In order for the lifetime of the electroweak vacuum to be larger than the age of the universe, the minimum of S must satisfy Notice that one needs to minimize S with respect to two variables only. Indeed, one of the variables ψ,ψ, H,H is fixed by Eq. (6.5), and the quantity (6.6) only depends on ratios of fields. The minimization is performed numerically. A point of the parameter space is admitted if the bounce action satisfies Eq. (6.7).

Constraint on A t
We now define: Due to the large value of the top quark Yukawa coupling, the bounce action can no longer be approximated by Eq. (6.6). One should instead minimize The minimization goes again over two variables (for example H u /t and H d /t), and is done numerically.

Results and discussion
We are now ready to address the question we asked in the introduction, namely whether the minimal renormalizable supersymmetric SU(5) model can be considered as a viable extension of the Standard Model. To answer this question, we must scan over the parameter space of the model and search for points passing all phenomenological and theoretical constraints 9 (precise gauge coupling unification, correct predictions for the Higgs boson and charged fermion masses, proton lifetime, experimental lower bounds on superpartner masses, metastability of the vacuum and perturbativity of the model). This is not a straightforward task, as the model involves a large number of parameters with boundary conditions defined at the GUT scale, while most constraints apply at low energy. In order to ease the whole analysis, we will use the semi-analytic solutions to the soft term RGEs obtained in Section 4. This will enable us to perform a much more efficient scan -even though it still involves a large number of parameters. Before going into this programme, let us first try to identify the region of the parameter space in which viable points are likely to be found. For a qualitative discussion of how each phenomenological or theoretical requirement constrains the model, we will consider only two parameters, tan β and the overall superpartner mass scale m susy : • a powerful constraint on the parameter space comes from accommodating the measured Higgs mass, which provides a tan β-dependent lower bound on m susy (the lower the value of tan β, the higher the value of m susy ). Sizable mass splittings among superpartners can change this bound, but not very drastically.
• another important constraint comes from the non-observation of proton decay, which provides another lower bound on m susy , with a different dependence on tan β. The actual value of the bound depends on the details of the superpartner spectrum.
• requiring the masses of the heavy GUT states derived from gauge coupling unification to remain below the cut-off scale M P lanck bounds m susy from above. The main constraint comes from the Higgs colour triplet mediating proton decay, whose mass m T strongly depends (through the gauge coupling unification condition) on the higgsino mass µ.
• the parameter space is also bounded by vacuum metastability constraints associated with large values of the A-terms. These are unavoidably present due to the sizable threshold corrections needed in the down-type quark sector. Furthermore, a large A t is necessary to accommodate the observed Higgs mass in some regions of the parameter space. Although the required values of the A-terms do not necessarily threaten the metastability of the electroweak vacuum, significant regions of the parameter space where the lifetime of the universe would be too short are excluded.
• the requirement that the third generation Yukawa couplings should remain perturbative up to the GUT scale excludes some portions of the parameter space in the small tan β region (top quark) and potentially also in the large tan β region (bottom quark and tau Yukawa couplings).
• the perturbativity of the parameters of the SU(5) superpotential, reflected in the condition m T , m 3 , m 8 ∼ < m V , can easily be satisfied by using the freedom allowed by gauge coupling unification. Indeed, for fixed M GU T and superpartner masses, m T and   One can therefore increase m V and decrease m 3 = m 8 by diminishing λ 5 and µ 5 in the superpotential (2.1) in such a way that σ 0 = µ 5 /λ 5 increases. Simultaneously, η H should be made smaller so that m T = 5η H σ 0 stays constant.
One can illustrate how these constraints restrict the parameter space of the model by plotting them in the (tan β, m susy ) plane, in the spirit of Fig. 1 in the introduction. To be able to do this, we assume a simplified superpartner spectrum with a common scale m susy for all sfermion masses, the µ parameter, m Hu and the SU(5) gaugino mass parameter M 1/2 . The down-type squark A-terms and A t are chosen to fit the fermion masses and the Higgs mass, while the slepton A-terms are taken to be 2/3 of their downtype quark counterparts 10 , and A u and A c are assumed to vanish. With m 2 H d and the B parameter determined by the electroweak symmetry breaking conditions, the whole MSSM spectrum, including the heavy Higgs masses, can be computed. While these inputs are not fully consistent with SU(5)-invariant boundary conditions at the GUT scale, they make it possible to display all the constraints discussed above in a two-dimensional plot. The result can be seen in Fig. 3, a simplified version of which was shown in the introduction. Let us now comment it: • the measured Higgs boson mass excludes the region below the red curve (which corresponds to maximal stop mixing); • the region below the green line is ruled out by the experimental lower bound on the proton lifetime (assuming the high-energy phases appearing in the proton decay amplitude, see Eq. (B.7), all vanish). The shape of this line can be understood by noting that the proton lifetime approximately scales as while given the assumptions made on the superpartner spectrum, the colour triplet mass is proportional to 11 m T ∝ m 5/6 susy (tan 2 β + 1) 1/12 , giving τ (p → K +ν ) ∝ m 11/3 susy tan 2 β (tan 2 β + 1) −11/6 ; (7.3) • the region above the purple curve is excluded by the vacuum metastability constraint on A t , while the region above the brown curve (which is almost entirely due to A b ) is ruled out by the constraints on all other A-terms. In drawing these curves we used the metastability conditions derived in Appendix D, but they can be approximated to a very good degree by: for A t and A b , respectively. Note that the constraint on |A b | is more easily satisfied for large tan β values due to m 2 H d = m 2 susy (2 tan 2 β − 1); • perturbativity of the top quark Yukawa coupling excludes the region to the left of the blue curve (which corresponds to λ t (M GU T ) = 1). There is no similar constraint for λ b and λ τ , which are always in the perturbative regime in the region of the parameter space shown in Fig. 3; • the constraint m T < M P lanck /10 (resp. m T < M P lanck ) rules out m susy values above the solid black line (resp. the dashed black line), while there are no additional constraints from m 3 , m 8 , m T < m V < M P lanck /10; • the pale green area is the region of the parameter space that is allowed by all the above constraints.
The tentative conclusion one can draw from Fig. 3, even though the assumptions made on the soft terms are not consistent with SU(5)-invariant boundary conditions at the GUT scale, is that viable points satisfying all phenomenological and theoretical constraints are likely to be found in the region bounded by 2 ∼ < tan β ∼ < 5 and 100 TeV ∼ < m susy ∼ < 1000 TeV. This definitely needs confirmation from a more careful investigation of the parameter space. Namely, one should scan over the parameters of the model, which besides tan β, sign(µ) and the heavy state masses include 15 soft terms: 8 sfermion (m 10 1,2,3 , m5 1,2,3 ) and Higgs (m 5 , m5) soft masses, 1 gaugino mass parameter (M 1/2 ) and 6 A-terms (A 10 1,2,3 , A5 1,2,3 ). Since these soft parameters are defined at the GUT scale, one needs to run them down to the scale m susy , where most constraints apply. In order to simplify the problem, we shall use the semi-analytic approximate solutions to the soft term RGEs derived in Section 4. This will allow us to trade the GUT-scale soft parameters for low-energy ones (namely 11 Indeed, given the choice µ 2 = m 2 Hu = m 2 susy , one has from the electroweak symmetry breaking conditions m 2 H d = m 2 susy (2 tan 2 β −1) and m 2 A = 2m 2 susy (tan 2 β +1). Plugging this, together with the assumption of equal sfermion masses, into Eq.        (5) symmetry at M GU T that passed all phenomenological constraints (the number of solutions found increases from light yellow to dark green). Also shown is a particular solution (magenta dot) characterized by a low value of the matching scale m susy ≡ mQ 3 (m susy ) mũc 3 (m susy ) and described in greater detail in the text below. In both panels, the red, green, purple and black lines represent the naive constraints from Fig. 3 associated with the measured Higgs mass, the proton lifetime, the metastability of the electroweak vacuum and the condition m T < M P lanck /10, respectively. SU(5) model can be seen in Fig. 4. The plot in the left panel displays all the points of the (tan β, m susy ) plane that have been checked; the ones for which at least one configuration of the superpartner spectrum passed all constraints are in green, while the ones that failed this test are in red. The plot in the right panel shows how frequently a solution was found in the model parameter space for each green point. The more vivid the colour of the point, the more spectrum configurations survived all the constraints. It turns out that the naive estimate of the allowed parameter space (the green region in Fig. 3), despite not respecting the SU(5) symmetry, tells us something about how likely it is for a randomly chosen point in the parameter space with given values of tan β and m susy to be compatible with all experimental and theoretical constraints discussed at the beginning of this section. In the part of the green region of Fig. 3 around several hundreds of TeV (close to the upper limit given by the requirement m T ≤ M P lanck /10), a randomly chosen point in the parameter space has a few percent probability of success, while the probability decreases when one exits the green region. One can see that the naive constraints associated with the Higgs mass (red line) and with the proton lifetime (green line) are very robust, which in the case of the Higgs mass can be traced back to the fact that the leading term in the 1-loop matching condition for the Higgs quartic coupling, Eq. (3.37), only depends on m susy in the maximal stop mixing case. The naive vacuum metastability bound (purple line) is less robust but remains a reasonable approximation to the exact condition, contrary to the naive m T < M P lanck /10 constraint (black line). The main reason for this is that the black line assumed µ(m susy ) = m susy , while a sizable portion of the viable points of the parameter space feature smaller values of µ, thus effectively decreasing the value of m T by virtue of the gauge coupling unification condition (3.12).
While these results confirm the naive expectation from Fig. 3 that the allowed parameter space of the minimal renormalizable supersymmetric SU(5) model is restricted to the region of heavy superpartners, isolated points with lighter supersymmetric particles are likely to be missed in this random search. In fact we were able to find a viable point in the parameter space with m susy = 63 TeV and some squarks as light as 22 TeV, depicted by a magenta dot in Fig. 4. We give below the input parameters at the GUT scale as well as the values of all soft terms at the scale m susy : Other parameters and observables: lighter than the sfermions and the higgsinos, and the lightest supersymmetric particle (LSP) is the bino, with mass M 1 = 16.9 TeV (theũ R and thec R are the co-NLSPs, i.e. the next-to-lightest supersymmetric particles). The superpartners are therefore out of reach of present and next-generation colliders, and supersymmetric contributions to flavour physics observables are strongly suppressed, but proton decay will be easily accessible at future large detectors. The Higgs sector is far into the decoupling regime, with a standard-like lightest Higgs boson and all non-standard Higgs bosons around m A ≈ 500 TeV. Finally, there is no suitable dark matter particle in the observable sector, as the relic density of a 17 TeV bino by far exceeds Ω CDM h 2 0.12 [47]. If R-parity is conserved, a natural candidate is a gravitino in the GeV range (or lower), to which the bino NLSP would decay without affecting Big Bang nucleosynthesis. The cold dark matter density would then be in the form of gravitinos coming from NLSP decays [48] (giving a contribution ∆ΩG = ΩBmG/M 1 , from which the upper bound mG ∼ < 1 GeV follows) and from thermal production during reheating [49,50]. In order to avoid gravitino overproduction from the latter process, the reheating temperature should lie in the TeV range, or lower. In the case of R-parity violation (which may be invoked to generate neutrino masses, as an alternative to the seesaw mechanism), the only possible dark matter candidate within this model is again the gravitino. Since it decays into a photon and a neutrino, acting as a source of monochromatic photons, extragalactic gamma ray constraints put a bound on its mass.  For bilinear R-parity violation, one obtains mG ∼ < 1 GeV [51]. Here, as in the R-parity conserving case, the maximal reheating temperature is around the TeV scale.
One may wonder whether other points exist in the parameter space of the minimal renormalizable SU(5) model with lighter superpartners than in the above example. While performing an extensive scan would probably reveal the existence of such points, it appears difficult to lower significantly the average superpartner mass scale below a few tens of TeV (some supersymmetric particles may however be accidentally lighter). The reason for this is the proton lifetime, which in the limit where all superpartners have the same mass m susy scales as τ p ∝ m 2 T m 2 susy ∝ m 11/3 susy (where in the last step the approximate constraint (3.19) from gauge coupling unification was used). The actual proton decay constraint depends on the individual superpartner masses 12 , and does not prevent some of them to be much smaller than the naive lower bound on m susy , but it seems to be difficult to reconcile a significantly lighter superpartner spectrum with proton decay and all other constraints. In fact gauge coupling unification is the main obstacle here, which makes it hard to accommodate a low superpartner mass scale with a large colour triplet mass [7].
What one may still try is to split the spectrum, making just a part of it light. This is not easy though, because of the threshold corrections to gauge couplings at M GU T and to the bottom and Higgs masses at m susy . For example, making the higgsino light decreases the mass of the colour triplet, see Eq. (3.12), and therefore shortens the proton lifetime. The gluino and/or theb R cannot be much lighter than theb L (whose mass is of order m susy ) either, since this would suppress the loop function in Eq. (3.26) and require an enormous value of X b to reproduce the observed m b /m τ ratio. This large value of X b would in turn imply a large negative threshold correction to the Higgs quartic coupling, see Eq. (3.38), making it impossible to fit the measured Higgs mass for moderate values of m susy . All this motivates the choice of parameter intervals made in our scan, namely mQ 1,3 (m susy ), mdc 1,3 (m susy ) and M 3 (M GU T ) in the range [0.5 m susy , 2 m susy ]. The only remaining possibility is to have the first two sfermion generations lighter than the third one, an opposite situation to the one considered e.g. in Ref. [7]. The investigation of this case would require the use of 2-loop RGEs for soft terms, as a strong mass hierarchy between different sfermion generations enhances the effect of 2-loop running and may give rise to tachyons [52]. Such a study is beyond the scope of this paper, and we leave it for future work.

Conclusions
If one excepts the issue of neutrino masses, the minimal renormalizable supersymmetric SU(5) model suffers from two main problems. The first one concerns the predictions for charged fermion masses. Although the GUT-scale equality of down-type quark and charged lepton Yukawa couplings leads to a qualitatively successful prediction for the bottom to tau mass ratio, it quantitatively differs from the measured value by some 20-30%, while the discrepancy is much larger for the first and second generations. Second, proton decay is too fast by a factor 10 4 or so for typical soft terms in the TeV range, essentially because gauge coupling unification requires a relatively light colour triplet. Each of the two problems has been addressed separately in the literature: large supersymmetric threshold corrections were used to modify the fermion mass predictions (see for example Refs. [11,12]); specific flavour structures of the soft terms [3] and a heavy superpartner spectrum [53] were invoked to increase the proton lifetime.
In this paper, we performed a complete analysis of the minimal renormalizable supersymmetric SU(5) model, taking into account all relevant phenomenological and theoretical constraints: charged fermion masses, the proton lifetime, gauge coupling unification, the Higgs mass, experimental lower bounds on superpartner masses and flavour constraints, metastability of the vacuum and perturbativity of the model. We showed that the model is still alive, and that the allowed region of the parameter space spreads over a large domain with m susy ≡ √ mt 1 mt 2 ranging from around 50 TeV to 10 4 TeV and tan β from approximately 2 to 5. Viable points were also found outside this region of the parameter space, but they are less frequent. Particularly interesting are the ones featuring some (relatively) light superpartners. We studied in greater detail one such point in which the lightest supersymmetric particle is a bino with mass 16.9 TeV. A generic feature of the model is the metastability of the vacuum, which is not a concern since its lifetime is typically much larger than the age of the universe. A consequence of the heavy supertpartner spectrum is that the observable sector does not contain a suitable dark matter candidate; however, a gravitino with a mass around the GeV scale or below can play the role of cold dark matter, both in the presence and in the absence of R-parity.
The present analysis is a good starting point for further research. One of the limitations is that we used the central values of the input SM parameters; taking into account the experimental and theoretical uncertainties will extend the viable region of the parameter space.
We presented in a compact way all the ingredients that are needed for a more extensive scan of the parameter space of the model. What we have shown is that the minimal renormalizable supersymmetric SU(5) model is indeed a viable extension of the Standard Model, in the sense that its parameter space contains points that satisfy all relevant phenomenological and theoretical constraints. The price to pay is that the superpartner spectrum is heavy, typically in the 100 TeV region or above, although we were able to find particular cases in which some of the supersymmetric particles can be as light as 17 TeV, and it is not excluded that viable points with even more split spectra exist in some corners of the parameter space.
We did not address in this paper the issue of neutrino masses. One possibility is to add Standard Model singlets (right-handed neutrinos) to the model and to generate neutrino masses via the seesaw mechanism. Another possibility is to allow either bilinear or trilinear R-parity violating terms. Notice that integrating out higgsinos and colour triplets with R-parity violating couplings induces corrections to the Yukawa couplings [21], which opens the possibility that the SU(5) fermion mass relations are cured by the combined effect of R-parity violation and of supersymmetric threshold corrections.
The possibilities to test experimentally the minimal renormalizable supersymmetric SU(5) model are limited. Barring possible isolated points of the parameter space with light remnants, superpartners are too heavy to be detected at the next generation of colliders or to give sizable contributions to flavour-violating observables. Proton decay could be around the corner, but the allowed parameter space is too vast to give a definite prediction for the proton lifetime. It is, however, possible to rule out the model (or at least most of its allowed parameter space), either by discovering a light superpartner spectrum or, if the Higgs sector is light enough to be accessible and studied at the next generation of colliders, by measuring a relatively large value of tan β.

A Procedure for solving the soft term RGEs
In this appendix, we derive the expressions for the superpartner spectrum of the minimal renormalizable supersymmetric SU(5) model that we have used in our analysis. These formulae account for the splitting of sfermion masses within the same SU(5) multiplet due to renormalization group running. They are (approximate) semi-analytic solutions to the 1-loop MSSM RGEs for the soft terms obtained under certain assumptions (namely, the hierarchy among Yukawa couplings allows us to neglect some terms in the RGEs and to solve them in a sequential manner, as explained below). We have checked that they are accurate up to the few percent level by running the full set of RGEs. The advantage of these approximate solutions is that the low-energy soft terms can be written as quadratic functions of the initial (GUT-scale) values of the soft parameters, which makes it possible to perform a scan over the parameter space of the model without having to solve the full set of RGEs for each point. Furthermore, it is possible to scan directly over the low-energy values of the soft terms (rather than the GUT-scale ones), since these formulae implicitly respect the SU(5) boundary conditions. A similar approach has already been employed in the past, e.g. in Ref. [26] for mSUGRA and in Ref. [27] for SU (5). Here we generalize this procedure to generation-dependent soft terms with SU(5) boundary conditions.

A.1 The procedure
In order to perform a scan over the parameter space of the minimal renormalizable supersymmetric SU(5) model, one has to solve a system of entangled differential equations describing the running of gauge, Yukawa and Higgs quartic couplings, as well as of the soft terms, which involves a large number of parameters (only in the supersymmetry breaking sector, there are upon SU(5) unification 15 parameters, namely 6 A-terms, 1 gaugino mass and 8 soft scalar masses). In addition, these parameters are subject to various phenomenological constraints defined at different scales. This makes it hard to solve the problem by brute force, and motivates the use of approximate semi-analytical solutions to the soft term RGEs.
Let us first try to circumvent the issue by solving the RGEs in steps. We first integrate numerically the system of RGEs for gauge, Yukawa and Higgs quartic couplings at the 2loop level. Note that these couplings depend on the soft terms only through supersymmetric threshold corrections, so we can use this fact to determine the values of the relevant soft terms at the matching scale m susy . Then we can solve the 1-loop RGEs for the soft terms -first gaugino masses, then A-terms and finally soft scalar masses. By working them out in this particular order, we are able to use in each step the knowledge of the running parameters that we have computed in the previous steps. Unfortunately this method is not efficient since one must solve the RGEs for the soft terms every time one changes the input (GUT-scale) values of the soft parameters. What we would like is to numerically solve the soft term RGEs for symbolic input values. The computational procedure described below does precisely that: when the input values of the soft parameters are changed, only the last step (out of 3) has to be re-run.
Step Step 2 5. Solve the 1-loop MSSM RGEs for gaugino masses, taking into account the unification condition (3.46). 7. Solve the RGEs for the soft scalar masses in a similar manner.

Assume the hierarchy
Step 3 8. Choose the values of the input soft parameters. 9. Make sure that no sfermion becomes tachyonic. Check that the values of the Aterms and soft scalar masses at m susy reproduce the observed values of the down-type quark masses and of the Higgs mass. Verify that all experimental lower bounds on superpartner masses and flavour constraints are satisfied. 10. Check vacuum (meta)stability constraints and the proton lifetime. 13 This assumption is based on the observed hierarchy of fermion masses, taking into account the SU(5) boundary condition Λ D = Λ T E and the supersymmetric threshold corrections to down-type quark masses, and is better justified in the low to moderate tan β regime.

A.2 Input values
All input values of the SM parameters needed for the running were taken either from Ref. [54] or from Ref. [55]:

A.3 Approximate expressions for the soft terms
In order to be able to approximately solve the 1-loop RGEs for soft terms, we shall neglect all mixings. This means that in addition to assuming that the sfermion soft terms are aligned with fermion masses at the GUT scale (see Subsection 2.1), we shall neglect the effects of the CKM matrix in the RGEs. This may not be fully justified, as the impact of V CKM on the running of the first two generation parameters can be significant. However, this effect is suppressed either by small Yukawa couplings or by small CKM angles and is therefore never numerically important 14 , so we shall set V CKM = 1 in the RGEs and omit the subleading Yukawa contributions.
With these assumptions, we can derive approximate semi-analytic solutions to the 1loop RGEs for the soft terms.

A.3.1 Gaugino masses
The 1-loop RGEs for gaugino masses: (where the last equality is only approximate because we run gauge couplings at 2 loops) has the following simple solution respecting SU(5) symmetry: This solution becomes exact when the 1-loop RGEs for gauge couplings are used.
MSSM RGEs for the gauge and Yukawa couplings. As a result, one obtains the running A-terms at an arbitrary scale as linear combinations of the 7 SU(5) soft parameters M 1/2 , A 10 i , A5 i (i = 1, 2, 3), with numerical coefficients depending on the choice of m susy and tan β (and very mildly on M GU T , m T and m V ). In practice, it will prove convenient for the exploration of the parameter space of the model to trade these GUT-scale parameters for low-energy ones, so as to express the A n (t) as a function of M 3 (m susy ), A d i (m susy ) and A u i (m susy ) (i = 1, 2, 3). Note that if the SU(5) boundary conditions at M GU T had not been imposed, the A n (t) would depend on 12 initial parameters, namely M 1,2,3 , functions of the GUT-scale soft scalar masses m 10 i , m5 i , m Hu (M GU T ), m H d (M GU T ) and of the m susy values of M 3 , A u,c,t and A d,s,b , with t-dependent coefficients that can be computed numerically using the known solutions to the 2-loop MSSM RGEs for gauge and Yukawa couplings.
Another combination of masses that appears in the RGEs for soft scalar masses is for which the 1-loop RGE takes the simple form: The last equality is only approximate because we run gauge couplings at 2 loops. Working in the 1-loop approximation, one can solve Eq. (A.21) straightforwardly: where we have used the fact that S( due to the SU(5) boundary conditions on soft scalar masses.
We now have all the ingredients needed to write the solutions to the 1-loop RGEs for soft scalar masses as: , the numerical coefficients P α n , Q α i and R α are given in Table 3, Plugging the semi-analytic expressions for Σ n (t) and with β Σ n (t) equal to 12λ 2 n (t)/(4π) 2 for squarks and to 8λ 2 n (t)/(4π) 2 for sleptons. From this and from the definition (A.24) of I Σn , one immediately deduces that where the integrals on the RHS can be expressed in terms of I Σp , I Ap and I M i , so that eventually all I Σn 's can be written as linear combinations of Σ p (m) − Σ p (M GU T ), I Ap and I M i . This leads to the following expressions for the running soft scalar masses: with the numerical coefficients T α n , U α n , V α i (resp. R α ) given in Table 4 (resp. Table 3).  where C ijkl (µ) is the Wilson coefficient of the 4-fermion operator u i d j d k ν l and the hadronic parameter W ijk where u p is the proton spinor. The lattice computation of Ref. [59] gives W 112 0 (µ = 2 GeV) = (0.111 ± 0.027) GeV 2 and W 121 0 (µ = 2 GeV) = (0.036 ± 0.014) GeV 2 , with statistical and systematic errors added in quadrature. In order to minimize the dependence of the proton decay rate on the renormalization scale µ, the C ijkl 's are evaluated at the scale µ = 2 GeV rather than at µ = m p .
The Wilson coefficients C ijkl are computed at the matching scale m susy at which supersymmetric partners are integrated out by "dressing" the QQQL operators with loops containing a wino: in which V is the CKM matrix, the parametersκ pmnl ≡ κ pmnl − κ nmpl are expressed in the super-CKM basis in which the up quarks are mass eigenstates, and the possible misalignment between the fermion and the sfermion mass eigenstate bases is neglected. In Eq. (B.4), the CKM matrix entries and theκ pmnl 's are renormalized at the scale m susy . The loop function I(W , a, b) is given by: where the function I 3 (x, y, z) has been defined in Eq. (3.30). For equal sfermion masses (m a = m b ≡ m susy ), I(W , a, b) reads: The parametersκ ijkl depend on the details of the Yukawa sector of the Grand Unified Theory. In the minimal supersymmetric SU (5) model, they are given by (neglecting the running between M GU T and the triplet mass scale): where the Yukawa couplings and the CKM matrix are evaluated at the scale m T , and the φ i are high-energy phases satisfying the constraint φ 1 + φ 2 + φ 3 = 0. These parameters must then be evolved down to the scale m susy by solving the appropriate RGEs. Neglecting the Yukawa couplings, the running simply amounts to an overall rescaling: where in the minimal renormalizable supersymmetric SU (5) model: The fact that A 112 LD = A 121 LD is due to the RG-induced mixing between 4-fermion operators with different flavour indices; when this (numerically small) mixing is neglected, A 112 LD = A 121 LD = A LD .

C UFB constraints
In this appendix, we derive the constraints on the model parameters associated with the absence of directions along which the scalar potential is unbounded from below. In doing this we will follow Ref. [61].

D Decays into CCB vacua
In this appendix, we describe a method to estimate the lifetime of the electroweak vacuum in the presence of charge and colour breaking (CCB) vacua induced by large A-terms from any sector and generation. To this end, we will approximate the minimal value of the bounce action in Euclidean space [62,63,64]: for the solution of the equations of motion subject to the boundary conditions (describing fields sitting in the false vacuum at r = ∞ and tending to a lower minimum at r = 0): 3) The general problem can be solved only numerically [16,65,17,66,67,68,69]. We will simplify it by minimizing the bounce action along a constant direction in field space [71,72,70], which effectively reduces the problem to a single differential equation. This approximation provides an upper bound on the minimal value of the correct multi-field bounce action [73,72]. This means that the points of the parameter space ruled out in this way are indeed characterized by a too short vacuum lifetime, but the method retains points that after a more careful and complete analysis would be rejected.
What we have to do is therefore to minimize an Euclidean action of the form: where the field φ(r) parametrizes a direction in field space, and check whether its minimal value is large enough for the lifetime of the electroweak vacuum to be larger than the age of the universe, namely S > 400 .  For each case, there are several subcases to be checked according to which generation indices are carried by the ψ andψ fields. For a given generation ψ i (the 2 ψ fields belong to the same generation), one must consider all generation choices for theψ fields that do not give a substantial contribution to the F-terms due to the size of the corresponding Yukawa couplings (otherwise there would be no CCB minimum lower than the electroweak vacuum). Namely, • when a singleψ field is involved (Cases 1 and 3), all generation indices should be considered; • when twoψ fields are involved (Cases 2, 4, 5 and 6), all combinations of two different generation indices should be considered; • the cases of twoψ fields with the same generation index should also be considered when the Yukawa coupling of the corresponding lepton is much smaller than λ ψ (i.e. one needs to consider only generations ofψ's whose fermionic partners are lighter than the fermionic partner of ψ).
Finally, starting from the semi-analytic approximation (D.6), it is easy to show that: hereafter referred to as the metastability constraint. This condition must be satisfied for all six cases in Table 5 to be able to conclude (within our approximate method) that the lifetime of the electroweak vacuum is longer than the age of the universe. Notice that one needs to minimize S with respect to two variables only. Indeed, one of the variables ψ,ψ, H,H is fixed by the D-term constraint (D. 26), and the quantity (D. 28) only depends on ratios of fields. The minimization is performed numerically. A point of the parameter space is admitted if the bounce action satisfies Eq. (D.29) in all cases listed in Table 5.
The D-term constraint (D.26) is most easily taken into account in the minimization of S by choosing a suitable parametrization of its general solution, for instance:

D.3 The special case of the stop
In this case too we will make suitable approximations for estimating the minimal value of the bounce action (for more details on the subject, see Refs. [17,76,77,78,79]). As explained in Ref. [61], one cannot assume exact cancellation of the D-terms here, but we will nevertheless stick to the relatively good approximation of vanishing SU(3) D-terms, namelyt =t c . Keeping the contributions of all other fields, one can write: L kin = |h u (r)| 2 + |h d (r)| 2 + 2|t (r)| 2 , Assuming again all quantities to be real, we take the same ansatz as for the other CCB constraints: (h u , h d ,t)(r) = (H u , H d , t) φ(r) , (D. 37)