Precise determination of the Higgs mass in supersymmetric models with vectorlike tops and the impact on naturalness in minimal GMSB

We present a precise analysis of the Higgs mass corrections stemming from vectorlike top partners in supersymmetric models. We reduce the theoretical uncertainty compared to previous studies in the following aspects: (i) including the one-loop threshold corrections to SM gauge and Yukawa couplings due to the presence of the new states to obtain the $\bar{\text{DR}}$ parameters entering all loop calculations, (ii) including the full momentum dependence at one-loop, and (iii) including all two-loop corrections but the ones involving $g_1$ and $g_2$. We find that the additional threshold corrections are very important and can give the largest effect on the Higgs mass. However, we identify also parameter regions where the new two-loop effects can be more important than the ones of the MSSM and change the Higgs mass prediction by up to 10 GeV. This is for instance the case in the low $\tan\beta$, small $M_A$ regime. We use these results to calculate the electroweak fine-tuning of an UV complete variant of this model. For this purpose, we add a complete $\textbf{10}$ and $\bar{\textbf{10}}$ representation of $SU(5)$ to the MSSM particle content. We embed this model in minimal Gauge Mediated Supersymmetry Breaking and calculate the electroweak fine-tuning with respect to all important parameters. It turns out that the limit on the gluino mass becomes more important for the fine-tuning than the Higgs mass measurements which is easily to satisfy in this setup.


I. INTRODUCTION
The discovery of the Higgs boson with a mass of about 125 GeV [1,2] has a strong impact on the parameter range of supersymmetric (SUSY) models, especially as its mass value is turning into a precision observable with an uncertainty below 1%. In particular, in constrained versions of the Minimal Supersymmetric Standard Model (MSSM) large regions of the parameter space are not consistent with this mass range [3]. This is in particular the case for models where SUSY breaking is assumed to be transmitted from the hidden to the visible sector via gauge interactions like in minimal Gauge Mediated SUSY Breaking (GMSB). Even relaxing the predictive boundary conditions of a constrained model and considering the phenomenological MSSM with many more parameters at the SUSY scale, it is still rather difficult to find regions with the correct Higgs mass. Either, a very large mixing in the stop sector or heavy stop masses are needed to push the Higgs mass to the desired range [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. However, the large stop mixing with light stops turns out to be dangerous because of charge and colour breaking minima [19][20][21][22][23]. On the other side, very heavy stops introduce again a hierarchy problem which SUSY was supposed to solve. The question about naturalness and fine-tuning is even more pronounced in regions the small tan β region which recently gained some interest because of Higgs fits [15][16][17]: in these regions the tree-level Higgs mass is suppressed by a factor cos(2β) and even much bigger loop corrections are needed than for larger values of tan β.
The fine-tuning in these models is often better by a few orders compared to the MSSM.
Alternatively, one can also consider models which give new loop-corrections due to the presence of additional large couplings to push the Higgs mass. This happens for instance in inverse-seesaw models [40,41] or models with vector-like quarks [42][43][44][45][46][47][48][49][50][51][52][53] at the one-loop level, or in models with trilinear R-parity violation at the two-loop level [54]. We are going to concentrate here on models with vectorlike tops partners. In these models, the effects on the Higgs mass have been so far just studied in the effective potential approach at one-loop.
Also a careful analysis of the threshold corrections to the standard model (SM) gauge and Yukawa couplings has been not performed to our knowledge so far. However, it is well known from the MSSM that the SUSY threshold corrections and one-loop momentum dependent effects can alter the Higgs mass by several GeV [55]. Of course, also two-loop corrections involving coloured states are crucial in the MSSM and it wouldn't be possible to reach a mass of 125 GeV without them [56][57][58][59][60][61][62][63][64][65][66][67][68][69]. As soon as the Yukawa-like interactions of the new (s)tops become large, one should expect that effects of a similar size than in the MSSM sector appear. Therefore, we make a careful analysis of all three effects: we calculate the full one-loop threshold corrections to get an accurate prediction of the running gauge and Yukawa couplings at the SUSY scale, we include the entire dependence of external momenta at the one-loop level, and we add the all two-loop corrections which are independent of electroweak gauge couplings. In this context, all calculations are performed within the SARAH [70][71][72][73][74][75] -SPheno [76,77] framework which allows for two-loop calculations in SUSY models beyond the MSSM [78,79]. The obtained precision is comparable to the standard calculations usually employed for the MSSM based on the results of Refs. [65][66][67][68][69].
Finally, we extend the particle content to have a complete 10 and 10 of SU (5) in addition to the MSSM particle content to get a model which is consistent with gauge coupling unification. This model has already been studied to some extent after embedding it in minimal supergravity or GMSB [49,80,81]. We choose here the variant where SUSY breaking is transmitted via gauge mediation and check for the first time for the fine-tuning in regions which are consistent with the Higgs measurements. We show that this gives usually a fine-tuning which can easily compete with other attempts to resurrect natural GMSB by including non-gauge interactions between the messenger particles and MSSM states [82][83][84][85][86][87][88][89][90][91][92].
This manuscript is organized as follows. We first introduce the minimal SUSY model with vectorlike top partners as well as the UV complete variant embedded in GMSB in sec. II.
In sec. III we summary briefly the main features of the tree-level masses before we explain in large detail the calculation of the one-and two-loop corrections. The numerical results are given in secs. IV and V. In sec. IV we discuss the impact of the different corrections at one-and two-loop on the SM-like Higgs mass using a SUSY scale input, before we analyse in sec. V the fine-tuning of the GMSB embedding. We conclude in sec. VI.

II. THE MSSM WITH VECTORLIKE TOPS
A. The minimal model (1, 1, 1) We extend the particle content of the MSSM by a pair of right-handed vectorlike quark superfieldsT andT . The particle content of the model and the naming conventions for all chiral superfields and their spin-0 as well as 1 2 components are summarized in Tab. II A. In addition, we have the usual vector superfieldsB,Ŵ ,Ĝ which carry the gauge bosons for U (1) Y × SU (2) L × SU (3) C as well as the gauginos λ B , λ W , λ G . The full superpotential for the model reads: Here, we skipped colour and isospin indices. The Yukawa couplings Y e , Y d and Y u are in general complex 3 × 3 matrices. The new interaction Y t is a vector, but we concentrate only on cases where the third component Y 3 t has non-vanishing values. To simplify the notation, we define therefore When we speak about the top-Yukawa coupling Y t , we refer to Y 33 u . The dimensionful parameters in the superpotential are the µ-parameter known from the MSSM, as well as the mass term M T for the vectorlike top quark superfields, and a bilinear term m t mixing the new states and the MSSM ones even before electroweak symmetry breaking (EWSB).
The soft-SUSY breaking terms for the model are In general, the T -and Bparameters are complex tensors of appropriate dimension, while the mass soft-terms for scalars are hermitian matrices, or vectors or scalars. The gaugino mass terms are complex scalar. However, we are going to neglect CP violation in the softsector, i.e. all parameters are taken to be real. For the trilinear soft-term of Y t we use a similar short-hand notation T 3 t ≡ T t in the following.
B. UV completion and fine-tuning

Gauge coupling unification
If we just include the right-handed top superfields, the model is not consistent with gauge coupling unification. To cure this problem, additional fields have to be added. The minimal choice is to add a pair of complete 10-plets under SU (5) which contain the states we are interested in, but also vectorlike left-handed quarks (Q ,Q ) and vector-like right-handed leptons (E ,Ē ). To generate mass terms for all components of the 10 and 10, the following extension of the superpotential is needed: Here, the Q-fields have quantum numbers ( 1 6 , 2, 3), (− 1 6 , 2,3), while the vector-like leptonŝ E ,Ê carry quantum numbers (±1, 1, 1) with respect to U (1) Y × SU (2) L × SU (3) c . We are going to assume that no further interactions between these additional states and the MSSM sector are present, i.e. these particles are only spectators when calculating the SUSY mass corrections. Nevertheless, because of their impact on the SUSY RGEs and also on the threshold corrections to the SM gauge couplings they can play an important role. We can see this already at the one-loop RGEs of the gauge couplings for the minimal model and the UV complete version: where we parametrized the β function as For δ U V = 0 we obtain the minimal model, while δ U V = 1 describes the UV complete version.
In Fig. 1 the re-established gauge unification can be observed. The one-loop β functions of the Yukawa couplings are the same in both model variants and read corrections. These effects will be included in our numerical analysis. Nevertheless, one can already see in Fig. 2 that the cut-off scale M C at which the Landau pole arises, given as a function of Y t , is pushed towards higher scales in the UV complete version.
The additional soft-terms which appear because of the extended particle content are the following: We can now embed the UV complete version in a constrained setup to relate the SUSY breaking parameters. We are going to choose the setup of gauge mediated SUSY breaking (GMSB) which we introduce now briefly.

Gauge mediated SUSY breaking and boundary conditions
The mediation of the SUSY breaking from the secluded to the visible sector happens in GMSB by messenger particles charged under SM gauge groups. The minimal model provides a pair of 5-plets under SU (5) which don't have any interaction with the MSSM sector but due to the gauge couplings. The necessary ingredients to break SUSY are the interaction of the messengers, called Φ,Φ, and a spurion field S described by S is a gauge singlet and acquires a vacuum expectation value (VEV) along its scalar and auxiliary component due to hidden sector interactions, which we leave here unspecified The coupling λ of eq. (14) can be absorbed into the redefinitions of M ≡ λM and F ≡ λF .
With these conventions, we find that the fermionic components of the messengers have a mass M , while the scalars get masses This gives the condition M 2 > F . The soft breaking masses of the MSSM fields are generated via loop diagrams involving the messenger particles. The gauginos receive masses Mλ at one-loop level while the scalar masses m 2 f are generated at the two-loop. The leading approximations for the soft breaking masses are α i (t) = g 2 i /(4π) are the running coupling constants at the scale t and C r is the Casimir of the representation r. The SUSY soft breaking scales Λ G and Λ S depend on F and M as follows: It is convenient to define For F M 2 this leads to Λ G = Λ S = Λ. Applying the general results to our (UV complete) model, we have the following boundary conditions at the messenger scale M for the scalar soft masses with j = 1, 2, 3. All off-diagonal entries are staying zero at the messenger scale. For the gaugino mass terms, we have the MSSM results while all other soft-terms vanish up to two-loop Furthermore, we assume that the bilinear mass terms for the vector states unify at the messenger scale We make no attempt to explain the size of µ or B µ in this setup. There are several proposals how these parameters receive numerical values needed for phenomenological reasons [93][94][95].
We take it as given that one of these ideas is working and calculate the µ and B µ from the vacuum conditions. Similarly, we are also agnostic concerning the cosmological gravitino problem usually introduced in GMSB by the Gravitino LSP and possible solutions for it [96][97][98][99][100][101][102].
Thus, our full set of input parameters in this setup is In this setup, the sensitivity of the Z mass on the fundamental parameters at the UV scale is calculated. α is a set of independent parameters at this scale and ∆ −1 α gives an estimate of the accuracy to which the parameter α must be tuned to get the correct electroweak breaking scale [105]. The smaller ∆ F T , the more natural is the model under consideration.
We use the messenger scale M in GMSB as a reference scale and calculate the FT with respect to The practical calculation of the FT in our numerical calculation works as follows: we vary these parameters at the messenger scale M and run the two-loop RGEs down to the SUSY scale. At the SUSY scale, the electroweak VEVs are calculated numerically using the minimization conditions of the potential and the resulting variation in the Z mass is derived.

III. THE MASS SPECTRUM OF THE MINIMAL MODEL
To get a good estimate of the fine-tuning by including the Higgs constraint, it is necessary to reduce the theoretical uncertainty of the Higgs mass prediction. Our aim is to get the same uncertainty as for the MSSM, namely to consider the Higgs mass in the range This precision can only be reached if a full one-loop calculation is done, and the dominant two-loop corrections are included. Since this has not been done before in literature for the considered model, we discuss our calculation of the mass spectrum, in particular of the threshold corrections and two-loop Higgs corrections, in detail.

A. Tree-level properties
When electroweak symmetry gets broken, the neutral Higgs states receive VEVs v d and v u and split in their CP even and odd components: We GeV. Using these conventions, the tree-level mass matrix squared for the scalar Higgs particles is the same as in the MSSM. It reads in This matrix is diagonalized by Z H : Two of the parameters in this matrix can be eliminated by the tadpole conditions for EWSB: We are going to solve these equations for the squared soft-masses m 2 H d and m 2 Hu when we consider a SUSY scale input. That leaves three free parameters in the Higgs sector at treelevel: tan β, µ and B µ . The last one is related to the tree-level mass squared M 2 A of the physical pseudo-scalar via However, when we consider the UV completion, m 2 H d and m 2 Hu are fixed at the SUSY scale and we are going to solve the above equations (39) and (40) for µ and B µ . Also, the mass matrices for the CP-odd and charged Higgs bosons, for down (s)quarks, charged and neutral (s)leptons, as well as for neutralino and charginos are identical to the MSSM. Only in the up (s)quark sector things change because of the additional top-like states. The scalar mass matrix that links the left-and right-handed MSSM up-squarks and the new vector-like states is given in the basis of ũ L,i ,ũ R,i ,t ,t * by with the diagonal entries This matrix is diagonalized by Z U : and we have eight mass eigenstates calledũ i in the following. Similarly, in the fermionic counterpart we choose the basis (u L,i ,t * ) / u * R,i , t * β 2 . The mass matrix in this basis reads Here, we need two rotation matrices U u L and U u R to diagonalize this matrix, The four generations of mass eigenstates are called u i where the first three generations correspond to the up, charm and top quark. is again a generalization of the renormalization procedure presented in Ref. [55]. We explain this calculation and the difference to the MSSM more detailed in sec. III B 2.
3. At the two-loop level, new corrections O(α t (α S + α t + α b + α t )) arise. The importance of these corrections was unknown up to now. However, with the generic results of Ref. [106] for the two-loop effective potential implemented into SARAH [78], a numerical derivation in analogy to Ref. [107] allows to obtain the two-loop self-energies at vanishing external momentum for the scalars which get a VEV. Moreover, since Ref. [79], a fully equivalent and diagrammatic calculation in the limit p 2 = 0 can also be performed by SARAH and SPheno. Both approaches are used to cross-check the two-loop results. We give more details about this calculation in sec. III B 3.

Threshold corrections
The presence of additional vectorlike states change the relations between the running DR parameters and the measured SM parameters. In the gauge sector, the relation between the SM couplings (MS scheme with five flavours) and the DR ones are Here, α We absorbed all corrections which don't change with respect to the MSSM in ∆α MSSM

S (µ)
and ∆α MSSM (µ). Note, this does not include the up-squark sector, now consisting of 8 squarks, to prevent double counting. In the case of the UV complete model, additional terms of the same form show up.
To relate α to the running couplings g 1 and g 2 , the running Weinberg angle sin Θ and the electroweak VEV in DR scheme are needed. Also here the vector-like tops enter because of the new loop corrections to the mass shifts δM 2 Z and δM 2 W of the gauge bosons. The corrections from the extended (s)top sector to the transversal self-energies are with The appearing vertices are given in appendix A 1. All other contributions are identical to the MSSM and given for instance in Ref. [55]. With that information, v and sin 2 Θ DR W are calculated by Here, G F is the Fermi constant and δ r doesn't receive new corrections compared to the MSSM (Expressions for δ r can be found in [108]). Also here the spectator fields in the UV complete version will show up in a similar way because their contributions don't vanish even in the limit that all superpotential and soft-breaking interactions of those are assumed to vanish.
The running Yukawa couplings are also calculated in an iterative way. We concentrate on the quark sector, because the leptons don't get new contributions from the new vector-like quarks at one-loop. This is also true for the UV complete model because these contributions are proportional to the superpotential interactions which we assume to vanish for the E and Q states. The starting point are the running fermion masses in DR obtained from the pole masses given as input: The two-loop parts are taken from Ref. [109,110]. The DR masses are matched to the eigenvalues of the loop-corrected fermion mass matrices calculated as Here, the pure QCD and QED corrections are dropped in the self-energiesΣ because they are already absorbed in the running DR masses. The self-energy contributions from the extended (s)top sector to down-quarks are The full self-energies in the up-quark sector read now Because of the length of the expressions eqs. (71)(72)(73), the sums over internal generation indices a and b are understood. All necessary vertices are listed in Appendix A 2 1 . The with the DR-masses taken from eqs. (61)(62)(63). In addition, the rotation matrices diagonalizing m values of M T : 1 and 3 TeV. In addition, we fixed tan β = 3 and all soft-masses to 1.5 TeV.
In total, this effect can be as large as a few percent and is larger for smaller M T because the t − t mixing becomes larger. This already gives an important change in the MSSM-like corrections to the Higgs states which turn out to be of order of a few GeV, as we will see. We fixed here M T = 1 TeV.
While a study of flavour physics in this model is beyond the scope of this paper, we want to briefly comment on the expected effects. The CKM matrix in this model is a 4 × 3 matrix and we adjust the Yukawa couplings Y d and Y u in our study in a way that the 3 × 3 sub-matrix assigning the couplings between SM-quarks is in agreement with measurements.
The last column of the CKM matrix carries the elements V t q which define the size of the flavour changing charged currents between the vectorlike top and the SM down-quarks. The size of |V t q | is constrained by the measurements of flavour violating processes which are known to a high precision and which are in agreement with SM predictions. In Ref. [111] the following limits were derived at 3σ: We show the prediction of these elements as a function of Y t in Fig. 4 for M T = 1 TeV.
One can see that the obtained values are well below the current bounds. The main reason for this is that we assume Y 1 t and Y 2 t to vanish.

One-loop corrections
A generic one-loop calculation with SARAH and SPheno was introduced in Ref. [112]. The With these values the tree-level masses are re-calculated and the calculation of the one-loop corrections is started. Here, first the one-loop corrections δt with i = u, d. All other corrections are identical to the results of Ref. [55]. Afterwards, we need the one-loop corrections to the scalar Higgs mass matrix. Here, the vectorlike top quarks contribute to the scalar self-energy Π(p 2 ) The necessary vertices to calculate δ t,t t (1) and Π t,t (p 2 ) are given in Appendix A 3. We can now express the one-loop corrected mass matrix of the scalar Higgs by Here, Π MSSM (m 2 h i ) for each eigenvalue is found. Previously, the one-loop corrections in this model have been calculated in the effective potential approach [46]. This calculation is equivalent to ours in the limit p 2 → 0. Thus, by checking this limit we can easily estimate the error introduced in these calculations by that approximation. Since the additional fermions and the scalars are usually heavier than the desired Higgs mass of 125 GeV, one can expect that the momentum effects are rather moderate. However, before we discuss this in detail, we go one step further to the two-loop corrections. O(α t α s ) with α t = (Y 33 u ) 2 /4π, α t = (Y 3 t ) 2 /4π. The next important contributions from the MSSM are those of O(α 2 t ). These come from diagrams involving (s)tops and Higgs states respectively Higgsinos. Also here, the diagrams shown in Fig. 6 are the same as in the MSSM, but the sums over (s)fermion generations

Two-loop corrections
The index ranges are: However, this involves a numerical derivation which sometimes suffers from numerical problems and rather large uncertainties. Thus, the second method implemented in SARAH and Given the two-loop corrections, the loop-corrected Higgs mass can be expressed by Here, we have no longer distinguished between corrections involving vectorlike tops or not, but used Π (XL) and δt (XL) for the sum of all contributions. The eigenvalues m 2 h i fulfilling Eig(m 2,(2L) h (m 2 h i )) = m 2 h i are associated with the scalar pole masses. In the following, the smaller value m 2 h 1 corresponds to the SM-like Higgs boson and we are going to use the short notation m h ≡ m 2 h 1 for it. Before we turn to the full calculation, we want to discuss briefly the importance of the different contributions at two-loop. For this purpose we depict in Fig. 7 the different two-loop contributions to the Higgs mass matrix:  However, we will not go into details in these aspects of this model here. We are just using the FlavorKit results [113] to double check that all points are in agreement with current bounds from flavour observables. This is, of course, expected as we already discussed in sec. III B 1. The Fortran code written by SARAH was compiled together with SPheno version 3.3.6. For all parameter scans in the following we have used the Mathematica package SSP [114]. We have identified in sec. III B 3 two regions where the new two-loop corrections are expected to be even more important. The first region is the one with non-vanishing T t . This is studied in Fig. 9 where we set T t = 2000 GeV · Y t . In addition, we check also the The other region we identified where the two-loop corrections can be important is the one where the SM-like Higgs has a larger down-type fraction. This happens if M 2 A becomes small. We discuss this case in Fig. 10 for zero and non-zero B T again. In particular for the As a next step we want to understand the dependence of the loop corrections on the involved masses a bit more. We start with the dependence on the vectorlike mass parameter M T and B T and show in Fig. 11 the Higgs mass at the one-and two-loop level. At oneloop we have the well-known picture that the corrections quickly decrease with increasing There is also another, very interesting observation: even for Y t = 0 the fine-tuning in The running gaugino mass at the SUSY scale is related to the one at the messenger scale by the ratio of the corresponding gauge coupling at both scales: We show the minimal fine-tuning in the (mg, m h ) plane in Fig. 15. It is interesting that the fine-tuning for m h = 125 GeV can be smaller than for m h = 122 GeV and m h = 128 GeV when the gluino mass is sufficiently large. For very large Y t where the FT becomes the best, the theory is not perturbative up to the GUT scale. Since there is a cut-off anyway in the theory, there is no real need to maintain gauge coupling unification by adding the spectator fields at the SUSY scale. Therefore, one might wonder what the FT of the minimal model is. This is depicted in Fig. 16. In this setup, the squarks are lighter for the same values of M and Λ because of the smaller strong coupling at the messenger scale. Thus, in general larger Λ is needed to increase the Higgs mass. This leads also to larger gluino masses. This is shown in Fig. 17 where we compare the minimal value of Λ to get a Higgs mass larger than 122 GeV in the (tan β, Y t ) plane for a messenger scale of again 10 7 GeV, and the resulting stop and gluino masses triggered by . We found that the fine-tuning can be reduced significantly compared to minimal GMSB with only the MSSM particle content. Often, those regions with the best fine-tuning which are in agreement with the Higgs mass measurement are ruled out by gluino searches. Interestingly, we find that for heavy gluino masses the fine-tuning for heavier Higgs masses can be even better. In particular, for mg 1400 GeV, the best fine-tuning is found for a Higgs mass of roughly 125 GeV.

ACKNOWLEDGEMENTS
We thank Mark D. Goodsell for a fruitful collaboration to automatize the two-loop calculations with SARAH and SPheno and many interesting discussions in this context. This has been crucial to facilitate this project.