Corrections to Yukawa couplings from higher dimensional operators in a natural SUSY $\mathsf{SO(10)}$ and LHC implications

We consider a class of unified models based on the gauge group $\mathsf{SO(10)}$ which with appropriate choice of Higgs representations generate in a natural way a pair of light Higgs doublets needed to accomplish electroweak symmetry breaking. In this class of models higher dimensional operators of the form matter-matter-Higgs-Higgs in the superpotential after spontaneous breaking of the GUT symmetry generate contributions to Yukawa couplings which are comparable to the ones from cubic interactions. Specifically we consider an $\mathsf{SO(10)}$ model with a sector consisting of $\mathsf{126+\overline{126} + 210}$ of heavy Higgs which breaks the GUT symmetry down to the standard model gauge group and a sector consisting of $2\times \mathsf{10+120}$ of light Higgs fields. In this model we compute the corrections from the quartic interactions to the Yukawa couplings for the top and the bottom quarks and for the tau lepton. It is then shown that inclusion of these corrections to the GUT scale Yukawas allows for consistency of the top, bottom and tau masses with experiment for low $\tan\beta$ with a value as low as $\tan\beta$ of 5$-$10. We compute the sparticle spectrum for a set of benchmarks and find that satisfaction of the relic density is achieved via a compressed spectrum and coannihilation and three sets of coannihilations appear: chargino-neutralino, stop-neutralino and stau-neutralino. We investigate the chargino-neutralino coannihilation in detail for the possibility of observation of the light chargino at the high luminosity LHC (HL-LHC) and at the high energy LHC (HE-LHC) which is a possible future 27 TeV hadron collider. It is shown that all benchmark models but one can be discovered at HL-LHC and all would be discoverable at HE-LHC. The ones discoverable at both machines require a much shorter time scale and a lower integrated luminosity at HE-LHC.


Introduction
Grand unified models based on SO(10) [1,2] are the most desirable of grand unified models as they provide unification of the standard model gauge group and a unification of one generation of matter consisting of quarks and leptons in a single irreducible representation. The Higgs sector of SO(10) models is very rich consisting of several possible representations which can be used to break the grand unified symmetry down to the standard model gauge group. Some of these consist of 16 + 16, 45, 54, 126 + 126, 210 among others. In this work we will focus on large Higgs representations to break the grand unified theory (GUT) symmetry for reasons explained below.
Large representations have been used in the literature for quite some time, a small sample of which are [3][4][5] and for some more recent works see, e.g., [6][7][8][9][10][11][12] and the references therein. However, in grand unified models with small as well as with large Higgs represenations the Higgs doublets lie in irreducible representations of the unified gauge group along with other components which carry color, such as color triplets. The super-partners of these enter in proton decay (for a review see [13]) and they must be very heavy, i.e., of the GUT scale size, which makes the Higgs doublets also superheavy and thus unsuitable for electroweak symmetry breaking. One can, of course, manufacture a light Higgs doublet pair by fine tuning which, however, is rather large.
It is more appealing to have models where some higher symmetry, a group theoretic constraint, or a vacuum selection constraint leads to a pair of light Higgs doublets. Such unified models may be viewed as natural, and GUT models which exhibit this property may be viewed as natural GUTs.
Natural GUT models may also be realized in the framework of field theory. Thus the Dimopoulos-Wilczek mechanism allows for generation of light Higgs doublets in SO(10) [18]. Another possibility to generate a light vectorlike Higgs doublet is by a combination of Higgs representations. In SU (5) one finds [19,20] that a combination of 5 +5, 50 + 50 and 75 of Higgs conspire to make the color Higgs triplets all heavy but leaves one pair of Higgs doublets light. A similar phenomenon occurs in SO(10) [21,22] where a pair of light Higgs doublets can arise purely by a proper combination of heavy and light Higgs multiplets. Models of this type are referred to as missing partner models and they belong to the larger class of natural models as defined above. This last class of models involve Higgs fields in large tensor and spinor representations 1 .
The mechanism that operates in natural field theoretic GUT models is the following: Suppose the GUT model consists of two types of Higgs fields, where one set is heavy and the other set is light.
Let us further suppose the heavy sector possesses n H D number of Higgs doublet pairs, and the light sector possesses n L D number of Higgs doublet pairs and n L D > n H D . In this case if the light and the 1 An example of a natural GUT model with spinor Higgs representations is the case when the heavy Higgs consists of 560 + 560 and the light Higgs consists of 2 × 10 + 320 [22]. heavy sectors mix, n H D number of light Higgs doublet pairs will become heavy leaving n L D − n H D number of Higgs doublet pairs light. In the class of models we consider n L D − n H D = 1 and thus one naturally produces one pair of light Higgs doublets which is desired for electroweak symmetry breaking. At the same time we need to make sure that the number of color triplets/anti-triplets n H T in the heavy sector and the number of color triplets/anti-triplets n L T in the light sector match, i.e., n L T − n H T = 0 which makes all the color Higgs triplets/anti-triplets heavy when the light and the heavy sectors mix.
It is of interest to investigate physics implications of SO(10) models of this type. Thus proton stability in these models has been discussed in [23]. Here we will discuss quark-charged lepton masses and the sparticle spectrum in a class of these models and also investigate the implications for supersymmetry (SUSY) discovery at the HL-LHC and HE-LHC. In this work we will consider one specific model where the heavy sector consists of 126 + 126 + 210 of Higgs fields and the light sector consists of 2 × 10 + 120 of Higgs fields. In this case using the counting discussed above only one Higgs doublet pair remains light while all the color triplet/anti-triplet pairs become heavy. An important result of our analysis is to show that in models of this type, higher dimensional operators can generate contributions to Yukawa couplings which are comparable to the contributions from the cubic interactions. The reason for this is the following: the quartic interactions of the type mattermatter-light Higgs-heavy Higgs suppressed by a heavy mass produce contributions comparable to those from the cubic interactions after spontaneous breaking of the GUT symmetry because the heavy Higgs have vacuum expectation values (VEVs) which are the same size as the heavy mass by which these interactions are suppressed.
The outline of the rest of the paper is as follows: In section 2 we give a description of the model.
In section 3 we give computations of the Yukawa couplings which arise from the cubic mattermatter-Higgs interactions and from the quartic matter-matter-Higgs-Higgs interactions where one of the Higgs fields belongs either to the 10-plets or to the 120-plet while the other Higgs field is heavy and is either a 126-plet or a 210-plet. After spontaneous symmetry breaking at the GUT scale these quartic interactions contribute to the Yukawa couplings. We show that the quartic superpotential corrections to the Yukawa couplings can be substantial and can modify the well known constraint that for the t − b − τ unification one needs a large tan β [24]. In section 4 we give a numerical estimate of the VEVs of the heavy fields which break the SO(10) symmetry down to the standard model gauge group. This is done for a set of benchmarks for the parameters involving the heavy fields. In this section we also give the numerical computations of the contributions of the quartic operators in the Yukawa couplings. LHC implications of the model regarding the possible observation of supersymmetry in this model is also discussed. Conclusions are given in section 5.
Several appendices are also included. Thus notation of the model is given in Appendix A where we also give a decomposition of the relevant irreducible representations of SO(10) in irreducible representations of SU (5). In Appendix B we discuss spontaneous breaking of the SO(10) symmetry by the heavy Higgs fields 126 + 126 and 210. In Appendix C we exhibit for completeness the 7 × 7 Higgs doublet mass matrix as a result of the mixing of the heavy fields 126 + 126 + 210 with the light fields 2 × 10 + 120 discussed in section 2. In Appendix D details of the computation of the contributions of quartic interactions to Yukawa couplings are given.

Corrections to Yukawa couplings from higher dimensional operators
Here we will consider only the Yukawa couplings for the third generation of matter as the analysis of all three generations is more complex 2 . Since the product 16×16 = 10 s +120 a +126 s , the 16-plet Here B and Γ's are the SO(10) charge conjugation and gamma matrices [6]. The decomposition of an SO(10) vertex in the SU(5) basis using the oscillators [26] and the techniques developed in [6][7][8]27] allow us to compute particle content in the SU(3) C × SU(2) × U(1) Y basis. Thus for W 3 in SU(5) decomposition we get The third generation Yukawas arising from Eq. (7) are given by where U dr1 and V dr1 are defined by Eq. (5) and evaluated numerically in Tables (2) and (3).
We first discuss the possible origin of quartic coupling appearing in Eq. (9). To that end, consider the following interactions in the superpotential: 16 × 16 × 126 + 210 × 10 × 126 + 126 + 126 × 126 and in tensor notation After spontaneous breaking, the heavy fields give corrections to the Yukawa couplings at the GUT scale δh We refer to appendix D for further details of the computation.
Quartic coupling Eq. (10) arises from the interactions 16 × 16 × 126 + 210 × 120 × 126 + 126 + 126 × 126 . This interaction in tensor notation takes the form Again after spontaneous breaking, the heavy fields give the following contributions to the third generation Yukawas Further details of the computation are given in appendix D.
The total Yukawas are the sum of the contributions from the cubic and from the quartic terms at the GUT scale. Thus we have In the renormalization group (RG) evolution, Eq. (23) acts as the boundary condition which produces the effective Yukawas at the electroweak scale Q so that at this scale the top, bottom, and tau lepton masses are related to the effective Yukawa couplings so that where we used the relations H d = v

Analysis of model implications
In this section we discuss the implications of the model discussed above. Here we will give numer- perimental discovery that the Higgs boson mass at 125 GeV [28,29] requires the size of weak scale supersymmetry to lie in the TeV region, the sparticle spectrum for the scalars is typically in the TeV region, and the current experimental limits on the gluino mass also lie in the TeV region. The RG evolution of the Yukawas is sensitive to the sparticle spectrum and thus both the GUT boundary conditions and the sparticle spectrum enter in a significant way in achieving consistency with the data on the third generation masses for which currently the experimental limits are [30] m t (pole) = 172.25 ± 0.08 ± 0.62 GeV, Thus in this analysis we give a specific set of benchmarks where consistency with the data of  Table 1.
The massless mode is identified as the Higgs doublet pair that enters in the electroweak symmetry breaking. The Higgs doublets in this pair do not involve components from 126 + 126 + 210 heavy Higgs and have components only from 2 × 10 + 120 light Higgs. For that reason the non-vanishing parts of U d are the components U d11 , U d21 , U d31 , U d61 and similarly for V d . These are recorded in Table 2 and Table 3. Here the parameters a, b 1,2 , c andc are as defined in appendix C and are taken to be in the range 0.1 − 2.0.      for the model points (a)−(j). The masses are in GeV.
Next we give a computation of the Yukawa couplings at the GUT scale. As discussed in section 3, contributions to the Yukawa couplings arise from cubic interactions of Eq. (7) and from quartic interactions of Eq. (15) and Eq. (19). The couplings that enter here are: f 10r (r = 1, 2), f 126 , ξ, λ r (r = 1, 2). We take them in the range 0.1 ≤ f 126 , λ r , f 10r ≤ 2.5 and 0.1 ≤ ξ ≤ 3.0. Using the set of parameters in Table 4 we exhibit in Table 5 the contribution to the Yukawa couplings from the cubic interactions, from the quartic interactions, and their sum. Table 5 Table 7.   Table 7: Low scale SUSY mass spectrum showing the Higgs boson, the stop, the gluino, the stau and the light electroweakino masses and the LSP relic density for the benchmarks of Table 6.
The sparticle spectrum of benchmarks (a)−(g) contains light electroweakinos, i.e., of mass less than 1 TeV while stops and gluinos are much heavier. Those points will be of interest in the next section where we discuss the LHC implications. The dark matter relic density is calculated using micrOMEGAs-5.0.9 [40] and we use as an upper limit the experimental value reported by the Planck collaboration [41] (Ωh 2 ) PLANCK = 0.1198 ± 0.0012 .
As seen from Table 7 some model points do not saturate the relic density and thus these models can accommodate more than one dark matter component, e.g., a hidden sector Dirac fermion [42][43][44] or an axion [45,46]. We have checked that the spin-independent proton-neutralino cross-sections The kink in the evolution of the Yukawas is due to sparticle mass threshold effects.

Electroweakino pair production at the LHC and their decay channels
The low energy sparticle spectrum of the benchmarks in Tables 6 and 7 contain light electroweakinos (charginos and neutralinos). In this section we investigate the potential of discovering light electroweakinos with small mass splittings at the LHC. According to Table 7, points (a)−(f) possess the property of a small mass splitting between the lightest chargino and the lightest neutralino (LSP). Note that the second lightest neutralino has the same mass as the lightest chargino. Points (g) and (i) have very small mass splittings (less than 8 GeV) and require special treatment [47].
Point (h) is an example of a stop coannihilation scenario where the stop lies close in mass to the LSP while point (j) points to a stau coannihilation region. We will not consider these scenarios here (for previous works involving stop and stau coannihilation, see, e.g. [48,49] [50] as well as light charged and CP odd Higgs [51,52].
Constraints on the electroweakino mass spectrum from the LHC have been taken into consideration when selecting the benchmarks under study. CMS has excluded charginos up to 230 GeV with a mass splitting of ∼ 20 GeV while lighter masses were excluded for larger mass splitting (down to 100 GeV for 35 GeV splitting) [53,54]. More recent searches [55] in the zero and one lepton channels excluded charginos up to 200 GeV for a larger range of mass splittings, up to 50 GeV. ATLAS has put more stringent constraints on charginos and neutralinos. For the small and intermediate mass splittings [56] chargino mass up to 345 GeV has been excluded and up to 200 GeV also ruled out for an almost degenerate spectrum. The limit on charginos reach a mass ∼ 1.1 TeV associated with a massless neutralino [57,58]. For chargino mass of more than 350 GeV, a mass splitting with the LSP of up to 50 GeV is still allowed and that mass gap increases for heavier spectra. The benchmarks (a)−(f) are in accordance with those constraints from ATLAS and CMS.
We consider electroweakino pair production,χ 0  Table 8 along with the branching ratios ofχ 0 2 andχ ± 1 into the different final states of interest.
Branching ratios   Table 6. Also shown are the branching ratios to quarks and leptons for the electroweakinos of the same benchmarks. Note that q ∈ {u, d, c, s} and ∈ {e, µ}.
The second neutralino three-body decays into two light leptons (electrons and muons) proceed through an off-shell Z and Higgs bosons. Light leptons may also come from the decay of taus.
This three-body decay (shown in the last column of Table 8) can also proceed via the exchange of a stau. We note that the branching ratio to two taus is particularly enhanced for benchmark (b) and this is because of a relatively light stau (983 GeV, see Table 7). The three-body decay of a chargino into quarks is mediated by an off-shell W boson and is the dominant decay channel as seen in Table 8.

Signal and background simulation and event selection
The signal which consists of electroweakino pair production can be reconstructed based on specific final states of our choice. Here we look for a pair of same flavor and opposite sign (SFOF) light leptons (electron or muons), at least two jets and a large missing transverse energy (MET). The leptons are expected to be soft as a result of the small mass splitting between the LSP and the NLSP (chargino or second neutralino). However, the lepton and MET systems receive a kick in momentum as they recoil against a hard initial state radiation (ISR). This ISR-assisted topology is crucial in extracting the signal from the large standard model (SM) background. The signal region (SR) will be denoted as SR 2 Nj with N ≥ 2 as the number of jets required in the final state. The dominant SM backgrounds come from diboson production, Z/γ+jets, dilepton production from off-  cut around that value will remove most of the dominant backgrounds especially the Z+jets which has a peak around the Z boson mass. A veto on b-tagged jets will reduce the tt background and further preselection criteria on MET will reduce the rest of the SM backgrounds. The dominant background remaining is from dilepton production via off-shell vector bosons. More analysis cuts are required to reduce such a background. We summarize the preselection and selection criteria in Table 9.

Cut implementation and the estimated integrated luminosity
Selection criteria are optimized per mass range and for each collider, i.e. for HL-LHC and HE-LHC.
Starting with HL-LHC, the two signal regions we consider are SR 2 N j-A and SR 2 N j-B. They have the same preselection criteria but differ in terms of the analysis cuts on the variables E miss T and m as shown in Table 9. Signal regions pertaining to HE-LHC are termed SR 2 N j-C and SR 2 N j-D and as HL-LHC, the only differences are in the same two variables mentioned before.
For HE-LHC, harder cuts on E miss T , p ISR T and E miss T /p Z T are applied. Another variable used in the analysis cuts is ∆φ( p miss T , Z) which is the opening angle between the MET and p Z T ensuring that no jets constructed from W bosons fake the dilepton system. The variable E miss T /p Z T is a powerful discriminant since, unlike the backgrounds, the signal has the most MET and the softest of leptons so we expect the signal to have a larger value of this variable compared to the backgrounds. In order to design the optimal cuts on this variable, we plot the distributions in E miss   (e) require an integrated luminosity ranging between ∼ 1200 fb −1 to ∼ 2500 fb −1 for HL-LHC and ∼ 390 fb −1 to ∼ 800 fb −1 for HE-LHC. Point (f) which is only discoverable at HE-LHC require ∼ 6300 fb −1 . Note that despite being heavier than point (c), point (d) requires less integrated luminosity for discovery. The reason is that point (c) has a small branching ratio to dileptons (see Table 8) and so the overall cross-section to the required final states is smaller. Note also that the branching ratio to leptons for point (f) is very small (7%) compounded with the fact that it is the heaviest makes it very difficult to detect and that is why even at HE-LHC, which could potentially collect around 15 ab −1 of data [67,68], the required integrated luminosity is large.

Conclusion
In this work we consider a class of SO(10) models which lead to a pair of light Higgs doublets without the necessity of a fine tuning needed in generic grand unified models. In this class we consider a model with 126 + 126 + 210 of heavy Higgs and a 2 × 10 + 120 of light Higgs. The focus of this work is to show that significant contributions from the higher dimensional operators to the Yukawa couplings arise from matter-matter-Higgs-Higgs interactions in the superpotential where one of the Higgs fields is light and the other heavy, even though the interactions are suppressed by a heavy mass. This occurs because the heavy fields, after spontaneous symmetry breaking of the GUT symmetry, develop VEVs which are order the GUT scale which overcomes the suppression of the higher dimensional operator by the heavy mass. In this work we focused on computing the corrections to the third generation Yukawas using quartic couplings of Eq. (15) and of Eq. (19).
The analysis shows that the contribution of the quartic terms to the Yukawas can produce substantial corrections to the GUT boundary conditions for the Yukawas. The RG evolution using the modified boundary conditions shows that a consistency with the third generation quarks and the charged leptons masses can be achieved even with a low value of tan β, i.e., a tan β as low as 5−10 consistent with gauge coupling unification. The sparticle spectrum for the models considered was investigated and it is found that the relic density as an upper limit constraint can be satisfied in three coannihilation regions that arise in the models investigated, i.e., coannihilations involving chargino-neutralino, stau-neutralino, and stop-neutralino. Further, LHC implications for some of the chargino-neutralino coannihilation models was carried out for the possibility of SUSY discovery via the detection of a light chargino at HL-LHC and at a possible future collider HE-LHC at 27 TeV. It is shown that most of the models investigated can be discovered at HL-LHC using up to its optimal integrated luminosity while all of the models are discoverable at HE-LHC with a significantly smaller integrated luminosity and on a much shorter time scale. Discovery of a chargino, a stau or a stop which appear as the lightest sparticles in the analysis along with a determination of tan β which indicates a low value for it would lend support to this class of unified models. We note in passing that in the models of the type discussed the LSP can both saturate the relic density or be only a fraction of it. This implies that dark matter could be either a one component WIMP (neutralino) dark matter, or a multicomponent one where the WIMPs comprise only a fraction and the rest arises from other sources such as axions or matter from the dark sector.
Here the fields S

Appendix C Higgs doublet mass matrix
A computation of the 7 × 7 doublet mass matrix was given in [25]. Here we record the matrix for completeness using the constraint of Eq. (37). We have