Probing Unification With Precision Higgs Physics

We propose a novel approach of probing grand unification through precise measurements on the Higgs Yukawa couplings at the LHC, which is well motivated by the appearance of effective operators not suppressed by the mass scale of unification $M_{\rm{U}}$ in realistic models of unification with minimal Yukawa sector. These operators modify the Higgs Yukawa couplings in correlated patterns at scale $M_{\rm{U}}$ that hold up to higher-order corrections. The coherences reveal that, the weak-scale effect on tau Yukawa coupling is the largest among the third generation, which if verified by the future LHC, can serve as a hint of unification.


I. INTRODUCTION
The Higgs Yukawa couplings to the standard model (SM) fermions such as top [1], bottom [2], and tau [3] have been verified at the LHC. With upcoming data at the future LHC, we will enter into an era of precision tests on the Higgs scalar, which may shed light on the fundamental laws underlying the electroweak symmetry breaking. While the minimal version of supersymmetry is not satisfactory such as in explaining the observed Higgs mass, it is still on the short list of frameworks that address some well-known big questions such as the hierarchy problem.
Since supersymmetry is advocated to solve the hierarchy problem, there are limited tools to probe the underlying grand unification theory (GUT), as the unification scale M U is far larger than the weak scale. Until now, proton decay is the most important tool to detect unification (For a review on unification and proton decay, see ref. [4].). According to the dependence of proton decay lifetime on M U , i.e. τ p ∼ 1/M 4 U , a large amount of models can evade the Super-Kamiokande limits [5,6] by adjusting the value of M U in the mass range 10 15−17 GeV. A delay of update on those experimental limits postpones our exploration along this line.
In this Letter, we propose a novel method of probing GUT through precision measurements on the Higgs Yukawa couplings. Similar to high-dimensional operators that lead to proton decay through interactions between the GUT-scale states and the SM fermions, there are analogies which modify the Higgs Yukawa couplings due to interactions between GUT-scale states φ and the Higgs doublets: We will show that a). ǫ i terms are less than unity but not suppressed by 1/M U for where φ ∼ M U represents the vacuum expectation value (vev) of SM singlet responsible for breaking the GUT gauge group. b) All of ǫ i terms are always correlated in specific patterns rather than independent parameters. c). The coherences lead to specific patterns of derivations in the Higgs Yukawa couplings from SM predictions, which can be verified by the future LHC. This is the subject of this Letter.

II. MINIMAL YUKAWA SECTOR
Let us briefly review the realistic models of unification with the minimal Yukawa sectors.
For the SU(5) unification, the Higgs fields in the minimal Yukawa sector are composed of a 5, a5 and a 45. The 45 [7] is added to the Yukawa sector in order to adjust the lepton and down quark Yukawa couplings at the scale M U . Under this Yukawa structure, the Yukawa couplings at scale M U are given by, where υ 5 u , υ5 d and υ 45 d refers to the vevs of doublets in 5, 5 and 45, respectively; υ u = υ sin β, υ d = υ cos β, with υ = 174 GeV; and Y u , Y d and Y 45 are 3× 3 matrixes in generation space, with i, j = 1 − 3. We recall that with m 45 ∼ M U , the proton decay mediated by the component fields in 45 is small. What matters [8] in this Yukawa sector is the generation of a small vev υ 45 d , compared to a large mass m 45 .
For the SO(10) unification, the minimal Yukawa sector [9] is composed of a 10 and an 126. The purpose of 126 [10] closely follows from that of 45 in the SU(5). The Yukawa couplings at scale M U read as, where υ 10 u,d and υ 126 u,d refers to the vevs of doublets in 10 and 126, respectively, and y ν is the neutrino Yukawa coupling. Similar to 45, for m 126 ∼ M U the proton decay due to component fields of 126 is small.
Fitting the values of Yukawa couplings at scale M U to their SM values at the scale m Z in terms of the renormalization group equations (RGEs), one can fix all of the input parameters in Eq.(3) or Eq.(4).

III. UNSUPPRESSED EFFECTIVE OPERATORS
It is well known that we will obtain the effective operators [11,12] which contribute to Eq.(1) after one integrates out heavy freedoms with characteristic mass scale M . One is also aware of that the ability of testing those effective operators dramatically declines as the value of M increases. In the situation M ∼ M U , the effects on SM observables due to those operators are supposed to be tiny (e.g., τ p ), unless they aren't suppressed by power laws of 1/M U . We will show that there are indeed unsuppressed effective operators in Eq.(1) by integrating out the heavy Higgs field45 or 126 in the minimal Yukawa structure as discussed in the preceding section.
We show in the left plot of Fig.1 the generation of effective operator after integrating out the vectorlike Higgs fields 45 in the minimal Yukawa sector 1 . Here, φ is a 75-dimensional Higgs that spontaneous breaks the SU(5) gauge group into the SM gauge group, which is often considered as the most economic solution to the doublet-triplet problem in the literature [13]. In Eq. (5), one finds the coefficients in Eq.(1) where the overall scale ǫ = φ(75) /M U , with M U referring to the effective VL mass m 45 . For simplicity, all Yukawa coefficients in the vertexes of the Feynman diagram are absorbed into ǫ. The coherence ǫ ij d ≃ ǫ ij e in Eq.(6) is a result of the GUT representation, which is independent of GUT-scale parameters such as ǫ, Y 45 and the ratio of two vevs. This coherence can be a key to reveal the SU(5) unification through the precision measurements on relevant Higgs Yukawa couplings.
Similarly, we can analysis the Feynman diagrams in the where the vectorlike Higgs is 126 instead of 45, φ is 210 instead of 75, and a 16 supermultiplet represents a whole generation of SM fermions. Unlike in the SU(5), in this minimal Yukawa sector the light Higgs doublets can be dynamically obtained as in the benchmark model studied below. In Eq. (7), one obtains where ǫ = φ(210) /M U , with M U referring to the effective VL mass m 126 . Once again, one observes that the coherences for the same generation in Eq.(8) are independent of the GUT-scale parameters such as ǫ and Y 126 .

IV. PRECISION MEASUREMENTS
Either in Eq.(6) or Eq. (8), the corrections to Higgs Yukawa couplings dominate the next-leading order contributions, as long as the Yukawa sector is minimal and ǫ is less than unity. Their impacts at the scale m Z rely on the magnitudes of orders of these corrections. In individual situation, there are small hierarchies among the matrix elements of Y ij 45 or Y ij 126 [9], which arise from the SM fermion mass hierarchies. As a result, the largest effects always occur in the third-generation Yukawa couplings y α (α = t, b, τ ). Since precision measurements on Yukawa couplings y α are prior to the first two generations at the LHC, we will mainly focus on y α as what follows.
Given an explicit ǫ, the weak-scale effects on y α can be derived as follows. First, one uses the central values [14] of SM observables at the scale m Z to determine all input parameters at the scale M U through the RGEs from m Z to M U . During this process, one has to deal with various intermediate effective theories. Second, we add correlated ǫ-corrections to y α at the scale M U , then perform the RGEs reversely from the scale M U to m Z , which gives rise to the dependences of δy α on ǫ at the scale m Z . During the RGEs, there are certain uncertain-ties are similar to those of proton decay 2 .
In the literature, there is a lack of "complete" fit to the SU(5) model with the minimal Yukawa sector. The "completeness" means that all SM fermion masses and mixings are addressed, with important constraints such as proton decay taken into account. Otherwise, the theoretical uncertainty is too large to invalidate the RGE analysis. For earlier studies on this model, see. e.g. refs. [15,16].
On the contrary, there are extensive studies on the SO(10) model with the minimal Yukawa sector. We will use the latest results in ref. [9], while earlier studies can be found, e.g. in refs. [17][18][19][20]. In this benchmark model, the effective theories are composed of the SM between the scale m Z to the gaugino mass threshold of order 1 TeV, the split-supersymmetry between 1 TeV and the sfermion mass threshold of order 10 2 TeV, the complete MSSM from 10 2 TeV to the right-hand neutrino mass threshold m νR , and finally the MSSM with U B−L between the scale m νR and M U .
Compared to the SU(5), there are more intermediate RGE steps in the SO(10). The main reasons for this are that the SO(10) has larger structure of gauge group than the SU(5), and the fit to SM neutrino observables also plays a role in affecting the RG trajectory. In this situation, the uncertainties include the explicit values of M U , m νR and intermediate mass threshold corrections. Since M U and m νR are restricted to narrow mass ranges 10 16−17 GeV and 10 12−13 GeV respectively, and δy α are logarithm dependent on them, one expects a theoretical uncertainty to δy α at most of order ∼ 1 − 3% .
The left plot in Fig.2 shows the values of δy α in the case of type-II correction in Eq.(8), with r = (υ 10 u /υ u ) · (υ 10 d /υ d ) −1 = 8.73 [9], where the one-loop RGEs of the SM [21,22], the split supersymmetry and the MSSM [23] are used. The splitting soft mass spectrum is chosen in order to avoid the constraint from proton decay [24]. In this plot, it is clear that the parameter ranges | ǫ |≥ 0.2 and 0.1 ≤| ǫ |≤ 0.2 can be tested through δy b and δy τ by the LHC with luminosity 300 fb −1 and 3000 fb −1 , respectively. Compared to y b and y τ , y t receives smallest correction but faces largest experimental uncertainties [25,26].
One can perform similar analysis in the right plot in Fig.2, which shows the values of δy α in the case of type-III correction in Eq.(8), with s = (υ 126 u /υ 126 d ) · (υ 10 d /υ 10 u ) [9]. Compared to type II, where ǫ t is the largest input value due to the enhancement factor r, ǫ τ is the largest input value in the case of type III. In this case, one expects larger value of δy τ , which indicates that the same LHC limits can reach smaller parameter region | ǫ |∼ 0.01 − 0.02, as shown in the figure. The parameter region | ǫ |≥ 0.02 can be fully covered by the LHC limit with luminosity 300 fb −1 . Whenever the corrections to δy α are roughly of same order, the magnitude of δy τ at the scale m Z is always the largest.
While modifying y α , the ǫ-corrections also contribute to off-diagonal elements of y u,d that lead to flavor violation. They appear even though ǫ ij is diagonal at the scale M U because of RGE effects [27]. In the context of type-II two Higgs doublets as we study here, the most stringent constraint in the parameter region with moderate or large tan β arises from BR(B s,d → µ + µ − ). A partial reason for it is that they are enhanced by the factor tan β, unlike in the other cases such as Br(t → u i h) (u i = {u, c}) that are suppressed by cos 2 (α − β). Because of the feature above, BR(B s,d → µ + µ − ) is actually more sensitive to parameters tan β and the neutral Higgs boson mass rather than the deviations of a few percent level in the Yukawa couplings in Fig.2. Typically, the ǫ corrections only yield a deviation of order ∼ 0.13% relative to the SM prediction BR(B s → µ + µ − ) SM = 3.26 × 10 −9 for tan β = 10 and m A = m H = 1 TeV, which is consistent with the LHCb limits BR(B s → µ + µ − ) exp = (2.8 +0.7 −0.6 ) × 10 −9 [28,29].

V. CONCLUSIONS
Unification is an important theoretical idea of new physics beyond SM. Yet, there are limited ways in testing it except proton decay experiment in the last a few decades. Unfortunately, the advance along this line is delayed due to the experimental status. In this Letter, we proposed the novel approach of probing unification through precision measurements on the Higgs Yukawa couplings especially of the third generation. Our analysis shows that a large deviation in y τ but small in y t and y b , if favored by the future LHC data, can serve as a smoking gun of any realistic model of unification with the minimal Yukawa sector.
Our results are supported by three observations. The first observation is the appearance of unsuppressed effective operators through integrating out the heavy Higgs freedom 45 or 126 in any realistic model with the minimal Yukawa sector. These operators dominate the nextleading-order corrections to Yukawa couplings. The second observation is that the corrections to y α at the scale M U are in three specific patterns, which are results of GUT representations. Lastly, the deviations to y α at the scale m Z can be verified by the future LHC limits (see Fig.2), although there are subject to certain uncertain-ties in an explicit RG trajectory between the scales m Z and M U .