Finite Naturalness and Quark-Lepton Unification

We study the implications of finite naturalness in Pati-Salam models where $SU(3)_C$ is embedded in $SU(4)$. For the minimal realization at low-scale of quark-lepton unification, which employs the inverse seesaw for neutrino masses, we find that radiative corrections to the Higgs boson mass are at least $\delta m_h^2 / m_h^2 \sim {\cal O}(10^4)$. The one-loop contributions to the Higgs mass are suppressed by four powers of the hypercharge gauge coupling. We find that for the vector leptoquarks the naively leading part of the two-loop corrections cancel. We assume the Dirac Yukawa couplings for neutrinos are equal to the up-type quark Yukawa couplings as predicted in the minimal theory for quark-lepton unification. Despite these findings, the two-loop corrections still dominate the finite naturalness bound. We mention a way to relax the lower bound on the vector leptoquark mass and have $\delta m_h^2 / m_h^2 \sim {\cal O}(10^2)$.


I. INTRODUCTION
The hierarchy problem (for some recent reviews see [1,2]) has motivated many proposed extensions of the standard model (SM), including technicolor and low energy supersymmetry.This problem arises from the quadratic dependence of the Higgs mass parameter on the momentum cut-off.The absence of experimental evidence for new particles at the Large Hadron Collider (LHC) has cast doubt on the relevance of this issue.The original description of the hierarchy problem involved integrating out momentum shells [3].Although this is a compelling physical picture, the quadratic divergences do not appear in all regulator approaches, for example in dimensional regularization.
In extensions of the SM that contain particles with masses (or fields with expectation values) much greater than the weak scale there can be a quadratic dependence of the Higgs mass on these quantities that is independent of the regularization and subtraction scheme.Demanding that these are not too large compared to the Higgs mass is the principle of finite naturalness [4].Since the hierarchy problem is not a mathematical inconsistency, it might be a red herring associated with how we approach the theory.Nonetheless, in this paper we take finite naturalness as a serious constraint on extensions of the SM.
An appealing extension of the SM is the SU(5) grand unified theory [5].Here, the strongest finite naturalness constraints occur at tree level (the doublet-triplet splitting problem).There are also problematic radiative corrections to the Higgs mass from virtual super heavy gauge bosons.Since the gauge couplings are known, these radiative corrections give δm 2 h /m 2 h ∼ O(10 24 ), which is clearly unacceptable if finite naturalness is to be taken seriously.
Quark-lepton unification, introduced by Pati and Salam [6,7], is also a very attractive extension of the SM.They proposed that the standard model SU(3) C is embedded in SU(4) where the leptons are interpreted as the fourth color.In this theory, baryon number is conserved and therefore the vector leptoquarks do not have to be super heavy to satisfy the strong constraints on baryon number violating processes.However, the vector leptoquarks do give rise to new flavor-violating processes that are very constrained by experimental data.For example, in the absence of large mixings, K L → µ ± e ∓ constrains the mass of the vector leptoquarks to be above 1000 TeV [8].Using the freedom in the unknown mixings between quarks and leptons this bound can be reduced to around 100 TeV [9,10].
In this article, we study the implications of finite naturalness on the minimal theory of quarklepton unification based on the gauge group SU(4) ⊗ SU(2) L ⊗ U(1) R [11] (see also Ref. [12]).To achieve a low-scale breaking of SU(4), we use the inverse seesaw mechanism for neutrino masses [12].For 100 TeV vector leptoquarks, we find that δm 2 h /m 2 h ∼ O(10 4 ), which is problematic from the perspective of finite naturalness.In the two-loop corrections to the Higgs mass involving the vector leptoquark we show that the naively leading contributions cancel.However, we find that the remaining parts still dominate the overall radiative contribution to the Higgs mass.An analogous effect happens in the two-loop contributions of the new neutral gauge boson, in which the naively leading corrections cancel.Finally, we discuss a way to relax the bounds on the vector leptoquark mass and have δm 2 h /m 2 h ∼ O(10 2 ).

II. MINIMAL THEORY FOR QUARK-LEPTON UNIFICATION
We consider the simplest Pati-Salam type extension of the SM, based on the gauge group [11,12] The quarks and leptons are unified in the following representations: Each of these comes in three copies.These representations contain the SM fermions plus (three) righthanded neutrinos.Since G PS is a product of three gauge groups, this theory has three gauge couplings, g 4 , g 2 , and g R , which are determined by the SM gauge couplings g 3 , g 2 , and g Y .
The gauge group G PS is connected to the SM gauge group through one breaking step, where an electrically neutral component of a non-trivial representation of SU(4) gets a vacuum expectation value (vev).This representation should be a singlet under SU(2) L , which is the same as the SU(2) L in the SM.We will focus on the simplest scalar representation able to trigger such a breaking, χ = (χ α , χ 4 ) ∼ (4, 1, 1/2) PS , and take Here α is a color index (α = 1, 2, 3) and A is a SU(4) index (A = 1, 2, 3, 4).The standard model hypercharge is given by where R is the U(1) R charge, and T 15 is the SU(4) generator A. The quark-lepton unification angle In this section, we summarize some aspects of minimal quark-lepton unification.Once χ gets a vev, the new massive vectors associated with the broken generators of SU(4) acquire mass through the covariant derivative of χ with T a being the generators of SU(4), normalized in the standard fashion, i.e.Tr(T a T b ) = 1 2 δ ab .The mass of the vector leptoquarks, X µ ∼ (3, 1, 2/3) SM , is given by The new neutral massive gauge boson, Z ′ µ , is a linear combination of the V 15µ gauge boson associated to the broken SU(4) generator T 15 (see Eq. ( 4)) and the U(1) R gauge boson, B Rµ .Its mass comes from the χ kinetic term, The rotation matrix that brings the gauge bosons to their mass eigenbasis is where the quark-lepton unification angle θ S is given by The hypercharge gauge coupling is g 1 = g R cos θ S , while the strong gauge coupling is g 3 = g 4 .Thus, the quark-lepton unification angle can be written as a function of standard model gauge couplings and the weak angle The mass of the gauge boson Z ′ is given by the trace of the mass matrix (the hypercharge gauge boson is massless in the electroweak symmetric phase), Therefore,
The standard model Higgs couples to the Z ′ µ gauge boson through the Higgs covariant derivative1 resulting in the following Feynman rules Therefore, the Z ′ µ contributes to the Higgs boson mass through the following one-loop diagrams and where we used Feynman gauge and set the subtraction point equal to the Higgs mass.In this paper, we neglect scheme-dependent terms that do not contain the large logarithm.Adding up both contributions, we find that the one-loop correction to the Higgs boson mass is given by Note that the above result is suppressed by four powers of the hypercharge coupling since C. Two-loop corrections to the Higgs mass The dominant two-loop corrections to the Higgs mass contain interactions of the vector leptoquarks and the Z ′ gauge boson with the fermions.We will give an estimate of the size of these contributions by computing only the diagrams involving the vector leptoquarks.
Because of the vertex of the Z ′ gauge boson with the Higgs, there are extra two-loop diagrams contributing to the corrections involving the neutral gauge boson.We do not consider these contributions given that they are proportional to powers of the weak coupling rather than the strong coupling.
In quark-lepton unification, the Higgs Yukawa couplings predict the following mass matrix relations The main corrections to the Higgs mass will involve the top quark and the neutrinos, and therefore we focus on the M u = M D ν prediction.The other relation in Eq. (19) will be addressed in the next section.
To correct the M u = M D ν identity, we assume the neutrinos get mass through the inverse seesaw [13,14], adding three left-handed singlets N L , which allows for the following interactions [12] where In the broken phase, the mass matrix in the basis (ν L , (ν R ) c , N L ) is given by where , and v = 246 GeV is the electroweak vev.We assume the following hierarchy µ ≪ M D ν ≪ M D χ which ensures that the light neutrinos are mostly ν L .We work in the limit µ → 0 and M D ν ≪ M D χ , where the heavy neutrino mass eigenstates are approximately Dirac and are obtained by diagonalizing M D χ .In the mass eigenbasis, choosing the left-handed neutrinos to align with the left-handed up-type quarks, the relevant interactions and mass terms are where , and the diagonal Dirac neutrino mass matrix In the above equation, Now, we can proceed with the discussion of the twoloop contributions of the X leptoquarks to the Higgs mass.We will first take all the couplings equal to one and ignore combinatorial factors.We identify two diagram topologies involving the vector leptoquarks and the fermions that give radiative corrections to the Higgs mass.We call I A the two-loop integral corresponding to topology A, given by while the two-loop integral associated with topology B is given by (24) We work in the limit where the relevant mass is the gauge boson mass (M ), and consider the fermions massless.Notice that we include a projector in the trace to account for the chirality characterizing the fermionic line.For the purposes of this paper we treat γ 5 as anticommuting with all other γ matrices.We evaluate the above expressions in d = 4 − ǫ dimensions.To assess the finite naturalness of this model we need to compute the leading scheme-independent finite parts of the radiative corrections to the Higgs mass.
By dimensional analysis, we expect all these diagrams to be proportional to The 1/ǫ 2 contribution from each diagram multiplying the above generates a divergent term of the form (1/ǫ) M 2 log(M 2 ).Should such term survive in the sum of all the diagrams it would require non-analytic counterterms.Hence we expect no residual 1/ǫ 2 dependence in the overall amplitude.As a consequence, the leading finite part will depend linearly rather than quadratically on log M 2 .We explicitly checked that this cancellation occurs.First, consider the diagrams The diagram where the left-handed neutrino ν 3 L pairs with the top quark gives where the factor 3 comes from color, and the 2 is a combinatorial factor that arises from possible ways of contracting the fields.The diagram involving righthanded neutrinos ν Rj leads to the following amplitude Similarly, the Higgs couples to the neutrinos, as the following diagrams display The diagram where the ν 3 L couples with the vector leptoquark gives where we sum over the right-handed neutrinos (i = 1, 2, 3).The diagram where the right-handed neutrinos couple to the vector leptoquark also gives where we sum over right-handed neutrinos and righthanded up-type quarks.Adding all contributions of topology A, i.e.Diag.( 25) and (28), One also expects the following diagrams, where, as in the previous cases, the factor 3 comes from color, a factor 2 comes from field contractions, and a factor 2 comes from the two contributions in Diag.(32).The overall contribution involving the X vector leptoquarks from both topologies is then proportional to As expected by general arguments and explicitly shown above, the leading scheme-independent piece scales linearly with the logarithm.Therefore, we find Hence, .
(36) Analogously to the vector leptoquark case, it can be shown that the leading contribution (proportional to g2 3 ) coming from the same topologies involving the Z ′ scales linearly with the logarithm of the heavy neutral gauge boson mass.However, this is slightly smaller than the X-boson radiative correction.The heavier mass of the Z ′ boson is indeed compensated by the prefactors from the SU (4) generators (see Eq. ( 4)) and the cosine of the quark-lepton unification angle.

D. Fermion masses
As manifest in Eq. ( 19), quark-lepton unification (with one Higgs) predicts the same mass matrix for the down-type quarks and charged leptons, which we have not addressed yet.Within the context of the G PS gauge group and demanding quark-lepton unification, there are two ways to correct the mass relation above at the renormalizable level M u = M D ν .Additionally, the lower bound on the vector leptoquark mass m X , and consequently m Z ′ (see Eq. ( 12)), can be considerably relaxed, as the unitarity condition on the mixing matrices is lifted.See Ref. [18] for a recent study on this possibility.The bound on the vector leptoquark mass in this case can be as low as a few TeV, which would also relax the tuning from the two-loop contributions

III. CONCLUDING REMARKS
The hierarchy problem, arising from the quadratic dependence of the Higgs mass parameter on the momentum cut-off, has motivated many proposed extensions of the SM.However, the absence of new particle discoveries at the LHC has raised doubts about the relevance of this issue.While quadratic divergences do not appear in all regulator approaches, certain extensions of the SM exhibit a quadratic dependence of the Higgs mass on the other masses of the theory, independent of the regularization scheme.This motivates the principle of finite naturalness, which demands that these quantities are not too much larger than the Higgs mass.
In this article, we explored whether Pati-Salam models with SU(3) C embedded in SU(4) respect the principle of finite naturalness.We showed that the minimal realization of quark-lepton unification leads to radiative contributions to the Higgs boson mass squared δm 2 h /m 2 h O(10 4 ), which is problematic regarding finite naturalness.We computed the two-loop corrections to the Higgs mass involving the vector leptoquark and showed that the naively leading contributions from these corrections cancel in the minimal theory for quark-lepton unification.Something analogous happens for the expected dominant contributions from the two-loop corrections involving the new neutral gauge boson, which cancel out.We also briefly mentioned extensions of the minimal model containing extra fermions where the level of tuning can be relaxed to δm 2 h /m 2 h ∼ O(10 2 ) by lowering the bound on the heavy gauge boson masses.