Dark Matter from $SU(6) \to SU(5) \times U(1)_N$

Matter and dark matter are unified under the framework of $SU(6)\to SU(5) \times U(1)_N$. A dark-matter candidate is possible, not because it is stable, but because it has a very long lifetime, in analogy to that of the proton in theories of grand unification. A specific example is presented.

Introduction : The existence of dark matter appears to be indisputable [1].Instead of taking it for granted as an ad hoc addition to the Standard Model (SM) of quarks and leptons, a more fundamental question may be asked as to its relationship with visible matter.One possible answer is that both belong to the same organizing symmetry, such as SO (10), but are distinguished by a marker symmetry such as U (1) χ [2,3] in SO (10) → SU (5) × U (1) χ .
Fermions and scalars which are odd and even under U (1) χ belong to the visible sector, whereas fermions and scalars which are even and odd under U (1) χ belong to the dark sector.
They are distinguished by (−1) Qχ+2j where j is the particle's spin.The lightest dark particle is assumed to be neutral and is stable because of this odd-even symmetry.
Another possible answer is that there is no marker symmetry and both visible and dark matter coexist in multiplets of an organizing symmetry such as SU (6) [4,5], but the darkmatter candidate itself has a very long lifetime, just as the proton has a very long lifetime in theories of grand unification.A specific complete model of SU (6) → SU (5) × U (1) N is presented here for the first time, where a dark fermion decays to SM particles through a superheavy gauge boson.SU(6) Unification of Visible and Dark Matter : Consider the extension of the well-known SU (5) model [6] of grand unification to SU (6) with SU (6) → SU (5) × U (1) N .Although one family of fundamental fermions under SU (5) is contained in the anomaly-free combination of 5 * and 10, the analogous case for SU (6) [7,8] is two 6 * = (5 * , −1) + (1, 5) and one 15 = (10, 2) + (5, −4).Let The scalar sector consists of from (24, 0) to  Neutrinos obtain Majorana seesaw masses proportional to v 2 4 /v 2 , with N 1 acting as the usual right-handed neutrino in left-right models.This shows that SU (6) may be used for seesaw neutrino masses in lieu of the customary SO (10).The 3 × 3 mass matrix spanning (N 2 , E 0 , Ē0 ) is of the form ( Gauge Boson Masses and Interactions : The gauge bosons belonging to the adjoint 35 representation of SU (6) are superheavy with masses proportional to v 3 except for those corre- and that of SU (2 The charged W ± mass is given by the massless photon is A = (e/g L )W 3 + (e/g Y )B, where B is the U (1) Y gauge boson and where For simplicity, v 5 = 2v 4 may be assumed, so that Z and Z N do not mix, thereby preserving all electroweak precision measurements involving the Z boson.
The gauge interactions of Z are given by and those of Z N by As such, Z N may be produced at the collider through its couplings to u and d quarks, and be discovered through its couplings to charged leptons.The present collider limit [9] is estimated to be a few TeV.
The Higgs potential is Note that the dimension-three µ 1 term breaks the Z 2 symmetry softly.Together with the µ 2 term, they ensure that there would be no extra accidental U(1) symmetry in V beyond The minimum of V is determined by The 4 × 4 mass-squared matrix spanning √ 2Im(η 0 1 , η 0 2 , φ 0 4 , φ 0 5 ) is given by Two zero eigenvalues appear, corresponding to )] and (0, 0, v 4 , −v 5 ), becoming the longitudinal components of Z N and Z respectively.The remaining two massive pseudoscalar components span (2v 2 , v 1 , 0, 0) and divided by It shows explicitly that µ 1 = 0 or µ 2 = 0 implies one zero eigenvalue, and µ 1 = µ 2 = 0 implies two.In the limit v 4,5 << v 1,2 , it reduces to The 4 × 4 mass-squared matrix spanning √ 2Re(η 0 1 , η 0 2 , φ 0 4 , φ 0 5 ) is given by , then its mass is given by It is the only linear combination of the four neutral scalar fields which has no v 1,2 contribution to its mass, and acts as the SM Higgs boson in its interactions.The other three scalar bosons are much heavier and have suppressed mixing with h, assuming again v 4,5 << v 1,2 , Consider now the linear combinations Then This shows that if µ 2 >> v 1,2 , then S 2 is much heavier than S 1 and their mixing is supressed.
This scenario is useful for the dark matter phenomenology to be discussed later.
The remaining scalar has mass given by It mixes with h through the term This shows that µ 1 , v 1 >> v 4,5 guarantees that H is heavier than h and their mixing is suppressed, as remarked earlier.
Dark Matter : Of the new particles beyond those of the Standard Model (SM), N 1 acts as the seesaw anchor of ν as already explained.It replaces the usually assumed righthanded neutrino.The others are the color-triplet fermions D of charge −1/3, the vectorlike electroweak doublet fermions (E − , E 0 ) and the neutral singlet N 2 fermion.At the level of the SM extension to U (1) N , as is clear from ( 7) and ( 8), this latter set of particles are distinguished from those of the SM by a discrete Z 2 symmetry under which they are odd.
The lightest, presumably N 2 , could then be dark matter.This is analogous to the stability of the proton from baryon number conservation in the SM.However, once it is realized that these particles are embedded into SU (6), it is clear that N 2 must also decay, just as the proton.
Together with (2), they allow the decay N 2 → e + W − through the superheavy neutral gauge boson X 0 in (5,6) as shown in Fig. 1.This amplitude is proportional to and is of similar magnitude to that of proton decay.This establishes the notion that dark matter stability is akin to proton stability in the context of the unification of matter and dark matter, in parallel to that of quarks and leptons.
As for the heavy E leptons, it is clear that E 0 decays to N 2 h from ( 8) and E − decays to E 0 W − .The heavy D quark decays through the scalar (3, 1, −1/3, −4) component of (5, −4) in 15 S or 21 S to N 2 dN 1 , with N 1 decaying to νh.Note that these interactions do not violate the Z 2 symmetry separating matter from dark matter at low energy.They serve the purpose of allowing the heavier dark particles to decay to N 2 rapidly, assuming that the mediating scalars are not too heavy.Note also that proton decay is possible through scalar exchange as in SU (5).Here it occurs through the mixing of (3, 1, −1/3, −4) with (3, 1, −1/3, 1) through the term 6 S × 15 * S × 84 S , and may be suppressed with a large mass for (3, 1, −1/3, 1) as in SU (5).
Relic Abundance and Direct Search : The relic abundance of the very long-lived N 2 is determined by its annihilation to scalar bosons which are in thermal equilibrium with SM particles.
In particular, the dominant process is shown in Fig. 2, assuming m , and the S 1 S 1 S 1 coupling is dominated by . Hence the annihilation cross section at rest multiplied by relative velocity is As an example, let v 1 = 2v 2 = 5 TeV, m N 2 = 1 TeV, m S 1 = 800 GeV, and µ 2 = 14.7 TeV, then σ ann × v rel = 3 × 10 −26 cm 3 /s, the canonical value for the correct dark matter relic abundance of the Universe.
As for direct search, since N 2 is a Majorana fermion, it does not couple to the Z N gauge boson at rest, so its only interaction with matter at underground experiments is through the SM Higgs boson.The mixing of S 1 with h comes from the term 2 ), whereas h couples to quarks by m q / 2(v 2 4 + v 2 5 ).The spin-independent elastic scattering cross section of N 2 off a Xenon nucleus per nucleon is given by where [10] For m N 2 = 1 TeV, v 1 = 2v 2 = 5 TeV, v 2 4 + v 2 5 = 174 GeV, m h = 125 GeV, the upper limit on θ from [11] σ 0 < 10 −45 cm 2 is 1.84 × 10 −3 .Without fine tuning, θ is of order 2 ) ∼ 1/60.Hence a fine tuning of order 1/10 is required.This is possible by adjusting the value of µ 1 in the range of v 1,2 relative to other free parameters in (25).with the appropriate addition of some particle multiplets [12,13,14,15], unification of gauge couplings is possible at a high energy scale.

Gauge Coupling Unification
Consider the one-loop renormalization-group equations where α i = g 2 i /4π and the coefficients b i are determined by the particle content between M 1 and M 2 .In the SM with one Higgs doublet, these are given by SU By incorporating dark matter as an essential component of grand unification, it is shown that just as the conservation of baryon number and lepton parity in the SM is violated in the unification of quarks and leptons, the conservation of dark parity at low energy is violated in the unification of matter and dark matter.The longevity of dark matter is then linked to the longevity of the proton, as a natural explanation of the former's existence.
× 15 F couples to 6 S and only 6 * F 1 × 15 F couples to 84 S .Thus D c D and E − E + + E 0 Ē0 masses are proportional to v 1 , whereas d c d and ee c masses are proportional to v 5 .Similarly, and νN 1 , E 0 N 2 masses to v 4 .Finally, 15 F × 15 F couples to 15 * S , with u c u masses proportional to v 4 as well.