Anomalous leptonic U(1) symmetry: Syndetic origin of the QCD axion, weak-scale dark matter, and radiative neutrino mass
UUCRHEP-T579June 2017
Anomalous Leptonic U(1) Symmetry:
Syndetic Origin of the QCD Axion,
Weak-Scale Dark Matter,and Radiative Neutrino Mass
Ernest Ma , Diego Restrepo , and ´Oscar Zapata Department of Physics and Astronomy,University of California, Riverside, California 92521, USA Instituto de F´ısica, Universidad de Antioquia,Calle 70 No. 52-21, Apartado A´ereo 1226, Medell´ın, Colombia
Abstract
The well-known leptonic U(1) symmetry of the standard model of quarks and lep-tons is extended to include a number of new fermions and scalars. The resulting theoryhas an invisible QCD axion (thereby solving the strong CP problem), a candidate forweak-scale dark matter, as well as radiative neutrino masses. A possible key connec-tion is a color-triplet scalar, which may be produced and detected at the Large HadronCollider. a r X i v : . [ h e p - ph ] J un ntroduction : In the standard model (SM) of quarks and leptons, there are four automaticglobal symmetries: baryon number B = 1 / L e = 1 for theelectron and its neutrino ν e , L µ = 1 for µ and ν µ , and L τ = 1 for τ and ν τ . As such, allneutrinos are massless. Given that we now know that neutrinos are massive and mix amongthemselves, the SM must be extended and L e,µ,τ must be replaced with L = L e + L µ + L τ .Hence L is still a valid global U (1) symmetry if neutrinos are strictly Dirac fermions, butoften than not, they are assumed to be Majorana fermions so that L is broken to ( − L , i.e.lepton parity.Theoretical mechanisms for obtaining Majorana neutrino masses are many [1], but thereis no experimental evidence for any one of them. Then there is the dark matter (DM)of the Universe. The SM has no explanation for it, but the intriguing idea exists that itmay be connected to the neutrino’s mass generating mechanism. In 2006, a simple one-loopradiative mechanism was proposed [2] with dark matter in the loop, called “scotogenic” fromthe Greek “scotos” meaning darkness. In 2015, it was shown [3] that the dark parity of thismodel, as well as many others, is derivable from lepton parity. This demonstrates how theleptonic U (1) L symmetry may be extended to include particles beyond those of the SM.In 2013, it was shown [4] that the well-known spontaneously broken anomalous Peccei-Quinn U (1) P Q symmetry [5], which solves the strong CP problem [6] and creates the QCDaxion [7, 8], has a residual Z symmetry which may in fact be dark parity. In this paper, wecombine all these ideas to show that, with the proper choice of fermions and scalars beyondthe SM, we can have U (1) P Q = U (1) L with the residual dark parity [3] given by ( − B + L +2 j ,i.e. the well-known R parity of supersymmetry, but not in a supersymmetric context.To implement this important new insight, i.e. U (1) P Q = U (1) L , in a specific model,we choose the singlet-doublet-fermion dark-matter scenario with three additional scalars toobtain scotogenic neutrino masses [9, 10]. This framework is however also adaptable for2adiative quark and lepton masses [11, 12]. Particles Beyond the SM : The new particles of our model are assigned under U (1) L asshown in Table 1. Note that the only new fermion which transforms under U (1) L is D . AsTable 1: Particle assignments under P Q = L .Particle SU (3) C SU (2) L U (1) Y P Q = L B RD L − / − D R − / − − N L − ( E , E − ) L,R − / − ζ / − χ , , − σ U (1) L is broken by (cid:104) σ (cid:105) , the QCD axion appears, together with the residualsymmetry R = ( − B + L +2 j , which is even for SM particles as well as σ , but odd for allthe other new particles. The axion is thus of the KSVZ type [13, 14] and the domain wallnumber is 1, so it is cosmologically safe [15].The axion decay constant F A , i.e. (cid:104) σ (cid:105) , is known to be large [16]: F A > × GeV.Hence the singlet D quark is expected to be heavy, unless the Yukawa coupling for σ ¯ D L D R isvery small. If this is indeed the case, then D may be produced in pairs at the Large HadronCollider (LHC), and the ¯ D L d R χ term [4] would allow it to be discovered. Alternatively, if adark scalar doublet ( η + η ) exists as in the original scotogenic model [2], then the ¯ D R ( u L η − + d L ¯ η ) term works as well [17]. On the other hand, if D is very heavy (of order F A ) as expected,then it is impossible for it to be produced at the LHC. In this study, we will consider insteadthe dark scalar quark ζ with charge 2 /
3. Its mass is not constrained and may well be withinthe reach of the LHC and be produced copiously in pairs through its gluon interaction.3 wo-Component Dark Matter : As shown in Ref. [4], the coexistence of the QCD axion witha stable weak-scale particle allows for a much more flexible two-component theory of darkmatter. It relaxes the severe constraints imposed on either component if considered alone. Itallows for a solution of the strong CP problem, without having the QCD axion as observabledark matter. Regarding the weak-scale DM particle, it is the lightest particle charged underthe residual dark parity ( − B + L +2 j , and can be either a real scalar [21, 22, 23] or anadmixture of a singlet-doublet fermion [24, 25, 26, 27].In the scalar case, since χ , , carry lepton number, they are complex with invariant( m χ ) ij χ ∗ i χ j terms. However, a 6 × σ ∗ χ i χ j terms. There are thus six real scalar eigenstates. Since the mass splittingsof the real and imaginary parts of the complex χ scalars are proportional to (cid:104) σ (cid:105) , they arepresumably large. Hence fine tuning is required [4] to make one component light and theother heavy, if we want the lightest χ (call it χ ) to be dark matter.In the fermion case, there are invariant mass terms m E ¯ E R E L and m N N L N L , as well asthe allowed mixing terms between N and E which are proportional to (cid:104) φ (cid:105) . The 3 × N L , E L , ¯ E R ) is then of the form M NE = m N m L m R m L m E m R m E (1)resulting in 3 Majorana fermion eigenstates, the lightest (call it N ) is dark matter. Assumefor example m L = m R = m V / √ > m N = m E >
0, then the three mass eigenvalues(in increasing magnitude) are m E − m V , − m E , m E + m V , (2)4orresponding to the three mass eigenstates N = N L / √ − ( E L + ¯ E R ) / , (3) N = ( E L − ¯ E R ) / √ , (4) N = N L / √ E L + ¯ E R ) / . (5)In either case, it may only account for part of dark matter, the rest coming from axions.In direct-search experiments, the exchange of Z is irrelevant because χ is a singlet, and N is Majorana. However, the exchange of h (the SM Higgs boson) will contribute. As for relicabundance, beyond those interactions of the minimal models mentioned earlier, we have alsothe Yukawa terms ¯ ν i N χ , which may also contribute. There are many free parameters inour model to make this work, but it is not our goal to examine them in any detail. Afterall, these issues have been dealt with thoroughly in those previous studies. Instead, we willfocus on the feasibility of finding the scalar quark ζ which connects the high scale (10 to10 GeV) of the axion to the much lower scale (100 GeV) of the dark-matter candidates χ and N . Scotogenic Neutrino Mass : Using the Yukawa terms χ (¯ ν L E R + ¯ e L E − R ), ¯ N L ( φ E R − φ + E − R ), N L ( φ E L − φ + E − L ), and σ ∗ χ i χ j , the one-loop diagram of Fig. 1 is obtained, thereby generatingFigure 1: One-loop generation of neutrino mass with U (1) L .three radiative neutrino masses through the spontaneous breaking of U (1) L [9, 10]. This ideawas previously applied directly to the canonical seesaw mechanism with singlet right-handed5eutrinos [18, 19], thus equating the axion scale to that of the neutrino seesaw. Here theaxion scale enters through (cid:104) σ (cid:105) . Our model differs conceptually from the previous use ofFig. 1 because we equate U (1) L with U (1) P Q and let it be spontaneously broken. However,the resulting neutrino mass matrix has the same structure as previous studies and the detailsare available in those references [9, 10].In this scotogenic model, the family index is carried by χ , so a possible family symmetrymay be considered. Let ν , , and χ , , transform as 3 under the non-Abelian discretesymmetry T for example [20]. The group multiplcation rule of T is3 × ∗ + 3 ∗ , (6)and since σ ∼ T , it does not close the loop of Fig. 1. We now need to add three extrascalars ρ , , ∼ ∗ to couple to χ i χ j to complete the loop of Fig. 1. It is then possiblefor (cid:104) ρ , , (cid:105) to be much smaller than (cid:104) σ (cid:105) , in which case the lightest χ may be naturally of theelectroweak scale as a dark-matter candidate. Possible Hadronic Connection : Whereas the heavy color-triplet fermion D connects withthe SM through the Yukawa term ¯ D L d R χ , another possible way is through the color-tripletscalar ζ , with the important terms f D ζ ¯ D L e R + H.c. as well as f N ζ ¯ u R N L + f E ζ ∗ ( ¯ E R u L + ¯ E + R d L ) + H.c. = f N ζ ¯ u R ( N + N ) / √ f E ζ ∗ ( − N / − N / √ N / u L + f E ζ ¯ d L E − R + H.c. (7)We assume that D is very heavy, so it decays away quickly in the early Universe to either eζ or dχ . Subsequently, either χ or N becomes a component of the dark matter of theUniverse, together with the axion.To test our hypothesis, we propose a search for ζ at the LHC. It is easily produced,because it is a scalar quark. We assume first that the Majorana fermion N is dark matter.6f f N is dominant, then ζ decays equally to N and N , with a quark jet in each case. Whereas N is stable and invisible, N will decay, i.e. N → ν i χ j + E ± W ∓ (8)with the subsequent decay or conversion (if χ j is virtual) χ j → ν k N + (cid:96) ± k E ∓ , (9)and E ± → W ± N . (10)Most events are then of the type 2 jets + missing energy. They are thus analogous to scalarquark pair production with decays to a quark and a neutralino in supersymmetry. We canthus borrow from the existing studies of supersymmetric scenarios to put a bound on m ζ asa function of m N and m N .If f E is dominant, then ζ decays equally to N , , and E + , with a quark jet in each case, asshown in Eq. (7). Whereas most N , , decays are invisible, E + decays according to Eq. (10).This is analogous to a squark decaying to a quark and a chargino which then decays to a W and a neutralino in supersymmetry. If we focus on the leptonic decay of W , then the finalstates of ζζ ∗ production at the LHC may also include 2 jets + (cid:96) ± + missing energy and 2jets + (cid:96) +1 (cid:96) − + missing energy.Consider the alternative case that the real scalar χ is dark matter. This means that thefermions N i are heavier. The decay of ζ to u + N i will have another step, i.e. N i → ν j χ , (11)which are invisible. The signature is again 2 jets + missing energy, but now there are tworelevant masses, m N i and m χ . If m N i is close to m ζ , the kinematics will be quite differentfrom the supersymmetric analog discussed previously where N is dark matter. The quark7ets will be soft and could miss the cut on their momenta. In that case, if f N dominates inEq. (7), the signal is just missing energy.If f E dominates, we have again the decay of ζ equally to N i and E + . For the latter, thesecond step is now E + → χ (cid:96) + i . (12)The final states of ζζ ∗ production at the LHC will again include 2 jets + (cid:96) ± + missing energyand 2 jets + (cid:96) +1 (cid:96) − + missing energy. However, since the charged leptons come directly from E decay, their numbers are not diminished by the branching fraction of W to leptons as inthe case where N is dark matter. Also, if m E is close to m ζ , the jets may be too soft to beobservable. In that case, we will only find leptons + missing energy. LHC Signatures : We will discuss first the case of fermion dark matter, where we have themain collider signature of two jets plus missing transverse energy: 2 j + E missT . The furtherdecay of E + to E via a W -boson only increases the number of soft objects so that the mainsignal is still just 2 j + E missT . This contribution, through the corresponding quasi-degenerate E + - E states, are usually already included in the analyses to be discussed below.Regarding the signal 2 j + E missT , it has been already studied by ATLAS and CMS in thecontext of simplified supersymmetric scenarios searching for squarks decaying into a quarkand neutralino. In the case of fermion dark matter, our branching fraction is approximately100% . The results for the production of a single squark reported by CMS [28] based on 35.9fb − at 13 TeV are thus fully applicable in our case and reproduced in Fig. 2 (solid line).They allow us to exclude, for example, m ζ up to 1.0 (0.8) TeV for m N = 100 (400) GeV .The exclusion limit at 95% confidence level on the cross section of direct production of ζ pairs(color bar) from [28] is shown in Fig. 2, where the region below the solid line corresponds to It is worth noticing that contrary to the standard scenario of singlet-doublet fermion dark matter wecan have now doublet-like fermion dark matter component less that 1 TeV while still being compatible withdirect detection constraints.
00 400 500 600 700 800 900 1000 1100 m (GeV) m D M ( G e V ) % C L u pp e r li m i t o n c r o ss s e c t i o n ( pb ) Figure 2: Exclusion limit at 95% CL on the cross section for ζ pair production from theCMS data [28] based on 35.9 fb − at 13 TeV. The region below the solid line corresponds tothe excluded region for the case of fermion DM ( N is the DM particle) whereas the regionbelow the dashed line is the excluded region for the case of scalar DM (when χ is the DMparticle).the current excluded region in the ( m N , m ζ ) plane. At 13 TeV with 36.1 fb − the ATLASresults [29] are reported taking into account the production of 8 squark states of the firstand second generation. Since the results are similar to the ones from CMS [28], we expectsimilar lower bounds on m ζ .In the case of scalar dark matter , E subsequently decays to ν + χ and E + to (cid:96) + + χ .This leads to the following collider signatures: 2 j + E missT , 2 j + (cid:96) ± + E missT and 2 j + (cid:96) +1 (cid:96) − + E missT ,with a branching fraction of 25%, 50% and 25%, respectively. These kinds of signals werestudied in Run-I by ATLAS at 8 TeV with luminosities of around 20 fb − in the contextof simplified supersymmetric models for squark production, assuming 100% branching frac- We assume m N (cid:29) m E . q ˜ q ∗ → ( q ˜ χ )( q ˜ χ ) with ˜ χ → ˜ (cid:96) ∓ (cid:96) ± / ˜ νν and ˜ (cid:96) ∓ / ˜ ν → (cid:96) ∓ ˜ χ /ν ˜ χ .By comparing the results from ATLAS for 2 j + E missT at 8 TeV [33] which uses a leptonveto, with the results of [31] which uses additional leptons, we can check that the searchwithout leptons has a greater sensitivity. For example, for a neutralino mass of 100 GeV, theexcluded mass of a eight-fold degenerate squark was 900 GeV without leptons, 860 GeV withopposite sign dileptons, and 800 GeV with one-lepton. In this way, the larger exclusion forthe production of squarks is in the signal without further leptons. In the case of scalar darkmatter, the branching for two jets and zero leptons is 25%, therefore the bounds discussedabove at 13 TeV for this signal become weaker. In particular, we have found that theexclusion for m ζ goes up to ∼
800 (600) GeV for a DM mass of 100 (400) GeV. The fullrecast is presented in the lower exclusion curve (dashed line) of Fig. 2.In our model the greater exclusion happens when the mass of E + is close to the ζ mass,such that the jets are sufficiently soft, so that the signal becomes effectively (cid:96) + (cid:96) − + E missT without jets. In such a case, we can recast the searches for simplified supersymmetric modelswith slepton pair production. The results of searches for the first two generations of sleptonsat Run-II in signals with opposite sign dileptons and missing transverse momentum for36 . − are reported by ATLAS in [34]. Taking into account the 25% branching into thecharged lepton and the scalar dark matter particle, the exclusion region at 13 TeV coversmass values up to m ζ ∼ m χ (cid:46)
800 GeV. In Fig. 3, we present the full recasted10
00 400 600 800 1000 1200 1400 1600 m (GeV) m ( G e V )
13 TeV @35.9/fb
Figure 3: Excluded region (below the solid line) for a singlet scalar quark decaying into asignal which becomes effectively of opposite-sign dileptons and missing transverse momentun.This is obtained from a recast of preliminary results of data for search of dileptons and missingtransverse momentun at 13 TeV by ATLAS [34] by using a luminosity of 35 . − .exclusion region at 13 TeV.The limits with one additional lepton studied in [31] are applicable to the case where N L is the lightest or the next to lightest neutral fermion, since this would correspond to anintermediate ˜ χ which decays into χ with further gauge bosons. Therefore, we would expectsofter bounds in this case. Conclusion : A new insight as to the nature of lepton number has been proposed. It isidentified with the Peccei-Quinn symmetry which solves the strong CP problem, with theappearance of an invisible axion. A residual symmetry remains, i.e. ( − L , which servesalso as dark parity, i.e. ( − B + L +2 j . New particles which are odd under this Z allow theone-loop radiative generation of neutrino masses, and provide a weak-scale component of11ark matter in addition to the axion. We show how the two sectors may be connected witha new singlet scalar quark ζ , which may be easily probed (or discovered in the future) at theLHC through its subsequent decays to either the fermion or scalar dark-matter candidate. Acknowledgements : The work of E. M. has been supported in part by the U. S. Departmentof Energy under Grant No. de-sc0008541. The work of D. R. and O. Z. has been partlysupported by the Grants Sostenibilidad-GFIF and CODI-IN650CE, and by COLCIENCIASthrough the Grant No. 111-565-84269.
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