Minimal renormalizable simplified dark matter model with a pseudoscalar mediator
MMinimal renormalizable simplified dark matter model with a pseudoscalar mediator
Seungwon Baek, ∗ P. Ko, † and Jinmian Li ‡ School of Physics, Korea Institute for Advanced Study, 85 Hoegiro, Seoul 02455, Korea
We consider a minimal renormalizable and gauge invariant dark matter (DM) model, in which thesinglet fermion DM has only axial couplings to a new pseudoscalar mediator. The mixing between thepseudoscalar mediator and the standard model (SM) Higgs boson induces the interactions betweenthe DM and SM particles. The DM candidate in this model can provide the correct thermal relicdensity and evades all direct detections, while it can produce observable signals in indirect detectionexperiments due to its large annihilation cross section. A comparative study for DM phenomenologyat the LHC is conducted for models with scalar mediators that have either scalar or pseudoscalarcouplings to SM particles and the DM. We find that the three scenarios have distinguishable featuresin scalar decay branching ratio, DM pair production cross section as well as the signal reaches atthe LHC. The LHC searches for some visible signals related to the scalar sector are also discussed.
I. INTRODUCTION
The existence of non-baryonic Dark Matter (DM)has been established only by astrophysical observationsthrough its gravitational effects [1]. Since the correctabundance of DM via thermal production could be gener-ically obtained if the DM is in the mass range of O (100) GeV and interacts with SM particles via electroweakforce, the so-called Weakly-Interacting-Massive-Particle(WIMP) paradigm has been one of the most interest-ing scenarios for thermal DM. Given that the DM in-teractions with the SM particles or among themselvesare unknown, effective field theory (EFT) is one viableway to simplify the study of DM phenomenology. TheEFT descriptions [2–4] of DM interactions are valid onlywhen momentum transfer is much smaller than the massof the mediator, which is usually not true for DM pro-ductions at high energy colliders [5–8], especially sincethe mediator mass scale is completely unknown. Simpli-fied DM model frameworks have been used extensively inDM searches at the LHC [9–11]. Here, the DM is neutralunder the Standard Model (SM) gauge group and inter-acting with the SM particles via the portal of a singleparticle [12–15].However, simplified DM models with a single media-tor can often violate the SM gauge symmetry [16–18],thus may become invalid for describing UV-completemodels . There are growing interests in simplified DMmodel that respect the gauge symmetry [16, 19–25]. Inparticular, the gauge invariant and renormalizable DMmodel with scalar mediators are constructed in its min-imal form [26] and two Higgs doublet model (2HDM)extended form [27]. Models with pseudoscalar mediatorsare more interesting, owing to the fact that stringent con-straints from DM direct detection can be evaded intrinsi-cally, while being able to explain some anomalies in DM ∗ [email protected] † [email protected] ‡ [email protected] Importance of SM gauge symmetry within the DM EFT waspointed out in Ref. [19]. indirect detection [28–30]. The collider phenomenologyof UV-complete DM models with pseudoscalar portal hasbeen studied in Ref. [31–35].In this work, a minimal renormalizable model withpseudoscalar mediator is proposed (analogy to the modelin Ref. [36] which focuses on the DM indirect detectionsignal). Compared to the models in Refs. [37, 38], thepseudoscalar mediator of this model only has an axialcoupling to DM particles. We show that there is largeportion of parameter space that is consistent with DMconstraints while giving measurable predictions in futureexperiments. At the LHC, this model can be searchedthrough signatures both with and without DM in thefinal state. The most remarkable DM signal is pro-duced by recoiling the DM pair against energetic ini-tial state radiated jet, i.e. mono-jet. We will compar-atively study these signatures for models with scalar me-diators that have either scalar or pseudoscalar couplingsto SM particles and the DM. The pseudoscalar can alsoproduce beyond SM (BSM) signatures without includ-ing DM. We will discuss the constraints on the signalsof A → V V → ( f ¯ f )( f ¯ f ) , H → AA → ( f ¯ f )( f ¯ f ) and A → H H at current stage of the LHC. II. MINIMAL RENORMALIZABLE MODELWITH PSEUDOSCALAR MEDIATOR
We propose a minimal renormalizable DM model witha pseudoscalar mediator assuming DM χ is a SM singletDirac fermion that couples to a pseudoscalar a which isalso a SM singlet scalar with a negative parity: L = ¯ χ ( i∂ · γ − m χ − ig χ aγ ) χ + 12 ∂ µ a∂ µ a − m a a − ( µ a a + λ Ha a ) (cid:18) H † H − v h (cid:19) − µ (cid:48) a a − λ a a − λ H (cid:18) H † H − v h (cid:19) . (1)Note that the parity is broken by the dim-3 µ a and µ (cid:48) a terms. We remove the tadpole for a and assume (cid:104) a (cid:105) = 0 . a r X i v : . [ h e p - ph ] A p r This model is unique, since the mediator a has a pseu-doscalar coupling to the DM χ , and scalar couplings tothe SM fields through its mixing with the SM Higgs bo-son (see Eq. (7) below), unlike most other renormalizablepseudoscalar mediator models based on 2HDMs and itsextensions.The µ a term induces the mixing between the pseu-doscalar a and the SM Higgs boson h after electroweaksymmetry breaking, making two mass eigenstates H and A : H = h cos α + a sin α , (2) A = − h sin α + a cos α . (3)So the variables λ H , m a and µ a in Eq. 1 can be expressedby physical parameters in mass eigenstate: λ H = 12 v h ( m H cos α + m A sin α ) , (4) m a = m H sin α + m A cos α , (5) µ a = sin α cos αv h ( m H − m A ) , (6)where v h is the vacuum expectation value of H .Then the interaction Lagrangian of H and A with theSM particles and DM will be given by L int = − ig χ ( H sin α + A cos α ) ¯ χγ χ − ( H cos α − A sin α ) × (cid:88) f m f v h ¯ f f − m W v h W + µ W − µ − m Z v h Z µ Z µ (7)The mass eigenstates of scalar fields have only scalar cou-plings to SM particles and have only axial couplings toDM, so we can expect that such model setup will notlead to any CP-violation effects in the SM.On the other hand, the extended Higgs sector couldaffect the electroweak precision test (EWPT) [39, 40] bygiving extra contributions to the SM gauge boson self-energy. Since the new pseudoscalar boson couples to theSM particles only through mixing with the SM Higgsdoublet, constraints from the oblique parameters andthe perturbative unitarity bound are exactly the samewith the scalar Higgs portal case considered in Ref. [26].Taking m H = 125 GeV, the measurements exclude themodels with scalar mixing angle α (cid:38) . . Similar con-straint is also obtained from the precision measurementsof SM Higgs boson signal strengths at the LHC run-I [41, 42], which indicate sin α (cid:46) . [43–45]. Moreover,if m χ < m H / , the stringent limit from the Higgs in-visible decay search Br ( H → χχ ) < . [46] requires g χ sin α (cid:46) . . III. DARK MATTER PHENOMENOLOGY
The measurements of anisotropy of the cosmic mi-crowave background (CMB) and of the spatial distribu-tion of galaxies find the relic density for cold non-baryonic matter to be Ω h = 0 . ± . [1]. In order not tooverclose the universe, the DM candidate in our modelshould annihilate effectively into SM particles. There aremainly three different DM annihilation mechanisms inour model framework: (1) DM mass is around the half of H /A mass so the annihilation cross section is resonantlyenhanced; (2) DM annihilate to SM gauge bosons/heavyfermions especially when g χ sin 2 α is large; (3) DM massis larger than H and/or A so the annihilation cross sec-tion can be enhanced by setting large scalar self-coupling.The micrOMEGAs [47] is used to calculate the ob-servables in DM phenomenology, with the model files forEq. (1) generated by Feynrules [48]. Taking H as theHiggs state with mass of 125 GeV, the model has sevenfree parameters: m A , g χ , α, m χ , λ Ha , µ (cid:48) a , λ a . (8)In DM annihilation, varying the g χ and α can only lead toa total rescaling of the cross section, while its dependenceon the m χ is more complicate, due to the opening ofnew annihilation channels with increasing m χ . Furthermore, as discussed in Sec. II, α should be smaller than 0.4according to the Higgs precision measurement, but nottoo small to guarantee sufficient signal rate at collider.So we will choose g χ = 1 and α = 0 . for the discussionsof this section and scan m χ ∈ [5 , GeV. The m A determines position of the pole that is due to resonantenhancement in DM annihilation. Scanning m A will leadto overlapped peaks in annihilation cross section thussmear out the peak structure. For clarification, we alsofix m A = 400 GeV. The rest of parameters are scannedin the ranges listed as following. λ Ha ∈ ± [10 − , √ π ] , µ (cid:48) a ∈ [5 , GeV , λ a ∈ [10 − , √ π ] (9)We will adopt the exponential scan over the λ Ha and λ a in order to have more points with small λ i , i = Ha, a .That is we define | λ i | = √ π R and perform uniform scanover R between [-5.5, 1].The relic density for models in the chosen parameterspace are illustrated in Fig. 1. In the region where DMannihilating into Higgs bosons are kinematically forbid-den, m χ is the only parameter that control the relic den-sity. The relic density becomes smaller when DM mass isapproaching half of the H mass. There is also a signif-icant drop at m χ ∼ GeV where the DM annihilatinginto gauge bosons are opening. When m χ (cid:38) m H /A , DMcan annihilate into scalar bosons through H /A medi-ation. So the scalar self-couplings are important. Es-pecially, for our parameter choice, χχ → AA is kine-matically disfavored, the relic density is monotonicallydecreased with increasing | λ Ha |.The DM has been searched actively by many under-ground experiments through its recoiling against nuclei[49, 50]. Following the notations of Ref. [51], the DM-SMparticles interaction can be written in terms of DM bilin-ear M χ , SM bilinear M f as well as form factor F ( s, t, u ) λ Ha λ Ha m χ [GeV]10 − − − − − − Ω h − − − − m χ [GeV]10 − − − − − − Ω h − − − − FIG. 1. Relic density with varying DM mass, for m A = 400 GeV, g χ = 1 and α = 0 . . Color code indicates the value of λ Ha . which includes the details of the model and nuclear formfactor: M = M χ M f · F ( s, t, u ) . (10)In our model, in the limit of low momentum transfer, theDM-SM fermion scattering matrix element is M ∝ M χ · M f = − q i ( ξ † χ ˆ S i ξ χ ) × [2 m f ( ξ † f ξ f ) + i µm f (cid:15) ijk q i v j ( ξ † f ˆ S k ξ f )] , (11)where q i is the momentum transfer, ξ f/χ are two com-ponent spinors for nucleon and DM, v is the relativevelocity of the dark matter and the target nucleon, µ = m χ m f / ( m χ + m f ) is the reduced mass of the darkmatter-nucleon system. Note the g χ and m A depen-dences are absorbed in F ( s, t, u ) . Eq. (11) is showing thatthe spin-independent (SI) DM-nucleon cross section issuppressed by the | (cid:126)q | while the spin-dependent cross sec-tion is even smaller ( ∝ | (cid:126)q | ). The results from the abovesemi-quantitative estimate can be seen more clearly inthe full formula for the SI direct detection cross section, σ SIχN = 2 π µ m χ λ N v , (12)where λ N = g χ sin α cos αm N v h (cid:18) m H − m A (cid:19) f N , (13)with N denoting nucleon and f N ≈ . . Assuming therelative velocity between the DM and nuclear is givenby the orbital speed of the Sun ∼ O (10 − ) , the typical σ SIχN of our model is around O (10 − ) of that in the scalarmediator model [26] as also have been justified by com-paring the scattering rates of ˆ O and ˆ O opeartors inRef. [52]. This means the DM of our model will not leaveany signals in direct detection experiments. However, the s-wave annihilation is still permitted: M χ = ¯ χ γ χ = − ( E + m )( E + m ) + (cid:126)k (cid:112) ( E + m )( E + m ) ξ † χ ξ χ , (14)with (cid:126)k is the DM momentum. So the non-relativisticDM particles that concentrated at the center of galaxiesmay still have relatively large annihilation cross section.Thus they can be observed in final state of photons [53],positron/anti-proton [54, 55] or neutrinos [56]. . . . . − − − − − − − − − − − − − − − − − − N γ / N γ , b ¯ b m χ [GeV] < σ v > c m / s b ¯ bWWZZt ¯ tH i H i h σv i ′ tot < σ v > c m / s b ¯ bWWZZt ¯ tH i H i h σv i ′ tot FIG. 2. Upper panel: The cross sections for different DMannihilation (at rest) channels. The dashed black curve cor-responds to the 95% CL exclusion limit on b ¯ b channel ob-tained from Milky Way Dwarf Spheroidal Galaxies with SixYears of Fermi-LAT data [57]. The weighted total annihila-tion cross sections are presented by black cross points, whichcan be compared with the Fermi-LAT data directly. Lowerpanel: the ratio between the number of photons within en-ergy E γ ∈ [1 , GeV per annihilation in our model and insimplified model where DM only annihilates to b ¯ b . In upper panel of Fig. 2, we plot the cross sections forall DM annihilation channels with varying m χ , where wehave chosen appropriate g χ such that the correct relicabundance ( Ω h = 0 . ) is obtained for each point inthe scanning. The exclusion bounds from the Fermi-LAT data are available only for some pure final states,e.g b ¯ b , τ + τ − , u ¯ u and W + W − . In order to obtain theFermi-LAT bound to our model, especially when DM isheavy ( m χ (cid:38) GeV) so that it dominantly annihilatesto heavy particles (
W/Z/h/t ), we assume that for a givenDM mass the gamma spectra of the b quark and heavyparticle final state have similar shape while their normal-izations can be different [58, 59]. So we can calculate theweighted total annihilation cross section by (cid:104) σv (cid:105) (cid:48) tot = (cid:104) σv (cid:105) tot N γ N γ,b ¯ b (15)where the (cid:104) σv (cid:105) tot is the DM total annihilation crosssection, N γ is the number of photons within energy E γ ∈ [1 , GeV per annihilation for a point in ourmodel and N γ,b ¯ b is the corresponding number in simpli-fied model where DM has the same mass as the pointand only annihilates to b ¯ b . Similar methodology was alsopursued in Ref. [60]. We plot the ratio N γ N γ,b ¯ b in the lowerpanel of Fig. 2, from which we can see that the ratio isclose to 1 when χχ → b ¯ b annihilation is dominant. How-ever, the gauge (Higgs) boson final state can produce less(more) photons in the range E γ ∈ [1 , GeV than the b quark final state. This also leads to a double enhance-ment of the ratio at m χ ∼ GeV, where multiple Higgsfinal state is kinematically opened. Then, the weightedtotal annihilation cross section can be compared to theFermi-LAT bound on the b ¯ b final state directly. We canconclude that the Fermi-LAT data from dwarf galaxiescan exclude the light DM mass region ( m χ < GeV)as well as the resonant region ( m χ ∼ m A / ), while allof our points are close to the bound and are expected tobe discovered/excluded in the near future. It has to benoted that this limit will be weakened if our DM particleonly constitutes a fraction of the total amount of DM. IV. LHC PHENOMENOLOGYA. Invisible channel: mono-jet
In this section, we discuss the DM phenomenology atthe LHC in terms of decay of scalar, production of DMand current limits from the LHC searches. To show themerit of our model setup, results are presented alongsidewith those of conventional theoretical frameworks for DMat collider: L AAint = − ig χ ( a sin α + A cos α ) ¯ χγ χ − i ( a cos α − A sin α ) (cid:88) f m f v h ¯ f γ f (16) L SSint = − g χ ( H sin α + H cos α ) ¯ χχ − ( H cos α − H sin α ) × (cid:88) f m f v h ¯ f f − m W v h W + µ W − µ − m Z v h Z µ Z µ (17)In the following, we denote the models of Eq. (17),Eq. (16) and Eq. (7) as SS, AA and SA respectively, sincethey are distinguished by the scalar/axial couplings be-tween SM particles and DM. For simplicity, in the discus-sion of this section α = 0 . and g χ = 1 are chosen. Andthe DM mass is fixed to m χ = 80 GeV to avoid SM Higgsinvisible decay while we keep relatively large DM pro-duction cross section. The mass of lighter scalar (pseu-doscalar) in SS (AA) scenario is chosen as m H /a = 125 GeV for comparison purpose. Then, assuming the H /A only decay into SM particles and DM, the only param-eter relevant in collider phenomenology is m H /A . This minimal decay width for H /A (denoted by A hereafter)can be written as Γ min ( A ) = Γ( A → χχ ) + Γ( A → V V ) + Γ( A → f f )= cos α · g χ m A π (1 − m χ m A ) i/ + sin α · G µ m A √ π δ V (cid:115) − m V m A (1 − m V m A + 12 m V m A )+ sin α · ( m f v ) m A π (1 − m f m A ) j/ , (18)where ( i, j ) = (1 , , (3 , , (1 , for SA, SS, AA scenar-ios respectively, Γ( A → V V ) = 0 for AA scenario and δ V = 1(2) for Z ( W ± ) . . . . . . . . . . . . . . . . . . . B r ( A → ¯ χχ ) m A [GeV] SASSAA B r ( A → ¯ χχ ) m A [GeV] SASSAA FIG. 3. The decay branching ratios of the second scalarboson into DM pair in three models. We have chosen m χ = 80 GeV, g χ = 1 and α = 0 . . The branching ratios of A → ¯ χχ are given in Fig. 3.When the m A is not much larger than m χ , the factor (1 − m χ m A ) i/ is important. So the Br ( A → ¯ χχ ) of SSscenario is smaller than that of SA scenario. As for m A (cid:29) m χ , both scenarios give the similar branching ratios.The AA scenario always has the largest Br ( A → ¯ χχ ) because of the absence of A - V - V coupling.The dominant DM production channel at theLHC is gluon-gluon fusion (ggF) through the topquark loop. The effective couplings for gluon-gluon-scalar/pseudoscalar after integrating the top quark are L scalar = α s π g v v τ [1 + (1 − τ ) f ( τ )] G µν G µν φ (19) L pseudoscalar = α s π g v v τ f ( τ ) G µν ˜ G µν A (20)where τ = 4 m t /m H/A , g v = sin α and f ( τ ) = (cid:40) arcsin √ τ , τ ≥ − (log √ − τ −√ − τ − iπ ) , τ < . (21)However, the ggF process itself does not produce anyobservable signals at detectors. Extra energetic jets ra-diating from either initial state gluon or top quark in theloop can circumvent this issue, which raise the mono-jetsignature. The leading order cross section for DM pairproduction in association with a jet is computed withinthe FeynRules/MadGraph5_aMC@NLO [61, 62] frame-work, where the jet is required to have p T ( j ) > GeV.Meanwhile, the higher order corrections to the ggF crosssection of Higgs production are found to be quite sig-nificant. Using the SusHi program [63], the NNLO K-factors for Higgs mass ∈ [100 , GeV are calculatedto be around 2.5. So the production cross section forthe DM pair associating with a jet is given by the LOcross section in MadGraph5_aMC@NLO multiplying auniversal K-factor of 2.5. − − − − − − − − − − − − σ ( ¯ χχ j ) [ pb ] m A [GeV] SASSAA σ ( ¯ χχ j ) [ pb ] m A [GeV] SASSAA FIG. 4. The mono-jet signal production cross section ingluon-gluon fusion channel at the 13 TeV LHC, where the jetis required to have p T ( j ) > GeV. Parameters are chosenas m χ = 80 GeV, g χ = 1 and α = 0 . . The resulting cross sections for all three scenarios arepresented in Fig. 4. The contributions of two propaga-tors that mediate the DM production will interferencewith each other [64], leading to different degree of sup-pressions for different scenarios in the light m A region.In particular, the cross sections drop dramatically whentwo propagators are close in mass. Models with heavier A are more interesting because of their larger produc-tion cross section. In this region, the DM productionsare dominated by the on-shell A production with subse-quent decay. The interference effect becomes importantonly for m A (cid:38) GeV, where the on-shell A productionis kinematically suppressed to some extent. This leadsto deviation in the production cross sections of SS andSA scenarios. Note the small bumps around m t for allscenarios are from the top quark mass effect.The mono-jet signature has been searched by ATLAScollaboration at 13 TeV with integrated luminosity of 3.2fb − [65]. The non detection of the signal could puta constraints on our model parameters. We adopt theCheckMATE2 program [66] to calculate the LHC searchconstraints on our model, in which the ATLAS mono-jet search has been implemented and validated. Check-MATE2 provide the R max -value at the final stage of its analysis, defined as R max = max i N model i N up i (22)where N model i and N up i is the number of signal events ofour model and number of new physics upper limit at 95%CL in the signal region i , respectively. S i g n a l s t r e n g t h ( = / R m a x ) m A [GeV]Exclusion limit SASSAASA10SS10AA10 S i g n a l s t r e n g t h ( = / R m a x ) m A [GeV]Exclusion limit SASSAASA10SS10AA10 FIG. 5. The 95% CL exclusion limits from the ATLAS mono-jet search at 13 TeV with integrated luminosity of 3.2 fb − .The dashed curves correspond to models with ten times largertotal width of A than Γ min due to the opening of new decaychannels. In Fig. 5, we present the LHC search limit with signalstrength ( = 1 /R max ) which gives the size of the crosssection that is probable at current stage of the LHC.Even in the region of m A (cid:38) m χ where the productioncross section is largest, the signal rate is at least one orderof magnitude below the current reach. Note that in thisregion, since A is mostly on-shell and Br ( A → ¯ χχ ) isalready close to one, taking larger g χ will not enhance thesignal rate. We would expect higher luminosity of LHCto probe/exclude this region. Among three scenarios, theAA scenario has the best search sensitivity. We find thatthe differences are mainly originated from the productionrate of mono-jet signals as shown in Fig. 4, while thekinematic distributions of final states are similar for allscenarios, i.e. similar cut efficiencies.In a realistic model, some new decay channels of A might be important, such as A → H H . This will leadto suppressed production rate of DM pair, meanwhile,the interference effect can become remarkable because ofthe wide width of A . In Fig. 5, we also plot the signalreaches for models with ten times larger total width of A than Γ min due to the opening of new decay channels. Inthe region with negligible interference, the signal reachesshould be one order of magnitude weaker than that ofmodels with Γ min , e.g. m A ∈ [2 m χ , GeV ] . The inter-ference effect is significant when off-shell A contributionis large, e.g. in the regions m A > GeV. It shrinks thedifference in signal reaches for models with narrow andbroad width of A , mainly because of the enhancementin production cross section. Moreover, the large interfer-ence effect can lead to distinguished signal reaches for SSand SA scenarios. B. Visible channels
Our model also predicts BSM signals without DM inthe final states. In this section, we will focus on thenon-DM signals of the SA scenario as we can expect thatthe exclusion bounds obtained for SA scenario can be di-rectly applied to SS scenario, since their differences onlyexist in DM sector. But the corresponding bounds in AAscenario could be quite different, due to different produc-tion cross section of A as well as the absence of tree level AZZ/AW W couplings.According to the Eq. (18), the heavy pseudoscalardominantly decays into top quarks and vector bosonsapart from the DM pair. The process of top quark pairproduction through the pseudoscalar resonance decay in-terferes strongly with the QCD t ¯ t background, leadingto difficulties in its searches at hadron colliders [67–69].However, the diboson final state may still be detectable.To survey the production cross sections of visible signalsin our model, we fix m χ = 80 GeV, g χ = 1 and varying m A ∈ [0 , GeV, α ∈ [0 , . , with the rest of param-eters scanned in the range as given in Eq. (9). We notethat varying m χ and g χ which is important in obtainingcorrect relic density and evading the DM indirect detec-tions will not affect the results in the following discussionsmuch. − − −
300 400 500 600 700 800 900 100010 − − −
300 400 500 600 700 800 900 1000 pp → A → VV [ f b ] m A [GeV] ZZ (4 l ) ZZ ( qqνν ) W W ( eνµν )ZZWW pp → A → VV [ f b ] m A [GeV] ZZ (4 l ) ZZ ( qqνν ) W W ( eνµν )ZZWW FIG. 6. Bounds correspond to the LHC searches for twovector boson resonance. The production cross sections of ZZ ( W W ) at 13 TeV in our model are shown by red (blue)points. For our parameter choice, the A → χχ is always impor-tant when it is kinematically allowed. So the vector bosonpair production cross section is suppressed by ∼ sin α ,from both A production and decay. We calculate theNNLO gluon-gluon fusion A production cross section at13 TeV by using SusHi and obtain decay branching ra-tios of A → V V from micrOMRGAs. The results areshown in Fig. 6. At 13 TeV, the vector boson pairproduction cross section in our model is only around [0 . , fb for m A ∈ [200 , GeV. The ATLAS col-laboration searches the high mass diboson resonance in ZZ → (cid:96) [70], ZZ → ννqq [71] and W W → eνµν [72] final states respectively with LHC run-II data. Their ex-clusion bounds at 95% confident level (CL) are shown inthe Fig. 6 as well. It can be seen that the signal of vectorboson pair production is at least two order of magnitudebelow the current LHC search sensitivities.On the other hand, the production rates of scalar pairs( AA/H H ) do not suffer from the sin α suppression asmuch as those of vector boson pair, because the couplingin scalar to scalar decay is controlled by the scalar-scalarmixing and scalar self-couplings: λ AH H = − µ a cos α + 2(3 λ H − λ Ha ) v h cos α sin α + 2 λ Ha v h sin α + (2 µ a − µ (cid:48) a ) cos α sin α (23) λ H AA = − µ a sin α − λ H − λ Ha ) v h sin α cos α − λ Ha v h cos α + (2 µ a − µ (cid:48) a ) sin α cos α (24)They can be either large or small. In the parameter spaceof our interest, the H → AA and A → H H can evenbecome dominant. − − − − − − − − − − − − σ H σ h S M · B r ( H → AA ) m A [GeV] 4 µ τ b µ τ µ σ H σ h S M · B r ( H → AA ) m A [GeV] 4 µ τ b µ τ µ FIG. 7. Bounds correspond to the LHC searches for lightboson pair from the SM Higgs decay. The shaded region isexcluded by the Higgs precision measurement. Our modelsare shown by dark green points.
When the m A < m H / , the pseudoscalar pair can beproduced from the SM Higgs decay, which will lead tofour fermion final states after A → f ¯ f . The cross sectionof this process can be quite large. Ref. [73] summarizesthe recent searches for light bosons from 125 GeV Higgsdecay in the final states of µ , τ , b µ and τ µ atLHC run-I. The bounds are presented on the produc-tion cross section of each final states normalized to theSM Higgs production cross section. In our model, for m A ∈ [0 , GeV, the decay branching fractions of thepseudoscalar are only determined by a single parame-ter m A . So those experimental bounds for different finalstates can be projected to the same plane, m A versus σ H σ h SM · Br ( H → AA ) , where σ H σ h SM = cos α . The pro-jected bounds are presented by lines in different colorsin Fig. 7. Further more, the precision measurements onHiggs coupling strength constrain the BSM Higgs bosondecay to be Br BSM (cid:46) [74] as shown by the shaded re-gion of the same figure (it will change slightly for varying α ). Finally, we plot the normalized cross section of pseu-doscalar pair production of our model by dark-green dots.We can see from the Fig. 7 that the µ search is quitesensitive to the region m A ∈ [2 m µ , m c ] where other de-cay modes are kinematically suppressed while searchesfor other final states do not have any sensitivities to ourmodel. The bound of BSM Higgs boson decay will ex-clude large portion of the parameter space where the cou-pling λ H AA is not suppressed. In the limit of small sin α ,Eq. 24 can be simplified to λ H AA ∼ λ Ha v h cos α . Wefind the visible points with Br BSM (cid:46) should have λ Ha (cid:46) . . − − −
300 400 500 600 700 800 900 100010 − − −
300 400 500 600 700 800 900 1000 pp → A → H H [ f b ] m A [GeV] 4 b b γ b τ pp → A → H H [ f b ] m A [GeV] 4 b b γ b τ FIG. 8. Bounds correspond to the LHC di-Higgs searchesin different final states. The production cross section of ourmodels at 13 TeV are shown by dark green points.
In the region m A ∈ [2 m H , GeV ] , the H pair canbe produced through A resonance decay. The cross sec-tion of A production is proportional to sin α , while theBr ( A → H H ) can be large for appropriate choice ofparameters in the scalar sector. The cross section of res-onant H pair production from gluon-gluon fusion in ourmodel are shown by dark green points in Fig. 8. The linesin the figure correspond to the 95% CL LHC searches con-straints from b [75], bbγγ [76] and bbτ τ [77] channels re-spectively. As have been done for Fig. 7, the known decaybranching ratios of H → b ¯ b/τ τ /γγ have been projectedout. It can been seen that the search for b final stateprovides the best sensitivity, and the search for bbγγ isbetter than b only in the low m A region. For a moderatemass of the pseudoscalar m A ∼ GeV, some parame-ter points are already close to the LHC search limit. Wewould expect those points can be probed/excluded in thenear future when larger data sample is obtained.
V. CONCLUSION
In this paper, we propose a minimal renormalizableand gauge invariant DM model with a pseudoscalar me- diator. The singlet fermion DM has only axial couplingsto the pseudoscalar, while the mixing between the pseu-doscalar and SM Higgs doublet leads to the interactionsof DM and SM fermions and gauge bosons. Owing tothe s-wave annihilation, the DM relic density can be eas-ily obtained and the DM indirect detection signals areremarkable. The momentum suppression in DM-nucleonscattering matrix leads to null signal in all DM directdetection experiments.We study the most up-to-date LHC search constraintson signals of the model both with and without DM in thefinal state. The mono-jet signature of our model is stud-ied comparatively with that of models with pure scalarand pure axial couplings between the mediator and SMparticles/DMs. Three scenarios give different predictionson the decay branching ratio of pseudoscalar/scalar toDM and the DM pair production cross section. As a re-sult, different mono-jet search sensitivities are obtainedin different scenarios. Among them, the AA scenario hasthe best search sensitivity at the LHC. And the sensitiv-ity of SA is slightly better than that of AA scenario whenthe inference effect between two propagators is consider-able. Due to the sin α suppression in resonant vectorboson pair production, the typical production cross sec-tion of resonant vector boson pair is at least two orderof magnitude below the current LHC search sensitivity.The searches for resonant scalar pairs are more promis-ing. For light m A ∈ [0 , . GeV, the stringent limitson the BSM Higgs boson decay branching ratio obtainedfrom Higgs precision measurements as well as the searchfor light bosons from 125 GeV Higgs boson decay in µ final state exclude very large portion of the parameterspace. As for heavy m A ∈ [250 , GeV, the produc-tion rate is suppressed by sin α while the A → H H can vary freely. A much better sensitivity is obtained forthis channel than that for resonant V V channel. Someof the parameter points are less than one order of mag-nitude away from the current search sensitivity, thus canbe probed/excluded in the near future.
Note Added:
After we submitted this paper on thearXiv.org, we came to learn that the same or similarmodel has been considered in Ref. [36]. We thank KarimGhorbani for bringing his paper to our attention.
ACKNOWLEDGMENTS
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