Motivated by the result of the Muon g-2 experiment and the long-standing anomalies in semileptonic 퐵 meson decays, we systematically build a class of minimal models that can address both experimental results thanks to the contributions of a set of new fields that include a thermal Dark Matter candidate. This talk is mainly based on Refs

Arcadi, G.

Calibbi, L.

Fedele, Marco

Mescia, F.

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arXiv

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PoS(EPS-HEP2021)182H anomalies and muon g − 2 from Dark MatterGiorgio Arcadi,0 Lorenzo Calibbi,1,∗ Marco Fedele2 and Federico Mescia30Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universitadegli Studi di Messina, Via Ferdinando Stagno d’Alcontres 31, I-98166 Messina, Italy1School of Physics, Nankai University, Tianjin 300071, China2Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany3Dept. de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos (ICCUB), Universitat deBarcelona, Martí i Franquès 1, E-08028 Barcelona, SpainE-mail: giorgio.arcadi@unime.it, calibbi@nankai.edu.cn,marco.fedele@kit.edu, mescia@ub.eduMotivated by the result of the Muon g-2 experiment and the long-standing anomalies in semilep-tonic  meson decays, we systematically build a class of minimal models that can address bothexperimental results thanks to the contributions of a set of new fields that include a thermal DarkMatter candidate. This talk is mainly based on Refs. [1, 2].*** The European Physical Society Conference on High Energy Physics (EPS-HEP2021), ****** 26-30 July 2021 ****** Online conference, jointly organized by Universität Hamburg and the research center DESY ***∗Speaker© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/PoS(EPS-HEP2021)182 anomalies and muon 6 − 2 from Dark Matter Lorenzo Calibbi1. Introduction and motivationsThe FNAL Muon g-2 experiment [3] confirmed the long-standing discrepancy between theo-retical prediction [4] of the anomalous magnetic moment of the muon, (6 − 2)`, and the previousmeasurement performed at BNL [5]. The combination of the two experiments deviates from theSM prediction by 4.2f. The compatibility of the two measurements convincingly excludes thepossibility that the discrepancy is due to a statistical fluctuation or some overlooked systematicaleffects in the old BNL experiment [5]. The only possible explanations are (i) an underestimation ofthe leading hadronic contribution (from hadronic vacuum polarisation) using the data-driven dis-persive approach, as the recent BWM lattice result may suggest [6]; (ii) the presence of additionalnew physics (NP) contributions.The persistent and coherent pattern of anomalies reported in semileptonic  meson decays ofthe kind 1 → B`` also seems to point to a NP sector coupling preferably to muons. In particular,LHCb has recently released an updated measurement of the theoretically clean lepton flavouruniversality (LFU) ratio ' = BR( →  `+`−)/BR( →  4+4−) reporting a 3.1f discrepancywith the SM prediction [7].Assuming that they are hints of NP, both the muon 6 − 2 discrepancy and the  anomaliesrequire new fields coupling to muons at scales . O(100) TeV [8, 9]. Therefore, we find it naturalto seek a common explanation of the two phenomena. Since Dark Matter (DM) is probably themost compelling call for NP, we are interested in extensions of the Standard Model (SM) that canalso account for it. Our goal is to systematically build the simplest models that can, simultaneously,(i) address the  anomalies, (ii) explain the muon 6 − 2 discrepancy, (iii) provide a candidate ofthermal-relic DM.We aim at models that are minimal in terms of the number of new fields and theirproperties—quantum numbers, number of sizeable couplings, etc.—but also in the sense that theNP contributions to semileptonic  decays and to the muon 6 − 2 are “induced” by DM. In otherwords, the DM field is not added ad hoc but directly enters the relevant Feynman diagrams.In order to fulfill the above assumptions in combination with the requirement of DM stability,our NP fields should not mix with SM fields, a condition that can be enforced by a discrete orcontinuous symmetry. Hence NP contributions to both muon 6 − 2 and 1 → B`` arise at one loop(with only NP fields running in the loops) as in the general framework considered in [10–12].bLs̄LbLs̄Lµ̄L µ̄LµL µLΨΨΦΦΦq ΨqΦℓ ΨℓClass F − Fermion mediator Class S − Scalar mediatorFigure 1: Basic diagrams providing a contribution to 1 → B`` involving only left-handed SM fields.2PoS(EPS-HEP2021)182 anomalies and muon 6 − 2 from Dark Matter Lorenzo CalibbiFigure 2: Summary of constraints for the model dubbed F  in Ref. [1] with DM being the Majoranafermion Ψ. The coloured regions are excluded by DM relic density (red), direct detection (blue), LHCsearches (orange) and  physics (green). The green hatched region corresponds to a SM-like contribution to1 → B``, while in the white area the  anomalies can be explained at the 2f level. See Ref. [1] for details.2. Systematic approach to DM and  anomaliesGlobal fits to the 1 → B`` data indicate that a satisfactory fit of the  anomalies is possible inpresence of NP coupling to left-handed (LH) fermions only, see e.g. [13–16]. Thus, an elegant andminimal setup to account for the  anomalies consists into extending the SM spectrum with threenew fields—either two scalars and one fermion or the other way round—whose quantum numbersunder the SM gauge group allow for couplings with LH leptons (!8) and LH quarks (&8), as shownin Figure 1. This class of models is described by either of the following Lagrangians:LF ⊃ Γ&8 &̄8 %'ΨΦ@ + Γ!8 !̄8 %'ΨΦℓ , LS ⊃ Γ&8&̄8 %'Ψ@ Φ + Γ!8 !̄8 %'Ψℓ Φ , (1)where ΨG are new vectorlike fermions, ΦG are new scalars, and the labels F , S indicate whetherthe field coupled with both quarks and leptons— that we dub “flavour mediator”— is a fermionΨ or a scalar Φ. In order to highlight the minimal ingredients required by the  anomalies, weonly consider couplings to muons, and second- and third-generation quarks. Besides SM gaugesymmetries, the above Lagrangians are also invariant under a /2 —or a global * (1)—symmetryin order to ensure DM stability provided it is the lightest NP particle.Requiring that, in order to provide a viable DM candidate, at least one of the new fields featuresa colourless and electrically neutral component (with null hypercharge if fermionic), a limitednumber of gauge quantum numbers assignments for the three new fields, and hence of possiblemodels, is found. These were listed in Ref. [1], where we performed a systematic study, consideringboth spin alternatives, and both cases of real/complex (Dirac/Majorana) scalar (fermion) DM. Aviable fit of the  anomalies not in conflict with bounds from DM phenomenology is obtained ifthe following conditions are satisfied.• DMhas to belong to a field directly interacting withmuons with a sizeable coupling, |Γ!` | & 2.This ensures an efficient DM annihilation into muons through t-channel diagrams, while a3PoS(EPS-HEP2021)182 anomalies and muon 6 − 2 from Dark Matter Lorenzo Calibbigood fit of the  anomalies can be achieved for a moderate value of the couplings to quarks,Γ&B Γ&1, thus evading stringent constraints from B mixing.• The DM should be an (* (2)! singlet. Otherwise, DM annihilations into gauge bosonpairs would be so efficient that the correct relic density would be achieved for multi-TeVDM masses, thus outside the region where a viable fit of  anomalies can be achieved.Furthermore, stringent bounds from searches for disappearing tracks at LHC exist, expeciallyin the case DM belongs to a fermion triplet.• The DM particle must be a Majorana fermion or a real scalar. If this were not the case, aviable fit of the  anomalies would be ruled out by direct detection searches for DM, due toa very large photon penguin contribution to the DM-nucleon scattering cross section.As an illustration of these general conclusions, we show in Figure 2 the combined constraints ona model with Majorana DM, featuring the following (* (3)2 × (* (2)! ×* (1). quantum numbers:Φ@ (3, 2, 1/6), Φℓ (1, 2,−1/2), Ψ (1, 1, 0). This is one of the simplest examples of a model—previously studied in Ref. [17]— that successfully addresses the  anomalies via interactions of aparticle that can account for 100% of the observed DM abundance.3. Adding the muon 6 − 2The minimal set of models illustrated in the previous section does also contribute to (6 − 2)`,as penguin diagrams involving the subset of NP fields coupled to the muons induce dipole operatorsof the type:L ⊃ 4 E8c2``(¯̀!f`a`')`a + h.c. ⇒ 0NP` =<`E2c2Re(``), (2)where 0` ≡ (6 − 2)`/2. The proportionality of the operator’s coefficient to the Higgs vev E makesit explicit that, following from gauge invariance, a flip of the chirality of the muon—hence aHiggs insertion— is necessary. Being the NP fields coupled only with LH muons, an externalmass insertion on a muon line would be the only option, causing 0NP` to be suppressed by the smallmuon Yukawa coupling, i.e. `` ∝ H`. While a value of 0NP` compatible with the experimentalmeasurement could be still achieved, this would come at the price of low masses of the NP fields—in the 100-200 GeV range— that would not be compatible with the constraints from LHC searchesµRµLΨΦℓΨ′〈H〉 µRµLΨℓΦaH〈H〉Φ′Figure 3: Diagrams giving chirally-enhanced contributions to (6 − 2)`.4PoS(EPS-HEP2021)182 anomalies and muon 6 − 2 from Dark Matter Lorenzo CalibbiLabel Φ@/Ψ@ Φℓ/Ψℓ Ψ/Φ Φ′ℓ/Ψ′ℓΨ′/Φ′FIa/SIa (3, 2, 1/6) (1, 2,−1/2) (1, 1, 0) (1, 1,−1) –FIb/SIb (3, 2, 1/6) (1, 2,−1/2) (1, 1, 0) – (1, 2,−1/2)FIc/SIc (3, 2, 7/6) (1, 2, 1/2) (1, 1,−1) (1, 1, 0) –FIIa/SIIa (3, 1, 2/3) (1, 1, 0) (1, 2,−1/2) (1, 2,−1/2) –FIIb/SIIb (3, 1, 2/3) (1, 1, 0) (1, 2,−1/2) – (1, 1,−1)FIIc/SIIc (3, 1,−1/3) (1, 1,−1) (1, 2, 1/2) – (1, 1, 0)FVa/SVa (3, 3, 2/3) (1, 1, 0) (1, 2,−1/2) (1, 2,−1/2) –FVb/SVb (3, 3, 2/3) (1, 1, 0) (1, 2,−1/2) – (1, 1,−1)FVc/SVc (3, 3,−1/3) (1, 1,−1) (1, 2, 1/2) – (1, 1, 0)Table 1: Minimal sets of fields fulfilling all requirements. The fields are denoted by their transformationproperties under, respectively, ((* (3)2 , (* (2)! ,* (1). ). Models in cyan feature singlet DM, models in redhave singlet-doublet mixed DM.and DM density outside very tuned regions of the parameter space [18]. To overcome this problemwe need to raise the minimal number of NP fields from three to four, requiring that two of them mixthrough a Higgs insertion. This induces diagrams such as those shown in Figure 3 that can lead to‘chirally-enhanced’ contributions to (6 − 2)`, that is, not suppressed by a chirality flip ∝ H`. Wethen redefine our classes of models as:Class F : either {Φ@, Φℓ , Φ′ℓ , Ψ} or {Φ@, Φℓ , Ψ, Ψ′}Class S : either {Ψ@, Ψℓ , Ψ′ℓ , Φ} or {Ψ@, Ψℓ , Φ, Φ′}Following again criteria of gauge invariance and DM stability, and considering the conditions listedat the end of the previous section, the analysis in Ref. [2] showed that the candidates for a combinedexplanation of (6 − 2)` and  anomalies are limited to the models listed in Table 1.In Figure 4, we show, as illustrative examples, the two models F  (Majorana DM) and F  (real scalar DM). In the first case, the chirality flip is realised by mixing of the singlet fermion Ψwith a (* (2)! doublet Ψ′. The DM candidate is the lightest neutral mass eigenstate, so that themodel is an extended versions of so-called singlet-doublet DM model, see e.g. [19]. As shown inthe left panel of the figure, it is possible to achieve the correct relic density (red isocontour) and,at the same time, a viable fit of (6 − 2)` and  anomalies (green and orange regions respectively).Further experimental constraints from direct detection (hatched blue region) and the invisible widthof the Higgs (hatched gray region) are evaded for "Ψ/"Ψ′  1, corresponding to the case ofsinglet-like DM. In the case of the F   model the DM candidate is instead a pure singlet state (areal scalar) and the mass mixing needed for (6−2)` is realised by two fermionic states that mediateits annihilations. As apparent from the right panel of Figure 4, also for this second model we canhave a viable interpretation of the anomalies as well as the correct relic density. In this case we needto invoke sizable coannihilations between the DM and the NP fermions. As DM is a real scalarwithout coupling with coloured charged states, direct detection DM searches do not constrain thismodel. A sizable portion of the viable parameter spaces is instead excluded by LHC searches formuons and missing energy [20], interpreted in terms of the production of the electrically-chargednew fermions, followed by decays into muons and DM.5PoS(EPS-HEP2021)182 anomalies and muon 6 − 2 from Dark Matter Lorenzo CalibbiFigure 4: Combined results for two benchmark choices of the parameters of the models FIB (left plot)and F   (right plot). The red isocontours indicate the correct DM relic density obtained from thermalfreeze-out. The green and orange regions provide good fits of, respectively, the 6 − 2 and  anomalies. Thehatched regions are excluded by direct detection (blue), the Higgs invisible width (gray) and LHC searches(purple). See [2] for details.4. Conclusions and outlookFollowing Refs. [1, 2], we showed that one can systematically build minimal models simul-taneously addressing the muon 6 − 2 and the  anomalies through loops involving a thermal DMcandidate that can account for 100% of the observed DM abundance. This can be achieved byintroducing only four new fields, at the price of a large coupling to LH muons (& 2) and a (mod-erate) chiral enhancement of the (6 − 2)` contribution. We remark that these are not meant tobe “realistic” models, rather the minimal ingredients that a fully-fledged theory may need to in-corporate—e.g. large muon couplings imply a Landau pole below ≈ 2500 TeV. Interestingly, asillustrated by the examples shown in Figures 2 and 4, these minimal solutions seem to be testableby future direct detection experiments and/or LHC searches, and, in the long run, they are definitelyin the reach of a high-energy muon collider [21].AcknowledgmentsLC would like to thank the organisers of EPS-HEP2021 and the conveners of the Dark Mattersession for the opportunity of presenting our work, and acknowledges support by the NationalNatural Science Foundation of China under the grant No. 12035008. FM acknowledges financialsupport from the State Agency for Research of the Spanish Ministry of Science and Innovationthrough the “Unit of Excellence María de Maeztu 2020-2023” award to the Institute of CosmosSciences (CEX2019-000918-M) and from PID2019-105614GB-C21 and 2017-SGR-929 grants.The work of MF is supported by the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) under grant 396021762 - TRR 257, “Particle Physics Phenomenology after the HiggsDiscovery”.6PoS(EPS-HEP2021)182 anomalies and muon 6 − 2 from Dark Matter Lorenzo CalibbiReferences[1] G. Arcadi, L. Calibbi, M. Fedele and F. Mescia, Systematic approach to -physics anomaliesand C-channel dark matter, 2103.09835.[2] G. Arcadi, L. Calibbi, M. Fedele and F. Mescia, Muon 6 − 2 and -anomalies from DarkMatter, Phys. Rev. Lett. 127 (2021) 061802 [2104.03228].[3] Muon g-2 collaboration, Measurement of the Positive Muon Anomalous Magnetic Momentto 0.46 ppm, Phys. Rev. Lett. 126 (2021) 141801 [2104.03281].[4] T. 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B anomalies and muon g - 2 from Dark Matter

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B anomalies and muon g - 2 from Dark Matter

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