Coloured Scalars Mediated Rare Charm Meson Decays to Invisible Fermions
CColoured Scalars Mediated Rare Charm Meson Decays to Invisible Fermions
Svjetlana Fajfer ∗ and Anja Novosel † Joˇzef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia andFaculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia (Dated: January 27, 2021)We consider effects of coloured scalar mediators in decays c → u invisibles . In particular, inthese processes, as invisibles, we consider massive right-handed fermions. The coloured scalar ¯ S ≡ (¯3 , , − / χ ),is particularly interesting. Then, we consider ˜ R ≡ (¯3 , , / χ mass is considered in the range ( m K − m π ) / ≤ m χ ≤ ( m D − m π ) /
2. We determine branching ratios for D → χ ¯ χ , D → χ ¯ χγ and D → πχχ for several χ masses, using most constraining bounds. For ¯ S , the most constraining is D − ¯ D mixing, while inthe case of ˜ R the strongest constraint comes from B → K /E . We find in decays mediated by ¯ S thatbranching ratios can be B ( D → χ ¯ χ ) < − for m χ = 0 . B ( D → χ ¯ χγ ) ∼ − for m χ = 0 . B ( D + → π + χ ¯ χ ) can reach ∼ − for m χ = 0 .
18 GeV. In the case of ˜ R these decayrates are very suppressed. We find that future tau-charm factories and Belle II experiments offergood opportunities to search for such processes. Both ¯ S and ˜ R might have masses within LHCreach. I. INTRODUCTION
Low-energy constraints of physics beyond StandardModel (BSM) are well established for down-like quarks bynumerous searches in processes with hadrons containingone b or/and s quark. However, in the up-quark sector,searches are performed in top decays, suitable for LHCstudies, while in charm hadron processes at b-factoriesor/and τ -charm factories. Recently, an extensive studyon c → uν ¯ ν appeared in Ref. [1], pointing out that ob-servables very small in the Standard Model (SM) offerunique (null) tests of BSM physics. Namely, for charmflavour changing neutral current (FCNC) processes, se-vere Glashow-Iliopoulos-Maiani (GIM) suppression oc-curs. The decay D → ν ¯ ν amplitude is helicity sup-pressed in the SM. The authors of [2] made very detailedstudy of heavy meson decays to invisibles, assuming thatthe invisibles can be scalars or fermions with both he-licites. They found out that in the SM branching ratio B ( D → ν ¯ ν ) = 1 . × − . Then the authors of [3]found that the decay width of D → invisibles in the SMis actually dominated by the contribution of D → ν ¯ νν ¯ ν .These studies’ main message is that SM provides no ir-reducible background to analysis of invisibles in decaysof charm (and beauty) mesons. They also suggested [2],that in searches for a Dark Matter candidate, it mightbe important to investigate process with χ ¯ χγ in the fi-nal state, since a massless photon eliminates the helicitysuppression. We also determine branching ratios for suchdecay modes. The authors of Ref. [1] computed the ex-pected event rate for the charm hadron decays to a finalhadronic state and neutrino - anti-neutrino states. Theyfound out that in experiments like Belle II, which canreach per-mile efficiencies or better, these processes can ∗ Electronic address:[email protected] † Electronic address:[email protected] be seen. In addition future FCC-ee might be capable ofmeasuring branching ratios of O (10 − ) down to O (10 − ),in particular D , D +( s ) and Λ + c decay modes.On the other hand, the Belle collaboration alreadyreached bound of the branching ratio for B ( D → invisibles ) = 9 . × − and the Belle II experiment isexpected to improve it [4]. The other e + e − machines asBESSIII [5] and future FCC-ee running colliders at the Z energies [6, 7] with a significant charm production with B ( Z → c ¯ c ) (cid:39) .
22 [7] provide us with excellent tools forprecision study of charm decays.In this work we focus on the particular scenarios withcoloured scalars or leptoquarks as mediators of the invisi-ble fermions interaction with quarks. The coloured scalarmight have the electric charge of 2 / − / D − ¯ D oscillations and inthe case of weak doublets, we include constraints fromother flavour processes.Motivated by previous works of Refs. [1, 2, 9–12], weinvestigate c → u ¯ χχ with χ being a massive SU (2) L singlet. Coloured scalars carry out interactions betweeninvisible fermions and quarks. Namely, leptoquarks usu-ally denote the boson interacting with quarks and lep-tons. However, the state ¯ S does not interact with theSM leptons and, therefore, it is more appropriate to call itcoloured scalar. Our approach is rather minimalistic dueto only two Yukawa couplings and the mass of colouredscalar. The effective Lagrangian and coloured scalar me-diators are introduced in Sec. II. In Sec. III we describeeffects of ¯ S mediator in rare charm decays, while in Sec.IV we give details of ˜ R mediation in the same processes.Sec. V contains conclusions and outlook. a r X i v : . [ h e p - ph ] J a n II. COLOURED SCALARS IN c → uχ ¯ χ In experimental searches, the transition c → u invisibles might be approached in processes c → u /E with /E being missing energy. Therefore, invisibles canbe either SM neutrinos or new right-handed neutralfermions (having quantum numbers of right-handed neu-trinos), or scalars/vectors as suggested in Ref. [2]. Theauthors of Refs. [1, 9] considered in detail general frame-work of New Physics (NP) in c → u invisibles , relying on SU (2) L invariance and data on charged lepton processes[9]. They found that these assumptions allow upper lim-its as large as few 10 − , while in the limit of lepton uni-versality branching ratios can be as large as 10 − . Toconsider invisible fermions, having quantum numbers ofright-handed neutrinos, and being massive, we extend theeffective Lagrangian by additional operators as describedin Refs. [3, 13] L eff = √ G F (cid:20) c LL ( u L γ µ c L )( ν L γ µ ν (cid:48) L )+ c RR ( u R γ µ c R )( ν R γ µ ν (cid:48) R ) + c LR ( u L γ µ c L )( ν R γ µ ν (cid:48) R )+ c RL ( u R γ µ c R )( ν L γ µ ν (cid:48) L ) + g LL ( u L c R )( ν L ν (cid:48) R )+ g RR ( u R c L )( ν R ν (cid:48) L ) + g LR ( u L c R )( ν R ν (cid:48) L )+ g RL ( u R c L )( ν L ν (cid:48) R ) + h LL ( u L σ µν c R )( ν L σ µν ν (cid:48) R )+ h RR ( u R σ µν c L )( ν R σ µν ν (cid:48) L ) (cid:21) + h. c. . (1)In Ref. [1] right-handed massless neutrinos are consid-ered. Also, in Ref. [12] authors considered charm me-son decays to invisible fermions, which have negligiblemasses. In the following, we consider massive right-handed fermions and use further the notation ν R ≡ χ R .Following [13], we write in Table I interactions of thecoloured scalar ¯ S and ˜ R with the up quarks and ˜ R and S with down quarks. Cloured Scalar Invisible fermion S = (¯3 , , /
3) ¯ d C iR χ j S ¯ S = (¯3 , , − /
3) ¯ u C iR χ j ¯ S ˜ R = (¯3 , , /
6) ¯ u iL χ j ˜ R / ˜ R = (¯3 , , /
6) ¯ d iL χ j ˜ R − / Table I. The coloured scalars ¯ S , S and ˜ R interactions withinvisible fermions and quarks. Here we use only right-handedcouplings of S . Indices i, j refer to quark generations. We concentrate only on coloured scalar and scalarleptoquark due to difficulties with vector leptoquarks.Namely, the simplest way to consider vector leptoquarksin an ultra-violet complete theory is when they play therole of gauge bosons. For example, U is one of the gaugebosons in some of Pati-Salam unification schemes [14, 15].However, other particles with masses close to U with many new parameters in such theories, making it ratherdifficult to use without additional assumptions.Coloured scalars contributing to transition c → uχ ¯ χ have following Lagrangians, as already anticipated in in[13] L ( ¯ S ) ⊃ ¯ y RR ij ¯ u C iR χ jR ¯ S + h.c. . (2) L ( ˜ R ) ⊃ ( V ˜ y LR ) ij ¯ u iL χ jR ˜ R / + ˜ y LR ij ¯ d iL χ jR ˜ R − / + h.c. . (3)Here, we give only terms containing interactions ofquarks with right-handed χ . The S scalar leptoquark,in principle, might mediate c → uχ ¯ χ on the loop level,with one W boson changing down-like quarks to u and c .Obviously, such a loop process is also suppressed by loopfactor 1 / (16 π ) and G F making it negligible. Also, dueto the right-handed nature of χ , one can immediately seethat in the case of ¯ S , the effective Lagrangian has onlyone contribution L eff = √ G F c RR (cid:0) ¯ u R γ µ c R (cid:1) ( ¯ χ R γ µ χ R ) , (4)with c RR = v M S ¯ y RR cχ ¯ y RR ∗ uχ . (5)In the case of ˜ R L eff = √ G F c LR (cid:0) ¯ u L γ µ c L (cid:1) ( ¯ χ R γχ R ) , (6)with c LR = − v M R (cid:16) V ˜ y LR (cid:17) uχ (cid:16) V ˜ y LR (cid:17) ∗ cχ . (7)For the mass of χ , kinematically allowed, in the c → uχ ¯ χ decay, one can relate this amplitude to b → sχ ¯ χ or in s → dχ ¯ χ . However, it was found [16] that the exper-imental rates for K → πν ¯ ν are very close to the SMrate [17], leaving very little room for NP contributions.Therefore, we avoid this kinematic region and considermass of χ to be m χ ≥ ( m K − m π ) /
2, while the charmdecays allow m χ ≤ ( m D − m π ) /
2. For our further studyit is very important that χ is a weak singlet and there-fore LHC searches of high- p T lepton tails [18, 19] are notapplicable for the constraints of interactions in the caseswe consider. However, further study of final states con-taining mono-jets and missing at LHC and future Highluminosity colliders will shed more light on these pro-cesses. III. ¯ S IN c → uχ ¯ χ Due to its quantum numbers, the coloured scalar ¯ S and χ can interact only with up-like quarks. Most gener-ally, the number of χ ’s can be three and the matrix y RR ij can have 9 × χ , thatcan couple to both u and c quarks. These two couplingsmight enter in amplitudes for processes with down-likequarks at loop-level, as discussed in [20]. Obviously, dueto the right-handed nature of χ , one can immediately seethat in the case of ¯ S , the effective Lagrangian has onlythe contribution L eff = √ G F v M ¯ y RR cχ ¯ y RR ∗ uχ ( u R γ µ c R )( χ R γ µ χ R ) . (8)First, we discuss constraints from D − ¯ D mixing andthen consider exclusive decays D → χ ¯ χ , D → ¯ χχγ ,and D → πχ ¯ χ . The authors of Ref. [12] consideredscalar leptoquarks allowing each up-quark can couple todifferent flavour of lepton or right-handed neutrino. Insuch a way, they avoid constraints from the D − ¯ D mixing.
1. Constraints from D − ¯ D The strongest constraints on χ interactions with u and c comes from the D − ¯ D oscillations. The interactionsin Eqs. (2) and (3) can generate transition D − ¯ D .Coloured scalar ¯ S contributes to the operator enteringthe effective Lagrangian [13, 21] L Dmix eff = − C (cid:0) ¯ cγ µ P R u (cid:1) (¯ cγ µ P R u ) , (9)with the Wilson coefficient given by C = 164 π M S (cid:16) ¯ y RR cχ (cid:17) (cid:16) ¯ y RR ∗ uχ (cid:17) . (10)The standard way to write the hadronic matrix ele-ment is (cid:10) ¯ D | (¯ uγ µ P R c )(¯ uγ µ P R c ) | D (cid:11) = m D f D B D , withthe bag parameter B D (3 GeV) = 0 . (cid:10) | ¯ uγ µ γ c ) | D ( p ) (cid:11) = if D p µ , with f D = 0 . x = 2 | M | / Γ = (0 . +0 . − . )% by HFLAV [24]. Thebound on this Wilson coefficient can be derived following[20, 25] (cid:12)(cid:12) rC ( M ¯ S ) (cid:12)(cid:12) m D f D B D D < x, (11)with a renormalisation factor r = 0 .
76 due to running of C from scale M ¯ S (cid:39) . | C | < . × − GeV − or (cid:12)(cid:12)(cid:12) ¯ y RR cχ ¯ y RR ∗ uχ (cid:12)(cid:12)(cid:12) < . × − M ¯ S /GeV. (12) c RR < .
363 GeV M ¯ S (GeV) . (13) D → χ ¯ χ The amplitude for this process can be written as M ( D → χ ¯ χ ) = √ G F f D c RR m χ ¯ u χ ( p ) γ v χ ( p ) , (14)giving the branching ratio B ( D → χ ¯ χ ) = 1Γ D G F f D m D π (cid:12)(cid:12)(cid:12) c RR (cid:12)(cid:12)(cid:12) m χ (cid:115) − m χ m D . (15)Using Belle bound B ( D → χ ¯ χ ) < . × − [8], one canfind easily the bound on Wilson coefficient (cid:12)(cid:12) c RR (cid:12)(cid:12) Belle < . m χ = 0 . S , allowingthe mass of χ to be ( m K − m π ) / < m χ < ( m D − m π ) / B ( D → χ ¯ χ ) < − , 10 − and 10 − , with (cid:12)(cid:12)(cid:12) ¯ y RR cχ ¯ y RR ∗ uχ (cid:12)(cid:12)(cid:12) = 1. We presentour result in Fig. 2 and find that mass of ¯ S , using thesereasonable assumptions, can be within LHC reach. m χ (GeV) B ( D → χ ¯ χ ) D − ¯ D < . × − < . × − < . × − Table II. Branching ratios for B ( D → χ ¯ χ ) for three selectedvalues of m χ . The constraints from the D − ¯ D mixing isused, with c RR ≤ . × − , assuming M ¯ S = 1000 GeV. M S - [ GeV ] y RR c χ y RR * χ BelleD - mixing Figure 1. The product of Yukawa couplings (cid:12)(cid:12)(cid:12) ¯ y RR cχ ¯ y RR ∗ uχ (cid:12)(cid:12)(cid:12) as afunction of the ¯ S mass. The pink line denotes the bound de-rived from Belle result [8], while the turquoise one is obtainedwith the bound from D − ¯ D oscillations. ���� �� m χ [ GeV ] M S _ [ G e V ] BR ( D → χχ )< - BR ( D → χχ )< - BR ( D → χχ )< - Figure 2. The allowed mass region for ¯ S in the range( m K − m π ) / < m χ < ( m D − m π ) /
2. The regions are ob-tained assuming B ( D → χ ¯ χ ) < − , 10 − and 10 − , forthe product (cid:12)(cid:12)(cid:12) ¯ y RR cχ ¯ y RR ∗ uχ (cid:12)(cid:12)(cid:12) = 1. D → χ ¯ χγ The authors of Ref. [2] suggested, that the helicitysuppression, present in the D → χ ¯ χ amplitude for m χ =0, is lifted by an additional photon in the final state andtherefore D → χ ¯ χγ might bring additional informationon detection of invisibles in the final state. They foundthat the branching decay is B ( D → χ ¯ χγ ) = G F F DQ f D | c RR | m D α π Γ D (cid:113) − x χ Y ( x χ ) . (16)In the above equations x χ = m χ /m D , F DQ =2 / − / ( m D − m c ) + 1 /m c ), f D = 0 . Y ( x χ ) is given in Appendix. Coefficient c RR is con-strained by Eq. (13). Comparing these results with m χ (GeV) B ( D → χ ¯ χγ ) D − ¯ D B ( D → χ ¯ χγ ) Belle < . × − < . × − < . × − < . × − < . × − < . × − Table III. Bounds on the branching ratio for B ( D → χ ¯ χγ ).In the second column the constraints from the D − ¯ D mixingis used, assuming M ¯ S = 1000 GeV. In the third column Bellebound B ( D → /E ) < . × − is used. the SM result presented in Ref. [2] B ( D → ν ¯ νγ ) SM =3 . × − , we see that the existing Belle bound al-lows significant branching ratio, while the bounds fromthe D − ¯ D mixing, for larger values of m χ , lead to thebranching ratio to be close to the SM results. Due tothe mass of χ , the photon energy can be in the range0 ≤ E γ ≤ ( m D − m χ ) / (2 m D ), which in principle woulddistinguish the SM contribution from the contributionswith massive invisible fermions. D → πχ ¯ χ The rare charm decays due to GIM-mechanism can-cellation are usually dominated by long distance contri-butions. Long distance contributions to exclusive de-cay channel D → πν ¯ ν were considered in Ref. [26].For example, the branching ratio BR ( D + → π + ρ → π + ν ¯ ν ) < × − . The authors of [26] discussed an-other possibility D + → τ + ν → π + ¯ νν and found thatthe branching ratio should be smaller than 1 . × − .An interesting study of these effects was done in Ref.[27], implying that in order to avoid these effects oneshould make cuts in the invariant χ ¯ χ mass square, q cut < ( m τ − m π )( m D − m τ ) /m τ .The amplitude for D → πχ ¯ χ can be written as M ( D → πχ ¯ χ ) = √ G F c RR ¯ u χ ( p ) γ µ P R v χ ( p ) (cid:10) π ( k ) | ¯ uγ µ P R | D ( p ) (cid:11) , (17)with the standard form-factors definition (cid:10) π ( k ) | ¯ uγ µ (1 ± γ ) | D ( p ) (cid:11) = f + ( q ) (cid:34) ( p + k ) µ − m D − m π q q µ (cid:35) + f ( q ) m D − m π q q µ , (18)with q = p − k . We follow the update of the form-factorsin Ref. [28]. This enables us to write the amplitudes inthe form given in Ref. [21] M ( D ( p ) → π ( k ) χ ( p ) ¯ χ ( p )) = √ G F [ V ( q )¯ u χ ( p ) /pv χ ( p )+ A ( q )¯ u χ ( p ) /pγ v χ ( p ) + P ( q )¯ u χ ( p ) γ v χ ( p )] , (19)with the following definitions V ( q ) = A ( q ) ≡ c RR f + ( q ) P ( q ) ≡ − c RR m χ (cid:34) f + ( q ) − m D − m π q ( f ( q ) − f + ( q )) (cid:35) . (20)We can the differential decay rate distribution as d B ( D → π ¯ χχ ) dq = 1Γ D N λ / β (cid:20) a ( q ) + 23 c ( q ) (cid:21) . (21)with notation λ ≡ λ ( m D , m π , q ), ( λ ( x, y, z ) = ( x + y + z ) − xy + yz + zx )), β = (cid:113) − m χ /q and N = G F π ) m D . Note that in case of charged charm mesonthere is a multiplication by 2 in the differential decayrate compared to neutral D . ���� �� × - × - × - × - × - m χ [ GeV ] ( B R ( D → π χχ ) m χ ) BR ( D + →π + χχ ) BR ( D →π χχ ) Figure 3. Branching fraction for D + → π + χ ¯ χ and D → π χ ¯ χ as a function of m χ . The integration bounds should be 4 m χ ≤ q ≤ ( m D − m π ) in the case of m χ = 0 . , .
8, while instead of m χ = 0 .
18 GeV, q cut is used from Ref. [27], givingthe lowest mass of the invisibles should be searchedin the region m χ ≥ (cid:112) q cut / (cid:39) .
29 GeV. This en-ables us to avoid the region in which the effects ofthe long distance dynamics dominates. One can use m χ (GeV) B ( D → π χ ¯ χ ) D − ¯ D B ( D + → π + χ ¯ χ ) D − ¯ D < . × − < . × − < . × − < . × − < . × − < . × − Table IV. Branching ratios for B ( D → πχ ¯ χ ). In the secondand the third columns the constraint from the D − ¯ D mixingis used, assuming the mass of M ¯ S = 1000 GeV. In the case m χ = 0 .
18, the cut in integration variable is done by taking q cut , as described in the text. the Belle bound [8] for B ( D → /E ) and determine c RR from D → χ ¯ χ for each χ mass. We obtain B ( D → π χ ¯ χ ) Belle ≤ . × − , . × − , . × − and B ( D + → π + χ ¯ χ ) Belle ≤ . × − , . × − , . × − for m χ = 0 . , . , . − might be possi-ble to observe at the future tau-charm factories and BelleII experiment. IV. ˜ R IN c → uχ ¯ χ The ˜ R leptoquark is a weak doublet and it inter-acts with quark doublets (3). Therefore, the appropri-ate couplings, ˜ y LR sχ ˜ y LR ∗ bχ can be constrained from the b → sχ ¯ χ and s → dχ ¯ χ decays, as well as from observ-ables coming from the B s − ¯ B s , B d − ¯ B d , K − ¯ K os-cillations as in [20]. We consider the most constrain-ing bounds coming from decays B → K /E and fromthe oscillations of B s − ¯ B s , relevant for the χ mass re-gion ( m K − m π ) / < m χ < ( m D − m π ) /
2. The de-cay B → K /E was recently studied by the authors of Ref. [29]. They pointed out that current bound onthe rate B → K /E when the SM branching ratio for B → Kν ¯ ν is subtracted from the experimental bound on B ( B + → K + /E ) is the most constraining. They derived B ( B → K /E ) < . × − as the strongest bound among B → H s /E ( H s is a hadron containing the s quark).
1. Constraints from B → K /E and B s − ¯ B s oscillations The amplitude for B → Kχ ¯ χ can be written as M ( B → Kχ ¯ χ ) = √ G F c LRB ¯ u χ ( p ) γ µ P R v χ ( p ) (cid:10) K ( k ) | ¯ uγ µ P L | B ( p ) (cid:11) . (22)In the case of Wilson coefficient c LRB it is easy to find [13] c LRB = − v M R ˜ y LR sχ ˜ y LR ∗ bχ . (23)The integration over the phase space depends on the massof m χ we chose. Here we can choose a mass χ , which weused in D decays ( m K − m π ) / < m χ < ( m D − m π ) / | c LRB | < . × − , < . × − and < . × − for m χ = 0 . , . , .
80 GeV.There are two box diagrams with χ within the boxcontributing to the B s − ¯ B s oscillations. The contributionof ˜ R box diagrams to the effective Lagrangian for the B s − ¯ B s oscillation is L NP ∆ B =2 = − π (cid:16) ˜ y LR sχ (cid:17) (cid:16) ˜ y LR ∗ bχ (cid:17) M R × (cid:0) ¯ sγ µ P R b (cid:1) (¯ sγ µ P R b ) . (24)We can understand this result in terms of the recentstudy of new physics in the B s − ¯ B s oscillation in [30].The authors of [30] introduced the following notation ofthe New Physics (NP) contribution containing the right-handed operators as L NP ∆ B =2 ⊃ − G F √ V tb V ∗ ts ) C RRbs (cid:0) ¯ sγ µ P R b (cid:1) (¯ sγ µ P R b ) . (25)Following their notation, one can write the modificationof the SM contribution by the NP as in Ref. [30]∆ M SM + NPs ∆ M SMs = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η / R SMloop C RRbs (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (26)They found that R SMloop = (1 . ± . × − and η = α s ( µ NP ) /α s ( µ b ). Relying on the Lattice QCD results ofthe two collaborations FNAL/MILC [31], HPQCD [32],the FLAG averaging group [33] published following re-sults, which we use in our calculations∆ M F LAG s = (20 . +1 . − . ) ps − = (1 . +0 . − . ) ∆ M exps , (27)From these results, one can easily determine bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) ˜ y LR sχ (cid:17) (cid:16) ˜ y LR ∗ bχ (cid:17) M R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ . × − GeV − , (28)The same couplings ˜ y LR sχ ˜ y LR ∗ bχ enter in the D − ¯ D mix-ing (9) and condition (12), and one can derive (cid:20)(cid:16) V us ˜ y LR sχ (cid:17) (cid:16) V cb ˜ y LR bχ (cid:17) ∗ + (cid:16) V cs ˜ y LR sχ (cid:17) (cid:16) V ub ˜ y LR bχ (cid:17) ∗ (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D − ¯ D < . × − M ˜ R / GeV . (29)The bound on coefficients in (28) lead to the one or-der of magnitude stronger constraint then one in (29),˜ y LR sχ ˜ y LR ∗ bχ < . × − M ˜ R / GeV. In our numeri-cal calculations we use this bound and do not specifythe mass of ˜ R . However, one can combine these con-straints and determined the ˜ R mass, which can sat-isfy both conditions. In Fig. (4) we present depen-dence of the couplings ˜ y LR sχ ˜ y LR ∗ bχ as a function of mass M ˜ R for masses using constrain from B s − ¯ B s mixingand from the bound B ( B + → K + /E ) < . × − for m χ = 0 . , . , . M R ∼ [ GeV ] y ˜ L R s χ y ˜ L R * χ B s mixing m χ = m χ = m χ = Figure 4. The allowed mass for ˜ R . Constraints are de-rived from the B s − ¯ B s mixing and from the bound B ( B + → K + /E ) < . × − for m χ = 0 . , . , . that the largest mass of ˜ R , which satisfies both con-ditions is M ˜ R (cid:39) , , m χ = 0 . , . , .
18 GeV respectively. All ˜ R masses be-low these limiting values are allowed and interestingly,they are within LHC reach. ˜ R in B ( D → χ ¯ χ ) , B ( D → χ ¯ χγ ) and B ( D + → π + χ ¯ χ ) Using the same expressions as in the previous sec-tion, we calculate branching ratios for D → χ ¯ χ , D → χ ¯ χγ and present them in Table V. The resultsfor D → πχ ¯ χ are presented in Table VI. The Wil-son coefficient c LRD is obtained using the constraint from B → K missing energy . For m χ = 0 . , . , . m χ (GeV) B ( D → χ ¯ χ ) B ( D → χ ¯ χγ )0 . < . × − < . × − . < . × − < . × − . < . × − < . × − Table V. Branching ratios for B ( D → χ ¯ χ ) and B ( D → χ ¯ χγ ). The bounds on the Wilson coefficient c LRD derived fromthe B ( B → K /E ) < . × − for selected masses of χ fromthe range ( m K − m π ) / < m χ < ( m D − m π ) / m χ (GeV) B ( D → π χ ¯ χ ) B ( D + → π + χ ¯ χ )0 . < . × − < . × − . < . × − < . × − . < . × − < . × − Table VI. Branching ratios for B ( D → π χ ¯ χ ) and B ( D + → π + χ ¯ χ ). The bounds on the Wilson coefficient c LRD derivedfrom B ( B → K /E ) < . × − . In the case m χ = 0 . q cut , asdescribed in the text. they are c LRD = | ( V us V ∗ cb + V cs V ∗ ub ) c LRB | = 4 . × − , . × − , . × − . Compared with the coloured scalar ¯ S mediation, the branching ratios for all three decay modesare suppressed for several orders of magnitude, indicatingthe important role of constraints from B mesons. Suchsuppressed branching ratios of the all rare charm decaysmediated by ˜ R is almost impossible to observe. On theother hand, decays of hadrons containing b quarks , medi-ated by ˜ R a much more suitable for searches of invisiblefermions. V. SUMMARY AND OUTLOOK
We have presented a study on rare charm decays withinvisible massive fermions χ in the final state. The massof χ is taken to be in the range ( m K − m π ) / < m χ < ( m D − m π ) /
2, since the current experimental results on B ( K → πν ¯ ν ) are very close to the SM result, almost ex-cluding the presence of New Physics. We considered twocases with coloured scalar mediators of the up-quarksinteraction with χ . The simplest model is one with¯ S = (¯3 , , − / R = (¯3 , , / S , the relevant constraint comes fromthe D − ¯ D oscillations. We have calculated branch-ing ratios for D → χ ¯ χ , D → χ ¯ χγ and D → πχ ¯ χ . Thecharm meson mixing severely constrain the branching ra-tio D → χ ¯ χ in comparison with the experimental resultfor the branching ratio of D → /E . For our choice of m χ the branching ratio for D → χ ¯ χγ can be calculatedusing experimental bound on the rate for D → /E . Inthis case, there is an enhancement factor up to three or-ders of magnitude smaller, depending on the mass of χ in comparison with the constraints from the D − ¯ D os-cillations. The branching ratios for D → πχ ¯ χ , based oncharm mixing constraint, are of the order 10 − − − ,suitable for searches at future tau-charm factories, BE-SIII and Belle II experiments.In the case of ˜ R , for the mass range of χ relevant forcharm meson rare decays, we rely on constraints comingfrom B ( B → K /E ) and from the B s − ¯ B s mixing. We findthat the all three decay modes D → χ ¯ χ , D → χ ¯ χγ and D + → π + χ ¯ χ are now having branching ratios fora factor 3 − S mediation, making them verydifficult for the observation.Interestingly, the mass of both mediators ¯ S and ˜ R are in the range of LHC reach, and hopefully, searchesfor mono-jets and missing energy might put constraintson their masses. VI. ACKNOWLEDGMENT
The work of SF was in part financially supported bythe Slovenian Research Agency (research core fundingNo. P1-0035). The work of AN was partially supportedby the Advanced Grant of European Research Council(ERC) 884719 — FAIME.
VII. APPENDIXA. Phase space factors
In eq. (16) phase space function Y ( x χ ) is used Y ( x χ ) = 1 − x χ + 3 x χ (3 − x χ + 4 x χ ) √ − x χ × log x χ (cid:113) − x χ − x χ + 12 x χ . (30)In Eq. (21) a ( q ) and c ( q ) are introduced denoting a ( q ) = λ | V ( q ) | + | A ( q ) | ) + 8 m χ m D | A ( q ) | +2 q | P ( q ) | + 4 m χ ( m D − m π + q )Re[ A ( q ) P ( q ) ∗ ] ,c ( q ) = − λβ | V ( q ) | + | A ( q ) | ) . (31) B. D → π form factors Following [34] one can use z − expansion with z = (cid:112) t + − q − √ t + − t (cid:112) t + − q + √ t + − t , (32)with t + = ( m D + m π ) and t = ( m D + m π )( √ m D −√ m π ). The form factors can be written as f D → π + ( q ) = f D → π (0) + c D → π + ( z − z )(1 + ( z + z ))1 − P V q , (33) f D → π ( q ) = f D → π (0) + c D → π ( z − z )(1 + ( z + z ))1 − P S q , (34)where z = z (0 , t π ). The fit parameters are given in Table(VII). For the most recent discussion on form-factors seealso [35]. f (0) c + P V (GeV) − c P S (GeV) − f , f + in the z -series expansionfor D → π [34]. C. B → K form factors Most recent results are presented in FLAG report [33] f BK + ( q ) = r − q m R + r − q m R , (35) f BK ( q ) = r − q m (cid:48) R . (36)The parameters are r = 0 . r = 0 . m R = 5 . m R (cid:48) = 6 .
12 GeV, as in [33]. [1] R. Bause, H. Gisbert, M. Golz, G. Hiller, Rare charm c → u ν ¯ ν dineutrino null tests for e + e − -machines (102020). arXiv:2010.02225 .[2] A. Badin, A. A. Petrov, Searching for light Dark Matterin heavy meson decays, Phys. Rev. D 82 (2010) 034005. arXiv:1005.1277 , doi:10.1103/PhysRevD.82.034005 . [3] B. Bhattacharya, C. M. Grant, A. A. Petrov, Invisiblewidths of heavy mesons, Phys. Rev. D 99 (9) (2019)093010. arXiv:1809.04606 , doi:10.1103/PhysRevD.99.093010 .[4] W. Altmannshofer, et al., The Belle II Physics Book,PTEP 2019 (12) (2019) 123C01, [Erratum: PTEP 2020,029201 (2020)]. arXiv:1808.10567 , doi:10.1093/ptep/ ptz106 .[5] M. Ablikim, et al., Future Physics Programme of BESIII,Chin. Phys. C 44 (4) (2020) 040001. arXiv:1912.05983 , doi:10.1088/1674-1137/44/4/040001 .[6] A. Abada, et al., FCC Physics Opportunities: FutureCircular Collider Conceptual Design Report Volume 1,Eur. Phys. J. C 79 (6) (2019) 474. doi:10.1140/epjc/s10052-019-6904-3 .[7] A. Abada, et al., FCC-ee: The Lepton Collider: FutureCircular Collider Conceptual Design Report Volume 2,Eur. Phys. J. ST 228 (2) (2019) 261–623. doi:10.1140/epjst/e2019-900045-4 .[8] Y.-T. Lai, et al., Search for D decays to invisible fi-nal states at Belle, Phys. Rev. D 95 (1) (2017) 011102. arXiv:1611.09455 , doi:10.1103/PhysRevD.95.011102 .[9] R. Bause, H. Gisbert, M. Golz, G. Hiller, Exploiting CP -asymmetries in rare charm decays, Phys. Rev. D101 (11) (2020) 115006. arXiv:2004.01206 , doi:10.1103/PhysRevD.101.115006 .[10] E. Golowich, J. Hewett, S. Pakvasa, A. A. Petrov, Re-lating D0-anti-D0 Mixing and D0 — > l+ l- with NewPhysics, Phys. Rev. D 79 (2009) 114030. arXiv:0903.2830 , doi:10.1103/PhysRevD.79.114030 .[11] J. Martin Camalich, M. Pospelov, P. N. H. Vuong,R. Ziegler, J. Zupan, Quark Flavor Phenomenology of theQCD Axion, Phys. Rev. D 102 (1) (2020) 015023. arXiv:2002.04623 , doi:10.1103/PhysRevD.102.015023 .[12] G. Faisel, J.-Y. Su, J. Tandean, Exploring charm de-cays with missing energy in leptoquark models (12 2020). arXiv:2012.15847 .[13] I. Dorˇsner, S. Fajfer, A. Greljo, J. Kamenik, N. Koˇsnik,Physics of leptoquarks in precision experiments and atparticle colliders, Phys. Rept. 641 (2016) 1–68. arXiv:1603.04993 , doi:10.1016/j.physrep.2016.06.001 .[14] M. Bordone, C. Cornella, J. Fuentes-Mart´ın, G. Isidori,Low-energy signatures of the PS model: from B -physicsanomalies to LFV, JHEP 10 (2018) 148. arXiv:1805.09328 , doi:10.1007/JHEP10(2018)148 .[15] L. Di Luzio, A. Greljo, M. Nardecchia, Gauge lepto-quark as the origin of B-physics anomalies, Phys. Rev.D 96 (11) (2017) 115011. arXiv:1708.08450 , doi:10.1103/PhysRevD.96.115011 .[16] G. Ruggiero, New Result on K + → π + ν ¯ ν from the NA62Experiment, KAON2019, Perugia, Italy, 10-13 Septem-ber 2019.[17] A. J. Buras, D. Buttazzo, J. Girrbach-Noe, R. Knegjens, K + → π + νν and K L → π νν in the Standard Model:status and perspectives, JHEP 11 (2015) 033. arXiv:1503.02693 , doi:10.1007/JHEP11(2015)033 .[18] A. Angelescu, D. A. Faroughy, O. Sumensari, LeptonFlavor Violation and Dilepton Tails at the LHC, Eur.Phys. J. C 80 (7) (2020) 641. arXiv:2002.05684 , doi:10.1140/epjc/s10052-020-8210-5 .[19] J. Fuentes-Martin, A. Greljo, J. Martin Camalich, J. D.Ruiz-Alvarez, Charm physics confronts high-p T leptontails, JHEP 11 (2020) 080. arXiv:2003.12421 , doi:10.1007/JHEP11(2020)080 .[20] S. Fajfer, D. Susiˇc, Coloured Scalar Mediated NucleonDecays to Invisible Fermion (10 2020). arXiv:2010.08367 . [21] S. Fajfer, N. Koˇsnik, Prospects of discovering newphysics in rare charm decays, Eur. Phys. J. C 75 (12)(2015) 567. arXiv:1510.00965 , doi:10.1140/epjc/s10052-015-3801-2 .[22] N. Carrasco, P. Dimopoulos, R. Frezzotti, V. Lubicz,G. C. Rossi, S. Simula, C. Tarantino, ∆S=2 and ∆C=2bag parameters in the standard model and beyond fromN f =2+1+1 twisted-mass lattice QCD, Phys. Rev. D92 (3) (2015) 034516. arXiv:1505.06639 , doi:10.1103/PhysRevD.92.034516 .[23] P. Zyla, et al., Review of Particle Physics, PTEP 2020 (8)(2020) 083C01. doi:10.1093/ptep/ptaa104 .[24] Y. S. Amhis, et al., Averages of b -hadron, c -hadron, and τ -lepton properties as of 2018 (9 2019). arXiv:1909.12524 .[25] S. Fajfer, N. Kosnik, Leptoquarks in FCNC charm de-cays, Phys. Rev. D 79 (2009) 017502. arXiv:0810.4858 , doi:10.1103/PhysRevD.79.017502 .[26] G. Burdman, E. Golowich, J. L. Hewett, S. Pakvasa, Rarecharm decays in the standard model and beyond, Phys.Rev. D 66 (2002) 014009. arXiv:hep-ph/0112235 , doi:10.1103/PhysRevD.66.014009 .[27] J. F. Kamenik, C. Smith, Tree-level contributions to therare decays B+ — > pi+ nu anti-nu, B+ — > K+ nu anti-nu, and B+ — > K*+ nu anti-nu in the Standard Model,Phys. Lett. B 680 (2009) 471–475. arXiv:0908.1174 , doi:10.1016/j.physletb.2009.09.041 .[28] R. Fleischer, R. Jaarsma, G. Koole, Testing LeptonFlavour Universality with (Semi)-Leptonic D ( s ) Decays,Eur. Phys. J. C 80 (2) (2020) 153. arXiv:1912.08641 , doi:10.1140/epjc/s10052-020-7702-7 .[29] G. Li, T. Wang, Y. Jiang, J.-B. Zhang, G.-L. Wang, Spin-1 / arXiv:2004.10942 .[30] L. Di Luzio, M. Kirk, A. Lenz, T. Rauh, ∆ M s theory pre-cision confronts flavour anomalies, JHEP 12 (2019) 009. arXiv:1909.11087 , doi:10.1007/JHEP12(2019)009 .[31] A. Bazavov, et al., B s ) -mixing matrix elements from lat-tice QCD for the Standard Model and beyond, Phys.Rev. D 93 (11) (2016) 113016. arXiv:1602.03560 , doi:10.1103/PhysRevD.93.113016 .[32] R. Dowdall, C. Davies, R. Horgan, G. Lepage, C. Mona-han, J. Shigemitsu, M. Wingate, Neutral B-meson mix-ing from full lattice QCD at the physical point, Phys.Rev. D 100 (9) (2019) 094508. arXiv:1907.01025 , doi:10.1103/PhysRevD.100.094508 .[33] S. Aoki, et al., FLAG Review 2019: Flavour Lat-tice Averaging Group (FLAG), Eur. Phys. J. C 80 (2)(2020) 113. arXiv:1902.08191 , doi:10.1140/epjc/s10052-019-7354-7 .[34] V. Lubicz, L. Riggio, G. Salerno, S. Simula, C. Tarantino,Scalar and vector form factors of D → π ( K ) (cid:96)ν decayswith N f = 2+1+1 twisted fermions, Phys. Rev. D 96 (5)(2017) 054514, [Erratum: Phys.Rev.D 99, 099902 (2019),Erratum: Phys.Rev.D 100, 079901 (2019)]. arXiv:1706.03017 , doi:10.1103/PhysRevD.96.054514 .[35] D. Beˇcirevi´c, F. Jaffredo, A. Pe˜nuelas, O. Sumensari,New Physics effects in leptonic and semileptonic decays(12 2020). arXiv:2012.09872arXiv:2012.09872