a r X i v :h e p -p h /0409218v 2 9 M a r 2005LPHEP-04-03September 2004Higgs bosons decay into bottom-strangein two Higgs Doublets ModelsAbdesslam ArhribD´e partement de Math´e matiques,Facult´e des Sciences et TechniquesB.P 416Tanger,Morocco.andLPHEA,D´e partement de Physique,Facult´e des Sciences-Semlalia,B.P.2390Marrakech,Morocco.AbstractWe analyze the decays {h 0,H 0,A 0}→¯s b within two Higgs Doublet Models with Natural Flavor Conservation (2HDM)type I and II.It is found that the Higgs bosons decay into bottom-strange can lead to a branching ratio in the range 10−5→10−3for small tan β≈0.1→0.5and rather light charged Higgs in the 2HDM type I.Whentan β>∼1,one can easily reach a branching ratio of the order 10−5.In 2HDM typeII,without imposing b →sγconstraint,the situation is the same as in 2HDM type I.If b →sγconstraint on charged Higgs mass (M H ±≥350GeV)is imposed,we obtain Br (h 0→¯s b )in the range 10−5–10−6.A comparison between the rates of h 0→¯s b and h 0→γγis made.It is found that in the fermiophobic scenario,h 0→γγis still the dominant decay mode.1IntroductionOne of the goals of the next generation of high energy colliders,such as the large hadroncollider LHC[1]or the linear collider LC[2]or muon colliders,is to probe top Flavor-Changing Neutral Couplings‘top FCNC’as well as the Higgs Flavor-Changing NeutralCouplings‘Higgs FCNC’.FCNC of heavy quarks have been intensively studied both from the theoretical and experimental point of view.Such processes are being well established inthe Standard Model(SM)and are excellent probes for the presence of new physics effectssuch as Supersymmetry,extended Higgs sector and extra fermions families.Within the SM,with one Higgs doublet,the FCNC Z¯t c vanishes at tree-level by theGIM mechanism,while theγ¯t c and g¯t c couplings are zero as a consequence of the unbrokenSU(3)c×U(1)em gauge symmetry.The Higgs FCNC H¯t c and H¯s b couplings also vanish due to the existence of only one Higgs doublet.Both top FCNC and Higgs FCNC aregenerated at one loop level by charged current exchange,but they are very suppressed by the GIM mechanism.The calculation of the branching ratios for top decays yields the SM predictions[3],[4]:Br(t→Zc)=1.3×10−13,Br(t→γc)=4.3×10−13,Br(t→gc)=3.8×10−11,Br(t→Hc)=5.6→3.2×10−14for M H=115→130GeV.(1) While for Higgs FCNC,calculation within SM leads to:Br(H→¯s b)≈10−7(resp10−9)m H=100(resp200)GeVBr(H→¯t c)≈1.5×10−16(resp3×10−13)m H=200(resp500)GeV(2) Many SM extensions predict that these top and Higgs FCNC can be orders of magnitude larger than their SM values(see[5]for an overview).For the Higgs FCNC,an important class of models where Higgs FCNC appear at tree level are the so called Two Higgs Doublet Model without Natural Flavor Conservation(NFC)2HDM-III[6,7,8,9].In this class of models,the branching ratio of h→¯t c can be larger than10%in some cases[7].In the framework of2HDM with NFC type I and II,top and Higgs FCNC have been studied in [10,11].It was shown that in2HDM-II the Br(Φ→¯t c),Φ=h0or H0,may reach10−5 for CP-even states[11].This rate is almost eight orders of magnitude larger than the SM one.Top and Higgs FCNC couplings have been addressed also in supersymmetry[12,13,14,15,16].In those studies it has been shown that Br(h0→¯s b)can be in the range of10−4-10−3. This rate originates mainly from theflavor violation interactions mediated by the gluino [12,14].In case of MSSM with R parity conservation,the top FCNC coupling t→ch0, can reach10−4branching ratio[15]in case offlavour violation induced by gluino.Hence,Higgs and top FCNC offer a good place to search for new physics,which maymanifest itself if those couplings are observed in future experiments such as LHC or LC [1,2].Therefore,models which can enhance those FCNC couplings are welcome.The aim of this paper is to study Higgs FCNC couplings such asΦ→¯s b,Φ=h0,H0,A0,in the framework of NFC two Higgs Doublet Models type I and II.It is found that the branching ratios of Br(Φ→¯s b),Φ=h0,H0,A0,can be greater than>∼10−5in quite asubstantial region of the2HDM parameters space.Br(Φ→¯t c)requires large tanβand light charged Higgs[11]while Br(Φ→¯s b)requires rather small tanβtogether with light charged Higgs and large soft breaking termλ5.We would like to mention here that due to the isolated top quark signature,Higgs FCNC Φ→¯t c event may be easy to search for experimentally.However,it is very difficult to isolate Higgs FCNCΦ→¯s b events from the background.The paper is organized as follows.In the next section,the2HDM is introduced.Rel-evant couplings are given,theoretical and experimental constraints on2HDM parameters are discussed.In the third section,we will study the effects of2HDM on Br(Φ→¯s b) which are evaluated in2HDM-I and2HDM-II.A comparison between Br(h0→¯s b)and Br(h0→γγ)is also discussed.Our conclusion is given in section4.2The2HDMTwo Higgs Doublet Models(2HDM)are formed by adding an extra complex SU(2)L⊗U(1)Y scalar doublet to the SM Lagrangian.Motivations for such a structure include CP–violation in the Higgs sector and the fact that some models of dynamical electroweak symmetry breaking yield the2HDM as their low-energy effective theory[17].The most general2HDM scalar potential which is both SU(2)L⊗U(1)Y and CP in-variant is given by[18]:V(Φ1,Φ2)=λ1(|Φ1|2−v21)2+λ2(|Φ2|2−v22)2+λ3((|Φ1|2−v21)+(|Φ2|2−v22))2+λ4(|Φ1|2|Φ2|2−|Φ+1Φ2|2)+λ5(ℜ(Φ+1Φ2)−v1v2)2+λ6[ℑ(Φ+1Φ2)]2(3) whereΦ1andΦ2have weak hypercharge Y=1,v1and v2are respectively the vacuum expectation values ofΦ1andΦ2and theλi are real–valued parameters.Note that this potential violates the discrete symmetryΦi→−Φi softly by the dimension two term λ5ℜ(Φ+1Φ2).The above scalar potential has8independent parameters(λi)i=1,...,6,v1andv2.After electroweak symmetry breaking,the combination v21+v22is thusfixed by the√electroweak scale through v21+v22=(2H0H+H−=−ig2M Wsin(β−α)(M2H0−M2H±)(5) h0H+H−=−ig2M Wcos(β−α)(M2h0−M2H±)(7) A0H+G−=−g g2(8)We need also the couplings of scalar boson to a pair of fermions both in2HDM-I and 2HDM-II.In those couplings,the relevant terms are as follows:h0¯t t∝M t cosαsinβ,A0¯t t∝M tsinβ,H0¯bb∝M bsinαtanβ2HDM−I(10)h0¯bb∝M b sinαcosβ,A0¯bb∝M b tanβ2HDM−II(11)(H−¯bt)L∝M b tanβ2HDM−I(12) (H−¯bt)L∝M b tanβ,(H−¯bt)R∝M t1718 Figure1:Generic contribution toΦ→f1f2in SM d1→d10,in2HDM d11→d18iv)Unitarity and perturbativity constraints on scalar parameters:It is well known that the unitarity bounds coming from a tree-level analysis[25,26]put severe constraints on all scalar trilinear and quartic couplings.The tree level unitarity bounds are derived with the help of the equivalence theorem,which itself is a high-energy approximation where it is assumed that the energy scale is much larger than the Z0and W±gauge-boson masses.We will use,instead of unitarity constraints,the perturbativity constraints by assuming that allλi satisfy:|λi|≤4π.(14) Those perturbative constraints on theλi allow us to investigate a larger parameter space than the one allowed by unitarity constraints.We would like to mention also that when performing the scan over the2HDM parameters space,we realize that for some points the widthsΓΦof the scalar particles become bigger than their corresponding masses:ΓΦ≥MΦ(Φ=h0,H0,A0,H±).This happens both when we impose tree level unitarity constraints and/or perturbativity constraints.Thewidth becomes large specially when the pure scalar decays like H0→h0h0,H0→H+H−, h0→H+H−,H0→A0A0and h0→A0A0are open.Wefind it is natural to add to the above constraints the requirement that the width of the scalar particles remains smaller than the mass of the corresponding particles:ΓΦ<MΦ(15) From the experimental point of view,the combined null–searches from all four CERN LEP collaborations derive the lower limit M H±≥78.6GeV(95%CL),a limit which applies to all models in which Br(H±→τντ)+Br(H±→c¯s)=1.For the neutral Higgs bosons, OPAL collaboration has put a limit on h0and A0masses of the2HDM.They conclude1e-171e-161e-15 1e-14 1e-13 1e-12 1e-11 1e-10 1e-09M H (GeV)2 Γ(H-->tc)(GeV)2 x Br(H-->tc)1e-101e-091e-08 1e-07 1e-06 1e-05 0.0001 0.001M H (GeV)2 Γ(H-->sb)(GeV)2 Br(H-->sb)Br(H-->µ µ)Figure 2:SM width and Branching ratio for H →¯t c (left)and H →¯s b (right)as a functionof Higgs mass.that the regions 1<∼M h <∼44GeV and 12<∼M A <∼56GeV are excluded at 95%CL independent of αand tan β[27].For simplicity we will assume that all scalar particles masses are >∼90GeV.3Higgs FCNC in 2HDM3.1Higgs FCNC in SMBefore presenting our results in 2HDM,we would like to give the Branching ratio of H →¯t c and H →¯s b in the SM.To our best knowledge,the first calculation for Br (H →¯s b )has been carried out in [28].However,in [28],numerical results have been given only for a very light Higgs boson M H =9GeV.Recently a new estimation,using dimensional analysisand power counting,has appeared both for Br (H →¯s b )[14]and Br (H →¯tc )[11].We refer the reader to [11,14]for more details on those estimations.Here we present exactresult based on diagrammatic calculations both for Br (H →¯s b )and Br (H →¯tc ).We give numerical results for the width as well as for the branching ratio.The Feynman diagrams contributing to those process in SM are depicted in Fig .(1)d 1→d 10.In the case of H →¯t c ,in Fig.(1)(f 1,f 2)=(t,c )and f ′i =d,s,b ,while for H →¯s b(f 1,f 2)is (b,s )and f ′i =u,c,t .The full loop calculation presented here is done with the help of FormCalc [29].FF and LoopTools packages [30]are used in numerical analysis.The numerical results shown in eqs.(1,2)is derived by FormCalc [29].In the SM,as expected,the branching ratio of H →¯t c and H →¯s b are very suppresseddue to GIM mechanism.The branching ratio is very small in both cases for higher Higgs mass M H ≥2M Z where H →W +W −and H →Z 0Z 0are open.Both in SM and 2HDM,the decay widths ΓSM Φand Γ2HDMΦof scalar particles:Φ=H SM ,h0,H0,A0and H±are computed at tree level as follows:ΓSMΦ=fΓ(Φ→f¯f)+Γ(Φ→V V)Γ2HDMΦ=fΓ(Φ→f¯f)+Γ(Φ→V V)+Γ(Φ→V H i)+Γ(Φ→H i H j)(16)QCD corrections toΦ→f¯f andΦ→{gg,γγ,γZ,V∗V∗,V V∗,V∗H i}decays are not included in the widths.The decay widths of the Higgs bosons are taken from[31]. For a Higgs mass heavier than250GeV,we get branching ratio of the order10−14→10−12(resp10−10→10−9)for H→¯t c(resp H→¯s b).In the case of H→¯s b,the branching ratio is enhanced for Higgs boson mass of the order M H≈100→120GeV where the width of the Higgs is very narrow.We have plotted in Fig.(2)both the decay width and the branching ratios of H→¯t c(left plot)and H→¯s b (right plot)as well as the branching ratio of H→µ+µ−.As it can be seen from the right plot Br(H→¯s b)is two orders of magnitude smaller than Br(H→µ+µ−).Since the decay width of H→¯t c is very suppressed,the threshold for t¯t production is absent in Fig.2(left).The situation is slightly different for H→¯s b where the decay width of H→¯s b is about6order of magnitude larger than decay width of H→¯t c.From the right plot of Fig.(2)one can see that the Br of H→¯s b is smaller once the t¯t threshold has been passed.3.2h0→¯s bTurning now to the2HDM Higgs bosons FCNC couplingsΦ→¯s b,Φ=h0,H0,A0.The Feynman diagrams are depicted in Fig.(1).The amplitude is sensitive to theΦH+H−and ΦH±G∓couplings through diagrams d12,13,14as well as to theΦt¯t and(H−¯bt)L,R couplings through diagrams d11,12,13,14.In2HDM,it is expected that the dominant contribution to the amplitude ofΦ0→¯s b comes from diagram d12.The amplitude of d12is proportional to the trilinear Higgs couplingΦ0H+H−and is given by(Φ=h0,H0):M d12=Φ0H+H−αV tstan2βM b2u(M b)(17)where we have neglected the strange quark mass.In the conventions of[29],the arguments of the Passarino-Veltman functions C i are{M2b,M2s,M2Φ,M2H±,M2t,M2H±}.The Yukawa coupling Y b of the bottom is model dependant and is given by Y b=−1/tanβ(resp Y b= tanβ)for2HDM-I(resp2HDM-II).In2HDM-I,1+tanβY b=0,the amplitude of d12is enhanced by M2tλ5−5−6H +M (GeV)H +M (GeV)λ5Figure 3:Contours for 2×Br (h 0→¯s b )in 2HDM-II tan β=0.3(left),tan β=1.5(right)in the (M H ±,λ5)plane with M h =110GeV,M H =180GeV,sin α=0.1and M A 0=M H ±We also give other numerical results for specific 2HDM parameters where Br (h 0→¯s b )and Br (H 0→¯s b )get their maximum values without violating δρand perturbativity constraints.We show in Fig.(3)contour plots for Br (h 0→¯s b )in 2HDM-II tan β=0.3(left)and tan β=1.5GeV (right)in the (M H ±,λ5)plane.λ5is varied in the perturbative range |λ5|<4π.The other inputs are M h =110GeV,M H =180GeV,sin α=0.1and M A 0=M H ±.The width Γh 0is computed at tree level according to eq.(16).Since the mass of h 0is taken at 110GeV,only light fermions contribute to Γh 0and so the width is very narrow and is of the order 57×10−4(resp 83×10−5GeV)at tan β=0.3(resp tan β=1.5).Such narrow width could enhance the branching ratio Br (h 0→¯s b ).We would like to mention first that for this set of parameters,the perturbativity of scalar quartic couplings λi is violated around λ5>∼5.5.We get |λ1|>4πfor tan β=0.3,while for tan β=1.5there is no such bound.Large branching ratios can be obtained for light charged Higgs mass.This can be seen in the left panel black and blue areas of Fig.(3)which correspond to small tan β=0.3and large |λ5|.In those areas the coupling h 0H +H −gets its largest value (see also Fig.(4)).In this case one can obtain branching ratio in the range:10−4<Br (h 0→¯s b )<6×10−4for M H ±<200GeV ,λ5<∼−1.2and λ5>∼3.For charged Higgs mass greater than 200GeV,there is also a region where the branching ratio can be in the range 10−5→10−4.This can be achieved by taking large and negative λ5<∼−1.In the case of positive λ5andM H ±>∼250GeV,the branching ratio decreases to a value <∼10−5.When tan β=1.5,the coupling h 0t ¯tis reduced,and we are left only with a small region where the branching ratio Br (h 0→¯s b )is of the order 10−5→10−4for M H ±<∼250GeV and large |λ5|>∼5.In both plots (left and right),the coupling h 0H +H −reaches itsM h M H M H±M Aλ52×Γ¯s b2×Γγγ−.98.210−37×10−46×10−36×10−35×10−35×10−3140 34011010064×10−62×10−610−410−4−.98.4610−32×10−47×10−32×10−32×10−37×10−3115 2501101905×10−45×10−510−410−4.18.19×10−410−32×10−43×10−3.525×10−2Table1:Maximum Branching ratios of h0→¯s b in2HDM-I and II and corresponding 2HDM parameters,all masses and decay width are in GeV.In Br and widthsΓcolumns, the upper row is for2HDM-I and the down row is for2HDM-IIminimal value in the region whereλ5≈0→2,which explains why the branching ratio is so small in this region.Now we turn to the case where M H±=M A,δρ=0.We have performed a system-atic scan over the full2HDM parameters space taking into accountδρand perturbativity constraints.The maximum branching ratios found for h0→¯s b in2HDM-I and II are displayed in table1.We show not only width and Br of h0→¯s b but also the width and Br of h0→γγfor comparison.The total width of the HiggsΓh0is also given.WhenΓh0 becomes comparable to the width of h0→¯s b and/or h0→γγ,those decays widths have to be included in the total widthΓh0in order to compute the Br¯s b and Brγγ.Thefirst three columns of table1are for2HDM parameters.From4th to8th columns we give Br and widths.In those columns,the upper row is for2HDM-I and the down row is for2HDM-II.In2HDM-I,Br(h0→¯s b)of the order10−3can be reached in the limit sinα→−0.98 (α→−π/2)and small tanβ≤0.5.In fact,this limit(α→−π/2)is very close to fermiophobic scenarioα=±π/2.In the fermiophobic limit,all couplings of h0to down quarks and leptons are suppressed eq.(10).In this limit,h0¯t t is also suppressed eq.(9).The width of light Higgs h0(M h<160GeV)is then very tiny in the limit sinα→−0.98.This tiny width together with large h0H+H−are the sources of enhancement of the Br(h0→¯s b) to10−3level.This can be seen in thefirst,second and third lines of table1In2HDM-II,the couplings of h0to down quarks and leptons are suppressed for sinα≈0.1 eq.(11).Hence,the width of light Higgs(M h<160GeV)is very tiny in the limit sinα≈0.1.Moreover,in this limit,the coupling h0¯t t is enhanced in both models2HDM-I and II.The decay widthΓ(h0→¯s b)which was≈10−6for sinα=−0.98is of the order ≈10−5for sinα=0.1.Consequently,the Br(h0→¯s b)reaches10−3.1e-081e-071e-06 1e-05 0.0001 0.001 0.01 0.1B r λ5h 0-->sb: 2HDMI h 0-->sb: 2HDMII h 0-->γγ: 2HDMI h 0-->γγ: 2HDMII1e-111e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01B rλ5h 0-->sb: 2HDMI h 0-->sb: 2HDMIIh 0-->γγ: 2HDMIh 0-->γγ: 2HDMII Figure 4:2×Br (h 0→γγ)and 2×Br (h 0→¯s b )in 2HDM-I and II,m H ±=100GeV,tan β=0.3(left)and tan β=5(right).All the other parameters are the same as in Fig.(3).In this scenario,as one can see from table 1,the Br(h 0→γγ)in 2HDM-I is 2×10−4which is smaller than Br(h 0→¯s b )=9×10−4.This is mainly due to the fact that the trilinear coupling h 0H +H −is very suppressed in this scenario (see more details in next section).As one can see from the last line of the table 1,there exist also values of sin α=0.18,far from fermiophobic scenario but with small λ5=0,where Br(h 0→¯s b )can be of the order 10−3.3.3Can h 0→¯s b compete with h 0→γγ?It is well known that the decay h 0→γγis loop induced and so is suppressed.In the SM,the branching ratio Br (H SM →γγ)is about ≈10−3for Higgs mass in the range M H =100→160GeV.Hence,with maximum branching ratio for h 0→¯s b of the order 1×10−4→6×10−4in 2HDM-I or II,it is legitimate to compare h 0→γγand h 0→¯s b in 2HDM-I or II.Of course,even if h 0→¯s b and h 0→γγhas a competitive branching ratio,we should keep in mind that h 0→γγdecay has a clear signature while the FCNC decay h 0→¯s b has not.We illustrate in Fig.(4)the branching ratio for h 0→¯s b and h 0→γγboth in 2HDM-I and II.The charged Higgs mass is fixed to 100GeV.It is clear that in the case tan β=0.3h 0→γγis about one order of magnitude bigger than h 0→¯s b .While,in the case of tan β=5h 0→γγis more than four orders of magnitude bigger than h 0→¯s b .This is because at tan β=0.3(resp tan β=5)the W loop are suppressed by a factor h 0W +W −∝sin(β−α)≈0.2(resp enhanced by h 0W +W −∝sin(β−α)≈0.96).All the dips observed in the plots correspond to the minimum of the coupling h 0H −H +.Those dips are not located at the same λ5,this is due to a destructive interference with others diagrams.H +M (GeV)λ5H +M (GeV)Figure 5:Contours for 2×Br (H 0→¯s b )in 2HDM-II in the plan (M H ±,λ5)M H =140GeV (left),(M H ±,M H )λ5=5(right)with tan β=0.3,M h =110GeV,sin α=0.95and M A 0=M H ±When h 0H −H +coupling is very suppressed,it may be possible that the Br(h 0→¯s b )could be higher than Br(h 0→γγ)as it can be seen both in the left plot of Fig.(4)for λ5=2.5and in table 1for sin α=0.1in 2HDM-II.However,even if Br(h 0→γγ)and Br(h 0→¯s b )become comparable,we should keep in mind that h 0→γγhas a very clear signature while h 0→¯s b does not.An interesting feature of the 2HDM-I,is its fermiophobic scenario.The light CP-even Higgs h 0of the 2HDM-I is fermiophobic in the limit α→π/2,all h 0couplings to fermions vanishes for α=π/2[32,18].If h 0,with a mass in the range 100→160GeV,is fermiophobic the dominant decay mode is h 0→γγ.It has been shown in Ref.[33]that in the fermiophobic limit,the branching ratio of the one loop induced decay h 0→¯bb ∗is below 10%→30%.As the decay h 0→¯s b is concerned,we have checked by systematic scan that in the fermiophobic limit,the decay width of h 0→γγis more than one order of magnitude bigger than the width of h 0→¯s b .3.4H 0→¯s bWe now discuss the heavy CP-even decay H 0→¯s b .Our numerical results are shown inFig.(5).To maximize the coupling H 0¯tt ,we choose of course small tan β≈0.3and large sin α≈0.95.In the right plot of Fig.(5),we show contour plots for Br (H 0→¯s b )in the plane (M H ±,λ5)for M H =140GeV.For CP-even Higgs mass 140GeV,H 0→W +W −,H 0→ZZ ,H 0→¯tt ,H 0→A 0Z and H 0→H i H j are not yet open,and so the width is narrow.In particular,for the set of parameters fixed here:M h =110GeV,sin α=0.95M h M H M H±M Aλ52×Γ¯s b2×Γγγ.1.510−34×10−53×10−210−34×10−48×10−3100 155115103-610−66×10−72×10−52×10−5.08.452×10−38×10−5.733×10−23×10−48×10−3100 1251151035×10−46×10−52×10−42×10−4.9.110−310−32×10−42×10−3.515×10−2100 1451151032×10−43×10−52×10−43×10−41e-121e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-050.00010.001B r (A 0-->s b )M Atan β=0.2tan β=0.3tan β=1.5tan β=60 0.0010.010.1 1 10 100 1000ΓA 0 (G e V )M Atan β=0.2tan β=0.3tan β=1.5tan β=60Figure 6:2×Br (A 0→¯s b )(left)and CP-odd A 0width ΓA 0(right)as function of M A for several values of tan βto down quarks and leptons are suppressed for sin α≈0.1(resp |sin α|≈0.9).In those cases the total Higgs width is very tiny and so the branching ratio of H 0→¯s b is enhanced.Of course,Br(H 0→¯s b )reach 10−3only for light charged Higgs,which is strongly disfavored by b →sγconstraint [24]in 2HDM-II.From table 2,one can see also that in 2HDM-II and for sin α≈{0.9,−0.98}the Br(H 0→¯s b )and Br(H 0→γγ)are of comparable size.This is again mainly due to the suppression of the coupling H 0H +H −in those limits.As in the case of light CP-even Higgs h 0,there exist values of sin α=−0.58far from fermiophobic limit with small λ5=0where Br(H 0→¯s b )can reach 10−3.3.5A 0→¯s bLet us now look at 2HDM contribution to A 0→¯s b .Since A 0is CP-odd,it does not couple to a pair of charged Higgs.The only pure trilinear scalar coupling which contributes to A 0→¯s b is A 0H ±G ∓eq.(8).Unlike the couplings H 0H +H −and h 0H +H −eqs (4,6),which depend both on Higgs masses,tan βas well as the soft breaking term λ5,the couplingA 0H ±G ∓depends only on the splitting M 2H ±−M 2A .As mentioned above,such splitting should not be too large,otherwise the δρconstraint is not satisfied.As one can readfrom eqs.(9,13),the couplings A 0¯t t and (H −¯bt )R are proportional to M t /tan β.Henceenhancement is expected at small tan β.As we stressed before,our 2HDM parameters in this case are:tan β,M A and M H ±.For simplification,we use the MSSM sum-rules to fix charged Higgs mass and αby using tan β,CP-odd mass M A and a SUSY scale which we take at 1TeV.CP-odd mass will be varied from 100GeV to 600GeV without worrying about perturbativity.tan βis taken to be >∼0.1.We present our numerical results for A 0→¯s b in 2HDM-II in Fig.(6).As can be seenfrom the left plot,the Branching ratio Br(A0→¯s b)is greater than10−5only for small tanβ≈0.1→0.35and light M A and M H±.For light M A<∼200GeV and low tanβ<∼1, the width of A0is still small and so the branching ratio is enhanced.For M A>∼200GeV, the decay A0→h0Z is open and the decay widthΓA0increases.Therefore,the branching ratio is reduced.Note that for tanβ=0.2,we cut offthe curve at M A≈400GeV where the widthΓA0starts to be greater than M A.At large tanβ,due to the bottom Yukawa coupling,both the partial widthΓ(A0→¯s b)and total widthΓA0are enhanced,and the branching ratio is saturated in the range[10−6,10−5].The situation is almost the same in2HDM-I.4ConclusionsIn the framework of the2HDM with naturalflavor conservation,we have studied various Higgs FCNCΦ→¯s b.The study has been carried out taking into account the experimental constraint on theρparameter and also perturbativity constraints on all the scalar quartic couplingsλi.Numerical results for the branching ratios have been discussed.We empha-sized the effect coming from both top and bottom Yukawa couplings and pure trilinear scalar couplings such as h0H+H−and H0H+H−.We have shown that,in2HDM-I and2HDM-II,the branching ratios of Higgs FCNC {h0,H0,A0}→¯s b are enhanced to the range of10−4→7×10−4for small tanβ,rather light charged Higgs boson and large soft breaking termλ5.The branching ratio of Br({h0,H0}→¯s b)can be pushed to10−3level when sinαis close to fermiophobic limit(sinα≈−0.98) or sinα≈0.1and even for sinαfar from those limits but with smallλ5=0. Charged Higgs mass of2HDM-I is not constrained by b→sγ,Br({h0,H0}→¯s b)can be of the order10−4→10−3for light charged Higgs which is comparable to size of SUSY pre-dictions[12,14].Those branching ratios rates,could still leads to large number of events at LHC[11].In2HDM-II with b→sγconstraint,branching ratios of{h0,H0}→¯s b are smaller than 10−5(resp10−4)for tanβ>1(resp tanβ<1).In the case of light CP-even m h0≈100→160GeV,we have also shown that the branching ratio of Br(h0→¯s b)is well below Br(h0→γγ)in most of the case.This is also the case in the fermiophobic scenario of2HDM-I.One interesting scenario is that both Br(h0→γγ)and Br(h0→¯s b)develop a dips for someλ5(see Fig.4).Those dips are not located at the sameλ5due to the presence of diagrams which contribute to h0→¯s b but not to h0→γγ.The dip for Br(h0→¯s b)is located forλ5=1while for Br(h0→γγ)it is located forλ5≈2.5.Forλ5≈2.5,we are already away from Br(h0→¯s b)dip,the Br(h0→¯s b)is slightly higher than Br(h0→γγ).Acknowledgments This work was done within the framework of the Associate Scheme of ICTP.Thanks to Thomas Hahn for his help.We also want to thank Andrew Akeroyd for discussions and for reading the manuscript.References[1]M.Beneke,I.Efthymipopulos,M.L.Mangano,J.Womersley(conveners)et al.,reportin the Workshop on Standard Model Physics(and more)at the LHC,Geneva,hep-ph/0003033[2]J.A.Aguilar-Saavedra et al.[ECFA/DESY LC Physics Working Group Collabora-tion],arXiv:hep-ph/0106315.[3]G.Eilam,J.L.Hewett and A.Soni,Phys.Rev.D44(1991)1473[Erratum-ibid.D59(1999)039901].[4]B.Mele,S.Petrarca and A.Soddu,Phys.Lett.B435,401(1998)[5]J.A.Aguilar-Saavedra,Acta Phys.Polon.B35(2004)2695[arXiv:hep-ph/0409342].E.W.N.Glover et al.,Acta Phys.Polon.B35(2004)2671[arXiv:hep-ph/0410110].0409342,0410110[6]T.P.Cheng and M.Sher,Phys.Rev.D35,3484(1987).[7]W.S.Hou,Phys.Lett.B296,179(1992).[8]D.Atwood,L.Reina and A.Soni,Phys.Rev.D55,3156(1997)[9]D.Atwood,L.Reina and A.Soni,Phys.Rev.D53,1199(1996)[arXiv:hep-ph/9506243].W.S.Hou,G.L.Lin and C.Y.Ma,Phys.Rev.D56,7434(1997). 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