a rXiv:h ep-ph/65335v417Oct27FCNC Top Quark Decays in Extra Dimensions G.A.Gonz´a lez-Sprinberg ∗Instituto de F´ısica,Facultad de Ciencias,Universidad de la Rep´u blica Montevideo,Uruguay R.Mart´ınez †and J.-Alexis Rodriguez ‡Departamento de F´ısica,Universidad Nacional de Colombia Bogot´a ,Colombia Abstract The flavor changing neutral top quark decay t →cX is computed,where X is a neutral standard model particle,in a extended model with a single extra dimension.The cases for the photon,X =γ,and a Standard Model Higgs boson,X =H ,are analyzed in detail in a non-linear R ξgauge.We find that the branching ratios can be enhanced by the dynamics originated in the extra dimension.In the limit where 1/R >>m t ,we have found Br (t →cγ)≃10−10for 1/R =0.5T eV .For the decay t →cH ,we have found Br (t →cH )≃10−10for a low Higgs mass value.The branching ratios go to zero when 1/R →∞.I.INTRODUCTIONFlavor changing neutral currents(FCNC)are very suppressed in the standard model (SM):there are no tree level contributions and at one loop level the charged currents operate with the Glashow-Iliopoulos-Maiani(GIM)mechanism.The branching ratio for top quark FCNC decays into charm quarks are of the order of10−11for t→cg and10−13for t→cγ,(Z)in the framework of the SM[1,2].This suppression can be traced back to the loop amplitudes:they are controlled by down-type quarks,mainly by the bottom quark, resulting in a m4b/M4W factor which can be compared to the enhancement factor that appears in the b→sγprocess where the top quark mass m t is involved instead of m b in this factor. This fourth power mass ratio is generated by the GIM mechanism and is responsible for the suppression beyond naive expectations based on dimensional analysis,power counting and Cabibbo-Kobayashi-Maskawa(CKM)-matrix elements involved.The top quark decay into the SM Higgs boson is even more suppressed[1,2]:Br(t→cH)∼10−13−10−15forM Z≤M H≤2M W.These rates are far below the reach of any foreseen high luminosity collider in the future.The highest FCNC top quark rate in the SM is t→cg,but this value is still six orders of magnitude below the possibility of observation at the LHC.The discovery of these FCNC effects would be a hint of new physics because of the large suppression in the SM.These FCNC decay modes can be strongly enhanced in scenarios beyond the SM,where some of them could be even observed at the LHC or ILC.New physics effects in extended Higgs sector models,SUSY and left-right symmetric models were studied in references[1-5].For example,in various SUSY scenarios the branching ratios can go up to the value10−5for the decay t→cg.Also,virtual effects of a Z′gauge boson on these rare top quark decays were studied.The decay t→cγhas been analyzed in reference[6],it has been shown that B(t→cγ)is at the10−6level in topcolor assisted technicolor models for m′Z=1T eV,which would allow the detection of this process at future colliders.On the other hand,the use of effective Lagrangians in parameterizing physics beyond the SM has been studied extensively in FCNC top quark couplings and decays[7,8,9].This formalism generates a model-independent parameterization of any new physics characterized by higher dimension operators.Under this approach,several FCNC transitions have been also significantly constrained:t→cγ[10,11],t→cg[11,12],l i→l jγ[13]and H→l i l j [14].New physics effects have also been introduced in models with large extra dimensions(ED)[15].In recent years,these models have been a major source of inspiration for beyond theSM physics in the ongoing research.In these scenarios the four dimensional SM emergesas the low energy effective theory of models living in more than four dimensions,wherethese extra dimensions are orbifolded.The presence of infinite towers of Kaluza-Klein(KK)modes are the remanent of the extended dimensional dynamics at low energies.The sizeof the extra dimensions can be unexpectedly large,with1/R at the scale of a few T eVwithout contradicting the present experimental data[16].Then,if these KK-modes arelight enough,they could be produced in the near future at the next generation of colliders.Scenarios where all the SMfields,fermions as well as bosons,propagate in the bulk areknown as”universal extra dimensions”[17,18].In these theories the number of KK-modesis conserved at each elementary vertex and the coupling of any excited KK-mode to twozero modes is prohibited.Then the constraints on the size of the extra dimensions obtainedfrom the SM precision measurements are less stringent than in the case where there is noconservation of the KK particles(non universal extra dimensions).The impact of the new physics coming from UED models has been widely studied andconstraints on the parameter1/R have been obtained.Analysis of the precision electroweakobservables led to the lower bound1/R 700−800GeV for a light Higgs boson mass andto1/R 300−400GeV for a heavy Higgs boson mass[19].On the other hand,using theprocess b→sγ,the resulting bound on the inverse compactification radius is1/R 250GeV[20].Moreover,a recent analysis making use of the exclusive branching ratio B→K∗γshowsthat under conservative assumptions1/R 250GeV[21].And from the inclusive radiative ¯B→Xγdecay,a lower bound on1/R 600GeV at95%C.L.can be obtained and it is sindependent of the Higgs boson mass value[22].Contributions from UED models have beenconsidered on several FCNC processes,reference[23]has found that the processes K L→π0e+e−,∆M s,K+→π+ν¯ν,K L→π0ν¯ν,B d,s→X d,sν¯ν,K L→µ+µ−,B d,s→µ+µ−,B→X sµ+µ−for1/R≈300GeV are enhanced relative to the SM expectation and the processes B→X sγ,B→X s g,ǫ′/ǫare suppressed respect to the SM.In general,the present data on FCNC processes are consistent with1/R 300GeV[23,24].Exclusive B→K∗l+l−,B→K∗ν¯νand B s decays[21,25]have been studied in the framework of the UED scenario and also rare semileptonicΛb decays[26].In this paper we study the FCNC decays of the top quark in a universal extra dimensiontheory where all the SMfields live infive dimensions.In particular,we compute the t→cγand t→cH decay modes in a non linear Rξ.This gauge has the advantage of a reduced number of Feynman diagrams as well as simplified Ward identities.These facts facilitates and clarify the calculation.The paper is organized as follows:in section II we present the general framework for the five dimensional Lagrangian and derive the corresponding four dimensional Lagrangian and Feynman rules.In section III and IV we compute the decays mode t→cγand t→cH respectively,and discuss the hypothesis implicit in the calculation.Finally,in section V we present some conclusions.In the Appendix(section VI)we show the terms in the Lagrangian that are important for the Feynman rules in our calculation.II.THE MODELWe begin presenting the SM Lagrangian infive dimensions;let x=0,1,2,3be the normal coordinates and x4=y thefifth one.Thefifth extra dimension is compactified on the orbifold S1/Z2orbifold of size R which is the compatification radius.We consider a generalization of the SM where the fermions,the gauge bosons and the Higgs doublet propagate in thefive dimensions.The Lagrangian L can be written asL= d4x dy(L A+L H+L F+L Y)(1) withL A=−14B MN B MN,L H=(D MΦ)†D MΦ−V(Φ),L F= U(iΓM D M)U+Q Y uΦc U−gamma matricesΓM areΓµ=γµandΓ4=iγ5with the metric tensor given by g MN=(+,−,−,−,−).The matterfields Q,D and U are fermionic four components spinors withthe same quantum numbers as the corresponding SMfields.To simplify the notation wehave suppressed the SU(2)and color indices.The standard and charge conjugate doubletstandard Higgsfields are denoted byΦ(x,y)andΦc(x,y)=iτ2Φ∗(x,y); Y u,d are the Yukawa matrices in thefive dimensional theory responsible for the mixing of different families whoseindices were suppressed in the notation for simplicity.We have not included in Eq.(2)theleptonic sector nor the SU(3)c dynamics because it is not relevant for our proposes.The lowenergy theory will only have zero modes forfields that are even under Z2symmetry:this isthe case for the Higgs doublet that we choose to be even under this symmetry in order tohave a standard zero mode Higgsfield.The Fourier expansions of thefields are given by:Bµ(x,y)=1πRB(0)µ(x)+√πR∞n=1B(n)µ(x)cos ny2R ,Q(x,y)=1πRQ(0)L(x)+√πR∞n=1 Q(n)L(x)cos ny R ,U(x,y)=1πRU(0)R(x)+√πR∞n=1 U(n)R(x)cos ny R ,D(x,y)=1πRD(0)R(x)+√πR∞n=1 D(n)R(x)cos ny R .(3)The expansions for Bµand B5are similar to the expansions for the gaugefields and the Higgs doublet(but this last one without theµor5Lorentz index).It is by integrating thefifth y component in Eq.(1)that we obtain the usual interaction terms and the KK spectrum for ED models.The interaction terms relevant for our calculation will be written in a non-linear Rξgauge(see for details[27]and thefirst reference in[5]).For example,in this gauge there is no mixing between the gauge bosons and the charged and neutral unphysical Higgsfields. Besides,the interaction terms are simplified in such a way that there are no trilinear termssuch as W+µG−W Aµ,where G−Wis an unphysical Higgsfield.We are interested in the thirdfamily of quarks and Q(n)t and Q(n)bare the upper and lower parts of the doublet Q.Similarly,the U(n)t and D(n)bare the KK modes of the usual right-handed singlet top and bottom quarks, respectively.There is a mixing between the mass and gauge eigenstates of the KK top quarks(Q(n)t and U(n)t)where the mixing angle is given by tan(2αn t)=m t/m n with m n=n/R.For the b quark the mixing is quite similar,but at leading order the only masses that remain are m t and m n and in this limit the mixing angle is zero.This leads to the spectrumm Q nb =m n and m Q nt=m U n=M W 41−m2VΓ(t→cγ)SM≃( n(M W/m n)2)2The sum on the KK tower of excited states can be evaluated as we will explain later in the text and we obtainΓ(t →cγ)ED36 (M W /(1/R ))26c 2W π2m 2tǫ∗αk µ(2 F R 2m t )¯u c σαµP R u t (11)where c W is the cosine of the weak mixing angle and the form factors F R 2and F R 2are related by F R 2m t =eg 229π2s 4w c 4w|F R 2|2≈4.8×10−7|F R 2|2.(13)In order to perform the one loop calculation,we consider two scenarios.Thefirst one, when the mass of the excited states associated to the quarks from the three low-energy families are quasi-degenerated at tree level,without any kind of radiative corrections to KK masses.In this case,when the excitations coming from the other quarks are taken into account,the transition amplitude for the process t→cγtakes the form∝ i=d,s,b V ti V∗ci1m2b(n/R)2FIG.1:Diagrams contributing to the t→cγdecay in the non linear Rξgauge.Diagrams with G−(n)W ¯D(n)bt(0)Lin the loop are proportional toλ2b and then to m2b and can beneglected.The leading contributions of type1diagrams(seefigure1)to the decay come from the following particles circulating in the loop:W(n)5Q(n)bQ(n)b,W(n)µQ(n)bQ(n)b,G(n)WQ(n)bQ(n)b,G(n)WD(n)bD(n)b.(15)where the external photon is coupled to the fermion in the loop.The sum of all these diagrams gives,for the form factor F R2,the following expression2F R2=216π2 10dz1−z0dw1−w−zM2W ,(16)where the factor X comes from the dimensional regularization tricks of the product of inverse propagators of the particles circulating in the loopX=m2t z2+m2t wz−m2t z+m2n.(17) The main contribution of type2diagrams(seefigure1)is the one with KK excitation of the standard model gauge boson or a scalarfield circulating in the loop which are coupled to the external photon:W(n)µW(n)µQ(n)b,W(n)5W(n)5Q(n)b,G(n)WG(n)WD(n)b,G(n)WG(n)WQ(n)b,(G(n)W W(n)µQ(n)b+W(n)µG(n)WQ(n)b).(18)These terms contribute to the F R 2form factor with the following expression2F R 2=−g 2e iX4(2−3w −4z +2wz +2z 2)−(1−w −z ) (w +z )−(w +z −1/2)m 2bm 2n :m 2b M 2W .(20)The numerical estimation of all these contributions is straightforward.All the excited mass terms are proportional to n/R ,except for the electroweak correction coming from the sym-metry breaking.From the numerical point of view this correction does not change the results and can be neglected without modifying the final estimates.Based on these hypothesis,wecan also take X ≃m 2nand,then,the sum over all the KK excited states can be easily done,asn 1 n 61R 2(21)where in any numerical estimate 1/R ≃O (1)TeV.Thus,within this approximation,the sum over all the excited KK states is equivalent to multiply the results obtained for the first KK excited state by the factor π2/6.The sum of all contributions using equations(16)and (19)gives,2F R 2=g 2e 16π2 10dz 1−z 0dw 12M 2W ≈g 2e16π2 −5M 2W .(22)By using equations (13)and (12),the numerical value for the decay width isΓ(t →cγ)=1.65×10−10GeV ,(23)for R−1=0.5TeV,and the branching fraction isBr(t→cγ)≡Γ(t→cγ)u c(p)(F L P L+F R P R)u t(k),(25) where F L and F R are form factors.In our notation we identify the external scalar Higgs H with the zero mode Higgsfield h(0).From this amplitude we can compute the decay widthΓ(t→cH)=m tm2t 2 |F L|2+|F R|2 .(26)FIG.2:Feynman diagrams for the t→c H decay in extra dimensions.The leading diagrams that contribute to the decay are shown in Figure2.The leading group of type1diagrams infigure2is the one with KK excitations of the SM quarks circulating in the loop which are coupled to the external higgs:W(n)µQ(n)bD(n)b,G(n)WQ(n)bQ(n)b,W(n)5Q(n)bD(n)b.(27)In this case,the external Higgs is coupled to the excited quark Q b generating a flavorchanging ¯Q(n )D b h (0)(see the appendix),which is proportional to the bottom quark mass m b .The contributions to the F L ,F R form factors are of the order of zero at leading order,i .e .F L =F R ≈0.The leading diagrams of type 2in figure 2is the one with KK excitations of the standard model gauge bosons and scalar fields circulating in the loop which are coupled to the external higgs boson h (0):W (n )5W (n )5Q (n )b ,G (n )W G (n )W Q (n )b ,W (n )µW (n )µQ (n )b ,W (n )µG (n )W Q (n )b ,G (n )W W (n )µQ (n )b ,(W (n )5G (n )W Q (n )b +G (n )W W (n )5Q (n )b ),(28)and these contribute to the form factor with the following expressions:F L =g 3V tb V ∗cb iX{2(1−w )m t M W − (z −1)(w +z )m 2t +w (z −2)m 2hm t16π2 10dz 1−z 0dw 16m 2n V tb V ∗cb i M W+7m 2h m t6m 2n V tb V ∗cb iimportant contribution to the loop correction comes from the excited KK states associated to the third generation.The results show a branching ratio that is above the SM one.The branching ratios for these two decay widths are of the order of10−10.There is a strong dependence on top quark mass m t in the amplitude of the t→cH process,it is coming from the type2diagrams infigure2with an excited scalar in the loop, resulting in a m3t/M3W factor.When we take the limit m n≫m t,wefind that the decaywidths for the t→cγ,(H)processes are decoupled respect to the new scale1/R and they go to zero.Considering that the excited states of the quarks are quasi-degenerated and the unitarity of the CKM matrix,the amplitudes for theflavor changing decay t−c due to the contribution of the excited states of the quarks,are suppressed by the factor(m b/(n/R))2, and the predicted values are smaller than the SM predictions.VI.ACKNOWLEDGEMENTSR.Martinez acknowledge thefinancial support from Fundacin Banco de la Republica. 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