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Effective theories and constraints on new physics R. Martinez, N. Poveda and J-Alexis Rodriguez DepartamentodeFísica,UniversidadNacionaldeColombia Bogota,Colombia 3 0 Abstract. 0 2 Anomalousmomentsofthetopquarkarisesfromoneloopcorrectionstotheverticest¯tgandt¯tg . We study these anomalouscouplingsin differentframeworks:effective theories, Standard Model n and 2HDM. We use available experimental results in order to get bounds on these anomalous a couplings. J 1 2 INTRODUCTION 1 v 6 The top quark is the heaviest fermion in the standard model (SM) with a mass of 7 174.3 5.1GeV1.In theframeworkoftheSM,thecouplingsofthetopquark arefixed 1 ± 1 bythegaugesymmetry.Anomalouscouplingsbetween thetopquark andgaugebosons 0 might affect the top quark production at high energies and also its decay rate. Precisely 3 measured quantities with virtual top quark contributions will yield further information 0 / regardingthesecouplings. h Since the top quark mass is so heavy, it is expected that its physics may be different p - from the lighter fermions and that the top quark might couple quite strongly to the p electroweak symmetry breaking sector. This suggests the SM is just an effective theory e h andthatthephysicsbeyondtheSMmaybemanifestedthroughan effectiveLagrangian : v involvingthetopquark. Xi Theframeworkofeffectivetheories,asameantoparametrizephysicsbeyondtheSM inamodelindependentway,hasbeenusedrecently.Twocaseshavebeenconsiderinthe r a literature,thedecouplingcase,whichincludestheHiggsboson,andthenon-decoupling case,wherethereisnotanyHiggsboson.Weshallconsideronlythefirstcase,inwhich the SM is a low-energy limit of a renormalizable theory. In this approach, the effective theoryparametrizestheeffectsatlowenergyofthefullrenormalizabletheorybymeans ofhighorder dimensionalnon-renormalizableoperators2. The effective Lagrangian approach is a convenient model independent parametriza- tion of the low-energy effects of the new physics that may show up at high energies. Effective Lagrangians, employed to study processes at a typical energy scale E can be writtenasapowerseriesin1/L ,wherethescaleL isassociatedwiththeheavyparticles massesoftheunderlyingtheory3.Thecoefficientsofthedifferenttermsintheeffective Lagrangian arise from integrating out the heavy degrees of freedom. In order to define an effective Lagrangian it is necessary to specify the symmetry and the particle content of the low-energy theory. In our case, we require the effective Lagrangian to be CP- conserving,invariantunderSMsymmetrySU(2) U(1) ,andtohaveasfundamental L Y ⊗ fieldsthesameonesappearingintheSMspectrum.ThereforeweconsideraLagrangian intheform L =L +(cid:229) a On (1) eff SM n n wheretheoperators On areofdimensiongreater thanfour. THE ANOMALOUS MAGNETIC MOMENT Theaimnowistoextractindirectinformationonthemagneticdipolemomentofthetop quarkfrom LEPdata, specificallyweusetheratiosR and R defined by b l G (Z bb¯) G (Z hadron) R = → , R = → , G (2) b G (Z hadron) l G (Z ll¯) Z → → inthecontextofaneffectiveLagrangianapproach.TheobliqueandQCDcorrectionsto thebquarkand hadronicZ decay widthscancel offintheratioR .Thispropertymakes b R very sensitive to direct corrections to the Zbb vertex, specially those involving the b heavytop quark, whileG and R are moresensitivetotheobliquecorrections. Z l In the present work, we consider the following dimension six and CP-conserving operators, OauWb =Q¯aLs mn Wmni t if˜URb , OaubB =Q¯aLs mn YBmn f˜URb, (3) where Qa is the quark isodoublet, Ub is the up quark isosinglet, a , b are the fam- L R ily indices, Bmn and Wmn are the U(1)Y and SU(2)L field strengths, respectively, and f˜ = it f . We use the notation introduced by Buchmüller and Wyler4. After spon- 2 ∗ taneous symmetry breaking, these fermionic operators generate also effective vertices proportional to the anomalous magnetic moments of quarks. The above operators for the third family give rise to the anomalous tt¯g vertex and the unknown coefficients e ab uB ande ab are related withtheanomalousmagneticmomentofthetopquark through uW m g dk = √2 t (s e 33 +c e 33), (4) t − m eQ W uW W uB W t wheres denotes thesineoftheweak mixingangle. W Theexpressionfor R isgivenby b R =RSM(1+(1 RSM)d ), (5) b b b b − whereRSM isthevaluepredicted bytheSM andd isthefactorwhichcontainsthenew b b physicscontribution,andit isdefined as follows 2 gSMgNP+gSMgNP + gNP 2+ gNP 2 d = V V A A V A (6) b (cid:0) (gSM)2+(cid:1)(gS(cid:0)M)2 (cid:1) (cid:0) (cid:1) V A and gSM and gSM are the vector and axial vector couplings of the Zbb¯ vertex. The V A contributions from new physics, eq. (3), to R and G are of two classes. One from l Z vertex correction to Zbb¯ in the G and the other from the oblique correction through hadr D k in thesin2q . Thesecan bewrittenas W R = RSM(1 0.1851D k +0.2157d ), l l b − R = RSM(1 0.03D k +0.7843d ), b b b − G = G SM(1 0.2351D k +0.1506d ) (7) Z Z b − whereD r isequal tozero for theoperators thatweare considering. WewillconsidertheoneloopcontributionoftheaboveeffectiveoperatorstotheZbb vertex.AfterevaluatingtheFeynmandiagrams,withinsertionsoftheeffectiveoperators Oab andOab weobtain uB uW m2 gNP = 4√2e 33 G m3 m 3c (C˜ C˜ ) t (C C +C ) V uW F W t W 12− 11 −√2m2 12− 11 0 (cid:8) W (1+a) 1 + (C +C +C )+ (C C C ) 11 12 0 12 11 0 8cW √2 − − 3a (B B ) , −4c m2 1− 0 W Z (cid:9) a gNP = 4√2e 33 G m3 m (C +C C ) A uW f W t −2c 0 12− 11 W (cid:8) 1 m2 (C C C )+ t (C C +C ) 12 0 11 12 11 0 − √2 − − √2m2 − W 2m s2 3 t W(C˜ +C˜ C˜ ) (B B ) (8) 0 12 11 1 0 − m − −4c m 2 − W W Z (cid:9) fortheoperatorO and, uW 4√2 s m2 gNP = gNP = e 33G m3 m W [ t (C C +C ) V A 3 uB F W tcW √2m2 12− 11 0 W 1 ( C +C C )], (9) 11 12 0 −√2 − − for the O . In the above equations a = 1 8s2 while C =C (m ,m ,m ), C˜ = uB − 3 W ij ij W t t ij C˜ (m ,m ,m ) and B =B (0,m ,m ) are thePassarino-Veltman scalar integralfunc- ij t W W i i t W tions5. The combination B B has a pole in d =4 dimensions that is identified with 0 1 thelogarithmicdependence−and can bereplaced by lnL 2/m2. Z Now wehavevariousposibilitiesto explorethespaceof theparameters e 33 , e 33 and uW uB dk . Using experimental values for G , R and R we find the allowed region in the t Z b l planee 33 e 33.Ifwedonotneglectthetermoftheorder gNP 2,wegetthefollowing uW uB − expressions (cid:0) (cid:1) 0.057e +0.058e 2+0.053e +0.0015e 2 +0.01e e = 1 (G exp/G SM), − B B W W B W − Z Z 0.023e +0.026e 2+0.016e +0.0007e 2 +0.0048e e = 1 (Rexp/RSM), − B B W W B W − b b 0.656e +0.686e 2+0.536e +0.018e 2 +0.1264e e = 1 (Rexp/RSM), − B B W W B W − l l wherewehaveomittedthesuperindex33. TheSM valuesfortheparametersthat wehaveused are G = 2.4963GeV, R =20.743, Z l R = 0.21572,G =17427MeV, G =84.018MeV, b hadr l withtheinputparameters: m = 174.3GeV, a (m )=0.118, t s Z m = 91.1861GeV, m =100GeV, L =1TeV. Z H Andtheexperimentalvaluesare G = 2.4952 0.0023GeV, R =20.804 0.050, Z l ± ± R = 0.21653 0.00069. b ± After doing a c 2 analysis at 95% C.L. we find the allow region for e 33 e 33 pa- uW uB − rameters. In this kind of scenarios new physics is explored assuming that its effects are smallerthantheSM effects, consequentlyoneexpectthat gNP/gSM 1 and, thenwe (cid:12) V,A V,A(cid:12)≪ get the inequalities e 33 0.11, e 33 0.48. By usingth(cid:12)e eq. (4)(cid:12)and the boundsgot inthenumericalana(cid:12)lyusWis(cid:12),≤weobta(cid:12)inufBo(cid:12)r≤dk t thefollowing(cid:12)values (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 2.94 dk 1.3, 0.76 dk 1.9, t t − ≤ ≤ − ≤ ≤ 1.3 dk 1.7 (10) t − ≤ ≤ which correspond to G , R and R , respectively. Therefore for these observables G , Z b l Z R and R , weget theallowedregion b l 2.94 dk 1.9. (11) t − ≤ ≤ ANOMALOUS CHROMOMAGNETIC DIPOLE MOMENT I We are interested in studying possible deviations from the SM on the decay b sg → withinthecontextoftheeffectiveLagrangianapproach.Severalauthorshavebeenused theCLEO results on radiativeB decays to set boundson theanomalous couplingof the t-quark.Wewillusedimension-sixoperatorwhicharefullstrongandelectroweakgauge invariantandcontributetob sg inordertoboundthechromomagneticdipolemoment → ofthetopquark. Since theanomalouschromomagneticdipolemomentofthetop quark appears in the top quark cross section, it is possible, due to uncertainties, to estimate the constraints thatitwouldimposeontheD k t.FortheLHC,theanomalouscouplingisconstrainedto g lieinthe range 0.09 D k t 0.16. Similarrange is obtainedfor thefutureNLC. The g influence of an−anomal≤ous D k≤on the cross section and associated gluon jet energy for tt¯g has been also analyzed. Events produced at 500 GeV in e+e colliders, with a cut − on the gluon energy of 500 GeV and integrated luminosity of 30fb 1, lead to a bound − of 2.1 D k t 0.66. g − ≤ ≤ >From the experimental information it is possible to get a limit on the D k t from g Tevatron. Following the reference by F. del Aguila7 and assuming that the only non- zero couplingispreciselythechromomagneticdipolemomentofthetopquark, wefind fromthecollecteddatathat theallowedregionis D k t 0.45. g | |≤ We consider the following dimension-six, CP-conserving operators, which are SU(3) S(2) U(1) gaugeinvariant C L Y ⊗ ⊗ l i Oab =Q¯as mn Gmn i f˜ub , (12) uG L 2 R where Gmn i is the gluon field strength tensor and a,b are the family indices. The above operator gives rise to the anomalous tt¯g vertex and its respective unknown coefficient e uG isrelated withtheanomalouschromomagneticmomentofthetopquark by ab g m dk t =√2 t e 33 . (13) g g M uG s W where by means of the dimension 5 coupling to an on-shell gluon, the anomalous chromomagneticdipolemomentofthetopquark isdefined as D k t g L5 =i( g) s u¯(t)s mn qn Tau(t)Gm ,a (14) 2 2m t withg andTa aretheSU(3) couplingandgenerators, respectively. s c TheeffectiveHamiltonianusedto computetheb s transitionis givenby8 → H = 4GFV V (cid:229) 8 c(m )O (m ), (15) eff t∗s tb i i − √2 i=1 where m is the energy scale at which H is applied. For i=1 6, O (m ) correspond eff i to four-quark operators, O (m ) is the electromagnetic dipole m−oment and O (m ) is 7 8 the chromomagnetic dipole operator. At low energy, m m , the only operator that b contributestob stransitionisO (m )whichresultsfrom≈amixingamongtheO (M ), 7 2 W O (M ) and O→(M )operators. 7 W 8 W The total contribution of the effective operator (12) to the O (M ) operator can be 8 W writtenas: c (M )=c (M )SM+dk tD c (M ), (16) 8 W 8 W g 8 W where 1 L 2 1 x x2+x(2 x)ln(x) D c (M ) = ln + − − 8 W 4V (cid:18)M2 (cid:19) V 8(1 x)2 ts W ts − 3x 4x2+x3+2xln(x) + − (17) 8(1 x)3 − andx=m2/M2 . t W Using the recent data from CLEO collaboration for the branching fraction of the process B(b sg ) = (3.21 0.43 0.27) 10 49, we get an allowed region for the − anomalousch→romomagnetic±dipole±moment×ofthetopquark tobe10 0.03 D k t 0.01 (18) g − ≤ ≤ ANOMALOUS CHROMOMAGNETIC DIPOLE MOMENT II Ournextobjectiveisto evaluatethecontributionat theone loop-levelto theanomalous chromomagnetic dipole moment of the top quark in different scenarios with the gluon boson on-shell. Beginning with the SM, the typical QCD correction through gluon exchange implies two different Feynman diagrams. After the explicit calculation of the loops,wefind that 1a (m ) D k t = s t . (19) g −6 p We note that its natural size is of the order of a /p similar to the QED anomalous s coupling,butnowincombinationwithafactor 1/6comingfromthecolorstructurein thediagram i.e.TaTbTa = Tb/6withTa bein−gthegenerators ofSU(3) . C − The otherpossiblecontributionin theframework ofthe SM comes from electroweak interactions.TherelevantcontributionsoccurwhenneutralHiggsbosonandthewould- beGoldstonebosonofZ are involvedintheloop.Thiscontributionreads √2G m2 D k t = F t [H (m )+H (m )], (20) g − 8p 2 1 h 2 Z where 1 x x3 H (m) = dx − , 1 Z0 x2 (2 m2/mt2)x+1 − − 1 x+2x2 x3 H (m) = dx − − . 2 Z0 x2 (2 m2/mt2)x+1 − − TheSM contributionisshowedin figure1 wherewe haveadded theQCD contribution. Itisworthnotingthat thebehaviourofthecurveforalargeHiggsbosonmassindicates decouplingandthatthevaluesofD k t liewithintheallowedregionforD k t comingfrom g g b sg . → FIGURE 1. Standar Model contribution to the anomalous cromomagnetic dipole moment of the top quarkvsthehiggsbosonmass Thecontributionswithinageneral2HDMwillbedifferentfromtheSMcontributions because of the presence of the virtual five physical Higgs bosons which appear in any two Higgs doublet model after spontaneous symmetry breaking: H0, A0, h0, H 11. ± Therefore, 2HDM predictions depend on their massses and on the two mixing angles a and b . For small b , the charged Higgs boson contribution is suppressed due to its largemassand thesmallbottomquark mass. Theexpressionfor thecontributionoftheneutralHiggsbosonsisgivenby √2G D k t = F[l 2 H (M2)+l 2 H (M2) g 8p 2 H0tt 1 H h0tt 1 h + l 2 H (M2)+l 2 H (M2)] (21) A0tt 2 A G0tt 2 Z where l are the Yukawa couplings in the so-called models of type I, II and III . Table itt Ishowsthecouplingsin theusualconvention. TABLE1. CouplingsoftheHiggseigenstateswiththetop quarkin2HDM.Weomitthefactorg/2m andinthemodel W type III we use the Sher-Cheng approach for the flavour changingcouplings. l modeltypeI modeltypeII modeltypeIII itt l mtsina mtcosa (1+h tUt )m sina H0tt sinb cosb √2 t l mtcosa mtsina (1+h tUt )m cosa h0tt sinb cosb √2 t l cotb m tanb m h tUtmt A0tt t t √2 l m m m G0tt t t t The Yukawa couplings of a given fermion to the Higgs scalars are proportional to the mass of the fermion and they are therefore naturally enhanced in this case. In the model type III appears flavour changing neutral couplings at tree level which can be parametrizedintheSher-Chengapproachwhereanaturalvaluefortheflavourchanging couplingsfromdifferentfamiliesshouldbeoftheorderofthegeometricaverageoftheir Yukawacouplings,hij =gh ij√mimj/(2mW)withh ij oftheorderofone. In order to show the behaviour of the contribution of the 2HDM to the anomalous chromomagneticdipolemomentofthetopquark,weevaluateexplicitlythecontribution FIGURE 2. Contour plot for the contribution of 2HDM to the anomalous cromomagnetic dipole moment of the top quark in the plane tanb m for M =200 GeV(solid line) and M =400 GeV H A A (dashedline)usingtheboundfromtheb sg−process.Theallowedregionisabovethecurve → for couplings type II. We show in figure 2 the allowed region (above the curve) for the plane tanb vs m using equation (21) and assuming that 0.03 D k t 0.01 from H g − ≤ ≤ b sg . We fix the following parameters: m = m , m = 200(400) GeV solid line H h A → (dashed line). The solid line for the scalar Higgs mass smaller (bigger) than 240 GeV correspondstothecut betweenequation(21)and theupper(lower)limitfrom b sg . → ACKNOWLEDGMENTS WethankCOLCIENCIAS forfinantialsupport REFERENCES 1. K.Hagiwaraet.al.(ParticleDataGroup), Phys.Rev.D66,1(2002). 2. H. Georgi, Nucl. Phys. 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