hep-ph/0407050 ¯ B − B mass difference in Little Higgs model d d S. RAI CHOUDHURY1∗, NAVEEN GAUR1†, ASHOK GOYAL1‡, NAMIT MAHAJAN2§ 5 0 1 DepartmentofPhysics&Astrophysics, 0 UniversityofDelhi,Delhi-110007,India. 2 n a 2 Harish-ChandraResearchInstitute, J ChattnagRoad,Jhunsi,Allahabad-211019,India. 1 4 v 0 5 0 7 0 Abstract 4 0 Analternate solution of hierarchy problem in the Standard Model namely, the Little Higgs model, h/ hasbeenproposedlately. InthisworkB0 B¯0massdifferenceintheframeworkoftheLittleHiggs p d− d modelisevaluated. Theexperimentallimitsonthemassdifferenceisshowntoprovidemeaningful - p constraints ontheparameterspaceofthemodel. e h : v i X r a ∗[email protected] †[email protected] ‡[email protected] §[email protected] Note:Inourearlierversionofthepaperamixupoftwosymbolsinournumericalcomputationlead toincorrectvaluesoftheB B¯ massdifference.ApaperthatappearedsubsequentlybyBuraset.al. d d − [1],presentedthecorrectcalculationofB B¯ massdifferenceinLittleHiggsmodel.Inthisversion d d − wehavecorrectedournumericalmistake.OurresultsforB B¯ massdifferencenowagreeswiththe d d − resultsofBuraset.al.[1].Sometyposinourearlierversionhavealsobeencorrected. Our understanding of the Standard Model (SM) is plagued by a major issue, the “hierarchyprob- lem”, arising out of the enormous difference between the electroweak and the Plank scale. For quite some time, Supersymmetry had provided an elegant framework for solving this problem although to- datethere is no compellingexperimental evidencein its support. During the last two years, an alterna- tivepossibilityhasbeenintroducedintheliteraturewheretheHiggsmassremainssmallbyvirtueofit being a Goldstoneboson of a global symmetrywhich is broken at a scale abovethe electroweak scale. These models are generically called the “LittleHiggs”models and the simplest of these, the “Littlest Higgs”(LH)model[2], has theleast numberofadditionalparticlesinvolved. In the gauge sector, the LH model contains weakly coupled gauge bosons with masses in the TeV scale in addition to the SM W± and Z [3,4]. These mix amongst themselves causing modification of SM gaugecouplingsofW±,Z withfermionsand amongthemselves. Inthequark sector,avector-like heavy top quark comes into play with mass in TeV range, which has trilinear coupling with SM gauge bosons. Once again the heavy top quark has mixing possibility with the SM top quark, resulting in modificationofcouplingstructureofquarkswithW± andZ. Inaddition,themodelhaschargedHiggs bosonswhichintroducescalarcouplingswithquark. Also,aheavierphotonwithmassintheTeVrange emerges, whichcouples bothtoleptonsandquarks. The presence of these new particles as well as changes in the SM interaction vertices, can cause changes in a variety of measurable parameters. Some of them have already been calculated in the literature [3,5–8]. These results provide good constraints on the parameters entering the LH model. DirectexperimentalconfirmationofseveralaspectsofLH,e.g.,themassesoftheheavyt-quarkandthe doubly charged Higgs, would require sharper estimates of the parameters of the theory. It is desirable therefore,toworkouttheconsequencesoftheLH-modelforasmanyobservablequantitiesaspossible in orderto sharpen theconstraintson theparameterspace ofsuch a model. In thisnote, wereport on a calculationofB B¯ andK K¯ mixingin thecontextofLH model. 0 0 0 0 − − In SM, there is one basic box diagram responsible for generating the effective Hamiltonian for the mixingofB B¯ and K K¯ . In LH, thereare manymore boxdiagrams (as shownin Figure 1)to 0 0 0 0 − − beevaluated. Thecouplingsandpropagatorsrequired forcalculatingthesediagramsarelistedin [3]. TheeffectiveHamiltonianresultingforthegraphs inFig.1 hasthestructure: G2 = F M2 S (q¯d) (q¯d) (1) Heff 16π2 W q V−A V−A with q = b,s for (B B¯ ) and (K K¯ ) mixing respectively. The invariant function S has the 0 0 0 0 q − − followingform: S = SSM +SLH (2) q q q 1 where in both S and S , the first term represents the SM contribution along with QCD corrections b s which are givenin detail in [9]. The second term gives the LH contributionto the mass difference. As these are the corrections to the SM contribution, we do not consider QCD corrections to them which would arise from gluonic loops added to the diagrams of Fig 1. The effective Lagrangian in the LH modeltoorder v21 is wellapproximatedby: f2 G2 △J = F M2 SLHQ( J = 2) (3) Leff 16π2 W j △ whereJ = B,S and j = b,s forB B¯ and K0 K¯0 respectively. Theyare givenas : d d − − ¯ ¯ Q( B = 2) = (b d ) (b d ) α α V−A β β V−A △ Q( S = 2) = (s¯ d ) (s¯ d ) (4) α α V−A β β V−A △ and v2 SLH = ξ2E(x ,W )+ ξ E(x ,x ,W ) b f2"i=Xu,c,t i i L i6=j=Xu,c,t,T ij i j L  +2c2 ξ′2E(x ,W ,W )+ ξ′ E(x ,x,W ,W ) s2  i i L H ij i j L H  i=Xu,c,t i6=j=Xu,c,t,T  + 1 2s+f λ2E(x ,W ,Φ)+ λ λ E(x ,x ,W,Φ) (5) − v !i=Xu,c,t i i L i6=jX=u,c,t i j i j L # whereξ ,ξ andfunctionsE aredefinedinappendixA. x = m2/m2 ,λ’saretheCKMfactorsdefined i ij i i W as λ = V V∗ (i = u,c,t)and λ = λ1λ and V ’sare theCKMmatrixelements. i id ib T λ2 t ij b WL;WH;(cid:8) d b u; ;t;T d u; ;t;T u; ;t;T WL;WH;(cid:8) WL;WH;(cid:8) d WL;WH;(cid:8) b d u; ;t;T b Figure1: Box diagramsinLH. We notethat despitetheoccurrence ofspinlessHiggscouplingsto quarks,theultimatestructureof 2 theeffectiveHamiltonianinLHretainsthesame(V A)formasinSMtoorder v . Giventheform − f 1f is the scale atwhich theglobalSU(5)symmetryis spontaneouslybrokenvia a vacuume(cid:16)xpe(cid:17)ctationvaluewhichis expectedto be in the TeV range and roughlyof the order of masses of heavy bosonsand v is the vev of standard model Higgs 2 oftheeffectiveHamiltonian,wecanproceedexactlyasinSMandcalculateitsmatrixelementbetween K K¯ or B B¯ states using the vacuum saturation approximation. There are no divergences in 0 0 0 0 − − theSMamplitudebecauseoftheunitarityoftheCKMmatrix;thisstatementholdseveninLHmodel2 whereonceagaintheunitarityofCKMensuresthatalldivergencesvanishtoorder(v/f)2. Neglecting QCD corrections andlongdistancecontributionswecan getthemassdifferenceto be: G2 M(B B¯ )LH = F M M2 f2SLH (6) △ 0 − 0 6π2 B W B b and G2 M(K K¯ )LH = F M M2 f2SLH (7) △ 0 − 0 6π2 K W K s whereM ,f are themasses anddecay constantsofBand K mesonsrespectively3. B,K B,K Itshouldbementionedthattherenormalizationgroupevolutionofthematrixelementshasbeenthe subjectofmuchworkandhasbeensummarizedin[10]andisfarfromtrivialsincethematrixelements are controlled by long distance dynamics and are generally parameterized by a “Bagfactor”B . K,B HoweverfortheneutralBmesoncase,thelongrangeinteractionsarisingfromtheintermediatevirtual statesare negligiblebecauseofthelarge Bmass,beingfar fromtheregionofhadronicresonances. The LH involves not only heavy vector bosons and quarks but also a large number of parameters over and above those in the SM. The global symmetry in the theory is broken at TeV range scale Λ (Λ = 4πf); the scalar bosons, doublets and triplets, acquire vacuum expectation values v and v′ s s respectively at the EW-scale, providing the convenient small parameters v/f and v′/f. The mixing of the charged and neutral vector bosons results in two mixing angle parameters θ and θ′ (with c = Cosθ,s = Sinθ,c′ = Cosθ′ and s′ = Sinθ′). Finally the Yukawa coupling of the fermions involves two parameters λ and λ with thecombinationx = λ21 occurring frequently. To the leadingorder 1 2 L λ21+λ22 in (v/f), the masses of all the heavy particles in LH can be expressed in terms of SM masses m and W m as : Z m 1 f WH . m ≈ scv W m λ2 +λ2f T 1 2 . m ≈ λ λ v t 1 2 m f Φ √2 . m ≈ v H The coupling of all heavy particles to SM particles as well among themselves are expressible in termsoftheseparameters withtheSM ones. The parameter space is obviouslytoo large. Requiring that the heavy particles havemasses in TeV range results in the condition 1 < 10. There is another restriction arising out of the requirement that sc themassofthetripletscalars bepositivedefinite[3]: v′2 v2 < (8) v2 16f2 2theCKMmatrixisunitaryinLHuptoorderv2/f2 3TheQCDcorrectionstothesehavebeenworkedoutinliterature 3 0.53 0.54 s = 0.2 0.525 ss == 00..58 xL = 0.2 ss == 00..25 xL = 0.4 0.53 s = 0.8 0.52 0.515 0.52 -1M (ps) 0.51 -1M (ps) ∆ ∆ 0.51 0.505 0.5 0.5 0.495 0.49 0.49 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 v/f v/f 0.55 0.57 0.54 sss === 000...258 xL = 0.6 0.56 ss == 00..25 xL = 0.8 s = 0.8 0.55 0.53 0.54 -1M (ps) 0.52 -1M (ps) 0.53 ∆ ∆ 0.52 0.51 0.51 0.5 0.5 0.49 0.49 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 v/f v/f Figure2: M(B B¯ )inps−1 withv/f. Fortheseplotswehaveuseds′ = s. Shadedareaindicates d d △ − theexperimentalbounds. We have varied v/f in the range 0 to 0.1. s,s′ in range 0.2 to 0.8 and x in range 0.2 to 0.8 in our L numericalanalysis. Otherparameters usedare givenintheAppendixB. Our results for the B0 B¯0 case are shown in Figure 2. Varying s′ doesn’t significantly change d − d our conclusions. The corresponding K0 K¯0 results too have similartrend. However, since there are − largeerrorbarsinthembecauseofQCDcorrectionsinvolved,itmakesitdifficulttodrawanydefinitive conclusions. Hence we haven’t shown them. In the plots shown in Fig.2 the shaded area corresponds tothemassdifference M(B B¯ ) = 0.5 0.01ps−1 whichisconsistentwithcurrentexperimental d d △ − ± bounds[11]. From Fig 2, it is easy to note that for low values of x , there is a region of parameter space (in L termsoftheparameters sand v/f)thatisconsistentwiththeexperimentallimits. Very specificallyfor x = 0.2, the bound on the scale f can be very close to 1 TeV for almost all the s values. However, L as x is increased, the LH contribution starts deviating significantly from the SM results. Therefore, L in principle the experimentally allowed band for the B0 B¯0 mass difference provides significant d − d constraints for the parameter space of generic Little Higgs models, and in particular the Littlest Higgs model. It will be fruitful to compare the limits on the parameters coming from precision electroweak data [12]. TothisendwerecallthattheLittlestHiggsmodeldoesnothavethecustodialsymmetryinherently 4 built into it and can therefore, in principle, lead to large corrections, arising both from heavy gauge bosonexchangediagramsandthetripletVEV.Anaivewayoutwouldbetohavetheextragaugeboson massesraisedbysomemeans. However,thiswouldspoilthemotivationofcircumventingthehierarchy problem and would also bring in the issueof fine tuning. It has been found that global fits to precision electroweak data imply the following bound (at 95% C.L.) on the scale f for any generic coupling (specifically varyingcand c′ between 0.1 and 0.995): f > 4 TeV (9) It is worth noting that this very stringent bound is almost (practically) independent of any variation of Higgs mass upto 200 GeV. Further, the bound still holds for any order unity value of the parameters encapsulatingthephysicsduetoproper UV-completionofthetheory. To benoted is thefact that these are very strong limits and the origin can simply be traced back to the absence of custodial symmetry. In the second reference of [12] it was noted that considering only precision electroweak data allows for a small region in parameter space where the bound on the scale f can be lowered to about 1 TeV. However, electroweak data combined with Drell-Yan production excludes this region and a combined analysis implies a bound very similar to the one quoted above. Constraints from low energy precision datalike(g 2) andatomicweak charge ofCesiumalsoindicatesimilarbounds,thoughitshouldbe µ − remembered thattherathersmall(g 2) correctionsmay notservetoputany meaningfulbounds. µ − Interestingly enough, in almost all the variations of the Littlest Higgs model [13], the constraints remain quitestrong and generically very similarto the minimalversion, though for some very specific choices of the parameters the constraints on f are relaxed to 1 2 TeV. This can be understood as − arisingduetosmallmixingbetweenthetwo setsofgaugebosonsand alsosmallcouplingbetween the fermions and heavy U(1) gauge boson. Nevertheless, these arise only in very specific models and for veryspecialchoicesoftheparametersandarenotagenericfeatureofLHmodels. Itmayalsobeuseful tokeepinmindthatthepositivityoftripletmasssquaredimposessevereconstraintsonthetripletVEV and therefore the parameters or combination of parameters entering the mass squared relation. This strong constraint considerably reduces the allowed parameter space. Therefore, it is natural to expect that in the variation of the minimal model where there is no triplet Higgs, the bounds are partially relaxed asis thecase inmodelswhichhavecustodialsymmetrybuiltintothem. Turning to contribution to M(B B¯ ), we would indeed get a bound similar to eqn (9) above d d △ − if we require that the LH contribution be no more than the experimental band for M(B B¯ ). d d △ − However,as has been pointedout by Buraset.al.[1] thatthere is ahadronicuncertainity ofabout 10% incalculationof M(B B¯ ). In viewofthisareasonableconstrainton(v/f)could begivenifthe d d △ − variation in the mass difference could be more than 10%. For a contribution of about 10% , the LH modelwould requirev/f 0.2, which isnot very useful in view ofthestrongerconstraint likeeqn(9) ≤ above. However, should it become possible for hadronic uncertainities to be reduced by a factor of 2 or 3, then the bound on (v/f) values becomes much lower, leading to constraints on the value of f comparablewiththevaluein equation(9)above. 5 Acknowledgments We thank Heather Logan for useful discussions. We thank Andrzej Buras for useful communications regarding their paper. We thank him and Selma Uhlig for discussions regarding this revised version of our paper. The work of SRC, AG and NG is supported under the SERC scheme of Department of Science &Technology(DST), Indiaundertheproject no. SP/S2/K-20/99. A Loop Functions W ,W ineqn (5)refer tothelight&heavy W-bosonin LH;c,sare themixingangles inLH.Various L H ξ’sare ξ2 = 2c2(c2 s2)λ2, i = u,c (A.1) i − i ξ2 = 2 c2(c2 s2)+x2 λ2 (A.2) t − L t n x2o ξ = 2 c2(c2 s2)+ L λ λ (A.3) it i t ( − 2 ) ξ = x2λ λ (A.4) iT − L i t ξ = ξ = ξ ξ (A.5) ij ji i j ξ = x2λ2 (A.6) tT − L t ξ′2 = (f/v)2λ2 (A.7) i − i ξ′ = (f/v)2λ (A.8) ij ij − thefunctions(E) usedineqn. (5)are : x x 1 3 1 3 1 i j E(x ,x ,W ) = + log(x ) i j L −xi xj (4 2(1 xi) − 4(1 xi)2) i − − − x x 1 3 1 3 1 j i + log(x ) −xj xi (4 2(1 xj) − 4(1 xj)2) j − − − 3 x x i j + (A.9) 4(1 x )(1 x ) i j − − 3 x 3 1 9 1 3 1 i E(x ,W ) = log(x ) x + (A.10) i L −2 (cid:18)xi −1(cid:19) i − i(4 4(1−xi) − 2(1−xi)2) x x 1 x2 E(x ,x ,W ,W ) = i j 1 1+ x + i log(x ) i j L H −xWH(xi −xj)(1−xi)(1− xWxiH) ( − xWH! i 4xWH) i x x 1 x2 j i j 1 1+ x + log(x ) −xWH(xj −xi)(1−xj)(1− xWxjH) ( − xWH! j 4xWH) j 3 1 + log(x ) (A.11) 4 1 1 1 xi 1 xj WH (cid:18) − xWH(cid:19)(cid:18) − xWH(cid:19)(cid:18) − xWH(cid:19) 6 3 x3 1+x E(x ,W ,W ) = i 2 WHx logx i L H 4x2 (1 x )2 1 xi 2 ( − xWH i) i WH − i (cid:18) − xWH(cid:19) 3 1 + log(x ) 4 2 WH 1 1 1 xi (cid:18) − xWH(cid:19)(cid:18) − xWH(cid:19) x 1 x2 i 1 1+ x + i (A.12) i −xWH(1−xi)(cid:18)1− xWxiH(cid:19) ( − xWH! 4xWH) E(x ,x ,x ,x ) = xixj xi 1− x4i log(x ) xj 1− x4j log(x ) i j WL φ 2 −(x x )(1(cid:16) x )((cid:17)x x ) i − (x x )(1(cid:16) x )((cid:17)x x ) j i j i φ i j i j φ j  − − − − − − x 1 xφ +  φ − 4 log(x ) (A.13) (x x )(x(cid:16) x )(cid:17)(1 x ) φ  φ i φ j φ − − −  E(xi,xWL,xφ) = x22i "− (1 (cid:16)x1i)−(xxφ4i(cid:17) xi) − (cid:26)xφ(cid:16)(11−x2xii(cid:17))2+(x3φx42i (cid:16)xx3φi)2−1(cid:17)(cid:27)log(xi) − − − − x 1 xφ + φ − 4 log(x ) (A.14) (xφ (cid:16)xi)2(1 (cid:17)xφ) φ # − − B Input parameters G = 1.16 10−5GeV−2 , f = 0.21, m = 5.3GeV, F B B × m = 80.4GeV , m = 91.2GeV WL ZL References [1] A.J. 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