HindawiPublishingCorporation AdvancesinHighEnergyPhysics Volume2012,ArticleID129879,43pages doi:10.1155/2012/129879 Review Article (cid:2) (cid:3) × (cid:2) (cid:3) × (cid:2) (cid:3) × (cid:2) (cid:3) U 3 Sp 1 U 1 U 1 C L L R Luis Alfredo Anchordoqui DepartmentofPhysics,UniversityofWisconsin-Milwaukee,P.O.Box413,Milwaukee,WI53201,USA CorrespondenceshouldbeaddressedtoLuisAlfredoAnchordoqui,[email protected] Received30August2011;Revised17October2011;Accepted20October2011 AcademicEditor:IraRothstein Copyrightq2012LuisAlfredoAnchordoqui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproductioninanymedium,providedtheoriginalworkisproperlycited. WeoutlinethebasicsettingoftheU(cid:2)3(cid:3) ×Sp(cid:2)1(cid:3) ×U(cid:2)1(cid:3) ×U(cid:2)1(cid:3) gaugetheoryandreviewthe C L L R associatedphenomenologicalaspectsrelatedtoexperimentalsearchesfornewphysicsathadron colliders. In this construction, there are two massive Z(cid:2)-gauge bosons, which can be naturally associatedwithbaryonnumberB andB−L(cid:2)Lbeingleptonnumber(cid:3).Wediscussthepotential signals which may be accessible at the Tevatron and at the Large Hadron Collider (cid:2)LHC(cid:3). In particular,weprovidetherelevantcrosssectionsfortheproductionofZ(cid:2)-gaugebosonsintheTeV region,leadingtopredictionsthatarewithinreachofthepresentorthenextLHCrun.Afterthatwe directattentiontoembeddingthegaugetheoryintotheframeworkofstringtheory.Weconsider extensionsofthestandardmodelbasedonopenstringsendingonD-branes,withgaugebosons duetostringsattachedtostacksofD-branesandchiralmatterduetostringsstretchingbetween intersectingD-branes.AssumingthatthefundamentalstringmassscaleisintheTeVrangeand thetheoryisweaklycoupled,weexploretheLHCdiscoverypotentialforReggeexcitations. 1. General Idea Therecentdevelopmentinhighenergyphysicshasputagreatemphasisongaugetheories; indeed the general theory of fundamental interactions, rather unimaginatively named the Standard Model (cid:2)SM(cid:3), is completely formulated in this framework. The SM agrees remarkably well with current data but has rather troubling weaknesses and appears to be asomewhatadhoctheory.ItisthoughtthattheSMmaybeasubsetofamorefundamental gaugetheory.Severalmodelshavebeenexplored,usingthefundamentalprincipleofgauge invarianceasguidepost.Thepurposeofthispaperistooutlinethemainphenomenological aspectsofonesuchmodels:U(cid:2)3(cid:3) ×Sp(cid:2)1(cid:3) ×U(cid:2)1(cid:3) ×U(cid:2)1(cid:3) .Thefirstaimistosurveythebasic C L L R featuresofthegaugetheory’spredictionregardingthenewmasssectorandcouplings.These features lead to new phenomena that can be probed using data from the Tevatron and the LargeHadronCollider(cid:2)LHC(cid:3).Inparticularthetheorypredictsthatadditionalgaugebosons 2 AdvancesinHighEnergyPhysics thatwewillshowareaccessibleatLHCenergies.Havingsoidentifiedthegeneralproperties of the theory, we focus on the potential to embed this model into a string theory. We show thiscanbeaccomplishedwithinthecontextofD-braneTeV-scalecompactifications.Finally, weexplorepredictionsinheritedfrompropertiesoftheoverarchingstringtheory. The SM is a spontaneously broken Yang-Mills theory with gauge group SU(cid:2)3(cid:3) × C SU(cid:2)2(cid:3) ×U(cid:2)1(cid:3) .Matterintheformofquarksandleptons(cid:2)i.e.,SU(cid:2)3(cid:3) tripletsandsinglets, L Y C resp.(cid:3) is arranged in three families (cid:2)i (cid:4) 1,2,3(cid:3) of left-handed fermion doublets (cid:2)of SU(cid:2)2(cid:3) (cid:3) L and right-handed fermion singlets. Each family i contains chiral gauge representations of left-handed quarks Qi (cid:4) (cid:2)3,2(cid:3)1/6 and leptons Li (cid:4) (cid:2)1,2(cid:3)−1/2 as well as right-handed up and down quarks, Ui (cid:4) (cid:2)3,1(cid:3)2/3 and Di (cid:4) (cid:2)3,1(cid:3)−1/3, respectively, and the right-handed lepton Ei (cid:4) (cid:2)1,1(cid:3)−1. The hypercharge Y is shown as a subscript of the SU(cid:2)3(cid:3)C × SU(cid:2)2(cid:3)L gauge representation (cid:2)A,B(cid:3). The neutrino is part of the left-handed lepton representation L and i doesnothavearight-handedcounterpart. TheSMLagrangianexhibitsanaccidentalglobalsymmetryU(cid:2)1(cid:3) ×U(cid:2)1(cid:3) ×U(cid:2)1(cid:3) × B e μ U(cid:2)1(cid:3) ,whereU(cid:2)1(cid:3) isthebaryonnumbersymmetryandU(cid:2)1(cid:3) (cid:2)α (cid:4) e,μ,τ(cid:3)arethreelepton τ B α flavor symmetries, with total lepton number given by L (cid:4) L (cid:5)L (cid:5)L . It is an accidental e μ τ symmetry because we do not impose it. It is a consequence of the gauge symmetries and the low energy particle content. It is possible (cid:2)but not necessary(cid:3), however, that effective interaction operators induced by the high energy content of the underlying theory may violatesectorsoftheglobalsymmetry. TheelectroweaksubgroupSU (cid:2)2(cid:3)×U (cid:2)1(cid:3)isspontaneouslybrokentotheelectromag- L Y neticU(cid:2)1(cid:3) bytheHiggsdoubletH (cid:4) (cid:2)1,2(cid:3) whichreceivesavacuumexpectationvalue EM 1/2 v/(cid:4)0inasuitablepotential.ThreeofthefourcomponentsofthecomplexHiggsare“eaten” by the W± and Z bosons, which are superpositions of the gauge bosons Wa of SU(cid:2)2(cid:3) and μ L B ofU(cid:2)1(cid:3) , μ Y W± (cid:4) √1 W1∓ √i W2, μ μ μ 2 2 (cid:2)1.1(cid:3) Z (cid:4)cosθ W3−sinθ B , μ W μ W μ with masses M2 (cid:4) παv2/sin2θ , M2 (cid:4) M2 /cos2θ , and α (cid:5) 1/128 at Q2 (cid:4) M2 . The W W Z W W W fourthvectorfield, A (cid:4)sinθ W3(cid:5)cosθ B , (cid:2)1.2(cid:3) μ W μ W μ persistsmassless,andtheremainingHiggscomponentisleftasaU(cid:2)1(cid:3) neutralrealscalar. EM The measured values M (cid:5) 80.4GeV and M (cid:5) 91.2GeV fix the weak mixing angle at W Z sin2θ (cid:5)0.23andtheHiggsvacuumexpectationvalueat(cid:6)H(cid:7)(cid:4)v (cid:5)246GeV(cid:7)1(cid:8). W Fermion masses arise from Yukawa interactions, which couple the right-handed fermionsingletstotheleft-handedfermiondoubletsandtheHiggsfield, L(cid:4)−Yij QHD −Yij(cid:6)abQ H†U −YijLHE (cid:5)h.c., (cid:2)1.3(cid:3) d i j u ia b j (cid:7) i j where (cid:6)ab is the antisymmetric tensor. In the process of spontaneous symmetry breaking, √ these interactions lead to charged fermion masses, mij (cid:4) Yijv/ 2, but leave the neutrinos f f AdvancesinHighEnergyPhysics 3 massless(cid:7)2(cid:8).(cid:2)Onemightthinkthatneutrinomassescouldarisefromloopcorrections.This, however, cannot be the case, because the only possible neutrino mass term that can be constructedwiththeSMfieldsisthebilinearLLC whichviolatesthetotalleptonsymmetry i j by two units (cid:2)LC (cid:4) CLT(cid:3). As mentioned above, total lepton number is a global symmetry i i of the model and therefore L-violating terms cannot be induced by loop corrections. Furthermore, the U(cid:2)1(cid:3)B−L subgroup is nonanomalous, and therefore B −L violating terms cannotbeinducedevenbynonperturbativecorrections.ItfollowsthattheSMpredictsthat neutrinosarestrictlymassless.(cid:3)Experimentalevidenceforneutrinoflavoroscillationsbythe mixing of different mass eigenstates implies that the SM has to be extended (cid:7)3(cid:8). The most economic way to get massive neutrinos would be to introduce the right-handed neutrino states (cid:2)having no gauge interactions, these sterile states would be essentially undetectable(cid:3) andobtainaDiracmasstermthroughaYukawacoupling. TheSMgaugeinteractionshavebeentestedwithunprecedentedaccuracy,including someobservablesbeyondevenonepartinamillion(cid:7)1(cid:8).Nevertheless,thesagaoftheSMis stillexhilaratingbecauseitleavesallquestionsofconsequenceunanswered.Themostevident of unanswered questions is why there is a huge disparity between the strength of gravity and of the SM forces. This hierarchy problem suggests that new physics could be at play at the TeV-scale and is arguably the driving force behind high energy physics for several decades.Muchofthemotivationforanticipatingtheexistenceofsuchnewphysicsisbased onconsiderationsofnaturalness.ThenonzerovacuumexpectationvalueofthescalarHiggs doubletcondensatesetsthescaleofelectroweakinteractions.However,duetothequadratic sensitivity of theHiggsmass toquantum corrections fromanarbitrarily high mass scaleΛ, withnonewphysicsbetweentheenergyscaleofelectroweakunificationandthevicinityof the Planck mass, the bare Higgs mass and quantum corrections have to cancel at a level of onepartin∼1030.Thisfine-tunedcancellationseemsunnatural,eventhoughitisinprinciple self-consistent.ThuseitherthescaleofnewphysicsΛismuchsmallerthanthePlanckscale or there exists a mechanism which ensures this cancellation, perhaps arising from a new symmetryprinciplebeyondtheSM;minimalsupersymmetry(cid:2)SUSY(cid:3)isatextbookexample (cid:7)4(cid:8).Ineithercase,anextensionoftheSMappearsnecessary. Inthispaperweexaminethephenomenologyofanewfangledextensionofthegauge sector, U(cid:2)3(cid:3) ×Sp(cid:2)1(cid:3) ×U(cid:2)1(cid:3) ×U(cid:2)1(cid:3) , which has the attractive property of elevating the C L L R two major global symmetries of the SM (cid:2)B and L(cid:3) to local gauge symmetries (cid:7)5(cid:8). (cid:2)The fundamental principles of the model are summarized in (cid:7)6–8(cid:8). Herein though we replace at full length the U(cid:2)2(cid:3) doublets by Sp(cid:2)1(cid:3) doublets. Besides the fact that this reduces the L L number of extra U(cid:2)1(cid:3)’s, one avoids the presence of a problematic Peccei-Quinn symmetry (cid:7)9–11(cid:8),associatedingeneralwiththeU(cid:2)1(cid:3)ofU(cid:2)2(cid:3) underwhichHiggsdoubletsarecharged L (cid:7)12(cid:8).Apointworthnotingatthisjuncture:thecompactsymplecticgroupSp(cid:2)1(cid:3)isequivalent toSU(cid:2)2(cid:3);ourchoiceofnotationwillbecomeclearinSection5.(cid:3)TheU(cid:2)1(cid:3) bosonY ,which Y μ gaugestheusualelectroweakhyperchargesymmetry,isalinearcombinationoftheU(cid:2)1(cid:3)of U(cid:2)3(cid:3) gauge boson C , the U(cid:2)1(cid:3) boson B , and a third additional U(cid:2)1(cid:3) field B(cid:2) . The Q , C μ R μ L μ 3 Q ,Q contentofthehyperchargeoperatorisgivenby 1L 1R Q (cid:4)c Q (cid:5)c Q (cid:5)c Q , (cid:2)1.4(cid:3) Y 1 1R 3 3 4 1L withc (cid:4) 1/2, c (cid:4) 1/6,andc (cid:4) −1/2(cid:7)13(cid:8).ThecorrespondingfermionandHiggsdoublet 1 3 4 quantumnumbersaregiveninTable1.ThecriteriaweadoptheretodefinetheHiggscharges are to make the Yukawa couplings (cid:2)HUQ, H†DQ, H†EL, HN L(cid:3) invariant under all i i i i i i i i 4 AdvancesinHighEnergyPhysics Table1:QuantumnumbersofchiralfermionsandHiggsdoublet. Name Representation Q Q Q Q 3 1L 1R Y 1 Q (cid:2)3,2(cid:3) 1 0 0 i 6 2 − Ui (cid:2)3,1(cid:3) −1 0 −1 3 1 Di (cid:2)3,1(cid:3) −1 0 1 3 1 − Li (cid:2)1,2(cid:3) 0 1 0 2 E (cid:2)1,1(cid:3) 0 −1 1 1 i N (cid:2)1,1(cid:3) 0 −1 −1 0 i 1 H (cid:2)1,2(cid:3) 0 0 1 2 three U(cid:2)1(cid:3)’s. From Table1, UQ has the charges (cid:2)0,0,−1(cid:3) and DQ has (cid:2)0,0,1(cid:3); therefore, i i i i the Higgs H has Q (cid:4) Q (cid:4) 0, Q (cid:4) 1, Q (cid:4) 1/2, whereas H† has opposite charges 3 1L 1R Y Q (cid:4)Q (cid:4)0,Q (cid:4)−1,Q (cid:4)−1/2.ThetwoextraU(cid:2)1(cid:3)’sarethebaryonandleptonnumber; 3 1L 1R Y theyaregivenbythefollowingcombinations: Q 1 1 1 B (cid:4) 3, L(cid:4)Q , Q (cid:4) Q − Q (cid:5) Q , (cid:2)1.5(cid:3) 1L Y 3 1L 1R 3 6 2 2 orequivalentlybytheinverserelations Q (cid:4)3B, Q (cid:4)L, Q (cid:4)2Q −(cid:2)B−L(cid:3). (cid:2)1.6(cid:3) 3 1L 1R Y EventhoughBisanomalous,withtheadditionofthreefermionsingletsN,thecombination i B−Lisanomalyfree.OnecanverifybyinspectionofTable1thattheseN havethequantum i numbers of right-handed neutrinos, that is, singlets under hypercharge. Therefore, this is a firstinterestingpredictionoftheU(cid:2)3(cid:3) ×Sp(cid:2)1(cid:3) ×U(cid:2)1(cid:3) ×U(cid:2)1(cid:3) gaugetheory:right-handed C L L R neutrinosmustexist. Beforediscussingthefavorablephenomenologicalimplicationsofthemodel,wedetail somedesirablepropertieswhichapplytogenericmodelswithmultipleU(cid:2)1(cid:3)symmetries. 2. Running of the Abelian Gauge Couplings We begin with the covariant derivative for the U(cid:2)1(cid:3) fields in the “flavor” 1,2,3,... basis in whichitisassumedthatthekineticenergytermscontainingXi arecanonicallynormalized: μ (cid:3) D (cid:4)∂ −i g(cid:2)QXi. (cid:2)2.1(cid:3) μ μ i i μ AdvancesinHighEnergyPhysics 5 The relations between the U(cid:2)1(cid:3) couplings g(cid:2) and any nonabelian counterparts are left open i (cid:4) fornow.WecarryoutanorthogonaltransformationofthefieldsXi (cid:4) O Yj.Thecovariant μ j ij μ derivativebecomes (cid:3)(cid:3) D (cid:4)∂ −i g(cid:2)QO Yj μ μ i i ij μ i j (cid:3) (cid:2)2.2(cid:3) (cid:4)∂ −i g Q Yj, μ j j μ j whereforeachj (cid:3) gjQj (cid:4) gi(cid:2)QiOij. (cid:2)2.3(cid:3) i Next,supposeweareprovidedwithnormalizationforthehypercharge(cid:2)takenasj (cid:4)1(cid:3) (cid:3) QY (cid:4) ciQi, (cid:2)2.4(cid:3) i hereafterweomitthebarsforsimplicity.Rewriting(cid:2)2.3(cid:3)forthehypercharge (cid:3) gYQY (cid:4) gi(cid:2)QiOi1 (cid:2)2.5(cid:3) i andsubstituting(cid:2)2.4(cid:3)into(cid:2)2.5(cid:3),weobtain (cid:3) (cid:3) gY Qici (cid:4) gi(cid:2)Oi1Qi. (cid:2)2.6(cid:3) i i One can think about the charges Qi,p as vectors with the(cid:4)components labeled by particles p. Let us fir(cid:4)st take the charges to be orthogonal, that is, pQi,pQk,p (cid:4) 0 for i/(cid:4)k. Multiplying(cid:2)2.6(cid:3)by Q , p k,p (cid:3) (cid:3) (cid:3) (cid:3) Qk,pgY Qi,pci (cid:4) Qk,p gi(cid:2)Oi1Qi,p, (cid:2)2.7(cid:3) p i p i weobtain g c (cid:4)g(cid:2)O , (cid:2)2.8(cid:3) Y i i i1 orequivalently g c Oi1 (cid:4) Y(cid:2)i. (cid:2)2.9(cid:3) g i 6 AdvancesinHighEnergyPhysics (cid:4) Orthogonalityoftherotationmatrix, O2 (cid:4)1,implies i i1 (cid:5) (cid:6) (cid:3) 2 g2 ci (cid:4)1. (cid:2)2.10(cid:3) Y g(cid:2) i i Then,thecondition (cid:5) (cid:6) (cid:3) 2 P ≡ 1 − ci (cid:4)0 (cid:2)2.11(cid:3) g2 g(cid:2) Y i i encodes the orthogonality of the mixing matrix connecting the fields coupled to the flavor charges Q ,Q ,Q ,... and the fields rotated, so that one of them, Y, couples to the 1 2 3 hypercharge Q . Therefore, for orthogonal charges, as the couplings run with energy, the Y conditionP (cid:4)0needstostayintact(cid:7)5(cid:8). A very important point is that the couplings that are running are those of the U(cid:2)1(cid:3) fields; hence the β functions receive contributions from fermions and scalars, but not from gauge bosons. As a consequence, if we start with a set of couplings at a high mass scale Λ satisfyingP (cid:4) 0,thisconditionwillbemaintainedatoneloopasthecouplingsrundownto lowerenergies(cid:2)Q(cid:3).Theone-loopcorrectiontothevariouscouplingsis (cid:7) (cid:8) 1 1 b Q (cid:4) − Y ln , (cid:2)2.12(cid:3) α (cid:2)Q(cid:3) α (cid:2)Λ(cid:3) 2π Λ Y Y (cid:7) (cid:8) 1 1 b Q (cid:4) − i ln , (cid:2)2.13(cid:3) α(cid:2)Q(cid:3) α(cid:2)Λ(cid:3) 2π Λ i i where 2 1 b (cid:4) TrQ2 (cid:5) TrQ2 , Y 3 Y,f 3 Y,s (cid:2)2.14(cid:3) 2 1 b (cid:4) TrQ2 (cid:5) TrQ2 , i 3 i,f 3 i,s withf andsindicatingcontributionfromferm(cid:4)ionandscalar(cid:4)loops,respectively. Recall that the charges are orthogonal, sQi,sQk,s (cid:4) fQi,fQk,f (cid:4) 0 for i/(cid:4)k. Then (cid:2)2.4(cid:3)implies (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) QY2,s (cid:4) ci2 Qi2,s, QY2,f (cid:4) ci2 Qi2,f, (cid:2)2.15(cid:3) s i s f i f hence, (cid:3) bY (cid:4) ci2bi. (cid:2)2.16(cid:3) i AdvancesinHighEnergyPhysics 7 Ontheother,theRG-inducedchangeofP definedin(cid:2)2.11(cid:3)reads (cid:7) (cid:8) (cid:7) (cid:8) (cid:3) 1 1 ΔP (cid:4)Δ − c2Δ α i α Y i i (cid:5) (cid:6) (cid:7) (cid:8) (cid:2)2.17(cid:3) (cid:3) 1 Q (cid:4) b − c2b ln . 2π Y i i Λ i Thus,P (cid:4)0staysvalidtooneloopifthechargesareorthogonal(cid:7)5(cid:8). Shouldthechargesnotbeorthogonal,itisinstructivetowrite(cid:2)2.6(cid:3)asV·Q(cid:4)0,where g c Vi (cid:4)Oi1− Y(cid:2)i. (cid:2)2.18(cid:3) g i CertainlyV (cid:4) 0stillholdsasapossiblesolution.Butasthechargesdonotformamutually i orthogonal basis, one can ask whether other solutions exist. This will be the case if, for nonzeroV, (cid:3) ViQiα (cid:4)0 (cid:2)2.19(cid:3) i for each α, where Qα is the U(cid:2)1(cid:3) charge of the particle α. In the U(cid:2)3(cid:3) × U(cid:2)2(cid:3) × U(cid:2)1(cid:3) i gauge group of (cid:7)12(cid:8), the right-handed electron is charged only with respect to one of the abelian groups. From (cid:2)2.19(cid:3), this sets one of the V’s (cid:2)say V (cid:3) equal to zero. For α (cid:4) Q,U, 1 i i D,L,E,N,H, there remain at least 4 additional equations satisfied by the remaining i i i i componentsV andV .TheresultingovercompletenessleadstoV (cid:4)V (cid:4)0. 2 3 2 3 Although in most models the condition P (cid:4) 0 holds in spite of the nonorthogonality oftheQ’s,theRGequationscontrollingtherunningofthecouplingslosetheirsimplicity.In i particular,since (cid:3) TrQY2 /(cid:4) ci2TrQi2, (cid:2)2.20(cid:3) i theRGequationsbecomecoupled.Inaddition,kineticmixingisgeneratedatonelooplevel evenifitisabsentinitially(cid:7)14,15(cid:8).Removalofthemixingterminordertorestorecanonical gaugekineticenergyrequiresanadditionalO(cid:2)3(cid:3)rotation,greatlycomplicatingtheanalysis. Here,weareconsideringmodelswheretheunderlyingsymmetryathighenergiesis U(cid:2)N(cid:3)ratherthanSU(cid:2)N(cid:3).Following(cid:7)12(cid:8),wenormalizeallU(cid:2)N(cid:3)generatorsaccordingto (cid:9) (cid:10) Tr TaTb (cid:4) 1δab, (cid:2)2.21(cid:3) 2 √ andmeasurethecorrespondingU(cid:2)1(cid:3) chargeswithrespecttothecouplingg / 2N,with N N g as the SU(cid:2)N(cid:3) coupling constant. Hence, the fundamental representation of SU(cid:2)N(cid:3) has N U(cid:2)1(cid:3) chargeunity.AnotherimportantelementoftheRGanalysisisthattheU(cid:2)1(cid:3)couplings N (cid:2)g(cid:2),g(cid:2),g(cid:2)(cid:3) run different from the nonabelian SU(cid:2)3(cid:3) (cid:2)g (cid:3) and SU(cid:2)2(cid:3) (cid:2)g (cid:3). This implies that 1 2 3 3 2 8 AdvancesinHighEnergyPhysics the p√revious relation for normalization of abelian and nonabelian coupling constants, g(cid:2) (cid:4) N g / 2N, holds only at the scale of U(cid:2)N(cid:3) unification (cid:7)5(cid:8). The SM chiral fermion charges N in Table1 are not orthogonal as given (cid:2)TrQ1LQ1R/(cid:4)0,(cid:3). Orthogonality can be completed by includingaright-handedneutrino. An obvious question is whether each of the fields on the rotated basis couples to a singlechargeQ.Let i L(cid:4)XTGQ (cid:2)2.22(cid:3) betheLagrangianinthe1,2,3,...basis,withXi andQ vectorsandGadiagonalmatrixin μ i N-dimensional“flavor”space.Nowrotatetoneworthogonalbasis(cid:2)Q(cid:3)forQ: Q(cid:4)RQ, (cid:2)2.23(cid:3) equation(cid:2)2.22(cid:3)becomes L(cid:4)XTGRQ. (cid:2)2.24(cid:3) Asitstands,eachXi doesnotcoupletoauniquechargeQ;hencewerotateX, μ i X(cid:4)OY, (cid:2)2.25(cid:3) toobtain L(cid:4)YTOTGRQ. (cid:2)2.26(cid:3) Wewishtoseeif,forgivenOandG,wecanfindanRsothat (cid:11) (cid:12) OTGR(cid:4)G diagonal . (cid:2)2.27(cid:3) i ThisallowseachY tocoupletoauniquechargeQ withstrengthg .Toseetheproblemwith μ i i this,werewrite(cid:2)2.27(cid:3)intermsofcomponents (cid:9) (cid:10) OT gjRjk (cid:4)giδik, (cid:2)2.28(cid:3) ij fori/(cid:4)k,(cid:2)2.28(cid:3)leadsto (cid:9) (cid:10) OT gjRjk (cid:4)0. (cid:2)2.29(cid:3) ij AdvancesinHighEnergyPhysics 9 In general, in (cid:2)2.29(cid:3) there are N(cid:2)N −1(cid:3) equations, but only N(cid:2)N −1(cid:3)/2 independent O ij generatorsinSO(cid:2)N(cid:3);thereforethesystemisoverdetermined(cid:7)16(cid:8).Ofcourse,ifG (cid:4) gI,the equationbecomes OTR(cid:4)I, (cid:2)2.30(cid:3) andsoO(cid:4)R. WeillustratewiththecaseN (cid:4)2;let (cid:5) (cid:6) C S R(cid:4) ϕ ϕ , −S C ϕ ϕ (cid:5) (cid:6) (cid:2) g 0 G(cid:4) 1 , (cid:2)2.31(cid:3) (cid:2) 0 g 3 (cid:5) (cid:6) C S O(cid:4) ϑ ϑ , −S C ϑ ϑ then, (cid:5) (cid:6) (cid:5) (cid:6) g(cid:2)C C (cid:5)g(cid:2)S S g(cid:2)C S −g(cid:2)S C g(cid:2) 0 OGR(cid:4) 1 ϑ ϕ 3 ϑ ϕ 1 ϑ ϕ 3 ϑ ϕ (cid:4) 1 . (cid:2)2.32(cid:3) g(cid:2)S C (cid:5)g(cid:2)C S g(cid:2)S S −g(cid:2)C C 0 g(cid:2) 1 ϑ ϕ 3 ϑ ϕ 1 ϑ ϕ 3 ϑ ϕ 3 Fromtheoff-diagonalterms,weobtain (cid:2) g g(cid:2)C S −g(cid:2)S C (cid:4)0(cid:4)⇒tanϑ(cid:4) 1 tanϕ, 1 ϑ ϕ 2 ϑ ϕ g(cid:2) 2 (cid:2)2.33(cid:3) (cid:2) g g(cid:2)S C −g(cid:2)C S (cid:4)0(cid:4)⇒tanϑ(cid:4) 2 tanϕ 1 ϑ ϕ 2 ϑ ϕ g(cid:2) 1 whichimpliesthatg(cid:2) (cid:4) g(cid:2) (cid:4) g orequivalentlythatGisamultipleoftheunitmatrix.Next, 1 2 weconsiderthediagonalelementsusingg(cid:2) (cid:4)g(cid:2) toobtain 1 2 (cid:11) (cid:12) cos ϑ−ϕ (cid:4)0(cid:4)⇒ϑ(cid:4)ϕ. (cid:2)2.34(cid:3) NotethatthematrixRhasoneindependentvariable,andtherearetwoindependenthomo- geneousequations. (cid:2) (cid:2) Any vector boson Y , orthogonal to the hypercharge, must grow a mass M in μ order to avoid long range forces between baryons other than gravity and Coulomb forces. The anomalous mass growth allows the survival of global baryon number conservation, preventingfastprotondecay(cid:7)17(cid:8).Itisthisthatwenowturntostudy. 10 AdvancesinHighEnergyPhysics 3. Premises of the Anomalous Sector OutsideoftheHiggscouplings,therelevantpartsoftheLagrangianarethegaugecouplings generatedbytheU(cid:2)1(cid:3)covariantderivativesactingonthematterfieldsandthe(cid:2)mass(cid:3)2matrix oftheanomaloussector L(cid:4)QTGX(cid:5) 1XTM2X, (cid:2)3.1(cid:3) 2 where Xi are the three U(cid:2)1(cid:3) gauge fields in the D-brane basis (cid:2)B ,C ,B(cid:2) (cid:3), G is a diagonal μ μ μ μ couplingmatrix(cid:2)g(cid:2),g(cid:2),g(cid:2)(cid:3),andQarethe3chargematrices. 1 3 4 Again,performarotationX (cid:4) OYandrequirethatoneoftheY’s(cid:2)sayY (cid:3)couplesto μ hypercharge.WethenobtaintheconstraintonthefirstcolumnofOgivenin(cid:2)2.9(cid:3).However, there is now an additional constraint: the field Y is an eigenstate of M2 with zero eigenvalue. μ UndertheOrotation,themasstermbecomes 1XTM2X(cid:4) 1YTM2Y, (cid:2)3.2(cid:3) 2 2 withM2 (cid:4) OTM2O.WeknowthatatleastY isaneigenstatewitheigenvaluezero.Wealso μ know that Poincare invariance requires the complete diagonalization of the mass matrix in order to deal with observables. However, further similarity transformations will undo the couplingofthezeroeigenstatetohypercharge.Thereseemsnowayofeventuallyfulfillingall theseconditionsexcepttorequirethatthesameOwhichrotatestocoupleY tohypercharge μ simultaneouslydiagonalizesM2sothat (cid:9) (cid:10) M2 (cid:4)diag 0,M(cid:2)2,M(cid:2)(cid:2)2 . (cid:2)3.3(cid:3) ThisimpliesthattheoriginalM2intheflavorbasisisgivenby (cid:9) (cid:10) M2 (cid:4)Odiag 0,M(cid:2)2,M(cid:2)(cid:2)2 OT, (cid:2)3.4(cid:3) whichresultsinthefollowingbaroquematrix: ⎛ ⎞ a b c ⎜ ⎟ ⎜ ⎟ M2 (cid:4)⎜b d e⎟, (cid:2)3.5(cid:3) ⎝ ⎠ c e f