Luis Alfredo Anchordoqui

44 Pages·2014·1.77 MB·English

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