JournalofModernPhysics,2019,10,35-42
http://www.scirp.org/journal/jmp
ISSNOnline:2153-120X
ISSNPrint:2153-1196
ATwo-Higgs-DoubletModelwithout
Flavor-ChangingNeutralCurrentsatTree-Level
ChilongLin
NationalMuseumofNaturalScience,Taiwan
Howtocitethispaper:Lin,C.L.(2019)
ATwo-Higgs-DoubletModelwithoutFlavor-
ChangingNeutralCurrentsatTree-Level.
JournalofModernPhysics,10, 35-42.
https://doi.org/10.4236/jmp.2019.101004
Received:December21,2018
Accepted:January20,2019
Published:January23,2019
Copyright
c
2019byauthor(s)andScientific
Research PublishingInc.
This work is licensed under the Creative Com-
monsAttributionInternationalLicense(CC
BY4.0).
http://creativecommons.org/licenses/by/4.0/
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lentesquehabitantmorbitristiquesenectuset
netusetmalesuadafamesacturpisegestas.
Phasellusidmagna.Duismalesuadainter-
dumarcu.Integer metus.Morbipulvinarpel-
lentesquemi.Suspendissesedesteumagna
molestieegestas.Quisquemilorem,pulvinar
eget,egestasquis,luctusat,ante.Proinauc-
torvehiculapurus.Fusceac nislaliquamante
hendreritpellentesque.Classaptenttacitiso-
ciosquadlitoratorquentperconubianostra,
perinceptoshymenaeos.Morbiwisi.Etiam
arcumauris,facilisissed,eleifendnon,non-
ummyut,pede.Crasutlacustempormetus
mollisplacerat.Vivamuseutortorvelmetus
interdum malesuada.
Sedeleifend,erossitametfaucibuselemen-
tum,urnasapienconsectetuermauris,quis
egestasleojustononrisus.Morbinonfelisac
liberovulputatefringilla.Maurisliberoeros,
lacinianon,sodalesquis,dapibusporttitor,
pede.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Morbi dapibus mauris condimentum
nulla.Cum sociisnatoque penatibus etmagnis
disparturient montes,nasceturridiculusmus.
Etiamsitameterat.Nullavarius.Etiamt-
inciduntduivitaeturpis.Donecleo.Morbi
vulputateconvallisest.Integeraliquet.Pel-
lentesque aliquetsodalesurna.
Nullameleifendjustoinnisl.Inhachabitasse
plateadictumst.Morbinonummy.Aliquam
utfelis.Invelitleo,dictumvitae,posuereid,
vulputatenec,ante.Maecenasvitaepedenec
dui dignissim suscipit.Morbimagna.Vestibu-
lumidpurusegetvelitlaoreetlaoreet.Prae-
sentsedleovelnibhconvallisblandit.Ut
rutrum.Donecnibh.Donecinterdum.Fusce
sed pede sit amet elitrhoncus ultrices.Nullam
atenimvitaepedevehiculaiaculis.
Classaptenttacitisociosquadlitoratorquen-
tperconubianostra, perinceptoshymenaeos.
Aeneannonummyturpisidodio.Integereu-
ismodimperdietturpis.Utnecleonecdi-
amimperdietlacinia.Etiamegetlacuseget
miultriciesposuere.Inplacerattristiquetor-
tor.Sedportavestibulummetus.Nullaia-
culissollicitudinpede.Fusceluctustellusin
dolor.Curabiturauctorvelitasem.Mor-
bisapien.Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Donecadipiscingurnavehic-
ulanunc.Sedornareleoinleo.Inrhoncus
leoutdui.Aeneandolorquam,volutpatnec,
fringillaid,consectetuervel,pede.
Nullamalesuadarisusuturna.Aenean
pretiumvelitsitametmetus.Duisiaculis.In
hachabitasseplateadictumst.Nullammo-
lestieturpisegetnisl.Duisamassaidpede
dapibusultricies.Sedeuleo.Inatmaurissit
amettortorbibendumvarius.Phasellusjusto
risus,posuerein,sagittisac,variusvel,tor-
tor.Quisqueidenim.Phasellusconsequat,
liberopretium nonummy fringilla,tortor lacus
vestibulumnunc,utrhoncusligulanequeid
justo.Nullamaccumsaneuismodnunc.Proin
vitae ipsumac metusdictum tempus.Nam ut
wisi.Quisque tortor felis, interdum ac, sodales
a,sempera,sem.Curabiturinvelitsitamet
duitristiquesodales.Vivamusmaurispede,
laciniaeget,pellentesquequis,scelerisqueeu,
est.Aliquamrisus.Quisquebibendumpede
eudolor.
Donec tempusneque vitaeest.Aenean egestas
odio sed risus ullamcorper ullamcorper.Sed in
nullaatortortinciduntegestas.Namsapien
tortor,elementumsitamet,aliquamin,port-
titorfaucibus,enim.Nullamconguesuscipit
nibh.Quisqueconvallis.Praesentarcunibh,
vehicula eget,accumsaneu,tincidunta,nibh.
Suspendissevulputate,tortorquisadipiscing
viverra, lacus nibh dignissim tellus, eu suscipit
risusantefringilladiam.Quisquealiberov-
elpedeimperdietaliquet.Pellentesquenunc
nibh,eleifenda,consequatconsequat,hen-
dreritnec,diam.Sedurna.Maecenaslaoreet
eleifendneque.Vivamuspurusodio,eleifend
non,iaculisa,ultricessitamet,urna.Mau-
risfaucibusodiovitaerisus.Innisl.Praesent
purus.Integeriaculis,semeuegestaslacinia,
lacuspedescelerisqueaugue,inullamcorper
dolorerosaclacus.Nuncinlibero.
Fuscesuscipitcursussem.Vivamusrisusmi,
egestasac,imperdietvarius,faucibusquis,
leo.Aeneantincidunt.Donecsuscipit.Cras
idjustoquisnibhscelerisquedignissim.Ali-
quamsagittiselementumdolor.Aeneancon-
sectetuerjustoinpede.Curabiturullamcorper
ligula nec orci.Aliquam purus turpis,aliquam
id, ornarevitae, porttitor non,wisi.Maecenas
luctusportalorem.Donecvitae ligulaeuante
pretiumvarius.Prointortormetus,convallis
et,hendreritnon,scelerisquein,urna.Cras
quisliberoeuligulabibendumtempor.Viva-
mustellusquam,malesuadaeu,tempussed,
temporsed,velit.Doneclaciniaauctorlibero.
Praesentsednequeidpedemollisrutrum.
Vestibulumiaculisrisus.Pellentesquelacus.
Ut quisnunc sedodio malesuada egestas.Duis
amagnasitametligulatristiquepretium.Ut
pharetra.Vestibulumimperdietmagnanec
wisi.Maurisconvallis.Sedaccumsansollic-
itudinmassa.Sedidenim.Nuncpedeenim,
laciniaut,pulvinarquis,suscipitsemper,elit.
Crasaccumsaneratvitaeenim.Crassollici-
tudin.Vestibulumrutrumblanditmassa.
Sedgravidalectusutpurus.Morbilaoreet
magna.Pellentesque eu wisi.Proin turpis.In-
tegersollicitudinauguenecdui.Fuscelectus.
Vivamusfaucibusnullaneclacus.Integerdi-
am.Pellentesquesodales,enimfeugiatcursus
volutpat,semmaurisdignissimmauris,quis
consequatsemestfermentumligula.Nullam
justo lectus, condimentum sit amet, posuere a,
fringillamollis, felis.Morbinulla nibh,pellen-
tesqueat,nonummyeu,sollicitudinnec,ip-
sum.Crasneque.Nuncaugue.Nullamvitae
quamidquampulvinar blandit.Nunc sitamet
orci.Aliquameratelit,pharetranec,aliquet
a,gravidain,mi.Quisqueurnaenim,viverra
quis,suscipitquis,tinciduntut,sapien.Cras
placeratconsequatsem.Curabituracdiam.
Curabiturdiamtortor,molliset,viverraac,
tempusvel,metus.
Curabituraclorem.Vivamusnonjustoin
duimattisposuere.Etiamaccumsanligula
idpede.Maecenastinciduntdiamnecvelit.
Praesentconvallissapienacest.Aliquamul-
lamcorpereuismodnulla.Integermollisenim
veltortor.Nullasodalesplaceratnunc.Sed
tempusrutrumwisi.Duisaccumsangravi-
dapurus.Nuncnunc.Etiamfacilisisduieu
sem.Vestibulumsemper.Praesenteueros.
Vestibulumtellusnisl,dapibusid,vestibulum
sit amet,placerat ac, mauris.Maecenas etelit
uteratplaceratdictum.Namfeugiat,turpis
etsodalesvolutpat,wisi quamrhoncusneque,
vitaealiquamipsumsapienvelenim.Maece-
nassuscipitcursusmi.
Quisque consectetuer.In suscipit mauris a do-
lor pellentesque consectetuer.Mauris convallis
nequenonerat.Inlacinia.Pellentesqueleo
eros,sagittisquis,fermentumquis,tincidun-
tut,sapien.Maecenassem.Curabitureros
odio,interdumeu,feugiateu,portaac,nis-
l.Curabiturnunc.Etiam fermentum convallis
velit.Pellentesquelaoreetlacus.Quisquesed
elit.Namquistellus.Aliquamtellusarcu,
adipiscingnon,tincidunteleifend,adipiscing
quis,augue.Vivamuselementumplaceraten-
im.Suspendisseuttortor.Integerfaucibus
adipiscingfelis.Aeneanconsectetuermattis
lectus.Morbimalesuadafaucibusdolor.Nam
lacus.Etiamarcu libero,malesuada vitae,ali-
quamvitae,blandittristique,nisl.
Maecenasaccumsandapibussapien.Duis
pretiumiaculisarcu.Curabiturutlacus.Ali-
quamvulputate.Suspendisseutpurussedsem
tempor rhoncus.Utquam dui, fringillaat, dic-
tum eget,ultricies quis,quam.Etiam semest,
pharetra non, vulputate in, pretium at, ipsum.
Nunc semper sagittis orci.Sed scelerisquesus-
cipitdiam.Utvolutpat,doloratullamcorp-
ertristique,erospurusmollisquam,sitamet
ornareantenuncetenim.
Phasellusfringilla,metusidfeugiatcon-
sectetuer,lacuswisiultricestellus,quis lobor-
tisnibhloremquistortor.Donecegestas
ornarenulla.Maurismitellus,portafau-
cibus,dictumvel,nonummyin,est.Ali-
quameratvolutpat.Intellusmagna,port-
titorlacinia,molestievitae,pellentesqueeu,
justo.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Sedorcinibh,scelerisquesitamet,
suscipitsed,placeratvel,diam.Vestibulum
nonummy vulputate orci.Donec et velit ac ar-
cuinterdumsemper.Morbipedeorci,cursus
ac,elementumnon,vehiculaut,lacus.Cras
volutpat.Namvelwisiquisliberovenenatis
placerat.Aeneansedodio.Quisqueposuere
purusacorci.Vivamusodio.Vivamusvar-
ius,nullasitametsemperviverra,odiomau-
risconsequatlacus,atvestibulumnequearcu
eutortor.Doneciaculistincidunttellus.Ali-
quameratvolutpat.Curabiturmagnalorem,
dignissimvolutpat,viverraet,adipiscingnec,
dolor.Praesentlacusmauris,dapibusvitae,
sollicitudinsitamet,nonummyeget,ligula.
Crasegestasipsumanisl.Vivamusvariusdo-
lorutdolor.Fuscevelenim.Pellentesque
accumsanligulaeteros.Crasidlacusnon
tortorfacilisisfacilisis.Etiamnislelit,cur-
sussed,fringillain,conguenec,urna.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.In-
tegeratturpis.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Duisfringilla,ligulasedporta
fringilla,ligulawisicommodofelis,utadipisc-
ingfelisduiinenim.Suspendissemalesuada
ultricesante.Pellentesquescelerisqueaugue
sitameturna.Nullavolutpataliquettortor.
Crasaliquam,tellusataliquetpellentesque,
justosapiencommodoleo,idrhoncussapien
quamaterat.Nullacommodo,wisiegetsol-
licitudinpretium,orciorcialiquamorci,ut
cursusturpisjustoetlacus.Nullaveltor-
tor.Quisque eratelit, viverrasit amet, sagittis
eget,portasitamet,lacus.
In hac habitasse platea dictumst.Proin at est.
Curabitur tempusvulputate elit.Pellentesque
sem.Praesent eusapien.Duiselitmagna, ali-
quet at,tempus sed, vehicula non, enim.Mor-
biviverraarcunecpurus.Vivamusfringilla,
enimet commodomalesuada,tortor metusel-
ementumligula,necaliquetestsapienutlec-
tus.Aliquammi.Utnecelit.Fusceeuismod
luctustellus.Curabiturscelerisque.Nullam
purus.Namultricies accumsanmagna.Morbi
pulvinarloremsitametipsum.Donecutjus-
tovitaenibhmolliscongue.Fuscequisdiam.
Praesent tempuserosutquam.
Donecinnisl.Fuscevitaeest.Vivamusante
ante,mattislaoreet,posuereeget,conguevel,
nunc.Fuscesem.Namvelorcieuerosviver-
raluctus.Pellentesquesitametaugue.Nunc
sitametipsumetlacusvariusnonummy.In-
teger rutrum sem egetwisi.Aenean eu sapien.
Quisqueornaredignissimmi.Duisaurnavel
risus pharetraimperdiet.Suspendisse potenti.
Morbijusto.Aeneannecdolor.Inhac
habitasseplateadictumst.Proinnonummy
porttitorvelit.Sedsitametleonecmetus
rhoncusvarius.Crasante.Vestibulumcom-
modosemtinciduntmassa.Namjusto.Ae-
neanluctus,felisetcondimentumlacinia,lec-
tusenimpulvinarpurus,nonportavelitnisl
sed eros.Suspendisse consequat.Mauris a dui
et tortormattispretium.Sed nulla metus,vo-
lutpatid, aliquameget, ullamcorperut, ipsum.
Morbieununc.Praesentpretium.Duisali-
quampulvinarligula.Ut blanditegestasjusto.
Quisqueposueremetusviverrapede.
Vivamussodaleselementumneque.Viva-
musdignissimaccumsanneque.Sedatenim.
Vestibulumnonummyinterdumpurus.Mau-
ris ornarevelitid nibh pretiumultricies.Fusce
temporpellentesqueodio.Vivamusauguepu-
rus,laoreetin,scelerisquevel,commodoid,
wisi.Duisenim.Nullainterdum,nunceusem-
pereleifend,enimdolorpretiumelit,utcom-
modoligulanislaest.Vivamusante.Nulla
leomassa,posuerenec,volutpatvitae,rhon-
cuseu,magna.
Quisquefacilisisauctorsapien.Pellentesque
gravidahendreritlectus.Maurisrutrumso-
dalessapien.Fuscehendreritsemvellorem.
Integerpellentesquemassavelaugue.Integer
elit tortor, feugiat quis, sagittis et, ornare non,
lacus.Vestibulumposuerepellentesqueeros.
Quisquevenenatisipsumdictumnulla.Ali-
quam quis quam nonmetus eleifend interdum.
Nam eget sapien ac mauris malesuadaadipisc-
ing.Etiameleifendnequesedquam.Nulla
facilisi.Proinaligula.Sedidduieunibh
egestastincidunt.Suspendissearcu.
Maecenasdui.Aliquamvolutpatauctorlorem.
Crasplaceratestvitaelectus.Curabitur
massalectus,rutrumeuismod,dignissimut,
dapibus a, odio.Ut eros erat, vulputate ut, in-
terdumnon,portaeu,erat.Crasfermentum,
felis inportacongue, velit leo facilisis odio,vi-
taeconsectetuerloremquamvitaeorci.Sed
ultrices,pedeeuplaceratauctor,anteligula
rutrum tellus,vel posuere nibh lacusnec nibh.
Maecenaslaoreetdoloratenim.Donecmo-
lestie dolornec metus.Vestibulum libero.Sed
quiserat.Sedtristique.Duispedeleo,fer-
mentumquis,consectetuereget,vulputatesit
amet,erat.
Donecvitaevelit.Suspendisseportafermen-
tummauris.Utvelnuncnonmaurisphare-
travarius.Duisconsequatliberoquisurna.
Maecenasatante.Vivamusvarius,wisised
egestastristique,odiowisiluctusnulla,lobor-
tisdictumdolorligulainlacus.Vivamusali-
quam, urna sed interdum porttitor,metus orci
interdum odio,sit ameteuismod lectusfelis et
leo.Praesentacwisi.Namsuscipitvestibu-
lum sem.Praesent eu ipsumvitae pede cursus
venenatis.Duis sed odio.Vestibulum eleifend.
Nullautmassa.Proinrutrummattissapien.
Curabiturdictumgravidaante.
Phasellusplaceratvulputatequam.Maece-
nasattellus.Pellentesquenequediam,dig-
nissim ac, venenatisvitae, consequat ut, lacus.
Nam nibh.Vestibulum fringilla arcu mollis ar-
cu.Sedetturpis.Donecsemtellus,volut-
pat et,varius eu,commodo sed,lectus.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisqueenimarcu,suscipitnec,tempus
at,imperdietvel,metus.Morbivolutpatpu-
rusaterat.Donecdignissim,semidsemper
tempus, nibh massa eleifendturpis,sedpellen-
tesquewisipurussedlibero.Nullamlobortis
tortorvelrisus.Pellentesqueconsequatnulla
eutellus.Donecvelit.Aliquamfermentum,
wisiacrhoncusiaculis,tellusnuncmalesuada
orci,quisvolutpat duimagnaidmi.Nuncvel
ante.Duisvitaelacus.Crasnecipsum.
Morbinunc.Aliquamconsectetuervariusnul-
la.Phaselluseros.Crasdapibusporttitorris-
us.Maecenasultricesmiseddiam.Praesent
gravida velit at elit vehicula porttitor.Phasel-
lusnislmi,sagittisac, pulvinarid,gravidasit
amet,erat.Vestibulumest.Loremipsumdo-
lorsit amet,consectetueradipiscingelit.Cur-
abitur idsem elementum leo rutrumhendrerit.
Utatmi.Donectinciduntfaucibusmassa.
Sed turpis quam, sollicitudin a, hendrerit eget,
pretiumut,nisl.Duishendreritligula.Nunc
pulvinarcongueurna.
Nuncvelit.Nullamelitsapien,eleifendeu,
commodo nec, semper sit amet, elit.Nulla lec-
tus risus,condimentum ut,laoreet eget,viver-
ra nec, odio.Proin lobortis.Curabitur dictum
arcuvelwisi.Crasidnullavenenatistortor
congueultrices.Pellentesqueegetpede.Sed
eleifendsagittiselit.Namsedtellussitamet
lectusullamcorper tristique.Maurisenim sem,
tristiqueeu,accumsanat,scelerisquevulpu-
tate, neque.Quisquelacus.Donec etipsum sit
amet elitnonummy aliquet.Sed viverranisl at
sem.Namdiam.Maurisutdolor.Curabitur
ornaretortorcursusvelit.
Morbitinciduntposuerearcu.Crasvenenatis
estvitaedolor.Vivamusscelerisquesemper
mi.Donecipsumarcu,consequatscelerisque,
viverraid,dictumat,metus.Loremipsum
dolor sit amet, consectetuer adipiscing elit.Ut
pede sem, tempus ut, porttitor bibendum, mo-
lestieeu,elit.Suspendissepotenti.Sedidlec-
tus sitamet purus faucibusvehicula.Praesen-
tsedsemnonduipharetrainterdum.Nam
viverra ultricesmagna.
Aeneanlaoreetaliquamorci.Nuncinterdum
elementumurna.Quisqueerat.Nullamtem-
porneque.Maecenasvelitnibh,scelerisquea,
consequatut,viverrain,enim.Duismagna.
Donecodioneque,tristiqueet,tincidunteu,
rhoncusac, nunc.Maurismalesuada malesua-
da elit.Etiam lacus mauris, pretium vel, blan-
ditin, ultriciesid,libero.Phasellusbibendum
eratutdiam.Incongueimperdietlectus.
Aeneanscelerisque.Fuscepretiumporttitor
lorem.Inhachabitasseplatea dictumst.Nulla
sitametnislatsapienegestaspretium.Nunc
nontellus.Vivamusaliquet.Namadipiscing
euismoddolor.Aliquameratvolutpat.Nul-
lautipsum.Quisquetinciduntauctorau-
gue.Nuncimperdietipsumegetelit.Ali-
quamquamleo,consectetuernon,ornaresit
amet,tristique quis,felis.Vestibulumante ip-
sumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Pellentesqueinterdum
quamsitametmi.Pellentesquemaurisdui,
dictuma,adipiscingac,fermentumsitamet,
lorem.
Utquiswisi.Praesentquismassa.Vivamus
egestasrisusegetlacus.Nunctincidunt,risus
quisbibendumfacilisis,lorempurusrutrum
neque,necportatortorurnaquisorci.Ae-
neanaliquet,liberosempervolutpatluctus,
pedeeratlaciniaaugue,quisrutrumsemip-
sumsitametpede.Vestibulumaliquet,nibh
sediaculissagittis,odiodolorblanditaugue,
egetmollisurnatellusidtellus.Aeneanali-
quetaliquamnunc.Nullaultriciesjustoeget
orci.Phasellustristiquefermentumleo.Sed
massametus,sagittisut,semperut,pharetra
vel,erat.Aliquamquamturpis,egestasvel,
elementumin,egestassitamet,lorem.Duis
convallis,wisisitametmollismolestie,libero
mauris portadui, vitaealiquam arcu turpisac
sem.Aliquamaliquetdapibusmetus.
Vivamuscommodoeroseleifenddui.Vestibu-
luminleoeuerattristiquemattis.Crasat
elit.Craspellentesque.Nullamidlacussit
ametliberoaliquethendrerit.Proinplacerat,
minonelementumlaoreet,eroselittincidun-
tmagna,arhoncussemarcuidodio.Nul-
laegetleoaleoegestasfacilisis.Curabitur
quisvelit.Phasellusaliquam,tortornecornare
rhoncus,purusurnaposuerevelit,etcommo-
dorisustellusquistellus.Vivamusleoturpis,
tempussitamet,tristiquevitae,laoreetquis,
odio.Proinscelerisquebibendumipsum.Eti-
amnisl.Praesentveldolor.Pellentesquevel
magna.Curabitururna.Vivamuscongueur-
na in velit.Etiam ullamcorper elementum dui.
Praesentnon urna.Sed placeratquam non mi.
Pellentesquediammagna,ultricieseget,ultri-
cesplacerat,adipiscingrutrum,sem.
Morbisem.Nullafacilisi.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Nullafacilisi.Mor-
bisagittisultriceslibero.Praesenteuligula
sedsapienauctorsagittis.Classaptenttaciti
sociosquadlitoratorquentperconubianos-
tra,perinceptoshymenaeos.Donecvelnunc.
Nuncfermentum,lacusidaliquamporta,dui
tortoreuismoderos,velmolestieipsumpurus
eulacus.Vivamuspedearcu, euismod ac,tem-
pusid,pretiumet,lacus.Curabitursodales
dapibusurna.Nunceusapien.Doneceget
nuncapededictumpretium.Proinmauris.
Vivamusluctusliberovelnibh.
Fuscetristiquerisusidwisi.Integermolestie
massaidsem.Vestibulumveldolor.Pel-
lentesquevelurnavelrisusultricieselemen-
tum.Quisquesapien urna, blanditnec, iaculis
ac,viverrain,odio.Inhachabitasseplatea
dictumst.Morbinequelacus,convallisvitae,
commodoac,fermentumeu,velit.Sedinor-
ci.Infringillaturpisnonarcu.Donecinante.
Phasellustemporfeugiatvelit.Aeneanvar-
iusmassanonturpis.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;
Aliquamtortor.Morbiipsum massa,imperdi-
etnon,consectetuervel,feugiatvel,lorem.
Quisqueeget loremnec elitmalesuadavestibu-
lum.Quisquesollicitudinipsum vel sem.Nul-
laenim.Proinnonummyfelisvitaefelis.Nul-
lampellentesque.Duisrutrumfeugiatfe-
lis.Maurisvelpedesedliberotinciduntmol-
lis.Phasellussedurnarhoncusdiameuismod
bibendum.Phasellussednisl.Integercondi-
mentumjusto id orciiaculis varius.Quisque et
lacus.Phaselluselementum, justoat dignissim
auctor,wisi odio lobortisarcu,sed sollicitudin
felisfeliseuneque.Praesentatlacus.
Vivamussitametpede.Duisinterdum,nunc
egetrutrumdignissim,nisldiamluctusleo,
ettinciduntvelitnislidtellus.Inloremtel-
lus,aliquetvitae,portain,aliquetsed,lectus.
Phasellussodales.Utvariusscelerisqueerat.
Invelnibheuerosimperdietrutrum.Donec
ac odio nec neque vulputatesuscipit.Nam nec
magna.Pellentesquehabitantmorbitristique
senectus et netus et malesuada fames acturpis
egestas.Nullamporta,odioetsagittisiaculis,
wisi neque fringilla sapien, vel commodo lorem
loremidelit.Utsemlectus,scelerisqueeget,
placeratet,tinciduntscelerisque,ligula.Pel-
lentesque nonorci.
Etiamvelipsum.Morbifacilisisvestibulum
nisl.Praesentcursuslaoreetfelis.Integer
adipiscingpretiumorci.Nullafacilisi.Quisque
posuerebibendumpurus.Nullaquam mauris,
cursuseget,convallisac,molestienon,enim.
Aliquamcongue.Quisquesagittisnonummy
sapien.Proinmolestiesemvitaeurna.Mae-
cenas lorem.Vivamus viverra consequat enim.
Nuncsedpede.Praesentvitaelectus.Prae-
sentnequejusto,vehiculaeget,interdumid,
facilisiset,nibh.Phasellusatpurusetlibero
laciniadictum.Fuscealiquet.Nullaeuante
placeratleosemperdictum.Maurismetus.
Curabiturlobortis.Curabitursollicitudin hen-
dreritnunc.Donecultriceslacusidipsum.
Donecanibhutelitvestibulumtristique.In-
tegeratpede.Crasvolutpatvariusmagna.
Phaselluseuwisi.Praesentrisusjusto,lobor-
tis eget, scelerisqueac, aliquet in, dolor.Proin
idleo.Nunciaculis,mivitaeaccumsancom-
modo, nequesemlacinianulla,quisvestibulum
justosemineros.Quisquesedmassa.Morbi
lectus ipsum, vulputate a, mollis ut,accumsan
placerat,tellus.Nullaminwisi.Vivamuseu
ligulaanuncaccumsancongue.Suspendisse
aclibero.Aliquameratvolutpat.Donecau-
gue.Nuncvenenatisfringillanibh.Fusceac-
cumsan pulvinar justo.Nullam semper, duiut
dignissim auctor, orcilibero fringilla massa,b-
landitpulvinarpedetortoridmagna.Nunc
adipiscing justo sed velit tincidunt fermentum.
Integerplacerat.Pellentesquehabitantmorbi
tristique senectuset netus etmalesuada fames
acturpisegestas.Sedinmassa.Classaptent
tacitisociosquadlitoratorquentperconubi-
anostra,perinceptoshymenaeos.Phasellus
tempusaliquamrisus.Aliquamrutrumpurus
atmetus.Donecposuereodioaterat.Nam
nonnibh.Phasellusligula.Quisquevene-
natislectusinaugue.Sedvestibulumdapibus
neque.
Mauris tempus eros at nulla.Sed quisdui dig-
nissimmaurispretiumtincidunt.Maurisac
purus.Phasellusaclibero.Etiamdapibusia-
culisnunc.Inlectus wisi,elementumeu, sollic-
itudinnec,imperdietquis,dui.Nullaviverra
nequeaclibero.Maurisurnaleo,adipiscing
eu,ultricesnon,blanditeu,dui.Maecenas
duineque,suscipitsitamet, rutruma,laoreet
in,eros.Uteunibh.Fuscenecerattempus
urnafringillatempus.Curabituridenim.Sed
ante.Crassodalesenimsitametwisi.Nunc
fermentum consequatquam.
Utauctor,augueportadignissimvestibulum,
arcudiamlobortisvelit,velscelerisqueris-
usauguesagittisrisus.Maecenaseujusto.
Pellentesquehabitant morbitristique senectus
etnetusetmalesuadafamesacturpiseges-
tas.Mauriscongueligulaegettortor.Nul-
lam laoreeturnased enim.Doneceget erosut
erosvolutpatconvallis.Praesentturpis.Inte-
germaurisdiam,elementumquis,egestasac,
rutrumvel,orci.Nullafacilisi.Quisqueadip-
iscing,nullavitaeelementumporta,semurna
volutpatleo,sedportaenimrisussedmassa.
Integeracenimquisdiamsodalesluctus.Ut
eget eros a ligula commodo ultricies.Donec eu
urnaviverradolorhendreritfeugiat.Aliquam
acorcivelerosconguepharetra.Quisque
rhoncus,justoeuvolutpatfaucibus,augueleo
posuerelacus,arhoncuspuruspedevelest.
Proinultricesenim.
Aenean tincidunt laoreet dui.Vestibulum ante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Integeripsumlec-
tus,fermentumac,malesuadain,eleifendut,
lorem.Vivamusipsumturpis,elementumvel,
hendrerit ut,semper at,metus.Vivamus sapi-
entortor,eleifendid,dapibusin,egestaset,
pede.Pellentesquefaucibus.Praesentlorem
neque, dignissim in,facilisis nec, hendrerit vel,
odio.Namatdiamacnequealiquetviver-
ra.Morbidapibusligulasagittismagna.In
lobortis.Donecaliquetultricieslibero.Nunc
dictumvulputate purus.Morbivarius.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Intempor.Phaselluscommodoporttitor
magna.Curabiturvehiculaodioveldolor.
Praesentfacilisis,augueaadipiscingvene-
natis,liberorisusmolestieodio,pulvinarcon-
sectetuerfeliseratacmauris.Namvestibu-
lumrhoncusquam.Sedveliturna,pharetra
eu,eleifendeu,viverraat,wisi.Maecenasul-
tricesnibhatturpis.Aeneanquam.Nulla
ipsum.Aliquamposuereluctuserat.Cur-
abiturmagnafelis,laciniaet,tristiqueid,ul-
tricesut,mauris.Suspendissefeugiat.Cras
eleifendwisivitaetortor.Phasellusleopurus,
mattissitamet,auctorin,rutrumin,magna.
Inhachabitasseplateadictumst.Phasellus
imperdietmetusinsem.Vestibulumacenim
nonsemultriciessagittis.Sedveldiam.
Integervelenimsedturpisadipiscingbiben-
dum.Vestibulumpededolor,laoreetnec,po-
suerein,nonummyin,sem.Donecimperdiet
sapienplaceraterat.Donecviverra.Aliquam
eros.Nunc consequat massaid leo.Sed ullam-
corper,loreminsodalesdapibus,risusmetus
sagittislorem,non porttitorpurusodioneco-
dio.Sedtinciduntposuereelit.Quisqueeu
enim.Donecliberorisus,feugiatac,dapibus
eget, posuere a,felis.Quisque vellectusutme-
tustincidunteleifend.Duis utpede.Duis velit
erat,venenatis vitae,vulputatea, pharetrasit
amet,est.Etiamfringillafaucibusaugue.
Aeneanvelitsem,viverraeu,tempusid,
rutrumid, mi.Nullamnec nibh.Proinullam-
corper,dolorincursustristique,erosaugue
tempornibh,atgravidadiamwisiatpurus.
Donec mattis ullamcorper tellus.Phasellus vel
nulla.Praesentinterdum,erosinsodalessol-
licitudin,nuncnulla pulvinar justo,a euismod
erossemnecnibh.Nullamsagittisdapibus
lectus.Nullamegetipsumeutortorlobortis
sodales.Etiampurusleo,pretiumnec,feu-
giatnon,ullamcorpervel,nibh.Sedvelelit
etquamaccumsanfacilisis.Nuncleo.Sus-
pendissefaucibuslacus.
Pellentesqueinterdumsapiensednulla.Proin
tincidunt.Aliquamvolutpatestvelmassa.
Seddolorlacus,imperdietnon,ornarenon,
commodoeu,neque.Integerpretiumsem-
perjusto.Proinrisus.Nullamidquam.
Namneque.Duisvitaewisiullamcorperdi-
amcongueultricies.Quisqueligula.Mauris
vehicula.
Curabiturnuncmagna,posuereeget,vene-
natiseu,vehiculaac,velit.Aeneanornare,
massaaaccumsanpulvinar,quamlorem
laoreetpurus, eusodalesmagnarisus molestie
lorem.Nunceratvelit,hendreritquis,male-
suadaut,aliquamvitae,wisi.Sedposuere.
Suspendisseipsumarcu,scelerisquenec,ali-
quameu,molestietincidunt,justo.Phasellus
iaculis.Sedposuereloremnonipsum.Pel-
lentesquedapibus.Suspendissequamlibero,
laoreeta,tincidunteget,consequatat,est.
Nullamutlectusnonenimconsequatfacili-
sis.Maurisleo.Quisquepedeligula,auctor
vel,pellentesquevel,posuereid,turpis.Cras
ipsumsem,cursuset,facilisisut,tempuseu-
ismod,quam.Suspendissetristiquedoloreu
orci.Maurismattis.Aeneansemper.Viva-
mustortormagna,facilisisid,variusmattis,
hendreritin,justo.Integerpurus.
Vivamusadipiscing.Curabitur imperdiet tem-
pusturpis.Vivamussapiendolor,congueve-
nenatis,euismodeget,portarhoncus,magna.
Proincondimentumpretiumenim.Fusce
fringilla,liberoetvenenatisfacilisis,erosen-
imcursusarcu, vitaefacilisisodioaugue vitae
orci.Aliquamvariusnibhutodio.Sedcondi-
mentumcondimentumnunc.Pellentesque
egetmassa.Pellentesquequismauris.Donec
utligulaacpedepulvinarlobortis.Pellen-
tesqueeuismod.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentelit.Utlaoreet
ornareest.Phasellusgravidavulputatenulla.
Donecsitametarcuutsemtempormalesua-
da.Praesenthendreritaugueinurna.Proin
enimante,ornarevel,consequatut,blandit
in,justo.Donecfeliselit,dignissim sed,sagit-
tisut,ullamcorpera,nulla.Aeneanpharetra
vulputateodio.
Quisqueenim.Proinvelitneque,tristiqueeu,
eleifendeget,vestibulumnec,lacus.Vivamus
odio.Duisodiourna,vehiculain,elementum
aliquam,aliquetlaoreet,tellus.Sedvelit.Sed
velmiacelitaliquetinterdum.Etiamsapi-
enneque,convalliset,aliquetvel,auctornon,
arcu.Aliquamsuscipitaliquamlectus.Proin
tinciduntmagnasedwisi.Integerblanditla-
cusutlorem.Sedluctusjustosedenim.
Morbimalesuadahendreritdui.Nuncmauris
leo, dapibus sitamet, vestibulum et, commodo
id,est.Pellentesquepurus.Pellentesquetris-
tique,nuncacpulvinaradipiscing,justoeros
consequatlectus, sitamet posuerelectusneque
velaugue.Cras consectetuerlibero aceros.Ut
egetmassa.Fuscesitametenimeleifendsem
dictum auctor.In egetrisus luctus wisiconva-
llis pulvinar.Vivamus sapien risus,temporin,
viverra in, aliquetpellentesque,eros.Aliquam
euismodliberoasem.
Nunc velit augue,scelerisquedignissim, lobor-
tiset,aliquamin,risus.Ineueros.Vestibu-
lumanteipsumprimisinfaucibusorciluctus
etultricesposuerecubiliaCurae;Curabitur
vulputateelitviverraaugue.Maurisfringilla,
tortorsitametmalesuadamollis,sapienmi
dapibusodio,acimperdietligulaenimeget
nisl.Quisquevitaepedeapedealiquetsus-
cipit.Phasellustelluspede,viverravestibu-
lum,gravidaid,laoreetin,justo.Cumsoci-
isnatoquepenatibusetmagnisdisparturient
montes,nasceturridiculusmus.Integercom-
modoluctuslectus.Maurisjusto.Duisvar-
iuseros.Sedquam.Craslacuseros,rutrum
eget,variusquis,convallisiaculis,velit.Mau-
ris imperdiet, metus at tristique venenatis, pu-
rusnequepellentesquemauris,aultriceselit
lacusnectortor.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentmalesuada.Nam
lacuslectus,auctorsitamet,malesuadavel,
elementumeget,metus.Duisnequepede,fa-
cilisiseget,egestaselementum,nonummyid,
neque.
Proinnonsem.Donecnecerat.Proinlibero.
Aliquamviverraarcu.Donecvitaepurus.
Donecfelismi,semperid,scelerisqueporta,
sollicitudinsed,turpis.Nullainurna.Integer
variuswisinonelit.Etiamnecsem.Mau-
risconsequat,risusnecconguecondimentum,
ligulaligula suscipiturna,vitaeportaodioer-
at quissapien.Proinluctus leoid erat.Etiam
massametus,accumsanpellentesque,sagittis
sitamet,venenatisnec,mauris.Praesentur-
na eros, ornarenec, vulputate eget, cursussed,
justo.Phasellus nec lorem.Nullam ligula ligu-
la,mollissitamet,faucibusvel,eleifendac,
dui.Aliquameratvolutpat.
Fusce vehicula,tortor etgravidaporttitor, me-
tusnibhconguelorem,uttempuspurusmau-
risapede.Integertinciduntorcisitamet
turpis.Aeneanametus.Aliquamvestibulum
lobortisfelis.Donecgravida.Sedsedurna.
Maurisetorci.Integerultricesfeugiatligu-
la.Seddignissimnibhamassa.Donecorci
dui,temporsed,tinciduntnonummy,viverra
sit amet, turpis.Quisque lobortis.Proin vene-
natistortornecwisi.Vestibulumplacerat.In
hachabitasseplateadictumst.Aliquampor-
tamiquisrisus.Donecsagittisluctusdiam.
Nam ipsumelit, imperdiet vitae,faucibus nec,
fringillaeget,leo.Etiamquisdolorinsapien
porttitorimperdiet.
Craspretium.Nullamalesuadaipsumut
libero.Suspendissegravidahendrerittellus.
Maecenas quislacus.Morbifringilla.Vestibu-
lumodioturpis,temporvitae,scelerisquea,
dictumnon,massa.Praesenteratfelis,por-
tasitamet,condimentumsitamet,placer-
atet,turpis.Praesentplaceratlacusaen-
im.Vestibulumnoneros.Utcongue.Donec
tristiquevariustortor.Pellentesquehabitant
morbi tristique senectus et netus et malesuada
famesacturpisegestas.Namdictumdictum
urna.
Phasellusvestibulumorcivelmauris.Fusce
quamleo,adipiscingac,pulvinareget,mo-
lestiesitamet,erat.Seddiam.Suspendisse
erosleo,tempuseget,dapibussitamet,tem-
puseu,arcu.Vestibulumwisimetus,dapibus
vel,luctussitamet,condimentumquis,leo.
Suspendissemolestie.Duisinante.Utso-
dales sem sit amet mauris.Suspendisseornare
pretiumorci.Fuscetristiqueenimegetmi.
Vestibulumeroselit,gravidaac,pharetra sed,
lobortisin,massa.Proinatdolor.Duisac-
cumsanaccumsanpede.Nullamblanditelit
inmagnalaciniahendrerit.Utnonummyluc-
tuseros.Fusceegettortor.
Utsitametmagna.Crasaligulaeuurna
dignissimviverra.Nullamtemporleoporta
ipsum.Praesentpurus.Nullamconsequat.
Maurisdictumsagittisdui.Vestibulumsol-
licitudinconsectetuerwisi.Insitametdiam.
Nullammalesuadapharetrarisus.Proinlacus
arcu, eleifendsed, vehiculaat, conguesit amet,
sem.Sedsagittispedea nisl.Sedtincidunt o-
dioapede.Seddui.Nameuenim.Aliquam
sagittislacusegetlibero.Pellentesquediam
sem,sagittismolestie,tristiqueet,fermentum
ornare,nibh.Nulla ettellusnon felisimperdi-
etmattis.Aliquameratvolutpat.
Vestibulumsodalesipsumidaugue.Integer
ipsumpede,convallissitamet,tristiquevi-
tae,temporut,nunc.Namnonligulanon
lorem convallis hendrerit.Maecenas hendrerit.
Sedmagnaodio,aliquamimperdiet,portaac,
aliqueteget,mi.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes,nascetur
ridiculusmus.Vestibulumnislsem,dignissim
vel,euismodquis,egestasut,orci.Nuncvi-
taerisusvelmetuseuismodlaoreet.Crassit
ametnequeaturpislobortisauctor.Sedali-
quamsemacelit.Crasvelitlectus,facilisis
id,dictumsed,portarutrum,nisl.Namhen-
dreritipsumsedaugue.Nullamscelerisque
hendreritwisi.Vivamusegestasarcusedpu-
rus.Utornarelectussederos.Suspendisse
potenti.Maurissollicitudinpedevelvelit.In
hachabitasseplateadictumst.
Suspendisseeratmauris,nonummyeget,
pretiumeget,consequatvel,justo.Pellen-
tesqueconsectetuereratsedlacus.Nullam
egestasnullaacdui.Doneccursusrhoncus
ipsum.Nuncetsemeumagnaegestasmale-
suada.Vivamusdictummassaatdolor.Mor-
biestnulla,faucibusac,posuerein,interdum
ut,sapien.Proinconsectetuerpretiumurna.
Donecsit ametnibhnec purusdignissimmat-
tis.Phasellusvehiculaelitatlacus.Nullafa-
cilisi.Crasutarcu.Sedconsectetuer.Integer
tristiqueelitquisfelisconsectetuereleifend.
Crasetlectus.
Utcongue malesuadajusto.Curabiturcongue,
felisathendreritfaucibus,maurislacusport-
titor pede, nec aliquam turpis diamfeugiat ar-
cu.Nullamrhoncusipsumatrisus.Vestibu-
lumadolorseddolorfermentumvulputate.
Sednecipsumdapibusurnabibendumlobor-
tis.Vestibulum elit.Namligula arcu, volutpat
eget,laciniaeu,lobortisac,urna.Nammol-
lisultricesnulla.Crasvulputate.Suspendisse
atrisus atmetuspulvinarmalesuada.Nullam
lacus.Aliquamtempusmagna.Aliquamut
purus.Prointellus.
Vestibulumante ipsumprimisinfaucibusorci
luctus et ultrices posuere cubilia Curae; Donec
scelerisquemetus.Maecenasnonmiutmetus
portahendrerit.Nuncsemper.Crasquiswisi
utloremposueretristique.Nuncvestibulum
scelerisquenulla.Suspendissepharetrasollic-
itudinante.Praesentatauguesitametante
interdum porta.Nunc bibendumaugue luctus
diam.Etiam nec sem.Sed eros turpis,facilisis
nec,vehiculavitae,aliquamsed,nulla.Cur-
abiturjustoleo,vestibulumeget,tristiqueut,
tempusat,nisl.
Nullavenenatisloremidarcu.Morbicur-
susurnaaipsum.Donecporttitor.Integer
eleifend,estnonmattismalesuada,minulla
convallismi,etauctorlectussapienutpu-
rus.Aliquamnullaaugue,pharetrasitamet,
faucibussemper,molestievel,nibh.Pellen-
tesquevestibulummagnaetmi.Sedfringilla
dolorveltellus.Nuncliberonunc,venenatis
eget,convallishendrerit,iaculiselementum,
mi.Nullamaliquam,felisetaccumsanvehic-
ula,magnajustovehiculadiam,eucondimen-
tumnislfelisetnunc.Quisquevolutpatmauris
avelit.Pellentesquemassa.Integeratlorem.
Nammetuserat,laciniaid,convallisut,pul-
vinarnon,wisi.Crasiaculismaurisutneque.
Crassodales,semvitaeimperdietconsequat,
pede purussollicitudin urna,ac aliquammetus
orciinleo.Utmolestieultricesmauris.Viva-
musvitaesem.Aliquameratvolutpat.Prae-
sentcommodo,nislacdapibusaliquet,tortor
orci sodales lorem, non ornare nulla lorem quis
nisl.
Sedatsemvitaepurusultricesvestibulum.
Vestibulumtinciduntlacusetligula.Pel-
lentesquevitaeelit.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;Duisornare,erateget
laoreetvulputate,lacusipsumsuscipitturpis,
etbibendumnislorcinonlectus.Vestibu-
lumnecrisusnecliberofermentumfringilla.
Morbinonvelitinmagnagravidahendrerit.
Pellentesquequislectus.Vestibulumeleifend
lobortisleo.Vestibulumnonaugue.Vivamus
dictumtempordui.Maecenasatligulaidfe-
liscongueporttitor.Nullaleomagna,egestas
quis,vulputatesitamet,viverraid,velit.
Utlectuslectus,ultriciessitamet,semper
eget,laoreetnon,ante.Proinatmassaquis
nuncrhoncusmattis.Aliquamlorem.Cur-
abiturpharetraduiatneque.Aliquameutel-
lus.Aeneantempus,felisvitaevulputateia-
culis,estdolorfaucibusurna,inviverrawisi
nequenonrisus.Fusceveldolornecsapien
pretiumnonummy.Integerfaucibusmassaac
nulla ornarevenenatis.Nullaquis sapien.Sed
tortor.Phasellusegetmi.Crasnunc.Crasa
enim.
Quisque nisl.In dignissim dapibus massa.Ae-
neansemmagna,scelerisquenec,ullamcorper
quis,porttitorut,lectus.Fuscedignissimfa-
cilisistortor.Vivamusgravidafelissitamet
nunc.Nampulvinarodiovelenim.Pellen-
tesquesitametest.Vivamuspulvinarleonon
sapien.Aliquameratvolutpat.Utelementum
auctormetus.Maurisvestibulumnequevitae
eros.Pellentesquealiquamquam.Donecve-
nenatistristiquepurus.Innisl.Nullavelit
libero,fermentumat,portaa,feugiatvitae,
urna.Etiamaliquetornareipsum.Proinnon
dolor.Aeneannuncligula,venenatissuscipit,
porttitorsit amet, mattissuscipit, magna.Vi-
vamusegestasviverraest.Morbiatrisussed
sapiensodalespretium.
Morbicongueconguemetus.Aeneansedpu-
rus.Nampedemagna,tristiquenec,porta
id,sollicitudinquis,sapien.Vestibulumblan-
dit.Suspendisse ut augue ac nibh ullamcorper
posuere.Integereuismod,nequeateleifend
fringilla,augueelitornaredolor,veltincidunt
purusest idlacus.Vivamusloremdui, commo-
doquis,scelerisqueeu,tinciduntnon,magna.
Cras sodales.Quisque vestibulum pulvinar di-
am.Phasellustincidunt, leovitaetristique fa-
cilisis,ipsum wisiinterdum sem,dapibussem-
pernulla velit vel lectus.Cras dapibusmauris
etaugue.Quisquecursusnullainlibero.Sus-
pendisseetloremsitametmaurismalesuada
mollis.Nullamidjusto.Maecenasvenenatis.
Doneclacusarcu,egestasac,fermentumcon-
sectetuer,tempuseu,metus.Proinsodales,
seminpretiumfermentum,arcusapiencom-
modomauris,venenatisconsequataugueur-
nainwisi.Quisquesapiennunc,variuseget,
condimentumquis,laciniain,est.Fuscefacil-
isis.Praesentnecipsum.
Suspendisseadolor.Namerateros,congue
eget,sagittisa,laciniain,pede.Maecenas
inelit.Proinmolestievariusnibh.Vivamus
tristiquepurussedaugue.Proinegestassem-
pertortor.Vestibulumanteipsumprimisin
faucibusorciluctusetultricesposuerecubil-
iaCurae;Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Vestibulumorcienim,sagittis
ornare,eleifendut,mattisat,ligula.Nulla
molestieconvallisarcu.Uterostellus,condi-
mentum at,sodalesin,ultricesvel, nulla.
Duismagnaante,bibendumeget,eleifend
eget,suscipitsed,neque.Vestibuluminmi
sedmassacursuscursus.Pellentesquepulv-
inarmollisneque.Fusceutenimvitaemauris
malesuadatincidunt.Vivamusaneque.Mau-
rispulvinar,sapienidcondimentumdictum,
quamarcurhoncusdui,idtemporlacusjusto
etjusto.Proinsitametorcieudiameleifend
blandit.Nunceratmassa,luctusac,fermen-
tumlacinia,tinciduntultrices,sapien.Prae-
sentsed orci vitae dolorsollicitudin adipiscing.
Crasaneque.Utrisusdui,interdum at,plac-
eratid,tristiqueeu,enim.Vestibulumante
ipsum primisin faucibusorci luctus etultrices
posuerecubiliaCurae;Etiamadipiscingeros
vestibulumdolor.Pellentesquealiquam,diam
eget eleifendposuere,augueeros porttitor lec-
tus,acdignissimduimetusnecfelis.Quisque
lacinia.Vestibulumtellus.Suspendissenec
wisi.Aeneanacfelis.Aliquamultricesmetus
etnulla.
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ametneque.Sedcommodomalesuadalectus.
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aliquameu,quam.Aeneanquisquamquis
wisigravidavehicula.Pellentesqueamassa
aleopretiumrhoncus.Suspendisseultrices.
Doneclaciniamalesuadamassa.Classapten-
ttacitisociosquadlitoratorquentperconu-
bianostra,perinceptoshymenaeos.Donec
pretiumornaremauris.Phasellusauctorer-
ategetenim.Integerscelerisque,feliseucon-
sequatfringilla,loremwisiultriciesvelit,id
vehiculapurusnullaegetodio.Nullammat-
tis,diamarutrumfermentum,odiosapien
tristiquequam,idmollistellusquaminodi-
o.Mauriseusapien.Donecaliquamloremsit
ametlorempharetralobortis.
Donecacvelit.Sedconvallisvestibulumsapi-
en.Vivamustemporlacussedlacus.Nunc
utlorem.Utettortor.Nullamvariuswisiat
diam.Etiamultricies,dolorsitametfermen-
tumvulputate,nequeliberovestibulumorci,
vitaefringillanequearcualiquetante.Lorem
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elit.Quisquevenenatislobortisaugue.Sed
tempor,tellusiaculispellentesquepharetra,
pededuimalesuadamauris,velultricesurna
maurisacnibh.Etiamnibhodio,ultriciesve-
hicula,vestibulumvitae,feugiateleifend,fe-
lis.Vivamus pulvinar.Aliquameratvolutpat.
Nullaegestasvenenatismetus.Namfeugiat
nuncquiselitegestassagittis.Sedvitaefe-
lis.Inliberoarcu,rhoncusin,commodoeget,
auctorin,enim.Vivamussuscipitest.Nul-
ladapibus,magnavelaliquetegestas,massa
massahendreritlacus,acrutrumtellustellus
sitametfelis.Crasviverra.
Suspendisseeununc.Aliquamdignissimurna
sit ametmauris.Cras commodo, urna ut port-
titor venenatis, arcu metussodalesrisus, vitae
gravidasapienligulainest.Donecvulputate
sollicitudinwisi.Donecvehicula,estidinter-
dumornare,nibhtellusconsectetuerjusto,a
ultrices felis erat at lectus.In est massa, male-
suadanon,suscipitat,ullamcorpereu,elit.
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tissed,temporeget,libero.Praesentligula.
Suspendissenulla.Etiamdiam.Nullaante
diam,vestibulumet,aliquetac,imperdietvi-
tae,urna.Fuscetinciduntlacusvelelit.Mae-
cenasdictum,tortornoneuismodbibendum,
pedenibhpretium tellus,at dignissimleoeros
egetpede.Nullavenenatiseleifenderos.Ae-
nean ut odio dignissim augue rutrum faucibus.
Fusceposuere,tellusegetviverramattis,er-
attellusportami,atfacilisissemnibhnon
urna.Phasellusquisturpisquismaurissus-
cipitvulputate.Sedinterdumlacusnonvelit.
Vestibulumante ipsumprimisinfaucibusorci
luctusetultricesposuerecubiliaCurae;
Vivamus vehicula leoa justo.Quisquenec au-
gue.Morbimauriswisi,aliquetvitae,dignis-
simeget,sollicitudinmolestie,ligula.Indic-
tumenim sitametrisus.Curabiturvitaevelit
eudiamrhoncushendrerit.Vivamusutelit.
Praesentmattisipsumquisturpis.Curabitur
rhoncusnequeeudui.Etiamvitaemagna.
Namullamcorper.Praesentinterdumbiben-
dummagna.Quisqueauctoraliquamdolor.
Morbieuloremetestporttitorfermentum.
Nuncegestasarcuattortorvariusviverra.
Fusceeu nulla utnulla interdum consectetuer.
Vestibulumgravida.Morbimattisliberosed
est.
Abstract
The flavor-changing neutral current (FCNC)problem attree-level is
a very critical defect of thetwo Higgs doubletextension of standard
model(SM).Inthisarticle,atwo-Higgs-doubletmodel(2HDM)in
whichsuchdefectsdonotexistatallistobedemonstrated.The
generalpatternofmatrixpairswhichcanbediagonalizedsimul-
taneouslybyasameunitarytransformationisproposedwithout
extraconstraintslikesymmetriesorzerosinMmatrices.Onlyan
assumptionofthehermiticityofmassmatricesisemployedinthe
derivation.Withthisassumption,numberofparametersinthemass
matrixofaspecificfermiontypeisreducedfromeighteendownto
five.Eigenvaluesandeigenvectorsareanalyticallyderivedanditis
surprising thatunitary transformationmatrix thus derived depend-
sononlytwoparameters.Itisaverygeneralandelegantwayto
solve thetree-level FCNC problemradicallyanditincludes previous
similarmodelsasspecialcaseswithspecificparametervalues.
Keywords
Two-Higgs-DoubletModel,CPViolation,2HDM
1.Introduction
Inthestandardmodel(SM)ofelectro-weakinteractionsCPsymmetry
canbeviolatedexplicitlybyrankingtheYukawacouplingsbetween
fermionsandHiggsfieldssuitablyandexpectthemtogeneratecom-
plexphases intheCabibbo-Kobayashi-Maskawa(CKM)matrix.But,
noone knows how theseYukawacouplings shouldbearrangedto give
asatisfactoryCKMmatrix.ThoughinSMthevacuumexpectation
value(VEV) ofits Higgsdoublet alsoprovided acomplex phasewhen
thegaugesymmetrywasspontaneouslybroken.However,CPsym-
metrycan’tbeviolatedspontaneouslyinSMsincetheonlyphase
generatedinthiswaycanalwaysberotatedaway.Thus,anexten-
sionofSMwithoneextraHiggsdoubletwasproposed[1],through
which,oneexpectsthephasesintheVEVsmayunlikelyberotated
DOI:10.4236/jmp.2019.101004Jan.23,201935JournalofModernPhysics
C.L.Lin
awaysimultaneouslyandanon-zerophasedifferencebetweenthem
mightsurvivesoastobringinCP-violationspontaneously.However,
adefinitewayregardinghowCP-violationcanbegeneratedinsuch a
modelisstillmissing.
BesidesfailingtosolvetheCPproblem,thisextraHiggsdoublet
alsobroughtinanextraproblemofflavor-changingneutralcurrents
(FCNCs)attreelevel.Thisproblemarisesifthosetwocomponents
M
1
andM
2
ofthequarkmassmatrixM=M
1
+M
2
corresponding
to Higgs doublets Φ
1
and Φ
2
respectively were not diagonalized simul-
taneouslybyasameunitarytransformationU.Respectivenon-zero
off-diagonal elements of diagonalized UM
1
U
and UM
2
U
in the mass
eigenstateleadtoflavor-changing-neutral(FCN) interactionsmediat-
edbyneutralHiggsscalarsattreelevel.
Atthebeginning,peoplewerenotawareofthedangerofsuchin-
teractionstoomuch.But,astheenergyandaccuracyofexperiments
increaseandnosucheffectsweredetecteduntil2005[2],theneedofhy-
potheses to explainthe smallnessof such interactions emerged.Owing
tothesmallnessofdetectedsuchinteractions,itisnaturalforpeople
toconsiderthemasloopcorrections.Ifoneconsidersthemasloop
corrections,thatmeanssuchinteractionswon’tappearattree-level.
Insuchamanner,thereshouldbematrixpairswhichcanbediago-
nalized simultaneously and how to findsuch matrix pairsbecomesthe
keytosolvetheFCNCproblem.
In[3]twoNatural-Flavor-Conservation(NFC)modelswerepro-
posed by employinga Z
2
discrete symmetry to forbidbothHiggs dou-
bletstocouplewithasamequarktypesimultaneously.Thesetwo
models areusually referredtoas theType-I andType-II NFC models
orjustType-IandType-II2HDMs.IntheType-Imodel,onlyoneof
theHiggsdoubletscoupleswithbothquarktypesandtheothercom-
pletelydoesnot.IntheType-IImodel,up-anddown-typequarks
couplewithdifferent Higgsdoubletsrespectively.Surelytheyarefree
ofFCNCssinceforaspecificquarktypeeitherM
1
=0orM
2
=0
guaranties the absence of FCNCs.Besides these two models,there are
alsosimilarmodelswhichareusuallyreferredtoastheType-IIIand
-IVmodels,sometimesthe-Xand-Ymodels,orthelepton-specific
andflipped2HDMs.However,thesemodelsarejustextensionsof
Type-I and-II modelsto includeleptons [4–7,andreferences therein].
Besidesmodelsmentionedinlastparagraph,therearealsosever-
al2HDMsinwhichbothM
1
andM
2
areassumednon-zero.But
thesemodelsdonothaveproperdynamicexplanationsfortheiras-
sumptions.Forinstance,inFritzsch-ansatz[8][9]anditsconsequent
developmentsseveral elementsofthemassmatriceswereassumedze-
rotosimplifythepatternofmassmatricesdowntoananalytically
manageable level.In theAligned Two-Higgs-Doublet model (ATHDM
orA2HDM)[5],anassumptionofM
1
andM
2
areproportionalwas
employedasanadhocconstraint.However,theyareinfactspecial
casesof themodeltobepresentedinthis article.Itisnoticeablethat
mostofthesemodelsalsoassumetheMmatricesareHermitianin
additiontotheseimposedconstraints.
BesidestheFCNC-free2HDMsmentionedabove,thereisanother
typeof2HDMwhichisalsoreferredtoasType-III2HDM[10–15,
andreferencestherein]andusuallycausesconfusionswiththeone
mentioned above.Such models do not deny the existence of FCNCs at
tree level.It just assumesthat tree-level FCNCs are highlysuppressed
orcanbecanceledbyloopcorrectionsdowntoempiricalvalues.
DOI:10.4236/jmp.2019.10100436JournalofModernPhysics
C.L.Lin
In2HDMs mentioned above,except fortheoneallows fortree-level
FCNCs,assumptionsorforbiddancewereemployedtosimplifythe
patternsoffermionmassmatrices.However,theoretically,amodel
withfeweradhocconstraintsisbettersinceitwillbemoregeneral
and natural.Forbiddingcouplingsbetween specificfermiontypesand
Higgsdoubletsisaverystrongconstraint.Inusual,physicistsprefer
ageneralmodelappliestowiderfieldstospecialmodelsconstrained
byspecialconditionsorsymmetries.Thus,amodelinwhichneither
M
1
,M
2
noranyelementsinthemhastobeassumedzerowillbea
betterone.
Theoretically,a2HDMinwhichbothM
1
andM
2
arenonzeroand
can bediagonalizedsimultaneously shouldexist.Ifsuch amodeldoes
exist, itis obviously moregeneralthan theZ
2
-symmetric onesorthose
assumingzerosinmassmatrices.Atthemeantime,assumptionsof
highlysuppressedtree-levelFCNCsorloopcorrectionsasstrongas
zeroth-orderinteractionsinType-IIImodelsarenolongerneeded.In
suchamodel,FCNCsattree-levelarenomoreproblemssincethey
donotexistatall.Nowtheonlyproblemishowtofindthemand
whatwilltheylookslike.
Infact,suchmatrixpairsdoexistandthefirstonehadbeendis-
coveredinaS
3
-symmetric2HDM[16–18]decadesago.Thematrix
pairderivedtherewere
M
1
=
ABB
BAB
BBA
,M
2
= i
0DD
D0D
DD0
,(1)
andtheunitarytransformationwhichdiagonalizethemsimultaneously
wasderivedanalyticallyas:
U=
1/
3(1i
3)/2
3(1+i
3)/2
3
1/
3(1+i
3)/2
3(1i
3)/2
3
1/
31/
31/
3
.(2)
Inrecentyears,threemoresuchFCNC-freematrixpairswerede-
rivedbyextendingtheS
3
-symmetricpatterntothree(S
3
+S
2
)-
symmetricpatterns[19].Inthisarticle,averygeneralpatternof
suchFCNC-freematrixpairsisderivedwithoutanysymmetriesor
assumptionsexceptthehermiticityofmassmatrices.Itnotonlyin-
cludestheS
3
-and(S
3
+S
2
)-symmetricmatrixpairsasspecialcases
init,butalsotheZ
2
-symmetric,FritzschansatzandA2HDM.
AtthebeginningofsectionII,aFCNC-freeconditionfora2HDM
whichwasfirstlygivenin[20]isdiscussedandimproved.Withthis
condition, number of free parametersin a mass matrix is substantially
reducedfromeighteendowntofive.Thus,analyticalsolutionsofthe
eigenvaluesandeigenvectorsarederivable.Itisamazingthatderived
unitarytransformationmatrixUwhichdiagonalizebothM
1
andM
2
simultaneously isextremely simple.It dependson onlytwo ofthe five
parametersineachquarktype.Conclusionsandsomediscussionson
itsapplicationongeneratingCPviolationwillbegiveninsectionIII.
2.A GeneralPatternfor FCNC-Free Mass
Matrices
Theoretically,findingmatrixpairswhichcanbediagonalizedbya
sameUmatrixsimultaneouslyisabetterwaytosolvetheFCNC
DOI:10.4236/jmp.2019.10100437JournalofModernPhysics
C.L.Lin
problemsincethatwillbemoregeneralthanthoseimposingsymme-
tries[16–19],forbiddance[3]or zerosinmassmatrices[8,9].Inusual,
physicistsprefer ageneralmodel appliestowiderfieldstospecialmod-
elsapplyonlyundercertainconditions.
Followingthishypothesis,wewouldliketostartthestudyfrom
averygeneralbasiswithoutanyimposedsymmetries,forbiddance
orzeros.Wetrytokeepitasgeneralaspossibleinthefollowing
derivations.However,astobeshownbelow,aHermitianassumption
oftheMmatrixisstillneededforsimplifyingtheMmatrixdown
toamanageablelevel.ThisHermitianassumptionfirstlyreducethe
number ofparameters inamost generalMmatrix from eighteen down
tonine.Then,withthehelpofaninterestingconditionbetweenM
1
andM
2
whichistoappearinEquation(7),theparameternumberis
furtherreduceddowntofive.Thus,eigenvaluesandcorrespondingU
matrix are now achievable analytically.In what follows, theprocedure
ofderivationswillbepresentedstepbystep.
Fora2HDM,theYukawacouplingsofQquarkscanbewrittenas
−L
Y
=
¯
Q
L
(Y
d
1
Φ
1
+Y
d
2
Φ
2
)d
R
+
¯
Q
L
(Y
u
1
Φ
1
+Y
u
2
Φ
2
)u
R
+h.c.,(3)
whereY
q
i
are3×3Yukawa-couplingmatricesforquarktypesq=
u,dandHiggsdoubletsi=1,2,respectively,andisthe2×2
antisymmetric tensor.Q
L
are left-handed quark doublets, and d
R
and
u
R
areright-handeddown-andup-typequarksinglets,respectively,
intheirweakeigenstates.Themassmatricescanthenbeexpressed
as
M
(u,d)
= M
(u,d)
1
+M
(u,d)
2
= Y
(u,d)
1
hΦ
1
i+Y
(u,d)
2
hΦ
2
i,(4)
wherehΦ
i
i≡v
i
e
i
/
2,i=1and2,arevacuumexpectationvalues
oftheHiggsdoublets.Here,abasisinwhichθ
1
=0willbechosen
andonlythephaseθ
2
isleftnon-zero.
Neglectingthehyper-indicesofM,themostgeneralpatternofa
3×3massmatrixcanalwaysbewrittenas
M=M
1
+M
2
=
A
1
+iD
1
B
1
+iC
1
B
2
+iC
2
B
4
+iC
4
A
2
+iD
2
B
3
+iC
3
B
5
+iC
5
B
6
+iC
6
A
3
+iD
3
,(5)
whereA,B,CandDareallrealandeachofthemmayreceive
contributionsfrom bothM
1
andM
2
arbitrarily.Such apatternisthe
most general one for a 3×3 matrix since it contains nine elements and
each ofthemhas onerealand oneimaginarycomponents.Thus, ifno
constraintswereimposed,thereareeighteenparametersinsuchaM
matrixintotal.
Itisobviousthateighteen parametersaretoomanytohavesuch a
matrixbediagonalizedanalytically.That’swhyphysicistsemployed
variousconstraintsinprevious2HDMstosimplifytheMpattern.
However,astobeshownbelow,aHermitianassumptionisalready
enough to simplify the Mpattern down toan analytically manageable
level.ConstraintslikeZ
2
,S
N
symmetriesorimposedzerosinM
matrixareinfactunnecessary.
IfoneassumesthemassmatrixMwereHermitian,theDparame-
terswillbeallzeroandEquation(5)becomes
M=
A
1
B
1
+iC
1
B
2
+iC
2
B
1
iC
1
A
2
B
3
+iC
3
B
2
iC
2
B
3
iC
3
A
3
,(6)
DOI:10.4236/jmp.2019.10100438JournalofModernPhysics
C.L.Lin
withB
4
=B
1
,B
5
=B
2
,B
6
=B
3
,C
4
=C
1
,C
5
=C
2
and
C
6
= C
3
.
In[20],acondition
M
1
M
2
M
2
M
1
= 0,(7)
wasgivenforamatrixpairwhichcanbediagonalizedbyasameU
matrixiftheywerebothHermitian.Thiscanbeeasilyprovedsince
UM
1
U
= UM
1
U
= M
diag.
1
and UM
2
U
= UM
2
U
= M
diag.
2
, where
M
diag.
1
andM
diag.
2
arediagonal.IfoneappliesUontoEquation(7),
onewillreceive
(UM
1
U
)(UM
2
U
)(UM
2
U
)(UM
1
U
)
= M
diag.
1
M
diag.
2
M
diag.
2
M
diag.
1
= 0.(8)
Thus,if onecanfind amatrixpair which satisfiesEquation (7),the
FCNCproblemattree-levelissolvedautomatically.
Basedonthis, itis instinctive foronetodividethe Hermitianmatrix
inEquation(6)intotwoHermitiancomponentsandsubstitutethem
intoEquation(7).Thesimplestwayfordoingsoistodividethem
intoonepurelyrealcomponentandonepurelyimaginarycomponent
as
M
R
=
A
1
B
1
B
2
B
1
A
2
B
3
B
2
B
3
A
3
,M
I
= i
0C
1
C
2
C
1
0C
3
C
2
C
3
0
.(9)
SubstituteEquation(9)intoEquation(7),wereceive
M
I
M
R
= i
B
1
C
1
+B
2
C
2
A
2
C
1
+B
3
C
2
B
3
C
1
+A
3
C
2
B
2
C
3
A
1
C
1
B
3
C
3
B
1
C
1
A
3
C
3
B
2
C
1
A
1
C
2
B
1
C
3
B
1
C
2
A
2
C
3
B
2
C
2
B
3
C
3
,
M
R
M
I
= i
B
1
C
1
B
2
C
2
A
1
C
1
B
2
C
3
A
1
C
2
+B
1
C
3
A
2
C
1
B
3
C
2
B
1
C
1
B
3
C
3
B
1
C
2
+A
2
C
3
B
3
C
1
A
3
C
2
B
2
C
1
A
3
C
3
B
2
C
2
+B
3
C
3
(10)
Thediagonalelementsgiveusfollowingconditions
B
1
C
1
= B
2
C
2
= B
3
C
3
(11)
andtheoff-diagonalonesgiveusotherthree
(A
1
A
2
)=(B
3
C
2
+B
2
C
3
)/C
1
,(12)
(A
3
A
1
)=(B
1
C
3
B
3
C
1
)/C
2
,(13)
(A
2
A
3
)=(B
2
C
1
+B
1
C
2
)/C
3
.(14)
But,substitutingEquation(11)intothesumofEquation(12)and
Equation (13)willreceive Equation(14).Sowe havein factonlyfour
equationstoreducethenumberofindependentparametersdownto
five.
Forsimplicity,onemayleaveA
3
,B
3
,C
3
,B
1
andB
2
independent
andreplacesA
1
,A
2
,C
1
andC
2
by
A
1
=A
3
+B
2
(B
2
1
B
2
3
)/B
1
B
3
,
A
2
=A
3
+B
3
(B
2
1
B
2
2
)/B
1
B
2
,
C
1
=B
3
C
3
/B
1
,C
2
= B
3
C
3
/B
2
,(15)
DOI:10.4236/jmp.2019.10100439JournalofModernPhysics
C.L.Lin
andEquation(5)nowbecomes
M=
A+xB(y
1
y
)yB+i
C
y
xBi
C
x
yBi
C
y
A+B(
y
x
x
y
)B+iC
xB+i
C
x
BiCA
,(16)
ifoneletsA A
3
,BB
3
,CC
3
andx B
2
/B
3
,y
B
1
/B
3
.
Subsequently,themasseigenvaluescanbederivedanalyticallyas
M
diag.
=
AB
x
y
C
x
2
+y
2
+x
2
y
2
xy
00
0AB
x
y
+C
x
2
+y
2
+x
2
y
2
xy
0
00A+B
(x
2
+1)y
x
,(17)
withtheUmatrixgivenas
U
(u)
=
x
2
+y
2
2(x
2
+y
2
+x
2
y
2
)
x(y
2
i
x
2
+y
2
+x
2
y
2
)
2
x
2
+y
2
x
2
+y
2
+x
2
y
2
y( x
2
+i
x
2
+y
2
+x
2
y
2
)
2
x
2
+y
2
x
2
+y
2
+x
2
y
2
x
2
+y
2
2(x
2
+y
2
+x
2
y
2
)
x(y
2
+i
x
2
+y
2
+x
2
y
2
)
2
x
2
+y
2
x
2
+y
2
+x
2
y
2
y(x
2
i
x
2
+y
2
+x
2
y
2
)
2
x
2
+y
2
x
2
+y
2
+x
2
y
2
xy
x
2
+y
2
+x
2
y
2
y
x
2
+y
2
+x
2
y
2
x
x
2
+y
2
+x
2
y
2
(18)
ifweconsiderheretheup-typequarks.
Surprisingly, all elementsof thismatrix areindependentof A,B and
C.Theydependonly ontwoparametersxandy.Similarly, thematrix
U
(d)
fordown-typequarksshouldhasthesamepatternandonemay
expressitsimplybyreplacingallparametersinM
(u)
andU
(u)
with
primedonesA
0
,B
0
,C
0
,x
0
B
0
2
/B
0
3
andy
0
B
0
1
/B
0
3
,respectively.
ThoughthispatternwasachievedbydividingMintotworealand
imaginarycomponents.However,anycombinationsofM
R
andM
I
like(pM
R
+qM
I
)withpandqarbitrarynumbers,even complex,can
beassignedtoM
1
orM
2
andstillbediagonalizedbythesameU.
Thus,anymatrixpairs
M
1
= pM
R
+qM
I
andM
2
= rM
R
+sM
I
,(19)
wherep,q,randsarearbitrarynumbers,willalsobediagonalized
bythesameUandfreeoftree-levelFCNCsnaturally.Besides,the
derivation demonstrated above didnot employ any symmetrieslikeZ
2
in [3],S
3
in [16–18] or(S
3
+S
2
) in [19]except for theonly assumption
of hermiticity of quark mass matrices.Such a matrix pattern is avery
generalone.Even, itincludesallprevious NFC models asspecial cases
init.
Forinstance,inType-Iand-IImodelssomeoftheM
q
i
components
areassignedzeroby theZ
2
discretesymmetry.Thatcanbeachieved
bylettingeitherA=B=0orC=0forcorrespondingquarktype.
TheS
3
patterninEquation(1)canbeachievedbylettingx= y= 1
(firstly achieved in [16–18]or thecase-1 in[19]).Thosethree patterns
achievedinthe(S
3
+S
2
)modelcorrespondtox=y=1(case-2),
x=y=1(case-3)andx=y=1(case-4),respectively.Even
theFritzschansatz[8,9]andthealignedtwo-Higgsdoubletmodel
(ATHDM)arealsoincluded.OnemayachievetheFritzschansatz
simplybylettingA
1
=A
2
=B
2
=0.ForATHDM,thatisthe
p·s= q·rcaseinthismodel.
DOI:10.4236/jmp.2019.10100440JournalofModernPhysics
C.L.Lin
3.ConclusionsandDiscussions
Thematrixpatternachievedinthis articleis averygeneral onesince
no symmetry isemployed during thederivation.Theonly assumption
employedisthehermiticityofquarkmassmatrices.Besides,itis
derivedanalyticallyfromanveryfundamentaltheoryofelectro-weak
interactions.Thus,itincludesalmostprevious2HDMsinit,except
forthetype-IIImodels.Thatprovidesuswithaveryrationaland
generalaspecttorealizethenatureofFCNCsandCPviolation.
In this modelall possible freedomin the Yukawa sectorgets param-
eterized interms offive parameters A,B, Cxandyfor a quarktype.
But,correspondingunitarytransformationmatrixU
(u)
dependsonly
ontwoofthem.AssumingU
(d)
hassimilarFCNC-freepatternand
assigningitscorrespondingparametersasx
0
andy
0
,theCKMmatrix
thus derived will depend on merely three of the four x, y, x
0
and y
0
pa-
rameters if oneconsiders the unitarity of CKMmatrix.That provides
uswithanewaspecttorealizetheexplicitoriginofCPviolation.
Theoretically,ofcourse,therecouldbeotherFCNC-freematrix
pairsthanthosepresentedinthisarticle.Forinstance,therecould
bematrixpairswhicharenotrespectivelyHermitianbutstillcanbe
diagonalized simultaneously.Surelytheywill notsatisfyEquation (7)
andthenumberoffreeparametersinthemwillbemanymorethan
fiveforeachfermiontype.
Besides,themassmatrixMitselfcouldbenon-Hermitian,too.In
thatcase, thenumberoffreeparameterssurelywillbemanymorethan
five.Bothcasesarefarbeyondourabilityofanalyticalderivationfor
now.
Beyondthe2HDMscopeofthisarticle.Itisinterestingthatthese
matrixpatternsalsoapplytoStandardModelandanevenextended
modelwiththreeHiggsdoublets.Thedetailedstudyonthemisnow
stillunderway.
ConflictsofInterest
Theauthordeclaresnoconflictsofinterestregardingthepublication
ofthispaper.
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