JournalofHighEnergyPhysics,GravitationandCosmology,2020,6,590-608
https://www.scirp.org/journal/jhepgc
ISSNOnline:2380-4335
ISSNPrint:2380-4327
InterestingQFTProblemsTackledinNew
Fashion
A.Plastino
1,2,3
,M.C.Rocca
1,2,4
1
DepartamentodeF´ısica,UniversidadNacionaldeLaPlata,LaPlata,Argentine
2
IFLP-ConsejoNacionaldeInvestigacionesCient´ıficasyTecnol´ogicas(IFLP-CCT-CONICET)-C.C.,LaPlata,
Argentina
3
SThAR-EPFL,Lausanne,Switzerland
4
DepartamentodeMatem´atica,UniversidadNacionaldeLaPlata,LaPlata,Argentine
Howtocitethispaper:Plastino,A.and
Rocca,M.C.(2020)InterestingQFTProblems
TackledinNewFashion.JournalofHighEn-
ergyPhysics,GravitationandCosmology,6,
590-608.
https://doi.org/10.4236/jhepgc.2020.64040
Received:July10,2020
Accepted:August14,2020
Published:August17,2020
Copyright
c
2020byauthor(s)andScientific
ResearchPublishingInc.
ThisworkislicensedundertheCreativeCom-
monsAttributionInternationalLicense(CC
BY4.0).
http://creativecommons.org/licenses/by/4.0/
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laoreetquam.Nullamidwisialiberotris-
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egestassemper,mollisid,leo.Nullaacmas-
saeurisusblanditmattis.Maurisutnunc.
Inhachabitasseplateadictumst.Aliquam
egettortor.Quisquedapibuspedeinerat.
Nuncenim.Induinulla,commodoat,con-
sectetuernec,malesuadanec,elit.Aliquam
ornaretelluseuurna.Sednecmetus.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.Pel-
lentesquehabitantmorbitristiquesenectuset
netusetmalesuadafamesacturpisegestas.
Phasellusidmagna.Duismalesuadainter-
dumarcu.Integermetus.Morbipulvinarpel-
lentesquemi.Suspendissesedesteumagna
molestieegestas.Quisquemilorem,pulvinar
eget,egestasquis,luctusat,ante.Proinauc-
torvehiculapurus.Fusceacnislaliquamante
hendreritpellentesque.Classaptenttacitiso-
ciosquadlitoratorquentperconubianostra,
perinceptoshymenaeos.Morbiwisi.Etiam
arcumauris,facilisissed,eleifendnon,non-
ummyut,pede.Crasutlacustempormetus
mollisplacerat.Vivamuseutortorvelmetus
interdummalesuada.
Sedeleifend,erossitametfaucibuselemen-
tum,urnasapienconsectetuermauris,quis
egestasleojustononrisus.Morbinonfelisac
liberovulputatefringilla.Maurisliberoeros,
lacinianon,sodalesquis,dapibusporttitor,
pede.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Morbidapibusmauriscondimentum
nulla.Cumsociisnatoquepenatibusetmagnis
disparturientmontes,nasceturridiculusmus.
Etiamsitameterat.Nullavarius.Etiamt-
inciduntduivitaeturpis.Donecleo.Morbi
vulputateconvallisest.Integeraliquet.Pel-
lentesquealiquetsodalesurna.
Nullameleifendjustoinnisl.Inhachabitasse
plateadictumst.Morbinonummy.Aliquam
utfelis.Invelitleo,dictumvitae,posuereid,
vulputatenec,ante.Maecenasvitaepedenec
duidignissimsuscipit.Morbimagna.Vestibu-
lumidpurusegetvelitlaoreetlaoreet.Prae-
sentsedleovelnibhconvallisblandit.Ut
rutrum.Donecnibh.Donecinterdum.Fusce
sedpedesitametelitrhoncusultrices.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,
liberopretiumnonummyfringilla,tortorlacus
vestibulumnunc,utrhoncusligulanequeid
justo.Nullamaccumsaneuismodnunc.Proin
vitaeipsumacmetusdictumtempus.Namut
wisi.Quisquetortorfelis,interdumac,sodales
a,sempera,sem.Curabiturinvelitsitamet
duitristiquesodales.Vivamusmaurispede,
laciniaeget,pellentesquequis,scelerisqueeu,
est.Aliquamrisus.Quisquebibendumpede
eudolor.
Donectempusnequevitaeest.Aeneanegestas
odiosedrisusullamcorperullamcorper.Sedin
nullaatortortinciduntegestas.Namsapien
tortor,elementumsitamet,aliquamin,port-
titorfaucibus,enim.Nullamconguesuscipit
nibh.Quisqueconvallis.Praesentarcunibh,
vehiculaeget,accumsaneu,tincidunta,nibh.
Suspendissevulputate,tortorquisadipiscing
viverra,lacusnibhdignissimtellus,eususcipit
risusantefringilladiam.Quisquealiberov-
elpedeimp erdietaliquet.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
ligulanecorci.Aliquampurusturpis,aliquam
id,ornarevitae,porttitornon,wisi.Maecenas
luctusportalorem.Donecvitaeligulaeuante
pretiumvarius.Prointortormetus,convallis
et,hendreritnon,scelerisquein,urna.Cras
quisliberoeuligulabibendumtempor.Viva-
mustellusquam,malesuadaeu,tempussed,
temporsed,velit.Doneclaciniaauctorlibero.
Praesentsednequeidpedemollisrutrum.
Vestibulumiaculisrisus.Pellentesquelacus.
Utquisnuncsedodiomalesuadaegestas.Duis
amagnasitametligulatristiquepretium.Ut
pharetra.Vestibulumimperdietmagnanec
wisi.Maurisconvallis.Sedaccumsansollic-
itudinmassa.Sedidenim.Nuncpedeenim,
laciniaut,pulvinarquis,suscipitsemper,elit.
Crasaccumsaneratvitaeenim.Crassollici-
tudin.Vestibulumrutrumblanditmassa.
Sedgravidalectusutpurus.Morbilaoreet
magna.Pellentesqueeuwisi.Prointurpis.In-
tegersollicitudinauguenecdui.Fuscelectus.
Vivamusfaucibusnullaneclacus.Integerdi-
am.Pellentesquesodales,enimfeugiatcursus
volutpat,semmaurisdignissimmauris,quis
consequatsemestfermentumligula.Nullam
justolectus,condimentumsitamet,posuerea,
fringillamollis,felis.Morbinullanibh,pellen-
tesqueat,nonummyeu,sollicitudinnec,ip-
sum.Crasneque.Nuncaugue.Nullamvitae
quamidquampulvinarblandit.Nuncsitamet
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
sitamet,placeratac,mauris.Maecenasetelit
uteratplaceratdictum.Namfeugiat,turpis
etsodalesvolutpat,wisiquamrhoncusneque,
vitaealiquamipsumsapienvelenim.Maece-
nassuscipitcursusmi.
Quisqueconsectetuer.Insuscipitmaurisado-
lorpellentesqueconsectetuer.Maurisconvallis
nequenonerat.Inlacinia.Pellentesqueleo
eros,sagittisquis,fermentumquis,tincidun-
tut,sapien.Maecenassem.Curabitureros
odio,interdumeu,feugiateu,portaac,nis-
l.Curabiturnunc.Etiamfermentumconvallis
velit.Pellentesquelaoreetlacus.Quisquesed
elit.Namquistellus.Aliquamtellusarcu,
adipiscingnon,tincidunteleifend,adipiscing
quis,augue.Vivamuselementumplaceraten-
im.Suspendisseuttortor.Integerfaucibus
adipiscingfelis.Aeneanconsectetuermattis
lectus.Morbimalesuadafaucibusdolor.Nam
lacus.Etiamarculibero,malesuadavitae,ali-
quamvitae,blandittristique,nisl.
Maecenasaccumsandapibussapien.Duis
pretiumiaculisarcu.Curabiturutlacus.Ali-
quamvulputate.Suspendisseutpurussedsem
temporrhoncus.Utquamdui,fringillaat,dic-
tumeget,ultriciesquis,quam.Etiamsemest,
pharetranon,vulputatein,pretiumat,ipsum.
Nuncsempersagittisorci.Sedscelerisquesus-
cipitdiam.Utvolutpat,doloratullamcorp-
ertristique,erospurusmollisquam,sitamet
ornareantenuncetenim.
Phasellusfringilla,metusidfeugiatcon-
sectetuer,lacuswisiultricestellus,quislobor-
tisnibhloremquistortor.Donecegestas
ornarenulla.Maurismitellus,portafau-
cibus,dictumvel,nonummyin,est.Ali-
quameratvolutpat.Intellusmagna,port-
titorlacinia,molestievitae,p ellentesqueeu,
justo.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Sedorcinibh,scelerisquesitamet,
suscipitsed,placeratvel,diam.Vestibulum
nonummyvulputateorci.Donecetvelitacar-
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.Quisqueeratelit,viverrasitamet,sagittis
eget,portasitamet,lacus.
Inhachabitasseplateadictumst.Proinatest.
Curabiturtempusvulputateelit.Pellentesque
sem.Praesenteusapien.Duiselitmagna,ali-
quetat,tempussed,vehiculanon,enim.Mor-
biviverraarcunecpurus.Vivamusfringilla,
enimetcommodomalesuada,tortormetusel-
ementumligula,necaliquetestsapienutlec-
tus.Aliquammi.Utnecelit.Fusceeuismod
luctustellus.Curabiturscelerisque.Nullam
purus.Namultriciesaccumsanmagna.Morbi
pulvinarloremsitametipsum.Donecutjus-
tovitaenibhmolliscongue.Fuscequisdiam.
Praesenttempuserosutquam.
Donecinnisl.Fuscevitaeest.Vivamusante
ante,mattislaoreet,posuereeget,conguevel,
nunc.Fuscesem.Namvelorcieuerosviver-
raluctus.Pellentesquesitametaugue.Nunc
sitametipsumetlacusvariusnonummy.In-
tegerrutrumsemegetwisi.Aeneaneusapien.
Quisqueornaredignissimmi.Duisaurnavel
risuspharetraimperdiet.Suspendissepotenti.
Morbijusto.Aeneannecdolor.Inhac
habitasseplateadictumst.Proinnonummy
porttitorvelit.Sedsitametleonecmetus
rhoncusvarius.Crasante.Vestibulumcom-
modosemtinciduntmassa.Namjusto.Ae-
neanluctus,felisetcondimentumlacinia,lec-
tusenimpulvinarpurus,nonportavelitnisl
sederos.Suspendisseconsequat.Maurisadui
ettortormattispretium.Sednullametus,vo-
lutpatid,aliquameget,ullamcorperut,ipsum.
Morbieununc.Praesentpretium.Duisali-
quampulvinarligula.Utblanditegestasjusto.
Quisqueposueremetusviverrapede.
Vivamussodaleselementumneque.Viva-
musdignissimaccumsanneque.Sedatenim.
Vestibulumnonummyinterdumpurus.Mau-
risornarevelitidnibhpretiumultricies.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
elittortor,feugiatquis,sagittiset,ornarenon,
lacus.Vestibulump osuerepellentesqueeros.
Quisquevenenatisipsumdictumnulla.Ali-
quamquisquamnonmetuseleifendinterdum.
Namegetsapienacmaurismalesuadaadipisc-
ing.Etiameleifendnequesedquam.Nulla
facilisi.Proinaligula.Sedidduieunibh
egestastincidunt.Suspendissearcu.
Maecenasdui.Aliquamvolutpatauctorlorem.
Crasplaceratestvitaelectus.Curabitur
massalectus,rutrumeuismod,dignissimut,
dapibusa,odio.Uteroserat,vulputateut,in-
terdumnon,portaeu,erat.Crasfermentum,
felisinportacongue,velitleofacilisisodio,vi-
taeconsectetuerloremquamvitaeorci.Sed
ultrices,pedeeuplaceratauctor,anteligula
rutrumtellus,velposuerenibhlacusnecnibh.
Maecenaslaoreetdoloratenim.Donecmo-
lestiedolornecmetus.Vestibulumlibero.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,urnasedinterdumporttitor,metusorci
interdumodio,sitameteuismodlectusfeliset
leo.Praesentacwisi.Namsuscipitvestibu-
lumsem.Praesenteuipsumvitaepedecursus
venenatis.Duissedodio.Vestibulumeleifend.
Nullautmassa.Proinrutrummattissapien.
Curabiturdictumgravidaante.
Phasellusplaceratvulputatequam.Maece-
nasattellus.Pellentesquenequediam,dig-
nissimac,venenatisvitae,consequatut,lacus.
Namnibh.Vestibulumfringillaarcumollisar-
cu.Sedetturpis.Donecsemtellus,volut-
patet,variuseu,commodosed,lectus.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisqueenimarcu,suscipitnec,tempus
at,imperdietvel,metus.Morbivolutpatpu-
rusaterat.Donecdignissim,semidsemper
tempus,nibhmassaeleifendturpis,sedp ellen-
tesquewisipurussedlibero.Nullamlobortis
tortorvelrisus.Pellentesqueconsequatnulla
eutellus.Donecvelit.Aliquamfermentum,
wisiacrhoncusiaculis,tellusnuncmalesuada
orci,quisvolutpatduimagnaidmi.Nuncvel
ante.Duisvitaelacus.Crasnecipsum.
Morbinunc.Aliquamconsectetuervariusnul-
la.Phaselluseros.Crasdapibusporttitorris-
us.Maecenasultricesmiseddiam.Praesent
gravidavelitatelitvehiculaporttitor.Phasel-
lusnislmi,sagittisac,pulvinarid,gravidasit
amet,erat.Vestibulumest.Loremipsumdo-
lorsitamet,consectetueradipiscingelit.Cur-
abituridsemelementumleorutrumhendrerit.
Utatmi.Donectinciduntfaucibusmassa.
Sedturpisquam,sollicitudina,hendreriteget,
pretiumut,nisl.Duishendreritligula.Nunc
pulvinarcongueurna.
Nuncvelit.Nullamelitsapien,eleifendeu,
commodonec,sempersitamet,elit.Nullalec-
tusrisus,condimentumut,laoreeteget,viver-
ranec,odio.Proinlobortis.Curabiturdictum
arcuvelwisi.Crasidnullavenenatistortor
congueultrices.Pellentesqueegetpede.Sed
eleifendsagittiselit.Namsedtellussitamet
lectusullamcorpertristique.Maurisenimsem,
tristiqueeu,accumsanat,scelerisquevulpu-
tate,neque.Quisquelacus.Donecetipsumsit
ametelitnonummyaliquet.Sedviverranislat
sem.Namdiam.Maurisutdolor.Curabitur
ornaretortorcursusvelit.
Morbitinciduntposuerearcu.Crasvenenatis
estvitaedolor.Vivamusscelerisquesemper
mi.Donecipsumarcu,consequatscelerisque,
viverraid,dictumat,metus.Loremipsum
dolorsitamet,consectetueradipiscingelit.Ut
pedesem,tempusut,porttitorbibendum,mo-
lestieeu,elit.Suspendissepotenti.Sedidlec-
tussitametpurusfaucibusvehicula.Praesen-
tsedsemnonduipharetrainterdum.Nam
viverraultricesmagna.
Aeneanlaoreetaliquamorci.Nuncinterdum
elementumurna.Quisqueerat.Nullamtem-
porneque.Maecenasvelitnibh,scelerisquea,
consequatut,viverrain,enim.Duismagna.
Donecodioneque,tristiqueet,tincidunteu,
rhoncusac,nunc.Maurismalesuadamalesua-
daelit.Etiamlacusmauris,pretiumvel,blan-
ditin,ultriciesid,libero.Phasellusbibendum
eratutdiam.Incongueimperdietlectus.
Aeneanscelerisque.Fuscepretiumporttitor
lorem.Inhachabitasseplateadictumst.Nulla
sitametnislatsapienegestaspretium.Nunc
nontellus.Vivamusaliquet.Namadipiscing
euismoddolor.Aliquameratvolutpat.Nul-
lautipsum.Quisquetinciduntauctorau-
gue.Nuncimperdietipsumegetelit.Ali-
quamquamleo,consectetuernon,ornaresit
amet,tristiquequis,felis.Vestibulumanteip-
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
maurisportadui,vitaealiquamarcuturpisac
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-
nainvelit.Etiamullamcorperelementumdui.
Praesentnonurna.Sedplaceratquamnonmi.
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,euismodac,tem-
pusid,pretiumet,lacus.Curabitursodales
dapibusurna.Nunceusapien.Doneceget
nuncapededictumpretium.Proinmauris.
Vivamusluctusliberovelnibh.
Fuscetristiquerisusidwisi.Integermolestie
massaidsem.Vestibulumveldolor.Pel-
lentesquevelurnavelrisusultricieselemen-
tum.Quisquesapienurna,blanditnec,iaculis
ac,viverrain,odio.Inhachabitasseplatea
dictumst.Morbinequelacus,convallisvitae,
commodoac,fermentumeu,velit.Sedinor-
ci.Infringillaturpisnonarcu.Donecinante.
Phasellustemporfeugiatvelit.Aeneanvar-
iusmassanonturpis.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;
Aliquamtortor.Morbiipsummassa,imperdi-
etnon,consectetuervel,feugiatvel,lorem.
Quisqueegetloremnecelitmalesuadavestibu-
lum.Quisquesollicitudinipsumvelsem.Nul-
laenim.Proinnonummyfelisvitaefelis.Nul-
lampellentesque.Duisrutrumfeugiatfe-
lis.Maurisvelpedesedliberotinciduntmol-
lis.Phasellussedurnarhoncusdiameuismod
bibendum.Phasellussednisl.Integercondi-
mentumjustoidorciiaculisvarius.Quisqueet
lacus.Phaselluselementum,justoatdignissim
auctor,wisiodiolobortisarcu,sedsollicitudin
felisfeliseuneque.Praesentatlacus.
Vivamussitametpede.Duisinterdum,nunc
egetrutrumdignissim,nisldiamluctusleo,
ettinciduntvelitnislidtellus.Inloremtel-
lus,aliquetvitae,portain,aliquetsed,lectus.
Phasellussodales.Utvariusscelerisqueerat.
Invelnibheuerosimp erdietrutrum.Donec
acodionecnequevulputatesuscipit.Namnec
magna.Pellentesquehabitantmorbitristique
senectusetnetusetmalesuadafamesacturpis
egestas.Nullamporta,odioetsagittisiaculis,
wisinequefringillasapien,velcommodolorem
loremidelit.Utsemlectus,scelerisqueeget,
placeratet,tinciduntscelerisque,ligula.Pel-
lentesquenonorci.
Etiamvelipsum.Morbifacilisisvestibulum
nisl.Praesentcursuslaoreetfelis.Integer
adipiscingpretiumorci.Nullafacilisi.Quisque
posuerebibendumpurus.Nullaquammauris,
cursuseget,convallisac,molestienon,enim.
Aliquamcongue.Quisquesagittisnonummy
sapien.Proinmolestiesemvitaeurna.Mae-
cenaslorem.Vivamusviverraconsequatenim.
Nuncsedpede.Praesentvitaelectus.Prae-
sentnequejusto,vehiculaeget,interdumid,
facilisiset,nibh.Phasellusatpurusetlibero
laciniadictum.Fuscealiquet.Nullaeuante
placeratleosemperdictum.Maurismetus.
Curabiturlobortis.Curabitursollicitudinhen-
dreritnunc.Donecultriceslacusidipsum.
Donecanibhutelitvestibulumtristique.In-
tegeratpede.Crasvolutpatvariusmagna.
Phaselluseuwisi.Praesentrisusjusto,lobor-
tiseget,scelerisqueac,aliquetin,dolor.Proin
idleo.Nunciaculis,mivitaeaccumsancom-
modo,nequesemlacinianulla,quisvestibulum
justosemineros.Quisquesedmassa.Morbi
lectusipsum,vulputatea,mollisut,accumsan
placerat,tellus.Nullaminwisi.Vivamuseu
ligulaanuncaccumsancongue.Suspendisse
aclibero.Aliquameratvolutpat.Donecau-
gue.Nuncvenenatisfringillanibh.Fusceac-
cumsanpulvinarjusto.Nullamsemper,duiut
dignissimauctor,orciliberofringillamassa,b-
landitpulvinarpedetortoridmagna.Nunc
adipiscingjustosedvelittinciduntfermentum.
Integerplacerat.Pellentesquehabitantmorbi
tristiquesenectusetnetusetmalesuadafames
acturpisegestas.Sedinmassa.Classaptent
tacitisociosquadlitoratorquentperconubi-
anostra,perinceptoshymenaeos.Phasellus
tempusaliquamrisus.Aliquamrutrumpurus
atmetus.Donecposuereo dioaterat.Nam
nonnibh.Phasellusligula.Quisquevene-
natislectusinaugue.Sedvestibulumdapibus
neque.
Mauristempuserosatnulla.Sedquisduidig-
nissimmaurispretiumtincidunt.Maurisac
purus.Phasellusaclibero.Etiamdapibusia-
culisnunc.Inlectuswisi,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
fermentumconsequatquam.
Utauctor,augueportadignissimvestibulum,
arcudiamlobortisvelit,velscelerisqueris-
usauguesagittisrisus.Maecenaseujusto.
Pellentesquehabitantmorbitristiquesenectus
etnetusetmalesuadafamesacturpiseges-
tas.Mauriscongueligulaegettortor.Nul-
lamlaoreeturnasedenim.Donecegeterosut
erosvolutpatconvallis.Praesentturpis.Inte-
germaurisdiam,elementumquis,egestasac,
rutrumvel,orci.Nullafacilisi.Quisqueadip-
iscing,nullavitaeelementumporta,semurna
volutpatleo,sedportaenimrisussedmassa.
Integeracenimquisdiamsodalesluctus.Ut
egeterosaligulacommodoultricies.Doneceu
urnaviverradolorhendreritfeugiat.Aliquam
acorcivelerosconguepharetra.Quisque
rhoncus,justoeuvolutpatfaucibus,augueleo
posuerelacus,arhoncuspuruspedevelest.
Proinultricesenim.
Aeneantinciduntlaoreetdui.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Integeripsumlec-
tus,fermentumac,malesuadain,eleifendut,
lorem.Vivamusipsumturpis,elementumvel,
hendreritut,semperat,metus.Vivamussapi-
entortor,eleifendid,dapibusin,egestaset,
pede.Pellentesquefaucibus.Praesentlorem
neque,dignissimin,facilisisnec,hendreritvel,
odio.Namatdiamacnequealiquetviver-
ra.Morbidapibusligulasagittismagna.In
lobortis.Donecaliquetultricieslibero.Nunc
dictumvulputatepurus.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.Nuncconsequatmassaidleo.Sedullam-
corper,loreminsodalesdapibus,risusmetus
sagittislorem,nonporttitorpurusodioneco-
dio.Sedtinciduntposuereelit.Quisqueeu
enim.Donecliberorisus,feugiatac,dapibus
eget,posuerea,felis.Quisquevellectusutme-
tustincidunteleifend.Duisutpede.Duisvelit
erat,venenatisvitae,vulputatea,pharetrasit
amet,est.Etiamfringillafaucibusaugue.
Aeneanvelitsem,viverraeu,tempusid,
rutrumid,mi.Nullamnecnibh.Proinullam-
corper,dolorincursustristique,erosaugue
tempornibh,atgravidadiamwisiatpurus.
Donecmattisullamcorpertellus.Phasellusvel
nulla.Praesentinterdum,erosinsodalessol-
licitudin,nuncnullapulvinarjusto,aeuismod
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,eusodalesmagnarisusmolestie
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.Curabiturimperdiettem-
pusturpis.Vivamussapiendolor,congueve-
nenatis,euismodeget,portarhoncus,magna.
Proincondimentumpretiumenim.Fusce
fringilla,liberoetvenenatisfacilisis,erosen-
imcursusarcu,vitaefacilisisodioauguevitae
orci.Aliquamvariusnibhutodio.Sedcondi-
mentumcondimentumnunc.Pellentesque
egetmassa.Pellentesquequismauris.Donec
utligulaacpedepulvinarlob ortis.Pellen-
tesqueeuismod.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentelit.Utlaoreet
ornareest.Phasellusgravidavulputatenulla.
Donecsitametarcuutsemtempormalesua-
da.Praesenthendreritaugueinurna.Proin
enimante,ornarevel,consequatut,blandit
in,justo.Donecfeliselit,dignissimsed,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,dapibussitamet,vestibulumet,commodo
id,est.Pellentesquepurus.Pellentesquetris-
tique,nuncacpulvinaradipiscing,justoeros
consequatlectus,sitametposuerelectusneque
velaugue.Crasconsectetuerliberoaceros.Ut
egetmassa.Fuscesitametenimeleifendsem
dictumauctor.Inegetrisusluctuswisiconva-
llispulvinar.Vivamussapienrisus,temporin,
viverrain,aliquetpellentesque,eros.Aliquam
euismodliberoasem.
Nuncvelitaugue,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-
risimperdiet,metusattristiquevenenatis,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,
ligulaligulasuscipiturna,vitaeportaodioer-
atquissapien.Proinluctusleoiderat.Etiam
massametus,accumsanpellentesque,sagittis
sitamet,venenatisnec,mauris.Praesentur-
naeros,ornarenec,vulputateeget,cursussed,
justo.Phasellusneclorem.Nullamligulaligu-
la,mollissitamet,faucibusvel,eleifendac,
dui.Aliquameratvolutpat.
Fuscevehicula,tortoretgravidaporttitor,me-
tusnibhconguelorem,uttempuspurusmau-
risapede.Integertinciduntorcisitamet
turpis.Aeneanametus.Aliquamvestibulum
lobortisfelis.Donecgravida.Sedsedurna.
Maurisetorci.Integerultricesfeugiatligu-
la.Seddignissimnibhamassa.Donecorci
dui,temporsed,tinciduntnonummy,viverra
sitamet,turpis.Quisquelobortis.Proinvene-
natistortornecwisi.Vestibulumplacerat.In
hachabitasseplateadictumst.Aliquampor-
tamiquisrisus.Donecsagittisluctusdiam.
Namipsumelit,imperdietvitae,faucibusnec,
fringillaeget,leo.Etiamquisdolorinsapien
porttitorimperdiet.
Craspretium.Nullamalesuadaipsumut
libero.Suspendissegravidahendrerittellus.
Maecenasquislacus.Morbifringilla.Vestibu-
lumodioturpis,temporvitae,scelerisquea,
dictumnon,massa.Praesenteratfelis,por-
tasitamet,condimentumsitamet,placer-
atet,turpis.Praesentplaceratlacusaen-
im.Vestibulumnoneros.Utcongue.Donec
tristiquevariustortor.Pellentesquehabitant
morbitristiquesenectusetnetusetmalesuada
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-
dalessemsitametmauris.Suspendisseornare
pretiumorci.Fuscetristiqueenimegetmi.
Vestibulumeroselit,gravidaac,pharetrased,
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,conguesitamet,
sem.Sedsagittispedeanisl.Sedtincidunto-
dioapede.Seddui.Nameuenim.Aliquam
sagittislacusegetlibero.Pellentesquediam
sem,sagittismolestie,tristiqueet,fermentum
ornare,nibh.Nullaettellusnonfelisimperdi-
etmattis.Aliquameratvolutpat.
Vestibulumsodalesipsumidaugue.Integer
ipsumpede,convallissitamet,tristiquevi-
tae,temporut,nunc.Namnonligulanon
loremconvallishendrerit.Maecenashendrerit.
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.
Donecsitametnibhnecpurusdignissimmat-
tis.Phasellusvehiculaelitatlacus.Nullafa-
cilisi.Crasutarcu.Sedconsectetuer.Integer
tristiqueelitquisfelisconsectetuereleifend.
Crasetlectus.
Utconguemalesuadajusto.Curabiturcongue,
felisathendreritfaucibus,maurislacusport-
titorpede,necaliquamturpisdiamfeugiatar-
cu.Nullamrhoncusipsumatrisus.Vestibu-
lumadolorseddolorfermentumvulputate.
Sednecipsumdapibusurnabibendumlobor-
tis.Vestibulumelit.Namligulaarcu,volutpat
eget,laciniaeu,lobortisac,urna.Nammol-
lisultricesnulla.Crasvulputate.Suspendisse
atrisusatmetuspulvinarmalesuada.Nullam
lacus.Aliquamtempusmagna.Aliquamut
purus.Prointellus.
Vestibulumanteipsumprimisinfaucibusorci
luctusetultricesposuerecubiliaCurae;Donec
scelerisquemetus.Maecenasnonmiutmetus
portahendrerit.Nuncsemp er.Crasquiswisi
utloremposueretristique.Nuncvestibulum
scelerisquenulla.Suspendissepharetrasollic-
itudinante.Praesentatauguesitametante
interdumporta.Nuncbibendumaugueluctus
diam.Etiamnecsem.Sederosturpis,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,
pedepurussollicitudinurna,acaliquammetus
orciinleo.Utmolestieultricesmauris.Viva-
musvitaesem.Aliquameratvolutpat.Prae-
sentcommodo,nislacdapibusaliquet,tortor
orcisodaleslorem,nonornarenullaloremquis
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
nullaornarevenenatis.Nullaquissapien.Sed
tortor.Phasellusegetmi.Crasnunc.Crasa
enim.
Quisquenisl.Indignissimdapibusmassa.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,
porttitorsitamet,mattissuscipit,magna.Vi-
vamusegestasviverraest.Morbiatrisussed
sapiensodalespretium.
Morbicongueconguemetus.Aeneansedpu-
rus.Nampedemagna,tristiquenec,porta
id,sollicitudinquis,sapien.Vestibulumblan-
dit.Suspendisseutaugueacnibhullamcorper
posuere.Integereuismod,nequeateleifend
fringilla,augueelitornaredolor,veltincidunt
purusestidlacus.Vivamusloremdui,commo-
doquis,scelerisqueeu,tinciduntnon,magna.
Crassodales.Quisquevestibulumpulvinardi-
am.Phasellustincidunt,leovitaetristiquefa-
cilisis,ipsumwisiinterdumsem,dapibussem-
pernullavelitvellectus.Crasdapibusmauris
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-
mentumat,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-
sentsedorcivitaedolorsollicitudinadipiscing.
Crasaneque.Utrisusdui,interdumat,plac-
eratid,tristiqueeu,enim.Vestibulumante
ipsumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Etiamadipiscingeros
vestibulumdolor.Pellentesquealiquam,diam
egeteleifendposuere,augueerosporttitorlec-
tus,acdignissimduimetusnecfelis.Quisque
lacinia.Vestibulumtellus.Suspendissenec
wisi.Aeneanacfelis.Aliquamultricesmetus
etnulla.
Praesentsedestnonnibhtempusvenenatis.
Praesentrhoncus.Curabitursagittisestsit
ametneque.Sedcommodomalesuadalectus.
Phasellusenimtellus,temporut,tristiqueeu,
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.Vivamustemp orlacussedlacus.Nunc
utlorem.Utettortor.Nullamvariuswisiat
diam.Etiamultricies,dolorsitametfermen-
tumvulputate,nequeliberovestibulumorci,
vitaefringillanequearcualiquetante.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisquevenenatislobortisaugue.Sed
tempor,tellusiaculispellentesquepharetra,
pededuimalesuadamauris,velultricesurna
maurisacnibh.Etiamnibhodio,ultriciesve-
hicula,vestibulumvitae,feugiateleifend,fe-
lis.Vivamuspulvinar.Aliquameratvolutpat.
Nullaegestasvenenatismetus.Namfeugiat
nuncquiselitegestassagittis.Sedvitaefe-
lis.Inliberoarcu,rhoncusin,commodoeget,
auctorin,enim.Vivamussuscipitest.Nul-
ladapibus,magnavelaliquetegestas,massa
massahendreritlacus,acrutrumtellustellus
sitametfelis.Crasviverra.
Suspendisseeununc.Aliquamdignissimurna
sitametmauris.Crascommodo,urnautport-
titorvenenatis,arcumetussodalesrisus,vitae
gravidasapienligulainest.Donecvulputate
sollicitudinwisi.Donecvehicula,estidinter-
dumornare,nibhtellusconsectetuerjusto,a
ultricesfeliseratatlectus.Inestmassa,male-
suadanon,suscipitat,ullamcorpereu,elit.
Namnullalacus,bibendumsitamet,sagit-
tissed,temporeget,libero.Praesentligula.
Suspendissenulla.Etiamdiam.Nullaante
diam,vestibulumet,aliquetac,imperdietvi-
tae,urna.Fuscetinciduntlacusvelelit.Mae-
cenasdictum,tortornoneuismodbibendum,
pedenibhpretiumtellus,atdignissimleoeros
egetpede.Nullavenenatiseleifenderos.Ae-
neanutodiodignissimauguerutrumfaucibus.
Fusceposuere,tellusegetviverramattis,er-
attellusportami,atfacilisissemnibhnon
urna.Phasellusquisturpisquismaurissus-
cipitvulputate.Sedinterdumlacusnonvelit.
Vestibulumanteipsumprimisinfaucibusorci
luctusetultricesposuerecubiliaCurae;
Vivamusvehiculaleoajusto.Quisquenecau-
gue.Morbimauriswisi,aliquetvitae,dignis-
simeget,sollicitudinmolestie,ligula.Indic-
tumenimsitametrisus.Curabiturvitaevelit
eudiamrhoncushendrerit.Vivamusutelit.
Praesentmattisipsumquisturpis.Curabitur
rhoncusnequeeudui.Etiamvitaemagna.
Namullamcorper.Praesentinterdumbiben-
dummagna.Quisqueauctoraliquamdolor.
Morbieuloremetestporttitorfermentum.
Nuncegestasarcuattortorvariusviverra.
Fusceeunullautnullainterdumconsectetuer.
Vestibulumgravida.Morbimattisliberosed
est.
Abstract
TheDimensionalRegularizationtechniqueofBolliniandGiambiagi
(BG)[Phys.Lett.B40,566(1972);IlNuovoCim.B12,20
(1972);Phys.Rev.D53,5761(1996)]cannotbeemployedfor
allSchwartzTemperedDistributionsExplicitlyLorentzInvariant
(STDELI)S
L
.Weliftedsuchlimitationin[J.Phys.Comm.2
115029(2018)],whichopensnewQFTpossibilities,centeringin
theuseofSTDELIthatallowsonetoobtainaproductinaring
withzerodivisors.Thisinturn,overcomesallproblemsregrading
QFTinfinities.Weprovideherethreeexamplesoftheapplication
ofourSTDELI-extensiontoquantumfieldtheory(A)theexact
evaluationofanelectron’sselfenergytooneloop,(B)theexact
evaluationofQED’svacuumpolarization,andC)theλ
ϕ
4
4!
theory
forsixdimensions,thatisnon-renormalizable.
Keywords
DimensionalRegularizationGeneralization,ElectronSelfEnergy,
VacuumPolarization,Six-DimensionalNonRenormalizableλ
ϕ
4
4!
Theory
1.Introduction
Thecurrentparadigminquantumfieldtheory(QFT)assertsthatnon
renormalizabletheoriesareintractable.Thisissobecauseaninfinite
numberofcountertermsareneededtodealwiththeinfinitiesthat
ariseintheconvolutionoftwoquantumpropagators.
Anascentnewparadigm,basedonUltrahyperfunctions(UHF),a
generalizationofthewellknownSchwartz’distributions,usesthefact
DOI:10.4236/jhepgc.2020.64040Aug.17,2020590JournalofHighEnergyPhysics,GravitationandCosmology
A.Plastino,M.C.Rocca
thatquantumpropagatorscanberepresentedbyUHF-distributions.
Ifonedoesit,thenconvolutionsarealwaysfiniteandinfinitiesdisap-
pear.Nonrenormalizabletheoriescanthenbetackledwithnoundue
effort.Thisfactishighlyunpalatableforworkersoftheoldparadigm
thatwillfighttothedeaththepropagationofthenewone.
Thus,thiseffortisbasedonquantumfieldtheoretical(QFT)results
ofthenewparadigm,exhaustivelydiscussedwithallpertinentdetails
in[1–6].TheyrevolvearoundUltrahyperfunctionstheory(UHT),a
generalizationofSchwartz’distributiontheory.Specifically,oneuses
aparticularkindofSchwartz’sdistributionscalledexplicitlyLorentz
invariantones(STDELI).WewillrevisithereviaSTDELItheexact
evaluationofanelectron’sselfenergytooneloop,theexactevaluation
ofQED’svacuumpolarization,andtheλ
ϕ
4
4!
theoryinsix(nonrenor-
malizable)dimensions.Weuseextensivelythemathematicsof[1–6],
withoutgivingalltheintermediateresultsforthesakeofsavingspace.
Forthesakeofcompleteness,though,somemathematicalsketchis
givenbelow,whichmaybeskipped.
1.1.BriefSummaryoftheMathematicalResultsto
BeEmployed
Thissubsectionmaybeomittedatafirstreading.Ourworkrevolves
aroundtheproblemofusingaworkableversionoftheproductoftwo
distributions(aproductinaringwithdivisorsofzero).Thisisan
oldconundrumofhardfunctionalanalysis.Inquantumfieldtheory
(QFT),theproblemofevaluatingtheproductofdistributionswith
coincidentpointsingularitiesisrelatedtotheasymptoticbehaviour
ofloopintegralsofpropagators,becomingthusanobstaclethatisto
theessencetoovercome.Wediditinreferences[1–5].
Fromamathematicalpointofview,practicallyalldefinitionsofthat
productofdistributionsleadtolimitationsonthesetofdistributions
thatcanbemultipliedbyeachothertogiveanotherdistributionof
thesametype.Infact,LaurentSchwartzhimselfwasunabletodefine
aproductofdistributionsregardedasanalgebra,insteadofasaring
withdivisorsofzero.
Inreferences[2–5]itwasdemonstratedthatitispossibletode-
fineageneralconvolutionbetweenultradistributions,introducedby
themathematicianJ.SebastiaoeSilvaandcalledanultrahyperfunc-
tion[7].Thisnewconvolutionyieldsthenanultrahyperfunction.
Therefore,wehaveaproductinaringwithzerodivisors.Sucharing
isthespaceofdistributionsofexponentialtype,orultradistributionsof
exponentialtype,obtainedapplyingtheanti-Fouriertransformtothe
spaceoftemperedultradistributionsorultradistributionsofexponen-
tialtype.Rememberthatultrahyperfunctionsarethegeneralization
andextensiontothecomplexplaneoftheSchwartz’tempereddistri-
butionsandofthedistributionsofexponentialtype[2–5].Thatis,the
temperatedistributionsandthoseofexponentialtypeareasubsetof
thesetofultrahyperfunctions.Onethenfacestheproblemofformu-
latingtheconvolutionbetweenultradistributions.Thisisacomplex
issue,difficulttomanage,evenifitexhibitstheadvantageofallowing
onetoattemptingtodealwithnon-renormalizableQFT’s.
Fortunatelyenough,wehavefound[1]thatamethodsimilarto
thatusedtodefinetheconvolutionofultradistributionscanalsobe
employedtodefinetheconvolutionofLorentzInvariantdistribution-
s[8],usingageneralizationBollini-Giambiaggi’s(BG)dimensionalreg-
ularization(DR)([9–11])inmomentumspace[1].Asaconsequence,
DOI:10.4236/jhepgc.2020.64040591JournalofHighEnergyPhysics,GravitationandCosmology
A.Plastino,M.C.Rocca
ultradistributionsneednottobeusedinourpresentcalculations,which
considerablysimplifiesthings.Takingadvantageoftheregularization
techniqueof[1]onecanalsoworkinconfigurationspace.Thus,oneis
ableobtainaconvolutionofLorentzInvarianttempereddistributions
inmomentumspaceandthecorrespondingproductinconfiguration
space[1].
Usingtheresultsof[1]wehavemanagedtodotheQuantumField
TheoryofEinstein’sGravity[12],theNon-RelativisticQuantumField
TheoryofVerlinde’sGravity[6]andtheNewton’sGravity[13].
Itisoftheessencetohaveaclearpictureoftheconsequencesof
theadvancesmadeinreference[1].Asastartingpoint,remarkthat
DRisoneofthemostimportantadvancesintheoreticalphysics,be-
ingusedinseveraldisciplines[14,15].IfoneaddstoitourDR-
generalization[1],theabovereferredtoconvolutionhappenstobe
bothoneofSTDELIinmomentumspaceandaproductinaringwith
divisorsofzeroinconfigurationspace.Ourworkof[1]canthenbe
usedtoquantizenon-renormalizableQFT’s.Thiscouldberegarded
asaworthwhileachievement.
Moretothepoint,letusemphasizethatour[1]-workisconcerned
withdeeperissuesthanthoseregardingQFTaxiomaticsinEuclidean
spaceandQFTrenormalization.Reference[1]generalizedBGdi-
mensionalregularizationtoallSchwartztempered,explicitlyLorentz
invariant,distributions(STDELI),somethingthatBGwereunableto
achieve.Permitustoinsistonthefactthatthiswouldpermitoneto
dealwithnon-renormalizableQFT’s.
Agreatadvantagethereofliesinthefactthatwedonothave
tousecountertermsinarenormalizationprocessdevotedtoelimi-
nateinfinities.Thisisexactlywhatwewouldnotwanttodo,since
anon-renormalizabletheoryinvolvesaninfinitenumberofcountert-
erms.Thecentralpurposeofourworkistoavoidcounterterms.In
addition,wedonotappealtoasimplecorrelation-functions’convolu-
tion(notdefinedforallSTDELI).Atthesametime,weconserve
allextantsolutionstotheproblemofrunningcouplingconstants
andtherenormalizationgroup.TheSTDELIconvolution,onceob-
tained,convertsconfigurationspaceintoaringwithzero-divisors.In
it,onehasnowdefinedaproductbetweenthering-elements.Thus,
anyunitary-causal-Lorentzinvarianttheoryquantifiedinsuchaman-
nerbecomespredictive.Thedistinctionbetweenrenormalizableon
not-renormalizableQFT’sbecomesunnecessarynow.WithourBG
generalization,thatusesLaurent’sexpansionsinthedimension,al-
lfiniteconstantsoftheconvolutionsbecomecompletelydetermined,
eliminatingarbitrarychoicesoffiniteconstants.Thisistantamount
toeliminatingallfiniterenormalizationsofthetheory.Whatisthe
importanceofusingonlythattermindependentofthedimensionin
Laurent’sexpansion?Thattheresultobtainedforfiniteconvolutions
willcoincidewithsuchaterm.Thisfacttranslatestoconfiguration
spacetheproduct-operationonaringwithdivisorsofzero.
1.2.PresentGoals
Wewishheretoillustratethepowerofourapproachwithreference
totwointerestingexamples.Insuchaveinwecalculatetwoconvo-
lutionsofdistributionsusedinQFT,inconnectionwiththeelectron
selfenergyandthevacuumpolarizationinQED.
DOI:10.4236/jhepgc.2020.64040592JournalofHighEnergyPhysics,GravitationandCosmology
A.Plastino,M.C.Rocca
2.PreliminaryMaterials
2.1.TheBG-OriginalCalculationofanElectron
SelfEnergytoOneLoop
Theselfenergyoftheelectronisdefinedas
Σ(p,ν)=
e
2
(2π)
ν
γ
µ
[·(pk)m]
[(pk)
2
+m
2
]k
2
γ
µ
d
ν
k.(2.1.1)
BGevaluatedthisintegralforthefirsttimein[10].BG’sresult,
obtainedusingtheirdefinitionofDR,is
Σ(p,ν)=
e
2
(4π)
ν
2
(m
2
ρ)
ν
2
2
Γ
2
ν
2
[(·p+m)(2ν)2m]
2
ν2
·
F
2
ν
2
,
ν
2
1,
ν
2
;1
1
ρ
+[m(·p+m)]
4
ν
2
·
F
2
ν
2
,
ν
2
,
ν
2
+1;1
1
ρ

,(2.1.2)
wherethevariableρisdefinedasρ=(p
2
+m
2
)/m
2
.Toobtainthe
finitepartoftheselfenergy,BGdecomposeditintheform
Σ(p,ν)=A+(·p+m)B+(·p+m)
2
Σ
f
(p,ν),(2.1.3)
whereA,B,andΣ
f
(p,ν)weredefinedas
A=[Σ(p,ν)]
·p+m=0
B=
Σ(p,ν)A
·p+m
·p+m=0
Σ
f
(p,ν)=
Σ(p,ν)A(·p+m)B
(·p+m)
2
·p+m=0
,(2.1.4)
with(·p+m)
1
=( m·p)/(p
2
+m
2
).Σ
f
(p,ν)thusturnsout
tobethefinitepartoftheselfenergy.Asaresultofthesedefinitions
weget
A=
e
2
m
ν3
(4π)
ν
2
ν1
ν3
Γ
2
ν
2
,(2.1.5)
B=
A
m
,(2.1.6)
Σ
f
(p,ν)=
e
2
m
ν4
(4π)
ν
2
Γ
2
ν
2
·

(2ν)
m·p
m
2
ρ
4
m·p
m
2
ρ
2
+
2
·
2
ν2
F
1,2
ν
2
,
ν
2
;1ρ
+
2
m·p
m
2
ρ
2
m·p
m
2
ρ
1

4
ν
2
·
F
1,2
ν
2
,
ν
2
+1;1ρ
+
ν1
ν3
2
m·p
m
2
ρ
2
1
+
m·p
m
2
ρ

.
(2.1.7)
ItshouldbenotedthatAandBareindependentofp.Thethree
aboveequationshavenotbeenmodifiedtillthepresenttime.However,
DOI:10.4236/jhepgc.2020.64040593JournalofHighEnergyPhysics,GravitationandCosmology
A.Plastino,M.C.Rocca
andthisisacrucialissue,BG’sdecompositionexhibitsanunwanted
facet.ThereareaninfinitenumberofwaystochooseAandB.The
causeisfoundinproductsofthegammafunctionwiththehypergeo-
metricfunctionatapoleofthegammafunction.Wemusttherefore
isolatesuchpoleinproperfashion,soastoavoidthisproblem.The
rigorouswaytodothisistousetheconvolutionofLorentzInvari-
antdistributionsobtainedin[1].Ifthisisdonestraightforwardly,the
resultobtainedturnsouttobetooinvolvedtobefacednumerical-
ly,asonefacesquitedifficulttoevaluatefunctions.Thesefunctions
arisewhenwederivethehypergeometricfunctions(HF)thatappear
in(2.1.2)withrespecttothedimensionνsoastoperformthecor-
respondingLaurent’sseriesexpansion,typicalofDR.Instead,before
HF-derivingasexplainedabove,onemustfirstisolatethegammafunc-
tionatthep ole(fromthehypergeometricfunction)andsubsequently
tackletheLaurentexpansion.Thiswillbeillustratedbelow.
2.2.VacuumPolarization
BGalsocalculatedthevacuumpolarizationinQED[10].Theintegral
thatdefinesitisgivenby
Π
µ,v
(k,ν)=
ie
2
(2π)
ν
Tr
γ
µ
·pm
p
2
+m
2
γ
v
·(pk)m
(pk)
2
+m
2
d
ν
p. (2.2.1)
Toevaluateit,thefollowingresultsshouldbeused
Trγ
µ
γ
v
=d(ν)η
µv
,(2.2.2)
Trγ
µ
γ
v
γ
ρ
γ
σ
=d(ν)(η
µv
ηρση
µρ
η
v σ
+η
µσ
η
v ρ
),(2.2.3)
whered(ν)isananalyticfunctionofthedimensionν,which,forνa
positiveinteger,matchesthenumberofcomponentsoftheassociated
spinorinaνdimensionalspaceTheresultoftheintegral(2.2.1)is
Π
µv
(k,ν)=
e
2
(4π)
ν
2
d(ν)
3
Γ
2
ν
2
m
ν4
(k
µ
k
v
η
µv
k
2
)F
2
ν
2
,2,
5
2
;
k
2
4m
2
.
(2.2.4)
Notethatinallterms,exceptforthefirstone,onefacesthemulti-
plicationofthegammafunctionbyazeroofthehypergeometricfunc-
tion.Therefore,thefinitepartofthevacuumpolarizationΠ
µvF
(k,ν)
isgivenby(noteacritical1subtractioninthelastbracket)
Π
µvF
(k,ν)=
e
2
(4π)
ν
2
d(ν)
3
Γ
2
ν
2
m
ν4
(k
µ
k
v
η
µv
k
2
)·
F
2
ν
2
,2,
5
2
;
k
2
4m
2
1
,(2.2.5)
andthevacuumpolarizationiscastas
Π
µv
(k,ν)=
e
2
(4π)
ν
2
d(ν)
3
Γ
2
ν
2
m
ν4
(k
µ
k
v
η
µv
k
2
)+Π
µv F
(k,ν)
(2.2.6)
DOI:10.4236/jhepgc.2020.64040594JournalofHighEnergyPhysics,GravitationandCosmology
A.Plastino,M.C.Rocca
3.SimplePreliminaryExampletoBetter
UnderstandOurProcedure
Soastobetterunderstandourapproachwewillconfrontnowwith
twosimpleexamples:theself-energyandthevacuumpolarization,
correspondingtoamasslessscalarfieldϕandamassivescalarfieldψ
withaninteractiongivenbygϕψ
2
3.1.TheScalarFieldSelfEnergy
Inthiscasetheselfenergyisgivenby:
Σ(p,ν)=
d
ν
k
k
2
[(pk)
2
+m
2
]
(3.1.1)
Theresultofthisintegralis:(see[9])
Σ(p,ν)=
ν
2
m
ν4
Γ
1
ν
2
F
1,2
ν
2
,
ν
2
;
p
2
m
2
(3.1.2)
Thislastequationcanbere-writtenintheform:
Σ(p,ν)=
2
ν
2
m
ν4
2ν
Γ
2
ν
2
F
1,2
ν
2
,
ν
2
;
p
2
m
2
(3.1.3)
Weusenowtheequality:
Γ
2
ν
2
F
1,2
ν
2
,
ν
2
;
p
2
m
2
=
Γ
2
ν
2
2
ν
Γ
3
ν
2
F
1,3
ν
2
,
ν
2
+1;
p
2
m
2
(3.1.4)
Withitweobtain:
Σ(p,ν)=
2
ν
2
m
ν4
2ν
Γ
2
ν
2
4
ν
2
m
ν6
p
2
ν(2ν)
Γ
3
ν
2
F
1,3
ν
2
,
ν
2
+1;
p
2
m
2
(3.1.5)
Theselfenergycanbethendecomposedas:
Σ(p,ν)=A
f
(p,ν)(3.1.6)
where
A=
2
ν
2
m
ν4
2ν
Γ
2
ν
2
(3.1.7)
and
Σ
f
(p,ν)=
4
ν
2
m
ν6
p
2
ν(2ν)
Γ
3
ν
2
F
1,3
ν
2
,
ν
2
+1;
p
2
m
2
(3.1.8)
ThisistheprocedurefollowedbyBolliniandGiambiagi.Weare
goingtoproceedwithourgeneralizationofdimensionaldimensional.
Forthiswearegoingtodeveloptheself-energyinLaurent’sseries.
Letfbegivenby:
f(ν)=
π
ν
2
m
ν4
(2ν)
(3.1.9)
DOI:10.4236/jhepgc.2020.64040595JournalofHighEnergyPhysics,GravitationandCosmology
A.Plastino,M.C.Rocca
Itsdevelopmentis:
f(ν)=
π
2
2
+
π
2
2
[1ln(π)2ln(m)](ν4)+
k=2
b
k
(ν4)
k
(3.1.10)
Asaconsequencewehave:
f(ν
2
ν
2
=
π
2
ν4
+
π
2
2
[ln(π)+2ln(m)+C1](ν4)+
k=2
d
k
(ν4)
k
(3.1.11)
Developmentofselfenergythenis:
Σ(p,ν)=
2
2
ν4
+
2
[ln(π)+2ln(m)+C1+
p
2
2m
2
F
1,1,3;
p
2
m
2

+
k=1
a
k
(ν4)
k
(3.1.12)
Wecanwriteagain:
Σ(p,ν)=A
f
(p,ν)(3.1.13)
with
A=
2
2
ν4
(3.1.14)
and
Σ
f
(p,ν)=
2
[ln(π)+2ln(m)+C1+
p
2
2m
2
F
1,1,3;
p
2
m
2

+
k=1
a
k
(ν4)
k
(3.1.15)
Thereforetheexactvalueofselfenergyinfourdimensionsis:
Σ
P
(p,4)=Σ
f
(p,4)=
2
[ln(π)+2ln(m)+C1+
p
2
2m
2
F
1,1,3;
p
2
m
2

(3.1.16)
3.2.TheVacuumPolarizationofaScalarField
Asanexampleweconsiderthevacuumpolarization,givenby
Π(p,ν)=
d
ν
k
(k
2
+m
2
)[(pk)
2
+m
2
]
.(3.2.1)
Theresultofthisintegralis(see[9])
Π(p,ν)=
ν
2
m
ν4
Γ
2
ν
2
F
1,2
ν
2
,
3
2
;
p
2
4m
2
.(3.2.2)
Weusetheequality
Γ
2
ν
2
F
1,2
ν
2
,
3
2
;
p
2
4m
2
=
DOI:10.4236/jhepgc.2020.64040596JournalofHighEnergyPhysics,GravitationandCosmology
A.Plastino,M.C.Rocca
Γ
2
ν
2
p
2
6m
2
Γ
3
ν
2
F
1,3
ν
2
,
5
2
;
p
2
4m
2
.(3.2.3)
Wehavethen
Π(p,ν)=
ν
2
m
ν4
Γ
2
ν
2
ν
2
m
ν6
p
2
6
Γ
3
ν
2
F
1,3
ν
2
,
5
2
;
p
2
4m
2
.(3.2.4)
Wemakethedecomposition
Π(p,ν)=A
f
(p,ν).(3.2.5)
Thus,weobtain
A=
ν
2
m
ν4
Γ
2
ν
2
,(3.2.6)
and
Π
f
(p,ν)=
ν
2
m
ν6
p
2
6
Γ
3
ν
2
F
1,3
ν
2
,
5
2
;
p
2
4m
2
.(3.2.7)
ThisistheresultobtainedbyBG.Weproceednowtogeneralizeit.
Wedefine
f(ν)=π
ν
2
m
ν4
(3.2.8)
Asaconsequence,weobtaintheLaurentexpansion
f(ν
2
ν
2
=
2π
2
ν4
π
2
[C+2ln(m)+ln(π)]+
k=1
b
k
(ν4)
k
.(3.2.9)
Theexpressionofthevacuumpolarizationisnow
Π(p,ν)=
2
2
ν4
π
2
[C+2ln(m)+ln(π)+
p
2
6m
2
F
1,1,
5
2
;
p
2
4m
2

+
k=1
a
k
(ν4)
k
.(3.2.10)
Wecanre-writeitas
Π(p,ν)=A
f
(p,ν),(3.2.11)
with
A=
2
2
ν4
,(3.2.12)
and
Π
f
(p,ν)=π
2
[C+2ln(m)+ln(π)+
p
2
6m
2
F
1,1,
5
2
;
p
2
4m
2

+
k=1
a
k
(ν4)
k
.(3.2.13)
Theexactvalueforthefourdimensionalvacuumpolarizationis
then
Π
P
(p,4)=Π
f
(p,4)=π
2
[C+2ln(m)+ln(π)+
DOI:10.4236/jhepgc.2020.64040597JournalofHighEnergyPhysics,GravitationandCosmology
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p
2
6m
2
F
1,1,
5
2
;
p
2
4m
2

.(3.2.14)
Asasecondexamplewegivethecalculationoftheself-energyofa
well-knownQFTmodel,intherenormalizablecaseν=4andinthe
non-renormalizablecaseν=6.
4.TheMasslessCasefor
λ
4!
ϕ
4
Model
4.1.Theν-DimensionalSelf-Energy
Asitiswellknown,inMinkowski’sspacethesecondorderselfenergy
isdefinedas(allremainingterms,tosecondorder,vanishusingour
method)
Σ(ρ)=(ρi0)
1
(ρi0)
1
(ρi0)
1
,(4.1.1)
whereρ=k
2
1
+k
2
2
+k
2
3
+....+k
2
ν1
k
2
0
.
Goingbeyondthemeredefinition(4.1.1),wewillactuallyevaluate
Σnowusingourmethodology.Inνdimensionswehave
Σ(ρ,ν)=[(ρi0)
1
(ρi0)
1
(ρi0)
1
]
ν
.(4.1.2)
WearethusconfrontedwiththeconvolutionofthreemasslessFeyn-
man’spropagators.Infacingsuchproblem,wetakeintoaccountthat
F
1
{f
1
f
2
f
3
}=(2π)
2ν
ˆ
f
1
ˆ
f
2
ˆ
f
3
.(4.1.3)
Accordingtoreference[16],wethishave
F
1
(ρi0)
1
=
e
2
(2π)
ν
2
ν2
π
ν
2
Γ
ν
2
1
(x+i0)
1
ν
2
,(4.1.4)
andaccordingly,
F
1
(ρi0)
1
(ρi0)
1
(ρi0)
1
=
e
i3π
2
(2π)
ν
2
3ν6
π
3ν
2
Γ
ν
2
1

3
(x+i0)
3
3ν
2
.(4.1.5)
Usingagainreference[16]wefacenow
F
(x+i0)
3
3ν
2
=
e
2
Γ
3ν
2
3
2
62ν
π
ν
2
Γ(3ν)(ρi0)
ν3
(4.1.6)
withwhichwecanwriteforΣ(ρ,ν)
Σ(ρ,ν)=[(ρi0)
1
(ρi0)
1
(ρi0)
1
]
ν
=
e
(3ν
3ν
2
3
π
ν
Γ
ν
2
1

3
Γ(4ν)(ρi0)
ν3
,(4.1.7)
whichsuccessfullycompletesourtask.
DOI:10.4236/jhepgc.2020.64040598JournalofHighEnergyPhysics,GravitationandCosmology
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4.2.AConfirmationoftheValidityoftheAbove
Result
Tosuchaneffect,weproceednowtothetraditionaltechniquefor
evaluatingΣviathewellknownFeynman’sintegralsandparameters.
Thus,wehave
Σ(k)=(ρi0)
1
(ρi0)
1
(ρi0)
1
=
d
4
pd
4
q
(p
2
i0)(q
2
i0)[(p+qk)
2
i0]
,(4.2.1)
sothat,inνdimensions
Σ(k,ν)=[(ρi0)
1
(ρi0)
1
(ρi0)
1
]
ν
=
d
ν
pd
ν
q
(p
2
i0)(q
2
i0)[(p+qk)
2
i0]
.(4.2.2)
AfterperformingthecorrespondingWickrotationswegetthen
d
ν
pd
ν
q
p
2
q
2
(p+qk)
2
.(4.2.3)
Letusevaluatefirst
Σ
1
(k,ν)=
d
ν
p
p
2
(p+qk)
2
.(4.2.4)
Wedefineherekq=k
1
andthen
Σ
1
(k
1
)=
d
ν
p
p
2
(pk
1
)
2
.(4.2.5)
WenowuseFeynman’sparametersandobtain
Σ
1
(k
1
)=i
1
0
d
ν
pdx
[(pk
1
x)
2
+a]
2
(4.2.6)
where
a=k
2
1
x(1x).(4.2.7)
Changevariablesbysettingu=pkxandwrite
Σ
1
(k
1
)=
1
0
d
ν
udx
[u
2
+a]
2
.(4.2.8)
Weevaluatetheintegraloveruusing[17]andhave
(u
2
)
n
(u
2
+a)
m
d
ν
u=
π
ν
2
Γ
ν
2
Γ
ν
2
+n
Γ
mn
ν
2
Γ(m)a
mn
ν
2
,(4.2.9)
whichleadsto:
Σ
1
(k
1
)=
ν
2
γ
2
ν
2
k
ν4
1
1
0
x
ν
2
2
(1x)
ν
2
2
dx.(4.2.10)
DOI:10.4236/jhepgc.2020.64040599JournalofHighEnergyPhysics,GravitationandCosmology
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Using[17]onceagainleadsto
1
0
x
ν
2
2
(1x)
ν
2
2
dx=
2
ν2
F
2
ν
2
,
ν
2
1,
ν
2
;1
.(4.2.11)
Alsofrom[17]onefinds
F
2
ν
2
,
ν
2
1,
ν
2
;1
=
Γ
ν
2
Γ
ν
2
1
Γ(ν2)
,(4.2.12)
sothat
Σ
1
(k
1
)=
1
Γ(ν2)
π
ν
2
Γ
ν
2
1

2
Γ
2
ν
2
k
ν4
1
.(4.2.13)
Usingthisresult,wehavefortheself-energy:
Σ(k,ν)=
1
Γ(ν2)
π
ν
2
Γ
ν
2
1

2
Γ
2
ν
2
×
d
ν
q
q
2
[(kq)
2
]
2
ν
2
.(4.2.14)
WenowturnourattentiontoFeynman’sgeneralizedparameters:
1
A
α
B
β
=
Γ(α+β)
Γ(α)Γ(β)
1
0
x
α1
(1x)
β1
dx
[Ax+B(1x)]
α+β
(4.2.15)
Theself-energythushastheexpression
Σ(k,ν)=
1
Γ(ν2)
π
ν
2
Γ
ν
2
1

2
Γ
3
ν
2
×
1
0
x
1
ν
2
d
ν
q
[(kq)
2
x+q
2
(1x)]
3
ν
2
,(4.2.16)
or,equivalently:
Σ(k,ν)=
1
Γ(ν2)
π
ν
2
Γ
ν
2
1

2
Γ
3
ν
2
×
1
0
x
1
ν
2
dx
d
ν
q
[(qkx)
2
+a]
3
ν
2
(4.2.17)
Wedefinenowa=k
2
x(1x),u=qkx,andthenthelastintegral
canbeeasilyevaluatedas
d
ν
u
[(u
2
+a]
3
ν
2
=
π
ν
2
Γ
ν
2
a
ν
2
3
B
ν
2
,3ν
.(4.2.18)
Thus,wehavefortheself-energy
Σ(k,ν)=
1
Γ(ν2)
π
ν
Γ
ν
2
1

2
Γ(3ν)k
2ν6
×
1
0
x
ν
2
2
(1x)
ν3
dx.(4.2.19)
DOI:10.4236/jhepgc.2020.64040600JournalofHighEnergyPhysics,GravitationandCosmology
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Finally,weobtain
Σ(k,ν)=
1
Γ
3ν
2
3
π
ν
Γ
ν
2
1

3
Γ(3ν)k
2ν6
.(4.2.20)
ReturningnowtoMinkowski’sspace,wehavethefollowingexpres-
sionfortheν-dimensionalself-energy
Σ(ρ,ν)=
1
Γ
3ν
2
3
π
ν
Γ
ν
2
1

3
Γ(3ν)(ρi0)
ν3
.(4.2.21)
Wewillnextgivethedimensionνfixedvalues.
4.3.TheFour-DimensionalSelf-Energy
Thisisaknownproblem,thatwediscusshereonlyfordidacticpur-
poses.Infourdimensionsthetheoryλ
ϕ
4
4!
isrenormalizable.
Asthereadershouldexp ect,weexpandΣ(ν)inLaurentseries
aroundthepoleatν=4[1–6,12,13]andfind
Σ(ρ,ν)=
π
4
ρ
2(4ν)
π
4
ρ
2
ln(ρi0)+lnπ+C
13
4
+
m=1
a
m
(ν4)
m
.
(4.3.1)
Thetermindependentofν4inourexpansionyieldstheself-
energy[1–6,12,13]:
Σ(ρ)=
π
4
ρ
2
ln(ρi0)+lnπ+C
13
4
.(4.3.2)
4.4.SuccessfullyDealingwiththeNon
RenormalizableCaseof6Dimensions
Weposeourselvesnowacriticaltest,byshowingthatwecandealwith
ayetunsolvedinstancewithoutunduehardship.Whenν=6,theλ
ϕ
4
4!
theoryisnon-renormalizable.Thus,ifweappealtotheBGtechnique,
wecannotproceedaswedidabove.Theachievementthatwewill
boastaboutnowisthatourpresentLaurentbasedapproachremains
undauntedbythesedifficulties.Thiswillbeappreciatedbelow.
OurLaurentexpansionis
Σ(ρ,ν)=
π
6
ρ
3
6!(6ν)
π
6
ρ
3
6!
[ln(ρi0)
+lnπ+C
187
60
+
m=1
a
m
(ν6)
m
.[(4.4.1)
Thetermindependentofν6inourexpansionyieldstheself-
energy[1–6,12,13]
Σ(ρ)=
π
6
ρ
3
6!
ln(ρi0)+lnπ+C
187
60
(4.4.2)
Aspromised,nounduehardshipwasinvolved.Nonrenormaliz-
ableQFTtheoriescanbeeasilydealtwithbyappealtoour
techniquesof[1–6,12,13].
DOI:10.4236/jhepgc.2020.64040601JournalofHighEnergyPhysics,GravitationandCosmology
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5.OurExactResults
5.1.TheElectronSelfEnergytoOneLoop
Toevaluateselfenergyweusethefollowingequalities
F
2
ν
2
,
ν
2
1,
ν
2
;1
1
ρ
=ρ
2
ν
2
F
1,2
ν
2
,
ν
2
;1ρ
,(5.1.1)
F
2
ν
2
,
ν
2
,
ν
2
+1;1
1
ρ
=ρ
2
ν
2
F
1,2
ν
2
,
ν
2
+1;1ρ
.
(5.1.2)
Wecanwrite(2.1.2)intheform
Σ(p,ν)=
e
2
(4π)
ν
2
m
ν4
Γ
2
ν
2
[(·p+m)(2ν)2m]
2
ν2
·
F
1,2
ν
2
,
ν
2
;1ρ
+[m(·p+m)]
4
ν
2
·
F
1,2
ν
2
,
ν
2
+1;1ρ

.(5.1.3)
Usingthefollowingrelations,wecanisolatethegammafunctionat
thepole
Γ
2
ν
2
F
1,2
ν
2
,
ν
2
;1ρ
=
Γ
2
ν
2
+
2(1ρ)
ν
Γ
3
ν
2
F
1,3
ν
2
,
ν
2
+1;1ρ
,(5.1.4)
Γ
2
ν
2
F
1,2
ν
2
,
ν
2
+1;1ρ
=
Γ
2
ν
2
+
2(1ρ)
ν+2
Γ
3
ν
2
F
1,3
ν
2
,
ν
2
+2;1ρ
.(5.1.5)
Replacingtheseresultsin(5.1.3)weobtain
Σ(p,ν)=
e
2
(4π)
ν
2
m
ν4

[(·p+m)(2ν)2m]
2
ν2
+
[m(·p+m)]
42ν
ν
Γ
2
ν
2
+
[(·p+m)(2ν)2m]
4(1ρ)
ν(ν2)
·
F
1,3
ν
2
,
ν
2
+1;1ρ
+[m(·p+m)]
(84ν)(1ρ)
ν(ν+2)
·
F
1,3
ν
2
,
ν
2
+2;1ρ

Γ
3
ν
2

,(5.1.6)
orequivalently
Σ(p,ν)=
e
2
(4π)
ν
2
m
ν4

4
ν2
2ν4
ν
m
4
ν
(·p+m)
Γ
2
ν
2
+
DOI:10.4236/jhepgc.2020.64040602JournalofHighEnergyPhysics,GravitationandCosmology
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[(·p+m)(2ν)2m]
4(1ρ)
ν(ν2)
·
F
1,3
ν
2
,
ν
2
+1;1ρ
+[m(·p+m)]
(84ν)(1ρ)
ν(ν+2)
·
F
1,3
ν
2
,
ν
2
+2;1ρ

Γ
3
ν
2

.(5.1.7)
Notethat1ρ=p
2
/m
2
.Wecandecomp osetheselfenergyin
theform
Σ(p,ν)=A+B(·p+m)+Σ
f
(p,ν),(5.1.8)
whereA,B,andΣ
f
(p,ν)aregivenby
A=
e
2
(4π)
ν
2
m
ν3
4
ν2
2ν4
ν
Γ
2
ν
2
,(5.1.9)
B=
e
2
(4π)
ν
2
m
ν4
4
ν
Γ
2
ν
2
,(5.1.10)
Σ
f
(p,ν)=
e
2
(4π)
ν
2
m
ν4
[(·p+m)(2ν)2m]
4(1ρ)
ν(ν2)
·
F
1,3
ν
2
,
ν
2
+1;1ρ
+[m(·p+m)]
(84ν)(1ρ)
ν(ν+2)
·
F
1,3
ν
2
,
ν
2
+2;1ρ

Γ
3
ν
2
.(5.1.11)
Thisdecompositionisconsistent,sinceboththeselfenergyandits
finitepartdependlinearlyon·p+m.NotethatΣ
f
(p.ν)canbe
re-writtenas
Σ
f
(p,ν)=Σ
f1
(p,ν)+(·p+m
f2
(p,ν),(5.1.12)
with
Σ
f1
(p,ν)=
e
2
(4π)
ν
2
m
ν3
8(1ρ)
ν(ν2)
·
F
1,3
ν
2
,
ν
2
+1;1ρ
(84ν)(1ρ)
ν(ν+2)
·
F
1,3
ν
2
,
ν
2
+2;1ρ

Γ
3
ν
2
,(5.1.13)
Σ
f2
(p,ν)=
e
2
(4π)
ν
2
m
ν4
(2ν)
4(1ρ)
ν(ν2)
·
F
1,3
ν
2
,
ν
2
+1;1ρ
(84ν)(1ρ)
ν(ν+2)
·
F
1,3
ν
2
,
ν
2
+2;1ρ

Γ
3
ν
2
.(5.1.14)
NotethatΣ
f1
(p,ν)andΣ
f2
(p,ν)areindependentof·p+m.
Althoughthisresultseemstobecorrect,toobtainthetrueconvolution
wemustperformthepertinentLaurent’sexpansion.Tosuchanend
wedefine
f(ν)=
m
ν4
(4π)
ν
2

4
ν2
2ν4
ν
m
4
ν
(·p+m)
.(5.1.15)
DOI:10.4236/jhepgc.2020.64040603JournalofHighEnergyPhysics,GravitationandCosmology
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Wethenusethefollowingexpressionforthegammafunction
Γ
2
ν
2
=
2
ν4
C+
k=1
c
k
(ν4)
k
(5.1.16)
andthusobtainthecorrespondingLaurentexpansion
f(ν
2
ν
2
=
·p
(4π)
2
(ν4)
+
1
(4π)
2
·p
C+2ln(m)ln(4π)
1
2
+2m+
k=1
b
k
(ν4)
k
.
(5.1.17)
Usingthepreviousresult,wehavethefollowingexpressionforthe
selfenergy
Σ(ν,p)=
e
2
(4π)
2
·p
ν4
+
·p
C+2ln(m)ln(4π)
1
2
+2m+
k=1
b
k
(ν4)
k
+
e
2
(4
π
)
ν
2
[(·p+m)(2ν)2m]
4(1ρ)
ν(ν2)
·
F
1,3
ν
2
,
ν
2
+1;1ρ
+[m(·p+m)]
(84ν)(1ρ)
ν(ν+2)
·
F
1,3
ν
2
,
ν
2
+2;1ρ

Γ
3
ν
2

.(5.1.18)
CompletingtheLaurentexpansionwehave
Σ(ν,p)=
e
2
(4π)
2
·p
ν4
+
·p
C+2ln(m)ln(4π)
1
2
+2m+
k=1
b
k
(ν4)
k
+
e
2
(4π)
ν
2
·p(1ρ)
1
3
F(1,1,4;1ρ)+
F(1,1,3;1ρ)]+2m(1ρ)F(1,1,3;1ρ)+
k=1
c
k
(ν4)
k
,(5.1.19)
orequivalently
Σ(ν,p)=
e
2
(4π)
2
·p
ν4
+
·p
C+2ln(m)ln(4π)
1
2
+2m+
e
2
(4π)
ν
2
·p(1ρ)
1
3
F(1,1,4;1ρ)+
F(1,1,3;1ρ)]+2m(1ρ)F(1,1,3;1ρ)+
DOI:10.4236/jhepgc.2020.64040604JournalofHighEnergyPhysics,GravitationandCosmology
A.Plastino,M.C.Rocca
k=1
a
k
(ν4)
k
.(5.1.20)
Thislastresultcanberecastas
Σ(ν,p)=
e
2
(4π)
2
·p+m
ν4
m
ν4
+
·p
C+2ln(m)ln(4π)
1
2
+2m+
e
2
(4π)
ν
2
·p(1ρ)
1
3
F(1,1,4;1ρ)+
F(1,1,3;1ρ)]+2m(1ρ)F(1,1,3;1ρ)+
k=1
a
k
(ν4)
k
.(5.1.21)
Wearenowallowedtoeffectthenthefollowingdecomposition
Σ(p,ν)=A+(·p+m)B
f
(p,ν),(5.1.22)
wheretheconstantAandBaregivenby
A=
e
2
m
(4π)
2
(ν4)
,(5.1.23)
B=
e
2
(4π)
2
(ν4)
,(5.1.24)
andthefinitepartoftheselfenergyis
Σ
f
(ν,p)=
e
2
(4π)
2
{·p
C+2ln(m)ln(4π)
1
2
+2m+
e
2
(4π)
ν
2
·p(1ρ)
1
3
F(1,1,4;1ρ)+
1
2
F(1,1,3;1ρ)]+2m(1ρ)F(1,1,3;1ρ)+
k=1
a
k
(ν4)
k
.(5.1.25)
Wehavethenthefourdimensionalresult(thesub-indexPbelow
means”physical”)
Σ
P
(p)=Σ
f
(p,4)=
e
2
(4π)
2
·p
C+2ln(m)ln(4π)
1
2
+2m+
·p(1ρ)[
1
3
F(1,1,4.1ρ)+
1
2
F(1,1,3,1ρ)]
2m(1ρ)F(1,1,3,1ρ)},(5.1.26)
whichistheexactresultoftheconvolution(see[1]).Wearrivethusto
theconclusionthatΣ
P
(p,ν)isthetruephysicalselfenergy,arather
niceconclusion.
DOI:10.4236/jhepgc.2020.64040605JournalofHighEnergyPhysics,GravitationandCosmology
A.Plastino,M.C.Rocca
5.2.VacuumPolarizationEvaluation
Thevacuump olarizationcanbewrittenintheform
Π
µv
(k,ν)=
e
2
(4π)
ν
2
d(ν)
3
Γ
2
ν
2
m
ν4
(k
µ
k
v
η
µv
k
2
)+Π
µvF
(k,ν),
(5.2.1)
whereΠ
µv F
(k,ν)isgivenby(2.2.5).TomaketheLaurentexpansion
wedefine
f(ν)=
1
(4π)
ν
2
d(ν)
3
m
ν4
(k
µ
k
v
η
µv
k
2
).(5.2.2)
Thus,weobtain
f(ν
2
ν
2
=
1
(4π)
2
d(ν)
3
k
µ
k
v
η
µv
k
2
4ν
+(k
µ
k
v
η
µv
k
2
)·
d(4)
3
[2ln(m)ln(4π)C]+
d
(4)
3
+
k=1
b
k
(ν4)
k
.(5.2.3)
Usingthisresult,weobtainforthevacuumpolarization
Π
µv
(k,ν)=
e
2
(4π)
2
d(ν)
3
k
µ
k
v
η
µv
k
2
4ν
e
2
(4π)
2
(k
µ
k
v
η
µv
k
2
)·
d(4)
3
[2ln(m)ln(4π)C]+
d
(4)
3
+
k=1
b
k
(ν4)
k
µv F
(k,ν).
(5.2.4)
Thefinitepartisthen
Π
µvf
(k,ν)=
e
2
(4π)
2
(k
µ
k
v
η
µv
k
2
)·
d(4)
3
(2ln(m)ln(4π)C]+
d
(4)
3
}
µv F
(k,4)+
k=1
a
k
(ν4)
k
.
(5.2.5)
Consequently,wehavefortheconvolution(infourdimensions)the
result
Π
µvP
(k)=Π
µvf
(k,4)=
e
2
(4π)
2
(k
µ
k
v
η
µv
k
2
)·
d(4)
3
[2ln(m)ln(4π)C]+
d
(4)
3

µvF
(k,4).(5.2.6)
6.Discussion
WeusedthenewQFT-paradigmbasedonSTEDELI-distributions
thatovercomestheoldproblemofQFT-infinities.Withthisnew
paradigm,wehaveheresolved,forinstance,thenonrenormaliz-
ableproblemposedbyλ
ϕ
4
4!
theoryinsixdimensions.TheoldQFT-
paradigmcandonothinghere.Instead,wesolvedit.
UsingSTEDELI,wewerethusableheretosuccessfullyhandlein
novelfashionseveraloldQFTproblems.
DOI:10.4236/jhepgc.2020.64040606JournalofHighEnergyPhysics,GravitationandCosmology
A.Plastino,M.C.Rocca
Itisoftheessencetorealizethat,inQFT,whenoneusespertur-
bativeexpansions,wearedealingwithproductsofdistributionsin
configurationspaceor,analogously,withconvolutionsofdistributions
inmomentumspace.
Usingsuchconvolution,wehaveexactlyevaluated,forthefirsttime,
theelectronselfenergyandthevacuumpolarizationinQED.
ConflictsofInterest
Theauthorsdeclarenoconflictsofinterestregardingthepublication
ofthispaper.
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