!Re+$e%$$and$the$infrared$value$of$the$QCD$coupling$constant$

$

!!!!A.$A.$Natale$IFT/Unesp$%$UFABC$

$$$$$$$$$$$$$$$$$$$$$$$$$$$! $$$$$$$$$

$$$$$$$$1)$Phenomenology$and$$the$IR$QCD$coupling$!$$$$$$$$2)$SDE$and$an$infrared$finite$QCD$coupling$!$$$$$$$$3)$Re+$e%$$$$$$$$$$4)$“Smearing“$$$Re+$e%$$$$$$$$$$$$5)$Conclusion$$!

John!Gomez!&!AN!

1)$Phenomenology$and$$the$IR$QCD$coupling!

Strong$interacOon$at$low$energies$$ Models$

Evidences$for$a$finite$coupling:$

1)  Jet$shapes$observables$→$α(0)$≈$0.63$$!!!!!!!!!!!!!![Dokshitzer,!Weber,...!PLB!352!(1995)451,!Nucl.Phys.B!469(1996)93]!!2)  Quarkonium$potenOal$models$→α(0)$≈$0.6$!!!!!!!!!!!![Godfrey,!Isgur,!PRD!32(1985)189;!Eichten!et!al.!PRL!34(1975)369;!PRD!21(1980)203;!...]!!3)  Optmized$perturbaOon$theory$(Re+$e%)$→α(0)$≈$0.8$!!!!!!!!!!!![MaUngly!and!Stevenson,!PRL!69(1992)1320,!PRD!49(1994)437]!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

4)  Quarkonium$fine$structure$→$α(0)$≈$0.4$$$$$$$$$$$$$$[$Badalian,$Simonov,$Baker,...PRD$60(1990)116008,$PRD$62(2000)094031]$$5)  QCD%inspired$models$(total$hadronic$cross$secOons)$→$α(0)$≈$0.5$!!!!!!!!!!![Block,!Halzen,!Margolis,!Pancheri,...!PRD!45(1992)!839;!PRD!60(1999)054024]!!6)  Unpolarized$proton$structure$funcOon$→$α(0)$≈$0.4$–$0.56$!!!!!!!!!!![Courtoy,!LiuZ,!PLB!726(2013)320]!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!●!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!●!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!●!!

HERA$data$→$“...$fronOer$between$perturbaOve$and$non%perturbaOve$physics$$$$$$$$$$$$$may$occur$at$relaOvely$small$momenta....”$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![E.!Levin!and!I.!Potashnikova,!JHEP!1402(2014)089]!

TheoreZcal!arguments:!

FuncOonal$Schrodinger$equaOon$$$$$$$$$α(0)$≈$0.5$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![J.!Cornwall,!PRD!76(2007)025012]!!Gribov$papers$on$confinement$$$$$$$α(0)$≈$0.44$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$[see!Eur.Phys.J.!C!10(1999)91]!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!●!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!●!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!●!!

QCD$%%$If$any$dynamical$mass$scale$is$generated$for$the$gluons$$$$$$$$$$$$$$$$$$$$$$$$$$$$$(or...$scale$in$the$gluon$propagator)$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$→$the$coupling$freezes$in$the$IR$$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![Aguilar,!Silva,!AN,!PRL!90(2003)152001]!

Laice$QCD$and$Schwinger%Dyson$equaOons$$$$$consistent$with$IR$finite$gluon$propagator$

→$non%perturbaOve$IR$fixed$point$

2)$SDE$and$an$infrared$finite$QCD$coupling!

SDE$+$Pinch$Technique$→$gauge$invariant$soluOons$for$propagators$and$verOces$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!![Cornwall,!Aguilar,!Binosi,!Papavassiliou,!et!al....!......!......!......!]!

an$effecOve$infrared$finite$QCD$charge:$

12

222

02 )]

)(4ln(4[)( −

Λ

+=

QCD

gs

QmQQ πβα

Cornwall,!PRD!26(1982)1453!

m$g$$=$500$MeV$→$ρ1$=$4.5$$,$$ρ2$=$%2$$$m$g$$=$600$MeV$→$ρ1$=$2.2$$,$$ρ2$=$%1.25$$$$$

1

2

2222

02 ))(,(ln4)(

−

""#

$

%%&

'

(()

*++,

-

Λ

+=

QCD

qmqhqq πβα

evoluOon$of$the$SDE$calculaOon....$

Aguilar,!Binosi,!Papavassiliou,!Quintero,!PRD!80(2009)085018!

)()()())(,( 222

24

222

1222

qmqqmqmqmqh

++= ρρ

simple$approximaOon$for$αs$$

12

222

02 )]

)(ln(4[)( −

Λ

+=

QCD

gs

QmQQ

ρπβα

$$$$$ρ$→$3$–$4$phenomenology!Courtoy,!LiuZ,...!

22

422 )(

g

gg mQ

mQm

+≈

Aguilar,!AN,!JHEP!0408!(2004)!057!

NOTE:$

This$is$not$your$favorite$αs$?$!Worried$about$non%perturbaOve$$αs$definiOon?$$

→!this!expression!maps!!!!!!any!IR!frozen!coupling...!!$$$$$$(is$the$possibility$αs$(0)$→$0$$alive?)$

3)$$Re+$e%$$

∑=→

→≡

−+−+

−+

−+

fn

iieeq

eehadronsee

sR 23)()()(

µµσσ

Parton!model!

QCD$)1(3)( 2 R(s)+= ∑−+

fn

iiee qsR

...)1()( 21 +++= ararasR

fn →!!!number!of!flavours!

21 1153.0986.1 fnr −=

22 00518.0200.1637.6 ff nnr −−−=

MSπα sa =

Re+$e%$$$$$ddd!!!Parton!model!!

Experimental$data$(PDG)$

$$$$$$$$$$$$$$$$$$[$OPT$→Maingly$and$Stevenson,$PRL$69(1992)1320,$PRD$49(1994)437,$recent$Stevenson,$NPB$875(2013)63]$

4)$$“Smearing“$$$Re+$e%$$$

Experiment$X$Theory$→$Smearing$

Poggio,$Quinn$and$Weinberg,$PRD$13(1976)1958$

( )∫∞

Δ+−%

%%

Δ=Δ

−+

0222

2 )();(

Qs

sRsdQR ee

pqwπ

smooth$curves$

Data$fit$

ρ,!φ,!J/ψ,!ψ(3686),!ψ(3770)!included!laier!(BW!→!δ!funcZons)!

PDG!data!

Red!curve!→!( )

24

1222

DQCAMQ

BAi ii

ii +++−

∑=

( )∫∞

Δ+−%

%%

Δ=Δ

−+

0222

2 )();(

Qs

sRsdQR ee

pqwπ

Exp.$$$→$$fit$$+$$narrow$resonances$

Th.→$QCD$$with$effecOve$charge$up$α3$

Exp.!data!–!Δ!=!1.5!GeV2!

!shaded!area!d!±!7%!TheoreZcal!curves!–!different!mg/Λ!!(0.7!magenta!–!excluded)!!

7%!d!ad!hoc!!!!!//!!!different!Δ!!!

Best$fit$→χ2$$test$$

( )∑Δ

−=

i i

thii

OOO

2exp

2exp2 ))(()( θθχ

12

222

02 )]

)(ln(4[)( −

Λ

+=

QCD

gs

QmQQ

ρπβα

22

422 )(

g

gg mQ

mQm

+≈

ρ!=!3!–!4!!!;!!!!Δ!=!!1.5!–!3!GeV2!!!;!!!mg/Λ!=!!0.5!–!2.5!

Considering:$

RPQW!!!!!up!to!Q2!=!2!GeV2!

χ2$$$$exp$X$th$$$$$(different$curves,$∆,$...)$$

ρ$$=$$4$$,$different$Δ$

Minimum$→$mg$/Λ$$≈$1.2$$$

Δ!=!1.5!GeV2!

ρ!!→!!3!!d!!!4!

Minimum→$$mg$/Λ$≈$1.4!! ρ!=!3!

!!Δ!=!2!GeV2! ρ!!→!!3!!d!!!4!

Minimum→$$mg$/Λ$≈$1.4$$ ρ!=!3!

If$$mg$/Λ$<$0.8$$$→$bad$χ2$$

Above$$mg$/Λ$=$1.6$,$slow$increase$of$$χ2$value$

Minimum$$$$$$$$$$$3$$<$ρ$<$$4$$$$$$$$→$$$$$$$$$$$$$1.2$$<$mg$/Λ$$<$$1.4$

12

222

02 )]

)(ln(4[)( −

Λ

+=

QCD

gs

QmQQ

ρπβα

20 48/)233( πβ fn−=

approximately$constant$with$the$product$$dd!!ρ$mg$/Λ$$

Global$duality$ [Bertlmann,!Launer,!de!Rafael,!NPB!250(1985)61]!

pQCD

ss

tdttdt )(Im1)(Im100

0exp

0

Π=Π ∫∫ ππ

$$matching$between$long$and$short$distances$$$$(s0=$1.5$GeV2)$

Imply:!

∫∫ =00

00

exp

s

th

s

RdsRds

Surface$of$$s0$$$$$and$ρ,$$mg$/Λ$

What$to$expect?$$Smaller$s0$

Other$consequences...!

(perturbaZve!α)!

Define!! ( ) ∫∫ −=ΔΛ00

00

exp02 ,/,,

s

th

s

g RdsRdsmsQε

Δ=1.5!;!mg/Λ=1.2!ddddddddddddddddddddddd!!!!!s0!!!!!!!!!!!!!ε(0)!!!!0.6!!!!!!!!d0.0987!!!!0.8!!!!!!!!d0.0276!!!!0.87!!!!!!d0.0060!!!!1.0!!!!!!!!!d0.0291!!!!1.2!!!!!!!!!d0.0812!!!!1.4!!!!!!!!!d0.1264!!!!!!!!

5)$Conclusion$

Re+$e%$$$$→$informaOon$about$the$IR$effecOve$QCD$charge!

Assuming$a$simplified$expression$for$the$effecOve$charge:$

12

222

02 )]

)(ln(4[)( −

Λ

+=

QCD

gs

QmQQ

ρπβα

we$obtain:$

α(0)$$$$$basically$dependent$on$the$product$$$$ρ$mg$/Λ$$

α$(0)$≈$0.7$

In!agreement!with!MaUngly!and!Stevenson!Improved!(or!different)!expressions!for!αs!...!?!How!it!will!change!at!next!order?!.......!

Smaller$scale$s0$$→$$fronOer$between$pert.$and$non%pert.$physics$!