How the Recent BABAR Data for P ! a Ect the Standard Model Predictions

Pi sma v ZhETF vol i s s pp February

How the recent BABAR data for P ! aect the Standard Mo del

pre dictions for the rare decays P ! l l

A E Dorokhov

Joint Institute for Nucle ar Re s e arch Bogoliub ov Lab oratory of Theoretical Phys ics

Dubna Mo scow region Rus s ia

Institute for Theoretical Problems of Microphys ics Mo scow State Univers ity

Mo scow Rus s ia

Submitte d January

Me asur ing the lepton anomalous magnetic moments g and the rare decays of light p s eudo scalar

me sons into lepton pairs P ! l l s erve as imp ortant te sts of the Standard Mo del To re duce the theoretical

uncertainty in the standard mo del pre dictions the data on the charge and trans ition form f actors of the light

p s eudo scalar me sons play a s ignicant role Recently new data on the b ehavior of the trans ition form f actors

P ! at large momentum transfer were supplie d by the BABAR collab oration There are s everal problems

with the theoretical interpretation of the s e data An unexp ecte dly slow decre as e of the pion trans ition form

f actor at high momenta the qualitative dierence in the b ehavior of the pion form f actor and the and

form f actors at high momenta the incons i stency of the me asure d ratio of the and form f actors with the

pre dicte d one We comment on the inuence of the new BABAR data on the rare decay branchings

i s s imilarly large deviation b etween the recent preci s e Mo der n co smology tells us that of the matter

exp er imental re sult on the rare decay into e e pair in the univers e i s not de scr ib e d in terms of the Stan

and the SM pre diction dard Mo del SM matter New excitement app e are d

At the e arly stage of the study of the lepton anom after the satellite exp er iments Fermi LAT PAMELA

ATIC HESS and WMAP that provide d data which give

alous magnetic moments a g enter ing the

l l

an indication interprete d as Dark Matter andor Pulsar

structure of the vector vertex

s ignals Thus the s e arch for hints of Phys ics b eyond

e a q

SM i s the fundamental problem of mo der n elementary

particle phys ics There are two strategie s to solve thi s

problem

playe d a fundamental role in the foundations of quantum

Firstly in high energy exp er iments an enormous

mechanics and in particular of quantum eld theory

amount of energy i s applie d in order to excite the he avy

At pre s ent the accent of the study i s shifte d to the te st of

degree s of f ree dom exp ecte d to b e as so ciate d with the

the SM and the s e arch of the phys ics b eyond it Within

new phys ics At the moment there i s not any evidence

the SM the dominant contr ibution to the lepton AMM

on a deviation b etween the SM pre dictions and the high

i s due to the QED supplemente d by small but vi s ible

energy data and we are urgently waiting for the re sults

corrections f rom the strong and we ak interactions

of the phys ical program at LHC

The electron AMM i s me asure d with one of the b e st

Another strategy i s to carry out low energy exp er

accuracie s obtaine d for phys ical ob s ervable s

iments where the lack of energy i s comp ensate d by a

exp

ppt a

huge stati stics pro ducing rare pro ce s s e s relate d to the

new interactions The low energy exp er iments are not

In the SM it i s given by

only supplement to the high energy program allowing

to get combine d re str ictions on the parameters of hy

we ak hadron QED SM

a a a a

e e e e

p othetical interactions but they are also unique instru

ment for the di scovery of the phys ics b eyond the SM

QED

C a

containing low mas s particle s At the moment there are

some problems with the matching of the exp er imental

where the rst three co ecients are known analytically data with the pre dictions of the SM The mo st f amous

and the two others in some approximations If the ne one i s the de screpancy by three standard deviations b e

structure constant would b e known f rom other inde tween exp er iment and SM theory for the muon

p endent source s the me asurement of the electron AMM anomalous magnetic moment AMM Another example

A E Dorokhov

would b e a str ingent te st of QED However the theoret which i s b elow the KTeV re sult

ical error i s dominate d by the uncertainty in the input

The main limitation on re ali stic pre dictions for the s e

value for the QED coupling constant and the prob

pro ce s s e s or iginate s f rom the large di stance contr ibu

lem i s revers e d to that of obtaining a b e st e stimate of

tions of the strong s ector of the SM where p erturbative

the QED coupling constant

QCD do e s not work In order to dimini sh the theoreti

cal uncertaintie s the us e of the exp er imental data on the

ppb

pion charge and trans ition form f actors are of crucial im

p ortance The rst one me asure d in e e

The gre at fe ature of the study of the leptonic AMM

by CMD SND KLOE and BABAR

i s that they are very s ens itive to the manife station of

provide s an e stimate for the hadron vacuum p olar iza

new phys ics Any interaction with character i stic scale

tion contr ibution to muon g with accuracy b etter

contr ibute s to the leptonic AMM like m An al

than The s econd one me asure d in e e e e P

ter native mechani sm for a contr ibution by new phys ics

for spacelike photons by CELLO CLEO and

was prop o s e d in It o ccurs at the tree level and

BABAR collab orations and in e e P for time

exhibits a line ar rather than quadratic dep endence on

like photons by the BABAR collab oration i s e s s en

m Therefore the he avier the lepton the more vi s ible

tial to re duce the theoretical uncertaintie s in the e sti

the interaction In thi s way the contr ibution of an in

mate s of the contr ibutions of the hadronic lightbylight

teraction to the muon AMM i s bigger than that to the

pro ce s s to the muon g and in the e stimate s of the de

electron AMM by a f actor m m Even big

cay widths of P l l The BABAR data on

ger would b e the eect for the lepton AMM However

the large momentum b ehavior of the form f actors caus e

the lepton i s highly unstable and the me asurement of

the following problems for the ir theoretical interpreta

its AMM i s problematic s ee for di scus s ions At

tion an unexp ecte dly slow decre as e of the pion tran

moment there are only very rough exp er imental limita

s ition form f actor at high momenta the qual

tions on the lepton AMM s et by the L OPAL

itative dierence in the b ehavior of the pion and

and DELPHI collab orations

form f actors at high momenta incons i stency of

a L

the me asure d ratio of the form f actors with the pre

dicte d one

a OPAL

In Figs the data for the and trans ition

a DELPHI

form f actors f rom the CELLO CLEO and BABAR col

f rom Z and e e e e pro ce s s e s while

he SM pre diction i s t 0.35

CELLO

SM CLEO

a 0.30

BABAR

e theoretical studie s of the muon AMM g s ee

Th 0.25

review the rare decays of light p s eudo scalar

for 0.20 2fp

me sons into lepton pairs and the compar i son

( )(GeV)

h the exp er imental re sults oer an imp ortant low wit 0.15

pgg

energy te sts of the SM The di screpancy b etween the

2 0.10

pre s ent SM pre diction of the muon AMM and its exp er

QFQ

ental determination i s

im 0.05

The s ituation with the rare decays of the light

me sons into lepton pairs b ecame more in

p s eudo scalar 0.1 1 10 100

g after the recent KTeV EI I exp er iment at tr iguin 2 2

Q (GeV )

FermiLab in which the pion decay into an electron

p o s itron pair was me asure d with high accuracy us ing

Fig The trans ition form f actor ! The data are

the K pro ce s s as a source of tagge d neutral pi

f rom the CELLO CLEO and BABAR Col

lab orations The dotte d line i s mas sle s s QCD asymptotic

ons R P l l P l l

tot

limit The notation for curve s i s explaine d in the text

KTeV

e e R

lab orations are pre s ente d In Figs and the BABAR

The standard mo del pre diction give s

p oint me asure d in the timelike region i s drawn at

Theor

e e R

Q GeV as suming that the spacelike and time

How the recent BABAR data for P aect

0.3

t zero momentum transfer the trans ition form f ac

CELLO A

or i s xe d by the twophoton decay width CLEO t

BABAR

0.2

2fh P

where i s the QED coupling constant M i s the re s P

( )(GeV)

onance mas s and P i s the twophoton partial

hgg 0.1

width of the me son P The axial anomaly pre dicts

QFQ

f 0 P

0.1 1 10 100

where f i s the me son decay constant Under as sump

2 2 P

ion of f actor ization p erturbative QCD pre dicts the as

Q (GeV ) t

ymptotic b ehavior of the F Q trans ition form

Fig The trans ition form f actor ! The data are

f actors as Q

f rom the CELLO and CLEO Collab orations The

CLEO re sults obtaine d in dierent decay mo de s are aver

Q F

age d The BABAR p oint me asure d in the timelike region

2 2

! i s drawn at Q GeV as suming that

the spacelike and timelike asymptotics of the form f actor

The p erturbative QCD corrections to thi s expre s s ion at

are equal The notation for curve s i s explaine d in the

large momentum transfer are extremely small

text

To de scr ib e the soft nonp erturbative region of Q

a s imple interp olation b etween Q and Q

s has b een prop o s e d by Bro dsky and Lepage BL 0.3 limit

CELLO CLEO

BABAR

F Q

0.2 P

P P 2 2fh’

( )(GeV)

’

where the value s of f are e stimate d f rom and P

h gg 0.1

MeV f MeV f MeV f

QFQ

The CLEO and CELLO collab oration parameter

ize d the ir data by a formula s imilar to but with the

le mas s b e ing a f ree tting parameter 0 p o

0.1 1 10 100

2 2

CLEO

F Q

Q (GeV )

f Q

Fig The trans ition form f actor ! The data

where MeV MeV and

are f rom the CELLO and CLEO Collab orations

MeV

The CLEO re sults obtaine d in dierent decay mo de s

In Figs the asymptotics are shown by dotte d

are average d The BABAR p oint me asure d in the time

0 2 2

line s the BL interp olations are given by dotdot

like region ! i s drawn at Q GeV

as suming that the spacelike and timelike asymptotics of

dashe d line s and the CLEO parametr izations extrap o

the form f actor are equal The dashe d line i s the p ertur

late d to higher momentum transfer are shown by

bative QCD asymptotic limit The notation for curve s i s

dotdashe d line s We s ee that the QCD inspire d expre s

explaine d in the text

s ion works well only for the me son form f actor

with f MeV Fig where as the

like asymptotics of the form f actor are equal It i s s een CLEO parametr ization undere stimate s the large Q

f rom the Figs and that the spacelike and timelike b ehavior On the other hand the CLEO parametr ization

p oints are conjugate d We hop e that new data f rom de scr ib e s the me son form f actor Fig well but the

the BABAR and BELLE collab orations conrm thi s as BL expre s s ion strongly undere stimate s the large Q b e

sumption havior We still have a go o d de scr iption of the me son

A E Dorokhov

meter in or m in to the fundamental QCD MeV form f actor by BL formula if one take s f

parameters i s uncle ar Secondly the or igin of the at short dashe d line in Fig but then the normalization

di str ibution amplitude i s not well justie d Mo st of the i s incorrect

QCD sum rule and instanton mo del calculations le ad For the and me sons the parametr izations

to endp oint suppre s s e d amplitude s s ee eg and correctly reect the exp er imental data at large

Thi s i s a s imple cons equence of the nonlo cal structure Q on the qualitative level Thi s i s not the cas e for the

of the QCD vacuum The piontoquarks vertex has its pion form f actor showing the growth at large Q which

own hadronic form f actor with the character i stic scale of i s unexp ecte d f rom the QCD f actor ization approach

the vacuum nonlo cality the instanton s ize It le ads to Fig However thi s growth i s e asy to explain

suppre s s ion if re ali stic value s of parameters are us e d in the context of the quark mo del Within thi s

Only under the as sumption that vacuum nonlo calitie s mo del the pion form f actor i s given by the quarklo op

di sapp e ar a at pion di str ibution amplitude i s obtaine d tr iangle diagram with momentum indep endent quark

Note that the atne s s i s a natural prop erty of the pho mas s s erving as an inf rare d regulator The form

ton di str ibution amplitude b ecaus e the photon has no f actor has double logar ithmic asymptotics at large Q

intr ins ic hadronic form f actor ln Q M and i s given by

The p o s s ible or igin of the dierence of the asymp

totic b ehavior of the pion and me son form f actors

Q F

m Q

arcs in

i s the avor comp o s ition of the s e me sons The pion

cons i sts of almo st mas sle s s u d quarks while the

f arcs in ln g

me sons include also an s quark The s quark with mas s

Q Q

m of order may b e cons idere d as a he avy one

s QC D

Recently a s imilar b ehavior to that pre dicte d by

The solid line in Fig i s where

2 Q

was found for the trans ition form f actor

the pion trans ition form f actor calculate d f rom Eq

me asure d by the BABAR collab oration for the range

with the parameter M MeV The advantage of

Q GeV The corre sp onding tte d mas s

thi s mo del i s that it dene s the form f actor for arbi

GeV parameter i s

trary virtuality of photons and has correct normaliza

Let us cons ider the p o s s ible inuence of the BABAR

tion at zero virtualitie s One of di sadvantage s i s that

data on the rare leptonic decays of the light p s eudo scalar

the corre sp onding integral for the decay constant f i s

me sons Remind that the imaginary part of the ampli

divergent and should b e regular ize d Another fe ature of

tude of the s e decays i s xe d unambiguously by the two

the mo del i s that the vertice s and propagators are lo cal

photon decay widths of the s e me sons Neglecting the

In particular the piontoquarks vertex i s lo cal just like

re al part of the amplitude the so calle d unitary b ound

in the NambuJonaLas inio mo del It i s known that in

i s obtaine d In particular one has

thi s cas e the pion di str ibution amplitude and di str ibu

tion function are constants

R e e

The at almo st constant pion di str ibution ampli

unitary

R e e

tude b ecame p opular in the context of the explanation of

the BABAR data within dierent f actor ization scheme s

which i s b elow the branching

For example in the mo del the pion tran

KTeV

R e e obtaine d by

s ition form f actor i s

the KTeV collab oration

Q F

The structure of the re al part of the amplitude was

cons idere d in detail in It cons i sts of the log

xQ dx

ar ithmic mo del indep endent terms lnm M and

exp

l P

x x

lnm a constant term relate d to the invers e mo

ment of the pion trans ition form f actor in symmetr ic

and has logar ithmically enhance d asymptotic b ehavior

kinematics

log Q Q In the kinematical range of Fig

the mo del practically coincide s with the mo del

A q ln

if the parameter GeV Note that the s e log

ar ithmically enhance d mo dels are not able to de scr ib e

the form f actors

In general the unitary b ound relate d to the twophoton in

However the s e approache s still contain some unan

terme diate state i s violate d in the cas e of me son where new

thre sholds app e ar swere d que stions First of all the relation of the para

How the recent BABAR data for P aect

+ +

Value s of the branchings R P ! l l P ! l l obtaine d in our approach and compare d with the

tot

available exp er imental re sults

R Unitary CLEOBABAR CLEOBABAR With mas s Exp er iment

b ound b ound OPE corrections

0 + 8

R ! e e

+ 6

R !

+ 9 4

R ! e e

0 + 7

R !

10 0 +

R ! e e

data corre sp onds to h and switching on with

the small co ecient h re sp ons ible for the as

t t F

ymptotics ob s erve d by BABAR dashe d line in Fig

Compar ing the constant calculate d with one

or another parameter ization of the data with the

t t F

integral in taken in the region f rom to GeV

we nd only small di screpancy and thus no change s for

n n

M and of small mas s corrections m m

pre dictions of the e e decay Table A s imilar

e e

M where M i s character i stic parameter of

conclus ion was found for the inuence of the BABAR

the form f actor As was explaine d in the data on the

data on the hadronic lightbylight scatter ing con

Q provide improve d trans ition form f actors F

tr ibution to the muon anomalous moment

lower b ounds for electronic decay mo de s b ecaus e in thi s

The new scale for the me son form f actor ob

cas e the logar ithmic terms in the amplitude dominate

taine d f rom the inclus ion of the BABAR data

over the constant term s ee Table On the contrary

slightly change s the numb ers in the s econd and third

for the muonic decay mo de s of the and me sons the

line s of the Table The fth column of Table contains

logar ithmic and constant terms are comparable and by

the pre dictions when the mas s corrections to the am

Q one gets upp er b ounds us ing the data on F

plitude are taken into account The s e corrections

for the branchings Table The analys i s of the CELLO

have some inuence on the pre dictions for the and

and CLEO data on the pion trans ition form f actor le ads

me sons decays Thus it i s cle ar that more preci s e data

to an improve d b ound

at low energy would le ad to stronger re str ictions on the

leptonic decays widths of the light p s eudo scalar me sons

CLEO

R e e R e e

Further indep endent exp er iments for e e at

WASAatCOSY and for l l KLOE

and BES III and other f acilitie s will b e crucial for

which i s b elow the KTeV re sult

re solution of the problem with the rare leptonic decays

Let us check how s ens itive the s e improve d b ounds

of light p s eudo scalr me sons It i s also imp ortant to con

are to the recent BABAR data at large momentum trans

rm the theoretical bas i s for a maximally mo del indep en

fer As s een f rom Figs there are two main change s

dent pre diction of the branchings s ee Table by getting

after the app e arance of the BABAR data compare d with

more preci s e data on the pion trans ition form f actor in

the CLEO parametr ization Firstly the tail of the pion

asymmetr ic as well as in symmetr ic kinematics in wider

form f actor has unexp ecte d asymptotics and s econdly

region of momentum transfer Such data are exp ecte d

MeV i s clo s er to the corre sp onding parame

so on f rom the BABAR BELLE at large momentum

ter In order to e stimate the inuence on the pion decay

transfer and KEDR at small momentum transfer col

we cho o s e the parametr ization sugge ste d in

lab orations

There are quite few attempts in the literature to

h Q h Q h f

Q F

explain the exce s s of the exp er imental data on the

M M Q M Q M

0 0

V V V V

where M MeV M MeV and the para

V V

However the parametr ization can not b e cons idere d as

meter h i s xe d by the anomaly The b e st

phys ical one and us e d for an extrap olation to higher Q b ecaus e it

t to the CELLO and CLEO data i s given by h

Q contradicts the TerazawaWe st inequality F

and h The b e st t including the BABAR Q following f rom unitar ity

A E Dorokhov

J P Miller E de Raf ael and B L Rob erts Rept Prog e e decay over the SM pre diction as a manife sta

Phys 70 arXivhepph

tion of phys ics b eyond the SM In Ref it was shown

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that thi s exce s s could b e explaine d within the currently

arXivhepph

p opular mo del of light dark matter involving a low mas s

A E Dorokhov Acta Phys Polon B 36

MeV vector b o sons U which pre sumably cou

arXivhepph

ple to the axialvector currents of quarks and leptons

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Another p o s s ibility was prop o s e d in Ref inter

arXiv hepph

preting the same exp er imental eect as the contr ibution

J Prade s arXiv hepph

of the light CPo dd Higgs b o son app e ar ing in the next

R R Akhmetshin et al CMD Collab oration Phys

tominimal sup ersymmetr ic SM

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The author thanks SB Geras imov VP Druzhinin

M N Achasov et al SND Collab oration J Exp

NI Ko chelev EA Kuraev SV Mikhailov AA Pivo

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varov AV Radyushkin for di scus s ions on the inter

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pretation of the high momentum transfer data for the

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p s eudo scalar me son trans ition form f actors

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