rho^0-gamma mixing in the neutral channel pion form factor |F_pi|^2 and its role in comparing e^+ e^- with tau spectral functions

0.43 MB·English

by  Fred Jegerlehner

DESY 11-008
HU-EP-11/04
1
1
0
2 ρ0 − γ mixing in the neutral channel pion form factor F(e)(s) and
π
n
a its role in comparing e+e− with τ spectral functions
J
9 Fred Jegerlehnera,b,∗, Robert Szafronc
2
aHumboldt-Universita¨t zu Berlin, Institut fu¨rPhysik, Newtonstrasse 15, D-12489 Berlin, Germany

bDeutsches Elektronen-Synchrotron (DESY), Platanenallee 6, D-15738 Zeuthen, Germany
]
h cInstitute of Physics, University of Silesia, ul. Uniwersytecka4, PL-40007 Katowice, Poland
p
-
p
e
h Abstract
[
We study the effect of ρ0−γ mixing in e+e− →π+π− and its relevance for the comparison of the square modulus

2 of the pion from-factor |F(e)(s)|2, as measured in e+e− annihilation experiments, and |F(τ)(s)|2 thecorresponding
π π
v
quantityobtainedafteraccountingforknownisospinbreakingeffectsbyanisospinrotationfromtheτ-decayspectra.
2
After correcting the τ data for the missing ρ−γ mixing contribution, besides the other known isospin symmetry
7
violating corrections, theππ I=1 part of thehadronic vacuumpolarization contribution tothemuon g−2 are fully
8
compatible between τ based and e+e− based evaluations. τ data thus confirm result obtained with e+e− data. Our
2
. evaluationoftheleadingordervacuumpolarizationcontribution,basedonalle+e−dataincludingmorerecentBaBar
1 andKLOEdata,yieldsahad,LO[e]=690.75(4.72)×10−10 (e+e−based),whileincludingτ datawefindahad,LO[e,τ]=
0 690.96(4.65)×10−10 (e+eµ−+τ based).Thisbacksthe∼3σdeviationbetweenaexperimentandatheory.Forµtheτ di-pion
11 branching fraction we find BπCπV0C = 25.20±0.0.17±0.28 from e+e−+CVC, wµhile Bππ0 = 2µ5.34±0.0.06±0.08 is
evaluated directly from theτ spectra.
:
v
i
X
Key words: γ ρmixing,ρ-mesonproperties,e+e−-annihilation,τ-decaypionformfactor,muonanomalousmagnetic
r moment. −
a
PACS: 13.66.Bc,13.35.Dx 14.60.Ef
1. Introduction
Isovectordata forthe pionformfactorobtainedfromhadronicτ-decayspectracanbe comparedwiththe
mixedisovector-isoscalardatameasuredinthe e+e− channelbymeansoftheoryinput [1].Inparticular,we
needsomemodelinordertobeabletodisentangleρ−ωmixingaswellasotherisospinbreaking(IB)effects.
The general problem in confronting measured quantities like |F(τ)(s)[I =1]|2 and |F(e)(s)|2 =|F(e)(s)[I =
π π π
1]+F(e)(s)[I = 0]|2 is the fact that the latter object is subject to quantum interference between the two
π
∗ Correspondingauthor.
Email addresses: [email protected] (FredJegerlehner), [email protected] (RobertSzafron).
URL: www-com.physik.hu-berlin.de/~fjeger/ (FredJegerlehner).
1February2011
amplitudes and in generalmay not be wellapproximatedby |F(e)(s)[I =1]|2+|F(e)(s)[I =0]|2. Without a
π π
specific model for the complex amplitudes one cannot get the precise relationship.
Commonly, pion form factors measured in the neutral channel in e+e− → π+π− and in the charged
channelinτ− →ν π−π0 decay(oritschargeconjugate)areparametrizedbyanextendedGounaris-Sakurai
τ
(GS) formula
BWGS (s)· 1+δ s BW (s) +βBWGS (s)+γBWGS (s)
ρ(770) M2 ω ρ(1450) ρ(1700)
F (s)= ω , (1)
π (cid:16) 1+(cid:17)β+γ
whichresultsasasumofmixingisovectorstates,eachdescribedbyaBreit-Wigner(BW)typeofamplitude.
The pion form factor is related to the corresponding cross section by1
πα2 β3 4πα2
σ(e+e− →π+π−)= π |F(e)(s)|2 = v (s), (2)
3 s π s 0
for point-like pions F (s) ≡ 1, where β is the pion velocity in the c.m. frame: β = 1−4m2/s. The
π π π π
spectral function v (s) is related to the form factor by
i
p
β (s)
v (s)= i |F(i)(s)|2 ; (i=0,−)↔(e,τ) , (3)
i 12π π
for the neutral (0) e+e−- and charged (-) τ-channel. The spectral function v−(s) can be measured very
precisely in τ-decay:
1 dΓ 6|V |2S B s 2s
Γ ds(τ− →ντπ−π0)= udm2 EW B e 1− m2 1+ m2 v−(s) , (4)
τ ππ (cid:18) τ(cid:19)(cid:18) τ(cid:19)
with m = (1776.84±0.17) MeV the τ mass, |V | = 0.9418±0.00019 the CKM matrix element, B =
τ ud e
(17.818±0.032)% the electron branching fraction, B = (25.51±0.09)% the di-pion branching fraction
ππ
and S =1.0235±0.0003 the short distance electroweak correction.
EW
Note that a single standard Breit-Wigner resonance yields
36 Γ(ρ→e+e−) s sΓ2
|F (s)|2 = ρ .
π α2 β3Γ(ρ→π+π−) M2 (s−M2)2+M2Γ2
π ρ ρ ρ ρ
Denoting the γ−ρ transition coupling by eM2/g the branching fraction at resonance reads
ρ ρ
R =. Γ(ρ→e+e−) = α2 gρ 2 Mρ 2 β3 ,
ρ Γ(ρ→π+π−) 36 g Γ ρ
(cid:18) ρππ(cid:19) (cid:18) ρ (cid:19)
.
with β =β (s=M2). In the case of complete ρ dominance g =g 2.
ρ π ρ ρ ρππ
The GS formula(1)alsodescribesthe chargedisovectorchannelprovidedδ =0,since thereis no charged
versionoftheω.IntheneutralchanneltheGSformuladoesnotfullyincludeρ0−γ mixing,whichisknown
since the early 1960’s,when the ρ had been discovered.A direct consequence of ρ0−γ mixing is the vector
meson dominance (VMD) model characterized by an effective Lagrangian[4]
eM2
L =− ρ ρ Aµ . (5)
γρ µ
g
ρ
1 QED corrections to e+e− π+π− have been summarized in [2,3]. They will not be considered in the following and we
→
assumethemtobetakenintoaccountintheextraction oftheformfactorfromtheexperiments.
2 AtresonancethesingleBWpionformfactorisgivenby
|Fπ(Mρ2)|2= α362 βρ3ΓΓeeππ ,
andforPDGvaluesoftheparametersyields|Fπ(Mρ2)|2≈39areasonablevalue(seebelow).
2
However,thisformdoesnotpreserveelectromagneticgaugeinvarianceandthephotonwouldacquireamass
unless we add a photon mass counterterm to the Lagrangian which is fine tuned appropriately. The pion
form factor here takes the form
M2 g
ρ ρππ
F (s)=− (6)
π s−M2 g
ρ ρ
and the condition of electromagnetic current conservation F (0) = 1 is satisfied only if g = g , which
π ρππ ρ
is called universality condition and corresponds to complete ρ dominance. In fact electromagnetic gauge
invariance can be implemented by writing the effective VMD Lagrangianin the form [5]
e
L = ρ Fµν , (7)
γρ µν
2g
ρ
in terms of the field strength tensors. As it satisfies gauge invariance, the form factor calculated here reads
s g
ρππ
F (s)=1− (8)
π s−M2 g
ρ ρ
and satisfies the current conservation condition F (0) = 1 in any case, irrespective of the universality
π
constraint g = g (for a recent discussion also see [6]). Obviously, this simple model is not able to
ρππ ρ
describe the pion form factor measured in e+e− → π+π− at low energies, unless we take into account
energy dependent finite widths effects of the ρ as it is done in the GS model [7]3. The energy dependence
of the ρ-width has to reflect the off-shell ρ∗ → ππ process. So we have to model effectively a “rho-pion-
photon” system, discarding the ω and its mixing with the ρ, which is well understood and will be taken
into consideration in a second step. Our focus here is to work out the difference in the relation between the
chargedchannelandtheneutralchannel,whichresultsfromtheρ0−γ mixing.Thelatterlikeρ0−ωmixing,
hasnocounterpartinthechargedchannel.Thepurposeofthisstudyistounderstandbetterthediscrepancy
betweenτ ande+e− di-pionspectra,whichhasbeenclearlyestablishedin[16]undertheassumptionthatall
possibleIBcorrectionswereaccountedfor.MorerecentdataformBelle[17]andKLOE[18,19],andapplying
improved IB corrections, confirmed a significant discrepancy [20]. Although the new π+π− spectrum from
BaBar [21], measured via the radiative return mechanism, is closer to the corresponding spectra obtained
from τ-decays, a discrepancy persists.
2. A ρ−γ mixing model and related self-energy effects
As already said, the VMD ansatz has to be replaced by a more realistic model which must take into
account
– the finite ρ-width, related to its decay ρ→π+π−,
– the ρ−γ mixing, which leads to non-diagonal propagationof the ρ−γ system, and
– the ρ−ω mixing, which we will consider in a second step.
This has to be implemented in an appropriate effective field theory (EFT). In a first step we consider the
interaction of the ρ with the pions together with their electromagnetic interaction, assuming the pions to
by point-like (scalarQED). As suggestedlong agoby Sakurai[22],the ρ may be treatedas a massivegauge
boson. The effective Lagrangianthus reads
L=L +L ; L =D π+D+µπ−−m2π+π− ; D =∂ −ieA −ig ρ . (9)
γρ π π µ π µ µ µ ρππ µ
The corresponding Feynman rules in momentum space are
3 Other models have been reviewed and investigated recently with emphasis on ρ ω mixing in Ref. [8]. Frequently used
−
descriptionsofthelowenergyππformfactorincludetheChPT-basedGuerrero-Pichformulation[9],theLeutwyler-Colangelo
approach [10], the resonance Lagrangian approach [11] (see e.g. [12]) or the related Hidden Local Symmetry (HLS) model as
appliedin[13,14],andthephenomenological Ku¨hn-Santamaria(KS)model[15].Aswewillseeourmodeliscloselyrelatedto
theGSmodel,andweadoptthelatterforcomparisonsandfits.
3
Aµππ =ˆ −ie(p+p′)µ ; ρµππ =ˆ −ig (p+p′)µ
ρππ
AµAνππ =ˆ 2ie2gµν ; ρµρνππ =ˆ 2ig2 gµν
ρππ
Aµρνππ =ˆ 2ieg gµν ; Aµρν =ˆ −ie/g (p2gµν −pµpν) .
ρππ ρ
ThemodelshouldbeunderstoodasasimplifiedversionofthebetterjustifiedeffectiveresonanceLagrangian
approach[11],whichextendsthe chiralstructureoflowenergyQCD(chiralperturbationtheory)toinclude
spin 1 resonances in a consistent way. A variant is the HLS model, which in the same context has been
applied to investigate the (ρ,ω,φ) mixing effects in [13]. Actually in [13] too, V −γ (V = ρ,ω,φ) mixing
amplitudes have been included (more on that below). The main difference to the GS model is that we take
our EFT Lagrangianserious in the sense that we include all relevant contributions to e+e− →π+π−, while
in the GS model some of the contributions have been neglected. In fact the GS model is incomplete in the
sense of a quantum field theory.
In sQEDthe contributionof a pion loopto the photonvacuum polarizationis givendiagrammaticallyby
−iΠµν(π)(q)= +
γγ .
and one then obtains the bare γ − ρ transverse self-energy functions
e2 eg g2
Π = f(q2) , Π = ρππ f(q2) and Π = ρππ f(q2) , (10)
γγ 48π2 γρ 48π2 ρρ 48π2
where
2
f(q2)≡q2h(q2)= B (m ,m ;q2)(q2−4m2)−4A (m )−4m2 + q2 , (11)
0 π π π 0 π π 3
(cid:18) (cid:19)
intermsofthestandardscalarone-loopintegralsA (m)andB (m,m;s)[23].Explicitly4,intheMSscheme
0 0
(µ the MS renormalizationscale)
µ2
h(q2)≡f(q2)/q2=2/3+2(1−y)−2(1−y)2G(y)+ln , (12)
m2
π
wherey =4m2/s andG(y)= 1 (ln1+βπ −iπ),forq2 >4m2.Note thatallcomponentsofthe (γ, ρ)2×2
π 2βπ 1−βπ π
matrix propagator are proportional to the same function f(q2). The renormalization conditions are such
that the matrix is diagonal and of residue unity at the photon pole q2 =0 and at the ρ resonance s=M2,
ρ
hence the renormalized self-energies read (see e.g. [24])
Πren(q2)=Π (q2)−q2Π′ (0)=. q2Π′ren(q2) (13)
γγ γγ γγ γγ
q2
Πren(q2)=Π (q2)− ReΠ (M2) (14)
γρ γρ M2 γρ ρ
ρ
dΠ
Πren(q2)=Π (q2)−ReΠ (M2)−(q2−M2)Re ρρ(M2) (15)
ρρ ρρ ρρ ρ ρ ds ρ
where Π (0) = Π (0) = Π (0) = 0 and Π′ (q2) = Π (q2)/q2, has been used. Note, that the tree level
γγ γρ ρρ γγ γγ
mixing term in the Lagrangian contributes to the bare γρ self-energy as Π(0) = q2(e/g ), which does not
γρ ρ
4 The standard Gounaris-Sakurai parametrization differs from our sQED model and utilizes h(q2)= 2(1 y)2G(y) which
− −
forq2→0behavesash(q2)→2y=8m2π/q2i.e.q2h(q2)→8m2π,whichinDγγ representsanon-vanishingphotonmass.While
the constant terms in (12) drop out by renormalization, the +2(1 y) term is required by electromagnetic gauge invariance
−
andinfactrendersh(q2) const.regularinthestaticlimit.
→
4
affect the renormalized self energies, however. In particular, δΠren = q2 e − q2 M2 e = 0. The ρ wave
γρ gρ Mρ2 ρgρ
function renormalization reads Z =1/(1+ dΠρρ(s=M2)) with
ρ ds ρ
dΠ g2 1 1+β
ρρ(s=M2)= ρππ 8/3−β2 1+(3−β2) ln ρ . (16)
ds ρ 48π2 ρ ρ 2β 1−β
(cid:26) (cid:20) ρ (cid:18) ρ(cid:19)(cid:21)(cid:27)
Numerically, Z ≃1.1289 at µ=m .
ρ π
It is crucialto observethat vacuumpolarizationeffects affect mass renormalizationofthe ρ as wellas γ -
ρ mixing, in spite of the fact that photon vacuum polarization has to be subtracted in the definition of F .
π
In other words, in sQED we would still have F (s)=1 in (2) while vacuum polarization is absorbed into a
π
running fine structure constant
α
α→α(s)= , (17)
1+Π′ren(s)
γγ
which mean that in calculating F (s) we have to multiply the result by 1+Π′ren(s).
π γγ
A convenient representation of Πren is given by
ρρ
Γ dh
Πren(s)= ρ s h(s)−Reh(M2) −(s−M2)M2 Re (18)
ρρ πM β3 ρ ρ ρ ds
ρ ρ ( (cid:12)s=Mρ2)
(cid:0) (cid:1) (cid:12)
(cid:12)
with (cid:12)
dh
s (s)=3y−1−3y(1−y)G(y) . (19)
ds
In particular5:
M Γ dh M Γ
Πren(0)= ρ ρ M2 = ρ ρ 3y −1−3y β2G(y ) ; y =4m2/M2 . (20)
ρρ πβ3 ρ ds πβ3 ρ ρ ρ ρ ρ π ρ
ρ (cid:12)s=Mρ2 ρ
(cid:12) (cid:0) (cid:1)
(cid:12)
Without mixing, pion produc(cid:12)tion mediated by the ρ resonance, yields the GS type pion form factor, nor-
malized to F (0)=1,
π
−M2+Πren(0)
FGS(s)= ρ ρρ . (21)
π s−M2+Πren(s)
ρ ρρ
The renormalized mixing self-energy may be written in a form
eg
Πren(s)= ρππ s h(s)−Reh(M2) . (22)
γρ 48π2 ρ
(cid:8) (cid:0) (cid:1)(cid:9)
Note that while the inverse propagator matrix is diagonal at the two propagator poles, off the poles it is
not diagonal. This is the main effect we are going to discuss now6.
The propagatorsare obtained by inverting the symmetric 2×2 self energy matrix
5 IncontrasttosQED,inthestandardGSformula
dh
h(s)= 2(1 y)2G(y) ; s (s)=y 1 3y(1 y)G(y) ,
− − ds − − −
whichissingularfors 0.Thefirsttermin(18)inthiscaseyieldsafinitecontributionsh(s) 8m2 andthus
→ → π
Πrρeρn(0)= MπρβΓρ3ρ (cid:18)8Mmρ22π +Mρ2 ddhs(cid:12)s=Mρ2(cid:19)= MπρβΓρ3ρ 3yρ−1−3yρβρ2G(yρ) ,
6i.eT.,hΠerρseρen(e0ff)ec≡ts−adreΓρveMryρsiismaicltauratlolyZn0otmγ(cid:12)(cid:12)omdiifixeidn,gin[25sp]iwtehiocfh(cid:0)lahcaksinbgeemnainnivfeessttiggaautegdetinh(cid:1)veaorrieatniccael.ly as well as experimentally at
−
LEPwithhighprecision(seeRefs.[26,27]andreferencestherein).Typically,theseeffectsatlowenergyorneartheZ poleare
5
q2+Π (q2) Π (q2)
Dˆ−1 = γγ γρ (23)
 Π (q2) q2−M2+Π (q2)
γρ ρ ρρ
 
with the result:
1 1
D = ≃
γγ q2+Π (q2)− Π2γρ(q2) q2+Πγγ(q2)
γγ q2−Mρ2+Πρρ(q2)
−Π (q2) −Π (q2)
γρ γρ
D = ≃
γρ (q2+Π (q2))(q2−M2+Π (q2))−Π2 (q2) (q2+Π (q2))(q2−M2+Π (q2))
γγ ρ ρρ γρ γγ ρ ρρ
1 1
D = ≃ . (24)
ρρ q2−Mρ2+Πρρ(q2)− q2Π+2γΠργ(γq2(q)2) q2−Mρ2+Πρρ(q2)
These expressions sum correctly all the irreducible self-energy bubbles7. The approximationsindicated are
the one-loop results. The extra terms are higher order contributions and are particularly relevant near the
resonance, characterizedby the location s of the pole of the propagator,which is given by the zero of the
P
inverse propagator:
Π2 (s )
s −m2 −Π (s )− γρ0 P =0 , (25)
P ρ0 ρ0ρ0 P s −Π (s )
P γγ P
with s =M˜2 complex. The usual (no mixing) considerations in determining the physical mass and width
P ρ0
of a resonance remain true if we denote the self-energies by Π (V =ρ0,ρ±) with
V
Πρ±(p2,···)=Πρ+ρ−(p2,···)
and
Π2 (p2,···)
Π (p2,···)=Π (p2,···)+ γρ0 .
ρ0 ρ0ρ0 p2−Π (p2,···)
γγ
Thus the location of the pole may be written as
M˜2−m2+Π(M˜2,m2,···)=0, (26)
for both the ρ± and the ρ0, where
M˜2 ≡ q2 =M2−iM Γ
ρ pole ρ ρ ρ
is characteri(cid:0)zed(cid:1)by mass and width of the ρ8. Note that the imaginary part of the self-energy function is
energydependent,whichimpliesanenergydependentwidth,ofcoursewiththecorrectphase-spacebehavior
of ρ→ππ decay.
How do off-diagonalelements ofthe γ − ρ propagatoraffectthe line-shape ofthe ρ? We assumewe know
the ρ mass M and the ρ width Γ for the unmixed ρ as it is seen e.g. in the isovector τ decay spectra, i.e.
ρ ρ
we comparethe resultwith a chargedρ± assumingequalmass andwidth. We therefore compareresultfirst
with the Belle data [17]. Of course our model does not fit the data, because a more sophisticated extended
expected tobesmallbecauseofthesmallnessoftheelectromagnetic finestructureconstant, andattheZ resonance, because
of the large mass and very small width of the Z0 boson. In case of the ρ witch’s mass lies not very far above the hadronic
ππ-threshold (which is very low by the fact that pions are quasi Nambu-Goldstone bosons) and due to the relatively large
(hadronic)widthweexpectcorrespondingmixingeffects tobemuchmorerelevant.Inthechargedchannel,inprinciple,there
is W± ρ± mixing, with some effective W+ρ−+h.c. coupling term. However, this produces a negligible effect because the
−
W propagator pole is far away from the ρ propagator pole and from the energy range of interest. In fact the mixing matrix,
diagonalizedattheρ-pole,remainsessentiallydiagonalinthewholerangeofinterest(<2GeV).
7 ItisofcursewellknownthatthisDysonsummationiscrucialforaproperdescriptionoftheparticle/resonancestructurein
particularnearthepoles,wherenaiveperturbationtheoryinanycasebreaksdown.
8 Forγ -Z0 mixingintheelectroweakStandardModelexplicitresultsuptotwoloopshavebeenworkedoutin[28].
6
Fig. 1. GSfits of the Belledata andthe effects of including higher states ρ′ and ρ′′ at fixed Mρ and Γρ.Doubts are inorder
whetherthehigherresonancesreallyaffecttheρresonanceinthewaysuggestedbythecommonlyadoptedGSparametrization.
Gounaris-Sakurai model (1) has been used to extract the ρ parameters. If we switch off the contributions
from ρ′ and ρ′′ by setting γ = 0 and β = γ = 0 we observe a substantial change in F (s) as illustrated in
π
Fig.1.NotethatinanEFTonewouldexpectthe heavierstatestodecouple,while inthe GStype modeling
the low energy tail is normalizedawayby the 1+β+γ normalizationfactor.In field theory in place of this
normalization a factor s/M2 would imply automatic decoupling. But that is not the way mass and width
res
ofthe ρaredeterminedusually.Evidently,inthe GSmodel,inthe ρ-regionthe higherresonancesserveasa
continuumbackgroundwithoutwhichgoodfitsingeneralarenotpossible.Soifwestickwithoursimplified
model we cannotexpect to get a goodrepresentationof the data without correspondingextensions. On the
other hand, the simplified model allows us to work out more clearly the effect of γ - ρ mixing. Above we
have diagonalized the mixing propagator matrix at the poles, this allows to make precise the meaning of
mass and width of the heavy unstable state. This is achieved if renormalized mixing self-energy is given by
q2
Π (q2)=Π (q2)−Π (0)− ReΠ (M2)−Π (0) . (27)
γρren γρ γρ M2 γρ ρ γρ
ρ
(cid:0) (cid:1)
This can be achieved by two subsequent transformations of the bare fields:
i) Infinitesimal (perturbative) rotation
′
A 1 −∆ A
b 0
=
    ′ 
ρ ∆ 1 ρ
b 0
    
diagonalizing the mass matrix at one-loop (n+1-loop) order given that the mass matrix has been diago-
nalized at tree (n-loop) level.
ii) Upperdiagonalmatrixwavefunctionrenormalizationinducingakineticmixingterm(thiscannotbedone
by an orthogonaltransformation)
′
A Z −∆ A
γ ρ r
=
 ′    
ρ p 0 Zρ ρr
    
which allows to normalizepthe residues to one for the γ- and ρ-propagator, respectively, and to shift to
zero the mixing propagator at the ρ-pole.
Thus the relationship between the bare and the renormalized (LSZ) fields is (expanded to linear order)
A = Z A − (∆ +∆ )ρ
b γ r ρ 0 r
ρb=pZρρr+∆0 Ar , (28)
p
7
generalizing the usual multiplicative field renormalizationrepresented by the first term for both fields. The
counter-terms ∆ and ∆ are determined by the condition (27)
0 ρ
Π (0)
γρ
∆ =
0 M2
ρ
ReΠ (M2)−Π (0)
γρ ρ γρ
∆ = . (29)
ρ M2
ρ
For our model ∆ = 0 and ∆ = e/g to leading order. The field transformations of course induce mixing
0 ρ ρ
counter terms at the vertices, which are absorbed into the definition of the physical couplings. In principle,
this non-symmetric transformation only affects the bookkeeping such that the propagator pole structure
becomesobvious.Itdoesnotchangethe valueofthe functionalintegrali.e.themixing countertermscancel
intheinteriorofFeynmandiagrams,unlessthephotonand/ortherhoareinvolvedasexternalfields(states).
As a consequence of the diagonalizationthe physical ρ acquires a directcoupling to the electron:starting
as usual from the bare Lagrangian
L =ψ¯ γµ(∂ −ie A )ψ (30)
QED e µ b bµ e
we obtain
L =ψ¯ γµ(∂ −ieA +ig ρ )ψ (31)
QED e µ µ ρee µ e
with g =e(∆ +∆ ), where in our case ∆ =0.
ρee ρ 0 0
The e+e− →π+π− matrix element in sQED is given by
M=−ie2v¯γµu(p −p ) F (q2) (32)
1 2 µ π
with F (q2)=1. In our extended VMD model we have the four terms shown in Fig. 2 and thus
π
e+ π+
γ γ ρ ρ γ ρ
+ + +
e− π−
Fig.2.Diagramscontributingtotheprocesse+e− π+π−.
→
F (s)∝e2D +eg D −g eD −g g D ,
π γγ ρππ γρ ρee ργ ρee ρππ ρρ
where the first term properly normalized must be unity9. Thus
F (s)= e2D +e(g −g )D −g g D / e2D . (33)
π γγ ρππ ρee γρ ρee ρππ ρρ γγ
Notethesigno(cid:2)ftheinducedcouplingg in(31),whichlea(cid:3)ds(cid:2)tothesi(cid:3)gnsasgivenin(33).Typicalcouplings
ρee
read g =5.8935, g =6.1559, g =0.018149and x=g /g =1.15128.
ρππbare ρππren ρee ρππ ρ
RealpartsandmodulioftheindividualtermsnormalizedtothesQEDphotonexchangetermaredisplayed
in Fig. 3.
An improved theory of the pion form factor has been developed in [10]. One of the key ingredients in
this approach is the strong interaction phase shift δ1(s) of ππ (re)scattering in the final state. In Fig. 4
1
we compare the phase of F (s) in our model with the one obtained by solving the Roy equation with ππ-
π
scattering data as input. We notice that the agreement is surprisingly good up to about 1 GeV. It is not
difficult to replace our phase by the more precise exact one.
We note that the precise s-dependence of the effective ρ-width is obtained by evaluating the imaginary
part of the ρ self-energy:
9 Notethat the conserved vector current (CVC)condition Fπ(0)=1inour model isgivenandsaturated bythesQEDterm
Dγγ alone,whileintheGSmodelFπ(0)=1isimposedbyforceonthetermDρρ,theonlyonepresentintheGScase.
8
Fig.3.Therealpartsandmoduliofthethreetermsof(33),individualandaddedup.
Fig.4.Thephaseof Fπ(E)as afunctionof thec.m.energyE.Wecompare theresultof theelaborate Royequation analysis
ofRef.[10]withtheoneduetothesQEDpion-loop.ThesolutionoftheRoyequationdependsonthenormalizationatahigh
energypoint(typically1GeV).Inourcalculationwecouldadjustitbyvaryingthecouplinggρππ.
9
g2
ImΠ = ρππ β3s≡M Γ (s) , (34)
ρρ 48π π ρ ρ
which yields
g2 s g2
Γ (s)/M = ρππ β3 ; Γ /M = ρππ β3 . (35)
ρ ρ 48π π M2 ρ ρ 48π ρ
ρ
In our model, in the given approximation, the on ρ-mass-shell form factor reads
g g M
F (M2)=1−i ρee ρππ ρ , (36)
π ρ e2 Γ
ρ
and the square modulus may be written as
36 Γ
|F (M2)|2 =1+ ee , (37)
π ρ α2β3Γ
ρ ρ
with
1 g2
ρee
Γ = M or g = 12πΓ /M . (38)
ρee ρ ρee ρee ρ
3 4π
q
It is interesting to note that the GS formula (21) does not involve Γ in any direct way, since the nor-
ρee
malization is fixed by applying an overall factor 1+dΓ /M ≡ 1−Πren(0)/M2 to enforce F (0) = 1. The
ρ ρ ρρ ρ π
leptonic width is then given by
2α2β3M2
ΓGS = ρ ρ (1+dΓ /M )2 . (39)
ρee 9Γ ρ ρ
ρ
In the CMD-2 fit 1+dΓ /M ≃1.089.
ρ ρ
The result for |F (s)|2 (using mass and width as before) is displayed in Fig. 5. We compare the results
π
obtained when ρ - γ mixing is properly taken into account with the one obtained by ignoring mixing and
with a GS fit with just the ρ takeninto account.At firstlook,the resultsagreefairly wellbut do notfit the
Belle data as expected if we do not include the higher resonances.
A detailed comparison, in terms of the ratio
|F (s)|2
π
r (s)≡ , (40)
ργ |F (s)|2
π Dγρ=0
shown in Fig. 6, however reveals substantial differences and proves the relevance of the mixing. We also
plotted the same ratio for the I=1 part of the GS fit, which exhibits a similar behavior as the true F (s).
π
This is not really surprising as one fits the same data just in a different way, i.e. with different parameters
for the ρ. This mixing affects, however,the relationship to the τ channel, which does not exhibit this effect.
Of course at higher energies, not to far above the ρ, it is not known whether the simple EFT model can
be trusted. Note that dropping the Π2 terms (approximation indicated in (24)) in the Dyson resummed
γρ
propagators does not affect the result. We have checked that ω−γ, φ−γ or Υ(4S)−γ mixing effects are
tiny away from the resonances and thus should not affect the interpretation of radiative return spectra as
measured at KLOE and BaBar.
WehavetocomparethemodelwiththeI =1partofthee+e−-data.TothisendwemaytaketheCMD-2
fitofthe CMD-2data[29]andsetthe mixingparameterδ =0asillustratedinFig.7.Inthis wayweobtain
the isovector part of the square of the pion form factor |F(e)[I =1](s)|2.
π
In order to compare F(τ)(s) extracted from τ-decay spectra with F(e)(s) measured in e+e−-annihilation
π π
we have to apply isospin breaking corrections as investigated in [30] and [31] (see also [16,32,20]):
– Massshift:theCottinghamformula,whichallowsusforaratherprecisecalculationoftheelectromagnetic
pion mass shift δmπ = mπ± −mπ0 ≃ 4.6 MeV, suggest the relation ∆m2π ≃ ∆Mρ2, which yields a shift
between changed an neutral ρ by δMρ =Mρ± −Mρ0 ≃ 12 ∆Mmρ02π =0.814 MeV.
10