Constraints on the two-pion contribution to hadronic vacuum polarization

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Constraints on the two-pion contribution to hadronic vacuum polarization

Gilberto Colangelo

^a

, Martin Hoferichter

^a

, Peter Stoffer

^b,c,∗

aAlbertEinsteinCenterforFundamentalPhysics,InstituteforTheoreticalPhysics,UniversityofBern,Sidlerstrasse5,3012Bern,Switzerland bDepartmentofPhysics,UniversityofCaliforniaatSanDiego,LaJolla,CA92093,USA

cUniversityofVienna,FacultyofPhysics,Boltzmanngasse5,1090Vienna,Austria

a rt i c l e i n f o a b s t r a c t

Articlehistory:

Received19October2020

Receivedinrevisedform13December2020 Accepted8January2021

Availableonline13January2021 Editor:B.Grinstein

Atlow energieshadronic vacuumpolarization (HVP)is strongly dominatedby two-pionintermediate states,whichareresponsibleforabout70% oftheHVPcontributiontotheanomalousmagneticmoment ofthemuon,a^HVP_μ .Lattice-QCDevaluationsofthelatterindicatethatitmightbelargerthancalculated dispersivelyon thebasisof e⁺e⁻→ ^{hadrons data,atalevel whichwould contest the}long-standing discrepancywith the a_μ measurement.In thisLetterwe study towhichextentthis2π contribution can be modified without, at the same time, producing a conflict elsewhere in low-energy hadron phenomenology.Tothisendweconsideradispersiverepresentationofthee⁺e⁻→²πprocessandstudy thecorrelationswhichtherebyemergebetweena^HVP_μ ,thehadronicrunningofthefine-structureconstant, the P-waveπ π phaseshift,andthechargeradiusofthepion.Inelasticeffectsplayanimportantrole, despitebeingconstrainedbytheEidelman–Łukaszukbound.Weidentifyscenariosinwhicha^HVP_μ canbe alteredsubstantially,drivenbychangesinthephaseshiftand/ortheinelasticcontribution,andillustrate theensuingchangesinthee⁺e⁻→²π crosssection.Inthecombined scenario,whichminimizesthe effectinthecrosssection, auniform shiftaround 4% isrequired.Atthe sametimeboththeanalytic continuationintothespace-likeregionandthepionchargeradiusareaffected atalevelthatcouldbe probedinfuturelattice-QCDcalculations.

©^{2021TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

The uncertainty in the Standard Model prediction for the anomalousmagneticmomentofthemuon [1–28]

a^SM_μ

=

116 591 810

(

43

) ×

10⁻¹¹ (1) is currentlydominatedby HVP, whoseleading-order contribution asderivedfrome⁺e⁻→hadrons crosssectionsreads [1,6–12]

a^HVP_μ

e⁺e⁻

=

6 931

(

40

) ×

10⁻¹¹

.

(2) TheresultingSMprediction (1) differsfromexperiment [29]

a^exp_μ

=

116 592 089

(

63

) ×

^{10−11 (3)}

by 3.7

σ

. Ifthis discrepancy were all to be blamed on an incor- rect evaluationoftheHVPcontribution,thiswouldhaveto beas

*

Correspondingauthor.

E-mailaddresses:gilberto@itp.unibe.ch(G. Colangelo),hoferichter@itp.unibe.ch (M. Hoferichter),peter.stoffer@univie.ac.at(P. Stoffer).

large as 7200×^{10−11 to reconcile the central values of the SM} andexperiment.Thatsuchapossibilityshouldinfactbeseriously consideredhasbecome apressingissueinview ofrecentlattice- QCDevaluations. Thelattice averageperformedinRef. [1] (based onRefs. [30–38])

a^HVP_μ

lattice average

=

^{7 116}

(

184

) ×

^{10−11 (4)}

isconsistent withboth the e⁺e⁻ value (2) (within 1

σ

), butalso with the experimental value (3). The more recent calculation of Ref. [39],a^HVPμ =⁷⁰⁸⁷(53)×^{10−11,quotes aslightlysmallercen-} tralvalue,butduetotheincreasedprecisionliesabove thee⁺e⁻ valueby2.3

σ

,whilereducingthetensionwithEq. (3) to1.5

σ

.

Forthesecond-most-importantclassofhadroniccontributions, hadroniclight-by-lightscattering(HLbL),thephenomenologicales- timatea^HLbLμ =⁹²(19)×¹⁰⁻¹¹[1,14–26,40–45] agreeswitha^HLbLμ = 82(35)×^{10−11 fromlattice}QCD [27] (includingthephenomeno- logicalestimateforthecharmcontribution),insuchawaythatan averageofthetwohasbeenusedinEq. (1).

This situation has triggered renewed interest in the conse- quencesof large changes to HVPelsewhere, especially for global https://doi.org/10.1016/j.physletb.2021.136073

0370-2693/©^{2021TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

electroweakfitsduetoits impactonthehadronicrunningofthe fine-structureconstant

α

[46–49].Theseanalyseshaveshownthat toavoidasignificanttensionwithelectroweakprecisiondata,the changestothehadroniccrosssectionsneedtobeconcentratedat lowenergies,atleastbelow2GeV,ascenarioindeedindicatedby Ref. [39].

Inpreviouswork [47–49] changestothehadroniccrosssections wereconsideredasawhole,withspecificassumptionsontheen- ergydependence.However,ifthechangesareconcentratedinthe low-energyregion,itisclearthatthemostrelevantabsoluteeffect willoccurinthedominant2

π

channel,sincetherequiredrelative changes in the subleading channels would become prohibitively large.Inthisregion,the2

π

channelisessentiallyelasticanddom- inatedbythe

ρ

resonance.Therelevanthadronicmatrixelement, thepion vectorformfactor(VFF),isstrongly constrainedby ana- lyticityandunitarity,whichimplythatbelow1GeV itisessentially determinedbytheP-wave

π π

phaseshift [8],whichisagaincon- strainedbyanalyticity,unitarity,andcrossingsymmetry,takingthe formof Royequations [50–53].The mainconclusion ofthe anal- ysis in Ref. [8] is that the VFF below1GeV can be described in termsofahandfulofparameters,whichcanallbedeterminedby afittothee⁺e⁻→²

π

data.Thefactthat thesedata,whichhave now reacheda remarkable levelof precision, typically below1%, canbewelldescribedbythishighlyconstrainedrepresentation,is anontrivialtestontheirquality.

Within this framework it is possible to address the question which changes becomepossible withoutviolating analyticity and unitarityandwithoutincurringother tensionselsewhere—besides thosewiththee⁺e⁻→²

π

cross-sectiondata.Tothisend,wefirst ofalldeterminewhatchangesintheparametersofthedispersive representationmaygeneratethedesiredchangeina^HVPμ .Withthe same setofparameters we thencalculate the P-wave

π π

phase shifts,the hadronicrunningof

α

, aswellasthe charge radiusof thepion,andtherebyestablishcorrelationsamongallthesequan- tities.

Finally, we identify scenarios in which significant changes to HVP remain possible despite these independent constraints on the pionVFF.The comparisonoftheresultingpredictions forthe e⁺e⁻→²

π

cross section to data allows usto quantify by how much theexperimental crosssections wouldneedto bechanged toaccommodatesuchanincreaseina^HVPμ .

2. Thepionvectorformfactor

The HVP contribution to the anomalous magnetic moment of the muon, expressedin termsof the e⁺e⁻→hadrons cross sec- tion,reads [54,55]

a^HVP_μ

= α

m_μ

3

π

2

ˆ

∞

s_thr

dsK

ˆ (

s

)

s² R_had

(

s

),

R_had

(

s

) =

^3s

4

π α

²

σ (

e⁺e⁻

→

^hadrons

),

(5) withaknownkernelfunction Kˆ(s).WiththepionVFF Fπ^V(s)de- finedasthematrixelementoftheelectromagneticcurrent jμ

em,

π

^±

(

p

) |

^jμem

(

0

) | π

^±

(

p

) = ± (

p

+

^p

)

^μF_π^V

((

p

−

^p

)

²

),

(6) the2

π

contributionbecomes

σ (

e⁺e⁻

→ π

⁺

π

⁻

) = π α

²

3s

σ

_π³

(

s

)

_Fπ^V

(

s

)

²

,

(7) where

σ

π(s)=

1−^4M2_π/s. Similarly, thetwo-pion contribution tothehadronicrunningof

α

,evaluatedatM²_Z,

α

⁽_had⁵⁾

(

M²_Z

) = α

M²_Z 3

π ⁻

ˆ

∞ sthr

ds R_had

(

s

)

s

(

M²_Z

−

s

) ,

(8)

is determined by Fπ^V(s). In both cases, the integration threshold becomes s_thr=^4M2_π^{, and radiative} corrections to the cross sec- tion are implemented in such a way that vacuum polarization isremoved, but final-stateradiation (FSR)included.Since Eq. (6) defines the matrix element in pure QCD, this implies that FSR correctionsneed tobe included inthe final step,see Ref. [8] for further details. In addition,we consider thecorrelation withthe pionchargeradius

^r2_π

=

^6dFπV

(

s

)

ds

s=⁰

=

⁶

π

ˆ

∞

4M²π

dsImF_π^V

(

s

)

s²

,

(9)

which,contrary toa^HVPμ and

α

⁽_had⁵⁾, isalso explicitlysensitive to thephaseofFπ^V(s).

Inthe elastic region, where2

π

is againthe only relevantin- termediate state, Fπ^V(s)is stronglyconstrainedby analyticity and unitarity.Iftheelasticregionextendedall thewaytoinfinity,the solutiontotheunitarityandanalyticityconstraintswouldbegiven bytheOmnèsfactor [56]

¹₁

(

s

) =

exp

⎧ ⎪

⎨

⎪ ⎩

s

π ˆ

∞

4Mπ²

ds

δ

₁¹

(

s

)

s

(

s

−

s

)

⎫ ⎪

⎬

⎪ ⎭ ,

(10) with the P-wave

π π

scattering phase shift δ₁¹(s). This phase shift,in turn, is stronglyconstrained by

π π

Roy equations [50–

53], which further limits the permissible changes in Fπ^V(s), see Refs. [57–64] forrepresentationsthatexploitthisintimateconnec- tion between the VFF and

π π

scattering. Below 1GeV inelastic effectsaresmall,butatthelevelofprecisionnecessaryhere,have tobe takenintoaccount. Todothiswe multiplythe fullyelastic Omnèsfactor (10) bytwoadditionalfactors,asinRefs. [8,58,59]

F_π^V

(

s

) =

¹₁

(

s

)

G_ω

(

s

)

G_in^N

(

s

),

(11) where Gω(s) accounts for theisospin-violating 3

π

cut, which is completelydominatedby

ρ

–

ω

mixing,andthe4

π

cutisexpanded intoaconformalpolynomial

G_in^N

(

s

) =

1

+

N k=1

ck

(

z^k

(

s

) −

z^k

(

0

)),

(12)

wheretheconformalvariable z

(

s

) =

√

sin

−

sc

− √

sin

−

s

√

sin

−

sc

+ √

sin

−

s (13)

permits inelastic phases above the

π ω

threshold sin=(Mπ⁰ + Mω)². The parameter sc is the value of s mapped to the origin, z(s_c)=^{0, and is varied around} −^{1GeV2. To ensure the correct} thresholdbehavior, the c_k are relatedby an additional constraint thatremovestheS-wavesingularity.

In total, the dispersive representation from Ref. [8] then in- volves the following free parameters: first, the solution of the

π π

Roy equations is determined once the phase shifts at s0= (0.8GeV)² ands1=(1.15GeV)² are specified, so that δ₁¹(s0) and δ₁¹(s1) are free fit parameters. Second, Gω(s) dependson the

ω

pole parameters as well as the overall strength of

ρ

–

ω

mixing.

Third,thereareN−^{1 freeparametersinG}_in^N(s)todescribeinelas- ticeffects.

TheresultsforthephaseshiftsfromafittoVFFdataare [8]

δ

₁¹

(

s0

) =

110

.

4

(

7

)

^◦

, δ

¹₁

(

s1

) =

165

.

7

(

2

.

4

)

^◦

,

(14) but for the purpose of this work it is crucial to understand to within which ranges they can be constrained withoutrelying on e⁺e⁻→²

π

data (or

τ

→

π π ν

τ). In principle, one could even considerindirectconstraintsthatarise,viatheRoyequations,from low-energydataincrossedchannels,suchasK4data [65–67],but herewesimplyquotetheresultsfromthepartial-waveanalyses Ref. [68]:

δ

₁¹

(

0

.

79 GeV

) =

⁹⁷

.

5

(

1

.

5

)

^◦

[

¹⁰³

.

9

(

6

)

^◦

] , δ

₁¹

(

0

.

81 GeV

) =

¹¹²

.

1

(

8

)

^◦

[

¹¹⁶

.

2

(

7

)

^◦

] , δ

₁¹

(

1

.

15 GeV

) =

167

.

7

(

3

.

3

)

^◦

[

165

.

7

(

2

.

4

)

^◦

],

Ref. [69]:

δ

₁¹

(

0

.

795 GeV

) =

105

.

0

(

1

.

5

)

^◦

[

107

.

2

(

6

)

^◦

], δ

¹₁

(

0

.

81 GeV

) =

114

.

0

(

1

.

4

)

^◦

[

116

.

2

(

7

)

^◦

],

δ

¹₁

(

1

.

15 GeV

) =

¹⁶⁴

(

6

)

^◦

[

¹⁶⁵

.

7

(

2

.

4

)

^◦

],

⁽¹⁵⁾

whereourvalues,extractedfromtheglobalfittoe⁺e⁻→²

π

data, areshowninbracketsforcomparison.

Theparameters inGω(s)donotneedtobeconsidered further becauseeitheronewouldhavetobechangedbeyondanyplausible range toproduce a relevant effectina^HVPμ .Finally, ifseveralfree parameters in the conformal polynomial are introduced, the re- sultinginelasticphaseshiftingeneralleads tounacceptablylarge violationsofWatson’sfinal-statetheorem [70].Aquantitativephe- nomenologicalboundcanbeformulatedbasedontheratio r

= σ

^I=1

(

e⁺e⁻

→

^hadrons

)

σ (

e⁺e⁻

→ π

⁺

π

⁻

) −

^{1 (16)}

ofnon-2

π

to2

π

hadroniccrosssectionsforisospinI=^1,e.g.,for thetotalphaseψ oftheVFF [71,72]

sin²

(ψ − δ

¹₁

) ≤

¹ 2

1

−

1

−

^r2

.

(17)

This EŁboundshowsthat inelastic effectsbelowthe

π ω

thresh- oldareindeednegligible,andlimitsthesizeoftheinelasticphase above. In practice, we use the implementation of the EŁ bound fromRef. [8],butnotethatthesedetailsareoflimitedimportance inthepresentcontext:oncetheEŁboundbecomesactive,thein- crease inthe

χ

² is rathersteep, so that the excluded parameter spaceisessentiallyinsensitivetotheexactimplementationofthe EŁbound.

3. ChangingHVP

WestartfromthemainresultsofRef. [8],wheretherepresen- tation (11) isfittoacombinationofthedatasetsofRefs. [73–85], leadingtoatwo-pioncontributiontoa^HVPμ below1GeV of [8]

a^{π π}_μ

≤^{1 GeV}

=

⁴⁹⁵

.

0

(

1

.

5

)(

2

.

1

) ×

¹⁰⁻¹⁰

=

⁴⁹⁵

.

0

(

2

.

6

) ×

¹⁰⁻¹⁰

,

(18) where thefirst erroristhe fituncertainty (inflatedby

χ

²/dof) and thesecond errorincludes all systematic uncertainties of the representation (11). Thecentralconfigurationuses N−¹=^{4 free} parameters inthe conformalpolynomial.Duetothesensitivityof theradiussumrule (9) tothephaseoftheVFF,fitswithtoomany free parametersinthe conformalpolynomialtend tobecomeun- stablefor^r2_π^{,becausethephaseneeds tobe}extrapolatedabove

theenergyforwhichtheEŁboundcanbeusedinpracticetocon- strainthesizeoftheimaginarypart.Forthisreason,inRef. [8] the centralevaluationofr²πwasobtainedwithN−1=1,butthefull variationwithN waskeptasasystematicuncertainty,whichdom- inatestheuncertaintyassignedtothefinalresult [8]

^r2_π

=

⁰

.

429

(

1

)(

4

)

fm²

=

⁰

.

429

(

4

)

fm²

.

(19) Here,weuseasreferencepointthevalueforN−¹=^{4 [8]}

^r2_π

N−1=4

=

⁰

.

426

(

1

)

fm²

,

(20)

wherethe error refers to the fit uncertainty only. Finally,the fit configurationwith N−¹=^{4 leadsto a two-pion}contributionto thehadronicrunningof

α

,

α

⁽_{π π}⁵⁾(M²_Z),of

α

⁽_{π π}⁵⁾

(

M²_Z

)

_≤

1 GeV

=

³²

.

62

(

10

)(

11

) ×

¹⁰⁻⁴

=

³²

.

62

(

15

) ×

¹⁰⁻⁴

.

(21) Starting fromthe central fit results, we now modify the con- tribution to a^HVPμ by including in the fit a hypothetical “lattice”

observationofaπ π

μ _{≤1 GeV}intheformofanadditionalcontribution tothe

χ

² functionthatweminimize.Thefitoutputforaπ π

μ

≤^{1 GeV} is then pulled away from the central fit result in Eq. (18), de- pending onthe input aπ π

μ

≤^{1 GeV} and its uncertaintythat acts as a weight. We find it convenient to adopta tiny uncertainty, be- cause it forces the output for aπ π

μ _{≤1 GeV} to essentially coincide withthe input. With a larger uncertainty(i.e., a smaller weight) thefitoutputforaπ π

μ _{≤1 GeV}willbesomewherebetweentheinput andEq. (18).However,thechoiceoftheweightsisimmaterialbe- causethefollowingstudiesareallbasedontheoutputaπ π

μ _≤

1 GeV. Foragivenoutput aπ π

μ _{≤1 GeV}, thefitalways finds theparameter valuesthatminimize thetensionwiththe cross-sectiondata.We considerthefollowingthreescenarios:

(1) “Low-energy” scenario: we fix all parameters of the disper- sive representation ofthe VFF to the central fit results with N−¹=^{4 without “lattice”input foraπ π}_{μ ≤1 GeV}, apart from thetwo phase-shiftparameters δ₁¹(s0) and δ¹₁(s1),which are usedasfreeparametersinafittodataand“lattice”inputfor aπ π

μ _≤

1 GeV.

(2) “High-energy”scenario: we fixall parameters apartfromthe parametersc_kintheconformalpolynomial.

(3) Combinedscenario:allparametersareusedasfreefitparam- eters.

Weareinterestedintheregionoftheparameterspacethatallows forasignificantupwardshiftinaπ π

μ .Fordefiniteness,wetake

a^{π π}_μ

≤^{1 GeV}

¹⁸

.

5

×

^{10−10 (22)}

asreference point, which corresponds to the difference between Eqs. (2) and (4).

Thedependenceofthe VFFon thetwo freephase parameters δ₁¹(s0)andδ¹₁(s1)isintertwinedwiththesolutionoftheRoyequa- tions for the phase δ¹₁(s), which in turn determines the Omnès function (10).Incontrast,thedependenceontheparametersinthe conformalpolynomial ismuchmoredirect,astheconstraintthat removestheS-wavesingularityisalinearrelationbetweenthepa- rametersc_k.Therefore,theVFFislinearintheparametersc_k and thesame istrue forthe contributionto the charge radius,while aπ π

μ and

α

_{π π}⁽⁵⁾(M²_Z)arequadraticintheconformalparametersck. However,intherelevantparameterrangethenon-linearitiesprove tobeverysmall.

Fig. 1.ImpactoftheEŁboundontheχ²forN−1=1. . .4 whenvaryinga^{π π}_μ

≤1 GeV awayfromthecentralfitresult.Theshadedareacorrespondsto0≤a^{π π}_μ

≤1 GeV≤ 18.5×10⁻¹⁰forN−1=4.

Inordertofurtherrestrictpossiblevariantsinscenarios(2)and (3),wefirstinvestigatetheroleoftheEŁboundinthecontextof variationsofaπ π

μ

≤^{1 GeV}.

4. ConstraintsduetotheEŁbound

The EŁ bound (17) provides an additional restriction on the permissible parameterspacethat is independentofthe two-pion cross-sectionmeasurements. UsingtheimplementationofRef. [8]

and the data compilation of Ref. [72], this constraint leads to a steepriseofthe

χ

²functionunlesstheinelasticphasestayssmall.

Toillustratethiseffect,weconsiderscenario(2)andfitconfigura- tionswithN−¹=¹. . .4 freeparametersintheconformalpolyno- mial.Starting fromthecentralfitresults,wevarytheinputvalue foraπ π

μ _{≤1 GeV}.TheimpactoftheEŁboundonthe

χ

²isshownin Fig.1,asafunctionofthefitoutputaπ π

μ

≤^{1 GeV}.Wefindthatthe boundseverelyrestrictsthepossiblechangesinaπ π

μ forN−¹=^1:

inducinglargershiftswithonlyasinglefreeparameterinthecon- formalpolynomialautomaticallyleadstoasignificanteffectinthe inelastic phasethat violatesthe EŁbound, thus excluding sucha scenario. Withtwo free parameters inthe conformal polynomial, the EŁ bound permitslarger changes in aπ π

μ , butstill imposes a restriction.ToevadetheEŁboundforlarge changesinaπ π

μ ,more freedom intheparameterizationisrequired,andindeedthesitu- ationchangesifweconsiderthreeormorefreeparametersinthe conformalpolynomial,seeFig.1.

Inorder tobetterunderstand thiseffect,we considerinsome detailthecaseofN−¹=^{2.Thefittodataaloneleadsto} a^{π π}_μ

^N_≤⁻¹⁼²

1 GeV

=

⁴⁹⁷

.

0

(

1

.

4

) ×

¹⁰⁻¹⁰

.

(23)

Varyingthetwoparametersc2,3 awayfromthecentralfitresults, wefindthattheEŁboundgivesacontributiontothe

χ

² thatre- sults in a strong anti-correlation between permissible values for the two free parameters. This is illustrated in Fig. 2, where we show thecontours for

χ

_EŁ² ∈ {⁰.1,1,10}^inthec2–c3 plane.Inthe close-up plot,wealso overlaya heatmap forthe resultingvalue ofaπ π

μ _≤

1 GeV.Accordingly,fortwofree parametersintheconfor- malpolynomialtheEŁboundalonenolongerexcludesvery large shifts in aπ π

μ , asshown by the ellipses in Fig. 2.However, large partsofthe

χ

_EŁ² ellipsisareinstrongtensionwiththecross-section data.Minimizingthetotal

χ

² inscenario(2)resultsinthebrown dashed path in Fig. 2, which corresponds to the brown curve showninFig.1.Forevenmorefreeparameters N−¹>2,thesit- uation remains qualitatively similar: the EŁbound again strongly

correlatesthefreeparametersoftheconformalpolynomial,essen- tiallyimposing one linear constraint, but the valuesof aπ π

μ that canbereachedarenolongerbounded.Therefore,inthefollowing wewill onlyconsider fitvariantswith N−¹=^{3 and N}−¹=^4, wheretheEŁboundiseasilyfulfilledevenforlargeshiftsinaπ π

μ . 5. Correlationswith

α

⁽_had⁵⁾ and^r2π

Wenowturnourattentiontothecorrelationsamongthethree quantities derived from HVP—the two-pion contribution to the anomalous magnetic momentof the muon aπ π

μ ,the pion charge radius ^r2_π^{, and the two-pion} contribution to the hadronic run- ningof

α

,

α

_{π π}⁽⁵⁾(M²_Z).Wevarythehypothetical“lattice”inputfor aπ π

μ

≤^{1 GeV}, perform thefits according to the three scenarios de- finedin Sect.3,andcompute the resultingoutput valuesforthe threequantities.TheresultsinFigs.3and4showthecorrelations ofaπ π

μ with^r_π^2and

α

⁽_{π π}⁵⁾(M²_Z),respectively,asinducedineach ofthescenarios.

Ifthechanges inaπ π

μ _{≤1 GeV} are inducedonly by variations of thetwo phase-shiftparameters δ₁¹(s₀) andδ₁¹(s₁),they haveonly littleimpactonthechargeradius^r2π^{,seeFig.3.Hence,inpractice} changes ofaπ π

μ induced by these parameters cannot be detected by a precision measurement of ^r2_π^{. However, a scenario where} the changes in aπ π

μ _{≤1 GeV} are induced by shifts in the parame- tersck oftheconformalpolynomial generateslargeshifts in^r2π andcouldbe constrainedbyadditionalinformationonthecharge radiusof the pion,atleastin principle. Atpresent, lattice deter- minations of the charge radius [86,87] have not yet reachedthe precisionthatcouldexcludetheseshifts:thecurrentlatticeuncer- taintiescovertheentireplotrangeinFig.3,butfutureprogresson thedeterminationofthechargeradiuscouldfurtherconstrainthe allowedparameterrange.Interestingly,thecombinedscenario(3) whereall parameters are allowedto vary leads tothe largestef- fectinthepioncharge radius,evenslightlylargerthantheeffect inthescenarios (2).By definition,thisisthe scenariowithmini- maltensionwiththecross-sectiondata,butFig.3showsthatthis comesattheexpenseofthelargestshiftinthechargeradius.

Incontrasttothepionchargeradius,allscenariosleadtovery similarcorrelations withthehadronicrunningof

α

,asshownin Fig.4.Ashiftinaπ π

μ

≤^{1 GeV}by18.5×¹⁰⁻¹⁰correspondstoashift in

α

⁽_{π π}⁵⁾(M²_Z)_{≤1 GeV}between1.2×^10−4and1.4×^{10−4,asshown} inFig.4.¹ Theexistenceofsuchacorrelationemergesbecausewe donot allow forarbitrary changes inthe hadroniccross section:

whileingeneralthetwoquantitiesneednot becorrelateddueto thedifferentenergydependenceoftheirkernelfunctions,wefind that a correlation doesarise ifonly changes in the

π π

channel areconsideredasallowedby analyticityandunitarityconstraints, while trying to minimize the tension withthe

π π

cross-section data.

6. Impactonthephaseshiftandcrosssection

Inscenario (1) we only allowthe two phase-shift parameters δ₁¹(s₀)andδ¹₁(s₁)todeviatefromthecentralfitresultstodata.If onlythe phase at s0=(0.8GeV)² were varied, a huge changein

1 Thisshiftisslightlysmallerthanthe1.8×^{10−4estimatedin}Ref. [47] iftherel- ativechangesoccurbelow1.94GeV butareotherwiseenergyindependent.Shiftsof thissizeviolatetheboundonα⁽_had⁵⁾(M²_Z)derivedinRef. [88].Sincethisboundwas derivedonthebasisofassumptions(dim-6 operatorassoleoriginoftheshiftin α⁽_had⁵⁾(M²_Z)andanarbitraryscalechoicewhenconvertingthederivativeoftheHVP functiontoα⁽_had⁵⁾(M²_Z)),wehavetoconcludethattheseassumptionsarenotten- able.Theresultforα⁽_had⁵⁾(M²_Z)indicatedbyRef. [39] leadstothesameconclusion.

Fig. 2.ImpactoftheEŁboundontheχ²forN−¹=^{2 whenvaryingthefreeparametersc}2andc3awayfromthecentralfitresult(denotedbydottedlines).Shownare theregionscorrespondingtoχEŁ²∈ {⁰.1,1,10}^{.Therightplotshowsinmoredetailtheparameterregionofinterestandtheresultsforaπ π}μ _{≤1 GeV}asaheat-mapoverlay.The browndashedlineshowsthepathofthefitinscenario(2).

Fig. 3.Correlationsbetweena^{π π}_μ and^r2π^{asinducedinthreedifferentscenarios:a}

“low-energy”scenario(1),whereshiftsina^{π π}_μ areinducedbychangesinthephase- shiftparametersδ¹₁(s0),δ¹₁(s1);two“high-energy”scenarios(2),wheretheshifts areduetochangesintheconformalpolynomialwithN−¹=^{3 orN}−¹=^4;anda combinedscenario(3)withN−¹=^{4,whereallfreeparametersinthedispersive} representationofthepionVFFareallowedtovary.

Fig. 4.Correlationsbetweena^{π π}_μ andα⁽π π⁵⁾(M²_Z)forthesamescenariosasinFig.3.

thephaseshiftofaboutδ¹₁(s0)=^{10◦ wouldbenecessarytoob-} tainashiftinaπ π

μ

≤^{1 GeV}by18.5×^{10−10.Ontheotherhand,such} a changeinaπ π

μ couldbe induced bytheparameter δ¹₁(s1) alone witha shiftby1.8^◦.Ifwe fitthetwo parameters simultaneously toacombinationofthespace- andtime-likedataontheVFFand thehypothetical“lattice”inputonaπ π

μ

≤^{1 GeV},ashiftinaπ π μ

≤^{1 GeV} by18.5×^10−10thencorrespondstomodestchangesinthephase

Fig. 5.Changeinthephaseshiftδ¹₁ ats0=(0.8GeV)² ands1=(1.15GeV)² asa functionofa^{π π}_μ .Inscenario(1)onlythesetwoparametersareusedtoachieve thechangeina^{π π}_μ ,whileinthecombinedscenario(3)allparametersarechanged simultaneously.

byδ¹₁(s0)=⁰.8^◦andδ¹₁(s1)=¹.7^◦,seeFig.5.Wenotethatthe partial-wave solutions giveninEq. (15) wouldactually favorval- uesslightlybelowour referencepoint (14),butcertainly exclude the requiredchange in δ¹₁(s0) if the shiftin aπ π

μ _{≤1 GeV} were in- ducedbythisparameteralone.

AsdiscussedinSect.5,indirectconstraintsonscenario(1)from adeterminationofthepionchargeradiusseemoutofreach.How- ever, direct constraints on δ₁¹(s0) and δ¹₁(s1) could be obtained fromlatticedeterminationsoftheelastic

π π

phaseshift [93–102], notonlyattheseexactpoints inenergy,butinthewhole

ρ

res- onance region: giventhephase valuesδ₁¹(s_0,1), theRoy solutions determinethemodified phase shiftoverthewhole energyrange.

However, theprecision oflattice data isnot yet sufficientto add meaningfulconstraintstotheparameterspace,andonlyasignifi- cantincreaseinprecisionwillhaveanimpactona^HVPμ determina- tions.

Fig.5 alsoshowsthe resultingshifts inthe phase parameters for scenario (3), δ₁¹(s0)= −⁰.2^◦ and δ₁¹(s1)= −⁰.3^◦. As dis- cussedinSect.5,itismostpromisingtoindirectlyconstrainsuch ascenariowithanimproveddeterminationofthepionchargera- dius.Infact,notonlytheradiusisrelevantinthisregard,butthe VFFin thewhole space-likeregion,asshowninFig. 6.Scenarios (2) and (3) move the curve outside the error band of the cen- tralfittodata.Preciselattice-QCDdeterminationsofthespace-like VFF [87] couldstart to discriminatebetweenthe centralsolution andthese shifted variants. Consistently with the small effect on theradius,scenario(1)withshiftsonlyinthetwophase-shiftpa-

Fig. 6.Close-upviewofthespacelikeregion.TheJLabdata [89–92] arenotusedin thefit.

Fig. 7.Close-up view of theρ–ωinterference region.

rametershasanegligibleeffectonthespace-likeVFF:theshifted solutionremainswellwithintheuncertaintiesofthecentralfitre- sult.

Finally,we take acloserlook atthepionVFF inthetime-like region.ThedispersiverepresentationoftheVFFallowsustoquan- tifyin detailhow thecrosssectionswould needto bealteredto achieveagivenchangeinaπ π

μ _≤

1 GeV,ineachofthethreescenar- ios. In Fig. 7, a close-up view ofthe

ρ

–

ω

interference region is shown. Itrevealsthat ifthe changeinaπ π

μ

≤^{1 GeV} were explained withthehelpofδ₁¹(s0,1),adramaticshiftofupto8% ofthecross sectionwouldbenecessary.Iftheshiftwereobtainedbychanging the parameters ck, the effectin the cross section at the

ρ

reso- nance would be only abouthalf as large, although the resulting cross section would still lie far outside the combined fit to the data.Thecombinedscenarioisveryclosetotheonewhereshifts areonlyallowedintheparametersc_k.

InFig. 8,wecompare boththedata setsandtheshifted vari- antsoftheVFFtothecentralfitresult,astherelativedifferences normalizedtothefitresult.Weagainseethatbyusingtheconfor- malpolynomialtoinducetheshift,theeffectonthecrosssections issmalleraroundthe

ρ

resonancethaninthescenariowithashift inδ₁¹(s0,1),whiletheeffectislarger belowabout0.72GeV.Com- paredto thespreadofthedatapoints, thenecessary shiftinthe crosssectionsisagainsignificant,althoughlessdrasticthaninsce- nario(1),wherethechangesareconcentratedinthe

ρ

region.This isconsistentwiththefactthattheconformalpolynomialparame- terizestheeffectsofinelasticitiesabovethe

π ω

threshold.

Fig. 8.ComparisonofthedatasetsandtheshiftedvariantsoftheVFF,relativeto thecentralfitsolution.

Fig. 9.Increaseintheχ² asafunctionofthefitoutputa^{π π}_{μ ≤1 GeV} inthethree scenarios,excludingthecontributionofthe“lattice”input(sincethisdependson thearbitraryuncertaintythatactsasaweight,seeSect.3).

While Figs. 7 and 8 make it evident that the changes in the crosssectionthatwouldgeneratethedesiredchangeinaπ π

μ _{≤1 GeV} are incompatible with the data, Fig. 9 shows the corresponding changein

χ

² asafunctionofaπ π

μ _≤

1 GeV,andprovidesaquantita- tivemeasureofthediscrepancy.Themostdramaticclashwiththe datawouldbe inscenario (1),butevenintheothertwoanysig- nificantchangeinaπ π

μ

≤^{1 GeV}comesatthepriceofhugeincreases in

χ

².Theseincreasescanbecomparedtothewell-knowntension betweenindividuale⁺e⁻datasets.ThecentralfitresultsofRef. [8]

reachatotal

χ

² of776 with627 degreesoffreedom.Thetension isreflectedbyanerrorinflationincludedinEq. (18) of

χ

²/dof= 1.11.Forthetargetshiftofaπ π

μ

≤^{1 GeV}=¹⁸.5×^{10−10,evensce-} nario(3)leadstoatotal

χ

² of941.

The results in Figs. 7 and 8 show that to minimize the ef- fect in the cross section, the changes mainly affect the inelastic partoftheVFFparameterizationandthusenergiesabovethe

π ω

threshold.Inprinciple, theseinelasticcontributions couldbe fur- ther constrained by e⁺e⁻→²

π

data above 1GeV [81,83,103],

τ

_→

π π ν

_τ [104],andexplicitinputontheinelasticchannels, but thisrequiresanextensionofourdispersiveformalismthatwillbe left forfuture work. We remark that anychanges in the physics above1GeV willalsohaveanimpacton

α

_{π π}⁽⁵⁾(M²_Z),whichisnot yet accounted for here: the higher in energy these changes are pushed, the higher the risk to exacerbate tensions in the global electroweakfit [46–49].