-
The anomalous magnetic moment of a muon, denoted as
aμ=(gμ−2)/2 , plays a crucial role in the precision tests of the Standard Model (SM) [1, 2]. The long-standing discrepancy between the SM prediction ofaμ and its experimental measurement has recently been updated to 4.2 standard deviations [3, 4], and it has sparked numerous theoretical investigations. The SM uncertainty onaμ is dominated by hadronic vacuum polarization (HVP), with the largest contribution originating from theππ intermediate states, accounting for over 70% of the HVP contribution. Theoretically, the two-pion low-energy contribution toaμ is expressed as an integral over the modulus squared of the pion electromagnetic form factor, which can be extracted frome+e− -annihilation experiments. In principle, the two-pion contribution toaμ can be evaluated accurately as long as the experimental data ofe+e−→π+π− are available everywhere at the required level of precision. Although it is known that a tension exists between the two most precise measurements by BaBar and KLOE Collaborations, the BaBar data lie systematically above the KLOE results in the dominant ρ region. Consequently, considerable efforts have been dedicated to finely describing the pion electromagnetic form factor [5−11]. In the dominant ρ region of thee+e−→π+π− process, the isospin-breaking effect due toρ−ω mixing, which becomes enhanced by the small mass difference between ρ and ω mesons, plays a significant role and should be considered appropriately.Usually, the momentum dependence of
ρ−ω mixing amplitude is neglected, and a constant mixing amplitude is used to describee+e−→π+π− data due to the narrowness of the ω resonance. The first study on the momentum dependence ofρ−ω mixing amplitude was conducted by Ref. [12]. Based on a quark loop mechanism ofρ−ω mixing, it was determined that the mixing amplitude significantly depends on momentum. Subsequently, the investigation of various loop mechanisms forρ−ω mixing was initiated in different models, such as the global color model [13], extended Nambu-Jona-Lasinio (NJL) model [14, 15], chiral constituent quark model [16, 17], and hidden local symmetry model [18−20]. In our pervious study [21], we examinedρ−ω mixing using a model independent approach through Resonance Chiral Theory (RχT) [22]. Guided by chiral symmetry and largeNC expansion, RχT provides us a reliable theoretical framework to study the dynamics with light flavor resonances and pseudo-Goldstone mesons in the intermediate energy region [23−28], and it has been successfully applied in the calculation ofaμ in the SM [9, 29−36]. In Ref. [21], we calculated the one-loop contributions toρ−ω mixing, which are at the next-to-leading order (NLO) in the1/NC expansion [28, 37−40]. In this study, we update the previous study [21] by incorporating the contribution arising from the kaon mass splitting in the kaon loops.Moreover, we focus on analysing the impact of the momentum dependence of
ρ−ω mixing on describing the pion vector form factor data and its contribution toaμ . Specifically, we perform two types of fits (with momentum-independent or momentum-dependentρ−ω mixing amplitude) describinge+e−→π+π− and andτ→ντ2π data in the energy region of 600–900 MeV, decay width ofω→π+π− , and compare their results. The fit results demonstrate that the momentum-independent and momentum-dependentρ−ω mixing schemes can effectively describe the data, while the momentum-dependent scheme exhibits higher self-consistency due to the reasonable imaginary part of the extracted mixing matrix elementΠρω . Regarding the contribution to the anomalous magnetic moment of a muon,aHVP,LOμ[π+π−] , which is evaluated between 0.6 GeV and 0.9 GeV, the results obtained from fits considering the momentum-dependentρ−ω mixing amplitude are in good agreement with those from fits that do not include the momentum dependence ofρ−ω mixing, within the margin of errors.This paper is organized as follows. In Sec. II, we introduce the description of
ρ−ω mixing and elaborate on the calculation ofρ−ω mixing amplitude up to the next-to-leading order in the1/NC expansion. In Sec. III, the fit results are shown and related phenomenologies are discussed. A summary is provided in Sec. IV. -
In the isospin basis
|I,I3⟩ , we define the pure isospin states|ρI⟩≡|1,0⟩ and|ωI⟩≡|0,0⟩ . The mixing between the isospin states of|ρI⟩ and|ωI⟩ can be implemented by considering the self-energy matrixΠμν=Tμν(Πρρ(s)Πρω(s)Πρω(s)Πωω(s)),
(1) with
Tμν≡gμν−pμpνp2 ands≡p2 . The none-zero off-diagonal matrix elementΠρω(s) contains information onρ−ω mixing. The mixing between the physical states ofρ0 and ω, is obtainable by introducing the following relation(ρ0ω)=C(ρIωI),C=(1−ϵ1ϵ21),
(2) where
ϵ1 andϵ2 denote the mixing parameters. The matrix of dressed propagators corresponding to physical states is diagonal [41],(1/sρ001/sω)=C(1/sρΠρω/sρsωΠρω/sρsω1/sω)C−1,
(3) where abbreviations
sρ andsω are defined by the following:sρ≡s−Πρρ(s)−m2ρ,sω≡s−Πωω(s)−m2ω.
(4) The information of
ρ−ω mixing is encoded in the off-diagonal element of the self-energy matrix, decomposed as follows:Πρω(s)=ΔudSρω(s)+4παEρω(s) ,
(5) where
Δud=mu−md denotes the mass difference between u and d quarks, and α denotes the fine-structure constant.Sρω(s) andEρω(s) denote the structure functions of the strong and electromagnetic interactions, respectively. In this study, the diagrams in Fig. 1 are calculated in RχT up to NLO in1/NC expansion.In RχT, the vector resonances can be described in terms of antisymmetric tensor fields with normalization:
⟨0|Vμν|V,p⟩=iM−1V{pμϵν(p)−pνϵμ(p)},
(6) where
ϵμ denotes the polarization vector. Here, the vector mesons are collected in a3×3 matrix as follows:Vμν=(1√2ρ0+1√2ωρ+K∗+ρ−−1√2ρ0+1√2ωK∗0K∗−¯K∗0−ϕ)μν.
(7) The effective Lagrangian for the leading order strong isospin-breaking effect, corresponding to the tree-level contribution diagram (a) in Fig. 1, is as follows [42, 43]:
Lρω2=λVV6⟨VμνVμνχ+⟩,
(8) with
χ+=u+χu++uχ+u andχ=2B0(s+ip) . The pseudo-Goldstone bosons originating from the spontaneous breaking of chiral symmetry can be filled nonlinearly into:uμ=i(u†∂μu−u∂μu†),u=exp(iΦ√2F),
(9) with the Goldstone fields
Φ=(1√2π0+1√6ηπ+K+π−−1√2π0+1√6ηK0K−ˉK0−2√6η).
(10) where F denotes the pion decay constant. Considering the mass relations of the vector mesons at
O(p2) in terms of the quark counting rule, the value of the coupling constant is determined as follows:λVV6=1/8 [42, 43]. Thus, the tree-level strong contribution can be expressed as follows:S(a)ρω=2MV .
(11) The Lagrangian describing the interactions between
Vμν and electromagnetic fields or Goldstone bosons are as follows:L2(V)=FV2√2⟨Vμνfμν+⟩+iGV√2⟨Vμνuμuν⟩ ,
(12) with the relevant building blocks defined as follows:
uμ=i[u+(∂μ −irμ)u−u(∂μ−ilμ)u+],fμν±=uFμνLu+±u+FμνRu.
(13) Here,
FμνL,R denote field strength tensors composed of the left and right external sourceslμ andrμ , andFV andGV denote real resonance couplings constants. The tree-level electromagnetic contribution from diagram (b) in Fig. 1 can be calculated using the Lagrangian in Eq. (12):E(b)ρω=FρFω3.
(14) The physical decay constants
Fρ andFω have been employed in the amplitude, and are differentiated by means of isospin breaking.The loop contributions of diagrams (d)–(i) in Fig. 1 have been extensively discussed in our previous study [21]. However, a noteworthy distinction in our current study is the inclusion of the contribution from diagram (c), which arises from the kaon mass splitting within the kaon loops. To ensure comprehensiveness, we present the expressions for the loop contributions in the Appendix A. Furthermore, it should be noted that the ultimate expression for the renormalized mixing amplitude
Πρω(p2) is presented in Eq. (A23). -
The mass and width of the ρ meson are conventionally determined by fitting to the experimental data of
e+e−→π+π− andτ→ντ2π [44], where various mechanisms are used to describeρ−ω mixing effect. To prevent interference due to theirρ−ω mixing mechanisms, we treat massMρ and relevant couplingsGρ andFρ as free parameters in our fit. Regarding its width, the energy-dependent form is constructed in a similar manner as introduced in [45]:Γρ(s)=sMρ96πF2[σ3πθ(s−4m2π)+12σ3Kθ(s−4m2K)],
(15) where
σP≡√1−4m2P/s andθ(s) is the step function.With respect to the ω mass, it has been indicated in Refs. [5, 7] that the result determined from
e+e−→π+π− is inconsistent with that from particle data group (PDG) [44], primarily determined by experiments involvinge+e−→3π ande+e−→π0γ . Therefore, we treated the ω mass and width as free parameters and estimated them by fitting in our programme. The physical couplingFω can be determined from the decay width ofω→e+e− . Using the Lagrangian formula in Eq. (12), the decay width can be derived as follows:Γe+e−ω=4α2πF2ω(2m2e+M2ω)√M2ω−4m2e27M4ω,
(16) Hence, the expression for
Fω can be obtained. Based on the decay widths provided above,sρ andsω in Eq. (4) can be rewritten assρ≃s−M2ρ+iMρΓρ(s) ,sω≃s−M2ω+iMωΓω .
(17) The pion form-factor in
τ→ντ2π decay, irrelevant toρ−ω mixing effect, were thoroughly examined in Refs. [36, 46−48]:Fτπ(s)=(1−GρFρsF21sρ)×exp{−s96π2F2(Re[A[mπ,Mρ,s]+12A[mK,Mρ,s]])}.
(18) Furthemorme, function
A(mP,μ,s)=ln(m2P/μ2)+8m2Ps−53+σ3Pln(σP+1σP−1).
(19) To incorporate isospin-breaking effects, one approach involves multiplying
|Fτπ(s)|2 by factorSEWGEM(s) , whereSEW=1.0233 corresponds to the short distance correction [49]. Additionally,GEM(s) accounts for the long-distance radiative correction, as described in [50]. Specifically, in our fit ofτ→ντ2π decay data, we perform the following substitution.|Fτπ(s)|2⇒SEWGEM(s)|Fτπ(s)|2.
(20) The pion form-factor in
e+e− annihilation is as follows:Feeπ(s)=[1−GρFρsF21sρ−GρFωs3F21sωΠρω1sρ−4√2aB0Fω(mu−md)s3F21sω]×exp{−s96π2F2(Re[A[mπ,Mρ,s]+12A[mK,Mρ,s]])}.
(21) As defined in Appendix A.2, parameter a is associated with the combined coupling constant of the direct
ωππ interaction. In the first bracket of Eq. (21), the second term corresponds to the contribution fromρππ coupling, third term represents the contribution ofρ−ω mixing, and fourth term corresponds to contribution from the direct isospin-breaking coupling of ω to the pion pair.The leading order contribution of
ππ(γ) intermediate state to the anomalous magnetic moment of the muon is as follows [51]:aππ(γ),LOμ=(αmμ3π)2∫∞4m2πdsˆK(s)s2Rππ(γ)(s),
(22) where
Rππ(γ)(s)=3s4πα2σ(0)(e+e−→ππ(γ)) ,
(23) and the kernel function is defined as follows:
ˆK(s)=3sm2μ[(1+x2)(1+x)2x2(ln(1+x)−x+x22)+x22(2−x2)+1+x1−xx2lnx] ,
(24) with
x=1−βμ(s)1+βμ(s),βμ(s)=√1−4m2μs .
(25) It should be noted that in the formula for
aππ(γ),LOμ in Eq. (22), the integration is performed from 4m2π to∞ . In this study, we focus on the momentum dependence ofρ−ω mixing. Therefore, we only describe the pion vector form factor up to 900 MeV. To extend the study by considering higher energies, we must consider the effects of excited resonances, includingρ′(1450) andρ′′(1700) . However, these effects are beyond the scope of this study. It is interesting to note that the1/s2 enhancement factor in Eq. (22) provides higher weight to the lowest lying resonanceρ(770) that couples strongly toπ+π− .The bare cross section, including final-state radiation, takes the following form [5, 52−54]:
σ(0)(e+e−→γ∗→π+π−(γ))=[1+απη(s)]π|α(s)|23sσ3π(s)|Feeπ(s)|2s+2m2esσe(s),
(26) where
η(s)=3(1+σ2π(s))2σ2π(s)−4logσπ(s)+6log1+σπ(s)2+1+σ2π(s)σπ(s)F(σπ(s))−(1−σπ(s))(3+3σπ(s)−7σ2π(s)+5σ3π(s))4σ3π(s)log1+σπ(s)1−σπ(s),F(x)=−4Li2(x)+4Li2(−x)+2logxlog1+x1−x+3Li2(1+x2)−3Li2(1−x2)+π22,Li2(x)=−∫x0dtlog(1−t)t. (27) The experimental data considered in this study are the pion form factor
Feeπ(s) of thee+e−→π+π− process measured by the OLYA [55], CMD [56], BaBar [57], BESIII [58], KLOE [59], CLEO [60], and SND [61] Collaborations, the form factorFτπ(s) ofτ→ντ2π decay measured by the ALEPH [62] and CLEO [63] Collaborations, and the decay width ofω→π+π− [44]. It should be noted that in the experimentally published form factor dataFeeπ(s) , the vacuum polarization effects have been excluded through the subtraction of the hadronic running ofα(s) . Thus, in our fitting of the form factor dataFeeπ(s) , the one-photon-reducible Fig. 1(b) should not be considered. Given that we focus on the analysis of theρ−ω mixing effect, we only take into account the form factorsFeeπ(s) andFτπ(s) data in the energy region of 600–900 MeV. It should be noted that for the pion form factorFeeπ(s) , a tension is observed between the two most precise measurements from BaBar and KLOE in theρ peak region. However, other measurements align with theirs within the stated uncertainties. This highlights the impact of momentum dependence of ρ−ω mixing, and to avoid the tension between BaBar and KLOE data, we conduct four separate fits. Specifically, in Fits Ia and Ib, we fit all data sets excluding BaBar with momentum-independentΠρω and momentum-dependentΠρω , respectively. In Fits IIa and IIb, we fit all data sets excluding KLOE with momentum-independentΠρω and momentum-dependentΠρω , respectively. Fits Ia and IIa involve eight free parameters:Mρ ,Gρ ,Fρ ,Mω ,Γω , a, and the real and imaginary part of constantΠρω . There are nine free parameters in Fits Ib and IIb:Mρ ,Gρ ,Fρ ,Mω ,Γω , a,XrW ,XrZ , andXrR . As defined in the Appendix A,XrW ,XrZ , andXrR are the corresponding parameters for the counterterms.In Fig. 2, the fitted results of the fits using momentum-independent
Πρω (Fits Ia and IIa) and momentum-dependentΠρω (Fits Ib and IIb) are shown as red dotted lines and black solid lines, respectively. The fitted parameters as well asχ2/d.o.f. are listed in Table 1. It is intriguing to compare the results obtained from fits utilising momentum-independentΠρω and momentum-dependentΠρω for the same datasets. Specifically, we compare Fit Ia and Ib and Fit IIa and IIb. When examining pion form factors|Feeπ(s)|2 and|Fτπ(s)|2 , we observe that the differences between the theoretical predictions of the fits using momentum-independentΠρω and the corresponding ones using momentum-dependentΠρω are low. Furthermore, it should be noted that for the pion form factor|Feeπ(s)|2 in Fits Ia and Ib, the theoretical predictions are much higher than the KLOE data at ρ peak, and these deviations contribute significantly to their value ofχ2 . Thus, we conclude that the momentum-independentΠρω and momentum-dependentΠρω can describe the data well, and the discordances among different collaborations contribute significantly toχ2 values in the fits.Figure 2. (color online) Fit results of the pion form factor in the
e+e−→π+π− process (left panel) andτ→ντ2π process (right panel) in the energy region of 600–900 MeV. The data ofe+e− annihilation are considered from OLYA (Gray) [55], CMD (Yellow) [56], BaBar (Blue) [57], BESIII (Green) [58], KLOE (Cyan) [59], CLEO (Magenta) [60], and SND (Orange) [61] Collaborations. The τ decay data are taken from the ALEPH (Orange) [62], and CLEO (Green) [63] collaborations. Fits Ia and Ib fit all data sets excluding BaBar (top), and Fits IIa and IIb fit all data sets excluding KLOE (bottom). Fits Ia and IIa use momentum-independentΠρω and are denoted by the red dashed lines. Fits Ib and IIb use momentum-dependentΠρω and are denoted by the black solid lines. The vertical lines lie atMρ ,Mω , and2Mω−Mρ (from left to right).Fit Ia Fit Ib Fit IIa Fit IIb Mρ /MeV775.35±0.10 775.68±0.12 775.45±0.10 775.55±0.11 Gρ /MeV55.25±0.09 55.74±0.08 54.21±0.09 55.03±0.07 Fρ /MeV152.65±0.29 151.40±0.21 155.65±0.23 153.38±0.31 Mω /MeV782.59±0.13 782.68±0.12 782.39±0.11 782.45±0.11 Γω /MeV8.97±0.27 9.03±0.26 8.04±0.16 8.16±0.17 a/GeV−1 −0.0020±0.0150 −0.0054±0.0010 −0.1066±0.0152 −0.0067±0.0009 Re(Πρω)/MeV2 −3372±112 − −3799±85 − Im(Πρω)/MeV2 296±669 − −4544±704 − XrW/GeV−6 − −0.141±0.013 − −0.177±0.008 XrZ/GeV−4 − 0.195±0.016 − 0.303±0.007 XrR/GeV−2 − −0.081±0.006 − −0.133±0.003 χ2/d.o.f. 410.7(238−8)=1.79 405.8(238−9)=1.77 392.2(341−8)=1.18 394.6(341−9)=1.19 aππμ|[0.6,0.9]GeV[×1010] 367.72±1.07 367.80±2.92 375.41±1.03 375.29±2.21 Table 1. Fit results of the parameters. Fits Ia and Ib fit all data sets excluding BaBar, and Fits IIa and IIb fit all data sets excluding KLOE. Fits Ia and IIa use momentum-independent
Πρω , while Fits Ib and IIb use momentum-dependentΠρω .In the last line of Table 1, we provide the results of
aHVP,LOμ[π+π−] , evaluated between 0.6 GeV and 0.9 GeV. The differences between the results using the momentum-independentΠρω and the results using the momentum-dependentΠρω for the same datasets, namely the differences between Fits Ia and Ib and Fits IIa and IIb, respectively, are negligible.In Fig. 3, we plot the real and imaginary parts of the mixing amplitudes
Πρω(s) in Fits Ib and IIb. It is determined that the real part is dominant withinρ−ω mixing region. The real part in Fit IIb demonstrates a significant momentum dependence, whereas the real part in Fit Ib displays a smooth momentum dependence. Additionally, it should be noted that the real parts of the two fits nearly reach the same point ats1/2=Mω . In comparison to the real part, the imaginary part is rather small. Ats=M2ω , in Fit Ib the mixing amplitudeΠρω(M2ω)=(−3405.0+62.1i) MeV2 , and in Fit IIbΠρω(M2ω)=(−3316.3+113.7i) MeV2 . The minimal magnitude of the imaginary part aligns with the findings presented in Refs. [64, 65]. However, therein the effect of directωI→π+π− was not considered. It is worth mentioning that larger imaginary part is obtained in [13, 17] by using global color model and a chiral constituent quark model, respectively. By utilising our fitted parameter results, we proceed to calculate the ratio of the two-pion couplings associated with the isospin-pure ω and ρ.Figure 3. (color online) Momentum dependence of the mixing amplitudes
Πρω(s) . The black solid and red dot-dashed lines correspond to the real part ofΠρω(s) in Fits Ib and IIb, respectively. The blue dashed and magenta dash-dot-dotted lines correspond to the imaginary part ofΠρω(s) in Fits Ib and IIb, respectively. The vertical line lies atMω .G=gωIππgρIππ=4√2aB0(mu−md)Gρ.
(28) The results are
G=(2.1±1.1)×10−3 in Fit Ib, andG=(2.6±1.2)×10−3 in Fit IIb. It should be noted that the value of G is expected to be of the orderα=1/137 in Ref. [65]. The central values of our results of G are in good agreement with the expectation in Ref. [65], while they are lower than other two estimations, namelyG=(5.0±1.7model±1.0data)×10−2 in [66] andG=(3.47±0.64)×10−2 in [67]. As listed in Table 1, the differences inχ2/d.o.f. between the momentum-dependent fits and momentum-independent fits for the same data sets are minimal. Theχ2 of Fit IIa is slightly lower than theχ2 of Fit IIb. However, Fit IIa contains one less fitting parameter than Fit IIb. It can be observed that the magnitude of the imaginary part ofΠρω in Fit IIa is significantly greater than those in other three fits. In our framework, the imaginary part ofΠρω arises fromπ0γ andππ real intermediate states. By considering the decay widths ofω→π0γ andρ→π0γ , the imaginary part ofΠρω , contributed fromπ0γ intermediate state, can be estimated to be approximately −150 MeV2 [7, 65]. If the estimated ratio of the two-pion couplings of the isospin-pure ω and ρ are used:G∼α=1/137 [65], then theππ intermediate state contribution to the imaginary part ofΠρω can be obtained in the order of several hundred MeV2. In our momentum-dependent scheme, the imaginary part ofΠρω due toπ0γ andππ intermediate states are explicitly computed, and the numerical results of ImΠρω in Fits Ib and IIb are of the order of one hundred MeV2, as expected. However, in the momentum-independent Fits Ia and IIa, the imaginary part ofΠρω is a free fitting parameter. As listed in Table 1, the fitted results of ImΠρω and parameter "a" in Fit IIa are unreasonably high. The fitted results of ImΠρω and parameter "a" in Fit Ia exhibit large error bars. Consequently, we conclude that both momentum-independent and momentum-dependentρ−ω mixing schemes can describee+e−→π+π− data well. However, the momentum-dependentρ−ω mixing scheme is more self-consistent, especially given the reasonable imaginary part ofΠρω , which is extracted.We wish to emphasize that the direct
ωI→π+π− coupling is generally an unknown quantity, and it impactsFeeπ(s) in two ways, both through the third term in the first bracket of Eq. (21), appearing as real intermediate state in the contributions toΠρω and through the fourth term in that bracket. Conventionally, the directωI→π+π− is assumed to be neutralized ine+e−→π+π− due to the fact that ω and ρ are quasidegenerate and that 2π channel dominates the ρ decay [65]. Theoretical models that do not neglect directωI→π+π− coupling may be more comprehensive, especially given the availability of high-precision data available currently. It should be noted that in Refs. [5, 7], the pion form factor has been examined in a model-independent way using dispersion theory. Specifically,ρ−ω mixing is subsumed in one parameterϵω , which should contain a small imaginary part originating from the radiative intermediate states (with an estimated phase of approximately 4 degrees). Furthemrore, given that directωI→π+π− coupling is not considered in Refs. [5, 7],ϵω term is actually a combination ofρ−ω mixing and directωI→π+π− . Therefore, it cannot be directly compared toΠρω discussed in this context. (Ats=M2ω , ourΠρω(M2ω) in Fits Ib and IIb contains negative phase.) Nevertheless, the ratio between the on-ω-mass-shellγ∗→ω→ππ transition amplitude andγ∗→ρ→ππ transition amplitude (withoutππ final state interaction) should be model independent. Withs=M2ω , the ratio between the second term and first term in Eq. (2.5) of [7] yieldsRωρ=Amplitude (γ∗→ω→ππ)/Amplitude (γ∗→ρ→ππ)=(0.178±0.003)×ei(4.66±1.13)∘ , using Reϵω=(1.97±0.03)×10−3 andδϵ=(4.5±1.2)∘ obtained therein. It can be observed that the difference between the phase ofRρω(M2ω) andδϵ is minimal. In this study, the ratio between the sum of the third term and fourth terms and the sum of the first and second terms in the first bracket of Eq. (21) predictsRωρ=(0.155±0.002)×ei(5.80±1.71)∘ andRωρ=(0.150±0.002)×ei(3.67±1.71)∘ in Fit Ib and IIb, respectively. It can be observed that our results ofRωρ approximately agree with that in [7].Using the central values of the fitted parameters of our best fit (Fit IIb) in Table 1, we calculate the decay width of
ω→π+π− Γω→π+π−=1192πF4(M2ω−4m2π)32×(exp[−M2ω96π2F2(Re[A[mπ,Mρ,M2ω]+12A[mK,Mρ,M2ω]])])2×|8√2B0(mu−md)a+2GρΠρω(M2ω)M2ω−M2ρ−i(MωΓω−MρΓρ)|2=0.013∣(0.29)+(−0.22+3.35i)∣2. (29) Based on Eq. (29), we can determine that the first term due to direct
ωI→π+π− is smaller than the second term due toρ−ω mixing by an order of magnitude. Within 1σ uncertainties, our theoretical value of the branching fraction isB(ω→π+π−)=(1.48±0.10)×10−2 , which is consistent with the values provided in PDG [44] and with those reported in the recent dispersive analysis [68].Regarding the mass of the ω meson, previous studies [5, 7, 57] indicated that the result extracted from
e+e−→π+π− is substantially lower than the current PDG average [44], which primarily relies one+e−→3π ande+e−→π0γ experiments. The discrepancy amounts to approximately 1 MeV, corresponding to around 5 σ considering the current precision. It has been observed that the fitted value forMω and phase ofϵω are strongly correlated [5, 7, 57]. It should be noted that directωI→π+π− coupling has not been considered in [5, 7, 57]. As indicated in Table 1 above, our fitted results for the mass of ω are in good agreement with the value in PDG:Mω=782.66±0.13 MeV, and this agreement remains unaffected by the inclusion or exclusion of the momentum dependence ofΠρω . Furthermore, we can observe that a strong correlation (80%) exists between parameter "a," which quantifies the directωI→π+π− coupling, and the mass of ω. As mentioned earlier, the directωI→π+π− coupling influences the imaginary part and real part of the amplitude, and thereby, affects the phase ofRρω(M2ω) . It should be noted that the phase ofRρω(M2ω) approximately agrees with the phase ofϵω . Thus, our observations align with with those in Refs. [5, 7, 57]. Hence, a strong correlation exists between the mass of the omega meson and phase ofϵω . Our findings suggest that the inclusion of directωI→π+π− coupling is likely crucial in the analysis aimed at extracting the ω mass from thee+e−→π+π− process. -
We utilized the resonance chiral theory to examine
ρ−ω mixing. Specifically, we analyzed the impact of the momentum dependence ofρ−ω mixing on describing the pion vector form factor in thee+e−→π+π− process and its contribution to the anomalous magnetic moment of muonaμ . The incorporation of momentum dependence arises from the calculation of loop contributions, which corresponds to the next-to-leading orders in1/NC expansion. Based on fitting to the data ofe+e−→π+π− andτ→ντ2π processes within the energy range of 600–900 MeV and decay width ofω→π+π− , we determine that theρ−ω mixing amplitude is dominated by its real part, and its imaginary part is relatively small. Although momentum-independent and momentum-dependentρ−ω mixing schemes yield satisfactory data descriptions, the latter proves to be more self-consistent due to the reasonable imaginary part of the mixing matrix elementΠρω . Regarding the contribution to anomalous magnetic moment of muonaππμ|[0.6,0.9]GeV , the results obtained from fits considering the momentum-dependentρ−ω mixing amplitude align well with those from corresponding fits that exclude the momentum dependence ofρ−ω mixing, within the margin of error. Additionally, we provide the ratio of the isospin-pure ω and ρ two-pion couplings, denoted asG=gωIππ/gρIππ , and observe thatρ−ω mixing plays a crucial role in the decay width ofω→π+π− . Furthermore, we ascertain that including directωI→π+π− coupling is essential in analyzing the extraction of the mass of the ω meson from thee+e−→π+π− process. -
We are grateful to Pablo Roig for helpful discussions and valuable suggestions.
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Using
ρKˉK andωKˉK vertexes obtained via the Lagrangian in Eq. (12):iGV/√2⟨Vμνuμuν⟩=iGV/F2ρ0μν(∂μK+∂νK−−∂μK0∂νˉK0)+iGV/F2ωμν(∂μK+∂νK−+∂μK0∂νˉK0)+... , we can calculate the charged and neutral kaon loops contribution to amplitudeΠkaon,chargedρω=G2Vp4192F4π2{(1−6m2K+p2)(λ∞−lnm2K+μ2)+53−8m2K+p2−σ3K+ln(σK++1σK+−1)},
and
Πkaon,neutralρω=−G2Vp4192F4π2{(1−6m2K0p2)(λ∞−lnm2K0μ2)+53−8m2K0p2−σ3K0ln(σK0+1σK0−1)},
where
σP≡√1−4m2P/p2 andλ∞≡1ϵ−γE+1+ln4π withϵ=2−d2 andγE is the Euler constant.The persistence of a non-zero structure function arises from the mass difference between the charged and neutral kaons as described:
S(c)ρω=1mu−md(Πkaon,chargedρω+Πkaon,neutralρω).
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For the isospin-violating vertex of
ωI→π+π− , we construct the Lagrangian:LωI→π+π−=a1i⟨Vμν{χ+,uμuν}⟩+a2i⟨Vμνuμχ+uν⟩=(a1−12a2)8√2B0iF2Δudωαβπ+απ−β.
For convenience, we define the combination
a≡a1−12a2. Theππ -loop contribution to the structure function can be calculated as follows:S(d)ρω=√2GVB0a12F4π2p4{(1−6m2πp2)(λ∞−lnm2πμ2)+53−8m2πp2−σ3πln(σπ+1σπ−1)}.
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According to the Lorentz, P and C invariances, the Lagrangian corresponding to the interaction of
ωIρIππ can be constructed as follows:LωIρIPP=b1⟨VμνVμν(uαuαχ++χ+uαuα)⟩+b2⟨VμνVμνuαχ+uα⟩+b3⟨Vμνχ+Vμνuαuα⟩+b4⟨VμνuαVμν(χ+uα+uαχ+)⟩+b5⟨VμαVναuμuνχ++VναVμαχ+uνuμ⟩+b6⟨VμαVναuμχ+uν⟩+b7⟨Vμαχ+Vναuμuν⟩+b8⟨VμαVναuνuμχ++VναVμαχ+uμuν⟩+b9⟨VμαVναuνχ+uμ⟩+b10⟨Vμαχ+Vναuνuμ⟩+b11⟨VμαuαVμβuβχ++VμβuαVμαχ+uβ⟩+b12⟨VμαuαVμβχ+uβ+VμβuαVμαuβχ+⟩+b13⟨VμαuβVμβuαχ++VμβuβVμαχ+uα⟩+b14⟨VμαuβVμβχ+uα+VμβuβVμαuαχ+⟩+g1i⟨VμνVμν(uα∇αχ−+∇αχ−uα)⟩+g2i⟨VμνuαVμν∇αχ−⟩+g3i⟨VμβVμαuβ∇αχ−+VμαVμβ∇αχ−uβ⟩+g4i⟨VμβVμα∇αχ−uβ+VμαVμβuβ∇αχ−⟩+g5i⟨VμβuβVμα∇αχ−+VμαuβVμβ∇αχ−⟩+λVV6⟨VμνVμνχ+⟩.
For simplicity, we define the combinations,
h1≡6b1−b2+3b3+b4−2g1−g2,h2≡4b5−b6+3b7+4b8−b9+3b10+2b11+2b12+2b13+2b14−2g3−2g4−2g5.
The mass difference between the charged and neutral pions in the internal lines of loops can be disregarded due to its higher-order magnitude beyond our scope of consideration. Consequently, the expanded expression of Lagrangian (A6) can be simplified as follows:
LωIρIππ=4B0F2h1(mu−md)ρIμνωμνπαπα−2B0F2λVV6(mu−md)ρIμνωμνπ2+4B0F2h2(mu−md)ρIμαωναπμπν.
With the aforementioned Lagrangian, the π-tadpole contribution to
ρ−ω mixing can be derived as follows:S(e)ρω=−m2πB08π2F2{(−16λVV6+4h1m2π+h2m2π)×(λ∞−lnm2πμ2)+h22m2π}.
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In the loop diagrams (f)-(i), the resonance chiral effective Lagrangian describing vector-photon-pseudoscalar (VJP) and vector-vector-pseudoscalar (VVP) vertices are provided in Ref. [69]:
LVJP=c1MVϵμνρσ⟨{Vμν,fρα+}∇αuσ⟩+c2MVϵμνρσ⟨{Vμα,fρσ+}∇αuν⟩+ic3MVϵμνρσ⟨{Vμν,fρσ+}χ−⟩+ic4MVϵμνρσ⟨Vμν[fρσ−,χ+]⟩+c5MVϵμνρσ⟨{∇αVμν,fρα+}uσ⟩+c6MVϵμνρσ⟨{∇αVμα,fρσ+}uν⟩+c7MVϵμνρσ⟨{∇σVμν,fρα+}uα⟩,
and
LVVP=d1ϵμνρσ⟨{Vμν,Vρα}∇αuσ⟩+id2ϵμνρσ⟨{Vμν,Vρσ}χ−⟩
+d3ϵμνρσ⟨{∇αVμν,Vρα}uσ⟩+d4ϵμνρσ⟨{∇σVμν,Vρα}uα⟩.
The couplings involved, or their combinations, can be estimated by matching the leading operator product expansion of
⟨VVP⟩ Green function to the same quantity evaluated within RχT. This procedure leads to high energy constraints on the resonance couplings [69]:4c3+c1=0,c1−c2+c5=0,c5−c6=Nc64π2MV√2FV,d1+8d2=−Nc64π2M2VF2V+F24F2V,d3=−Nc64π2M2VF2V+F28F2V.
Using the the effective vertices stated in Eqs. (A10) and (A11), the
πγ loop contribution, i.e., the summation of the loops diagrams (f)-(i), can be expressed asiΠρωϵρμϵμω=1p2∫dnk(2π)n−ik2i(p−k)2−m2π×[(k⋅p)2ϵμρϵωμ−k2p2ϵμρϵωμ+p2k⋅ϵρk⋅ϵω]×{−32e23M2VF2[c1(p−k)⋅k−c2(p−k)⋅p−4c3m2π−c5p⋅k+c6p2]2−16√2FVe23MVF2[1M2ω−k2+1M2ρ−k2]×[d1(p−k)2+8d2m2π+2d3p⋅k]×[c1(p−k)⋅k−c2(p−k)⋅p−4c3m2π−c5p⋅k+c6p2]−16F2Ve23F2(M2ρ−k2)(M2ω−k2)×[d1(p−k)2+8d2m2π+2d3p⋅k]2}.
The subsequent calculation is straightforward. However, the result of the extracted electromagnetic structure function
Eπγρω≡E(f)ρω+E(g)ρω+E(h)ρω+E(i)ρω is too extensive to present here. It should be noted that in our numerical computation we employ the high energy constraints in Eq. (A12) along with the fitted parameters provided in Ref. [25]. Therefore, all the parameters involved inEπγρω are known. -
Given that the ω meson predominantly decays into the three-pion state, its two-loop self energy diagram contributes beyond NLO in
1/NC and is not relevant for our current consideration. The self-energy diagrams for ρ meson are depicted in Fig. A1. The Lagrangian required to renormalize ρ meson one-loop self-energy has been provided in Ref. [38],L4Y=XY12⟨∇2Vμν{∇ν,∇σ}Vμσ⟩+XY24⟨{∇ν,∇α}Vμν{∇σ,∇α}Vμσ⟩+XY34⟨{∇σ,∇α}Vμν{∇ν,∇α}Vμσ⟩.
Specifically, only the combination of couplings
XY≡XY1+XY2+XY3≡XrY+δXY is relevant for this purpose. Using Lagrangians in Eqs. (12) and (A14), ρ self-energy takes the form:Σρ(p2)=−G2V48F4π2p4{(1−6m2πp2)(λ∞−lnm2πμ2)−8m2πp2−σ3πln(σπ+1σπ−1)+(1−6m2Kp2)(λ∞−lnm2Kμ2)−8m2Kp2−σ3Kln(σK+1σK−1)+103}−XYp4.
The renormalized ρ mass fulfills:
M2ρ=M2V+Σρ(M2ρ).
Given that physical
Mρ is finite, the following holds:δXY=−G2V48F4M2ρπ2(1−6m2πM2ρ−6m2KM2ρ)λ∞.
The wave-function renormalization constant of ρ meson is obtained from:
Zρ=1+∂Σρ(p2)∂p2|p2=M2ρ.
In our calculation of
ρ−ω mixing, the tree amplitudes can only absorb the ultraviolet divergence that is proportional top0 . To neutralizeO(p2) ,O(p4) , andO(p6) ultraviolet divergence originating from loop contributionS(c)ρω ,S(d)ρω ,S(e)ρω , andEπγρω , we construct the counterterms as follows:Lct=YA⟨VμνVμνχ+⟩−12YB⟨∇λVλμ∇νVνμχ+⟩+YC12⟨∇2Vμν{χ+,{∇ν,∇σ}Vμσ}⟩+YC24⟨{∇ν,∇α}Vμν{χ+,{∇σ,∇α}Vμσ}⟩+YC34⟨{∇σ,∇α}Vμν{χ+,{∇ν,∇α}Vμσ}⟩+ZAFV2√2⟨Vμνfμν+⟩+ZBFV2√2⟨Vμν∇2fμν+⟩+ZCFV2√2⟨Vμν∇4fμν+⟩+ZDFV2√2⟨Vμν∇6fμν+⟩.
We adopt
¯MS−1 subtraction scheme and absorb the divergent pieces proportional toλ∞ by the bare couplings in the counterterms. Consequently, the remanent finite pieces of counterterms can be expressed asΠctρω=XrWp6+XrZp4+XrRp2 ,
with
XrW≡8παFρFω3(ZrD+ZrBZrC) ,XrZ≡4παFρFω3(2ZrC+ZrB2)+16Mρ(mu−md)(YrC1+YrC2+YrC3) ,XrR≡8παFρFω3ZrB−4Mρ(mu−md)YrB .
In summary, at the NLO in
1/NC , the UV-renormalized mixing amplitude is as follows:Πrρω(p2)=S(a)ρω√Zρ+ˉS(c)ρω+ˉS(d)ρω+ˉS(e)ρω+¯Eπγρω(p2)+XrWp6+XrZp4+XrRp2,
where a bar denotes that the divergences are subtracted.
As discussed in Ref. [41], the mixing amplitude should vanish as
p2→0 . Thus, the final expression of the renormalized mixing amplitude is obtained as follows:Πρω(p2)=Πrρω(p2)−Πrρω(0) ,
where an additional finite shift is imposed to guarantee that constraint
Πρω(0)=0 is satisfied. It should be noted that due to the finite shift performed in Eq. (A23), our numerical calculation is actually independent of the couplingXrY . In our numerical computation, scale μ will be set toMρ , and we use(mu−md)=−2.49 MeV provided by PDG [44].
