# π

Authors: M. I. Krivoruchenko
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

The generalized Ward-Takahashi identity (gWTI) in the pion sector for broken isotopic symmetry is derived and used for the model-independent calculation of the longitudinal form factorf- of the πe3 vector vertex. The on-shell f- is found to be proportional to the mass difference of the pions and the difference between the vector isospin T=1 and scalar isospin T=2 pion radii. A numerical estimate of the form factor yields a value two times higher than the previous estimate from the quark model. Off-shell form factors are known to be ambiguous because of the gauge dependence and the freedom in the parameterization of the fields. The near-mass-shell f- appears to be an exception, allowing for experimental verification of the consequences of the gWTI. We calculate the near-mass-shell f- using the gWTI and dispersion techniques. The results are discussed in the context of the conservation of vector current (CVC) hypothesis.


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## Body

## 1. Introduction

Theπ+→π0e+νe decay (πe3) is one of the main semileptonic electroweak processes. The vector nature of the transition, its simple kinematics, and the precise measurement of the partial width make this decay particularly attractive for testing the Standard Model.The decay amplitude is proportional to theVud element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Radiative corrections and pion structure effects in the πe3 decay have been calculated with high accuracy [1–6], sufficient for verification of the unitarity of the CKM matrix. The experimental data, however, are not yet sufficiently precise for this purpose [7, 8].Measurements of theπe3 decay are also motivated by the possibility of testing the conservation of vector current (CVC) in the meson sector. The CVC hypothesis [9–11] suggests that the isovector component of the electromagnetic current and the charged components of the weak vector current belong to the same isospin triplet. In the limit of exact isotopic symmetry, conservation of the electromagnetic current implies conservation of the weak vector current.Off the mass shell, the CVC is equivalent to the Ward-Takahashi identity (WTI) for the isospinSU(2) group. The WTI, however, is of greater generality and leads to useful relationships between off-shell form factors, including those that vanish when some of the external legs are on shell. The violation of isotopic symmetry, associated with the small mass difference between the up and down quarks and the electromagnetic and weak interactions, results in the nonconservation of the charge-changing components of the weak vector current. For the broken symmetry, the CVC condition and the WTI are replaced by partial CVC and the generalized WTI (gWTI), which are especially sensitive to the pattern of isotopic symmetry breaking.Off-shell form factors enter the descriptions of nucleon knockout reactions [12, 13] and bremsstrahlung reactions [14]. As a result of shifting away from the mass shell, the electromagnetic currents of bound nucleons are different from the free currents, which also leads to observable effects [15].The parameterization of the degrees of freedom associated with the pion field can be performed in various ways, which produces off-shell ambiguity in the amplitudes. The on-shell form factors are related to the asymptotic states and are uniquely defined. This statement is known as the equivalence theorem (ET) [16–18]. The off-shell form factors, although they contribute to the physical amplitudes, depend on this parameterization and cannot be measured experimentally [19, 20].We report a notable exception to this rule. The longitudinal part of theπe3 vertex is shown to be uniquely defined near the mass shell and fundamentally accessible to measurements, and therefore, it can be investigated for its consistency with the gWTI.In quark models, pions are usually poorly described because of their Goldstone boson nature. As an application of the gWTI, we derive a model-independent expression and provide a numerical estimate for the longitudinal form factorf-. The structure of this paper is as follows. Section 2 begins with a discussion of the constraints imposed by the WTI of the symmetry group U(1) on the electromagnetic pion form factors. In Section 3, a generalization of the WTI for broken isotopic symmetry is derived, and in Section 4, it is used to find the relationships between weak vector form factors. The near-mass-shell form factor f- is then expressed in terms of the pion mass difference and the pion radii in the isospin T=1 and 2 channels. In Section 5, we focus on extracting the isospin T=2 pion radius from the experimental data on ππ-scattering phase shifts and numerically evaluate f-. Finally, we conclude in Section 6 with a summary and a discussion.


## 2.U(1) Vector Vertex

The on-shell conserved vector current of a charged scalar particle is determined by one form factor. Off the mass shell, there are two form factors. In the most general case, the current can be written as follows:(1)Γμp′,p=p′+pμF1+qμp′2-p2F2,where q=p′-p is the momentum transfer. The form factors Fi are symmetric functions of p′2 and p2 and arbitrary functions of q2 and the physical mass m of the charged pion. The factor p′2-p2 in the second term is added to ensure the negative C parity of the current. The form factor decomposition (1) arises in scalar quantum electrodynamics (QED) (see, e.g., [21]) and in chiral perturbation theory (χPT) (see, e.g., [22]). The WTI of the U(1) symmetry group establishes a relationship between F1 and F2 that is identical to that of (4).Isotopic symmetry implies that the mass difference of nonstrange quarks can be neglected and that the electromagnetic and weak interactions are switched off beyond tree level. In this limit, the weak interaction is described by an isovector currentΓμa(p′,p)=TαΓμ(p′,p), where the Tα are isospin generators. As a result, exact CVC occurs: a dressed vertex forms an isospin triplet, and the weak vector current is conserved.The WTI associated with theU(1) symmetry group imposes a constraint on the vertex (see, e.g., [23]):(2)Δ-1p′-Δ-1p=qμΓμp′,p,where Δ(p)=p2-m2-Σ(p2,m) is the renormalized pion propagator. The self-energy operator satisfies(3)Σm2,m=0,∂∂p2Σp2,mp2=m2=0.Combining (2) with (1) yields(4)q2F2p′2,p2,q2=Δ-1p′-Δ-1pp′2-p2-F1p′2,p2,q2.In the limit p′2=p2=m2, we obtain(5)F2m2,m2,q2=1-F1m2,m2,q2q2.In the vicinity of q2=0, the form factor F1 can be expanded to give(6)F2m2,m2,0=-16r2v,where r2v is the vector charge radius.The pion form factorF1(p′2,p2,q2), which describes strong interaction effects, is an analytic function of the complex variable p2 at least in the circle p2-m2<8m2. The nearest to the mass shell singularity at p2=9m2 is caused by the three-pion threshold. A Taylor series in powers of p2-m2 converges within that circle. The same is true for the variable p′2. A Taylor series in powers of q2 converges for q2<4m2, where the radius of convergence is determined by the two-pion threshold.The equivalence of the Coulomb and Lorentz gauges in QED was rigorously proved in [24, 25]. On shell, the amplitudes are gauge invariant, whereas the off-shell dependence on the gauge persists. F1 and, by virtue of (5), F2 are thus uniquely defined, when both legs of the charged pion are on the mass shell. To first order in the displacement from the mass shell, the longitudinal component of the vertex is also gauge invariant, as is evident from (1). Aside from these cases, Γμ(p′,p) depends on the gauge and cannot be measured.F
2 with two on-shell legs p′2=p2=m2 does not contribute to the current. Off the mass shell, however, F2 does contribute, and its contribution is uniquely determined by the WTI. Isotopic rotation of F2 is not sufficient to obtain a full weak-interaction vertex. We show that the violation of isotopic symmetry generates an isospin T=2 contribution that is unrelated to isotopic rotation.


## 3. Generalized Ward-Takahashi Identity for Broken Isotopic Symmetry

In order to find the longitudinal form factor of the weak vector current, we derive a generalization of the WTI that is associated with the replacement of the exactU(1) symmetry by the broken SU(2) symmetry.Let us consider the variation of the pion propagator(7)iΔαβx′,x=0Tφαx′φβx0under the SU(2) transformation(8)φ⟶φ′=e-iχφ.In terms of a Cartesian basis, the pion field φα(x) is Hermitian, and χ is an infinitesimal imaginary antisymmetric matrix that can be expanded in the group generators with real coefficients χ=∑a‍χaTa. In the conventional normalization, TrTaTb=2δab. The variation of the pion propagator is given by(9)iδΔx′,x=χx′Δx′,x-Δx′,xχx.In matrix notation, Δ-1iδΔΔ-1=-iδΔ-1=[Δ-1,χ].Independent calculation ofδΔ involves the effective Lagrangian, which we consider to be of the form(10)Leff=LeffDμφ†,Dμφ,φ†,φ,where iDμ=i∂μ-eAμ-Bμ, Aμ is the electromagnetic field, e=T3, and Bμ=BμaTa is the weak vector field associated with Z0 and W± bosons. In a Cartesian basis, Dμ is real. The isotopic invariance in Leff is broken by the mass term φ†m2φ with [m2,Ta]≠0 and by the electromagnetic and weak interactions. In other respects, Leff is an arbitrary function, which reflects the ambiguity in the parameterization of the pion field.The combination ofeAμ+Bμ ensures the implementation of the CVC condition at a bare interaction vertex: the electromagnetic current coincides with the third component of the weak vector current by construction. If the isotopic symmetry in Leff would not have been broken, the CVC condition could also hold for a dressed vertex, and the weak vector current could also be conserved. In the case of this broken symmetry, however, the dressed vertex acquires an admixture of tensor components, while the charge-changing weak vector current is no longer conserved. The WTI can be viewed as a generalization of the CVC for an off-shell vertex. It permits generalizations for broken symmetries; thus, when the isotopic symmetry is broken, the CVC becomes partial CVC, whereas the WTI of the exact isotopic symmetry turns into the WTI of the broken isotopic symmetry.Transformation (8) is equivalent to the compensating transformation in Leff:(11)φ⟶φ′=eiχφ,which generates variations of the pion mass term and the vector fields in Leff:(12)m2⟶m′2=m2+m2,iχ,eAμ⟶eAμ′=eAμ+e,iχAμ,Bμ⟶Bμ′=Bμ+∂μχ+Bμ,iχ.As a result, Leff acquires the correction(13)δLeff=-TrJμ∂μχ+e,iχAμ+Bμ,iχ-TrJm2,iχ,where(14)Jμβα=-∂Leff∂Bαμβ,Jβα=-∂Leff∂m2αβare the vector and scalar currents of the pion.The response of the pion propagator to the compensating transformation is(15)iδΔx′,x=0Tφx′φ~xi∫d4yδLeffy0=∫d4yd4z′d4ziΔx′,z′·-iΓμaz′,z,y∂μχay+Θaz′,z,yχay+Ωaz′,z,yχay·iΔz,x,with vertex functions defined by(16)Γμaz′,z,y=-∫d4x′d4xΔ-1z′,x′0Tφx′·φ~xTrJμyTa0Δ-1x,z,Θaz′,z,y=-∫d4x′d4xΔ-1z′,x′0Tφx′·φ~xTrJym2,Ta0Δ-1x,z,Ωaz′,z,y=-∫d4x′d4xΔ-1z′,x′0Tφx′·φ~xTrJμyeAμy+Bμy,Ta0·Δ-1x,z.Now, let us consider momentum space:(17)2π4δ4p′-p-qΓμap′,p=∫d4zd4z′d4yeip′z′-ipz-iqyΓμaz′,z,y,2π4δ4p′-p-qΘap′,p=∫d4zd4z′d4yeip′z′-ipz-iqyΘaz′,z,y,2π4δ4p′-p-qΩap′,p=∫d4zd4z′d4yeip′z′-ipz-iqyΩaz′,z,y.Combining (9) and (15) leads to(18)Δ-1p′Ta-TaΔ-1p=qμΓμap′,p-Θap′,p-Ωap′,p.This equation generalizes the WTI for the broken SU(2) symmetry. It is shown graphically in Figure 1.Figure 1
Schematic representation of the gWTI for the pion propagator with broken isotopic symmetry. The blobs denote the dressed vertices. The first one is a Lorentz vector entering the pionβ decay. The last three are Lorentz scalars; they are responsible for the symmetry breaking by the pion masses and the electromagnetic and weak interactions.


## 4.SU(2) Vector Vertex

In a Cartesian basis,Δ(x,y)=Δ~(y,z), which means that Δ(p)=Δ~(-p). Given that Δ-1(p)=p2-m2-Σ(p2), the pion propagator is symmetric under transposition of the isospin indices: Δ(p)=Δ~(p).The isospin content of a3×3 matrix M can be determined using projection operators:(19)MT=0=12TrM,MaT=1=12TrMTa,MabT=2=12TrMTaTb+TbTa-23δabTT+1.Among symmetric matrices, there are only isospin-0 or isospin-2 matrices that are proportional to the unit matrix or to quadratic forms of Ta. The allowed pion mass operator is therefore m2+Σ(m2)=C0+C1(T3)2, where the Ci are numbers. The charged pions are degenerate in mass; therefore, C invariance holds.The vertex functions obey(20)Γμap′,p=Γ~μa-p,-p′,Θap′,p=Θ~a-p,-p′,Ωap′,p=Ω~a-p,-p′.The functions Θa(p′,p) and Ωa(p′,p) depend on the variables p′2,p2, and q2. Symmetry (antisymmetry) in the isospin indices implies that vertex function is symmetric (antisymmetric) under the permutation of p′2 and p2. Functions that are symmetric in isospin describe states with isospin T=0 and 2, whereas those that are antisymmetric describe isospin T=1 states. Similar properties characterize the form factors of Γμa(p′,p).The vertex functions can be expanded in scalar functionsFi±a that are symmetric in p′2 and p2. The lower index ± indicates the symmetry with respect to permutation of the isospin indices: F~i±a=±Fi±a (i=1,2,3). The expansion takes the form(21)Γμap′,p=p′+pμF1-a+p′2-p2F1+a+qμp′2-p2F2-a+F2+a,(22)Θap′,p+Ωap′,p=F3+a+p′2-p2F3-a.If there was no isospin symmetry breaking, we could haveFi-a=TaFi-, as there are no other SU(2) generators, and Fi+a=0, implying that Γμa(m2,m2,q2)∝Ta and Θa(m2,m2,q2)+Ωa(m2,m2,q2)=0.The gWTI (18) can be split into isospin-symmetric and isospin-antisymmetric parts:(23)-m2+12Σp′+Σp,Ta=p′2-p22F1+a+q2F2+a-F3+a,(24)Ta-12Σp′-Σpp′2-p2,Ta=F1-a+q2F2-a-F3-a,where [,] is a commutator and {,} is an anticommutator.Consider some of the consequences of (23). For p′=p, q=0, we obtain(25)m2+Σp2,Ta=F3+ap2,p2,0.In a cyclic basis, the SU(2) generators are equal to T±1=∓(T1±iT2)/2 and T0=T3. The left-hand side of (25), when sandwiched between the initial and final states in the pion β decay, gives the pion mass difference:(26)π0m2+Σp2,T-1π+=mf2-mi2≡-Δmπ2,where mπ0=mf and mπ+=mi. In the more general case of p′2=p2 and q≠0, one finds(27)m2+Σp2,Ta=-q2F2+ap2,p2,q2+F3+ap2,p2,q2.The trace of the commutator of finite-dimensional matrices vanishes, and therefore, the left-hand side of (25) and (27) describes a state of total isospin 2. The isospin-zero component of (27) satisfies the equation q2(F2+a)T=0-(F3+a)T=0=0. At zero momentum transfer, (F3+a)T=0=0, and the overall normalization of the T=0 form factors is not fixed. In the t channel of the pion β decay, the state π+π0 has the T3=1 isospin projection, so the T=0 channel is of no interest.As a consequence of elastic unitarity and analyticity, theq2 dependence of the on-mass-shell form factor is determined by the Jost function DJT(q2) that can be constructed in terms of the phase shift in the corresponding channel (see, e.g., [23]). We consider an S-wave because the last three vertices in δLeff induced by the transformation (11) do not contain derivatives of χa, and thus, the S-wave T=2 channel of the pion-pion scattering is relevant. We therefore write(28)F3+amf2,mi2,q2T=2=F3+amf2,mi2,0DJT=2q2,where DJT=2(0)=1. The value of F3+a(mf2,mi2,0) is proportional to the mass splitting in the pion multiplet. We restrict ourselves to the first order in Δmπ2. Accordingly, the physical pion masses in the arguments of the vertex functions can be replaced by a mean value, for example, μ=(mf+mi)/2, provided that these functions, such as F2+a and F3+a, are already small. Using (25)–(28), we obtain (29)F3+aμ2,μ2,q2T=2=m2+Σμ2,TaDJT=2q2,F2+aμ2,μ2,q2T=2=m2+Σμ2,TaF2+μ2,μ2,q2,where(30)F2+μ2,μ2,q2=1/DJT=2q2-1q2.Let us consider the consequences of (24). Equation (21) shows that F2-a can be evaluated with the required accuracy in the limit of exact isotopic symmetry with [m2+Σ(p),Ta]=Fi+a=F3±a=0 and F1-a=TaF1-. On the mass shell, moreover, Σ(p)=0, and thus,(31)F2-μ2,μ2,q2=1-F1-μ2,μ2,q2q2.To the leading order of the expansion in Δmπ2, the form factor F2-a is determined by the isotopic rotation of the electromagnetic form factor (5).The form factorsF2±, as defined by (30) and (31), are not singular at q2=0. For small q2, they are determined by the pion radii. The on-shell weak vector current is usually parameterized in the form(32)π0p′d¯γμ1-γ5uπ+p=2p′+pμf++qμf-,where qμ=(p′-p)μ.The exact CVC condition implies(33)f-=0.Partial CVC, as follows from the comparison of (21) and (32), yields(34)f-=mπ02-mπ+2F2-μ2,μ2,0+F2+μ2,μ2,0=mπ+2-mπ026r2vT=1-r2sT=2.The isovector part is recovered from the electromagnetic vertex, whereas the isotensor part is independent of it. Remarkably, for a dressed vertex, the W± boson, being a member of the weak isospin triplet, is coupled to both the strong isospin triplet and the strong isospin quintet.Equation (34) can also be derived in a simpler and more direct way, analogously to the standard analysis of Kl3 decays [26].First, we introduce the form factors(35)fvq2=F1-mf2,mi2,q2,(36)fsq2=F1-mf2,mi2,q2+q2F2-mf2,mi2,q2+F2+mf2,mi2,q2,with f+=fv(0)=fs(0)=1. In the t channel, the space-like component of the current is proportional to fv(q2), whereas the time-like component is proportional to fs(q2). By applying the t-channel unitarity condition for the form factors, we find that fv is determined by the J=T=1 scattering amplitude, whereas fs is determined by the J=0,T=2 scattering amplitude, where J is the angular momentum of the pions. fv and fs can thus be identified as vector and scalar form factors, respectively, which is consistent with the gWTI.Second, we assume an expansion inq2 for both fv(q2) and fs(q2). This assumption conveys the analyticity of the form factors. Equation (36) leads to(37)1+16r2sT=2q2=1+16r2vT=1q2+q2mf2-mi2f-+Oq4,such that we recover (34) and, in addition, observe that it is exact for the on-shell case. However, the connection to the CVC condition remains hidden in this simplified approach.The nature of the form factorfs can also be clarified with the aid of (32). Taking the scalar product of both sides of the equation with the vector qμ yields(38)π0p′i∂μd¯γμ1-γ5uπ+p=-Δmπ22fsq2.On the left-hand side, we obtain the form factor, which is a Lorentz scalar. Equation (38) is the formal statement of the nonconservation of the lowering component of the weak vector current.Equation (34) can therefore be derived using either the gWTI or the kinematic decomposition of (35) and (36). Because recourse to the gWTI is optional, (34) is less interesting for testing the gWTI. We therefore analyze the form factors off the mass shell.The WTI of exact symmetry implies, to first order in the displacementp′2-p2 and for low momentum transfers,(39)f-=-p′2-p26r2vT=1.The WTI of broken symmetry implies, to first order inΔmπ2 and the displacement p′2-p2 and for low momentum transfers,(40)f-=-p′2-p26r2vT=1+mπ02-mπ+26r2sT=2.Theπ+→π0e+νeγ decay rate depends on the near-mass-shell behavior of the form factor f-. Any such reaction may serve for testing the gWTI.For large momentum transfers, the pion radii in (34), (39), and (40) should be replaced with the form factors of (30) and (31). Expressions (33), (34), (39), and (40) correspond to the various versions of the CVC condition.The on-shell form factorF1-a is independent of both the gauge and the parameterization. By virtue of (30) and (31), the on-shell form factors F2±a are also uniquely defined. The longitudinal component of the vertex (21) contains the factor p′2-p2 in F2-a, whereas F2+a has smallness of O(Δmπ2). We thus conclude that the longitudinal component of Γμa(p′,p) is uniquely defined in the neighborhood of the mass shell to first order in the pion mass splitting and the displacement p′2-p2. In the neighborhood of the mass shell, the form factor f- thus avoids the general rule [16–20, 24, 25] that states that off-shell amplitudes are ambiguous.The results obtained for the near-mass-shell representation off- in terms of the physical masses and radii of the pions exhibit explicit independence of the gauge and the parameterization of the pion field.From a mathematical point of view, one can speak of the uniqueness of the longitudinal component ofΓμa(p′,p) in an infinitesimal neighborhood of the mass shell or, equivalently, of the uniqueness of the longitudinal component of Γμa(p′,p) and its first derivatives with respect to pν′ and pν on the mass shell to first order in the pion mass splitting.The transverse component of the vertex does not exhibit any specific behavior, and thus, we considerf+ to be ambiguous off shell.The generality of the relationships (34) and (40) is influenced by only the CVC condition at a bare interaction vertex. This condition is satisfied in the Standard Model, so any violation of these relationships can be interpreted as an indication for new physics at or above the electroweak scale.


## 5. Numerical Estimates

In this section, we employ the dispersion techniques for numerical estimation of the pion radius〈r2〉sT=2, entering (34) and (40), and of the form factor f-.In the scattering ofT=2 pions, the inelastic channels do not become relevant until an invariant mass of 1.2 GeV [27–29]. The elastic scattering amplitude is determined by the phase shift through the Jost function:(41)DJTs=exp-sπ∫4μ2+∞δTs′ds′s′s′-s.The unitarity relation for the form factor implies (cf. (28))(42)F3+sT=PsDJTs,where P(s) is a finite-degree polynomial. The degree of P(s) can be fixed by the quark-counting rules, which, in our case, take the form F2±(s)~1/s for s→-∞. The asymptotes of the Jost function are determined by value of the phase at infinity. The experimental phase is known for energies up to smax=2.1 GeV. At this energy, δ(smax)≈0. If one takes δ(s)=0 for s>2.1 GeV, then the quark-counting rules are not satisfied, even for the zero-degree polynomial P(s)=1. The minimal required modification is to introduce a hypothetical resonance with mass mX>smax. Such a resonance should have a large width because it is not observed experimentally. Note that, in the Veneziano model, there are no resonances in the T=2 channel. The contribution of a large s is suppressed in the dispersion integral by the inverse of the second power of s, and one can assume that this contribution is not dominant. The contribution of the large-s region can be evaluated by comparing the contributions of the intervals s=2μ-1.5 GeV and 1.5-2.1 GeV, for which experimental data are available. A polynomial P(s) of degree n requires n+1 resonances. If there are primitives, then the number of resonances increases. The numbers of resonances and primitives are correlated with the number of Castillejo-Dalitz-Dyson poles [30].Expanding the Jost function in the vicinity ofs=0, we obtain(43)r2sT=6π∫4μ2+∞δTs′s′2ds′.For numerical estimates, the upper limit of the integral is replaced by smax; the scattering phase shift between the experimental points is interpolated linearly. The results are shown in Table 1 for the data sets from [27–29]. A further refinement of the estimate can be done using more theoretical inputs from semiphenomenological models of pion-pion scattering [31] and lattice QCD [32].Table 1
Lorentz scalar isospinT=2 mean square radius of the pion, as determined using (43). The phase-shift analyses of [27–29] are used; (a) and (b) denote nonequivalent datasets of scattering phases. smax⁡⁡ is the maximum energy up to which the phase analysis is performed.
〈
r
2
〉
s
T
=
2 [fm2]
s
max
⁡
⁡ [GeV]
References
-
0.09
±
0.05
1.4
[27]
-
0.10
±
0.01
2.1
[28] (a)
-
0.10
±
0.03
2.1
[28] (b)
-
0.11
±
0.02
1.5
[29] (a)
-
0.13
±
0.01
1.5
[29] (b)The negative “mean square radius” occurs because of the negativet-channel scattering phase, which is a signature of repulsion. In the general case, the mean square radius is defined as the first expansion coefficient of the corresponding form factor in powers of q2, and the sign of this expansion coefficient is not fixed a priori. Vector boson exchange produces repulsive forces between identical particles. The T=2 channel with T3=2 contains a pair of positively charged pions, and thus, the repulsive nature of the phase can be attributed to the dominance of the t-channel ρ-meson exchange between the pions.Equation (43) has a variety of applications. In the zero-width approximation with δT=1(s)=πθ(s-mρ2), where mρ is the mass of the ρ meson, one arrives at the well-known expression for the pion charge radius, 〈r2〉vT=1=6/mρ2. In the T=1 channel with an attraction, 〈r2〉vT=1 becomes positive and approaches the experimental value, thereby supporting the vector meson dominance model. The pion scalar radius can be determined using the data for pion-pion scattering in the T=0 channel, yielding r2sT=0=0.61±0.04fm2 [33]. The positive value of r2sT=0 reflects the dominance of the attractive interaction, which is consistent with the existence of the σ meson.In the dispersion theory, the contribution from the high energy part of the dispersion integrals is considered to be the important source of theoretical uncertainties. The representation (41) is justified in an approximation of neglecting all the inelastic channels. In our case, the elasticity holds for s=2μ-1.2 GeV [27–29], so the dispersion integral from this region accurately assesses the contribution to the pion radius. With increasing the energy above 1.2 GeV, the parameter of inelasticity increases smoothly up to 0.5 for s=2.1 GeV. The part of the integral (43) corresponding to this region gives therefore an approximate estimate of the contribution. Using results of [28], we find that this contribution is numerically small and equals only -0.007fm2. The uncertainty associated with higher energies s>2.1 GeV can be estimated by assuming the existence of a hypothetical resonance X with mass mX>2.1 GeV. In the zero-width approximation, the error equals 6/mX2~0.03fm2 for mX=3 GeV, which is comparable to the experimental uncertainty of the contribution from the interval s=2μ-1.2 GeV.Experimental value of theπ+ charge radius equals r2vT=1=0.672±0.008fm2 [34]. Using the value r2sT=2=-0.10±0.03±0.03fm2, we obtain with the help of (34) (44)f-=2.97±0.25×10-3,which is a factor of two greater than the light-front quark model prediction [5]. The estimated error corresponds to the experimental errors in the ππ-scattering phases and the uncertainty of the high energy part of the integral representation (43).Contribution of the longitudinal form factor to theπ+→π0e+νe decay rate can be estimated to give ΔB/B=-0.94×10-3f-; the additional small factor in f- arises for kinematic reasons. The experimental error in B is 0.6% [34]. We thus reaffirm earlier conclusions that f- is currently beyond the capabilities of the experimental study. The possibility of measuring the longitudinal weak vector current in the neutron β decay [35], muon capture [36], and in the τ-→π-π0ντ and τ-→K-K0ντ decays is, perhaps, more promising.A favorable experimental situation exists in theKl3 decays. The large difference in the masses of π- and K-mesons hampers the near-mass-shell expansion, so the Kl3 decays seem to be not suitable for the measuring f- off shell and testing the gWTI. At the same time the dispersion techniques allowing determining f- on shell have found in these decays a successful application [37, 38].The dominant part ofΔmπ2 is electromagnetic in origin [39–41]. Small mass difference between up and down quarks leads to the π0-η mixing, which also contributes to Δmπ2. Equation (34) describes therefore the electromagnetic contribution of order O(α) to the longitudinal weak vector current and the small π0-η mixing effect.χPT has been very successful in describing properties of light mesons in the nonperturbative regime [40–43]. Predictions for the ππ-scattering lengths, first obtained by Weinberg [44] and improved by the development of χPT, agree with experiment to a high degree of accuracy [45, 46]. The pion radii 〈r2〉sT=0 and 〈r2〉vT=1 are also well understood [33, 40, 47, 48], while the pion radius 〈r2〉sT=2 has not been discussed so far. The embedding of isotopic symmetry breaking into the χPT framework on line with [3, 4, 41, 49] could allow also describing the near-mass-shell form factor f-, starting from the first principles of QCD.


## 6. Conclusion

In this paper, the longitudinal component of the weak vector current in theπe3 decay was calculated. This component arises due to the violation of isotopic symmetry by the pion masses and the electromagnetic and weak interactions. A generalization of the WTI was derived in the pion sector to account for the isotopic symmetry breaking. It was shown that the isovector T=1 part of the current can be reconstructed through isotopic rotation of the off-shell pion electromagnetic form factors, whereas the isotensor T=2 part has no analogs but rather is uniquely determined by the gWTI combined with the elastic unitarity, analyticity, and the pion-pion scattering data. Because of the low momentum transfer, the on-shell form factor is proportional to the pion mass difference and the difference of the vector-isovector and scalar-isotensor pion radii. The relationship between the near-mass-shell form factor and the pion radii (40) is a consequence of the CVC condition extended beyond the approximation of exact isotopic symmetry.The various versions of the CVC condition are distinguished depending on whether the isotopic symmetry is exact or broken and whether the outer legs in the vertex are on or off shell. The corresponding predictions for the longitudinal form factorf- are given in (33), (34), (39), and (40). In the case of the exact isotopic symmetry, the bare and dressed weak vertices are pure isospin triplets; the CVC and the WTI hold on and off shell, respectively. In the case of the broken isotopic symmetry, the dressed weak vertex is no longer a pure isospin triplet; partial CVC and the gWTI hold on and off shell.Favorable conditions for testing the gWTI by measuring the longitudinal weak vector current appear to exist in theτ-→π-π0ντγ and τ-→K-K0ντγ decays with soft photon emission because of the involvement of the near-mass-shell form factors, the relatively large momentum transfer in the weak vertex, and the small mass differences within the pion and kaon multiplets.A nontrivial consequence of partial CVC occurs in the off-shell case, in which the vertex in general is not gauge invariant and depends on the parameterization of the pion field. The only exception is the longitudinal component of the vertex in the neighborhood of the mass shell. To first order in the pion mass splitting and the displacement from the mass shell, the longitudinal component is independent of both the gauge and the parameterization, and therefore, the near-mass-shell form factorf- appears to be a unique object whose properties are unambiguously determined by the partial CVC condition (the gWTI) and also fundamentally allow for experimental verification.



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*Source: 656239-2015-05-27.xml*