Baryonic content of the pion

The baryon form factor of charged pions arises since isospin symmetry is broken with unequal up and down quark masses, $m_d>m_u$, as well as electromagnetic effects. We obtain estimates for this basic property in two phenomenological ways: from simple constituent quark models, as well as from fitting the $e^+e^- \to \pi^+ \pi^-$ data. All our methods yield the result that the baryon mean square radius, extracted from the slope of the form factor, is positive for $\pi^+$, hence a picture where the outer region has a net baryon, and the inner region a net antibaryon density, both compensating each other such that the total baryon number is zero. For $\pi^-$ the effect is equal and opposite. We estimate the corresponding mean squared baryon radius as $\langle r^2 \rangle_B^{\pi^{+}} = (0.03-0.04~{\rm fm})^2$.


Introduction
Since its prediction and subsequent discovery, the pion has been scrupulously investigated as the basic lightest hadron and the pseudo-Goldstone boson of the dynamically broken chiral symmetry. Many of its electroweak and mechanical properties have been studied and determined both experimentally [1] as well as theoretically from a first principles point of view, with notable recent advances from lattice QCD. In this Letter, we draw attention to the baryonic structure of the charged pions, π + and π − , a remarkable property which, to the best of our knowledge, has not been studied in an explicit manner before. Despite being a zero baryon number state, the composition of the charged pion is not baryonless. We show that simple quark models and data analyses imply a characteristic pattern where the matter and antimatter radial distributions are separated at a distance of r ∼ 0.5 fm. For π + , the inner (outer) region carries a net antibaryon (baryon) density, and opposite for π − (cf. Fig. 1). The situation is reminiscent of the well-known case of the electric form factor of the neutron which carries no charge, nevertheless possesses a non-zero electric form factor, such that (in the Email addresses: psanchez@ifae.es (Pablo Sanchez-Puertas), earriola@ugr.es (Enrique Ruiz Arriola), Wojciech.Broniowski@ifj.edu.pl (Wojciech Broniowski) Breit frame) the inner (outer) region has positive (negative) charge density, with the mean squared radius (msr) r 2 n Q = −0.1161(22) fm 2 . Similarly, the neutral Kaon K 0 has r 2 K 0 Q = −0.077(10) fm 2 [1] despite its null charge. Even more striking, the nucleon is known to have a nonvanishing strange form factor despite being strangeless [2,3].

Current conservation
To see how the effect arises, let us first recall for completeness some very basic facts. In QCD, one has the conservation laws with q j (x) denoting the quark field with N c colors and flavor j = u, d, s, c, b, t, which for equal quark masses corresponds to the conservation of the vector current and ensures the quark number conservation for any species. For the specific case of the pion, we neglect s and heavier flavors, as they represent corrections suppressed by the Okubo-Zweig-Iizuka (OZI) rule and are subleading in the large-N c limit of QCD. In this case the baryon current (isosinglet) and the third isospin component of the isovector current are where N c ≡ 3 is assumed in the following. With these definitions, the Gell-Mann-Nishijima formula provides the electromagnetic current J µ The baryon, isospin, and charge form factors are defined via the on-shell matrix elements of the corresponding currents, namely with F a Q (q 2 ) = F a 3 (q 2 ) + 1 2 F a B (q 2 ).

Charge conjugation and Isospin violation
Now come the standard symmetry arguments. Since J µ B is odd under charge conjugation C, it implies that for the C-even neutral pion F π 0 B (q 2 ) = 0 identically, while for the charged pions it provides the relation . Similarly, for the case of an exact isospin symmetry, i.e., with m u = m d and neglecting small electromagnetic effects, G-parity symmetry yields F π ± B (q 2 ) = 0. However, this is no longer the case in the real world where the isospin is broken with m d > m u . G-parity ceases to be a good symmetry and F π ± B (q 2 ) may be -and in fact is -non-zero, with F π + B (q 2 ) = −F π − B (q 2 ) 0. Moreover, if we take (say, for π + ) Zdiγ 5 u as an interpolating field, then a direct application of the Ward-Takahashi-Green identities for the conserved B and Q currents implies (for the canonical pion field) F π ± B (0) = 0, F π ± Q (0) = ±1.

Coordinate space interpretation
A popular interpretation of form factors is based on choosing the Breit reference frame, where there is no energy transfer. Then the form factor in the space-like region q 2 = − q 2 ≡ −Q 2 ≤ 0 allows one to construct the (naive) 3-dimensional baryon density as As ambiguities stemming from relativity arise [4,5], it has been argued that a frame-independent interpretation can be formulated in terms of a transverse density in the 2-dimensional impact-parameter b [6], where instead of Eq. (4) one takes the Fourier integral with exp(i q ⊥ · b). (see, e.g., [7] for a review and Ref. [8]). Here, for our illustrative purpose, we choose to show the r-space densities, as the b-space results are simply related and qualitatively the same. We have no obvious sources coupling the charged pions solely to the baryon current, hence a direct experimental measurement of F π ± B (q 2 ) is not possible. This is also in common with the neutron electric form factor, where a direct determination is hampered by the absence of free neutron targets and its extraction requires scattering on bound neutrons in the deuteron. 1 The analysis requires an accurate deuteron wave function as well as meson exchange current effects [10], hence the need for additional theoretical input. Returning to the novel case of the so far disregarded pion baryonic form factor, we will content ourselves with rather unsophisticated but complementary and realistic estimates. They are based on dimensional analysis, quark models, and an extraction from e + e − → π + π − data analysis. Within uncertainties, our results are consistent.

Dimensional analysis
A generic order of magnitude estimate of the discussed isospin violating effect can be obtained at the leading order in the pion momenta and the quark mass splitting ∆m ≡ m d − m u = 2.8(2) MeV (in this work we use m u = 2.01 (14) MeV and m d = 4.79(16) MeV [11]).
We expect the two-pion contribution to the baryon current to be of the effective form with c an undetermined dimensionless number and Λ a typical low energy hadronic scale (say, m ρ ∼ 770 MeV).
As it should, this current is odd under C, is trivially conserved, and its contribution vanishes for q 2 = 0, providing the form factor F π + B (q 2 ) = q 2 c∆m/Λ 3 + . . . , with msr r 2 π + B = 6c∆m/m 3 ρ c 0.002 fm 2 c(0.04 fm) 2 , a small number compared to the electric charge radius r 2 π + Q = (0.659(4) fm) 2 = 0.434(5) fm 2 [1]. One may seek further guidance in Chiral Perturbation Theory. In particular, the term in Eq. (5) arises starting from the O(p 6 ) chiral Lagrangian [12], from where we find an explicit relation It involves two C i coefficients, for which currently there are no independent estimates. The naturalness condition yields C i F −4 π ∼ m −4 ρ , hence c ∼ 1, as previously argued.

Yukawa quark model
Next, we come to our quark model estimates for ∆m 0 effects. To start, we explore the fact that the coordinate representation motivates a toy constituentquark model based on the familiar impulse approximation in nuclear physics (cf. Fig. 2). In this framework Motivated by Vector Meson Dominance model (VMD), that provides a reasonable description for the isovector channel, F π + 3 (q 2 ) = M 2 ρ /(M 2 ρ −q 2 ) (and allows for a simple extension to other pseudoscalar mesons), we take the normalized Yukawa- 2 ∆m and M denoting the constituent quark mass. This ensures that, for ∆m = 0, the resulting isovector and charge form factors  Figure 3: The baryon form factor of π + in the space-like region. The bands correspond to the extraction from the data, and the lines to various models described in the text.
reproduce the VMD phenomenology provided we take M = 1 2 m ρ 385 MeV, while for the baryon form factor we find Eq. (7) yields r 2 π + B 3∆m/N c M 3 (0.04 fm) 2 . Our result for the baryon density is plotted in Fig. 1

Chiral quark model
We now pass to a quark model where the pion is described with a fully relativisticqq dynamics. The Nambu-Jona-Lasinio (NJL) model with consitutent quarks (see [13] and Refs. therein) implements the spontaneously broken chiral symmetry, providing the quarks dynamically with a constituent quark mass M. A leading-N c diagrammatic representation of the baryon form factor is given in Fig. 4, where we indicate the momenta, and the π + ud coupling constant of the point-like coupling is (M u + M d )/

√
2F π , with F π denoting the pion weak decay constant. The Lorentz, gauge, and chiral symmetries are preserved by implementing the Pauli-Villars regularization [13], imposed in these effective models to suppress the hard momentum contribution from the quark loop.
The result of a standard evaluation has the compact form for the expressions leading in the quark mass splitting ∆m: with the basic one-loop integral evaluated in the Euclidean space as where the quark propagator is G(l) ≡ 1/(l 2 − M 2 + i ). The subscript 'reg' indicates the regularization, imposed in these effective models to suppres the hard momentum contribution from the loop. Explicit calculation in the chiral limit yields the result with the short-hand notation s = 1/ 1 − 4M 2 /t introduced. The low-Q 2 expansion is The baryon radius is r 2 π + B = (0.03 fm) 2 with M = 0.3− 0.35 GeV and the Pauli-Villars cut-off Λ 0.7 GeV, adjusted to give the physical value of F π . Actually, since the one-loop calculation of Fig. 4 yields a finite result for the electric charge msr [14] and for the baryon msr, without regularization we obtain a numerically very similar (though not identical) value, r 2 π + B = (∆m/N c M) r 2 π + Q ∼ (0.03 fm) 2 . The baryon form factor in the NJL model with PV regularization (NJL) and in the unregularized case (unreg.) are plotted in Fig. 3 for Q 2 Λ 2 ∼ 0.5 GeV 2 . Momenta higher than the cut-off are hard and are outside of the fiducial range of the effective quark models, hence the functions are not plotted there.

Vector meson dominance
Alternatively to quark models, on the basis of the quark-hadron duality we may adopt a purely hadronic description that shall illustrate the significance of the ρ − ω mixing in the context of the baryon form factor. In VMD, the physical ω and ρ 0 mesons are linear combinations of the isoscalar ω 0 and isovector ρ 3 states with a mixing angle θ. With the current-field identities [15,16] and the matrix elements 0|J µ

the form factors in the space-like region read
with g ωππ and g ρππ the couplings for ω → π + π − and ρ → π + π − decays and f B,3 related to ρ/ω → + − decays (see for instance Ref. [17]). The conditions F B (0) = 0 and F 3 (0) = 1 imply g ρππ = 2m 2 ρ cos θ/ f 3 , g ωππ = 2m 2 ω sin θ/ f 3 , and . (13) The above formula nicely illustrates basic physical features: the association of emergence of F B (−Q 2 ) with the ρ − ω mixing, and its vanishing value at Q 2 = 0. However, Eqs. (13) hold literally for narrow-width mesons only, which is certainly not the case for the broad ρ resonance and precludes building a successful phenomenology. For that reason we do not elaborate numerically Eqs. (13), treating them only as a guideline for a more sophisticated analysis of the next section, where the width of resonances is properly incorporated.

Data analysis
We now use the available high statistics data in the time-like region to extract the baryon form factor. Actually, the Gell-Mann-Nishijima formula for the form factors would allow for a direct determination if it were not for the fact that, unlike for |F π ± Q (q 2 )| accessible from the e + e − → π + π − reaction from BaBar [18] and KLOE [19,20,21,22]), the F π ± 3 (q 2 ) form factor remains unknown. One could in principle consider the flavor-changing current J + µ =ūγ µ d and the corresponding form factor F + (q 2 ) appearing in the matrix element π 0 |J + µ |π + , determined to a high precision in τ → π + π 0 ν τ decays by Belle [23]. In the strict isospin limit (∆m = 0), the latter is simply related to F π ± 3 (q 2 ) via isospin rotation, a relation that has been exploited in the context of the muon (g − 2) [24,25,26,27,28,29,30,31,32,33,34], but without paying an effort to extract F π ± B (q 2 ). Moreover, the isospin relation no longer holds if ∆m 0 or the electromagnetic effects are accounted for. Actually, while the isospin version of the Ademollo-Gatto nonrenormalization theorem [35] implies F π ± 3 (0) = F + (0) + O(∆m 2 ), this is no longer true at finite momentum transfer, where F π ± 3 (q 2 ) = F + (q 2 ) + O(∆m), comparable itself to the effect we aim to extract, F π ± B (q 2 ) = O(∆m). Indeed, our attempts to do so with the aid of dispersion relations provided noisy results. These can be ascribed to the isospin violating corrections, as we detail in the analysis below. The points indicate the experimental data from e + e − → π + π − (BaBar and KLOE for the Q form factor), and from τ − → π 0 π − ν τ (Belle for the + form factor). Our fits are represented with the solid bands following the points, where the widths reflect the statistical errors. We also give the resulting fits to the 3 form factors, indicated with the dashed bands.
The results are shown in Fig. 5. The two highestreaching bands correspond to our fits to F π ± Q (q 2 ) from BaBar [18] and KLOE [19,20,21,22] with QED effects from vacuum polarization and final state radiation removed. The corresponding F π ± 3 (q 2 ) form factors resulting from these fits are shown as the two dashed bands. These can be compared to the fit to the F + (q 2 ) form factor from Belle, shown as a band. The mild difference is clear and can be ascribed to the mentioned isospin-breaking corrections. The obtained value of the mixing parameter from the BaBar/KLOE data is c ρω = [36(1)/37(2)] × 10 −4 GeV −2 .
To extract the behavior in the spacelike region, we make use of analyticity and the perturbative high-energy behavior at large q 2 , F π ± B (q 2 ) = O(1/q 2 ) [38], that allows one to write down the unsubtracted (and subtracted) dispersion relations .
The results of Fig. 3 show that despite a discrepancy between BaBar and KLOE in the time-like region (cf. Fig 5), the obtained baryonic form factors in the space-like region are compatible. Likewise, the baryonic radius computed from Eq. (16) yields r 2 π + B = (0.0411(7) fm) 2 for BaBar and (0.0412(12) fm) 2 for KLOE, are in a remarkable agreement.

Conclusions and outlook
We summarize in Table 1 all the obtained estimates for the baryonic msr of the charged pion, which fall in the range r 2 π + B = ((0.03 − 0.04) fm) 2 = (0.001 − 0.002) fm 2 . The agreement with the Yukawa model complies to the natural understanding on the positive sign of the radius for π + (ud), where the lighter u component is more extended than the heavierd component. Comparing the estimates of Table 1 to the accuracy of the experimental charge radius, r 2 π Q = (0.659(4) fm) 2 = 0.434(5) fm 2 [1], or to the most recent ab initio lattice QCD calculations with physical averaged quark masses, r 2 π Q = (0.648(15) fm) 2 = 0.42(2) fm 2 [39] and r 2 π Q = 0.430(5)(12) fm 2 [40], we note that the signal is a factor of 4 − 10 too small to affect current charge radius determinations. However, the small baryonic msr is also coming from the 1/N c prefactor preceding the baryon current. Without this factor, one finds N c r 2 π + B = r 2 π + u − r 2 π + d = (0.003 − 0.005) fm 2 , that might be within reach in lattice QCD.
The case of the baryonic content of the Kaon is much more promising. Since ford and s quarks the baryon number equals minus the charge, in models with structureless constituent quarks (such as our toy Yukawa model) r 2 K 0 B = − r 2 K 0 Q . The Yukawa model yields r 2 K 0 B (0.22 fm) 2 0.05 fm 2 , whereas PDG [1] quotes r 2 K 0 Q = −(0.28(2) fm) 2 = −0.077(10) fm 2 , with the uncertainty 5 times smaller than the Yukawa model estimate for the baryon msr. Similarly, VMD predicts F K 0 B (q 2 ) = N −1 c (D ω (q 2 ) − D φ (q 2 )), thus r 2 K 0 B = (0.23 fm) 2 = 0.052 fm 2 , while a data-driven analysis such as that in Sect. 4 would be more involved. In particular, the KK threshold, well above the ρ, ω mesons, would require a more elaborated analysis as outlined in Refs. [41,42].
Hopefully, more accurate experiments and their corresponding analyses of the vector form factors would provide a better understanding of the fundamental issue of the matter-antimatter distribution in pseudoscalar mesons.
Supported by the European H2020 MSCA-COFUND