Soft-pion Theorem Research Articles

Pion GPDs: Constraints, modelling and experimental access

The study of Generalised Parton Distributions (GPDs) is a hot-topic in hadron physics. Due to its connection with chiral symmetry breaking, assessing those of pions is of great interest. For this reason, we present a novel model-building strategy for pion GPDs capable of fulfilling all of the theoretical constraints required by QCD: support, polynomiality, positivity and soft-pion theorem. The methodology is illustrated with a simple model, and exploited afterwards for the calculation of pions' deeply virtual Compton Scattering (DVCS) Compton Form Factors (CFFs) at NLO. The results show gluon-content to play the dominant role in pions' response to DVCS at EIC kinematics.

Revisiting gamma ^*rightarrow gamma pi ^0eta near the phi (1020) using analyticity and the left-cut structure

Amplitudes of the form gamma ^*(q^2)rightarrow gamma P_1P_2 appear as sub-processes in the computation of the muon g-2. We test a proposed theoretical modelling against very precise experimental measurements by the KLOE collaboration at q^2=m^2_phi . Starting from an exact, parameter-free dispersive representation for the S-wave satisfying QCD asymptotic constraints and Low’s soft photon theorem we derive, in an effective theory spirit, a two-channel Omnès integral representation which involves two subtraction parameters. The discontinuities along the left-hand cuts which, for timelike virtualities, extend both on the real axis and into the complex plane are saturated by the contributions from the light vector mesons. In the case of P_1P_2=pi eta , we show that a very good fit of the KLOE data can be achieved with two real parameters, using a T-matrix previously determined from gamma gamma scattering data. This indicates a good compatibility between the two data sets and confirms the validity of the T-matrix. The resulting amplitude is also found to be compatible with the chiral soft pion theorem. Applications to the I=1 scalar form factors and to the a_0(980) resonance complex pole are presented.

Analysis of the nonleptonic two-body decays of the Λ hyperon

We systematically study two-body nonleptonic decays of light lambda hyperon $\Lambda \to p \pi^- (n\pi^0)$ with account for both short- and long-distance effects. The short-distance effects are induced by five topologies of external and internal weak $W^\pm$ exchange, while long-distance effects are saturated by an inclusion of the so-called pole diagrams with intermediate $\frac{1}{2}^+$ and $\frac{1}{2}^-$ baryon resonances. The contributions from $\frac12^+$ resonances are calculated straightforwardly by account for nucleon and $\Sigma$ baryons whereas the contributions from $\frac{1}{2}^-$ resonances are calculated by using the well-known soft-pion theorem in the current-algebra approach. It allows one to express the parity-violating $S$-wave amplitude in terms of parity-conserving matrix elements. From our previous analysis of heavy baryons we know that short-distance effects induced by internal topologies are not suppressed in comparison with external $W$-exchange diagram and must be included for description of data. Here, in the case of $\Lambda$ decays we found that the contribution of external and internal $W$-exchange diagrams is sizably suppressed, e.g., by one order of magnitude in comparison with data, which are known with quite good accuracy. Pole diagrams play the major role to get consistency with experiment.

Second Gegenbauer moment of a ρ -meson distribution amplitude

Using the soft pion theorem, crossing, and the dispersion relations for the two pion distribution amplitude ($2\pi$DA) we argue that the second Gegenbauer moment the $\rho$-meson DA ($a_2^{(\rho)}$) most probably is negative. This result is at variance with the majority of the model calculations for $a_2^{(\rho)}$. Using the instanton theory of the QCD vacuum, we computed $a_2^{(\rho)}$ at a low normalisation point and obtain for the ratio $ a_2^{(\rho)}/M_3^{(\pi)}$ {\it definitely negative value} in the range of $a_2^{(\rho)}/M_3^{(\pi)}\in [-2, -1]$. The range of values corresponds to a generous variation of the parameters of the instanton vacuum. The value of the second Gegenbauer moment of pion DA is positive in the whole range and is compatible with its the most advanced lattice measurement. It seems that the topologically non-trivial field configurations in the QCD vacuum (instantons) lead to qualitatively different shapes of the pion and the $\rho$-meson DAs.

Effective field theory in the study of long range nuclear parity violation on lattice

A non-zero signal A_gamma ^mathrm {np}=(-3.0pm 1.4pm 0.2)times 10^{-8} of the gamma-ray asymmetry in the neutron-proton capture was recently reported by the NPDGamma Collaboration which provides the first determination of the Delta I=1 parity-odd pion-nucleon coupling constant h_pi ^1=(2.6pm 1.2pm 0.2)times 10^{-7}. The ability to reproduce this value from first principles serves as a direct test of our current understanding of the interplay between the strong and weak interaction at low energy. To motivate new lattice studies of h_pi ^1, we review the current status of the theoretical understanding of this coupling, which includes our recent work that relates it to a nucleon mass-splitting by a soft-pion theorem. We further investigate the possibility of calculating the mass-splitting on the lattice by providing effective field theory parameterizations of all the involved quark contraction diagrams. We show that the lattice calculations of the easier connected diagrams will provide information of the chiral logarithms in the much harder quark loop diagrams and thus help in the chiral extrapolation of the latter.

Nuclear axial currents from scale-chiral effective field theory**Supported by National Science Foundation of China (NSFC) (11475071, 11547308) and the Seeds Funding of Jilin University

By incorporating hidden scale symmetry and hidden local symmetry in the nuclear effective field theory, combined with the double soft-pion theorem, we predict that the Gamow-Teller operator coming from the space component of the axial current should remain unaffected by the QCD vacuum change caused by the baryonic density, whereas the first forbidden beta transition operator coming from the time component should be strongly enhanced. While the latter has been confirmed for some time, the former was given support by a powerful recent ab initio quantum Monte Carlo calculation for light nuclei, which also confirmed the old “chiral filter hypothesis.” Formulated in terms of the Fermi-liquid fixed point structure of strong-coupled nuclear interactions, we offer an extremely simple resolution to the long-standing puzzle of the “quenched gA,” [], found in nuclear Gamow-Teller beta transitions, giant Gamow-Teller resonances, and double beta decays.

Novel Soft-Pion Theorem for Long-Range Nuclear Parity Violation.

The parity-odd effect in the standard model weak neutral current reveals itself in the long-range parity-violating nuclear potential generated by the pion exchanges in the ΔI=1 channel with the parity-odd pion-nucleon coupling constant h_{π}^{1}. Despite decades of experimental and theoretical efforts, the size of this coupling constant is still not well understood. In this Letter, we derive a soft-pion theorem relating h_{π}^{1} and the neutron-proton mass splitting induced by an artificial parity-even counterpart of the ΔI=1 weak Lagrangian and demonstrate that the theorem still holds exact at the next-to-leading order in the chiral perturbation theory. A considerable amount of simplification is expected in the study of h_{π}^{1} by using either lattice or other QCD models following its reduction from a parity-odd proton-neutron-pion matrix element to a simpler spectroscopic quantity. The theorem paves the way to much more precise calculations of h_{π}^{1}, and thus a quantitative test of the strangeness-conserving neutral current interaction of the standard model is foreseen.

A Nakanishi-based model illustrating the covariant extension of the pion GPD overlap representation and its ambiguities

A systematic approach for the model building of Generalized Parton Distributions (GPDs), based on their overlap representation within the DGLAP kinematic region and a further covariant extension to the ERBL one, is applied to the valence-quark pion's case, using light-front wave functions inspired by the Nakanishi representation of the pion Bethe–Salpeter amplitudes (BSA). This simple but fruitful pion GPD model illustrates the general model building technique and, in addition, allows for the ambiguities related to the covariant extension, grounded on the Double Distribution (DD) representation, to be constrained by requiring a soft-pion theorem to be properly observed.

Soft pion theorem, asymptotic symmetry and new memory effect

It is known that soft photon and graviton theorems can be regarded as the Ward-Takahashi identities of asymptotic symmetries. In this paper, we consider soft theorem for pions, i.e., Nambu-Goldstone bosons associated with a spontaneously broken axial symmetry. The soft pion theorem is written as the Ward-Takahashi identities of the S-matrix under asymptotic transformations. We investigate the asymptotic dynamics, and find that the conservation of charges generating the asymptotic transformations can be interpreted as a pion memory effect.

From Bethe–Salpeter Wave functions to Generalised Parton Distributions

We review recent works on the modelling of Generalised Parton Distributions within the Dyson-Schwinger formalism. We highlight how covariant computations, using the impulse approximation, allows one to fulfil most of the theoretical constraints of the GPDs. Specific attention is brought to chiral properties and especially the so-called soft pion theorem, and its link with the Axial-Vector Ward-Takahashi identity. The limitation of the impulse approximation are also explained. Beyond impulse approximation computations are reviewed in the forward case. Finally, we stress the advantages of the overlap of lightcone wave functions, and possible ways to construct covariant GPD models within this framework, in a two-body approximation.

B→πℓνat zero recoil from lattice QCD with physicalu/dquarks

The exclusive semileptonic decay $B \rightarrow \pi \ell \nu$ is a key process for the determination of the Cabibbo-Kobayashi-Maskawa matrix element $V_{ub}$ from the comparison of experimental rates as a function of $q^2$ with theoretically determined form factors. The sensitivity of the form factors to the $u/d$ quark mass has meant significant systematic uncertainties in lattice QCD calculations at unphysically heavy pion masses. Here we give the first lattice QCD calculations of this process for u/d quark masses going down to their physical values, calculating the $f_0$ form factor at zero recoil to 3\%. We are able to resolve a long-standing controversy by showing that the soft-pion theorem result $f_0(q^2_{max}) = f_B/f_{\pi}$ does hold as $m_{\pi} \rightarrow 0$. We use the Highly Improved Staggered Quark formalism for the light quarks and show that staggered chiral perturbation theory for the $m_{\pi}$ dependence is almost identical to continuum chiral perturbation theory for $f_0$, $f_B$ and $f_{\pi}$. We also give results for other processes such as $B_s \rightarrow K \ell \nu$.

Soft-Pion theorems for large scale structure

Consistency relations — which relate an N-point function to a squeezed (N+1)-point function — are useful in large scale structure (LSS) because of their non-perturbative nature: they hold even if the N-point function is deep in the nonlinear regime, and even if they involve astrophysically messy galaxy observables. The non-perturbative nature of the consistency relations is guaranteed by the fact that they are symmetry statements, in which the velocity plays the role of the soft pion. In this paper, we address two issues: (1) how to derive the relations systematically using the residual coordinate freedom in the Newtonian gauge, and relate them to known results in ζ-gauge (often used in studies of inflation); (2) under what conditions the consistency relations are violated. In the non-relativistic limit, our derivation reproduces the Newtonian consistency relation discovered by Kehagias & Riotto and Peloso & Pietroni. More generally, there is an infinite set of consistency relations, as is known in ζ-gauge. There is a one-to-one correspondence between symmetries in the two gauges; in particular, the Newtonian consistency relation follows from the dilation and special conformal symmetries in ζ-gauge. We probe the robustness of the consistency relations by studying models of galaxy dynamics and biasing. We give a systematic list of conditions under which the consistency relations are violated; violations occur if the galaxy bias is non-local in an infrared divergent way. We emphasize the relevance of the adiabatic mode condition, as distinct from symmetry considerations. As a by-product of our investigation, we discuss a simple fluid Lagrangian for LSS.

Consistent model forπNtransition distribution amplitudes and backward pion electroproduction

The extension of the concept of generalized parton distributions leads to the introduction of baryon to meson transition distribution amplitudes (TDAs), non-diagonal matrix elements of the nonlocal three quark operator between a nucleon and a meson state. We present a general framework for modelling nucleon to pion ($\pi N$) TDAs. Our main tool is the spectral representation for \pi N TDAs in terms of quadruple distributions. We propose a factorized Ansatz for quadruple distributions with input from the soft-pion theorem for \pi N TDAs. The spectral representation is complemented with a D-term like contribution from the nucleon exchange in the cross channel. We then study backward pion electroproduction in the QCD collinear factorization approach in which the non-perturbative part of the amplitude involves \pi N TDAs. Within our two component model for \pi N TDAs we update previous leading-twist estimates of the unpolarized cross section. Finally, we compute the transverse target single spin asymmetry as a function of skewness. We find it to be sizable in the valence region and sensitive to the phenomenological input of our \pi N TDA model.

πNtransition distribution amplitudes: Their symmetries and constraints from chiral dynamics

Baryon to meson transition distribution amplitudes (TDAs) extend the concept of generalized parton distributions. Baryon to meson TDAs appear as building blocks in the collinear factorized description of amplitudes for a class of hard exclusive reactions, prominent examples of which being hard exclusive meson electroproduction off a nucleon in the backward region and baryon-antibaryon annihilation into a meson and a lepton pair. We study the general properties of these objects following from the underlying symmetries of QCD. In particular, the Lorentz symmetry results in the polynomiality property of the Mellin moments in longitudinal momentum fractions. We present a detailed account of the isotopic and permutation symmetry properties of nucleon to pion (\pi N) TDAs. This restricts the number of independent leading twist \pi N TDAs to eight functions, providing description of all isotopic channels. Using chiral symmetry and the crossing relation between \pi N TDAs and \pi N generalized distribution amplitudes we establish soft pion theorems for \pi N TDAs, which determine the magnitude of \pi N TDAs. Finally, we build a simple resonance exchange model for \pi N TDAs considering N and \Delta(1232) exchange contributions into the isospin-1/2 and isospin-3/2 \pi N TDAs.

The pion transition form factor and the pion distribution amplitude

Recent BaBaR data on the pion transition form factor, whose Q^2 dependence is much steeper then predicted by asymptotic Quantum Chromodynamics (QCD), have caused a renewed interest in its theoretical description. We present here a formalism based on a model independent low energy description and a high energy description based on QCD, which match at a scale Q_0. The high energy description incorporates a flat pion distribution amplitude, phi(x)=1, at the matching scale Q_0 and QCD evolution from Q_0 to Q>Q_0. The flat pion distribution is connected, through soft pion theorems and chiral symmetry, to the pion valance parton distribution at the same low scale Q_0. The procedure leads to a good description of the data, and incorporating additional twist three effects, to an excellent description of the data.

Measurement of the branching fractions for [formula omitted] and [formula omitted

Using 58×106J/ψ events collected with the Beijing Spectrometer (BESII) at the Beijing Electron Positron Collider (BEPC), the branching fractions of J/ψ to pp¯η and pp¯η′ are determined. The ratio Γ(J/ψ→pp¯η)Γ(J/ψ→pp¯) obtained by this analysis agrees with expectations based on soft-pion theorem calculations.

Dissociation of theJ/ψby light mesons, and chiral symmetry

The implication of chiral symmetry for the pion-induced dissociation of the $J/\ensuremath{\psi}$ is examined in detail. It is shown how the low-energy dynamics of pions, constrained by chiral symmetry, affect the dissociation cross section. The derived soft-pion theorem is then integrated into a Lagrangian model that includes also abnormal parity content and chiral-symmetric form factors. Dissociation by the \ensuremath{\rho} meson is also considered.

Contribution of the DK continuum in the QCD sum rule for DsJ(2317)

Using the soft-pion theorem and the assumption on the final-state interactions, we include the contribution of the DK continuum into the QCD sum rules for the DsJ(2317) meson. We find that this contribution can significantly lower the mass and the decay constant of the Ds(0+) state. For the value of the current quark mass mc(mc) = 1.286 GeV, we obtain the mass of Ds(0+) M=2.33 ± 0.02 GeV in the interval s0 = 7.5–8.0 GeV2, being in agreement with the experimental data, and the vector current decay constant of Ds(0+) f0=0.128 ± 0.013 GeV, much lower than those obtained in the previous literature.

Generalized parton distributions of the pion in chiral quark models and their QCD evolution

We evaluate Generalized Parton Distributions of the pion in two chiral quark models: the Spectral Quark Model and the Nambu-Jona-Lasinio model with a Pauli-Villars regularization. We proceed by the evaluation of double distributions through the use of a manifestly covariant calculation based on the alpha representation of propagators. As a result polynomiality is incorporated automatically and calculations become simple. In addition, positivity and normalization constraints, sum rules and soft pion theorems are fulfilled. We obtain explicit formulas, holding at the low-energy quark-model scale. The expressions exhibit no factorization in the t-dependence. The QCD evolution of those parton distributions is carried out to experimentally or lattice accessible scales. We argue for the need of evolution by comparing the Parton Distribution Function and the Parton Distribution Amplitude of the pion to the available experimental and lattice data, and confirm that the quark-model scale is low, about 320 MeV.

Factorization in Multibody Radiative B Decays

We study the radiative decays B → K π γ and B → K π e + e − , including both K ∗ resonant contributions and non-resonant ones. We describe new soft pion theorems with which we compute certain non-resonant multibody amplitudes. We present results for CP asymmetry in B → K π γ and for the forward-backward asymmetry in B → K π e + e − .