We perform a dispersive analysis of the $ωπ$ electromagnetic transition form factor, using as input the discontinuity provided by unitarity below the $ωπ$ threshold and including for the first time experimental data on the modulus measured from $e^+e^-\toωπ^0$ at higher energies. The input leads to stringent parameterization-free constraints on the modulus of the form factor below the $ωπ$ threshold, which are in disagreement with some experimental values measured from $ω\to π^0γ^*$ decay. We discuss the dependence on the input parameters in the unitarity relation, using for illustration an $N/D$ formalism for the P partial wave of the scattering process $ωπ\to ππ$, improved by a simple prescription which simulates the rescattering in the crossed channels. Our results confirm the existence of a conflict between experimental data and theoretical calculations of the $ωπ$ form factor in the region around 0.6 GeV and bring further arguments in support of renewed experimental efforts to measure more precisely the $ω\toπ^0γ^*$ decay.
I try to describe the extremely fruitful interaction I had with Harry Lehmann and the results which came out of the analyticity unitarity programme, especially the proof of the Froissart bound, which, with recent and future measurements of total cross-sections and real parts, remains topical.
We show how the combination of analyticity properties derived from local field theory and the unitarity condition (in particular positivity) leads to non-trivial physical results, including the proof of the "Froissart bound" from first principles and the existence of absolute bounds on the pion-pion scattering amplitude.
The experimental data on the processes $ππ\toππ,K\bar{K},ηη,ηη^\prime$ in the $I^GJ^{PC}=0^+0^{++}$ channel have been jointly analyzed to study the status and nature of the $f_0$. The method of analysis is based on analyticity and unitarity and uses an uniformization procedure. Some spectroscopic implications from results of the analysis are discussed.
Using the analytic properties of two-point functions in QCD, as well as unitarity, bounds on the B meson form factor F(q^2) can be derived. Heavy quark spin symmetry, correctly taken into account, is shown to improve these bounds significantly.
These notes are a fuller version of four lectures given at the 2015 International Summer Workshop in Reaction Theory held at Indiana University, Bloomington. The aim is to provide a simple introduction to how the tools of "the S-matrix era" - i.e. the constraints of unitarity, analyticity and crossing symmetry - can be incorporated into analyses of final state interactions in two- and three-hadron systems. The main focus is on corrections to the isobar model in three-hadron final states, which may be relevant once more as much larger data sets become available.
In $1/N_c$ expansion, analyticity and crossing lead to crossing even and odd ($C=\pm 1$) vacuum exchanges at high-energy, the {\em Pomeron} and the {\em Odderon}. We discuss how, using {\em String/Gauge duality}, these can be identified with a reggeized {\em Graviton} and the anti-symmetric {\em Kalb-Ramond fields} in $AdS$ background. With confinement, these Regge singularities interpolate with glueball states. We also discuss unitarization based on eikonal sum in $AdS$.
We investigate the scalar K pi form factor at low energies by the method of unitarity bounds adapted so as to include information on the phase and modulus along the elastic region of the unitarity cut. Using at input the values of the form factor at t=0 and the Callan-Treiman point, we obtain stringent constraints on the slope and curvature parameters of the Taylor expansion at the origin. Also, we predict a quite narrow range for the higher order ChPT corrections at the second Callan-Treiman point.
A method of deriving bounds on the weak meson form factors, based on perturbative QCD, analyticity and unitarity, is generalized in order to fully exploit heavy quark spin symmetry in the ground state $(L=0)$ doublet of pseudoscalar $(B)$ and vector $(B^*)$ mesons. All the relevant form factors of these mesons are taken into account in the unitarity sum. They are treated as independent functions along the timelike axis, being related by spin symmetry only near the zero recoil point. Heavy quark vacuum polarisation up to three loops in perturbative QCD and the experimental cross sections $σ(e^+…
We study the shape parameters of the $Dπ$ scalar and vector form factors using as input dispersion relations and unitarity for the moments of suitable heavy-light correlators evaluated with Operator Product Expansions, including $O(α_s^2)$ terms in perturbative QCD. For the scalar form factor, a low energy theorem and phase information on the unitarity cut are implemented to further constrain the shape parameters. We finally determine points on the real axis and isolate regions in the complex energy plane where zeros of the form factors are excluded.
Analyticity and unitarity techniques are employed to obtain bounds on the shape parameters of the scalar and vector form factors of semileptonic $K_{l3}$ decays. For this purpose we use vector and scalar correlators evaluated in pQCD, a low energy theorem for scalar form factor, lattice results for the ratio of kaon and pion decay constants, chiral perturbation theory calculations for the scalar form factor at the Callan-Treiman point and experimental information on the phase and modulus of $Kπ$ form factors up to an energy $\tin=1 {\rm GeV}^2$. We further derive regions on the real axis and i…
We use the recently measured accurate BaBaR data on the modulus of the pion electromagnetic form factor, $F_π(t)$, up to an energy of 3 GeV, the I=1 $P$-wave phase of the $ππ$ scattering amplitude up to the $ω-π$ threshold, the pion charge radius known from Chiral Perturbation Theory, and the recently measured JLAB value of $F_π$ in the spacelike region at $t=-2.45 {\rm GeV}^2$ as inputs in a formalism that leads to bounds on $F_π$ in the intermediate spacelike region. We compare our constraints with experimental data and with perturbative QCD along with the results of several theoretical mode…
We use unitarity and analyticity of scattering amplitudes to constrain fermionic operators in the standard model effective field theory. For four-fermion operators at mass dimension 8, we scatter flavor superpositions in fixed standard model representations and find the Wilson coefficients to be constrained so that their contraction with any pair of pure density matrices is positive. These constraints imply that flavor-violating couplings are upper-bounded by their flavor-conserving cousins. For instance, LEP data already appears to preclude certain operators in upcoming $μ\to 3e$ measurements…
Applying the three axiomatic criteria of Lorentz invariance, analyticity and unitarity to scattering amplitudes involving the Goldstone bosons and the Higgs boson, we derive a general sum rule for the Strongly Interacting Light Higgs Lagrangian. This sum rule connects the IR coefficient $c_H$ to the UV properties of the theory, and can be used, for instance, to capture the role of resonances in processes like $V_{\rm L}V_{\rm L}\to hh$ and $V_{\rm L}V_{\rm L}\to V_{\rm L}V_{\rm L}$, with $V=W^{\pm},Z$.
We derive rigorous bounds on corrections to Einstein gravity using unitarity and analyticity of graviton scattering amplitudes. In $D\geq 4$ spacetime dimensions, these consistency conditions mandate positive coefficients for certain quartic curvature operators. We systematically enumerate all such positivity bounds in $D=4$ and $D=5$ before extending to $D\geq 6$. Afterwards, we derive positivity bounds for supersymmetric operators and verify that all of our constraints are satisfied by weakly-coupled string theories. Among quadratic curvature operators, we find that the Gauss-Bonnet term in …
We study a model-independent parameterization of the vector pion form factor that arises from the constraints of analyticity and unitarity. Our description should be suitable up to sqrt(s) ~ 1.2 GeV and allows a model-independent determination of the mass of the rho(770) resonance, M(rho) = (775.1 +- 0.5) MeV. We analyse the experimental data on tau(-) -> pion(-) pion(0) neutrino(tau), in this framework, and its consequences on the low-energy observables worked out by chiral perturbation theory. An evaluation of the two-pion contribution to the anomalous magnetic moment of the muon, a_muon, an…
We consider the application of the DRA method to the case of several master integrals in a given sector. We establish a connection between the homogeneous part of dimensional recurrence and maximal unitarity cuts of the corresponding integrals: a maximally cut master integral appears to be a solution of the homogeneous part of the dimensional recurrence relation. This observation allows us to make a necessary step of the DRA method, the construction of the general solution of the homogeneous equation, which, in this case, is a coupled system of difference equations.
Leading (large) logarithms in non-renormalizable theories have been investigated in the recent past. Besides some general considerations, explicit results for the expansion coefficients (in terms of leading logarithms) of partial wave amplitudes and of scalar and vector form factors have been given. Analyticity and unitarity constraints haven been used to obtain the expansion coefficients of partial waves in massless theories, yielding form factors and the scalar two-point function to five-loop order in the O(4)/O(3) model. Later, the all order solutions for the partial waves in any O(N+1)/O(N…
We consider the vector and scalar form factors of the charm-changing current responsible for the semileptonic decay D\rightarrow πl ν. Using as input dispersion relations and unitarity for the moments of suitable heavy-light correlators evaluated with Operator Product Expansions, including O(α_s^2) terms in perturbative QCD, we constrain the shape parameters of the form factors and find exclusion regions for zeros on the real axis and in the complex plane. For the scalar form factor, a low energy theorem and phase information on the unitarity cut are also implemented to further constrain the s…
The shape of the vector and scalar K_{l3} form factors is investigated by exploiting analyticity and unitarity in a model-independent formalism. The method uses as input dispersion relations for certain correlators computed in perturbative QCD in the deep Euclidean region, soft-meson theorems, and experimental information on the phase and modulus of the form factors along the elastic part of the unitarity cut. We derive constraints on the coefficients of the parameterizations valid in the semileptonic range and on the truncation error. The method also predicts low-energy domains in the complex…
We derive non-linear recursion equations for the leading infrared logarithms in massless non-renormalizable effective field theories. The derivation is based solely on the requirements of the unitarity, analyticity and crossing symmetry of the amplitudes. That emphasizes the general nature of the corresponding equations. The derived equations allow one to compute leading infrared logarithms to essentially unlimited loop order without performing a loop calculation. For the implementation of the recursion equation one needs to calculate tree diagrams only. The application of the equation is demo…
A model for baryonic Regge trajectories compatible with the threshold behavior required by unitarity and asymptotic behavior in agreement with analyticity constraints is given in explicit form. Widths and masses of the baryonic resonances on the N and $Δ$ trajectories are reproduced. The MacDowell symmetry is exploited and an application is given.
Motivated by the discrepancies noted recently between the theoretical calculations of the electromagnetic $ωπ$ form factor and certain experimental data, we investigate this form factor using analyticity and unitarity in a framework known as the method of unitarity bounds.We use a QCD correlator computed on the spacelike axis by operator product expansion and perturbative QCD as input, and exploit unitarity and the positivity of its spectral function, including the two-pion contribution that can be reliably calculated using high-precision data on the pion form factor. From this information, we…
Bounds on the shape parameters of K_{l3} form factors are investigated in a general model independent framework. Using the method of unitarity bounds for correlators evaluated in pQCD, chiral symmetry and experimental information on the phase and modulus of πK form factors up to an energy t_{in}=1 GeV^2, we constrain the shape parameters of the scalar and vector form factors, the value of the scalar form factor at t=M_π^2-M_K^2, and exclude regions where zeros can exist on the real energy line and in the complex energy plane.
We compute spin-flip cross section for graviton photoproduction on a spin-1/2 target of finite mass. Using this tree-level result, we find one-loop graviton correction to the spin-flip low-energy forward Compton scattering amplitude by using Gerasimov-Drell-Hearn sum rule. We show that this result agrees with the corresponding perturbative computations, implying the validity of the sum rule at one-loop level, contrary to the previous claims. We discuss possible effects from the black hole production and string Regge trajectory exchange at very high energies. These effects seem to soften the UV…
If the Electroweak Symmetry Breaking Sector turns out to be strongly interacting, the actively investigated effective theory for longitudinal gauge bosons plus Higgs can be efficiently extended to cover the regime of saturation of unitarity (where the perturbative expansion breaks down). This is achieved by dispersion relations, whose subtraction constants and left cut contribution can be approximately obtained in different ways giving rise to different unitarization procedures. We illustrate the ideas with the Inverse Amplitude Method, one version of the N/D method and another improved versio…
We consider the evaluation of the $ηπ$ isospin-violating vector and scalar form factors relying on a systematic application of analyticity and unitarity, combined with chiral expansion results. It is argued that the usual analyticity properties do hold (i.e. no anomalous thresholds are present) in spite of the instability of the $η$ meson in QCD. Unitarity relates the vector form factor to the $ηπ\to ππ$ amplitude: we exploit progress in formulating and solving the Khuri-Treiman equations for $η\to 3π$ and in experimental measurements of the Dalitz plot parameters to evaluate the shape of the …
The two-pion contribution from low energies to the muon magnetic moment anomaly, although small, has a large relative uncertainty since in this region the experimental data on the cross sections are neither sufficient nor precise enough. It is therefore of interest to see whether the precision can be improved by means of additional theoretical information on the pion electromagnetic form factor, which controls the leading order contribution. In the present paper we address this problem by exploiting analyticity and unitarity of the form factor in a parametrization-free approach that uses the p…
High-statistics data on the $e^+e^-\to π^+π^-$ cross section and the pion vector form factor have been obtained recently by several collaborations. Unfortunately, there are some tensions between different datasets, especially the most precise ones, which have not been resolved so far. Additional independent constraints on the data are therefore of interest. We consider a parametrization-free method of analytic extrapolation proposed recently, which is based on a mixed phase and modulus extremal problem and combines rigorous upper and lower bounds with numerical simulations to account for the s…
The two-pion low-energy contribution to the anomalous magnetic moment of the muon, $a_μ\equiv(g-2)_μ/2$, expres sed as an integral over the modulus squared of the pion electromagnetic form fac tor, brings a relatively large contribution to the theoretical error, since the low accuracy of experimental measurements in this region is amplified by the drastic increase of the integration kernel. We derive stringent constraints on the two-pion contribution by exploiting analyticity and unitarity of the pion electromagnetic form factor. To avoid the poor knowledge of the modulus of this function, we …
The availability of a reliable bound on an integral involving the square of the modulus of a form factor on the unitarity cut allows one to constrain the form factor at points inside the analyticity domain and its shape parameters, and also to isolate domains on the real axis and in the complex energy plane where zeros are excluded. In this lecture note, we review the mathematical techniques of this formalism in its standard form, known as the method of unitarity bounds, and recent developments which allow us to include information on the phase and modulus along a part of the unitarity cut. We…
We present a general formalism for deriving bounds on the shape parameters of the weak and electromagnetic form factors using as input correlators calculated from perturbative QCD, and exploiting analyticity and unitarity. The values resulting from the symmetries of QCD at low energies or from lattice calculations at special points inside the analyticity domain can beincluded in an exact way. We write down the general solution of the corresponding Meiman problem for an arbitrary number of interior constraints and the integral equations that allow one to include the phase of the form factor alo…
It is shown that the fundamental properties of gauge-independence, gauge-invariance, unitarity, and analyticity of the $S$-matrix lead to the unambiguous generalization of the pinch technique algorithm to two loops.
The problem of $γπ\to ππ$ is studied using the axial anomaly, elastic unitarity, analyticity and crossing symmetry. The solution of the integral equation for this amplitude is given by an iteration procedure.The final solution disagrees with vector meson dominance models.
In the approach, based on analyticity and unitarity and assuming an influence of coupled channels, experimental data on the isovector P-wave of pion-pion scattering have been analyzed to study rho-like mesons below 1.9 GeV.
From the properties of analyticity and unitarity it has been recently obtained an exponentiated expression for the pion form factor. In this work I show the validity of this expression comparing its order p^6 term with the one exactly calculated in ChPT.
Using our present knowledge on effective hadronic theories, short-distance QCD information, the $1/N_C$ expansion, analyticity and unitarity, we derive an expression for the pion form factor, in terms of $m_π, m_K, M_ρ$ and $f_π$. This parameter-free prediction provides a surprisingly good description of the experimental data up to energies of the order of 1 GeV.
The form factors of the light pseudoscalar mesons are investigated in a dispersive formalism based on hadronic unitarity, analyticity and the OPE expansion of the QCD Green functions. We propose generalizations of the original mathematical techniques, suitable for including additional low energy information provided by experiment or Chiral Perturbation Theory (CHPT). The simultaneous treatment of the electroweak form factors of the $π$ and $K$ mesons allows us to test the consistency with QCD of a low energy CHPT theorem. By applying the formalism to the pion electromagnetic form factor, we de…
Bounds on the expansion coefficients of the strangeness changing $Kπ$ form factors were derived recently from analyticity and unitarity, using as input suitable correlation functions calculated by perturbative QCD in the Euclidian region. We investigate two types of invariant amplitudes and their corresponding dispersion relations, and show that they lead to similar results for the shape of the vector and scalar $K_{\ell 3}$ form factors.
Starting from hyperbolic dispersion relations, we derive a closed system of Roy-Steiner equations for pion-nucleon scattering that respects analyticity, unitarity, and crossing symmetry. We work out analytically all kernel functions and unitarity relations required for the lowest partial waves. In order to suppress the dependence on the high-energy regime we also consider once- and twice-subtracted versions of the equations, where we identify the subtraction constants with subthreshold parameters. Assuming Mandelstam analyticity we determine the maximal range of validity of these equations. As…
We test the consistency of the data on the nucleon structure functions with analyticity and the Regge asymptotics of the virtual Compton amplitude. By solving a functional extremal problem, we derive an optimal lower bound on the maximum difference between the exact amplitude and the dominant Reggeon contribution for energies $ν$ above a certain high value $ν_h(Q^2)$. Considering in particular the difference of the amplitudes $T_1^\inel(ν, Q^2)$ for the proton and neutron, we find that the lower bound decreases in an impressive way when $ν_h(Q^2)$ is increased, and represents a very small frac…
I present a method for evaluating the hadronic vacuum polarization contribution below 1 GeV to a_μwhich relies on analyticity, unitarity and chiral symmetry, as well as on data. The main advantage is that in the region just above threshold, where data are either scarce or have large errors, these theoretical constraints are particularly strong, and therefore allow us to reduce the uncertainties with respect to a purely data-based evaluation. Some preliminary numerical results are presented as illustration of the method.
We propose a new parametrization of the $B\toπ$ vector form factor, $f_+(q^2)$, as an expansion in powers of a conformal mapping variable, which satisfies unitarity, analyticity and perturbative QCD scaling. The unitarity constraint is used also for defining the systematic error of the expansion. We fit with the new parametrization the available experimental and theoretical information on exclusive $B\toπlν$ decays, making a conservative estimate of the effects of correlations in the systematic and statistical errors of the lattice results. With four parameters to describe $f_+(q^2)$, the syst…
We discuss the consistency of fermion mass generation by boundary conditions and brane localized terms in higher dimensional Higgsless models of gauge symmetry breaking. The sum rules imposed by tree-level unitarity and Ward Identities are applied to check the consistency of mass generation by orbifold projections and more general boundary conditions consistent with the variational principle. We find that the sum rules are satisfied for boundary conditions corresponding to brane localized mass and kinetic terms consistent with the reduced gauge symmetry on the brane.
We reevaluate the two form factors relevant for the $ηπ$ second-class $τ$ decay mode, making systematic use of analyticity, unitarity, combined with updated inputs to the NLO chiral constraints. We focus, in particular, on the shape of the $ρ$ resonance peak which is a background-free signature of a second-class current. Its dispersive construction requires the $ηπ\toππ$ scattering amplitude which we derive from a family of Khuri-Treiman equations solutions constrained with accurate recent results on the $η\to3π$ Dalitz plot.
We present a detailed numerical study of the K_{e3} decays to O(p^6,(m_d-m_u)p^2,e^2p^2) in chiral perturbation theory with virtual photons and leptons. We describe the extraction of the CKM matrix element |V_{us}| from the experimental K_{e3} decay parameters. We propose a consistency check of the K^+_{e3} and K^0_{e3} data that is largely insensitive to the dominating theoretical uncertainties, in particular the contributions of order p^6. Our analysis is highly relevant in view of the recent high statistics measurement of the K^+_{e3} branching ratio by E865 at Brookhaven which does not ind…
A model for a Regge trajectory compatible with the threshold behavior required by unitarity and asymptotics in agreement with Mandelstam analyticity is analyzed and confronted with the experimental data on the spectrum of the $ρ$ trajectory as well as those on the $π^- p\to π^0 n$ charge-exchange reaction. The fitted trajectory deviates considerably from a linear one both in the space-like and time-like regions, matching nicely between the two.
In this work we discuss in detail the non-perturbative determination of the momentum dependence of the form factors entering in semileptonic decays using unitarity and analyticity constraints. The method contains several new elements with respect to previous proposals and allows to extract, using suitable two-point functions computed non-perturbatively, the form factors at low momentum transfer $q^2$ from those computed explicitly on the lattice at large $q^2$, without any assumption about their $q^2$-dependence. The approach will be very useful for exclusive semileptonic $B$-meson decays, whe…
The combined 3-channel analysis of experimental data on the coupled processes $ππ\toππ,K\bar{K},ηη$ is carried out in the channel with the vacuum quantum numbers. An approach, using only first principles (analyticity and unitarity) and the uniformizing variable, is applied. Definite indications of the QCD nature of the $f_0$ resonances below 1.9 GeV are obtained, among them a surprising indication for $f_0(980)$ to be the $ηη$ bound state. An assignment of the scalar mesons below 1.9 GeV to lower nonets is proposed.
Nuclear beta decays between (J^pi,T) = (0^+,1) analog states yield the best value for the Vud element of the Cabibbo-Kobayashi-Maskawa matrix. Current world data establish the corrected Ft values of 14 separate superallowed transitions to a precision of order 0.1% or better. The validity of the small theoretical correction terms is confirmed by excellent consistency among the 14 Ft values and by recent measurements that compare pairs of mirror superallowed transitions. With consistency established, the results now yield |Vud| = 0.97420(21). This value is consistent with the considerably less p…
We use a multiply-subtracted Omnes dispersion relation for the form factor f^+ in B->pi semileptonic decay, allowing the direct input of experimental and theoretical information to constrain its dependence on q^2. Apart from these inputs we use only unitarity and analyticity properties. We obtain |V(ub)|=(4.02 \pm 0.35)\times 10^{-3}, improving the agreement with the value determined from inclusive methods, and competitive in precision with them.
We analyze the quantum consistency of anomalous abelian models and of their effective field theories, rendered anomaly-free by a Wess-Zumino term, in the case of multiple abelian symmetries. These models involve the combined Higgs-Stuckelberg mechanism and predict a pseudoscalar axion-like field that mixes with the goldstones of the ordinary Higgs sector. We focus our study on the issue of unitarity of these models both before and after spontaneous symmetry breaking and detail the set of Ward identities and the organization of the loop expansion in the effective theory. The analysis is perform…
We present estimates of the hyperon elastic form factors for the baryon octet and the $Ω^-$ baryon for large four-momentum transfer squared, $q^2$, in the timelike region ($q^2>0$). Experimentally, those form factors can be extracted from the $e^+ e^- \to B \bar B$ and $p \bar p \to B \bar B$ processes, where $B$ stands for a general baryon. Our results are based on calculations of the elastic electromagnetic form factors in the spacelike region ($Q^2 = - q^2 > 0$) within a covariant quark model. To connect the results in the spacelike region to those in the timelike region, we use asymptotic …
We analyse Belle data on the decay $τ^- \to π^- π^0 ν_τ$ using a dispersive representation of the vector form factor which is consistent with chiral symmetry and preserves analyticity and unitarity exactly. We fit the unknown theoretical parameters from the data, determining the values of the related low-energy observables $\left\langle r^2\right\rangle^π_V$ and $c_V^π$. The implementation of isospin breaking effects is also discussed.
