Subtraction Constants Research Articles

Non-perturbative methods for a chiral effective field theory of finite density nuclear systems

Recently we have developed a novel chiral power counting scheme for an effective field theory of nuclear matter with nucleons and pions as degrees of freedom [1]. It allows for a systematic expansion taking into account both local as well as pion-mediated multi-nucleon interactions. We apply this power counting in the present study to the evaluation of the pion self-energy and the energy density in nuclear and neutron matter at next-to-leading order. To implement this power counting in actual calculations we develop here a non-perturbative method based on Unitary Chiral Perturbation Theory for performing the required resummations. We show explicitly that the contributions to the pion self-energy with in-medium nucleon–nucleon interactions to this order cancel. The main trends for the energy density of symmetric nuclear and neutron matter are already reproduced at next-to-leading order. In addition, an accurate description of the neutron matter equation of state, as compared with sophisticated many-body calculations, is obtained by varying only slightly a subtraction constant around its expected value. The case of symmetric nuclear matter requires the introduction of an additional fine-tuned subtraction constant, parameterizing the effects from higher order contributions. With that, the empirical saturation point and the nuclear matter compressibility are well reproduced while the energy per nucleon as a function of density closely agrees with sophisticated calculations in the literature.

Dual parametrization and Abel transform tomography for twist-3 DVCS

We derive the dual parametrization for the DVCS amplitude to the twist-3 accuracy. The relation of the dual parametrization to the Abel transform tomography for amplitudes of hard exclusive processes is elaborated. We prove the dispersion relations for the twist-3 DVCS amplitude in the Wandzura–Wilczek approximation and show that the subtraction constant is given by the D-term form factor.

Vector meson-vector meson interaction in a hidden gauge unitary approach

The formalism developed recently to study vector meson-vector meson interaction, and applied to the case of $\ensuremath{\rho}\ensuremath{\rho}$, is extended to study the interaction of the nonet of vector mesons among themselves. The interaction leads to poles of the scattering matrix corresponding to bound states or resonances. We show that 11 states (either bound or resonant) get dynamically generated in nine strangeness-isospin-spin channels. Five of them can be identified with those reported in the PDG, i.e., the ${f}_{0}(1370)$, ${f}_{0}(1710)$, ${f}_{2}(1270)$, ${f}_{2}^{\ensuremath{'}}(1525)$, and ${K}_{2}^{*}(1430)$. The masses of the latter three tensor states have been used to fine-tune the free parameters of the unitary approach, i.e., the subtraction constants in evaluating the vector meson-vector meson loop functions in the dimensional regularization scheme. The branching ratios of these five dynamically generated states are found to be consistent with data. The existence of the other six states should be taken as predictions to be tested by future experiments.

Improved dispersion relations for [formula omitted

We perform a dispersive theoretical study of the reaction γγ→π0π0 emphasizing the low energy region. The large source of theoretical uncertainty to calculate the γγ→π0π0 total cross section for s≳0.5 GeV within the dispersive approach is removed. This is accomplished by taking one more subtraction in the dispersion relations, where the extra subtraction constant is fixed by considering new low energy constraints, one of them further refined by taking into consideration the f0(980) region. This allows us to make sharper predictions for the cross section for s≲0.8 GeV, below the onset of D-wave contributions. In this way, were new more precise data on γγ→π0π0 available one might then distinguish between different parameterizations of the ππ isoscalar S-wave. We also elaborate on the width of the σ resonance to γγ and provide new values.

Rare decayπ0→e+e−: Theory confronts KTeV data

Within the dispersive approach to the amplitude of the rare decay ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}$ the nontrivial dynamics is contained only in the subtraction constant. We express this constant, in the leading order in $({m}_{e}/\ensuremath{\Lambda}{)}^{2}$ perturbative series, in terms of the inverse moment of the pion transition form factor given in symmetric kinematics. By using the CELLO and CLEO data on the pion transition form factor given in asymmetric kinematics, the lower bound of the decay branching ratio is found. The restrictions following from QCD allow us to make a quantitative prediction for the branching $B({\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}})=(6.2\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\cdot\else\textperiodcentered\fi{}{10}^{\ensuremath{-}8}$ which is $3\ensuremath{\sigma}$ below the recent KTeV measurement. We confirm our prediction by using the quark models and phenomenological approaches based on the vector meson dominance. The decays $\ensuremath{\eta}\ensuremath{\rightarrow}{l}^{+}{l}^{\ensuremath{-}}$ are also discussed.

Finite-bandwidth Kramers-Kronig relations for acoustic group velocity and attenuation derivative applied to encapsulated microbubble suspensions

Kramers-Kronig (KK) analyses of experimental data are complicated by the conflict between the inherently bandlimited data and the requirement of KK integrals for a complete infinite spectrum of input information. For data exhibiting localized extrema, KK relations can provide accurate transforms over finite bandwidths due to the local-weighting properties of the KK kernel. Recently, acoustic KK relations have been derived for the determination of the group velocity (cg) and the derivative of the attenuation coefficient (alpha') (components of the derivative of the acoustic complex wave number). These relations are applicable to bandlimited data exhibiting resonant features without extrapolation or unmeasured parameters. In contrast to twice-subtracted finite-bandwidth KK predictions for phase velocity and attenuation coefficient (components of the undifferentiated wave number), these more recently derived relations for cg and alpha' provide stricter tests of causal consistency because the resulting shapes are invariant with respect to subtraction constants. The integrals in these relations can be formulated so that they only require the phase velocity and attenuation coefficient data without differentiation. Using experimental data from suspensions of encapsulated microbubbles, the finite-bandwidth KK predictions for cg and alpha' are found to provide an accurate mapping of the primary wave number quantities onto their derivatives.

MASS AND WIDTH OF THE σ

I report on recent work done in collaboration with Irinel Caprini and Gilberto Colangelo1. We observe that the Roy equations lead to a representation of the ππ scattering amplitude that exclusively involves observable quantities, but is valid for complex values of s. At low energies, this representation is dominated by the contributions from the two subtraction constants, which are known to remarkable precision from the low energy theorems of chiral perturbation theory. Evaluating the remaining contributions on the basis of the available data, we demonstrate that the lowest resonance carries the quantum numbers of the vacuum and occurs in the vicinity of the threshold. Although the uncertainties in the data are substantial, the pole position can be calculated quite accurately, because it occurs in the region where the amplitude is dominated by the subtractions. The calculation neatly illustrates the fact that the dynamics of the Goldstone bosons is governed by the symmetries of QCD.

On the lowest resonance of QCD

I report on recent work done in collaboration with Irinel Caprini and Gilberto Colangelo. We observe that the Roy equations lead to a representation of the ππ scattering amplitude that exclusively involves observable quantities, but is valid for complex values of s. At low energies, this representation is dominated by the contributions from the two subtraction constants, which are known to remarkable precision from the low energy theorems of chiral perturbation theory. Evaluating the remaining contributions on the basis of the available data, we demonstrate that the lowest resonance carries the quantum numbers of the vacuum and occurs in the vicinity of the threshold. Although the uncertainties in the data are substantial, the pole position can be calculated quite accurately, because it occurs in the region where the amplitude is dominated by the subtractions. The calculation neatly illustrates the fact that the dynamics of the Goldstone bosons is governed by the symmetries of QCD.

An improved study of the kappa resonance and the non-exotic s wave πK scatterings up to [formula omitted] of LASS data

We point out that the dispersion relation for the left-hand cut integral presented in one of our previous paper [H.Q. Zheng, et al., Nucl. Phys. A 733 (2004) 235] is actually free of subtraction constant, even for unequal mass elastic scatterings. A new fit to the LASS data [D. Aston, et al., LASS Collaboration, Nucl. Phys. B 296 (1988) 493] is performed and firm evidence for the existence of κ pole is found. The correct use of analyticity also put strong constraints on threshold parameters—which are found to be in good agreement with those obtained from chiral theories. We also determined the pole parameters of K 0 ∗ ( 1430 ) on the second sheet, and reconfirm the existence of K 0 ∗ ( 1950 ) on the third sheet. We stress that the LASS data do not require them to have the twin pole structure of a typical Breit–Wigner resonance.

S wave Kπ scattering and effects of κ in [formula omitted

Kπ S wave scattering is studied using a chiral unitary approach (ChUT) taking into account coupled channels. With the amplitudes derived from the lowest order chiral Lagrangian as the kernel of a set of coupled channel Bathe–Salpeter equations, the I = 1 / 2 S wave Kπ scattering phase shifts below 1.2 GeV can be fitted by one parameter, a subtraction constant, and a scalar resonance corresponding to the controversial κ ( K 0 ∗ ( 800 ) ) can be generated dynamically. A good description of the I = 3 / 2 S wave Kπ scattering phase shifts below 1.2 GeV can also be obtained. An artificial singularity in the cut-off method of the 2-meson loop integral of the ChUT is found. The formalism is applied to deal with the S wave Kπ final state interaction (FSI) in the decay J / ψ → K ¯ ∗ 0 ( 892 ) K + π − , and a qualitatively good fit to the data is achieved. The role of κ in the decay is discussed.

The Fubini-Furlan-Rosetti sum rule and related aspects in light of covariant baryon chiral perturbation theory⋆

We analyze the Fubini-Furlan-Rosetti sum rule in the framework of covariant baryon chiral perturbation theory to leading one-loop accuracy and including next-to-leading-order polynomial contributions. We discuss the relation between the subtraction constants in the invariant amplitudes and certain low-energy constants employed in earlier chiral perturbation theory studies of threshold neutral pion photoproduction off nucleons. In particular, we consider the corrections to the sum rule due to the finite pion mass and show that below the threshold they agree well with determinations based on fixed-t dispersion relations. We also discuss the energy dependence of the electric dipole amplitude E0+.

Finite‐bandwidth Kramers–Kronig relations for the determination of acoustic group velocity and frequency derivative of attenuation

Kramers–Kronig (K–K) analyses of experimental data are complicated by the extrapolation problem, which is concerned with how the unexamined spectral bands impact K–K calculations. In this talk the causal linkages in resonant‐type data are demonstrated without extrapolation through acoustic K–K relations for the group velocity and the derivative of the attenuation coefficient (components of the derivative of the complex wave number). Due to their shape‐invariant predictions with respect to subtraction constants, these new relations provide stricter tests of causal consistency than previously established K–K relations for the phase velocity and attenuation coefficient. For both the group velocity and attenuation derivative, three forms of the relations are derived. These relations are equivalent for bandwidths covering the infinite spectrum, but differ when restricted to finite spectral windows. Using experimental data from microsphere suspensions, the accuracy of finite‐bandwidth K–K predictions for group velocity, and the attenuation derivative are demonstrated. Of the multiple methods, the most accurate forms were found to be those whose integrals were expressed only in terms of the phase velocity, and attenuation coefficient themselves requiring no differentiated quantities.

Causal determination of acoustic group velocity and frequency derivative of attenuation with finite-bandwidth Kramers-Kronig relations

Kramers-Kronig (KK) analyses of experimental data are complicated by the extrapolation problem, that is, how the unexamined spectral bands impact KK calculations. This work demonstrates the causal linkages in resonant-type data provided by acoustic KK relations for the group velocity (c(g)) and the derivative of the attenuation coefficient (alpha') (components of the derivative of the acoustic complex wave number) without extrapolation or unmeasured parameters. These relations provide stricter tests of causal consistency relative to previously established KK relations for the phase velocity (c(p)) and attenuation coefficient (alpha) (components of the undifferentiated acoustic wave number) due to their shape invariance with respect to subtraction constants. For both the group velocity and attenuation derivative, three forms of the relations are derived. These relations are equivalent for bandwidths covering the entire infinite spectrum, but differ when restricted to bandlimited spectra. Using experimental data from suspensions of elastic spheres in saline, the accuracy of finite-bandwidth KK predictions for c(g) and alpha' is demonstrated. Of the multiple methods, the most accurate were found to be those whose integrals were expressed only in terms of the phase velocity and attenuation coefficient themselves, requiring no differentiated quantities.

Causality and acoustic group velocity in dispersive media

This talk examines two specific issues related to causality and the group velocity of acoustic pulses in dispersive media. First, the causal prediction of group velocity from attenuation with finite-bandwidth Kramers-Kronig (K-K) relations is discussed. Without extrapolating beyond the measurement bandwidth, the causal linkages of resonant-type data are established using expressions derived from the acoustic K-K relations to predict group velocity and slope of attenuation (frequency derivative of attenuation). These predictions provide a stricter test of causal consistency than the determination of the phase velocity and attenuation coefficient due to their shape-invariance with respect to subtraction constants. Secondly, conditions under which the group velocity is the velocity of the peak in the envelope of a acoustic pulse are described. The spatial and temporal signatures of acoustic pulses with arbitrarily large (e.g., superluminal) and negative group velocities are also demonstrated. These signatures are shown to be consistent with both simple and relativistic causality.

Detailed Analysis of the Chiral Unitary Model for Meson-Baryon Scattering with Flavor SU (3) Breaking Effects

We study s-wave meson-baryon scattering using the chiral unitary model. We consider $1/2^{-}$ baryon resonances as quasibound states of the low lying mesons ($\pi,K,\eta$) and baryons ($N,\Lambda,\Sigma,\Xi$). In previous works, the subtraction constants which appear in loop integrals were found to largely depend on the channels, and it was necessary to fit these constants to reproduce the data. In order to extend this model to all channels with fewer parameters, we introduce flavor SU(3) breaking interactions in the framework of chiral perturbation theory. It is found, however, that the observed SU(3) breaking in meson-baryon scattering cannot be explained by the present SU(3) breaking interactions. The role and importance of the subtraction constants in the present framework are discussed.

Extrema bounds for the soft Pomeron intercept

By using an extended Regge parametrization and taking into account the discrepancies in the high-energy pp and p̄p total cross section data in both accelerator and cosmic-ray regions, we estimate extrema bounds for the soft Pomeron intercept. First we consider two ensembles of data with either the CDF or the E710 and E811 results for σtotp̄p at 1.8 TeV, from which we obtain the bounds 1.102 and 1.081, respectively. These ensembles are then combined with the highest and lowest estimations for σtotpp from cosmic-ray experiments (6–40 TeV), leading to the upper and lower bounds 1.109 and 1.082, respectively. The effects of simultaneous fits to σtot and ρ, individual fits to σtot, and the influence of the subtraction constant in the dispersion relations are also presented. Our global results favor the E710 and E811 data.

Non-perturbative renormalisation of four fermion operators and B0 − B¯0 mixing with Wilson fermions

We present new results for the renormalisation and subtraction constants for the four fermionΔ F = 2 operators, computed non-perturbatively in the RI-MOM scheme (in the Landau gauge). From our preliminary analysis of the lattice data at β = 6.45, for the B 0 − B¯ 0 mixing bag-parameter we obtain B B RGI = 1.46(7)(1).

Analytic models and forward scattering from accelerator to cosmic-ray energies

Analytic models for hadron-hadron scattering are characterized by analytical parametrizations for the forward amplitudes and the use of dispersion relation techniques to study the total cross section $\sigma_{tot}$ and the $\rho$ parameter. In this paper we investigate four aspects related to the application of the model to $pp$ and $\bar{p}p$ scattering, from accelerator to cosmic-ray energies: 1) the effect of different estimations for $\sigma_{tot}$ from cosmic-ray experiments; 2) the differences between individual and global (simultaneous) fits to $\sigma_{tot}$ and $\rho$; 3) the role of the subtraction constant in the dispersion relations; 4) the effect of distinct asymptotic inputs from different analytic models. This is done by using as a framework the single Pomeron and the maximal Odderon parametrizations for the total cross section. Our main conclusions are the following: 1) Despite the small influence from different cosmic-ray estimations, the results allow us to extract an upper bound for the soft pomeron intercept: $1 + \epsilon = 1.094$; 2) although global fits present good statistical results, in general, this procedure constrains the rise of $\sigma_{tot}$; 3) the subtraction constant as a free parameter affects the fit results at both low and high energies; 4) independently of the cosmic-ray information used and the subtraction constant, global fits with the odderon parametrization predict that, above $\sqrt s \approx 70$ GeV, $\rho_{pp}(s)$ becomes greater than $\rho_{\bar{p}p}(s)$, and this result is in complete agreement with all the data presently available. In particular, we infer $\rho_{pp} = 0.134 \pm 0.005$ at $\sqrt s = 200$ GeV and $0.151 \pm 0.007$ at 500 GeV (BNL RHIC energies).

Dispersion relations and soft pion theorems for K→ππ

We propose a new method to obtain the K -> pi pi amplitude from K -> pi which allows one to fully account for the effects of final state interactions. The method is based on a set of dispersion relations for the K -> pi pi amplitude in which the weak Hamiltonian carries momentum. The soft pion theorem, which relates this amplitude to the K -> pi amplitude, can be used to determine one of the two subtraction constants - the second constant is at present known only to leading order in chiral perturbation theory. We solve the dispersion relations numerically and express the result in terms of the unknown higher order corrections to this subtraction constant.

Scalar gluonium and instantons

The impact of QCD instantons on scalar glueball properties is studied in the framework of an instanton-improved operator product expansion (IOPE) for the ${0}^{++}$ glueball correlation function. Direct instanton contributions are found to strongly dominate over those from perturbative fluctuations and soft vacuum fields. All IOPE sum rules, including the one involving a subtraction constant, show a high degree of stability and are, in contrast to previous glueball sum rules, consistent with the low-energy theorem for the zero-momentum correlator. The predicted glueball mass ${m}_{G}=1.53\ifmmode\pm\else\textpm\fi{}0.2 \mathrm{GeV}$ is less sensitive to the instanton contributions then the glueball coupling (residue) ${f}_{G}=1.01\ifmmode\pm\else\textpm\fi{}0.25 \mathrm{GeV},$ which increases by about half an order of magnitude. Both glueball properties are shown to obey scaling relations as a function of the average instanton size and density.