Unsubtracted Dispersion Relations Research Articles

Electromagnetic form factors of the transition from the spin-3/2 Σ to the Λ hyperon

The three electromagnetic form factors for the transition from a $3/{2}^{+}\phantom{\rule{4pt}{0ex}}{\mathrm{\ensuremath{\Sigma}}}^{*}$ hyperon to the ground-state $\mathrm{\ensuremath{\Lambda}}$ hyperon are studied. At low energies, combinations of the transition form factors can be deduced from Dalitz decays of the ${\mathrm{\ensuremath{\Sigma}}}^{*}$ hyperon to $\mathrm{\ensuremath{\Lambda}}$ plus an electron-positron pair. It is pointed out how more information can be obtained with the help of the self-analyzing weak decay of the $\mathrm{\ensuremath{\Lambda}}$. In particular, it is shown that these transition form factors are complex quantities already in this kinematical region. Such measurements are feasible at hyperon factories such as, for instance, the Facility for Antiproton and Ion Research (FAIR). At higher energies, the transition form factors can be measured in electron-positron collisions. The transition form factors are related to decay distributions and differential cross sections. Using dispersion theory, the low-energy electromagnetic form factors for the ${\mathrm{\ensuremath{\Sigma}}}^{*}$-to-$\mathrm{\ensuremath{\Lambda}}$ transition are related to the pion vector form factor. The additionally required input, i.e., the two-pion--${\mathrm{\ensuremath{\Sigma}}}^{*}$--$\mathrm{\ensuremath{\Lambda}}$ amplitudes, is determined from relativistic next-to-leading-order (NLO) baryon chiral perturbation theory, including the baryons from the octet and the decuplet. A poorly known NLO parameter is fixed to the experimental value of the ${\mathrm{\ensuremath{\Sigma}}}^{*}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Lambda}}\ensuremath{\gamma}$ decay width. Pion rescattering is taken into account by using dispersion theory and solving a Muskhelishvili-Omn\`es equation. Subtracted and unsubtracted dispersion relations are discussed. However, in view of the fact that the transition form factors are complex quantities, the current data situation does not allow for a full determination of the subtraction constants. To reduce the number of free parameters, unsubtracted dispersion relations are used to make predictions for the transition form factors in the low-energy space- and timelike regions.

Finiteness of two- and three-point functions and the renormalization group

Two and three point functions of composite operators are analysed with regard to (logarithmically) divergent contact terms. Using the renormalisation group of dimensional regularisation it is established that the divergences are governed by the anomalous dimensions of the operators and the leading UV-behaviour of the $1/\epsilon$-coefficient. Explicit examples are given by the $<G^2G^2>$-, $<\Theta \Theta>$-trace of the energy momentum tensor) and $<\bar q q \bar q q>$- correlators in QCD-like theories. The former two are convergent when all orders are taken into account but divergent at each order in perturbation theory implying that the latter and the the $\epsilon \to 0$ limit do not generally commute. Finite correlation functions obey unsubtracted dispersion relations which is of importance when they are directly related to physical observables. As a byproduct the $R^2$-anomaly is extended to NNLO ($O(\alpha^5)$) using a recent $<G^2G^2>$-computation.

Dispersion representation of the D-term form factor in deeply virtual Compton scattering

We review the dispersion relation formalism for the deeply virtual Compton scattering amplitudes, and derive a dispersive representation of the D-term form factor, using unsubtracted t-channel dispersion relations. Results for the D-term form factor as function of t as well as at t = 0 are shown in comparison with available model predictions and phenomenological parametrizations.

Dispersion Representation of Deeply Virtual Compton Scattering

We discuss the dispersive representation of the D-term form factor for hard exclusive reactions, using unsubtracted t-channel dispersion relations. This representation provides a microscopical interpretation of the physical content of the D-term form factor in terms of t-channel exchanges with the appropriate quantum numbers. The contribution from two-pion intermediate states is explicitly evaluated, and the corresponding results for the D-term form factor as function of t as well as at t = 0 are discussed in comparison with available model predictions and phenomenological parametrizations.

Electromagnetic form factors of nucleons with QCD constraints: Systematic study of space- and time-like regions

Elastic electromagnetic form factors of nucleons are investigated for both the time-like and the space-like momenta by using the unsubtracted dispersion relation with QCD constraints. It is shown that the calculated form factors reproduce the experimental data reasonably well; they agree with recent experimental data for the neutron magnetic form factors for the space-like data obtained by the CLAS Collaboration and are compatible with the ratio of the electric and magnetic form factors for the time-like momentum obtained by the BABAR Collaboration.

New evaluation of proton structure corrections to hydrogen hyperfine splitting

We consider the proton structure corrections to the hydrogen ground-state hyperfine structure, focusing on a state-of-the-art evaluation of the inelastic nucleon corrections — the polarizability corrections — using analytic fits to the most recent data. We find a value for the fractional correction Δpol of 1.3 ± 0.3 ppm. This is 1–2 ppm smaller than the value of Δpol one would deduce using hyperfine-splitting data and elastic proton structure corrections obtained from modern form factor fits. In addition, we discuss the derivations of the relevant formulas, paying attention to lepton mass effects and to questions surrounding the use of unsubtracted dispersion relations. PACS Nos.: 31.30.Gs, 32.10.Fn, 14.20.Dh, 13.40.Gp

On the precision of the theoretical predictions forππscattering

In a recent paper, Pelaez and Yndurain evaluate some of the low energy observables of pi pi scattering and obtain flat disagreement with our earlier results. The authors work with unsubtracted dispersion relations, so that their results are very sensitive to the poorly known high energy behaviour of the scattering amplitude. They claim that the asymptotic representation we used is incorrect and propose an alternative one. We repeat their calculations on the basis of the standard, subtracted fixed-t dispersion relations, using their asymptotics. The outcome fully confirms our earlier findings. Moreover, we show that the Regge parametrization proposed by these authors for the region above 1.4 GeV violates crossing symmetry: Their ansatz is not consistent with the behaviour observed at low energies.

A NON-SM CHARGE RADIUS AND NON-SM ANAPOLE MOMENT OF A DIRAC NEUTRINO AND THE HIGH PRECISION MEASUREMENT OF THE MUON (g - 2)

We assume that the Dirac neutrino possesses a physical non-standard model charge radius [Formula: see text] and a physical non-standard model anapole moment [Formula: see text] and calculate the contribution of these quantities to the anomalous magnetic moment of the muon aμ using an unsubtracted dispersion relation. We find that these contributions can be important and allow us to set the bound [Formula: see text].

CONTRIBUTION OF A NEUTRINO ANAPOLE MOMENT TO THE MUON ANOMALOUS MAGNETIC MOMENT

We consider a Dirac neutrino and compute the contribution of its anapole moment κ anap to the anomalous magnetic moment of the muon using an unsubtracted dispersion relation and assuming a small neutrino structure. We find that this contribution could be important and may allow us to put a bound on κ anap .

Calculation of the QED couplingα^(MZ) in the modified minimal-subtraction scheme

I calculate the QED coupling, alpha, directly in the MS-bar scheme using an unsubtracted dispersion relation for the three light quarks, and perturbative QCD for charm and bottom quarks. Compact analytical expressions are presented, making this approach particularly suitable for electroweak fits. After alpha^(-1) (m_tau) = 133.513 +- 0.025 is obtained in a first step, I perform a 4-loop renormalization group evolution with 3-loop matching conditions to arrive at alpha^(-1) (M_Z) = 127.934 +- 0.026 for alpha_s (M_Z) = 0.120. The corresponding hadronic contribution to the on-shell coupling is Delta alpha_had^(5) (M_Z) = 0.02779 +- 0.00019. The error is mainly from m_c, and from experimental uncertainties in e^+ e^- annihilation into unflavored and strange hadrons and tau decay data.

What can new time-like data tell us about the electromagnetic structure of the proton and neutron?

Unsubtracted dispersion relations are used to link new data obtained in the time-like region with the electromagnetic structure of the nucleons as measured in the space-like region. Simple analytic representations of the magnetic form factor of the proton G p M ( Q 2) and the electric form factor of the neutron G n E ( Q 2) are derived which have the following properties built-in: the correct normalization and slope at Q 2 = 0 and correct asymptotic behavior at | Q 2| → ∞. The pred of perturbative QCD at asymptotic Q 2 and meson physics at low Q 2 are combined to fit the existing data in the entire range of the four-momentum transfer squared. This leads to the following observations: (i) Measurements of cross sections for the reaction p p → e +e − taken at the Fermilab E-760 match measurements made at the SLAC-136 experiment only if the logarithmic falloff of Q 4 G p M ( Q 2), expected from leading twist predictions of perturbative quantum chromodynamics, is included. (ii) Analytic continuation of the neutron time-like form factor into the space-like interval 0.1 GeV 2 ≤ Q 2 ≤ 1 GeV 2 using the limited FENICE data gives a factor of ∼ 2 inconsistency in G n E ( Q 2) with the recent Saclay and Mainz measurements.

Possible violation of sum rules for higher-twist parton distributions

Precise measurements of polarized electro-production and Drell-Yan in the deep inelastic limit will soon provide first information on the higher twist parton distributions $g_T(x)$ and $h_L(x)$. Sum rules for higher-twist structure functions are only valid provided the corresponding Compton amplitudes satisfy un-subtracted dispersion relations. Subtracted dispersion relations have to be used when the (real part of the) forward scattering amplitudes does not fall off rapidly enough for $\nu \rightarrow \infty$ (fixed $Q^2$). Formally, such subtractions lead to $\delta$-functions at the origin in the parton distributions, which are not accessible to experiment, and the integral over the data fails to satisfy the sum rule. The $\delta$-functions in the parton distributions can be identified with the zero-modes that appear in light-front quantization. An explicit {\it infinite momentum boost} identifies these soft quark modes with low momentum contributions to the self-energy.

Heavy-meson electromagnetic mass differences from QCD.

We compute the electromagnetic mass differences of mesons containing a single heavy quark in terms of measurable data using QCD-based arguments in heavy-quark effective theory. We derive an unsubtracted dispersion relation that shows that the mass differences are caculable in terms of the properties of lowest-lying physical intermediate states. We then consider the problem in the large-N limit, where N is the number of QCD colors. In this limit, we can write a kind of double-dispersion relation for the amplitude required to determine the electromagnetic mass difference. We use this to derive analogues of the Weinberg sum rules for heavy-meson matrix elements valid to leading order in 1/N and to O(1/${\mathit{m}}_{\mathit{Q}}$) in the heavy-quark expansion. In order to obtain our final result, we assume that the electromagnetic mass differences and sum rules are dominated by the lowest-lying states in analogy with the situation for the ${\mathrm{\ensuremath{\pi}}}^{+}$-${\mathrm{\ensuremath{\pi}}}^{0}$ mass difference. Despite the fact that some of the matrix elements appearing in our final result have not yet been accurately measured, we can obtain useful estimates of the electromagnetic mass differences: for example, we obtain (${\mathit{M}}_{\mathit{B}}^{+}$-${\mathit{M}}_{\mathit{B}}^{0}$${)}^{\mathrm{EM}}$\ensuremath{\simeq}+1.8 MeV. We argue that our results are accurate to about 30%.

Remarks on the dispersion relations used to calculate Delta rho.

We use the operator product expansion to show that nonperturbative QCD corrections to \ensuremath{\Delta}\ensuremath{\rho} can be calculated using unsubtracted dispersion relations for either the transverse or the longitudinal vacuum polarization functions. Recent calculations of the nonperturbative contribution to \ensuremath{\Delta}\ensuremath{\rho} based on a nonrelativistic calculation of corrections to the tt\ifmmode\bar\else\textasciimacron\fi{} threshold are inconsistent with this result.

U(1) axial charge in the chiral limit.

The U (1) axial-vector form factor is studied using an unsubtracted dispersion relation. The associated axial charge is shown to be analytic in quark masses in the chiral limit although individual terms in the Cheng-Li separation exhibit large isospin and flavor-SU (3) symmetry breaking. A new separation of the U (1) axial charge is proposed, based on the wave functions of the physical states entering in the spectral density

Uncorrelated Two Pion Effects in the Superconvergent Dispersion Relation for the Nucleon Electromagnetic Form Factors

We incorportate uncorrelated two pion effects in the superconvergent dispersion relations for the electromagnetic form factors of nucleon which have been proposed by us to synthesize the hadron physics in the low energy and the perturbative QCD in the high energy region. Inclusion of the uncorrelated two pion effects improves the result appreciably as compared with the previous calcula­ tion where the vector dominance is assumed for the low energy region. In the previous paper we investigated the elastic electromagnetic form factors of nucleons with recourse to the unsubtracted dispersion relations with the superconver­ gence conditions. 1) We were able to synthesize the hadron and quark systems in the following respect: The calculated form factors reproduce the result of vector domi­ nance model (VDM) in the low t region, t being the squared momentum transfer, ,and asymptotically they realize the prediction of the perturbative QCD. We shall call the paper, Ref. 1), I hereafter. Good agreement was obtained with the experimental data but we still had the following problems: First, the theoretical result for the electric form factor of proton becomes slightly larger than the experimental data in the low t region. Secondly, we must take the p meson mass mp much smaller than the experimental one; the best fit was obtained for mp;;::; 0.63 Ge V while experimentally it is 0.77 GeV. It is the purpose of this paper to solve the problems by taking into account the uncorrelated two pions in the superconvergent dispersion relations as we did in the analysis of low energy hadronic processes for which thep wave two pions contrib­ ute. 2 ),3) We break up the dispersion integral into three parts at t = A12 and.t = A22 as in 1. In the region 4,u2< t < A1 2 , ,u being the pion mass, the iso-vector absorptive parts are given in terms of the helicity amplitudes 7r7r-NN with] = 1 by using the formulae given in Ref. 2). For the iso-scalar parts the vector dominance is assumed with the (f) and ¢ mesons. The higher t regions are treated in the same way as was done in, I; for the intermediate region A12< t<Al the vector dominance is also-assumed witn; heavy vector bosons phenomenologically introduced. The absorptive parts in the high t region, A22< t, are taken so as to reproduce the asymptotic form factors derived via the perturbative QCD. 4 )

One-particle contributions to [formula omitted] mixing

We calculate one-particle contributions to K 0 K 0 mixing using an approach based on unsubtracted dispersion relations. Our results indicate a value of the factor B in the range 0.9–1.2.

Contribution of a neutrino magnetic coupling to the muon magnetic moment

We consider a massive Dirac neutrino and compute the contribution of its magnetic coupling to the anomalous magnetic moment of the muon using an unsubtracted dispersion relation and assuming a small neutrino structure. We find a result negligibly small.

Note on low-energy proton Compton scattering

The unpolarized differential cross section for spin-1/2 Compton scattering is shown to be completely specified up to terms of the sixth power of the photon frequency by the mass, charge, anomalous magnetic moment, and dynamic electromagnetic polarizabilities of the target and four other sturcture dependent constants. The two polarizabilities and the four new constants are expressed in terms of the threshold values and threshold derivatives of the invariant amplitudes of the process. Their parts which can be expected to satisfy fixed-$t=0$ unsubtracted dispersion relations are evaluated in terms of single-pion photoproduction data. The extraction of the proton polarizabilities from low-energy elastic $\ensuremath{\gamma}$-proton scattering is discussed in the light of the results obtained.

Gparity and the breaking of exchange degeneracy

Exact exchange degeneracy, a property of the planar $S$ matrix, is not a feature of the physical world. $G$ parity, an exact symmetry of strong interactions, fails to be maintained by the discontinuities of the planar $S$ matrix. This paper examines the connection between these two conflicting concepts and shows how calculations of exchange-degeneracy breaking may be based on $G$ parity. The relation of $G$ parity to the cylinder and torus components of the topological expansion is a central feature. We apply our arguments to $\ensuremath{\rho}\ensuremath{-}{A}_{2}$ splitting and justify the use of an unsubtracted dispersion relation for ${\ensuremath{\alpha}}_{A2}\ensuremath{-}{\ensuremath{\alpha}}_{\ensuremath{\rho}}$. Our analysis of the dispersion relation reveals a dominant role for the $\ensuremath{\pi}\ensuremath{\pi}$ channel.