Improvements on hadronic form factors using analyticity and unitarity constraints

Abstract

Form factors are of central importance in strong interaction dynamics. While electromagnetic form factors are an important source of information about the internal structure of hadrons, the knowledge of the weak transition form factors are crucial for a proper ex traction of the quark mixing parameters, in particular the Cabbibo-Kobayashi-Maskawa matrix elements. An accurate knowledge of the CKM elements are in turn essential for a precision test of the Standard Model. The form factors have been studied in various low- energy effective theories like Chiral Perturbation Theory. Heavy Quark Effective Theory, QCD sum rule approach and on the lattice where various quantities of interest have been computed. In order to accommodate the vast amount of experimental information now available, analyticity turns out to be the ideal tool to correlate the different pieces of in formation to make predictions on the form factors. In this work, we present a formalism based on analyticity, unitarity, dispersion relation techniques as well as the method of analytic continuation in order to obtain information on the form factors. Technically, this method uses the fact that the bound on an integral involving the modulus squared of the form factors along the unitarity cut is known from certain QCD observables. We employ analyticity to transform the problem, use values of the form factor and its derivatives at t = 0 and/or exploit knowledge of the same at various points in the analyticity region. In addition, various theoretical and experimental inputs are used, in particular the phase and modulus of the form factors, low-energy theorems and values of the form factor at different spacelike points. Using this technique, we obtain stringent constraints the low-energy expansion coefficients or the shape parameters of various hadronic form factors and also isolate regions in the complex t-plane where the zeros are excluded. We further present new alternate parameterizations for the heavy-light Dn form factor and also discuss the implication of our method on the spacelike pion electromagnetic form factor. In Chapter 1. we give an account of the form factors and their relevance for study in various phenomenological analysis after which we present in detail the formalism of the method of unitarity bounds which presents the basis for the investigations carried out in this thesis. In Chapter 2, we study the K —> n form factor which is crucial for the determination of \VUS\, one of the elements of the CKM matrix. In this case, the integral is bounded by means of a dispersion relation satisfied by a strangeness-changing QCD correlator which on the other hand is calculated upto five loops in QCD. The available phase and modulus information of the Ktt form factors and the low-energy theorems, namely the Callan-Treiman relations are incorporated into a formalism which involves the Omnes function in order to to obtain stringent constraints on our results. This section mainly concentrates on the zeros of the scalar and vector Kix form factors after a brief highlight of the results on the shape parameters. The knowledge of the zeros have several phenomenological implications. The dispersive representations assume apriori that the form factors do not have zeros in the complex energy-plane which is in turn crucial for their analyses. In addition, the absence of zeros is assumed in the recent analysis of the KTeV data. Using our technique, we are able to isolate a narrow region in the low energy complex t-plane where the zeros are excluded which provides confidence for such phenomenological analysis. In Chapter 3. we study the D —* 7r form factor asociated with a heavy-light decay which is particularly useful in extracting another element of the CKM matrix, namely the \Vcd\. Here, we use a charm-changing QCD correlator satisfying a dispersion relation to bound the integral. This correlator receives contribution from both perturbative and non-perturbative QCD. The perturbative calculations have been performed upto three- loops including for moments higher in the Q2expansion. With these integral bounds and using as inputs the relativistic Breit-Wigner phase shift and the Callan-Treimanrelation, we perform an analysis of the shape parameters of the Dn form factor. Due to poor experimental information on the Dtt sector, we do not yet have a knowledge on the modulus. In this case we use an optimal formalism using the method of Lagrange multipliers which takes into account only the phase and the Callan-Treiman relation. This amounts to solving a set of integral equations which have been generalised for including N spacelike points. We observe that our results are considerably more stringent when we take an overlap of the different moments of the correlators. An analysis of the Dtt zeros has also been performed. Even in this case, we find a narrow domain in the low-energy region where the zeros are excluded. We also propose an alternative and more efficient parameterization in terms of the con formal variable z which properly implements the singularities related to the lowest charmed resonances and is useful for description of semilcptonic data. The above entire Dtt analyses, in the framework of the improved unitarity bound technique, have been done for the first time and can prove useful in the near future for consistency checks when there is more experimental and theoretical information available. In Chapter 4, we revisit the pion electromagnetic form factor which is motivated in part by an improvement in the experimental data that goes as input into our analysis. Recently, BABAR has carried out a precision higli-statistics determination of the modulus of the pion electromagnetic form factor. This in turn has been used for an accurate evaluation of the two pionic hadronic contribution to the muon anomalous magnetic moment (g — 2)tl. Compared to prior works, which used less precise values due to poor information from the experimental side, we now have a more precise bound on the integral coming from the above mentioned muon (g — 2)fl. Moreover, compared to earlier works which have not used the modulus information in their formalism (except for some work by Caprini). we now fully exploit a more sophisticated framework which uses the modulus information along with spacelike data which gives us much better constraints on our results. For the spacelike data, we use the recent precise determination by Huber et.al coming from the Jefferson Lab (JLab). As regards the phase, we use the parameterization of Garcia-Martin et.al which is valid even at considerably higher energies compared to the phase from Roy equations. Using these as inputs and the pion charge radius, (r2) which is known from experiment and Chiral Perturbation theory, we find that our most stringent constraints come when we include a spacelike datum measured at JLab. We compare the shape parameters c and d of the pion form factor with those available in the literature and find that they are consistent with most of the results. Turning to the issue of zeros for the pion form factor, not much has been done in the recent past (last being for work done by Raszillier et.al). We present modern results and systematically address the issue of zeros for the pion form factor. In Chapter 5, we study the spacelike pion electromagnetic form factor which serves as an excellent observable to study the onset of perturbative QCD. While at low Q2, F~ has been well determined by light-cone sum rules and at high Q2 by lattice QCD, the intermediate Q2 region has not yet been well studied. This region is complicated by the fact that there is an interplay of both hard and soft contributions which is poorly understood. Using our technique, we study the behaviour of F~ in this region using suitably choosen weight functions and compare them with that of perturbative QCD which is now calculated at NLO and with various experimental data and non-perturbative models available in the literature. As inputs, we use the phase and modulus from Garcia- Martin et.al and BABAR respectively as well as the spacelike datum from JLab to obtain bounds on Fv in the spacelike region. We find that our method gives rise to stringent bounds in the region Q2 < 10 GeV2 that exclude the onset of the asymptotic perturbative QCD regime for Q2 < 7 GeV2. Finally in Chapter 6, we summarize the results emerging from the analysis carried out in this thesis. We present our outlook for the future and a brief discussion on the application of our method to other related problems.