A theoretical study of meson transition form factors
AA theoretical study of mesontransition form factors
Dissertationzur Erlangung des Grades,,Doktor der Naturwissenschaften” am Fachbereich Physik, Mathematik und Informatikder Johannes Gutenberg-Universit¨at Mainz
Pablo Sanchez-Puertas geboren in Granada (Spanien)Mainz, 2016 a r X i v : . [ h e p - ph ] M a r bstract This thesis studies the lightest pseudoscalar mesons, π , η , and η (cid:48) , throughtheir transition form factors. Describing the underlying structure of hadronsis still a challenging problem in theoretical physics. These form factors, whichcan be experimentally measured, provide valuable information on the pseu-doscalar meson inner structure and are of fundamental interest for describingtheir elementary interactions. Obtaining a precise description for these formfactors has become a pressing subject given their role in one of the finesttests of our understanding of particle physics: the anomalous magnetic mo-ment of the muon. The foreseen experimental precision for this observablechallenges the available theoretical descriptions so far.Still, incorporating the available experimental information into a theore-tical framework becomes increasingly difficult as the experimental precisionimproves, challenging simplified frameworks. In this work, we propose to usethe framework of Pad´e theory in order to precisely describe these form fac-tors in the space-like region, which provides a well-founded mathematically-based and data-driven approach for this task.The first part of our study is devoted to extract the parameters rele-vant to our approach using the available single-virtual space-like data. Theaccuracy of the method, beyond that of previous approaches, has been la-ter confirmed in experiments performed in the low-energy time-like regionfor the η and η (cid:48) cases. To give consideration to these new results, we in-corporated the corresponding data into our analysis. The extension of theformalism to the most general double-virtual case is subsequently discus-sed, which requires the introduction, for the first time in this context, ofCanterbury approximants, the bivariate version of Pad´e approximants.As a direct application of our results, the η − η (cid:48) mixing parametershave been extracted from the single-virtual transition form factors. Theemployed method provides an alternative to the traditional ones, obtainingcompetitive results while minimizing modeling errors.Besides, our double-virtual description is employed for describing the ra-re decays of the pseudoscalar mesons into a lepton pair. The latter processoffers an opportunity to test the doubly virtual pseudoscalar mesons transi-tion form factors as well as an opportunity to discuss possible new physicscontributions in light of the present discrepancies.Finally, our approach is used to obtain a precise calculation for thepseudoscalar-pole contribution to the hadronic light-by-light piece of theanomalous magnetic moment of the muon. This includes, for the first time,a systematic error and meets the required precision in foreseen experiments.v usammenfassung Die vorliegende Dissertation befasst sich mit dem Studium der leichtestenpseudoskalaren Mesonen π , η , and η (cid:48) via deren ¨Ubergansformfaktoren. Ei-ne Beschreibung der zugrunde liegenden Struktur der Hadronen stellt in dertheoretischen Physik immer noch eine Herausforderung dar. Diese Formfak-toren, die experimentell bestimmt werden k¨onnen, stellen eine wichtige In-formationsquelle ¨uber die innere Struktur pseudoskalarer Mesonen dar undsind von grundlegendem Interesse f¨ur die Beschreibung ihrer elementarenWechselwirkungen. Der Erhalt einer pr¨azisen Beschreibung f¨ur diese Form-faktoren ist, mit Blick auf ihre Rolle in einem der genauesten Tests unseresVerst¨andnisses der Teilchenphysik: dem anomalen magnetischen Momentdes Myons, zu einem dringlichen Thema geworden.Die Einarbeitung der verf¨ugbaren, experimentell ermittelten, Informa-tionen in einen theoretischen Rahmen wird nach wie vor mit zunehmenderGenauigkeit der Experimente schwieriger, was vereinfachte Modelle auf dieProbe stellt. Im Rahmen dieser Arbeit schlagen wir vor, sich der Pad´e-Approximation, ein sowohl mathematisch als auch auf Daten basierenderund somit wohlbegr¨undeter Zugang zu diesem Problem ist, zu bedienen, umdiese Formfaktoren in raumartigen Bereichen pr¨azise beschreiben zu k¨onnen.Im ersten Teil unserer Betrachtungen widmen wir uns unter Ausnut-zung von Messwerten raumartiger Prozesse mit einem virtuellen Photon,der Extraktion der f¨ur unseren Zugang relevanten Parameter. Die Genauig-keit dieser Methode, die ¨uber bisherige Versuche hinausgeht, wurde sp¨aterdurch Experimente die im raumartigen Niedrigenergiesektor f¨ur die F¨allevon η und η (cid:48) durchgef¨uhrt wurden, best¨atigt. In der Folge wird die Auswei-tung des Formalismus auf den allgemeinsten Fall zweier virtueller Photonendiskutiert, was die in diesem Kontext erstmalige Einf¨uhrung der Canterbury-Approximation, der zweidimensionalen Pad´e-Approximation, erfordert.Als eine direkte Anwendung unserer Ergebnisse, wurden die Parameterder Mischung η − η (cid:48) aus dem ¨Ubergangsformfaktor eines virtuellen Photonsermittelt. Die verwendete Methode bietet eine Alternative zu traditionellverwendeten, wobei wir konkurrenzf¨ahige Ergebnisse erhalten und zugleichmodellbezogene Fehler minimieren.Zudem wird unsere Beschreibung von Prozessen mit zwei virtuellen Pho-tonen auf die Beschreibung der seltenen Zerf¨alle eines pseudoskalaren Me-sons in ein Leptonen-Paar angewendet. Der genannte Prozess bietet die Gele-genheit die ¨Ubergansformfaktoren pseudoskalarer Mesonen f¨ur zwei virtuellePhotonen zu testen.Schlussendlich wird unsere Vorgehensweise dazu verwendet, eine genaueBerechnung f¨ur den Beitrag des pseudoskalaren Pols zum Anteil der hadroni-schen Licht-Licht-Streuung des anomalen magnetischen Moments des Myonszu erhalten. In dieser mitinbegriffen ist erstmalig ein systematischer Fehlerund sie entspricht der f¨ur Experimente geforderten, ben¨otigten Genauigkeit.i cknow ledgements I would like to express my gratitude to Pere Masjuan for his guidanceduring these four years of doctoral studies. His door was always open whenlooking for help and I have profited from his knowledge in physics in count-less and interesting discussions. This work would not have been possiblewithout his guidance and great enthusiasm. The realization of this the-sis would have not been possible either without the help of Marc Vander-haeghen, to whom I thank for his support, reading and interesting commentsconcerning this manuscript.During my time at Mainz, I could profit as well from the help and com-ments from many people at the Nuclear Physics department; a very specialthanks goes to Tobias Beranek, Mikhail Gorchtein, Nikolay Kivel and Olek-sandr Tomalak, for many and valuable discussions and for sharing theirwisdom with me. Besides, I am particularly indebted to my office mates,Patricia Bickert and Nico Klein for the hours stolen in trying to understandthe basic concepts of chiral perturbation theory and to Hans Christian Langefor helping with the German translation. In addition, and besides Mainz, Iwas lucky to have a pleasant collaboration with R. Escribano, without whomone chapter of this thesis could not have been possible, and his PhD studentSergi Gonzalez-Sol´ıs, whom which I was luck to share my office and manydiscussions.Finally, the greatest acknowledgement goes to my wife, for her support,love and encouragment, and for reading this manuscript since the earlytimes.iii ontents
Preface xi1 Quantum Chromodynamics and related concepts 1 χ PT . . . . . . . . . . . . . . . . . . . . . 31.4 Closing the gap: large- N c QCD . . . . . . . . . . . . . . . . . 101.5 Pad´e approximants . . . . . . . . . . . . . . . . . . . . . . . . 151.6 The pseudoscalar transition form factors . . . . . . . . . . . 21 η and η (cid:48) LEPs . . . . . . . . . . . . . . . . . 342.5 Time-like data: η and η (cid:48) LEPs . . . . . . . . . . . . . . . . . 412.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 η − η (cid:48) mixing 75 η − η (cid:48) mixing from the TFFs . . . . . . . . . 814.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89ix Contents4.6 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . 95 F P γ ∗ γ ∗ ( Q , Q ) . . . . . . . . . . . 1055.4 Final results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.5 Implications for χ PT . . . . . . . . . . . . . . . . . . . . . . . 1165.6 Implications for new physics contributions . . . . . . . . . . . 1205.7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . 127 g − a µ . . . . . . . . . . . . . . 1306.3 Generic HLbL contribution to a µ . . . . . . . . . . . . . . . . 1366.4 The pseudoscalar-pole contribution . . . . . . . . . . . . . . . 1386.5 Beyond pole approximation . . . . . . . . . . . . . . . . . . . 1546.6 Final results for a HLbL µ . . . . . . . . . . . . . . . . . . . . . . 1616.7 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . 162 A.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169A.2 Feynman rules and spinors . . . . . . . . . . . . . . . . . . . . 169A.3 S -matrix, cross sections and decay rates . . . . . . . . . . . . 170 B Supplementary material 171
B.1 Formulae for the g V P γ couplings . . . . . . . . . . . . . . . . 171B.2 Cutcosky rules for additional vector states in P → ¯ (cid:96)(cid:96) . . . . . 172B.3 Fierz transformations . . . . . . . . . . . . . . . . . . . . . . 174 reface The past decades of fundamental research in particle physics have estab-lished the standard model (SM) of particle physics as the microscopic the-ory of fundamental interactions, encompassing the strong, weak and elec-tromagnetic forces in a SU (3) c × SU (2) L × U (1) Y gauge theory . Even ifits formulation fits in a few lines, it provides the most successful theory everformulated in the history of particle physics, and still stands in a good shapeafter thorough tests over the years —some of them standing to astonishingprecision.However, the SM as it is, was known to provide an incomplete descrip-tion of nature even before its last piece remaining, the Higgs boson, wasdiscovered in 2012 at the LHC experiment [3, 4]. Firstly, the SM fails toincorporate Einstein’s theory of general relativity —quantizing gravity isstill a fundamental problem in theoretical physics. Secondly, there is greatevidence that the ordinary matter which is described within the SM cannotexplain the rotational galaxy curves, which seems to require the existenceof dark matter —actually, the SM only describes the visible matter, whichcorresponds with around 5% of the energy content of the universe, whereasdark matter [5] would account for 26% [6, 7]. At larger cosmological scales,it is hard to explain the observed curvature of the universe without the pres-ence of dark energy [8] —accounting for the remaining 69% energy contentof the universe— for which the vacuum energy is a possible candidate; theSM however provides a number which is far too large as compared to theobservational requirements. Furthermore, the SM is not able to describebaryogenesis —the CP violation within the SM is not large enough— norinflation. Asides, the SM does not contain a mass term for the neutrinos,which is required to explain neutrino oscillations; the origin of neutrinomass and whether neutrinos are Dirac or Majorana particles is still an openquestion. There exist in addition theoretical reasons for which the SM isthought to be just the low-energy manifestation of an ultraviolet (UV) com-pletion including, at the very least, gravity —known to be important atenergies around the Planck scale Λ Planck ∼ GeV. Furthermore, one See Refs. [1, 2] for a detailed review. xiii Prefaceof the most renowned issues has to deal with the so called hierarchy prob-lem [9]. This is related to the large radiative corrections which the Higgsmass receives. These make natural to expect a mass close to the next scaleof new-physics —say Λ
Planck — in contrast with its (now) well known mass m H = 125 . GeV, which is inconceivable to test at any collider. Additional ex-amples of these tiny contributions appear as well within the SM, such asin flavor physics, where the effective contribution from the charm and topquark in ¯ K − K [16] and ¯ B − B [17, 18] mixing, respectively, already pre-iiidicted the order of the charm quark mass as well as a heavy top mass priorto their discoveries. Even preciser estimates were obtained both for the topand the Higgs masses based on electroweak precision observables [19] be-fore they were discovered. The list of processes potentially sensitive to newphysics effects is long, especially in flavor physics. Still at far lower energies,there is a world-famous observable which, given its experimental precision,plays an important role in looking for new physics and constraining BSMtheories: the anomalous magnetic moment of the muon ( g µ −
2) [20]. Thelatter is related to the magnetic dipole moment of the muon µ m governingits interaction with a classical magnetic field B through the Hamiltonian H = − µ m · B ( x ) , µ m = g µ (cid:18) e Q m µ (cid:19) S , (1)with S ( Q ) the muon spin(charge) and g µ its gyromagnetic ratio, which clas-sical value can be predicted from Dirac theory, obtaining g µ = 2. Suchquantity receives however quantum corrections —arising within the SM ofparticle physics— implying deviations from the ( g µ −
2) = 0 value. Similarly,new kind of physics would produce additional corrections to this observable.Therefore, a very precise measurement of ( g µ −
2) would allow to test BSMphysics provided we are able to calculate the SM contributions to this observ-able, including quantum electrodynamics (QED), quantum chromodynamics(QCD) and electroweak (EW) contributions, to the astonishing precision towhich a µ ≡ ( g µ − / a exp µ = 116592091(63) × − . (2)The current theoretical estimation for the SM contribution reads a th µ = 116591815(57) × − , (3)and leads to a 3 . σ discrepancy among theory and experiment. This hasmotivated speculations on BSM physics contributing to this observable. Forthis reason, two new experiments have been projected at Fermilab [22] andJ-PARC [23] aiming for an improved precision δa µ = 16 × − in orderto sort out the nature of the discrepancy —note that even a negative resultin the search for new physics effects would provide then a valuable con-straint on BSM theories. However, this effort will be in vain unless a similartheoretical improvement is achieved, the current limiting factor being theSM hadronic corrections, among which, the leading order hadronic vacuumpolarization (HVP) and hadronic light-by-light (HLbL) contributions domi-nate. Improving the current errors represents however an extremely difficulttask, as these calculations involve non-perturbative hadronic physics whichcannot be obtained from a first principles calculation —the exception is of See Chapter 6 for detailed numbers and references. iv Prefacecourse lattice QCD which nevertheless requires some advances, specially forthe HLbL, in order to improve current theoretical estimates.The main objective of this thesis is to improve on a particularly largecontribution dominating the HLbL —the pseudoscalar-pole contribution—at the precision required for the new projected ( g µ −
2) experiments. Tothis end, it is necessary to carefully describe the pseudoscalar meson inter-actions with two virtual photons. These are encoded in their transition formfactors (TFFs), that must be described as precisely and model-independentas possible —including an accurately defined error— in order to achieve aprecise and reliable result. Their study concerns the first part of this the-sis. To this end, the methodology of Pad´e approximants and multivariateextensions relevant for the double-virtual TFFs description are considered.Closely related to the HLbL, we address the calculation of the rare P → ¯ (cid:96)(cid:96) decays, where P = π , η, η (cid:48) and (cid:96) = e, µ . These processes, showinga similar dependence on the pseudoscalar TFFs as the HLbL, not only offera valuable check on our TFF description, but represent the only source ofexperimental information on the double-virtual TFF up to day —describingthe double-virtual TFF behavior is very important in order to achieve aprecise HLbL determination. Beyond that, the large suppression of theseprocesses within the SM offers an opportunity to search for possible newphysics effects in these decays. In the light of the present experimentaldiscrepancies, we carefully describe and discuss them together with theirimplications in ( g µ − N c arguments, this is notclear at all as the recent discovery of a new plethora of the so-called XY Z exotic states [25], or even the possible pentaquark states in the charm quarksector [26] shows. Even more elusive is the question of gluonium states —purely gluonic quarkless bound states— for which several candidates exist.Given their quantum numbers, it is possible that some gluonium admix-ture exists in the η (cid:48) . Elucidating the η and η (cid:48) structure has been a veryinteresting and controversial topic, which relevance is not only theoreticalbut phenomenological, as it enters a number of heavy mesons decays. Theelectromagnetic interactions, encoded in their TFFs, offer a probe to testthe internal η and η (cid:48) structure which we use in order to obtain a new deter-mination for the η − η (cid:48) mixing parameters.v Outline
The thesis is structured as follows: the fundamental concepts on QCD andTFFs employed in this thesis are presented in Chapter 1 along with thetheory of Pad´e approximants, which we adopt to describe the pseudoscalarTFFs. In Chapter 2, we use the available data for the η and η (cid:48) in order toextract the required low-energy parameters (LEPs). The excellent predic-tion that the method provides for the low-energy time-like region —basedon space-like data and proving the power of the approach— is discussed andlater incorporated into our analysis. This allows for an improvement in ourLEPs extraction and provides a single description for the whole space-likeand low-energy time-like regions. In Chapter 3, the generalization of Pad´eapproximants to the bivariate case is introduced for the first time in thiscontext and carefully discussed, thus providing a framework to reproducethe most general doubly-virtual TFF. In Chapter 4, we discuss as a firstapplication from our outcome an alternative extraction for the η − η (cid:48) mixingparameters, which overcomes some problematics of previous approaches andincorporates subleading large- N c and chiral corrections. In Chapter 5, wediscuss a first application based on our TFF parameterization: the calcu-lation of P → ¯ (cid:96)(cid:96) decays, which are of interest given current experimentaldiscrepancies. Our method improves upon previously existing VMD-basedmodels, specially for the η and η (cid:48) . As a closure, a careful discussion onpossible new-physics effects is presented. Finally, in Chapter 6, we use ourapproach to calculate the pseudoscalar-pole contribution to the hadroniclight-by-light ( g µ −
2) contribution. For the first time, a systematic methodproperly implementing the theoretical constraints, not only for the π , butfor the η and η (cid:48) mesons and including a systematic error is achieved. Be-sides, the resulting calculation succeeds in obtaining a theoretical error inaccordance to that which is foreseen in future ( g µ −
2) experiments, whichis the main goal of this thesis.vi Preface hapter Quantum Chromodynamics andrelated concepts
Contents χ PT . . . . . . . . . . . . . 31.4 Closing the gap: large- N c QCD . . . . . . . . . 101.5 Pad´e approximants . . . . . . . . . . . . . . . . 151.6 The pseudoscalar transition form factors . . . 21
In this chapter, we introduce the essential concepts of Quantum Chromody-namics (QCD) that will be required along this thesis. First, we introduceQCD, the quantum-field theory (QFT) of the strong interactions. We dis-cuss then one of its central properties, asymptotic freedom. This feature,allowing to perform a perturbative expansion at high-energies, forbids atthe same time a similar application at low energies. For this reason, we in-troduce chiral perturbation theory ( χ PT), the effective field theory of QCDat low energies, which is our best tool to describe the physics of pions ( π ),kaons ( K ) and eta ( η ) mesons at low-energies, providing the relevant frame-work for discussions in this thesis. None of the previous descriptions areable to describe the intermediate energy region at around 1 GeV though.A successful framework providing some insight in this intermediate energyregime, encompassing both the chiral expansion and perturbative QCD lim-its, is the limit of large number of colors, large- N c . We argue that the1 Chapter 1. Quantum Chromodynamics and related conceptssuccess of resonant approaches inspired from such limit may be connectedto the mathematical theory of Pad´e approximants (PAs), which is subse-quently introduced. Finally, we briefly describe the pseudoscalar transitionform factors. QCD is the microscopic theory describing the strong interactions in terms ofquarks and gluons. The former are the matter building blocks of the theory,whereas the latter represent the force carriers. It consists of a Yang-Mills SU (3) c — c standing for color— theory which Lagrangian is given as L QCD = (cid:88) f q f ( i /D − m q ) q f − G cµν G c,µν , (1.1)where q f represent the quark spinor fields transforming under the funda-mental SU (3) c representation; as such, they are said to come in N c = 3colors. Quarks come in addition in n f = 6 different species or flavors f , up( u ), down ( d ), strange ( s ), charm ( c ), bottom ( b ) and top ( t ), with differentmasses m q spanning over five orders of magnitude. The symbol /D = γ µ D µ ,with γ µ the Dirac matrices (see Appendix A) and D µ the covariant derivative D µ = ∂ µ − ig s A cµ t c , (1.2)with g s the strong coupling constant, t c = λ c / A cµ the N c − SU (3) c representation. Finally, the G cµν term stands for thefield strength tensor G cµν = ∂ µ A aν − ∂ ν A aµ + g s f abc A bµ A cν . (1.3)The central property promoting QCD as the theory of the strong inter-actions is asymptotic freedom. In any QFT, renormalization effects lead to anon-constant coupling which is said to run with the energy. Such dependenceis described with the help of the renormalization group (RG) equations forthe coupling constant α s = g s / (4 π ) [27], µ dα s dµ = β ( α s ) = − α s (cid:18) β α s π + β (cid:16) α s π (cid:17) + β (cid:16) α s π (cid:17) + ... (cid:19) , (1.4)where β = ( N c − n f ), being n f the number of active flavors; addi-tional β , ,... terms can be found, up to β , in Refs. [27, 28]. The re-markable property in Eq. (1.4) is the overall negative sign for β > Section based in Refs. [1, 27]. The structure constants f abc are defined from [ t a , t b ] = if abc t c . .3. Low energy QCD: χ PT 3i.e. for n f < N c <
17, deserving a Nobel prize in 2004 to D. J. Gross,H. D. Politzer and F. Wilczek . This sign implies the decreasing of the strongcoupling constant at high energies —asymptotic freedom— and allows foran easy and standard perturbative expansion in terms of quarks and gluonsdegrees of freedom. This property will be used in Section 1.6.1 to derive thehigh energy behavior for the pseudoscalar transition form factors (TFFs).In contrast, at low energies α s increases, leading to a non-perturbative be-havior and a strong-coupling regime, which is thought to be responsible forconfinement, this is, the fact that free quarks and gluons are not observedin nature; instead, they bind together to form color-singlet states knownas hadrons —the pions and proton among them. It must be emphasizedat this point that confinement cannot be strictly explained on the basis ofEq. (1.4), which is based on a perturbative calculation. Indeed, describingconfinement represents a still unsolved major theoretical challenge in math-ematical physics as formulated for instance by the Clay Math institute [29].Describing QCD at low-energies therefore represents a formidable task. Sofar, a first principles calculation based on Eq. (1.1) has only been achievedthrough Lattice QCD [30], an expensive computational numerical methodbased on the ideas from K. Wilson [31] consisting in a four dimensionaleuclidean space-time discretization of the QCD action. Additional, Dyson-Schwinger equations provides for a continuum non-perturbative approachto quantum field theories, which have been solved within some further ap-proximation schemes. However, even if lattice calculations have shown atremendous progress in the recent years, not all type of observables are atpresent accessible in lattice QCD. Furthermore, they are extremely costlyand require some guidance when performing the required extrapolations.A viable and successful analytic approach comes by the hand of χ PT, thelow-energy effective field theory of QCD. χ PT At the Lagrangian level, Eq. (1.1) is invariant by construction under Lorentzand local SU (3) c transformations. Eq. (1.1) is invariant too under the dis-crete charge conjugation ( C ), parity ( P ), and time reversal ( T ) transforma-tions. In addition, there exists on top an almost-exact accidental symme-try which is not obvious or explicit in the construction, this is, the chiralsymmetry; using the left-handed P L = − γ and right-handed P R = γ projectors, the QCD Lagrangian may be written as L QCD = iq L /Dq L + iq R /Dq R − q L M q R − q R M q L − G cµν G c,µν , (1.5) Interesting enough, at order α s , Eq. (1.4) leads to the solution α s ( µ ) =2 π/ ( β ln( µ/ Λ QCD )), which defines an intrinsic (certainly non-perturbative) scale Λ
QCD . Most of the notations and concepts in this section are taken from Ref. [32].
Chapter 1. Quantum Chromodynamics and related conceptswhere M = diag( m u , m d , m s , m c , m b , m t ) and q L ( R ) = P L ( R ) q with q =( u, d, s, c, b, t ) T . If the quark masses were left apart, the Lagrangian wouldbe symmetric as well under the chiral global transformations q L ( R ) → U L ( R ) q L ( R ) , where U L (cid:54) = U R represents a unitary matrix in flavor space,mixing then different flavors. This is, QCD does not distinguish amongchiral quark flavors. Whereas the massless quark limit would represent abad approximation for the heavy ( c , b , t ) quarks, this is not the case forthe light ( u , d , s ) ones; the fact that the light hadrons are much heavierthan the light quark masses points that the light quark masses should havelittle, if anything, to do with the mechanism conferring light hadrons theirmasses. The origin of the latter should be traced back to confinement, andis responsible for generating most of the visible particle masses in the uni-verse. Chiral symmetry should be therefore a good approximation for thelight-quarks sector.Consequently, at the low energies where the heavy quarks do not play arole, we should find an approximate U (3) L × U (3) R symmetry. Through theuse of Noether theorem, this would imply a set of 18 conserved currents andassociated charges. These are conveniently expressed in terms of the vectorand axial currents J aµ = L aµ + R aµ = qγ µ λ a q J a µ = R aµ − L aµ = qγ µ γ λ a q, (1.6)where λ a / λ a Gell-Mann matrices and a singlet which, for later convenience, wedefine as λ = (cid:112) / × . The symmetry group may be rewritten then as U (3) L × U (3) R = U (1) V × U (1) A × SU (3) V × SU (3) A . However, the previoussymmetry group holds only at the classical level; quantum corrections breakthe axial U (1) A symmetry. Precisely, the axial current divergence is givenas [1] ∂ µ J a µ = {P a , M} − g s π (cid:15) αβµν G bαβ G cµν tr (cid:18) λ a t b t c (cid:19) , (1.7)where the pseudoscalar current P a = qiγ λ a q has been used, t b,c are the SU (3) c generators associated to the strong interactions and λ a / SU (3) A , the associated generatorsare traceless matrices in flavor space, producing a vanishing trace for therightmost term; this contrasts with the (flavor singlet) U (1) A transforma-tions, which generator is proportional to the unit matrix in flavor space .Consequently, the singlet axial current is not conserved even in the chirallimit of vanishing quark masses M = 0; it is called therefore an anomaloussymmetry. Global means that, unlike in gauge theories, U L,R (cid:54) = U L,R ( x ), i.e., the transformationdoes not depend on the space-time coordinate. Note that tr( t a t b ) = (1 / δ ab . .3. Low energy QCD: χ PT 5All in all, at the quantum level we should have an approximate U (1) V × SU (3) V × SU (3) A symmetry. The U (1) V symmetry is related to the baryonnumber conservation in the SM and is as important as to forbid the protondecay. The SU (3) V symmetry would imply the existence of degenerate-mass flavor multiplets in the hadronic spectrum, whereas the SU (3) A wouldimply analogous multiplets with opposite parity. However, the latter is notrealized in nature: degenerate opposite parity multiplets are not found, indi-cating that the axial symmetry is spontaneously broken. This is thought tobe related to the fact that, whereas the QCD Lagrangian is invariant underthese transformations, the vacuum of the theory is not —the complex struc-ture of the QCD vacuum is thought to be the ultimate responsible for thespontaneous breaking of the chiral symmetry. An important consequenceof this feature comes by the hand of Goldstone’s theorem. Goldstone’s the-orem dictates that, whenever a global symmetry is spontaneously broken,massless goldstone bosons with the quantum number of the broken gen-erators appear. In nature, it seems that the symmetry breaking pattern is U (1) V × SU (3) V × SU (3) A → U (1) V × SU (3) V and 8 pseudoscalar Goldstonebosons should appear in correspondence with the 8 broken SU (3) A genera-tors. In nature, there are no massless particles to which such hypotheticalstates could be associated to. There exists however, an octet of pseudoscalarparticles much lighter than the standard mesons. These are the π ’s, K ’s and η mesons. It is believed that in the chiral limit m u,d,s → χ PT, it is worth to take a brief detouranticipating some of the consequences of the large- N c limit of QCD. Partic-ularly, we are interested in the U (1) A axial anomaly. As we will comment inSection 1.4, ’t Hooft showed that in the large- N c limit, the strong couplingconstant should be replaced as g s → ˜ g s / √ N c , where ˜ g s is to be fixed as N c → ∞ [33]. Consequently, in the chiral limit ( M → ∂ µ J a µ = − ˜ g s π N c (cid:15) αβµν G bαβ G cµν tr (cid:18) λ a t b t c (cid:19) N c →∞ −−−−→ , (1.8)and the singlet axial current is conserved too as N c → ∞ . In such limit,the U (1) A anomalous symmetry would be recovered, and the spontaneouslybreaking of the chiral symmetry would come with an additional Goldstoneboson, the η (cid:48) . Consequently, considering N c as a parameter large enough,the η (cid:48) could be incorporated to the χ PT Lagrangian in a combined chiral andlarge- N c expansion, which is known as large- N c chiral perturbation theory( (cid:96)N c χ PT). More formal arguments for the vanishing η (cid:48) mass in the large- N c chiral limit can be found in Ref. [34]. Chapter 1. Quantum Chromodynamics and related concepts In the chiral limit of QCD, we believe in the existence of 8 masless Goldstonebosons associated to the breaking of the chiral symmetry —9 if the large- N c limit is considered. Above this, there is a mass gap below the intrinsicscale that is generated in QCD, call it Λ χ , where the full zoo of hadronicparticles appears. Quantitatively, this spectrum starts around 0 . . QCD are integratedout from the theory, which is effectively described in terms of the relevantdegrees of freedom, the Goldstone bosons. The effect of the physics aboveΛ χ are encoded in a plethora of terms appearing in the effective Lagrangian—actually, as many of them as the underlying symmetries allow to include.Of course, writing down the most general effective Lagrangian allowed bythe assumed symmetry principles of the theory represents a formidable —ifnot impossible— task, as it contains an infinite number of terms. The secondingredient for constructing a useful effective field theory is the presence ofan expansion parameter, according to which only a finite number of termsis required in order to achieve a prescribed precision. For effective fieldtheories of spontaneously broken symmetries, this is an expansion in termsof small momenta p / Λ χ . In the real world, the small quark masses arenon-zero, giving mass to the pseudo-Goldstone bosons. Still, these are muchsmaller than Λ χ , which allows to systematically incorporate additional termsaccounting for the explicit symmetry breaking as an expansion in terms of m q / Λ χ — χ PT is therefore an effective field theory description of QCD interms of small momenta and quark masses.The theoretical framework to describe such theories was initiated byWeinberg [35], Coleman, Wess and Zumino [36] and in collaboration withCallan in [37]. It generally implies that the Goldstone boson fields, φ ( x ),transform non-linearly upon the symmetry group; they are described thenin terms of the U ( x ) matrix U ( x ) = exp (cid:18) iφ ( x ) F (cid:19) = 1 + i φ ( x ) F + ... (1.9)with F a parameter required to obtain a dimensionless argument and φ ( x )the matrix associated to the Goldstone bosons φ ( x ) = (cid:88) a =1 φ a λ a = π + √ η √ π + √ K + √ π − − π + √ η √ K √ K − √ K − √ η , (1.10)which serves as a building block of the theory. In this way, one can writethe most general Lagrangian according to the powers of momentum p n —what is equivalent, the number of derivatives ∂ n — and powers of the quark.3. Low energy QCD: χ PT 7masses (accounting that m q ∼ p ). Due to Lorentz invariance, derivativesappears in even numbers, 2 n , leading to the decomposition L = L + L + L + ... + L n + ... . (1.11)In addition, any of the pieces displayed above produces an infinite number ofcontributions —Feynman diagrams— to some given specific process. Con-sequently, an additional scheme classifying these pieces according to theirrelevance is required. This is achieved using Weinberg’s power counting [38],which assigns a chiral dimension D to every amplitude M (see Appendix A)arising from a particular diagram according to its properties upon momenta, p , and quark masses, m q , scaling M ( p i , m q ) → M ( tp i , t m q ) = t D M ( p i , m q ) . (1.12)The final result is given, in four space-time dimensions, in terms of thenumber of internal pseudo-Goldstone boson propagators, N I , number ofloops, N L , and the number of vertices N k from L k (see Eq. (1.11)) as D = 4 N L − N I + ∞ (cid:88) k =1 kN k . (1.13) The most general Lagrangian at leading order, L , reads [39, 40] L = F (cid:16) D µ U D µ U † (cid:17) + F (cid:16) χU † + U χ † (cid:17) , (1.14)where F is known as the pion decay constant in the chiral limit due to itsrelation at LO with the π ± decay. The covariant derivative is defined as D µ U = ∂ µ U − ir µ U + iU l µ = ∂ µ U − i [ v µ , U ] − i { a µ , U } (1.15)and allows to couple the pseudo-Goldstone bosons to external left ( l µ ) andright ( r µ ) handed —alternatively vector ( v µ ) and axial ( a µ )— currents. Fi-nally χ = 2 B ( s + ip ), where B is related to the quark condensate (cid:104) ¯ qq (cid:105) in thechiral limit and s ( p ) are the external (pseudo)scalar currents . This allowsto introduce the finite quark masses effects via s → M = diag( m u , m d , m s ).The most general Lagrangian construction at the next order, L , wasdiscussed in the seminal papers from Gasser and Leutwyler [39, 40].Finally, the large- N c limit allows to include the η (cid:48) as a ninth degree offreedom, giving birth to (cid:96)N c χ PT, a low-energy description of QCD in termsof small momenta, quark masses and the large number of colors. In this The elements v µ , a µ , s, p are defined in terms of generating functional external currents L ext = v aµ ¯ qγ µ λ a q + a aµ ¯ qγ µ γ λ a q − s a ¯ qλ a q + p a ¯ qiγ λ a q ≡ ¯ qγ µ ( v µ + γ a µ ) q − ¯ q ( s − iγ p ) q . Chapter 1. Quantum Chromodynamics and related conceptsframework, the expansion parameters are p ∼ m q ∼ N − c ∼ O ( δ ) and theexpansion reads L = L (0) + L (1) + L (2) + ... + L ( δ ) + ... . (1.16)In addition, the N c scaling has to be incorporated to Eq. (1.12). As a result,it can be obtained among others that F ∼ O ( N / c ), or that loop processesas well as additional flavor traces are N c -suppressed in this framework. Theleading order Lagrangian is given as [41] L (0) = F (cid:16) D µ U D µ U † (cid:17) + F (cid:16) χU † + U χ † (cid:17) − τ ( ψ + θ ) , (1.17)where Eq. (1.10) is to be replaced by φ ( x ) = (cid:88) a =0 φ a λ a = π + √ η + F ψ √ π + √ K + √ π − − π + √ η + F ψ √ K √ K − √ K − √ η + F ψ (1.18)with λ = (cid:112) / × and ψ ≡ √ F φ , being φ the field to be related tothe singlet Goldstone boson in the chiral large- N c limit. The τ term inEq. (1.17) is connected with the (cid:104) | T { ω ( x ) ω (0) } | (cid:105) two-point function inthe pure gluonic theory and θ , the vacuum angle [41], represents an externalcurrent —similar to the 2 Bs term in χ . The Lagrangians described above —including the higher order in Eqs. (1.11)and (1.16)— can be shown to be invariant under φ → − φ transformationsif no external currents are considered, meaning that they always containinteractions with an even number of pseudo-Goldstone bosons. This re-mains the case even if vector currents are included in the formalism. Thepreceding Lagrangians cannot describe the π → γγ and related decays.The π → γγ decay has indeed been a fascinating process in the history ofparticle physics, the underlying mechanism driving this decay remaining amystery until the independent discovery of the anomalies —the breaking ofclassical symmetries in QFT— in 1969 by Adler [42] and Bell-Jackiw [43](ABJ anomaly). The ABJ anomaly can be used then to predict the π → γγ decay in the chiral limit of QCD —see for instance [44]. The systematic in-corporation of anomalies into chiral Lagrangians is accomplished by the useof the Wess-Zumino-Witten (WZW) action [45, 46], which introduces addi-tional terms involving an odd number of Goldstone bosons as well as terms The winding number density is defined as ω = − g s π (cid:15) µνρσ G cµν G cρσ = − α s π G c ˜ G c withthe dual tensor ˜ G c,µν = (cid:15) µνρσ G cρσ . .3. Low energy QCD: χ PT 9such as φγγ, φ γ , etc (see Ref. [32, 41]). For our case of interest, we refer tothe leading term inducing P → γγ decays [47] L WZW = N c α π (cid:15) µνρσ F µν F ρσ tr (cid:0) Q φ (cid:1) (1.19)which is valid both, for χ PT and (cid:96)N c χ PT, where it appears at order L and L (1) , respectively. In the expression above, Q = diag(2 / , − / , − /
3) isthe charge operator and φ = λ a φ a . As an example, we outline here the LO results for the pseudoscalar massesand decay constants in (cid:96)N c χ PT. From the Lagrangian Eq. (1.17), and taking χ → B M , we obtain for the kinetic terms at LO L (0)kin = ∂ µ π + ∂ µ π − − B ˆ mπ + π − + 12 (cid:0) ∂ µ π ∂ µ π − B ˆ mπ π (cid:1) + ∂ µ K + ∂ µ K − + ∂ µ ¯ K ∂ µ K − B ( ˆ m + m s ) (cid:0) K + K − + ¯ K K (cid:1) + 12 (cid:0) ∂ µ η ∂ µ η − B (cid:0) m +4 m s (cid:1) η η (cid:1) −
12 ( η η + η η ) √ ( ˆ m − m s )+ 12 (cid:0) ∂ µ η ∂ µ η − B (cid:0) m +2 m s (cid:1) η η − τF η η (cid:1) . (1.20)In the expression above, the isospin-symmetric limit m u = m d ≡ ˆ m hasbeen used. Eq. (1.20) allows to identify the pions and kaons masses at LO m π ± = m π ≡ ˚ M π = 2 B ˆ m m K ± = m K ≡ ˚ M K = B ( ˆ m + m s ) . (1.21)The η and η (cid:48) masses require additional work since the terms from the thirdline in Eq. (1.20) are non-diagonal, leading to the η − η (cid:48) mixing. This willbe discussed in more detail in Chapter 4. For the moment, let us note thatin standard χ PT η = η , which receives mass from the quarks alone. Thesinglet component η acquires a large topological mass M τ = 6 τ /F absentin the octet terms.Finally, we define the pseudoscalar decay constants, which are of majorinterest for discussing the η − η (cid:48) mixing in Chapter 4 as well as for calculatingnew physics contributions to P → ¯ (cid:96)(cid:96) decays in Chapter 5, where P = π , η, η (cid:48) . The pseudoscalar decay constants are defined in terms of the matrixelements of the pseudoscalars with the axial current (cid:104) | J a µ | P ( p ) (cid:105) ≡ ip µ F aP , J a µ = ¯ qγ µ γ λ a q. (1.22)They can be obtained at LO from Eq. (1.17) taking an external axial current a µ ≡ a aµ λ a , see Eq. (1.15). The relevant term reads − F tr ( ∂ µ φ a µ ) = − F ∂ µ φλ a ) a aµ . (1.23)0 Chapter 1. Quantum Chromodynamics and related conceptsIdentifying λ a with the relevant SU (3) matrix, i.e., λ for the π , one obtainsin χ PT that F π ± = F π = F K ± = F K = F η ≡ F . In (cid:96)N c χ PT, the η − η (cid:48) mixing makes this picture more complicated for the η and η (cid:48) mesons.The success obtained in χ PT at higher orders (state of the art is O ( p ))in predicting different observables shows a good performance of the the-ory, which is to day our best tool to produce analytical calculations forlow-energy hadronic physics. Still, the theory is not expected to be validabove some scale, often defined as Λ χ ≡ πF , which is below the pQCDapplicability range. For a particular process, the natural scale at which onecan expect a poor performance is given by the closest relevant hadronic res-onance which has not been included in the theory as an active degree offreedom. Unfortunately, this avoids to match the theory with pQCD. N c QCD
Describing all the QCD phenomenology with its great complexity representsa challenging task. The complex analytic structure which QCD requires —think about reproducing all nuclear physics as a part— makes an analyticdescription nonviable. Consequently, so far, only perturbative expansionshave reached success in analytically describing particular sectors of QCD,but the lack of an apparent perturbative parameter of the theory at all scalesavoids the whole QCD description within a single framework. However, ’tHooft pointed out that there might be such a candidate for an expansionparameter in QCD, this is, the limit of large number of colors, large N c [33].Its phenomenological success and the fact that it is the only framework jus-tifying some known features of QCD, such as Regge theory or the OZI ruleamong others, makes this approximation to QCD very useful even if so far itonly produces a qualitative picture of QCD rather than a quantitative one .The large- N c limit of QCD is based on the combinatorics SU ( N c ) groupfactors arising in diagrammatic calculations. Recall for instance the RGequation for the strong coupling constant α s in Eq. (1.4). There, N c plays arelevant role in the leading coefficient for the β -function β = (11 N c − n f ).In the large- N c limit, the first part dominates. Actually, if a smooth andnon-trivial behavior is desired in such a limit, the strong coupling constantshould be taken as g s → ¯ g s / √ N c , where ¯ g s is kept fixed as N c → ∞ . Then,the RG equation for ¯ α s ≡ ¯ g s / π would resemble that in Eq. (1.4) with β defined as β = (cid:16) − N c n f (cid:17) —otherwise, ¯ g s would tend to 0 inducinga trivial theory . This means that any interacting process in the large- N c limit will not survive unless the combinatoric factors of the relevant dia- This introduction is mainly based on Refs. [48–50]. In addition, this guarantees that the induced QCD scale, Λ
QCD , as well as the hadronmasses, remain N c -independent. .4. Closing the gap: large- N c QCD 11 i ¯ j i ¯ j i ¯ j i ¯ jj ¯ j iii ¯ i ¯ j j ¯ kk i ¯ i j ¯ jk ¯ kl ¯ l Figure 1.1:
The different QCD vertices and propagators (gray) in the color-lines notation(black). The indices i, j, k, l stand for color indices. ¯ g s N − c ¯ g s N − c ¯ g s N − c ¯ g s N − c N c N c N c Figure 1.2:
Examples of diagrams contributing to the gluon self-energy. Upper graphsshow the vertex suppression, ¯ g s / √ N c , and the lower ones the combinatoric SU ( N c ) en-hancement arising from closed color lines ∼ N c . grams are large enough to compensate for the ¯ g s / √ N c factors. It turns outthat only a certain class of diagrams, which can be classified according theirtopology, survive in this limit (in the purely gluonic theory these are theso called planar diagrams). To figure this out, it is convenient to employthe color-line notation introduced by ’t Hooft [33], according to which thequarks propagators can be illustrated as color lines, the gluons propagatorsas color-anticolor lines and a similar representation holds for the vertices,see Fig. 1.1.As an example, we show in Fig. 1.2 different contributions to the vacuumpolarization appearing in the α s running together with their N c counting.From those diagrams, only the first and second ones have a combinatoricfactor arising from closed color lines large enough to counteract the verticessuppression; the third and fourth are suppressed with respect to the previousones by factors of N − c and N − c , respectively. The leading diagrams belongto the so called planar diagrams. In contrast to the third one, they can bedrawn in such a way that color lines do not cross each other and are leadingin the large- N c expansion. Contrary, non-planar diagrams and quark loopsare N − c and N − c suppressed, respectively.Therefore, in order to obtain the gluon self-energy, it would be sufficient,at leading order in the large- N c expansion, to take the planar diagramscontributions. The resummation of all the planar diagrams has only beenachieved so far in a 1 + 1 space-time dimensions [51]. Therefore, it is difficultto obtain a quantitative answer in large N c . Still, it is possible to obtaina qualitative picture for a variety of QCD phenomena. In this thesis, it is2 Chapter 1. Quantum Chromodynamics and related concepts ¯ g s N − c ¯ g s N − c ¯ g s N − c N c N c N c N c Figure 1.3:
Different contributions to quark bilinear correlation functions, where theinsertion is marked by a cross. Upper graphs indicate the strong-coupling suppressionand the lower ones the combinatoric SU ( N c ) enhancement. of interest what concerns Green’s functions involving ¯ q Γ q bilinear currents,where Γ is a Dirac bilinear matrix. It turns out that planarity is not enoughthen. For the case of bilinear currents, the leading diagrams are the planardiagrams with only a single quark loop which runs at the edge of the dia-gram [48]. To see this, we refer to Fig. 1.3, where crosses refer to bilinearcurrents insertions. The first diagram is of order N c ; the second, with agluon at the edge, is N − c suppressed; the third one, with an internal quarkloop, is N − c suppressed; the fourth is N − c suppressed.These observations have far reaching consequences once confinement isassumed: take a typical leading diagram such as that in Fig. 1.4 and cutit through to search for possible intermediate states, this is, intermediatequarks and gluons color singlet combinations. First of all, as quark loopsare N c suppressed, any intermediate state contains one and only one q ¯ q pair.Second, a closer look to Fig. 1.4 reveals that it is not possible to have two ormore singlet configurations, say q ¯ q and some gluonic state —more precisely,these configurations are N c suppressed. In conclusion, all the quarks andgluons must bind together to form one particle color-singlet states; the dia-gram in Fig. 1.4 represents thereby a perturbative approximation to a singlehadron. As a conclusion, bilinear two-point correlation functions —suchas the vacuum polarization— can be expressed in terms of single particleintermediate meson states with the appropriate quantum numbers:1 i (cid:90) d xe ik · x (cid:104) | T { J ( x ) J (0) } | (cid:105) ≡ (cid:104) J ( k ) J ( − k ) (cid:105) = (cid:88) n a n k − m n + iε , (1.24)where the meson masses, m n , are N c independent. Furthermore, it is knownthat, in the perturbative regime, such function behaves logarithmically, re-quiring then an infinite number of mesons. In the large- N c limit, correlationfunctions are given in terms of an infinite sum of narrow-width (stable)meson states. In addition, since the correlation function is of order N c , a n = (cid:104) | J | n (cid:105) = √ N c ..4. Closing the gap: large- N c QCD 13 ¯ i ij ¯ j k ¯ kl ¯ l X n a n a n k − m n + iε Figure 1.4:
A typical N c -leading contribution to a bilinear two-point function. Multiplesinglet color intermediate states cannot appear at the leading order. Two-point functionscan be understood then in the large- N c limit as a sum over single meson states (right). X n X n ++ + √ N c √ N c √ N c √ N c √ N c √ N c √ N c √ N c √ N c √ N c √ N c √ N c √ N c √ N c √ N c √ N c N − / c N − / c N − / c N − / c N − c Figure 1.5:
A typical N c -leading contribution to the three- and four- point function(upper and lower row, respectively) and possible meson-exchange decomposition (crossedchannels are implied). The same reasoning can be extrapolated to higher order correlation func-tions for quark bilinears. As an example, we illustrate this for the three-and four-point functions in Fig. 1.5. Again, the only possible intermedi-ate color singlet states are single particles —multiparticle states being N c suppressed. In addition, the large- N c counting allows to obtain that thethree(four) meson vertex is 1 / √ N c (1 /N c ) suppressed, leading to a large- N c estimate for the meson decay widths. In general terms, it is found that anyGreen’s function must contain, at the leading order, single-pole contribu-tions alone —multiparticle states are suppressed. Actually, using crossingsymmetry and unitarity arguments, the large- N c limit implies that, at theleading order, every amplitude can be expressed as if arising from the treelevel calculation from some local Lagrangian with the following properties: • Green’s functions for bilinear quark currents can be expressed as sumsover single meson states —an analogous result holds for purely gluoniccurrents which can be expressed as a sum over purely gluonic boundstates (glueballs). Actually, these are of relevance for this thesis, as the pseudoscalar transition formfactors can be defined in terms of the (cid:104)
V V
P(cid:105)
Green’s function and the hadronic light-by-light tensor is related to the (cid:104)
V V V V (cid:105) one. • The amplitude for a bilinear current to create m mesons from the vac-uum (cid:104) | J | n m (cid:105) is O ( N − m/ c ) —similarly, the amplitude for creating g glueball states (cid:104) | J | n g (cid:105) is O ( N − gc ). • Vertices involving m mesons are O ( N − m/ c ) —for g glueball statesthey are of O ( N − gc ). • Similarly, a meson-glueball vertex —and thereby meson-glueball mixing—can be obtained to be O (1 / √ N c ).The properties above, even qualitative, allow to understand many QCDphenomenological observations supporting the applicability of the large- N c limit: • Most of the observed mesons are (mainly) q ¯ q states and additional q ¯ q content seems suppressed. • The dominance of narrow resonances over multiparticle continuum. • Hadronic decays proceed, dominantly, via resonant states. • It provides a natural explanation (the only one so far) for Regge phe-nomenology. • It is the only framework justifying the Okubo-Zweig-Iizuka (OZI) rule. It explains for instance why φ → ¯ KK dominates over φ → ρπ orthe approximate nonet symmetry in meson multiplets .Note that in the combined chiral and large- N c limit not only may one expectthe η (cid:48) to be degenerate in mass with the pseudo-Goldstone bosons, but todecouple from glueball mixing effects, providing thus an ideal framework toimplement the η (cid:48) into the chiral description as previously said.We concluded the previous sections observing that χ PT and pQCD couldnot provide a complete description of QCD at all scales. The large- N c limitdoes not provide a quantitative answer either —as we do not know how tosolve it yet— but it provides a qualitative description. In the region of in-terest between χ PT and pQCD, the relevant physics is provided by the roleof intermediate resonances. One could interpolate the QCD Green’s func-tions from the low energies —calculable within χ PT— to the high-energiesusing a rational function incorporating the minimum number of resonancesrequired to reproduce the pQCD behavior. This approach has been knownas the minimal hadronic approximation (MHA) [52] and has provided withsuccessful and reasonable descriptions involving different phenomena. The OZI rule refers to the suppression of quark-disconnected contributions such asthat in the fourth diagram in Fig. 1.3. Note that the mass difference from singlet against octet mesons would arise fromdiagrams such as the last one in Fig. 1.3, which are N c suppressed. .5. Pad´e approximants 15 The large- N c limit dictates that, to leading order, the QCD Green’s func-tions are characterized in terms of the different poles arising from interme-diate resonance exchanges and their residues, motivating the constructionof some rational ansatz for them. Specializing to two-point functions, say,the hadronic vacuum polarization (HVP) (cid:90) d xe iq · x (cid:104) | T { J ( µ ) ( x ) J ( ν ) (0) } | (cid:105) ≡ i ( q g µν − q µ q ν )Π( q ) , (1.25)the large- N c limit suggests that, at leading order, (cid:98) Π( q ) ≡ Π( q ) − Π(0) = A ( q )( q − M V )( q − M V ) ... ( q − M V n ) ... , (1.26)with (cid:98) Π( q ) the renormalized HVP, M V n the n -th vector meson resonancemass and A ( q ) a polynomial . Moreover, the residues from (cid:98) Π( q ) can beexpressed in term of the vector resonances’ decay constants F V n aslim q → M Vn ( q − M V n ) (cid:98) Π( q ) ≡ − F V n , (cid:104) | J µ | V n (cid:105) ≡ M V n F V n (cid:15) µ . (1.27)Phenomenologically, one may adjust the required number of vector reso-nances to reproduce the pQCD behavior (MHA) to the physical ( N c =3)resonance masses and even to determine some coefficients in Eq. (1.27)from the physical vector meson decays. Alternatively, as in the real worldthe vector mesons have a finite width, one may obtain the parameters inEq. (1.26) through a data-fitting procedure instead. The procedure out-lined often provides a reasonable description, which, in some cases, may gobeyond the large- N c expectations. Still, when aiming for precision, large N c is not enough and it would be desirable to be able to implement allthe information at hand: the well-known low-energy behavior from χ PT —including multiparticle intermediate states— and the high-energy behaviorfrom pQCD —via the operator product expansion (OPE). In this section,we introduce Pad´e approximants (PAs), which can be precisely used for thistask. Pad´e theory defines then a rigorous mathematical approach which isapplicable, at least, in the space-like region. As an outcome, this theory isable to justify the reason why, sometimes, the MHA provides such a goodperformance beyond large- N c expectations . the lim q →∞ Π( q ) ∼ ln( q ) behavior requires an inifnite number of resonances andtherefore A ( q ) should be an infinite degree polynomial as well. This is possible with a finite number of resonances if the anomalous dimensions vanishso no logarithmic corrections appear or, approximately, if they appear as correction to theleading Q behavior [53]. This is not the case for the HVP; it is the case however for theΠ LR function [54] or the TFFs. This section is based on Ref. [55]. A thorough discussion of PAs can be found inRefs. [56, 57].
Given a function f ( z ) of complex variable z with a well defined power ex-pansion around the origin and a radius of convergence | z | = R , f ( z ) = ∞ (cid:88) n =0 f n z n , (1.28)the Pad´e approximant [56, 57] is a rational function P NM ( z ) = Q N ( z ) R M ( z ) = (cid:80) Nn =0 a n z n (cid:80) Mm =0 b m z m (1.29)with coefficients a n , b m defined to satisfy the accuracy-through-order condi-tions up to order N + MP NM ( z ) = f + f z + ... + f M + N z M + N + O ( z L + M +1 ) . (1.30)Note that, without loss of generality, one can always choose b = 1.If the original function f ( z ) has a radius of convergence R → ∞ , f ( z )is said to be an entire function and is given by its power series expansion,Eq. (1.28), everywhere in the complex plane. Employing PAs for this kind offunctions may accelerate the convergence rate with respect to the series ex-pansion, but the gain may not be dramatic. The situation changes for seriesexpansions with a finite radius of convergence R : the power series Eq. (1.28)represents a divergent series beyond R and convergence deteriorates as oneapproaches this point. It is in this case where PAs become a powerful tool;they cannot only dramatically improve the convergence rate within | z | < R with respect to Eq. (1.28), but may provide convergence in a larger domain D ⊂ C ( {| z | < R } ⊂ D ), which in some cases could extend (almost) tothe whole complex plane —in such case, PAs would provide in a sense aformal tool to perform an analytic continuation of a given series expansion.This is very important, as the functions we want to deal with in QCD, arenot analytic in the whole complex plane; as we saw, in the large- N c limitthese functions are characterized by an infinite set of resonances or poles,whereas in the real N c = 3 world, multiparticle intermediate states implythe existence of branch cuts. In that sense, the applicability of Eq. (1.28)would be very limited ( R would be given by the lowest (multi)particle pro-duction point). The study of convergence properties for PAs is much morecomplicated than for the cases of power expansions and represents an ac-tive field of research in applied mathematics. Nevertheless, there are someclasses of functions for which convergence properties are very well-known. Inthe following, we describe the convergence properties for meromorphic andStieltjes functions, which are representative cases of QCD Green’s functions. The definition is not special for the origin ( z = 0) and generally applies to any point z in the complex plane as long as the series expansion is well-defined around z = z . In the mathematical literature P NM ( z ) is commonly noted as [ N/M ], [ N | M ] or N/M . .5. Pad´e approximants 17 The large- N c limit of QCD: meromorphic functions A particular class of functions we are interested in are meromorphic func-tions, this is, functions which are analytic in the whole complex plane exceptfor a set of isolated poles and, therefore, represents the case of interest oflarge- N c QCD. The convergence properties of PAs to this kind of functionsare very well-known and can be summarized in terms of Montessus’ andPommerenke’s theorems as given in Ref. [57]:
Montessus’ theorem
Let f ( z ) be a function which is meromorphic in the disk | z | ≤ R , with m poles at distinct points z , z , ..., z m , where | z | ≤ | z | ... ≤ | z m | ≤ R . Let thepole at z k have multiplicity µ k and let the total multiplicity (cid:80) mk =1 µ k = M .Then, f ( z ) = lim L →∞ P LM ( z ) (1.31)uniformly on any compact subset of D m = { z, | z | ≤ R, z (cid:54) = z k , k + 1 , , ..., m } . (1.32)When dealing with Green’s functions in the large- N c limit of QCD, thismeans that a sequence of approximants P LM ( z ) will provide an accurate de-scription within a disk | z | < R englobing the first M poles as long as L → ∞ .The advantage of the theorem is that it provides uniform convergence, whichis a strong property as it implies that no spurious poles or “defects” —seethe theorem below— will appear. In particular, the position of the m polesand their residues will be correctly determined as L → ∞ (see Refs. [53, 54]).The disadvantage is that the theory does not say anything outside | z | < R and the number of poles within must be anticipated —an information whichmight be unknown. If these requirements were too strong for some specificapplication, one may resort to Pommerenke’s theorem instead. Pommerenke’s theorem
Let f ( z ) be a function which is analytic at the origin and analytic in theentire complex plane except for a countable number of isolated poles andessential singularities. Suppose ε, δ > M exists suchthat any P LM sequence with L/M = λ (0 < λ < ∞ ) satisfies | f ( z ) − P λMM | ≤ ε (1.33)for any M ≥ M , on any compact set of the complex plane except for a set E M of measure less than δ . Consequently, convergence is found as M → ∞ .As an interesting corollary, previous theorem can be generalized to P N + kN ( z )sequences with k ≥ − , for which convergence is not uniformly guar-anteed but in measure. This means that the region in the complex planewhere Eq. (1.33) is not satisfied becomes arbitrarily small. See Ref. [54] fora nice illustration of this feature and the use of Pommerenke’s theorem forthe (cid:104) V V − AA (cid:105) QCD Green’s function.
Back to the N c = 3 real world: Stieltjes functions As a consequence of the previous theorems, convergence of PAs to meromor-phic functions can be guaranteed, and thereby, the convergence of PAs toQCD Green’s functions in the large- N c limit of QCD follows. This allows toreconstruct and to extend the otherwise divergent series defined in Eq. (1.28)—which may be obtained from χ PT— up to an arbitrary large domain aslong as enough terms in the power-series expansion are known. Of course,this does not guarantee an analogous performance in the real world with N c = 3. For instance, the hadronic vacuum polarization (cid:98) Π( q ), Eq. (1.25),does no longer consists of an infinite number of resonances; multiparticlechannels starting at the ππ threshold manifest themselves instead as a cutalong the real axis, allowing to express the vacuum polarization through aonce-substracted dispersion relation [58] (cid:98) Π( q ) = q (cid:90) ∞ s th dtt ( t − q − iε ) 1 π Im Π( t + iε )= z (cid:90) du − uz − iε π Im Π (cid:16) s th u + iε (cid:17) , (1.34)where s th = 4 m π is the lowest threshold for particle production, z = q /s th ,and a change of variables t = s th u − has been performed in the second lineof Eq. (1.34). The fact that Im Π( q ) is related through the optical theoremto the σ ( e + e − → hadrons) cross section, a positive quantity, guarantees thatsuch a function is of the Stieltjes kind.Stieltjes functions are defined in terms of a Stieltjes integral [57], f ( z ) = (cid:90) ∞ dφ ( u )1 + zu , | arg( z ) | < π, (1.35)where φ ( u ) is a bounded non-decreasing function with finite and real- Defects are regions of the complex plane featuring a pole and a close-by zero —theireffect is nevertheless limited to a neighborhood around it and not the whole complex plane. Note that the function φ ( u ) is not even required to be continuous. As an example, φ ( u ) = θ ( u − u ) → dφ ( u ) = δ ( u − u ) du , which is meromorphic and Stieltjes. .5. Pad´e approximants 19valued moments defining a formal expansion around the origin f j = (cid:90) ∞ u j dφ ( u ) , j = 0 , , , ... ⇒ f ( z ) = ∞ (cid:88) j =0 f j ( − z ) j . (1.36)Note that, given a continuous non-zero dφ ( u ) function non-vanishing along0 ≤ u ≤ /R , the Stieltjes function is not well defined in the real −∞ 1. In addition, thepoles (and zeros) of the approximant are guaranteed to lie along the negativereal axis and to have positive residues.An additional property that Stieltjes functions can be shown to obeyis that the diagonal(subdiagonal) P NN (+1) ( z ) sequence decreases(increases)monotonically as N increases, having a lower(upper) bound. Indeed, if f ( z )is a Stieltjes function,lim N →∞ P NN +1 ≤ f ( z ) ≤ lim N →∞ P NN ( z ) , ( | arg( z ) | < π ) . (1.37)More generally, any P N + JN ( J ≥ − 1) sequence is monotonically increas-ing(decreasing) for J odd(even).The condition that a function is Stieltjes is a very strong one and guar-antees the possibility to reconstruct such a function through the use of PAs.Moreover, poles and zeros from PAs are guaranteed to pile along the nega-tive real axis, excluding the possibility of defects. This allows to reconstructcertain hadronic functions, like the vacuum polarization, in the whole cutcomplex plane. This reconstruction excludes nevertheless the threshold andresonance region (which is ill-defined as well in the original function) andPAs poles cannot be associated therefore to physical resonances but to ana-lytic properties of the underlying function. The PA zeros and poles conspirethereby to mimic the effects from the discontinuity at the cut. We illustratesuch effect in Fig. 1.6 for the Stieltjes function z − ln(1+ z ). These propertiesexplain therefore the excellent performance of rational approaches beyondthe naive large- N c estimation. As a final remark, let us note that a functioncould be meromorphic and Stieltjes at the same time (i.e., if every pole hasa positive-defined residue). In such a case, Stieltjes properties would applyas well. In addition, Stieltjes functions can be shown to obey certain determinantal condi-tions [55, 57]. See Ref. [58] for an application of them. Figure 1.6: The z − ln(1 + z ) function (first column) is compared to the P ( z ) and P ( z ) PAs (second and third column). Upper(lower) row illustrates the real(imaginary)parts. So far, we have only discussed the implementation of PAs based on the low-energy expansion Eq. (1.28). However, in the large- N c approximation, oreven in the real N c = 3 world, one may wish to include the information aboutsome resonances’ position. Additionally, further information away from theorigin could be available —the high-energy expansion among others. In thissection, further extensions of PAs are presented allowing to incorporate thiskind of information. Pad´e type and partial Pad´e approximants As said, from Montessus’s and Pommerenke’s theorems, it follows that, even-tually, the poles and residues of the underlying function are reproduced bythe approximant. However, it would be interesting to incorporate this in-formation from the beginning whenever this is known. This possibility isbrought by Pad´e type and partial Pad´e approximants. Partial Pad´e approximants If the lowest-lying K poles at z = z , z , ..., z K from the underlyingfunction are known in advance, this information could be incorporated fromthe beginning using the so called Partial Pad´e approximants defined as P NM,K ( z ) = Q N ( z ) R M ( z ) T K ( z ) , (1.38)where Q N ( z ), R M ( z ) are degree N and M polynomials and T K ( z ) = ( z − z )( z − z ) ... ( z − z K ) is a degree K polynomial defined as to have all the.6. The pseudoscalar transition form factors 21zeros exactly at the first K -poles location. Pad´e type approximants Pad´e type approximants is another kind of rational approximant T NM ( z ) = Q N ( z ) T M ( z ) (1.39)in which all the poles of the approximant are fixed in advance to the originalfunction lowest-lying poles. This is, T M ( z ) = ( z − z )( z − z ) ... ( z − z M ).This requires however the knowledge of every pole of the original functionif one is aiming to construct an infinite sequence ( N, M → ∞ ).An interesting discussion and illustration of partial Pad´e and Pad´e typeapproximants is illustrated for a physical case, the (cid:104) V V − AA (cid:105) function, inRefs. [54, 59]. Here we only note that these approximants could justify whythe MHA has often such a good performance—and a slower convergence—wrt PAs that offer an improvement based on a mathematical framework. N-point Pad´e approximants Eventually, one could have analytical information of a particular function,not only at the origin, but at different points, say, z and z f ( z ) = ∞ (cid:88) n =0 a n ( z − z ) n , f ( z ) = ∞ (cid:88) n =0 b n ( z − z ) n , (1.40)which belongs to what is known as the rational Hermite interpolation prob-lem. Typical cases is when low-energy, high energy or threshold behavior areknown in advance. It is possible then to construct an N-point PA, P NM ( z ),in which J ( K ) terms are fixed from the series expansion around z ( z ) fromEq. (1.40), where J + K = N + M + 1. Note that, for N + M + 1 points,this would correspond to a fitting function interpolating between the givenpoints. In general, N-point PAs will produce an improved overall picturewith respect to typical (one-point) PAs of the same order, whereas the lat-ter will provide a more precise description around their expansion point. The central object of interest in this thesis are the transition form factors(TFFs) describing the interactions of the lowest-lying pseudoscalar mesons( P ) with two (virtual) photons and as such characterize the internal pseu-2 Chapter 1. Quantum Chromodynamics and related conceptsdoscalar structure. From the S -matrix element (cid:104) γ ∗ γ ∗ | S | P (cid:105) ≡ i M ( P → γ ∗ γ ∗ )(2 π ) δ (4) ( q + q − p ) (1.41)= ( ie ) (cid:90) d x (cid:90) d y (cid:104) γ ∗ γ ∗ | T { A µ ( x ) j µ em ( x ) , A ν ( y ) j ν em ( y ) } | P (cid:105) = − e (cid:90) d x e iq · x (cid:90) d y e iq · y (cid:104) | T { j µ em ( x ) , j ν em ( y ) } | P (cid:105) = − e (cid:90) d x e iq · x (cid:104) | T { j µ em ( x ) , j ν em (0) } | P (cid:105) (2 π ) δ (4) ( q + q − p )where p, q and q represent the pseudoscalar and photon momenta, therelevant amplitude defining the pseudoscalar TFF can be extracted: i M ( P → γ ∗ γ ∗ ) = − e (cid:90) d x e iq · x (cid:104) | T { j µ em ( x ) , j ν em (0) } | P ( p ) (cid:105)≡ ie (cid:15) µνρσ q ρ q σ F P γ ∗ γ ∗ ( q , q ) , (1.42)which represents a purely hadronic object. For the case of real photons, theTFFs can be related in the chiral (and, for the η (cid:48) , combined large N c ) limitto the ABJ anomaly [1], obtaining for F P γ ∗ γ ∗ (0 , ≡ F P γγ F P γγ = N c π F tr( Q λ P ) ⇒ M ( P → γγ ) = e (cid:15) µνρσ (cid:15) ∗ µ (cid:15) ∗ µ q ρ q σ F P γγ ⇒ Γ( P → γγ ) = πα m P F P γγ , (1.43)where F is the decay constant in the chiral limit defined in Eq. (1.22) and λ P = λ , , for the π , η and η , respectively. For an elementary particle,the TFF would be constant, whereas for composite particles is expected toexhibit a q -dependency providing valuable information on the pseudoscalarmeson structure. To study the TFF from first principles in the most general q regime poses a formidable task, for which the only firm candidate so faris lattice QCD —there exist some promising results in Refs. [60–62] within alimited energy range. Still, there exists some knowledge at some particularenergy regimes where different approaches apply. At large space-like energies, the TFF can be calculated as a convolution of aperturbatively calculable hard-scattering amplitude T H and a gauge invari-ant meson distribution amplitude (DA) φ ( a ) P encoding the non-perturbativedynamics of the pseudoscalar bound state [63] (summation over flavor a =3 , , a = 3 , q, s in the flavor basis, see Chapter 4), F P γ ∗ γ ∗ ( Q , Q ) = tr (cid:0) Q λ a (cid:1) F aP (cid:90) dx T H ( x, Q , , µ ) φ ( a ) P ( x, µ ) , (1.44) j µ em = ¯ uγ µ u − ¯ dγ µ d − ¯ sγ µ s ≡ Q ¯ qγ µ q defines the electromagnetic current —sumover quarks and colors is implicit. .6. The pseudoscalar transition form factors 23 φ ( a ) P xp (1 − x ) p q q q q (1 − x ) pxpφ ( a ) P φ ( a ) P Figure 1.7: Left: leading order diagrams in pQCD contributing to the hard-scatteringamplitude T H . Right: gluon exchanges inducing a gauge link or Wilson line. with ¯ x = 1 − x . The hard scattering amplitude at LO (see Fig. 1.7) reads T LO H = 1¯ xQ + xQ + ( x → ¯ x ) , (1.45)whereas the DA can be defined in terms of the matrix element [67] (cid:104) | ¯ q ( z ) γ µ γ [ z , z ] λ a q ( z ) | P ( p ) (cid:105) = ip µ F aP (cid:90) dx e − iz · p φ ( a ) P ( x, µ ) , (1.46)where z = ¯ xz + xz and obeys φ ( a ) P ( x ) = φ ( a ) P (¯ x ). As a non-perturbativeobject, its particular shape is unknown from first principles at an arbitrary(renormalization) scale µ . However, its asymptotic behavior at large energiesis well-known: the DA follows the ERBL evolution [63, 68] which allows fora convenient decomposition in terms of Gegenbauer polynomials φ ( a ) P ( x, µ ) = 6 x (1 − x ) (cid:32) ∞ (cid:88) n =1 c ( a )2 n,P ( µ ) C / n (2 x − (cid:33) , (1.47)with coefficients evolving at LO as c ( a ) n ( µ ) = (cid:18) α s ( µ ) α s ( µ ) (cid:19) γ n /β c ( a ) n ( µ ) , (1.48)As a result, asymptotic freedom implies that at large µ ∼ Q + Q theDA tends to the asymptotic one, φ as ( x ) = 6 x (1 − x ). Consequently, thehigh-energy behavior follows trivially from Eqs. (1.44) and (1.45), implyinglim Q →∞ F P γ ∗ γ ∗ ( Q , 0) = 6 F aP Q tr( Q λ a ) , (1.49)lim Q →∞ F P γ ∗ γ ∗ ( Q , Q ) = 2 F aP Q tr( Q λ a ) . (1.50) The NLO result was calculated in Refs. [64, 65]. See also Ref. [66]. [ z , z ] represents a gauge link or Wilson line, see Fig. 1.7, right. The LO anomalous dimensions read γ n = C F (cid:16) ψ ( n + 2) + γ E ] − (cid:104) n +1)( n +2) (cid:105)(cid:17) , C F = N c − N c = and α s ( µ ) evolution should be at LO. β has been defined below Eq. (1.4). An additional effect has to be accounted for the singlet component —a careful de-scription can be found in Ref. [67]. Whereas it has a non-negligible effect, we postponeits discussion to Chapter 4 as it does not change the conclusions outlined below. Figure 1.8: The different contributions up to O ( p ) to the P γγ process. First is LO O ( p ) contributions, while the other are NLO. The first one is known as the Brodsky-Lepage (BL) asymptotic behavior,whereas the second one can be obtained independently from the OPE of twoelectromagnetic currents [69] which are solid pQCD predictions.Even if the DA shape is largely unknown —the c coefficient has beenestimated from lattice QCD for the π [70–72]— it can be modeled toreproduce the available experimental data for the space-like single-virtualTFF at Q large enough (the double-virtual TFF has not been measuredso far). This has been studied for the π in light-cone pQCD [73], us-ing light-cone sum rules, both for the π [66, 74, 75] and η, η (cid:48) [67], or us-ing flat DAs —which became popular after the B A B AR data release for the π [76]— among others [77–80]. In addition, transverse momentum effectshave been studied [81, 82]. Alternatively, the TFF has been analyzed usingDyson-Schwinger equations [83], from Holographic models [84] and employ-ing anomaly sum rules [85, 86]. The agreement among different parame-terizations and the conclusions drawn from different authors is not clear atall, except for the solid results Eqs. (1.49) and (1.50). Particularly, there isno consensus on the range on applicability of pQCD and the onset of theasymptotic behavior. In addition, the pQCD approach cannot be extendeddown to Q → χ PT. χ PT At low-energies, χ PT can be used to provide the TFF behavior. At leadingorder O ( p ) ( O (1) in (cid:96)N c χ PT), this is described via the WZW LagrangianEq. (1.19), which exactly reproduces the ABJ result Eq. (1.43). Remarkably,this is a free-parameter prediction once the decay constant F has been fixedfrom other processes. In order to probe the pseudoscalar structure, higherorders bringing mass and q (and large- N c ) corrections are required. AtNLO in χ PT, O ( p ), the TFF result arises from the diagrams in Fig. 1.8 andwave-function renormalization, and can be found in Refs. [87–89]. Using the.6. The pseudoscalar transition form factors 25 L ,(cid:15) Lagrangian from Ref. [90], the TFF reads ( p , is a time-like quantity) F P γ ∗ γ ∗ ( p , p ) = N c tr( Q λ P )4 π F P (cid:18) − π (cid:104) ,(cid:15) c P + 2L ,(cid:15) c P + L ,(cid:15) ; r ( p + p ) (cid:105) + 196 π F (cid:20) − (cid:18) ln (cid:16) m π µ (cid:17) + ln (cid:16) m K µ (cid:17) + 23 (cid:19) p + ( p − m π ) × H (cid:16) p m π (cid:17) +( p − m K ) H (cid:16) p m K (cid:17) +( p → p ) (cid:21)(cid:19) . (1.51)At q = q = 0, corrections arise from the L ,(cid:15) , counterterms which arenevertheless not necessary to render the P → γγ amplitude finite since thedivergence is reabsorbed in the wave-function renormalization upon F → F P replacement, a result which holds only at NLO [91]. Furthermore, thesecorrections vanish in the chiral limit and they are commonly dismissed. Forfinite virtualities, an additional counterterm, L ,(cid:15) ; r is required to absorb thedivergencies , incorporating a p , -dependency together with the H ( s ) loopfunction (see Eq. (3.10) in Ref. [2]) H ( s ) = β ( s ) ln (cid:16) β ( s ) − β ( s ) (cid:17) , s ≤ 02 + | β ( s ) | (cid:0) − ( | β ( s ) | ) − π (cid:1) , < s < 42 + iπβ ( s ) + β ln (cid:16) − β ( s )1+ β ( s ) (cid:17) , s ≥ β ( s ) = √ − s − . A naive extrapolation to incorporate the η singletstate would yield an analogous result to that in Eq. (1.51) with an extra1 / N c . It turns out that, in the (cid:96)N c χ PT counting,loops and the L ,(cid:15) contributions are N c suppressed. Consequently, the chirallogarithms and loop function should be absent together with L ,(cid:15) . Moreover,an additional purely singlet OZI-violating term Λ [47] appears , which incontrast to L ,(cid:15) cannot be avoided in order to cancel the F QCD scaledependency. Describing the physical η and η (cid:48) TFFs requires though tointroduce the mixing, which we discuss in Chapter 4. It is well known thatthe TFF p , dependency in Eq. (1.51) is fully dominated by L ,(cid:15) instead ofthe (a priori large) chiral logarithms [91, 92] —a sign that such a processis dominated from vector resonance effects, with the consequent breakdownof the chiral expansion at energies close to the resonance. Given the lowest-lying ρ and ω resonances, one cannot expect the chiral theory to work beyond0 . —even if including an infinite number of terms— and this cannotbe matched to pQCD to provide a full-energy range description. c π = ˚ M π and c π = 0; c η = M π − M K and c η = 8( ˚ M π − ˚ M K ); in a naive η implementation, c η = M π + ˚ M K and c η = ( ˚ M π + ˚ M K ). L ,(cid:15) → L ,(cid:15) ; r + δ π F with δ = ( (cid:15) + ln(4 πµ ) + γ E + 1). Which amounts to replace Q λ )4 π F (1 − [ ... ]) → Q λ )4 π F (1 + Λ − [ ... ]) in Eq. (1.51). As previously stated, the presence of resonances limits the applicability ofthe chiral effective field theory which begs for the presence of additionaldegrees of freedom. One possibility is to parametrize these contributions intothe chiral theory in terms of pseudoscalar mesons rescattering effects whichare experimentally known [89]. Actually, this can be generalized advocatingfor a fully a dispersive framework [93–96] incorporating different time-likeinformation. Note however that such approaches have in practice either alimited range of applicability or require some modeling assumptions.Alternatively, the situation can be analyzed within the large- N c limitof QCD in which the resonances are far more important than those effectswhich may be accounted for in χ PT or pQCD. From this point of view,one could describe the TFFs through modeling the infinite tower of vectorresonances [97]. Alternatively, it has been customary to employ the MHA tosaturate the TFF with a minimal finite amount of well-known resonances [98,99]. Furthermore, there have been attempts to incorporate the resonances(within large N c ) explicitly into χ PT in what is known as resonance chiralperturbation theory [100, 101].From an orthogonal point of view, PAs can be used to directly addressthe problem posed at the end of Sections 1.6.1 and 1.6.2, this is, to providean interpolation between χ PT and pQCD (at least in the full space-likeregion) without the necessity of invoking large N c —which is ultimatelyan approximation and requires some modeling. Recall that PAs do notonly apply in the large- N c limit of meromorphic functions, but offer anopportunity to go beyond this and to apply them to the real world, as it wasshown for the case of Stieltjes functions. In this way, PAs allow to improveupon ideas as old as the MHA or the Brodsky-Lepage (BL) interpolationformula [102]. Moreover, having a limited amount of information, theyprovide improved convergence properties with respect to typical resonantapproaches used nowadays. For the case of the TFF, the analytic propertiesof the function are much more intricate than for two-point Green’s functions,and therefore we cannot anticipate convergence —note however that thesalient features such as the ππ elastic rescattering and different resonancesare of the Stieltjes kind. We can however check this a posteriori and estimatea systematic error from the convergence pattern, which we anticipate to beexcellent, which provides an advantage with respect to previous methods.From this point of view, all the required information is encapsulated in theTFF series expansion F P γ ∗ γ ( Q ) = F P γγ (0 , (cid:18) − b P Q m P + c P Q m P − d P Q m P + ... (cid:19) , (1.53)which can be determined from data as it is explained in the next chapter. hapter Data analysis with Pad´eapproximants Contents η and η (cid:48) LEPs . . . . . . . . . 342.5 Time-like data: η and η (cid:48) LEPs . . . . . . . . . 412.6 Conclusions . . . . . . . . . . . . . . . . . . . . 52 For the phenomenological applications of pseudoscalar transition form fac-tors (TFFs) covered in this thesis, we find that a very accurate descriptionof these TFFs at very low energies —where no available experimental dataexists— is required. For this reason, and regarding our approach based onPad´e approximants (PAs) to reconstruct the TFFs, it is extremely impor-tant to our work to know the series expansion for the TFF at zero energies.For the moment, we will restrict ourselves to the simpler single virtual case F P γ ∗ γ ( Q ) ≡ F P γ ∗ γ ∗ ( Q , 0) = F P γγ (cid:18) − b P Q m P + c P Q m P − d P Q m P + ... (cid:19) , (2.1)where b P , c P and d P are referred to as slope, curvature and third deriva-tive, respectively. The value for F P γγ ≡ F P γ ∗ γ (0) is well known for everypseudoscalar, as it is related to the Adler [42]-Bell-Jackiw [43] anomaly andcan be theoretically related in χ PT to the meson decay constants for the278 Chapter 2. Data analysis with Pad´e approximants π , η and η (cid:48) (the mixing parameters are required for the last two though, seeChapter 4). Furthermore, they can be experimentally extracted from themeasured P → γγ two-photon decays[10, 103–105]. In contrast, the addi-tional low-energy parameters (LEPs) b P , c P , ... cannot be obtained from firstprinciples in QCD or predicted from χ PT, as their values are given in termsof unknown low-energy constants. Moreover, they are not directly relatedto any experimental quantity. Consequently, these parameters have alwaysbeen obtained after modelization. For instance, with quark-loop models [88],Brodsky-Lepage interpolation formula [88, 102], resonance models [101] or χ PT supplied with vector meson dominance (VMD) ideas [87, 106].A possible venue to address this problem would be to use low-energyexperimental data so that χ PT or the series expansion, Eq. (2.1), apply.Then, the above parameters could be extracted from a fitting procedure ina model-independent way. However, these data at very low energies are, ingeneral, not available, rather scarce, or not precise, and one relies then onfits to models from high-energy data to extract these parameters. Such pro-cedure is model-dependent and implicitly includes a systematic error whichhas never been considered. Actually, depending on the fitted data set, in-consistencies seem to appear in some cases, for instance, when comparingspace-like and time-like data-based extractions for the slope parameter b η .In this chapter, we show how PAs can be used as a data-fitting tool toextract valuable information of the underlying function —the single-virtualTFF— including, among others, the desired LEPs in Eq. (2.1). We illus-trate that current inconsistencies cannot only be understood, but actuallysolved within a Pad´e framework. The results from this chapter representsthe starting point of the following ones, as it provides the basic inputs forreconstructing the TFFs which are used in our calculations, as well as for ex-tracting the η − η (cid:48) mixing parameters. We proceed as follows: in Section 2.2,we outline the procedure to obtain the LEPs from a fitting procedure. Thecorresponding systematic error is estimated in Section 2.3 through the useof different well-motivated models. Then, we apply our approach to thereal case for the η and η (cid:48) mesons using space-like data in Section 2.4. InSection 2.5, we argue, in view of the recent Dalitz decay measurements, whyPAs could be applied to the low-energy time-like region as well, reevalu-ating our LEPs extraction. We give our conclusions and main results inSection 2.6. Traditionally, the lack of low-energy data for the TFF has implied that theLEPs have been determined from phenomenological fits to high-energy data..2. Pad´e approximants as a fitting tool 29There, the vector meson dominance (VMD) fitting function F V MDP γ ∗ γ ( Q ) = F P γγ Λ Λ + Q (2.2)has been employed [107, 108], which then —upon expansion— allowed toextract a determination for the slope b P parameter, which for this model isgiven by b P = m P / Λ . Additional LEPs were not discussed in this contextthough as they are all fixed in the ansatz above (i.e. c P = b P ). More-over, given the quality and precision of previous data, this discussion wasirrelevant then, a situation which has changed with the recent release ofnew and more precise data in a wider energy regime, which makes timely astudy of this kind. The possible deficiencies and model dependencies fromthis approach can be easily understood from Pad´e theory, where the oldVMD determinations can be understood as the simplest step in a system-atic and convergent expansion [54, 109]. As such, this implies that previousfitting approaches —implying large systematic uncertainties as we illustratebelow— can be systematically improved, which makes possible not only amore accurate determination for b P , but a meaningful extraction for ad-ditional parameters such as c P and d P in a model-independent way afterperforming the expansion of Eq. (2.1) for the fitted approximants.Certainly, previous assertion relies on the assumption that the underly-ing function is such that some convergence to a given PA sequence exists,so our fits and the LEPs extracted from them will converge to the real ones.Having incomplete analytical information about the TFFs, this cannot beguaranteed beforehand. Note however that the prominent features around Q = 0 and the space-like region seem dominated by the role of the lowest-lying resonances —of almost meromorphic nature— and the ππ rescatteringeffects, essentially accounting for the ρ width —basically of Stieltjes nature.The existing convergence theorems, see Section 1.5.1, would justify then anexcellent performance, at least, in the space-like region, of main interestfor our applications. Finally, even in the case where convergence to theunderlying function cannot be guaranteed —nor disproved—, the PAs prac-titioner can still judge on the convergence of a given sequence a posteriori after the fitting procedure. We illustrate this in the following with the helpof three different well-motivated models, where the different scenarios de-scribed above apply. This exercise will provide not only helpful to describeand get familiar with the procedure we use to extract the LEPs, but toassess a systematic error that we will employ when determining the LEPsfrom real data.The last point to discuss is the kind of sequences that should be usedthen for the fitting procedure. A glance at time-like data reveals that thefirst resonance effects are dominating the low-energy time- and space-like0 Chapter 2. Data analysis with Pad´e approximantsdescription, meaning that including a single pole is enough to achieve aprecise description and motivates the use of the P N sequence. It is im-portant to note however that such description violates unitarity at highenergies as it diverges as ( Q ) N − . This motivates the use of a second se-quence, P NN +1 , which can be thought of as a two-point PA (see Section 1.5.2)and incorporates the appropriate high-energy BL behavior, see Eq. (1.49), F P γ ∗ γ ( Q ) ∼ Q − . Given the uncertainty about convergence, cross-checkingthe results from both sequences will reassure the consistency of the method.In what follows, we restrict our attention to these two sequences. Alterna-tive choices exist, as for instance, Pad´e- or partial Pad´e-type approximants,see Section 1.5.2. Their arbitrariness in choosing a pole, and their slowerconvergence as compared to the previous ones make them less attractive,and we do not further consider their study. For testing the convergence of our chosen P N and P NN +1 sequences, we pro-pose the use of three different motivated theoretical models out of the vastliterature, which will illustrate the performance of our approach in differ-ent representative situations. These are, the large- N c Regge model fromRefs. [97, 110], the logarithmic model in Ref. [111] —which finds inspirationin flat distribution amplitudes [77] and quark models [112]— and the holo-graphic model proposed in Refs. [84, 113]. For studying the convergencepattern, we generate a set of pseudo-data points in a similar manner to howreal experimental data are distributed. As this is to represent the ideal casewhere the function is known up to arbitrary precision, we don’t ascribe anyerror to the data for our fitting procedure, and therefore it does not makesense to give the χ from the fits. Finally, we perform the expansion inEq. (2.1) to extract the LEPs from our fits and compare. N c Regge model The large- N c model from Refs. [97, 110] consists of an infinite sum of vectorresonances, which sum can be expressed in terms of the polygamma function ψ ( n ) = d n +1 z n +1 ln Γ( z ) with Γ( z ) the Gamma function, F P γ ∗ γ ( Q ) = aF P γγ Q ψ (1) (cid:16) M a (cid:17) (cid:20) ψ (0) (cid:18) M + Q a (cid:19) − ψ (0) (cid:18) M a (cid:19)(cid:21) . (2.3)The parameters above have been slightly renamed for convenience with re-spect to those appearing in Refs. [97, 110]. To reproduce the physical case,we choose the experimental F P γγ ≡ F P γγ (0 , 0) together with a = 1 . and M = λ × . 64 GeV [111, 114] where λ = 1 , . , . 05 for the π , η, η (cid:48) ,.3. Estimation of a systematic error 31 P P P P P P P P P P Exact F Pγγ . 268 0 . 273 0 . 274 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . b P . 332 0 . 373 0 . 394 0 . 404 0 . 411 0 . 416 0 . 419 0 . 421 0 . 413 0 . 425 0 . c P — 0 . 143 0 . 163 0 . 173 0 . 182 0 . 188 0 . 192 0 . 195 0 . 185 0 . 201 0 . d P — — 0 . 067 0 . 074 0 . 081 0 . 085 0 . 088 0 . 091 0 . 083 0 . 097 0 . Table 2.1: LEPs determination from the space-like pseudo-data set for the large- N c Regge model. F Pγγ is expressed in GeV − ; additional quantities are dimensionless. respectively . In what follows, we focus in the η case, though very similarresults are obtained for the η (cid:48) as it shares a similar TFF and available datasets. Actually, the results for the π are similar too, see Ref. [111]. Forthis model convergence is expected as it is a meromorphic function, whichin addition represents the interesting case in which the large- N c limit applies.Adopting the points defined in Ref. [114] —10 points in the region0 . < Q < . , 15 points in the region 2 . < Q < . and 10 points in the region 8 . < Q < 34 GeV — which resembles theexperimental situation, we obtain the results in Table 2.1. We find theexpected convergence pattern we anticipated (note that the curvature andthird derivative are not extracted up to P and P , respectively). Moreover,we find an hierarchy: there is a faster convergence for F P γγ ≡ F P γ ∗ γ (0), thenfor b P , and so on. An important observation at this point is that no mat-ter whether strong correlations and, possibly, a tiny χ value in the realcase appear, the highest the element within a sequence, the better the ex-traction for the LEPs becomes —a feature common to all the models andcharacteristic of PAs. Therefore, we should aim for the largest possibleelement in our sequence when fitting real data for extracting our desiredparameters. In addition, we find that the P NN +1 sequence has the betterperformance—note though that this sequence increases its number of pa-rameters in units of two, so the P should be compared with the P andso on. This can be understood from the fact that, even if at these ener-gies it is the influence of the first pole that dominates, there are additionalhigher resonances. Not less, this sequence implements as well the appro-priate high-energy behavior, relevant for the data range we are using. InSection 2.5, we will employ also some very low-energy time-like data pointsin addition to the space-like ones. Given their small q values, we expectthat they significantly improve the accuracy from our determination, whichdemands a new systematic error evaluation. For this, we add to our previ-ous pseudo-data set 8 points in the (0 . < q < (0 . GeV region,15 points in the (0 . < q < (0 . GeV region and 31 points in the(0 . < q < (0 . region in order to reproduce the experimental The parameter a is taken from the analysis of different Regge trajectories in Ref. [115].For the π , M is roughly the ρ and ω meson masses. For the η and η (cid:48) there is an interplayof ρ, ω and φ resonances [116] which effectively translates in the λ parameter. P P P P P P P P P P Exact F Pγ . 279 0 . 276 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . b P . 415 0 . 433 0 . 437 0 . 437 0 . 436 0 . 435 0 . 435 0 . 434 0 . 435 0 . 433 0 . c P — 0 . 196 0 . 204 0 . 207 0 . 209 0 . 210 0 . 210 0 . 210 0 . 210 0 . 210 0 . d P — — 0 . 095 0 . 098 0 . 101 0 . 102 0 . 102 0 . 103 0 . 102 0 . 104 0 . Table 2.2: LEPs determination from the space- and time-like pseudo-data set for thelarge- N c Regge model. F Pγγ is expressed in GeV − ; additional quantities are dimension-less. P P P P P P P P P P Exact F Pγ . 268 0 . 273 0 . 274 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . b P . 159 0 . 196 0 . 217 0 . 227 0 . 233 0 . 238 0 . 241 0 . 243 0 . 233 0 . 247 0 . c P — 0 . 040 0 . 052 0 . 058 0 . 063 0 . 068 0 . 071 0 . 073 0 . 064 0 . 078 0 . d P — — 0 . 012 0 . 015 0 . 017 0 . 020 0 . 021 0 . 023 0 . 018 0 . 026 0 . Table 2.3: LEPs determination from the space-like pseudo-data set for the logarithmicmodel. F Pγγ is expressed in GeV − ; additional quantities are dimensionless. situation [117]. Our results are displayed in Table 2.2. We find very similarconclusions together with an improved accuracy —to be expected from theincreased amount of low-energy data points. This model finds inspiration in quark models [112] or flat distribution am-plitudes [77], which have been proposed ever since the puzzling B A B AR datafor the π TFF [76] were released. The model includes a logarithmic en-hancement with respect to the BL asymptotic behavior, F P γ ∗ γ ∗ ( Q ) = F P γγ M Q ln (cid:18) Q M (cid:19) , (2.4)with M = 0 . [77] to reproduce B A B AR data [76] and F P γγ to re-produce the physical value. This second model is known to belong to theclass of Stieltjes functions, which guarantees the performance of the methodand allows to test the effects of perturbative logarithms as well, representingtherefore an interesting case of study. Taking the pseudo-data points dis-cussed above, we find the results in Table 2.3. Again, we can reach to verysimilar conclusions as those in the Regge model. Moreover, we find that inthis case the P NN +1 sequence has even better performance with respect tothe P N than in the previous case. This can be understood from the conver-gence theorems for Stieltjes functions existing for the P NN +1 sequence (seeSection 1.5.1) and from the much more involved analytic structure whicha cut implies with respect to a single pole. Once more, we reanalyze thesystematic error for the case where we include the time-like data points ontop and display the results in Table 2.4.3. Estimation of a systematic error 33 P P P P P P P P P P Exact F Pγ . 281 0 . 278 0 . 276 0 . 276 0 . 275 0 . 275 0 . 275 0 . 275 0 . 281 0 . 275 0 . b P . 199 0 . 230 0 . 245 0 . 251 0 . 253 0 . 253 0 . 253 0 . 253 0 . 253 0 . 251 0 . c P — 0 . 057 0 . 068 0 . 075 0 . 079 0 . 081 0 . 082 0 . 083 0 . 081 0 . 084 0 . d P — — 0 . 019 0 . 022 0 . 025 0 . 027 0 . 028 0 . 029 0 . 027 0 . 031 0 . Table 2.4: LEPs determination from the space- and time-like pseudo-data set for thelogarithmic model. F Pγγ is expressed in GeV − ; additional quantities are dimensionless. P P P P P P P P P P Exact F Pγ . 279 0 . 277 0 . 276 0 . 276 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . b P . 376 0 . 357 0 . 342 0 . 334 0 . 327 0 . 322 0 . 319 0 . 316 0 . 307 0 . 311 0 . c P — 0 . 126 0 . 114 0 . 107 0 . 101 0 . 096 0 . 092 0 . 090 0 . 078 0 . 083 0 . d P — — 0 . 038 0 . 034 0 . 031 0 . 028 0 . 026 0 . 025 0 . 018 0 . 017 0 . Table 2.5: LEPs determination from the space-like pseudo-data set for the holographicmodel. F Pγγ is expressed in GeV − ; additional quantities are dimensionless. Finally, we take a model based on light-front holographic QCD from Ref. [84]and we restrict ourselves, for simplicity, to the simplest leading twist result(though similar patterns are found for the other models in [84]) F P γ ∗ γ ∗ ( Q ) = P q ¯ q π F π (cid:90) dx (1 + x ) x Q P q ¯ q / (8 π F π ) (2.5)with P qq = 1 / To summarize, we find after comparing to different well-motivated modelsthat, both, P N and P NN +1 sequences have a great performance for extractingthe LEPs through a fitting procedure from experimental data. We emphasizethat this may be the case even when convergence is not guaranteed, seeSection 2.3.3. As an important result, we show that in order to have the mostaccurate prediction, we should reach the highest element in each sequenceregardless of correlations or χ ν (cid:28) P NN +1 sequence whichhas the better performance, though it increases its number of parametersin units of two, making the fitting procedure more complicated than for4 Chapter 2. Data analysis with Pad´e approximants P P P P P P P P P P Exact F Pγ . 271 0 . 273 0 . 274 0 . 274 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . b P . 334 0 . 323 0 . 316 0 . 313 0 . 310 0 . 310 0 . 310 0 . 310 0 . 311 0 . 311 0 . c P — 0 . 102 0 . 095 0 . 091 0 . 087 0 . 085 0 . 084 0 . 083 0 . 082 0 . 083 0 . d P — — 0 . 028 0 . 026 0 . 024 0 . 023 0 . 022 0 . 022 0 . 020 0 . 020 0 . Table 2.6: LEPs determination from the space- and time-like pseudo-data set for theholographic model. F Pγγ is expressed in GeV − ; additional quantities are dimensionless. P P P P P P P P P P F Pγ / / / . / . / / / / . / / b P / 20 20 / 10 15 / / / / / / . / / c P — 50 / 30 40 / 20 30 / 10 25 / / / / / / d P — — 60 / 40 50 / 30 45 / 20 40 / 15 30 / 10 30 / / 15 15 / Table 2.7: Our final systematic errors in % for the SL/(SL+TL) data sets the P N sequence. In order to estimate the systematic errors, we adopt aconservative approach and take those arising from the quark model, whichshows the slowest convergence pattern, obtaining the results in Table 2.7—such table represents the main result from this section. At this point, wefind that it is possible to have a meaningful extraction for the slope andcurvature parameters in both data sets, whereas an accurate extraction forthe third derivative is only possible and considered in our combined space-and time-like data set study. In addition, an analogous procedure shows thatexperimental determinations for b P based on time-like data alone should beascribed an additional 5% systematic error. η and η (cid:48) LEPs Having demonstrated the excellent performance of Pad´e approximants asfitting functions to extract the LEPs, and having estimated a systematicerror for the procedure, we proceed to their extraction in the real case.This was done for the π in Ref. [111] and we extend this approach to the η and η (cid:48) below [114]. For that, we start using all the available data inthe space-like region in which convergence is expected. This comprises themeasurements from CELLO [107], CLEO [108] and B A B AR [118] for the η and η (cid:48) , and the additional L3 Collaboration [105] data-set at low-energiesfor the η (cid:48) . In addition, we use the measured two-photon decay widthsΓ η → γγ = 0 . η (cid:48) → γγ = 4 . Q F η ( (cid:48) ) γ ∗ γ ( Q ) rather than F η ( (cid:48) ) γ ∗ γ ( Q ), since this is the standard way in which experimental data hasbeen published, with the exception of L3 and CELLO collaborations. For The CELLO Collaboration does not report a systematic error for each bin of data.While for the η (cid:48) case such error is 16% of the total number of events (which we translate .4. Space-like data: η and η (cid:48) LEPs 35these, we transform their results into Q F η ( (cid:48) ) γ ∗ γ ( Q ). Moreover, we relatethe two-photon decay widths to F η ( (cid:48) ) γ ∗ γ (0) using the relation | F P γγ | = 64 π (4 πα ) Γ P → γγ m P , (2.6)obtaining F ηγγ = 0 . − and F η (cid:48) γγ = 0 . − .For the fitting procedure, we employ the P N and P NN +1 sequences mo-tivated in the previous sections, which translate into the P N and P NN se-quences for the Q F η ( (cid:48) ) γ ∗ γ ( Q ) published data. Then, we must reach thehighest possible element within a sequence as to maximally reduce the sys-tematic uncertainty for the LEPs extraction as shown in the previous section.However, when using real data, it is not possible to go all the way up to anarbitrary large N element. At some point, some of these parameters fromwhich our PAs are built become statistically compatible with zero, meaningthat its extraction is meaningless. We must stop at this point and take thisresult as our better extraction, and ascribe a systematic error as estimatedfrom our results in the previous section.In order to show the performance of our method, we employ a bottom-upapproach. We start fitting the Q F η ( (cid:48) ) γ ∗ γ ( Q ) space-like data without anyinformation at Q = 0. This means in particular that the mathematicallimit lim Q → Q F η ( (cid:48) ) γ ∗ γ ( Q ) = 0 is not imposed but extracted from data.In a second step, we impose such limit making use of PAs whose numeratorstarts at order Q (i.e. there is no constant term). This study allows thento extract the TFFs at zero, and therefore predict the two-photon partialdecay widths in addition to the slope and curvature parameters. Finally,as a last step, we incorporate the measured two-photon partial widths inour set of data, to be fitted together with the space-like data points. Thisapproach will show the robustness of our results.Starting then without constraining the lim Q → Q F η ( (cid:48) ) γ ∗ γ ( Q ) = 0 limit,we find that our fits “see the zero” for the η and η (cid:48) cases within two andone standard deviations for the η and η (cid:48) , respectively. Particularly, we find P (0) = 0 . P (0) = − . η and η (cid:48) , respectively.Once this is seen to be zero, the next coefficient in its series expansion isassociated with the TFF normalization. We find F η (cid:48) γγ (0) = 0 . − ,which translates into Γ η (cid:48) → γγ = 5 . . 7) keV. This illustrates the potentialof space-like data, which are ranging from 0 . in the η case and into 32% for each bin), for the η case only 12% for the two-photon channel is reported.Accounting for all the different systematic sources we could find in the publication, weascribe a 12% of systematic error for the hadronic η decay which leads to a 6% error forthe global number of events (implying 12% systematic error for each bin). η η (cid:48) N b η c η F ηγγ , GeV − χ ν N b η (cid:48) c η (cid:48) F ηγγ , GeV − χ ν P N ( Q ) 2 0 . . . . 79 5 1 . . . . P NN ( Q ) 1 0 . . . . 78 1 1 . . . . Table 2.8: LEPs for the η and η (cid:48) TFFs obtained from our fits without including infor-mation on Γ P → γγ . The first column indicates the type of sequence used for the fit and N is the highest order achieved. We also present the quality of the fits in terms of χ ν .Errors are only statistical and symmetrized. ææææ òòòòòòòòòòò ò àà à à à à à à à à à P H Q L P H Q L P H Q L Q @ GeV D Q F Η Γ * Γ H Q L @ G e V D ììì ì ìææææ æ òòòòòòò ò ò àà à à à à à à à à à P H Q L P H Q L P H Q L Q @ GeV D Q F Η ’ Γ * Γ H Q L @ G e V D Figure 2.1: η (left panel) and η (cid:48) (right panel) TFFs best fits. Blue-dashed lines showour best P L ( Q ) without including the Γ P → γγ information in our fits; green-dot-dashedlines show our best P L ( Q ) when including the Γ P → γγ information in our fits; black-solidlines show our best P NN ( Q ) in the latter case, which extrapolation down to Q = 0 and Q → ∞ is shown as a black-dashed line. Experimental data points are from CELLO(red circles) [107], CLEO (purple triangles) [108], L3 (blue diamonds) [105], and B A B AR (orange squares) [118] collaborations. from 0 . 06 to 35 GeV for the η (cid:48) , to predict LEPs, which are our main aimfor further applications in this work [114].Next, we make use of lim Q → Q F η ( (cid:48) ) γ ∗ γ ( Q ) = 0, meaning that the PAsnumerator starts already at order Q . This simple constraint allows for animproved LEPs determination, shown in Table 2.8. In this case, we reachup to the second and fifth elements of the P L sequence for the η and η (cid:48) ,respectively, which TFFs are shown in Fig. 2.1. Unfortunately, it is notpossible to go beyond the first element for the P NN sequence in both cases:for higher elements, the fit places poles in the space-like region, mimickingstatistical fluctuations in the data —such results should not be consideredand the sequence should be truncated at this point. Remarkably, in thisapproach we obtain Γ η → γγ = 0 . η (cid:48) → γγ = 4 . . . P L sequence, up to the fifth andsixth element for the η and η (cid:48) , respectively. On the other hand, including .4. Space-like data: η and η (cid:48) LEPs 37 η η (cid:48) N b η c η χ ν N b η (cid:48) c η (cid:48) χ ν P N ( Q ) 5 0 . . . 80 6 1 . . . P NN ( Q ) 2 0 . . . 77 1 1 . . . . . . . Table 2.9: LEPs for the η and η (cid:48) TFFs obtained from our fits when including informationon Γ P → γγ . The first column indicates the type of sequence used for the fit and N is thehighest order achieved. The last row shows our final result for each LEP —find details inthe text. We also present the quality of the fits in terms of χ ν . Errors are only statisticaland symmetrized. (cid:232) (cid:232) (cid:232) (cid:232) (cid:232)(cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:225)(cid:225)(cid:225)(cid:225) P11 P21 P31 P41 P51 CELLO0.20.30.40.50.60.7 b Η (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:225)(cid:225)(cid:225)(cid:225) P11 P21 P31 P41 P51 P61 CELLO1.01.52.0 b Η ' Figure 2.2: Slope predictions for the η (left panel) and η (cid:48) (right panel) TFFs using the P L ( Q ) sequence (blue circles). The inner error bars correspond to the statistical error ofthe different fits. The outer error bars are the combination of statistical and systematicerrors determined as explained in the main text. The CELLO determination is also shownfor comparison (empty-red squares). Γ P → γγ allows to reach up to the second element in the P NN sequence for the η , whereas this is not possible for the η (cid:48) . The obtained TFFs are shown inFig. 2.1. The LEPs obtained for these cases are shown in Table 2.9. Thesimilarity for these results and those found previously without including thetwo-photon decay widths, Table 2.9, are quite reassuring. On top, we showour convergence results for the slope b P and curvature c P parameters withinthe P L sequence in Figs. 2.2 and 2.3, where the systematic errors are takenfrom Table 2.7. The observed pattern shows an excellent convergence. Inthese plots, we show in addition the results from CELLO for b η ( η (cid:48) ) obtainedfrom a VMD model fit [107]. To perform an appropriate comparison, weadd to their determinations an additional 40% error corresponding to the P element as determined in Table 2.7.In addition, we comment on the fitted poles obtained from the P N se-quence, which we show in Fig. 2.4, and range from √ s p = (0 . − . 77) GeVand √ s p = (0 . − . 86) GeV for the η and η (cid:48) respectively. We note thatsuch pole does not correspond to a particular physical resonance. It cor-responds instead to an effective parameter which effectively accounts forthe presence of different resonances, threshold effects and analytic structure8 Chapter 2. Data analysis with Pad´e approximants (cid:232) (cid:232) (cid:232) (cid:232) (cid:232)(cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:225)(cid:225)(cid:225)(cid:225) P11 P21 P31 P41 P51 CELLO0.00.10.20.30.40.5 c Η (cid:225)(cid:225)(cid:225)(cid:225) P11 P21 P31 P41 P51 P61 CELLO0.51.01.52.02.53.03.54.0 c Η ' Figure 2.3: Curvature predictions for the η (left panel) and η (cid:48) (right panel) TFFs usingthe P L ( Q ) sequence (blue circles). The inner error bars correspond to the statistical errorof the different fits. The outer error bars are the combination of statistical and systematicerrors determined as explained in the main text. The CELLO determination is also shownfor comparison (empty-red squares). P11 P21 P31 P41 P510.600.650.700.750.80 s p (cid:64) G e V (cid:68) P11 P21 P31 P41 P51 P610.650.700.750.800.850.900.95 s p (cid:64) G e V (cid:68) Figure 2.4: Pole-position predictions for the η (left panel) and η (cid:48) (right panel) TFFsusing the P L ( Q ) sequence. For comparison, we also display (orange and blue bands) therange m eff ± Γ eff / of the whole function in general. For comparison, we show as orange andblue bands what would correspond to the effective VMD meson resonance, m eff [116] using m ρ = 0 . 775 GeV, Γ ρ = 0 . 148 GeV, m ω = 0 . 783 GeV,Γ ω = 0 . 008 GeV, m φ = 1 . 019 GeV and Γ φ = 0 . 004 GeV and a mixing anglein the flavor basis φ = 39 ◦ (see Chapter 4). Alternatively, see Ref. [119] orthe updated values of Ref. [120] in [114]). The bands represent the range ofsuch mass implied by the half-width rule [115, 121, 122], i.e., m eff ± Γ eff / N c corrections to typical resonanceapproaches. We obtain m eff = 0 . m eff = 0 . η and η (cid:48) . As already indicated in Refs. [54, 59, 109, 111], fitting space-like data in resonant models does not produce an accurate determinationfor resonance parameters. We do not recommend then this method for suchdetermination. For an alternative model-independent method, we refer tothe interested readers to Refs. [123, 124].Finally, it is possible from the P NN sequences to extrapolate beyond theavailable data up to arbitrary large Q values (dashed lines in Fig. 2.1),.4. Space-like data: η and η (cid:48) LEPs 39which allows for extracting the asymptotic behavior. For the η , we reach upto the second element, while for the η (cid:48) we reach only up to the first elementdue to the appearance, once more, of space-like poles mimicking statisticalnoise in the data. We obtain [114]lim Q →∞ Q F ηγ ∗ γ ( Q ) = 0 . , (2.7)lim Q →∞ Q F η (cid:48) γ ∗ γ ( Q ) = 0 . . (2.8)We emphasize here that previous errors —which are statistical alone— couldbe deceptive. While the results for the η , arising from a higher element, sug-gest a larger error than for the η (cid:48) counterpart, the last has an intrinsic largersystematic error and it would be desirable as well for the η (cid:48) to reach a higherelement, an achievement which is not possible with the available data so far.For completeness, if we would have used the P element for extracting theasymptotic behavior of the η , we would have obtained 0 . including the information on Γ P → γγ , we ob-tain [114] b η = 0 . stat (3) syst c η = 0 . stat (7) syst , (2.9) b η (cid:48) = 1 . stat (7) syst c η (cid:48) = 1 . stat (34) syst , (2.10)where the second error is systematic, of the order of 5% and 20% for b P and c P , respectively. When the spread of the central values for the weightedaveraged result is larger than the error after averaging, we enlarge this errorto cover the spread . For the η (cid:48) case, we could only reach the first elementwithin the P NN sequence. Since the first error of each sequence has a largesystematic uncertainty, this should not be used, and consequently, we donot include it in our averaged result.For the η , the slope of the TFF obtained in Eq. (2.9) can be comparedwith b η = 0 . b η = 0 . b η = 0 . b η = 0 . b η = 0 . b η = 0 . b η = 0 . P ) to space-like data. Our We thank C.F. Redmer for discussions on the average procedure. ææææææææ æ æææ àà à àà àààà à àà ààòòòò Χ TDisp . Rel.Disp . Rel. 2Axial Anom .Lepton - GCELLOCLEONA60A2WASAA2 newOur Work b Η ææææææ æææ ææ ààà à ààà àòòòò Χ TDisp . Rel.Axial Anom .Lepton - G’CELLOCLEOBESIIIOur Work b Η ’ Figure 2.5: Slope determinations for η (left panel) and η (cid:48) (right panel) TFFs fromdifferent theoretical (red circles) and experimental (blue squares) references discussed inthe text. Inner error is the statistical one and larger error is the combination of statis-tical and systematic errors. ChPT [87, 106], VMD, Quark Loop, BL [88], R χ T [101],Disp.Rel [95], Disp.Rel 2 [96, 131], Axial Anom. [86], Lepton-G [125], Lepton-G’ [116],CELLO [107], CLEO [108], NA60 [126], A2 [127], WASA [128], A2 new [129], BESIII [130],Our Work [114]. rigorous mathematical and systematical approach improves on this issue.For the η (cid:48) , the slope in Eq. (2.10) can be compared with b η (cid:48) = 1 . b η (cid:48) = 1 . b η (cid:48) = 1 . b η (cid:48) = 1 . b η = 0 . 51 and b η (cid:48) = 1 . 47 from χ PT [87, 106] forsin θ P = − / θ P the η − η (cid:48) mixing angle in the octet-singlet basisdefined at lowest order; b η = 0 . 53 and b η (cid:48) = 1 . 33, from vector meson domi-nance (VMD) [88]; b η = 0 . 51 and b η (cid:48) = 1 . 30, from constituent-quark loops; b η = 0 . 36 and b η (cid:48) = 2 . 11, from the Brodsky-Lepage interpolation formula[102]; b η = 0 . b η (cid:48) = 1 . b η = 0 . +0 . − . and b η (cid:48) = 1 . +0 . − . from a dispersive analysis [95] .Eventually, we want to comment on the effective single-pole mass de-termination Λ P which Eq. (2.9) implies for the P reconstruction. Using b P = m P / Λ P and the values in Eq. (2.9) , we obtain Λ η = 0 . 706 GeV andΛ η (cid:48) = 0 . 833 GeV. These values together with Λ π = 0 . 750 GeV obtained inRef. [111] lead to Λ η < Λ π < Λ η (cid:48) , in agreement with constituent-quark loops The dispersive results [95] neglected the a tensor meson contribution [131]. Afteraccounting for this, they obtain b η = 0 . +6 − ) [96, 131]. .5. Time-like data: η and η (cid:48) LEPs 41 t t t t t t r η . 274 0 . − . × − . × − − . × − — 1 . η (cid:48) . 343 0 . − . × − . × − − . × − . × − . Table 2.10: Fitted coefficients for our best P L ( Q ) for the η and η (cid:48) TFFs in units ofGeV − i for t ( r ) i and GeV − for t . and VMD model approaches [88].Notoriously, our results for the LEPs would not be affected to the quotedprecision if the additional high-energy data points measured by B A B AR Col-laboration at q = 112 GeV [132] are included through the duality as-sumption that lim q →∞ F P γ ∗ γ ( q ) = lim Q →∞ F P γ ∗ γ ( Q ) extends to largebut finite energies. One would expect similarly that this is the case for thespace-like B A B AR data in the (4 − 35) GeV range [118]. However, this isnot the case: the high-energy data are relevant in order to reach higher PAsequences leading to more constrained values of the LEPs. In the case athand, only the B A B AR Collaboration provides precise measurements in theregion between 5 and 35 GeV . For instance, the value of the η slope pa-rameter shown in Eq. (2.9), b η = 0 . b η = 0 . B A B AR data are not included in the fits. In view of this behaviorand having in mind the π TFF controversy after the measurements of the B A B AR [76] and Belle [133] collaborations, a second experimental analysisby the Belle Collaboration covering this high-energy region would be verywelcome. Remarkably, we will find in Section 2.5 that even when includingvery low-energy time-like data, the B A B AR data points are still of relevance.For convenience, we also provide our parametrization of the highest P L fits, which can be used to predict the TFF low-energy behavior. Definingthe P L ( Q ) for F η ( (cid:48) ) γ ∗ γ ( Q ) as P L ( Q ) = t + t Q + ...t L ( Q ) L r Q , (2.11)the corresponding coefficients are given in Table 2.10 η and η (cid:48) LEPs Our space-like data-based description above [114] provides an accurate de-scription for the TFF in the low-energy range, which is the reason why wecould obtain such an accurate extraction for the LEPs. Of course, there isno special analytic property at Q = 0 which prevents us to make a predic-tion for low time-like energies. It remains the question then on what lowmeans here. It is well known that at larger time-like energies the appear-ance of production thresholds, starting with π + π − , imply the appearance2 Chapter 2. Data analysis with Pad´e approximants ] ) [GeV/c l + m(l | η | F (a) This Work: DataThis Work: Fit (p0=1)A2, 2011TL calculation approxim.ePad ] ) [GeV/c l + m(l | η | F (b) This Work: DataThis Work: Fit (p0=1)NA60, In InDT calculation Figure 2.6: The normalized η TFF results obtained from A2 Collaboration at MAMI.Plot taken from Fig. 10 in [129]. Their results displayed as solid squares [129] are comparedto NA60 [126] (open squares in (b)) and former A2 results [127] (open circles in (a)). Thedata and their fit is compared to different theoretical calculations: TL [134], dispersivetheory (DT) [95] and our results [114] from Eq. (2.11) (red line in (a) with gray errorband). of additional singularities and cuts. The analytic structure of PAs in turnis given by a set of isolated poles, which would in principle forbid its useabove threshold production and would question then the applicability of ourapproach to the η ( (cid:48) ) → ¯ (cid:96)(cid:96)γ Dalitz decays above threshold.Very recently, the A2 Collaboration at MAMI [129] reported a new mea-surement of the η → e + e − γ Dalitz decay with the best statistical preci-sion up to date, which allowed them to extract the (normalized) η TFF,˜ F ηγ ∗ γ ( q ), in the low-energy time-like region, q ∈ (4 m e , m η ). In their study,they performed a comparison with different theoretical models, obtaining theresults in Fig. 2.6. The agreement with our parameterization, Eq. (2.11), isexcellent (we note that this would not have been the case for the simplest P element). Moreover, we can see that our parameterization is superiorcompared with the different theoretical models considered in [129], thoughthe precision from data does not allow to discard any of them.Furthermore, new time-like data are also available from BESIII. Theyhave been able to measure, for the first time, the η (cid:48) → e + e − γ Dalitz decay ,allowing them to extract the normalized η (cid:48) TFF in the q ∈ (4 m e , m η (cid:48) ) re-gion [130]. Since their last bin is at 0 . 75 GeV and our approximant pole,Eq. (2.11), lies at 0 . 83 GeV, we can extrapolate up to their last point, ob-taining again an excellent agreement —though the current precision is notcomparable to that in the η Dalitz decay— see Fig. 2.7. The η (cid:48) → µ + µ − γ was measured before [135, 136] though with less precision, and inthe higher range q ∈ (4 m µ , m η (cid:48) ). .5. Time-like data: η and η (cid:48) LEPs 43 s (cid:64) GeV (cid:68) (cid:200) F (cid:142) Η ' BESIII data on Η ' (cid:174) Γ e (cid:45) e (cid:43) Our prediction Figure 2.7: Our space-like P N prediction (blue line), Eq. (2.11), for the η (cid:48) TFF includingstatistical errors (blue-band) compared with the recent BESIII results [130]. The excellent agreement displayed above challenged our understandingof PAs and the underlying reason behind these findings [117, 137, 138]. SincePAs are analytic functions in the whole complex plane except at their poleslocation, they cannot reproduce the analytical structure which a branch cutrequires. As an example, our construction above would not allow to openthe second Riemann sheet and, consequently, it cannot be used to determineresonance parameters. The latter would be possible if constructing the ap-proximant above the threshold [123, 124], which however would forbid theLEPs determination. For the particular case when the original function tobe approximated is Stieltjes with a finite radius R of convergence aroundthe origin, it is a well-known result in the theory of Pad´e approximants thatthe sequence P N + JN ( z ) (with J ≥ − 1) converges to the original function as N → ∞ on any compact set in the complex plane, excluding the cut at R ≤ z < ∞ , see Section 1.5.1 —where the poles of the approximant locateto emulate the cut effects, cf. Fig. 1.6. In other words, even though the ππ unitary cut driving the decay is of Stieltjes nature, there is a priori no rea-son why the PA should work above the branch cut. The surprising situationis, however, that at least the P L ( s ) sequence does seem to work well abovethe cut (cf. Figs. 2.6 and 2.7) for the two observables. One could speculateabout the good agreement found above.To qualitatively understand the situation, as a first approximation, itwould be fair to say that the TFF is a meromorphic function —as it wouldin the large- N c limit of QCD. In such scenario, PAs are an excellent approxi-mation tool [54]. Particularly, if the TFF contains a single and isolated pole,the P L ( s ) sequence reproduces the pole of the TFF with infinite precision.As soon as the width is again switched on, the ππ threshold opens a branchcut responsible for that width. Then, at first, no mathematical theorem willguarantee convergence on this scenario. On the contrary, if the convergencetheorem is to be satisfied, one would expect the single pole of the P L ( s ) to4 Chapter 2. Data analysis with Pad´e approximantsbe located closer and closer to the threshold point as soon as L → ∞ , sincethis is the first singular point the PA is going to find. However, the behav-ior of this ππ branch cut at threshold is well known as it comes from the ππ P-wave, implying the imaginary part expansion at threshold to behaveas ( s − m π ) / —such behavior can be easily obtained from Eqs. (1.51)and (1.52) and gives an estimate for the discontinuity size. Beyond, thewell-studied ππ P-wave rescattering will be responsible to modulate suchdiscontinuity, which is related to the well-studied ππ vector form factor. Itis the smoothness of such discontinuity that explains the excellent perfor-mance found above. More precisely, taking the definition of a P L ( s ) givenby P L ( s ) = L − (cid:88) k =0 a k ( s ) k + a L ( s ) L − a L +1 ( s ) a L ( s ) , (2.12)we would expect the PA pole to effectively account for the TFF pole, whereasthe polynomial part would accurately reproduce the induced ππ P-wave ef-fects subthreshold. The latter would guarantee a reasonable approximationabove threshold as long as the discontinuity is mild, this is, as this does notbecome resonant. This happens basically at a distance of the pole given bythe half-width rule [121], which can provide a simple estimate of the PAsapplicability range. In a realistic situation with multiple cuts, the picturewill develop new features, but the final result would be similar. The PApole becomes an effective pole resulting from the combination of the abso-lute values of the different resonances entering the process, closer to the onewith larger coupling in the particular reaction and with shifts produced bytheir respective widths.For a quantitative discussion, we focus on the particular case of the η TFF. To illustrate our statements, we choose the dispersive approach fromRef. [95], which has the appropriate ππ branch cut implementation alongwith ππ rescattering effects . We generate then a space-like data set analogto that in Section 2.3 with such model and perform a fit using the P N ( Q )sequence of approximants. The results are shown in Fig. 2.8 left and displaya perfect agreement below threshold with respect to the dispersive model anda smooth offset above. Both the dispersive model and the PA extrapolationsto the time-like region can be compared to real experimental data for thatchannel. Interestingly enough, the observed offset is below the experimen-tal resolution as can be inferred from Fig. 2.8 left, supporting our previouscomments and justifying the observed performance of PAs. In addition, therelative difference of the fitted approximants with respect to the dispersivemodel is plotted in Fig. 2.8 right. The latter suggests that a precision around Our study requires an unsubstracted version of [95]; though this may deteriorate theaccuracy to which the data is reproduced, it does not affect our discussion. .5. Time-like data: η and η (cid:48) LEPs 45 m Π ç ç ç ç ç ç ç ç ç ç çà à à à à à à à à à à à àæææææææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ q @ GeV D È F Η ΓΓ * H q L (cid:144) F Η ΓΓ H L m Π - - - q @ GeV D R e l . D i ff . H % L Figure 2.8: Dispersive model for the η TFF in the time-like region (thick-black line)compared to the fit to that model in the space-like region with P N ( q ) approximants,which is extrapolated to the time-like region (light to dark blue lines for N = 0 , , , ππ threshold. The data corresponds to A22011 (empty orange circles) [127], NA60 (yellow squares) [126] and A2 2013 (red circles)[129] and are included to provide a context for the differencies among the model and PAs. 5% and 10% could be achieved from our results at energies above thresholdand close to the η mass, respectively; below this precision, it seems unlikelythat experiments could spot deviations from our approach predictions. Still,PAs cannot differentiate among the different weights of the different contri-butions appearing in the TFF. However, being fitted to experimental data,all the possible pieces are included —as they are in the data. An interest-ing exercise would be to compare our predictions below threshold againstdispersive approaches, where each contribution must be explicitly included.Incorporating every single contribution represents though a formidable task,for which only those expected to play the main role are included. In thisrespect, our approach would help on identifying if relevant pieces should beincluded, as well as potential model dependencies in such formalisms. Morecomments later in this section.The discussion above already excludes the generalization of our resultsto any arbitrary Stieltjes function since one can immediately conclude thatthe clue feature of the function that would allow the PA to provide a goodperformance above the branch cut is its behavior around the threshold point.As an example, for a scalar resonance the effects would be larger. Particu-larly, the imaginary part behavior at threshold starts at order ( s − m π ) / .This, together with the broadness of scalar resonances [10], would anticipatean early and large disagreement above threshold between data and PAs.In the light of the excellent prediction that PAs provide for the availabletime-like data and the discussion above, we proceed to include the time-likedata in our study [117, 138] . We take, on top of the previous space-like data The size of current errors for the TFFs in the time-like region played a relevant role in η and η (cid:48) Dalitz-decays.For the first, this includes the η → γe + e − results from A2 Collaboration in2011 [127], together with the more recent ones [129], as well as the NA60 Col-laboration results [126] obtained from the η → γµ + µ − Dalitz decay . Thesecollaborations include as well their fitted VMD Λ parameter (cf. Eq. (2.2))which includes both, statistic and systematic errors. Unfortunately, suchsystematic error is not included in the data. In order to obtain the combinedstatistical and systematic published error, one can define a new source oferror defined in the following way: ∆ final = (cid:112) ∆ + ( (cid:15) | F ( Q i ) | ) for each Q i datum, with (cid:15) some percentage. The specific value for (cid:15) is chosen asto reproduce their combined statistical and systematical error . We findthat for the different collaborations, Λ − = (1 . ± stat ± syst ) [127],Λ − = (1 . ± stat ± syst ) [126] and Λ − = (1 . ± stat ± syst ) [129],require (cid:15) = 6 . , . 9% and 4 . η (cid:48) , the time-likedata comprise only the BESIII results [130] . Fortunately, this time theyprovided a systematic error for the data points. For the fitting procedure,we employ the χ function χ = (cid:88) SL (cid:32) Q P N − M ( Q ) − Q F exp P γ ∗ γ ( Q ) σ exp (cid:33) + (cid:88) TL (cid:32) ˜ P N − M ( Q ) − ˜ F exp P γ ∗ γ ( Q ) σ exp (cid:33) + (cid:32) P N − M (0) − F P γ ∗ γ (0) σ exp (cid:33) , (2.13)where P NM ( Q ) is the PA to fit Q F exp P γ ∗ γ ( Q ) and ˜ f ( Q ) means that ˜ f (0) = 1.We next report on our results. We start by fitting with a P L ( Q ) se-quence. We reach up to L = 7 both for η and η (cid:48) , which is shown in Fig. 2.9as a green-dashed line. The smaller plot in Fig. 2.9 is a zoom into the time-like region. The obtained LEPs are collected in Table 2.11 and shown inFigs. 2.10 and 2.11 together with our previous results in Figs. 2.2 and 2.3when only space-like data were included in our fits. The stability observedfor the LEPs with the P L ( Q ) sequence is remarkable, and the impact ofthe inclusion of time-like data is clear since it not only allows us to reachhigher precision on each PA but also enlarges our PA sequence by two andone elements for the η and η (cid:48) , respectively. The stability of the result is also the previous discussion. If in the near future more precise data with discriminating powerenough to discern branch cut effects become available, it may be necessary to carefullyreconsider which data points could be used. More recently, NA60 presented an improved preliminary result, Λ − = (1 . ± . stat ± . syst GeV − [139], but the corresponding data are not yet published. We thank Marc Unverzagt for discussions on this subject. As said, previous results from Lepton-G from η (cid:48) → µ + µ − γ have rather large errorsand are not available in their publication [135, 136]. .5. Time-like data: η and η (cid:48) LEPs 47 òòòòòòòòòòò ò àà à à à à à à à à àææææéééééééééééééééééééééééé ŸŸŸŸŸŸŸŸŸŸŸ øøøøøøøøøøøøø éééééééééééééééééééééééé ŸŸŸŸŸŸŸŸŸŸŸ øøøøøøøøøøøøø - - - - - - - P H Q L P H Q L 0. 10. 20. 30. 40. - Q @ GeV D Q È F Η ΓΓ * H Q L È @ G e V D æææ æ æàà à à à à à à à à à òòòòòòò ò ò ææææ æ È F Ž Η ’ ΓΓ * H q L P H Q L P H Q L 0. 5. 10. 15. 20. 25. 30. 35. 40.0.0.050.10.150.20.25 Q @ GeV D Q È F Η ’ ΓΓ * H Q L È @ G e V D Figure 2.9: η and η (cid:48) TFF best fits. Green-dashed line shows our best P L ( Q ) fit andblack line our best P NN ( Q ) fit. Experimental data points in the space-like region are fromCELLO (red circles) [107], CLEO (purple triangles) [108], L3 (green points) [105], and B A B AR (orange squares) [118] collaborations. Experimental data points in the time-likeregion are from NA60 (blue stars) [126], A2 2011 (dark-green squares) [127], A2 2013(empty-green circles) [129], and BESIII (blue points) [130]. The inner plot shows a zoominto the time-like region. η η (cid:48) N b η c η d η χ ν N b η (cid:48) c η (cid:48) d η (cid:48) χ ν P N ( Q ) 7 0 . . . . . . . . P NN ( Q ) 2 0 . . . . . . . . . . . . . . Table 2.11: Low-energy parameters for the η and η (cid:48) TFFs obtained from the PA fits toexperimental data. The first column indicates the type of sequence used for the fit and N is its highest order. The last row shows the weighted average result for each LEP. We alsopresent the quality of the fits in terms of χ ν . Errors are only statistical and symmetrical. reached earlier, the systematic error is reduced and our method allows toextract, for the first time, the LEPs from a combined fit to all the availabledata . In order to reproduce the asymptotic behavior of the TFF, we havealso considered the P NN ( Q ) sequence (second row in Table 2.11). The re-sults obtained are in very nice agreement with our previous determinations.The best fit is shown as black-solid line in Fig. 2.9. We reach N = 2(1) forthe η ( η (cid:48) ). Since these approximants contain the correct high-energy behav-ior built-in, they can be extrapolated up to infinity (black-dashed line inFig. 2.9) and then predict the leading 1 /Q coefficient [117, 138]lim Q →∞ Q F ηγ ∗ γ ( Q ) = 0 . +0 . − . GeV , (2.14)lim Q →∞ Q F η (cid:48) γ ∗ γ ( Q ) = 0 . . (2.15)Even though the prediction for the η is larger —but compatible withinerrors— than our previous result from the space-like data, Eq. (2.7), it The only exception is the Novosibirsk data [140–144] in the resonant region around0 . − . 400 GeV. (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) P11 P21 P31 P41 P51 P61 P710.500.550.600.65 b Η Figure 2.10: Slope (top), curva-ture (bottom-left), and third derivative(bottom-right) predictions for the η TFFusing the P L ( Q ) (blue points). Previousspace-like data results, Fig. 2.2, are alsoshown (empty-orange squares). Only sta-tistical errors are shown. (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) P11 P21 P31 P41 P51 P61 P710.250.300.350.40 c Η (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) P11 P21 P31 P41 P51 P61 P710.100.150.200.25 d Η is still far below the B A B AR time-like measurement at q = 112 GeV , F ηγ ∗ γ (112) = 0 . η (cid:48) is on the otherhand similar to the previous space-like determination Eq. (2.8). See morediscussions on B A B AR time-like measurements below.Our combined weighted average results from Table 2.11, taking into ac-count both types of PA sequences, give [117, 138] b η = 0 . stat (4) sys b η (cid:48) = 1 . stat (1) sys (2.16) c η = 0 . stat (5) sys c η (cid:48) = 1 . stat (3) sys (2.17) d η = 0 . stat (18) sys d η (cid:48) = 2 . stat (21) sys (2.18)where the first error is statistic and the second systematic, see Table 2.7.These results can be compared to our previous results from space-like data,Eqs. (2.9) and (2.10), which shows the great improvement not only on thestatistical error, but on the systematic one as well, both by an order ofmagnitude. Our results, Eqs. (2.16) to (2.18), represent the most precisedetermination to date for the LEPs. As a further check, for the η (cid:48) , we havechecked the relevance of including the last data points in the time-like region.We have found that omitting them yields very similar results. Therefore,we believe this justifies their inclusion in our fitting procedure. For compar-ison, we update in Fig. 2.12 our previous Fig. 2.5 to include this additionaldetermination . Note as said, that previous dispersive results [95] stands atone standard deviation from ours, both for η and η (cid:48) . The reason being theomission of the a tensor meson contribution, which was observed in [131]and recently included in their later analysis for the η [96], bringing theirresult closer to our value and confirming thereby our determination —the.5. Time-like data: η and η (cid:48) LEPs 49 á á á á á á è è è è è è è P P P P P P P b Η ' Figure 2.11: Slope (top), curva-ture (bottom-left), and third derivative(bottom-right) predictions for the η (cid:48) TFFusing the P L ( Q ) (blue points). Previousspace-like data results, Fig. 2.2, are alsoshown (empty-orange squares). Only sta-tistical errors are shown. á á á á á á è è è è è è è P P P P P P P c Η ' è è è è è è è P P P P P P P d Η ' η (cid:48) modified result has not been reported— which could have been predictedfrom our b P determination and shows the potential of our method to esti-mate unaccounted effects in dispersive approaches. In addition, this resultcould be used as an input to perform further subtractions in their method.After showing the excellent precision achieved in our study, we wouldlike to comment on the role of data in our results. The models studied inSection 2.2 suggest that, due to the large amount of low-energy data, thepresence of new data will not improve on the systematic errors achievedso far (except for the d η ( η (cid:48) ) parameter if higher elements are reached, seeTable 2.7). However, since the current limitation, except for d η ( η (cid:48) ) , is thestatistical one, new precise data will be very welcome. In principle, onemay think that it is the low-energy data which may be preferred. We no-tice however, that in order to reach large PA sequences —which allow formore accurate extractions— the high-energy data, which from 5 to 35 GeV is dominated by B A B AR , is also very important. To show the role of eachcollaboration, we report for the η case (similar results are obtained for the η (cid:48) ) on the different results for the slope and asymptotic values arising fromeach one in Table 2.12. We find that a fit exclusively to B A B AR data yieldssimilar results both for the slope and asymptotic values than other space-likeconfigurations. This contrast for instance for the asymptotic value obtainedwhen only CELLO or time-like data is used. The role of B A B AR data is thentwofold, allowing to reach larger approximants, such as P ( Q ) and deter-mining basically the asymptotic value. In view of the π puzzle between B A B AR [76] and Belle [133] results, a second experimental measurement cov-0 Chapter 2. Data analysis with Pad´e approximants ææææææææ æ ææææ àà à àà àààà à àà ààòòòò Χ TPA SL - dataDisp . Rel.Disp . Rel. 2Axial Anom .Lepton - GCELLOCLEONA60A2WASAA2 newOur Work b Η ææææææ ææææ ææ ààà à ààà àòòòò Χ TPA SL - dataDisp . Rel.Axial Anom .Lepton - G’CELLOCLEOBESIIIOur Work’ b Η ’ Figure 2.12: Slope determinations for the η from different theoretical (red circles) andexperimental (blue squares) references discussed in the text. Inner error is the statisticalone and larger error is the combination of statistical and systematic errors. ChPT [87, 106],VMD, Quark Loop, BL [88], R χ T [101], Disp. Rel. [95], Disp. Rel. 2 [96, 131], AxialAnom. [86], Lepton-G [125], Lepton-G’ [116], CELLO [107], CLEO [108], NA60 [126],A2 [127], WASA [128], A2 new [129], BESIII [130], PA SL-data [114], Our Work [117],Our Work’ [138]. ering the high-energy region would be very welcome here. In the future, theBelle II Collaboration may be able to provide such measurements.To complete our previous discussion, we comment as well on the roleof Γ η → γγ in our extractions given the current discrepancy among e + e − col-lider results and Primakoff measurements for this quantity. We find thatour previous results are rather stable though mildly depend on this in-put. For instance, if we would have used the value measured through thePrimakoff mechanism omitted in the PDG average [10] (i.e., Γ Primakoff ηγγ =0 . b η = 0 . , specially given thatboth, Γ ηγγ and b η , play a central role in our following calculations: η − η (cid:48) mixing, P → ¯ (cid:96)(cid:96) decays and ( g − B A B AR Collaboration atvery large time-like energies [132]. As already mentioned before, B A B AR measured the process e + e − → γ ∗ → η ( (cid:48) ) γ at the center of mass energies √ s = 10 . 58 GeV. Its relation to the TFF [146], σ ( e + e − → P γ ) = 2 π α (cid:18) − m P s (cid:19) | F P γ ∗ γ ( s ) | , (2.19)where s the center of mass energy squared, allowed them to extract ameasurement for the TFF absolute value in the time-like region for q = This kind of measurement is part of the experimental programme of GlueX Collabo-ration at CLAS in Jefferson Lab [145]. .5. Time-like data: η and η (cid:48) LEPs 51 Data range P L ( Q ) P NN ( Q )(GeV ) L b η N b η η ∞ CELLO [107] 0.62–2.23 2 0 . . . . . . B A B AR [118] 4.47–34.38 4 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . TL,SL -0.221 – 34.38 7 0 . ( ) . ( ) . ( ) Table 2.12: Role of the different sets of experimental data in determining slope andasymptotic values ( η ∞ ) of the η TFF. SL refers the the space-like data set, i.e., data fromCELLO,CLEO, B A B AR [107, 108, 118] collaborations, and TL refers to the time-like dataset, i.e., data from NA60+A2-11+A2-13 [126, 127, 129] collaborations. Bold numbers areour final result. No systematic errors included. 112 GeV , obtaining q F ηγ ∗ γ ( q ) = 0 . q F η (cid:48) γ ∗ γ ( q ) =0 . − m P /s ) (seeRef. [146]) that was missing in the B A B AR expression, and assuming that du-ality F P γ ∗ γ ( Q ) = F P γ ∗ γ ( q ) [132] holds at large but finite energies, implies | Q F ηγ ∗ γ ( Q ) | Q =112 GeV = 0 . , | Q F η (cid:48) γ ∗ γ ( Q ) | Q =112 GeV = 0 . . (2.20)This suggests to include these data points in our fitting procedure, assumingthat at this high-momentum transfer, the duality between space- and time-like region holds, and no extra error should be included. For the η (cid:48) , givenour results in Eqs. (2.8) and (2.15), in excellent agreement with B A B AR results, it is clear that this won’t change much. For the η case, its inclusionwill mainly modify the asymptotic prediction from P increasing its valueup to lim Q →∞ Q F ηγ ∗ γ ( Q ) = 0 . 247 GeV, higher than the B A B AR resultand with a good χ ν < 1. Curiously enough, the fit function at Q =112 GeV is Q F ηγ ∗ γ ( Q ) = 0 . 219 GeV, below Eq. (2.20). Even worseis the prediction (assuming duality) for the time-like counterpart at q =112 GeV , q F ηγ ∗ γ ( q ) = 0 . 307 GeV. One may speculate in light of theseresults on the validity of duality assumptions and whether the asymptoticregime is reached or not. Actually, a recent analysis of the η and η (cid:48) TFFsbased on perturbative corrections [67] concludes that the difference betweenthe time- and space-like form factors at Q = 112 GeV can be of the order(5 − σ ( e + e − → P γ ) (fb) P J/ψ ψ (2 S ) ψ (3770) π η η (cid:48) Table 2.13: The continuum cross sections for σ ( e + e − → P γ ) processes in fb at the centerof mass energies of different charmonium resonances. large error on duality assumptions at these energies. Notice however that,even at these high-energies, the TFFs are sensitive to soft scales for x (cid:39) π and η cases, whichTFFs, definitely not VMD-like, seems to require a broad DA [67]. Similarresults are found from CLEO results [147], which measured cross sectionsat q = 14 GeV —assuming continuum contribution and duality— lead to | Q F ηγ ∗ γ ( Q ) | Q =14 GeV = 0 . , | Q F η (cid:48) γ ∗ γ ( Q ) | Q =14 GeV = 0 . , (2.21)even though with potentially larger corrections being at lower energies. Onthe other hand, the η (cid:48) seems not that affected, which may suggest a muchnarrower DA less sensitive to the end-point behavior. This would be rea-sonable given its heavier singlet nature, introducing an explicit scale thatwould drive the DA away from a flat shape. Still, to draw firmer conclu-sions, further and more precise experimental results are required. There isat the moment an ongoing analysis at BES III to measure such processes at q = 18 . [148].Alternatively, we can use our TFF description to extract the cross sec-tion which duality arguments would imply for these processes when usingEq. (2.19). This contribution is of relevance when estimating backgroundcontribution to ψ ( nS ) → γη ( (cid:48) ) decays. We obtain at the center of massenergies of the different resonances, the cross sections quoted in Table 2.13,where, for completeness, we include the π results obtained from the workin Ref. [111]. This represents an improvement with respect to Ref. [146]as the latter assumes the asymptotic behavior to extrapolate down to thecharmonium energies. Still, we note that these predictions are only valid inthe case that duality holds (strictly as Q → ∞ ) and would require a morerefined analysis in line of [67] in order to estimate for these corrections. In this chapter we have described how PAs can be used as fitting func-tions in order to extract relevant information from the pseudoscalar TFFs,.6. Conclusions 53 F Pγγ b P c P d P P ∞ (GeV − ) (GeV) π [111] 0 . . . F π η [117] 0 . . . . . η (cid:48) [138] 0 . . . . . η SL [114] 0 . . . . η (cid:48) SL [114] 0 . . . . Table 2.14: The main results from our work in this chapter. The numbers come fromthe combined space- and time-like data, Sections 2.4 and 2.5. We include the π resultsfrom Ref. [111] and the TFFs at zero energies implied by experiments. In addition, wequote what would be obtained from space-like data alone, which is labelled as P SL . namely the LEPs and the asymptotic behavior. We have demonstrated thisusing three different models for the TFF, illustrating the PAs performancein cases where convergence theorems exist or not, that has allowed on topto estimate a systematic error, an unique property of our approach. Theproposed method has been applied then to the real η and η (cid:48) cases, obtainingan excellent performance in the space-like region. Moreover, we have dis-cussed that our previous description can be extrapolated for these TFFs intothe low-energy time-like region up to an excellent accuracy, allowing for thefirst combined description as well as an improved LEPs determination. Allin all, our method has allowed a systematic and model-independent robustextraction for the central quantities that we need for later reconstructingthe (single-virtual) pseudoscalar TFFs. Moreover, we were able to explainthe existing discrepancies among space- and time-like data analysis fromdifferent collaborations on the basis of a systematic error. Our main resultsare the low-energy parameters for the TFF expansion F P γ ∗ γ ( Q ) = F P γγ (cid:18) − b P Q m P + c P Q m P − d P Q m P + ... (cid:19) , (2.22)as well as the asymptotic behavior, P ∞ ≡ lim Q →∞ Q F P γ ∗ γ ( Q ). We reca-pitulate them together with the π results from space-like data, which werenot analyzed here, but in Ref. [111], in Table 2.14. We expect to reanalyzethe π TFF as well in the near future once the new data from BESIII [149]in the low-energy space-like (0 . ≤ Q ≤ 10) GeV range and time-like datafrom NA62 [150] and A2 [148] collaborations from the π → γe + e − decaybecome available. Moreover, there are prospects to measure the π TFF ateven lower space-like energies at KLOE-2 [151] and GlueX [152] collabo-rations. This would allow for a statistical and systematic improvement forthe π LEPs. Additional data for the η and η (cid:48) mesons is expected too in asimilar range. Although this would not improve much the systematic error,an improvement on the statistical one —the dominant at the moment— isto be expected. For completeness, we also show the η and η (cid:48) results using4 Chapter 2. Data analysis with Pad´e approximantsspace-like data alone, labelled as η ( (cid:48) ) SL , in order to compare the effects ofincluding the time-like data. We remark that the value shown for the TFFat zero energies, F P γγ in Table 2.14, is the experimental one obtained fromthe Γ P γγ decay widths from PDG [10]. Actually, this result has changed forthe π with respect to Ref. [111], where the Γ PrimEx π γγ [103] value was used.We include however the subsequent PDG combination [10] including, amongothers, the value from Ref. [103]. In addition, the asymptotic behavior wasnot extracted there but included, since its theoretical prediction, π ∞ = 2 F π ,is a clean one as compared to the η and η (cid:48) , where the mixing and effectsrelated to their singlet component obscure their calculation. This representsthe first step in order to reconstruct our PAs describing the pseudoscalarTFFs in next chapters. hapter Canterbury Approximants Contents So far, we have carefully described how to reconstruct the single-virtualtransition form factor (TFF) from the theory of Pad´e approximants (PAs).However, for almost every practical application in this thesis, see Chapters 5and 6, it is the double-virtual TFF that is required. From the very basic prin-ciple of Bose symmetry, we know that F P γ ∗ γ ∗ ( Q , Q ) = F P γ ∗ γ ∗ ( Q , Q ).Such symmetry principle certainly simplifies the most general form that thedouble-virtual TFF could have, but it is not constrictive enough as to fullypredict the double-virtual TFF from its single-virtual version alone. Weillustrate this assertion using two simple ans¨atze. A simple extension ofthe single-virtual TFF, which respects Bose symmetry, is the factorizationapproach F fact P γ ∗ γ ∗ ( Q , Q ) = F P γ ∗ γ ∗ ( Q , × F P γ ∗ γ ∗ ( Q , F P γγ . (3.1)This construction was proposed back in the 60’s based on vector mesondominance ideas [116, 153, 154] —and recently reconsidered in [96]. There,the form factor was given through vector resonance exchanges as depictedin Fig. 3.1 left, which implicitly uses factorization. Note however that556 Chapter 3. Canterbury Approximantsin a large- N c framework additional diagrams exist —see Fig. 3.1 right orRef. [100]— which break factorization. Still, from the study in Ref. [91], itseems that the leading logarithms in χ PT support the factorization approachat low energies, corrections appearing one loop higher than expected —andeven two loops higher in the chiral limit. However, Eq. (3.1) cannot repro-duce at the same time the high-energy single- and double-virtual behaviorwhich is implied from pQCD, see Section 1.6.1. Namely, if the single-virtualTFF falls as Q − —as the BL, Eq. (1.49), implies— the double-virtual fac-torized version, Eq. (3.1), necessarily falls as Q − , in conflict with the OPEwhich predicts Q − , Eq. (1.50). This implies that, even if factorizationwould be appropriate at low-energies, it must fail at energies large enough. P J µ J ν J µ J ν P P P γ ∗ γ ∗ γ ∗ γ ∗ γ ∗ γ ∗ V V V V V Figure 3.1: Left: standard vector meson dominance conception; factorization is implied.Right: resonant approach to the TFF; factorization is not implied. The graphics on toparise from the large- N c pseudoscalar pole contribution to the (large- N c ) Green’s functionssketched below (cf. Fig. 1.5). An alternative idea, which would keep Bose symmetry without spoilingthe high energy behavior, would be to extend the TFF as F P γ ∗ γ ∗ ( Q , Q ) = F P γ ∗ γ ∗ ( Q + Q , λQ − , its double-virtual coun-terpart would read ( λ/ Q − , whereas pQCD requires ( λ/ Q − instead,see Eqs. (1.49) and (1.50). These examples illustrate that the TFF double-virtual extension cannot be trivially reconstructed from the single-virtualone, but will require a dedicated effort. From a Pad´e theory point of view,this amounts to the observation that, given the most general double-virtualTFF series expansion, F P γ ∗ γ ∗ ( Q , Q ) = F P γγ (cid:18) − b P Q + Q m P + c P Q + Q m P + a P ;1 , Q Q m P + ... (cid:19) , (3.2)Bose symmetry only dictates that a P ; i,j = a P ; j,i , but does not enforce ad-ditional relations among the single-virtual parameters, b P , c P , ... , and thedouble-virtual ones, a P ; i,j , which therefore must be provided as an addi-tional input. In this chapter, we explore how to consistently generalize inthe spirit of Pad´e theory our previous approach, which would provide thena model-independent framework to reconstruct the most general double-virtual TFF from the parameters in Eq. (3.2). Our method is described inSection 3.2, while its performance and properties are explored along Sec-.2. Canterbury approximants 57tion 3.3 using practical examples. Once more, experimental data, whenavailable, would provide then the external required input to reconstruct theTFF. We investigate this possibility, in analogy to Chapter 2, in Section 3.4.Finally, we summarize the main results in Section 3.5. To extend the PAs to the bivariate case, we follow the approach from theCanterbury Group, started by Chisholm in Refs. [155, 156] and giving birthto what is known as Canterbury approximants (CAs) [57, 157]. This ap-proach requires symmetrizing some equations, which is ideal in our case ofstudy given the symmetry of our function. In this section, we review thebasics of the method when applied to symmetric functions. Let’s define afunction f ( x, y ) = f ( y, x ) analytic in a certain domain around x = y = 0,which series expansion reads f ( x, y ) = (cid:88) α,β c α,β x α y β , ( c α,β = c β,α ) . (3.3)The Canterbury approximant is constructed from the rational function C NM ( x, y ) = P N ( x, y ) Q M ( x, y ) = (cid:80) Ni,j =0 a i,j x i y j (cid:80) Mk,l =0 b k,l x k y l ( b , = 1) . (3.4)Note that the rational function is constructed as to have the maximumpower in each variable rather than a total maximum power in x i y j with i + j ≤ N ( M ), essential for the construction [155]. Next, we need to set thedefining equations for the bivariate approximant in analogy to Eq. (1.30).A natural extension from the univariate case would be M (cid:88) i,j b i,j x i y j ∞ (cid:88) α,β c α,β x α y β = N (cid:88) k,l a k,l x k y l + O (cid:0) x γ y n + m +1 − γ (cid:1) , (3.5)with γ ∈ (0 , n + m + 1). Such set of equations define (Bose symmetry isimplied) N + M (cid:88) i =0 = N + M + 1 ( c i, terms) (3.6) i + j = N + M (cid:88) ( i ≥ j )=1 = (cid:40) ( N + M ) , N + M ∈ even ( N + M ) − , N + M ∈ odd ( c i,j terms) (3.7)constraints for the single and double-virtual parameters, respectively, thefirst of which are reminiscent from the univariate case. To obtain the number8 Chapter 3. Canterbury Approximantsof equations for the double-virtual terms, note that each order O ( L ) ≡O ( x L − i y i ) involves, after using Bose symmetry, L/ L − / 2) coefficientsfor L ∈ even(odd), implying (cid:80) L/ i =1 i + (cid:80) L/ − i =1 i = L / L ∈ evenand 2 (cid:80) ( L − / i =1 i = ( L − / L ∈ odd. In turn, Eq. (3.4) involves N (cid:88) i =0 + M (cid:88) j =1 = N + M + 1 ( a i, , b i, terms) (3.8) N (cid:88) ( i ≥ j )=1 + M (cid:88) ( i ≥ j )=1 = 12 N ( N + 1) + 12 M ( M + 1) ( a i,j , b i,j terms) (3.9)terms for the single-virtual and double-virtual parameters, respectively —toobtain the number of double-virtual terms, note that (cid:80) L ( i ≥ j )=1 = (cid:80) Li =1 i = L ( L + 1) / 2. Expressing Eq. (3.9) as ( N + M ) / N − M ) / N + M ) / C NM ( x, y ) approximant forwhich N ≥ M (an identical procedure applies for M ≥ N ). Its P N ( x, y )numerator polynomial involves N ( N + 1) / ∼ x N y M ≤ N are present and need to be included therefore in the defining equations.However, we find that for a given order O ( L ≤ N ) not all the terms needto be filled in Table 3.1; the additional terms up to O ( N + M + 1) repre-sent M ( M + 1) / Q M ( x, y )polynomial double-virtual parameters, fixing every coefficient in Eq. (3.4).These represent the defining equations for CAs, which can be summarizedas M (cid:88) i,j b i,j x i y j ∞ (cid:88) α,β c α,β x α y β − N (cid:88) k,l a k,l x k y l = ∞ (cid:88) γ,δ d γ,δ x γ y δ , (3.10) d γ,δ = 0 0 ≤ γ + δ ≤ M + Nd γ,δ = 0 0 ≤ γ ≤ max( M, N ) , ≤ δ ≤ max( M, N ) d γ,δ = 0 1 ≤ γ ≤ min( M, N ) ,δ = M + N +1 − γ. (3.11)The defining equations, Eqs. (3.10) and (3.11), represent the most impor-tant definition in this chapter as it is the basis to reconstruct the bivariateapproximants. The definition above corresponding to the Canterbury groupfulfills several properties [155, 157, 158]: • If either x or y is taken to vanish, CAs reduce to PAs..2. Canterbury approximants 59 O (2) O (3) O (4) O (5) O (6) O (7) O (8) C M − C M − − , − C M 11 21 22 , − − , − C M 11 21 22 , 31 32 , − , − , − − , − , − − , − , − , − C M 11 21 22 , 31 32 , 41 33 , , − , − , − , − , − , − C M 11 21 22 , 31 32 , 41 33 , , 51 43 , , − , , − , − C M 11 21 22 , 31 32 , 41 33 , , 51 43 , , 61 44 , , , − C M 11 21 22 , 31 32 , 41 33 , , 51 43 , , 61 44 , , , Table 3.1: Coefficients b i,j = b j,i ≡ i,j appearing in the degree N polynomial P N ( x, y )from C NM ( x, y ). The order O stands for i + j . • If the original function is symmetric, this is f ( x, y ) = f ( y, x ), theresulting CAs preserve this symmetry as well. • If the original function can be written f ( x, y ) = g ( x ) h ( y ), the resultingCAs factorize in terms of the PAs for g ( x ) and h ( y ). • The C NM ( x, y ) approximant for 1 /f ( x, y ) is identical to 1 / ˜ C MN ( x, y ),being ˜ C MN ( x, y ) the approximant for f ( x, y ). • The diagonal approximants are invariant under the group of homo-graphic transformations, this is, if C NN ( x, y ) is the approximant for f ( Ax − Bx , Ay − Cy ), this is identical to ˜ C NN ( Ax − Bx , Ay − Cy ), where ˜ C NN ( x, y ) isthe approximant to f ( x, y ) —a well known property of diagonal PAs.These properties are of relevance for us. In particular, reduction to PAsallows us to connect to our previous work; the second condition guaranteesBose symmetry; the third one is interesting regarding factorization discus-sions, whereas the last properties are reassuring in the sense that they ex-tend important and well known properties of PAs to the bivariate case. Inaddition, Montessus theorem (cf. Section 1.5.1) as well as convergence toStieltjes functions have been proved for CAs as well [57, 159, 160]. Note thatthe former guarantees convergence of CAs for the pseudoscalar TFFs in thelarge- N c limit of QCD. As a final comment, there exist additional extensionsof PAs to the multivariate case. Their relevance can be understood for exam-ple if considering non-symmetric functions, which substantially complicatesthe procedure outlined above (for more details see Ref. [161] and referencestherein). Note however that alternative approaches may not respect severalof the properties quoted above.0 Chapter 3. Canterbury Approximants In this section, we illustrate the performance and operation of CAs for theparticular cases of two functions already discussed in Chapter 2 in theirunivariate case (i.e., one of their variables is taken to be zero) in the contextof PAs, where excellent results were obtained . These are the Regge andlogarithmic models discussed in Chapter 2.The first one reads in its bivariate (double-virtual) form [97] F Regge P γ ∗ γ ∗ ( Q , Q ) = aF P γγ Q − Q (cid:104) ψ (0) (cid:16) M + Q a (cid:17) − ψ (0) (cid:16) M + Q a (cid:17)(cid:105) ψ (1) (cid:16) M a (cid:17) , (3.12)and we take M = 0 . a = 1 . , see Section 2.3.1. We notethat, whereas QCD evolution is necessary to restore the BL asymptoticbehavior for one large virtuality [97], the asymptotic behavior for two equaland large virtualities is already built-in in the model. To see this, takelim Q → Q ≡ Q F Regge P γ ∗ γ ∗ ( Q , Q ) = F P γγ ψ (1) (cid:16) M a (cid:17) ψ (1) (cid:18) M + Q a (cid:19) , (3.13)which asymptotic behavior Eq. (3.13) readslim Q →∞ F Regge P γ ∗ γ ∗ ( Q , Q ) = aF P γγ ψ (1) (cid:16) M a (cid:17) Q − + O ( Q − ) . (3.14)The second (logarithmic) model is generalized to the bivariate (double-virtual) version as F log P γ ∗ γ ∗ ( Q , Q ) = F P γγ M Q − Q ln (cid:18) Q /M Q /M (cid:19) , (3.15)with M = 0 . , see Section 2.3.2. We note that this function arises asa natural extension of flat distribution amplitudes, in the line of [77, 78], tothe double-virtual case. To see this, consider the representation F log P γ ∗ γ ∗ ( Q , Q ) = F P γγ M (cid:90) dx xQ + (1 − x ) Q + M , (3.16)which essentially corresponds to a flat DA φ P ( x ) ≡ F (1 , , , − Q /M , − Q /M ). As an additional source for practical applications and discussions, the reader is referredto a similar study of the Euler’s Beta function in [158]. .3. Practical examples 61This function has a singularity at Q = Q = − M and branch cut disconti-nuities for Q < − M , disappearing whenever both virtualities meet suchcondition at the same time. A nice feature from this model is again obtainedin the limit lim Q → Q ≡ Q F log P γ ∗ γ ∗ ( Q , Q ) = F P γγ M M + Q , (3.17)which fulfills the appropriate asymptotic behavior, even if the BL limit wasnot reproduced. A final interesting property, is that Eq. (3.16) can be re-expressed as F P γγ M ( Q + Q ) + M (cid:90) + − duuz + 1 ; z = Q − Q ( Q + Q ) + M , (3.18)which represents an extended Stieltjes function —see section 5.6 from Ref. [57] The Stieltjes theorem for PAs proved to be a powerful tool in physical ap-plications [58, 162]. It provides convergence for the whole complex plane—except for the cut, where the original function itself is ill-defined— as wellas bounds ( P NN +1 ( x ) ≤ f ( x ) < P NN ( x )) for the (Stieltjes) function to be ap-proximated, Section 1.5.1. In this subsection, we illustrate its performancefor the bivariate case through the use of the logarithmic model in Eq. (3.15),which corresponds to a generalized Stieltjes function, for which convergenceis guaranteed [57, 160].As a first analysis, we check the convergence for the diagonal C NN ( Q , Q )and subdiagonal C NN +1 ( Q , Q ) sequences. The lowest order elements read C ( Q , Q ) = F P γγ Q + Q M + Q Q M , (3.19) C ( Q , Q ) = F P γγ (1 + Q + Q M + Q Q M )1 + Q + Q )3 M + Q Q M , (3.20) C ( Q , Q ) = F P γγ (1 + Q + Q M + Q Q M )1 + Q + Q M + Q Q M + Q + Q M + Q Q ( Q + Q )15 M + Q Q M . (3.21)The performance for these sequences is excellent up to large Q valuesas it is illustrated in Fig. 3.2, where the relative deviation, defined as C NM ( Q , Q ) /F log P γ ∗ γ ∗ ( Q , Q ) − 1, is shown for two selected cases. There,we observe —as anticipated— that the diagonal and subdiagonal sequencesapproach the original function from above and below, respectively. Recall in2 Chapter 3. Canterbury Approximants - %- %- %- % ¥ ¥ Q H GeV L Q H G e V L C % % % % ¥ ¥ Q H GeV L Q H G e V L C Figure 3.2: Convergence of the C NN +1 ( Q , Q ) and C NN ( Q , Q ) sequences to the logarith-mic model. We show the C (left) and C (right) elements, respectively. The first, second,third, and fourth contours, from light to dark red, stand for the relative ∓ , ∓ , ∓ 10 and ∓ 20% deviations. Both axis have been scaled as Q / (1 + Q ). See discussion in the text. this respect that the C NN ( Q , Q ) sequence behaves as a constant for large Q values, the C NN +1 ( Q , Q ) falls as Q − ( Q − ) for one (two) large virtuali-ties, and the original function, as ln( Q ) Q − and Q − , respectively, for oneand two large virtualities.An interesting implication from Stieltjes theorem is that the poles andzeros from the approximant must be located along the branch cut discon-tinuity, where the function itself is ill-defined. We check as a second stepthis property, and illustrate the poles and zeros for some elements of thediagonal and subdiagonal sequences in Fig. 3.3. There is no pole or zeroin the space-like region and, in addition, these approach to the branch cutlocations, as expected from the univariate case. There is an interestingremark though. As observed, there exist poles and zeros in the time-likeregion where no cut exists (light shaded time-like region in Fig. 3.3). Still,these poles and zeros are spurious in the sense that they approach the gray-shaded regions in Fig. 3.3 —where these should be located— as the orderof the approximant increases, but indicate a slower convergence within thisregion. It would be interesting in this respect to find whether it is possibleto accelerate such convergence. We note in this respect that the logarithmicmodel in Eq. (3.15) enjoys an additional symmetry, F log P γ ∗ γ ∗ ( − Q − M , − Q − M ) = − F log P γ ∗ γ ∗ ( Q , Q ) , (3.22)which actually relates the two space- and time-like light-shaded regions inFig. 3.3. It is intuitive that, constraining such symmetry into the approxi-mant, the excellent convergence which is obtained for the space-like regionwill be translated into the time-like one. We find that such symmetry canonly be implemented —at least for the lowest approximants— for the sub-.3. Practical examples 63 - - - - - - Q H GeV L Q H G e V L - - - - - - Q H GeV L Q H G e V L Figure 3.3: The poles (left) and zeros (right) for the C , C and C elements as dashed,dash-dotted and full lines, respectively. The gray-shaded areas represent the regions forwhich a branch cut exists. diagonal sequence, which lowest elements read C ( Q , Q ) = F P γγ ( Q + Q )2 M , (3.23) C ( Q , Q ) = F P γγ (cid:16) ( Q + Q )2 M (cid:17) ( Q + Q ) M + ( Q + Q )6 M + Q Q M , (3.24) C ( Q , Q ) = F P γγ (cid:16) ( Q + Q ) M + Q + Q )60 M + Q Q M (cid:17) Q + Q )2 M + Q + Q )5 M + ( Q + Q )20 M + Q Q M + Q Q ( Q + Q )20 M . (3.25)It is amusing to check that, in addition, for Q = Q the equal-virtual behav-ior Eq. (3.17) is exactly reproduced in Eqs. (3.23) to (3.25) even if this wasnot imposed. Incidentally, we find that the polynomials in our approximants,Eqs. (3.23) to (3.25), can be constructed as (cid:80) Ni + j =0 c i,j Q i Q j , missing theelements Q i Q j with i + j > N . We remark that this is a particular featurefor this model, which cannot be generalized to other functions [160].To end our discussion, we show the poles and zeros of Eqs. (3.23) to (3.25)in Fig. 3.4. In contrast to Fig. 3.3, there are no poles or zeros in the region( x, y ) < − ( M , M ) (time-like light shaded region in Fig. 3.4), which cannow be described —as anticipated— to the same precision as the space-likeone. As a conclusion, whenever a symmetry principle exists, its inclusionimproves convergence. We shall not forget that such symmetry necessarilyimplies a connection among the single- and double-virtual parameters inEq. (3.2). An interesting discussion along these lines is found in Ref. [163].4 Chapter 3. Canterbury Approximants C & C C C - - - - - - Q H GeV L Q H G e V L C C - - - - - - Q H GeV L Q H G e V L Figure 3.4: From lighter to darker full red lines, the poles (left) and zeros (right) for the C , C and C elements once the symmetry of the original function has been constrained.The dotted-dashed lines represent the original logarithmic function branch cuts. We notethat the pole for the C approximant overlaps with one pole of the C approximant. Theygray-shaded areas represent the regions for which a cut is opened. N c limit and meromorphic functions theo-rems: Montessus and Pommerenke In this subsection, we employ the Regge model in Eq. (3.12) to discuss ad-ditional convergence theorems which apply to the large- N c limit of QCD, inwhich the Green’s functions become meromorphic. These are the Montessus’theorem and Pommerenke’s theorem. As a brief summary from Section 1.5.1,we recall that, for the special case of meromorphic functions, Montessus the-orem implies convergence within a disk containing M poles for the P NM ( x )sequence, whereas Pommerenke’s theorem implies convergence in the wholecomplex plane for the P N + MN ( x ) sequence. We shall not forget that Monte-sus theorem has been obtained already for the multivariate case [57, 159].In addition, we recall that, if a meromorphic function have only positiveresidues (the same applies if all are negative), this is of the Stieltjes kind.As such condition is fulfilled for the Regge model, Stieltjes theorem applieshere as well.To discuss Montessus theorem, we reconstruct the C N sequence for theRegge model, which for the first elements read C ( Q , Q ) F P γγ = 11 − ( Q + Q ) ψ (2) aψ (1) − Q Q a ( ψ (3) ψ (1) − ( ψ (2) ψ (1) ) ) , (3.26) C ( Q , Q ) F P γγ = 1 − Q + Q a ( ψ (3) ψ (2) − ψ (2) ψ (1) ) + Q Q a ( ( ψ (3) ψ (2) ) − ψ (3) ψ (1) − ψ (4) ψ (2) )1 − ( Q + Q ) ψ (3) aψ (2) − Q Q a ( ψ (4) ψ (2) − ( ψ (3) ψ (2) ) ) , (3.27)where ψ ( n ) ≡ ψ ( n ) ( M /a ). The performance, as expected, resembles thatof the univariate case. As an example, we show how the poles of the.3. Practical examples 65 C C C - - - - - - Q H GeV L Q H G e V L C C C - - - - - - Q H GeV L Q H G e V L Figure 3.5: The poles from the C N ( Q , Q ) (left) and C N ( Q , Q ) (right) sequencesfor the C , C , C and C , C , C elements, respectively, (light to dark red lines). Theoriginal first and second poles are displayed as dashed-dotted lines. C N ( Q , Q ) sequence approach those of the original function at Q ( Q ) = − M in Fig. 3.5 left. As we move either further from the first pole, or farinto the space-like region, convergence deteriorates and is eventually lost aswe move away from the convergence disk. This is in accordance to Montes-sus theorem, and can be easily understood for this particular case from thepower-like behavior of the approximant, which rapidly diverges as N is in-creased, in contrast to the original function. To enlarge such convergencedisk beyond the second pole from the model, we need to go to the C N se-quence. The poles from such approximant are illustrated in Fig. 3.5 (rightpanel), where it can be observed the hierarchical convergence for the poles,which approach faster to those closer to the expansion point. This is to beexpected, as the imprint from the poles far from the origin should be small.Eventually, our goal is to reproduce the function in the whole complexplane or, at least, in the whole space-like region. To this aim, and dealingwith meromorphic functions, we can appeal to Pommerenke’s theorem andcheck if this seems to extend to the bivariate case too. As an example, weuse the subdiagonal C N − N ( Q , Q ) sequence, for which the theorem applies.We show the relative error, defined as in the previous subsection in Fig. 3.6,obtaining excellent results and suggesting that Pommerenke’s applies to thebivariate case too. Moreover, there we find that the original function isalways approached from below in this sequence. The opposite would havebeen found for the diagonal sequence. This was to be anticipated as thisfunction is not only meromorphic but Stieltjes, which places stronger con-traints.Given the observed ability of the approximants to reproduce the originalpole, it is natural to ask ourselves whether its residue is approached at asimilar convergence rate. Actually, this quantity is of physical relevance too.6 Chapter 3. Canterbury Approximants - %- %- %- % ¥ ¥ Q H GeV L Q H G e V L C ¥ ¥ Q H GeV L Q H G e V L C ¥ ¥ Q H GeV L Q H G e V L C Figure 3.6: Convergence of the C N − N ( Q , Q ) sequence to the Regge model for differentelements. The first, second, third, and fourth contours, from light to dark red, stand forthe relative − , − , − 10 and − 20% deviations. Both axis have been scaled as Q / (1+ Q ). C C C - - - Q H GeV L R e s H Q L Figure 3.7: From lighter to darker full-red lines, the residue associated to the C , C , C approximants whenever some virtuality hits a pole. The original residue, overlapping withthe C element, is plotted as dotted-dashed black line. As an example, in our Regge model for the TFF, this would represent somevector meson form factor, say, the ωπ γ ∗ TFF —of course, in the real worldwith finite-width resonances, this identification is misleading, and wouldonly hold, approximately, for extremely narrow resonances. To this end, wetake the residue from our C N ( Q , Q ), which is illustrated in Fig. 3.7. Wefind an excellent convergence too, even if the accuracy is smaller than thatfound for the pole position. If we would repeat the same exercise for the C N sequence, we would find an excellent convergence for extracting the firstpole —see Fig. 3.5— and its residue. For the second pole, as illustrated inFig. 3.5, the convergence is slower and an even slower convergence rate isfound for its residue. We conclude that, as in the univariate case of PAs,Canterbury approximants provide an excellent description for meromorphicfunctions in the space-like region, they are able to predict the poles posi-tion and, eventually, describe their residues as well, this is, they provide a.3. Practical examples 67 ¥ ¥ Q H GeV L Q H G e V L C T - %- %- %- % ¥ ¥ Q H GeV L Q H G e V L C T ¥ ¥ Q H GeV L Q H G e V L C T Figure 3.8: Convergence of the C T NN +1 ( Q , Q ) sequence to the Regge model for differentelements. The first, second, third, and fourth contours, from light to dark red, stand forthe relative − , − , − 10 and − 20% deviations. Both axis have been scaled as Q / (1+ Q ). complete description of the original function. From the previous discussion, it seems that if the poles would have beenknown a priori , these could have been used from the very beginning, bring-ing additional parameters to our approach. This is interesting, as in the realsituation we often know several poles from our function , but not its seriesexpansion. In this section, we study the implications from this approach,in which the poles of the approximant are given in advance, and are in cor-respondence with the lowest-lying poles from the original function. This isknown in the univariate case as Pad´e-Type approximants, see Section 1.5.2,and have been implicitly used in the past years in resonant approaches. Forreconstructing these approximants, we build in our case the denominatorfrom our Canterbury-Type approximant, C T NM , as M − (cid:89) n =0 ( Q + M + na )( Q + M + na ) , (3.28)whereas the remaining parameters from the P N ( Q , Q ) polynomial, Eq. (3.4),are fixed from the series expansion. The obtained results for the first approx-imants are shown in Fig. 3.8. Comparing with Fig. 3.6, it is easy to see thatthe achieved convergence rate is not as satisfactory as in the previous case,and the resulting systematic error from the approach is larger. This waseasy to anticipate, as the position of the poles —specially those far from theexpansion point— did not exactly correspond to the original pole locationin our previous examples. This kind of approach may better reproduce theresonant region which is close to the fixed poles, but this comes at cost of the It must be noted that, in the real world, many of these poles may have a significantwidth. Including them as real zero-width poles implies then an additional error. In the previous subsections, we found that the convergence from our approx-imants deteriorated at very large Q values. This was easy to anticipate, asour models (Eqs. (3.12) and (3.15)) approached 0 for Q = Q ≡ Q → ∞ as Q − (cf. Eqs. (3.14) and (3.17)), whereas none of the approximantsconstructed above implemented such behavior. In this subsection, we dis-cuss how such behavior —which could have been anticipated from the OPEexpansion— can be implemented into our approximant. To this object, wereview the concept of two-point PAs, see Section 1.5.2, applied to CAs, whichin our case allows to describe both, the low- and the high-energy expansions,providing then a tool to unify our knowledge from χ PT and pQCD.Our two expansions of interest for the Regge and logarithmic models are F Model π γ ∗ γ ∗ ( Q , Q ) = ∞ (cid:88) n,m =0 c n,m Q n Q m ( Q , → , (3.29) F Model π γ ∗ γ ∗ ( Q , Q ) = ∞ (cid:88) n =0 c OPE n Q − n ( Q = Q → ∞ ) . (3.30)The first one represents the expansion at the origin of energies used in previ-ous sections, whereas the second one represents the OPE expansion for equallarge virtualities Q = Q ≡ Q → ∞ . For illustrating the construction oftwo-point CAs, we make use of the diagonal and subdiagonal sequences,which high-energy behavior expansion reads (see Eq. (3.4)) C NN ( Q , Q ) = a N,N b N,N + 2 a N,N − b N,N − b N,N − a N,N b N,N Q − + ... , (3.31) C NN +1 ( Q , Q ) = a N,N b N +1 ,N +1 Q − + ... ( b N +1 ,N +1 (cid:54) = 0) , (3.32) C NN +1 ( Q , Q ) = a N,N b N +1 ,N Q − + ... ( b N +1 ,N +1 = 0) . (3.33)For both of our models c OPE0 = 0, the first non-vanishing term in thehigh-energy expansion Eq. (3.30) being c OPE1 . This implies a N,N = 0 and b N +1 ,N +1 = 0 for the diagonal and subdiagonal sequences, respectively (cf.Eqs. (3.31) to (3.33)). If additional terms from the high-energy expansionare to be included in our two-point CA, say c OPE1 , additional constraints are.3. Practical examples 69 - % %- %- %- % % % ¥ ¥ Q H GeV L Q H G e V L C ¥ ¥ Q H GeV L Q H G e V L C ¥ ¥ Q H GeV L Q H G e V L C Figure 3.9: Convergence of the C NN +1 ( Q , Q ) sequence with the appropriate high-energybehavior to the Regge model for different elements. The first, second, third, and fourthouter(inner) contours, from light to dark red, stand for the relative ∓ , ∓ , ∓ 10 and ∓ Q / (1 + Q ). C C C C C C Exact c OPE1 . 172 0 . 234 0 . 246 0 . 374 0 . 276 0 . 264 0 . Table 3.2: The prediction for the leading c OPE1 term in the high-energy expansion for thediagonal and subdiagonal sequences compared to the exact result for the Regge model. present. The resulting equations are taken instead those arising from thehigher order terms in the low-energy expansion. However, in contrast to PAs,for the bivariate case there are many different terms c N,M Q N Q M of thesame order L = N + M . For our models, we find that the best convergence isachieved when the most asymmetric terms are replaced for the high-energyones, this is, the terms c L − , , c L − , , ... are replaced by c OPE0 , c OPE1 , ... .As an illustration, we show the result from matching c OPE0 = 0 alone.The resulting equation replaces the c N, ( c N +1 , ) matching condition forthe diagonal (subdiagonal) sequence, respectively. The results obtained forthe Regge model are illustrated in Fig. 3.9, and show the expected improvedconvergence along the Q = Q region. A similar improvement is achievedfor the logarithmic model as well. Actually, we find that in this case theequal-virtual behavior Eq. (3.17) is exactly satisfied, reproducing then allthe terms in Eq. (3.30) and reaching an infinite precision along Q = Q .For the Regge model this is no longer possible, as its equal-virtual behav-ior Eq. (3.13) is not represented by a rational function, requiring then aninfinite sequence to reproduce it. Still, additional terms in the high-energyexpansion may be predicted even if these were not matched. As an example,we show the prediction for the c OPE1 term in Eq. (3.30) in Table 3.2. As aconclusion, we find that CAs are able as well to use the information at zeroand infinity, providing a reliable description of the underlying function inthe whole-energy range.0 Chapter 3. Canterbury Approximants Our knowledge about the double-virtual TFF is rather scarce. Theoretically,the situation resembles that of the single-virtual TFF. At high-energies,pQCD can be used to predict the leading Q = Q ≡ Q behavior insimilarity to the BL limit, see Eq. (1.50). At low energies, χ PT can beused to obtain the TFF series expansion at zero virtualities, leading to aclear prediction for F P γγ . A higher order calculation could be performedto obtain the single-virtual leading Q behavior, however, some unknownlow-energy constants were required to regularize the theory, thus losing pre-dictive power. The situation does not ameliorate for the double-virtualexpansion, where an even higher order calculation is required to obtain thecoefficients for Q and Q Q with the consequent proliferation of addi-tional unknown low-energy constants.The experimental situation for the double-virtual case is even more com-plicated. Whereas for the single-virtual case the theoretical ignorance wasalleviated with a rich experimental knowledge of the TFF in a wide kine-matical regime, there is at the moment not a single measurement for thedouble-virtual TFF. As a result, it is difficult to assess the different theo-retical ideas. This situation is related to the particular kinematics of theprocesses in which the double-virtual TFF can be experimentally accessed.In the space-like region, such measurement can be accessed at e + e − colliders in the e + e − → e + e − γ ∗ γ ∗ → e + e − P reaction. Such cross sec-tion is two-fold suppressed. On the one hand, the photon emission fromthe e ± is suppressed for large photon virtualities. On the other hand, theTFF F P γ ∗ γ ∗ ( Q , Q ) receives an additional Q suppression with respect to F P γ ∗ γ ∗ ( Q , e ± scattering angles.However, this kinematic regime is experimentally extremely challenging dueto the detector geometry and Bhabha scattering background. Remarkably,there is an ongoing effort at BES III to measure this process at low ener-gies [149], which will provide valuable information.In the time-like region, the double-virtual TFF can be accessed at ener-gies below the pseudoscalar mass in the double Dalitz decay process P → γ ∗ γ ∗ → (cid:96) + (cid:96) − (cid:96) (cid:48) + (cid:96) (cid:48)− . However, its large suppression due to the additionalelectromagnetic couplings with respect to the two photons and Dalitz de-cays, makes such process very challenging. In addition, even though its BRwould provide valuable information, it is the differential decay width whichgives direct access to the TFF, which measurement requires even higherstatistics. Moreover, the presence of the photon propagators greatly en-hances low energies relative to the high energies, hiding the double-virtualeffects, of order ( O ( q q )), as compared to the single-virtual ones, of order O ( q ), encoded in the slope parameter b P and playing the main role in.4. Canterbury approximants as a fitting tool 71 C C C C C C C C C Exact F Pγγ . 270 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . b P . 614 0 . 743 0 . 787 0 . 806 0 . 805 0 . 828 0 . 743 0 . 821 0 . 832 0 . c P . 377 0 . 609 0 . 727 0 . 792 0 . 793 0 . 893 0 . 609 0 . 856 0 . 914 0 . a P ;1 , . 612 0 . 798 0 . 861 0 . 887 0 . 881 0 . 918 0 . 797 0 . 906 0 . 925 0 . a P ;2 , . 520 0 . 827 0 . 964 1 . . 019 1 . 124 0 . 827 1 . 087 1 . 150 1 . Table 3.3: Convergence for the logarithmic model parameters. The parameters aredefined according to Eq. (3.2) with m P = 1 GeV. this decay . It is evident that a first measurement on the double-virtual TFF is requiredto improve our current knowledge, but is equally important to perform anappropriate and reliable theoretical analysis from these data. In this sectionwe discuss, in analogy to Chapter 2, how CAs provide an excellent toolto perform such analysis and extract the relevant low- (and high-) energyparameters in Eq. (3.2) in a systematic and model-independent fashion, andassess on the precision which would be achieved.For this purpose, we speculate about a possible measurement for thedouble-virtual TFF corresponding to 36 points in the [(0 , × (0 , region and investigate what could be obtained for the double-virtual pa-rameters from a fitting procedure similar to that in Chapter 2 for the single-virtual case. We emphasize that 11 of the 36 data points, correspondingto the single-virtual TFF, are already available at even finer gridding, andwill be improved in the future thanks to BESIII [149], NA62 [150], A2 [148],KLOE-2 [151] and GlueX [152] collaborations. The purely double-virtualdata-points are then reduced to 25. Moreover, only 15 of them are trulyindependent data points which need to be measured, as half of the squaregrid can be obtained by reflection from Bose symmetry.To show the performance of the method, we employ the different se-quences which have been revised in this chapter, C N ( Q , Q ) , C NN ( Q , Q )and C NN +1 ( Q , Q ). We quote the extracted values for the different pa-rameters of the series expansion, Eq. (3.2), in Tables 3.3 and 3.4 for thelogarithmic and Regge model, respectively. The agreement and conver-gence obtained is excellent, meaning that we have the chance to have adecent extraction once the first measurement for the double-virtual TFF isperformed. Naturally, the systematic accuracy that may be achieved de-pends on whether the quantity of data points is larger or smaller than that This handicap would be alleviated using the (cid:96) = µ channel for the η and η (cid:48) , which isinsensitive to the very low-energy dynamics given the µ mass [137]. We take a square grid with 1 GeV spacing starting at (0 , 0) GeV and ending at(5 , 5) GeV . C C C C C C C C C Exact F Pγγ . 273 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . 275 0 . b P . 085 1 . 246 1 . 295 1 . 315 1 . 316 1 . 334 1 . 246 1 . 330 1 . 335 1 . c P . 186 1 . 632 1 . 813 1 . 900 1 . 911 2 . 005 1 . 632 1 . 978 2 . 009 2 . a P ;1 , . 824 2 . 038 2 . 050 2 . 042 2 . 627 2 . 017 2 . 034 2 . 020 2 . 017 2 . a P ;2 , . 690 3 . 239 3 . 267 3 . 238 3 . 177 3 . 124 3 . 239 3 . 141 3 . 122 3 . Table 3.4: Convergence for the Regge model parameters. The parameters are definedaccording to Eq. (3.2) with m P = 1 GeV. c OPE1 C C C C C C ExactLog 0 . 191 0 . 174 0 . 171 0 . 137 0 . 152 0 . 148 0 . . 103 0 . 078 0 . 074 0 . 043 0 . 061 0 . 063 0 . Table 3.5: The c OPE1 coefficient in Eq. (3.30) extracted from different approximants foreach model. The last column represents the exact value. used here, but equally important is the measured energy range. On the onehand, if we would have enlarged the interval beyond 5 GeV , but keepingthe same number of data points, the quality of the extraction would havedeteriorated. On the other hand, taking a smaller interval —while keepingthe number of data points— would improve the result and convergence ofthe sequence. Once more, we emphasize that the systematic error from thefirst element, the C ( Q , Q ), is not negligible, which stress the necessity ofusing larger approximants. In particular, this means that, even if we employthe C ( Q , Q ) approximant to describe the TFF in some calculation, weshould not take the parameters which are obtained from a direct fit to thislast, but those obtained for the highest approximants. This is a well-knownfeature in PAs and its oversight would result in a large systematic error. Given the large amount of unknowns in our approximants, it may be useful,specially regarding the real case in which data contain non-negligible sta-tistical errors, to reduce the quantity of free parameters. One possibility isto implement the high-energy double-virtual behavior, which is dictated bypQCD as explained in Section 3.3.4. In this way, we do not only get rid ofone parameter, but we can extract the high-energy expansion as well, seeSection 3.3.4. We find that this approach results in an improved extractionof the low-energy parameters as compared to Tables 3.3 and 3.4, with theexception of the C ( Q , Q ) approximant, which often involves a poor de-scription. In addition, we extract the c OPE1 parameter from the high-energyexpansion Eq. (3.30), which result is shown in Table 3.5 for the logarithmicand Regge models for different approximants. In particular, we find that thediagonal (subdiagonal) sequence seems to provide a lower (upper) bound for.5. Conclusions 73this value —in accordance with Section 3.3.4—, offering a powerful methodto obtain an estimate for the systematic error.The method presented here provides a powerful mathematical approachnot only to reconstruct or extract the TFF, but for the experimentalists toanalyze their data without any theoretical prejudice and to estimate reliablesystematic errors in an easy way. This is actually not only of relevance forthe double-virtual measurement projected at BESIII, but for those collab-orations measuring the single-virtual TFF. In this sense, we have to recallthat these experiments always involve a deeply virtual photon together witha quasi-real one; the virtuality from the latter is certainly small but doesnot need to vanish. As an example, for the Belle π measurement [133] thisis mainly less than 0 . 01 GeV , whereas for B A B AR it may be as large as0 . [76, 118]. To assess the corrections from the quasi-real photon ef-fects, the experimental community requires then some model parametrizingthe double-virtual TFF. The chosen parametrization is not unique, for in-stance, Belle uses a factorized approach, whereas B A B AR takes a 1 / ( Q + Q )parametrization. Our method would be of help for these experiments in or-der to improve in precision and systematics. In addition, this may allow toextract some information about the double-virtual TFF. Finally, there areongoing lattice studies for the π TFF [164]; such approaches do requrie aswell some function to fit their results. Our apporach would provide then avaluable tool for them as well. In this chapter, we have introduced a generalization of PAs to the bivariatecase. This generalization extends the previous ideas on Pad´e theory forthe single-virtual to the most-general double-virtual TFF. For the case ofsymmetric functions, as the TFF, the use of Canterbury approximants isnatural and straightforward, it guarantees the convergence to meromorphicfunctions (representing the large- N c limit of QCD), respects factorizationwithout imposing it (which may approximately holds at low-energies for theTFF), reproduces well-known properties from PAs and provides convergenceto Stieltjes functions.In addition, the performance of the approach has been illustrated throughthe use of two different models previously employed in the univariate case.We have found that the intuition from PAs when dealing with poles andcuts can be extrapolated to this case. Moreover, in similarity to PAs, thepoles may be given in advance, though this implies again larger systematicerrors. Once more, the method allows to implement not only the low-, butthe high-energy information as well. As a final remark, we have shown thatthe underlying symmetries of the original function may help to improve on4 Chapter 3. Canterbury Approximantsconvergence. Regretfully, there is no clear symmetry or relation among thelow-energy expansion parameters for the TFF beyond that imposed fromBose symmetry, though a deeper study along this line would be of interest.In analogy to PAs, our method allows then to extract the (theoreti-cally unknown) low- and high-energy parameters entering the TFF fromexperimental data through a fitting procedure in a systematic and model-independent fashion. The ongoing experimental effort at BESIII to performsuch a measurement would provide then the last required piece of informa-tion to reconstruct the double-virtual TFF. As an outcome, our method maybe of interest for the experimental community (which often has to deal withthe double-virtual TFF even if measuring the single-virtual one) and for thelattice community.This chapter closes the theoretical framework which has been developedfor describing the pseudoscalar TFFs. With all the required ingredients athand, we proceed to discuss in the next chapters different applications inwhich these TFFs represent the main input in the calculation. hapter η − η (cid:48) mixing Contents η − η (cid:48) mixing from the TFFs . 814.5 Applications . . . . . . . . . . . . . . . . . . . . 894.6 Conclusions and outlook . . . . . . . . . . . . . 95 The η − η (cid:48) mixing has been a subject of deep investigation since the advent ofthe quark model. Early attempts to describe the η − η (cid:48) structure through theuse of SU (3) F symmetry and Gell-Mann-Okubo (GMO) mass formulas ap-peared in Refs. [165, 166], which obtained a mixing angle θ P ≈ − ◦ . Lateron, as χ PT was established as the low-energy effective field theory of QCDand calculations at NLO became available, it was realized that correctionsto the GMO mass formula shifted the mixing angle to θ P ≈ − ◦ , whichwas in better agreement with experimental results [167, 168]. However,in the years to come, different phenomenological analysis appeared, ques-tioning such result and suggesting values from θ P = − ◦ to θ P = − ◦ ,depending on the observables taken into account and on the models as-sumptions [169–173]. This situation was understood after the developmentof large- N c χ PT ( (cid:96)N c χ PT), which provides a framework to bring the η (cid:48) me-son into χ PT. It was clear after the publication of [47, 174], and subsequentworks [41, 175, 176], that the η − η (cid:48) mixing requires two angles to parametrizetheir decay constants as a consequence of SU (3) F breaking. This feature has756 Chapter 4. η − η (cid:48) mixingbeen incorporated in subsequent phenomenological analysis [119, 120, 177–181] resulting in different values depending on the modeling procedure.In the following, we take our previous results from Chapter 2 in order toprovide a new alternative determination for the η − η (cid:48) mixing parameters. Asan advantage, this approach is free of the simplifying assumptions requiredin previous approaches. In Section 4.2, we provide a brief reminder of themixing at LO in (cid:96)N c χ PT, whereas the necessity of a two-angle descriptionat NLO is discussed in Section 4.3, where we introduce the octet-singletand quark-flavor basis. Our novel approach for determining the mixingparameters is discussed in Section 4.4. As an innovation, we sequentiallyinclude the effects of OZI-violating parameters and, in general, the full NLOcorrections in a comprehensive way. Applications concerning the mixing arediscussed in Section 4.5. Finally, we present our conclusions in Section 4.6. From the (cid:96)N c χ PT Lagrangian L (0) Eq. (1.17), we extract the LO result forthe kinetic and mass terms for the (bare) η and η fields, η B ≡ ( η , η ) T [182,183], L (0) = 12 ∂ µ η TB K ∂ µ η B − η TB M η B , (4.1) K = × , M = (cid:18) M M M M + M τ (cid:19) , (4.2)which entries can be expressed in terms of the LO π and K masses, Eqs. (1.20)and (1.21), as M = 2 B m + 2 m s ) = 13 (4 ˚ M K − ˚ M π ) , (4.3) M = 2 B m + m s ) = 13 (2 ˚ M K + ˚ M π ) , (4.4) M = 2 √ 23 ( ˆ m − m s ) = − √ 23 ( ˚ M K − ˚ M π ) , (4.5)and M τ = τF . It is clear then from ˆ m (cid:54) = m s —equivalently, ˚ M K (cid:54) = ˚ M π —,that the η and η fields will mix among each other into the physical η and η (cid:48) . At this order, Eq. (4.1) can be diagonalized through the rotation matrix R ( θ P ) = (cid:18) cos θ P − sin θ P sin θ P cos θ P (cid:19) , (4.6)allowing to express the physical η P = ( η, η (cid:48) ) T fields in terms of the bare onesin Eq. (4.1) as η P = R ( θ P ) η B , where [182, 183]sin(2 θ P ) = 2 M M η (cid:48) − M η , (4.7).3. Two-angle mixing schemes 77and M η , M η (cid:48) are the eigenvalues solving the previous system, this is, theprediction for the physical masses. The mixing introduced above when di-agonalizing the mass term M is referred to as the state-mixing and in-volves a single angle θ P , not only at this order, but at any order. At LO in (cid:96)N c χ PT, one obtains the result θ P = − . ◦ [183]. However, non-negligiblecorrections are found at higher orders in the systematic (cid:96)N c χ PT expan-sion [183, 184] shifting this value towards θ P ≈ − ◦ [183].Of special interest for our later discussions are the pseudoscalar decayconstants. These are defined in terms of the QCD axial current as (cid:104) | J a µ | P (cid:105) = ip µ F aP J a µ = qγ µ γ λ a q Tr( λ a λ b ) = 2 δ ab , (4.8)where λ a is a Gell-Mann matrix in flavor space and λ = (cid:112) / × . We re-mark that our normalization for the axial current yields F π = 92 . (cid:96)N c χ PT, one finds F π = F K = F . For the η and η (cid:48) , due to themixing, the decay constants are conveniently expressed, following Ref. [47]at LO as F P ≡ (cid:18) F η F η F η (cid:48) F η (cid:48) (cid:19) = F (cid:18) cos θ P − sin θ P sin θ P cos θ P (cid:19) = R ( θ P ) (cid:18) F F (cid:19) ≡ R ( θ P ) ˆ F , (4.9)where ˆ F = diag( F , F ) and F = F = F at LO. Consequently, the η and η (cid:48) couple both, to the octet and singlet axial currents. It follows then that,at LO, their couplings to these currents ( F P ) can be expressed in terms ofthe octet and singlet ˆ F decay constants using the same rotation matrix weused for the state mixing, this is, F P = R ( θ P ) ˆ F , cf. Eq. (4.6) and commentsbelow. In the jargon of η − η (cid:48) mixing, the decay constants follow the statemixing. This situation is particular to the LO case. As we illustrate below, athigher orders, SU (3) F breaking effects destroy this simple picture, requiringa two-angle description to express the decay constants. When moving on to NLO, the η − η (cid:48) mixing becomes more involved as nowthe kinetic matrix K in Eq. (4.1) becomes non-diagonal too [47, 120, 175,178], a fact which was pointed out for the first time in [47, 175]. Actually, K and M cannot be simultaneously diagonalized within a single rotation.The diagonalization is performed then, perturbatively, in two sequentialsteps [182, 183]. First, a field redefinition for the bare fields η B = Z / T ˆ η allows to diagonalize the kinetic term K . Then, the resulting mass matrix, Z / M Z / T , is diagonalized through a rotation η P = R ( θ P )ˆ η ; the required8 Chapter 4. η − η (cid:48) mixingangle in this rotation defines the state-mixing angle in analogy to Eq. (4.6).Note however that the overall transformation η P = R ( θ P )( Z / T ) − η B in-cludes the non-diagonal Z / matrix. For these reasons, the pseudoscalardecay constants cannot be expressed in a simple form analog to Eq. (4.9) asfour parameters are now required. Instead, they are defined as F P ≡ (cid:18) F η F η F η (cid:48) F η (cid:48) (cid:19) ≡ (cid:18) F cos θ − F sin θ F sin θ F cos θ (cid:19) (cid:54) = R ( θ P ) (cid:18) F F (cid:19) . (4.10)We emphasize again that the state-mixing involves a single mixing angle, θ P , at any order. It is the decay constants F aP description that requires two-angles or, alternatively, four independent quantities. (cid:96)N c χ PT provides thenthe appropriate framework to relate these decay constants to other quan-tities in the mesonic sector of QCD. Among others, the mixing-angle andadditional decay constants F π and F K . Particularly, at NLO, the followingrelations hold [178, 182] F = 4 F K − F π , F = 2 F K + F π F π Λ , (4.11) F F sin( θ − θ ) = − √ (cid:0) F K − F π (cid:1) , (4.12) θ + θ = 2 θ P , θ − θ = − √ (cid:18) F K F π − (cid:19) , (4.13)with Λ an OZI-violating parameter. Eq. (4.10) defines the so-called octet-singlet mixing scheme and relations (4.11) to (4.13) hold up to NNLOcorrections in the combined (cid:96)N c χ PT expansion. Given that F K /F π =1 . SU (3) F break-ing implies θ (cid:54) = θ . It was the neglected SU (3) F breaking encoded in theGMO formula and F /F π —not included up to [167, 168]— that lead to badresults in the earlier years [165, 166]. The same effect, this time encoded in( θ − θ ) (cid:54) = 0, lead to different extractions for the decay constants from dif-ferent observables [169–173], which often require the decay constants ratherthan the state-mixing.At this point, there is a further property which must be discussed. Giventhe anomalous dimension of the singlet axial current, the singlet decay con-stants defined via (cid:104) | J µ | P (cid:105) = ip µ F P will inherit the scale-dependencywhich is dictated from QCD [41, 47, 175] µ dF dµ = γ A ( µ ) F = − C ( r ) N F α s π F + O ( α s ) = − N F (cid:18) α s ( µ ) π (cid:19) F . (4.14) To obtain these relations, the relevant LECs defining these quantities have been tradedfor F π and F K . Moreover, multiplicative factors such as ( F K /F π − 1) have been neglectedas they can be understood as NNLO effects. .3. Two-angle mixing schemes 79Here, µ is the renormalization scale, γ A ( µ ) the axial current anomalousdimension [185] given in terms of the group invariant C ( r ) —for the fun-damental representation C ( r ) = ( N c − / (2 N c )— and N F is the numberof active flavors ( u, d, s, ... ) at that scale. The solution to this equation isgiven, at O ( α s ) as [47, 175, 185] F ( µ ) = F ( µ ) (cid:18) N F β (cid:18) α s ( µ ) π − α s ( µ ) π (cid:19)(cid:19) ≡ F ( µ )(1 + δ RG ( µ )) , (4.15)where µ is some reference scale and we have used the LO result for the α s running, involving at this order the beta function coefficient β = 11 N c / − N F / 3. Of course, physical observables are scale independent, and F ( µ )-dependent terms will be accompanied by additional terms in such a way thatthe scale-dependency is cancelled. In the (cid:96)N c χ PT Lagrangian, this is easyto see, as these (Λ i OZI-violating) terms are explicitly included in order tomake the (bare) Lagrangian scale-independent. As an example, the WZWpart requires an additional term [47] L (2)WZW ⊃ N c α Λ √ πF (cid:15) µνρσ F µν F ρσ η , (4.16)where Λ is a scale-dependent OZI-violating parameter with running Λ ( µ ) =Λ ( µ )(1 + δ RG ( µ )) analogous to that in Eq. (4.15) and renders the two-photon decays in Eqs. (4.27) and (4.28) scale-independent. Alternatively,heavy processes involving η ( η (cid:48) ) in final states are often expressed in termsof (cid:80) i C i ( µ ) (cid:104) | O i | P (cid:105) matrix elements, where O i is a local operator —forinstance, O j = qγ µ γ q — and C i ( µ ) is the so-called Wilson coefficient, whichaccounts for the operator evolution from the heavy ( µ = M H ) to the low( µ = µ ) scale. The latter should match that of F ( µ ), implying that anyshift µ → µ (cid:48) would not alter the result. This is the case for the TFFasymptotic behavior discussed in Section 4.4. The features outlined above make the description of any physical process in-volving the singlet sector much involved. For this reason, later on, the quark-flavor mixing scheme was proposed in Ref. [119]. This scheme was motivatedby the fact that vector and tensor singlet mesons —where the axial anomalyplays no role— can be pretty well described in terms of light and strangequark singlet components. Actually, we show below that such assumptionagrees with NLO (cid:96)N c χ PT provided that OZI-violating effects are obviated.In such approximation, the physical states and decay constants follow thesame mixing and can therefore be described in terms of one angle alone,which greatly simplifies our description. Defining the light and strange axial Note our (cid:15) = 1 convention and the replacement with respect to [47] ψ → ( √ /F ) η . η − η (cid:48) mixingcurrents J q,s µ = qγ µ γ λ q,s q , with λ q = diag(1 , , 0) and λ s = diag(0 , , √ (cid:104) | J q,s µ | P ( p ) (cid:105) ≡ ip µ F q,sP read( F qsP ) ≡ (cid:18) F qη F sη F qη (cid:48) F sη (cid:48) (cid:19) ≡ (cid:18) F q cos φ q − F s sin φ s F q sin φ q F s cos φ s (cid:19) . (4.17)Relating the decay constants in both basis is rather simple as it only amountsto a rotation of our fundamental QCD currents. From the above definition,it is easy to check that the octet-singlet and quark-flavor basis are relatedvia rotation matrix (cid:18) J µ J µ (cid:19) = 1 √ (cid:18) −√ √ (cid:19) (cid:18) J q µ J s µ (cid:19) ⇒ ( J µ ) α = U ( θ ideal ) αa ( J qs µ ) a . (4.18)Here, the equation on the left-hand side has been expressed in matricialform in the right one with obvious identifications. The indices α and a denote octet-singlet and flavor indices, respectively (summation assumed ifrepeated indices). Then, the decay constants in Eqs. (4.10) and (4.17) canbe related as ( F qsP ) P a = ( F P ) P α U ( θ ideal ) αa , (4.19)where the index P = { η, η (cid:48) } and, again, summation over repeated indicesis assumed. Relation (4.19) will be our dictionary when relating results indifferent basis. In this way, we can translate Eqs. (4.11) and (4.12) to theiranalogues in the quark-flavor basis obtaining [120, 178] F q = F π + 23 F π Λ , F s = 2 F K − F π + 13 F π Λ , (4.20) F q F s sin( φ q − φ s ) = √ F π Λ . (4.21)It is clear, as anticipated, that neglecting the OZI-violating Λ i parametersimplies φ q = φ s ≡ φ , achieving a simpler one-angle description for the decayconstants, F qsP = R ( φ )diag( F q , F s ). Indeed, there is a strong phenomenolog-ical success supporting this idea [120, 178]. Under the assumption φ q = φ s —that is commonly known as the FKS scheme [119, 177, 178]—, this basishas become a standard choice given its simplicity and the predictive powerwith respect to the octet-singlet one. This assumption is specially useful forstudying the TFFs [67] within pQCD.An alternative approach to understand this situation follows from thepQCD picture in Ref. [177] when considering the Fock state description ofthe η and η (cid:48) . Given that ( m u (cid:39) m d ) (cid:28) m s , it seems reasonable that η and η (cid:48) may be described in terms of light and strange quarks degrees of freedom | η q (cid:105) = Ψ q √ | uu + dd (cid:105) + ..., | η s (cid:105) = Ψ s | ss (cid:105) + ..., (4.22).4. Determining the η − η (cid:48) mixing from the TFFs 81where the ellipses stand for additional Fock states including gluons and seaquarks, and Ψ q and Ψ s stand for the wave-functions, which are in generaldifferent from each other, i.e., Ψ q (cid:54) = Ψ s . Finally, F q,sP is related to the Ψ q,s wave function normalization, cf. Eq. (1.46). Assuming further that | η (cid:105) = cos φ | η q (cid:105) − sin φ | η s (cid:105) , | η (cid:48) (cid:105) = sin φ | η q (cid:105) + cos φ | η s (cid:105) , (4.23)implies that, when rotating back to the octet-singlet basis, an analogous η − η (cid:48) description along the lines of Eq. (4.23), | η (cid:105) = cos θ P | η (cid:105) − sin θ P | η (cid:105) , | η (cid:48) (cid:105) = sin θ P | η (cid:105) + cos θ P | η (cid:105) , (4.24)would require defining the corresponding Fock states as | η (cid:105) = Ψ q + 2Ψ s | uu + dd − ss (cid:105)√ √ q − Ψ s )3 | uu + dd + ss (cid:105)√ , (4.25) | η (cid:105) = √ q − Ψ s )3 | uu + dd − ss (cid:105)√ q + Ψ s | uu + dd + ss (cid:105)√ , (4.26)so what has been defined as the octet(singlet) | η (cid:105) component is an admix-ture of the octet and singlet Fock states unless SU (3) F -symmetry representsa good approximation and Ψ q = Ψ s holds. This represents a result analogousto that in Eq. (4.12). Conversely, in such SU (3) F -symmetric case, where θ = θ = θ P , we could start with an analogous single-octet description.Rotating back to the flavor basis, we would find an analogous result to thatin Eq. (4.25), namely, that the light(strange) quark state is an admixtureof light and strange quark Fock states unless Ψ = Ψ . In this language,this is easy to see, as | qq (cid:105) -like states get mixed via the QCD anomaly, anOZI-violating effect analogous to the result in Eq. (4.21).To summarize, the quark-flavor basis provides a simpler choice —in termsof a single angle— whenever the precision we aim for does not require toinclude OZI-violating effects in our framework and has become the most pop-ular choice in phenomenological analyses [119, 120, 177, 178, 180, 181]. Inthe case where the required precision may become sensitive to OZI-violatingeffects, both basis involve the use of two-angles —alternatively, four inde-pendent decay constants— and the octet-singlet basis may become simplerfor incorporating such effects. η − η (cid:48) mixing from the TFFs The different analyses used in the literature to extract the mixing parame-ters defined in the previous section — F , F , θ , θ in the octet-singlet basisor, alternatively, F q , F s , φ q , φ s in the quark-flavor basis— find often non-compatible values among their extractions. As an illustration, we refer to2 Chapter 4. η − η (cid:48) mixingthe approaches from Refs. [47, 119, 120, 179] which are depicted in Fig. 4.1.It would be desirable then to have an alternative approach which is definedin terms of (cid:96)N c χ PT quantities alone —the decay constants— and has con-trol over the OZI-violating parameters. This requires avoiding, for instance,models for the V P γ transitions —more comments on them in Section 4.5.1—which are widely used to extract the mixing parameters, or, eventually, thepopular J/ Ψ → γη ( η (cid:48) ) decays —further comments on this point in Sec-tion 4.5.2. We suggest that this is possible using the available informationon the η and η (cid:48) TFFs from Chapter 2. Moreover, it is possible to accountfor the OZI-violating parameters, whose impact we discuss below. Actuallyour approach does not only allow to extract the above-mentioned mixingparameters but the additional OZI-violating parameter Λ , cf. Eq. (4.16).The starting point in our approach is the remarkable observation that,not only the low-energy behavior for the η and η (cid:48) TFFs —related to theirtwo photon decays—, but their high-energy behavior lim Q →∞ F P γ ∗ γ ( Q )dictated by pQCD Eq. (1.49) is given, essentially, in terms of the desiredmixing parameters. Particularly, at NLO, the two-photon decays can becalculated from (cid:96)N c χ PT, obtaining [41, 138, 178] F ηγγ ≡ F ηγγ (0) = 14 π ˆ c (1 + K ) F η (cid:48) − ˆ c (1 + K + Λ ) F η (cid:48) F η (cid:48) F η − F η (cid:48) F η , (4.27) F η (cid:48) γγ ≡ F η (cid:48) γγ (0) = 14 π − ˆ c (1 + K ) F η + ˆ c (1 + K + Λ ) F η F η (cid:48) F η − F η (cid:48) F η , (4.28)where ˆ c = 1 / √ c = 2 √ / √ K ≡ K M π − M K and K ≡ K M π + ˚ M K are related to the LEC K in the (cid:96)N c χ PT Lagrangian [41] . The latter appear as well in the π TFF via F πγγ ≡ F πγγ (0) = 1 + K ˚ M π π F π . (4.29)From the experimental π → γγ result [10], we obtain K = − . OZI-violating parameter from Eq. (4.16) must be included to render the resultscale-independent. To see this, note that both F P and Λ , unlike F P and K , scale as (1 + δ RG ( µ )). This produces overall factors in the numeratorand denominator canceling the scale-dependency. To obtain the expressionfor the high-energy behavior, we have first to take into account the running The K LEC represents the (cid:96)N c χ PT version for the SU (3) F χ PT L (cid:15) LEC, seeRef. [41]. Particularly, it compares to Eq. (1.51) via K → − (1024 π / L (cid:15) . .4. Determining the η − η (cid:48) mixing from the TFFs 83of the axial current, Eq. (4.14), which implies an additional running effecton top of that of the Gegenbauer coefficients, Eq. (1.47). From Eq. (4.15),and taking as the reference scale for the ( η ) η (cid:48) → γγ decays µ = 1 GeV, weobtain for F P at Q → ∞ the relation F P ( ∞ ) = F P (cid:18) − N F β α s π (cid:19) = F P (1 + δ RG ( ∞ )) ≡ F P (1 + δ ) , (4.30)where α s is to be evaluated at 1 GeV and F P is the decay constant appearingin the η ( η (cid:48) ) → γγ decays, to be taken at µ = 1 GeV. Taking into accountcorrections from higher orders by using the α s -running to four-loops accu-racy [186] as well as considering threshold effects, we obtain that δ = − . η ∞ ≡ lim Q →∞ Q F ηγ ∗ γ ( Q ) = 2(ˆ c F η + ˆ c (1 + δ ) F η ) , (4.31) η (cid:48)∞ ≡ lim Q →∞ Q F η (cid:48) γ ∗ γ ( Q ) = 2(ˆ c F η (cid:48) + ˆ c (1 + δ ) F η (cid:48) ) . (4.32)The resulting effect is by no means negligible and, to our best knowledge,was implemented for the first time in Ref. [67].We have at this stage a set of four equations at our disposal (Eqs. (4.27),(4.28), (4.31) and (4.32)) to extract the four mixing parameters we areinterested in. It seems then a straightforward task to determine the mixingparameters —at least, if we neglect the a priori small parameter Λ andeither neglect or take K from the π → γγ decay. However, there is asubtle connection among the different equations which avoids for such aneasy solution. As noted for the first time in our work in Refs. [114, 117], thesystem of equations is degenerate. To see this, we can obtain an expressionfor F η and F η (cid:48) from Eqs. (4.31) and (4.32). Then, substituting in Eqs. (4.27)and (4.28), we can linearize the system, which may be expressed in matrixform as A (cid:0) F η , F η (cid:48) , F η , F η (cid:48) (cid:1) T = (cid:0) η ∞ , η (cid:48)∞ , , (cid:1) T , (4.33)where the A matrix is defined as A = c c (1 + δ ) 00 2ˆ c c (1 + δ )0 ˜ c − π ˆ c η (cid:48)∞ F ηγγ π ˆ c η ∞ F ηγγ − ˜ c − ˜ c c − π ˆ c η (cid:48)∞ F η (cid:48) γγ π ˆ c η ∞ F η (cid:48) γγ , (4.34)where ˜ c = ˆ c (1 + K ) and ˜ c = ˆ c (1 + K + Λ ). Then, the degeneracy isinferred from the determinant, which is proportional to (cid:0) ˆ c (1 + K ) + ˆ c (1 + δ )(1 + K + Λ ) (cid:1) − π (cid:0) F ηγγ η ∞ + F η (cid:48) γγ η (cid:48)∞ (cid:1) . (4.35)4 Chapter 4. η − η (cid:48) mixingIt may look that Eq. (4.35) is in general non-vanishing. However, it turnsout that F ηγγ η ∞ + F η (cid:48) γγ η (cid:48)∞ = ˆ c (1 + K ) + ˆ c (1 + δ )(1 + K + Λ )2 π = 32 π (cid:18) (cid:2) K + 8 (cid:0) δ + ( K + Λ )(1 + δ ) (cid:1)(cid:3)(cid:19) , (4.36)yields a vanishing value for Eq. (4.35), where in the last term we havereplaced the charge factors ˆ c i . As an alternative approach, we can find thatthere is a null space for the system in Eq. (4.34), (cid:0) ˆ c F η (cid:48) γγ (1 + δ ) , − ˆ c (1 + δ ) F ηγγ , − ˆ c F η (cid:48) γγ , ˆ c F ηγγ (cid:1) T . (4.37)All in all, we have to deal with a degenerate system, which may look like adead-end for our approach. However, contrary to the expectations, it turnsout that one can take advantage of Eq. (4.36) to solve all these problems.Curiously enough, the OZI-violating Λ i parameters play a central role inthis discussion. In order to illustrate their impact and conceptual relevance,we first set K = 0 and sequentially include these parameters one by one.First, we set Λ = Λ = 0 and discuss the results. Second, we let Λ (cid:54) = 0but, still, Λ = 0. Third, we let Λ , Λ (cid:54) = 0 and obtain them through afitting procedure. Finally, we include the parameter K , which completesthe full list of NLO LECs which are relevant to our study. The latter is themain result from this chapter and represents, to our best knowledge, thefirst result fully consistent with (cid:96)N c χ PT at NLO. Finally, we discuss ourfindings and compare to previous phenomenological approaches. η − η (cid:48) mixing: K = Λ = Λ = 0 The simplest choice one can take to solve for the mixing parameters, seeRef. [117], is to set all the OZI-violating Λ i parameters present in our equa-tions to 0, this is Λ = Λ = 0 (as well as K = 0). This choice implies,via Eq. (4.21), that φ q = φ s ≡ φ . This does not only break the degeneracyof our system, but reduces the number of free parameters down to 3, whichallows to solve the system using a set of three equations out of Eqs. (4.27),(4.28), (4.31) and (4.32). We call the attention however, that obtainingthe same solution for any set is not guaranteed unless relation Eq. (4.36), F ηγγ η ∞ + F η (cid:48) γγ η (cid:48)∞ = π (cid:0) δ (cid:1) , is fulfilled. In our case, taking the inputvalues from Table 2.14, we obtain 0 . π for the left hand side, whereasthe right hand side yields 0 . π for δ = − . 17. Therefore, it seems thatneglecting the OZI-violating parameters has not a tremendous impact. Notehowever that, to reach such agreement, we need to introduce the runningparameter δ from Eq. (4.30), which in the FKS scheme should be zero.In any case, since the condition Eq. (4.36) is not exactly fulfilled, everyset of equations will yield only marginally-compatible solutions. In order to.4. Determining the η − η (cid:48) mixing from the TFFs 85solve the system, we decide to take the result which makes use of F ηγγ , F η (cid:48) γγ and η ∞ alone. The reason is motivated in two-fold way. On the one hand, F ηγγ and F η (cid:48) γγ have been directly measured to an excellent precision. Onthe other hand, among the asymptotic values, η ∞ is the one with the mostreliable extraction, see Chapter 2. Finally, we expect that the η parametersare theoretically cleaner, as they are less sensitive to the singlet effects weare neglecting at this stage. As a result, taking the F ηγγ , F η (cid:48) γγ and η ∞ valuesfrom Table 2.14, we obtain [117] F q F π = 1 . , F s F π = 1 . , φ = 38 . . ◦ , (4.38) F F π = 1 . F F π = 1 . θ = − . . ◦ θ = − . . ◦ , (4.39)where in the second line we have used Eq. (4.19) to translate the result intothe octet-singlet basis. As an illustration, had we used η (cid:48)∞ instead of η ∞ , wewould have obtained F q /F π = 1 . , F s /F π = 1 . , φ = 41 . . ◦ . Hadwe obviated RG-effects, we would find some deviations in sets containingthe η ∞ , while big deviations would be found for those containing η (cid:48)∞ , as thesinglet content is more important for the η (cid:48) , see Ref. [114]. Our result isin line with previous findings [47, 114, 119, 120, 179] and has competitiveerrors. For comparison, see Fig. 4.1, Option I. η − η (cid:48) mixing: K = Λ = 0 , Λ (cid:54) = 0 As illustrated before, the previous approach suffers from the fact that so-lutions from different sets yield different results which are only marginallycompatible. This was easy to anticipate given that the degeneracy condi-tion (4.36) was only marginally fulfilled for Λ = 0. In this second approach,we assume that, still, Λ = 0, but Λ is a free parameter, which is fixedas to fulfill Eq. (4.36), obtaining [117] Λ = 0 . = − . V P γ decays. They differ in sign, but agree on its small magnitude, even beyondwhat is expected from the naive 1 /N c counting. Still, as Λ = 0, we stick tothe one-angle quark-flavor scheme, whereby any set of three equations canbe used with the same result. Taking the same inputs as in previous sectionfrom Table 2.14, we obtain F q F π = 1 . , F s F π = 1 . , φ = 38 . . ◦ , (4.40) F F π = 1 . F F π = 1 . θ = − . . ◦ θ = − . . ◦ . (4.41)As an advantage, choosing Λ (cid:54) = 0, we can obtain analog results for any cho-sen set of equations, which improves with respect to the previous situation.Our results are displayed under the label Option II in Fig. 4.1 and show theimpact of including the Λ parameter.6 Chapter 4. η − η (cid:48) mixing η − η (cid:48) mixing: K = 0 , Λ , Λ (cid:54) = 0 The approaches adopted in Sections 4.4.1 and 4.4.2 present, at the formallevel, some theoretical inconsistencies. Namely, we found that running ef-fects —neglected in the common FKS scheme— encoded in δ , see Eq. (4.30),were important in our determination. However, these require, formally, thepresence of the Λ parameter if the scale-dependency for the asymptotic be-havior is to be cancelled —see Eqs. (4.31) and (4.32). Similarly, including Λ requires the presence of Λ to cancel the scale-dependency in the two photondecays —see Eqs. (4.27) and (4.28). Besides, at the phenomenological level,there is further evidence pointing to Λ (cid:54) = 0 effects. Particularly, our previ-ous results —and basically every phenomenological estimate, see Fig. 4.1—indicate that F q > F π with around 3 σ significance. This, via Eq. (4.20),implies a non-vanishing positive value for Λ , which in our simplified ap-proach was taken to be zero. This in turn, would imply via Eq. (4.21) that φ q (cid:54) = φ s , invalidating then our previous assumptions and pointing out thenecessity of using a general scheme with two different angles and non-zeroΛ , parameters for describing the η and η (cid:48) decay constants, an approachthat we adopt in this section (but still retaining K = 0).In order to solve our system, and focusing on the octet-singlet basis, wehave at disposal four equations —Eqs. (4.27), (4.28), (4.31) and (4.32)—and five unknowns — F , F , θ , θ and Λ . In order to cure this situation,we can resort, as in the previous section, to the Eq. (4.36), which wouldprovide the required constraint to fix Λ , but we still have to face the factthat our system is linear dependent. In order to overcome this problem,we notice that NLO (cid:96)N c χ PT provides a clean prediction for both, F and F F sin( θ − θ ) in terms of the well-known value for F K /F π [10]. Takingeither of them as a constraint, one would add an additional equation tothe previous system, which would provide a unique solution. Taking both,would lead to an overdetermined system, which in general has no solution.For this reason, we adopt a democratic procedure[138] in which we performa fit including both F and F F sin( θ − θ ) constraints together withEqs. (4.27), (4.28), (4.31), (4.32) and (4.36). In addition, we ascribe a 3%theoretical uncertainty for the (cid:96)N c χ PT predictions by noticing that F K /F π typically receives 3% corrections from the NNLO . Consequently, we addthis error in quadrature on top of the one from [10] for our fitting procedure.As in the previous section, we take the inputs in Table 2.14. We obtain a We use preciser relations than those from Section 4.3.1: ( F /F π ) = 1+ F K F π ( F K F π − F F sin( θ − θ ) = − √ F π ( F K F π − F K F π + (Λ → To see this, consider F K /F π = 1 . (cid:39) (cid:15) + (cid:15) . This leads to the estimate for theNNLO correction (cid:15) = 0 . 03. Explicit results in Ref. [187] leads to similar values too. .4. Determining the η − η (cid:48) mixing from the TFFs 87 æ ææ æàà à à - - - - - - Opt.IVOpt.IIIOpt.IIOpt.IEFBDOFKSL Θ æææ æà ààà Opt.IVOpt.IIIOpt.IIOpt.IEFBDOFKSL F (cid:144) F Π æ ææ æà ààà - - - Θ æ ææ æà ààà F (cid:144) F Π æ ææ æà à àà 36 38 40 42 44 46 48 Opt.IVOpt.IIIOpt.IIOpt.IEFBDOFKSL Φ q æ ææ æà ààà Opt.IVOpt.IIIOpt.IIOpt.IEFBDOFKSL F q (cid:144) F Π æ ææ æà àà à 36 38 40 42 44 46 48 Φ s æææ æà ààà F s (cid:144) F Π Figure 4.1: Our mixing parameters Eqs. (4.38) to (4.43) (blue squares) compared todifferent theoretical results (orange circles), see description in the text. The upper(lower)pannel displays our results in the octet-singlet(quark-flavor) basis. The references standfor L [47], FKS [119], BDO [179] EF [120]. fit with χ ν = 0 . 35 and the following results for the mixing parameters [138] F F π = 1 . , F F π = 1 . , θ = − . . ◦ , θ = − . . ◦ , (4.42) F q F π = 1 . , F s F π = 1 . , φ q = 40 . . ◦ , φ s = 38 . . ◦ . (4.43)These results are labelled as Option III in Fig. 4.1, where the impact ofincluding Λ (cid:54) = 0 can be appreciated. In addition, we obtain for the OZI-violating parameters and the state-mixing angleΛ = 0 . , Λ = 0 . , θ P = − . . ◦ . (4.44)Here, Λ and θ P are not directly fitted parameters, but can be obtained by8 Chapter 4. η − η (cid:48) mixingmeans of Eq. (4.11) and Eq. (4.13), respectively. η − η (cid:48) mixing: K , Λ , Λ (cid:54) = 0 Finally, to quantify the impact of a non-zero K parameter and to have afully consistent description at NLO in (cid:96)N c χ PT, we include the former inthe last step. To do so, and given the poor extraction from π → γγ decays( K = − . F πγγ in our fitting procedure. We obtain a fit with χ ν = 0 . 54 and thefollowing values for the mixing parameters F F π = 1 . , F F π = 1 . , θ = − . . ◦ , θ = − . . ◦ , (4.45) F q F π = 1 . , F s F π = 1 . , φ q = 40 . . ◦ , φ s = 38 . . ◦ . (4.46)In addition, we findΛ = 0 . , K = − . , Λ = 0 . , θ P = − . . ◦ . (4.47)The results for the mixing parameters, Eqs. (4.45) to (4.47), represent themain result from this chapter. We remind that we have used in our proce-dure a renormalization scale µ = 1 GeV. Consequently, our values shouldbe understood at such scale. This applies to the OZI-violating param-eters Λ , and, in the octet-singlet basis, to the singlet decay constants F P . Whereas this may be adequate for pQCD studies such as those inRefs. [67, 188], the (cid:96)N c χ PT practitioner may find more helpful the scale-independent Λ − = 0 . = − . = 0 . − = 0 . η − η (cid:48) mixingparameters purely based on (cid:96)N c χ PT Lagrangian quantities —to this day,the only consistent framework to describe the η − η (cid:48) system. Our approachfully incorporates the required OZI-violating parameters (necessary to ren-der scale-independent results) as well as the K LEC, which are neglectedin most of the previous phenomenological approaches [119, 120, 177, 178,180, 181]. In addition, our approach does not rely on a phenomenologicalmodel involving further assumptions, as required for instance when using V → P γ transitions —find further details in Section 4.5.1. We note in thisrespect that previous approaches following the FKS scheme should have used Again, we use a preciser relation ( F /F π ) = (1 + Λ ) + ( F K F π − F K F π + Λ ) [183]. .5. Applications 89 F q = F π to be consistent. Finally, we emphasize that our approach makesuse of 4 independent quantities alone to determine the mixing parameters.This contrasts with previous approaches requiring a larger amount of inputin their fits and often with a large χ ν value [120]. The extraction of the mixing parameters provides an important input tounderstand the structure of the η − η (cid:48) , which is still a matter of debateand research nowadays due to its complexity —for the most recent stud-ies, see [183, 184]. However, its interest lies beyond unravelling the struc-ture of these pseudoscalars, as these parameters enter in a large varietyof phenomenological applications. See for instance those in Refs. [67, 168,178, 188–190], involving processes at low energies, such as pp → π η ( (cid:48) ) ,mid-energies, such as B → J/ Ψ η ( (cid:48) ) , or as energetic as Z → η ( (cid:48) ) γ decays.Consequently, our parameter extraction could be further tested using theseprocesses. We do not pursue here such an ambitious programme, but merelydescribe two selected applications, namely, V → P γ and P → V γ transitionswhere P = η ( (cid:48) ) and V = ρ, ω, φ , as well as J/ Ψ → η ( (cid:48) ) γ decays. g V P γ couplings As a first application, we provide in this section the g V P γ couplings de-scribing the interaction of the lowest-lying nonet of vector mesons with thepseudoscalar mesons and a photon. As such, they describe ρ, ω, φ → ηγ , η (cid:48) → ρ ( ω ) γ and φ → η (cid:48) γ decays, from which they can be experimentallyextracted. Alternatively, these parameters can be theoretically related tothe QCD-anomalous Green function (cid:104) P | T (cid:8) J EMµ ( x ) , J aν (0) (cid:9) | (cid:105) which, forvanishing virtualities, is given in terms of the triangle anomaly. The g V P γ couplings appear then when a dispersive representation saturated with thelowest-lying vector resonances is adopted [120, 170, 178]. The resultingexpressions are given in Appendix B.1, which include the OZI violating pa-rameter Λ as appearing in Ref. [178] and K as an additional novelty. Ourresults found for the g V P γ couplings are displayed in Tab. 4.1 together withthe experimental values; the different outcomes for the methods employedin Sections 4.4.1 to 4.4.4 are labelled as Option I, II, III and IV, respectively.Though the agreement is not excellent, it has to be taken into account thathigher resonances and continuum has been neglected in the employed disper-sive representation, which implies non-negligible modeling associated errors,to some extent common both to the η and η (cid:48) [170]. Therefore, it may bemore adequate to take the ratio g V ηγ /g V η (cid:48) γ instead [178], which is displayedin Table 4.1 as well. Actually, the agreement among our predictions and the The coupling is defined as (cid:104) P | J EMµ | V ν (cid:105) | ( p P − p V ) =0 = − g V Pγ (cid:15) µνρσ p ρP p σV [170]. η − η (cid:48) mixing Option I Option II Option III Option IV Experiment g ρηγ . . . . . g ρη (cid:48) γ . . . . . g ωηγ . . . . . g ωη (cid:48) γ . . . . . g φηγ − . − . − . − . − . g φη (cid:48) γ . . . . . g ρηγ /g ρη (cid:48) γ . . . . . g ωηγ /g ωη (cid:48) γ . . . . . g φηγ /g φη (cid:48) γ − . − . − . − . − . R J/ψ . . . . . Table 4.1: Summary of g V Pγ couplings together with R J/ Ψ , see description in the text.Experimental determinations are from Ref. [10]. experiment in these ratios is excellent for the ρ and ω cases and reasonablefor the φ . The predictive power for these decays, which are used as inputs intraditional approaches instead, should be considered as an advantage fromour approach. R J/ψ It has been argued in Refs. [189, 190] that the η − η (cid:48) mixing parameters couldbe used as well to calculate decays in the charmonium region. Note that allthese processes need to change flavor, which —neglecting electromagneticeffects— necessarily happens through OZI violating mechanisms, where thesinglet sector plays a central role. Specially popular, and widely used inphenomenological analyses [119, 120, 170] are the J/ Ψ → η ( (cid:48) ) γ decays, inparticular its ratio R J/ Ψ defined in Eq. (4.48) below. It is thought that thedominant mechanism underlying these decays is given by an intermediatetwo gluon state as depicted in Fig. 4.2 (see Ref. [191]) which allows to expressthe ratio as R J/ψ = BR ( J/ψ → η (cid:48) γ ) BR ( J/ψ → ηγ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) η (cid:48) | G µν,c ˜ G cµν | (cid:105)(cid:104) η | G µν,c ˜ G cµν | (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:32) m J/ψ − m η (cid:48) m J/ψ − m η (cid:33) , (4.48)where the first factor is the matrix element required from the process asoutlined in Ref. [170] and the second factor is pure phase space. Note thata factorization formalism is implicit, assuming as well that everything elsebut the above matrix elements cancels out in the ratio.Remarkably, even though (cid:96)N c χ PT does not incorporate gluons as explicitdegrees of freedom, it allows to calculate Green’s functions involving them.This possibility is brought by Ward identities, which in this case via Eq. (1.7)relate the purely gluonic current in Eq. (4.48) to quark currents and theirdivergencies. Particularly, for each individual flavor q = u, d, s, ... , Eq. (1.7),.5. Applications 91 J/ Ψ γη ( ′ ) Figure 4.2: Expected main contribution to J/ Ψ → γη ( (cid:48) ) processes. reads ∂ µ ( qγ µ γ q ) = 2 m q qiγ q − g s π (cid:15) αβµν G cαβ G cµν ≡ m q qiγ q + ω. (4.49)As an interesting academic exercise, we can further explore this relation,which under certain simplifying assumptions, allows to calculate the required (cid:104) P | G µν,c ˜ G cµν | (cid:105) matrix elements in terms of the mixing parameters [119,120, 170]. To show this, note that the divergence of the singlet axial current in the limit in which m u,d → − α s π G µν,c ˜ G cµν m u,d → = √ (cid:16) √ ∂ µ J µ − (2 / √ m s ¯ siγ s (cid:17) . (4.50)Fortunately, for m u,d → 0, the pseudoscalar strange quark current appearingabove can be connected to the divergence of the octet axial current which, insuch limit , reads ∂ µ J µ = − (2 / √ m s ¯ siγ s . As a consequence, the followingexpression has been obtained in the literature [119, 120, 170] − α s π G µν,c ˜ G cµν m u,d (cid:28) m s (cid:39) √ (cid:16) √ ∂ µ J µ + ∂ µ J µ (cid:17) (4.51)which holds up to light quark mass corrections or, equivalently, m π /m K effects [178]. The relation above allows to express the R J/ Ψ ratio in termsof the axial currents matrix elements as defined in Eqs. (4.8) and (4.10): R J/ψ (cid:39) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m η (cid:48) ( F sin θ + √ F cos θ ) m η ( F cos θ − √ F sin θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:32) m J/ψ − m η (cid:48) m J/ψ − m η (cid:33) (4.52)where (cid:39) stands for m u,d (cid:54) = 0 effects, which will be estimated below. To checkwhat is expected in different regimes of the theory as well as the accuracyof the approximation in Eq. (4.51), we take the LO results in (cid:96)N c χ PT. Forthe ˆ m → m π → 0) the equality in Eq. (4.51) holds exactly, andthe gluonic matrix elements read, at LO, √ m P ( F P − √ F P ) ˆ m → = (cid:104) P | ω | (cid:105) = √ F P M τ , (4.53) The singlet axial current reads J µ = (1 / √ (cid:0) ¯ uγ µ γ u + ¯ dγ µ γ d + ¯ sγ µ γ s (cid:1) ; for com-pletness, J µ = (1 / √ (cid:0) ¯ uγ µ γ u + ¯ dγ µ γ d − sγ µ γ s (cid:1) . η − η (cid:48) mixingwhereas the ratio itself reads, again at LO, (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) η (cid:48) | ω | (cid:105)(cid:104) η | ω | (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) cos θ P − sin θ P (cid:12)(cid:12)(cid:12)(cid:12) . (4.54)For the special case where m s effects are negligible as compared to thetopological ones, this is, m K (cid:28) M τ , the η and η (cid:48) would become purelyoctet and singlet, respectively, with masses and mixing angle at LO m η = 43 m K (cid:18) − (cid:15) (cid:19) , m η (cid:48) = M τ (cid:18) (cid:15) (cid:19) , θ P = − √ (cid:15), (4.55)with (cid:15) = m K /M τ . As a consequence, the η would not receive a singletadmixture and would not couple to the gluons, with the ratio in Eq. (4.54)diverging as | − / (2 √ (cid:15) − | .An opposite scenario would be that in which the large- N c limit repre-sents an excellent approximation, whereby M τ → F M τ = 6 τ /F ∼ / √ N c → 0, but m s > M τ (cid:28) m K ). In such a case, the η wouldbe become a massive η s meson, whereas the η (cid:48) would become a massless η q ,with masses and mixing angle at LO m η = 2 m K (1 + 16 ˜ (cid:15) ) , m η (cid:48) = 23 M τ (1 − 16 ˜ (cid:15) ) , θ P = π − θ ideal + 13 √ (cid:15), (4.56)with now ˜ (cid:15) = M τ /m K . In this case, both matrix elements would vanish as M τ → 0, but its ratio in Eq. (4.54) would be kept fixed at |−√ − (1 / (cid:15) ) | ,with 2 its limiting value. Consequently, as far as LO results are concerned,a result R J/ Ψ > M τ > m s .Finally, but still at LO, we discuss the accuracy of the approximationof neglecting the light quark masses in Eq. (4.51). From the LO results inRef. [183], for which M τ = 0 . 82 GeV and θ P = − . ◦ , we obtain for theleft hand side of Eq. (4.53) 0 . 60 and 1 . 58 for the η and η (cid:48) . For the righthand side, the results read 0 . 55 and 1 . 55, respectively. As a consequence,we obtain that the equality Eq. (4.51) holds at around 5% precision for thematrix elements, implying a 10% systematic uncertainty for the R J/ Ψ result.After this discussion, we proceed to our determination. From Eq. (4.52)and our mixing parameters determination from Sections 4.4.1 to 4.4.4, weobtain the results quoted in the last row from Table 4.1. We find a differenceof 1 . σ among our final result for the mixing parameters prediction (OptionIV) and experiment. Yet this is not large, it would be interesting to havea preciser theoretical and experimental prediction, as this process couldbe sensitive to non-standard phenomena such as gluonium admixtures or c ¯ c content in the η (cid:48) . However, to confirm such eventual discrepancy may requirea more detailed analysis, including the light-quark mass effects neglectedabove, that could be around a 10% effect and would involve additional Λ i OZI-violating parameters. In addition, it would be interesting to retain.5. Applications 93additional OZI-suppressed contributions to the hard process non consideredin Fig. 4.2 and which may be non-negligible in the light of ψ (2 S ) decays—see discussions in [192]. Finally, it has to be mentioned that previousanalysis did not include the RG effects which would appear in such process,necessary to render the amplitude scale-independent. A similar argumentto that above Eq. (4.30) would imply (1 − δ RG ( m J/ Ψ )) = (1 − . F terms, we obtain R J/ Ψ = 4 . As explained in Sec. 4.3.2, under the assumption that large- N c OZI-violatingeffects are negligible, the η and η (cid:48) Fock states may be described through theuse of a single angle in terms of the light and strange quarks wave func-tions Ψ q , Ψ s , common to the η and η (cid:48) . These define the meson distributionamplitudes φ qη = φ qη (cid:48) ≡ φ q and φ sη = φ sη (cid:48) ≡ φ s , Eq. (1.46), which are usedto calculate the η and η (cid:48) TFFs. Such distribution amplitudes can be usedto obtain the unphysical —i.e., non measurable— light- and strange-quarkTFF, F qγ ∗ γ ( Q ) and F sγ ∗ γ ( Q ), respectively, in terms of which the physical η and η (cid:48) TFF can be expressed as F ηγ ∗ γ ( Q ) = cos φF qγ ∗ γ ( Q ) − sin φF sγ ∗ γ ( Q ) , (4.57) F η (cid:48) γ ∗ γ ( Q ) = sin φF qγ ∗ γ ( Q ) + cos φF sγ ∗ γ ( Q ) . (4.58)The light- and strange-quark TFFs are related to the physical ones viarotation F qγ ∗ γ ( Q ) = cos φF ηγ ∗ γ ( Q ) + sin φF η (cid:48) γ ∗ γ ( Q ) , (4.59) F sγ ∗ γ ( Q ) = − sin φF ηγ ∗ γ ( Q ) + cos φF η (cid:48) γ ∗ γ ( Q ) . (4.60)Our mixing parameters extraction would allow to find such a decomposition,which represents an interesting theoretical result. In order to reconstructthem, we take our averaged result φ ≡ ( φ q + φ s ) / . . ◦ from the mix-ing angles obtained in Section 4.4.3 together with our fits from the TFFs inChapter 2. In addition, as a consequence of assuming a mild large- N c OZI vi-olating effects, which seems a reasonable estimation according to our results,it is theoretically expected that the π distribution amplitude φ π should bethe same as that from the light-quarks φ q . This is easy to understand as, inthis limit, the U (3) F symmetry would be recovered, guaranteeing then theequality of all distribution amplitudes —symmetry breaking effects shouldbe accounted though for the strange quark, which does not represent a prob-lem for the arguments above. Consequently, the resulting TFF should be, This should not be a bad approximation given our results in the previous section; wenote however that Λ (cid:54) = 0 implies that this is not a strict result but an approximate one. η − η (cid:48) mixing áááááá ááááá á á á á á á á áèèèèèèèèèèèèèèèèèèè èèèèèè è è è è è è è è è è è èè è è è è è è è è è è è è è èòòòòòò òòòòò ò ò ò ò ò ò ò ò ΠΗ q Η S Q @ GeV D Q F P ΓΓ * H Q L @ G e V D áááááá ááááá á á á á á á á áèèèèèèèèèèèèèèèèèèè èèèèèè è è è è è è è è è è è èè è è è è è è è è è è è è è èòòòòòò òòòòò ò ò ò ò ò ò ò ò ΠΗ q Η S Q @ GeV D Q F P ΓΓ * H Q L @ G e V D Figure 4.3: The light(strange)-quark TFF in orange(dotted-red) together with the π TFF (blue). The left plot shows the P NN -based description, whereas the right one rep-resents the P N one. The former TFFs have been multiplied by a charge factor 3 / / √ 2, respectively (see details in the text). When possible, the η and η (cid:48) TFF data pointshave been combined to extract what would be the light- and strange-quark TFF data asorange triangles and open-red squares, respectively. The data for the π appears as bluepoints. up to a charge factor 5/3, equivalent ( F qγ ∗ γ ( Q ) = (5 / F πγ ∗ γ ( Q )). Forthis reason, we plot in Fig. 4.3, the results for the π , η q and η s TFF ob-tained from Eqs. (4.59) and (4.60) and normalized to the π charge. Thisamounts to multiply the light- and strange-quark TFF by the charge factors3 / / √ 2, respectively. We find that actually the light-quark and the π TFFs match each other up to the Q ∼ scale, where the con-troversial Belle- B A B AR discrepancy manifests [76, 133]. Provided φ q (cid:39) φ π ,our approach supports Belle data against B A B AR and strongly calls for anew preciser measurement at Belle II. In addition, the results above show abehavior beyond the simplest VMD ( P approximant) approach and shouldwarn therefore against oversimplified descriptions. Finally, we give the re-sulting (dimensionful, i.e. m P = 1 in Eq. (2.1)) slope for these TFFs b η q = 1 . − = (0 . − , (4.61) b η s = 0 . +0 . − . ) GeV − = (1 . +0 . − . ) GeV) − , (4.62)which has been obtained from our values in Table 2.14. These could becompared with the results for the π , η , and η (cid:48) results from Chapter 2 , b π = 1 . − = (0 . − , (4.63) b η = 1 . − = (0 . − , (4.64) b η (cid:48) = 1 . − = (0 . − , (4.65)which shows again the expected similarity among the π and the light-quarkquantities. To obtain them, the results from Table 2.14 should be multiplied by m − P . .6. Conclusions and outlook 95 In this chapter, we have presented a new and alternative determination forthe η − η (cid:48) mixing parameters using information on the TFFs exclusively. Asan advantage, our formulation allows for a straightforward connection to thequantities arising in the (cid:96)N c χ PT Lagrangian —up to day, the only consistentframework to describe the η − η (cid:48) system— and avoids thereby the use ofmodels and approximations as those taken in studies using V → P γ and P → V γ processes or J/ Ψ decays. Moreover, besides implementing the fullNLO (cid:96)N c χ PT expressions including the relevant OZI-violating parameters,we have been able to provide a determination for them. Even if we findsmall values for them, their role is not negligible and plays a crucial rolein the TFFs asymptotic behavior —the role of the LEC K is by contrastnegligible. We remark that including them is necessary to achieve formallya consistent picture. This is a disadvantage from previous approaches, inwhich these parameters were kept finite for some quantities and vanishingin others. To illustrate their impact, we used a sequential approach inwhich the different OZI-violating effects and finally K were sequentiallyincluded one by one. Remarkably, we achieve a competitive prediction withrespect to existing approaches, that required a large amount of inputs intheir fits and usually obtained a large χ ν value, highlighting possible model-dependencies. This put us in a perfect position to test the mixing-schemein different observables.Possible venues to improve and extend our work would be a thoroughand detailed calculation of the RG-equation for the singlet axial current,including higher orders. In addition, it would be interesting to see if ongoingstudies of the η − η (cid:48) provide additional insights which may help in extractingthe mixing parameters [183, 184, 193]. Lattice studies such as [194] wouldhelp in this point as well —note however that they obtain the pseudoscalar,rather than the axial current matrix element. A final point of interest wouldbe the application of our results to the calculation of additional charmoniumand weak decays in lines of Refs. [189, 190] with a proper account of OZI-violating effects.6 Chapter 4. η − η (cid:48) mixing hapter Pseudoscalar to lepton pair decays Contents F P γ ∗ γ ∗ ( Q , Q ) . . . 1055.4 Final results . . . . . . . . . . . . . . . . . . . . 1125.5 Implications for χ PT . . . . . . . . . . . . . . . 1165.6 Implications for new physics contributions . . . 1205.7 Conclusions and outlook . . . . . . . . . . . . . 127 The psedusocalar decays into lepton pairs, P → ¯ (cid:96)(cid:96) , are a beautiful place tokeep track of the evolution of our understanding of QCD, which is behindthe mechanism driving these processes. Its pioneering study was initiatedby Drell [195] back in 1959, well before the time where the pseudoscalardecays into photons were properly understood on basis of the Adler [42]-Bell-Jackiw [43] (ABJ) anomaly. Still, he was able to set a lower bound forthe π → e + e − decay. Further studies (some of them rather qualitative) ap-peared in the 60’s with the advent of VMD ideas [196–200] which were justbeing develeoped at that time. Later on, the development of perturbativeQCD stimulated different approaches in the 80’s. Among them, quark loopmodels based on duality ideas [112, 201–205] and phenomenological mod-els based on the novel understanding of exclusive reactions in pQCD [206]—which were improved through the use of data [207–209]. More recently,the development of χ PT, the low-energy effective field theory of QCD, pro-vided an alternative approach to study these decays [210, 211], which inaddition may be complemented with large- N c and resonant ideas [99, 212].978 Chapter 5. Pseudoscalar to lepton pair decays P ( q ) ℓ ( p, s ) ℓ ( p ′ , s ′ ) kq − k p − k Figure 5.1: The leading order contribution to P → ¯ (cid:96)(cid:96) processes. The shadowed blobstands for the QCD dynamics in the P → γ ∗ γ ∗ transition encoded in F Pγ ∗ γ ∗ ( k , ( q − k ) ). The motivation for this continuous study has been undoubtedly bound tothe different experimental anomalies appearing in these processes along theyears, stimulating a continuous revision and speculation about new-physicseffects [213–217].In this chapter, we apply all the machinery developed for reconstructingthe TFFs and benefit from our novel ideas, gaining on precision and obtain-ing, for the first time, a reliable systematic error estimation, taking specialcare of the η and η (cid:48) cases. In this way, we want to update the status of thesedecays to the standards of precision met nowadays —required for testing thelow-energy frontier of the SM [218, 219]. The calculation details of these pro-cesses together with their relevant features are outlined in Section 5.2. Thesystematic error assessment is described in Section 5.3, including a carefuldescription of some particular features —previously overlooked— present forthe η and η (cid:48) but not for the π . Our results, discussed in Section 5.4, showinteresting features when compared to χ PT as we describe in Section 5.5.Finally, we discuss new physics implications in Section 5.6. The leading order QED contribution to P → ¯ (cid:96)(cid:96) decays is mediated throughan intermediate two-photon state as sketched in Fig. 5.1. The gray blobappearing there stands for the hadronic effects encoded in the P → γ ∗ γ ∗ transition. For real photons, such process is theoretically well known interms of the ABJ anomaly, and can be obtained as well in the odd-paritysector of χ PT, see Section 1.6.2. For deeply virtual photons, the lim-its lim Q →∞ F P γ ∗ γ ( Q ) [102] and lim Q →∞ F P γ ∗ γ ∗ ( Q , Q ) [191] (see Sec-tion 1.6.1) are known as well. However, the interpolation in between thesetwo regimes is a theoretically unknown territory, what has been amendedthrough wise and different modeling procedures, explaining the large amountof studies on these processes. Parametrizing such interaction in terms of the An additional but subleading tree-level Z boson electroweak contribution exists too,cf. Section 5.6. .2. The process: basic properties and concepts 99most general TFF, F P γ ∗ γ ∗ ( q , q ), we obtain for the matrix element i M = (cid:90) d k (2 π ) ( − ie F P γ ∗ γ ∗ ( k , ( k − q ) )) (cid:15) µνρσ k µ ( q − k ) ρ − ig νν (cid:48) k − ig σσ (cid:48) ( q − k ) × u p,s ( − ieγ ν (cid:48) ) i ( /p − /k ) + m (cid:96) ( p − k ) − m (cid:96) ( − ieγ σ (cid:48) ) v p (cid:48) s (cid:48) = (cid:90) d k (2 π ) e (cid:15) µνρσ k µ q ρ [ u p,s γ ν (( /p − /k ) + m (cid:96) ) γ σ v p (cid:48) s (cid:48) ] k ( q − k ) (( p − k ) − m (cid:96) ) F P γ ∗ γ ∗ ( k , ( k − q ) ) , (5.1)where k is the momentum running through the loop and must be inte-grated over all energies. The definitions for the different elements follow theconventions in [1] and can be found in Appendix A. At this stage of thecalculation, it is convenient to evaluate the spinor contractions. This can bedone using the pseudoscalar projector defined in Eq. (A16) from Ref. [220].We recall it here adapted to our conventions —which amounts to shift theantisymmetric tensor sign with respect to [220]— for completeness, v p (cid:48) ,s (cid:48) u p,s | out , P = 12 (cid:112) q (cid:104) − m (cid:96) /qγ − i(cid:15) αβγδ γ α γ β p γ p (cid:48) δ + q γ (cid:105) . (5.2)The subindex out means that such equality holds for the final state particles,while subindex P means that it is in a pseudoscalar state. Using standardtrace techniques together with Eq. (5.2), we find that the spinorial part insquare brackets from Eq. (5.1) yields − i (2 √ m (cid:96) /m P ) (cid:15) νσαβ q α k β . Insertingback into Eq. (5.1) and using (cid:15) νσµρ (cid:15) νσαβ = − δ µα δ ρβ − δ µβ δ ρα ), we obtain thefinal result i M = 2 √ m (cid:96) m P α F P γγ iπ q (cid:90) d k [ k q − ( k · q ) ] ˜ F P γ ∗ γ ∗ ( k , ( k − q ) ) k ( q − k ) [( p − k ) − m (cid:96) ] , (5.3)where the √ m P term can be traced back to the effective pseudoscalar uγ v interaction and m (cid:96) to the helicity flip. F P γγ ≡ F P γγ (0 , 0) and so˜ F P γ ∗ γ ∗ ( k , ( k − q ) ) is the normalized TFF, ˜ F P γ ∗ γ ∗ (0 , 0) = 1. The decaywidth reads then (see Appendix A.3)Γ( P → ¯ (cid:96)(cid:96) ) = 116 πm P β (cid:96) |M| , (5.4)with β (cid:96) = (cid:113) − m (cid:96) /m P the lepton velocity. It is customary in the lit-erature to express Eq. (5.4) in terms of the Γ( P → γγ ) result, so the Here it may worth to stress that in our convention (cid:15) = +1. To see this, note that, from Eq. (5.2), Tr( uγ v ) = √ m P . In addition, this allows toeffectively express i M = − i (¯ uiγ v )2 m (cid:96) α F Pγγ A ( q ), with A ( q ) defined in Eq. (5.6) The two photon decay-width reads Γ( P → γγ ) = e m P π | F Pγγ | . 00 Chapter 5. Pseudoscalar to lepton pair decaysnormalization for the TFF dependency disappears, which is the reason thatit was factored out in Eq. (5.3). In such a way, the final result readsBR( P → ¯ (cid:96)(cid:96) )BR( P → γγ ) = 2 (cid:18) αm (cid:96) πm P (cid:19) β (cid:96) |A ( q ) | . (5.5)The prefactor in Eq. (5.5) already predicts tiny BRs for these processes,which are known as rare decays. This is due to the electromagnetic α andthe helicity flip suppression m (cid:96) /m P factors with respect to the P → γγ decay. The last parameter, A ( q ), is related to the loop amplitude andencode the QCD dynamics encapsulated in the TFF, A ( q ) = 2 iπ q (cid:90) d k (cid:0) k q − ( k · q ) (cid:1) ˜ F P γ ∗ γ ∗ ( k , ( q − k ) ) k ( q − k ) (cid:0) ( p − k ) − m (cid:96) (cid:1) . (5.6)The formulae in Eqs. (5.5) and (5.6) represent the main standard resultsnecessary to calculate the BRs. At this point, it may seem hopeless to sayanything about Eq. (5.6) without any information on the TFF, which isactually required to render the —otherwise divergent— loop integral finite.However, it is still possible to derive some important general results. Amongthem, the unitary bound obtained by Drell [195], the result for a constantTFF (of relevance for χ PT) and the relevant regimes in which a preciseTFF determination is required. The latter is an essential prerequisite forany proper discussion on systematic errors and how to reconstruct the TFF. To derive the imaginary part associated to these processes, we use the Cut-cosky rules, relating the imaginary part of the diagram to its discontinu-ities [1]. The latter are computed replacing the propagators which can beput on-shell as p − m + i(cid:15) → − πiδ ( p − m ) θ ( p ). For the π —being thelightest hadronic particle— the only possible intermediate state appearingin the loop is the two photon one. Following Cutcosky and replacing thephoton propagators in Eq. (5.6), one obtains Im A γγ = ( − πi ) π q (cid:90) d k ( q k − ( q · k ) ) ˜ F P γγ ( k , ( q − k ) )(( p − k ) − m ) δ ( k ) δ (( q − k ) ) , = − m P (cid:90) d Ω dk m P ( k ) ˜ F P γγ (0 , m P k (1 − β (cid:96) cos θ )) 12 m P δ ( k − m P , = π (cid:90) d Ω β (cid:96) cos θ − π β (cid:96) ln (cid:18) − β (cid:96) β (cid:96) (cid:19) . (5.7) The prefactor in Eq. (5.5) is O (10 − ) for the π → e + e − , O (10 − (10 − )) for the η → e + e − ( µ + µ − ) and O (10 − (10 − )) for the η (cid:48) → e + e − ( µ + µ − ). We use polar coordinates dk = d Ω dk k d k and specialize to the pseudoscalar restframe, where (cid:126)q = ( m P ,(cid:126) 0) and (cid:126)p = m P / , (cid:126)β (cid:96) ). To perform integration over d k we use δ ( k − m i ) = δ (( k ) − k − m i ). .2. The process: basic properties and concepts 101Remarkably, this observation allowed Drell [195] to put already a lowerbound in 1959, which is known as the unitary bound, |A ( m π ) | ≥ (Im A γγ ( m π )) = (cid:18) π β (cid:96) ln (cid:18) − β (cid:96) β (cid:96) (cid:19)(cid:19) = ( − . . (5.8)Quite often, this bound has been extended to the heavier η, η (cid:48) and K L pseu-doscalar states. This generalization is however incorrect, as all of theseparticles will have intermediate π + π − γ states in addition, cf. Fig. 5.3. Thisis specially important for the η (cid:48) , where such π + π − state becomes resonantat the ρ peak, besides the additional ω resonance. This feature is carefullyillustrated for the η and η (cid:48) in Section 5.3.3 in order to assess the systematicerror. We find small corrections for the η , but large deviations for the η (cid:48) . Asa further illustration, we derive in Appendix B.2 the additional contributionsto the imaginary part that a narrow-width vector meson would produce.Repeatedly, this result has been used in the literature for estimating thewhole amplitude using Cauchy’s integral formula, which is often referred toas a dispersion relation. This consists in reconstructing the original functionEq. (5.6) from its γγ discontinuity above q = 0. As the imaginary part,Eq. (5.7), does not fall rapidly enough at infinity —which is related tothe divergent character of Eq. (5.6) for a constant TFF— a subtraction isrequired, so the final result reads [201, 207, 221]Re A ( q ) = A (0) + q π (cid:90) ∞ ds Im A γγ ( s ) s ( s − q ) . (5.9)Still, the value for A (0) must be calculated from Eq. (5.6), which representsthough a simpler calculation. The result from the dispersive integral leadsexactly to the terms in brackets in Eq. (5.12). We note here that such cal-culations are approximate. For a general pseudoscalar mass the additionalcontributions to the imaginary part coming from the TFF must be specified—actually these would allow to write an unsubtracted dispersion relation,cf. Appendix B.2. Consequently, such calculations are approximate as theywould neglect all kinds of m P,(cid:96) / Λ corrections, where Λ is some TFF charac-teristic scale [201, 209, 221]. Before continuing, it will be useful in view of the next discussion and Sec-tion 5.5, to estimate the result which is obtained when taking a constant(WZW) TFF. Obviously, the result will include some divergent term —to becancelled once the TFF is switched on— which needs regularization. Taking˜ F P γ ∗ γ ∗ ( k , ( k − q ) ) = 1, the loop integral Eq. (5.6) can be expressed using02 Chapter 5. Pseudoscalar to lepton pair decaysdimensional regularization in terms of known scalar integrals A WZW ( q ) = 2 q µ q ν q C µν ( q , m (cid:96) , m (cid:96) ; 0 , , m (cid:96) ) − B ( m (cid:96) ; 0 , m (cid:96) )= 12 (cid:0) q C ( q , m (cid:96) , m (cid:96) ; 0 , , m (cid:96) ) − B ( m (cid:96) ; 0 , m (cid:96) ) + 1 (cid:1) . (5.10)Note here that if we were to use some cut-off in our integrals for the regular-ization procedure —which is particularly useful for deriving the approximateformula— the peculiarities of dimensional regularization must be accountedfor carefully. As an example, from the first line in Eq. (5.10), the divergentpart arises from2 4 d Div[ C ] − B ] = 2 d ∆ (cid:15) − (cid:15) = − 32 ∆ (cid:15) + 14 , (5.11)where we have used d = 4 − (cid:15) and ∆ (cid:15) = (cid:15) − γ E + ln 4 π . The additionalfinite extra-term which is found should be subtracted from Eq. (5.10) if notusing dimensional regularization. Performing the calculation for the scalarfunctions C and B , we find, in dimensional regularization, A WZW ( q ) = iπ β (cid:96) L + 1 β (cid:96) (cid:20) L + π 12 + Li (cid:18) β (cid:96) − 11 + β (cid:96) (cid:19)(cid:21) − 52 + 32 ln (cid:18) m (cid:96) µ (cid:19) , (5.12)where L = ln (cid:16) − β (cid:96) β (cid:96) (cid:17) , β (cid:96) = (cid:113) − m (cid:96) q is the lepton velocity and Li ( x ) isthe dilogarithm function . If we were using a cut-off regularization µ → ∞ ,from Eq. (5.10), and accounting for the last piece in Eq. (5.11) , we wouldfind similar results but replacing the last terms in Eq. (5.12) by − − ln(1+ µ m (cid:96) ). Before providing any input for the TFF, it is very convenient to analyzethe loop-integral. This allows to identify the relevant scales involved in theproblem, which is extremely important in order to achieve the most appro-priate TFF description. For this task, it is very convenient to carry out anapproximate calculation in terms of m P,(cid:96) / Λ , where Λ is some characteristicscale encoded in the form factor. Following [212], we take A ( q ) = A WZW ( q ) + 2 iπ q (cid:90) d k (cid:0) k q − ( k · q ) (cid:1) ( F P γ ∗ γ ∗ ( k , ( q − k ) ) − k ( q − k ) (cid:0) ( p − k ) − m (cid:96) (cid:1) , (5.13)where we have added and subtracted a constant term —precisely, that inEq. (5.12). The remaining integral is essentially zero at scales k ∼ m P , m (cid:96) The dilogarithm or Spence’s function is defined as Li ( x ) = − (cid:82) x dt ln(1 − t ) t . .2. The process: basic properties and concepts 103below Λ , as the TFF remains constant. Above, all the terms O ( p , q , m (cid:96) )can be neglected. At such scales, the leading term from the tensor k µ k ν q µ q ν part is given by k µ k ν ∼ (1 /d ) k g µν , as additional terms are m P / Λ sup-pressed. We are left then with A ( q ) (cid:39) A WZW ( q ) + 2 iπ (1 − d ) (cid:90) d k F P γ ∗ γ ∗ ( k , k ) − k ) , = A WZW ( q ) − (cid:90) µ dQ F P γ ∗ γ ∗ ( Q , Q ) − Q . (5.14)The first line corresponds, essentially, to the result Eq. (12) in [212], whereasin the second one, we have Wick-rotated and introduced a cut-off regulariza-tion. The obtained integral is still divergent for µ → ∞ , which is expectedas it must cancel the divergency in A WZW ( q ), see Eq. (5.12). In order toremove it, we identify the origin of the UV divergent term in Eq. (5.12),subtract there, and plug into Eq. (5.14), obtaining A app ( q ) = iπ β (cid:96) L + 1 β (cid:96) (cid:20) L + π 12 + Li (cid:18) β (cid:96) − 11 + β (cid:96) (cid:19)(cid:21) − (cid:90) ∞ dQ Q (cid:18) m (cid:96) Q + m (cid:96) − F P γ ∗ γ ∗ ( Q , Q ) (cid:19) . (5.15)This kind of approximation, obtained in many different ways, has beenwidely used in the literature, see explicitly in Refs. [207, 212] and implicitin most of the quoted references. Exceptions are the full calculation inRef. [222], and those including partial corrections in Refs. [206, 208, 209].While these are relevant to the precision we are aiming, specially for the η and η (cid:48) cases, the approximation in Eq. (5.15) is enough, at least for the π ,to understand the relevant dynamics in this process. To illustrate this, weplot in Fig. 5.2 the integrand of Eq. (5.15), K ( Q ), for the electron case.As one can see, it involves the space-like symmetric ( Q = Q ) kinematics.In addition, the integrand is peaked at very low-energies close to the leptonmass, where the TFF essentially remains constant. The TFF effects becomevisible and specially relevant in the (0 . − . 4) GeV region, where the slopeparameter is roughly enough to describe the TFF; the effects from additionalparameters appear roughly above this region —where the two black lines inFig. 5.2 separate— and represent a minor contribution to the integral. Thehigh-energy tail plays though a non-negligible role too. Given the sensitivityto the double virtual regime, this challenging process would represent thefirst experimental probe to the TFF double-vitrtual kinematics. From thefeatures enumerated above, any serious approach developed to deal with thisprocess should implement: That amounts to remove the − ln(1 + µ /m (cid:96) ) term from A WZW ( q ). Note that weare using a cut-off regularization, so the comments below Eq. (5.12) apply. 04 Chapter 5. Pseudoscalar to lepton pair decays m e ¬- - - - - - - @ GeV D K H Q L @ G e V - D % % % % % Contribution - ‰ - - - - Q @ GeV D K H Q L @ G e V - D > % % % % % % Contribution Figure 5.2: