[1.7cm] Dynamical generation of hadronic resonances in effective models with derivative interactions
[2.3cm] Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften
[0.5cm] vorgelegt beim Fachbereich Physik der Johann Wolfgang Goethe-Universität in Frankfurt am Main
aus Oppeln (Polen)
[0.5cm] Frankfurt am Main (2016)
[0.5cm] (D 30)
vom Fachbereich Physik der
Johann Wolfgang Goethe-Universität als Dissertation angenommen
Prof. Dr. René Reifarth
PD Dr. Francesco Giacosa
Prof. Dr. Dirk H. Rischke
Datum der Disputation: 10.08.2016
“When I was young, I observed that nine out of ten things I did were failures.
So I did ten times more work.”
– George Bernard Shaw
Dies ist, um mein Versprechen einzuhalten…
Light scalar mesons can be understood as dynamically generated resonances. They arise as ’companion poles’ in the propagators of quark-antiquark seed states when accounting for hadronic loop contributions to the self-energies of the latter. Such a mechanism may explain the overpopulation in the scalar sector – there exist more resonances with total spin than can be described within a quark model.
Along this line, we study an effective Lagrangian approach where the isovector state
couples via both non-derivative and derivative interactions to pseudoscalar mesons. It is demonstrated that the propagator has two poles:
a companion pole corresponding to and a pole of the seed state . The positions of these
poles are in quantitative agreement with experimental data. Besides that, we investigate similar models for the isodoublet state by performing a fit to
phase shift data in the channel. We show that, in order to fit the data accurately, a companion
pole for the , that is, the light , is required. A large- study confirms that both resonances below GeV are predominantly four-quark states, while the heavy states are quarkonia.
This thesis is based on the following publications:
T. Wolkanowski, M. Sołtysiak, and F. Giacosa, as a companion pole of , Nucl. Phys. B 909, 418 (2016) arXiv:1512.01071 [hep-ph]
T. Wolkanowski and F. Giacosa, as a companion pole of , PoS CD15, 131 (2016) arXiv:1510.05148 [hep-ph]
T. Wolkanowski, F. Giacosa, and D. H. Rischke, revisited, Phys. Rev. D 93, 014002 (2016) arXiv:1508.00372 [hep-ph]
T. Wolkanowski, Dynamical generation of hadronic resonances, Acta Phys. Polon. B Proceed. Suppl. 8, 273 (2015) arXiv:1410.7022 [hep-ph]
J. Schneitzer, T. Wolkanowski, and F. Giacosa, The role of the next-to-leading order triangle-shaped diagram in two-body hadronic decays, Nucl. Phys. B 888, 287 (2014) arXiv:1407.7414 [hep-ph]
T. Wolkanowski and F. Giacosa, The scalar-isovector sector in the extended Linear Sigma Model, Acta Phys. Polon. B Proceed. Suppl. 7, 469 (2014)
Die QCD ist die Theorie der starken Wechselwirkung. Sie beschreibt im Allgemeinen die Kraft zwischen farbgeladenen Quarks als einen Austausch von ebenfalls farbigen Gluonen. Wegen des Phänomens des Confinements kommen keine farbgeladenen Teilchen isoliert in der Natur vor, so dass Quarks und Gluonen in Form von Hadronen gebunden vorliegen müssen. Im Speziellen müsste man aus der QCD diese gebundenen Zustände als Lösungen erhalten, es ist allerdings bis heute nicht möglich, die Theorie ohne Gebrauch von Näherungsmethoden vollständig zu lösen. Erschwerend kommt hinzu, dass die meisten Hadronen instabil sind und somit relativ schnell zerfallen. In der Regel können sie anhand ihrer Zerfallscharakteristika identifiziert und wahlweise direkt oder indirekt vermessen werden; Quarkmodelle waren imstande, viele der bekannten Hadronen qualitativ wie quantitativ zu erklären. Weitere Experimente haben in der Vergangenheit aber gezeigt, dass es mehr Teilchen mit gleichen Quantenzahlen zu geben scheint, als mit einfachen Quarkmodellen konstruierbar sind. Insbesondere im skalaren Sektor () spricht man hierbei von Überbevölkerung.
Ein Beispiel mag dies verdeutlichen: Es ist weitläufig akzeptiert, dass es zwei Mesonen mit Isospin gibt, das schwere und das leichte . Da beide jeweils ein Isotriplet bilden, existieren je drei Zustände mit unterschiedlicher elektrischer Ladung. Sofern das positive ein Quark-Antiquark-Paar darstellt, weist das Quarkmodell ihm die Zusammensetzung zu. Damit sind die Möglichkeiten ausgeschöpft, die geforderten Quantenzahlen korrekt wiederzugeben, es gibt also keine Freiheit mehr. Unklar bleibt nun aber, in welchem der beiden Isotriplets der obige Zustand vorzufinden ist. Und selbst wenn man diese Frage beantworten könnte, verblieben drei geladene Teilchen einer Sorte ohne Erklärung.
In den letzten Jahren hat sich zunehmend gezeigt, dass die leichten Skalare, also auch das , durch hadronische Schleifen-Beiträge erzeugt werden könnten. Letztere sind quantenfeldtheoretische Korrekturen, die in effektiven Modellen der QCD berücksichtigt werden können. Während die schweren Skalare als Quark-Antiquark-Paare angenommen werden, werden die leichten Partner nicht explizit berücksichtigt, sondern stattdessen als Mischzustände gedeutet, die durch die Schleifen-Beiträge generiert werden. Wie kann man das verstehen?
Strenggenommen setzt sich der Zustandsvektor eines Mesons aus mehreren Beiträgen zusammen, weil das Teilchen die Möglichkeit hat zu zerfallen. Wenn es zum Beispiel in zwei andere Mesonen zerfallen kann, dann beinhaltet der Zustandsvektor neben dem -Anteil unter anderem Vier-Quark- bzw. Zwei-Meson-Beiträge. Bei Vektormesonen sind die erstgenannten dominant und bestimmen hauptsächlich ihre Eigenschaften; die anderen Beiträge entstammen aus den Schleifen und verschieben den jeweiligen Propagatorpol nur geringfügig weg von der reellen Achse. Im Gegensatz dazu glaubt man, dass bei Skalaren der Sachverhalt anders ist. Zum einen sind die zusätzlichen Anteile oft relativ größer als im Fall der Vektoren und ihr Einfluss auf Masse und Zerfallsbreite (also auf den Resonanzpol) ist somit nicht zu vernachlässigen. Daneben ist die Idee aber, dass sie weitere Pole auf die komplexe Ebene führen, die (manchmal) als neue Teilchen identifiziert werden können. Es ist genau dieses Bild, mit dem man versucht, die Überbevölkerung im skalaren Sektor zu erklären.
Die Idee einer solchen dynamischen Erzeugung von skalaren Resonanzen ist in der Literatur auf unterschiedliche Weise im Rahmen von effektiven Modellen verfolgt worden. In der vorliegenden Arbeit haben wir uns diesen Bemühungen angeschlossen. Hierzu wurden durch das sogenannte “erweiterte Lineare Sigma Modell” (eLSM) [1, 2, 3] inspirierte effektive Theorien mit derivativen Kopplungen eingesetzt, um den eingangs erwähnten Isovektor und das Isodoublet (auch genannt) zu beschreiben.
Zunächst wurde in Kapitel \thechapter der Formalismus zur Berechnung von Schleifen-Beiträgen insbesondere mit derivativen Kopplungen erarbeitet. Dabei konnte gezeigt werden, dass der Zugang über Dispersionsrelationen nicht identisch ist mit den üblichen Feynman-Regeln. Während nämlich im zweiten Fall die Quantisierung eines Wechselwirkungsterms mit Ableitungen vor den Zerfallsprodukten zu Kaulquappen-Diagrammen führt, die nach korrekter Behandlung der Ableitungen im weiteren Verlauf sich gegen gleiche Beiträge mit umgekehrtem Vorzeichen wegheben, verbleiben im ersten Fall diese überzähligen Terme.
Ein gravierender Effekt tritt auf, wenn zusätzlich eine Ableitung vor dem zerfallenden Teilchen vorhanden ist. Auch hier werden Kaulquappen-Diagramme erzeugt, die jetzt außerdem energieabhängig sind; bei korrekter Behandlung werden sie ähnlich wie gerade beschrieben unwirksam gemacht. Darüber hinaus wird aber die Normierung der Spektralfunktion zerstört und muss durch eine Renormierung der Felder wiederhergestellt werden.
In Kapitel \thechapter widmeten wir uns dann der Erweiterung einiger älterer Arbeiten von Törnqvist und Roos [4, 5], sowie Boglione und Pennington  zum Thema der dynamischen Erzeugung im skalaren Sektor. Nach erfolgreicher Reproduktion der dortigen Ergebnisse erweiterten wir das zugrundeliegende Modell und brachten Licht in einige damals getätigte Aussagen. Boglione und Pennington argumentierten beispielsweise, sie hätten das schwere durch eine Analyse der Breit–Wigner-Massen gefunden. Tatsächlich aber haben wir in ihrem Modell keinen entsprechenden Pol finden können, sondern lediglich einen mit zu hoher Masse. Obwohl auch insgesamt die Polstruktur in quantitativer Hinsicht nur schlecht den experimentellen Befunden entsprach, erzeugte das Modell tatsächlich zusätzliche Pole. Dies deuteten wir so, dass ein verbessertes Modell vielleicht imstande wäre, auch quantitativ zu überzeugen.
Deshalb untersuchten wir anschließend eine eigene effektive Theorie daraufhin, ob zwei Isotriplets gleichzeitig beschreibbar sind. Wir forderten, dass die beiden Resonanzen als Pole im Propagator existieren und die experimentellen Verzweigungsverhältnisse von richtig wiedergegeben wurden. Dadurch konnte der Parameterbereich unserer freien Modellparameter hinreichend gut eingeschränkt werden; aus dem relevanten Fenster ließen sich Werte entnehmen, die zum gewünschten Ergebnis führten. Genauere Resultate sind nicht möglich, da das Modell mehr Parameter besitzt, als Gleichungen seitens des Experiments zu lösen wären bzw. es keine adäquaten Daten für einen besseren Fit gibt (z.B. keine Daten zu Phasenverschiebungen). Bemerkenswert ist allerdings, dass unsere Abschätzung der relativen Kopplungsstärken von zu seinen Zerfallskanälen darauf hindeutet, dass der Kaon-Kaon-Kanal dominant ist. Außerdem machten wir Vorhersagen für die Phasenverschiebungen und Inelastizität im Sektor mit Isospin . Beide Untersuchungen bestätigten andere frühere Arbeiten.
Zuletzt führten wir eine Untersuchung unseres Modells für eine große Anzahl an QCD-Farben durch: Wenn der Zustand durch die hadronischen Wechselwirkungen zustande kommt, dann muss er verschwinden, sobald die Kopplungsstärke zu diesen Kanälen klein genug wird. Im Grenzfall einer großen Anzahl an QCD-Farben konnten wir genau das beobachten. Der Pol für näherte sich für kleiner werdende Kopplungen der reellen Achse an, verschwand aber für einen kritischen Wert. Der Pol für dagegen verschwand nicht, sondern wurde für kleiner werdende Kopplungen zum Pol eines stabilen Teilchens. All das bestätigte unsere Annahme, dass die Resonanz unter GeV eine Form von Vier-Quark-Zustand ist, während wir für den schweren Partner genau das Gegenteil fanden.
In Kapitel \thechapter verfolgten wir die gleiche Idee für den Sektor mit Isospin , wendeten unser effektives Modell aber auf andere Weise an. Anstatt einen Satz von Parametern zu suchen, der zwei Pole für die beiden benötigten Zustände und generiert, führten wir einen Fit der experimentell ermittelten -Phasenverschiebung durch . Dabei wurden vier verschiedene Varianten unseres Modells benutzt:
Nicht-derivative und derivative Kopplungen: Dieser Fall entsprach der Situation wie bei Isospin und lieferte die beste Beschreibung der Daten. Wir konnten hieraus zwei Pole extrahieren, deren Position sehr gut mit den vorhandenen Ergebnissen aus dem PDG  übereinstimmt – unsere Fehler sind aber deutlich kleiner. Die Untersuchung in einer großen Anzahl an QCD-Farben zeigte das gleiche Bild wie für , nämlich dass die leichte Resonanz ein dynamisch generiertes Vier-Quark-Objekt ist, während einen gewöhnlichen Quark-Antiquark-Zustand darstellt.
Nur nicht-derivative Kopplungen: Es zeigte sich, dass diese Version des Modells nicht imstande ist, die Daten wiederzugeben. Hinzu kommt allerdings, dass es nicht möglich war, überhaupt einen Pol für das zu generieren. Wir schlossen daraus, dass zumindest für unser Modell die Ableitungsterme im Allgemeinen sehr wichtig für die akkurate Beschreibung der experimentellen Befunde, speziell für die Anwesenheit des leichten skalaren Kaons aber essentiell notwenig sind. Ähnliche Aussagen lassen sich auch für den Fall des Isotriplets machen.
Nur derivative Kopplungen: Hierbei war der Fit zwar deutlich besser als vorher, wurde aber nach statistischer Auswertung als nicht adäquat verworfen. Das Weglassen nicht-derivativer Kopplungen hatte im Vergleich zur Version des Modells unter nur geringen Einfluss auf die Position des Pols von . Ein dynamisch generierter zusätzlicher Pol für wurde aber in der komplexen Ebene weiter nach rechts und näher an die reelle Achse geführt, was eine zu geringe Zerfallsbreite lieferte als erwartet. Insgesamt ist dadurch klar geworden, dass beide Arten von Kopplungen notwendig sind.
Wie unter , nun aber mit abgewandeltem Formfaktor: Auch dieser Fit stellte sich als nicht akzeptabel heraus. Des Weiteren lieferte der dynamisch generierte Pol für eine verhältnismäßig große Masse. Das Verhalten des Pols war an sich auch deutlich anders als in den Fällen zuvor: Der Pol startete im Grenzfall einer großen Anzahl an QCD-Farben tief in der komplexen Ebene und es konnte kein kritischer Wert der Kopplungskonstanten für sein Erscheinen bestimmt werden. Da der Formfaktor eine Ausprägung der Modellabhängigkeit unseres Ansatzes darstellt, konnten wir mit unserer Studie die These stärken, dass der Gauß’sche Formfaktor in der Tat eine hervorragende Wahl ist.
Im Anschluss änderten wir unser Modell so ab, dass es nur Terme mit zusätzlich derivativen Kopplungen vor den zerfallenden Teilchen enthielt. Die Qualität des Fits war vergleichbar mit dem unter Bemerkenswert ist der Umstand, dass die beiden darin existierenden Pole ziemlich genau die gleiche Position haben wie unter Dies lässt den Schluss zu, dass im Rahmen unseres Ansatzes ein guter Fit nur dann möglich ist, wenn ein akzeptabler Pol für das erzeugt wird. Wie dem auch sei, als problematisch empfinden wir, dass – entweder durch die Präsenz der spezifischen Wechselwirkung oder wegen eines numerischen Problems – eine Kopplungskonstante deutlich ausgeprägtere Fehler erhält, als in allen anderen Fällen zuvor.
Das gleiche Phänomen beobachteten wir beim letzten untersuchten Modell, dem eLSM mit freien Parametern. Der Fit lieferte ein Ergebnis erneut vergleichbar mit dem Fall unter Auch die Lage der Pole zeigte sich sehr ähnlich, jedoch kam es bei mindestens zwei Kopplungskonstanten zu deutlich größeren Fehlern. Viel auffälliger war aber der Umstand, dass der Parameter , welcher die nackte Quark-Antiquark-Paar Masse widerspiegelt, stark herabgesetzt wurde. Das ist deswegen sonderbar, weil in vergleichbaren Modellen die Hinzunahme eines Strange-Quarks (wie für den Sektor mit gegeben) diesen Wert normalerweise erhöht.
Die Hauptaussage der vorliegenden Arbeit ist, dass einige der leichten skalaren Mesonen im Rahmen von spezifischen hadronischen Modellen tatächlich als dynamisch generierte Resonanzen auftreten können – was eine mögliche Lösung für das eingangs beschriebene Problem der Überbevölkerung wäre. Es konnte sogar gezeigt werden, dass ein dem eLSM äquivalentes Modell dieses Phänomen enthalten kann. Der wesentliche nächste Schritt wäre deshalb, zunächst unseren Mechanismus an den Isoskalaren mit zu testen, um letztlich die elf freien Parameter des eLSM durch einen simultanen Fit einer größeren Auswahl von experimentellen Daten zu fixieren. Ein positives Ergebnis würde eine Antwort auf die Frage liefern, ob das eLSM seine Erfolgsgeschichte weiter schreiben kann. Die nötige Vorarbeit, um das zu überprüfen, wurde hier geleistet.
- \thechapter Introduction
- \thechapter Resonances
- \thechapter Derivative interactions and dispersion relations
- \thechapter Dynamical generation: The
- \thechapter Dynamical generation: The
- \thechapter Summary and conclusions
- \thechapter Mathematical formulas
- \thechapter Conventions
- \thechapter Kinematics of two-body decays
- \thechapter Multi-valued complex functions and Riemann sheets
- A Introductory example
- B Riemann sheets
- C Analytic continuation
List of Figures:
- 1 Tree-level vertices of QCD: The solid…
- 2 Left panel: LO diagram of the…
- 3 Pictorial representation of an unstable meson’s…
- 4 Visualization of the simplest possible way…
- 5 Born expansion of the RSE’s transition…
- 6 Spectral functions (left panels) and position…
- 7 In the left panel we show…
- 8 Inelasticity parameter and phase shifts…
- 9 Pole structure of our effective model…
- 10 The solid (red) curve shows our…
- 11 In the left panel we show…
- 12 The solid (red) curve shows our…
- 13 In the left panel we show…
- 14 The left panel shows the case…
- 15 The first row is for the…
- 16 The solid (red) curve shows our…
- 17 In the left panel we show…
- 18 The solid (red) curve shows our…
- 19 In the left panel we show…
- 20 The solid (red) curve shows our…
- 21 In the left panel we show…
- 22 Schematic decay process .
- 23 Multi-valued character of with paths (dashed…
- 24 Riemann surface of the complex root…
- 25 Analytic continuation of the complex root…
List of Tables:
Chapter \thechapter Introduction
1 Historical remarks
The beginning of the th century was a fascinating time of confusion. Physicists all around the world became puzzled by some unexpected experimental observations and new ideas concerning the microscopic structure of nature (later on incorporated in the theory
Besides philosophical and fundamental challenges after finding the appropriate mathematical formalism, (non-relativistic) quantum mechanics faced a huge problem in establishing a theory of nuclear forces. In , it was Yukawa who applied field-theoretical methods to derive the nucleon-nucleon force as an interaction through one-pion exchange . Although this description finally turned out to be not the right path to follow, it was the motivation for a vast amount of new approaches in particle physics during the next decades. We will not try to review all those ideas, failures, and milestones. However, one principle can lead us to an understanding of this time: physicists usually believe that every description of nature should be made as simple as possible – but no simpler.
After seminal works by Gell-Mann , Ne’eman, and Zweig , a classification scheme for the new and already known particles was established, as well as a unified theory for explaining hadrons and their interactions. Gell-Mann and Zweig proposed a solution using group-theoretical methods, namely, they treated all the different hadronic states as manifestations of multiplets within the (flavor) group. This required the existence of quarks, that is, elementary particles with spin as building blocks of hadrons, which interact via an octet of vector gauge bosons, the gluons. The fundamental theory of the interaction between quarks and gluons is quantum chromodynamics (QCD) . One main property of QCD, known as confinement, is the fact that the strong force between the particles does not decrease with distance. It is therefore believed that quarks and gluons can never be separated from hadrons. This is related to the technical problem that the whole theory is non-perturbative in the low-energy regime, which is relevant for describing hadrons and also atomic nuclei.
Despite huge efforts in recent years, it was up to now not possible to solve QCD analytically. In particular, lattice QCD is under continuous growth, where one tries to map the fundamental theory on a discretized space-time grid and performs specific calculations by using a large amount of computational power. Even the treatment of dynamical issues like the application of a coupled-channel scattering formalism seems to be coming within range, see e.g. Ref. . Besides many not yet solved problems, lattice QCD definitely has become a well-established non-perturbative approach for QCD. Other strategies have been found by using holographic models and the gauge/gravity correspondence, for instance to extract meson masses with good accuracy [15, 16]. As will be discussed later, another very successful approach to QCD relies on the concept of effective field theories (EFTs). There, one maps the fundamental theory onto a low-energy description by following a very general prescription – as a consequence, the relevant degrees of freedom become hadrons and their interactions. Chiral perturbation theory (chPT) [17, 18] as a prototype of this concept has been applied e.g. to meson-meson scattering.
2 The quark model and QCD
As already mentioned, one motivation for a new fundamental theory of hadronic particles was the lack of a classification scheme. Concerning dynamics there was another important question: why do most of the new unstable particles not decay into all other particles when their decays would be kinematically allowed? This suggested that there must be some ’rules’ at work, restricting the amount of allowed decay channels. Strictly speaking, composite hadrons would possess some quantum numbers that are conserved under the strong interaction – leading us to symmetries.
Today we know that one can interpret the lightest hadrons, the pion isotriplet, in terms of quark content as , , and . This is a natural consequence of isospin or flavor symmetry which is (nearly) exact in QCD, because the difference in mass between up and down quarks is very small compared to the hadronic scale. Consequently, all hadrons built from those quarks will be arranged within an multiplet, like in the case of the pion isotriplet, and have (nearly) the same mass. Adding a strange quark, slightly heavier than the up and down quark but still light enough, gives rise to multiplets like the pseudoscalar octet. The general mathematical formalism can be introduced by using the basis of strong isospin and hypercharge , yielding the state vectors of those three quarks:
The multiplets are then constructed from this fundamental triplet and the antitriplet formed by the corresponding antiquarks. Here, the substructure of the resulting mesons obeys a pattern. The physical mesons form a singlet and an octet, while for the baryons ( states) we find a singlet, two octets, and a decuplet. We also know today that, in addition to the three light quarks, there exist three heavy quarks: the charm, bottom, and top quark. This highly increases the number of physical particles . The properties of all six quarks can be found in Table 1.
The upper classification scheme for hadrons in terms of their valence quarks is the famous quark model. After the discovery of the baryon it was possible to assign the correct spin and flavor content to its state vector by using the quark model – the only way to do so and obtain a charge state is by having three up quarks. This leads to a symmetric flavor, spin, and spatial wave function, in particular . Therefore, the total (many-body) wave function is also symmetric. But this result is in contradiction to the fact that a fermionic many-body wave function has to be antisymmetric. In order to resolve this a new color degree of freedom for quarks was introduced: they carry either red, green, or blue color charge. Assuming that the baryon is an antisymmetric superposition in color space, it is straightforward to construct its total antisymmetric wave function, which is also ’white’, i.e., invariant under rotations in color space: . Here, is the Levi-Civita symbol and the summation runs over the three colors ( red etc.). The corresponding expression for a meson like the pion would be . Note that the number of colors can be determined from the experiment either from the neutral decay or the ratio of the cross sections for and . The best correspondence with experimental data is unambiguously obtained if the number of colors is .
Now, the Lagrangian of QCD is constructed by starting from the Dirac version for massive spin- particles, where the quarks are incorporated as spinors with flavors, each in the fundamental representation of the (color) gauge group. The Lagrangian is then invariant under global transformations. For the same reason as in QED, we postulate the transformations to depend on the space-time coordinate, hence one requires the Lagrangian to be invariant under local transformations. This is only possible if one includes some further pieces transforming in such a way as to cancel the additional terms caused by the derivative in the Dirac operator. Since the latter brings in a Lorentz index, the required modification introduces eight new spin- fields, the gluons, living in the adjoint representation of the symmetry group, the octet. It is generated from the direct product of the color triplet and the antitriplet; thus gluons do carry color charge which is one main difference to QED. However, as the photon they are massless.
As mentioned, the QCD Lagrangian fulfills gauge invariance because a quark field in the fundamental representation transforms as
where the denote the generators, the Gell-Mann matrices, and the group parameters (here, ). In analogy to the Dirac Lagrangian we therefore have
with implied summation over the flavor index . The covariant derivative
contains the eight gluon gauge fields . They transform under the gauge group according to
such that the covariant derivative transforms as
making the first term in Eq. (3) invariant under transformations. The second part of the QCD Lagrangian represents the kinetic term for the gluons, given by the square of the field-strengths associated with the gauge fields.
with the totally antisymmetric structure constants.
It is worthwhile to look at the tree-level vertex structure of the QCD Lagrangian
The latter shows first the interaction vertex between quarks and gluons which is induced by the covariant derivative. In the second and third panel one recognizes three- and four-gluon interactions, where the former is momentum-dependent. These two vertices are a consequence of the non-abelian group structure of . Note that the four-gluon vertex is of order in the gauge coupling constant .
3 Aim of this work
Intense research during the past decades has demonstrated that the majority of mesons can be understood as being predominantly states . However, the quark model is not the end of the story. Most important for us in this thesis is the phenomenon of overpopulation in the scalar sector: it is not possible to assign all known mesons as quarkonia. For example, the state with , lives in the isotriplet of the scalar meson octet, meaning that there exist three resonances with different electric charge. They have the same quark content as the pseudoscalar pions; since isospin symmetry is nearly exact in QCD, they are also nearly degenerated in mass. This isotriplet can now be identified with either the resonance or the . The quark model cannot explain which is the correct assignment, but can give however an interpretation of one isovector state (see also Table 2).
|Particle||Quark content||Mass [MeV]|
|, ,||, ,||,|
|, , ,||, , ,||,|
|, ,||, ,|
|, , ,||, , ,|
In the literature, many suggestions have been discussed to solve this problem, such as the introduction of various unconventional mesonic states such as glueballs, hybrids, and four-quark states . Along this line, a specific concept of dynamically generated states was put forward e.g. in Refs. [20, 21, 4, 6]. The main idea is that these states are not constructed, as in the quark model, from some building blocks and a confining potential, but rather arise from interactions between conventional mesons – they appear as companion poles in the relevant process amplitude. We will present a more detailed explanation of this idea at the end of Chapter \thechapter. Our aim will be to describe some of the physical mesons as dynamically generated states. This will be successfully performed for the isovector () and isodoublet () sectors, that is, we will show that for the heavy quarkonia states and the couplings to their decay channels are capable of dynamically generating the light states (Chapter \thechapter) and , also known as (Chapter \thechapter), respectively. To this end, we will apply a hadronic model that includes meson-meson interactions via derivative and non-derivative terms. In order to cope with these, Chapter \thechapter is dedicated to work out the formalism of such interactions.
Organization of the thesis:
Chapter \thechapter: After a short introduction on resonances, we present the framework where we want to study scalar resonances: the extended Linear Sigma Model (eLSM) as an example for an effective model of QCD. We also illustrate the idea of dynamical generation via hadronic loop contributions in other effective theories.
Chapter \thechapter: Since derivative and non-derivative interaction terms play an important role in our models, we present in detail how they are incorporated in order to calculate hadronic loop contributions. We also show that there is an apparent discrepancy between using ordinary Feynman rules and dispersion relations.
Chapters \thechapter and \thechapter: We apply the idea of dynamical generation introduced in the second chapter by discussing and extending previous calculations in the isovector sector with . Then, we construct effective Lagrangians where and couple to pseudoscalar mesons by both non-derivative and derivative interactions. For both cases we look for companion poles that can be assigned to the corresponding resonances below GeV, i.e., the and the .
Chapter \thechapter Resonances
4 Unstable particles and resonances
The ideal quark model introduced in the previous chapter demonstrated that it is in principle capable of describing some of the most important aspects of nature, i.e., the baryonic and mesonic ground states which are arranged as an octet and a decuplet, and a nonet. All of them can be considered as built from quarks and antiquarks, where the specific composition depends on some very few quantum numbers like spin and angular momentum . States with higher total spin such as the vector mesons decay to pseudoscalars by the strong interaction, unveiling their constituent nature by decay patterns. One distinguishes hadrons between particles and resonances. In the framework of quantum field theory, the first term is assigned to quanta of some fields; they are able to propagate over sufficiently large time scales (e.g. from a creation reaction to a detector) and hence can be identified in experiments. In particular, they possess distinct measurable properties and consequently should satisfy the energy dispersion relation.
The further terminology can be fixed in the following way: a stable particle is able to propagate over an indefinite amount of time specific for the relevant interactions the particle obeys (for example, pions are stable for what concerns the strong interaction). This holds true until interactions with other particles occur. If the former does not hold true, we speak of unstable particles. For example, we know that charged pions as part of the particle shower in secondary cosmic rays have a mean life time of about s. They can be described as nearly stable as long as the propagation and interaction time is much smaller than the mean life time (including relativistic time-dilation effects). Nevertheless, when considering time scales of some seconds those particles decay into other particles, namely muons and neutrinos.
The baryon decuplet with total spin contains the baryon which possesses an extremely short mean life time on the order of s, the time scale of the strong interaction. Though clearly an unstable particle, in this case it makes more sense to treat this particle like an excitation emerging when investigating nuclear matter and when performing high-energy collision experiments, respectively. The correct term would therefore be resonance. When traveling nearly at the speed of light those resonant states could only overcome distances of about m before decaying. Yet, formally they can nevertheless be interpreted as fluctuations of some underlying field and so we may use the terms ’unstable particle’ and ’resonance’ interchangeably.
In general, by treating a particle decay as a Poisson process one usually defines
where is called the decay width of the resonance associated with a specific set of final states, namely its decay products. As a direct consequence, an exponential decay law for the survival probability of the particle in its rest frame is obtained,
One can show that this in fact is only a simplified picture, valid for narrow resonances with relatively large mean life times only . For example, positively charged pions with dominant leptonic decay channel and a mass of about MeV possess a mean life time of about s. The ratio yields , while for neutral pions with and a mean life time of about s with MeV one obtains . This measure can be implemented within a rule of thumb: whenever mass and decay width become comparable, a resonance leaves the realm of the exponential decay law.
Among such and other difficulties, very short-lived unstable particles in particular cannot be directly observed. Their existence is established from some scattering processes, like the inelastic reaction of two incoming (stable) particles and , and a set of outgoing particles and , where the subset contains an intermediate resonance such that without detection. Another possibility may be the elastic process , where a resonance is created during the fusion of the incoming particles and finally decays without detection. A huge area of research is the extraction of resonance information from the corresponding scattering data.
5 Parameterization of experimental data
In the old days when QCD was not yet (fully) developed, a framework called -matrix theory was applied to interpret the experimental data. It was founded on the very basic understanding of quantum mechanics and some few postulates that mainly consist of unitarity, relativistic invariance, conservation of energy-momentum and angular momentum, and analyticity. The whole field, rich by its own history and methods, cannot be summarized appropriately in this work. For classic literature see for example Refs. [23, 24], though the foundations were formulated much earlier by Wheeler  and Heisenberg . However, it may be possible to give a very interesting quote made by Chew and Frautschi  in the context of this theory. While pointing out a definition for ’pure potential scattering’ they stated, it is plausible “[…] that none of the strongly interacting particles are completely independent but that each is a dynamical consequence of interactions between others.” This remark shall guide us in some sense throughout the thesis at hand.
For the moment let us recall that, in context of scattering theory, the general expression for the decay width has nearly the same formal structure as the differential cross section [28, 29]. By performing a scattering experiment, e.g. of the type with intermediate resonance , and measuring the invariant mass distribution of the outgoing particles, one may find a peak in the differential cross section located around a value , the mass of the resonance . This is because the elastic differential cross section is obtained as the squared scattering amplitude,
Here, is the absolute value of three-momentum of one of the outgoing particles in the rest frame of the resonance , while represents the angular momentum of the partial wave amplitude , and is the corresponding phase shift. For a resonance with total spin the relevant partial wave has a maximum at . One finds by Taylor expansion that near the resonance mass the total elastic cross section is
with and as the total spins of the incoming particles. The last factor is called the relativistic Breit–Wigner distribution. The above expression holds true for a single separated resonant state with only one decay channel and total decay width . The obtained curve is a good approximation of the rate in the region of the resonance only; its mass simply corresponds to the maximum, while the physical width is the full width at half maximum.
One should note that this parameterization in principle introduces a pole on the complex energy plane according to . However, the physical mass and width of a resonance are found from the position of the nearest pole on the appropriate unphysical Riemann sheet
a procedure going back to Peierls . The corresponding pole mass and width in general do not agree with the values a Breit–Wigner parameterization imposes on data, but they do for a narrow and well-separated resonance, in particular, far away from the opening of decay channels. The realization of this mere fact was crucial: compared to the vector and tensor mesonic states, the issue of scalar mesons has been the subject of a vivid debate among the physical community for a long time. Their identification and explanation in terms of quarks and gluons turned out to be very difficult and furthermore, some of those particles possess large decay widths, several decay channels, and a huge background.
Hence, one should remark that Eq. (12) only describes a non-interfering production cross section of a single resonant state with two incoming (stable) particles, while usually background reactions and other multi-channel effects distort the pure contribution from the resonance, such that it is harder to observe if there is really something or not. For instance, one can be faced with very broad structures that cannot be separated from the background, the same as with line shapes partly deformed because of nearby decay opening channels. In such cases only the presence of a pole and its real part provides a good definition of a resonance mass. Furthermore, the existence and position of the pole is independent of the specific reaction studied. The general procedure of extracting the pole would then be to construct the -matrix and partial wave amplitude, respectively, which is then applied directly to fit experimental data or from which a suitable function can be derived to perform the fit (like the phase shift).
6 The extended Linear Sigma Model in a nutshell
A substantial progress in hadron physics was achieved when the concept of an effective field theory (EFT) was applied to the low-energy regime of QCD. Weinberg has pointed out the general ideas in Ref. , i.e., the key point is to identify the appropriate degrees of freedom and to write down the most general Lagrangian consistent with the assumed symmetries. As a consequence, it is not necessary anymore to solve the underlying fundamental theory due to the fact that within the new framework the degrees of freedom (’the basis’) are not quarks and gluons, but composite particles, namely hadrons.
In this thesis, our Lagrangians will be inspired by an effective model called the extended Linear Sigma Model (eLSM) [1, 2, 3], in which a linear representation of chiral symmetry is incorporated [35, 36, 37] and where both the scalar and pseudoscalar degrees of freedom are present. This allows to introduce -parity, conserved by the strong interaction, and corresponding eigenvectors for the pions. The chiral partner of the pion was found to be the state (and not the ). The eLSM was formulated for quark flavors, vanishing temperatures and densities, and includes vector and axial-vector mesons, in some versions also candidates for the lowest lying scalar and pseudoscalar glueballs [38, 39]. Further extensions can be found in Refs. [40, 41, 42].
The main ingredients of the eLSM are composite fields, all assigned as states. This can be proven by using large- arguments [43, 44]: the masses and decay widths obtained within the model scale as and , respectively. The assignment of the required meson matrices for all sectors is summarized by
|• (Pseudo-)Scalars :|
|• Left-handed :|
|• Right-handed :|
The first matrix represents the scalar and pseudoscalar mesons, the other two combine left- and right-handed vector and axial-vector mesons. Note that such an assignment restricts the number of possible quark-antiquark states, for instance there is only one state that forms the scalar isotriplet with . Since it is known that two isotriplets exist , it may be realized as either the or the . The eLSM in fact gives an answer which of the two it addresses (namely, the one above GeV), but the general problem of overpopulation (in the scalar sector) is not solved in the present form of the model. As mentioned at the end of the previous chapter, we will present a possible solution in this work.
For dimensional reasons, the meson matrices are not identical to the perturbative quark currents; the sign just states that both sides transform in the same way under global chiral transformations:
with the chiral rotations
The are the ordinary Gell-Mann matrices (here, ). This brings us to a short discussion of symmetries: where are the QCD symmetries hidden and where are they broken in the eLSM? The mesonic part of the eLSM Lagrangian in its ’full glory’ has the following form:
Here, the field-strength tensors
have been defined together with
The constants , , , , , , , , and are model parameters with specific large- behavior. For instance, the bare mass is directly related to the shift of the gluonic field in the dilaton part of the model (not shown here), which goes like , while scales as because it is associated with quartic meson interaction vertices. For a detailed discussion see Ref. .
Now, the Lagrangian (20) must implement the QCD symmetries and their breaking:
The gauge symmetry is exact in QCD. Since the degrees of freedom in the eLSM are colorless hadrons and confinement is trivially fulfilled, this symmetry is present from the very beginning by construction.
The chiral symmetry is exact for vanishing bare quark masses in QCD and in fact realized there as a global one. Because of the transformation behavior (18) of our meson matrices, most of the terms shown in Eq. (20) are invariant under chiral rotations. For example,
where the unitarity property of the chiral rotations was used together with the fact that the trace in flavor space is invariant under cyclic permutations. Chiral symmetry breaking needs to be modeled separately for the different mesonic sectors; this is accounted for by the remaining non-invariant terms.
In the (pseudo)scalar sector, the term generates explicit chiral symmetry breaking due to non-vanishing quark masses. The term contains the matrix with diagonal entries , with flavor index , where the entries are proportional to the -th quark mass (with for exact isospin symmetry for up and down quarks).
In the (axial-)vector sector, the term containing the matrix is responsible for explicit symmetry breaking since
and hence introduces terms proportional to the squared quark masses as required.
Chiral symmetry is also spontaneously broken in QCD because of a non-vanishing expectation value of the quark condensate, . For , this leads to the emergence of eight Nambu–Goldstone bosons which should be massles. However, they are not massless, because the symmetry is also explicitly broken by the term (for instance, it is ). This results in eight light pseudo-Nambu–Goldstone bosons, the inhabitants of the well-known octet of pseudoscalar mesons. The eLSM incorporates spontaneous chiral symmetry breaking due to the sign of .
The chiral symmetry is broken by quantum effects, too; in QCD this is known as the chiral or anomaly that induces a mass splitting between the pion and the meson, as well as the exceptional higher mass of the singlet state around GeV. This becomes evident via an extra term in the divergence of the axial-vector singlet current even when all quark masses vanish. The eLSM accounts for this by the term proportional to , see also Refs. [38, 39]. It is invariant under but not under .
The gauge sector of QCD in the classical limit (strictly speaking, the classical action) is invariant under dilatation transformations, which is also true for the quark sector in the chiral limit. This symmetry is therefore explicitly broken for finite quark masses, but it is also anomalously broken when quantum corrections are considered: the trace of the energy-momentum tensor, which represents the conserved current, picks up a term proportional to the -function of QCD. The running of the strong coupling constant then renders this term unequal to zero. The eLSM describes the trace anomaly by including a dilaton field with a convenient potential , such that dilatation symmetry is broken explicitly in the chiral limit. The corresponding part is not displayed in Eq. (20).
All terms in the effective Lagrangian are invariant. This is evident from the construction of the meson matrices and the transformation behavior of the quark fields. When applying charge conjugation on the (pseudo)scalar meson matrix, , one finds for example
where we used that a matrix and its transpose have the same trace, together with the fact that the trace in flavor space is invariant under cyclic permutations.
Effective descriptions (of QCD) have their own issues. The eLSM Lagrangian contains only terms up to order four in dimension. This is not because one would like to preserve renormalizability, since an effective model can in principle not be valid up to arbitrarily large scales.
Although such new characteristics may complicate the handling of the model, it turns out that perturbative calculations can be applied in order to calculate tree-level masses and decay widths of resonances.
It was demonstrated in Ref.  that the next-to-leading order (NLO) triangle diagram of a hadronic decay, depicted in Figure 2, can be safely neglected in the case of a simple scalar theory without derivatives. This approximation has been used to study the well-known isoscalar resonances , , , and , the and decay channels of , and the decay of the isovector state . Except for , one can therefore justify a posteriori all studies in which triangle diagram contributions were not taken into account. Since in the field of hadron physics there are usually other (and even larger) sources of uncertainties due to various (and sometimes subtle) approximations and simplifications, the restriction to the leading-order tree-level diagram and to the (resummed) one-loop quantum corrections is reasonable and usually sufficient. In this work quantum corrections will therefore be only considered up to one-loop level.
The eLSM turned out to be quite successful. However, if hadronic resonances are constructed as states, then the problem of the overpopulation in the scalar sector remains unsolved. The scalar sector is described by the predominantly quark-antiquark states , , , and the , while the is predominantly gluonic. Then, the light resonances below GeV, namely , , , and , should not be part of the eLSM and form a nonet of predominantly some sort of four-quark objects. Some very different model approaches that try to include those particles are presented shortly in the following. The key message will be that at least some (scalar) mesons cannot just be predominantly quarkonia but, since they are highly influenced by the dynamics of their hadronic decay channels, they could be rather dynamically generated objects. This is based on the idea mediated by the prior quote of Chew and Frautschi . Please note that the following presentation can neither be sufficient nor complete; further details, as well as other approaches or general achievements, will only be mentioned via citations in suitable places within this thesis.
7 Dynamical generation: Different approaches
7.1 Unitarized Quark Model (UQM)
In quark models the quarks and antiquarks are assumed to be confined by the strong interaction. Then, the constituent quark masses are the result of absorbing the main interaction with the gluons – what remains in dynamics is transformed into some confining potential which is used to form hadronic particles. Among others, Törnqvist and Roos [47, 4, 5] and later also Boglione and Pennington [48, 6] studied extensions by including meson-loop contributions to preexisting quark-antiquark states – this method was used to unitarize the amplitudes which was called unitarization. They came to the conclusion that it could indeed be possible to generate more (scalar) states than actually feasible in a quark model, when starting from those preexisting mesons as bare seed states.
How this can be understood? Let us consider as an intuitive example the meson which possesses the main decay channel . It fits very well into the ideal octet of vector states with predominant flavor configuration of , while the pseudoscalar kaons are and , respectively. In terms of QCD, the decay is described by creating a pair out of the vacuum, while the decay into three pions is suppressed by the OZI rule. Since there is indeed the possibility to decay, i.e., to end up with a configuration of four quarks, the Fock space of the initial particle must contain at least four-quark components . Consequently, the meson is better described as the combination of all its contributions:
where and . For the vector mesons, however, the component is dominant, see also Figure 3. In contrast, for the scalar resonances below GeV one can imagine the situation where the four-quark components dominate – which is the case because of the nature of the -wave coupling – and would bind. The latter need not necessarily be the actual microscopic picture: such states would not be pure molecules but contain some residue of their quarkonia seeds.
This reasoning is what leads to the mechanism called dynamical generation. Since the unstable particle’s propagator represents the probability amplitude for propagating from one space-time point to another, all intermediate interactions in the form of hadronic loops can occur. This establishes access to the four-quark contributions. As usual, they shift the seed state pole from the real energy axis into the complex plane of an unphysical Riemann sheet, but in the case of the scalar sector it furthermore may create new poles. Those poles can be extracted from scattering data and some of them could be identified with physical resonances – see Figure 4 for a visualization of this agument.
It became apparent that in order to obtain additional resonance poles, the UQM needs to include Adler zeros and a further
-dependence in the amplitudes, respectively. With these modifications it was possible to find at least some extra poles,
in particular a putative pole for the states and one for . This is one of the reasons why the corresponding publications [4, 5] are very famous. On the other hand, no pole was found that could have been assigned to the
, while two poles were obtained in the isovector sector.
7.2 Resonance Spectrum Expansion model (RSE)
This quark-meson model has quite a long history because the original version was published already in the ’s [52, 20], while further developments have been achieved during the past decades, see e.g. Refs. [53, 54, 55] and references therein. As in the case of the UQM, the RSE model is based on the unitarization of bare (scalar) states by their strong coupling to -wave two-meson channels, in particular it is a coupled-channel model that describes elastic meson scattering of the form . The transition operator (entering the scattering formalism) contains an effective two-meson potential which is assumed to contain only intermediate -channel exchanges of infinite towers of seed states, see also Figure 5. This corresponds to the spectrum of a confining potential which is chosen as a harmonic oscillator with constant frequency. The power of the formalism lies in the separable form of the interaction matrix elements, resulting in a closed form of the off-shell -matrix, and giving the possibility to study for example resonance poles.
Although in general there is no need for any approximation, for pedagogical reasons one can state that for low energies the infinity tower can be reduced to an effective constant; one is left with a contact term that dynamically generates exactly one pole in the case of scalar mesons . The tower still can be approximated for somewhat higher energies by its leading term and an effective constant for the remaining sum. Then one obtains, apart from the dynamically generated resonance, another pole associated with the leading seed state, that is, the leading propagator mode. While the specific elaboration of the RSE model is very different from the UQM, both rely on the incorporation of hadronic loop contributions.
The RSE model was applied to different flavors, including charm and bottom, and needs only one elementary set of parameters. Surprising and most interesting for us is the fact that after the parameters are fixed by the vector and pseudoscalar spectra, all the low-lying scalar states are fully generated as resonance poles. It was therefore suggested to assign them to another distinct nonet of low-mass scalars purely obtained from dynamics.
7.3 Coupled-channel unitarity approach by Oller, Oset, and Peláez
Oller and Oset generated low-lying scalar mesons dynamically in the framework of a coupled-channel Lippmann–Schwinger (LS) approach [57, 58]. The starting point here is the standard chiral Lagrangian in lowest order of chPT [17, 18]. It contains the most general low-energy interactions of the pseudoscalar mesons at this order. From this Lagrangian the tree-level amplitudes for scattering are obtained and consequently the meson-meson potential terms needed for the coupled-channel analysis. It turns out that it is possible to reduce the LS equation (where relativistic meson propagators are applied) to pure algebraic relations, yielding a simple form for the -matrix. Unitarizing this amplitude creates for instance the pole of the in the isovector sector.
One advantage of this approach is that it requires the use of just one free parameter, namely a cutoff in the loop integrals coming from the LS equation, which is fixed to experimental data. In the further extension of Ref. , the next-order Lagrangian of chPT is taken into account, where additional parameters entering by this procedure are also fitted to data. In this study also a pole for the was obtained in collaboration with Peláez  (the case was not investigated in the previous works). Later, in Ref. , it was demonstrated that for the scalar sector the unitarization of the chPT amplitude is strong enough to dynamically generate the low-lying resonances including the . It was a priori not possible to say if the generated states in this approach are in fact quark-antiquark or four-quark resonances, and if they can be linked to the heavier mesons or not, see Ref.  for a detailed discussion of this issue. Yet, in Ref.  a preexisting octet (and singlet) of bare resonances around GeV (and GeV) was included as a set of CDD poles . It was then found that e.g. the physical in fact originates from the octet, giving a clear statement about its nature.
Quite interestingly, in Ref.  Oller et al. estimated the influence of the unphysical cuts for the elastic and -waves with and , respectively. Such cuts have been included in their previous works only in a perturbative sense (they were absorbed for example in one free parameter mentioned above) – strictly speaking, no loop effects in the - and -channels were considered. It was argued that this kind of simplification is indeed justified due to the quality of the results: the relevant infinite series were summed up in the -channel. In Ref. , however, the unphysical cuts were incorporated in terms of chPT up to , together with the exchange of resonances in the - and -channels. It was shown that the contributions from the former are rather small, because of cancellations with contributions coming from the latter, supporting the view of treating the unphysical cuts in a perturbative way.
7.4 Jülich mesonic -channel exchange model
The Jülich meson-exchange model  was extended in Ref.  to account for further meson-meson interactions. The approach was first based on a coupled-channel analysis of the - and -channels, where it was found that the can be generated by vector-meson exchanges in the -channel, that are strong enough to produce a bound state pole in the -matrix. After extending this consistently to the system, it was possible to demonstrate that the isovector sector is governed by the same dynamics, that is, by the coupling to the kaon-kaon channel. However, it was stressed that the was rather a dynamically generated threshold effect with a relatively low-lying pole, because the important exchange between the two kaons becomes repulsive, not allowing to form a molecule.
One should note that a putative pole for the was also obtained, generated from a strong -channel exchange in the potential. Nevertheless, Harada et al. observed that neglecting this contribution did not remove such a pole from the scattering amplitude of their own model, but only changed its position slightly . It was also argued elsewhere that the pole generated by the Jülich group was considerably lighter and broader than generally accepted.
8 Concluding remarks
We have presented some very different models that try to generate additional (scalar) resonances. In simplified terms, except for the Jülich model, the generation mechanisms are exploiting hadronic loop contributions either by dressing seed states or by relying on meson-meson loops in the scattering amplitude only – the overall dynamics are thus highly influenced by the decay channels. We therefore ask in the following if it is possible to generate low-lying scalar mesons by a similar mechanism within the eLSM.
Chapter \thechapter Derivative interactions and dispersion relations
In this thesis we exploit the idea of dynamical generation in order to study two types of mesonic resonances: the scalar–isovector states and , and the isodoublet resonances and . It was mentioned at the end of the previous chapter that models of dynamical generation focus on the unitarization of bare scalar (seed) states via strong couplings to intermediate (hadronic) states. To this end, it becomes necessary to compute such loop contributions. If a 3d form factor (or regularization function) is applied, usually any one-loop diagram can be obtained from a dispersion relation. However, it will be demonstrated in the following that care is needed in the case of derivative interactions which naturally appear in our effective Lagrangians: there is an apparent discrepancy between ordinary Feynman rules and dispersion relations.
9 Dispersion relations
Multi-valued complex functions are typically manipulated by introducing branch cuts when performing contour integrations in the complex plane. Assuming the following properties of a function defined on the complex plane,
holomorphy except at the cut, where for real it is
and vanishing faster than (with ),
the function can be expressed due to Cauchy’s integral formula in the limit by using its imaginary part right above the cut :
Here, and mark the branch points on the real axis. The full function is determined by the discontinuity only and can be calculated by evaluating the dispersion integral in Eq. (29).
As an instructive example we take the complex root function . Since it does not decrease for , we need to modify the dispersion integral by using a slightly modified function :
This new function has the same branch cut structure as (note that there is no simple pole at ). The discontinuity of the pure root function at the cut is simply two times itself. For the new function this means
The dispersion integral can then be computed by using a Hankel contour path of integration with left open end:
Finally, the original function can be denoted as
We have used the identity
The dispersion integral in Eq. (29) is not valid on the real -axis; this limit cannot be performed in a naive way since the integration contour could collide with a pole. We either use the Sokhotski–Plemelj theorem to identify the real part as the Cauchy principal value integral or simply obtain it from