Implementation of Τ-Lepton Decays Into the Event-Generator SHERPA

Institut für Theoretische Physik Fakultät Mathematik und Naturwissenschaften Technische Universität Dresden

Implementation of τ-lepton Decays into the Event-Generator SHERPA

Diplomarbeit zur Erlangung des akademischen Grades Diplom-Physiker

vorgelegt von Thomas Laubrich geboren am 7. August 1980 in Karl-Marx-Stadt

Dresden 2006 Ä Eingereicht am 31. M¨arz 2006

1. Gutachter: Prof. Dr. R¨udiger Schmidt 2. Gutachter: Dr. Frank Krauss Kurzfassung

In der vorliegenden Diplomarbeit wird die Implementierung von τ-Lepton-Zerfällen in den Ereignis-Generator SHERPA beschrieben. Dafür wird das Modul HADRONS++ entwickelt, wel- ches unstabile Leptonen und Hadronen in einer Kette zerfallen lässt. Die τ-Lepton-Zerfälle werden als erstes System vollständig in das leicht-erweiterbare Modul eingebaut. Auf diese Weise wird zum einen dessen Infrastruktur getestet und zum anderen der Grundstein für die Implementierung von Hadronenzerfällen in SHERPA gelegt. Bei der Implementierung der Zerfallsketten spielen Spinkorrelationen eine große Rolle. Ein Algorithmus, der die korrekten Spinkorrelationen beim Zerfall berücksichtigt, wird im- plementiert und dargestellt. Um die Zerfallsamplituden in HADRONS++ zu berechnen, wird der Helizitätsformalismus verwendet. Die Grundlagen, auf denen dieser Formalismus basiert, werden gezeigt. Desweiteren wird die numerische Integration beschrieben, die eine wichtige Rolle für das Erstellen der Zerfallskinematik spielt. Die Parametrisierungen der einzelnen τ-Lepton-Zerfallskanäle werden detailliert beschrieben. Insbesondere wird die Chirale Störungstheorie und deren Extrapolation zu höheren Energien betrachtet. Die Beschreibung vorkommender Resonanzen wird mittels des phäno- menologischen Kühn-Santamar´ıa-Modells und einer effektiven Feldtheorie, der Resonanten Chiralen Theorie, vorgenommen. Beide Ansätze werden verglichen.

Abstract

In this diploma thesis the implementation of τ-lepton decays into the event-generator SHERPA is described. The module HADRONS++ which develops a decay chain for unstable leptons and hadrons is implemented. The τ-lepton decays are the first system being implemented into the easily-extendable module. On the one hand, the module’s infrastructure is tested in this way. On the other, the foundation for the implementation of hadron decays in SHERPA is laid. Spin correlation effects play a major role when implementing decay chains. An algorithm which takes account of the correct spin correlations during the decay is implemented and displayed. In order to compute the decay matrix elements in HADRONS++, the helicity formalism is used. The fundamentals on which this formalism is based are presented. Furthermore, the numerical integration algorithm is explained. It plays an important role for choosing the correct decay kinematics. The parameterisation of each τ-lepton decay channel is described in detail. Particularly, the Chiral Perturbation Theory and its extrapolation to higher energies are discussed. Oc- curring resonances are implemented according to the phenomenological Kühn-Santamar´ıa model and an effective field theory, the Resonance Chiral Theory. Both approaches are compared.

Contents

1 Introduction...... 1

2 Chiral Perturbation Theory...... 3 2.1 Motivation: Semileptonic τ-leptonDecays ...... 3 2.2 QCDandChiralSymmetry ...... 4 2.3 TheGoldstone-Theorem ...... 7 2.4 SymmetryRequirements...... 8 2.5 Construction of an Eﬀective Lagrangian ...... 9 2.6 Extrapolation to Higher Energies ...... 13 2.7 Breit-Wigner Parameterisation ...... 13 2.8 ResonanceChiralTheory ...... 14 2.8.1 TheVielbeinField ...... 14 2.8.2 Resonance Chiral Theory Lagrangians ...... 14

3 Implementation...... 17 3.1 HelicityAmplitudes ...... 17 3.1.1 SpinorBasisandMassiveSpinors...... 17 3.1.2 SpinorProducts ...... 18 3.1.3 BasicBuildingBlocks ...... 19 3.2 NumericalIntegration ...... 20 3.2.1 Monte Carlo Integration ...... 21 3.2.2 ChoosingDecayKinematics ...... 22 3.3 SpinCorrelation ...... 23 3.3.1 Production of Particles in the Hard Process ...... 23 3.3.2 DecayofParticles ...... 23

4 Parameterisation of Tau Decay Channels ...... 27 4.1 LeptonicChannels ...... 27 4.2 Semileptonic Channels: General Remarks ...... 28 4.3 TheOne-PseudoscalarMode ...... 30 4.4 The Two-Pseudoscalar Mode with ∆S =0...... 30 4.4.1 Kühn-Santamar´ıaModel...... 31 4.4.2 ResonanceChiralTheory ...... 32 4.5 The Two-Pseudoscalar Mode with ∆S = 1...... 34 ± 4.5.1 Kühn-Santamar´ıaModel...... 34 4.5.2 ResonanceChiralTheory ...... 35 4.6 TheThree-PseudoscalarMode ...... 37 4.6.1 Kühn-Santamar´ıaModel...... 39 4.6.2 ResonanceChiralTheory ...... 42 4.7 TheFour-PionMode...... 46 4.7.1 TheOne-ProngDecay ...... 47 4.7.2 TheThree-ProngDecay ...... 48 4.8 TheRemainingModes ...... 51 5 Results and Discussion...... 53 5.1 LeptonicChannels ...... 53 5.2 SemileptonicChannels ...... 54 5.2.1 ThePion/KaonChannel...... 55 5.2.2 TheTwo-PionChannel ...... 55 5.2.3 ThePion-KaonChannel...... 57 5.2.4 TheThree-PionChannel ...... 59 5.2.5 TheKaon-Two-PionChannel ...... 61 5.2.6 ThePion-Two-KaonChannel ...... 63 5.2.7 TheFour-PionMode...... 63 5.3 SpinCorrelations...... 65 5.3.1 Intermediate Z-Boson ...... 65 5.3.2 Intermediate W ±-Boson...... 65

6 Summary and Outlook...... 69

A More on Chiral Perturbation Theory ...... 73 A.1 Axial and Vector Current at Order (p2) ...... 73 O A.1.1 The Left- and Right-Handed Currents ...... 73 A.1.2 TheOne-GoldstoneMode ...... 75 A.1.3 TheTwo-GoldstoneMode...... 75 A.2 Eﬀective Lagrangian at (p4)...... 76 O B The Two-Body Decay...... 77

Bibliography...... 79 List of Tables

2.1 Definition of the Gell-Mann matrices...... 5 2.2 Values of the non-vanishing structure constants f abc...... 5 2.3 Notation for external gauge fields...... 10 2.4 Transformationproperties...... 11 2.5 Notation of meson resonance fields in antisymmetric tensor notation...... 14

3.1 , , -functions for diﬀerent helicity combinations...... 19 X Y Z 3.2 Estimated error of various integration schemes ...... 21

4.1 Resonances in the τ π π0ν , τ K K ν channel...... 32 − → − τ − → − S,L τ 4.2 Resonances in the τ π K ν , τ K π0ν channel...... 35 − → − S,L τ − → − τ 4.3 Parameters for the respective three-pseudoscalar channels...... 37 4.4 Notation of invariant masses in the three-pseudoscalar channel...... 39 i 4.5 Parameters of the resonances in the three-pseudoscalar channels (FA, FS). . . 40 4.6 Three-meson channels with an anomaly term...... 41 4.7 Parameters of the resonances in the three-pseudoscalar channels (FV ). . . . . 42 4.8 Parameters of the resonances in the three-pseudoscalar channels (RχT). . . . 45 4.9 Notation of invariant masses in τ 4πν mode...... 47 → τ 4.10 Parameters of the resonances in the four-pion channel (one-prong)...... 47 4.11 Contributions to the three-prong four-pion channel...... 48 4.12 Parameters of the resonances in the four-pion channel (three-prong)...... 49

6.1 List of all implemented τ-lepton decay channels...... 71

List of Figures

2.1 Feynman diagram for the semi-leptonic channel...... 3

3.1 Decay algorithm for a particle with a spin density matrix...... 24

4.1 Feynman diagram for the leptonic channel...... 27 4.2 Feynman diagram for the semi-leptonic channel...... 28 4.3 Examples for strangeness-conserving and strangeness-changing channels. . . . 29 4.4 Feynman diagram for the strangeness-conserving two-meson channel...... 31 4.5 Feynman diagram for the loop contribution to the two-meson channel. . . . . 32 4.6 Feynman diagram for the three-meson channel...... 39 4.7 Feynman diagram for the parity-violating three-meson channel...... 42 4.8 All contributions to the RχT form factors of the 3π-mode...... 44

5.1 Energy spectrum of the outgoing electron...... 53 5.2 The invariant mass distribution of the two-pion final state...... 56 5.3 The invariant mass distribution of the two-pion final state...... 56 5.4 The invariant mass distribution of the two-kaon final state...... 57 5.5 The branching ratio as a function of cd...... 58 5.6 The invariant mass distribution of the pion-kaon final state...... 59 5.7 Invariant mass distribution of the three outgoing pions...... 60 5.8 Invariant mass distribution of the two outgoing pions...... 61 5.9 Invariant mass distribution of the three outgoing mesons...... 62 5.10 Invariant mass distribution of the two outgoing mesons...... 62 5.11 Invariant mass distributions of the 4π channel...... 64 5.12 Energy distribution of the outgoing pion...... 66 5.13 Invariant mass distribution of the outgoing two-pion final state...... 66 5.14 Energy distribution of the outgoing pion...... 67

1 Introduction

In 1975, the τ-lepton was discovered by M.L. Perl [1] when he and his colleagues discovered the reaction + e + e− µ± + e∓ + missing energy. (1.1) → His interpretation was that a heavy lepton-antilepton pair must have been produced and decayed into the electron and muon. It was concluded that this heavy lepton has a mass between 1.6 . . . 2.0 GeV. It was named the τ-lepton originating from the Greek word for “the third”. The years 1975 to 1978 were a period of confirmation for the existence of the τ-lepton by independent experiments at SLAC and DESY. These measurements were in general agreement with the calculations by Y.S. Tsai [2]. His paper was the “bible” of theoretical predictions for the pattern of heavy lepton decays expected if the third generation was a repeat of the pattern of the first two. This confirmation was important in order to exclude the possibility that the recently discovered charm quark had been rediscovered: In 1974 the existence of the J/Ψ particle, which lies in a similar mass range, was proven by independent experiments at SLAC and Brookhaven. In 1995 M.L. Perl together with F. Reines was awarded the Nobel prize for their pioneering work on lepton physics. Nowadays, the Standard Model [3] contains three generations of quarks and leptons:

generation quarks mass mq leptons mass ml lifetime τl 1 d 4 . . . 8 MeV e 0.511 MeV > 4.6 1026 yr − ≈ × u 1.5 . . . 4.0 MeV νe - - 2 s 80 . . . 130 MeV µ 105.7 MeV 2.19703 (4) µs − ≈ c 1.15 . . . 1.35 GeV νµ - - 3 b 4.60 . . . 4.90 GeV τ 1.77699+0.29 GeV 290.6 1.1 fs − 0.26 ± t 178.0 4.3 GeV ν -− - ± τ The τ-lepton is the only lepton being heavy enough to decay into lighter hadrons such that it provides an excellent laboratory for measuring hadronic currents. These can be measured by analysing scattering processes, also called events, in high energy physics experiments. A common feature these experiments is the immense number of particles produced in the final state. This makes a theoretical description complicated and unmanageable for analytical calculations. Simulation programs like SHERPA are therefore indispensable: SHERPA [4] is a multi-purpose event-generator which up to now invokes PYTHIA [5] to handle the hadron decays. The first topic of this diploma thesis is to implement a new module with the name HADRONS++ into SHERPA handling the hadron decays with full spin correlations. The τ-lepton decays are used for testing the module. Present τ decay libraries such as TAUOLA [6] use a phenomenological parameterisation for the decay amplitudes: the Kühn-Santamar´ıa model [7]. This model will be scrutinised and compared with another model: the Resonance Chiral Theory (RχT) [8]. However, this thesis focuses on the theoretical aspect of the τ decays rather than the software architecture. The outline of the thesis is as follows:

Chap. 2 is about the Chiral Perturbation Theory (χPT) used for calculating the decay • amplitudes for the semileptonic decays. It will be shown that the χPT needs to be 2 1 Introduction

modiﬁed in order to describe τ decays accurately. Two approaches will be presented: the K¨uhn-Santamar´ıa model and RχT.

The subsequent Chap. 3 describes the helicity formalism used for calculating the decay • matrix elements, the numerical integration algorithm used for choosing the correct decay kinematics, and the spin correlation algorithm.

Chap. 4 intends to describe and explain the parameterisation for each τ decay channel • individually.

Chap. 5 is dedicated to the presentation of results which have been obtained by em- • ploying the new HADRONS++ module. It also compares the results produced by using the K¨uhn-Santamar´ıa model and the Resonance Chiral Theory.

The ﬁnal chapter gives a summary of the presented diploma thesis. • 2 Chiral Perturbation Theory

The mass of the τ-lepton is roughly 1.8 GeV so that it is not heavy enough to produce a charmed hadron in its decay. Its decay products are either leptons or light hadrons consisting merely of up, down, and strange quarks. Consequently, the τ-lepton decays can be described by using electroweak theory for the leptonic decay channels and • QCD with 3 flavours (up, down, strange) for the semileptonic channels. • QCD is a theory that describes the interaction of quarks, anti-quarks, and gluons. At low energies (E < Mρ 770 MeV) an SU(3)Flavour octet of pseudoscalar particles can be found: 0 0 ≈0 (π±,π , K±, K , K¯ , η). It is possible to construct an effective field theory with the correct symmetry requirements whose only degrees of freedom are these eight particles: Chiral Perturbation Theory (χPT). This chapter intends to show the construction of χPT starting from massless QCD, which bears a chiral symmetry. However, chiral symmetry is broken because quarks are not massless. The Goldstone-Theorem then predicts the existence of some states which are related to the breakdown of this chiral symmetry. These states can be associated with the eight lightest hadrons and, thus, their transformation laws can be used in order to construct an effective Lagrangian, in this case for χPT. At the end of this chapter, the extrapolation to higher energies will be discussed because resonances, such as ρ(770), a1(1260), etc., play an important role in semileptonic τ-lepton decays, but they are not described by χPT in its original form.

2.1 Motivation: Semileptonic τ-lepton Decays

Consider a τ-lepton decaying into a ﬁnite number of hadrons, see Fig. 2.1,

− τ − ν + W − ν h + ...h . (2.1) → τ → τ 1 n

ντ GF √ 2 hn

. τ − .

Figure 2.1: Feynman diagram for the semi-leptonic channel with n outgoing hadrons. The graph shows Fermi’s pointlike interaction between the leptonic and the hadronic current. 4 2 Chiral Perturbation Theory

The multi-hadron ﬁnal state must have a total negative charge due to the conservation of electric charge. Using Fermi’s pointlike interaction, i.e. neglecting the contribution from the W -boson propagator, the decay amplitude reads

GF µ = L Hµ, (2.2) M √2 with GF denoting the Fermi constant

5 2 G = 1.16639 10− GeV− . (2.3) F × Lµ corresponds to the electroweak current

Lµ = ν γµ(1 γ ) τ (2.4) h τ | − 5 | i QCD with γ5 = iγ0γ1γ2γ3. Hµ denotes the hadronic current

HQCD = V h h ...h V A 0 , (2.5) µ CKM × h 1 2 n | µ − µ| i which creates the n-hadron final state from the vacuum. Vµ and Aµ are the vector and axialvector quark current, respectively. VCKM is the corresponding CKM matrix element. QCD In order to obtain an expression for Hµ , QCD needs to be solved. Alternatively, an effective strong interaction field theory being dual to QCD, the Chiral Perturbation Theory, can be constructed. The hadronic current

HχPT = V h h ...h V A 0 (2.6) µ CKM × h 1 2 n | µ − µ| i can be obtained by computing perturbatively the vector and axialvector current Vµ and Aµ, respectively, using the χPT Lagrangian.

2.2 QCD and Chiral Symmetry

Consider q denoting a vector in ﬂavour space that collects the three lightest quark ﬂavours:

u q := d . (2.7)   s   The QCD Lagrangian [9] describing the interaction of the three lightest quarks is given by

8 1 = Ga Ga;µν +qiγ ¯ µD q +q ¯ q + , (2.8) LQCD −4 µν µ M Lgf Xa=1 where stands for gauge ﬁxing terms, which do not play any role in the context of this Lgf discussion. The gluon ﬁeld strength tensor is

8 Ga = ∂ Ga ∂ Ga + f abcGb Gc (2.9) µν µ ν − ν µ µ ν b,cX=1 with a = 1 . . . 8 denoting the colour index. The colour-gauge covariant derivative Dµ reads

8 λa D = ∂ i Ga . (2.10) µ µ − µ 2 a=1 X 2.2 QCD and Chiral Symmetry 5

0 1 0 0 i 0 1 0 0 − λ := 1 0 0 λ := i 0 0 λ := 0 1 0 1   2   3  −  0 0 0 0 0 0 0 0 0 0 0 1 0 0 i   − λ := 0 0 0 λ := 0 0 0 4   5   1 0 0 i 0 0 0 0 0 0 0 0  1 0 0 1 λ := 0 0 1 λ := 0 0 i λ := 0 1 0 6   7  −  8 √   0 1 0 0 i 0 3 0 0 2 −      

Table 2.1: Deﬁnition of the Gell-Mann matrices.

abc 123 147 156 246 257 345 367 458 678 f abc 1 1 1 1 1 1 1 1 √3 1 √3 2 − 2 2 2 2 − 2 2 2

Table 2.2: Values of the non-vanishing totally antisymmetric structure constants f abc of SU(3).

λa stands for the Gell-Mann matrices, which are deﬁned in Tab. 2.2. These are hermitian, traceless 3 3 matrices with × trace λaλb = 2δab and λa, λb = 2 f abcλc. (2.11) c n o h i X The values of the structure constants f abc can be calculated explicitly through

i f abc = − trace λa, λb λc . (2.12) 4 nh i o They are given in Tab. 2.2. The quark masses are described by the quark mass matrix

mu 0 0 = 0 m 0 (2.13) M  d  0 0 ms   in flavour space. The right- and left-handed parts of the quark fields are defined as

1+ γ 1 γ q := 5 q, q¯ =q ¯ − 5 , R 2 R 2 1 γ 1+ γ q := − 5 q, andq ¯ =q ¯ 5 . (2.14) L 2 L 2

They obey the following transformation properties under application of the operator γ5:

γ q = q and γ q = q . (2.15) 5 R R 5 L − L Neglecting the gauge fixing terms , the QCD Lagrangian in terms of the chiral fields q Lgf R and qL reads 1 = Ga Ga;µν +q ¯ iγµD q +q ¯ iγµD q +q ¯ q +q ¯ q , (2.16) LQCD −4 µν R µ R L µ L RM L LM R 6 2 Chiral Perturbation Theory which in case of = 0 leads to M 1 0 = Ga Ga;µν +q ¯ iγµD q +q ¯ iγµD q . (2.17) LQCD −4 µν R µ R L µ L This identity shows that the left- and right-handed parts of the quark fields decouple if the quarks are treated as being massless. Obviously, the massless QCD Lagrangian in Eq. (2.17) is invariant under the global transformations

q V q and • R → R R q V q • L → L L with the unitary constraint

VR†VR = VL†VL = ½. (2.18)

This is called the chiral symmetry. Both transformations, VR and VL, belong to an SU(3) group: V SU(3)Flavour. (2.19) R/L ∈ R/L Therefore, the symmetry group of massless QCD is

g = SU(3)Flavour SU(3)Flavour. (2.20) R ⊗ L a;µ According to Noether’s theorem, there must exist two conserved currents, namely JR and a;µ a JL . Since the colour index runs from a = 1 . . . 8, eight right-handed currents JR and eight a left-handed currents JL are expected. It is convenient to introduce vector and axial currents being deﬁned as superpositions of the left- and right-handed currents: λa Aa;µ := J a;µ J a;µ =qγ ¯ µγ q, R − L 5 2 λa V a;µ := J a;µ + J a;µ =qγ ¯ µ q. (2.21) R L 2 In massive QCD with a quark mass matrix , it can be shown, see e.g. Ref. [10], that the M axial currents Aa;µ are only conserved if all quark masses vanish. On the other hand, the vector currents V a;µ are conserved if the quarks have identical masses. In other words:

λa ∂ Aa;µ = iq¯ , q = 0 m = m = m = 0, µ M 2 ⇐⇒ u d s λa ∂ V a;µ = iq¯ , q = 0 m = m = m . (2.22) µ M 2 ⇐⇒ u d s a The Noether charges corresponding to the axial and vector currents are denoted by QA and a QV , respectively. a Only in massless QCD, the axial charges QA commute with the Hamiltonian because only massless quarks imply conserved axial currents and therefore conserved axial charges. If h is a single hadron state with energy E, H0 h = E h , then the states Qa h have | i QCD | i | i A | i the same energy as h , H0 Qa h = EQa h , but opposite parity, | i QCD A | i A | i a a 1 a P h = η h (η = 1) PQ h = PQ P − P h = ηQ h . (2.23) | i | i ± ⇒ A | i A | i − A | i The listings of the Particle Data Group [3], however, do not show any trace of degenerate multiplets with opposite parity at low energies. Instead, there exists a state of almost the same energy and opposite parity as h which | i contains a hadron and a pion [9], which is clearly not a single particle state. Consequently, chiral symmetry must be broken. The small values of the up, down and strange quark mass 2.3 The Goldstone-Theorem 7

compared to the energy region of interest imply that the quark masses mq can be used as breaking terms in a suitably formulated perturbation theory. The axial charges a 3 a;0 QA(t)= d ~xA (t, ~x) (2.24) Z would be conserved if chiral symmetry was unbroken. An inﬁnitesimal transformation generated by an axial charge reads

a ϕ(x) ϕ′(x)= ϕ(x)+ iǫ [ϕ(x),Q ] (a = 1 . . . 8), (2.25) → A where ϕ denotes an hermitian scalar ﬁeld. Non-conserved axial charges leads to a non- vanishing vacuum expectation value

0 [ϕ(x),Qa ] 0 = 0, (2.26) h | A | i 6 which can only be fulﬁlled if Qa 0 = 0. (2.27) A | i 6 From Eq. (2.26) follows that even the vacuum expectation value of the scalar ﬁeld changes under this transformation: 0 ϕ′ (x) 0 = 0 ϕ (x) 0 . (2.28) i 6 h | i | i This is referred to as spontaneous chiral symmetry breaking (SCSB).

2.3 The Goldstone-Theorem

This section intends to show the relevance of Goldstone’s Theorem [11] to χPT. It states that, if a theory has an exact symmetry, such as a gauge symmetry, which is not a symmetry of the vacuum, the theory must contain a zero mass particle, the Goldstone-particle. As the vacuum does not show the symmetry generated by the axial charges, there must exist some Goldstone-particles. More generally, consider an expression like 0 [ ,Qa ] 0 h | O A | i where stands for any operator rather than ϕ. It will be shown that this kind of expression O acts as order parameter of SCSB. According to the deﬁnition of Noether charges the vacuum expectation value is

0 [ ,Qa ] 0 = d3~y 0 n n Aa;0(y) 0 h | O A | i h |O| i Z n X n 0 Aa;0(y) n n 0 = 0, (2.29) − h |O| i 6 o which needs to be non-zero. The inserted complete set of eigenstates fulﬁls the completeness relation

½ = n n . (2.30) | i h | n X To ensure that this expression does not vanish, there must be some states N , the Goldstone- | i states, among this complete set of eigenstates with

N : 0 N N Aa;0(y) 0 = 0. (2.31) ∃ | i h |O| i 6

The product of two numbers is non-zero if and only if both factors are non-zero. N Aa;0(y) 0 = 0 then implies that N must be a pseudoscalar ﬁeld associated with Aa;0. 6 | i Hence, N has the name Goldstone-Boson. It is shown in [9] that it has vanishing mass. | i Furthermore, the operator has to be a pseudoscalar operator because of 0 N = 0. O h |O| i 6 8 2 Chiral Perturbation Theory

The simplest choice [12] is to take =q ¯ γ λbq = b. It can be shown that the vacuum O i 5 j O expectation value of the commutator [12, 13] is 2 0 q¯ γ λbq ,Qa 0 = δabδ 0 q¯ q 0 . (2.32) i 5 j A 3 ij h | i j| i D h i E

As a result, the quark condensate, 0 uu¯ 0 = 0 dd¯ 0 = 0 ss¯ 0 = 0, (2.33) h | | i h | | i 6 is the natural order parameter of SCSB. a There are eight broken axial generators of the chiral symmetry group g: QA (a = 1 . . . 8). Consequently, there should be eight pseudoscalar Goldstone-states which are denoted by

ϕa := N a . (2.34) | i | i It is the quark mass matrix , cf. Eq. (2.13), that both generates the mass of the eight M lightest hadrons and explicitly breaks the global chiral symmetry, cf. Eq. (2.16). For this 0 reason the Goldstone-Bosons can be identiﬁed with the eight lightest hadron states π±, π , 0 0 K±, K , K¯ , and η. The (small) masses of these eight states turn the Goldstone-Bosons into Pseudo-Goldstone-Bosons not being massless states anymore.

2.4 Symmetry Requirements

This section intends to derive the symmetry of the (Pseudo-)Goldstone-Bosons. This symmetry determines the transformation behaviour of the effective Lagrangian. The spontaneous chiral symmetry breaking reduces the chiral symmetry group g of the QCD Lagrangian to Flavour Flavour h = SU(3)V = SU(3)R+L . (2.35) It is shown in [13] that h is a subgroup of g. The (Pseudo-)Goldstone-fields are described by the coordinates of the coset g/h. A transformationg ˆ g transforms the left- and the right- ∈ handed parts of a quark field independently, i.e.g ˆ = (VL, VR). In contrast, a transformation hˆ h transforms the left- and right-handed part in exactly the same way, i.e. V = V . ∈ L R The coset g/h is defined as the setgh ˆ withg ˆ g butg ˆ / h. Consider a transformation ∈ ∈ gˆ with V = V and multiply it by an hˆ = (V, V ) h. This way a representation for the L 6 R ∈ transformations of the coset g/h can be obtained:

ˆ gˆh = (V V, V V ) = (V V, V V †V V ) = (½, V V †)(V V, V V ) g/h. (2.36) L R L R L L R L L L ∈ /h h ∈ ∈

˜ | {z }| ˜ {z } ½ Hence, representators Φ of the coset g/h can be written as Φ = (½, Φ) with Φ = . A 6 transformation law for Φ˜ can be constructed such that it remains a representator of this coset after application of the transformation. Ifg ˆ = (V , V ) g theng ˆΦ˜ = (V , V Φ). A L R ∈ L R transformation hˆ h can be chosen such that hˆ = (V †, V †), which leads to ∈ L L ˜ ˜ˆ Φ′ =g ˆΦh = (VLVL†, VRΦVL†) = (½, VRΦVL†). (2.37) Consequently, the transformation law of the coset reads

Φ′ = VRΦVL†. (2.38) Introducing a space-time dependence, the required symmetry of Φ is associated with the transformation law Φ(x) V Φ(x)V † with (V , V ) g. (2.39) 7→ R L R L ∈ 2.5 Construction of an Eﬀective Lagrangian 9

Since the Goldstone-Bosons appear as a result of SCSB, it should be them, which are rep- resented by Φ. Additionally, linear combinations of the Pseudo-Goldstone-Bosons can be 0 0 0 identiﬁed with the eight lightest pseudoscalar mesons π±, π , K±, K , K¯ , and η8 in the following way:

0 3 1 2 π := ϕ , π± := √2(ϕ iϕ ), ∓ 4 5 0 0 6 7 K± := √2(ϕ iϕ ), K /K¯ := √2(ϕ iϕ ), and ∓ ∓ 8 η8 := ϕ . (2.40)

Since Φ SU(3)Flavour, it can be written as ∈ i√2 Φ(x) = exp φ(x) , ( F ) 1 0 1 + + 8 π + η8 π K 1 √2 √6 a a π 1 π0 + 1 η K0 φ(x):= λ ϕ (x)=  − √2 √6 8  (2.41) √2 − 0 2 a=1 K− K¯ η8 X  − √6    with F denoting a normalisation constant.

2.5 Construction of an Eﬀective Lagrangian

An effective Lagrangian describing the interaction of the (Pseudo-)Goldstone-Bosons ϕa, collected in Φ, can now be constructed: The Lagrangian must be invariant under the transformation (2.39). Starting point is the massless QCD Lagrangian because of its chiral symmetry. The method of choice is the external fields treatment [14], because the inclusion of external fields promotes the global chiral symmetry g to a local one; • additionally, Green’s functions of quark currents are induced by the corresponding • external field and can be obtained through functional derivatives of the generating functional Z with respect to the corresponding external fields. Four external hermitian matrix fields need to be introduced: the vector and axialvector field v and a , respectively, and • µ µ the scalar and pseudoscalar field s and p, respectively. • Additionally, two more gauge fields are introduced, the right- and left-handed fields:

r := v + a and l := v a . (2.42) µ µ µ µ µ − µ ex Tab. 2.3 gives a summary of the gauge fields. External photons Aµ and W -boson fields ;ex Wµ± are among the right- and left-handed gauge fields: r = e Aex + ..., µ − Q µ ex e +;ex lµ = e Aµ Wµ T+ +h.c. , (2.43) − Q − sin θW √2 where e denotes the electromagnetic coupling and θW the Weinberg angle. The quark charge matrix and the operator T+ are defined as 2 0 0 0 V V 1 ud us = 0 1 0 and T := 0 0 0 (2.44) Q 3  −  +   0 0 1 0 0 0 −     10 2 Chiral Perturbation Theory

symbol name

vµ vector field aµ axialvector field s scalar field p pseudoscalar field rµ right-handed field lµ left-handed field

Table 2.3: Notation for external gauge fields. with Vij as relevant CKM matrix elements. The Green’s functions for electromagnetic and semileptonic weak currents can be obtained by deriving the generating functional Z with respect to the external photon and the external W boson field, respectively. According to [15, 16] the massless QCD Lagrangian 0 can be extended by coupling LQCD the quarks q = (u, d, s)⊤ to the external hermitian matrix fields vµ, aµ, s, and p: ext = 0 +qγ ¯ µ(v + a γ )q q¯(s ipγ )q. (2.45) LQCD LQCD µ µ 5 − − 5 It can be seen that the scalar and pseudoscalar field provide an adequate way of incorporating explicit chiral symmetry breaking by means of the quark mass matrix . This is because s M couples to the quark fields in the same way as does, cf. Eq. (2.8): M s = + . . . (2.46) M The extended QCD Lagrangian, Eq. (2.45), remains invariant under the following (local) transformations: q V q , R 7→ R R q V q , L 7→ L L s + ip V (s + ip)V †, 7→ R L l V (l + i∂ )V †, and µ 7→ L µ µ L r V (r + i∂ )V †. (2.47) µ 7→ R µ µ R The basic idea is to use these symmetry properties to build an effective Lagrangian describing the dynamics of the Goldstone-Bosons. It is the generating functional that links this effective theory with QCD:

0; out 0; in = eiZ[a,v,s,p] = q q¯ G exp i d4x ext h | iv,a,s,p D D D µ LQCD Z Z = Φexp i d4x . (2.48) D Leff Z Z Since the effective Lagrangian only depends on the field Φ, it has to be invariant under the local chiral transformation g Φ(x) V (x)Φ(x)V †(x). (2.49) 7→ R L Hence, the kinetic part of the Lagrangian must contain covariant derivatives which can be written as D Φ := ∂ Φ ir Φ+ iΦl . (2.50) µ µ − µ µ Note that the order of the matrix fields, i.e. Φ, rµ and lµ, matters. In the last term Φ must be to the left of lµ to ensure the correct transformation property under g: g D Φ V (D Φ)V †, µ 7→ R µ L g D Φ† V (D Φ†)V †. (2.51) µ 7→ L µ R 2.5 Construction of an Effective Lagrangian 11

chiral charge parity transformation conjugation transformation element g C P Φ VRΦVL† Φ⊤ Φ† DµΦ VR(DµΦ)VL† (DµΦ)⊤ (DµΦ)† χ VRχVL† χ⊤ χ†

Table 2.4: Transformation properties of building blocks for an eﬀective Lagrangian.

Consequently, the covariant derivative acting on the adjoint ﬁeld Φ† must be of the form

D (Φ†)= ∂ Φ† + iΦ†r il Φ†. (2.52) µ µ µ − µ

Dµ(Φ†) behaves under chiral transformation like (DµΦ)†. This leads to the transformation law µ g µ D Φ†D Φ V D Φ†D ΦV †. (2.53) µ 7→ L µ L Using the fact that a value of a trace remains invariant under cyclic permutation of its arguments, it can be seen that

µ g µ µ D Φ†D Φ V D Φ†D ΦV † = D Φ†D Φ , (2.54) µ 7→ L µ L µ D E D E D E where denotes the trace in flavour space. Moreover, the constructed Lagrangian has to h·i be invariant under charge and parity transformation as well. Tab. 2.5 shows the transformation properties under different gauge groups for all relevant building blocks [17]. The term in Eq. (2.54) does not change under charge conjugation because of the invariance of the trace under transposing its argument. In addition, parity transformation leaves this term unchanged, too. As a result, the kinetic part of the effective Lagrangian can be written as

2 kin F µ = D Φ†D Φ . (2.55) Leff 4 µ D E F 2 The global factor 4 ensures the proper normalisation of the kinetic part. This can be seen as follows: Expanding the field matrix Φ, see Eq. (2.41), in (Pseudo-)Goldstone-fields

i√2

Φ(x)= ½ + φ(x)+ ..., (2.56) F the kinetic term becomes

2 F µ 1 µ D Φ†D Φ = D φ†D φ + . . . . (2.57) 4 µ 2 µ D E D E In order to include chiral symmetry breaking terms, expressions like

(s + ip)Φ† and Φ(s + ip)† (2.58) D E D E have to be considered because, as mentioned above, the scalar field s incorporates the chiral symmetry breaking. These terms are invariant under chiral transformation. It is beneficial to define χ := 2B(s + ip) = 2B + ..., (2.59) M 12 2 Chiral Perturbation Theory which contains the quark mass matrix in the same way as s does. Hereby, B is another normalisation constant. Charge conjugation and parity transformation reads

χ†Φ C (χ†Φ)⊤ = χ†Φ , 7→ D E D E D E Φ†χ C (Φ†χ)⊤ = Φ†χ , 7→ D E D E D E χ†Φ Φ†χ P (χ†Φ Φ†χ)† = (χ†Φ Φ†χ). (2.60) ± 7→ ± ± ± It can be seen that the term χ†Φ+Φ†χ (2.61) is invariant under chiral, charge, and parityD transformatiE on. As a consequence, an effective Lagrangian may look like 2 F µ ′ = D Φ†D Φ+ χ†Φ+Φ†χ (2.62) Leff 4 µ D E where χ contains the quark mass matrix. The Lagrangian is primed because this effective Lagrangian is not unique. Obviously, terms with e.g. four covariant derivatives can be constructed as well. It is therefore convenient to expand the general effective Lagrangian into a series = (0) + (2) + (4) + ..., (2.63) Leff Leff Leff Leff where (n) denotes the most general Lagrangian of order (pn), i.e. it contains terms with n Leff O covariant derivatives. Lorentz invariance forces n to be even. To maintain bookkeeping the chiral counting rule is defined such that one order of quark mass matrix, i.e. χ, is equivalent to two orders of derivatives. The theory described by this effective Lagrangian is known as Chiral Perturbation Theory (χPT). The Lagrangian of order (p0) can only depend on Φ but, by definition, not on its O derivative. Still, it must be invariant under chiral transformation, i.e.

(0) ! (0) (Φ) = (V ΦV †) V , V (2.64)

Leﬀ Leﬀ R L ∀ R L ½ with VR†VR = VL†VL = ½. Taking VR = and VL = Φ yields (0) (0)

(Φ) = (½) (2.65) Leff Leff implying that (0) = const. (2.66) Leff For the generating functional Z[v,a,s,p], Eq. (2.48), (0) merely contributes a constant term Leff whose functional derivatives are zero. Hence, it is (2) that gives the leading contribution to Leff the generating functional Z[v,a,s,p]. Clearly, the expression given in Eq. (2.62) is the most general Lagrangian of order (p2): O 2 (2) F µ = D Φ†D Φ+ χ†Φ+Φ†χ . (2.67) Leff 4 µ D E This Lagrangian can be used in order to calculate hadronic currents of the form 0 V a Aa h h ...h (2.68) µ − µ 1 2 n with h h ...h being an n-hadron state. This is explicitly done in App. A.1. It is also | 1 2 ni shown that χPT currents are parity conserving: It is an axial (vector) current that pro- duces or annihilates an odd (even) number of pseudoscalar hadrons. The pion decay can be calculated using the techniques shown in the appendix: 0 V A π+(p) = iF √2p . (2.69) | µ − µ| µ

2.6 Extrapolation to Higher Energies 13

This expression gives F a physical meaning: at order (p2) F can be interpreted as the pion O decay constant F = fπ = 92.4 MeV. (2.70) Furthermore, the other still arbitrary constant B, which was introduced in the deﬁnition of χ = 2B + . . . , gains a meaning, too: It is shown in [18] that M 0 q¯iqj 0 = F 2Bδij. (2.71) − That means that B is a measure for the quark condensate, which is the natural order parameter of SCSB. These hadronic currents couple to the electroweak currents so that a τ-lepton decay amplitude with Fermi’s pointlike interaction, see Sec. 2.1, can be obtained.

2.6 Extrapolation to Higher Energies

χPT is only successful in the low-energy region E < M 770 MeV. The ρ-meson, a vector ρ ≈ resonance, is the lightest hadron that is not included in the χPT framework. For semileptonic τ decays

− τ − ν + Resonances ν h + ...h , (2.72) → τ → τ 1 n the ρ-meson plays an important role due to its occurrence as reson ance. However, χPT does not give any information about resonances such as ρ, a1,(1260), etc. If the hadronic current H = V h h ...h V A 0 (2.73) µ CKM × h 1 2 n | µ − µ| i is given in a closed expression, form factors Fi describing the phenomenology of the resonance i are introduced. Clearly, Fi must depend on momentum transfer of the corresponding resonance. That is H = i F (q2) (2.74) µ Oµ i Xi with i denoting operators that give the Lorentz structure of the hadronic current. Basically, Oµ the i are obtained from an eﬀective χPT Lagrangian. Oµ 2.7 Breit-Wigner Parameterisation

The most intuitive form factors are of the Breit-Wigner type. The ansatz of K¨uhn and Santamar´ıa [7] is to use terms like

2 2 2 MX q MX BWX (q )= ≪ 1 (2.75) M 2 q2 i q2Γ (q2) −→ X − − X 2 with an energy dependent width ΓX (q ) for constructingp the relevant form factors. These terms arise from the propagator of the resonance X in the channel q2: M 2 X . (2.76) M 2 q2 X − This model is constructed to satisfy the chiral symmetry of massless QCD. That is, if q2 ≪ M 2 , BW (q2) 1. Furthermore, it often happens that a resonance X comes along with X X −→ higher resonances X′, X′′, etc.. This can be handled by replacing

2 1 2 2 2 BW (q ) BW (q )+ β BW ′ (q )+ γ BW ′′ (q ) (2.77) X → 1+ β + γ X X X with arbitrary constants β and γ that have to be fitted to experimental data. Nearly every τ-lepton decay has own specific resonances so that the Kühn-Santamar´ıa form factors will be described for each decay channel individually in Chap. 4. 14 2 Chiral Perturbation Theory

name J P name J P + vector Vµν 1− axialvector Aµν 1 + scalar S 0 pseudoscalar P 0−

Table 2.5: Notation of meson resonance ﬁelds in antisymmetric tensor notation.

2.8 Resonance Chiral Theory

On the other hand, an effective field theory describing the interaction between the lightest pseudoscalar hadrons and resonances can be constructed as well: Resonance Chiral Theory (RχT). Its construction closely follows the construction of χPT. χPT is constructed to describe the interaction of pseudoscalar mesons. However, resonances can be vector, axialvector, and scalar particles. They are not restricted to be pseudoscalar. Therefore, transformation laws need to be fixed first. To do so, it is beneficial to consider another field rather than Φ.

2.8.1 The Vielbein Field The field matrix u(x) [14] defined by Φ(x)= u(x)2 obeys the following chiral transformation law: 1 1 u(x) V u(x)H− = Hu(x)V − . (2.78) 7→ R L It induces a compensator field H which represents an element of the conserved subgroup Flavour h = SU(3)V . The vielbein fields [19] are defined by

u := i[u†(∂ ir )u u(∂ il )u†] and µ µ − µ − µ − µ χ := u†χu† uχ†u. (2.79) ± ± Their chiral transformation law reads

uµ g uµ 1 H H− . (2.80) χ 7−→ χ ± ± The advantage of the vielbein field notation is that Lorentz invariant terms are automatically chiral invariant. As an example, the effective Lagrangian at order (p2) simply reads, cf. O Eq. (2.67), F 2 (2) = u uµ + χ . (2.81) Leff 4 h µ +i Moreover, The transformation law of any resonance field R V,A,S,P , see Tab. 2.5, ∈ { } reads 1 R HRH− , (2.82) 7→ i.e. it transforms like a vielbein field.

2.8.2 Resonance Chiral Theory Lagrangians RχT is an effective theory for the resonance fields in Tab. 2.5. It describes their interaction with pseudoscalar mesons. It is sufficient to consider the lowest order coupling in the chiral expansion which is linear in the resonance fields. The vector and axialvector mesons are described in terms of antisymmetric tensor fields [19, 20, 21], i.e. V = V . The covariant µν − νµ derivatives, which are necessary for the construction of the kinetic terms, are defined by

▽µX := ∂µX + [Γµ, X] (2.83) 2.8 Resonance Chiral Theory 15 with 1 Γ := [u†(∂ ir )u + u(∂ il )u†]. (2.84) µ 2 µ − µ µ − µ In matrix notation the lightest multiplet of resonances can be written as

8 1 a a 1 R(x)= λ R (x)+ R (x)½, (2.85) √ √ 1 2 a=1 3 X where the singlet state is denoted by R1. The multiplet of lightest vector resonances [22] is explicitly given through

0 ρ + ω8 + ω1 ρ+ K + √2 √2 √3 ∗ 0 ρ ω8 ω1 0 Vµν =  ρ + + K  , (2.86) − − √2 √6 √3 ∗ 0 0 2ω8 ω1  K∗ K∗ +   − √6 √3 µν   0 0 0 where the meson fields (ρ±, ρ , K∗±, K∗ , K¯ ∗ ,ω8,ω1) are linear combinations of the fields Ra, cf. Eqs. (2.40). In analogy to the construction of the effective Lagrangian in Sec. 2.5, the kinetic and (resonance-pseudoscalar) interaction terms for a meson resonance with mass MR can be constructed:

2 1 ▽λ ▽ νµ MR µν kin 2 Rλµ νR 2 Rµν R ; R V, A , R = − − ∈ { } (2.87) L ( 1 ▽DµR▽ R M 2 R2 ; E R S, P , 2 µ − R ∈ { } (2) FV µν GV µ ν = Vµν f + i V µνu u , (2.88) LV 2√2 + √2 h i (2) FA µν A = Aµν F , (2.89) L 2√2 − (2) = c Su uµ + c Sχ , (2.90) LS d h µ i m h +i (2) = id Pχ , (2.91) LP m h ii where all coupling constants are real. Note that the kinetic term are written in form similar to the right hand side of Eq. (2.57) for the pseudoscalar mesons, i.e. no additional normalisation constant needs to be introduced. The fields f µν are defined as ± µν µν µν f := uFL u† u†FR u (2.92) ± ± with FR,L being the field strength tensors associated with the right- and left-handed fields: F µν = ∂µrν ∂νrµ i [rµ, rν] , R − − F µν = ∂µlν ∂ν lµ i [lµ, lν] . (2.93) L − − A decay channel that only involves a single type of resonances R V,A,S,P is de- ∈ { } scribed by the RχT Lagrangian

= (2) + kin + (2). (2.94) LRχT Lχ LR LR There are also channels that, for instance, contain both axial and vector resonances with an axial-vector-pseudoscalar coupling. The most general (vector-axialvector-pseudoscalar) interaction Lagrangian allowed by the symmetry constraints is

(2) µν µν α = λ1 [V , Aµν ] χ + iλ2 [V , Aµα] hµ LVAP h −i +iλ [▽µV , Aνα] u + iλ [▽αV , Aν ] uµ 3 h µν αi 4 h µν α i +iλ [▽αV , Aµν ] u (2.95) 5 h µν αi 16 2 Chiral Perturbation Theory

with non-dimensional real constants λi and

hµν := ▽µuν + ▽νuµ. (2.96)

The RχT Lagrangian [23] for such a channel is then given by

= (2) + kin + kin + (2) + (2) + (2) . (2.97) LRχT Lχ LV LA LV LA LVAP In any case, the RχT Lagrangian can be used to compute the hadronic current including resonances. After rearranging the expression, the form factors can be identified by compar- ison with Eq. (2.74). In the low-energy region the results of RχT reproduce the results of χPT at order (p4). O In contrast, it is shown in [23] that the Kühn-Santamar´ıa model fails to reproduce χPT results at next-to-leading order (p4). This is of no surprise as it is only defined to have a O correct lowest order behaviour. The results of RχT and the corresponding form factors will be discussed for each τ decay channel individually in Chap. 4. 3 Implementation

This chapter deals with three important aspects of the HADRONS++ module:

the helicity amplitudes, • the numerical integration, and • the spin correlation algorithm. •

3.1 Helicity Amplitudes

For computing decay matrix elements it is adequate to use helicity amplitudes. It has the following advantages:

Amplitudes for certain helicity combinations can be evaluated as complex numbers, • and

it is straightforward to implement spin correlations. • This section intends to describe the helicity formalism, which is used to calculate decay matrix elements in HADRONS++.

3.1.1 Spinor Basis and Massive Spinors The spinors for (anti-)particles u with momentum p and helicity λ are solutions of the Dirac equation of motion

(p/ m)u(p,λ) = 0, (3.1) ∓ u¯(p,λ)(p/ m) = 0, (3.2) ∓ where the -sign refers to particles and the +-sign to anti-particles. m denotes the mass of − the (anti-)particle: m2 := p/2 = p2. (3.3) Summing over the two helicities, the spinors obey the completeness relation

+; particle, u(p,λ)u¯(p,λ) = p/ m (3.4) ± ; anti-particle. λ= ( X± − Their normalisation is chosen by

u¯(p,λ)u(p,λ) = 2m. (3.5) ± ( , ) It is useful to deﬁne chiral spinors w · · by introducing an arbitrary light-like four-vector k0: 1 w(k0,λ)w¯(k0,λ) := (1 + λγ )k/ (3.6) 2 5 0 18 3 Implementation

(k0,λ) (k0,λ) such that λ= w w¯ = k/0. Chiral spinors with opposite helicity can be deﬁned by introducing another± arbitrary four-vector k with k2 = 1 and k k = 0: P 1 1 − 0 · 1

(k0,λ) (k0, λ) w =: λk/1w − . (3.7)

The idea is to write spinors u(p,λ) in terms of chiral spinors w(k0,λ) acting as a spinor basis. They behave like massless spinors with momentum k0. Consequently, massive spinors [24] can be written as

(p,λ) p/ m (k0, λ) u := ± w − , (3.8) √2p k · 0 (p,λ) (k0, λ) p/ m u¯ : =w ¯ − ± , (3.9) √2p k · 0 where the +-sign refers to particles and the -sign to anti-particles. These spinors satisfy − the Dirac Eq. (3.1), and its conjugated Eq. (3.2). Although they are often referred to as helicity states, λ actually does only label an eigenstate of the helicity operator if m = 0. For m = 0, it rather refers to eigenstates of the operator γ5s/, where sµ is the polarisation given 6 through pµ m sµ := kµ (3.10) m − p k 0 · 0 such that s p = 0 and s2 = 1. However, the completeness relation (3.4) is satisﬁed by the · − spinors deﬁned in (3.8) and (3.9). That means that even though sµ depends on an arbitrary vector k0, this issue is irrelevant for calculating decay probabilities with unpolarised outgoing particles.

3.1.2 Spinor Products

The S-function is deﬁned as

(p1,λ1= ) (p2,λ2= ) S( ; p ,p ) :=u ¯ ± u ∓ . (3.11) ± 1 2 Using the abbreviation η := 2p k , (3.12) i i · 0 S can be calculated for both particles (+) andp anti-particles ( ): −

(p1,λ) (p2, λ) S(λ; p1,p2) =u ¯ u −

(k0, λ) p/1 m1 p/2 m2 (k0,λ) =w ¯ − ± ± w η1 η2

1 (k0,λ) (k0, λ) = trace w w¯ − (p/1 m1)(p/2 m2) η1η2 ± ± (k ,−λ) n=λk/1w 0 o

λ (|k0,{zλ)} (k0, λ) = trace w − w¯ − (p/1 m1)(p/2 m2)k/1 η1η2 ± ± 1 = (1 λγ5)k/0 n 2 − o | {z } λ 1 1 = trace (1 λγ5)k/0p/1p/2k/1 + m1m2trace (1 λγ5)k/0k/1  η1η2  2 − 2 −   n o n o =2k0 k1=0 · 2  µ ν ρ σ  = [λ(p1 k0)(p2 k1) λ(p1 k1)(p2 k0) | iǫµνρσp1 p{z2k0k1 ] . }(3.13) η1η2 · · − · · − 3.1 Helicity Amplitudes 19

λ λ [p , λ ; p , λ ; c , c ] 1 2 Y 1 1 2 2 R L ++ cRµ1η2 + cLµ2η1 + c S(+; p ,p ) − L 1 2 λ λ [p , λ ; p ; p , λ ; c , c ] 1 2 X 1 1 2 3 3 R L ++ µ µ η2c + µ2η η c + c S(+; p ,p )S( ; p ,p ) 1 3 2 L 2 1 3 R R 1 2 − 2 3 + c µ η S(+; p ,p )+ c µ η S(+; p ,p ) − L 1 2 2 3 R 3 2 1 2 λ λ λ λ [p , λ ; p , λ ; p , λ ; p , λ ; c1 , c1 ; c2 , c2 ] 1 2 3 4 Z 1 1 2 2 3 3 4 4 R L R L ++++ 2[S(+; p ,p )S( ; p ,p )c1 c2 + µ µ η η c1 c2 + µ µ η η c1 c2 ] 3 1 − 2 4 R R 1 2 3 4 L R 3 4 1 2 R L +++ 2η c1 [S(+; p ,p )µ c2 + S(+; p ,p )µ c2 ] − 2 R 1 4 3 L 1 3 4 R ++ + 2η c1 [S( ; p ,p )µ c2 + S( ; p ,p )µ c2 ] − 1 R − 3 2 4 L − 4 2 3 R ++ 2[S(+; p ,p )S( ; p ,p )c1 c2 + µ µ η η c1 c2 + µ µ η η c1 c2 ] −− 4 3 − 2 3 R L 1 2 3 4 L L 3 4 1 2 R R + ++ 2η c2 [S(+; p ,p )µ c1 + S(+; p ,p )µ c1 ] − 4 R 1 3 2 R 2 3 1 L + + 0 − − + + 2[µ µ η η c1 c2 + µ µ η η c1 c2 µ µ η η c1 c2 µ µ η η c1 c2 ] −− − 1 4 2 3 L L 2 3 1 4 R R − 1 3 2 4 L R − 2 4 1 3 R L + 2η c2 [S(+; p ,p )µ c1 + S(+; p ,p )µ c1 ] −−− 3 L 4 2 1 L 1 4 2 R

Table 3.1: , , -functions for diﬀerent helicity combinations. Combinations, which are X Y Z not listed, can be obtained by swapping + and L R simultaneously. ↔− ↔

On the other hand,

(p1,λ) (p2,λ) (k0, λ) p/1 m1 p/2 m2 (k0, λ) u¯ u =w ¯ − ± ± w − η1 η2

1 (k0, λ) (k0, λ) = trace w − w¯ − (p/1 m1)(p/2 m2) η1η2 ± ± 1 = (1 λγ5)k/0 n 2 − o 1 | 1 {z } = trace (1 λγ5)(k/0p/1m2 + k/0p/2m1) ±η1η2 2 − n o = η1µ2 + η2µ1. (3.14)

Here, 1 λγ trace ± 5 k/ p/ = 2k p = η2 (3.15) 2 0 i 0 · i i n o was used. The µi are deﬁned by mi µi := , (3.16) ± ηi where the positive sign refers to particles, whereas the negative sign refers to anti-particles.

3.1.3 Basic Building Blocks

The matrix elements of hadron and τ decays can be constructed using more complex structures than S-functions. These building blocks are

[p , λ ; p , λ ; c , c ] :=¯u(p1,λ1)[c P + c P ]u(p2,λ2), (3.17) Y 1 1 2 2 R L R R L L [p , λ ; p ; p , λ ; c , c ] :=¯u(p1,λ1)p/ [c P + c P ]u(p3,λ3), (3.18) X 1 1 2 3 3 R L 2 R R L L [p , λ ; p , λ ; p , λ ; p , λ ; c1 , c1 ; c2 , c2 ] :=¯u(p1,λ1)γµ[c1 P + c1 P ]u(p2,λ2) Z 1 1 2 2 3 3 4 4 R L R L R R L L (p3,λ3) 2 2 (p4,λ4) u¯ γµ[cRPR + cLPL]u . (3.19) 20 3 Implementation

The projectors PR and PL are deﬁned as 1 γ P := ± 5 . (3.20) R/L 2

Each building block can be evaluated as a function of the momenta pi, the helicities λi and the coupling constants cL/R. The value of these building blocks can be written in terms of cL/R, ηi, µi and S-functions. The results are given in Tab. 3.1. The calculation is similar to the calculation of the S-function in Eq. (3.13) but more involved. As an example the evaluation of an -function is given: Y [p , +; p , +; c , c ] =u ¯(p1,+)(c P + c P )u(p2,+) Y 1 2 R L L L R R ( ) p/1 + µ1η1 p/2 + µ2η2 ( ) =w ¯ − (cLPL + cRPR) w − η1 η2 1 ( ) ( ) = trace w − w¯ − (p/1 + µ1η1)(cLPL + cRPR)(p/2 + µ2η2) η1η2 n =PLk/0 o 1 = trace P| Lk/{z0p/1(c}LPL + cRPR)µ2η2 + PLk/0µ1η1(cLPL + cRPR)p/2 η1η2 1 n o = trace cLPLk/0p/1µ2η2 + cRPLk/0p/2µ1η1 η1η2 n o 1 2 2 = cLη1µ2η2 + cRη2µ1η1 η1η2 h i = cLη1µ2 + cRη2µ1. (3.21)

In a similar fashion, the -function of other helicity combinations and the -function can Y X be computed. For evaluating the -function two tricks are needed: Z the line-reversal trick • (p1,λ1) (p2,λ2) (p2, λ2) rev (p1, λ1) u¯ Γ u = λ1λ2u¯ − Γ u − ,

where Γ stands for any string of γ-matrices and Γ rev for the same string in reversed order, and

the Chisholm identity, which reads •

(p1,λ1) µ (p2,λ2) (p2,λ2) (p1,λ1) (p1, λ1) (p2, λ2) u¯α (γ )αβu (γµ)γδ = 2 u u¯ + 2 u − u¯ − , β γδ γδ where α,β,γ,δ are spinor indices.

3.2 Numerical Integration

The diﬀerential decay rate [25] of a particle with momentum P decaying into n particles with momenta pi is given by

2 dΓ(P p ...p )= |M| dΦ (p ,...,p ) . (3.22) → 1 n 2P f 1 n S denotes a combinatorial factor being necessary to avoid overcounting of identical config- S urations in case there are identical particles in the final state: If la denotes the number of identical final state particles of type a, the symmetry factor is 1 = . (3.23) S l ! a a Y 3.2 Numerical Integration 21

integration scheme error Monte Carlo ( 1 ) √N O 2 trapezoidal (N − d ) O 4 Simpson rule (N − d ) O

Table 3.2: Estimated error of various d-dimensional integration schemes using N points

dΦf (p1,...,pn) stands for the Lorentz invariant phase space factor of the n particle final state. In case spin effects are of no relevance, it is appropriate to average over incoming helicities and sum over outgoing helicities, i.e. 1 T := 2 (3.24) 2s + 1 |M| × S λ0,λX1,...,λn with s being the spin of the decaying particle. The differential decay width becomes 1 dΓ(P p1 ...pn)= T dΦf (p1,...,pn) . (3.25) → 2P0 phase space factor flux factor | {z } The Lorentz invariant phase space element is|{z} given by n n d4p dΦ (p ,...,p )= i δ p2 m2 Θ(p0) δ(4) P p . (3.26) f 1 n (2π)4 i − i i − i "i=1 # i=1 ! Y X 3.2.1 Monte Carlo Integration The integration of a differential decay width requires a multidimensional integration. A Monte Carlo integration [26] is preferred because the error does not depend on the dimension of the phase space. In contrast, the error of a classical integration scheme does, see Tab. 3.2. Hence, for high-dimensional integration, d 8, Monte Carlo techniques are more suited. ≥ The integral of a multidimensional function f in a volume V is given by

I = fdV. (3.27) Z Picking randomly N points xi in the volume V , the function f is calculated at every point xi (i = 1 ...N) to obtain an estimate of the integral

N V I V f = f(x ). (3.28) ≈ h i N i Xi=1 The standard deviation leads to the error estimate: Var[f] 1 E = V ( ) with Var[f]= f 2 f 2 . (3.29) r N ∼ O √N − h i

A naive approach would be to choose phase space points isotropically: Rambo [27] is such an algorithm, which generates uniformly distributed four-momenta. Decay amplitudes, however, usually contain resonances such that the structure has sin- gular behaviour at the resonant phase space points. The convergence of the Monte Carlo integration can be improved by choosing the phase space points unisotropically. This method is called Importance Sampling. Introducing the density of points g such that

gdV = 1, (3.30) Z 22 3 Implementation the integral in Eq. (3.27) can be rewritten as

f I = gdV. (3.31) g Z This can be interpreted as an integral over f/g with the nonuniform distribution of phase space points gdV . Its variance is therefore Var[ f ] becoming zero if f g. In order to g ≡ calculate the integral in Eq. (3.31), a transformation of variables requiring the integral over g has to be done. In case of f g, the integral over g would be the one to be calculated. ≡ Therefore, the best choice for g is an easy to integrate approximation to f. As a consequence, after transforming the variables, a weight factor needs be to introduced that consists of the Jacobian. For example, the peaking contribution [26] of a resonance with momentum transfer s and mass (width) MR (ΓR) is proportional to

1 g = . (3.32) (M 2 s)2 + M 2 Γ2 R − R R The value of s (s s s ) is determined according to min ≤ ≤ max

s = M 2 + M Γ tan ρ y(s ) y(s ) + y(s ) , (3.33) R R R × max − min min h i where ρ is a random number in the interval [0, 1]. The function y is deﬁned as

x M 2 tan y(x) := − R . (3.34) MRΓR

So the weight factor is y(s ) y(s ) w = max − min . (3.35) MRΓRg The integration algorithm, especially the density of points, plays an important role for choosing the decay kinematics.

3.2.2 Choosing Decay Kinematics

The kinematics for the decay P p ...p is chosen according to the following algorithm: → 1 n 1. Choose a phase space point x = (p ,...,p ) Φ isotropically (Rambo) or using a 1 n ∈ f nonuniform distribution of points.

1 2. Calculate the diﬀerential decay rate dΓ(x) = 2P T (x)dV (x) at the phase space point x.

3. Calculate the ratio dΓ(x) . (3.36) max dΓ(y) y Φ ∈ f

4. Pick a random number r with 0 r< 1. ≤ 5. Take this phase space point x if r is less than the ratio in (3.36). Otherwise reject this point and start again at step 1. 3.3 Spin Correlation 23

3.3 Spin Correlation

The spin of the τ-lepton is not directly observable but translates into correlations among its decay products. The spin correlation algorithm [28] for particle decay chains is therefore an important feature and has been fully implemented into SHERPA. The idea is to use spin density and decay matrices to pass spin information from mother to daughter particles and mutually among the daughters. Note that throughout these sections repeated indices are summed over.

3.3.1 Production of Particles in the Hard Process Consider a 2 n process where the helicities of the incoming (spin- 1 ) particles are denoted → 2 by κ1 and κ2. The helicities of the outgoing particles are λ1, . . . , λn. The matrix element is denoted by . Such a process might be Mκ1κ2;λ1...λn + + e e− τ τ − . (κ1) (κ2) → (λ1) (λ2) The kinematics of the outgoing particles is produced according to the distribution

n 1 2 ′ ′ ′ ′ ′ ′ T = ρκ κ ρκ κ κ1κ2;λ1...λn κ∗ κ ;λ ...λ′ δλiλ , (3.37) 1 1 2 2 M M 1 2 1 n i Yi=1 using the algorithm described in Sec. 3.2.2. For polarised beam particles the polarisation density matrix is 1 i i 2 (1 + ) 0 ρ ′ = P 1 i (i = 1, 2) (3.38) κiκi 0 (1 ) 2 − P with i denoting the polarisation of particle i. In case of an unpolarised particle i ( i = 0), P 1 P i ′ this reduces to ρ ′ = δκiκ . If both incoming particles are unpolarised, (3.37) reduces to κiκi 2 i 1 T (unpol) = 2. (3.39) 4 |Mκ1κ2;λ1...λn | κXi;λi 3.3.2 Decay of Particles After producing the n outgoing particles, the unstable ones need to decay. The decay of any particle with a spin density matrix σ(k) is handled in a decay algorithm depicted in Fig. 3.1. The spin density matrix for a particle that was produced in the hard process is given through

n (k) 1 (1) (2) (i) σλ λ′ = ρκ κ′ ρκ κ′ κ1κ2;λ1...λk...λn κ∗′ κ′ ;λ′ ...λ′ ...λ′ Dλ λ′ , (3.40) k k N 1 1 2 2 M M 1 2 1 k n i i σ i=1 Yi=6 k where n (1) (2) (i) Nσ = trace ρκ κ′ ρκ κ′ κ1κ2;λ1...λk...λn κ∗′ κ′ ;λ′ ...λ′ ...λ′ Dλ λ′ . (3.41) 1 1 2 2 M M 1 2 1 k n i i i=1 n Yi=6 k o (i) The decay matrices D ′ are unit matrices for stable and for yet to be decayed particles. λiλi The decay algorithm of a particle returns the corresponding decay matrix so that the decay matrices for unstable particles being already developed are not necessarily unit matrices. The algorithm, see Fig. 3.1, is recursive to ensure the development of a complete decay chain, in which all unstable hadrons are decayed with full spin correlation. In SHERPA two spin correlation treatments are implemented: 24 3 Implementation

(0) Decay algorithm; input: σ ′ λ0λ0

Select decay mode (1 m) and choose kinematics acc. to (0) → m ′ T = σλ λ′ λ0;α1...αm λ∗′ ;α′ ...α′ i=1 δαiα . 0 0 M M 0 1 m i Q

Initialise the decay matrices of the daughters (1 i m): (i) ≤ ≤ ′ D ′ = δαiα . αiαi i

Choose unstable daughter k; calculate its spin density matrix (k) 1 (0) m (i) σ ′ = σ ′ λ ;α ...α ...αm ∗′ ′ ′ ′ D ′ . αkα Nσ λ0λ 0 1 k λ ;α ...α ...αm i=1 αiα k 0 M M 0 1 k i=6 k i Q

Perform decay of k by calling the decay algorithm with σ(0) = σ(k).

Get decay matrix from the decay algorithm D(k) = D(0).

PP PP PP Are all unstable PP PP no PP daughters developed? PP PP PP PP yes

(0) 1 m (k) return Dλ λ′ = N λ0;α1...αm λ∗′ ;α′ ...α′ k=1 Dα α′ 0 0 D M M 0 1 m k k Q Figure 3.1: Decay algorithm for a particle with a spin density matrix σ(0). The helicity of the decaying particle is denoted by λ0, whereas the helicities of the daughters are denoted by α . denotes the decay amplitude. The normalisation constants N and N i Mλ0;α1...αm σ D are chosen such that the spin density matrices and decay matrices have unit trace. 3.3 Spin Correlation 25

full spin correlations and • “diagonal” spin correlations, which means that the oﬀ-diagonal elements of the spin • density and decay matrices are neglected.

For most spin-sensitive observables in the τ-systems, the oﬀ-diagonal elements can be neglected, which will be shown when discussing the results in Chap. 5.

4 Parameterisation of Tau Decay Channels

This chapters intends to describe and explain the parameterisation of a variety of τ-lepton decay channels in a form used for implementing matrix elements in HADRONS++.

4.1 Leptonic Channels

Consider a decay of the form

τ −(P ) ν (k )+ l−(p) +ν ¯ (k ) (l = e, µ), (4.1) −→ τ 1 l 2 where the momenta are written in brackets. Due to momentum conservation, the momentum transfer is given by q = P k = p + k . (4.2) − 1 2 According to the Feynman rules for electroweak interactions, the decay amplitude is given µ µ by two electroweak currents Lτ and Ll which are coupled by the W -boson. Its Feynman diagram is shown in Fig. 4.1 (a). The matrix element therefore reads

q q 2 µ ν ig gµν M 2 = − Lµ − W Lν (4.3) M 2√2 τ q2 M 2 l − W with g being the electroweak coupling constant. The leptonic currents are given by electroweak theory:

Lµ =u ¯(k1,λντ )γµ(a b γ )u(P,λτ ) and τ 1 − 1 5 µ (p,λ ) µ (k ,λ ) L =u ¯ l γ (a b γ )u 2 νl . (4.4) l 2 − 2 5 Purely left-handed currents are realised by setting the constants

ai = bi = 1 (i = 1, 2). (4.5)

ντ (k1) ντ (k1) GF √2

τ − (P ) W − l− (p) τ − (P ) l− (p)

ν¯l (k2) ν¯l (k2) (a) (b)

Figure 4.1: Feynman diagram for the leptonic channel (a) with a propagating W −-boson and (b) neglecting the propagation of W −. 28 4 Parameterisation of Tau Decay Channels

ντ (k1) GF √2 hn (pn)

. τ − (P ) .

h1 (p1)

Figure 4.2: Feynman diagram for the semi-leptonic channel with n outgoing mesons. The graph shows an eﬀective coupling between the leptonic and hadronic currents, where the propagation of the W −-boson is neglected.

Since the W -boson mass is 80.425 GeV, the momentum transfer in τ-lepton decays is much 2 smaller than MW : q2 = P 2 2P k < M 2 M 2 . (4.6) − · 1 τ ≪ W Hence, the contributions arising from the propagator term can be neglected. In the limit q2 0 the matrix elements reads → qµqν 2 gµν 2 2 ig µ − MW ν g µ gµν ν GF µ = lim − Lτ 2 Ll = − Lτ 2 Ll = Lτ Ll,µ, (4.7) M q2 0 2√2 q2 M 8 M √2 → − W − W where GF denotes the Fermi constant deﬁned by

2 GF g := 2 . (4.8) √2 8MW The corresponding Feynman diagram is shown in Fig. 4.1 (b) and known as Fermi’s pointlike interaction. The amplitude then reads G F (k1,λντ ) µ (P,λτ ) (p,λl) (k2,λν ) = u¯ γ (a1 b1γ5)u u¯ γµ(a2 b2γ5)u l M √2 − − GF 1 1 2 2 = [k1, λν ; P, λτ ; p, λl; k2, λν ; c , c ; c , c ] (4.9) √2 Z τ l R L R L with the coupling constants ci = a b (i = 1, 2). R/L i ∓ i It can be shown that the contribution from the propagator [25] to the total decay width is less than 0.03%:

Γ 3 M 2 m2 1.00029; electron, =1+ τ 2 l = (4.10) (q2 M 2 ) 2 2 Γ ≪ W 5 MW − MW (1.00028; muon, where Γ denotes the total decay width without neglecting q2 M 2 , see Fig. 4.1 (a). ≪ W 4.2 Semileptonic Channels: General Remarks

Consider a τ-lepton decaying into a ﬁnite number of hadrons

− τ −(P ) ν (k)+ h (p )+ ...h (p ) , (4.11) → τ 1 1 n n n o 4.2 Semileptonic Channels: General Remarks 29

ντ (k) ντ (k) V GF V GF ud √2 us √2

0 0 τ − (P ) ρ− π (p2) τ − (P ) K∗− π (p2)

π− (p1) K− (p1)

(a) (b)

Figure 4.3: Examples for strangeness-conserving (a) and strangeness-changing (b) channels. The double line represents resonances: the ρ resonance for ∆S = 0 and K∗-meson the ∆S = 1. ± where the momenta are written in brackets. The momentum transfer equals the sum of the hadrons’ momenta: n q = P k = p . (4.12) − i Xi=1 The general decay amplitude using Fermi’s pointlike interaction is

GF µ = L Hµ. (4.13) M √2

Hµ denotes the hadronic current, which can be obtained from the χPT Lagrangian and extrapolated to higher energies, K¨uhn-Santamar´ıa model, • or

from the RχT Lagrangian directly. • It is deﬁned as the current

H = V h h ...h V A 0 . (4.14) µ CKM × h 1 2 n | µ − µ| i χPT+extrapolation; RχT The corresponding Feynman diagram is depicted| in Fig.{z 4.2. There} are two types of semileptonic decay channels: strangeness-conserving, ∆S = 0, and • strangeness-changing, ∆S = 1, channels. • ± Some examples are shown in Fig. 4.3. The corresponding CKM matrix elements VCKM µ are Vud and Vus, respectively. L denotes the leptonic current given by electroweak theory (a = b = 1 corresponds to a purely left-handed current)

Lµ =u ¯(k,λν )γµ(a bγ )u(P,λτ ), (4.15) − 5 which has to be used for writing the matrix element in terms of the block functions , , X Y and . Note that the -function is linear in its second argument p , i.e. Z X 2 [p , λ ; αp + βp′ ; p , λ ; c , c ]=α [p , λ ; p ; p , λ ; c , c ] X 1 1 2 2 3 3 R L X 1 1 2 3 3 R L +β [p , λ ; p′ ; p , λ ; c , c ]. (4.16) X 1 1 2 3 3 R L 30 4 Parameterisation of Tau Decay Channels

Consequently, if the hadronic current is of the form

n H = V A p (4.17) µ CKM × i iµ Xi=1 with Ai being Lorentz invariant then the matrix element is given through n GF = Ai ν ,i,τ VCKM, (4.18) M √2 X τ × Xi=1 where is an abbreviation for Xντ ,i,τ : =u ¯(k,λν )p/ (a bγ )u(P,λτ ) Xντ ,i,τ i − 5 = [k, λ ; p ; P, λ ; c = a b, c = a + b]. (4.19) X ν i τ R − L As a result, in order to write down the matrix element using block functions, one merely has to write the hadronic current in a form that matches Eq. (4.17), replace piµ by its corresponding -function and multiply the global factor GF . Therefore, only the X Xντ ,i,τ √2 hadronic currents will be of interest in the following sections.

4.3 The One-Pseudoscalar Mode

The hadronic current of the one-pseudoscalar mode

τ − h−ν (4.20) → τ with h π , K is simply given by a χPT current (2.69) − ∈ { − −} µ µ µ µ H = h−(p) V A 0 V = if √2p V (4.21) | − | × CKM − h × CKM with fh being the h decay constant

fπ = 92.4 MeV,

fK = 113 MeV, and Vud = 0.9744(1); h− = π−, VCKM = (4.22) (Vus = 0.2205(11); h− = K−. The values for the CKM matrix elements are taken from Ref. [25].

4.4 The Two-Pseudoscalar Mode with ∆S =0

The hadronic χPT current of the two-pseudoscalar mode

0 τ − h−h ν (4.23) → τ with h π, K is given by a current of the form ∈ { } µ 0 µ µ µ H = V h−(p )h (p ) V A 0 = √2(p p ) V C , (4.24) χPT ud 1 2 | − | 1 − 2 ud h where Ch is a Clebsch-Gordon coeﬃcient [22]

1; h = π, Ch = (4.25) 1 ; h = K. ( √2 4.4 The Two-Pseudoscalar Mode with ∆S = 0 31

ντ (k) GF √2

0 τ − (P ) ρ− h (p2)

h− (p1)

Figure 4.4: Feynman diagram for the strangeness-conserving two-meson channel. The line double line represents a resonance modelled in the form factors.

Due to the parity conservation property of χPT currents, the hadronic current is given by the vector current V only. Furthermore, since m 0 m 0 and m − m 0 , the vector µ π ≈ π K ≈ K current is conserved such that the hadronic current is transverse to qµ:

(p p ) qµ = (p p ) (p + p )µ = p2 p2 = 0. (4.26) 1 − 2 µ 1 − 2 µ 1 2 1 − 2 Therefore, only vector meson resonances are expected in this channel: ρ, ρ′, and ρ′′, see Tab. 4.1. Fig. 4.4 shows the vector resonance that needs to be introduced into the decay 2 amplitude by introducing a vector form factor FV (q ) [29]. As a result, the hadronic current reads Hµ = √2(p p )µF (q2)V C . (4.27) 1 − 2 V ud h 4.4.1 K¨uhn-Santamar´ıa Model The ansatz of K¨uhn-Santamar´ıa [7] is to consider the properties of a propagating ρ-meson, see Fig. 4.4. The energy dependence of the resonance width is incorporated in its propagator M 2 ρ (4.28) M 2 q2 iM Γ ρ − − ρ ρ through the substitution iM Γ i q2Γ (q2). (4.29) ρ ρ → ρ Since the ρ-meson decays into two pions withp a branching ratio of

BR(ρ 2π) 100%, (4.30) → ≈ its oﬀ-shell width [7] is

M 2 ~p(q2) 3 Γ (q2)=Γ (M 2) ρ | | Θ(q2 4m2 ) (4.31) ρ ρ ρ q2 ~p − π | ρ| with Θ denoting the Heavyside function and 1 1 ~p(q2) = q2 4m2 , ~p = M 2 4m2 . (4.32) | | 2 − π | ρ| 2 ρ − π p q Thus, the vector form factor in the K¨uhn-Santamar´ıa model is simply

2 1 2 2 2 F (q )= BW (q )+ β BW ′ (q )+ γ BW ′′ (q ) (4.33) V 1+ β + γ ρ ρ ρ 32 4 Parameterisation of Tau Decay Channels

2 Resonance X MX [GeV] ΓX (MX ) [GeV] rel. strength

ρ− (ρ(770)− ) 0.7749 0.1490 ρ (ρ− ) 1.408 0.502 β = 0.167 ′− (1450) − ρ′′− (ρ(1700)− ) 1.700 0.235 γ = 0.050

Table 4.1: Resonances in the τ π π0ν , τ K K ν channel. The parameters [30] − → − τ − → − S,L τ are tuned input parameters to the model and do not necessarily represent physical masses or widths.

ντ (k) GF √2 π−, K−

0 τ − (P ) h (p2)

h− (p1)

Figure 4.5: Feynman diagram for the loop contribution to the strangeness-conserving two- meson channel in χPT. Particles running in the loop are the charged pion and kaon. with 2 2 Mρ,ρ′,ρ′′ BWρ,ρ′,ρ′′ (q )= (4.34) 2 2 2 2 M ′ ′′ q i q Γ ′ ′′ (q ) ρ,ρ ,ρ − − ρ,ρ ,ρ and β, γ some real constants. The used parameters arep displayed in Tab. 4.1. Note that the oﬀ-shell width is deﬁned such that far below the resonance region, i.e. q2 M 2, the vector form factor reproduces the χPT result at leading order (p2): ≪ ρ O

q2 M 2 F (q2) ≪ ρ 1. (4.35) V −→ 4.4.2 Resonance Chiral Theory Near the production threshold of the outgoing two hadron state, χPT at order (p4) well O describes the vector form factor. At one loop [16], see Fig. 4.5, it takes the form

2Lr(µ2) q2 m2 m2 1 m2 m2 F (q2)χPT =1+ 9 q2 Re A( π , π )+ A( K , K ) (4.36) V f 2 − 96π2f 2 q2 M 2 2 q2 M 2 π π ρ ρ with the function A containing the loop contribution:

m2 m2 m2 m2 5 σ + 1 A( , )=ln + 8 + σ3 ln , (4.37) q2 µ2 µ2 q2 − 3 σ 1 − where σ denotes the phase-space factor

4m2 σ := 1 . (4.38) s − q2 4.4 The Two-Pseudoscalar Mode with ∆S = 0 33

Lr(µ2) denotes an (p4) chiral counterterm renormalised at scale µ2, see App. A.2 for the 9 O (p4) χPT Lagrangian with the low energy constant L . O 9 Using the RχT Lagrangian [19, 31] in Eq. (2.94) for a vector resonances, the leading eﬀect that is induced by the ρ resonance is

F G q2 F (q2)RχT =1+ V V , (4.39) V f 2 M 2 q2 π ρ − where the couplings FV and GV characterise the strength of the ργ and ρππ couplings, respectively. It is suggested that F (q2) vanishes suﬃciently fast for q2 . This leads to V →∞

FV GV 2 = 1 (4.40) fπ at leading order in 1/NC and if higher mass states, such as ρ′ and ρ′′, are not considered. This expression, also known as vector meson dominance model, is therefore

M 2 F (q2)VMD = ρ . (4.41) V M 2 q2 ρ −

2 2 2 VMD q2 At low momentum transfer (q Mρ ) FV (q ) 1+ 2 . As a consequence, the vector ≪ ≈ Mρ meson dominance model predicts the chiral coupling L9 to be

2L9 1 2 = 2 , (4.42) fπ Mρ which is in agreement with the phenomenologically extracted value [32]. Obviously, Eqs. (4.36) and (4.41) can be combined to get an improved RχT form factor. Additionally, making a Dyson summation of the ρ self-energy corrections, i.e. introducing a ρ-width in the denomi- nator, leads to the improved RχT vector form factor [32]:

M 2 F (q2)= ρ V M 2 q2 iM Γ (q2) ρ − − ρ ρ q2 m2 m2 1 m2 m2 exp − Re A( π , π )+ A( K , K ) (4.43) × 96π2f 2 q2 M 2 2 q2 M 2 π ρ ρ with an oﬀ-shell ρ-meson width, i.e. the inverse of the imaginary part of the propagator,

2 Mρ 2 2 3 1 2 2 3 Γρ(q )= 2 2 Θ(q 4mπ)σπ + Θ(q 4mK )σK 96π fπ − 2 − n o M m2 m2 1 m2 m2 = ρ Im A( π , π )+ A( K , K ) . (4.44) −96π2f 2 q2 M 2 2 q2 M 2 π ρ ρ It can be seen in Eq. (4.37) that the imaginary part of the loop function is given by the phase space factor, i.e. m2 m2 ImA( , )= σ3Θ(q2 4m2). (4.45) q2 µ2 −

2 2 2 At q = Mρ , the resonance width Γρ(Mρ ) = 144 GeV, which provides quite a good approxi- exp mation to the experimental meson width, Γρ = 150.3 1.6 MeV [3]. ± Higher resonances, ρ′ and ρ′′, are not included in RχT, see Eq. (2.86). 34 4 Parameterisation of Tau Decay Channels

4.5 The Two-Pseudoscalar Mode with ∆S = 1 ± The χPT hadronic current for the strangeness-changing two pseudoscalar mode

τ − (Kπ)−ν (4.46) → τ is given by a vector current due to parity conservation

Hµ = V K(p )π(p ) V µ 0 χPT us h K π | | i V = us pµ pµ . (4.47) √2 K − π Since the pion and kaon mass are quite distinct, the vector current is not conserved anymore as it is the case in the strangeness-conserving two meson channel. In other words, the µ µ µ µ hadronic current is not transverse to q = p + pπ. A transverse vector of q is (p p K K − π − ∆Kπ q)µ with ∆ = m2 m2 > 0 because q2 Kπ K − π ∆ ∆ (p p Kπ q) q = q (p p ) Kπ q2 K − π − q2 · · K − π − q2 = p2 p2 ∆ (4.48) K − π − Kπ = 0. (4.49)

The hadronic current can be written as V ∆ ∆ Hµ = us pµ pµ Kπ qµ + Kπ qµ (4.50) χPT √2 K − π − q2 q2 transverse longitudinal such that two form factors [33] can be| introduced:{z } | {z } a vector form factor F (q2) carrying spin J = 1 and • V a scalar form factor F (q2) carrying spin J = 0. • S The hadronic current is then V ∆ µ ∆ Hµ = us p p Kπ q F (q2)+ Kπ qµF (q2) √ K − π − q2 V q2 S 2 V ∆ ∆ = us Kπ (F F )+ F pµ + Kπ (F F ) F pµ . (4.51) √ q2 S − V V K q2 S − V − V π 2 4.5.1 Kühn-Santamar´ıa Model The discussion for the Kühn-Santamar´ıa model form factor is essentially identical to the strangeness-conserving two-meson channel. The occurring vector resonance is the K(892)∗− - meson, whereas the scalar resonance is the spin-0 meson K0(1430)∗− . Their parameters are listed in Tab. 4.2. The influence of higher resonances, K(1410)∗ and K(1680)∗ , is neglected. The second-lightest K0∗-meson is the K0(1950)∗ -meson being heavier than the τ-lepton such that it can not occur as resonance. The vector and scalar form factors can be written directly:

2 2 FV (q ) = BW ∗− (q ) and (4.52) K(892) 2 2 FS(q ) = BW ∗− (q ) (4.53) K0(1430) with 2 MK∗ BWK∗ (q )= . (4.54) M ∗ q2 i q2Γ ∗ (q2) K − − K p 4.5 The Two-Pseudoscalar Mode with ∆S = 1 35 ±

2 Resonance X MX [GeV] ΓX (MX ) [GeV] form factor 2 K(892)∗− 0.89166 0.0508 vector, FV (q ) 2 K0(1430)∗− 1.412 0.294 scalar, FS(q )

Table 4.2: Resonances in the τ π K ν , τ K π0ν channel. The parameters are − → − S,L τ − → − τ input parameters to the model and do not necessarily represent physical masses or widths.

The expression for the oﬀ-shell resonance width has to be calculated from the K Kπ ∗ → decay:

2 2 3 2 2 MK∗ ~p(q ) 2 2 ΓK∗ (q )=ΓK∗ (MK∗ ) | | Θ(q (mK + mπ) ), (4.55) q2 ~p ∗ − | K | where

2 1 2 2 2 ~p(q ) = λ(q ,m ,mπ) and | | 2 q2 K q 1 ∗ p 2 2 2 ~pK = λ(MX ,mK ,mπ). (4.56) | | 2MX q

The function λ(a, b, c) is deﬁned as

λ(a, b, c):= a2 + b2 + c2 2ab 2ac 2bc − − − = [a (√b + √c)2] [a (√b √c)2]. (4.57) − × − −

4.5.2 Resonance Chiral Theory

In RχT the vector form factor can be derived in a similar fashion to the vector form factor for the ∆S = 0 case. For simplicity, K∗ stands for K(892)∗− . Hence, the vector form factor [33, 34] is given by

2 2 MK∗ FV (q )= 2 2 2 M ∗ q iM ∗ Γ (q ) K − − K V 3 2 r 2 r 2 2 2 exp q Re M ∗ (q )+ M ∗ (q ) Re L ∗ (q )+ L ∗ (q ) × 2f 2 K π K η − K π K η π h i(4.58) using

s 2Σ ∆2 k 1 M r (s)= − J¯(s)+ J¯(s) + and (4.59) PQ 12s 3s2 − 6 288π2 ∆2 L (s)= J¯(s) (4.60) PQ 4s 36 4 Parameterisation of Tau Decay Channels with the abbreviations

2 2 Σ:=ΣPQ = MP + MQ, ∆:=∆ = M 2 M 2 , PQ P − Q 2 2 2 2 ν := λ(q ,mP ,mQ), 1 ∆ M 2 Σ M 2 ν (s + ν)2 ∆2 J¯(s):= 2+ ln Q ln Q ln − , 32π2 s M 2 − ∆ M 2 − s (s ν)2 ∆2 ( P P − − ) M 2 M 2 M 2 ¯ 1 Σ P Q Q J ′(0) = 2 2 + 2 2 ln 2 , 32π (∆ ∆ MP ) ¯ J¯(s):= J¯(s) sJ¯′(0), and − 2 M 2 2 MP 2 Q 1 MP ln 2 MQ ln 2 k := µ − µ . 32π2 ∆ The constant µ2 appearing in the deﬁnition of k corresponds to the renormalisation constant. 2 2 In order to be in accordance with Ref. [31] the renormalisation scale is set to µ = Mρ . The 2 vector oﬀ-shell width ΓV (q ) [33] reads

2 MK∗ 1 3/2 2 2 2 2 ΓV (q )= 2 2 λ (q ,mK ,mπ)Θ s (mK + mπ) 128πfπ q − h +λ3/2(q2,m2 ,m2)Θ s (m + m )2 . (4.61) K η − K η i 2 2 2 At q = KK∗ , the resonance width ΓV (MK∗ ) = 55 MeV, which provides quite a good ap- exp proximation to the experimental meson width, Γ ∗ = 50.8 0.9 MeV [3]. K ± The scalar form factor [33] is found to be (K0∗ stands for K0(1430)∗− )

2 M ∗ 2 2 2 2 K0 fπ mπ + mK FS (q )= 2 2 2 1 1 2 2 M ∗ q iM ∗ Γ (q ) − − 4c M ∗ K K0 S " d K # 0 − − 0 i ImF exp ReF + 4 (4.62) × 4 1+(ImF )2 4 using the abbreviation

1 ∆2 F = 5q2 2Σ 3 Kπ J¯ (q2) 4 8f 2 − Kπ − q2 Kπ π 1 ∆2 + 3q2 2Σ Kπ J¯ (q2). (4.63) 24f 2 − Kπ − q2 Kη π The scalar oﬀ-shell width reads 2 2 3 2 ΓS(q )= 4 2 cd(q ΣKπ)+ cmΣKπ 32πfπMK∗ q − 0 h i λ3/2(q2,m2 ,m2 )Θ(q2 (m + m )2) × K π − π K 2 1 2 2 2 2 2 + 4 2 cd(q 7mK mπ)+ cm(5mK 3mπ) 864πfπ MK∗ q − − − 0 h i λ3/2(q2,m2 ,m2)Θ(q2 (m + m )2). (4.64) × K η − η K

The two constants cd and cm were introduced in the RχT Lagrangian for scalar resonances, Eq. (2.90). They have to fulﬁl 4cdcm 1= 2 (4.65) fπ 4.6 The Three-Pseudoscalar Mode 37

ﬁnal hadron state parameters resonances (123) (123) 2 (123) h1h2h3 A G m(123) X A V13 V23 + 2 2 0 0 π−π−π Vud 1 mπ 2mπ a1− ρ ρ 0 0 2 2 π π π− Vud 1 mπ mπ a1− ρ− ρ− + 1 2 2 2 0 0 K−π−K 2 Vud 1 mπ mπ + mK a1− ρ K∗ − 1 2 2 2 0 KS,Lπ−KS,L 2 Vud 1 mπ mπ + mK a1− ρ K∗− 0 −3 2 K π K V 0 m 0 a− ρ - − S,L 2√2 ud π 1 − 0 0 1 2 2 2 π π K− 4 Vus 1 mK 2(mπ + mK) K1− K∗− K∗− + 1 2 − 2 2 0 0 K−π−π 2 Vus 1 mK mπ + mK K1− K∗ ρ 0 −3 2 π K π V 0 m 0 K− ρ - − S,L 2√2 us K 1 − + 2 2 0 0 K−K−K Vus 1 mK 2mK K1− ρ ρ 1 2 2 K K K V 1 m 2m K− ρ ρ − S,L S,L − 2 us K K 1 − −

Table 4.3: Parameters for the respective three-pseudoscalar channels [35]. The last three columns display the resonances that are modelled in the form factors. A 2 stands for the three-particle axial resonance in q , i.e. a1(1260)− or K1(1400)− . Vij denotes for 2 the vector resonances in (pi + pj) , i.e. s and t. K∗ stands for K(892)∗ . by arguments similar to Eq. (4.40). Unfortunately, no more model independent constraints can be found so that one of the constants cd and cm needs to be ﬁxed by integrating over the phase-space and comparing the total decay width with experimental data. It will be shown in Sec. 5.2.3 that c 0.012 . . . 0.0125 GeV (4.66) d ≈ yields reasonable results.

4.6 The Three-Pseudoscalar Mode

Consider the three-pseudoscalar channel

τ − (h h h )−ν . (4.67) → 1 2 3 τ In the strict chiral limit, i.e. massless hadrons, the hadronic current becomes transverse due to the conservation of axial current in massless χPT. It is given by (the vector current V µ does not contribute due the conservation of parity)

Hµ = V h (p )h (p )h (p ) V µ Aµ 0 χPT,0 CKM × h 1 1 2 2 3 3 | − | iχPT,0 (123) 2√2A µ µ = V1 + V2 , (4.68) 3fπ =:C | {z } where A(123) are the coeﬃcients for the diﬀerent decay channels, see Tab. 4.3, and

qµqν V µ := gµν (p p ) (i = 1, 2), qµ := pµ + pµ + pµ. (4.69) i − q2 i − 3 ν 1 2 3 For convenience, the global constant C was introduced. When including one order of quark masses, the axial current is not conserved anymore. As a result, in χPT at leading order (p2), a contribution to the parity conserving hadronic current which is longitudinal with O 38 4 Parameterisation of Tau Decay Channels respect to q is expected:

µ µ 1 (q + p2) (p1 p3) µ HχPT = C (p1 p3) 2 · 2 − q × " − − 2 q m # n − (123) 1 (q + p ) (p p ) + (p p )µ 1 · 2 − 3 qµ 2 − 3 − 2 q2 m2 " − (123) # 1 X(123) + qµ (4.70) 2 q2 m2 − (123) o (123) 2 with the parameters X and m(123) given in Tab. 4.3. The terms in this expression can be rearranged to yield µ µ µ µ HχPT = C V1 + V2 + q F (4.71) × k with the longitudinal contribution denoted by F . To build up the decay amplitude, the chiral result has to be modiﬁed by resonances in allk possible channels, i.e. writing

Hµ = C V µF 1 + V µF 2 + qµF (4.72) × 1 A 2 A S 1 2 with the axial form factors FA and FA as well as the scalar form factor FS. The pioneering work [36] shows that an anomaly term has to be implemented in chiral Lagrangians violating the rule that the weak vector and axialvector current are responsible for the production of an even and odd number of pseudoscalars, respectively. It is shown in [37] that not all channels need an anomaly term. For instance, G-parity forbids a contribution to the three- 0 0 pion channel and Bose symmetry to the decay into π π K− because the anomaly term is antisymmetric. Therefore, the hadronic current can be parameterised in terms of four independent form factors: (123) µ 2√2A µ 1 µ 2 µ µαβγ H = V1 FA + V2 FA + q FS + iε p1αp2βp3γFV (4.73) 3fπ with FV being the vector form factor. At this stage, a rule that translates the anomaly term in the current into a helicity µαβγ amplitude needs to be introduced because a term like iε p1αp2βp3γ does not match with Eq. (4.17). The product µαβγ µ ε p1αp2βp3γ =: p (4.74) is of course a four-momenta, which can be computed explicitly:

0 p p11p22p33 + p12p23p31 + p13p21p32 p11p23p32 p13p22p31 p12p21p33 1 − − − p p10p22p33 p12p23p30 p13p20p32 + p10p23p32 + p13p22p30 + p12p20p33   = − − −  . p2 p p p + p p p + p p p p p p p p p p p p 10 21 33 11 23 30 13 20 31 − 10 23 31 − 13 21 30 − 11 20 33 p3  p p p p p p p p p + p p p + p p p + p p p    − 10 21 32 − 11 22 30 − 12 20 31 10 22 31 12 21 30 11 20 32    (4.75) The idea is to use this four-momenta directly as second argument of the -function [24]. X The corresponding -function is X

p = [k, λ ; p; P, λ ; c , c ], (4.76) Xν, ,τ X ν τ R L which can be computed using the formulae in Tab. 3.1 with

p2 η2 = 2k0 p and µ2 = . (4.77) · pη2 p 4.6 The Three-Pseudoscalar Mode 39

ντ (k) ντ (k) GF GF √2 √2

h3 (p3) h3 (p3) V23 V13 τ − (P ) A τ − (P ) A h2 (p2) h1 (p1)

h1 (p1) h2 (p2) (a) (b)

Figure 4.6: Feynman diagram for the three-meson channel. A stands for a modelled axial resonance whereas Vij stands for a modelled vector resonance in the axial form factors. There are two amplitudes to this process (a) and (b). They are distinct if h = h . The two vector 1 6 2 resonances V13 and V23 may be diﬀerent, see Tab. 4.3.

invariant mass deﬁnition 2 2 q (p1 + p2 + p3) 2 s (p1 + p3) 2 t (p2 + p3)

Table 4.4: Notation of invariant masses in the three-pseudoscalar channel.

4.6.1 K¨uhn-Santamar´ıa Model The Three- and Two-Body Resonances

This channel is dominated by a three-particle axial resonance A and two two-particle vector resonances V13 and V23 in the s and t channel, respectively, cf. Fig. 4.6.

It is assumed that the form factors are dominated by the axial resonance a1(1260)− or K− depending on whether the current has ∆S = 0 or ∆S = 1. The authors of 1(1400) ± [38] suggested to distinguish between the Breit-Wigner terms of the axial resonances in the strangeness-conserving and strangeness-changing current:

2 2 Ma1 BWa (q )= and 1 M 2 q2 i q2Γ (q2) a1 − − a1 2 M iMK ΓK 2 K1 − p1 1 BWK1 (q )= 2 2 . (4.78) M q iMK ΓK K1 − − 1 1 There are some strangeness-changing channels which are best described by two axial resonances. This can be done by replacing

BWK− +α BWK− 2 2 1(1400) 1(1270) BWK1 (q ) = BWK− (q ) . (4.79) 1(1400) → 1+ α

The oﬀ-shell width Γ (q2) is calculated in [7] by computing dΓ (a 3π). It is found that a1 dq2 1 → the imaginary part of the inverse a1 propagator is ﬁxed through

g(q2) q2Γ (q2)= M Γ (M 2 ) (4.80) a1 a1 a1 a1 2 g(Ma1 ) p 40 4 Parameterisation of Tau Decay Channels

2 resonance X MX [GeV] ΓX (MX ) [GeV] rel. strength

A: a1(1260)− 1.254 0.599 K1(1400)− 1.402 0.174 K1(1270)− 1.240 0.090 α = 2.5 V13, V23: ρ 0.773 0.145 ρ 1.370 0.510 β = 0.145 ′ − K(892)∗ 0.892 0.051

Table 4.5: Parameters of the resonances in the three-pseudoscalar channels as they are used in the axial and scalar form factors of the K¨uhn-Santamar´ıa model [38]. A, V13, and V23 stand for the axial and vector resonances in Fig. 4.6, respectively. with the three meson phase space factor

2 1 2 2 2 2 g(q )= − dsdt V BW (s) + V BW ′ (t) + 2V V Re[BW (s) BW ′ (t)∗] (4.81) q2 1 | ρ | 2 | ρ | 1 · 2 ρ ρ Z using

2 2 (√q mπ ) tmax(s) dsdt = − ds dt 2 Z Z4mπ Ztmin(s) 1 2 t (s):= (q2 m2 )2 λ1/2(q2,s,m2 ) λ1/2(s,m2 ,m2 ) . (4.82) min/max 4s − π − π ± π π h i The two-body channels are dominated by the vector mesons V ρ, K . The ij ∈ { (892)∗ } corresponding parameters are displayed in Tab. 4.5. Their Breit-Wigner terms are

BWρ(x) in Eq. (4.34); Vij = ρ BWVij (x)= (4.83) (BWK∗ (x) in Eq. (4.54); Vij = K∗ with x s,t . ∈ { }

The Axial and Scalar Form Factors

The vector meson dominance model [35], i.e. using Breit-Wigner propagators for the resonances, yields the hadronic current

2√2 Hµ = A(123) BW (q2)ζµν(q)Γ BW (s)ζαβ (p + p )(p p ) VMD A A να V13 V13 1 3 1 3 β 3fπ − n 2 µν αβ + BWA(q )ζ (q)Γνα BWV (t)ζ (p1 + p3)(p2 p3)β A 23 V23 − 1 X(123) + qµ , (4.84) 2 q2 m2 − (123) o (123) where the non-resonant contribution ( X ) was added by hand. Γµν is proportional to ∝ µν the vertex describing the axialvector(A)-vector(V )-pseudoscalar(P ) coupling and ζX (q) is µν qµqν an abbreviation for g 2 . It is shown in [35] that the coupling A(q) V (k)P can be − MX → written as q q 1 q k Γ = g µ ν + µ ν . (4.85) µν µν − q2 m2 2 q2 m2 − (123) − (123) 4.6 The Three-Pseudoscalar Mode 41

ﬁnal hadron state parameter resonances (123) h1h2h3 B V V13 V23 + 0 0 K−π−K 2 ρ− ρ K∗ 0 KS,Lπ−KS,L 2 ρ− ρ K∗− + − 0 0 K−π−π 2 K∗− K∗ ρ 0 −4 π−KS,Lπ 3 K∗− ρ− -

Table 4.6: Three-meson channels with an anomaly term. The parameter B(123) is used for the vector form factor. The last three columns display the resonances whose parameters are used in the vector form factor. K∗ stand for K(982)∗ .

Allowing two vector resonances in s, V13 and V13′ with a relative strength deﬁned by β13, and similarly in t, V23 and V23′ with β23, BWVij in Eq. (4.81) has to be replaced by

1 ′ BWVij BWVij +βij BWV . (4.86) → 1+ βij ij n o µ Finally, HVMD can be compared with the hadronic current in Eq. (4.73) in order to obtain the general formulae for the axial form factors:

2 2 β BW ′ (s) 2 2 1 2 BWV13 (s) 1 m1 m3 13 V13 1 m1 m3 FA = BWA(q ) 1 −2 + 1 −2 1+ β13 − 3 M 1+ β13 − 3 M ′ ( V13 ! V13 ! 2 2 β BW ′ (t) 2 2 BWV23 (t) 2 m2 m3 23 V23 2 m2 m3 + −2 + −2 , (4.87) 1+ β23 × 3 M 1+ β23 × 3 M ′ V23 V23 ) 2 2 β BW ′ (t) 2 2 2 2 BWV23 (t) 1 m2 m3 23 V23 1 m2 m3 FA = BWA(q ) 1 −2 + 1 −2 1+ β23 − 3 M 1+ β23 − 3 M ′ ( V23 ! V23 ! 2 2 β BW ′ (s) 2 2 BWV13 (s) 2 m1 m3 13 V13 2 m1 m3 + −2 + −2 . (4.88) 1+ β13 × 3 M 1+ β13 × 3 M ′ V13 V13 )

− 0 − 0 Note that for those channels with G(123) = 0, i.e. G(K π KS,L) = G(π KS,Lπ ) = 0, the Breit-Wigner terms BW and BW ′ have to vanish: V23 V23

(123) ′ G = 0 BWV23 BWV 0. (4.89) ⇐⇒ ≡ 23 ≡ The general formula for the scalar form factor is

1 q2 M 2 F = X(123) + BW (q2) − A S 2(q2 m2 ) A q2M 2 − (123) ( A BWV13 (s) 2 2 2 2 m(123) q 2t s + 2m1 + m2 × 1+ β13 − − h m2 m2 1 − 3 m2 (q2 + s m2) q2(s M 2 ) − M 2 (123) − 2 − − V13 V13 i β BW ′ (s) 13 V13 + (V13 V13′ ) + (1 2), (s t) . (4.90) 1+ β13 ↔ ↔ ↔ ) h i h i 42 4 Parameterisation of Tau Decay Channels

ντ (k) ντ (k) GF GF √2 √2

h3 (p3) h3 (p3) V23 V13 τ − (P ) V τ − (P ) V h2 (p2) h1 (p1)

h1 (p1) h2 (p2) (a) (b)

Figure 4.7: Feynman diagram for the parity-violating three-meson channel. V , V13 and V23 label vector resonances that are displayed in Tab. 4.6.

2 resonance X MX [GeV] ΓX (MX ) [GeV] rel. strength V : ρ− 0.773 0.145 ρ 1.500 0.220 β = 0.25 ′− − ρ 1.750 0.120 γ = 0.04 ′′− − K(892)∗ 0.892 0.051 K 1.412 0.227 β = 0.25 (1410)∗ − K 1.714 0.323 γ = 0.04 (1680)∗ − V13, V23: ρ 0.773 0.145 ρ 1.370 0.510 β = 0.145 ′ − K 0.892 0.051 α = 0.2 (892)∗ −

Table 4.7: Parameters of the resonances in the three pseudoscalar channels as they are used in the vector form factor of the K¨uhn-Santamar´ıa model [39]. V , V13, and V23 denote the vector resonances in Fig. 4.7. The Vij are identical with the Vij resonances in Fig. 4.6. In the − 0 0 (π KS,Lπ ) π−KS,Lπ channel α = 0 because the K∗ resonance does not occur as V23: G = 0.

The Vector Form Factor Fig. 4.7 shows the parity violating vector-vector-pseudoscalar coupling which determines the vector form factor (123) 3B 2 FV = 2 2 TV (q )TV13V23 (s,t). (4.91) 8π fπ × Tab. 4.6 displays the resonances V , V13, and V23 whose parameters, see Tab. 4.7, are used for TV and TV13V23 , respectively. The required functions suggested by the authors of [39] are

2 1 2 2 2 T (q )= BW (q )+ β BW ′ (q )+ γ BW ′′ (q ) , ρ 1+ β + γ ρ ρ ρ 2 2 n o TK∗ (q ) = BWK∗ (q ),

1 BWρ(s)+ β BWρ′ (s) T ∗ (s,t)= + α BW ∗ (t) , and ρK 1+ α 1+ β K n o 1 BWρ(t)+ β BWρ′ (t) T ∗ (s,t)= + α BW ∗ (s) . (4.92) K ρ 1+ α 1+ β K n o 4.6.2 Resonance Chiral Theory In RχT there are only three modes being investigated so far: 4.6 The Three-Pseudoscalar Mode 43

τ π 2π0ν , • − → − τ τ π+2π ν , and • − → − τ τ K+K π ν . • − → − − τ The three-pion channels were studied in [23] and the pion-two-kaon channel is currently being worked out [40].

The Three-Pion Channel As mentioned in Sec. 2.8, the RχT Lagrangian in Eq. (2.97) is used to calculate the hadronic current in order to identify the form factors. The number of coupling constants is quite large: FV , GV , FA, λ1,...,λ5. To reduce this number, consider the form factors in the large Nc limit (approximated with only one octet of resonances). By similar arguments as in the two-pion channel, cf. Eq. (4.40), it is found that FV GV 2 1 2 = 0 and 2FV GV FV = 0. (4.93) − fπ − Additionally, the ﬁrst Weinberg sum rule [41] leads to F 2 F 2 = f 2. (4.94) V − A π In other words, these coupling constants can be written in terms of only one constant, the pion decay constant fπ: 1 FV = fπ√2, GV = fπ, and FA = fπ. (4.95) √2

It is also shown in Ref. [23] that the λi (i = 1 . . . 5) are not independent. They appear in one combination only, namely 1 1 λ0 := 4λ1 + λ2 + λ4 + λ5 , (4.96) −√2 2 h i which can not be determined from ﬁrst principles. The authors of [23] obtained a value for λ0 by ﬁtting the resulting form factors to data from ALEPH [42]: λ = 11.9 0.4. (4.97) 0 ± The relevant form factors are from Fig. 4.8 (a)-(d) from Fig. 4.8 (e) 3 s q2 F 1 = 1 β(q2,s,t), A − − 2 M 2 s iM Γ (s) − M 2 q2 iM Γ (q2) z ρ −}| − ρ ρ { z a1 − − a}|1 a1 { 3 t q2 F 2 = 1 β(q2,t,s), (4.98) A − − 2 M 2 t iM Γ (t) − M 2 q2 iM Γ (q2) ρ − − ρ ρ a1 − − a1 a1 where 3 x β(q2,x,y) := 2 M 2 x iM Γ (x) ρ − − ρ ρ 2q2 + x u F (q2,x) − − M 2 x iM Γ (x) ρ − − ρ ρ 2 u x F (q ,y) 2 − and − Mρ y iMρΓρ(y) x m − − F (q2,x) := λ π . (4.99) 2q2 − 0 q2 44 4 Parameterisation of Tau Decay Channels

ντ ντ GF GF √2 √2 π π π

τ − τ − π π

π π (a) (b)

ντ ντ GF GF √2 √2 ρ π π ρ τ − τ − π π π

π π (c) (d)

ντ GF √2 π ρ τ − a1 π

π (e)

Figure 4.8: Feynman diagrams of all contributions to the RχT form factors of the 3π-mode. (a) and (b) display the χPT diagrams. (c) and (d) are RχT diagrams with only one (vector) resonance, whereas (e) shows the two resonance RχT contribution.

The used values of the parameters are listed in Tab. 4.8. The ﬁrst part of the form factors comes from the sum of the diagrams in Fig. 4.8 (a) - (d), whereas the last term comes from the amplitude in Fig. 4.8 (e). The contribution from the scalar form factor is neglected because it vanishes with the square of the pion mass due to the conservation of vector and axial currents in the chiral limit: FS 0. (4.100) mπ=0 ≡

As stated above, the three-pion channels do not have any contribution from the anomaly terms. In other words, the vector form factor

F 0. (4.101) V ≡ The oﬀ-shell widths of the axial and vector resonances have to be computed as well because the resonances, especially the a1(1260), are rather wide. The energy-dependent widths 4.6 The Three-Pseudoscalar Mode 45

2 resonance X MX [GeV] ΓX (MX ) [GeV]

a1(1260)− 1.204 0.480 ρ 0.7751 0.145

K(892)∗ 0.892 0.051

Table 4.8: Parameters of the resonances in the three pseudoscalar channels as they are used in the RχT parameterisation [23]. The values do not necessarily represent physical masses and widths. are to handle the oﬀ-shell character of the resonances. The vector oﬀ-shell width does not deviate from the one obtained for the two-pion (two-kaon) channel:

Mρ 2 3 1 2 3 Γρ(s)= 2 2 Θ(s 4mπ)σπ + Θ(s 4mK )σK (4.102) 96π fπ − 2 − n o with the phase space factor

4mP σP = 1 . (4.103) r − s

On the other hand, the construction of the a1(1260)-meson width is much more involved. It would amount to a non-trivial two-loop calculation within a theory whose regularisation is still not well deﬁned. The authors of [23] therefore suggested to introduce a chiral-based oﬀ-shell width that endows the appropriate kinematical features, cf. Eq. (4.80):

α φ(q2) M 2 Γ (q2)=Γ (M 2 ) a1 Θ(q2 9m2 ), (4.104) a1 a1 a1 φ(M 2 ) q2 − π a1 where the function φ(q2) is given by, cf. Eq. (4.81),

2 2 2 2 2 2 φ(q )= q dsdt V BW (s) +V BW (t) +2(V V ) Re[BW (s) BW (t)∗] . (4.105) 1 | ρ | 2 | ρ | 1 · 2 ρ ρ Z n o

The introduced parameter α ruling the q2 behaviour in Eq. (4.104) can not be specified from first principles and has to be used as fit parameter determined from experimental data [42]. The authors of [23] found

α = 2.45 0.15. (4.106) ±

The Pion-Two-Kaon Channel

The channel τ K π K+ν has been implemented in the RχT parameterisation as well. − → − − τ The calculations are very involved but similar to the three-pion case. Using

∆ := m2 m2 (4.107) Kπ K − π 46 4 Parameterisation of Tau Decay Channels as before, the axial form factors are

1 1 3s ∆Kπ FA = 1+ 2 2 2 M s iMρΓρ(s) − M ∗ t iM ∗ Γ ∗ ρ − − K − − K K 2 3s 2 2 2 q + (2q + s u +∆Kπ)F (s,m ,q ) √2 − 2 − π − M 2 q2 iM Γ (q2) M 2 s iM Γ (s) a1 − − a1 a1 ρ − − ρ ρ h 2 2 ∆Kπ + (u s ∆Kπ)F (t,mK ,q ) + 2 − − , (4.108) MK∗ t iMK∗ ΓK∗ − − i 2 1 3t +∆Kπ FA = 1+ 2 2 M ∗ t iM ∗ Γ ∗ K − − K K q2 3t+∆Kπ + (2q2 + t u)F (t,m2 ,q2) √ 2 K 2 2 2 2 − 2 − − M q iMa Γa (q ) M ∗ t iM ∗ Γ ∗ a1 − − 1 1 K − − K K h 2 2 (u t)F (s,mπ,q ) + 2− (4.109) Mρ s iMρΓρ(s) − − i with a b F (a, b, c) := λ . (4.110) 2c − 0 c The scalar form factor is

2 2 3 mπ mK u FS = 2 2 1+ 2− 2 q mπ q − n 2 2 1 s(t u) t(s u) (q mK)∆Kπ + 2 2 − + −2 − − (4.111) 2q Mρ s iMρΓρ(s) MK∗ t iMK∗ ΓK∗ − − − − o and the vector form factor reads 3 FV = 2 2 −4π fπ 9 s t 2 2 2 + 2 −32π f × M s iM Γ M ∗ t iM ∗ Γ ∗ π ω − − ω ω K − − K K n 2 2 o2 2 3 1 c (q + s)+ mπ c (q + t)+ mK + 2 2 2 ·2 + 2 · (4.112) 4 Mρ q iMρΓρ(q ) × Mω s iMωΓω MK∗ t iMK∗ ΓK∗ − − n − − − − o with 2 3Mρ c := 1 2 2 . (4.113) − 8π fπ

4.7 The Four-Pion Mode

There are two 4π modes of the τ-lepton:

(i) the one-prong mode τ π (p ) π0(p ) π0(p ) π0(p ) ν and − → − 1 2 3 4 τ (ii) the three-prong mode τ π+(p ) π0(p ) π (p ) π (p ) ν . − → 1 2 − 3 − 4 τ The 4π channels have not been studied under RχT. Therefore, the form factors in HADRONS++ so far are parameterised according to the K¨uhn-Santamar´ıa model only. The construction of the 4π hadronic currents and its form factors is similar to the 3π case in the previous section. Hence, the discussion for the 4π channels is kept rather short. There are several subchannels carrying a resonance. Tab. 4.9 lists the used notation for invariant masses and displays the possible resonances. 4.7 The Four-Pion Mode 47

invariant mass deﬁnition invariant mass of ... 2 2 0 + 0 q (p1 + p2 + p3 + p4) π−3π (ρ−) π π 2π− (ρ−) 2 2 0 + r2 (q p2) π−2π ( ) π π−π− (a1−) 2 − 2 0 − + 0 0 r3 (q p3) π−2π ( ) π π π− (ω, a1) 2 − 2 0 − + 0 0 r4 (q p4) π−2π ( ) π π π− (ω, a1) − 2 0 − + 0 + s2 (p1 + p2) π−π (ρ−) π π (ρ ) 2 0 + 0 s3 (p1 + p3) π−π (ρ−) π π− (ρ ,σ,f0) 2 0 + 0 s4 (p1 + p4) π−π (ρ−) π π− (ρ ) t (p + p )2 2π0 ( ) π0π (ρ ) 3 2 3 − − − t (p + p )2 2π0 ( ) π0π (ρ ) 4 2 4 − − −

Table 4.9: Notation of invariant masses in τ 4πν subchannels. The last four columns → τ show the multi-hadron state that corresponds to the invariant mass for both the one-prong mode (i) and three-prong mode (ii) as well as corresponding resonances that might occur (in brackets). ω, f0 and a1 stand for ω(782), f0(980), and a1(1260), respectively. The σ particle is also known as f . A “ ” means that this subchannel does not resonate. 0(600) − 2 resonance X MX [GeV] ΓX (MX ) [GeV] rel. strength ρ 0.773 0.145 ρ 1.500 0.220 β = 0.25 ′ − ρ 1.750 0.120 γ = 0.04 ′′ −

Table 4.10: Parameters of the resonances in the four-pion channel (one-prong) [6]. The values do not necessarily represent physical masses and widths.

4.7.1 The One-Prong Decay 2 The only channels that might carry a resonance are the s2,3,4- and q -subchannels because the ρ0-meson does not decay into a 2π0 ﬁnal state due to isospin. Consequently, the hadronic 2 current contains the form factors Tρ(q ) and Tρ(sk) with k = 2, 3, 4 and 1 T (s)= BW (s)+ β BW ′ (s)+ γ BW ′′ (s) (4.114) ρ 1+ β + γ ρ ρ ρ n o using the usual Breit Wigner term Eq. (4.34) with the parameters displayed in Tab. 4.10. The hadronic current reads

4 2√3 2 k ν Hµ = 2 VudTρ(q ) Aµν Tρ(sk)(pk p1) , (4.115) fπ − Xk=2 ( ) ∗ 4 k (q| 2pl)µ(q {zpl)ν } Aµν = gµν − −2 . (4.116) − (q pl) l=2,l=k X6 − The expression ( ) in Eq. (4.115) can then be computed: ∗ 4 4 r (p p ) ( )= T (s ) p p (q 2p ) l · k − 1 . (4.117) ∗ ρ k  kµ − 1µ − − l µ r2  k=2 l=2,l=k l X  X6  This gives an expression of the hadronic current that is useful for the matrix element construction using helicity amplitudes, Eqs. (4.17) and (4.18). 48 4 Parameterisation of Tau Decay Channels

+ 0 the ωπ-contribution τ ω π ν π π π−π ν − → (782) − τ → − τ ω 0 + 0 the a π-contributions τ a− π ν π π−π−π ν 1 − → 1,(1260) τ → τ | {z − } a1 0 + 0 τ a π ν π π π−π ν − → 1,(1260) − τ →| {z } − τ 0 a1 + 0 the σρ-contribution τ σρ ν π π−π π−ν − → − τ → | {z τ } σ ρ− + 0 the f ρ-contribution τ f ρ ν π π−π π−ν 0 − → 0,(980) − |τ {z→}| {z } τ f0 ρ− | {z }| {z } Table 4.11: Contributions to the three-prong four-pion channel.

4.7.2 The Three-Prong Decay Various contributions, see Tab. 4.11, have to be taken into account for the three-prong mode. The hadronic current [43] is written in such a way that is takes all contributions into consideration:

1 ω a1 2 σρ 2 f0ρ 2 Hµ = αωJµ + αa1 Jµ Fa1 (q )+ ασρJµ Fσρ(q )+ αf0ρJµ Ff0ρ(q ) , (4.118) k αk n o where theP Breit-Wigner amplitudes are given by

2 1 0 1 2 2 2 3 2 F (q )= β + β BW (q )+ β BW ′ (q )+ β BW ′′ (q ) (4.119) k 3 i × k k ρ k ρ k ρ i=0 βk P i with k standing for the corresponding contribution. In general, both αk and βk can be complex and seen as free parameters. In SHERPA all four contributions have been fully implemented.

The ωπ-Current The main contribution is assumed to come from a three-body resonance, the ω-meson, in the following way: + 0 τ − ω π−ν π π π−π−ν . (4.120) → (782) τ → τ ω ω χ a The ωπ-current Jµ = Jµ + Jµ originates from | {z } χ a chiral part Jµ , i.e. the current from the chiral Lagrangian extended with resonance • propagators,

and an anomalous part J a containing the ρωπ coupling. • µ The chiral part is similar to the one in the one-prong mode, Eq. (4.115),

4 χ 2√3 2 k ν Jµ = 2 VudTρ(q ) RkAµν Tρ(sk)(pk p1) (4.121) fπ − Xk=2 with 1; k = 3, 4 Rk := (4.122) ( 2; k = 2 − and Tρ(s) given in Eq. (4.114) using the parameters in Tab. 4.12. The anomalous part is 4.7 The Four-Pion Mode 49

2 resonance X MX [GeV] ΓX (MX ) [GeV] rel. strength ρ 0.773 0.145 ρ 1.500 0.220 β = 0.25, σ = 0.02 ′ − − ρ′′ 1.750 0.120 γ = 0 ω(782) 0.790 0.0085 a1(1260) 1.230 0.600 σ 0.800 0.800 f0(980) 0.980 0.600

Table 4.12: Parameters of the resonances in the four-pion channel (three-prong) [6]. The values do not necessarily represent physical masses and widths. given by

J a =T˜ (q2)V G g F µ ρ ud ω3π ρωπ ρ× BW˜ (r2) p (r p p p r p p p ) ω 4 3µ 4 · 2 1 · 4 − 4 · 1 2 · 4 n +p (r p p p r p p p ) 2µ 4 · 1 3 · 4 − 4 · 3 1 · 4 +p (r p p p r p p p ) 1µ 4 · 3 2 · 4 − 4 · 2 3 · 4 +BW˜ (r2) p (r p p p r p p p ) ω 3 4µ 3 · 2 1 · 3 − 3 · 1 2 · 3 +p (r p p p r p p p ) 2µ 3 · 1 4 · 3 − 3 · 4 1 · 3 +p (r p p p r p p p ) (4.123) 1µ 3 · 4 2 · 3 − 3 · 2 4 · 3 o with the coupling constants [6]

3 Gω3π = 1476 GeV− , 1 gρωπ = 12.924 GeV− , 2 Fρ = 0.266Mρ . (4.124)

The introduced Breit-Wigner terms are

2 2 2 T˜ρ(q )= BW˜ ρ(q )+ σBW˜ ρ′ (q ), 1 ˜ ′ BW ρ,ρ ,ω(s)= 2 . (4.125) M ′ s iM ′ Γ ′ ρ,ρ ,ω − − ρ,ρ ,ω ρ,ρ ,ω It is convenient to split such an expression into smaller bits when actually implementing it. Deﬁning the function

D := r p −1 p +1 p r p +1 p −1 p , (4.126) k,l k · l l · k − k · l l · k where l 1 denotes the successor/predecessor of l in the (ordered) set l 1, 2, 3, 4 k . The ± ∈ { }\{ } successor of the last element is defined to be the first element and the predecessor of the first element is defined to be the last element. Eq. (4.123) can then be written as

4 4 a ˜ 2 ˜ 2 Jµ = Tρ(q )VudGω3πgρωπFρ BW ω(rk) plµDk,l, (4.127) k=3 l=1,l=k X X6 which perfectly ﬁts Eq. (4.17). 50 4 Parameterisation of Tau Decay Channels

The a1π-, σρ-, and f0ρ-Current

Another contributing three-body resonance is the a1(1260)-meson:

0 + 0 τ − a− π ν π π−π−π ν , → 1,(1260) τ → τ − a1 0 + 0 τ − a π−ν |π {zπ π−}π−ν . (4.128) → 1,(1260) τ → τ 0 a1 | {z } Additionally, the two-body resonances f0(980) and σ, which is also known as f0(600), contribute in the following way: + 0 τ − σ/f ρ−ν π π−π π−ν . (4.129) → 0,(980) τ → τ − σ/f0 ρ

Using the transverse projector for a four-vector pµ | {z }| {z } p p T (p)= g + µ ν , (4.130) µν − µν p2 the corresponding currents read

J a1 =T (q) µ µν × T νκ(r )BW (r2) BW (s )(p p ) + BW (s )(p p ) 2 a1 2 ρ 3 1 − 3 κ ρ 4 1 − 4 κ h T νκ(r )BW (r2)BW (s )(p p ) + BW (t )(p p ) − 4 a1 4 ρ 2 1 − 2 κ ρ 3 3 − 2 κ T νκ(r )BW (r2)BW (s )(p p ) + BW (t )(p p ) , (4.131) − 3 a1 3 ρ 2 1 − 2 κ ρ 4 4 − 2 κ J σρ =T (q) i µ µν × BW (s )BW (t )(p p )ν + BW (s )BW (t )(p p )ν , and (4.132) σ 3 ρ 3 4 − 2 σ 4 ρ 4 3 − 2 J f0ρ =hT (q) i µ µν × BW (s )BW (t )(p p )ν + BW (s )BW (t )(p p )ν . (4.133) f0 3 ρ 3 4 − 2 f0 4 ρ 4 3 − 2 h i The introduced Breit-Wigner terms BWσ(s) and BWf0 (s) are given by the usual term M BW = σ,f0 , σ,f0 M s i√sΓ (s) σ,f0 − − σ,f0 M 2 s 4m2 Γ (s)=Γ (M 2 ) σ,f0 − π Θ(s 4m2 ). (4.134) σ,f0 σ,f0 σ,f0 s M 2 4m2 − π σ,f0 − π !

a1 These expressions have to be written in a form that matches Eq. (4.17). The current Jµ consists of three terms of the form

νκ Tµν (q)T (r)C Apκ + Bqκ , (4.135) where A, B, and C are Lorentz invariant whereas rµ, pµ, and qµ denote four-momenta. After νκ contracting over Tµν(q)T (r), this term becomes Ar p + Br q q p q q q r Ar p + Br q C p A + q B r · · q A · + B · · · · (4.136) µ µ − µ r2 − µ q2 q2 − q2 r2 h i so that the decay amplitude can easily be constructed using Eqs. (4.17) and (4.18). The σρ f0ρ currents Jµ and Jµ contain terms like

AT (q)(p p )ν (4.137) µν i − 2 4.8 The Remaining Modes 51 which are equivalent to q (p p ) A p p + q · i − 2 , (4.138) 2µ − iµ µ q2 where A denotes a Lorentz invariant variable. Again, this is of great use when translating the current into a decay amplitude.

4.8 The Remaining Modes

The remaining channels, see Tab. 6.1, are implemented with an isotropic decay amplitude: the kinematics is only distributed according to the phase space. This approximation is justiﬁed because the branching ratio for each mode is very small.

5 Results and Discussion

This chapter is dedicated to the presentation of some results which were obtained by em- ploying the new HADRONS++ module.

5.1 Leptonic Channels

The leptonic channels can be treated analytically. So comparing the analytical solution of a distribution with the simulation provides an excellent sanity check if the implementation is correct. Setting a2 = b2 = 1 in Eq. (4.4), it can be shown that the energy spectrum [25] of the outgoing lepton is

2 2 2 dΓ(τ − l−ν¯lντ ) a1 + b1 GF ~p E 2 2 → = | | 3 12 Mτ + ml 2Mτ E dE 2 3(2π) × ( − (a + b )2 2M + 1 1 8M E 3M 2 m2 3+ τ (5.1) a2 + b2 τ − τ − l E 1 1 )

right-handed current (V+A)

2.5 a =-b =1

1 1

left-handed current V-A

a =b =1

1 1

analytical solution

2.0 ] -1

1.5 [GeV E /d d

1.0 1/

0.5

0.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

electron energy E [GeV] Figure 5.1: Energy spectrum of the outgoing electron in the τ rest frame. 54 5 Results and Discussion

with Mτ denoting the τ-lepton mass. The meaning of a1 and b1 is explained in Eq. (4.4). The lepton momentum is ~p = E2 m2, (5.2) | | − l q where ml is the lepton mass. The energy spectrum is generated for the two configurations µ leptonic current Le line in Fig. 5.1 a = b = 1 right-handed V µ + Aµ blue line 1 − 1 ⇔ a = b = 1 left-handed V µ Aµ red line 1 1 ⇔ − The corresponding histograms are shown in Fig. 5.1. It can be seen that the numerical histogram perfectly fits the analytical solution which is drawn with a black line. Using purely left-handed electroweak currents, i.e. a1 = b1 = 1, Eq. (5.1) simplifies to

dΓ G2 ~p = F | l| M E(3M 4E) m2(2M 3E) , (5.3) dE 12π3 τ τ − − l τ − h i which can be integrated such that the total decay width is

G2 M 5 m2 Γ= F τ f l , (5.4) 192π3 M 2 τ where the function f(x) is deﬁned as

f(x) := 1 8x + 8x3 x4 12x2 ln x. (5.5) − − − Using numerical integration, the total decay widths are

13 SHERPA Eq. (5.4) PDG [3] [10− GeV] τ e ν¯ ν Γ = 4.043 (9) 4.049 4.04 (1) − → − e τ τ µ ν¯ ν Γ = 3.853 (8) 3.938 3.93 (1) − → − µ τ The calculated decay widths are in accordance with the theoretical value from Eq. (5.4) and the experimental data published by the PDG [3].

5.2 Semileptonic Channels

The implementation of the resonances is the most important feature to be tested. Addi- tionally, the results of the K¨uhn-Santamar´ıa model are compared with the results obtained by the RχT parameterisation. The best way to see resonances is to measure the invariant masses of the outgoing hadrons. + Using the event-generator SHERPA, a τ τ −-pair is produced with the process

+ + e−e τ −τ (5.6) →

e+ τ +

Z/γ∗

e− τ − with a CMS energy of 92 GeV. 5.2 Semileptonic Channels 55

5.2.1 The Pion/Kaon Channel The channels τ π ν and τ K ν are two-body decays, which can be treated − → − τ − → − τ analytically. The hadronic current does not contain any resonances and can therefore be calculated directly from χPT, see Eq. (2.69). The total decay width [25] is

2 2 2 2 GF Vud 2 3 mh Γ(τ − h−ν )= | | f M 1 , h− π−, K− . (5.7) → τ 8π h τ − M 2 ∈ { } τ Numerical integration yields

13 SHERPA Eq. (5.7) PDG [3] [10− GeV] τ π ν Γ = 2.432 2.528 2.505 (25) − → − τ τ K ν Γ = 0.1604 0.1606 0.1554 (52) − → − τ The calculated decay widths are in agreement with the theoretical values from Eq. (5.7) and the experimental data published by the PDG [3]. The numerical integration does not give an error-estimate because the integrand does not depend on the phase space point and is therefore constant, see Eq. (B.6) in the appendix. Most kinematical observables of the one-meson channel are spin-sensitive and will be discussed in Sec. 5.3.

5.2.2 The Two-Pion Channel The two-pion channel, τ π π0ν , is considered for demonstrating the effect of a vector − → − τ resonance. Fig. 5.2 displays the two-pion final state invariant mass distribution using the Kühn-Santamar´ıa model (blue line) and the RχT parameterisation (red line). It can be seen that both the Kühn-Santamar´ıa model and the RχT parameterisation fits the experimental data [30] obtained by the CLEO collaboration. In the energy region above the resonance, see Fig. 5.3, the Kühn-Santamar´ıa model fits the data better. This is of no surprise as it includes higher resonances such as ρ′ and ρ′′ whereas the RχT only describes a single ρ resonance. Additionally, the Kühn-Santamar´ıa model with a single ρ resonance, i.e. using β = γ = 0 in Eq. (4.33), is depicted (dashed blue line). As expected, it has a similar shape to the RχT histogram describing only the influence of a single ρ resonance. Nevertheless, it can be concluded that it is the ρ resonance that dominates. The following table shows the integrated decay width for different parameterisations:

13 Γ [10− GeV] BR [%] KS model 4.92 (1) 21.7 KS model (β = γ = 0) 3.80 (3) 16.8 RχT 5.14 (4) 22.7 PDG [3] 5.75 25.4

The calculated branching ratio using the Kühn-Santamar´ıa model with three resonances, the ρ, ρ′, ρ′′-meson, has a similar value to the one obtained by RχT with only one resonance, the ρ-meson. The Kühn-Santamar´ıa calculation with only one resonance, i.e. β = γ = 0, yields a branching ratio of 4.9% smaller then the one obtained with three resonances. The branching ratio calculated from RχT is about 2.7% smaller in absolute value than the PDG value. Including higher resonances into the RχT framework might increase the branching ratio such that it fits the PDG value. This indicates that the RχT approach may prove more fruitful than the Kühn-Santamar´ıa model. The two-kaon decay channel, τ K K ν , is considered, too. The invariant mass − → − S,L τ distribution of the outgoing two-kaon final state is shown in Fig. 5.4. Clearly, the production threshold of the two-kaon final state is higher than Mρ so that the ρ-meson can not resonate. 56 5 Results and Discussion

2 mode

Kühn-Santamaría model

Kühn-Santamaría model = =0

Resonance Chiral Theory

CLEO ] 4 -1 [GeV m

3 /d d 1/

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

invariant mass m [GeV]

2 Figure 5.2: The invariant mass distribution of the two-pion ﬁnal state. The experimental data are published in [30].

2 mode

Kühn-Santamaría model

Kühn-Santamaría model = =0

Resonance Chiral Theory

CLEO

0 ] 10 -1 [GeV m /d d 1/

-1

-2

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

invariant mass m [GeV]

2 Figure 5.3: The invariant mass distribution of the two-pion final state in a logarithmic plot. In the energy region above 1 GeV, the Kühn-Santamar´ıa model fits the experimental data better due to its inclusion of higher resonances. 5.2 Semileptonic Channels 57

3.5

2K mode

Kühn-Santamaría model

3.0

Resonance Chiral Theory

2.5 ] -1

2.0 [GeV m /d

1.5 d 1/

1.0

'' resonance

0.5

0.0

0.8 1.0 1.2 1.4 1.6 1.8

invariant mass m [GeV]

2K Figure 5.4: The invariant mass distribution of the two-kaon ﬁnal state.

It can be seen that the K¨uhn-Santamar´ıa model predicts a resonant structure in the region close to the τ-lepton mass: the ρ′′ resonance. As a result, the K¨uhn-Santamar´ıa model is to be preferred in channels with non-negligible higher resonances.

5.2.3 The Pion-Kaon Channel There are two contributions to the pion-kaon decay width:

a “normal” contribution Γ resulting from the vector form factor and • n a scalar contribution Γ resulting from the scalar form factor. • s

Before using RχT, the constant cd needs to be ﬁxed, see Eq. (4.65). The calculated branching ratio of the τ π K ν /π0K ν decay as a function of c is plotted in Fig. 5.5 (left/right). − → − S τ − τ d The PDG values are 0.445% and 0.45%, respectively, so

c 0.012 . . . 0.0125 GeV (5.8) d ≈ is a good choice. The τ π K ν channel has been studied thoroughly by the CLEO collaboration [44] − → − S τ and will be discussed in more detail. The constant cd needed by RχT parameterisation is set to cd = 0.0125. The integrated decay width is:

KS model [GeV] RχT [GeV] vector form factor Γ = 6.97 (2) 10 15 7.62 (2) 10 15 n × − × − scalar form factor Γ = 2.76 (3) 10 16 2.38 (3) 10 15 s × − × − total Γ +Γ = 7.25 (2) 10 15 1.004 (5) 10 14 n s × − × − both form factors Γ = 7.23 (2) 10 15 1.002 (4) 10 14 × − × − 58 5 Results and Discussion

K mode 0 -

K mode 0.54

with error estimate

n s

0.52

0.50

0.48 BR [%] BR

0.46

0.44 branching ratio branching

0.42

0.40

0.38

0.010 0.012 0.014 0.010 0.012 0.014 0.016

constant c [GeV]

d Figure 5.5: The branching ratio of (left) τ π K ν and (right) τ π0K ν as a − → − S τ − → − τ function of cd. In addition, the left histogram contains the branching ratio corresponding to Γn +Γs drawn with a dashed line.

The ﬁrst (second) row displays the integrated width using only the vector (axial) form factor, i.e. F 0 (F 0). The sum is shown in the third row (“total”). The last row contains S ≡ V ≡ the integrated decay width using both form factors. It can be seen that for both models

Γn +Γs = Γ (5.9) as it is expected. Fig. 5.5 (left) shows that in RχT this identity holds true for a greater range of cd: The dashed line corresponds to Γn +Γs and lies within the error band of Γ. The scalar contribution to the total decay width is 3.8% in the Kühn-Santamar´ıa model and 24% in RχT. This difference between KS model and RχT is because cd is chosen such that the calculated branching ratio is correct. Only the RχT scalar form factor, Eq. (4.62), depends on cd, so that the choice of cd has a strong impact on the scalar contribution. The invariant mass distribution of the channel τ π K ν is depicted in Fig. 5.6. − → − S τ It clearly shows a K(892)∗− resonance. The relatively high scalar contribution of RχT makes the distribution fit the experimental data better. This can be seen by considering the RχT histogram with F 0 (dashed red line), which nearly coincides with the Kühn-Santamar´ıa S ≡ model histogram. Additionally, the other pion-kaon channels are considered. The total decay widths Γ(τ π0K ν ), Γ(τ π K ν ), and Γ(τ π K ν ) differ slightly due to the mass − → − τ − → − S τ − → − L τ difference between charged and neutral pions or kaons. Since m ± m 0 and m ± m , π ≈ π K ≈ KS,L the deviations are very small:

KS model [GeV] RχT [GeV] with cd [GeV] τ π0K ν Γ = 7.52 (2) 10 15 1.020 (4) 10 14 0.012 − → − τ × − × − τ π K ν Γ = 7.23 (2) 10 15 1.002 (4) 10 14 0.0125 − → − S τ × − × − τ π K ν Γ = 7.24 (6) 10 15 1.01 (1) 10 14 0.0125 − → − L τ × − × − 5.2 Semileptonic Channels 59

K mode

12 Kühn-Santamaría model

Resonance Chiral Theory

Resonance Chiral Theory F =0

CLEO ] -1

8 [GeV m /d

6 d 1/

0.4 0.6 0.8 1.0 1.2 1.4 1.6

invariant mass m [GeV] -

S Figure 5.6: The invariant mass distribution of the pion-kaon ﬁnal state.

The calculated branching ratio for all channels is about 0.32% (KS) and 0.45% (RχT), 0 respectively. The PDG value is 0.45% for the π K−-mode and 0.445% for the π−KS,L-mode. Clearly, the Kühn-Santamar´ıa model lacks a sufficient high scalar contribution whereas RχT is fitted to give a good result.

5.2.4 The Three-Pion Channel The one-prong and three-prong mode of the three-pion channel have only a small diﬀerence (123) in the scalar form factor FS, Eq. (4.90): The constant X diﬀers, see Tab. 4.3. However, the contribution from the scalar form factor is very small so that only the three-prong mode is discussed. The total decay width has two contributions:

the “normal” contribution Γ resulting from the axial form factors and • n the scalar contribution Γ resulting from the scalar form factor. • s The decay width for the τ π+2π ν channel using the K¨uhn-Santamar´ıa model nor- − → − τ malised to the electron decay width Γ = 4.043 10 13 GeV is: e × − SHERPA Ref. [35]

normal contribution Γn/Γe = 0.3634 (2) 0.356 scalar contribution Γ /Γ = 7.446 (6) 10 6 7.30 10 6 s e × − × − total Γ/Γe = 0.3634 (2) 0.356 The SHERPA results are in good agreement with the results published in Ref. [35], where the authors used slightly diﬀerent global constants explaining the small deviations between SHERPA and [35]. The scalar contribution to the total decay width is only 0.002% and can thus be neglected. The decay width corresponds to a branching ratio of

+ (KS) BR(τ − 2π−π ν ) 6.5%. → τ ≈ 60 5 Results and Discussion

1.2

3 mode

Kühn-Santamaría model

1.0

Kühn-Santamaría model =0

Resonance Chiral Theory

ALEPH

0.8 ]

0 -2

10 [GeV 2

-1

10 0.6 m /d d

-2

10 1/

0.4

-3

0.2

-4

0.25 0.7

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

invariant mass squared

2 2

m [GeV ]

2 Figure 5.7: Invariant mass distribution of the three outgoing pions. The experimental data are published in [42]. The inset shows the invariant mass distribution of the three outgoing pions near the production threshold.

The total decay width using the RχT parameterisation is

Γ(RχT) = 0.5101 (4), Γe which corresponds to a branching ratio of

+ (RχT) BR(τ − 2π−π ν ) 9.1%. → τ ≈ This value coincides with the measured value [3]

+ (PDG) BR(τ − 2π−π ν ) = (9.16 0.10)%. → τ ±

Note that RχT involves two constants, λ0 (4.97) and α (4.106), which were ﬁtted to experimental data. In the three-pion channel two kinds of resonances are present:

an axial resonance in the 3π invariant mass and • a vector resonance in the π+π final state. • − The invariant mass of the three-pion final state is shown in Fig. 5.7 for both the RχT parameterisation (red line) and the Kühn-Santamar´ıa model (blue line). The Kühn-Santa- mar´ıa model describes two vector resonances: the ρ- and ρ′-meson with relative strength β. However, the influence of the ρ′-meson is rather small: the dashed blue line in Fig. 5.7 corresponds to the Kühn-Santamar´ıa model with β = 0. It can be seen that both models are in good agreement with experimental data [42]. The distribution clearly shows resonant behaviour at the mass of the axial resonance. The low-energy region for both models is 5.2 Semileptonic Channels 61

4.0

3 mode

3.5 Kühn-Santamaría model

Kühn-Santamaría model =0

Resonance Chiral Theory

3.0 ] -1 - +

2.5 2.0

state

- -

1.6 state [GeV

2.0 m

1.2 /d d 0.8

1.5 1/

0.4

1.0 0.0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.5

0.0

0.4 0.8 1.2 1.6

invariant mass m [GeV]

2 Figure 5.8: Invariant mass distribution of the two outgoing pions. The main histogram shows the mπ−π+ distribution and the inset shows the non-resonant mπ−π− histogram. shown in the inset of Fig. 5.7. In the first bins RχT fits the experimental data better. This is expected, since in contrast to the Kühn-Santamar´ıa model, it is a next-to-leading order parameterisation, i.e. (p4). O + As expected, the invariant mass of the π−π final state shows a resonant behaviour at the ρ-meson mass region, see Fig. 5.8. The inset of Fig. 5.8 shows the invariant mass distribution of the other two-pion final state, π−π−, which does not have any resonant regions.

5.2.5 The Kaon-Two-Pion Channel The total decay width in K¨uhn-Santamar´ıa parameterisation of the τ K π π+ν chan- − → − − τ nel can be compared with the original work in Ref. [35]. In contrast to the three-pion channel, this mode has an additional anomalous contribution Γa resulting from the vector form factor: SHERPA Ref. [35]

normal contribution Γn/Γe = 0.03144 (2) 0.0313 scalar contribution Γ /Γ = 3.236 (3) 10 6 3.30 10 6 s e × − × − anomalous contribution Γa/Γe = 0.00202 (1) 0.0014 total Γ/Γe = 0.03346 (3) 0.0327

SHERPA reproduces the results from Ref. [35]. The scalar contribution to the total decay width is less than 0.01% so that the axial current is conserved. There are experimental data from the CLEO collaboration published in [44], where the 0 authors measured the KSπ−π invariant mass. The data show that not only one axial resonance needs to be considered but two:

a K - and • 1(1400) a K -meson with a relative strength α. • 1(1270) 62 5 Results and Discussion

- 0

K mode

Kühn-Santamaría model

CLEO

Kühn-Santamaría model ]

-1 Kühn-Santamaría model

with only K resonance

1(1400) with only K resonance

1(1270)

4 [GeV m /d

3 d 1/

0.8 1.0 1.2 1.4 1.6 1.8

invariant mass m [GeV] - 0

− 0 Figure 5.9: Invariant mass distribution mπ KSπ . The experimental data are published in [42]. This invariant mass has two resonant regions: a K1(1400)- and a K1(1270)-meson resonance.

0 3

K state

- 0

state 1

] 0 4 -1

0.6 0.8 1.0 1.2 [GeV

m K state

S /d

2 d

1/ 1

0.6 0.8 1.0 1.2

0.2 0.4 0.6 0.8 1.0 1.2 1.4

invariant mass m [GeV] 0 0 Figure 5.10: Invariant mass distribution of the two-particle ﬁnal states π−π , KSπ , and KSπ−. Only the two-pion ﬁnal state has a resonant behaviour (ρ resonance). 5.2 Semileptonic Channels 63

The distribution for the three-meson invariant mass is shown in Fig. 5.9. It shows two resonant regions. This is indicated by the two hatched areas that correspond to a simulation with only one axial resonance, either the K1(1270)- or the K1(1400)-meson. Fig. 5.10 shows that the 2π-ﬁnal state carries a ρ−-meson resonance. On the other hand, 0 it can be seen in the inset of Fig. 5.10 that the π KS ﬁnal state does not show any resonant − 0 behaviour, which is realised by G(K π KS,L) = 0, see Tab. 4.3.

5.2.6 The Pion-Two-Kaon Channel

The τ K π K+ν channel using the K¨uhn-Santamar´ıa model can be compared with − → − − τ the results published in the original article:

SHERPA Ref. [35]

normal contribution Γn/Γe = 0.00352 (2) 0.0037 scalar contribution Γ /Γ = 1.312 (2) 10 6 1.3 10 6 s e × − × − vector form factor Γa/Γe = 0.00242 (1) 0.0023 total Γ/Γe = 0.00594 (3) 0.0061

The SHERPA-results are in accordance with the results published in Ref. [35]. It can be seen that the relative contribution of the scalar part is about 37%.

5.2.7 The Four-Pion Mode

The 4π channel (three-prong mode) contains a variety of resonances listed in Tab. 4.9. Experimental data, i.e. invariant mass distributions of subchannels are published by the CLEO collaboration in Ref. [45]. The ωπ-current, Eqs. (4.121) and (4.123), is assumed to be the main contribution. The following resonances are parameterised:

invariant mass corresponding state resonances in the ωπ-current 2 + 0 q π π 2π− 2 + r2 π π−π− 2 + 0 a r3,4 π π π− ω in Jµ + 0 + χ s2 π π ρ in Jµ + 0 χ s3,4 π π− ρ in Jµ 0 t3,4 π π−

Fig. 5.11 shows the histogram generated by a simulation using only the ωπ-contribution, i.e. setting αω = 1 and αa1 = ασρ = αf0σ = 0 (see Eq. (4.118)), with no vector resonance in the 2 0 1 2 3 q -subchannel, i.e. setting βω = 1 and βω = βω = βω = 0 (see Eq. (4.119)). It can be seen 2 thatp the resonant subchannels √s2,3,4 and r3,4 are in accordance with the experimental 2 data. Ref. [45] states that the r2-subchannelq contains an a1-meson resonance. However, the a -contribution is rather small because the ωπ-current alone leads to a reasonable r2- 1 p 2 histogram. Only the √t -histogram does not coincide with the experimental data: It does 3,4 p not show resonant behaviour because the ωπ-current does not contain the ρ resonance in the √t3,4-subchannel. Resonances in the √t3,4-subchannel are described by the a1-, σρ-, and f0ρ-current, which need to be taken into account as well. However, this amounts to a ﬁt i of the independent complex parameters αk and βk with i, k = 1 . . . 4, see Eqs. (4.118) and (4.119). So only the ωπ-contribution is taken into account for generating the histograms in Fig. 5.11. 64 5 Results and Discussion

2 2 q r3,4 p q 8

2.5

+ - - 0 + - - 0

mode mode

SHERPA SHERPA ]

2.0 6 -1 CLEO

CLEO ] -1

+ - 0

+ - - 0 state [GeV 1.5

state [GeV m

4 q /d /d

1.0 3 d d 1/ 1/

0.5

0 0.0

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.4 0.6 0.8 1.0 1.2 1.4 1.6

invariant mass q [GeV] invariant mass m [GeV]

2 r2 √t3,4

2.0 p 3.0

+ - - 0

+ - - 0 mode

mode

SHERPA

2.5 SHERPA

CLEO

CLEO ] ]

1.5 -1 -1

resonance

2.0

- 0 + - -

state state [GeV [GeV m m

1.0 1.5 /d /d d d

1.0 1/ 1/

0.5

0.0 0.0

0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.2 0.4 0.6 0.8 1.0 1.2 1.4

invariant mass m [GeV] invariant mass m [GeV]

√s2 √s3,4

3.0

+ - - 0

mode

resonance

+ - - 0

mode 2.5 SHERPA

2.5 ] SHERPA

CLEO -1 ]

CLEO -1

2.0

+ 0

+ - state [GeV

state [GeV resonance m

1.5 m

1.5 /d /d d d

1.0

1.0 1/ 1/

0.5

0.0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

invariant mass m [GeV] invariant mass m [GeV]

Figure 5.11: Invariant mass distributions of the 4π channel considering only the ωπ- contribution. All relevant invariant masses are shown. The ωπ-current nearly describes all resonant structures apart from the √t3,4-subchannel. The experimental data is published in Ref. [45]. 5.3 Spin Correlations 65

5.3 Spin Correlations

The τ-lepton polarisation can be measured from the energy distribution of its decay products. The discussion of spin eﬀects is done for the channel τ − π−ντ using diﬀerent intermediate + → + particles produced in a e e− collision and decaying into a τ τ −-pair.

5.3.1 Intermediate Z-Boson + A τ τ −-pair is produced via a Z-boson with a centre of mass energy ECM = MZ so that the full process is e+ ντ

τ − π Z/γ∗ −

+ τ + π

ν¯ 2 e− τ using sin θW = 0.23147. It is shown in Ref. [46] that the spectrum of the variable 2E z := π , (5.10) MZ where the pion energy is calculated in the rest frame of the Z-boson, has negative slope. Its value is a measure for the longitudinal τ polarisation, d 1 dΓ = 2 = 2 ( 0.15), (5.11) dz Γ dz Pτ × − 2 and strongly depends on the value of sin θW . This is because the Weinberg angle is a measure for the mixing between the intermediate Z-boson and a virtual photon, which does not polarise the τ-lepton. The corresponding distribution is shown in Fig. 5.12. The black line corresponds to the histogram published in Ref. [46], where the authors used the event generator PYTHIA [5] and the τ decay library TAUOLA [6] for their analyses. The red line shows the histogram when the spin correlation algorithm is switched off, whereas the blue line show the histogram using the full spin correlation algorithm (solid blue line) and neglecting the off-diagonal elements (dashed blue line), see Sec. 3.3.2, respectively. The invariant mass distribution is depicted in Fig. 5.13 and it can be seen that the shape of the distribution depends on whether the spin correlation algorithm is switched on or off (blue and red line, respectively). The histograms produced by SHERPA coincide with those published in Ref. [46], which are again drawn with black lines. As a result, Figs. 5.12 and 5.13 show that SHERPA reproduces the results published in Ref. [46]. Furthermore, the effect of the the off-diagonal elements in the spin correlation algorithm can be neglected because the shape of the dashed blue histograms (without off- diagonal elements) coincide with the shape of the solid blue histograms (full spin correlation).

5.3.2 Intermediate W ±-Boson + + Consider the Z-boson producing a W W −-pair resulting in a µ τ − pair such that the full process is e+ ντ

τ − π− W − ν¯τ Z W + µ+

e− ν¯µ 66 5 Results and Discussion

1.4

1.2

1.0

z 0.8 /d d

0.6 1/

full spin correlation

0.4

no off-diagonal elements

no spin correlation

0.2 PYTHIA+TAUOLA

0.0

0.0 0.2 0.4 0.6 0.8 1.0

pion energy z

Figure 5.12: Energy distribution of the outgoing π− in the rest frame of the Z-boson. The black line corresponds to a simulation with PYTHIA [5] and TAUOLA [6] published in Ref. [46].

0.018

0.016

0.014

] 0.012 -1

0.010 [GeV m /d

0.008 d 1/

0.006

- +

state

full spin correlation

0.004

no off-diagonal elements

no spin correlation

0.002

PYTHIA+TAUOLA

0.000

0 20 40 60 80 100

invariant mass m [GeV] - +

Figure 5.13: Invariant mass distribution of the outgoing two-pion ﬁnal state. The shape of the invariant mass distribution depends on whether the spin correlation algorithm is switched on or oﬀ. 5.3 Spin Correlations 67

2.5

full spin correlation

no off-diagonal elements

2.0

no spin correlation

PYTHIA+TAUOLA

1.5 z /d d 1/

1.0

0.5

0.0

0.0 0.2 0.4 0.6 0.8 1.0

pion energy z

Figure 5.14: Energy distribution of the outgoing π− in the rest frame of the W −-boson. The black line corresponds to a simulation with PYTHIA [5] and TAUOLA [6] published in Ref. [46].

The τ polarisation is then = 1 [46] so that the distribution slope is maximally negative Pτ − and touches zero at z = 1. The respective distribution is shown in Fig. 5.14.

6 Summary and Outlook

The results of this thesis, mainly related to the implementation of a variety of τ-lepton decay channels in the new module HADRONS++, can be summarised in the following way:

1. The HADRONS++ module was developed. That is:

All necessary parameters, i.e. branching ratios, coupling constants, name of the • integrator, etc., are read from plain text ﬁles and stored in a separate folder. So the HADRONS++ module is easy to steer and can be equipped with a Web interface or some other user-friendly application. SHERPA decides for each hadron whether the hadron decay is already implemented • in HADRONS++ or not. If HADRONS++ knows its decay channels, branching ratios, etc., SHERPA calls HADRONS++ to perform the decay. Otherwise SHERPA calls the interface to PYTHIA to handle the decay. The spin correlation algorithm was fully implemented. • A variety of importance sampling integration schemes were implemented for in- • dividual decays.

2. 38 τ decay channels, see Tab. 6.1 and Ref. [3], were implemented to test the software architecture of the new module and to test the recently developed RχT against the popular K¨uhn-Santamar´ıa model. New decay channels can easily be implemented by the users of SHERPA and are automatically equipped with correct spin correlations.

3. The construction of χPT starting from massless QCD was shown in detail. The main part was the construction of the eﬀective Lagrangian using only the symmetry requirements from Goldstone’s theorem.

4. The thesis showed how χPT can be used to compute the decay amplitudes for τ-lepton decays. χPT in its original form is not able to describe the decays accurately because it does not contain the inﬂuence of resonances. This requires an extrapolation of the theory to higher energies. Two approaches were presented, implemented and tested:

the K¨uhn-Santamar´ıa model, which introduces form factors that were determined • using meson propagators (Breit-Wigner terms), and RχT, which is an improved theory that reproduces χPT in the low-energy limit. • It was shown that both approaches have their own advantages and disadvantages.

5. The parameterisation of a variety of τ-lepton decay amplitudes was described. Not only the amplitudes were presented but also how they emerge from ﬁrst principles. The amplitudes were written in terms of the -, -, and -functions, i.e. using the X Y Z helicity formalism, which was presented as well.

Tab. 6.1 displays all implemented τ-lepton decay channels. 70 6 Summary and Outlook

Still, a lot of theoretical work needs to be done in the ﬁeld of τ-lepton decays:

1. The majority of the decay channels are not parameterised using RχT: Work on RχT is still ongoing. New RχT parameterised channels are to be implemented, yet. This also means that for certain decay channels higher resonances need to be introduced into the RχT framework.

2. The authors of Ref. [47] recently developed the 5π amplitude using the K¨uhn-Santa- mar´ıa model. This amplitude has not been implemented, yet.

3. However, many semileptonic decay channels, especially those with 5 or more mesons, are still not parameterised – neither in K¨uhn-Santamar´ıa nor in RχT parameterisation. 71

τ − ντ + branching ratio [%] τ − ντ + branching ratio [%] → → 0 e−ν¯e 17.8400 π−4π 0.0950 µ−ν¯µ 17.3600 K−KS 0.0770 0 π−π 25.4200 K−KL 0.0770 0 π− 11.0600 K−KLπ 0.0755 0 0 π−2π 9.1700 K−KSπ 0.0755 + 0 2π−π 9.1600 K−2π 0.0580 + 0 π 2π−π 4.2500 π−KSKL 0.0550 0 + 0 π−3π 1.0800 K−K π−π 0.0420 0 K− 0.6860 K−3π 0.0380 + 0 0 2π−π 2π 0.5500 KSπ−2π 0.0312 0 0 K−π 0.4500 KLπ−2π 0.0312 π−KL 0.4450 K−η 0.0270 π−KS 0.4450 π−2KS 0.0240 + π−π K− 0.3300 π−2KL 0.0240 + 0 + 0 2π−π 3π 0.2748 3π−2π π 0.0181 0 π−π KS 0.1850 π−η 0.0130 0 π−π KL 0.1850 π−2η 0.0100 0 0 π−π η 0.1740 K−4π 0.0050 + + π−K K− 0.1550 K 2K− 0.0037

Table 6.1: List of all implemented τ-lepton decay channels.

Appendix A More on Chiral Perturbation Theory

A.1 Axial and Vector Current at Order (p2) O The χPT Lagrangian at order (p2) is given by O 2 (2) F µ = D Φ†D Φ+ χ†Φ+Φ†χ , (A.1) Leff 4 µ D E see Eq. (2.67), with the covariant derivatives D Φ= ∂ Φ ir Φ+ iΦl , µ µ − µ µ D Φ† = ∂ Φ† + iΦ†r il Φ†, (A.2) µ µ µ − µ and the field χ = 2B(s + ip) = 2B + . . . (A.3) M The field matrix Φ can be written in terms of the hermitian fields ϕa:

8 i Φ = exp λaϕa . (A.4) (F ) Xa=1 Additionally, the right- and left-handed external fields can be expanded in terms of Gell- Mann matrices: 8 a 8 a a λ a λ rµ = r and lµ = l . (A.5) µ √2 µ √2 Xa=1 Xa=1 The goal is to compute the axial and vector current in terms of the hermitian fields ϕa in order to obtain a closed form of the currents 0 V a Aa ϕb(p) and 0 V a Aa ϕb(p )ϕc(p ) . (A.6) µ − µ µ − µ b c D E a bD c d E The current for a three-particle state 0 Aµ ϕ (pb )ϕ (pc)ϕ (pd) can be computed in similar fashion. Therefore, it suffices to calculate the one- and two-particle current in order to show the principle. It will be shown that the χPT currents are parity conserving, i.e. a state with an odd (even) number of Goldstone fields is produced or annihilated by an axial (vector) current.

A.1.1 The Left- and Right-Handed Currents The generating functional Z[v,a,s,p] is deﬁned by the path-integral

iZ[v,a,s,p] i R d4x e = Φe Leff . (A.7) D Z At lowest order in momenta, = (2), the generating functional reduces to the classical Leff Leff action (2) iZ[v,a,s,p] i R d4x 4 (2) e = Φe Leff Z[v,a,s,p]= d x = S . (A.8) D ⇒ Leff 2 Z Z 74 A More on Chiral Perturbation Theory

The left- and right-handed currents are obtained by merely taking the corresponding derivatives with respect to the external ﬁelds δS δSkin J µ(x)= 2 = 2 and L δl (x) δl (x) µ l r 0 µ l r 0 ≡ ≡ kin ≡ ≡ µ δS2 δS2 J (x)= = . (A.9) R δr (x) δr (x) µ l r 0 µ l r 0 ≡ ≡ ≡ ≡ It is suﬃcient to consider only the kinetic part of the action

2 kin F 4 µ S = d x D Φ†(x)D Φ(x) (A.10) 2 4 µ Z D E because the breaking term in the Lagrangian, Φ χ + χ Φ , does not contain left- and ∝ † † right-handed gauge ﬁelds, see Eq. (A.3). Hence, the currents are given through

2 a;µ F δ 4 ν J (x)= d x′ D Φ†(x′)D Φ(x′) L 4 δla(x) ν µ Z l r 0 2 D E ≡ ≡ F ∂ ν = D Φ†(x)D Φ(x) , 4 ∂la ν µ l r 0 2 ≡ ≡ a;µ F δ 4 ν J (x)= d x′ D Φ†(x′ )D Φ(x′) R 4 δra(x) ν µ Z l r 0 2 D E ≡ ≡ F ∂ ν = D Φ†(x)D Φ(x) , (A.11) 4 ∂ra ν µ l r 0 ≡ ≡ respectively. The covariant derivatives have to be written in terms of ra and la by using µ µ Eq. (A.5): a a a λ a λ DνΦ= ∂ν Φ i r Φ l Φ and − ν √2 − ν √2 a X a a a λ a λ D Φ† = ∂ Φ† + i r Φ† l Φ† . (A.12) ν ν ν √ − ν √ a 2 2 X This leads to ν a ∂Dν Φ†D Φ λ µ µ = i (∂ Φ†)Φ Φ†∂ Φ and ∂la √2 − µ l r 0 ν ≡ ≡ a h i ∂Dν Φ†D Φ λ µ µ = i (∂ Φ)Φ† Φ∂ Φ† . (A.13) ∂ra √2 − µ l r 0 ≡ ≡ h i µ µ These expressions can be simpliﬁed by using ∂ (Φ†Φ) = ∂ (ΦΦ†) = 0, i.e. µ µ µ µ (∂ Φ†)Φ = Φ†∂ Φ and (∂ Φ)Φ† = Φ∂ Φ†, (A.14) − − such that ν ∂Dν Φ†D Φ a µ = i√2 λ (∂ Φ†)Φ and ∂la µ l r 0 ν ≡ ≡ D E ∂Dν Φ†D Φ a µ = i√2 λ (∂ Φ)Φ† (A.15) ∂ra µ l r 0 ≡ ≡ D E resulting in

2 a;µ F √2 a µ J = i λ (∂ Φ†)Φ and L 4 2 D E a;µ F √2 a µ J = i λ (∂ Φ)Φ† . (A.16) R 4 D E A.1 Axial and Vector Current at Order (p2) 75 O

A.1.2 The One-Goldstone Mode The aim is to evaluate the current

a a b a a b 0 V A ϕ (p) = 0 V A ϕ †(p) 0 (A.17) µ − µ µ − µ D E D E b b with ϕ †(p) being the creation operator of the state ϕ (p) and p its momentum. In order to compute the current, the ﬁeld matrix Φ has to expanded in terms of the hermitian ﬁelds ϕa:

i a a 1 a b a b Φ= ½ + λ ϕ λ λ ϕ ϕ + ..., F − 2F 2 a X Xa,b i 1 ∂ Φ= λa∂ ϕa λaλb∂ (ϕaϕb)+ . . . (A.18) µ F µ − 2F 2 µ a X Xa,b Eqs. (A.16) yield

a;µ F √2 a a′ µ a′ i a′ a′ JL = λ λ ∂ ϕ . . . ½ + λ ϕ . . . 4 * ′ − ! F ′ − !+ Xa Xa F √2 ′ ′ F √2 = λaλa ∂µϕa + = ∂µϕa + . . . and 4 · · · 2 a′ D E X =2δaa′ F √2 J a;µ = ∂µ|ϕa {z+ . .} . (A.19) R − 2 It is suﬃcient to consider only one order of ϕa in the currents. Higher orders do not contribute since any ϕa acts as an annihilation operator in Eq. (A.17). Therefore, an expression with more than one annihilation operator yields zero because (A.17) contains only one creation b operator, ϕ †(p). It can easily be seen that the vector current has no contribution:

a;µ µ µ V (x)= JR + JL = 0. (A.20)

It is only the axial current that contributes:

Aa;µ(x)= J µ J µ = F √2∂µϕa(x)= iF √2pµϕa(x) (A.21) R − L − − a where ϕa(x) eipa x was used. This shows the parity conserving property of the χPT ∝ · currents. As a result, the annihilation of a Goldstone ﬁeld with momentum p is given by

a b a b ab 0 A ϕ (p) = 0 A ϕ †(p) 0 = iF √2p δ , (A.22) − µ − µ µ D E D E where ϕaϕb 0 = δab 0 was used. † | i | i A.1.3 The Two-Goldstone Mode The aim is now to evaluate the current

a a b c a a b c 0 V A ϕ (p )ϕ (p ) = 0 V A ϕ †(p )ϕ †(p ) 0 . (A.23) µ − µ b c µ − µ b c D E D E In contrast to the two-Goldstone mode, the ﬁeld matrix Φ has to be expanded such that both a;µ a;µ JL and JR contain two annihilation operators instead of only one. Applying Eq. (A.16) 76 A More on Chiral Perturbation Theory yields

a;µ F √2 a b′ µ b′ i b′ c′ µ b′ c′ JL = λ λ ∂ ϕ λ λ ∂ (ϕ ϕ )+ . . . 4 *  ′ − 2F ′ ′  Xb Xb ,c   i c′ c′ ½ + λ ϕ . . . . (A.24) F ′ − ! + Xc Taking into account only those terms that contain two annihilation operators,

a;µ i√2 a b′ c′ µ b′ c′ 1 µ b′ c′ JL = λ λ λ (∂ ϕ )ϕ ∂ (ϕ ϕ ) 4 ′ ′ − 2 Xb ,c D E i√2 ′ ′ 1 ′ ′ ′ ′ = λaλb λc (∂µϕb )ϕc ϕb ∂µϕc 4 2 − b′,c′ D E X =2if ab′c′

i√2 | ab{z′c′ µ} µ b′ c′ = − f (pb′ pc′ )ϕ ϕ . (A.25) 4 ′ ′ − Xb ,c In a similar way, it can be shown that

a;µ a;µ i√2 ab′c′ µ µ b′ c′ JR = JL = − f (pb′ pc′ )ϕ ϕ . (A.26) 4 ′ ′ − Xb ,c This shows that for the two Goldstone state the axial current does not have any contribution,

a Aµ = 0, (A.27) whereas the vector current is given by

a i√2 ab′c′ µ µ b′ c′ Vµ = − f (pb′ pc′ )ϕ ϕ . (A.28) 2 ′ ′ − Xb ,c As result, the annihilation of two Goldstone ﬁelds is

a b c a b c i√2 abc µ µ 0 V ϕ (p )ϕ (p ) = 0 V ϕ †(p )ϕ †(p ) 0 = − f (p p ). (A.29) µ b c µ b c 2 b − c D E D E

A.2 Effective Lagrangian at (p4) O At the next-to-leading order in momenta, (p4), the most general Lagrangian [16] which is O invariant under chiral, charge, and parity transformation is given by 2 (4) µ µ ν =L D Φ†D Φ + L D Φ†D Φ D Φ†D Φ Leff 1 µ 2 µ ν D µ E νD ED µ E +L3 DµΦ†D ΦDνΦ†D Φ + L4 DµΦ†D Φ Φ†χ + χ†Φ 2 2 D µ E D ED E +L D Φ†D Φ(Φ†χ + χ†Φ) + L Φ†χ + χ†Φ + L Φ†χ χ†Φ 5 µ 6 7 − D E µνD E µν D E +L χ†Φχ†Φ+Φ†χΦ†χ iL F D ΦD Φ† + F D Φ†D Φ 8 − 9 R µ ν L µ ν D µν E D µν µν E +L10 Φ†FR ΦFL;µν + H1 FR;µν FR + FL;µνFL + H2 χ†χ . (A.30) D E D E The last two terms do not contain the pseudoscalar fields. In other words, the two coupling constants H1 and H2 are not direct measurable. The other ten coupling constants L1 ...L10 determine the low-energy behaviour of the Green’s functions. Appendix B The Two-Body Decay

Consider the τ (P ) π (p) ν (k) decay in the τ-lepton rest frame, i.e. P = (M ,~0). − → − τ τ Averaging over the initial state τ-spin, the amplitude squared can be written as 1 T = 2 = G2 f 2 V 2trace P k/p/P/p/ 2 |M| F π| ud| L λXτ ,λν n o = 4G2 f 2 V 2 2(k p)(P p) p2(P k) . (B.1) F π| ud| · · − · n =Mτ Eπ =Mτ Eν o The conservation of four-momentum | {z } | {z }

M E E τ = ν + π (B.2) ~0 ~k ~p leads to µ2 ~k = ~p and E = ± , (B.3) − π/ν 2M 2 2 2 where µ := Mτ mπ. Consequently, ± ± 2 2 2 2 2 T = 2GF fπ Vud 2µ+(k p) mπµ (B.4) | | · − − n o with 2 2 2 2 ~ µ (µ+ + µ ) µ k p = Eν Eπ k ~p = − 2 − = − (B.5) · − · 4Mτ 2 such that

2 2 2 2 2 2 2 T = 2GF fπ Vud µ+µ mπµ | | − − − 2 2 2n2 2 2 o = 2GF fπ Vud µ µ+ mπ = const.. (B.6) | | − − n 2 o =Mτ

This means, that a numerical integration over| a constant{z } is performed leading to a vanishing variance. Hence, the error-estimate becomes zero. Integrating over a two-body ﬁnal state phase space gives 2 2 2 2 GF Vud 2 3 mπ Γ(τ − π−ν )= | | f M 1 . (B.7) → τ 8π π τ − M 2 τ

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At this place I want to thank those people who have helped me in various ways and whose support has been invaluable during the period of my diploma studies.

First, I would like to thank my supervisors Prof. Dr. R. Schmidt and Dr. F. Krauss for the great opportunity of working in the SHERPA group and the support they gave me.

I am very grateful to all members of the Theoretical High Energy Physics group, Dr. Frank Krauss, Dr. Radoslaw Matyszkiewicz, Dipl.-Phys. Tanju Gleisberg, Dipl.-Phys. Stefan Höche, Dipl.-Phys. Steffen Schumann, Dipl.-Phys. Jan-Christopher Winter, Caroline Semm- ling, Timo Fischer, and Frank Siegert for all the help they have provided me and the pleasant working atmosphere. I am much obliged to Dr. Frank Krauss and Dipl.-Phys. Tanju Gleis- berg for teaching me a lot about C++-programming, software architecture and quantum field theory while I was working on this thesis. I also would like to thank the members of the group for carefully proofreading my thesis.

I want to thank especially Dr. Jorge Portolés Ibáñez and Pablo Roig Garcés of the University of Valencia, Spain. I have to thank for the kind hospitality and the stimulated discussions we had when I was in Valencia. I thank the DAAD and the University of Valencia for funding.

I would like to thank the staff of the Institute for Theoretical Physics, especially the secre- taries G. Schädlich, G. Latus, and U. Wächtler for the pleasant atmosphere.

Finally, I thank my friends for their encouragement and my family, especially my girlfriend, for their love and support. Without them I would never have got so far at all. Versicherung

Hiermit versicherere ich, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe; die aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht. Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde vorgelegt.

Thomas Laubrich Dresden, 31. M¨arz 2006