The Rare Decay of the Neutral Pion into a Dielectron

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Uppsala University

Department of Physics and Astronomy

Master’s Thesis

The Rare Decay of the Neutral Pion into a Dielectron

Author:

Hazhar Ghaderi

Supervisor:

Prof. Dr. Stefan Leupold

π ⁰

e ⁺

γ

^∗

e ⁻

γ

^∗

F

_π⁰_γ∗γ^∗

Nov 16, 2013

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Acknowledgments

Working on this project has been one of the best experiences of my life. I felt welcomed to the group from the very start, much thanks to the initiatives taken by my supervisor Stefan Leupold. He has been very supportive and understanding not only on a personal level but of course also on a professional level, taking time and very patiently explaining things when needed. It has been a pleasure to see him work through the problems I have presented to him. His work ethic is unmatched and he truly is a great role model as a scientist and as a human being.

I also would like to thank everyone at the division of Nuclear Physics and the division of High Energy Physics for making me feel at home in the building. In particular Gunnar Ingelman for all the interesting talks and for spreading his enthusiasm for physics. I would like to thank my officemate Carla Terschl¨usen for all the learning conversations. Thanks to Magnus Wolke (thanks for hadronm¨ote), Glenn Wouda, Carlos Granados, Carl Niblaeus, Aman Steinberg and many others for making my stay fun.

Finally I would like to thank my family and my extended family for their continuous support and encouragement.

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Abstract

We give a rather self-contained introduction to the rare pion to dielectron decay which in nontrivial leading order is given by a QED triangle loop. We work within the dispersive framework where the imaginary part of the amplitude is obtained via the Cutkosky rules.

We derive these rules in detail.

Using the twofold Mellin-Barnes representation for the pion transition form factor, we derive a simple expression for the branching ratio B(π⁰ → e⁺e⁻) which we then test for various models. In particular a more recent form factor derived from a Lagrangian for light pseudoscalars and vector mesons inspired by effective field theories.

Comparison with the KTeV experiment at Fermilab is made and we find that we are more than 3σ below the KTeV experiment for some of the form factors. This is in agreement with other theoretical models, such as the Vector Meson Dominance model and the quark- loop model within the constituent-quark framework. But we also find that we can be in agreement with KTeV if we explore some freedom of the form factor not fixed by the low-energy Lagrangian.

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List of Figures

1 Feynman diagram for the process π⁰ → e⁺e⁻. The dashed line represents the incoming neutral pion, which via the two intermediate virtual photons (wiggly lines) goes to the electron- positron pair (bold directed lines). The grayish blob represents the pion (to two photons) transition form factor which is not known from first principles. . . 3 2 QED Feynman diagram for the process π⁰ → e⁺e⁻ with all

the indicated momenta and vertices. The π⁰ (dashed line) decaying into the dielectron (bold directed lines) via two intermediate photons (wiggly lines). . . 6 3 Feynman diagram for the process π⁰ → e⁺e⁻ with two inter-

mediate photons. The grayish blob represents the transition form factor Fπ⁰γ^∗γ^∗. . . 15 4 Higher order contributions to π⁰ → e⁺e⁻[28]. (a) A particular

two-loop contribution with (b) as counterterm. (c) This is a counterterm corresponding to another two loop contribution not shown here. All fermion lines extend to the right. . . 27 5 Feynman diagram for the process π⁰ → e⁺e⁻ with two inter-

mediate photons. The grayish blob represent the transition form factor Fπ⁰γ^∗γ^∗. . . 28 6 Feynman diagram for the process π⁰ → e⁺e⁻ with the two

intermediate photons lines cut, indicated by the dashed line.

The grayish blob represent the transition form factor Fπ⁰γ^∗γ^∗. . 34 7 Feynman diagrams for the direct decay π⁰ → γγ. . . 37 8 The integration contour and the pole structure related to the

integral (5.1). . . 41 9 The branch cut of P (s = Q²) in the complex s-plane ranging

from [s0,∞), in our case s0 = 0. The unphysical threshold Q² = 0 corresponds to the two-photon intermediate state. . . 49 10 Two identical Feynman diagrams for the rare decay π⁰ → e⁺e⁻

with different notation for the loop momenta. . . 55 11 The singularities of the integrand in Eq. (7.12). The lines

Lj represents the half-planes where the arguments of the Γ functions produce singularities. The point γ belonging to the triangle {x1 > 0, x₂ > 0, x₁ + x₂ < 1/2} is also shown. The line l∆ is explained in the text. Figure inspired by [44]. . . . 68

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12 The singularities of the integrand in Eq. (7.12). The line l∆of figure 11 has been rotated so we can read off the degeneration.

Figure inspired by [44]. . . 71 13 Tree-level Feynman diagram for the process P + P → P + P

via an intermediate virtual vector meson V^∗. . . 83 14 Loop diagrams for the process P + P → V^∗ → P + P . These

should be subleading, relative to the tree-level diagram of figure 13. . . 84 15 Feynman diagram for the pseudoscalar Goldstone boson decay

P → V^∗V^∗ → γγ to two photons via two virtual vector mesons. 85 16 Feynman diagrams for the processes taken into account in de-

riving the form factor. . . 87 17 The form factor with mV = 1. . . 92 18 The subtraction constant for the form factor (9.4), plotted

here as a function of a cutoff point k = Λ/mV given in units of mV ≈ 775 MeV. The middle horizontal line indicates the experimentally obtained value together with its uncertainties (lines above and below it) [13].. . . 95 19 The branching ratio B(π⁰ → e⁺e⁻) for the form factor (9.4).

See caption of figure 18 for more details. . . 96 20 The subtraction constant for the form factor (9.13). See cap-

tion of figure 18 for more details. . . 97 21 The branching ratio B(π⁰ → e⁺e⁻) for the form factor (9.13).

See caption of figure 18 for more details. . . 99 22 Feynman diagrams for π⁰ → γ^∗γ^∗ in a quark-loop model. . . 100 23 The modified form factors FI and FII together with the orig-

inal form factor Four. Also included are FsVMD and FQL for comparison. . . 101 24 The subtraction constant A(0) for the form factor (9.24). See

caption of figure 18 for more details. . . 102 25 The branching ratio B(π⁰ → e⁺e⁻) for the form factor (9.24).

See caption of figure 18 for more details. . . 103 26 The subtraction constant A(0) for the form factor (9.27). See

caption of figure 18 for more details. . . 104 27 The branching ratio B(π⁰ → e⁺e⁻) for the form factor (9.27).

See caption of figure 18 for more details. . . 105 28 Feynman diagram for the process e⁺e⁻ → e⁺e⁻π⁰. . . 110

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29 Feynman diagram for the process φ3 → φ²φ2 with the momenta and masses indicated. The decaying particle φ3 (dou- ble line) has mass M . The two outgoing particles φ2 and φ2

are identical with mass m. The dashed lines represents virtual particles of mass η. . . 115 30 The integration path and the poles in the complex l0-plane. . 118 31 The real part of the integral (A.28) plotted for different values

of −C with A, B and m² fixed. . . 123 32 A cut diagram with the cut indicated by the diagonal dots. . 125 33 The poles of the retarded propagator in the complex k0-plane.

Both lie in the lower half-plane. . . 129 34 A cut diagram with the cut indicated by the diagonal dots. . 140 35 Feynman diagram for the process e⁺e⁻→ 3π via intermediate

ω and ρ⁰ vector mesons. The blob represents further decays of the ρ⁰ meson to two pions. . . 149 36 Feynman diagram showing an electron scattering off of a heavy

target. The blob represent the vertex function. . . 155 37 The pole structure of the two Γ functions and the contour of

integration. Closing the contour to the right we pick up all the residues at z = 0, 1, 2, . . . and obtain a series. . . 158 38 One-loop self-energy Feynman diagram. The dashed line rep-

resents a massless propagator. . . 159

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List of Tables

1 The subtraction term A(0) and the branching ratio B(π⁰ → e⁺e⁻) for different models of the pion transition form factor Fπ⁰γ^∗γ^∗. The cutoff point Λ is also given. . . 106 2 The subtraction term A(0) and the branching ratio B(π⁰ →

e⁺e⁻) for different models of the pion transition form factor Fπ⁰γ^∗γ^∗. The form factors F1− F⁴ evaluated at Λ = mV = 775 MeV. . . 167

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Popul¨ arvetenskaplig sammanfattning p˚ a svenska

Naturvetenskapen är byggd p˚a ett samspel mellan teori och experiment. I princip s˚a är en teori naturvetenskaplig endast om dess förutsägelser kan testas av experiment.

S˚a vitt vi vet idag finns det fyra fundamentala s˚a kallade krafter, eller kanske mer korrekt, växelverkan omkring oss. Till dessa hör de mycket bekanta: gravitationella och elektromagnetiska växelverkan. Men det finns ytterligare tv˚a, mindre kända, växelverkan som p˚averkar oss p˚a ett indirekt sätt. Till dessa hör den svaga växelverkan som förklarar radioaktivt sönderfall och den starka växelverkan som bl.a. h˚aller ihop protonen.

Precis som i andra teorier s˚a förekommer modellering inom naturve- tenskapliga teorier. En framg˚angsrik modell är den s˚a kallade partikelfy- sikens standardmodell (SM). Denna modell beskriver hur naturens minsta best˚andsdelar växelverkar med varandra¹ och är sedan länge, efter rader av korrekta förutsägelser, accepterad som standard.

Den senaste i raden av SM:s framg˚angar var upptäckten av Higgsparti- keln vars existens förutsp˚addes inom SM redan p˚a 60-talet. Men precis som namnet antyder s˚a är SM en modell och som av olika anledningar har sina begränsningar. En s˚adan är att SM ej inkluderar gravitationen. Detta är i sig inte en katastrof p˚a en praktiskt niv˚a ty i dagens experiment vid v˚ara mest kraftfulla acceleratorer s˚a är gravitationskraften mellan elementarpar- tiklar s˚a liten att den är totalt försumbar jämfört med de andra krafterna som verkar.

Ett annat problem är att som det ser ut nu s˚a best˚ar den större delen av universums materia av n˚agon sorts materia som ej beskrivs av standardmodellen. Denna materia kallas p.g.a. olika skäl för mörk materia och det finns flera teorier för den. Teorier för mörk materia tillhör allts˚a ny fysik bortom standardmodellen (BSM).

Med upptäckten av Higgspartikeln s˚a är standardmodellen i det stora he- la ett avslutat kapitel, men fortfarande ˚aterst˚ar det en hel drös med fr˚agor inom SM. Standardmodellen är en kvantfältteori och en kvantfältteori är ex- akt lösbar endast i sällsynta fall. Som tur är kan man i de flesta fall tillämpa en approximationsmetod, t.ex. störningsteori och lösa ekvationerna approx-

1 Svaga, starka och elektromagnetiska växelverkan är beskrivna av standardmodellen medan gravitationell växelverkan är (ännu) ej.

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imativt till den precision man önskar. Problemet är att störningsteori ej är tillämpbar för den starka växelverkan i de lägre energiregionerna. Här f˚ar man istället införa effektiva fältteorier eller n˚agon sorts hadronmodell.² Att först˚a den starka växelverkan i de lägre energiregionerna, d.v.s. p˚a hadronniv˚a är mycket önskvärt och ligger i framkanten av teoretisk kärnfysik.

Andra fr˚agor som gäckar är att i vissa fall s˚a överensstämmer inte SM:s förutsägelse med experiment till den grad vi skulle vilja. Detta kvantifierar man genom att bestämma och tala om hur mycket teori och experiment avviker fr˚an varandra. En s˚adan kvantitet är det som kallas en standardavvikelse och som betecknas med den grekiska bokstaven sigma σ. Allmänt gäller det att om avvikelsen mellan teori och experiment ej överstiger tv˚a σ s˚a är man ganska säker p˚a att man är p˚a rätt sp˚ar. ˚A andra sidan s˚a är man ganska säker p˚a att man är fel ute om den överstiger fem σ. Däremellan hittar man dock en del reaktioner eller förutsägelser som avviker fr˚an experiment med cirka tre σ. Dessa är av intresse d˚a de säger n˚agot om v˚art först˚aelse av SM.

Händer det att dessa reaktioner ocks˚a är väldigt sällsynta s˚a är de av ännu större intresse, d˚a dessa är känsliga för fysik bortom standardmodellen.

En s˚adan reaktion är den väldigt sällsynta sönderfallet

π⁰ → e⁻e⁺ (1)

där π⁰ st˚ar för en partikel som kallas pimesonen och som best˚ar av kvark och antikvarkar. P˚a högerledet av reaktionen (1) hittar vi en elektron betecknad med e⁻ och dess antipartikel (positron) betecknad med e⁺. En π⁰-meson sönderfaller nästan alltid till tv˚a fotoner d.v.s. tv˚a ljuspartiklar, men cirka en p˚a 100000000 sönderfaller till ett elektron-positron par given av reaktionen (1).

Nyligen mättes reaktionen (1) i ett experiment vid Fermilab med en bra precision. Problemet är att det verkar som att standardmodellens förutsägelse om reaktionen skiljer sig med cirka 3.3σ fr˚an vad detta experiment mätte.

Som vi antydde ovan s˚a involverar hadronteori en viss modellering och detta stämmer in p˚a reaktionen (1). Denna reaktion har modellerats av flera olika grupper som resulterat i liknande resultat, d.v.s. cirka 3σ avvikelse fr˚an experiment. Som sagt s˚a ligger en 3σ-avvikelse p˚a gränslandet mellan ‘rätt’

och ‘fel’ och det ¨ar d¨ar detta examensarbete kommer in i bilden.

2Subatomära partiklar uppbyggda av kvarkar kallas för hadroner. Tv˚a välkända hadroner är protonen och neutronen. En delmängd av hadroner kallas för mesoner. Dessa är sy- stem uppbyggda av kvark och antikvarkar, som t.ex. π⁰-mesonen (pimesonen). Pimesonens

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I detta examensarbete har vi studerat sönderfallet (1) grundligt och g˚att igenom och jämfört olika modeller som använts just för det. Vi har dessutom introducerat en ganska färsk modell för reaktionen.

V˚ara mest konservativa förutsägelser ang˚aende ovan reaktion är i linje med de resultat andra teoretiker tagit fram. Men vi har ocks˚a visat att v˚ara förutsägelser, i motsats till andra teoretikers, kan faktiskt vara i överensstämmelse med experiment om vi till˚ater oss att vara lite liberala med v˚ar modell. Detta skulle i s˚adana fall tala emot tillskott fr˚an BSM-fysik till reaktionen ovan.

Men för att vara säker p˚a denna slutsats s˚a behövs djupare studier i teoretisk hadronfysik, vilket vi även argumenterat för. Vi har ocks˚a argumenterat för behovet av experimentella resultat för att guida oss vidare i mörkret.

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1 Introduction

Having most likely found the final particle of the Standard Model (SM) [1,2,3], it is no overstatement that the SM has been an extremely successful theory up to date.¹ But as is well known, the SM also has its shortcomings.

Apart from the failure to include gravity,² it is also well known from astronomical observations that our current understanding of the universe only covers approximately 5% of everything. Aside from the obscure so called dark energy, a large portion of the unknown is believed to be some kind of matter that does not interact or interacts weakly³ with matter and other particles described by the SM. Due to this lack of interaction, in particular with electromagnetism, this unknown matter or dark matter as it is popu- larly named, is not directly visible in astronomical observations, but rather reveal its presence indirectly through its gravitational interaction with SM matter inside of galaxies (see e.g. [4,5, 6]).

There are at least two different approaches one can take in the search of physics beyond the standard model (BSM): One can either look for new degrees of freedom in high-energy reactions, for this we have to await the new LHC era. Alternatively one can look at high-precision measurements of low- energy observables like rare decays or electromagnetic properties of particles.

The basic idea is that a reaction that is very unlikely or even forbidden by the SM can get a significant contribution from a BSM process like the exchange of a new particle. As far as leptons are concerned, the SM calculations are straightforward since the involved coupling constants are small and a pertur- bative treatment is technically feasible to a large number of loops. Hence a high accuracy in the SM prediction can be achieved and possible BSM contributions can be isolated even if they are small. However, sooner or later in the loop expansion, each SM process obtains contributions from quarks.

Quark interactions are well described by quantum chromodynamics (QCD) where the coupling constant varies strongly with energy. For large energies, in the asymptotically free region, the couplings are small and perturbation theory can be applied [7]. At low energies on the other hand, the quarks are confined to bound states called hadrons and perturbation theory is not

1 Some of the people behind the theory of the Higgs field just got rewarded a Nobel prize the other week, [3].

2Due to the weakness of gravity, it is not a disaster to neglect it in present-day particle collisions in our largest colliders.

3 Weak as in the technical sense of the word: the weak interaction.

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valid. Here one needs to use some kind of effective field theory or hadronic model and in general it is a nontrivial task to calculate with a reasonable uncertainty the quark/hadronic contribution to an SM process. On the one hand, if the uncertainty estimate is too large, one could not differentiate any significant BSM contribution by comparing experiment to theory. If on the other hand the uncertainty estimate is too small, one might misinterpret a discrepancy between an SM prediction and experiment as a sign of new physics. The QED contributions being in principle under control, one needs to get a better understanding and control over the hadronic contributions.

This is where this thesis comes into the picture.

We will here consider the rare decay

π⁰ → e⁺e⁻ (1.1)

which is influenced by the pion transition form factor

π⁰ → γ^∗γ^∗ (1.2)

where γ^∗ denotes a virtual photon. When the meson π⁰ (mass ∼ 1/7 proton

∼ 135 MeV) couples to photons, the photons might directly interact with it. Or first form other hadrons, so called ‘vector mesons’ having the same quantum numbers as a photon.⁴ Here, we will mainly be concerned with the vector mesons ρ and ω, which to a first approximation have a common mass of ∼ 775 MeV [8].

The rare decay (1.1) is a classical problem in the sense that the first prediction [9] was already made in 1959 but the problem persists to be relevant to this very day. This is because apparently there still exists a sizable discrepancy between SM prediction (cf. e.g. [10, 11, 12]) and experimental results. In particular the result from 2007 of the KTeV E799-II collaboration at Fermilab [13].

On the more technical side of things, we will mainly work with the leading order contribution which⁵ is given by the QED one-loop triangle diagram shown in figure 1. All other SM contributions e.g. from weak interactions are negligible here [14].

The pion transition form factor Fπ⁰γ^∗γ^∗6 essentially parametrizes our ignorance of the details of the decay and as we will show in the following section,

4 No electric charge, spin=1.

5 Due to the fact that no spinless current coupling exist between quarks and leptons.

6 Henceforth the pion transition form factor will simply be referred to as the form

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π ⁰

e ⁺

γ

^∗

e ⁻

γ

^∗

F

_π⁰_γ∗γ^∗

Figure 1: Feynman diagram for the process π⁰ → e⁺e⁻. The dashed line represents the incoming neutral pion, which via the two intermediate virtual photons (wiggly lines) goes to the electron-positron pair (bold directed lines).

The grayish blob represents the pion (to two photons) transition form factor which is not known from first principles.

is needed to make the diagram convergent. Throughout the years, several different models have been used for the form factor starting with a simple sharp cutoff-function used by Drell [9]. Other models among many include the Vector Meson Dominance model (VMD) [15] and a quark-loop within the constituent quark model [16].

In this thesis we will consider a more recent model for the form factor which we will derive from a Lagrangian for light pseudoscalars and vector mesons inspired by effective field theories. This Lagrangian will be presented in section 8 but before that, we will in detail go through and derive all the necessary ingredients needed to calculate the branching ratio

R = B(π⁰ → e⁺e⁻)

B(π⁰ → γγ) . (1.3)

To this end we start off section 2, as mentioned above, by showing that the diagram of figure 1with a constant form factor is indeed divergent and find the divergent form. In section3, we will introduce a general form factor and derive an expression for the pertinent decay amplitude. Next, in section4we will derive a unitarity bound on the ratio (1.3) which is a general result and does not depend on the form factor. In doing this we will use several results,

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many of which are proven in appendix A. The dispersion representation or the Kramers-Kronig relations are introduced in section 5 where a couple of examples are also given in order to make the reader familiar with the subject.

Section 6is a short section where we explicitly calculate the real part of the amplitude and in doing so demonstrate the technique of rewriting an integral in terms of Spence or dilogarithm functions defined in appendix F. Next in section 7, we give our full attention to the form factor by introducing the twofold Mellin-Barnes representation for it. We also derive a quite simple expression involving the form factor, given at the end of the same section. For the reader who is not very familiar with the Mellin-Barnes technique, we refer to appendix E where we have given some examples and a short introduction to the subject. Section 8 is devoted to a recently proposed Lagrangian we use to derive the form factor from. Having derived everything we need, we present the results for different form factors and compare with experiment and with other theoretical models in section 9. Finally, we give a summary and an outlook in section 10.

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2 The Process π

⁰

→ e

⁺

e

⁻

with F

_π⁰_γ∗γ^∗

= 1

In this section we will consider the triangle loop introduced in the previous section but now we will work without any form factor. In other words we will put F ≡ 1. We will show that the pertinent triangle loop is divergent for constant form factors⁷ and isolate the divergent part.

We emphasize that the content of this section is more of a qualitative kind which for historical reasons has a different metric than the rest of the thesis. However, wherever things get quantitative in this thesis, the same metric is always used.

2.1 The Amplitude

Let us consider the decay π⁰ → e⁺e⁻ shown in figure 2 where we have put the form factor to unity. We will now show that this diagram diverges. The π⁰γγ-coupling is given by the Wess-Zumino-Witten (WZW) term which we can write for our purpose here as [18]

LWZW= e²

32π²fε^µναβFµν(x)Fαβ(x) π⁰(x) +· · · , (2.1) where f is the pion decay constant. The WZW vertex factor can be obtained by introducing the Fourier transform of each field in

iSWZW = i Z

d⁴xL^WZW

as explained in appendix C. This yields a vertex factor of

−ie²· 4 · 2

32π²f ε^µναβ· pµ· qα (2.2) where pµand qαare the momenta carried by each photon line. The factor of 2 comes from doing the same calculation with the two photon lines exchanged.

7 We will, without loss of generality, set F = 1 instead of a generic constant.

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ν β

β⁰

ν⁰

P

p⁰₁+ k

−p

⁰1

k

p

⁰₂

p⁰₂− k

Figure 2: QED Feynman diagram for the process π⁰ → e⁺e⁻ with all the indicated momenta and vertices. The π⁰ (dashed line) decaying into the dielectron (bold directed lines) via two intermediate photons (wiggly lines).

The factor of four comes from the fact that

−ε^µναβ∂νAµ= {rename µ ↔ ν }

=−ε^νµαβ∂µAν

={let µ ↔ ν in the ε}

= ε^µναβ∂µAν

and similarly for the α, β-part. Thus

ε^µναβFµνFαβ = ε^µναβ(∂µAν− ∂^νAµ)(∂αAβ− ∂^βAα)

= 2× 2 × ε^µναβ(∂µAν)(∂αAβ)

and since ∂µ→ ip^µ and ∂α → iq^α (see appendix B and C) we get (2.2).

The Feynman diagram for the process is shown in figure2, so let us write down the analytic expression for the diagram. Using Feynman rules for QED

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(see e.g. appendix B or [19, 20]) we find that the amplitude is given by iM =

Z d⁴k

(2π)⁴us⁰₂(p⁰₂)ieγ^ν⁰−i(−/k + m)

k²+ m²− iieγ^β⁰vs⁰₁(p⁰₁)

× −igββ⁰

(p⁰₁+ k)²− i

−igνν⁰

(p⁰₂− k)²− i

× −ie²· 8

32π²f ε^µναβ(p⁰₁+ k)α(p⁰₂− k)µ,

(2.3)

where m denotes the electron mass. For later use we also introduce the pion mass M . Cleaning up a little bit we can write

M = i 8e⁴

32π²fε^µναβ

Z d⁴k (2π)⁴

× us⁰₂γν(m− /k)γβvs⁰₁(p⁰₁+ k)α(p⁰₂− k)µ

[k²+ m²− i][(p⁰1+ k)²− i][(p⁰2− k)²− i].

(2.4)

The denominator can be rewritten by introducing Feynman parameters (see e.g. [19, 20]) so that

M = i8e⁴

32π²f × 2ε^µναβ Z 1

0

dxdydzδ(Σ− 1)

Z d⁴k (2π)⁴

Nµναβ

D , (2.5)

where Σ = x + y + z. Here, the numerator is given by

Nµναβ = us⁰₂γν(m− /k)γβvs⁰₁(p⁰₁+ k)α(p⁰₂− k)µ, while D is given by

D = (k + yp⁰₁ − zp⁰2)²− (yp⁰1− zp⁰2)²+ xm²+ yp⁰²₁ + zp⁰²₂ − i.

We can immediately throw away terms proportional to kµkα because they are symmetric under µ↔ α while the ε is totally antisymmetric. Doing this and shifting k such that

k + yp⁰₁− zp2 := l⇒ d⁴k = d⁴l and

D→ D(l²) = l²− (yp⁰1− zp⁰2)²+ xm²+ yp⁰²₁ + zp⁰²₂ − i

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we find⁸

Nµναβ = uγν

"

(m + y/p⁰₁− z/p⁰₂)γβv· x · p⁰1αp⁰_2µ+ /lγβv(p⁰_1αlµ− p⁰2µlα)

#

= N_µναβ⁽⁰⁾ + N_µναβ⁽²⁾ . In the above we have defined N_µναβ⁽⁰⁾ := uγν

m + y/p⁰₁− z/p⁰₂

γβv· x · p⁰1αp⁰_2µ and the term proportional to l²

N_µναβ⁽²⁾ := uγν/lγβv p⁰_1αlµ− p⁰2µlα .

We have used several constraints in the intermediate steps of the above calculations such as x + y + z = 1 and ε^µναβpµpν = 0 etc.. Finally we have used the fact that terms linear in l integrate to zero.

2.2 The Finite Part

Thus we see that the amplitude splits up into two parts: M = M1 +M2

where

M1 = i8e⁴· 2 32π²f

Z 1 0

dxdydz δ(Σ− 1)ε^µναβN_µναβ⁽⁰⁾

Z d⁴l (2π)⁴

1 D³

= i8e⁴· 2 32π²f × J1

with

J1 :=

Z 1 0

Z d⁴l (2π)⁴

1

D³ (2.6)

and

M2 = i8e⁴ · 2

32π²f × J2 (2.7)

with J₂ :=

Z 1 0

dxdydz δ(Σ− 1)

Z d⁴l (2π)⁴

ε^µναβuγν/lγβv p⁰_1αlµ− p⁰2µlα

D³ . (2.8)

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The integral in (2.6) looks convergent and indeed it is. To see this one can go over to Euclidean space and do the l-integral in spherical coordinates or one can just simply look up the integral from a table (e.g. appendix of [19]).

We choose the former because it’s more fun.

Thus consider the denominator D = D(x, y, z) in (2.6)

D(x, y, z) = −l²0+ l²+ xyp⁰²₁ + xzp⁰²₂ + yzP²+ xm²− i

where we have simplified by using

2p⁰₁· p⁰2 = (p⁰₁+ p⁰₂)² − p⁰²1 − p⁰²2

= P²− p⁰²1 − p⁰²2.

The integrand has poles (with respect to the l₀-integration variable) when D = 0 i.e. when

l²₀ = l²+ xyp⁰²₁ + xzp⁰²₂ + yzP²+ xm²

| {z }

A

−i.

Now, ±√

A− i ≈ ±√

A∓ i which means that the poles lie in the second and fourth quadrant in the complex l0-plane provided that we choose A such that A≥ 0. But

A = l²+ xyp⁰²₁ + xzp⁰²₂ + yzP²+ xm²

which is nonnegative provided that all the p-squares are positive⁹ hence for the time being this is what we choose. The physical values can later be obtained by analytic continuation after we have done the integral.

Now let us go to Euclidean space by making a Wick rotation letting¹⁰ l⁰ = il4,

and

lj = lj

9 Recall that in this metric the physical i.e. on-shell values are such that p⁰²₁ =−m² and so on for the other p’s.

10We refer the interested reader to appendixAfor a similar but more detailed calculation.

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where l = (lj, l4) and d⁴l = id⁴l. Hence Z d⁴l

(2π)⁴ 1 D³ → i

Z d⁴l (2π)⁴

1 (l²+ ∆)³

= i

(2π)⁴ Z

dΩ4

Z _∞

0

|l|³ (|l|²+ ∆)³

= i

(2π)⁴S4· 1

4∆ = i

(2π)⁴2π²· 1 4∆

= 1 2

i (4π)²

1

∆

where ∆ = xyp⁰²₁ + xzp⁰²₂ + yzP²+ xm² and we have used that S4 = 2π², the area of the 3-sphere.

Collecting all this, the integral (2.6) becomes J1 = 1

2 i

(4π)² × I1

where we have defined I₁ : =

Z 1 0

∆

= ε^µναβuγν

Z 1 0

dxdydz δ(Σ− 1)mx + yx/p⁰₁− xz/p⁰₂

∆

| {z }

:=I

γβvp⁰_1αp⁰_2µ. (2.9)

The integral in the large parentheses I = I(p²₁, p²₂, P²) defined above is an analytic function of the complex variables p²₁, p²₂ and P². Hence by analytic continuation we can in particular consider it near a region or even at the

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physical on-shell values I(−m²,−m²,−M²) which is given by Ion shell =

Z 1 0

dxdydz δ(Σ− 1) mx + yx/p⁰₁− xz/p⁰₂ xm²(1− y − z) − yM²z

= Z 1

0

dxdydz δ(Σ− 1)mx + yx/p⁰₁− xz/p⁰₂ xm²− yzM²

= m Z 1

0

dxdydz δ(Σ− 1) x xm²− yzM² + (/p⁰₁− /p⁰₂)

Z 1 0

dxdydz δ(Σ− 1) xy xm²− yzM² : = m· IA+ (/p⁰₁− /p⁰₂)· IB.

(2.10)

The two integrals IA,B can be computed (numerically) as follows M²· I^A:=

Z 1 0

dx x Z 1−x

0

dy 1

y²− y(1 − x) + xm²/M²

= Z 1

0

dx 2x

y⁺− y⁻log y⁻ y⁺

≈ − 50.044,

(2.11)

where y^±(x) are the zeros of y²− y(1 − x) + xm²/M² = 0. Similarly M²· I^B :=

Z 1 0

dxx(1− x)

y⁺− y⁻ logy⁻ y⁺

≈ − 4.3487,

(2.12)

where we have used the relation m²/M² = (0.511/135)² ≈ 1.43 · 10⁻⁵, for the electron and pion mass, m and M respectively. So it seems that this part of the amplitude is finite.

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2.3 The Divergent Part

Let us go back to the second term of the amplitude namely eq. (2.7), the one proportional to l² and which definitely looks divergent. Notice first that

J2 ∝ J =

Z d⁴l

(2π)⁴ F (l²)/llµ

=

Z d⁴l

(2π)⁴ F (l²)γ^ρlρlµ

= Const.× γ^ρgρµ

Z d⁴l

(2π)⁴ F (l²)l²

where the third equality follows from Lorentz invariance. To find the constant we contract both sides with g^ρµ and find that¹¹ Const. = 1/4, thus we can write M2 as

M² = i8· 2e⁴ 32π²f

1 4

Z 1 0

dxdydz δ(Σ− 1)

× ε^µναβuγνγ^ρ gρµγβvp⁰_1α− g^ραγβvp⁰_2µ

| {z }

ζ

Z d⁴l (2π)⁴

l² D³

| {z }

∼Γ()

. (2.13)

We recall that the Feynman measure is given by Z

dFn = (n− 1)!

Z 1 0

dx1. . . dxnδ(x1+· · · + xⁿ− 1), (2.14) normalized such that Z

dFn1 = 1. (2.15)

Now we see that the final integral is indeed divergent and goes like Γ(0) = lim

→0Γ() with

Γ() = 1/− γ^E +O() where γE ≈ 0.577 is the Euler-Mascheroni constant.

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2.3.1 The Form of the Divergent Part

To find the explicit form of the divergent part we need to simplify the sandwiched structure of equation (2.13)

ζ = uε^µναβγν γµγβp⁰_1α− γ^αγβp⁰_2µ v.

This can be rewritten by using the identity¹²

γµγνγλ ≡ (terms ∝ g^µνγλ cyclic order ) + iεµνλσγ^σγ5, so that we get:

ζ = uε^µναβγνγµγβp⁰_1α− ε^µναβγνγαγβp⁰_2µ v

= iu ενµβσε^µναβp⁰_1α− εναβσε^µναβp⁰_2µ γ^σγ5v

= iu −6δσ^αp⁰_1α− 6δσ^µp⁰_2µ γ^σγ5v

= +i6uγ5P v,/

where we have moved the γ₅ to the left to remove the negative sign and we have used the identity εσµνβε^αµνβ =−6δ^ασ.

Thus we have found that M = −2 · 6e⁴

32π²f uγ₅P v/ 1

− γ^E +O()

+ finite part (2.16) which is divergent and we see from the explicit form of the divergency that it contains a γ5. The implications of the appearance of the γ5 will be dis- cussed in a moment but first let us discuss why this divergency is only a product of our ignorance and unrealistic modeling of the decay. Physically, this divergency is (obviously) not present.

2.3.2 Physical Explanation of the Non-Divergency

The divergency of the amplitude with Fπ⁰γ^∗γ^∗ ≡ 1 can be understood as follows: On the one hand we have set the form factor to unity which means that we have treated the pion as a totally structureless object. On the other

12We will contract with ε anyway so the terms involving the metric tensor vanish hence we omit them here.

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hand we know that the pion is a meson consisting of a quark antiquark pair.

But we are integrating over all momenta and consequently one probes the pion to smaller and smaller distances so that at some point the structure of the pion i.e. the quark degrees of freedom becomes relevant and our simple model breaks down.

2.3.3 Regularizing the Divergent Integral

One could perhaps try to regularize this by cutting the integral off at some energy level but the standard way is dimensional regularization [21]. The problem in this case is that our amplitude contains γ5, which is a genuinely 4-dimensional object and it is not clear at all how one is supposed to pro- ceed in treating γ5 in more general n dimensions? The same goes for the Levi-Civita symbol ε^µναβ. There is a scheme though called HVBM regularization [22] which remedies these problems. What one does there is that one splits up the n-dimensional metric g^µν into a 4-dimensional and an (n− 4)- dimensional one with ε^µναβ defined to have non-vanishing components in the 4-dimensional subspace only. One also have that the γ5 anticommutes with the γµ’s as usual in d = 4 but commutes in (n− 4) space, see e.g. [23]. To facilitate computations one can use computer softwares such as Mathemat- ica and there is actually a package for Mathematica called Tracer that can handle the γ’s in arbitrary (integer) n-dimensions ([24, 25]).

Another alternative often used in Chiral Perturbation Theory (χP T ) is dimensional reduction where one (tries to) projects out the form factor from the amplitude by some projection (see e.g. [26]). The subsequent Dirac algebra is then done in four dimensions so that one is left with scalar integrals to which the dimensional regularization technique can be applied.

Finally we conclude that the amplitude (2.16) corresponds to a Lagrangian of the form

L ∼ uγ⁵γ^µ∂µπ⁰v(· · · )

which is of the same form as the counter term Lagrangian used in e.g. [27], one of the earlier attempts on this problem in χP T .

We will now leave this divergency problem and reintroduce the form factor in the next section where we also will lay out the plan to our approach.

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3 The Process π

⁰

→ e

⁺

e

⁻

with a General Form Factor F

_π⁰_γ∗γ^∗

We saw in section 2that we need to introduce a more realistic model for the process π⁰ → e⁺e⁻. One way to do that is to introduce a form factor that effectively kills off the divergent integrals at large energies/loop momenta.

In the present section we will show how to decompose the off shell amplitude Mπ⁰→e⁺e⁻ in terms of four scalar form factors which then reduces to a single form factor when on shell. We will derive an expression for the generic 1→ 2 particles decay rate and then evaluate it for our case, i.e. derive an expression for the decay rate Γ(π⁰ → e⁺e⁻) which turns out to depend solely on one single scalar form factor.

Q

−q⁺ l− q−

q₋

l + q+

Fπ⁰γ^∗γ^∗ l

Figure 3: Feynman diagram for the process π⁰ → e⁺e⁻with two intermediate photons. The grayish blob represents the transition form factor Fπ⁰γ^∗γ^∗.

3.1 Four Scalar Form Factors

The Feynman diagram for the decay of interest is shown in figure 3 and its amplitude is defined by the matrix element

e⁺(q+, s+)e⁻(q₋, s₋); out

π⁰(Q); in = i(2π)⁴δ⁽⁴⁾(Q− q−− q⁺)M^π⁰→e⁺e⁻

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where strong and electromagnetic interactions are taken into account. As we saw in section 2 equation (2.4) we can write the amplitude as

M^π⁰→e⁺e⁻ = u(q₋, s₋)(structure of γ-matrices and momenta)v(q+, s+) so let us denote this sandwiched structure by T so that we write

M^π⁰→e⁺e⁻ = u(q₋, s₋) T v(q+, s+). (3.1) Instead of taking the route of section 2 where we ended up with having to regularize terms involving γ5 etc. we will now show how to decompose T into four scalar form factors.

3.1.1 Most General Parametrization of the Vertex Function Let us go back to our one particle irreducible vertex defined in equation (3.1) T (π⁰ → e⁺e⁻; q₋, q+) and expand it (parametrize it) as follows

T = a1 + bγ5+ c/q₊+ d/q₋+ e/q₊γ5+ f /q₋γ5+ gq₊^µq^ν₋Sµν

+ hεµναβq^µ₊q₋^νS^αβ

(3.2) with Lorentz invariant coefficients a, b, . . . , h. This parametrization is too general, we have some physical constraints we can put on the coefficients a, b . . . g, h. For instance, since the pion and the positron are negative under parity change and the electron is positive, the overall amplitude must be invariant under parity change. Thus let us check if all the coefficients are nonzero. Under a parity transformation P we have that (t, x) −→ (t, −x) so^P that p→p := (pe ⁰,−p) and s±→ s±. We will in the following also make use

of u(p) = +γ⁰u(p),e

v(p) = −γ⁰v(p),e

⇒

u(p) =γ⁰u(p)e†

γ⁰

= u(p)γe ⁰ and

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Then under parity with P = diag(1, −1, −1, −1) we have that Mπ⁰→e⁺e⁻ = u(q₋, s₋) T (π⁰ → e⁺e⁻; q₋, q+) v(q+, s+)

→ u(eP q₋, s₋)T (Pq−,Pq⁺)v(qe+, s+)

= u(q₋, s₋)γ⁰T (Pq−,Pq⁺) (−γ⁰)v(q+, s+) which implies

γ⁰T (q₋, q+) γ⁰ =−T (Pq−,Pq+). (3.3) Thus let us use this and check wether all the coefficients in equation (3.2) are needed to parametrize T :

(a) γ⁰1γ⁰ = +1 ⇒ a = −a = 0.

(b) bγ⁰γ₅γ⁰ =−bγ5 ⇒ b OK.

(c) γ⁰/q_±γ⁰ = γ⁰γ^µγ⁰q^±_µ = γ⁰γ⁰γ⁰q₀^±+ γ⁰γⁱγ⁰q^±_i = γ⁰q^±₀ − γⁱq_i^± = γ^µ(Pq±)µ

which implies that c = −c = 0, d = −d = 0.

(d) d = 0 see above.

(e) γ⁰/q_±γ5γ⁰ =−γ⁰/q_±γ⁰γ5 =−γ^µ(Pq±)µγ5 ⇒ e and f OK.

(f) f OK see above.

(g) For this one we need the following: γ⁰S0iγ⁰ = −S⁰ⁱ and γ⁰Sijγ⁰ = (−)(−)Sij = Sij, then

q₊^µq^ν₋γ⁰Sµνγ⁰ = q₊⁰q₋^νγ⁰S0νγ⁰+ q₊ⁱ q^ν₋γ⁰Siνγ⁰

= q₊⁰q₋⁰γ⁰S00γ⁰+ q⁰₊q₋ⁱ γ⁰S0iγ⁰ + qⁱ₊q₋⁰γ⁰S_i0γ⁰+ qⁱ₊q₋^jγ⁰Sijγ⁰

= (−q+⁰qⁱ₋+ qⁱ₊q₋⁰)S0i+ q₊ⁱ q^j₋Sij

= (+q₊⁰)(−q₋ⁱ )S0i+ (−qⁱ+)(q₋⁰)Si0 + (−qⁱ+)(−q^j₋)Sij

= (Pq+)^µ(Pq−)^νSµν

⇒ g = −g = 0.

The Rare Decay of the Neutral Pion into a Dielectron

Uppsala University

Department of Physics and Astronomy

Master’s Thesis

The Rare Decay of the Neutral Pion into a Dielectron

Author:

Hazhar Ghaderi

Supervisor:

Prof. Dr. Stefan Leupold

π 0

e +

γ

e −

γ

F

Nov 16, 2013

Acknowledgments

Contents

List of Figures

List of Tables

Popul¨ arvetenskaplig sammanfattning p˚ a svenska

1 Introduction

π 0

e +

γ

e −

γ

F

2 The Process π

→ e

e

with F

= 1

2.1 The Amplitude

P

−p

p

2.2 The Finite Part

2.3 The Divergent Part

3 The Process π

→ e

e

with a General Form Factor F

3.1 Four Scalar Form Factors

π ⁰

e ⁺

e ⁻

π ⁰

e ⁺

e ⁻