Hadron Physics Lectures for the 19Th UK Nuclear Physics Summer School, Queen’S University Belfast D

Home , C parity, Hadron, Multiplet, Nuclear physics

Hadron Physics Lectures for the 19th UK Nuclear Physics Summer School, Queen’s University Belfast D. G. Ireland (University of Glasgow) 30 August, 31 August and 1 September, 2017 2 d g ireland

Introduction

Hadrons are the particles that feel the strong nuclear force. This force is described by the theory of Quantum ChromoDynamics (QCD), a ﬁeld theory whose constituents are quarks (the particles) and gluons (the force carriers). QCD forms the strongly interacting sector of the standard model of particle physics, but has a unique character compared to the other forces of nature.

One of the main features of the strong interaction is that the constituents have never been observed in isolation. Hadrons are composite particles, made from quarks and bound by gluons. They are the only physical manifestations of QCD that we can study.

Nuclei are built from protons and neutrons (and very occasionally hyperons!) and are held together by pions, all of which are hadrons. An understanding of nuclear physics therefore rests on an understanding of hadrons, even if the details of hadronic interactions at high energies are not relevant for understanding collective phenomena of heavy nuclei.

With only three lectures on the topic, there is barely time to scratch the surface. However, rather than give a superﬁcial overview of as much as possible, beyond a general introduction, I have cho- sen a few topics to look at in sufﬁcient detail so as to stretch the participants during their time at the summer school.

This document contains material presented on the lecture slides. As you can see, it does not look exactly the same, but these notes contain all the information on the slides, plus more. There are occa- sional extra comments (like this one) in these printed notes that do 1 1 There are also notes placed in the mar- not appear on the slides. To help you follow the lectures, the slide gins, like this one. The side notes are number is written, underlined, in the margin. used to highlight important information or provide some background. Please contact me ([email protected]) if you spot any mistakes, or would like further explanation. hadron physics 3

The Big Picture

The study of Hadron Physics is part of a wide spectrum of re- search that aims to be able to describe the nature of the matter that we observe in the universe. It sits at the interface between particle-, or high-energy physics, and nuclear physics. From particle physics it shares a “reductionist” philosophy - a desire to understand ev- erything from basic constituents. On the other hand it involves the study of the structure of composite particles, and thus shares a great deal of common ground with nuclear structure physics, such as a study of effects that are “emergent properties” due to the interaction of several constituents. In hadron physics, we study the hadrons, the particles in nature that feel the strong nuclear force. These are categorised in to mesons (bosons, with integer spin quantum numbers) and baryons (fermions, with half-integer spin). We want to know how many hadrons there are, how they interact with each other, how they decay, etc.. We also want to know about their properties: static quantities such as mass, shape, size, magnetic moment; dynamic quantities such as the distribution of charge and current that are measured with structure functions. 4 d g ireland

The Nuclear Chart

Part of the reason for studying hadrons is that the basic constituents of nuclei (protons, neutrons, pions,...) are such particles. Without a full understanding of hadrons, therefore, we cannot really say that we understand nuclear matter.

Quantum ChromoDynamics

The theory underlying the strong nuclear reaction is quantum chromodynamics (QCD), a quantum ﬁeld theory that describes ﬁelds of “matter” particles, quarks, that interact via the exchange of force- carrying bosons called gluons. It is somewhat like quantum electro- dynamics (QED), the theory underlying electromagnetism, but there are some important differences that lead to quite unique phenomena. Due to the mathematical structure of the interaction, gluons carry “colour” charge, and can therefore interact with each other. This is quite different to QED, where the force carriers, the photons, do not hadron physics 5

interact with each other. In QCD this leads to many of the features that we are quite used to, such as superposition of photon states (e.g. lasers).

QCD Running Coupling Constant

Figure 1: Nucleons become more com- plex, the closer one looks at them.

Figure 2: Models that retain features of QCD are needed for practical calculations.

The Strong Coupling constant, which measures the strength of the interaction, varies with energy and at the scale of hadron masses, its value approaches unity. Any scheme that relies on a small param- eter to use perturbation theory for calculations is therefore bound to fail for hadrons. 6

Conﬁnement

Figure 3: If mesons are “pulled” apart, the energy stored in the gluon string is enough to create a quark-antiquark pair if the string breaks. 7

Since gluons interact with each other, this gives rise to an observed phenomenon know as confinement: quarks and gluons have 2 2 never been observed in isolation. In relativistic heavy ion collisions the quark-gluon plasma has been shown to form as a new state of matter. This is sometimes referred to as a de-confined phase. However at the energy scales relevant to ordinary hadronic matter, all hadronic states are confined. 6 d g ireland

The Origin of Mass

The Higgs Mechanism is only responsible for about 1-2% of the mass of nucleons, though the generation of quark mass. That means that around 98% of the visible mass in the universe must be gener- ated by an additional mechanism. Recent calculations have shown that even if quarks were to have no mass from the interaction with the Higgs field, it is possible via a mechanism called dynamical chiral symmetry breaking (DCSB) for QCD to give mass to quarks. The simple picture is that quarks moving through the “glue” acquire inertia due to the interaction with the “stickiness”. The situation is a little more complicated than that, but it has been proven that DCSB is a feature of QCD, and is thought to be related to confinement. However it has not yet been proved that confinement is a predictable consequence of QCD.

Light and Heavy Particles

In these lectures we will concentrate on what is referred to as the light quark sector, which generally encompasses hadrons that contain hadron physics 7

combinations of up, down or strange quarks. The heavy quark sector (charm, bottom and top) is still very much the purview of high- energy physics, although the boundary between the two is bot fuzzy and arbitrary.

Scattering Experiments

Figure 4: Conceptually, most mea- surements are identical to Rutherford’s original experiment

As with other areas of nuclear physics, most of the information about hadrons comes from scattering experiments. We will primarily discuss experiments that involve beams of photons and electrons, although will note in passing that a large amount of information has been derived from hadron beam experiments.

Cross Sections

• Select beam (type and current), target (type and thickness)

• Measure number of particles in detector, beam charge

Reaction a + b → X We imagine an experiment where a beam of particles a hits a target containing particles b. We want to know how a reacts with b. 8 d g ireland

• Flux: 1 dN Φ = a = n v a A dt a a

where na is beam particle density, A is the illuminated target area and va is the beam velocity.

• Number of target particles

Nb = nb Ad

where nb is target particle density, A is the illuminated target area and d is the target thickness.

• Luminosity: dN L = Φ N = a n d a b dt b which has dimensions L2T−1. 11

Differential Cross Section Consider a detector, area A, angle θ and distance r from beam

A ∆Ω = r2 Number of reactions seen by detector in time ∆t is

dσ (E, θ) N (E, θ) = L ∆Ω∆t dΩ where the piece dσ (E, θ) dΩ is the differential cross section dσ (E, θ) 12 An angular distribution is the variation of with angle θ. dΩ If the detector determines energy E0 of scattered particles

d2σ (E, E0, θ) dΩdE0 is the double differential cross section The total cross section is the integral

ZZ d2σ (E, E0, θ) σ = dΩdE0 tot dΩdE0 13 hadron physics 9

Scattering Theory

Assumptions Wavefunction of beam represented by plane waves. i.e.

∝ ψi exp (iki.r)

14 The momentum transfer is given by

q = h¯ (ki − k f ) Using quantum collision theory, it can be shown that the differential cross-section is equal to the square magnitude of a function called the scattering amplitude:

dσ = | f (θ)|2 dΩ where

• m Z iq.r f (θ) = − V(r) exp dr 2πh¯ 2 h¯

• m = mass of electron

• V(r) = potential between target and electron 15 Now

q.r = qr cos θ0 10 d g ireland

dr = r2 sin θ0drdθ0dφ so the expression for the scattering amplitude becomes

2m Z ∞ f (θ) = V (r) r sin (qr/¯h) dr hq¯ 0 16

Coulomb Scattering As an example of how to calculate scattering amplitudes, we start with a point charge scattering from a point charge (Rutherford scattering). Screened Coulomb potential:

Ze2 V (r) = exp (−r/a) r the scattering amplitude becomes:

2mZe2 Z ∞ qr −r f (θ) = sin exp dr hq¯ 0 h¯ a

[N.B. without the screening term, this integral become ill-deﬁned]

2mZe2 = 2 2 + h¯ q a2 and as a → ∞

dσ 2 2 4 2 1 ∝ 1 = | f (θ)| = 4m e Z . dΩ q4 4 θ sin 2 which is the Rutherford scattering cross-section. N.B. for full experimental analyses, a more sophisticated analysis is required which takes into account the effects of relativity and the spin of the electron, but the main concepts are maintained in this 17 simpler derivation. hadron physics 11

Rutherford scattering cross-section

18 Using the theoretical framework of the last section, we can use this to work out how to “measure” the size and shape of nuclei.

Elastic Electron Scattering

For elastic scattering: E E0 = + E ( − ) 1 Mc2 1 cosθ where M is the mass of the target nucleus. 19

Form Factors

The potential due to an element of the nuclear volume Zeρ (r) dr is Ze2 dV = ρ(r)dr s 12 d g ireland

To calculate the effect of electrons scattering from a charge distribution, we must integrate over the nuclear volume:

mZe2 ZZ ρ (r) i f (θ) = exp q. (r + s) drds 2πh¯ 2 s h¯ Z ∞ sin (qr/¯h) 2mZe2 Z ∞ = ρ (r) 4πr2dr. sin (qs/¯h) ds 0 (qr/¯h) hq¯ 0 20 So the cross-section becomes a product of two terms:- dσ dσ 2 ⇒ Ω = Ω .F (q) d d Ruth dσ where Ω is the Rutherford scattering cross-section, and d Ruth F2 (q) is the square of the elastic scattering Form Factor. The form factor is nothing other than the Fourier-Bessel transform of the charge distribution, so by measuring the cross-section, one can try to perform an inverse transform to extract the real charge distribution:

1 Z sin (qr/¯h) ρ (r) = F (q) 4πr2dr 2π (qr/¯h) In practice a lack of knowledge of data points at large values of q, means that some model of the charge distribution must be introduced, from which a theoretical form factor can be calculated and used to compare with the data points. The form of the charge distribution is often parameterised, and the parameters are adjusted to 21 give a best ﬁt to the measured data. The ﬁgure below shows the form factors that would be obtained with simple charge distributions: hadron physics 13

Electron Scattering Exercise 1 Using the material in the previous section (and anything you can ﬁnd online!), estimate the size of accelerators and detectors that are required to carry out hadron physics experiments.

Hall A, Jefferson Laboratory, USA (example of electron scattering facility) 23 14 d g ireland

Measured Spectra of Hadrons

Particle Data Group Entries - Mesons These listings, together with other very useful information is also 3 3 Particle Data Group website. http : / / online at the Particle Data Group website . pdg.lbl.gov/ 20024 4 The following tables indicate the difference between the and K. Hagiwara et al. “Review of Parti- 20175 cle Physics”. In: Physical Review D 66 listings: (2002), pp. 010001+. url: http://pdg. lbl.gov 5 C. Patrignani et al. “Review of Particle Physics”. In: Chin. Phys. C40.10 (2016), p. 100001. doi: 10.1088/1674- 1137/ 40/10/100001 24 hadron physics 15

Figure 5: PDG Meson table, 2002 16 d g ireland

Figure 6: PDG Meson table, 2017 hadron physics 17

Observations

• There are o(100) identiﬁed meson states. Symbol Quantum Number • Mass is listed in brackets if the decay is a strong interaction decay. I Isospin G G-Parity • List is grouped according to quark ﬂavour. J Spin P Parity • The number has increased over the 15-year period. C C-Parity

• Most new states are in the heavy-ﬂavour / high-energy physics sector.

• Each state has several associated quantum numbers.

25 Parity The Parity operator reﬂects a wavefunction in the origin:

Pˆ (ψ (r)) = ψ (−r) = ηpψ (r)

Wavefunction of a system can be separated into radial and angular part:

ψ (r) = R (r) Ylm (θ, φ) Spherical harmonics have property

l Ylm (π − θ, φ + π) = (−1) Ylm (θ, φ)

Fermions and anti-fermions have opposite parity: + Pˆ (qq) = (−1)L 1 26

Charge Conjugation Charge conjugation transforms particle to antiparticle âA¸Sneutral˘ particles are eigenstates of this operator, e.g.: E E ˆ 0 0 C π = ηC π Combination of position and spin wavefunctions lead to the rule: + Cˆ (qq) = (−1)L S 27

G-Parity A charged particle is not an eigenstate of charge conjugation, e.g.: + − Cˆ π = η π

However, if we combine an isospin operator, such that

Rˆ |I, Izi = |I, −Izi

Then charged particles will be eigenstates of a combined operator, + + + Gˆ π = Cˆ Rˆ π = π and + + Gˆ (qq) = (−1)L S 1 28 18 d g ireland

Combinations of Quantum Numbers

Exercise 2a Work out the possible combinations of the quantum numbers JPC that are allowed for mesonic states containing only a quark and an anti-quark. 29

Particle Data Group Entries - Baryons The following tables indicate the difference between the 2002 and 2017 listings (same references as before)

Figure 7: PDG Baryon table, 2002

Nomenclature

Symbol Isospin Quark Content 1 N 2 Three u and/or d quarks ∆ 1 2 Three u and/or d quarks Λ 0 Two u and/or d quarks Σ 1 Two u and/or d quarks Ξ 1 2 One u and/or d quark Ω 0 No u and/or d quarks hadron physics 19

Figure 8: PDG Baryon table, 2017 20 d g ireland

• Mass is listed in brackets if the decay is a strong interaction decay.

• Subscript to main symbol denotes heavy (usually c or b) quark content

Pre-2012 Notation Other symbols Letter Angular momentum of π − N ﬁnal state First subscript isospin content (integer or half integer) Second subscript total spin of state (×2)

Post-2012 Notation States labelled JP. Isospin and orbital angular momentum deduced from other labels.

Star Rating ???? Existence certain; properties explored ??? Existence likely / certain some properties undetermined ?? Evidence of existence fair ? Evidence of existence poor 31

Observations

• There are o(100) identiﬁed baryon states.

• The number has increased over the 15-year period.

• There are new states in both the light and heavy-ﬂavour sectors. 32

Combinations of Quantum Numbers Exercise 2b Deduce the quantum numbers of:

1. The Σ(1915)

2. The N(2250) 33 hadron physics 21

Building Hadrons from Quarks

A fundamental guiding principle in physics is that of symmetry. The powerful theory of Emmy Noether states that if the Hamilto- nian of a system is invariant under a group of transformations, then there exist corresponding conserved quantities.

Exact Conservation Laws

Symmetry Conservation Law Translation in time Energy Translation in space Linear Momentum Rotation in space Angular Momentum Local gauge invariance Charge

Transformation in Colour space Colour 34

Approximate Conservation Laws

Approximate Symmetry Conservation Law Spatial Inversion Parity Particle-antiparticle interchange Charge Conjugation Temporal Inversion Time-reversal invariance Transformations in isospace Isospin

Transformation in ﬂavour space Flavour 35

The Use of Quarks

Gell-Mann used the symmetry of the masses of particles and the ideas from group theory, and postulated ﬂavour symmetry that has SU(3) as its symmetry group. This is an approximate symmetry, Figure 9: Murray Gell-Mann since the masses of the particles are not exactly equal, but serves as 36 a guide to the physics underlying strongly interacting particles. It meant that predictions could be made based on the construction of multiplets, which can be seen as representations of the symmetry group. The group theory behind this is beyond the scope of three lectures, but a method of calculating the number of states in multiplets, using Young diagrams will be introduced. 22 d g ireland

The Quark Picture of Hadrons

SU(N) Multiplets Figure 10: The quark picture of baryons 1 and mesons. Uniquely identiﬁed by string of (N- ) integers, (α, β, γ...). For 37 SU(3), the two labels are the number of steps:-

Octet is (1,1), decuplet is (3,0) For N>3, visualisation more difﬁcult N.B. multiplet = representation For SU(N) describing N ﬂavour quarks, N quarks together form 38 a (1,0,...,0) multiplet, N antiquarks form (0,...,0,1) conjugate multiplet.

Young Diagrams for SU(N) Method for calculating the number associated with each multiplet. Young diagrams (or tableau) are:-

• Arrays of boxes or other symbol, left justiﬁed

• Each row at least as long as the row below

• At most N rows

For SU(3) the diagrams:-

, , , ,

39 represent the multiplets (1,0), (0,1), (0,0), (1,1), (3,0) How to read the boxes Number of overhanging boxes in each row corresponds to the labels (α, β, γ, ...) Depends on which group (SU(N)), so hadron physics 23

represents (2,3,1) in SU(4),(2,3) in SU(3) and is not valid for SU(2) (why?) 40 For N ﬂavours of quarks, the fundamental representation is given by a single box . The conjugate representation for antiquarks is given by a column of N − 1 boxes, e.g.

for SU(3) 41

Calculating Multiplicities Work out two factors: Numerator Place N in top left box, and number as shown. Take product of all numbers.

Denominator Take the product of hooks:

43 24 d g ireland

Product of Two Multiplets 1 e.g. two isospin- particles (u and d; symmetry group SU(2)): 2

⊗ = ⊕

These can be identiﬁed with the states:

2 × 3 1 = 3 |uui , √ (|udi + |dui), |ddi 2 × 1 2 44 2 × 1 1 = 1 √ (|udi − |dui) 2 × 1 2

Multiplets and Particle States We can now derive the multiplicities for mesons and baryons in the quark model. Using ﬂavour SU(3)

Mesons

Contain a quark ( ) and an anti-quark ( )

⊗ = ⊕

resulting in a singlet and an octet

3 ⊗ 3 = 1 ⊕ 8 45

Baryons We already have the direct product of two fundamental representations: ⊗ = ⊕

In SU(2) this gave 2 ⊗ 2 = 3 ⊕ 1

for SU(3) it is 3 ⊗ 3 = 6 ⊕ 3 46 Now make a direct product with each of the previous multiplets:

⊗ = ⊕

and ⊗ = ⊕

We thus have

(3 ⊗ 3) ⊗ 3 = (10 ⊕ 8) ⊕ (8 ⊕ 1) 47 hadron physics 25

Flavour and Spin Symmetry One can combine the symmetry of ﬂavour with spin with a direct product of the symmetry groups:

SU(3) f lavour ⊗ SU(2)spin = SU(6) which generates the “super-multiplets” for mesons:

6 ⊗ 6 = 1 ⊕ 35

and baryons:

6 ⊗ 6 ⊗ 6 = 56 ⊕ 70 ⊕ 70 ⊕ 20

A further direct product includes radial excitations via the couple of angular momentum

SU(6) f lavour−spin ⊗ O(3)a.m. 48 We can now associate the observed resonances with this scheme:

Exercise 3 Use Young diagrams to show that the number of members of the “super-multiplets” are as given above. 50 26 d g ireland

Finding Hadronic States

Resonance Hunting

Pion Scattering

Most information about baryon resonances up to the year 2000 was obtained through pion-nucleon scattering experiments. Pi- ons excite the nucleon to higher mass states, which then decay to detected pions or nucleons. However, since the decay is a strong interaction the widths of the peaks on mass spectra overlap with each other, which results in three or four “bumps”, referred to as resonance regions. Careful measurement of angular distributions are required, to enable partial wave analysis (PWA) to identify states with different spins and parities. hadron physics 27

Quark Model Baryon Spectrum

By comparing the measured states with the multiplets predicted by the simple quark model, it could be seen that many of the predicted states had not been detected. This was, and still is, known as the missing resonance problem.

Quark Model Calculations of the Baryon Spectrum

6 6 Even state-of-the-art quark model calculations predicted that Simon Capstick and W. Roberts. more states should exist, and so the rationale for using the quark “Quark models of baryon masses and decays”. In: Prog. Part. Nucl. Phys. 45 model (which is after all only a simple model to describe QCD) was (2000), S241–S331 called into question. 28 d g ireland

Lattice QCD Baryon Spectrum

As the capabilities of high performance computing have improved, the ability to carry out complicated calculations using the technique of Lattice QCD has significantly advanced. This technique is a method of simulating QCD from first principles by performing the path integrals required in the field theory by picking out sample paths through a discretised version of space-time. In the limit of infinite samples, these calculations will approach pure QCD. 7 7 Robert G. Edwards et al. “Excited state Landmark calculations in the last few years have shown that the baryon spectroscopy from lattice QCD”. spectrum of nucleon excited states can be calculated. Until about 15 in: Phys. Rev. D 84.7 (Oct. 2011), p. 074508. doi: 10.1103/PhysRevD.84. years ago, this was thought be be beyond the reach of computational 074508. url: http://link.aps.org/ resources. One of the surprising findings from these calculations doi/10.1103/PhysRevD.84.074508 was that the pattern of excitations bore a startling resemblance to the spectrum predicted by the each quark models. In other words, the quark model is an emergent property of QCD. However, this gave further weight to the idea that several nucleon resonances are indeed “missing”. It was one of the spurs that drove a large effort to identify baryon resonance, primarily using photon and electron beams at facilities such as Jefferson Lab (USA), Bonn (Germany), SPRing-8 (Japan), ... Examination of the differences between the 2002 and 2017 PDG tables shows that there are indeed a few more resonant states that are now well-established. The story continues... hadron physics 29

Finding a Resonant State

Dalitz Plots

If a measured final state contains three particles, one can use the technique of Dalitz plots to identify possible resonances. Two particles decaying from a resonance will show up as a band (horizontal, vertical or diagonal) on the plot. This can be more revealing than a simple mass plot.

Example - the LHCb Pentaquark

8 R. "Aaij and others". “Observation of J/ψp Resonances Consistent with Pen- Λ0 − taquark States in b → J/ψK p De- cays”. In: Phys. Rev. Lett. 115 (7 2015), p. 072001. doi: 10.1103/PhysRevLett. 115.072001. url: https://link.aps. org/doi/10.1103/PhysRevLett.115. 072001

8 9 9 As an example the recent result from LHCb illustrates this. . You This is into the charm sector, which I can just about make out a horizontal band in the plot. said I would not cover - I lied! 30 d g ireland

58 The projection onto the mass of the J/ψ − p system shows a peak.

59 “Proof” of a resonance is that it exhibits phase motion as it goes through a resonant mass. This can be plotted on an Argand diagram. Unfortunately, limited time prevents further explanation! The left hand plot looks fairly convincing. The right hand plot is left to the reader to assess. hadron physics 31

Meson Spectrum and Exotics

Hadrons are Complicated!

Nature’s “real” states are more likely to be a superposition of several states with different numbers of quarks and anti-quarks. An intriguing possibility is that one of the states is one in which the “string” of gluons binding the quarks is in an excited states, leading to a hybrid (qq)g state or even a pure gluon gg state, referred to as a glueball. 32 d g ireland

Non-Quark Model Hadrons

The leading component in a particular hadron’s wavefunction may therefore be one of these non-quark model states. If the states contains quantum number combinations that cannot be derived from a quark model conﬁguration, that might be a “smoking-gun” for a new state of matter.

Exotic Hybrid Mesons

Several calculations predict that the lowest mass exotic hybrid mesons should exist at masses of a few GeV. hadron physics 33

Lattice QCD Meson Spectrum

The simple model calculations have again been supported from 10 10 Lattice QCD calculations . These predictions have been a major Jozef J. Dudek et al. “Toward the excited meson spectrum of dynamical driver behind the upgrade of the Jefferson Lab accelerator, CEBAF, 82 3 12 QCD”. in: Phys. Rev. D . (Aug. to energies of GeV. There is an exciting prospect of these new states 2010), p. 034508. doi: 10 . 1103 / being discovered in the next couple of years. PhysRevD . 82 . 034508. url: http : / / link.aps.org/doi/10.1103/PhysRevD. 82.034508 GlueX at Jefferson Lab

65 34 d g ireland

CLAS12 at Jefferson Lab

67 End of Lectures hadron physics 35

References

[1] R. "Aaij and others". “Observation of J/ψp Resonances Consis- Λ0 − tent with Pentaquark States in b → J/ψK p Decays”. In: Phys. Rev. Lett. 115 (7 2015), p. 072001. doi: 10.1103/PhysRevLett. 115 . 072001. url: https : / / link . aps . org / doi / 10 . 1103 / PhysRevLett.115.072001. [2] Simon Capstick and W. Roberts. “Quark models of baryon masses and decays”. In: Prog. Part. Nucl. Phys. 45 (2000), S241–S331. [3] Jozef J. Dudek et al. “Toward the excited meson spectrum of dynamical QCD”. In: Phys. Rev. D 82.3 (Aug. 2010), p. 034508. doi: 10.1103/PhysRevD.82.034508. url: http://link.aps. org/doi/10.1103/PhysRevD.82.034508. [4] Robert G. Edwards et al. “Excited state baryon spectroscopy from lattice QCD”. In: Phys. Rev. D 84.7 (Oct. 2011), p. 074508. doi: 10.1103/PhysRevD.84.074508. url: http://link.aps. org/doi/10.1103/PhysRevD.84.074508. [5] K. Hagiwara et al. “Review of Particle Physics”. In: Physical Re- view D 66 (2002), pp. 010001+. url: http://pdg.lbl.gov. [6] Particle Data Group website. http://pdg.lbl.gov/. [7] C. Patrignani et al. “Review of Particle Physics”. In: Chin. Phys. C40.10 (2016), p. 100001. doi: 10 . 1088 / 1674 - 1137 / 40 / 10 / 100001.