The Particle World

CERN Summer School Lectures 2021

Lectures 1 and 2: The Standard Model

David Tong Further Reading

around 240 pages! (Sorry) What are we made of?

"If we consider protons and neutrons as elementary particles, we would have three kinds of elementary particles [p,n,e].... This number may seem large but, from that point of view, two is already a large number.”

Paul Dirac 1933 Solvay Conference The Standard Model

12 particles + 4 forces + Higgs boson The Standard Model

ContentsContentsContentsContents Contents

1 Introduction1 Introduction11 Introduction Introduction1 Introduction 1 1 1 1 1

1 Introduction1 Introduction11 Introduction Introduction1 Introduction The numbers in the table are the masses of the particles, written as multiples of the electron mass. (Hence the electron itself is assigned mass 1.) The masses of the neutrinosGravity are ElectromagnetismGravity knownGravity ElectromagnetismGravity to be very Electromagnetism Weak Electromagnetism smallGravity Strong but, Weak otherwise Electromagnetism Strong Weak Weak are Strong onlyStrong constrained Weak Strong within a window and not yet established individually.

Each horizontal line of this diagram is called a generation. Hence, each generation consists of an electron-like particle, twoHiggs quarks, and a neutrino. The statement that each generation behaves the same means that, among other things, the electric charges of all electron-like particles in the first column are 1(inappropriateunits);theelectric 1 charges of all quarks in the second column are 3 and all those in the third column 2 + 3 . All neutrinos are electrically neutral. We understand aspects of this horizontal pattern very well. In particular, various mathematical consistency conditions tell us that the particles must come in a collective of four particles, and their properties are largely fixed. In particular, we understand why the particles have the electric charges that they do: this is forced upon us by the mathematics and they simply can’t be anything else. Moreover if, one day, we were to find a fourth species of electron-like particle, then we can be sure that there are also two further quarks and a neutrino to discover as well. We’ll describe this more in Section 4.

We don’t, however, understand the observed pattern of masses. More importantly, we don’t understand the vertical direction in the pattern at all. We don’t understand why there are 3 generations in the world and not, say, 17. Nonetheless, we know from both particle physics and from cosmological observations that there are no more than 3

–9–

–1– –1– –1––1– –1– Contents The Standard Model 1 Introduction 1

1 Introduction

ContentsContentsContentsContents GravityContents Electromagnetism Weak Strong

1 Introduction1 Introduction11 Introduction Introduction1 Introduction 1 1 1 1 1

Higgs

1 Introduction1 Introduction11 Introduction Introduction1 Introduction The numbers in the table are theHypercharge masses of the particles, written as multiples of the electron mass. (Hence the electron itself is assigned mass 1.) The masses of the neutrinosGravity are ElectromagnetismGravity knownGravity ElectromagnetismGravity to be very Electromagnetism Weak Electromagnetism smallGravity Strong but, Weak otherwise Electromagnetism Strong Weak Weak are Strong onlyStrong constrained Weak Strong within a window and not yet established individually.

–9–

–1– –1– –1––1– –1– Our First Unification: Fields

• Particles are excitations of underlying quantum fields

• Forces are also due to fields and have associated particles

• Electromagnetism = photon • Strong = gluon • Weak = W and Z bosons • Gravity = graviton Intrinsic Angular Momentum = Spin Contents

1 Introduction 1

These are fermions.

1 Introduction The Pauli exclusion Contents Spin ½ ContentsContents Contents principle applies to fermions. 1 Introduction 11 Introduction IntroductionGravity1 Electromagnetism Introduction Weak Strong 1 1 1 1

Higgs Spin 1 1 Introduction11 Introduction Introduction1 Introduction Spin 2 The numbers in the table are the masses of the particles, written as multiples of the electron mass. (Hence the electron itself is assigned mass 1.) The masses of the neutrinosGravity are Electromagnetism knownGravityGravity toHypercharge be very Electromagnetism Weak Electromagnetism smallGravity Strong but, otherwise Electromagnetism Weak Weak are Strong onlyStrong constrained Weak Strong within a window and not yet established individually.

Each horizontal line of this diagram is called a generation. Hence, each generation consistsThese are ofall an(gauge) electron-like bosons particle, twoHiggs quarks, andSpin a 0 neutrino. The statement that each generation behaves the same means that, among other things, the electric charges of all electron-like particles in the first column are 1(inappropriateunits);theelectric 1 charges of all quarks in the second column are 3 and all those in the third column 2 + 3 . All neutrinos are electrically neutral. We understand aspects of this horizontal pattern very well. In particular, various mathematical consistency conditions tell us that the particles must come in a collective of four particles, and their properties are largely fixed. In particular, we understand why the particles have the electric charges that they do: this is forced upon us by the mathematics and they simply can’t be anything else. Moreover if, one day, we were to find a fourth species of electron-like particle, then we can be sure that there are also two further quarks and a neutrino to discover as well. We’ll describe this more in Section 4.

–1– –9–

–1– –1––1– –1– A Remarkable Fact We won’t explain the mathematics behind the Dirac equation in these lectures, but it’s so beautiful that it would be a shame not to show it to you. I’ve put it in a picture All spin ½ particles are described by the same equation, discovered by Dirac frame to highlight that it’s here for decoration as much as anything else.

Here is the quantum ﬁeld; it depends on space and time.Here m Itis th alsoe mass. has four components, so it’s similar to a vector but di↵ers in a subtle way. It is know as a spinor. For what it’s worth, the parameter mConsequencis thee: all mass matter of particles the come particle, with anti-particles while @µ denotes derivatives and µ are a bunch of 4 4matrices.IfyouwanttounderstandwhattheDiracequation ⇥ really means, you can ﬁnd details in the lectures on Quantum Field Theory.

Dirac originally wrote his equation to describe the electron. But, rather wonderfully, it turns out that this same equation describes muons, taus, quarks and neutrinos. This 1 is part of the rigid structure of quantum ﬁeld theory. Any particle with spin 2 must be described by the Dirac equation: there is no other choice. It is the unique equation consistent with the principles of quantum mechanics and special relativity.

1 The Dirac equation encodes all the properties of particles with spin 2 . Given such aparticle,onceyouﬁxtheorientationofthespintherearetwopossiblestatesin which the particle can sit. Roughly speaking, it can spin clockwise or it can spin anti-clockwise. We call these two states “spin up” and “spin down”.

The Pauli exclusion principle states that no two fermions can sit in the same quantum state. But the quantum state is determined by both the position and the spin of the electron. This means that an electron with spin down can be in the same place as an electron with spin up, because their spins di↵er. If you’ve done some basic chemistry, this should be familiar: both the hydrogen and helium atoms have electrons sitting in the orbit that sits closest to the nucleus. The two electrons in helium necessarily have di↵erent spins to satisfy the exclusion principle. But, by the time you get to the third element in the periodic table, lithium, there is no longer room for an additional electron in the closest orbit and the third electron is forced to sit in the next one out.

2.1.3 Anti-Matter The real pay-o↵ from the Dirac equation comes when you solve it. The most general solution has an interesting property: there is a part which describes the original particles, like the electron. But there is a second part that describes particles with the same

–33– Mass

The numbers in the table are the masses of the particles, written as multiples of the electron mass. (Hence the electron itself is assigned mass 1.) The masses of the neutrinosAn aside: are In known the Standard to be very Model, small the but, masses otherwise of all particles are only are constrained determined within a window and notby yetthe establishedstrength of interaction individually. with the Higgs field.

Each horizontal line of this diagram is called a generation. Hence, each generation consists of an electron-like particle, two quarks, and a neutrino. The statement that each generation behaves the same means that, among other things, the electric charges of all electron-like particles in the first column are 1(inappropriateunits);theelectric 1 charges of all quarks in the second column are 3 and all those in the third column 2 + 3 . All neutrinos are electrically neutral. We understand aspects of this horizontal pattern very well. In particular, various mathematical consistency conditions tell us that the particles must come in a collective of four particles, and their properties are largely fixed. In particular, we understand why the particles have the electric charges that they do: this is forced upon us by the mathematics and they simply can’t be anything else. Moreover if, one day, we were to find a fourth species of electron-like particle, then we can be sure that there are also two further quarks and a neutrino to discover as well. We’ll describe this more in Section 4.

–9– A similar story arises when we consider quantum mechanics. The fundamental constant in quantum mechanics is Planck’s constant,

34 ~ =1.054571817 10 Js ⇥ It has units of energy time. What this constant is really telling is us that, at the ⇥ fundamental level there is a close connection between energy and time. A process with energy E will typically take place in a time T = ~/E. In this way, we can translate between units of energy and units of time, and these concepts are not as distinct as our ancestors believed. To highlight this, we choose units so that

~ =1

The choice c = ~ =1isreferredtoasnatural units. It means that there’s only one dimensionfull quantity left, which we usually take to be energy. Any measurement — whether it’s of length, time or mass — can be expressed in terms of energy.

A Sense of Scale Units for Mass The SI unit of energy is a Joule and is not particularly appropriate for the sub-atomic world.The Instead, entries for we neutrinos use the are electronvolt upper bounds (eV) on which the mass. is the Meanwhile, energy an the electron masses of picks the up 2 when5force-carryingparticlesareWe accelerated measure mass acrossin terms 1of volt.energy, This using isE=mc . The unit of choice is the electronvolt

19 1eV 1.6 10 J ⇡ ⇥ TheOr electronvolt MeV = 106 eV is or the GeV energy = 109 eV scale or TeV appropriate= 1012 eV. for atomic physics. For example, the energy that binds an electron to a proton to form a hydrogen atom is E 13.6eV. ⇡ For elementary particles, we will need a somewhat larger unit of energy. We typically For each of these6 particles, there9 is an associated length scale. We get this by trans- use MeVMore confusingly, = 10 eVwe oralso GeVmeasure = 10distanceeV. in The terms LHC, of inverse our energy, best using current collider, runs at an energyforming scale energy measuredE into in a TeV length = using 1012 theeV. fundamental constants of Nature c and ~, ~c The masses of the 12 matter particles = cover a range from eV to GeV. They(1.2) are: E This is known as the Compton wavelength. Roughly speaking, it can be viewed as For a particle of mass E, this is the “Compton wavelength”, or the size of the particle. the size of the particle. For example, for the electron the Compton wavelength is 12 2 10 m. Perhaps somewhat surprisingly, theNote: heavierheavier a particleparticles is,are thesmaller smaller! e ⇡ ⇥ its Compton wavelength.

The Biggest and the Smallest There are two further length scales that we should mention before delving into details of the subatomic world. One is associated to the strength of the gravitational force. Newton’s constant is given by

11 3 1 2 G 6.67 10 m kg s N ⇡ ⇥ –17–

–18– The Masses of Particles

The entries for neutrinos are upper bounds on the mass. Meanwhile, the masses of the 5force-carryingparticlesare

For each of these particles, there is an associated length scale. We get this by trans- forming energy E into a length using the fundamental constants of Nature c and ~, ~c = (1.2) E This is known as the Compton wavelength. Roughly speaking, it can be viewed as the size of the particle. For example, for the electron the Compton wavelength is 12 Compton wavelength of electron 2 10 m. Perhaps somewhat surprisingly, the heavier a particle is, the smaller e ⇡ ⇥ its Compton wavelength.

11 3 1 2 G 6.67 10 m kg s N ⇡ ⇥

–18– The Masses of Particles

Note: photon and graviton both massless.

(The gluon is a little subtle...see later.) 2 In natural units, Newton’s constant has dimensions of [Energy] . Putting back the factors of ~ and c, we can derive an energy scale knownThe as Masses the Planck massof Particles

~c 18 Mpl = 2 10 GeV r8⇡G ⇡ ⇥ 2 (TheIn natural factor units, of 8⇡ Newton’sis conventional, constant and has is dimensions sometimes of dropped.) [Energy] This. Putting is an enormous back the energyfactors scale,of ~ and ﬁfteenc, we orders can derive of magnitude an energy larger scale known than the as the scales Planck that mass appear in the Standard Model. The corresponding length scale is the Planck length

~c 18 Mpl = 2 10 GeV 88⇡⇡~GG⇡ ⇥ 35 Lpl = r 8 10 m c3 ⇡ ⇥ (The factor of 8⇡ is conventional,r and is sometimes dropped.) This is an enormous This,energy in scale, turn, fifteen is a tiny orders length of magnitude scale, 15 orders larger of than magnitude the scales smaller that than appear the in scale the thatStandard we have Model. explored The corresponding at our best particle length colliders. scale is the The Planck Planck length scale is where both quantum mechanics and gravity become important. It seems likely that space and time cease to make sense at these scales, although8⇡ G it’s not clear exactly what this means. ~ 35 2 Lpl = 8 10 mIn natural units, Newton’s constant has dimensions of [Energy] . Putting back the c3 ⇡ ⇥ On the other end of the spectrum,r the largest size thatfactors we of can~ talkand c about, we can is the derive an energy scale known as the Planck mass entireThis, inobservable turn, is universe, a tiny length scale, 15 orders of magnitude smaller than the scale that we have explored at our best particle colliders. The Planck scale is where both ~c 18 26 Mpl = 2 10 GeV Luniverse 9 10 m 8⇡G ⇡ ⇥ 2 quantum mechanics and gravity become⇡ important.⇥ It seemsIn natural likely that units, space Newton’s and time constantr has dimensions of [Energy] . Putting back the cease to make sense at these scales, althoughThe biggest it’s not clearfactors exactly of whatand thisc, we means. can deriveThe an energysmallest scale known as the Planck mass The corresponding energy scale is (The factor~ of 8⇡ is conventional, and is sometimes dropped.) This is an enormous On the other end of the spectrum, the largest size thatenergy we scale, can talk fifteen about orders is the of magnitude larger than the scales that appear in the 33 ~c 18 entire observable universe, H 2 10 eV Standard Model. The correspondingMpl = length scale2 10 is theGeV Planck length ⇡ ⇥ r8⇡G ⇡ ⇥ This energy scale is closely related to the so-called26 “Hubble(The constant” factor of that 8⇡ is measures conventional, and8⇡~G is sometimes35 dropped.) This is an enormous Luniverse 9 10 m L = 8 10 m ⇡ ⇥ pl 3 how fast the universe is expanding, albeit written in the unusualenergy units scale, of fifteen electronvolts. orders of magnituder c ⇡ larger⇥ than the scales that appear in the Clearly,The corresponding it’s a tiny energy energy scale. scale is A particle with the massThis,Standard of H inwould turn, Model. have is a The tiny the corresponding same length scale, length 15 orders scale of is magnitude the Planck smaller length than the scale size as the entire universe. 33 that we have explored at our best particle colliders. The Planck scale is where both H 2 10 eV 8⇡~G 35 This, then, is the playground of physics.⇡ ⇥ The goal of physicsquantum is to mechanics understand and every- gravityLpl = become important.8 10 It seemsm likely that space and time c3 ⇡ ⇥ thingThis energy that can scale happen is closely at lengths related in to the the range so-called “Hubblecease constant” to make sense that atmeasures these scales,r although it’s not clear exactly what this means. how fast the universe is expanding, albeit written in the unusualThis, in units turn, of is electronvolts. a tiny length scale, 15 orders of magnitude smaller than the scale 34 26 On the other end of the spectrum, the largest size that we can talk about is the 10 m

It turns out that there is quite a lot of interesting things in this window! And, under- –19– lying many of them, is the Standard Model.

–19– Electric Charge

Charge = -1 -1/3 +2/3 0

The numbers in the table are the masses of the particles, written as multiples of the electron mass. (Hence the electron itself is assigned mass 1.) The masses of the Theneutrinos electric charge are known characterizes to be verythe (relative) small but, strength otherwise of the are electromagnetic only constrained interaction within a window and not yet established individually.

–9– The equations that describe the dynamics of the electric and magnetic ﬁelds are Contents known as the Maxwell equations. We won’t need them in these lectures but, for com- Contents pleteness, here they are: ⇢ @B 1 Introduction E = , E = 1 1 Introduction r · ✏0 r⇥ @t 1 @E B =0 , B = µ J + ✏ (2.3) r · r⇥ 0 0 @t ✓ ◆ The ﬁelds E and B react to the presence electric charge density ⇢ and electric currents

J, while ✏0 and µ0 are two constants that characterise the strength of the electric and 1 Introduction magnetic forces in a way that we will describe more below. 1 Introduction The equations, as written above, hide the full beauty of the Maxwell equations. A better formulation encodes both the electric and magnetic fields in a 4 4anti- ⇥ Gravitysymmetric Electromagnetismmatrix called the field strength, Weak which Strong takes the form F = @ A @ A . Electromagnetism (or QED) µ⌫ µ ⌫ ⌫ µ GravityOnly then Electromagnetism do the Maxwell equations Weak reveal their Strong true simplicity, in a way that deserves hanging in a frame, The equations that describe the dynamics of the electric and magnetic fields are known as the Maxwell equationsThe Maxwell. We won’t Equations need them in these lectures but, for com- pleteness, here they are: Higgs ⇢ @B E = , E = Higgs r · ✏0 r⇥ @t or @E B =0 , B = µ J + ✏ (2.3) r · r⇥ 0 0 @t ✓ ◆ The fields E and B react to the presence electricYou charge can density learn⇢ moreand electric about currents the Maxwell equations and their classical solutions in the lec- J, while ✏0 and µ0 are two constants that characterisetures on theElectromagnetism strength of theHypercharge electric. and Famously, among the solutions to these equations are elec- magnetic forces in a way that we will describe more below. Hypercharge This implies the Coulombtromagneticforce which, waves, in including natural units, visible reads light. When you look at these solutions through The equations, as written above, hide thethe full beauty lens of of quantum the Maxwell mechanics, equations. you find that they decompose into particles, known as A better formulation encodes both the electric and magnetic fields in a 4 4anti- ⇥ symmetric matrix called the field strength, whichphotons takes the. form F = @ A @ A . µ⌫ µ ⌫ ⌫ µ Only then do the Maxwell equations reveal their true simplicity, in a way that deservesQ1Q2 A photon can comeF in= two↵ di↵2erent states that we call polarisation. These are entirely hanging in a frame, ↵Q1rQ2 analogous to the “spinF = up” and “spin down” states of the electron. (The fact that both r2 with the fine structurespin constant 1/2andspin1particleshavetwointernalstatesissomethingofacoincidence.For example, it’s only truee in2 three spatial1 dimensions; the counting is di↵erent in other dimensions.) ↵ = 2 4⇡✏e 0~c ⇡ 1371 ↵ = You can learn more about the Maxwell equations andThe their theory classical describing solutions in the the lec- electromagnetic field interacting with the electron field is tures on Electromagnetism. Famously, among the solutions to these equations4⇡✏ are0~ elec-c ⇡ 137 tromagnetic waves, including visible light. Whenknown you look as quantum at these solutions electrodynamics through ,orQED for short. It is the theory describing light the lens of quantum mechanics, you find that theyinteracting decompose with into particles, matter, known and as ultimately underpins large swathes of science, including photons. condensed matter physics and chemistry. Happily, it is also the simplest component of A photon can come in two di↵erent states thatthe we call Standard polarisation. Model. These are entirely analogous to the “spin up” and “spin down” states of the electron. (The fact that both spin 1/2andspin1particleshavetwointernalstatesissomethingofacoincidence.For example, it’s only true in three spatial dimensions; the counting is di↵erent in other dimensions.) –39– The theory describing the electromagnetic field interacting with the electron field is known as quantum electrodynamics,orQED for short. It is the theory describing light interacting with matter, and ultimately underpins large swathes of science, including condensed matter physics and chemistry. Happily, it is also the simplest component of the Standard Model.

–39–

–1–

–1– Contents

1 Introduction 1

1 Introduction

ContentsGravity Electromagnetism Weak Strong

Contents 1 Introduction 1

1 Introduction Higgs 1

1 Introduction

1 Introduction Hypercharge Gravity Electromagnetism Weak Strong

Gravity Electromagnetism Weak Strong Q Q F = ↵ 1 2 r2 Higgs 2.3.12.3.1 The TheLong, Long, Confusing Confusing History History of Renormalisation of Renormalisation Higgs WhileWhile the description the description of renormalisation of renormalisation described described above above is fairly is fairly straightforward, straightforward, the the Feynmanmathematicsmathematics2 underlying underlyingDiagrams it is not. it is For not. this For reason, this reason, our forefathers our forefathers had to had travel to travel a long a long and tortuousande tortuous road roadto make1 to make sense sense of quantum of quantum ﬁeld theoryﬁeld theory in general, in general, and the and issue the issue of of ↵ = Hypercharge renormalisation4⇡✏renormalisation0~c ⇡ in particular.137 in particular. An important fact: quantum fieldThe theory storyThe story starts is starts inhard! the in late the 1920s, late 1920s, soon soonafter after the original the original development development of quantum of quantum

Hypercharge mechanics.mechanics. The quantum The quantum pioneers pioneers — Heisenberg, — Heisenberg, Dirac, Dirac, Pauli Pauli and others and others — tried — tried to to

+ applyapply their theirideas ideas to the to interaction the interaction of light of lightand matter. and matter. They They tried tried to ask to very ask simplevery simple

2.3.1 The Long, Confusing+ History of Renormalisationquestions,questions, like the like probability the probability forQ a photonforQ a photon to scatter to scatter o↵ an o electron.↵ an electron. While While they didn’tthey didn’t + + 1 2 While the description of renormalisation describedhave abovehave the1 diagrammatic the is fairly diagrammatic straightforward,F tools= tools later↵ later introduced the introduced by Feynman, by Feynman, they theydid understand did understand that that We are saved in+ QED because ↵ 1 . This allowsr2 us to write down an approximate

mathematics underlying it is not. For this+ reason,we could ourwe forefathers could approach approach had the to problem the travel problem a in long a perturbative in a perturbative expansion, expansion, starting starting with with a process a process

+ 137 + ⇡ ⌧ and tortuous road to make sense of quantum fieldwhich theorywhich we now inwe general, draw now draw like and this: like the this: issue of solution Q1Q2 renormalisation in particular.F += ↵+ r2+ e2 1 The story starts in the late 1920s, soon after the original development↵ = of quantum mechanics. The quantum pioneers — Heisenberg, Dirac, Pauli and others4⇡✏ —0 tried~c to⇡ 137 Figure 9. The renormalisation of electric charge. e.g.applywhat their is ideas the to the probability interaction2 of light for and a matter. photon They triedto toscatter ask very simpleoff an electron in some direction? questions, like the probabilitye for a photon1 to scatter o↵ an electron. While they didn’t While it’s challenging to talk↵ = about quantum fields, we canThey buildThey found some found that intuition this that calculation this by calculation gave pretty gave pretty good good agreement agreement with thewith experiments. the experiments. But But have the diagrammatic4 tools⇡✏0 later~c ⇡ introduced137 by Feynman, they did understand that reverting towe the could language approach of particles. the problem As we get in acloser perturbative to thethen electron, expansion,then they the theytried electric starting tried to do field to better, with do better, a and1 process compute and compute the leading the leading corrections. corrections. In diagrammatic In diagrammatic gets strongerwhich and, we as now a result, draw the like energy this: density stored inlanguage, thelanguage, field this gets means largerthis means evaluatingand↵ evaluating diagrams diagrams1 like this like this 12 2 larger. At some point — a distance of around 10 m — the energy density is so large⇡ 137 ⌧ that an electron-positron pair can be produced from the vacuum. 1 There is aProbability general rule in quantum ↵= field theory that1 anything that+ can happen does, +++ + +...... +... in fact, happen. This means a single⇡ electron137 is⌧ surrounded by a swarm of particle-anti- particle pairs, continually popping in and out of the vacuum. As we get closer still, (↵2) muons, tausThey and found even quarks that this will calculation also appear gave in pretty the mix. goodHowever, We agreement learnHowever, herethat with anythey here the simple ranthey experiments. into ranO a into problem. a But problem. Each Each of these of these subsequent subsequent diagrams diagrams was pro- was pro- picture youthen may have they of tried a single to do particle better, giving and risecompute to the the electricportional leadingportional field corrections. to is really↵4, to as↵ very expected.4, as In far expected. diagrammatic But the But proportionality the proportionality constant constant was infinity. was infinity. That That made made from the truth:language, it is impossible this means to evaluating enforce any2 diagrams kind of social like thisit distancing veryit hard very in hardto the argue quantum to argue that the that contributions the4 contributions from from these these diagrams diagrams was smaller was smaller than than the the world. (↵ ) (↵ ) O first. first. O This story should really be viewed in terms of quan- left-handed right-handed The quantumThe quantum heroes heroes worked worked on this on problem this problem for well for over well a over decade, a decade, but made but made little little tum fields. When we talk of a swarm of particle-anti-+ +... 4 progress.progress. Looking Looking back, back, many many of their of their ideas ideas were weresimply simply too crazy. too crazy. Having Having forged forged one one particle pairs, it is really a metaphor for( the↵ ) quantum spin revolutionrevolution they were,they were, like Che like Guevara, Che Guevara, all keen all keenfor the for next. the next. Bohr Bohr wanted wanted to get to rid get rid field being excitedspin in a tangled and elaborateO fashion. of energyof energy conservation. conservation. Heisenberg Heisenberg wanted wanted to make to make spacetime spacetime non-commutative. non-commutative. Pauli Pauli Just asMore theHowever, vacuumcomplicated is here somethingmomentum they ran complicated intodiagrams,momentum a problem. in quan- Each like of these –1– subsequent , are diagrams less was and pro- less important. wantedwanted to invade to invade Bolivia. Bolivia. Yet the Yet answer the answer they werethey seekingwere seeking did not, did ultimately, not, ultimately, require require tum field theory,portional so too to is↵ the4, as notion expected. of a single But the particle. proportionality constant was infinity. That made an overhaulan overhaul of the of foundations the foundations of physics. of physics. It needed It needed a di↵erent a di↵erent approach. approach. In the languageit very of hard Feynman to argue diagrams, that the these contributions particle- from these diagrams was smaller than the anti-particlefirst. pairs are captured by the diagrams that include loops of particles, like the one shown on the right. The quantum heroes worked on this problem for well over a decade, but made little The excitedprogress. quantum Looking field back,has an many important of their consequence ideas were forsimply the too strength crazy. of Having the forged one –1– –51––51– electromagneticrevolution interaction. they were, Again, like we Che can understand Guevara, all this keen in the for language the next. of Bohr particles. wanted to get rid The swarmof of energy particle-anti-particle conservation. Heisenberg pairs will notwanted be oriented to make randomlyspacetime around non-commutative. the Pauli electron. Instead,wanted the to invade positrons, Bolivia. which Yet carry the positive answer they charge, were will seeking be attracted did not, to ultimately, the require an overhaul of the foundations–1– of physics. It needed a di↵erent approach.

–49–

–51– We can easily extend our Feynman diagrams to include these extra particles. By convention, we depict any fermion with a forward arrow and any anti-fermion with a backward arrow , but now we must label these lines with the particle name to show what particle we’re talking about. Every particle with electric charge will have an interaction vertex of the form . When evaluating Feynman diagrams, these vertices contribute a factor of Q2↵ to the probabilities, where Q is the charge of the particle, in units where the electron has Q = 1. This brings new opportunities. For example, we can collide an electron and positron, but now produce new particles such as a muon-anti-muon pair as shown in the diagram below.

e+ µ+

e µ

We still have to worry about energy and momentum conservation when evaluating this diagram. If, in the centre of mass frame, the energy of the incoming electron-positron pair is less than 2mc2, the rest mass of the muon-anti-muon pair, then the muons cannot be produced. However, when the incoming energy exceeds this threshold, then we can start to produce new particles from old ones. The same kind of process allows us to produce any charged particles from the collision of electrons and positrons. This,

of course, is the basis for particle colliders.

Renormalisation+ 2.3 Renormalisation + + +

At the fundamental level our world is built not from particles, but from ﬁelds. Moreover,+ +

as we stressedLook in theclose introduction, at the electron. these ﬁelds froth and foam in the uncertain+ quantum + world. This gives rise to an important phenomenon known as renormalisation+ . +

Let’s consider a single electron. It gives rise to an electric field which, like the force, varies as an inverse square law, Figure 9. The renormalisation of electric charge. e Large energy density near electron, This allows E = While2 ˆr it’s challenging to talk about quantum fields, we can build some intuition by 4⇡✏0r for the creation of particle-anti-particle pairs reverting to the language of particles. As we get closer to the electron, the electric field where ˆr is the unit vector pointing radiallygets strongeroutwards. and, Clearly as a result, the the electric energy density field gets stored in the field gets larger and 12 bigger and bigger as we approach the positionlarger. Atr some=0oftheparticle.Butwhatis point — a distance of around 10 m — the energy density is so large

happening to the electron ﬁeld near thisthat point? an electron-positron It turns out, pair that can both be produced electric from and the vacuum.

There is a general+ rule in quantum ﬁeld theory that anything that can happen does, electron ﬁelds start thrashing wildly as we get near to r =0. +

in fact, happen. This means+ a single electron is surrounded by a swarm of particle-anti- particle pairs, continually+ popping+ in and out of the vacuum. As we get closer still,

muons, taus+ and even quarks will also appear in the mix. We learn that any simple

picture you may have of a single+ particle giving rise to the electric field is really very far –48– + from the truth: it is impossible+ to enforce any kind of social distancing in the quantum world. + + This story should really be viewed in terms of quan- left-handed right-handed tum fields. When we talk of a swarm of particle-anti- particle pairs, it is really a metaphor for the quantum Figure 9. The renormalisation of electric charge. spin field being excitedspin in a tangled and elaborate fashion. This is the physics behind Justthe asincreasingly the vacuum iscomplicated somethingmomentum complicated diagramsmomentum in quan-like this While it’s challengingtum to fieldtalk theory, about so quantum too is the fields, notion of we a cansingle build particle. some intuition by reverting to the languageIn of the particles. language Asof Feynman we get closer diagrams, to the these electron, particle- the electric field gets stronger and, as aanti-particle result, the pairs energy are densitycaptured stored by the diagramsin the field that gets larger and 12 larger. At some point —include a distance loops of of particles, around 10 like them one — shown the energy on the density right. is so large that an electron-positronThe pair excited can be quantum produced field from has the an important vacuum. consequence for the strength of the electromagnetic interaction. Again, we can understand this in the language of particles. There is a general rule in quantum field theory that anything that can happen does, The swarm of particle-anti-particle pairs will not be oriented randomly around the in fact, happen. This meanselectron. a single Instead, electron the positrons, is surrounded which carry by a positive swarm charge, of particle-anti- will be attracted to the particle pairs, continually popping in and out of the vacuum. As we get closer still, muons, taus and even quarks will also appear in the mix. We learn that any simple picture you may have of a single particle giving rise to the electric field is really very far from the truth: it is impossible to enforce any kind of social–49– distancing in the quantum world.

This story should really be viewed in terms of quan- left-handed right-handed tum fields. When we talk of a swarm of particle-anti- particle pairs, it is really a metaphor for the quantum spin field being excitedspin in a tangled and elaborate fashion. Just as the vacuum is somethingmomentum complicatedmomentum in quantum field theory, so too is the notion of a single particle. In the language of Feynman diagrams, these particle- anti-particle pairs are captured by the diagrams that include loops of particles, like the one shown on the right.

The excited quantum ﬁeld has an important consequence for the strength of the electromagnetic interaction. Again, we can understand this in the language of particles. The swarm of particle-anti-particle pairs will not be oriented randomly around the electron. Instead, the positrons, which carry positive charge, will be attracted to the

–49– Renormalisation

As you look more closely, the charge of an electron gets bigger!

Figure 10. The renormalisation of ﬁne structure constant. Constants of nature are not constant!

electron in the centre, while the electrons will be repelled, as shown in Figure 9. The net e↵ect is that as you get closer to the electron, you ﬁnd more and more negative charges outside you. Which must mean that the original charge must have been bigger than we see. This is an e↵ect known as screening. This swarm of particle-anti-particle pairs is actually hiding the true charge of the electron in the centre.

In fact, an excellent analogy of this phenomenon arises in metals. Take a positive charge and place it in a metal. The mobile electrons will enthusiastically cluster around, screening the positive charge so that it can’t be detected at large distances. This is very similar to what happens with the electron in the vacuum and is one of many situations in which ideas in particle physics are mirrored in condensed matter physics.

Because the e↵ective charge of the electron gets bigger at shorter distances, so too does the interaction strength as captured by the fine structure constant (2.5). In fact, we learn that the fine structure constant is very badly named, since it’s not in fact 12 constant at all. At distances larger than r & 10 m, it plateaus to the usually quoted value of ↵ 1/137. But as you go to smaller scales, the strength of the electromagnetic ⇡ interaction increases logarithmically. For example, the strength of the interaction has 17 been well measured at the scale of the weak force, which is roughly r 10 m, where ⇡ it is found to be ↵ 1/127. A sketch of the variation of the fine structure constant — ⇡ often referred to as running — is shown in Figure 10.

The lesson of renormalisation as described above is a general one. It turns out that none of the dimensionless physical constants of nature are, in fact, constant. All of them depend on the distance scale you’re looking at.

–50– Yang-Mills theory is,The like itsStrong predecessor,and basedWeak on electricForc ﬁeldse E(x,t)andmagnetic ﬁelds B(x,t). Each of these is again a 3-dimensional vector,

Yang-MillsYang-Mills theory theoryE is, is,=( like likeE itsx its,E predecessor, predecessor,y,Ez)and based basedB on on=( electric electricBx,B fieldsy fields,Bz)EE(x(x,t,t)andmagnetic)andmagnetic Both nuclear forces have associated “electric” and “magnetic” fields fieldsfieldsBB((xx,t,t).). Each Each of of these these is is again again a a 3-dimensional 3-dimensional vector, vector, This is the essence of what it means for a field to be spin 1. The novelty in Yang-Mills

theory is that each componentEE=(=(EExx,E,E ofyy,E these,Ezz)and)and vectorsBB=( is=( nowBBxx,B,B ayy,B matrix,Bzz)) at each point in space and time, rather than just a number. There are di↵erent versions of Yang-Mills theory ThisThis is is the the essence essence of of what what it it means means for for a a field field to to be be spin spin 1. 1. The The novelty novelty in in Yang-Mills Yang-Mills for di↵erent kinds of matrices. There is a Yang-Mills theory based on 2 2matrices, theorytheory is is that that each each component component of of these these vectors vectorsBut iseach is now now component a a matrix matrix is at atnow each each itself point point a ⇥matrix. in in space space and one based on 3 3matricesandsoonforeachintegerN. However, once you andand time, time, rather rather than than⇥ just just a a number. number. There There are are di di↵↵erenterent versions versions of of Yang-Mills Yang-Mills theory theory decidedforfor di di↵ on↵erenterent the kinds size kinds of of of the matrices. matrices. matrix, There There everything is is a a Yang-Mills Yang-Mills else is fixed.theory theory based based on on 2 2 2matrices,2matrices, • 1 x 1 matrix Electromagnetism ⇥⇥ andand one one based based on on 3 3 3matricesandsoonforeachinteger3matricesandsoonforeachintegerNN.. However, However, once once you you To• write2 x 2 thematrix classical⇥⇥Weak equationsforce of motion, it’s bestor U(1) to x again SU(2) bundlex SU(3) the “electric” decideddecided• 3 x 3 on on matrix the the size size of of the theStrong matrix, matrix, force everything everything else else is is fixed. fixed. and “magnetic” fields into a 4 4matrixF , each component of which is now itself ⇥ µ⌫ an NToToN write writematrix: the the classical classical i.e a matrix equations equations of ofmatrices. of motion, motion, it’s The it’s best best Yang-Mills to to again again bundle equations bundle the the of “electric” “electric” motion are ⇥ andand “magnetic” “magnetic” fields fields into into a a 4 4 4matrix4matrixFFµ⌫,, each each component component of of which which is is now now itself itself oneThese of fields the keyare governed equations by ofthe physics, Yang⇥⇥-Mills and equations fullyµ⌫ deserving of their place in a frame ananNN NNmatrix:matrix: i.e i.e a a matrix matrix of of matrices. matrices. The The Yang-Mills Yang-Mills equations equations of of motion motion are are ⇥⇥ oneone of of the the key key equations equations of of physics, physics, and and fully fully deserving deserving of of their their place place in in a a frame frame

Here is something like a partial derivative with respect to space and time, but one Dµ thatHereHere also includesµ isis something something some likecommutator like a a partial partial derivative of derivative matrices. with with (It’s respect respect known to to space space as a and covariant and time, time, but derivative.) but one one DDµ Onthat thethat also right-hand also includes includes side some some sits commutator commutatorJ ⌫, the analog of of matrices. matrices. of the (It’s (It’s electric known known current as as a a covariant covariant which, derivative.) derivative.) as we will see ⌫ shortly,OnOn the the arises right-hand right-hand from quarks. side side sits sits TheJJ ⌫,, the theYang-Mills analog analog of of equations the the electric electric are current current very which, which, similar as as to we we the will will Maxwell see see equations.shortly,shortly, arises arises Indeed, from from if quarks. quarks. you choose The The Yang-Mills Yang-Mills 1 1 matrices, equations equations which are are very very are similar similar just numbers, to to the the Maxwell Maxwell then the equations.equations. Indeed, Indeed, if if you you choose choose 1 1⇥11 matrices, matrices, which which are are just just numbers, numbers, then then the the Yang-Mills equations reduce to the Maxwell⇥⇥ equations. Yang-MillsYang-Mills equations equations reduce reduce to to the the Maxwell Maxwell equations. equations. To specify any force described by Yang-Mills theory, we just need to say how big the ToTo specify specify any any force force described described by by Yang-Mills Yang-Mills theory, theory, we we just just need need to to say say how how big big the the matricesmatricesmatrices are. are. are. Nature Nature Nature is is is kind kind kind to to to us: us: us: she she she has has has chosen chosen chosen to to to make make make use use use only only only of of the the of simplestthe simplest simplest matrices.matrices.matrices.

• Electromagnetism:••Electromagnetism:Electromagnetism: 1 1 1 1matrices1matrices1matrices ⇥⇥⇥ • Weak••WeakWeak Nuclear Nuclear Nuclear Force: Force: Force: 2 2 2 2matrices2matrices2matrices ⇥⇥⇥ • StrongStrong Nuclear Nuclear Force: Force: 3 3 3matrices3matrices • Strong• Nuclear Force: 3 ⇥ 3matrices ⇥⇥ Isn’t that nice! Isn’tIsn’t that that nice! nice!

–66––66– –66– The Strong Force (or QCD)

No Yes Yes No

TheEach numbers quark in comes the table in three are colours the masses, which of we the take particles, to be red, written green as and multiples blue. of the electron mass. (Hence the electron itself is assigned mass 1.) The masses of the neutrinos are known to be very small but, otherwise are only constrained within a window(Note: and a better not yet counting established is that individually. each generation contains 1+3+3+1=8 particles.)

–9– Contents

1 Introduction 1

with c the speed of light. This means that if the strength of the electric force gets 1 Introduction weaker, then the magnetic force necessarily gets stronger, and vice versa.

There is a similar story for Yang-Mills. But now the gluons are doing the screening, and couple to both the chromoelectric and the chromomagnetic fields. It turns out that, Gravity Electromagnetismbecause Weak they are Strong spin 1 particles, they screen the chromomagnetic fields more strongly than they screen the chromoelectric fields. In other words, the chromomagnetic part of the Yang-Mills interaction gets weaker as we go to larger distances. The relation (3.1) then tells us that the chromoelectric part of the interaction necessarily gets stronger. HiggsThe upshot is that gluons anti-screen. If you want the gory details, the calculation can be found in Section 2.4 of the lectures on Gauge Theory.

So what is the strength of the strong force? At the energy scale E 100 GeV, 17 ⇡ corresponding to a distance scale of r 10 m, we have Hypercharge ⇠ ↵ 0.1whenE 100 GeV s ⇡ ⇡ Even at these fairly high energies, the strength of the force is an order of magnitude

larger than QED. If we go to higher energies, or shorter distances, ↵s decreases. In fact, Q1Q2 Fwith= ↵c theas we speed go to of arbitrarily light. Thishigh energies, means the that strength if the of strength the strong of force the vanishes, electric↵ forces 0. gets r2 ! weaker,This then phenomenon the magnetic is known force asnecessarilyasymptotic gets freedom stronger,: it means and that vice at versa. high energies, or Whyshort distanceis the scales, Strong the strong Force force essentially Strong? disappears! Theree2 is a similar1 story for Yang-Mills. But now the gluons are doing the screening, ↵ = However, outside of particle colliders, everything that we observe in the world takes 17 and4⇡✏ couple0~cplace⇡ to137 at both distance the chromoelectric scales significantly and larger the than chromomagnetic 10 m, and the fields. strong It force turns only out gets that, becausestronger they are as spin we go 1 to particles, larger distances. they screen But here the there chromomagnetic is another surprise. fields According more strongly to 15 than theynaive screen calculations, the chromoelectric by the time you fields. get In to around other words,r 10 them, chromomagnetic or an energy scale part of of ⇠ 1 E 100 MeV, the coupling constant appears to get infinitely large! strong coupling↵the constant Yang-Mills⇠1 interaction gets weaker as we go to larger distances. The relation (3.1) then⇡ 137 tells⌧Above, us that I theused chromoelectric the phrase “naive” part calculations, of the interaction because I have necessarily in mind gets the kind stronger. of The upshotperturbative is that gluons Feynman anti-screen. diagram calculations If you want that the we described gory details, in the the previous calculation section. can be foundBut, in Section as we stressed, 2.4 of these the diagrams lectures only on Gauge make sense Theory when. the coupling is small. As soon 2as the coupling is around ↵s 1, the very complicated Feynman diagrams, involving (↵ ) ⇡ SoO whatlots is of theloops, strength are just as of important the strongenergy as theforce? = simple 1/distance At Feynman the energy diagrams scale andE we have100 no GeV, Figure 19. The running of coupling for the strong force, now plotted against17 energy scale. ⇡ correspondingcontrol to over a the distance calculation. scale But of r this10 is exactlym, wewhat have happens in QCD! At very short This figure is taken from the review of QCD by the Particle Data Group.⇠ (↵4distances,) we’re fine and we can do calculations. But at long distances, the theory At high energy,which sayO it’s E=100 measured.becomes GeV, This very iswe the challenging.have story of ↵ renormalisation s 0 The . 1when . But separation that wethe met strong previously betweenE force in100 “easy”gets GeV stronger and “hard” as we turns out to be Section 2.3. But here is where there’s14 a crucial⇡ di↵erence15 between electromagnetism⇡ go to larger distances. (Asymptoticaround r 10freedom.) mto10 m, but is usually expressed in terms of an energy scale and the strong force: as we⇠ go to larger distances, the strong force gets stronger,not Evenweaker. at A sketchknown these of the as fairly coupling the isstrong high shown in energies, coupling Figure 18. (This scale the should,or strength beLambda-QCD contrasted ofwith the, force is an order of magnitude Finally,the running and of most the fine importantly, structure constant, the as shown existence in Figure of10 a.) mass The experimental gap is consistent with experiment,largerdata where for the than running no QED.massless of the strong If gluon we coupling go is to seen. is shown higher Moreover, in Figure energies,19, now while plotted or the against shorter strong force distances, is strong,↵s itdecreases. In fact, Taken naively, energy ↵ s scale E , which at the is inversely energy related scale: to length by E =1/r⇤.QCD 200 MeV as we!1 go to arbitrarily high energies, the strength⇡ of the strong force vanishes, ↵s 0. is also short-ranged. The characteristic energy scale ⇤QCD corresponds to a distance ! scale whichHow can in we natural understand units this intuitively? (remember For electromagnetism,= c =1)is there was a simple Thisphysical phenomenon picture in which the electric is known charge gets as screened~asymptotic by particle-anti-particle freedom pairs,: it means that at high energies, or shortand so appears distance smaller scales, as we go to the longer strong distances. Forforce the strong essentially force, the gluons disappears! themselves are doing the screening. Except they1anti-screen, meaning15 that they cause This corresponds to a distance scale RQCD = 5 10 –70–m the force to get stronger the further out we go!⇤ ⇡ ⇥ However, outside of particleQCD colliders, everything that we observe in the world takes In fact there is an intuitive way to understand this, although it’s rather subtle. To understand–1– how this scale a↵ects the world around us, we first17 need to throw in the placeA clue canat bedistance found lurking scales back in significantly the theory of electromagnetism. larger than Recall 10 that m, and the strong force only gets finalthe key Maxwell ingredient: equations quarks. contain two parameters: 1/✏0 characterises the strength of the strongerelectric force, as while we 1/µ go0 characterises to larger the strength distances. of the magnetic But force. here But there these is another surprise. According to are not independent. They are related by 15 3.2naive Quarks calculations, by the time you get to around r 10 m, or an energy scale of 1 ✏ µ = (3.1) ⇠ E 100 MeV, the coupling0 0 2 constant appears to get infinitely large! Usually⇠ in physics, we can get by withoutc memorising lots of random names. The strong force is the exception, leaving us looking more like botanists than physicists. First,Above, there are I the used many the hundreds phrase of names“naive” of di calculations,↵erent particles. because But more I important have in mind the kind of areperturbative names of groups Feynman of particles, diagram–69– each classifying calculations a di↵erent that property. we described in the previous section. But, as we stressed, these diagrams only make sense when the coupling is small. As soon To kick things o↵, the fermions in the Standard Model are divided into two di↵erent as the coupling is around ↵ 1, the very complicated Feynman diagrams, involving types s ⇡ lots of loops, are just as important as the simple Feynman diagrams and we have no Quarks: These are particles that feel the strong force. control• over the calculation. But this is exactly what happens in QCD! At very short distances,• Leptons: These we’re are fine particles and we that can don’t. do calculations. But at long distances, the theory Thebecomes leptons are very the challenging. electron, muon, The tau separation and three species between of neutrino. “easy” and (The “hard” name turns out to be 14 15 comesaround from ther Greek10 ✏⇡⌧mto10ó ⇣ meaning m, small.) but Leptons is usually don’t expressed interact with in the termsSU(3) of an energy scale ⇠ Yang-Millsknown field, as the andstrong we will coupling ignore them scale for,or theLambda-QCD rest of this section., In contrast, the six quarks — up, down, strange, charm, bottom and top — do feel the strong force. ⇤ 200 MeV In the language of Feynman diagrams, weQCD denote⇡ the quarks as a solid line, with an arrow distinguishing quark from anti-quark. This is the same kind of line that we previously used to denote leptons, so we add a label q to show that it’s a quark. The interaction between quarks and gluons is then described by the interaction vertex –70–

When evaluating these Feynman diagrams, each vertex contributes a factor of the

strong coupling, ↵s. We can then use the Feynman diagrams to compute, say, the force between a quark and anti-quark. This comes from the following diagram:

–72– If the quark and anti-quark are separated by a distance r R , then evaluating If the quark and anti-quark are separated by a distance ⌧r QCDR , then evaluating this diagram results in an attractive force that is very similar⌧ to theQCD Coulomb force this diagram results in an attractive force that is very similar to the Coulomb force (2.4), (2.4), ↵s F (r) ↵ when r RQCD (3.2) F (r)⇠ r2 s when r⌧ R (3.2) ⇠ r2 ⌧ QCD Here ↵s itself also depends on r, albeit logarithmically. We already saw a sketch of this Heredependence↵s itself in also Figure depends18. on r, albeit logarithmically. We already saw a sketch of this This is a characteristic energy scale ofdependence QCD. At energies in FigureE18. ⇤ , the strong force However, there’s a catch: ifQCD the distance between the quarks is too big — bigger than is not particularly strong and we can trust Feynman diagrams. But by the time we get RQCDHowever,— then there’s the language a catch: ifof the Feynman distance diagrams between stops the quarks working. is too As webig increase — bigger the than to energies ⇤QCD, the strong force livesRseparation upQCD to— its then between name. the Most quarks language phenomena to distances of Feynman greater that diagrams are than dueR stopsQCD, the working. Coulomb-like As we expression increase the If the quark and anti-quark areIf separated the quark by and a distance anti-quarkr areRQCD separated, then evaluating by a distance r RQCD, then evaluating to the strong force have an energy somewhereseparation(3.2)stopsbeingtherightone,anditinsteadchangesto in between the ballpark quarks of to⇤ distancesQCD greater⌧ than RQCD, the Coulomb-like⌧ expression this diagram results in an attractivethis diagram forceConfinement that results is very in similar an attractive to the force Coulomb that force is very similar to the Coulomb force (3.2)stopsbeingtherightone,anditinsteadchangesto (2.4), (2.4), F (r) constant when r R (3.3) Before we describe some of these phenomena, it’s worth pausing⇠ to mention that QCD something unusual has happened here. The strength↵s of the strongF (r) force,constant like QED, when↵sr is RQCD (3.3) At short distances,A constant F ( r ) force may butwhen not at seemlongr ⇠ likedistancesRQCD much. F But ( r ) itbecomes gets exhausting. whenconstant.r (3.2) This RQCD is better seen if (3.2) ⇠ r2 ⌧ ⇠ r2 ⌧ characterised by a dimensionless couplingAwe constant look↵s. at But the force the associated may phenomena not energy seem like needed of renormalisation much. to separate But it gets a quark exhausting. and anti-quark This is by better distance seen if means that this couplingHere ↵ dependss itself also on depends scale,weR. look For and on short at ther the, albeitHere distances, upshot associated↵ logarithmically.s itself of the this energy also energy is depends that needed takes We we already the on to ultimately separater same, albeit saw form a alogarithmically. sketch quark as in electrostatics, andof this anti-quark We already by distance saw a sketch of this exchange a dimensionlessdependence number, in Figure↵s, forR18. For a. dimensionful shortdependence distances, scale the in⇤ Figure energyQCD↵s . 18 takes. the same form as in electrostatics, In terms of the potential energy, V ( r ) +constantat short distances, when rbut atRQCD long distances ⇠r ⌧ However, there’s a catch: if the distanceHowever, between there’s↵ thes a catch: quarks if is the too distance big — bigger between than the quarks is too big — bigger than 3.1.2 The Mass Gap But when the quarksV experience(r) a+constant constant force, when the energyr R growsQCD linearly RQCD — then the language of FeynmanRQCD — diagrams then⇠ the stopsr language working. of Feynman As we increase diagrams⌧ the stops working. As we increase the When we try to studyseparation the strong between force quarks on energy to distancesseparation scales greaterE< between⇤ thanQCDR, quarks corresponding2 , the to Coulomb-like distances to greater expression than R , the Coulomb-like expression But when the quarks experienceV (r) ⇤QCD a constantr when force,r theRQCD energy growsQCD linearly 14 ⇠ QCD distance scales, r>(3.210)stopsbeingtherightone,anditinsteadchangestom, we have a problem.(3.2 The)stopsbeingtherightone,anditinsteadchangesto strong coupling means that 2 Clearly if you want to separateV (r) the⇤ quark-anti-quarkr when r pairRQCD by a long distance, then it the fields are wildly fluctuating on these scales, and our favourite method⇠ QCD of Feynman costsF an(r) increasingconstant amount when of energy.r RQCD InF particular,(r) constant it costs anwhen infinite(3.3)r amountRQCD of energy (3.3) diagrams is no longer useful. This alsoClearly means if⇠ that you want the classical to separate equations the quark-anti-quark of⇠ motion pair by a long distance, then it to separate them an infinite distance. But taking, say, the anti-quark a long way away A constant force may not seem likeA much.constant But force it gets may exhausting. not seem like This much. is better But it seen gets if exhausting. This is better seen if are no guide at all for what the quantumcostsis tantamount theory an increasing might to leaving look amount like. the of quark energy. on its In own. particular, In other it costswords, an a infinitesolitary amount quark requires of energy we look at the associatedto separate energy them neededwe look an to infiniteat separate the associated distance. a quark energy But and taking, anti-quark needed say, to bythe separate distance anti-quark a quark a long and way anti-quark away by distance The gluon is the first casualty of stronginfinite coupling. energy! As Quarks we explained do not want at to the be alone: beginning they only occur in bound states with R. For short distances,isother tantamount the quarks energyR or. to takes anti-quarks. For leaving short the same the distances, This quark form phenomenon on as the its in energy own. electrostatics, is In takes called other theconfinement words, same a form solitary. as in quark electrostatics, requires of this section, the classical Yang-Millsinfinite equations energy!↵ suggest Quarks that do not the want gluon to be should alone:↵ bethey only occur in bound states with This is confinementV (r) . Wes don’t+constant see isolated when quarks.rV (r)R s +constant whenr R massless. But the strong coupling e↵ectsother change⇠ quarksr this. or anti-quarks. Instead, the This gluon phenomenon⌧ —⇠QCD whichr is is called a confinement⌧. QCD ripple of the Yang-Mills fieldsAlso, —hasthe mass,force carrying given byfield is not massless. The But when the quarks experienceBut a constant when the force, quarks the energy experience–73– grows a constantlinearly force, the energy grows linearly gluons stick together to form glueballs, with mass around mgluon ⇤QCD . This2 is the “mass gap” 2 ⇡V (r) ⇤QCD r when r RQCDV (r) ⇤QCD r when r RQCD problem. ⇠ –73–⇠ This is sometimes referredClearly if to you as want the Yang-Mills to separate theClearly mass quark-anti-quark gap if you. The want “gap” to pair separate by here a long isthe one distance,quark-anti-quark then it pair by a long distance, then it costs an increasing amount of energy.costs In an particular, increasing it amount costs an of infinite energy. amount In particular, of energy it costs an infinite amount of energy between the ground state and the first excited state. For theories ofFigure massless 20. The particles, chromoelectric flux tube between a quark and anti-quark in a meson state, there is no gap becauseto separate we can havethem particles an infinite of distance. arbitrarilyto separate But low taking, them energy. say, anfrom infinite the But the QCD anti-quark for distance. simulation massivea of ButDerek long taking, Leinweber way away say, the anti-quark a long way away particles, the minimumis tantamount amount of to energy leaving needed the quark isis onE tantamount= itsmc own.2. In to other leaving words, the quark a solitary on its quark own. requires In other words, a solitary quark requires infinite energy! Quarks do not wantinfinite to be energy! alone: Quarks theyConfinement, only do occur not like want in the bound Yang-Millsto be states alone: mass with gap, they has only so-far occur resisted ina rigorous bound mathemat- states with ical derivation. But we again have very clear evidence from numerical simulations, To say that the Yang-Millsother quarks mass or anti-quarks. gap is dicult Thisother to phenomenon prove quarks would or is anti-quarks. called betogether somethingconfinement with This a handful of phenomenon. an of less-than-rigorous is called mathematicalconfinement arguments,. that confine- understatement. Demonstrating the mass gap is generally regarded asment one occurs. of the Moreover, major this also gives us some intuition for what’s going on. open problems in theoretical physics. Indeed, a million dollar Clay mathematicsFirst, let’s recall what prize happens in electrostatics. If we separate a positive and negative electric charge by awaits anyone who succeeds. Although we do not have–73– any rigoroussome (ordistance evenr, then semi- an electric–73– field is set up between rigorous) derivations of the mass gap, there is no doubt that it is athe property two. The form of Yang- of this electric field is shown on the right, and ultimately is responsible for the F 1/r2 ⇠ Mills. Our best theoretical evidence comes from computer simulationsCoulomb which, force law in that this the charges experience. context, are called lattice simulations, reflecting the fact that spacetimeIn is Yang-Mills approximated theory, the chomoelectric field takes a by a grid, or lattice, or points. These simulations show unambiguouslysimilar that form if the the quark gluon and anti-quark are separated by a distance r R . But as you increase the separation of the quark and anti- ⌧ QCD is massive. You can read more about the lattice, and other approachesquark to beyond Yang-Mills, the critical distance RQCD, the form of the field changes. Instead of in the lectures on Gauge Theory. the field lines spreading out, the mass of the gluon forces them to bunch together into string-like configurations called flux tubes. This can be clearly seen in the computer simulation of QCD shown in Figure 20. It’s as if the quark and anti-quark are joined by a piece of string. If you want to separate them further, you have to stretch the flux tube and this costs an energy V (r) r proportional to its length. This is responsible ⇠ for the confinement of quarks. –71–

–74– m<⇤QCD necessarily has its size >RQCD. That means that there’s no way to bring two of these quarks closer than RQCD, so these light quarks will only experience the conﬁning force (3.3).

In contrast, the three heavier quarks have masses

mcharm =1.3GeV

mbottom =4.2GeV

mtop =170GeV

These are all much heavier than ⇤QCD. These three quarks all have Compton wavelength R , so it makes sense for them to come close enough to experience the ⌧ QCD Coulomb-like force (3.2). This means that we might expect the spectrum of hadrons containing charm, bottom and top quarks to be a little di↵erent from those containing only the lighter quarks. Indeed, this turns out to be the case.

3.3 Baryons We’ll kick thing o↵ with baryons, containing three quarks. To start, suppose that we have only the up and down quark to work with. There are various ways that we can 1 combine these quarks. First, recall that the each quark has spin 2 . When combining quarks, we need to ﬁgure out what to do with their spins.

3.3.1 Protons andHadrons Neutrons (Stuff Made of Quarks) Suppose that we have two spins in one direction, and the third spin in the opposite direction. This will result in a baryon of spin 1 + 1 1 = 1 . There are two choices, 2 2 2 2 which• Baryons: result in three the quarks. two most For familiar example baryons: the proton (p)andtheneutron(n). Their quark content and masses are

n (ddu) m 939.57 MeV n ⇡ p (uud) m 938.28 MeV p ⇡ These are the two lightest spin 1 baryons. Recall that the down quark has charge 1/3 2 andA puzzle: the up m quarkdown= charge 5 meV +2and/3, m soup=2 the MeV. proton Where has chargedoes the +1 mass whileFigure come 21 the. The from?neutron flux tube between has three no quarks in a baryon state, from the QCD simulation of Derek Leinweber electric charge. • Baryons: These contain three quarks. If the three quarks have colour vectors !1, We start by putting the spin of the quark and anti-quark in opposite! and directions.! then they combine This as the triple product ! (! ! )toformacolour Already, there is something of a surprise here. The up and down2 quarks3 each have 1 · 2 ⇥ 3 results• Mesons: in mesons quark with-anti spin-quark 0. pair. There For are example, three such pions mesons that weneutral can state. build Schematically, from the this looks like mass of a few MeV. Yet the proton and neutron each have mass of around 1000 MeV. up and down quarks, known as pions. Their masses and quark content are given by baryon = rbg How is this possible given that the proton and neutron are supposedThe fact to that contain the baryon contains three 3 quarks, rather than any other number, can + ¯ be traced to the 3 3matricesthatdescribethestrongforce.Thefluxtubefor quarks each? ⇡ (du) m 139 MeV ⇥ ⇡ a baryon is shown in Figure 21. ⇡0 1 (¯uu dd¯ ) m 135 MeV p2 ⇡ To understand the collection of hadrons that emerges after confinement, we first need to look at the masses of quarks. In particular, we should compare the masses to the ⇡ (¯ud) m 139 MeV characteristic scale of the strong interactions, ⇤ 200 MeV. ⇡ QCD ⇡ + Three of the quarks have masses smaller than ⇤ . The ⇡ is the anti-particle of ⇡ and the–77– two have exactly the sameFigure mass. 20. The chromoelectric The neutral flux tube between a quarkQCD and anti-quark in a meson state, 0 from the QCD simulation of Derek Leinweber Note:pion, P⇡ionsis ahave combination spin 0 and of so up should and down be thought quarks asof as shown. “force It carrying” has no electric particles! charge, Som ...down =5MeV Confinement, like the Yang-Mills massmup gap,=2MeV has so-far resisted a rigorous mathemat- but a very similar mass to the ⇡±. The similar masses reflect the isospin symmetry ical derivation. But we again havemstrange very=95MeV clear evidence from numerical simulations, which says that, as far as the strong force is concerned, the up andtogether down with a handful quarks of less-than-rigorous have mathematical arguments, that confine- Thement up and occurs. down Moreover, quark have this masses also gives significantly us some intuition smaller for than what’s⇤QCD going, while on. the strange the same properties. quark is only slightly smaller. Recall from our discussion in 1.2 that the Compton First, let’s recall what happens in electrostatics. If wavelength, = ~/mc, can be thought of as the size of a particle. Any particle with we separate a positive and negative electric charge by Despite their similar masses, the neutral and charged pionssome have distance ratherr, then an di electric↵erent field is set up between the two. The form of this electric field is shown on the lifetimes. The neutral pion decays through the electromagnetic forceright, and to ultimately two photons is responsible for–76– the F 1/r2 ⇠ Coulomb force law that the charges experience. 0 ⇡ + In Yang-Mills theory, the chomoelectric field takes a ! similar form if the quark and anti-quark are separated by a distance r R . But as you increase the separation of the quark and anti- 17 ⌧ +QCD It has a lifetime of around 10 seconds. In contrast, the chargedquark beyond pions the⇡ criticaland distance⇡RQCD, the form of the field changes. Instead of decay through the weak force. We’ll see in Section 4 that theythe typically field lines spreading decay out, the to mass a of the gluon forces them to bunch together into string-like configurations called flux tubes. This can be clearly seen in the computer muon and a neutrino simulation of QCD shown in Figure 20. It’s as if the quark and anti-quark are joined by a piece of string. If you want to separate them further, you have to stretch the flux + + tube and this costs an energy V (r) r proportional to its length. This is responsible ⇡ µ + ⌫ and ⇡ µ +¯⌫ ⇠ ! µ ! µ for the confinement of quarks. 8 They live for 10 seconds, an eternity in the subatomic world and much longer than any of the baryons except the proton and neutron.

–74– There is one, very important characteristic that distinguishes mesons from baryons. Mesons, made of a quark and anti-quark, have integer spin and are therefore bosons. Baryons, made of three quarks, have half integer spin and are therefore fermions.

Back at the beginning of Section 2, we explained that fermions are “matter particles” while bosons are “force particles”. It should come as no surprise to learn that baryons, like protons and neutrons, are matter particles. After all, you’re made up of them. But it may be less familiar to hear that mesons, like the pion, are force particles. What force do they mediate?

The answer to this is quite lovely: the pions give rise to an attractive force between the baryons. In particular, they give rise to an attractive force between any collection of protons and neutrons. It is this force that binds the protons and neutrons together inside the nucleus.

–85– Contents

1 Introduction 1

is important because it means that beta decay proceeds by a neutron decaying into a proton, rather than the other way around. We’ll learn more about this in Section 4 1 Introductionwhere we discuss the weak force. 3.3.2 Delta Baryons Gravity Electromagnetism Weak Strong We could ask: why can’t we have a baryon with, say, three up quarks? The answer to this lies in the Pauli exclusion principle. The full explanation is a little subtle, but the upshot is that we can have three up quarks in a baryon, but only if all their spins point in the same direction. (AtHiggs first glance this might seem the wrong way around since, in chemistry, the Pauli exclusion principle dictates that electrons in the same orbital state have opposite spins. But quarks have that additional colour degree of freedom, and there is an anti-symmetryHypercharge there which, in turn, requires a symmetric alignment of spins.) is important because it means that beta decay proceeds by a neutron decaying into a Q1Q2 1 1 1 3 proton,When rather all than the the spins other point way around. in the same We’ll direction, learn more the about baryon this itselfin Section has spin4 2 + 2 + 2 = 2 F = ↵ 2 Here “almost conserved” means conserved by the strongwhere interaction.baryon. we discuss Now The the there weak strong are force. four choices,r all of which are known as Delta ()baryons.These interactions cannot change the number of strange quarks and so particles like ⌃±, 0 0, particles have more or less equal2 mass, but di↵erent charges ⇤ and ⌅ do not decay straight away. The decay4.2.3 only3.3.2 Weak proceeds Decays Delta through Baryons the weak e 1 ↵ = interaction, and this takes significantly longer. We’ll describe how these decays occur EnricoWe Fermi could was the ask: first why person can’t to we understand have a4 baryon⇡✏ beta++0~c decay.⇡ with,137 In say, 1934, three just up 18 quarks?months The answer to in Section 4. In contrast, particles like the baryons decay directly through the strong (uuu) after thethis discovery lies in of the the Pauli neutron,3.3.3 exclusion he Strangeness proposed principle. a simple The quantum full explanation field theory is in a which little subtle, but the interaction, and this happens much faster. a neutron can decay into a proton, an electron and+ an(uud anti-neutrino.) 9 The associated upshot is that we can have three up quarks1 in a baryon, but only if all their spins point Feynman diagram is Let’s now consider baryons that we> canm construct1232 from MeV the first three quarks: up, down in the same direction. (At first glance↵ 0 this(udd might1 ) > seem the⇡ wrong way around since, (As an aside: there’s always one complication. It turns out that, among theand collection strange. ⇡ 137 ⌧ => of strange baryons, there is one which is unstable: the ⌃0in chemistry,⇤0 + with the a Pauli lifetime exclusion of principle dictates that electrons in the same orbital (ddd) 1 20 state! have oppositeWe’ll spins. first But consider quarks baryonshave that with additional spin . Incolour addition degree to of the freedom, proton and neutron, we around 10 seconds. But this is allowed by the3.3.3 strong Strangeness force because the number of > 2 0 Decay now10 have four further baryons2 that> contain a single strange quark, called sigma (⌃) strange quarks is unchanged. TheHere⇤ “almost, as we’ve conserved” seen, thenand means waits thereHere another conserved is the an 10 anti-symmetrysuperscripts, byseconds the strong there ++, interaction. which,+,(↵ 0) and in The turn, - specify strong;> requires the a symmetric electric charge alignment of the of particle. Let’s now consider baryonsbaryons that we can constructO from the first three quarks: up, down before it too decays.) interactions cannot changespins.) the number of strange quarks and so particles like ⌃±, and strange. ⇤0 and ⌅0, do not decay straight away.We don’t The decay see onlybaryons proceeds floating through4 around the weak in the world. They have a lifetime of around As a general rule of thumb, hadrons can decay through oneWhen of the all three the forces: spins point strong in the same( direction,↵ ) ⌃ the(dds baryon) m itself1197 has MeV spin 1 + 1 + 1 = 3 Thus, right from the24 beginning, the story of1 theO weak force was one of decay.⇡ 2 2 2 2 interaction,0 and this takesWe’ll significantly first0 consider10 longer.seconds, baryons We’ll with after describe spin which2 . how In they addition these decay, decays to0 the typically occur proton and into neutron, a proton we or neutron together with All (likehadrons the other’s), electromagneticthan the proton (like are⌃ )orweak(like unstable. They⌃±,baryon. decay.⇤ and Now⌅.). The there lifetimes are four of choices, all of which⌃ are(dus known) m as Delta1193 ( MeV)baryons.These in Section 4. In contrast,nowFermi’s have particles four theory like further was the one baryons ofbaryons the great that decay breakthrough’s contain directly a single through of particle strange the physics. strong quark, Not called⇡ only did sigma it (⌃) these particles reflect the decay process: particlesamesoncalledapion.Forexample, have more or less equal mass, but di↵⌃erent+(uus charges) m 1189 MeV interaction, and thisbaryons happensgive a correct much explanation faster. for beta decay, but it was the first time that a quantum⇡ field 22 24 theory was written down in which a particle++ of++ one↵s type can transmute+ into something • Strong decay: 10 to 10 seconds. and the lambda (uuu (⇤)baryon) p + ⇡ and n + ⇡ ⇠ (As an aside: there’selse. always Any one idea complication. that the neutron It⌃ turns was(dds composed out) m that,1197 among of!1 a MeVproton the collection plus electron (or, as was + ⇡ ! ! 16 21 0 0 (uud0 ) 9 0 • Electromagnetic decay:of strange10 to baryons, 10 seconds. therealso is suggested, one which the is proton unstable: was⌃ composed the(dus⌃24) m of⇤ a neutron+1193 withwit MeV +mh a positron) lifetime⇤ 1232(dus were)MeV of m consigned1116 MeV to ⇠ 20 The lifetime of 10 0!seconds⇡ > is much shorter than⇡ we can measure and particles with around 10 seconds.the But waste this bin. is allowed by the strong+ force(udd because) > the⇡ number of 7 13 Again,⌃ (uus the) superscript+m 1189=> labels MeV+ the electric charge of the baryon. You may have noticed • Weak decay: 10 to 10 seconds. such0 short lifetimes are⇡ ( called,⇡ud) quiteµ +10 reasonably,⌫µ unstable. Even when moving close to ⇠ strange quarks is unchanged.It took The several⇤ , more as we’ve decades seen, to then understand waits(ddd another what) ! things 10 look0seconds like if0 we zoom in a and the lambdathe (⇤ speed)baryon ofthat light the quarkbaryons content don’t> of the travel⌃ and far⇤ enoughare the same. to register The di↵ directlyerence lies in in particle the details de- Where you sit within each rangebefore depends it too decays.) on otherlittle factors, further. such The as the structure relative of the masses electroweak theory> in its essentially complete form Here the superscripts,of the ++, wavefunctions +, 0 and - specify for;> the the up and electric down charge quarks. of the (Technically, particle. some di↵erent minus was first understoodtectors. by10 Instead, Steven⇤ Weinberg,0 (dus they) revealm but many1116 themselves others MeV got close in includingmore indirect Sheldon means as so-called resonances of the parent4 and The daughter Weak particles. ForceAs a general Particles rule of that thumb, live hadrons for up to can 10 decaysecondssigns through mean are one that of the all ⌃ threebaryons forces: have strong isospin 1, while the ⇤ has isospin 0.) Glashow, Abdus Salam and John Ward. ⇡ referred to4 (I The think, Weak somewhat Force tongue in cheek) as stable.We Inin don’t contrast, certain0 see any experiments.baryons particle floating0 We’ll around describe in the world. this further They have in Interlude a lifetime ofC.1 around. (like the ’s), electromagneticAgain, the (likesuperscript⌃ )orweak(like labels the electric⌃±, ⇤ chargeand ⌅.). of The the baryon. lifetimes You of may have noticed 20 24 There are also two types of baryons that contain two strange quarks, called cascade, Thethat most lasts famousIt 10 is now weakseconds time decay toor shorterturn is how to is, the we like weak first the force. discoveredthatbaryon,W-bosons the For10 quark many referred theseconds, content years, weak to as there offorce afterunstable the was which⌃0 and justor they a⇤ one0 are decay, manifes- the–1– same. typically The into di↵ aerence proton lies or in neutron the details together with It is now time to turnthese to particles the weak reflect force. the For decay many process: years,The lifetime thereor was xi of ( just⌅)baryons the one manifes-baryons is actually the characteristic timescale of the strong resonancetation. of the weak force, namely betaof decay the wavefunctionsamesoncalledapion.Forexample, for the up and down quarks. (Technically, some di↵erent minus tation of the weak force, namely beta decayWe know22 that24 if we zoom into the neutron and proton24 we find quarks. Beta decay • Strong decay: 10 to 10force: seconds.TQCD = RQCD/c 10 seconds. If anything happens due to the strong force, It should be clear from the discussion, however,signs⇠ thatoccurs there’s mean when that nothing a down all ⌃ very quarkbaryons qualitatively changes have++ into isospin an⇡ up 1, quark. while+ the However,⇤⌅has(dss this isospin) processm 0.)1322 doesn’t MeV n p + e +¯⌫ p + ⇡ and n + ⇡⇡ happen throughite usually a direct interaction happens of on four! roughly fermions: this it is timescale. mediated0 ! by the W-boson, di↵erent between a stable particle like the ⇤ andn ! a resonancep + e +¯ like⌫ the16 . Both21 will decay ⌅ (uss) m 1315 MeV • ElectromagneticThere decay: are also10e twoto types10 ofseconds. baryons that contain two strange quarks,⇡ called cascade, ! and10 looks⇠ like this. 24 in less than the blink of an eye. But a lifetime of 10 secondsThe lifetime mean of that, 10 withseconds good is much shorter than we can measure and particles with We now know that this can be understoodor xi (7⌅ in)baryons terms13 of the downNone quark of these decaying baryons into are familiar from our everyday experience. This is because technology,We you now can know take that a photograph this• Weak can be decay: of understood the particle’s10 intosuch track terms10 short inseconds. of a the lifetimescloud down chamber are quark called, or decaying quite reasonably, intou unstable. Even when moving close to Or, if you lookan upmore quark, closely, electron and anti-electron⇠ neutrino theyd again decay, typically to protons and pions. However, here there is a surprise: bubble chamber.an up quark, You can electron see many and anti-electron such photographs neutrino inthe Interludes speed ofB lightandC⌅baryons. When(dss) don’tm travel1322 MeV far enough to register directly in particle de- Where you sit within each range depends on otheralthough factors, these such newW⇡ as baryons the relative have masses a mass in the same ballpark as the ’s, they live for tectors. Instead, they0 reveal themselves in more indirect–79– means as so-called resonances aparticleleavessuchavividtrace,it’shardtodenyitsexistence.Incontrast,we’red u + e +¯⌫e ⌅ (uss) m 1315 MeV10 0 0, of the parent and daughter particles. Particlessignificantly that live for longer. up⇡ to In 10 particular, seconds the are⌃±,the⇤ and the ⌅ all live for a whopping d ! u + e +¯⌫e 20 never going to take a photograph of something that lastsin certain 10 seconds. experiments. But10 We’ll that describe thise further in Interlude C.1. referred to (I think,!None somewhat of these tongue baryons in cheek) are10 familiar as seconds.stable from. In our contrast, everyday any experience. particle This is because doesn’t meanThere that is nothing it’s any less in the real! strong It just force20 leaves or its electromagnetism signature in more that subtle would ways. allow one type of There is nothing inthat the lasts strong 10 forceseconds orthey electromagnetism againor shorterThe decay, lifetime is, typically like that the of would the tobaryon, protons allowbaryons referred one and is type pions. actually⌫ to¯e of as However,unstable the characteristic hereor a there istimescale a surprise: of the strong quark to morph into another. We need to invoke something new.Now, 10 10 seconds may not sound like much. Indeed, it’s dicult to imagine having although these new baryons have a mass 24 in the same ballpark as the ’s, they live for quark to morph intoresonance another.. We need to invokeforce: somethingTQCD = R new.QCD/c 10 seconds. If anything happens due to the strong force, 24 a rich⇡ and fulfilling life0 in this time.0, But it’s an aeon compared to the 10 seconds That “something new” turns outsignificantly to be somethingit usually longer. happens old In particular, and on familiar. roughly the ⌃ Nature this±,the timescale.⇤ hasand the ⌅ all live for a whopping That “somethingIt new” should turns be clear out from to10 be the something discussion, old however, andthat familiar. that the there’s baryons Nature nothing has live. very This qualitatively is a puzzle: why do these new baryons have a such a atendencytore-usehergoodideasoverandoveragain,andtheweakforceisno10 seconds. atendencytore-usehergoodideasoverandoveragain,andtheweakforceisnodi↵erent between a stable particle like the ⇤ andcomparatively a resonance like long the life,. evenBoth though will decay their masses are comparable to the ? 10 exception. Like the strong force, it too isNow, described 10 byseconds Yang-Mills may not theory. sound10 The like di much.↵erence Indeed, it’s dicult to imagine having exception. Like thein strong less than force, the it blink too is of described an eye. But by Yang-Mills a lifetime of theory. 10 seconds The–129– di↵erence mean that, with good is that the matrices are now 2 2insteadof3a rich and3. fulfilling However, life inas this we’llA time. partial see in But this answer it’s section, an to aeonthis is compared to invoke to a new the 10 conservation24 seconds law. We know that electric is that the matricestechnology, are now 2 you⇥2insteadof3 can take a photograph⇥3. However, of the as we’ll particle’s see in track this section, in a cloud–79– chamber or it’s not just the strengths of the forces that di↵er and the weakcharge and is conserved strong forces in all interactions. It turns out that there is another quantity – bubble chamber.–81–⇥ Youthat can the see⇥ manybaryons such live. photographs This is a puzzle: in Interludes why doB theseand C new. When baryons have a such a it’s not just the strengths of the forces that di↵er and the weakstrangeness and strong– which forces is conserved. Or, at the very least, almost conserved. Strangeness manifest themselvesaparticleleavessuchavividtrace,it’shardtodenyitsexistence.Incontrast,we’re in a very di↵erentcomparatively manner. long life, even though their masses are comparable to the ? manifest themselves in a very di↵erent manner. is simply a count of the number of strange quarks. 20 The three forcesnever of Nature going together to take a provideA photograph partial the answer foundation of something to this of is tothe that invoke Standard lasts a 10 new Model. conservationseconds. But law. that We know that electric The three forces of Nature together provide the foundation of the Standard Model. In mathematical languagedoesn’t mean these that forces it’scharge are any characterised isless conserved real! It just inby all a leaves group interactions. its signature It turns in more out subtle that there ways. is another quantity – In mathematical language these forcesstrangeness are characterised– which by is conserved. a group Or, at the very least, almost conserved. Strangeness G = SU(3)is simplySU(2) a countU(1) of the number of strange quarks. –80– G = SU(3) ⇥SU(2) ⇥U(1) ⇥ ⇥ where the 3 3matrixfieldsofSU(3) describe the strong force, and the 2 2ma- where the 3 ⇥3matrixfieldsofSU(3) describe the strong force, and the 2 ⇥ 2ma- trix fields of⇥SU(2) describe the weak force. However, rather surprisingly the⇥ fields trix fields of SU(2) describe the weak force. However, rather surprisingly–80– the fields of U(1) do not describe the force of electromagnetism! Instead, they describe an of U(1) do not describe the force of electromagnetism! Instead, they describe an “electromagnetism-like” force that is called hypercharge. The combination of SU(2) “electromagnetism-like” force that is called hypercharge.–81– The combination of SU(2) and U(1) is sometimes referred to as electroweak theory. We will learn in Section 4.2 and U(1) is sometimes referred to as electroweak theory. We will learn in Section 4.2 how electromagnetism itself lies within. how electromagnetism itself lies within. The weak force has few obvious manifestations in our everyday life and, in many ways, The weak force has few obvious manifestations in our everyday life and, in many ways, is the most intricate and subtle of all the forces. It is intimately tied to the Higgs boson is the most intricate and subtle of all the forces. It is intimately tied to the Higgs boson and, through that, the way in which elementary particles get mass. Moreover, both and, through that, the way in which elementary particles get mass. Moreover, both the most beautiful parts of the Standard Model, and those aspects that we understand the most beautiful parts of the Standard Model, and those aspects that we understand least, are to be found in the weak force. least, are to be found in the weak force.

–111– –111– Here “almost conserved” means conserved by the strong interaction. The strong

interactions cannot change the number of strange quarks and so particles like ⌃±, 0 0, ⇤ and ⌅ do not decay straight away. The decay only proceeds through the weak interaction, and this takes signiﬁcantly longer. We’ll describe how these decays occur in Section 4. In contrast, particles like the baryons decay directly through the strong interaction, and this happens much faster.

(As an aside: there’s always one complication. It turns out that, among the collection of strange baryons, there is one which is unstable: the ⌃0 ⇤0 + with a lifetime of 20 ! around 10 seconds. But this is allowed by the strong force because the number of 0 10 strange quarks is unchanged. The ⇤ , as we’ve seen, then waits another 10 seconds before it too decays.)

AsParticles a general rule of thumb, vs hadronsResonances can decay through one of the three forces: strong 0 0 (like the ’s), electromagnetic (like ⌃ )orweak(like⌃±, ⇤ and ⌅.). The lifetimes of these particles reﬂect the decay process:

22 24 • Strong decay: 10 to 10 seconds. ⇠ 16 21 • Electromagnetic decay: 10 to 10 seconds. ⇠ 7 13 • Weak decay: 10 to 10 seconds. ⇠ Where you sit within each range depends on other factors, such as the relative masses 10 of the parent and daughter particles. Particles that live for up to 10 seconds are If a particle decays throughreferredthe weak to (I force, think, we somewhat can take tongue a photograph in cheek) asof stableit! . In contrast, any particle 20 that lasts 10 seconds or shorter is, like the baryon, referred to as unstable or a resonance.

It should be clear from the discussion, however, that there’s nothing very qualitatively di↵erent between a stable particle like the ⇤ and a resonance like the . Both will decay 10 in less than the blink of an eye. But a lifetime of 10 seconds mean that, with good technology, you can take a photograph of the particle’s track in a cloud chamber or bubble chamber. You can see many such photographs in Interludes B and C. When aparticleleavessuchavividtrace,it’shardtodenyitsexistence.Incontrast,we’re 20 never going to take a photograph of something that lasts 10 seconds. But that doesn’t mean that it’s any less real! It just leaves its signature in more subtle ways.

If it decays through the strong force, or EM, then we see it more indirectly

–81– The Weak Force

half of each particle!

The numbers in the table are the masses of the particles, written as multiples of the electron mass. (Hence the electron itself is assigned mass 1.) The masses of the neutrinos are known to be very small but, otherwise are only constrained within a window and not yet established individually.

–9– Figure 30. Parity violation of Chien-Shiung Wu.

4.1 The Structure of the Standard Model When describing the strong force, we saw that it a↵ects some particles (we call these quarks) while leaving other untouchedParity (we Violation call these leptons). Our ﬁrst task now should be to describe which particles are a↵ected by the weak force.

You might think that we could simply list those particles that feel the weak force. But, as we will see, things are not quite so straightforward. It turns out that the weak force acts on exactly half of the particles in the universe. But it does so by acting on exactly half of each and every particle!

4.1.1 Parity Violation There is one deﬁning characteristic of the weak force and hypercharge that di↵erentiates them from the strong force (and from electromagnetism). They do not respect the symmetry of parity.

This fact was discovered by Chien-Shiung Wu, on a cold winters day, in New York City, in December 1956. Wu’s experiment was technically challenging, but conceptually very simple. She placed a bunch of Cobalt atoms in a magnetic ﬁeld and watched them die. Cobalt undergoes beta decay

60 60 Co Ni + e +¯⌫ +2 ! e with a half-life of around 5.3 years. The two photons arise because cobalt ﬁrst decays to an excited state of the nickel nucleus, which subsequently decays down to its ground state emitting two gamma rays. The whole point of the magnetic ﬁeld was to make sure that the nucleon spins of the atoms were aligned. Wu discovered that the electrons were preferentially emitted in the opposite direction to the nucleon spin

–112– Chiral Fermions

left-handed fermion right-handed fermion

Left-handed particles experience the weak force, right-handed do not. The Forces of the Standard Model

Each generation splits into 8x2 sets of particles

Figure 32. The jigsaw of the Standard Model. Only quarks experience the strong force. Only left-handedNote: particlesWe don’t experience yet know the if weak. the right-handed neutrino exists. shows that each generation really contains 8 particles (strictly 8 Dirac spinors). There are 3 + 3 from the two quarks, and a further 1 + 1 from the two leptons.

1 The next step in deconstructing the Standard Model is to note that each spin 2 particle should really be decomposed into its left-handed and right-handed pieces. Only the left-handed pieces then experience the weak force SU(2). If we were to follow the path of the strong force, you might think that we should introduce some new degree of freedom, analogous to colour, on which the weak force would act. A sort of weak colour. Like pastel. In fact, that’s not necessary. The “weak colour” is already there in the particles we have.

This is illustrated in Figure 32. The right-hand particles are the collection of coloured quarks and colourless leptons. The left-handed particles are the same, except now the weak force acts between the up and down quark, and between the electron and neutrino. In other words, the names of distinct particles — up/down for quarks and electron/neutrino for leptons — are precisely the “weak colour” label we were looking for! We’ve denoted this in the ﬁgure by placing the weak doublets in closer proximity.

This should strike you as odd. For the strong force, the red, blue and green quarks all act in the same way. We say that there is a symmetry between them. However, it’s very hard to make the same argument for the “weak colour” label. The electron and neutrino are very di↵erent beasts. If we’re really introducing “weak colour” in analogy with actual colour, surely there should a symmetry between them. What’s going on?

–115– To write down a theory which violates parity is then straightforward: we simply need to ensure that the left-handed particles experience a di↵erent force from the right-handed particles.

The weak force accomplishes this in the most extreme way possible: only left-handed particles experience the weak force. Right-handed particles do not feel it at all. For reasons that we now explain, this is the key property of the weak force and one of the key properties of the Standard Model.

There are quite a few things that we will need to unpick regarding the weak force. Not least is the fact that, as stressed above, the distinction between left-handed and right-handed particles is only valid when the particles are massless. A remarkable and shocking consequence of parity violation is that, at the fundamental level, all elemen- 1 tary spin 2 particles are indeed massless! The statement that elementary particles – like electrons, quarks and neutrinos – are fundamentally massless seems to be in sharp contradiction with what we know about these particles! We learn in school that electrons and quarks have mass. Indeed, in the introduction to these lecture notes we included a table with the masses of all elementary particles. How can this possibly be reconciled with the statement that they are, at heart, massless? Clearly we have a little work ahead of us to explain this! We’ll do so in Section 4.2 where we introduce the Higgs boson.

4.1.2 A Weak Left-Hander We’re now in a positionThe Structure to explain how of thethe three Standard forces of Model the Standard Model act on the matter particles. The short-hand mathematical notation for the forces is

G = SU(3) SU(2) U(1) ⇥ ⇥ Let’s ﬁrst recall some facts from the previous chapter. The strong force is associated to the “SU(3)” term in the equationstrong above. Asweak we explainedhypercharge in Chapter 3,theanalogof the electric and magnetic ﬁelds for the strong force are called gluons, and are described by 3 3matrices.(Thisiswhatthe“3”inSU(3) means.) Correspondingly, each quark ⇥ Particles Strong Weak Hypercharge carries an additional label, that wequarks call colouryes thatyes comes+1/6 in one of three variants which Left-handed we take to be red, green or blue. leptons no yes -1/2 up quark yes no +2/3 down quark yes no -1/3 While quarks comeRight-handed in three, colour-coded varieties, the leptons – i.e. the electron and neutrino – do not experienceelectron the strongno forceno and hence-1 they come in just a single, colourless variety. In the introduction,neutrino weno saidno that each0 generation contains four particles: twoTable quarks 2. The Standard and two Model leptons. forces acting However, on each of the a fermions better in a counting, single generation. including colour,

A perfectfor the jigsaw: right-handed Anomaly particlescancellation coincidemeans with theirthat electromagneticit could hardly charge.be any This,other asway we ! shall see, is no coincidence. However, the hypercharges for the left-handed particles are rather di↵erent.

Before we move, I should mention that–114– there is one caveat. (Isn’t there always!) We don’t yet have direct evidence for the existence of the right-handed neutrino and there is a possibility that it doesn’t exist! Indeed, many people would say that the right-handed neutrino should not be included in the list of particles in the Standard Model. From the table you can see that the right-handed neutrino is neutral under all three of the forces in the Standard Model and this makes it very challenging to detect. We’ll see the indirect evidence for its existence in Section 4.4 where we describe more about neutrinos in general.

4.1.3 A Perfect Jigsaw The particles and forces listed in Table 2 summarise 150 years of work (dated from R¨ontgen’s discovery of X-rays), dedicated to understanding the structure of matter at the most fundamental level. The ﬁrst thing that comes to mind when you see it is: what a mess! The individual elements comprise some of the most gorgeous objects in theoretical physics – the Dirac, Maxwell and Yang-Mills equations. And yet any semblance of elegance would seem to have been jettisoned at the last, with the di↵erent components thrown together in this strange higgledy-piggledy fashion. Why this collection of forces and particles? In particular, why this strange collection of hypercharges?

Happily, there is an a wonderful and astonishing answer to these questions. The beautiful truth is simply: it could barely have been any other way.

–118– In this equation, V () is the Higgs potential while the terms on the right-hand-side describeIn the this couplings equation, to theV ( fermions) is the in Higgs the theory. potential Both while of these the termswill be on described the right-hand-side in more detaildescribe below. the couplings to the fermions in the theory. Both of these will be described in more detail below. As we’ve alluded to earlierThe in these Higgs lectures, the Field Higgs field plays a number of roles and we’llAs elaborate we’ve alluded on these to as earlier we go in along. these For lectures, now we the mention Higgs only field that plays the a Higgs number of roles field doesand not we’ll experience elaborate the on strong these as force, we go but along. it does For feel now both we the mention weak forceonly that and the Higgs This is both the simplest and most complicated field in the Standard Model! hypercharge.field does We not should experience therefore the augment strong Table force,2 butlisting it the does forces feel both experienced the weak by force and each particlehypercharge. with one We further should entry: therefore augment Table 2 listing the forces experienced by each particle with one further entry: Particle Strong Weak Hypercharge Higgs Particleno Strongyes Weak+1/2 Hypercharge Table 3. The forcesHiggs experiencedno byyes the Higgs boson+1/2

Note that, because the HiggsTable field 3. has The no forces spin, experienced it doesn’t decompose by the Higgs further boson into left- and right-handed pieces. It just is. Note that, because the Higgs field has no spin, it doesn’t decompose further into left- Likeand all other right-handed fields, ripples pieces. of the It just Higgs is. field give rise to particles. This is the Higgs boson, the last of the Standard Model particles to be discovered. In weighs in at a mass TheLike all Higgs other fields,Expectation ripples of Valuethe Higgs field give rise to particles. This is the Higgs boson, the last of the Standard Model particles to be discovered. In weighs in at a In this equation, V () is the HiggsThe upshot potential of while the discussion them terms125 above on GeV the is right-hand-side that, even in the vacuum, the Higgs field averages mass H ⇡ describe the couplings to the fermionsto something in the theory. non-vanishing. Both of these That will be something described in non-vanishing turns out to have the making it the secondFigure heaviest 34. Two possible particle shapes in the for Standard the potential Model, for a after scalar the field. top The quark. Higgs field has a more detail below. Two dimensionsrelevant scales: of energy. • Mass It is mH 125 GeV potential like that shown on the right, so⇡ that the field condenses, with = 0 in the vacuum. However, the real importance of the Higgs lies not in the particle (althoughh i6 that’s As we’ve alluded to earliermaking in these it lectures, the second the heaviest Higgs field particle plays in a thenumber Standard of roles Model, after the top quark. certainly interesting!)The fate but, of as a scalar we’ll• Condensate now field explain, is not something more in that246 a property we GeV get to of chose. the field It isitself determined (4.2) and we’ll elaborate on these as we go along. For now we mention onlyh thati⇡ the Higgs However,dynamically the real by importance the theory. Any of scalar the Higgs field experiences lies not in a potential the particle energy (although that we call that’s field does not experience4.2.1 the strong The Higgs force, Potential but it does feel both the weak force and certainlyThisV is( interesting!) known). This is as a the function but, Higgs as that we’llvacuum tells now us explain, how expectation much more energy in value ita propertycosts. It for is the one of field the of thefieldto take key itself fundamental hypercharge. We should therefore augment Table 2 listing the forces experienced by It is theGiven condensate thatscales the thatcertain Higgs in thegives values. field universe. the is Roughly Higgs simpler speaking,its than Midas all there thetouch: are others, two everything di why↵erent does shapes that it play thatit touches these such potentials an gets a mass each particle with one furtherimportant4.2.1 entry: role?take The Well, in Higgs theories there’s Potential of something particle physics. that These a spin are 0 shown field can in Figure do that34. higher spin fields cannot:So they whatIn thecan are vacuum, “turn the on” the consequences in scalar the field vacuum. sits of at thethis? minimum Well, of they the potential. are pretty If the dramatic: potential the vacuum ParticleGivenStrong that theWeak HiggsHypercharge field is simpler than all the others, why does it play such an expectationhas the shape value shown (4.2 on) theturns left outof Figure to give34, then a mass = 0to in everything the vacuum. However, that it touches. This important role? Well, there’s something that a spinh i 0 field can do that higher spin Higgs noif the potentialyes has+1/2 the shape shown on the right of Figure 34, then =0inthe is known as the Higgs mechanism or the Higgs e↵ect. Theh particlesi6 that get masses fields cannot:vacuum, they and canmore “turn interesting on” inthings the happen. vacuum. It turns out that the potential for the Table 3. Theinclude forcesHiggs experienced both field in the the by W-bosons Standard the Higgs–122– Model boson and has Z-bosons the shape shown that on mediate the right, the and this weak is what and hypercharge, togetherendows with the allHiggs the with spin its power. 1/2 matter (This statement particles is of roughly the Standard true. A more Model. accurate We’ll postpone Note that, because the Higgs fieldadiscussionoffermionmassestoSection hasdepiction no spin, of the it doesn’t Higgs potential decompose will be further given in into Section4.3 left-, but4.3.2.) the punchline is simply that the and right-handed pieces. It just is. –122– quarks,This, electrons of course, and brings neutrinos up another get interesting the masses question: that why does we the advertised Higgs potential back in Section 1. Like all other fields, ripples ofHere, the Higgsin we our will world field begin givehave the rise by shape focussing to particles. on the right, on This the and massesis not the the Higgs shape of the on W-bosons the left? We don’t and knowZ-bosons and some the answer to that. At present, it is an input into the Standard Model and, hopefully, boson, the last of the Standardof Model theirwill consequences.particles be explained to by be some discovered. But more first complete we In will weighs theory attempt, in in the at future. anda largely fail, to give some intuition mass for why the Higgs field gives mass to particles at all. Here we are focussing on the question of whether the minimum of V ()sitsat =0, or =0.ButanotherquestionthatwecouldaskisthevalueofV ()itselfatthe mH 1256 GeV Analogiesminimum.⇡ for In the the context Higgs of particle Mechanism physics, this plays no role: it is just like any other making it the second heaviest particleThepotential result in the of energy, Standard the where Higgs Model, only expectation potential after the di↵erences top value quark. really matter= 0 and is youutterly can always startling. add The Higgs aconstanttoV () without changing the physics.h However,i6 once we include gravity However, the real importancefield of theinto is like Higgs the mix, the lies the ancient not value in of kingthe the particle potential Midas,(although energy but instead becomes that’s very of turning important everything and contributes to gold it makes certainly interesting!) but, as we’lleverything nowtowards explain, massive. the cosmological more (Both in a constant. property make We’ll things of the say field heavier.)more itself about this Why? in Section 5.3.1.

4.2.1 The Higgs Potential This is not an easy question to answer at the level of these lectures. What’s perhaps Given that the Higgs field is simplerunusual than about all the this others, is that why it’s does not at it all play di suchcult an to understand the Higgs e↵ect at the –124– important role? Well, there’s somethinglevel of equations. that a spin Indeed, 0 field it’s can one do that of the higher simpler spin calculations in quantum field theory, fields cannot: they can “turn on”but in that the vacuum. doesn’t change the fact that you do first need to learn quantum field theory. Largely, the diculty in translating from equations to everyday language lies in the fact that the Higgs e↵ect is really a phenomenon that is to do with fields rather than particles–122– and we simply don’t have much intuition for how these objects behave. If we want to avoid the mathematics, we’re obliged to rely on analogy. And, sadly, good analogies that relate the Higgs boson to more familiar, everyday phenomena are hard to come by. Here, for example, is a bad analogy. We could say that the Higgs field is like some kind of treacle. If you drag a spoon through treacle, you experience more resistance than if you drag it through water. Something similar happens with the

–125– Contents

1 Introduction 1

1 Introduction

Gravity Electromagnetism Weak Strong

Higgs

Hypercharge

Q Q F = ↵ 1 2 r2

e2 1 ↵ = 4⇡✏0~c ⇡ 137

1 ↵ 1 ⇡ 137 ⌧

(↵2) O (↵4) O

↵ How Particles Get a Mass s !1

In the Standard Model, all fermions and gauge bosons are obliged to be fundamentally massless + + ⇡ (ud) µ + ⌫µ They get a mass !by interaction with the Higgs. The Higgs Expectation Value The upshot of the discussionMass = aboveg is that, even in the vacuum, the Higgs ﬁeld averages to something non-vanishing. That⇥h somethingi non-vanishing turns out to have the dimensions of energy. It is

some dimensionless coupling 246 GeV (4.2) h i⇡ This is known as the Higgs–1–vacuum expectation value. It is one of the key fundamental scales in the universe. • The Higgs gives mass to the W-boson and Z-boson and all fermions. So what are the consequences of this? Well, they are pretty dramatic: the vacuum expectation value• (4.2The) turnsphoton out is remains to give amassless: mass to it everything is the one that that got it touches.away! This is known as the Higgs mechanism or the Higgs e↵ect. The particles that get masses include both the• W-bosonsRecall: the andmass Z-bosonsof the proton that mediateand neutron the weakdo not and come hypercharge, from the Higgs! together with all the spin 1/2 matter particles of the Standard Model. We’ll postpone adiscussionoffermionmassestoSection4.3, but the punchline is simply that the quarks, electrons and neutrinos get the masses that we advertised back in Section 1. Here, we will begin by focussing on the masses of the W-bosons and Z-bosons and some of their consequences. But ﬁrst we will attempt, and largely fail, to give some intuition for why the Higgs ﬁeld gives mass to particles at all.

Analogies for the Higgs Mechanism The result of the Higgs expectation value = 0 is utterly startling. The Higgs h i6 ﬁeld is like the ancient king Midas, but instead of turning everything to gold it makes everything massive. (Both make things heavier.) Why?

This is not an easy question to answer at the level of these lectures. What’s perhaps unusual about this is that it’s not at all dicult to understand the Higgs e↵ect at the level of equations. Indeed, it’s one of the simpler calculations in quantum field theory, but that doesn’t change the fact that you do first need to learn quantum field theory. Largely, the diculty in translating from equations to everyday language lies in the fact that the Higgs e↵ect is really a phenomenon that is to do with fields rather than particles and we simply don’t have much intuition for how these objects behave.

If we want to avoid the mathematics, we’re obliged to rely on analogy. And, sadly, good analogies that relate the Higgs boson to more familiar, everyday phenomena are hard to come by. Here, for example, is a bad analogy. We could say that the Higgs ﬁeld is like some kind of treacle. If you drag a spoon through treacle, you experience more resistance than if you drag it through water. Something similar happens with the

–125– How should we think about this? It seems to say that if, for example, a charm quark decays by emitting a W-boson, then the end product is both a strange quark and a down quark, in some combination. But, of course, we’re in a quantum world here. And just as a particle can be in two places at the same time, or a cat both dead and alive, the decay product of a charm is indeed a quantum superposition of a down quark and a strange. As usual, this manifests itself in our experiments as probability. We get probabilities by taking the square of a Feynman diagram: so the probability of c s+W + is proportional to cos2 ✓ while the probability of c d+W + is proportional ! ! to sin2 ✓.

The phenomenon of quark mixing resolves our earlier puzzle: it’s now quite possible for a meson like the kaon to decay, because there is an escape route for the strange quark, with Feynman diagrams like this now allowed: u s

One Last Thing: QuarkW Mixing

The only price we pay is that the probability for such events to happen is reduced byThere sin2 ✓ is a 0misalignment.05. This results between in the an interactions increased with lifetime the Higgs for mesons and the containinginteraction strange with the⇡ weak force. quarks.

ThisIt turns story out repeatsthat you withcan choose the addition to have the of up an-sector extra aligned. generation. But then Now the theredown sector is mixing betweenis not. theThe down,result is strange a superposition and bottom of particles. quarks, and the simple rotation (4.7)isreplaced by a morechange complicated their type, so matrix it must be equation a weak interaction that does the job. But the weak interaction that we’ve described above doesn’t allow quarks to change generation. For astrangequark,wehavetheFeynmandiagramd0 Vud Vus Vub d

0 s0 1 = 0 Vcd Vcs Vcb c1 0 s 1 (4.8) s These particles interact B b0 C B Vtd Vts Vtb C B b C with weak force B C B C B C These particles interact @ A @ W A @ A with Higgs, and so definite The 3 3matrixisknownastheCabbibo-Kobayashi-Maskawa(orCKM)matrix.The ⇥ mass. upper-leftBut then 2 we’re2 sub-matrix left with a agrees charmed with quark the and Cabbibo there’s nowhere mixing for (4.7 that), butto go. now Instead, we have the This is how,⇥ for example, mesons with strange quarks decay possibilityfor a kaon of mixing to decay between we would all need three an interaction quarks. that mixes the generations, like this: u You might reasonably ask: whats made the up-sector special? Why is the up-sector aligned, as in (4.6), while the down-sector has the complicated CKM matrix? The answer is that this is just a choice. There’s someW freedom in the equations to guarantee a partial alignment between the weak and Higgs forces. The convention is to pick the If such a decay was possible then the resulting up quark could annihilate with theu ¯ in up-sector to be aligned because then the misalignment looks simple for the relatively the kaon, while the W can decay into, say, an electron and anti-neutrino in the usual way.

So what are we missing? Is it possible for the weak force to mediate decays that change quarks from one generation into–144– another? The answer to the second question is: yes. The answer to the ﬁrst question is that we’re missing something rather subtle about the meaning of the word “particle”!

So far, we have been using two, somewhat di↵erent meanings of the word “particle” and tacitly assuming that they coincide. These are:

• A particle is an excitation of the field that has a fixed energy. Or, because E = mc2, an equivalent way of saying this is that we can assign a specific mass to the particle. In the language of quantum mechanics, we say that it is an energy eigenstate.

• A particle is the object that interacts with a particular force. This is really pertinent only for the weak force which, as we’ve seen, turns one particle into a di↵erent particle: say the down quark into an up quark.

The subtlety comes about because, for the weak force, these two ideas of what it means to be a particle don’t quite agree. The excitations of the field with a fixed energy aren’t the same thing as the excitations of the field that have a specific interaction with the weak force. Another way of saying this is to recall that the mass of the particle comes

–142– One Last Thing: and Lepton Mixing

NeutrinoThere is a Mixing similar statement Angles for neutrinos With three generations, neutrino mixing is described by introducing a 3 3matrix, ⇥ entirely analogous to the CKM matrix that we met for quarks in (4.8). This is

⌫e Ue1 Ue2 Ue3 ⌫1

0 ⌫µ 1 = 0 Uµ1 Uµ2 Uµ3 1 0 ⌫2 1 (4.13)

These particles interact B ⌫⌧ C B U⌧1 U⌧2 U⌧3 C B ⌫3 C B C B C B C with weak force and are @ A @ A @ A These particles have definite producedOn thein, left-handsay, beta deca sidey we have neutrinos ⌫e, ⌫µ and ⌫⌧ that interactmass. These with theirare energy coun- terpart electrons through the weak force; On the right-hand sideeigenstates we have that neutrinostravel unchanged through space. ⌫ , ⌫ and ⌫ that have deﬁnite mass. Relating them is a 3 3 matrix is known as 1 2 3 ⇥ the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, or simply the neutrino mixing matrix. This gives rise to neutrino oscillations The components of the PMNS matrix have now been measured to reasonable accu- racy. The absolute values are roughly

U U U 0.80.50.1 | e1|| e2|| e3| 0 U U U 1 0 0.30.50.7 1 | µ1|| µ2|| µ3| ⇡ B U⌧1 U⌧2 U⌧3 C B 0.40.60.6 C B | || || | C B C @ A @ A Some values are known fairly well; others less well. There are, for example, error bars of 0.1onU . ± ⌧2 The ﬁrst thing to note is that the PMNS matrix is strikingly di↵erent from the CKM matrix describing the mixing of quarks15. In the quark sector, the CKM matrix was close to being the unit matrix, with just small o↵-diagonal elements. This meant that there was close alignment between the masses and the weak force.

But we see no such thing in the neutrino sector. The mixing is pretty much as big

as it can be! The only exception seems to be the mixing between ⌫1 and ⌫⌧ , which is smallish at about 0.1. Once again, we see that, in quantitative detail, the neutrinos really behave nothing like the charged fermions.

We do not have an explanation for the structure of the PMNS matrix. Indeed, its form came as a surprise to theorists. Surely it is telling us something important. It’s just we don’t yet know what!

d0 V V V d V V V 0.97 0.22 0.004 ud us ub | ud|| us|| ub| 15 Recall that 0 s0 1 = 0 V V V 1 0 s 1 with 0 V V V 1 0 0.22 0.97 0.04 1. cd cs cb | cd||cs||cb| ⇡ B b0 C B Vtd Vts Vtb C B b C B Vtd Vts Vtb C B 0.009 0.04 0.999 C B C B C B C B | || || | C B C @ A @ A @ A @ A @ A

–163– light strange mesons, with no need to invoke the charm quark in the argument. But if you were feeling a little perverse, there’s nothing to stop you redeﬁning everything withNeutrino the down-sector Mixing Angles aligned and the up-sector askew, or even some combination of theWith two. three generations, neutrino mixing is described by introducing a 3 3matrix, The Mixing Matrices ⇥ entirely analogous to the CKM matrix that we met for quarks in (4.8). This is The components of the CKM matrix have been accurately measured experimentally.

It turns out that some of the⌫e elementsU cane1 U bee2 U complexe3 ⌫1 numbers and we’ll explain the For quarks,significance we have of the this CKM in Section matrix4.3.4. For now, we give just the absolute values of each 0 ⌫µ 1 = 0 Uµ1 Uµ2 Uµ3 1 0 ⌫2 1 (4.13) element which are roughly B ⌫⌧ C B U⌧1 U⌧2 U⌧3 C B ⌫3 C B C B C B C @ A @ A @ A On the left-hand sideV weud haveVus neutrinosVub ⌫ , ⌫0.97and 0.⌫22that 0.004 interact with their coun- | || || | e µ ⌧ terpart electrons through0 V theV weakV force;1 0 On0. the22 right-hand 0.97 0.04 1 side we have neutrinos | cd|| cs|| cb| ⇡ ⌫1, ⌫2 and ⌫3 that have definite mass. Relating them is a 3 3 matrix is known as B Vtd Vts Vtb C B 0.009 0.04 0.999⇥C the Pontecorvo-Maki-Nakagawa-SakataB | || || | C (PMNS)B matrix, or simplyC the neutrino mixing @ A @ A Youmatrix. can see the Cabbibo angle sitting there in V =sin✓ 0.22. The full CKM us ⇡ For neutrinos,matrixThe extends we components have the the Cabbibo ofPMNS the PMNS matrix angle matrix to 10 have parameters now been – measured the 9 above, to reasonable together withaccu- a complexracy. The phase absolute that we’ll values discuss are roughly in Section 4.3.4. Just like we have no understanding of why the Cabbibo angle takes its particular Ue1 Ue2 Ue3 0.80.50.1 value, nor do we have any| good|| understanding|| | of the CKM matrix. As you can see, 0 Uµ1 Uµ2 Uµ3 1 0 0.30.50.7 1 it’s not far from a diagonal| matrix,|| || with| the⇡ Cabbibo terms being the only ones that B U⌧1 U⌧2 U⌧3 C B 0.40.60.6 C aren’t tiny. We don’t knowB | why.|| || | C B C @ A @ A Some values are known fairly well; others less well. There are, for example, error bars It’s worth pausing to take in a bigger perspective here. In the first part of this of 0.1onU . ± ⌧2 We onlychapter, know wethese described parameters how by the experimental matter content measurement. of the Standard Why do Model they take interacts these values? with Why arethe th diThee↵erentmatrices first forces. thingso todifferent? There note is we that found the PMNS a beautiful matrix consistent is strikingly picture di↵erent – a perfect from the jigsaw CKM – 15 inmatrix which the describing interactions the mixing were largely of quarks forced. In upon the quark us by sector, the consistency the CKM requirements matrix was ofclose the theory. to being For the a theoreticalunit matrix, physicist, with just it small is really o↵-diagonal the dream elements. scenario. This This, meant however, that contraststhere was starkly close with alignment the story between of flavour. the masses Even and focussing the weak solely force. on the quarks, we find thatBut there we are see 6 no Yukawa such thing couplings in the that neutrino determine sector. their The mass, mixing plus is pretty a further much 10 as entries big

ofas the it CKM can be! matrix The only that exception determine seems their to mixing. be the mixing And none between of these⌫1 and parameters⌫⌧ , which areis ﬁxed,smallish or understood at about 0.1. at a Once deeper again, level. we see that, in quantitative detail, the neutrinos really behave nothing like the charged fermions. Somewhat ironically, much of this complexity can be traced to the simplicity of the Higgs.We The do not strong have and an explanation electroweak for forces the structure are described of the by PMNS Yang-Mills matrix. type Indeed, theories, its andform these came come as a with surprise mathematical to theorists. subtleties Surely it that is telling are ultimately us something responsible important. for It’s the quantumjust we don’t consistency yet know conditions what! that constrain their interactions. But there are no such such subtle constraints for the Higgs boson. It is a simple, spin 0 particle, that d0 V V V d V V V 0.97 0.22 0.004 ud us ub | ud|| us|| ub| can15 do as it pleases and the result is the plethora of extra parameters that we’ve seen. Recall that 0 s0 1 = 0 V V V 1 0 s 1 with 0 V V V 1 0 0.22 0.97 0.04 1. cd cs cb | cd||cs||cb| ⇡ B b0 C B Vtd Vts Vtb C B b C B Vtd Vts Vtb C B 0.009 0.04 0.999 C B C B C B C B | || || | C B C @ A @ A @ A @ A @ A

–145––163– To write down a theory which violates parity is then straightforward: we simply need to ensure that the left-handed particles experience a di↵erent force from the right-handed particles.

There are quite a few things that we will need to unpick regarding the weak force. Not least is the fact that, as stressed above, the distinction between left-handed+ and+ ⇡ (ud) µ + ⌫µ right-handed particles is only valid when the particles are massless. A remarkable! and shocking consequence of parity violation is that, at the fundamental level, all elementary spin 1 particles are indeed massless! The statement that elementary particles – 2 Mass = g like electrons, quarks and neutrinos – are fundamentally massless seems to be in sharp⇥h i contradiction with what we know about these particles! We learn in school that electrons and quarks have mass. Indeed, in the introduction to these lecture notes we 1 included a table with the masses of all elementary particles. How can this↵ possibly QED ⇡ 127 be reconciled with the statement that they are, at heart, massless? Clearly we have a little work ahead of us to explain this! We’ll do so in Section 4.2 where we introduce1 ↵ the Higgs boson. strong ⇡ 10

4.1.2 A Weak Left-Hander 1 ↵weak We’re now in a positionSummary: to explain The how theGreatest three forces Theory of the Standard of All ModelTime act⇡ on30 the matter particles. The short-hand mathematical notation for the forces is

3 4 G = SU(3) SU(2) U(1) ⇢⇤ (10 eV) ⇥ ⇥ ⇡ Let’s ﬁrst recall some facts from the previous chapter. The strong force is associated to the “SU(3)” term in the equation above. As we explained in Chapter 3,theanalogof ⇢⇤ = ⇢SM + something else the electric and magnetic ﬁelds for the strong force are called gluons, and are described by 3 3matrices.(Thisiswhatthe“3”inSU(3) means.) Correspondingly, each quark ⇥ carries an additional label, that we call colour that comes in one of three variants which three generations we take to be red, green or blue. ⇥ While quarks come in three, colour-coded varieties, the leptons – i.e. the electron and neutrino – do not experience the strong force and hence they come in just a single, colourless variety. In the introduction, we said that each generation contains four particles:Figure two 32 quarks. The jigsaw and of the two Standard leptons. Model. However, Only quarks experiencea better the counting, strong force. including colour, Only left-handed particles experience the weak.

with all complications coming from interactions with Higgs! shows that each generation really contains 8 particles (strictly 8 Dirac spinors). There are 3 + 3 from the two quarks, and a further 1 + 1 from the two leptons.

1 The next step in deconstructing the Standard–114– Model is to note that each spin 2 particle should really be decomposed into its left-handed and right-handed pieces. Only the left-handed pieces then experience the weak force SU(2). If we were to follow the path of the strong force, you might think that we should introduce some new degree of freedom, analogous to colour, on which the weak force would act. A sort of weak colour. Like pastel. In fact, that’s not necessary. The “weak colour” is already there in the particles we have.

This is illustrated in Figure 32. The right-hand particles are the collection of coloured quarks and colourless leptons. The left-handed particles are the same, except now –2– the weak force acts between the up and down quark, and between the electron and neutrino. In other words, the names of distinct particles — up/down for quarks and electron/neutrino for leptons — are precisely the “weak colour” label we were looking for! We’ve denoted this in the ﬁgure by placing the weak doublets in closer proximity.

–115–