ParticleParticle PhysicsPhysics

OrhanOrhan CAKIRCAKIR AnkaraAnkara UniversityUniversity

ISTAPP2011 OverviewOverview

● units in particle physics ● historical introduction ● elementary particles ● four forces ● symmetries ● hadrons ● particle decays ● interaction cross sections ● standard model and beyond

O.Cakir 1 UnitsUnits inin ParticleParticle PhysicsPhysics

● S.I. Units: kg, m, s are a natural choice for “everyday” objects, but they are inconveniently large in particle physics.

● Atomic physicists introduced the electron volt – the energy of an electron when accelerated through a potential difference of 1 volt: 1 eV=1.6x10-19 joules.

● However, in particle physics we use Natural Units – From Quantum Mechanics – the unit of action: ħ – From Relativity – speed of light: c – From Particle Physics – unit of energy: GeV (proton rest-mass energy~938 MeV/c2=1.67x10-24 g)

● Natural units are used throughout these lectures. O.Cakir UnitsUnits inin ParticleParticle PhysicsPhysics -- 22 ● Units become (with the dimensions) – Energy: GeV Time: (GeV/ħ)-1 – Momentum: GeV/c Length: (GeV/ħc)-1 – Mass: GeV/c2 Area: (GeV/ħc)-2 ● Algebra can be simplified by setting ħ=c=1. Now all quantities are given in powers of GeV.

● To convert back to S.I. Units, one need to keep missing factors of ħ and c.

● µ In Heaviside-Lorentz units we set ħ=c=ε0= 0=1, and Coulomb's law takes the form 1 q2 F = 4 r2 ● Electric charge q has dimensions: (FL2)1/2=(EL)1/2=(ħc)1/2 O.Cakir HistoricalHistorical IntroductionIntroduction People have long asked: ● What is the world made of? ● What holds it together? ● Why do so many things in this world share the same character- istics? Empedocles 492-432 BC People have come to realize that the matter of the world is made from aa fewfew fundamentalfundamental buildbuild-- inging blocks blocks (simple and structureless objects-not made of anything smaller) of nature. "...in reality there are atoms and space." (Democritus 400 BC).

O.Cakir HistoricalHistorical IntroductionIntroduction -- 22 The study (by Mendeleev, 1869) to categorize atoms into groups that shared similar chemical properties (Periodic Table of the Elements). Due to Moseley's work, thethe modernmodern periperi-- odicodic table table is is based based on on the the atomic atomic num num-- bersbers of of the the elements elements rather than atomic mass. Moreover, the experiments helped scientists determ- ine that atoms have a tiny but dense, positive nucleus (N+) and a cloud of negative electrons (e-).

O.Cakir HistoricalHistorical IntroductionIntroduction -- 33 Elementary particle physics was born in 1897 with J.J. Thomson's discovery of the electron (“corpus- cules”). ● The cathode rays (beam of particles) emitted by a hot filament could be deflected by a magnet, and this suggested they carried electric charge, and the direction of the curvature required that the charge be negative.

● The word 'electron' first used by G. Johnstone Stoney in 1891 to denote the unit of charge in an electro-chem- istry experiment. O.Cakir HistoricalHistorical IntroductionIntroduction -- 44 In 1909, Ernest Rutherford set up an experiment to test the validity of the prevailing theory. In doing so he established a way that for the first time physicists could "look into" tiny particles they couldn't see with microscopes.

Some of the alpha particles were deflected at large angles to the foil; some even hit the screen in front of the foil! Obvi- ously some other explanation was needed. ● Rutherford concluded that there must be something inside an atom for the alpha particles to bounce off of that is small, dense, and positively charged: thethe nucleus.nucleus. O.Cakir HistoricalHistorical IntroductionIntroduction -- 55

● PhotonPhoton (1900-1924)(1900-1924) ✔ M.Planck explains blackbody radiation (1900), radiation is quantized. ✔ A.Einstein proposes a quantum of light which behaves like a particle (1905), photoelectric effect (E≤hν-w), equivalence of mass and energy, special relativity. ● A.H.Compton found that the light scattered from a particle at rest is shifted in wavelength (1923), λ λ λ θ λ according to '= + c(1-cos ), where c is the Compton wavelength of the target particle. ✔ The name photon is suggested by the chemist Gilbert Lewis (1926). O.Cakir HistoricalHistorical IntroductionIntroduction -- 66

● MesonsMesons (1934-1947)(1934-1947) ➢ Yukawa theory for strong force, m~300me ➔ meson: middle-weight, baryons: heavy-weight, lepton: light-weight. ➢ Powel et al. (1947) discovers two middle-weight particles, pion and muon. ➢ I.I.Rabi (1946) comments "who ordered that?" for the muon. ● Anti-particlesAnti-particles (1930-1956)(1930-1956)  Dirac equation gives positive and negative energy solutions (particle / antiparticle) - Feynman- Stuckelberg formulation (1940). ● Anderson discovers positron (1932). ● Anti-proton and anti-neutron was discovered at Berkeley (1955-56). O.Cakir HistoricalHistorical IntroductionIntroduction -- 77 ● NeutrinosNeutrinos (1930-1962)(1930-1962) ➢ W.Pauli (1930) suggest neutral particle to explain continuous electron spectrum in beta decay. ➢ In nuclear beta decay, a neutral missing particle, E.Fermi called it “neutrino” (1933) and introduction of weak interaction theory. ➢ J.Chadwick discovers neutron (1931) ➢ H.Yukawa (1934) meson (pion) theory of nuclear forces ● StrangeStrange ParticlesParticles (1947-1960)(1947-1960) ➔ Pions in cosmic rays, calculation of electromagnetic processes, introduction of Feynman diagrams, The Berkeley synchro-cyclotron produces the first lab. pions. O.Cakir HistoricalHistorical IntroductionIntroduction -- 88 StrangeStrange ParticlesParticles (1947-1960)(1947-1960) ➔ Discovery of K+ meson (1949) via its decay. ➔ Neutral pion is discovered in 1950. ➔ Discovery of particles called “delta” baryon (s=3/2). ➔ C.N.Yang and R.Mills develop a new class of theories called "gauge theories" in 1954. Unification/ClassificationUnification/Classification IdeasIdeas (1957)(1957) ➔ J.Schwinger (1957) writes a paper proposing unification of weak and electromagnetic interactions. ➔ The group SU(3) to classify and organize particles (1961)

➔ ν ν Experiments verify two types of neutrinos, e and m (1962). O.Cakir HistoricalHistorical IntroductionIntroduction -- 99 QuarkQuark ModelModel (1964)(1964) ➔M.Gell-Mann and G.Zweig (1964) suggest that mesons and baryons are composed of quarks (flavors u,d,s) ➔Omega particles discovered (1964, BNL) , as Gell- Mann predicted within baryon decuplet. ➔O.W.Greenberg, M.Y.Han and Y.Nambu introduce the quark property of color charge (1965). All observed hadrons are color neutral. EWEW TheoryTheory (1967-70)(1967-70) ➔S.Weinberg and A.Salam (1967) propose a theory that unifies electromagnetic and weak interactions. ➔J.Bjorken and R.Feynman (1969) analyze the SLC data in terms of a model of constituent particles inside the proton (provided evidence for quarks!) O.Cakir HistoricalHistorical IntroductionIntroduction -- 1010

EWEW TheoryTheory (1970-74)(1970-74) ➔ S.Glashow, J.Iliopoulos, L.Maiani (1970) recognize the critical importance of a fourth type of quark in the context of a standard model, Z0 has no flavor-changing weak interactions. ➔ H.Fritzsch and M.Gell-Mann suggested the strong interaction theory QCD (1973) ➔ D.Politzer, D.Gross, F.Wilczek discover that the color theory of the strong interaction has a special property, now called "asymptotic freedom" ➔ In 1974, J.Iliopoulos presents the view of particle physics called the Standard Model (SM).

O.Cakir HistoricalHistorical IntroductionIntroduction -- 1111

● NewNew MembersMembers (1974-78)(1974-78) ➢ S.Ting et al. at BNL and B.Richter et al. at SLAC announce on the same day that they discovered the same new particle. Since the discoveries are given equal weight, the particle is commonly known as the J/Psi (cc) meson. ➢ G.Goldhaber, F.Pierre find the D0 meson (anti-up and charm quarks) (76) ➢ The tau lepton (lepton of third generation) is discovered by M.Perl et al. at SLAC (76). ➢ L.Lederman et al. at Fermilab discover yet another quark (and its antiquark) (77), the "bottom" quark → this leads to the search for sixth quark – "top". O.Cakir HistoricalHistorical IntroductionIntroduction -- 1212 StandardStandard ModelModel (1978-)(1978-) ● C.Prescott, R.Taylor (1978) observe a violation of parity conservation, as predicted by the SM. ● Strong evidence for a gluon (1979) radiated by the initial quark or antiquark at the DESY. ● In 1983, the W± and Z0 intermediate bosons demanded by the electroweak theory are observed at CERN SpS by C.Rubbia et al. ● SLAC and CERN experiments suggest (1989) three generations of fundamental particles (inferring from Z0-boson lifetime) with three very light neutrinos. ● In 1995, CDF and D0 experiments at Fermilab discover the top quark (175 GeV). O.Cakir …… andand thethe timelinetimeline summarysummary

O.Cakir LowLow EnergyEnergy -- HighHigh EnergyEnergy Small

ClassicalClassical QuantumQuantum MechanicsMechanics MechanicsMechanics Fast QuantumQuantum RelativisticRelativistic FieldField MechanicsMechanics TheoryTheory

● For things that are both fast and small (particles), we require a theory that incorporates relativity and quantum principles: Quantum Field Theory (QFT).

● Most of our experimental information comes from three main sources: 1) scattering events, 2) decays, 3) bound states. O.Cakir OurOur curiositycuriosity andand toolstools forfor researchresearch

bacteria virus atoms in DNA 0.01 mm 0.0001 mm 0.000001 mm

● We need particleparticle acceleratorsaccelerators to look into the deepest structure of matter! O.Cakir ProbingProbing thethe elementaryelementary buildingbuilding blocksblocks ofof naturenature

reflection bending around

Higher Beam Energy Shorter Wave Length Better resolution

electron

nucleus α particle α particle

Rutherford’s Experiment, 1909

O.Cakir TheThe ParticleParticle PhysicsPhysics isis thethe studystudy ofof

● Matter (spin-1/2 particles): the fundamental constituents of the universe - thethe elementaryelementary particlesparticles ● Force (spin-1 particles): the fundamental forces of nature - thethe interactionsinteractions betweenbetween thethe elementaryelementary particlesparticles

● Try to cat- egorize the Particles and Forces in as simple and funda- mental man- ner possible.

O.Cakir ParticleParticle PhysicsPhysics

O.Cakir O.Cakir FourFour ForcesForces ofof NatureNature Forces can be explained as the boson exchange between fermions, and the type of boson defines the force.

O.Cakir DynamicsDynamics ofof ParticleParticle PhysicsPhysics

Quantum Electrodynamics (QED), Weak Interactions (WI), Quantum Chromodynamics (QCD) Forces Electromagnetic Weak Strong Bosons Photon (γγ ) W±, Z Gluon (g) Matter l± , quarks l± , νν, quarks quarks involved

Feynman diagrams: creation and annihilation of matter

O.Cakir Symmetry/ConservationSymmetry/Conservation The laws of physics are symmetrical with re- spect to translation in time (they work the same today as they did yesterday): Noether's (1971) theorem relates this invariance to con- servation of energy. The symmetries are asso- ciated with conservation laws.

SymmetrySymmetry ConservationConservation lawlaw TranslationTranslation inin timetime ↔↔ EnergyEnergy TranslationTranslation inin spacespace ↔↔ MomentumMomentum RotationRotation ↔↔ AngularAngular momentummomentum GaugeGauge transformationtransformation ↔↔ ChargeCharge

O.Cakir SystematicSystematic StudyStudy ofof SymmetriesSymmetries Symmetry operations has the following properties:

● If Si and Sj are in the set, then the product SiSj=Sk.

● There is an element I such that ISi=SiI=Si for all Si.

● -1 For every element Si there is an inverse Si , such -1 -1 that SiSi =Si Si=I.

● There is associativity such that Si(SjSk)=(SiSj)Sk. o S: x-->-x S+: rotation 120 S : flips around a f(x)=f(-x) a A

c b

B C a O.Cakir SystematicSystematic StudyStudy ofof SymmetriesSymmetries -- 22 ● Translations in space and time form an Abelian group

(SiSj=SjSi), while rotations in 3D do not (SiSj≠SjSi). ● If the elements depend on continuous parameters (angle of rotation) → continuous groups ● If the elements are labelled by an index that takes only integer values → discrete groups ● In elementaryelementary particleparticle physicsphysics, the most common groups are of the type UU((nn)): a collection of unitary (U-1=U*) n x n matrices. Unitary matrices with “determinant 1” called SUSU((nn)): special unitary. For real unitary matrices (O-1=O), the group is OO((nn)). SOSO((nn)) is the group of special orthogonal n x n matrices. O.Cakir SymmetriesSymmetries inin ParticleParticle PhysicsPhysics

● One of the aim of the particle physics is to discover the fundamental symmetries of our universe. ● Introduce the ideas of the SU(2) and SU(3) symmetry groups which play a major role in particle physics. – Suppose physics is invariant under the transformation (like the rotation of coordinate axes) Conserving the probability normalization require

has to be unitary

O.Cakir SymmetriesSymmetries inin ParticleParticle PhysicsPhysics -- 22

● For physical predictions to be unchanged by the symmetry transformation, also require all QM matrix elements unchanged

Consider the infinitesimal transformation (ε small) Well, “do- ing noth- ing” (11) is also a here unitarity requires symmetry operation! O.Cakir DiscreteDiscrete SymmetriesSymmetries Charge Conjugation (C)

● Classical electrodynamics is invariant under C, potentials and fields reverse sign under C, but there is a compensating charge factor so the force remain the same.

● C|p>=|p>=±|p>, it changes the sign of all “internal quantum numbers” such as charge, baryon number, lepton number, strangeness, etc.– while leaving mass, energy, momentum and spin untouched. It has limited eigenstates (photon, rho, eta etc.).

● ν It is not a symmetry of the weak interactions (no L!)

● An extended transformation “G-parity”, G=CR2 iπI(2) O.Cakir where R2=e . Ex: all pions are eigenstates of G. DiscreteDiscrete SymmetriesSymmetries -- 22 Parity (P)

● Lee and Yang (56) propose a test for parity in weak 60 60 ν interactions. Beta decay in Co--> Ni+e+ e where most electrons are emitted opposite to nuclear spin. It is a symmetry of strong and electromagnetic interactions, but it is violated in weak interactions. ScalarScalar P(s)=sP(s)=s PseudoscalarPseudoscalar P(p)=-pP(p)=-p VectorVector P(P(vv)=-)=-vv PseudovectorPseudovector P(P(aa)=)=aa (or(or axialaxial vector)vector)

● neutrinos are left-handed; antineutrinos are right-handed.

O.Cakir DiscreteDiscrete SymmetriesSymmetries -- 33 Time Reversal (T) and the TCP ● Like “running the movie backward”, say n+p --> d+γ process and its reverse order process d+γ -->n+p. No experiment has shown direct evidence of T violation. The most sensitive experimental tests are the upper limits on the -26 electric dipole moments of neutron (dn<6x10 -27 cm e) and electron (de<1.6x10 cm e). ● TCP theorem states that combined operation (TCP) is an exact symmetry of any interaction.

O.Cakir GaugeGauge GroupsGroups inin ParticleParticle PhysicsPhysics

charge

anti- charge

complex three 2x2 eight 3x3 number Matrices matrices (Pauli) (Gell-Mann)

O.Cakir HadronsHadrons Hadrons are color singlet bound states of quarks. ● Mesons are qiqj states of quarks and antiquarks.

● Baryons are qiqjqk states of quarks. ● Quarks are confined into hadrons.

Electric field lines are spread out as the Color force lines charges are separated. collimated into a tube as the quarks are sep- arated, it will break into two when the force applied as re- quired.

O.Cakir MesonsMesons If the orbital angular momentum of the states is l, then the parity P=(-1)l+1. For qq mesons the charge conjugation parity C=(-1)l+s and the G-parity (-1)I+l+s. In an SU(4) classification we have 2s+1 PC 44 xx 44=15=15 ++ 11. Spectrum: n lJ (J ), l=0: pseudoscalars (0-+) and vectors (1--) l=1:scalars(0++), axial vectors (1++) and (1+-), tensors (2++). 1 -+ ex:ex: 1 S0(0 )-->K-mesonK-meson

η c

O.Cakir Baryons Baryons are fermions with baryon number B=1, and they are in color singlet state.

|qqq> A=|color > A×|space, spin, flavor >S It is useful to classify baryons into bands having the same quantum number of excitation.

O.Cakir SymmetriesSymmetries inin ParticleParticle PhysicsPhysics -- 33

Symmetries play a crucial role in our life! In the QFT they play very special role: every force can be derived from the principle of the internal symmetry – local gauge invariance predicts gauge bosons

ΨΨ''→→ ee-iθψψ ΨΨ''→→ ee-iθ(x)ψψ O.Cakir FourFour VectorsVectors Four vector notation

In particle physics we deal with the relativistic particles → all calculations should be Lorentz in- variant. These calculations can be performed via four-vector production. Lorentz transformation

O.Cakir Ex:Ex: TopTop QuarkQuark InvariantInvariant MassMass

Energy-momentum conservation gives

pt = pb pl  p 2 2 2 2 mt ≡mbl =EbEl E − pb pl p

here we want to calculate pz of neutrino via W-boson mass con- ● opening angle straint p⋅p cos = l  2 2 2 | pl || p | pW = pl  p ; ml =E lE − pl p  2 mW ≈2El E −| pl || p |cos 2 mW ≈2| pl || p |− pl T⋅p T − pl z p z 

2 2 2 2 2 mW ≈2 pl T  pl z  p T  p z− pl T⋅p T − pl z p z

O.Cakir W-W-bosonboson massmass constraintconstraint

2 2 2 2 2 mW ≈2 pl T  pl z  p T  p z− pl T⋅p T − pl z p z

2 2 2 2 1/2  pl z ±[ pl  −4 pl T p T ] p z = 2 2 pl T 2 =mW2 pl T⋅p T where we choose the solution well reconstruct the top mass 2 2 2 1/ 2 2 2 mt = pbmb | pl || p | − pb pl  p 2 2 2 1/ 2 2 2 1/ 2 2 2 1/2 2 mt =[ pbmb  pl T  pl z   pT  p z  ] 2 2 2 2 −[ pb pl  p T  p z 2 pb⋅pl2 pbT⋅p T  pbT p z ]

−2 pl T⋅pT  pl z p z 

O.Cakir InteractionInteraction LagrangiansLagrangians →→ VertexVertex FactorsFactors ● QED interaction term ● In a similar way, incoming quark – outgoing quark – where three fields – gluon (q,‾q,g) incoming fermion - interacts at one point outgoing fermion - via QCD interaction ψ ψ photon ( ,‾ , A) term interacts at one point, and the vertex is defined. After removing the fields the remain vertex factor is part will give vertex -ig-ig λλ/2/2γγ µ . γγ µ s factor -ig-igeqq . O.Cakir InteractionsInteractions Interactions of gauge bosons with fermions are described by the vertices. Type of gauge bosons and nature of interactions determine the proper- ties of interactions.

O.Cakir InteractionInteraction →→ AmplitudeAmplitude

Fourier trans- formation

elastic scattering

O.Cakir FeynmanFeynman RulesRules

● Free Lagrangian ● Interaction terms → propagator → vertex factors

O.Cakir FeynmanFeynman DiagramsDiagrams

Processes of high energy physics are complex, they include radiation, loops, etc. However, the LO processes can be considered as a first approximation to the interactions between elementary particles (leptons, quarks and gauge bosons). Feynman diagrams are graphical presentations of the physics processes.

e+e--Zqq ggH0bb

O.Cakir ElectroweakElectroweak ProcessesProcesses

SM

For a certain type of process the amplitude and the differential cross section via diagrams

O.Cakir Example:Example: ee+ee-γγ /Z/Z0µµ +µµ − processprocess

SembolikSembolik Calculation:Calculation: Total amplitude: M=M γ +M Z REDUCE, FORM, Mathematica etc.

O.Cakir InteractionInteraction CrossCross SectionSection The differential cross section is given by 24 d = | M |2 d   p  p ; p p .. , p  2 2 2 fi n 1 2 3, 4,. n2 4 p1⋅p2 −m1 m2 in the center of mass frame 2 2 2  p1⋅p2 −m1 m2= p1cm s and it is useful to define the Mandelstam variables s= p  p 2= p  p 2 1 2 3 4 ● the two-body =m2m22 E E −2 p⋅p 1 2 1 2 1 2 cross section t= p − p 2= p − p 2 1 3 2 4 is given by 2 2 =  − − ⋅ 2 m1 m3 2E1 E 3 p1 p3 d  1 |M | 2 2 u= p − p  = p − p  = 2 1 4 2 3 d t 64 s | p | 2 2 1cm =m1m4−2 E1 E 42 p1⋅p4 O.Cakir ToolsTools forfor CalculationsCalculations

CompHEP/CalcHEP[*] program packages are used for calculation of decay and collision processes of element- ary particles in the tree level approximation. They make available passing from lagrangian to the final distribu- tions with the high level of automatization. There are also some other packages for different purposes. O.Cakir Example: e+e-→W+W- process

CompHEP, calculates |amplitude|2 expressions via symbolicsymbolic session session, and gives the output for Reduce/Mathematica/Reduce input codes. NumerNumer-- icalical session session is used for cross section calculations, decay width calculations, and event generations. The distributions can be analyzed further by using Root program. Event information can be in the LHE format.

O.Cakir Kinematical Distributions

The threshold en- ergy scan can be e+e-→W+W- performed for some energy range. After a threshold energy (~160 GeV) the cross section in- creases and reaches maximum at √s~190 GeV. In- creasing the energy

, GeV will not be more be- neficial with the cross section for this process.

O.Cakir PairPair ProductionProduction atat CollidersColliders At the colliders, one can study matter and forces, also create the new and heavy matter, via the forces, converting the energy of collid- ing particles into mass.

O.Cakir MissingMissing TransverseTransverse EnergyEnergy (MET)(MET)

● An important portion of incoming hadron ν energy goes into beam pipe. For the particles (neutrinos) undetectable directly, transverse momentum component in the per- pendicular (to the Sum over transverse beam direction) plane can be calculated. momentum of all final state detectable particles.

O.Cakir ParticleParticle DecaysDecays The decay rate of a particle of mass M into n bodies in its rest frame is given by 24 d  = |M |2 d  P ; p p .. , p  2 M fi n 1, 2,. n with 3 n n d p d  P ; p p .., p =4P− p  i n 1, 2,. n ∑i =1 i ∏i=1 3 2 2 Ei

Mfi is the Lorentz invariant amplitude specific to the process for the initial to final state transition.

O.Cakir StandardStandard ModelModel ofof ParticleParticle PhysicsPhysics

Standard Model includes quantum field theories, Quantum ElectroDynamics (QED), Quantum ChromoDynamics (QCD) and Weak interactions (WI). These are based on the relativistic quantum mechanics. In quantum mechanics, an s state of a physical ψ system is defined by s wave function (Schrodinger) or ket |s> (Dirac). Observables =∫ dnxψ*Oψ remain invariant under global (independent from position) phase transformation.

O.Cakir TheThe StandardStandard ModelModel (SM)(SM) The gauge group of An important feature of standard model is weak interactions is the SU(3) x SU(2) x U(1) violation of the “parity”. C W Y This suggests that it is where C denotes color, natural to study with W for weak isospin, and two-component spinors Y for hypercharge. The (1x2 or 2x1). These corresponding gauge a spinors also comprise fields are Gµ (a=1,8), the basics of the four- i . Wµ (i=1,3) and Bµ component spinor representation of the Lorentz group.

O.Cakir SM Summary

*Standard Model is based on the gauge

symmetry: SU(3)SU(3)C xx SU(2)SU(2)W xx U(1)U(1)Y. However, this symmetry is broken

spontaneously to SU(3)SU(3)CxU(1)xU(1)em. *Matter: there are three leptons and quarks families. *Higgs field have an important rol:

● one Higgs doublet interacts the other fields

● gain a vacuum expectation value (~246 GeV)

● Quarks and leptons, W/Z bosons and Higgs itself get their masses from this mechanism. O.Cakir StatusStatus ofof PrecisionPrecision MeasurementsMeasurements

EW fit - http://lepewwg.web.cern.ch/LEPEWWG/

O.Cakir QuantumQuantum numbersnumbers ofof ParticlesParticles

 Quantum numbers of elementary particles according to SU(3) X SU(2) X U(1)

 Lagrangian: – gauge interactions – Matter fermions – Yukawa interactions – Higgs potential

O.Cakir FieldField TheoriesTheories

● Lagrangian in classical ● In fieldfield theorytheory we study mechanics is given by with a field φ (x,y,z,t) as the L(q,q.,t) and L=T-V. Equation of motion is function of space and time. given by In the relativistic theory (4D space-time) Euler-Lagrange equation conjugate momentum corresponding to the coordinates which is not → spin-0: Klein- present explicitly in the Gordon equation; Lagrangian is conserved. spin-1/2: Dirac → Newton's laws equation; spin-1: Proca equation. O.Cakir GlobalGlobal veve LocalLocal PhasePhase TransformationsTransformations

Free Dirac lagrangian LocalLocal phasephase transformationtransformation

Derivation of wave function GlobalGlobal phasephase transformationtransformation ψ(x)−−>e-iqαψ(x) leads to an extra term ● Dirac lagrangian remains The Lagrangian reads invariant under this transformation

● Absolute phase of wave Total lagrangian should remain function is not measurable invariant under this transforma- (arbitrary) tion, then we should add both ● The relative phases in the kinetic term and interaction interference are not affected by term with the gauge boson to phase transformation. the free Dirac Lagrangian. ThisThis willwill automaticallyautomatically generategenerate ● Symmetry->charge thethe gaugegauge bosonboson ofof thethe interaction.interaction. O.Cakirconservation U(1)U(1) GaugeGauge SymmetrySymmetry

Electromagnetic Lagrangian remains invariant under local U(1) gauge transformation.

where the vector field transfroms and the covariant de- rivative becomes

The type of the interaction is obtained from the local phase transformation. Quantum electrodynamics is a gauge theory with U(1) phase symmetry.

O.Cakir QEDQED LagrangianLagrangian

Vector boson kinetic term Mass term

Interaction term Fermion kinetic term

O.Cakir Yang-MillsYang-Mills TeoriTeori Yang and Mills extended the local symmery to non- abelian case. Transformation matrix S has determinant 1. Lagrangian is invariant under SU(2) global phase transformation. There will be extra terms for local trans- formation, in order to remove these terms we should add extra fields and interaction terms.

Covariant derivative can be read

scalar product transforms as

Lagrangian invariant under local SU(2) gauge trans- formation

O.Cakir BrokenBroken SymmetrySymmetry

● Some symmetries are not exact - they are broken! very important feature!

● Spontaneous symmetry breaking predicts masses for fermions and W, Z gauge bosons ● Predict one more massive particle: the Higgs boson, responsible for mass generation - the only missing and the most wanted Standard Model particle!

O.Cakir SpontaneousSpontaneous SymmetrySymmetry BreakingBreaking

● φ Lagrangian for scalar field, V(V(φ))

For φ −−>-φ , Lagrangian remains φ invariant. Potential have minimum φ at φ = ± µ/λ. New variable in terms of the deviation from minimum is η= φ ± µ/λ. In this case the Lagrangian

● The new Lagrangian is not symmetrical for η−−>-η; the symmery is spontaneously broken (SSB).

O.Cakir HiggsHiggs MechanismMechanism

● Complex field

and the Lagrangian

We want it to be invariant under local gauge transformation, make a transformation that the system will be at minimum energy state

Due to choice φ '=φ and φ '=φ −µ/λ Due to choice φ1 '=φ1 and φ2 '=φ1 −µ/λ 1 1 2 φ 1 massless Goldstone boson field φ1 ' dis- massless Goldstone boson field 1 ' dis- appears and gives the mass to boson appearsφ and gives the mass to boson Before SSB After SSB A' . φ2' field (Higgs boson) has mass. A' . 2' field (Higgs boson) has mass.

O.Cakir GaugeGauge BosonBoson MassesMasses

Gauge boson masses are aobtained from the term φ 2 |Dµ | . Covariant derivative scalar field

and gauge fields mass eigenstates , ,

mass terms,

O.Cakir FermionFermion massesmasses Fermion masses is obtained from the couplings of

left-handed fermion (fL) and right-handed fermion φ (fR) with scalar field ( )

fermion masses are given in terms of Yukawa coupling and the vacuum expectation value (v=246 GeV) for top quark, for top quark, For an explanation of neut- For an explanation of neut- √ ≈ rino masses and mixings yt= √2mt/v ≈1. rino masses and mixings yt= 2mt/v 1. one needs to go beyond the one needs to go beyond the Standart Model! Standart Model! O.Cakir NeutrinosNeutrinos In the SM formulated in 1970's neutrinos are assumed to be massless, then there is only one helicity state (left-handed) for neutrinos. In 1960's Pontecorvo, Maki, Nakagaya and Sakata (PMNS) suggested that neutrinos can be produced and ν ν ν annihilated in flavor eigenstates ( e, µ, τ ), in the processes, and they can propagate in space in the ν ν ν mass eigenstates ( 1, 2, 3).

e U 11 U 12 U 13 1 =  U 21 U 22 U 23 2  U 31 U 32 U 333

O.Cakir NeutrinoNeutrino MixingMixing ν ν Considering the mixing of µ and τ in terms of ν ν 2 and 3 (atmospheric neutrinos) with one mix- ing angle θ, the wave amplitudes

=2cos 3 sin 

=−2sin 3 cos where E denotes neutrino energy, mass eigen- states depend on time

2t=20exp−i E 2t 

3t=30exp−i E3 t 

O.Cakir NeutrinoNeutrino MixingMixing -- 22 Study on an example of muon-type neutrinos in the initial state

20=0cos 

30=0sin 

time dependency  t=2tcos3tsin  Amplitude for muon-neutrinos 2 2 A t=t / 0=cos  exp−iE 2 tsin  exp−iE3 t and intensity 2 2 I t/ I  0=1−sin 2sin [E 3−E 2t /2]

O.Cakir NeutrinoNeutrino MassesMasses If neutrinos are DiracDirac particlesparticles: ● neutrino and anti-neutrino are distinct particles ● left-handed state and massless IfIf MajoranaMajorana particlesparticles: ● Particle and anti-particle are the same ν=νc.

In general, lepton masses can originate from both Dirac and Majorana mass terms.

Here m and m , Majorana masses for left-handed mL mD L R and right-handed states, respectively. m denotes m D m R D Dirac mass.

O.Cakir NeutrinoNeutrino MassesMasses -- 22

● We can diagonalize the mass matrix, in this case eigenvalues 2 2 m1,2=[mRm L±m R−mL  4 mD ]/2

here we assume that mL is much smaller; and mR=M is much greater than Dirac scale and it is around GUT scale. Physical neutrino mass is given by

m2 ThisThis mechanism mechanism (see-saw), (see-saw), taking taking D right-handed neutrino mass very large , m1≈ , m2≈M right-handed neutrino mass very large , M givesgives left-handed left-handed Majorana Majorana neutrino neutrino massmass very very small. small.

O.Cakir SMSM ParametersParameters

● 3 gauge couplings (g1, g2, g3) ● 2 Higgs parameters (µ, λ)

● 6 quark masses

p

p

a a

● F

F r

3 quark mixing angle + 1 phase r

a

a l

CKM l

a

m a

● matrix m

v

v e

3 (+3) lepton masses e

o

o

t

t

r

e r

● e r

(3 lepton mixing angles + 1 phase) r s PMNS s ()=Dirac neutrino mass case matrix

O.Cakir FlavorFlavor ProblemProblem

● Mass hierarchy Heavy quarks forming hadrons are b and c quarks. The hadrons (mesons/baryons) from these can 1.family 2.family 3.family 1.family 2.family 3.family be detected effectively.

Charged weak current leads to flavor mixing.

1/1000 Proton 1000 mass

Electroweak symmetry breaking vor Fla s nge may explain how they get their cha masses, but it does not explain what the value of their masses are.

O.Cakir

HeavyHeavy FlavorFlavor PhysicsPhysics L

-6 L

(

(

i

L i

● L ●

N

N g

m ≈3 MeV m ≤10 MeV g i

i ν1

h

e

g

h

(

u e

g

(

m

m

u

t

u

h

t

h

t

n

t

t

n

≤

t

≤

r

r

Λ

e

q

Λ

e

i q

-5 i

n

n u

● ● u

u

u

Q Q

m ≈5 MeV m ≤10 MeV o

o

t C

ν2 t

C a

d a

r

r

-

-

D

D

r

r

i

p

i

p

)

)

n

k

n

k

h

h

s

o

s

o

)

) s ● ● -4 s

ms≈100 MeV mν3 ≤10 MeV

L

L

(

(

i

i

E

E

g

g

D

D h

● ● h

M

t M

m ≈1270 MeV M ≈0.5 MeV t

c e

l

l

/

/

e

e

M

M

p

p

D

D

t

t

o

o M

● ● M

n

n V

m ≈4200 MeV V m ≈100 MeV )

µ ) s

b s

e

e

r

r

q

q

l

l

y

y

e

e

u

u

T

T

p

p

h

h

a

a a

● ● a t

t

e

e

r r

m ≈1800 MeV o u

m ≈172000 MeV o

τ u

k

a k

t a

n

n

v

v

y

y

O.Cakir CPCP ViolationViolation inin SMSM Complex coupling constants in the Lagrangian terms,

Except the charged current couplings, all couplings in the mass basis of the SM (3 families and 1 Higgs doublet) can be made real. An important property is

In the SM, 1 phase in the mixing matrix is responsible from the CP violation in weak interactions.

O.Cakir CKMCKM

CP violation is small in the SM δ ( exp=0.0001), flavor physics and CP violation re- quires precision calculations/measurements.

O.Cakir SMSM ParametersParameters Mixing matrix for the charged current W+/- interac-

tions coupling with quarks uL and dL

Standard choice of parameters

≈ ≈ Magnitude of the elements: |Vud| 0.97425, |Vus| 0.2252, | ≈ ≈ ≈ ≈ ≈ Vub| 0.00389,|Vcd| 0.230, |Vcs| 1.023, |Vcb| 0.0406, |Vtd| 0.0084, | ≈ ≈ Vts| 0.0387, |Vtb| 0.88. O.Cakir CPCP veve BAUBAU

● Baryon asymetry in the ● Mass parameter at universe (BAU) can be electroweak scale calculated from KM CP O(100 GeV), case: calculated asymmetry -17 (n -n )/n ≈n /n ∼ JP P /M12 O(10 ) is well below B B γ B γ u d the observed value ● Jarlskog parameter O(10-10). (J~O(10-5) is a ● Therefore we need parametrization of CP more sources for CP violation in quark sector. violation.

O.Cakir . HomeworkHomework 1) Classify elementary particles - for each case - accord- ing to their (a) colors, (b) electric charges, (c) isospins, (d) hypercharges, (e) masses, (f) spins.

2) For the leptonic decay of W-boson (a) define the in- variant mass of the leptonic system, (b) calculate the transverse mass of the system in terms of their mo- mentum components.

3) A π0 meson (rest mass 135 MeV/c) with momentum 270 GeV/c decays into two photons. If the mean lifetime of π0 is 8.5x10-17 sec, calculate to 10% accuracy how far the pion will travel prior to decay. What will be the minim- um value of opening angle of its two decay photons in the laboratory.

O.Cakir