Introduction to the Standard Model of particle physics

Transcription

1 Introduction to the Standard Model of particle physics I. Schienbein Univ. Grenoble Alpes/LPSC Grenoble Laboratoire de Physique Subatomique et de Cosmologie Summer School in Particle and Astroparticle physics Annecy-le-Vieux, July 2017

2 III. The Standard Model of particle physics (2nd round)

3 The general procedure Introduce Fields & Symmetries Construct a local Lagrangian density Describe Observables How to measure them? How to calculate them? Falsify: Compare theory with data

4 The general procedure Introduce Fields & Symmetries Construct a local Lagrangian density Describe Observables How to measure them? How to calculate them? Falsify: Compare theory with data

5 The general procedure Introduce Fields & Symmetries Construct a local Lagrangian density Describe Observables How to measure them? How to calculate them? Falsify: Compare theory with data

6 The general procedure Introduce Fields & Symmetries Construct a local Lagrangian density Describe Observables How to measure them? How to calculate them? Falsify: Compare theory with data

7 Fields & Symmetries

8 Matter content of the Standard Model (including the antiparticles) Q = Matter Higgs Gauge u LC B A (3, 2) 1/3 L LC B A (1, 2) -1 H = d L e h+ h 0 C A (1, 2) 1 A (1, 1) 0 u c R (3, 1) -4/3 e c R (1, 1) 2 W (1, 3) 0 d c R (3, 1) 2/3 c R (1, 1) 0 G (8, 1) 0 Q c = 0 1 uc L d c L C A (3, 2) -1/3 L c = 0 c L e c L 1 C A (1, 2) 1 H = 0 h h 0 1 C A (1, 2) 1 A (1, 1) 0 u R (3, 1) 4/3 e R (1, 1) -2 W (1, 3) 0 d R (3, 1) -2/3 R (1, 1) 0 G (8, 1) 0

9 Matter content of the Standard Model Left-handed up quark u L: LH Weyl fermion: ulα~(1/2,0) of so(1,3) a color triplet: u Li~3 of SU(3)c Indices: (ul)iα with i=1,2,3 and α=1,2 Similarly, left-handed down quark d L ul and dl components of a SU(2)L doublet: Qβ = (ul, dl) ~ 2 Q carries a hypercharge 1/3: Q ~ (3,2) 1/3 of SU(3)c x SU(2)L x U(1)Y Indices: Qβiα with β=1,2 ; i=1,2,3 and α=1,2

10 Matter content of the Standard Model There are three generations: Q k, k =1,2,3 Lot s of indices: Qkβiα(x) We know how the indices β,i,α transform under symmetry operations (i.e., which representations we have to use for the generators)

11 Matter content of the Standard Model Right-handed up quark ur: RH Weyl fermion: urα.~(0,1/2) of so(1,3) a color triplet: uri~3 of SU(3)c a singlet of SU(2)L: ur~1 (no index needed) ur carries hypercharge 4/3: ur ~ (3,1)4/3 Indices: (ur)iα. with i=1,2,3 and α.=1,2 (Note the dot) Note that ur c ~ (3*,1)-4/3

12 Matter content of the Standard Model Again there are three generations: u Rk, k =1,2,3 Lot s of indices: urkiα.(x) And so on for the other fields...

13 Terms for the Lagrangian

14 How to build Lorentz scalars? Scalar field (like the Higgs) Real field 1 µ 1 2 m2 2 Note: The mass dimension of each term in the Lagrangian has to be 4 Complex field = 1 p 2 (' 1 + i' 2 µ m 2

15 How to build Lorentz scalars? Fermions (spin 1/2) Left-handed Weyl spinor i L µ L µ =(1, i ) Right-handed Weyl spinor µ =(1, i ) i R µ R Mass term mixes left and right i µ µ L + i µ µ R m( L R + R L) Dirac spinor in chiral basis L = R i µ m with = 0 and µ = 0 µ µ 0

16 How to build Lorentz scalars? Vector boson (spin 1) U(1) gauge boson ( Photon ) 1 4 F µ F µ m2 A µ A µ where F µ µ A µ Mass term allowed by Lorentz invariance; forbidden by gauge invariance In principle, there is a second invariant 1 4 F µ e F µ with e Fµ = 1 2 µ F F F / ~ E ~B Violates Parity, Time reversal, and CP symmetry; prop. to a total divergence doesn t contribute in QED BUT strong CP problem in QCD

17 Gauge symmetry Idea: Generate interactions from free Lagrangian by imposing local (i.e. α = α(x)) symmetries Does not fall from heavens; generalization of minimal coupling in electrodynamics Final judge is experiment: It works

18 Local gauge invariance for a complex scalar µ m 2 is invariant under e i. What if now α = α(x) depends on the µ (e i (x) µ (e i (x) ) m 2 (e i (x) ) (e i (x) ) =[@ µ e i (x) + e i µ ] [@ µ e i (x) + e i µ ] m 2 =[ie i µ (x) + e i µ ] [ie i µ (x) + e i µ ] m 2 =[ ie i µ (x) + e i µ ][ie i µ (x) + e i µ ] m 2 = ie i µ (x) ie i µ (x) ie i µ (x) e i µ + e i µ ie i µ (x) + e i µ e i µ m 2 µ m 2 +non-zeroterms Not invariant under U(1)

19 Local gauge invariance for a complex scalar field Can we find a derivative operator that commutes with the gauge transformation? Define D µ µ + ia µ, where the gauge field A µ transforms as A µ A µ D µ (@ µ + i[a µ (x)])[e i (x) ] µ [e i (x) ]+i[a µ (x)][e i (x) ] = ie i µ (x) + e i µ + ia µ e i (x) i@ µ (x)e i (x) = e i µ + ia µ e i (x) = e i (x) [@ µ + ia µ ] = e i (x) D µ Nota bene: We call D µ the covariant derivative, because it transforms just like itself: e i (x) and D µ e i (x) D µ D µ D µ m 2 e i (x) D µ e i (x) D µ m 2 e i (x) e i (x) = D µ D µ m 2

20 Local gauge invariance for a complex scalar field Can we find a derivative operator that commutes with the gauge transformation? Define D µ µ + ia µ, where the gauge field A µ transforms as A µ A µ D µ (@ µ + i[a µ (x)])[e i (x) ] µ [e i (x) ]+i[a µ (x)][e i (x) ] = ie i µ (x) + e i µ + ia µ e i (x) i@ µ (x)e i (x) = e i µ + ia µ e i (x) = e i (x) [@ µ + ia µ ] = e i (x) D µ Nota bene: We call D µ the covariant derivative, because it transforms just like itself: e i (x) and D µ e i (x) D µ D µ D µ m 2 e i (x) D µ e i (x) D µ m 2 e i (x) e i (x) = D µ D µ m 2

21 Local gauge invariance for a complex scalar field Can we find a derivative operator that commutes with the gauge transformation? Define D µ µ + ia µ, where the gauge field A µ transforms as A µ A µ D µ (@ µ + i[a µ (x)])[e i (x) ] µ [e i (x) ]+i[a µ (x)][e i (x) ] = ie i µ (x) + e i µ + ia µ e i (x) i@ µ (x)e i (x) = e i µ + ia µ e i (x) = e i (x) [@ µ + ia µ ] = e i (x) D µ Nota bene: We call D µ the covariant derivative, because it transforms just like itself: e i (x) and D µ e i (x) D µ D µ D µ m 2 e i (x) D µ e i (x) D µ m 2 e i (x) e i (x) = D µ D µ m 2

22 Local gauge invariance for a complex scalar field Can we find a derivative operator that commutes with the gauge transformation? Define D µ µ + ia µ, where the gauge field A µ transforms as A µ A µ D µ (@ µ + i[a µ (x)])[e i (x) ] µ [e i (x) ]+i[a µ (x)][e i (x) ] = ie i µ (x) + e i µ + ia µ e i (x) i@ µ (x)e i (x) = e i µ + ia µ e i (x) = e i (x) [@ µ + ia µ ] = e i (x) D µ Nota bene: We call D µ the covariant derivative, because it transforms just like itself: e i (x) and D µ e i (x) D µ D µ D µ m 2 e i (x) D µ e i (x) D µ m 2 e i (x) e i (x) = D µ D µ m 2

23 Local gauge invariance Expanding the Lagrangian for a complex scalar field D µ D µ m 2 invariant under local U(1) transformations D µ D µ m 2 µ + ia µ µ )+ A µ A µ m 2 Demand symmetry Generate interactions Generated mass for gauge boson (after acquires a vacuum expectation value) Explicit mass term forbidden by gauge symmetry (although otherwise allowed): m 2 A µ A µ m 2 (A µ )(A µ ) 6= m 2 A µ A µ (2.20) Simplest form of Higgs mechanism Vector-scalar-scalar interaction

24 Local gauge invariance Non-Abelian gauge symmetry for a complex scalar field Abelian Non-Abelian: component notation Non-Abelian: vector notation U = e i (x) U = e i a (x)t a R U = e i a (x)t a R U i U i k k U A µ A a µtr a A µ A µ A µ A a µt a UA a µt a U i (@ g µu)u A µ UA µ U i (@ g µu)u F µ µ A µ Fµ a µ A A a µ gf abc A b µa c F µ µ A µ + ig[a µ, A ] F µ F µ F µ UF µ U F µ invariant Fµ F a aµ invariant Tr(F µ F µ )invariant D µ µ + iga a µt a R

25 Conjecture All fundamental internal symmetries are gauge symmetries. Global symmetries are just accidental and not exact.

26 Spontaneous Symmetry Breaking

27 One page summary of the world Gauge group SU(3) c SU(2) L U(1) Y Particle content Q = 0 C A d L 1 C A Matter Higgs Gauge u LC B A (3, 2) 1/3 L LC B A (1, 2) -1 H = e h+ h 0 C A C A (1, 2) 1 B (1, 1) 0 u c R (3, 1) -4/3 e c R (1, 1) 2 W (1, 3) 0 d c R (3, 1) 2/3 c R (1, 1) 0 G (8, 1) 0 Lagrangian (Lorentz + gauge + renormalizable) SSB L = 1 4 G µ G µ +...Q k /DQ k +...(D µ H) (D µ H) µ 2 H H 4 (H H) Y k`q k H(u R )` H H 0 + p v SU(2) L U(1) Y U(1) Q B,W 3,Z 0 and W 1 µ,w 2 µ W +,W Fermions acquire mass through Yukawa couplings to Higgs

28 The Higgs mechanism The Higgs potential: V =μ 2 ϕ ϕ + λ (ϕ ϕ) 2 Vacuum = Ground state = Minimum of V: If μ 2 >0 (massive particle): ϕmin = 0 (no symmetry breaking) If μ 2 <0: ϕmin = ±v = ±(-μ 2 /λ) 1/2 These two minima in one dimension correspond to a continuum of minimum values in SU(2). The point ϕ = 0 is now instable. Choosing the minimum (e.g. at +v) gives the vacuum a preferred direction in isospin space spontaneous symmetry breaking Perform perturbation around the minimum

29 Higgs self-couplings In the SM, the Higgs self-couplings are a consequence of the Higgs potential after expansion of the Higgs field H~(1,2)1 around the vacuum expectation value which breaks the ew symmetry: V H = µ 2 H H + (H H) m2 hh 2 + r 2 m hh h4 with: m 2 h =2 v 2,v 2 = µ 2 / Note: v=246 GeV is fixed by the precision measures of GF In order to completely reconstruct the Higgs potential, on has to: Measure the 3h-vertex: via a measurement of Higgs pair production SM 3h = r 2 m h h h h h Measure the 4h-vertex: more difficult, not accessible at the LHC in the high-lumi phase h h h

30 Higgs self-couplings In the SM, the Higgs self-couplings are a consequence of the Higgs potential after expansion of the Higgs field H~(1,2)1 around the vacuum expectation value which breaks the ew symmetry: V H = µ 2 H H + (H H) m2 hh 2 + r 2 m hh h4 with: m 2 h =2 v 2,v 2 = µ 2 / Note: v=246 GeV is fixed by the Measuring the 3h-couplings: In order to completely reconstruct the Higgs potential, on has to: Measure the 3h-vertex: via a measurement of Higgs pair production SM 3h = major goal for the high-lumi phase r 2 m h at the LHC The Higgs particle is just the messenger Need to reconstruct the potential Measure the 4h-vertex: more difficult, not accessible at the LHC in the high-lumi phase precision measures of GF h h h h h h h

31 Not so compact anymore L SM = i ψ l γ µ µ ψ l + l=e,µ,τ l =ν e,ν µ,ν τ i ψ l γ µ µ ψ l + 3 i ψ qi γ µ µ ψ qi + i q=u,c,t 3 i ψ q i γ µ µ ψ q i i q =d,s,b 1 2 ( µw + ν νw + µ )( µ W ν ν W µ ) 1 4 ( µz ν ν Z µ )( µ Z ν ν Z µ ) 1 4 ( µa ν ν A µ )( µ A ν ν A µ ) 1 4 l=e,µ,τ λ l v 2 ψl ψ l 3 i q=u,c,t 8 ( µ G a ν νg a µ )( µ G aν ν G aµ )+ 1 2 µh µ h a=1 λ q v 2 ψqi ψ qi 3 i q =d,s,b λ q v 2 ψq i ψ q i ( gv ) 2 W + µ 2 W µ 1 ( ) 2 gv Z µ Z µ 1 ( ) 2m 2 h 2 2 2cosθ W 2 g + ψ l γ µ (4 sin 2 θ W 1+γ 5 )ψ l Z µ + ψl γ µ (1 γ 5 )ψ l Z µ 4cosθ W l=e,µ,τ l =ν e,ν µ,ν τ ( 3 ) g + ψ qi γ µ ( cosθ W 3 sin2 θ W γ 5 )ψ qi Z µ + ψ q i γ µ ( 4 3 sin2 θ W 1+γ 5 )ψ q i Z µ i q=u,c,t i q =b,s,b ( + g 2 ψ νl γ µ (1 γ 5 )ψ l W µ + + ) ψ l γ µ (1 γ 5 )ψ νl Wµ 2 l=e,µ,τ l=e,µ,τ + g V qq ψq γ µ (1 γ 5 )ψ q i W µ + + Vqq ψ q i γ µ (1 γ 5 )ψ qi Wµ +g em ( +g s ( 3 i,j l=e,µ,τ i l=e,µ,τ 8 a q=u,c,t q =d,s,b ψ l γ µ ψ l A µ q=u,c,t λ l 2 ψl ψ l h q=u,c,t ψ qj γ µ ψ qi G a µt a ij + 3 i q=u,c,t ψ qi γ µ ψ qi A µ i,j 8 a λ q 2 ψqi ψ qi h i q =d,s,b 3 i q=u,c,t q =d,s,b q =d,s,b q =d,s,b ψ q j γ µ ψ q i G a µt a ij ψ q i γ µ ψ q i A µ ) λ q 2 ψq i ψ q i h +ig em [ µ A ν W µ W +ν + µ W + ν W ν A µ + µ W ν W +µ A ν µ A ν W ν W +µ µ W + ν W µ A ν µ W ν W +ν A µ] +ig cos θ W [ µ Z ν W µ W +ν + µ W + ν W ν Z µ + µ W ν W +µ Z ν µ Z ν W ν W +µ µ W ν + W µ Z ν µ Wν W +ν Z µ] + g2 v 2 W µ + W µ h + g2 v Z 4cos 2 µ Z µ h λvh 3 θ [ W +gem 2 W + ν W µ A ν A µ W µ + W µ A ν A ν] [ + g 2 cos 2 θ W W + ν W µ Z ν Z µ W µ + W µ Z ν Z ν] [ +g 2 cos θ W sin θ W 2W + µ W µ Z ν A ν W µ + W ν A ν Z µ W µ + W ] ν A µ Z ν + g2 [ W 2 µ W µ W ν + W +ν Wµ W +µ Wν W +ν] + g2 4 W µ + W µ h 2 g 2 + Z 8cos 2 µ Z µ h 2 λ θ W 4 h4 8 8 f abc f ade G b µ Gc ν Gµd G νe g s 2 a,b,c f abc ( µ G aν ν G a µ )Gµb G νc g2 s 4 a,b,c d,e,f ) g em = g sin θ W, v 2 = m2 λ (m 2 < 0,λ>0), m l = λ lv 2,m q = 3λqv 2, m W = gv 2,m Z = gv 2cosθ W, m h = 2m 2

32 IV. From the SM to predictions at the LHC

33 Scattering theory Cross sections can be calculated as = 1 Z dps (n) 2 M fi F We integrate over all final state configurations (momenta, etc.). The phase space (dps) only depend on the final state particle momenta and masses Purely kinematical We average over all initial state configurations This is accounted for by the flux factor F Purely kinematical The matrix element squared contains the physics model Can be calculated from Feynman diagrams Feynman diagrams can be drawn from the Lagrangian The Lagrangian contains all the model information (particles, interactions)

34 Cross section The differential cross section: d = 1 F M 2 d n The Lorentz-invariant phase space: d n =(2 ) 4 (4) (p a + p b n X q f=1 p f ) ny f=1 d 3 p f (2 ) 3 2E f The flux factor: F = (p a p b ) 2 p 2 ap 2 b

35 Decay width The differential decay width: d = 1 2E a M 2 d n The Lorentz-invariant phase space: d n =(2 ) 4 (4) X n (p a p f ) ny f=1 f=1 d 3 p f (2 ) 3 2E f Rest frame of decaying particle: E a = M a

36 Life time and branching ratio Life time: =1/ Branching ratio: BR(i f) = (i f) (i all)

37 The model All the model information is included in the Lagrangian Before electroweak symmetry breaking: very compact L = 1 4 B µ B µ 1 4 W µ W i µ i 3X + + f=1 3X f=1 After electroweak symmetry breaking: quite large Example: electroweak boson interactions with the Higgs boson: 1 4 Ga µ G µ a h Lf i µ D µ L f +ē Rf i µ D µ e f R h Qf i µ D µ Q f +ū Rf i µ D µ u fr + d i Rf i µ D µ d f R + D µ ' D µ ' V (') i

38 Feynman diagrams and Feynman rules I Diagrammatic representation of the Lagrangian Electron-positron-photon (q = -1) e + From the Lagrangian e Muon-antimuon-photon (q = -1) µ + µ The Feymman rules are the building blocks to construct Feynman diagrams...

39 Loop diagrams Loops exist, but their contribution is often small two interactions... four interactions

40 Feynman diagrams and Feynman rules II From Feynman diagrams to Mfi : External particles im fi = Interactions Propagator We construct all possible diagrams with the set of rules at our disposal We can then calculate the squared matrix element and get the cross section e + e µ + µ Feynman rules h i v sa (p a )( ie µ i h i µ ) u sb (p b ) (p a + p b ) 2 ū s2 (p 2 )( ie ) v s1 (p 1 )

41 Feynman rules for the Standard Model Z QED QED qq qqz l l llz W W W W Z Almost all the building blocks necessary to draw any SM diagrams W +- g QED QCD qq W qqg l W ggg WWWW gggg QCD coupling much stronger than QED coupling dominant diagrams h QED (m) qqh llh W W h ZZh Partial li

42 Drawing Feynman diagrams I Z W +- QED QED QED qq qqz qq W l l llz l W W W W W Z WWWW We can now combine building blocks to draw diagrams This ensures to focus only on the allowed diagrams We must only consider the dominant diagrams Process 0. uū t t g QCD qqg ggg gggg QCD 2 h QED (m) qqh llh W W h ZZh Partial li QED 2 (subdominant)

43 Drawing Feynman diagrams II Z W +- g h QED QED QED QCD QED (m) qq qqz qq W qqh qqg l l llz l W llh W W W W ggg Z W W h WWWW gggg ZZh Partial li Find out the dominant diagrams for Process 1. gg t t Process 2. gg t th Process 3. uū t t b b What is the QCD/QED order? (keep only the dominant diagrams)

44 Check your answer online: Requires registration

45 Web process syntax Initial state u u~ > b b~ t t~ Final state u u~ > b b~ t t~ QED=2 Minimal coupling order u u~ > h > b b~ t t~ Required intermediate particles Excluded particles u u~ > b b~ t t~ / z a u u~ > b b~ t t~, t~ > w- b~ Specific decay chain

46 MadGraph output User requests a process g g > t t~ b b~ u d~ > w+ z, w+ > e+ ve, z > b b~ etc. 1 d~ w+ u diagram 1 b~ z 6 w+ ve 4 e+ 3 QCD=0, QED=4 1 u d diagram 2 z w+ b~ 6 ve 4 e+ 3 QCD=0, QED= b d~ b : SUBROUTINE SMATRIX(P1,ANS) C C Generated by MadGraph II Version Updated 06/13/05 C RETURNS AMPLITUDE SQUARED SUMMED/AVG OVER COLORS C AND HELICITIES C FOR THE POINT IN PHASE SPACE P(0:3,NEXTERNAL) C C FOR PROCESS : g g -> t t~ b b~ C C Crossing 1 is g g -> t t~ b b~ IMPLICIT NONE C C CONSTANTS C Include "genps.inc" INTEGER NCOMB, NCROSS PARAMETER ( NCOMB= 64, NCROSS= 1) INTEGER THEL PARAMETER (THEL=NCOMB*NCROSS) C C ARGUMENTS C REAL*8 P1(0:3,NEXTERNAL),ANS(NCROSS) C 2 1 u d~ u diagram 3 w+ z ve b e+ b~ QCD=0, QED=4 MADGRAPH returns: Feynman diagrams Self-contained Fortran code for Mf i 2 Still needed: What to do with a Fortran code? How to deal with hadron colliders?

47 Proton-Proton collisions I The master formula for hadron colliders = 1 X Z F We sum over all proton constituents (a and b here) We include the parton densities (the f-function) ab dps (n) dx a dx b f a/p (x a ) f b/p (x b ) M fi 2 a b They represent the probability of having a parton a inside the proton carrying a fraction xa of the proton momentum

48 PDFs: x-dependence Valence quarks p= uud Up Down

49 PDFs: x-dependence Gluon Valence quarks p= uud Gluons carry about 40% of momentum

50 PDFs: x-dependence Valence quarks p= uud Gluons carry about 40% of momentum Sea Sea quarks light quark sea, strange sea

51 PDFs: Q-dependence Altarelli-Parisi evolution equations Valence quarks p= uud Gluons carry about 40% of momentum Sea Sea quarks light quark sea, strange sea

52 PDFs: Q-dependence Altarelli-Parisi evolution equations Valence quarks p= uud Gluons carry about 40% of momentum Sea quarks light quark sea, strange sea

53 PDFs: Q-dependence Altarelli-Parisi evolution equations Valence quarks p= uud Gluons carry about 40% of momentum Sea quarks light quark sea, strange sea

54 PDFs: Q-dependence Altarelli-Parisi evolution equations Valence quarks p= uud Gluons carry about 40% of momentum Sea quarks light quark sea, strange sea

55 PDFs: Q-dependence Altarelli-Parisi evolution equations Valence quarks p= uud Gluons carry about 40% of momentum Sea quarks light quark sea, strange sea

56 Proton-Proton collisions II This is not the end of the story... At high energies, initial and final state quarks and gluons radiate other quark and gluons The radiated partons radiate themselves And so on... Radiated partons hadronize We observe hadrons in detectors

57 Input parameters The SM Lagrangian has 26 input parameters (of course not all are equally important) They need to be fixed in order to make predictions The values and patterns of these parameters are quite bizarre

58 Quantum Corrections Quantum corrections have to be considered (otherwise some predictions very rough) UV divergences appear Renormalization of Lagrangian parameters and fields This leads to running parameters Scale-dependence governed by renormalization group equations (RGEs)

59 Asymptotic Freedom Renormalization of UV-divergences: Running coupling constant a s := α s /(4π) Gross, Wilczek ( 73); Politzer ( 73) a s (µ) = 1 β 0 ln(µ 2 /Λ 2 ) 0.4 α s (µ) NLO, MSbar upper: α s (M Z )=0.121 α s (M Z )= lower: α s (M Z )= α s (M Z )=0.118 Non-abelian gauge theories: negative beta-functions da s d ln µ 2 = β 0a 2 s where β 0 = 11 3 C A 2 3 n f µ (GeV) asympt. freedom: a s for µ Nobel Prize 2004