Summary SM Lagrangian [1. Generation Leptons]

Teilchenphysik 2 — W/Z/Higgs an Collidern

Sommersemester 2019

Matthias Schr ¨oder und Roger Wolf | Vorlesung 6

INSTITUT FUR¨ EXPERIMENTELLETEILCHENPHYSIK(ETP)

Summary SM Lagrangian [1. Generation Leptons]

L

= L

+ L

+ L

+ L

+ L

+ L

+ L

L

= i νγ

∂

ν + ie γ

∂

e L

= −

2sinθ_W

(νγ

e

) W

+ ( e

γ

ν) W

L

= −

WcosθW

[(νγ

ν) + ( e

γ

e

)] Z

+ e ( e γ

e ) [ A

+ tan θ

Z

] L

= −

W

W

−

B

B

L

=

(∂

H ) (∂

H ) + m

W

W

+

m

Z

Z

+ 2

2 W

v

HW

W

+

HZ

Z

+

v²

H

W

W

+

v²

H

Z

Z

L

= −

m

H

+

2 H

2v

H

−

H

L

= − m

ee −

Hee

Summary SM Lagrangian

Programme

3. From Theory to Experiment (and Back)

3.1 From theory to observables

◦ Cross-section calculation: basic picture

◦ Fermion propagator and perturbation theory

◦ Scattering matrix and Feynman rules 3.2 Reconstruction of experimental data

◦ Reminder: accelerators and particle detectors

◦ Trigger

◦ Reconstruction of physics objects 3.3 Measurements in particle physics

◦ Basic tools (PDFs, Histograms, Likelihood)

◦ Parameter estimation

◦ Hypothesis testing

◦ Determination of physics properties (confidence intervals)

◦ Search for new physics (exclusion limits) 3.4 Experimental techniques

◦ Efficiency measurements

◦ Background estimation

3. From Theory to Experiment (and Back)

3.1 From theory to observables

◦ Cross-section calculation: basic picture

◦ Fermion propagator and perturbation theory

◦ Scattering matrix and Feynman rules 3.2 Reconstruction of experimental data

◦ Reminder: accelerators and particle detectors

◦ Trigger

◦ Reconstruction of physics objects 3.3 Measurements in particle physics

◦ Basic tools (PDFs, Histograms, Likelihood)

◦ Parameter estimation

◦ Hypothesis testing

◦ Determination of physics properties (confidence intervals)

◦ Search for new physics (exclusion limits) 3.4 Experimental techniques

◦ Efficiency measurements

◦ Background estimation

3.1 From theory to observables

From Theory to Observables

Lagrange density (“Lagrangian”)

L = ψ i γ ^µ ∂ µ ψ − ^m ψψ

↓ Euler–Lagrange eq. ← ^{from d S = 0}

eq. of motion

i γ ^µ ∂ µ ψ − ^m ψ = 0

↓

quantum-mechanical state ψ (e. g. plane wave)

↓

observables

3.1.1. Cross-section calculation: basic picture

Cross Section

◦ ^{Measure of} transition rate initial → ^final state for given process

◦ Follows from Fermi’s golden rule:

σ = | matrix element | ² · phase space flux of colliding particles

◦ matrix element: probability amplitude, encodes process dynamics

◦ phase space: number of available final states

◦ Link between

◦ theory: compute cross section

◦ experiment: measure cross section

Model for Particle Scattering

Scattering matrix S ^transforms initial state ψ i into scattering wave

ψ

= S · ψ i

Observation: projection of plane wave ψ

out of

scattering wave ψ

initial particle:

plane wave ψ

localised potential A

spherical scattering wave ψ

Transition probability amplitude S

= ψ _f ^† · ψ

= ψ _f ^† · S · ψ i

Example: QED

◦ Electron in electromagnetic field L = ψ i γ ^µ ∂ µ ψ − ^q ψγ ^µ A _µ ψ

| {z }

covariant deriv.

− ^m ψψ

| {z }

Yukawa coupl.

Euler–Lagrange

−→ eqs. ( i γ ^µ ∂ _µ − ^m ) ψ = q γ ^µ A _µ ψ inhomogeneous Dirac eq.

◦ ^{Formal solution} of inhomogeneous Dirac equation ψ( x ) = ψ

( x ) + q

Z

d ⁴ x ⁰ K ( x , x ⁰ )γ ^µ A _µ ( x ⁰ )ψ( x ⁰ ) with

Plane wave (free electron) ψ

: ( i γ ^µ ∂ µ − ^m )ψ 0 ( x ) = 0

Green’s function K : ( i γ ^µ ∂ _µ − ^m ) K ( x , x ⁰ ) = δ ⁴ ( x − ^{x 0} )

Solution of Inhomogeneous Dirac Equation

◦ The function

ψ( x ) = ψ

( x ) + q Z

d

x

K ( x , x

)γ

A

( x

)ψ( x

)

is a formal solution of the inhomogeneous Dirac equation:

( i γ

∂

− m ) ψ( x ) = ( i γ

∂

− m ) ψ

( x )

| {z }

=0

+ q R

d

x

( i γ

∂

− m ) K ( x , x

)

| {z }

=δ⁴(x−x⁰)

γ

A

( x

)ψ( x

)

= q γ

A

( x )ψ( x ) X

∂µacts onx, not onx⁰

But not a direct solution: ψ appears on LHS and RHS Turns differential equation into integral equation:

propagates the solution from x ⁰ to x

3.1.2. Fermion propagator and perturbation theory

Green’s Function K

◦ Best way to find Green’s function K is via its Fourier transform _K ˜

K ( x , x ⁰ ) ≡ ^K ( x − ^{x 0} ) = _{( 2} ¹ _π)

Z

d ⁴ p _K ˜ ( p ) e ⁻

⁽

⁻

⁾

◦ Applying Dirac equation:

( i γ ^µ ∂ _µ − ^m ) K ( x − ^{x 0} )

| {z }

k per definition

= _{( 2} ¹ _π)

R

d ⁴ p (γ ^µ p _µ − ^m ) ˜ K ( p )

| {z }

k

e ⁻

⁽

⁻

⁾

δ ⁴ ( x − ^{x 0} ) = _{( 2} ¹ _π)

R d ⁴ p 1

e ⁻

⁽

⁻

⁾

Thus it is: (γ ^µ p _µ − ^m ) ˜ K ( p ) = 1

Fermion Propagator

◦ Fourier transform of the Green’s function is called fermion propagator (γ ^µ p _µ − ^m ) ˜ K ( p ) = 1 4

(γ ^µ p _µ + m ) · (γ ^µ p _µ − ^m ) ˜ K ( p ) = (γ ^µ p _µ + m ) · 1 4

K ˜ ( p ) = (γ ^µ p _µ + m ) p ² − ^{m 2}

◦ Fermion propagator is a 4 × 4 matrix, acts in the spinor space

◦ Only defined for virtual fermions since p

− ^m

= E

− ~ p

− ^m

6 = 0

Green’s Function ↔ Fermion Propagator

◦ The Green’s function can be obtained from the propagator by an inverse Fourier transformation

K ( x − ^{x 0} ) = _{( 2 π)} ¹

R d ³ ~ p e ⁻

^~

^(~

^−~

⁾ R +∞

−∞ dp 0 (γ

+

)

p²

−

e ⁻

⁽

⁻

⁾

Green’s Function ↔ Fermion Propagator

◦ The Green’s function can be obtained from the propagator by an inverse Fourier transformation

K ( x − ^{x 0} ) = _{( 2 π)} ¹

R d ³ ~ p e ⁻

^~

^(~

^−~

⁾ R +∞

−∞ dp 0 (γ

+

)

(

−

)(

+

) e ⁻

⁽

⁻

⁾

↓

E ² = ~ p ² + m ²

◦ ^K ( x − ^{x 0} ) has 2 poles in the integration plane at p

= ± ^E

◦ Can be solved with methods of function theory (see e. g. Schm ¨user)

◦ Correct expression for the fermion propagator ( infinitesimal):

K ˜ ( p ) = (γ ^µ p _µ + m )

p ² − ^{m 2} + i > 0

Green’s Function

(see e. g. Schm ¨user)

◦ ^{For t} > t ⁰ (forward evolution) K ( x − ^{x 0} ) = _{( 2} ⁻ _π)

Z

d ³ ~ p +γ ⁰ E − ~γ~ p + m

2E e ⁻

⁽

⁻

⁾⁺

^~

^(~

^−~

⁾

◦ ^{For t} < t ⁰ (backward evolution) K ( x − ^{x 0} ) = _{( 2} ⁻ _π)

Z

d ³ ~ p − γ ⁰ E − ~γ~ p + m

2E e ⁺

⁽

⁻

⁾⁺

^~

^(~

^−~

⁾

Propagator and Time Evolution

(see e. g. Schm ¨user)

◦ ^{K describes} time evolution of free fermion

◦ General solution to Dirac equation ψ( t , ~ x ) = i

Z

d ³ ~ x ⁰ K ( x − ^{x 0} )γ ⁰ ψ( t ⁰ , ~ x ⁰ ) for t > t ⁰ particle with

> 0 going forward in time

ψ( t , ~ x ) = i Z

d ³ ~ x ⁰ ψ( t ⁰ , ~ x ⁰ )γ ⁰ K ( x − ^{x 0} ) for t > t ⁰ particle with

< 0

going backward in time

Solution of Inhomogeneous Dirac Equation

◦ The function

ψ( x ) = ψ

( x ) + q Z

d

x

K ( x , x

)γ

A

( x

)ψ( x

)

is a formal solution of the inhomogeneous Dirac equation

But not a direct solution: ψ appears on LHS and RHS Turns differential equation into integral equation:

propagates the solution from x ⁰ to x

The Perturbative Series

◦ Integral equation can be solved iteratively by expansion in coupling ψ( x ) = ψ

( x ) + q

Z

d

x

K ( x , x

)γ

A

( x

)ψ( x

)

ψ

( x ) = ψ

( x )

◦ 0th order: neglect external field (no scattering)

The Perturbative Series

◦ Integral equation can be solved iteratively by expansion in coupling ψ( x ) = ψ

( x ) + q

Z

d

x

K ( x , x

)γ

A

( x

)ψ( x

)

ψ

( x ) = ψ

( x )

ψ

( x ) = ψ

( x ) + q R

d

x

K ( x , x

) γ

A

( x

) ψ

( x

)

◦ 1st order: assume ψ ^{( 0 )} ( x ) is close to actual solution

The Perturbative Series

◦ Integral equation can be solved iteratively by expansion in coupling ψ( x ) = ψ

( x ) + q

Z

d

x

K ( x , x

)γ

A

( x

)ψ( x

)

ψ

( x ) = ψ

( x )

ψ

( x ) = ψ

( x ) + q R

d

x

K ( x , x

) γ

A

( x

) ψ

( x

)

ψ

( x ) = ψ

( x ) + q R

d

x

K ( x , x

)γ

A

( x

)ψ

( x

)

◦ ^{2nd order:} ψ ^{( 1 )} ( x ) as better approximation at RHS

The Perturbative Series

◦ Integral equation can be solved iteratively by expansion in coupling ψ( x ) = ψ

( x ) + q

Z

d

x

K ( x , x

)γ

A

( x

)ψ( x

)

ψ

( x ) = ψ

( x )

ψ

( x ) = ψ

( x ) + q R

d

x

K ( x , x

) γ

A

( x

) ψ

( x

)

ψ

( x ) = ψ

( x ) + q R

d

x

K ( x , x

)γ

A

( x

)ψ

( x

) + q

R R

d

x

d

x

K ( x , x

)γ

A

( x

) K ( x

, x

)γ

A

( x

)ψ

( x

)

◦ ^{2nd order:} ψ ^{( 1 )} ( x ) as better approximation at RHS

The Perturbative Series

◦ Integral equation can be solved iteratively by expansion in coupling ψ( x ) = ψ

( x ) + q

Z

d

x

K ( x , x

)γ

A

( x

)ψ( x

)

ψ

( x ) = ψ

( x )

ψ

( x ) = ψ

( x ) + q R

d

x

K ( x , x

) γ

A

( x

) ψ

( x

)

ψ

( x ) = ψ

( x ) + q R

d

x

K ( x , x

)γ

A

( x

)ψ

( x

) + q

R R

d

x

d

x

K ( x , x

)γ

A

( x

) K ( x

, x

)γ

A

( x

)ψ

( x

)

◦ ^{Terms in} ψ ^{( 2 )} ( x ) correspond to 0, 1, 2 scatterings at potential A _µ

The Perturbative Series

◦ Integral equation can be solved iteratively by expansion in coupling ψ( x ) = ψ

( x ) + q

Z

d

x

K ( x , x

)γ

A

( x

)ψ( x

)

ψ

( x ) = ψ

( x )

ψ

( x ) = ψ

( x ) + q R

d

x

K ( x , x

) γ

A

( x

) ψ

( x

)

ψ

( x ) = ψ

( x ) + q R

d

x

K ( x , x

)γ

A

( x

)ψ

( x

) + q

R R

d

x

d

x

K ( x , x

)γ

A

( x

) K ( x

, x

)γ

A

( x

)ψ

( x

)

◦ RHS in inhomogeneous Dirac equation treated as small “perturbation”

The Perturbative Series

◦ Integral equation can be solved iteratively by expansion in coupling ψ( x ) = ψ

( x ) + q

Z

d

x

K ( x , x

)γ

A

( x

)ψ( x

)

ψ

( x ) = ψ

( x )

ψ

( x ) = ψ

( x ) + q R

d

x

K ( x , x

) γ

A

( x

) ψ

( x

)

ψ

( x ) = ψ

( x ) + q R

d

x

K ( x , x

)γ

A

( x

)ψ

( x

) + q

R R

d

x

d

x

K ( x , x

)γ

A

( x

) K ( x

, x

)γ

A

( x

)ψ

( x

)

◦ Expansion in coupling justified since q = e = √

4 πα em ¹

3.1.3. Scattering matrix and Feynman rules

The Matrix Element S ^fi

◦ Scattering matrix S transforms initial state ψ i into scattered state ψ scat

◦ Matrix element S

(“scattering amplitude”) given by projection of final state ψ f out of ψ scat

S ^fi = R

d ³ x f ψ _f ^† ( x f ) ψ scat ( x f )

= R

d ³ x _f ψ _f ^† ( x _f )

ψ i ( x _f ) + q R

d ⁴ x ⁰ K ( x _f , x ⁰ )γ ^µ A _µ ( x ⁰ )ψ i ( x ⁰ ) + . . .

= δ fi + S

⁽

⁾ + S

⁽

⁾ + . . .

◦ δ

: no scattering (undisturbed initial state)

◦ S

: one scattering process, “leading order” (LO)

◦ S

: two scattering processes, “next-to-leading order” (NLO)

LO Matrix-Element S fi ^{( 1 )}

◦ Matrix element at 1st order perturbation theory S

= q

Z

d

x

ψ

( x

) Z

d

x

K ( x

, x

)γ

A

( x

)ψ

( x

)

LO Matrix-Element S fi ^{( 1 )}

◦ Matrix element at 1st order perturbation theory S

= q

Z d

x

Z

d

x

ψ

( x

) K ( x

, x

)

| {z }

=−iψf(x⁰)(see 3.1.2)

γ

A

( x

)ψ

( x

)

◦ ^{Propagator K} ( x f , x ⁰ ) extrapolates the state ψ f measured in the detector at x f = ( t f , ~ x f ) back to the scattering target at x ⁰ = ( t ⁰ , ~ x ⁰ )

S fi ^{( 1 )} = − ^iq Z

d ⁴ x ⁰ ψ _f ( x ⁰ )γ ^µ ψ i ( x ⁰ ) A _µ ( x ⁰ )

LO Matrix-Element S fi ^{( 1 )}

◦ Matrix element at 1st order perturbation theory S

= q

Z d

x

Z

d

x

ψ

( x

) K ( x

, x

)

| {z }

=−iψf(x⁰)

γ

A

( x

)ψ

( x

)

◦ ^{Propagator K} ( x f , x ⁰ ) extrapolates the state ψ f measured in the detector at x f = ( t f , ~ x f ) back to the scattering target at x ⁰ = ( t ⁰ , ~ x ⁰ )

S fi ^{( 1 )} = − ^iq Z

d ⁴ x ⁰ ψ

( x ⁰ )γ ^µ ψ

( x ⁰ ) A _µ ( x ⁰ )

Corresponds exactly to the IA term in L (see lecture 2)

ψf

−iqγ^µ

LO Matrix-Element S fi ^{( 1 )}

◦ Matrix element at 1st order perturbation theory S

= q

Z d

x

Z

d

x

ψ

( x

) K ( x

, x

)

| {z }

=−iψf(x⁰)

γ

A

( x

)ψ

( x

)

◦ ^{Propagator K} ( x f , x ⁰ ) extrapolates the state ψ f measured in the detector at x f = ( t f , ~ x f ) back to the scattering target at x ⁰ = ( t ⁰ , ~ x ⁰ )

S fi ^{( 1 )} = − ^iq Z

d ⁴ x ⁰ ψ

( x ⁰ )γ ^µ ψ

( x ⁰ ) A _µ ( x ⁰ )

◦ Incoming fermion ψ

◦ Outgoing fermion ψ

◦ Interaction with potential A

at x

◦ LO matrix-element: sum of contributions at all x

Aµ

ψi

ψf

−iqγ^µ

LO Matrix-Element S fi ^{( 1 )}

◦ Matrix element at 1st order perturbation theory S

= q

Z d

x

Z

d

x

ψ

( x

) K ( x

, x

)

| {z }

=−iψf(x⁰)

γ

A

( x

)ψ

( x

)

◦ ^{Propagator K} ( x f , x ⁰ ) extrapolates the state ψ f measured in the detector at x f = ( t f , ~ x f ) back to the scattering target at x ⁰ = ( t ⁰ , ~ x ⁰ )

S fi ^{( 1 )} = − ^iq Z

d ⁴ x ⁰ ψ _f ( x ⁰ )γ ^µ ψ i ( x ⁰ ) A _µ ( x ⁰ )

◦ ^{But also A} µ evolves: photon is back-scattered

◦ Evolution according to inhomogeneous wave equation ^A

= J

(Lorentz gauge)

◦ Solution via Green’s function: photon propagator

Aµ

ψi

ψf

−iqγ^µ

Fermion-Fermion Scattering (LO)

Institute of Experimental Particle Physics (IEKP)

14

The matrix element ( complete picture )

target projectile virtual photon

exchange

S

= i ( 4 π

q )

Z

d

q δ

( p

− p

− q ) u ( p

)γ

u ( p

) − g

q

+ i δ

( p

− p

+ q ) u ( p

)γ

u ( p

)

Fermion-Fermion Scattering (LO)

Institute of Experimental Particle Physics (IEKP)

15

The matrix element ( complete picture )

target projectile virtual photon

exchange

S

= i ( 4 π

q )

Z

d

q δ

( p

− p

− q ) u ( p

)γ

u ( p

) − g

q

+ i δ

( p

− p

+ q ) u ( p

)γ

u ( p

)

Fermion-Fermion Scattering (LO)

Institute of Experimental Particle Physics (IEKP)

16

The matrix element ( complete picture )

target projectile virtual photon

exchange

S

= i ( 4 π

q )

Z

d

q δ

( p

− p

− q ) u ( p

)γ

u ( p

) − g

q

+ i δ

( p

− p

+ q ) u ( p

)γ

u ( p

)

Fermion-Fermion Scattering (LO)

Institute of Experimental Particle Physics (IEKP)

17

The matrix element ( complete picture )

target projectile virtual photon

exchange

S

= i ( 4 π

q )

Z

d

q δ

( p

− p

− q ) u ( p

)γ

u ( p

); − g

q

+ i δ

( p

− p

+ q ) u ( p

)γ

u ( p

)

Feynman Rules (QED)

Matrix element calculation can be represented with Feynman diagrams

Institute of Experimental Particle Physics (IEKP)

18

Feynman Rules ( QED )

●

Feynman diagrams are a way to represent the elements of the matrix element calculation:

●

Incoming ( outgoing ) fermion.

●

Incoming ( outgoing ) photon.

●

Fermion propagator.

●

Photon propagator.

●

Lepton-photon vertex.

Legs:

Vertices:

Propagators:

Four-momenta of all virtual particles have to be integrated out.

Higher Orders

◦ Scattering amplitude S ^fi only known in perturbation theory

◦ ^Works the better the smaller the perturbation is

◦ ^QED: α

≈

◦ ^EWK: α

=

W

≈ ⁴ α

◦ ^QCD: α

( m

) ≈ ⁰ . 12

◦ If well in perturbative regime, first-order contribution already sufficient to describe main features of scattering process

◦ Contribution of order “ α ”

◦ Called “leading order”, “tree level”, or “Born level”

Higher Orders

◦ So far, discussed contributions to S ^{fi at order} α ¹

◦ ^{e. g. LO e}

^e

→ ^e

^e

^scattering

◦ Contributions at order α

:

Institute of Experimental Particle Physics (IEKP)

22

Order diagrams ( QED )

●

We have only discussed contributions to , which are of order in QED. (e.g. LO scattering) .

●

Diagrams which contribute to order would look like this:

Additional legs: Loops:

( in propagators or legs ) ( in vertices ) NLO contribution to the

2 → 2 process LO contribution to a 2 → 4 process

→ Opens phase space

Modifies (effective) masses of particles

→ Running masses

Modifies (effective) couplings of particles

→ Running couplings

Examples for Running Constants

10 100 1000

Q [GeV]

0.05 0.10 0.15 0.20 0.25 α

( Q )

αs(MZ) = 0.1171±^0.00750.0050(3-jet mass) αs(MZ) = 0.1185±0.0006 (World average)

CMSR32ratio CMS tt prod.

CMS incl. jet CMS 3-jet mass

HERA LEP PETRA SPS Tevatron

Eur.Phys.J.C75(2015)186

◦ Running of the constants can be observed

◦ Value needs to be measured at one scale

→ evolution predicted (DGLAP equations)

Effect of Higher Order Corrections

◦ Change of overall normalisation of cross sections

◦ Change of coupling and kinematic opening of phase space

◦ Change of kinematic distributions

◦ For example, softer or harder p

spectrum of final-state particles

◦ Effects (correction to LO) usually

◦ “small” in QED O ( 1 %)

◦ “large” in QCD O ( 10 %) : reliable predictions often require (N)NLO

◦ Higher order corrections can be mixed, e. g. O (α em α ² _s )

◦ ^Challenge: number of diagrams quickly explodes for higher-order

calculations

Summary

◦ Cross section link between theory and experiment

◦ Computed from quantum-mechanical amplitude of scattering process: obtained from perturbation theory

◦ ^Propagator as formal solution of equation of motions

◦ ^Amplitude expanded in coupling constant: each term corresponds to distinct process

◦ Feynman rules: compact recipe to compute amplitudes