1/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Introduction to Theoretical Particle Physics
Jurgen¨ R. Reuter
DESY Theory Group
Hamburg, 07/2015 2/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Literature
I Georgi: Weak Interactions and Modern Particle Physics, Dover, 2009
I Quigg: Gauge Theories of the Strong, Weak, and Electromagnetic Interactions, Perseus 1997
I Peskin: An Introduction to Quantum Field Theory Addison Wesly, 1994
I Weinberg, The Quantum Theory of Fields, Vol. I/II/(III) Cambridge Univ. Press, 1995-98
I Itzykson/Zuber, Quantum Field Theory, McGraw-Hill, 1980
I Bohm/Denner/Joos,¨ Gauge Theories of Strong and Electroweak Interactions, Springer, 2000
I Kugo, Eichtheorie, Springer, 2000 (in German)
I ...and many more 3/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
”I have nothing to offer but blood, toil, tears and sweat.” 4/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 ...but it is fun, though.... 5/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Part I (Vorabend)
From Lagrangians to Feynman Rules: Quantum Field Theory with Hammer and Anvil 6/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Why Quantum Field Theory? Subatomic realm: typical energies and length scales are of order • (~c) 200MeV fm use of both special relativity and quantum mechanics∼ mandatory· ⇒ Particles (quantum states) are created and destroyed, hence particle • number not constant: beyond unitary time evolution of a single QM system Schrodinger¨ propagator (time-evolution operator) violates • microcausality Scattering on a potential well for relativistic wave equation leads to • unitarity violation Use quantized fields: can be viewed as continuos limit of QM • many-body system with many (discrete) degrees of freedom Least Action Principle leads to classical equations of motion • (Euler-Lagrange equations)
∂ ∂ S = dtL = dtd3x = d4x (φ, ∂ φ) ∂ L = L L L µ ⇒ µ ∂(∂ φ) ∂φ Z Z Z µ 7/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 What kind of fields? Classical wave equations must be Lorentz-covariant • Action and Lagrangian (density) are Lorentz scalars • Fields classified according to irreps of Lorentz group • Simplest case: Lorentz scalars (real/complex), Klein-Gordon equation • = (∂ φ)∗(∂µφ) m2 φ 2 ( + m2)φ (∂2 ~ 2 + m2)φ = 0 L µ − | | ⇒ ≡ t − ∇
Spin 1/2 particles: Dirac equation • = iΨ(i/∂ + m)Ψ (i/∂ m)Ψ = 0 where /a a γµ L ⇒ − ≡ µ 0 σµ γµ = σµ = (1, ~σ)σ ¯µ = (1, ~σ) γµ, γν = 2ηµν 1 σ¯µ 0 − { } γ5 = iγ0γ1γ2γ3 is the chiral projector: = 1 (1 γ5) PL/R 2 ∓ 8/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Dawn of gauge theories
Spin 1 particles: Maxwell’s equations • 1 = F F µν with F = ∂ A ∂ A +g [A ,A ] L −4 µν µν µ ν − ν µ µ ν ⇒
Invariant under (local) gauge transformation, G: 1 A A0 = GA G−1 (∂ G)G−1 µ −→ µ µ − g µ Electric and magnetic fields are defined via:
˙ E~ = A~ ~ A g A , A~ 0 0 µν µν − − ∇ − ∂µF = 0 i [Aµ,F ] B~ = ~ A~ g A~ h, A~ i − → ∇ × − 2 × h i µ0 0 = DµF = ~ E~ + g A~ , E~ ∇ · i 0 = D F µi = E˙ i + (~h B~ )ii g A ,Ei + g A~ , B~ µ − ∇ × − 0 × h i 8/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Dawn of gauge theories
Spin 1 particles: Maxwell’s equations • 1 = F F µν with F = ∂ A ∂ A +g [A ,A ] L −4 µν µν µ ν − ν µ µ ν ⇒
Invariant under (local) gauge transformation, G: 1 A A0 = GA G−1 (∂ G)G−1 µ −→ µ µ − g µ Electric and magnetic fields are defined via:
˙ E~ = A~ ~ A g A , A~ 0 0 µν µν − − ∇ − ∂µF = ig [Aµ,F ] B~ = ~ A~ g A~ h, A~ i − → ∇ × − 2 × h i µ0 0 = DµF = ~ E~ + g A~ , E~ ∇ · i 0 = D F µi = E˙ i + (~h B~ )ii g A ,Ei + g A~ , B~ µ − ∇ × − 0 × h i 9/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 ...for the curious... Spin 3/2 particles: Rarita-Schwinger equation • 1 1 = µνρσΨ γ γ ∂ Ψ mΨ [γµ, γν ]Ψ L 2 µ 5 ν ρ ν − 4 µ ν ⇒ m (/∂γν Ψ γν /∂Ψ ) = 0 ν − ν γµΨ = 0 ∂µΨ = 0 (i/∂ m)Ψ = 0 µ µ − µ Leads only to sensible theory in supergravity
Spin 2 particles: de Donder equation • 4πG = N det g R with R = Rµ ν and L − c4 | | µ ν p Rρ = ∂ Γρ ∂ Γρ + Γρ Γλ Γρ Γλ σµν µ νσ − ν µσ µλ νσ − νλ µσ Γρ = 1 gρσ (∂ g ∂ g ∂ g ) define g = η + 16πG h µν 2 µ νσ − ν µσ − ρ µν µν µν N µν h 1 η hρ = ∂ ∂ρ h 1 η hσ + (µ pν) µν − 2 µν ρ µ ρν − 2 ρν σ ↔ 10/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 How to quantize a field (Analogous to QM) Field and its canonically conjugate momentum: π = ∂ /∂φ˙ • L Canonically transform to the Hamiltonian of the system: • ∂ 2 = π L 1 π2 + 1 ~ φ + 1 m2φ2 H ˙ − L → 2 2 ∇ 2 ∂φ φ˙=φ˙(φ,π)
Impose equal-time commutation relations: • [φ(~x), π(~y)] = iδ3(~x ~y)[φ(~x), φ(~y)] = [π(~x), π(~y)] = 0 − Creation and annihilation operators to diagonalize the Hamiltonian: • † 3 3 0 [a , a 0 ] = 2E (2π) δ (~p ~p )[a , a 0 ] = 0 p p p − ~p ~p Heisenberg/interaction picture: (field) operators time dependent • Creation/annihilation operators Fourier coefficients of field operators: • 3 −ik·x † +ik·x d k φ(x) = dk ake + ake dk 3 ≡ (2π) (2Ek) Z Z Z f f 11/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Hamiltonian is a sum of harmonic oscillators: • d3p = E a† a + const. H (2π)3 p p p Z Infinite constant is abandoned by normal ordering renormalization • d3p : := E : a† a : Note: 0 : : 0 = 0 H (2π)3 p p p h O i Z Creation operator creates a 1-particle state with well-defined • momentum (plane wave): a† 0 = p p | i | i Field operator creates a 1-particle state at x: φ(x) 0 = dp eip·x p • | i | i Consider a complex field: R • f −ik·x † +ik·x † −ik·x † +ik·x φ(x) = dk ake + bke , φ (x) = dk bke + ake Z Z Field operator:f positve frequency modes (e−ik·x) havef annihilation operator for particles, negative frequency modes (e+ik·x) have creation operator for anti-particles a, b are independent, commute (independent Fourier components!) Got rid of negative energies as in classical wave equations • 12/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Microcausality and the Feynman propagator Amplitude for a particle to propagate from y to x: • 4 d p |~x−~y|→∞ D(x y) 0 φ(x)φ(y) 0 = e−ip(x−y) e−m|~x−~y| − ≡ h i (2π)4 ∼ Z falls off exponentially outside light cone, but is non-zero Measurement is determined through the commutator: • [φ(x), φ(y)] = D(x y) D(y x) = 0 (for (x y)2 < 0) − − − − Cancellation of causality-violating effects by Feynman prescription: • I Particles propagated into the future (retarded)
I Antiparticles propagated into the past (advanced) D(x y) for x0 > y0 D (x y) = − = θ(x0 y0) 0 φ(x)φ(y) 0 F − D(y x) for x0 < y0 − h i − + θ(y0 x0) 0 φ(y)φ(x) 0 0 T [φ(x)φ(y)] 0 − h i ≡ h i Feynman propagator (time-ordered product, causal Green’s function) • p y d4p i x −→ D (x y) = e−ip(x−y) = F − (2π)4 p2 m2 + i Z − 13/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 i prescription Solution of wave equation by Fourier transform: •
( + m2)φ(x) = iδ4(x y) F.T. − −→ i i φ˜(p) = = p2 m2+i (p0 E +i)(p0 + E i) − − p p− 2 2 0 with Ep = + ~p + m p p Prescription tells you +ip0x0 • where to propagate in time: neg. frequ.: e E + i − P
+E i P − 0 0 pos. frequ.: e−ip x 13/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 i prescription Solution of wave equation by Fourier transform: •
( + m2)φ(x) = iδ4(x y) F.T. − −→ i i φ˜(p) = = p2 m2+i (p0 E + i)(p0+E i) − − p p − 2 2 0 with Ep = + ~p + m p p Prescription tells you +ip0x0 • where to propagate in time: neg. frequ.: e E + i − P
+E i P − 0 0 pos. frequ.: e−ip x 13/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 i prescription Solution of wave equation by Fourier transform: •
( + m2)φ(x) = iδ4(x y) F.T. − −→ i i φ˜(p) = = p2 m2+i (p0 E + i)(p0+E i) − − p p − 2 2 0 with Ep = + ~p + m p p Prescription tells you +ip0x0 • where to propagate in time: neg. frequ.: e E + i − P
+E i P − 0 0 pos. frequ.: e−ip x 14/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Quantization of the Dirac field Spin-statistics theorem (Fierz/Luders/Pauli,¨ 1939/40): • I Spin 0, 1, 2,...: bosons ⇒ commutators 1 3 I Spin 2 , 2 ,...: fermions ⇒ anticommutators equal-time anticommutators: ψ(x) , ψ†(y) = δ(~x ~y)δ • α β − αβ Field operators: • Z X s s −ipx s † s +ipx ψ(x) = dpf apu (p)e + bp v (p)e s Z X s † s +ipx s s −ipx ψ(x) = dpf ap u (p)e + bpv (p)e s
Feynman propagator (time-ordered product, causal Green’s function) • 0 ψ(x)ψ(y) 0 for x0 > y0 SF (x y) = − = 0 T ψ(x)ψ(y) 0 − 0 ψ(y)ψ(x) 0 for x0 < y0 −
p y d4p i(/p + m) x S (x y) = e−ip(x−y) = F − (2π)4 p2 m2 + i Z − 15/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The rocky road from the S matrix to cross sections
4D QFTs without interactions exactly solvable, otherwise not •
Idea: For scattering process sharply located in space-time, use: • I Asymptotically free quantum state for t → −∞
I Interaction described completely local in space-time
I Asymptotically free quantum state for t → +∞ General axioms of QFT: • 1. All eigenvalues of P µ are in the forward lightcone 2. There is a Poincare-invariant´ vacuum state |0i 3. For every particle there is 1-particle state |pi 4. Asymptotic fields are free fields whose creation operators span a Fock space (asymptotic completeness) Use the (Kallen-Lehmann)¨ spectral representation: • ∞ i F.T. 0 T [Φ(x)Φ(y)] 0 = dµ2ρ(µ2) h i p2 µ2 + i Z0 − 16/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
∞ iZ 2 2 i 2 2 + dµ ρ(µ ) 2 2 −→ p m + i 2 p µ + i − Z4m − Z: Wave function renormalization, m: renormalized mass
ρ Bound states
1P Continuum of ≥2P states 2 2 2 p state m 4m
m2 4m2 µ2 Asymptotic LSZ (Lehmann-Symanzik-Zimmermann) condition: • Heisenberg fields of full theory are asymptotically free fields (up to wave function and mass renormalization)
0→−∞ 0→ ∞ Φ(x) x √Zφ (x) Φ(x) x + √Zφ (x) −→ in −→ out Asymptotic fields obey free field equations! Simple Fock spaces • ⇒ † † † in = a a . . . a 0 V in,k1 in,k2 in,kn | i † † † Assumption: in out out = na a . . . a 0 o V ≡ V ≡ V V in,k1 in,k2 in,kn | i n o 17/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
The S-Matrix Wheeler, 1939; Heisenberg 1943
Scattering probability from initial to final state: Prob. = β α 2 • |h out ini| S-Matrix transforms in into out states: β α = β S α • h out ini h in ini 1. S-Matrix is unitary (prob. conserv.): S†S = SS† = 1 † 2. Transforms asymptotic field operators: φout(x) = S φin(x)S 3. S-matrix invariant under symmetries: [Q, S] = 0
I Four major steps to calculate scattering cross sections
1. LSZ formula: S-matrix as n-point Green’s functions of full theory 2. Gell-Mann–Low formula: Green’s functions of full theory expressed by perturbation series of free field Green’s functions 3. Wick’s theorem, Feynman rules (and elimination of vacuum bubbles) 4. Phase space integration: scattering cross section from S-matrix elements 18/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Step 1: LSZ formula Lehmann/Symanzik/Zimmermann, 1955 Idea:
I For t → ±∞ asymptotic fields are free
I Project out creation operators from free field operators by inverse Fourier transform
I S-Matrix element is (upto normaliz.) Green’s function multi-particle pole
I One gets tis pole by amputation of external legs of the Green’s function
' $ hp1, . . . pn, out q1, . . . ql, ini = hp1, . . . pn, in S q1, . . . ql, ini = n+l n Z ! i Y 4 ipiyi 2 (disconn. terms) + √ d yie (yi + mi ) · Z i=1 l Z Y 4 −iqj xj 2 d xj e (xj + mj ) h0 T [Φ(y1)Φ(y2) ... Φ(xl)] 0i = j=1 amputated
Yields Feynman rules for external particles: 1P on-shell wavefunctions • & % 18/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Step 1: LSZ formula Lehmann/Symanzik/Zimmermann, 1955 Idea:
I For t → ±∞ asymptotic fields are free
I Project out creation operators from free field operators by inverse Fourier transform
I S-Matrix element is (upto normaliz.) Green’s function multi-particle pole
I One gets tis pole by amputation of external legs of the Green’s function
' $ hp1, . . . pn, out q1, . . . ql, ini = hp1, . . . pn, in S q1, . . . ql, ini = n+l n Z ! i Y 4 ipiyi 2 (disconn. terms) + √ d yie (yi + mi ) · Z i=1 l Z Y 4 −iqj xj 2 d xj e (xj + mj ) h0 T [Φ(y1)Φ(y2) ... Φ(xl)] 0i = j=1 amputated
Yields Feynman rules for external particles: 1P on-shell wavefunctions • & % 19/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Step 2: Gell-Mann–Low formula Gell-Mann/Low, 1951
I Use: field and time evolution operators in the interaction picture
I Transform fields of the full theory into asymptotically free fields asymptotically free fields
I Solve the Schrodinger¨ equation of IA picture time-evolution operator as a (time-ordered) perturbation series
t 0 0 t→+∞ U(t) = T exp i dt Hint(t ) S − −→ Z−∞
I Time evolution of field operators and also vacuum state (vacuum polarization!) = + L Lkinetic Lint # 0 T [Φ(x1) ... Φ(xn)] 0 = h i 4 0 T φin(x1) . . . φin(xn) exp i d x int[φin(x)] 0 4 L 0 T exp i d x int[φin(x)] 0 L R R " ! 19/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Step 2: Gell-Mann–Low formula Gell-Mann/Low, 1951
I Use: field and time evolution operators in the interaction picture
I Transform fields of the full theory into asymptotically free fields asymptotically free fields
I Solve the Schrodinger¨ equation of IA picture time-evolution operator as a (time-ordered) perturbation series
t 0 0 t→+∞ U(t) = T exp i dt Hint(t ) S − −→ Z−∞
I Time evolution of field operators and also vacuum state (vacuum polarization!) = + L Lkinetic Lint # 0 T [Φ(x1) ... Φ(xn)] 0 = h i 4 0 T φin(x1) . . . φin(xn) exp i d x int[φin(x)] 0 4 L 0 T exp i d x int[φin(x)] 0 L R R " ! 20/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Step 3: Wick’s theorem, Feynman Rules Wick, 1950 Task is to calculate VEV of time-ordered product of free fields: • 0 T [φ(x1)φ(x2) . . . φ(xn)] 0 (Note: Lint [φ(x)] is a polynomial of free fields!) h i Decompose fields in annihilator (pos. freq.) and creator (neg. freq.) • part to show T [φ(x)φ(y)] = : φ(x)φ(y) : + DF (x y) − Wick’s theorem (proof by induction) φ := φ(x ) etc. • i i # T [φ1 . . . φn] = : φ1 . . . φn : + : φ1 . . . φn−2 : DF,n−1,n + permut. h : φ1 . . . φn−4 : DF,n−3,n−2DF,n−1,n + DF,n−3,n−1DF,n−2,n i + DF,n−3,nDF,n−2,n−1 + perm. + ... + product of only Feynman propagators
0 T [φ1 . . . φn] 0 = DF,i,j for example: •⇒"h i all perm. ! 0 T [φ1φ2φ3φ4] 0 = DF, DF, h i Q12 34 1 2 1 2 1 2
+ DF,13DF,24 + DF,14DF,23 = + +
3 4 3 4 3 4 Signs from fermion anticommutations arise from Wick’s theorem! • 21/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Feynman rules (position space) Example for = 1 (∂ φ)(∂µφ) 1 m2φ2 λ φ4 L 2 µ − 2 − 4!
x y – Propagator D (x y) F − 4 – Vertex z ( iλ) d z − R – external point x exp[ ipx sgn(in/out)] − · – divide by the symmetry factor
Note: Position integrals correspond to QM superposition principle • Example for symmetry factor: •
yields a symmetry factor 3! 22/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Feynman rules (momentum space)
– Propagator i/(p2 m2 + i) p −→ −
– Vertex iλ − – external point 1 (or u, v etc.) – momentum conservation at each vertex – integrate over d4p/(2π)4 undetermined momenta R – divide by symmetry fac.
Note: Momentum integrals correspond to QM superposition principle • 3 2 2 2 Example in λφ theory: Mandelstam variables: s ≡ (p1 + p2) t ≡ (p1 − p3) u ≡ (p1 − p4)
+ + =
2 i i i ( iλ) 2 + 2 + 2 − s−m t−m u−m 23/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Bubbles, bubbles, vacuum bubbles Vacuum bubbles are infinite diagrams (Fermion loops: (-1)) •
2 δ4(p) (2T) (Vol) ∼ −→ · Vacuum bubbles in the numerator and denominator of Gell-Mann–Low • formula Symmetry factors vacuum bubbles exponentiate • ⇒ Just a normalization factor: cancels out •
0 T φ(x)φ(y) exp( i dtHint(t)) 0 0 T [Φ(x)Φ(y)] 0 = − = h i 0 T exp( i dtHint(t))) 0 − R x y x y y R + x + + exp + + + ··· ··· exp + + + ··· 23/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Bubbles, bubbles, vacuum bubbles Vacuum bubbles are infinite diagrams (Fermion loops: (-1)) •
2 δ4(p) (2T) (Vol) ∼ −→ · Vacuum bubbles in the numerator and denominator of Gell-Mann–Low • formula Symmetry factors vacuum bubbles exponentiate • ⇒ Just a normalization factor: cancels out •
0 T φ(x)φ(y) exp( i dtHint(t)) 0 0 T [Φ(x)Φ(y)] 0 = − = h i 0 T exp( i dtHint(t))) 0 − R x y x y R x y + + + exp + + + ··· ··· exp + + + ··· 24/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Fermion lines
Signs from disentangling field operator contractions: • 0 bψ(x)ψ(x)ψ(y)ψ(y)ψ(z)ψ(z)b† 0 h | | i
k3 k2 k1 p + k1 + k2 + k3 = q p = p1 p2 i(/p + /k + m) i(/p + /k + /k m) u(p) 1 1 2 u(q) (p + k )2 m2 + i (p + k + k )2 m2 + i 1 − 1 2 − External fermions: • us(p) vs(p) us(p) vs(p) 25/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Step 4: Phase Space Integration and Cross Sections Number N of scattering events for • 2 particle beams with particle densities ρ1,2, relative velocity v (V scattering volume, T scattering time)
N = V T ρ ρ v σ · · 1 · 2 · · Constant of proportionality: cross section • effective scattering area (e.g. geometric scattering σ = πr2) • How to get the cross section? • P 2 1. Probability to get any final state |ni from initial state |αi: n |hn|S|αi| 2. Project on a specific final state 3. Use Fermi’s Golden Rule 4. Momentum integral from projection becomes phase space integral over final-state momenta Flux factor, invariant matrix element, phase space measure
2 n n βα 4 4 dσα→β = |M | dpi (2π) δ (p1 + p2 qi) 2 2 2 − 4 (p1 p2) m1m2 i=1 ! i=1 · − Y X p f 26/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Analogous decay width Γ of a particle (Life time τ = 1/Γ) • 2 n n βα 4 4 dΓ → = |M | dp (2π) δ (p q ) α β 2m i − i α i=1 ! i=1 Y X 2 2 scattering depends only on √fs and cos θ: • → dσ 1 ~q = | out| 2 all masses equal dΩ 64π2s ~p |Mβα| | in| Simple example, λφ4 theory: •
λ2 1 = −iλ 2 = λ2 σ = Mβα ⇒ |Mβα| ⇒ 4π · s
σ
∝ 1/s
√ s 26/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Analogous decay width Γ of a particle (Life time τ = 1/Γ) • 2 n n βα 4 4 dΓ → = |M | dp (2π) δ (p q ) α β 2m i − i α i=1 ! i=1 Y X 2 2 scattering depends only on √fs and cos θ: • → dσ 1 ~q = | out| 2 all masses equal dΩ 64π2s ~p |Mβα| | in| Simple example, λφ4 theory: •
λ2 1 = −iλ 2 = λ2 σ = Mβα ⇒ |Mβα| ⇒ 4π · s
σ
∝ 1/s
√ s 27/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Quantization of the Electromagnetic Field
• 1 = F F µν j Aµ L −4 µν − µ µ µ µ Euler-Lagrange equation A ∂ (∂ A)= j invariant under gauge • transformation: Aµ(x) Aµ(x)− + ∂µf(·x) → Gauge invariance makes life hard (but for the wise easy!) • ˙ 0 I A0 absent ⇒ no Π I Kinetic term is singular, hence not invertible 3 I [Aµ(~x),Aν (~y)] = −iηµν δ (~x − ~y) leads to negative and zero norm states on Fock space!
3 Z † ∗ X (λ) (λ) −ipx (λ) (λ) +ipx Aµ(x) = dkf ak µ (k)e + ak µ (k)e λ=0 Solution: gauge fixing: ∂ A = 0 • · I Physical states |αi: Transversal polarizations ((k) · k = 0)
I Unphysical states |χi: longitudinal (space-like) pol./ scalar (time-like) pol.
I Physical Fock space {|αi} contains only positive-norm states
I |αi → |αi + |χi just corresponds to a gauge transformation
I Gupta-Bleuler quantization: hα|(∂ · A)|βi = 0 28/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Quantum Electrodynamics (QED)
I Local (gauge) transformations 1 ψ(x) exp[ieQ θ(x)]ψ(x) A (x) A (x) + ∂ θ(x) → el. µ → µ e µ
I Derivative terms are no longer invariant covariant derivatives ∂ ψ D ψ = (∂ ieQ A ) ψ ⇒ µ f → µ,f f µ − f µ f 1 = ψ (i /D m ) ψ F F µν LQED f f − f f − 4 µν Xf
s µ µ u (p) ieQf γ vs(p) us(p)
µ ν −iηµν s 2 v (p) → k k +i µ ∗ µ(k) i(/p+m) µ µ(k) → p p2−m2+i 29/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Ward identities: gauge invariance for the wise
I Noether theorem: continuos symmetry implies a conserved current where the conserved charge is the symmetry generator
d ∂ µ = 0 Q := d3x 0(~x) with Q = 0 µJ ⇒ J dt Z [iQ, φ(x)] = δφ(x) symmetry of the Lagrangian
I Current conservation implies Ward identities: (contact terms vanish on-shell) µ 0 = ∂ 0 T [ (x)φ1(x1) . . . φn(xn)] 0 x h J i
J µ kµ = 0 I Generalization for off-shell amplitudes (and also non-Abelian gauge theories): Slavnov-Taylor identities 30/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 + + e e− µ µ : A sample calculation → − vI(p2) u(q1)
2 µ 1 = ie QeQµ (u(q1)γ v(q2)) (v(p2)γµu(p1)) −ieQeγµ −ieQµγν M s −iηµν 2 (p1+p2)
u(p1) v(q2)
I Square the matrix element, sum over final state spins, average over initial spins
X X Spin sums: us (p)us (p) = (/p + m1) vs (p)vs (p) = (/p m1) α β αβ α β − αβ s=± s=±
X r s r s ∗ X r s r s Use (u (p1)γµv(p2) )(u (p1)γν v (p2)) = (u (p1)γµv (p2))(v (p2)γν u (p1)) r,s r,s X r s r s = tr [(u (p1)γµv (p2))(v (p2)γν u (p1))] = tr [(/p1 + m)γµ(/p2 m)γν ] r,s −
= 4(p1,µp2,ν + p1,ν p2,µ) 2sηµν − 31/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
I Neglect all masses:
[4(p p + p p ) 2sη ] [4(q q + q q ) 2sη ] 1,µ 2,ν 1,ν 2,µ − µν 1,µ 2,ν 1,ν 2,µ − µν 2 2 2 2 = 32(q1p1)(q2p2) + 32(q1p2)(q2p1) = 8(t + u ) = 4s (1 + cos θ)
I
dσ 1 1 σ = dΩ = 2 dΩ 64π2s 2 |M| s,r Z X e4 1 1 4πα = (2π) d(cos θ)(1 + cos2 θ) = 32π2s 2 3s Z−1 2 I Sommerfeld’s fine structure constant α = e /(4π) 1/137 ∼ I Result: 4πα2 σ(e+e− µ+µ−) = → 3s 32/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Part II (1. Abend)
The Mighty Valkyrie: Non-Abelian Gauge Theories and Renormalization 33/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Quantum (a.k.a. radiative) corrections Real corrections: radiation of photons etc. • Virtual (loop) corrections • Vacuum Self energies polarization
Vertex corrections Box diagrams
Example: e− anomalous magnetic moment g ψ i [γµ, γν ] F ψ • 2 µν Dirac theory tree level: g = 2 α := e2/4π
α + g = 2 1 + = 2.00232282 ⇒ 2π Experimentally: g = (2.00231930436222 0.00000000000148) ± 34/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Ultraviolet and infrared (mass) singularities
I Ultraviolet singularities (in loops)
q
k → 4 2 d q tr[γµ(/q+m)γν (/q+/k+m)] := iΣµν (k) = e 4 (q2−m2)((q+k)2−m2) k → − − (2π) Z q + k d4q 1, q, q2 dq q−1, 1, q −→ q4 ∼ Z Z Logarithmic, linear, quadratic divergencies ⇒
I Collinear and soft (mass or infrared) singularities
% k 1 1 2 2 = → p −→ (p + k) m 2p k → p + k − · 1 = with β = p /E 1 2E E (1 βcos θ) e e ∼ e γ − Collinear singularity: cos θ 1 Soft singularity: E 0 → γ → 35/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Dimensional Regularization
Divergent integrals need to be cast in a finite form to be treated • Analytical continuation of • loop integrals to D = 4 2< 4 for ultraviolet singularities − phase space integrals to D = 4 + 2> 4 for soft/collinear singularities Keep dimensionality of integrals unphysical scale µ • ⇒ d4q dDq µ4−D (2π)4 −→ (2π)D Z Z E.g. QED vacuum polarization: • 2α 1 Σ (k) = k2η k k dxx(1 x) µν π µν − µ ν − · Z0 1 x(1 x)k2 + m2 γ + ln(4π) ln − − − E − µ2
(γE = 0.577 ... Euler constant) ∆ := 1/ γ + ln(4π) − E 36/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Power Counting Question: Which Feynman diagrams do contain UV divergencies? •
4 4 # Numerator d k1d k2 . . . dkL k ∼ (/k m) ... (k2) ... (k2 ) ∼ k# Denominator Z i − j n Define superficial degree of divergence (in D dimensions) • D = (# Numerator) (# Denominator) = D L I 2I D − · − e − γ
± ± Ee = # external e , Eγ = # extern. γ, Ie = # internal e , Iγ = # intern. γ, V = # vertices, L = # loops −1 −2 I Fermion propagators contribute k , bosons k n I Vertices with n derivatives contribute k
I Euler identity: L = Ie + Iγ − V + 1 (by induction)
1 I Simple line count: V = 2Iγ + Eγ = 2 (2Ie + Ee) Power counting master formula: • = D(I + I V + 1) I 2I = D E 3 E § D e γ − − e − γ − γ − 2 e ¤ Depends only on number of external lines ¦ ¥ 37/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 QED Singularities – Classification of QFTs
= 4 = 3 D D (irrelevant, normalization) = 0 (Furry’s theorem)
= 2 = 1 D D 2 D = 0, Ward id., k ηµν − kµkν = 0 (Furry’s theorem) = 0 = 1 D D finite, Ward identity D = 0 (chirality)
= 0 = 4 E 3 E D D − γ − 2 e
Only finite # of (sup.) Superrenormalizable Theory I D ∼ − divergent graphs Finite # of (sup.) div. ampl., Renormalizable Theory 0 I D ∼ · but at all orders of pert. series all amplitudes of sufficiently Non-Renormalizable Theory +I D ∼ high order diverge 37/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 QED Singularities – Classification of QFTs
= 4 = 3 D D (irrelevant, normalization) = 0 (Furry’s theorem)
= 2 = 1 D D 2 D = 0, Ward id., k ηµν − kµkν = 0 (Furry’s theorem) = 0 = 1 D D finite, Ward identity D = 0 (chirality)
= 0 = 4 E 3 E D D − γ − 2 e
Coupling constants have foo o Superrenormalizable Theory I D ∼ − positive mass dimension Coupling constants are Renormalizable Theory 0 I D ∼ · dimensionless Coupling constants have Non-Renormalizable Theory +I D ∼ negative mass dimension 38/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Renormalization In a renormalizable theory infinities cancel in the calculation of • physical observables! Example: UV divergence of vertex correction and Z cancel • e Infinities cancel: there are finite shifts in physical parameters (g 2!) • − Easier calculational prescription: Renormalized perturbation theory • I Express Lagrangian of bare fields and parameters 1 2 1 2 2 λ0 4 L(φ0, m0, λ0) = (∂φ0) − m0φ0 − φ0 by renormalized ones: √ 2 2 4! 0 I φ = Zφren. eliminates wave function factor 2 2 2 I Expand parameters/fields δZ = Z − 1, δm = m0Z − m , δλ = λ0Z − λ
1 2 1 2 2 λ 4 1 2 1 δλ L = (∂φ ) − m φ − φ + δ (∂φ ) − δ φ 2 − φ 4 2 ren 2 ren 4! ren 2 Z ren 2 m ren 4! ren
I This generates counterterms, e.g.
= −iλ −→ = −iδλ
Loop diagrams and counterterms added up are finite! • Counterterms are not unique: on-shell scheme, MS scheme • 39/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 KLN: Discarding mass singularities Soft/collinear singularities only appear for (quasi) massless particles • They appear both in real diagrams( Q2 = ~k2 ) • ⊥,max k Z 1 2 e→γR α Q2 1 + x e→R p dσ (p, pR) ln 2 dx dσ (xp, pR) + p − k ∼ 2π me 1 x ··· 0 − and virtual diagrams Z 1 α Q2 δZe ln m2 dx(1 x) + non-log terms ∼ − 2π e 0 − k Z 1 e→γR α Q2 e→R x p p − k dσvirt. (p, pR) ln 2 dσ (xp, pR) dx + ∼ − 2π me 1 x ··· 0 − soft singularity: me 0 collinear singularity: x 1 Soft-coll. singularities→ cancel between the three contributions→ Splitting function: P = (1 + x2)/(1 x) (cf. later) • ee − KLN Theorem Bloch/Nordsieck, 1937; Kinoshita 1962, Lee/Nauenberg, 1964 Unitarity guarantees that transition amplitudes are finite when summing over all degenerate states in initial and final state, order by order in perturbation theory (or for renormalization schemes free of mass sing.)
40/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Renormalization group and running couplings High-momentum modes short-distance quantum fluctuations • ≡ Fourier integrating over these λΛ < k < Λ modes (path integral) • Consider S = dDx 1 (∂φ)2 1 m2φ| 2| Bφ4 Rescale: 2 − 2 − q x0 = λxR k0 = k/λ µφ 0 = λ2−D(1 +δZ)φ (0 < λ < 1) ∼ Integrating out generates higher-dimensional operators, e.g.
D φ6 ∼ · Effect is a shift in the masses, parameters, and field normalization • (hence a renormalization) Z Z D D 0 1 0 1 0 0 0 0 0 0 0 0 d x = d x (∂φ )2 m 2φ 2 B φ 4 C (∂φ) 4 D φ 6 + ... L µ<λΛ 2 − 2 − − −
2 I Close to a fix point: m ,B,C,D,... = 0 0 2 2 2 −1 −2 m = (m + δm )(1 + δZ) λ I Keep only linear terms B0 = (B + δB)(1 + δZ)−2λD−4 C0 = (C + δC)(1 + δZ)−2λD I Relevant operators 0 −3 2D−6 D = (D + δD)(1 + δZ) λ I Marginal operators
I Irrelevant operators 40/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Renormalization group and running couplings High-momentum modes short-distance quantum fluctuations • ≡ Fourier integrating over these λΛ < k < Λ modes (path integral) • Consider S = dDx 1 (∂φ)2 1 m2φ| 2| Bφ4 Rescale: 2 − 2 − q x0 = λxR k0 = k/λ µφ 0 = λ2−D(1 +δZ)φ (0 < λ < 1) ∼ Integrating out generates higher-dimensional operators, e.g.
D φ6 ∼ · Effect is a shift in the masses, parameters, and field normalization • (hence a renormalization) Z Z D D 0 1 0 1 0 0 0 0 0 0 0 0 d x = d x (∂φ )2 m 2φ 2 B φ 4 C (∂φ) 4 D φ 6 + ... L µ<λΛ 2 − 2 − − −
2 I Close to a fix point: m ,B,C,D,... = 0 m0 2 = m2λ−2 grows for E 0 → I Keep only linear terms B0 = BλD−4 const. for E 0 → C0 = CλD shrinks for E 0 I Relevant operators 0 2D−6 → D = Dλ shrinks for E 0 I Marginal operators → I Irrelevant operators 40/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Renormalization group and running couplings High-momentum modes short-distance quantum fluctuations • ≡ Fourier integrating over these λΛ < k < Λ modes (path integral) • Consider S = dDx 1 (∂φ)2 1 m2φ| 2| Bφ4 Rescale: 2 − 2 − q x0 = λxR k0 = k/λ µφ 0 = λ2−D(1 +δZ)φ (0 < λ < 1) ∼ Integrating out generates higher-dimensional operators, e.g.
D φ6 ∼ · Effect is a shift in the masses, parameters, and field normalization • (hence a renormalization) Z Z D D 0 1 0 1 0 0 0 0 0 0 0 0 d x = d x (∂φ )2 m 2φ 2 B φ 4 C (∂φ) 4 D φ 6 + ... L µ<λΛ 2 − 2 − − −
2 I Close to a fix point: m ,B,C,D,... = 0 m0 2 = m2λ−2 grows for E 0 → I Keep only linear terms B0 = BλD−4 const. for E 0 → C0 = CλD shrinks for E 0 I Superrenormalizable theory 0 2D−6 → D = Dλ shrinks for E 0 I Renormalizable theory → I Nonrenormalizable theory 41/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Renormalization Group Equation Wilson, 1971; Callan, Symanzik, 1970 Bare Green’s function do not depend on renormalization scale µ • d (n) d h −n/2 (n) i 0 = µ G (g0, m0, reg.) = µ Z Gren (g(g0, m0, µ), m(g0, m0, µ), µ, reg.) dµ dµ ·
∂ ∂g ∂ 1 ∂m ∂ 1 dZ (n) µ + µ µ m n µ Gren = 0 ∂µ ∂µ ∂g − − m ∂µ ∂m − − 2Z dµ
β(g, m, µ) = ∂g/∂(ln µ) β (RG) function • β(g, m, µ) = ∂(ln m)/∂(ln µ) anomalous mass dimension • − γ(g, m, µ) = 1 d(ln Z)/d(ln µ) anomalous dimension • − 2 3 2 2 Typical one-loop values: g g g β(g) = β0 16π2 γm(g) = γm,0 16π2 γ(g) = γ0 16π2 Solution of RG equation: • α
3 dg β0g β0 < 0 = 2 d ln µ 16π ⇒
2 β > 0 2 g (µ0) 0 g (µ) = 2 g (µ0) µ 1− β0 ln( ) 8π2 µ0 ln µ 41/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Renormalization Group Equation Wilson, 1971; Callan, Symanzik, 1970 Bare Green’s function do not depend on renormalization scale µ • d (n) d h −n/2 (n) i 0 = µ G (g0, m0, reg.) = µ Z Gren (g(g0, m0, µ), m(g0, m0, µ), µ, reg.) dµ dµ ·
∂ ∂ ∂ (n) µ + β(g, m, µ) γm(g, m, µ)m nγ(g, m, µ) Gren = 0 ∂µ ∂g − ∂m −
β(g, m, µ) = ∂g/∂(ln µ) β (RG) function • β(g, m, µ) = ∂(ln m)/∂(ln µ) anomalous mass dimension • − γ(g, m, µ) = 1 d(ln Z)/d(ln µ) anomalous dimension • − 2 3 2 2 Typical one-loop values: g g g β(g) = β0 16π2 γm(g) = γm,0 16π2 γ(g) = γ0 16π2 Solution of RG equation: • α
3 dg β0g β0 < 0 = 2 d ln µ 16π ⇒
2 β > 0 2 g (µ0) 0 g (µ) = 2 g (µ0) µ 1− β0 ln( ) 8π2 µ0 ln µ 41/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Renormalization Group Equation Wilson, 1971; Callan, Symanzik, 1970 Bare Green’s function do not depend on renormalization scale µ • d (n) d h −n/2 (n) i 0 = µ G (g0, m0, reg.) = µ Z Gren (g(g0, m0, µ), m(g0, m0, µ), µ, reg.) dµ dµ ·
∂ ∂ ∂ (n) µ + β(g, m, µ) γm(g, m, µ)m nγ(g, m, µ) Gren = 0 ∂µ ∂g − ∂m −
β(g, m, µ) = ∂g/∂(ln µ) β (RG) function • β(g, m, µ) = ∂(ln m)/∂(ln µ) anomalous mass dimension • − γ(g, m, µ) = 1 d(ln Z)/d(ln µ) anomalous dimension • − 2 3 2 2 Typical one-loop values: g g g β(g) = β0 16π2 γm(g) = γm,0 16π2 γ(g) = γ0 16π2 Solution of RG equation: • α
3 dg β0g β0 < 0 = 2 d ln µ 16π ⇒
2 β > 0 2 g (µ0) 0 g (µ) = 2 g (µ0) µ 1− β0 ln( ) 8π2 µ0 ln µ 41/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Renormalization Group Equation Wilson, 1971; Callan, Symanzik, 1970 Bare Green’s function do not depend on renormalization scale µ • d (n) d h −n/2 (n) i 0 = µ G (g0, m0, reg.) = µ Z Gren (g(g0, m0, µ), m(g0, m0, µ), µ, reg.) dµ dµ ·
∂ ∂ ∂ (n) µ + β(g, m, µ) γm(g, m, µ)m nγ(g, m, µ) Gren = 0 ∂µ ∂g − ∂m −
β(g, m, µ) = ∂g/∂(ln µ) β (RG) function • β(g, m, µ) = ∂(ln m)/∂(ln µ) anomalous mass dimension • − γ(g, m, µ) = 1 d(ln Z)/d(ln µ) anomalous dimension • − 2 3 2 2 Typical one-loop values: g g g β(g) = β0 16π2 γm(g) = γm,0 16π2 γ(g) = γ0 16π2 Solution of RG equation: • 1/α
3 dg β0g = 2 β0 > 0 d ln µ 16π ⇒
α(µ) = α(µ0) α(µ0) µ 1− β0 ln( ) β0 < 0 2π µ0 ln µ 42/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Non-Abelian Gauge (Yang-Mills) theories Yang/Mills,1954 Local symmetry based on a (semi-)simple Lie group or direct product: • a a h a bi c a a φi(x) exp [igθ (x)T ] ψj (x) T ,T = ifabcT Dµ,ij = ∂µδij +igT A → ij ij µ Matter-gauge boson vertex contains a non-Abelian generator: •
µ, a gψT a /Aaψ igγ T a − ij ⇒ − µ ij Non-Abelian field strength tensor: • [D ,D ] = igF a T a = ig ∂ Aa ∂ Aa gf Ab Ac T a µ ν µν − µ ν − ν µ − abc µ ν does contain gauge boson self interactions: p, µ τ σ −gf abc(p − k)ν ηµσ −ig2f abcf ade(ηµσ ηντ − ηµτ ηνσ ) +(q − p)σ ηµν +f abdf ace(ηµν ηστ − ηµτ ηνσ ) a a d k, σ µ νσ abe acd µν στ µσ ντ c +(k − q) η b +f f (η η − η η ) b c q, ν µ ν
Dynkin index T : tr T aT b = T δ T = 1 Quadratic R R R R ab → F 2 a a 4 Casimir: T T = CR1 Cadj. := CA 3 Cfund. := CF R R → → 3 a X 43/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Quantization of Yang-Mills theories Use path integral representation of Green’s functions • Summing over physically equivalent field configurations (gauge orbits) • Gauge fixing: Choose one configuration per space-time point • Can be written as a functional determinant Faddeev/Popov, 1967 • Leads to gauge-fixing/ghost Lagrangian: •
1 2 a µ ab b LGF +FP = − (∂ · A) − c¯ ∂ D c 2ξ µ
First term leads to invertible gluon propagator • Faddeev-Popov ghosts c, c¯: fermionic scalars!!! cancel unphysical • longitudinal and scalar gluon modes, preserve S-Matrix unitarity • k a µ, a ν, b −i kµkν 2 ηµν − (1 − ξ) 2 δab k +i k p µ, c − gfabcpµ a b i δ k2+i ab k b Ghosts decouple in QED • After gauge fixing: gauge invariance lost, remainder global, non-linear • BRST symmetry Becchi/Rouet/Stora,1976, Tyutin, 1975, Batalin/Vilkoviskiy, 1976 44/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 QCD: Asymptotic freedom, Confinement
I Quantum Chromodynamics (QCD) is SU(3) Yang-Mills theory of strong interactions
I QCD β function is negative! Gross,Politzer,Wilczek, 1973
11 2 β = C N T C N 11 < 0 0 F f R− 3 A → 3 f −
I Asymptotic Freedom: αs 0 for µ Quarks quasi-free particles→ (Antiscreening→ ∞ of YM field)
I Confinement/Infrared Slavery: Landau pole: α for µ Λ 0.2 GeV s → ∞ ∼ QCD ∼ I QCD forms bound states at scale (1 3) ΛQCD: mesons (qq¯) and baryons (qqq) − × 45/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Discovery of QCD: Deep Inelastic Scattering (DIS)
1969 SLAC electron beam to hadronic • fixed target Cross section const., no 1/s drop-off •
dσ R 1 P 2 2 (P ) = 0 dx f Ff (x, Q )× Q2 dQ dσ (e−f → e−f, xP ) xP dQ2
P
I Bjorken scaling Ff (x) const.: scattering at quasi-free∼ partons
Bjorken, Feynman, 1969
I Scaling violations: 2 2 Ff (x) := Ff (x, Q ) ln Q described by logarithmically enhanced∼ higher order QCD radiative corrections (Altarelli-Parisi equations) 46/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Proofs: R Ratio, Jet Events, and all that...
e+e− annihilation into hadrons
+ − + − 4πα2 σ(e e µ µ ) := σ0 = 3s + − → P 2 σ(e e hadrons) = σ0 Nc Q → · · f f σ(e+e− hadrons) 5 R = → = below 2ms σ(e+e− µ+µ−) 3 →
I Gluon discovery: 3 jets @ DORIS/PETRA Timm/Wolf (DESY), 1979
I x1 = Eq/Ee, x2 = Eq¯/Ee, x3 = Eg/Ee
dσ + − (e e → qqg¯ ) = σ0× dx1dx2 2 2 X 2 2αs x + x 3 Q 1 2 f − − f 3π (1 x1)(1 x2) 46/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Proofs: R Ratio, Jet Events, and all that...
e+e− annihilation into hadrons
+ − + − 4πα2 σ(e e µ µ ) := σ0 = 3s + − → P 2 σ(e e hadrons) = σ0 Nc Q → · · f f σ(e+e− hadrons) R = → = 2 below 2mc σ(e+e− µ+µ−) →
I Gluon discovery: 3 jets @ DORIS/PETRA Timm/Wolf (DESY), 1979
I x1 = Eq/Ee, x2 = Eq¯/Ee, x3 = Eg/Ee
dσ + − (e e → qqg¯ ) = σ0× dx1dx2 2 2 X 2 2αs x + x 3 Q 1 2 f − − f 3π (1 x1)(1 x2) 46/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Proofs: R Ratio, Jet Events, and all that...
e+e− annihilation into hadrons
+ − + − 4πα2 σ(e e µ µ ) := σ0 = 3s + − → P 2 σ(e e hadrons) = σ0 Nc Q → · · f f σ(e+e− hadrons) 10 R = → = below 2mb σ(e+e− µ+µ−) 3 →
I Gluon discovery: 3 jets @ DORIS/PETRA Timm/Wolf (DESY), 1979
I x1 = Eq/Ee, x2 = Eq¯/Ee, x3 = Eg/Ee
dσ + − (e e → qqg¯ ) = σ0× dx1dx2 2 2 X 2 2αs x + x 3 Q 1 2 f − − f 3π (1 x1)(1 x2) 46/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Proofs: R Ratio, Jet Events, and all that...
e+e− annihilation into hadrons
+ − + − 4πα2 σ(e e µ µ ) := σ0 = 3s + − → P 2 σ(e e hadrons) = σ0 Nc Q → · · f f σ(e+e− hadrons) 11 R = → = below 2mt σ(e+e− µ+µ−) 3 →
I Gluon discovery: 3 jets @ DORIS/PETRA Timm/Wolf (DESY), 1979
I x1 = Eq/Ee, x2 = Eq¯/Ee, x3 = Eg/Ee
dσ + − (e e → qqg¯ ) = σ0× dx1dx2 2 2 X 2 2αs x + x 3 Q 1 2 f − − f 3π (1 x1)(1 x2) 46/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Proofs: R Ratio, Jet Events, and all that...
e+e− annihilation into hadrons
+ − + − 4πα2 σ(e e µ µ ) := σ0 = 3s + − → P 2 σ(e e hadrons) = σ0 Nc Q → · · f f σ(e+e− hadrons) 11 R = → = below 2mt σ(e+e− µ+µ−) 3 →
I Gluon discovery: 3 jets @ DORIS/PETRA Timm/Wolf (DESY), 1979
I x1 = Eq/Ee, x2 = Eq¯/Ee, x3 = Eg/Ee
dσ + − (e e → qqg¯ ) = σ0× dx1dx2 2 2 X 2 2αs x + x 3 Q 1 2 f − − f 3π (1 x1)(1 x2) 46/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Proofs: R Ratio, Jet Events, and all that...
e+e− annihilation into hadrons
+ − + − 4πα2 σ(e e µ µ ) := σ0 = 3s + − → P 2 σ(e e hadrons) = σ0 Nc Q → · · f f σ(e+e− hadrons) 11 R = → = below 2mt σ(e+e− µ+µ−) 3 →
I Gluon discovery: 3 jets @ DORIS/PETRA Timm/Wolf (DESY), 1979
I x1 = Eq/Ee, x2 = Eq¯/Ee, x3 = Eg/Ee
dσ + − (e e → qqg¯ ) = σ0× dx1dx2 2 2 X 2 2αs x + x 3 Q 1 2 f − − f 3π (1 x1)(1 x2) 47/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Factorization Theorem Cornerstone of perturbative QCD: M γ * • L Φ Meson distribution Factorization theorem Sterman, 1979; Collins/Soper, 1981 amplitude H Hard scattering Hadronic cross sections can be split into process • x −xx I Perturbative part: hard scattering process 1 1 I Non-perturbative part: f Generalized parton distribution parton distribution functions I Non-perturbative part: jet or fragmentation functions t Hard scattering cross sections perturbatively calculable • Parton distribution and fragmentation functions from experimental fits • Perturbative evolution of PDFs/fragmentation func. to different scales: • Z 1 d αs(Q) dz X x x Fg (x, Q) = Pg←q (z) Fq ( ,Q) + Fq¯( ,Q) d ln Q π x z f z z x x + Pg←g ( ,Q)Fg ( ,Q) z z 1 d αs(Q) Z dz x x x Fq (x, Q) = Pq←q (z)Fq ( ,Q) + Pq←g ( ,Q)Fg ( ,Q) d ln Q π x z z z z d dz x x x Fq (x, Q) = Pq¯←q¯(z)Fq¯( ,Q) + Pq¯←g ( ,Q)Fg ( ,Q) d ln Q z z z z 48/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Hadronic Cross Sections, PDFs
Parton Distribution Functions (PDFs): Ff (x)dx = prob. of finding • constituent f with longitudinal momentum fraction x Momentum sum rule: 1 dxx F (x) + F (x) + F (x) = 1 • 0 q q q¯ q¯ g h i • 1 R P P 1 Charge sum rules: dx [F (x) F (x)] = 2, dx [F (x) F ¯(x)] = 1 0 u − u¯ 0 d − d • PDFs Have to beR fitted from experiments R • Hadronic cross section are calculated according to the factorization theorem:
σ(p(P1) + p(P2) Y + X) = → Z 1 Z 1 X dx1 dx2 Ff (x1)F ¯(x2) f · 0 0 f
σshowered(f(x1P1) + f¯(x2P2) Y ) → · Y F˜(Y M,B) → Y • Parton shower describes QCD radiation 48/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Hadronic Cross Sections, PDFs
Parton Distribution Functions (PDFs): Ff (x)dx = prob. of finding • constituent f with longitudinal momentum fraction x Momentum sum rule: 1 dxx F (x) + F (x) + F (x) = 1 • 0 q q q¯ q¯ g h i • 1 R P P 1 Charge sum rules: dx [F (x) F (x)] = 2, dx [F (x) F ¯(x)] = 1 0 u − u¯ 0 d − d • PDFs Have to beR fitted from experiments R • Hadronic cross section are calculated according to the factorization theorem:
σ(p(P1) + p(P2) Y + X) = → Z 1 Z 1 X dx1 dx2 Ff (x1)F ¯(x2) f · 0 0 f
σshowered(f(x1P1) + f¯(x2P2) Y ) → · Y F˜(Y M,B) → Y • Parton shower describes QCD radiation 49/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Lattice QCD
Discretize space and time • Solve Yang-Mills equation of motion numerically • Gluon fields are links between lattice points • Fermions sit on the sites of the lattice • Problem: artefacts from continuum limit •
I Hadron spectra are ”measured” on the lattice
I mπ is the usual input
I Old days: ”quenched” (no fermions)
I One sees V (r) r confinement potential ∼ 49/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Lattice QCD
Discretize space and time • Solve Yang-Mills equation of motion numerically • Gluon fields are links between lattice points • Fermions sit on the sites of the lattice • Problem: artefacts from continuum limit •
I Hadron spectra are ”measured” on the lattice
I mπ is the usual input
I Old days: ”quenched” (no fermions)
I One sees V (r) r confinement potential ∼ 50/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Part III (2. Abend)
The Noble Hero: Hidden Symmetries (formerly known as Spontaneous Symmetry Breaking) 51/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Basics of Hidden Symmetries Hidden symmetry is obeyed by the Lagrangian (and the E.O.M.) • It is not respected by the spectrum, especially the ground state • In principle only possible in a system of infinite volume • • Nambu-Goldstone Theorem Goldstone, 1961; Nambu, 1960; Goldstone/Salam/Weinberg, 1962 For any broken symmetry generator of a global symmetry there is a massless boson (Nambu-Goldstone boson) in the theory. Two cases: i) Qa 0 = 0 a unbroken or Wigner-Weyl phase | i ∀ ii) Qa 0 = 0 for at least one a Nambu-Goldstone phase | i 6 ⇒ I Simple proof:
a a ∂ a φi iθ Tikφk V Tijφj = 0 → ⇒ ∂φi ⇒ ∂2 ∂ V T a 0 φ 0 + V T a = 0 ∂φ ∂φ jk h k i ∂φ ji i j h0 φ 0i j h0 φ 0i
2 =0 =(m )ij | {z } | {z } 52/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Nambu-Goldstone Theorem N-component real scalar field, possesses O(N) symmetry • 1 µ2 g = (∂ φT )(∂µφ) φT φ (φT φ)2 with φ = (φ , . . . , φ ) L 2 µ − 2 − 4 1 N
V V µ > 0 µ < 0 Minimizing the potential: φ = 0 (metastable) or h i φT φ φ T φ = µ2/g > 0 h i ∼ h i h i −
φ φ Without loss of generality: φ = (0, 0,..., 0, φ ) VEV in n-th comp. • h ii h N i Mass squared matrix: • 2 ! 2 ∂ V (φ) 0(N−1)×(N−1) 01×(N−1) (M )ij = = 2g φi φj = ∂φ ∂φ h i h i 0 2g φ 2 i j φ=hφi (N−1)×1 h i O(N) symmetry group broken down to O(N 1) symmetry group • − # broken symmetry generators = # Goldstone bosons = • 1 N(N 1) 1 (N 1)(N 2) = N 1 2 − − 2 − − − 53/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Chiral Symmetry Breaking in QCD
I Light quarks (almost) massless SU(2) SU(2) global symmetry ⇒ L × R u Q := = Q /DQ = Q /DQ + Q /DQ d L L L R R (mass term: mQQ = mQ Q + Q Q − − L R R L I Rewrite left- and right-handed rotations into vector and axial transformations: u u u exp i ~σ θ~ exp i ~σ θ~ exp i ~σ θ~ exp i ~σ θ~ γ5 d → 2 LPL 2 RPR d ⇒ 2 V 2 A d h i h i h i h i I For massive quarks, the axial Noether current is not conserved: → ∂ J~µ = 0 ∂ J~µ = 2mQ ~σ γ5Q m 0 0 µ V µ A − 2 −→ I If SU(2)A were exact: h TA h degenrate opposite parity pairs of hadrons, not seen in Nature| i ⇒ | i I SU(2) hidden symmetry T~ h = h + ~π A A | i | i I Pions are the Nambu-Goldstone bosons (NGB) of spontaneously broken SU(2)A 54/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 What breaks chiral symmetry?
I Strong interactions (QCD) make quark-antiquark pairs condens (like Cooper pairs in a BCS superconductor)
3 3 I Quark condensate: (300 MeV) Λ qq is invariant under ∼ QCD ∼ h i SU(2)V , but breaks SU(2)A
I Axial current generates pion states:
0 J µ,a π = iF δ pµeipπ x F = 184 MeV from pion decay h | A | bi π ab π π
I Explicit breaking of SU(2)A by finite quark masses
I Pions only approximate NGBs, i.e. pseudo NGBs (pNGBs):
1 2 4(mu + md) 2 (uu + dd) mπ = h2 i Fπ
± 0 I Difference m(π ) m(π ) 5 MeV from electromagnetic quantum corrections − ≈
I Include strange quark: SU(3) SU(3) stronger broken ms 95 MeV vs. m 2 6 MeV × ∼ u,d ∼ − 55/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Hidden Local Symmetries Anderson, 1961; Higgs, 1964; Brout/Englert, 1964; Kibble 1964 Consider scalar electrodynamics: • 1 λ = F F µν + (D φ)†(Dµφ) V (φ) V (φ) = µ2 φ 2 + ( φ 2)2 L −4 µν µ − − | | 2 | |
• Remember the gauge trafos: A A + ∂ θ(x), φ(x) exp[ ieθ(x)]φ(x) µ µ µ √ → → − iα I Minimize√ the√ potential ⇒ hφi = v/ 2e where v/ 2 = µ/ λ
I Radial excitation: ”Higgs field”
I Phase is the NGB
i π(x) φ(x) = √1 (v + h(x))e v 2 I Evaluating the kinetic term
1 e2 1 2 D φ 2 = (∂h)2 + (v + h)2 A ∂ π A | µ | 2 2 µ − ev µ µ Mixture between gauge boson and NGB. Define B := A 1 ∂ π • µ µ − ev µ Field strength term does not change under this redefinition • 55/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Hidden Local Symmetries Anderson, 1961; Higgs, 1964; Brout/Englert, 1964; Kibble 1964 Consider scalar electrodynamics: • 1 λ = F F µν + (D φ)†(Dµφ) V (φ) V (φ) = µ2 φ 2 + ( φ 2)2 L −4 µν µ − − | | 2 | |
• Remember the gauge trafos: A A + ∂ θ(x), φ(x) exp[ ieθ(x)]φ(x) µ µ µ √ → → − iα I Minimize√ the√ potential ⇒ hφi = v/ 2e where v/ 2 = µ/ λ
I Radial excitation: ”Higgs field”
I Phase is the NGB
i π(x) φ(x) = √1 (v + h(x))e v 2 I Evaluating the kinetic term
1 e2 1 2 D φ 2 = (∂h)2 + (v + h)2 A ∂ π A | µ | 2 2 µ − ev µ µ Mixture between gauge boson and NGB. Define B := A 1 ∂ π • µ µ − ev µ Field strength term does not change under this redefinition • 55/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Hidden Local Symmetries Anderson, 1961; Higgs, 1964; Brout/Englert, 1964; Kibble 1964 Consider scalar electrodynamics: • 1 λ = F F µν + (D φ)†(Dµφ) V (φ) V (φ) = µ2 φ 2 + ( φ 2)2 L −4 µν µ − − | | 2 | |
• Remember the gauge trafos: A A + ∂ θ(x), φ(x) exp[ ieθ(x)]φ(x) µ µ µ √ → → − iα I Minimize√ the√ potential ⇒ hφi = v/ 2e where v/ 2 = µ/ λ
I Radial excitation: ”Higgs field”
I Phase is the NGB
i π(x) φ(x) = √1 (v + h(x))e v 2 I Evaluating the kinetic term
1 e2 1 2 D φ 2 = (∂h)2 + (v + h)2 A ∂ π A | µ | 2 2 µ − ev µ µ Mixture between gauge boson and NGB. Define B := A 1 ∂ π • µ µ − ev µ Field strength term does not change under this redefinition • 56/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Higgs Mechanism VEV generates mass term for the gauge boson • Gauge boson mass: only consistent (renormalizable) way ‘t Hooft/Veltman, 1971 •
• 1 µν 1 2 m 1 2 1 2 3 4 = Fµν F + M BµB u + (∂h) mhh gh,3h gh,4h L − 4 2 B 2 − 2 − − 2 2 2 2 mh 2 mh with m = λv MB = ev gh,3 = m = h 2v h 8v2 Higgs field generates particle masses proportional to its VEV and ¨ • its coupling to that particle
2 2 MB MB © 2 v 2 v2 Feynman rules
2 2 mh mh 3 v 3 v2
Hey, what happened to the Nambu-Goldstone theorem?? • Longitudinal polarisation now becomes physical, Goldstone boson takes over its place in cancelling unphysical degrees of freedom. ¨
© 57/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Electroweak Standard Model
I Standard Model (SM) is SU(3) SU(2) U(1) gauge theory c × L × Y I Nuclear forces known since 1930s
I QCD (SU(3)c) proven to be the correc theory in 1968-1980 (DIS, e+e− jets at SLAC/DESY) →
I Weak interactions known since 1895 (beta decay)
I Charged current weak processes, e.g. − − muon decay µ e ν¯ ν Fermi, 1934 → e µ I Weak interactions couple only to left-handed particles Wu, 1957; Goldhaber, 1958
I Discovery of neutral currents in ν-nucleus scattering 1973, discrepancy in strength to charged current weak mixing angle ⇒ I Production of W , Z bosons (CERN, 1983) 58/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Lagrangian and its particles in totaliter
Building blocks (SU(3) ,SU(2) ) quantum numbers: • c L U(1)Y
QL uR dR LL eR H νR
(2, 3) 1 (1, 3) 4 (1, 3) 2 (2, 1)1 (1, 1)−2 (2, 1)1 (1, 1)0 3 3 − 3
I All renormalizable interactions possible with these fields:
1 1 = ψ /Dψ tr [G Gµν ] tr [W W µν ] LSM − 2 µν − 2 µν ψ=Q,u,d,L,e,H,νX 1 B Bµν + Y uQ H†u + Y dQ Hd − 4 µν L R L R + Y eL He +Y nL H†ν + µ2H†H λ(H†H)2 L R L R − 59/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Electroweak Symmetry Breaking
Higgs vev H = (0, v/√2) breaks SU(2)L U(1)Y U(1)em. • h ~i ~σ 0 × → Dµφ = (∂µ + igWµ 2 + ig YBµ)φ Electroweak gauge boson mass term: • 0 0 1 ~σ g µ ~σ g µ 0 ∆ = (0, v) gW~ µ + Bµ gW~ + B L 2 2 2 2 2 v Three massive vector bosons W ±, Z • ± 1 1 2 1 1 3 0 v p 2 0 2 Wµ = √ Wµ iWµ mW = gv Zµ = gWµ g Bµ mZ = g + g 2 ∓ 2 √g2+g0 2 − 2 Orthogonal combination remains massless photon • 1 0 3 Aµ = g Wµ + gBµ mA = 0 √g2+g0 2
Rewrite the covariant derivative: σ± = 1 (σ1 σ2) • 2 ± 0 g + + − − 1 2 3 0 2 gg 3 Dµ = ∂µ+i √ (Wµ σ +Wµ σ )+i p Zµ(g T g Y )+i p Aµ(T +Y ) 2 g2 + g0 2 − g2 + g0 2 g g0 Weak mixing angle: cos θW = , sin θW = √g2+g0 2 √g2+g0 2 Gell-Mann–Nishijima relation: Q = T 3 + Y 59/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Electroweak Symmetry Breaking
Higgs vev H = (0, v/√2) breaks SU(2)L U(1)Y U(1)em. • h ~i ~σ 0 × → Dµφ = (∂µ + igWµ 2 + ig YBµ)φ Electroweak gauge boson mass term: • 0 0 1 ~σ g µ ~σ g µ 0 ∆ = (0, v) gW~ µ + Bµ gW~ + B L 2 2 2 2 2 v Three massive vector bosons W ±, Z • ± 1 1 2 1 1 3 0 v p 2 0 2 Wµ = √ Wµ iWµ mW = gv Zµ = gWµ g Bµ mZ = g + g 2 ∓ 2 √g2+g0 2 − 2 Orthogonal combination remains massless photon • 1 0 3 Aµ = g Wµ + gBµ mA = 0 √g2+g0 2
Rewrite the covariant derivative: σ± = 1 (σ1 σ2) • 2 ± g + + − − 1 3 2 Dµ = ∂µ + i √ (Wµ σ + Wµ σ ) + i Zµ(T sin θW Q) + ieAµQ 2 cos θW − g g0 Weak mixing angle: cos θW = , sin θW = √g2+g0 2 √g2+g0 2 Gell-Mann–Nishijima relation: Q = T 3 + Y 60/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Electroweak Feynman Rules (all momenta outgoing)
f¯ f¯ f¯ g −i γ 2 cos θ µ g 5 W −ieγµQ −i √ γµ(1 − γ ) (T 3 − 2 sin2 θ Q ) f 2 2 f W f 5 3 Aµ ± Zµ −γ T Wµ f f f f
W + µ Zµ H
m2 m2 m2 2i W η 2i Z η −3i H v µν v µν v H H H
− Zν H Wν
W + µ H Zµ H H H
2 2 2 mW mW mH 2i ηµν 2i ηµν −3i v2 v2 v2
− H Zν H H H Wν
2 + −µ 1 2 2 H 2 Easily derivable: mW Wµ W + 2 mZ Z 1 + v 61/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Electroweak Feynman Rules (all momenta outgoing) + Wµ
h ρ µν µ νρ ν µρi −igWWX (k− − k+) η + (q − k−) η + (k+ − q) η gWWZ = g cos θW gW W γ = e Xµ
− Wν
+ µ W X h i σ 1 −ig 2ηµν ηστ − ηµσ ηνσ − ηµσ ηντ WWX1X2 2 gW W γγ = e 2 2 gWWZZ = g cos θW 2 gW W γZ = g cos θW sin θW 2 − Xν gWWWW = −g Wτ 2 62/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Fermion masses: Yukawa terms
Fermion mass terms mf (f LfR + f RfL) forbidden by U(2)L U(1)Y • gauge invariance − × Yukawa coupling is gauge invariant dimension-4 operator: • vYe H ∆ Y uk. = Ye(LL φ)eR eLeR 1 + L − · → − √2 v Again, Higgs boson couples proportional to mass: • f¯
mf 1 −i mf = Yf v v √2 H f
Hierarchy of Yukawa couplings according to fermion masses: Yt 1, • Y 10−2, Y 10−3, Y 10−5 ≈ c,τ,b ≈ µ,s ≈ e,ν,d ≈ Y 10−10, but Majorana mass term = 1 m νc ν possible • ν . LMajorana − 2 ν R R • 1 Y g2 Hγγ, Hgg couplings: c HF F µν v 16π2 · · µν 63/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Higgs: Properties and Search
Production: gluon/vector boson fusion decays predominantly into the heaviest particles b¯b hopeless: background! Detection of rare decays
102 pp → H (NNLO+NNLL QCD + NLO EW) s= 14 TeV
Complicated search: many LHC HIGGS XS WG 2010 H+X) [pb] 10 channels → pp → qqH (NNLO QCD + NLO EW) (pp
σ pp pp → → WH (NNLO QCD + NLO EW) high statistics necessary ZH (NNLO QCD +NLO EW) 1 pp → ttH (NLO QCD) γγ: mass determination
∗ 10•1 MH & 125 GeV: ZZ ```` → 100 200 300 400 500 1000 MH [GeV] 63/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Higgs: Properties and Search 1 WW bb Production: gluon/vector boson fusion •1 ττ gg tt 10 LHC HIGGS XS WG 2013 ZZ
cc decays predominantly into the heaviest 10•2 particles
γγ Zγ Higgs BR + Total Uncert [%] Uncert Total + BR Higgs ¯ 10•3 bb hopeless: background!
µµ Detection of rare decays 10•4 90 200 300 400 1000 MH [GeV]
102
Complicated search: many pp LHC HIGGS XS WG 2012 →
H+X) [pb] 10 H+X at
channels → s=14 TeV pp
(pp →
σ H+X at pp high statistics necessary 1 → s=8 TeV H+X at γγ: mass determination s=7 TeV 10•1 ∗ MH & 125 GeV: ZZ ```` → 100 200 300 400 500 600 700 800 900 1000 MH [GeV] 63/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Higgs: Properties and Search 1 WW bb Production: gluon/vector boson fusion •1 ττ gg ZZ 10 LHC HIGGS XS WG 2013
cc decays predominantly into the heaviest 10•2 particles
γγ Zγ Higgs BR + Total Uncert [%] Uncert Total + BR Higgs ¯ 10•3 bb hopeless: background!
µµ Detection of rare decays 10•4 80 100 120 140 160 180 200 MH [GeV]
→ γ γ ∫ -1 h L dt = 30 fb tth (h → bb) (no K-factors) h → ZZ(*) → 4 l Complicated search: many ATLAS h → WW(*) → lνlν 10 2 qqh → qq WW(*) qqh → qq ττ channels Total significance Signal significance high statistics necessary
10 γγ: mass determination
∗ MH & 125 GeV: ZZ ```` 1 → 100 120 140 160 180 200 2 Mh (GeV/c ) 63/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Higgs: Properties and Search 1 WW bb Production: gluon/vector boson fusion •1 ττ gg ZZ 10 LHC HIGGS XS WG 2013
cc decays predominantly into the heaviest 10•2 particles
γγ Zγ Higgs BR + Total Uncert [%] Uncert Total + BR Higgs ¯ 10•3 bb hopeless: background!
µµ Detection of rare decays 10•4 80 100 120 140 160 180 200 MH [GeV]
Complicated search: many channels high statistics necessary γγ: mass determination M 125 GeV: ZZ∗ ```` H & → 64/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 4.7.2012: The Discovery of the Higgs (?)
−1 I After roughly 5 fb data from 2011 and 2012: 4.7.2012 CERN-Seminar: We have found a scalar boson at 125.3 0.6 GeV ± 65/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
I Rolf Heuer (CERN DG, 4.7.12): As a layman I would say we have it! 66/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 2015: It’s the/a/sort of Higgs(-like) WTF
−1 I 2012 data: 25 fb compatible with EW precision ⇒
I Higgs compatible with EW precision measurements:
I Higgs measurement by far not precise enough!!! 67/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Digitization Science Fiction 68/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Flavor, the CKM matrix, and CP violation Three generations of fermions in Nature • Diagonalization of fermion mass matrices: • 2 † 2 2 2 † 2 † 2 2 2 † v YuYu = Ludiag(mu, mc , mt )Lu v YdYd = Lddiag(md, ms, mb )Ld
Rotation of quark fields leaves a trace in the charged current: • † uL /W (LuLd)dL = uL /WVCKM dL
CKM matrix: unitary, experimentally • almost diagonal
Three angles θ12, θ13, θ23, • one phase Phase violates CP (charge • conjugation and parity) After discovery of neutrino • oscillations: MNS matrix CKM describes flavor incredibly well • 69/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Part IV (3. Abend)
Gotterd¨ ammerung:¨ Beyond the Standard Model 70/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Standard Model of Particle Physics – Doubts
Measurement Fit |Omeas−Ofit|/σmeas 0 1 2 3 – describes microcosm (too well?) (5) ∆αhad(mZ) 0.02758 ± 0.00035 0.02768
mZ [GeV] 91.1875 ± 0.0021 91.1875
ΓZ [GeV] 2.4952 ± 0.0023 2.4957 σ0 [nb] 41.540 ± 0.037 41.477 had 28 free parameters Rl 20.767 ± 0.025 20.744 0,l Afb 0.01714 ± 0.00095 0.01645
Al(Pτ) 0.1465 ± 0.0032 0.1481
Rb 0.21629 ± 0.00066 0.21586
Rc 0.1721 ± 0.0030 0.1722 0,b Afb 0.0992 ± 0.0016 0.1038 0,c Afb 0.0707 ± 0.0035 0.0742
Ab 0.923 ± 0.020 0.935
Ac 0.670 ± 0.027 0.668
Al(SLD) 0.1513 ± 0.0021 0.1481 2 lept sin θeff (Qfb) 0.2324 ± 0.0012 0.2314
mW [GeV] 80.398 ± 0.025 80.374
ΓW [GeV] 2.140 ± 0.060 2.091
mt [GeV] 170.9 ± 1.8 171.3
0 1 2 3 form of Higgs potential ?
Hierarchy Problem chiral symmetry: δm v ln(Λ2/v2) f ∝ no symmetry for quantum corrections to the Higgs mass δM 2 Λ2 Λ M = 1019 GeV H ∝ ∼ Planck 20000 GeV2 = ( 1000000000000000000000000000000020000 –
1000000000000000000000000000000000000 ) GeV2 70/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Standard Model of Particle Physics – Doubts – describes microcosm (too well?) – 28 free parameters
– form of Higgs potential ?
Hierarchy Problem M [GeV] H chiral symmetry: δm v ln(Λ2/v2) 750 f ∝ no symmetry for quantum corrections to the 500 Higgs mass
250 2 2 2 19 2 2 δMH Λ MPlanck = (10 ) GeV Λ[GeV] ∝ ∼ 20000 GeV2 = ( 1000000000000000000000000000000020000 – 103 106 109 1012 1015 1018 1000000000000000000000000000000000000 ) GeV2 71/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Open questions 60 – Unification of all interactions (?) 50 U(1) 40 -1 −9 i 30 – Baryon asymmetry ∆N ∆N ¯ 10 α SU(2) B − B ∼ missing CP violation 20 Standard Model 10 SU(3)
– Flavour: three generations 0 102 104 106 108 1010 1012 1014 1016 1018 2 µ (GeV) – Tiny neutrino masses: m v ν ∼ M – Dark Matter:
I stable
I weakly interacting
I mDM ∼ 100 GeV
– Quantum theory of gravity
– Cosmic inflation
– Cosmological constant 72/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 73/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Ideas for New Physics since 1970
(1) Symmetry for elimination of quantum corrections – Supersymmetry: Spin statistics ⇒ corrections from bosons and fermions cancel each other – Little Higgs Models: Global symmetries ⇒ corrections from particles of like statistics cancel each other
(2) New Building Blocks, Substructure – Technicolor/Topcolor: Higgs bound state of strongly interacting particles
(3) Nontrivial Space-time structure eliminates Hierarchy – Extra Space Dimensions: Gravitation appears only weak – Noncommutative Space-time: space-time coarse-grained
(4) Ignoring the Hierarchy – Anthropic Principle: Parameters are as we observe them, since we observe them 74/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Supersymmetry (SUSY) Gelfand/Likhtman, 1971; Akulov/Volkov, 1973; Wess/Zumino, 1974
Q
FeÖÑiÓÒ – connects gauge and space-time symmetries |BÓ×ÓÒi | i Q – Multiplets with equal-mass fermions and
bosons J1 r
SUSY broken in Nature 1 r r ⇒ 2 0 r r
– Every particle gets a superpartner – Minimal Supersymmetric Standard Model (MSSM) – Mass eigenstates: Charginos: χ˜± = H˜ ±, W˜ ± Neutralinos: χ˜0 = H,˜ Z,˜ γ˜ 75/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 SUSY: Success and Side-Effects MSSM: spontaneous SUSY breaking E (SUSY partners in MeV range) Λ(?) Breaking in “hidden sector” Breaking mechanism induces 100 free F = O(1 TeV) parameters solves hierarchy problem: v = 246 GeV δM F log(Λ2) H ∝ 60 I Existence of fundamental scalars 50 U(1) I Form of Higgs potential
40 I light Higgs (M = 90 50 GeV) H ± -1 i 30 SU(2) I discrete R parity α 20 I SM particles even, SUSY partners odd Standard Model I prevents a proton decay too rapid 10 SU(3) I lightest SUSY partner (LSP) stable 0 0 Dark Matter χ˜1 102 104 106 108 1010 1012 1014 1016 1018 µ (GeV) I Unification of coupling constants 75/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 SUSY: Success and Side-Effects MSSM: spontaneous SUSY breaking E (SUSY partners in MeV range) Λ(?) Breaking in “hidden sector” Breaking mechanism induces 100 free F = O(1 TeV) parameters solves hierarchy problem: v = 246 GeV δM F log(Λ2) H ∝ 60 I Existence of fundamental scalars 50 MSSM U(1) I Form of Higgs potential 40 I light Higgs (M = 90 50 GeV) H ± -1 i 30 SU(2) I discrete R parity α 20 I SM particles even, SUSY partners odd SU(3) I prevents a proton decay too rapid 10 I lightest SUSY partner (LSP) stable 0 0 Dark Matter χ˜1 102 104 106 108 1010 1012 1014 1016 1018 µ (GeV) I Unification of coupling constants 75/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 SUSY: Success and Side-Effects MSSM: spontaneous SUSY breaking E (SUSY partners in MeV range) Λ(?) Breaking in “hidden sector” Breaking mechanism induces 100 free F = O(1 TeV) parameters solves hierarchy problem: v = 246 GeV δM F log(Λ2) H ∝ 800 m [GeV] I Existence of fundamental scalars 700 I Form of Higgs potential 600 g˜ t˜2 u˜ , d˜ L R ˜ I light Higgs (MH = 90 50 GeV) u˜R, d˜L b2 500 ˜ b1 ± I discrete R parity 400 H0,A0 H± 0 χ˜ ˜ χ˜04 2± t1 χ˜3 I SM particles even, SUSY partners odd
300 I prevents a proton decay too rapid
200 ˜lL τ˜2 I lightest SUSY partner (LSP) stable ν˜l χ˜0 χ˜ 2 1± 0 ˜lR τ˜1 h0 Dark Matter χ˜1 100 0 χ˜1 I Unification of coupling constants 0 76/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
”SUSY will be discovered, even if non-existent” 76/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
What, if not SUSY? experimentally constrained, but not ruled out
77/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Higgs as Pseudo-Goldstone boson: Technicolor
Nambu-Goldstone Theorem: Spontaneous breaking of a global sym- metrie: spectrum contains massless (Goldstone) bosons 1960/61
Color: Adler/Weisberger, 1965; Weinberg, 1966-69 Light pions as (Pseudo-)Goldstone bosons of spontaneously broken chiral symmetry
O(1 GeV) Λ Skala Λ: chiral symmetry breaking, Quarks, SU(3)C O(150 MeV) v Scale v: pions, kaons, . . . 77/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Higgs as Pseudo-Goldstone boson: Technicolor
Nambu-Goldstone Theorem: Spontaneous breaking of a global sym- metrie: spectrum contains massless (Goldstone) bosons 1960/61
Technicolor: Georgi/Pais, 1974; Georgi/Dimopoulos/Kaplan, 1984 Light Higgs as (Pseudo)-Goldstone boson of a new spontaneously bro- ken chiral symmetry
O(1 TeV) Λ Skala Λ: chiral symmetry breaking, techni-quarks, SU(N)TC O(250 GeV) v Skala v: Higgs, techni-pions
experimentally constrained, but not ruled out 78/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Collective Symmetry Breaking, Moose Models Collective Symmetry Breaking:
Arkani-Hamed/Cohen/Georgi/Nelson/. . . , 2001
2 different global symmetries; if one were unbroken Higgs exact Goldstone boson ⇒
Higgs mass only by quantum corrections of 2. order: M (0.1)2 Λ H ∼ ×
O(10 TeV) Λ Scale Λ: chiral SB, strong interaction O(1 TeV) F Scale F : Pseudo-Goldstone bosons, new gauge bosons O(250 GeV) v Scale v: Higgs 79/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Little-Higgs Models – Economic implementation of collective symmetry breaking
– New Particles: I Gauge bosons: Littlest Higgs Arkani-Hamed/Cohen/Katz/Nelson, 2002 γ0,Z 0,W 0 ± M [GeV] Z0 Φ 0 ± 1250 ±± W ± Φ I Heavy Fermions: Φ T T , U,C,... 1000 ΦP 0 U, C 750 γ I Quantum corrections to 500 MH cancelled by h 250 particles of like η t statistics W ± Z – “Little Big Higgs”: Higgs heavy (300 500 GeV) − – discrete T -(TeV scale) parity:
I allows for new light particles 0 I Dark matter: LTOP (lightest T-odd), often γ 80/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Extra Dimensions & Higgsless Models Motivation: String theory 30 −17 3 + n Space dimensions: Radius R 10 n ∼ cm Antoniadis, 1990; Arkani-Hamed/Dimopoulos/Dvali, 1998 Gravitation strong in higher dimensions Particles in quantum well: Kaluza-Klein tower Production of mini Black Holes at LHC
I “Higgsless Models”: Higgs component of higher-dim. gauge field
I “Large Extra Dimensions”: continuum of states
I “Warped Extra Dimensions”: discrete, resolvable resonances Randall/Sundrum, 1999
I “Universal Extra Dimensions”: also fermions/gauge bosons in higher dimensions 80/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Extra Dimensions & Higgsless Models Motivation: String theory 30 −17 3 + n Space dimensions: Radius R 10 n ∼ cm Antoniadis, 1990; Arkani-Hamed/Dimopoulos/Dvali, 1998 Gravitation strong in higher dimensions Particles in quantum well: Kaluza-Klein tower Production of mini Black Holes at LHC
I “Higgsless Models”: Higgs component of higher-dim. gauge field
I “Large Extra Dimensions”: continuum of states
I “Warped Extra Dimensions”: discrete, resolvable resonances Randall/Sundrum, 1999
I “Universal Extra Dimensions”: also fermions/gauge bosons in higher dimensions 81/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 KK parity and Dark Matter
I typical Kaluza-Klein spectra
700 1050 GH1,0L 1R = 500 GeV Μ GH1,1L 1R = 500 GeV 1000 Μ 650 3 H1,0L Q+ H1,0L 950 H1,0L T- 3 1,1 Q+ Q H L H1,0L + H1,1L 600 H1,0L U- 900 H1,1L T- D- Q+ H1,1L H1,1L U- 850 D- 550 mass
H1,0L mass WΜ H1,0L H1,0L 800 H L+ H1,1L H1,1L GH H1,0L H1,0L W 500 G E- 750 Μ H1,1L H1,1L H1,0L H H L+ BΜ EH1,1L H1,0L 700 H1,1L - WH BΜ H1,1L 450 WH BH1,0L 650 H H1,1L BH 400 600
I Spectrum structure similar to SUSY, but shifted in spin
I Dark matter: lightest KK-odd particle (LKP) Photon resonance γ0 (in 5D vector, in 6D scalar)
I Quote from SUSY orthodoxy: “This is a strawman’s model invented with the only purpose to be inflamed to shed light on the beauty of supersymmetry!” 82/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015
Noncommutative Space-time Wess et al., 2000 – Assumption: non-commuting Space-time coordinates [ˆxµ, xˆν ] = iθµν – Classical analogue: charged particle in lowest Landau level: x , x = 2c(B−1) /e { i j}P ij – Low energy limit of string theory Seiberg/Witten, 1999 – Yang-Landau-Theorem violated: Z γγ, gg possible → 13000 – Special direction in the Universe: -1 LHC ( s = 14 TeV; L = 10 fb ) standard model 12000 Λ = 100 GeV NC B = (1,0,0) broken rotational invariance K =-0.021; K =-0.340 11000 ZZ γ Z γγ E = (1,0,0) – Cross sections depend 10000 on azimuth 9000 8000 Varying signals as Number of Events 7000 ⇒ Earth rotates 6000 5000 0 1 2 3 4 5 6 Φ(γ) – Dark Matter, cosmology, theoretical problems E 83/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Which model? A Conspiracy Unmasked 84/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Challenge of the LHC QCD
Partonic subprocesses: qq, qg, gg WW no fixed partonic energy
proton - (anti)proton cross sections QCD
109 109
8 8 10 σ 10 tot 107 107 Tevatron LHC 34 −1 −1 106 106 R = σ = 10 cm s 105 105 LL σ b 4 4 High rates for t, W/Z, H, huge 10 10 1 - s
2
- ⇒ 3 3
10 10 m
c backgrounds
jet 33 2 σ (E > √s/20) 2
10 jet T 10 10 = )
1 σ 1 L W
10 10 r nb o ( σ Z f 0 0 c
ÂeØ p σ T 10 jet 10 e s
σ (E > 100 GeV) /
jet T s -1 -1 t n
10 10 e v e 10-2 10-2
-3 σ -3 10 t 10 p p jet -4 σ (E > √s/4) -4 ÏÏ 10 jet T 10 η -5 σ (M = 150 GeV) -5 10 Higgs H 10
-6 -6 10 σ 10 Higgs(MH = 500 GeV) 10-7 10-7 0.1 1 10 √s (TeV) e− 84/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 The Challenge of the LHC
Partonic subprocesses: qq, qg, gg no fixed partonic energy
proton - (anti)proton cross sections
109 109
8 8 10 σ 10 tot 107 107 Tevatron LHC 34 −1 −1 106 106 R = σ = 10 cm s 105 105 LL σ b 4 4 High rates for t, W/Z, H, huge 10 10 1 - s
2
- ⇒ 3 3
10 10 m
c backgrounds
jet 33 2 σ (E > √s/20) 2
10 jet T 10 10 = )
1 σ 1 L W
10 10 r nb o ( σ Z f 0 0 c σ 10 jet 10 e s
σ (E > 100 GeV) /
jet T s
-1 -1 t
n
10 10 e v e
10-2 10-2
-3 σ -3 10 t 10 jet -4 σ √ -4 ÏÏ 10 jet(ET > s/4) 10
-5 σ (M = 150 GeV) -5 η = 5 η = +5 10 Higgs H 10 − -6 -6 10 σ 10 Higgs(MH = 500 GeV) 10-7 10-7 η =0 0.1 1 10 √s (TeV) 85/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Search for New Particles 105 evt/10 GeV 1 = 100 fb− L Decay products of heavy particles: R
I high-pT Jets 104
I many hard leptons
Production of coloured particles 1000 weakly interacting particles only in decays
100 0 100 200 300 400 500 Dark Matter discrete parity (R, T , KK) ¯ ⇔ pT (b/b) [GeV] I only pairs of new particles ⇒ high energies, long decay chains
I Dark Matter ⇒ large missing energy in detector ( /ET ) Different Models/Decay Chains — same signatures 85/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Search for New Particles Decay products of heavy particles: q ℓ q ℓ 0 I high-pT Jets χ˜1 ℓ˜R q˜L I many hard leptons
ℓ˜R 0 Production of coloured particles χ˜1 q˜ q R ℓ weakly interacting particles only in decays q ℓ Dark Matter discrete parity (R, T , KK) ⇔ I only pairs of new particles ⇒ high energies, long decay chains
I Dark Matter ⇒ large missing energy in detector ( /ET ) Different Models/Decay Chains — same signatures 86/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Model Discrimination – A Journey to Cross-Roads
I Mass of new particles: end points of decay spectra
0.04
1500 0.035
-1 0.03 ) / 0.05
* § ¦ 0.025
θ ` q1 ` 1000 2 1 0.02 0.015 Events/1 GeV/100 fb
500 dp / d(cos 0.01 SUSY UED 0.005 PS 0 -1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 0 ˜§ 0 0 *
χ ` χ q˜L ˜ ˜ 0 20 40 60 80 100 cos θ 2 R 1 m(ll) (GeV)
I Spin of new particles: Spin of new particles: angular correlations, . . .
I Model determination: measuring coupling constants
Precise predictions for signals and backgrounds ⇒ – kinematic cuts – Exclusive multi particle final states 2 → 4 up to 2 → 10 – Quantum corrections: real and virtual corrections 87/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Outlook pedo mellon a minno.
I LHC: new era of physics
I New Particles, new symmetries, new interactions
I Dark Matter
I Interesting times!
“Will man nun annehmen, dass das abstrakte Denken das Hochste¨ ist, so folgt daraus, dass die Wissenschaft und die Denker stolz die Existenz verlassen und es uns anderen Menschen uberlassen,¨ das Schlimmste zu erdulden. Ja es folgt daraus zugleich etwas fur¨ den abstrakten Denker selbst, dass er namlich,¨ da er ja doch selbst auch ein Existierender ist, in irgendeiner Weise distrait sein muss.”
Søren Kierkegaard 88/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 Outlook pedo mellon a minno.
I LHC: new era of physics
I New Particles, new symmetries, new interactions
I Dark Matter
I Interesting times!
“Will man nun annehmen, dass das abstrakte Denken das Hochste¨ ist, so folgt daraus, dass die Wissenschaft und die Denker stolz die Existenz verlassen und es uns anderen Menschen uberlassen,¨ das Schlimmste zu erdulden. Ja es folgt daraus zugleich etwas fur¨ den abstrakten Denker selbst, dass er namlich,¨ da er ja doch selbst auch ein Existierender ist, in irgendeiner Weise distrait sein muss.”
Søren Kierkegaard 89/89J urgen¨ R. Reuter Theoretical Particle Physics DESY, 07/2015 One Ring to Find Them, One Ring to Rule Them Out?