Download Introduction to Theoretical Particle Physics

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Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Introduction to Theoretical Particle Physics
Jürgen R. Reuter
DESY Theory Group
Hamburg, 07/2015
2/89
Jürgen 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 Böhm/Denner/Joos, Gauge Theories of Strong and Electroweak Interactions, Springer,
2000
I Kugo, Eichtheorie, Springer, 2000 (in German)
I
...and many more
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Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
”I have nothing to offer but blood, toil, tears and sweat.”
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Jürgen R. Reuter
Theoretical Particle Physics
...but it is fun, though....
DESY, 07/2015
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Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Part I (Vorabend)
From Lagrangians to Feynman
Rules: Quantum Field Theory with
Hammer and Anvil
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Jürgen 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
• Schrödinger 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=
Z
dtL =
Z
3
dtd xL =
Z
4
d xL(φ, ∂µ φ)
⇒
∂µ
∂L
∂(∂µ φ)
=
∂L
∂φ
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Jürgen 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
L = (∂µ φ)∗ (∂ µ φ) − m2 |φ|2
⇒
~ 2 + m2 )φ = 0
( + m2 )φ ≡ (∂t2 − ∇
• Spin 1/2 particles: Dirac equation
L = iΨ(i/
∂ + m)Ψ
γµ =
0
σ̄ µ
σµ
0
⇒ (i/
∂ − m)Ψ = 0
σ µ = (1, ~σ ) σ̄ µ = (1, −~σ )
where
a
/ ≡ aµ γ µ
{γ µ , γ ν } = 2η µν 1
γ 5 = iγ 0 γ 1 γ 2 γ 3 is the chiral projector: PL/R = 21 (1 ∓ γ 5 )
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Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
The Dawn of gauge theories
• Spin 1 particles: Maxwell’s equations
1
L = − Fµν F µν with Fµν = ∂µ Aν − ∂ν Aµ +g [Aµ , Aν ]
4
⇒
Invariant under (local) gauge transformation, G:
1
Aµ −→ A0µ = GAµ G−1 − (∂µ G)G−1
g
Electric and magnetic fields are defined via:
h
i
~ = −A
~˙ − ∇A
~ 0 − g A0 , A
~
E
h
i
∂µ F µν = 0−i [Aµ , F µν ]
g ~
~
~
~
~
B = ∇ × A − 2 A×, A
h
i

~E
~ + g A·,
~ E
~
 0 = Dµ F µ0 = ∇
h
ii
 0 = D F µi = −Ė i + (∇
~ × B)
~ i −g A0 , E i + g A×,
~ B
~
µ
→
8/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
The Dawn of gauge theories
• Spin 1 particles: Maxwell’s equations
1
L = − Fµν F µν with Fµν = ∂µ Aν − ∂ν Aµ +g [Aµ , Aν ]
4
⇒
Invariant under (local) gauge transformation, G:
1
Aµ −→ A0µ = GAµ G−1 − (∂µ G)G−1
g
Electric and magnetic fields are defined via:
h
i
~ = −A
~˙ − ∇A
~ 0 − g A0 , A
~
E
h
i
∂µ F µν = −ig [Aµ , F µν ]
g ~
~
~
~
~
B = ∇ × A − 2 A×, A
h
i

~E
~ + g A·,
~ E
~
 0 = Dµ F µ0 = ∇
h
ii
 0 = D F µi = −Ė i + (∇
~ × B)
~ i −g A0 , E i + g A×,
~ B
~
µ
→
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Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
...for the curious...
• Spin 3/2 particles: Rarita-Schwinger equation
L=
1
1 µνρσ
Ψµ γ5 γν ∂ρ Ψν − mΨµ [γ µ , γ ν ] Ψν
2
4
⇒
m (/
∂ γ ν Ψν − γ ν ∂/ Ψν ) = 0
γ µ Ψµ = 0
∂ µ Ψµ = 0
(i/
∂ − m) Ψµ = 0
Leads only to sensible theory in supergravity
• Spin 2 particles: de Donder equation
L=−
4πGN p
| det g| R
with R = Rµµν ν and
c4
Rρσµν = ∂µ Γρνσ − ∂ν Γρµσ + Γρµλ Γλνσ − Γρνλ Γλµσ
p
Γρµν = 12 g ρσ (∂µ gνσ − ∂ν gµσ − ∂ρ gµν ) define gµν = ηµν + 16πGN hµν
hµν − 21 ηµν hρρ = ∂µ ∂ ρ hρν − 12 ηρν hσσ + (µ ↔ ν)
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Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
How to quantize a field (Analogous to QM)
• Field and its canonically conjugate momentum: π = ∂L/∂ φ̇
• Canonically transform to the Hamiltonian of the system:
2
∂L
~
− L
+ 12 m2 φ2
H= π
→ 12 π 2 + 12 ∇φ
∂ φ̇
φ̇=φ̇(φ,π)
• Impose equal-time commutation relations:
[φ(~x), π(~y )] = iδ 3 (~x − ~y )
[φ(~x), φ(~y )] = [π(~x), π(~y )] = 0
• Creation and annihilation operators to diagonalize the Hamiltonian:
[ap , a†p0 ] = 2Ep (2π)3 δ 3 (~
p − p~0 )
[ap~ , ap~0 ] = 0
• Heisenberg/interaction picture: (field) operators time dependent
• Creation/annihilation operators Fourier coefficients of field operators:
Z
Z
Z
d3 k
f ak e−ik·x + a† e+ik·x
f≡
φ(x) = dk
dk
k
(2π)3 (2Ek )
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Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
• Hamiltonian is a sum of harmonic oscillators:
Z
d3 p
Ep a†p ap + const.
H=
3
(2π)
• Infinite constant is abandoned by normal ordering renormalization
Z
d3 p
: H :=
Ep : a†p ap :
Note:
h0 : O : 0i = 0
(2π)3
• Creation operator creates a 1-particle state with well-defined
momentum (plane wave): a†p |0i = |pi
R
f eip·x |pi
• Field operator creates a 1-particle state at x: φ(x) |0i = dp
• Consider a complex field:
Z
f ak e−ik·x + b† e+ik·x ,
φ(x) = dk
k
φ† (x) =
Z
f bk e−ik·x + a† e+ik·x
dk
k
Field operator: positve frequency modes (e−ik·x ) have 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
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Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Microcausality and the Feynman propagator
• Amplitude for a particle to propagate from y to x:
Z
d4 p −ip(x−y)
D(x − y) ≡ h0 φ(x)φ(y) 0i =
e
(2π)4
|~
x−~
y |→∞
∼
e−m|~x−~y|
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
I
Particles propagated into the future (retarded)
Antiparticles propagated into the past (advanced)
DF (x − y) =
D(x − y) for x0 > y 0
D(y − x) for x0 < y 0
= θ(x0 − y 0 ) h0 φ(x)φ(y) 0i
+ θ(y 0 − x0 ) h0 φ(y)φ(x) 0i ≡ h0 T [φ(x)φ(y)] 0i
• Feynman propagator (time-ordered product, causal Green’s function)
Z
x p −→ y
d4 p
i
−ip(x−y)
DF (x − y) =
e
=
(2π)4 p2 − m2 + i
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Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
i prescription
• Solution of wave equation by Fourier transform:
F.T.
( + m2 )φ(x) = iδ 4 (x − y)
−→
i
i
= 0
φ̃(p) = 2
p − m2 +i
(p − Ep +i)(p0 + Ep −i)
p
with Ep = + p~ 2 + m2
p0
• Prescription tells you
where to propagate in time:
neg. frequ.:
0
e+ip
x0
−EP + i
+EP − i
pos. frequ.:
e−ip
0
x0
13/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
i prescription
• Solution of wave equation by Fourier transform:
F.T.
( + m2 )φ(x) = iδ 4 (x − y)
−→
i
i
= 0
φ̃(p) = 2
p − m2 +i
(p −Ep + i)(p0 +Ep − i)
p
with Ep = + p~ 2 + m2
p0
• Prescription tells you
where to propagate in time:
neg. frequ.:
0
e+ip
x0
−EP + i
+EP − i
pos. frequ.:
e−ip
0
x0
13/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
i prescription
• Solution of wave equation by Fourier transform:
F.T.
( + m2 )φ(x) = iδ 4 (x − y)
−→
i
i
= 0
φ̃(p) = 2
p − m2 +i
(p −Ep + i)(p0 +Ep − i)
p
with Ep = + p~ 2 + m2
p0
• Prescription tells you
where to propagate in time:
neg. frequ.:
0
e+ip
x0
−EP + i
+EP − i
pos. frequ.:
e−ip
0
x0
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Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Quantization of the Dirac field
• Spin-statistics theorem (Fierz/Lüders/Pauli, 1939/40):
I
I
Spin 0, 1, 2, . . .: bosons ⇒ commutators
Spin 21 , 32 , . . .: fermions ⇒ anticommutators
• equal-time anticommutators: ψ(x)α , ψ † (y)β = δ ( ~x − ~y )δαβ
• Field operators:
Z
ψ(x) =
f
dp
X
f
dp
X
asp us (p)e−ipx + bsp † v s (p)e+ipx
asp † us (p)e+ipx + bsp v s (p)e−ipx
s
Z
ψ(x) =
s
• Feynman propagator (time-ordered product, causal Green’s function)
SF (x−y) =
SF (x − y) =
− 0 ψ(x)ψ(y) 0 for x0 > y 0
= 0 T ψ(x)ψ(y) 0
0
0
− 0 ψ(y)ψ(x) 0 for x < y
Z
d4 p
i(/
p + m) −ip(x−y)
e
=
4
2
(2π) p − m2 + i
x
p
y
15/89
Jürgen 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
I
I
Asymptotically free quantum state for t → −∞
Interaction described completely local in space-time
Asymptotically free quantum state for t → +∞
• General axioms of QFT:
1.
2.
3.
4.
All eigenvalues of P µ are in the forward lightcone
There is a Poincaré-invariant vacuum state |0i
For every particle there is 1-particle state |pi
Asymptotic fields are free fields whose creation operators span a Fock
space (asymptotic completeness)
• Use the (Källen-Lehmann) spectral representation:
Z ∞
i
dµ2 ρ(µ2 ) 2
F.T. h0 T [Φ(x)Φ(y)] 0i =
2 + i
p
−
µ
0
16/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Z ∞
i
iZ
+
dµ2 ρ(µ2 ) 2
2 + i
p2 − m2 + i
p
−
µ
2
4m
Z: Wave function renormalization, m: renormalized mass
−→
ρ
Bound states
Continuum of ≥2P states
1P
state
m2
4m2
m2
4m2
p
2
µ2
• Asymptotic LSZ (Lehmann-Symanzik-Zimmermann) condition:
Heisenberg fields of full theory are asymptotically free fields (up to wave function and
mass renormalization)
Φ(x)
x0 →−∞
−→
√
Zφin (x)
x0 →+∞
Φ(x) −→
√
Zφout (x)
• Asymptotic fields obey free field equations! ⇒ Simple Fock spaces
n
o
Vin = a†in,k1 a†in,k2 . . . a†in,kn |0i
n
o
Assumption: Vin ≡ Vout ≡ V
Vout = a†in,k1 a†in,k2 . . . a†in,kn |0i
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Jürgen R. Reuter
The S-Matrix
Theoretical Particle Physics
DESY, 07/2015
Wheeler, 1939; Heisenberg 1943
• Scattering probability from initial to final state:
2
Prob. = |hβout αin i|
• S-Matrix transforms in into out states: hβout αin i = hβin S αin i
1. S-Matrix is unitary (prob. conserv.):
2. Transforms asymptotic field operators:
3. S-matrix invariant under symmetries:
I
S † S = SS † = 1
φout (x) = S † φin (x)S
[Q, S] = 0
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
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Jürgen R. Reuter
Theoretical Particle Physics
Step 1: LSZ formula
DESY, 07/2015
Lehmann/Symanzik/Zimmermann, 1955
Idea:
I
I
I
I
For t → ±∞ asymptotic fields are free
Project out creation operators from free field operators by inverse Fourier
transform
S-Matrix element is (upto normaliz.) Green’s function multi-particle pole
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 Y
n Z
i
4
ip y
2
(disconn. terms) + √
d yi e i i (yi + mi ) ·
Z
i=1


Z
l
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
%
18/89
Jürgen R. Reuter
Theoretical Particle Physics
Step 1: LSZ formula
DESY, 07/2015
Lehmann/Symanzik/Zimmermann, 1955
Idea:
I
I
I
I
For t → ±∞ asymptotic fields are free
Project out creation operators from free field operators by inverse Fourier
transform
S-Matrix element is (upto normaliz.) Green’s function multi-particle pole
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 Y
n Z
i
4
ip y
2
(disconn. terms) + √
d yi e i i (yi + mi ) ·
Z
i=1


Z
l
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
%
19/89
Jürgen R. Reuter
Theoretical Particle Physics
Step 2: Gell-Mann–Low formula
DESY, 07/2015
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 Schrödinger equation of IA picture time-evolution operator
as a (time-ordered) perturbation series
Z t
t→+∞
dt0 Hint (t0 )
−→
S
U (t) = T exp −i
−∞
I
Time evolution of field operators and also vacuum state (vacuum
polarization!)
L = Lkinetic + Lint
#
h0 T [Φ(x1 ) . . . Φ(xn )] 0i =
R
0 T φin (x1 ) . . . φin (xn ) exp i d4 xLint [φin (x)] 0
R
0 T exp i d4 xLint [φin (x)] 0
"
!
19/89
Jürgen R. Reuter
Theoretical Particle Physics
Step 2: Gell-Mann–Low formula
DESY, 07/2015
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 Schrödinger equation of IA picture time-evolution operator
as a (time-ordered) perturbation series
Z t
t→+∞
dt0 Hint (t0 )
−→
S
U (t) = T exp −i
−∞
I
Time evolution of field operators and also vacuum state (vacuum
polarization!)
L = Lkinetic + Lint
#
h0 T [Φ(x1 ) . . . Φ(xn )] 0i =
R
0 T φin (x1 ) . . . φin (xn ) exp i d4 xLint [φin (x)] 0
R
0 T exp i d4 xLint [φin (x)] 0
"
!
20/89
Jürgen 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:
(Note: Lint [φ(x)] is a polynomial of free fields!)
h0 T [φ(x1 )φ(x2 ) . . . φ(xn )] 0i
• 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) φi := φ(xi ) etc.
T [φ1 . . . φn ] = : φ1 . . . φn : + : φ1 . . . φn−2 : DF,n−1,n + permut.
h
: φ1 . . . φn−4 : DF,n−3,n−2 DF,n−1,n + DF,n−3,n−1 DF,n−2,n
i
+ DF,n−3,n DF,n−2,n−1 + perm. + . . . + product of only Feynman propagators
"
Q
• ⇒ h0 T [φ1 . . . φn ] 0i = all perm. DF,i,j
h0 T [φ1 φ2 φ3 φ4 ] 0i = DF,12 DF,34
1
!
for example:
2
+ DF,13 DF,24 + DF,14 DF,23 =
1
2
+
3
4
1
2
3
4
+
3
4
• Signs from fermion anticommutations arise from Wick’s theorem!
21/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Feynman rules (position space)
Example for L = 12 (∂µ φ)(∂ µ φ) − 12 m2 φ2 −
–
Propagator
–
Vertex
–
–
external point
divide by the
symmetry factor
y
x
z
x
λ 4
4! φ
DF (x − y)
R
(−iλ) d4 z
exp[−ipx · sgn(in/out)]
• Note: Position integrals correspond to QM superposition principle
• Example for symmetry factor:
yields a symmetry factor 3!
22/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Feynman rules (momentum space)
i/(p2 − m2 + i)
–
Propagator
–
Vertex
−iλ
–
–
–
external point
momentum conservation
integrate over
undetermined momenta
divide by symmetry fac.
1 (or u, v etc.)
at each vertex
R 4
d p/(2π)4
–
p −→
• Note: Momentum integrals correspond to QM superposition principle
Example in λφ3 theory: Mandelstam variables: s ≡ (p1 + p2 )2 t ≡ (p1 − p3)2 u ≡ (p1 − p4 )2
+
+
=
(−iλ)2
i
s−m2
+
i
t−m2
+
i
u−m2
23/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Bubbles, bubbles, vacuum bubbles
• Vacuum bubbles are infinite diagrams
∼
(Fermion loops: (-1))
2
δ 4 (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
R
0 T φ(x)φ(y) exp(−i dtHint (t)) 0
R
=
h0 T [Φ(x)Φ(y)] 0i =
0 T exp(−i dtHint (t))) 0

y
y
x
x
y
x
+
+
+
+
+ · · · exp 

exp 
+
+

+ ···

+ ···
23/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Bubbles, bubbles, vacuum bubbles
• Vacuum bubbles are infinite diagrams
∼
(Fermion loops: (-1))
2
δ 4 (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
R
0 T φ(x)φ(y) exp(−i dtHint (t)) 0
R
=
h0 T [Φ(x)Φ(y)] 0i =
0 T exp(−i dtHint (t))) 0


y
y
x
x
y
x
+
+
+ · · · exp 
+
+
+ ···


exp 
+
+
+ ···
24/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Fermion lines
• Signs from disentangling field operator contractions:
h0|bψ(x)ψ(x)ψ(y)ψ(y)ψ(z)ψ(z)b† |0i
p + k1 + k2 + k3 = q
u(p)
• External fermions:
k3
k2
p1
p2
k1
=
p
i(/
p + k/1 + m)
i(/
p + k/1 + k/2 m)
u(q)
2
2
(p + k1 ) − m + i (p + k1 + k2 )2 − m2 + i
us (p)
v s (p)
us (p)
v s (p)
25/89
Jürgen 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 · ρ1 · ρ2 · v · σ
• Constant of proportionality:
cross section
• effective scattering area (e.g. geometric scattering σ = πr2 )
• How to get the cross section?
1.
2.
3.
4.
P
2
Probability to get any final state |ni from initial state |αi:
n |hn|S|αi|
Project on a specific final state
Use Fermi’s Golden Rule
Momentum integral from projection becomes phase space integral over
final-state momenta
Flux factor, invariant matrix element, phase space measure
|Mβα |
2
dσα→β = p
4 (p1 · p2 )2 − m21 m22
n
Y
i=1
fi
dp
!
(2π)4 δ 4 (p1 + p2 −
n
X
i=1
qi )
26/89
Jürgen R. Reuter
Theoretical Particle Physics
• Analogous decay width Γ of a particle
2
dΓα→β
|Mβα |
=
2mα
n
Y
i=1
• 2 → 2 scattering depends only on
fi
dp
√
DESY, 07/2015
(Life time τ = 1/Γ)
!
(2π)4 δ 4 (p −
n
X
qi )
i=1
s and cos θ:
1 |~qout |
dσ
=
|Mβα |2
dΩ
64π 2 s |~
pin |
all masses equal
• Simple example, λφ4 theory:
Mβα =
−iλ
⇒
|Mβα |2 = λ2
σ
∝ 1/s
√
s
⇒
σ=
λ2 1
·
4π s
26/89
Jürgen R. Reuter
Theoretical Particle Physics
• Analogous decay width Γ of a particle
2
dΓα→β
|Mβα |
=
2mα
n
Y
i=1
• 2 → 2 scattering depends only on
fi
dp
√
DESY, 07/2015
(Life time τ = 1/Γ)
!
(2π)4 δ 4 (p −
n
X
qi )
i=1
s and cos θ:
1 |~qout |
dσ
=
|Mβα |2
dΩ
64π 2 s |~
pin |
all masses equal
• Simple example, λφ4 theory:
Mβα =
−iλ
⇒
|Mβα |2 = λ2
σ
∝ 1/s
√
s
⇒
σ=
λ2 1
·
4π s
27/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Quantization of the Electromagnetic Field
•
1
L = − Fµν F µν − jµ Aµ
4
• Euler-Lagrange equation Aµ − ∂ µ (∂ · A) = j µ invariant under gauge
transformation: Aµ (x) → Aµ (x) + ∂ µ f (x)
• Gauge invariance makes life hard (but for the wise easy!)
I
I
I
Ȧ0 absent ⇒ no Π0
Kinetic term is singular, hence not invertible
[Aµ (~
x), Aν (~
y )] = −iηµν δ 3 (~
x−~
y ) leads to negative and zero norm states
on Fock space!
Z
Aµ (x) =
f
dk
3 X
(λ) (λ)
(λ) † (λ) ∗
µ (k)e+ipx
ak µ (k)e−ipx + ak
λ=0
• Solution: gauge fixing: ∂ · A = 0
I
I
I
I
I
Physical states |αi: Transversal polarizations ((k) · k = 0)
Unphysical states |χi: longitudinal (space-like) pol./ scalar (time-like) pol.
Physical Fock space {|αi} contains only positive-norm states
|αi → |αi + |χi just corresponds to a gauge transformation
Gupta-Bleuler quantization:
hα|(∂ · A)|βi = 0
28/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Quantum Electrodynamics (QED)
I
Local (gauge) transformations
ψ(x) → exp[ieQel. θ(x)]ψ(x)
I
1
Aµ (x) → Aµ (x) + ∂µ θ(x)
e
Derivative terms are no longer invariant ⇒ covariant derivatives
∂µ ψf → Dµ,f ψf = (∂µ − ieQf Aµ ) ψf
LQED =
µ
X
f
1
ψ f (iD
/ f − mf ) ψf − Fµν F µν
4
ieQf γ µ
→ k
−iηµν
k2 +i
→ p
i(/
p+m)
p2 −m2 +i
µ
ν
µ
µ
us (p)
v s (p)
us (p)
v s (p)
∗µ (k)
µ (k)
29/89
Jürgen 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
∂µ J µ = 0
I
I
⇒
Z
d
Q=0
dt
[iQ, φ(x)] = δφ(x) symmetry of the Lagrangian
Q :=
d3 x J 0 (~x)
with
Current conservation implies Ward identities: (contact terms vanish on-shell)
0 = ∂xµ h0 T [J (x)φ1 (x1 ) . . . φn (xn )] 0i
Generalization for off-shell amplitudes
(and also non-Abelian gauge theories):
Slavnov-Taylor identities
kµ
Jµ
=0
30/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
e+ e− → µ+ µ−: A sample calculation
I
v(p2 )
u(q1 )
−ieQe γµ
−ieQµ γν
−iη µν
(p1 +p2 )2
u(p1 )
I
M = ie2 Qe Qµ (u(q1 )γ µ v(q2 ))
1
(v(p2 )γµ u(p1 ))
s
v(q2 )
Square the matrix element, sum over final state spins, average over
initial spins
Spin sums:
X
X
p + m1)αβ
usα (p)usβ (p) = (/
s=±
s=±
Use
X
(ur (p1 )γµ v(p2 )s )(ur (p1 )γν v s (p2 ))∗ =
r,s
=
X
r,s
X
s
vα
p − m1)αβ
(p)v sβ (p) = (/
(ur (p1 )γµ v s (p2 ))(v r (p2 )γν us (p1 ))
r,s
tr [(ur (p1 )γµ v s (p2 ))(v r (p2 )γν us (p1 ))] = tr [(/
p1 + m)γµ (/
p2 − m)γν ]
= 4(p1,µ p2,ν + p1,ν p2,µ ) − 2sηµν
31/89
I
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Neglect all masses:
[4(p1,µ p2,ν + p1,ν p2,µ ) − 2sηµν ] [4(q1,µ q2,ν + q1,ν q2,µ ) − 2sηµν ]
= 32(q1 p1 )(q2 p2 ) + 32(q1 p2 )(q2 p1 ) = 8(t2 + u2 ) = 4s2 (1 + cos2 θ)
I
σ=
Z
dΩ
1 1X
dσ
=
|M|2
dΩ
64π 2 s 2 s,r
=
e4 1
(2π)
32π 2 s 2
I
Sommerfeld’s fine structure constant
I
Result:
Z
1
d(cos θ)(1 + cos2 θ) =
−1
α = e2 /(4π) ∼ 1/137
σ(e+ e− → µ+ µ− ) =
4πα2
3s
4πα
3s
32/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Part II (1. Abend)
The Mighty Valkyrie:
Non-Abelian Gauge Theories and
Renormalization
33/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Quantum (a.k.a. radiative) corrections
• Real corrections: radiation of photons etc.
• Virtual (loop) corrections
Vacuum
polarization
Self energies
Vertex corrections
Box diagrams
• Example: e− anomalous magnetic moment g ψ 2i [γ µ , γ ν ] Fµν ψ
Dirac theory tree level: g = 2
α := e2 /4π
+
Experimentally:
⇒
α
= 2.00232282
g =2 1+
2π
g = (2.00231930436222 ± 0.00000000000148)
34/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Ultraviolet and infrared (mass) singularities
I
Ultraviolet singularities (in loops)
q
k →
k →
q+k
Z
d4 q tr[γµ (/q+m)γν (/q+/k+m)]
:= −iΣµν (k) = −e
2
2
2
2
(2π)4 (q −m )((q+k) −m )
Z
Z 4
d q
2
1,
q,
q
∼
dq q −1 , 1, q
−→
4
q
2
⇒ Logarithmic, linear, quadratic divergencies
I
Collinear and soft (mass or infrared) singularities
% k
→ p+k
→ p
1
1
=
(p + k)2 − m2
2p · k
1
=
with
2Ee Eγ (1 − βcos θ)
−→
Collinear singularity: cos θ → 1
β = pe /Ee ∼ 1
Soft singularity: Eγ → 0
35/89
Jürgen 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
phase space integrals
to D = 4 − 2 < 4
to D = 4 + 2 > 4
for ultraviolet singularities
for soft/collinear singularities
• Keep dimensionality of integrals ⇒ unphysical scale µ
Z
Z
dD q
d4 q
4−D
−→
µ
4
(2π)
(2π)D
• E.g. QED vacuum polarization:
Σµν (k) =
2α
π
Z
1
k 2 ηµν − kµ kν
dxx(1 − x)·
0
1
−x(1 − x)k 2 + m2
− γE + ln(4π) − ln
µ2
(γE = 0.577 . . . Euler constant)
∆ := 1/ − γE + ln(4π)
36/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Power Counting
• Question: Which Feynman diagrams do contain UV divergencies?
∼
Z
d4 k1 d4 k2 . . . dkL
k # Numerator
∼ # Denominator
2
2
(/
k i − m) . . . (kj ) . . . (kn )
k
• Define superficial degree of divergence D
(in D dimensions)
D = (# Numerator) − (# Denominator) = D · L − Ie − 2Iγ
Ee = # external e± , Eγ = # extern. γ, Ie = # internal e± , Iγ = # intern. γ, V = # vertices, L = # loops
I
Fermion propagators contribute k−1 , bosons k−2
Vertices with n derivatives contribute kn
Euler identity:
L = Ie + Iγ − V + 1
(by induction)
I
Simple line count:
I
I
V = 2Iγ + Eγ = 12 (2Ie + Ee )
• Power
counting master formula:
D = D(Ie + Iγ − V + 1) − Ie − 2Iγ = D − Eγ − 32 Ee
Depends only on number of external lines
37/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
QED Singularities – Classification of QFTs
D=4
D=3
(irrelevant, normalization)
= 0 (Furry’s theorem)
D=2
D=1
D = 0, Ward id., k2 ηµν − kµ kν
= 0 (Furry’s theorem)
D=0
D=1
finite, Ward identity
D = 0 (chirality)
D = 4 − Eγ − 32 Ee
D=0
Superrenormalizable Theory
D ∼ −I
Renormalizable Theory
D ∼0·I
Non-Renormalizable Theory
D ∼ +I
Only finite # of (sup.)
divergent graphs
Finite # of (sup.) div. ampl.,
but at all orders of pert. series
all amplitudes of sufficiently
high order diverge
37/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
QED Singularities – Classification of QFTs
D=4
D=3
(irrelevant, normalization)
= 0 (Furry’s theorem)
D=2
D=1
D = 0, Ward id., k2 ηµν − kµ kν
= 0 (Furry’s theorem)
D=0
D=1
finite, Ward identity
D = 0 (chirality)
D = 4 − Eγ − 32 Ee
D=0
Superrenormalizable Theory
D ∼ −I
Renormalizable Theory
D ∼0·I
Non-Renormalizable Theory
D ∼ +I
Coupling constants have foo o
positive mass dimension
Coupling constants are
dimensionless
Coupling constants have
negative mass dimension
38/89
Jürgen 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 Ze cancel
• Infinities cancel: there are finite shifts in physical parameters (g − 2!)
• Easier calculational prescription: Renormalized perturbation theory
I
I
I
Express Lagrangian of bare fields and parameters
L(φ0 , m0 , λ0 ) = 12 (∂φ0 )2 − 21 m20 φ20 − λ4!0 φ40 by renormalized ones:
√
φ0 = Zφren. eliminates wave function factor
Expand parameters/fields δZ = Z − 1, δm = m20 Z − m2 , δλ = λ0 Z 2 − λ
L=
I
1
1
λ
1
1
δλ
(∂φren )2 − m2 φ2ren − φ4ren + δZ (∂φren )2 − δm φren2 − φren4
2
2
4!
2
2
4!
This generates counterterms, e.g.
= −iλ
−→
= −iδλ
• Loop diagrams and counterterms added up are finite!
• Counterterms are not unique: on-shell scheme, MS scheme
39/89
Jürgen 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 = ~k 2
)
⊥,max
k
p
p−k
dσ e→γR (p, pR ) ∼
and virtual diagrams
2
α
Q
ln m
2
e
2π
1
Z
dx
0
1 + x2 e→R
dσ
(xp, pR ) + · · ·
1−x
2 Z 1
α
Q
ln m
dx(1 − x) + non-log terms
2
e
2π
0
Z 1
α
x
e→γR
Q2
dσvirt.
(p, pR ) ∼ −
ln m
dσ e→R (xp, pR )
dx
+ ···
2
e
2π
1−x
0
δZe ∼ −
k
p
p−k
soft singularity: me → 0
collinear singularity: x → 1
Soft-coll. singularities cancel between the three contributions
• Splitting function: Pee = (1 + x2 )/(1 − x) (cf. later)
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.)
40/89
Jürgen 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)
R
Rescale:
Consider S = dD x 12 (∂φ)2 − 12 m2 φ2 − Bφ4
x0 = λx
k0 = k/λ ∼ µ
φ0 =
q
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
dD xL
Z
=
µ<λΛ
m0 2
B0
C0
D0
=
=
=
=
d D x0
1
1
(∂φ0 )2 − m02 φ0 2 − B 0 φ0 4 − C 0 (∂φ)0 4 − D0 φ0 6 + . . .
2
2
(m2 + δm2 )(1 + δZ)−1 λ−2
(B + δB)(1 + δZ)−2 λD−4
(C + δC)(1 + δZ)−2 λD
(D + δD)(1 + δZ)−3 λ2D−6
m2 , B, C, D, . . . = 0
I
Close to a fix point:
I
Keep only linear terms
I
Relevant operators
I
Marginal operators
I
Irrelevant operators
40/89
Jürgen 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)
R
Rescale:
Consider S = dD x 12 (∂φ)2 − 12 m2 φ2 − Bφ4
x0 = λx
k0 = k/λ ∼ µ
φ0 =
q
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
dD xL
Z
=
µ<λΛ
m0 2
B0
C0
D0
=
=
=
=
d D x0
1
1
(∂φ0 )2 − m02 φ0 2 − B 0 φ0 4 − C 0 (∂φ)0 4 − D0 φ0 6 + . . .
2
2
m2 λ−2 grows for E → 0
BλD−4 const. for E → 0
CλD shrinks for E → 0
Dλ2D−6 shrinks for E → 0
m2 , B, C, D, . . . = 0
I
Close to a fix point:
I
Keep only linear terms
I
Relevant operators
I
Marginal operators
I
Irrelevant operators
40/89
Jürgen 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)
R
Rescale:
Consider S = dD x 12 (∂φ)2 − 12 m2 φ2 − Bφ4
x0 = λx
k0 = k/λ ∼ µ
φ0 =
q
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
dD xL
Z
=
µ<λΛ
m0 2
B0
C0
D0
=
=
=
=
d D x0
1
1
(∂φ0 )2 − m02 φ0 2 − B 0 φ0 4 − C 0 (∂φ)0 4 − D0 φ0 6 + . . .
2
2
m2 λ−2 grows for E → 0
BλD−4 const. for E → 0
CλD shrinks for E → 0
Dλ2D−6 shrinks for E → 0
m2 , B, C, D, . . . = 0
I
Close to a fix point:
I
Keep only linear terms
I
Superrenormalizable theory
I
Renormalizable theory
I
Nonrenormalizable theory
41/89
Jürgen R. Reuter
Theoretical Particle Physics
Renormalization Group Equation
DESY, 07/2015
Wilson, 1971; Callan, Symanzik, 1970
• Bare Green’s function do not depend on renormalization scale µ
0=µ
µ
i
d h −n/2
d (n)
(n)
Z
· Gren (g(g0 , m0 , µ), m(g0 , m0 , µ), µ, reg.)
G (g0 , m0 , reg.) = µ
dµ
dµ
∂
∂g ∂
+µ
−
∂µ
∂µ ∂g
−
1 ∂m
µ
m ∂µ
m
∂
1 dZ
(n)
−n −
µ
Gren = 0
∂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) = − 12 d(ln Z)/d(ln µ)
anomalous dimension
Typical one-loop values:
3
g
β(g) = β0 16π
2
• Solution of RG equation:
dg
d ln µ
g 2 (µ) =
2
g
γm (g) = γm,0 16π
2
2
g
γ(g) = γ0 16π
2
α
=
β0 g 3
16π 2
1−
g 2 (µ0 )
g 2 (µ0 )
β0 ln( µµ
8π 2
0
β0 < 0
⇒
β0 > 0
)
ln µ
41/89
Jürgen R. Reuter
Theoretical Particle Physics
Renormalization Group Equation
DESY, 07/2015
Wilson, 1971; Callan, Symanzik, 1970
• Bare Green’s function do not depend on renormalization scale µ
0=µ
i
d h −n/2
d (n)
(n)
Z
· Gren (g(g0 , m0 , µ), m(g0 , m0 , µ), µ, reg.)
G (g0 , m0 , reg.) = µ
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) = − 12 d(ln Z)/d(ln µ)
anomalous dimension
Typical one-loop values:
3
g
β(g) = β0 16π
2
• Solution of RG equation:
dg
d ln µ
g 2 (µ) =
2
g
γm (g) = γm,0 16π
2
2
g
γ(g) = γ0 16π
2
α
=
β0 g 3
16π 2
1−
g 2 (µ0 )
g 2 (µ0 )
β0 ln( µµ
8π 2
0
β0 < 0
⇒
β0 > 0
)
ln µ
41/89
Jürgen R. Reuter
Theoretical Particle Physics
Renormalization Group Equation
DESY, 07/2015
Wilson, 1971; Callan, Symanzik, 1970
• Bare Green’s function do not depend on renormalization scale µ
0=µ
i
d h −n/2
d (n)
(n)
Z
· Gren (g(g0 , m0 , µ), m(g0 , m0 , µ), µ, reg.)
G (g0 , m0 , reg.) = µ
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) = − 12 d(ln Z)/d(ln µ)
anomalous dimension
Typical one-loop values:
3
g
β(g) = β0 16π
2
• Solution of RG equation:
dg
d ln µ
g 2 (µ) =
2
g
γm (g) = γm,0 16π
2
2
g
γ(g) = γ0 16π
2
α
=
β0 g 3
16π 2
1−
g 2 (µ0 )
g 2 (µ0 )
β0 ln( µµ
8π 2
0
β0 < 0
⇒
β0 > 0
)
ln µ
41/89
Jürgen R. Reuter
Theoretical Particle Physics
Renormalization Group Equation
DESY, 07/2015
Wilson, 1971; Callan, Symanzik, 1970
• Bare Green’s function do not depend on renormalization scale µ
0=µ
i
d h −n/2
d (n)
(n)
Z
· Gren (g(g0 , m0 , µ), m(g0 , m0 , µ), µ, reg.)
G (g0 , m0 , reg.) = µ
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) = − 12 d(ln Z)/d(ln µ)
anomalous dimension
Typical one-loop values:
3
g
β(g) = β0 16π
2
• Solution of RG equation:
dg
d ln µ
α(µ) =
=
1−
β0 g 3
16π 2
2
g
γm (g) = γm,0 16π
2
2
g
γ(g) = γ0 16π
2
1/α
⇒
α(µ0 )
α(µ0 )
β0 ln( µµ
2π
0
β0 > 0
)
β0 < 0
ln µ
42/89
Jürgen 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:
h
φi (x) → exp [igθa (x)T a ]ij ψj (x)
i
T a , T b = ifabc T c
a a
Dµ,ij = ∂µ δij +igTij
Aµ
• Matter-gauge boson vertex contains a non-Abelian generator:
/ aψ
−gψTija A
µ, a
⇒
− igγµ Tija
• Non-Abelian field strength tensor:
a
[Dµ , Dν ] = igFµν
T a = −ig ∂µ Aaν − ∂ν Aaµ − gfabc Abµ Acν T a
does contain gauge boson self interactions:
p, µ
τ
a
k, σ
c
−gf abc (p − k)ν η µσ
+(q − p)σ η µν +(k − q)µ η νσ
σ
a
d
b
−ig 2 f abc f ade (η µσ η ντ − η µτ η νσ )
+f abd f ace (η µν η στ − η µτ η νσ )
+f abe f acd (η µν η στ − η µσ η ντ )
b
c
q, ν
µ
ν
tr TRa TRb = TR δab → TF =
Dynkin index TR :
X
Casimir:
TRa TRa = CR 1
a
Cadj. := CA → 3
1
2
Quadratic
Cf und. := CF →
4
3
43/89
Jürgen 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:
LGF +F P = −
1
2
a µ ab b
(∂ · A) − c̄ ∂ Dµ c
2ξ
• First term leads to invertible gluon propagator
• Faddeev-Popov ghosts c, c̄: fermionic scalars!!! cancel unphysical
longitudinal and scalar gluon modes, preserve S-Matrix unitarity
•
k
µ, a
a
ν, b
a
b
k
−i
k2 +i
ηµν − (1 − ξ)
kµ kν
k2
δab
p
i
δ
k2 +i ab
µ, c
− gfabc pµ
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
44/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
QCD: Asymptotic freedom, Confinement
I
I
Quantum Chromodynamics (QCD) is SU (3) Yang-Mills theory of
strong interactions
QCD β function is negative!
Gross,Politzer,Wilczek, 1973
β0 = CF Nf TR −
11
2
CA → Nf −11 < 0
3
3
I
Asymptotic Freedom:
αs → 0 for µ → ∞
Quarks quasi-free particles
(Antiscreening of YM field)
I
Confinement/Infrared Slavery:
Landau pole: αs → ∞ for µ ∼ ΛQCD ∼ 0.2 GeV
I
QCD forms bound states at scale (1 − 3) × ΛQCD : mesons (q q̄) and
baryons (qqq)
45/89
Jürgen 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
Q2
xP
dσ
dQ2
(P ) =
dσ
dQ2
R1
0
−
dx
P
f
Ff (x, Q2 )×
(e f → e− f, xP )
P
I
Bjorken scaling Ff (x) ∼ const.:
scattering at quasi-free partons
Bjorken, Feynman, 1969
I
Scaling violations:
Ff (x) := Ff (x, Q2 ) ∼ ln Q2 described
by logarithmically enhanced higher
order QCD radiative corrections
(Altarelli-Parisi equations)
46/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Proofs: R Ratio, Jet Events, and all that...
e+ e− annihilation into hadrons
σ(e+ e− → µ+ µ− )
σ(e+ e− → hadrons)
R=
I
Gluon discovery: 3 jets @
DORIS/PETRA Timm/Wolf (DESY), 1979
I
x1 = Eq /Ee , x2 = Eq̄ /Ee , x3 = Eg /Ee
dσ
+ −
(e e → q q̄g) = σ0 ×
dx1 dx2
X 2 2αs
x21 + x22
3
Qf
3π
(1
−
x1 )(1 − x2 )
f
2
:= σ0 = 4πα
3s
P
= σ0 · Nc · f Q2f
σ(e+ e− → hadrons)
5
=
σ(e+ e− → µ+ µ− )
3
below 2ms
46/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Proofs: R Ratio, Jet Events, and all that...
e+ e− annihilation into hadrons
σ(e+ e− → µ+ µ− )
σ(e+ e− → hadrons)
R=
I
Gluon discovery: 3 jets @
DORIS/PETRA Timm/Wolf (DESY), 1979
I
x1 = Eq /Ee , x2 = Eq̄ /Ee , x3 = Eg /Ee
dσ
+ −
(e e → q q̄g) = σ0 ×
dx1 dx2
X 2 2αs
x21 + x22
3
Qf
3π
(1
−
x1 )(1 − x2 )
f
2
:= σ0 = 4πα
3s
P
= σ0 · Nc · f Q2f
σ(e+ e− → hadrons)
=2
σ(e+ e− → µ+ µ− )
below 2mc
46/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Proofs: R Ratio, Jet Events, and all that...
e+ e− annihilation into hadrons
σ(e+ e− → µ+ µ− )
σ(e+ e− → hadrons)
R=
I
Gluon discovery: 3 jets @
DORIS/PETRA Timm/Wolf (DESY), 1979
I
x1 = Eq /Ee , x2 = Eq̄ /Ee , x3 = Eg /Ee
dσ
+ −
(e e → q q̄g) = σ0 ×
dx1 dx2
X 2 2αs
x21 + x22
3
Qf
3π
(1
−
x1 )(1 − x2 )
f
2
:= σ0 = 4πα
3s
P
= σ0 · Nc · f Q2f
σ(e+ e− → hadrons)
10
=
σ(e+ e− → µ+ µ− )
3
below 2mb
46/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Proofs: R Ratio, Jet Events, and all that...
e+ e− annihilation into hadrons
σ(e+ e− → µ+ µ− )
σ(e+ e− → hadrons)
R=
I
Gluon discovery: 3 jets @
DORIS/PETRA Timm/Wolf (DESY), 1979
I
x1 = Eq /Ee , x2 = Eq̄ /Ee , x3 = Eg /Ee
dσ
+ −
(e e → q q̄g) = σ0 ×
dx1 dx2
X 2 2αs
x21 + x22
3
Qf
3π
(1
−
x1 )(1 − x2 )
f
2
:= σ0 = 4πα
3s
P
= σ0 · Nc · f Q2f
σ(e+ e− → hadrons)
11
=
σ(e+ e− → µ+ µ− )
3
below 2mt
46/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Proofs: R Ratio, Jet Events, and all that...
e+ e− annihilation into hadrons
σ(e+ e− → µ+ µ− )
σ(e+ e− → hadrons)
R=
I
Gluon discovery: 3 jets @
DORIS/PETRA Timm/Wolf (DESY), 1979
I
x1 = Eq /Ee , x2 = Eq̄ /Ee , x3 = Eg /Ee
dσ
+ −
(e e → q q̄g) = σ0 ×
dx1 dx2
X 2 2αs
x21 + x22
3
Qf
3π
(1
−
x1 )(1 − x2 )
f
2
:= σ0 = 4πα
3s
P
= σ0 · Nc · f Q2f
σ(e+ e− → hadrons)
11
=
σ(e+ e− → µ+ µ− )
3
below 2mt
46/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Proofs: R Ratio, Jet Events, and all that...
e+ e− annihilation into hadrons
σ(e+ e− → µ+ µ− )
σ(e+ e− → hadrons)
R=
I
Gluon discovery: 3 jets @
DORIS/PETRA Timm/Wolf (DESY), 1979
I
x1 = Eq /Ee , x2 = Eq̄ /Ee , x3 = Eg /Ee
dσ
+ −
(e e → q q̄g) = σ0 ×
dx1 dx2
X 2 2αs
x21 + x22
3
Qf
3π
(1
−
x1 )(1 − x2 )
f
2
:= σ0 = 4πα
3s
P
= σ0 · Nc · f Q2f
11
σ(e+ e− → hadrons)
=
σ(e+ e− → µ+ µ− )
3
below 2mt
47/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
The Factorization Theorem
• Cornerstone of perturbative QCD:
Factorization theorem Sterman, 1979; Collins/Soper, 1981
• Hadronic cross sections can be split into
I
I
I
Perturbative part: hard scattering process
Non-perturbative part:
parton distribution functions
Non-perturbative part:
jet or fragmentation functions
000
111
111
000
000
111
Φ
000
111
000
111
0000
1111
000
111
0000
1111
000
111
0000
1111
γ*
L
H
x1
M
Meson distribution
amplitude
Hard scattering
process
x 1 −x
000000000
111111111
111
000
Generalized
000000000
111111111
0000
1111
f
parton distribution
000
111
000000000
111111111
0000
1111
000
111
0000
1111
000
111
0000
1111
t
• Hard scattering cross sections perturbatively calculable
• Parton distribution and fragmentation functions from experimental fits
• Perturbative evolution of PDFs/fragmentation func. to different scales:
αs (Q)
d
Fg (x, Q) =
d ln Q
π
1
x
x
Fq ( , Q) + Fq̄ ( , Q)
z
z
x
f
x
x
+ Pg←g ( , Q)Fg ( , Q)
z
z
Z 1
d
αs (Q)
dz
x
x
x
Fq (x, Q) =
Pq←q (z)Fq ( , Q) + Pq←g ( , Q)Fg ( , Q)
d ln Q
π
z
z
z
z
x
dz
x
x
x
d
Fq (x, Q) =
Pq̄←q̄ (z)Fq̄ ( , Q) + Pq̄←g ( , Q)Fg ( , Q)
d ln Q
z
z
z
z
Z
dz
z
Pg←q (z)
X
48/89
Jürgen 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
hP
i
R1
P
• Momentum sum rule: 0 dxx
q Fq (x) +
q̄ Fq̄ (x) + Fg (x) = 1
•
Charge sum rules:
R1
0
dx [Fu (x) − Fū (x)] = 2,
• PDFs Have to be fitted from experiments
R1
• Hadronic cross section are calculated according
to the factorization theorem:
σ(p(P1 ) + p(P2 ) → Y + X) =
Z 1
Z 1
X
dx2
Ff (x1 )Ff¯(x2 )·
dx1
0
0
f
σshowered (f (x1 P1 ) + f¯(x2 P2 ) → Y )·
Y
F̃ (Y → M, B)
Y
• Parton shower describes QCD radiation
0
dx [Fd (x) − Fd¯(x)] = 1
48/89
Jürgen 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
hP
i
R1
P
• Momentum sum rule: 0 dxx
q Fq (x) +
q̄ Fq̄ (x) + Fg (x) = 1
•
Charge sum rules:
R1
0
dx [Fu (x) − Fū (x)] = 2,
• PDFs Have to be fitted from experiments
R1
• Hadronic cross section are calculated according
to the factorization theorem:
σ(p(P1 ) + p(P2 ) → Y + X) =
Z 1
Z 1
X
dx2
Ff (x1 )Ff¯(x2 )·
dx1
0
0
f
σshowered (f (x1 P1 ) + f¯(x2 P2 ) → Y )·
Y
F̃ (Y → M, B)
Y
• Parton shower describes QCD radiation
0
dx [Fd (x) − Fd¯(x)] = 1
49/89
Jürgen R. Reuter
Theoretical Particle Physics
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
DESY, 07/2015
49/89
Jürgen R. Reuter
Theoretical Particle Physics
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
DESY, 07/2015
50/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Part III (2. Abend)
The Noble Hero:
Hidden Symmetries
(formerly known as Spontaneous
Symmetry Breaking)
51/89
Jürgen 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 |0i = 0∀a unbroken or Wigner-Weyl phase
ii) Qa |0i =
6 0 for at least one a ⇒ Nambu-Goldstone phase
I Simple proof:
a
φi → iθa Tik
φk
∂V a
T φj = 0 ⇒
∂φi ij
∂ 2 V ∂V a
T
h0
φ
0i
+
k
∂φi ∂φj h0 φ 0i jk
∂φj h0
|
{z
}
|
{z
⇒
=(m2 )ij
=0
a
Tji
=0
φ 0i
}
52/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
The Nambu-Goldstone Theorem
• N -component real scalar field, possesses O(N ) symmetry
L=
µ2 T
g
1
(∂µ φT )(∂ µ φ) −
φ φ − (φT φ)2
2
2
4
V
with φ = (φ1 , . . . , φN )
V
µ > 0
µ < 0
Minimizing the potential:
hφi = 0 (metastable) or
T
hφT φi ∼ hφi hφi = −µ2 /g > 0
φ
φ
• Without loss of generality: hφi i = (0, 0, . . . , 0, hφN i) VEV in n-th comp.
• Mass squared matrix:
2
(M )ij
∂ 2 V (φ) = 2g hφi i hφj i =
=
∂φi ∂φj φ=hφi
0(N −1)×(N −1)
0(N −1)×1
01×(N −1)
2g hφi2
!
• O(N ) symmetry group broken down to O(N − 1) symmetry group
• # broken symmetry generators = # Goldstone bosons =
1
1
2 N (N − 1) − 2 (N − 1)(N − 2) = N − 1
53/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Chiral Symmetry Breaking in QCD
I
Light quarks (almost) massless ⇒ SU (2)L × SU (2)R global symmetry
Q :=
u
d
/ = QL DQ
/ L + QR DQ
/ R
L = QDQ
(mass term: − mQQ = −mQL QR + QR QL
I
I
Rewrite left- and right-handed rotations into vector and axial
transformations:
h
i
h
i u
h
i
h
i u
u
5
~
σ~
~
σ~
~
σ~
~
σ~
→ exp i 2 θL PL exp i 2 θR PR
⇒ exp i 2 θV exp i 2 θA γ
d
d
d
For massive quarks, the axial Noether current is not conserved:
∂µ J~Vµ = 0
I
I
I
m→0
µ
∂µ J~A
= −2mQ ~σ2 γ 5 Q −→ 0
If SU (2)A were exact: |hi ⇒ TA |hi degenrate opposite parity pairs of
hadrons, not seen in Nature
SU (2)A hidden symmetry T~A |hi = |h + ~π i
Pions are the Nambu-Goldstone bosons (NGB) of spontaneously
broken SU (2)A
54/89
Jürgen 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)
I
Quark condensate: (300 MeV)3 ∼ Λ3QCD ∼ hqqi is invariant under
SU (2)V , but breaks SU (2)A
I
Axial current generates pion states:
µ,a
h0|JA
|πb i = iFπ δab pµπ eipπ x
Fπ = 184 MeV from pion decay
I
Explicit breaking of SU (2)A by finite quark masses
I
Pions only approximate NGBs, i.e. pseudo NGBs (pNGBs):
m2π =
4(mu + md ) h 12 (uu + dd)i
Fπ2
I
Difference m(π ± ) − m(π 0 ) ≈ 5 MeV from electromagnetic quantum
corrections
I
Include strange quark: SU (3) × SU (3) stronger broken ms ∼ 95 MeV
vs. mu,d ∼ 2 − 6 MeV
55/89
Jürgen R. Reuter
Theoretical Particle Physics
Hidden Local Symmetries
DESY, 07/2015
Anderson, 1961; Higgs, 1964; Brout/Englert, 1964; Kibble 1964
• Consider scalar electrodynamics:
1
L = − Fµν F µν + (Dµ φ)† (Dµ φ) − V (φ)
4
•
V (φ) = −µ2 |φ|2 +
Remember the gauge trafos: Aµ → Aµ + ∂µ θ(x),
Evaluating the kinetic term
|Dµ φ|2 =
φ(x) → exp[−ieθ(x)]φ(x)
I
√ iα
Minimize
the
√ potential ⇒ hφi = v/ 2e where
√
v/ 2 = µ/ λ
I
Radial excitation: ”Higgs field”
I
Phase is the NGB
φ(x) =
I
λ
(|φ|2 )2
2
√1 (v
2
i
+ h(x))e v π(x)
2
1
e2
1
(∂h)2 + (v + h)2 Aµ − ∂µ π Aµ
2
2
ev
• Mixture between gauge boson and NGB. Define Bµ := Aµ −
• Field strength term does not change under this redefinition
1
ev ∂µ π
55/89
Jürgen R. Reuter
Theoretical Particle Physics
Hidden Local Symmetries
DESY, 07/2015
Anderson, 1961; Higgs, 1964; Brout/Englert, 1964; Kibble 1964
• Consider scalar electrodynamics:
1
L = − Fµν F µν + (Dµ φ)† (Dµ φ) − V (φ)
4
•
V (φ) = −µ2 |φ|2 +
Remember the gauge trafos: Aµ → Aµ + ∂µ θ(x),
Evaluating the kinetic term
|Dµ φ|2 =
φ(x) → exp[−ieθ(x)]φ(x)
I
√ iα
Minimize
the
√ potential ⇒ hφi = v/ 2e where
√
v/ 2 = µ/ λ
I
Radial excitation: ”Higgs field”
I
Phase is the NGB
φ(x) =
I
λ
(|φ|2 )2
2
√1 (v
2
i
+ h(x))e v π(x)
2
1
e2
1
(∂h)2 + (v + h)2 Aµ − ∂µ π Aµ
2
2
ev
• Mixture between gauge boson and NGB. Define Bµ := Aµ −
• Field strength term does not change under this redefinition
1
ev ∂µ π
55/89
Jürgen R. Reuter
Theoretical Particle Physics
Hidden Local Symmetries
DESY, 07/2015
Anderson, 1961; Higgs, 1964; Brout/Englert, 1964; Kibble 1964
• Consider scalar electrodynamics:
1
L = − Fµν F µν + (Dµ φ)† (Dµ φ) − V (φ)
4
•
V (φ) = −µ2 |φ|2 +
Remember the gauge trafos: Aµ → Aµ + ∂µ θ(x),
Evaluating the kinetic term
|Dµ φ|2 =
φ(x) → exp[−ieθ(x)]φ(x)
I
√ iα
Minimize
the
√ potential ⇒ hφi = v/ 2e where
√
v/ 2 = µ/ λ
I
Radial excitation: ”Higgs field”
I
Phase is the NGB
φ(x) =
I
λ
(|φ|2 )2
2
√1 (v
2
i
+ h(x))e v π(x)
2
1
e2
1
(∂h)2 + (v + h)2 Aµ − ∂µ π Aµ
2
2
ev
• Mixture between gauge boson and NGB. Define Bµ := Aµ −
• Field strength term does not change under this redefinition
1
ev ∂µ π
56/89
Jürgen 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 2
1
1
1
Bµ B m u + (∂h)2 − mh h2 − gh,3 h3 − gh,4 h4
L = − Fµν F µν + MB
4
2
2
2
with
m2h = λv 2
MB = ev
gh,3 =
m2h
2v
m2h =
m2h
8v 2
Higgs field generates particle masses proportional to its VEV and
•
its coupling to that particle
2
2
MB
v
2
3
m2h
v
3 v2h
Feynman rules
2
MB
v2
m2
• 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.
57/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
The Electroweak Standard Model
I
Standard Model (SM) is
SU (3)c × SU (2)L × U (1)Y gauge theory
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− ν̄e νµ Fermi, 1934
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)
58/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
The Lagrangian and its particles in totaliter
• Building blocks (SU (3)c , SU (2)L )U (1)Y quantum numbers:
I
QL
uR
dR
LL
eR
H
νR
(2, 3) 13
(1, 3) 34
(1, 3)− 32
(2, 1)1
(1, 1)−2
(2, 1)1
(1, 1)0
All renormalizable interactions possible with these fields:
LSM =
X
ψ=Q,u,d,L,e,H,ν
ψ Dψ
/ −
1
1
tr [Gµν Gµν ] − tr [Wµν W µν ]
2
2
1
− Bµν B µν + Y u QL H † uR + Y d QL HdR
4
+ Y e LL HeR +Y n LL H † νR + µ2 H † H − λ(H † H)2
59/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Electroweak Symmetry Breaking
√
• Higgs vev hHi = (0, v/ 2) breaks SU (2)L × U (1)Y → U (1)em.
0
σ
~µ~
Dµ φ = (∂µ + ig W
2 + ig Y Bµ )φ
• Electroweak gauge boson mass term:
∆L =
1
(0, v)
2
~µ
gW
~
σ
g0
+ Bµ
2
2
1
√
2
Wµ1 ∓ iWµ2
mW =
~µ
gW
1
gv
2
Zµ = √
1
g 2 +g 0 2
• Orthogonal combination remains massless
Aµ = √
1
g 2 +g 0 2
g 0 Wµ3 + gBµ
0
v
Dµ = ∂µ +i √g (Wµ+ σ + +Wµ− σ − )+i p
1
g2
Weak mixing angle: cos θW = √
gWµ3 − g 0 Bµ
+
g0 2
g
,
g 2 +g 0 2
3
v
2
p
g2 + g0 2
mA = 0
Zµ (g 2 T 3 −g 0 2 Y )+i p
sin θW = √
Gell-Mann–Nishijima relation: Q = T + Y
mZ =
photon
σ ± = 12 (σ 1 ± σ 2 )
• Rewrite the covariant derivative:
2
g0
~
σ
+ Bµ
2
2
W ±, Z
• Three massive vector bosons
Wµ± =
g0
g 2 +g 0 2
gg 0
g2
+ g0 2
Aµ (T 3 +Y )
59/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Electroweak Symmetry Breaking
√
• Higgs vev hHi = (0, v/ 2) breaks SU (2)L × U (1)Y → U (1)em.
0
σ
~µ~
Dµ φ = (∂µ + ig W
2 + ig Y Bµ )φ
• Electroweak gauge boson mass term:
∆L =
1
(0, v)
2
~µ
gW
g0
~
σ
+ Bµ
2
2
1
√
2
Wµ1 ∓ iWµ2
mW =
1
gv
2
~µ
gW
Zµ = √
1
g 2 +g 0 2
• Orthogonal combination remains massless
Aµ = √
1
g 2 +g 0 2
g0
~
σ
+ Bµ
2
2
0
v
W ±, Z
• Three massive vector bosons
Wµ± =
g 0 Wµ3 + gBµ
Dµ = ∂µ + i √g (Wµ+ σ + + Wµ− σ − ) + i
Weak mixing angle: cos θW = √
mZ =
v
2
p
g2 + g0 2
photon
mA = 0
σ ± = 12 (σ 1 ± σ 2 )
• Rewrite the covariant derivative:
2
gWµ3 − g 0 Bµ
1
Zµ (T 3 − sin2 θW Q) + ieAµ Q
cos θW
g
,
g 2 +g 0 2
3
sin θW = √
Gell-Mann–Nishijima relation: Q = T + Y
g0
g 2 +g 0 2
60/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Electroweak Feynman Rules (all momenta outgoing)
f¯
f¯
f¯
±
Wµ
Aµ
f
f
+
Wµ
Zµ
2i
H
m2
W η
µν
v
−
Wν
+
Wµ
2i
−
Wν
Zµ
f
H
2i
H
m2
Z η
µν
v
−3i
H
Zν
Zµ
H
m2
W η
µν
v2
H
Easily derivable:
g
−i
γ
2 cos θW µ
3 − 2 sin2 θ
(Tf
3
−γ 5 Tf
g
−i √ γµ (1 − γ 5 )
2 2
−ieγµ Qf
H
H
2i
Zν
m2
H
v
H
H
m2
W η
µν
v2
−3i
H
H
m2W Wµ+ W −µ + 21 m2Z Z 2
1+
H
H 2
v
m2
H
v2
61/89
Jürgen R. Reuter
Theoretical Particle Physics
Electroweak Feynman Rules (all momenta outgoing)
+
Wµ
h
i
−igW W X (k− − k+ )ρ η µν + (q − k− )µ η νρ + (k+ − q)ν η µρ
gW W Z = g cos θW
gW W γ = e
Xµ
−
Wν
+
Wσ
−
Wτ
µ
X1
ν
X2
h
i
−igW W X X 2η µν η στ − η µσ η νσ − η µσ η ντ
1 2
gW W γγ = e2
gW W ZZ = g 2 cos2 θW
gW W γZ = g 2 cos θW sin θW
gW W W W = −g 2
DESY, 07/2015
62/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Fermion masses: Yukawa terms
• Fermion mass terms −mf (f L fR + f R fL ) forbidden by U (2)L × U (1)Y
gauge invariance
• Yukawa coupling is gauge invariant dimension-4 operator:
vYe
H
∆LY uk. = −Ye (LL · φ)eR → − √ eL eR 1 +
v
2
• Again, Higgs boson couples proportional¯ to mass:
f
1
mf = √ Yf v
2
−i
mf
v
H
f
• Hierarchy of Yukawa couplings according to fermion masses: Yt ≈ 1,
Yc,τ,b ≈ 10−2 , Yµ,s ≈ 10−3 , Ye,ν,d ≈ 10−5
• Yν . 10−10 , but Majorana mass term LMajorana = − 21 mν ν c R νR possible
•
Hγγ, Hgg couplings:
1 Y g2
· c · HFµν F µν
v 16π 2
63/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Higgs: Properties and Search
Production: gluon/vector boson fusion
decays predominantly into the heaviest
particles
bb̄ hopeless: background!
Complicated search: many
channels
high statistics necessary
σ(pp → H+X) [pb]
102
pp →
NLO
+N
N
LL
QC
D+
NLO
EW
)
10
pp →
pp
pp
ZH
1
O QC
D + NL
O EW
)
H
(N
O
→
ttH
(NNL
W
NL
pp
qqH
→
→
γγ: mass determination
MH & 125 GeV: ZZ ∗ → ````
s= 14 TeV
H (N
LHC HIGGS XS WG 2010
Detection of rare decays
(N
N
LO
QC
D
QC
D
+N
LO
(N
LO
QC
+
NL
O
EW
EW
)
)
D)
10-1
100
200
300
400 500
1000
MH [GeV]
63/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
1
WW
bb
ττ
gg
tt
decays predominantly into the heaviest
particles
cc
10-2
γ γ Zγ
bb̄ hopeless: background!
10-3
Detection of rare decays
µµ
90
200
300
400
1000
MH [GeV]
102
Complicated search: many
channels
high statistics necessary
10
pp
→
H+
Xa
t
pp
→
H+
Xa
t
pp
1
LHC HIGGS XS WG 2012
10-4
Production: gluon/vector boson fusion
ZZ
σ(pp → H+X) [pb]
10-1
LHC HIGGS XS WG 2013
Higgs BR + Total Uncert [%]
Higgs: Properties and Search
→
H+
X
γγ: mass determination
MH & 125 GeV: ZZ ∗ → ````
at
s=
1
4T
eV
s=
8
Te
V
10-1
100
Te
V
s=
7
200
300
400
500
600
700
800
900 1000
MH [GeV]
63/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
1
LHC HIGGS XS WG 2013
WW
bb
gg
ττ
10-1
ZZ
Production: gluon/vector boson fusion
decays predominantly into the heaviest
particles
cc
10-2
γγ
Zγ
bb̄ hopeless: background!
10-3
Detection of rare decays
µµ
10-4
80
100
120
140
160
180
200
MH [GeV]
Complicated search: many
channels
Signal significance
Higgs BR + Total Uncert [%]
Higgs: Properties and Search
h → γγ
tth (h → bb)
h → ZZ(*) → 4 l
(*)
h → WW
→ lνlν
∫ L dt = 30 fb
(no K-factors)
-1
10
ATLAS
2
qqh
qqh
→ qq WW(*)
→ qq ττ
Total significance
high statistics necessary
10
γγ: mass determination
MH & 125 GeV: ZZ ∗ → ````
1
100
120
140
160
180
200
2
Mh (GeV/c )
63/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
1
LHC HIGGS XS WG 2013
Higgs BR + Total Uncert [%]
Higgs: Properties and Search
WW
bb
gg
ττ
10-1
ZZ
Production: gluon/vector boson fusion
decays predominantly into the heaviest
particles
cc
10-2
γγ
Zγ
bb̄ hopeless: background!
10-3
Detection of rare decays
µµ
10-4
80
100
120
140
160
180
200
MH [GeV]
Complicated search: many
channels
high statistics necessary
γγ: mass determination
MH & 125 GeV: ZZ ∗ → ````
64/89
Jürgen R. Reuter
Theoretical Particle Physics
4.7.2012: The Discovery of the Higgs (?)
I
After roughly 5 fb−1 data from 2011 and 2012:
4.7.2012 CERN-Seminar: We have found a scalar boson at
125.3 ± 0.6 GeV
DESY, 07/2015
65/89
I
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Rolf Heuer (CERN DG, 4.7.12): As a layman I would say we have it!
66/89
Jürgen R. Reuter
Theoretical Particle Physics
2015: It’s the/a/sort of Higgs(-like) WTF
I
2012 data: 25 fb−1 ⇒ compatible with
EW precision
I
Higgs compatible with EW precision
measurements:
I
Higgs measurement by far not precise
enough!!!
DESY, 07/2015
67/89
Jürgen R. Reuter
Theoretical Particle Physics
Digitization Science Fiction
DESY, 07/2015
68/89
Jürgen 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:
v 2 Yu Yu† = Lu diag(m2u , m2c , m2t )L†u
v 2 Yd Yd† = Ld diag(m2d , m2s , m2b )L†d
• Rotation of quark fields leaves a trace in the charged current:
uL W
/ (L†u Ld )dL = uL W
/ VCKM 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
69/89
Jürgen R. Reuter
Theoretical Particle Physics
Part IV (3. Abend)
Götterdämmerung:
Beyond the Standard Model
DESY, 07/2015
70/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
The Standard Model of Particle Physics – Doubts
Measurement
(5)
∆αhad(mZ)
mZ [GeV]
ΓZ [GeV]
0
σhad
[nb]
0.02768
91.1875
2.4952 ± 0.0023
2.4957
41.540 ± 0.037
20.767 ± 0.025
Rl
0,l
0.01714 ± 0.00095
Afb
Al(Pτ)
Rb
Fit
0.02758 ± 0.00035
91.1875 ± 0.0021
0.1465 ± 0.0032
0.21629 ± 0.00066
0.1481
0.21586
0.1721 ± 0.0030
0.1722
0.0992 ± 0.0016
0.1038
0.0707 ± 0.0035
Afb
0.923 ± 0.020
Ab
0.670 ± 0.027
Ac
Al(SLD)
2
lept
sin θeff (Qfb)
mW [GeV]
ΓW [GeV]
mt [GeV]
(too well?)
28 free parameters
41.477
20.744
0,c
Afb
– describes microcosm
0.01645
0,b
Rc
meas
fit
meas
|O
−O |/σ
0
1
2
3
0.0742
0.935
0.668
0.1513 ± 0.0021
0.1481
0.2324 ± 0.0012
0.2314
80.398 ± 0.025
80.374
2.140 ± 0.060
2.091
170.9 ± 1.8
171.3
0
1
2
3
form of Higgs potential ?
Hierarchy Problem
chiral symmetry:
δmf ∝ v ln(Λ2 /v 2 )
no symmetry for quantum corrections to the
Higgs mass
2
δMH
∝ Λ2
Λ ∼ MPlanck = 1019 GeV
20000 GeV2 = ( 1000000000000000000000000000000020000 –
1000000000000000000000000000000000000 ) GeV2
70/89
Jürgen 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
MH [GeV]
chiral symmetry:
750
no symmetry for quantum corrections to the
Higgs mass
500
250
Λ[GeV]
103
δmf ∝ v ln(Λ2 /v 2 )
106
109 1012 1015 1018
2
2
δMH
∝ Λ2 ∼ MPlanck
= (1019 )2 GeV2
20000 GeV2 = ( 1000000000000000000000000000000020000 –
1000000000000000000000000000000000000 ) GeV2
71/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Open questions
60
50
– Unification of all interactions (?)
U(1)
– Baryon asymmetry ∆NB − ∆NB̄ ∼ 10−9
missing CP violation
αi-1
40
30
10
– Flavour: three generations
– Tiny neutrino masses: mν ∼
– Dark Matter:
I
I
I
stable
weakly interacting
mDM ∼ 100 GeV
– Quantum theory of gravity
– Cosmic inflation
– Cosmological constant
SU(2)
20
SU(3)
Standard Model
0 2 4 6 8 10 12 14 16 18
10 10 10 10 10 10 10 10 10
2
v
M
µ (GeV)
72/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
73/89
Jürgen 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
74/89
Jürgen R. Reuter
Theoretical Particle Physics
Supersymmetry (SUSY)
DESY, 07/2015
Gelfand/Likhtman, 1971; Akulov/Volkov, 1973; Wess/Zumino, 1974
Q
– connects gauge and space-time symmetries
|Bosoni
|Fermioni
Q
– Multiplets with equal-mass fermions and
bosons
⇒ SUSY broken in Nature
J
1
r
1
2
r
0
r
r
r
– Every particle gets a superpartner
– Minimal Supersymmetric
Standard Model (MSSM)
– Mass eigenstates:
Charginos: χ̃± = H̃ ± , W̃ ±
Neutralinos: χ̃0 = H̃, Z̃, γ̃
75/89
Jürgen 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
parameters
v = 246 GeV
solves hierarchy problem:
δMH ∝ F log(Λ2 )
60
50
U(1)
αi-1
40
30
I
Existence of fundamental scalars
I
Form of Higgs potential
I
light Higgs (MH = 90 ± 50 GeV)
discrete R parity
I
SU(2)
I
20
10
SU(3)
I
Standard Model
I
0 2 4 6 8 10 12 14 16 18
10 10 10 10 10 10 10 10 10
µ (GeV)
F = O(1 TeV)
I
SM particles even, SUSY partners odd
prevents a proton decay too rapid
lightest SUSY partner (LSP) stable
Dark Matter χ̃01
Unification of coupling constants
75/89
Jürgen 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
parameters
v = 246 GeV
solves hierarchy problem:
δMH ∝ F log(Λ2 )
60
50
MSSM
U(1)
αi-1
40
30
I
Existence of fundamental scalars
I
Form of Higgs potential
I
light Higgs (MH = 90 ± 50 GeV)
discrete R parity
I
SU(2)
I
20
SU(3)
I
10
I
0 2 4 6 8 10 12 14 16 18
10 10 10 10 10 10 10 10 10
µ (GeV)
F = O(1 TeV)
I
SM particles even, SUSY partners odd
prevents a proton decay too rapid
lightest SUSY partner (LSP) stable
Dark Matter χ̃01
Unification of coupling constants
75/89
Jürgen 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
parameters
800
v = 246 GeV
solves hierarchy problem:
δMH ∝ F log(Λ2 )
m [GeV]
I
Existence of fundamental scalars
I
Form of Higgs potential
light Higgs (MH = 90 ± 50 GeV)
discrete R parity
700
600
g̃
ũL , d˜R
ũR , d˜L
500
400 H 0 , A0
H±
χ̃040
χ̃3
χ̃±
2
t̃2
b̃2
b̃1
I
I
t̃1
I
300
I
200
100
h0
l̃L
ν̃l
τ̃2
l̃R
τ̃1
I
χ̃02
χ̃±
1
χ̃01
I
0
F = O(1 TeV)
SM particles even, SUSY partners odd
prevents a proton decay too rapid
lightest SUSY partner (LSP) stable
Dark Matter χ̃01
Unification of coupling constants
76/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
”SUSY will be discovered, even if non-existent”
76/89
Jürgen R. Reuter
Theoretical Particle Physics
What, if not SUSY?
DESY, 07/2015
77/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Higgs as Pseudo-Goldstone boson: Technicolor
Nambu-Goldstone Theorem: Spontaneous breaking of a global symmetrie: 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
Λ
O(1 GeV)
O(150 MeV)
v
Skala Λ: chiral symmetry
breaking,
Quarks, SU (3)C
Scale v: pions, kaons, . . .
77/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Higgs as Pseudo-Goldstone boson: Technicolor
Nambu-Goldstone Theorem: Spontaneous breaking of a global symmetrie: 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 broken chiral symmetry
Λ
O(1 TeV)
O(250 GeV)
v
Skala Λ: chiral symmetry
breaking, techni-quarks,
SU (N )T C
Skala v: Higgs, techni-pions
experimentally constrained, but not ruled out
78/89
Jürgen 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:
Λ
O(10 TeV)
O(1 TeV)
F
O(250 GeV)
v
MH ∼ (0.1)2 × Λ
Scale Λ: chiral SB, strong
interaction
Scale F : Pseudo-Goldstone
bosons, new gauge bosons
Scale v: Higgs
79/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Little-Higgs Models
– Economic implementation of
collective symmetry breaking
– New Particles:
I
I
Gauge bosons:
γ0, Z0, W 0 ±
Heavy Fermions:
T , U , C, . . .
Littlest Higgs
M [GeV]
1250
Φ
±
Φ
Arkani-Hamed/Cohen/Katz/Nelson, 2002
Z0
Φ±±
Quantum corrections to
MH cancelled by
particles of like
statistics
T
ΦP
1000
γ0
750
I
W0±
U, C
500
250
h
η
– “Little Big Higgs”: Higgs heavy (300 − 500 GeV)
– discrete T -(TeV scale) parity:
I
allows for new light particles
I
Dark matter: LTOP (lightest T-odd), often γ 0
W±
Z
t
80/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Extra Dimensions & Higgsless Models
Motivation: String theory
30
3 + n Space dimensions: Radius R ∼ 10 n −17
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
80/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Extra Dimensions & Higgsless Models
Motivation: String theory
30
3 + n Space dimensions: Radius R ∼ 10 n −17
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
“Universal Extra Dimensions”: also
fermions/gauge bosons in higher
dimensions
I
81/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
KK parity and Dark Matter
I
typical Kaluza-Klein spectra
700
1050
GH1,0L
Μ
1R = 500 GeV
1000
1R = 500 GeV
GΜH1,1L
650
mass
600
DH1,0L
-
550
WΜH1,0L
500
450
400
BH1,0L
Μ
HH1,0L
LH1,0L
+
GH1,0L
H
EH1,0L
-
T-H1,0L
950
UH1,0L
-
900
mass
Q+3 H1,0L
QH1,0L
+
Q+3 H1,1L
QH1,1L
+
DH1,1L
-
850
800
750
WH1,1L
Μ
WH1,0L
H
700
BH1,1L
Μ
BHH1,0L
650
TH1,1L
UH1,1L
-
GH1,1L
H
HH1,1L
L+H1,1L
EH1,1L
-
WH1,1L
H
BH1,1L
H
600
I
Spectrum structure similar to SUSY, but shifted in spin
I
Dark matter: lightest KK-odd particle (LKP)
Photon resonance γ 0
(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!”
82/89
Jürgen R. Reuter
Theoretical Particle Physics
Noncommutative Space-time
DESY, 07/2015
Wess et al., 2000
– Assumption: non-commuting
Space-time coordinates [x̂µ , x̂ν ] = iθµν
– Classical analogue: charged particle in lowest Landau level:
{xi , xj }P = 2c(B −1 )ij /e
– Low energy limit of string theory
Seiberg/Witten, 1999
– Yang-Landau-Theorem violated: Z → γγ, gg possible
– Cross sections depend
on azimuth
⇒ Varying signals as
Earth rotates
13000
-1
12000
11000
Number of Events
– Special direction in the Universe:
broken rotational invariance
LHC ( s = 14 TeV; L = 10 fb )
Λ NC = 100 GeV
standard model
B = (1,0,0)
K ZZ γ =-0.021; K Z γγ =-0.340
E = (1,0,0)
10000
9000
8000
7000
6000
5000
0
– Dark Matter, cosmology, theoretical problems
1
2
E
3
Φ(γ )
4
5
6
83/89
Jürgen R. Reuter
Which model?
Theoretical Particle Physics
DESY, 07/2015
A Conspiracy Unmasked
84/89
Jürgen R. Reuter
Theoretical Particle Physics
The Challenge of the LHC
DESY, 07/2015
QCD
Partonic subprocesses: qq, qg, gg
WW
no fixed partonic energy
QCD
proton - (anti)proton cross sections
9
9
10
10
8
10
8
10
σtot
7
7
10
Tevatron
LHC
6
10
5
10
5
σb
4
10
3
3
2
σ (nb)
10
jet
σjet(ET
> √s/20)
1
10
σZ
0
10
2
10
σW
1
10
jet
σjet(ET
0
10
> 100 GeV)
-1
-1
10
10
-2
10
-3
10
jet
σjet(ET
-4
10
-5
-6
10
High rates for t, W/Z, H, ⇒ huge
backgrounds
Jet
pT
-2
10
10
L = 1034 cm−1 s−1
33
10
-2 -1
4
10
events/sec for L = 10 cm s
10
10
R = σL
6
10
-3
σt
10
> √s/4)
10
-4
σHiggs(MH = 150 GeV)
-5
p
WW
p
η
10
-6
10
σHiggs(MH = 500 GeV)
-7
10
10
-7
0.1
1
√ s (TeV)
10
10
e−
84/89
Jürgen 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
9
9
10
10
8
10
8
10
σtot
7
7
10
Tevatron
LHC
6
10
5
10
5
σb
4
10
3
3
2
10
jet
σjet(ET
> √s/20)
1
10
σZ
0
10
2
10
σW
1
10
jet
σjet(ET
0
10
> 100 GeV)
-1
10
-2
10
-1
10
-2
10
-3
jet
σjet(ET
-4
10
-5
10
-6
10
-3
σt
10
10
> √s/4)
10
σHiggs(MH = 150 GeV)
-5
10
entral
forward
η = −5
WW
-6
10
σHiggs(MH = 500 GeV)
-7
0.1
High rates for t, W/Z, H, ⇒ huge
backgrounds
-4
-7
10
L = 1034 cm−1 s−1
33
10
-2 -1
4
10
events/sec for L = 10 cm s
10
10
R = σL
6
10
σ (nb)
10
1
√ s (TeV)
10
10
η=0
η = +5
85/89
Jürgen R. Reuter
Theoretical Particle Physics
Search for New Particles
DESY, 07/2015
105
R
Decay products of heavy particles:
I
I
high-pT Jets
many hard leptons
104
Production of coloured particles
weakly interacting particles only in decays
Dark Matter ⇔ discrete parity
I
I
evt/10 GeV
L = 100 fb−1
1000
100
(R, T , KK)
0
100
200
300
400
500
pT (b/b̄) [GeV]
only pairs of new particles ⇒ high energies, long decay chains
Dark Matter ⇒ large missing energy in detector (E
/T)
Different Models/Decay Chains — same signatures
85/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
Search for New Particles
Decay products of heavy particles:
I
I
q
q
high-pT Jets
many hard leptons
ℓ
ℓ
Production of coloured particles
weakly interacting particles only in decays
Dark Matter ⇔ discrete parity
I
I
χ̃01
ℓ̃R
q̃L
ℓ̃R
q
q̃R
q
ℓ
(R, T , KK)
only pairs of new particles ⇒ high energies, long decay chains
Dark Matter ⇒ large missing energy in detector (E
/T)
Different Models/Decay Chains — same signatures
χ̃01
ℓ
86/89
Jürgen 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
`2
`1
dp / d(cos θ*) / 0.05
q1
Events/1 GeV/100 fb-1
1500
1000
500
0.035
0.03
0.025
0.02
0.015
0.01
SUSY
UED
PS
0.005
q̃L
χ̃20
`˜
R
χ̃10
0
0
0
20
40
60
m(ll) (GeV)
80
-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1
100
cos θ*
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
87/89
Jürgen R. Reuter
Outlook
Theoretical Particle Physics
DESY, 07/2015
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 Höchste ist, so folgt daraus, dass die Wissenschaft
und die Denker stolz die Existenz verlassen und es
uns anderen Menschen überlassen, das Schlimmste
zu erdulden. Ja es folgt daraus zugleich etwas für den
abstrakten Denker selbst, dass er nämlich, da er ja doch
selbst auch ein Existierender ist, in irgendeiner Weise
distrait sein muss.”
Søren Kierkegaard
88/89
Jürgen R. Reuter
Outlook
Theoretical Particle Physics
DESY, 07/2015
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 Höchste ist, so folgt daraus, dass die Wissenschaft
und die Denker stolz die Existenz verlassen und es
uns anderen Menschen überlassen, das Schlimmste
zu erdulden. Ja es folgt daraus zugleich etwas für den
abstrakten Denker selbst, dass er nämlich, da er ja doch
selbst auch ein Existierender ist, in irgendeiner Weise
distrait sein muss.”
Søren Kierkegaard
89/89
Jürgen R. Reuter
Theoretical Particle Physics
DESY, 07/2015
One Ring to Find Them, One Ring to Rule Them Out?