t8

8. El Modelo Estándar de las interacciones
electrodébiles (QFD) y fuertes (QCD).
El bosón de Higgs
452
1. Gauge Theories
B The gauge symmetry principle
B Quantization of gauge theories
2. Spontaneous Symmetry Breaking
B Discrete symmetry
B Continuous symmetry: global vs gauge
3. The Standard Model
B Gauge group and particle representations
4. Electroweak interactions
B Case of one family
B Electroweak SSB: Higgs sector, gauge boson and fermion masses
B Additional generations: fermion mixings (quark sector)
B Complete Lagrangian and Feynman rules
B Phenomenology
5. Strong interactions
453
1. Gauge Theories
454
The symmetry principle
free Lagrangian
Lagrangian of a free fermion field ψ( x ):
(Dirac)
L0 = ψ(i/
∂ − m)ψ
∂/ ≡ γµ ∂µ ,
ψ = ψ † γ0
⇒ Invariant under global U(1) phase transformations:
ψ( x ) 7→ ψ0 ( x ) = e−iqθ ψ( x ) ,
q, θ (constants) ∈ R
⇒ By Noether’s theorem there is a conserved current:
jµ = q ψγµ ψ ,
∂µ jµ = 0
and a Noether charge:
ˆ
Q=
d3 x j 0 ,
∂t Q = 0
455
The symmetry principle
free Lagrangian
A quantized free fermion field:
ˆ
ψ( x ) =
d3 p
q
(2π )3 2E~p
∑
a~p,s u(s) (~p)e−ipx + b~†p,s v(s) (~p)eipx
s=1,2
– is a solution of the Dirac equation (Euler-Lagrange):
(i/
∂ − m)ψ( x ) = 0 ,
(/
p − m)u(~p) = 0 ,
(/
p + m)v(~p) = 0 ,
– is an operator from the canonical quantization rules (anticommutation):
†
{ a~p,r , a~†k,s } = {b~p,r , b~k,s
} = (2π )3 δ3 (~p −~k)δrs ,
{ a~p,r , a~k,s } = · · · = 0 ,
that annihilates/creates particles/antiparticles on the Fock space of fermions
456
The symmetry principle
free Lagrangian
For a quantized free fermion field:
⇒ Normal ordering for fermionic operators (H spectrum bounded from below):
: a~p,r a~†q,s : ≡ − a~†q,s a~p,r ,
⇒ The Noether charge is an operator:
ˆ
ˆ
: Q : = q d3 x : ψγ0 ψ : = q
Q a~†k,s |0i = +q a~†k,s |0i (particle) ,
†
†
b~p,r
: ≡ −b~q,s
: b~p,r b~q,s
d3 p
(2π )3
∑
s=1,2
a~†p,s a~p,s − b~†p,s b~p,s
†
†
Q b~k,s
|0i = −q b~k,s
|0i (antiparticle)
457
The symmetry principle
gauge invariance dictates interactions
To make L0 invariant under local ≡ gauge transformations of U(1):
ψ( x ) 7→ ψ0 ( x ) = e−iqθ ( x) ψ( x ) ,
θ = θ (x) ∈ R
perform the minimal substitution:
∂µ → Dµ = ∂µ + ieqAµ
(covariant derivative)
where a gauge field Aµ ( x ) is introduced transforming as:
1
Aµ ( x ) 7→ A0µ ( x ) = Aµ ( x ) + ∂µ θ ( x )
e
⇐
Dµ ψ 7→ e−iqθ ( x) Dµ ψ
ψ Dψ
/ inv.
⇒ The new Lagrangian contains interactions between ψ and Aµ :
(
coupling e
Lint = −eq ψγµ ψAµ
∝
charge q
(= −e jµ Aµ )
458
The symmetry principle
gauge invariance dictates interactions
Dynamics for the gauge field ⇒ add gauge invariant kinetic term:
(Maxwell)
1
L1 = − Fµν F µν
4
⇐
Fµν = ∂µ Aν − ∂ν Aµ 7→ Fµν
The full U(1) gauge invariant Lagrangian for a fermion field ψ( x ) reads:
Lsym
1
= ψ (i D
/ − m)ψ − Fµν F µν
4
(= L0 + Lint + L1 )
The same applies to a complex scalar field φ( x ):
1
Lsym = ( Dµ φ)† D µ φ − m2 φ† φ − λ(φ† φ)2 − Fµν F µν
4
459
The symmetry principle
non-Abelian gauge theories
A general gauge symmetry group G is an N-dimensional compact Lie group
g∈G,
θ a = θ a (x) ∈ R ,
−iT a θ a
~
g( θ ) = e
,
T a = Hermitian generators ,
Tr{ T a T b } ≡ 21 δab ,
a = 1, . . . , N
[ T a , T b ] = i f abc T c
structure constants:
(Lie algebra)
f abc = 0 Abelian
f abc 6= 0 non-Abelian
⇒ Finite-dimensional irreducible representations are unitary:

d-multiplet :
d × d matrices :
Ψ( x ) 7→ Ψ0 ( x ) = U (~θ )Ψ( x ) ,

ψ
 1
 . 
Ψ =  .. 
 
ψd
U (~θ ) [given by { T a } algebra representation]
460
The symmetry principle
Examples:
non-Abelian gauge theories
G
N
Abelian
U(1)
1
Yes
SU(n)
n2 − 1
No
(n × n unitary matrices with det = 1)
• U(1): 1 generator (q), one-dimensional irreps only
• SU(2): 3 generators
f abc = e abc (Levi-Civita symbol)
◦ Fundamental irrep (d = 2): T a = 12 σ a (3 Pauli matrices)
a )bc = −i f abc
◦ Adjoint irrep (d = N = 3): ( Tadj
• SU(3): 8 generators
√
= 1,
=
= 23 , f 147 = f 156 = f 246 = f 247 = f 345 = − f 367 =
◦ Fundamental irrep (d = 3): T a = 12 λ a (8 Gell-Mann matrices)
a )bc = −i f abc
◦ Adjoint irrep (d = N = 8): ( Tadj
f 123
f 458
f 678
1
2
(for SU(n): f abc totally antisymmetric)
461
The symmetry principle
non-Abelian gauge theories
To make L0 invariant under local ≡ gauge transformations of G:
Ψ( x ) 7→ Ψ0 ( x ) = U (~θ )Ψ( x ) ,
~θ = ~θ ( x ) ∈ R
substitute the covariant derivative:
eµ ,
∂µ → Dµ = ∂µ − igW
e µ ≡ T a Wµa
W
where a gauge field Wµa ( x ) per generator is introduced, transforming as:
i
0
†
e
e
e
Wµ ( x ) 7→ Wµ ( x ) = U Wµ ( x )U − (∂µ U )U †
g
⇐
Dµ Ψ 7→ UDµ Ψ
Ψ DΨ
/ inv.
⇒ The new Lagrangian contains interactions between Ψ and Wµa :
(
coupling g
Lint = g Ψγµ T a ΨWµa
∝
charge T a
(= g ja Wµa )
µ
462
The symmetry principle
non-Abelian gauge theories
Dynamics for the gauge fields ⇒ add gauge invariant kinetic terms:
(Yang-Mills)
LYM
1 n e e µν o
1 a a,µν
= − Tr Wµν W
= − Wµν
W
2
4
⇐
e µν 7→ U W
e µν U †
W
e µν ≡ ∂µ W
e ν − ∂ν W
e µ − ig[W
e µ, W
e ν]
W
⇒
a
= ∂µ Wνa − ∂ν Wµa + g f abc Wµb Wνc
Wµν
⇒ LYM contains cubic and quartic self-interactions of the gauge fields Wµa :
1
Lkin = − (∂µ Wνa − ∂ν Wµa )(∂µ W a,ν − ∂ν W a,µ )
4
1
Lcubic = − g f abc (∂µ Wνa − ∂ν Wµa )W b,µ W c,ν
2
1
Lquartic = − g2 f abe f cde Wµa Wνb W c,µ W d,ν
4
463
Quantization of gauge theories
propagators
The (Feynman) propagator of a scalar field:
ˆ
†
D ( x − y) = h0| T {φ( x )φ (y)} |0i =
i
d4 p
−ip·( x −y)
e
(2π )4 p2 − m2 + ie
is a Green’s function of the Klein-Gordon operator:
2
4
(x + m ) D ( x − y) = −iδ ( x − y)
⇔
e ( p) =
D
i
p2 − m2 + ie
The propagator of a fermion field:
ˆ
S( x − y) = h0| T {ψ( x )ψ(y)} |0i = (i/
∂ x + m)
d4 p
i
−ip·( x −y)
e
(2π )4 p2 − m2 + ie
is a Green’s function of the Dirac operator:
4
(i∂
/ x − m)S( x − y) = iδ ( x − y)
⇔
Se( p) =
i
/
p − m + ie
464
Quantization of gauge theories
propagators
BUT the propagator of a gauge field cannot be defined unless L is modified:
(e.g. modified Maxwell)
Euler-Lagrange:
1
1
L = − Fµν F µν − (∂µ Aµ )2
4
2ξ
∂L
∂L
− ∂µ
=0
∂Aν
∂(∂µ Aν )
⇒
gµν − 1 −
1
ξ
∂µ ∂ν Aµ = 0
– In momentum space the propagator is the inverse of:
k
k
1
i
µ ν
e µν (k ) =
kµ kν ⇒ D
−k2 gµν + 1 −
−
g
+
(
1
−
ξ
)
µν
ξ
k2 + ie
k2
⇒ Note that (−k2 gµν + kµ kν ) is singular!
⇒ One may argue that L above will not lead to Maxwell equations . . .
unless we fix a (Lorenz) gauge where:
∂µ Aµ = 0
⇐
Aµ 7→ A0µ = Aµ + ∂µ Λ with ∂µ ∂µ Λ ≡ −∂µ Aµ
465
Quantization of gauge theories
gauge fixing (Abelian case)
The extra term is called Gauge Fixing:
LGF = −
1 µ
( ∂ A µ )2
2ξ
⇒ modified L equivalent to Maxwell Lagrangian just in the gauge ∂µ Aµ = 0
⇒ the ξ-dependence always cancels out in physical amplitudes
Several choices for the gauge fixing term (simplify calculations): Rξ gauges
(’t Hooft-Feynman gauge)
ξ=1:
(Landau gauge)
ξ=0:
igµν
k2 + ie
kµ kν
i
e
Dµν (k) = 2
− gµν + 2
k + ie
k
e µν (k) = −
D
466
Quantization of gauge theories
gauge fixing (non-Abelian case)
For a non-Abelian gauge theory, the gauge fixing terms:
LGF = − ∑
a
1
(∂µ Wµa )2
2ξ a
allow to define the propagators:
ab
e µν
(k) =
D
kµ kν
iδab
− gµν + (1 − ξ a ) 2
k2 + ie
k
BUT, unlike the Abelian case, this is not the end of the story . . .
467
Quantization of gauge theories
Faddeev-Popov ghosts (?)
Add Faddeev-Popov ghost fields c a ( x ), a = 1, . . . , N:
adj
LFP = (∂µ c a )( Dµ ) ab cb = (∂µ c a )(∂µ c a − g f abc cb Wµc )
⇐
adj
c
Dµ = ∂µ − igTadj
Wµc
Computational trick: anticommuting scalar fields, just in loops as virtual particles
e ab (k ) =
D
iδab
k2 + ie
[(−1) sign for closed loops! (like fermions)]
⇒ Faddeev-Popov ghosts needed to preserve gauge symmetry:
= i( gµν k2 − k µ k ν )Π(k2 )
⇒ Faddeev-Popov ghosts needed to preserve unitarity at the loop level:
qq̄ → qq̄
468
Quantization of gauge theories
complete Lagrangian
Then the complete quantum Lagrangian is
Lsym + LGF + LFP
⇒ Note that in the case of a massive vector field
(Proca)
1
1 2
µν
L = − Fµν F + M Aµ Aµ
4
2
is not gauge invariant
– The propagator is:
e µν (k) =
D
i
k2 − M2 + ie
− gµν +
µ
ν
k k
M2
469
2. Spontaneous Symmetry Breaking
470
Spontaneous Symmetry Breaking
discrete symmetry
Consider a real scalar field φ( x ) with Lagrangian:
1
1
λ
L = (∂µ φ)(∂µ φ) − µ2 φ2 − φ4 invariant under φ 7→ −φ
2
2
4
⇒
1 2
H = (φ̇ + (∇φ)2 ) + V (φ)
2
1 2 2 1 4
V = µ φ + λφ
2
4
µ2 , λ ∈ R (Real/Hermitian Hamiltonian) and λ > 0 (existence of a ground state)
(a) µ2 > 0: min of V (φ) at φcl = 0
r
(b)
µ2
< 0: min of V (φ) at φcl = v ≡ ±
– A quantum field must have v = 0
a |0i = 0
− µ2
, in QFT h0| φ |0i = v 6= 0 (VEV)
λ
⇒
φ( x ) ≡ v + η ( x ) ,
h0| η |0i = 0
471
Spontaneous Symmetry Breaking
discrete symmetry
At the quantum level, the same system is described by η ( x ) with Lagrangian:
1
λ 4
µ
2 2
3
L = (∂µ η )(∂ η ) − λv η − λvη − η
2
4
√
(mη = 2λ v)
not invariant under
η 7→ −η
⇒ Lesson:
L(φ) had the symmetry but the parameters can be such that the ground state of
the Hamiltonian is not symmetric
(Spontaneous Symmetry Breaking)
⇒ Note:
One may argue that L(η ) exhibits an explicit breaking of the symmetry. However
this is not the case since the coefficients of terms η 2 , η 3 and η 4 are determined by
just two parameters, λ and v
(remnant of the original symmetry)
472
Spontaneous Symmetry Breaking
continuous symmetry
Consider a complex scalar field φ( x ) with Lagrangian:
L = (∂µ φ† )(∂µ φ) − µ2 φ† φ − λ(φ† φ)2 invariant under U(1): φ 7→ e−iqθ φ
r
− µ2
v
2
λ > 0, µ < 0 : h0| φ |0i ≡ √ , |v| =
λ
2
Take v ∈ R+ . In terms of quantum fields:
1
φ( x ) ≡ √ [v + η ( x ) + iχ( x )],
2
h0| η |0i = h0| χ |0i = 0
1
1
λ
1
L = (∂µ η )(∂µ η ) + (∂µ χ)(∂µ χ) − λv2 η 2 − λvη (η 2 + χ2 ) − (η 2 + χ2 )2 + λv4
2
2
4
4
Note: if veiα (complex) replace η by (η cos α − χ sin α) and χ by (η sin α + χ cos α)
⇒ The actual quantum Lagrangian L(η, χ) is not invariant under U(1)
√
U(1) broken ⇒ one scalar field remains massless: mη = 2λ v, mχ = 0
473
Spontaneous Symmetry Breaking
continuous symmetry
Another example: consider a real scalar SU(2) triplet Φ( x )
a a
1
λ
1
L = (∂µ ΦT )(∂µ Φ) − µ2 ΦT Φ − (ΦT Φ)2 inv. under SU(2): Φ 7→ e−iT θ Φ
2
2
4
that for λ > 0, µ2 < 0 acquires a VEV h0| ΦT Φ |0i = v2
(µ2 = −λv2 )


ϕ1 ( x )


1

√
( ϕ1 + iϕ2 )
Assume Φ( x ) =  ϕ2 ( x ) 
and
define
ϕ
≡

2
v + ϕ3 ( x )
λ
1
µ
2 2
†
2
†
2 2 1
L = (∂µ ϕ )(∂ ϕ) + (∂µ ϕ3 )(∂ ϕ3 ) − λv ϕ3 − λv(2ϕ ϕ + ϕ3 ) ϕ3 − (2ϕ ϕ + ϕ3 ) + λv4
2
4
4
†
µ
⇒ Not symmetric under SU(2) but invariant under U(1):
ϕ 7→ e−iqθ ϕ
(q = arbitrary)
ϕ3 7 → ϕ3
SU(2) broken to U(1) ⇒ 3 − 1 = 2 broken generators
( q = 0)
⇒ 2 (real) scalar fields (= 1 complex) remain massless: m ϕ = 0, m ϕ3 =
√
2λ v
474
Spontaneous Symmetry Breaking
continuous symmetry
⇒ Goldstone’s theorem:
[Nambu ’60;
Goldstone ’61]
The number of massless particles (Nambu-Goldstone bosons) is equal to the number of
spontaneously broken generators of the symmetry
⇒
Hamiltonian symmetric under group G
⇒
By definition: H |0i = 0
[T a , H ] = 0 ,
a = 1, . . . , N
H ( T a |0i) = T a H |0i = 0
– If |0i is such that T a |0i = 0 for all generators
⇒ non-degenerate minimum: the vacuum
– If |0i is such that
0
a
T
|0i 6= 0 for some (broken) generators a0
⇒ degenerate minimum: chose one (true vacuum) and
⇒ excitations (particles) from |0i to
0
a a
e−iT θ
0
0
a a
e−iT θ
0
|0i 6 = |0i
|0i cost no energy: massless!
475
Spontaneous Symmetry Breaking
gauge symmetry
Consider a U(1) gauge invariant Lagrangian for a complex scalar field φ( x ):
1
L = − Fµν F µν + ( Dµ φ)† ( D µ φ) − µ2 φ† φ − λ(φ† φ)2 ,
4
Dµ = ∂µ + ieqAµ
1
inv. under φ( x ) 7→
=
, Aµ ( x ) 7→
= Aµ ( x ) + ∂µ θ ( x )
e
If λ > 0, µ2 < 0, the L in terms of quantum fields η and χ with null VEVs:
φ0 ( x )
e−iqθ ( x) φ( x )
1
φ( x ) ≡ √ [v + η ( x ) + iχ( x )] , µ2 = −λv2
2
1
1
1
L = − Fµν F µν + (∂µ η )(∂µ η ) + (∂µ χ)(∂µ χ)
4
2
2
λ
1
− λv2 η 2 − λvη (η 2 + χ2 ) − (η 2 + χ2 )2 + λv4
4
4
+ eqvAµ ∂µ χ + eqAµ (η∂µ χ − χ∂µ η )
1
1
2
µ
+ (eqv) Aµ A + (eq)2 Aµ Aµ (η 2 + 2vη + χ2 )
2
2
A0µ ( x )
Comments:
√
(i) mη = 2λ v
mχ = 0
(ii) M A = |eqv| (!)
(iii) Term Aµ ∂µ χ (?)
(iv) Add LGF
476
Spontaneous Symmetry Breaking
gauge symmetry
Removing the cross term (no mixing in propagators) and new gauge fixing:
LGF = −
1
( ∂ µ A µ − ξ M A χ )2
2ξ
total deriv.
⇒
L + LGF
}|
{
z
1
1
1 2
µ
µ
µν
µ 2
= − Fµν F + M A Aµ A − (∂µ A ) + M A ∂µ ( A χ)
4
2
2ξ
1
1
µ
+ (∂µ χ)(∂ χ) − ξ M2A χ2 + . . .
2
2
and the propagators of Aµ and χ are:
"
e µν (k ) =
D
e (k) =
D
kµ kν
i
− gµν + (1 − ξ ) 2
2
2
k − M A + ie
k − ξ M2A
#
i
k2 − ξ M2A + ie
⇒ χ has a gauge-dependent mass: actually it is not a physical field!
477
Spontaneous Symmetry Breaking
gauge symmetry
A more transparent parameterization of the quantum field φ is
iqζ ( x )/v
φ( x ) ≡ e
1
√ [v + η ( x )] ,
2
h0| η |0i = h0| ζ |0i = 0
1
φ( x ) 7→ e−iqζ ( x)/v φ( x ) = √ [v + η ( x )]
2
1
1
µν
L = − Fµν F + (∂µ η )(∂µ η )
4
2
λ 4
1 4
2 2
3
− λv η − λvη − η + λv
4
4
1
1
+ (eqv)2 Aµ Aµ + (eq)2 Aµ Aµ (2vη + η 2 )
2
2
⇒
ζ gauged away!
Comments:
√
(i) mη = 2λ v
(ii) M A = |eqv|
(iii) No need for LGF
⇒ This is the unitary gauge (ξ → ∞): just physical fields
478
Spontaneous Symmetry Breaking
⇒ Brout-Englert-Higgs mechanism:
gauge symmetry
[Higgs ’64;
Englert, Brout ’64;
[Anderson ’62]
Guralnik, Hagen, Kibble ’64]
The gauge bosons associated with the spontaneously broken generators become massive,
the corresponding would-be Goldstone bosons are unphysical and can be absorbed,
the remaining massive scalars (Higgs bosons) are physical (the smoking gun!)
• The would-be Goldstone bosons are ‘eaten up’ by the gauge bosons (‘get fat’)
and disappear (gauge away) in the unitary gauge (ξ → ∞)
⇒ Degrees of freedom are preserved
Before SSB: 2 (massless gauge boson) + 1 (Goldstone boson)
After SSB:
3 (massive gauge boson)
+ 0 (absorbed would-be Goldstone)
• For loops calculations, ’t Hooft-Feynman gauge (ξ = 1) is more convenient:
⇒ Gauge boson propagators are simpler, but
⇒ Goldstone bosons must be included in internal lines
479
Spontaneous Symmetry Breaking
gauge symmetry
Comments:
• After SSB the FP ghost fields (unphysical) acquire a gauge-dependent mass,
due to interactions with the scalar field(s):
e ab (k ) =
D
iδab
k2 − ξ M2A + ie
• Gauge theories with SSB are renormalizable
[’t Hooft, Veltman ’72]
UV divergences appearing at loop level can be removed by renormalization of
parameters and fields of the classical Lagrangian ⇒ predictive!
480
3. The Standard Model
481
[Glashow ’61;
SM: Gauge group and particle reps
Weinberg ’67;
[D. Gross, F. Wilczek;
Salam ’68]
D. Politzer ’73]
The Standard Model is a gauge theory based on the local symmetry group:
SU(3)c ⊗ SU(2) L ⊗ U(1)Y → SU(3)c ⊗ U(1)Q
{z
}
| {z } |
| {z }
strong
em
electroweak
with the electroweak symmetry spontaneously broken to the electromagnetic
U(1)Q symmetry by the Brout-Englert-Higgs mechanism
(ingredients: 12 flavors + 12 gauge bosons + H)
The particle (field) content:
I
II
III
Q
f
uuu
ccc
ttt
f0
ddd
sss
bbb
2
3
− 13
f
νe
νµ
ντ
0
f0
e
µ
τ
−1
Fermions
spin
1
2
Quarks
Leptons
Qf = Qf0 + 1
Bosons
spin 1
8 gluons
W± , Z
γ
spin 0
Higgs
strong interaction
weak interaction
em interaction
origin of mass
482
SM: Gauge group and particle representations
The fields lay in the following representations (color, weak isospin, hypercharge):
Multiplets
SU(3)c ⊗ SU(2) L ⊗ U(1)Y
I

Quarks
(3, 2, 61 )

uL
dL
Higgs


 
c
 L
sL
III
 
t
 L
bL
Q = T3 + Y
2
3
− 13
=
=
1
2
− 21
+
1
6
1
6
+
(3, 1, 32 )
uR
cR
tR
2
3
=
0+
2
3
(3, 1, − 31 )
dR
sR
bR
− 13 =
0−
1
3

Leptons
II
(1, 2, − 21 )


νeL

eL


νµ L



ντL



0=
µL
τL
−1 =
1
2
− 12
−
−
1
2
1
2
(1, 1, −1)
eR
µR
τR
−1 =
0−1
(1, 1, 0)
νeR
νµR
ντR
0=
0+0
(1, 2, 21 )
(3 families of quarks & leptons)
⇒ Electroweak (QFD): SU(2) L ⊗U(1)Y
Strong (QCD): SU(3)c
483
4. Electroweak interactions
484
The EWSM with one family (of quarks or leptons)
Consider two massless fermion fields f ( x ) and f 0 ( x ) with electric charges
Q f = Q f 0 + 1 in three irreps of SU(2) L ⊗U(1)Y :
L0F
0
= i f ∂/f + i f ∂/f
0
f R,L
= iΨ1 ∂/Ψ1 + iψ2 ∂/ψ2 + iψ3 ∂/ψ3 ;
1
= (1 ± γ5 ) f ,
2 

fL
,

Ψ1 =
f L0
| {z }
(2, y1 )
0
f R,L
1
= (1 ± γ5 ) f 0
2
ψ2 = f R ,
|{z}
(1, y2 )
ψ3 = f R0
|{z}
(1, y3 )
To get a Langrangian invariant under gauge transformations:
Ψ1 ( x ) 7 → U L ( x )e
−iy1 β( x )
Ψ1 ( x ),
UL ( x ) = e
−iT i αi ( x )
,
σi
T =
2
i
(weak isospin gen.)
ψ2 ( x ) 7→ e−iy2 β( x) ψ2 ( x )
ψ3 ( x ) 7→ e−iy3 β( x) ψ3 ( x )
485
The EWSM with one family
covariant derivatives
⇒ Introduce gauge fields Wµi ( x ) (i = 1, 2, 3) and Bµ ( x ) through covariant derivatives:

i
e µ + ig0 y1 Bµ )Ψ1 , W
e µ ≡ σ Wµi 

Dµ Ψ1 = (∂µ − igW


2
⇒ LF
Dµ ψ2 = (∂µ + ig0 y2 Bµ )ψ2




0
Dµ ψ3 = (∂µ + ig y3 Bµ )ψ3
where two couplings g and g0 have been introduced and
e µ ( x ) 7 → U L ( x )W
e µ ( x )U † ( x ) − i (∂µ UL ( x ))U † ( x )
W
L
L
g
1
Bµ ( x ) 7→ Bµ ( x ) + 0 ∂µ β( x )
g
⇒ Add gauge invariant kinetic terms for the gauge fields
LYM
1 i
1
i,µν
= − Wµν W
− Bµν Bµν ,
4
4
j
i
Wµν
= ∂µ Wνi − ∂ν Wµi + geijk Wµ Wνk
(include self-interactions of the SU(2) gauge fields) and Bµν = ∂µ Bν − ∂ν Bµ
486
The EWSM with one family
mass terms forbidden
⇒ Note that mass terms are not invariant under SU(2) L ⊗U(1)Y , since LH and RH
components do not transform the same:
m f f = m( f L f R + f R f L )
⇒ Mass terms for the gauge bosons are not allowed either
⇒ Next the different types of interactions are analyzed
487
The EWSM with one family
charged current interactions

e µ Ψ1 ,
L F ⊃ gΨ1 γµ W
Wµ3
1
e µ = √
W
2
2Wµ
√
2Wµ†

−Wµ3

⇒ charged current interactions of LH fermions with complex vector boson field Wµ :
LCC
g
√
f γµ (1 − γ5 ) f 0 Wµ† + h.c. ,
=
2 2
ℓ
W
1
√
Wµ ≡
(Wµ1 + iWµ2 )
2
d
W
ν
u
ν
u
W
W
ℓ
d
488
The EWSM with one family
neutral current interactions
The diagonal part of
e µ Ψ1 − g0 Bµ (y1 Ψ1 γµ Ψ1 + y2 ψ γµ ψ2 + y3 ψ γµ ψ3 )
L F ⊃ gΨ1 γµ W
2
3
⇒ neutral current interactions with neutral vector boson fields Wµ3 and Bµ
We would like to identify Bµ with the photon field Aµ but that requires:
y1 = y2 = y3
and
g0 y j = eQ j
⇒
impossible!
⇒ Since they are both neutral, try a combination:
  
 
Wµ3
c
− sW
Z
sW ≡ sin θW , cW ≡ cos θW
 ≡ W
  µ
θW = weak mixing angle
Bµ
sW cW
Aµ
3
LNC =
with
T3
∑ ψj γ
j =1
µ
n
h
3
0
i
h
3
0
i
o
− gsW T + g cW y j Aµ + gcW T − g sW y j Zµ ψj
σ3
=
(0) the third weak isospin component of the doublet (singlet)
2
489
The EWSM with one family
neutral current interactions
To make Aµ the photon field:
e = gsW = g0 cW
Q = T3 + Y

where the electric charge operator is:
Q1 = 
Qf
0
0
Qf0

 ,
Q2 = Q f ,
Q3 = Q f 0
⇒ Electroweak unification: g of SU(2) and g0 of U(1) are related
⇒ The hyperchages are fixed in terms of electric charges and weak isospin:
y1 = Q f −
1
1
= Qf0 + ,
2
2
y2 = Q f ,
LQED = −e Q f f γµ f Aµ
y3 = Q f 0
+ ( f → f 0)
⇒ RH neutrinos are sterile: y2 = Q f = 0
490
The EWSM with one family
neutral current interactions
The Zµ is the neutral weak boson field:
Z
LNC
= e f γµ (v f − a f γ5 ) f Zµ
with
vf =
2
T 3f − 2Q f sW
L
2sW cW
,
af =
+ ( f → f 0)
T 3f
L
2sW cW
The complete neutral current Lagrangian reads:
Z
LNC = LQED + LNC
f
γ
f
Z
f = u, d, ℓ
f = u, d, ν, ℓ
491
The EWSM with one family
gauge boson self-interactions
Cubic:
LYM
o
iecW n µν †
†
⊃ L3 = −
W µ Z ν − Wµ† Wν Z µν
W Wµ Zν − Wµν
sW
o
n
†
W µ Aν − Wµ† Wν F µν
+ ie W µν Wµ† Aν − Wµν
with
Fµν = ∂µ Aν − ∂ν Aµ
Zµν = ∂µ Zν − ∂ν Zµ
W
γ
Wµν = ∂µ Wν − ∂ν Wµ
W
Z
W
W
492
The EWSM with one family
gauge boson self-interactions
Quartic:
LYM ⊃ L4 = −
e2
2
2sW
Wµ† W µ
2
− Wµ† W µ† Wν W ν
o
2 n
e 2 cW
ν
ν
† µ
† µ
− 2
Wµ W Zν Z − Wµ Z Wν Z
sW
o
e 2 cW n
+
2Wµ† W µ Zν Aν − Wµ† Z µ Wν Aν − Wµ† Aµ Wν Z ν
sW
n
o
2
† µ
ν
† µ
ν
− e Wµ W Aν A − Wµ A Wν A
W
W W
γ
W
γ
W
Z
W
W W
γ
W
Z
W
Z
Note: even number of W and no vertex with just γ or Z
493
Electroweak symmetry breaking
setup
Out of the 4 gauge bosons of SU(2) L ⊗U(1)Y with generators T 1 , T 2 , T 3 , Y we
need all to be broken except the combination Q = T 3 + Y so that Aµ remains
massless and the other three gauge bosons get massive after SSB
⇒ Introduce a complex SU(2) Higgs doublet
 
 
φ+
1 0


, h0| Φ |0i = √
Φ=
0
2 v
φ
with gauge invariant Lagrangian (µ2 = −λv2 ):
L Φ = ( Dµ Φ ) † D µ Φ − µ 2 Φ † Φ − λ ( Φ † Φ ) 2 ,
take yΦ =
1
2
⇒
e µ + ig0 yΦ Bµ )Φ
Dµ Φ = (∂µ − igW
 
0
3

=0
( T + Y ) |0i = Q
v
{ T 1 , T 2 , T 3 − Y } |0i 6 = 0
494
Electroweak symmetry breaking
gauge boson masses
Quantum fields in the unitary gauge:
σi i
Φ( x ) ≡ exp i θ ( x )
2v




0
1

√ 
2 v + H (x)
0
σi i
1

Φ( x ) 7→ exp −i θ ( x ) Φ( x ) = √ 
2v
2 v + H (x)
⇒
1 physical Higgs field
H (x)
3 would-be Goldstones
θ i ( x ) gauged away
– The 3 dof apparently lost become the longitudinal polarizations of W± and Z that
get massive after SSB:
LΦ ⊃ L M
g2 v2 † µ g2 v2
=
Wµ W + 2 Zµ Z µ
4}
8cW
| {z
|
{z }
2
MW
⇒
MW = MZ cW
1
= gv
2
1 2
2 MZ
495
Electroweak symmetry breaking
Higgs sector
⇒ In the unitary gauge (just physical fields):
LΦ = L H + L M + L HV 2 + 14 λv4
1 2 2 M2H 3 M2H 4
1
µ
H − 2H ,
L H = ∂µ H∂ H − M H H −
2
2
2v
8v
MH =
q
H
H
H
H
H
H
−2µ2
=
√
2λ v
H
2
L M + L HV 2 = MW
Wµ† W µ
W
W
2
H2
1+ H+ 2
v
v
H
H
2
1 2
2
H
+ MZ Zµ Z µ 1 + H + 2
2
v
v
Z
Z
H
Z
Z
H
H
W
W
H
496
Electroweak symmetry breaking
Higgs sector
Quantum fields in the Rξ gauges:


φ+ ( x )
 ,
Φ( x ) =  1
√ [ v + H ( x ) + iχ ( x )]
φ− ( x ) = [φ+ ( x )]∗
2
LΦ = L H + L M + L HV 2
1 4
+ λv
4
1
+ (∂µ φ+ )(∂µ φ− ) + (∂µ χ)(∂µ χ)
2
+ iMW (Wµ ∂µ φ+ − Wµ† ∂µ φ− ) + MZ Zµ ∂µ χ
+ trilinear interactions [SSS, SSV, SVV]
+ quadrilinear interactions [SSSS, SSVV]
497
Electroweak symmetry breaking
gauge fixing
To remove the cross terms Wµ ∂µ φ+ , Wµ† ∂µ φ− , Zµ ∂µ χ and define propagators add:
LGF = −
1
1
1
( ∂ µ Z µ − ξ Z M Z χ )2 −
( ∂ µ A µ )2 −
|∂µ W µ + iξ W MW φ− |2
2ξ γ
2ξ Z
ξW
⇒ Massive propagators for gauge and (unphysical) would-be Goldstone fields:
k
k
i
µ ν
γ
e µν
− gµν + (1 − ξ γ ) 2
D
(k) = 2
k + ie
k
k
k
i
µ ν
Z (k) =
e µν
D
−
g
+
(
1
−
ξ
)
;
µν
Z 2
k2 − M2Z + ie
k − ξ Z M2Z
k
k
i
µ ν
W (k) =
e µν
D
−
g
+
(
1
−
ξ
)
;
µν
W
2 + ie
2
k2 − MW
k2 − ξ W MW
e χ (k) =
D
i
k2 − ξ Z M2Z + ie
e φ (k) =
D
i
2 + ie
k2 − ξ W MW
(’t Hooft-Feynman gauge: ξ γ = ξ Z = ξ W = 1)
498
Electroweak symmetry breaking
Faddeev-Popov ghosts (?)
The SM is a non-Abelian theory ⇒ add Faddeev-Popov ghosts ci ( x ) (i = 1, 2, 3)
1
c1 ≡ √ ( u + + u − ) ,
2
i
c2 ≡ √ ( u + − u − ) ,
2
c 3 ≡ cW u Z − sW u γ
LFP = (∂µ ci )(∂µ ci − geijk c j Wµk ) + interactions with Φ
|
{z
} |
{z
}
U kinetic + [UUV]
U masses + [SUU]
⇒ Massive propagators for (unphysical) FP ghost fields:
e uγ ( k ) =
D
i
,
k2 + ie
e uZ (k) =
D
i
,
2
2
k − ξ Z MZ + ie
e u± ( k ) =
D
i
2 + ie
k2 − ξ W MW
(’t Hooft-Feynman gauge: ξ Z = ξ W = 1)
499
Electroweak symmetry breaking
Faddeev-Popov ghosts (?)
LFP = (∂µ uγ )(∂µ uγ ) + (∂µ u Z )(∂µ u Z ) + (∂µ u+ )(∂µ u+ ) + (∂µ u− )(∂µ u− )

iecW µ
µ
µ
µ

u
)
u
−
(
∂
u
)
u
]
A
−
[(
∂
u
)
u
−
(
∂
u− )u− ] Zµ
+
ie
[(
∂
+
+
−
−
µ
+
+


s

W


iecW µ
µ
µ
†
[UUV] − ie[(∂ u+ )uγ − (∂ uγ )u− ]Wµ +
[(∂ u+ )u Z − (∂µ u Z )u− ]Wµ†

sW




 + ie[(∂µ u )u − (∂µ u )u ]W − iecW [(∂µ u )u − (∂µ u )u ]W
− γ
γ +
µ
− Z
µ
Z +
sW
2
2
− ξ Z M2Z u Z u Z − ξ W MW
u+ u+ − ξ W MW
u− u−

1
1

+
−

Hu
−
φ
u
+
φ
u+
u
−
eξ
M

−
Z
Z
Z
Z

2sW cW
2sW



!#
"


2 − s2
cW
1
+
W
[SUU] − eξ W MW u+
( H + iχ)u+ − φ
uγ −
uZ

2sW
2sW cW



"
!#


2
2

cW − sW
1

−
− eξ W MW u−
( H − iχ)u− − φ
uγ −
uZ
2sW
2sW cW
500
Electroweak symmetry breaking
fermion masses
We need masses for quarks and leptons without breaking gauge symmetry
⇒ Introduce Yukawa interactions:

 

φ+
φ 0∗
 uR
LY = − λ d u L d L   d R − λ u u L d L 
φ0
−φ−

 

φ+
φ 0∗
 νR + h.c.
− λ` ν L ` L   ` R − λν ν L ` L 
φ0
−φ−


 
0∗
+
φ
φ
 transforms under SU(2) like Φ =  
where Φc ≡ iσ2 Φ∗ = 
−φ−
φ0
⇒ After EW SSB, fermions acquire masses:
o
n
1
LY ⊃ − √ (v + H ) λd dd + λu uu + λ` `` + λν νν
2
⇒
v
mf = λf √
2
501
Additional generations
Yukawa matrices
There are 3 generations of quarks and leptons in Nature. They are identical copies
with the same properties under SU(2) L ⊗ U(1)Y differing only in their masses
⇒ Take a general case of nG generations and let u jI , d jI , νjI , ` jI be the members of
family j (j = 1, . . . , nG ). Superindex I (interaction basis) was omitted so far
⇒ General gauge invariant Yukawa Lagrangian:

 




+
0∗
φ
φ
I
(
d
)
(u) I 
I
I





LY = − ∑
λ
d
+
λ
u jL d jL
kR
jk
jk ukR
0
−

φ
−φ
jk



 

+
0∗
φ
φ
I
(`)
(
ν
)
I
I
 λ ν I  + h.c.
+
+ ν jL
` jL  0  λ jk `kR
kR 
jk
−
−φ
φ
(d)
(u)
(`)
(ν)
where λ jk , λ jk , λ jk , λ jk are arbitrary Yukawa matrices
502
Additional generations
mass matrices
After EW SSB, in nG -dimensional matrix form:
n
o
H
I
I
I
I
I
I
I
I
LY ⊃ − 1 +
d L Md d R + u L Mu u R + l L M` l R + ν L Mν ν R + h.c.
v
with mass matrices
(d) v
(Md )ij ≡ λij √
2
(Mu )ij ≡
(u)
λij
v
√
2
(M` )ij ≡
(`)
λij
v
√
2
(Mν )ij ≡
(ν)
λij
v
√
2
⇒ Diagonalization determines mass eigenstates d j , u j , ` j , νj
in terms of interaction states d jI , u jI , ` jI , νjI , respectively
⇒ Each M f can be written as
M f = H f U f = S†f M f S f U f ⇐⇒ M f M†f = H2f = S†f M2f S f
q
with H f ≡ M f M†f a Hermitian positive definite matrix and U f unitary
– Every H f can be diagonalized by a unitary matrix S f
– The resulting M f is diagonal and positive definite
503
Additional generations
fermion masses and mixings
In terms of diagonal mass matrices (mass eigenstate basis):
Md = diag(md , ms , mb , . . .) ,
M` = diag(me , mµ , mτ , . . .) ,
LY ⊃ − 1 +
n
H
v
Mu = diag(mu , mc , mt , . . .)
Mν = diag(mνe , mνµ , mντ , . . .)
d Md d + u Mu u + l M` l + ~ν Mν ~ν
o
where fermion couplings to Higgs are proportional to masses and
d L ≡ Sd d LI
d R ≡ Sd Ud d RI
u L ≡ Su u LI
u R ≡ Su Uu u RI
l L ≡ S` l LI
l R ≡ S` U` l RI
νL ≡ Sν νLI
νR ≡ Sν Uν νRI

Neutral Currents preserve chirality 
⇒ LNC does not change flavor
I
I
I
I

f f = f f and f f = f f
L L
L L
R R
R R
⇒ GIM mechanism
[Glashow, Iliopoulos, Maiani ’70]
504
Additional generations
quark sector
However, in Charged Currents (also chirality preserving and only LH):
u LI d LI = u L Su S†d d L = u L Vd L
with V ≡ Su S†d the (unitary) CKM mixing matrix
[Cabibbo ’63;
Kobayashi, Maskawa ’73]
g
LCC = √ ∑ ui γµ (1 − γ5 ) Vij d j Wµ† + h.c.
2 2 ij
⇒
ui
dj
W
Vij
W
ui
V∗ij
dj
⇒ If ui or d j had degenerate masses one could choose Su = Sd (field redefinition)
and flavor would be conserved in the quark sector. But they are not degenerate
⇒ Su and Sd are not observable. Just masses and CKM mixings are observable
505
Additional generations
quark sector
How many physical parameters in this sector?
• Quark masses and CKM mixings determined by mass (or Yukawa) matrices
• A general nG × nG unitary matrix, like the CKM, is given by
n2G real parameters = nG (nG − 1)/2 moduli + nG (nG + 1)/2 phases
Some phases are unphysical since they can be absorbed by field redefinitions:
ui → eiφi ui ,
d j → eiθ j d j
⇒
Vij → Vij ei(θ j −φi )
Therefore 2nG − 1 unphysical phases and the physical parameters are:
(nG − 1)2 = nG (nG − 1)/2 moduli + (nG − 1)(nG − 2)/2 phases
506
Additional generations
quark sector
⇒ Case of nG = 2 generations: 1 parameter, the Cabibbo angle θC :


cos θC sin θC


V=
− sin θC cos θC
⇒ Case of nG = 3 generations: 3 angles + 1 phase. In the standard parameterization:

Vud Vus Vub

V=
 Vcd
Vtd

Vcs
Vts


1
0
0

c13
0 s13
 




Vcb  = 0 c23 s23 
0
1

Vtb
0 −s23 c23
−s13 eiδ13 0
c12 c13
s12 c13


= −s12 c23 − c12 s23 s13 eiδ13 c12 c23 − s12 s23 s13 eiδ13

s12 s23 − c12 c23 s13 eiδ13 −c12 s23 − s12 c23 s13 eiδ13
with cij ≡ cos θij ≥ 0,
sij ≡ sin θij ≥ 0
e−iδ

0


 −s

  12 c12 0
0
0 1
c13

s13 e−iδ13
s23 c13
c23 c13
(i < j = 1, 2, 3)
c12
s12 0


δ13 only source

 ⇒ of CP violation

in the SM !
and 0 ≤ δ13 ≤ 2π
507
Complete SM Lagrangian
fields and interactions
L = L F + LYM + LΦ + LY + LGF + LFP
• Fields: [F] fermions
[S] scalars
[V] vector bosons
• Interactions: [FFV]
[U] unphysical ghosts
[FFS]
[SSV]
[SVV]
[VVV]
[VVVV]
[SSS]
[SSSS]
[SUU]
[UUVV]
[SSVV]
508
Complete SM Lagrangian
Feynman rules
Feynman rules for generic couplings normalized to e (all momenta incoming):
(iL)
[FFVµ ]
[FFS]
[SVµ Vν ]
[S( p1 )S( p2 )Vµ ]
[Vµ (k1 )Vν (k2 )Vρ (k3 )]
[Vµ (k1 )Vν (k2 )Vρ (k3 )Vσ (k4 )]
ie( gS − gP γ5 ) = ie(c L PL + c R PR )
ieKgµν
ieG ( p1 − p2 )µ
ieJ gµν (k2 − k1 )ρ + gνρ (k3 − k2 )µ + gµρ (k1 − k3 )ν
2
ie C 2gµν gρσ − gµρ gνσ − gµσ gνρ
ie2 C2 gµν
also [UUVV]
[SSS]
ieC3
also [SUU]
[SSSS]
ie2 C4
[SSVµ Vν ]
Note: g L,R = gV ± g A
ieγµ ( gV − g A γ5 ) = ieγµ ( g L PL + gR PR )
Attention to symmetry factors!
c L,R = gS ± gP
http://www.ugr.es/local/jillana/SM/FeynmanRulesSM.pdf
509
Phenomenology
Input parameters
Parameters:
17 +9 =
1
1
1
1
9 +3
formal:
g
g0
v
λ
λf
MW
MZ
practical:
α
MH
mf
4
6
VCKM
UPMNS
where e = gsW = g0 cW and
e2
α=
4π
|
1
MW = gv
2
{z
g, g0 , v
M
MZ = W
cW
}
MH =
√
2λ v
v
mf = √ λf
2
⇒ Many (more) experiments
⇒ After Higgs discovery, for the first time all parameters measured!
510
Phenomenology
Input parameters
Experimental values
• Fine structure constant:
α−1 = 137.035 999 074 (44)
[Particle Data Group ’15]
from Harvard cyclotron ( ge )
• The SM predicts MW < MZ in agreement with measurements:
MZ = (91.1876 ± 0.021) GeV
MW = (80.387 ± 0.016) GeV
from LEP1/SLD
from LEP2/Tevatron/LHC
• Top quark mass:
mt = (173.24 ± 0.95) GeV
from Tevatron/LHC
• Higgs boson mass:
M H = (125.6 ± 0.4) GeV
from LHC
• ...
511
Phenomenology
Observables and experiments
Low energy observables
• ν-nucleon (NuTeV) and νe (CERN) scattering:
asymmetries CC/NC and ν/ν̄
2
⇒ sW
• atomic parity violation (Ce, Tl, Pb):
asymmetries eR,L N → eX due to Z-exchange between e and nucleus
• muon decay (PSI):
lifetime
GF2 m5µ
1
2
2
= Γµ =
/m
f
(
m
µ)
e
τµ
192π 3
f ( x ) ≡ 1 − 8x + 8x3 − x4 − 12x2 ln x
iM =
√
ie
2sW
2
2
⇒ sW
⇒ GF
2 )
Fermi theory (−q2 MW
z
}|
{
−igρδ
4GF
GF
πα
ρ
δ
ρ
eγ νL 2
νL γ µ ≡ i √ (eγ νL )(νL γρ µ) ; √ = 2 2
2
q − MW
2sW MW
2
2
512
Phenomenology
Observables and experiments
Low energy observables
⇒ Fermi constant provides the Higgs VEV (electroweak scale):
√
−1/2
v=
2GF
≈ 246 GeV
⇒ Consistency checks: e.g.
From muon lifetime:
GF = 1.166 378 7(6) × 10−5 GeV−2
If one compares with (tree level result)
G
πα
πα
√F = 2 2 =
2 /M2 ) M2
2sW MW
2(1 − MW
2
Z
W
using measurements of MW , MZ and α there is a discrepance that disappears
when quantum corrections are included
513
Phenomenology
Observables and experiments
e+ e− → f¯ f
nh
i
2
dσ
fα
= Nc β f 1 + cos2 θ + (1 − β2f ) sin2 θ G1 (s)
dΩ
4s
o
+2( β2f − 1) G2 (s) + 2β f cos θ G3 (s)
G1 (s) = Q2e Q2f + 2Qe Q f ve v f Reχ Z (s) + (v2e + a2e )(v2f + a2f )|χ Z (s)|2
G2 (s) = (v2e + a2e ) a2f |χ Z (s)|2
G3 (s) = 2Qe Q f ae a f Reχ Z (s) + 4ve v f ae a f |χ Z (s)|2
⇐ A FB
s
f
with χ Z (s) ≡
,
N
c = 1 (3) for f = lepton (quark), β f = velocity
2
s − MZ + iMZ Γ Z
σ(s) =
f
Nc
i
2πα2 h
β f (3 − β2f ) G1 (s) − 3(1 − β2f ) G2 (s) ,
3s
βf =
q
1 − 4m2f /s
514
Phenomenology
Observables and experiments
Z production (LEP1/SLD)
MZ , Γ Z , σhad , A FB , A LR , Rb , Rc , R`
2
⇒ M Z , sW
from e+ e− → f¯ f at the Z pole (γ − Z interference vanishes). Neglecting m f :
σhad
Γ(e+ e− )Γ(had)
= 12π
M2Z Γ2Z
Γ(had)
Γ(`+ `− )
f αMZ 2
Γ(Z → f f¯) ≡ Γ( f f¯) = Nc
(v f + a2f )
3
Rb =
Γ(bb̄)
Γ(had)
Rc =
Γ(cc̄)
Γ(had)
R` =
Forward-Backward and (if polarized e− ) Left-Right asymmetries due to Z:
A FB
σ − σB
3
Ae + Pe
= F
= Af
σF + σB
4 1 + Pe Ae
A LR
σ − σR
= Ae Pe
= L
σL + σR
with A f ≡
2v f a f
v2f + a2f
515
Phenomenology
Observables and experiments
W-pair production (LEP2)
e+ e− → WW → 4 f (+γ)
W production (Tevatron/LHC)
pp̄/pp → W → `ν` (+γ)
⇒ MW
⇒ MW
Top-quark production (Tevatron/LHC)
pp̄/pp → tt̄ → 6 f
⇒ mt
516
Observables and experiments
Higgs production (LHC)
pp → H + X and H decays to different channels ⇒ M H
[VBF]
gg fusion
WW, ZZ fusion
[ggF]
[VBF]
[VH]
[VH]
[ttH]
Higgs-strahlung
Higgs BR + Total Uncert [%]
[ggF]
tt̄ fusion
1
10-1
WW
bb
gg
oo
LHC HIGGS XS WG 2013
Phenomenology
ZZ
cc
10-2
aa
Za
120
140
10-3
[ttH]
µµ
10-4
80
100
160
180
200
MH [GeV]
517
5. Strong interactions
518
Strong interactions
Properties
Quantum Chromodynamics (QCD) is the theory of strong interactions
Quarks and gluons are the fundamental dof but they never show up as free states:
they are bound in hadrons (confinement):
Baryons (q1 q2 q3 or q1 q2 q3 )
name
Mesons
content
Q [e]
p
proton
uud
+1
p
antiproton
uud
−1
n
neutron
ddu
n
antineutron
ddu
Λ
lambda
uds
Λ
antilambda
uds
. . . ∼ 120 . . .
0
m [GeV]
name
π0
0,938
π+
π−
0,939
K+
K−
0
1,116
(q1 q2 )
K0
K
0
content
neutral pion
charged pion
charged kaon
neutral kaon
uu, dd
Q [e]
0
ud
+1
du
−1
us
+1
su
−1
ds
sd
0
m [GeV]
0,135
0,140
0,494
0,498
. . . ∼ 140 . . .
and exotics (glueballs, tetraquarks, pentaquarks, . . . )
519
Strong interactions
Properties
Strong interactions are responsible for:
• Stability of nuclei (nucleon-nucleon interaction is a residual strong force)
strong attraction is greater than electric repulsion
• ∼ 99 % of nucleon mass is binding energy, i.e. most of the mass in everything!
520
QCD
Lagrangian
SU(3) gauge symmetry
LQCD = ψ f i i D
/ij − mδij
{z
|
quarks
1 a a,µν
ψ f j − Fµν F
} |4 {z }
(flavor diagonal)
gluons
a
= ∂µ Aνa − ∂ν Aµa + gs f abc Abµ Acν
Fµν
(Anti–)quarks ψ f come in Nc = 3 colors (anticolors) and there are n f = 6 flavors:

 f = u, d, s, c, b, t (flavor index)
ψfi
fundamental irrep 3 (3̄)
 i = 1, . . . , Nc = 3 (color index)
Gluons Aµa come in Nc2 − 1 = 8 combinations of color and anticolor:
Aµa
a = 1, . . . , Nc2 − 1 = 8
(color index)
adjoint irrep 8
521
QCD
Lagrangian
SU(3) gauge symmetry
LQCD = ψ f i i D
/ij − mδij
{z
|
quarks
1 a a,µν
ψ f j − Fµν F
} |4 {z }
(flavor diagonal)
gluons
a
= ∂µ Aνa − ∂ν Aµa + gs f abc Abµ Acν
Fµν
Quark kinetic terms and quark-gluon interactions come from covariant derivative:
( Dµ )ij =
δij ∂µ − igs tija Aµa
,
tija
1 a
= λij
2
(8 Gell-Mann matrices 3 × 3)
Gluon kinetic terms and self-interactions fixed by SU(3) structure constants f abc :
Lkin
Lcubic
Lquartic
1
= − (∂µ Aνa − ∂ν Aµa )(∂µ A a,ν − ∂ν A a,µ )
4
1
= − gs f abc (∂µ Aνa − ∂ν Aµa )Ab,µ Ac,ν
2
1
= − gs2 f abe f cde Aµa Abν Ac,µ Ad,ν
4
522
QCD
Lagrangian
Feynman rules
Quark and gluon external legs and propagators are as usual
Vertices:
(interactions with would-be Goldstones and Faddeev-Popov ghosts omitted here)
523
QCD
About color charges
Quarks carry color charge:
Antiquarks carry anticolor charge:
 
R
 
ψ = ψ( x ) ⊗  G 
B
ψ = ψ( x ) ⊗ R G B
Gluons carry color and anticolor. A gluon emission repaints the quark:
 

a, µ
1
0 1 0




e.g.
ψi t1ij ψj ∼ 12 0 1 0 1 0 0 0
0 0 0
0
i
j
524
QCD
About color charges
3 ⊗ 3̄ = 1 ⊕ 8
8 gluons:
(1 in 9 combinations is color-neutral)
(
RR − GG
RR + GG − 2BB
If the color-singlet massless gluon state existed:
RR + GG + BB
it would give rise to a strong force of infinite range!
Likewise, only color-singlet states can exist as free particles:
q1 q̄2
q1 q2 q3
3 ⊗ 3̄ = 1 ⊕ 8
3 ⊗ 3 ⊗ 3 = 1 ⊕ 8 ⊕ 8 ⊕ 10
(here color-singlets are mesons)
(here color-singlets are baryons)
but q1 q2 color singlets do not exist, since 3 ⊗ 3 = 3̄ ⊕ 6
525
QCD
About color charges
t a = 12 λ a
Color algebra (useful identities):
1
(convention)
Tr(t t ) = TR δab , TR =
2
Nc2 − 1
a a
tik tkj = CF δij , CF =
2Nc
a b
f acd f bcd = C A δab ,
a
tija tkl
C A = Nc
1
1
= δil δjk −
δ δ
2
2Nc ij kl
In QCD:
Nc = 3 ,
(Fierz)
4
CF = ,
3
CA = 3
⇒ Since C A > CF , gluons have a larger color charge than quarks and therefore
gluons interact more strongly
526
QCD
About color charges
Color flow:
e.g.
qq̄ → qq̄
jk → il
(1)
(2)
(3)
xx → xx
xx → yy
xy → xy
xy → yx
qq̄ → qq̄
1
3
(1)
f =
(2)
1
f =
2
qq → qq
jl → ik
xx → xx
xy → yx
xy → xy
xx → yy
f (ijkl )
f ( xxxx ) =
f (yxxy) =
f ( xxyy) =
f (yxyx ) =
1
3
1
2
− 16
(3)
f =−
1
6
0
527
QED vs QCD
running coupling
All coupling constants run:
g2
α≡
= α( Q2 ), where Q is the momentum scale of the process
4π
Q2
∂α
= β(α) ,
2
∂Q
β ( α ) ≡ − β 0 α2 (1 + β 1 α + β 2 α2 + . . . )
2
α( Q ) =
α( Q20 )
1 + β 0 α( Q20 ) ln
Q2
(Leading Order)
Q20
Physically, this is related to the (anti-)screening of the fundamental charges by
quantum fluctuations, depending on the sign of β 0 :
e2
1
• In QED: αem =
, β 0,QED (αem ) = −
4π
3π
33 − 2n f
gs2
• In QCD: αs =
, β 0,QCD (αs ) =
4π
12π
(< 0)
(> 0 for n f ≤ 16)
528
QED
running coupling
In QED, the fluctuating vacuum behaves like a dielectric medium,
screening the bare electric charge e0 at increasing distances R ∼ 1/Q:
e.g. α( M2Z ) ≈ 1/128
529
QCD
running coupling
Contributions to the QCD beta function β(αs ) (from QCD vacuum polarization):
(1) screening
(2), (3)
anti-screening (non-abelian!)
530
QCD
running coupling
⇒ There is a scale ΛQCD where αs → ∞ (dimensional transmutation) given at LO by
1
1
2
Λ2QCD = Q2 exp −
⇔
α
(
Q
)
=
s
β 0 α s ( Q2 )
Q2
β 0 ln 2
ΛQCD
ΛQCD ≈ 200 MeV, that is R ∼ 1/Q ≈ 1 fm (the size of a proton!)
Asymptotic freedom:
At short distances (Q ΛQCD ) quarks and gluons are almost free,
they interact weakly: perturbative regime
Infrared slavery:
At long distances (Q ∼ ΛQCD ) the coupling diverges (Landau pole),
quarks and gluons interact very strongly (confinement into hadrons):
non-perturbative regime
⇒ Strong interactions are short-range, despite of gluon being massless
531
QCD
running coupling
532
QCD
Perturbative methods
Only when αs ( Q2 ) 1, i.e. when Q ΛQCD
Starting point: diagrams involving quarks and gluons at a given order
e.g.
q
t
q̄
t̄
Then: many sophisticated techniques to match real life . . .
533
QCD
real life (e.g. at LHC)
534
QCD
real life (e.g. at LHC)
535
QCD
real life (e.g. at LHC)
536
QCD
real life (e.g. at LHC)
537
QCD
real life (e.g. at LHC)
538
QCD
real life (e.g. at LHC)
539
Hadron structure
Deep Inelastic Scattering
Consider the following hadron scattering, described by the invariant quantities:
k0
k
q = k − k0
P
Q2 = − q2
Q2
x=
2( Pq)
y=
( Pq)
( Pk)
(momentum transfer squared)
(hadron’s momentum fraction
carried by the struck quark, as shown later)
(inelasticity)
Some have an easier interpretation in the lab frame, where the hadron is at rest:
P = ( M, 0, 0, 0)
k = ( E, 0, 0, E)








0
0
0
0
k = ( E , E sin θ, 0, E cos θ ) 
⇒
E − E0
y=
E
with s = ( P + k )2 = 2ME + M2 the CM energy squared.
Notice that
Q2 = 2( Pk) xy = xy(s − M2 )
(usually s M2 )
540
Hadron structure
Deep Inelastic Scattering
Varying Q2 changes the resolution of our microscope:
– Al low Q2 (long wavelength) one sees a pointlike hadron
(unresolved structure)
– At high Q2 (short wavelength) the probe interacts with a hadron constituent
resolving its structure
⇒ This is the deep inelastic scattering (DIS)
541
Hadron structure
DIS
Lepton–quark scattering
To understand the underlying processes, consider first the case of eq → eq
mediated by a photon exchange:
k
γ
p
k0
e
q = k − k0
p0
q
iM =
µ
0 −igµν ν
je,r ,r0 (k, k ) 2 jq,r2 ,r0 ( p,
1 1
2
q
µ
je,r
p0 )
0
0
(r1 ) 0
µ (r1 )
(
k,
k
)
=
u
(
k
)(
ieγ
)u (k)
0
,r
1 1
ν
jq,r
0 ( p,
2 ,r2
0
p)=u
(r20 )
( p0 )(−ieeq γν )u(r2 ) ( p)
The (unpolarized) differential cross section is:
dσ
1 1
=
16π ŝ2 4 r
dt̂
0
1 ,r2 ,r1 ,r2
|M| =
2πα2em eq2
ŝ2 + û2
ŝ2 t̂2
i1
dσ
4πα2em h
2
2
=
1
+
(
1
−
y
)
e
q
2
dQ2
Q4
or
where
∑0
2
pk − pk0
û
ŝ = ( p + k) , t̂ = (k − k ) = − Q , û = ( p − k ) ⇒ y =
= 1+
ŝ
pk
2
0 2
2
0 2
542
Hadron structure
DIS
Lepton–quark scattering
Assuming that the struck quark carries a fraction ξ of the hadron momentum P,
pµ = ξPµ
taking an on-shell (massless) quark,
p02 = ( p + q)2 = q2 + 2( pq) = −2( Pq)( x − ξ ) = 0
⇒
x=ξ
[as promised: x is the hadron’s momentum fraction carried by the struck quark]
and introducing
ˆ
0
1
dx δ( x − ξ ) = 1
we have:
i1
d2 σ
4πα2em h
2
2
=
1
+
(
1
−
y
)
e
δ( x − ξ )
q
2
dxdQ2
Q4
(1)
543
Hadron structure
DIS
Lepton–hadron scattering
Consider now the case of eh → eh:
k0
k
γ
q = k − k0
P
Notice that before, in the case of eq → eq, the cross section can be written as the
µν
contraction of a leptonic tensor (Lµν ) and a hadronic tensor (Wq ):
dσ
1 1
=
dQ2
16π ŝ2 4 r
∑0
0
1 ,r2 ,r1 ,r2
4πα2em
µν
0
0
|M| ≡
L
(
k,
k
)
W
(
p,
p
)
µν
q
4
Q
2
Lµν (k, k0 ) ≡ [kµ k0ν + kν k0µ − (kk0 ) gµν ]
1
ŝ
Wq ( p, p0 ) ≡ [ pµ p0ν + pν p0µ − ( pp0 ) gµν ]
µν
eq2
ŝ
where an appropriate normalization has been chosen
544
Hadron structure
DIS
Lepton–hadron scattering
µν
Now, replace Wq by the most general tensor built from momenta P and q:
W µν ≡ −W1 gµν +
W2 µ ν W4 µ ν W5 µ ν
µ ν
P
P
+
q
q
+
(
P
q
+
q
P )
2
2
2
M
M
M
compatible with parity conservation. For weak interactions (W ± ,Z) add:
eµναβ Pα q β
µνρσ k k 0 in Lµν )
iW3
(to
be
contracted
with
an
extra
ie
ρ σ
2M2
The form factors Wi = Wi ( x, Q2 ) depend on two (scalar) variables at a given s.
Since Lµν is symmetric, antisymmetric contributions are not introduced in W µν .
Furthermore, current conservation qµ W µν = qν W µν = 0 implies that:
2
( Pq)
M2
( Pq)
W5 = − 2 W2 , W4 =
W2 + 2 W1
2
q
q
q
The resulting hadronic tensor is:
µ
ν
q q
W
( Pq)
( Pq)
W µν = W1 − gµν + 2
+ 22 Pµ − 2 qµ
Pν − 2 qν
q
M
q
q
545
Hadron structure
DIS
Lepton–hadron scattering
Contracting the leptonic tensor with our generalized hadronic tensor one gets:
ŝ Lµν W
µν
W2
W3
0
0
= 2(kk )W1 + [2( Pk)( Pk ) − (kk ) M ] 2 − [( Pk)(qk ) − ( Pk )(qk)] 2
M
M
0
0
0
2
It is customary to redefine the form factors:
F1 =
( Pk)
W1 ,
( Pq)
F2 =
( Pk)
( Pk)
W
,
F
=
W3
2
3
2
2
M
M
so that, introducing x and y with pµ = xPµ and neglecting M2 s (then ŝ = xs),
1 h 2
y i
µν
Lµν W =
xy F1 + (1 − y) F2 + xy 1 −
F3
x
2
where we have used
(kk0 )
(kk0 )
0
( Pq) = y( Pk) =
, ( Pk ) = ( Pk) − ( Pq) =
(1 − y ) ,
x
xy
ŝ
(qk0 ) = −(qk) = (kk0 ) , (kk0 ) = y
2
546
Hadron structure
DIS
Lepton–hadron scattering
The differential cross section eh → eh (photon exchange) then reads:
2
2
4παem
1−y
d σ
2
=
[
1
+
(
1
−
y
)
]
F
+
( F2 − 2xF1 )
1
2
4
x
dxdQ
Q
that can be compared to the cross section eq → eq in (1):
i1
d2 σ
4πα2em h
2
2
=
1
+
(
1
−
y
)
δ( x − ξ )
e
q
2
dxdQ2
Q4
⇒ If hadron constituents were free quarks (spin
– Callan-Gross relation:
1
2
particles with charge eq ), then:
F2 = 2xF1
[ F2 = eq2 xδ( x − ξ )]
– Bjorken scaling. The structure functions would not depend on Q2 :
Fi ( x, Q2 ) = Fi ( x )
(the constituents are pointlike particles)
Both properties are modified by gluon corrections
547
Hadron structure
DIS
Neutrino-nucleon scattering (small detour)
From previous expressions one can easily obtain cross sections for νN scattering,
CC (W-exchange) and NC (Z-exchange), replacing the photon propagator by
1
1
− 2 →− 2
,
2
Q
Q + MV
MV = MW,Z
and extracting/absorbing some constants from/in the form factors, as
GF2
π 2 α2em
g4
= 4 4 =
4
2sW MW
32MW
The νN cross sections are then:
CC,NC
CC,NC
d2 σνN
d2 σνN
= xs
dxdy
dxdQ2
=
GF2 s
2
MV
π
2
Q2 + MV
!2
h
i
y
xy2 F1CC,NC + (1 − y) F2CC,NC + xy 1 −
F3CC,NC
2
548
Hadron structure
DIS
Parton Model
The hadron is composed of partons: quarks and/or antiquarks, in principle.
Then, one can introduce the Parton Distribution Functions (PDF)
F2 ( x ) = 2xF1 ( x ) =
∑ eq2 x fq/h (x)
q,q̄
where f q/h dx expresses the probability to find a quark q inside hadron h carrying
a fraction of the hadron longitudinal momentum in [ x, x + dx ].
In e-proton and e-neutron scattering one probes the nucleon structure functions
2
2
2
1 ep
2
1
1
p
F2 =
( up + up ) + −
( dp + d ) + −
( sp + sp ) + . . .
x
3
3
3
2
2
2
1 en
2
1
1
n
F2 =
( un + un ) + −
( dn + d ) + −
( sn + sn ) + . . .
x
3
3
3
where the PDFs ( f u/p ( x ) ≡ up ( x ), etc.) are related by isospin symmetry:
up = dn ≡ u ,
dp = un ≡ d ,
sp = sn ≡ s ,
...
549
Hadron structure
DIS
Parton Model
The quantum numbers of a proton must be those of a uud combination of
valence quarks ( f v ). The rest are sea quarks ( f s ):
u = uv + us ,
d = dv + ds ,
u = us = us ,
d = ds = ds ,
s = ss = ss ,
...
u
d
u
550
Hadron structure
DIS
Parton Model
Then the following sum rules apply:
ˆ
ˆ
1
1
dx [u( x ) − u( x )] = 2
dx uv ( x ) =
0
ˆ
0
ˆ
1
dx dv ( x ) =
0
0
1
dx [d( x ) − d( x )] = 1
And, assumimg just u, d and s and taking S as the total sea contribution:
1 ep
1
F2 = (4uv + dv ) +
x
9
1 en
1
F2 = (uv + 4dv ) +
x
9
4
S
3
4
S
3
ep
One can now guess what the proton structure function F2 looks like . . .
551
Hadron structure
If the proton is
DIS
Parton Model
ep
then F2 ( x ) is
552
Hadron structure
If the proton is
DIS
Parton Model
ep
then F2 ( x ) is
553
Hadron structure
DIS
Parton Model (Where are the gluons?)
Another sum rule: sum of partons must carry all hadron momentum,
ˆ 1
dx x [u( x ) + u( x ) + d( x ) + d( x ) + s( x ) + s( x ) + . . . ] = 1
0
ˆ
Actually gluons carry about 50 % of hadron momentum:
0
1
dx xg( x ) ≈ 0.5
554