Post on 10-Jun-2020
transcript
Introduction to the Standard Model
1. e+e- annihilation and QCD
M. E. PeskinPiTP Summer SchoolJuly 2005
In these lectures, I will describe the phenomenology of the Standard Model of particle physics.
I will discuss mainly processes seen in e+e- annihilation at CM energies from 10 to 200 GeV.
Later in the school, you will hear about proton-proton collisions at very high energy. But e+e- reactions are much simpler, so this is a good place to start.
Even at 10 GeV, the energy is high enough that we can ignore the masses of most of the quarks and leptons. Then we can see the Standard Model gauge interactions in a pure form and relate phenomena directly to the Standard Model Lagrangian.
You know the Standard Model as
a 4-d SU(3) x SU(2) x U(1) non-supersymmetric gauge theory
that is chiral but anomaly-free
in which masses are forbidden by SU(2) x U(1) symmetry
and therefore arise only by spontaneous symmetry breaking.
Fine. But, what do the particles of the Standard Model actually look like in the lab ?
Begin with the leptons:
Begin, in fact, with the simplest reaction involving leptons
in QED
At high energy, ignore all masses. Then,
helicity
is a Lorentz-invariant. Fermions are described as
left-handed right-handed
The is the antiparticle of the and vice versa.
e−
, µ−
, τ−
+ νe, νµ, ντ
e+e−
→ µ+µ−
h = !S · P̂
e−
Re+
L
The amplitudes for between states of definite helicity are very simple:
that is,
iM(e−Le+R → µ−
Lµ+R)
= 2ie2
q2[2p1 · k22p2 · k1]
1/2
= ie2(1 + cos θ)
(1)
iM(e−L
e+R→ µ−
Lµ+
R) = iM(e−
Re+L→ µ−
Rµ+
L) = ie2(1 + cos θ)
iM(e−L
e+R→ µ−
Rµ+
L) = iM(e−
Re+L→ µ−
Lµ+
R) = ie2(1 − cos θ) (1)
e+e−
→ µ+µ−
p1
p2
k1
k2
These formulae lead to the regularities:
with . The second formula sets the size of cross sections for all QED and electroweak processes.
Here are some examples of e+e- annihilation to leptons
and the related process of Bhabha scattering:
dσ
d cos θ∼ (1 + cos2 θ) σ =
4πα2
3s=
87. fb
(ECM TeV)2
s = q2 = (ECM )2
e+e−
→ µ+µ−
e+e−
→ τ+τ−
e+e−
→ e+e−
+
A relativistic muon is a “minimum ionizing” particle, giving a continuous small energy loss in matter:
The muon is long-lived and essentially stable in high energy experiments.
Electrons emit hard gamma rays, and gammas convert to electron-positron pairs in a characteristic distance called a radiation length . In heavy materials,
The result is an electromagnetic shower:
An electromagnetic calorimeter collects the resulting ionization, which gives a measure of the total energy of the original particle.
τ = 2.2 × 10−6
sec cτ = 660 m
dE
dx≈ 1 GeV/m
X0 X0 ∼ cm
e e + _e e + _
Z0
SLD
e+e−
→ µ+µ−
BaBar
SLD
e e + _ + _Z
0
For the lepton,
The rapidly decays through weak interactions
This produces a variety of events
τ
τ
τ
τ = 2.9 × 10−13
sec cτ = 0.087 mm
τ−
→ ντνee−
→ ντνµµ−
→ ντud , ντus
→ ντπ− , ντρ− , ντa−
1, · · · (1)
π−
π0
3π
e+e−
→ τ+τ−
→ eµνν
Leptonic reactions are modified by radiative corrections.
I would like to examine in particular the diagrams giving initial-state radiation (ISR)
becomes singular when
0 ← (p + q)2 = 2p · q + p2∼ −2p
2
⊥p
q
In the limit in which the and are almost collinear, we can isolate the sigularity for emission. Its coefficient has the form:
and this leads to the following approximate expression
with
f(x) =α
2π
∫dp2
⊥
p2
⊥
1 + (1 − x)2
x=
α
2π· log
s
m2e
·
1 + (1 − x)2
x
e−
γ
γ
p
xp
(1-x)p
e−
L=
ie
√2(1−x)
x(1−x) p⊥ γL
ie
√2(1−x)
xp⊥ γR
σ(e−(p1)e+(p2) → γ + e−e+
→ γ + X)
=
∫ 1
0
dx f(x) σ(e−((1 − x)p1) + e+(p2) → X)
+
∫ 1
0
dx f(x) σ(e−(p1) + e+((1 − x)p2) → X) (1)
These singular terms give enhanced radiation in directions parallel to the momenta of the initial state particles (ISR) and also parallel to the momenta of the final state particles (FSR).
Here is a BaBar hadronic event with ISR
In addition to the leptonic processes, we also have
How do these reactions appear ?
Quarks are not seen in isolation. If isolated quarks could be produced in e+e- annihilation, it would be obvious, because quarks carry fractional charge.
e+e−
A property of the strong coupling phase of gauge theories is that the gauge charge is permanently confined into gauge singlet states.
A way to picture this is that the electric flux of the gauge field cannot spread into the vacuum. Since this flux is conserved, we have states such as
This object is a relativistic string. It has
As the string becomes longer, it becomes energetically favorable to create an additional pair.
E ∼ L
ud
_
This leads to a first picture of the time development of a state created in e+e- annihilation:
Asymptotic freedom implies that all of these nonperturbative effects take place with no large momentum transfer.
Then, the final momenta are approximately collinear and parallel to the original quark directions.
u
u
uu
u
d
d
d
d d
ddss
_
_
_ _
_
_
_
Thus, we expect
the final states in should look like jets of mesons and baryons
the angular distribution of the axes of the jets should be
the cross section for such jet-like events should be
e+e−
dσ
d cos θ∼ (1 + cos2 θ)
σ =4πα2
3s· 3 ·
∑
f
Q2
f
e+e−
BaBar
SLD
SLD
θ
's 's
0
1
2
3
4
5
6
7
s in GeV
Bacci et al.
Cosme et al.
PLUTO
CESR, DORIS
MARK I
CRYSTAL BALL
MD-1 VEPP-4
VEPP-2M ND
DM2
BES 1999
BES 2001BES 2001
CMD-2 2004
KLOE 2005
Burkhardt, Pietrzyk 2005
15 5.9 6 1.4 0.9
rel. err. cont.
0 1 2 3 4 5 6 7 8 9 10
Rhad
What do we see in such a jet ? The components are:
Long-lived charged hadrons: ‘s : Long-lived neutral hadrons: Short-lived hadrons with :
Charged particles appear as tracks.
s appear as energy deposited in an electromagnetic calorimeter
Hadrons appear as energy deposition in a hadron calorimeter thick enough to allow many hadron interaction lengths
so by choosing different materials, it possible to separately measure the various components of the jet
π . K , p cτ(K+) = 3.7 mπ
0→ 2γ with cτ = 2.5 × 10
−6cm
n , K0
L
cτ ∼ few cm K0
S , Λ , Σ , Ξ
X0 = 0.56 cm (Pb) , 1.76 cm (Fe) , 35.3 cm (Be)
π0
γ
λI = 17.1 cm (Pb) , 16.8 cm (Fe) , 40.7 cm (Be)
Up to now we have ignored the asymptotically free QCD coupling
Let’s now compute the effects of order :
These include diagrams with virtual gluons
and diagrams with real gluon emission:
setting
the gluon emission diagrams predict:
αs = g2
s/4π
Eq = xq
√s
2Eq = xq
√s
2Eg = xg
√s
2
dσ
dxqdxq
= σ0 ·
2αs
3π·
x2q
+ x2
q
(1 − xq)(1 − xq)
αs
Notice that the gluon emission is singular as These are configurations in which the gluon is collinear with the antiquark or quark.
It is possible integrate up the total cross section by working in
Then the real gluon emission gives
The virtual diagrams give
in all
xq , xq → 1
d = 4 + ε
σ = σ0 ·
2αs
3π·
(8
ε2+
1
ε[4 log
s
M2− 6]
+ log2 s
M2− 3 log
s
M2−
7π2
6+
57
6
)(1)
σ = σ0 ·
{1 +
2αs
3π·
(−
8
ε2−
1
ε[4 log
s
M2− 6]
− log2 s
M2+ 3 log
s
M2+
7π2
6− 8
)}(1)
σ = σ0 · (1 +2αs
3π
·
3
2) = σ0 · (1 +
αs
π
)
The final result, and its generalization to higher orders, gives a small positive modification of the cross section for
Gluon emission at wide angles gives
3-jet events at a rate proportional to
4-jet events at a rate proportional to
etc.
e+e−
σ = σ0 · (1 +αs(s)
π
+ 1.441
(αs(s)
π
)2
− 12.8
(αs(s)
π
)3
+ · · ·)
αs
α2
s
SLD 3-jet event
SLD 4-jet event
3–968121A3
00
2
4
1/N
dn/d
x3
1/N
dn
/dx
1
x3
0.2 0.4 0.6
0
5
10
15
x1
0.6 0.8 1.0
1/N
dn/d
x2
0
2
4
6
0.80.6 1.0
00
11
/N d
n/d
co
sE
K
cos EK
0.4 0.8
2
3
(c)
(a)
(d)
(b)
SLD HERWIG 5.7
x2
If a large fraction of events contain gluon emissions, we need a systematic way to identify jets and to count the number of jets in an event.
Actually, since we only observe hadrons and not quarks, we need such a procedure anyway, to give a systematic way to compute, e.g., the orientation of the jet axis.
Among many possible ways to do this, a particularly useful one is based on thrust
The axis that gives the maximum (the thrust axis) is a reasonable definition of the event axis.
QCD predicts in leading order
T = maxn̂
∑j |n̂ · !pj |
∑j |!pj |
dσ
dT= σ0 ·
2αs
3π
{2(3T 2
− 3T + 2)
T (1 − T )log
2T − 1
1 − T−
3((3T − 2)(2 − T )
(1 − T )
}
e+e−
Particle distribution from the thrust axis, from the BaBar experiment ( ECM = 10.58 GeV)
For multi-jet events, we need a way to partition the hadrons in the event into jets. Here is a particular method; later in the school, we will discuss other possible algorithms.
Let
Choose the 2 particles in the final state s.t. is minimal. Merge them and replace them with 1 particle with . Continue until
Then the resulting can be taken as the operationally defined jet momenta.
Jets in QCD have a fractal structure, with more structure revealed as is decreased.
yij =(pi + pj)2
s≈
2pi · pj
s≈
1
2xixj(1 − cos θij)
(pi + pj)2
(pi + pj)2
s> ycut
p = (pi + pj)
pi
ycut
OPAL (91 GeV)Durham
2-jet3-jet4-jet5-jet
PYTHIAHERWIG
ycut
Jet
Fra
ctio
n
0
0.2
0.4
0.6
0.8
1
10-4
10-3
10-2
10-1
OPAL (197 GeV)Durham
2-jet3-jet4-jet5-jet
PYTHIAHERWIG
ycut
Jet
Fra
ctio
n
0
0.2
0.4
0.6
0.8
1
10-4
10-3
10-2
10-1
Thinking about how multiple gluons are emitted by a quark, we can arrive at a less naive model of how a state in e+e- annihilation evolves into a state of many hadrons.
Define the fragmentation function:
For a hadron in the final state of e+e- annihilation, let
longitudinal fraction
Then let
be the probability of finding a hadron in the final state with longitudinal fraction .
h
z = Eh/(1
2
√s)
fh(z)dz
zh
p
zp
xp
(1/2
Ne
vts
) d
n/d
xp
+
K K+ (x0.1)
p p (x0.02)
SLD u,d,s Jets102
101
100
10-1
10-2
10-3
10-4
0.04 0.1 0.4 18-20038675A13
fragmentationfunctions
measured bySLD
The variable z gives the fraction of the original quark energy that ends up in a final handron. With QCD, we have studied the first stage of the degradation of the energy of the quark, due to gluon emission.
Let’s look again at the formula for gluon radiation
In the limit (q+g collinear; ), integrating over , this becomes
This is the same radiation formula that we saw in QED.
xq → 1 xq ≈ (1 − xg)xq
dσ
dxg
≈ σ0 ·
4
3
αs
2π· log
s
m2(qg)min
·
1 + (1 − xg)2
xg
dσ
dxqdxq
= σ0 ·
2αs
3π·
x2q
+ x2
q
(1 − xq)(1 − xq)
This calculation describes the emission of one approximately collinear gluon. By repeating the process, we can describe multiple gluon emissions.
These multiple gluons build up the structure of a quark jet.
The quark first emits hard gluons. As we go to smaller or smaller virtualities, the quarks and gluons emit further quarks and gluons.
p⊥
Each emission degrades the energy of the final hadrons. This effect of collinear gluon emission from the quark is described by the Gribov-Lipatov equation
Here P(z) is the splitting function
where A is determined by the condition of quark number conservation
The singularity is normally treated in QCD by defining a distribution that agrees with 1/(1-z) for z < 1 such that
Pq←q(z) =4
3
(1 + z2
(1 − z)− Aδ(1 − z)
)
dfq(x, Q)
d log Q=
αs
π
∫ 1
0
dz
zPq←q(z) fq(
x
z, Q)
∫ 1
0
dz1
(1 − z)+= 0
∫ 1
0
dzPq←q(z) = 0
The full dynamics of QCD contains a number of possible collinear splitting processes. Each leads to a characteristic kernel, the splitting functions
Putting these effects together, we obtain the Altarelli-Parisi equations. For example
Pq←q(z) =4
3
(1 + z2
(1 − z)++
3
2δ(1 − z)
)
Pg←q(z) =4
3
(1 + (1 − z)2
z
)
Pq←g(z) =1
2
(z2 + (1 − z)2
)
Pg←g(z) = 6
(1 − z
z+
z
(1 − z)++ z(1 − z) + (
11
12−
nf
18)δ(1 − z)
)(1)
dfh←q(x, Q)
d log Q=
αs
π
∫ 1
0
dz
z
{Pq→q(z) fh←q(
x
z, Q) + Pg←q(z) fh←g(
x
z, Q)
}
According to this equation, fragmentation functions evolve as a function of s. This effect is observed.
PDG (Biebel and Webber) compilation
Finally, we can ask about the value of needed to explain the various QCD effects we have discussed in this lecture.
The hallmark of a gauge theory is that there is a single unified coupling constant. Is experiment consistent with this ?
αs
e+e- event shape variables only
All techniques
Bethke
We have now seen that the strong interactions at high energy show very directly the quark-gluon structure of an SU(3) gauge theory - QCD.
In the next lecture, we will examine the weak interactions at high energy. We will see that these also reflect the gauge stuctures of the Standard Model.