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Particle Physics Phenomenology3. Evolution equations and final-state showers
Torbjorn Sjostrand
Department of Astronomy and Theoretical PhysicsLund University
Solvegatan 14A, SE-223 62 Lund, Sweden
Lund, 30 January 2018
Multijets – the need for showers
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 2/50
QED bremsstrahlung – 1
Accelerated electric charges radiate photons ,
see e.g. J.D. Jackson, Classical Electrodynamics.A charge ze that changes its velocity vector from β to β′ radiatesa spectrum of photons that depends on its trajectory. In thelong-wavelength limit it reduces to
limω→0
d2I
dωdΩ=
z2e2
4π2
∣∣∣∣ε∗( β′
1− nβ′− β
1− nβ
)∣∣∣∣2where n is a vector on the unit sphere Ω, ω is the energy of theradiated photon, and ε its polarization.
1 For fast particles radiation collinear with the β and β′
directions is strongly enhanced.
2 dN/dω = (1/ω)dI/dω ∝ 1/ω so infinitely many infinitely softphotons are emitted, but the net energy taken away is finite.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 3/50
QED bremsstrahlung – 2
An electrical charge, say an electron,is surrounded by a field:
For a rapidly moving chargethis field can be expressed in terms ofan equivalent flux of photons:
dnγ ≈2αem
π
dθ
θ
dω
ω
Equivalent Photon Approximation,or method of virtual quanta (e.g. Jackson)(Bohr; Fermi; Weiszacker, Williams ∼1934)
e!
e!
e!
e!
.
θ: collinear divergence, saved by me > 0 in full expression.
ω: true divergence, nγ ∝∫
dω/ω =∞, but Eγ ∝∫
ω dω/ω finite.
These are virtual photons: continuously emitted and reabsorbed.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 4/50
QED bremsstrahlung – 3
When an electron is kicked into a new direction,the field does not have time fully to react:
e!
Initial State Radiation (ISR):part of it continues ∼ in original direction of eFinal State Radiation (FSR):the field needs to be regenerated around outgoing e,and transients are emitted ∼ around outgoing e direction
Emission rate provided by equivalent photon flux in both cases.Approximate cutoffs related to timescale of process:the more violent the hard collision, the more radiation!
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 5/50
Divergences
Emission rate q→ qg diverges whencollinear: opening angle θqg → 0
soft: gluon energy Eg → 0
Almost identical to e → eγbut QCD is non-Abelian so additionally
g → gg similarly divergent
αs(Q2) diverges for Q2 → 0
(actually for Q2 → Λ2QCD)
Big probability for one emission =⇒ also big for several=⇒ with ME’s need to calculate to high order and with manyloops =⇒ extremely demanding technically (not solved!), andinvolving big cancellations between positive and negativecontributions.Alternative approach: parton showers
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 6/50
The Parton-Shower Approach
2 → n = (2 → 2) ⊕ ISR ⊕ FSR
FSR = Final-State Radiation = timelike showerQ2
i ∼ m2 > 0 decreasingISR = Initial-State Radiation = spacelike showersQ2
i ∼ −m2 > 0 increasing
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 7/50
Why “time”like and “space”like?
Consider four-momentum conservation in a branching a → b c
p⊥a = 0 ⇒ p⊥c = −p⊥b
p+ = E + pL ⇒ p+a = p+b + p+c
p− = E − pL ⇒ p−a = p−b + p−c
Define p+b = z p+a, p+c = (1− z) p+a
Use p+p− = E 2 − p2L = m2 + p2
⊥
m2a + p2
⊥a
p+a=
m2b + p2
⊥b
z p+a+
m2c + p2
⊥c
(1− z) p+a
⇒ m2a =
m2b + p2
⊥z
+m2
c + p2⊥
1− z=
m2b
z+
m2c
1− z+
p2⊥
z(1− z)
Final-state shower: mb = mc = 0⇒ m2a =
p2⊥
z(1−z) > 0 ⇒ timelike
Initial-state shower: ma = mc = 0⇒ m2b = − p2
⊥1−z < 0 ⇒ spacelike
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 8/50
Showers and cross sections
Shower evolution is viewed as a probabilistic process,which occurs with unit total probability:the cross section is not directly affectedHowever, more complicated than so
PDF evolution ≈ showers ⇒ enters in convoluted crosssection, e.g. for 2 → 2 processes
σ =
∫∫∫dx1 dx2 dt fi (x1,Q
2) fj(x2,Q2)
dσij
dt
Shower affects event shapeE.g. start from 2-jet event with p⊥1 = p⊥2 = 100 GeV.ISR gives third jet, plus recoil to existing two, sop⊥1 = 110 GeV, p⊥2 = 90 GeV, p⊥1 = 20 GeV:
hardest p⊥jet spectrum goes uptwo-jets with both jets above some p⊥min comes downthree-jet rate goes upinclusive p⊥jet spectrum goes up (steeply falling slope!)
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 9/50
Doublecounting
Do not doublecount: 2 → 2 = most virtual = shortest distanceConflict: theory derivations assume virtualities strongly ordered;interesting physics often in regions where this is not true!
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 10/50
Timelike shower: the prototype
For the rest of this lecture, consider specifically
Final State Radiation
Standard process: e+e− → qqg (or qq→ γ∗/Z0/W± → qqg).
Contributions from two Feynman diagrams:
For simplicity neglect quark masses.Flavour dependence and mix of γ∗ and Z 0 propagators divides outin ratio σ(e+e− → qqg)/σ(e+e− → qq).
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 11/50
Timelike shower: the prototype
For the rest of this lecture, consider specifically
Final State Radiation
Standard process: e+e− → qqg (or qq→ γ∗/Z0/W± → qqg).
Contributions from two Feynman diagrams:
For simplicity neglect quark masses.Flavour dependence and mix of γ∗ and Z 0 propagators divides outin ratio σ(e+e− → qqg)/σ(e+e− → qq).
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 11/50
The phase space – 1
Define p0 = p1 + p2 + p3, and
xi =2p0pi
p20
=2EcmEi
E 2cm
=2Ei
Ecm
so that x1 + x2 + x3 = 2. Also note that
yjk =m2
jk
E 2cm
=(pj + pk)2
p20
=(p0 − pi )
2
p20
=p20 − 2p0pi + p2
i
p20
= 1− xi
with i , j , k permutation of 1, 2, 3.Hence y23 + y13 + y12 = 3− (x1 + x2 + x3) = 1.
Limit x1, x2 → 1 ⇔ y23, y13 → 0 ⇔ m2qg,m
2qg → 0,
i.e. propagator for q→ qg, q→ qg goes on shell.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 12/50
The phase space – 2
Since 0 ≤ yjk ≤ 1 and xi = 1− yjk it follows that 0 ≤ xi ≤ 1.Also obvious from momentum conservation:pi + pj + pk = 0⇒ |pi | = | − (pj + pk)| ≤ |pj |+ |pk |⇔ Ei ≤ Ej + Ek ⇔ xi ≤ xj + xk
⇒ xi ≤ (xi + xj + xk)/2 = 1.
Allowed triangular region of phase space:
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 13/50
The phase space – 2
Since 0 ≤ yjk ≤ 1 and xi = 1− yjk it follows that 0 ≤ xi ≤ 1.Also obvious from momentum conservation:pi + pj + pk = 0⇒ |pi | = | − (pj + pk)| ≤ |pj |+ |pk |⇔ Ei ≤ Ej + Ek ⇔ xi ≤ xj + xk
⇒ xi ≤ (xi + xj + xk)/2 = 1.
Allowed triangular region of phase space:
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 13/50
The matrix elements – 1
σ0 = σ(e+e− → qq) =4πα2
em
3Nc
q2f
s
− 2qf vf
4 sin2θW
s −m2Z
(s −m2Z)2 + m2
ZΓ2Z
+v2f + a2
f
(4 sin2θW )2s
(s −m2Z)2 + m2
ZΓ2Z
dσMEσ0
= αs2π
43
x21+x2
2(1−x1)(1−x2)
dx1 dx2
Note on Phase Space:
dPS2 ∝ d cos θq dφq
dPS3 ∝ d cos θq dφq dχqg dx1 dx2
Since x1, x2 → 1 dominates, x21 + x2
2 ≈ 2, and
dσME
σ0≈ αs
2π
4
3
2 dx1 dx2
(1− x1)(1− x2)=
αs
2π
8
3
dy23 dy13
y23y13=
αs
2π
8
3
dm223 dm2
13
m223m
213
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 14/50
The matrix elements – 2
Two “kinds” of singularities, collinear and soft, closely interlinked.
Convenient (but arbitrary) subdivision:
1
(1− x1)(1− x2)
(1− x1) + (1− x2)
x3=
1
(1− x2)x3+
1
(1− x1)x3
so split into radiation “from” q and “from” q
dσME
σ0≈ αs
2π
8
3
(dm2
13
m213
dx3
x3+
dm223
m223
dx3
x3
)Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 15/50
The matrix elements – 3
Divergences necessitates introduction of cuts for physical rate:
(ε, δ) cuts (Sterman–Weinberg)xi < 1− ε, θij > δ“all but a fraction ε of the energy is withincones of size δ”
y cut: yij > ymin
Also called Thrust cut, whereT = max(x1, x2, x3) = 1−min(y12, y13, y23)for three-jets.∫
ymin
dσME
σ0≈ αs
2π
8
3
∫ymin
dy23 dy13
y23 y13≈ 4αs
3πln2 ymin
i.e. double logarithmic divergence.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 16/50
From matrix elements to parton showers
Rewrite for x2 → 1, i.e. q–g collinear limit:
1− x2 =m2
13
E 2cm
=Q2
E 2cm
⇒ dx2 =dQ2
E 2cm
define z as fraction q retainsin branching q→ qg
x1 ≈ z ⇒ dx1 ≈ dz
x3 ≈ 1− z
⇒ dP =dσ
σ0=
αs
2π
dx2
(1− x2)
4
3
x22 + x2
1
(1− x1)dx1 ≈
αs
2π
dQ2
Q2
4
3
1 + z2
1− zdz
In limit x1 → 1 same result, but for q→ qg.
dQ2/Q2 = dm2/m2: “mass (or collinear) singularity”
dz/(1− z) = dω/ω “soft singularity”
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 17/50
The DGLAP equations
Generalizes to
DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi)
dPa→bc =αs
2π
dQ2
Q2Pa→bc(z) dz
Pq→qg =4
3
1 + z2
1− z
Pg→gg = 3(1− z(1− z))2
z(1− z)
Pg→qq =nf
2(z2 + (1− z)2) (nf = no. of quark flavours)
Universality: any matrix element reduces to DGLAP in collinear limit.
e.g.dσ(H0 → qqg)
dσ(H0 → qq)=
dσ(Z0 → qqg)
dσ(Z0 → qq)in collinear limit
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 18/50
The iterative structure
Generalizes to many consecutive emissions if strongly ordered,Q2
1 Q22 Q2
3 . . . (≈ time-ordered).To cover “all” of phase space use DGLAP in whole regionQ2
1 > Q22 > Q2
3 . . ..
Iteration givesfinal-stateparton showers:
Need soft/collinear cuts to stay away from nonperturbative physics.Details model-dependent, but around 1 GeV scale.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 19/50
The ordering variable
In the evolution with
dPa→bc =αs
2π
dQ2
Q2Pa→bc(z) dz
Q2 is a “temporal” variable: ordering of emissions (memory)
z is a “spatial” variable: no ordering (no memory)
If Q2 = m2 is one possible evolution variablethen Q ′2 = f (z)Q2 is also allowed, since∣∣∣∣d(Q ′2, z)
d(Q2, z)
∣∣∣∣ =∣∣∣∣∣ ∂Q′2
∂Q2∂Q′2
∂z∂z
∂Q2∂z∂z
∣∣∣∣∣ =∣∣∣∣ f (z) f ′(z)Q2
0 1
∣∣∣∣ = f (z)
⇒ dPa→bc =αs
2π
f (z)dQ2
f (z)Q2Pa→bc(z) dz =
αs
2π
dQ ′2
Q ′2 Pa→bc(z) dz
Q ′2 = E 2a θ2
a→bc ≈ m2/(z(1− z)); angular-ordered shower
Q ′2 = p2⊥ ≈ m2z(1− z); transverse-momentum-ordered
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 20/50
The Sudakov form factor – 1
(Back to radioactive decay:) Conservation of total probability:P(no decay) = 1− P(decay)“multiplicativeness” in “time” evolution:Pno(0 < t ≤ T ) = Pno(0 < t ≤ T1) Pno(T1 < t ≤ T )
Subdivide further, with Ti = (i/n)T , 0 ≤ i ≤ n:
Pno(0 < t ≤ T ) = limn→∞
n−1∏i=0
Pno(Ti < t ≤ Ti+1)
= limn→∞
n−1∏i=0
(1− Pdecay(Ti < t ≤ Ti+1))
= exp
(− lim
n→∞
n−1∑i=0
Pdecay(Ti < t ≤ Ti+1)
)
= exp
(−∫ T
0
dPdecay(t)
dtdt
)=⇒ dPfirst(T ) = dPdecay(T ) exp
(−∫ T
0
dPdecay(t)
dtdt
)
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 21/50
The Sudakov form factor – 2
Correspondingly, with Q ∼ 1/t (Heisenberg)
dPa→bc =αs
2π
dQ2
Q2Pa→bc(z) dz
exp
−∑b,c
∫ Q2max
Q2
dQ ′2
Q ′2
∫αs
2πPa→bc(z
′) dz ′
where the exponent is (one definition of) the Sudakov form factor
A given parton can only branch once, i.e. if it did not already do so
Note that∑
b,c
∫ ∫dPa→bc ≡ 1 ⇒ convenient for Monte Carlo
(≡ 1 if extended over whole phase space, else possibly nothinghappens before you reach Q0 ≈ 1 GeV).
More common/strict definition of Sudakov form factor based on∫ Q2
Q20
instead of∫ Q2
max
Q2 .
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 22/50
The Sudakov form factor – 3
Sudakov form factor provides“time” ordering of shower:lower Q2 ⇐⇒ longer timesQ2
1 > Q22 > Q2
3
Q21 > Q2
4 > Q25
etc.
P. S k a n d s
1 1
i
j
k
I
ij
k
I
m+1 m+1
K
K
VINCIA
12
Giele, Kosower, Skands, PRD 78 (2008) 014026, PRD 84 (2011) 054003Gehrmann-de Ridder, Ritzmann, Skands, PRD 85 (2012) 014013
Virtual Numerical Collider with Interleaved Antennae
Written as a Plug-in to PYTHIA 8C++ (~20,000 lines)
Based on antenna factorization- of Amplitudes (exact in both soft and collinear limits)
- of Phase Space (LIPS : 2 on-shell → 3 on-shell partons, with (E,p) cons)
Resolution TimeInfinite family of continuously deformable QE
Special cases: transverse momentum, dipole mass, energy
Radiation functionsArbitrary non-singular coefficients, anti
+ Massive antenna functions for massive fermions (c,b,t)
Kinematics mapsFormalism derived for arbitrary 2→3 recoil maps, κ3→2
Default: massive generalization of Kosower’s antenna maps
Mass-Ordering p?-ordering Energy-Ordering(m2
min) (
m2↵geometric
) (
m2↵arithmetic
)
Line
arin
y0.2
0.2
0.4
0.4
0.60.6 0.8
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yijy jk
(a) Q2
E
= m2
D
= 2 min(yij
, yjk
)s
0.2
0.4
0.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(b) Q2
E
= 2p?p
s = 2
py
ij
yjk
s
0.2
0.4
0.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(c) Q2
E
= 2Eps = (yij
+ yjk
)s
Qua
drat
icin
y
0.2
0.2
0.4
0.4
0.6
0.60.8
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(d) Q2
E
=
m
4Ds
= 4 min(y2
ij
, y2
jk
)s
0.2
0.4
0.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yijy jk
(e) Q2
E
= 4p2
? = 4yij
yjk
s
0.2
0.40.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(f) Q2
E
= 4E2= (y
ij
+ yjk
)
2s
Figure 2: (Q23, ) constant value for evolution variables linear (top) and quadratic (bottom) in the branch-
ing invariants, for mass-ordering (left), p?-ordering (middle), and energy-ordering (right). Note that theenergy-ordering variables intersect the phase-space boundaries, where the antenna functions are singu-lar, for finite values of the evolution variable. They can therefore only be used as evolution variablestogether with a separate infrared regulator, such as a cut in invariant mass, not shown here.
To obtain simple forms for the antenna integrals carried out below, we work with two possible definitions. For each antenna integral, we simply choose whichever definition gives the simplestintegrals,
1 =
yij
yij
+ yjk
(23)
2 = yij
. (24)
The corresponding Jacobian factors, for each of the evolution-variable choices we shall consider, aretabulated in tab. 1.
Note that, for the special case of the m2D
and m4D
variables, which contain the non-analytic functionmin(y
ij
, yjk
), the definitions in eqs. (23) and (24) apply to the branch with yij
> yjk
. For the otherbranch, y
ij
and yjk
should be interchanged. With this substitution, the Jacobians listed in tab. 1 apply to
9
Mass-Ordering p?-ordering Energy-Ordering(m2
min) (
m2↵geometric
) (
m2↵arithmetic
)
Line
arin
y
0.2
0.2
0.4
0.4
0.60.6 0.8
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(a) Q2
E
= m2
D
= 2 min(yij
, yjk
)s
0.2
0.4
0.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(b) Q2
E
= 2p?p
s = 2
py
ij
yjk
s
0.2
0.4
0.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(c) Q2
E
= 2Eps = (yij
+ yjk
)s
Qua
drat
icin
y
0.2
0.2
0.4
0.4
0.6
0.60.8
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(d) Q2
E
=
m
4Ds
= 4 min(y2
ij
, y2
jk
)s
0.2
0.4
0.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(e) Q2
E
= 4p2
? = 4yij
yjk
s
0.2
0.40.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(f) Q2
E
= 4E2= (y
ij
+ yjk
)
2s
Figure 2: (Q23, ) constant value for evolution variables linear (top) and quadratic (bottom) in the branch-
ing invariants, for mass-ordering (left), p?-ordering (middle), and energy-ordering (right). Note that theenergy-ordering variables intersect the phase-space boundaries, where the antenna functions are singu-lar, for finite values of the evolution variable. They can therefore only be used as evolution variablestogether with a separate infrared regulator, such as a cut in invariant mass, not shown here.
To obtain simple forms for the antenna integrals carried out below, we work with two possible definitions. For each antenna integral, we simply choose whichever definition gives the simplestintegrals,
1 =
yij
yij
+ yjk
(23)
2 = yij
. (24)
The corresponding Jacobian factors, for each of the evolution-variable choices we shall consider, aretabulated in tab. 1.
Note that, for the special case of the m2D
and m4D
variables, which contain the non-analytic functionmin(y
ij
, yjk
), the definitions in eqs. (23) and (24) apply to the branch with yij
> yjk
. For the otherbranch, y
ij
and yjk
should be interchanged. With this substitution, the Jacobians listed in tab. 1 apply to
9
Mass-Ordering p?-ordering Energy-Ordering(m2
min) (
m2↵geometric
) (
m2↵arithmetic
)
Line
arin
y
0.2
0.2
0.4
0.4
0.60.6 0.8
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(a) Q2
E
= m2
D
= 2 min(yij
, yjk
)s
0.2
0.4
0.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(b) Q2
E
= 2p?p
s = 2
py
ij
yjk
s
0.2
0.4
0.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(c) Q2
E
= 2Eps = (yij
+ yjk
)s
Qua
drat
icin
y
0.2
0.2
0.4
0.4
0.6
0.60.8
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(d) Q2
E
=
m
4Ds
= 4 min(y2
ij
, y2
jk
)s
0.2
0.4
0.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(e) Q2
E
= 4p2
? = 4yij
yjk
s
0.2
0.40.6
0.8
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
yij
y jk
(f) Q2
E
= 4E2= (y
ij
+ yjk
)
2s
Figure 2: (Q23, ) constant value for evolution variables linear (top) and quadratic (bottom) in the branch-
ing invariants, for mass-ordering (left), p?-ordering (middle), and energy-ordering (right). Note that theenergy-ordering variables intersect the phase-space boundaries, where the antenna functions are singu-lar, for finite values of the evolution variable. They can therefore only be used as evolution variablestogether with a separate infrared regulator, such as a cut in invariant mass, not shown here.
To obtain simple forms for the antenna integrals carried out below, we work with two possible definitions. For each antenna integral, we simply choose whichever definition gives the simplestintegrals,
1 =
yij
yij
+ yjk
(23)
2 = yij
. (24)
The corresponding Jacobian factors, for each of the evolution-variable choices we shall consider, aretabulated in tab. 1.
Note that, for the special case of the m2D
and m4D
variables, which contain the non-analytic functionmin(y
ij
, yjk
), the definitions in eqs. (23) and (24) apply to the branch with yij
> yjk
. For the otherbranch, y
ij
and yjk
should be interchanged. With this substitution, the Jacobians listed in tab. 1 apply to
9
pT
mD
Eg
vincia.hepforge.org
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 23/50
The Sudakov form factor – 4
Sudakov regulates singularity for first emission . . .
. . . but in limit of repeated softemissions q→ qg (but no g → gg)one obtains the same inclusiveQ emission spectrum as for ME,i.e. divergent ME spectrum⇐⇒ infinite number of PS emissionsProof: as for veto algorithm (what isprobability to have an emission at Qafter 0, 1, 2, 3, . . . previous ones?)
More complicated in reality:
energy-momentum conservation effects big since αs big,so hard emissions frequent
g → gg branchings leads to accelerated multiplicationof partons
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 24/50
Matrix elements and the Sudakov
Recall e+e− → qqg, with phase space slicing at y :
σR(y)
σ0≈ 4αs
3πln2 y
σV(y)
σ0≈ αs
π− 4αs
3πln2 y
σ0 + σR(y) + σV(y) =(1 +
αs
π
)σ0
If small finite term αs/π neglected then σV(y) = −σR(y)
If shower emissions ordered in y then
dPdy
=1
σ0
dσR
dyexp
(−∫ 1
y
1
σ0
dσR
dy ′dy ′)
=1
σ0
dσR
dyexp
(σV(y)
σ0
)=
1
σ0
dσR
dy
(1 +
σV(y)
σ0+ · · ·
)Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 25/50
Shower evolution generation – 1
Veto algorithm crucial for practical implementation
Allowed Q2-dependent range zmin(Q2) ≤ z ≤ zmax(Q
2)overestimated by maximum range zmin(Q
20 ) ≤ z ≤ zmax(Q
20 ).
Splitting functions overestimated to simplify, e.g. 1 + z2 ≤ 2.
For now assume αs fixed, so combine to upper estimateIz ' (αs/2π)
∫P(z) dz
Splitting probability overestimated by
IzdQ2
Q2exp
(−∫ Q2
max
Q2
IzdQ ′2
Q ′2
)
“Radioactive decays” formula gives solution
exp
(−Iz ln
Q2max
Q2
)=
(Q2
Q2max
)Iz
= R ⇒ Q2 = Q2maxR
1/Iz
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 26/50
Shower evolution generation – 2
If Q < Q0 then stop evolution
Pick z in overestimated range with overestimated P, e.g.∫ z
zmin(Q20 )
2 dz ′
1− z ′= R
∫ zmax(Q20 )
zmin(Q20 )
2 dz ′
1− z ′
If z outside allowed range for chosen Q2 or ifPtrue(z)/Poverestimation(z) < R then put Q2
max = Q2 andcontinue downwards evolution (veto algorithm)
If two possible channels, like g → gg and g → qq,then combined overestimation or use “the winner take it all”,i.e. pick one Q2 scale for each channel and retain the largest
If αs = αs(Q2) ∝ 1/ ln(Q2/Λ2) then use that∫ Q2
max
Q2
1
ln Q′2
Λ2
dQ ′2
Q ′2 =
∫ Q2max
Q2
d ln Q′2
Λ2
ln Q′2
Λ2
= ln lnQ2
max
Λ2− ln ln
Q2
Λ2
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 27/50
Coherence – 1
QED: Chudakov effect (mid-fifties)
QCD: colour coherence for soft gluon emission
solved by • requiring emission angles to be decreasingor • requiring transverse momenta to be decreasing
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 28/50
Coherence – 2
Simple illustration of principles: consider emission of soft gluonoff fast moving colour singlet qq system:
recall1
σ
dσ
dx1dx2∝ x2
1 + x22
(1− x1)(1− x2)∼ 2
(1− x1)(1− x2)
1−x1 = y23 =(p2 + p3)
2
(p1 + p2 + p3)2≈ 2p2p3
2p1p2=
E2E3(1− cos θ23)
E1E2(1− cos θ12)=
E3a23
E1a12
with aij = 1− cos θij .Recasting matrix element in gluon momentum variables:
dσ
σ∝ dE3
E3dΩ3
a12
a13a23
which is singular both for a13 → 0 and a23 → 0.Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 29/50
Coherence – 3
Rewrite angular dependence
2a12
a13a23=
2a12 − a13 − a23
a13a23+
1
a13+
1
a23
=1
a13
(1 +
a12 − a13
a23
)+
1
a23
(1 +
a12 − a23
a13
)where (a12 − a13)/a23 is not singular:a23 → 0 ⇒ 2 and 3 become parallel ⇒ a12 − a13 → 0.
cos θ23 = sin θ12 sin θ13 cos ϕ
+cos θ12 cos θ13
Also θij → 0 ⇒ aij = 1− cos θij ≈ θ2ij/2.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 30/50
Coherence – 4
1+a12 − a13
a23≈ 1+
θ212 − θ2
13
θ212 + θ2
13 − θ12θ13 cos ϕ= 1+
1− r2
1 + r2 − 2r cos ϕ
with r = θ13/θ12.
∫ 2π
0
(1 +
1− r2
1 + r2 − 2r cos ϕ
)dϕ = 2π
(1 + sign(1− r2)
)= 4πΘ(1−r)
i.e. = 4π for r < 1 but = 0 for r > 1 .Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 31/50
Coherence – 5
Thus, averaged over ϕ
dσ
σ∝ dE3
E3dΩ3
2
θ213
Θ(θ12 − θ13) +2
θ223
Θ(θ12 − θ23)
Reasoning above generalizes also when more than two partonsmay radiate.Key consequences of angular ordering:
slower multiplicity growth of partons (and thus of hadrons),
depletion of particle production at small x = E/Ejet.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 32/50
The eikonal approximation – 1
Derivation of soft radiation pattern for generic topology:Soft gluons = long wavelength.Thus internal lines of hard processes are not resolved,i.e. only external lines contribute,and these may be treated as classical currents:
dσn+1 = dσnd3k
(2π)32ω
∣∣∣∣∣n∑
i=1
gs Tipi
pik
∣∣∣∣∣2
where Ti are colour charge matrices, ω = |k| and
pi
pik=
Ei (1,βi )
Ei (1,βi ) ω (1,n)=
1
ω
(1,βi )
1− nβi
is ∝ (φ,A) of a moving charge (cf. Jackson).
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 33/50
The eikonal approximation – 2
Expand square
dσn+1 = dσnd3k
2πω
αs
2π
∑i ,j
(−TiTj)pipj
(pik)(pjk)
Note that for p2i = m2
i = 0 the i = j terms vanish,
the radiation is not by a single charge but by a dipole .E.g.
dσ3
σ2=
d3p3
2πE3
αs
2π
4
3
p1p2
(p1p3)(p2p3)2
=αs
2π
4
3
s(1− x3)/2
s2(1− x2)(1− x1)/42
d3p3
2πE3
≈ αs
2π
4
3
2
(1− x1)(1− x2)dx1 dx2
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 34/50
The eikonal approximation – 3
Generalizes to higher multiplicities,e.g. e+e− → q(1) q(2) g(3) g(k):
dσ4
σ3∝ αs
2π
(−T1T3) 13 + (−T2T3) 23 + (−T1T2) 12
d3k
2πω
∝ αs
2π
13 + 23− 1
N2C
12
d3k
2πω
whereij =
pipj
(pik)(pjk)
encode radiating dipoles.
Neglecting 1/N2c -suppressed terms, radiation can be viewed
as coming from two “independent” dipoles, qg and qg.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 35/50
The eikonal approximation – 4
Compare with e+e− → q(1) q(2) γ(3) g(k):the γ does not carry colour, so here only qq dipole.
x axis: angular range between q and q.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 36/50
Scale choice
Branching rate of a → bc is proportional to αs(Q2).
Naive guess Q2 = m2a.
Then next-to-leading branching kernel in limit z → 1 (soft g)
αs
2πPq→qg(z) → αs
2π
Pq→qg(z) +
αs
2πP ′q→qg(z)
≈ αs(m
2a)
2π
1− ln(1− z)
ln(m2a/Λ2)
Pq→qg(z)
≈ αs((1− z)m2a)
2πPq→qg(z) ≈
αs(p2⊥)
2πPq→qg(z)
Corresponding results hold for g → gg, z → 0 and z → 1,
so natural to pick generically Q2 = p2⊥ = z(1− z)m2
a .
Does not absorb finite corrections for non-singular z ,so not complete NLO, but part of the way.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 37/50
The Common Showering Algorithms (LEP era)
Three main approaches to showering in common use:
Two are based on the standard shower languageof a → bc successive branchings:
HERWIG: Q2 ≈ E 2(1− cos θ) ≈ E 2θ2/2PYTHIA: Q2 = m2 (timelike) or = −m2 (spacelike)
One is based on a picture of dipole emission ab → cde:
ARIADNE: Q2 = p2⊥; FSR mainly, ISR is primitive
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 38/50
Ordering variables in final-state radiation (LEP era)
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 39/50
Data comparisons (LEP)
All three algorithms do a reasonable job of describing LEP data,but typically ARIADNE (p2
⊥) > PYTHIA (m2) > HERWIG (θ)
det.
cor. statistical uncertainty
had.
cor
.1/m
dm
/dT
ALEPH Ecm = 91.2 GeV
PYTHIA6.1
HERWIG6.1
ARIADNE4.1
datawith statistical systematical errors
(dat
a-M
C)/d
ata
T
total uncertainty
0.50.75
11.25
1.5
0.50.75
1.01.25
10-3
10-2
10-1
1
10
-0.5
-0.25
0.0
0.25
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
. . . and programs evolve to do even better . . .
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 40/50
The HERWIG algorithm
Basic ideas, to which much has been added over the years:
1 Evolution in Q2a = E 2
a ξa with ξa ≈ 1− cos θa, i.e.
dPa→bc =αs
2π
d(E 2a ξa)
E 2a ξa
Pa→bc(z) dz =αs
2π
dξa
ξaPa→bc(z) dz
Require ordering of consecutive ξ values, i.e. (ξb)max < ξa and(ξc)max < ξa.
2 Reconstruct masses backwards in algorithmm2
a = m2b + m2
c + 2EbEcξa
Note: ξa = 1− cos θa only holds for mb = mc = 0.
3 Reconstruct complete kinematics of shower (forward again).
+ angular ordering built in from start− total jet/system mass not known beforehand (⇒ boosts)− some wide-angle regions never populated, “dead zones”
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 41/50
The ARIADNE algorithm – 1
partonic picture equivalent dipole picture
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 42/50
The ARIADNE algorithm – 2
Dual description of topology:a dipole connects two partonsa gluon connects two dipoles
Dual description of particle productionshower: parton → parton + partondipole: dipole → dipole + dipole
Use effective 3-jet matrix elementsin rest frame of each new dipole
Evolve downwards in p⊥⇒ angular ordering for free
No need to assign virtualities to partons;(E ,p) conserved inside dipole
(qg) → (qg) + (gg) and (gg) → (gg) + (gg) standard fare;generalization of g → qq nontrivial
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 43/50
The ARIADNE algorithm – 3
Transform 3-jet variables from (x1, x2) to (κ, y) where
eκ = p2⊥ = E 2
3 =(E1E3)(E2E3)
E1E2≈ (p1p3)(p2p3)
s/4= s(1− x1)(1− x2)
(in frame with q and q opposite and g at 90)
y ≈ 1
2ln
(E + pz)g(E − pz)g
≈ 1
2ln
(1− x1)(E + pz)q(1− x2)(E − pz)q
=1
2ln
1− x1
1− x2
dσ
σ≈ 8
3
αs
2π
dx1 dx2
(1− x1)(1− x2)=
8
3
αs(p2⊥)
2π
dp2⊥
p2⊥
dy=4αs(p
2⊥)
3πdκ dy
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 44/50
Comments on dipole showers – 1 (Frank Krauss)
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 45/50
Comments on dipole showers – 2 (Frank Krauss)
Nowadays also Herwig++/Herwig7 has a dipole shower as anoption.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 46/50
Dipoles and recoils
Consider dipole emission qq→ qqg.Given internal topology, e.g. (x1, x2), how is event oriented,i.e. how is recoil of emission shared?
ARIADNE: minimize p2⊥q + p2
⊥q of new q and qwith respect to original qq axis.
PYTHIA and others: keep either q or q direction fix,e.g. with P = m2
qg/(m2qg + m2
qg) for q to keep its direction.
(In addition isotropic ϕ angle around original qq axis.)Such differences carry over to subsequent emissions,with further ambiguities, and can affect final results.Exemplifies “subleading effects”.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 47/50
Prompt photon production in showers
In shower evolution:
Pq→qg =αs
2π
4
3
1 + z2
1− z
Pq→qγ =αem
2πe2q
1 + z2
1− z
Pq→qγ
Pq→qg=
αem e2q
αs43
≈ 1
200
but
no direct emission from g ⇒ dilution
γ uncharged ⇒ no coherence ⇒ more, but negligibly
at small p⊥ overwhelming background from π0 → γγ
competition with g emission reduces by factor 2–3relative to if only γ emission from quark
reasonably well tested/understood at LEP
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 48/50
Leading Log and beyond
Neglecting Sudakovs, rate of one emission is:
Pq→qg ≈∫ Q2
max
Q2max
dQ2
Q2
∫ zmax
zmin
dzαs
2π
4
3
1 + z2
1− z∼ αs ln2
Rate for n emissions is of form:
Pq→qng ∼ (Pq→qg)n ∼ αn
s ln2n
Next-to-leading log (NLL): include all corrections of type αns ln2n−1
No existing generator completely NLL (?), but• energy-momentum conservation (and “recoil” effects)• coherence• 2/(1− z) → (1 + z2)/(1− z)• scale choice αs(p
2⊥) absorbs singular terms ∝ ln z , ln(1− z)
in O(α2s ) splitting kernels Pq→qg and Pg→gg
• . . .⇒ far better than naive, analytical LL
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 49/50
On the road to NLL showers
Key ingredient to NLL is NLO splitting kernels:• 1 → 3, i.e q→ qgg, q→ qq′q′, g → ggg, g → qqg;• virtual corrections to 1 → 2 (not positive definite).Are known in terms of energy sharing, but less well defined in p⊥.
NLLJET (Kato, Munehisa, ∼ 1990): first with NLO splittingkernels for e+e−, but other problems.
VINCIA (Skands et al., 2017): combine O(α2s )-corrected
iterated 2 → 3 kernels for ordered emissions with tree-level2 → 4 kernels for unordered emissions, but only for e+e−.
DIRE (Hoche, Prestel, 2017): dipole shower with NLOsplitting kernels, including handling of negative weights, alsofor hadron–hadron.
Further efforts under way, e.g. for Herwig7.
Also colour effects relevant, including 1/N2C -suppressed terms,
studied by Platzer, Sjodahl and Thoren.
Torbjorn Sjostrand PPP 3: Evolution equations and final-state showers slide 50/50