Particle Physics Phenomenology 3. Evolution equations and final...

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.



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.



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!


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


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


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


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!)


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!


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).


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.


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:


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



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


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.


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”


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


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.


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


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

)



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 .



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


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




1 =

yij

yij

+ yjk

(23)

2 = yij

. (24)



and m4D


ij

, yjk


> yjk


ij

and yjk


9


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




1 =

yij

yij

+ yjk

(23)

2 = yij

. (24)



and m4D


ij

, yjk


> yjk


ij

and yjk


9

pT

mD

Eg

vincia.hepforge.org



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


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


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


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


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.


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.


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).



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



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.



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.


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.


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


Ordering variables in final-state radiation (LEP era)


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 . . .


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”


The ARIADNE algorithm – 1

partonic picture equivalent dipole picture



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



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


Comments on dipole showers – 1 (Frank Krauss)


Comments on dipole showers – 2 (Frank Krauss)

Nowadays also Herwig++/Herwig7 has a dipole shower as anoption.


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”.


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


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


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.


Particle Physics Phenomenology 3. Evolution equations and final...

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