Monte Carlo Tools for the LHC

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Monte Carlo Tools for the LHC

Gennaro Corcella

CERN, Department of Physics Theory Division

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Outline

First lecture:

Introduction to MC methods Parton shower algorithms

General-purpose event generators Second lecture:

Matrix-element corrections to parton showers MC@NLO and matrix-element generators

Comparisons and concluding remarks

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Introduction to Monte Carlo methods

(A. Pellissetto, Lectures at the the 2nd Seminar of Theoretical Physics, Parma, 1992; S. Weinzierl, hep-ph/0006269, M.H.

Seymour, Lectures at CTEQ Summer School, 2000)

Monte Carlo algorithms are based on the generation of random numbers

A random number generator is a program which produces numbers which approach the properties of purely random numbers

Example: a sample of instable nuclei and a periodic counter which gives 0 if the number of decays is odd, 1 if the number of decays is even

Problem: storage of a huge amount of bits is not manageable

Practical random number generators try to approximate random number sets Example: with a computer able to store words with n bits, generate integer numbers x₁, . . . x_n, with 0 ≤ xⁿ ≤ 2ⁿ, and x_n+1 = f (x_n)

‘Random’ numbers U_n, 0 ≤ Uⁿ ≤ 1, can be obtained setting Uⁿ = x_n/2ⁿ As x_n+1 and x_n are related, they cannot be random numbers!

However, properly choosing the function f , we can get a good approximation of random numbers, in the sense that they appear random to somebody who does not know the applied algorithm

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Many random number generators follow the linear congruential method (LCG), by P. L’Ecuyer, Comp. Phys. Comm. 60 (1990) 329:

x_n+1 = (ax_n + c) mod m 0 ≤ a < m ; 0 ≤ c < m

a is called multiplier, m modulus, c increment

Typically: c = 0 (multiplicative linear congruential generator), and m = 2^w, with w available number of bits

The best approximation of a truly random number generator is achieved if x_n has the maximum possible period, which is m

A necessary condition is that a and c are relatively prime to m

Preliminary applications of random numbers: generation of a probability distribution and Monte Carlo integration

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Generation of probability distributions: inverse transform method

Problem: given a generator of uniform random numbers u ∈ [0, 1], generate the variable x according to the probability distribution f (x), which is known analitycally

Z +∞

−∞

f (x)dx = 1 ; Prob(x₁ < x < x₂) =

Z x₂ x₁

f (x)dx The cumulative distribution F (a) gives the probability that x ≤ a:

F (a) =

Z a

−∞ f (x)dx ; 0 ≤ F (a) ≤ 1

F (a) is a random variable, uniformly distributed in [0, 1], for any x in the domain of f (x). We can set it equal to the generated random number and obtain x inverting the function F (x):

u = F (x) ; x = F⁻¹(u) Example: generate f (x) = 2/x² for x ∈ [1, 2] , R 2

1 f (x)dx = 1

F (x) =

Z x 1

2

t² dt = 2 − 2

x ; 2 − 2

x = u ⇒ x = 2

2 − u ; 1 ≤ x ≤ 2

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Acceptance-rejection method

It is used if the function f (x) is known only numerically

Enclose f (x) in a function g(x) = Ch(x), with f (x) and h(x) normalized to one, and C > 1

Compare f (x) with ug(x), with u uniform random number ug(x) ≤ f(x): accept x ; ug(x) ≥ f(x): reject x

Better results if g(x) is fairly close to f (x) and C ^>_∼ 1 a ≤ x ≤ b; 0 ≤ y ≤ c

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Monte Carlo integration:

I = Z b

a g(x)dx = (b − a)hg(x)i

The integral I is computed as the average value of a random variable

One generates u₁,. . . u_N random numbers in [0, 1], evaluates x_i = a + u_i(b − a) and g(x_i)

The integral is approximated to the following sample mean:

I ' I^N = (b − a) 1 N

N

X

i=1

g(x_i) The variance reads:

σ² = (b − a) Z b

a

[g(x)]²dx −

"

Z b a

g(x)dx

#2

Central limit theorem: I_N is a statistic variable with mean I and variance σ/√ N The Monte Carlo estimate for the variance σ_N is obtained using the same sample x₁, . . . x_N :

σ_N² = b − a N

N

X

i=1

[g(x_i)]² − IN² ; I ' I^N ± σ^N/√ N

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Importance sampling: change of integration variables ([a, b] = [0, 1]) Z

dxf (x) =

Z f (x)

p(x)p(x)dx =

Z f (x)

p(x)dP (x) ; p(x) = ∂^d

∂x₁ . . . ∂x_dP (x)

If p(x) ≥ 0 and R p(x)dx = 1, we can generate a sample of random numbers x₁ . . . x_N according to P (x) and get the following estimates:

I_N = 1 N

N

X

i=1

f (x_i)

p(x_i) ; σ² f p

= 1 N

N

X

i=1

f (x_i) p(x_i)

2

− IN²

Example (M.H. Seymour):

R 1

0 dx cos ^π₂x = 0.637 ± 0.307/√ N R 1

0 dx(1 − x²)^cos(^π₂^x)

1−x² = Rρ₂

ρ₁ dρ^cos(^π₂^x(ρ))

1−x(ρ)²

= 0.637 ± 0.031/√ N

Better efficiency after change of variables

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VEGAS: learn where the function is large and approximate it to a step function The integration region is divided into a rectangular grid; an integral is performed in each region to understand where the integrand is largest and approximate the optimal probability p_best = |f(x)|/R |f(x)|dx to a step function

The grid is optimized and frozen; each j-th integration gives the estimate:

I_j = 1 N_j

N_j

X

n=1

f (x_n)

p(x_n) ; σ_j² = 1 N_j

N_j

X

n=1

f (x_n) p(x_n)

2

− Ij²

The final result reads: I =





m

X

j=1

N_j σ_j²





−1



m

X

j=1

N_jI_j σ_j²





Example with f (x) = cos ^π₂x

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Multi-channel integration

If the integrand has several peaks in different regions, we need a variable transformation for each peak domain

Each channel will have probability p_i(x), according to which we generate x:

Channel i is selected with probability α_i, with Pm

i=1 α_i = 1, p(x) = P

iα_ip_i(x) The integral and its estimate read (N_i = α_iN ):

I = Z

dxf (x) =

m

X

i=1

α_i

Z f (x)

p(x)dP_i(x) ; p_i(x) = ∂P_i

∂x ; I_N = 1 N

m

X

i=1 N_i

X

n_i=1

f (x_n_i) p(x_n_i)

The error is given by:

σ² = 1

N[W (α) − I²] ; W (α) =

m

X

i=1

α_i

Z f (x) p(x)

2

dP_i(x)

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Phase-space integration (ab → 1 . . . n)

Phase-space factor for n particles of momenta p₁ . . . p_n and masses m₁ . . . m_n:

dΦ_n(P, p₁, . . . p_n) =

n

Y

i=1

d⁴p_i

(2π)³2E_i(2π)⁴δ⁴ P −

n

X

i=1

p_i

!

Factorization: Q = Pj

k=1 p_k, Q → p¹ . . . p_j, P → Qp^j . . . p_n Φ_n(P, p₁, . . . p_n) = 1

2πdQ²dΦ_j(Q, p₁, . . . , p_j)dΦ_n−j+1(P, Q, p_j+1, . . . , p_n) ;

Sequential approach: n-body phase space as a sequence of 1 → 2 decays dΦ_n = 1

(2π)ⁿ⁻²dM_n−1² . . . dM₂²dΦ₂(n) . . . dΦ₂(2)

with q_i = Pi

j=1 p_j, M_i² = q_i², (m₁ + . . . m_i)² ≤ Mi² ≤ (Mⁱ⁺¹ − mⁱ⁺¹)² M₂² = (p₁ + p₂)², . . . M_n−1² = (p₁ + . . . p_n−1)²

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In the rest frame of q_i, the phase space for q_i(M_i) → pⁱ(m_i)q_i−1(M_i−1) reads:

dΦ₂(q_i, q_i−1, p_i) =

qλ(q_i², q_i−1² , m²_i)

(2π)² 8q_i² dϕ_i d cos θ_i

λ(x, y, z) = x² + y² + z² − 2xy − 2yz − 2xz

Algorithm for phase space generation: (u_i random numbers)

1) Rest frame of q_i: q_i = P = p₁ + . . . p_n, M_i = pq_i²; ϕ_i = 2πu₁, cos θ_i = u₂; M_i−1 = (m₁ + . . . m_i−1) + u₃(M_i − mⁱ)

2) Set

|~pⁱ⁰| =

qλ(M_i², M_i−1² , m²_i)

2M_i ; ~p_i⁰ = |~pⁱ⁰|(sin θⁱsin ϕ_i, sin θ_i cos ϕ_i, cos θ_i)

p⁰_i = “q

| ~p_i⁰| + m²_i, ~p_i⁰”

, q⁰_i−1 = “q

| ~p_i⁰| + M_i−1² , − ~p_i⁰”

3) Transform back to the original Lorentz system;

4) Set i → n − 1: if n ≥ 2 repeat all steps The weight reads:

w = 1

(2π)³ⁿ⁻⁴2²ⁿ⁻¹M_n

n

Y

i=2

qλ(M_i², M_i−1² , m²_i) M_i

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Monte Carlo techniques can be used to simulate multi-parton radiation in high-energy processes

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& &&&&

''''

( (((( (

)))) ))

**** *

+++ ++

, ,,,

----

---- . ..

. ..

///

0000

11 11

11 11 22 2

22 2

333

44

55

66 6

777

777 88 88

88 88

9 999

9 999:: :::

:: :::

;;; ;;

;;; ;; << <<

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= ==

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AA AA

Figure by Frank Krauss

Standard Monte Carlo event generators (HERWIG/PYTHIA):

Hard 2 → 2 subprocess: leading-order (LO) matrix element Parton showers in the soft or collinear approximation

Matrix-element corrections for hard and large-angle parton radiation

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A simpler case: e⁺e⁻ → γ(q) → q(p¹)¯q(p₂)g(p₃) (see also M.Grazzini’s lectures)

____^^^^

____^^^^^- γ(q)

@@

@@@

I q(p¯ ₂)

g(p₃)

q(p₁)

____^^^^

____^^^^^- γ(q)

q(p₁)

@I@@

g(p₃)

@I@@ q(p¯ ₂)

d²σ

dx₁dx₂ = σ₀C_F α_s 2π

x²₁ + x²₂

(1 − x¹)(1 − x²) ; x_i = 2p_i · q

q² = 2E_i

√s ; x₁ + x₂ + x₃ = 2

x₁ → 1: g k ¯q ; x² → 1: g k q ; x³ → 0: soft gluon radiation x²₁ + x²₂

(1 − x¹)(1 − x²) = −2 + 1 + (1 − x³)² x₃

1

1 − x¹ + 1 1 − x²

Collinear approximation: θ = θ₂₃ → 0 , x¹ → 1

z = E₃

E₂ + E₃ = x₃

2 − x¹ ' x³ ; θ² ' 2(1 − cos θ) ' 4(1 − x¹) x₃(1 − x³)

P (z) = C_F1 + (1 − z)²

z : Altarelli − Parisi splitting function x²₁ + x²₂

(1 − x¹)(1 − x²) ' 1 C_F

4

x₃(1 − x³)

P (z)

θ² , J = x₃(1 − x³) 4

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x²₁ + x²₂

(1 − x¹)(1 − x²) ' 1 C_F

4

x₃(1 − x³)

P (z)

θ² , J = x₃(1 − x³) 4

Differential cross section in terms of z and θ:

d

²

σ = σ

₀

α

_S

2π P (z)dz dθ

²

θ

²

Universal in the collinear limit

- - @

@@ 1 − z

R

z

- - @

@@ z

R

1 − z

P_gq(z) = C_F ^1+(1−z)_z ² P_qq(z) = C_F ^1+z_1−z²

@

@@ 1 − z

R z

1 − z

z

P_qg(z) = C_F 1 + (1 − z)²

P_gg(z) = C_A^z⁴^+1+(1−z)_z(1−z) ⁴

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Soft approximation: γ(q) → q(p¹)¯q(p₂)g(p₃, ) ω = E₃ E^1,2 ; |~p³| |~p^1,2|

M^aµ = −ieg^ST^au(p¯ ₁)

γ^µ i

6 p³+ 6 p¹ 6 + 6 i

6 p²+ 6 p³γ^µ

v(p₂)

= g_ST^a_ν

p^ν₁

p₁ · p³ − p^ν₂ p₂ · p³

e¯u(p₁)γ^µv(p₂)

Eikonal factorization: universal in the soft limit

d²σ = σ₀C_Fα_S π

2dω ω

d cos θ

(1 − cos θ)(1 + cos θ) = σ₀C_Fα_S 2π

2dω ω

d cos θ

1 − cos θ + d cos θ

1 − cos(π − θ)

Collinear limit : d²σ = σ₀ α_S

2π P (z)dz dθ²

θ² P (z)dz = C_F 1 + (1 − z)²

z dz ' C^F 2 ωdω Soft and/or collinear limit :

d²σ = σ₀ α_S

2π P (z)dz d cos θ 1 − cos θ

Starting point to simulate multiple radiation: need of ordering variable

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In the collinear approximation, any Q² ∝ θ² is feasible to order multiple radiation

- θ

p, E

zE

@@

@@ R

(1 − z)E

Gluon transverse momentum: k_T² = z²(1 − z)²E²θ² Invariant mass: p² = z(1 − z)E²θ²

Collinear limit: ln k_T² ∼ ln p² ∼ ln θ² dθ²

θ² = dk_T²

k_T² = dp² p²

Soft gluons can be emitted anywhere, at any angle

-

Angular ordering allows one to implement probabilistically multiple soft emissions

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Angular ordering

__^^

__^^^- _@

@@

I q(p¯ ₂)

g(p₃)

q(p₁)

__^^

__^^^-

q(p₁)

@@I @

g(p₃)

@@

I@ q(p¯ ₂)

__^^

__^^^- ^@

@@

@@ I

-

θ₁₂ θ₁₃

__^^

__^^^-

R@@

- R@@

θ₁₂

θ₂₃

|M|² ∼ W = ω² 2

p₁

p₁ · p³ − p₂ p₂ · p³

2

= 1 − cos θ¹²

(1 − cos θ¹³)(1 − cos θ²³) (soft limit)

W = W₁ + W₂ ; W₁ = 1 2

W + 1

1 − cos θ¹³ − 1

1 − cos θ²³

Z 2π 0

dφ₁₃

2π W₁ = 1

1 − cos θ¹³, if θ₁₃ < θ₁₂

= 0 if θ₁₃ < θ₁₂

Colour coherence: a parton radiates up to its colour partner

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Implementation of angular ordering

dP =

^d_σ²^σ

0

=

^α_2π^S

P (z)dz

_{1−cos θ}^{d cos θ} ____^^^^^u θ_max

@@ I@

θ < θ_max

-

Need to evaluate probability of no branching at larger angles Analogy with nuclear decay:

dP = λdt dN = −N⁰λdt N (t) = N₀ exp(−λt) = N⁰ exp[−R t

0 dP ] Probability of no decay in [0, t]: ^{N (t)}_N

0

= exp h

− R

t

0

dP i

Probability of no branching in [θ, θ_max]:

∆_S(θ_max, θ) = exp

"

−α_S 2π

Z θ_max θ

d cos θ⁰ 1 − cos θ⁰

Z z_max z_min

dzP (z)

#

Sudakov form factor

dP = α_S

2π P (z)dz d cos θ

1 − cos θ ∆_S(θ_max, θ)

Unitarity: 1=R+V ∆_S sums virtual and unresolved emissions

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Angular-ordered parton showers

__^^

__^^^- _@

@@

@@@ I

@@

@ R

__^^

__^^^- ^@^θ

@I@@

@-

@I@

@@

@ R

@@I@

- @

@@ R

θ₁ θ₂

θ₂⁰ θ⁰₁

Parton shower ⇒ colour flow ⇒ angular ordering:

θ₁ < θ; θ₂ < θ₁; θ₁⁰ < θ; θ₂⁰ < θ₁⁰

dP₁ = α_S

2π P (z₁)dz₁ d cos θ₁

1 − cos θ¹∆_S(θ, θ₁)

dP₂ = α_S

2π P (z₂)dz₂ d cos θ₂

1 − cos θ²∆_S(θ₁, θ₂)dP₁

Iterating dP one construct the multiple-radiation algorithm

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Evolution variable in general-purpose event generators

__^^

__^^^u_@

@@

p₁(E₁)

I

p(E) θ

k(ω)

- p₂(E₂), z = ω/E

HERWIG (G.C., I.Knowles,G. Marchesini, S. Moretti, K. Odagiri, P. Richardson, M.H. Seymour, B.R. Webber) :

Q² = E²(1 − cos θ) ' E²θ²/2 ; Q_max = √p · p¹ ; E = Q_max ; θ < π/2 Soft approximation: angular ordering

HERWIG++ (S.Gieseke, A. Ribon, M. Seymour, P. Stevens, B. Webber): Q⁰² = Q²+^max(k_z₂²^,p²⁾+_z₂_(1−z)^k² ₂ (only e⁺e⁻ at the moment)

Angular ordering, better treatment of soft phase space and heavy quark masses PYTHIA (up to 6.2 version) (T. Sjostrand, L. Lonnblad, S. Mrenna, P. Skands): Q² = p²

It includes angular ordering only by an additional veto (see CDF PRD 50 (1994) 5562)

PYTHIA 6.3: Q² = k_T²

ARIADNE (L. Lonnblad): Q² = k_T²

The Sudakov form factor will be: ∆_S(Q²_max, Q²) = ∆_S(Q²_max, Q²₀)/∆_S(Q², Q²₀)

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Implementation of parton shower algorithm in a Monte Carlo program

Main step: given a generator of random numbers R and branching with evolution variable Q²₁, generate another branching at Q²₂

The Sudakov form factor gives the probability of evolution between two values of Q² with no resolvable emission

∆_S(Q²₁, Q²₂) = R

The variable z is to be distributed according to the splitting function P (z) The cumulative distribution will be ( : infrared cutoff)

F (z) =

Rz

P (z)dz R 1−

P (z)dz z will be obtained by solving the following equation:

Z z

dzP (z) = R⁰

Z 1−

dzP (z)

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Initial-state radiation

Hard-scattering quantities are fixed (e.g. m_W for W production, x_Bj for Deep Inelastic Scattering): better efficiency with backward evolution

~ -

h

Q²₁, x₁

@@

R Q²₂, x₂ = zx₁

__^^

__^{u -}^^

z = Eg/E₁

Q²₁ < Q²₂ (ordering variable); x₁ > x₂

Step 1: evolution Q²₂ → Q²1 at fixed x₂ with no branching

∆_S(Q²₂, Q²₁)f_q(x₂, Q²₁)

f_q(x₂, Q²₂) = ∆_S(Q²₂, Q²₀) f_q(x₂, Q²₂)

f_q(x₂, Q²₁)

∆_S(Q²₁, Q²₀) = R Step 2: generate z = x₂/x₁ at given Q²₁:

Z z

dz

z P (z)f x₂

z , Q²₁

= R⁰

Z 1−

dz

z P (z)f x₂

z , Q²₁

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The algorithm is analogous to the final-state radiation, with ∆_S → ∆^S/f f /∆_S satisfies the DGLAP evolution equation

(see Grazzini’s lectures and ‘QCD and Collider Physics’, pp.163–173 for proof) :

Q² ∂

∂Q²

f

∆_S

= 1

∆_S

Z dz z

α_S

2π f (x/z, Q²)P (z)

Solving iteratively for an evolution Q²₁ → Q²: f (x, Q²)

∆_S(Q²₁, Q²) = f (x, Q²₁)

∆_S(Q²₁, Q²₁) + α_S 2π

Z Q²₁ Q²

dQ⁰² Q⁰²

Z dz

z P (z) f (x/z, Q⁰²)

∆_S(Q²₁, Q⁰²)

f (x, Q²) = ∆_S(Q²₁, Q²)f (x, Q²₁) + α_S 2π

Z Q² Q²₁

dQ⁰² Q⁰²

∆_S(Q²₁, Q²)

∆_S(Q²₁, Q⁰²)

Z dz

z P (z)f (x/z, Q⁰²) Interpretation :

∆_S(Q²₁, Q²)f (x, Q²₁): no branching in [Q²₁, Q²]

P (z) f (x/z, Q⁰²) : branching at Q⁰²; ∆_S(Q²₁, Q²)/∆_S(Q²₁, Q⁰²) evolution Q⁰² → Q²

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Colour coherence in the hard scattering (example: q(p₁)¯q(p₂) → q⁰(p₃)¯q⁰(p₄))

-

@@

@

¯ I

q⁰ q⁰

q¯

@@

@

R q ^@^@_@

@@ R

-

@@

@@@

R

Mandelstam variables: ˆs = (p₁ + p₂)², ˆt = (p₁ − p³)², ˆu = (p₂ − p³)²

Double-differential cross section (massless appoximation):

d²σˆ

dˆsdˆt = 4

9 α²_Sˆt² + ˆu² ˆ

s³ δ(ˆs + ˆt + ˆu)

Generate ˆs, ˆt and ˆu according to d²σ/(dˆsdˆt)

In a given frame, determine components of p₁, p₂, p₃ and p₄ in terms of ˆs, ˆt, ˆu Boost back to laboratory frame

Subsequent emissions:

q and q⁰ radiate at θ < θ₁; ¯q and ¯q⁰ at θ⁰ < θ₂

~ -

θ₁

@@ I@@θ₂

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Branching algorithm for the final-state radiation (forward evolution)

__^^

__^^^u_@

@I@

q(p) θ

g(p₁)

- q(p₂)

z =

^E_E¹

dP =

^α_2π^S

P (z)dz

^dQ_Q₂²

∆

_S

(Q

²_max

, Q

²

)

Initial-state radiation (backward evolution)

~ -

h

q(p)^@_@

@@@ R q(p₂)

__^^

__^{u -}^^

g(p₁) θ

z =

^E_E²

dP =

^α_2π^S

P (z)dz

^x/z_x ^f^b_f^(x/z,Q²⁾

a(x,Q²)

dQ² Q²

∆_S,a(Q²_max,Q²)

∆_S,b(Q²,Q²₀)

Scale of α_S in parton showers: using the transverse momentum of the emitted parton allows one to resum a class of soft/collinear logarithms

(D. Amati et al. NPB (1980) 173 )

α_S(k_T²) = α_S(Q²)

1 + α_S(Q²)b₀ ln(Q²/k_T²) ' α_S(Q²)

"

1 − α_S(Q²)b₀ lnQ²

k_T² + . . .

#

b₀ = 33 − 12n_f

12π (LO)

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Parton showers and resummation: expansion of the Sudakov form factor

∆_S(Q²_max, Q²) = exp

"

−α_S 2π

Z Q²max Q²

dQ⁰² Q⁰²

Z z_max z_min

dzP (z)

#

Example: all q → qg, soft and collinear limit (z → 0, Q² → 0) P (z) = C_F 1 + (1 − z)²

z ' 2C_F

z z_min = Q₀

Q z_max = 1 − z^min (HERWIG)

∆_S ' exp

−α_SC_F

4π log² Q_max Q²₀

1 + α_S

4π log² Q²

Q²₀ + α²_SC_F²

32π² log⁴ Q² Q²₀ . . .

dP ∝ α_S π

dQ² Q²

dz z

1 + α_SC_F

4π log² Q²

Q²₀ + α²_SC_F²

32π² log⁴ Q²

Q²₀ + . . .

Probability of one emission at Q²:

P₁ ∝ Z

dP = α_S

4π log² Q²

Q²₀ + α²_S

256π² log⁴ Q²

Q²₀ + . . . Resummation of double logarithms ∼ αⁿSL²ⁿ

Soft or collinear limit: resummation of single logarithms ∼ αⁿ_SLⁿ