Event Generator Physics Part 2: Hard processes and parton showers

Event Generator Physics

Part 2: Hard processes and parton showers

Torbj¨ orn Sj¨ ostrand

Theoretical Particle Physics

Department of Astronomy and Theoretical Physics Lund University

S¨olvegatan 14A, SE-223 62 Lund, Sweden

DK–PI Summer School 2022, Neusiedl, Austria

Multijets – the need for Higher Orders



2→ 6 process or 2 → 2 dressed up by bremsstrahlung!?

Perturbative QCD

Perturbative calculations⇒ Matrix Elements.

Improved calculational techniques allows

? more legs (= final-state partons)

? more loops (= virtual partons not visible in final state) but with limitations, especially for loops.

Parton Showers:

approximations to matrix element behaviour,

most relevant for multiple emissions at low energies and/or angles.

Main topic of this lecture.

Matching and Merging:

methods to combine matrix elements (at high scales) with parton showers (at low scales),

with a consistent and smooth transition.

Huge field at LHC.

The basic QCD processes

Six basic 2→ 2 QCD processes:

qq⁰ → qq⁰ qq→ q⁰q⁰ qq→ gg qg→ qg gg→ qq gg→ gg Mandelstam variables

ˆ

s = (p1+ p2)² = (p3+ p4)² tˆ = (p₁− p3)² = (p₂− p4)² ˆ

u = (p₁− p4)² = (p₂− p3)²

In rest frame, massless limit: m1 = m2 = m3 = m4 = 0 ˆ

s = E_CM² ˆt =−sˆ

2(1− cos ˆθ) ˆ

u =−sˆ

2(1 + cos ˆθ) ˆ

s + ˆt + ˆu = 0

Closeup: qg → qg

Considerq(1) g(2)→ q(3) g(4):

t : p_g^∗ = p₁− p3 ⇒ m²g^∗ = (p₁− p3)² = ˆt ⇒ dˆσ/dˆt ∼ 1/ˆt² u : p_q^∗ = p₁− p4 ⇒ m²q^∗ = (p₁− p4)²= ˆu⇒ dˆσ/dˆt ∼ −1/ˆsˆu s : p_q^∗ = p1+ p2 ⇒ mq^2∗ = (p1+ p2)² = ˆs ⇒ dˆσ/dˆt ∼ 1/ˆs² Contribution of each sub-graph is gauge-dependent,

only sum is well-defined:

dˆσ

dˆt = πα²_s ˆ s²

sˆ²+ ˆu² tˆ² +4

9 ˆ s (−ˆu)+4

9 (−ˆu)

ˆ s



Composite beams

In reality all beams are composite:

p : q, g, q, . . . e⁻:e⁻, γ, e⁺, . . . γ : e^±, q, q, g

Factorization σ^AB =X

i,j

ZZZ

f_i^(A)(x1, Q²) f_j^(B)(x2, Q²)dˆσij

dˆt dx1dx2dˆt x : momentum fraction, e.g. p_i = x₁p_A; p_j = x₂p_B

Q²: factorization scale, “typical momentum transfer scale”

Factorization only proven for a few cases, likeγ^∗/Z⁰ prodution, and strictly speaking not correct e.g. for jet production,

but good first approximation and unsurpassed physics insight.

In the beginning: Electrodynamics

An electrical charge, say an electron, is surrounded by a field:

For a rapidly moving charge

this field can be expressed in terms of an equivalent flux of photons:

dn_γ ≈ 2α_em π

dθ θ

dω ω Equivalent Photon Approximation, or method of virtual quanta (e.g. Jackson) (Bohr; Fermi; Weisz¨acker, Williams∼1934)

e⁻

e⁻

e⁻

e⁻

.

θ: collinear divergence, saved by m_e> 0 in full expression. ω: true divergence, n_γ∝R

dω/ω =∞, but Eγ ∝R

ω dω/ω finite.

These are virtual photons: continuously emitted and reabsorbed.

In the beginning: Electrodynamics

An electrical charge, say an electron, is surrounded by a field:

For a rapidly moving charge

this field can be expressed in terms of an equivalent flux of photons:

dnγ ≈ 2α_em π

dθ θ

dω ω Equivalent Photon Approximation, or method of virtual quanta (e.g. Jackson) (Bohr; Fermi; Weisz¨acker, Williams∼1934)

e⁻

e⁻ e⁻

e⁻

.

θ: collinear divergence, saved by m_e> 0 in full expression. ω: true divergence, n_γ∝R

dω/ω =∞, but Eγ ∝R

ω dω/ω finite. These are virtual photons: continuously emitted and reabsorbed.

In the beginning: Electrodynamics

An electrical charge, say an electron, is surrounded by a field:

For a rapidly moving charge

this field can be expressed in terms of an equivalent flux of photons:

dnγ ≈ 2α_em π

dθ θ

dω ω Equivalent Photon Approximation, or method of virtual quanta (e.g. Jackson) (Bohr; Fermi; Weisz¨acker, Williams∼1934)

e⁻

e⁻ e⁻

e⁻

.

θ: collinear divergence, saved by m_e> 0 in full expression.

ω: true divergence, n_γ ∝R

dω/ω =∞, but Eγ ∝R

ω dω/ω finite.

These are virtual photons: continuously emitted and reabsorbed.

In the beginning: Bremsstrahlung

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 e Final 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!

In the beginning: Exponentiation

AssumeP

Eγ Ee such that energy-momentum conservation is not an issue. Then

dPγ =dn_γ ≈ 2α_em π

dθ θ

dω ω is the probability to find a photon atω and θ, irrespectively of which other photons are present.

Uncorrelated⇒ Poissonian number distribution:

Pi = hnγiⁱ i ! e^−hnγi with

hnγi = Z θmax

θmin

Z ωmax

ωmin

dn_γ ≈ 2α_em π ln

θ_max θmin

 ln

ω_max ωmin



Note thatR

dPγ =R

dn_γ> 1 is not a problem:

proper interpretation is that many photons are emitted.

Exponentiation: reinterpretation of dPγ into Poissonian.

So how is QCD the same?

A quark is surrounded by a gluon field dPg =dng ≈ 8αs

3π dθ

θ dω

ω i.e. only differ by substitution α_em→ 4αs/3.

An accelerated quark emits gluons with collinear and soft divergences, and as Initial andFinal State Radiation.

e⁻

q

Typically hngi =R

dn_g  1 since αs  αem

⇒ even more pressing need for exponentiation.

So how is QCD different?

QCD is non-Abelian, so a gluon is charged and is surrounded by its own field:

emission rate 4α_s/3→ 3αs, field structure more complicated, interference effects more important.

αs(Q²) diverges for Q² → Λ²_QCD, with Λ_QCD ∼ 0.2 GeV = 1 fm⁻¹. Confinement: gluons below Λ_QCD not resolved ⇒ de facto cutoffs.

.

Unclear separation between

“accelerated charge” and “emitted radiation”:

many possible Feynman graphs≈ histories.

The Parton-Shower Approach

2→ n = (2 → 2) ⊕ ISR ⊕ FSR

FSR = Final-State Radiation = timelike shower Q_i²∼ m² > 0 decreasing

ISR = Initial-State Radiation = spacelike showers Q_i²∼ −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 + p_L ⇒ 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₋= E²− p_L² = m²+ p_⊥² m²_a+ p²_⊥a

p+a

= m²_b+ p_⊥b² z p+a

+ m²_c+ p²_⊥c (1− z) p+a

⇒ ma²= m_b²+ p²_⊥

z +m²_c+ p²_⊥ 1− z = m²_b

z + m²_c

1− z + p_⊥² z(1− z) Final-state shower: mb= mc = 0⇒ ma²= ^p

2

⊥

z(1−z) > 0⇒ timelike Initial-state shower: m_a = m_c = 0⇒ m²b=−₁^p_−z^2⊥ < 0⇒ spacelike

Doublecounting

Do not doublecount: 2→ 2 = most virtual = shortest distance (detailed handling of borders⇒ match & merge)

Final-state radiation

Standard process e⁺e⁻→ qqg by two Feynman diagrams:

x_i = 2Ei

E_cm x₁+x₂+x₃= 2

dσ_ME

σ₀

=

1)(1−x2)

dx

dx

Convenient (but arbitrary) subdivision to “split” radiation: 1

(1− x1)(1− x2)

(1− x¹) + (1− x²)

x₃ = 1

(1− x2)x₃ + 1 (1− x1)x₃

Final-state radiation

Standard process e⁺e⁻→ qqg by two Feynman diagrams:

x_i = 2Ei

E_cm x₁+x₂+x₃= 2

dσ_ME

σ₀

=

1)(1−x2)

dx

dx

Convenient (but arbitrary) subdivision to “split” radiation:

1

(1− x1)(1− x2)

(1− x¹) + (1− x²)

x₃ = 1

(1− x2)x₃ + 1 (1− x1)x₃

From matrix elements to parton showers

Rewrite for x2 → 1, i.e. q–g collinear limit:

1− x2 = m²₁₃ E_cm² = Q²

E_cm² ⇒ dx2= dQ² E_cm² define z as fraction q retains

in branchingq→ qg

x₁ ≈ z ⇒ dx1 ≈ dz x3 ≈ 1 − z

⇒ dP =dσ σ0

= αs

2π dx2

(1− x2) 4 3

x₂²+ x₁²

(1− x1) dx1 ≈ αs

2π dQ²

Q² 4 3

1 + z² 1− z dz 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 = α_s 2π

dQ²

Q² P_a→bc(z)dz P_q→qg = 4

3 1 + z²

1− z

Pg→gg = 3(1− z(1 − z))² z(1− z) P_g→qq = n_f

2 (z²+ (1− z)²) (n_f =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, Q₁² Q2² Q3². . . (≈ time-ordered).

To cover “all” of phase space use DGLAP in whole region Q₁²> Q₂² > Q₃². . ..

Iteration gives final-state parton showers:

Need soft/collinear cuts to stay away from nonperturbative physics.

Details model-dependent, but around 1 GeV scale.

The Sudakov form factor – 1

Time evolution, conservation of total probability:

P(no emission) = 1 − P(emission).

Multiplicativeness, with T_i = (i/n)T , 0≤ i ≤ n:

Pno(0≤ t < T ) = lim

n→∞

nY−1 i =0

Pno(Ti ≤ t < Ti +1)

= lim

n→∞

nY−1 i =0

(1− Pem(T_i ≤ t < Ti +1))

= exp − lim_n

→∞

n−1

X

i =0

Pem(T_i ≤ t < Ti +1)

!

= exp



− Z T

0

dPem(t) dt dt



=⇒ dPfirst(T ) = dPem(T )exp



− Z T

0

dPem(t) dt dt



cf. radioactive decay in lecture 1.

The Sudakov form factor – 2

Expanded, with Q ∼ 1/t (Heisenberg) dPa→bc = α_s

2π dQ²

Q² P_a→bc(z)dz

× exp



−X

b,c

Z Q_max² Q²

dQ⁰² Q⁰²

Z α_s

2πP_a→bc(z⁰)dz⁰





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 thatP

b,c

R R dPa→bc ≡ 1 ⇒ convenient for Monte Carlo (≡ 1 if extended over whole phase space, else possibly nothing happens before you reach Q0 ≈ 1 GeV).

The Sudakov form factor – 3

Sudakov regulates singularity for first emission . . .

. . . but in limit of repeated soft emissionsq→ qg (but no g → gg) one obtains the same inclusive Q emission spectrum as for ME, i.e. divergent ME spectrum

⇐⇒ infinite number of PS emissions

More complicated in reality:

energy-momentum conservation effects big since α_s big, so hard emissions frequent

g→ gg branchings leads to accelerated multiplication of partons

The ordering variable

In the evolution with

dPa→bc = α_s 2π

dQ²

Q² P_a→bc(z)dz Q² orders the emissions (memory).

If Q²= m² is one possible evolution variable then Q⁰²= f (z)Q² is also allowed, since

d(Q⁰², z) d(Q², z)   =

∂Q⁰²

∂Q²

∂Q⁰²

∂z ∂z

∂Q² ∂z

∂z

  =

 f (z) f⁰(z)Q²

0 1

 = f (z)

⇒ dPa→bc = α_s 2π

f (z)dQ²

f (z)Q² P_a→bc(z)dz = α_s 2π

dQ⁰²

Q⁰² P_a→bc(z)dz Q⁰²= E_a²θ_a²_→bc ≈ m²/(z(1− z)); angular-ordered shower Q⁰²= p_⊥² ≈ m²z(1− z); transverse-momentum-ordered

Coherence

QED: Chudakov effect (mid-fifties)

QCD: colour coherence for soft gluon emission

solved by • requiringemission angles to be decreasing or • requiringtransverse momenta to be decreasing

Coherence

QED: Chudakov effect (mid-fifties)

QCD: colour coherence for soft gluon emission

solved by • requiringemission angles to be decreasing or • requiringtransverse momenta to be decreasing

The dipole picture

1→ 2 branching = replace m = 0 parton by pair with m > 0.

Breaks energy–momentum conservation.

Herwig angular-ordered shower: post-facto rescaling machinery.

Alternative: dipole picture (most common, 3 variants in PYTHIA).

2→ 3 parton branching, or 1 → 2 colour dipole branching.

Can be viewed as radiator a→ bc with recoiler r.

Quark vs. gluon jets

Pg→gg

P_q→qg ≈ Nc

C_F = 3 4/3 = 9

4 ≈ 2

⇒ gluon jets are softer and broader than quark ones (also helped by hadronization models, lecture 4).

(GeV/c) Jet PT

50 100 200 300 1000

〉chN〈

5 10 15 20 25

= 7 TeV s pp

Data |y| < 1 Data 1 < |y| < 2 Gluon Jets (Pythia Tune Z2) Quark Jets (Pythia Tune Z2)

L dt = 36 pb-1

∫

CMS

(GeV/c) Jet PT

50 100 200 300 1000

〉2Rδ〈

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

= 7 TeV s pp

Data |y| < 1 Data 1 < |y| < 2 Gluon Jets (Pythia Tune Z2) Quark Jets (Pythia Tune Z2)

L dt = 36 pb-1

∫

CMS

Note transition g jets→ q jets for increasing p⊥.

Parton Distribution Functions

Hadrons are composite, with time-dependent structure:

f_i(x, Q²) = number density of partons i at momentum fraction x and probing scale Q².

Linguistics (example):

F₂(x, Q²) = X

i

e_i²xf_i(x, Q²) structure function parton distributions

PDF evolution

Initial conditions at small Q₀² unknown: nonperturbative.

Resolution dependence perturbative, by DGLAP:

DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi) df_b(x, Q²)

d(ln Q²) =X

a

Z 1 x

dz

z fa(y, Q²) α_s 2πP_a→bc

 z = x

y



DGLAP already introduced for (final-state) showers:

dPa→bc = α_s 2π

dQ²

Q² P_a→bc(z)dz Same equation, but different context:

dPa→bc is probability for the individual parton to branch; while df_b(x, Q²) describes how the ensemble of partons evolve by the branchings of individual partons as above.

Initial-State Shower Basics

• Parton cascades in p are continuously born and recombined.

• Structure at Q is resolved at a time t ∼ 1/Q before collision.

• A hard scattering at Q² probes fluctuations up to that scale.

• A hard scattering inhibits full recombination of the cascade.

• Convenient reinterpretation:

Forwards vs. backwards evolution

Event generation could be addressed by forwards evolution:

pick a complete partonic set at low Q0 and evolve, consider collisions at different Q² and pick byσ of those.

Inefficient:

1 have to evolve and check for all potential collisions, but 99.9. . . % inert

2 impossible (or at least very complicated) to steer the production, e.g. of a narrow resonance (Higgs)

Backwards evolution is viable and∼equivalent alternative:

start at hard interaction and trace what happened “before”

Backwards evolution master formula

Monte Carlo approach, based on conditional probability : recast df_b(x, Q²)

dt =X

a

Z ₁

x

dz

z f_a(x⁰, Q²) α_s

2π P_a→bc(z) with t = ln(Q²/Λ²) andz = x/x⁰ to

dPb= df_b

f_b =|dt| X

a

Z

dz x⁰fa(x⁰, t) xf_b(x, t)

α_s

2π P_a→bc(z) then solve for decreasing t, i.e. backwards in time,

starting at high Q² and moving towards lower, with Sudakov form factor exp(−R

dPb).

Extra factor x⁰f_a/xf_b relative to final-state equations.

Initial- vs. final-state showers

Both controlled by same evolution equations dPa→bc = α_s

2π dQ²

Q² P_a→bc(z)dz · (Sudakov) but

Final-state showers:

Q² timelike (∼ m²)

decreasing E, m², θ both daughters m² ≥ 0 physics relatively simple

⇒ “minor” variations:

Q², shower vs. dipole, . . .

Initial-state showers:

Q² spacelike (≈ −m²)

decreasing E , increasing Q², θ one daughter m² ≥ 0, one m² < 0 physics more complicated

⇒ more formalisms:

DGLAP, BFKL, CCFM, GLR, . . .

Combining FSR with ISR

Separate processing of ISR and FSR misses interference (∼ colour dipoles)

ISR+FSR add coherently in regions of colour flow and destructively else in “normal” shower by azimuthal anisotropies automatic in dipole (by proper boosts)

Combining FSR with ISR

Separate processing of ISR and FSR misses interference (∼ colour dipoles)

ISR+FSR add coherently in regions of colour flow and destructively else in “normal” shower by azimuthal anisotropies automatic in dipole (by proper boosts)

Next-to-leading log showers

dPg=dng ≈ 8α_s 3π

dθ θ

dω

ω 7→ αsL² gives leading-log answer P_n ∝ (αsL²)ⁿ=αⁿ_sL²ⁿ.

Resummation/exponentiation gives Sudakov P0 ∝ exp(−αsL²).

(Transverse momentum cuts bothθ and ω ⇒ αⁿsLⁿ.)

More careful handling of kinematics,α_s running, splitting kernels (alsog→ ggg), etc., give subleading corrections ∝αⁿ_sL²ⁿ⁻¹. All showers have some elements of NLL, e.g. momentum conservation, but some dedicated ongoing projects:

PanScales (Salam et al.) Deductor (Nagy, Soper) Herwig 7 (Pl¨atzer et al.) Vincia (Skands et al.) Dire (H¨oche, Prestel)

PanScales

PanScales is a dipole shower, but with several special choices.

Set of possible evolution scales, withβ = 0 like p_⊥. Recoil local inside dipole or global in whole system.

Careful splitting of emission on two sides of dipole.

Higher-order splitting kernels and α_s.

Some more. 4

FIG. 2. Left: ratio of the cumulative y23distribution from several showers divided by the NLL answer, as a function of

↵sln y23/2, for ↵s! 0. Right: summary of deviations from NLL for many shower/observable combinations (either ⌃^shower(↵s! 0, ↵sL = 0.5)/⌃NLL 1 or (Nshower^subjet(↵s! 0, ↵sL²= 5)/NNLL^subjet 1)/p↵s). Red squares indicate clear NLL failure; amber triangles indicate NLL fixed-order failure that is masked at all orders; green circles indicate that all NLL tests passed.

Fig. 1.

The left-hand plot of Fig. 1 shows the Pythia8 dipole algorithm (not designed as NLL accurate), while the middle plot shows our PanGlobal shower with = 0.

The dipole result is clearly not independent of 12

for ↵s! 0, with over 60% discrepancies, extending the fixed-order conclusions of Ref. [37]. The discrepancy is only' 30% for gg events (not shown in Fig. 1), and the di↵erence would, e.g., skew machine learning [67] for quark/gluon discrimination. PanGlobal is independent of 12. The right-hand plot shows the ↵s! 0 limit for multiple showers. The overall pattern is as expected:

PanLocal works for = 0.5, but not = 0, demon- strating that with ktordering it is not sufficient just to change the dipole partition to get NLL accuracy. Pan- Global works for = 0 and = 0.5. (Showers that coincide for ↵s! 0, e.g. Dire v1 and Pythia8, typically di↵er at finite ↵s, reflecting NNLL di↵erences.)

Next, we consider a range of more standard observ- ables at NLL accuracy. They include the Cambridge py23resolution scale [68]; two jet broadenings, BTand BW [69]; fractional moments, FC1 obs, of the energy- energy correlations [47]; the thrust [70, 71], and the max- imum ui= kti/Qe ^obs|⌘i|among primary Lund declus- terings i. Each of these is sensitive to soft-collinear ra- diation as kt/Qe ^obs|⌘|, with the obs values shown in Fig. 2 (right). Additionally, the scalar sum of the trans- verse momenta in a rapidity slice [72], of full-width 2, is useful to test non-global logarithms (NGLs). These ob- servables all have the property that their distribution at NLL can be written as [47, 53, 72–74]

⌃(↵s, ↵sL) = exp⇥

↵_s¹g1(↵sL) + g2(↵sL) +O ↵ⁿsL^{n 1}⇤ , (6) where ⌃ is the fraction of events where the observable is smaller than e^L (g1 = 0 for the rapidity slice kt).

We also consider the k-algorithm [75] subjet multiplic-

ity [76], [51]§ 5.

Fig. 2 (left) illustrates our all-order tests of the shower for one observable, py23. It shows the ratio of the ⌃ as calculated with the shower to the NLL result, as a function of ↵sln py23in the limit of ↵s! 0. The stan- dard dipole algorithms disagree with the NLL result, by up to 20%. This is non-negligible, though smaller than the disagreement in Fig. 1, because of the azimuthally averaged nature of the py23observable. In contrast the PanGlobal and PanLocal( = 0.5) showers agree with the NLL result to within statistical uncertainties.

Fig. 2 (right) shows an overall summary of our tests. The position of each point shows the result of

⌃shower(↵s! 0, ↵sL = 0.5)/⌃NLL 1 or (N_shower^subjet(↵s! 0, ↵sL²= 5)/N_NLL^subjet 1)/p↵s. If it di↵ers from 0, the point is shown as a red square. In some cases (amber tri- angles) it agrees with 0, though an additional fixed-order analysis in a fixed-coupling toy shower [37] [51]§ 2 re- veals issues a↵ecting NLL accuracy, all involving hitherto undiscovered spurious super-leading logarithmic terms.¹ Green circles in Fig. 2 (right) indicate that the shower/observable combination passes all of our NLL tests, both at all orders and in fixed-order expansions.

The four shower algorithms designed to be NLL accurate pass all the tests. These are the PanLocal shower (dipole and antenna variants) with = ¹₂ and the PanGlobal shower with = 0 and =¹₂.

1Such terms, (↵sL)ⁿ(↵sL²)^pin ln ⌃, starting typically for n = 3 (sometimes 2), p 1, appear for traditional ktordered dipole showers for global (obs> 0) and non-global observables [51]§ 3.

Terms of this kind can generically exist [77–79], but not at leading-colour or for pure final-state processes with rIRC [47]

safe observables. In many cases, the spurious super-leading log- arithms appear to resum to mask any disagreement with NLL.

Correct NLL answer from analytical resummation variable by variable.

Torbj¨orn Sj¨ostrand Event Generator Physics 2 slide 34/44

Matrix elements vs. parton showers

ME : Matrix Elements

+ systematic expansion in α_s (‘exact’) + powerful for multiparton Born level + flexible phase space cuts

− loop calculations very tough

− negative cross section in collinear regions

⇒ unpredictive jet/event structure

− no easy match to hadronization PS : Parton Showers

− approximate, to LL (or NLL)

− main topology not predetermined

⇒ inefficient for exclusive states + process-generic ⇒ simple multiparton + Sudakov form factors/resummation

⇒ sensible jet/event structure + easy to match to hadronization

Matrix elements and parton showers

Recall complementary strengths:

• ME’s good for well separated jets

• PS’s good for structure inside jets Marriage desirable! But how?

Problems: • gaps in coverage?

• doublecounting of radiation?

• Sudakov?

• NLO consistency?

Much work in last 20 – 30 years; impossible to cover it all.

Three main areas, in ascending order of complication:

1) Matrix-element correction with lowest-order nontrivial process 2) Merge leading-order multiparton process — vetoed showers 3) Match to next-to-leading order process —MC@NLO, POWHEG

Note unintuitive terminology match vs. merge⇒ M&M

Matrix Element Corrections (MEC)

= cover full phase space with smooth transition ME/PS.

Want to reproduce W^ME= 1 σ(LO)

dσ(LO + g) d(phasespace) by shower generation withW^PS>W^ME + correction procedure

wanted

z }| { W^ME=

generated

z }| { W^PS

correction

z }| { W^ME

W^PS

• Exponentiate ME correction by shower Sudakov form factor:

W_actual^PS (Q²)=W^ME(Q²)exp − Z Q_max²

Q²

W^ME(Q⁰²)dQ⁰²

!

• Memory of shower remains in Q² choice, i.e. “time” ordering.

• ME regularized: probability ≤ 1 instead of divergent.

• NLO correction simple for FSR, more messy for ISR:

replaceσ(LO)→ σ(NLO) in prefactor.

Vetoed Parton Showers (Merging)

Generic method to combine ME’s of several different orders to NLL accuracy; has become a standard tool for many studies Basic idea:

consider (differential) cross sectionsσ0, σ1, σ2, σ3, . . ., corresponding to a lowest-order process (e.g. W production), with more jets added to describe more complicated topologies, in each case to the respective leading order

σi, i ≥ 1, are divergent in soft/collinear limit

absent virtual corrections would have ensured “detailed balance”, i.e. an emission that adds toσ_{i +1} subtracts from σ_i such virtual corrections correspond (approximately) to the Sudakov form factors of parton showers

so use shower routines to provide missing virtual corrections

⇒ rejection of events (especially) in soft/collinear regions

Simple merging example

Considere⁺e⁻→ qq, qqg, qqgg, with Qmax> Q > Qmin. Early matching without ME correction would read

W_qqg=W^ME(Q²)exp − Z Q²_max

Q²

W^PS(Q⁰²)dQ⁰²

!

For fixed-multiplicity topologies at scale Q_min: W_qq = exp −

Z Q_max² Q²_min

W_qq^PS

!

W_qqg = exp − Z Q_max²

Q²

W_qq^PS

!

W_qqg^ME(Q²)exp − Z Q²

Q_min²

W_qqg^PS

!

Wqqgg = exp − Z Q_max²

Q²₁

W_qq^PS

!

exp − Z Q²₁

Q₂²

W_qqg^PS

!

W_qqgg^ME(Q₁², Q2²)

× exp − Z Q₂²

Q²_min

W_qqgg^PS

!

(with intermediate scales Q1> Q2) Regular shower allowed only below Q (unlike matching).

Veto scheme

1) Pick hard process, mixing according toσ0:σ1 :σ2 :. . ., above some ME cutoff(e.g. all p_⊥i > p_⊥0, all Rij > R0), with large fixedα_s0

2) Reconstruct imagined shower history (in different ways) 3) Weight W_α =Q

branchings(α_s(k_⊥i² )/α_s0) ⇒ accept/reject CKKW-L:

4) Sudakov factor for non-emission on all lines above ME cutoff W_Sud =Q

“propagators⁰⁰

Sudakov(k_⊥beg² , k_⊥end² ) 4a) CKKW : use NLL Sudakovs 4b) L: use trial showers

5) W_Sud ⇒ accept/reject 6) do shower, vetoing emissions above cutoff

MLM:

4) do parton showers 5) (cone-)cluster showered event

6) match partons and jets 7) if all partons are matched, and n_jet = n_parton, keep the event, else discard it

Matching

Objectives:

Total rate should be accurate to NLO.

NLO results are obtained for all observables when (formally) expanded in powers of αs.

Hard emissions are treated as in the NLO computations.

Soft/collinear emissions are treated as in shower MC.

The matching between hard and soft emissions is smooth.

The outcome is a set of “normal” events.

Two commonly used schemes — MC@NLO and POWHEG — but intermediate or quite different ones could be possible.

Matching: MC@NLO

Madgraph5 aMC@NLO:

automated MC@NLO calculations for “arbitrary” process.

Matching: POWHEG

dσ = ¯B(v )dΦv

R(v, r ) B(v ) exp



− Z

p⊥

R(v, r⁰) B(v ) dΦ⁰_r

 dΦr



where

B(v ) = B(v ) + V (v ) +¯ Z

dΦr[R(v, r )− C (v, r)] . and

v, dΦ_v Born-level n-body variables and differential phase space r, dΦr extra n + 1-body variables and differential phase space B(v ) Born-level cross section

V (v ) Virtual corrections

R(v, r ) Real-emission cross section

C (v, r ) Conterterms for collinear factorization of parton densities.

Note thatR

[· · · dΦ^r]≡ 1 andR B(v )dΦ¯ v ≡ σNLO.

Summary

Overall production rate approximately set by lowest-order process.

Parton showers give multijets and subjet structure.

Sudakov factor is the crucial element for shower construction.

Recall “resummation” 1− x + O(x²)7→ exp(−x).

Dipole approach convenient for coherence and energy–momentum conservation.

ISR technically more complicated than FSR.

Matching&merging current frontline in the struggle to provide highest possible perturbative accuracy of event properties.