Non-perturbative hadronisation corrections with the ARES

Non-perturbative hadronisation corrections with theARES method

Andrea Banfi1, Basem Kamal El-Menoufi2, Ryan Wood1

1Department of Physics and AstronomyUniversity of Sussex

2Consortium for Fundamental Physics, School of Physics and AstronomyUniversity of Manchester

QCD Master Class 2021

Ryan Wood QCD Master Class 2021 1st September 2021 1 / 22

Introduction

Broader Context

• Event shape observables (infrared and collinear safe) in e+e− annihilationare often used to perform precise determinations of the strong couplingconstant, αs .

Why do we wish to perform precise determinations of αs?

• αs is the least well-known coupling in the gauge sector of the StandardModel.

• 2019 Particle Data Group world average of αs had an uncertainty of about1%, (αs = 0.1179± 0.0010).

• The uncertainty on αs is becoming increasingly critical for precision colliderphenomenology.


Why use these event shape observables?

• In principal they are the ideal testing ground for perturbative QCD.

• They can be computed order-by-order in perturbation theory.

• αs small due to Q ∼ MZ, therefore a well-behaved perturbation series.

• Non-perturbative (hadronisation) effects should be suppressed by inversepowers of Q.

Problems:

• Hadrons (in Measurements) vs Quarks/Gluons (in Theoretical Calculations).

• Non-perturbative effects turn out to be significant even at Q ∼ MZ.

• How do we estimate these hadronisation effects?


• In the 1990’s it was discovered that Hadronisation Corrections to collinearand infrared-safe two-jet event shape observables in e+e− annihilation couldbe described in terms of an expansion in negative powers of the centre ofmass energy, Q.

• The leading correction in this sense leads to a shift of the cumulativedistribution of the form:

Σ (V ) = ΣPT (V − ⟨δV ⟩)

with:

⟨δV ⟩ ∝ 1

Q

• The mean change in the observable’s value is due to the emission of softnon-perturbative radiation.


• Illustrated by way of an example (τ = 1− T ):

• To reiterate - Leading Hadronisation Corrections provide a shift of theperturbative event-shape distribution.


• In 2019, a collaboration of theorists and experimentalists presentedstate-of-the-art extractions of the strong coupling based on N3LO + NNLLaccurate predictions for the two-jet rate in the Durham clustering algorithmin e+e− annihilation.

[arXiv: 1902.08158v1]

• This was the first time in history that hadronisation uncertainties for thesetypes of observables were larger than QCD uncertainties.

• Our aim is to introduce a new method to compute leading hadronisationcorrections to two-jet event shapes in e+e− annihilation.


Set-up

• We begin by considering a generic recursive infrared and collinear safe(rIRC) observable in e+e− annihilation:

V ({p̃}, k1, ..., kn) ⩾ 0

where:

• {p̃} = {p̃1, p̃2} are the momenta of a hard quark-antiquark pair

• k1, ..., kn are the subsequent emissions


• In the Born limit V ({p̃}) = 0, whereas in general V ({p̃}) ⩾ 0.

• We shall consider the region in which:

V ({p̃}, k1, ..., kn) = v ≪ 1

• In this limit all secondary emissions are soft &/or collinear.

• We shall only consider soft & collinear secondary emissions widely separatedin angle.

• The observable cumulant is given by:

Σ(v) = ΣPT(v) + δΣNP(v)


• At NLL accuracy we can write:

ΣPT(v) = e−RNLL(v)FNLL(v)

where:

• RNLL(v) is the Radiator (can be calculated analytically)

• The transfer function is given by:

FNLL(v) =

⟨Θ

(1− Vsc ({p̃}, k1, ..., kn)

v

)⟩and can be computed by a Monte Carlo method.


Approach

• Leading hadronisation corrections are due to the contribution of a very softgluon (aka gluer):

• We define:δVNP ≡ V ({p̃}, k , {ki})− V ({p̃}, {ki})


• We find that:

δΣNP = −⟨δVNP⟩dΣPT

dv

• This gives us that:

Σ(v) = ΣPT(v) + δΣNP(v)

= ΣPT(v)− ⟨δVNP⟩dΣPT

dv= ΣPT (v − ⟨δVNP⟩)

• The gluer produces a shift in the perturbative event-shape distribution


• We shall restrict ourselves to event shapes for which:

δVNP ({p̃}, k , {ki}) =ktQfV (η, ϕ, {ki})

with η = − ln tan θ2


• Therefore:

⟨δVNP ({p̃}, k , {ki})⟩ =⟨kt⟩Q

cV

with:

cV =

⟨fV (η, ϕ, {ki}) δ

(1− Vsc({p̃},{ki})

v

)⟩⟨δ(1− Vsc({p̃},{ki})

v

)⟩• The denominator may be written as R ′F(R ′) with:

R ′ ≡ −vdR

dv


Additive Observables

• For additive observables we have:

V ({p̃}, k1, ..., kn) =n∑

i=1

ζi with ζi ≡ V ({p̃}, ki )

• Therefore we trivially find that:

δVNP ({p̃}, k , {ki}) = V ({p̃}, k)

and therefore:fV (η, ϕ, k1, ..., kn) = fV (η, ϕ)

• This results in a trivial cancellation, giving:

cV =

∫dη

dϕ

2πfV (η, ϕ)

• Let us consider the following examples (Thrust, C-parameter, Heavy JetMass)


Thrust

• The Thrust is given by:

T ≡ maxn⃗

∑i |p⃗i · n⃗|Q

, τ ≡ 1− T

• We find that:fτ (η, ϕ) = e−|η|

• Therefore:

cτ =

∫dη

dϕ

2πfτ (η, ϕ)

=

∫ ∞

−∞dη e−|η|

= 2


C-parameter

• The C-parameter is given by:

C ≡ 3

1− 1

2

∑i,j

(pi · pj)2

(pi · Q) (pj · Q)

• We find that:

fC (η, ϕ) =3

cosh η

• Therefore:

cC =

∫dη

dϕ

2πfC (η, ϕ)

=

∫ ∞

−∞dη

3

cosh η

= 3π


Heavy Jet Mass• The Heavy Jet Mass is given by:

ρH ≡ maxi=1,2

M2i

Q2, M2

i ≡

∑j∈H(i)

pj

2

• A non-zero hadronisation correction to the heavy-jet mass arises only whenthe NP gluon is emitted in the heavier hemisphere.

• We find that:

fρH(η, ϕ, k1, . . . , kn) = e−ηΘ(η)Θ(ρ1 − ρ2) + eηΘ(−η)Θ(ρ2 − ρ1)

• Therefore:

cρH =

∫dη

dϕ

2πfρH (η, ϕ)

=1

2

∫ 0

−∞dη eη +

1

2

∫ ∞

0

dη e−η

= 1


Non-additive Observables

• For non-additive observables life becomes more tricky.

• To illustrate, let us consider Broadening-like observables:

• The Single Jet Broadenings are given by:

BL ≡∑

i∈H(1)

|p⃗i × n⃗T|2Q

& BR ≡∑

i∈H(2)

|p⃗i × n⃗T|2Q

• The Wide Jet Broadening is given by:

BW ≡ max{BL,BR}

• The Total Jet Broadening is given by:

BT ≡ BL + BR


• The difficulty presents itself when the shape-functions are calculated:

• We find that:

fBW(η, ϕ, k1, . . . , kn)

= Θ(η(1))

√1 + 2eη(1) pt,1Q

cosϕ1 + e2η(1)

(pt,1Q

)2

− eη(1) pt,1

Q

Θ(B1 − B2)

+ 1 ↔ 2

and

fBT(η, ϕ, k1, . . . , kn)

=∑ℓ

Θ(η(ℓ))

√1 + 2eη(ℓ) pt,ℓQ

cosϕℓ + e2η(ℓ)

(pt,ℓQ

)2

− eη(ℓ) pt,ℓ

Q


• Analytical expressions for cBWand cBT

have been computed[hep-ph/9812487v3]:

cBW =1

2

[−2− ψ(1)− lnB + η0 + χ

(R ′

2

)− ρ

(R ′

2

)+ ψ

(1 +

R ′

2

)]

cBT= 2cBW

− ψ

(1 +

R ′

2

)+ ψ (1 + R ′) +

1

R ′

• N.B cBT holds in the limit R ′ ≫√2R ′′

• We have successfully been able to reproduce these results both analyticallyand numerically using our method.


Thrust Major

• Having used the Broadening-like Observables as an ideal ’test of concept’the next step is to apply our approach to suitable observables for which noanalytic calculation for the shift has been computed (nor is currentlybelieved to be possible).

• The Thrust Major is given by:

TM ≡ maxn⃗·n⃗T

∑i |p⃗i · n⃗|Q

• We find that:

fTM(η, ϕ, k1, . . . , kn)

=∑ℓ

Θ(η(ℓ))

[∣∣∣∣∣cosϕℓ + eη(ℓ)

Q|py ,ℓ|

∣∣∣∣∣− eη(ℓ)

Q|py ,ℓ|

]


Next Steps

• We are currently completing our computation for cTM

• On completion we shall compare our results to parton-shower eventgenerators.

• We shall use results for the shifts of Broadening-like observables and theThrust Major to perform a simultaneous fit for αs and α0.

• Thank you for listening


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Non-perturbative hadronisation corrections with the ARES