Parton Branching Algorithms & Improved Parton Showers Simon - - PowerPoint PPT Presentation
Parton Branching Algorithms & Improved Parton Showers Simon Pltzer Particle Physics University of Vienna at the CERN QCD Lunch Remote locations | 3 April 2020 Mainly based on recent work with Forshaw & Holguin QCD, Event
Simon Plätzer Particle Physics — University of Vienna at the CERN QCD Lunch Remote locations | 3 April 2020 Mainly based on recent work with Forshaw & Holguin
QCD description of collider reactions: Complexity challenges precision. Hard partonic scattering: NLO QCD routinely Jet evolution — parton showers: NLL sometimes, mostly unclear Multi-parton interactions Hadronization
Parton shower algorithms Lack a systematic expansion, obstruct fully differential NNLO for the hard process, open questions regarding mass effects and unstable particles. Hadronization models Lack constraints from perturbative evolution: Hiding perturbative corrections? Genuine uncertainties/constraints? Rethink foundations of parton showers.
Resummation of observables which globally measure deviations from n-jet limit. Use for analytic resummation and basis of parton shower algorithms.
Parton branching algorithm: Formulates iterative structure of contributions to cross sections. Could mean iterative structure of amplitudes or ‘density operators’. Parton shower: Numerical implementation of a parton branching algorithm. Used to solve evolution equations stemming from a parton branching algorithm.
No global measure of deviation from jet configuration: Coherent branching fails, full complexity of amplitudes strikes back. Even with a dipole approach 1/N effects possibly become comparable to subleading logarithms, and intrinsically 10% effects. Cannot ignore in the quest for higher precision.
Formulate iterative structures quite generally, with the goal of systematically approximating the iteration, not “iterating an approximation”.
Similar in spirit to Nagy & Soper
Theoretical control Actual predictions Guiding principle to incremental improvements
Explore new methods and paradigms in their
Non-global observables and accuracy for global
An(q⊥; {p}n) = Z dRnVq⊥,qn ⊥DnAn−1(qn ⊥; {p}n−1)D†
nV† q⊥,qn ⊥Θ(q⊥ qn ⊥).
Va,b = Pexp ✓
a
dq⊥ q⊥ Γn(q⊥) ◆
Γ1 Γ† D1 D2 D†
2D†
1H |Mi hM| A0 b 2 1 a a 2 1 b A1 A2
density operator
phase space integration
[Angeles, De Angelis, Forshaw, Plätzer, Seymour – JHEP 05 (2018) 044] [Forshaw, Holguin, Plätzer – JHEP 1908 (2019) 145]
Dn(qn ⊥; qn ∪ {˜ p}n−1) O D†
n(qn ⊥; qn ∪ {˜
p}n−1) = X
in,jn
Z δq(in,jn)
n ⊥
(qn ⊥) Sin
n O Sjn † n
+ X
in
Z δq(in,~
n) n ⊥
(qn ⊥) Cin
n O Cin † n ,
collinear contributions soft contributions
[Angeles, De Angelis, Forshaw, Plätzer, Seymour – JHEP 05 (2018) 044]
E ∂Gn(E) ∂E = −ΓGn(E) − Gn(E)Γ† + Dµ
n Gn−1(E) D† nµ u(E, ˆ
kn).
1 2 3 4 1 2 3 ¯ 1 ¯ 2 ¯ 3
|123i |213i |312i h123|123i h123|213i h123|312i
Leading(l)
τσ [A] = l
X
k=0
Aτσ
Leading(0)
τσ
h VnAnV†
n
i = δτσ
σ
Leading(0)
τσ [An]
σ
= exp @−N X
i,j c.c. in σ
λi¯ λjW (n)
ij
1 A
Leading(0)
τσ
h DnAn−1D†
n
i = δτσ X
i,j c.c. in σ\n
λi¯ λjR(n)
ij
Leading(0)
τ\n,σ\n [An−1]
Primary application: Non-global observables Re-derive BMS equation: Prototype of constructing a dipole shower
colour connected dipoles
Utilise colour flow basis, and expand around large-N:
[Angeles, De Angelis, Forshaw, Plätzer, Seymour – JHEP 05 (2018) 044]
[τ|Γ|σi = NδτσΓσ + Στσ + 1 N δτσρ
1 Γ Σ ρ1 Γ2 Γ3 ΣΓ ΣΓ2 Σ2Γ Σ2 ρΓ2 ρΓ Σ3 ρΣΓ ρΣ ρ21 ρ2Γ ρ2Σ ρ31 ρΣ2 (0 flips) × 1 × (αsN)n (1 flip) × αs × (αsN)n (0 flips) × αsN−1 × (αsN)n (t[...]t|0 flips)r (t[...]t|0 flips)r−1 t[...]s|1 flip × N−1 (t[...]t|0 flips)r (t[...]t|0 flips)r−1 t[...]s|1 flip × N−1 (t[...]t|0 flips)r−1 s[...]s|0 flips × N−2 (0 flips) × α2
s × (αsN)n
(t[...]t|0 flips)r (t[...]t|0 flips)r−1 t[...]t|2 flips × 1 virtuals reals α0
s
α1
s
α2
s
α3
s
N3 N2 N1 N0 N−1 N−2 N−3 (2 flips) × α2
s × (αsN)n
(t[...]t|0 flips)r−1 t[...]t|2 flips
Σ(n)
στ = N e−W (n)
σ
e−W (n)
τ
W (n)
σ
W (n)
τ
⇥ X
i,k c.c. in σ j,l c.c. in σ
⇣ λiλjW (n)
ij
+ ¯ λk¯ λlW (n)
kl
λi¯ λlW (n)
il
¯ λkλjW (n)
kj
⌘ δ i,l c.c. in τ
k,j c.c. in τ
VLC+NLC
n
|σi = V (n)
σ
|σi 1 N X
τ
δ#transpositions(τ,σ),1Σ(n)
στ |τi
[Plätzer – EPJ C 74 (2014) 2907]
dipole flips at next-to-leading colour
Systematically sum colour enhanced terms
ng αsN ∼ 1. easing powers
lnWab = ↵s 2⇡ X
i<j
Tg
i · Tg j
Z b2
a2
dq2 q2 Z d3k 2E 1 ⇡ (S · k)2 K2(pi, pj; k) ni · nj ni · n n · nj (q2 − K2(pi, pj; k)) ✓ij(k) − K2(pi; k) ni · n (q2 − K2(pi; k))✓i(k) − K2(pj; k) nj · n (q2 − K2(pj; k))✓j(k) !
ln Wab
⇡ X
i<j
Tg
i · Tg j
Z b
a
dE E Z dΩ 4⇡ ✓ni · nj − ni · n − nj · n ni · n n · nj ◆ = ↵s ⇡ X
i<j
Tg
i · Tg j
Z b
a
dE E ln ni · nj 2
ln Kab
⇡ X
i
(Tg
i )2
Z b
a
dE E Z dΩ 4⇡ 2 ni · n
ln Kab
= αs 2π X
i
(Tg
j)2
Z b
a
dk⊥ k⊥ Z 1
α
dz 1 − z + α Z dφ 2π = αs 2π X
i
(Tg
j)2
Z b
a
dk⊥ k⊥ Z 1−α dz 1 − z Z dφ 2π
Identify and subtract collinear singularities in soft evolution
collinear evolution softness
[Forshaw, Holguin, Plätzer – JHEP 1908 (2019) 145]
ln Wab
= αs π X
i<j
Tg
i · Tg j
Z b
a
dk⊥ k⊥ Z dy dφ 2π (θij(k) − θi(k)) Z Z
Energy ordering (Dipole) pt ordering
lnWab = ↵s 2⇡ X
i<j
Tg
i · Tg j
Z b2
a2
dq2 q2 Z d3k 2E 1 ⇡ (S · k)2 K2(pi, pj; k) ni · nj ni · n n · nj (q2 − K2(pi, pj; k)) ✓ij(k) − K2(pi; k) ni · n (q2 − K2(pi; k))✓i(k) − K2(pj; k) nj · n (q2 − K2(pj; k))✓j(k) !
hQ| Q Q p1? p2? 1 2 3 p3?
hM(Q)|
Identify and subtract collinear singularities in soft evolution
collinear evolution softness
[Forshaw, Holguin, Plätzer – JHEP 1908 (2019) 145]
[Forshaw, Holguin, Plätzer – arXiv:2003:06400]
Combine insight from soft evolution, large-N expansions and collinear subtractions:
q⊥ ∂An(q⊥; {p}n) ∂q⊥ = − Γn(q⊥) An(q⊥; {p}n) − An(q⊥; {p}n) Γ†
n(q⊥)
+ Z dRn Dn(qn ⊥) An−1(qn ⊥; {p}n−1) D†
n(qn ⊥) q⊥ δ(q⊥ − qn ⊥).
dσn(µ) = n Y
i=1
dΠi ! Tr An(µ),
Σ(µ; {p}0, {v}) = Z X
n
dσn(µ) u({p}n, {v}),
Start from amplitude evolution equations
[Forshaw, Holguin, Plätzer – arXiv:2003:06400]
nin · njn nin · n njn · n = Pinjn + Pjnin, where 2Pinjn = nin · njn nin · n nin · n njn · n + 1 nin · n
θin,jn θn,jn in jn φn
⌦ |Mn|2 u({p}n) ↵
1,...,n =
⌦ |Mn|2↵
1,...,n hu({p}n)i1,...,n
+
n
X
m=1
σm( ⌦ |Mn|2↵
1,...,n) σm(hu({p}n)i1,...,n) Corm(
⌦ |Mn|2↵
1,...,n , hu({p}n)i1,...,n)
X ⌦ ↵ + higher order correlations,
Collinear subtractions within a dipole? Recall angular ordering and coherent branching: Azimuthal average will result in angular ordering and simplify colour structures.
irrelevant for global observables at NLL
[Forshaw, Holguin, Plätzer – arXiv:2003:06400]
nin · njn nin · n njn · n = Pinjn + Pjnin, where 2Pinjn = nin · njn nin · n nin · n njn · n + 1 nin · n
Collinear subtractions within a dipole? Recall angular ordering and coherent branching: Azimuthal average will result in angular ordering and simplify colour structures.
ζ ∂ ⌦ |Mn(ζ)|2↵
1,...,n
∂ζ ⇡
jn+1
X
υ
αs π Z dz Pυυjn+1(z) hΘon shellin+1 ⌦ |Mn(ζ)|2↵
1,...,n +
X
υ
αs π Pυυjn(zn) ⇥ hΘon shellin Z d4pjn δ4(pjn z−1
n ˜
pjn) ⌦ |Mn−1(ζn,jn)|2↵
1,...,n−1 ζn,jn δ(ζ ζn,jn)
includes momentum mapping and physical phase space boundaries for on-shell partons
[Forshaw, Holguin, Plätzer – arXiv:2003:06400]
Colour in dipole shower evolution follows the BMS derivation.
q⊥Leading(0)
τσ
" ∂ ˆ An(q⊥) ∂q⊥ # ⇡ αs π Z dS(qn+1)
2
4π X
in+1,jn+1 c.c. in σ
⇥ 4λin+1¯ λjn+1Nc Z δq(in+1,jn+1)
n+1 ⊥
(q⊥) Θon shell δτσ Leading(0)
τσ
h ˆ An(q⊥) i + Z ✓ Y
in
d4pin ◆ X
in,jn c.c. in σ
λi¯ λjNc Z δq(in,jn)
n ⊥
(qn ⊥) Rsoft
injn
⇥ δτσ Leading(0)
τ\n σ\n
h ˆ An−1(qn ⊥) i q⊥ δ(q⊥ qn ⊥). (A.
Combination with collinear contributions: partition using coherent branching logic
✓ pin · pjn pin · qn pjn · qn T · pjn T · qn 1 pjn · qn + T · pin T · qn 1 pin · qn ✓ pin · pjn pin · qn pjn · qn T · pjn T · qn 1 pjn · qn + T · pin T · qn 1 pin · qn
[Forshaw, Holguin, Plätzer – arXiv:2003:06400]
(q(ic
n,icn)
n ?
)2 = 2(pic
n · qn)(p icn · qn)
pic
n · p icn
Rdipole
ic
n
= ✓1 2 + Asymic
nicn(qn)
◆ Ric
n,
Asymic
nicn(qn) =
" T · pic
n
4T · qn (q(ic
nicn)
n ⊥
)2 pic
n · qn
− T · p icn 4T · qn (q(ic
nicn)
n ⊥
)2 p icn · qn #
q? ∂|M(σ)
n (q?)|2
∂q? ≈ − αs π X
ic
n+1
Z dq
(ic
n+1,icn+1)
?
δ(q
(ic
n+1,icn+1)
?
− q?) Z dz Θon shell Pυinυin(z) |M(σ)
n (q?)|2
+ αs π Z ✓ Y
jn
d4pjn ◆ Rdipole
ic
n
Pυinυin(zn) q?δ(q(ic
n,icn)
n ?
− q?)|M(σ/n)
n1 (q(ic
n,icn)
n ?
)|2, (2.10)
Evolution now per colour flow matrix element: New dipole shower evolution, reduces to coherent branching upon azimuthal average and BMS evolution for large-angle soft.
[Forshaw, Holguin, Plätzer – arXiv:2003:06400]
Recoil needs to be addressed separately.
pic
n
pic
n
zpic
n + k⊥ + O(k2)
(1 − z)pic
n
O(k⊥) unbalanced momentum p¯
icn
PJ ˆ PJ
Boost to ZMF to conserve energy rescale momentum
˜ PJ
ˆ qn = (1 − zn)pic
n + k? + (q(ic nicn)
n ?
)2 1 − zn p icn 2pic
n · p icn
, ˆ pic
n = znpic n,
(q(ic
nicn)
n ?
)2 = −k2
?,
k? · pic
n = k? · p icn = 0.
Redistribute recoil globally, but per emission, inspired by Herwig’s kinematic reconstruction.
recent discussion in: [Bewick, Ferrario, Richardson, Seymour — arXiv:1904:11866]
balance only longitudinal fractions
This scheme is free of the issues encountered for local dipole recoils.
[Dasgupta, Dreyer, Hamilton, Monni, Salam — JHEP 09 (2018) 033]
Dipole branching algorithms can be supplemented by correction factors for real emission, but still lack virtual contributions beyond leading-N. Different from amplitude level evolution.
[Plätzer, Sjödahl – JHEP 1207 (2012) 042] [Plätzer, Sjödahl, Thoren – JHEP 11 (2018) 009]
Correct real emission by exact colour correlations,
Available in Herwig 7.2, including collinear contributions.
Mn+1 = X
i6=j
X
k6=i,j
4παs pi · pj Vij,k(pi, pj, pk) T2
˜ ij
T˜
k,nMnT † ˜ ij,n
Pij,k(p2
⊥, z; p ˜ ij, p˜ k) = J (p2 ⊥, z; p ˜ ij, p˜ k)Vij,k(p2 ⊥, z; p ˜ ij, p˜ k) ⇥ 1
T2
˜ ij
hMn|T ˜
ij · Tk|Mni
|Mn|2
[Holguin, Forshaw, Plätzer – arXiv:2003:06399]
Σ(L) =
∞
(Ncαs)n
n+1
Cn,m(L)
Cn,m = C(0)
n,m LCΣ
+ 1 Nc C(1)
n,m
+ 1 N 2
c
C(2)
n,m
+...
1 Γ Σ ρ1 Γ2 Γ3 ΣΓ ΣΓ2 Σ2Γ Σ2 ρΓ2 ρΓ Σ3 ρΣΓ ρΣ ρ21 ρ2Γ ρ2Σ ρ31 ρΣ2 (0 flips) × 1 × (αsN)n (1 flip) × αs × (αsN)n (0 flips) × αsN−1 × (αsN)n (t[...]t|0 flips)r (t[...]t|0 flips)r−1 t[...]s|1 flip × N−1 (t[...]t|0 flips)r (t[...]t|0 flips)r−1 t[...]s|1 flip × N−1 (t[...]t|0 flips)r−1 s[...]s|0 flips × N−2 (0 flips) × α2 s × (αsN)n (t[...]t|0 flips)r (t[...]t|0 flips)r−1 t[...]t|2 flips × 1 virtuals reals α0 s α1 s α2 s α3 s N3 N2 N1 N0 N−1 N−2 N−3 (2 flips) × α2 s × (αsN)n (t[...]t|0 flips)r−1 t[...]t|2 flipsdσn+k+1 σn+k = dΦ+18παs mn+k| Γn+k(1) |mn+k mn+k|mn+k
Colour matrix element corrections reconsidered. Cross-section unitarity is not sufficient to produce correct subleading-N virtual evolution.
Trnorm
= eTrnorm(V) +
O
s N n−2 c
(TrnormδV2 − (TrnormδV)2)
Compare to “exponent counting”
Γn(Γ) = −
n
i=j
Ti Γ Tj ωij, [Hoeche, Reichelt – arXiv:2001.11492v1]
[De Angelis, Forshaw, Plätzer — in progress]
[Plätzer – EPJ C 74 (2014) 2907]
|τ3 [τ3| D |τ2 [τ2| V |τ1 [τ1| H |σ1] σ1| V † |σ2] σ2| D† |σ3] σ3|
1 ¯ 1 2 3 ¯ 3 2 ¯ 1 1 ¯ 2 ¯ 2 1 ¯ 1 2 ¯ 2 1 ¯ 3 3 ¯ 1 2 ¯ 2
τ3 = (312) τ2 = (21) τ1 = (12) σ1 = (21) σ2 = (21) σ3 = (231)
Full evolution needs to sample colour flows differently in amplitude and conjugate amplitude, and account for differences in virtual evolution at the amplitude level.
amplitude conjugate amplitude
CVolver library implements numerical evolution in colour space
−0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.01 0.1 1 Σ(ρ) ρ singlet → gg N k composition at n = 3 total d=2 k = 5 k = 3 k = 1 k = −1 × 101 total d=0 0.05 0.1 0.15 0.2 0.25 0.3 0.01 0.1 1 Σ(ρ) ρ singlet → q¯ q N k composition at n = 3 total d=2 k = 4 k = 2 k = 0 × 101 k = −2 × 102 total d=0
preliminary
Use parton branching at amplitude level as a theoretical tool and as a framework for new algorithms in their own right. Derive parton shower algorithms covering global and non-global observables. Possibly the most we can do with incremental improvements to existing approaches. Implementation in Herwig ongoing. Formulate new parton shower algorithms featuring full-colour evolution, requires careful analysis of accuracy in subleading-N contributions.