Phenomenology of TMD Fragmentation using the Winner-Take-All axis - - PowerPoint PPT Presentation
Phenomenology of TMD Fragmentation using the Winner-Take-All axis Lorenzo Zoppi Based on ongoing work with Andreas Papaefstathiou, Duff Neill, Wouter Waalewijn Outline 1. Framework a) Ingredients: jets, hadron TMD,
Based on ongoing work with Andreas Papaefstathiou, Duff Neill, Wouter Waalewijn
a) Ingredients: jets, hadron TMD, winner-take-all axis. b) Technical details: factorization, resummation.
2 Work in progress!
“Jets live in-between theory and experiment”
hadrons are produced into collimated sprays.
dominated by soft/collinear dynamics.
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that prescribes how to build the jet from a collection of (observed/predicted) particles.
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Inclusive Jet(hadron) production e+e− → Jet(h)X pp → Jet(h)X
k⊥ = EJ sin(ϑ) We study the differential distribution of hadron transverse momentum with respect to the jet axis. EJ Jet energy Eh EJ ≡ z Hadron energy fraction zk⊥ Hadron transverse momentum
Baumgart, Leibovich, Mehen, Rothstein ’14, Bain, Dai, Hornig, Leibovich, Makris, Mehen ’16 Kaufmann, Mukherjee, Vogelsang ’15 Kang, Ringer, Vitev ’16 Dai, Kim, Leibovich ’16
Fragmentation Functions
Arleo, Fontannaz, Guillet, Nguyen ’13 Kaufmann, Mukherjee, W. Vogelsang ’15
Chien, Vitev ’15 Chang, Quin ’16 Wang, Wei, Zhang ‘16 Kang, Ringer, Vitev ’17
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~ psoft + ~ pcoll = 0
Practical realization of recoil-free axis with a simple implementation.
Run a recombination algorithm:
(Define an distance measure between particles)
Merge them;
By construction, the axis lie on a particle. Soft particles are doomed to lose when recombined with collinear ones.
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E1 > E2 ⇒ ( E = E1 + E2 ˆ n = ˆ n1
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d(h) dEJdzhdk2
⊥
(EJ, zh, ~ k⊥) = H(µ) ⊗
EJ
J(~ k⊥, R, µ) ⊗
zh d(h)(µ)
When the energy scales are separated, factorization follows (Soft Collinear Effective Theory) Integrated FFs Universal, thanks to soft insensitivity. TMD Jet functions Depend on jet defintion (radius, algorithm).
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d(h) dEJdzhdk2
⊥
(EJ, zh, ~ k⊥) = H(µ) ⊗
EJ
J(~ k⊥, R, µ) ⊗
zh d(h)(µ)
When , a further factorization takes place. d(h) dEJdzhdk2
⊥
(EJ, zh, ~ k⊥) = H(µ) ⊗
EJ
B(R, µ) ⊗
zh C(~
k⊥, µ) ⊗
zh d(h)(µ)
k⊥ ⌧ EJR Boundary functions Do not know about TMD. Matching coefficients Do not know about jet size.
10 Leading Log (LL) DGLAP equations can be derived imposing strong angular ordering This corresponds to studying the branching tree of a parton shower. Nodes of the branching tree splitting probability P(z) Final probability of hadron integrate over with z energy fraction relevant branching history WTA: axis and hadron travel together until some splitting of order . at each node, the axis follows the daughter particle with energy fraction . After the splitting, further divisions do not alter in a significative way: standard DGLAP Before the splitting, the axis has to track the hadron: the function obeys a modified DGLAP
ϑi+1 ⌧ ϑi
ϑ ' k⊥ EJ
dDi d ln µ = αs 2π Z 1
z
dz0 z0 P 0
ij( z
z0 )Dj(z0) P 0
ij(z) = P(z)θ(z − 1
2) ϑ0 = R
ϑ z > 1
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The same picture does not generalize to other recoil-free axis. E.g. broadening axis A. Larkoski, D. Neill, J. Thaler ’14 Minimizes Even in the planar case, the broadening axis always lie on a
contribute to determine which particle takes control of the axis.
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b = min
ˆ n
X
i∈Jet
2 Ei EJ sin ϑi,ˆ
n
2
Region I Region II Region III
0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x z
The non-planar case is further complicated by a non-trivial interplay of angles and energy fractions. The broadening axis does not necessarily lie on a particle. Evolution of TMD fragmenting jet functions with respect to the broadening axis cannot be cast in a DGLAP-like form.
Axis goes leftwards ⇔ zl + zL > 1/2
independent from set of Fragmentation Functions thanks to sum rule
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e+e− → Jet(hadron) Z 1 dz z d(h)(µ) = 1 resum ln k⊥ EJR
f(~ k⊥) ≡ dN=1
e+e−→Jet(h)
dk⊥ = Z Q
Emin
dEJ Z 1 dz z n He+e−(µ) ⊗ B(R, µ) ⊗ C(~ k⊥, µ) ⊗ d(h)(µ)
k⊥) = Z Q
Emin
dEJ Z 1 dz z n He+e−(µ) ⊗ G(R,~ k⊥, µ)
k⊥, µ) = J(R,~ k⊥, µ) ⊗ d(h)(µ) k⊥ ∼ EJR
The angular distributions behave remarkably as a very definite power law
Shower cutoff Jet radius causes discontinuity
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Analytic
HW7, partons Analytic
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J(~ k⊥, R, µ) = A + B ln ⇣ k⊥ EJR ⌘ ✓(EJR − k⊥)
Gradually switches down the cross section when approaching the jet boundary. The logarithmic term appears only at NLO: NLO is the first significative fixed order.
EJ,min < EJ < Q 2
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Energy recombination scheme: p = p1 + p2 Using traditional jet definitions, Sudakov double logarithms arise as a consequence of soft/ collinear overlap. Insensitivity to soft radiation removes such double logs, resulting in the definite power law.
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Quark and gluon have different jet shapes. In addition, they are not a pure power law. Linearity of the transverse momentum/angular distributions is a non-trivial fact.
0.010 0.050 0.100 0.500 0.05 0.10 0.50 1 5 10 θ
1 G (R,R) dG dθ (θ,R)
EJ > 50 GeV, Q=250, Tan R
2 =0.4
Gluon Initiated Quark Initiated
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Probe of the factorization of jet boundary: Single hadrons and whole subjets yield the same when .
Shower cut off Jet radius
ϑ > r
sensitivity to soft radiation.
show a definite power-law behavior. Results of the simulations agree with the factorization framework.
potential applications:
sensitivity to soft radiation.
show a definite power-law behavior. Results of the simulations agree with the factorization framework.
potential applications:
Jqq(x, EJR,~ k⊥, z, µ) = ↵sCF 2⇡ ⇣ 1 ⇡µ2 1
⊥/µ2 +
✓
1 2 − z 1 + z2 1 − z − 2(~ k⊥)(1 − z) n 2(1 + x2) h⇣ 1 1 − x ⌘
+ ln
⇣EJR µ ⌘ + ⇣ln(1 − x) 1 − x ⌘
+
i + 1 − x
k⊥)(1 − x)✓
2 hn 2(1 + z2) h⇣ 1 1 − z ⌘
+ ln
⇣EJRz µ ⌘ + ⇣ln(1 − z) 1 − z ⌘
+
i + 1 − z i + ✓ 1 2 − z h 21 + z2 1 − z ln
io⌘
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A(z) = δ(1 − z) +
∞
X
n=0 n
Y
I=1
h Z 1 dzi 2P(zi) Z ϑi−1
ϑmin
dϑi ϑi i δ
Y
j
zj
∞
X
n=1
1 n! lnn R ϑmin
Y
i=1
h Z 1 dzi 2P(zi) i δ
Y
j
zj
d ln µ = dA d ln R = Z 1
z
dz0 z0 2P z z0
bl = (ϕl + ϕr)zr + X
i∈L
zi(ϕi − ϕl) + X
i∈R
zi(ϕi + ϕl) br = (ϕl + ϕr)zl + X
i∈L
zi(ϕi + ϕr) + X
i∈R
zi(ϕi − ϕr) bl < br ⇔ zl + zL > zr + zR
A(z) = δ(1 − z) +
∞
X
n=0
1 n! lnn ⇣ R ϑmin ⌘ An(z) = 1 2n X
{(J1,...,Jn)} n
Y
i=1
h Z ∞ dzi 2P(zi) i δ
Y
j
zj
2 − zJn)θ(z + zJn − 1/2)