TMD splitting functions unabridged: real contributions Mirko Serino - - PowerPoint PPT Presentation

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TMD splitting functions unabridged: real contributions Mirko Serino - - PowerPoint PPT Presentation

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions TMD splitting functions unabridged: real contributions Mirko Serino Ben Gurion

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

TMD splitting functions unabridged: real contributions

Mirko Serino

Ben Gurion University of the Negev, Be’er Sheva, Israel & Institute of Nuclear Physics, Cracow, Poland Resummation, Evolution, Factorization 2017, Madrid, Spain, November 13–16 2017

Work in collaboration with Martin Hentschinski, Aleksander Kusina and Krzysztof Kutak

arXiv: 1711.04587 !

Supported by NCN grant DEC-2013/10/E/ST2/00656 & the Kreitman School of the Ben Gurion University

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions 1 Gauge invariant amplitudes in High Energy Factorisation 2 The C.F.P. approach to splitting kernels 3 Transverse Momentum Dependent splitting functions Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

What is your purpose: the TMD splitting kernels

Short term: connecting DGLAP and low-x evolutions. Long-term: Monte Carlo evolution bridging between DGLAP and BFKL Squared matrix elements for the determination of the real contributions to the splitting functions à-la Curci-Furmanski-Petronzio (CFP)

k p0 q

kµ = ypµ + kµ

⊥

qµ = xpµ + qµ

⊥ + q2 + q2

2xpn nµ p′ = q − k pµ = (1, 0, 0, 1) nµ = (1, 0, 0, −1) First three computed and consistent with DGLAP Catani and Hautmann Nucl.Phys. B427 (1994) Gituliar, Hentschinski, Kutak JHEP 1601 (2016) 181 So far, a computation of the ˜ Pgg kernel which could successfully reproduce the DGLAP AND BFKL limit not done: let me show one !

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Gauge invariant amplitudes in High Energy Factorisation

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

High Energy Factorization: more degrees of freedom

High Energy Factorization (Catani,Ciafaloni,Hautmann, 1991 / Collins,Ellis, 1991) σh1,h2→q¯

q =

  • d2k1⊥d2k2⊥

dx1 x1 dx2 x2 Fg(x1, k1⊥, µ2) Fg(x2, k2⊥, µ2) ˆ σgg (m, x1, x2, s, k1⊥, k2⊥) Fg’s: unintegrated gluon densities,

  • d2kT Fg(x, kt, µ2) = fg(x, µ2).

Non negligible transverse momentum ⇔ small x physics. Exact initial state kinematics ⇒ collinear higher order effects ab initio. Progress to connect DGLAP and low-x evolution (this approach) Momentum parameterization: kµ

1 = x1 lµ 1 + kµ 1⊥

, kµ

2 = x2 lµ 2 + kµ 2⊥

l2

i = 0,

li · ki = 0, k2

i = −k2 i ⊥,

i = 1, 2

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

High Energy Factorization in Lipatov’s approach

High Energy Factorization (Catani,Ciafaloni,Hautmann, 1991 / Collins,Ellis, 1991) σh1,h2→q¯

q =

  • d2k1⊥d2k2⊥

dx1 x1 dx2 x2 Fg(x1, k1⊥, µ2) Fg(x2, k2⊥, µ2) ˆ σgg (m, x1, x2, s, k1⊥, k2⊥) Usual tool for hard matrix elements: Lipatov’ effective action Lipatov, Nucl.Phys. B721 (1995) 111-135 Antonov, Cherednikov, Kuraev, Lipatov, Nucl.Phys. B452 (2005) 369-400 Seff = SQCD +

  • d4x
  • tr
  • (W−[v] − A−) ∂2

⊥A+ + (W+[v] − A+) ∂2 ⊥A−

  • W±[v] = − 1

g ∂±U[v±] = v± − g v± 1 ∂± v± + g2 v± 1 ∂± v± 1 ∂± v± + . . . vµ ≡ −i ta Aa

µ , gluon field

A± ≡ −i ta Aa

± ,

reggeized gluon fields U[v±] = P exp

  • − g

2 x±

−∞

dz± v±(z±, x⊥)

  • ,

x⊥ = (x±, x) Sudakov parameterisation of initial state for for HEF: kµ

1 = x1 lµ 1 + kµ 1⊥

, kµ

2 = x2 lµ 2 + kµ 2⊥ ,

l2

i = 0,

li · ki = 0, k2

i = −k2 i ⊥,

i = 1, 2

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Gauge invariant amplitudes with off-shell gluons

Kutak, Kotko, van Hameren, JHEP 1301 (2013) 078

Problem: general partonic processes must be described by gauge invariant amplitudes ⇒ ordinary Feynman rules are not enough ! THE IDEA:

  • n-shell amplitudes are gauge invariant, so off-shell gauge-invariant amplitudes could

be got by embedding them into on-shell processes... ...first result...: 1) For off-shell gluons: represent g∗ as coming from a ¯ qqg vertex, with the quarks taken to be on-shell

pA pA′ pB pB′ k2 pA pA′ pB pB′ + + pA pA′ pB pB′ k1 k2 = pA pA′ pB pB′ + · · ·

embed the scattering of the off-shell gluons in the scattering of two quark pairs carrying momenta pµ

A = kµ 1 , pµ B = kµ 2 , pµ A′ = 0, pµ B′ = 0

Assign the spinors |p1, |p1] to the A-quark and

ik / k2 → i p /1 p1·k to the A-propagators;

same for the B-quark line.

  • rdinary Feynman elsewhere and factor x1
  • −k2

⊥/2 to match to the collinear limit

Big advantage: Spinor helicity formalism

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Prescription for off-shell quarks

Kutak, Salwa, van Hameren, Phys.Lett. B727 (2013) 226-233

... and second result: 2) for off-shell quarks: represent q∗ as coming from a γA ¯ qAq vertex, where γA and ¯ qA are on shell; γA is an artificial flavour-changing neutral boson coupling only to q !

+ + = + · · · qA γA u X g g γA u qA u(k1) g qA γA u g qA γA u

embed the scattering of the quark with whatever set of particles in the scattering

  • f an auxiliary quark-photon pair, qA and γA carrying momenta

pµ

qA = kµ 1 , pµ γA = 0

Let qA-propagators of momentum k be

i p /1 p1·k and assign the spinors |p1, |p1] to

the A-quark. Assign the polarization vectors ǫµ

+ = q|γµ|p1] √ 2p1q , ǫµ − = p1|γµ|q] √ 2[p1q]

to the auxiliary photon, with q a light-like auxiliary momentum. Multiply the amplitude by x1

  • −k2

1 ⊥/2 and use ordinary Feynman rules

everywhere else.

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Novel results on Amplitudes in High Energy Factorization QCD

With growing number of legs, it is necessary to figure out practical ways to compute amplitudes efficiently. A promising possibility is the BCFW (Britto-Cachazo-Feng-Witten) recursion relation, originally discovered for on-shell QCD amplitudes and extended to off-shell gluon amplitudes in A. van Hameren, JHEP 1407 (2014) 138 A general analysis extending the modified BCFW to amplitudes with fermion pairs has been developed in A. van Hameren, MS JHEP 1507 (2015) 010 and

  • A. van Hameren, K. Kutak, MS, JHEP 1702 (2017) 009

Numerical implementation and cross-checks are done and always successful. A program exists implementing Berends-Giele recursion relation, A. van Hameren,

  • M. Bury, Comput.Phys.Commun. 196 (2015) 592-598

Loops in HEF, a proof of concept: Andreas van Hameren, arXiv:1710.07609;

In order to apply CFP, we need to switch amplitudes ⇒ open-index vertices Let us extract them !

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Open-index vertices in HEF: Γµ

g∗q∗q(q, k, p′) Simple Idea: write down Feynman diagrams and then "deconstruct" by removing the "polarisation vectors", i.e. ǫµ(p) for an on-shell particle vs. pµ for an off-shell particle

+

HEF gluon q = y p+q⊥ , HEF quark k = x p+k⊥ , radiated quark p′ = k−q p′2 = 0 A(q, k, p′) = p|γµ|p] √ 2 dµν(q) q2 p| ǫ /p+ √ 2 k / k2 γν √ 2 − γν √ 2 p / 2p · p′ ǫ /p+ √ 2 |n] = nµ dµν(q) q2 [n|

  • γν −

pν p · p′ k / k / k2 |p] = nµ dµν(q) q2 [n| Γµ

g∗q∗q(q, k, p′) |p]

dµν(q) = −gµν+ nµqν + nνqµ q2 , N.B. not invertible for n2 = 0, i.e in light-cone gauge

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Open-index vertices in HEF: Γµ

g∗q∗q(q, k, p′)

+ +

HEF gluon q = y p+q⊥ , HEF gluon k = x p+k⊥ , radiated gluon p′ = k−q p′2 = 0 A(q, k, p′) = ( √ 2) pµ1 nµ2 ǫµ3(p′) q2 k2

  • Vλκµ3(q, k, p′) dµ1 λ(q) dµ2 κ(k)

+ dµ1µ2(k) q2nµ3 n · p′ − dµ1µ2(q) k2pµ3 p · p′

  • ≡

( √ 2) pµ1 nµ2 ǫµ3(p′) q2 k2 Γµ1µ2µ3(q, k, p′) Vλκµ3(q, k, p′): ordinary 3-gluon vertex dµν not invertible in light-cone gauge ⇒ it has to be kept everywhere !

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Open-index vertices in HEF: all of them, with color

Full set of gauge invariant 3-point off-shell vertices Γµ

q∗g∗q(q, k, p′)

= i g ta dµν(k)

  • γν − nν

k · n q /

  • Γµ

g∗q∗q(q, k, p′)

= i g ta dµν(q)

  • γν − pν

p · q k /

  • Γµ

q∗q∗g(q, k, p′)

= i g ta

  • γµ −

pµ p · p′ k / + nµ n · p′ q /

  • Γµ

g∗g∗g(q, k, p′)

= i g f abc

  • Vλκµ3(q, k, p′) dµ1 λ(q) dµ2 κ(k)

+ dµ1µ2(k) q2nµ3 n · p′ − dµ1µ2(q) k2pµ3 p · p′

  • We still need a suitable procedure

to extract splitting functions from these vertices... to which we turn next !

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

The C.F.P. approach to splitting kernels

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

The fundamental result to start from

Why does the parton model work so well ? Ellis, Georgi, Machacek, Politzer, Ross, Nucl. Phys. B152 (1979) Proof of collinear factorisation based on 2 Particle Irreducible expansion of the scattering process In light-cone gauge, ALL the IR divergences come only from the convolution integral over the intermediate momenta connecting the 2PI kernels K How to put at work this result to get an evolution equation ?

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

The CFP approach to factorisation: projectors

Usual, sketchy factorization formula σ(0) = I Γ , ˜ f (x, µF ) ≡ Γ ˜ f (0)(x, µF ) C: renormalised hard matrix element; ˜ f : physical PDFs σ(0) = I(0)

  • 1 + K (0) + K (0) K (0) + . . .
  • ≡ I(0) G(0) .

G(0) ≡

  • 1 + K (0) + K (0) K (0) + . . .
  • =

1 1 − K (0) K (0): 2PI emission kernel, propagators on upper legs integrated ver

  • ddl

⇒ source of IR singularities Curci, Furmanski and Petronzio’s idea: isolate the divergent piece through a projector, K (0) = (1 − PC ) K (0) + PC K (0) and use it recursively show that G(0) = G Γ , G = 1 1 − (1 − PC ) K (0) , Γ = 1 1 − PC K , K ≡ K (0) G , I ≡ I(0) G

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

The CFP approach to factorisation: isolating the IR poles

General considerations on the tensor decomposition of the kernel lead to

k q l k l q A B α β α0 β0

Intermediate quark: A(q, l)...αα′ Ps αα′

ββ′ B...ββ′(l, k)

≡ A(q, l)...αα′ (l /)αα′ 2 (n /)ββ′ 2n · l B...ββ′(l, k) , Intermediate gluon A(q, l)...µ′ν′ Ps µ′ν′

µν

B...µν(l, k) ≡ A(q, l)...µ′ν′ dµ′ν′(l) d − 2 (−gµν) B...µν(l, k) , We can split the spin projectors into an "in" and "out" component Ps ≡ Ps

in ⊗ Ps

  • ut

The momentum Pǫ projector extracts the poles in ǫ from the

  • ddl after

setting the incoming particle on-shell

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

The CFP approach: results for the collinear projectors

Collinear spin projectors: Ps µν

g, in

= 1 d − 2

  • −gµν + lµnν + nµlν

l · n

  • ,

Ps µν

g, out = −gµν

Ps

q, in

= / l 2 , Ps

q, out =

/ n 2 n · l Important to keep in mind: The "In" projector is a d-dim average over incoming particle helicities Projectors are defined only modulo finite terms (renormalization scheme) In order to move on to TMDs, one needs gauge-invariant generalisation of the collinear vertices need some projectors with something retaining TMD dependence and reducing to the CFP projectors (modulo finite terms) in the collinear limit and squares to itself all the way through Someone did this already (more on this later): Catani and Hautmann, Nucl.Phys. B427 (1994) 475-524

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Transverse Momentum Dependent splitting functions

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

The Catani-Hautmann proposal

The modified projectors for HEF gluons: Catani and Hautmann, Nucl.Phys. B427 (1994) 475-524 quarks: Gituliar, Hentschinski, Kutak - JHEP 1601 (2016) 181 l = y p + l⊥ , ⇒    Ps µν

g, in = − lµ

⊥ lν ⊥

l2

⊥

Ps µν

g, out = −gµν

Ps

q, in = y p / 2

Ps

q, out = / n 2 n·l

CH prescription derived from analysis of heavy quark production ⇒ both numerators of gluon propagators factorize Mg∗g∗→q¯

q(k1, k2; p3, p4) = 2 x1 x2 pµ1 1

pµ2

2

  • k2

1 ⊥ k2 2 ⊥

dµ1ν1(k1) dµ2ν2(k2) ˆ Mg∗g∗→q¯

q µ1,µ2

(k1, k2; p3, p4) y pµ dµν(k) = k⊥ ν if kµ = y pµ + kµ

⊥

Ps

g ⊗ Ps g

= kµ′

⊥ kν′ ⊥

k2

⊥

(−gµν) kµ

⊥ kν ⊥

k2

⊥

(−gρσ) = − kµ′

⊥ kν′ ⊥

k2

⊥

(−gρσ) = Ps

g

kµ

⊥ kν ⊥

k2

⊥

  • φ

k⊥→0

= dµν(k = z p) d − 2

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Modifying the Catani-Hautmann projectors

Please remember: no factorisation of both dµν’s in Ag∗g∗g(q, k, p′) = √ 2 pµ1 nµ2 ǫµ3(p′) q2 k2

  • Vλκµ3(q, k, p′) dµ1

λ (q) dµ2 κ (k)+ dµ1µ2(k) q2nµ3

n · p′ −dµ1µ2(q) k2pµ3 p · p′

  • ⇒ Ps

g in = −y2 pµpν

k2

⊥

⇒ Ps

g out = −gµν + kµnν + kνnµ

k · n − k2 nµnν (k · n)2 Notice: Ps

g out enforced by P2 = P

Next to check is the collinear limit: CH projectors and ours turn out to be equivalent for k⊥ → 0 ! kµ1 Γµ1µ2µ3

g∗g∗g (k, q, p′) k⊥→0

= 0 ⇒ y pµ1 Γµ1µ2µ3

g∗g∗g (k, q, p′) k⊥→0

= k⊥µ1 Γµ1µ2µ3

g∗g∗g (k, q, p′)

Final set of projectors Ps µν

g, in

= −y2 pµpν k2

⊥

, Ps µν

g, out = −gµν + kµnν + kνnµ

k · n − k2 nµnν (k · n)2 , Ps

q, in

= y p / 2 , Ps

q, out =

/ n 2 n · l

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Extracting splitting functions with the CFP method

General kinematics kµ = ypµ + kµ

⊥ ,

qµ = xpµ + qµ

⊥ + q2 + q2

2xpn nµ , ˜ q = q − zk , IR divergent part of the kernel ˆ Kij

  • z, k2

µ2 , ǫ, αs

  • = z

dq2d2+2ǫq 2(2π)4+2ǫ Θ(µ2

F + q2)Pj, in ⊗ ˆ

K (0)

ij

(q, k) ⊗ Pi, out g2 2π δ

  • (k − q)2

Wij = Pj, in ⊗ ˆ K (0)

ij

(q, k) ⊗ Pi, out ˆ Kij

  • z, k2

µ2 , ǫ, αs

  • =

z g2 2(2π)3+2ǫ

  • d2+2ǫq

z 1 − z Wij

  • q2=− ˜

q2+z(1−z)k2 1−z

× Θ

  • µ2

F − ˜

q2 + z(1 − z)k2 1 − z

  • Transform transverse space measure, q → ˜

q ˆ Kij

  • z, k2

µ2 , ǫ, αs

  • = αs

2π z e−ǫγE µ2ǫ

  • d2+2ǫ ˜

q π1+ǫ ˜ q2 ˜ P(0)

ij

Θ

  • µ2

F − ˜

q2 + z(1 − z)k2 1 − z

  • Mirko Serino

TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

The quark TMDs

We exactly recover the result of JHEP 1601 (2016) 181 ˜ P(0)

qg

= TR

  • ˜

q2 ˜ q2 + z(1 − z)k2 2 ×

  • 1 + 4z2(1 − z)2 k2

˜ q2 + 4z(1 − z)(1 − 2z) k · ˜ q ˜ q2 − 4z(1 − z) (k · ˜ q)2 k2 ˜ q2

  • ˜

P(0)

gq

= CF

  • ˜

q2 ˜ q2 + z(1 − z)k2 2 ˜ q2 (˜ q − (1 − z)k)2 × 2 z − 2 + z + 2(1 − z)(1 + z − z2) k2 ˜ q2 + z(1 − z)2(1 + z2) k4 ˜ q4 + 4z2(1 − z)2 k2 k · ˜ q ˜ q4 + 4(1 − z)2 k · ˜ q ˜ q2 + 4z(1 − z)2 (k · ˜ q)2 ˜ q4

  • ˜

P(0)

qq

= CF

  • ˜

q2 ˜ q2 + z(1 − z)k2 2 ˜ q2 (˜ q − (1 − z)k)2 × 1 + z2 1 − z + (1 + z + 4z2 − 2z3) k2 ˜ q2 + z2(1 − z)(5 − 4z + z2) k4 ˜ q4 + 2z(1 − 2z) k · ˜ q ˜ q2 + 2z(1 − z)(1 − 2z)(2 − z) k2 k · ˜ q ˜ q4 − 4z(1 − z)2 (k · ˜ q)2 ˜ q4

  • Mirko Serino

TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

The ˜ Pgg TMD

The central new result of this work: ˜ P(0)

gg (z, ˜

q, k) = CA

  • ˜

q2 ˜ q2 + z(1 − z)k2 2 ˜ q2 (˜ q − (1 − z)k)2 ×

  • − 4z2 − 4z + 2

z(1 − z) − z(1 − z)(4z4 − 12z3 + 9z2 + 1) k4 ˜ q4 −4z(1 − z) k · ˜ q2 k2 ˜ q2 + 2(4z3 − 6z2 + 6z − 3) k · ˜ q ˜ q2 −4z(1 − z)2(3 − 5z) k · ˜ q2 ˜ q4 − (4z4 − 8z3 + 5z2 − 3z − 2) k2 ˜ q2 +8z(1 − z)2 k · ˜ q3 k2 ˜ q4 − 2z2(1 − z)(3 − 4z)(3 − 2z) k2 k · ˜ q ˜ q4

  • −ǫ CA z(1 − z) ˜

q2 k2 (2z − 1)k2 + 2k · ˜ q ˜ q2 + z(1 − z)k2 2

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Kinematical limits of the ˜ Pgg TMD: DGLAP and BFKL

  • 1. Introduce angular averaging

¯ P(0)

ij

= 1 π π dφ sin2ǫ φ ˜ P(0)

ij

and easily get the DGLAP limit lim

| k⊥|→0

¯ P(0)

gg = 2 CA

  • z

1 − z + 1 − z z + z (1 − z)

  • 2. In the high energy limit z → 0:

lim

z→0

ˆ Kgg

  • z, k2

µ2 , ǫ, αs

  • =

αsCA π(eγE µ2)ǫ d2+2ǫ ˜ p π1+ǫ Θ

  • µ2

F − (k − ˜

p)2 1 ˜ p2 = d2+2ǫq π1+ǫ Θ

  • µ2

F − q2

αsCA π(eγE µ2)ǫ 1 (q − k)2 , (1) the real part of the LO BFKL kernel

Mirko Serino TMD splitting functions unabridged: real contributions

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Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Kinematical limits of the ˜ Pgg TMD, CCFM

Last we are interested in the limit ˜ p → 0, i.e. vanishing transverse momentum of the produced gluon Slicing parameter λ: (|˜ p| < λ) infinite vs (|˜ p| > λ) finite integral For λ → 0 we find for the divergent part of the TMD kernel ˆ K div.

gg

  • z, k2

µ2 , ǫ, αs

  • =

Θ(µF − k2) αsCA π e−γE ǫ ǫΓ(1 + ǫ) λ2 µ2 ǫ 1 (1 − z)1−2ǫz = Θ(µF − k2) αsCA π e−γE ǫ ǫΓ(1 + ǫ) λ2 µ2 ǫ 1 2ǫ δ(1 − z) + 1 (1 − z)1−2ǫ

+

z

  • .

(2) Then ˜ p → 0 and get the real CCFM kernel ˆ Kgg

  • z, k2

µ2 , 0, αs

  • = z ·
  • d ˜

p2 ˜ p2 αsCa π 1 z + 1 1 − z + O ˜ p2 k2 Comes "for free"...no ad hoc assumptions made a priori

Mirko Serino TMD splitting functions unabridged: real contributions

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SLIDE 26

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Conclusions and perspectives

The method by Curci, Furmanski and Petronzio was successfully extended to the TMD case using gauge invariant vertices. The essential subtleties which prevent the Catani-Hautmann generalisation from being directly extended to the Pgg case were uncovered and worked out. All the gauge invariant real contributions to the QCD TMD splitting functions are now available. All the three limits, DGLAP, BFKL and CCFM are consistently satisfied: very non trivial consistency check ! For other early attempts, see Ciafaloni, Colferai, Salam, Stasto Phys. Lett. B587 (2004) Kwiecinski, Martin, Stasto Phys. Rev. D56 (1997) One next step: understand connection to Balitsky,Tarasov, JHEP 10 (2015) 017 The next step are the virtual corrections, for which a systematic method has recently been proposed in A. van Hameren, arXiv:1710.07609...so stay tuned !

Mirko Serino TMD splitting functions unabridged: real contributions

slide-27
SLIDE 27

Outline Gauge invariant amplitudes in High Energy Factorisation The C.F.P. approach to splitting kernels Transverse Momentum Dependent splitting functions

Conclusions and perspectives

The method by Curci, Furmanski and Petronzio was successfully extended to the TMD case using gauge invariant vertices. The essential subtleties which prevent the Catani-Hautmann generalisation from being directly extended to the Pgg case were uncovered and worked out. All the gauge invariant real contributions to the QCD TMD splitting functions are now available. All the three limits, DGLAP, BFKL and CCFM are consistently satisfied: very non trivial consistency check ! For other early attempts, see Ciafaloni, Colferai, Salam, Stasto Phys. Lett. B587 (2004) Kwiecinski, Martin, Stasto Phys. Rev. D56 (1997) One next step: understand connection to Balitsky,Tarasov, JHEP 10 (2015) 017 The next step are the virtual corrections, for which a systematic method has recently been proposed in A. van Hameren, arXiv:1710.07609...so stay tuned !

Muchas gracias por su atención y... hasta pronto !!!

Mirko Serino TMD splitting functions unabridged: real contributions