Soft and Non-global Observables Mike Seymour – / CERN TH on behalf of Rene Angeles Martinez, Matthew De Angelis, Jeffrey R. Forshaw, Simon Plätzer, Michael H. Seymour based on JHEP 05 (2018) 044 (:1802.08531) Soft evolution

• In particular, how does soft gluon emission affect subsequent soft gluon evolution? Soft Gluons and Non-global Observables

• Soft gluon emission evolution algorithm • An IR-finite reformulation • The ordering variable • Colour flow basis • (Towards) an exact numerical implementation • (“amplitude-level parton shower”) et al [14], as well as Weigert and Caron-Huot [7, 11, 30]. In Appendix B we make the can- cellation of infrared divergences explicit for observables that are inclusive below a resolution scale, and in Appendix C we explicitly calculate contributions in a fixed-order expansion. Finally, Appendix D sets up the machinery to deal with the fact that most colour bases are not orthogonal (see [31] for a notable exception).

2Thegeneralalgorithm

Our starting point is the cross section for emitting n soft gluons. At this stage we will assume that it is ok to order successive emissions in energy. This assumption is ok for processes that are insensitive to Coulomb gluon exchanges but appears not to be valid otherwise [29, 32]. We have that

=Tr(V H(Q)V† ) Tr A (µ) 0 µ,Q µ,Q ⌘ 0 µ =Tr(V D V H(Q)V† D† V† ) ⇧ d 1 µ,E1 1 E1,Q E1,Q 1µ µ,E1 d 1 Tr A (µ) d⇧ ⌘ 1 1 ⌫ µ =Tr(V D V D V H(Q)V† D† V† D† V† ) ⇧ ⇧ d 2 µ,E2 2 E2,E1 1 E1,Q E1,Q 1µ E2,E1 2⌫ µ,E2 d 1d 2 Tr A (µ) d⇧ d⇧ ⌘ 2 1 2 etc. (2.1) et al [14], as well as Weigert and Caron-Huot [7, 11, 30]. In Appendix B we make the can- wherecellationH( ofQ) infraredis the hard divergences scattering explicit matrix, forH observables= thatand, are in inclusive the eikonal below approximation, a resolution |MihM| scale, and in Appendixµ C we explicitly calculate contributions in a fixed-order expansion. µ pj Finally,Di = AppendixTj Ei Detsets al [,14 up], as the well machinery as Weigert and to deal Caron-Huot with the [7, 11 fact, 30]. that In Appendix most colourB we make bases the are can- pj qi not orthogonalXj (see [cellation·31] for of a infrared notable divergences exception). explicit for observables that are inclusive below a resolution 2✏ 3 2✏ ↵s µ dEi scale,d⌦i and in Appendix C we explicitly calculate contributions in a fixed-order expansion. Softd⇧i = gluon, emission evolution(2.2) – general algorithm 1+2✏ 2✏ 2Thegeneralalgorithm ⇡ Ei Finally,4⇡(2⇡) Appendix D sets up the machinery to deal with the fact that most colour bases are not2 orthogonal✏ b (see [31] for a notable exception). 3 2✏ 2 2✏ Our starting point↵ issµ the crossd sectionEk for emitting n softd⌦k gluons. At this⌦ stage we will Va,b = P exp ( Ti Tj) !ij(kˆ) i⇡ ij , assume that it2 is⇡ ok(2⇡ to) 2 order✏ E successive1+2✏ emissions· in energy.4⇡ This assumption 2⇡ is ok3 for where2Thegeneralalgorithm ⇥(Ea Ekn) isi

In Appendix A,weshowthatEq.(2.10) is–3– the same as the leading-logarithmic accuracy RG equations considered in [7, 11, 14, 30].

2For simplicity, we work in d =4spacetime dimensions unless otherwise stated.

–4– et al [14], as well as Weigert and Caron-Huot [7, 11, 30]. In Appendix B we make the can- cellation of infrared divergences explicit for observables that are inclusive below a resolution scale, and in Appendix C we explicitly calculate contributions in a fixed-order expansion. Finally, Appendix D sets up the machinery to deal with the fact that most colour bases are not orthogonal (see [31] for a notable exception).

2Thegeneralalgorithm

Our starting point is the cross section for emitting n soft gluons. At this stage we will assume that it is ok to order successive emissions in energy. This assumption is ok for processes that are insensitive to Coulomb gluon exchanges but appears not to be valid otherwise [29, 32]. We have that

=Tr(V H(Q)V† ) Tr A (µ) 0 µ,Q µ,Q ⌘ 0 µ =Tr(V D V H(Q)V† D† V† ) ⇧ d 1 µ,E1 1 E1,Q E1,Q 1µ µ,E1 d 1 Tr A (µ) d⇧ ⌘ 1 1 ⌫ µ =Tr(V D V D V H(Q)V† D† V† D† V† ) ⇧ ⇧ d 2 µ,E2 2 E2,E1 1 E1,Q E1,Q 1µ E2,E1 2⌫ µ,E2 d 1d 2 Tr A (µ) d⇧ d⇧ ⌘ 2 1 2 etc. (2.1) where H(Q) is the hard scattering matrix, H = and, in the eikonal approximation, |MihM| µ µ pj Di = Tj Ei , pj qi Xj · 2✏ 3 2✏ ↵s µ dEi d⌦i d⇧i = 1+2✏ 2✏ , (2.2) ⇡ Ei 4⇡(2⇡) 2✏ b 3 2✏ 2 2✏ ↵sµ dEk d⌦k ˆ ⌦ Va,b = P exp 2✏ 1+2✏ ( Ti Tj) !ij(k) i⇡ ij , 2⇡(2⇡) a Ek · ( 4⇡ 2⇡ )3 Z Xi

(pi pj) ! (kˆ)=E2 · . (2.3) ij k (p k)(p k) Soft gluon emissioni · evolutionj · – general algorithm If partons i and j are both in the initial state or they are both in the final state then ' µ e.g. ! % = & + & -(/)&1 +1 &1 ij =1otherwise ij =0" . Note', that)* " the)*, sum, over)* partons,, "' ',j)*in the definition Di is context- specific, i.e. it runs over any prior soft gluon emissions in addition1 to the1 partons1 in the hard e e !" = 2 3 2 $ 2 3 2 scattering. Likewise, the colour charge operators, Tj, and the Sudakov operators, Va,b,are in a context-specific representation of SU(3)c. The operators An satisfy the recurrence equation simple iteration gives each extra emission:

µ An(E)=VE,En Dn An 1(En) Dnµ† V† ⇥(E En) , (2.4) E,En  1Note: ⌦d =2⇡d/2/(d/2).

–3– Soft gluon emission evolution – general algorithm

dσ# = Tr '# ( dΠ# In principle, ( = 0, but if observable fully inclusive for 6 < 89, can set ( = 89.

Σ = + , d-# .#(01, … 0#) #

Measurement function

c.f. Weigert, hep-ph/0312050, Caron-Huot, arxiv:1501.03754, 1604.07417, Becher et al., arxiv:1605.02737 An IR-finite reformulation

• In general !" = ! $%, $% !"'( $% in soft $% limit

• Suppose ! ), $ = Θ+,- ) + Θ/0 ) Θ 12 < 45

• Write Γ = Γ7 + Γ87 9 dΩ2 (1 − ! ) )>?%()) = 9 dΩ2 /0 “In” region

9 dΩ2 !())>?%()) = 9 dΩ2 +,- • Keep Γ87 exponentiated, expand in Γ7 “Out” region → IR (real+virtual) cancellation, order by order → ”out-of-gap gluon” expansion / “dressed gluon” expansion An IR-finite reformulation

Σ" = &% $ &% = global contribution

Σ' = ( ,& - ,& $ &% - &% )*+ ' , /0 , % = one gluon out of gap − & . & $ &

' , % /0 % − & $ & . & . C 78 : . ? . 78 ' C ? Σ' = −2324 5 2 log + 223 24 5 3 log + ⋯ 9 0! ?@ 9 C! ?@ c.f. Khelifa-Kerfa & Delenda arXiv:1501.00475 Leading NC

• Σ" is iterative solution to BMS equation

#$%&(() dΩ1 6%161& = + 4%&(5) $%1 ( $1& ( − $%&(() #( ,-. 43 6%&

• with " 9:;< A LMN Σ" 8 = 6%&(()$%& ((), ( = log , 6DE = exp −( ∫ 4DE(5) = BC JK O=

• but contains full colour Exchange between hard partons The Ordering Variable i and j is ordered by transverse momentum wrt i and j q n 2 '". ! '#. ! ! "# $ = '". '# i (ij) qn dkT q(ij) kT Z n+1 j But exchange between hard

qn+1 parton j and gluon emitted by

i hard parton i is also ordered by transverse momentum wrt i, j

qn (ij) qn dkT q([n]j) kT Z n+1 j NB: NOTHING TO DO WITH transverse momentum of real dipoles at amplitude- squared level (c.f. Neill & Vaidya, arXiv:1803.02372, Höche & Prestel, …) qn+1

Figure 1: Illustrating how the ‘dipole transverse momentum’ serves to limit the virtual loop integration. The index [n] refers to the gluon with momentum qn. the explicit calculation reveals that the relevant dipole momentum is that of the parent of the parton which couples to the virtual gluon, i.e. parton i in the lower graph of Figure 1. The corresponding differential cross section has a similar structure to Eq. (2.1)but with

A (˜q ) H(Q)+ I (q ,Q) H(Q)+H(Q) I† (q ,Q) , 0 1 ⌘ ij 1 ij 1 Xi

eb ↵s dkT d⌦ I (a, b)= T T ! (kˆ) ✓ (k) i⇡ , (2.27) ij ⇡ i · j k 4⇡ ij ij ij Za T ⇢Z where ✓ (k)=⇥(p (p k) > 0)⇥(p (p k) > 0) restricts the regione of the angular ij j i i j integration to be the same as in the phase space integral for a real gluon with the same transverse momentum. This can be written

✓ 2pi pj 2kT ln tan < ln · , or sin ✓ > . (2.28) 2 k | | 2p p p T ⇠ i j · p

–9– The Ordering Variable • qn can be derived from one-loop Once again,emission the result in limits operator 2 and 3 can be deduced (Catani by taking & theGrazzini corresponding…) collinear limit of the leading expression in limit 1, Eq. (4.28). i (ij) The• leading cuts in limits 1–3 are presented in Fig. 12 and can be expressed in terms of qn dk or from exact calculation of all T the two colour tensors in Eq. (4.28), which are illustrated in the final column of the figure. q(ij) kT There arecontributing additional graphs, other than diagrams the ones shown, that involve the four-gluon vertex Z n+1 (Ángeles Martínez, JRF & MHS, arxiv: 1510.07998, and j but, along with the ghost graphs, these are sub-leading. In limit 1 all cuts in this figure are Ángeles Martínez PhD )

i i i i i i q n+1 1 1 + + + + i 1 2 2 j j j 2 j j j

i i 1 i 1 i 2 + + + qn (ij) qn dkT j j j j i i i i i i ([n]j) kT qn+1 2 2 Z + + + + j 2 1 1 j j j 1 j j j

i i i 2 i qn+1 1 + + + + Figure 1: Illustrating how the ‘dipole transverse momentum’ servesj to limitj the virtualj j loop integration. The index [n] refers to the gluon with momentum qn. Figure 3: Caption Figure 12. Leading graphs in limits 1–3. Their contributions are projected onto the two colour structures in the final column. the explicit calculation reveals that the relevant dipole momentum is that of the parent of leading except that with a four-gluon vertex. The non-trivial way in which these graphs the parton which couples to the virtual gluon, i.e. parton i in the lower graph of Figure 1. combine to deliver Eq. (4.28) is illustrated by considering, as an example, the graphs that The corresponding differential cross section has a similar structuregive rise to theto term Eq. with (2.1 Lorentz)but structure with i⇡ pj "1 q1 "2 · · (4.36) A0(˜q1) H(Q)+ Iij (q1,Q) H(Q)+H(Q) I† (q1,Q) , 8⇡2 p q q q ⌘ ij j · 1 1 · 2 Xi 0)⇥(p (p k) > 0) restricts the regione of the angular ij j i i j integration to be the same as in the phase space integral for a real gluon with the same transverse momentum. This can be written –19–

✓ 2pi pj 2kT ln tan < ln · , or sin ✓ > . (2.28) 2 k | | 2p p p T ⇠ i j · p

–9– Jeff Forshaw, PSR 2018, Lund Jeff Forshaw, PSR 2018, Lund Jeff Forshaw, PSR 2018, Lund Jeff Forshaw, PSR 2018, Lund Jeff Forshaw, PSR 2018, Lund Jeff Forshaw, PSR 2018, Lund Jeff Forshaw, PSR 2018, Lund dijet veto = ⇡/4, =0.1

NLC n 12 n=0 1 n =1 n =2 n =3 )

⇢ 2 ( 10 jet mass = ⇡/2, =0.1 ⌃

NLC n 12 n=0 1 n =1 4 10 CVolver 0.x n =2 n =3

10 3 10 2 10 1 ) ⇢ 2 ( 10 ⇢ ⌃

4 10 CVolver 0.x

de Angelis, Forshaw, Plätzer, preliminary 3 2 1 10 10 10 ⇢ Summary

• Soft gluon emission evolution algorithm shown to be equivalent to: • Loop expansion of Catani & Grazzini • Renormalization-based of Weigert, Caran-Huot, Becher, … • Out-of-gap / dressed gluon expansion • Important role of ordering variable at amplitude level • Transverse momentum of emitted gluon in dipole frame of its emission • Colour flow basis allows numerical implementation to arbitrary order in colour expansion • For any observable (amplitude-level parton shower…) • In progress (de Angelis, Forshaw, Plätzer…)