Soft Gluons and Non-Global Observables Mike Seymour – University of Manchester / CERN TH on Behalf of Rene Angeles Martinez, Matthew De Angelis, Jeffrey R
Total Page:16
File Type:pdf, Size:1020Kb
Soft Gluons and Non-global Observables Mike Seymour – University of Manchester / 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 (arxiv:1802.08531) Soft gluon 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 σ =Tr(V H(Q)V† ) Tr A (µ) 0 µ,Q µ,Q ⌘ 0 µ σ =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 σ =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⇡ δij , 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<j the Heaviside function.( A general observable, ⌃, can be computed) Z X Z processes that4 are insensitiveusing to Coulomb gluon exchanges but appears not to bee valid5 • Explicit1 calculation of amplitude otherwiseand [29, 32]. WeOur have starting that point is the cross section for emitting n soft gluons. At this stage we will assume that it is⌃ ok(µ to)= order successived(pσn unp(k emissions)1,k2, ,k inn) energy., This assumption is ok(2.5) for (-squared) ˆ 2 i j ··· |ℳ⟩ = = processes that! areij(k insensitive)=Ek n to Coulomb· gluon. exchanges but appears not to be(2.3) valid Z X(pi k)(pj k) σ0 =Tr(Vµ,QH(Q)V† ) Tr A0(µ)n· · otherwise [29µ,Q, 32].⌘ We have that If partons i and j are bothµ in the initial= state ord⇧ theyi Tr A aren(µ) bothun(k1 in,k2, thekn final) , state then dσ1 =Tr(Vµ,E D VE ,QH(Q)V† D† V† ) d⇧1 ··· 1 1 1 E1n,Q i=11µZ µ,E!1 µ δij =1otherwise δij =0. Note that the sumX overY partons j in the definition Di is context- Tr A1(µ) d⇧σ 1 =Tr(V H(Q)V† ) Tr A (µ) specific, i.e.⌘ it runs overwhere any the0 priorun are soft theµ,Q observable gluon emissionsµ,Q dependent in0 measurement addition functionsto the partons and the k ini are the soft hard gluon µ ⌘ ⌫ µ e dσ2 =Tr(Ve µ,E2 D2VE2,E1 D1 VE1,QH(Q)V† D1†µV† D2†⌫V† ) d⇧1d⇧2 | ⟩⟨ | scattering. Likewise,momenta. the colourσ =Tr( We charge suppressV D operators, dependenceV H(Q onT)EVj1 the†,,Q and hardD† theV partons†E Sudakov2,E) 1 ⇧ and integration operators,µ,E2 overV theira,b,are phase ℳ ℳ = & = & d 1 µ,E1 1 E1,Q E1,Q 1µ µ,E1 d 1 in a context-specificTr A2space.(µ representation) d⇧1d⇧2 of SU(3) . The operators A satisfy the recurrence ⌘ Tr A1(µ) d⇧1 c n ⌘ µ 0 equation etc. In the above, we should⌫ take theµ limit , though we will consider non-zero(2.1) values dσ2 =Tr(Vµ,E2 D2VE2,E1 D1 VE1,QH(!Q)VE† ,QD1†µVE† ,E D2†⌫Vµ,E† ) d⇧1d⇧2 in what follows. The carat on kˆ reminds us that !ij(kˆ1) is dependent2 1 only upon2 the direction Tr Aµ(µ) d⇧ d⇧ where H(Q) is theA hardnof(E the)= scattering vectorVE,Ek ninD matrix, then2 Aijnrest11H(E frame.2=n) D Thenµ† V path-ordering,E,E† and,⇥( inE the P,E ineikonaln) the, definition approximation, of V (2.4)is not ⌘ − |MihM| n a,b 1 d d/2 actuallyµ neededetc. here, because the expression in curly brackets in Eq. (2.2) is independent(2.1) µNote: ⌦ =2⇡ /pΓj(d/2). Di = Tj Ei of the, ordering variable, Ek. The cross sections in Eq. (2.1) are the general building blocks pj whereqi H(Q) is the hard scattering matrix, H = and, in the eikonal approximation, • emissionXj for· any and observable and exchange they can be used as the|M basisihM| forclosely a Monte Carlo computer code to 2✏ 3 2✏ µ ↵ µ dE generatedµ⌦ − partonic events.pj s i D =i T E , d⇧i = 1+2✏ i 2✏ j, i 2 (2.2) related− ⇡ E 4⇡(2We⇡) can− also writepj qi an evolution equation : i Xj · –3– 2✏ @bA2✏(E) 3 2✏ 3 2✏ 2 2✏ ↵ µ ↵s µdn EdEi d⌦i − d⌦ µ− ⌦ sd⇧i = E k = ΓAn(E, ) An(E)Γ† + Dkn An 1(E)ˆDnµ† E δ(E −En) , (2.6)(2.2) Va,b = P exp 2✏ − ⇡ @E1+21+2✏ ✏4⇡(2−⇡() 2T✏ i −Tj) −!ij(k) i⇡ − δij , 2−⇡(2⇡) Ei −− · 4⇡ − 2⇡ 3 − Za k i<j (Z ) where X2✏ b 3 2✏ 2 2✏ * ↵ µ E ⌦ − ⌦ 4 s ↵s d k d⌦k d k ˆ e− 5 d0 1 Va,b = P exp Γ = ( T T () Ti Tj) ! (kˆ) i⇡ δ!ij(k.) i⇡ δij , and 2−⇡(2⇡) 2✏ E1+2i✏ j − · ij 4⇡ ij − 2⇡ (2.7)3 − ⇡ Za −k ·i<j 4⇡ (Z − ) '(,* = exp . Γ i<j (p Xp )⇢Z 4 ˆ X2 i j e 5 0 1 !ij(k)=Ek · . e (2.3) ( Whenand the measurement function(p factorizes,k)(p i.e.k) un(k1,k2, kn)=u(E1, kˆ1) u(En, kˆn), i · j · ··· ··· we can define (p p ) If partons i and j are both in the initial state! (kˆ)= orE they2 arei j both. in the final state then ij nk · (2.3) (pi k)(pj k) µ δij =1otherwise δij =0. Note that the sum over partonsd·jEiin the· definition Di is context- Gn(E)= An(E) (2.8) specific, i.e. it runs over any priori softj gluon emissions in additionEi to the partons in the hard If partons and are both in the initiali=1 Z state or they are both in the final state then e e Y µ scattering. Likewise,δij the=1 colourotherwise chargeδij =0 operators,. Note that theTj sum, and over the partons Sudakovj in the operators, definition DVi isa,b context-,are and Eq. (2.6) becomes in a context-specificspecific, representation i.e. it runs over of any SU(3) priorc. soft The gluon operators emissions inA additionn satisfy to the the partons recurrence in the hard e e equation scattering. Likewise,@Gn(E) the colour charge operators, Tj,µ and the Sudakov operators,ˆ Va,b,are E = ΓGn(E) Gn(E)Γ† + Dn Gn 1(E) Dnµ† u(E,kn). (2.9) in a context-specific@E representation− − of SU(3) . The operators− A satisfy the recurrence µ c n An(E)=VE,En Dn An 1(En) Dnµ† V† ⇥(E En) , (2.4) Thisequation can be re-written as− E,En 1Note: ⌦d =2⇡d/2/Γ(d/2). µ A (E)=V D A (E ) D V† ⇥(E E ) , @Gn(E) ↵ns E,Ed⌦nk n n 1 n nµ† E,En n (2.4) E = !ij(kˆ) −Ti Tj Gn + Gn Ti Tj (2.10) 1 @Ed d/2⇡ " 4⇡ · · Note: ⌦ =2⇡ /Γ(Xi<jd/2). Z ⇣ ⌘ ˆ ˆ !ij(kn)(Ti Gn 1Tj† + Tj Gn 1Ti†) u(E,kn) + Coulomb terms.