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CERN-PH-TH/2007-261 PreprinttypesetinJHEPstyle-HYPERVERSION 11 January 2008 Parton showers with quantum interference: leading color, spin averaged 8 0 Zoltán Nagy 0 2 Theory Division, CERN n CH-1211 Geneva 23, Switzerland a E-mail: [email protected] J 2 Davison E. Soper 1 Institute of Theoretical Science ] h University of Oregon p Eugene, OR 97403-5203, USA - p E-mail: [email protected] e h [ Abstract: We have previously described a mathematical formulation for a parton shower 1 v based on the approximation of strongly ordered virtualities of successive parton splittings. 7 Quantum interference, including interference among different color and spin states, is in- 1 cluded. In this paper, we add the further approximations of taking only the leading color 9 1 limit and averaging over spins, as is common in parton shower Monte Carlo event gener- . 1 ators. Soft gluon interference effects remain with this approximation. We find that the 0 leading color, spin averaged shower in our formalism is similar to that in other shower 8 0 formulations. We discuss some of the differences. : v Xi Keywords: perturbative QCD, parton shower. r a Contents 1. Introduction 1 2. Direct spin-averaged splitting functions 2 2.1 Final state q → q+g splitting 6 2.2 Initial state q → q+g splitting 8 2.3 Final state g → g+g splitting 11 2.4 Initial state g → g+g splitting 14 2.5 Other cases 15 3. Interference diagrams 16 4. Spin-averaged interference graph splitting functions 18 5. The leading color limit 21 6. Evolution equation 22 7. Other approaches 27 7.1 Dipole shower 27 7.2 Antenna shower 28 7.3 Angular ordering approximation 30 8. Conclusions 31 A. The remaining splitting functions 32 1. Introduction In Ref. [1], we presented a formalism for a mathematical representation of a parton shower thatincorporatesinterferenceinbothspinandcolor. Inthispaper, weanalyzethisformal- ism in the approximation that we average over parton spins at each step and keep only the leading contributions in an expansion in powers of 1/N2, where N = 3 is the number of c c colors.1 Our interest is to elucidate the structure of the full shower formulation of Ref. [1] by examining what happens when the spin-averaged and leading color approximations are imposed. We also anticipate that the approximate shower may be of use in implementing successively better approximations to the full shower including spin and color. 1More precisely, we average over the spins of incoming partons at each step and sum over the spins of the outgoing partons. – 1 – Our main focus is on the splitting functions that would be used to generate the shower in the spin averaged approximation (which is a customary approximation in current parton shower event generators). In our formalism, there are two sorts of splitting functions. The direct splitting functions correspond to the squared amplitude for a parton l to split into daughter partons that, in our notation, carry labels l and m+1, where m+1 is the total number of final state partons after the splitting. In this paper, we use the spin dependent splittingfunctionsfromRef.[1]andsimplyaverageoverthespinsofthemotherpartonand sum over the spins of the daughter partons. We analyze some of the important properties of these functions. We also need interference splitting functions. These correspond to the interference between the amplitude for a parton l to split into partons with labels l and m + 1 and the amplitude for another parton k to split into partons with labels k and m+1. These functions generate leading singularities when parton m+1 is a soft gluon. WeimprovethespecificationsofRef.[1]forthisbydefiningausefulformforcertainweight functionsA andA thatwereassignedthedefaultvalues1/2inRef.[1]. Wewillseethat lk kl with the improved form for A , the total splitting probabilities acquire useful properties ij in the soft gluon limit. Wewillseethatwhenwemakethespin-averagedandleadingcolorapproximations,the parton shower formalism of Ref. [1] amounts to something quite similar to standard parton showereventgenerators. Onesignificantpointincommonisthatthesplittingfunctionsare positive. One difference with some standard event generators is that an angular ordering approximationisnotneededbecausethecoherenceeffectsthatleadtoangularorderingare built into the formalism from the beginning, both for initial state and final state splittings. This coherence feature is a natural consequence of a dipole based shower, as in the final state showers of Ariadne [2] and the k option of Pythia [3] or the showers [4, 5] based T on the Catani-Seymour dipole splitting formalism [6]. Additionally, our formalism differs from others in its splitting functions and its momentum mappings. 2. Direct spin-averaged splitting functions We begin with the splitting functions that correspond to the amplitude for a parton to split times the complex conjugate amplitude for that same parton to split. We follow the notation of Ref. [1]. Before the splitting, there are partons that carry the labels {a,b,1,...,m}, where a and b are the labels of the initial state partons. The momenta and flavors of these partons are denoted by {p,f} = {p ,f ;...;p ,f }. The flavors m a a m m are {g,u,u¯,d,...}, with the initial state flavors f and f denoting the flavors coming out a b of the hard interaction and thus the opposite of the flavors entering the hard interaction. We let l be the label of the parton that splits. After the splitting, there are m+1 final state partons. The momenta and flavors of the partons are {pˆ,fˆ} . We use the label l m+1 for one of the daughter partons and the label m+1 for the other daughter parton.2 The partons that do not split keep their labels. However, they donate some of their momenta to the daughter partons so that the daughter partons can be on shell. Thus pˆ (cid:54)= p in i i 2For a final state q → qg splitting, we use m+1 for the label of the gluon. For a final state g → qq¯ splitting, we use m+1 for the label of the q¯. – 2 – general for a spectator parton. The momenta and flavors after the splitting, {pˆ,fˆ} , are m+1 determined by the momenta and flavors before the splitting, {p,f} and variables {ζ ,ζ } m p f that describe the splitting.3 A certain mapping {pˆ,fˆ} = R ({p,f} ,{ζ ,ζ }) (2.1) m+1 l m p f defined in Ref. [1] gives the relation. Figure 1: Illustration Eq. (2.2) for a qqg final state splitting. The small filled circle represents the splitting amplitude v . The mother parton has momentum pˆ +pˆ , but in the amplitude l l m+1 (cid:12) (cid:11) (cid:12)M({p,f}m) , this off-shell momentum is approximated as an on-shell momentum pl. The splitting functions in Ref. [1] are based on spin dependent splitting amplitudes v . l (cid:12) (cid:11) One starts with the amplitude (cid:12)M({p,f}m) to have m partons. The amplitude is a vector (cid:12) (cid:11) in spin⊗color space. After splitting parton l, we have a new amplitude (cid:12)Ml({p,f}m+1) of the form illustrated in Fig. 1 (cid:12)(cid:12)Ml({pˆ,fˆ}m+1)(cid:11) = t†l(fl → fˆl +fˆm+1)Vl†({pˆ,fˆ}m+1)(cid:12)(cid:12)M({p,f}m)(cid:11) . (2.2) Here t† is an operator on the color space that simply inserts the daughter partons with l the correct color structure. The factor V† is a function the momenta and flavors and l is an operator on the spin space. It leaves the spins of the partons other than parton l undisturbed and multiplies by a function v that depends on the mother spin and the l daughter spins:   (cid:10){sˆ}m+1(cid:12)(cid:12)Vl†({pˆ,fˆ}m+1)(cid:12)(cid:12){s}m(cid:11) =  (cid:89) δsˆj,sjvl({pˆ,fˆ}m+1,sˆm+1,sˆl,sl) . (2.3) j∈/{l,m+1} Thus the splitting is defined by the splitting amplitudes v , which are derived from the l QCD vertices. 3Whenagluonsplits,ζ determineswhetherthedaughtersarea(g,g)pair,a(u,u¯)pair,etc. InRef.[1], f we defined the splitting variables ζ in a rather abstract way, but one could imagine using for ζ the p p virtuality of the daughter parton pair, a momentum fraction, and an azimuthal angle. – 3 – We can illustrate this for the case of a final state q → q +g splitting, for which we define v ({pˆ,fˆ} ,sˆ ,sˆ,s ) = l m+1 m+1 l l √4πα ε (pˆ ,sˆ ;Qˆ)∗ U(pˆl,sˆl)γµ[p/ˆl +p/ˆm+1+m(fl)]n/lU(pl,sl) . (2.4) s µ m+1 m+1 [(pˆ +pˆ )2−m2(f )]2p ·n l m+1 l l l There are spinors for the initial and final quarks. There is a polarization vector for the daughter gluon, defined in timelike axial gauge so that pˆ ·ε = Qˆ·ε = 0. Here Qˆ is the m+1 total momentum of the final state partons, which is the same before and after the splitting. There is a vertex γµ for the qqg interaction. There is a propagator for the off shell quark with momentum pˆ +pˆ . So far, this is exact. Finally, there is an approximation that l m+1 applies when the splitting is nearly collinear or soft. We approximate pˆ +pˆ by p in l m+1 l the hard interaction and insert a projection n/ /2p ·n onto the “good” components of the l l l Dirac field. This projection uses a lightlike vector n that lies in the plane of Qˆ and p , l l Q2 n = Q− p . (2.5) l (cid:112) l Q·p + (Q·p )2−Q2m2(f ) l l l Figure 2: Illustration of how v times v∗ appears in the calculation of the approximate matrix l l element (cid:12)(cid:12)Ml({pˆ,fˆ}m+1)(cid:11) times its complex conjugate, (cid:10)Ml({pˆ,fˆ}m+1)(cid:12)(cid:12). If we average over spins, (cid:12) (cid:11) (cid:10) (cid:12) we need to multiply (cid:12)M({p,f}m) times M({p,f}m)(cid:12) by Wll, Eq. (2.8). With one exception, the direct splitting functions in Ref. [1] are products of a splitting amplitude, v , times a complex conjugate splitting amplitude, v∗, l l v ({pˆ,fˆ} ,sˆ ,sˆ,s ) v ({pˆ,fˆ} ,sˆ(cid:48) ,sˆ(cid:48),s(cid:48))∗ . (2.6) l m+1 m+1 l l l m+1 m+1 l l The calculation of (cid:12)(cid:12)Ml({pˆ,fˆ}m+1)(cid:11) times (cid:10)Ml({pˆ,fˆ}m+1)(cid:12)(cid:12) using vl × vl∗ is illustrated in Fig. 2. In this calculation, in general, we have to keep track of two spin indices, s and s(cid:48) for each parton in order to describe quantum interference in the spin space. However, in this paper we make an approximation. We set s(cid:48) = s for each parton, sum over the daughter – 4 – parton spins and average over the mother parton spins. Thus we use a splitting function4 1 (cid:88) W = |v ({pˆ,fˆ} ,sˆ ,sˆ,s )|2 ll l m+1 m+1 l l 2 sˆl,sˆm+1,sl foranyflavorcombinationallowedwithourconventionsforassigningthelabelsl andm+1 except for a final state g → g+g splitting, for which we do something slightly different because the two gluons are identical. We make manifest the definition of which flavor combinations are allowed by defining 1/2 , l ∈ {1,...,m}, fˆ = fˆ = g  l m+1  1 , l ∈ {1,...,m}, fˆ (cid:54)= g,fˆ = g  l m+1   0 , l ∈ {1,...,m}, fˆ = g,fˆ (cid:54)= g S ({fˆ} ) = l m+1 . (2.7) l m+1 1 , l ∈ {1,...,m}, fˆ = q,fˆ = q¯  l m+1  0 , l ∈ {1,...,m}, fˆl = q¯,fˆm+1 = q   1 , l ∈ {a,b} This is 1 for the allowed combinations, 0 otherwise, with a statistical factor 1/2 for a final state g → g+g splitting. The complete definition of W is then ll (cid:26) 1 (cid:88) W = S ({fˆ} ) |v ({pˆ,fˆ} ,sˆ ,sˆ,s )|2 ll l m+1 l m+1 m+1 l l 2 sˆl,sˆm+1,sl +θ(l ∈ {1,...,m},fˆ = fˆ = g) (2.8) l m (cid:27) (cid:104) (cid:105) × |v ({pˆ,fˆ} ,sˆ ,sˆ,s )|2−|v ({pˆ,fˆ} ,sˆ ,sˆ,s )|2 . 2,l m+1 m+1 l l 3,l m+1 m+1 l l The second term applies for a final state g → g+g splitting and is arranged to keep the total splitting probability the same but associate the leading soft gluon singularity with gluon m+1 rather than gluon l. The functions v and v are defined in Sec. 2.3. 2,l 3,l Theformofthesplittingamplitudev dependsonthetypeofpartonsthatareinvolved. l However, thereisacommonresultinthelimitpˆ → 0wheneverpartonm+1isagluon. m+1 In this limit, v is given by the eikonal approximation, l √ ε(pˆ ,sˆ ;Q)∗·pˆ veikonal({pˆ,fˆ} ,sˆ ,sˆ,s ) = 4πα δ m+1 m+1 l . (2.9) l m+1 m+1 l l s sˆl,sl pˆ ·pˆ m+1 l The soft gluon limit of W is then ll pˆ ·D(pˆ ;Qˆ)·pˆ Weikonal = 4πα l m+1 l . (2.10) ll s (pˆ ·pˆ)2 m+1 l Here D is the sum over sˆ of ε ε∗, µν m+1 µ ν pˆµ Qˆν +Qˆµpˆν Qˆ2pˆµ pˆν D (pˆ ,Qˆ) = −g + m+1 m+1 − m+1 m+1 . (2.11) µν m+1 µν pˆ ·Qˆ (pˆ ·Qˆ)2 m+1 m+1 4The function W here is the same as w in Ref. [1]. ll ll – 5 – eikonal ThefunctionW anditsapproximateformW givethedependenceofthesplitting ll ll operator on momentum and spin for a given set of parton flavors. The partons also carry color. In Ref. [1] there is a separate factor that gives the color dependence. This factor is an operator on the color space that we can call t†⊗t , where t† is the operator in Eq. (2.2), l l l which inserts the proper color matrix into the amplitude, and t inserts the proper color l matrixintothecomplexconjugateamplitude.5 Sofar, wedonotmakeanyapproximations withrespecttocolor. InSec.5, wewillmaketheapproximationofkeepingonlytheleading color conributions. We now turn to a more detailed discussion of W for particular cases. ll 2.1 Final state q → q+g splitting Let us look at W for a final state q → q+g splitting, ll 4πα 1 W = s D (pˆ ,Qˆ) ll 2(p ·n )2 (2pˆ ·pˆ )2 µν m+1 l l l m+1 (2.12) 1 (cid:104) (cid:105) × Tr (p/ˆ +m)γµ(p/ˆ +p/ˆ +m)n/ (p/ +m)n/ (p/ˆ +p/ˆ +m)γν . 4 l l m+1 l l l l m+1 Here m = m(f ) is the quark mass, the lightlike vector n is given by Eq. (2.5), and D is l l µν given by Eq. (2.11). It will be convenient to examine the dimensionless function pˆ ·pˆ l m+1 F ≡ W . (2.13) ll 4πα s The limiting behavior of F as the gluon m+1 becomes soft, pˆ → 0, is simple. Then m+1 the eikonal approximation applies and we obtain from Eq. (2.10) pˆ ·D(pˆ ,Qˆ)·pˆ l m+1 l F = . (2.14) eikonal pˆ ·pˆ l m+1 The full behavior of F is more complicated, pˆ ·n m+1 l F = [1+h(y,a ,b )]F + . (2.15) l l eikonal p ·n l l Here 1+y+λr 2a y l l h(y,a ,b ) = + −1 , (2.16) l l 1+r λr (1+r ) l l l where 2pˆ ·pˆ l m+1 y = , 2p ·Qˆ l Qˆ2 a = , l 2p ·Qˆ l m2 (2.17) b = , l 2p ·Qˆ l (cid:112) r = 1−4a b , l l l (cid:112) (1+y)2−4a (y+b ) l l λ = . r l 5InRef.[1],wewritet†(f →f +g)fortheoperatorthatweherecalljustt† andwedenotetheoperator l l l l t†⊗t by G(l,l). l l – 6 – The eikonal approximation to F will turn out to be significant in our analysis when we incorporate the effect of soft-gluon interference graphs. We will find that it is of some importanceforthenumericalgoodbehaviorofthesplittingfunctionsincludinginterference that F −F ≥ 0 . (2.18) eikonal To see that this property holds we note first that pˆ ·n /p ·n is non-negative. Remark- m+1 l l l ably, h(y,a ,b ) ≥ 0 also. To prove this, we first note that l l (λ−1+y)2+4y h(y,a ,0) = , (2.19) l 4λ so that h(y,a ,0) ≥ 0. Then we show that h(y,a ,b )−h(y,a ,0) ≥ 0 by simply making l l l l plots of this function. This establishes the positivity property Eq. (2.18). WenowexamineF furtherundertheassumptionthatm = 0. WewriteF asafunction of the dimensionless virtuality variable y, and a momentum fraction6 pˆ ·n m+1 l z = . (2.20) (pˆ +pˆ)·n m+1 l l It is also convenient to use an auxiliary momentum fraction variable pˆ ·Qˆ λ 2a y m+1 l x = = z+ , (2.21) (pˆ +pˆ)·Qˆ 1+y (1+y)(1+y+λ) m+1 l (cid:112) where, for m = 0, λ = (1+y)2−4a y. Using these variables, l 1−x 2a y l F = 2 − , (2.22) eikonal x x2(1+y)2 and (cid:20) (λ−1+y)2+4y(cid:21) 1 F = 1+ F + z[1+y+λ] . (2.23) eikonal 4λ 2 As y → 0, F must turn into the Altarelli-Parisi function for this splitting, 1+(1−z)2 F (z) = . (2.24) AP z Indeed, the derivation given above is one way to derive the Altarelli-Parisi function. We illustrate how F(z,y,a ,b ) at b = 0 approaches F (z) in Fig. 3. l l l AP 6Note that there are many different ways to define a momentum fraction variable. The value of the splitting function for a given choice of daughter parton momenta does not depend on the momentum fraction variable that one uses to label these momenta. We have taken a simple definition of z in order to display results in a graph. – 7 – Figure 3: The spin averaged splitting function F defined in Eq. (2.23) for a final state q →q+g splitting, plotted versus the momentum fraction z of the gluon, as defined in Eq. (2.20). The quark is taken to be massless and we set a = 4. The four curves are for, from bottom to top, l y =0.03,0.01,0.001,0. For y =0, the result is the Altarelli-Parisi function, [1+(1−z)2]/z. 2.2 Initial state q → q+g splitting Here we consider an initial state q → q+g splitting, as illustrated in Fig. 4. For notational convenience, we let it be parton “a” that splits, so l = a. We allow both parton “a” and parton “b” to have masses, m ≡ m(f ) and m ≡ m(f ). One could, of course, choose a a b b thesemassestobezero. Partonm+1isa(massless)gluon. Theshowerevolutionforinitial state particles runs backwards in physical time. Parton “a”, which carries momentum p a into the hard interaction, splits into the final state gluon with momentum p and an m+1 initial state parton that carries momentum pˆ into the splitting. For a nearly collinear a splitting, p ≈ pˆ −pˆ . In physical time, it is the initial state parton with momentum a a m+1 pˆ that splits. a Figure 4: An initial state q →q+g splitting. – 8 – Following Ref. [1], we define the kinematics using lightlike vectors p and p that are A B lightlikeapproximationstothemomentaofhadronsAandB,respectively,with2p ·p = s. A B The momenta of the partons that enter the hard scattering, p and p , are defined using a b momentum fractions η and η . After the splitting, the momentum fractions are ηˆ and ηˆ . a b a b Because parton “a” splits, ηˆ (cid:54)= η . However, with our kinematics, the momentum fraction a a of parton “b” remains unchanged: ηˆ = η . The initial state parton momenta are defined b b to be m2 p = η p + a p , a a A B η s a m2 p = η p + b p , (2.25) b b B A η s b m2 pˆ = ηˆ p + a p . a a A B ηˆ s a Themomentumofthefinalstatespectatorpartonschangesinordertomakesomemomen- tum available to allow both p and pˆ to be on shell with zero transverse momenta. We a a denote the total momentum of the final state partons before the splitting by Q = p +p a b and after the splitting by Qˆ = pˆ +p . In the splitting function, we make use of a lightlike a b vector n in the plane of p and Q. With a convenient choice of normalization, n = p . a a a B In the following formulas, it will be convenient to define P = pˆ −pˆ . a a m+1 Using the definition Eq. (2.8) with the splitting amplitudes v from Table 1 of Ref. [1], a we write the spin averaged splitting function as 4πα 1 W = s D (pˆ ,Qˆ) aa 2(p ·p )2 (2pˆ ·pˆ )2 µν m+1 a B a m+1 (2.26) 1 (cid:104) (cid:105) × 4Tr (p/â+ma)γµ(P/a+ma)n/a(p/a+ma)n/a(P/a+ma)γν . Here D is given in Eq. (2.11). µν The spin averaged splitting function can be simplified. Let us adopt the notation m2 m2 Φ = a , Φ = b . (2.27) a b ηˆ η s ηˆ η s a b a b Then the result can conveniently be displayed in terms of the dimensionless function pˆ ·pˆ a m+1 F ≡ W . (2.28) aa 4πα s The result is pˆ ·p Φ Φ (ηˆ −η )2 F = F + m+1 B + a b a a F . (2.29) eikonal p ·p (1−Φ Φ )η2 eikonal a B a b a Here the first term is the simple eikonal approximation for soft gluon emission, pˆ ·D(pˆ ,Qˆ)·pˆ a m+1 a F = . (2.30) eikonal pˆ ·pˆ a m+1 – 9 –