Parton showers from the dipole formalism
aa r X i v : . [ h e p - ph ] S e p MZ-TH/07-14
Parton showers from the dipole formalism
Michael Dinsdale, Marko Ternick and Stefan Weinzierl
Institut für Physik, Universität Mainz,D - 55099 Mainz, Germany
Abstract
We present an implementation of a parton shower algorithm for hadron colliders and electron-positron colliders based on the dipole factorisation formulæ. The algorithm treats initial-statepartons on equal footing with final-state partons. We implemented the algorithm for masslessand massive partons.
Introduction
Event-generators like Pythia [1, 2], Herwig [3, 4] or Sherpa [5] are a standard tool in high energyparticle physics. In these tools the physics of particle collisions is modelled by a simulation withdifferent stages – hard scattering, parton showering, hadronisation – to name the most importantones. The hard scattering process is calculable in perturbation theory. The same holds – intheory at least – for the parton showering process, the relevant scales are still large enough forperturbation theory to be applicable. In practise however, one is forced into approximations dueto the large parton multiplicities. These approximations are derived from the behaviour of thematrix elements in singular regions. The matrix elements become singular in phase space regionscorresponding to the emission of collinear or soft particles. The first showering algorithms startedfrom the collinear factorisation of the matrix elements and approximated colour interferenceeffects through angular ordering [6,7]. An exception is the algorithm implemented in Ariadne [8–12], which is based on a dipole cascade picture. Most shower algorithms are in the collinear limitaccurate to the leading-logarithmic approximation. Extensions to the next-to-leading logarithmicapproximation have been studied in [13–16].Recent years have witnessed significant developments related to shower algorithms, includ-ing procedures to match parton showers to fixed-order tree-level matrix elements [17–21] andmethods to combine parton showers with next-to-leading order matrix elements [22–45]. Theshower algorithms in Pythia, Herwig and Ariadne have been improved [46–48] and new pro-grams like the shower module Apacic++ [49, 50] of Sherpa have become available. Other im-provements include the study of uncertainties in parton showers [51–53], as well as showers inthe context of the soft-collinear effective theory [54].Of particular importance is the matching of parton showers with next-to-leading order ma-trix elements. The pioneering project MC@NLO [31, 55–58] used an existing shower program(Herwig) and adapted the NLO calculation to the shower algorithm, at the expense of sacrificingthe correctness in certain soft limits. It is clear that a better but more labour-intensive approachwould adapt the shower algorithm to NLO calculations. Nowadays in NLO computations thedipole subtraction method [59–63] is widely used. Nagy and Soper [35, 36] proposed to build ashower algorithm from the dipole subtraction terms.In this paper we report on an implementation of a shower algorithm based on the dipoleformalism as suggested by Nagy and Soper. We take the dipole splitting functions as the splittingfunctions which generate the parton shower. In the dipole formalism, a dipole consists of anemitter-spectator pair, which emits a third particle, soft or collinear to the emitter. The formalismtreats initial- and final-state partons on the same footing. In contrast to other shower algorithms,no distinction is made between final- and initial-state showers. The only difference betweeninitial- and final-state particles occurs in the kinematics. In the implementation we have the fourcases final-final, final-initial, initial-final and initial-initial corresponding to the possibilities ofthe particles of the emitter-spectator-pair to be in the initial- or final-state. All four cases areincluded, therefore the shower can be used for hadron colliders and electron-positron colliders.We implemented the shower for massless and massive partons. Initial-state partons are howeveralways assumed to be massless. We use spin-averaged dipole splitting functions. The showeralgorithm is correct in the leading-colour approximation. As the evolution variable we use the2ransverse momentum in the massless case, and a variable suggested in [47, 64] for the massivecase. The variable for the massive case reduces to the transverse momentum in the masslesslimit. Schumann and Krauss report on a similar but separate implementation of a parton showeralgorithm based on the dipole formalism [65].This paper is organised as follows: In section 2 we review basic facts about the colour de-composition of QCD amplitudes and the dipole formalism. In section 3 we discuss the showeralgorithm. In section 4 we present numerical results from the parton shower simulation program.Finally, section 5 contains the summary. Technical details can be found in the appendix. Ap-pendix A discusses the case of a massless final-state emitter and a massless final-state spectatorin detail. Appendix B describes the construction of the four-momenta of the ( n + ) -particle statein all cases. This appendix is also useful in the context of a phase-space generator for the realemission part of NLO computations. In this section we briefly review the colour decomposition of QCD amplitudes and the dipoleformalism.
In this paper we use the normalisationTr T a T b = d ab (1)for the colour matrices. Amplitudes in QCD may be decomposed into group-theoretical factors(carrying the colour structures) multiplied by kinematic functions called partial amplitudes [66–70]. The partial amplitudes are gauge-invariant objects. In the pure gluonic case tree levelamplitudes with n external gluons may be written in the form A n ( , , ..., n ) = (cid:18) g √ (cid:19) n − (cid:229) s ∈ S n / Z n d i s j s d i s j s ... d i s n j s A n ( s , ..., s n ) , (2)where the sum is over all non-cyclic permutations of the external gluon legs. The quantities A n ( s , ..., s n ) , called the partial amplitudes, contain the kinematic information. They are colour-ordered, e.g. only diagrams with a particular cyclic ordering of the gluons contribute. The choiceof the basis for the colour structures is not unique, and several proposals for bases can be foundin the literature [71, 72]. Here we use the “colour-flow decomposition” [72, 73]. This basisis obtained by replacing every contraction over an index in the adjoint representation by twocontractions over indices i and j in the fundamental representation: V a E a = V a d ab E b = V a (cid:16) T ai j T bji (cid:17) E b = (cid:16) √ T ai j V a (cid:17) (cid:16) √ T bji E b (cid:17) . (3)3s a further example we give the colour decomposition for a tree amplitude with a pair of quarks: A n + ( q , , , ..., n , ¯ q ) = (cid:18) g √ (cid:19) n (cid:229) S n d i q j s d i s j s ... d i s n j ¯ q A n + ( q , s , s , ..., s n , ¯ q ) , (4)where the sum is over all permutations of the gluon legs. The tree amplitude with a pair ofquarks, n gluons and an additional lepton pair has the same colour structure as in eq. (4). Insquaring these amplitudes a colour projector d ¯ ii d j ¯ j − N c d ¯ i ¯ j d ji (5)has to applied to each gluon. In these examples we have two basic colour structures, a colourcluster described by the “closed string” d i s j s d i s j s ... d i s n j s (6)and a colour cluster corresponding to the “open string” d i q j s d i s j s ... d i s n j ¯ q . (7)Born amplitudes with additional pairs of quarks have a decomposition in colour factors, whichare products of the two basic colour clusters above. The colour factors in eq. (2) and eq. (4) areorthogonal to leading order in 1 / N c . The starting point for the calculation of an observable O in hadron-hadron collisions in perturba-tion theory is the following formula: h O i = Z dx f ( x ) Z dx f ( x ) K ( ˆ s ) ( J + ) ( J + ) n n Z d f n ( p , p ; p , ..., p n + ) O ( p , ..., p n + ) | A n + | . (8)This equation gives the contribution from the n -parton final state. The two incoming particles arelabelled p and p , while p to p n + denote the final state particles. f ( x ) gives the probabilityof finding a parton a with momentum fraction x inside the parent hadron h . A sum over allpossible partons a is understood implicitly. 2 K ( s ) is the flux factor, 1 / ( J + ) and 1 / ( J + ) correspond to an averaging over the initial helicities and n and n are the number of colourdegrees of the initial state particles. d f n is the phase space measure for n final state particles,including (if appropriate) the identical particle factors. The matrix element | A n + | is calculatedperturbatively. At leading and next-to-leading order one has the following contributions: h O i LO = Z n O n d s B , h O i NLO = Z n + O n + d s R + Z n O n d s V + Z n O n d s C . (9)4ere we used a rather condensed notation. d s B denotes the Born contribution, while d s R de-notes the real emission contribution, whose matrix element is given by the square of the Bornamplitudes with ( n + ) partons | A ( ) n + | . d s V gives the virtual contribution, whose matrix el-ement is given by the interference term of the one-loop amplitude A ( ) n + with ( n + ) partonswith the corresponding Born amplitude A ( ) n + . d s C denotes a collinear subtraction term, whichsubtracts the initial-state collinear singularities. Within the subtraction method one constructs anapproximation term d s A with the same singularity structure as d s R . The NLO contribution isrewritten as h O i NLO = Z n + (cid:16) O n + d s R − O n d s A (cid:17) + Z n (cid:16) O n d s V + O n d s C + O n d s A (cid:17) , (10)such that the terms inside the two brackets are separately finite. The matrix element correspond-ing to the approximation term d s A is given as a sum over dipoles [59–63]: (cid:229) pairs i , j (cid:229) k = i , j D i j , k + " (cid:229) pairs i , j D ai j + (cid:229) j (cid:229) k = j D a jk + (cid:229) j D a j , b + ( a ↔ b ) . (11)In eq. (11) the labels i , j and k denote final-state particles, while a and b denote initial-stateparticles. The first term describes dipoles where both the emitter and the spectator are in thefinal-state. D ai j denotes a dipole where the emitter is in the final-state, while the spectator is inthe initial-state. The reverse situation is denoted by D a jk : Here the emitter is in the initial-stateand the spectator is in the final-state. Finally, D a j , b denotes a dipole where both the emitter andthe spectator are in the initial-state. The full complexity is only needed for hadron colliders;for electron-positron annihilation the subtraction terms inside the square bracket are absent. Thedipole subtraction terms for a final-state emitter-spectator pair have the following form: D i j , k = A ( ) ∗ n + (cid:0) p , ..., ˜ p ( i j ) , ..., ˜ p k , ... (cid:1) ( − T k · T i j ) T i j V i j , k p i · p j A ( ) n + (cid:0) p , ..., ˜ p ( i j ) , ..., ˜ p k , ... (cid:1) . (12)The structure of the dipole subtraction terms with initial-state partons is similar. Here T i denotesthe colour charge operator for parton i and V i j , k is a matrix in the spin space of the emitterparton ( i j ) . In general, the operators T i lead to colour correlations, while the V i j , k ’s lead to spincorrelations. The colour charge operators T i for a quark, gluon and antiquark in the final stateare quark : A ∗ ( ... q i ... ) (cid:0) T ai j (cid:1) A (cid:0) ... q j ... (cid:1) , gluon : A ∗ ( ... g c ... ) (cid:16) i f cab (cid:17) A (cid:16) ... g b ... (cid:17) , antiquark : A ∗ ( ... ¯ q i ... ) (cid:0) − T aji (cid:1) A (cid:0) ... ¯ q j ... (cid:1) . (13)The corresponding colour charge operators for a quark, gluon and antiquark in the initial stateare quark : A ∗ ( ... ¯ q i ... ) (cid:0) − T aji (cid:1) A (cid:0) ... ¯ q j ... (cid:1) , A ∗ ( ... g c ... ) (cid:16) i f cab (cid:17) A (cid:16) ... g b ... (cid:17) , antiquark : A ∗ ( ... q i ... ) (cid:0) T ai j (cid:1) A (cid:0) ... q j ... (cid:1) . (14)In the amplitude an incoming quark is denoted as an outgoing antiquark and vice versa.In this paper we neglect spin-correlations and work to leading-order in 1 / N c . Therefore wereplace the splitting functions V i j , k by the spin-averaged splitting functions: V i j , k → h V i j , k i (15)In the leading-colour approximation we only have to take into account emitter-spectator pairs,which are adjacent inside a colour cluster. For those pairs we obtain for the colour charge opera-tors ( − T k · T i j ) T i j = (cid:26) / ( i j ) is a gluon,1 emitter ( i j ) is a quark or antiquark. (16)We introduce the notation P i j , k = h V i j , k i ( p i + p j ) − m i j · q (cid:0) h V i j , k i (cid:1) , P i j , a = h V ai j i ( p i + p j ) − m i j · x · q (cid:0) h V ai j i (cid:1) , P a j , k = h V a jk i (cid:12)(cid:12) p a · p j (cid:12)(cid:12) · x · q (cid:16) h V a jk i (cid:17) , P a j , b = h V a j , b i (cid:12)(cid:12) p a · p j (cid:12)(cid:12) · x · q (cid:16) h V a j , b i (cid:17) . (17)The functions P will govern the emission of additional particles in the shower algorithm. Thespin-averaged dipole splitting functions h V i can be found in [59, 63]. The Heavyside theta-functions ensure that the functions P will be non-negative. They are needed for splittings be-tween an initial- and a final-state particle, since the dipole splitting functions h V ai j i and h V a jk i may take negative values in certain regions of phase space. In addition, the spin-averaged dipolesplitting functions for massive partons are slightly modified: Terms related to the soft singularityare re-arranged between the two dipoles forming an antenna, in order to ensure positivity of theindividual dipole splitting functions in the singular limit. In this section we describe the shower algorithm. We first discuss the colour treatment in sec-tion 3.1. The shower algorithm for massless final-state partons is discussed in section 3.2. Thenecessary modifications for initial-state partons are discussed in section 3.3. Finally, massivepartons are discussed in section 3.4.
Before starting the parton showers, the partons from the hard matrix element have to be assignedto colour clusters. For the simplest matrix elements, like e + e − → q ¯ q , the choice is unique: The6uark-antiquark pair forms a colour cluster. For the parton shower we work in the leading-colourapproximation. In the leading-colour approximation we have to take into account only emitter-spectator pairs, which are adjacent inside a colour cluster. We have implemented two options:In the first one, which we call the “strict leading-colour approximation”, we take exactly theterms which are leading in an expansion in 1 / N c and only those. As a consequence, all splittings g → q ¯ q are ignored, as they are colour-suppressed compared to g → gg . In this approximation C F is replaced by C F → . (18)For the second option, which we call the “modified leading-colour approximation”, we includethe splitting g → q ¯ q and keep C F as ( N c − ) / / N c . In this case, if a gluon in a closed stringsplits into a quark-antiquark pair, the closed string becomes an open string. If a gluon in an openstring splits into a quark-antiquark pair, the open string splits into two open strings. We first describe the shower algorithm for electron-positron annihilation. The extension toinitial-state partons is treated in section 3.3. For the shower algorithm we use as an evolutionvariable t = ln − k ⊥ Q , (19)where Q is a fixed reference scale and k ⊥ is the transverse momentum of a splitting. During theshower evolution we move towards smaller (more negative) values of t . We start from a given n -parton configuration. In the dipole formalism, emission of additional partons is described byan emitter-spectator pair. In the leading colour approximation emitter and spectator are alwaysadjacent in the cyclic order. The probability to evolve from t to t (with t > t ) without anyresolvable branching is given by the Sudakov factor. For the algorithm considered here, theSudakov factor is given as a product of factors corresponding to the no-emission probabilities forindividual dipoles’ emissions: D ( t , t ) = (cid:213) ˜ i , ˜ k D ˜ i , ˜ k ( t , t ) . (20)If parton ˜ i can emit different partons, D ˜ i , ˜ k ( t , t ) factorises in turn into different contributions: D ˜ i , ˜ k ( t , t ) = (cid:213) j D i j , k ( t , t ) , (21)An example is the possibility of a gluon to split either into two gluons or into a ¯ qq -pair. Wedenote the emitter before the splitting by ˜ i , while the emitter after a splitting is denoted by i . Thisnotation takes into account that the emitter might change its “flavour” due to a splitting, like in7he case of a g → ¯ qq splitting. D i j , k ( t , t ) is the probability that the dipole formed by the emitter˜ i and spectator ˜ k does not emit a parton j . It is given by D i j , k ( t , t ) = exp − t Z t dt C ˜ i , ˜ k Z d f unres d (cid:16) t − T ˜ i , ˜ k (cid:17) P i j , k , (22)where C ˜ i , ˜ k is a colour factor. In the leading colour approximation this factor is non-zero only if ˜ i and ˜ k are adjacent in a colour cluster. Then C ˜ i , ˜ k is obtained from eq. (16) and given by C ˜ i , ˜ k = (cid:26) for ˜ i = g , i = q , ¯ q . (23)The dipole phase space is given by Z d f unres = ( p ˜ i + p ˜ k ) p Z d k z + ( k ) Z z − ( k ) dz z ( − z ) (cid:18) − k z ( − z ) (cid:19) , (24)with z ± ( k ) = (cid:16) ± √ − k (cid:17) . (25)The variable k is proportional to the transverse momentum of the splitting k = ( − k ⊥ )( p ˜ i + p ˜ k ) . (26) T ˜ i , ˜ k depends on the dipole invariant mass ( p ˜ i + p ˜ k ) and the phase space variable k for the emis-sion of an additional particle and is given by T ˜ i , ˜ k = ln k ( p ˜ i + p ˜ k ) Q (27)With the help of the delta-function we may perform the integration over k , while keeping theintegration over t and z . Then k ( t ) = Q e t ( p ˜ i + p ˜ k ) . (28) P i j , k is the dipole splitting function. As an example we quote the splitting function for the q → qg splitting: P q → qg = C F pa s ( µ )( p ˜ i + p ˜ k ) y (cid:20) − z ( − y ) − ( + z ) (cid:21) , y = k ( t ) z ( − z ) . (29)8 s is evaluated at the scale µ = − k ⊥ = k ( p ˜ i + p ˜ k ) . The probability that a branching occurs at t is given by (cid:229) ˜ i , ˜ k (cid:229) j C ˜ i , ˜ k Z d f unres d (cid:16) t − T ˜ i , ˜ k (cid:17) P i j , k D ( t , t ) . (30)We can now state the shower algorithm. Starting from an initial evolution scale t we proceed asfollows:1. Select the next dipole to branch and the scale t at which this occurs. This is done asfollows: For each dipole we generate the scale t , i j , k of the next splitting for this dipolefrom a uniformly distributed number r , i j , k in [ , ] by solving (numerically) the equation D i j , k ( t , t , i j , k ) = r , i j , k . (31)We then set t = max (cid:0) t , i j , k (cid:1) . (32)The dipole which has the maximal value of t , i j , k is the one which radiates off an additionalparticle.2. If t is smaller than a cut-off scale t min , the shower algorithm terminates.3. Next we have to generate the value of z . Again, using a uniformly distributed randomnumber r in [ , ] we solve z Z z − ( t ) dz ′ J ( t , z ′ ) P i j , k = r z + ( t ) Z z − ( t ) dz ′ J ( t , z ′ ) P i j , k , (33)where the Jacobian factor J ( t , z ) is given by J ( t , z ) = k ( t ) z ( − z ) (cid:18) − k ( t ) z ( − z ) (cid:19) . (34)4. Select the azimuthal angle f . Finally we generate the azimuthal angle from a uniformlydistributed number r in [ , ] as follows: f = p r . (35)5. With the three kinematical variables t , z and f and the information, that parton ˜ i emits aparton j , with parton ˜ k being the spectator, we insert the new parton j . The momenta p ˜ i and p ˜ k of the emitter and the spectator are replaced by new momenta p i and p k . The detailshow the new momenta p i , p j and p k are constructed are given in the appendix B.9. Set t = t and go to step 1.Remark: Step 1 of the algorithm is equivalent to first generating the point t from a uniformlydistributed number r in [ , ] by solving (numerically) the equation for the full Sudakov factor D ( t , t ) = r , (36)and then selecting an individual dipole with emitter ˜ i , emitted particle j and spectator k withprobability [74] P i j , k = C ˜ i , ˜ k R d f unres d (cid:16) t − T ˜ i , ˜ k (cid:17) P i j , k (cid:229) ˜ l , ˜ n (cid:229) m C ˜ l , ˜ n R d f unres d (cid:16) t − T ˜ l , ˜ n (cid:17) P lm , n . (37) In this subsection we discuss the necessary modifications for the inclusion of initial-state partons.In the presence of initial-state partons there is no separation into final-state showers and initial-state showers. Initial-state radiation is treated on the same footing as final-state radiation. Thealgorithm generates initial-state radiation through backward evolution, starting from a hard scaleand moving towards softer scales. Therefore the shower evolves in all cases from a hard scaletowards lower scales.
Final-state emitter and initial-state spectator
For an initial-state spectator we modify the Sudakov factor in eq. (22) to D i j , a ( t , t ) = exp − t Z t dt C ˜ i , ˜ a Z d f unres d (cid:16) t − T ˜ i , ˜ a (cid:17) x a f ( x a , t ) x ˜ a f ( x ˜ a , t ) P i j , a , (38)where x ˜ a is the momentum fraction of the initial hadron carried by ˜ a , while x a is the momentumfraction carried by a . The initial parton of the n -particle state is denoted by ˜ a , while the initialparton of the ( n + ) -particle state is denoted by a . We set x = x ˜ a x a . (39)The unresolved phase space is given by Z d f unres = | p ˜ i p ˜ a | p Z x ˜ a dxx Z dz . (40)The transverse momentum between i and j is expressed as − k ⊥ = ( − x ) x z ( − z ) ( − p ˜ i p ˜ a ) (41)10nd T ˜ i , ˜ a is therefore given by T ˜ i , ˜ a = ln − k ⊥ Q = ln ( − p ˜ i p ˜ a ) ( − x ) z ( − z ) xQ . (42)A subtlety occurs for the emission between a final-state spectator and an initial-state emitter. Wediscuss this for the splitting q → qg . The spin-averaged splitting function for the q → qg splittingis given by h V kqg i = pa s C F (cid:20) − z + ( − x ) − ( + z ) (cid:21) . (43)In contrast to the final-final case this function is not a positive function on the complete phase-space. It can take negative values in certain (non-singular) regions of phase-space. This is noproblem for its use as a subtraction terms in NLO calculations, but prohibits a straightforwardinterpretation as a splitting probability for a shower algorithm. However, since negative val-ues occur only in non-singular regions, we can ensure positiveness by modifying the splittingfunctions through non-singular terms. The simplest choice is to set P i j , a = h V ai j i ( p i + p j ) · x · q (cid:0) h V ai j i (cid:1) . (44)For a final-state emitter we eliminate the x -integration with the help of the delta-function: Z x ˜ a dxx d (cid:16) t − T ˜ i , ˜ a (cid:17) = + z ( − z ) k ( t ) , x = + k ( t ) z ( − z ) , k ( t ) = Q e t ( − p ˜ i p ˜ a ) . (45)For the boundaries we obtain k ( t ) < − x ˜ a x ˜ a , z − ( t ) < z < z + ( t ) , z ± ( t ) = (cid:18) ± r − k ( t ) x ˜ a − x ˜ a (cid:19) . (46)The modifications to the shower algorithm are as follows: The dipoles for the emission from afinal-state emitter with an initial-state spectator are included in the Sudakov factor in eq. (20).With this modification steps 1 and 2 are as above. Let us define f lm , n = , if l and n are final-state particles , x a f ( x a , t ) x ˜ a f ( x ˜ a , t ) , if l = a is an initial-state particle , x b f ( x b , t ) x ˜ b f ( x ˜ b , t ) , if n = b is an initial-state particle and l is a final-state particle . (47)In step 2 we replace formula (33) by z Z z − ( t ) dz ′ J ( t , z ′ ) f i j , a P i j , a = r z + ( t ) Z z − ( t ) dz ′ J ( t , z ′ ) f i j , a P i j , a , (48)with the Jacobian J ( t , z ) = + z ( − z ) k ( t ) . (49)Steps 4 to 6 proceed as in the case described above.11 nitial-state emitter and final-state spectator For an initial-state emitter ˜ a with a final-state spectator ˜ i the Sudakov factor is given by D a j , i ( t , t ) = exp − t Z t dt C ˜ a , ˜ i Z d f unres d (cid:16) t − T ˜ a , ˜ i (cid:17) x a f ( x a , t ) x ˜ a f ( x ˜ a , t ) P a j , i . (50)The unresolved phase space is again given by eq. (40). The transverse momentum between a and j is given by − k ⊥ = ( − x ) x ( − z ) ( − p ˜ i p ˜ a ) (51)and T ˜ a , ˜ i is given by T ˜ a , ˜ i = ln ( − p ˜ i p ˜ a ) ( − x )( − z ) xQ . (52)For a initial-state emitter we eliminate the z -integration with the help of the delta-function: Z dz d (cid:16) t − T ˜ a , ˜ i (cid:17) = k ( t ) x ( − x ) , z = − k ( t ) x ( − x ) , k ( t ) = Q e t ( − p ˜ i p ˜ a ) . (53)For the boundaries we obtain k ( t ) < − x ˜ a x ˜ a , x < x + ( t ) , x + ( t ) = + k ( t ) . (54)There are no new modifications to the shower algorithms compared to the case for a final-stateemitter and an initial-state spectator, except that in step 3 we now generate the value of x accord-ing to x Z x ˜ a dx ′ J ( t , x ′ ) f a j , i P a j , i = r x + ( t ) Z x ˜ a dx ′ J ( t , x ′ ) f a j , i P a j , i , (55)with the Jacobian J ( t , x ) = k ( t ) ( − x ) . (56) Initial-state emitter and initial-state spectator
For an initial-state emitter ˜ a with an initial-state spectator ˜ b the Sudakov factor is given by D a j , b ( t , t ) = exp − t Z t dt C ˜ a , ˜ b Z d f unres d (cid:16) t − T ˜ a , ˜ b (cid:17) x a f ( x a , t ) x ˜ a f ( x ˜ a , t ) P a j , b . (57)12n this case we do not rescale the momentum of the spectator, but transform all final-state mo-menta. Therefore no factor x b f ( x b , t ) x ˜ b f ( x ˜ b , t ) (58)appears in the Sudakov factor. The unresolved phase space is given by Z d f unres = (cid:12)(cid:12) p ˜ a p ˜ b (cid:12)(cid:12) p Z x ˜ a dxx − x Z dv . (59)The transverse momentum between a and j is given by − k ⊥ = ( − x ) x v (cid:0) p ˜ a p ˜ b (cid:1) (60)and T ˜ a , ˜ b is given by T ˜ a , ˜ b = ln (cid:0) p ˜ a p ˜ b (cid:1) ( − x ) vxQ . (61)We integrate over v with the help of the delta-function: − x Z dv d (cid:16) t − T ˜ a , ˜ b (cid:17) = k ( t ) x ( − x ) , v = k ( t ) x ( − x ) , k ( t ) = Q e t (cid:0) p ˜ a p ˜ b (cid:1) . (62)For the boundaries we obtain k ( t ) < ( − x ˜ a ) x ˜ a , x < x + ( t ) , x + ( t ) = + k ( t ) − r k ( t ) + k ( t ) ! . (63)In step 3 of the shower algorithm we again select x according to x Z x ˜ a dx ′ J ( t , x ′ ) f a j , b P a j , b = r x + ( t ) Z x ˜ a dx ′ J ( t , x ′ ) f a j , b P a j , b , (64)with the Jacobian J ( t , x ) = k ( t ) ( − x ) . (65)13 .4 The shower algorithm for massive partons In this subsection we discuss the modifications of the shower algorithms due to the presence ofmassive partons. We first address the issue of a splitting of a gluon into a heavy quark pair. Thismainly concerns the splitting of a gluon into b -quarks. We will always require that initial-stateparticles are massless. Therefore for processes with initial-state hadrons we do not consider g → Q ¯ Q splittings. Calculations for initial-state hadrons should be done in the approximation ofa massless b -quark. In the case of electron-positron annihilation the parton shower affects onlythe final state. Here we can consistently allow splittings of a gluon into a pair of massive quarks.As evolution variable we use in the massive case t = ln − k ⊥ + ( − z ) m i + z m j Q . (66)This choice reduces to eq. (19) in the massless limit and is suggested by dispersion relations forthe running coupling [47, 64]. Final-state emitter and final-state spectator
The unresolved phase space is given by Z d f unres = ( p ˜ i + p ˜ k ) p (cid:0) − µ i − µ j − µ k (cid:1) (cid:2) l ( , µ i j , µ k ) (cid:3) − y + Z y − dy ( − y ) z + ( y ) Z z − ( y ) dz , (67)where the reduced masses µ l and the boundaries on the integrations are defined in appendix B ineqs. (119)-(122). T ˜ i , ˜ k is given by T ˜ i , ˜ k = ln (cid:16) ( p ˜ i + p ˜ k ) − m i − m j − m k (cid:17) yz ( − z ) Q . (68)Again, we have to ensure that the splitting functions are positive. The original spin-averageddipole splitting functions can take negative values in certain regions of phase-space. In the mas-sive case the negative region can extend into the singular region. The problem is related to thesoft behaviour of the dipole splitting functions. Since a squared Born matrix element is positivein the soft gluon limit, the negative contribution from a particular dipole is compensated by thecontribution from the dipole, where emitter and spectator are exchanged. The sum of the twocontributions is positive in the singular region. Therefore we can cut out the negative region fromthe first dipole and add it to the second dipole. The second dipole will stay positive.As in the massless case we eliminate the y -integration: y + Z y − dy ( − y ) z + ( y ) Z z − ( y ) dz d (cid:16) t − T ˜ i , ˜ k (cid:17) = z max Z z min dz y ( − y ) , (69) y = k ( t ) z ( − z ) , k ( t ) = Q e t ( p ˜ i + p ˜ k ) − m i − m j − m k . (cid:18) − k z ( − z ) (cid:19) h k − ( − z ) ¯ m i − z ¯ m j i − (cid:18) k z ( − z ) (cid:19) ¯ m k + m i ¯ m j ¯ m k ≥ , (70)with ¯ m l = m l ( p ˜ i + p ˜ k ) − m i − m j − m k for l ∈ { i , j , k } . (71)This equation is solved numerically for z min and z max . Then z is generated according to z Z z min ( t ) dz ′ J ( t , z ′ ) P i j , k = r z max ( t ) Z z min ( t ) dz ′ J ( t , z ′ ) P i j , k , (72)with the Jacobian J ( t , z ) = (cid:0) − µ i − µ j − µ k (cid:1) (cid:2) l ( , µ i j , µ k ) (cid:3) − k ( t ) z ( − z ) (cid:18) − k ( t ) z ( − z ) (cid:19) . (73) Final-state emitter and initial-state spectator
The unresolved phase space is given by Z d f unres = | p ˜ i p ˜ a | p Z x ˜ a dxx Z z − ( x ) dz = | p ˜ i p ˜ a | p Z z − ( x ˜ a ) dz x + ( z ) Z x ˜ a dxx (74)The integration boundary is given by z − ( x ) = x ˜ µ − x ( − ˜ µ ) , x + ( z ) = z ˜ µ + z ( − ˜ µ ) , ˜ µ = m i | p ˜ i p ˜ a | . (75) T ˜ i , ˜ a is given by T ˜ i , ˜ a = ln − k ⊥ + ( − z ) m i Q = ln ( − p ˜ i p ˜ a ) ( − x ) z ( − z ) xQ . (76)For a final-state emitter we eliminate the x -integration with the help of the delta-function: Z x ˜ a dxx d (cid:16) t − T ˜ i , ˜ a (cid:17) = + z ( − z ) k ( t ) , x = + k ( t ) z ( − z ) , k ( t ) = Q e t ( − p ˜ i p ˜ a ) . (77)15or the boundaries we obtain z + ( t ) = (cid:18) + r − k ( t ) x ˜ a − x ˜ a (cid:19) , z − ( t ) = max x ˜ a ˜ µ − x ˜ a ( − ˜ µ ) , (cid:18) − r − k ( t ) x ˜ a − x ˜ a (cid:19) , − s k ( t ) µ ! . (78)The boundary on k ( t ) is given for ˜ µ < ( − x ˜ a ) / x ˜ a by k ( t ) < − x ˜ a x ˜ a . (79)For ( − x ˜ a ) / x ˜ a < ˜ µ we have k ( t ) < − x ˜ a x ˜ a − − − x ˜ a x ˜ a ˜ µ + − x ˜ a x ˜ a ˜ µ ! = µ (cid:16) + x ˜ a ˜ µ − x ˜ a (cid:17) . (80) z is generated according to z Z z − ( t ) dz ′ J ( t , z ′ ) f i j , a P i j , a = r z + ( t ) Z z − ( t ) dz ′ J ( t , z ′ ) f i j , a P i j , a , (81)with the Jacobian J ( t , z ) = + z ( − z ) k ( t ) . (82) Initial-state emitter and final-state spectator T ˜ a , ˜ i is given by T ˜ a , ˜ i = ln ( − p ˜ i p ˜ a ) ( − x )( − z ) xQ . (83)For an initial-state emitter we eliminate the z -integration with the help of the delta-function: Z z − ( x ) dz d (cid:16) t − T ˜ a , ˜ i (cid:17) = k ( t ) x ( − x ) , z = − k ( t ) x ( − x ) , k ( t ) = Q e t ( − p ˜ i p ˜ a ) . (84)For the boundaries we obtain k ( t ) < ( − x ˜ a ) x ˜ a [ − x ˜ a ( − ˜ µ )] , x < x + ( t ) , x + ( t ) = + k ( t ) − q k ( t ) + ˜ µ k ( t ) (cid:16) + k ( t ) ( − ˜ µ ) (cid:17) , (85)16he value of x is generated according to x Z x ˜ a dx ′ J ( t , x ′ ) f a j , i P a j , i = r x + ( t ) Z x ˜ a dx ′ J ( t , x ′ ) f a j , i P a j , i , (86)with the Jacobian J ( t , x ) = k ( t ) ( − x ) . (87) In this section we show numerical results obtained from the parton shower. We first discussin section 4.1 observables related to electron-positron annihilation and then in section 4.2 theshower in hadron collisions. The shower algorithm depends on two parameters, the strong cou-pling a s and the scale Q min . For the strong coupling we use the leading-order formula a s ( µ ) = pb ln µ L , b = − N f . (88)The cut-off scale Q min gives the scale at which the shower terminates. As our shower is correctin the leading-colour approximation, we also study the effects of different treatments of sub-leading colour contributions. As described in section 3.1 we have implemented two options: Thestrict leading-colour approximation and the modified leading-colour approximation. Numericaldifferences from these two options will give an estimate of uncertainties due to subleading-coloureffects. For electron-positron annihilation we use a s ( m Z ) = .
118 corresponding to L =
88 MeV. Westart the shower from the 2 → e + e − → q ¯ q . We first study the event shapevariables thrust, the C-parameter and the D-parameter. The distributions of the first momentsof these observables are shown in figure 1 for two choices of the cut-off parameter: Q min = Q min = → y cut = .
008 and the E -scheme for the recombination. Then events with exactly four jetsare selected. We consider the modified Nachtmann-Reiter angle [76], the Körner-Schierholz-Willrodt angle [77], the Bengtsson-Zerwas angle [78] and the angle a between the jets withthe smallest energy [79]. In the plots we show the results from the different options for the colourtreatment for Q min = .2 Hadron colliders For the Tevatron and the LHC we study Z / g ∗ -production. We start from the 2 → q ¯ q → Z / g ∗ → l + l − . As parton distribution functions we use the CTEQ 6L1 set [80, 81].For consistency we use here a s ( m Z ) = .
130 corresponding to L =
165 MeV. The centre-of-mass energy we set to √ s = .
96 TeV for the Tevatron and to √ s =
14 TeV for the LHC. Werequire a cut on the invariant mass of the lepton pair of m l + l − >
80 GeV . (89)As cut-off parameter for the parton shower we use Q min = In this paper we presented an implementation of a shower algorithm based on the dipole for-malism. The formalism treats initial- and final-state partons on the same footing. The showercan be used for hadron colliders and electron-positron colliders. We also included in the showeralgorithm massive partons in the final state. We studied numerical results for electron-positronannihilation, the Tevatron and the LHC.
Acknowledgments
We would like to thank Zoltan Nagy for discussions and useful comments on the manuscript.
A Sudakov factors for massless final-state partons
In this appendix we discuss in more detail the Sudakov factors for massless final-state partons.This case is simple enough that one integration can be done analytically. The spin-averageddipole subtraction terms in four dimensions are P q → qg = C F pa s ( µ ) s i jk y (cid:20) − z ( − y ) − ( + z ) (cid:21) , P g → gg = C A pa s ( µ ) s i jk y (cid:20) − z ( − y ) + − ( − z )( − y ) − + z ( − z ) (cid:21) , P g → q ¯ q = T R pa s ( µ ) s i jk y [ − z ( − z )] , (90)with s i jk = (cid:0) p i + p j + p k (cid:1) = ( p ˜ i + p ˜ k ) . (91)18he dipole phase space measure is Z d f unres = s i jk p Z d k z + ( k ) Z z − ( k ) dz z ( − z ) (cid:18) − k z ( − z ) (cid:19) , (92)with z ± ( k ) = (cid:16) ± √ − k (cid:17) . (93)The strong coupling is evaluated at the scale µ = − k ⊥ : a s ( µ ) = a s (cid:18) k s i jk (cid:19) . (94)The Sudakov factor is given by D i j , k ( t , t ) = exp − t Z t dt C ˜ i , ˜ k Z d f unres d (cid:16) t − T ˜ i , ˜ k (cid:17) P i j , k , (95)For the splitting q → qg we obtain D i j , k ( t , t ) = exp − C ˜ i , ˜ k C F k + Z k − d kk a s ( µ ) p z + ( k ) Z z − ( k ) dz ( − y ) (cid:20) − z ( − y ) − ( + z ) (cid:21) , (96)with k − = Q s i jk e t , k + = min (cid:18) , Q s i jk e t (cid:19) , y = k z ( − z ) , µ = k s i jk . (97)The integration over z can be done analytically: Z dz ( − y ) (cid:20) − z ( − y ) − ( + z ) (cid:21) = − z − z + k [ ln z − ( − z )] − + k (cid:20) k ln z + ln (cid:0) k + ( − z ) (cid:1) + √ k arctan (cid:18) √ k ( − z ) (cid:19)(cid:21) . (98)The same holds for the other splittings. Therefore we obtain for the Sudakov factors D i j , k ( t , t ) = exp − C ˜ i , ˜ k C k + Z k − d kk a s (cid:0) k s i jk (cid:1) p (cid:0) V i j , k ( k , z + ) − V i j , k ( k , z − ) (cid:1) , (99)19here C is a colour factor and equal to C = C F for q → qg , C A for g → gg , T R for g → q ¯ q . (100)The functions V i j , k ( k , z ) are given by V qg , k ( k , z ) = − z − z + k [ ln z − ( − z )] − + k (cid:20) k ln z + ln (cid:0) k + ( − z ) (cid:1) − √ k arctan (cid:18) √ k ( − z ) (cid:19)(cid:21) , V gg , k ( k , z ) = − z + z − z − k z + k ln z − z + + k (cid:20) k ln 1 − zz + ln k + z k + ( − z ) − √ k arctan (cid:18) z √ k (cid:19) + √ k arctan (cid:18) ( − z ) √ k (cid:19)(cid:21) , V gq , k ( k , z ) = z − z + z + k z − k z − z . (101) B Insertion of emitted particles
In this appendix we list the relevant formulæ for the insertion of one additional four-vector intoa set of n four-vectors. This insertion satisfies momentum conservation and can be consideredas the inverse of the ( n + ) → n phase space mapping of Catani and Seymour. These insertionmappings are also useful for an efficient phase-space integration of the real emission contributionin NLO calculations. Therefore we quote in addition the relevant phase space weights. Forthe shower algorithm, these weights are not needed, as they are taken into account through thegeneration of the shower. B.1 Insertion for final-state particles
The massless case
We start with the simplest case, where both the emitter and the spectator are in the final state andall particles involved in the dipole splitting are massless. The insertion procedure is identical tothe one used in [82]. Given the four-vectors ˜ p i j and ˜ p k together with the three variables y , z and f s we would like to construct p i , p j and p k , such that p i + p j + p k = ˜ p i j + ˜ p k , p i = p j = p k = . (102)In four dimensions we have for the phase space measure d f unres = s i jk p Z dy ( − y ) Z dz p Z d f s , (103)20here s i jk = ( ˜ p i j + ˜ p k ) = ( p i + p j + p k ) . It is convenient to work in the rest frame of P = ˜ p i j + ˜ p k = p i + p j + p k . We shall orient the frame in such a way, that the spatial components of˜ p k are along the z -direction. When used as a phase space generator we set y = u , z = u f s = p u , (104)where u , u and u are three uniformly distributed random numbers in [ , ] . From y = s i j s i j + s ik + s jk , z = s ik s ik + s jk (105)we obtain s i j = yP , s ik = z ( − y ) P , s jk = ( − z )( − y ) P . (106)If s i j < s jk we want to have p ′ k → p k as s i j →
0. Define E i = s i j + s ik √ s i jk , E j = s i j + s jk √ s i jk , E k = s ik + s jk √ s i jk , (107) q ik = arccos (cid:18) − s ik E i E k (cid:19) , q jk = arccos (cid:18) − s jk E j E k (cid:19) . (108)In our coordinate system we have p ′ i = E i ( , sin q ik cos ( f s + p ) , sin q ik sin ( f s + p ) , cos q ik ) , p ′ j = E j ( , sin q jk cos f s , sin q jk sin f s , cos q jk ) , p ′ k = E k ( , , , ) . (109)The momenta p ′ i , p ′ j and p ′ k are related to the momenta p i , p j and p k by a sequence of Lorentztransformations back to the original frame p i = L boost L xy ( f ) L xz ( q ) p ′ i (110)and analogously for the other two momenta. The explicit formulæ for the Lorentz transforma-tions are obtained as follows : Let | P | = p ( ˜ p i j + ˜ p k ) and denote by ˆ p k the coordinates of thehard momentum ˜ p k in the centre of mass system of ˜ p i j + ˜ p k . ˆ p k is given byˆ p k = E P | P | ˜ E k − ~ ˜ p k · ~ P | P | ,~ ˜ p k + ~ ˜ p k · ~ P | P | ( E P + | P | ) − ˜ E k | P | ! ~ P ! (111)The angles are then given by q = arccos E k E ′ k − p k · p ′ k (cid:12)(cid:12)(cid:12) ˆ ~ p k (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) ~ p ′ k (cid:12)(cid:12) , f = arctan (cid:18) ˆ p yk ˆ p xk (cid:19) . (112)21or the case considered here particle k is massless and the formula for q reduces to q = arccos − p k · p ′ k p tk p t ′ k ! . (113)The explicit form of the rotations is L xz ( q ) = q q − sin q q , L xy ( f ) = f − sin f
00 sin f cos f
00 0 0 1 . (114)The boost p = L boost q is given by p = E P | P | E q + ~ q · ~ P | P | ,~ q + ~ q · ~ P | P | ( E P + | P | ) + E q | P | ! ~ P ! . (115)The weight is given by w = s i jk p ( − y ) . (116) The massive case
We now consider the case of final state particles with arbitrary masses:˜ p i j = m i j , p i = m i , p j = m j , ˜ p k = p k = m k . (117)The dipole phase space reads [63] d f unres = s i jk p (cid:0) − µ i − µ j − µ k (cid:1) (cid:2) l (cid:0) , µ i j , µ k (cid:1)(cid:3) − y + Z y − dy ( − y ) z + Z z − dz p Z d f s , (118)where s i jk = (cid:0) ˜ p i j + ˜ p k (cid:1) , µ l = m l √ s i jk , l ( x , y , z ) = x + y + z − xy − yz − zy . (119)The integration boundaries are given by y + = − µ k ( − µ k ) − µ i − µ j − µ k , y − = µ i µ j − µ i − µ j − µ k . (120) z ± = µ i + (cid:16) − µ i − µ j − µ k (cid:17) y h µ i + µ j + (cid:16) − µ i − µ j − µ k (cid:17) y i (cid:0) ± v i j , i v i j , k (cid:1) . (121)22he general formula for the relative velocities is v p , q = p − p q / ( pq ) . In our case the relativevelocities are given by v i j , k = rh µ k + (cid:16) − µ i − µ j − µ k (cid:17) ( − y ) i − µ k (cid:16) − µ i − µ j − µ k (cid:17) ( − y ) , v i j , i = r(cid:16) − µ i − µ j − µ k (cid:17) y − µ i µ j (cid:16) − µ i − µ j − µ k (cid:17) y + µ i . (122)For the phase space generation we set y = ( y + − y − ) u + y − , z = ( z + − z − ) u + z − , f s = p u . (123)We again work in the rest frame of P = ˜ p i j + ˜ p k = p i + p j + p k , such that the spatial componentsof ˜ p k are along the z -direction:˜ p i j = (cid:0) ˜ E i j , , , − (cid:12)(cid:12) ~ ˜ p k (cid:12)(cid:12)(cid:1) , ˜ p k = (cid:0) ˜ E k , , , (cid:12)(cid:12) ~ ˜ p k (cid:12)(cid:12)(cid:1) . (124)For the invariants we have2 p i p j = y (cid:0) P − m i − m j − m k (cid:1) , p i p k = z ( − y ) (cid:0) P − m i − m j − m k (cid:1) , p j p k = ( − z ) ( − y ) (cid:0) P − m i − m j − m k (cid:1) . (125)The invariants are related to y and z as follows: y = p i p j p i p j + p i p k + p j p k , z = p i p k p i p k + p j p k . (126)In our chosen frame p ′ i = | ~ p i | ( E i | ~ p i | , sin q ik cos ( f s + p ) , sin q ik sin ( f s + p ) , cos q ik ) , p ′ j = (cid:12)(cid:12) ~ p j (cid:12)(cid:12) ( E j (cid:12)(cid:12) ~ p j (cid:12)(cid:12) , sin q jk cos f s , sin q jk sin f s , cos q jk ) , p ′ k = | ~ p k | ( E k | ~ p k | , , , ) . (127)The energies are obtained from the invariants as follows: E i = s i jk − p j p k + m i − m j − m k √ s i jk , E j = s i jk − p i p k − m i + m j − m k √ s i jk , E k = s i jk − p i p j − m i − m j + m k √ s i jk . (128)23or the angles we have q ik = arccos (cid:18) E i E k − p i p k | ~ p i | | ~ p k | (cid:19) , q jk = arccos E j E k − p j p k (cid:12)(cid:12) ~ p j (cid:12)(cid:12) | ~ p k | ! . (129)The momenta p ′ i , p ′ j and p ′ k are related to the momenta p i , p j and p k by the same sequence ofLorentz transformations as in eq. (110). The weight is w = s i jk p (cid:0) − µ i − µ j − µ k (cid:1) (cid:2) l (cid:0) , µ i j , µ k (cid:1)(cid:3) − ( − y ) ( y + − y − ) ( z + − z − ) . (130) B.2 Insertion for an antenna between an initial-state and a final state
The massless case
Here the ( n + ) -particle phase space is given by a convolution: d f n + = Z dx d f n ( xp a ) d f dipole . (131)The dipole phase space reads: d f dipole = (cid:12)(cid:12) p i j p a (cid:12)(cid:12) p Z dz p Z d f s . (132)The angle f s parametrises the solid angle perpendicular to ˜ p i j and xp a . Therefore we can treatthe case of a final-state emitter with an initial-state spectator as well as the case of an initial-state emitter with a final-state spectator at the same time. x and z are related to the invariants asfollows: x = − p i p a − p j p a − p i p j − p i p a − p j p a , z = − p i p a − p i p a − p j p a . (133)For the phase space generation we set x = − u , z = u , f s = p u . (134)We denote Q = ˜ p i j + xp a = p i + p j + p a . It is convenient to work in the rest frame of P = p i + p j = Q − p a and to orient the frame such that p a is along the z -axis. For the invariants wehave 2 p i p j = (cid:0) − Q (cid:1) − xx , p i p a = zx Q , p j p a = − zx Q . (135)24n this frame p ′ i = E i ( , sin q ia cos f s , sin q ia sin f s , cos q ia ) , p ′ j = E i ( , − sin q ia cos f s , − sin q ia sin f s , − cos q ia ) , p ′ a = ( − | E a | , , , | E a | sign ( p za ′ )) . (136)We have E i = | P | , E a = | P | ( P · p a ) , q ia = arccos (cid:20) sign ( p za ′ ) (cid:18) − + p i p a E i E a (cid:19)(cid:21) . (137)The momenta p ′ i , p ′ j are again related to the momenta p i , p j by a sequence of Lorentz transfor-mations as in eq. (110). The weight is given by w = (cid:12)(cid:12) Q (cid:12)(cid:12) p x . (138) The massive case
The dipole phase space now reads: d f dipole = (cid:12)(cid:12) p i j p a (cid:12)(cid:12) p z + Z z − dz p Z d f s . (139)The integration boundaries are given by z + = , z − = µ − x + µ . (140)where µ = m i (cid:12)(cid:12) p i j p a (cid:12)(cid:12) = xm i (cid:12)(cid:12) Q − m i (cid:12)(cid:12) . (141)We consider only the case where m ˜ i j = m i = m and all other masses are zero. For the phase spacegeneration we set x = − u , z = ( z + − z − ) u + z − , f s = p u . (142)For the invariants we have now2 p i p j = (cid:0) − Q + m i (cid:1) − xx , p i p a = zx (cid:0) Q − m i (cid:1) , p j p a = − zx (cid:0) Q − m i (cid:1) . (143)We parametrise the momenta as p ′ i = | ~ p i | ( E i | ~ p i | , sin q ia cos f s , sin q ia sin f s , cos q ia ) , p ′ j = | ~ p i | ( , − sin q ia cos f s , − sin q ia sin f s , − cos q ia ) , p ′ a = ( − | E a | , , , | E a | sign ( p za ′ )) . (144)25hen E i = P + m i | P | , E a = | P | ( P · p a ) , q ia = arccos (cid:20) sign ( p za ′ ) ( E i E a − p i p a ) | ~ p i | ( − E a ) (cid:21) . (145)The momenta p ′ i , p ′ j are again related to the momenta p i , p j by a sequence of Lorentz transfor-mations as in eq. (110). The weight is given by w = (cid:12)(cid:12) Q − m i (cid:12)(cid:12) p x ( z + − z − ) . (146) B.3 Insertion for an initial-state antenna
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