Three-particle correlations in QCD jets and beyond
Redamy Perez Ramos, Vincent Mathieu, Miguel-Angel Sanchis-Lozano
aa r X i v : . [ h e p - ph ] A p r Report: IFIC/11-18, FTUV-11-0419
Three-particle correlations in QCD jets and beyond
Redamy P´erez-Ramos , Vincent Mathieu and Miguel-Angel Sanchis-Lozano Departament de F´ısica Te`orica and IFIC, Universitat de Val`encia - CSICDr. Moliner 50, E-46100 Burjassot, Spain
Abstract : In this paper, we present a more detailed version of our previous work for three-particlecorrelations in quark and gluon jets [1]. We give theoretical results for this observable in the doublelogarithmic approximation and the modified leading logarithmic approximation. In both resummationschemes, we use the formalism of the generating functional and solve the evolution equations analyticallyfrom the steepest descent evaluation of the one-particle distribution. In addition, in this paper we includepredictions beyond the limiting spectrum approximation and study this observable near the hump of thesingle inclusive distribution. We thus provide a further test of the local parton hadron duality (LPHD)and make predictions for the LHC. The computation of higher rank correlators is presented in the doublelogarithmic approximation and shown to be rather cumbersome. e-mail: [email protected] e-mail: vincent.mathieu@ific.uv.es e-mail: miguel.angel.sanchis@ific.uv.es Introduction
The observation of quark and gluon jets has played a crucial role in establishing Quantum Chromody-namics (QCD) as the theory of strong interaction within the Standard Model of particle physics [2, 3].Jets, narrowly collimated bundles of hadrons, reflect configurations of quarks and gluons at short dis-tances [4, 5].The evolution of gluon and quark initiated jets is dominated by soft gluon bremsstrahlung. Powerfulschemes, like the Double Logarithmic Approximation (DLA) and the Modified Leading Logarithmic Ap-proximation (MLLA), which allow for the perturbative resummation of soft-collinear and hard-collineargluons before the hadronization occurs, have been developed over the past thirty years (for a reviewsee [6]). In the frame of high energy jets, one of the strikest predictions of perturbative QCD (pQCD),which follows as a consequence of Angular Ordering (AO) within the MLLA and the Local PartonHadron Duality (LPHD) hypothesis [7], is the existence of the hump-backed shape [8] of the inclu-sive energy distribution of hadrons, later confirmed by experiments at colliders like the LEP [9, 10] andthe Tevatron [11]. Within the same formalism, the transverse momentum distribution, or k ⊥ -spectra ofhadrons produced in p ¯ p collisions at center of mass energy √ s = 1 . TeV at the Tevatron [12], was welldescribed by MLLA [13] and next-to-MLLA (NMLLA) [14, 15] predictions inside the validity rangesprovided by such schemes, both supported by the LPHD. Thus, the study and tests of enough inclusiveobservables like the inclusive energy distribution and the inclusive transverse momentum k ⊥ spectra ofhadrons have shown that the perturbative stage of the process, which evolves from the hard scale orleading parton virtuality Q ∼ E to the hadronization scale Q , is dominant. In particular, these issuessuggest that the hadronization stage of the QCD cascade do not affect pQCD predictions and therefore,that the LPHD hypothesis is successful while treating one-particle inclusive observables.The study of particle correlations in intrajet cascades, which are less inclusive observables, provide arefined test of the partonic dynamics and the LPHD. In [16], the two-particle correlations inside quarkand gluon jets were first computed at DLA. In [17, 18], this observable was computed for the first time atMLLA for such particles, whose energy or x (energy fraction of the jet carried away by one parton) staysclose to the maximum of the one-particle distribution. In [19], the previous solutions were extended, atMLLA, to all possible values of x by exactly solving the QCD evolution equations. This observable wasmeasured by the OPAL collaboration in the e + e − annihilation at the Z peak, that is for √ s = 91 . GeVat LEP [20]. Though the agreement with predictions presented in [19] turned out to be rather good for thedescription of the data [20], a discrepancy still subsists pointing out a possible failure of the LPHD forless inclusive observables. However, these measurements were redone by the CDF collaboration in p ¯ p collisions at the Tevatron for mixed samples of quark and gluon jets [11]. The agreement with predictionspresented in [17, 18] turned out to be rather good, in particular for very soft particles ( x ≪ . ) havingvery close energy fractions ( x ≈ x ). A discrepancy between the OPAL and CDF analysis showed upand still stays unclear. That is why, the measurement of two-particle correlations at the LHC becomescrucial.By going one step beyond, in this paper we give predictions for three-particle correlations inside quarkand gluon jets. This observable together with two-particle correlations can be measured in equal footingat the LHC. Such tests will provide further verifications of the LPHD for less inclusive observablesand shed more light on the role of confinement in jet evolution. Further issues on the importance of1orrelations versus single-particle distributions studies have been presented in [21, 22].The paper is organized as follows. • in section 2 we recall the formalism of jet generating functionals and their evolution equations; • the kinematics and the process for the inclusive production of three particles inside the jet arespecified in subsection 2.1 and 2.1.1 respectively; • in subsection 2.2, we obtain the MLLA exact system of integro-differential evolution equations forthe three-particle correlations and in subsection 2.3, the single logarithms (SLs) contributions areobtained from the exact evolution equations; • in subsection 2.4, we obtain the DLA solution of the evolution equations and study the shape andoverall normalization of this observable; • in subsection 2.5 the evolution equations are solved iteratively and the solution are expressed interms of the logarithmic derivatives of the one-particle distribution and the two-particle correla-tions; • in subsection 2.6, we finally give the analytical predictions which will be displayed in order toprovide predictions for the Tevatron and the LHC; • in subsection 2.7, the hump approximation is applied to this observable; • in subsection 2.8, the region in x where the emission of three correlated particles becomes domi-nant is discussed; • in subsection 2.9, we give the analytical solution of the DLA four-particle correlator and showthat including higher order corrections for differential higher rank correlators would become acumbersome task; • in subsection 3, the predictions are displayed and the phenomenology is applied to the Tevatronand the LHC; • a conclusion summarizes this work; the appendices are written as complements of the main coreof the paper. A generating functional Z ( E, Θ; { u } ) can be constructed [23] that describes the azimuth averaged partoncontent of a jet of energy E with a given opening half-angle Θ ; by virtue of the exact angular ordering(MLLA), it satisfies the following integro-differential evolution equation [6] dd ln Θ Z A ( p, Θ; { u } ) = 12 X B,C Z dz Φ B [ C ] A ( z ) α s (cid:0) k ⊥ (cid:1) π (cid:16) Z B (cid:0) zp, Θ; { u } (cid:1) Z C (cid:0) (1 − z ) p, Θ; { u } (cid:1) − Z A (cid:0) p, Θ; { u } (cid:1)(cid:17) ; (1)2n (1), z and (1 − z ) are the energy-momentum fractions carried away by the two offspring in the A → BC parton decay described by the standard one loop splitting functions [24] Φ q [ g ] q ( z ) = C F z − z , Φ g [ q ] q ( z ) = C F − z ) z , (2) Φ q [¯ q ] g ( z ) = T R (cid:0) z + (1 − z ) (cid:1) , Φ g [ g ] g ( z ) = 2 C A (cid:18) − zz + z − z + z (1 − z ) (cid:19) , (3) C A = N c , C F = ( N c − / N c , T R = 1 / , (4)where N c is the number of colors; Z A in the integral in the r.h.s. of (1) accounts for 1-loop virtualcorrections, which exponentiate into Sudakov form factors. α s ( q ) is the running coupling constant ofQCD α s ( q ) = 4 π N c β ln q Λ QCD , (5)where Λ QCD ≈ a few hundred MeV’s is the intrinsic scale of QCD, and β = 14 N c (cid:16) N c − n f T R (cid:17) (6)is the first term in the perturbative expansion of the β function, n f the number of light quark flavors.If the radiated parton with 4-momentum k = ( k , ~k ) is emitted with an angle Θ with respect to thedirection of the jet, one has ( k ⊥ is the modulus of the transverse trivector ~k ⊥ orthogonal to the directionof the jet) k ⊥ ≃ | ~k | Θ ≈ k Θ ≈ zE Θ when z ≪ or k ⊥ ≈ (1 − z ) E Θ when z → , and a collinearcutoff k ⊥ ≥ Q is imposed.In (1) the symbol { u } denotes a set of probing functions u a ( k ) with k the 4-momentum of a secondaryparton of type a . The jet functional is normalized to the total jet production cross section such that Z A ( p, Θ; u ≡
1) = 1; (7)for vanishingly small opening angle it reduces to the probing function of the single initial parton Z A ( p, Θ → { u } ) = u A ( k ≡ p ) . (8)To obtain exclusive n -particle distributions one takes n variational derivatives of Z A over u ( k i ) withappropriate particle momenta, i = 1 . . . n , and sets u ≡ afterwards; inclusive distributions are gener-ated by taking variational derivatives around u ≡ . We introduce the n-particle differential inclusivedistribution, also known as parton densities, as [6] x . . . x n D ( n ) A ( x , . . . , x n , Y ) = E . . . E n δ n δu ( k ) . . . δu ( k n ) Z A ( k , . . . , k n , Θ; { u ( k ) } ) (cid:12)(cid:12)(cid:12)(cid:12) u =1 . (9)Accordingly, we introduce the following notations for gluon and quark jets A = G, Q, ¯ QA ( n )1 ...n ( z ) ≡ x z . . . x n z D ( n ) A ( x z , . . . , x n z , Y + ln z ) , A ( n )1 ...n ≡ x . . . x n D ( n ) A ( x , . . . , x n , Y ) , (10)which we will use hereafter; x i corresponds to the Feynman energy fraction of the jet taken away by oneparticle “ i ”. In the case of three-particle correlations n = 3 , the observable to be measured experimen-tally reads C (3) A = A (3)123 A A A . .1 Kinematics and variables The probability of soft gluon radiation off a color charge (moving in the z direction) has the polar angledependence sin Θ d Θ2(1 − cos Θ) = d sin(Θ / / ≃ d ΘΘ ; therefore, we choose the angular evolution parameter to be Y = ln 2 E sin(Θ / Q ⇒ dY = d sin(Θ / /
2) ; (11)note that this choice accounts for finite angles O (1) up to the full opening half-angle Θ = π , at which Y Θ= π = ln 2 EQ , where E is the center-of-mass annihilation energy of the process e + e − → q ¯ q . For small angles (11)reduces to Y ≃ ln QQ , Θ ≪ , ddY = dd ln Θ , (12)where Q = E Θ , defined as the virtuality of the jet, is the maximal transverse momentum of a partoninside the jet with opening half-angle Θ . Moreover, we make use of variables known from previousworks [19, 25], ℓ = ln zx i , y = ln x j E Θ Q , λ = Q Λ QCD , (13) ℓ i = ln 1 x i , y j = ln x j E Θ Q , η ij = ln x i x j , Y = ℓ i + y j + η ij . (14)Since ddy = dd ln Θ , y could also be used as the evolution-time variable in forthcoming quark and gluonjet evolution equations. Accordingly, the anomalous dimension, related to the coupling constant (5), canbe parametrized as follows γ ( q ) = 2 N c α s ( q ) π ⇒ γ ( ℓ + y ) = 1 β ( ℓ + y + η ij + λ ) , (15)such that, • for one particle [6], the denominator in (15) is simply ℓ + y + λ , with [26] ℓ = ln zx , y = ln xE Θ Q , η = 0 ; • for two-particle correlation [19, 25], ℓ + y + η , with ℓ = ln zx , y = ln x E Θ Q , η = ln x x ; • for three-particle correlation, ℓ + y + η , with ℓ = ln zx , y = ln x E Θ Q , η = η + η = ln x x . The production of three hadrons is displayed in Fig.1 after a quark or a gluon (A) jet of energy E ,half opening angle Θ and virtuality Q = E Θ has been produced in a high energy collision. Thekinematical variable characterizing the process is given by the transverse momentum k ⊥ = zE Θ ≥ Q (or (1 − z ) E Θ ≥ Q ) of the first splitting A → BC . The parton C fragments into three offspring such4 x 1x23 Θ 0 Θ 1 Θ 2 Θ3
E (1−z)EA zECB
Figure 1: Three-particle yield and angular ordering inside a high energy jet.that three hadrons of energy fractions x , x and x can be triggered from the same cascade followingthe condition: Θ ≥ Θ ≥ Θ ≥ Θ , (16)which arises from the exact AO in MLLA [6]. In particular, the condition Θ ≥ Θ is kinematical ratherthan supported by the AO; it states that every collinear gluon is emitted inside the jet of half openingangle Θ . The two variables entering the evolution equations are z and Θ , such that x ≤ z ≤ ⇒ ≤ ℓ ≤ ℓ . (17)From (16) and the initial condition at threshold x E Θ ≥ x E Θ ≥ x E Θ ≥ Q , one has Q x E ≤ Θ ≤ Θ ⇒ ≤ y ≤ y . (18) The evolution equations satisfied by (9) are derived from the MLLA master equation for the generatingfunctional Z A ( u ( k i )) (1). In this case, one takes the first δZ A δu ( k ) , second δ Z A δu ( k ) δu ( k ) , and finally third δ Z A δu ( k ) ...δu ( k ) functional derivatives of Z A ( u ( k i )) over the probing functions u ( k i ) so as to obtain thesystem of evolution equations for 3-particle correlations. Following from (1), after applying the oper-ator δ δu ( k ) ...δu ( k ) to both members of the equation, according to (9) and (10) together with the initialcondition (7), it is straightforward to get the coupled system of evolution equations Q (3) y = Z x dz α s π Φ gq ( z ) h G (3) ( z ) + (cid:16) Q (3) (1 − z ) − Q (3) (cid:17) + G (2)12 ( z ) Q (1 − z ) + G ( z ) Q (2)12 + G (2)13 ( z ) Q (1 − z ) + G ( z ) Q (2)13 + G (2)23 ( z ) Q (1 − z ) + G ( z ) Q (2)23 i , (19a) G (3) y = Z x dz α s π Φ gg ( z ) h G (3) ( z ) − zG (3) + G (2)12 ( z ) G (1 − z ) + G (2)13 ( z ) G (1 − z )+ G (2)23 ( z ) G (1 − z ) i + Z x dz α s π n f Φ qg ( z ) h(cid:16) Q (3) ( z ) − G (3) (cid:17) + 2 Q (2)12 ( z ) Q (1 − z )+ 2 Q (2)13 ( z ) Q (1 − z ) + 2 Q (2)23 ( z ) Q (1 − z ) i . (19b)The l.h.s. of the equations (19a) and (19b) can be written in the convenient form ˆ A (3) = A (3) − A A A − ( A (2)12 − A A ) A − ( A (2)13 − A A ) A − ( A (2)23 − A A ) A , (20)5here A = G, Q, ¯ Q is the leading parton of the jet. Moreover, we have introduced the notations A ( n )1 ...n = A ( n )1 ...n (1) , where A ( n )1 ...n ≡ A ( n )1 ...n (1) = x . . . x n D ( n ) ( x , . . . , x n , Y ) , for the sake of simplicity. The evolution equations for the single inclusive distribution and the two-particle correlation are written in [19] in the form Q y = Z x dz α s π Φ gq ( z ) (cid:20)(cid:16) Q (1 − z ) − Q (cid:17) + G ( z ) (cid:21) , (21a) G y = Z x dz α s π (cid:20) Φ gg ( z ) (cid:16) G ( z ) − zG (cid:17) + n f Φ qg ( z ) (cid:16) Q ( z ) − G (cid:17)(cid:21) , (21b)and ( Q (2) − Q Q ) y = Z x dz α s π Φ gq ( z ) (cid:20) G (2) ( z ) + (cid:16) Q (2) (1 − z ) − Q (2) (cid:17) + (cid:16) G ( z ) − Q (cid:17)(cid:16) Q (1 − z ) − Q (cid:17) + (cid:16) G ( z ) − Q (cid:17)(cid:16) Q (1 − z ) − Q (cid:17)(cid:21) , (22a) ( G (2) − G G ) y = Z x dz α s π Φ gg ( z ) (cid:20)(cid:16) G (2) ( z ) − zG (2) (cid:17) + (cid:16) G ( z ) − G (cid:17)(cid:16) G (1 − z ) − G (cid:17)(cid:21) + Z x dz α s π n f Φ qg ( z ) (cid:20) (cid:16) Q (2) ( z ) − Q ( z ) Q ( z ) (cid:17) − (cid:16) G (2) − G G (cid:17) + (cid:16) Q ( z ) − G (cid:17)(cid:16) Q (1 − z ) − G (cid:17)(cid:21) , (22b)respectively. Making use of the equations (21a,21b) and (22a,22b), one can then construct the totalderivatives [ A A A ] y , h ( A (2)12 − A A ) A i y , h ( A (2)13 − A A ) A i y , h ( A (2)23 − A A ) A i y as they ap-pear in (20), which are to be subtracted, term by term from the system of equations (19a,19b). Therefore,we get the equivalent system for the three-particle correlations inside quark and gluon jets: ˆ Q (3) y = Z x dz α s π Φ gq ( z ) h G (3) ( z ) + (cid:16) Q (3) (1 − z ) − Q (3) (cid:17) (23a) + (cid:16) Q (2)12 (1 − z ) − Q (2)12 (cid:17) ( G ( z ) − Q ) + (cid:16) G (2)12 ( z ) − Q (2)12 (cid:17) ( Q (1 − z ) − Q )+ (cid:16) Q (2)13 (1 − z ) − Q (2)13 (cid:17) ( G ( z ) − Q ) + (cid:16) G (2)13 ( z ) − Q (2)13 (cid:17) ( Q (1 − z ) − Q )+ (cid:16) Q (2)23 (1 − z ) − Q (2)23 (cid:17) ( G ( z ) − Q ) + (cid:16) G (2)23 ( z ) − Q (2)23 (cid:17) ( Q (1 − z ) − Q )+ (( Q − G ( z )) ( Q (1 − z ) − Q ) + ( Q − G ( z )) ( Q (1 − z ) − Q )) Q + (( Q − G ( z )) ( Q (1 − z ) − Q ) + ( Q − G ( z )) ( Q (1 − z ) − Q )) Q + (( Q − G ( z )) ( Q (1 − z ) − Q ) + ( Q − G ( z )) ( Q (1 − z ) − Q )) Q ] , ˆ G (3) y = Z x dz α s π Φ gg ( z ) h(cid:16) G (3) ( z ) − zG (3) (cid:17) + (cid:16) G (2)12 ( z ) − G (2)12 (cid:17) ( G (1 − z ) − G ) (23b) + (cid:16) G (2)13 ( z ) − G (2)13 (cid:17) ( G (1 − z ) − G ) + (cid:16) G (2)23 ( z ) − G (2)23 (cid:17) ( G (1 − z ) − G )+ ( G − G ( z ))( G (1 − z ) − G ) G + ( G − G ( z ))( G (1 − z ) − G ) G + ( G − G ( z ))( G (1 − z ) − G ) G ] + Z x dz α s π n f Φ qg ( z ) h(cid:16) Q (3) ( z ) − G (3) (cid:17) + 2 (cid:16) Q (2)12 ( z ) − G (2)12 (cid:17) ( Q (1 − z ) − G ) + (2 Q ( z ) Q ( z ) − G G ) G + 2 (cid:16) Q (2)13 ( z ) − G (2)13 (cid:17) ( Q (1 − z ) − G ) + (2 Q ( z ) Q ( z ) − G G ) G (cid:16) Q (2)23 ( z ) − G (2)23 (cid:17) ( Q (1 − z ) − G ) + (2 Q ( z ) Q ( z ) − G G ) G + ( G − Q ( z ))(2 Q (1 − z ) − G ) G + ( G − Q ( z ))(2 Q (1 − z ) − G ) G + ( G − Q ( z ))(2 Q (1 − z ) − G ) G ] . The system of evolution equations (23a,23b), which appears as a consequence of the exact AO in intra-jetcascades, provides the complete theoretical picture of the three-particle correlations as a function of x i and the characteristic hardness of the jet Q ; this is the first new result of this paper. However, since theseequations could only be solved numerically, we will extract the SLs contributions O ( √ α s ) in order toprovide an approximated analytical solution in the following. Let us start with equation (23a). We proceed to cast all SLs contributions corresponding to hard-collinearparton splittings in the shower. In the hard parton fragmentation region one has z ∼ (1 − z ) ∼ , such thatthe second contribution in (23a) can be approximated through a Taylor series for ln z ∼ ln(1 − z ) ≪ ℓ ,written in the appendix A. Therefore, one obtains the simplified system of evolution equations ˆ Q (3) y = Z x dz α s π Φ gq ( z ) G (3) ( z ) , (24) ˆ G (3) y = Z x dz α s π (1 − z )Φ gg ( z ) G (3) ( z ) + Z x dz α s π n f Φ qg ( z ) h(cid:16) Q (3) − G (3) (cid:17) + 2 (cid:16) Q (2)12 − G (2)12 (cid:17) (25) × ( Q − G ) + (2 Q Q − G G ) G + 2 (cid:16) Q (2)13 − G (2)13 (cid:17) ( Q − G ) + (2 Q Q − G G ) G + 2 (cid:16) Q (2)23 − G (2)23 (cid:17) ( Q − G ) + (2 Q Q − G G ) G + ( G − Q )(2 Q − G ) G + ( G − Q )(2 Q − G ) G + ( G − Q )(2 Q − G ) G ] , where we have kept all terms of order O ( √ α s ) , which contribute to MLLA. In addition, from the DLArelation Z A = Z C A /N c G [27], and Eqs.(9-10), one has the useful expressions for the single inclusivedistribution, two- and three-particle correlations: Q i = C F N c G i , Q (2) ij = C F N c G (2) ij + C F N c (cid:18) C F N c − (cid:19) G i G j , i = j, (26) Q (3) = C F N c G (3) + C F N c (cid:18) C F N c − (cid:19)(cid:16) G (2)12 G + G (2)13 G + G (2)23 G (cid:17) + C F N c (cid:18) C F N c − (cid:19)(cid:18) C F N c − (cid:19) × G G G , (27)which in turn can be replaced in (25). The two expressions written in (26) are known from previousworks at DLA [16, 27], while (27) will be used for the first time in this context. After integrating overthe regular part of the splitting functions (2), (3) and (4), one obtains the integro-differential system ofequations ( η = η + η ), ˆ Q (3) y = C F N c Z ℓ dℓγ ( ℓ + y ) G (3) ( ℓ, y ; η ) − C F N c γ ( ℓ + y ) G (3) ( ℓ , y ; η ) , (28) ˆ G (3) y = Z ℓ dℓγ ( ℓ + y ) G (3) ( ℓ, y ; η ) − aγ ( ℓ + y ) G (3) ( ℓ , y ; η ) + ( a − b ) γ ( ℓ + y ) (29) × h(cid:16) G (2)12 ( ℓ , y + η ; η ) − G ( ℓ , y + η ) G ( ℓ + η , y + η ) (cid:17) G ( ℓ + η , y )+ (cid:16) G (2)13 ( ℓ , y ; η ) − G ( ℓ , y + η ) G ( ℓ + η , y ) (cid:17) G ( ℓ + η , y + η ) (cid:16) G (2)23 ( ℓ + η , y ; η ) − G ( ℓ + η , y + η ) G ( ℓ + η , y ) (cid:17) G ( ℓ , y + η ) i + ( a − c ) γ ( ℓ + y ) G ( ℓ , y + η ) G ( ℓ + η , y + η ) G ( ℓ + η , y ) , with the following hard constants, a ( n f ) = 14 N c (cid:20) N c + 43 n f T R (cid:18) − C F N c (cid:19)(cid:21) n f =3 = 0 . , (30) b ( n f ) = 14 N c " N c − n f T R (cid:18) − C F N c (cid:19) n f =3 = 0 . , (31) c ( n f ) = 14 N c " N c + 43 n f T R (cid:18) − C F N c (cid:19) n f =3 = 0 . , (32)where n f = 3 corresponds to the number of light active flavors of quarks u, d, s . As an example of suchprocedure, one could write the example, a ( n f ) = Z dz (cid:20) (1 − z ) (cid:16) − z (1 − z ) (cid:17) + n f T R C A (cid:16) z + (1 − z ) (cid:17) (cid:18) − C F N c (cid:19)(cid:21) . The first integral terms of the equations in (28) and (29) are of classical origin and therefore, universal.Corrections ∝ − , a , ( a − b ) and ( a − c ) , which are O ( √ α s ) suppressed, better account for energyconservation at each vertex of the splitting process, as compared with the DLA. Notice that the form ofthe quark initiated jet equation (28) is universal at MLLA (see (80) and (82 in the appendix A.1 for thesingle inclusive distribution and two-particle correlation respectively), that is, invariant with respect tothe number of particles considered in the cascade. In the equation for the gluon initiated jet (29), thefirst and second constants a ( n f ) and b ( n f ) were obtained in the frame of the single inclusive distributionand two-particle correlations respectively [17, 18]. The third constant c ( n f ) appears in this paper forthe first time for the three-particle correlation. In particular, notice that a certain recurrency shows upin the coefficients combining the colour factors ( − n − (cid:16) − C F N c (cid:17) n , as a function of the number n ofparticles considered in the shower. In this subsection we compute the leading order DLA contributions in order to provide general featuresconcerning the the shape and overall normalization of three-particle correlations. This procedure isequivalent to cast the leading order (LO) solution of the equations (28,29). We differentiate (28) and(29) with respect to “ ℓ ”, such that after setting hard corrections ∝ / , a, b, c = 0 , the MLLA evolutionequations are reduced to the new DLA compact differential equation h ˜ A (3) i ℓy = C A N c γ G (3) , (33)with h ˆ A (3) i ℓy = nh(cid:16) C (3) A − (cid:17) − (cid:16) C (2) A − (cid:17) − (cid:16) C (2) A − (cid:17) − (cid:16) C (2) A − (cid:17)i A A A o ℓy , (34)after having set A (3) = C (3) A A A A for the three-particle correlator and A (2) ij = C (2) A ij A i A j for the two-particle correlator. We fix the anomalous dimension to the characteristic hardness of the jet Q ≈ E Θ γ ( E Θ ) = const ) and solve this equation iteratively by derivating the r.h.s. of (34) with respect to ℓ and y , such that the solution of (33) reads (cid:16) ˙ C (3) A − (cid:17) − (cid:16) ˙ C (2) A − (cid:17) − (cid:16) ˙ C (2) A − (cid:17) − (cid:16) ˙ C (2) A − (cid:17) (35) = N c C A (cid:16) ˙ C (2) A − (cid:17) + (cid:16) ˙ C (2) A − (cid:17) + (cid:16) ˙ C (2) A − (cid:17) + ˜∆ + ˜∆ + N c C A
12 + ˜∆ + ˜∆ + ˜∆ , which have been written in terms of the logarithmic derivatives of the one-particle spectrum, ˜∆ ij = γ − (cid:0) ψ A i ,ℓ ψ A j ,y + ψ A i ,y ψ A j ,ℓ (cid:1) , ψ A i ,ℓ = 1 A i ∂A i ∂ℓ , ψ A i ,y = 1 A i ∂A i ∂y (36)and the DLA two-particle correlator [6, 16] (for a review see also [28]) ˙ C (2) A ij − N c C A
11 + ∆ ij . (37)The dot over C ( n ) differentiates the DLA correlators from the MLLA correlators obtained below. In DLAhowever, since the single inclusive distribution satisfies Q = C F N c G [27], one has ψ Q i ,ℓ = ψ G i ,ℓ ≡ ψ i,ℓ , ψ Q i ,y = ψ G i ,y ≡ ψ i,y . That is why, we will use the much simplest notation ψ G i ,ℓ = ψ i,ℓ , ψ G i ,y = ψ i,y . It is worth giving theorder of magnitude of some quantities that will be considered in forthcoming calculations. In DLA, theone-particle inclusive distribution can be written as A i ( ℓ, y ) ∝ exp (cid:0) γ √ ℓy (cid:1) asymptotically for fixedrunning coupling γ = const [27]. Though the solution with fixed coupling constant provides generalfeatures of the single inclusive distribution, it is not enough for the description of a more realistic pictureat colliders. However, from its simplicity, it can be used to give the order of magnitude of terms involvedin the solution of the DLA and MLLA evolution equations. Therefore, making use of (36), one has ψ A i ,ℓ = O ( γ ) , ψ A i ,y = O ( γ ) , ψ A i ,ℓℓ = O ( γ ) , ψ A i ,ℓy = O ( γ ) , ψ A i ,yy = O ( γ ) , (38) ˜∆ ij = O (1) , ˜∆ ij,ℓ = O ( γ ) , ˜∆ ij,y = O ( γ ) , (39)where ψ A i ,ℓℓ , ψ A i ,ℓy and ψ A i ,yy are double derivatives of ψ A i = ln A i ( ℓ, y ) . The DLA solution (35)describes the following picture: the first term (= − in the l.h.s. translates the independent or decor-related emission of three hadrons in the shower like depicted by Fig.2a. After inserting the two-particlecorrelator (37) in the l.h.s. of (35), terms ∝ N c C A correpond to the case where two partons are correlatedinside the same subjet, while the other one is emitted independently from the rest like in Fig.2b. Next, re-placing (37) in the r.h.s. of (35) one obtains a contribution ∝ N c C A described by Fig.2c, where two partonsare emitted independently inside the same subjet. The last term ∝ N c C A depicted by Fig.2d, involves threeparticles strongly correlated inside the same partonic shower and corresponds to the cumulant of genuinecorrelations. Actually, this interpretation has been given after computing the color factors of such Feyn-man diagrams describing the process, normalized by C A in the end. Notice that diagrams displayed inFig.2c and Fig.2d present the same color factors but different Lorentz structure. In both cases, the DLAstrong AO Θ ≫ Θ ′ ≫ Θ ′′ and strong energy ordering x ≫ x ≫ x are necessary conditions satisfiedby (33) [29]. 9 Figure 2: Three particles emitted inside the shower with color factors for the square of the amplitudes: C A , C A N c , C A N c and C A N c for a, b, c and d respectively.Performing the steepest descent evaluation of the DLA single inclusive distribution from an integralrepresentation, which was written in Mellin space in the form [16, 27], G ( ℓ, y ) = ( ℓ + y + λ ) Z Z dωdν (2 πi ) e ωℓ + νy Z ∞ dsν + s (cid:18) ω ( ν + s )( ω + s ) ν (cid:19) /β ( ω − ν ) e − λs , Q = C F N c G. (40)and which accounts for the running of the coupling α s , the energy of most particles inside the jet wasproved to be close to the maximum of the distribution, which shapes like a Gaussian in this region [27], A i ( ℓ i , Y ) ≃ exp (cid:20) − √ β ( ℓ i − ℓ max ) Y / (cid:21) , ℓ max ≈ Y . (41)From this method [16], the expressions of the logarithmic derivative of the one particle distribution werewritten as, ψ i,ℓ ( µ i , ν i ) = γ e µ i , ψ i,y ( µ, ν ) = γ e − µ i . (42)such that ∆ ij and the correlator were given in the form [16], ∆ ij = 2 cosh( µ i − ν j ) , ˙ C (2) A ij = 1 + N c C A
11 + 2 cosh( µ i − µ j ) (43)respectively, where ( µ i , ν i ) were related to ( ℓ i , y i ) through the 2x2 non-linear system of equations [16], y i − ℓ i y i + ℓ i = (sinh 2 µ i − µ i ) − (sinh 2 ν i − ν i )2(sinh µ i − sinh ν i ) , sinh ν i √ λ = sinh µ i √ ℓ i + y i + λ . (44)Therefore, the DLA three-particle correlator reads in this approximation ˙ C (3) A = 1 + (cid:16) ˙ C (2) A − (cid:17) + (cid:16) ˙ C (2) A − (cid:17) + (cid:16) ˙ C (2) A − (cid:17) (45) + N c C A (cid:16) ˙ C (2) A − (cid:17) + (cid:16) ˙ C (2) A − (cid:17) + (cid:16) ˙ C (2) A − (cid:17) µ − µ ) + cosh( µ − µ ) + cosh( µ − µ )+ N c C A
11 + cosh( µ − µ ) + cosh( µ − µ ) + cosh( µ − µ ) . with ˙ C (2) A ij extracted from (43). Taking | ℓ i − ℓ max |≪ σ ∝ Y / for i = 1 , , , one has in thisapproximation (see appendix C.2) ∆ ij ≈ (cid:18) ℓ i − ℓ j Y (cid:19) = 2 + 9 (cid:20) ln( x j /x i )ln( Q/Q ) (cid:21) , (46)10o that, ∆ + ∆ + ∆ ≈ (cid:18) ℓ − ℓ Y (cid:19) + 9 (cid:18) ℓ − ℓ Y (cid:19) + 9 (cid:18) ℓ − ℓ Y (cid:19) = 6 + 9 (cid:20) ln( x /x )ln( Q/Q ) (cid:21) + 9 (cid:20) ln( x /x )ln( Q/Q ) (cid:21) + 9 (cid:20) ln( x /x )ln( Q/Q ) (cid:21) . (47)Therefore, the shape of the three-particle correlator can be expected to be quadratic as a function of thedifference ( ℓ i − ℓ j ) , as for the two-particle correlator. Thus, the correlator is strongest when particleshave the same energy x = x = x .Moreover, the decreasing behavior of the correlator as one parton gets much harder than the others x i ≫ x j shows that QCD coherence effects dominate this region of the phase space as interferencesbetween such gluons occur. New kinds of contributions like the one in the first term of the r.h.s. of (35)appear in this context.The overall normalization of the n -particle correlator is fixed by that of the same rank multiplicity-correlator determining the multiplicity fluctuations inside the jet [16], C ( k ) A ( x , . . . , x k ) ≤ h n ( n − . . . ( n − k + 1) ih n i k . Then, one has C (2) A ( x , x ) − ≤ N c C A , C (3) A ( x , x , x ) − ≤ N c C A + N c C A . (48)These bounds can also be obtained by setting ∆( x i , x j ) = 2 (for x i = x j ) in (37) and (35) respectively.Since DLA neglects the energy balance, it is not realistic and does not provide the real physical pictureof any jet process in the frame of jet calculus. As we can see, the computation of three-particle correlations requires a mastering knowledge of theone-particle inclusive energy distribution and two-particle correlations. The behavior of the two-particlecorrelators as shown by these solutions was proved to be quadratic as a function of ( ℓ i − ℓ j ) and increasingas a function of ( ℓ i + ℓ j ) like in the Fong-Webber approximation [17,18]. However, the solutions (92,93)(see appendix A.1) were shown to better account for soft gluon coherence effects, by describing theflatting of the slopes as ( ℓ i + ℓ j ) increases. In [25], the solution was obtained by the steepest descentevaluation of the spectrum G i ( ℓ, y ) , while in [19], the evaluation was performed by taking the expressionof G i ( ℓ, y ) given by (89) in the appendix A.1. In [19], the solution of the evolution equations for two-particle correlation were obtained from the differential version of the equations (90,91) over ℓ and y written in the appendix A.1. Therefore, in this subsection, we will make some transformations in orderto simplify this cumbersome task without adding further information. In the appendix A.1, we brieflysummarize what should be known in order to complete the solution of the evolution equations for thethree-particle correlations.Differentiating (28) and (29) with respect to “ ℓ ” , one has the differential system of evolution equations11or three-particle correlations, ˆ Q (3) ℓy = C F N c γ G (3) − C F N c γ (cid:16) G (3) ℓ − β γ G (3) (cid:17) , (49) ˆ G (3) ℓy = γ G (3) − aγ (cid:16) G (3) ℓ − β γ G (3) (cid:17) +( a − b ) γ nh(cid:16) G (2)12 − G G (cid:17) G (50) + (cid:16) G (2)13 − G G (cid:17) G + (cid:16) G (2)23 − G G (cid:17) G i ℓ − β γ h(cid:16) G (2)12 − G G (cid:17) G + (cid:16) G (2)13 − G G (cid:17) G + (cid:16) G (2)23 − G G (cid:17) G io +( a − c ) γ (cid:2) ( G G G ) ℓ − β γ G G G (cid:3) , which is written in this paper for the first time. The equation (50) is self-contained and can be solvediteratively like (33). For this purpose, one sets G (3) = C (3) G G G G and G (2) ij = C (2) G ij G i G j in theleft and right hand sides of (50), such that the solution obtained in the appendix B can be written in thecompact form C (3) G − (cid:16) C (2) G − (cid:17) F (2)12 + (cid:16) C (2) G − (cid:17) F (2)13 + (cid:16) C (2) G − (cid:17) F (2)23 + F (3)123 , (51)where, F (2) ij = 1 + N (2) G ij D (2) G , F (3)123 = N (3) G D (3) G , (52)with N (2) G ij = 1 − b (cid:0) ψ ,ℓ + ψ ,ℓ + ψ ,ℓ − β γ (cid:1) − aζ ℓ + ( a − b ) χ ijℓ + ξ ij + δ ij − ǫ − ǫ , (53a) D (2) G = 2 + ∆ + ∆ + ∆ + aζ ℓ + 2 aβ γ + ǫ + ǫ , (53b) N (3) G = 1 − c (cid:0) ψ ,ℓ + ψ ,ℓ + ψ ,ℓ − β γ (cid:1) − aζ ℓ + ( a − b )( χ ℓ + χ ℓ + χ ℓ ) + ( ξ + δ ) (53c) + ( ξ + δ ) + ( ξ + δ ) − ǫ − ǫ ,D (3) G = D (2) G = 2 + ∆ + ∆ + ∆ + aζ ℓ + 2 aβ γ + ǫ + ǫ . (53d)The solution (51) can be checked to recover the DLA result (35) inside a gluon jet, that is for C A = N c .Since DLA neglects recoil effects at each splitting inside the cascade, one should expect the DLA three-particle correlation to be much larger than MLLA predictions and therefore to overestimate the data. Weintroduce the following notations and give the order of magnitude of each contribution following from(38) and (39), ζ = ln ˙ C (3) G , ζ ℓ = ˙ C (3) G ,ℓ ˙ C (3) G = O ( γ ) , ζ y = ˙ C (3) G ,y ˙ C (3) G = O ( γ ) , (54a) χ ijℓ = ˙ C (2) G ij ,ℓ ˙ C (2) G ij = O ( γ ) , χ ijy = ˙ C (2) G ij ,y ˙ C (2) G ij = O ( γ ) , (54b) ξ ij = 1 γ h χ ijℓ ( ψ ,y + ψ ,y + ψ ,y ) + χ ijy ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) i = O ( γ ) , (54c) δ ij = 1 γ (cid:16) χ ijℓ χ ijy + χ ijℓ,y (cid:17) = O ( γ ) , (54d) ǫ = 1 γ [ ζ ℓ ( ψ ,y + ψ ,y + ψ ,y ) + ζ y ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ )] = O ( γ ) , (54e) ǫ = 1 γ ( ζ ℓ ζ y + ζ ℓ,y ) = O ( γ ) . (54f)12he solution of the gluon evolution equation for the correlator can be either obtained numerically bysolving (50) or by performing the evaluation from the previous solution (51). However, in this paper,we will directly compute the solution (51) from the steepest descent method introduced in [25] andmake some approximations in subsection 2.6. Accordingly, the solution of (49) is also obtained inthe appendix B by setting Q (3) = C (3) Q Q Q Q and Q (2) ij = C (2) Q ij Q i Q j in the l.h.s. of (49) and G (3) = C (3) G G G G in the r.h.s. of the same equation, such that, C (3) Q − (cid:16) C (2) Q − (cid:17) ˜ F (2)12 + (cid:16) C (2) Q − (cid:17) ˜ F (2)13 + (cid:16) C (2) Q − (cid:17) ˜ F (2)23 + ˜ F (3)123 , (55)where, ˜ F (2) ij = 1 + N (2) Q ij D (2) Q , ˜ F (3)123 = N (3) Q D (3) Q , (56)with N (2) Q ij = ˜ ξ ij + ˜ δ ij − ˜ ǫ − ˜ ǫ , (57a) D (2) Q = ˜∆ + ˜∆ + ˜∆ + X i Q iℓy γ Q i + ˜ ǫ + ˜ ǫ , (57b) N (3) Q = C F N c C (3) G (cid:20) − (cid:0) ψ ,ℓ + ψ ,ℓ + ψ ,ℓ + ζ ℓ − β γ (cid:1)(cid:21) G G G Q Q Q + ( ˜ ξ + ˜ δ ) (57c) + ( ˜ ξ + ˜ δ ) + ( ˜ ξ + ˜ δ ) − ˜ ǫ − ˜ ǫ ,D (3) Q = D (2) Q = ˜∆ + ˜∆ + ˜∆ + X i Q iℓy γ Q i + ˜ ǫ + ˜ ǫ , (57d)where one find the list of corrections, ˜ ζ = ln ˙ C (3) Q , ˜ ζ ℓ = ˙ C (3) Q ,ℓ ˙ C (3) Q = O ( γ ) , ˜ ζ y = ˙ C (3) Q ,y ˙ C (3) Q = O ( γ ) , (58a) ˜ χ ijℓ = ˙ C (2) Q ij ,ℓ ˙ C (2) Q ij = O ( γ ) , ˜ χ ijy = ˙ C (2) Q ij ,y ˙ C (2) Q ij = O ( γ ) , (58b) ˜ ξ ij = 1 γ h ˜ χ ijℓ ( ψ Q ,y + ψ Q ,y + ψ Q ,y ) + ˜ χ ijy ( ψ Q ,ℓ + ψ Q ,ℓ + ψ Q ,ℓ ) i = O ( γ ) , (58c) ˜ δ ij = 1 γ (cid:16) ˜ χ ijℓ ˜ χ ijy + ˜ χ ijℓ,y (cid:17) = O ( γ ) , (58d) ˜ ǫ = 1 γ h ˜ ζ ℓ ( ψ Q ,y + ψ Q ,y + ψ Q ,y ) + ˜ ζ y ( ψ Q ,ℓ + ψ Q ,ℓ + ψ Q ,ℓ ) i = O ( γ ) , (58e) ˜ ǫ = 1 γ (cid:16) ˜ ζ ℓ ˜ ζ y + ˜ ζ ℓ,y (cid:17) = O ( γ ) . (58f)The order of magnitude of these terms follows from (38) and (39). Setting all corrections to zero, onerecovers the DLA solution (35) for C A = C F . The solutions (51) and (55) of the evolution equations en-tangle corrections of order O ( γ ) and O ( γ ) , which are MLLA and NMLLA respectively. Furthermore,every term in (51) and (55) can be associated to a Feynman diagram of Fig.2 as was explained in subsec-tion 2.4. The functions F (3)123 and ˜ F (3)123 in (51) and (55) correspond respectively to the cumulant of genuinecorrelations associated to the process displayed in Fig.1 and Fig.2d. These contributions, (54a-54f) and(58a-58f) are small corrections arising from the iterative solution of the evolution equations because one13akes the derivatives over the functions ζ = ln ˙ C (3) G , ˜ ζ = ln ˙ C (3) Q and χ ij = ln ˙ C (2) G ij , ˜ χ ij = ln ˙ C (2) Q ij forboth quark and gluon jets. For the evaluation of such corrections one needs to take the DLA expressionsof ˙ C (3) A and ˙ C (2) A ij written in (35) and (37) respectively. In [19], the exact solutions of the two-particle evolution equations were compared with the MLLA so-lutions from the steepest descent method for the one particle distribution. The agreement between bothapproaches was successful and made possible the fast computation of the correlators from the steepestdescent. That is the reason for in this paper, we limit ourselves to this method. Making use of the ratio(87), it is easy to demonstrate that, ψ Q,ℓ = ψ ℓ + O ( γ ) , ψ Q,y = ψ y + O ( γ ) , ˜∆ ij mlla = ∆ ij + O ( γ ) . (59)Dropping corrections of order O ( γ ) , which go beyond the MLLA approximation, we obtain for thegluon jet F (2) ij mlla = 1 + 1 − b ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) + ξ ij − ǫ + ∆ + ∆ + ǫ , (60) F (3)123 mlla = 1 − c ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) + ξ + ξ + ξ − ǫ + ∆ + ∆ + ǫ (61)and for the quark jet ˜ F (2) ij mlla = 1 + ˜ ξ ij − ˜ ǫ + ∆ + ∆ − a ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) + ˜ ǫ , (62) ˜ F (3)123 mlla = N c C F C (3) G [1 − a ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ )] + ˜ ξ + ˜ ξ + ˜ ξ − ˜˜ ǫ + ∆ + ∆ − a ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) + ˜ ǫ . (63)The subtracted terms ∝ − a in the denominators of (62) and (63) appear after having replaced (87) and(88) in (57b) and (57c) respectively. Such simplified expressions are useful for the steepest descent eval-uation that proved successful while describing the single inclusive distribution and two-particle correla-tions in [25]. Except the MLLA corrections ǫ and ξ ij , all the other corrections and functions appearingin the solutions of the evolution equations were obtained in [25], which will allow for the straightforwardcomputation of the three-particle correlators in quark and gluon jets. We write some of these formulæfor the evaluation in the appendix C. Integrating the equation (81) over “ y ”, the solution for the singleinclusive distribution is given by the following integral representation in Mellin space [25], G ( ℓ, y ) = ( ℓ + y + λ ) Z Z dωdν (2 πi ) e ωℓ + νy Z ∞ dsν + s (cid:18) ω ( ν + s )( ω + s ) ν (cid:19) /β ( ω − ν ) (cid:18) νν + s (cid:19) a/β e − λs . (64)The integral representation (64) was estimated by the steepest descent method at small x ≪ andhigh energy scale Q ≫ ; the approached solution was compared with the exact solution (89) (seethe appendix A.1) in the limiting spectrum ( λ = 0 ) and beyond ( λ = 0 ). In particular, (64) was alsodemonstrated to be equivalent to (89) for λ = 0 [19]. The agreement between the approached andexact solutions turned out to be good, such that the following expressions of the approached logarithmic14erivatives from the steepest descent method were suited for the evaluation of the two-particle correlators[25], ψ i,ℓ ( µ i , ν i ) = γ e µ i + 12 aγ h e µ i ˜ Q ( µ i , ν i ) − tanh ν i − tanh ν i coth µ i (cid:16) e µ i ˜ Q ( µ i , ν i ) (cid:17)i (65) − β γ h ν i (cid:16) K ( µ i , ν i ) (cid:17) + C ( µ i , ν i ) (cid:16) e µ i ˜ Q ( µ i , ν i ) (cid:17)i + O ( γ ) ,ψ i,y ( µ, ν ) = γ e − µ i − aγ h e − µ i ˜ Q ( µ i , ν i ) + tanh ν i − tanh ν i coth µ i (cid:16) e − µ i ˜ Q ( µ i , ν i ) (cid:17)i − β γ h ν i (cid:16) K ( µ i , ν i ) (cid:17) − C ( µ i , ν i ) (cid:16) e − µ i ˜ Q ( µ i , ν i ) (cid:17)i + O ( γ ) , (66)where the functions ˜ Q ( µ i , ν i ) , C ( µ i , ν i ) and K ( µ i , ν i ) are defined in the appendix C. The term ∝ a in(65) and (66) accounts for energy conservation while that ∝ β accounts for the running of the coupling α s . The variables ( µ i , ν i ) are related to ( ℓ i , y i ) through the same 2x2 non-linear system of equations (44).After inverting (44) numerically, µ i ( ℓ i , y i ) and ν i ( ℓ i , y i ) can be plugged into (65) and (66) so as to get thelogarithmic derivatives of the single inclusive spectrum as a function of the original kinematical variables ℓ i and y i as it was done in [25]. The MLLA two-particle correlators involved in (51) and (55) are (108)and (109) and are written in the appendix C. These expressions have been taken from reference [25].Corrections ξ ij , ˜ ξ ij and ǫ , ˜ ǫ are new for three-particle correlations. Such expressions are explicitlywritten in the appendix C.1 from the steepest descent evaluation of the single inclusive distribution(64). They are small and decrease the three-particle correlator for ℓ i = ℓ j , that is when one partonis much harder than the other. Notice that the steepest descent method constitutes the only way for thedisentanglement between MLLA O ( √ α s ) and NMLLA O ( α s ) corrections appearing in the solution ofthe evolution equations for the two and three-particle correlations. It makes also possible to distinguishbetween corrections following from the energy balance and the running effects of the coupling constant α s . Finally, this method also allows for the application of the hump approximation or Fong-Webberexpansion of the solutions with MLLA O ( √ α s ) accuracy [17, 18].In this frame, the role of MLLA corrections should be expected to be larger than for the two-particlecorrelations. Indeed, higher order corrections increase with the rank of the correlator, which is knownfrom the Koba-Nielsen-Olesen (KNO) problem for intra-jet multiplicity fluctuations [28, 30, 31]. For the2-particle for instance one has ∝ − b ( ψ ,ℓ + ψ ,ℓ ) and for the three-particle correlator one gets the largercorrection ∝ − c ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) . From the steepest descent evaluation introduced in [25], near the hump of the single inclusive distribution | ℓ − Y / |≪ σ ∝ Y / for i = 1 , , , corrections ξ ij , ˜ ξ ij and ǫ , ˜ ǫ could be written in the symbolicform (see appendix C.2), ξ ij , ˜ ξ ij ≃ (cid:18) ℓ i − ℓ j Y (cid:19) γ + O ( γ ) , (67) ǫ , ˜ ǫ ≃ (cid:18) ℓ − ℓ Y (cid:19) γ + (cid:18) ℓ − ℓ Y (cid:19) γ + (cid:18) ℓ − ℓ Y (cid:19) γ + O ( γ ) , (68)such that both can be neglected ξ ij ≈ , ǫ ≈ in this approximation, like δ ij was also in [25]. In theappendix C.2, following from the steepest descent method, the expressions of (53a-53d) are given and1557a-57d) expanded in √ α s . In particular, the expressions (128e) and (128g), after being expanded in γ , can be demonstrated to recover the Fong-Webber results for the two-particle correlations [17, 18].Replacing the expressions (128a-128j) into (51,52) and (55,56), one finds those for the three-particlecorrelators in the Fong-Webber approximation [17, 18]. This solution will be compared with that from(61) and (63) after making use of (65) and (66) in subsection 3. x region In [19], the sign of the two-particle correlator ( C (2) A − ≥ ) was studied as a function of x in the regionof the phase space where the two partons (hadrons after assuming the LPHD) are strongly correlated.From the previous inequality, it turned out that two partons with ℓ i & . ( x i . . ) at LHC energyscales (i.e. Q = 450 GeV, see subsection 3) are correlated as they are emitted from the same cascadefollowing the QCD AO. Asymptotically Y → ∞ , one has ℓ i & . ( x i . . ).For three-particle correlations we study the sign of the cumulant of the genuine correlator F (3)123 > anddetermine the approximate region in x where diagrams displayed in Fig.1 and Fig.2d become dominant.One has, − c ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) + ξ + ξ + ξ − ǫ > . However, corrections ξ ij , ǫ have been shown to be negligible and to vanish for particles having the sameenergy momentum. Thus, we rather study the sign of − c ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) > . Making use of ψ ℓ = γ q yℓ = γ q Y − ℓℓ for the sake of simplicity, one has, − cγ r Y − ℓℓ > ⇔ ℓ > M MY , M = 9 c β = 10 . . Thus, for LHC energy Y = 7 . , the value of ℓ ( x ) where the cumulant becomes positive turns out tobe ℓ & . , which in x corresponds to x . . . Asymptotically Y → ∞ , one has ℓ i & . ( x i . . × − ). Therefore, there exists a range in x where the observable C (3)123 is dominated by theemission of two correlated partons emitted independently from the third one, that is . . x . . for diagrams Fig.2b and Fig.2c; for x . . , the process will be dominated by three particles emittedfrom the same partonic cascade following the QCD AO described in Fig.2d. Asymptotically Y → ∞ ,one has . × − . x . . for diagrams Fig.2b and Fig.2c, and x . . × − for Fig.2d. Thesevalues will indeed justify our choices for the representation of the three-particle correlations as functionof ( x , x , x ) in subsection 3. It is worth reminding that the LPHD hypothesis has also been confronted to multi-particle factorialmoments up to the 5th order in the experimental studies of ep and e + e − collisions at HERA [34] and LEP[35] respectively, where it was found that the LPHD hypothesis faces difficulties when it is applied to softmulti-particle fluctuations. In this work the studies are carried out by using the momentum and transversemomentum cuts in order to test the MLLA soft limit calculations [33]. The theoretical computation of16ultiplicity correlators or multiplicity fluctuations h n ( n − . . . ( n − k + 1) i was performed in [32] atMLLA up to the rank k = 5 of the correlator.However, performing these calculations for higher rank differential inclusive correlators, related to theprevious ones by the integral h n ( n − . . . ( n − k + 1) i A = Z dx . . . dx k x . . . x k D ( k ) A ( x , . . . , x k , Y ) becomes rather cumbersome. As an example, in this subsection, we display the DLA equation andsolution of the 4-particle correlator. The DLA equation reads, ˆ A (4)1234 = C A N c γ G (4)1234 , (69)where ˆ A has been defined in the appendix D in (129). The solution of (69) with the definition of ˆ A (129)reads, C (4) A − N c C A H (cid:16) ˙ C (2) (cid:17) + N c C A H (cid:16) ˙ C (3) , ˙ C (2) (cid:17) + N c C A H (cid:16) ˙ C (3) , ˙ C (2) (cid:17) + ∆ + ∆ + ∆ + ∆ + ∆ , (70)where the functions H , H and H are written in the appendix D in (130), (131) and (132) respectively.The solution (70) can also be interpreted in terms of Feynman diagrams contributing to the emission offour hadrons inside the jet. Accordingly, the term ∝ N c C A correspond to the case A → wheretwo offspring are correlated while the other two are emitted independently; as a consequence it dependsonly on the two-particle correlator. The second term ∝ N c C A is associated to the cases A → (12)(34) and A → (123)4 , which translates into either emitting two sub-jets with two-particles correlated withineach, or emitting three correlated partons like in Fig.1 with another independent emission. Finally, theterm ∝ N c C A after setting H = 1 + . . . corresponds to the full correlated emission of four offspring insidethe same shower. The inclusion of SLs corrections to (70) would be cumbersome and stays beyond thescope of this paper. On the other hand, the computation of differential higher order rank ( k ) correlatorsat MLLA would imply the failure of the perturbative approach because of the increasing size of higherorder corrections ∝ ( ψ ,ℓ + . . . ψ k,ℓ ) = O ( √ α s ) . Hence, for higher order k correlators, the small x rangewhere MLLA predictions stay valid gets reduced even at high energy scales, such that (see subsection2.8) M k = k c k β , ℓ k > M k M k Y with c k = 14 N c " N c + ( − k n f T R (cid:18) − C F N c (cid:19) k . In this section, we perform theoretical predictions for three-particle correlations for the LHC. We displaythe MLLA solutions (51) and (55) of the evolution equations (50) and (49) respectively. We comparethe DLA solution of the evolutions equations from section 2.4 with the MLLA solution from the steepestdescent evaluation of the one-particle distribution in subsection 2.6 and the solution from the humpapproximation in 2.7. Thus, 17 the DLA solution is computed by plugging (43) into (45); • the MLLA solution from the steepest descent will be displayed by substituting the MLLA two-particle correlators (108), (109) and the functions (60), (61), (62) and (63) into (51) and (55) forgluon and quark jets respectively; • the MLLA hump approximation will be displayed by plugging (128a)-(128j) into (52) and (56)and finally (51) and (55).In particular, the computation of the DLA and MLLA solutions from the steepest descent needs the priorinversion of the system of equations (44) in order to obtain ( µ i , ν i ) as functions of the original kinematicalvariables ( ℓ i , y i ). The correlators are functions of the variables ( ℓ i , y i ) and the virtuality of the jet Q = E Θ . After setting y i = Y − ℓ i with fixed Y = ln( Q/Q ) in the arguments of the solutions (51)and (55) the dependence can be reduced to the following: C (3) G ( ℓ , ℓ , ℓ , Y ) and C (3) Q ( ℓ , ℓ , ℓ , Y ) . λ ≈ In this subsection we give predictions within the limiting spectrum λ . . for charged hadrons mostlycomposed by pions and kaons.In Fig.3, the DLA (35), MLLA hump approximation from subsection 2.7 and MLLA (51) three-particlecorrelators are displayed, as a function of the difference ( ℓ − ℓ ) = ln( x /x ) for two fixed values of ℓ = ln(1 /x ) = 4 . , . , fixed sum ( ℓ + ℓ ) = | ln( x x ) | = 10 and finally fixed Y = 7 . (virtuality Q = 450 GeV and Λ QCD = 250
MeV), which is realistic for the LHC phenomenology [13]. The values ℓ = ln(1 /x ) = 4 . , . ( x = 0 . , x = 0 . ) have been chosen according to the range of theenergy fraction x i ≪ . , where the MLLA scheme can only be applied and in particular, the range x . . , where the cumulant correlator F (3)123 is dominant (see subsection 2.8).In Fig.4, the DLA (35), MLLA hump approximation from subsection 2.7 and MLLA (51) three-particlecorrelators are displayed, in this case, as a function of the sum ( ℓ + ℓ ) = | ln( x x ) | for the samevalues of ℓ = ln(1 /x ) = 4 . , . , for x = x and Y = 7 . . The range . ≤ | ln( x x ) | ≤ . hasbeen chosen according to the condition x . . discussed in 2.8.As expected in both cases, the DLA and MLLA three-particle correlators are larger inside a quark thanin a gluon jet. Of course, these plots will be the same and the interpretation will apply to all possible per-mutations of three particles (123). As observed and written above, the difference between the DLA andMLLA results is quite important pointing out that overall corrections in O ( √ α s ) are quite large. Indeed,the last behavior is not surprising as was already observed on the treatment of multiplicity fluctuationsof the third kind, where [32] h n ( n − n − i G h n i G = 2 .
25 [1 − (1 . − . n f ) √ α s ] , h n ( n − n − i Q h n i Q = 4 .
52 [1 − (2 . − . n f ) √ α s ] . For instance, for one quark jet produced at the Z peak of the e + e − annihilation ( Q = 45 . GeV), onehas α s = 0 . . Replacing this value into the previous formula for a quark jet multiplicity correlator,one obtains a variation from 4.52 (DLA) to 0.83 (MLLA). That is one of the reasons for DLA has been18 Y=7.5 ; λ =0.1 ; |Ln(x x )|=10 C ( ) G Ln(x /x ) DLA Ln(1/x )=4.5DLA Ln(1/x )=5.5Hump Ln(1/x )=4.5Hump Ln(1/x )=5.5MLLA Ln(1/x )=4.5MLLA Ln(1/x )=5.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.512345678 Y=7.5 ; λ =0.1 ; |Ln(x x )|=10 C ( ) Q Ln(x /x ) DLA Ln(1/x )=4.5DLA Ln(1/x )=5.5Hump Ln(1/x )=4.5Hump Ln(1/x )=5.5MLLA Ln(1/x )=4.5MLLA Ln(1/x )=5.5 Figure 3: Three-particle correlations inside a gluon jet (left) and a quark jet (right) as a function of ℓ − ℓ = ln( x /x ) for ℓ + ℓ = | ln( x x ) | = 10 , ℓ = ln(1 /x ) = 4 . , . , fixed Y = 7 . in thelimiting spectrum approximation λ ≈ .known to provide unreliable predictions which should not be compared with experiments. From Fig.3,the correlation are observed to be the strongest when particles have the same energy x i = x j for fixed x k and to decrease when one parton is much harder the others. Indeed, in this region of the phasespace two competing effects should be satisfied: on one hand, as a consequence of gluon coherence andAO, gluon emission angles should decrease and on the other hand, the convergence of the perturbativeseries k ⊥ = x i E Θ i ≥ Q should be guaranteed. That is why, as the collinear cut-off parameter Q is reached, gluons are emitted at larger angles and destructive interferences with previous emissionsoccur. Moreover, the observable increases for softer partons with x decreasing, which is for partons lesssensitive to the energy balance. In Fig.4 the MLLA correlations increase for softer partons, then flattenand decrease as a consequence of soft gluon coherence, reproducing for three-particle correlations, thehump-backed shape of the one-particle distribution. Because of the limitation of the phase space, onehas C (3) ≤ for harder partons. Finally, in Fig.5, we display the three-particle correlators as functionof the sum | ln( x x x ) | , for x = x = x ; when compared with Fig.4 and Fig.3, the correlators areshown to be larger. That is why, and as expected, the correlations are the strongest for particles havingthe same energy-momentum x = x = x . In these figures, the MLLA hump approximation is seento become larger than the DLA correlator for smaller values of x than those close to the hump region,which is unphysical. This is due to the fact that this approximation should not be trusted beyond thehump region | ℓ − Y / |≪ σ ∝ Y / , Y / . in this case.The MLLA hump approximation from subsection 2.7 is observed to be larger than the MLLA solutionfrom the steepest descent of the one-particle distribution but one should bear in mind that this is onlyan approximation made for the sake of clarity in the interpretation of the solutions. In particular, fromFig.3 one can observe a smoother descent for the slope of the correlators in this case than that given fromthe more exact steepest descent. This difference comes from the role played by the iterative correctionsdisplayed in Fig.8, which decrease the correlators away from the hump region when one of the partonsbecomes harder than the others. Near the maximum x i = x j of the correlators, the difference between19 Y=7.5 ; λ =0.1 ; Ln(x /x )=0 C ( ) G |Ln(x x )| DLA Ln(1/x )=4.5DLA Ln(1/x )=5.5Hump Ln(1/x )=4.5Hump Ln(1/x )=5.5MLLA Ln(1/x )=4.5MLLA Ln(1/x )=5.5 7 8 9 10 11 12 13012345678910 Y=7.5 ; λ =0.1 ; Ln(x /x )=0 C ( ) Q |Ln(x x )| DLA Ln(1/x )=4.5DLA Ln(1/x )=5.5Hump Ln(1/x )=4.5Hump Ln(1/x )=5.5MLLA Ln(1/x )=4.5MLLA Ln(1/x )=5.5 Figure 4: Three-particle correlations inside a gluon jet (left) and a quark jet (right) as a function of ℓ + ℓ = | ln( x x ) | for x = x , ℓ = ln(1 /x ) = 4 . , . , fixed Y = 7 . in the limiting spectrumapproximation λ ≈ .the two approaches is O (cid:16) ℓ k Y γ (cid:17) and should decrease for x i → , according to (68). λ = 0 The approximated evaluation of the one-particle distribution from the steepest descent method madepossible the evaluation of the two-particle correlations beyond the limiting spectrum approximation, thatis for Q = Λ QCD . Accordingly, it makes also possible the evaluation of the three-particle correlators C (3) G ( ℓ , ℓ , ℓ , Y ) and C (3) Q ( ℓ , ℓ , ℓ , Y ) beyond this limit λ = 0 . This parameter, also known ashadronization parameter, guarantees in particular the convergence of the perturbative approach α s ≪ .In Fig.6 and Fig.7 we display the same set of curves beyond the limiting spectrum ( λ = 1 . ) as inFig.3 and Fig.4 in the limiting spectrum ( λ ∼ ), with the exception of curves coming from the humpapproximation. The value of λ in this case was evaluated for Q ∼ GeV, which corresponds to theproton mass, and Λ QCD = 250
MeV. As observed the correlation increases with λ and the range where C (3) ≥ becomes larger in this case. In this paper we provide the first full pQCD treatment of three-particle correlations in parton showersand a further refined test of the LPHD within the limiting spectrum approximation and beyond. Theevolution equations satisfied by this differential observable have been obtained for the first time andthe differential version of the equations has been solved iteratively. It has been possible to interpretthe analytical solution in terms of Feynman diagrams describing the process and to evaluate it from thesteepest descent method applied to the single inclusive distribution. The correlations have been displayedin the range x . . , where the process is dominated by three particles emitted from the same partoniccascade following the QCD AO described in Fig.1 and Fig.2d. Furthermore, four-particle correlations20 Y=7.5 ; λ =0.1 ; x =x =x C ( ) G )| DLAMLLAHump 11 12 13 14 15 16 17 18 19 20 210123456
Y=7.5 ; λ =0.1 ; x =x =x C ( ) Q )| DLAMLLAHump
Figure 5: Three-particle correlations inside a gluon jet (left) and a quark jet (right) as a function of ℓ + ℓ + ℓ = | ln( x x x ) | for x = x = x , fixed Y = 7 . in the limiting spectrum approximation λ ≈ . −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.511.251.51.7522.252.5 Y=7.5 ; λ =1.5 ; |Ln(x x )|=10 C ( ) G Ln(x /x ) DLA Ln(1/x )=4.5DLA Ln(1/x )=5.5MLLA Ln(1/x )=4.5MLLA Ln(1/x )=5.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.511.522.533.544.55 Y=7.5 ; λ =1.5 ; |Ln(x x )|=10 C ( ) Q Ln(x /x ) DLA Ln(1/x )=4.5DLA Ln(1/x )=5.5MLLA Ln(1/x )=4.5MLLA Ln(1/x )=5.5 Figure 6: Three-particle correlations inside a gluon jet (left) and a quark jet (right) as a function of ℓ − ℓ = ln( x /x ) for ℓ + ℓ = | ln( x x ) | = 10 , ℓ = ln(1 /x ) = 4 . , . , fixed Y = 7 . in thelimiting spectrum approximation λ = 1 . . 21 Y=7.5 ; λ =1.5 ; Ln(x /x )=0 C ( ) G |Ln(x x )| DLA Ln(1/x )=4.5DLA Ln(1/x )=5.5MLLA Ln(1/x )=4.5MLLA Ln(1/x )=5.5 7 8 9 10 11 12 130.511.522.533.544.55 Y=7.5 ; λ =1.5 ; Ln(x /x )=0 C ( ) Q |Ln(x x )| DLA Ln(1/x )=4.5DLA Ln(1/x )=5.5MLLA Ln(1/x )=4.5MLLA Ln(1/x )=5.5 Figure 7: Three-particle correlations inside a gluon jet (left) and a quark jet (right) as a function of ℓ + ℓ = | ln( x x ) | for x = x , ℓ = ln(1 /x ) = 4 . , . , fixed Y = 7 . in the limiting spectrumapproximation λ = 1 . .have been computed at DLA so as to show that the inclusion of higher order corrections for more thanthree particles would rather be a cumbersome task. The correlations have been shown to be strongestfor the softest hadrons having the same energy x = x = x in both quark and gluon jets, increasingas a function of ln( x i /x j ) and | ln( x i x j ) | when x k softens, that is for partons being less sensitive to theenergy balance.Coherence effects appear when one or two of the partons involved in the process is harder than theothers, thus reproducing for this observable the hump-backed shape of the one particle distribution.Away from the maximum at x i = x i , because of limitation of the phase space, one has C (3) ≤ .Predictions beyond the limiting spectrum for heavier charged hadrons as compared with pions and kaonsshow that the correlations should increase as the parameter Q equals the mass of such hadrons and therange where C (3) ≥ has been enlarged beyond this limit. The last statement is not surprising becausesoft gluon emission gets suppressed between the two scales Q and Λ QCD for λ = 0 , thus decreasingthe particle yield inside the whole jet. This measurement would in particular provide an additionaland independent check of the LPHD for massive charged hadrons. As was shown in 2.4, the DLAsolution of the evolution equations provide general features of the observable showing its unreliabilityto be compared with the experiment. That is why, the MLLA shape and overall normalization of thisobservable should be compared with the data. In the case of p ¯ p collisions at the Tevatron, since dietevents consist of both gluon and quark jets, in order to compare data to theory, a parameter f g for mixedsamples of quark and gluon jets was chosen [11]. In pp collisions at the LHC, the same procedure can beapplied so as to measure the two- and three-particle correlations. Furthermore, MLLA corrections havebeen shown to be larger for three than for two particles, that is to increase as the number of particlesincreases.As was the case for two particles, the three-particle correlations are larger inside a quark than in a gluonjet. Same trends have been observed in HERA and LEP data for soft multi-particle fluctuations in [34,35].Finally, we give the first analytical predictions for intra-jet three-particle correlations in view of forth-22oming measurements by ATLAS, CMS and ALICE at the LHC. Acknowledgements
We gratefully acknowledge enlightening discussions with W. Ochs and E. Sarkisyan-Grinbau as wellas support from Generalitat Valenciana under grant PROMETEO/2008/004 and M.A.S. from FPA2008-02878 and GVPROMETEO2010-056. V.M acklowledges support from the grant HadronPhysics2, aFP7-Integrating Activities and Infrastructure Program of the European Commission under Grant 227431,by UE (Feder) and the MICINN (Spain) grant FPA 2010-21750-C02-01.
A MLLA approximation
In (23a), for ln(1 − z ) ≪ ln x and ln z ≪ ln x , we perform the following Taylor expansions: Q (3) (1 − z ) − Q (3) ≈ ln(1 − z ) dQ (3) dℓ + O ( α s ) , (71) (cid:16) Q (2) ij (1 − z ) − Q (2) ij (cid:17) ( G k ( z ) − Q k ) + (cid:16) G (2) ij ( z ) − Q (2) ij (cid:17) ( Q k (1 − z ) − Q k )= ln(1 − z ) " dQ (2) ij dℓ ( G k − Q k ) + (cid:16) G (2) ij − Q (2) ij (cid:17) dQ k dℓ + O ( α s ) , (72) ( Q i − G i ( z )) ( Q j (1 − z ) − Q j ) Q k ≈ ln(1 − z )( Q i − G i ) dQ j dℓ Q k + O ( α s ) . (73)Since none of these terms contribute to MLLA O ( √ α s ) , they will be dropped hereafter. In equation(23b), we perform the following approximations in the hard fragmentation region, (cid:16) G (2) ij ( z ) − G (2) ij (cid:17) ( G k (1 − z ) − G k ) ≈ ln z ln(1 − z ) dG (2) ij dℓ G k dℓ + O ( α s ) , (74) ( G i − G i ( z ))( G j (1 − z ) − G j ) G k ≈ − ln z ln(1 − z ) dG i dℓ dG j dℓ G k + O ( α s ) . (75)Neither (74) nor (75) contribute to MLLA. The other terms in (23b) can be written as, Q (3) ( z ) − G (3) ≈ (2 Q (3) − G (3) ) + 2 ln z dQ (3) dℓ + O ( α s ) (76) (cid:16) Q (2) ij ( z ) − G (2) ij (cid:17) ( Q k (1 − z ) − G k ) ≈ (cid:16) Q (2) ij − G (2) ij (cid:17) ( Q k − G k ) (77) +2 ln(1 − z ) (cid:16) Q (2) ij − G (2) ij (cid:17) dQ k dℓ + 2 ln z ( Q k − G k ) dQ (2) ij dℓ + O ( α s ) , (2 Q i ( z ) Q j ( z ) − G i G j ) G k ≈ (2 Q i Q j − G i G j ) G k + ln z (cid:18) Q i dQ j dℓ + dQ i dℓ Q j (cid:19) + O ( α s ) , (78) ( G i − Q i ( z ))(2 Q j (1 − z ) − G j ) G k ≈ ( G i − Q i )(2 Q j − G j ) G k − Q j − G j ) G k ln z dQ i dℓ G k G i − Q i ) ln(1 − z ) dQ j dℓ G k + O ( α s ) , (79)such that only the first terms in (76), (77), (78) and (79) will be kept in the following. Furthermore, wemake use of the identity [19] Z dz Φ gg ( z ) (cid:16) G (3) ( z ) − zG (3) (cid:17) = Z dz (1 − z )Φ gg ( z ) (cid:16) G (3) ( z ) + (cid:16) G (3) ( z ) − G (3) (cid:17)(cid:17) , such that G ( n ) ( z ) − zG ( n ) can be replaced by, G ( n ) ( z ) − zG ( n ) → (1 − z ) h G ( n ) ( z ) + (cid:16) G ( n ) ( z ) − G ( n ) (cid:17)i ≈ (1 − z ) " G ( n ) ( z ) + ln z dG ( n ) dℓ , ( n = 1 , , ) in the r.h.s. of equations (21b), (22b) and (23b). Indeed, terms ∝ ln z, ln(1 − z ) provideNMLLA corrections O ( α s ) which improve energy conservation; however, their inclusion goes beyondthe scope of the present paper. A.1 One and two particle distributions at small x The MLLA integro-differential version of equations (21a,21b) and (22b,22a) is obtained after integratingover the regular part of the splitting functions, such that [6, 18, 19] Q i,y = C F N c Z ℓ dℓ ′ γ ( ℓ ′ + y ) G i ( ℓ ′ , y ) − C F N c γ ( ℓ + y ) G i ( ℓ, y ) , (80) G i,y = Z ℓ dℓ ′ γ ( ℓ ′ + y ) G i ( ℓ ′ , y ) − aγ ( ℓ + y ) G i ( ℓ, y ) , (81)with γ ( ℓ + y ) = β ( ℓ + y + λ ) , and the two-particle correlations ( ˆ A (2) ij = A (2) ij − A i A j ) [18, 19], ˆ Q (2) ij,y = C F N c Z ℓ i dℓγ ( ℓ + y j ) G (2) ij ( ℓ, y j , η ij ) − C F N c γ ( ℓ i + y j ) G (2) ij ( ℓ i , y j , η ij ) , (82) ˆ G (2) ij,y = Z ℓ i γ ( ℓ + y j ) G (2) ( ℓ, y j , η ij ) − aγ ( ℓ i + y j ) G (2) ij ( ℓ i , y j , η ij )+ ( a − b ) γ ( ℓ i + y j ) G ( ℓ i , y j + η ij ) G ( ℓ i + η ij , y j ) , (83)with γ ( ℓ i + y j ) = β ( ℓ i + y j + η ij + λ ) , after accounting for hard corrections O ( √ α s ) . After differentiating(80,81) and (82,83) with respect to “ ℓ ” , one has [19] Q i,ℓy = C F N c γ G i − C F N c γ ( G i,ℓ − β γ G i ) , (84) G i,ℓy = γ G i − aγ ( G i,ℓ − β γ G i ) , (85)from where the following useful relations hold in MLLA [19], Q i,ℓy γ Q i = (cid:20) − ψ i,ℓ (cid:21) C F N c G i Q i + O ( γ ) , (86) G i Q i = N c C F (cid:20) − (cid:18) a − (cid:19) ψ i,ℓ (cid:21) + O ( γ ) , (87) Q i,ℓy γ Q i = 1 − aψ i,ℓ + O ( γ ) . (88)24orrections ∝ β in (84) and (85), which are NMLLA, account for the running of the coupling constant α s and those ∝ , a, ( a − b ) account for energy conservation in the hard parton splitting region. TheMLLA gluon inclusive spectrum is given by the solution of (85) [6] and can be written in the form [14]: G i ( ℓ, y ) = 2 Γ( B ) β Z π dτπ e − Bα F B ( τ, y, ℓ ) , (89)where the integration is performed with respect to τ defined by α = 12 ln yℓ + iτ and with F B ( τ, y, ℓ ) = cosh α − y − ℓy + ℓ sinh αℓ + yβ α sinh α B/ I B (2 p Z ( τ, y, ℓ )) ,Z ( τ, y, ℓ ) = ℓ + yβ α sinh α (cid:18) cosh α − y − ℓy + ℓ sinh α (cid:19) ,B = a/β and I B is the modified Bessel function of the first kind. The formula in (89) correspondsindeed to the so-called hump-backed plateau, which describes the energy spectrum of soft hadrons in thelimiting spectrum approximation Q = Λ QCD [6, 28]. This result is well known and constitutes one ofthe strikest predictions of pQCD. The corresponding solution of (84) for Q i ( ℓ, y ) can be obtained from(87) with accuracy O ( √ α s ) . The system of differential evolution equations for two-particle correlationsfollows from (82) and (83), such that [19] h Q (2) ij − Q i Q j i ℓy = C F N c γ G (2) ij − C F N c γ (cid:16) G (2) ij,ℓ − β γ G (2) ij (cid:17) , (90) h G (2) ij − G i G j i ℓy = γ G (2) ij − aγ (cid:16) G (2) ij,ℓ − β γ G (2) ij (cid:17) +( a − b ) γ (cid:2) ( G i G j ) ℓ − β γ G i G j (cid:3) . (91)In [19], the system (82,83) was solved iteratively after replacing G (2) ij = C (2) G,ij G i G j and Q (2) ij = C (2) Q,ij Q i Q j in (91) and (90) respectively. The MLLA solutions of (90) and (91), which are to be used inthe present paper read [19] C (2) G ij − − δ ij − b ( ψ i,ℓ + ψ j,ℓ )1 + ∆ ij + δ ij , (92) C (2) Q ij − C (2) G ij − N c C F (cid:20) b − a )( ψ i,ℓ + ψ j,ℓ ) 1 + ∆ ij ij (cid:21) , (93)which were evaluated by the steepest descent method over the single inclusive distribution in [25]. Wehave introduced the following notations and functions [19], ∆ ij = γ − ( ψ i,ℓ ψ j,y + ψ i,y ψ j,ℓ ) = O (1); (94) χ ij = ln ˙ C (2) G ij = O (1) , χ ijℓ = ∂χ ij ∂ℓ = O ( γ ) , χ y = ∂χ ij ∂y = O ( γ ); (95) δ ij = γ − h χ ijℓ ( ψ i,y + ψ j,y ) + χ ijy ( ψ j,ℓ + ψ i,ℓ ) i = O ( γ ) , (96)where, following from (38) and (39), we have evaluated the corresponding order of magnitude of thesequantities in powers of the anomalous dimension γ ∝ √ α s . The solution is iterative with respect tocorrections χ and δ , which need the prior evaluation of the DLA solution ˙ C (2) G ij of the equations.25 Iterative solution of the evolution equations
Let us first solve the equation (50). For the sake of simplicity, it is much easier to solve the equivalentequation: ˆ G (3) ℓy = γ G (3) − aγ (cid:16) G (3) ℓ − β γ G (3) (cid:17) +( a − b ) γ nh G (2)12 G + G (2)13 G + G (2)23 G i ℓ (97) − β γ h G (2)12 G + G (2)13 G + G (2)23 G io + (2 a − b + c ) γ (cid:2) ( G G G ) ℓ − β γ G G G (cid:3) . One has to substitute the following in the l.h.s. of the equation (97): G (3) = C (3) G G G G , G (2) ij = C (2) G ij G i G j . Thus, after normalizing by γ G G G , one finds, h ( C (3) G − G G G i ℓy γ G G G = C (3) G ( ǫ + ǫ ) + ( C (3) G −
1) [3 + ∆ + ∆ + ∆ (98) − a ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) + 3 aβ γ (cid:3) , while for the other terms in the r.h.s. of the same equation one finds, h ( C (2) G ij − G G G i ℓy γ G G G = ( C (2) G ij − X i =1 G i,ℓy γ G i + ∆ + ∆ + ∆ ! + C (2) G ij ξ ij + C (2) G ij δ ij = ( C (2) G ij − (cid:2) + ∆ + ∆ − a ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) + 3 aβ γ + ξ ij + δ ij i + ξ ij + δ ij . (99)The r.h.s. provides the following contribution r.h.s.γ G G G = C (3) G − a C (3) G ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ + ζ ℓ − β γ ) + ( a − b ) h C (2) G ( χ ℓ + ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) + C (2) G ( χ ℓ + ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) + C (2) G ( χ ℓ + ψ ,ℓ + ψ ,ℓ + ψ ,ℓ ) − β γ ( C (2) G + C (2) G + C (2) G ) i + (3 b − a − c )( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ − β γ ) . (100)After adding (98) and (99) and equating with (100) together with some algebra in between, one finds thesolution written in (51). Following the same iterative procedure Q (3) = C (3) Q Q Q Q , Q (2) ij = C (2) Q ij Q i Q j ; G (3) = C (3) G G G G , for the quark jet evolution equation written in (49), one has, (cid:16) C (3) Q − (cid:17) ˜∆ + ˜∆ + ˜∆ + X i =1 Q i,ℓy γ Q i + ˜ ǫ + ˜ ǫ ! (101) − (cid:16) C (2) Q − (cid:17) ˜∆ + ˜∆ + ˜∆ + X i =1 Q i,ℓy γ Q i + ˜ ξ + ˜ δ ! − (cid:16) C (2) Q − (cid:17) ˜∆ + ˜∆ + ˜∆ + X i =1 Q i,ℓy γ Q i + ˜ ξ + ˜ δ ! − (cid:16) C (2) Q − (cid:17) ˜∆ + ˜∆ + ˜∆ + X i =1 Q i,ℓy γ Q i + ˜ ξ + ˜ δ ! C F N c C (3) G (cid:20) −
34 ( ψ ,ℓ + ψ ,ℓ + ψ ,ℓ + ζ ℓ − β γ ) (cid:21) G G G Q Q Q +( ˜ ξ + ˜ δ ) + ( ˜ ξ + ˜ δ ) + ( ˜ ξ + ˜ δ ) − ˜ ǫ − ˜ ǫ . Finally by adding and subtracting (˜ ǫ + ˜ ǫ ) in every term ∝ (cid:16) C (2) Q ij − (cid:17) in the l.h.s. of (101) one finds(55). C Steepest descent evaluation: reminder from [25]
The evaluation of the integral representation by the steepest descent method at small x ≪ (or large ℓ ≫ ) and very high energy Y ≫ leads to the result, G ( ℓ, y ) ≈ N ( µ, ν, λ ) exp (cid:20) β (cid:16)p ℓ + y + λ − √ λ (cid:17) µ − ν sinh µ − sinh ν + ν − aβ ( µ − ν ) (cid:21) , (102)where N ( µ, ν, λ ) = 12 ( ℓ + y + λ ) (cid:16) β λ (cid:17) / p π cosh νDetA ( µ, ν ) , with DetA ( µ, ν ) = β ( ℓ + y + λ ) (cid:20) ( µ − ν ) cosh µ cosh ν + cosh µ sinh ν − sinh µ sinh ν sinh µ cosh ν (cid:21) . The logarithmic derivatives of the spectrum given in (65) and (66) were derived from (102) and it was alsoshown that (102) reproduces the Gaussian shape of the inclusive distribution near the hump ℓ max ≈ Y / .From (102), one has indeed, G ( ℓ, y ) ≈ (cid:18) π √ β [( ℓ + y + λ ) / − λ / ] (cid:19) / exp (cid:18) − √ β ℓ + y + λ ) / − λ / ( ℓ − Y / (cid:19) , (103)where the MLLA ℓ max reads, ℓ max ≈ Y aβ (cid:16) √ Y + λ − √ λ (cid:17) . Setting a = 0 and λ = 0 in the previous expressions one recovers the DLA results, which are needed forsubsection 2.4. The functions entering as a function of ( µ, ν ) in (65) and (66) are the following, ˜ Q ( µ, ν ) = cosh µ sinh µ cosh ν − ( µ − ν ) cosh ν − sinh ν ( µ − ν ) cosh µ cosh ν + cosh µ sinh ν − sinh µ cosh ν , (104) K ( µ, ν ) = −
12 sinh ν ( µ − ν ) cosh µ − sinh µ ( µ − ν ) cosh µ cosh ν + cosh µ sinh ν − sinh µ cosh ν , (105) L ( µ, ν ) = 32 coth µ −
12 ( µ − ν ) cosh ν sinh µ + sinh ν sinh µ ( µ − ν ) cosh µ cosh ν + cosh µ sinh ν − sinh µ cosh ν , (106) C ( µ, ν ) = L ( µ, ν ) + tanh ν coth µ (1 + K ( µ, ν )) . (107)The expressions for the two particle correlations follow from (92) and (93) [25], C (2) G ij = 1 + 1 − bγ ( e µ i + e µ j ) − δ ij µ i − µ j ) + ∆ ′ ( µ i , ν i , µ j , ν j ) + δ ij , (108) C (2) Q ij = 1 + N c C F (cid:20) C (2) G ij − b − a ) γ e µ i + e µ j µ i − µ j ) (cid:21) , (109)27here, δ ij = β γ (cid:16) µ i − µ j (cid:17) (cid:16) µ i − µ j (cid:17) (cid:16) ˜ Q ( µ i , ν i ) + ˜ Q ( µ j , ν j ) (cid:17) , (110)and ∆ ′ ( µ i , ν i , µ j , ν j ) = − aγ h e µ i + e µ j − sinh( µ i − µ j )( ˜ Q i − ˜ Q j ) + cosh µ tanh ν + cosh µ tanh ν − sinh µ i tanh ν j coth µ j − sinh µ j tanh ν i coth µ i + sinh( µ i − µ j ) (cid:16) tanh ν i coth µ i ˜ Q i − tanh ν j coth µ j ˜ Q j (cid:17)i − β γ h cosh µ i − sinh µ i C j + cosh µ j − sinh µ j C i + sinh( µ i − µ j )( C i ˜ Q i − C j ˜ Q j )+ cosh µ i tanh ν j (1 + K j ) + cosh µ j tanh ν i (1 + K i )] . (111)The solutions (108) and (109) are the ones to be used in this paper for the evaluations of the three-particlecorrelations and will be directly inserted in the solutions (51) and (55) respectively. C.1 Corrections ξ ij , ˜ ξ ij and ǫ , ˜ ǫ For the computation of these corrections, one only needs to take the DLA part of the logarithmic deriva-tives of the one-particle distribution ψ i,ℓ = γ e µ i and ψ i,y = γ e − µ i , such that after replacement in (54c)and (58c) one finds, ξ ij = 1 γ h χ ijℓ (cid:0) e − µ + e − µ + e − µ (cid:1) + χ ijy ( e µ + e µ + e µ ) i , (112) ˜ ξ ij = 1 γ h ˜ χ ijℓ (cid:0) e − µ + e − µ + e − µ (cid:1) + ˜ χ ijy ( e µ + e µ + e µ ) i , (113)where χ ijℓ = β γ tanh µ i − µ j µ i − µ j ) e µ i ˜ Q i − e µ j ˜ Q j , ˜ χ ijℓ = − N c C F ˙ C (2) G ij ˙ C (2) Q ij χ ijℓ , (114) χ ijy = − β γ tanh µ i − µ j µ i − µ j ) e − µ i ˜ Q i − e − µ j ˜ Q j , ˜ χ ijy = − N c C F ˙ C (2) G ij ˙ C (2) Q ij χ ijy ; (115)with ˙ C (2) G ij = 1 + 11 + 2 cosh( µ i − µ j ) , ˙ C (2) Q ij = 1 + N c C F
11 + 2 cosh( µ i − µ j ) . (116)Accordingly, replacing ψ i,ℓ = γ e µ i and ψ i,y = γ e − µ i in (54e) and (58e), one has ǫ = 1 γ (cid:2) ζ ℓ (cid:0) e − µ + e − µ + e − µ (cid:1) + ζ y ( e µ + e µ + e µ ) (cid:3) , (117) ˜ ǫ = 1 γ h ˜ ζ ℓ (cid:0) e − µ + e − µ + e − µ (cid:1) + ˜ ζ y ( e µ + e µ + e µ ) i , (118)where ζ ℓ , ˜ ζ ℓ and ζ y , ˜ ζ y should be found from the DLA expression of C (3) written in (35), for C A = N c ina gluon jet and C A = C F in a quark jet. Introducing the parametrization in ( µ, ν ), one has respectively, ˙ C (3) G = 1 + (cid:16) ˙ C (2) G − (cid:17) + (cid:16) ˙ C (2) G − (cid:17) + (cid:16) ˙ C (2) G − (cid:17) (119) + 12 (cid:16) ˙ C (2) G − (cid:17) + (cid:16) ˙ C (2) G − (cid:17) + (cid:16) ˙ C (2) G − (cid:17) µ − µ ) + cosh( µ − µ ) + cosh( µ − µ )
12 11 + cosh( µ − µ ) + cosh( µ − µ ) + cosh( µ − µ ) , and ˙ C (3) Q = 1 + (cid:16) ˙ C (2) Q − (cid:17) + (cid:16) ˙ C (2) Q − (cid:17) + (cid:16) ˙ C (2) Q − (cid:17) (120) + N c C F (cid:16) ˙ C (2) Q − (cid:17) + (cid:16) ˙ C (2) Q − (cid:17) + (cid:16) ˙ C (2) Q − (cid:17) µ − µ ) + cosh( µ − µ ) + cosh( µ − µ )+ N c C F
11 + cosh( µ − µ ) + cosh( µ − µ ) + cosh( µ − µ ) . Thus, in order to get ζ ℓ and ζ y , one should start from (119,120) and make use of ∂µ i ∂ℓ − ∂µ j ∂ℓ = − β γ e µ i ˜ Q i − e µ j ˜ Q j , ∂µ i ∂y − ∂µ j ∂y = β γ e − µ i ˜ Q i − e − µ j ˜ Q j . Therefore, everything is ready for the computation of ζ ℓ = 1˙ C (3) G ˙ C (3) G ,ℓ , ζ y = 1˙ C (3) G ˙ C (3) G ,y ; ˜ ζ ℓ = 1˙ C (3) Q ˙ C (3) Q ,ℓ , ˜ ζ y = 1˙ C (3) Q ˙ C (3) Q ,y . (121)For instance, ˙ C (3) G ,ℓ = χ ℓ ˙ C (2) G + χ ℓ ˙ C (2) G + χ ℓ ˙ C (2) G + 12 χ ℓ ˙ C (2) G + χ ℓ ˙ C (2) G + χ ℓ ˙ C (2) G µ − µ ) + cosh( µ − µ ) + cosh( µ − µ ) − (cid:16) ˙ C (2) G − (cid:17) + (cid:16) ˙ C (2) G − (cid:17) + (cid:16) ˙ C (2) G − (cid:17) [1 + cosh( µ − µ ) + cosh( µ − µ ) + cosh( µ − µ )] (cid:20) sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19) + sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19) + sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19)(cid:21) −
12 1[1 + cosh( µ − µ ) + cosh( µ − µ ) + cosh( µ − µ )] (cid:20) sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19) + sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19) + sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19)(cid:21) , (122)and ˙ C (3) Q ,ℓ = ˜ χ ℓ ˙ C (2) Q + ˜ χ ℓ ˙ C (2) Q + ˜ χ ℓ ˙ C (2) Q + N c C F ˜ χ ℓ ˙ C (2) Q + ˜ χ ℓ ˙ C (2) Q + ˜ χ ℓ ˙ C (2) Q µ − µ ) + cosh( µ − µ ) + cosh( µ − µ ) − N c C F (cid:16) ˙ C (2) Q − (cid:17) + (cid:16) ˙ C (2) Q − (cid:17) + (cid:16) ˙ C (2) Q − (cid:17) [1 + cosh( µ − µ ) + cosh( µ − µ ) + cosh( µ − µ )] (cid:20) sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19) + sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19) + sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19)(cid:21) − N c C F µ − µ ) + cosh( µ − µ ) + cosh( µ − µ )] (cid:20) sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19) + sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19) + sinh( µ − µ ) (cid:18) ∂µ ∂ℓ − ∂µ ∂ℓ (cid:19)(cid:21) . (123)For derivatives with respect to y , it is enough to replace ℓ by y in the previous expressions. In Fig.8, wedisplay ǫ ( ℓ , ℓ , ℓ , Y ) as a function of the sum | ln( x x ) | and the difference ( ℓ − ℓ ) = ln( x /x ) for two fixed values of ℓ = ln(1 /x ) = 4 . , . , x = x and fixed sum ( ℓ + ℓ ) = | ln( x x ) | = 10 and fixed Y = 7 . . As expected, this correction decreases the correlations away from the hum regionand for harder particles. 29 Y=7.5 ; λ =0.1 ; Ln(x /x )=0 ε |Ln(x x )| Ln(1/x )=4.5Ln(1/x )=5.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.500.050.10.150.20.250.30.350.40.45 Y=7.5 ; λ =0.1 ; |Ln(x x )|=10 ε Ln(x /x ) Ln(1/x )=4.5Ln(1/x )=5.5 Figure 8: Correction ǫ ( ℓ , ℓ , ℓ , Y ) as a function of ℓ − ℓ = ln( x /x ) for ℓ + ℓ = | ln( x x ) | = 10 , ℓ = ln(1 /x ) = 4 . , . , fixed Y = 7 . in the limiting spectrum approximation λ ≈ . C.2 Hump approximation
In this approximation, we consider the energy of the three partons to be close to the maximum of thesingle inclusive distribution | ℓ − Y / |≪ σ ∝ Y / for i = 1 , , . In [25], it was demonstrated that, ψ i,ℓ ℓ i ∼ Y/ ≈ γ (1 + µ i + 12 µ i ) , ψ i,y ℓ i,j ∼ Y/ ≈ γ (1 − µ i + 12 µ j ) , µ i ℓ i ∼ Y/ ≈ y − ℓy + ℓ , (124)for a, β , λ = 0 , which is DLA. In the same approximation one has the following for a, β = 0 and λ = 0 , ∆ ij ℓ i,j ∼ Y/ ≈ µ i − µ j ) − aγ (2 + µ i + µ j ) − β γ , (125)where ( µ i − µ j ) ℓ i,j ∼ Y/ ≈ (cid:18) ℓ i − ℓ j Y (cid:19) , µ i + µ j ℓ i,j ∼ Y/ ≈ (cid:18) − ℓ i + ℓ j Y (cid:19) . (126)Moreover, δ ij ℓ i ∼ Y/ ≈ β γ ( µ i − µ j ) = 2 β γ (cid:18) ℓ i − ℓ j Y (cid:19) , (127)since γ (cid:16) ℓ i − ℓ j Y (cid:17) ≪ (cid:16) ℓ i − ℓ j Y (cid:17) , δ was neglected in this approximation.Applying the previous expansions to (53a-53d) and (57a-57d), it is easy to find: N (2) Q ij = 0 , (128a) N (3) G = 1 − c √ β (cid:18) − | ln( x x x ) | ln( Q/Q ) (cid:19) p ln( Q/Q ) = 1 − c (cid:18) − ℓ + ℓ + ℓ Y (cid:19) γ , (128b) D (3) G = D (2) G = 8+9 (cid:20) ln( x /x )ln( Q/Q ) (cid:21) + 9 (cid:20) ln( x /x )ln( Q/Q ) (cid:21) + 9 (cid:20) ln( x /x )ln( Q/Q ) (cid:21) − β p β ln( Q/Q ) , (128c) − a √ β (cid:18) − | ln( x x x ) | ln( Q/Q ) (cid:19) p ln( Q/Q ) , = 8+9 (cid:18) ℓ − ℓ Y (cid:19) +9 (cid:18) ℓ − ℓ Y (cid:19) +9 (cid:18) ℓ − ℓ Y (cid:19) − β γ − a (cid:18) − ℓ + ℓ + ℓ Y (cid:19) γ , (2) G ij = 1 − b √ β (cid:18) − | ln( x x x ) | ln( Q/Q ) (cid:19) p ln( Q/Q ) = 1 − b (cid:18) − ℓ + ℓ + ℓ Y (cid:19) γ , (128d) C (2) G ij = 1 + 1 − b √ β (cid:16) − | ln( x i x j ) | ln( Q/Q ) (cid:17) √ ln( Q/Q ) h ln( x i /x j )ln( Q/Q ) i − q β ln( Q/Q ) − a √ β (cid:16) − | ln( x i x j ) | ln( Q/Q ) (cid:17) √ ln( Q/Q ) , (128e) = 1 + 1 − b (cid:16) − ℓ i + ℓ j Y (cid:17) γ (cid:16) ℓ i − ℓ j Y (cid:17) − β γ − a (cid:16) − ℓ i + ℓ j Y (cid:17) γ , (128f) C (2) Q ij = 1 + N c C F " C (2) G ij − b − a ) γ − | ln x i x j | p ln( Q/Q ) ! (128g) = 1 + N c C F (cid:20) C (2) G ij − b − a ) γ (cid:18) − ℓ i + ℓ j Y (cid:19)(cid:21) ,D (3) Q = 9 + 9 (cid:20) ln( x /x )ln( Q/Q ) (cid:21) + 9 (cid:20) ln( x /x )ln( Q/Q ) (cid:21) + 9 (cid:20) ln( x /x )ln( Q/Q ) (cid:21) − β p β ln( Q/Q ) (128h) − a √ β (cid:18) − | ln( x x x ) | ln( Q/Q ) (cid:19) p ln( Q/Q ) , = 9+9 (cid:18) ℓ − ℓ Y (cid:19) +9 (cid:18) ℓ − ℓ Y (cid:19) +9 (cid:18) ℓ − ℓ Y (cid:19) − β γ − a (cid:18) − ℓ + ℓ + ℓ Y (cid:19) γ , (128i) N (3) Q = N c C F C (3) G " − a √ β (cid:18) − | ln( x x x ) | ln( Q/Q ) (cid:19) p ln( Q/Q ) (128j) = N c C F C (3) G " − a √ β (cid:18) − ℓ + ℓ + ℓ ln( Q/Q ) (cid:19) p ln( Q/Q ) . D DLA solution of the 4-particle correlations
Below, we display the expressions related to subsection 2.9. In the l.h.s. of the evolution equation (69),we define ˆ A (4)1234 = A (4)1234 − (cid:16) A (3)123 − A A A (cid:17) A − (cid:16) A (3)134 − A A A (cid:17) A − (cid:16) A (3)234 − A A A (cid:17) A (129) − (cid:16) A (3)124 − A A A (cid:17) A − (cid:16) A (2)12 − A A (cid:17)(cid:16) A (2)34 − A A (cid:17) − (cid:16) A (2)13 − A A (cid:17)(cid:16) A (2)24 − A A (cid:17) − (cid:16) A (2)14 − A A (cid:17)(cid:16) A (2)23 − A A (cid:17) + (cid:16) A (2)12 − A A (cid:17) A A + (cid:16) A (2)13 − A A (cid:17) A A + (cid:16) A (2)14 − A A (cid:17) A A + (cid:16) A (2)23 − A A (cid:17) A A + (cid:16) A (2)24 − A A (cid:17) A A + (cid:16) A (2)34 − A A (cid:17) A A − A A A A . In the DLA solution (70) of the equation (69), we have introduced the expressions: H = (cid:16) ˙ C (2)12 − (cid:17) + (cid:16) ˙ C (2)13 − (cid:17) + (cid:16) ˙ C (2)14 − (cid:17) + (cid:16) ˙ C (2)23 − (cid:17) + (cid:16) ˙ C (2)24 − (cid:17) + (cid:16) ˙ C (2)34 − (cid:17) , (130) H = (cid:16) ˙ C (3)123 − (cid:17) + (cid:16) ˙ C (3)124 − (cid:17) + (cid:16) ˙ C (3)134 − (cid:17) + (cid:16) ˙ C (3)234 − (cid:17) + (cid:16) ˙ C (2)14 − (cid:17) (cid:16) ˙ C (2)23 − (cid:17) (131) + (cid:16) ˙ C (2)34 − (cid:17) (cid:16) ˙ C (2)12 − (cid:17) + (cid:16) ˙ C (2)13 − (cid:17) (cid:16) ˙ C (2)24 − (cid:17) − (cid:16) ˙ C (2)12 − (cid:17) − (cid:16) ˙ C (2)13 − (cid:17) − (cid:16) ˙ C (2)14 − (cid:17) − (cid:16) ˙ C (2)23 − (cid:17) − (cid:16) ˙ C (2)24 − (cid:17) − (cid:16) ˙ C (2)34 − (cid:17) , = 1 + (cid:16) ˙ C (3)123 − (cid:17) + (cid:16) ˙ C (3)124 − (cid:17) + (cid:16) ˙ C (3)134 − (cid:17) + (cid:16) ˙ C (3)234 − (cid:17) + (cid:16) ˙ C (2)14 − (cid:17) (cid:16) ˙ C (2)23 − (cid:17) (132) + (cid:16) ˙ C (2)34 − (cid:17) (cid:16) ˙ C (2)12 − (cid:17) + (cid:16) ˙ C (2)13 − (cid:17) (cid:16) ˙ C (2)24 − (cid:17) − (cid:16) ˙ C (2)12 − (cid:17) − (cid:16) ˙ C (2)13 − (cid:17) − (cid:16) ˙ C (2)14 − (cid:17) − (cid:16) ˙ C (2)23 − (cid:17) − (cid:16) ˙ C (2)24 − (cid:17) − (cid:16) ˙ C (2)34 − (cid:17) . References [1] Redamy Perez Ramos, Vincent Mathieu, and Miguel-Angel Sanchis-Lozano. Three-particle corre-lations in QCD parton showers. arXiv:1104.1973 [hep-ph].[2] Harald Fritzsch and Murray Gell-Mann. Current algebra: Quarks and what else? eConf ,C720906V2:135–165, 1972.[3] D. J. Gross and Frank Wilczek. ULTRAVIOLET BEHAVIOR OF NON-ABELIAN GAUGE THE-ORIES.
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