Soft anomalous dimension matrices in heavy quark-antiquark hadroproduction in association with a gluon jet
aa r X i v : . [ h e p - ph ] S e p Soft anomalous dimension matrices in heavyquark-antiquark hadroproduction in associationwith a gluon jet
E. Szarek Institute of Physics, Jagiellonian UniversityŁojasiewicza 11, 30-348 Kraków, Poland
Abstract
We compute the soft anomalous dimension (SAD) matrices for production of mas-sive quarks Q and ¯ Q in association with a gluon jet, from massless quarks q andantiquarks ¯ q : q ¯ q → Q ¯ Qg , and in the gluon scattering gg → Q ¯ Qg . To analyzethe behaviour of the eigenvalues of SAD matrices we perform numerical studies oftheir eigensystems at two special kinematical configurations. [email protected] Introduction
In QCD one finds infrared divergences in perturbative corrections: soft collinear,collinear and soft non-collinear. Soft divergences appear for the energy of a gluon E → , and the collinear ones when the angle θ → between a massless partonand a gluon. After the procedure of regularization the IR divergences cancel outin observables (like cross sections), but they leave logarithmic terms dependingon scales characterizing virtual and real corrections. The logarithmic remnantsbecome very large near the absolute threshold, so they are important in processesof heavy particles production. In the absolute threshold limit the characteristicvelocity of the outgoing partons β is very small, which means that total energy √ ˆ s of partons in center of mass system is very close to m th where m th is the sum ofmasses of products in the process. The characteristic scale of the real corrections,that come from the collinear gluon radiation is m th β , and the characteristic scaleof the virtual corrections is proportional to m th . The real and virtual correctionscombined give a leading contribution to cross section proportional to α s log β .Such logarithms appear in every order of perturbative expansion contributing with (cid:0) α s log β (cid:1) n in the leading logarithmic (LL) approximation, α ns log n − β in thenext–to–leading logarithmic (NLL) approximation, and so on. When β ≪ , α s log β may be close or grater than one, and one needs to resum those correc-tions to all orders. The remnant logarithms are reordered in a new perturbative1xpansion due to the resummation procedure. The resummation formalism is de-scribed in [1, 2]. The fundamental object used in the resummation procedure isthe soft anomalous dimension (SAD) matrix.The soft gluon resummation technique that employs the SAD matrices havenumerous applications in modern particle physics, in particular in estimates ofsuperparticles hadroproduction. The SAD matrices carry information about colourflow between particles in the studied processes. The soft gluon resummation effectsbecome very important in cross sections near the threshold for heavy particlesproduction.The SAD matrices were calculated for various types of processes. Firstly, cal-culations were performed for the Drell – Yan processes → with two incomingcoloured particles and one colour-neutral [3, 4]. Then, there were considered pro-cesses → , like q ¯ q → q ¯ q and gg → q ¯ q for massless and massive products infinal state, in one-loop approximation [2]. This approach was extended to all re-actions containing light quarks and gluons [5]. It allowed to obtain predictionsfor cross sections for production of heavy quarks (especially for the top quark)[6, 7, 8, 9, 10, 11, 12, 13], and compare with experimental data. The SAD matricesalso play an important role in predictions for squarks and gluinos hadroproduc-tion cross sections. Soft anomalous dimension matrices were calculated at one loop[14, 15, 16] and two loops [17, 18, 19] for such processes. Recently a lot of effort2as been made to obtain accurate predictions for reactions involving the Higgsboson. Firstly, there was obtained the hadroproduction cross section improved bythe soft gluon resummation at the NNLL approximation [20] and then at the N LL level [21] for the → process: gg → H . For the supersymmetric charged Higgsboson hadroproduction the soft gluon resummation was performed at two loopsfor the process bg → tH − [22]. Next the soft gluon resummation was extended to anew class of processes: → containing 4 coloured and 1 colour neutral particles,which gives more accurate predictions for the Higgs boson hadroproduction crosssection in association with the top and antitop quarks [23, 24, 25, 26].In this paper the SAD matrices are derived for → processes with 5 colouredparticles at one loop in the perturbation theory, q ¯ q → Q ¯ Qg and gg → Q ¯ Qg . Thequark and antiquark in the final state are both massive. Earlier calculations ofthe SAD matrices in similar reactions have been performed by M. Sjödahl [27, 28],however only for massless final state protons. In this paper we consider the following scattering processes: q α ( p )¯ q β ( p ) → Q γ ( p ) ¯ Q δ ( p ) g a ( p ) , (1)3nd g a ( p ) g b ( p ) → Q α ( p ) ¯ Q β ( p ) g c ( p ) , (2)where α , β , γ , δ , a , b and c stand for colour indices (Greek letters are used fordescription of a fundamental representation of SU( N c ) and Roman letters for anadjoint representation) and p i , i = 1 , . . . , , denote the momenta of particles.Due to hard factorization theorems in QCD one can distingiush soft and hardpart of sufficiently inclusive processes. The soft function S IJ fulfills the renormal-ization group equation: (cid:18) µ ∂∂µ + β ( g ) ∂∂g (cid:19) S IJ = − (cid:16) Γ † S (cid:17) IB S BJ − S IA (Γ S ) AJ , (3)where S IJ is a matrix in colour space and carries information about soft wideangle gluon emissions, indices I , J corresponds to colour tensors constructed from SU( N c ) representations. They depend on a studied proccess: the colour chargesof participating particles and the exchange channel. For example if we consider aquark-antiquark annihilitation, I and J tensors correspond to a flow of a coloursinglet or octet in the s-channel. At one loop, the soft anomalous dimension matrixare defined as follows [2, 5]: 4 ′ S ( g ) = − g ∂∂g Res ǫ → Z S ( g, ǫ ) , (4)where g is the coupling constant for QCD, Z S ( g, ǫ ) – a renormalization matrix ofthe soft matrix S IJ . Z S receives contributions from the soft gluons. The generalfor of the SAD matrix can be derived from the paper presented in [29, 30, 31].However, in our explicit calculations we apply the method elaborated in [2, 5].To get Z S one needs to sum over the contributions Z ( D ) S of relevant Feynmandiagrams. Each contribution Z ( D ) S to Z S coming from a single Feynman diagram D , can be factorized into a colour factor and a kinematic factor: Z ( D ) S ∝ colour factor × kinematic factor . (5)The colour part of every diagram is represented by SU( N c ) tensors decomposed inan orthogonal and normalized basis. The vectors from colour basis are connectedwith a soft gluon line which is represented by colour tensor if abc . The form of Z S depends on wheather the partons between which there is an exchange of the softgluon, are massive or massless (see figure 1). For massive particles i and j [2, 5]: Z ( D ) S ( g, ǫ ) = c ij s ij απ ǫ (cid:16) L ( ij ) β + L i + L j − (cid:17) . (6)For a massive particle i and a massless particle j :5 ( D ) S ( g, ǫ ) = − c ij s ij α π ǫ (cid:18) ln (cid:20) v ij s m i (cid:21) − L i − ln ν j + 1 (cid:19) . (7)For massless particles i and j : Z ( D ) S ( g, ǫ ) = − c ij s ij απ ǫ (cid:18) ln (cid:20) δ i δ j v i · v j (cid:21) −
12 ln ( ν i ν j ) + 1 (cid:19) . (8)In the above equations factors c ij stand for colour factors, the number s ij is relatedto the type of particles and the direction of the momentum flow in a diagram.Namely: s ij = ∆ i ∆ j δ i δ j . (9)The factors ∆ i depend on the type of particles between which the exchange ofthe gluon occurs, they have values: +1( − for a quark (antiquark). The factors δ i = +1( − , for the same (opposite) direction of momentum flow between a partonand the soft gluon. Vectors v i are rescaled momenta of the particles v µi = p µi Q ,where Q = q ˆ s , and v ij = v i · v j . The factors ν i = ( v i · n ) | n | depend on a choice of thereference vector n µ of the axial gauge. In the axial gauge A = 0 in the center ofmass system of the colliding partons one has ν i = . The function L ( ij ) β dependson the relative velocity β ij of the outgoing partons:6igure 1. Feynman diagrams contributing to Z S for process q ¯ q → Q ¯ Qg (for gg → Q ¯ Qg the topologies are analogous). The soft gluon is indicated by g inthe diagrams.7 ( ij ) β = 1 − m / ˆ s ij β ij (cid:18) ln 1 − β ij β ij + iπ (cid:19) , (10)where β ij = p − m / ˆ s ij and ˆ s ij = ( p i + p j ) . In the processes considered themassive particles are labelled by and , hence in what follows β will be used. L i are dependent on the choice of gauge: L i = [ L i (+ n ) + L i ( − n )] , where L i ( ± n ) = 12 | v i · n | q ( v i · n ) − m n /s (11) ln δ ( ± n ) 2 m /s − | v i · n | − q ( v i · n ) − m n /sδ ( ± n ) 2 m /s − | v i · n | + q ( v i · n ) − m n /s + ln δ ( ± n ) n − | v i · n | − q ( v i · n ) − m n /sδ ( ± n ) n − | v i · n | + q ( v i · n ) − m n /s . Contributions L i also appear in the self-interaction terms for the heavy quarks (an-tiquarks). The contribution from the self-interaction of heavy quarks (antiquarks)is α s π T R N c − N c ( L i + L j − , (12)where is the identity matrix in the colour space and the factor T R comes fromthe normalization of generators and it equals . The contribution from the self-interaction of heavy quarks (antiquarks) is added to the soft anomalous dimension8atrix and the dependence of Γ ′ S on L i is canceled out. Following refs. [2, 5] theDrell – Yan contribution is subtracted from the soft anomalous dimension matrix.At one loop the Drell – Yan SAD matrix takes the form α s π C F ( α s π C A ) for thepartons in the colour triplet (octet) state and C A = T R N c and C F = T R N c − N c .The final form of the soft anomalous dimension matrix Γ S ( g, ǫ ) is the following: Γ S ( g, ǫ ) = Γ ′ S ( g, ǫ ) + α s π T R N c − N c ( L + L − − X i C iA,F , (13)where i - all massless particles in the examined process.In this paper processes with five interacting particles are considered (see figure2). Figure 2. An example of a particle collision in a process → .To fully describe the phase space of such physical system one needs five inde-9endent variables: the global azimuthal angle φ , which carries information aboutthe rotation symmetry of the reaction and four Mandelstam–type variables: t = ( p − p ) , (14) t = ( p − p ) ,u = ( p − p ) ,u = ( p − p ) . The remaining scalar products of particle momenta p i · p ( i = 1 , , , may beexpressed in the terms of above variables: p · p = 12 (cid:0) t + u + s − m − m (cid:1) , (15) p · p = 12 (cid:0) t + u + s − m − m (cid:1) ,p · p = 12 (cid:0) t + u + s − m − m (cid:1) ,p · p = 12 (cid:0) t + u + s − m − m (cid:1) , where m , is the mass of heavy quark (antiquark).10 Results
In this section we collect results for the soft anomalous dimension matrices fortwo processes q ¯ q → Q ¯ Qg and gg → Q ¯ Qg , where q , ¯ q – denote the masslessquark/antiquark, and Q , ¯ Q – the massive quark/antiquark. Calculations of thecolour factors were obtained in the s -channel basis, using the package ColorMath[28] for Mathematica. The colour factors were combined with formulas (4), (6) –(8), also (13), and Γ S was obtained. It is convenient to introduce new variables T , T , U , U and then Λ , Ω , Γ , Σ , which are defined in the following way: T = ln (cid:18) p · p m √ s (cid:19) − − iπ , (16) T = ln (cid:18) p · p m √ s (cid:19) − − iπ ,U = ln (cid:18) p · p m √ s (cid:19) − − iπ ,U = ln (cid:18) p · p m √ s (cid:19) − − iπ . In this study the case is considered of the mass of quark and antiquark that havethe same value m = m = m and we introduce variables,11 = T + T + U + U , (17) Ω = T + T − U − U , Γ = T − T + U − U , Σ = T − T − U + U . In what follows, the SAD matrices will be presented in terms of the independentvariables Λ , Ω , Γ , Σ and the variables v i = 2 p i · p /s that can be expressed by Λ , Ω , Γ , Σ with equalities (11), (12) and (13). The variables v i are kept in order tosimplify the form of the matrices. q ¯ q → Q ¯ Qg The following orthogonal and normalized colour basis was used in the calculations[27]: T αβγζa = 1 p N c ( N c − T R δ αβ t aγζ , (18) T αβγζa = 1 p N c ( N c − T R δ γζ t aβα ,T αβγζa = 1 q N c ( N c −
1) ( T R ) t bβα t cγζ if bca , αβγζa = √ N c q N c −
4) ( N c −
1) ( T R ) t bβα t cγζ d bca . Notice that indices of the adjoint representation are a , b , c and indices of thefundamental representation are α , β , γ and ζ . The following results are valid for N c ≥ [27].The soft anomalous dimension matrix Γ q ¯ q → Q ¯ Qg can be split into two parts: Γ q ¯ q → Q ¯ Qg = Γ (1) q ¯ q → Q ¯ Qg (Λ , Ω , Γ , Σ) + Γ (2) q ¯ q → Q ¯ Qg ( v i ) , (19)where Γ (1) receives contributions from the soft gluon exchanges between particles , , , , and Γ (2) from exchanges between particles i and , with i = 1 , , , .Hence: Γ (1) q ¯ q → Q ¯ Qg = α s π T R × ( − N c ) L β N c Ω N c √ N c − √ N c Ω Γ √ N c L β + N c N c √ N c − √ N c Ω Σ √ √ N c − √ N c Ω √ N c − √ N c Ω L β + ( N c − ) Ω+ N c (2+Λ)4 N c p N c − Γ √ √ p N c − L β + ( N c − ) Ω+ N c (2+Λ)4 N c , (20)and 13 (2) q ¯ q → Q ¯ Qg = α s π T R × N c ln ( v v ) 0 0 √ ln (cid:16) v v (cid:17) N c ln ( v v ) 0 ln (cid:16) v v (cid:17) √
20 0 N c ln (cid:0) v v v v (cid:1) ln (cid:16) v v v v (cid:17) p N c − √ ln (cid:16) v v (cid:17) ln (cid:16) v v (cid:17) √ ln (cid:16) v v v v (cid:17) p N c − N c ln (cid:0) v v v v (cid:1) + α s π T R × diag (cid:18) iπN c , iπ − N c N c , iπ − N c N c , iπ − N c N c (cid:19) . (21) The SAD matrices for q ¯ q → Q ¯ Qg were calculated in parallel in [33] . Note that,the obtained SAD matrix is complex symmetric. This property has been provedto hold in general in an orthonormal basis [32]. The same feature will be foundalso for the gg -channel. For clarity we denoted L (34) β as L β in all the matrices. Thegeneral form of the soft anomalous dimension matrix is rather complicated, hencein order to provide more insight into its properties we consider special kinematicalconfigurations for which the matrix simplifies. First, we consider the case when themomenta of the outgoing quark and antiquark are equal, p µ = p µ . Then variables Λ , Ω , Γ , Σ reduce to Λ → Λ ′ = 2 T + 2 U , Σ → Σ ′ = 2 T − U , Γ → , Ω → .In this special case it is convenient to introduce a variable β = q − m ˆ s . In thislimit the form of the soft anomalous matrix becomes significantly simpler: Several differences were found between results of [33] and the results of this paper and theirorigin was clarified in correspondence with R. Schäfer. (1) q ¯ q → Q ¯ Qg ( p µ = p µ ) = α s π T R × ( − N c ) L β N c L β + N c N c Σ ′ √ L β + N c (2+Λ ′ )4 N c Σ ′ p N c − Σ ′ √ ′ p N c − L β + N c (2+Λ ′ )4 N c (22) Γ (2) q ¯ q → Q ¯ Qg ( p µ = p µ ) = α s π T R × N c ln ( v v ) 0 0 00 N c ln v (cid:16) v v (cid:17) √
20 0 N c ln ( v v v ) ln (cid:16) v v (cid:17) p N c −
40 ln (cid:16) v v (cid:17) √ ln (cid:16) v v (cid:17) p N c − N c ln ( v v v ) + α s π T R × diag (cid:18) iπN c , iπ − N c N c , iπ − N c N c , iπ − N c N c (cid:19) . (23) It can be seen that the soft anomalous dimension matrix can be divided in twoblocks: × and × .Next the limit Σ ′ = 0 is performed that corresponds to p · p = p · p . Theobtained matrix has a diagonal form: 15 q ¯ q → Q ¯ Qg (Σ ′ = 0) = α s π T R × (cid:26) N c × diag (cid:18) (cid:0) − N c (cid:1) L β , L β + N c , L β + N c ′ ) , L β + N c ′ ) (cid:19) + diag (cid:18) N c ln v , N c ln v , N c ln (cid:0) v v (cid:1) , N c ln (cid:0) v v (cid:1)(cid:19) + diag (cid:18) iπN c , iπ − N c N c , iπ − N c N c , iπ − N c N c (cid:19)(cid:27) . (24) q ¯ q → Q ¯ Qg In this section we consider the behaviour of the SAD eigenvalues for p = p and two different scattering angles θ ( ◦ and ◦ ) where θ is an angle betweenthe incoming and outgoing partons in the CMS frame. θ = 90 ◦ represents themost symmetrical case, and the choice of θ = 30 ◦ represents a less symmetricalconfiguration. The limit Σ ′ → corresponds to the case of θ = 90 ◦ . This analysismust be done carrefully because L (34) β in the limit β → gives singular terms ∝ iβ . We need to execute three steps. While performing the limit β → onesubtracts the singular terms from the SAD matrix and then the limit β → maybe studied. Finally we present results after a subtraction of the asymptotic small β behaviour that is treated analytically. Numerical calculations were performedfor the N c = 3 case. The eigenvalues of ˜Γ S does not contain the prefactor α s π .16he relation between the full SAD matrix Γ S and ˜Γ S is Γ S = α s π ˜Γ S . The singularmatrix subtracted from ˜Γ S has a form iπβ × diag (cid:0) − , , , (cid:1) .For a general β there is a degeneracy of eigenvalues for θ = 90 ◦ , there arethree different eigenvalues. For θ = 30 ◦ there is no degeneracy. All eigenvaluesare complex with non-trival real and imaginary part. For θ = 90 ◦ there are twodifferent values of the imaginary part of eigenvalues instead of four, which is thecase of ◦ .For β → one finds a singular term proportional to log β which gives a con-tribution to the real part of the eigenvalues. Each eigenvalue of Γ S has the sameleading behaviour in β → for both scattering angles. One finds one asymptoticform of the eigenvalues of small β : λ sing = 6 log β. (25)In Figures 3 and 4 we show regularized eigenvalues. They are defined as λ reg i = λ i − λ sing . One observes a quite similar behaviour of the regularized eigensystem for θ = 90 ◦ and θ = 30 ◦ . In Figure 3 one can see that all Re( λ reg , ◦ ) are either constant( λ ) or slightly increasing ( λ , λ ) up to β ≈ . . Im( λ ◦ ) are constant in β . Inthe case of θ = 30 ◦ , the real and imaginary part of λ shows a constant behaviour.The real parts of λ , λ and λ exhibit a similar behaviour for small β – they areslowly varying for moderate β , then for β > . they are rapidly increasing. The17maginary parts of these remaining eigenvalues are nearly constant for β < . ,and then start to slowly decrease ( λ and λ ) or slowly increase ( λ ).18 . . . . . . . . . − − − β R e (cid:0) λ r e g , ◦ (cid:1) Re (cid:0) λ reg , ◦ (cid:1) for q ¯ q → Q ¯ Qg Re (cid:16) λ reg , ◦ (cid:17) Re (cid:16) λ reg , ◦ (cid:17) Re (cid:16) λ reg , ◦ (cid:17) . . . . . . . . . − − − − β I m (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ (cid:1) for q ¯ q → Q ¯ Qg Im (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ , , (cid:1) Figure 3. The real (top) and imaginary (bottom) parts of the eigenvalues of ˜Γ S for q ¯ q → Q ¯ Qg at θ = 90 ◦ .19 . . . . . . . . . − − β R e (cid:0) λ r e g , ◦ (cid:1) Re (cid:0) λ reg , ◦ (cid:1) for q ¯ q → Q ¯ Qg Re (cid:16) λ reg , ◦ (cid:17) Re (cid:16) λ reg , ◦ (cid:17) Re (cid:16) λ reg , ◦ (cid:17) Re (cid:16) λ reg , ◦ (cid:17) . . . . . . . . . − − − − β I m (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ (cid:1) for q ¯ q → Q ¯ Qg Im (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ (cid:1) Figure 4. The real (top) and imaginary (bottom) parts of the eigenvalues of ˜Γ S for q ¯ q → Q ¯ Qg at θ = 30 ◦ .20 .2 gg → Q ¯ Qg The following orthogonal and normalized colour basis was used in the calculations[27]: T abαβc = 1( N c − √ T R t cαβ δ ab , (26) T abαβc = 1 N c p N c − T R if abc δ αβ ,T abαβc = 1 p N c −
4) ( N c − T R d abc δ αβ ,T abαβc = 12 N c q ( N c −
1) ( T R ) if abn if mcn t mαβ ,T abαβc = 1 q N c −
4) ( N c −
1) ( T R ) d abn if mcn t mαβ ,T abαβc = 1 q N c −
4) ( N c −
1) ( T R ) if abn d mcn t mαβ ,T abαβc = 1 q N c − ( N c −
1) ( T R ) d abn d mcn t mαβ ,T abαβc = 1 q N c −
4) ( N c −
1) ( T R ) P
10+ ¯10 abmc t mαβ ,T abαβc = 1 q N c −
4) ( N c −
1) ( T R ) P − ¯10 abmc t mαβ ,T abαβc = − q N c ( N c + 3) ( N c −
1) ( T R ) P abmc t mαβ ,T abαβc = 1 q N c ( N c −
3) ( N c + 1) ( T R ) P abmc t mαβ , where 21
10+ ¯10 abcd = 12 ( δ ac δ bd − δ ad δ cb ) − N c f abg f cdg , (27) P − ¯10 abcd = 12 d acg if bgd − d bgd if acg ,P abcd = N c N c + 2) d abg d cdg + 12 f adg f cbg − f abg f cdg + 14 δ ad δ bc + 14 δ ac δ bd + 12 ( N c + 1) δ ab δ cd ,P abcd = − N c N c + 2) d abg d cdg − f adg f cbg + 14 f abg f cdg + 14 δ ad δ bc + 14 δ ac δ bd −
12 ( N c + 1) δ ab δ cd . As for the q ¯ q → Q ¯ Qg case the soft anomalous dimension matrix Γ gg → Q ¯ Qg issplit in two parts: Γ gg → Q ¯ Qg = Γ (1) gg → Q ¯ Qg (Λ , Ω , Γ , Σ) + Γ (2) gg → Q ¯ Qg ( v i ) , (28)where 22 (1) gg → Q ¯ Qg = α s π T R × ( − N c ) L β N c − Ω 0 − Ω √ √ . . . − Ω L β + N c N c Γ2 . . . ( − N c ) L β N c . . . − Ω √ L β + N c (2+Λ)4 N c − Σ N c . . . Γ √ − Σ N c L β + N c (2+Λ)4 N c − Ω √ . . . Γ2 − Ω √ L β + N c (2+Λ)2 N c . . . Ω2 √ q ( N c − N c − N c +2)( N c − − Ω q N c +12( N c −
1) Ω2 q ( N c − N c +2)( N c − N c − Σ2 √ q ( N c − N c − N c +2)( N c − . . . − Ω √ √ N c − − Ω √ − Ω ( N c − ) √ N c − p N c − N c Σ √ √ N c − . . . Ω2 √ q ( N c +3)( N c +1)( N c − N c +2) − Ω q N c +32( N c +1) − Ω2 q ( N c − N c +3)( N c +1)( N c +2) − Σ2 √ q ( N c − N c +3)( N c +1)( N c +2) . . . Γ √ p N c − − Γ4 p N c − . . . − Ω √ √ N c − Ω q N c − N c − . . .. . . . . . Ω2 √ q ( N c − N c − N c +2)( N c − − Ω √ √ N c − √ q ( N c +3)( N c +1)( N c − N c +2) . . . − Ω q N c +12( N c − − Ω √ − Ω q N c +32( N c +1) Γ √ − Ω √ √ N c − . . . Ω2 q ( N c − N c +2)( N c − N c − − Ω ( N c − ) √ N c − − Ω2 q ( N c +3)( N c − N c +2)( N c +1) − Γ4 p N c − q N c − N c − . . . Γ4 p N c − − Ω4 p N c − . . . − Σ N c q N c +3 N c +1 − Σ2 q N c − N c − . . . L β + N c (3+Λ)4 N c Σ N c √ N c −
4) Ω √ √ N c − . . . L β + N c [2 N c +1+Λ( N c +1)]2 N c − Σ2 q ( N c − N c +1)( N c +3)2( N c −
2) Ω2 q ( N c − N c +1)( N c +3)2( N c − . . . L β + N c ( N c +1)(2+Λ)2 N c Σ2 q N c +3 N c +1 . . . − Σ2 q N c − N c − − N c Σ4 Σ2 q N c +3 N c +1 4+4 L β + N c (2+Λ)4 N c − N c Σ √ N c − . . . − N c Σ √ N c − L β N c , (29) (2) gg → Q ¯ Qg = α s π T R × N c ln ( v v ) 0 0 0 √ ln (cid:16) v v (cid:17) . . . N c ln ( v v ) 0 0 0 ln (cid:16) v v (cid:17) . . . N c ln ( v v ) 0 0 0 0 . . . N c ln (cid:0) v v v v (cid:1) N c ln (cid:16) v v (cid:17) . . . √ ln (cid:16) v v (cid:17) N c ln (cid:16) v v (cid:17) N c ln (cid:0) v v v v (cid:1) . . . ln (cid:16) v v (cid:17) N c ln ( v v ) . . . N c −
1) ln ( v v ) + ln ( v v ) . . . ln (cid:16) v v (cid:17) p N c − N c ln (cid:16) v v (cid:17) √ √ N c − . . . (cid:16) v v (cid:17) q ( N c +3)( N c +1)( N c − N c +2 . . . √ ln (cid:16) v v (cid:17) ln (cid:16) v v (cid:17) p N c − (cid:16) v v (cid:17) q N c − N c − . . . . . .. . . . . . . . . (cid:16) v v (cid:17) √ . . . ln (cid:16) v v (cid:17) p N c − . . . ln (cid:16) v v (cid:17) p N c − . . . ln (cid:16) v v (cid:17) N c √ √ N c − ln (cid:16) v v (cid:17) q ( N c +3)( N c +1)( N c − N c +2 1 √ . . . (cid:16) v v (cid:17) q N c − N c − . . . N c ln (cid:0) v v v v (cid:1) N c ln (cid:16) v v (cid:17) . . . N c + 1) ln ( v v ) − ln ( v v ) ln (cid:16) v v (cid:17) q N c +3 N c +1 . . . N c ln (cid:16) v v (cid:17) ln (cid:16) v v (cid:17) q N c +3 N c +1 N c ln (cid:0) v v v v (cid:1) N c √ N c − ln (cid:16) v v (cid:17) . . . N c √ N c − ln (cid:16) v v (cid:17) N c ln ( v v ) + diag (cid:18) − iπN c , − iπN c , − iπN c , − iπN c , − iπN c , − iπN c , − iπN c , − iπ ( N c − , − iπ ( N c + 1) , − iπN c , − iπN c (cid:19) . (30) n the next step a special case is considered p µ = p µ . The obtained matrix hasa block – diagonal form: Γ gg → Q ¯ Qg ( p µ = p µ ) = α s π T R × Γ × Γ × Γ × (31)where Γ × = 1 N c × diag (cid:8) (cid:0) − N c (cid:1) L β + N c ln ( v v ) − iπN c , (32) L β + N c (cid:18) − iπ + Λ ′ v v ) (cid:19) , (cid:0) − N c (cid:1) L β + N c ln ( v v ) − iπN c , (cid:9) , Γ × = L β + N c (2 − iπ +Λ ′ +2 ln( v v v ))4 N c − N c (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) − N c (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) L β + N c (2 − iπ +Λ ′ +2 ln( v v v ))4 N c , and 25 × = L β + N c (cid:16) − iπ + Λ ′ +ln( v v ) (cid:17) N c √ q ( N c − N c − N c +2) N c − (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) N c √ √ N c − (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) . . . √ q ( N c − N c − N c +2) N c − (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) L β + N c ( N c − ( ′ +2 ln( v v ) ) − iπN c ( N c +1)+2 N c ln v N c . . . N c √ √ N c − (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) L β + N c ( − iπ +Λ ′ +2 ln( v v v ) ) N c . . . − √ q ( N c +3)( N c +1)( N c − N c +2 (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) . . . − q N c − N c − (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) − N c (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) . . . . . .. . . − √ q ( N c +3)( N c +1)( N c − N c +2 (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) . . . − q N c − N c − (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) . . . − N c (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) . . . L β + N c ( N c +1) ( ′ +2 ln( v v ) ) − iπN c ( N c − − N c ln v N c q N c +3 N c +1 (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) . . . q N c +3 N c +1 (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) L β + N c ( − iπ +Λ ′ +2 ln( v v v ) ) N c − N c √ N c − (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) . . . − N c √ N c − (cid:16) Σ ′ − (cid:16) v v (cid:17)(cid:17) L β + N c (ln v − iπ ) N c . (33) For the case N c = 3 , the last block becomes even simpler: Γ × = Γ × ⊗ Γ × .After performing the limit Σ ′ = 0 matrices take the following form: gg → Q ¯ Qg (cid:0) Σ ′ = 0 (cid:1) = α s π T R × N c × diag (cid:26) (cid:0) − N c (cid:1) L β + N c ln ( v v ) − iπN c , L β + N c (cid:18) − iπ + Λ ′ v v ) (cid:19) , (cid:0) − N c (cid:1) L β + N c ln ( v v ) − iπN c , L β + N c (cid:0) − iπ + Λ ′ + 2 ln ( v v v ) (cid:1) , L β + N c (cid:0) − iπ + Λ ′ + 2 ln ( v v v ) (cid:1) , L β + N c (cid:18) − iπ + Λ ′ v v ) (cid:19) , L β + N c ( N c − (cid:0) ′ + 2 ln ( v v ) (cid:1) − iπN c ( N c + 1) + 2 N c ln v , L β + N c (cid:0) − iπ + Λ ′ + 2 ln ( v v v ) (cid:1) , L β + N c N c + 1) (cid:0) ′ + 2 ln ( v v ) (cid:1) − iπN c ( N c − − N c ln v , L β + N c (cid:0) − iπ + Λ ′ + 2 ln ( v v v ) (cid:1) , L β + N c (ln v − iπ ) (cid:27) . gg → Q ¯ Qg In this subsection, we perform an analogous analysis of the eigensystem for gg → Q ¯ Qg to the case of q ¯ q → Q ¯ Qg . The set of the eigenvalues is richer then in thescattering process of the quark and antiquark due to the larger colour basis. For θ = 90 ◦ the real parts of the regularized eigenvalues are shown in fig. 5 andthe imaginary parts are shown in fig. 6. The singular matrix in β has a form iπβ × diag (cid:0) − , − , , , , , , , , , (cid:1) in this case. One finds also onevalue of the leading small β behaviour of the eigenvalues, which is the same as inthe quark channel: λ sing = 6 log β. (34)After the procedure of regularization (analogous to the q ¯ q scattering case) onecan see some similarities for both the scattering angles. In the case θ = 90 ◦ theeigensystem consists of 6 different eigenvalues. The degenerate eigenvalues are27 = λ , λ = λ = λ = λ and λ = λ . The real parts of the eigenvaluesare nearly constant up to β ≈ . and all the imaginary parts are constant inwhole range of β . The results for θ = 30 ◦ are shown in fig. 7 (the real partsof eigenvalues) and in fig. 8 (the imaginary parts of eigenvalues). The singularpart of the eigenvalues at θ = 30 ◦ is the same as for θ = 90 ◦ . The degeneracy ofthe eigensystem is lower (the degeneracy between eigenvalues 4,5,6,7 is reduced tothe separate degeneracy λ = λ and λ = λ ). The real parts of eigenvalues arenearly flat for β < . then they grow rapidly. Im( λ ◦ ) ( Im( λ ◦ ) ) is a growing(decreasing) function of β . The imaginary parts of the remaining eigenvalues areconstant.Moreover, comparing the behaviour of the eigensystem of SAD matrices forprocesses q ¯ q → Q ¯ Qg and gg → Q ¯ Qg one finds some similarities. For example, at θ = 90 ◦ there is a constant behaviour in β for the imaginary part of eigenvalues inboth reactions. When the kinematic configuration becomes less symmetrical (the θ = 30 ◦ case) the set of eigenvalues with a flat β -dependence is reduced. In this paragraph we compare the calculated regularized eigenvalues λ reg i of theSAD matrices to the SAD eigenvalues for processes q ¯ q → Q ¯ Q and gg → Q ¯ Q inthe small β region. Note that the full eigenvalues for → processes containin addition to the regular parts a negative singular term β for the q ¯ q and gg channel. The logarithmic terms combine with the dominant regular terms into28 . . . . . . . . . − − − β R e (cid:0) λ r e g , ◦ (cid:1) Re (cid:0) λ reg , ◦ (cid:1) for gg → Q ¯ Qg Re (cid:16) λ reg , ◦ , (cid:17) Re (cid:16) λ reg , ◦ (cid:17) Re (cid:16) λ reg , ◦ , , , (cid:17) . . . . . . . . . − − − − β R e (cid:0) λ r e g , ◦ (cid:1) Re (cid:0) λ reg , ◦ (cid:1) for gg → Q ¯ Qg Re (cid:16) λ reg , ◦ (cid:17) Re (cid:16) λ reg , ◦ (cid:17) Re (cid:16) λ reg , ◦ (cid:17) Figure 5. The real parts of the regularized eigenvalues of ˜Γ S for gg → Q ¯ Qg at θ = 90 ◦ .29 . . . . . . . . . − − − − − β I m (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ (cid:1) for gg → Q ¯ Qg Im (cid:0) λ ◦ , , , , , (cid:1) Im (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ (cid:1) Figure 6. The imaginary parts of the eigenvalues of ˜Γ S for gg → Q ¯ Qg at θ = 90 ◦ .even larger negative terms of the eigenvalues, that is they lead to stronger effects ofgluon radiation. For the process q ¯ q → Q ¯ Q the real parts of the two at β → tendto − . α s π and . Recall that for the case of q ¯ q → Q ¯ Qg the largest (negative) SADeigenvalue reads Re (cid:0) λ reg , ◦ (cid:1) = − . α s π (for θ = 90 ◦ ) and Re (cid:0) λ reg , ◦ (cid:1) = − α s π (for θ = 30 ◦ ). It means that the effect of soft gluon radiation for q ¯ q → Q ¯ Qg isalmost two times stronger (the ◦ case) or three times larger (the ◦ case). In thegluonic case the radiation effects are even stronger. For gg → Q ¯ Qg we obtained Re (cid:0) λ reg , ◦ (cid:1) = − α s π (for θ = 90 ◦ ) and Re (cid:0) λ reg , ◦ (cid:1) = − α s π (for θ = 30 ◦ ), sothe radiation is enhanced by factors three and five correspondingly with respectto the process gg → Q ¯ Q . The imaginary parts of eigenvalues cancel out in theregime β → , so we will not discuss them. These results imply that the soft gluonradiation is a source of enhanced corrections for the heavy quark pair productionin association with a gluon jet. 30 . . . . . . . . . − − − β R e (cid:0) λ r e g , ◦ (cid:1) Re (cid:0) λ reg , ◦ (cid:1) for gg → Q ¯ Qg Re (cid:16) λ reg , ◦ , (cid:17) Re (cid:16) λ reg , ◦ , (cid:17) Re (cid:16) λ reg , ◦ , (cid:17) Re (cid:16) λ reg , ◦ , (cid:17) . . . . . . . . . − − β R e (cid:0) λ r e g , ◦ (cid:1) Re (cid:0) λ reg , ◦ (cid:1) for gg → Q ¯ Qg Re (cid:16) λ reg , ◦ (cid:17) Re (cid:16) λ reg , ◦ (cid:17) Re (cid:16) λ reg , ◦ (cid:17) Figure 7. The real parts of the regularized eigenvalues of ˜Γ S for gg → Q ¯ Qg at θ = 30 ◦ .31 . . . . . . . . . − − − − β I m (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ (cid:1) for gg → Q ¯ Qg Im (cid:0) λ ◦ , (cid:1) Im (cid:0) λ ◦ , , , (cid:1) Im (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ , (cid:1) Im (cid:0) λ ◦ (cid:1) Im (cid:0) λ ◦ (cid:1) Figure 8. The imaginary parts of the eigenvalues of ˜Γ S for gg → Q ¯ Qg at θ = 30 ◦ .In this paper we have derived the one-loop soft anomalous dimension matricesfor q ¯ q → Q ¯ Qg and gg → Q ¯ Qg . We presented the SAD matrices for an arbitraryscattering angle θ of a clustered pair of heavy quark and antiquark with respect tothe incoming parton axis in the CMS frame. We also analyzed the spectrum of theeigenvalues of the SAD matrices in details for two kinematic configurations θ = 90 ◦ and ◦ , performing explicit numerical calculations of the SAD eigenvalues. Theobtained results are a step towards implementing the soft resummation procedurefor Q ¯ Q - jet production in hadron colliders, and improving accuracy of theoreticalpredictions. Acknowledgments
The author would like to thank Prof. L. Motyka for the help of preparing thispaper, Prof. M. Praszałowicz for valuable comments on the manuscript, Prof. A.32ulesza for the discussion, Prof. M. Sjödahl and R. Schäfer for the correspondence.This work was supported by the Polish NCN grant DEC-2014/13/B/ST2/02486.