The Spin Alignment of Vector Mesons in High Energy pp Collisions
TThe Spin Alignment of Vector Mesons in High Energy pp Collisions
Kai-bao Chen, Zuo-tang Liang, Yu-kun Song, and Shu-yi Wei School of Science, Shandong Jianzhu University, Jinan, Shandong 250101, China Institute of Frontier and Interdisciplinary Science,Key Laboratory of Particle Physics and Particle Irradiation (MOE),Shandong University, Qingdao, Shandong 266237, China School of Physics and Technology, University of Jinan, Jinan, Shandong 250022, China European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*)and Fondazione Bruno Kessler, Strada delle Tabarelle 286, I-38123 Villazzano (TN), Italy
The spin alignment of vector mesons produced in high energy reactions is determined by the spin-dependent fragmentation function D LL ( z, µ f ) that is shown to be independent of the polarizationof the fragmenting quark. In this paper, we extract the spin-dependent fragmentation function D LL ( z, µ f ) from data on the spin alignment of K ∗ in e + e − annihilation at LEP in two differentscenarios and apply them to make predictions in pp collisions. We make detailed analysis of contri-butions from different sub-processes and show that the spin alignment should be quite significantalso in high energy pp collisions. I. INTRODUCTION
The spin dependence of fragmentation functions (FFs)is one of the important aspects in high energy spinphysics and plays an important role in studying the prop-erties of Quantum Chromodynamics (QCD) in generaland the hadronization mechanism in particular. So faras the polarization of produced hadrons is concerned,two classes of polarizations have been often studied, i.e.,the vector and the tensor polarizations. The former canbe studied by measuring the polarization of hyperonsvia their spin self analyzing weak decays, and the lat-ter is studied via strong decays of vector mesons into twopseudo-scalar mesons. The tensor polarization is usu-ally decomposed into five components. Among them, the S LL -component is directly related to the probability forthe third component of spin to take zero that is calledthe spin alignment. The spin alignment of vector mesonshas been measured in e + e − -annihilations and other highenergy reactions [1–8].Comparing with that of parton distribution functions(PDFs), we know even less about the spin dependence ofFFs. Among different aspects, hyperon polarizations arebest studied both experimentally [9–26] and phenomeno-logically [27–47]. Parameterizations of the correspondingspin dependent FFs have been proposed [48].For the tensor polarization of vector mesons, the studyhas in fact two advantages: (1) there is little contami-nation from decay processes; (2) no decay parameter isinvolved in the two-body strong decay of the vector me-son so that there is no uncertainty caused by the decayparameter [26] and the measurement efficiency is high.Measurements have been carried out on the spin align-ment and also the off-diagonal components in high en-ergy reactions [1–8]. We have in particular data on thespin alignment with relatively high accuracy from exper-iments at LEP [1–4]. The data show an evident spinalignment of vector mesons produced in e + e − annihila-tions and triggered many phenomenological studies [49–56]. Since the collision energy is at the Z pole, the fragmenting quark and anti-quark are highly polarized.Therefore, it was quite natural to attribute the spin align-ment to the polarization of the parent quark and/or anti-quark. Most of the phenomenological efforts have beenaccomplished following such a perception [53–56].Recently, progresses in the theoretical study have beenachieved in particular in the formal QCD description ofthe spin dependence of FFs [57–67]. In the QCD fieldtheory, FFs are defined via Lorentz decompositions ofthe quark-quark correlator. A systematic study of such adecomposition has been accomplished [63, 64] and the re-sults show in particular that the spin alignment is deter-mined solely by the S LL -dependent FF D LL and D LL is independent of the spin of the fragmenting quark. Cor-respondingly the first attempt to extract D LL ( z ) fromthe LEP data [1–3] has been performed in [65].Although it might be counter-intuitive, this conclusionis actually expected by the parity invariance. This canbe seen clearly in the helicity base. As a component ofthe polarization tensor, S LL is a scalar that is invariantunder the space inversion. Hence, one cannot establish arelation between S LL and the helicity of the quark in aparity conserved manner. This is quite different from thecase for the longitudinal polarization of Λ, where λ q λ Λ is a parity-invariant structure that should be included inthe decomposition of fragmentation function, where λ q and λ Λ are helicities of the quark and Λ respectively.Though the prediction is very solid, it is however quitedifficult to understand why the fragmentation of an un-polarized quark leads to vector meson with a larger prob-ability at the helicity zero state. Experimental check ofthe quark polarization independence of the vector me-son spin alignment should be a very basic test of thefragmentation picture and deep studies in this directionshould lead to new insights on the hadronization mecha-nism. In this connection, it might be also interesting tomention that spin effects have also attracted much atten-tion recently in heavy ion collisions. Here, a very specialstate of hadronic matter – the quark gluon plasma (QGP)is formed and the hadronization mechanism is different. a r X i v : . [ h e p - ph ] F e b Both hyperon polarization and vector meson spin align-ment have been studied at RHIC as well as at the LHCin this connection. The studies have been inspired bythe theoretical predictions [68, 69] and the experimentalconfirmation [70] on the global polarization of QGP withrespect to the reaction plane. The vector meson spinalignment was predicted [69] to be strongly dependenton the global polarization of quarks and anti-quarks be-cause they are produced via the quark coalescence ratherthan the quark fragmentation mechanism.Currently, both RHIC and the LHC provide good op-portunities in experiments to study vector meson spinalignment in pp collisions. In particular at RHIC thequark polarization independence can easily be testedsince RHIC is also a polarized pp collider. It is thustimely and important to make predictions for such mea-surements.In this paper, we study the spin alignment of vectormesons in pp → V X . We extract the S LL -dependentFF D LL from the LEP data and make predictions for pp collisions. In Sec. II, we present the basic formulaeneeded for such numerical calculations. In Sec. III, wepresent parameterizations of D LL and numerical resultsin Sec. IV. A short summary is given in Sec. V. II. THE FORMALISM
In this section, we present the differential cross sec-tion of vector meson production in pp collisions neededto calculate the spin alignment. We do the calculationsup to the order where the first order of pQCD evolutionof FFs is included, and present the formulae needed forsuch calculations. A. The differential cross section
We consider pp → V X in the high p T region wherecollinear factorization is applicable and study the spinalignment of produced vector meson V . Since the spinalignment is independent of the polarization of the frag-menting quark, the calculations are similar in the polar-ized or unpolarized collisions. We simply take unpolar-ized pp as an example.To calculate the spin alignment of V , we need to con-sider the spin dependent differential cross section. Werecall that the polarization of spin-1 particles is describedby a 3 × ρ . In the rest frame of theparticle, ρ is usually decomposed as [63, 64, 71], ρ = 13 ( + 32 S i Σ i + 3 T ij Σ ij ) , (1)where Σ i is the spin operator of a spin-1 particle, andΣ ij = (Σ i Σ j + Σ j Σ i ) − δ ij . T ij = Tr( ρ Σ ij ) is the polarization tensor and is parameterized as, T = 12 − S LL + S xxT T S xyT T S xLT S xyT T − S LL − S xxT T S yLT S xLT S yLT S LL . (2)Here, the polarization vector S is similar to that for spin-1/2 hadrons. The polarization tensor T is further de-composed into a Lorentz scalar S LL , a Lorentz vector S µLT = (0 , S xLT , S yLT , S µνT T thathas two nonzero independent components S xxT T = − S yyT T and S xyT T = S yxT T . It has in total five independent com-ponents. The spin alignment ρ is directly related to S LL by ρ = (1 − S LL ) /
3, where ρ takes the physicalmeaning of the probability for the third component m ofspin of V to take zero while S LL = ( ρ ++ + ρ −− ) / − ρ is the difference of m to take ± m is just the helicity λ V of the vector meson V .To calculate the spin alignment ρ of the producedvector meson V , we need to consider the S LL -dependentpart of the cross section and sum over all other compo-nents of polarization. Since S LL is a Lorentz scalar, the S LL -dependent part takes the same form as that of theunpolarized part. In this way, we obtain the differentialcross section in the collinear factorization form as [72], dσ pp → V X dyd p T = (cid:88) abcd (cid:90) dy (cid:90) dzz x f a ( x , µ f ) x f b ( x , µ f ) × π d ˆ σ ab → cd d ˆ t [ D V c ( z, µ f ) + S LL D V LLc ( z, µ f )] , (3)where f a,b ( x i , µ f ) is the parton distribution function [73]with x i the longitudinal momentum fraction and µ f thefactorization scale, D V c ( z, µ f ) and D V LLc ( z, µ f ) are thespin averaged and S LL -dependent FFs of c → V X re-spectively; y and p T denote the rapidity and transversemomentum of V and they are related to x , x and z by x = p T ( e y + e y ) /z √ s , x = p T ( e − y + e − y ) /z √ s ; y isthe rapidity of parton d after the scattering; d ˆ σ ab → cd /d ˆ t is the cross section of the partonic process ab → cd atthe leading order. The partonic process includes all theelementary processes at the parton level such as q q → q q , q ¯ q → q ¯ q , q q → q q , q g → q g , gg → gg , q ¯ q → q ¯ q , q ¯ q → q ¯ q , q ¯ q → gg , and gg → q ¯ q .We consider the unpolarized reaction and the cross sec-tions for these elementary processes are available in lit-erature [72]. Here, we note in particular that in Eq. (3),FFs are defined for a given polarization state followingthe same convention as that in [61] where D V c ( z, µ f ) isthe spin-averaged FF and is related to the spin-summedFF D Vc ( z, µ f ) by D Vc ( z, µ f ) = 3 D V c ( z, µ f ).Besides presenting the differential cross section interms of y and p T , we can also make predictions in termsof other variables such as ( x F , p T ) where x F ≡ p z / √ s =2 m T sinh y/ √ s , m T = (cid:112) m + p T , and dyd p T = dx F d p T / (cid:113) x F + 4 m T /s. (4) B. The spin alignment
The spin-alignment of V is then given by, ρ V = dσ λ V =0 (cid:46) (cid:88) λ V = ± , dσ λ V . (5)For the helicity λ V = ± S LL = 1 /
2, while for λ V = 0 state, S LL = −
1. Hence, we obtain, ρ V ( y, p T ) = 13 − dσ S LL pp → V X dyd p T (cid:46) dσ spin-summed pp → V X dyd p T , (6)where the spin-summed cross section is given by, dσ spin-summed pp → V X dyd p T = 3 (cid:88) abcd (cid:90) dy (cid:90) dzz x f a ( x , µ f ) × x f b ( x , µ f ) 1 π d ˆ σ ab → cd d ˆ t D c ( z, µ f ) , (7)while the S LL -dependent part is, dσ S LL pp → V X dyd p T = (cid:88) abcd (cid:90) dy (cid:90) dzz x f a ( x , µ f ) × x f b ( x , µ f ) 1 π d ˆ σ ab → cd d ˆ t D V LLc ( z, µ f ) . (8)From the definition of S LL in particular its relationto ρ we see that its value range is − ≤ S LL ≤ / − ≤ D LL ( z, µ f ) /D ( z, µ f ) ≤
1. In this way0 ≤ ρ ≤ C. The QCD evolution of D LL The QCD evolution of collinear FFs is given by corre-sponding DGLAP equations [74–77] with time-like split-ting functions [78–80]. The evolution equation of the S LL -dependent FF D LL is the same as that for unpo-larized FF D , i.e., ∂∂ ln Q D h LLa ( z, Q )= α s ( Q )2 π (cid:88) b (cid:90) z dξξ D h LLb ( zξ , Q ) P ba ( ξ ) , (9)where a or b denotes different types of partons includingdifferent flavors of quarks, anti-quarks and gluon. P ba ( ξ )is just the leading order splitting function. III. THE PARAMETERIZATION OF THEFRAGMENTATION FUNCTION
Even in the unpolarized case, we do not have an appro-priate parameterization for the fragmentation function ofvector mesons. Hence, we take the form of parameteriza-tions based on symmetry properties, models and conjunc-tions and fix the free parameters using data available.
A. The unpolarized fragmentation function
Currently, there is no parameterization of the fragmen-tation function of vector meson production available inthe market even for the unpolarized case. However, wehave parameterizations of the pseudo-scalar meson ( K ± )production e.g. AKK08 [81] and DHESS [82]. Also asimple relation between the yields of K ∗ / ¯ K ∗ and K ± has been observed [83] that leads to a linear dependenceof z for the ratio D K ∗ + u /D K + u approximately [65], i.e., D K ∗ + u ( z, µ ) = A (2 z + 1) D K + u ( z, µ ) , (10)where µ = 2 GeV is the initial scale and A ≈ . D K ∗ a ( z, µ ) = A (2 z + 1) D K a ( z, µ ) , (11)where a stands for u, d, s, ¯ u, ¯ d, ¯ s and gluon g ; K ∗ standsfor K ∗± , and ¯ K ∗ and K for the corresponding pseudo-scalar mesons.For FFs of pseudo-scalar mesons, we apply isospin andcharge conjugation symmetries and take, D K u = D ¯ K u = D K + d = D K − d , (12) D K d = D ¯ K d = D K + u = D K − u , (13) D K s = D ¯ K s = D K + s = D K − s , (14) D K u = D ¯ K u = D K − d = D K + d , (15) D K d = D ¯ K d = D K − u = D K + u , (16) D K s = D ¯ K s = D K − s = D K + s . (17)Here, for clarity, we omit arguments of fragmentationfunctions in Eqs. (12-17).For the unpolarized FF of ρ meson, we take it similarto that of K ∗ besides the strangeness suppression factorin the fragmentation process. As usual, we differenti-ate between the favored and unfavored fragmentations.For the favored FF, we separate it into the leading andnon-leading parts. The leading part is for hadron thatcontains the fragmenting quark and the non-leading partis the rest, i.e., we take, D ρ, favored1 a ( z, µ ) = D ρ, favored , leading1 a ( z, µ )+ D ρ, favored , nonleading1 a ( z, µ ) , (18) D ρ, favored , nonleading1 a ( z, µ ) = D ρ, unfavored1 b ( z, µ ) . (19)We relate those for ρ to K ∗ by, D ρ, favored , leading1nonstrange ( z, µ ) = D K ∗ , favored , leading1strange ( z, µ ) , (20) D ρ, unfavored1 a ( z, µ ) = D K ∗ , unfavored1 a ( z, µ ) /λ s , (21)where λ s is the strangeness suppression factor and is sim-ply taken as λ s = 1 / D ρ + u ( z, µ ) = D K ∗ s ( z, µ ) + 1 − λ s λ s D K ∗ u ( z, µ ) , (22) D ρ + d ( z, µ ) = D K ∗ u ( z, µ ) /λ s , (23) D ρ u ( z, µ ) = D ρ d ( z, µ )= 12 D K ∗ s ( z, µ ) + 2 − λ s λ s D K ∗ u ( z, µ ) , (24) D ρ + s ( z, µ ) = D ρ s ( z, µ ) = D ρ + d ( z, µ ) . (25) B. The S LL -dependent fragmentation function We take two different scenarios for the parameteriza-tions of S LL -dependent FFs. In the first scenario, wefollow the same strategy employed in [65] and differenti-ate between favored and unfavored fragmentations, i.e., D unfavored1 LL ( z, µ ) = c D unfavored1 ( z, µ ) , (26) D favored1 LL ( z, µ ) = c ( a z + 1) D favored1 ( z, µ ) , (27)where c and a are two free parameters.In the second scenario, we adopt the same form of pa-rameterizations for both favored and unfavored fragmen-tations. In this case, we find that the linear factor az + 1does not offer a good description to the data available [2]and we change the power of z to 1 /
2, i.e., D LL ( z, µ ) = c ( a z / + 1) D ( z, µ ) , (28)where c and a are free parameters.From the condition that − ≤ D LL /D ≤
1, we ob-tain constraints for the parameters in the parameteriza-tions given by Eqs. (26-28). They should be taken inthe range − ≤ c i ≤ { /c i , − /c i } ≤ a i + 1 ≤ max { /c i , − /c i } . It should be mentioned that by choos-ing the parameters in this region, we obtain the FF D LL satisfying the positivity condition − ≤ D LL /D ≤ D LL is the same as thatfor D , the results obtained at other scales remain inthis physical range. We also check this constantly in thenumerical calculations.We note that as we usually do for the spin dependentFFs, here we choose to parameterize the relationship be-tween the S LL -dependent FFs and unpolarized FFs inboth scenarios. In this way, we take the standpoint thatthe unpolarized FFs are known to much higher accura-cies than those for the polarized ones. However, in reality,there are still very high uncertainties in the unpolarizedFFs. We have also different parameterizations available.They can influence the values of the parameters in theparameterizations in both scenarios given by Eqs. (26-28)and also our numerical results for pp → V X . In this pa-per, we choose two sets of parameterizations, AKK08 [81]and DHESS [82], to show the influences.
C. Fits to the LEP data and results of D LL We fix the parameters in the parameterizations givenby Eqs. (26-28) by applying a χ analysis with thedata available [1, 2]. For each value of parameters inEqs. (26-28), we evolve the corresponding FFs utilizingthe DGLAP equation given by Eq. (9) to obtain a datasetof FFs at different factorization scales. Then we calculatethe spin alignment of K ∗ with this dataset and comparethe results obtained with the LEP data [1, 2] to get thecorresponding χ value. We note that, to be consistentwith the LEP data [1, 2], here as well as in the follow-ing of the paper, K ∗ represents the sum of K ∗ and itsanti-particle ¯ K ∗ . AKK08Scenario I
DHESSScenario I
FIG. 1. The χ plot in scenario I with AKK08 and DHESSparameterizations. Here, as well as in all figures in the fol-lowing of this paper, K ∗ denotes the sum of K ∗ and ¯ K ∗ . Shown in Fig. 1 are the χ plots in scenario I withAKK08 and DHESS parameterizations for unpolarizedFFs. The minimal values are χ = 0 .
88 at ( c , a ) =(0 . , − .
6) with AKK08 and χ = 0 .
83 at ( c , a ) =(0 . , − .
7) with DHESS respectively. With these valuesof c and a , we obtain the spin alignments of K ∗ and ρ mesons and compare the results with data [1, 2] inFig. 2. We also see that the data can be well describedin both cases.From Fig. 2, we see that the scale dependence is moreobvious in the small z region but quite small at large z .It is also more obvious for K ∗ than that for ρ . To seewhere this difference comes from, we take AKK08 as anexample to look at the corresponding results for FFs.In Fig. 3, we show the ratios D K ∗ LLc /D K ∗ c for differentflavors of quarks and that of gluon with AKK08 FFs.The corresponding S LL -dependent FFs D K ∗ LLc are shownin Fig. 4,We note that for the production of K ∗ , u -quark frag-mentation is unfavored while d and s fragmentations arefavored. From Fig. 3, we see that, in scenario I, the ratio D LL /D is almost the same for favored fragmentationsof different flavors of quarks but it is very different fromthat for the unfavored quark fragmentation. It is nega-tive and relatively larger in magnitude in most of the z region in the favored case, but is positive and relativelysmaller in the unfavored case. The scale dependence inthe favored case is quite weak but seems much strongerin the unfavored case. We see also that, though start-ing from the same ratio at the initial scale, the gluon AKK08Scenario I e + e − → K ∗ X AKK08Scenario I e + e − → ρ X . . . . . . . z ρ OPAL √ s = 2 GeV √ s = 91 . √ s = 500 GeV . . . . z DELPHI √ s = 2 GeV √ s = 91 . √ s = 500 GeV DHESSScenario I e + e − → K ∗ X DHESSScenario I e + e − → ρ X . . . . . . . z ρ OPAL √ s = 2 GeV √ s = 91 . √ s = 500 GeV . . . . z DELPHI √ s = 2 GeV √ s = 91 . √ s = 500 GeV FIG. 2. (Color online) The spin alignments of K ∗ and ρ in e + e − → V X at the Z -pole calculated in Scenario I withAKK08 and DHESS unpolarized FFs compared with exper-imental data [1, 2]. In the calculations, we have chosen thecenter-of-mass energy of e + e − as the factorization scale. -0.9-0.6-0.30 -0.9-0.6-0.300.1 0.3 0.5 0.7 0.90.260.280.30.32 0.1 0.3 0.5 0.7 0.9-0.4-0.200.20.4 FIG. 3. (Color online) The ratio of the spin dependent frag-mentation function D LL ( z, µ f ) to that of the correspondingspin averaged D ( z, µ f ) at different scales in Scenario I withAKK08 FFs. fragmentation function behaves quite differently from theunfavored quark fragmentation function after the QCDevolution. It becomes even negative at large z . This isbecause in QCD evolution to the first order, gluon split-ting to a q ¯ q -pair g → q ¯ q and gluon radiation of a quark q → qg are considered. For the gluon fragmentation, af-ter the gluon splitting g → q ¯ q , different flavors of quarkscan be produced so that favored quark fragmentation cancontribute thus brings large change to gluon FF. In con-trast, for the unfavored quark fragmentation, after thegluon radiation of the quark q → qg , the flavor of q is -100102030 10 -3 -100102030 10 -3 -3 -3 FIG. 4. (Color online) The spin dependent fragmentationfunction D LL ( z, µ f ) at different scales in Scenario I withAKK08 FFs. unchanged and the fragmentation remains unfavored.From Fig. 4, we see similar behaviors as those forthe corresponding ratios in Fig. 3. We see again sim-ilar behaviors for the favored FFs that are very differ-ent from the unfavored FF and also different from gluonFF. Here, we see explicitly that favored FFs dominate atlarger z while unfavored and gluon fragmentations playimportant roles at small z . We also see that because ofthe strangeness suppression in fragmentation, the leadingcontributions from s -quark fragmentation is much largerthan that from d -quark.From the results shown in Figs. 3 and 4, we can nowunderstand why there is a slight difference between thescale dependence of the spin alignment of K ∗ and ρ as shown in Fig. 2. Because of the strangeness suppres-sion in the favored d -fragmentation, contributions fromunfavored quark and gluon fragmentations are relativelylarger for the production of K ∗ than that of ρ . Thestronger scale dependence of D LL /D for the unfavoredand gluon fragmentation leads to a slightly stronger scaledependence of the spin alignment of K ∗ than that of ρ .The corresponding results of scale dependence withDHESS FFs are similar, and we skip the detailed dis-cussions here .The calculation in the second scenario is similar. Theobtained χ distributions are shown in Fig. 5. Here, wesee that with this scenario for polarized FFs, the χ dis-tributions in the case with AKK08 unpolarized FFs andthat with DHESS are quite similar to each other. Theminimal χ value flows in the valley where a ∼ − . χ = 2 .
68 in phys-ical regions of c and a at ( c , a ) = (1 . , − .
9) and χ = 1 .
78 at the same values of c and a with DHESS.We then calculate the spin alignment of K ∗ and ρ mesons with these parameters and compare with data in AKK08Scenario II
DHESS Scenario II
FIG. 5. The χ plot in scenario II with AKK08 and DHESSparameterizations. Fig. 6. We see that a reasonable agreement with the datacan also be achieved in this case.We again take AKK08 as an example and show our re-sults obtained in this scenario of the ratios D K ∗ LLc /D K ∗ c and the corresponding S LL -dependent FFs D K ∗ LLc inFigs. 7 and 8 respectively.
AKK08Scenario II e + e − → K ∗ X AKK08Scenario II e + e − → ρ X . . . . . . . z ρ OPAL √ s = 2 GeV √ s = 91 . √ s = 500 GeV . . . . z DELPHI √ s = 2 GeV √ s = 91 . √ s = 500 GeV DHESSScenario II e + e − → K ∗ X DHESSScenario II e + e − → ρ X . . . . . . . z ρ OPAL √ s = 2 GeV √ s = 91 . √ s = 500 GeV . . . . z DELPHI √ s = 2 GeV √ s = 91 . √ s = 500 GeV FIG. 6. (Color online) The spin alignment of K ∗ (left panel)and that of ρ (right panel) in e + e − → V X at the Z -polecalculated in Scenario II with AKK08 and DHESS FFs com-pared with experimental data [1, 2]. From Fig. 7, we see that the ratios D LL /D in thisscenario for favored, unfavored and gluon fragmentationsare quite similar with each other. By starting with thesame parameterization at the initial scale, we obtain sim-ilar results after the QCD evolution. The tiny differencesare resulted from the differences in the corresponding un-polarized FFs. Also because there is no large difference inthe ratios D LL /D between the favored and unfavoredfragmentations in this scenario, we do not see similar dif-ference in Fig. 6 between the spin alignment of K ∗ andthat of ρ in this scenario as that shown in Fig. 2 inscenario I.Because of the differences in the corresponding unpo- -1-0.500.50.1 0.3 0.5 0.7 0.9-1-0.50 0.3 0.5 0.7 0.9 FIG. 7. (Color online) The ratio of the spin dependent frag-mentation function D LL ( z, µ f ) to that of the correspondingspin averaged D ( z, µ f ) at different scales in Scenario II withAKK08 FFs. -3 -3 -3 -3 FIG. 8. (Color online) The spin-dependent fragmentationfunction D LL ( z, µ f ) at different scales in Scenario II withAKK08 FFs. larized FFs, the obtained D LL ( z, µ f ) shown in Fig. 8exhibits also quite large differences between the favoredand unfavored quark fragmentation and that of gluon.Here we see that, similar to those in scenario I, the fa-vored FFs also dominate at larger z but the unfavoredand gluon FFs may have large contribution in the small z -region. The gluon FF D LLg ( z, µ f ) is negative andquite large in magnitude for small z and should play animportant role in this region.Comparing the FFs obtained in the two different sce-narios, we see quite large differences. Nevertheless theobtained spin alignments in both cases can describethe LEP data [1, 2]. This is because the freedom tochoose different parameterizations is quite large, the LEPdata [1, 2] alone can not fix them to high accuracy. Inthis connection, we note that we have not considered theflavor dependence of the ratio between the unpolarizedand the S LL -dependent FFs besides different choices forthe favored and unfavored fragmentation in scenario I. Itis clear that more data in different reactions are necessaryin order to determine these FFs to high precisions. IV. NUMERICAL RESULTS FOR pp → V X
In this section, we apply the FFs obtained in Sec. IIIto pp → V X and calculate the spin alignment of vectormesons numerically. To have a better understanding ofthe results in such a complicated process, we first presentthe fractional production rate of different flavor of par-tons. After that, we show our predictions on the spinalignment of K ∗ and ρ mesons in both scenarios. Werecall that, for all the results presented throughout thepaper, to be consistent with the LEP data [1, 2], K ∗ represents the sum of K ∗ and its anti-particle ¯ K ∗ . A. Contributions of different flavors
From Eq. (3), we can calculate contributions from dif-ferent subprocesses to the cross section separately. Thefractional contribution from a given type of parton c tojet production is given by, R jet c ( y, p T ) = dσ pp → cX dyd p T (cid:46) (cid:88) c dσ pp → cX dyd p T , (29) dσ pp → cX dyd p T = (cid:88) abd (cid:90) dy x f a ( x , µ f ) × x f b ( x , µ f ) 1 π d ˆ σ ab → cd d ˆ t . (30)Similarly, the fractional contribution to vector mesonproduction is given by, R V c ( y, p T ) = dσ pp → cX → V X dyd p T (cid:46) dσ pp → V X dyd p T , (31) dσ pp → cX → V X dyd p T = (cid:88) abd (cid:90) dy (cid:90) dzz x f a ( x , µ f ) × x f b ( x , µ f ) 1 π d ˆ σ ab → cd d ˆ t D V c ( z, µ f ) . (32)In Fig. 9, we show the results of R jet c ( y, p T c ) calculatedfrom Eqs. (29) and (30) at the RHIC and LHC energies inthe middle rapidity as functions of p T . Taking K ∗ as anexample, we show the corresponding results of R V c ( y, p T )calculated with Eqs. (31) and (32) in Fig. 10.From Fig. 9, we see that in the presented p T regions,the gluon contribution dominates at both RHIC and theLHC energies for jet productions. The u/ ¯ u contribution | y | < . √ s = 200 GeV pp → jet + X | y | < . √ s = 5 .
02 TeV pp → jet + X − p T (GeV) R j e t c ( y , p T ) gluon u/ ¯ ud/ ¯ d s/ ¯ s
50 100 150 200 p T (GeV) gluon u/ ¯ ud/ ¯ d s/ ¯ s FIG. 9. (Color online) Fractional contributions R jet c ( y, p T )to jet production from different flavors of quarks/anti-quarksand gluon at | y | < . p T in pp collisions atRHIC energy √ s = 200 GeV (left) and LHC energy √ s = 5 . | y | < . √ s = 200 GeV pp → K ∗ X , AKK08 | y | < . √ s = 5 .
02 TeV pp → K ∗ X , AKK08 − p T (GeV) R V c ( y , p T ) g → K ∗ u/ ¯ u → K ∗ d/ ¯ d → K ∗ s/ ¯ s → K ∗
20 40 60 80 100 120 140 p T (GeV) g → K ∗ u/ ¯ u → K ∗ d/ ¯ d → K ∗ s/ ¯ s → K ∗ | y | < . √ s = 200 GeV pp → K ∗ X , DHESS | y | < . √ s = 5 .
02 TeV pp → K ∗ X , DHESS − − p T (GeV) R V c ( y , p T ) g → K ∗ u/ ¯ u → K ∗ d/ ¯ d → K ∗ s/ ¯ s → K ∗
20 40 60 80 100 120 140 p T (GeV) g → K ∗ u/ ¯ u → K ∗ d/ ¯ d → K ∗ s/ ¯ s → K ∗ FIG. 10. (Color online) Fractional contributions R V c ( y, p T ) tothe production of K ∗ from different flavors of quarks/anti-quarks and gluon at | y | < . p T in pp colli-sions at RHIC energy √ s = 200 GeV (left) and LHC energy √ s = 5 .
02 TeV (right) obtained with AKK08 and DHESSFFs respectively. is the largest among the three flavors of quarks while s/ ¯ s is the smallest. This results from the differences inPDFs [73] for different flavors of partons.However, when FFs are taken into account, fromFig. 10, we see that the gluon contribution becomes lessdominate. With AKK08 FFs the d/ ¯ d contribution is evenlarger than the gluon contribution at the RHIC energy for p T >
12 GeV while u/ ¯ u contribution becomes the small-est one. With DHESS FFs the d/ ¯ d and s/ ¯ s contributionsdominate at almost all p T range at the RHIC energy, andthey dominate at the LHC energy for p T >
50 GeV. Theexact value of R Vc depends on the specific FF parame-terizations, but the overall trends are similar. This isbecause the differential cross section for the productionof parton c decreases very fast with increasing p T , muchfaster than the FF of c → V X decreases with increas-ing z . Usually the z -dependence of FF is much smoothercompared with the p T -dependence of the cross section.As a result, in the large p T region for hadron produc-tion, contributions from relatively large z (say z > . p T .From Fig. 10, we also see that, by studying the p T de-pendence in the central rapidity region, we can study theinterplay of contributions from gluon and favored quarkfragmentation, while at the LHC, we mainly study thecontribution from gluon fragmentation. Quark fragmen-tations should dominate the fragmentation regions in thecollision processes, i.e., at very forward or very backwardrapidities.To see the behaviors at the fragmentation regions ex-plicitly, in Figs. 11 and 12, we show the correspondingresults at the RHIC energy with p T > p T >
10 GeV as functions of x F . pp → jet + X pp → jet + Xp T > √ s = 200 GeV p T >
10 GeV √ s = 5 .
02 TeV . . . − − − x F R j e t c ( x F ) gluon u/ ¯ ud/ ¯ d s/ ¯ s . . . x F gluon u/ ¯ ud/ ¯ d s/ ¯ s FIG. 11. (Color online) Fractional contributions R jet c ( y, p T )to jet production from different flavors of quarks/antiquarksand gluon as functions of x F in pp collisions at RHIC energy √ s = 200 GeV with p T ≥ √ s = 5 .
02 TeV p T ≥
10 GeV (right).
From Fig. 11 and 12, we see clearly that in the large x F region quark contribution dominates. For jet production, u/ ¯ u plays the most important role. Taking the FFs intoaccount, for K ∗ -production, the favored fragmentationfrom d/ ¯ d dominates. Hence, by studying hadron produc-tion at larger x F , we study predominately the favoredquark fragmentation.At the end of this part, we emphasize that, by study-ing vector meson production in pp → V X for large p T at RHIC and LHC energies, even in the central rapidityregions, contributions from FFs at relatively large z dom-inate. From the results for FFs obtained in Sec. III B, wefind D LL is significantly different from zero as well inthe relatively large z region. This leads us to the expec-tation that the vector meson spin alignment should bequite significant in pp collisions. pp → K ∗ X , AKK08 pp → K ∗ X , AKK08 p T > √ s = 200 GeV p T >
10 GeV √ s = 5 .
02 TeV . . . . . − x F R V c ( x F ) g → K ∗ u/ ¯ u → K ∗ d/ ¯ d → K ∗ s/ ¯ s → K ∗ . . . . . x F g → K ∗ u/ ¯ u → K ∗ d/ ¯ d → K ∗ s/ ¯ s → K ∗ pp → K ∗ X , DHESS pp → K ∗ X , DHESS p T > √ s = 200 GeV p T >
10 GeV √ s = 5 .
02 TeV . . . . . − − x F R V c ( x F ) g → K ∗ u/ ¯ u → K ∗ d/ ¯ d → K ∗ s/ ¯ s → K ∗ . . . . . x F g → K ∗ u/ ¯ u → K ∗ d/ ¯ d → K ∗ s/ ¯ s → K ∗ FIG. 12. (Color online) Fractional contributions R V c ( y, p T )to the production of K ∗ from different flavors ofquarks/antiquarks and gluon as functions of x F in pp col-lisions at RHIC energy √ s = 200 GeV with p T ≥ √ s = 5 .
02 TeV with p T ≥
10 GeV(right) obtained with AKK08 and DHESS FFs respectively.
B. The spin alignment in pp → V X
Using the spin-dependent FFs obtained in Sec. III B,we calculate the spin alignment of vector meson in pp → V X using Eqs. (6) and (8). We present our predictionsin the following.
AKK08 pp → K ∗ X , √ s = 200 GeV AKK08 pp → ρ X , √ s = 200 GeV . . . . p T (GeV) ρ | y | < .
5: Sce. I Sce. II1 < | y | <
2: Sce. I Sce. II p T (GeV) | y | < .
5: Sce. I Sce. II1 < | y | <
2: Sce. I Sce. II
DHESS pp → K ∗ X , √ s = 200 GeV DHESS pp → ρ X , √ s = 200 GeV . . . . p T (GeV) ρ | y | < .
5: Sce. I Sce. II1 < | y | <
2: Sce. I Sce. II p T (GeV) | y | < .
5: Sce. I Sce. II1 < | y | <
2: Sce. I Sce. II
FIG. 13. (Color online) Spin alignments of vector mesons in pp collisions at RHIC energy √ s = 200 GeV for K ∗ and ρ in two rapidity regions as functions of p T . In Fig. 13, we show the spin alignments for K ∗ and ρ at the RHIC energy in two rapidity regions as functionsof p T . From Fig. 13, we see the following distinct featuresfor the spin alignment in pp → V X at RHIC energy.First, both the results for K ∗ and those for ρ aresignificantly different from 1 /
3, i.e., they show quite sig-nificant spin alignments in both cases. The deviationsof ρ from 1 / p T . This is just consistent with the qualitative expec-tation mentioned at the end of Sec. IV A. The increaseswith increasing p T are mainly due to increasing relativecontributions from the quark fragmentation in particu-lar those in the large z region where D LL /D is moresignificant. We emphasize that these qualitative featuresare essential properties of the results and they are in-dependent of the details of the parameterizations of thepolarized FFs and unpolarized FFs.Second, there is a significant difference between theresults of the precise magnitudes obtained in scenario Iand those in scenario II and there is also a quite obviousdifference between results obtained using the two differ-ent sets of unpolarized FFs. We see in particular a largedifference between the results obtained with AKK08 andDHESS FFs in scenario I. These differences are mainlydue to the large difference in gluon fragmentation func-tions in the two scenarios and in the two sets of unpolar-ized FFs. With AKK08, the gluon contribution to vec-tor meson production is very large (see Fig. 12) and thisleads to a large difference between the results obtainedwith polarized FFs in scenario I and those in scenario II(upper panel in Fig. 13); while with DHESS, the gluoncontribution is much smaller than that with AKK08 (seeFig. 12), the difference becomes much smaller (lowerpanel in Fig. 13).Third, there is also a quite significant difference be-tween the results obtained in the two different rapidityregions. This is mainly because there are more contribu-tions from the quark jets in the forward/backward rapid-ity. This leads to a larger vector meson spin alignmentin a more forward/backward rapidity.Forth, there is no distinct difference between the re-sults for K ∗ and those for ρ . This is because thatwe have not considered the flavor dependence in our pa-rameterizations of D LL /D . The small difference comesmainly from strangeness suppression in the unpolarizedfragmentation functions.In Fig. 14, we show the results obtained at the LHCenergy. From Fig. 14, we see quite similar qualitativefeatures as those seen from Fig. 13 at the RHIC energy.Here, we have the advantage to study a much wider p T range so that we can study the p T -dependence more in-tensively. As mentioned above, the increase with p T ofthe spin alignment is caused by the increasing contribu-tions from quarks fragmentations relative to the gluonfragmentation. It is also because the gluon contributionbecomes more dominate at LHC energy in the relativesmall p T region in Fig. 14, the spin alignment in that re-gion is closer to 1 / AKK08 pp → K ∗ X , √ s = 5 .
02 TeV AKK08 pp → ρ X , √ s = 5 .
02 TeV
20 40 60 80 100 120 1400 . . . . . p T (GeV) ρ | y | < .
5: Sce. I Sce. II2 < | y | <
3: Sce. I Sce. II
20 40 60 80 100 120 140 p T (GeV) | y | < .
5: Sce. I Sce. II2 < | y | <
3: Sce. I Sce. II
DHESS pp → K ∗ X , √ s = 5 .
02 TeV DHESS pp → ρ X , √ s = 5 .
02 TeV
20 40 60 80 100 120 1400 . . . . . p T (GeV) ρ | y | < .
5: Sce. I Sce. II2 < | y | <
3: Sce. I Sce. II
20 40 60 80 100 120 140 p T (GeV) | y | < .
5: Sce. I Sce. II2 < | y | <
3: Sce. I Sce. II
FIG. 14. (Color online) Spin alignments of vector mesons in pp collisions at the LHC energy √ s = 5 .
02 TeV for K ∗ and ρ in two rapidity regions as functions of p T . I and II are also more significant.In Figs. 15 and 16, we show results for p T -integratedspin alignments of K ∗ and ρ as functions of x F at RHICand LHC energies respectively. p T > √ s = 200 GeV pp → K ∗ X p T > √ s = 200 GeV pp → ρ X . . . . . . . . . x F ρ AKK08 Sce. I Sce. IIDHESS Sce. I Sce. II . . . . . x F AKK08 Sce. I Sce. IIDHESS Sce. I Sce. II
FIG. 15. (Color online) Spin alignments of vector mesons in pp collisions at the RHIC energy √ s = 200 GeV for K ∗ and ρ at p T > x F . Here, from Figs. 15 and 16, we see rapid increases of thespin alignment with increasing x F , quite similar to thatobserved in e + e − shown in Fig. 2 and such a behavioris more obvious in scenario I. The increase reflects againthe increasing relative contributions from favored quarkfragmentations to the gluon fragmentation and also z -dependence of the favored S LL -dependent FF D LL rel-ative to the corresponding unpolarized FF D . The rel-ative larger values in the small x F region in scenario IIare due to the quite large D LL of gluon fragmentationin the small z region. We recall that gluon fragmentationis even less known in the unpolarized case, this providesalso a good opportunity to study gluon fragmentationmechanism.0 p T >
10 GeV √ s = 5 .
02 TeV pp → K ∗ X p T >
10 GeV √ s = 5 .
02 TeV pp → ρ X . . . . . . . . . x F ρ AKK08 Sce. I Sce. IIDHESS Sce. I Sce. II . . . . . x F AKK08 Sce. I Sce. IIDHESS Sce. I Sce. II
FIG. 16. (Color online) Spin alignments of vector mesons in pp collisions at the LHC energy √ s = 5 .
02 TeV for K ∗ and ρ at p T >
10 GeV as functions of x F . From all the results shown in Figs. 13-16, we see clearlythat spin alignments of vector mesons are in generalquite significant in pp → V X at high energies. Study-ing these spin alignments should provide a good test toQCD fragmentation mechanism in general and differen-tiate between different parameterizations scenarios, pro-vide precise information on quark or gluon fragmentationin different kinematic regions in particular.
V. SUMMARY
In the QCD description of high energy reactions, thespin alignment of vector meson in a fragmentation pro-cess is described by the S LL -dependent fragmentationfunction D LL defined via the Lorentz decomposition ofthe quark-quark correlator. A systematic study of the Lorentz decomposition show that D LL is independentof the polarization of the fragmenting quark. The firstattempt to extract D LL for K ∗ from the LEP data [1, 2]on e + e − -annihilations has been made in [65].In this paper, we follow the same procedure of [65] andmake parameterizations of D LL in two different scenar-ios for K ∗ and ρ from different flavors of quarks, anti-quarks and gluon and evolve them using DGLAP equa-tion. We apply the results obtained to pp → V X andmake predictions for the spin alignment of vector mesonsat RHIC and LHC energies.The results show that the data [1, 2] available is farfrom enough to determine the precise forms of D LL fordifferent vector mesons from different flavors of quarks,anti-quarks and gluon. Nevertheless, we predict very sig-nificant spin alignments for vector mesons in pp collisionsat high energies. The results show a number of distinctfeatures so that measurements of vector meson spin align-ments in different kinematic regions in pp collisions arenot only able to check the quark polarization indepen-dence of D LL but also sensitive to study the favoredquark fragmentation and/or gluon fragmentation respec-tively. ACKNOWLEDGEMENTS
This work was supported in part by the NationalNatural Science Foundation of China (approval Nos.11675092, 11947055, 11890713, 11505080) and Shan-dong Province Natural Science Foundation Grant No.ZR2018JL006. [1] P. Abreu et al. [DELPHI Collaboration], Phys. Lett. B , 271 (1997). doi:10.1016/S0370-2693(97)00758-2[2] K. Ackerstaff et al. [OPAL Collaboration], Phys. Lett.B , 210 (1997) doi:10.1016/S0370-2693(97)01077-0[hep-ex/9708022].[3] G. Abbiendi et al. [OPAL Collaboration], Eur. Phys.J. C , 61 (2000) doi:10.1007/s100520050003 [hep-ex/9906043].[4] K. Ackerstaff et al. [OPAL Collaboration], Z. Phys. C ,437 (1997).[5] A. Chukanov et al. [NOMAD Collaboration], Eur. Phys.J. C , 69 (2006) doi:10.1140/EPJC/S2006-02500-4[hep-ex/0604050].[6] B. I. Abelev et al. [STAR Collaboration], Phys. Rev. C , 061902 (2008).[7] C. Zhou, Nucl. Phys. A , 559 (2019).[8] S. Acharya et al. [ALICE Collaboration],arXiv:1910.14408 [nucl-ex].[9] A. Lesnik et al. , Phys. Rev. Lett. , 770 (1975).[10] G. Bunce et al. , Phys. Rev. Lett. , 1113 (1976).[11] J. Bensinger et al. , Phys. Rev. Lett. , 313 (1983).[12] S. A. Gourlay et al. , Phys. Rev. Lett. , 2244 (1986).[13] M. I. Adamovich et al. [WA89 Collaboration], Z.Phys. A , 379 (1995) doi:10.1007/BF01291194 [hep- ex/9409001].[14] M. Althoff et al. [TASSO Collaboration], Z. Phys. C ,27 (1985). doi:10.1007/BF01642477[15] D. Buskulic et al. [ALEPH Collaboration], Phys. Lett. B , 319 (1996). doi:10.1016/0370-2693(96)00300-0[16] K. Ackerstaff et al. [OPAL Collaboration], Eur. Phys.J. C , 49 (1998) doi:10.1007/s100520050123 [hep-ex/9708027].[17] A. Airapetian et al. [HERMES Collabo-ration], Phys. Rev. D , 112005 (2001)doi:10.1103/PhysRevD.64.112005 [hep-ex/9911017].[18] A. Airapetian et al. [HERMES Collabo-ration], Phys. Rev. D , 072004 (2006)doi:10.1103/PhysRevD.74.072004 [hep-ex/0607004].[19] P. Astier et al. [NOMAD Collaboration], Nucl. Phys. B , 3 (2000). doi:10.1016/S0550-3213(00)00503-4[20] P. Astier et al. [NOMAD Collaboration], Nucl. Phys. B , 3 (2001) doi:10.1016/S0550-3213(01)00181-X [hep-ex/0103047].[21] M. Alekseev et al. [COMPASS Collaboration], Eur. Phys.J. C , 171 (2009) doi:10.1140/epjc/s10052-009-1143-7[arXiv:0907.0388 [hep-ex]].[22] B. I. Abelev et al. [STAR Collaboration], Phys. Rev.D , 111102 (2009) doi:10.1103/PhysRevD.80.111102 [arXiv:0910.1428 [hep-ex]].[23] J. Adam et al. [STAR Collaboration], Phys. Rev.D , 112009 (2018) doi:10.1103/PhysRevD.98.112009[arXiv:1808.07634 [hep-ex]].[24] J. Adam et al. [STAR Collaboration], Phys. Rev.D , 091103 (2018) doi:10.1103/PhysRevD.98.091103[arXiv:1808.08000 [hep-ex]].[25] Y. Guan et al. [Belle Collaboration], Phys. Rev. Lett. , 042001 (2019) doi:10.1103/PhysRevLett.122.042001[arXiv:1808.05000 [hep-ex]].[26] M. Ablikim et al. [BES III Collaboration], Nat. Phys.15, 631–634 (2019). https://doi.org/10.1038/s41567-019-0494-8.[27] G. Gustafson and J. Hakkinen, Phys. Lett. B , 350(1993). doi:10.1016/0370-2693(93)91444-R[28] Z. t. Liang and C. Boros, Phys. Rev. Lett. ,3608 (1997) doi:10.1103/PhysRevLett.79.3608 [hep-ph/9708488].[29] C. Boros and Z. t. Liang, Phys. Rev. D , 4491 (1998)doi:10.1103/PhysRevD.57.4491 [hep-ph/9803225].[30] C. x. Liu and Z. t. Liang, Phys. Rev. D , 094001 (2000)doi:10.1103/PhysRevD.62.094001 [hep-ph/0005172].[31] C. x. Liu, Q. h. Xu and Z. t. Liang, Phys. Rev.D , 073004 (2001) doi:10.1103/PhysRevD.64.073004[hep-ph/0106184].[32] Liang Zuo-tang and Liu Chun-xiu, Phys. Rev. D ,057302 (2002). doi:10.1103/PhysRevD.66.057302[33] Q. h. Xu, C. x. Liu and Z. t. Liang, Phys. Rev.D , 114008 (2002) doi:10.1103/PhysRevD.65.114008[hep-ph/0204318].[34] H. Dong, J. Zhou and Z. t. Liang, Phys. Rev. D ,033006 (2005) doi:10.1103/PhysRevD.72.033006 [hep-ph/0506207].[35] Q. h. Xu, Z. t. Liang and E. Sichtermann, Phys. Rev.D , 077503 (2006) doi:10.1103/PhysRevD.73.077503[hep-ph/0511061].[36] Y. Chen, Z. t. Liang, E. Sichtermann, Q. h. Xuand S. s. Zhou, Phys. Rev. D , 054007 (2008)doi:10.1103/PhysRevD.78.054007 [arXiv:0707.0534 [hep-ph]].[37] J. Zhou, F. Yuan and Z. T. Liang, Phys. Rev.D , 114008 (2008) doi:10.1103/PhysRevD.78.114008[arXiv:0808.3629 [hep-ph]].[38] S. s. Zhou, Y. Chen, Z. t. Liang and Q. h. Xu, Phys. Rev.D , 094018 (2009) doi:10.1103/PhysRevD.79.094018[arXiv:0902.1883 [hep-ph]].[39] B. Q. Ma and J. Soffer, Phys. Rev. Lett. , 2250 (1999)doi:10.1103/PhysRevLett.82.2250 [hep-ph/9810517].[40] B. Q. Ma, I. Schmidt and J. J. Yang, Phys. Lett.B , 107 (2000) doi:10.1016/S0370-2693(00)00167-2[hep-ph/9906424].[41] B. Q. Ma, I. Schmidt and J. J. Yang, Phys. Rev.D , 034017 (2000) doi:10.1103/PhysRevD.61.034017[hep-ph/9907224].[42] B. Q. Ma, I. Schmidt and J. J. Yang, Nucl. Phys.B , 331 (2000) doi:10.1016/S0550-3213(00)00021-3[hep-ph/9907556].[43] B. Q. Ma, I. Schmidt, J. Soffer and J. J. Yang, Eur.Phys. J. C , 657 (2000) doi:10.1007/s100520000447[hep-ph/0001259].[44] B. Q. Ma, I. Schmidt, J. Soffer and J. J. Yang, Phys. Rev.D , 114009 (2000) doi:10.1103/PhysRevD.62.114009[hep-ph/0008295]. [45] Y. Chi and B. Q. Ma, Phys. Lett. B , 737 (2013)doi:10.1016/j.physletb.2013.09.044 [arXiv:1310.2005[hep-ph]].[46] X. Liu and B. Q. Ma, Eur. Phys. J. C ,no. 5, 409 (2019) doi:10.1140/epjc/s10052-019-6916-z[arXiv:1905.02360 [hep-ph]].[47] J. R. Ellis, A. Kotzinian and D. V. Naumov, Eur. Phys.J. C , 603 (2002) doi:10.1140/epjc/s2002-01025-2 [hep-ph/0204206].[48] D. de Florian, M. Stratmann and W. Vogelsang, Phys.Rev. D , 5811 (1998) doi:10.1103/PhysRevD.57.5811[hep-ph/9711387].[49] M. Anselmino and P. Kroll, Phys. Rev. D , 36 (1984).[50] M. Anselmino, M. Bertini, F. Murgia and P. Quintairos,Eur. Phys. J. C , 539 (1998) doi:10.1007/s100520050159[hep-ph/9704420].[51] M. Anselmino, M. Bertini, F. Murgia and B. Pire,Phys. Lett. B , 347 (1998) doi:10.1016/S0370-2693(98)00978-2 [hep-ph/9805234].[52] M. Anselmino, M. Bertini, F. Caruso, F. Murgiaand P. Quintairos, Eur. Phys. J. C , 529 (1999)doi:10.1007/s100529900200, 10.1007/s100520050652[hep-ph/9904205].[53] Q. h. Xu, C. x. Liu and Z. t. Liang, Phys. Rev.D , 111301 (2001) doi:10.1103/PhysRevD.63.111301[hep-ph/0103267].[54] Q. h. Xu and Z. t. Liang, Phys. Rev. D , 017301 (2002)doi:10.1103/PhysRevD.66.017301 [hep-ph/0205291].[55] Q. h. Xu and Z. t. Liang, Phys. Rev. D , 114013 (2003)doi:10.1103/PhysRevD.67.114013 [hep-ph/0304125].[56] Q. h. Xu and Z. t. Liang, Phys. Rev. D , 034023 (2003)doi:10.1103/PhysRevD.68.034023 [hep-ph/0307327].[57] D. Boer, R. Jakob and P. J. Mulders, Nucl. Phys.B , 345 (1997) doi:10.1016/S0550-3213(97)00456-2[hep-ph/9702281].[58] D. Boer, R. Jakob and P. J. Mulders, Phys. Lett.B , 143 (1998) doi:10.1016/S0370-2693(98)00136-1[hep-ph/9711488].[59] D. Boer, Nucl. Phys. B , 23 (2009) doi:10.1016/j.nuclphysb.2008.06.011 [arXiv:0804.2408[hep-ph]].[60] D. Pitonyak, M. Schlegel and A. Metz, Phys. Rev. D ,no. 5, 054032 (2014) doi:10.1103/PhysRevD.89.054032[arXiv:1310.6240 [hep-ph]].[61] S. y. Wei, Y. k. Song and Z. t. Liang, Phys. Rev. D ,no. 1, 014024 (2014) doi:10.1103/PhysRevD.89.014024[arXiv:1309.4191 [hep-ph]].[62] S. Y. Wei, K. b. Chen, Y. k. Song andZ. t. Liang, Phys. Rev. D , no. 3, 034015 (2015)doi:10.1103/PhysRevD.91.034015 [arXiv:1410.4314[hep-ph]].[63] K. b. Chen, S. y. Wei, W. h. Yang and Z. t. Liang,arXiv:1505.02856 [hep-ph].[64] K. b. Chen, W. h. Yang, S. y. Wei andZ. t. Liang, Phys. Rev. D , no. 3, 034003 (2016)doi:10.1103/PhysRevD.94.034003 [arXiv:1605.07790[hep-ph]].[65] K. b. Chen, W. h. Yang, Y. j. Zhou andZ. t. Liang, Phys. Rev. D , no. 3, 034009 (2017)doi:10.1103/PhysRevD.95.034009 [arXiv:1609.07001[hep-ph]].[66] S. y. Wei, Y. k. Song, K. b. Chen andZ. t. Liang, Phys. Rev. D , no. 7, 074017 (2017)doi:10.1103/PhysRevD.95.074017 [arXiv:1611.08688[hep-ph]]. [67] K. b. Chen, S. y. Wei and Z. t. Liang, Front. Phys. (Bei-jing) , no. 6, 101204 (2015) doi:10.1007/s11467-015-0477-x [arXiv:1506.07302 [hep-ph]].[68] Z. T. Liang and X. N. Wang, Phys. Rev. Lett. , 102301 (2005) Erratum: [Phys. Rev. Lett. , 039901 (2006)]. doi:10.1103/PhysRevLett.94.102301,10.1103/PhysRevLett.96.039901 [nucl-th/0410079].[69] Z. T. Liang and X. N. Wang, Phys. Lett. B , 20 (2005)doi:10.1016/j.physletb.2005.09.060 [nucl-th/0411101].[70] L. Adamczyk et al. [STAR Collaboration], Nature ,62 (2017). doi:10.1038/nature23004. [arXiv:1701.06657[nucl-ex]].[71] A. Bacchetta and P. J. Mulders, Phys. Rev. D , 114004(2000) [hep-ph/0007120].[72] J. F. Owens, Rev. Mod. Phys. , 465 (1987).doi:10.1103/RevModPhys.59.465[73] S. Dulat et al. , Phys. Rev. D , no. 3, 033006 (2016)doi:10.1103/PhysRevD.93.033006 [arXiv:1506.07443[hep-ph]].[74] Y. L. Dokshitzer, Sov. Phys. JETP , 641 (1977) [Zh.Eksp. Teor. Fiz. , 1216 (1977)].[75] V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. ,438 (1972) [Yad. Fiz. , 781 (1972)]. [76] V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. ,675 (1972) [Yad. Fiz. , 1218 (1972)].[77] G. Altarelli and G. Parisi, Nucl. Phys. B , 298 (1977).doi:10.1016/0550-3213(77)90384-4[78] J. F. Owens, Phys. Lett. B , 85 (1978).doi:10.1016/0370-2693(78)90108-9[79] H. Georgi and H. D. Politzer, Nucl. Phys. B , 445(1978). doi:10.1016/0550-3213(78)90269-9[80] T. Uematsu, Phys. Lett. B , 97 (1978).doi:10.1016/0370-2693(78)90444-6[81] S. Albino, B. A. Kniehl and G. Kramer, Nucl. Phys.B , 42 (2008) doi:10.1016/j.nuclphysb.2008.05.017[arXiv:0803.2768 [hep-ph]].[82] D. de Florian, M. Epele, R. Hernandez-Pinto,R. Sassot and M. Stratmann, Phys. Rev. D (2017) no.9, 094019 doi:10.1103/PhysRevD.95.094019[arXiv:1702.06353 [hep-ph]].[83] P. V. Shlyapnikov, Phys. Lett. B512