Gluon Sivers function in a lightcone spectator model
Abstract
We calculate the gluon Sivers function of the proton in the valence region using a lightcone spectator model with the presence of the gluon degree of freedom. We obtain the values of the parameters by fitting the model resulting gluon density distribution to the known parametrization. We find that our results agree with the recent phenomenological extraction of the gluon Sivers function after considering the evolution effect. We also estimate the mean transverse momentum of the gluon in a transversely polarized proton and find that it is within the range implied by the Burkardt sum rule.
pacs:
12.38.t, 13.60.r, 13.88.+eI Introduction
The Sivers function sivers is a leadingtwist transverse momentum dependent (TMD) distribution function which describes the asymmetric distribution of unpolarized partons in a transversely polarized proton. It is of great interest because it can give rise to azimuthal asymmetries of finalstate particles in various high energy process involving a transversely polarized nucleon, also because it encodes nontrivial partonic structure in the transverse plane through spinorbital correlation.
In recent years, the quark Sivers function has been extensively studied from both theoretical and experimental sides and much progress has been made. Significant Sivers effect in semiinclusive deep inelastic scattering was measured by the HERMES Airapetian:2004tw ; Airapetian:2009ae , COMPASS compass05 ; Alekseev:2010rw ; Bradamante:2011xu , and the JLab Hall A Collaborations Qian:2011py . The data on the Sivers single spin asymmetries (SSAs) were further utilized by different groups anselmino05b ; efr05 ; cegmms ; vy05 ; Anselmino:2008sga to extract the quark Sivers functions of the proton within the TMD factorization Ji:2004xq . On the other hand, there are a number of calculations on the quark Sivers function using various QCDinspired models Brodsky:2002plb ; Yuan:2003plb ; Burkardt:2003je ; Lu:2004au ; Cherednikov:2006zn ; Lu:2006kt ; Bacchetta:2008prd ; Courtoy:2008vi ; Courtoy:2008dn ; Pasquini:2010af . Furthermore, TMD evolution collinsbook ; Idilbi:2004vb ; Aybat:2011ge ; Aybat:2011zv of the quark Sivers function is found to be important to consistently describe the SSA data measured at different energy scales Aybat:2011ta .
Compared to the quark Sivers function, the knowledge of the gluon Sivers function (for a review, see Ref. Boer:2015vso ) is still limited. Even so, model calculations of have been performed in literature, mainly focusing on the small region by means of the dipole formalism Boer:2015pni ; Boer:2016xqr , or by employing a quark target model Goeke:2006ef which is different from the realistic case of proton target. Besides, Burkardt Burkardt:2004ur derived a useful constraint on , the socalled Burkardt sum rule, which states that the total transverse momentum of all partons in a transversely polarized proton should vanish. In terms of the Sivers function, it means that the sum of the first transverse moments of all the quark, antiquark and gluon Sivers functions is zero. Very recently, the authors of Ref. D'Alesio:2015uta performed a phenomenological estimate on using the midrapidity data on the transverse SSA measured in process at RHIC Adare:2013ekj .
In this work, we study the gluon Sivers function of the proton from an intuitive model concerning the gluon structure of the nucleon. The purpose of the study is to provide information on in the valence region, which is complementary to the phenomenological analysis on the experimental data as well as the dipole calculation. The main difficulty in the calculation is how to generate the gluon degree of freedom, since in the naive parton model the proton is composed by three valence quarks. As a first estimate, here we consider a Fock state for a transversely polarized proton that contains a gluon, and we group the three valance quark as a spectator particle. We then present the wavefunctions for the Fock state in the lightcone formalism Brodsky:2000ii . The underlying model is used to reproduce the collinear gluon density distribution and to obtain the values of the parameter. Based on this, we calculate the Sivers function using the overlap representation of the lightcone wavefunctions. The finalstate interaction necessary to produce nonzero phase for the gluon Sivers function is properly taken into account through an interaction kernal. As a check, we also compare our numerical result with the extracted in Ref. D'Alesio:2015uta and estimate the mean transverse momentum of the gluon in a transversely polarized proton.
Ii calculation of the gluon Sivers function in an overlap representation
The unpolarized gluon TMD distribution and the gluon Sivers function appear in the decomposition of the correlation function Burkardt:2004 ; Goeke:2006ef ; Meissner:2007rx
(1) 
where is the field strength tensor of the gluon, and is the Wilson line ensuring the gauge invariance of the correlator. The symbol “” in the subscript denotes that the Wilson line in the operator definition of the correlator is futurepointing, which is appropriate for defining TMD distributions in SIDIS.
In Refs. Goeke:2006ef ; Meissner:2007rx , the authors calculated the gluon TMD distributions for a quark target using perturbative QCD, in which the gluon is produced from the radiation off the parent quark. In the case the target is a proton, the presence of the gluon degree of freedom is not obvious. The minimum Fock state for the proton that containing gluon is . As the fourbody system is very complicated, here we resort to a more phenomenological approach to assume that the three quarks can be grouped into a spectator particle. Thus, in this model in which the degree of freedom of a gluon is present, the proton can be viewed as a composite system formed by a gluon and a spectator particle :
(2) 
with and the spin indices for the gluon and the spectator particle. In principle the spectator has the spin quantum number or . In this work we only consider the spin1/2 component, that is, we assume that the contribution from the spin component is negligible for simplicity. Therefore, in the case the gluon is the active parton, the Fockstate expansion of proton with has the following possible form:
(3) 
where are the wavefunctions corresponding to the twoparticle states . Here and denote the components of the spins of the constituent gluon and spectator, respectively, and is the longitudinal momentum fraction of the gluon. Motivated by the wavefunction of the electron Fock state Brodsky:2000ii , the Fock state of which is composed of a spin1 photon and a spin electron, we propose that the lightcone wavefunctions appearing in Eq. (3) have the following forms
(4) 
where , and are the masses of the proton and the spectator state, respectively, and is the wavefunction in the momentum space
(5) 
with the coupling of the nucleongluonspectator vertex, and the gluon mass. In principle gluon is a massless gauge boson. Here we keep in our formula following the convention used in Ref. Brodsky:2000ii . As shown in the next section, we fix GeV in our numerical calculation.
Although the wavefunctions in Eq. (4) are similar to those of the electron, there are several differences between them. The first one is that the mass of the spectator particle could be different from the mass of the proton, while in the electron case, the spectator fermion is the same as the electron. The second one is that coupling for the electron wavefunction is a constant, whereas the coupling is not necessary to be a constant, since there is nonperturbative color interaction involved in the nucleongluonspectator vertex. In order to simulate the nonperturbative physics for the vertex, we adopt the BrodskyHwangLepage prescription Brodsky:1980vj for the coupling :
(6) 
Here is a constant parameter which represents the strength of the protongluonspectator vertex, a cutting off parameter, and the invariant mass of the two particle system:
(7) 
Similarly, the Fockstate expansion for a proton with has the form
(8) 
where
(9) 
Having the lightcone wavefunctions of the proton, we can directly calculate the unpolarized gluon TMD distribution using the overlap representation
which yields the following result
(10) 
with
After the transverse momentum is integrated out, the unpolarized distribution of the gluon has the form
(11) 
where , and
(12) 
is the incomplete gamma function.
In the overlap representation, the gluon Sivers function may be calculated from the expression Lu:2006kt ; Bacchetta:2008prd
(13) 
where is the interaction kernel which simulates the gluon rescattering between the active parton and the spectator. Originally the overlap representation is applied to calculate various form factors of the nucleon, as well as the nucleon anomalous magnetic moment. Recently it has also been adopted to calculate the quark Sivers functions Lu:2006kt ; Bacchetta:2008prd and quark BoerMulders function Bacchetta:2008prd in the spectator model. In this work we adopt the form of as follows:
(14) 
which is extracted from the calculation of the gluon sivers function in the quark target model Goeke:2006ef ; Meissner:2007rx . Of course the final state interaction kernel should be model dependent. Here we assume that the kernel in our spectator model is the same as that in the quark target model, since in the quark target model the spectator is also a spin1/2 particle.
Substituting Eq. (14) into Eq. (13) and performing the integration over , we arrive at the result of the gluon Sivers function in the spectator model:
(15) 
where we have used the following integration formula:
(16) 
Parameters  Fit 1 (LO)  Fit 2 (NLO) 

(GeV)  
(GeV)  
(GeV)  0 (fixed)  0 (fixed) 
(GeV)  0.80  0.85 
Iii Numerical results
To present the numerical result for the gluon Sivers function, we need to specify the values of the parameters , , and in our model. As the unpolarized gluon distribution in the valence region is fairly known, we fit the spectator model result of to the existed parametrization for to determine the values of the above parameters. We will choose the GRV98 parametrization Gluck:1998xa to perform two different fits for comparison. We note that the same parametrization has been applied in Ref. D'Alesio:2015uta to extract the gluon Sivers function. We select 80 data points in the interval with a step value of . We have tried to include the gluon PDF in the smaller region or in the larger region in the fit, However, we find that in this case a good fit cannot be achieved. This indicates that our model is applicable in the region which is not so small and not so large. In the first fit (denoted as fit 1) we apply the leading order (LO) set of GRV98 gluon PDF, while in the second fit (denoted as fit 2) we adopt the nexttoleading order (NLO) set of GRV98 gluon PDF.
Furthermore, when doing the fit, we have to choose an energy scale at which our model can be compared to the parametrization. Here we consider the scale as a special parameter of the model, which means that we also search the region of the GRV98 PDF to find the lowest . The best results for the parameters from the two fits are shown in Table. 1. In both fits we fix the parameter to be 0 GeV, since the gluon should be massless. For the mass of the spectator, we required , which is necessary to form a stable proton state. In fit 1 we find that the lower possible scale is always preferred by the fit. Therefore, in this fit we choose the lowest allowed scale of GRV98 parametrization as the model scale, which is . In fit 2, the model scale is slightly larger than the lowest allowed scale.
In Fig. 1, we plot the fitted gluon PDF (solid line) in our model and compare it with the GRV98 parametrization (dashed line). The left and right panels show the results from fit 1 and fit 2, respectively. As the GRV98 parametrization does not provide the uncertainties for PDFs, we can not deduce from the fit the corresponding errors of the model parameters. Here we assume that the error for our model resulting is 30% and show the error band in Fig. 1. Although our spectator model is simple, we find that in both fits, the GRV98 gluon PDF can be well described by our model with 4 parameters. One can also see that a better agreement between the model results and the parametrization is obtained in fit 1.
Using the values of the parameters obtained in the fits, we calculate the numerical result of the Sivers function at the model scale, and show versus at and in Fig. 2. For comparison, we also show the dependence of (timed with ). For the strong coupling needed in the calculation, we adopt its value provided by the GRV98 code for consistency. That is, we choose in fit 1 and in fit 2, respectively. We find that the dependence of the gluon TMD distribution functions changes when changes. More specifically, the peak of the curves shifts from lower region to higer when increases.
The first transverse moment of the gluon Sivers function is defined as
(17) 
Here the notion is the one used in Ref. D'Alesio:2015uta . In the left and right panels of Fig. 3 we plot as a function calculated form the parameters in fit 1 and fit 2, respectively. We find that the first transverse moment of the gluon Sivers function in our model is negative, which is consistent with the result extracted in Ref. D'Alesio:2015uta , and calculated from fit 2 at the model scale is around 30% larger than that from fit 1.
We also compare our model with the extracted gluon Sivers function in Ref. D'Alesio:2015uta . Since the result in Ref. D'Alesio:2015uta is given at the scale , which is higher than our model scale, it is necessary to evolve the gluon Sivers function given at different scales to that at the same scale for comparison. At the tree level, the can be related to the twist3 trigluon function
(18) 
whereas the complete QCD evolution for is given in Ref. Braun:2009mi . However, its evolution is rather complicated, i.e., it also mix with the more general function with , the quarkgluon QiuSterman function qs and so on. Here we only consider the homogenous term in the evolution to assume the following evolution kernel:
(19) 
where is the LO evolution kernel of the gluon to gluon splitting function for . We expect that the approximation adopted here will not change the result qualitatively. In Fig. 4, we show the evolved at from fit 1 and fit 2, and compare it with the gluon Sivers function extracted in Ref. D'Alesio:2015uta at the same scale. We find that the evolution effect for is substantial. In particular, the evolution from lower scale to higher scale increases the magnitude of in the region , while it decreases the magnitude of in the larger region. However, the scale dependence of is weaker than that of , because of the additional terms in Eq. (19). We also find that the in the region , the gluon sivers function from fit 1 is comparable with the SIDIS2 set in in Ref. D'Alesio:2015uta after the 10% tolerance band is considered. In the region , the gluon sivers function from both fits qualitatively agrees with the SIDIS1 set in Ref. D'Alesio:2015uta .
Finally, we calculate the average transverse momentum of the gluon inside a transversely polarized proton:
(20) 
The quantity can be constrained by the Burkardt sum rule and the average transverse momentum of quarks and antiquarks. In Ref. Anselmino:2008sga , using the extraction of the Sivers distribution functions for quarks and antiquarks, the authors provided a determination on the allowed range of at GeV:
(21) 
We evolve the gluon Sivers function in our model to the scale GeV and estimate in the two fits
Our results show that the average transverse momentum of the gluon from fit 1 agrees with the bound on given in Ref. Anselmino:2008sga .
A very important theoretical constraint on the Sivers function is the positivity bound Bacchetta:1999kz ; Mulders:2000sh
(22) 
We have checked that the gluon Sivers function from our model satisfies the above inequality in the region GeV. This is similar to the quark sector of the Sivers function in the quarkdiquark model, for which a violation of the positivity bound is observed Kotzinian:2008fe at high . As explained in Ref Pasquini:2011tk , the violation of the inequality for Todd TMDs may be due to the fact that Todd TMDs is evaluated to , while Teven TMD distributions is truncated at in model calculations. However, we find that in our model the gluon TMD distributions are very small (less than ) at the region the positivity bound is violated. Besides, GeV is much larger than the mean transverse momentum of the gluon in our model. Therefore, the fact that in our model the inequality only holds in the region GeV is an acceptable result, as our model is assumed to be valid in the region where is not so large.
Iv Conclusion
In this work, we studied the gluon Sivers function using a lightcone spectator model. We treat the Fock state of the proton state as a composite system formed by a gluon and a spin spectator particle, in the case the active parton is a gluon. Using the overlap representation, we calculated the unpolarized gluon distribution function and the gluons Sivers function . In the calculation, we adopte the wavefunctions of the proton Fock state motivated by the wavefunctions of the electron Fock state. Besides, we choose the BrodskyHuangLepage prescription for the wavefunctions in the momentum space. Furthermore, in the case of the Sivers function, we adopt an interaction kernel that simulates the finalstate interaction between the gluon and the spectator. The values of the parameters in the model are determined by fitting the model resulting with the GRV98 parametrization. Specifically, the LO and NLO gluon PDF sets are adopted to obtain two sets of fit. Our numerical calculations show that the first transverse moment of the gluon Sivers function in our model is negative, and is several percent in magnitude. We also compared our model results with the recent extraction of and find that our result can coincide with the parametrization of in Ref. D'Alesio:2015uta . In the estimate we taken into account the evolution of the gluon Sivers function, and find that it is important to include the evolution effect in order to compare model calculation with phenomenological analysis given at different energy scales. Finally, our model can be suitably extended to estimate the gluon helicity distribution and we leave it as a future study. In conclusion, our study may provide useful information of the gluon Sivers function from an intuitive model concerning the gluon structure of the nucleon.
Acknowledgements
This work is partially supported by the National Natural Science Foundation of China (Grants No. 11575043, No. 11120101004 and No. 11475006), and by the Qing Lan Project.
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