The \cos 2\phi azimuthal asymmetry of unpolarized p\bar{p} collisions at Tevatron

# The cos2ϕ azimuthal asymmetry of unpolarized p¯p collisions at Tevatron

Tianbo Liu School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China    Bo-Qiang Ma School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Center for High Energy Physics, Peking University, Beijing 100871, China
July 13, 2019
###### Abstract

We calculate the azimuthal asymmetry of the unpolarized Drell-Yan dilepton production process in the -resonance region at the Tevatron kinematic domain. Such an azimuthal asymmetry can provide additional information about a spin-related new parton distribution function, i.e., the Boer-Mulders function of the proton, compared to the process. Therefore the available data of unpolarized proton-antiproton collision at Tevatron can contribute to our study on the spin structure of the nucleon.

###### pacs:
13.85.Qk, 13.88.+e, 14.70.Hp

The study of the intrinsic transverse momentum dependent (TMD) distribution functions has received much attention in recent years Barone:2001sp (). Such new quantities of the nucleon provide us a significant perspective on understanding the spin structure of hadrons and the non-perturbative properties of quantum chromodynamics (QCD). The intrinsic transverse momentum of partons may cause special effects in high energy scattering experiments Cahn:1978se (). It was naively speculated that the polarization of at least one incoming hadron is necessary to investigate the spin-related structure and properties of hadrons, however the situation will change if the transversal motions of quarks inside the hadron will take into account. The Drell-Yan process is an ideal ground for testing perturbative QCD and probing TMD distribution functions, and its cross section is well described by next-to-leading order QCD calculations Stirling:1993gc (). Surprisingly, the first measurement of the Drell-Yan angular distribution, performed by NA10 Collaboration for at 140, 194 and 286 GeV, indicates a sizable azimuthal asymmetry Falciano:1986wk (); Guanziroli:1987rp () which cannot be described by leading and next-to-leading order perturbative QCD Brandenburg:1993cj (). Furthermore, the subsequent result by the Fermilab E615 Collaboration reveals that the Lam-Tung relation Lam:1978pu (), which is analogous to the Callan-Gross relation Callan:1969uq () in deep-inelastic scattering, obtained as a consequence of the spin- nature of the quarks, is clearly violated Conway:1989fs (). The violation has also been tested in recent and Drell-Yan dimuon processes measured by E866/NuSea Collaboration Zhu:2006gx (); Zhu:2008sj ().

Several attempts were made to interpret this asymmetry, such as the factorization breaking QCD vacuum effect Brandenburg:1993cj () which is possible the helicity flip in the instanton model Boer:2004mv (), higher twist effect Brandenburg:1994wf (); Eskola:1994py (); Heinrich:1991zm () and the coherent states Blazek:1989kt (). Boer pointed out that the azimuthal asymmetry could be due to a non-vanished TMD distribution function  Boer:1999mm (), named as the Boer-Mulders function later, as one of the eight leading-twist TMD distribution function contained in Mulders:1995dh (); Boer:1997nt ()

 Φ=12{f1⧸n+−f⊥1TϵijTpTiSTjM⧸n++h1T[⧸ST,⧸n+]γ52+(SLg1L+pT⋅STMg1T)γ5⧸n++(SLh⊥1L+pT⋅STMh⊥1T)[⧸pT,⧸n+]γ52M+ih⊥1[⧸pT,⧸n+]2M}, (1)

where is the quark-quark correlation matrix, defined as

 (2)

The Boer-Mulders function is another time-reversal odd (-odd) distribution function which characterizes the correlation between quark transverse momentum and quark transverse spin, analogous to the Sivers function which signifies the correlation between quark transverse momentum and hadron transverse spin Sivers:1989cc (). The non-vanished -odd distribution functions can arise from the initial-state or final-state interaction Collins:2002kn (); Brodsky:2002rv (); Gamberg:2003ey (); Boer:2002ju (). In general, the path-order Wilson line arising from the requirement of QCD gauge invariance for quark correlation functions provides non-trivial phases and leads to non-vanished -odd distribution functions Ellis:1982wd (); Efremov:1979qk (); Collins:1981uw (); Ji:2002aa (). Due to the present of Wilson line, opposite sign of the Boer-Mulders function or Sivers function in semi-inclusive deep inelastic scattering (SIDIS) and Drell-Yan processes is expected Boer:2003cm (); Collins:2004nx (),

 h⊥1(x,p2T)|SIDIS=−h⊥1(x,p2T)|DY. (3)

The existence of -odd distribution function can cause azimuthal asymmetries in SIDIS at leading twist level Boer:1997nt (), and the product of two Boer-Mulders functions of two incoming hadrons may give a sizable azimuthal asymmetry in unpolarized Drell-Yan processes by establishing a preferred transverse momentum direction from the spin-transverse momentum correlation, which is called the Boer-Mulders effect Boer:1999mm (). Thus, the measurement of the Boer-Mulders function will promote our understanding of QCD. Many theoretical and phenomenological studies are carried out along this direction Lu:2004hu (); Lu:2005rq (); Bianconi:2005bd (); Sissakian:2005vd (); Sissakian:2005yp (); Lu:2006ew (); Barone:2006ws (); Lu:2007kj (); Gamberg:2005ip (); Zhang:2008nu (); Zhang:2008ez (); Barone:2010gk (); Lu:2009ip (); Lu:2011mz (); Yuan:2003wk (); Pasquini:2006iv (); Gockeler:2006zu (); Burkardt:2007xm ().

Recently, the Collider Detector at Fermilab (CDF) Collaboration first measured the angular distribution coefficients of Drell-Yan pairs in the mass region from unpolarized collisions at  Aaltonen:2011nr (). This indicates that it is feasible to investigate spin physics at Tevatron. In this paper, we calculate the azimuthal asymmetry caused by the Boer-Mulders effect in the -pole region with the kinematic conditions at Tevatron.

The angular distribution coefficients are generally frame dependent. We choose the Collins-Soper (CS) frame Collins:1977iv () to perform the calculation. It is the center of mass of the lepton pair with the axis defined as the bisector of and beams. The polar angular is defined as the angular of the positive lepton with respect to the axis direction, and the azimuthal angular is defined as the angular of the lepton plane with respect to the proton plane. In this frame the Lam-Tung relation is insensitive to the higher fixed-order perturbative QCD Mirkes:1994eb () or the QCD resummation Boer:2006eq (); Berger:2007si (); Berger:2007jw (). The angular differential cross section for unpolarized Drell-Yan process has the general form:

 1σdσdΩ= 34π1λ+3(1+λcos2θ+μsin2θcosϕ +ν2sin2θcos2ϕ), (4)

where is the solid angle and , , and are angular distribution coefficients. For azimuthal symmetrical scattering, the coefficients . It can also be written as Oakes:NCA44 (); Lam:1978pu ():

 dσdΩ= WT(1+cos2θ)+WL(1−cos2θ) +WΔsin2θcosϕ+WΔΔsin2θcos2ϕ. (5)

When taking into account both virtual photon and -boson contribution, the leading order unpolarized Drell-Yan cross section is Boer:1999mm ()

 dσ(h1h2→l¯lX)dΩdx1dx2d2qT=α23Q2∑a{K1(θ)F[f1af1a]+[K3(θ)cos2ϕ+K4(θ)sin2ϕ]×F[(2^h⋅pT^h⋅kT−pT⋅kT)h⊥1ah⊥1aM2]}, (6)

where , are the Bjorken variables standing for the longitudinal momentum fractions carried by the partons in the proton and antiproton, and , , , and are the fine structure constant, the mass of proton, the transverse momentum, and invariant mass of respectively. The structure function notation in this equation is defined as

 F[⋯]=∫d2pTd2kTδ2(pT+kT−qT)[⋯], (7)

where , are the transverse momenta of quarks in proton and antiproton, and is the direction of the transverse momentum of . The coefficients , and are expressed as:

 K1(θ) = 14(1+cos2θ)[e2a+2glVeagaVχ1+cl1ca1χ2] (8) +cosθ2[2glAeagaAχ1+cl3ca3χ2], K3(θ) = 14sin2θ[e2a+2glVeagaVχ1+cl1ca2χ2], (9) K4(θ) = 14sin2θ[2glVeagaAχ3], (10)

where is the charge of quarks (antiquarks), and and are the vector and axial-vector coupling constants to the -boson. We take their values in Ref.Nakamura:2010zzi (). The is defined as:

 cj1=(gjV2+gjA2),cj2=(gjV2−gjA2),cj3=2gjVgjA, (11)

where or . The -boson propagator is given by:

 χ1 = 1sin2θWQ2(Q2−M2Z)(Q2−M2Z)2+Γ2ZM2Z, (12) χ2 = 1sin2θWQ2Q2−M2Zχ1, (13) χ3 = −ΓZMZQ2−M2Zχ1, (14)

where is the Weinberg angle. In Eq.(6), we assume that the TMD distribution functions for antiquarks (quarks) in the antiproton are the same as those for quarks (antiquarks) in proton, and the summation over the index are for different flavors with , , , and .

In our calculation, we take the Boer-Mulders functions extracted from and Drell-Yan data Zhang:2008nu (); Lu:2009ip (). The parametrizations for is Lu:2009ip ():

 h⊥1q(x)=Hqxcq(1−x)bf1q(x), (15)

and the TMD part is parametrized with a Gaussian form:

 h⊥1q(x,p2T) = h⊥1q(x)exp(−p2Tp2bm)πp2bm, (16) f1q(x,p2T) = f1q(x)exp(−p2Tp2un)πp2un. (17)

This parametrization is based on the assumption that the asymmetry comes only from the Boer-Mulders effect in the region , and in this region the following relation hold:

 x1=Q√sey,x2=Q√se−y, (18)

where is the rapidity of the . We can also express the cross section of the Drell-Yan process depending on and with an additional Jacobian determinant:

 dσdydQ2d2qTdΩ=1sdσdx1dx2d2qTdΩ. (19)

From Eq.(6), the azimuthal dependent terms are the second and the third terms with and forms respectively. However, the term is suppressed, which can be found from (10) and (14). As shown in Ref.Lu:2011mz (), we can write the coefficient of term in Eq.(The azimuthal asymmetry of unpolarized collisions at Tevatron) into two parts, the perturbative QCD effect and the Boer-Mulders effect . Then using an approximate Lam-Tung relation , one can give the asymmetry caused by the Boer-Mulders effect:

 2νBM=4WBMΔΔWT+WL≈2ν+λ−1. (20)

Comparing (The azimuthal asymmetry of unpolarized collisions at Tevatron) and (6), and neglecting the in the denominator because at low region, we can get the following relation:

 νBM(qT,y,Q)=∑a1Q2K3(θ)F[(2^h⋅pT^h⋅kT−pT⋅kT)h⊥1ah⊥1aM2]∑a1Q2K1(θ)F[f1af1a]. (21)

In the numerical calculation, we choose the values of parameters in the Boer-Mulders function as those in Ref.Lu:2009ip (); Lu:2011cw (). There is still an unsettled factor which might be flavor dependent in the parametrization, because it will be canceled in the product of two Boer-Mulders functions of quark and antiquark. It can range in the region , which is limited by the positivity bounds Bacchetta:1999kz (); Zhang:2008nu (); Lu:2009ip (). However, in the Drell-Yan process, it has the product of two Boer-Mulders functions of two quarks or two antiquarks which will not cancel the factor . The cross section has different behavior with different values for . Therefore, we can learn additional information of the Boer-Mulders function from Drell-Yan processes. In this work, we choose three different values for , and to calculate and show their different behavior.

In order to give with respect to a parameter , or , we should integrate for the other parameters of the numerator and the denominator in Eq.(21) respectively. The integral over need to be cut off at , because intrinsic transverse momentum plays a significant role at low and the fitting for the parameters has excluded the data with .

Comparing (The azimuthal asymmetry of unpolarized collisions at Tevatron) with the angular distribution form taken by CDF Aaltonen:2011nr ():

 dσdϕ∝1+β3cosϕ+β2cos2ϕ+β7sinϕ+β5sin2ϕ, (22)

will contribute to caused by the Boer-Mulders effect at low .

In summary, we calculated the cos2 azimuthal asymmetry in the unpolarized Drell-Yan dilepton production processes in the mass region at CDF kinematic domain. It can be measured by experimental detection of the Lam-Tung relation violation. It is possible to study the spin structure of hadrons in unpolarized collision processes around mass region at Tevatron. In addition, the processes can give more significant information of the Boer-Mulders function than processes. It can help us to settle the factor in the parametrization, and the prediction that the Boer-Mulders function have different signs in SIDIS and Drell-Yan processes Collins:2002kn () also awaits experimental confirmation. Therefore the available data of Tevatron are ideal to investigate the spin structure of nucleons via the unpolarized process at the pole. Besides, the GSI-PANDA experiment Lutz:2009ff () will run unpolarized Drell-Yan processes with colliding at , and PAX experiment Barone:2005pu () may preform unpolarized Drell-Yan process with the fixed target mode at . They will provide us an environment to study the Boer-Mulders effect at and peaks and to understand the structure of nucleons. All these Drell-Yan experiments will give us significant promotion in understanding the hadron structure and non-perturbative QCD properties.

###### Acknowledgements.
We are greatly indebted to Prof. Liang Han for the stimulating discussion about possible experimental analysis on spin physics at Tevatron. This work is partially supported by National Natural Science Foundation of China (Grants No. 11021092, No. 10975003, No. 11035003, and No. 11120101004), by the Research Fund for the Doctoral Program of Higher Education (China).

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