A first determination of the unpolarized quark TMDs from a global analysis
Cristian Pisano Alessandro Bacchetta Filippo Delcarro Marco Radici Andrea Signori
hep-ph




§ INTRODUCTION



Transverse momentum dependent parton distribution and fragmentation functions (TMDs) are fundamental objects that encode information on the intrinsic motion of quarks and gluons inside hadrons. As such, they provide a three-dimensional picture of hadrons in momentum space. In analogy to the more common collinear  parton distribution and fragmentation functions (integrated over transverse momentum), TMDs are not purely perturbative quantities, which could be fully derivable from first principles. Only their dependence on the energy scale can be calculated in perturbative QCD. However, in contrast to DGLAP equations which govern the evolution of collinear distributions,  the TMD evolution kernels contain a nonperturbative part as well. Hence, the complete determination of TMDs is not at all trivial, and requires a continuos interplay between theory and experiment. 

In this contribution to the proceedings, based on Ref. <cit.> to which we refer for further details, we present a first attempt to extract unpolarized quark TMDs through a global fit. To this aim, we consider only those processes for which factorization has been established <cit.> and data are already available, namely semi-inclusive deep-inelastic scattering (SIDIS), Drell-Yan (DY) and Z boson production in proton-proton collisions. We note that electron-positron annihilation into two hadrons is an important reaction that is, however, currently missing. Its future analysis could allow for an independent study of transverse momentum dependent fragmentation functions <cit.>.

Since the data included in the present study come from many experiments, carried out at different energies, our results not only shed light on the partonic transverse momentum dependence, but they represent a test of TMD evolution over a large energy range. Furthermore,  at the same time, we are able to probe the universality properties of the TMDs among different processes.

This contribution is organized as follows. In section <ref>, we briefly outline the adopted TMD framework.  In section <ref> we clarify our approximations in the phenomenological implementation of TMD evolution equations and describe the nonperturbative input of the fit. Our results are presented and discussed in section <ref>. A summary and suggestions for future possible improvements are provided in section <ref>.




§ THEORETICAL FRAMEWORK


The cross section  σ_N^h and transverse structure function  F_UU,T for the SIDIS process ℓ(l) + N(P) →ℓ(l') + h(P_h) + X are commonly expressed in terms of the variables

    x    = Q^2/2 P· q ,
       
    y    = P · q/P · l ,
       
    z    = P · P_h/P· q ,

with Q^2=-q^2 = -(l-l^')^2. If we denote the corresponding quantities for the fully inclusive DIS process ℓ(l) + N(P) →ℓ(l')  + X by σ_DIS and F_T, it is possible to define the hadron multiplicity as follows, 

    m_N^h (x,z,|P_h|, Q^2) = d σ_N^h / ( dx   dz  d|P_h|  dQ^2) /dσ_DIS / ( dx  dQ^2 ) ≈2 π |P_h| F_UU ,T(x,z,P_h^2, Q^2)/F_T(x,Q^2)  .

We point out that the last approximation in (<ref>) is only valid in the kinematic region under study, i.e  P_hT^2 ≪ Q^2 and  M^2 ≪ Q^2, with  P_hT being the component of the three-momentum P_h transverse to q and M the nucleon mass.

The structure function F_UU,T can be conveniently expressed in terms of Fourier transforms of the distribution  f_1^a(x, k_^2;Q^2) and fragmentation function  D_1^a h(z, P_^2; Q^2 ) for a quark with flavor a and electric charge e_a in units of the proton charge, which are defined by

    f̃_1^a(x, ^2;Q^2)    =
    ∫_0^∞ d |k_| 
                    |k_|J_0( |k_|) 
           f_1^a(x, k_^2;Q^2) ,
    D̃_1^a h(z, ^2; Q^2 )    =
    ∫_0^∞d |P_|/z^2 |P_| 
                                                 J_0( |P_|/z)
           D_1^a h(z, P_^2; Q^2 ) .
  
To leading order (LO) in perturbative QCD and in the region P_hT^2 ≪ Q^2, F_UU,T reads

    F_UU,T(x,z, P_h ^2, Q^2)    ≈ 2π∑_a e_a^2 x 
           ∫_0^∞d  J_0( |P_hT|/z)
          f̃_1^a(x, ^2;Q^2) D̃_1^a h(z, ^2;
          Q^2 ) .
 

Similarly, the  cross sections for the DY and Z production processes, h_A(P_A) + h_B(P_B) →ℓ^+(l) + ℓ^-(l^') + X, can be written as

    dσ/dQ^2  dq_T^2 dη   = σ_0^γ,Z(F_UU^1 + 1/2 F_UU^2) ,
 
where q and η are respectively the four-momentum and rapidity (w.r.t. the direction defined by P_A) of the virtual photon or Z boson exchanged in the reaction, while Q^2 = q^2. The explicit expressions  for the elementary cross sections σ_0^γ,Z can be found  in Ref. <cit.>. At LO, in the kinematic limit q_T^2 ≪ Q^2, F_UU^2 ≈ 0 and 

    F_UU^1(x_A,x_B, q_T^2, Q^2)    ≈
     2π∑_a  H_UU^1 a ∫_0^∞ d   J_0(  |q_T|) f̃_1^a(x_A, ^2;Q^2)  f̃_1^a̅(x_B, ^2;Q^2 )    ,
 
where  H_UU, γ^1 a(Q^2) ≈e_a^2/N_c, H_UU, Z^1 a(Q^2) ≈[ (I_3a - 2 e_asin^2 θ_W)^2 + I_3a^2]/N_c, 
with N_c being the number of colors, θ_W the Weinberg angle,  and I_3a the weak isospin. The longitudinal momentum fractions x_A, x_B can be expressed in terms of the rapidity η by the relation  x_A/B = Q/√(s) e^±η.



§ TMD EVOLUTION



The Fourier transforms of the TMDs, evolved at the scale Q^2, at LO can be written as <cit.>

    f_1^a (x,  ^2; Q^2)    = f_1^a (x ; μ_b^2) 
      e^S (μ_b^2, Q^2)  e^g_K() ln (Q^2 / Q_0^2) f_1  NP^a (x, ^2)  ,
    
    D_1^a→ h (z, ^2; Q^2)    = D_1^a→ h (z; μ_b^2)   e^S (μ_b^2, Q^2)  e^g_K( ) ln (Q^2 / Q_0^2) D_1  NP^a→ h (z, ^2)   ,

where f_1^a (x ; μ_b^2) and D_1^a→ h (z; μ_b^2) are the common  collinear distribution and fragmentation functions, evaluated at an initial scale μ_b. The evolution is driven by the Sudakov exponent[We point out that a factor 1/2 is missing in the definition of the Sudakov factor  in (2.3) of <cit.>.]

    S(μ_b^2,Q^2)=- 1/2∫_μ_b^2^Q^2dμ^2μ^2[A(α_S(μ^2))ln(Q^2μ^2) 
    + B(α_S(μ^2)) ]  ,
 
where, at the next-to-leading logarithmic (NLL) accuracy, 

    A    = C_F(α_S/π) +  1/2 C_F
    (α_S/π)^2 [
    C_A ( 67/18 - π^2/6)
    - 5/9 N_f ],
       
    B    = - 3/2C_F
    (α_S/π) ,
 
with C_F = (N^2_c-1)/2N_c, C_A=N_c  and N_f the number of active flavors. 

The  initial scale is chosen to be μ_b = 2 e^-γ_E/_∗, where γ_E is the Euler constant and

    _∗   ≡_∗(;_ min,_ max) = _ max(1-e^- ^4 / _ max^4/1-e^- ^4 / _ min^4)^1/4 .
  
When performing the integrals in (<ref>) and (<ref>), the variable _∗ replaces  in the perturbative parts of the TMDs defined in (<ref>)-(<ref>), because at large  perturbative calculations are not reliable any more. The
  _∗ is assumed to saturate on the maximum value _ max, as suggested also in Refs. <cit.>. Numerical values are taken to be

    _ max   = 2 e^-γ_E  GeV^-1≈ 1.123   GeV^-1, 
       _ min   = 2 e^-γ_E/Q  .
 
In this way,  the scale μ_b is constrained to be always between 1 GeV and Q, such that the collinear distributions and fragmentation functions are never computed at a scale lower than 1 GeV.  Furthermore, in the definition of the perturbative Sudakov exponent (<ref>), the lower integration bound is always smaller than the upper one. When Q= Q_0=1 GeV, _ min  = _ max and no evolution effects are present: at this scale TMDs are simply given by the corresponding collinear function multiplied by a nonperturbative contribution.

In (<ref>) and (<ref>), the nonperturbative Sudakov factor is chosen to be g_K () = - g_2 ^2 / 4,
where g_2 is a free parameter. 





The nonperturbative part of the distribution function in (<ref>) is parametrized as 

    f_1  NP^a (x, ^2)    = 1/2π
            e^-g_1a^2/4( 1 - λ  g_1a^2/1+λ g_1a^2/4)  ,

where λ is a free parameter and the Gaussian width g_1 is taken to be dependent on the longitudinal momentum fraction x as follows,

    g_1 (x) = N_1  (1-x)^α  x^σ/ (1 - x̂)^α x̂^σ  ,

with x̂ = 0.1, while α,  σ,  N_1 ≡  g_1 (x̂) are also free parameters.
Similarly, the nonperturbative fragmentation function in (<ref>) is given by  the combination of two Gaussians,

    D_1  NP^a → h (z, ^2)    = 
         g_3 a→ h   e^-g_3 a→ h^2/4 z^2
            + (λ_F/z^2)    g_4 a→ h^2(1 - g_4 a→ h^2/4 z^2)
               e^- g_4 a→ h^2/4z^2/2 π z^2 (g_3 a→ h + (λ_F/z^2)    g_4 a→ h^2) ,
 
with λ_F has to be fitted to the data. The z-dependent Gaussian widths g_3,4 are parametrized according to 

    g_3,4 (z) = N_3,4  (z^β + δ) (1-z)^γ/ (ẑ^β + δ)   (1 - ẑ)^γ  ,

where ẑ = 0.5, and  β,  γ,  δ, N_3,4≡ g_3,4 (ẑ)  are the free parameters.  Hence, the average transverse momentum squared of the distribution and fragmentation functions at the initial scale Q = Q_0 read, respectively, 

    ⟨k_⊥^2 ⟩ (x)    = g_1(x) + 2 λ g_1^2(x)/1+ λ g_1(x),
       ⟨P_⊥^2 ⟩ (z)    = g_3^2(z) + 2 λ_F
      g_4^3(z)/g_3(z) + λ_F g_4^2(z) .











§ NUMERICAL RESULTS



The following data sets are included in our analysis:  SIDIS events off proton <cit.> and deuteron targets <cit.>, DY bins at low energy <cit.> and Z boson production at the Tevatron <cit.>. 

Concerning the kinematical cuts applied, we impose Q^2 > 1.4 GeV^2 and consider only the small transverse momentum region by selecting the maximum value of the transverse momenta measured in each process on the basis of phenomenological  considerations <cit.>. In this way, it is possible to identify two different momentum scales, as required by TMD factorization. Furthermore, the current fragmentation region in SIDIS is isolated by implementing the cut 0.2 < z < 0.7. After this selection, the total number of analyzed experimental bins is 8059. 






For the collinear distributions f_1^a in (<ref>) we adopt the GJR08FFnloE parametrization <cit.>, while 
for the collinear fragmentation functions D_1^a→ h in (<ref>) the DSS14 NLO set for pions <cit.> and the DSS07 NLO set for kaons <cit.> are used.
  
The ten nonperturbative parameters of the TMDs f_1^a and D_1^a→ h at Q = 1  GeV are fitted to the data using a replica methodology <cit.>. The outcome is presented in Table <ref>, where we give the avarage values of the parameters over the full set of 200 replicas, with a standard deviation based on the  68% confidence level (C.L.) <cit.>. In addition, we find that the nonperturbative parameter of the evolution, also fitted to the data,  is g_2 =  (0.13 ± 0.01) GeV^2. 

The average χ^2/d.o.f = 1.55 ± 0.05 is rather good. This value can be further reduced to 1.02 without changing any free parameter, just by restricting the kinematical cuts such that it is possible to better identify the region where TMD factorization is supposed to hold.

For illustration, our resulting multiplicities for the SIDIS process ep→ e π^+X at  are presented in Fig. <ref> as a function of  the transverse momentum of the detected  pion P_hT, at different values of the ⟨ x⟩, ⟨ z ⟩, ⟨ Q^2⟩ bins. The bands are computed as the 68% C.L. envelope of the full sets of curves from all the replicas.
Similarly, in Fig. <ref> we show the multiplicities for positive hadron production off a deuteron target at . Notice that, since  measurements are affected by normalization errors <cit.>,  we have fitted normalized  multiplicities. These are obtained by dividing the data in each bin in (x,z,Q^2)  by the data point with the lowest P_hT in the bin, which is not  included in the analysis. 

In Figs. <ref> and <ref> we compare, respectively,  the cross sections for the DY and Z boson production processes with the measurements from different experiments. The results are differential in the transverse momentum of the exchanged virtual boson q_T. It is interesting to note that  the position of the peak moves from q_T∼ 1 GeV for DY events to q_T∼ 7 GeV for Z boson production, as an effect of TMD evolution.

Finally, we comment on the  average transverse momentum squared obtained for the incoming parton, ⟨k_^2 ⟩, and the one acquired during the fragmentation process, ⟨P_⊥^2 ⟩. They are defined by (<ref>) at the initial scale Q = 1 GeV, where the TMDs coincide with their nonperturbative input, in terms of the fit parameters.  Their dependence on the longitudinal momentum fractions, respectively x and z, are presented in Fig. <ref>.  

In Fig. <ref> we show the values of ⟨k_^2 ⟩  at x=0.1 in the horizontal axis,  and of ⟨P_⊥^2 ⟩ at  z=0.5 in the vertical axis. The white square (label 1) denotes the average values of these two quantities obtained in the present analysis,  both calculated at the same scale Q=1 GeV, while each black dot around the white square is the outcome of one replica. The red region represents the envelope of the 68% of the replicas that are closest to the average value. In the figure, our results are compared with other extractions. In particular, the white circle and the orange region around it (label 2) refer to the flavor-independent scenario in <cit.>, which was obtained neglecting TMD evolution and fitting only  SIDIS data from the  experiment at an average ⟨ Q^2 ⟩= 2.4 GeV^2. The older analysis shows a strong anticorrelation between the transverse momenta, which is reduced in the present study. This effect is due to the inclusion of Drell–Yan and Z production data, that add new physical information about TMD parton distributions without the influence of fragmentation functions. In the two analyses, the average values of ⟨k_^2 ⟩ at  x=0.1 are similar and compatible within error bands, while 
the present values of ⟨P_⊥^2 ⟩ at z=0.5  turn out to be larger than in the older study,  mainly as a consequence of the  data. In general, the x and z dependence of the transverse momentum squared are also different in the two extractions. 







§ CONCLUSIONS



We have performed the first extraction of unpolarized quark TMDs from a simultaneous fit of SIDIS, Drell-Yan and Z boson production data, taken in several experiments at different energies.  Our analysis is restricted to the the small transverse momentum region and implements TMD evolution effects up to the NLL accuracy. Our χ^2/d.o.f. = 1.55 ± 0.05 can be further improved to 1.02 without changing the parameters, only by restricting the kinematic cuts.  It turns out  that most of the discrepancies with the measurements come from the normalization rather than from the transverse momentum shape. Hopefully, this tension could be reduced by a more precise treatment from the perturbative point view <cit.>.  

In future studies, we plan to improve our analysis by exploring different functional forms for all the nonperturbative quantities, possibly including a  flavor dependence of the intrinsic transverse momenta. Furthermore, one should try to properly match the description at low transverse momentum with the fixed-order results at high transverse momentum, obtained within the framework of collinear factorization. 



This work is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 647981, 3DSPIN). AS acknowledges support from U.S. Department of Energy contract DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, manages and operates Jefferson Lab. Preprint number: JLAB-THY-18-2628.






99


Bacchetta:2017gcc
  A. Bacchetta, F. Delcarro, C. Pisano, M. Radici and A. Signori,
  Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production,
  JHEP 1706 (2017) 081

  [arXiv:1703.10157 [hep-ph]].
  


Collins:2011zzd
  J. Collins,
  Foundations of perturbative QCD,
  Cambridge University Press (2011).


Bacchetta:2015ora
  A. Bacchetta, M. G. Echevarria, P. J. G. Mulders, M. Radici and A. Signori,
  Effects of TMD evolution and partonic flavor on e^+ e^- annihilation into hadrons,
  JHEP 1511 (2015) 076

  [arXiv:1508.00402 [hep-ph]].
  


Aybat:2011zv
  S. M. Aybat and T. C. Rogers,
  TMD Parton Distribution and Fragmentation Functions with QCD Evolution,
  Phys. Rev. D  83 (2011) 114042

  [arXiv:1101.5057 [hep-ph]].
  


Airapetian:2012ki
  A. Airapetian et al. [ Collaboration],
  Multiplicities of charged pions and kaons from semi-inclusive deep-inelastic scattering by the proton and the deuteron,
  Phys. Rev. D 87 (2013) 074029
 
  [arXiv:1212.5407 [hep-ex]].
  


Adolph:2013stb
  C. Adolph et al. [ Collaboration],
  Hadron Transverse Momentum Distributions in Muon Deep Inelastic Scattering at 160 GeV/c,
  Eur. Phys. J. C 73 (2013)  2531 [Erratum: Eur. Phys. J. C 75 (2015)  94]

  [arXiv:1305.7317 [hep-ex]].
  


Ito:1980ev
  A. S. Ito et al.,
  Measurement of the Continuum of Dimuons Produced in High-Energy Proton - Nucleus Collisions,
  Phys. Rev. D 23 (1981) 604.
  
  


Moreno:1990sf
  G. Moreno et al.,
  Dimuon production in proton - copper collisions at √(s) = 38.8-GeV,
  Phys. Rev. D 43 (1991) 2815.



Affolder:1999jh
  T. Affolder et al. [CDF Collaboration],
  The transverse momentum and total cross section of e^+e^- pairs in the Z boson region from pp̅ collisions at √(s) = 1.8 TeV,
  Phys. Rev. Lett. 84 (2000) 845

  [hep-ex/0001021].


Abbott:1999wk
  B. Abbott et al. [D0 Collaboration],
  Measurement of the inclusive differential cross section for Z bosons as a function of transverse momentum in p̅p collisions at √(s) = 1.8 TeV,
  Phys. Rev. D 61 (2000) 032004
 
  [hep-ex/9907009].
  
 

Aaltonen:2012fi
  T. Aaltonen et al. [CDF Collaboration],
  Transverse momentum cross section of e^+e^- pairs in the Z-boson region from pp̅ collisions at √(s)=1.96 TeV,
  Phys. Rev. D 86 (2012) 052010

  [arXiv:1207.7138 [hep-ex]].
  
  
 
Abazov:2007ac
  V. M. Abazov et al. [D0 Collaboration],
  Measurement of the shape of the boson transverse momentum distribution in p p̅→ Z / γ^*→ e^+ e^- + X events produced at √(s)=1.96-TeV,
  Phys. Rev. Lett.  100 (2008) 102002
 
  [arXiv:0712.0803 [hep-ex]]. 


Gluck:2007ck
  M. Glück, P. Jimenez-Delgado and E. Reya,
  Dynamical parton distributions of the nucleon and very small-x physics,
  Eur. Phys. J. C 53 (2008) 355

 [arXiv:0709.0614 [hep-ph]].
  


deFlorian:2014xna
  D. de Florian, R. Sassot, M. Epele, R. J. Hernández-Pinto and M. Stratmann,
  Parton-to-Pion Fragmentation Reloaded,
  Phys. Rev. D 91 (2015),  014035
 
  [arXiv:1410.6027 [hep-ph]].
  


deFlorian:2007aj
  D. de Florian, R. Sassot and M. Stratmann,
  Global analysis of fragmentation functions for pions and kaons and their uncertainties,
  Phys. Rev. D 75 (2007) 114010
  
  [hep-ph/0703242 [hep-ph]].


Signori:2013mda
  A. Signori, A. Bacchetta, M. Radici and G. Schnell,
  Investigations into the flavor dependence of partonic transverse momentum,
  JHEP 1311 (2013) 194

  [arXiv:1309.3507 [hep-ph]].


Schweitzer:2010tt
  P. Schweitzer, T. Teckentrup and A. Metz,
  Intrinsic transverse parton momenta in deeply inelastic reactions,
  Phys. Rev. D 81 (2010) 094019
 
  [arXiv:1003.2190 [hep-ph]].
  
  
 
Anselmino:2013lza
  M. Anselmino, M. Boglione, J. O. Gonzalez Hernandez, S. Melis and A. Prokudin,
  Unpolarised Transverse Momentum Dependent Distribution and Fragmentation Functions from SIDIS Multiplicities,
  JHEP 1404 (2014) 005

  [arXiv:1312.6261 [hep-ph]].
  
  

Echevarria:2014xaa
  M. G. Echevarria, A. Idilbi, Z. B. Kang and I. Vitev,
  QCD Evolution of the Sivers Asymmetry,
  Phys. Rev. D 89 (2014) 074013

  [arXiv:1401.5078 [hep-ph]].
  


Scimemi:2017etj 
  I. Scimemi and A. Vladimirov,
  Analysis of vector boson production within TMD factorization,
  [arXiv:1706.01473 [hep-ph]].
  



