Asmita Mukherjee Indian Institute of Technology Bombay, Mumbai, India

Transcription

1 . p.1/26 Sivers Asymmetry in e + p e + J/ψ + X Asmita Mukherjee Indian Institute of Technology Bombay, Mumbai, India Single spin asymmetry Model for J/ψ production Formalism for calculating the asymmetry Models for the Sivers function used Results Effect of TMD evolution Summary and conclusions QCD Evolution Workshop JLab, May 212 In collaboration with R. Godbole (IISc, Bangalore), A. Misra (Mumbai U.), V. Rawoot (Mumbai U.) arxiv : [hep-ph], PRD 85, 9413 (212)

2 . p.2/26 Single Spin Asymmetry in J/ψ production Aim is to study the asymmetry in low virtuality electroproduction of charmonium by scattering unpolarized electrons off transversely polarized proton Transverse momentum dependent Sivers function describes the probability of finding an unpolarized parton inside a transversely polarized hadron : correlation between transverse momentum of the unpolarized quarks and gluons and the nucleon spin related to orbital angular momentum : Sivers asymmetry gives access to the orbital angular momentum of the partons SSAs involving the transverse momentum dependent pdfs and fragmentation functions: very often two or more of these functions contribute to the same physical observable In this process that we are considering, at LO, there is contribution only from a single partonic subprocess γg c c : can be used as a clean probe of gluon Sivers function May throw some light on the charmonium production mechanism as well

3 . p.3/26 Model for charmonium production that we consider Color evaporation model : first proposed by Halzen and Matsuda (1978), H. Fritsch (1977) CEM predicts a cross section for the J/ψ production from the cross section of the c c pair Statistical treatment of color : color can evaporate by multiple soft gluon emission Cross section for charmonium production is proportional to the rate of production of c c pair integrated over the mass range 2m c to 2m D σ = 1 9 Z 2mD 2m c dm c c dσ c c dm c c where m c is the charm quark mass and 2m D is the D D threshold; 1/9 is the statistical probability for the production of a color singlet state; Mc c 2 is the squared invariant mass of c c In later versions data are better fitted by the inclusion of a phenomenological factor in differential cross section formula, which depends on a Gaussian distribution of the transverse momentum of the charmonium Eboli, Gregores, Halzen (23)

4 . p.4/26 Cross section for charmonium production Cross section for low-virtuality electroproduction of J/ψ : σ ep e+j/ψ+x = Z 4m 2 D 4m 2 c dm 2 c c Z dˆσγg c c dy dx f γ/e (y) f g/p (x). dm 2 c c The photon flux in the electron is approximated by the distribution (Weizsacker-Williams approximation) f γ/e (y, E) = α π {1 + (1 y)2 y ln E m 1 «+ y 2 2» «2 ln y 2 (2 y)2 2 2y + ln 2y 2 y + 1 «}. Brodsky, Kinoshita,Terazawa (1971); Kniehl (1991) where y is the energy fraction of electron carried by the photon, E is the energy of the electron and m is the mass

5 . p.5/26 Single Spin Asymmetry Generalization of CEM expression by taking into account the transverse momentum dependence dσ e+p e+j/ψ+x dm 2 = Z dx γ dx g [d 2 k γ d 2 k g ] f g/p (x g, k g )f γ/e (x γ, k γ ) dˆσγg c c dm 2 where M 2 M 2 c c Single spin asymmetry for a transversely polarized target is defined as The numerator is of the form = dσ dσ dσ + dσ dσ dσ = Z dx γ dx g d 2 k γ d 2 k g N f g/p (x g, k g ) f γ/e (x γ, k γ ) dˆσ γg c c

6 . p.6/26 Single Spin Asymmetry dˆσ is the elementary cross section for the process γg c c given by dˆσ = 1 2ŝ d 3 p c 2E c d 3 p c 2E c 1 (2π) 2 δ4 (p γ + p g p c p c ) M γg c c 2. Changing the variable to q = p c + p c we get d 3 p c 2E c = d 4 q δ((q p c ) 2 m 2 c). Using the expression for total partonic cross section ˆσ γg c c (M 2 ) = 1 2ŝ Z d 3 p c 2E c 1 (2π) 2 δ((q p c) 2 m 2 c ) M γg c c 2 We also change the variables from q and q L to M 2 and rapidity y so that dm 2 dy = 2dq dq L

7 And similarly for the denominator of the asymmetry. p.7/26 Single Spin Asymmetry We finally obtain d 4 σ dydm 2 d 2 q T d 4 σ dydm 2 d 2 q T = 1 2 Z [dx γ d 2 k γ dx g d 2 k g ] N f g/p (x g, k g ) f γ/e (x γ, k γ )δ 4 (p g + p γ q) ˆσ γg c c (M 2 ) N f g/p (x g, k g ) is related to the gluon Sivers function N f g/p (x, k g ) by N f g/p (x g, k g ) = N f g/p (x g, k ) Ŝ ( ˆP ˆk g ) One can perform the x γ and x g integrations to obtain d 4 σ dydm 2 d 2 q T d 4 σ dydm 2 d 2 q T = 1 s Z [d 2 k γ d 2 k g ] N f g/p (x g, k g )f γ/e (x γ, k γ ) δ 2 (k γ + k g q T )ˆσ γg c c (M 2 ) x g,γ = M s e ±y.

8 . p.8/26 Sivers Asymmetry Calculate the weighted asymmetry to extract the Sivers function Vogelsang and Yuan (25) A sin(φ q T φ S ) N = R dφqt [dσ dσ ] φ S ) R dφqt [dσ + dσ ] φ qt and φ S are the azimuthal angles of the J/ψ and proton spin respectively Asymmetry in the rapidity distribution will involve dσ dσ = Z dφ qt Z Z 4m 2 Z D q T dq T [dm 2 ] [d 2 k g ] N f g/p (x g, k g ) 4m 2 c f γ/e (x γ, q T k g ) ˆσ (M 2 ) φ S ) SSA depends on Weizsacker-Williams function, gluon distribution function and gluon Sivers function

9 . p.9/26 Models for WW Function and Sivers function We choose a kinematical configuration in which proton with momentum P is moving along z axis and is transversely polarized in y direction Ŝ ( ˆP ˆk g ) = ˆk gz = cos φ k where, k g = k (cos φ k, sin φ k, ) For k g dependence of the unpolarized pdf s, we use a simple factorized and Gaussian form f g/p (x g, k ) = f g/p (x g ) 1 π k g 2 g / k2 g. e k 2 The WW distribution has k dependence. We have used two choices for WW function for the photon

10 . p.1/26 Models for WW Function and Sivers function We have used two choices for WW function for the photon 1) A simple Gaussian form as above : f γ/e (x γ, k γ ) = f γ/e (x γ ) 1 π k γ 2 γ / k2 γ e k 2 2) Second form : f γ/e (x γ, k γ ) = f γ/e (x γ ) 1 2π N k 2 γ + k 2 N is a normalization constant, which gets canceled in the asymmetry For the Sivers function we use two models M. Anselmino, M. Boglione, U. D Alesio, S. Melis, F. Murgia and A. Prokudin (29) M. Anselmino, M. Boglione, U. D Alesio, E. Leader and F. Murgia (24) Model I has been used in analysis of SSA in SIDIS and DY process Model II has been used for quark Sivers function to estimate SSA in D meson production at RHIC

11 . p.11/26 Models for Sivers Function Model I : N f g/p (x g, k g) = N 1 f g/p (x g ) π k g 2 h(k g) e k 2 g / k2 g cos(φ k ) where the gluon Sivers function, N f g/p (x g ) is defined as N f g/p (x g ) = 2 N g (x g ) f g/p (x g ) N g (x g ) is an x-dependent normalization for gluon to be chosen so that the gluon Sivers function obeys the positivity bound N f g/p (x g, k g) 2 ˆf g/p (x g, k g ) 1, x g, k g

12 . p.12/26 Models for Sivers Function h(k g) = 2e k g e k 2 M 1 g /M2 1 M 1 is parameter obtained by fitting the recent experimental data corresponding to pion and kaon production at HERMES and COMPASS Parametrization for quark Sivers function N f (x) = N f x a f (1 x) b f (a f + b f ) (a f+b f ) a f a f b f b f. where a f, b f, N f for u and d quarks are free parameters obtained by fitting the data. However, there is no information available on N g (x) We use (Boer and Vogelsang (24)) (a) N g (x) = (N u (x) + N d (x)) /2, (b) N g (x) = N d (x).

13 . p.13/26 Models for Sivers Function Model II : N f g/p (x g, k g ) = N f g/p (x g ) 1 π k 2 g e k 2 g / k2 g 2k gm k g 2 + M2 cos(φ k ), and M = q k g 2 where the gluon Sivers function is given as in Model I Best fit parameters of Sivers functions (Anselmino et al (211) N u =.4, a u =.35, b u = 2.6, N d =.97, a d =.44, b d =.9, M 2 1 =.19 GeV 2. Other parameters we use are k 2 g = k2 γ =.25 GeV 2. Will comment on this choice later

14 . p.14/26 SSA for J/ψ production at JLab.25.2 (a) (b).25.2 (a) (b) y q T (GeV) Single spin asymmetry at JLab as a function of y (left panel) and q T (right panel). The plots are for model I with two parametrizations (a) (solid red line) and (b) (dashed blue line). The integration ranges are ( q T 1) GeV and ( y.25). The results are given at s = 4.7 GeV

15 . p.15/26 SSA for J/ψ production at HERMES.25.2 (a) (b).25.2 (a) (b) y q T (GeV) Single spin asymmetry at HERMES as a function of y (left panel) and q T (right panel). The plots are for model I with two parametrizations (a) (solid red line) and (b) (dashed blue line). The integration ranges are ( q T 1) GeV and ( y.6). The results are given at s = 7.2 GeV

16 . p.16/26 SSA for J/ψ production at COMPASS.25.2 (a) (b).25.2 (a) (b) y q T (GeV) Single spin asymmetry at COMPASS as a function of y (left panel) and q T (right panel). The plots are for model I with two parametrizations (a) (solid red line) and (b) (dashed blue line). The integration ranges are ( q T 1) GeV and ( y 1). The results are given at s = GeV

17 . p.17/26 SSA for J/ψ production at erhic.25.2 (a) (b).25.2 (a) (b) y q T (GeV) Single spin asymmetry at erhic as a function of y (left panel) and q T (right panel). The plots are for model I with two parametrizations (a) (solid red line) and (b) (dashed blue line). The integration ranges are ( q T 1) GeV and ( y 1). The results are given at s = 31.6 GeV

18 . p.18/26 SSA for J/ψ production at erhic (a) (b) y (a) (b) q T (GeV) Single spin asymmetry at erhic as a function of y (left panel) and q T (right panel). The plots are for model I with two parametrizations (a) (solid red line) and (b) (dashed blue line). The integration ranges are ( q T 1) GeV and ( y 1). The results are given at s = GeV

19 . p.19/26 SSA for J/ψ production at COMPASS.12.1 I (a) II (a).12.1 I (a) II (a) y q T (GeV) The single spin asymmetry at COMPASS as a function of y (left panel) and q T (right panel). The plots are for two models I (solid red line) and II (dashed blue line) with parametrization(a). The integration ranges are ( q T 1) GeV and ( y 1). The results are given at s = GeV

20 . p.2/26 SSA for J/ψ production at COMPASS.35.3 I (b) II (b).35.3 I (b) II (b) y q T (GeV) Single spin asymmetry at COMPASS as a function of y (left panel) and q T (right panel). The plots are for two models I (solid red line) and II (dashed blue line) with parametrization (b). The integration ranges are ( q T 1) GeV and ( y 1). The results are given at s = GeV

21 . p.21/26 Compare I (a) -JLab I (a) -HERMES I (a) -COMPASS I (a) -erhic-1 I (a) -erhic I (a) -JLab I(a) -HERMES I(a) -COMPASS I(a) -erhic-1 I(a) -erhic y q T (GeV) Asymmetry in model I with parameterization (a) compared for JLab ( s = 4.7 GeV) (solid red line), HERMES ( s = 7.2 GeV) (dashed green line), COMPASS ( s = GeV) (dotted blue line), parametrization-1 ( s = 31.6 GeV) (long dashed pink line) and erhic-2 ( s = GeV) (dot-dashed black line)

22 . p.22/26 Dependence on the photon distribution Gaussian dependence, I (a) Dipole dependence, I (a) Gaussian dependence, I (b) Dipole dependence I (b) y Single spin asymmetry A sin(φ q T φ S ) N at COMPASS as a function of y : plots are for model I with two parametrization (a) (solid red line and dashed green line) and (b) (dotted blue line and long dashed pink line) compared for Gaussian and dipole k dependence of WW function. The results are given at s = GeV

23 Scale Evolution of TMDs TMD evolution Aybat, Rogers; Aybat, Collins, Qiu, Rogers, (211) It has been found that TMD evolution affects the single spin asymmetry in SIDIS Aybat, Prokudin, Rogers (211); Anselmino, Boglione, Melis (212) QCD scale evolution effectively changes the width of the gaussian in the parametrization k2 f(x, k, Q) = f(x, Q )R(Q, Q ) e w 2 πw 2 R(Q, Q ) : overall scale dependent factor, f(x, Q ) : integrated pdf at scale Q w 2 (Q, Q ) =< k 2 > +2g 2log Q Q TMD evolution is governed by the overall factor R(Q, Q ) and by the gaussian width and is faster than DGLAP evolution Also, the average value of k changes with energy. p.23/26

24 . p.24/26 Effect of scale evolution of TMDs In the color evaporation model for charmonium production, the numerator and denominator of the asymmetry become d 3 σ dyd 2 q T d3 σ dyd 2 q T = 1 s Z 4m 2 D 4m 2 c dm 2 N f g/p (x g )f γ/e (x γ ) 2e q T M 1 k2 S 2 exp[ q 2 T /( k2 S + k2 γ )] π[ k 2 S + k2 γ ]2 k 2 g cos(φ qt ) ˆσ γg c c (M 2 ) d 3 σ dyd 2 q T + d3 σ dyd 2 q T = 2 s Z 4m 2 D 4m 2 c dm 2 f g/p (x g )f γ/e (x γ ) exp[ q2 T /( k2 g + k2 γ )] π[ k 2 g + k2 γ ] ˆσ γg c c (M 2 ) 1 ks 2 = 1 M k 2 g.

25 . p.25/26 Effect of scale evolution of TMDs Charmonium production in this model depends only on the scale M 2 = ŝ; which is the invariant mass squared of the c c pair Cross section is integrated over M 2 from 4m 2 c to 4m 2 D, which is a rather narrow region The TMDs in the differential cross section can be sensitive to this scale : and this range of scale is independent of the CM energy of the experiment concerned So the parametrization of the Sivers function at a low scale can be used to predict the asymmetry at a high scale (for example erhic) for charmonium production in this model We take the average k to be the same in all the experiments

26 . p.26/26 Summary and Conclusion We have calculated SSA for J/ψ electroproduction using the simple color evaporation model : at leading order this involves only photon-gluon fusion and can give information on gluon Sivers function We use Weizsacker-Williams approximation for the photon distribution Introduce transverse momentum dependence in the distribution functions and use the existing parametrization of the TMDs : use gaussian k dependence of the WW function The relevant scale for charmonium production in this model is M 2 which is the squared invariant mass of the c c pair and independent of the CM energy of the experiment : use of the parametrization justified Photoproduction : have to consider higher order and resolved photon contributions; also the role of gauge links will become important at higher orders Would be interesting to see how sensitive the SSA is on the charmonium production model and the effect of scale evolution of the TMDs SSA is sizable and does not depend much on the choice of the k dependence of the photon distribution; can be used as a probe for the gluon Sivers function