Transverse Momentum Distributions of Partons in the Nucleon

Transcription

1 Lattice 2008, Williamsburg Transverse Momentum Distributions of Partons in the Nucleon Bernhard Musch Technische Universität München presenting work in collaboration with LHPC and Philipp Hägler (TUM), Andreas Schäfer, Meinulf Göckeler (Univ. Regensburg), John Negele (MIT), Dru Renner (DESY Zeuthen) supported by

2 motivation: parton picture Fast nucleon: Quarks look like partons. Distribution depends on momentum fraction x k + /P + of the nucleon momentum P, intrinsic transverse momentum k, transverse position b (impact parameter).

3 motivation: parton picture How are the quarks distributed with respect to x and k? TMDPDFs transverse momentum dependent parton distribution functions e.g. f 1 (x, k ) Fast nucleon: Quarks look like partons. Distribution depends on momentum fraction x k + /P + of the nucleon momentum P, intrinsic transverse momentum k, transverse position b (impact parameter).

4 motivation: parton picture How are the quarks distributed with respect to x and k? TMDPDFs transverse momentum dependent parton distribution functions e.g. f 1 (x, k ) Fast nucleon: Quarks look like partons. Distribution depends on momentum fraction x k + /P + of the nucleon momentum P, PDFs intrinsic transverse momentum k, TMDPDFs transverse position b (impact parameter). GPDs

5 k -dependence and factorization example: (SIDIS) semi inclusive deep inelastic scattering experiment hadron γ * k quark k P nucleon P

6 k -dependence and factorization example: (SIDIS) semi inclusive deep inelastic scattering experiment hadron γ * k quark k P P nucleon factorization = hard process + soft blobs (non-perturbative) [Collins, Soper, Sterman PLB 83, NPB 85] [Ji, Ma, Yuan PRD (2005)], [Mulders, Tangerman NPB (1996)]

7 k -dependence and factorization hadron g * k quark Dirac matrix Γ k P nucleon Φ [Г] (k,p,s) TMDPDFs f 1 (x,k ),... P non-perturbative correlator, defined as Φ [Γ] (k, P, S) P q(k) Γ q(k) P

8 k -dependence and factorization hadron g * k quark Dirac matrix Γ k P nucleon Φ [Г] (k,p,s) TMDPDFs f 1 (x,k ),... P non-perturbative correlator, defined as Φ [Γ] (k, P, S) 1 d 4 l e ik l 2 (2π) 4 P, S q(l) Γ U q(0) P, S

9 gauge link operator U P q(l) Γ U q(0) P is gauge invariant. continuum ( U P exp ig l along path from 0 to l 0 dξ µ A µ (ξ) ) factorization in SIDIS : path runs to infinity and back here (up to now): straight path retains probability interpretation! e.g., [Bacchetta et al., PRL85,712 (2000)]

10 gauge link operator U continuum ( U P exp ig P q(l) Γ U q(0) P is gauge invariant. lattice l along path from 0 to l 0 dξ µ A µ (ξ) ) product of link variables factorization in SIDIS : path runs to infinity and back here (up to now): straight path retains probability interpretation! e.g., [Bacchetta et al., PRL85,712 (2000)]

11 extracting nucleon structure from the lattice Ingredients Output : 3-point correlator C 3pt gauge configs. quark propagators nucleon sequential propagators [We neglect disconnected contributions (absent for up minus down).]

12 extracting TMDPDFs from the lattice We use the Chroma library [Edwards, Joo (2005)] to process MILC gauge configurations staggered Asqtad action, 2+1 flavors, a fm, m π 500, 610, and 760 MeV [Orginos, Toussaint PRD (1999)] LHPC propagators domain wall valence fermions, m π adjusted to staggered sea, nucleon momenta: P = 0 and P = 500 MeV e.g., [Hägler et al. PRD (2008)]

13 extracting TMDPDFs from the lattice ratio of correlators far away from nucleon source and sink C 3pt(τ, t sink, P,...) C 2pt(t sink, P,...) 0 τ t sink P, S q(l) Γ U q(0) P, S C3 pt C2 pt matrix element extracted from plateau value

14 extracting TMDPDFs from the lattice ratio of correlators far away from nucleon source and sink C 3pt(τ, t sink, P,...) C 2pt(t sink, P,...) 0 τ t sink P, S q(l) Γ U q(0) P, S isolation of Lorentz-invariant amplitudes compare [Mulders, Tangerman NPB (1996)] P, S q(l) γ µ U q(0) P, S = 4 Ã2 Pµ + 4i mn 2 à 3 l µ P, S q(l) γ µγ 5 U q(0) P, S = 4 m N à 6 S µ 4i m N à 7 P µ(l S) The amplitudes fulfill à i (l 2, l P ) = + 4 m N 3 à 8 l µ(l S) ]. [Ãi (l 2, l P ) Lattice restriction: l 0 = l 4 = 0 l 2 0, l P P l 2

15 First Results (Renormalization is preliminary.)

16 Re Ã2(l 2, l P ) from the lattice Re 2A P fm 4 P q(l) γ 4 U q(0) P Ã2(l 2, l P ) 0 Fourier trans. f lat 1 (x, k ) }{{} TMDPDF

17 Re Ã2(l 2, l P =0) and the TMDPDF f 1 (x, k ) f 1 (x, k ) dk Φ [γ+] (k, P, S) 2 Re A up quarks m π = 500 MeV renormalization: W.line taxi W.loop taxi 1 loop PT

18 Re Ã2(l 2, l P =0) and the TMDPDF f 1 (x, k ) 1 st Mellin moment f (1) 1 ( k ) dx dk Φ [γ+] (k, P, S) 2 Re A up quarks m π = 500 MeV renormalization: W.line taxi W.loop taxi 1 loop PT

19 Re Ã2(l 2, l P =0) and the TMDPDF f 1 (x, k ) 1 st Mellin moment f (1)lat 1 (k ) dx = dk Φ [γ+] (k, P, S) d 2 l (2π) 2 eik l 2 Ã2( l 2, 0) 2 Re A up quarks m π = 500 MeV renormalization: W.line taxi W.loop taxi 1 loop PT

20 Re Ã2(l 2, l P =0) and the TMDPDF f 1 (x, k ) 1 st Mellin moment f (1)lat 1 (k ) dx = dk Φ [γ+] (k, P, S) d 2 l (2π) 2 eik l 2 Ã2( l 2, 0) 2 Re A up quarks m π = 500 MeV renormalization: W.line taxi W.loop taxi 1 loop PT fit function C 1 exp( l 2 /σ 2 1) + C 2 exp( l 2 /σ 2 2)

21 Re Ã2(l 2, l P =0) and the TMDPDF f 1 (x, k ) 1 st Mellin moment f (1)lat 1 (k ) dx = dk Φ [γ+] (k, P, S) d 2 l (2π) 2 eik l 2 Ã2( l 2, 0) up density GeV f (1)lat 1 (k x, k y =0) up quarks m π = 500 MeV renormalization: W.line taxi W.loop taxi 1 loop PT

22 Re Ã2(l 2, l P =0) and the TMDPDF f 1 (x, k ) 1 st Mellin moment f (1)lat 1 (k ) dx = dk P q(k)γ + q(k) P d 2 l (2π) 2 eik l 2 Ã2( l 2, 0) up density GeV f (1)lat 1 (k x, k y =0) up quarks m π = 500 MeV renormalization: W.line taxi W.loop taxi 1 loop PT f (1)lat 1 (k ) gives the density of quarks with an intrinsic transverse momentum k = (k x, k y )

23 Re Ã2(l 2, l P =0) and the TMDPDF f 1 (x, k ) 1 st Mellin moment f (1)lat 1 (k ) dx = dk P q(k)γ + q(k) P d 2 l (2π) 2 eik l 2 Ã2( l 2, 0) ky GeV up f (1)lat 1 (k ) gives the density of quarks with an intrinsic transverse momentum k = (k x, k y ) k x GeV

24 linear extrapolation k 2 1/2 to physical pion mass k GeV renormalization: W.line taxi 0.2 W.loop taxi 1 loop PT up-down RMS transverse momentum k 2 1/2 = (649 ± 18 stat ) MeV based on double Gaussian Ansatz compare phenomenology [Anselmino et al., PRD71, (2005)]: k 2 1/2 500 MeV based on single Gaussian Ansatz

25 polarized quark densities: the TMDPDF g 1T (x, k ) In a transversely spin polarized nucleon ( S P ): 1 dx dk Φ [γ+ 1 2 (1+γ5)] (k, P, S) = 2 ( 1 f (1)lat 1 (k ) + k S g (1)lat 1T (k ) 2 m N ) 2 Re A up quarks m π = 500 MeV renormalization: W.line taxi W.loop taxi 1 loop PT g (1)lat 1T is obtained from amplitude Ã7

26 polarized quark densities: the TMDPDF g 1T (x, k ) In a transversely spin polarized nucleon ( S P ): 1 dx dk P, S q(k)γ (1 + γ 5 )q(k) P, S = ( 1 f (1)lat 1 (k ) + k S g (1)lat 1T (k ) 2 m N ) y z x up density of quarks with positive helicity in a proton with spin pointing in x direction ky GeV net transverse momentum k x k x = (135±10 stat ±6 renorm. ) m π = 500 MeV k x GeV

27 polarized quark densities: the TMDPDF g 1T (x, k ) In a transversely spin polarized nucleon ( S P ): 1 dx dk Φ [γ+ 1 2 (1+γ5)] (k, P, S) = 2 ( 1 f (1)lat 1 (k ) + k S g (1)lat 1T (k ) 2 m N ) y z x down density of quarks with positive helicity in a proton with spin pointing in x direction ky GeV net transverse momentum k x k x = ( 24 ± 5 stat ± 3 renorm. ) m π = 500 MeV k x GeV

28 polarized quark densities: the TMDPDF g 1T (x, k ) In a transversely spin polarized nucleon ( S P ): 1 dx dk Φ [γ+ 1 2 (1+γ5)] (k, P, S) = 2 ( 1 f (1)lat 1 (k ) + k S g (1)lat 1T (k ) 2 m N ) ky GeV y z x u-d density of quarks with positive helicity in a proton with spin pointing in x direction k -densities analogous to impact parameter densities [Diehl, Hägler EPJC44 (2005)], [QCDSF PRL98, (2007)] see also [G. Miller PRC76, (2007)] k x GeV

29 Link Renormalization Thanks to Gunnar Bali and Vladimir Braun (Univ. Regensburg) for helpful discussions

30 continuum renormalization and Taxi Driver Method continuum renormalization of Wilson lines [Craigie, Dorn NPB185,204 (1981)] smooth path U ren = Zz 1 exp( δm L ) U L is the total length of the Wilson line Zz 1, δm, ν(θ) are renormalization constants δm µ ˆ= 1 a removes linear divergence This linear divergence is a long-standing problem in heavy-light calculations.

31 continuum renormalization and Taxi Driver Method continuum renormalization of Wilson lines [Craigie, Dorn NPB185,204 (1981)] θ U ren = Zz 1 exp( δm L ν(θ) ) U L is the total length of the Wilson line Zz 1, δm, ν(θ) are renormalization constants δm µ ˆ= 1 a removes linear divergence This linear divergence is a long-standing problem in heavy-light calculations.

32 continuum renormalization and Taxi Driver Method continuum renormalization of Wilson lines [Craigie, Dorn NPB185,204 (1981)] θ U ren = Zz 1 exp( δm L ν(θ) ) U L is the total length of the Wilson line Zz 1, δm, ν(θ) are renormalization constants δm µ ˆ= 1 a removes linear divergence This linear divergence is a long-standing problem in heavy-light calculations. link renormalization on the lattice: Taxi Driver Method 5 links 7 links working hypothesis: like continuum theory U lat ren =Z 1 z exp( aδm #links ν #corners ) U lat `=5 Idea: Evaluate straight and step like link paths Tr 0 U lat 0 on Landau gauge fixed ensemble. Adjust aδm, ν such that Tr 0 Uren lat 0 depends smoothly on l only.

33 Taxi Driver Renormalization 1 3Tr 0 U 0 on Landau gauge fixed ensemble (no link smearing) Tr 0 U

34 Taxi Driver Renormalization 1 3Tr 0 U ren 0 renormalized requiring smoothness Tr 0 U 0 ren a

35 Taxi Driver Renormalization on different lattices fine: MILC a = fm, m π 760 MeV coarse: MILC a = fm, m π 790 MeV with and without HYP smearing (reduces aδm drastically) fine fine, smeared coarse coarse, smeared Still a-dependence. Renormalization incomplete or O(a 2 )-effects?

36 Summary Results: First lattice calculation of quark distributions f lat 1 and g lat 1T as a function of transverse momentum. Densities of longitudinally polarized quarks in a transversely polarized proton are deformed. Outlook: Analysis of further amplitudes and TMDPDFs. Need for improved renormalization of the non-local operators. Study of non-straight gauge links similar as in SIDIS.

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38 Backup Slides

39 Renormalization constants on different lattices a (fm) method aδm ν Z z 0.12 taxi (17) (80) 1.107(19) 0.12 perturb taxi (17) (69) 1.098(23) 0.08 perturb smeared taxi (42) (16) 1.021(17) 0.12 smeared perturb smeared taxi (32) (11) 1.017(16) 0.08 smeared perturb Coarse lattice: a = fm, m π 790 MeV, m u = m d = m s Fine lattice: a = fm, m π 760 MeV, m u = m d = m s

40 Amplitude Ã2 compared to VEV of gauge link P = 0 q(l) γ + U q(0) P = Tr 0 U 0 turns out constant

41 (x, k ) factorization hypothesis

42 l P - dependence of Ã2(l 2, l P ) 2 Re Ã2(l 2, l P ) 2 Im Ã2(l 2, l P ) Re 2A Im 2A 2 P P fm fm

43 l P - dependence of Ã2(l 2, l P ) 2 Re Ã2(l 2, l P ) 2 Im Ã2(l 2, l P ) Re 2A fm 0.37 fm 0.62 fm fm 1.5 fm fm 0.25 fm 0.5 fm 0.74 fm 1.2 fm Im 2A 2 offset fm 1.2 fm 0.99 fm 0.74 fm 0.62 fm 0.5 fm 0.37 fm 0.25 fm 0.12 fm 0. fm

44 (x, k )-factorization hypothesis factorization hypothesis f lat 1 (x, k ) = ˆf lat 1 (x) f (1)lat 1 ( k ) as in phenomenological applications, e.g., [Anselmino PRD (2005)] Then Ã2 factorizes, too: à 2 (l 2, l P ) =  2 (l P ) Ã2(l 2, 0). To test this, we define a scaled amplitude  2 (l 2, l P ) Ã2(l 2, l P ) Re Ã2(l 2, 0) If factorization holds,  2 should be l 2 -independent.

45 (x, k )-factorization hypothesis factorization hypothesis f lat 1 (x, k ) = ˆf lat 1 (x) f (1)lat 1 ( k ) as in phenomenological applications, e.g., [Anselmino PRD (2005)] Then Ã2 factorizes, too: à 2 (l 2, l P ) =  2 (l P ) Ã2(l 2, 0). To test this, we define a scaled amplitude  2 (l 2, l P ) Ã2(l 2, l P ) Re Ã2(l 2, 0) If factorization holds,  2 should be l 2 -independent. within statistics Im 2A 2 offset P fm

46 comparison to CTEQ parton distributions All our data for  2 (l 2, l P ) at m π 610 MeV compared to a Fourier transform of f 1 (x) from CTEQ5 [Lai et al., EPJ C12, 375 (2000)] Re 2A P small offsets

47 comparison to CTEQ parton distributions All our data for  2 (l 2, l P ) at m π 610 MeV compared to a Fourier transform of f 1 (x) from CTEQ5 [Lai et al., EPJ C12, 375 (2000)] Im 2A P small offsets