Transverse momentum-dependent parton distributions from lattice QCD. Michael Engelhardt New Mexico State University

Transcription

1 Transverse momentum-dependent parton distributions from lattice QCD Michael Engelhardt New Mexico State University In collaboration with: B. Musch P. Hägler J. Negele A. Schäfer

2 Lattice theorists go shopping... Looking for: Definition of TMDs in terms of matrix elements of a QCD operator There should exist a Lorentz frame in which the operator is at one fixed time separations in operator should be generically space-like (light cone approached as a limit)

3 ... and find pictures of SIDIS (and DY)... suggesting such things can actually be defined, Factorization? Inclusion of final/initial state interactions?

4 Want to appropriately count quarks of momentum k... Φ [Γ] (x,k T,P,S,...) = dk P,S q(k) Γ q(k) P,S k + xp + = d 2 b T (2π) 2 d(b P) (2π)P + exp (ix(b P) ib T k T ) Φ [Γ] unsubtr.(b,p,s,...) S(b 2,...) b + =0 Φ [Γ] unsubtr.(b,p,s,...) P,S q(0) Γ U[0,...,b] q(b) P,S 2 Soft factor S (typically a combination of vacuum expectation values of Wilson line structures) required to subtract divergences of Wilson line U Here, will consider only ratios in which soft factors cancel

5 Decomposition into TMDs All leading twist structures: Φ [γ+] = f ǫ ij k i S j m N f T odd Φ [γ+ γ 5] = Λg + k T S T m N g T Φ [iσi+ γ 5] = S i h + (2k ik j k 2 T δ ij)s j 2m 2 N h T + Λk i m N h L + ǫ ij k j m N h odd

6 Gauge link structure... dictated by the physical process: In matrix element Φ [Γ] unsubtr.(b,p,s,...) 2 P,S q(0) Γ U[0,...,b] q(b) P,S Staple-shaped gauge link U[0, ηv, ηv + b, b] incorporates SIDIS final state effects

7 Gauge link structure Gauge link roughly follows direction of ejected quark, (close to) light cone Effective resummed description of gluon exchanges between ejected quark and remainder of nucleon in evolving final state Take staple direction off light-cone into space-like region cf. scheme advanced by Aybat, Collins, Qiu, Rogers in order to control rapidity divergences. Parametrize in terms of Collins-Soper-type parameter ˆζ P v P v Light-like staple for ˆζ. Perturbative evolution equations for large ˆζ? This scheme appears to fit the requirements of a lattice calculation!

8 Return to decomposition into TMDs (leading twist): Φ [γ+] = f ǫ ij k i S j m N f T odd Φ [γ+ γ 5] = Λg + k T S T m N g T Φ [iσi+ γ 5] = S i h + (2k ik j k 2 T δ ij)s j 2m 2 N h T + Λk i m N h L + ǫ ij k j m N h odd Without initial/final state effects, T-odd Sivers and Boer-Mulders functions f T and h would vanish Modified universality f T-odd, SIDIS = f T-odd, DY. SIDIS: ηv P, DY: ηv P.

9 Decomposition into TMDs (leading twist) q N U L T U f h Boer-Mulders (T-odd) L g h L T f T g T h h T Sivers (T-odd)

10 Fourier-transformed TMDs f(x,b 2 T,...) d 2 k T exp(ib T k T )f(x,k 2 T,...) In limit b T 0, formally recover k T -moments: f (n) (x,b 2 T,...) n! 2 m 2 b 2 N T n f(x,b 2 T,...) f (n) (x, 0,...) d 2 k T k 2 T 2m 2 N n f(x,k 2 T,...) f (n) (x) However, b T 0 limit delicate will not attempt to extrapolate to b T = 0, but give results at finite b T ; b T acts as regulator. Also, we can only access limited range of b P, so cannot Fourier-transform to obtain x-dependence. Therefore, consider only first x-moments (accessible at b P = 0): f [] (k 2 T,...) dxf(x,k 2 T,...)

11 Return to correlator Invariant amplitudes Φ [Γ] unsubtr.(b,p,s, ˆζ,µ) P,S q(0) Γ U[0,ηv,ηv + b,b] q(b) P,S 2 Decompose in terms of invariant amplitudes; at leading twist, 2P + Φ [γ+ ] unsubtr. = A 2B + im N ǫ ij b i S j A 2B 2P + Φ [γ+ γ 5 ] unsubtr. = ΛA 6B + i[(b P)Λ m N (b T S T )] 2P + Φ [iσi+ γ 5 ] unsubtr. = im N ǫ ij b j A 4B S i A 9B A 7B im N Λb i A 0B + m N [(b P)Λ m N (b T S T )]b i A B

12 Invariant amplitudes and TMDs Amplitudes related to TMDs via Fourier transform (showing just the ones relevant for Sivers, Boer-Mulders shifts): f (x,k 2 T, ˆζ,...,ηv P) = 2 A 2B f T(x,k 2 T, ˆζ,...,ηv P) = 4m 2 N k 2 T F F A 2B where F A i h (x,k 2 T, ˆζ,...,ηv P) = 4m 2 N k 2 T d 2 b T (2π) 2 exp( ib T k T ) S(b 2,...) d(b P) 2π F A 4B exp(ix(b P)) A i ( b 2 T,b P, ˆζ,ηv P)

13 Invariant amplitudes and TMDs Conversely, invariant amplitudes directly give selected x-integrated TMDs in Fourier (b T ) space (again showing just the ones relevant for Sivers, Boer-Mulders shifts): f [](0) (b 2 T, ˆζ,...,ηv P) = 2 A 2B ( b 2 T, 0, ˆζ,ηv P)/ S(b 2,...) f []() T (b 2 T, ˆζ,...,ηv P) = 2A 2B ( b 2 T, 0, ˆζ,ηv P)/ S(b 2,...) h []() (b 2 T, ˆζ,...,ηv P) = 2 A 4B ( b 2 T, 0, ˆζ,ηv P)/ S(b 2,...)

14 Generalized shifts Form ratios in which soft factors, (Γ-independent) multiplicative renormalization factors cancel Sivers shift: f []() k y TU m T N f [](0) = dx d2 k T k y Φ [γ+] (x,k T,S T = (, 0)) dx d 2 k T Φ [γ+] (x,k T,S T = (, 0)) Average transverse momentum of unpolarized ( U ) quarks orthogonal to the transverse ( T ) spin of nucleon; normalized to the number of valence quarks. Since b T 0 limit delicate, consider more generally ratio of Fourier-transformed TMDs at nonzero b 2 T, k y TU (b 2 T,...) m N f []() T (b 2 T,...) f [](0) (b 2 T,...) (remember b T 0 limit corresponds to taking k T -moment). Generalized shift.

15 Generalized shifts from amplitudes Now, can also express this in terms of invariant amplitudes: k y TU (b 2 T,...) m N f []() T (b 2 T,...) f [](0) (b 2 T,...) = m N Analogously, Boer-Mulders shift: k y UT (b 2 T,...) = m N A 4B ( b 2 T, 0, ˆζ,ηv P) A 2B ( b 2 T, 0, ˆζ,ηv P) A 2B ( b 2 T, 0, ˆζ,ηv P) A 2B ( b 2 T, 0, ˆζ,ηv P)

16 Lattice setup Evaluate directly Φ [Γ] unsubtr.(b,p,s, ˆζ,µ) 2 P,S q(0) Γ U[0,ηv,ηv + b,b] q(b) P,S τ Euclidean time: Place entire operator at one time slice, i.e., b, ηv purely spatial Since generic b, v space-like, no obstacle to boosting system to such a frame! Parametrization of correlator in terms of A i invariants permits direct translation of results back to original frame Form desired ratios of A i invariants Extrapolate η, ˆζ numerically

17 Lattice setup Use three MILC 2+-flavor gauge ensembles with a 0.2 fm: m π = 369 MeV ; ; 284 samples m π = 369 MeV ; ; 5264 samples m π = 58 MeV ; ; 3888 samples Sink momenta P: (0, 0, 0), (, 0, 0), ( 2, 0, 0), (,, 0) Variety of b, ηv; note b P, b v (lowest x-moment, kinematical choices/constraints) Largest ˆζ = 0.78

18 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T Ζ 0.39, DY Sivers Shift, u d quarks b T 0.2 fm, m Π 58 MeV SIDIS Η v lattice units

19 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T Ζ 0.39, DY Sivers Shift, u d quarks b T 0.24 fm, m Π 58 MeV SIDIS Η v lattice units

20 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T Ζ 0.39, DY Sivers Shift, u d quarks b T 0.36 fm, m Π 58 MeV SIDIS Η v lattice units

21 Results: Sivers shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn f T Ζ 0.39, DY Sivers Shift, u d quarks b T 0.47 fm, m Π 58 MeV SIDIS Η v lattice units

22 Results: Sivers shift Dependence of SIDIS limit on b T GeV f 0 mn f T Sivers Shift SIDIS, u d quarks Ζ 0.39, m Π 58 MeV b T fm

23 Dependence on staple extent; flavor separated Results: Sivers shift GeV f 0 m N f T Sivers Shift DY preliminary down quarks, Ζ 0.38, l 0.36 fm SIDIS GeV f 0 m N f T Sivers Shift DY preliminary up quarks, Ζ 0.38, l 0.36 fm SIDIS staple extentη v lattice units staple extentη v lattice units

24 Results: Sivers shift Dependence on staple extent; sequence of panels at different ˆζ GeV f 0 mn f T Sivers Shift SIDIS, u d quarks Ζ 0, b T 0.36 fm, m Π 58 MeV Η v lattice units

25 Results: Sivers shift Dependence on staple extent; sequence of panels at different ˆζ GeV f 0 mn f T Ζ 0.39, DY Sivers Shift, u d quarks b T 0.36 fm, m Π 58 MeV SIDIS Η v lattice units

26 Results: Sivers shift Dependence on staple extent; sequence of panels at different ˆζ GeV f 0 mn f T Ζ 0.78, DY Sivers Shift, u d quarks b T 0.36 fm, m Π 58 MeV SIDIS Η v lattice units

27 Results: Sivers shift Dependence of SIDIS limit on ˆζ GeV f 0 m N f T total contrib. A 2 b T 0.36 fm m Π 58 MeV Sivers Shift SIDIS, u d quarks Ζ

28 Dependence of SIDIS limit on ˆζ, all three ensembles Results: Sivers shift GeV f 0 mn f T m Π 58MeV 20 3 m Π 369MeV 20 3 m Π 369MeV 28 3 b T 0.36 fm Sivers Shift SIDIS, u d quarks Ζ

29 Results: Boer-Mulders shift Dependence on staple extent GeV f 0 mn h DY Boer Mulders Shift, u d quarks Ζ 0.39, b T 0.36 fm, m Π 58 MeV SIDIS Η v lattice units

30 Results: Boer-Mulders shift Dependence of SIDIS limit on b T GeV f 0 mn h Boer Mulders Shift SIDIS, u d quarks Ζ 0.39, m Π 58 MeV b T fm

31 Dependence on staple extent; flavor separated Results: Boer-Mulders shift 0.4 Boer Mulders Shift total 0.4 Boer Mulders Shift total GeV 0.2 contrib. from a 4 GeV 0.2 contrib. from a 4 f f m N h down quarks, Ζ 0.39, l 0.36 fm DY preliminary SIDIS m N h up quarks, Ζ 0.39, l 0.36 fm DY preliminary SIDIS staple extentη v lattice units staple extentη v lattice units

32 Results: Boer-Mulders shift Dependence of SIDIS limit on ˆζ GeV total contrib. A 4 f 0 m N h b T 0.36 fm m Π 58 MeV Boer Mulders Shift SIDIS, u d quarks Ζ

33 Dependence of SIDIS limit on ˆζ, all three ensembles Results: Boer-Mulders shift GeV f m Π 58MeV 20 3 m Π 369MeV 20 3 m Π 369MeV 28 3 mn h b T 0.36 fm 0.2 Boer Mulders Shift SIDIS, u d quarks Ζ

34 Conclusions Exploratory study of TMDs using staple-shaped gauge link structures Accessed T-odd Sivers, Boer-Mulders observables; SIDIS, DY limits distinguished by sign of v P. For u-d quark combination, SIDIS Sivers and Boer-Mulders TMDs both sizeable and negative. To avoid soft factors, multiplicative renormalization constants, constructed appropriate ratios of Fouriertransformed TMDs ( shifts ). v taken off light cone: Dependence on Collins-Soper parameter ˆζ. In addition to ηv, need to also consider ˆζ. ηv seems under good control; plateaux reached at moderate values. ˆζ remains a challenge. No clear trends seen in the data sets available. Need much larger ˆζ. Presently investigating pion with this in mind. No significant volume dependence, pion mass dependence detected within the limited set of (three) cases considered

35 Results: Transversity Dependence on staple extent; sequence of panels at different b T.4 h, u d quarks f 0 0 h DY Ζ 0.39, b T 0.2 fm, m Π 58 MeV SIDIS Η v lattice units

36 Results: Transversity Dependence on staple extent; sequence of panels at different b T.4 h, u d quarks f 0 0 h DY Ζ 0.39, b T 0.24 fm, m Π 58 MeV SIDIS Η v lattice units

37 Results: Transversity Dependence on staple extent; sequence of panels at different b T.4 h, u d quarks f 0 0 h DY Ζ 0.39, b T 0.36 fm, m Π 58 MeV SIDIS Η v lattice units

38 Dependence of SIDIS/DY limit on b T Results: Transversity.5 h, u d quarks f 0 h Ζ 0.39, m Π 58 MeV b T fm

39 Results: Transversity Dependence of SIDIS/DY limit on ˆζ f 0 h total contrib. A 9 m b T 0.36 fm m Π 58 MeV h, u d quarks Ζ

40 Results: Transversity Dependence of SIDIS/DY limit on ˆζ, all three ensembles f 0 h m Π 58MeV 20 3 m Π 369MeV 20 3 m Π 369MeV 28 3 b T 0.36 fm h, u d quarks Ζ

41 Results: g T worm gear shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn g T g T Shift, u d quarks Ζ 0.39, b T 0.2 fm, m Π 58 MeV DY SIDIS Η v lattice units

42 Results: g T worm gear shift Dependence on staple extent; sequence of panels at different b T GeV f 0 mn g T g T Shift, u d quarks Ζ 0.39, b T 0.36 fm, m Π 58 MeV DY SIDIS Η v lattice units

43 Dependence of SIDIS/DY limit on b T Results: g T worm gear shift GeV f 0 mn g T g T Shift, u d quarks Ζ 0.39, m Π 58 MeV b T fm

44 Dependence of SIDIS/DY limit on ˆζ 0.30 Results: g T worm gear shift GeV f 0 mn g T 0.25 total contrib. A b T 0.36 fm m Π 58 MeV g T Shift, u d quarks Ζ

45 Results: g T worm gear shift Dependence of SIDIS/DY limit on ˆζ, all three ensembles GeV f m Π 58MeV 20 3 m Π 369MeV 20 3 m Π 369MeV 28 3 mn g T 0.0 b T 0.36 fm 0.05 g T Shift, u d quarks Ζ

46 Conclusions Exploratory study of TMDs using staple-shaped gauge link structures Accessed T-odd Sivers, Boer-Mulders observables; SIDIS, DY limits distinguished by sign of v P. For u-d quark combination, SIDIS Sivers and Boer-Mulders TMDs both sizeable and negative. To avoid soft factors, multiplicative renormalization constants, constructed appropriate ratios of Fouriertransformed TMDs ( shifts ). v taken off light cone: Dependence on Collins-Soper parameter ˆζ. In addition to ηv, need to also consider ˆζ. ηv seems under good control; plateaux reached at moderate values. ˆζ remains a challenge. No clear trends seen in the data sets available. Need much larger ˆζ. Presently investigating pion with this in mind. No significant volume dependence, pion mass dependence detected within the limited set of (three) cases considered