(Bessel-)weighted asymmetries

Transcription

1 QCD Evolution Workshop, Jefferson Lab (Bessel-)weighted asymmetries Bernhard Musch (Jefferson Lab) presenting work in collaboration with Daniël Boer (University of Groningen), Leonard Gamberg (Penn State University-Berks), Alexei Prokudin (JLab), Philipp Hägler (Johannes Gutenberg-Universität Mainz), John Negele (MIT), Andreas Schäfer (Univ. Regensburg)

2 Part I (Bessel-) Weighted Asymmetries in Semi-Inclusive Deep Inelastic Scattering (SIDIS)

3 Semi-inclusive Deep Inelastic Scattering (SIDIS) 3 }{{} l + N(P ) }{{} lepton nucleon S? Á S l + h(p h ) }{{} hadron P h? + X }{{} rest P h Á h () P l' P l q P x B = q2 2P q x x B : longitudinal momentum fraction of the struck quark in the nucleon z h = P P h P q z z h : longitudinal momentum fraction deposited in the measured hadron

4 Semi-inclusive Deep Inelastic Scattering 4 lepton hadron P h photon quark p gluons TMD FFs fragmentation functions P nucleon TMDs transverse momentum dependent parton distribution functions P

5 SIDIS cross section, see, e.g., [Bacchetta et al. JHEP (2007)] 5 dσ { dx B dz h dφ S dφ h dph 2 dy α2 x B Q 2 F UU,T + S sin(φ h φ S )F sin(φ h φ S ) UT,T } + 6 further sturctures F sin(φ h φ S ) UT,T = x B H sin(φ h φ S ) UT,T (Q 2 ) e 2 a a d 2 p T d 2 K T d 2 l T δ (2)( ) zp T + K T + l T P h p T cos(φ h φ p ) M f a T (x, p 2 T ) }{{} TMD S(l 2 T ) }{{} softf. D a (z, K 2 T ) }{{} FF 3 non-perturbative ingredients: TMD: momentum distribution of quarks in nucleon TMD FF: fragmentation function soft factor S (soft gluons), (absorbed into TMD/FF in [Aybat, Rogers (20)][Collins tbp])

6 What s a weighted asymmetry? 6 weighted asymmetry for general weights w 0, (φ h, P h ) d P A w h dφ h dφ S w (φ h, P h ) dσ = 2 d P h dφ h dφ S w 0 (φ h, P h ) dσ polarized, dσ dσ unpolarized, dσ + dσ traditional weighed asymmetry: w 0 =, w sin(nφ h +...) P h n e.g., A P h zm sin(φ h φ s) UT = 2 where f (n) (x) d 2 p T a e2 a f ()a T (x) D a(0) (z) (z), a e2 a f a(0) (x) D a(0) ( ) p 2 n T 2M 2 f(x, p 2 T ) [Kotzinian, Mulders PLB (997)] [Boer, Mulders PRD (998)] generalized to Bessel weights: w 0, sin(nφ h +...) J n ( P h B T ) 2J ( P h B T ) zmb sin(φ h φ A T s) a UT = 2 e2 a now f, f () T a e2 a f ()a T (x, z 2 BT 2 (0)a f (x, z 2 BT 2 ) D (0)a (z, B 2 T ) ) D (0)a (z, B 2 T ), and D are Fourier-transforms of TMDs/FFs (see later).

7 Why weighted asymmetries? 7 advantages deconvolution : simple products instead of convolutions of TMDs and FFs soft factor S cancels Problem: The p T -moments f (0), f (),... are ill-defined. T example: f (x, p 2 T ) /p2 T for large p2 [Bacchetta et al. JHEP T (2008)]. f (0) (x) d 2 p T f(x, p 2 T ) undefined without regularization Why Bessel-weights? natural generalization naturally more sensitive to low P h circumvent the problem of ill-defined p T -moments

8 warm-up: Fourier decomposition in polar coord. 8 d 2 b T (2π) 2 e ip T b T f(b T ) d bt 2π = 2π b T = n= 0 dφ b 2π e i p T b T cos(φ p φ b ) n= e inφp d bt 2π b T ( i) n J n ( p T b T ) f n ( b T ) e inφ b f n ( b T ) J n (x) : Bessel function J n p T b T J 0 J 2Π p T

9 Fourier-transformed TMDs (and FFs) 9 definition f(x, b 2 T ) d 2 p T e ib T p T f(x, p 2 T ) = 2π d p T p T J 0 ( b T p T ) f(x, p 2 T ) 0 ( f (n) (x, b 2 T ) n! 2 ) n M 2 b 2 T f(x, b 2 T ) = 2πn! ( ) n pt (M 2 ) n d p T p T J n ( b T p b T T ) f(x, p 2 T ) connection to p T -moments f (n) (x, 0) = d 2 p T ( ) p 2 n T 2M 2 f(x, p 2 T ) f (n) (x) At b T = 0, the Fourier-transformed TMDs f (n) (x, b 2 T ) are equivalent to p T -moments of TMDs, and can be UV divergent [Ji,Ma,Yuan PRD (2005)].

10 Rewriting the SIDIS cross section in Fourier space 0 dσ dx B dy dψ dz h dφ h dph 2 where α2 x B Q 2 { d bt (2π) b 2 T S(b T ) + J 0 ( b T P h ) H UU,T (Q 2 (0) ) P[ f D (0) ] + S T sin(φ h φ S ) J ( b T P h ) H sin(φ h φ S ) UT,T (Q 2 () ) P[ f T D (0) ] + ɛ cos(2φ h ) J 2 ( b T P h ) H cos(2φ h) UU (Q 2 ) P[ h () H () ] further structures... + Ỹ }{{} assume small P[ f (n) D(m) ] x B (zm b T ) n (zm h b T ) m a e 2 a f a(n) (x, z 2 b 2 T ) D a(m) (z, b 2 T ). }, similar as in [Idilbi,Ji,Ma,Yuan PRD (2004)]

11 Bessel-weighted asymmetries d P A w h dφ h dφ S w (φ h, P h ) dσ = 2 d P h dφ h dφ S w 0 (φ h, P h ) dσ Choose weights that project onto the desired Fourier-mode, e.g., w 0 = J 0 ( P h B T ), w = J ( P h B T ) zmb T 2 sin(φ h φ S ) use d P h P h J n ( P h b T ) J n ( P h B T ) = δ( b T B T )/B T 2J ( P h B T ) zmb sin(φ h φ A T s) UT = 2 S(B T 2 ) Hsin(φ h φ S ) UT,T (Q 2 ) a e2 a S(B T 2 ) H UU,T (Q 2 ) a e2 a ()a f T (x, z 2 BT 2 (0)a ) D (z, B 2 T ) (0)a f (x, z 2 BT 2 (0)a ) D (z, BT 2 ) accessible window [BT min, Bmax T ] [ P h max, P h resolution ] traditional weighted asymmeties at B T = 0 UV divergences.

12 Bessel-weighted asymmetries 2 d P A w h dφ h dφ S w (φ h, P h ) dσ = 2 d P h dφ h dφ S w 0 (φ h, P h ) dσ Choose weights that project onto the desired Fourier-mode, e.g., w 0 = J 0 ( P h B T ), w = J ( P h B T ) zmb T 2 sin(φ h φ S ) w 0 B T 0 w B T 0 P h /(zm) sin(φ h φ S ) use d P h P h J n ( P h b T ) J n ( P h B T ) = δ( b T B T )/B T 2J ( P h B T ) zmb sin(φ h φ A T s) UT = 2 S(B T 2 ) Hsin(φ h φ S ) UT,T (Q 2 ) a e2 a S(B T 2 ) H UU,T (Q 2 ) a e2 a ()a f T (x, z 2 BT 2 (0)a ) D (z, B 2 T ) (0)a f (x, z 2 BT 2 (0)a ) D (z, BT 2 ) accessible window [BT min, Bmax T ] [ P h max, P h resolution ] traditional weighted asymmeties at B T = 0 UV divergences.

13 Part II Fourier-transformed TMDs at the level of matrix elements

14 The TMD correlator 4 lightcone coordinates w ± = 2 (w 0 ±w 3 ), so w = w +ˆn + + w ˆn + w T ; P + large, P T = 0 Several factorization frameworks have been proposed. Here we choose [Ji,Ma,Yuan PRD (2005)] as an example: Φ [Γ] 2 Φ [Γ] unsubtr. (b, P, S, v, µ) {}}{ d 4 b P, S q(0) Γ U[0, v, v + b, b] q(b) P, S (2π) 4 0 U[0, v, v+b T, b T, b T ṽ, ṽ, 0] 0 } {{ } S(b 2 T, ρ, µ) parametrization in terms of TMDs, example Γ = γ + dp Φ [γ+ ] = f (x, p 2 p + =xp + T ; ˆζ, ρ, µ) ɛ ijp i S j ft (x, p 2 T ; m ˆζ, ρ, µ) N [Ralston, Soper NPB 979], [Mulders, Tangerman NPB 996], [Goeke, Metz, Schlegel PLB 2005] where ˆζ2 (P v)2 (v ṽ)2 P 2 v 2, ρ ˆ= v 2 ṽ 2

15 gauge links 5 U[a, b, c,...] P exp ( ig b a dξµ A µ (ξ) ig ) c b dξµ A µ (ξ) +...? transverse link can be omitted in covariant gauge v? v + b + b T 0 v 0 v }{{}}{{} Φ [Γ] unsubtr. (b, P, S, v, µ) S(b 2 T, ρ, µ) ˆζ 2 (P v)2 (v ṽ)2 P 2 v 2, ρ ˆ= v 2 ṽ 2. v, ṽ lightlike for ˆζ, ρ ζ = 4M 2 ˆζ 2 : Collins-Soper evolution param. [CS NPB (98)] evolution eqns. for large ˆζ, ρ [Idilbi,Ji,Ma,Yuan PRD (2004)] v~ v

16 decomposition: Lorentz-invariant amplitudes 6 as in [Goeke,Metz,Schlegel PLB (2005)] but in Fourier-space 2 [γ Φ µ ] unsubtr. (b, P, S, v, µ) = P, S q(0) γµ U[0, v, v + b, b] q(b) P, S = P µ Ã 2 im 2 b µ Ã 3 imɛ µναβ P ν b α S β Ã 2 + M 2 (v P ) vµ B + M v P ɛµναβ P ν v α S β B7 im 3 v P ɛµναβ b ν v α S β B8 M 3 v P (b S)ɛµναβ P ν b α v β B9 im 3 (v P ) 2 (v S)ɛµναβ P ν b α v β B0 in total 32 amplitudes Ã(±) i (b 2 (±), b P, b v/(v P ), ˆζ), B i (...), denoted (+) for v P > 0 (SIDIS), ( ) for v P < 0 (Drell-Yan) time-reversal even : Ã (+) i = Ã( ) i time-reversal odd : Ã (+) i = Ã( ) i

17 decomposition: Lorentz-invariant amplitudes 7 as in [Goeke,Metz,Schlegel PLB (2005)] but in Fourier-space 2 [γ Φ + ] unsubtr. (b, P, S, v, µ) = P, S q(0) γ+ U[0, v, v + b, b] q(b) P, S + imp + ɛ ij b i S j (Ã2 R(ˆζ) B ) 8 where ( = P + Ã 2 + R(ˆζ) B ) } {{ } Ã 2B } {{ } Ã 2B R(ˆζ) + ˆζ 2, note that lim R(ˆζ) = 0 ˆζ f (x, b 2 T ; ζ, ρ, µ) ( 2 d(b P ) = e ix(b P ) Ã S( b 2 2B b 2 T, µ 2 T, b P,, ρ) (2π) ) (b P )R(ˆζ) M 2, ˆζ, µ f () T (x, b2 T ; ζ, ρ, µ) = 2 S( b 2 T, µ 2, ρ) d(b P ) (2π) e ix(b P ) Ã 2B ( b 2 T, b P, ) (b P )R(ˆζ) M 2, ˆζ, µ

18 average transverse momentum shift (here: Sivers) 8 unpolarized quark density in a transversely polarized nucleon ρ T U (x, p T, S T ) = f (x, p 2 T ) ɛ ijp i S j M f T (x, p 2 T ) = dp Φ [γ+ ] p y T U p y dx d 2 p T p y ρ T U (x, p T, S T = (, 0)) dx d2 p T ρ T U (x, p T, S T = (, 0)) () dx f T = M (x) (0) dx f (x) p y T U := average quark momentum in transverse y-direction measured in a proton polarized in transverse x-direction. u u d S? dipole moment, shift attention divergences from high-p T -tails! generalized average transverse momentum shift () dx f T p y T U (B T ) M (x, B2 T ) (0) dx f (x, B2 T ) = S( B T 2,...) Ã2B( BT 2, 0, 0, ˆζ, µ) S( B T 2,...) Ã2B( BT 2, 0, 0, ˆζ, µ)

19 lattice calculations (more at DIS 20 ) 9 v b p y T U (B T ; ζ) M Ã lat dx f () T (x, B2 T ) dx f (0) (x, B2 T ) 2 ( BT 2 = lim, 0, 0, ˆζ, µ, η) R(ˆζ) η Ã lat 2 ( B2 T, 0, 0, ˆζ, µ, η) + R(ˆζ) lattice limitations link spacelike, finite length look for plateau at large η ˆζ max = P lat M B T a (at least a few lattice spacings a) + typical lattice limitations B lat 8 ( BT 2, 0, 0, ˆζ, µ, η) B lat ( B2 T, 0, 0, ˆζ, µ, η) preliminary lattice results MILC lattices (staggered) LHPC propagators (domain wall) pion mass m π 500 MeV box 20 3, spacing a 0.2 fm

20 preliminary lattice results: Sivers shift 20 GeV f 0 m N f T Sivers Shift DY preliminary down quarks, Ζ 0.39, B T 0.36 fm SIDIS Η () dx f T M (x, B2 T ) (0) dx f (x, B2 T ) = Ã 2 R(ˆζ) B 8 Ã 2 + R(ˆζ) B

21 preliminary lattice results: Boer-Mulders shift 2 GeV Boer Mulders Shift 0.4 total contrib. from A f 0 m N h down quarks, Ζ 0.39, B T 0.36 fm DY preliminary SIDIS Η () dx h (x, BT 2 M ) (0) dx f (x, B2 T ) = Ã 4 R(ˆζ) B 3 Ã 2 + R(ˆζ) B R(ˆζ) ˆζ 0 expect numerator dominated by Ã4 close to lighcone. (We are still far from that region.)

22 preliminary lattice results: Boer-Mulders shift 22 GeV Boer Mulders Shift 0.4 total contrib. from A f 0 m N h down quarks, Ζ 0.78, B T 0.36 fm DY preliminary SIDIS Η () dx h (x, BT 2 M ) (0) dx f (x, B2 T ) = Ã 4 R(ˆζ) B 3 Ã 2 + R(ˆζ) B R(ˆζ) ˆζ 0 expect numerator dominated by Ã4 close to lighcone. (We are still far from that region.)

23 preliminary lattice results: Boer-Mulders shift Boer Mulders Shift SIDIS GeV f 0 m N h preliminary total contrib. from A 4 down quarks, B T 0.36 fm () dx h (x, BT 2 M ) (0) dx f (x, B2 T ) = Ã 4 R(ˆζ) B 3 Ã 2 + R(ˆζ) B R(ˆζ) ˆζ 0 expect numerator dominated by Ã4 close to lighcone. Ζ

24 preliminary lattice results: Boer-Mulders shift Boer Mulders Shift SIDIS GeV f 0 m N h ? preliminary total contrib. from A 4 down quarks, B T 0.36 fm () dx h (x, BT 2 M ) (0) dx f (x, B2 T ) = Ã 4 R(ˆζ) B 3 Ã 2 + R(ˆζ) B R(ˆζ) ˆζ 0 expect numerator dominated by Ã4 close to lighcone. Ζ

25 Conclusions 25 Soft factors cancel in weighted asymmetries. generalization to Bessel-weights avoid divergences from high p T -tails similar advantages found in ratios of Fourier-transformed TMDs first lattice calculations for the Sivers- and the Boer-Mulders shift albeit at a relatively low Collins-Soper parameter ζ towards observables with minimal model-dependence. Thanks to the MILC and LHP lattice collaborations for providing gauge configurations and propagators.