Lattice TMD observables at the physical pion mass Michael Engelhardt - PowerPoint PPT Presentation

Lattice TMD observables at the physical pion mass Michael Engelhardt New Mexico State University In collaboration with: B. Musch, P. H¨ agler, J. Negele, A. Sch¨ afer J. R. Green, N. Hasan, S. Krieg, S. Meinel, A. Pochinsky, S. Syritsyn T. Bhattacharya, R. Gupta, B. Yoon S. Liuti, A. Rajan

Fundamental TMD correlator unsubtr. ( b, P, S, . . . ) ≡ 1 Φ [Γ] � 2 � P, S | ¯ q (0) Γ U [0 , . . . , b ] q ( b ) | P, S � � � Φ [Γ] d 2 b T � � � � d ( b · P ) unsubtr. ( b, P, S, . . . ) � � Φ [Γ] ( x, k T , P, S, . . . ) ≡ � (2 π ) P + exp ( ix ( b · P ) − ib T · k T ) � � (2 π ) 2 S ( b 2 , . . . ) � � � � � � b + =0 � • “Soft factor” S required to subtract divergences of Wilson line U � • S is typically a combination of vacuum expectation values of Wilson line structures • Here, will consider only ratios in which soft factors cancel

Gauge link structure motivated by SIDIS � �� � � � �� � � �� � Beyond tree level: Rapidity divergences suggest taking staple direction slightly off the light cone. Approach of Aybat, Collins, Qiu, Rogers makes v space-like. Parametrize in terms of Collins-Soper parameter ζ ≡ P · v ˆ | P || v | Light-like staple for ˆ ζ → ∞ . Perturbative evolution equations for large ˆ ζ . “Modified universality”, f T-odd, SIDIS = − f T-odd, DY

Fundamental TMD correlator unsubtr. ( b, P, S, . . . ) ≡ 1 Φ [Γ] � 2 � P, S | ¯ q (0) Γ U [0 , ηv, ηv + b, b ] q ( b ) | P, S � � � Φ [Γ] d 2 b T � � � � d ( b · P ) unsubtr. ( b, P, S, . . . ) � � Φ [Γ] ( x, k T , P, S, . . . ) ≡ � (2 π ) P + exp ( ix ( b · P ) − ib T · k T ) � � (2 π ) 2 S ( b 2 , . . . ) � � � � � � b + =0 � • “Soft factor” S required to subtract divergences of Wilson line U � • S is typically a combination of vacuum expectation values of Wilson line structures • Here, will consider only ratios in which soft factors cancel

Decomposition of Φ into TMDs All leading twist structures:   ǫ ij k i S j Φ [ γ + ] = f 1 − f ⊥      odd   1 T  m H Φ [ γ + γ 5 ] = Λ g 1 + k T · S T g 1 T m H Φ [ iσ i + γ 5 ] = S i h 1 + (2 k i k j − k 2   T δ ij ) S j 1 T + Λ k i ǫ ij k j h ⊥ h ⊥ h ⊥   1 L +    odd   1 2 m 2  m H m H H

TMD Classification All leading twist structures: q → H U L T ↓ h ⊥ U f 1 ← − Boer-Mulders 1 (T-odd) h ⊥ L g 1 1 L f ⊥ h ⊥ T g 1 T h 1 1 T 1 T ↑ Sivers (T-odd)

� Decomposition of Φ into amplitudes ζ, µ ) ≡ 1 Φ [Γ] unsubtr. ( b, P, S, ˆ � 2 � P, S | ¯ q (0) Γ U [0 , ηv, ηv + b, b ] q ( b ) | P, S � Decompose in terms of invariant amplitudes; at leading twist, 1 Φ [ γ + ] � � � unsubtr. = A 2 B + im H ǫ ij b i S j A 12 B 2 P + 1 Φ [ γ + γ 5 ] � � � unsubtr. = − Λ A 6 B + i [( b · P )Λ − m H ( b T · S T )] A 7 B 2 P + 1 Φ [ iσ i + γ 5 ] � � � = im H ǫ ij b j A 4 B − S i A 9 B unsubtr. 2 P + � � − im H Λ b i A 10 B + m H [( b · P )Λ − m H ( b T · S T )] b i A 11 B (Decompositions analogous to work by Metz et al. in momentum space)

� Relation between Fourier-transformed TMDs and invariant amplitudes A i Invariant amplitudes directly give selected x -integrated TMDs in Fourier ( b T ) space (showing just the ones relevant for Sivers, Boer-Mulders shifts), up to soft factors: f [1](0) ˜ ( b 2 T , ˆ A 2 B ( − b 2 T , 0 , ˆ S ( b 2 , . . . ) � � ζ, . . . , ηv · P ) = 2 ζ, ηv · P ) / 1 f ⊥ [1](1) ˜ ( b 2 T , ˆ A 12 B ( − b 2 T , 0 , ˆ S ( b 2 , . . . ) � � ζ, . . . , ηv · P ) = − 2 ζ, ηv · P ) / 1 T h ⊥ [1](1) ( b 2 A 4 B ( − b 2 S ( b 2 , . . . ) ˜ T , ˆ T , 0 , ˆ � � ζ, . . . , ηv · P ) = 2 ζ, ηv · P ) / 1

Generalized shifts Form ratios in which soft factors, (Γ-independent) multiplicative renormalization factors cancel Boer-Mulders shift: � h ⊥ [1](1) dx d 2 k T k y Φ [ γ + + s j iσ j + γ 5 ] ( x, k T , P, . . . ) � ˜ � � � � 1 � � � k y � UT ≡ m H = � � dx d 2 k T Φ [ γ + + s j iσ j + γ 5 ] ( x, k T , P, . . . ) f [1](0) � � � ˜ � � 1 � s T =(1 , 0) Average transverse momentum of quarks polarized in the orthogonal transverse (“ T ”) direction in an unpolarized (“ U ”) hadron; normalized to the number of valence quarks. “Dipole moment” in b 2 T = 0 limit, “shift”. Issue: k T -moments in this ratio singular; generalize to ratio of Fourier-transformed TMDs at nonzero b 2 T , h ⊥ [1](1) ˜ ( b 2 T , . . . ) 1 � k y � UT ( b 2 T , . . . ) ≡ m H f [1](0) ˜ ( b 2 T , . . . ) 1 (remember singular b T → 0 limit corresponds to taking k T -moment). “Generalized shift”.

Generalized shifts from amplitudes Now, can also express this in terms of invariant amplitudes: h ⊥ [1](1) ˜ T , 0 , ˆ ( b 2 A 4 B ( − b 2 � T , . . . ) ζ, ηv · P ) 1 � k y � UT ( b 2 T , . . . ) ≡ m H = m H T , 0 , ˆ A 2 B ( − b 2 f [1](0) ˜ � ( b 2 ζ, ηv · P ) T , . . . ) 1 Analogously, Sivers shift (in a polarized hadron): T , 0 , ˆ A 12 B ( − b 2 � ζ, ηv · P ) � k y � TU ( b 2 T , . . . ) = − m H T , 0 , ˆ A 2 B ( − b 2 � ζ, ηv · P ) Worm-gear ( g 1 T ) shift: A 7 B ( − b 2 T , 0 , ˆ � ζ, ηv · P ) � k x � TL ( b 2 T , . . . ) = − m N T , 0 , ˆ A 2 B ( − b 2 � ζ, ηv · P ) Generalized tensor charge (no k -weighting) : h [1](0) ˜ A 9 B ( − b 2 T , 0 , ˆ ζ, ηv · P ) − ( m 2 N b 2 / 2) A 11 B ( − b 2 T , 0 , ˆ � � ζ, ηv · P ) 1 = − T , 0 , ˆ A 2 B ( − b 2 f [1](0) � ˜ ζ, ηv · P ) 1

Lattice setup Φ [Γ] unsubtr. ( b, P, S, ˆ � • Evaluate directly ζ, µ ) ������� ������� ≡ 1 2 � P, S | ¯ q (0) Γ U [0 , ηv, ηv + b, b ] q ( b ) | P, S � ���� ������ ���������������� ���������������� • Euclidean time: Place entire operator at one time � � slice, i.e., b , ηv purely spatial � � • Since generic b , v space-like, no obstacle to boost- � � ing system to such a frame! � � ��� � � • Parametrization of correlator in terms of A i in- variants permits direct translation of results back � to original frame; form desired A i ratios. � � • Use variety of P , b , ηv ; here b ⊥ P , b ⊥ v (lowest ��������� x -moment, kinematical choices/constraints) � ��� � ��� ���� τ • Extrapolate η → ∞ , ˆ ζ → ∞ numerically.

✞ Results: Sivers shift Dependence on staple extent; sequence of panels at different | b T | 0.6 0.6 Sivers � Shift, u � d � quarks Sivers - Shift, u - d - quarks [ 1 ] ( 0 ) ( GeV ) � 1 � � 0 � � GeV � 0.4 0.4 0.2 0.2 1 1 ˜ � � � 1 � � 1 � � f 0.0 [ 1 ] ( 1 ) / f 0.0 ✄ = 0.24, � � 0.32, Ζ ✂ - 0.2 � 0.2 | b T | = 0.11 fm , 1T 1T � b T � � 0.11 fm, ˜ � m N f m N f ☎ = 139 MeV m m Π � 317 MeV - 0.4 � 0.4 ✁ DY SIDIS � DY SIDIS � � - 0.6 � 0.6 �� � - ✆ - 10 - 5 0 5 10 � 10 � 5 0 5 10 ✆ Η � v � � lattice units � ✝ | v | ( lattice units )

✞ Results: Sivers shift Dependence on staple extent; sequence of panels at different | b T | 0.6 0.6 Sivers � Shift, u � d � quarks Sivers - Shift, u - d - quarks [ 1 ] ( 0 ) ( GeV ) � 1 � � 0 � � GeV � 0.4 0.4 0.2 0.2 1 1 ˜ � � � 1 � � 1 � � f 0.0 [ 1 ] ( 1 ) / f 0.0 ✄ = 0.24, � � 0.32, Ζ ✂ - 0.2 � 0.2 | b T | = 0.23 fm , 1T 1T � b T � � 0.23 fm, ˜ � m N f m N f ☎ = 139 MeV m m Π � 317 MeV - 0.4 � 0.4 ✁ DY SIDIS � DY SIDIS � � - 0.6 � 0.6 �� � - ✆ - 10 - 5 0 5 10 � 10 � 5 0 5 10 ✆ Η � v � � lattice units � ✝ | v | ( lattice units )

✞ Results: Sivers shift Dependence on staple extent; sequence of panels at different | b T | 0.6 0.6 Sivers � Shift, u � d � quarks Sivers - Shift, u - d - quarks [ 1 ] ( 0 ) ( GeV ) � 1 � � 0 � � GeV � 0.4 0.4 0.2 0.2 1 1 ˜ � � � 1 � � 1 � � f 0.0 [ 1 ] ( 1 ) / f 0.0 ✄ = 0.24, � � 0.32, Ζ ✂ - 0.2 � 0.2 | b T | = 0.34 fm , 1T 1T � b T � � 0.34 fm, ˜ � m N f m N f ☎ = 139 MeV m m Π � 317 MeV - 0.4 � 0.4 ✁ DY SIDIS � DY SIDIS � � - 0.6 � 0.6 �� � - ✆ - 10 - 5 0 5 10 � 10 � 5 0 5 10 ✆ Η � v � � lattice units � ✝ | v | ( lattice units )

✞ Results: Sivers shift Dependence on staple extent; sequence of panels at different | b T | 0.6 0.6 Sivers � Shift, u � d � quarks Sivers - Shift, u - d - quarks [ 1 ] ( 0 ) ( GeV ) � 1 � � 0 � � GeV � 0.4 0.4 0.2 0.2 1 1 ˜ � � � 1 � � 1 � � f 0.0 [ 1 ] ( 1 ) / f 0.0 ✄ = 0.24, � � 0.32, Ζ ✂ - 0.2 � 0.2 | b T | = 0.46 fm , 1T 1T � b T � � 0.46 fm, ˜ � m N f m N f ☎ = 139 MeV m m Π � 317 MeV - 0.4 � 0.4 ✁ DY SIDIS � DY SIDIS � � - 0.6 � 0.6 �� � - ✆ - 10 - 5 0 5 10 � 10 � 5 0 5 10 ✆ Η � v � � lattice units � ✝ | v | ( lattice units )