Understanding the large pT spectrum in SIDIS Nobuo Sato In collaboration with: ODU/JLab Gonzalez-Hernandez, Rogers, NS, Wang - PRD98 2018 27th Workshop on Deep-Inelastic Scattering - arXiv:1903.01529 2019 and Related Subjects DIS 2019 Turin, Italy 1 / 20
SIDIS regions Breit frame identified hadron p µ outgoing lepton l ′ µ h p ⊥ p ⊥ h h incoming proton P µ q = l − l ′ exchanged photon Current fragmentation incoming lepton l µ Collinear factorization Current fragmentation Soft region Target region TMD factorization ???? Fracture functions y h 2 / 20
SIDIS regions small transverse detected momentum hadron outgoing p ⊥ quark h ⊗ incoming quark Current fragmentation Collinear factorization aka W Current fragmentation Soft region Target region TMD factorization ???? Fracture functions y h 3 / 20
SIDIS regions large transverse outgoing detected momentum quark hadron p ⊥ ⊗ h incoming quark Current fragmentation Collinear factorization aka FO (=fixed order) Current fragmentation Soft region Target region TMD factorization ???? Fracture functions y h 4 / 20
SIDIS regions W FO small transverse large transverse momentum momentum p ⊥ p ⊥ h h Current fragmentation Current fragmentation Collinear factorization Collinear factorization Current fragmentation Soft region Target region Current fragmentation Soft region Target region TMD factorization ???? Fracture functions TMD factorization ???? Fracture functions y h y h outgoing detected detected quark hadron hadron outgoing ⊗ quark ⊗ incoming quark incoming quark 5 / 20
SIDIS regions W FO small transverse large transverse momentum momentum p ⊥ p ⊥ h h Current fragmentation Current fragmentation Collinear factorization Collinear factorization matching region Current fragmentation Soft region Target region Current fragmentation Soft region Target region TMD factorization ???? Fracture functions TMD factorization ???? Fracture functions aka ASY (=asymptotic) y h y h outgoing detected detected quark hadron hadron outgoing ⊗ quark ⊗ incoming quark incoming quark 5 / 20
SIDIS regions dσ = W + FO − ASY + O ( m 2 /Q 2 ) dxdQ 2 dzdp ⊥ h ∼ W for q T ≪ Q ∼ FO for q T ∼ Q q T /Q = ( p ⊥ h /z ) /Q → scale separation 6 / 20
Toy example 10 0 FO | AY | Γ( x, z, Q, q T ) Y 10 − 1 | W | W + Y Q = 2 . 0 (GeV) 10 − 2 10 − 3 10 − 4 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 q T (GeV) 7 / 20
Existing phenomenology Anselmino et al Bacchetta et al These analyses used only W (Gaussian, CSS) → no FO nor ASY Samples with q T /Q ∼ 1 . 63 have been included BUT TMDs are only valid for q T /Q ≪ 1 ! 8 / 20
FO @ LO predictions (DSS07) Gonzalez, Rogers, NS, Wang PRD98 (2018) COMPASS 17 h + 10 8 6 20 . 0 data / theory(LO) vs . q T (GeV) 4 2 10 PDF : CJ15 FF : DSS07 8 6 8 . 3 4 p ⊥ 2 Q 2 (GeV 2 ) h q T > Q 10 2 4 6 8 ? 6 3 . 5 Current fragmentation 4 Collinear factorization 2 10 2 4 6 8 Current fragmentation Soft region Target region TMD factorization ???? Fracture functions 6 1 . 8 y h 4 2 < z > = 0 . 24 10 2 4 6 8 < z > = 0 . 34 6 1 . 3 4 < z > = 0 . 48 2 < z > = 0 . 68 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 0 . 007 0 . 010 0 . 016 0 . 03 0 . 04 0 . 07 0 . 15 0 . 27 x bj 9 / 20
Trouble with large transverse momentum � 1 dξ � e 2 ξ − x H ( ξ ) f q ( ξ, µ ) d q ( ζ ( ξ ) , µ ) + O ( α 2 S ) + O ( m 2 /q 2 ) FO = q 2 q xz T 1 − z + x q Q 2 + FFs needs to be updated? 10 / 20
FO @ LO predictions (DSS07) Gonzalez, Rogers, NS, Wang PRD98 (2018) COMPASS 17 h + 10 8 6 20 . 0 data / theory(LO) vs . q T (GeV) 4 2 10 PDF : CJ15 FF : DSS07 8 6 8 . 3 4 p ⊥ 2 Q 2 (GeV 2 ) h q T > Q 10 2 4 6 8 ? 6 3 . 5 Current fragmentation 4 Collinear factorization 2 10 2 4 6 8 Current fragmentation Soft region Target region TMD factorization ???? Fracture functions 6 1 . 8 y h 4 2 < z > = 0 . 24 10 2 4 6 8 < z > = 0 . 34 6 1 . 3 4 < z > = 0 . 48 2 < z > = 0 . 68 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 0 . 007 0 . 010 0 . 016 0 . 03 0 . 04 0 . 07 0 . 15 0 . 27 x bj 11 / 20
FO @ LO predictions (JAM18) Gonzalez, Rogers, NS, Wang PRD98 (2018) COMPASS 17 h + 10 8 6 20 . 0 data / theory(LO) vs . q T GeV data / theory(NLO) vs . q T (GeV) 4 2 10 PDF : JAM18 FF : JAM18 8 6 8 . 3 4 p ⊥ 2 Q 2 (GeV 2 ) h q T > Q 10 2 4 6 8 ? 6 3 . 5 Current fragmentation 4 Collinear factorization 2 10 2 4 6 8 Current fragmentation Soft region Target region TMD factorization ???? Fracture functions 6 1 . 8 y h 4 2 < z > = 0 . 24 10 2 4 6 8 < z > = 0 . 34 6 1 . 3 4 < z > = 0 . 48 2 < z > = 0 . 68 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 0 . 007 0 . 010 0 . 016 0 . 03 0 . 04 0 . 07 0 . 15 0 . 27 x bj 12 / 20
Trouble with large transverse momentum � 1 dξ � e 2 ξ − x H ( ξ ) f q ( ξ, µ ) d q ( ζ ( ξ ) , µ ) + O ( α 2 S ) + O ( m 2 /q 2 ) FO = q 2 q xz T 1 − z + x q Q 2 + O ( α 2 S ) corrections might be important 13 / 20
ff ff fi order α 2 S corrections to FO d s p /dp T (pb/GeV) 2 £ Q 2 £ 4.5 GeV 2 10 2 There are strong indications KKP NLO that order α 2 KKP LO S corrections are 10 K NLO K LO very important 1 10 2 4.5 £ Q 2 £ 15 GeV 2 An order of magnitude 10 correction at small p T . 1 10 2 15 £ Q 2 £ 70 GeV 2 As a sanity check, we need to have an independent calculation 10 1 3 4 5 6 7 8 9 10 15 p T (GeV) Daleo,et al. (2005) PRD.71.034013 14 / 20
O ( α 2 S ) calculation (Wang, Gonzalez-Hernandes, Rogers, NS - arXiv:1903.01529) � 1+ � 1+ dξ dζ ζ 2 ˆ W µν W µν ( P, q, P H ) = ij ( q, x/ξ, z/ζ ) f i/P ( ξ ) d H/j ( ζ ) ξ x − z − 1 � | 2 } dΠ ( N ) − Subtractions ˆ µν ; P µν P P ˆ { P µν W ( N ) W ( N ) {| M 2 → N | 2 ; | M 2 → N µν } ≡ g g pp (2 π ) 4 Born/Virtual � Generate all 2 → 2 and 2 → 3 squared amplitudes � Evaluate 2 → 2 virtual graphs Real (Passarino-Veltman) � Integrate 3-body PS analytically � Check cancellation of IR poles 15 / 20
FO @ LO predictions (JAM18) COMPASS 17 h + 10 8 6 20 . 0 data / theory(LO) vs . q T GeV data / theory(NLO) vs . q T (GeV) 4 2 10 PDF : JAM18 FF : JAM18 8 6 8 . 3 4 p ⊥ 2 Q 2 (GeV 2 ) h q T > Q 10 2 4 6 8 ? 6 3 . 5 Current fragmentation 4 Collinear factorization 2 10 2 4 6 8 Current fragmentation Soft region Target region TMD factorization ???? Fracture functions 6 1 . 8 y h 4 2 < z > = 0 . 24 10 2 4 6 8 < z > = 0 . 34 6 1 . 3 4 < z > = 0 . 48 2 < z > = 0 . 68 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 0 . 007 0 . 010 0 . 016 0 . 03 0 . 04 0 . 07 0 . 15 0 . 27 x bj 16 / 20
FO @ NLO (JAM18) COMPASS 17 h + 10 8 6 20 . 0 data / theory(NLO) vs . q T (GeV) 4 2 10 PDF : JAM18 FF : JAM18 8 6 8 . 3 4 p ⊥ 2 Q 2 (GeV 2 ) h q T > Q 10 2 4 6 ? � 8 6 3 . 5 Current fragmentation 4 Collinear factorization 2 10 2 4 6 8 Current fragmentation Soft region Target region TMD factorization ???? Fracture functions 6 1 . 8 y h 4 2 < z > = 0 . 24 10 2 4 6 8 < z > = 0 . 34 6 1 . 3 4 < z > = 0 . 48 2 < z > = 0 . 68 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 0 . 007 0 . 010 0 . 016 0 . 03 0 . 04 0 . 07 0 . 15 0 . 27 x bj 17 / 20
Understanding the large x (Wang, Gonzalez-Hernandes, Rogers, NS - arXiv:1903.01529) /F LO 6 1 Large corrections threshold 4 F NLO corrections are observed 2 1 z = 0 . 2 q T = Q z = 0 . 8 q T = Q The x at the minimum can be Q = 2 GeV used as an indicator of where 6 Q = 20 GeV such corrections are expected to 4 be large 2 z = 0 . 2 q T = 2 Q z = 0 . 8 q T = 2 Q x 0 . 01 0 . 1 0 . 01 0 . 1 18 / 20
Understanding the large x (Wang, Gonzalez-Hernandes, Rogers, NS - arXiv:1903.01529) COMPASS kinematics 7 7 7 x > x 0 6 6 6 x ≤ x 0 5 5 5 < z > = 0 . 24 < z > = 0 . 48 < z > = 0 . 69 q T /Q 4 4 4 3 3 3 2 2 2 1 1 1 x x x 0 . 01 0 . 1 0 . 01 0 . 1 0 . 01 0 . 1 The blue region might receive large threshold corrections This can potential explain why the O ( α 2 S ) fail to describe the data at large x 19 / 20
Summary and outlook O ( α 2 S ) corrections are important to describe SIDIS at COMPASS The large x region receives large threshold corrections which can explain the difficulty to describe the data Potential impact of SIDIS large p T data on PDFs/FFs global analysis 20 / 20
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