Matching between TMD and Collinear Factorizations Nobuo Sato - PowerPoint PPT Presentation

Matching between TMD and Collinear Factorizations Nobuo Sato ODU/JLab POETIC 19 LBL 1 / 32

A historical note on DY p T spectrum 2 / 32

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pp √ s = 27 . 4 GeV y = 0 3 / 32

pp √ s = 27 . 4 GeV y = 0 3 / 32

Fact check... 4 / 32

PDFs from 1978 (PhysRevD.18.3378) 5 / 32

PDFs from 1978 (PhysRevD.18.3378) 5 / 32

PDFs from 1978 (PhysRevD.18.3378) 5 / 32

u d g 0 . 4 0 . 6 0 . 75 0 . 3 xf ( x ) 0 . 4 0 . 50 0 . 2 0 . 2 0 . 25 0 . 1 0 . 0 0 . 0 0 . 00 0 . 2 0 . 4 0 . 6 0 . 8 0 . 2 0 . 4 0 . 6 0 . 8 0 . 2 0 . 4 0 . 6 0 . 8 5 5 5 Ratio to CJ15nlo HS µ = 5 . 5 GeV 4 4 4 CJ15nlo 3 3 3 2 2 2 1 1 1 0 0 0 0 . 2 0 . 4 0 . 6 0 . 8 0 . 2 0 . 4 0 . 6 0 . 8 0 . 2 0 . 4 0 . 6 0 . 8 x x x 6 / 32

1978 2015 Q = 5 . 5 GeV Q = 5 . 5 GeV T (nb GeV − 3 ) Q = 7 . 5 GeV Q = 7 . 5 GeV 10 − 3 10 − 3 Q = 9 . 5 GeV Q = 9 . 5 GeV 10 − 4 10 − 4 dσ/dQ/dy/dp 2 10 − 5 10 − 5 10 − 6 10 − 6 HS PDFs CJ15nlo PDFs 0 1 2 3 4 5 0 1 2 3 4 5 q T (GeV) q T (GeV) Data from Phys.Rev.Lett. 40 (1978) 1117 7 / 32

1978 2015 Q = 5 . 5 GeV Q = 5 . 5 GeV Edσ/d 3 p (cm 2 GeV − 2 ) Q = 7 . 5 GeV Q = 7 . 5 GeV 10 − 36 10 − 36 Q = 9 . 5 GeV Q = 9 . 5 GeV 10 − 37 10 − 37 10 − 38 10 − 38 10 − 39 10 − 39 10 − 40 10 − 40 HS PDFs CJ15nlo PDFs 0 1 2 3 4 5 0 1 2 3 4 5 q T (GeV) q T (GeV) Data from Phys.Rev. D23 (1981) 604-633 8 / 32

SIDIS p ⊥ h Lab frame outgoing lepton l ′ µ Current fragmentation incoming lepton l µ Collinear factorization X target P µ Current fragmentation Soft region Target region TMD factorization ???? Fracture functions identified hadron p µ y h h Breit frame identified hadron p µ outgoing lepton l ′ µ Key question : How is p ⊥ h h generated at p ⊥ short distances? h incoming proton P µ q = l − l ′ Different regions are sensitive to distinct exchanged photon physical mechanisms incoming lepton l µ 9 / 32

Nucleon structures accessible in SIDIS 18 dσ F i Standard label β i F i ( x, z, Q 2 , P 2 � ∼ hT ) β i F 1 F UU,T 1 dx dy d Ψ dz dφ h dP 2 F 2 F UU,L ε hT i =1 √ 1 − ε 2 F 3 F LL S || λ e F sin( φ h + φ S ) | � F 4 S ⊥ | ε sin( φ h + φ S ) UT Name Symbol meaning F sin( φ h − φ S ) | � F 5 S ⊥ | sin( φ h − φ S ) UT,T f q upol. PDF U. pol. quarks in U. pol. nucleon F sin( φ h − φ S ) | � 1 F 6 S ⊥ | ε sin( φ h − φ S ) UT,L g q pol. PDF L. pol. quarks in L. pol. nucleon F cos 2 φ h F 7 ε cos(2 φ h ) 1 UU h q Transversity T. pol. quarks in T. pol. nucleon F sin(3 φ h − ψ S ) | � F 8 S ⊥ | ε sin(3 φ h − φ S ) 1 UT √ f ⊥ (1) q F cos( φ h − φ S ) | � 1 − ε 2 cos( φ h − φ S ) F 9 S ⊥ | λ e Sivers U. pol. quarks in T. pol. nucleon LT 1 T F sin 2 φ h F 10 S || ε sin(2 φ h ) h ⊥ (1) q UL Boer-Mulders T. pol. quarks in U. pol. nucleon F cos φ S 1 | � F 11 S ⊥ | λ e � 2 ε (1 − ε ) cos φ S LT h ⊥ (1) q Boer-Mulders T. pol. quarks in U. pol. nucleon F cos φ h � F 12 S || λ e 2 ε (1 − ε ) cos φ h 1 LL . . . F cos(2 φ h − φ S ) | � . . . F 13 S ⊥ | λ e � 2 ε (1 − ε ) cos(2 φ h − φ S ) . . . LT F sin φ h � F 14 S || 2 ε (1 + ε ) sin φ h D q UL FF U. pol. quarks to U. pol. hadron 1 F sin φ h � F 15 λ e 2 ε (1 − ε ) sin φ h LU H ⊥ (1) q Collins T. pol. quarks to U. pol. hadron F cos φ h F 16 � 2 ε (1 + ε ) cos φ h 1 UU . . . F sin φ S | � � . . . F 17 S ⊥ | 2 ε (1 + ε ) sin φ S UT . . . F sin(2 φ h − φ S ) | � � F 18 S ⊥ | 2 ε (1 + ε ) sin(2 φ h − φ S ) UT 10 / 32

Regions in SIDIS 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 11 / 32

Regions in SIDIS 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 12 / 32

Regions in SIDIS small transverse W large transverse FO 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 13 / 32

Regions in SIDIS small transverse W large transverse FO 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 13 / 32

What is large or small transverse momentum? Scale separation p ⊥ h q T /Q Current fragmentation z = P · p h Collinear factorization q T = p ⊥ P · q , h /z Current fragmentation Soft region Target region Merging factorization theorems TMD factorization ???? Fracture functions y h dσ = W + FO − ASY + O ( m 2 /Q 2 ) dxdQ 2 dzdp ⊥ h ∼ W for q T ≪ Q ∼ FO for q T ∼ Q 14 / 32

Small transverse momentum Collins, Rogers PRD91 (2015) � d 2 b T � (2 π ) 2 e − i q T · b T F f/N ( x, b T , µ, ζ F ) D h/f ( z, b T , µ, ζ D ) + O ( q 2 T /Q 2 ) W = H f ( Q, µ ) f 15 / 32

Small transverse momentum Collins, Rogers PRD91 (2015) � d 2 b T � (2 π ) 2 e − i q T · b T F f/N ( x, b T , µ, ζ F ) D h/f ( z, b T , µ, ζ D ) + O ( q 2 T /Q 2 ) W = H f ( Q, µ ) f CSS evolution equation ∂ ln F f/N ( x, b T , ζ F , µ ) = ˜ ∂ ln √ ζ F K ( b T , µ ) + Related to vacuum matrix elements of products of Wilson Lines + Independent of flavor, target and spin + Independent of x + Universal across TMDs and processes 15 / 32

Small transverse momentum Collins, Rogers PRD91 (2015) � d 2 b T � (2 π ) 2 e − i q T · b T F f/N ( x, b T , µ, ζ F ) D h/f ( z, b T , µ, ζ D ) + O ( q 2 T /Q 2 ) W = H f ( Q, µ ) f CSS evolution equation RG equations d ˜ ∂ ln F f/N ( x, b T , ζ F , µ ) K ( b T , µ ) = ˜ = − γ K ( α S ( µ )) ∂ ln √ ζ F K ( b T , µ ) d ln µ d ln F f/N ( b T , µ ) = γ f ( α S ( µ ) , 1) − 1 2 γ K ( α S ( µ )) ln ζ F µ 2 d ln µ + Related to vacuum matrix elements of d ln µ ln H ( Q, µ ) = − 2 γ f ( α S ( µ ) , 1) + γ K ( α S ( µ )) ln Q 2 d products of Wilson Lines µ 2 + Independent of flavor, target and spin + Independent of x + Universal across TMDs and processes 15 / 32

Small transverse momentum Collins, Rogers PRD91 (2015) � d 2 b T Valid for 0 ≤ q T ≪ Q � (2 π ) 2 e − i q T · b T W = H f ( Q, µ ) f � 1 d ˆ x x, µ b ∗ ) ˜ × e − g f/N ( x,b T ,b max ) x, b ∗ , µ 2 x f f/N (ˆ C f/p ( x/ ˆ b ∗ , α S ( µ b ∗ )) ˆ x � 1 d ˆ z × e − g h/f ( z,b T ,b max ) z, µ b ∗ ) ˜ z, b ∗ , µ 2 z 3 d h/f (ˆ C h/f ( z/ ˆ b ∗ , α S ( µ b ∗ )) ˆ z � ˜ � − g K ( b T ,b max ) � K ( b ∗ ,µ b ∗ ) � Q 2 Q 2 × Q 2 µ 2 0 b ∗ �� µ Q � �� dµ ′ 2 γ ( α S ( µ ′ ) , 1) − ln Q 2 ( µ ′ ) 2 γ K ( α S ( µ ′ )) × exp µ ′ µ b ∗ Quantities in pink are non-perturbative 16 / 32

Large transverse momentum Valid for q T ∼ Q � 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 � � q 2 Q 2 xz + Attention : T 1 − z + x < ξ < 1 + large q T probes large ξ in PDFs + Can be useful in collinear global fits 17 / 32

Matching region Valid for 0 ≪ q T ≪ Q � 1 � 1 dξ dζ ASY ∼ d ( z ; µ ) ξ f ( ξ ; µ ) P ( x/ξ ) + f ( x ; µ ) ζ d ( ζ ; µ ) P ( z/ζ ) x z � � Q 2 � � − 3 + 2 C F f ( x ; µ ) d ( z ; µ ) ln q T 2 + Interpolates between W and FO + FO − ASY ≡ Y 18 / 32

Toy example: how we expect the W+FO-ASY to work 10 0 FO | AY | Y 10 − 1 dσ | W | dxdQ 2 dzdp ⊥ W + Y h 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) 19 / 32

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 ! 20 / 32

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 21 / 32