Threshold resummation at next-to-leading power Robert Szafron - PowerPoint PPT Presentation

Threshold resummation at next-to-leading power Robert Szafron Technische Universit¨ at M¨ unchen RADCOR 2019 9-13 September Avignon Robert Szafron 1/21

Outline ◮ Hard function ◮ Kinematic corrections ◮ Soft function ◮ Fixed order expansion ◮ Higgs threshold production � � ∞ dσ α n dz = c n δ (1 − z ) s n =0    � ln m (1 − z ) � 2 n − 1 �  c nm + d nm ln m (1 − z )  + . . .  + 1 − z + m =0 ◮ Leading power ◮ Next-to-leading power → α s ln(1 − z ) + α 2 s ln 3 (1 − z ) + . . . ◮ Leading Log: m = 2 n − 1 − Robert Szafron 1/21

The Drell-Yan process - the leading power threshold factorization A ( p A ) B ( p B ) → γ ∗ ( Q ) + X � 1 � dQ 2 = 4 πα 2 dσ DY em σ LP dx a dx b f a/A ( x a ) f b/B ( x b ) ˆ ab ( z ) 3 N c Q 4 0 a,b [G. P. Korchemsky, G. Marchesini, 1993] [T. Becher, M. Neubert, G. Xu, 0710.0680; S. Moch, A. Vogt, hep-ph/0508265] Factorization of partonic cross-section σ LP ( z ) = | C ( Q 2 ) | 2 Q S DY ( Q (1 − z )) ˆ z = Q 2 / ˆ s partonic threshold z → 1 Leading Power Soft function � dx 0 4 π e ix 0 Ω / 2 1 N c Tr � 0 | ¯ T ( Y † + ( x 0 ) Y − ( x 0 )) T ( Y † S DY (Ω) = − (0) Y + (0)) | 0 � Robert Szafron 2/21

Next-to-leading power SCET offers systematic and intuitive approach to power expansion. Sources of power suppression: ◮ Subleading operators → see talk by Martin [M. Beneke, M. Garny, R. S. and J. Wang, 1712.04416; M. Beneke, M. Garny, R. S. and J. Wang, 1808.04742 ] ◮ Time-ordered products of subleading Lagrangian (see also [M. Beneke, M. Garny, R. S. and J. Wang, 1907.05463 ] ); which are factorized into ◮ collinear functions → see talk by Sebastian ◮ generalized soft function – this talk ◮ Phase space expansion – kinematic corrections –this talk We work in position space SCET [M. Beneke and T. Feldmann, hep-ph/0211358] � � / n i + 1 L (0) = ¯ in i − D + i / in i + D i / ξ i D ⊥ i D ⊥ i 2 ξ i i � � � L (0) + L (1) L = + . . . i i i For example L (1) iξq = q s ( x i − ) W † i ( x ) i / D ⊥ i ξ i ( x ) + h.c. Robert Szafron 3/21

DY cross-section at NLP [M. Beneke, A. Broggio, M. Garny, S. Ja´ skiewicz, R. S., L. Vernazza, J. Wang, 1809.10631] A ( p A ) B ( p B ) → γ ∗ ( Q ) + X z = Q 2 threshold z → 1 ˆ s Ω ∼ Q (1 − z ) ≪ Q Factorization theorem valid at LL accuracy � � d 3 � q 1 s ) × Q 2 d 4 x e i ( x a p A + x b p B − q ) · x ˆ σ ( z ) = H (ˆ � (2 π ) 3 2 Q 2 + � 2 π q 2 � � � S 0 ( x ) + 2 · 1 � dω J ( O ) 2 ξ ( x a n + p A ; ω ) � × S 2 ξ ( x, ω ) + ¯ c -term 2 Scales: ◮ hard µ h ∼ Q ◮ collinear µ c ∼ √ Q Ω (no LL collinear contribution at NLP) ◮ soft µ s ∼ Ω Robert Szafron 1/21

Factorization of time-ordered products at NLP We separate the Lagrangian insertions into collinear and soft parts L ( n ) ( z ) = L ( n ) ( z ) ⊗ L ( n ) ( z − ) c s ◮ Soft fields are multipole expanded – convolution variable is one-dimensional ◮ We perform Fourier transform for each z − ◮ We gather all the collinear structures that correspond to a given soft structure, use equation of motion for the soft building blocks � �� � � � � ◮ Order λ 2 amplitude is A ∼ � T � q PDF d 4 z χ c ( tn + ) , L (2) ( z ) X s ◮ States factorize i.e. � X s | = � X s | s � 0 | c � � � �� � � � � � � � � � � � L (2) χ c , L (2) ◮ A ∼ dz − X s ( z − ) � 0 0 � q PDF � T s c s c This gives an NLP collinear function �� � � � n + z χ c ( tn + ) × L ( n ) d 4 z e i ω i × T ( z ) 2 c = J ( t ; ω ) χ PDF ( tn + ) c Collinear function is a non-local object See talk by Sebastian tomorrow Robert Szafron 2/21

Hard Function ˆ σ ( z ) = H (ˆ s ) � � d 3 � q 1 Q 2 d 4 x e i ( x a p A + x b p B − q ) · x � × (2 π ) 3 2 Q 2 + � 2 π q 2 � � � S 0 ( x ) + 2 · 1 dω J ( O ) � 2 ξ ( x a n + p A ; ω ) � S 2 ξ ( x, ω ) + ¯ c -term × 2 Robert Szafron 2/21

Hard function When considering power corrections we have to be careful about kinematic factors. Consider the hard function � ¯ C A 0 ( t, ¯ t ) J A 0 s, µ h ) = | C A 0 ( − ˆ s ) | 2 dt d ¯ t � µ ( t, ¯ ψγ µ ψ (0) = t ) , H (ˆ s around Q 2 We can obtain power corrections from expansion of ˆ s ) = H ( Q 2 ) + Q 2 (1 − z ) H ′ ( Q 2 ) + . . . H (ˆ Robert Szafron 3/21

Hard function When considering power corrections we have to be careful about kinematic factors. Consider the hard function � ¯ C A 0 ( t, ¯ t ) J A 0 s, µ h ) = | C A 0 ( − ˆ s ) | 2 dt d ¯ t � µ ( t, ¯ ψγ µ ψ (0) = t ) , H (ˆ s around Q 2 We can obtain power corrections from expansion of ˆ s ) = H ( Q 2 ) + Q 2 (1 − z ) H ′ ( Q 2 ) + . . . H (ˆ This correction modifies the LP factorization σ ( z ) = H ( Q 2 ) QS DY ( Q (1 − z )) + Q 2 (1 − z ) H ′ ( Q 2 ) QS DY ( Q (1 − z )) ˆ with � � µ �� α s ln 2 H (ˆ s ) = 1 + O and S DY (Ω) = δ (Ω) + O ( α s ) µ h s ln 2 � � µ This contributions starts at α 2 so at the LL accuracy it is enough µ h s ) by H ( Q 2 ) to replace H (ˆ Robert Szafron 3/21

Kinematic corrections ˆ σ ( z ) = H (ˆ s ) Q 2 � � d 3 � q 1 d 4 x e i ( x a p A + x b p B − q ) · x (2 π ) 3 2 √ × 2 π Q 2 + � q 2 � � � S 0 ( x ) + 2 · 1 dω J ( O ) � 2 ξ ( x a n + p A ; ω ) � S 2 ξ ( x, ω ) + ¯ c -term × 2 Robert Szafron 3/21

Kinematic corrections I At LP we only need the soft function at x = x 0 but for now consider the soft function for generic x S 0 ( x ) = 1 � T ( Y † + ( x ) Y − ( x )) T ( Y † N c Tr � 0 | ¯ − (0) Y + (0)) | 0 � Use partonic center-of-mass frame x a � p A + x b � p B = 0 Momentum � p X s of the soft hadronic final state is balanced by the lepton-pair � q + � p X s = 0 √ q 0 = q ∼ λ 2 , s + O ( λ 2 ) � ˆ Energy of the soft radiation √ � q 2 � λ 6 � q 2 = Ω ∗ 2 − � [ x 1 p 1 + x 2 p 2 − q ] 0 = p 0 Q 2 + � X s = ˆ s − 2 Q + O with Ω ∗ = 2 Q 1 − √ z � λ 6 � = Q (1 − z ) + 3 4 Q (1 − z ) 2 + O √ z Robert Szafron 4/21

Kinematic corrections II Expansion of the kinematic factors leads to � � d 3 � q 1 d 4 x e i ( x a p A + x b p B − q ) · x � Q S 0 ( x ) � (2 π ) 3 2 Q 2 + � 2 π q 2 � dx 0 � λ 4 �� 1 + ix 0 ∂ 2 � 4 π e ix 0 Ω ∗ / 2 S 0 ( x 0 , � � � x → + O x ) | � x =0 2 Q → S DY ( Q (1 − z )) + 1 QS K 1 ( Q (1 − z )) + 1 QS K 2 ( Q (1 − z )) + O ( λ 4 ) NLP kinematic soft functions ∂ ∂ Ω ∂ 2 S K 1 (Ω) = x S 0 (Ω , � x ) | � � x =0 3 4 Ω 2 ∂ S K 2 (Ω) = ∂ Ω S 0 (Ω , � x ) | � x =0 Robert Szafron 5/21

Kinematic corrections III It is more convenient to introduce ∆ ab ( z ) = ˆ σ ab ( z ) z ∆ LP σ LP ab ( z ) but ∆ NLP ab ( z ) = ˆ ( z ) receives additional NLP correction ab (1 − z ) × ˆ σ LP ( z ) which leads to S K 3 (Ω) = Ω S 0 (Ω , � x ) | � x =0 Factorization theorem for ∆( z ) = ∆ q ¯ q ( z ): H ( Q 2 ) ∆( z ) = � � 3 1 × Q S DY ( Q (1 − z )) + QS Ki ( Q (1 − z )) i =1 � � +2 · 1 dω J ( O ) 2 ξ ( x a n + p A ; ω ) � S 2 ξ ( x, ω ) + ¯ c -term 2 No further expansion in λ is needed! Robert Szafron 6/21

Example: expansion of the soft function RGE I In position space, renormalization of the LP soft function is multiplicative � � d � � S 0 ( x ) = 2Γ cusp L − 2 γ W S 0 ( x ) d ln µ � � − 1 4 n − xn + xµ 2 e 2 γ E L ≡ ln γ W = O ( α 2 s ) Robert Szafron 7/21

Example: expansion of the soft function RGE I In position space, renormalization of the LP soft function is multiplicative � � d � � S 0 ( x ) = 2Γ cusp L − 2 γ W S 0 ( x ) d ln µ � � − 1 4 n − xn + xµ 2 e 2 γ E L ≡ ln γ W = O ( α 2 s ) Expansion of the soft function, x = ( x 0 , 0 , 0 , z ) S 0 ( x 0 ) + . . . + 1 x =0 z 2 + . . . S 0 ( x ) = � � ∂ 2 � z � S 0 ( x ) | � 2 Robert Szafron 7/21

Example: expansion of the soft function RGE I In position space, renormalization of the LP soft function is multiplicative � � d � � S 0 ( x ) = 2Γ cusp L − 2 γ W S 0 ( x ) d ln µ � � − 1 4 n − xn + xµ 2 e 2 γ E L ≡ ln γ W = O ( α 2 s ) Expansion of the soft function, x = ( x 0 , 0 , 0 , z ) S 0 ( x 0 ) + . . . + 1 x =0 z 2 + . . . S 0 ( x ) = � � ∂ 2 � z � S 0 ( x ) | � 2 Expansion of the log generates inhomogeneous term � z 4 � z 2 L = L 0 − ( x 0 ) 2 + O ( x 0 ) 4 � � − 1 4( x 0 ) 2 µ 2 e 2 γ E L 0 ≡ ln Robert Szafron 7/21

Example: expansion of the soft function RGE II Coefficient of z 2 gives � � 1 d 1 2 ∂ 2 � � ∂ 2 z � z � ( x 0 ) 2 � S 0 ( x ) | � x =0 = 2Γ cusp L 0 − 2 γ W S 0 ( x ) | � x =0 − S 0 ( x 0 ) d ln µ 2 2 Define soft functions ix 0 � � ∂ 2 z � S 3 ( x 0 ) = S 0 ( x ) | � x =0 2 − 2 i � � S x 0 ( x 0 ) = S ( x 0 ) x 0 − iε Robert Szafron 8/21

Example: expansion of the soft function RGE II Coefficient of z 2 gives � � 1 d 1 2 ∂ 2 � � ∂ 2 z � z � ( x 0 ) 2 � S 0 ( x ) | � x =0 = 2Γ cusp L 0 − 2 γ W S 0 ( x ) | � x =0 − S 0 ( x 0 ) d ln µ 2 2 Define soft functions ix 0 � � ∂ 2 z � S 3 ( x 0 ) = S 0 ( x ) | � x =0 2 − 2 i � � S x 0 ( x 0 ) = S ( x 0 ) x 0 − iε Robert Szafron 8/21