Threshold resummation at NLP: Drell-Yan qq channel and gg H Robert - PowerPoint PPT Presentation

Threshold resummation at NLP: Drell-Yan qq channel and gg → H Robert Szafron Technische Universit¨ at M¨ unchen SCET 2019 25-28 March UC San Diego Robert Szafron 1/20

Leading-logarithmic threshold resummation of the Drell-Yan process at next-to-leading power Martin Beneke, Alessandro Broggio, Mathias Garny, Sebastian Ja´ skiewicz, Robert Szafron, Leonardo Vernazza and Jian Wang arXiv:1809.10631 ◮ Operator basis and renormalization ֒ → see talk by Martin ◮ Factorization theorem and evaluation of collinear functions ֒ → see talk by Sebastian Robert Szafron 1/20

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 1/20

DY cross-section A ( p A ) B ( p B ) → DY( Q ) + X z = Q 2 threshold z → 1 ˆ s Ω ∼ Q (1 − z ) 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 σ ( z ) ˆ = 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 Q ≫ Ω Robert Szafron 1/20

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 1/20

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 2/20

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 2/20

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 2/20

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 3/20

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 4/20

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 5/20

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 6/20

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 6/20

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 6/20

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 7/20

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 7/20

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 Soft functions mix � � d � S 3 ( x 0 ) + � � S 3 ( x 0 ) = 2Γ cusp L 0 − 2 γ W S x 0 ( x 0 ) d ln µ � � d � � S x 0 ( x 0 ) = 2Γ cusp L 0 − 2 γ W S x 0 ( x 0 ) d ln µ � � Note: � S 3 ( x 0 ) = O ( α s L 0 ) and � α s L 2 S x 0 ( x 0 ) = 1 + O 0 � S x 0 ( x 0 ) corresponds to θ -soft previously function defined in JHEP 1808 (2018) 013 by Ian Moult, Iain Stewart, Gherardo Vita, Hua Xing Zhu Robert Szafron 7/20

RGE for kinematic soft functions Proceeding like in the example we obtain � � d S ( x 0 ) � 1 � = 2Γ cusp L 0 − 2 γ W S ( x ) d ln µ   0 0 0 +1   0 0 − 6 +3    � S ( x 0 ) + Γ cusp  0 0 0 − 4 0 0 0 0 � � T with � S ( x 0 ) = S K 1 , � � S K 2 , � S K 3 , � S x 0 � � d S K 1+ K 2+ K 3 ( x 0 ) = � S K 1+ K 2+ K 3 ( x 0 ) − 6 Γ cusp � � S K 3 ( x 0 ) , 2Γ cusp L 0 − 2 γ W d ln µ Note: � S K 1+ K 2+ K 3 ( x 0 ) = O ( α s ) No LL kinematic corrections to all orders! Robert Szafron 8/20

Soft function ˆ σ ( z ) = H (ˆ s ) � � d 3 � 1 q 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 8/20

Time-ordered products Soft operator in position space � � � � − (0) Y + (0) i∂ ν � Y † Y † in − ∂ B + ⊥ S 2 ξ ( x, z − ) = ¯ + ( x ) Y − ( x ) ⊥ ν ( z − ) T T with decoupled soft fields B µ ± = Y † ± [ iD µ s Y ± ] Lagrangian is already multipole expanded → soft fields depend only on z − � / n + � 2 ξ = 1 L (2) χ c z µ ⊥ z ν i∂ ν in − ∂ B + 2 ¯ 2 χ c ⊥ µ Robert Szafron 9/20