Power corrections with SCET Robert Szafron Technische Universit¨ at M¨ unchen MTTD 2019 1-6 September Katowice Robert Szafron 1/24
Outline 1. Introduction 2. SCET formalism beyond LP ◮ N-jet operator ◮ Soft loops and KSZ theorem 3. Applications ◮ Threshold resummation for Drell-Yan ◮ gg → H 4. Summary Robert Szafron 1/24
Power expansion Consider expansion of a cross-section in some threshold variable z ∞ � � 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 QCD limits ◮ collinear p i · p j ≪ Q 2 QCD singular limits lead to the appearance of large logarithms of a ratio of different scales ◮ soft E s ≪ Q Before we discover New Physics, we must be sure that we understand the Standard Model! Robert Szafron 1/24
Power expansion Consider expansion of a cross-section in some threshold variable z ∞ � � 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 QCD limits ◮ collinear p i · p j ≪ Q 2 QCD singular limits lead to the appearance of large logarithms of a ratio of different scales ◮ soft E s ≪ Q Before we discover New Physics, we must be sure that we understand the Standard Model! Robert Szafron 1/24
Power expansion Consider expansion of a cross-section in some threshold variable z ∞ � � 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 − QCD limits ◮ collinear p i · p j ≪ Q 2 QCD singular limits lead to the appearance of large logarithms of a ratio of different scales ◮ soft E s ≪ Q Before we discover New Physics, we must be sure that we understand the Standard Model! Robert Szafron 1/24
Factorization at Next-to-Leading power � � N J ( i ) a J ( i ) C a C ∗ dσ = b ⊗ ⊗ S ab b a,b i =1 ◮ C a Hard functions; ∼ Q λ is a power-counting parameter, e.g. λ = 1 − z Robert Szafron 2/24
Factorization at Next-to-Leading power � � N J ( i ) a J ( i ) C a C ∗ dσ = b ⊗ ⊗ S ab b a,b i =1 ◮ C a Hard functions; ∼ Q ◮ J ( i ) i - collinear (jet) functions; ∼ Qλ a λ is a power-counting parameter, e.g. λ = 1 − z Robert Szafron 2/24
Factorization at Next-to-Leading power � � N J ( i ) a J ( i ) C a C ∗ dσ = b ⊗ ⊗ S ab b a,b i =1 ◮ C a Hard functions; ∼ Q ◮ J ( i ) i - collinear (jet) functions; ∼ Qλ a ◮ S ab soft functions ∼ Qλ 2 λ is a power-counting parameter, e.g. λ = 1 − z Robert Szafron 2/24
Factorization at Next-to-Leading power � � N J ( i ) a J ( i ) C a C ∗ dσ = b ⊗ ⊗ S ab b a,b i =1 ◮ C a Hard functions; ∼ Q ◮ J ( i ) i - collinear (jet) functions; ∼ Qλ a ◮ S ab soft functions ∼ Qλ 2 ◮ � ab – sum over various functions, related to different sources of power-suppression λ is a power-counting parameter, e.g. λ = 1 − z Robert Szafron 2/24
A bit of SCET formalism Robert Szafron 3/24
Soft Collinear Effective Field Theory (SCET) [C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, hep-ph/0011336] What is SCET? Effective field theory used to describe energetic particles. QCD SCET Collinear n 3 Collinear Collinear n 2 n 4 Collinear Collinear n 1 n 5 Soft ◮ Collinear sectors where energetic particles within a single region can interact with each other but not with particle in a different sector. ◮ Soft sector - mediates interactions between collinear sectors ◮ Every interaction has well-defined power-counting – allows for systematic expansion Robert Szafron 4/24
Position space formulation of 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 ξ i ∼ λ Soft modes are multipole expanded n i + D = n i + ∂ − ign + i A i ∼ 1 φ C ( x ) φ s ( x ) → D µ ⊥ i = ∂ µ ⊥ i − ign + i A µ ⊥ i ∼ λ n µ φ C ( x ) φ s ( x − ) + . . . ; x − = 2 n + x − n i − D = n i − ∂ − ign i − A i − ign i − A s ( x i − ) ∼ λ 2 Light-cone coordinates i = ( n i + p i ) n µ + p i ⊥ i + ( n i − p i ) n µ p µ i − i + p i p j ∼ Q 2 , p 2 i = 0 2 2 n i − p i ∼ λ 2 Q n i + p i , ∼ Q p i ⊥ i ∼ λQ, Robert Szafron 5/24
N-jet operator in SCET [M. Beneke, M. Garny, R. S. and J. Wang, 1712.04416] � � N � � N � � � O ( x ) = dt i C ( { t i } ) ψ i ( x + t i n i + ) i =1 i =1 ◮ C ( { t i } ): hard matching coefficient Robert Szafron 6/24
N-jet operator in SCET [M. Beneke, M. Garny, R. S. and J. Wang, 1712.04416] � � N � � N � � � O ( x ) = dt i C ( { t i } ) ψ i ( x + t i n i + ) i =1 i =1 ◮ C ( { t i } ): hard matching coefficient ◮ ψ i : collinear field (gauge invariant building blocks) ◮ collinear quark χ i ≡ W † i ξ i � � ◮ collinear gluon A µ ⊥ i = W † iD µ ⊥ i W i i Robert Szafron 6/24
N-jet operator in SCET [M. Beneke, M. Garny, R. S. and J. Wang, 1712.04416] � � N � � N � � � O ( x ) = dt i C ( { t i } ) ψ i ( x + t i n i + ) i =1 i =1 ◮ C ( { t i } ): hard matching coefficient ◮ ψ i : collinear field (gauge invariant building blocks) ◮ collinear quark χ i ≡ W † i ξ i � � ◮ collinear gluon A µ ⊥ i = W † iD µ ⊥ i W i i Power suppression: ◮ add derivatives ∂ ⊥ ∼ λ ◮ add extra fields in the same direction Robert Szafron 6/24
N-jet operator in SCET [M. Beneke, M. Garny, R. S. and J. Wang, 1712.04416] � � N � � N � � � O ( x ) = dt i C ( { t i } ) ψ i ( x + t i n i + ) i =1 i =1 ◮ C ( { t i } ): hard matching coefficient ◮ ψ i : collinear field (gauge invariant building blocks) ◮ collinear quark χ i ≡ W † i ξ i � � ◮ collinear gluon A µ ⊥ i = W † iD µ ⊥ i W i i Power suppression: ◮ add derivatives ∂ ⊥ ∼ λ ◮ add extra fields in the same direction Example 3-jet LP operator: � O A 0 dt 1 dt 2 dt 3 C A 0 ( t 1 , t 2 , t 3 ) χ 1 ( t 1 n 1+ ) γ µ χ 2 ( t 2 n 2+ ) A µ 3 (0) = ⊥ 3 ( t 3 n 3+ ) Example 3-jet NLP operators ( λ suppressed): � O A 1 dt 1 dt 2 dt 3 C A 1 ( t 1 , t 2 , t 3 ) χ 1 ( t 1 n 1+ ) γ µ γ ν ∂ ν ⊥ 2 χ 2 ( t 2 n 2+ ) A µ 3 (0) = ⊥ 3 ( t 3 n 3+ ) � O B 1 dt 1 dt 2 dt 3 C B 1 ( t 1 , t 2 , t 3 ) χ 1 ( t 1 n 1+ ) γ µ γ ν A ν ⊥ 2 χ 2 ( t 2 n 2+ ) A µ 3 (0) = ⊥ 3 ( t 3 n 3+ ) Robert Szafron 6/24
Leading power anomalous dimension [T. Becher, M. Neubert, 0901.0722] Simple structure up to two loop: � − s ij � � � Γ = − γ cusp ( α s ) T i · T j ln + γ i ( α s ) µ 2 i<j i s ij = 2 p i · p j + i 0 Soft and collinear parts are known at the three loop level [Ø. Almelid, C. Duhr, E. Gardi 1507.00047; S. Moch, J.A.M. Vermaseren, A. Vogt,hep-ph/0507039] ◮ governs the evolution of the hard functions C A 0 � d d ln µC P = Γ QP C Q Q ◮ QCD: log structure is determined by IR poles ◮ SCET: turns IR poles of QCD into UV poles of N-jet operator – RG technique can be used Robert Szafron 7/24
Next-to-Leading power anomalous dimension � − s ij x i k x j l � � � � Γ P Q ( x, y ) = δ P Q δ ( x − y ) − γ cusp ( α s ) T i k · T j l ln µ 2 i<j k,l � � � � � δ [ i ] ( x − y ) γ i δ ( x − y ) γ ij + γ i k ( α s ) + 2 P Q ( x, y ) + 2 P Q ( y ) i k i i<j New structures at NLP ◮ collinear mixing – fields at different positions along the light-cone mix under renormalization [M. Beneke, M. Garny, R. S. and J. Wang, 1712.04416; M. Beneke, M. Garny, R. S. and J. Wang, 1808.04742 ] q 2 q 2 p p p p t i 1 t i 1 t i 1 t i 1 t i 2 t i 2 t i 2 t i 2 q 2 t i 1 t i 1 t i 1 t i 1 t i 1 q 1 q 1 q 1 q 1 q 2 ( b, i ) F ( b, i ) B ( b, i ) V ( b, i ) J p p p p t i 2 t i 2 t i 2 q 2 t i 1 t i 1 q 2 q 2 t i 1 t i 1 t i 1 t i 2 t i 2 t i 2 t i 2 q 1 q 1 q 1 q 2 q 1 ( b, ii ) B ( b, ii ) V ( b, ii ) J ( b, iii ) F Robert Szafron 8/24
Next-to-Leading power anomalous dimension � − s ij x i k x j l � � � � Γ P Q ( x, y ) = δ P Q δ ( x − y ) − γ cusp ( α s ) T i k · T j l ln µ 2 i<j k,l � � � � � δ [ i ] ( x − y ) γ i δ ( x − y ) γ ij + γ i k ( α s ) + 2 P Q ( x, y ) + 2 P Q ( y ) i k i i<j New structures at NLP ◮ collinear mixing – fields at different positions along the light-cone mix under renormalization ◮ soft mixing – time-ordered products of NLP Lagrangian with the N-jet operator mix into N-jet operator [M. Beneke, M. Garny, R. S. and J. Wang, 1712.04416; M. Beneke, M. Garny, R. S. and J. Wang, 1808.04742 ] q 2 q 2 p p p p t i 1 t i 1 t i 1 t i 1 t i 2 t i 2 t i 2 q t i 2 q 2 q 1 t i 1 t i 1 t i 1 t i 1 t i 1 q 1 q 1 q 1 q 1 q 2 1 ( b, i ) F ( b, i ) B ( b, i ) V ( b, i ) J t i p p p p s t i 2 t i 2 t i 2 q 2 t i 1 t i 1 q 2 q 2 t i 1 t i 1 t i 1 t j t i 2 1 t i 2 t i 2 t i 2 q 1 q 1 q 1 q 2 q 1 q 2 ( b, ii ) B ( b, ii ) V ( b, ii ) J ( b, iii ) F Robert Szafron 8/24
Recommend
More recommend
