Towards Precision Jet Mass Calculations Randall S. Kelley Frontiers - PowerPoint PPT Presentation

Towards Precision Jet Mass Calculations Randall S. Kelley Frontiers in QCD (INT-11-3) Oct 5, 2011 R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 1 / 40

References Resummation of jet mass with a jet veto arXiv:1102.0561v2 RK, Matthew D. Schwartz, Hau Xing Zhu The two-loop hemisphere soft function arXiv:1105.3676 RK, Robert M. Schabinger, Matthew D. Schwartz, Hau Xing Zhu R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 2 / 40

Outline Introduction 1 2-loop Hemisphere Soft function 2 Inclusive R dependent Jet Shapes 3 Exclusive Jet Masses 4 Factorization of the Soft Function 5 R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 3 / 40

Outline Introduction 1 2-loop Hemisphere Soft function 2 Inclusive R dependent Jet Shapes 3 Exclusive Jet Masses 4 Factorization of the Soft Function 5 R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 4 / 40

Introduction Very large number of jets at the LHC. Jets provide a wealth of information about QCD and exploring new physics. - excess in the number of jets could be a sign of new physics Substructure may be critical in new physics searches. - massive boosted heavy particles can be found in jet Jet rate distributions have been calculated to NLO, but little has been said about structure of jets (i.e. m 2 , R , angularity, etc.). Predictions may be spoiled by large logarithms ( log n m 1 m 2 , log R , etc) Effective field theories provide a way to systematically improve calculations. R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 5 / 40

Factorization (a preview) τ = 1 − T ≈ m 2 L + m 2 R Q 2 dσ � ∼ H ( Q, µ h ) dk L dk R dm 2 L dm 2 R × J ( m 2 L − k L Q, µ j ) J ( m 2 R − k R Q, µ j ) S ( k L , k R , µ s ) ( Fleming et al., Schwartz) Factorization is achieved using Thrust LO Soft Collinear Effective theory 20 (SCET) 15 Use the LO results in SCET to predict the NLO singular piece d Σ d Τ 10 using renormalization group Σ 0 Τ SCET evolution (RGE). QCD 5 Compare α 2 s results to EVENT2 (Catani and Seymore) 0 0.0 0.1 0.2 0.3 0.4 0.5 Τ R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 6 / 40

Factorization (a preview) τ = 1 − T ≈ m 2 L + m 2 R Q 2 dσ � ∼ H ( Q, µ h ) dk L dk R dm 2 L dm 2 R × J ( m 2 L − k L Q, µ j ) J ( m 2 R − k R Q, µ j ) S ( k L , k R , µ s ) ( Fleming et al., Schwartz) Factorization is achieved using Thrust NLO Soft Collinear Effective theory 300 (SCET) 250 Use the LO results in SCET to 200 predict the NLO singular piece QCD d Σ d Τ 150 Σ 0 using renormalization group Τ SCET evolution (RGE). 100 Compare α 2 50 s results to EVENT2 (Catani and Seymore) 0 0.0 0.1 0.2 0.3 0.4 0.5 Τ R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 6 / 40

Factorization (a preview) τ = 1 − T ≈ m 2 L + m 2 R Q 2 dσ � ∼ H ( Q, µ h ) dk L dk R dm 2 L dm 2 R × J ( m 2 L − k L Q, µ j ) J ( m 2 R − k R Q, µ j ) S ( k L , k R , µ s ) ( Fleming et al., Schwartz) Factorization is achieved using Thrust NLO Soft Collinear Effective theory 0 (SCET) � 2000 � 4000 Use the LO results in SCET to � 6000 predict the NLO singular piece d Σ d Τ � 8000 Σ 0 Τ using renormalization group � 10000 evolution (RGE). � 12000 Compare α 2 s results to EVENT2 � 14000 (Catani and Seymore) � 12 � 10 � 8 � 6 � 4 � 2 Τ R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 6 / 40

What could go wrong? The factorization theorem is valid only when m L and m R are small (i.e. the small m 2 region is dominated by IR degrees of freedom) SCET does not guarantee log m 2 L /m 2 R are resummed by RGE (can be calculated by brute force) Produce non-global logarithms ( Dasgupta and Salem) � α s � 2 16 π 2 � m 2 � log 2 L − C F C A 4 π 3 Q 2 Hard emissions are not included in SCET degrees of freedom (type 1) Sharply divided phase space with separated scales m L ≪ m R (type 2) Finite jet size ( R ), and cutoff scales ( E out < ω ) complicate the problem considerably. R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 7 / 40

What could go wrong? � ρ R Q 2 � ∞ d 2 σ dm 2 Σ( ρ R ) = dm 1 2 dm 2 1 dm 2 0 0 2 Produce non-global logarithms ( Dasgupta and Salem) � α s � 2 16 π 2 � m 2 � log 2 L − C F C A 4 π 3 Q 2 Hard emissions are not included in SCET degrees of freedom (type 1) Sharply divided phase space with separated scales m L ≪ m R (type 2) Finite jet size ( R ), and cutoff scales ( E out < ω ) complicate the problem considerably. R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 7 / 40

What could go wrong? � ρ R Q 2 � ∞ d 2 σ dm 2 Σ( ρ R ) = dm 1 2 dm 2 1 dm 2 0 0 2 Produce non-global logarithms ( Dasgupta and Salem) � α s � 2 16 π 2 � m 2 � log 2 L − C F C A 4 π 3 Q 2 Hard emissions are not included in SCET degrees of freedom (type 1) Sharply divided phase space with separated scales m L ≪ m R (type 2) Finite jet size ( R ), and cutoff scales ( E out < ω ) complicate the problem considerably. R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 7 / 40

Main Points Seek to understand non-global logarithms and how to control them. Understand how different jet shapes and jet sizes ( R ) affect the observables. Consider first inclusive and then exclusive observables. We perform resummation for a 2-jet observable with jets of size R . τ ω = m 2 1 + m 2 2 , E 3 < ω Q 2 Demonstration of factorization of the soft function: S R ( k, ω, µ ) = S in R ( k, µ ) S out R ( ω, µ ) and discuss limitations. R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 8 / 40

Outline Introduction 1 2-loop Hemisphere Soft function 2 Inclusive R dependent Jet Shapes 3 Exclusive Jet Masses 4 Factorization of the Soft Function 5 R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 9 / 40

Hemisphere Jets try calculating: dσ � ∼ H ( Q, µ h ) dk L dk R dm 2 L dm 2 R × J ( m 2 L − k L Q, µ j ) J ( m 2 R − k R Q, µ j ) S ( k L , k R , µ s ) RG evolution only resums log m 2 Q 2 , but does not say anything about log m 2 L . m 2 R These logs come from k L /k R in the soft function. R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 10 / 40

Calculation of two-loop Soft function R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 11 / 40

Calculation of two-loop Soft function α 2 S ( k L , k R ) = δ ( k L ) δ ( k R ) + α s 4 π S (1) ( k L , k R , µ ) + 16 π 2 S (2) ( k L , k R , µ ) + · · · s NLO result S (2) = C 2 F S C F + C F C A S C A + C F n f T F S n f � k L � µ 2 ǫ µ 4 ǫ µ 2 ǫ � � S = ( k L k R ) 1+4 ǫ f k R , ǫ + δ ( k L ) + δ ( k R ) g ( ǫ ) k 1+2 ǫ k 1+2 ǫ R L There is a different f ( r, ǫ ) and g ( ǫ ) for each color factor, where r = k L /k R . f ( r, ǫ ) was calculated independently by (Hornig et al. 1105.4628) R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 12 / 40

Cumulative the Soft Function µ aǫ Terms of the form k 1+ aǫ , a = 2 , 4 , must be thought of as distributions and integrated. � X � Y R ( X, Y, µ ) = dk L dk R S ( k L , k R , µ ) 0 0 Result is used for integrated heavy jet mass and thrust distributions. The singular parts of the thrust and heavy jet mass distributions can be extracted (previously only known numerically ) � α s 1 dσ + α s � 2 dτ = δ ( τ ) D ( τ ) 4 π [ D (1) ( τ )] + + [ D (2) ( τ )] + + · · · δ σ 0 4 π Removes a source of theoretical uncertainty in N 3 LL result for heavy jet mass, improving fits to α s . R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 13 / 40

Cumulative the Soft Function µ aǫ Terms of the form k 1+ aǫ , a = 2 , 4 , must be thought of as distributions and integrated. � X � Y R ( X, Y, µ ) = dk L dk R S ( k L , k R , µ ) 0 0 Result is used for integrated heavy jet mass and thrust distributions. The singular parts of the thrust and heavy jet mass distributions can be extracted (previously only known numerically ) � α s � − 3 π 4 − 335 π 2 + 22 π 4 � 638 ζ 3 − 2140 � � 2 D ( τ ) 10 C 2 = F + C F C A δ 4 π 9 54 45 81 � � + 74 π 2 + 80 π 2 � − 232 ζ 3 + C F T F n f 9 27 81 Removes a source of theoretical uncertainty in N 3 LL result for heavy jet mass, improving fits to α s . R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 13 / 40

Asymptotic behavior and Non-Global Logarithms Non-global logs must come from the µ -independent part of the soft function. � X � � X � µ , Y R ( X, Y, µ ) = R µ + R f µ Y for z = X Y ≫ 1 , �� 8 ( z ) = π 4 3 − 16 π 2 81 + 154 π 2 � | log z | + − 136 + 184 ζ 3 � R z ≫ 1 2 C 2 F + C F n f T F f 9 27 9 3 + 44 π 2 + 8 π 4 − 871 π 2 � − 4 � − 8 ζ 3 − 4 � | log z | − 506 ζ 3 − 2032 � 3 π 2 log 2 z + + C F C A 9 9 5 54 81 R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 14 / 40

Asymptotic behavior and Non-Global Logarithms Non-global logs must come from the µ -independent part of the soft function. � X � � X � µ , Y R ( X, Y, µ ) = R µ + R f µ Y for z = X Y ∼ 1 , ( z ) = π 4 3 − 4 π 2 �� − 2 − 4 log 2 2 + 44 log 2 � � 1 � + 88 ζ 3 log 2 z − 32 Li 4 R z ∼ 1 2 C 2 F + f 3 3 2 9 − 4 log 4 2 − 871 π 2 + 16 π 4 − 28 ζ 3 log(2) − 2032 + 4 � 3 π 2 log 2 z C F C A 81 54 9 3 �� 4 log 2 z + 154 π 2 3 − 16 log 2 � − 136 81 − 32 ζ 3 � C F n f T F + O (log 3 z ) . + 3 27 9 Hoang-Kluth ansatz (0806.3852) only valid at small log z . R. S. Kelley (Harvard) Jet Mass Oct 5, 2011 14 / 40