Exploring minijets beyond leading power Piotr Kotko Penn State University based on: supported by: DEC-2011/01/B/ST2/03643 P .K., A. Stasto, M. Strikman DE-FG02-93ER40771 arXiv:1608.00523 San Cristobal de las Casas, 12/01/2016
Introduction Minijets in Pythia • The rise of the total cross section with energy in hadron-hadron collision is due to minijets; in collinear factorization � dx A dx B � � x A , x B ; µ 2 � � x A ; µ 2 � � x B ; µ 2 � d σ 2 jet = d ˆ σ ab → cd f a / A f b / B x A x B a , b , c , d • d σ 2 jet is divergent for p T → 0 � � p 2 α s d σ 2 jet T ∼ dp 2 p 4 T T • phenomenological regularization [T. Sjostrand, M. van Zijl, Phys.Rev.D 36 (1987) 2019] � � d σ ′ α s p 2 T + p 2 T 0 ( s ) 2 jet ∼ � 2 , dp 2 � p 2 T + p 2 T 0 ( s ) T with p T 0 ( s ) = p T 0 ( s / s 0 ) λ . 1
Introduction Goal: calculate p T 0 ( s ) from some simple approach • the p T 0 ( s ) regularization takes the collinear factorization out of the leading power approximation • idea: use frameworks that have power corrections by including transverse momenta of incoming partons: High Energy (or k T ) Factorization (HEF) (or Color Glass Condensate, but saturation is not essential here) 2
Introduction Goal: calculate p T 0 ( s ) from some simple approach • the p T 0 ( s ) regularization takes the collinear factorization out of the leading power approximation • idea: use frameworks that have power corrections by including transverse momenta of incoming partons: High Energy (or k T ) Factorization (HEF) (or Color Glass Condensate, but saturation is not essential here) Plan 1 High Energy Factorization 2 Non-leading-power extension of DDT (Diakonov-Dokshitzer-Troyan) formula for dijet in pp 3 Direct study of minijet suppression 4 Hard dijet observable sensitive to p T 0 ( s ) cutoff 5 Summary 2
High Energy Factorization (HEF) Gluon production in HEF [S. Catani, M. Ciafaloni, F. Hautmann, Nucl.Phys. B366 (1991) 135-188] [J.C. Collins, R.K. Ellis, Nucl.Phys. B360 (1991) 3-30] d σ AB → gg = F g ∗ / A ( x A , k T A ; µ ) ⊗ d ˆ σ g ∗ g ∗ → gg ( x A , x B , k T A , k T B ; µ ) ⊗ F g ∗ / B ( x B , k T B ; µ ) k A k A p 1 F g ∗ /A p A = k A p 2 k B k B k B p B F g ∗ /B [E. Antonov, L. Lipatov, E. Kuraev, I. Cherednikov, Nucl.Phys. B721 (2005) 111-135] [P .K. JHEP 1407 (2014) 128] F g ∗ / H ( x H , k T H ; µ ) – Unintegrated Gluon Distribution (UGD) d ˆ σ g ∗ g ∗ → gg – hard process with off-shell gauge invariant amplitude 3
High Energy Factorization (HEF) Gluon production in HEF [S. Catani, M. Ciafaloni, F. Hautmann, Nucl.Phys. B366 (1991) 135-188] [J.C. Collins, R.K. Ellis, Nucl.Phys. B360 (1991) 3-30] d σ AB → gg = F g ∗ / A ( x A , k T A ; µ ) ⊗ d ˆ σ g ∗ g ∗ → gg ( x A , x B , k T A , k T B ; µ ) ⊗ F g ∗ / B ( x B , k T B ; µ ) k A k A p 1 F g ∗ /A p A = k A p 2 k B k B k B p B F g ∗ /B [E. Antonov, L. Lipatov, E. Kuraev, I. Cherednikov, Nucl.Phys. B721 (2005) 111-135] [P .K. JHEP 1407 (2014) 128] F g ∗ / H ( x H , k T H ; µ ) – Unintegrated Gluon Distribution (UGD) d ˆ σ g ∗ g ∗ → gg – hard process with off-shell gauge invariant amplitude dp 2 T d φ � � � 1 z 1 dz 1 z 2 dz 2 d 2 � k TA d 2 � d σ AB → gg = k TB 64 π 2 ( z 1 + z 2 ) 2 � 2 + z 2 p 2 � 2 � � p T − � � z 1 K T T � 1 � 2 + � � 1 2 � � p T − � � z 1 , z 2 , � k TA , � � � z 1 S p 2 F g ∗ / A ( z 1 + z 2 , k TA ) F g ∗ / B K T T , k TB � � M � k TB � � z 2 S � g ∗ g ∗ → gg 3
Unintegrated Gluon Distributions (UGDs) • Kimber-Martin-Ryskin (KMR) [M. Kimber, A. D. Martin, and M. Ryskin, Phys.Rev. D63, 114027 (2001)] ∂ F g ∗ / H ( x , k T , µ ) = [ f g / H ( x , k T ) T g ( k T , µ )] ∂ k 2 T � T , µ 2 � K 2 T g – the Sudakov form factor 4
Unintegrated Gluon Distributions (UGDs) • Kimber-Martin-Ryskin (KMR) [M. Kimber, A. D. Martin, and M. Ryskin, Phys.Rev. D63, 114027 (2001)] ∂ F g ∗ / H ( x , k T , µ ) = [ f g / H ( x , k T ) T g ( k T , µ )] ∂ k 2 T � T , µ 2 � K 2 T g – the Sudakov form factor • Kwiecinski-Martin-Stasto (KMS) [J. Kwiecinski, Alan D. Martin, A.M. Stasto, Phys.Rev. D56 (1997) 3991-4006] BFKL + DGLAP corrections + kinematic constraint + running α s � x � k 2 � � x � � q 2 z , q 2 z − q 2 T − k 2 z , k 2 � x T F θ T F � 1 � ∞ � dq 2 k 2 z , k 2 T T T T F + α s N c dz � � � � T T x , k 2 x , k 2 F = F 0 + T T q 2 � � π z � q 2 T − k 2 � k 2 � � 4 q 4 T + k 4 x T T 0 T � T � x � x � 1 � � k 2 � � + α s P gg ( z ) − 2 N c T � dq 2 z , q 2 z , k 2 + zP gq ( z ) Σ dz T F T T 2 π k 2 z k 2 x T T 0 Fitted to HERA data. [K. Kutak, S. Sapeta, Phys. Rev. D 86 (2012) 094043] 4
Unintegrated Gluon Distributions (UGDs) • Kimber-Martin-Ryskin (KMR) [M. Kimber, A. D. Martin, and M. Ryskin, Phys.Rev. D63, 114027 (2001)] ∂ F g ∗ / H ( x , k T , µ ) = [ f g / H ( x , k T ) T g ( k T , µ )] ∂ k 2 T � T , µ 2 � K 2 T g – the Sudakov form factor • Kwiecinski-Martin-Stasto (KMS) [J. Kwiecinski, Alan D. Martin, A.M. Stasto, Phys.Rev. D56 (1997) 3991-4006] BFKL + DGLAP corrections + kinematic constraint + running α s � x � k 2 � � x � � q 2 z , q 2 z − q 2 T − k 2 z , k 2 � x T F θ T F � 1 � ∞ � dq 2 k 2 z , k 2 T T T T F + α s N c dz � � � � T T x , k 2 x , k 2 F = F 0 + T T q 2 � � π z � q 2 T − k 2 � k 2 � � 4 q 4 T + k 4 x T T 0 T � T � x � x � 1 � � k 2 � � + α s P gg ( z ) − 2 N c T � dq 2 z , q 2 z , k 2 + zP gq ( z ) Σ dz T F T T 2 π k 2 z k 2 x T T 0 Fitted to HERA data. [K. Kutak, S. Sapeta, Phys. Rev. D 86 (2012) 094043] • CCFM • ... 4
Non-leading-power extension of DDT Dokshitzer-Diakonov-Troyan (DDT) formula (leading power) [Y. Dokshitzer,D. Dyakonov, S. Troyan, Phys.Rep. 58 (1980) 269-395] � dx A d σ 2 jet dx B ∂ � x A , x B ; µ 2 � � � � � � � T , µ 2 �� x A ; K 2 x B ; K 2 T 4 K 2 = d ˆ σ gg → gg f g / A f g / B T T g dK 2 ∂ K 2 x A x B T T f g/A f g/A f g/A f g/A p A p A p A p A T g . . . . . . T g T g T g . . . . . . p B p B p B p B f g/B f g/B f g/B f g/B � T , µ 2 � K 2 – the Sudakov form factor. DDT applies when µ 0 ≪ K T ≪ µ . T g 5
Non-leading-power extension of DDT Dokshitzer-Diakonov-Troyan (DDT) formula (leading power) [Y. Dokshitzer,D. Dyakonov, S. Troyan, Phys.Rep. 58 (1980) 269-395] � dx A d σ 2 jet dx B ∂ � x A , x B ; µ 2 � � � � � � � T , µ 2 �� x A ; K 2 x B ; K 2 T 4 K 2 = d ˆ σ gg → gg f g / A f g / B T T g dK 2 ∂ K 2 x A x B T T f g/A f g/A f g/A f g/A p A p A p A p A T g . . . . . . T g T g T g . . . . . . p B p B p B p B f g/B f g/B f g/B f g/B � T , µ 2 � K 2 – the Sudakov form factor. DDT applies when µ 0 ≪ K T ≪ µ . T g Improved DDT (IDDT) formula (beyond leading power) [P .K., A. Stasto, M. Strikman, arXiv:1608.00523] d σ ( IDDT ) = d σ ( IS ) 2 jet + d σ ( FS ) 2 jet 2 jet d σ ( IS ) � x A , x B , � � � � � T , µ 2 � x B , K 2 ⊗ T 3 K 2 2 jet = 2 F g ∗ / A ( x A , K T , µ ) ⊗ d ˆ σ g ∗ g → gg K T ⊗ f g / B T g d σ ( FS ) � � � x A , x B , � � � � � T , µ 2 � � T , µ 2 � x A ; K 2 x B ; K 2 K 2 ⊗ T 3 K 2 2 jet = 2 f g / A ⊗ d ˆ σ gg → gg ∗ K T ⊗ f g / B ⊗ T g T T g � T , µ 2 � � T , µ 2 � K 2 = ∂ T g K 2 /∂ K 2 T . There is a restriction K T ≤ µ . where T g 5
Direct study of minijet suppression Inclusive dijets with Pythia: anti- k T with R = 0 . 5, rapidity [ − 4 , 4 ] 6
Direct study of minijet suppression Inclusive dijets with Pythia: anti- k T with R = 0 . 5, rapidity [ − 4 , 4 ] 1e+20 pythia PS (soft QCD) Inclusive dijets p T > 2 GeV pythia PS+HAD (soft QCD) 1e+18 LO Collinear Factorization 1e+16 � S = 30.0 TeV (x10 9 ) 1e+14 d � /dp Tj [nb/GeV] � S = 20.0 TeV (x10 6 ) 1e+12 1e+10 � S = 14.0 TeV (x10 3 ) 1e+08 � S = 7.0 TeV 1e+06 Using GRV98 PDFs 10000 gg->gg channel 100 5 10 15 20 25 30 p Tj [GeV] 6
Direct study of minijet suppression Inclusive dijets with HEF and IDDT 1e+20 HEF (KMR) Inclusive dijets p T > 2 GeV IDDT 1e+18 LO Collinear Factorization 1e+16 � S = 30.0 TeV (x10 9 ) 1e+14 d � /dp Tj [nb/GeV] � S = 20.0 TeV (x10 6 ) 1e+12 1e+10 � S = 14.0 TeV (x10 3 ) 1e+08 � S = 7.0 TeV 1e+06 Using GRV98 PDFs 10000 gg->gg channel 100 5 10 15 20 25 30 p Tj [GeV] 7
Direct study of minijet suppression Inclusive dijets with HEF and IDDT 1e+22 HEF (KMR) Inclusive dijets p T > 2 GeV IDDT 1e+20 LO Collinear Factorization 1e+18 � S = 30.0 TeV (x10 9 ) 1e+16 d � /dp Tj [nb/GeV] � S = 20.0 TeV (x10 6 ) 1e+14 1e+12 � S = 14.0 TeV (x10 3 ) 1e+10 � S = 7.0 TeV 1e+08 Using GRV98 PDFs 1e+06 gg->gg channel 10000 2.5 3 3.5 4 4.5 5 p Tj [GeV] 7
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