High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Towards new analyses How could we further and deeply probe BFKL? 1. Study less inclusive two-body final states... [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2016, 2017)] Di-hadron production ⋄ inclusive production of a pair of charged light hadrons well separed in rapidity ⋄ much smaller values of the transverse momentum than jets! ⋄ possibility to constrain not only the PDFs, but also the FFs! Heavy-quark pair photoproduction ⋄ quark masses play the role of hard scale ⋄ e + e − at LEP2 and future lepton colliders 2. Study three- and four-body final-state processes... [F. Caporale, F.G. C., G. Chachamis, A. Sabio Vera (2016)]; [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2016, 2017)] Multi-jet production ⋄ definition of new, suitable BFKL observables (talk by David Gordo Gómez ) Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 9 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Outline 1 Introductory remarks QCD and semi-hard processes BFKL resummation Towards new analyses Phenomenology 2 Mueller–Navelet jet production Inclusive di-hadron production Heavy-quark pair photoproduction Excursus: electroproduction of ρ mesons as UGD discriminator 3 Conclusions & Outlook Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 10 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Mueller–Navelet jet production Mueller–Navelet jets p 1 x 1 k J, 1 proton ( p 1 ) + proton ( p 2 ) → jet 1 ( k 1 ) + jet 2 ( k 2 ) + X x 2 k J, 2 p 2 large jet transverse momenta (hard scales): � 1 ∼ � k 2 k 2 2 ≫ Λ 2 QCD large rapidity gap between jets, ∆ y ≡ Y = y J 1 − y J 2 , which requires large c.m. energy of the proton collisions, s = 2 p 1 · p 2 ≫ � k 2 1 , 2 [A.H. Mueller, H. Navelet (1987)] Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 11 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Mueller–Navelet jet production Forward jet impact factor take the impact factors for colliding partons [V.S. Fadin, R. Fiore, M.I. Kotsky, A. Papa (2000)] [M. Ciafaloni and G. Rodrigo (2000)] quark vertex gluon vertex “open” one of the integrations over the phase space of the intermediate state to allow one parton to generate the jet ( x J p 1 , � ( x J p 1 , � k J ) k J ) xp 1 xp 1 q � � q quark jet vertex gluon jet vertex � use QCD collinear factoriz.: q f s ⊗ [quark vertex] + f g ⊗ [gluon vertex] s = q ,¯ Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 12 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Mueller–Navelet jet production BFKL cross section (Mueller–Navelet jets)... 1 1 � � d ˆ σ i , j ( x 1 x 2 s , µ ) d σ � = dx 1 dx 2 f i ( x 1 , µ ) f j ( x 2 , µ ) dx J 1 dx J 2 d 2 k J 1 d 2 k J 2 dx J 1 dx J 2 d 2 k J 1 d 2 k J 2 i , j = q ,¯ q , g 0 0 ◮ slight change of variable in the final p 1 state ◮ project onto the eigenfunctions of the x 1 k J, 1 LO BFKL kernel, i.e. transfer from the reggeized gluon momenta to the ( n , ν ) -representation ◮ suitable definition of the azimuthal coefficients � � ∞ d σ 1 � x 2 k J, 2 = C 0 + 2 cos ( n φ ) C n dx J 1 dx J 2 d | � k J 1 | d | � ( 2 π ) 2 k J 2 | d φ J 1 d φ J 2 n = 1 p 2 with φ = φ J 1 − φ J 2 − π Y = ln x J 1 x J 2 s s 0 ...useful definitions: , Y 0 = ln | � k J 1 || � | � k J 1 || � k J 2 | k J 2 | Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 13 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Mueller–Navelet jet production Observables and kinematics (MN-jets) Observables: C n � � �� φ -averaged cross section C 0 , � cos n φ J 1 − φ J 2 − π � ≡ C 0 , with n = 1 , 2 , 3 � cos [ 2 ( φ 1 − φ 2 − π )] � ≡ C 2 � cos [ 3 ( φ 1 − φ 2 − π )] � � cos [ 2 ( φ 1 − φ 2 − π )] � ≡ C 3 ≡ R 21 , ≡ R 32 . � cos ( φ 1 − φ 2 − π ) � C 1 C 2 ⋄ Integrated coefficients: � y 1 , max � y 2 , max � ∞ � ∞ � � C n = dy 1 dy 2 dk J 1 dk J 2 δ ( y 1 − y 2 − Y ) C n y J 1 , y J 2 , k J 1 , k J 2 y 1 , min y 2 , min k J 1 , min k J 2 , min Kinematic settings: ⋄ R = 0 . 5 and √ s = 7 , 13 TeV ⋄ y C max � | y J 1 , 2 | � 4 . 7 ⋄ symmetric and asymmetric choices for k J 1 and k J 2 ranges Numerical tools: F ORTRAN + NLO MSTW08 PDFs + CERNLIB [A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt (2009)] http:/ /cernlib.web.cern.ch/cernlib Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 14 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Mueller–Navelet jet production Theory versus experiment C 1 /C 0 C 2 /C 0 1.1 1.1 LLA LLA BLM a BLM a 1 1 BLM b BLM b 0.9 0.9 CMS data CMS data 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 3 4 5 6 7 8 9 Y 3 4 5 6 7 8 9 Y C 2 /C 1 1.1 LLA BLM a 1 R n 0 ≡ C n / C 0 = � cos [ n ( φ J 1 − φ J 2 − π )] � BLM b 0.9 CMS data R nm ≡ C n / C m = R n 0 / R m 0 0.8 0.7 vs Y = y J 1 − y J 2 0.6 small-cone approximation 0.5 BLM scale setting 0.4 0.3 � CMS (7 TeV; | � k 1 | , | � k 2 | � 35 GeV) 0.2 0.1 ( 7 TeV theory vs exp.) [F. Caporale, D.Yu. Ivanov, B. Murdaca, A. Papa (2014)] 0 3 4 5 6 7 8 9 Y ( 7 TeV BFKL vs DGLAP + asym ) [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2015)] ( 13 TeV predictions + C 0 ( Y ) ) [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2016)] Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 15 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Mueller–Navelet jet production High-energy DGLAP α 3 � � ! ⋄ NLA BFKL expressions for the observables truncated to O s Why asymmetric cuts? ◮ suppress Born contribution to φ -averaged cross section C 0 (back-to-back jets) ⋄ avoid instabilities observed in NLO fixed-order calculations [J.R. Andersen, V. Del Duca, S. Frixione, C.R. Schmidt, W.J. Stirling (2001)] [M. Fontannaz, J.P. Guillet, G. Heinrich (2001)] emphasize ⋄ enhance effects of additional hard gluons − − − − → BFKL effects Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 16 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook R nm for k J 1 > 35 GeV, k J 2 > 45 GeV at √ s = 7 TeV Mueller–Navelet jet production C 1 /C 0 C 2 /C 0 1.2 1 1 0.8 0.8 � cos ( 2 φ ) � � cos ( φ ) � 0.6 0.6 BFKL a 0.4 BFKL a 0.4 BFKL b BFKL b k J 2 > 45 GeV k J 2 > 45 GeV DGLAP a DGLAP a 0.2 0.2 DGLAP b DGLAP b 0 0 2 4 6 8 10 2 4 6 8 10 Y Y C 3 /C 0 C 2 /C 1 k J 2 > 45 GeV 1 k J 2 > 45 GeV 1 0.8 0.8 0.6 � cos ( 2 φ ) � 0.6 � cos ( 3 φ ) � � cos ( φ ) � 0.4 0.4 BFKL a BFKL a BFKL b BFKL b DGLAP a DGLAP a 0.2 DGLAP b 0.2 DGLAP b 0 0 2 4 6 8 10 2 4 6 8 10 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2015)] Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 17 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Inclusive di-hadron production Di-hadron production: theoretical setup Process: proton ( p 1 ) + proton ( p 2 ) → h 1 ( k 1 ) + h 2 ( k 2 ) + X ...LHC physics! p 1 π + , K + , p ( k 1 , θ 1 , y 1 ) x 1 x 2 π − , K − , ¯ p ( k 2 , θ 2 , y 2 ) p 2 Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 18 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Inclusive di-hadron production Di-hadron production Process: proton ( p 1 ) + proton ( p 2 ) → h 1 ( k 1 ) + h 2 ( k 2 ) + X ...LHC physics! 1 1 � � d σ � dx 1 dx 2 f i ( x 1 , µ ) f j ( x 2 , µ ) d ˆ σ ( x 1 x 2 s , µ ) = dy 1 dy 2 d 2 � k 1 d 2 � dy 1 dy 2 d 2 � k 1 d 2 � k 2 k 2 i , j = q , g 0 0 ⋄ large hadron transverse momenta: � 1 ∼ � k 2 k 2 2 ≫ Λ 2 QCD ⇒ pQCD allowed ⋄ QCD collinear factorization ⋄ large rapidity intervals between hadrons (high energies) ⇒ ∆ y = ln x 1 x 2 s | � k 1 || � k 2 | � � a ( 0 ) s ln n s + a ( 1 ) s ln n − 1 s � α n α n ⇒ BFKL resummation: n n n ⋄ Collinear fragmentation of the parton i into a hadron h ⇒ convolution of D h i with a coefficient function C h i 1 � α h � i ( z ) dz → d σ h = d α h d σ i = C h dz z D h C h � z , µ i ( z , µ ) i α h where α h is the momentum fraction carried by the hadron Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 19 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Inclusive di-hadron production Observables and kinematics (di-hadrons) Observables: C n φ -averaged cross section C 0 , � cos ( n φ ) � ≡ C 0 ≡ R n 0 , with n = 1 , 2 , 3 � cos ( 2 φ ) � � cos ( φ ) � ≡ C 2 � cos ( 3 φ ) � � cos ( 2 φ ) � ≡ C 3 ≡ R 21 , ≡ R 32 . C 1 C 2 ⋄ Integrated coefficients: � y 1 , max � y 2 , max � k 1 , max � k 2 , max C n = dy 1 dy 2 dk 1 dk 2 δ ( y 1 − y 2 − Y ) C n ( y 1 , y 2 , k 1 , k 2 ) y 1 , min y 2 , min k 1 , min k 2 , min Kinematic settings: ⋄ √ s = 7 , 13 TeV ⋄ | y i | � 2 . 4 , 4 . 7 , with i = 1 , 2 ⋄ k 1 , 2 � 5 GeV ...vs k MN − jets � 35 GeV! → more secondary gluon emissions! J 1 , 2 Phenomenological analysis: ⋄ full NLA BFKL ⋄ (MSTW08, MMHT14, CT14) PDFs ⊛ ( AKK , DSS , HKNS ) FFs [F.G. C., D.Yu Ivanov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 20 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook C 0 and R nm at √ s = 13 TeV, Y � 4 . 8 , µ F = µ BLM Inclusive di-hadron production R 5 LLA AKK 10 LLA HKNS NLA AKK NLA HKNS 1 4 10 0.8 C 0 [nb] <cos φ> MOM scheme 0.6 BLM µ F = µ R = µ R 3 2 10 s = (13 TeV) 0.4 LLA AKK LLA HKNS MOM scheme NLA kernel AKK NLA kernel HKNS BLM 0.2 µ F = µ R = µ R NLA AKK 2 2 10 NLA HKNS s = (13 TeV) 0 1 2 3 4 5 1 2 3 4 5 Y Y LLA AKK LLA AKK LLA HKNS LLA HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 1 1 0.8 0.8 <cos 2 φ> C 2 /C 1 0.6 0.6 0.4 0.4 MOM scheme MOM scheme 0.2 BLM 0.2 BLM µ F = µ R = µ R µ F = µ R = µ R 2 2 s = (13 TeV) s = (13 TeV) 0 0 1 2 3 4 5 1 2 3 4 5 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 21 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Heavy-quark pair photoproduction Heavy-quark pair photoproduction Process: γ ( p 1 ) + γ ( p 2 ) → Q ( q 1 ) + X + Q ( q 2 ) ... Q stands for a charm/bottom quark or antiquark q 1 p 1 photoproduction channel collision of (quasi-)real photons equivalent photon flux approximation quark masses play the role of hard scale X first predictions within partial NLA BFKL (NLA Green’s function + LO impact factors) ⋄ LEP2 and future e + e − colliders p 2 q 2 [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2017) arXiv:1709.10032 [hep-ph] ] Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 22 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Excursus: electroproduction of ρ mesons as UGD discriminator Electroproduction of ρ mesons and UGDs Process: γ ∗ + proton → ρ + proton ...exclusive process! ⋄ leading helicity amplitudes are known (Wandzura-Wilczek) → process solved in helicity � d 2 k x = Q 2 T λ ρ λ γ ( s ; Q 2 ) = is ( k 2 ) 2 Φ γ ∗ ( λ γ ) → ρ ( λ ρ ) ( k 2 , Q 2 ) F ( x , k 2 ) , s Interesting transitions: encoded by γ ∗ → Φ γ ∗ L → ρ L L → ρ L − − − − − encoded by γ ∗ → Φ γ ∗ T → ρ T T → ρ T − − − − − ⋄ HERA data available for T 11 / T 00 [H1 Collaboration (2010)] ◮ ideal testing ground to probe and constrain the proton UGD! Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 23 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Excursus: electroproduction of ρ mesons as UGD discriminator Electroproduction of ρ mesons - T 11 / T 00 (preliminary) ⋄ Different models of UGDs need to be tested... ⋄ ...and then compared with the standard definition ( à la BFKL) • example: unpolarized model [I.P. Ivanov and N.N. Nikolaev (2002)] HERA data 1.6 W = 100 GeV Full Soft CTEQ14 - CTEQ4L Hard 1.2 hard = 0.3 GeV k min soft = 0 GeV T 11 /T 00 k min 0.8 0.4 0 0 5 10 15 20 25 2 Q ( preliminary results ) [A.D. Bolognino, MD thesis (2017)] ⇒ Further, dedicated investigation is underway (...a new Ansatz ?) = [A.D. Bolognino, F.G. C., D.Yu. Ivanov, A. Papa (in progress)] Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 24 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Outline 1 Introductory remarks QCD and semi-hard processes BFKL resummation Towards new analyses Phenomenology 2 Mueller–Navelet jet production Inclusive di-hadron production Heavy-quark pair photoproduction Excursus: electroproduction of ρ mesons as UGD discriminator 3 Conclusions & Outlook Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 25 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook Conclusions... The BFKL approach offers a common basis for the description of semi-hard processes ; it relies on a remarkable property of perturbative QCD, the gluon Reggeization Physical amplitudes in NLA are written in terms of a universal Green’s function and of process-dependent impact factors of the colliding particles The number of reactions which can be investigated within NLA BFKL depends on the list of available NLO impact factors calculated so far Successful tests of NLA BFKL in the Mueller–Navelet channel with the advent of the LHC; nevertheless, new BFKL-sensitive observables as well as more exclusive final-state reactions are needed ( di-hadron , heavy-quark pair , multi-jet production processes,...) Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 26 / 27
High-energy resummation in pQCD Introductory remarks Phenomenology Conclusions & Outlook ...Outlook ⋄ Comparison with: fixed-order DGLAP predictions, Monte Carlo inspired calculations (all processes) ⋄ Comparison higher-twist predictions: final-state objects stemming from (two) independent gluon ladders (MPI) (all processes) (Mueller–Navelet jets) [R. Maciula, A. Szczurek (2014)] (Mueller–Navelet jets) [B. Ducloué, L. Szymanowski, S. Wallon (2015)] (Four-jets) [K. Kutak, R. Maciula, M. Serino, A. Szczurek, A. van Hameren (2016, 2016)] ⋄ Inclusion of other resummation effects ⋄ Probe the BFKL dynamics through other processes... ◮ hadron-jet correlations: FF dependence + asymmetric rapidity and transverse momenta ranges [A.D. Bolognino, F.G. C., D.Yu. Ivanov, M.M. Maher, A. Papa (in progress)] ◮ heavy-quark pair production: calculation of th NLO q ¯ q impact factor hadroproduction (process initiated by quarks and gluons) [A.D. Bolognino, F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (in progress)] Francesco Giovanni Celiberto Universidad Complutense de Madrid November 16th, 2017 27 / 27
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BACKUP slides
BACKUP slides ...and azimuthal coefficients (MN-jets) � + ∞ α 2 s ( µ R ) K ( 1 ) ( n , ν ) ] α 2 d ν e ( Y − Y 0 ) [ ¯ α s ( µ R ) χ ( n , ν )+ ¯ C n = s ( µ R ) − ∞ c ( 1 ) c 1 ( n , ν ) + c ( 1 ) � � �� ( n , ν ) ( n , ν ) 1 2 × c 1 ( n , ν ) c 2 ( n , ν ) 1 + α s ( µ R ) c 2 ( n , ν ) where � n � � n � 2 + 1 2 + 1 χ ( n , ν ) = 2 ψ ( 1 ) − ψ 2 + i ν − ψ 2 − i ν χ ( n , ν )+ β 0 � − χ ( n , ν ) + 10 3 + ı d � c 1 ( n , ν ) � �� K ( 1 ) ( n , ν ) = ¯ � µ 2 χ ( n , ν ) + 2 ln d ν ln R 8 N c c 2 ( n , ν ) � C F C A � c 1 ( n , ν , | � ( � k 2 ) i ν − 1 / 2 k | , x ) = 2 f g ( x , µ F ) + f a ( x , µ F ) C A C F a = q ,¯ q ...several NLA-equivalent expressions can be adopted for C n ! − → ...we use the exponentiated one [F. Caporale, D.Yu Ivanov, B. Murdaca, A. Papa (2014)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides On the scale optimization: BLM method NLA BFKL corrections to cross section with opposite sign with respect to the leading order (LO) result and large in absolute value... ⋄ ...call for some optimization procedure... ⋄ ...choose scales to mimic the most relevant subleading terms BLM [ S.J. Brodsky, G.P. Lepage, P.B. Mackenzie (1983)] � preserve the conformal invariance of an observable... � ...by making vanish its β 0 -dependent part * “Exact” BLM: suppress NLO IFs + NLO Kernel β 0 -dependent factors * Partial (approximated) BLM: � 2 = k 1 k 2 exp � µ BLM � � 1 + 2 � − f ( ν ) − 5 � a) 2 3 I ← NLO IFs β 0 R 3 � 2 = k 1 k 2 exp � µ BLM � � 1 + 2 � − 2 f ( ν ) − 5 3 + 1 � b) 2 3 I 2 χ ( ν , n ) ← NLO Kernel β 0 R f ( ν ) depends on the process [F. Caporale, D.Yu. Ivanov, B. Murdaca, A. Papa (2015)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides MN-jets: the BFKL BLM cross section � 2 = k 1 k 2 exp µ BLM 1 + 2 − f ( ν ) − 5 ∼ 5 2 k 1 k 2 � � � � � a) 2 3 I R 3 � 2 = k 1 k 2 exp µ BLM 1 + 2 − 2 f ( ν ) − 5 3 + 1 < ( 11 . 5 ) 2 k 1 k 2 � � � � � b) 2 3 I 2 χ ( ν , n ) R � � �� � + ∞ χ ( n , ν )− T β CA χ ( n , ν )− β 0 α 2 8 CA χ 2 ( n , ν ) x J 1 x J 2 ( Y − Y 0 ) α s ( µ R ) χ ( n , ν )+ ¯ ¯ s ( µ R ) ¯ BFKL ( a ) C = d ν e n | � k J 1 || � k J 2 | − ∞ × α 2 s ( µ R ) c 1 ( n , ν , | � k J 1 | , x J 1 ) c 2 ( n , ν , | � k J 2 | , x J 2 ) c ( 1 ) c ( 1 ) � � 1 ( n , ν , | � 2 ( n , ν , | � �� 1 − 2 ¯ k J 1 | , x J 1 ) + ¯ k J 2 | , x J 2 ) πα s ( µ R ) T β + α s ( µ R ) × c 1 ( n , ν , | � c 2 ( n , ν , | � k J 1 | , x J 1 ) k J 2 | , x J 2 ) � + ∞ � � �� χ ( n , ν )− T β x J 1 x J 2 α 2 ( Y − Y 0 ) α s ( µ R ) χ ( n , ν )+ ¯ ¯ s ( µ R ) ¯ CA χ ( n , ν ) BFKL ( b ) C = d ν e n | � k J 1 || � k J 2 | − ∞ × α 2 s ( µ R ) c 1 ( n , ν , | � k J 1 | , x J 1 ) c 2 ( n , ν , | � k J 2 | , x J 2 ) � β 0 c ( 1 ) c ( 1 ) � � 1 ( n , ν , | � 2 ( n , ν , | � �� 4 π χ ( n , ν ) − 2 T β � ¯ k J 1 | , x J 1 ) + ¯ k J 2 | , x J 2 ) × 1 + α s ( µ R ) + α s ( µ R ) c 1 ( n , ν , | � c 2 ( n , ν , | � π k J 1 | , x J 1 ) k J 2 | , x J 2 ) Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides MN-jets: the DGLAP BLM cross section � 2 = k 1 k 2 exp µ BLM 1 + 2 − f ( ν ) − 5 ∼ 5 2 k 1 k 2 � � � � � a) 2 3 I R 3 � 2 = k 1 k 2 exp µ BLM 1 + 2 − 2 f ( ν ) − 5 3 + 1 < ( 11 . 5 ) 2 k 1 k 2 � � � � � b) 2 3 I 2 χ ( ν , n ) R � + ∞ x J 1 x J 2 DGLAP ( a ) d ν α 2 s ( µ R ) c 1 ( n , ν , | � k J 1 | , x J 1 ) c 2 ( n , ν , | � C = k J 2 | , x J 2 ) n | � k J 1 || � k J 2 | − ∞ � 1 − 2 πα s ( µ R ) T β + ¯ × α s ( µ R ) ( Y − Y 0 ) χ ( n , ν ) � c ( 1 ) c ( 1 ) �� 1 ( n , ν , | � 2 ( n , ν , | � ¯ k J 1 | , x J 1 ) + ¯ k J 2 | , x J 2 ) + α s ( µ R ) c 1 ( n , ν , | � c 2 ( n , ν , | � k J 1 | , x J 1 ) k J 2 | , x J 2 ) � + ∞ x J 1 x J 2 DGLAP ( b ) d ν α 2 s ( µ R ) c 1 ( n , ν , | � k J 1 | , x J 1 ) c 2 ( n , ν , | � C = k J 2 | , x J 2 ) n | � k J 1 || � k J 2 | − ∞ � β 0 4 πχ ( n , ν ) − 2 T β � � × 1 + α s ( µ R ) + ¯ α s ( µ R ) ( Y − Y 0 ) χ ( n , ν ) π c ( 1 ) c ( 1 ) � 1 ( n , ν , | � 2 ( n , ν , | � �� ¯ k J 1 | , x J 1 ) + ¯ k J 2 | , x J 2 ) + α s ( µ R ) c 1 ( n , ν , | � c 2 ( n , ν , | � k J 1 | , x J 1 ) k J 2 | , x J 2 ) Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides MN-jets: the “exact" BLM cross section � � �� � + ∞ χ ( n , ν )+ T conf x J 1 x J 2 α MOM ( µ BLM α MOM ( µ BLM ( Y − Y 0 ) ¯ ) χ ( n , ν )+ ¯ ) ¯ χ ( n , ν ) s s C BLM R R Nc = d ν e n | � k J 1 || � k J 2 | − ∞ × ( α MOM ( µ BLM )) 2 c 1 ( n , ν , | � k J 1 | , x J 1 ) c 2 ( n , ν , | � k J 2 | , x J 2 ) s R � � � c ( 1 ) c ( 1 ) � 1 ( n , ν , | � 2 ( n , ν , | � + 2 T conf ¯ k J 1 | , x J 1 ) + ¯ k J 2 | , x J 2 ) 1 + α MOM ( µ BLM × ) , s R c 1 ( n , ν , | � c 2 ( n , ν , | � N c k J 1 | , x J 1 ) k J 2 | , x J 2 ) with the µ BLM scale chosen as the solution of the following integral equation... R � s ∞ � ¯ α MOM ( µ BLM � ) χ ( n , ν ) � x J 1 x J 2 s � 3 R C β α MOM ( µ BLM n ≡ d ν ) s R | � k J 1 || � s 0 k J 2 | − ∞ � 3 + ln ( µ BLM ) 2 � � × c 1 ( n , ν ) c 2 ( n , ν ) β 0 5 1 + 2 R − 2 3 I 2 N c Q 1 Q 2 � ��� 3 + ln ( µ BLM ) 2 ) ln s χ ( n , ν ) − χ ( n , ν ) + 5 � 1 + 2 ! α MOM ( µ BLM R + ¯ − 2 3 I = 0 s R s 0 2 2 Q 1 Q 2 Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides ...choosing the µ BLM scale (MN-jets) R ...which represents the condition that terms proportional to β 0 in C n disappear � � � α MOM = − π 1 + 4 α s ( µ R ) T 1 − , 2 T π with T = T β + T conf , T β = − β 0 � 1 + 2 � , 3 I 2 � 17 T conf = C A 2 I + 3 � 1 − 1 � ξ 2 − 1 � 6 ξ 3 2 ( I − 1 ) ξ + 3 I , 8 � 1 ln ( x ) where I = − 2 0 dx x 2 − x + 1 ≃ 2 . 3439 and ξ is a gauge parameter. Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides R nm for k J 1 > 35 GeV, k J 2 > 50 GeV at √ s = 7 TeV C 1 /C 0 C 2 /C 0 1.2 k J 2 > 50 GeV 1 1 0.8 0.8 � cos ( 2 φ ) � � cos ( φ ) � 0.6 0.6 BFKL a 0.4 BFKL a 0.4 k J 2 > 50 GeV BFKL b BFKL b DGLAP a DGLAP a 0.2 0.2 DGLAP b DGLAP b 0 0 2 4 6 8 10 2 4 6 8 10 Y Y C 3 /C 0 C 2 /C 1 1 1 k J 2 > 50 GeV 0.8 0.8 0.6 � cos ( 2 φ ) � 0.6 � cos ( 3 φ ) � � cos ( φ ) � 0.4 0.4 BFKL a BFKL a BFKL b BFKL b k J 2 > 50 GeV DGLAP a DGLAP a 0.2 DGLAP b 0.2 DGLAP b 0 0 2 4 6 8 10 2 4 6 8 10 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2015)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Exclusion of central jet rapidities (MN-jets) Motivation... ⋄ At given Y = y J 1 − y J 2 ... → | y J i | could be so small ( � 2 ), that the jet i is actually produced in the central region, rather than in one of the two forward regions → longitudinal momentum fractions of the parent partons x ∼ 10 − 3 → for | y J i | and | k J i | < 100 GeV ⇒ increase of C 0 by 25 % due to NNLO PDF effects [J. Currie, A. Gehrmann-De Ridder, E. W. N. Glover, J. Pires (2014)] ! Our BFKL description of the process could be not so accurate... ...let’s return to the original Mueller–Navelet idea! ⋄ remove regions where jets are produced at central rapidities... → ...in order to reduce as much as possible theoretical uncertainties Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
1 J y BACKUP slides Rapidity range (MN-jets) � 4 . 7 � 4 . 7 � � � � | y 1 | − y C | y 2 | − y C � � dy 1 dy 2 δ ( y 1 − y 2 − Y ) θ θ C n y J 1 , y J 2 , k J 1 , k J 2 max max − 4 . 7 − 4 . 7 0 4.7 9.4 9.4 y J 1 � y J 2 Y 4.7 4.7 0 � 4.7 0 4.7 Y = y J 1 − y J 2 y J 1 Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
y 1 J BACKUP slides Rapidity range (MN-jets) � 4 . 7 � 4 . 7 � � � � | y 1 | − y C | y 2 | − y C � � dy 1 dy 2 δ ( y 1 − y 2 − Y ) θ θ C n y J 1 , y J 2 , k J 1 , k J 2 max max − 4 . 7 − 4 . 7 � y max C y max C 0 4.7 9.4 9.4 4.7 � y max C y J 1 � y J 2 Y 4.7 4.7 C 4.7 � y max 0 C C � 4.7 � y max 0 y max 4.7 Y = y J 1 − y J 2 y J 1 Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
J y 1 BACKUP slides Rapidity range (MN-jets) � 4 . 7 � 4 . 7 � � � � | y 1 | − y C | y 2 | − y C � � dy 1 dy 2 δ ( y 1 − y 2 − Y ) θ θ C n y J 1 , y J 2 , k J 1 , k J 2 max max − 4 . 7 − 4 . 7 � y max C y max C 0 4.7 9.4 9.4 Y � 7.5 4.7 � y max C Y � 5.5 y J 1 � y J 2 Y 4.7 4.7 Y � 3.5 C 4.7 � y max Y � 1.5 0 C C � 4.7 � y max 0 y max 4.7 Y = y J 1 − y J 2 y J 1 Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides R nm for k J 1 > 20 GeV, k J 2 > 35 GeV at √ s = 13 TeV 5 1 × 10 k J 1 > 20 GeV 1 k J 2 > 35 GeV 4 1 × 10 0.8 3 1 × 10 C 1 0.6 __ C 0 C 0 k J 1 > 20 GeV k J 2 > 35 GeV 2 0.4 1 × 10 C C y max = 0 y max = 0 0.2 C C y max = 1.5 y max = 1.5 1 1 × 10 C C y max = 2.5 y max = 2.5 0 5 6 7 8 9 3 4 5 6 7 8 9 Y Y C C y max = 0 y max = 0 1 1 C C y max = 1.5 y max = 1.5 C C y max = 2.5 y max = 2.5 0.8 0.8 k J 1 > 20 GeV k J 1 > 20 GeV k J 2 > 35 GeV k J 2 > 35 GeV 0.6 0.6 C 3 C 2 __ __ C 0 C 0 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 0 vs Y = y J 1 − y J 2 - “exact" MOM BLM method 5 5 1 × 10 1 × 10 C y max = 0 C y max = 1.5 4 4 C 1 × 10 1 × 10 y max = 2.5 k J 1 > 35 GeV k J 2 > 35 GeV 3 3 1 × 10 1 × 10 C 0 C 0 k J 1 > 20 GeV k J 2 > 20 GeV 2 2 1 × 10 1 × 10 C y max = 0 C y max = 1.5 1 1 1 × 10 1 × 10 C y max = 2.5 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Y Y 5 5 1 × 10 1 × 10 k J 1 > 20 GeV k J 1 > 20 GeV k J 2 > 30 GeV k J 2 > 40 GeV 4 4 1 × 10 1 × 10 3 3 1 × 10 1 × 10 C 0 C 0 2 2 1 × 10 1 × 10 C C y max = 0 y max = 0 C C y max = 1.5 y max = 1.5 1 1 1 × 10 1 × 10 C C y max = 2.5 y max = 2.5 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 1 / C 0 vs Y - “exact" BLM method 1 1 0.8 0.8 0.6 0.6 C 1 C 1 __ __ C 0 C 0 k J 1 > 20 GeV k J 1 > 35 GeV k J 2 > 20 GeV k J 2 > 35 GeV 0.4 0.4 C C y max = 0 y max = 0 0.2 C 0.2 C y max = 1.5 y max = 1.5 C C y max = 2.5 y max = 2.5 0 0 5 6 7 8 9 5 6 7 8 9 Y Y 1 1 0.8 0.8 C 1 0.6 C 1 0.6 __ __ C 0 C 0 k J 1 > 20 GeV k J 1 > 20 GeV k J 2 > 30 GeV k J 2 > 40 GeV 0.4 0.4 C C y max = 0 y max = 0 0.2 C 0.2 C y max = 1.5 y max = 1.5 C C y max = 2.5 y max = 2.5 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 2 / C 0 vs Y - “exact" BLM method C C y max = 0 y max = 0 1 1 C C y max = 1.5 y max = 1.5 C C y max = 2.5 y max = 2.5 0.8 0.8 k J 1 > 20 GeV k J 1 > 35 GeV k J 2 > 20 GeV k J 2 > 35 GeV 0.6 0.6 C 2 C 2 __ __ C 0 C 0 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y C C y max = 0 y max = 0 1 1 C C y max = 1.5 y max = 1.5 C C y max = 2.5 y max = 2.5 0.8 0.8 k J 1 > 20 GeV k J 1 > 20 GeV k J 2 > 30 GeV k J 2 > 40 GeV C 2 0.6 C 2 0.6 __ __ C 0 C 0 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 3 / C 0 vs Y - “exact" BLM method C C y max = 0 y max = 0 1 1 C C y max = 1.5 y max = 1.5 C C y max = 2.5 y max = 2.5 0.8 0.8 k J 1 > 20 GeV k J 1 > 35 GeV k J 2 > 20 GeV k J 2 > 35 GeV 0.6 0.6 C 3 C 3 __ __ C 0 C 0 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y C C y max = 0 y max = 0 1 1 C C y max = 1.5 y max = 1.5 C C y max = 2.5 y max = 2.5 0.8 0.8 k J 1 > 20 GeV k J 1 > 20 GeV k J 2 > 30 GeV k J 2 > 40 GeV C 3 0.6 C 3 0.6 __ __ C 0 C 0 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 2 / C 1 vs Y - “exact" BLM method C C y max = 0 y max = 0 1 1 C C y max = 1.5 y max = 1.5 C C y max = 2.5 y max = 2.5 0.8 0.8 k J 1 > 20 GeV k J 1 > 35 GeV k J 2 > 20 GeV k J 2 > 35 GeV 0.6 0.6 C 2 C 2 __ __ C 1 C 1 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y C C y max = 0 y max = 0 1 1 C C y max = 1.5 y max = 1.5 C C y max = 2.5 y max = 2.5 0.8 0.8 k J 1 > 20 GeV k J 1 > 20 GeV k J 2 > 30 GeV k J 2 > 40 GeV C 2 0.6 C 2 0.6 __ __ C 1 C 1 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 3 / C 2 vs Y - “exact" BLM method C C y max = 0 y max = 0 1 1 C C y max = 1.5 y max = 1.5 C C y max = 2.5 y max = 2.5 0.8 0.8 k J 1 > 20 GeV k J 1 > 35 GeV k J 2 > 20 GeV k J 2 > 35 GeV 0.6 0.6 C 3 C 3 __ __ C 2 C 2 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y C C y max = 0 y max = 0 1 1 C C y max = 1.5 y max = 1.5 C C y max = 2.5 y max = 2.5 0.8 0.8 k J 1 > 20 GeV k J 1 > 20 GeV k J 2 > 30 GeV k J 2 > 40 GeV C 3 0.6 C 3 0.6 __ __ C 2 C 2 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides BLM comparisons of C 0 and R n 0 vs Y - y C max = 2 . 5 1 1000 0.8 0.6 C 1 k J 1 > 20 GeV __ k J 1 > 20 GeV C 0 100 C 0 k J 2 > 20 GeV k J 2 > 20 GeV 0.4 BLM a BLM a BLM b BLM b BLM exact BLM exact 10 0.2 0 5 6 7 8 9 5 6 7 8 9 Y Y 1 1 k J 1 > 20 GeV k J 1 > 20 GeV k J 2 > 20 GeV k J 2 > 20 GeV 0.8 0.8 BLM a BLM a BLM b BLM b C 2 0.6 C 3 0.6 __ __ BLM exact BLM exact C 0 C 0 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides BLM comparisons of C 2 / C 1 and C 3 / C 2 vs Y - y C max = 2 . 5 k J 1 > 20 GeV 1 1 BLM a BLM b k J 2 > 20 GeV BLM exact 0.8 0.8 BLM a BLM b BLM exact C 2 0.6 C 3 0.6 __ __ C 1 C 2 0.4 0.4 k J 1 > 20 GeV 0.2 0.2 k J 2 > 20 GeV 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Ivanov, B. Murdaca, A. Papa (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides The BFKL BLM cross section (di-hadrons) � y max � ∞ � ∞ � + ∞ � = e Y � C BLM α MOM ( µ ∗ dy 1 dk 1 dk 2 d ν exp ( Y − Y 0 ) ¯ R ) χ ( n , ν ) n s s y min k 1 , min k 2 , min − ∞ � i ν � � � � � χ ( n , ν ) + T conf k 2 � R )) 2 C F 1 α MOM ( µ ∗ 4 ( α MOM ( µ ∗ 1 + ¯ R ) ¯ χ ( n , ν ) s s C A C A | � k 1 || � � k 2 k 2 | 2 � x � α 1 � α 1 � 1 � 2 i ν − 1 dx C A � � � f g ( x ) D h f a ( x ) D h × + g a x α 1 C F x x α 1 a = q ,¯ q � z � 1 � α 2 � − 2 i ν − 1 dz C A � � α 2 � � f g ( z ) D h f a ( z ) D h × + g a z α 2 C F z z α 2 a = q ,¯ q � � c ( 1 ) c ( 1 ) �� c 2 ( n , ν ) + 2 T conf ¯ c 1 ( n , ν ) + ¯ 1 ( n , ν ) 2 ( n , ν ) α MOM ( µ ∗ × 1 + ¯ R ) , s C A with the µ ∗ R scale chosen as the solution of the following integral equation... Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Numerical specifics (di-hadrons) Numerical tools: F ORTRAN → w eak time dependence on multidim. integration ranges + NLO MSTW08 PDFs (comparison with MMHT14 and CTEQ14 ) [A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, (2009)] + three different FF parameterizations! ◮ AKK [S. Albino, B.A. Kniehl, G. Kramer, (2008)] ◮ DSS [D. de Florian, R. Sassot, M. Stratmann, (2007)] ◮ HKNS [M. Hirai, S. Kumano, T.-H. Nagai, K. Sudoh, (2007)] + CERNLIB http:/ /cernlib.web.cern.ch/cernlib Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides BLM values for µ R (di-hadrons) 50 50 45 45 40 40 35 35 1/2 1/2 30 30 BLM / (k 1 k 2 ) BLM / (k 1 k 2 ) 25 25 µ R 20 µ R 20 BLM scale for C 0 - AKK BLM scale for C 0 - AKK BLM scale for C 0 - HKNS BLM scale for C 0 - HKNS 15 15 BLM scale for C 1 - AKK BLM scale for C 1 - AKK BLM scale for C 1 - HKNS BLM scale for C 1 - HKNS BLM BLM 10 µ F = µ R 10 µ F = µ R BLM scale for C 2 - AKK BLM scale for C 2 - AKK BLM scale for C 2 - HKNS 2 BLM scale for C 2 - HKNS 2 s = (7 TeV) s = (13 TeV) 5 5 BLM scale for C 3 - AKK BLM scale for C 3 - AKK BLM scale for C 3 - HKNS BLM scale for C 3 - HKNS 0 0 5 5 1 2 3 4 1 2 3 4 Y Y 50 50 45 45 40 40 35 35 1/2 1/2 30 30 BLM / (k 1 k 2 ) BLM / (k 1 k 2 ) 25 25 20 20 µ R µ R BLM scale for C 0 - AKK BLM scale for C 0 - AKK BLM scale for C 0 - HKNS BLM scale for C 0 - HKNS 15 15 BLM scale for C 1 - AKK BLM scale for C 1 - AKK BLM scale for C 1 - HKNS BLM scale for C 1 - HKNS 10 10 BLM scale for C 2 - AKK ( µ F ) 1,2 = k 1,2 BLM scale for C 2 - AKK ( µ F ) 1,2 = k 1,2 BLM scale for C 2 - HKNS 2 BLM scale for C 2 - HKNS 2 s = (7 TeV) s = (13 TeV) 5 5 BLM scale for C 3 - AKK BLM scale for C 3 - AKK BLM scale for C 3 - HKNS BLM scale for C 3 - HKNS 0 0 1 2 3 4 5 1 2 3 4 5 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 0 at √ s = 7 , 13 TeV, Y � 4 . 8 , µ F = µ BLM R 5 5 10 10 4 4 10 10 C 0 [nb] C 0 [nb] MOM scheme MOM scheme BLM BLM µ F = µ R = µ R µ F = µ R = µ R 3 3 2 2 10 10 s = (7 TeV) s = (13 TeV) LLA AKK LLA AKK LLA HKNS LLA HKNS NLA kernel AKK NLA kernel AKK NLA kernel HKNS NLA kernel HKNS NLA AKK NLA AKK 2 2 10 NLA HKNS 10 NLA HKNS 1 2 3 4 5 1 2 3 4 5 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 0 at √ s = 7 , 13 TeV, Y � 4 . 8 , ( µ F ) 1 , 2 = | � k 1 , 2 | 5 5 10 10 4 4 10 10 C 0 [nb] C 0 [nb] MOM scheme MOM scheme BLM BLM ( µ F ) 1,2 = k 1,2 , µ R = µ R ( µ F ) 1,2 = k 1,2 , µ R = µ R 3 3 2 2 10 10 s = (7 TeV) s = (13 TeV) LLA AKK LLA AKK LLA HKNS LLA HKNS NLA kernel AKK NLA kernel AKK NLA kernel HKNS NLA kernel HKNS NLA AKK NLA AKK 2 2 NLA HKNS NLA HKNS 10 10 1 2 3 4 5 1 2 3 4 5 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides R nm at √ s = 13 TeV, Y � 4 . 8 , µ F = µ BLM R LLA AKK LLA AKK LLA HKNS LLA HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 1 1 0.8 0.8 <cos 2 φ> <cos φ> 0.6 0.6 0.4 0.4 MOM scheme MOM scheme BLM BLM 0.2 0.2 µ F = µ R = µ R µ F = µ R = µ R 2 2 s = (13 TeV) s = (13 TeV) 0 0 1 2 3 4 5 1 2 3 4 5 Y Y LLA AKK LLA AKK LLA HKNS LLA HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 1 1 0.8 0.8 <cos 3 φ> C 2 /C 1 0.6 0.6 0.4 0.4 MOM scheme MOM scheme 0.2 BLM 0.2 BLM µ F = µ R = µ R µ F = µ R = µ R 2 2 s = (13 TeV) s = (13 TeV) 0 0 1 2 3 4 5 1 2 3 4 5 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides R nm at √ s = 13 TeV, Y � 4 . 8 , ( µ F ) 1 , 2 = | � k 1 , 2 | LLA AKK LLA AKK LLA HKNS LLA HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 1 1 0.8 0.8 <cos 2 φ> <cos φ> 0.6 0.6 0.4 0.4 MOM scheme MOM scheme BLM BLM 0.2 0.2 ( µ F ) 1,2 = k 1,2 , µ R = µ R ( µ F ) 1,2 = k 1,2 , µ R = µ R 2 2 s = (13 TeV) s = (13 TeV) 0 0 1 2 3 4 5 1 2 3 4 5 Y Y LLA AKK LLA AKK LLA HKNS LLA HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 1 1 0.8 0.8 <cos 3 φ> C 2 /C 1 0.6 0.6 0.4 0.4 MOM scheme MOM scheme 0.2 BLM 0.2 BLM ( µ F ) 1,2 = k 1,2 , µ R = µ R ( µ F ) 1,2 = k 1,2 , µ R = µ R 2 2 s = (13 TeV) s = (13 TeV) 0 0 1 2 3 4 5 1 2 3 4 5 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides R nm at √ s = 7 TeV, Y � 4 . 8 , µ F = µ BLM R LLA AKK LLA AKK LLA HKNS LLA HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 1 1 0.8 0.8 <cos 2 φ> <cos φ> 0.6 0.6 0.4 0.4 MOM scheme MOM scheme BLM BLM 0.2 0.2 µ F = µ R = µ R µ F = µ R = µ R 2 2 s = (7 TeV) s = (7 TeV) 0 0 1 2 3 4 5 1 2 3 4 5 Y Y LLA AKK LLA AKK LLA HKNS LLA HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 1 1 0.8 0.8 <cos 3 φ> C 2 /C 1 0.6 0.6 0.4 0.4 MOM scheme MOM scheme 0.2 BLM 0.2 BLM µ F = µ R = µ R µ F = µ R = µ R 2 2 s = (7 TeV) s = (7 TeV) 0 0 1 2 3 4 5 1 2 3 4 5 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides R nm at √ s = 7 TeV, Y � 4 . 8 , ( µ F ) 1 , 2 = | � k 1 , 2 | LLA AKK LLA AKK LLA HKNS LLA HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 1 1 0.8 0.8 <cos 2 φ> <cos φ> 0.6 0.6 0.4 0.4 MOM scheme MOM scheme BLM BLM 0.2 ( µ F ) 1,2 = k 1,2 , µ R = µ R 0.2 ( µ F ) 1,2 = k 1,2 , µ R = µ R 2 2 s = (7 TeV) s = (7 TeV) 0 0 1 2 3 4 5 1 2 3 4 5 Y Y LLA AKK LLA AKK LLA HKNS LLA HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 1 1 0.8 0.8 <cos 3 φ> C 2 /C 1 0.6 0.6 0.4 0.4 MOM scheme MOM scheme 0.2 BLM 0.2 BLM ( µ F ) 1,2 = k 1,2 , µ R = µ R ( µ F ) 1,2 = k 1,2 , µ R = µ R 2 2 s = (7 TeV) s = (7 TeV) 0 0 1 2 3 4 5 1 2 3 4 5 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 0 at √ s = 7 , 13 TeV, Y � 9 . 4 , µ F = µ BLM R 5 5 10 10 MOM scheme BLM µ F = µ R = µ R 2 s = (7 TeV) 4 4 10 10 3 3 10 10 C 0 [nb] C 0 [nb] MOM scheme BLM µ F = µ R = µ R 2 2 10 10 2 s = (13 TeV) LLA AKK LLA AKK LLA HKNS LLA HKNS NLA kernel AKK NLA kernel AKK 1 1 10 10 NLA kernel HKNS NLA kernel HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 0 0 10 10 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 0 at √ s = 7 , 13 TeV, Y � 9 . 4 , ( µ F ) 1 , 2 = | � k 1 , 2 | 5 5 10 10 4 4 10 10 3 3 10 10 C 0 [nb] C 0 [nb] MOM scheme MOM scheme BLM BLM ( µ F ) 1,2 = k 1,2 , µ R = µ R ( µ F ) 1,2 = k 1,2 , µ R = µ R 2 2 10 2 10 2 s = (7 TeV) s = (13 TeV) LLA AKK LLA AKK LLA HKNS LLA HKNS NLA kernel AKK NLA kernel AKK 1 1 10 10 NLA kernel HKNS NLA kernel HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 0 0 10 10 5 7 5 7 6 8 9 6 8 9 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides R nm at √ s = 13 TeV, Y � 9 . 4 , µ F = µ BLM R 1 1 LLA AKK LLA AKK MOM scheme MOM scheme LLA HKNS LLA HKNS BLM NLA AKK BLM NLA AKK µ F = µ R = µ R µ F = µ R = µ R NLA HKNS NLA HKNS 2 2 0.8 s = (13 TeV) 0.8 s = (13 TeV) 0.6 0.6 <cos φ> <cos 2 φ> 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y 1 1 LLA AKK LLA AKK MOM scheme MOM scheme LLA HKNS LLA HKNS BLM NLA AKK BLM NLA AKK µ F = µ R = µ R µ F = µ R = µ R NLA HKNS NLA HKNS 2 2 0.8 s = (13 TeV) 0.8 s = (13 TeV) 0.6 0.6 <cos 3 φ> C 2 /C 1 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides R nm at √ s = 13 TeV, Y � 9 . 4 , ( µ F ) 1 , 2 = | � k 1 , 2 | 1 1 LLA AKK LLA AKK MOM scheme MOM scheme LLA HKNS LLA HKNS BLM NLA AKK BLM NLA AKK ( µ F ) 1,2 = k 1,2 , µ R = µ R ( µ F ) 1,2 = k 1,2 , µ R = µ R NLA HKNS NLA HKNS 2 2 0.8 s = (13 TeV) 0.8 s = (13 TeV) 0.6 0.6 <cos 2 φ> <cos φ> 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y 1 1 LLA AKK LLA AKK MOM scheme MOM scheme LLA HKNS LLA HKNS BLM NLA AKK BLM NLA AKK ( µ F ) 1,2 = k 1,2 , µ R = µ R ( µ F ) 1,2 = k 1,2 , µ R = µ R NLA HKNS NLA HKNS 2 2 0.8 s = (13 TeV) 0.8 s = (13 TeV) 0.6 0.6 <cos 3 φ> C 2 /C 1 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides R nm at √ s = 7 TeV, Y � 9 . 4 , µ F = µ BLM R 1 1 LLA AKK LLA AKK MOM scheme MOM scheme LLA HKNS LLA HKNS BLM NLA AKK BLM NLA AKK µ F = µ R = µ R µ F = µ R = µ R NLA HKNS NLA HKNS 2 2 0.8 s = (7 TeV) 0.8 s = (7 TeV) 0.6 0.6 <cos φ> <cos 2 φ> 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y 1 1 LLA AKK LLA AKK MOM scheme MOM scheme LLA HKNS LLA HKNS BLM NLA AKK BLM NLA AKK µ F = µ R = µ R µ F = µ R = µ R NLA HKNS NLA HKNS 2 2 0.8 s = (7 TeV) 0.8 s = (7 TeV) 0.6 0.6 <cos 3 φ> C 2 /C 1 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides R nm at √ s = 7 TeV, Y � 9 . 4 , ( µ F ) 1 , 2 = | � k 1 , 2 | 1 1 LLA AKK MOM scheme LLA HKNS BLM NLA AKK ( µ F ) 1,2 = k 1,2 , µ R = µ R NLA HKNS 2 0.8 0.8 s = (7 TeV) 0.6 0.6 MOM scheme <cos 2 φ> <cos φ> BLM ( µ F ) 1,2 = k 1,2 , µ R = µ R 2 s = (7 TeV) 0.4 0.4 0.2 LLA AKK 0.2 LLA HKNS NLA AKK NLA HKNS 0 0 5 6 7 8 9 5 6 7 8 9 Y Y 1 1 LLA AKK LLA AKK MOM scheme MOM scheme LLA HKNS LLA HKNS BLM NLA AKK BLM NLA AKK ( µ F ) 1,2 = k 1,2 , µ R = µ R ( µ F ) 1,2 = k 1,2 , µ R = µ R NLA HKNS NLA HKNS 2 2 0.8 s = (7 TeV) 0.8 s = (7 TeV) 0.6 0.6 <cos 3 φ> C 2 /C 1 0.4 0.4 0.2 0.2 0 0 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 0 at √ s = 7 , 13 TeV, µ R = � | � k 1 || � k 2 | , ( µ F ) 1 , 2 = | � k 1 , 2 | 6 6 10 10 5 5 10 10 C 0 [nb] C 0 [nb] 4 4 10 10 MS scheme MS scheme 1/2 1/2 ( µ F ) 1,2 = k 1,2 , µ R = (k 1 k 2 ) ( µ F ) 1,2 = k 1,2 , µ R = (k 1 k 2 ) 2 2 s = (7 TeV) s = (13 TeV) 3 3 10 LLA AKK 10 LLA AKK LLA HKNS LLA HKNS NLA kernel AKK NLA kernel AKK NLA kernel HKNS NLA kernel HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 2 2 10 10 1 2 3 4 5 1 2 3 4 5 Y Y 5 5 10 10 4 4 10 10 3 3 10 10 C 0 [nb] C 0 [nb] MS scheme MS scheme 1/2 1/2 ( µ F ) 1,2 = k 1,2 , µ R = (k 1 k 2 ) (µ F ) 1,2 = k 1,2 , µ R = (k 1 k 2 ) 2 2 10 2 10 2 s = (7 TeV) s = (13 TeV) LLA AKK LLA AKK LLA HKNS LLA HKNS NLA kernel AKK NLA kernel AKK 1 1 10 10 NLA kernel HKNS NLA kernel HKNS NLA AKK NLA AKK NLA HKNS NLA HKNS 0 0 10 10 5 6 7 8 9 5 6 7 8 9 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides C 0 , R 10 at √ s = 7 , 13 TeV, Y � 4 . 8 , µ F = r � | � k 1 || � k 2 | 6 10 r = 1/2 r = 1/2 r = 1 r = 1 r = 2 r = 2 r = 4 r = 4 1 5 10 0.8 C 0 [nb] 4 <cos φ> 10 0.6 0.4 3 HKNS FF parametrization HKNS FF parametrization 10 MOM scheme MOM scheme 1/2 , µ R = µ R BLM 0.2 1/2 , µ R = µ R BLM µ F = r(k 1 k 2 ) µ F = r(k 1 k 2 ) 2 2 s = (7 TeV) 2 s = (7 TeV) 10 0 1 2 3 4 5 1 2 3 4 5 Y Y 6 10 r = 1/2 r = 1/2 r = 1 r = 1 r = 2 r = 2 r = 4 1 r = 4 5 10 0.8 <cos φ> C 0 [nb] 4 10 0.6 0.4 HKNS FF parametrization 3 HKNS FF parametrization 10 MOM scheme MOM scheme 0.2 1/2 , µ R = µ R BLM µ F = r(k 1 k 2 ) 1/2 , µ R = µ R BLM µ F = r(k 1 k 2 ) 2 2 s = (7 TeV) s = (13 TeV) 2 10 0 1 2 3 4 5 1 2 3 4 5 Y Y [F.G. C., D.Yu. Iv anov, B. Murdaca, A. Papa (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Looking for new observables BFKL f eature: factorization between transverse and longitudinal (rapidities) degrees of freedom Usual “growth with energy" signal mainly probes the longitudinal degrees of freedom Mueller–Navelet correlation momenta mainly probe one of the transverse components, the azimuthal angles ! We would like to study observables for which the p T (any p T along the BFKL ladder) enters the game... ⋄ ...to probe not only the general properties of the BFKL ladder, but also “to peek into the interior"... ⋄ ...by studying azimuthal decorrelations where the p T of extra particles introduces a new dependence... ...multi-jet production! [R. Maciula, A. Szczurek (2014, 2015)] [K. Kutak, R. Maciula, M. Serino, A. Szczurek, A. van Hameren (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Looking for new observables BFKL f eature: factorization between transverse and longitudinal (rapidities) degrees of freedom Usual “growth with energy" signal mainly probes the longitudinal degrees of freedom Mueller–Navelet correlation momenta mainly probe one of the transverse components, the azimuthal angles ! We would like to study observables for which the p T (any p T along the BFKL ladder) enters the game... ⋄ ...to probe not only the general properties of the BFKL ladder, but also “to peek into the interior"... ⋄ ...by studying azimuthal decorrelations where the p T of extra particles introduces a new dependence... ...multi-jet production! [R. Maciula, A. Szczurek (2014, 2015)] [K. Kutak, R. Maciula, M. Serino, A. Szczurek, A. van Hameren (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Three- and four-jet production p 1 k A , ϑ A , Y A p 1 x 1 x 1 k A , θ A , Y A k 1 , ϑ 1 , y 1 k J , θ J , y J k 2 , ϑ 2 , y 2 x 2 x 2 k B , θ B , Y B k B , ϑ B , Y B p 2 p 2 [F. Caporale, F.G. C., G. Chachamis, A. Sabio Vera (2016)] [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2016)] [F. Caporale, G. Chachamis, B. Murdaca, A. Sabio Vera (2015)] [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
Three-jet production...
BACKUP slides An event with three tagged jets φ 1 k B k J φ 2 k A Beam axis Y B < y J < Y A Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides The three-jet partonic cross section S tarting point: differential partonic cross-section (no PDFs) d 3 ˆ � � σ 3 − jet α s ¯ p B δ ( 2 ) � � d 2 � d 2 � p A + � = � k J − � p A p B dk J d θ J dy J π k J � � � � � p B , � × ϕ k A , � p A , Y A − y J ϕ � k B , y J − Y B p 1 x 1 k A , θ A , Y A Multi-Regge kinematics Rapidity ordering: Y B < y J < Y A k J lie above the experimental resolution scale k J , θ J , y J ϕ is the LO BFKL gluon Green function α s = α s N c /π ¯ x 2 k B , θ B , Y B p 2 Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Three-jets: generalized azimuthal correlations Prescription : integrate over all angles after using the projections on the two azimuthal angle differences indicated below...to define: � 2 π � 2 π � 2 π �� d 3 ˆ σ 3 − jet � � �� � � d θ A d θ B d θ J cos M θ A − θ J − π cos N θ J − θ B − π dk J d θ J dy J 0 0 0 N � ∞ � 2 π d θ (− 1 ) M + N cos ( M θ ) cos (( N − L ) θ ) � L − 1 p 2 � N − L � � � � N dp 2 � k 2 2 2 = ¯ α s J L �� � N 0 0 � p 2 + k 2 L = 0 p 2 k 2 J + 2 J cos θ � � � p 2 + k 2 � � k 2 A , p 2 , Y A − y J p 2 k 2 J cos θ , k 2 × φ M J + 2 B , y J − Y B φ N Main observables: generalized azimuthal correlation momenta � � cos ( M ( θ A − θ J − π )) cos ( N ( θ J − θ B − π )) PQ = C MN R MN = C PR � � cos ( P ( θ A − θ J − π )) cos ( Q ( θ J − θ B − π )) ⋄ Remove the contribution from the zero conformal spin to − → drastically reduce the dependence on collinear configurations study R MN PQ with integer M , N , P , Q > 0 Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Partonic prediction of R 11 22 for k J = 30 , 45 , 70 GeV k A = 40, k B = 50, Y A = 10, Y B = 0 60 k J = 30 45 70 50 40 30 11 R 22 20 10 0 -10 -20 1 2 3 4 5 6 7 8 9 y J [F. Caporale, G. Chachamis, B. Murdaca, A. Sabio Vera (2015)] Y A − Y B is fixed to 10 ; y J varies beetwen 0 . 5 and 9 . 5 . Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Partonic prediction of R 21 22 for k J = 30 , 45 , 70 GeV k A = 40, k B = 50, Y A = 10, Y B = 0 30 k J = 30 45 70 25 20 15 21 R 22 10 5 0 -5 -10 1 2 3 4 5 6 7 8 9 y J [F. Caporale, G. Chachamis, B. Murdaca, A. Sabio Vera (2015)] Y A − Y B is fixed to 10 ; y J varies beetwen 0 . 5 and 9 . 5 . Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Next step: hadronic level predictions (3-jets) Introduce PDFs and running of the strong coupling: d σ 3 − jet = dk A dY A d θ A dk B dY B d θ B dk J dy J d θ J � � 8 π 3 C F ¯ α s ( µ R ) 3 x J A x J B p B δ ( 2 ) � � d 2 � d 2 � p A + � p A � k J − � p B N 3 k A k B k J C N C � × f g ( x J A , µ F ) + f r ( x J A , µ F ) C F r = q ,¯ q N C � × f g ( x J B , µ F ) + f s ( x J B , µ F ) C F s = q ,¯ q � � � � � p B , � × ϕ k A , � p A , Y A − y J � k B , y J − Y B ϕ Match the LHC kinematical cuts (integrate d σ 3 − jet on k T and rapidities): ⋄ 1. k A � 35 GeV; k B � 35 GeV; symmetric cuts 2. k A � 35 GeV; k B � 50 GeV; asymmetric cuts ⋄ a) Y A and Y B integrated on windows b) Y A − Y B ≡ Y fixed ⋄ binning on y J Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides a) Integrate over a forward, backward and central rapidity bin [ Y B bin ] [ y J bin ] [ Y A bin ] 2 π k B θ B k J θ J Azimuthal Angle → k A θ A 0 - 1.5 - 1 - 0.5 - ∞ ∞ max max min min Y B Y B Y B Y A Y A Y A Rapidity → Y max = − Y min = 4 . 7 A B Y min = − Y max = 3 A B Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides a) Integrate over a forward, backward and central rapidity bin [ Y B bin ] [ y J bin ] [ Y A bin ] 2 π k B θ B k J θ J Azimuthal Angle → k A θ A 0 - 0.5 0 0.5 - ∞ ∞ max max min min Y B Y B Y B Y A Y A Y A Rapidity → Y max = − Y min = 4 . 7 A B Y min = − Y max = 3 A B Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides a) Integrate over a forward, backward and central rapidity bin [ Y B bin ] [ y J bin ] [ Y A bin ] 2 π k B θ B k J θ J Azimuthal Angle → k A θ A 0 0.5 1 1.5 - ∞ ∞ max max min min Y B Y B Y B Y A Y A Y A Rapidity → Y max = − Y min = 4 . 7 A B Y min = − Y max = 3 A B Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides a) R 12 33 ( y i ) at 13 TeV k min = 35 GeV, k min = 50 GeV, k max = k max = 60 GeV (asymmetric) A B A B min = 50 GeV; s = 13 TeV; k B k J ∈ ● bin - 1, ● bin - 2, ● bin - 3 • [20, 35] GeV 6 LLA • [35, 60] GeV NLA MOM BLM • [60, 120] GeV 4 2 12 R 33 0 - 2 - 4 - 6 - 1.0 - 0.5 0.0 0.5 1.0 y i [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides a) R 12 33 ( y i ) at 7 TeV k min = 35 GeV, k min = 50 GeV, k max = k max = 60 GeV (asymmetric) A B A B min = 50 GeV; s = 7 TeV; k B k J ∈ ● bin - 1, ● bin - 2, ● bin - 3 • [20, 35] GeV 6 LLA • [35, 60] GeV NLA MOM BLM • [60, 120] GeV 4 2 12 R 33 0 - 2 - 4 - 6 - 1.0 - 0.5 0.0 0.5 1.0 y i [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides a) R 12 33 ( y i ) at 13 and 7 TeV min = 50 GeV; s = 13 TeV; k B k J ∈ ● bin - 1, ● bin - 2, ● bin - 3 min = 50 GeV; k J ∈ ● bin - 1, ● bin - 2, ● bin - 3 s = 13 TeV; k B 50 6 LLA NLA MOM BLM 4 40 2 30 12 R 33 0 δ x (%) 20 - 2 10 - 4 0 - 6 - 1.0 - 0.5 0.0 0.5 1.0 - 1.0 - 0.5 0.0 0.5 1.0 y i y i min = 50 GeV; min = 50 GeV; s = 7 TeV; k B k J ∈ ● bin - 1, ● bin - 2, ● bin - 3 s = 7 TeV; k B k J ∈ ● bin - 1, ● bin - 2, ● bin - 3 50 6 LLA NLA MOM BLM 40 4 2 30 12 δ x (%) R 33 0 20 - 2 10 - 4 0 - 6 - 1.0 - 0.5 0.0 0.5 1.0 - 1.0 - 0.5 0.0 0.5 1.0 y i y i [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides a) R 22 33 ( y i ) at 13 and 7 TeV min = 50 GeV; s = 13 TeV; k B k J ∈ ● bin - 1, ● bin - 2, ● bin - 3 min = 50 GeV; k J ∈ ● bin - 1, ● bin - 2, ● bin - 3 s = 13 TeV; k B 50 6 LLA NLA MOM BLM 4 40 2 30 22 R 33 0 δ x (%) 20 - 2 10 - 4 0 - 6 - 1.0 - 0.5 0.0 0.5 1.0 - 1.0 - 0.5 0.0 0.5 1.0 y i y i min = 50 GeV; min = 50 GeV; s = 7 TeV; k B k J ∈ ● bin - 1, ● bin - 2, ● bin - 3 s = 7 TeV; k B k J ∈ ● bin - 1, ● bin - 2, ● bin - 3 50 6 LLA NLA MOM BLM 40 4 2 30 22 δ x (%) R 33 0 20 - 2 10 - 4 0 - 6 - 1.0 - 0.5 0.0 0.5 1.0 - 1.0 - 0.5 0.0 0.5 1.0 y i y i [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides b) Integrate over a central rapidity bin [ y J bin ] 2 π k B θ B k J θ J Azimuthal Angle → k A θ A 0 - 0.5 0 0.5 - ∞ ∞ min max Y B Y A Rapidity → Y max = − Y min = 4 . 7 A B Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides b) R 12 33 vs Y at 13 TeV k min = 35 GeV, k min = 50 GeV, k max = k max = 60 GeV (asymmetric) A B A B ��� = �� ���� � = �� ���� � � � � ∈ ● ��� - �� ● ��� - �� ● ��� - � • [20, 35] GeV 4 • [35, 60] GeV • [60, 120] GeV 2 0 ��� R �� �� ��� ��� ��� - 2 - 4 - 6 6.5 7.0 7.5 8.0 8.5 9.0 � [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides b) R 12 33 vs Y at 7 TeV k min = 35 GeV, k min = 50 GeV, k max = k max = 60 GeV (asymmetric) A B A B ��� = �� ���� � = � ���� � � � � ∈ ● ��� - �� ● ��� - �� ● ��� - � • [20, 35] GeV 4 • [35, 60] GeV • [60, 120] GeV 2 0 ��� R �� �� ��� ��� ��� - 2 - 4 - 6 6.5 7.0 7.5 8.0 8.5 9.0 � [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides b) R 12 33 vs Y at 13 and 7 TeV ��� = �� ���� � = �� ���� � � � � ∈ ● ��� - �� ● ��� - �� ● ��� - � ��� = �� ���� � = �� ���� � � � � ∈ ● ��� - �� ● ��� - �� ● ��� - � 50 4 40 2 30 0 ��� �� R �� ��� ��� ��� δ x (%) 20 - 2 10 - 4 - 6 0 6.5 7.0 7.5 8.0 8.5 9.0 6.5 7.0 7.5 8.0 8.5 9.0 � � ��� = �� ���� ��� = �� ���� � = � ���� � � � � ∈ ● ��� - �� ● ��� - �� ● ��� - � � = � ���� � � � � ∈ ● ��� - �� ● ��� - �� ● ��� - � 50 4 40 2 30 0 ��� R �� �� ��� ��� ��� δ x (%) 20 - 2 10 - 4 0 - 6 6.5 7.0 7.5 8.0 8.5 9.0 6.5 7.0 7.5 8.0 8.5 9.0 � � [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides b) R 22 33 vs Y at 13 and 7 TeV ��� = �� ���� � = �� ���� � � � � ∈ ● ��� - �� ● ��� - �� ● ��� - � ��� = �� ���� � = �� ���� � � � � ∈ ● ��� - �� ● ��� - �� ● ��� - � 6 50 40 4 30 �� ��� R �� 2 δ x (%) ��� ��� ��� 20 10 0 0 - 2 6.5 7.0 7.5 8.0 8.5 9.0 6.5 7.0 7.5 8.0 8.5 9.0 � � ��� = �� ���� ��� = �� ���� � = � ���� � � � � ∈ ● ��� - �� ● ��� - �� ● ��� - � � = � ���� � � � � ∈ ● ��� - �� ● ��� - �� ● ��� - � 50 6 40 4 30 δ x (%) R �� �� ��� 2 20 ��� ��� ��� 10 0 0 6.5 7.0 7.5 8.0 8.5 9.0 - 2 6.5 7.0 7.5 8.0 8.5 9.0 � � [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
...four-jet production
BACKUP slides A four-jet primitive lego-plot 2 π k B ϑ B k 1 ϑ 1 Azimuthal Angle → k 2 ϑ 2 k A ϑ A 0 - ∞ ∞ max min Y B Y B Y A Y A y 2 y 1 Rapidity → Y max = − Y min = 4 . 7 A B Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Four-jets: generalized azimuthal coefficients - partonic level � 2 π � 2 π � 2 π � 2 π C MNL = d ϑ A d ϑ B d ϑ 1 d ϑ 2 cos ( M ( ϑ A − ϑ 1 − π )) 0 0 0 0 d 6 σ 4 − jet � � k A , � � k B , Y A − Y B cos ( N ( ϑ 1 − ϑ 2 − π )) cos ( L ( ϑ 2 − ϑ B − π )) dk 1 dy 1 d ϑ 1 dk 2 d ϑ 2 dy 2 = 2 π 2 ¯ α s ( µ R ) 2 (− 1 ) M + N + L ( ˜ Ω M , N , L + ˜ Ω M , N , − L + ˜ Ω M , − N , L k 1 k 2 + ˜ Ω M , − N , − L + ˜ Ω − M , N , L + ˜ Ω − M , N , − L + ˜ Ω − M , − N , L + ˜ Ω − M , − N , − L ) with � + ∞ � + ∞ � 2 π � 2 π ˜ Ω m , n , l = dp A p A dp B p B d φ A d φ B 0 0 0 0 � n � � n e − im φ A e il φ B � p A e i φ A + k 1 p B e − i φ B − k 2 �� � n �� � n p 2 A + k 2 p 2 B + k 2 1 + 2 p A k 1 cos φ A 2 − 2 p B k 2 cos φ B � � � � | � p B | , | � ϕ m k A | , | � p A | , Y A − y 1 ϕ l | � k B | , y 2 − Y B �� � � p 2 A + k 2 p 2 B + k 2 ϕ n 1 + 2 p A k 1 cos φ A , 2 − 2 p B k 2 cos φ B , y 1 − y 2 Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Four-jets: generalized azimuthal coefficients - partonic level � 2 π � 2 π � 2 π � 2 π C MNL = d ϑ A d ϑ B d ϑ 1 d ϑ 2 cos ( M ( ϑ A − ϑ 1 − π )) 0 0 0 0 d 6 σ 4 − jet � � k A , � � k B , Y A − Y B cos ( N ( ϑ 1 − ϑ 2 − π )) cos ( L ( ϑ 2 − ϑ B − π )) dk 1 dy 1 d ϑ 1 dk 2 d ϑ 2 dy 2 M ain observables: generalized azimuthal correlation momenta = � cos ( M ( ϑ A − ϑ 1 − π )) cos ( N ( ϑ 1 − ϑ 2 − π )) cos ( L ( ϑ 2 − ϑ B − π )) � PQR = C MNL R MNL C PRQ � cos ( P ( ϑ A − ϑ 1 − π )) cos ( Q ( ϑ 1 − ϑ 2 − π )) cos ( R ( ϑ 2 − ϑ B − π )) � Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Partonic prediction of C MNL vs k 1 , 2 (4-jets) [F. Caporale, F.G. C., G. Chachamis, A. Sabio Vera (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Partonic prediction of C MNL vs k 1 , 2 (4-jets) [F. Caporale, F.G. C., G. Chachamis, A. Sabio Vera (2016)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Partonic prediction of R MNL PQR vs k 1 , 2 (4-jets) [F. Caporale, F.G. C., G. Chachamis, A. Sabio Vera (2017)] Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
BACKUP slides Next step: hadronic level predictions (4-jets) Introduce PDFs and running of the strong coupling Use realistic LHC kinematical cuts: 1. k min = 35 GeV, k max ⋄ = 60 GeV A A k min = 45 GeV, k max = 60 GeV B B k min = 20 GeV, k max = 35 GeV 1 1 k min = 60 GeV, k max = 90 GeV 2 2 2. k min = 35 GeV, k max = 60 GeV A A k min = 45 GeV, k max = 60 GeV B B k min = 25 GeV, k max = 50 GeV 1 1 k min = 60 GeV, k max = 90 GeV 2 2 ⋄ Y = Y A − Y B fixed; Y A − y 1 = y 1 − y 2 = y 2 − Y B = Y / 3 ⋄ √ s = 7 , 13 TeV Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
8 6.5 9 8.5 7.5 7 BACKUP slides 221 at √ s = 7 TeV vs Y = Y A − Y B for two k 1 bins R 122 min = 35 GeV ; min = 45 GeV ; min = 60 GeV ; max = 90 GeV s = 7 TeV; k A k B k 2 k 2 8 20 � k 1 � GeV � 35 25 � k 1 � GeV � 50 6 4 R 221 122 2 0 � 2 6.5 7 7.5 8 8.5 9 Y [F. Caporale, F.G. C., G. Chachamis, D. Gordo Gómez, A. Sabio Vera (2016)] Y is the rapidity difference between the most forward/backward jet; Y A − y 1 = y 1 − y 2 = y 2 − Y B = Y / 3 . Francesco Giovanni Celiberto High-energy resummation in the semi-hard QCD sector November 16th, 2017
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