High energy effects in multi-jet production at LHC David Gordo G´ omez david.gordo@csic.es Instituto de F´ ısica Te´ orica UAM/CSIC Madrid, Spain in collaboration with F. Caporale, F. Celiberto, G. Chachamis, A. Sabio Vera based on Nuclear Physics B 910 (2016) 374-386 arXiv:1606.00574 V Postgraduate Meeting On Theoretical Physics November 17 th - 18 th , 2016 Oviedo, Spain
Introduction Multi-jet production Conclusions & Outlook Outline Introduction 1 Motivation BFKL Mueller Navelet jets Multi-jet production 2 A new way to probe BFKL Three-jet at partonic level Three-jet at hadronic level Conclusions & Outlook 3 David Gordo G´ omez High energy effects in multi-jet production at LHC 2 / 22
Introduction Multi-jet production Conclusions & Outlook Motivation High energy limit The high energy limit studies a limited part of the phase space, but allow us to compute things otherwise impractical Purely theoretical ⋄ CFT’s ⋄ AdS/CFT ⋄ Special Functions ⋄ Integrability Methods ⋄ Spin Chains David Gordo G´ omez High energy effects in multi-jet production at LHC 3 / 22
Introduction Multi-jet production Conclusions & Outlook Motivation High energy limit Phenomenology ⋄ Mueller-Navelet jets ⋄ Muellet-Tang jets ⋄ DIS at small x With the advent of LHC we have access to higher energies: opportunity to test pQCD in the high-energy limit and the applicability of BFKL resummation . David Gordo G´ omez High energy effects in multi-jet production at LHC 4 / 22
Introduction Multi-jet production Conclusions & Outlook BFKL BFKL BFKL does not cover all high energy energy scattering, but it is essential to understand some of its aspects. Consider quark-quark scattering in the Regge Limit . s >> | t | ∼ Q 2 >> Λ 2 QCD The amplitude at LO in α s is David Gordo G´ omez High energy effects in multi-jet production at LHC 5 / 22
Introduction Multi-jet production Conclusions & Outlook BFKL BFKL If we go to NLO large logarithms appear A ( 1 ) ∝ A ( 0 ) α s log s Q 2 Real corrections Virtual corrections p p 1 1 p p 2 2 At arbitrary order, we will have terms proportional to Q 2 ) q that are not negligible in the Regge limit. ( α s ) p ( α s log s Q 2 ) q terms LLA BFKL: ( α s log s Q 2 ) q terms NLLA BFKL: α s ( α s log s All orders result in perturbation thery! David Gordo G´ omez High energy effects in multi-jet production at LHC 6 / 22
Introduction Multi-jet production Conclusions & Outlook BFKL Rapidity variable p � tanh y = E For m=0 it coincides with the pseudo-rapidity η = y ( m = 0 ) = − log tan θ 2 Related to the angle of the Picture from momentum with the beam axis [D. Colferai, F. Schwennsen, L. Szymanowski, S. Wallon (2010)] ... 2 → 2 elastic scattering at high energies ⇒ Y ≡ y 1 − y 2 = log s | t | x 1 x 2 s ... Muller-Navelet jets ⇒ Y = ln | � k J ,1 || � k J ,2 | David Gordo G´ omez High energy effects in multi-jet production at LHC 7 / 22
Introduction Multi-jet production Conclusions & Outlook Mueller Navelet jets Warming up: Mueller–Navelet jets It has been the playground for BFKL tests since it was proposed in p 1 [ A. H. Mueller, H. Navelet (1987)] x 1 k J, 1 ⋄ At Y=0, no minijet radiation in the rapidity interval. Exact correlation d σ ∼ δ 2 ( � k J ,1 − � k J ,2 ) ⋄ At large Y the BFKL approach predicts decorrelations (minijets) x 2 k J, 2 Key observable: correlation in the azimuthal p 2 angle of the 2 tagged jets. ...large jet transverse momenta: � k 2 J ,1 ∼ � k 2 J ,2 ≫ Λ 2 QCD DGLAP evolution. pQCD applicable. x 1 x 2 s ...large rapidity interval between jets: Y = ln | � k J ,1 || � k J ,2 | BFKL resummation effects α Y ∼ 1 David Gordo G´ omez High energy effects in multi-jet production at LHC 8 / 22
Introduction Multi-jet production Conclusions & Outlook Mueller Navelet jets Warming up: Mueller–Navelet jets It has been the playground for BFKL tests since it was proposed in p 1 [ A. H. Mueller, H. Navelet (1987)] x 1 k J, 1 ⋄ At Y=0, no minijet radiation in the rapidity interval. Exact correlation d σ ∼ δ 2 ( � k J ,1 − � k J ,2 ) ⋄ At large Y the BFKL approach predicts decorrelations (minijets) x 2 k J, 2 Key observable: correlation in the azimuthal p 2 angle of the 2 tagged jets. � � ∞ d σ 1 ∑ = C 0 + 2 cos ( n θ ) C n ( 2 π ) 2 dx 1 dx 2 d | � k J ,1 | d | � k J ,2 | d θ 1 d θ 2 n = 1 David Gordo G´ omez High energy effects in multi-jet production at LHC 9 / 22
Introduction Multi-jet production Conclusions & Outlook Mueller Navelet jets Mueller–Navelet jets NLLA predictions against LHC data quite successful for large rapidities. David Gordo G´ omez High energy effects in multi-jet production at LHC 10 / 22
Introduction Multi-jet production Conclusions & Outlook Mueller Navelet jets Mueller–Navelet jets ⋄ Big dependence on high order corrections in C 0 due to collinear contamination, better to define ratios. ⋄ Focusing in azimuthal angle correlations is more fruitful than the usual ”growth with energy” behaviour. ⋄ Including more jets allow us to study azimuthal correlations and its dependence on transverse momentum. Less inclusive observables! Multi-jet production! David Gordo G´ omez High energy effects in multi-jet production at LHC 11 / 22
Introduction Multi-jet production Conclusions & Outlook A new way to probe BFKL Three- and four-jet production p 1 p 1 k A , ϑ A , Y A 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 k B , θ B , Y B x 2 k B , ϑ B , Y B p 2 p 2 [F. Caporale, G. Chachamis, B. Murdaca, A. Sabio Vera (2015)] [F. Caporale, F.G. Celiberto, G. Chachamis, D.G.G., A. Sabio Vera (2016)] [F. Caporale, F.G. Celiberto., G. Chachamis, A. Sabio Vera (2016)] [F. Caporale, F.G. Celiberto, G. Chachamis, D. G.G., A. Sabio Vera (2016)] David Gordo G´ omez High energy effects in multi-jet production at LHC 12 / 22
Introduction Multi-jet production Conclusions & Outlook Three-jet at partonic level An event with three tagged jets φ 1 k B k J φ 2 k A Beam axis 2 Π k B Θ B Y B < y J < Y A k J Θ J Azimuthal Angle � k A Θ A 0 �� � Y B y J � � Y A � Y B �� 2 Y A Rapidity � David Gordo G´ omez High energy effects in multi-jet production at LHC 13 / 22
Introduction Multi-jet production Conclusions & Outlook Three-jet at partonic level The three-jet partonic cross section Starting 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 , � × k A , � p A , Y A − y J � k B , y J − Y B ϕ ϕ p 1 Multi-Regge kinematics rapidity x 1 k A , θ A , Y A ordering: Y B < y J < Y A k J lie above the experimental resolution scale k J , θ J , y J ϕ is the BFKL gluon Green function (LLA or NLLA) α s = α s N c / π ¯ x 2 k B , θ B , Y B p 2 David Gordo G´ omez High energy effects in multi-jet production at LHC 14 / 22
Introduction Multi-jet production Conclusions & Outlook Three-jet at partonic level Generalized azimuthal correlations - partonic level 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 θ J cos ( M ( θ A − θ J − π )) cos ( N ( θ J − θ B − π )) d 3 ˆ σ 3 − jet d θ A d θ B dk J d θ J dy J 0 0 0 � ∞ � 2 π d θ ( − 1 ) M + N cos ( M θ ) cos (( N − L ) θ ) N � L − 1 p 2 � N − L � N � � k 2 dp 2 � ∑ 2 2 = ¯ α s L J �� � N 0 0 � L = 0 p 2 + k 2 p 2 k 2 J + 2 J cos θ � � p 2 + k 2 � k 2 A , p 2 , Y A − y J J cos θ , k 2 � � p 2 k 2 × φ M φ N J + 2 B , y J − Y B Main observables: generalized azimuthal correlation ratios (w/o the 0 component) PQ = C MN = � cos ( M ( θ A − θ J − π )) cos ( N ( θ J − θ B − π )) � R MN C PR � cos ( P ( θ A − θ J − π )) cos ( Q ( θ J − θ B − π )) � David Gordo G´ omez High energy effects in multi-jet production at LHC 15 / 22
Introduction Multi-jet production Conclusions & Outlook Three-jet at hadronic level Next step: hadronic level predictions 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 JA x JB � � p B δ ( 2 ) � � d 2 � d 2 � p A + � k J − � p A � p B N 3 k A k B k J C � � N C f g ( x JA , µ F ) + ∑ × f r ( x JA , µ F ) C F r = q , ¯ q � � N C f g ( x JB , µ F ) + ∑ × f s ( x JB , µ F ) C F s = q , ¯ q � � � � � p B , � × ϕ k A , � p A , Y A − y J ϕ � k B , y J − Y B Match the LHC kinematical cuts (integrate d σ 3 − jet on k T and rapidities Y A , Y B ): ⋄ 1. 35 GeV ≤ k A ≤ 60 GeV; 35 GeV ≤ k B ≤ 60 GeV; symmetric cuts 35 GeV ≤ k A ≤ 60 GeV; 50 GeV ≤ k B ≤ 60 GeV; 2. asymmetric cuts ⋄ Y = Y A − Y B fixed; y J = ( Y A + Y B ) / 2 √ s = 7, 13 TeV ⋄ David Gordo G´ omez High energy effects in multi-jet production at LHC 16 / 22
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