Resummation in PDF fj ts Luca Rottoli Rudolf Peierls Centre for - PowerPoint PPT Presentation

Resummation in PDF fj ts Luca Rottoli Rudolf Peierls Centre for Theoretical Physics, University of Oxford

LHC, New Physics, and the pursuit of Precision LHC as a discovery machine ‣ Higgs Boson ✓ 10 1 𐄃 ‣ BSM particles (never as of today) RadISH 2.0 10 0 8 TeV, pp → Z ( → l + l − ) + X 0.0 < η < 2.4, 66 < m ll < 116 GeV Focus in LHC run II NNPDF3.0 (NNLO) (1 / σ ) d σ /dp T 10 − 1 uncertainties with µ R , µ F , Q variations ‣ Measurement of the Standard Model parameters with very Fixed Order from arXiv:1610.01843 high precision 10 − 2 ‣ Signals of New Physics beyond the Standard Model NNLO 10 − 3 NNLO+NNLL NNLO+N 3 LL A theorist’s Quest: Data 10 − 4 ‣ New BSM scenarios to be tested 1 . 20 1 . 15 1 . 10 ‣ New techniques to enhance signal/background ratio and 1 . 05 1 . 00 0 . 95 isolate tiny deviations from SM predictions 0 . 90 0 . 85 ‣ Development of accurate and precise theoretical predictions 0 . 80 10 1 10 2 p T Goal: 1% accuracy in theoretical predictions [Bizon,Monni,Re,LR,Torielli et al, in preparation] 1 Università di Cagliari, November 22, 2017

LHC, New Physics, and the pursuit of Precision Large HADRON Collider A crucial ingredient in the physics precision programme at the LHC is the accurate understanding of the internal structure of the initial state hadrons 2 Università di Cagliari, November 22, 2017

Factorization X ˆ σ ab → X b a proton proton Proton’s dynamics occurs on a timescale ~ 1fm Production of a heavy particle e.g. Higgs Production ( hard process ) occurs on timescale 1/M X ~ 1/100 GeV ~ 0.002 fm Large separation between scales allows to separate the hard process and treat it independently from the hadronic dynamics: collinear factorization 3 Università di Cagliari, November 22, 2017

Factorization X √ centre-of-mass energy s hard scale of the process Q ˆ σ ab → X b a h 2 h 1 Z σ ( s , Q 2 ) = ∑ dx 1 dx 2 f a / h 1 ( x 1 , Q 2 ) f b / h 2 ( x 2 , Q 2 ) ˆ σ ab → X ( Q 2 , x 1 x 2 s ) X a , b 4 Università di Cagliari, November 22, 2017

Factorization X √ centre-of-mass energy s hard scale of the process Q ˆ σ ab → X b a h 2 h 1 Z σ ( s , Q 2 ) = ∑ dx 1 dx 2 f a / h 1 ( x 1 , Q 2 ) f b / h 2 ( x 2 , Q 2 ) ˆ σ ab → X ( Q 2 , x 1 x 2 s ) X a , b σ X ( Q 2 , s ) = ∑ f a / h 1 ( Q 2 ) ⊗ f b / h 2 ( Q 2 ) ⊗ ˆ σ ab → X ( Q 2 , s ) a , b 4 Università di Cagliari, November 22, 2017

Factorization X √ centre-of-mass energy s hard scale of the process Q ˆ σ ab → X b a h 2 h 1 Z σ ( s , Q 2 ) = ∑ dx 1 dx 2 f a / h 1 ( x 1 , Q 2 ) f b / h 2 ( x 2 , Q 2 ) ˆ σ ab → X ( Q 2 , x 1 x 2 s ) X a , b partonic cross-section short-distance: perturbative 5 Università di Cagliari, November 22, 2017

Factorization X √ centre-of-mass energy s hard scale of the process Q ˆ σ ab → X b a h 2 h 1 Z σ ( s , Q 2 ) = ∑ dx 1 dx 2 f a / h 1 ( x 1 , Q 2 ) f b / h 2 ( x 2 , Q 2 ) ˆ σ ab → X ( Q 2 , x 1 x 2 s ) X a , b partonic cross-section short-distance: perturbative QCD at short distance is perturbative ( asymptotic freedom ) σ = ˆ σ 0 ( 1 + . . . ) ˆ LO 5 Università di Cagliari, November 22, 2017

Factorization X √ centre-of-mass energy s hard scale of the process Q ˆ σ ab → X b a h 2 h 1 Z σ ( s , Q 2 ) = ∑ dx 1 dx 2 f a / h 1 ( x 1 , Q 2 ) f b / h 2 ( x 2 , Q 2 ) ˆ σ ab → X ( Q 2 , x 1 x 2 s ) X a , b partonic cross-section short-distance: perturbative QCD at short distance is perturbative ( asymptotic freedom ) σ 0 ( 1 + α s c 1 + α 2 σ = ˆ s c 2 + . . . ) ˆ NLO 5 Università di Cagliari, November 22, 2017

Factorization X √ centre-of-mass energy s hard scale of the process Q ˆ σ ab → X b a h 2 h 1 Z σ ( s , Q 2 ) = ∑ dx 1 dx 2 f a / h 1 ( x 1 , Q 2 ) f b / h 2 ( x 2 , Q 2 ) ˆ σ ab → X ( Q 2 , x 1 x 2 s ) X a , b partonic cross-section short-distance: perturbative QCD at short distance is perturbative ( asymptotic freedom ) σ 0 ( 1 + α s c 1 + α 2 σ = ˆ s c 2 + . . . ) ˆ NNLO 5 Università di Cagliari, November 22, 2017

Factorization X √ centre-of-mass energy s hard scale of the process Q ˆ σ ab → X b a h 2 h 1 Z σ ( s , Q 2 ) = ∑ dx 1 dx 2 f a / h 1 ( x 1 , Q 2 ) f b / h 2 ( x 2 , Q 2 ) ˆ σ ab → X ( Q 2 , x 1 x 2 s ) X a , b Parton Distribution Functions (PDFs) long-distance: non-perturbative Parton distribution functions (PDFs) are universal objects which encode information on the substructure of the proton and which describe the dynamics of quarks and gluons ( partons ) PDFs are currently extracted from experiments 6 Università di Cagliari, November 22, 2017

Parton Distribution Functions f ( x , Q 2 ) PDFs depend on two kinematic variables fraction of the momentum of the proton 7 Università di Cagliari, November 22, 2017

Parton Distribution Functions f ( x , Q 2 ) PDFs depend on two kinematic variables Scale of the process 7 Università di Cagliari, November 22, 2017

Parton Distribution Functions f ( x , Q 2 ) PDFs depend on two kinematic variables and are parametrized at an initial scale Q 0 7 Università di Cagliari, November 22, 2017

Parton Distribution Functions f ( x , Q 2 ) PDFs depend on two kinematic variables and are parametrized at an initial scale Q 0 10 7 Evolution in Q 2 is encoded in DGLAP equation 10 6 ⇣ x Z 1 ∂ dz ⌘ Q 2 ∂ Q 2 f i ( x , Q 2 ) = z , α s ( Q 2 ) f j ( z , Q 2 ) z P ij 10 4 x Q 2 [GeV] 10 3 10 2 10 1 1 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 1 x 7 Università di Cagliari, November 22, 2017

Parton Distribution Functions f ( x , Q 2 ) PDFs depend on two kinematic variables 10 7 Evolution in Q 2 is encoded in DGLAP equation 10 6 ⇣ x Z 1 ∂ dz ⌘ Q 2 ∂ Q 2 f i ( x , Q 2 ) = z , α s ( Q 2 ) f j ( z , Q 2 ) z P ij 10 4 x Q 2 [GeV] splitting functions 10 3 10 2 10 1 1 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 1 x 7 Università di Cagliari, November 22, 2017

Parton Distribution Functions f ( x , Q 2 ) PDFs depend on two kinematic variables 10 7 Evolution in Q 2 is encoded in DGLAP equation 10 6 ⇣ x Z 1 ∂ dz ⌘ Q 2 ∂ Q 2 f i ( x , Q 2 ) = z , α s ( Q 2 ) f j ( z , Q 2 ) z P ij 10 4 x Q 2 [GeV] splitting functions 10 3 10 2 ⇣ ⌘ = P ( 0 ) ij ( x ) + α s P ( 1 ) s P ( 2 ) x , α s ( Q 2 ) ij ( x ) + α 2 ij ( x ) + . . . P ij 10 1 1 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 1 x 7 Università di Cagliari, November 22, 2017

DGLAP equation ⇣ x Z 1 ∂ dz ⌘ Q 2 ∂ Q 2 f i ( x , Q 2 ) = z , α s ( Q 2 ) f j ( z , Q 2 ) z P ij x 2n f + 1 coupled di ff erential equation number of (active) fm avours However, strong interactions do not tell apart quarks and antiquarks ( charge conjugation and SU(n f ) fm avour symmetry ) P q i q j = P ¯ P q i ¯ q j = P ¯ P q i g = P ¯ q i g ≡ P qg , P gq i = P g ¯ q i ≡ P gq q j , q i q j , q i ¯ Σ ( x , Q 2 ) = ∑ [ q i ( x , t ) + ¯ q i ( x , t )] Only singlet combination couples to gluon i ✓ ◆ ✓ ◆ ✓ ◆ ∂ Σ P qq P qg Σ Q 2 ⊗ = ∂ ln Q 2 g P gq P gg g 8 Università di Cagliari, November 22, 2017

DGLAP equation Q 2 = 10 GeV 2 Q 2 = 10 4 GeV 2 1 1 NNPDF3.1 (NNLO) g/10 0.9 0.9 2 4 2 2 2 xf(x, =10 GeV ) xf(x, =10 GeV ) µ µ s 0.8 0.8 g/10 0.7 0.7 d Q 2 evolution 0.6 0.6 u v c 0.5 0.5 u v 0.4 0.4 u d v b s 0.3 0.3 d v 0.2 0.2 d u 0.1 0.1 c 0 0 3 3 − − 2 1 2 1 − − − − 10 10 10 10 10 10 1 1 x x growth of small- x gluon Parton lose momentum and shifts at smaller values of x 9 Università di Cagliari, November 22, 2017

Parton Distribution Function Fits σ X ( Q 2 , s ) = ∑ f a / h 1 ( Q 2 ) ⊗ f b / h 2 ( Q 2 ) ⊗ ˆ σ ab → X ( Q 2 , s ) a , b d Q 2 dQ 2 f i ( Q 2 ) = P ij ( α s ( Q 2 )) ⊗ f j ( Q 2 ) theoretical input theoretical prediction (to be compared with data) 10 Università di Cagliari, November 22, 2017

Parton Distribution Function Fits σ X ( Q 2 , s ) = ∑ f a / h 1 ( Q 2 ) ⊗ f b / h 2 ( Q 2 ) ⊗ ˆ σ ab → X ( Q 2 , s ) a , b d Q 2 dQ 2 f i ( Q 2 ) = P ij ( α s ( Q 2 )) ⊗ f j ( Q 2 ) theoretical input theoretical prediction (to be compared with data) PDF fj ts are typically based on fj xed-order theory… σ 0 ( 1 + α s c 1 + α 2 σ = ˆ s c 2 + . . . ) ˆ ⇣ ⌘ = P ( 0 ) ij ( x ) + α s P ( 1 ) s P ( 2 ) x , α s ( Q 2 ) ij ( x ) + α 2 ij ( x ) + . . . P ij …but is fj xed-order theory always good enough? 10 Università di Cagliari, November 22, 2017

Large logarithms Single (double) logarithmic enhancement s ln j α k 0 ≤ j ≤ ( 2 ) k Perturbative convergence is spoiled when Finite in the limit x → 0 α s ln ( 2 ) ∼ 1 e.g. small- x behaviour of splitting functions " # ∞ n ⇣ α s ⌘ n m − 1 ln m − 1 1 A ( n ) P ( n ) ( x ) x + x ¯ ∑ ∑ xP ( x , α s ) = 2 π n = 0 m = 1 Instability at small- x 11 Università di Cagliari, November 22, 2017

Large logarithms Single (double) logarithmic enhancement s ln j α k 0 ≤ j ≤ ( 2 ) k Perturbative convergence is spoiled when Finite in the limit x → 0 α s ln ( 2 ) ∼ 1 e.g. small- x behaviour of splitting functions " # ∞ n ⇣ α s ⌘ n m − 1 ln m − 1 1 A ( n ) P ( n ) ( x ) x + x ¯ ∑ ∑ xP ( x , α s ) = 2 π n = 0 m = 1 Instability at small- x All-order resummation of the logarithmically enhanced terms (n ≥ 0, m=n) leading-logarithm (LL x), (n ≥ 0, m=n,n-1) next-to-leading-logarithm (NLL x ), etc. 12 Università di Cagliari, November 22, 2017