Threshold resummation in direct photon production Nobuo Sato Florida State University In collaboration with: J. Owens D. Westmark
Motivation: ◮ Parton distribution functions (PDFs) - essential ingredients for hadron colliders. ◮ PDFs cannot be computed from first principles - extracted from experimental data. ◮ The uncertainties in the fitted PDFs are different among the parton species. ◮ In particular, gluon distribution is highly unconstrained at large x ! ◮ Production of a state with mass m and rapidity y probes PDFs at x ∼ ( m/ √ s ) e ± y which is relevant for BSM physics.
Motivation: How to constrain gluon PDF at large x ? → Single inclusive direct photon production at fixed target experiments. ◮ In the past, the data was used to constrain gluon PDF at large x ≤ 0 . 6 . ◮ It was removed from global fittings due to inconsistencies between the theory at NLO and the data of various fixed target experiments.
Motivation: 6 WA70 √ s = 23 . 0GeV pp CDF √ s = 1800 . 0GeV p¯ data / theory(NLO) vs . x T p D0 √ s = 1960 . 0GeV p¯ µ R , IF , FF = 0 . 5 ∗ p T p 5 E706 √ s = 31 . 5GeV pp FFs = BFG II E706 √ s = 38 . 7GeV pp PHENIX √ s = 200 . 0GeV pp 4 R110 √ s = 63 . 0GeV pp data / theory(NLO) R806 √ s = 63 . 0GeV pp R807 √ s = 63 . 0GeV pp UA6 √ s = 24 . 3GeV pp 3 UA6 √ s = 24 . 3GeV p¯ p 2 1 0 0 . 2 0 . 4 0 . 6 10 − 2 10 − 1 x T
Motivation: Can we improve theory at NLO? → threshold resummation for single inclusive direct photon production. ◮ Catani, Mangano, Nason, Oleari, Vogelsang, hep-ph/9903436 (direct contribution) ◮ de Florian, Vogelsang, hep-ph/0506150 (direct + jet fragmentation)
Theory of direct photons At LO: (a) direct contribution (b) jet fragmentation dσ ( x T ) � f a/A ( x a , µ IF ) ∗ f b/B ( x b , µ IF ) ∗ D γ/c ( z, µ F F ) ∗ ˆ p 3 = Σ(ˆ x T , ... ) T dp T a,b,c ◮ Direct contribution: D γ/γ = δ (1 − z ) ◮ Jet fragmentation: D γ/c ∼ α em /α S
Theory of direct photons Beyond LO: dσ ( x T ) � f a/A ( x a , µ IF ) ∗ f b/B ( x b , µ IF ) ∗ D γ/c ( z, µ F F ) ∗ ˆ p 3 = Σ(ˆ x T , ... ) T dp T a,b,c 1 LO α s L 2 α s L α s NLO α 2 s L 4 α 2 s L 3 α 2 s L 2 α 2 s L NNLO . . . . . . . . . . . . . . . α n s L 2 n α n s L 2 n − 1 α n s L 2 n − 2 N n LO ... LL NLL NNLL ... √ ˆ x T = 2 p T /z ˆ s ˆ s = x a x b S x 2 L = ln(1 − ˆ T ) “Threshold logs” ◮ Resummation: technique to find the exponential representation of threshold logs.
Theory of direct photons When are threshold logs important? � 1 � 1 � 1 x 2 dσ ( x T ) � � p 3 � T = dx a dx b dzf a ( x a ) f b ( x b ) D ( z )Σ T z 2 x a x b dp T x 2 xT x 2 T a,b,f T xa √ xaxb CDF:(collider) √ √ ˆ x T = 2 p T /z s ˆ S = 1 . 8 TeV s = x a x b S ˆ x T = [0 . 03 , 0 . 11] x 2 L = ln(1 − ˆ T ) UA6:(fixed target) √ x T ˆ x T = z √ x a x b = [ x T , 1] S = 24 GeV √ x T = [0 . 3 , 0 . 6] x T = 2 p T / S ◮ Threshold logs are more relevant for fixed target experiments. ◮ Due to trigger bias effect � z � → 1 which leads to enhancement of fragmentation component from theshold logs.
Theory of direct photons Key observation: D.de Florian,W.Vogelsang (Phys.Rev. D72 (2005)) Fractional Contribution 1 . 0 direct direct+fragment fragment direct+fragment 0 . 8 LO NLO 0 . 6 NLL ratio 0 . 4 ratio vs . p T pp → γ + X √ s = 24 . 3 GeV 0 . 2 PDFs = Cteq6 FFs = BFG µ R , IF , FF = p T 0 . 0 4 . 0 4 . 5 5 . 0 5 . 5 6 . 0 6 . 5 7 . 0 p T
Theory of direct photons ◮ Resummation is performed in “mellin space”: � 1 � c + i ∞ 1 dxx N − 1 f ( x ) dNx − N F N f N = f ( x ) = 2 πi 0 c − i ∞ ◮ The invariant cross section in N-space: dσ ( N ) p 3 � f a/A ( N + 1) f b/B ( N + 1) D γ/c (2 N + 3)ˆ = Σ( N ) T dp T a,b,f ◮ The resummed partonic cross section in N-space is given by: Σ NLL ( N ) = Ce S [ln( N +1)] ˆ ˆ Σ Born ( N ) e S (ln( N )) = ∆ a G i ∆ ( int ) N ∆ b N ∆ c N J d � N i,N i
Phenomenology NLO 10 2 NLO + NLL UA6 ζ = 0 . 5 10 1 ζ = 1 . 0 Ed σ/ d 3 p (pb) ζ = 2 . 0 10 0 Ed σ/ d 3 p (pb) vs p T (GeV) UA6 experiment 10 − 1 pp → γ + X √ s = 24 . 3 GeV PDFs = Cteq6 , FFs = BFGII 10 − 2 4 . 0 4 . 5 5 . 0 5 . 5 6 . 0 6 . 5 7 . 0 p T (GeV)
NLO+NLL 7000.0 µ R = µ IF = µ FF = 0 . 5 ∗ p T 7000.0 1960.0 NLO 1800.0 630.0 1960.0 1800.0 S √ 630.0 DOF vs 630.0 630.0 546.0 χ 2 200.0 63.0 DOF profile 63.0 63.0 38.7 38.7 31.5 31.5 Phenomenology: χ 2 24.3 24.3 23.75 23.0 25 20 15 10 5 0 DOF χ 2
NLO+NLL 7000.0 µ R = µ IF = µ FF = 1 . 0 ∗ p T 7000.0 1960.0 NLO 1800.0 630.0 1960.0 1800.0 S √ 630.0 DOF vs 630.0 630.0 546.0 χ 2 200.0 63.0 DOF profile 63.0 63.0 38.7 38.7 31.5 31.5 Phenomenology: χ 2 24.3 24.3 23.75 23.0 25 20 15 10 5 0 DOF χ 2
NLO+NLL 7000.0 µ R = µ IF = µ FF = 2 . 0 ∗ p T 7000.0 1960.0 NLO 1800.0 630.0 1960.0 1800.0 S √ 630.0 DOF vs 630.0 630.0 546.0 χ 2 200.0 63.0 DOF profile 63.0 63.0 38.7 38.7 31.5 31.5 Phenomenology: χ 2 24.3 24.3 23.75 23.0 25 20 15 10 5 0 DOF χ 2
S NLO+NLL √ 7000.0 µ R = µ IF = µ FF = 0 . 5 ∗ p T Optimal Normalization vs 7000.0 1960.0 NLO 1800.0 630.0 1960.0 1800.0 Phenomenology: Optimal normalization 630.0 630.0 630.0 546.0 200.0 63.0 63.0 63.0 38.7 38.7 31.5 31.5 24.3 24.3 23.75 23.0 6 5 4 3 2 1 0 Optimal Normalization
S NLO+NLL √ 7000.0 µ R = µ IF = µ FF = 1 . 0 ∗ p T Optimal Normalization vs 7000.0 1960.0 NLO 1800.0 630.0 1960.0 1800.0 Phenomenology: Optimal normalization 630.0 630.0 630.0 546.0 200.0 63.0 63.0 63.0 38.7 38.7 31.5 31.5 24.3 24.3 23.75 23.0 6 5 4 3 2 1 0 Optimal Normalization
S NLO+NLL √ 7000.0 µ R = µ IF = µ FF = 2 . 0 ∗ p T Optimal Normalization vs 7000.0 1960.0 NLO 1800.0 630.0 1960.0 1800.0 Phenomenology: Optimal normalization 630.0 630.0 630.0 546.0 200.0 63.0 63.0 63.0 38.7 38.7 31.5 31.5 24.3 24.3 23.75 23.0 6 5 4 3 2 1 0 Optimal Normalization
Phenomenology: Gluon constraints ◮ Bayesian reweighting technique. Watt and Thorne (1205.4024), NNPDF collaboration (1012.0836) ◮ The idea: ◮ Compute cross section for the available data sets using an ensemble of N random PDFs: { PDF K } . ◮ Compute the χ 2 DOF ( k ) of the combined data sets for each member PDF K in the ensemble. ◮ Compute weights as: e − 1 2 χ 2 DOF ( k ) w K = N k e − 1 2 χ 2 DOF ( k ) � ◮ The new constrained PDFs and its uncertainty can be written as: � PDF � ( x, Q ) = 1 � w k PDF ( x, Q ) k N k � 1 � w k ( PDF ( x, Q ) k − � PDF � ( x, Q )) 2 δ � PDF � ( x, Q ) = N k
Phenomenology: Gluon constraints (preliminary) eigen directions 1 . 4 reweighted 1 . 2 Ratio to CTEQ6 1 . 0 0 . 8 Gluon(x , Q = 10 . 0 GeV) 0 . 6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x
Conclusions: ◮ High- x PDFs important for production of a state with mass m at forward rapidities. ◮ Threshold resummation improves the theoretical prediction of direct photons at fixed target experiments → potential constrains on gluon PDF up to x ∼ 0 . 6 . To do: ◮ Constrain PDFs with threshold resummation in DIS and lepton-pair production (D. Westmark). ◮ Compare our results with SCET calculations by Becher and Schwarts (only resummation of direct part).
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