Resummation in PDF fits. Luca Rottoli Rudolf Peierls Centre for Theoretical Physics, University of Oxford

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1 Resummation in PDF fits Luca Rottoli Rudolf Peierls Centre for Theoretical Physics, University of Oxford

2 LHC, New Physics, and the pursuit of Precision LHC as a discovery machine Higgs Boson 10 1 BSM particles Focus in LHC run II (never as of today) Measurement of the Standard Model parameters with very high precision (1/ )d /dpt RadISH TeV, pp! Z(! l + l )+X 0.0< <2.4, 66< m ll <116 GeV NNPDF3.0 (NNLO) uncertainties with µ R,µ F,Q variations Fixed Order from arxiv: Signals of New Physics beyond the Standard Model A theorist s Quest: NNLO NNLO+NNLL NNLO+N 3 LL Data New BSM scenarios to be tested New techniques to enhance signal/background ratio and isolate tiny deviations from SM predictions Development of accurate and precise theoretical predictions p T Goal: 1% accuracy in theoretical predictions [Bizon,Monni,Re,LR,Torielli et al, in preparation] 1

3 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

4 Factorization X proton a ŝ ab!x b 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 ~ fm Large separation between scales allows to separate the hard process and treat it independently from the hadronic dynamics: collinear factorization 3

5 Factorization X p s Q centre-of-mass energy hard scale of the process a ŝ ab!x h 1 h 2 b s(s, Q 2 )=Â X a,b Z dx 1 dx 2 f a/h1 (x 1, Q 2 ) f b/h2 (x 2, Q 2 )ŝ ab!x (Q 2, x 1 x 2 s) 4

6 Factorization X p s Q centre-of-mass energy hard scale of the process h 1 a ŝ ab!x b h 2 s(s, Q 2 )=Â X a,b Z dx 1 dx 2 f a/h1 (x 1, Q 2 ) f b/h2 (x 2, Q 2 )ŝ ab!x (Q 2, x 1 x 2 s) s X (Q 2, s) =Â a,b f a/h1 (Q 2 ) f b/h2 (Q 2 ) ŝ ab!x (Q 2, s) 4

7 Factorization X p s Q centre-of-mass energy hard scale of the process h 1 a ŝ ab!x b h 2 s(s, Q 2 )=Â X a,b Z dx 1 dx 2 f a/h1 (x 1, Q 2 ) f b/h2 (x 2, Q 2 )ŝ ab!x (Q 2, x 1 x 2 s) partonic cross-section short-distance: perturbative 5

8 Factorization X p s Q centre-of-mass energy hard scale of the process h 1 a ŝ ab!x b h 2 s(s, Q 2 )=Â X a,b Z dx 1 dx 2 f a/h1 (x 1, Q 2 ) f b/h2 (x 2, Q 2 )ŝ ab!x (Q 2, x 1 x 2 s) partonic cross-section short-distance: perturbative QCD at short distance is perturbative (asymptotic freedom) ŝ = ŝ 0 (1 +...) 5 LO

9 Factorization X p s Q centre-of-mass energy hard scale of the process h 1 a ŝ ab!x b h 2 s(s, Q 2 )=Â X a,b Z dx 1 dx 2 f a/h1 (x 1, Q 2 ) f b/h2 (x 2, Q 2 )ŝ ab!x (Q 2, x 1 x 2 s) partonic cross-section short-distance: perturbative QCD at short distance is perturbative (asymptotic freedom) ŝ = ŝ 0 (1 + a s c 1 + a 2 s c ) 5 NLO

10 Factorization X p s Q centre-of-mass energy hard scale of the process h 1 a ŝ ab!x b h 2 s(s, Q 2 )=Â X a,b Z dx 1 dx 2 f a/h1 (x 1, Q 2 ) f b/h2 (x 2, Q 2 )ŝ ab!x (Q 2, x 1 x 2 s) partonic cross-section short-distance: perturbative QCD at short distance is perturbative (asymptotic freedom) ŝ = ŝ 0 (1 + a s c 1 + a 2 s c ) 5 NNLO

11 Factorization X p s Q centre-of-mass energy hard scale of the process h 1 a ŝ ab!x b h 2 s(s, Q 2 )=Â X a,b Z dx 1 dx 2 f a/h1 (x 1, Q 2 ) f b/h2 (x 2, Q 2 )ŝ ab!x (Q 2, x 1 x 2 s) 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

12 Parton Distribution Functions PDFs depend on two kinematic variables f (x, Q 2 ) fraction of the momentum of the proton 7

13 Parton Distribution Functions PDFs depend on two kinematic variables f (x, Q 2 ) Scale of the process 7

14 Parton Distribution Functions PDFs depend on two kinematic variables f (x, Q 2 ) and are parametrized at an initial scale Q 0 7

15 Parton Distribution Functions PDFs depend on two kinematic variables f (x, Q 2 ) and are parametrized at an initial scale Q Evolution in Q 2 is encoded in DGLAP equation 10 6 Q 2 Q 2 f i(x, Q 2 )= Z 1 x dz z P ij x z, a s(q 2 ) f j (z, Q 2 ) 10 4 Q 2 [GeV] x

16 Parton Distribution Functions PDFs depend on two kinematic variables f (x, Q 2 ) 10 7 Evolution in Q 2 is encoded in DGLAP equation 10 6 Q 2 Q 2 f i(x, Q 2 )= Z 1 x dz z P ij x z, a s(q 2 ) f j (z, Q 2 ) 10 4 splitting functions Q 2 [GeV] x

17 Parton Distribution Functions PDFs depend on two kinematic variables f (x, Q 2 ) 10 7 Evolution in Q 2 is encoded in DGLAP equation 10 6 Q 2 Q 2 f i(x, Q 2 )= Z 1 x dz z P ij x z, a s(q 2 ) f j (z, Q 2 ) 10 4 splitting functions Q 2 [GeV] 10 3 x, P ij a s (Q 2 ) = P (0) (x)+a ij s P (1) (x)+a 2 ij s P (2) ij 10 2 (x) x

18 DGLAP equation Q 2 Q 2 f i(x, Q 2 )= Z 1 x dz z P ij x z, a s(q 2 ) f j (z, Q 2 ) 2n f + 1 coupled differential equation number of (active) flavours However, strong interactions do not tell apart quarks and antiquarks (charge conjugation and SU(n f ) flavour symmetry) P qi q j = P qi q j, P qi q j = P qi q j, P qi g = P qi g P qg, P gqi = P g qi P gq Only singlet combination couples to gluon S(x, Q 2 )=Â[q i (x, t)+ q i (x, t)] i Q 2 ln Q 2 S g = Pqq P gq P qg P gg S g 8

19 DGLAP equation g/10 s Q 2 = 10 GeV 2 Q 2 = 10 4 GeV 2 NNPDF3.1 (NNLO) 2 xf(x,µ 2 =10 GeV ) d v u v Q 2 evolution s c b g/10 d u 4 xf(x,µ 2 =10 u v d v 2 GeV ) c x u d x growth of small-x gluon Parton lose momentum and shifts at smaller values of x 9

20 Parton Distribution Function Fits s X (Q 2, s) =Â a,b f a/h1 (Q 2 ) f b/h2 (Q 2 ) ŝ ab!x (Q 2, s) theoretical prediction (to Q 2 d dq 2 f i(q 2 )=P ij (a s (Q 2 )) f j (Q 2 ) theoretical input be compared with data) 10

21 Parton Distribution Function Fits s X (Q 2, s) =Â a,b f a/h1 (Q 2 ) f b/h2 (Q 2 ) ŝ ab!x (Q 2, s) theoretical prediction (to Q 2 d dq 2 f i(q 2 )=P ij (a s (Q 2 )) f j (Q 2 ) theoretical input be compared with data) PDF fits are typically based on fixed-order theory ŝ = ŝ 0 (1 + a s c 1 + a 2 s c ) x, P ij a s (Q 2 ) = P (0) (x)+a ij s P (1) (x)+a 2 ij s P (2) (x)+... ij but is fixed-order theory always good enough? 10

22 Large logarithms Single (double) logarithmic enhancement k s ln j 0 j (2)k Perturbative convergence is spoiled when Finite in the limit x 0 s ln (2) 1 e.g. small-x behaviour of splitting functions xp(x, a s )= Â n=0 as 2p n " nâ m=1 # A (n) m 1 lnm 1 1 x + x P (n) (x) Instability at small-x 11

23 Large logarithms Single (double) logarithmic enhancement k s ln j 0 j (2)k Perturbative convergence is spoiled when Finite in the limit x 0 s ln (2) 1 e.g. small-x behaviour of splitting functions xp(x, a s )= Â n=0 as 2p n " nâ m=1 # A (n) m 1 lnm 1 1 x + x P (n) (x) Instability at small-x All-order resummation of the logarithmically enhanced terms (n 0, m=n) leading-logarithm (LLx), (n 0, m=n,n-1) next-to-leading-logarithm (NLLx), etc. 12

24 Resummation in PDF fits Including resummation in PDF fits: provides consistent predictions when resummed computations are used improves the quality of the PDF fits helps in investigating the impact of missing higher orders it brings us closer to all-order PDFs 13

25 Global PDF fits Processes used in global PDF fits 14

26 Global PDF fits Processes used in global PDF fits Collider Drell-Yan Z differential top production Collider Deep-Inelastic Scattering Fixed-Target Drell-Yan Jets Fixed-Target Deep-Inelastic Scattering 14

27 Resummation in global PDF fits Large x: threshold resummation double logs due to soft gluon emission ln k (1 x) (1 x) + [Bonvini,Marzani,Rojo,LR,Ubiali,Ball,Bertone, Carrazza,Hartland ] 15

28 Resummation in global PDF fits Small x: high-energy resummation single logs due to high-energy gluon emission 1 x lnk x [Ball,Bertone,Bonvini,Marzani,Rojo,LR ] 15

29 Resummation in global PDF fits Resummation affects: Observable (coefficient functions) Evolution (splitting functions) = 0 C( s (µ) f (µ) [ f (µ)] µ 2 d dµ 2 f (µ) =P( s(µ)) f (µ) observable (coefficient function) evolution (splitting function) small x NLLx* NLLx large x (N)NNLL 16 *starts at NLLx

30 PDFs with large-x resummation [Bonvini,Marzani,Rojo,LR,Ubiali,Ball,Bertone, Carrazza,Hartland ]

31 PDFs with large-x resummation: NNPDF3.0res [Bonvini,Marzani,Rojo,LR,Ubiali,Ball,Bertone, Carrazza,Hartland ] Datasets considered in NNPDF3.0res process observable included? DIS dσ/(dxdq 2 ) (NC, CC, F2c ) DY Z/γ dσ/(dydm 2 ) DY W differential in lepton kinematics no public code available yet tt total σ jets inclusive dσ/(dydp T ) NLL known to be poor Accuracy is competitive with global fit, except for large-x gluon (jets not included) Resummation is included supplementing fixed-order computations with K-factors public code TROLL K Nk LO+N k LL = Nk LO+N k LL N k LO bonvini/troll 17

32 NNPDF3.0res: Impact on phenomenology Higgs Production Higgs cross section: gluon fusion SM Higgs is not affected by resummation of PDFs ratio to central NNLO (with baseline PDFs) NNLO, fixed order PDFs NNLO+NNLL, fixed order PDFs NNLO+NNLL, resummed PDFs LHC 13 TeV m H ~2 TeV NNLO+NNLL with resummed PDFs is similar to FO PDFs (larger uncertainty) m H [GeV] m H ~600 GeV cancellation of 1/2 of the enhancement 18

33 NNPDF3.0res: Impact on phenomenology Susy particles [Beenakker,Borschensky,Krämer,Kulesza,Laenen,Marzani,Rojo ] K NLO+NLL (pp! g g + X) p S = 13 TeV Predictions for MSSM particles are Global fit NLL/NLO DIS+DY+top Prescription (1) Prescription (2) However, PDF errors are very large modified when using 1.20 resummed PDFs m q = m g = m [GeV] 19

34 Outlook First ever global fit of PDFs with threshold resummation PDFs are suppressed in the large-x region; at intermediate values of x quark PDFs are slightly enhanced (sum rule); negligible effects at x<0.01 Inclusion of resummation compensates the enhancement from resummation in partonic cross sections Consistent resummed calculations might be closer to fixed order results Limitations: larger uncertainties due to reduced dataset. Methodology enables to have truly global resummed PDFs when calculations for missing processes will be available. New processes to be included: DY Z/γ (Zp T ), tt (differential) 20

35 PDFs with small-x resummation [Ball,Bertone,Bonvini,Marzani,Rojo,LR ]

36 Need for small-x resummation? Deep Inelastic Scattering HERA dataset H1 and ZEUS σ r,nc (x,q 2 ) x 2 i x = , i=21 x = , i=20 x = , i=19 x = , i=18 x = , i=17 x = , i=16 x = , i=15 x = , i=14 x = , i=13 x = , i=12 x = 0.005, i=11 x = 0.008, i=10 x = 0.013, i=9 x = 0.02, i=8 HERA I NC e + p Fixed Target HERAPDF1.0 x = 0.032, i=7 data collected down to very small x x = 0.05, i=6 10 x = 0.08, i=5 x = 0.13, i=4 1 x = 0.18, i=3 x = 0.25, i= x = 0.40, i=1 Very good agreement over vast range of x and Q x = 0.65, i= Q 2 / GeV 2 21 Figure 9: HERA combined NC e + p reduced cross section and fixed-target data as a function 2

37 Need for small-x resummation? Description of HERA data poorer when data points at smaller values of x are included and fixed-order theory is used Fixed order theory could be not sufficient to describe data points at small x and/or small Q 2 Effect is more pronounced if NNLO theory is used more points at small x included Courtesy of Juan Rojo This may indicate the need for small-x resummation 22

38 (very brief) Overview of small-x resummation Small-x resummation based on kt-factorization and BFKL. Developed mostly in the 90s-00s [Catani,Ciafaloni,Colferai,Hautmann,Salam, Stasto][Altarelli,Ball,Forte] [Thorne,White] Affects both evolution (LLx, NLLx) and coefficient functions (NLLx, lowest logarithmic order) in the singlet sector Splitting functions are resummed using ABF (Altarelli,Ball,Forte) procedure New formalism for coefficient function [Bonvini,Marzani,Peraro ] and further improvements on the ABF formalism [Bonvini,Marzani,Muselli,Peraro ] Resummed splitting functions and coefficient functions available through public code HELL Use in PDF fits possible thanks to the interface with APFEL bonvini/hell apfel.hepforge.org 23

39 Towards a global small-x resummed fit All ingredients for a PDF fit to DIS data are now available In principle, one should add additional processes: DY Q 2 x 1/( 0c) 2 Jets top Ongoing work in this direction (temporary) Exclusion region for hadronic data However, a global fit is possible if conservatives cuts on hadronic data are applied and points which may feature small-x enhancement are excluded s(q 2 ) log 1 x c 1 24 Value of c (slope of the line) selects the exclusion region

40 NNPDF31sx: impact on PDFs stabilization of the gluon with respect to the perturbative order NNPDF31sx DIS only, Q = 100 GeV g(x,q 2 ) /g(x,q 2 )[ref] NLO NNLO NNLO+NLLx x PDFs compatible within error at medium and large x 25

41 NNPDF31sx: fit quality 2 /N dat 2 2 /N dat 2 NLO NLO+NLLx NNLO NNLO+NLLx NMC SLAC BCDMS CHORUS NuTeV dimuon HERA I+II incl. NC HERA I+II incl. CC HERA c NC HERA F2 b DY E866 DY d / p DY DY E886 p DY E605 p CDF Z rap CDF Run II k t jets D0 Z rap D0 W! e asy D0 W! µ asy ATLAS total ATLAS W, Z 7TeV ATLAS HM DY 7 TeV ATLAS W, Z 7TeV ATLAS jets TeV ATLAS jets 2.76 TeV ATLAS jets TeV ATLAS Zp T 8TeV(p ll T,M ll) ATLAS Zp T 8TeV(p ll T,y ll) ATLAS tt tot ATLAS t t rap CMS total CMS Drell-Yan 2D CMS jets 7 TeV CMS jets 2.76 TeV CMS Zp T 8TeV(p ll T,y ll) CMS tt tot CMS t t rap Total χ 2 NNLO+NLLx smallest 1.10 vs ~1.12 (NLO & NLO+NLLx), 1.13 (NNLO) 26

42 NNPDF31sx: fit quality 2 /N dat 2 2 /N dat 2 NLO NLO+NLLx NNLO NNLO+NLLx NMC SLAC BCDMS CHORUS NuTeV dimuon HERA I+II incl. NC HERA I+II incl. CC HERA c NC HERA F2 b DY E866 DY d / p DY DY E886 p DY E605 p CDF Z rap CDF Run II k t jets D0 Z rap D0 W! e asy D0 W! µ asy sensible improvement in the χ 2 (χ 2 NNLO-χ 2 NNLO+NLLx)= -121 ATLAS total ATLAS W, Z 7TeV ATLAS HM DY 7 TeV ATLAS W, Z 7TeV ATLAS jets TeV ATLAS jets 2.76 TeV ATLAS jets TeV ATLAS Zp T 8TeV(p ll T,M ll) ATLAS Zp T 8TeV(p ll T,y ll) ATLAS tt tot ATLAS t t rap CMS total CMS Drell-Yan 2D CMS jets 7 TeV CMS jets 2.76 TeV CMS Zp T 8TeV(p ll T,y ll) CMS tt tot CMS t t rap Total

43 NNPDF31sx: fit quality 2 /N dat 2 2 /N dat 2 NLO NLO+NLLx NNLO NNLO+NLLx NMC SLAC BCDMS CHORUS NuTeV dimuon HERA I+II incl. NC HERA I+II incl. CC HERA c NC HERA F2 b DY E866 DY d / p DY DY E886 p DY E605 p CDF Z rap CDF Run II k t jets D0 Z rap D0 W! e asy D0 W! µ asy sensible improvement in the χ 2 mostly driven by HERA DIS data ATLAS total ATLAS W, Z 7TeV ATLAS HM DY 7 TeV ATLAS W, Z 7TeV ATLAS jets TeV ATLAS jets 2.76 TeV ATLAS jets TeV ATLAS Zp T 8TeV(p ll T,M ll) ATLAS Zp T 8TeV(p ll T,y ll) ATLAS tt tot ATLAS t t rap CMS total CMS Drell-Yan 2D CMS jets 7 TeV CMS jets 2.76 TeV CMS Zp T 8TeV(p ll T,y ll) CMS tt tot CMS t t rap Total

44 Small-x resummation and HERA data Compute the χ 2 removing data points in the region where resummation effects are expected cuts on DIS data a s (Q 2 ) ln 1 x D cut resummation effects might be important here fixed-order description should be good here 27

45 Small-x resummation and HERA data NNPDF3.1sx, HERA NC inclusive data NNLO NNLO+NLLx NLO NLO+NLLx NNLO worsens if small-x data are included /N dat χ NNLO+NLLx χ 2 flattens at larger values of D cut D cut NNLO+NLLx offers the best description 28

46 Small-x resummation and HERA data r,nc HERA NC p s = 920 GeV, Q 2 =4.5 GeV 2 NNLO NNLO+NLLx HERA data FL(x,Q 2 ) x =8.8e-5 x =1.3e-4 x =1.7e-4 x =2.2e-4 x = 3.2e-4 x = 4.0e-4 x = 5.4e-4 x = 6.9e-4 NNPDF3.1sx x = 9.6e-4 x = 1.2e-3 x = 1.6e-3 x = 2.4e-3 x = 3.0e-3 x = 4.0e-3 x = 5.4e-3 x = 7.4e-3 x = 9.9e-3 x = 1.8e-2 Ratio to data x NNLO NNLO+NLLx H Q 2 [GeV 2 ] improved description of data at small-x and 29 their slope

47 Small-x resummation and collider processes Need to include small-x resummation in hadronic cross section (especially DY) First estimate of impact of resummation can be obtained by computing approximate results with resummation only in PDF evolution s X (Q 2, s) =Â a,b f a/h1 (Q 2 ) f b/h2 (Q 2 ) ŝ ab!x (Q 2, s) Q 2 d dq 2 f i(q 2 )=P ij (a s (Q 2 )) f j (Q 2 ) 30

48 Small-x resummation and collider processes Need to include small-x resummation in hadronic cross section (especially DY) First estimate of impact of resummation can be obtained by computing approximate results with resummation only in PDF evolution 1.06 NNPDF31sx, CMS 7 TeV, 20 GeV <M ll < 30 GeV 1.04 Ratio to NNLO NNLO NNLO+NLLx Dilepton rapidity y ll precision LHC phenomenology in extreme kinematic regions would require small-x resummed PDFs 31

49 Conclusions & outlook First global fit with small-x resummation in the NNPDF framework Evidence that NNLO+NLLx improves with respect to NNLO Description of the data at small x/small Q 2 significantly improves when resummation effects are included Potential for reducing uncertainties for processes not necessarily related to small-x physics Computation of small-x resummation for other processes needed Motivation to explore further probes of small-x dynamics at the LHC, such as low-mass DY at LHCb PDF sets with joint (large-x + small-x) resummation? 32

50 Backup

51 Threshold resummation in a nutshell Convolution integral diagonalise in Mellin space (x, Q 2 )=x x 1 dz z L x z, Q2 ˆ (z, Q 2 ) z Double logarithmic enhancement due to soft gluon emission (N, Q 2 )=L(N, Q 2 ) 0 (N, Q 2 )C(N) C(N) =1 + n=1 s 2n k=0 c nk ln k N + O(1/N) N-soft Exponentiation The functions g i resum α s kln k N to all orders C(N) =g 0 ( s ) exp 1 s g 1 ( s ln N)+g 2 ( s ln N)+ s g 3 ( s ln N)+... LL NLL NNLL

52 Impact on PDFs NLO+NLL ) [ref] 2 ) [new] / g ( x, Q 2 g ( x, Q NNPDF3.0 DIS+DY+Top, Q =10 NLO NLO+NLL 2 GeV ) [ref] 2 ) [new] / Σ ( x, Q 2 Σ ( x, Q NNPDF3.0 DIS+DY+Top, Q =10 NLO NLO+NLL 2 GeV 1 10 x 1 10 x NNPDF3.0 DIS+DY+Top, Q =10 2 GeV NNPDF3.0 DIS+DY+Top, Q =10 2 GeV NNLO NNLO ) [ref] 1.2 NNLO+NNLL ) [ref] 1.2 NNLO+NNLL 2 ) [new] / g ( x, Q 2 2 NNLO+NNLL g ( x, Q ) [new] / Σ ( x, Q 2 Σ ( x, Q x 1 10 x

53 Parton Distribution Functions PDFs depend on two kinematic variables f (x, Q 2 ) 10 7 Evolution in x 2 is encoded in BFKL equation x x f +(x, Q 2 )= Z 0 dn 2 n K µ 2 n 2, a s(q 2 ) f + (x, n 2 ) Q 2 [GeV] x

54 Small-x resummation of DGLAP evolution ABF procedure based on duality with BFKL evolution symmetry of the BFKL kernel momentum conservation resummation of (subleading, but fundamental) running coupling effects LO NLO NNLO α s = 0.20, n f = 4, Q 0 MS LO NLO NNLO α s = 0.20, n f = 4, Q 0 MS x P gg (x) x P qg (x) x x Courtesy of Marco Bonvini

55 Small-x resummation of DGLAP evolution ABF procedure based on duality with BFKL evolution symmetry of the BFKL kernel Now matching at NNLO available! momentum conservation resummation of (subleading, but fundamental) running coupling effects x P gg (x) LO NLO NNLO LO+LL NLO+NLL NNLO+NLL α s = 0.20, n f = 4, Q 0 MS x P qg (x) LO NLO NNLO NLO+NLL NNLO+NLL α s = 0.20, n f = 4, Q 0 MS dip [Ciafaloni,Colferai,Salam,Stasto] x x Courtesy of Marco Bonvini

56 Small-x resummation of coefficient functions x C 2,g (x,α s ) α s = 0.20, n f = 4, Q 0 MS NLO NLO+NLL NNLO NNLO+NLL N3LO x C L,g (x,α s ) α s = 0.20, n f = 4, Q 0 MS NLO NLO+NLL NNLO NNLO+NLL N3LO x x massive DIS coefficient functions available and implemented in HELL VFNS (FONLL = S-ACOT) implementation resummed matching conditions in HELL C [n f +1] L,g (m) =C [n f ] L,g (m), C[n f +1] 2,g (m) =C [n f ] 2,g (m) K hg(m) f [n f +1] i (m) = Courtesy of Marco Bonvini j=g,q i...q n f K ij (m) f [n f ] j, i = g, q,...q n f +1 See also [Thorne,White]