Fragmentation Functions and Global QCD Fits

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1 Fragmentation Functions and Global QCD Fits The 32nd Annual Hampton University Graduate Studies Program (HUGS2017) Emanuele R. Nocera Rudolf Peierls Centre for Theoretical Physics - University of Oxford Emanuele R. Nocera (Oxford) (Oxford) FFs and Global QCD Fits June / 48

2 1 Lecture 1: Theoretical framework Table of contents foreword: why Fragmentation Functions? the basics: recap on factorisation and evolution single inclusive annihilation as a case study single hadron production in semi-inclusive DIS and proton-proton collisions 2 Lecture 2: Methodological aspects of global QCD fits determining the probability density in a space of functions goodness of fit parton parametrisation, uncertainty representation, parameter optimisation fit validation 3 Lecture 3: Phenomenology of Fragmentation Functions overview of recent determinations of Fragmentation Functions higher-order corrections, heavy quark mass effects applications of Fragmentation Functions DISCLAIMER These lectures contain a personal selection of topics and are certainly not exhaustive Emanuele R. Nocera (Oxford) (Oxford) FFs and Global QCD Fits June / 48

3 Bibliography 1 General textbooks on perturbative QCD R. K. Ellis, W. J. Stirling and B. R. Webber, QCD and Collider Physics, Cambridge (1996) J. C. Collins, Foundations of perturbative QCD, Cambridge (2011) 2 Reviews on Fragmentation Functions A. Metz and A. Vossen, Prog. Part. Nucl. Phys. 91 (2016) 136 S. Albino, Rev. Mod. Phys. 82 (2010) 2489 S. Albino et al. [arxiv:0804:2021] 3 Reviews on Parton Distribution Functions S. Forte and G. Watt, Ann. Rev. Nucl. Part. Sci. 63 (2013) 291 P. Jimenez-Delgado, W. Melnitchouk and J. F. Owens, J. Phys. G40 (2013) S. Forte, Acta Phys. Polon. B41 (2010) Specific topics not addressed above more journal references along the way as we proceed DISCLAIMER These lectures will focus on collinear leading-twist Fragmentation Functions for single-inclusive unpolarised hadron production Transverse-momentum-dependent/multi-hadron fragmentation not covered here Emanuele R. Nocera (Oxford) (Oxford) FFs and Global QCD Fits June / 48

4 Fragmentation Functions and Global QCD Fits Lecture 1: Theoretical framework Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

5 Outline 1 Foreword: why Fragmentation Functions? from partons in the initial state to partons in the final state 2 The basics definition of Fragmentation Functions, factorisation, evolution properties of splitting functions, theoretical constraints 3 SIA as a case study SIA cross sections/multiplicities: theoretical expectations and data sets flavours schemes: changing the number of active flavours higher-order QCD corrections 4 Other processes hadron production in SIDIS: multiplicities and data sets hadron production in pp collisions: cross section and data sets Focus the attention on single-inclusive annihilation as a paradigm for the illustration of features/issues of fragmentation in the cleanest theoretical framework Try to minimise overlap with J. W. Qiu s lectures Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

6 1.1 Foreword: why Fragmentation Functions? Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

7 Hadron physics, or the quest for the nucleon structure Nucleons make up all nuclei, and hence most of the visible matter in the Universe They are bound states with internal structure and dynamics Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

8 The QCD picture of the nucleon naive picture realistic picture three non-relativistic quarks QCD indefinite number of relativistic factorization,evolution quarks and gluons Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

9 QCD, a powerful theoretical framework 1 What are the general features of QCD? keywords: renormalisation; asymptotic freedom; infrared safety; factorisation addressed in Jianwei Qiu s lecture on the first week 2 How reliable is a theoretical QCD calculation? keywords: scale dependence; higher-order corrections; resummation(s) addressed in Jianwei Qiu s lecture on the first week 3 How can we relate QCD to experiment? keywords: factorisation; evolution; distribution functions; global analysis addressed in these lectures (with a focus on Fragmentation Functions) 4 QCD has proven to be successful in the last 35+ years GOALS Qualitative QCD: discovery physics Set up and check the framework Quantitative QCD: precision physics Understand and describe a large class of hard-scattering processes and phenomena Precision QCD: new physics Investigate possible deviations from the Standard Model Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

10 Lepton-hadron facilities Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

11 Hadron-hadron facilities Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

12 Nucleons in the initial state: Parton Distribution Functions 1 The densites of partons f = q, q, g with momentum fraction x f(x) f (x) + f (x) f(x) f (x) f (x) 2 Allow for a proper field-theoretic definition as matrix elements of bilocal operators proton parton P xp q(x) = q(x) = collinear transition of a massles proton h into a massless parton i with fractional momentum x local OPE = lattice formulation [Rev.Mod.Phys. 67 (1995) 157] 1 dy e iy xp + P ψ(0, y, 0 )γ + ψ(0) P 4π 1 dy e iy xp + P, S ψ(0, y, 0 )γ + γ 5 ψ(0) P, S 4π with light-cone coordinates y = (y +, y, y ), y + = (y 0 + y z )/ 2, y = (y 0 y z )/ 2, y = (v x, v y ) 3 The definitions can be generalised to include a transverse-momentum dependence Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

13 Nucleons in the initial state: Parton Distribution Functions Process Reaction Subprocess PDFs probed x l ± {p, n} l ± + X γ q q q, q, g x 0.01 l ± n/p l ± + X γ d/u d/u d/u x 0.01 ν( ν)n µ (µ + ) + X W q q q, q 0.01 x 0.5 νn µ µ + + X W s c s 0.01 x 0.2 νn µ + µ + X W s c s 0.01 x 0.2 e ± p e ± + X γ q q g, q, q x 0.1 e + p ν + X W + {d, s} {u, c} d, s x 0.01 e ± p e ± c c + X γ c c, γ g c c c, g x 0.1 e ± p jet(s) + X γ g q q g 0.01 x 0.1 l ± { p, d, n } l ± + X γ q q q + q, g x 0.8 pp µ + µ + X uū, d d γ q x 0.35 pn/pp µ + µ + X (u d)/(uū) γ d/ū x 0.35 p p(pp) jet(s) + X gg, qg, qq 2jets g, q x 0.5 p p (W ± l ± ν) + X ud W +, ū d W u, d, ū, d x 0.05 pp (W ± l ± ν) + X u d W +, dū W u, d, ū, d, (g) x p p(pp) (Z l + l ) + X uu, dd(uū, d d) Z u, d(g) x pp (W + c) + X gs W c, g s W + c s, s x 0.01 pp t t + X gg t t g x 0.01 p p W ± + X ul dr W +, d L ū R W u ū d d 0.05 x 0.4 p p π + X gg qg, qg qg g 0.05 x 0.4 l ± { p, d } l ± h + X γ u ū d d q q x 0.5 g l ± { p, d } l ± D + X γ g c c g 0.06 x 0.2 Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

14 ] 2 Unpolarised PDFs Accessed kinematic coverage Polarised PDFs 2 [ GeV T / p / M 2 Q 7 10 NMC SLAC BCDMS CHORUS NTVDMN EMCF2C HERACOMB HERAF2CHARM F2BOTTOM DYE886 DYE605 CDF D0 ATLAS CMS LHCb ] 2 [GeV /p 2 /M 2 Q T EMC SMC SMClowx E142 E143 E154 E155 COMPASS-D COMPASS-P COMPASS-P15 HERMES97 HERMES JLAB-E JLAB-EG1-DVCS JLAB-E COMPASS-OC ± STAR-W STAR-jets PHENIX-jets x O(4000) data points after cuts Q 2 cut = 1 GeV 2 W 2 cut = GeV x O(400) data points after cuts Q 2 cut = 1 GeV 2 W 2 cut = GeV 2 kinematic cuts: Q 2 Q 2 cut (pqcd) and W 2 = m 2 p + 1 x x Q2 Wcut 2 (no HT) Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

15 Unpolarised and polarised PDFs Unpolarised PDFs CT14 MMHT14 NNPDF3.1 ABM16 HERAPDF2.0 fixed-target DIS HERA fixed-target DY Tevatron (W, Z) Tevatron (jets) LHC (W, Z) latest update PRD 89 (2014) EPJC 75 (2015) EPJC 75 (2015) arxiv: arxiv: Polarised PDFs DSSV NNPDFpol1.1 JAM LSS BB DIS SIDIS pp (jets,π 0 ) (jets,w ± ) latest update PRL 113 (2014) NPB 887 (2014) PRD 93 (2016) PRD 82 (2010) NPB 841 (2010) Most of them are publicly available through the LHAPDF interface [EPJ C75 (2015) 132] Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

16 Polarised and unpolarised PDFs NNPDF3.1 (NNLO) g/10 xf(x,µ 2 =10 GeV ) u v s c g/10 d u v 4 2 xf(x,µ 2 =10 GeV ) u 0.3 s d v 0.3 b d v c x d u x Largely benefit from HERA and LHC programs Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

17 Polarised and unpolarised PDFs Are starting to benefit from JLAB and RHIC programs Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

18 1.2 Fragmentation Functions: the basics Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

19 Hadrons in the final state: Fragmentation Functions FFs allow for a proper field-theoretic definition as matrix elements of bilocal operators parton hadron P/z P D h i (z) = 1 12π X collinear transition of a massless parton i into a massless hadron h with fractional momentum z no local OPE = no lattice formulation [Rev.Mod.Phys. 67 (1995) 157] dy e i P + z y Tr [ γ + 0 ψ(y)p h(p ) X h(p ) X P ψ(0) 0 ] with light-cone coordinates and appropriate gauge links P, P y = (y +, y, y ), y + = (y 0 + y z )/ 2, y = (y 0 y z )/ 2, y = (v x, v y ) All these definitions have ultraviolet divergences which must be renormalized in order to define finite PDFs to be used in the factorization formulas (PDF/FFs are scheme dependent) All these definitions can be generalised to include longitudinal/transverse polarizations Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

20 Factorisation of physical observables [Adv.Ser.Direct.HEP 5 (1988) 1] 1 A variety of sufficiently inclusive processes allow for a factorised description short-distance part hard interaction of partons process-dependent kernels e + X factorisation scheme & scale µ N X long-distance part nucleon structure universal parton distributions 2 Physical observables are written as a convolution of coefficient functions and FFs O I = 1 ( ) C Ii (y, α s(µ 2 )) D i (y, µ 2 dy x )+p.s. corrections f g = i=q, q,g x y f g(y) y e h l l N 2 X h h e + + e h + X single-inclusive annihilation (SIA) l + N l + h + X semi-inclusive deepinelastic scattering (SIDIS) 3 Coefficient functions allow for a perturbative expansion N 1 X N 1 + N 2 h + X high-p T hadron production in proton-proton collisions (pp) C If (y, α s) = a k s C (k) If (y), as = αs/(4π) k=0 4 After factorization, all quantities (including FFs/PDFs) depend on µ Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

21 Evolution of FFs: DGLAP equations [NP B126 (1977) 298] 1 A set of (2n f + 1) integro-differential equations, n f is the number of active flavors n f 1 ln µ 2 D i(x, µ 2 ) = j x 2 Often written in a convenient basis of PDFs dz z P ( ji z, αs(µ 2 ) ) ( x D j z, µ2) n f n f D NS;± = (D q ± D q) (D q ± D q ) D NS;v = (D q D q) D Σ = (D q + D q) q q ln µ 2 ln µ 2 D NS;±,v(x, µ 2 ) = P ±,v (x, µ 2 F ) D NS;±,v(x, µ 2 ) ( DΣ (x, µ 2 ) ( ) ) P qq 2n f P gq D g(x, µ 2 = ) P gg 1 2n f P qg 3 With perturbative computable (time-like) splitting functions P ji (z, α s) = k=0 ( DΣ (x, µ 2 ) D g(x, µ 2 ) a k+1 s P (k) ji (z), a s = α s/(4π) ) P qq (0) P gq (0) P qg (0) P gg (0) Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

22 Splitting functions: LO and NLO Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

23 Splitting functions: NNLO Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

24 Properties of splitting functions 1 At LO: time-like and space-like splitting functions are equal, provided Pqg S,(0) Pgq T,(0) 2 At NLO [NPB 175 (1980) 27, PLB 97 (1980) 497, PRD 48 (1993) 116] time-like and space-like splitting functions are related by analytic continuation 3 at NNLO [PLB 638 (2006) 61, PLB 659 (2008) 290, NPB 854 (2012) 133] an uncertainty still remains on the exact form of P qg (2) (it does not affect its log behavior) Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

25 Properties of splitting functions 1 At LO: time-like and space-like splitting functions are equal, provided Pqg S,(0) Pgq T,(0) 2 At NLO [NPB 175 (1980) 27, PLB 97 (1980) 497, PRD 48 (1993) 116] time-like and space-like splitting functions are related by analytic continuation 3 at NNLO [PLB 638 (2006) 61, PLB 659 (2008) 290, NPB 854 (2012) 133] an uncertainty still remains on the exact form of P qg (2) (it does not affect its log behavior) Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

26 Properties of splitting functions Must be careful with fixed-order splitting functions as z 0 (m = 1,..., 2k + 1) space-like case P ji ak+1 s x log k+1 m 1 x P ji ak+1 s z time-like case log 2(k+1) m 1 z Soft gluon logarithms diverge more rapidly in the TL case than in the SL case: as z decreases, ( ) the unresummed SGLs spoil the convergence of the FO series for P (z, a s) if log 1 z O a 1/2 s Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

27 Properties of splitting functions Must be careful with fixed-order splitting functions as z 0 (m = 1,..., 2k + 1) space-like case P ji ak+1 s x log k+1 m 1 x P ji ak+1 s z time-like case log 2(k+1) m 1 z Soft gluon logarithms diverge more rapidly in the TL case than in the SL case: as z decreases, ( ) the unresummed SGLs spoil the convergence of the FO series for P (z, a s) if log 1 z O a 1/2 s Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

28 Solving DGLAP equations 1 Transform to Mellin space (convolutions become ordinary products) n f ln µ 2 D i(n, µ 2 ( ) = γ ji N, αs(µ 2 ) ) ( D j N, µ 2 ) 1 γ ji (N) = zz N 1 P ji (z) j 0 2 Solve evolution equations in Mellin space D i (N, µ 2 ) = j Γ ij ( N, αs(µ 2 ), α s(µ 2 0 )) D j (N, µ 2 0 ) 3 The evolution kernels Γ ij ( N, αs(µ 2 ), α s(µ 2 0) ) satisfy the evolution equations n f ln µ 2 Γ ij(n, µ 2 ( ) = γ ik N, αs(µ 2 ) ) ( Γ kj N, µ 2 ) k α s Γ ij ( N, αs(µ 2 ), α s(µ 2 0) ) = k dαs d ln Q 2 = β(αs) = n=0 R ik (N, α s)γ kj ( N, αs(µ 2 ), α s(µ 2 0 )) 4 The perturbative solution of the matrix R ij = R = R (0) + α sr (1) +... is α n+2 s β n R (0) = γ(0) β 0 R (k) = γ(k) β 0 k i=1 β i R (k i) γ = α n+1 s γ (0) β 0 n=0 Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

29 Solving DGLAP equations: LO example 1 For the flavour nonsinglet and valence quark FFs ( Γ NS,LO N, αs(µ 2 ), α s(µ 2 0) ) (0) = αs(µ2 R NS ) α s(µ 2 0 ) 2 For the flavour singlet, diagonalise the 2 2 R matrix ( αs(µ Γ S,LO (N, α s(µ 2 ), α s(µ 2 2 ) λ+ (N) ( ) αs(µ 2 ) λ (N) ) 0)) = e + (N) α s(µ 2 0 ) + e (N) α s9µ 2 0 ) [ λ ± (N) = 1 ] ( ) γ qq (0) (N) + γ gg (0) (N) ± γ qq (0) (N) γ gg (0) 2 (0) (N) + 4γ qg (N)γ gq (0) (N) 2β 0 1 e ± (N) = ± λ + (N) λ (N) (R(0) S (N) λ (N)I) ( 3 Determine the evolution factors Γ ij z, αs(µ 2 ), α s(µ 2 0) ) by inverse Mellin transform ( Γ ij z, αs(µ 2 ), α s(µ 2 0 )) dn ( = C 2πi x N Γ NS N, αs(µ 2 ), α s(µ 2 0 )) 4 Use public codes which solve the evolution equations numerically APFEL[CPC 185 (2014) 1647] MELA[JHEP 1503 (2015) 046] ffevol[cpc 183 (2012) 1002] QCDnum[CPC 182 (2011) 490] Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

30 1 Momentum sum rule 2 Charge sum rule Theoretical constraints 1 dzz Di h (z, µ2 ) = 1 parton i h 0 1 dz e h Di h (z, µ2 ) = e i parton i h 0 where e h(i) is the electric charge of the hadron h (parton i) 3 Positivity of cross sections implies that FFs should be positive-definite at LO 4 Charge conjugation symmetry Dq h+ ( q) = Dh q (q) Dg h+ = Dg h 5 Isospin symmetry of the strong interaction D π+ u = Dd π Dd π+ = Du π approximate, as m u m d, but no phenomenological evidences of violation Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

31 1.3 SIA as a case study Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

32 e (k 2) e + (k 1) γ, Z 0 SIA cross section: general structure h(p h ) X e + (k 1) + e (k 2) γ,z0 h(p h ) + X q = k 1 + k 2 q 2 = Q 2 > 0 z = 2P h q Q 2 dσ h dz = F h T (z, Q 2 ) + F h L(z, Q 2 ) = F h 2 (x, Q 2 ) { } Fk=T,L,2 h = 4πα2 em e 2 D h Q 2 Σ Ck,q S + n f Dg h Ck,g S + DNS h Ck,q NS e 2 = 1 n f n f ê 2 p p=1 n f ( ) DΣ h = Dp h + D h p p=1 n f ( ) ê2 ( ) DNS h = p e p=1 2 1 Dp h + D h p ê 2 p = e2 p 2epχ 1(Q 2 )v ev p + χ 2 (Q 2 )(1 + v 2 e )(1 + v2 p ) χ 1 (s) = χ 2 (s) = 1 s(s MZ 2 ) 16 sin 2 θ W cos 2 θ W (s MZ 2 )2 + Γ 2 Z M Z 2 1 s sin 4 θ W cos 4 θ W (s MZ 2 )2 + Γ 2 Z M Z 2 Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

33 SIA cross section: some remarks 1 Quark and antiquark FFs always appear throught the combination D q + = D q + D q no separation between quark and antiquark FFs 2 The leading contribution to C S 2,g is O(α s) direct sensitivity to D g only beyond LO indirect sensitivity to D g vis DGLAP evolution 3 The effective electroweak charges ê q depend on the energy scale ê 2 q / e2 (M Z ) 1; ê 2 q / e2 (10 GeV) 1 partial flavour flavour separation 4 Measurements are often multiplicities, i.e. cross sections normalised to σ tot n σ tot(q) = 4πα2 (Q) f Q 2 ê 2 q (Q) ( ( )) αs(q) s n C n π µ q n=1 2 with coefficients C 1 (1) = 1 C 2 (1) n f n f C 3 (1) n f 0.005n 2 f ê 2 q q 2 / n f ê 2 q q Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

34 Experimental data CERN-LEP: ALEPH [ZP C66 (1995) 353] DELPHI [EPJ C18 (2000) 203] OPAL [ZP C63 (1994) 181] KEK: BELLE (n f = 4) [PRL 111 (2013) ] TOPAZ [PL B345 (1995) 335] DESY-PETRA: TASSO [PL B94 (1980) 444, ZP C17 (1983) 5, ZP C42 (1989) 189] SLAC: BABAR (n f = 4) [PR D88 (2013) ] SLD [PR D58 (1999) ] TPC [PRL 61 (1988) 1263] dσ h dz = 4πα2 (Q 2 ) Q 2 F h 2 (z, Q2 ) h = π + + π, K + + K ; possibly normalised to σ tot Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

35 Scaling violations As the scale increases, multiplicites are shifted towards lower values Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

36 Flavour schemes 1 Assumption: the final-state hadron has no intrinsic heavy quark component 2 Fixed-flavour number sheme (FFNS) the only quarks treated as partons are the n L light quarks F (n L) 2 (z, Q 2, m 2 h ) = F L,(n L) 2 (z, Q 2 ) + F H,(n L) n L F L,(n L) 2 (z, Q 2 ) = C L,(n L) 2,i i ( n L F H,(n L) 2 (z, Q 2, m 2 h ) = C H,(n L) 2,i i (z, Q2 µ 2 z, Q2 m 2, µ2 h m 2 h 2 (z, Q 2, m 2 h ) ) D (n L) i (z, µ 2 ) ) Q 2 µ 2 D (n L) i (z, µ 2 ) is accurate in the quark mass threshold region and below suffers from unresummed ln(q 2 /m 2 h ) in the Wilson coefficients, large for Q2 m 2 h 3 Zero-mass variable flavour number scheme (ZM-VFNS) treat heavy quarks as massless partons below their threshold introduce a heavy quark FF the remormalisation of the FF resums logs due to parton splitting via DGLAP n L +1 F (n L+1) 2 (z, Q 2 ) = C (n L+1) 2,i i (z, Q2 µ 2 ) D (n L+1) i (z, µ 2 ) the reliability of the ZM-VFNS is reduced where powers of m 2 h /Q2 are significant Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

37 Changing the number of active flavours [JHEP 0510 (2005) 034] 1 All partons with mass m < µ = s must be assigned as active even though they are not active at µ = µ 0 the number of active partons depend on the scale of the process 2 Determine the matching conditions between a scheme with n f and n f + 1 flavours 3 Consider the production of a light hadron h dσ h hg (z) dz dσ dz = i I nl D (nl) i (z, µ 2 ) dˆσ(n L) i (z, µ 2 ) + D (n L) g (z, µ 2 ) dσ h hg (z) dz dz the first term involves the FF and the ˆσ i for all partons i excluding heavy flavours the second term is the O(α s) heavy-flavour contribution { α s = σ h h 2π C 1 + (1 y)2 F 2 ln y } s m 2 + ln(1 y) 1 h 4 In the MS scheme, treating all flavour as massless dσ dz = D (n) i (z, µ 2 ) dˆσ(n) i (z, µ 2 ) dz i I n I n = I nl {h, h} Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

38 Changing the number of active flavours [JHEP 0510 (2005) 034] The difference between the two cross sections should vanish 0 = i I nl,i g + [ ] D (n) i (z, µ 2 ) D (n L) i (z, µ 2 ) dˆσ i(z, µ 2 ) dx [ ] D g (n) (z, µ 2 ) dˆσ(n) (z, µ 2 ) D (n L) g (z, µ 2 ) dˆσ(n L) (z, µ 2 ) dz dz + i {h, h} D (n) i (z, µ 2 ) dˆσ(z, µ2 ) dz D (n L) g (z, µ 2 ) dσ h hg(z) dz ˆσ (n) i and ˆσ (n L) i replaced with ˆσ i, since these cross sections differ by terms O(α 2 s ) ˆσ g is O(α s), hence D (n L) g = D g (n) (they differ by O(α s)) dˆσ g (n) (z, µ 2 )/dz dˆσ (n L) g (z, µ 2 )/dz = dˆσ h hg (z, µ 2 )/dz (massless MS) D (n) are O(αs), so that only the Born hard cross section is needed h/ h dˆσ h/ h(z, µ 2 ) σ dz h hδ(1 z) + O(α s) dˆσ h hg (z, µ 2 } ) α s = σ dz h h 2π C 1 + (1 z)2 F 2 {2 ln z + ln(1 z) + ln Q2 z µ 2 Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

39 Changing the number of active flavours [JHEP 0510 (2005) 034] 1 The matching consitions for the light and heavy quarks are D (n) h D (n) i (z, µ 2 ) = D (n L) i (z, µ 2 ) for i I nl, i g [ (z, µ2 ) = D (n) h (z, µ2 ) = D g(z, µ 2 ) αs 2π C 1 + (1 z) 2 F z 2 Similarly one can obtain the matching condition for the gluon ( D (n) g (z, µ 2 ) = D (n L) g (z, µ 2 ) 1 T F α s 3π ) µ2 ln m 2 h ln µ2 m ln z h 3 Matching conditions for FFs are non-zero at NLO (at variance with PDFs) can be used to generate radiatively charm and bottom contributions to F h 2 are required when evolving the perturbative charm FF through the bottom threshold 4 In practice the FF heavy-quark intrinsic component (non-vanishing if not active) is non-negligible one usually treats (discontinously) heavy-quark FFs as unknown functions inconsistency: heavy-quark FF effects should not depend on the flavour scheme alleviated when the scale of the process is above the heavy quark threshold ] Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

40 General-mass flavour number scheme 1 Combine a FFN scheme (at Q m h ) with a ZM-VFN scheme (at Q m h ) a GM-VFN scheme operates as a tower of FFN schemes increasing n L as the scale increases over each quark mass threshold 2 Demand that physical observables are continuous across the thresholds F GM 2 (z, m 2 h, µ2 ) = n L j n L +1 = 3 Demand the matching conditions for the FFs D (n L+1) i (z, µ 2 ) = i C GM (n L) 2,j (z, m 2 h, µ2 ) D (n L) j (z, µ 2 ) C GM (n L+1) 2,i (z, m 2 h, µ2 ) D (n L+1) j (z, µ 2 ) n L 4 Combine the two to obtain the condition j A (n L) ij (µ 2 /m 2 h ) D(n L) j (z, µ 2 ) n L +1 C GM (n L) 2,j (z, m 2 h, µ2 ) = C GM (n L+1) i,2 (z, m 2 h, µ2 ) A (n L) ij (µ 2 /m 2 h ) i 5 Ensure that the condition above is satisfied at all orders Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

41 Comparing various flavour schemes [PRD 94 (2016) ] dσ ZM dz = i=q,g,h ˆσ ZM i (Q) Di ZM (Q) dσ GM dz = i=q,g,h ˆσ GM (Q, m h ) Di GM (Q) dσ M dz = i=q,g ˆσ M i (Q, m h) D M i (Q) + ˆσM h (Q, m h) D M h Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

42 Impact of higher-order QCD corrections [PRD 92 (2015) ] K-factor smaller at NNLO/NLO than at NLO/LO indication of large unresummed logarithms at small and large values of z renormalization scale variations: estimate of missing higher-order corrections at most 5% at NNLO at the lowest Q theoretical uncertainties under control Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

43 1.4 Other processes Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

44 SIDIS cross section l(m) + p(p ) γ l(m ) + h(p h ) + X 2F h 1 = e 2 q q, q F h L = αs 2π dσ h [ dxdydz = 2πα2 em 1 + (1 y) 2 Q 2 2F1 h y { q Dq h + αs 2π e 2 q q, q ] 2(1 y) + FL h y [ q C 1 qq D h q + q C 1 gq D h g + g C 1 qg D h q [ q C L qq D h q + q C L gq D h g + g C L qg D h q ] ]} separate information on D q and D q, no direct information on D g additional convolution with a PDF Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

45 SIDIS multiplicities 1 The quantity usually measured by experiments is the SIDIS multiplicity M h (x, z, Q) = dσh /dxdydz dσ h DIS /dxdy 2 Define two regions in the Breit frame (the frame in which the virtual photon energy vanishes and its spatial momentum is antiparallel with the initial proton s momentum) current fragmentation region: θ < π/2 target fragmentation region: θ > π/2 (θ, angle between the spatial momentum of the detected hadron and the virtual photon) 3 fracture functions describe the partonic structure of the initial hadron after it has produced the detected hadron (p h + i + X) fracture functions do not contribute to the cross section when the direction of the detected hadron s momentum is within the current fragmentation region Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

46 COMPASS multiplicities: pions [PLB 764 (2017) 1] 3-D kinematic binning in x, y, and z (317 kinematic bins) very weak y dependence, strong z dependence Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

47 COMPASS multiplicities: pions [PLB 764 (2017) 1] 3-D kinematic binning in x, y, and z (317 kinematic bins) very weak y dependence, strong z dependence Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

48 COMPASS multiplicities: kaons [PL B767 (2017) 133] 3-D kinematic binning in x, y, and z (317 kinematic bins) very weak y dependence, strong z dependence Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

49 COMPASS multiplicities: kaons [PL B767 (2017) 133] 3-D kinematic binning in x, y, and z (317 kinematic bins) very weak y dependence, strong z dependence Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

50 Hadron production in pp collisions p(p a) + p(p b ) h(p h ) + X d 3 σ E h = f dp 3 a f b ˆσ ab c Dc h h a,b,c = i,j,k,l dxa x a dxb x b dz z 2 f i/pa (x a)f j/p b D h/k (z)ˆσ ij kl δ(ŝ + ˆt + û) can verify and improve constraints on the charge-sign and flavour separation of quark FFs provided by SIA and SIDIS constrain gluon FFs significantly better than data from SIA and SIDIS, owing to the occurrence of the gluon FF at LO in the calculation the detected hadron h will sometimes be a soft remnant from one of the initial state hadrons (as in SIDIS) a current fragmentation region analogous to SIDIS is free of initial hadron remnants depending on the kinematical region, uncertainties from the PDFs can be sizeable available data sets are in a kinematical regime where higher order perturbative corrections (possibly requiring resummation) and/or higher twist effects are large Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

51 Hadron production in pp collisions Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

52 1.4 Summary of Lecture 1 Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48

53 Summary 1 Parton Distribution Functions are required in factorisation formulas if hadrons (typically nucleons) are prepared in the initial state unpolarised/polarised PDFs required depending on their polarisation unpolarised/polarised PDFs need to be determined from the data 2 Fragmentation Functions are required in factorisation formulas if hadrons are identified in the final state FFs are the time-like counterparts of PDFs inherit from PDFs evolution equations FFs need to be determined from the data 3 Variety of processes used to access FFs single hadron production in SIA, SIDIS and pp collisions SIA remains the theoretically cleanest process among the three SIA is also the less challenging, as it does not involve PDFs 4 Need to put together all this information in a global QCD analysis devise a suitable fitting methodology parametrisation, representation of uncertainties, fit validation fitting methodology valid for both PDFs and FFs Lecture 2: Methodological aspects of global QCD fits Emanuele R. Nocera (Oxford) FFs and Global QCD Fits June / 48