High energy scattering in QCD

Transcription

1 High energy scattering in QCD I Parton model, Bjorken scaling, François Gelis and CEA/Saclay

2 General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions General introduction François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 2/73

3 Infrared and collinear divergences General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions Calculation of some process at LO : x 1 x 2 (M, Y ) { x1 = M e +Y / s x 2 = M e Y / s François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 3/73

4 Infrared and collinear divergences General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions Calculation of some process at LO : x 1 x 2 (M, Y ) { x1 = M e +Y / s x 2 = M e Y / s Radiation of an extra gluon : z,k x 1 (M, Y ) = α s x 1 dz z M d 2 k k 2 x 2 François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 3/73

5 Infrared and collinear divergences General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions Large logs : log(m ) or log(1/x 1 ), under certain conditions these logs can compensate the additional α s, and void the naive application of perturbation theory resummations are necessary Logs of M = DGLAP. Important when : M Λ QCD x 1, x 2 are rather large Logs of 1/x = BFKL. Important when : M remains moderate x 1 or x 2 (or both) are small Physical interpretation : The physical process can resolve the gluon splitting if M k If x 1 1, the gluon that initiates the process is likely to result from bremsstrahlung from another parent gluon François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 4/73

6 Multiple scatterings General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions Single scattering : 2-point function in the projectile gluon number François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 5/73

7 Multiple scatterings General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions Single scattering : 2-point function in the projectile gluon number Multiple scatterings : 4-point function in the projectile higher correlation multiple scatterings in the projectile François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 5/73

8 Multiple scatterings General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions Power counting : rescattering corrections are suppresssed by inverse powers of the typical mass scale in the process :» µ 2 M 2 n The parameter µ 2 has a factor of α s, and a factor proportional to the gluon density rescatterings are important at high density Relative order of magnitude : 2 scatterings 1 scattering Q2 s M 2 with Q 2 s α s xg(x, Q 2 s) πr 2 When this ratio becomes 1, all the rescattering corrections become important These effects are not accounted for in DGLAP or BFKL François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 6/73

9 Heavy Ion Collisions General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions 99% of the multiplicity below p 2 GeV Q 2 s might be as large as 10 GeV 2 at the LHC ( s = 5.5 TeV) both the logs of 1/x and the multiple scatterings are important François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 7/73

10 Goals General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions Develop a framework for resumming all the [α s ln(1/x)] m [Q s /M ] n corrections Generalize the concept of parton distribution Due to the high density of partons, observables depend on higher correlations (beyond the usual parton distributions, which are 2-point correlation functions) These distributions should be universal, with non-perturbative information relegated into the initial condition of some evolution equation Develop techniques for describing the early stages of heavy ion collisions in this framework François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 8/73

11 Goals General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions t freeze out hadrons in eq. gluons & quarks in eq. gluons & quarks out of eq. strong fields z (beam axis) hydrodynamics kinetic theory classical EOMs calculate the initial production of semi-hard particles prepare the stage for kinetic theory or hydrodynamics François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 9/73

12 General outline General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions Lecture I : Parton model, Bjorken scaling, Lecture II : Parton evolution at small x, Saturation Lecture III : Hadron-hadron collisions in the CGC framework François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 10/73

13 Lecture I General introduction IR & Coll. divergences Multiple scatterings Heavy Ion Collisions General introduction Kinematics of Deep Inelastic Scattering Structure functions Light-cone behavior of a free field theory, DGLAP equation François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 11/73

14 Introduction Kinematical variables DIS cross-section Structure functions François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 12/73

15 Introduction to DIS Introduction Kinematical variables DIS cross-section Structure functions Basic idea : smash a well known probe on a nucleon or nucleus in order to try to figure out what is inside... Photons are very well suited for that purpose because their interactions are well understood Deep Inelastic Scattering : collision between an electron and a nucleon or nucleus, by exchange of a virtual photon e - p, A Variant : collision with a neutrino, by exchange of Z 0, W ± François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 13/73

16 Kinematical variables Introduction Kinematical variables DIS cross-section Structure functions k P k q θ X Note : the virtual photon is space-like: q 2 0 Other invariants of the reaction : ν P q s (P + k) 2 M 2 X (P + q) 2 = m 2 N + 2ν + q2 One uses commonly : Q 2 q 2 and x Q 2 /2ν In general M 2 m2, and we have : 0 x 1 X N (x = 1 corresponds to the case of elastic scattering) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 14/73

17 DIS cross-section Introduction Kinematical variables DIS cross-section Structure functions The simplest cross-section is the inclusive cross-section, obtained by measuring the momentum of the scattered electron and summing over all the hadronic final states X Z E dσ e N e X d 3 k = E dσ e N d 3 k = X states X E dσ e N e X d 3 k [dφ X ] 32π 3 (s m 2 N )(2π)4 δ(p+k k P X ) M X 2 spin M X = ie h u( q k )γ µ u( i k) X Jµ (0) N(P) 2 In the amplitude squared appears the leptonic tensor : D L µν u( k )γ µ u( k)u( k)γ ν u( E k ) spin = 2(k µ k ν + k ν k µ g µν k k ) (the electron mass has been neglected) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 15/73

18 DIS cross-section Introduction Kinematical variables DIS cross-section Structure functions The inclusive cross-section can be written as : E dσ e N d 3 k = 1 e 2 32π 3 (s m 2 ) q 4 4πLµν W µν N where W µν is the hadronic tensor, defined as: 4πW µν X Z [dφ X ](2π) 4 δ(p + q P X ) states X D N(P) J ν (0) X X Jµ (0) N(P) E spin François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 16/73

19 DIS cross-section Introduction Kinematical variables DIS cross-section Structure functions The inclusive cross-section can be written as : E dσ e N d 3 k = 1 e 2 32π 3 (s m 2 ) q 4 4πLµν W µν N where W µν is the hadronic tensor, defined as: 4πW µν = X Z Z [dφ X ] d 4 y e i(p +q P X ) y states X D N(P) J ν (0) X X Jµ (0) N(P) E spin François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 16/73

20 DIS cross-section Introduction Kinematical variables DIS cross-section Structure functions The inclusive cross-section can be written as : E dσ e N d 3 k = 1 e 2 32π 3 (s m 2 ) q 4 4πLµν W µν N where W µν is the hadronic tensor, defined as: 4πW µν = X Z Z [dφ X ] d 4 y e i(p +q P X ) y states X D N(P) J ν (0) X X Jµ (0) N(P) E spin François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 16/73

21 DIS cross-section Introduction Kinematical variables DIS cross-section Structure functions The inclusive cross-section can be written as : E dσ e N d 3 k = 1 e 2 32π 3 (s m 2 ) q 4 4πLµν W µν N where W µν is the hadronic tensor, defined as: 4πW µν = X Z Z [dφ X ] d 4 y e iq y states X D N(P) J ν (y) X X Jµ (0) N(P) E spin François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 16/73

22 DIS cross-section Introduction Kinematical variables DIS cross-section Structure functions The inclusive cross-section can be written as : E dσ e N d 3 k = 1 e 2 32π 3 (s m 2 ) q 4 4πLµν W µν N where W µν is the hadronic tensor, defined as: 4πW µν = X Z Z [dφ X ] d 4 y e iq y states X D N(P) J ν (y) X X Jµ (0) N(P) E spin François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 16/73

23 DIS cross-section Introduction Kinematical variables DIS cross-section Structure functions The inclusive cross-section can be written as : E dσ e N d 3 k = 1 e 2 32π 3 (s m 2 ) q 4 4πLµν W µν N where W µν is the hadronic tensor, defined as: Z 4πW µν = d 4 y e iq y D N(P) J ν(y)1j µ (0) N(P) E spin François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 16/73

24 DIS cross-section Introduction Kinematical variables DIS cross-section Structure functions The inclusive cross-section can be written as : E dσ e N d 3 k = 1 e 2 32π 3 (s m 2 ) q 4 4πLµν W µν N where W µν is the hadronic tensor, defined as: Z D N(P) J 4πW µν = d 4 y e iq y ν (y)j µ (0) N(P) E spin W µν contains all the informations about the properties of the nucleon under consideration that are relevant to the interaction with the photon This object cannot be calculated perturbatively It obeys: q µ W µν = q ν W µν = 0 (conservation of e.m. current) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 16/73

25 Structure functions Introduction Kinematical variables DIS cross-section Structure functions For a (spin-averaged) nucleon, the most general form of W µν is: W µν = W 1 g µν + W 2 P µ P ν m 2 N +W 4 q µ q ν m 2 N + W 5 P µ q ν m 2 N + W 3 ǫ µνρσ P ρ q σ + W 6 q µ P ν m 2 N m 2 N W 3 = 0 for parity conserving currents (like e.m. currents) W µν = W νµ from parity and time-reversal symmetry hence W 5 = W 6 From the Ward identities q µ W µν = q ν W µν = 0, one gets: W 5 = W 2 P q q 2 m 2 N W 4 = W 1 q + W (P q) q 4 François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 17/73

26 Structure functions Introduction Kinematical variables DIS cross-section Structure functions Therefore, for interactions with a photon, we have: W µν = W 1 g µν q «µq ν + W 2 q 2 m 2 N ««P q P q P µ q µ P q 2 ν q ν q 2 And the DIS cross-section in the nucleon rest frame reads: dσ e N de dω = α 2 em 4m N E 2 sin 4 (θ/2) ˆ2sin 2 (θ/2)w 1 + cos 2 (θ/2)w 2 where Ω is the solid angle of the scattered electron It is customary to define slightly rescaled structure functions: F 1 W 1, F 2 ν m 2 N Note: F 1 is proportional to the interaction cross-section between the nucleon and a transverse photon W 2 François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 18/73

27 Bjorken scaling Longitudinal F François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 19/73

28 Bjorken scaling Bjorken scaling : F 2 depends very weakly on Q 2 Bjorken scaling Longitudinal F SLAC F x François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 20/73

29 Longitudinal F F L F 2 2xF 1 is quite smaller than F 2 : Bjorken scaling Longitudinal F F L vs. F 2 for Q 2 = 20 GeV 2 F L F e x François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 21/73

30 e-mu cross-section Towards a field theory François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 22/73

31 Analogy with the e- mu- cross-section e-mu cross-section Towards a field theory In terms of F 1 and F 2, the DIS cross-section reads: dσ e N de dω = α 2 em 4m N E 2 sin 4 θ 2» 2F 1 sin 2 θ 2 + m2 N ν F 2 cos 2 θ 2 It is instructive to compare it to the e µ cross-section: dσ e µ de dω = α2 emδ(1 x) 4m µ E 2 sin 4 θ 2» sin 2 θ 2 + m2 µ θ ν cos2 2 If the constituents of the nucleon that interact in the DIS process were spin 1/2 point-like particles, we would have: 2F 1 = m N m c δ(1 x c ), F 2 = m c m N δ(1 x c ) where m c is some effective mass for the constituent (comparable to m N because it is trapped inside the nucleon) and x c Q 2 /2q p c with p µ c the momentum of the constituent François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 23/73

32 Analogy with the e- mu- cross-section e-mu cross-section Towards a field theory If p µ c = x F P µ, then x c = x/x F, and: 2F 1 δ(x x F ), F 2 δ(x x F ) The structure functions F 1 and F 2 would therefore not depend on Q 2, but only on x Conclusion : Bjorken scaling could be explained if the constituents of the nucleon that are probed in DIS are spin 1/2 point-like particles The variable x measured in DIS would have to be identified with the fraction of momentum carried by the struck constituent François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 24/73

33 e-mu cross-section Towards a field theory The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : Z 4πW µν d 4 p i = (2π) 4 2πδ(p 2 )(2π) 4 δ(x F P + q p ) D xf P Jµ E (0) p p Jν (0) xf P spin François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 25/73

34 e-mu cross-section Towards a field theory The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : 4πW µν i = 2πδ((x F P + q) 2 ) D xf P Jµ E (0) xf P + q x F P + q Jν (0) xf P spin François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 25/73

35 e-mu cross-section Towards a field theory The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : 4πW µν i = 2πδ((x F P + q) 2 ) D xf P J E µ (0) x F P + q x F P + q J ν (0) x F P spin François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 25/73

36 e-mu cross-section Towards a field theory The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : 4πW µν i = 2πδ((x F P + q) 2 ) e2 i 2 tr(x F / P γ µ (x F /P + q)γ ν ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 25/73

37 e-mu cross-section Towards a field theory The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : 4πW µν i = 2πδ((x F P + q) 2 ) e2 i 2 tr (x F / Pγ µ (x F /P + q)γ ν ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 25/73

38 e-mu cross-section Towards a field theory The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : 4πW µν i = 2πδ(2x F P q + q 2 ) e2 i 2 tr (x F / Pγ µ (x F /P + q)γ ν ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 25/73

39 e-mu cross-section Towards a field theory The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : 4πW µν i = 2π 1 2P q δ(x F x) e2 i 2 tr (x F / Pγ µ (x F /P + q)γ ν ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 25/73

40 e-mu cross-section Towards a field theory The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : 4πW µν i = 2π 1 2P q δ(x F x) e2 i 2 tr (x F / P γ µ (x F /P + q)γ ν ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 25/73

41 e-mu cross-section Towards a field theory The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : 4πW µν i = 2π 1 2P q δ(x F x) 2e 2 i (x 2 F P µ P ν + x F (P µ q ν + q µ P ν ) x F g µν P q) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 25/73

42 e-mu cross-section Towards a field theory The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : e 2 i 4πW µν i = 2πx F δ(x F x)» «g µν qµ q ν + 2x F q 2 P q ««P µ q µ P q P ν q ν P q q 2 q 2 François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 25/73

43 e-mu cross-section Towards a field theory The historical parton model describes the nucleon as a collection of point-like fermions, called partons A parton of type i, carrying the fraction x F of the nucleon momentum, gives the following contribution to the hadronic tensor : e 2 i 4πW µν i = 2πx F δ(x F x)» «g µν qµ q ν + 2x F q 2 P q ««P µ q µ P q P ν q ν P q q 2 q 2 If there are f i (x F )dx F partons of type i with a momentum fraction between x F and x F + dx F, we have W µν = X i Z 1 0 dx F f i (x x F ) W µν i F One obtains the following structure functions : F 1 = 1 2 X e 2 if i (x), F 2 = 2xF 1 i François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 25/73

44 e-mu cross-section Towards a field theory This model provides an explicit realization of Bjorken scaling The relation F 2 = 2xF 1 implies that the cross-section between a longitudinally polarized photon and the nucleon is suppressed compared to that of a transverse photon The observation of this property provides further support of the fact that the relevant constituents are spin 1/2 fermions If the partons were spin 0 particles, we would have W µν i (2x F P µ + q µ )(2x F P ν + q ν ) and it is easy to check that this leads to F 1 = 0 (σ transverse = 0) Caveats and puzzles : The parton model assumes that partons are free inside the nucleon. How can this be true in a strongly bound state? One would like to have a field theoretical description of what is going on, including the effect of interactions, quantum fluctuations, etc... François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 26/73

45 Field theory point of view e-mu cross-section Towards a field theory A nucleon at rest is a very complicated object... Contains fluctuations at all space-time scales smaller than its own size Only the fluctuations that are longer lived than the external probe participate in the interaction process The only role of short lived fluctuations is to renormalize the masses and couplings Interactions are very complicated if the constituents of the nucleon have a non trivial dynamics over time-scales comparable to those of the probe François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 27/73

46 Field theory point of view e-mu cross-section Towards a field theory Dilation of all internal time-scales for a high energy nucleon Interactions among constituents now take place over time-scales that are longer than the characteristic time-scale of the probe the constituents behave as if they were free Many fluctuations live long enough to be seen by the probe. The nucleon appears denser at high energy (it contains more gluons) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 28/73

47 What would we learn? e-mu cross-section Towards a field theory The field theory that describes the interactions among partons should be able to explain the evolution with x of the parton distributions, since it comes from bremsstrahlung This field theory should also describe the evolution with Q 2 (i.e. the deviations from Bjorken scaling), which is due to the fact that the probe resolves more quantum fluctuations when Q 2 increases For the picture to be predictive, one should be able to prove from first principles the factorization of hadronic cross-section into a hard process (calculable?) and the parton distributions (not calculable?) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 29/73

48 Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 30/73

49 Kinematics of the Bjorken limit Bjorken limit : Q 2, ν +, x = constant Go to a frame where the photon momentum is : Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Therefore : q µ = 1 q (ν,0, 0, ν m 2 + m 2 N Q2 ) N q + q0 + q 3 2 ν m N + q q0 q 3 2 m N x constant Since q y = q + y + q y + q y, the integration over y µ is dominated by : y m N ν 0, y + (m N x) 1 Hence : y 2 2y + y 1/Q 2 0 François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 31/73

50 Kinematics of the Bjorken limit W µν can be rewritten in terms of the commutator [J µ(y), J ν (0)]. Thus, y 2 0 (causality). Therefore, the Bjorken limit is dominated by : Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions 0 y 2 1 Q 2 0 i.e. by points very close to (and above) the light-cone t y 2 = 1/Q 2 z Note : in this limit, the components of y µ are not small François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 32/73

51 Time ordered correlator of currents Consider a time-ordered product of currents : Z 4πT µν i d 4 ye iq y D N(P) T(J µ(y)j ν (0)) N(P) E spin q q Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions T µν is a forward Compton amplitude P P François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 33/73

52 Time ordered correlator of currents Consider a time-ordered product of currents : Z 4πT µν i d 4 ye iq y D N(P) T(J µ(y)j ν (0)) N(P) E spin q q Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions T µν is a forward Compton amplitude W µν = 2 ImT µν At fixed Q 2, T µν (ν, Q 2 ) is analytic in ν, with cuts on the real axis starting at ±Q 2 /2 the branch point at ν = Q 2 /2 comes from (P + q) 2 m 2 N T µν is symmetric under (µ ν, q q) T µν has a tensor decomposition similar to W µν, with structure functions T 1 and T 2 : T µν = T 1 g µν q µq ν q 2 «+ T 2 P q ««P q P q P µ q µ P q 2 ν q ν q 2 F r is related to the discontinuity of T r across the cut P P François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 33/73

53 Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Operator Product Expansion Consider a correlator A(0)B(x)φ(x1 ) φ(x n ) where A and B are two local operators, possibly composite When x 0, this function is usually singular because products of operators at the same point are ill-defined These singularities do not depend on the nature and localization of the other fields φ(x i ) One can obtain them from an expansion of the form A(0)B(x) = x 0 X C i (x) O i (0) i the O i (0) are local operators with the quantum numbers of AB the C i (x) are numbers that contain the singular behavior When x 0, C i (x) behaves as C i (x) x 0 x d(o i) d(a) d(b) (up to logs) only the operators with a low mass dimension matter François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 34/73

54 Operator Product Expansion of T(JJ) Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions The local operators that may appear in the OPE of T(J µ(y)j ν (0)) can be classified according to the representation of the Lorentz group to which they belong. Denote them O µ 1 µ s s,i, where s is the spin of the operator, and the index i labels the various operators having the same structure. The OPE takes the form : X s,i C s,i µ 1 µ s (y) O µ 1 µ s s,i (0) The Wilson coefficients of these operators must have the following structure : C s,i µ 1 µ s (y) y µ1 y µs C s,i (y 2 ) The expectation values in the nucleon state are of the form : D N(P) O µ 1 µ s s,i (0) N(P) E = [P µ1 P µ s +trace terms] O s,i spin François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 35/73

55 Power counting and twist Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Let d s,i be the mass dimension of the operator O µ 1 µ s s,i Then, the dimension of C s,i (y 2 ) is 6 + s d s,i this function scales as (y 2 ) (d s,i s 6)/2 (up to logs) In a standard OPE, where y µ 0, the factor y µ1 y µs would bring an extra y s to this scaling behavior, making the coefficient of O µ 1 µ s s,i scale as y ds,i 6, and high-dimension operators would be suppressed But in the Bjorken limit, the components of y µ do not go to zero, and therefore the factor y µ1 y µs should not be counted. In this case, it is the difference d s,i s (called the twist ) that controls the scaling behavior of the coefficient The leading behavior of T(J µ(y)j ν (0)) is controlled by the operators having the smallest twist. There is an infinity of them, because the dimension d s,i can be compensated by a higher spin François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 36/73

56 Operator Product Expansion of T(JJ) Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Going back to the OPE of the structure functions T 1 and T 2, we can write generically : X s,i Os,i Z d 4 y e iq y C s,i (y 2 ) (P y) s François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 37/73

57 Operator Product Expansion of T(JJ) Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Going back to the OPE of the structure functions T 1 and T 2, we can write generically : X s,i Os,i Z d 4 y e iq y C s,i (y 2 ) (P y) s François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 37/73

58 Operator Product Expansion of T(JJ) Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Going back to the OPE of the structure functions T 1 and T 2, we can write generically : X s,i Os,i ip µ q µ «s Z d 4 y e iq y C s,i (y 2 ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 37/73

59 Operator Product Expansion of T(JJ) Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Going back to the OPE of the structure functions T 1 and T 2, we can write generically : X s,i Os,i ip µ q µ «s Z d 4 y e iq y C s,i (y 2 ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 37/73

60 Operator Product Expansion of T(JJ) Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Going back to the OPE of the structure functions T 1 and T 2, we can write generically : X s,i Os,i ip «s µ ecs,i( q µ q µ ) q µ François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 37/73

61 Operator Product Expansion of T(JJ) Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Going back to the OPE of the structure functions T 1 and T 2, we can write generically : X s,i Os,i ip µ q µ «s ecs,i( q µ q µ ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 37/73

62 Operator Product Expansion of T(JJ) Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Going back to the OPE of the structure functions T 1 and T 2, we can write generically : X Os,i ( 2iP q) s s,i e C (s) s,i ( q µq µ ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 37/73

63 Operator Product Expansion of T(JJ) Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Going back to the OPE of the structure functions T 1 and T 2, we can write generically : X x X s s i Os,i ( i) s Q 2s e C (s) s,i (Q2 ) {z } D s,i (Q 2 ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 37/73

64 Operator Product Expansion of T(JJ) Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Going back to the OPE of the structure functions T 1 and T 2, we can write generically : X x X s s i Os,i ( i) s Q 2s e C (s) s,i (Q2 ) {z } D s,i (Q 2 ) Note: from their definitions, T 1 and T 2 differ by a power of P. Having the same dimension, they differ in fact by a factor x : T 1 (x, Q 2 ) = X s x s X i Os,i D1;s,i (Q 2 ) T 2 (x, Q 2 ) = X s x 1 s X i Os,i D2;s,i (Q 2 ) Since all the powers of x and Q 2 have been counted explicitly, D 1;s,i and D 2;s,i can only differ by constant factors and logs Note : T 1 is even and T 2 is odd in x s is even in this sum François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 37/73

65 Operator Product Expansion of T(JJ) Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions The coefficient function C s,i (y 2 ) behaves like y d s,i s 6 Its Fourier transform C s,i (Q 2 ) scales as Q 2+s d s,i So does D r;s,i (Q 2 ) Q 2s C (s) s,i (Q2 ) Therefore, if the leading twist operators correspond to d s,i s = 2, we get Bjorken scaling automatically The coefficients D r;s,i (Q 2 ) are calculable in perturbation theory, and do not depend on the target The matrix elements O s,i are non perturbative, and contain all the information about the target At this stage, the predictive power of this approach is limited to scaling properties, because we do not know the target dependent factors O s,i François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 38/73

66 Moments of F1 and F2 Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions The OPE provides a Taylor expansion of T 1,2 in powers of x 1 (all the x dependence is in the factor x s ) : T r = X even s t r (s,q 2 ) x a r s = X even s t r (s,q 2 ) with a 1 = 0, a 2 = 1. From this, we get (for s even) : t r (s, Q 2 ) = 1 2πi «Q 2 s ar Z 2 C «s ar 2 ν s a r Q 2 dν ν νa r s T r (ν,q 2 ) Do the integration by wrapping the contour around the cuts, and use the relation between F r and the discontinuity of T r across the cut : ν C t r (s, Q 2 ) = 2 π Z 1 0 dx x xs a r F r (x, Q 2 ) the OPE gives the x-moments of the DIS structure functions François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 39/73

67 Bare Wilson coefficients Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Now, let us assume that the underlying field theory of strong interactions has spin 1/2 fermions (quarks) and vector bosons (gluons). The operators with the lowest twist are (dimension s + 2 and spin s, hence twist 2) : O µ 1 µ s s,f O µ 1 µ s s,g ψ f γ {µ 1 µ2 µ s} ψ f F α {µ 1 µ2 µ s 1 F µ s}α where the brackets { } denote a symmetrization of the indices µ 1 µ s and a subtraction of the trace terms on those indices In order to compute the Wilson coefficients, one can exploit the fact that they do not depend on the target: consider an elementary target (single fermion or vector boson) for which everything is calculable (including the O s,i ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 40/73

68 Bare Wilson coefficients Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Consider a quark state of a given flavor f and spin σ. At lowest order, one has : f, 1 µ s σ Oµ s,f f, σ = δff u σ (P)γ {µ 1 u σ (P)P µ2 P µ s} f, 1 µ s σ Oµ s,g f, σ = 0 Averaging over the spin of the quark, and comparing with P µ1 P µ s O s,i, leads to : Os,f f = δ ff, Os,g f = 0 On the other hand, one can calculate directly the expectation value of the current-current correlator in this quark state. This is simply done by taking the parton model results for F 1,2 and using dispersion relations to get T 1,2. For s even : t 1 (s, Q 2 ) = 1 π e2 f, t 2 (s, Q 2 ) = 2 π e2 f François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 41/73

69 Bare Wilson coefficients Therefore, the bare coefficient functions are : Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions D 1;s,f (Q 2 ) = 1 π e2 f, D 2;s,f (Q 2 ) = 2 π e2 f Repeating the same steps with a vector boson state gives : D 1;s,g (Q 2 ) = D 2;s,g (Q 2 ) = 0 if the vector bosons are assumed to be electrically neutral Going back to a nucleon target, it is convenient to define singlet quark distributions from their moments : Z 1 0 dx h i x xs f f (x) + f f(x) = O s,f so that : F 1 (x) = 1 2 X f e 2 f h i f f (x) + f f(x), F 2 (x) = 2xF 1 (x) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 42/73

70 Bare Wilson coefficients Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Note : by considering the Deep Inelastic Scattering of a neutrino on the same target, one would access the non singlet quark distribution. By repeating the same arguments, one would obtain : 1 0 dx [ ] x xs f f (x) f f(x) = O s,f for s odd François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 43/73

71 Learnings from free field theory Despite the fact that the result is embarrassingly similar to what we obtained in a much simpler way in the naive parton model, this exercise has taught us several things : Kinematics of the BJ limit Time-ordered correlator Operator Product Expansion OPE of T(JJ) Moments of F1 and F2 Bare Wilson coefficients Conclusions Bjorken scaling can be derived from first principles in a field theory of free fermions (somewhat disturbing given that these fermions are constituents of a strongly bound state) We now have an operatorial definition of the distribution f i (x) (not calculable perturbatively however) More importantly, the experimental observation of Bjorken scaling is telling us that the field theory of strong interactions must become a free theory in the limit Q 2 + asymptotic freedom As shown by Gross, Wilczek, Politzer in 1973, non-abelian gauge theories with a reasonable number of fermionic fields (like QCD with 6 flavors of quarks) have this property François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 44/73

72 Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2, DGLAP equation François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 45/73

73 Introduction Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 In the previous discussion, we have implicitly assumed that there is no scale dependence in the moments O s,i of the distribution functions In fact, they depend on the renormalization scale µ 2 the distribution functions are scale dependent as well The structure functions F 1 and F 2 being cross-sections, they cannot depend on the renormalization scale µ 2 there should also be a µ 2 dependence in the coefficient functions, in order to compensate the µ 2 dependence from Os,i The Wilson coefficients will be some trivial power of Q 2 imposed by their dimension (that alone would imply Bjorken scaling), times a function of the ratio Q 2 /µ 2. This corrective factor will violate Bjorken scaling François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 46/73

74 Callan-Symanzik equation Consider the following correlators : Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 G JJ (x) T(J(x)J(0)), G s,i (0) O s,i (0) G JJ (x) = X C s,i (x)g s,i (0) s,i The Callan-Symanzik equations for G JJ and G s,i are : [µ µ + β g + 2γ J ] G JJ = 0 [(µ µ + β g ) δ ij + γ s,ij ] G s,j = 0 where β is the beta function, γ J the anomalous dimension of the current J (in fact γ J = 0 for conserved currents), and γ s,ij the matrix of anomalous dimensions for the O s,i (the operator mixing is limited to operators with the same Lorentz structure) By combining the previous equations, one gets : [(µ µ + β g ) δ ij γ s;ji ] C s,j = 0 François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 47/73

75 Solution of the CS equation The dimensionless coefficients D r;s,i (Q, µ, g) are in fact functions D r;s,i (Q/µ, g). Under rescalings of Q, they obey : ˆ` Q Q + β(g) g δij γ s,ji (g) D r;s,j (Q/µ, g) = 0 Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 In order to solve this equation, let us first introduce the running coupling g(q, g) such that : ln(q/q 0 ) = Z g(q,g) g dg β(g ) (this is equivalent to Q Q g(q, g) = β(g(q, g)) and g(q 0, g) = g) Any function F(g(Q, g)) is a solution of ˆ Q Q + β(g) g F = 0 We also have ˆ Q Q + β(g) g e R Q Q 0 dm M γ(g(m,g)) =» e R Q Q 0 dm M γ(g(m,g)) γ(g) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 48/73

76 Solution of the CS equation Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 Therefore, the Wilson coefficients at scale Q can be expressed in terms of the Wilson coefficients at scale Q 0 by : D r;s,i (Q/µ,g) = D r;s,j (Q 0 /µ, g(q, g))» e R Q dm Q 0 M γ s (g(m,g)) If the underlying theory is asymptotically free, like QCD, then at large Q the coupling is small and we can approximate : γ s,ij (g) = g 2 A ij (s), g 2 (Q, g) = 8π 2 β 0 ln(q/λ QCD ) ji where the A ij (s) are given by a 1-loop perturbative calculation Finally, the solution can be rewritten as : 2 ln(q/λqcd ) D r;s,i (Q/µ, g) = D r;s,j (Q 0 /µ, g(q, g)) 4 ln(q 0 /Λ QCD ) «8π 2 A(s) β ji François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 49/73

77 in F1 and F2 Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 The moments of the structure function F 1 at scale Q 2 read : Z 1 0 dx x xs F 1 (x, Q 2 ) = X i,f e 2 f 2 2 ln(q/λqcd ) 4 ln(q 0 /Λ QCD ) «8π 2 A(s) β 0 F 1 keeps the same form F 1 (x, Q 2 ) = 1 [ ] 2 f e2 f ff +f f, provided we define singlet quark distributions by: Z 1 0 dx h i x xs f f (x, Q 2 )+f f(x, Q 2 ) X i 2 ln(q/λqcd ) 4 ln(q 0 /Λ QCD ) 3 5 fi «8π 2 A(s) β 0 O s,i Q0 3 5 fi O s,i Q0 The quark distribution is now Q 2 dependent It depends on the expectation value of operators involving gluons at LO preserve the Callan-Gross relation : F 2 (x, Q 2 ) = 2xF 1 (x, Q 2 ) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 50/73

78 Probabilistic interpretation Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 In order to make the interpretation of the Q dependence more transparent, let us introduce as well a gluon distribution, even though it is not probed directly in DIS : Z 1 0 dx x xs f g (x, Q 2 ) X i 2 ln(q/λqcd ) 4 ln(q 0 /Λ QCD ) «8π 2 A(s) β gi O s,i Q0 The derivative of the moments of the parton distributions with respect to ln(q 2 ) is (f f f f + f f, f g f g ) : Q 2 f i (s, Q 2 ) = g2 (Q, g) A Q 2 ji (s)f 2 j (s, Q 2 ) In order to go further, we need the following result : A(s)f(s) = Z 1 0 Z dx 1 x xs x dy y A(x/y)f(y) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 51/73

79 Probabilistic interpretation Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 Define the splitting functions P ij from their moments : Z 1 0 dx x xs P ij (x) 4π 2 A ij (s) Therefore, one has the following evolution equation for f i (x, Q 2 ) (DGLAP) : Q 2 f i (x, Q 2 ) = g2 (Q, g) Q 2 8π 2 Z 1 x dy y P ji(x/y)f j (y, Q 2 ) Interpretation : the resolution of the γ changes with Q Low Q: Large Q: g 2 P ji (z) describes the splitting j i, where the daughter parton takes the fraction z of the momentum of the original parton François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 52/73

80 Anomalous dimensions Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 The anomalous dimension of an operator O is given by : γ O = µ Z O Z O µ, where O renormalized = Z O 1 O bare At 1-loop, the operator O µ 1 µ s s,f has the following corrections : Moreover, to ensure gauge invariance, the operator O µ 1 µ s s,f should be defined as : O µ 1 µ s s,f ψ f γ {µ 1 D µ2 D µ s} ψ f Therefore, one has also the following 1-loop diagrams : For s even, there are mixings between O s,f and O s,g (for s odd, O s,g is a total derivative that does not play any role) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 53/73

81 Anomalous dimensions Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 At 1-loop, the coefficients A ij (s) in the anomalous dimensions are : ( " A gg (s)= π s(s 1) 1 sx (s+1)(s+2) + j=2 A gf (s) = 1 j ff 1 4π 2 s s(s + 1)(s + 2) A fg (s) = 1 j ff 1 3π 2 s s(s 1) ( ) A ff (s) = 1 2 sx 1 6π 2 s(s + 1) δ ff j j=2 # 1 + N f j 6 ) Since A gf (s) is flavor independent, the non-singlet linear combinations ( P f a f O s,f with P f a f = 0) are eigenvectors of the matrix of anomalous dimensions, with an eigenvalue A ff (s) These linear combinations do not mix with the remaining two P operators, f O s,f and O s,g, through renormalization François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 54/73

82 Valence sum rules (s=1) Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 In the case of s = 1, the anomalous dimension of the non-singlet quark operators is A ff (s = 1) = 0 Going back to the evolution equation for the moments of quark distributions, this means that we have : Q 2 8 < : Z 1 0 dx X f i a f hf 9 = f (x, Q 2 ) + f f(x, Q 2 ) ; = 0 for any linear combination such that P f a f = 0 For instance, for a nucleon, this implies that the number of u + u quarks minus the number of d + d quarks is independent of Q 2 Interpretation : the production of extra quarks by g q q produces quarks of all flavors in equal numbers François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 55/73

83 Momentum sum rule (s=2) Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 In the singlet sector, the matrix of anomalous dimensions for s = 2 reads : A ff(2) A fg (2) A = A N f A gf (2) A gg (2) π 2 N f 12 This matrix has a vanishing determinant, which means that a linear combination of the flavor singlet operators is not renormalized : O µν 2,g + f Oµν 2,f This leads also to a sum rule : Q 2 8 < : Z dxx 4 X f h i f f (x, Q 2 ) + f f(x, Q 2 ) N f = + f g (x, Q 2 ) 5 ; = 0 Interpretation : the total longitudinal momentum of the target is conserved, and the momentum that goes into the newly produced gluons must be taken from the quarks François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 56/73

84 Practical strategy Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 Due to the non-perturbative nature of the parton distributions at a given fixed scale Q, it does not make sense to try to predict the value of F r at a given Q out of nothing Instead, fit the parton distributions from the measurement of F r at a moderately low scale Q 0 using DGLAP, evolve them to a higher scale Q predict the values of the structure functions F r at the scale Q compare with DIS measurements This approach can be systematically improved by going to higher order, both for the hard subprocess, and for the splitting functions and beta function Current state of the art : NNLO program fully implemented (3-loop splitting functions : Moch, Vermaseren, Vogt (2004)) François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 57/73

85 HERA results for F2 HERA results and NLO DGLAP fit : Callan-Symanzik equation Solution of the CS equation Probabilistic interpretation Anomalous dimensions Valence sum rules Momentum sum rule Practical strategy HERA results for F2 p F 2 /(2 i) x= x= x= x= x= x= x=0.005 x=0.008 x=0.016 x=0.032 x=0.05 x=0.08 x= x= H1 ZEUS x= Q 2 François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 58/73

86 Deep Inelastic Scattering Drell-Yan process Collinear factorization Separation of timescales Initial state interactions Final state François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 59/73

87 in DIS Deep Inelastic Scattering Drell-Yan process Collinear factorization Separation of timescales Initial state interactions Final state The DIS structure functions can be written as : F r (x, Q 2 ) = X i Z 1 x dzf i (z, Q 2 )F r,i (x/z,q 2 ) + O F r,i is the structure function for a target parton i (at leading order, it is non-zero only for quarks) x/z is the Bjorken-x variable for the system γ i m 2 N Q 2 Schematically, one can represent this factorization as : «F r,i zp P f i François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 60/73

88 in DIS Deep Inelastic Scattering Drell-Yan process Collinear factorization Separation of timescales Initial state interactions Final state In perturbation theory, the terms included by the RG evolution correspond to factors of g 2 enhanced by large logarithms : g 2 ln `Q 2 /µ 2 where µ 2 is some soft cutoff The logs are due to collinear divergences in loop corrections to F r,i. The first power of g 2 ln(q 2 /µ 2 ) comes from : f q f q f q f q f g f g François Gelis 2007 Lecture I / III X Hadron Physics, Florianopolis, Brazil, March p. 61/73