Introduction to Quantum Chromodynamics

Transcription

1 Introduction to Quantum Chromodynamics Michal Šumbera Nuclear Physics Institute ASCR, Prague December 15, 2009 Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

2 QCD improved quark-parton mode 1 Introduction 2 General framework 3 Partons within partons Branching functions in QCD Multigluon emission and Sudakov formfactors of partons 4 Partons within hadrons Evolution equations at the leading order Moments of structure functions and sum rules Extraction of gluon PDF from scaling violations Evolution equations at the next-to-leading order Hard scattering and the factorization theorem 5 Brief survey of methods of solving the evolution equations 6 Exercises Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

3 Literature Our discussion is based on Quarks, partons and Quantum Chromodynamics by Jiří Chýla Available at chyla/lectures/text.pdf Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

4 Introduction In this lecture: How the naive QPM is modified within the framework of pqcd. Why pqcd modification to QPM is such that the basic concepts of the QPM maintain their meaning and are useful even in a theory which simultaneously aspires to describe the confinement of colored partons. We ll try to distinguish the effects which can be calculated in perturbation theory (in which partons are treated essentially as massless observable particles), from those where the color confinement plays a crucial role and perturbation theory is inapplicable. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

5 General framework Hard collision A + B F +..., where A,B are h, l or gauge bosons. hard process dominated by short distance interaction. pqcd description retains basic QPM strategy of dividing space time evolution of collision into three distinct stages. Figure 1: The general scheme of QCD improved QPM Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

6 General framework I: Initial evolution is represented by PDF D a/a (x 1, M, FS 1 ), D b/b (x 2, M, FS 2 ) of partons a, b and by factorization scheme. Incalculable in pqcd. Only their dependence on the factorization scale M is calculable. II: Hard scattering of partons : a + b c + d is described by parton level cross section σ ab cd (s, x 1, x 2, p c, p d, µ, M 1, FS 1, M 2, FS 2 ), (1) where s = (p a + p b ) 2, µ is the hard scattering scale, in general different from the factorization scales M i, and FS 1, FS 2 are factorization schemes defining the distribution functions D a/a, D b/b. (1) is calculable in pqcd Leftrightarrow some measure of hardness is large compared to the mass of proton. QPM is zeroth order approximation to (1) which is then systematically improved in pqcd. III: Hadronization of partonic state produced in hard parton scattering is the least understood stage of collision. pqcd is not applicable here and various sorts of models must be employed. In the independent fragmentation model, D h/p (z, M, FS) acquire, similarly to PDFs inside hadrons, dependence on the M and FS. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

7 General framework In this lecture only stages I and II will be discussed. General formula in pqcd is the same as in QPM: σ(a + B F + anything) = (2) dx 1 dx 2 D a/a (x 1, M 1 )D b/b (x 2, M 2 )σ ab cd (s, x 1, x 2, p c, p d, M 1, M 2 ) abcd D hadr (p c, p d, P F ) }{{} model dependent Sum runs over all parton combinations leading to required final hadronic state F and σ ab cd contains all appropriate δ functions expressing the overall momentum conservation on the partonic level. Convolution stands for further integrations over the momenta of final state partons p c, p d, which lead to the final hadronic state F. Initial PDFs depend on factorization scales M 1, M 2 which are, similarly to the hard scale µ, unphysical and the cross sections of physical processes must, if evaluated to all orders, be independent of them. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

8 Partons within partons Basic idea of partons within partons picture is contained already in Weizsäcker-Williams approximation (WWA) in QED, avoiding at the same time conceptual problem of parton confinement in hadrons. Consider e p in the region of small Q 2. This interaction cannot be described by an incoherent sum of cross-sections of interactions on individual partons. Our interest is now structure of the beam electron not of the proton. We stay within pqed and concentrate on the upper vertex in Fig.2. Figure 2: The relation between cross sections of ep (a) and γp (b) interactions in low Q 2 region. Kinematics of the branching e e + γ (c) and the vertex describing the branching γ e + e (d). Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

9 Equivalent photon approximation Basic idea of WWA: Interpret e + p e... as emission from the beam electron of a nearly real photon followed by its interaction with the target proton. Correspondingly express σ tot (ep) in terms of σ tot (γp) of the real photon proton collision γ + p.... Basic assumption: for small photon virtuality (mass) γ + p γ + p. Measure of smallness of the photon virtuality is not directly Q 2, but rather its ratio Q 2 /W 2, where W 2 s γp. Small virtuality: in the overall ep CMS the photon in Fig. 2 behaves as nearly real and parallel to the beam electron, which thus looks like being accompanied by the beam of photons with certain momentum distribution function. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

10 Equivalent photon approximation Determination of this distribution function proceeds in several steps: 1 In the proton rest frame (with proton mass M), dσ = (2π)4 v e 4 d 3 k 2E2M Lµν W µν (q, p) (2π) 3 2E k 1 Q 4 = = (2π)4 e 4 d 4 q 2E2M Lµν W µν (q, p) (2π) 3 Q 4 δ(k 2 ), (3) where L µν is the lepton tensor and W µν (q, p) is the hadronic tensor associated with the lower vertex in Fig. 2a. W µν is related to DIS structure functions F i, but for low Q 2 it cannot be expressed in terms of PDFs. Neglecting m e we set in the second part of (3) v = 1. 2 Similarly we express σ tot (γp) = (2π)4 e 2 2E γ 2M ( 12 g µν ) W µν (Q 2 = 0, pq), (4) where W µν (Q 2 = 0) is the same hadronic tensor as in (3) and 1 2 results from the averaging over the spin states of the incoming photon. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

11 Equivalent photon approximation 3 Most general form of the symmetric W µν (Q 2 ) for unpolarized e + p scattering consisten with Lorentz invariance and parity conservation is: W µν = C 1 (Q 2, pq)g µν +C 2 (Q 2, pq)p µ p ν +C 3 (Q 2, pq)q µ q ν +C 4 (Q 2, pq) (p µ q ν + p ν q µ ), (5) where C i (Q 2, pq) contain all information about the structure of the unpolarized proton. In γp rest frame as W 2 = 2pq Q 2 + M 2. In the following the dependence on pq will not be explicitly written out. 4 Requirement of gauge invariance imposes further restriction: q µ W µν (Q 2 ) = 0 q ν [ C1 + C 3 q 2 + C 4 (pq) ] + p ν [ C2 (pq) + C 4 q 2] = 0, (6) which implies that only two of the four functions C i are independent C 4 (Q 2 ) = C 1 (Q 2 1 ) (pq) C 3(Q 2 ) q2 (pq), (7) q 2 C 2 (Q 2 ) = C 1 (Q 2 ) (pq) 2 + C 3(Q 2 ) (pq) 2. (8) q 4 Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

12 Equivalent photon approximation 5 For the real photon only C 1 (Q 2 = 0) contributes to the contraction: 1 2 g µν W µν (Q 2 = 0)) = C 1 (Q 2 = 0). (9) 6 For the virtual photon only the terms proportional to C 1, C 2 contribute after contraction with the leptonic tensor L µν : L µν W µν = 2 [ 2C 1 (Q 2 )(kk 2m 2 ) + C 2 (Q 2 ) ( 2(kp)(k p) M 2 (kk ) )]. (10) 7 Recalling basic kinematical relations from lecture on QPM, taking into account (8) and keeping in (10) the first two leading terms in Q 2, we get: ( ) 2 ( ) ] kp kp L µν W µν = 2C 1 (Q 2 )Q [ m2 qp qp Q 2 [ ] 1 + (1 y) = 2C 1 (Q 2 )Q 2 2 y 2 2m2 Q 2, (11) where y qp/kp is the fraction of incoming electron energy, carried away by the exchanged virtual, nearly parallel, photon. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

13 Weizsäcker-Williams approximation 8 Express d3 k 2E in terms of y and the transverse (w.r.t. to the incoming e) k momentum q T of the emitted γ. In the collinear kinematics, i.e. for q T q (see Fig. 2c) we have: d 3 k = δ(k 2 )d 4 1 q = dφdq dq 2. π. T = 2E k 4E k 2(1 y) dydq2 T = π dydq 2 2 (12) where in the last equality approximation y q /E k was used. In the collinear kinematics we also have Q 2 = 4E k E k sin 2 (ϑ/2) =. q2 T 1 y, (13) i.e. photon virtuality for fixed y is proportional to qt 2. 9 [ (13) + (12) + (11) ] (3), leading to the following total cross section of e + p collision at low Q 2 : [ ( )] σ ep (S, Qmax) 2 = dy dq2 T α 1 + (1 y) 2 qt 2 2m2 y 2π y Q 2 σ γp (ys), (14) where integral is taken over values of y, q T satisfying the condition Q 2 < Qmax. 2 This is the mentioned Weizsäcker-Williams approximation. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

14 Weizsäcker-Williams approximation Several aspects of this formula are worth to comment. y 2 y in the denominator of (11) was used to convert E k E γ. (14) naturally suggests to interpret the function [ α (1 y) 2 2π Q 2 2m2 y y Q 2 as the probability to find a photon with virtuality Q 2 and momentum yp inside an electron of momentum P. ] (15) Although formally of the order O(m 2 ) the second term in (15) gives actually a finite contribution after integration over Q 2 (or qt 2 ) in some interval (Qmin 2, Q2 max) : σ ep (S, Qmax, 2 Qmin 2 ) =. ([ ] [ α 1 + (1 y) 2 dy ln Q2 max 1 2π y Qmin 2 2m 2 y Qmin 2 1 ]) Qmax 2 σ γp (ys), (16) }{{} f γ/e (y, Qmax, 2 Qmin) 2 where the lower limit follows from kinematics: Qmin 2 = m2 y 2 1 y. (17) Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

15 Weizsäcker-Williams approximation Considering Qmax 2 (relabeled now as Q 2 ), as a free parameter, we get: f γ/e (y, Q 2 ) = α [ ] 1 + (1 y) 2 ln Q2 (1 y) 2(1 y) 2π y m 2 y 2 + O(m 2 /Q 2 ) y = α [ 1 + (1 y) 2 ln Q2 1 + (1 y)2 + ln 1 y ] 2(1 y) 2π y m2 y y 2 (18) y where m 2 from matrix element got canceled by 1/m 2 coming from phase space integration, giving finite contribution to f γ/e (y, Q 2 ). This function is then interpreted as the distribution function of photons, carrying fraction y of electron energy and having virtuality up to Q 2. Crucial feature of WWA is the presence of the logarithmic term ln(q 2 /m 2 ), which is due to the fact that the matrix element (11) is proportional to Q 2. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

16 Weizsäcker-Williams approximation One can also introduce complementary distribution function probability of finding electrons inside an electron: f e/e (y, Q 2 ) = δ(1 y) + α [ ] 1 + y 2 ln Q2 2π 1 y m 2, (19) where the δ function corresponds to the case of no photon radiation. To include contribution of virtual diagrams in (19) one needs to replace [... ] [... ] +. This modification is crucial for the probability interpretation of f e/e (x, Q 2 ): 1 0 f e/e (x, Q 2 )dx = 1 (20) total number of electrons minus the number of positrons inside an electron is conserved (and equal to 1 for a physical electron). Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

17 Weizsäcker-Williams approximation: Real photon Same ideas can be applied also to the case that the beam particle is a real photon which can split into a e + e pair ( see Fig. 2d). The distribution function of electrons (or positrons) inside a photon, carrying fraction y of its energy and having virtuality up to Q 2, is given as f e/γ (y, Q 2 ) = α [ y 2 + (1 y) 2] ln Q2 2π m 2 (21) with the same provision about the neglected terms as for previous case. Notice that there is no IR singularity in (21) and consequently no + distribution in this expression. Smallness of Q 2 needed for validity of WWA depends on process in the lower vertex of Fig. 2ab, in which γ interacts with the target particle, as well as on the kinematical region considered. WWA works when photon virtuality is small compared to values of basic kinematical variables describing its interaction with the target particle. For the case of the total cross section in (3) required Q 2 /s γp = Q 2 /ys 1, for the photoproduction of jets in e + p (HERA) Q 2 should be much smaller than the transverse momentum squared of the produced jets. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

18 Weizsäcker-Williams approximation: Real photon The essence of the WWA is contained in the branching functions all of which reflect the basic QED vertex eγe. 1 + (1 x)2 P γe (x) =, (22) x [ ] 1 + x 2 P ee (x) =, (23) 1 x + P eγ (x) = x 2 + (1 x) 2, (24) For processes dominated by low Q 2 the incoming electron behaves as a beam of nearly real electrons and photons, described by f γ/e and f e/e. Similarly, a single photon can be viewed as a photon accompanied by a beam of nearly real electrons and positrons, distributed in the photon according to the function f e/γ (x, Q 2 ). Finally since all QED branching functions appear always in the product with QED couplant α, the admixture of photons inside an electron and electrons in a photon are small effects. whch are entirely due to the interaction of electrons and photons with their own electromagnetic field. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

19 Branching functions in QCD Straightforward generalization of WWA: e q and γ G and inclusion of the 3 gluon vertex, which leads to P GG (x) branching function having the same physical interpretation as the other branchings. Due to color QCD branchings acquire additional n f dependent factors: P qq (0) (x) = P (0) qq (x) = 4 [ ] 1 + x x {[ x GG (x) = 6 1 x P (0) P (0) Gq (x) = P(0) Gq (x) = 4 3 [ 1 + (1 x) 2 [ P (0) x 2 qg (x) = P(0) qg (x) = + (1 x) 2 2 ] ( 33 2nf x x + x(1 x) + x 36, (25) + ], (26) ], (27) ) } 1 δ(1 x) (28) These branching functions are the same for all quark flavors q i, q i. Parton branchings are now due to interaction of q and G with their own chromodynamic field. The n f dependent term in (28) comes from virtual qq loop correction to gluon propagator. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

20 Multigluon emission When discussing KLN theorem we saw how the integration over the transverse momentum of the gluon emitted from the target quark, together with the virtual corrections yields for large Q 2 the process independent contribution to the cross section of the process e + q e + q + g: σ α s P (0) qq (x) ln Q2 m 2 g (29) This equation is correct provided 0 ε < x < 1 ε 1. The logarithmic terms come from the integration over the transverse momentum around the singularity at q T = 0. In the case of two gluon emission the leading term proportional to α 2 s ln 2 (Q 2 /m 2 g ) comes from integration over the region of strongly ordered virtualities t 1, t 2 in Fig. 3a t 2 ɛ t 1, t 1 ɛq 2, (30) where ɛ is some small parameter introduced to quantify the meaning of strongly ordered. As we shall see the coefficients in front of the leading logarithms do not depend on this parameter. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

21 Multigluon emission Figure 3: Multigluon emission from incoming parton leg (a) in eq collision (a), and from outgoing quarks in e + e qq annihilation (b). Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

22 Multigluon emission Omitting the branching functions and concentrating on the transverse momenta of radiated gluons we find that the contribution of the diagram with two radiated gluons is proportional to (we define τ = t for all virtualities) ɛq 2 αs 2 dτ 1 (µ) m τ g 2 1 α 2 s (µ) ɛτ1 m 2 g dτ ɛq 2 2 = αs 2 dτ 1 (µ) τ 2 mg 2 τ 1 ln ɛτ 1 [ 1 Q 2 2 ln2 mg 2 + f 1 (ln ɛ) ln Q2 mg 2 + f 2 (ln ɛ) where f i (ln ɛ) are simple polynomials in ln ɛ. m 2 g = ], (31) Calculation can be generalized to n emitted gluons: coefficient A n in front of the term with the same power of α s and ln Q 2 (the so called leading log (LL) ) ( ( ) ( ) ) Q σ (n) αs n A n ln n 2 Q mg 2 + B n ln n 1 2 m f n (x) (32) g }{{}}{{} LL NLL comes entirely from the region (30) of strongly ordered virtualities, is ɛ independent and equals 1/n!. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

23 Multigluon emission In (32) the scale µ must to be used in α s (µ). The best µ in vertices of the ladder of Fig. 3a, where quark of virtuality τ i+1 emits a quark with virtuality τ i and a real gluon, is µ 2 = τ i, i.e., µ 2 should be identified with the highest virtuality of all partons interacting in this vertex. Scales of α s are different at different vertices along the ladder and increase when moving up from the proton to the qγq vertex. Consider the correction to the emission of the first gluon in Fig. 3a, brought about by the emission and subsequent reabsorption of another gluon. UV renormalization of this loop leads to the appearance of ln(µ/τ i ) where µ is the argument of α s (µ) at the loop vertices. Large UV logarithms can be avoided by setting µ τ i. For strongly ordered virtualities the smaller one in each ladder link can be neglected and thus the vertex effectively describes the interaction of one off mass shell and two massless partons. As in the mentioned logarithm µ enters scaled by something describing the kinematics of the vertex, the maximal virtuality at that vertex is essentially the scale available. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

24 Multigluon emission and Sudakov formfactors of partons Consider again e + e qq as discussed in connection with the KLN. Probability of emitting a single real gluon is ln 2 β, while the radiative corrections (virtual emissions) give negative contributions, which cancel both single and double logs of β. By the KLN theorem this mechanism operates at each order of α s. Probability of multiple gluon emission grows with the order of perturbation expansion as there are more powers of ln 2 β. As a result probability of no gluon emission must accordingly be suppressed. This important phenomenon is expressed quantitatively in the so called Sudakov formfactors of quarks and gluons. Imagine that the quark (or antiquark) produced in e + e annihilations has a positive virtuality τ and assume for simplicity that τ is small with respect to the CMS energy squared s = 4E 2. Kinematical bounds on the fraction z of the parent quark energy, carried by the daughter quark after the gluon emission in Fig. 3b are given as z min = 1 [ 1 ].= 1 τ/e 2 2 τ s ; z max = 1 [ 1 + ].= 1 τ/e 2 2 τ 1 s. (33) Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

25 Sudakov formfactors of partons The time like Sudakov formfactor gives the probability that there will be no gluon radiation with the virtuality of the outgoing off mass shell quark between τ 0 and τ and the fraction z between z min and z max S(τ 0, τ, E) exp { τ dτ τ 0 τ α s ( τ ) P qq (0) (z)dz z min 2π zmax where {... } gives probability of gluon radiation from the quark with τ (τ 0, τ) and will carry energy fraction 1 z (1 z max, 1 z min ). Note the analogy to usual non-decay probability of an unstable system. }, (34) Neglecting dependence α s ( τ) integration in (34) can be performed: [ S(τ 0, τ, E) exp α ( ) ( ) ] s 4 τ 4E 2 2π 3 ln ln + O(ln τ 0 ). (35) τ 0 τ 0 Notice that [... ] contains a double log of τ 0 and that the Sudakov formfactor vanishes as τ 0 0. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

26 Partons within hadrons Up to now quarks and gluons were treated as free particles before and after the collision, despite the experimental evidence that they exist merely inside hadrons and behave like free particles only if probed at short distances. Basic idea to get around this (Fig. 4) is reminiscent of the UV renormalization of electric or color charges. Figure 4: Graphical representation of the definition of dressed parton distribution functions inside a given hadron for the nonsinglet quark distribution function. τ i denotes absolute value of the virtuality of a given intermediate state. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

27 Evolution equations at the leading order 1 x Define primordial (sometimes also called bare ) distribution functions of partons inside hadrons, q 0 (x), G 0 (x), which depend on x only and are interpreted in the sense of the QPM. Introduce nonsinglet (NS) quark distribution functions, which at the LO coincide with the valence distribution functions of QPM. We need to consider only the effects of multiple gluon emissions off the primordial quarks, described by q NS,0 (x), as sketched in Fig. 4. Summing up contributions from ladders in Fig. 4 we define the renormalized, or dressed, NS quark distribution function [ 1 ( ) ] dy x M 2 q NS (x, M) q NS,0 (x) + P qq (0) dτ 1 α s (τ 1 ) q NS,0 (y) + (36) x y y m τ 2 1 2π dy 1 dw M 2 dτ τ1 [ ( ) 1 dτ 2 α s (τ 1 ) α s (τ 2 ) x ( y ) ] P qq (0) P qq (0) q NS,0 (w)+, y y w m τ 2 1 m τ 2 2 2π 2π y w where newly introduced scale M has the meaning of maximal virtuality of the quark interacting with γ in the upper, QED, vertex of Fig. 4a and coinciding with the factorization scale introduced at the beginning of this lecture. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

28 Factorization scale M vs. renormalization scale µ In both cases scales emerge when bare quantities are replaced with their dressed counterparts. Important difference between them are that: µ emerged in the process of ultraviolet renormalization, i.e. concerns short distance properties of the theory M has been introduced to deal with parallel singularities, i.e. concerns large distances. In evolution equations M is interpreted as the upper bound on parton virtualities included in the definition of dressed PDFs, without specification of its relation to kinematic variables of any physical process. In DIS it is common to set M 2 = Q 2 and thus include in the dressed PDFs gluon emission even if it is actually far from parallel. This is legal to do as there is no sharp dividing line between small and large virtualities, but for large virtualities the pole term 1/t, which, after integration, leads to ln(q 2 /m 2 ), no longer dominates the exact matrix element for the gluon radiation. In the LL approximation there are no good arguments in flavour of this, or any other, choice of M and all values of M are in principle equally good. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

29 Evolution equations at the leading order: DGLAP Similarly to step from bare to the renormalized electric charge in QED (or color charge in QCD) we take derivative of both sides in (36) with respect to the scale M: dq NS (x, M) d ln M 2 = α 1 ( ) s(m) dy x 2π x y P(0) qq (37) y [ 1 ] dw M 2 dτ 2 α s (τ 2 ) ( y ) q NS,0 (y) + y w m τ 2 2 2π P(0) qq q NS,0 (w) +, w }{{} q NS (y, M) i.e. α s 2π dq NS (x, M) d ln M 2 dz = α s(m) 2π 1 x dy y P(0) qq ( ) x q NS (y, M) = y dyp (0) qq (z)q NS (y)δ(x yz) α s 2π P(0) qq q NS, (38) which is the so called Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation for the dressed NS quark distribution function q NS (x, M). Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

30 Evolution equations at the leading order: DGLAP Derivation of DGLAP is based on two essential ingredients: 1 Strong ordering of virtualities in the ladder of Fig. 3a. 2 The fact that the argument of α s (µ) at each vertex of this ladder is given by the upper (i.e. the largest) virtuality. As a consequence the only dependence on M appears in the upper bound on the integration over the largest virtuality τ 1 in each ladder of Fig. 4. Note that parallel logarithms have completely disappeared in the process of taking derivative with respect to ln Q 2! Moreover, this equation effectively resums the LL series (36). There is a simple relation between particle densities in the intervals of z and x, where x is the daughter particle momentum fraction (see Fig. 4) when y x/z is held fixed: d ln x = d ln z. The contribution to the density D(x) of particles having after the branching x in the interval (x, x + dx) and coming from the interval (y, y + dy) of the primordial momentum y is equal to x = yz dx = ydz D(x) dn dx = 1 dn y dz = 1 D(z). (39) y Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

31 Evolution equations at the leading order: DGLAP Branching function P (0) qg (z), describing the probability to find a quark (of any flavor) inside the gluon, also contributes to the LL evolution equations for dressed quark/antiquark distribution functions dq i (x, M) d ln M dq i (x, M) d ln M = α s(m) π = α s(m) π [ 1 x [ 1 x dy y P(0) qq dy y P(0) qq ( x y ( x y ) 1 q i (y, M) + ) q i (y, M) + x 1 x dy y P(0) qg dy y P(0) qg ( x y ( x y ) ] G(y, M) (40) ) ] G(y, M) (41) where the gluon distribution function G(x, M) satisfies similar evolution equation dg(x, M) d ln M = (42) α s (M) π [ nf i=1 1 x dy y P(0) Gq ( ) x (q i (y, M) + q y i (y, M)) + 1 x dy y P(0) GG ( x y ) G(y, M) * From now on we drop the adjective dressed and write the derivative with respect to ln M instead of ln M 2. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58 ]

32 Evolution equations at the leading order: Kernels Figure 5: Graphical representation of the kernels of the evolution equations for the quark (a) and gluon(b) distribution functions inside the proton. Solution requires boundary conditions containing information on the x dependence of PDFs at some arbitrary initial scale M 0. Later cannot be calculated from pqcd and must be taken from experimental data. N.B. results truncated to any finite order do depend on the choice of M 0. The evolution equations therefore determine merely the M dependence of parton distribution functions, not their absolute magnitude. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

33 PDF evolution in x Q 2 plane: DGLAP, BFKL etc. Y = ln 1/x saturation region ln Q 2 s (Y) non-perturbative region BK/JIMWLK BFKL DGLAP ln Λ 2 ln Q 2 QCD α s ~ 1 α s << 1 Figure 6: Schematic representation of PDF evolution in x Q 2 plane Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

34 PDF at Q 2 = 10GeV 2 : at small x the sea is BIG! 1 HERA-I PDF (prel.) experimental uncertainty Q 2 = 10 GeV 2 20 HERA-I PDF (prel.) experimental uncertainty Q 2 = 10 GeV model uncertainty HERA Structure Functions Working Group Nucl. Phys. B (2008) model uncertainty HERA Structure Functions Working Group Nucl. Phys. B (2008) xu v xf xg ( 1/20) xf xg 0.4 xd v 8 xs ( 1/20) xs x Figure 7: Comparison between valence quark and sea PDF: sea distributions are scaled by a factor of 1/20. xu v xd v 1 x Figure 8: Comparison between valence quark and sea PDF: sea distributions are now unscaled. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

35 F 2 (x, Q 2 ) up to HERA em F 2 -log10 (x) 5 4 x= x= x= x= x= x= x= x= x= x= x= x=0.005 HERA F 2 ZEUS NLO QCD fit H1 PDF 2000 fit H H1 (prel.) 99/00 ZEUS 96/97 BCDMS E665 NMC 3 x=0.008 x=0.013 x= x=0.032 x=0.05 x=0.08 x=0.13 x=0.18 x=0.25 x=0.4 x= Q 2 (GeV 2 ) Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

36 Solving DGLAP Most straightforward method solves the system of coupled evolution equations by means of sophisticated numerical algorithms on powerful computers. The standard analysis of experimental data on DIS proceeds in the following steps: The initial M 0 is chosen. Some parameterization of the boundary condition for parton distribution function D i (x, M 0 ), i = q, q, G, for instance, D i (x, M 0 ) = A i x α i (1 x) β i (1 + γ i x) (43) where A i, α i, β i, γ i are free parameters, is chosen. These parameters, together with the value of the basic QCD parameter Λ, entering α s (M/Λ), are then varied within some reasonable ranges and for each such set the evolution equations (40)-(43) are solved. This yields D i (x, M) at all x and M. In this way obtained theoretical predictions are fitted to experimental data, allowing thereby the determination of the parameters Λ, A i, α i, β i, γ i for each flavor of the quarks as well as for the gluon. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

37 Moments of structure functions and sum rules DGLAP looks particularly simple in terms of moments: f (n) 1 0 x n f (x)dx. (44) dg(n, M) d ln M Instead of convolutions in (40)-(43) we get simple multiplications for the moments = α s(m) π dq i (n, M) d ln M dq i (n, M) d ln M ( = α s(m) π = α s(m) π nf P (0) Gq (n) i=1 ( ) P qq (0) (n)q i (n, M) + P (0) qg (n)g(n, M), (45) ( ) P qq (0) (n)q i (n, M) + P (0) qg (n)g(n, M), (46) (q i (n, M) + q i (n, M)) + P (0) GG (n)g(n, M) ) (47) Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

38 Moments of structure functions and sum rules dq NS (n, M) d ln M A particularly simple equation holds for moments of the nonsinglet quark distribution function (a α s /π) = α ( s(m) ca(m) P qq (0) (n)q NS (n, M) q NS (n, M) = A n π 1 + ca(m) A n are unknown constants, which play the same role as the boundary condition (43) on the distribution functions and which must also be determined from experimental data. ) P (0) qq (n)/b (48) Two features of the above solution are worth noting. Due to the fact that, trivially, P qq (0) (0) = 0 the integral over q NS is M independent. For n > 0 the moments of P qq (0) read: P qq (0) (n) = 4 [ 2S 1 (n) n n ] n 1, S 1 (n) 2 k = ψ(n+1)+γ E, k=1 which implies negative value of P (0) qq (n > 0). This provides the basis for one of the methods for solving the evolution equations. (49) Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

39 Moments of structure functions and sum rules Combining (45)-(47) we obtain evolution equation for the sum: n f S(n, M) (q i (n, M) + q i (n, M))+G(n, M) = q(n, M)+q(n, M)+G(n, M) (50) α s (M) π i=1 ds(n, M) which reads: d ln M = (51) [( ) ( ) ] P qq (0) (n) + P (0) Gq (n) (q(n, M) + q(n, M)) + 2n f P (0) qg (n) + P(0) GG (n) G(n, M) Since S(1, M) represents the fractional sum of the momenta carried by all partons inside the proton (or other hadrons) (51) for n = 1 should be equal to unity at any scale M and therefore the derivative (51) should vanish. This happens provided 1 ( ) P qq (0) (1) + P (0) Gq (1) = dzz P qq (0) (z) + P (0) Gq (z) = 0, (52) 2n f P (0) qg (1) + P(0) GG (1) = ( dzz 2n f P (0) qg (z) + P(0) GG (z) ) = 0. (53) Straightforward evaluation of the above integrals using (25)-(28) shows that in QCD these conditions are, indeed, satisfied. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

40 Extraction of gluon PDF from scaling violations Contrary to quark distribution functions measured directly in l + p DIS the gluon distribution function enters only indirectly as one of the sources (the other being quark distributions themselves) of scaling violations, described by the evolution equations (40) and (41). The relative importance of the two branchings q q + G and G q + q depends on x: while q q + G is dominant at large x G q + q is imporant at small (x 10 2 ) region. Dropping in small x region the first term in brackets of (40) and (41), the evolution equations for quark and antiquark distribution functions q i (x, M), q i (x, M) become identical and read dq i (x, M) d ln M = dq i(x, M) d ln M = α [ 1 s(m) dy π x y P(0) qg ( ) ] x G(y, M). (54) y Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

41 Extraction of gluon PDF from scaling violations Number of effectively massless quarks to be taken into account in (54) depends on the scale M. For n f = 4, these equations imply the following expression for the derivative of F ep 2 (x, M) ( ) df ep 2 (x, M) d ln M = α n s(m) f =4 1 2 ei 2 dz ((x/z)g(x/z)) P (0) qg (z). (55) π i=1 x }{{} 20/9 Eq. (55) is nonlocal, in the sense that the value of its left hand side at some x depends on gluon distribution function G(x) in the whole interval (x, 1). In order to get a local relation between df ep 2 (x, M)/dx and G(x). Prytz has suggested to expand H(x/z, M) (x/z)g(x/z, M) around z = 1 2, the symmetry point of P (0) qg (z) = (z2 + (1 z) 2 )/2: H(x/z, M) = H(2x, M)+H (2x, M)(z 1/2)+H (2x, M)(z 1/2) 2 /2!+ (56) and keeping only the first two terms in (56). Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

42 Extraction of gluon PDF: Prytz approximation At small x the lower integration bound in (55) can safely be set to zero. Second and higher order terms in (56) will vanish: df ep 2 (x, M) d ln M = 20α s(m) 9π 2xG(2x, M) 1 0 dzp (0) qg (z) } {{ } 1/3 = 20α s(m) 2xG(2x, M) (57) 27π The price for simplicity of (57): in practice data do not allow us to determine df 2 (x, M)/d ln M locally at each M, but only as some average for the measured range of M 2 = Q 2. Extracted gluon PDF cannot be attached to any well defined scale M. This simple approximate formula was used at HERA to determine average gluon distribution function from scaling violations, see Fig. 9a. Results of complete analysis at all x and Q 2 using the full set of coupled DGLAP evolution equations (40-43) is shown in Fig. 9b. Note the sizable variation of G(x, Q) as Q 2 increases from 5 to 20 GeV! Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

43 Extraction of gluon PDF: Prytz approximation Figure 9: a) Gluon distribution functions measured at HERA an in NMC experiment, using several different methods, including that based on (57) and denoted Prytz ; b) Gluon density xg(x, M) at two values of the scale M = Q. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

44 Higher orders contribution to gluon PDF: NNLO Figure 10: Valence quark dissociating into qg qgg qggq q ˇ Michal Sumbera (NPI ASCR, Prague) Figure 11: G-PDF as measured in higher orders of DIS Introduction to QCD December 15, / 58

45 Evolution equations at the next-to-leading order In the LL approximation (32) at each order of α s only the term with the same power of the parallel logarithm ln(m 2 /m 2 ), coming from the integration over the region of strongly ordered parton virtualities was kept. In the next-to-leading logarithm (NLL) approximation we extend the integration range by taking into account also the configuration where one of the emissions is not strongly ordered, thereby contributing one power of the parallel log less, i.e. at each order of α s we include also the terms ( ) M αs k ln k 1 2. (58) m 2 g Inclusion of these terms entails several novel features. The relation between PDFs in the NLL approximation and observable structure functions F i (x, Q 2 ) becomes more complicated. Branching functions P ij become perturbative expansions in powers of the couplant and thus functions of both z and α s (M) P ij (z, α s (M)) = P (0) ij (z) + α s(m) P (1) ij (z) +. (59) π Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

46 NLL Evolution equations Contrary to LL branching functions P (0) ij (z) which correspond directly to basic QCD interaction vertices qgq and 3g, evolution functions P (1) ij (z) come from the whole set of diagrams. For instance P q (1) j q i is associated with both diagrams in Fig. 12 and its flavor structure can be decomposed as follows: P (1) q j q i = P qq (1)V δ ji + P qq (1)S. (60) Figure 12: Diagrams contributing to the NLO branching functions P (1) q i q j and P qi q j. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

47 NLL Evolution equations Starting at NLL approximation we encounter branching functions P qi q j describing probability of finding antiquarks inside quarks and vice versa. They are associated with the diagram in Fig. 12a and can be decomposed similarly to (60): P (1) q j q i = P (1)V qq δ ji + P (1)S qq, (61) where the singlet parts in qq and qq channels, coming from the gluon branching in Fig. 12a, are equal, while the valence parts, originating from both diagrams, differ: P (1)S qq = P (1)S qq ; P(1)V qq P (1)V qq. (62) The nonequality in (62) comes from Fig. 12a when i = j, two identical fermions in the final state. Contrary to the LL branching functions P (0) ij, which are universal, (z) are similarly as the coefficients c i of the β function: P (1) ij ambiguous. da(µ, c i ) d ln µ = ba2 (µ, c i ) [ 1 + ca(µ, c i ) + c 2 a 2 (µ, c i ) + ] (63) Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

48 NLL Evolution equations: Factorization scheme The set FS={P (k) ij ; k 1} of nonuniversal higher order branching functions defines the factorization scheme. Corresponding evolution equation: dq NS (x, M) d ln M α 1 [ ( ) s(m) dy π y q NS(y, M) P (0) x NS + α ( ) ] s(m) P (1) x NS + (64) y π y x is thus a definition equation of q NS (x, M), similarly as (63) is a definition equation of the QCD couplant α s (M)! There are, however, two types of quark nonsiglet distribution functions 1 the valence type q ( ) NS,i q i q i ; i = u, d, s,, (65) 2 the nonvalence type q (+) NS q i + q i 1 n f which for n f = 2 reduces to u (+) = 1 2 n f k=1 (q k + q k ), (66) ( ) 1 [ ] u + u d d = (u d) (d u). (67) 2 Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

49 NLL Evolution equations: Factorization scheme In the LL approximation q PDFs are generated exclusively by gluon splitting we expect u = d. Consequently (67) coincides with the difference 1 2 (u( ) d ( ) ) of the valence-like NS distributions. In the NLL approximation the q (+) and q ( ) NS PDFs evolve according to different kernels, expressed as combinations of P qq (1)V and P (1)V qq : P (1) ( ) P(1)V qq P (1)V qq, P (1) (+) P(1)V qq + P (1)V qq. (68) In FS used in phenomenological analyses P (1)V qq is very small difference between these two types of NS distributions negligible. Converted into moments the NLL NS evolution equation (64) implies: dq NS (n, M) d ln M = α [ s π q NS(n, M) P (0) NS (n) + α s(m) π P (1) NS (n) ], (69) [ ] (0) P ca(m) NS (n)/b q NS (n, M) = A n (1 + ca(m)) P(1) NS (n)/bc, (70) 1 + ca(m) where a(m) α s (M)/π. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

50 Hard scattering and the factorization theorem Basic idea of factorization is to split range of virtualities τ t corresponding to Feynman diagram in Fig. 13a into two parts: τ τ 0 : The integral over nearly parallel configurations is absorbed in the dressed PDFs q NS (x, τ 0 ) and similarly for PDFs of q and G. τ > τ 0 : Integration yields finite result C NS (z, Q/M, FS) called hard scattering cross section in the NS channel. Beside Q 2, M 2 = τ 0 and z it also depends on the FS of PDFs, specified at NLL by P (1) ij (z), and hard scattering scale µ, which in general is different from factorization scale M. Figure 13: KLN in DIS. t in a) denotes virtuality of intermediate quark. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

51 Hard scattering and the factorization theorem Hard scattering cross section is itself given as expansion in α s (µ) ( C NS z, Q ) [ M, FS = δ (k) NS δ(1 z) + α s(µ) π C (1) NS (z, QM ) ], FS +, (71) where the superscript (k) distinguishes between different NS channels and the quantities δ (k) NS include electromagnetic (or weak) couplings of quarks. According to the factorization theorem a generic NS structure function F NS (x, Q 2 ) can be written as the convolution 1 ( F NS (x, Q 2 dy x ) = y q NS(y, M, FS)C NS y, Q ) M, FS, (72) x where all dependence Q resides in the hard-scattering cross section C NS. Note that both q NS (x, M, FS) and C NS (z, Q/M, FS) depend on the factorization scale M, as well as on the factorization convention FS. N.B. physical quantity F NS (x, Q 2 ) in (72) depends neither on the factorization scale M, nor on the renormalization scale µ. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

52 Hard scattering and the factorization theorem Factorization theorem guarantees that all parallel singularities can be absorbed into q NS and also that dependence on unphysical quantities M and P (k) ij, which define the FS, cancel in the convolution (72), provided perturbative expansions in (71) as well as (64) are summed to all orders. Cancelation of the dependences of q NS and C NS on the FS is guaranteed by: C (1) NS (z, QM ) ( ), FS = δ (k) NS P (0) NS (z) ln Q M + P(1) NS (z) + k(z), (73) b which expresses C (1) NS explicitly as a function of P(1) and where k(z) is a function of z, independent of M and P (1) NS. In terms of moments the relations (72) and (73) read C (1) NS (n, QM ) ( ), FS = δ (k) NS P (0) NS (n) ln Q M + P(1) NS (n) + k(n) (74) b ( and F NS (n, Q 2 ) = q NS (n, M, FS)C NS n, Q ) M, FS. (75) Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

53 Solving the evolution equations: Buras Gaemers method Historically first method used to solve evolution equations in the LO. It starts by assuming that the initial conditions on NS, i.e. valence u v and d v quark PDFs are given by so called beta distribution: which implies for the moments q v (x, M 0 ) = A q x ηq 1 (1 x) η q 2, (76) q v (n, M 0 ) = A q B(η q 1 + n + 1, ηq 2 + 1), Γ(x)Γ(y) B(x, y) Γ(x + y). (77) Using these initial conditions in DGLAP evolution equations (38) for q v (n, M) yields explicit analytic expressions for their scale dependence. * Note, however, that the simplicity of these explicit solutions is lost when we attempt to translate them back into the x space. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

54 Solving the evolution equations: Buras Gaemers method BG assumed that these solutions can be reasonably approximated at all M by the form (77) in which the coefficients A q, η q 1, ηq 2 are allowed to depend on M as: [ ] η q i (s) = ηq i +η q i s, i = 1, 2; A q δ q ln(m/λ) = B(η q 1 (s), 1 + s ln (78) ηq 2 (s)), ln(m 0 /Λ) where δ u = 2 and δ d = 1. These later conditions guarantee the validity of fundamental QPM sum rules, like the Gross-Llewellyn-Smith sum rule: 1 0 F νn 3 (x, Q 2 )dx = 1 0 (u v (x) + d v (x))dx, (79) The variable s defined in (78) is usually called the evolution variable. η q i, η q i were determined by fitting, for a chosen initial M 0, the approximation based on (78) to the exact LO solutions for the first 10 moments. Λ appearing in (78) can not be directly associated with any renormalization scheme and is usually denoted Λ LO. A number of experiments used this method to determine Λ LO from fits to experimental data. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

55 Numerical and Jacobi polynomials methods Sophisticated numerical methods developed by Duke, Owens and collaborators in 1980th. Due to the immense increase of CPU and memory of modern computers the original limits of this method have disappeared and now most of the groups use these methods. The numerical routines are usually made available by their authors, so that it is relatively easy and straightforward to use them even for nonexperts and experimentalists. Another technique is based on expansion of the convolution (72) into (orthogonal) Jacoby polynomials: k 1 Θ αβ k (x) c αβ kj x j, dxx α (1 x) β Θ αβ k (x)θαβ l (x) = δ kl, (80) j=0 In terms of the Jacobi moments a αβ k F NS (x, Q 2 ) = x α (1 x) β 0 (Q2 ),F NS (x, Q 2 ) can be expanded as: k=0 Θ αβ k (x)aαβ k (Q2 ) (81) former being given as linear combinations of conventional moments F NS (j, Q 2 ): 1 k a αβ k (Q2 ) dxf NS (x, Q 2 )Θ αβ k (x) = c αβ kj F NS (j, Q 2 ). 0 j=0 Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, 2009 (82) 55 / 58

56 Jacobi polynomials methods, PDFLIB Substituting (70) and (74) into (75) and this subsequently into (82) and (81), we get an explicit expression for F NS (x, Q 2 ) as a function of the unknown constants A n and parameter the Λ MS. Instead of A n we can also start with some initial distribution, like (76), specifying q NS (x, M 0 ) at some M 0, and use it to evaluate the constants A n by means of the expression (70). Having determined A n we can then proceed as outlined before. Over the last decade a large number of phenomenological analyses of experimental data have been carried out and PDFs of nucleons, pions and recently also photons, determined. Several groups of theorists and phenomenologists have been systematically improving these analyses by incorporating new and more precise data as well as by employing more sophisticated theoretical methods. All these distribution functions are now available, as functions of both the momentum fraction x and the factorization scale M, in the computerized form in the CERN library PDFLIB. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

57 F 2 (x, Q 2 ) up to HERA em F 2 -log10 (x) 5 4 x= x= x= x= x= x= x= x= x= x= x= x=0.005 HERA F 2 ZEUS NLO QCD fit H1 PDF 2000 fit H H1 (prel.) 99/00 ZEUS 96/97 BCDMS E665 NMC 3 x=0.008 x=0.013 x= x=0.032 x=0.05 x=0.08 x=0.13 x=0.18 x=0.25 x=0.4 x= Q 2 (GeV 2 ) Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58

58 Exercises 1 Argue on physical terms why P (0) GG (x) must contain the + distribution. 2 Show explicitly that the integration over the region of unordered virtualities doesn t produce the leading logs. 3 Derive (33) and evaluate the integrals in (34), neglecting the dependence of α s on τ. 4 Show what results if the argument µ of α s (µ) were the same and equal to M 2 for all the vertices in the ladder of Fig. 4a. 5 Show that the NS combinations in (66) and (65) do not mix under branching. 6 Working in the LL approximation and assuming the parametrization (43), express the constants A(n) as functions of the parameters A, α, β, γ, setting for simplicity γ = 0. 7 Determine the value of the coupling δ NS in (71) for the NS structure function defined as 1 ep 2 (F2 F2 en). 8 Derive the WW approximation for the scalar charged particle. Show that the corresponding branching function P γe (x) 1 x x. Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, / 58