The partonic structure of protons and nuclei: from current facilities to the EIC Alberto Accardi Hampton U. and Jefferson Lab “Frontiers in Nuclear and Hadronic Physics” Galileo Galilei Institute, Florence, Italy 20-24 February 2017
Plan of the lectures PART 1: QCD factorizatjon and global PDF fjttjng • Lecture 1 – Hadrons, partons and Deep Inelastjc Scatuering • Lecture 2 – Parton model • Lecture 3 – The QCD factorizatjon theorem • Lecture 4 – Global PDF fjts PART 2: Parton distributjons from nucleons to nuclei • Lecture 5 / 6 PART 3: The next QCD frontjer – The Electron-Ion collider • Lextures 7 / 8 2 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Lecture 2 – Parton Model Parton model – Heuristjc derivatjon DIS revisited – More kinematjcs – Collinear factorizatjon, defjnitjon of PDF – Parton model for DIS and its limitatjons 3 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Parton Model 4 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Parton model (see [Feynman]) We have evidence that a proton is a composite object made of spin ½ partjcles At high-energy, expect a “probe” to interact with these point-like objects – In DIS , the photon wave-length in rest frame, neglectjng masses: – E.g. , for x =0.1, Q 2 =4 GeV 2 (and puttjng back c and hbar ), l = 10 -17 m = 10 -2 fm to be compared with R p ≈ 1 fm 5 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Parton model (see [Feynman]) We have evidence that a proton is a composite object made of spin ½ partjcles (and should also expect some radiated gluons) At high-energy, expect a “probe” to interact with these point-like objects k k quark ' p p i proto proto n n In DIS, the photon scatuers on quasi-free quarks – Empirical evidence: F 2 = 2x B F 1 Seen by a high-energy probe, the nucleon seems a box of practjcally free “partons” sharing the proton’s momentum 6 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Parton model (see [Feynman]) So, the nucleon is a box of practjcally free “partons” sharing momentum How can we understand this (respectjng relatjvity, quantum mech., unitarity, etc...) ? – In fjeld theory, proton wave functjon is specifjed by amplitude to fjnd any number of partons moving with various momenta – This picture is however frame dependent – needs a good choice: • Rest frame : assume fjnite energy of interactjons, i.e. , fjnite interactjon tjmes • Infjnite momentum frame along z directjon: tjmes are dilated, interactjon is slower and slower, untjl as p z → ∞ they appear to not interact at all – this is the right frame for our intuitjve picture (and for precise realizatjon in fjeld theory) 7 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Parton model (see [Feynman]) What is a good variable to describe the partons? – A momentum fractjon, say x = k z / p z , is invariant under boost along z (and phenomenologically successful in many processes) – Partons have “intrinsic” transverse momentum k T 2 ≈ 0.4 GeV 2 (small compared to Q 2 , neglect in fjrst instance) In fjeld theory, amplitude for a state of energy E to be made of n partjcles of total energy E n = E 1 + E 2 +...+ E n is dominated in perturbatjon theory by – Using the amplitude for 2 partons is 8 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Parton model (see [Feynman]) Important consequences: – The proton wave functjon depends on x i , and k T | i (but transverse momenta are small compared to Q 2 : they are negligible in fjrst instance – but not uninterestjng, ask Alexei) – Partons cannot have negatjve x i (unless this is very small, see Feynman) • Imagine a 2 parton state, with x 1 < 0, then the denominator is much larger than for 2 positjve x partons, for which E 2 ≈ 0, and • States with parton of negatjve fractjonal momentum are very much suppressed compared to all x i >0 9 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Parton model (see [Feynman]) Defjne a parton distributjon Probability of fjnding a parton i with momentum Fractjon between x and x + dx Hard processes (e.g. a DIS cross sectjon) should be “factorized” 10 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Sum rules What can we expect in general? – There are gluons, expect also “sea” quark antj-quark pairs – Proton charge: +1 , but u : +2/3, d : -1/3, s : -1/3, g : 0 – Proton isospin: 1/2 , but u : +1/2, d : -1/2, s : 0, g : 0 – Proton strangeness: 0 , but u : 0, d : 0, s : 1, g : 0 11 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Sum rules Parton sum rules (for the proton) – Charge conservatjon – Momentum conservatjon 12 accardi@jlab.org GGI, Feb 2017 – Lecture 2
How to probe these “partons”? electron electron k k ' quark i p p proton proton DIS ≈ photon-quark elastjc scatuering Interpretatjon of x B – Parton carries fractjon x of proton's momentum: k m = x p m – 4-momentum conservatjon: k' = k+q – Partons have zero mass: k 2 = k' 2 = 0 The virtual photon probes quarks with x = x B 13 accardi@jlab.org GGI, Feb 2017 – Lecture 2
How to probe these “partons”? Beware: There is an inconsistency in the derivatjon: – From k = xp follows that quarks are massive, M q = xM !! A heuristjc way out is to work in the “infjnite momentum frame”, where so that one can neglect the proton's mass: – This frame is also important to betuer justjfy the parton model – But quarks should be massless in any frame • The problem lies in the defjnitjon of x • We'll see a betuer solutjon in tomorrow's lecture – Similarly, the vector 3-momentum is not a Lorentz-invariant scale • In fact, M can be neglected compared to Q , not , as we shall see 14 accardi@jlab.org GGI, Feb 2017 – Lecture 2
How to probe these “partons”? Caveats: – Tacit probabilistjc assumptjon: we are multjplying probabilitjes rather than amplitudes • justjfjed by tjme dilatjon / smallness of photon wavelength arguments • Can be broken by sofu (long wavelength) initjal state interactjons between the proton and quark lines – Likewise, we are assuming the same parton distributjon applies to other process: “universality” – The non-trivial proof in QCD is called “QCD factorizatjon theorem” 15 accardi@jlab.org GGI, Feb 2017 – Lecture 2
DIS revisited 16 accardi@jlab.org GGI, Feb 2017 – Lecture 2
More kinematics Light-cone coordinates : a natural coord. system for processes dominated by large momentum Cartesian 0 light-cone a a + a – 3 17 accardi@jlab.org GGI, Feb 2017 – Lecture 2
More kinematics Boosts of velocity b in the 3-directjon – Boost-invariant quantjtjes: 0 a a + a – Light-cone (Sudakov) vectors: 3 18 accardi@jlab.org GGI, Feb 2017 – Lecture 2
More kinematics Boosts of velocity b in the 3-directjon – Boost-invariant quantjtjes: 0 a a + “fractjonal momenta” a – Light-cone (Sudakov) vectors: 3 19 accardi@jlab.org GGI, Feb 2017 – Lecture 2
More kinematics Collinear frames : – a set of frames such that p, q lie in the (+,-) plane with “Nachtmann variable” • Parameter p + controls boost in 3 - directjon x → x B • “massless limit” : as Q 2 → ∞, • Bjorken x B intepreted as fractjonal momentum of the photon – Ex.1 (med): derive this imposing M 2 = p 2 , Q 2 =- q 2 , x B = Q 2 /(2 p ⋅ q ); try fjrst by settjng M =0. 20 accardi@jlab.org GGI, Feb 2017 – Lecture 2
More kinematics Special cases: – Proton rest frame: p q – Breit frame: p q this is an (important) example of an “infjnite momentum frame” – Ex.2 (easy): derive these formulae 21 accardi@jlab.org GGI, Feb 2017 – Lecture 2
More kinematics Special cases: – Proton rest frame: p q negligible only ≪ Q 2 if M 2 – Breit frame: p q this is an (important) example of an “infjnite momentum frame” – Ex.2 (easy): derive these formulae 22 accardi@jlab.org GGI, Feb 2017 – Lecture 2
More kinematics Special cases: – Proton rest frame: q “M=0” p p q q negligible only ≪ Q 2 if M 2 – Breit frame: p p p “M=0” q q this is an (important) example of an “infjnite momentum frame” – Ex.2 (easy): derive these formulae 23 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Collinear factorization in DIS at LO 24 accardi@jlab.org GGI, Feb 2017 – Lecture 2
see, Accardi, Qiu, JHEP 2008 (simple) Collinear factorization* [Collins] (full proof) Start from the handbag diagram Parton fractjonal momentum: Parton's Bjorken x : – Expand around on-shell ( k 2 = m q 2 = 0) and collinear ( k ⟂ =0) momentum lead to O ( L 2 / Q 2 ) Call this correctjon in s DIS *Note: will consider M 2 / Q 2 ≪ 1 for simplicity (but check the exercises) 25 accardi@jlab.org GGI, Feb 2017 – Lecture 2
Collinear factorization* Consequences: ≈ – Now, and the quark is massless in any frame! – Let's impose also that the fjnal state quark is on shell: in any collinear reference frame! – Ex.3 (easy): show that in general, x = x *Note: will consider M 2 / Q 2 ≪ 1 for simplicity (but check the exercises) 26 accardi@jlab.org GGI, Feb 2017 – Lecture 2
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