Tercera Sesión. XI Escuela de Física Fundamental. Universidad Veracruzana, Xalapa. 28 de Septiembre de 2016

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1 Tercera Sesión XI Escuela de Física Fundamental Universidad Veracruzana, Xalapa. 28 de Septiembre de / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 1/35 35

2 3 lectures: three quarks for Muster Mark! Lecture 1: QCD at the LHC Gauge invariance and Feynman rules for QCD Lecture 2: Renormalization and running α s pqcd in e + e -collisions: from partons to hadrons, jets, shape variables Lecture 3: pqcd in lepton-hadron collisions: DIS and parton evolution pqcd at the LHC Overlap with other lectures is unavoidable. As with other things in life, the U and everything: overlap is good :) 2 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 2/35 35

3 The parton model quarks binding forces that confine them are due to soft gluon exchange a hard virtual gluon exchange would break the proton apart! (Q > Q m p /Q and α s 0 ) time scale for qq interaction > 1/m p off-shell photon can probe the proton with limited lifetime photon can probe incoherent quark: free quark inner structure of the proton probed with off-shell photon is universal (wave function of the proton determined by soft-gluon dynamics) so simplest way to probe: deeply inelastic scattering (DIS) (what historically led to idea of partons) 3 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 3/35 35

4 The parton model G. Sterman, CTEQ (2004) 4 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 4/35 35

5 The parton model G. Sterman, CTEQ (2004) 5 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 5/35 35

6 The parton model 6 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 6/35 35

7 Deep Inelastic Scattering (DIS) D. de Florian, CLASHEP (2015) 7 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 7/35 35

8 The parton model and DIS x-sec for intn of the virtual photon with proton at LO: σ 0 = 1 0 dx i e 2 i f i (x)ˆσ 0 (γ q i q i, x) (1) f i (x) density of quarks of flavour i carrying a fraction x of the proton momentum ˆσ 0 intn between the photon and a free (massless) quark: ˆσ 0 = 1 M0 (γ q q ) 2 d 2 p (2π) 4 δ(p q p) flux (2π) 3 2p 0 = 1 M0 2 2πδ(p 2 ) (2) flux 8 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 8/35 35

9 The parton model and DIS Using p = xp + q, where P is the proton momentum, we get (p ) 2 = 2xP q + q 2 2xP q Q 2 (3) infinite momentum frame P µ (P, 0, 0, P) with P >> M. where x bj = Q2 2P q ˆσ 0 (γ q q ) = 2π M0 2 1 flux 2P q δ(x x bj) (4) σ 0 = 2π M0 2 flux Q 2 is the so-called Bjorken-x variable. Finally: i x bj ei 2 f i (x bj ) 2π M0 2 F flux Q 2 2 (x bj ) (5) So, measurement of inclusive ep x-sec as function of Q 2 and P q (= m p (E E) in the proton rest frame) probes the quark momentum distribution inside the proton. 9 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 9/35 35

10 contd. D. de Florian, CLASHEP (2015) 10 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 10/35 35

11 contd. D. de Florian, CLASHEP (2015) 11 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 11/35 35

12 contd. D. de Florian, CLASHEP (2015) 12 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 12/35 35

13 Parameter space exploration (PDG) 13 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 13/35 35

14 contd. D. de Florian, CLASHEP (2015) 14 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 14/35 35

15 contd. D. de Florian, CLASHEP (2015) 15 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 15/35 35

16 Parton evolution Go beyond LO parton-model and add real-emissions Real-emission corrections to the Born level process: q q k p + p (6) p k p The first diagram is proportional to 1/(p k) 2 = 1/(2pk), which diverges when k is emitted parallel to p: p k = p 0 k 0 (1 cos θ) 0 when cos θ 1 (7) The second diagram is also divergent, if k is emitted parallel to p : harmless! summing over all possible final states for inclusiveness coll div cancelled in final-state q self-energy corrections 16 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 16/35 35

17 Parton evolution: gauge fixing + parametrization The amplitude for the only diagram carrying the initial-state singularity is: M g = igλ a iju(p )Γ p k ˆɛ(k)u(p) (8) (p k) 2 Squaring the most singular part of the amplitude, and summing over colours and spins, we get: N C F { }}{ M g 2 = g tr(λ a λ a ) 1 t 2 g ɛ a with t = (p k) 2 = k 2 t /(1 z). tr[ p Γ( p k) ɛp ɛ ( p k)γ ] (9) 17 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 17/35 35

18 Parton evolution: AP splitting functions So the one-gluon emission process factorizes in the collinear limit into the Born process times a factor which is independent of the beams nature! If we add the gluon phase-space: [dk] we get where d 3 k (2π) 3 2k = dk dφ 0 k 0 2π 1 dk b ot 2 8π 2 2 = dz 1 1 z 16π dk 2 2 (10) Mg 2 [dk] = dk 2 ( αs ) dz P k 2 qq (z) M 0 2 (11) 2π 1 + z 2 P qq (z) = C F (12) 1 z is the so-called Altarelli-Parisi splitting function for the q q transition (z is the momentum fraction of the original quark taken away by the quark after gluon emission). 18 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 18/35 35

19 Parton evolution: correction to parton-model x-sec Ready to calculate the corrections to the LO parton-model x-sec: σ g = dx f (x) 1 dz dk 2 ( αs ) P flux k 2 qq (z) M 0 2 2πδ(p 2 ) 2π (13) Using (p 2 ) = (p k + q) 2 (zp + q) 2 = (xzp + q) 2 and So finally, σ g = 2π flux δ(p 2 ) = 1 1 ( 2P q z δ x x bj z ( ) σ M0 2 Q 2 i ) = x ( bj z δ( x x bj z ) (14) ei 2 α s dk 2 x dz ( x ) bj 2π k 2 z P qq(z)f i z (15) 19 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 19/35 35

20 Parton evolution: parton density and RGE Inclusion of the O(α s ) correction is equivalent to a contribution to the parton density: f i (x) f i (x) + α Q 2 s dk 2 1 dz ( x ) 2π µ 2 k 2 0 x z P qq(z)f i z (16) The renormalized parton density: ( ) Q f (x, Q 2 ) = f (x, µ αs dz ) + log µ 2 2π x z P qq(z)f ( x z ) (17) 20 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 20/35 35

21 Parton evolution: parton density and RGE RGE condition: df (x, Q 2 ) d ln µ 2 = µ 2 df (x, µ2 ) dµ 2 α s 2π and then 1 x dz z P qq(z)f µ 2 df (x, µ2 ) = α 1 s dz dµ 2 2π x z P qq(z)f ( x ) 0 (18) z ( x z, µ2 ) This equation is the DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) equation In analogy to R e + e where RGE induces resummation of leading logs, here the DGLAP equation resums full tower of leading logarithms of Q 2. (19) 21 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 21/35 35

22 Parton evolution: parton density evolution O(α s ) parton evolution equation for the density of the ith quark flavour: df q (x, t) = α 1 s dz ( x ) ( x )] [P qq (z)f i dt 2π x z z, t + P qg (z)f g z, t (20) O(α s ) parton evolution equation for the density of gluons: df g (x, t) dt with = α s 2π 1 x dz z [ P qg = 1 2 P gq (z) ( x ) ( x ) ] f i z, t + P gg (z)f g z, t i=q,q [ z 2 + (1 z) 2] 1 + (1 z) 2 P gq (z) = P qq (1 z) = C F [ z 1 z P gg (z) = 2C A + z ] + z(1 z) z 1 z (21) (22) 22 / 35 M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 22/35

23 Parton evolution: valence/singlet densities Define moments of an arbitrary function g(x) as: g n = 1 0 dx x x n g(x) (23) Evol eqns turn into ordinary linear differential equations: df (n) i dt df (n) g = α s 2π [P(n) qq f (n) i + P qg (n) f g (n) ] (24) + P (n) gq f (n) i ] (25) = α s dt 2π [P(n) gg f g (n) Take valence (V (x, t)) and singlet (Σ(x, t)) densities: V (x) = i f i (x) i f i (x) (26) Σ(x) = i f i (x) + i f i (x) (27) 23 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 23/35 35

24 contd. D. de Florian, CLASHEP (2015) 24 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 24/35 35

25 contd. D. de Florian, CLASHEP (2015) 25 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 25/35 35

26 contd. D. de Florian, CLASHEP (2015) 26 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 26/35 35

27 contd. D. de Florian, CLASHEP (2015) 27 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 27/35 35

28 . 28 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 28/35 35

29 Why do we keep QCDing? cannot be done with quantitative seriousness without pqcd beyond LO but also QCD is at the heart of everything! 29 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 29/35 35

30 QCD reviews in PDG 30 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 30/35 35

31 QCD reviews in PDG 31 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 31/35 35

32 Higgs at hadron colliders D. de Florian, CLASHEP (2015) 32 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 32/35 35

33 Invitación 33 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 33/35 35

34 Invitación 34 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 34/35 35

35 Gracias Maria Elena Tejeda-Yeomans Cuerpo Académico de Partículas y Cosmología Universidad de Sonora elena.tejeda@fisica.uson.mx unidusu@gmail.com Facebook: MaElena Tejeda-Yeomans Wordpress: particuleando.com 35 / M.E. Tejeda-Yeomans elena.tejeda@fisica.uson.mx Iniciación a la QCD 35/35 35