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(Perturbative) QCD, Jets & Colliders

(Perturbative) QCD, Jets & Colliders. Lecture 2: Calculating with QCD – e + e - Physics and Perturbation Theory (Correcting the Parton Model). Stephen D. Ellis University of Washington. TSI 06 TRIUMF July 2006. Outline. Introduction & (Pre) History – The Parton Model

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(Perturbative) QCD, Jets & Colliders

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  1. (Perturbative) QCD, Jets & Colliders Lecture 2: Calculating with QCD – e+e- Physics and Perturbation Theory(Correcting the Parton Model) Stephen D. Ellis University of Washington TSI 06 TRIUMF July 2006

  2. Outline • Introduction & (Pre) History – The Parton Model • pQCD - e+e- Physics and Perturbation Theory(Correcting the Parton Model) • pQCD - Hadrons in the Initial State and PDFs • pQCD - Hadrons and Jets in the Final State • Jets at Work S. D. Ellis TSI 2006 Lecture 2

  3. Parton Model Summary: • Observed quantum numbers of meson & baryons well described by SU(3) of Flavor Flavor 8 + 1Flavor 10 + 8 + 8 + 1 • Quarks and gluons ~ free to explain “scaling” in DIS and e+e-, no intrinsic dimensionful scale!! • Partons in a hadron described by “scaling” PDFs = probability of finding parton with momentum faction x (a function only of x, no dimensionful parameters!?) • Hadrons in a parton, i.e, in a jet – collinear shower arising from a parton, described by a scaling Fragmentation function of z, the momentum fraction (and only z, no dimensionful parameters) S. D. Ellis TSI 2006 Lecture 2

  4. ASIDE: Young Diagrams – Group Engineering (Phys 558 Lecture 1) • Each horizontal row of boxes is at least as long as the horizontal row below it. • We can think of the horizontal direction as symmetrization (with respect to some internal index) and the vertical direction as anti-symmetrization. There are at most n rows for the case of SU(n).For the SU(3) the simplest representations look like S. D. Ellis TSI 2006 Lecture 2

  5. The counting of states within a given representation involves filling in the boxes starting with the upper left hand corner. For SU(N) you put N in that box and then increase the number when moving to the left and decrease the number when moving down. An example is .Next we must define a “hook”. A hook is the set of boxes that form a “right hook”, moving first up and then right. For the previous example there are 3 possible hooks, 1 involving all three boxes, one involving only the right most box and one involving only the bottom box, . Without proof, we note that the number of states in the representation represented by a Young diagram is given by the product of all the boxes with numbers in them (i.e., the product of the numbers in the boxes) divided by the product of the lengths (number of boxes) of the hooks. For the example above for SU(3), we have as expected. S. D. Ellis TSI 2006 Lecture 2

  6. Two other examples to test your understanding are • To actually combine multiplets, i.e., define a product of representations, we need to carefully label things. Here we use the notation of the PDG. Consider the product of 2 octets, • where we use boxes to represent the first octet and letters for the second (with “a” for the first row, “b” for the second, etc.). Now we proceed to “add the boxes” with the following rules.Start with the left-hand Young diagram (the boxes) . S. D. Ellis TSI 2006 Lecture 2

  7. Add the “a’s” in all ways that produce a valid Young diagram, but with no more than a single “a” in each column (initially symmetric labels cannot be antisymmetrized) • Starting in the second row (where the “b’s” were initially) add the “b’s” subject to the constraint that, reading from right to left starting at the end of the first row and moving on to the second row, the number of “a’s” must be  the number of “b’s” ( the number of “c’s”). Thus the allowed Young diagrams are S. D. Ellis TSI 2006 Lecture 2

  8. Using the rules noted earlier we can work out the multiplicity of each of these irreducible representations (i.e., with the “hooks” formula) S. D. Ellis TSI 2006 Lecture 2

  9. The above applies to any SU(N), just replace 3 by N. Here is another feature useful for SU(3) whose representations are planar AND have the symmetries of the underlying fundamental triangle. Thus the boundary of the representation in terms of p, the number of steps across the top of the representation, and q, the number of steps down the diagonal. Thus we can label a represnetation by (p,q), • This allows us to separate the 3 = (1,0) while the anti-triplet is (0,1) . The symmetric octet is (1,1). The 6 is (2,0) and the 6-bar is(0,2) • The connection to the Young diagrams is that the first row has p more boxes than the second row and the second row has q more boxes than the third row. S. D. Ellis TSI 2006 Lecture 2

  10. Parton Model Summary: • Observed quantum numbers of meson & baryons well described by SU(3) of Flavor Flavor 8 + 1Flavor 10 + 8 + 8 + 1 • Quarks and gluons ~ free to explain “scaling” in DIS and e+e-, no intrinsic dimensionful scale!! • Partons in a hadron described by “scaling” PDFs = probability of finding parton with momentum faction x (a function only of x, no dimensionful parameters!?) • Hadrons in a parton, i.e, in a jet – collinear shower arising from a parton, described by a scaling Fragmentation function of z, the momentum fraction (and only z, no dimensionful parameters) S. D. Ellis TSI 2006 Lecture 2

  11. Parton Model makes perfect sense for FREE quarks and gluonsHow can it working for interacting partons??Use perturbative expansions to study this question because that is what we know how to do! (pQCD) S. D. Ellis TSI 2006 Lecture 2

  12. The (Classical) QCD Lagrangian Acting on the triplet and octet, respectively, the covariant derivative is The matrices for the fundamental (tabB) and adjoint (TCDB) representations carry the information about the Lie algebra (fBCD is the structure constant of the group) S. D. Ellis TSI 2006 Lecture 2

  13. Feynman Rules: Propagators – (in a general gauge represented by the parameter , Feynman gauge is  = 1; this form does not include axial gauges) Vertices – Quark – gluon 3 gluons S. D. Ellis TSI 2006 Lecture 2

  14. Feynman Rules II: 4 gluons S. D. Ellis TSI 2006 Lecture 2

  15. pQCD I - Use QCD Lagrangian to Correct the Parton Model • Naïve QCD Feynman diagrams exhibit infinities at nearly every turn, as they must in a conformal theory with no “bare” dimensionful scales (ignore quark masses for now).*** First consider life in the Ultra-Violet – short distance/times or large momenta (the Renormalization Group at work): • The UV singularities mean that the theory • does not specify the strength of the coupling in terms of the “bare” coupling in the Lagrangian • does specify how the coupling varies with scale [s() measures the “charge inside” a sphere of radius 1/] *** Typical of any renormalizable gauge field theory. This is one reason why String theorists want to study something else! We will not discuss the issue of choice of gauge. Typically axial gauges ( ) yield diagrams that are more parton-model-like, so-called physical gauges. S. D. Ellis TSI 2006 Lecture 2

  16. Consider a range of distance/time scales – 1/ • use the renormalization group below some (distance) scale 1/m (perhaps down to a GUT scale 1/M where theory changes?) to sum large logarithms ln[M/] • use fixed order perturbation theory around the physical scale 1/ ~ 1/Q (at hadronic scale 1/m things become non-perturbative, above the scale M the theory may change) Short distance Long distance S. D. Ellis TSI 2006 Lecture 2

  17. Diagrammatically • Corrections to the parton model come from adding gluon interactions, including LO 2 Loop 1 Loop Loops are UV divergent like dk4/k4 - keep (logarithmic) contributions from the range  to M as a “formal” series for the effective coupling in terms of the initial coupling. S. D. Ellis TSI 2006 Lecture 2

  18. Interpret as screening/anti-screening of color charge in volume (1/)3 + + 2/ + S. D. Ellis TSI 2006 Lecture 2

  19. Sum (reorganize) the (cutoff) calculation results as an effective (renormalized) coupling** This result is more compactly specified by the renormalization group equation, which can be evaluated order-by-order in perturbation (PDG notation) **Masses and wave functions also exhibit renormalization. S. D. Ellis TSI 2006 Lecture 2

  20. Must Sum Large Logarithms The “running” coupling illustrates typical features of QCD – • expanding to a fixed power of s is often not enough* • large logarithms (the remnants of the infinities) must be resumed to all orders by some technique • By measuring s at some scale 0 can define a dimensionful parameter QCD Dimensional transmutation !! * In any case is an asymptotic expansion, not convergent S. D. Ellis TSI 2006 Lecture 2

  21. Beyond 1-Loop The first form above is the “one-loop” solution for s (keeping only the 0 term). pQCD allows one to systematically include the higher loop corrections, as expansion in inverse powers of ln[]. S. D. Ellis TSI 2006 Lecture 2

  22. Asymptotic Freedom/Infrared Slavery Our knowledge of the behavior in the UV is now encoded in QCD. Note that the precise value of QCD will to depend on the order of the  function used (1-loop, 2-loop, etc.) and the scheme. The data does not change, only the internal theoretical parameters. • Experimentally QCD ~ 21625 MeV (using 5 “active” flavors at the Z pole) • The running of the coupling is clear in the data, as is the precision of our knowledge of s, e.g., s(mZ) = 0.1176  0.002. Look at the (amazing!) behavior of running s – As  increases, s decreases – asymptotic freedom! As  decreases, s increases – infrared slavery! Just what we wanted in the parton model!!!!! S. D. Ellis TSI 2006 Lecture 2

  23. s(Q) NOTE -1 • EM(Q) – only fermion loops contribute, runs the other way (0EM < 0!) S. D. Ellis TSI 2006 Lecture 2

  24. But Note!! • Physical quantities, (Q), cannot depend on  • This will be important! S. D. Ellis TSI 2006 Lecture 2

  25. Other Potential Singularities – Infrared(after renormalizing, i.e., removing, the UV singularities, formally with counter terms) • Soft & Collinear! (Massless) Propagators can go on shell due to emission of soft & collinear gluons • Infrared (soft) familiar from QED – e.g., since the photon is zero mass (in the gauge symmetric theory), the theory wants to emit an infinite number of zero energy photons and the exclusive (electron) cross section diverges. Fix with inclusive cross section that sums over soft photons over 0 < E < E leading to Ln[E/Q] dependence • Collinear, mq  0, still gives ln[Q/ mq] which is large for mq Q, and here we will think about mq → 0 S. D. Ellis TSI 2006 Lecture 2

  26. pQCD II - Perturbative Corrections to Parton Model – e+e- Annihilation • Revisit e+e‑ scattering (massless partons!) – real emission • Define handy variables (q2=Q2=s) S. D. Ellis TSI 2006 Lecture 2

  27. Sum & Square - • Sum amplitudes and square, visualized (ignoring the lepton part) as the (3-body) imaginary (absorptive) parts of the following loop diagrams (the vertical dashed line identifies the particles that are put on the mass shell, i.e., that are the “real” particles in the final state) + + + S. D. Ellis TSI 2006 Lecture 2

  28. e+e- Annihilation cont’d Phase Space • The cross section looks like (see HW) Hence the singular regions are: • Collinear gluon -- 130, x21230, x11 • Soft gluon – x30  (x11 and x21) with (1- x1)/(1- x2) fixed The red singularities arise from a propagator above going on-shell – either 1+3 or 2+3 ! S. D. Ellis TSI 2006 Lecture 2

  29. Long Distance  Collinear/Soft Singularities • On-shell propagators – long distance propagation – perturbative expansion fails – 1+ s x big + s2 x bigger … (No Surprise) • Still parton model-like picture – short distance/time simple, long distance complex. How do we proceed? • Ask questions that are insensitive to long distance structuree.g., TOT which receives contributions from all states – details cannot matter – in detail the singularities in the virtual graph (interfering with LO) cancel with those above  + S. D. Ellis TSI 2006 Lecture 2

  30. ASIDE: Dim(emsional) Reg(ulation) • Say we want – • Consider – [44-2, Wick rotate to Euclidean space] • Calculate – S. D. Ellis TSI 2006 Lecture 2

  31. Simplify • Using – • Find –singular bits, plus finite bits 0, plus log singularity as m0 • Define Scheme – subtract (absorb) 1/ , E and ln(4) bits**** **** You can hid anything in infinity! S. D. Ellis TSI 2006 Lecture 2

  32. pQCD Calculation III: Apply Dim-Reg to total e+e- cross section • Real emission Numerator -  Dependence of matrix element Denominator -  Dependence of phase space  Dependence of Born S. D. Ellis TSI 2006 Lecture 2

  33. Virtual - • Virtual emission (interference, < 0!) • Sum and set  0, R = (e+e-hadrons)/(e+e-+-) Parton Model NLO QCD Correction S. D. Ellis TSI 2006 Lecture 2

  34. Well behaved as promised ! • Finite and well behaved – more work, higher order corrections S. D. Ellis TSI 2006 Lecture 2

  35. Higher Orders Typical Behavior When -n Cancel - • Physical quantity is  INdependent !! Fixed Order pQCD is NOT!! • pQCD higher orders exhibit explicit ln(/Q) factors •  of higher orders exhibits reduced dependence on unphysical parameter  • at order sn the residual explicit ln() dependence is order sn+1 •  dependence is an artifact of the truncation of the perturbative expansion S. D. Ellis TSI 2006 Lecture 2

  36. Generalize the Cancellation with New Concept - InfraRed Safety!! • Define InfraRed Safe (IRS) quantities – insensitive to collinear and soft emissions, i.e., real and virtual emissions contribute to same value of quantity and the infinites can cancel! • Powerful tools exist to study the appearance of infrared poles (in dim reg) in complicated momentum integrals viewed as contour integrals in the complex (momentum) plane. For a true singularity the contour must be “pinched” between (at least) 2 such poles (else Cauchy will allow us to avoid the issue). We will not review these tools in detail here. • Here we consider some simple examples of IRS quantities S. D. Ellis TSI 2006 Lecture 2

  37. InfraRed Safety!! InfraRed Safe (IRS) quantities – insensitive to collinear and soft emissions, i.e., • Thrust - d/dT S. D. Ellis TSI 2006 Lecture 2

  38. Another IRS quantity - • EEC Energy-Energy Correlation Both quantities are insensitive to: • Ei,pi0 • Collinear split En,pn(1-)(En,pn )+(En,pn) even for the autocorrelation En2= En2(1-)2+ En22+2(1-)En2 • Jet cross sections also qualify and we will come back to them. S. D. Ellis TSI 2006 Lecture 2

  39. SUMMARY - For IRS Quantities • -n bits from real and virtual emissions contribute to the same values of the IRS quantity and CANCEL!! • Exhibit (reliable) perturbative expansions even when mass scales (quarks) are set to zero in perturbative calculation • Life is more complicated when there is more than 1 physical scale, e.g., Q1 & Q2, and 1  ln[Q1/Q2] – must sum large logarithms to all orders S. D. Ellis TSI 2006 Lecture 2

  40. Check Kinematic Limits • Thrust in Collinear & soft limit [~2 jets] - potentially 2 logs! • Back-to-Back EEC – Collinear & soft • Front-to-Front EEC – Collinear only S. D. Ellis TSI 2006 Lecture 2

  41. Double Leading Logarithm Approximation - DLLA • In the DLLA – sum the large double logarithms [soft & collinear] order by order (often in impact parameter space) – corresponding to ordered emission of gluons – • Let  be a dimensionless parameter controlling this limit and L = ln() – a typical perturbative series or factor This structure arises often , e.g., EEC – T - S. D. Ellis TSI 2006 Lecture 2

  42. DLLA associated with • the probability to not radiate soft and collinear gluons, which is very small if soft and collinear enough • the name Sudakov Form Factor • In the limit 0, subleading terms will be important! S. D. Ellis TSI 2006 Lecture 2

  43. Lessons – • Big improvement LO  LO + NLO, not so much NLO  NNLO • See stable region around p=0, ~Q (“natural” scale) • Sign of   0 divergence oscillates order by order • Prescription for ?Several – e.g. • PMS (Principle of Minimal Sensitivity) • FAC (Fastest Apparent Convergence) • BLM – absorb all nf dependence into s But there is no right answer! Residual dependence on  in the “physical range” (e.g., p=01) is simply a (crude) measure of the uncertainty due to the uncalculated higher order contributions! S. D. Ellis TSI 2006 Lecture 2

  44. Summary - IR & Long Distances • pQCD not well behaved at Long Distances (in the Infrared) • Long distances in final state handled with IRS quantities • Long distances in the initial state, e.g., hadrons are still a problem for pQCD!! See the NEXT lecture S. D. Ellis TSI 2006 Lecture 2

  45. Extra Detail Slides S. D. Ellis TSI 2006 Lecture 2

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