Introduction to Event Generators Le c tu re 3 : Ha dron isa tion a n d J e ts THEORY PHENOMENOLOGY EXPERIMENT time g a ( − ig s t a ij γ µ ) Figure by T. Sjöstrand q j q i QCD “Jets” INTERPRETATION Peter Skands (Monash University) 11th MCnet School, Lund 2017
MONTE CARLOS & FRAGMENTATION ๏ PYTHIA anno 1978 • (then called JETSET) LU TP 78-18 November, 1978 A Monte Carlo Program for Quark Jet Generation T. Sjöstrand, B. Söderberg A Monte Carlo computer program is presented, that simulates the fragmentation of a fast parton into a jet of mesons . It uses an iterative scaling scheme and is compatible with the jet model of Field and Feynman. Note : Field-Feynman was an early fragmentation model Now superseded by the String (in PYTHIA) and Cluster (in HERWIG & SHERPA) models. 2 Peter Skands Monash University
FROM PARTONS TO PIONS Here’s a fast parton It ends up Fast: It starts at a high It showers at a low effective factorization scale (bremsstrahlung) factorization scale Q = Q F = Q hard Q ~ m ρ ~ 1 GeV Q Q hard 1 GeV 3 Peter Skands Monash University
FROM PARTONS TO PIONS Here’s a fast parton It ends up Fast: It starts at a high It showers at a low effective factorization scale (bremsstrahlung) factorization scale Q = Q F = Q hard Q ~ m ρ ~ 1 GeV Q Q hard 1 GeV How about I just call it a hadron? → “Local Parton-Hadron Duality” 4 Peter Skands Monash University
PARTON → HADRONS? ๏ Early models: “Independent Fragmentation” • Local Parton Hadron Duality (LPHD) can give useful results for inclusive quantities in collinear fragmentation • Motivates a simple model: π q π “Independent Fragmentation” π ๏ But … • The point of confinement is that partons are coloured • Hadronisation = the process of colour neutralisation → Unphysical to think about independent fragmentation of a single ๏ parton into hadrons → Too naive to see LPHD (inclusive) as a justification for Independent ๏ Fragmentation (exclusive) → More physics needed ๏ 5 Peter Skands Monash University
COLOUR NEUTRALISATION A physical hadronization model ๏ Should involve at least TWO partons, with opposite color • charges (e.g., think of them as R and anti-R) * non-perturbative Late times (non-perturbative) anti-R moving along right lightcone R m o v i n g a l o n g l e f t l i g pQCD Time h t c o n e Early times (perturbative) Space Strong “confining” field emerges between the two charges when their separation >~ 1fm *) Really, a colour singlet state � R ¯ � G ¯ � B ¯ �� � � 1 ↵ ↵ ↵� function R + G + B √ 3 definition of a singlet). The other eight 6 Peter Skands Monash University
RECAP: COLOUR FLOW ๏ Colour flow in parton showers (leading-colour approximation) Example: Z 0 → qq 1 1 3 2 5 4 7 1 1 4 5 5 3 3 4 7 6 2 2 System #1 System #2 System #3 Coherence of pQCD cascades → not much “overlap” between systems → Leading-colour approximation pretty good ( LEP measurements in e + e - → W + W - → hadrons confirm this (at least to order 10% ~ 1/N c2 ) ) Note : (much) more color getting kicked around in hadron collisions. More tomorrow. 7 Peter Skands Monash University
-15 THE ULTIMATE LIMIT: WAVELENGTHS > 10 M ๏ Quark-Antiquark Potential What physical ! system has a ! • As function of separation distance SCALING. . . POTENTIAL: 2641 STATIC QUARK-ANTIQUARK 46 linear potential? Scaling plot LATTICE QCD SIMULATION. 2GeV- Bali and Schilling Phys Rev D46 (1992) 2636 Long Distances ~ Linear Potential (in “quenched” approximation) 1 GeV— 2 “Confined” Partons Short Distances ~ “Coulomb” (a.k.a. Hadrons) B = 6. 0, L=16 'V ~ B = 6. 0, L=32 ~ ~ I ~ B = 6. 2, L=24 B = 6. 4, L-24 “Free” Partons I B = 6. 4, L=32 -2 A k, I 4 2' t 3. 0. 5 1. 2. 5 5 5 1 fm l~ RK FIG. 4. All potential data of the five lattices have been scaled to a universal curve by subtracting Vo and measuring energies and to V(R) = R — units of &E. The dashed curve correspond ~/12R. Physical units are calculated distances in appropriate by exploit- ing the relation &cr =420 MeV. ~ Force required to lift a 16-ton truck AM~a=46. 1A~ &235(2)(13) MeV . we are aware that our lattice turbative results. Although suScient, dare to resolution is not yet really we might apply to to say, this value does not necessarily Needless of the the Coulomb-like previe~ continuum behavior 8 Peter Skands Monash University full QCD. In Fig. 6(a) [6(b)] we visualize term from our results. the behavior of the confining In addition to the long-range in the K-e plane from fits to various confidence regions it is of considerable interest to investigate its ul- lattices at P=6. 0 potential on- and off-axis potentials on the 32 into the weak cou- structure. As we proceed traviolet [6. 4]. We observe that the impact of lattice discretization on e decreases by a factor 2, as we step up from P = 6. 0 to are expected to meet per- pling regime lattice simulations 150 Barkai '84 o '90 MTC Our results:--- 140 130- 120- 110- 100- 80— 5. 6 5. 8 6. 2 6. 4 c = &E /(a AL ) ] as a function of P. Our results are combined FIG. 5. The on-axis string tension [in units of the quantity with pre- and Rebbi [11]. vious values obtained by the MTc collaboration [10] and Barkai, Moriarty,
FROM PARTONS TO STRINGS ๏ Motivates a model: • Let color field collapse into a narrow flux tube of uniform energy density κ ~ 1 GeV / fm String ๏ Worldsheet • Limit → Relativistic 1+1 dimensional worldsheet Schwinger Effect ÷ ๏ In “unquenched” QCD Non-perturbative creation of e + e - pairs in a strong • g → qq → The strings will break ๏ external Electric field ~ E 2 = m q 2 + p T 2 → Gaussian suppression of high m T Probability from e + Tunneling Factor Heavier quarks suppressed. Prob(d:u:s:c) ≈ 1 : 1 : 0.2 : 10 -11 ✓ − m 2 − p 2 e - ◆ ⊥ P ∝ exp time κ / π Pedagogical Review: B. Andersson, The Lund model. + Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol., 1997. ( κ is the string tension equivalent) 9 Peter Skands Monash University
THE (LUND) STRING MODEL Main implementation: PYTHIA. (EPOS also implements a string-based hadronisation model.) Map: • Quarks → String Endpoints • Gluons → Transverse Excitations (kinks) • Physics then in terms of string worldsheet evolving in spacetime • Probability of string break (by quantum tunneling) constant per unit area → → STRING EFFECT AREA LAW Simple space-time picture Details of string breaks more complicated (e.g., baryons, spin multiplets) 10 Peter Skands Monash University
FRAGMENTATION FUNCTION ๏ (see lecture notes for how selection is made ๏ Having selected a hadron flavor between different spin/excitation states) • How much momentum does it take? leftover string, further string breaks M Spacelike Separation Spacetime Picture time t The meson M takes a fraction z of the quark momentum, q z How big that fraction is, spatial z ∈ [0,1], is determined by the separation fragmentation function , f(z,Q 02 ) 11 Peter Skands Monash University
LEFT-RIGHT SYMMETRY • Causality → Left-Right Symmetry • → Constrains form of fragmentation function! z ๏ → Lund Symmetric Fragmentation Function q − b ( m 2 h + p 2 ✓ ◆ f ( z ) ∝ 1 ? h ) z (1 − z ) a exp z Small a Small b → “high-z tail” → “low-z enhancement” a=0.9 a=0.1 b=2 2.0 b=0.5 1.5 1.5 1.0 1.0 both curves using both curves 0.5 0.5 a=0.5, m T =1 using b=1, m T =1 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 Note: In principle, a can be flavour-dependent. In practice, we only distinguish between baryons and mesons 12 Peter Skands Monash University
ITERATIVE STRING BREAKS Causality → May iterate from outside-in Note: using light-cone coordinates: p + = E + p z u ( � p ⊥ 0 , p + ) shower Q UV � + ( � p ⊥ 1 , z 1 p + ) Q IR p ⊥ 0 − � · · · d ¯ d K 0 ( � p ⊥ 2 , z 2 (1 − z 1 ) p + ) p ⊥ 1 − � s ¯ s ... On average, expect energy of n th “rank” hadron ~ E n ~ <z> n E 0 13 Peter Skands Monash University
(NOTE ON THE LENGTH OF STRINGS) ๏ In Spacetime : • String tension ≈ 1 GeV/fm → a 5-GeV quark can travel 5 fm before all its kinetic energy is transformed to potential energy in the string. • Then it must start moving the other way ( → “yo-yo” model of mesons. Note: string breaks → several mesons) Rapidity is useful because it is additive under Lorentz boosts (along the rapidity axis) ✓ ( E + p z ) 2 ๏ In Rapidity : ✓ E + p z ◆ ◆ y = 1 = 1 2 ln 2 ln s E 2 − p 2 E − p z 1 − β (convenient variable y 0 = y + ln z in momentum space) 1 + β ( ) for m → 0 : 1 ✓ 1 + cos θ ◆ 2 ln = − ln tan( θ / 2) = η ➾ Δ y difference is invariant 1 − cos θ “Pseudorapidity” Particle Production: Scaling in lightcone p ± =E±p z ➾ flat central rapidity plateau (+ some endpoint effects) If the quark gives all its energy to a single pion traveling along the z axis ✓ 2 E q ◆ y max ∼ ln m π Increasing E q → logarithmic growth in rapidity range 14 Peter Skands Monash University
Recommend
More recommend
