Monte Carlo Methods in Particle Physics Bryan Webber University of - PowerPoint PPT Presentation

Monte Carlo Methods in Particle Physics Bryan Webber University of Cambridge IMPRS, Munich 19-23 November 2007 Monte Carlo Methods 2 Bryan Webber

Structure of LHC Events 1. Hard process 2. Parton shower 3. Hadronization 4. Underlying event Monte Carlo Methods 2 Bryan Webber

Lecture 2: Parton Showers QED: accelerated charges 1. annihilation to jets. radiate. 2. Universality of collinear QCD identical: accelerated emission. colours radiate. 3. Sudakov form factors. gluons also charged. 4. Universality of soft emission.  cascade of partons. 5. Angular ordering. = parton shower. 6. Initial-state radiation. 7. Hard scattering. 8. The Colour Dipole Model. Monte Carlo Methods 2 Bryan Webber

annihilation to jets Three-jet cross section: singular as Rewrite in terms of quark-gluon opening angle and gluon energy fraction : Singular as and . Monte Carlo Methods 2 Bryan Webber

can separate into two independent jets: jets evolve independently Exactly same form for anything eg transverse momentum: invariant mass: Monte Carlo Methods 2 Bryan Webber

Collinear Limit Universal: Dokshitzer-Gribov-Lipatov- Altarelli-Parisi splitting kernel: dependent on flavour and spin Monte Carlo Methods 2 Bryan Webber

Resolvable partons What is a parton? Collinear parton pair single parton Introduce resolution criterion, eg Virtual corrections must be combined with unresolvable real emission Resolvable emission: Finite Virtual + Unresolvable emission: Finite Unitarity: P(resolved) + P(unresolved) = 1 Monte Carlo Methods 2 Bryan Webber

Sudakov form factor Probability(emission between and ) Define probability(no emission between and ) to be . Gives evolution equation Sudakov form factor factor =Probability(emitting no resolvable radiation) Monte Carlo Methods 2 Bryan Webber

Multiple emission But initial condition? Process dependent Monte Carlo Methods 2 Bryan Webber

Monte Carlo implementation Can generate branching according to By choosing uniformly: If no resolvable radiation, evolution stops. Otherwise, solve for = emission scale Considerable freedom: } Evolution scale: Equivalent at this stage, but can be very z: Energy? Light-cone momentum? important numerically Massless partons become massive. How? Upper limit for ? Monte Carlo Methods 2 Bryan Webber

Parton Shower • Evolution in t (q 2 ) and x (DIS) Basic 2-step: ∆ ( t 2 , t 1 ) P ( x 2 /x 1 ) e + e - : same formula, opposite direction! Monte Carlo Methods 2 Bryan Webber

Running coupling Effect of summing up higher orders: k, 1 − z q + ... q ′ , z Scale is set by maximum virtuality of emitted gluon k 2 max = (1 − z ) q 2 Similarly in , scale is set by g → gg ′ min { k 2 max , k ′ 2 max } = min { z, (1 − z ) } q 2 ≃ z (1 − z ) q 2 ≡ k 2 T Scale change absorbed by replacing by α S ( q 2 ) α S ( k 2 T ) Faster parton multiplication Monte Carlo Methods 2 Bryan Webber

Soft limit Also universal. But at amplitude level… soft gluon comes from everywhere in event.  Quantum interference. Spoils independent evolution picture? Monte Carlo Methods 2 Bryan Webber

Angular Ordering NO: outside angular ordered cones, soft gluons sum coherently: only see colour charge of whole jet. Soft gluon effects fully incorporated by using as evolution variable: angular ordering First gluon not necessarily hardest! Monte Carlo Methods 2 Bryan Webber

Soft Gluon Emission • Propagator factor for emission from external line, energy E, mass m 1 ± 1 ± 1 ( p ± q ) 2 − m 2 = 2 p · q = 2 ω E (1 − v cos θ ) Including numerator, get universal eikonal factor in soft limit F soft = p · ε p · q No enhancement for emission from internal lines ( p + q ) 2 − m 2 → p 2 − m 2 � = 0 as ω → 0 Monte Carlo Methods 2 Bryan Webber

Enhancement factor in amplitude for each external line implies cross section enhancement is sum over all pairs of external lines: d ω d Ω α s � d σ n +1 = d σ n C ij W ij 2 π 2 π ω i,j where is element of solid angle for emitted gluon, d Ω C ij is a colour factor, and radiation function is given by W ij W ij = ω 2 p i · p j 1 − v i v j cos θ ij p i · q p j · q = (1 − v i cos θ iq )(1 − v j cos θ jq ) Colour-weighted sum of radiation functions is C ij W ij antenna pattern of hard process. Monte Carlo Methods 2 Bryan Webber

Angular Ordering Radiation function can be separated into two parts containing collinear singularities along lines i and j. Consider for simplicity massless particles, . Then v i,j = 1 ij + W j W ij = W i where ij � � ij = 1 1 1 W i W ij + . − 1 − cos θ iq 1 − cos θ jq 2 This function has the remarkable property of angular ordering. Write angular integration in polar coordinates w.r.t. direction of i, d Ω = d cos θ iq d φ iq Performing azimuthal integration, we find � 2 π d φ iq 1 2 π W i ij = if θ iq < θ ij , otherwise 0. 1 − cos θ iq 0 Monte Carlo Methods 2 Bryan Webber

To prove angular ordering property, write where , 1 − cos θ jq = a − b cos φ iq a = 1 − cos θ ij cos θ iq . Defining , we have z = exp( i φ iq ) b = sin θ ij sin θ iq � 2 π 1 1 d φ iq � dz I i = ij ≡ 2 π 1 − cos θ jq ( z − z + )( z − z − ) i π b 0 where z-integration contour is the unit circle and . Now only pole at can lie � z ± = a/b ± a 2 /b 2 − 1 z = z − inside unit circle, so � 1 1 I i ij = a 2 − b 2 = | cos θ iq − cos θ ij | Hence � 2 π 1 d φ iq 2 π W i 2(1 − cos θ iq )[1 + (cos θ iq − cos θ ij ) I i = ij ] ij 0 1 = if θ iq < θ ij , otherwise 0. 1 − cos θ iq Monte Carlo Methods 2 Bryan Webber

Coherent Branching Angular ordering provides basis for coherent parton branching formalism, which includes leading soft gluon enhancements to all orders. In place of virtual mass-squared variable t in earlier treatment, use angular variable ζ = p b · p c ≃ 1 − cos θ E b E c as evolution variable for branching , and impose angular a → bc ζ ′ < ζ ordering for successive branchings. Iterative formula for n-parton emission becomes d ζ ζ dz α s d σ n +1 = d σ n 2 π In place of virtual mass-squared cutoff, we must use angular cutoff for coherent branching. This is to some extent arbitrary, depending on how we classify emission as unresolvable. ζ 0 = t 0 /E 2 Simplest choice is . Monte Carlo Methods 2 Bryan Webber

With this cutoff, the most convenient definition of evolution t = E 2 ζ ≥ t 0 ˜ variable is not itself but rather ζ Angular ordering condition for timelike branching ζ b , ζ c < ζ a t b < z 2 ˜ ˜ t c < (1 − z ) 2 ˜ ˜ becomes a → bc t , t where and . Thus cutoff on becomes t = ˜ ˜ z = E b /E a t a z � � t 0 / ˜ t 0 / ˜ t < z < 1 − t Neglecting masses of & , virtual mass-squared of and b c a transverse momentum of branching are t = z (1 − z )˜ t = z 2 (1 − z ) 2 ˜ p 2 t , t Thus for coherent branching Sudakov form factor of quark becomes � 1 − √ � ˜ � � t t 0 /t ′ dt ′ dz ˜ 2 π α s ( z 2 (1 − z ) 2 t ′ ) ˆ ∆ q (˜ t ) = exp P qq ( z ) − t ′ √ 4 t 0 t 0 /t ′ This falls more slowly than without coherence, due to suppression of soft gluon emission by angular ordering. Monte Carlo Methods 2 Bryan Webber

Initial state radiation In principle identical to final state (for not too small x) In practice different because both ends of evolution fixed: Use approach based on evolution equations… Monte Carlo Methods 2 Bryan Webber

Backward Evolution DGLAP evolution: pdfs at as function of pdfs at Evolution paths sum over all possible events. Formulate as backward evolution: start from hard scattering and work down in up in towards incoming hadron. Algorithm identical to final state with replaced by Monte Carlo Methods 2 Bryan Webber

Note that for initial-state (spacelike) branching a → bc ( incoming, spacelike), angular ordering condition is b a ! c θ b > θ a > θ c c and so for we now have z = E b /E a ! ! a b a b t b > z 2 ˜ ˜ t c < (1 − z ) 2 ˜ ˜ t a , t a t a > ˜ ˜ t b > ˜ ˜ Thus we can have either or , especially t b t a at small z Spacelike branching becomes disordered at small x. Monte Carlo Methods 2 Bryan Webber

Hard Scattering Sets up initial conditions for parton showers. Colour coherence important here too. Emission from each parton confined to cone stretching to its colour partner Essential to fit Tevatron data… Monte Carlo Methods 2 Bryan Webber

Three-jet correlations (CDF) Monte Carlo Methods 2 Bryan Webber

Distributions of third-hardest jet in multi-jet events HERWIG has complete treatment of colour coherence, PYTHIA+ has partial Monte Carlo Methods 2 Bryan Webber