Amplitude Analysis An Experimentalists View 1 K. Peters Part II Kinematics and More
Overview 2 Kinematics and More Phasespace Dalitz-Plots Observables Spin in a nutshell Examples
Goal 3 For whatever you need the parameterization of the n-particle phase space It contains the static properties of the unstable (resonant) particles within the decay chain like mass width spin and parities as well as properties of the initial state and some constraints from the experimental setup/measurement The main problem is, you don t need just a good description, you need the right one Many solutions may look alike, but only one is right
n-particle Phase space, n=3 4 Dalitz plot
Phase Space Plot - Dalitz Plot 5 Q small Q large Energy conservation E 3 = E tot -E 1 -E 2 dn ~ (E 1 de 1 )(E 2 de 2 )(E 3 de 3 )/(E 1 E 2 E 3 ) Phase space density ρ = dn/de tot ~ de 1 de 2 Kinetic energies Q=T 1 +T 2 +T 3 Plot x=(t 2 -T 1 )/ 3 y=t 3 -Q/3 Flat, if no dynamics is involved
The first plots τ/θ-puzzle 6 Dalitz applied it first to K L -decays The former τ/θ puzzle with only a few events goal was to determine spin and parity And he never called them Dalitz plots
Scattering & decay regions 7
t Scattering regions 8 A C B D A D s A C B C B D u
Decay region 9 A C D B
Decay region 10 A C D B
Decay region 11 A C D B
Decay region 12 D 0 K 0 S π + π - D K
Dalitz plot 13 D 0 K 0 S π + π - π + π - (GeV) 2 K 0 S π - K 0 S π + (GeV) 2
Phase space 14 visual inspection of the phase space distribution are the structures? structures from signal or background? are there strong interferences, threshold effects, potential resonances? ф K + K - K*(892) K - π +
Dalitz Plot 2D Phase Space 15 How can resonances be studied in multi-body decays? Consider 3 body decay M m 1 m 2 m 3 (all spin 0) Degrees of freedom 3 Lorentz-vectors 12 3 Masses -3 Energy conserv. -1 Momentum conserv. -3 3 Euler angles -3 Remaining d.o.f 2 Dalitz-Plot available phase space Complete dynamics described by two variables! Usual choice
Kinematics in the Dalitz-Plot 16 3 M 2 1 3. 2. 6. 1. s 3,max 2. s 3,min 3. s 1,max 5. 4. 1. 4. s 1,min 5. s 2,max 6. s 2,min
Kinematics in the Dalitz-Plot 17 Phasespace limits (example: s 1 ): Pos 3: M rest system: s 1 maximum, if E 1 is at minimum Pos 4: m 23 rest system (= Jackson-Frame R 23 ), p 2 =-p 3 full picture but: not the whole cube is accessible!
Kinematics in the Dalitz-Plot 18 Example: need s 2,± (s 1 ); calculate in Restsystem R 23 (p 2 =-p 3, p 1 =p) kinematical function with and
Properties of Dalitz-Plots 19 Density distribution in the Dalitz Plot given by for Spin-0 Particles M, m 1, m 2, m 3 Dynamics is contained by the matrix element M non-resonant processes M=const., uniform distribution resonant processes bands (horizontal, vertical, diagonal) spins Density distribution along the bands
Dalitz Plot 2D Phase space 20 Possible decay patterns: Non-Resonant (or background) flat (homogeneous) distribution available phase space Resonance R in m 12, m 13, m 23 Band Structures Position: Mass of R Density: Spin of R
Dalitz-Plot Tool (Root) Examples. 21
Properties of Dalitz Plots 22 For the process M Rm 3, R m 1 m 2 the matrix element can be expressed like Winkelverteilung (Legendre Polyn.) Formfaktor (Blatt-Weisskopf-F.) Resonanz-Fkt. (z.b. Breit Wigner) decay angular distribution of R Form-(Blatt-Weisskopf)-Factor for M Rm 3, p=p 3 in R 12 J L+l Z 0 0 + 0 1 0 1 + 1 cos 2 θ 0 2+ 2 [cos 2 θ 1/3] 2 Form-(Blatt-Weisskopf)-Factor for R m 1 m 2, q=p 1 in R 12 Dynamical Function (Breit-Wigner, K-Matrix, Flatté)
Angular Distributions 23 density distribution along the band = decay angular distribution of R results from Spin of R, the spin configuration and polarization of initial and final state(s) Compare R=ρ and ф (both 1 -- ) angular distributions are different!!
Dalitz-Plot-Analysis 24 Simultaneous fit of all resonant structures in a Dalitz-Plot Takes into account interference between resonances! The Dalitz Plot The Ingredients Fit Results Reco Efficiency
It s All a Question of Statistics... 25 p 3π 0 with 100 events
It s All a Question of Statistics...... 26 p 3π 0 with 100 events 1000 events
It s All a Question of Statistics......... 27 p 3π 0 with 100 events 1000 events 10000 events
It s All a Question of Statistics............ 28 p 3π 0 with 100 events 1000 events 10000 events 100000 events
Observables, cont d 29 are there symmetries in the phase space? unique assignment of phase space coordinates is important to avoid double counting transformation necessary? Most Dalitz plots are symmetric: Problem: sharing of events Possible solution: transform DP f r f(r) f
Spin Part in a nutshell 30 just a few words usually this is not part of your job there are very many formalisms and packages finally it s just a decomposition of the phase space which obeys all the necessary symmetries of the reaction thus (in principle) straigth forward and usually done by other people
Formalisms an overview (very limited) 31 Non-relativistic Tensor formalisms in non-relativistic (Zemach) or covariant flavor Fast computation, simple for small L and S Spin-projection formalisms where a quantization axis is chosen and proper rotations are used to define a two-body decay Efficient formalisms, even large L and S easy to handle Relativistic Tensor Formalisms based on Lorentz invariants (Rarita-Schwinger) where each operator is constructed from Mandelstam variables only Elegant, but extremely difficult for large L and S
Spin-Projection Formalisms 32 Differ in choice of quantization axis Helicity Formalism parallel to its own direction of motion Transversity Formalism the component normal to the scattering plane is used Canonical (Orbital) Formalism the component m in the incident z-direction is diagonal
Zemach Formalism 33 For particle with spin S traceless tensor of rank S Similar for orbital angular momentum L with indices
The Original Zemach Paper 34
Tensors revisited 35 The Zemach amplitudes are only valid in the rest frame of the resonance. Thus they are not covariant Retain covariance by adding the time component and use 4-vectors Behavior under spatial rotations dictates that the time component of the decay momentum vanishes in the rest frame This condition is called Rarita Schwinger condition μ For Spin-1 it reads Su S p 0 μ with p = (p a +p b )/m the 4-momentum of the resonance The vector S μμ is orthogonal to the timelike vector p μ and is therefore spacelike, thus S 2 < 0
Covariant Tensor Formalism 36 The most simple spin-1 covariant tensor with above properties is S μ =q μ -(qp)p μ with q = (p a -p b ) The negative norm is assured by the equation S q ( qp) q R 2 2 2 2 where q R is the break-up three-momentum the general approach and recipe is a lecture of its own and you should refer to the primary literature for more information to calculate the amplitudes and intensities you may use qft++
qft++ Package 37 qft++ = Numerical Object Oriented Quantum Field Theory (by Mike Williams, Carnegie Mellon Univ.) Calculation of the matrices, tensors, spinors, angular momentum tensors etc. with C++ classes 37
qft++ Package 38 Example: Amplitude and Intensity given by and qft++: Declaration and Calculation Intensity Angular distribution of 38
Helicity Decay Amplitudes 39 Helicity From two-particle state Helicity amplitude
f 2 ππ (Ansatz) 40 Initial: f 2 (1270) I G (J PC ) = 0 + (2 ++ ) Final: π 0 I G (J PC ) = 1 - (0 -+ ) Only even angular momenta, since η f = η π2 (-1) l Total spin s =2s π =0 Ansatz JM J J* A N F D φθ, λλ J λλ Mλ 1 2 1 2 λ λ λ 0 J 2 1 2 2M 2 2* A N F D φθ, 00 2 00 M0 2 NF 5 20 0020 00 0000 a 5a 2 00 20 20 2M 2* A 5 a D φθ, 00 20 M0 1 1
f 2 ππ (Rates) 41 d d θ e 1M A 5a 2 00 20 d θ 00 2 2 1M 1 M'* I θ A ρ A MM, ' θ e 20 2 10 2 d10 θe d20 θe 2 2iφ 00 MM' 00 1 1 ρ 2J 1 1 iφ iφ iφ Amplitude has to be symmetrized because of the final state particles 2 6 2 d θ sin θ 4 2 3 d θ 10 sin θ cos θ 2 2 3 2 1 d θ cos θ 00 2 2 20 2 2 15 4 2 2 3 2 1 I θ a sin θ15sin θcos θ5 cos θ 20 4 2 2 a 20 2 const 1 4 2 2 3 4 1 2 1 15 sin θsin θcos θ cos θ cos θ 4 4 2 12
42 THANK YOU for today