Amplitude Analysis An Experimentalists View

Transcription

1 Amplitude Analysis An Experimentalists View Lectures at the Extracting Physics from Precision Experiments Techniques of Amplitude Analysis Klaus Peters GSI Darmstadt and GU Frankfurt Williamsburg, June 2012

2 Amplitude Analysis An Experimentalists View 2 K. Peters Part III Spin

3 Overview 3 Spin-Parity Basics Formalisms Examples

4 Formalisms an overview (very limited) 4 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

5 Zemach Formalism 5 For particle with spin S traceless tensor of rank S Similar for orbital angular momentum L with indices

6 Example: Zemach p (0 -+ ) f 2 π 0 6 Construct total spin 0 amplitude A=A f2 π x A ππ

7 Example: Zemach p (0 -+ ) f 2 π 0 7 Angular distribution (Intensity)

8 The Original Zemach Paper 8

9 Spin-Projection Formalisms 9 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

10 Spin-Projection Formalisms 10 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

11 How To Construct a Formalism 11 Key steps are Definition of single particle states of given momentum and spin component (momentum-states), Definition of two-particle momentum-states in the s-channel center-of-mass system and of amplitudes between them, Transformation to states and amplitudes of given total angular momentum (J-states), Symmetry restrictions on the amplitudes, Derive Formulae for observable quantities.

12 Generalized Single Particle State 12 In general all single particle states are derived from a lorentz transformation and the rotation of the basic state with the Wigner rotation

13 Properties 13 Helicity Transversity Canonical property possibility/simplicity partial wave expansion simple complicated complicated parity conservation no yes yes crossing relation no good bad specification of kinematical constraints no yes yes

14 Rotation of States Canonical System Helicity System

15 Rotations 15 Single particle states jm ' ' jm δ δ 1 jj' jm, mm' jm jm Rotation R Unitary operator U D function represents the rotation in the angular momentum space Valid in an inertial system Relativistic state URR UR UR iαj iβj iγj z y UR αβγ,, e e e * J,, ',, m', m m' * J D ', αβγ,, jm' URαβγ,, m m im' α j im' γ e dmm ' βe UR αβγ jm jm D αβγ 1 R Ω z pjm, URΩ L β j pjλ, D Ω pjm, mλ z jm jm

16 Single Particle State 16 Canonical 1) momentum vector is rotated via z-direction. 2) absolute value of the momentum is Lorentz boosted along z 3) z-axis is rotated to the momentum direction

17 Two-Particle State 17 Canonical constructed from two single-particle states (back-to-back) Couple s and t to S Couple L and S to J Spherical Harmonics

18 Single Particle State 18 Helicity 1) z-axis is rotated to the momentum direction 2) Lorentz Boost Therefore the new z-axis, z, is parallel to the momentum

19 Two-Particle State 19 Helicity similar procedure no recoupling needed normalization

20 Completeness and Normalization 20 Canonical Helicity completeness completeness normalization normalization

21 Canonical Decay Amplitudes 21 Canonical From two-particle state LS-Coefficients

22 Helicity Decay Amplitudes 22 Helicity From two-particle state Helicity amplitude

23 Spin Density and Observed # of Events 23 To finally calculate the intensity i.e. the number of events observed Spin density of the initial state Sum over all unobserved states taking into account

24 Relations Canonical Helicity 24 Recoupling coefficients Start with Canonical to Helicity Helicity to Canonical

25 Clebsch-Gordan Tables 25 Clebsch-Gordan Coefficients are usually tabled in a graphical form (like in the PDG) Two cases j 1 x j 2 J J M M coupling two initial particles with j 1 m 1 and j 2 m 2 to final system JM m 1 m 2 j 1 m 1 j 2 m 2 JM m 1 m 2 decay of an initial system JM to j 1 m 1 and j 2 m 2 j 1 and j 2 do not explicitly appear in the tables all values implicitly contain a square root Minus signs are meant to be used in front of the square root

26 Using Clebsch-Gordan Tables, Case x JM jm 1 1 jm 2 2 jm jm JM /2 1/ /2-1/ /6 1/2 1/ /3 0-1/ /6-1/2 1/ /2 1/ /2-1/

27 Using Clebsch-Gordan Tables, Case x JM jm 1 1 jm 2 2 jm jm JM /2 1/ /2-1/ /6 1/2 1/ /3 0-1/ /6-1/2 1/ /2 1/ /2-1/

28 Parity Transformation and Conservation 28 Parity transformation single particle two particles helicity amplitude relations (for P conservation)

29 f 2 ππ (Ansatz) 29 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 λ λ λ 0 J M 2 2* A N F D φθ, M0 2 NF a 5a M 2* A 5 a D φθ, M0 1 1

30 f 2 ππ (Rates) 30 d d θ e 1M A 5a d θ M 1 M'* I θ A ρ A MM, ' θ e d10 θe d20 θe 2 2iφ 00 MM' ρ 2J 1 1 iφ iφ iφ Amplitude has to be symmetrized because of the final state particles d θ sin θ d θ 10 sin θ cos θ d θ cos θ I θ a sin θ15sin θcos θ5 cos θ a 20 2 const sin θsin θcos θ cos θ cos θ

31 ω π 0 γ (Ansatz) 31 Initial: ω I G (J PC ) = 0 - (1 -- ) Final: π 0 I G (J PC ) = 1 - (0 -+ ) γ I G (J PC ) = 0(1 -- ) Only odd angular momenta, since η ω = η π η γ (-1) l Only photon contributes to total spin s = s π +s γ Ansatz JM J J* A N F D φθ, λλ J λλ Mλ λ λ λ λ λ J γ 1 1M 1 1* A N F D φθ, λ0 1 λ0 Mλ 1 3 NF 310 1λ Jλ 1λ 001λ a λ a λ M 3 1* A λ a D φθ, λ0 11 Mλ 2 λ 2 1

32 ω π 0 γ (Rates) 32 λ γ =±1 do not interfere, λ γ =0 does not exist for real photons Rate depends on density matrix Choose uniform density matrix as an example d θ e d θ e 1M A d θ d θ λ M 1 M'* I θ A ρ A δ MM, ', λλ, ' ρ iφ iφ 1 d θ 11 e 0 d11 θe λ0 MM' λ'0 λλ' cosθ 1cosθ I sin cos 2 2 θ sin θ 1 const 2 iφ iφ 1 cosθ d θ d θ sinθ d θ d θ d θ θ

33 f 0,2 γγ (Ansatz) 33 Initial: f 0,2 I G (J PC ) = 0 + (0,2 ++ ) Final: γ I G (J PC ) = 0(1 -- ) Only even angular momenta, since η f = η γ2 (-1) l Total spin s = 2s γ =2, l=0,2 (f 0 ), l=0,2,4 (f 2 ) Ansatz JM J J* A N F D φθ, λλ J λλ Mλ λ λ λ 1 2 J 0 00 A 0 N F 0* D φθ, λλ 1 λλ 0λ NF l sλ Jλ s λ s λ sλ a 0 λ λ ls 1a λ 1 λ 0λ a λ 1 λ 2λ a a ls

34 f 0,2 γγ (cont d) * A N F D φθ, λλ 1 λλ * a a D φθ, const const Ratio between a 00 and a 22 is not measurable Problem even worse for J=2 J 2 2M A 2 N F 2* D φθ, λλ 2 λλ Mλ NF l sλ Jλ s λ s λ sλ a 2 λ λ ls 5a λ 1λ 1 λ a 20 2λ2λ 1λ 1 λ 2λ a 40 2λ2λ 1λ 1 λ 2λ ls

35 f 0,2 γγ (cont d) 35 Usual assumption J=λ= ls ls 5a NF l s s s s a a M 2 2 2* A N F F D φθ, M2 2* N ' D φθ, A 00 N ' 2 M Symmetrization Comparison tbd

36 p (2 ++ ) ππ 36 Proton antiproton in flight into two pseudo scalars Initial: p J,M=0,±1 Final: π I G (J PC ) = 1 - (0 -+ ) Ansatz JM J J* A N F D φθ, λλ J λλ Mλ λ λ λ 0 J l JM J J* A N F D φθ, J 00 M0 J NF 2 J1 l0 00 J a 2 J1a l J 00 l0 J0 δ lj * * A 2J1 a D φθ, 2J1a d θ e JM J J imφ 00 J0 M0 J0 M0 1 Problem: d-functions are not orthogonal, if φ is not observed ambiguities remain in the amplitude polarization is needed

37 p π 0 ω 37 Two step process First step p π 0 ω - Second step ω π 0 γ Combine the amplitudes JM 1λω JM A Ω, Ω A Ω A Ω J* J J* N F D,1 0 Ω ω λ λ λ γn F D pp, J λ 0 Mλ Ωω λ λ ω γ λ γ λ ω ω γ γ ω γ ω γ ω ω J * * λ a D Ω l l λ Jλ a D Ω,11, 1 ω γ ω 2 ω λ λ γ ω ω pp l Mλ ω l J * * λ a D Ω D Ω l l λ Jλ a 2 ω γ ω ω,11 λ λ γ Mλ ω ω ω pp,1 l l helicity constant a ω,11 factorizes and is unimportant for angular distributions

38 p (0 -+ ) f 2 π 0 38 Initial: p I G (J PC ) = 1 - (0 -+ ) Final: f 2 (1270) I G (J PC ) = 0 + (2 ++ ) π 0 I G (J PC ) = 1 - (0 -+ ) is only possible from L=2 Ansatz JM J J* A N F D Ω λλ J λλ Mλ f π pp f f π A Ω A Ω N F D Ω N F D Ω * 2 2* 00 00, , N F a pp,0 00 pp, N F a f,2 00 f, * A Ω A Ω 5a a D 00 f 00 π pp,22 f Ω 5a a cos θ,20 00 π pp,22 f, I cosθ 5 a a cos θ pp,22 f, λ λ λ J 0 pp J f 2 2

39 General Statements 39 Flat angular distributions General rules for spin 0 initial state has spin 0 0 any both final state particles have spin 0 J 0+0 Special rules for isotropic density matrix and unobserved azimuth angle one final state particle has spin 0 and the second carries the same spin as the initial state J J+0

40 Moments Analysis 40 Consider reaction Total differential cross section expand H leading to

41 Moments Analysis cont d 41 Define now a density tensor the d-function products can be expanded in spherical harmonics and the density matrix gets absorbed in a spherical moment

42 Example: Where to start in Dalitz plot anlysis 42 Sometimes a moment-analysis can help to find important contributions best suited if no crossing bands occur D 0 K S K + K - L,,0 M0 L,,0 t LM D φθ I Ω D φθ dω M0

43 E791: D+ K-π+π+ Dalitz Striking K*(892) bands asymmetry implies strong S-wave interference (in Kπ) Dalitz plot analysis as an Interferometer Model-independent analysis by using interference to fix the S-wave 43

44 Recipe 44 Create slices in m 2 (K - π + ) S-wave is than a binned function with parameters c k and γ k S = c k e iγ k S Model well known P- and D-wave (K *, K 1 and K * 2) add form factors and put this into the fit P main uncertainty from K * and K 1 D

45 Comparison with Data 45 magnitude S phase S

46 Proton-Antiproton Rest 46 Atomic initial system formation at high n, l (n~30) slow radiative transitions de-excitation through collisions (Auger effect) Stark mixing of l-levels (Day, Snow, Sucher 1960) Advantages J PC varies with target density isospin varies with n (d) or p target incoherent initial states unambiguous PWA possible Disadvantages phase space very limited small kaon yield S P D F n=4 n=3 n=2 Ka (1%) n=1 S-Wave L b L a P-Wave (99% of 2P) rad. Transition Stark-Effect ext. Auger-Effect Annihilation

47 p Initial Rest 47 Quantumnumbers G=(-1) I+L+S P=(-1) L+1 C=(-1) L+S CP=(-1) 2L+S+1 I=0 J PC I G L S 1 S pseudo scalar 1 - ; S vector 1 + ; P axial vector 1 + ; P scalar 1 - ; I=1 3 P axial vector 1 - ; P tensor 1 - ;

48 Proton-Antiproton Annihilation in Flight 48 Annihilation in flight scattering process: no well defined initial state maximum angular momentum rises with energy Advantages larger phase space formation experiments Disadvantages many waves interfere with each other many waves due to large phase space 100 l [mb] 10 l=1 l=2 l=4 l=3 ann= l l l(p)=(2l+1) [1-exp(- l(p))] / p 2 2 l (p)=n(p) exp(-3l(l+1)/4p R ) P [GeV/c] lab ang. mom. l l ~ p /0.2 GeV/ c cms 2

49 Scattering Amplitudes in p in Flight (I) 49 p helicity amplitude M J 2L 1 H L 0 Sν Jν s ν s ν Sν JMLS JM νν 1 2 LS, 2J 1 J J 1 J H η H νν J ν ν CP transform only H ++ and H +- exist CP=(-1) 2L+S+1 S and CP directly correlated CP conserved in strong int. singlet and triplet decoupled C-Invariance C transform H ++ =0 L and if L+S-J P directly odd correlated CP-Invariance C conserved in strong int. (if total charge is q=0) H +- =0 if S=0 and/or J=0 odd and even L decouples 4 incoherent sets of coherent amplitudes

50 Scattering Amplitudes in p in Flight (II) 50 Singlett even L J PC L S H ++ H +- Singlett odd L J PC L S H ++ H +- 1 S Yes No 1 D Yes No 1 P Yes No 1 F Yes No 1 G Yes No 1 G Yes No Triplett even L J PC L S H ++ H +- 3 S Yes Yes 3 D Yes Yes 3 D Yes Yes 3 D Yes Yes Triplett odd L J PC L S H ++ H +- 3 P Yes No 3 P No Yes 3 P Yes Yes 3 F Yes No 3 F No Yes 3 F Yes Yes

51 Tensors revisited 51 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

52 Covariant Tensor Formalism 52 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 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++

53 qft++ Package 53 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 53

54 qft++ Package 54 Example: Amplitude and Intensity given by and qft++: Declaration and Calculation Intensity Angular distribution of 54

55 Differences to the Zemach formalism 55 it is possible to show, that for γ =E/m for the resonant system formed by (a+b)

56 Comparison γ=1 and γ= 56 γ=1 (non-relativistic case) γ= (ultra-relativistic) the angular distributions can be radically different it depends on the available phase space of a resonance, if this effect is actually measurable

57 Covariant extension of the helicity formalism 57 in non-covariant description we obtained to relationsship where a LS is a constant for each J in covariant description a LS depend on λ s,λ L!!!

58 Covariant extension of the helicity formalism 58 the formula for a LS reads then with n=1 if S+λ s+l+ λ L =odd and n=0 otherwise W = s of the two-body system and W 0 = W(m 0 ) q = two-body breakup momentum and q 0 = q(m 0 ) B L = Form-factor f λ (γ) = f-function for given daughter particle with Lorentz-factors γ Definition with

59 59 THANK YOU for today

60 Amplitude Analysis An Experimentalists View Lectures at the Extracting Physics from Precision Experiments Techniques of Amplitude Analysis Klaus Peters GSI Darmstadt and GU Frankfurt Williamsburg, June 2012

61 Amplitude Analysis An Experimentalists View 2 K. Peters Part IV Dynamics

62 Overview 3 Dynamics Scattering T-Matrix Breit-Wigner Blatt-Weisskopf

63 Properties of Dalitz Plots 4 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 cos 2 θ [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)

64 Interference problem 5 PWA The phase space diagram in hadron physics shows a pattern due to interference and spin effects This is the unbiased measurement What has to be determined? Analogy Optics PWA # lamps # level # slits # resonances positions of slits masses sizes of slits widths but only if spins are properly assigned Optics Dalitz plot bias due to hypothetical spin-parity assumption

65 Introducing Partial Waves, cont d 6 Schrödinger s Equation Scattering of particles on a spherical potential incoming planar wave outgoing spherical wave

66 Introducing Partial Waves... 7 Compose planar wave in terms of partial waves with given L spherical Besselfct. Legendre-Polyn. with

67 Introducing Partial Waves, cont d 8 wave without scattering outgoing incoming wave with scattering (only outgoing part is modified) for the scattered wave ψ S one gets Inelasticity + Phaseshift

68 Argand Plot 9

69 Standard Breit-Wigner 10 Argand Plot Intensity I=ΨΨ * Full circle in the Argand Plot Phase motion from 0 to π Phase δ Speed dφ/dm

70 Breit-Wigner in the Real World 11 e + e - ππ m ππ ρ-ω

71 Isobar Model 12 Generalization construct any many-body system as a tree of subsequent two-body decays the overall process is dominated by two-body processes the two-body systems behave identical in each reaction different initial states may interfere Isobar We need need two-body spin -algebra various formalisms need two-body scattering formalism final state interaction, e.g. Breit-Wigner

72 The Full Amplitude 13 For each node an amplitude f(i,i 3,s,Ω) is obtained. The full amplitude is the sum of all nodes. Summed over all unobservables

73 Dynamical Functions are Complicated 14 Search for resonance enhancements is a major tool in meson spectroscopy The Breit-Wigner Formula was derived for a single resonance appearing in a single channel But: Nature is more complicated Resonances decay into several channels Several resonances appear within the same channel Thresholds distort line shapes due to available phase space A more general approach is needed for a detailed understanding (see last lecture!)

74 S-Matrix 15 Differential cross section Scattering amplitude S-Matrix Total scattering cross section with and

75 Harmonic Oscillator (classics revisited) 16 Free oscillator Damped oscillator Solution External periodic force Oscillation strength and phase shift Lorentz function

76 Breit-Wigner Function 17 Wave function for an unstable particle Fourier transformation for E dependence Finally our first Breit-Wigner

77 Dressed Resonances T meets Field Theory 18 Suppose we have a resonance with mass m 0 We can describe this with a propagator But we may have a self-energy term leading to

78 T-Matrix Perturbation = We can have an infinite number of loops inside our propagator every loop involves a coupling b, so if b is small, this converges like a geometric series

79 T-Matrix Perturbation Retaining BW 20 So we get and the full amplitude with a dressed propagator leads to which is again a Breit-Wigner like function, but the bare energy E 0 has now changed into E 0 -<{b}

80 Relativistic Breit-Wigner 21 Argand Plot Phase δ Intensity I=ΨΨ * By migrating from Schrödinger s equation (non-relativistic) to Klein-Gordon s equation (relativistic) the energy term changes different energy-momentum relation E=p 2 /m vs. E 2 =m 2 c 4 +p 2 c 2 The propagators change to s R -s from m R -m

81 Barrier Factors - Introduction 22 At low energies, near thresholds but is not valid far away from thresholds -- otherwise the width would explode and the integral of the Breit-Wigner diverges It reflects the non-zero size of the object Need more realistic centrifugal barriers q known as Blatt-Weisskopf damping factors We start with the semi-classical impact parameter L and use the approximation for the stationary solution of the radial differential equation with we obtain

82 Blatt-Weisskopf Barrier Factors 23 The energy dependence is usually parameterized in terms of spherical Hankel-Functions we define F l (q) with the following features Main problem is the choice of the scale parameter q R =q scale

83 Blatt-Weisskopf Barrier Factors (l=0 to 3) 24 by Hippel and Quigg (1972) Usage

84 Form/Barrier factors Resonant daughters i 2 2 m m m m a b a b qi ρ 1 as m ; ρ 1 1 i m m m Scales and Formulae Breakup-momentum formula was derived from a cylindrical potential may become complex (sub-threshold) the scale (197.3 MeV/c) may be different for different processes need <F l (q)>= F l (q)dbw valid in the vicinity of the pole since F l (q) q l ρρ 1 π ρ 1 complex even above threshold meaning of mass and width are mixed up needs analytic continuation Imaginary K -ρ Real E (GeV)

85 Input = Output 26

86 Outline of the Unitarity Approach 27 The most basic feature of an amplitude is UNITARITY Everything which comes in has to get out again no source and no drain of probability Idea: Model a unitary amplitude Realization: n-rank Matrix of analytic functions, T ij one row (column) for each decay channel What is a resonance? A pole in the complex energy plane T ij (m) with m being complex Parameterizations: e.g. sum of poles Γ 0 /2 Im Re m 0

87 28 THANK YOU for today