An Elementary Introduction to Partial-Wave Analyses at BNL Part I

Transcription

1 An Elementary Introduction to Partial-Wave Analyses at Part I S. U. Chung Department, Brookhaven National Laboratory, Upton, NY e852/reviews.html S. U. Chung / p.1

2 Plan of Talk S. U. Chung / p.2

3 Introduction: Plan of Talk General References, Notations and Conventions S. U. Chung / p.2

4 Introduction: Plan of Talk General References, Notations and Conventions Reactions: a+b c+d π + p X + p and π + p X 0 + n S. U. Chung / p.2

5 Introduction: Plan of Talk General References, Notations and Conventions Reactions: a+b c+d π + p X + p and π + p X 0 + n General n-body Decay Modes of X where n 3: X π + π + π, X K + K + π, X π + π + π + π, X η + π + π + π S. U. Chung / p.2

6 Introduction: Plan of Talk General References, Notations and Conventions Reactions: a+b c+d π + p X + p and π + p X 0 + n General n-body Decay Modes of X where n 3: X π + π + π, X K + K + π, X π + π + π + π, X η + π + π + π ************* Simple Decay Modes of X where n = 2: X π + π, X η + π S. U. Chung / p.2

7 Introduction: Plan of Talk General References, Notations and Conventions Reactions: a+b c+d π + p X + p and π + p X 0 + n General n-body Decay Modes of X where n 3: X π + π + π, X K + K + π, X π + π + π + π, X η + π + π + π ************* Simple Decay Modes of X where n = 2: X π + π, X η + π Conclusions and Future Prospects S. U. Chung / p.2

8 General References: Introduction Poincaré Group to Rotation Group S. U. Chung / p.3

9 General References: Introduction Poincaré Group to Rotation Group S. Weinberg: The Quantum Theory of Fields, Cambridge, UK (1995), Volume I, Chapter 2: Relativistic Quantum Mechanics, p.49 The Poincaré Algebra One-Particle States (Mass> 0 and = 0) P-, T -, C-Operators Michele Maggiore: A Modern Introduction to Quantum Field Theory, Chapter 2 (Lorentz and Poincaré Symmetries in QFT) Sections 5 and 6 (Spinorial Representations and Dirac Fields) First published 2005 S. U. Chung / p.3

10 General References: Introduction Poincaré Group to Rotation Group S. Weinberg: The Quantum Theory of Fields, Cambridge, UK (1995), Volume I, Chapter 2: Relativistic Quantum Mechanics, p.49 The Poincaré Algebra One-Particle States (Mass> 0 and = 0) P-, T -, C-Operators Michele Maggiore: A Modern Introduction to Quantum Field Theory, Chapter 2 (Lorentz and Poincaré Symmetries in QFT) Sections 5 and 6 (Spinorial Representations and Dirac Fields) First published 2005 M. Jacob and G. C. Wick, Ann. Phys. (USA) 7, 404 (1959) Helicity... S. U. Chung / p.3

11 General References: Introduction Poincaré Group to Rotation Group S. Weinberg: The Quantum Theory of Fields, Cambridge, UK (1995), Volume I, Chapter 2: Relativistic Quantum Mechanics, p.49 The Poincaré Algebra One-Particle States (Mass> 0 and = 0) P-, T -, C-Operators Michele Maggiore: A Modern Introduction to Quantum Field Theory, Chapter 2 (Lorentz and Poincaré Symmetries in QFT) Sections 5 and 6 (Spinorial Representations and Dirac Fields) First published 2005 M. Jacob and G. C. Wick, Ann. Phys. (USA) 7, 404 (1959) Helicity... M. E. Rose, Angular Momentum..., Wiley, NY (1957) Clebsch-Gordan coefficients my notation: ( j 1 m 1 j 2 m 2 jm) The d-functions... S. U. Chung / p.3

12 General References: Introduction Poincaré Group to Rotation Group S. Weinberg: The Quantum Theory of Fields, Cambridge, UK (1995), Volume I, Chapter 2: Relativistic Quantum Mechanics, p.49 The Poincaré Algebra One-Particle States (Mass> 0 and = 0) P-, T -, C-Operators Michele Maggiore: A Modern Introduction to Quantum Field Theory, Chapter 2 (Lorentz and Poincaré Symmetries in QFT) Sections 5 and 6 (Spinorial Representations and Dirac Fields) First published 2005 M. Jacob and G. C. Wick, Ann. Phys. (USA) 7, 404 (1959) Helicity... M. E. Rose, Angular Momentum..., Wiley, NY (1957) Clebsch-Gordan coefficients my notation: ( j 1 m 1 j 2 m 2 jm) The d-functions... S. U. Chung, Spin Formalisms, CERN 71-8 (Updated Version) spinfm1.pdf Zemach amplitudes... S. U. Chung / p.3

13 Generators of the Poincaré Group P µ, J µν K µ J µ Derived quantities: W µ S and L A. J. Macfarlane, J. Math. Phys. 4, 490 (1963) A. McKerrell, NC 34, 1289 (1964) S. U. Chung, Quantum Lorentz Transformations, lorentzb.pdf S. U. Chung / p.4

14 Allowed quantum numbers for Quarkonia Consider a q q, where q = {u,d,s}, in a state of L and S L = Orbital angular momentum (= 0, 1, 2, 3, ) a S = Total intrinsic spin (= 0, 1) b P = ( ) L+1 for any q q state c C = ( ) L+S for a neutral q q state c L S J L+S Forbidden J PC s: (0 ) d, 0 +, 1 +, (2 + ) e, 3 +, etc. S. U. Chung / p.5

15 Allowed quantum numbers for Quarkonia Consider a q q, where q = {u,d,s}, in a state of L and S L = Orbital angular momentum (= 0, 1, 2, 3, ) a S = Total intrinsic spin (= 0, 1) b P = ( ) L+1 for any q q state c C = ( ) L+S for a neutral q q state c L S J L+S Forbidden J PC s: (0 ) d, 0 +, 1 +, (2 + ) e, 3 +, etc. a S. U. Chung, Spin Formalisms, CERN Yellow Report 71-8 (Updated) spinfm1.pdf b S. U. Chung, Quantum Lorentz Transformations lorentzb.pdf c S. U. Chung, C- and G-parity: a New Definition and Applications (Version IV) Cparity4.pdf d S. U. Chung, Quantum Numbers for Hybrid mesons in the Flux-tube Model (Version III) flxtb0.pdf e S. U. Chung, Photon-Pomeron Fusion Processes S. U. Chung / p.5

16 Phase motion of a Breit-Wigner form S. U. Chung / p.6

17 Phase motion of a Breit-Wigner form Im m=m 0 m 0 = Mass (constant) Γ 0 = Width (constant) 2δ 1/2 (m) = m 0 Γ 0 m 2 0 m2 im 0 Γ 0 = e iδ(m) sinδ(m) cotδ(m) = m2 0 m2 m 0 Γ 0 0 Re S. U. Chung / p.6

18 Phase motion of a Breit-Wigner form Im m=m 0 m 0 = Mass (constant) Γ 0 = Width (constant) (m) = m 0 Γ 0 m 2 0 m2 im 0 Γ 0 = e iδ(m) sinδ(m) cotδ(m) = m2 0 m2 m 0 Γ 0 0 2δ 1/2 Re ρ ππ: m 0 Γ 0 (m) = m 2 0 m2 im 0 Γ(m) Γ(m) = Γ 0 ( m0 m Γ 0 ( m0 m ) ( q q 0 ) ( q q 0 ) (2l+1), l = 1 )[ ] Fl (q) 2 F l (q 0 ) S. U. Chung / p.6

19 Euler Angles z, z x z, z y β x α γ y y, y (node) U[R(α,β,γ)] = exp[ iα J z ] exp[ iβ J y ] exp[ iγ J z ] = exp[ iγ J z ] exp[ iβ J y ] exp[ iα J z ] D j m m (α,β,γ) = jm U[R(α,β,γ)] jm M. E. Rose, Elementary Theory of Angular Momentum, John Wiley & Sons, Inc. (see Chapter IV) S. U. Chung / p.7

20 General two-body decay for J s 1 + s 2 : where λ = λ 1 λ 2. We have A Jλ 1λ 2 M (Ω) = pλ 1 ; pλ 2 M JM = N J F J λ 1 λ 2 D J Mλ (φ,θ,0), N J = 2J + 1 4π where ( ) w 1/2 Fλ J 1 λ 2 = 4π JM λ 1 λ 2 M JM p ( ) 1 2l+1 2 = G J ls ls 2J + 1 (l0sλ Jλ)(s 1λ 1 s 2 λ 2 Sλ) G J ls = 4π ( w p ) 1/2 JMlS M JM and F J 2 λ 1 λ 2 = λ 1 λ 2 ls G J 2 ls S. U. Chung / p.8

21 Helicity states: p, jλ = φ,θ, p, jλ = U[ R(φ,θ,0)]U[L z (p)] jλ = U[L( p )]U[ R(φ,θ,0)] jλ (b) z z h ŷh ẑ ˆp ˆp φ θ node y x Helicity coordinate system: ˆx h = ŷ h ẑ h ŷ h ẑ ˆp, ẑ h = ˆp S. U. Chung / p.9

22 Helicity states: p, jλ = φ,θ, p, jλ = U[ R(φ,θ,0)]U[L z (p)] jλ = U[L( p )]U[ R(φ,θ,0)] jλ (b) z z h ŷh ẑ ˆp ˆp φ θ node y x Helicity coordinate system: ˆx h = ŷ h ẑ h ŷ h ẑ ˆp, ẑ h = ˆp A cautionary note: See a note by S. U. Chung Sequential Decays involving ω γ + π 0 omega6.pdf S. U. Chung / p.9

23 Zemach Formalism: C. Zemach, Nuovo Cimento 32, 1605 (1964) C. Zemach, Phys. Rev. 140 B, 97 (1965) S. U. Chung, Spin Formalisms, CERN 71-8 (Updated Version) spinfm1.pdf G J ls = GJ ls pl where p is a breakup momentum in a suitable rest frame(rf). PWA Program: G J ls = GJ ls F l(p); F l (p) (p/p R ) l, for p/p R 0 F l (p) 1, for p/p R where F l (p) s are the Blatt-Weisskopf barrier factors with q R = GeV/c corresponding to 1.0 fermi. S. U. Chung / p.10

24 Blatt-Weisskopf centrifugal-barrier factors F. von Hippel and C. Quigg, Phys. Rev. 5, 624 (1972) S. U. Chung, Formulas for Angular-Momentum Barrier Factors, brfactor1.pdf The radial Schrödinger equation with the potential V(r) = 0 for r > R ( interaction radius ) is [ ] d 2 dρ l(l+1) ρ 2 U l (ρ) = 0, U l (ρ) = ρ h (1) l (ρ) where ρ = pr. The outgoing-wave solutions are given by U l (ρ) with h (1) l (ρ) = Sherical Hankel functions of the first kind [i(ρ π ] l 2 l ) ( 1) k U l (ρ) = ρ h (1) l (ρ) = i exp k=0 (l+k)! k!(l k)! (2iρ) k Note U l(ρ) ρ l, ρ 0 U l(ρ) = 1 ρ S. U. Chung / p.11

25 ln Ul ( pr) Radial Wave Functions ρ(770) ππ p = GeV r p = hc/p = fm l = r(fm) ln Ul ( pr) f 2 (1270) ππ p = GeV r p = hc/p = fm l = r(fm) r = R = 1fm; p R = GeV; ln U 1 (pr) = U 1 (p/p R ) = U 1 (1.840) = r = R = 1fm; p R = GeV; ln U 2 (pr) = U 2 (p/p R ) = U 2 (3.158) = S. U. Chung / p.12

26 Define Blatt-Weisskopf barrier factors via f l (x) = Ul (pr) r Ul (pr) r=r = x 1, x = pr = p/p R l (x) h (1) where R = 1fm and p R = GeV/c. Observe f 0 (x) = 1, f 1 (x) = x x 2 + 1, f 2(x) = x 2 (x 2 3) 2 + 9x 2 S. U. Chung / p.13

27 Barrier Factors: Examples f l (x) for l = x f l (x) for l = x f l (x) for l = x f l (x) for l = x S. U. Chung / p.14

28 Barrier Factors: Examples f l (x) for l = x f l (x) for l = x f l (x) for l = x f l (x) for l = x S. U. Chung / p.15

29 a(π )+b(p) c(x )+d(p): x x J X z J π θ 0 p z p S. U. Chung / p.16

30 a(π )+b(p) c(x )+d(p): x x J X z J π θ 0 p z p z J π (beam) in the X rest frame (XRF) t = t t min = 2 p i p f (1 cosθ 0 ) 0 Reflection Operator: Π y (π) S. U. Chung / p.16

31 General Angular Distributions in Reflectivity Basis: S. U. Chung and T. L. Trueman, Phys. Rev. D11, 633 (1975) Consider a(π )+b(p) c(x )+d(p), X (χ) π + π π, K 0 K π ± Let jm be the spin state for c where the quantization axis is defined in the production plane, i.e. one takes either the helicity or the Jackson frame for c. The amplitude for production and decay of the c is A χm p c χm, p d λ d T p a λ a, p b λ b D χ m(τ) where χ specifies the complete quantum state for c, which includes its spin j, its parity, its C-parity, its isotopic spin, and its decay products with the phase-space element given by τ; λ s refer to the helicities; T is the transition operator of the process ab cd; and D is the decay amplitude for c, which may consist of a product of the rotation functions as well as the Breit-Wigner forms. S. U. Chung / p.17

32 The distribution function follows immediately I(τ) λ a λ b λ d χm χ m p c χm, p d λ d T p a λ a, p b λ b p c χ m, p d λ d T p a λ a, p b λ b Dm(τ)D χ χ m (τ) Note that the helicities of a, b and d are the external unobserved variables and therefore summed over outside of the absolute square of the amplitude A. In terms of the generalized spin-density matrix ρ χ χ mm p c χm, p d λ d T p a λ a, p b λ b p c χ m, p d λ d T p a λ a, p b λ b λ a λ b λ d the distribution function assumes an elegant form χ χ I(τ) ρ mm Dm(τ)D χ χ m (τ) χm χ m S. U. Chung / p.18

33 We are now ready to introduce the reflection operator through the production plane for ab cd. Let this plane be defined to be the x-z plane, i.e. the production normal is along the y-axis. Then the reflection operator defined by Π y = U[R y (π)]π = ΠU[R y (π)] where U[R y (π)] is a unitary operator representing a rotation by π around the y-axis, i.e. U[R y (π)] = exp( iπ J y ) which is simply given by the standard d-function U m m[r y (π)] = d j m m (π) = ( ) j m δ m, m Note also that U m m[r y ( π)] = d j m m ( π) = ( ) j+m δ m, m Let Λ is a general operator in the xz-plane. Then, we see that [ Πy,U[Λ( p c )] ] = 0, Π y p c χ m = η c ( ) j m p c χ m, Π 2 y = ( )2 j I where η c is the intrinsic parity of the c. Also true for helicity states ( p i λ i, i = a,b,d ). Also true for massless particles (m λ = ± j). S. U. Chung / p.19

34 We move over to the reflection-basis states for c, i.e. p c ε χ m = θ(m) { p c χ m +ε η c ( ) j m p c χ m }, η c = intrinsic parity of c where θ(m) = 1, m > 0; θ(m) = 1, m = 0; θ(m) = 0, m < These basis states constitute eigenvectors of the reflection operator ε 2 = ( ) 2 j Π y p c ε χ m = ε ( ) 2 j p c ε χ m so that we have ε = ±1 for bosons and ε = ±i for fermions. Note, in addition, that εε = ε 2 = 1 for both bosons and fermions. The generalized density matrix in the reflectivity basis is, with m 0 and m 0, ε ε ρ χ χ mm ε p c χm, p d λ d T p a λ a, p b λ b ε p c χ m, p d λ d T p a λ a, p b λ b λ a λ b λ d S. U. Chung / p.20

35 We shall explore the consequences of a reflection operation applied to the transition matrices above. The key observation is that Π y leaves the transition operator T unperturbed, i.e. [Π y,t] = 0. The three-vectors p i (i = a,b,c,d ) are left unchanged under Π y, i.e. the boost operators and/or the rotations about the y-axis, which enter in the definitions of the helicity or the Jackson frames, remain invariant under Π y. Therefore, the parity conservation is relegated to exploring the consequences of Π y acting on the rest states. Insert Π y 1 Π y = Π y Π y = I next to each T and propagate Π y and Π y backwards and forwards, respectively, to find so that ε ε ρ χ χ mm εε ( ) 2( j j ) λ a λ b λ d ε p c χm, p d, λ d T p a, λ a, p b, λ b ε p c χ m, p d, λ d T p a, λ a, p b, λ b ε ε ρ χ χ mm = ε ε ε ε ρ χ χ mm So we see that ε ε = +1. Multiply it by ε from the right and noting that ε ε = ε 2 = +1, we find ε = ε. Here we have carefully handled the derivation, so that the formula above applies to both bosons and fermions. S. U. Chung / p.21

36 We shall explore the consequences of a reflection operation applied to the transition matrices above. The key observation is that Π y leaves the transition operator T unperturbed, i.e. [Π y,t] = 0. The three-vectors p i (i = a,b,c,d ) are left unchanged under Π y, i.e. the boost operators and/or the rotations about the y-axis, which enter in the definitions of the helicity or the Jackson frames, remain invariant under Π y. Therefore, the parity conservation is relegated to exploring the consequences of Π y acting on the rest states. Insert Π y 1 Π y = Π y Π y = I next to each T and propagate Π y and Π y backwards and forwards, respectively, to find so that ε ε ρ χ χ mm εε ( ) 2( j j ) λ a λ b λ d ε p c χm, p d, λ d T p a, λ a, p b, λ b ε p c χ m, p d, λ d T p a, λ a, p b, λ b ε ε ρ χ χ mm = ε ε ε ε ρ χ χ mm So we see that ε ε = +1. Multiply it by ε from the right and noting that ε ε = ε 2 = +1, we find ε = ε. Here we have carefully handled the derivation, so that the formula above applies to both bosons and fermions. S. U. Chung, General Spin-Density Matrix spinmtrx1.pdf S. U. Chung / p.21

37 The density matrix can be written, quite generally, in a block-diagonal form ε ρ χ χ mm ε p c χm, p d λ d T p a λ a, p b λ b ε p c χ m, p d λ d T p a λ a, p b λ b λ a λ b λ d a fundamental formula which incorporates parity conservation in the production process ab cd. The distribution function in the reflectivity basis is I(τ) 2 ε χm χ m ε ρ χ χ mm ε D χ m(τ) ε D χ m (τ), m 0, m 0 where ε D is the decay amplitude in the reflectivity basis. Consider a simple decay c s 1 (λ 1 )+s 2 (λ 2 ) In the crf, we have, with p c = 0 and τ = R(φ,θ,0), ε Dm(τ) χ = qλ 1 ; qλ 2 M ε jm = qλ 1 ; qλ 2 λ 1 λ 2 ε jm λ 1 λ 2 ε jm M ε jm { = N j F j θ(m) D j λ 1 λ 2 mλ (R)+ε η c ( ) j m D j }, mλ (R) λ = λ 1 λ 2 S. U. Chung / p.22

38 An Example: X ρ + π, ρ π + π Let s = 1 and λ be the ρ spin and helicity. Define W = Effective mass of the 3π system w = Effective mass for ρ ππ p = Breakup momentum between w and the bachelor π in XRF (Θ,Φ) = The orientation of p in XRF q = Breakup momentum for ρ ππ in ρrf (θ,φ) = The orientation of q in ρrf Then, the full decay amplitude is ε D χ m(τ) = λ ε D j η c m (τ) F l (p) (w) F s (q) D s λ 0 (φ,θ,0) { ε D j η c m (τ) = N j F j λ θ(m) D j mλ (Φ,Θ,0)+ε η c ( ) j m D j } mλ (Φ,Θ,0) ( ) 1 F j 2l+1 2 λ = (l0sλ Jλ) 2 j+ 1 Note the decay amplitude G j ls has been absorbed into the production amplitudes (the variables to be fitted experimentally). S. U. Chung / p.23

39 The rank of the density matrix is determined by the number of independent terms in the summation on helicities. Let n i = 2s i + 1 or n i = 2(photons) for i = a, b, d depending on whether a particle is massive or massless. The total number in the sum is N = n a n b n d So the rank of the density matrix is (N + 1)/2 if N is odd, and it is N/2 if N is even. Note that the reduction in the rank comes from parity conservation in the production process (to show this, apply Π y again to the amplitudes). S. U. Chung / p.24

40 Table I. Rank of Spin-Density Matrix for X Reaction Rank π p π X + 1 π p X p 2 π + n X ++ 4 pp X p 4 np X ++ 8 γ p X 0 p 4 γ p π + X 0 2 ν p e X ++ 1 e p e X + 1 φ p X 0 p 6 φ p π + X 0 3 π η π X 0 1 π φ π X 0 2 The electrons are assumed to come with one helicity. True in general for odd-half-integer spins in the limit of zero mass. S. U. Chung / p.25

41 Consider an N ε N ε density matrix, with i = {χ m} and j = {χ m }, ε ρ i j = K ε V ε ik V jk, = ε ρ = ε V ε V, = ε ρ = ε ρ k=1 where i, j = 1,,N ε ; k = 1,,K ε, and K ε (= 1,, ) is the rank of the density matrix. Note that ε ρ is an N ε N ε square matrix, whereas ε V is, in general, a retangular matrix N ε K ε. The Cholesky decomposition of ε V is, e.g. for N ε = 6 or for 6 6 ε ρ, ε V ε V ε 21 V ε V = { ε V ik } = ε V ε 31 V ε 32 V ε V ε 41 V ε 42 V ε 43 V ε V ε 51 V ε 52 V ε 53 V ε 54 V 55 0 ε V 61 ε V 62 ε V 63 ε V 64 ε V 65 ε V 66 where ε V ik is complex in general but ε V ii = real 0. There are 6 real diagonal elements and 15 complex off-diagonal elements of ε V, for a total of 36 parameters required to describe a 6 6 ε ρ. The rank is given by the number of columns counting from the left, with the rest being zero. For example, if the rank=2, then we must have ε V ik = 0, i k 3. In this case, there are 2 real diagonal elements and 9 complex off-diagonal elements, for a total of 20 parameters in the problem. S. U. Chung / p.26

42 Now go back to the notation {χ m} and {χ m } ε ρ χ K ε χ mm = ε V χ mk ε V χ m, m 0, m 0 k k=1 and define ε U k (τ) = N ε ε V χ mk ε Dm(τ), χ m 0 χ m The distribution function I(τ) 2 ε N ε χm χ m ε ρ χ χ ε mm Dm(τ) χ ε D χ m (τ) becomes I(τ) 2 ε K ε ε U k (τ) 2 k=1 S. U. Chung / p.27

43 Experimental Moments Let R(α,β,γ) describe the orientation of the 3π system in the XRF. Compare the unnormalized experimental moments with the predicted ones H x (LMN) = n D L MN(R i ) i H x (LMN) = η(r)i(r)d L MN(R)dR The body-fixed z axis is chosen as the normal to the decay plane. This is done for each region of the Dalitz plot. S. U. Chung / p.28

44 Strangeonium Excitations K η 2 (2150) K ± K S π ǫ = +1, K + (892), K + 2 (1430) ǫ = 1, K +, K + 1 (1270), K+ 1 (1400) p Λ s ū K η 2 (2150) s s u s u u d p Λ s u d S. U. Chung / p.29

45 Strangeonium Excitations K η 2 (2150) K ± K S π ǫ = +1, K + (892), K + 2 (1430) ǫ = 1, K +, K + 1 (1270), K+ 1 (1400) p Λ s ū K η 2 (2150) s s u s S. U. Chung: C- and G-parity:... Cparity4.pdf... K Kπ...pwafm0.pdf, kkbarpi1.pdf u u d p Λ s u d S. U. Chung / p.29

46 Maximum-Likelihood Method Introduce the so-called extended likelihood function for finding n events in a given mass bin L [ n n n! ] n n e i [ ] I(τ i ) I(τ) η(τ) φ(τ) dτ no phase-space factors ini(τ i ) where η(τ) is the experimental finite acceptance at τ and the invariant phase-space element given by ( ) dφ dφ = dτ = φ(τ)dτ dτ The first bracket in L represents the Poisson probability for finding n events in the mass bin, and the expectation value n is n I(τ) η(τ) φ(τ) dτ The likelihood function L can now be written, dropping the factors depending on n alone, L [ n i I(τ i ) ] [ exp ] I(τ) η(τ) φ(τ) dτ S. U. Chung / p.30

47 The log of the likelihood function now has the form, lnl = n i lni(τ i ) I(τ) η(τ) φ(τ) dτ We shall adopt the following shorthand notation α = {εk; χ m} and α = {εk; χ m } and write I(τ) = αα V α V α D α (τ) D α (τ) The so-called experimental normalization integral is given by Ψ x α α = [Dα (τ) D α (τ) ] η(τ) φ(τ) dτ so that lnl = n i ln [ αα V α V α D α (τ i ) D α (τ i ) ] αα V α V α Ψ x α α S. U. Chung / p.31

48 We explore the normalization for V s by setting V = xw, where x is independent of α, lnl = n i ln [ x 2 αα W α W α D α (τ i ) D α (τ i ) ] x 2 αα W α W α Ψ x α α At the maximum, we should have 0 = lnl x 2 = n i [ 1 x 2 ] αα W α W α Ψ x α α so that αα V α V α Ψ x α α = n We can define the theoretical normalization integral, with η(τ) = 1, Ψ α α = [Dα (τ) D α (τ) ] φ(τ) dτ S. U. Chung / p.32

49 The predicted number of events is N = αα V α V α Ψ α α αα N αα, N αα = V α V α Ψ α α So the predicted number of events for a partial wave α is N αα = V α 2 Ψ α α, Ψ α α = Dα (τ) 2 φ(τ) dτ The predicted number of events for the interference between the partial waves α and α is N αα + N α α = 2R { V α Vα } Ψ α α, α α S. U. Chung / p.33