an elementary introduction to partial-wave analyses at bnl …bes.ihep.ac.cn/conference/pwa/am...

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An Elementary Introduction to

Partial-Wave Analyses at BNL

—Part I—

S. U. Chung

Physics Department, Brookhaven National Laboratory,

Upton, NY 11973

http://cern.ch/suchung/http://www.phy.bnl.gov/ ˜e852/reviews.html

S. U. Chung / BNL – p.1

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Plan of Talk


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Plan of Talk• Introduction:

General References,Notations and Conventions


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• Reactions: a+b → c+d

π− + p → X−+ p and π− + p → X0 +n


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π− + p → X−+ p and π− + p → X0 +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


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π− + p → X−+ p and π− + p → X0 +n


*************

• Simple Decay Modes of X where n = 2:X → π +π, X → η +π


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π− + p → X−+ p and π− + p → X0 +n


*************

• Simple Decay Modes of X where n = 2:X → π +π, X → η +π

• Conclusions and Future Prospects


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IntroductionGeneral References: Poincaré Group to Rotation Group


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• S. Weinberg:‘The Quantum Theory of Fields,’ Cambridge, UK (1995), Volume I, Chapter 2:Relativistic Quantum Mechanics, p.49—

The Poincaré AlgebraOne-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


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• M. Jacob and G. C. Wick, Ann. Phys. (USA) 7, 404 (1959)‘ Helicity. . . ’


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• M. E. Rose, ‘Angular Momentum. . . ,’ Wiley, NY (1957)Clebsch-Gordan coefficients my notation: ( j1m1 j2m2| jm)

The d-functions. . .


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• M. E. Rose, ‘Angular Momentum. . . ,’ Wiley, NY (1957)Clebsch-Gordan coefficients my notation: ( j1m1 j2m2| jm)

The d-functions. . .

• S. U. Chung, ‘Spin Formalisms,’ CERN 71-8 (Updated Version)—spinfm1.pdfZemach amplitudes. . .


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Generators of the Poincare GroupPµ , Jµν → Kµ ⊕ Jµ

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


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Allowed quantum numbers for Quarkonia

• Consider a qq, 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 qq statec

• C = (−)L+S for a neutral qq statec

• |L−S| ≤ J ≤ L +S

• Forbidden JPC ’s: (0−−)d ,0+−, 1−+, (2+−)e, 3−+, etc.


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Allowed quantum numbers for Quarkonia

• Consider a qq, 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 qq statec

• C = (−)L+S for a neutral qq statec

• |L−S| ≤ J ≤ L +S

• Forbidden JPC ’s: (0−−)d ,0+−, 1−+, (2+−)e, 3−+, etc.

aS. U. Chung,“Spin Formalisms,” CERN Yellow Report 71-8 (Updated)—spinfm1.pdf

bS. U. Chung,“Quantum Lorentz Transformations”—lorentzb.pdf

cS. U. Chung,“C- and G-parity: a New Definition and Applications” (Version IV)—Cparity4.pdf

dS. U. Chung,“Quantum Numbers for Hybrid mesons in the Flux-tube Model” (Version III)—flxtb0.pdf

eS. U. Chung,“Photon-Pomeron Fusion Processes”


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Phase motion of a Breit-Wigner form


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Im

Re0

2δ 1/2

m=m0

m0 = Mass (constant)Γ0 = Width (constant)

∆(m) =m0 Γ0

m20−m2− im0 Γ0

= eiδ (m) sinδ (m)

cotδ (m) =m2

0−m2

m0 Γ0


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Im

Re0

2δ 1/2

m=m0

m0 = Mass (constant)Γ0 = Width (constant)

∆(m) =m0 Γ0

m20−m2− im0 Γ0

= eiδ (m) sinδ (m)

cotδ (m) =m2

0−m2

m0 Γ0

ρ → ππ:

∆(m) =m0 Γ0

m20−m2− im0 Γ(m)

Γ(m) = Γ0

(m0

m

)

(

qq0

)(2ℓ+1)

, ℓ = 1

→ Γ0

( m0

m

)

(

qq0

)[

Fℓ(q)

Fℓ(q0)

]2


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Euler Anglesz, z′

z′′, z′′′

x y

y′, y′′ (node)

α

β

γ

y′′′

−x′′′

U [R(α,β ,γ)] = exp[−iα Jz] exp[−iβ Jy] exp[−iγ Jz]

= exp[−iγ J′′z ] exp[−iβ J′y ] exp[−iα Jz]

D jm′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)


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Helicity states:

|~p, jλ 〉 = |φ ,θ , p, jλ 〉

= U [R(φ ,θ ,0)]U [Lz(p)] | jλ 〉 = U [L(~p )]U [

R(φ ,θ ,0)] | jλ 〉

z

p

x

y

node

φ

θ

zh

yh∝ z × p

(b)

Helicity coordinate system:

xh = yh × zh yh ∝ z× p, zh = p


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Helicity states:

|~p, jλ 〉 = |φ ,θ , p, jλ 〉

= U [R(φ ,θ ,0)]U [Lz(p)] | jλ 〉 = U [L(~p )]U [

R(φ ,θ ,0)] | jλ 〉

z

p

x

y

node

φ

θ

zh

yh∝ z × p

(b)

Helicity coordinate system:

xh = yh × zh yh ∝ z× p, zh = p

A cautionary note:See a note by S. U. Chung‘Sequential Decays involving ω → γ +π0’omega6.pdf


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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

GJℓS =⇒ GJ

ℓS · pℓ

where p is a breakup momentum in a suitable rest frame(RF).

BNL PWA Program:

GJℓS =⇒ GJ

ℓS ·Fℓ(p); Fℓ(p) ∝ (p/pR )ℓ, for p/pR → 0

Fℓ(p) ∝ 1, for p/pR → ∞

where Fℓ(p)’s are the Blatt-Weisskopf barrier factors with qR = 0.1973GeV/ccorresponding to 1.0 fermi.


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Blatt-Weisskopf centrifugal-barrier factorsF. 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

[

d2

dρ2 +1− ℓ(ℓ+1)

ρ2

]

Uℓ(ρ) = 0, Uℓ(ρ) = ρ h(1)ℓ (ρ)

where ρ = pr. The outgoing-wave solutions are given by Uℓ(ρ) with

h(1)ℓ (ρ) = Sherical Hankel functions of the first kind

Uℓ(ρ) = ρ h(1)ℓ (ρ) = −i exp

[

i(ρ − π2

ℓ )] ℓ

∑k=0

(−1)k (ℓ+ k)!k! (ℓ− k)!

(2iρ)−k

Note∣

∣

∣

∣

Uℓ(ρ)

∣

∣

∣

∣

ρ→0∝ ρ−ℓ,

∣

∣

∣

∣

Uℓ(ρ)

∣

∣

∣

∣

ρ→∞= 1


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Radial Wave Functions

00.20.40.60.8

11.21.41.6

0 0.5 1 1.5 2 2.5 3

r

r

r

r

r

r

r

r

r

r

r

r

rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

ρ(770) → ππp = 0.363GeV

rp = h c/p = 0.544 fm

ℓ = 1

ln∣ ∣ ∣U

ℓ(pr

)∣ ∣ ∣

r (fm) →0

0.51

1.52

2.53

3.54

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

r

r

r

r

r

r

r

r

r

r

r

r

r

rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

f2(1270) → ππp = 0.623 GeV

rp = h c/p = 0.317 fm

ℓ = 2

ln∣ ∣ ∣U

ℓ(pr

)∣ ∣ ∣

r (fm) →

r = R = 1fm;pR = 0.1973GeV;

ln∣

∣

∣U1(pR)

∣

∣

∣= 0.1282

∣

∣

∣U1(p/pR)

∣

∣

∣= 1.137

∣

∣

∣U1(1.840)

∣

∣

∣= 1.137

r = R = 1fm;pR = 0.1973GeV;

ln∣

∣

∣U2(pR)

∣

∣

∣= 0.1606

∣

∣

∣U2(p/pR)

∣

∣

∣= 1.174

∣

∣

∣U2(3.158)

∣

∣

∣= 1.174


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Define Blatt-Weisskopf barrier factors via

fℓ(x) =

∣

∣Uℓ(pr)∣

∣

r→∞∣

∣Uℓ(pr)∣

∣

r=R

=1

x∣

∣

∣h(1)

ℓ (x)∣

∣

∣

, x = pR = p/pR

where R = 1fm and pR = 0.1973GeV/c. Observe

f0(x) = 1, f1(x) =x√

x2 +1, f2(x) =

x2√

(x2−3)2 +9x2· · ·


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Barrier Factors: Examplesfℓ(x) for ℓ = 1

1.0

0.8

0.6

0.4

0.2

0 0.5 1 1.5 2 2.5 3b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

bbbbbbbbbbb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

x →

fℓ(x) for ℓ = 21.0

0.8

0.6

0.4

0.2

0 1 2 3 4 5 6b b

bbb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

bbbbbbbbbbb b b b b b b b b b b b b b b b b b b b b b b b b b b b

x →

fℓ(x) for ℓ = 31.0

0.8

0.6

0.4

0.2

0 1 2 3 4 5 6 7 8 9b b b b b

bb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

bbbbbbbbb b b b b b b b b b b b b b b b b b b b b b b b b b b b

x →

fℓ(x) for ℓ = 41.0

0.8

0.6

0.4

0.2

0 2 4 6 8 10 12bbbbbbbbb

bbbbbbbbb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

bbbbbbbbbbbbbbbbbbbbbbbbbb

bbbbbbb

bbbbbbbbbb

bbbbbbbbbbbbbbbbb

bbbbbb

x →


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Barrier Factors: Examplesfℓ(x) for ℓ = 5

1.0

0.8

0.6

0.4

0.2

0 2 4 6 8 10 12 14 16b b b b b b b b

bb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

bbbbbbb b

b b bb b b

b b b b b b b b b b b b b b b b b b b b b

x →

fℓ(x) for ℓ = 61.0

0.8

0.6

0.4

0.2

0 2 4 6 8 10 12 14 16 18bbbbbbbbbbbbb

bbbbbbb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

bbbbbbbbbbbbbbbbbbbbbb

bbbbbbb

bbbbbbbbb

bbbbbbbbbbbbbb

bbbbbbbbbbb

x →

fℓ(x) for ℓ = 71.0

0.8

0.6

0.4

0.2

0 5 10 15 20 25bbbbbbbbbbbbbb

bbbbbbb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b


bbbbbbb

bbbbbbbbbbb

bbbbbbbbbbbbbbbbbb

bbbb

x →

fℓ(x) for ℓ = 1001.0

0.8

0.6

0.4

0.2

0 50 100 150 200 250 300bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

bbbb

b

b

b

b

b

b

b

b

b

b

b

b

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

bbbbbbbbbbbbbbbbbbb

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb


x →


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a(π−)+b(p) → c(X−)+d(p):

π− p

X−

zJ

xJ

p

x

zθ0


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a(π−)+b(p) → c(X−)+d(p):

π− p

X−

zJ

xJ

p

x

zθ0

~zJ ∝ π− (beam) in the X rest frame (XRF)t ′ = |t|− |t|min

= 2pi p f (1−cosθ0) ≥ 0

Reflection Operator: Πy(π)


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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−, K0K∓π±

Let | jm〉 be the spin state for c where the quantization axis is defined in the productionplane, i.e. one takes either the helicity or the Jackson frame for c. The amplitude forproduction and decay of the c is

A ∝ ∑χm

〈~pc χm, ~pdλd |T |~paλa, ~pbλb〉Dχm(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 givenby 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’ aswell as the Breit-Wigner forms.


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The distribution function follows immediately

I(τ) ∝ ∑λaλbλd

∑χm χ ′m′

×〈~pc χm, ~pdλd |T |~paλa, ~pbλb〉〈~pc χ ′m′, ~pdλd |T |~paλa, ~pbλb〉∗

×Dχm(τ)Dχ ′ ∗

m′ (τ)

Note that the helicities of a, b and d are the ‘external’ unobserved variables and thereforesummed over outside of the absolute square of the amplitude A. In terms of thegeneralized spin-density matrix

ρχ χ ′mm′ ∝ ∑

λaλbλd

〈~pc χm, ~pdλd |T |~paλa, ~pbλb〉〈~pc χ ′m′, ~pdλd |T |~paλa, ~pbλb〉∗

the distribution function assumes an elegant form

I(τ) ∝ ∑χmχ ′m′

ρχ χ ′mm′ Dχ

m(τ)Dχ ′ ∗m′ (τ)


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We are now ready to introduce the reflection operator through the production planefor ab → cd. Let this plane be defined to be the x-z plane, i.e. the production normal isalong the y-axis. Then the reflection operator defined by

Πy = U [Ry(π)]Π = ΠU [Ry(π)]

where U [Ry(π)] is a unitary operator representing a rotation by π around the y-axis, i.e.

U [Ry(π)] = exp(−iπ Jy)

which is simply given by the standard d-function

Um′ m[Ry(π)] = d jm′ m(π) = (−) j−m δm′,−m

Note also that

Um′ m[Ry(−π)] = d jm′ m(−π) = (−) j+m δm′,−m

Let Λ is a general operator in the xz-plane. Then, we see that

[

Πy,U [Λ(~pc)]]

= 0, Πy |~pc χ m〉 = ηc (−) j−m |~pc χ −m〉, Π2y = (−)2 j I

where ηc is the intrinsic parity of the c. Also true for helicity states ( |~pi λi〉, i = a,b,d ).Also true for massless particles (m → λ = ± j).


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We move over to the reflection-basis states for c, i.e.

|~pc ε χ m〉 = θ (m)

|~pc χ m〉+ ε ηc (−) j−m |~pc χ −m〉

, ηc = intrinsic parity of c

where

θ (m) =1√2, m > 0; θ (m) =

12, m = 0; θ (m) = 0, m < 0

These basis states constitute eigenvectors of the reflection operator

ε2 = (−)2 j → Πy |~pc ε χ m〉 = ε (−)2 j |~pc ε χ m〉

so that we have ε = ±1 for bosons and ε = ±i for fermions. Note, in addition, thatεε∗ = |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 ′ρχ χ ′mm′ ∝

∑λaλbλd

〈ε ~pc χm, ~pdλd |T |~paλa, ~pbλb〉〈ε ′~pc χ ′m′, ~pdλd |T |~paλa, ~pbλb〉∗


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We shall explore the consequences of a reflection operation applied to the transitionmatrices above. The key observation is that Πy leaves the transition operator T

unperturbed, i.e. [Πy,T ] = 0. The three-vectors ~pi ( i = a,b,c,d ) are left unchanged underΠy, i.e. the boost operators and/or the rotations about the y-axis, which enter in thedefinitions of the helicity or the Jackson frames, remain invariant under Πy. Therefore,the parity conservation is relegated to exploring the consequences of Πy acting on the‘rest’ states. Insert Πy

−1 Πy = Πy† Πy = I next to each T and propagate Π†

y and Πy

backwards and forwards, respectively, to find

ε ε ′ρχ χ ′mm′ ∝ εε ′ ∗ (−)2( j− j′) ∑

λaλbλd

〈ε ~pc χm, ~pd ,−λd |T |~pa,−λa, ~pb,−λb〉〈ε ′~pc χ ′m′, ~pd ,−λd |T |~pa,−λa, ~pb,−λb〉∗

so thatε ε ′ρχ χ ′

mm′ = ε ε ′ ∗ × ε ε ′ρχ χ ′mm′

So we see that ε ε ′ ∗ = +1. Multiply it by ε ′ from the right and noting thatε ′ ε ′ ∗ = |ε ′|2 = +1, we find ε = ε ′. Here we have carefully handled the derivation, so thatthe formula above applies to both bosons and fermions.


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We shall explore the consequences of a reflection operation applied to the transitionmatrices above. The key observation is that Πy leaves the transition operator T

unperturbed, i.e. [Πy,T ] = 0. The three-vectors ~pi ( i = a,b,c,d ) are left unchanged underΠy, i.e. the boost operators and/or the rotations about the y-axis, which enter in thedefinitions of the helicity or the Jackson frames, remain invariant under Πy. Therefore,the parity conservation is relegated to exploring the consequences of Πy acting on the‘rest’ states. Insert Πy

−1 Πy = Πy† Πy = I next to each T and propagate Π†

y and Πy

backwards and forwards, respectively, to find

ε ε ′ρχ χ ′mm′ ∝ εε ′ ∗ (−)2( j− j′) ∑

λaλbλd

〈ε ~pc χm, ~pd ,−λd |T |~pa,−λa, ~pb,−λb〉〈ε ′~pc χ ′m′, ~pd ,−λd |T |~pa,−λa, ~pb,−λb〉∗

so thatε ε ′ρχ χ ′

mm′ = ε ε ′ ∗ × ε ε ′ρχ χ ′mm′

So we see that ε ε ′ ∗ = +1. Multiply it by ε ′ from the right and noting thatε ′ ε ′ ∗ = |ε ′|2 = +1, we find ε = ε ′. Here we have carefully handled the derivation, so thatthe formula above applies to both bosons and fermions.

S. U. Chung,‘General Spin-Density Matrix’spinmtrx1.pdf


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The density matrix can be written, quite generally, in a block-diagonal form

ε ρχ χ ′mm′ ∝

∑λaλbλd

〈ε ~pc χm, ~pdλd |T |~paλa, ~pbλb〉〈ε ~pc χ ′m′, ~pdλd |T |~paλa, ~pbλb〉∗

a fundamental formula which incorporates parity conservation in the production processab → cd. The distribution function in the reflectivity basis is

I(τ) ∝2

∑ε

∑χmχ ′m′

ε ρχ χ ′mm′

ε Dχm(τ) ε Dχ ′ ∗

m′ (τ), m ≥ 0, m′ ≥ 0

where ε D is the decay amplitude in the reflectivity basis. Consider a simple decay

c → s1(λ1)+ s2(λ2)

In the cRF, we have, with ~pc = 0 and τ = R(φ ,θ ,0),

ε Dχm(τ) = 〈~qλ1;−~qλ2|M |ε jm〉 = 〈~qλ1;−~qλ2|λ1λ2 ε jm〉〈λ1λ2 ε jm|M |ε jm〉

= N j F jλ1 λ2

θ (m)

D j ∗mλ (R)+ ε ηc (−) j−m D j ∗

−mλ (R)

, λ = λ1−λ2


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An Example: X → ρ +π, ρ → π +πLet 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 Dχm(τ) = ∑

λ

ε D j ηcm (τ) Fℓ(p) ∆(w) Fs(q) Ds ∗

λ 0(φ ,θ ,0)

ε D j ηcm (τ) = N j F j

λ θ (m)

D j ∗mλ (Φ,Θ,0)+ ε ηc (−) j−m D j ∗

−mλ (Φ,Θ,0)

F jλ =

(

2ℓ+12 j +1

) 12

(ℓ0sλ |Jλ )

Note the decay amplitude G jℓs has been absorbed into the production amplitudes (the

variables to be fitted experimentally).


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The rank of the density matrix is determined by the number of independent terms inthe summation on helicities. Let

ni = 2si +1 or ni = 2(photons) for i = a, b, d

depending on whether a particle is massive or massless. The total number in the sum is

N = na nb nd

So the rank of the density matrix is (N +1)/2 if N is odd, and it is N/2 if N is even. Notethat the reduction in the rank comes from parity conservation in the production process(to show this, apply Πy again to the amplitudes).


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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 → X0p 4

γ p → π+X0 2

ν p → e−X++ 1†

e−p → e−X+ 1†

φ p → X0p 6

φ p → π+X0 3

π−η → π−X0 1

π−φ → π−X0 2† The electrons are assumed to come with one helicity.

True in general for odd-half-integer spins in the limit of zero mass.


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Consider an Nε ×Nε density matrix, with i = χ m and j = χ ′ m′,

ε ρ i j =Kε

∑k=1

εV ikεV ∗

jk, =⇒ ε ρ = εV εV †, =⇒ ε ρ = ε ρ†

where i, j = 1, · · · ,Nε ; k = 1, · · · ,Kε , and Kε (= 1, · · · ,∞) is the rank of the density matrix.Note that ε ρ is an Nε ×Nε square matrix, whereas εV is, in general, a retangular matrixNε ×Kε . The ‘Cholesky’ decomposition of εV is, e.g. for Nε = 6 or for 6×6 ε ρ,

εV = εV ik =

εV 11 0 0 0 0 0εV 21

εV 22 0 0 0 0εV 31

εV 32εV 33 0 0 0

εV 41εV 42

εV 43εV 44 0 0

εV 51εV 52

εV 53εV 54

εV 55 0εV 61

εV 62εV 63

εV 64εV 65

εV 66

where εV ik is complex in general but εV ii = real ≥ 0. There are 6 real diagonal elementsand 15 complex off-diagonal elements of εV , for a total of 36 parameters required todescribe 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 εV ik = 0,i ≥ k ≥ 3. In this case, there are 2 real diagonal elements and 9 complex off-diagonalelements, for a total of 20 parameters in the problem.


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Now go back to the notation χ m and χ ′ m′

ε ρχ χ ′mm′ =

Kε

∑k=1

εV χmk

εV χ ′ ∗m′ k , m ≥ 0, m′ ≥ 0

and define

εUk(τ) =Nε

∑χ m

εV χmk

ε Dχm(τ), m ≥ 0

The distribution function

I(τ) ∝2

∑ε

Nε

∑χmχ ′m′

ε ρχ χ ′mm′

ε Dχm(τ) ε Dχ ′ ∗

m′ (τ)

becomes

I(τ) ∝2

∑ε

Kε

∑k=1

∣

∣

εUk(τ)∣

∣

2


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Experimental Moments

Let R(α,β ,γ) describe the orientation of the 3π system in the XRF. Compare theunnormalized experimental moments

Hx(LMN) =n

∑i

DLMN (Ri)

with the predicted ones

Hx(LMN) =∫

η(R) I(R)DLMN(R)dR

The body-fixed z axis is chosen as the normal to the decay plane. This is done for eachregion of the Dalitz plot.


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Strangeonium Excitations

K−

η′2(2150)K±

KS

π∓

ǫ = +1, K∗+(892), K∗+2 (1430)

ǫ = −1, K+, K+1 (1270), K+

1 (1400)

p Λ

K−s

u

p

uud

η′2(2150) s

s

Λ

sud

s ↑↑u


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Strangeonium Excitations

K−

η′2(2150)K±

KS

π∓

ǫ = +1, K∗+(892), K∗+2 (1430)

ǫ = −1, K+, K+1 (1270), K+

1 (1400)

p Λ

K−s

u

p

uud

η′2(2150) s

s

Λ

sud

s ↑↑u S. U. Chung:‘C- and G-parity: . . .’ Cparity4.pdf. . . KKπ . . .pwafm0.pdf, kkbarpi1.pdf


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Maximum-Likelihood MethodIntroduce the so-called extended likelihood function for finding ‘n’ events in a given

mass bin

L ∝[

nn

n!e−n

] n

∏i

[

I(τi)∫

I(τ) η(τ) φ(τ) dτ

]

no phase-space factors inI(τi)

where η(τ) is the experimental finite acceptance at τ and the invariant phase-spaceelement given by

dφ =

(

dφdτ

)

dτ = φ(τ)dτ

The first bracket in L represents the Poisson probability for finding ‘n’ events in the massbin, and the expectation value n is

n ∝∫


The likelihood function L can now be written, dropping the factors depending on n alone,

L ∝

[

n

∏i

I(τi)

]

exp

[

−∫

I(τ) η(τ) φ(τ) dτ]


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The ‘log’ of the likelihood function now has the form,

lnL =n

∑i

ln I(τi) −∫


We shall adopt the following shorthand notation

α = εk;χ m and α ′ = εk;χ ′ m′

and writeI(τ) = ∑

αα ′Vα V ∗

α ′ Dα (τ) D∗α ′ (τ)

The so-called ‘experimental’ normalization integral is given by

Ψxα α ′ =

∫

[

Dα (τ) D∗α ′ (τ)

]

η(τ) φ(τ) dτ

so that

lnL =n

∑i

ln[

∑αα ′

Vα V ∗α ′ Dα (τi) D∗

α ′ (τi)]

− ∑αα ′

Vα V ∗α ′ Ψx

α α ′


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We explore the normalization for V ’s by setting V = xW ,where x is independent of α,

lnL =n

∑i

ln[

x2 ∑αα ′

Wα W ∗α ′ Dα (τi) D∗

α ′ (τi)]

− x2 ∑αα ′

Wα W ∗α ′ Ψx

α α ′

At the maximum, we should have

0 =∂ lnL

∂x2 =n

∑i

[ 1x2

]

− ∑αα ′

Wα W ∗α ′ Ψx

α α ′

so that

∑αα ′

Vα V ∗α ′ Ψx

α α ′ = n

We can define the theoretical normalization integral, with η(τ) = 1,

Ψα α ′ =∫

[

Dα (τ) D∗α ′ (τ)

]

φ(τ) dτ


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The predicted number of events is

N = ∑αα ′

Vα V ∗α ′ Ψα α ′

≡ ∑αα ′

Nαα ′ , Nαα ′ = Vα V ∗α ′ Ψα α ′

So the predicted number of events for a partial wave α is

Nαα = |Vα |2 Ψα α , Ψα α =∫

∣

∣Dα (τ)∣

∣

2 φ(τ) dτ

The predicted number of events for the interference between the partial waves α and α ′

is

Nαα ′ +Nα ′α = 2ℜ

Vα V ∗α ′ Ψα α ′

, α 6= α ′


an elementary introduction to partial-wave analyses at bnl …bes.ihep.ac.cn/conference/pwa/am...

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