Angular distributions of the polarized photons and electron in the decays of the ^sup 3^D^sub 3^ state of charmonium

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

We calculate the combined angular-distribution functions of the polarized photons (...... and ......) and electron (......) produced in the cascade process ......, when the colliding ...... and ...... are unpolarized. Our results are independent of any dynamical models and are expressed in terms of the spherical harmonics whose coefficients are functions of the angular-momentum helicity amplitudes of the individual processes. Once the joint angular distribution of (......, ......) and that of (......, ......) with the polarization of either one of the two particles are measured, our results will enable one to determine the relative magnitudes as well as the relative phases of all the angular-momentum helicity amplitudes in the radiative decay processes ...... and .......

Eur. Phys. J. C (2014) 74:2723DOI 10.1140/epjc/s10052-014-2723-8

Regular Article - Theoretical Physics

Angular distributions of the polarized photons and electron in the decays of the 3 D3 state of charmonium

Alex W. K. Moka, Cheuk-Ping Wongb, Wai-Yu SitcDepartment of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

Received: 2 October 2013 / Accepted: 23 December 2013 / Published online: 4 February 2014 The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract We calculate the combined angular-distribution functions of the polarized photons (1 and 2) and electron (e) produced in the cascade process p p 3D3

3 P2 +1 ( +2)+1 (e+ +e)+1 +2, when the

colliding p and p are unpolarized. Our results are indepen

dent of any dynamical models and are expressed in terms of the spherical harmonics whose coefcients are functions of the angular-momentum helicity amplitudes of the individual processes. Once the joint angular distribution of (1, 2) and

that of (2, e) with the polarization of either one of the two particles are measured, our results will enable one to determine the relative magnitudes as well as the relative phases of all the angular-momentum helicity amplitudes in the radiative decay processes 3D3 3 P2 + 1 and 3 P2 + 2.

1 Introduction

Recently there has been great interest in charmonium spectroscopy above the open charm D D threshold of 3.73 GeV

[14]. Although the mass of the unobserved 13D3 state of charmonium is expected to lie slightly above the charm threshold [2], its Zweig-allowed strong decay to D D is sup

pressed by the F-wave angular-momentum barrier [5,6].

The total strong width of 13D3 is predicted to be just 0.5 MeV [1] and therefore other decay modes such as 3 P2 and J/ may be observable [3]. The measurement of the angular distributions in these prominent radiative and hadronic decays of the charmonium 13D3 state can provide valuable information on the true dynamics of the charmonium system above the charm threshold. In fact, the observation of the radiative decays of the charmonium states below and above the charm threshold is an important component of the planned PANDA experiments at FAIR

a e-mail: [email protected]

b e-mail: [email protected]

c e-mail: [email protected]

[7,8], which study charmonium spectroscopy in p p annihi

lation.

In our previous paper [9], it is shown that by measuring the combined angular distribution of the two photons (1, 2)

and that of the second photon and electron (2, e), regardless of their polarizations, in the sequential decay process originating from unpolarized p p collisions, namely, p p

3 D3 3 P2 + 1 ( + 2) + 1 (e+ + e) + 1 + 2,

one can extract the relative magnitudes as well as the cosines of the relative phases of all the angular-momentum helicity amplitudes in the radiative decay processes 3D3 3 P2 +1

and 3 P2 + 2. The sines of the relative phases of these

helicity amplitudes, however, cannot be determined uniquely. By including the measurement of the polarization of one of the decay particles, one may also obtain unambiguously the sines of the relative phases [10,11]. So in this paper we calculate the combined angular distributions of the nal particles (1, 2 and e) with the determination of the polarization of one particle in the above cascade process when p and p are

unpolarized.

In general, the helicity amplitudes are complex and their relative phases are nontrivial. It is important to obtain them from experiments because we can then learn about the true dynamics of the charmonium system from the decays of the charmonium states. Once the combined angular distribution of 1, 2 and e and the polarization of any one of the particles in unpolarized p p collisions are experimentally

measured, our expressions will enable one to calculate the relative magnitudes as well as the relative phases of all the angular-momentum helicity amplitudes in the two radiative decay processes 3D3

3 P2 + 1 and 3 P2 + 2.

As our calculation is based only on the general principles of quantum mechanics and symmetry, our results are independent of any dynamical models. In addition, our results on the partially integrated angular distributions where the combined angular-distribution function of 1, 2 and e is integrated over the direction of one of the three particles are quite interesting. They show that by measuring the two-

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2723 Page 2 of 13 Eur. Phys. J. C (2014) 74:2723

particle angular distribution of (1, 2) and that of (2, e) with the polarization of either one of the two particles, one can also get complete information on the helicity amplitudes.

The format of the rest of this paper is as follows. In Sect. 2, we give the calculations for the combined angular distribution with polarization determination of the electron and of the two photons in the cascade process p p 3D3 3 P2 + 1

( + 2) + 1 (e+ + e) + 1 + 2, when p and p

are unpolarized. We then show how the measurement of this joint angular distribution of polarized 1, 2 and e enables us to obtain complete information on the helicity amplitudes in the two radiative decay processes 3D3 3 P2 + 1 and

3 P2 + 2. We also present three different results for

the combined angular distribution, in which the polarization of only one of the three particles, 1, 2 and e, is measured.

In Sect. 3, we present the results for the partially integrated angular distributions in different cases where the combined angular-distribution function of the three particles is integrated over the direction of one particle. These results can all be expressed in terms of the orthogonal spherical harmonic functions. We point out how the measurement of these two-particle angular distributions will again give complete information on all the helicity amplitudes in the two radiative decay processes. Finally, in Sect. 4, we make some concluding remarks.

2 The polarized angular distributions of the photons and electron

We consider the cascade process, p(1) + p(2) 3D3() 3 P2()+1() [()+2()]+1() [e(1)+

e+(2)] + 1() + 2(), in the 3D3 rest frame or the p p

c.m. frame. The Greek symbols in the brackets represent the helicities of the particles except , which represents the z component of the angular momentum of the stationary 3D3

resonance. We choose the z axis to be the direction of motion of 3 P2 in the 3D3 rest frame. The x and y axes are arbitrary and the experimentalists can choose them according to their convenience. A symbolic sketch of the cascade process is shown in Fig. 1.

Following the conventions of our previous paper [9], the probability amplitude for the cascade process can be expressed in terms of the Wigner D-functions and the angular-momentum helicity amplitudes B12, A, E and

C12 for the individual sequential processes as

T 1212 =

715

162 C12 B12

Fig. 1 Symbolic sketch of p(1) + p(2) 3 D3() 3 P2() +

1() [() + 2()] + 1() [e(1) + e+(2)] + 1() +

2() showing different angles of the decay particles

In the D-functions, the angles (, ) giving the direction of

p, the angles ( , ) giving the direction of and the angles ( , ) giving the direction of e are measured in the 3D3, the 3 P2 and the rest frames, respectively. The angles of each decay particle observed in different rest frames can be calculated using the Lorentz transformation. The equations relating these angles are given in [12].

Because of the C and P invariances, the angular-momentum helicity amplitudes in (1) are not all independent. We have

B12

P

= B12,

B12

C

= B21,

A

P

= A,

E

(2)

P

= E,

C12

P

= C12,

C12

C

= C21.

Making use of the symmetry relations of (2), we now re-label the independent angular-momentum helicity amplitudes as follows:

B0 = 2B

1

2

33 [summationdisplay]

11 [summationdisplay]

1

2 , B1 = 2B

1

2

A+,E

D3(, , )D2+,( , , )

D1( , , ). (1)

12 ,

Ai = Ai2,1 = A2i,1 (i = 0, 1, 2, 3, 4), E j = E j1,1 = E1j,1 ( j = 0, 1, 2),

C0 = 2C

1

2

1

2 , C1 = 2C

1

2

12 . (3)

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Eur. Phys. J. C (2014) 74:2723 Page 3 of 13 2723

We will also make use of the following normalizations:

|B0|2 + |B1|2 = |C0|2 + |C1|2 = 1 (4) and

[summationdisplay]

i=0

4 |Ai|2 =

+ J1J2d [bracketleftBig] J3J2d +(D1 D1 D2 + D2)

+ J3J2d (D1 + D1 D2 D2)[bracketrightBig] , , (10)

J1J2d =

35

2 [summationdisplay]

s(d)

[parenleftbigg]A s+d

2 Asd

2

[parenrightbigg]

As+d2 A

sd

2

2 |E j |2 = 1 (5)

When p and p are unpolarized, the normalized function

describing the combined angular distribution of the electron and the two photons whose polarizations are also observed can be written as

W1(, ; , ; , )

= N

12

[summationdisplay]

1,2

s d 4

2 |J2; d

[angbracketrightbigg]

[angbracketleftbigg]22;

s + d 4

2 ,

s + d 6

2 ,

[angbracketrightbigg] , (11)

[angbracketleftbigg]33;

s d 6

2 |J1; d

J3J2d =

15

2 [summationdisplay]

[parenrightbigg]

Es +d

2

E s d

2

12

[summationdisplay] 2

s (d )

[parenleftbigg]E s +d

2 Es d

2

T 1212T 1212, (6)

where the subscripts , and 1 of W represent the polarizations of 1, 2 and e, respectively. The normalization constant N in (6) is determined by requiring that the integral of the angular-distribution function W1(, ; , ; , )

over all the directions of 1, 2 and e or over all the angles, (, ; , ; , ), is 1. In (6) we sum over the helicities

2 since e+ is not observed. Substituting (1) into (6) and performing the various sums will then give an expression for the angular-distribution function W1(, ; , ; , )

in terms of the Wigner D-functions. After very long algebra, we get

W1(, ; , ; , )

=

1 (4)3

[angbracketrightbigg]

[angbracketleftbigg]22;

s + d

2 ,

s d

2 |J2; d

[angbracketleftbigg]11;

s + d 2

2 ,

s d 2

2 |J3; d

[angbracketrightbigg] , (12)

0,2,4,6

[summationdisplay]

J1

B J1

2

[summationdisplay]

J3=0

C J31

J2+[parenleftBig] 1

2

4

[summationdisplay]

J2=0

J3

(1)

[parenleftBig] 2+

2

D1 = DJ1d,0(, , )DJ2d,d ( , , )

DJ3d ,0( , , ), (13)

D2 = DJ1d,0(, , )DJ2d,d ( , , )

DJ3d ,0( , , ), (14)

s(d) = |d|, |d| + 2, . . . , 8 |d|, (15) s (d ) = |d |, |d | + 2, . . . , 4 |d |, (16)

dm = min{4, J1, J2}, (17) d m = min{2, J2, J3}. (18)

The Wigner D-functions in (13) and (14) are given by [13]

D jm,m (, , ) = j, m|R(, , )| j, m (19) where , , are Euler angles and the rotation operator R(, , ) can be written as

R(, , ) = eiJzeiJyei Jz. (20)

The explicit expressions for all the coefcients in (7) are given in Appendix A. Making use of the orthogonal relation of the Wigner D-functions,

2

[integraldisplay]

0

d

dm

[summationdisplay]

d=0

d m

[summationdisplay]

d =0

[parenleftbigg]1

d0

2

[parenrightbigg][parenleftbigg]1

d 0 2

[parenrightbigg] I J1J2J3dd (7)

where

B J1 = 7

1 (1) 33; , |J1; 0 |B|2, (8)

C J31 = (1)

1+ 12 [parenrightBig]J33

2

[integraldisplay]

0

d

[integraldisplay]

0

D jmm (, , )D j

sin d

1 (1) 11; , |J3; 0 |C|2, (9)

I J1J2J3dd = J1J2d+ [bracketleftBig] J3J2d +(D1 + D1 + D2 + D2)

+ J3J2d (D1 D1 + D2 D2)[bracketrightBig]

=

822 j + 1

mm j j , (21)

we can obtain these coefcients as

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2723 Page 4 of 13 Eur. Phys. J. C (2014) 74:2723

(1)

J2

2 (2+)+ J32 (1) B J1C J31[braceleftbigg] J1J2d+

[bracketleftBig] J3J2d + + J3J2d (1 d

=

1 4

1

[summationdisplay]

12

[summationdisplay] 1

W1(, ; , ; , )

=

0)[bracketrightBig]

+ J1J2d [bracketleftBig] J3J2d +(1 d0) + J3J2d (1 d0)(1 d

0)[bracketrightBig] [bracerightbigg]

1

2(4)3

0,2,4,6

[summationdisplay]

J1

B J1

0,2

[summationdisplay] J3

C J3

4

[summationdisplay]

J2=0

J2

(1)

[parenleftBig] 1+

2

2 [parenrightbigg][braceleftbigg] J1J2d+

[braceleftBig] J3J2d +[1 + (1)J2](D 1 + D 1 + D 2 + D 2)

+ J3J2d [1 (1)J2](D 1 D 1 + D 2 D 2)[bracerightBig]

+ J1J2d[braceleftBig] J3J2d +[1 + (1)J2]

(D 1 D 1 D 2 + D 2)

+ J3J2d [1 (1)J2](D 1 + D 1 D 2 D 2)[bracerightBig]

[bracerightbigg] (23)

= (2J1 + 1)(2J2 + 1)(2J3 + 1)

[integraldisplay] W1(, ; , ; , )D1d d d . (22)

When we have sufcient experimental data for the angular-distribution function W, where the nal polarizations, , and , of all the three decay particles are measured, the integral on the right side of (22) can be determined numerically for all possible allowed values of J1, J2, J3, d

and d . Thus we can extract the different coefcients B J1, C J31, J1J2d and J3J2d on the left side of (22). From these

coefcients we can determine the relative magnitudes of the A, B, C and E helicity amplitudes as well as the cosines and sines of the relative phases of the A and E helicity amplitudes in the radiative decay processes 3D3

3 P2 + 1

and 3 P2 + 2, respectively. Let us illustrate more

clearly how the measurements of (J1 J2 J3dd ) coefcients can give all the information. First, the measurement of the (00100) and (00200) coefcients yields 100+ and 200+, and,

with the normalization |E0|2 + |E1|2 + |E2|2 = 1, the rel

ative magnitudes of E j are determined. Next the measurements of (01000), (02000), (03000) and (04000) coefcients yields 010+, 020+, 030+ and 040+, and with the normalization

|A0|2 +|A1|2 +|A2|2 +|A3|2 +|A4|2 = 1, the relative mag

nitudes of Ai are determined. Measuring (20000) yields B2 and with the normalization |B0|2 + |B1|2 = 1, the relative

magnitudes of B0 and B1 are obtained. The measurement of (00200) yields C2

dm

[summationdisplay]

d=0

d m

[summationdisplay]

d =0

[parenleftbigg]1

d0

2

[parenrightbigg][parenleftbigg]1

d 0

where

D 1 = D1( = 1) = DJ1d,0DJ2d,d DJ3d ,0 (24) and

D 2 = D2( = 1) = DJ1d,0DJ2d,d DJ3d ,0. (25) As J3 can only take the values 0 and 2 in (23), we have dened

C J3 = C J312 =

3

1 (1) 11; , |J3; 0 |C|2

(J3 = 0, 2). (26)

The coefcients of the Wigner D-functions in (23) can be obtained from

(1)

J2

2 (1+) B J1C J3[braceleftbigg] J1J2d+[braceleftBig] J3J2d +[1 + (1)J2]

+ J2J3d [1 (1)J2](1 d

12 and with the normalization |C0|2+|C1|2, the relative magnitudes of C0 and C1 are determined. After having obtained all the relative magnitudes, now measuring the (02101) and (02201) coefcients yields Re(E1E0),

Re(E2E1), Im(E1E0) and Im(E2E1). Hence the cosines and sines of the relative phases of E j are determined. Finally, by measuring the (22010), (24010), (42010) and (44010) coefcients, we can obtain the cosines and sines of the relative phases of Ai.

By summing over one or two helicity indices (, and 1) of (7), we can easily obtain different combined angular-distribution functions where the polarizations of only one or two decay products (1, 2 and e) are measured. Suppose we are interested in only measuring the polarization of 1,

the normalized combined angular distribution of 1, 2 and

e will then become

W(, ; , ; , )

0)[bracerightBig]

+ J1J2d[braceleftBig] J3J2d +[1 + (1)J2](1 d0)

+ J3J2d [1 (1)J2](1 d0)(1 d

0)[bracerightBig][bracerightbigg]

= 2(2J1 + 1)(2J2 + 1)(2J3 + 1)

[integraldisplay] W(, ; , ; , )D 1d d d . (27)

Similarly, if we only measure the polarization of 2, the normalized combined angular distribution of 1, 2 and e

will become

W(, ; , ; , )

=

1 4

1

[summationdisplay]

12

[summationdisplay] 1

W1(, ; , ; , )

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Eur. Phys. J. C (2014) 74:2723 Page 5 of 13 2723

=

1

2(4)3

0,2,4,6

[summationdisplay]

J1

B J1

0,2

[summationdisplay] J3

C J3

4

[summationdisplay]

J2=0

(1)

[parenleftBig] 1

2 (J2+J3)

d m

[summationdisplay]

d =0

[parenleftbigg]1

2 [parenrightbigg] [braceleftBig] J1J2d+

[1 + (1)J2] [bracketleftBig] J3J2d +(D 1 + D 1 + D 2 + D 2)

+ J3J2d (D 1 D 1 + D 2 D 2)[bracketrightBig] J1J2d

[1 (1)J2] [bracketleftBig] J3J2d +(D 1 D 1 D 2 + D 2)

+ J3J2d (D 1 + D 1 D 2 D 2)[bracketrightBig][bracerightBig] (28)

where

D 1 = D1( = 1) = DJ1d,0DJ2d,d DJ3d ,0 (29) and

D 2 = D2( = 1) = DJ1d,0DJ2d,d DJ3d ,0. (30)

The coefcients of the angular functions in (28) can be obtained from

(1)

[parenleftBig] 1

2 (J2+J3) B J1C J3[braceleftBig] J1J2d+[1 + (1)J2]

[bracketleftBig] J3J2d + + J3J2d (1 d

where

D 1 = D1( = = 1) = DJ1d,0DJ2d,d DJ3d ,0 (33)

and

D 2 = D2( = = 1) = DJ1d,0DJ2d,d DJ3d ,0. (34)

The coefcients in (32) can be obtained from

B J1C J31[braceleftbigg] J1J2d+[1 + (1)J2][braceleftBig] J3J2d +[1 + (1)J3]

+ J3J2d [1 (1)J3](1 d

dm

[summationdisplay]

d=0

d0

2

[parenrightbigg][parenleftbigg]1

d 0

0)[bracerightBig] + J1J2d

[1 (1)J2][braceleftBig] J3J2d +[(1)J3 1](1 d0)

J3J2d [1 + (1)J3](1 d0)(1 d

0)[bracerightBig][bracerightbigg]

= 4(2J1 + 1)(2J2 + 1)(2J3 + 1)

[integraldisplay] W1(, ; , ; , )D 1d d d . (35)

Here, J3 can take the values 0, 1 and 2.

If we now average over the polarizations 1 of e in (32) as well, we get

W(, ; , ; , )

=

1

2

0)[bracketrightBig] J1J2d [bracketleftBig]1 (1)J2[bracketrightBig]

[bracketleftBig] J3J2d +(1 d0) + J3J2d (1 d0)(1 d

0)[bracketrightBig][bracerightBig]

12

[summationdisplay] 1

W1(, ; , ; , )

=

= 2(2J1 + 1)(2J2 + 1)(2J3 + 1)

[integraldisplay] W(, ; , ; , )D 1d d d (31)

where again J3 can only take the values 0 and 2.

If we are interested in only measuring the polarization 1 of e, the combined angular distribution of 1, 2 and e

will become

W1(, ; , ; , )

=

1 4

1

2(4)3

0,2,4,6

[summationdisplay]

J1

B J1

0,2

[summationdisplay] J3

C J3

4

[summationdisplay]

J2=0

dm

[summationdisplay]

d=0

d m

[summationdisplay]

d =0

[parenleftbigg]1

d0

2

[parenrightbigg]

2 [parenrightbigg] [braceleftBig] J1J2d+ J3J2d + [bracketleftBig]1 + (1)J2[bracketrightBig]

+ J1J2d J3J2d [bracketleftBig]1 (1)J2[bracketrightBig][bracerightBig]

[bracketleftBig](1)J2(D 1 + D 1) + (D 2 + D 2)[bracketrightBig] . (36)

Using (10) and (11), we have

[parenleftbigg]1

d0

2

[parenleftbigg]1

d 0

1

[summationdisplay] 1 W1(, ; , ; , )

=

1 4(4)3

0,2,4,6

[summationdisplay]

J1

C J31

4

[summationdisplay]

J2=0

dm

[summationdisplay]

d=0

d m

[summationdisplay]

d =0

[parenleftbigg]1

d0

2

2 [parenrightbigg] [braceleftBig] J1J2d+ J3J2d +

[bracketleftBig]1 + (1)J2[bracketrightBig] + J1J2d J3J2d [bracketleftBig]1 (1)J2[bracketrightBig][bracerightBig]

=

J1J2d J3J2d

2 (37)

where

J1J2d = 35

[parenrightbigg]

[parenrightbigg][parenleftbigg]1

d 0

B J1

2

[summationdisplay]

J3=0

2 [parenrightbigg][braceleftbigg] J1J2d+[1 + (1)J2][braceleftBig] J3J2d +

[1 + (1)J3](D 1 + D 1 + D 2 + D 2) + J3J2d

[1 (1)J3](D 1 D 1 + D 2 D 2)[bracerightBig]

+ J1J2d[1 (1)J2][braceleftBig] J3J2d +[(1)J3 1]

(D 1 D 1 D 2 + D 2) J3J2d [1 + (1)J3]

(D 1 + D 1 D 2 D 2)[bracerightBig]

[bracerightbigg] (32)

[parenleftbigg]1

d 0

[parenleftbigg]1

d0

2

[parenrightbigg]

[summationdisplay]

s(d)

[bracketleftbigg]A s+d

2 Asd

2

[bracketrightbigg]

+ (1)J2 As+d2 A

sd

2

[angbracketleftbigg]22;

s + d 4

2 ,

s d 4

2 |J2; d

[angbracketrightbigg]

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2723 Page 6 of 13 Eur. Phys. J. C (2014) 74:2723

[angbracketleftbigg]33;

s + d 6

2 ,

s d 6

2 |J1; d

[angbracketrightbigg] , (38)

Case 1 We integrate over the angles ( , ) or the direction of e and then average over the polarization 1 of e. The combined angular distribution of 1 and 2, and the polarization of only one of the two particles are measured. The explicit expressions are given in the following.

Only is measured:

[tildewide][tildewide]

W(, ; , )

=

1 4

J3J2d = 15

[parenleftbigg]1

d 0 2

[parenrightbigg]

[summationdisplay]

s (d )

[bracketleftbigg]E s +d

2 Es d

2

[bracketrightbigg]

+ (1)J2 Es +d

2

E s d

2

s + d

2 ,

s d

2 |J2; d

[angbracketrightbigg]

[angbracketleftbigg]22;

[angbracketrightbigg] . (39)

By combining (37) and (36), we now recover our results in [9], where the polarizations of the decay particles are not measured.

Using (27), (31) or (35) it can be seen that once the combined angular distribution W(, ; , ; , ),

W(, ; , ; , ) or W1(, ; , ; , ) is mea

sured, one can also get the same information on the helicity amplitudes as one obtained from measuring the angular distribution function W1(, ; , ; , ) where the

polarizations of the three particles 1, 2 and e are observed. In other words, by measuring the combined angular distribution of the decay particles 1, 2 and e and the polarization of any one particle, we can get complete information on the helicity amplitudes in the radiative decay processes

3 D3 3 P2 + 1 and 3 P2 + 2. In addition, we can

also get the relative magnitudes of the helicity amplitudes in the production process p p 3D3 and in the nal decay

process e+e.

3 Partially integrated angular distributions

The partially integrated angular distributions obtained from (7) will look a lot simpler and we will gain greater insight from them. There are three different cases in which the polarization and the angular distribution of only one particle (1,

2 or e) are measured. We nd that these results are identical to the single-particle angular-distribution functions given in [9], where the polarizations of the individual particles are not measured. So including the measurement of the polarizations in the single-particle angular distributions does not give any further information. However, we will nd that the measurement of the polarizations of the decay particles can provide us more information on the helicity amplitudes when we measure the simultaneous angular distributions of two particles. We now consider three different cases of two-particle angular distributions. We will express our results in terms of the spherical harmonics by using the following relation:

DJM0 =

[radicalbigg] 4

[angbracketleftbigg]11;

s + d 2

2 ,

s d 2

2 |J3; d

1

12

[summationdisplay] [summationdisplay] [integraldisplay] W1(, ; , ; , )d

=

1 4

[braceleftbigg]

14 +

15 B0020+ 020+Y00(, )Y20( , )

+

13 B0040+ 040+Y00(, )Y40( , )

+

15 B2200+ 000+Y20(, )Y00( , )

+

15 B2220+ 020+Y20(, )Y20( , )

+

25 [bracketleftBig]B2221+ 020+ Re(Y2,(, )Y 2,( , ))

i B2221 020+ Im(Y2,(, )Y 2,( , ))

+ B2222+ 020+ Re(Y2,2(, )Y 2,2( , ))

i B2222 020+ Im(Y2,2(, )Y 2,2( , ))[bracketrightBig]

1

+ 35 B2240+ 040+Y20(, )Y40( , )

+ 35 [bracketleftBig]B2241+ 040+ Re(Y2,(, )Y 4,( , ))

i B2241 040+ Im(Y2,(, )Y 4,( , ))

+ B2242+ 040+ Re(Y2,2(, )Y 4,2( , ))

i B2242 040+ Im(Y2,2(, )Y 4,2( , ))[bracketrightBig]

+

13 B4400+ 000+Y40(, )Y00( , )

2

1

+ 35 B4420+ 020+Y40(, )Y20( , )

2

+ 35 [bracketleftBig]B4421+ 020+ Re(Y4,(, )Y 2,( , ))

i B4421 020+ Im(Y4,(, )Y 2,( , ))

+ B4422+ 020+ Re(Y4,2(, )Y 2,2( , ))

i B4422 020+ Im(Y4,2(, )Y 2,2( , ))[bracketrightBig]

+

19 B4440+ 040+Y40(, )Y40( , )

+

29 [bracketleftBig]B4441+ 040+ Re(Y4,(, )Y 4,( , ))

i B4441 040+ Im(Y4,(, )Y 4,( , ))

+ B4442+ 040+ Re(Y4,2(, )Y 4,2( , ))

2J + 1

Y J M. (40)

123

Eur. Phys. J. C (2014) 74:2723 Page 7 of 13 2723

i B4442 040+ Im(Y4,2(, )Y 4,2( , ))

+ B4443+ 040+ Re(Y4,3(, )Y 4,3( , ))

i B4443 040+ Im(Y4,3(, )Y 4,3( , ))

+ B4444+ 040+ Re(Y4,4(, )Y 4,4( , ))

i B4444 040+ Im(Y4,4(, )Y 4,4( , ))[bracketrightBig]

+

113 B6600+ 000+Y60(, )Y00( , )

+

+ B2222+ 020+ Re(Y22(, )Y 22( , ))[bracketrightBig]

+

2i(1)

35 [bracketleftBig]B2231 030+ Im(Y21(, )Y 31( , ))

+ B2232 030+ Im(Y22(, )Y 23( , ))[bracketrightBig]

1

+ 35 B2240+ 040+Y20(, )Y40( , )

12 (1)

2

165 B6620+ 020+Y60(, )Y20( , )

+

+ 35 [bracketleftBig]B2241+ 040+ Re(Y21(, )Y 41( , ))

+ B2242+ 040+ Re(Y22(, )Y 42( , ))[bracketrightBig]

+

13 B4400+ 000+Y40(, )Y00( , )

+

265 [bracketleftBig]B6621+ 020+ Re(Y6,(, )Y 2,( , ))

i B6621 020+ Im(Y6,(, )Y 2,( , ))

+ B6622+ 020+ Re(Y6,2(, )Y 2,2( , ))

i B6622 020+ Im(Y6,2(, )Y 2,2( , ))[bracketrightBig]

1

+ 313 B6640+ 040+Y60(, )Y40( , )

2i(1)

12 (1)

33 B4411 010+ Im(Y41(, )Y 11( , ))

1

+ 35 B4420+ 020+Y40(, )Y20( , )

+ 35 [bracketleftBig]B4421+ 020+ Re(Y41(, )Y 21( , ))

+ B4422+ 020+ Re(Y42(, )Y 22( , ))[bracketrightBig]

+

2i(1)

2

+ 313 [bracketleftBig]B6641+ 040+ Re(Y6,(, )Y 4,( , ))

i B6641 040+ Im(Y6,(, )Y 4,( , ))

+ B6642+ 040+ Re(Y6,2(, )Y 4,2( , ))

i B6642 040+ Im(Y6,2(, )Y 4,2( , ))

+ B6643+ 040+ Re(Y6,3(, )Y 4,3( , ))

i B6643 040+ Im(Y6,3(, )Y 4,3( , ))

+ B6644+ 040+ Re(Y6,4(, )Y 4,4( , ))

i B6644 040+ Im(Y6,4(, )Y 4,4( , ))[bracketrightBig]

[bracerightbigg]. (41)

2

12 (1)

37 [bracketleftBig]B4431 030+ Im(Y41(, )Y 31( , ))

+ B4432 030+ Im(Y42(, )Y 32( , ))

+ B4433 030+ Im(Y43(, )Y 33( , ))[bracketrightBig]

+

19 B4440+ 040+Y40(, )Y40( , )

+

Only is measured:

29 [bracketleftBig]B4441+ 040+ Re(Y41(, )Y 41( , ))

+ B4442+ 040+ Re(Y42(, )Y 42( , ))

+ B4443+ 040+ Re(Y43(, )Y 43( , ))

+ B4444+ 040+ Re(Y44(, )Y 44( , ))[bracketrightBig]

+

113 B6600+ 000+Y60(, )Y00( , )

+

W (, ; , )

=

1 4

[tildewide][tildewide]

1

12

[summationdisplay] [summationdisplay] [integraldisplay] W1(, ; , ; , )d

2i(1)

12 (1)

39 B6611 010+ Im(Y61(, )Y 11( , ))

+

[braceleftbigg]

=

1 4

14 +

15 B0020+ 020+Y00(, )Y20( , )

+

165 B6620+ 020+Y60(, )Y20( , )

+

265 [bracketleftBig]B6621+ 020+ Re(Y61(, )Y 21( , ))

+ B6622+ 020+ Re(Y62(, )Y 22( , ))[bracketrightBig]

+

2i(1)

13 B0040+ 040+Y00(, )Y40( , )

+

15 B2200+ 000+Y20(, )Y00( , )

+

112 [bracketleftBig]B6631 030+ Im(Y61(, )Y 31( , ))

+ B6632 030+ Im(Y62(, )Y 32( , ))

+ B6633 030+ Im(Y63(, )Y 33( , ))[bracketrightBig]

2i(1)

12 (1)

15 B2211 010+ Im(Y21(, )Y 11( , ))

+

12 (1)

15 B2220+ 020+Y20(, )Y20( , )

+

25 [bracketleftBig]B2221+ 020+ Re(Y21(, )Y 21( , ))

123

2723 Page 8 of 13 Eur. Phys. J. C (2014) 74:2723

1

+ 313 B6640+ 040+Y60(, )Y40( , )

Only 1 is measured:

2

+ 313 [bracketleftBig]B6641+ 040+ Re(Y61(, )Y 41( , ))

+ B6642+ 040+ Re(Y62(, )Y 42( , ))

+ B6643+ 040+ Re(Y63(, )Y 43( , ))

+ B6644+ 040+ Re(Y64(, )Y 44( , ))[bracketrightBig]

W1( , ; , )

=

1 4

1

1

[tildewide][tildewide] [summationdisplay] [summationdisplay] [integraldisplay] W1(, ; , ; , )d

=

[bracerightbigg]. (42)

An inspection of (41) and (42) shows that the magnitudes of the A, B and E helicity amplitudes as well as the cosines and sines of the relative phases of the A helicity amplitudes can be extracted from the measurement of either

[tildewide][tildewide]

1 4

[braceleftbigg]

14 +

15C2000+ 200+Y00( , )Y20( , )

+

15C0020+ 020+Y20( , )Y00( , )

+

2i15C11020+ 121Im(Y21( , )Y11( , ))

+

W . It

should be noted that the measurement of the polarization of one of the decay particles is essential for getting the sines of the relative phases among the A helicity amplitudes. This is not possible for the unpolarized case.

Case 2 We integrate over (, ) or the direction of 1 and average over the polarization of 1. The combined angular distribution of 2 and e and the polarization of either one of them are measured. The expressions are given in the following.

Only is measured:

[tildewide][tildewide]

W ( , ; , )

=

1 4

W or [tildewide][tildewide]

15C2020+ 220+Y20( , )Y20( , )

25C2020+ 221+Re(Y21( , )Y21( , ))

+

25C2020+ 222+Re(Y22( , )Y22( , ))

+

13C0040+ 040+Y40( , )Y00( , )

2i

+ 33C11040+ 141Im(Y41( , )Y11( , ))

1

+ 35C2040+ 240+Y40( , )Y20( , )

2

35C2040+ 241+Re(Y41( , )Y21( , ))

1

12

[bracerightbigg]. (44)

An examination of (43) and (44) shows that we can obtain the magnitudes of the E helicity amplitudes as well as both the cosines and the sines of the relative phases of the E helicity amplitudes when the simultaneous angular distribution of 2 and e with the polarization of either one particles is measured. As in case 1, the measurement of the polarization is essential for getting the sines of the relative phases of these helicity amplitudes uniquely. It is worth noting that we can now obtain all the information on the helicity amplitudes from the measurement of the joint angular distributions of only two particles.

Case 3 We integrate over ( , ) or the direction of 2 and then average over the polarization of 2. The combined angular distribution of 1 and e and the polarization of either one of them are measured. Since we cannot obtain any useful information from this case, we do not provide the long expressions here.

4 Concluding remarks

We have derived the model-independent expressions for the combined angular distribution of the nal photons (1 and 2)

[summationdisplay] [summationdisplay] [integraldisplay] W1(, ; , ; , )d

=

2

+ 35C2040+ 242+Re(Y42( , )Y22( , ))

1 4

[braceleftbigg]

1 4 +

15C2000+ 200+Y00( , )Y20( , )

+

15C0020+ 020+Y20( , )Y00( , )

+

15C2020+ 220+Y20( , )Y20( , )

+

25C2020+ [bracketleftBig] 221+Re(Y2,( , )Y2,( , ))

+ i 221Im(Y2,( , )Y2,( , ))

+ 222+Re(Y2,2( , )Y2,2( , ))

i 222Im(Y2,2( , )Y2,2( , ))[bracketrightBig]

+

13C0040+ 040+Y40( , )Y00( , )

1

+ 35C2040+ 240+Y40( , )Y20( , )

2

+ 35C2040+ [bracketleftBig] 241+Re(Y4,( , )Y2,( , ))

+ i 241Im(Y4,( , )Y2,( , ))

+ 242+Re(Y4,2( , )Y2,2( , ))

i 242Im(Y4,2( , )Y2,2( , ))[bracketrightBig]

[bracerightbigg]. (43)

123

Eur. Phys. J. C (2014) 74:2723 Page 9 of 13 2723

and electron (e) in the cascade process, p + p 3D3

3 P2 + 1 ( + 2) + 1 (e+ + e) + 1 + 2, when p and p are unpolarized and the polarization of any one of the three decay particles is measured. Our expressions are based only on the general principles of quantum mechanics and the symmetry of the problem. We have also derived the partially integrated angular-distribution functions which give the two-particle angular distributions of (1, 2) and (2, e) with the measurement of the polarization of one particle in each cases. Once these polarized angular distributions are experimentally measured, our expressions can be used to extract the information of all the independent helicity amplitudes in the radiative decay processes 3D3 3 P2 + 1 and

3 P2 +2. In fact, the analysis of the angular correlations

in the nal decay products will serve to verify the presence of the intermediate 3D3 state and its J PC quantum numbers in the cascade process. The experimentally determined values of the helicity amplitudes can then be compared with the predictions of various dynamical models.

The great advantage of measuring the angular distributions with the polarization of one particle is that one can obtain not only the relative magnitudes of the helicity amplitudes but also both the cosines and the sines of the relative phases of the helicity amplitudes in the decay processes

3 D3 3 P2 + 1 and 3 P2 + 2. This is important

because the helicity amplitudes are in general complex [14].

Therefore by measuring the combined angular distribution of 1, 2 and e with the polarization of any one of the three particles, we can obtain complete information on the helicity amplitudes in the two radiative decay processes. Alternatively, we can get the same information by measuring the two-particle angular distribution of 2 and e and that of 1 and 2 with the polarization of either one of the two particles.

Both the theorists and the experimentalists would like to express their results in terms of the multipole amplitudes in the radiative transitions 3D3 3 P2 + 1 and 3 P2 +

2. The relationship between the helicity and the multipole amplitudes are given by the orthogonal transformations [15, 16]:

Ai =

transformations of (45) and (46) are orthogonal,

[summationdisplay]

i=0

4 |Ai|2 =

5 |ak|2 = 1 ,

2 |E j |2 =

3 |ek|2 = 1 .

(47)

It is noteworthy that the decay process 3D3 3 P2 + 1 has

ve independent helicity amplitudes corresponding to ve multipole amplitudes E1, M2, E3, M4 and E5. In any potential model for heavy quarkonia, the M4 or higher multipole amplitudes is zero to order v2/c2 because in this approximation there is no fourth or higher rank tensor component in the transition operator [17]. So by measuring the angular distributions, one can further test the validity of the non-relativistic potential models.

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Funded by SCOAP3 / License Version CC BY 4.0.

Appendix A: Expressions of coefcients

A.1 Expressions of B J1

B0 = 1, (48) B2 =

2 3

[parenleftbigg]

|B0|2 +

3

4|B1|2

[parenrightbigg] , (49)

B4 = 3

[radicalbigg] 2 11

[parenleftbigg]

|B0|2 +

1 6|B1|2

[parenrightbigg] , (50)

B6 =

10 33

[parenleftbigg]

|B0|2

3

4|B1|2

[parenrightbigg] . (51)

A.2 Expressions of C J31

C0

1

2

5 ak[radicalbigg]2k + 1

5 k, 1; 3, (i 1)|2, (i 2) (45)

(i = 0, 1, 2, 3, 4),

and

E j =

= C0 = 1, (52)

C1

3

2|C1|2, (53)

C2

1

2

=

= C2 = 2

[parenleftbigg]

[parenrightbigg] . (54)

A.3 Expressions of J1J2d

000+ = 1, (55)

200+ =

1

2

|C0|2

1

2|C1|2

3 ek[radicalbigg]2k + 1

5 k, 1; 1, ( j 1)|2, j

( j = 0, 1, 2), (46)

where ak and ek are the radiative multipole amplitudes in

3 D3 3 P2 + 1 and 3 P2 + 2, respectively. Since the

[parenleftbigg]

5 |A0|2

3 5|A2|2

4

5|A3|2

3 5|A4|2

[parenrightbigg] , (56)

123

2723 Page 10 of 13 Eur. Phys. J. C (2014) 74:2723

400+ =

3 22

[parenleftbigg]

|A0|2

7 3|A1|2

2462(|A0|2 + 24|A1|2

+ 90|A2|2 + 80|A3|2 + 15|A4|2), (74)

211+ =

640+ =

1

5 [bracketleftbigg]Re(A1 A0) +

310 Re(A2 A1)

+

3

5 Re(A3 A2)

2

5 Re(A4 A3)

+

13|A2|2 + 2|A3|2 +

1 3|A4|2

[parenrightbigg] , (57)

600+ =

1

233(|A0|2 6|A1|2

+15|A2|2 20|A3|2 + 15|A4|2), (58)

01

0+ =

2 [parenleftbigg]|A0|

[bracketrightBigg] , (75)

2

+

1

2 |A1|

2

2 [parenrightbigg] , (59)

1

2 |A3|

2 |A4|

411+ =

15 [bracketleftbigg]Re(A1 A0)

410 Re(A2 A1)

3

2 Re(A3 A2) +

21

0+ =

5 6

[parenleftbigg]

|A0|

2

+

25 |A3|

2

2 [parenrightbigg] , (60)

41

0+ =

+

35 |A4|

12 Re(A4 A3)

[bracketrightBigg] , (76)

3 11

[parenleftbigg]

|A0|

2

76 |A1|

2 [parenrightbigg] , (61)

2 |A3|

2

13 |A4|

61

0+ =

166 (|A0|

2 3|A1|

2 + 10|A3|

2 15|A4|

2), (62)

611+ =

1

2

[radicalbigg] 7

33

[bracketleftBigg]Re(A1 A0) 6

5

02

0+ =

10 7

[parenleftbigg]

|A0|

2

1

2 |A1|

[parenrightbigg] ,

(63)

2 Re(A2 A1)

+ 53 Re(A3 A2) 52 Re(A4 A3)[bracketrightBig] , (77)

221+ =

2 |A2|

2

1

2 |A3|

2 + |A4|

2

22

0+ =

5[radicalbigg] 5

42

[parenleftbigg]

|A0|

2

+

35 |A2|

2

2

2 [parenrightbigg] , (64)

5 [bracketleftbigg]Re(A1 A0) +

110 Re(A2 A1)

+

25 |A3|

35 |A4|

5 2

1

53 Re(A3 A2) +

2

5 Re(A4 A3)

420+ = 3

[radicalbigg] 5 77

[parenleftbigg]

[bracketrightBigg] , (78)

|A0|2 +

7 6|A1|2

13|A2|2 |A3|2 +

1 3|A4|2

[parenrightbigg] , (65)

421+ =

15 77

[bracketleftbigg]Re(A1 A0)

4

310 Re(A2 A1)

12 Re(A4 A3)

[bracketrightbigg] , (79)

1

+ 23 Re(A3 A2)

620+ =

[radicalbigg] 5

462(|A0|2 + 3|A1|2

15|A2|2 + 10|A3|2 + 15|A4|2), (66)

030+ =

621+ =

1

2

[radicalbigg] 5

11

[bracketleftBigg]Re(A1 A0)

5

2 Re(A2 A1)

12 (|A0|2 2|A1|2 + 2|A3|2 |A4|2), (67)

230+ =

2 Re(A4 A3)[bracketrightbigg] , (80)

231+ =

[parenleftbigg]

5 |A0|2

85 |A3|2 +

35 |A4|2

[parenrightbigg] , (68)

430+ =

53 Re(A3 A2) + 5

[parenleftbigg]

3 |A0|2 +

14

3 |A1|2 + 4|A3|2

13 |A4|2

[parenrightbigg] , (69)

5 [bracketleftBigg]Re(A1 A0)

2 5 Re(A2 A1)

630+ =

1

266 (|A0|2 + 12|A1|2 40|A3|2 15|A4|2),

(70)

2

53 Re(A3 A2)

2

5 Re(A4 A3)

[bracketrightBigg] , (81)

040+ =

114 (|A0|2 4|A1|2 + 6|A2|2 4|A3|2 + |A4|2),

(71)

431+ =

3 2

10 [bracketleftBigg]Re(A1 A0) +

4 3

[parenleftbigg]

5 |A0|2

18

5 |A2|2 +

16

5 |A3|2

2 5 Re(A2 A1)

240+ =

35 |A4|2

[parenrightbigg] , (72)

+

13 Re(A3 A2) +

12 Re(A4 A3)

[bracketrightbigg] , (82)

[parenleftbigg]

[radicalbigg] 7

22 [bracketleftBig]Re(A1 A0) + 10 Re(A2 A1)

3 |A1|2

+ 2|A2|2 8|A3|2 +

1 3|A4|2

440+ =

3

277

|A0|2 +

28

631+ =

1

2

[parenrightbigg] , (73)

103 Re(A3 A2) 5

2 Re(A4 A3)[bracketrightbigg] , (83)

123

Eur. Phys. J. C (2014) 74:2723 Page 11 of 13 2723

241+ =

5 2

[radicalbigg] 5 42

[bracketleftBigg]Re(A1 A0) 3

2 5 Re(A2 A1)

431 =

3i

2

10 [bracketleftBigg]Im(A1 A0) +

4 3

2 5 Im(A2 A1)

23

5 Re(A3 A2) +

2

5 Re(A4 A3)

[bracketrightBigg] , (84)

+

13 Im(A3 A2) +

12 Im(A4 A3)

[bracketrightbigg] , (94)

441+ =

5 2

+ 6 [bracketleftBigg]Re(A1 A0) + 4

2 5 Re(A2 A1)

[radicalbigg] 7

22 [bracketleftBig]Im(A1 A0) + 10 Im(A2 A1)

631 =

i 2

3 Re(A3 A2)

12 Re(A4 A3)

[bracketrightbigg] , (85)

[radicalbigg] 5

66 [bracketleftBig]Re(A1 A0) + 310 Re(A2 A1)

+ 103 Re(A3 A2) + 52 Re(A4 A3)[bracketrightBig] , (86)

211 =

103 Im(A3 A2) 5

2 Im(A4 A3)[bracketrightbigg] , (95)

241 =

641+ =

1

2

5i

2

[radicalbigg] 5

42

[bracketleftBigg]Im(A1 A0) 3

2 5 Im(A2 A1)

5i

23

[bracketleftbigg]Im(A1 A0) +

310 Im(A2 A1)

+

3

5 Im(A3 A2)

2

5 Im(A4 A3)

[bracketrightBigg] , (96)

+

23

5 Im(A3 A2) +

2

5 Im(A4 A3)

[bracketrightBigg] , (87)

441 =

5i

2

6 [bracketleftBigg]Im(A1 A0) + 4

2 5 Im(A2 A1)

411 = i

15 [bracketleftbigg]Im(A1 A0)

410 Im(A2 A1)

3

2 Im(A3 A2) +

3 Im(A3 A2)

12 Im(A4 A3)

[bracketrightbigg] , (97)

12 Im(A4 A3)

[bracketrightBigg] , (88)

[radicalbigg] 5

66 [bracketleftBig]Im(A1 A0) + 310 Im(A2 A1)

+103 Im(A3 A2) + 52 Im(A4 A3)[bracketrightBig] , (98)

222+ =

641 =

i 2

611 =

i 2

[radicalbigg] 7 33

[bracketleftBigg]Im(A1 A0) 6

5

2 Im(A2 A1)

+53 Im(A3 A2) 52 Im(A4 A3)[bracketrightBig] , (89)

221 =

[bracketleftbigg] Re(A2 A0)

+3 Re(A3 A1) + 2

5 21

5i

2

5 [bracketleftbigg]Im(A1 A0) +

110 Im(A2 A1)

3 5 Re(A4 A2)

[bracketrightbigg], (99)

1

53 Im(A3 A2) +

2

5 Im(A4 A3)

[bracketrightBigg] , (90)

422+ = 3

30 [bracketleftbigg] Re(A2 A0)

421 =

15i 77

[bracketleftbigg]Im(A1 A0)

4

310 Im(A2 A1)

1

23 Re(A3 A1)

2

3

5 3 Re(A4 A2)

[bracketrightbigg], (100)

1

+ 23 Im(A3 A2)

12 Im(A4 A3)

[bracketrightbigg] , (91)

622+ =

10

5

2 Im(A2 A1)

33 [bracketleftbig]Re(A2 A0)

23 Re(A3 A1) + 15 Re(A4 A2)[bracketrightBig] , (101)

232+ =

621 =

i 2

[radicalbigg] 5 11

[bracketleftBigg]Im(A1 A0)

53 Im(A3 A2) + 5

2 Im(A4 A3)[bracketrightbigg] , (92)

231 =

5 [bracketleftBigg]Re(A2 A0) 2

3 5 Re(A4 A2)

[bracketrightBigg] , (102)

5i

22

[bracketleftBigg]Im(A1 A0)

2 5 Im(A2 A1)

432+ =

3 2

30 [bracketleftBigg]Re(A2 A0) +

2

3

5 3 Re(A4 A2)

[bracketrightBigg] , (103)

2

53 Im(A3 A2)

2

5 Im(A4 A3)

[bracketrightBigg] , (93)

70

632+ =

1

2

33 [bracketleftBig]Re(A2 A0) 15 Re(A4 A2)[bracketrightBig] , (104)

123

2723 Page 12 of 13 Eur. Phys. J. C (2014) 74:2723

242+ =

5 [bracketleftbigg] Re(A2 A0)

642 =

i 2

10 [bracketleftbigg] Im(A2 A0)

43 Re(A3 A1) + 2

3 5 Re(A4 A2)

[bracketrightbigg], (105)

83 Im(A3 A1) +

15 Im(A4 A2)[bracketrightbigg], (116)

433+ =

10 [bracketleftbigg] Re(A2 A0)

+

23

9 Re(A3 A1)

215

9 Re(A4 A2)

3 2

+ 35 [bracketleftBigg]Re(A3 A0) +

2

3 Re(A4 A1)

[bracketrightBigg] , (117)

442+ =

9 2

[bracketrightbigg], (106)

633+ =

1

2

70 [bracketleftbigg]Re(A3 A0)

32 Re(A4 A1)

[bracketrightbigg] , (118)

35 [bracketleftBigg]Re(A3 A0)

2

3 Re(A4 A1)

[bracketrightBigg] , (119)

10 [bracketleftbigg] Re(A2 A0)

+

443+ =

3 2

642+ =

1

2

643+ =

1

2

70 [bracketleftbigg]Re(A3 A0) +

32 Re(A4 A1)

[bracketrightbigg] , (120)

83 Re(A3 A1) +

15 Re(A4 A2)[bracketrightbigg], (107)

222 =

433 =

3i

35 [bracketleftBigg]Im(A3 A0) +

2

3 Im(A4 A1)

[bracketrightBigg] , (121)

[bracketleftbigg] Im(A2 A0)

+3 Im(A3 A1) + 2

5i 21

633 =

i 2

2 70 [bracketleftbigg]Im(A3 A0)

32 Im(A4 A1)

[bracketrightbigg] , (122)

3 5 Im(A4 A2)

[bracketrightbigg], (108)

35 [bracketleftBigg]Im(A3 A0)

2

3 Im(A4 A1)

[bracketrightBigg] , (123)

443 =

3i

30 [bracketleftbigg] Im(A2 A0)

643 =

i 2

2 70 [bracketleftbigg]Im(A3 A0) +

32 Im(A4 A1)

[bracketrightbigg] , (124)

422 = 3i

1

23 Im(A3 A1)

2

3

5 3 Im(A4 A2)

[bracketrightbigg], (109)

444+ =

105

11 Re(A4 A0), (125)

644+ =

5 2

14

11 Re(A4 A0), (126)

444 = i

105

11 Im(A4 A0), (127)

644 =

622 = i

10 [bracketleftbigg] Im(A2 A0)

23 Im(A3 A1) + 15 Im(A4 A2)

[bracketrightbigg], (110)

5i

2

14

11 Im(A4 A0). (128)

A.4 Expressions of J3J2d

000+ = 1, (129)

010+ =

232 =

5i

23

[bracketleftBigg]Im(A2 A0) 2

3 5 Im(A4 A2)

[bracketrightBigg] , (111)

432 =

3i

2

30 [bracketleftBigg]Im(A2 A0) +

2

3

5 3 Im(A4 A2)

[bracketrightBigg] , (112)

70

632 =

i 2

33 [bracketleftBig]Im(A2 A0) 15 Im(A4 A2)[bracketrightBig] , (113)

242 =

12(|E0|2 + 2|E2|2), (130)

020+ =

[parenleftbigg]

10 |E0|2 +

1

2|E1|2 |E2|2

[parenrightbigg] , (131)

030+ =

5i

27

[bracketleftbigg] Im(A2 A0)

2 [parenleftbigg]|E1|2 +

1

2|E2|2

[parenrightbigg] , (132)

040+ = 3

43 Im(A3 A1) + 2

3 5 Im(A4 A2)

[bracketrightbigg], (114)

[parenleftbigg]

2 |E0|2

2

3|E1|2 +

1 6|E2|2

[parenrightbigg] , (133)

100+ =

442 =

9i

2

10 [bracketleftbigg] Im(A2 A0)

3 2(|E0|2 |E2|2), (134) 110+ =

3|E2|2, (135)

120+ =

15

7 (|E0|2 + |E2|2), (136)

[bracketrightbigg], (115)

+

23

9 Im(A3 A1)

215

9 Im(A4 A2)

123

Eur. Phys. J. C (2014) 74:2723 Page 13 of 13 2723

130+ =

3

2 |E2|2, (137)

140+ = 3

15

221 =

i 2

7 [bracketleftBig]Im(E1E0) 6 Im(E2 E1)[bracketrightBig] , (157)

231 = i

3 7(|E0|2

16|E2|2), (138)

200+ =

3 [bracketleftBigg]Im(E1E0) +

[bracketrightBigg] , (158)

3 2 Im(E2E1)

12(|E0|2 2|E1|2 + |E2|2), (139) 210+ = (|E1|2 |E2|2), (140)

220+ =

241 =

3i

2

10 [bracketleftbigg]Im(E1E0) +

16 Im(E2E1)

[bracketrightbigg] , (159)

5 7(|E1|2 |E1|2 |E2|2), (141)

230+ = 2

[parenleftbigg]

|E1|2 +

222+ =

30

7 Re(E2E0), (160)

232+ =

15

2 Re(E2E0), (161)

242+ =

1 4|E2|2

[parenrightbigg] , (142)

240+ =

3 7

[parenleftbigg]

|E0|2 +

4 3|E1|2 +

1 6|E2|2

[parenrightbigg] , (143)

111+ =

3 2

10

7 Re(E2E0), (162)

222 = i

30

7 Im(E2E0), (163)

232 = i

15

2 Im(E2E0), (164)

242 =

3 2

[bracketleftBigg]Re(E1E0) +

[bracketrightBigg] , (144)

121+ =

2 3 Re(E2E1)

1

2

15

7 [bracketleftBig]Re(E1E0) + 6 Re(E2 E1)[bracketrightBig] , (145)

131+ =

3 [bracketleftBigg]Re(E1E0)

[bracketrightBigg] , (146)

3i

2

10

7 Im(E2E0). (165)

References

1. T. Barnes, S. Godfrey, E.S. Swanson, Phys. Rev. D 72, 054026 (2005)

2. E. Eichten, K. Lane, C. Quigg, Phys. Rev. D 73, 014014 (2006)3. E. Eichten, S. Godfrey, H. Mahlke, J. Rosner, Rev. Mod. Phys. 80, 1161 (2008)

4. T. Fernndez-Carams, A. Valcarce, J. Vijande, Phys. Rev. Lett. 103, 222001 (2009)

5. T. Barnes, S. Godfrey, Phys. Rev. D 69, 054008 (2004)6. E. Eichten, K. Lane, C. Quigg, Phys. Rev. D 69, 094019 (2004)7. D. Bettoni, Proc. CHARM 2007 workshop (New York, 2007). http://arxiv.org/abs/arXiv:0710.5664v1

Web End =arXiv:0710.5664v1 [hep-ex]

8. PANDA Collaboration, Physics performance report for PANDA: strong interaction studies with antiprotons, 2009. http://arxiv.org/abs/arXiv:0903.3905v1

Web End =arXiv:0903. http://arxiv.org/abs/arXiv:0903.3905v1

Web End =3905v1 [hep-ex]

9. A.W.K. Mok, C.P. Wong, W.Y. Sit, J. High Energy Phys. 1210, 083 (2012)

10. A.W.K. Mok, K.J. Sebastian, Eur. Phys. J. C 67, 125 (2010)11. A.W.K. Mok, M.F. Chow, Eur. Phys. J. C 71, 1792 (2011)12. A.W.K. Mok, K.J. Sebastian, Eur. Phys. J. C 56, 189 (2008)13. A.D. Martin, T.D. Spearman, Elementary Particle Theory (North-Holland, Amsterdam, 1970), pp. 113115

14. K.J. Sebastian, Phys. Rev. D 49, 3450 (1994)15. A.W.K. Mok, K.J. Sebastian, Eur. Phys. J. C 63, 101 (2009)16. F. Karl, S. Meshkov, J.L. Rosner, Phys. Rev. D 13, 1203 (1976)17. K.J. Sebastian, X.G. Zhang, Phys. Rev. D 55, 225 (1997)

3 2 Re(E2E1)

141+ =

3 2

10 [bracketleftbigg]Re(E1E0)

16 Re(E2E1)

[bracketrightbigg] , (147)

[bracketrightBigg] , (148)

221+ =

211+ =

3 2

[bracketleftBigg]Re(E1E0)

2 3 Re(E2E1)

1

2

15

7 [bracketleftBig]Re(E1E0) 6 Re(E2 E1)[bracketrightBig] , (149)

231+ =

3 [bracketleftBigg]Re(E1E0) +

[bracketrightBigg] , (150)

241+ =

3 2 Re(E2E1)

3 2

10 [bracketleftbigg]Re(E1E0) +

16 Re(E2E1)

[bracketrightbigg] , (151)

[bracketleftBigg]Im(E1E0) +

[bracketrightBigg] , (152)

121 =

111 =

3i

2

2 3 Im(E2E1)

i 2

15

7 [bracketleftBig]Im(E1E0) + 6 Im(E2 E1)[bracketrightBig] , (153)

131 = i

3 [bracketleftBigg]Im(E1E0)

[bracketrightBigg] , (154)

3 2 Im(E2E1)

141 =

3i

10 [bracketleftbigg]Im(E1E0)

16 Im(E2E1)

[bracketrightbigg] , (155)

211 =

3i

2

2 [bracketleftBigg]Im(E1E0)

[bracketrightBigg] , (156)

2 3 Im(E2E1)

123