Separation of different wave components in the Bethe–Salpeter wave function

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— journal of March 2011

physics pp. 397–405

Separation of different wave components in the Bethe–Salpeter wave function

JIAO-KAI CHEN

Department of Physics and Electronic Engineering, Weinan Teachers University, Weinan 714000, People’s Republic of China

E-mail: chenjk@wntc.edu.cn

MS received 1 July 2010; revised 23 September 2010; accepted 1 October 2010

Abstract. The scalar products of polarization tensor and unit vectors are presented explicitly in spherical coordinate system expanded in terms of spherical harmonic functions. By applying the obtained formulae, different wave components in the Salpeter wave function can be shown explicitly, and the results are consistent with the results obtained by L–S coupling analysis. The cancelation formula is given, by which the terms with pure L = J +1 wave components in the Salpeter wave function for the bound state withηP = (−1)^J can be obtained by separating the L = J−1 wave components from mixing terms. This separation provides the basis for studying higher-order contributions from the coupling of L= J−1 and J+1 wave states, and for solving the Salpeter equation exactly without approximation.

Keywords. Salpeter wave function; mixing; spherical coordinate system; pure L= J+1 term.

PACS Nos 11.10.St; 12.39.Ki

1. Introduction

Mixings are interesting phenomena. There are two types of mixings. One type is the mixing of the spin singlet and spin triplet for quarkonium Q¯1Q2composed of different kinds of quarks. If two quarks are similar, this kind of mixing will disappear. The other type is the mixing of L =J−1 and J+1 wave states for quarkoniumQ¯1Q2composed of two types of quarks and quarkoniumQ Q composed of the same type of quarks.¯

For the first type, the mixture of¹P1and³P1is a good example [1–3]. For the second type, ³S1−³D1 mixing is related to the properties ofψ, ψ [3–9]. For example, ψ (ψ(3770)) is assumed to be a pure³D1state, but the predicted DD width based on this assumption is not consistent with experimental results. The small total width implies the effect of³D1−2³S1mixing [3]. The state Y(4360), one of the new states discovered in recent years [4,10], may also be a D-wave state.

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Both mixture of¹P1and³P1 and the mixture of L = J −1 and J+1 wave states are phenomenological, and the mixing angles are fitted using experiment data. When higher accuracy is demanded, the effect of coupling of L =J−1 and J+1 wave states should be considered. Within the Bethe–Salpeter formalism [11,12], the L = J −1 and J +1 wave states are coupled. Therefore, separation of different wave components should be done before calculating the coupling of different wave states. It is shown in [13] that S-D coupling should be considered at the order ofv⁴for vector mesons 1⁻⁻, and the D-wave state contributions to decay constant should be considered also in the higher order by coupled eigenvalue equations, i.e., the Salpeter equations, even when only simple potential, the scalar plus the zero components of vector potential, is considered and the orbit–spin and tensor terms are neglected. In this paper, we concentrate on kinematics only, the separation of different wave components in the Salpeter wave function.

This paper is organized as follows: In §2 are the Salpeter wave functions with parity ηP =(−1)^J are presented. In §3, the explicit form of the scalar products of polarization tensor and unit vectors and its applications are presented. The conclusion is given in §4.

2. The Salpeter wave function

The general form of the Bethe–Salpeter wave function with parityηP =(−1)^J reads in CMS frame [15] as

χ^J(q)= μ1···μJqˆ^μ²· · · ˆq^μ^J ˆ

q^μ¹(g1+γ⁰g2) +γ^μ¹(g3+γ⁰g4)+ ˆq^μ¹ ˆq(g5+γ⁰f6) +σ^μ¹^νqˆ_νg7+i⁰^μ¹^αβqˆ_αγ_βγ⁵g8

, (1) where J is the spin of the bound state, qˆ^μ = (0,qˆ),qˆ = q/|q|, σ^μν = [γ^μ, γ^ν]/2, gi =gi(q). The polarization tensor^μ¹^μ²^···μ^J is total symmetric, traceless and transversal

^μ¹^μ²^...μ^J(P)=^μ²^μ¹^···μ^J(P), _μ^μ₁¹^μ²^···μ^J(P)=0,

P_μ₁^μ¹^μ²^···μ^J(P)=0. (2)

In CMS frame,⁰^μ²^···μ^J(P) = 0, P =(M,0), M is the mass of the bound state. The three-dimensional equal-time Salpeter wave function is defined as

ϕ^J(q)= dq0

2πχ^J(q). (3)

Applying the definition, the general form of the Salpeter wave function for the bound state composed of different quarks with parityηP =(−1)^Jis obtained from the Bethe–Salpeter wave function (eq. (1))

ϕ^J(q) = μ1···μJqˆ^μ²· · · ˆq^μ^J ˆ

q^μ¹(f1+γ⁰f2) +γ^μ¹(f3+γ⁰f4)+ ˆq^μ¹ ˆq(f5+γ⁰f6) +σ^μ¹^νqˆ_νf7+i⁰^μ¹^αβqˆ_αγ_βγ⁵f8

, (4) where fi = fi(|q|)= _dq₀

2πgi(q).

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Applying the constraintsϕ_P⁺⁺(q)=ϕ_P⁻⁻(q)=0 [11–14] on the Salpeter wave function (4), the constraints for the state with parityηP=(−1)^J are obtained as

f1= (ωa+ωb)|q|

maωb+mbωa(f3− f5), f2= − (ωa−ωb)|q|

maωb+mbωa(f4− f6),

f7= (ωa−ωb)|q| maωb+mbωa

f3, f8= (ωa+ωb)|q|

maωb+mbωa

f4, (5)

whereωa,b=

m²_a_,_b+ q², ma,bare the masses of constituents a,b.

If the two constituents of the bound state are the same kinds of quarks, wave function (4) will be categorized into two classes according to the charge conjugation parity. When parity and charge conjugation parity are ηP = ηC = (−1)^J, the general form of wave function (4) becomes

ϕ^J(q) = _μ1···μJqˆ^μ²· · · ˆq^μ^J ˆ

q^μ¹f1+γ^μ¹(f3+γ⁰f4) + ˆq^μ¹ ˆq(f5+γ⁰f6)+i0μ1αβqˆ^αγ^βγ⁵f8

, (6) and when parity and charge conjugation parity are ηP = (−1)^J andηC = (−1)^J⁻¹, respectively, the wave function (4) becomes

ϕ^J(q) = μ1···μJqˆ^μ²· · · ˆq^μ^J ˆ

q^μ¹γ⁰f2+σ^μ¹^νqˆ_νf7

. (7) Applying the constraintsϕ_P⁺⁺(q)= ϕ_P⁻⁻(q)= 0 on wave functions (6) and (7), the constraints for states withηP=ηC=(−1)^Jare

f1=|q|

m(f3− f5), f8= |q|

m f4, (8)

and the constraints for the states withηP=(−1)^J,ηC=(−1)^J⁻¹are

f2=0, f7=0. (9)

As expected (see table 1), it is impossible to construct a Salpeter wave function for a fermion–antifermion bound state with exotic quantum numbers ηP = (−1)^J and ηC=(−1)^J⁻¹.

Observing eqs (4)–(6) and (8), it is found that the terms f3,4corresponding to L=J−1 wave states and the terms f5,6corresponding to L = J +1 wave states play dominant roles in wave functions, while f2,7 and f1,8 are zeros or small terms suppressed by a factor 1/m which are corrections to dominant terms. The terms f5,6 are a mixture of L =J−1 and J+1 wave states. These results are consistent with L–S coupling analysis (see table 1). In table 1, the states are categorized according to J^PC(^2S⁺¹LJ), where J is the spin of the bound system, S is the total spin of the quark and antiquark, L is the orbital angular quantum number, P is the parity number(P =(−1)^L⁺¹), C is the charge conjugation quantum number(C=(−1)^L⁺^S). Further details can be found in [13].

To calculate the coupling of L=J−1 and J+1 wave states using the Salpeter equation [11–13], it is necessary to obtain pure L = J+1 wave components by separating the L =J−1 wave components from mixing terms f5,6. In Cartesian coordinate system this mixing cannot be separated clearly, especially for high spin bound states, for example 2⁺, 2⁺⁺, etc. We found that it can be done in spherical coordinate system [13,16,17].

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Table 1. The physically existing states for the bound states composed of fermion and antifermion: J^PC(²^∗^S⁺¹L_J).

Spin of bound Parity Wave state Charge parity Total spin of constituents state(J) (J^P) (L_J) (J^PC) (²^∗^S⁺¹L_J)

J=0 0⁻ S₀ 0⁻⁻ /

0⁻⁺ ¹S₀

0⁺ P₀ 0⁺⁻ /

0⁺⁺ ³P₀

J=1 1⁻ S₁−D₁ 1⁻⁻ ³S₁−³D₁

1⁻⁺ /

1⁺ P₁ 1⁺⁻ ¹P₁

1⁺⁺ ³P₁

J=2 2⁻ D₂ 2⁻⁺ ¹D₂

2⁻⁻ ³D₂

2⁺ P₂−F₂ 2⁺⁻ /

2⁺⁺ ³P₂−³F₂

/means that the states|qq¯with exotic quantum numbers cannot be constructed by the Salpeter wave function.

3. Formulae and applications

In spherical coordinate systems, the scalar products in eqs (4) and (6) can be rewritten by expanding spherical harmonic functions Ylm in explicit form as

^μ¹^...μ^Jqˆ_μ₁· · · ˆq_μ_J = J

m=−J

^μ¹^...μ^Jqˆ_μ₁· · · ˆq_μ_J[J,m]

= J

m=−J

Z[J,m]. (10)

The explicit expression for function Z[J,m]reads as

Z[J,m] =YJ m[J,m]z[J,m], (11) where −J ≤ m ≤ J . The function[J,m] can be found in eq. (A.20), and func- tion z[J,m] in eq. (A.22). The derivations of the above two equations are given in Appendix A. Applying the function Z[J,m], different wave components in wave functions (1), (4) and (6) can be shown explicitly. This expansion provides bases for the exact solutions to the Salpeter equation [16,17], and for the coupling of L =J−1 and J+1 wave states to obtain higher-order contributions [13].

Using eqs (10) and (11), the cancellation formula can be easily obtained as ^μ¹^μ²^...μ^Jqˆ_μ₂· · · ˆq_μ_J

ˆ

q_μ₁ ˆq+ J

2 J+1γμ1 . (12)

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In this equation, the L = J −1 wave components in the first term have been cancelled by the second term, and the results are all pure L = J +1 wave components. Similar formula was obtained in ref. [15].

Applying the cancellation formula (12) to separate the L = J −1 wave components from the main terms f5,6in eqs (4) and (6), the Salpeter wave function (4) for the bound states withηP =(−1)^J can be rewritten as

ϕ^J(q) = _μ1···μJqˆ^μ²· · · ˆq^μ^J

×

ˆ

q^μ¹(f1+γ⁰f2)+γ^μ¹(f₃+γ⁰f₄) +

ˆ

q^μ¹ ˆq+ J

2 J+1γ^μ¹ (f5+γ⁰f6) +σ^μ¹^νqˆ_νf7+i⁰^μ¹^αβqˆ_αγ_βγ⁵f8

, (13)

and the Salpeter wave function (6) for the bound states withηP = ηC =(−1)^J can be rewritten as

ϕ^J(q) = _μ1···μJqˆ^μ²· · · ˆq^μ^J

×

ˆ

q^μ¹f1+i⁰^μ¹^αβqˆ_αγ_βγ⁵f8+γ^μ¹(f₃+γ⁰f₄) +

ˆ

q^μ¹ ˆq+ J

2 J+1γ^μ¹ (f5+γ⁰f6)

. (14)

In eqs (13) and (14), the functions f₃= f3−J/(2 J+1)f5and f₄ = f4−J/(2 J+1)f6. In the above two rewritten wave functions, the main terms f5,6are pure L =J+1 wave components, and the main terms f₃_,₄are pure L=J−1 wave components, while f1,2,7,8

are zeros or small terms which are correction terms to main terms by the constraints (5) and (8).

In the wave functions (13) and (14), the expansions with spherical harmonic functions of the coefficients of f1,2,3,4 are apparent by applying eqs (10) and (11), and the ex- pansions of coefficients of f5,6,7,8can be easily obtained by some calculations. When calculating the expansions of coefficients of f8, the formulaeγβγ5 = ₃ⁱ_!βσρλγ^σγ^ργ^λ and iμ···⁰^μαβqˆ_αγβγ5=μ···γ^μ ˆqγ⁰−μ···qˆ^μγ⁰are needed.

As examples, the Salpeter wave functions for 3⁻ and 3⁻⁻ read respectively from eqs (13) and (14) as

ϕ³⁻(q) = _μ1μ2μ3qˆ^μ²qˆ^μ³

×

ˆ

q^μ¹(f1+γ⁰f2)+γ^μ¹(f₃+γ⁰f₄)+

ˆ

q^μ¹ ˆq+1 3γ^μ¹

×(f5+γ⁰f6)+σ^μ¹^νqˆ_νf7+i⁰^μ¹^αβqˆ_αγ_βγ⁵f8

, (15)

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and

ϕ³⁻⁻(q) = _μ1μ2μ3qˆ^μ²qˆ^μ³

ˆ

q^μ¹f1+i⁰^μ¹^αβqˆ_αγ_βγ⁵f8+γ^μ¹(f₃+γ⁰f₄) +

ˆ

q^μ¹ ˆq+1

3γ^μ¹ (f5+γ⁰f6)

. (16)

4. Conclusion

In this paper, the scalar products of polarization tensor and unit vectors are presented explicitly in spherical coordinate system expanded in terms of spherical harmonic functions.

Applying the obtained formulae, different wave components in the Salpeter wave function can be shown explicitly. The cancellation formula is given by which the terms with pure L =J+1 wave components in the Salpeter wave function for the bound state with ηP=(−1)^Jcan be obtained by separating the L =J−1 wave components from mixing terms. This separation provides basis for studying higher-order contributions from the coupling of L= J−1 and J+1 wave states, and for the approach to solve the Salpeter equation exactly without approximation.

Appendix A. The derivations of formulae (10) and (11)

In spherical coordinate system, the unit vectorqˆ^μ =(0,qˆ),qˆ= q/|q|can be rewritten as

ˆ

q⁺ = −(qˆ1+iqˆ2)

√2 =

4π

3 Y11(θq, φq), (A.1)

ˆ

q⁻ = (ˆq1−iqˆ2)

√2 = 4π

3 Y1−1(θq, φq), (A.2)

ˆ

q = ˆq3= 4π

3 Y10(θq, φq), (A.3)

where Ylmare spherical harmonic functions. The scalar product is represented in spherical coordinate system as

p·q = p⁰q⁰+p⁺q⁻+p⁻q⁺−pq. (A.4) By expanding with spherical harmonic functions, the scalar products of polarization tensor and unit vectorq can be written in explicit form asˆ

^μ¹^...μ^Jqˆ_μ₁· · · ˆq_μ_J = J

m=−J

^μ¹^...μ^Jqˆ_μ₁· · · ˆq_μ_J[J,m]

= J

m=−J

Z[J,m]. (A.5)

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Here the function Z[J,m]is given by

Z[J,m] =YJ m

4π 3

J[(J+m)/2] n=m

Cⁿ_JCⁿ_J⁻₋^m_n^s¹^s²^...^s^J

×

dY_{J m}^∗ (Y11)^m(−Y10)^J⁺^m⁻²ⁿ(Y11Y1−1)ⁿ⁻^m, (A.6) where m ≥ 0, s1 = · · · = sm = · · · = sn = −, sn+1 = · · · = sn+n−m = +, sn+n−m+1 = · · · = sJ = . Cⁿ_J are binomial coefficients. [(J +m)/2]takes integer part of(J+m)/2. The function Z[J,m]for m<0 can be obtained in a similar way.

The properties of polarization tensor (2) lead to

2^+−···=2^−+···=^···. (A.7)

In eq. (A.6), Y11Y1−1can be rewritten as Y11Y1−1=1

2Y10Y10+ · · ·. (A.8)

The remaining terms in the above equation cannot contribute to eq. (A.6). Using eqs (A.7) and (A.8), eq. (A.6) can be simplified as

Z[J,m] =YJ m[J,m]z[J,m], z[J,m] =B[J,m]y[J,m], (A.9) where

[J,m] =^s¹^s²^...^s^J, s1= · · · =sm= −, sm+1= · · · =sJ =, (A.10)

B[J,m] = (−1)^J⁺^m(2)^2m

[(J+m)/2] n=m

Cⁿ_JCⁿ_J⁻₋^m_n(2)⁻²ⁿ

=(−1)^J⁺^m (2)^J⁺^mJ![J+1/2]

√π(J+m)!(J−m)!, (A.11)

y[J,m] = 4π

3 J

dY_{J m}^∗ (Y11)^m(Y10)^J⁻^m. (A.12) Applying the relations

Y10YJ m= 3

4π (a[J,m]YJ+1m+a[J−1,m]YJ−1m) , Y11YJ m=

3 8π

b[J,m]YJ+1m+1−b[J−1,−m−1]YJ−1m+1

, Y1−1YJ m =

3 8π

b[J,−m]YJ+1m−1−b[J−1,m−1]YJ−1m−1

, (A.13)

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where

a[J,m] =

(J+1)²−m²

(2 J+1)(2 J+3), b[J,m] =

(J+m+1)(J+m+2) (2 J+1)(2 J+3) ,

(A.14) the following formulae are obtained

(Y11)^J = 3

2 3

8π

J−1

[2(J−1)+2]!!

[2(J−1)+3]!!YJ J, (A.15) and

(Y10)^kYJ m = 3

4π k

(J+m+k)!(J−m+k)!

(J+m)!(J−m)!

×

(2 J−1)!!(2 J+1)!!

[2(J+k)−1]!![2(J+k)+1]!!YJ+km

+ · · ·, (A.16)

where J!is a factorial function ( J! = J(J −1)· · ·). (2 J +3)!!is a double factorial function ((2 J+3)!! =(2 J+3)(2(J−1)+3)· · ·). The remaining terms in eq. (A.16) contribute nothing to final results. Applying eqs (A.15) and (A.16), the function y[J,m] (eq. (A.12)) can be simplified as

y[J,m] =2¹⁻^m^/²π¹^/²

(J+m)!(J−m)!

(2 J−1)!!(2 J+1)!!. (A.17)

Applying the results obtained above, the function Z[J,m]reads when m≥0 Z[J,m] =(−1)^J⁺^mYJ m[J,m](2)¹⁺^J⁺^m^/²J![J+1/2]

×[(J+m)!(J−m)!(2 J−1)!!(2 J+1)!!]⁻¹^/², (A.18) where[J+1/2]is the Euler gamma function. The function Z[J,m]can be extended easily to m<0 case. The final expression for function Z[J,m]reads as

Z[J,m] =(−1)^J^+|^m^|YJ m[J,m](2)¹⁺^J^+|^m^|/²J![J+1/2]

×[(J+m)!(J−m)!(2 J−1)!!(2 J+1)!!]⁻¹^/², (A.19) where−J≤m≤ J . The function[J,m]in eq. (A.10) should be replaced by

[J,m] =^s¹^s²^...^s^J, (A.20)

where

s1= · · · =s_|m|= −Sign[m], s_|m|+1= · · · =sJ =,if m=0,

s1= · · · =sJ =, if m=0.

(A.21)

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The function Sign[x]gives−1, 0 or 1 depending on whether x is negative, zero, or posi- tive. The function z[J,m]reads from eq. (A.19) as

z[J,m] =(−1)^J^+|^m^|(2)¹⁺^J^+|^m^|/²J![J+1/2]

×[(J+m)!(J−m)!(2 J−1)!!(2 J+1)!!]⁻¹^/². (A.22) It is an even function

z[J,−m] =z[J,m]. (A.23)

From eqs (A.19) and (A.22), the recurrence formula can be obtained as

√J+mz[J,m] +√ 2√

J−m+1z[J,m−1] = 0. (A.24) The results presented in Appendix A can also be obtained easily by replacing the scalar products in eq. (A.5) with(· ˆq)^J when calculating. This technique will make the calcu- lation very simple.

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