Electromagnetic properties of heavy mesons in heavy quark limit

6 September 2001

Physics Letters B 516 (2001) 65–71 www.elsevier.com/locate/npe

Electromagnetic properties of heavy mesons in heavy quark limit Chien-Wen Hwang Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan Received 19 June 2001; accepted 5 July 2001 Editor: T. Yanagida

Abstract The electromagnetic (EM) form factors of the pseudoscalar and vector mesons are calculated in the light-front framework. We extract these form factors from the relevant matrix elements directly, instead of choosing the Breit frame. The results show that the charge radius of the meson are related to both the first and second longitudinal momentum square derivative of the momentum distribution function. In addition, the static properties of the EM form factors and the heavy quark symmetry of the charge radii are checked analytically when we take the heavy quark limit.  2001 Published by Elsevier Science B.V. PACS: 12.39.Hg; 13.40.Gp

The understanding of the electromagnetic (EM) properties of hadrons is an important topic, and the EM form factors which are calculated within the nonperturbative methods are the useful tool for this purpose. There have been numerous experimental [1–6] and theoretical studies [7–9] of the EM form factors of the light pseudoscalar meson (π ana K). However, the EM form factors of the light vector meson (ρ and K ∗ ) have fewer investigations than their pseudoscalar counterparts [10–12], even though they could provide much information about the boundstate dynamics. As for the EM form factors of the heavy mesons (which containing one heavy quark), there are much fewer study than the light ones. In the heavy hadron investigation, however, the heavy quark symmetry (HQS) [13] is a fundamental and model-independent property. In this work, we will study the EM form factors of the heavy mesons with the light-front framework, and will check whether HQS is satisfied or not among these EM properties. As is well known [14], the light-front quark model (LFQM) allows an exact separation in momentum space between the center-of-mass motion and intrinsic wave functions, a consistent treatment of quark spins and the center-of-mass motion can also be carried out. Thus it has been applied in the past to calculate various form factors [15]. Let us begin with the light-front formalism. A meson bound state consisting of a heavy quark q1 and an antiquark q¯2 with total momentum P and spin S can be written as 
  SS  
  3  3  
M(P , S, Sz ) = d p1 d p2 2(2π)3 δ 3 P − p˜1 − p˜2 Ψ z p˜1 , p˜2 , λ1 , λ2
q1 (p1 , λ1 )q¯2 (p2 , λ2 ) , λ1 ,λ2

(1) E-mail address: [email protected] (C.-W. Hwang). 0370-2693/01/$ – see front matter  2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 9 0 4 - 2

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where p1 and p2 are the on-mass-shell light-front momenta,   p⊥ = p1 , p2 ,

  p˜ = p+ , p⊥ ,

p− =

2 m2 + p⊥ , p+

(2)

and
 dp+ d 2 p⊥ 
q(p1 , λ1 )q(p , ¯ 2 , λ2 ) = bλ†1 (p1 )dλ†2 (p2 )|0 , d 3p ≡ 3 2(2π)       †  bλ (p ), bλ (p) = dλ (p ), dλ† (p) = 2(2π)3 δ 3 p˜  − p˜ δλ λ .



(3)

In terms of the light-front relative momentum variables (x, k⊥ ) defined by p1+ = (1 − x)P + ,

p2+ = xP + ,

p1⊥ = (1 − x)P⊥ + k⊥ ,

p2⊥ = xP⊥ − k⊥ ,

(4)

the momentum-space wave-function Ψ SSz can be expressed as   SS Ψ SSz p˜1 , p˜ 2 , λ1 , λ2 = Rλ1 λz2 (x, k⊥ )φ(x, k⊥ ),

(5) SS

where φ(x, k⊥ ) describes the momentum distribution of the constituents in the bound state, and Rλ1 λz2 constructs a state of definite spin (S, Sz ) out of light-front helicity (λ1 , λ2 ) eigenstates. Explicitly,


SS λ1 |R†M (1 − x, k⊥ , m1 )|s1 λ2 |R†M (x, −k⊥ , m2 )|s2 12 s1 ; 12 s2
S, Sz , Rλ1 λz2 (x, k⊥ ) = (6) s1 ,s2

where |si are the usual Pauli spinors, and RM is the Melosh transformation operator [17]: RM (x, k⊥ , mi ) =

mi + xM0 + i σ · k⊥ × n  , 2 (mi + xM0 )2 + k⊥

(7)

with n = (0, 0, 1), a unit vector in the z-direction, and M02 =

2 2 m2 + k ⊥ m21 + k⊥ + 2 . (1 − x) x

(8) SS

In practice it is more convenient to use the covariant form for Rλ1 λz2 [16]:  p1+ p2+ SSz u(p ¯ 1 , λ1 )Γ v(p2 , λ2 ), Rλ1 λ2 (x, k⊥ ) = √ 0 2M where 0 ≡ M



M02 − (m1 − m2 )2 ,

Γ = −/εˆ (Sz ) + with



εˆ · (p1 − p2 ) M0 + m1 + m2

Γ = γ5

(pseudoscalar, S = 0),

(vector, S = 1),

2 ⊥ , 0, ε⊥ (±1) , ε  (±1) · P ⊥ P+   1 −M02 + P⊥2 + µ εˆ (0) = , P , P⊥ . M0 P+

εˆ µ (±1) =

(9)

√ ε⊥ (±1) = ∓(1, ±i)/ 2 , (10)

C.-W. Hwang / Physics Letters B 516 (2001) 65–71

We normalize the meson state as


   M(P  , S  , Sz )
M(P , S, Sz ) = 2(2π)3 P + δ 3 P − P δS  S δSz Sz ,

67

(11)

so that the normalization condition of the momentum distribution function can be obtained 
2 dx d 2 k⊥

φ(x, k⊥ )
= 1. (12) 3 2(2π) In principle, φ(x, k⊥ ) can be obtained by solving the light-front QCD bound state equation [18]. However, such first-principles solutions have not been done yet, and the phenomenological wave functions must be considered. In this Letter, we do not designate anyone to use but only ask the momentum distribution function has the scaling behavior [19] √ φQq¯ (x, k⊥ ) → mQ Φ(mQ x, k⊥ ), (13) √ in the infinite quark mass limit mQ → ∞. The factor mQ comes from the particular normalization we have assumed for the physical state in Eqs. (11) and (12). The reason why the light-front heavy-meson wave function should have such an asymptotic form is as follows. Since x is the longitudinal momentum fraction carried by the light antiquark, the meson wave function should be sharply peaked near x ∼ ΛQCD /mQ . It is thus clear that only terms of the form “mQ x” survive in the wave function as mQ → ∞; that is, mQ x ≡ X is independent of mQ in the mQ → ∞ limit. Now we consider the pseudoscalar meson first. The EM form factor of a meson P is determined by the scattering of one virtual photon and one meson. Since the momentum of the virtual photon q is space-like, it is always possible to orient the axes in such a manner that Q+ = 0. Thus the EM form factor of a pseudoscalar meson P is determined by the matrix element



  P (P  )
J +
P (P ) = eFP Q2 (P + P  )+ , (14) where J µ = qe ¯ q eγ µ q is the vector current, eq is the charge of quark q in e unit, and Q2 = (P  − P )2
0. This process is illustrated in Fig. 1. With LFQM, FP can be extracted by Eq. (14)   )    dx d 2 k⊥ φP (x, k⊥ ) φP  (x, k⊥    A2 + k⊥ · k⊥ FP Q2 = eq1 3 2(2π) 2 2 A2 + k⊥ A2 + k⊥   )   dx d 2 k⊥ φP (x, k⊥ ) φP  (x, k⊥    + eq¯2 (15) A2 + k⊥ · k⊥ , 3 2(2π) 2  2 A2 + k⊥ A2 + k⊥  √  = k + xQ , k  = k − (1 − x)Q , A = xm + (1 − x)m , and A2 + k 2 = M 0 x(1 − x). From where k⊥ ⊥ ⊥ ⊥ ⊥ ⊥ 1 2 ⊥  2 comes from the Melosh transformation. After Eqs. (6), (7), and (9), it is understandable that the term A2 + k⊥

Fig. 1. Diagrams contributing to the EM form factors of meson.

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C.-W. Hwang / Physics Letters B 516 (2001) 65–71

fixing the parameters which appear in the wave function, Eq.  (15) fits the experimental data well [20]. But this is not the whole story. We consider the term φ˜ P ≡ φP (x, k⊥ )/ A2 + k 2 and take the Taylor expansion around k 2 ⊥



  2  2  d φ˜ P 
 2   2  d 2 φ˜P 

2 2 2
φ˜ P  k⊥ − k + ···. = φ˜ P  k⊥ + k + k⊥ − k⊥ ⊥ ⊥

2 2 2 dk⊥ Q⊥ =0 2(dk⊥ ) Q⊥ =0 Then, by using the identity   1 2 2 Q⊥ , d 2 k⊥ (k⊥ · Q⊥ )(k⊥ · Q⊥ ) = d 2 k⊥ k⊥ 2 we can rewrite (15) to    2  2   A2 + 2k⊥ dx d 2 k⊥ 2 2 2 2 2  FP Q = (eq1 + eq¯2 ) − Q φ (x, k⊥ ) eq1 x + eq¯2 (1 − x) ΘP + ΘP k⊥ 2 2(2π)3 P A2 + k⊥   + O Q4 , where ΘM =

1 φ˜ M



 d φ˜ M , 2 dk⊥

P = Θ

1 φ˜ M



 d 2 φ˜ M . 2 )2 (dk⊥

⊥

(16)

(17)

(18)

(19)

The mean square radius r 2 P ≡ 6(dFP (Q2 )/dQ2 )|Q2 =0 can be obtained easily using r 2 P ≡ r 2 q1 + r 2 q¯2 , where      

2  2  d  dx d 2 k⊥ 2 d 2 2 2 2 ˜ P A2 + 2k⊥ φ˜ P , φ r q = eq1 −6 (20) x + A + k k ⊥ ⊥ 2 2 1 2(2π)3 dk⊥ dk⊥     

2  2  d   2 dx d 2 k⊥ d 2 2˜ 2 2 2 r q¯ = eq¯2 −6 (21) A φ˜ P . φ k (1 − x) + 2k + A + k P ⊥ ⊥ ⊥ 2 2 2 2(2π)3 dk⊥ dk⊥

It is worthwhile to mention that, first, the static property FP (0) = eP is quite easily checked in (18). Secondly, from Eqs. (20) and (21), we find that the mean square radius is related to the first and second longitudinal momentum square derivatives of φ˜ which contain the Melosh transformation effect. When we consider the heavy meson q1 → Q, we get X , A → A˜ ≡ X + m2 , (22) mQ and Eq. (13) in the heavy quark limit (mQ → ∞), then the mean square radius       2

2  d  −6 d 2 dX d 2 k⊥ 2  ˜ 2 2 2 2 ˜ + k⊥  → 0,  A + 2k⊥ Φ Φ X + A r Q = eQ (23) k ⊥ 2 2 2(2π)3 m2Q dk⊥ dk⊥     

2  2  2  d  dX d 2 k⊥ d 2 2 2 2 ˜ ˜  ,  r q¯ = eq¯2 −6 (24) Φ Φ A + 2k⊥ + A + k⊥ k⊥ 2 2 2 2(2π)3 dk⊥ dk⊥  2 + k 2 . Eq. (23) means that, in the heavy quark limit, the mean square radius r 2 P is blind to  = Φ/ A where Φ ⊥ the flavor of Q. This is the so-called flavor symmetry. Next, the vector meson case is considered. As in the pseudoscalar case, we set Q+ = 0, then the EM form factor of a vector meson V is parametrized by the matrix element

  
+
 V (P , - )
J
V (P , -)        1 = G1 Q2 - · -  (P + P  )+ − G2 Q2 P · -  - + + P  · - - + + G3 Q2 P · -  P  · -(P + P  )+ , (25) 2MV2 x→

C.-W. Hwang / Physics Letters B 516 (2001) 65–71

69

where MV is the V -meson mass and -(-  ) is the polarization of V (V  ). In the literature [21–24], these invariant form factors are usually obtained by choosing the Breit frame. For the vector meson there are only three independent form factors, whereas four matrix elements of J + are independent [7] since J + is (i) Hermitian, (ii) invariant under rotations about n , (iii) time reversal, and (iv) reflection on the plane perpendicular to n . An additional constraint, which called the angular condition, is related to the rotational invariance of the charge density [12]. This invariance is related to the unitary transformations based upon a subset of Poincaré generators. The authors of Ref. [23] had written the angular condition in the following form    ∆ Q2 ≡ (1 + 2η)J1,1 + J1,−1 − 8η J1,0 − J0,0 = 0,

(26)

where η = Q2 /4MV2 and J-  ,- ≡ V  (P  , -  )|J + |V (P , -) . In generally, this condition must be satisfied when the matrix elements of the many-body current exist, and breaks for the one-body. Thus, when only the one-body current operator qγ ¯ µ q is considered, Eq. (26) usually does not equal to zero and the calculation of these form factors depends upon the description used [12]. Following [24,25], J0,0 is expected to be the dominant matrix element in the pQCD regime. Thus, as the argument in [23], it is reasonable that we calculate the Gi form factor without using the matrix element J0,0 in the constituent quark model. In this Letter, we will extract the form factors directly, instead of choosing the Breit frame. The Q2 dependence of these form factors must be calculated order by order. In spite of the fact that we cannot obtain the exact form, the advantages, however, are evident and will be mentioned later. We choose the matrix elements J1,1 , J1,−1 , J1,0 and Eq. (17) to extract the form factors G1 , G2 , and G3 in the suitable order  2  2   2  k⊥ dx d 2 k⊥ ˜ 2 2 2 2 2 2 φ V eq1 x ΘV A + 2k⊥ + x ΘV k⊥ A + k⊥ − 3 2(2π) MV2 

4     xk⊥ + O Q4 , (27) + e + x ↔ (1 − x) q¯ 2 2 2 2(1 − x)M0 WV      2  xk⊥ dx d 2 k⊥ 2 d d G2 (Q2 ) + 2 2 ˜ = − c+ φ eq1 xa + x b + MV 2(2π)3 V WV 2WV (1 − x)M0 WV        d 2 + eq¯2 x ↔ (1 − x) + O Q2 , + xk⊥ ΘV b + (28) WV    2    Ak⊥ dx d 2 k⊥  G3 (Q2 ) A 2 2 ˜2 eq1 x + eq¯2 (1 − x) φV − = + (29) + O Q2 , 2 3 2 2(2π) WV 4MV 2x(1 − x)M0 WV

  G1 Q2 = (eq1 + eq¯2 ) − Q2



where WV = M0 + m1 + m2 and

 1−x 2       x  2 1 2 2 02 1 − 2x − 2A M02 − m21 + m22 , m1 + k ⊥ m2 + k ⊥ − +M M0 1 − x x WV

 1−x 2  m1 + m2  2  2  −4 x  2 2 2 b= m1 + k ⊥ m2 + k ⊥ − − M0 − m1 + m22 , M0 1 − x x WV

2 2 m + m2 M0 m1 − m2 − + 1 , c=4 M0 WV M0 WV 

  2  2  1−x 2   x  2 1−x 2 2 2 (m1 + m2 ) d =4 m + k⊥ − m2 + k⊥ + (M0 − m1 − m2 ) M0 − m1 + m2 . M0 1−x 1 x a=

(30)

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C.-W. Hwang / Physics Letters B 516 (2001) 65–71

These Gi (Q2 ) form factors are related to the charge, magnetic and quadrupole form factors as [11]     2 Q2 GQ , GM Q2 = −G2 , GC Q2 = G1 + 3 4MV2    2 Q2 GQ Q = G1 + G2 + 1 + G3 . 4MV2

(31)

These form factors are proportional to the static quantities of charge e, magnetic moment µ, and quadrupole moment Q when Q2 = 0, GC (0) = eV ,

GM (0) = µ

(the unit is e/(2MV )), GQ (0) = QMV2 .  
The mean square radius of the vector meson r 2 V = 6 dGC (Q2 )/dQ2
Q2 =0 is

2



 r V = r 2 q + r 2 q¯ 1 2      d   2 dx d 2 k⊥ 2 ˜  2 d 2 2 2 2 = eq1 −6 x φV A + 2k⊥ + A + k⊥ k⊥ 2 2 2(2π)3 dk⊥ dk⊥

 2 4 k⊥ k⊥ − + φ˜ V 2 (xWV ) 2x(1 − x)M02 WV2      2  d   2 dx d 2 k⊥ d 2 2˜ 2 2 2 (1 − x) φV A + 2k⊥ + A + k⊥ k⊥ + eq¯2 −6 2 2 2(2π)3 dk⊥ dk⊥

 2 4 k⊥ k⊥ − + φ˜ V ((1 − x)WV )2 2x(1 − x)M02 WV2

1 . + 2 (G1 + G2 + G3 )

2 M V

(32)

(33)

Q =0

Similar to the pseudoscalar case, in addition to Eqs. (13) and (22), there are some behaviors existing M0 → mQ + O(mQ x),

WV → 2mQ + O(mQ x),

(34)

if the heavy quark limit is taken. We also obtain the Eqs. (23) and (24) for vector meson case. Thus the mean square radii of pseudoscalar and vector meson are equal r 2 P = r 2 V . This means that they satisfy the spin symmetry. Combine this relation and Eq. (23), we find that the light degrees of freedom are blind to the flavor and spin orientation of the heavy quark. This is the so-called heavy quark symmetry (HQS). Up to now, we have not designated any wave function yet, this also satisfies the well-known property that HQS is model-independent. Reviewing the processes, we can realize that, in this approach, the static properties of the EM form factors and the heavy quark symmetry of the mean square radii can be checked much more easily than in the Breit frame. This is the major reason why we calculate the Q2 dependence of those form factors order by order. In conclusion, we have calculated the EM form factors of the pseudoscalar and vector mesons. Specially, the EM form factors of vector meson are extracted from the relevant matrix elements directly, instead of choosing the Breit frame. We found that the charge radius is related to both the first and second longitudinal momentum square derivative of the momentum distribution function. We also found the static properties of the EM form factors and the heavy quark symmetry of the mean square radii are checked analytically by evaluating the Q2 dependence of those form factors order by order. Therefore, in the heavy quark limit, the charge radii of pseudoscalar and vector mesons have both spin and flavor symmetries, and these properties are model-independent.

C.-W. Hwang / Physics Letters B 516 (2001) 65–71

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Acknowledgements This work was supported in part by the National Science Council of R.O.C. under the Grant No. NSC89-2112M-007-054.

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