Generalized structure of hadron-quark vertex function in Bethe-Salpeter framework Applicati
SDU-HEP200505 Generalized structure of hadron-quark vertex function in Bethe-
Salpeter framework: Applications to leptonic decays of V-mesons
Shashank Bhatnagar1
Department of Physics, Addis Ababa University, P.O.Box 101739, Addis Ababa,
Ethiopia
and
Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy
Shi-Yuan Li2
Department of Physics, Shandong University, Jinan, 250100, P.R. China
and
Institute of Particle Physics, Huazhong Normal University, Wuhan, 430079, P. R. China Abstract: We employ the framework of Bethe-Salpeter equation under Covariant Instantaneous Ansatz to study the leptonic decays of vector mesons. The structure of hadron-quark vertex function Γ is generalized to include various Dirac covariants (other γ.i) from their complete set. They are incorporated in accordance with a na?ve than ε
power counting rule order by order in powers of the inverse of the meson mass. The ρ, and φ mesons are calculated with the incorporation of leading decay constants for ω
order covariants.
PACS: 11.10.St, 12.39.Ki, 13.20.-v, 13.20.Jf
1. Regular Associate, ICTP, shashank_bhatnagar@https://www.wendangku.net/doc/7d1280442.html,
2. lishy@https://www.wendangku.net/doc/7d1280442.html,
1. Introduction:
Meson decays provide an important tool for exploring the structure of these simplest bound states in QCD and for studying the non-perturbative (long distance) behavior of strong interactions. Since the task of calculating hadron structure from QCD alone is very difficult, one relies on specific models of hadron dynamics to gain some understanding of hadronic structures at low energies. This study can most effectively be accomplished by applying a particular framework of hadron dynamics to a diverse range of phenomena. A number of studies in nonperturbative QCD have been carried out on pseudo-scalar mesons. However, the situation is different in case of vector mesons. Flavorless vector mesons play an important role in hadron physics due to their direct coupling to photons and thus provide an invaluable insight into phenomenology of electromagnetic couplings to hadrons. Thus a realistic description of vector mesons at the quark level of compositeness would be an important element in our understanding of hadron dynamics and reaction processes. There have been a number of recent studies [1-6] on processes involving strong decays, radiative decays and leptonic decays of vector mesons. Such studies offer a direct probe of hadron structure and help in revealing some aspects of the underlying quark-gluon dynamics that are complementary to what is learnt from pseudo-scalar mesons.
Thus in this paper we study leptonic decays of vector mesons (Hereafter, we use the short form V-mesons) which proceed via one photon annihilation of q q pair constituting V-meson through quark loop diagram (Fig.1). We employ a QCD oriented framework of Bethe-Salpeter Equation (BSE) under Covariant Instantaneous Ansatz (CIA) [7]. CIA is a Lorentz-invariant generalization of Instantaneous Approximation (IA). For q q system, CIA formulation [7] ensures an exact interconnection between 3D and 4D forms of the BSE. The 3D form of BSE serves for making contact with the mass spectra of hadrons,
whereas the 4D form provides the q Hq vertex function )(q )
Γfor evaluation of transition amplitudes. A BS framework under IA was also earlier suggested by Bonn group [8] where they have applied their formalism to various physical processes.
We had earlier employed the framework of BSE under CIA for calculation of decay constants [7, 9] of heavy-light pseudoscalar mesons and calculation of in process. In a recent work we had evaluated the leptonic decays of V-mesons (such as πf γπ20→φωρ,,) [10] in this framework. However, one of the simplified assumptions in all these calculations was the fact that the BS wave function was restricted to have a single Dirac structure (e.g., 5γ for P-mesons, εγ. for V-mesons etc.). However recent studies [2, 11-13] have revealed that various mesons have many different covariant structures in their wave functions whose inclusion was also found necessary to obtain quantitatively accurate observables [2].
Hence the present work grew out as a first attempt to develop and use a BS wave function for a hadron, which would incorporate more than a single Dirac structure for calculation of transition amplitudes for various processes. We first postulate and discuss a power counting rule for choosing various Dirac covariants from their complete set (see, e.g., [2,
11-13]) for V-mesons’ wave functions in Section 2. In section 3 we calculate leptonic decay constants of V-mesons using the wave function developed in section 2. We conclude with discussions in Section 4.
2. Structure of generalized vertex function )(q )
Γ in BSE under CIA:
We first outline the CIA framework taking the case of scalar “quarks” for simplicity. For a q q system with an effective kernel K and 4D wave function ),(q P Φ, the 4D BSE takes the form,
)',()',('),()2(4214q P q q K q d q P i Φ=ΦΔΔ∫π, (2.1) where are the inverse propagators of two scalar quarks. Hereafter, we use the free particle propagator, i.e., 2,1Δ
2
2,122,12,1p m +=Δ , (2.2) where are (effective) constituent masses of quarks. The 4-momenta of the quark and anti-quark, , are related to the internal 4-momentum of the hadron by 2,1m 2,1p μq
μμμq P m
p ±=2,12,1? . (2.3) Here is the total 4-momentum of the hadron with mass 21p p P +=M . 2/]/)(1[222212,1M m m m ?±=) are the Wightman-Garding (WG) definitions [9] of masses of individual quarks. We now use an ansatz on the BS kernel K in Eq. (2.1), which is
assumed to depend on 3D variables ',μμq q )
), i.e.,
)',()',(q q K q q K )
)= (2.4) where
μμμP P
P q q q 2.?=)
(2.5)
is observed to be orthogonal to the total 4-momentum P ( i.e., 0.=P q )
) irrespective of whether the individual quarks are on-shell or off-shell. A similar form of the BS kernel was also suggested in ref. [8]. Hence the longitudinal component of , μq
2.P
P
q M M =σ (2.6) does not appear in the form )',(q q K )
) of the kernel. For reducing Eq. (2.1) to the 3D form,
we define a 3D wave function )(q )
φ as ∫+∞
∞
?Φ=
),()(q P Md q σφ). (2.7)
Substituting Eq.(2.7) in Eq.(2.1) with the definition of kernel in Eq.(2.4), we get a covariant version of the Salpeter Equation [14],
)'()',(')()()2(33q q q K q d q q D )
)))))φφπ∫= , (2.8)
where )(q D )
is a 3D denominator function defined by
∫+∞
∞
?ΔΔ=2121
)(1σπMd i q D )
(2.9) whose value can be easily worked out by contour integration by noting the positions of the poles in the complex ?σplane (as shown in detail in Ref.[10]). We note that the RHS of Eq. (2.8) is identical to RHS of Eq. (2.1) by virtue of Eqs.(2.4) and (2.7). We thus have
an exact interconnection between 3D wave function )(q )
φand the 4D wave function : ),(q P Φ
)(2)()(),(21q i
q q D q P )
))Γ≡=ΦΔΔπφ (2.10)
where )(q )
Γ is the BS vertex function under CIA. The exact interconnection between 3D and 4D forms of BSE under CIA is thus brought out. The 3D form serves for making contact with the mass spectrum of hadrons, whereas the 4D form provides the
q Hq vertex function which satisfies a 4D BSE with a natural off-shell extension over the entire 4D space (due to the positive definiteness of the quantity
)?(q
Γ2
22
2).(?P P q q q
?= throughout the entire 4D space) and thus provides a fully Lorentz-invariant basis for evaluation of various transition amplitudes through various quark loop diagrams. Due to these properties, this framework can be profitably employed not only for mass spectral predictions but also for evaluation of various transition amplitudes [7,9,10] all the way from low energies to high energies.
We now outline the framework of BSE under CIA for the case of fermionic quarks
constituting a particular meson. The scalar propagators
in the above equation are replaced by the proper fermionic propagators . Now comes the problem of
incorporation of relevant Dirac structures in vertex function 1
?Δi F S )?(q
Γ. In this connection we wish to state that in applications of BSE under CIA until now, the q Hq vertex function was restricted to have a single Dirac structure. However recent studies [2,11-13] have revealed that various mesons have many different covariant structures in their wave functions and their inclusion was also found necessary to obtain quantitatively accurate observables [2]. Hence the present study attempts to incorporate other Dirac covariants in the structure of q Hq vertex function systematically. Thus to incorporate the relevant Dirac structures in the vertex function, we take guidance from some of the recent works [2, 11-13] where the transverse q Hq vertex function for a V-meson has been expressed as a linear combination of eight Dirac covariants [2, 11-13], (i=0,…,7), each multiplying a Lorentz scalar amplitude . Similarly the V i Γ),.,(22P P q q F i q Hq vertex
functions for both pseudo-scalar and scalar mesons are expressible as a linear combinations of four Dirac covariants and respectively [2, 11-13]. (The discussion on pseudoscalar and scalar mesons will be relegated to a separate paper.) However, the choice of these covariants is not unique, as can also be seen from the choice of the eight covariants employed in Ref. [2, 11-13] for a V-meson. What we want to do is to find a “criterion” so as to systematically choose among these eight Dirac covariants. In this connection we wish to state that in Ref. [2, 11] for a V-meson, it was noticed that the covariant )3,...,0(=Γi P i )3,...,0(=Γi S i μγ?1=Γ (where is the transverse projection of the four vector 2/).(?P P P γγγμμμ?=μγ) was considered to be the most important one. Such ‘leading order’ covariants (εγ.i for V-meson, 5γfor P-meson etc.) were used in our work so far for calculation of some transition amplitudes [7, 9, 10] (since it can be noticed that εγ. is the same as εγ.? on account of 0.=εP with με the polarization vector). These were also used in earlier calculations of various transition amplitudes and mass spectra of hadrons in BSE under Null-Plane Ansatz (NPA) [15]. On the other hand in Ref. [2, 11] only five of the eight covariants (ie. ) were considered to be important for calculation of vector meson masses and decay constants. Further in Ref. [2, 11], calculations have also been made for masses and decay constants of V-mesons for various other subsets of eight covariants (see table III of Ref.[2]). 51,...,ΓΓ
On lines of these studies, we try to generalize the structure of q Hq vertex function for a V-meson
)?()?().(21
)?(q q
D i N i
q V φεγπ=Γ (2.11) which we had used earlier [10]. For incorporating the Dirac structures in the expression
for we take their forms as in Ref. [12]. We notice that in the expression for the CIA vertex function, in Eq. (2.11), the factor )?(q
Γ)?(q
Γ)?()?(q q D φ multiplying εγ.i is nothing but the Lorentz invariant momentum dependent scalar which depends upon and (see Sec. 3 of this paper) and thus has a certain dimensionality of mass. However, the Lorentz scalar amplitudes , (22,P q P q . ),(P q i Γ,...2,1,0=i ) multiplying the various Dirac structures in Ref. [12] have different dimensionalities of mass. For adapting this decomposition to
write the structure of vertex function )?(q
Γ, we re-express the q Hq vertex functions for V-meson (see Eq.(10) of Ref. [12]) by making these coefficients ),(P q i Γ dimensionless, weighing each covariant with an appropriate power of M , the meson mass. Thus each
term in the expansion of is associated with a certain power of )?(q
ΓM . In detail, we can express as μμεV
Γ
)?()?(21
.
q q
D N i
V V φπεεμμμμΩ=Γ; (2.12)
3
72625423
2
1
0)].)(.().)(.)[(.().)(.().)(.().()].)(.(2).)(.)(.().)(.)(.[()].)(.(.[).)(.().(M
A P q q P q M A q q i M A P q i M A q P q i P q q P M
A i
M A q q M
A P A i γγγγεγεγεεγεγγεγγγεγγεγεγεγεγεμμ?+?+++?+?++=Ω where (7) are eight dimensionless and constant coefficients (why they can be constant will be explained in the following) to be determined. Now since we use constituent quark masses where the quark mass m is approximately half of the hadron mass i A ,...,0=i M , we can use the ansatz
M P q ~<< (2.13) in the rest frame of the hadron. Then each of the eight terms in Eq.(2.12) receives suppression by different powers of . Thus we can arrange these terms as an
expansion in powers of M /1)1
(M
O . We can then see in the expansion of μμεΩ that the
structures associated with the coefficients have magnitudes 10,A A )1
(0M
O and are of leading order. Those with are 5,432,,,A A A A )1(1M O , while those with are 76,A A )1
(2M
O .
This na?ve power counting rule suggests that the maximum contribution to the calculation of any vector meson observable should come from the Dirac structures εγ.i and
M
P 1
).)(.(γεγ associated with the constant coefficients and respectively. As a first
attempt we take the form of the 0A 1A q Hq vertex function incorporating these leading order
terms in expansion (2.12) and ignoring )1(1M O and )1
(2M
O terms for the moment and
try to calculate the V -meson decay constants taking only these leading order terms. Thus we take the modified form of q Hq vertex function
)?()?(21
].).)(.(.[)?(10q q
D N i
M A P A i q V φπγεγεγ+=Γ . (2.14) For the flavourless vector mesons, which are eigenstates of the charge parity, there is an extra restriction on the use of the Dirac structures [13]. In general, the coefficients of the Dirac structures could be functions of i A P q ?, hence can be written as a Taylor series in powers of . However, the coefficients here used are dimensionless. Hence the
various terms in the series should be powers of P q ?2M
P q ?, which is of order O(M 1
). If we
want to keep the leading contributions, as discussed above, we should only keep the zeroth-order terms of the Taylor series. This justifies the usage of and etc. as constant in the above equations. But now we choose only the C-even part of the coefficients, since only odd powers of q are C-odd. Hence only the proper C value Dirac
0A 1A
structures can be used. The Dirac structures used in (2.14) are consistent with the charge parity of the flavourless vector mesons (the second term originates from νμνσP ). In this paper we will investigate the numerical results to this order.
In a similar manner one can express the full hadron-quark vertex function for a pseudoscalar, scalar and axial vector mesons, etc., in BSE under CIA, taking guidance from Ref. [12]. Then we can incorporate the Dirac structures according to the power counting rule. At the same time, the restriction by charge parity should also be respected.
From the above analysis of the structure of q Hq vertex function (in Eq.(2.14)) we notice
that the structure of 3D wave function )?(q
φas well as the form of the 3D BSE (Eq.(2.8)) are left untouched and have the same form as in our previous works which justifies the usage of the same form of the input kernel we used earlier. Now we briefly mention some features of the BS formulation employed. The structure of BSE is characterized by a single effective kernel arising out of a four-fermion lagrangian in the Nambu-Jonalasino [15-16] sense. The formalism is fully consistent with Nambu-Jona-Lasino [16] picture of chiral symmetry breaking but is additionally Lorentz-invariant because of the unique
properties of the quantity , which is positive definite throughout the entire 4D space. The input kernel in BSE is taken as one-gluon-exchange like as regards color (2?q
)',(q q K 22
1.21λλr r ) and spin() dependence. The scalar function )2()1(μμγγ)'(q q V ? is a sum of one-gluon exchange and a confining term [9-10,15]. Thus
OGE V .Conf V )'(21.2
1)',()
2()1()2()1(q q V V V q q K ?=μμλλr r ;
)
2,1(2,1)2,1(2μμγm V ±=;
r q q i q q S e C r M m m a r r d q q q q V r r ).'??(20021
222102322
])??41([43)
'??(4)'??(???++?=?∫ωωπα; (2.15) 29.,028.00==C a .
The values of parameters have been calibrated to fit the meson mass spectra
obtained by solving the 3D BSE [15]. Here in the expression for , the constant term is designed to take account of the correct zero point energies, while term (1) simulates an effect of an almost linear confinement for heavy quark sectors (large ) , while retaining the harmonic form for light quark sectors (small ) [15]. This representation is thus asymptotically consistent with linear confinement (as is believed to be true for QCD) though the intervening length scales in light quark sectors give it an effectively harmonic appearance. Now comes to the problem of the 3D BS
wave function. The ground state wave function 00,C a )'??(q q
V ?2