The chiral transitions in heavy-light mesons

# The chiral transitions in heavy-light mesons

A.M.Badalian    Yu.A.Simonov    M.A.Trusov ITEP, Moscow, Russia
December 22, 2007
###### Abstract

The mass shifts of the -wave and mesons due to coupling to , and , channels are studied using the chiral quark-pion Lagrangian without fitting parameters. The strong mass shifts down MeV and MeV for and and MeV for and are calculated. Two factors are essential for large mass shifts: strong coupling of the and states to the -wave decay channel, containing a Nambu-Goldstone meson, and the chiral flip transitions due to the bispinor structure of both heavy-light mesons. The masses MeV and MeV,very close to and , are predicted. Experimental limit on the width MeV puts strong restrictions on admittable mixing angle between the and states, , which corresponds to the mixing angle between the and states, .

###### pacs:
14.40.Lb, 12.39.Fe, 12.40.Yx

## I Introduction

The heavy-light (HL) mesons play a special role in hadron spectroscopy. First of all, a HL meson is the simplest system, containing one light quark in the field of almost static heavy antiquark, and that allows to study quark (meson) chiral properties. The discovery of the and mesons 1 (); 2 () with surprisingly small widths and low masses has given an important impetus to study chiral dynamics and raised the question why their masses are considerably lower than expected values in different approaches: in relativistic quark model calculations 3I ()6I (), on the lattice 7I (), in QCD Sum Rules 8I (); 9I (), in chiral models 10I ()12I ()( for reviews see also 13I (); 14I ()). The masses of and in closed-channel approximation typically exceed by 140 and 90 MeV their experimental numbers.

Thus main theoretical goal is to understand dynamical mechanism responsible for such large mass shifts of the and levels (both states have the light quark orbital angular momentum and ) and explain why the position of other two levels (with remains practically unchanged. The importance of second fact has been underlined by S.Godfrey in 5I ().

The mass shifts of the mesons have already been considered in a number of papers with the use of unitarized coupled-channel model 15I (), in nonrelativistic Cornell model 16I (), and in different chiral models 17I ()19I (). Here we address again this problem with the aim to calculate also the mass shifts of the and states and the widths of the and states, following the approach developed in 18I (), for which strong coupling to the S-wave decay channel, containing a pseudoscalar () Nambu-Goldstone (NG) meson, is crucially important. Therefore in this approach principal difference exists between vector-vector () and (or ) channels. This analysis of two-channel system is performed with the use of the chiral quark-pion Lagrangian which has been derived directly from the QCD Lagrangian 20I () and does not contain fitting parameters, so that the shift of the state 140 MeV is only determined by the conventional decay constant .

Here the term ”chiral dynamics” implies the mechanism by which in the transition from one HL meson to another the octet of the NG mesons is emitted. The corresponding Lagrangian ,

 ΔLFCM=¯q(σr)exp(iγ5ϕ/fπ)q, (1)

contains the light-quark part , where is the octet of NG mesons and the important factor is present. In the lowest order in this Lagrangian coincides with well-known effective Lagrangian suggested in 21I (),22I (), where, however, an arbitrary constant is introduced . At large , as argued in 21I (), this constant has to be equal unity, . In 10I (); 17I (); 22I () this effective Lagrangian was applied to describe decays of HL mesons taking .

More general Lagrangian (1) was derived in the framework of the field correlator method (FCM) 20I (); 23I (), in which the constant in all cases, and which contains NG mesons to all orders, as seen from its explicit expression (1).

In Appendix A with the use of the Dirac equation we show that in the lowest order in , if indeed . In our calculations we always use with the and derive the nonlinear equation for the energy shift and width, , as in 18I (). We do not assume any chiral dynamics for the unperturbed levels, which are calculated here with the use of the QCD string Hamiltonian 24I (); 25I (), because the mass shift appears to be weakly dependent on the position of unperturbed level.

It is essential that resulting shifts of the levels are large only for the mesons, which lie close to the thresholds, but not for the mesons, in this way violating symmetry between them (this symmetry is possible in close-channel approximation). In our calculations shifted masses of the and practically coincide with those for the and , in agreement with the experimental fact that MeV 26I () is equal or even larger than MeV. The states with and have no mass shifts and for them the mass difference is MeV, that just corresponds to the mass difference between the and light quark dynamical masses.

For the and mesons calculated masses are also close to those of the and mesons. Therefore for given chiral dynamics the states cannot be considered as the chiral partners of the ground-state multiplet , as suggested in 11I ().

We also analyse why two other members of the 1P multiplet, with and , do not acquire the mass shifts due to decay channel coupling (DCC) and have small widths. Such situation occurs if the states and appear to be almost pure and states. Still small mixing angle between them, , is shown to be compatible with experimental restriction on the width of , admitting possible admixture of other component in the wave function (w.f.) .

In our analysis the 4-component (Dirac) structure of the light quark w.f. is crucially important. Specifically, the emission of a NG meson is accompanied with the factor which permutes higher and lower components of the Dirac bispinors. For the -wave and the -wave states it is exactly the case that this ”permuted overlap” of the w.f. is maximal because the lower component of the first state is similar to the higher component of the second state and vice versa. We do not know other examples of such a ”fine tuning”.On the other hand in the first approximation we neglect an interaction between two mesons in the continuum, like ,etc.

In present paper we concentrate on the -wave mesons and the effects of the channel coupling. While the 1P levels of the mesons are now established with good accuracy 1 (),2 (),26I (), for the mesons only relatively narrow states have been recently observed 27I (),28I (). According to these data the splitting between the and levels is small, MeV, while the mass difference between and states is again MeV, as for the and mesons.

The actual position of the levels is important for several reasons. Firstly, since dynamics of mesons is very similar to that of , the observation of predicted large mass shifts of the levels would give a strong argument in favour of the decay channel mechanism suggested here and in 18I (). Secondly, observation of all -wave states for the , mesons could clarify many unclear features of spin-orbit and tensor interactions in mesons. Understanding of the decay channel coupling (DCC) mass shifts could become an important step in constructing chiral theory of strong decays with emission of one or several NG particles.

The paper is organized as follows. In the next Section we discuss the formalism from 18I (), extending that to the case of the and mesons and also to the states, and discuss the mixing between the and states. In Section 3 the masses of HL mesons, calculated in closed-channel approximation, are given. The Section 4 is devoted to the mechanism of chiral transitions while in Section 5 our calculations of the mass shifts due to DCC are presented.The predictions of the masses and discussion of our results are given in Section 6, while Section 7 contains the Conclusions. In Appendix A a connection between the lowest order of and the effective Lagrangian is illustrated. In Appendix B the details of our calculations of the masses are given, while in Appendix C the connection between FS splittings and the mixing matrix (and angle) of the states is discussed.

## Ii Mixing of the 1+ and 1+′ states

It is well known that in single-channel approximation, due to spin-orbit and tensor interactions the -wave multiplet of a HL meson is splitted into four levels with 29I (). Here for the states we use the notation H(L) for the higher(lower) eigenstate of the mixing matrix because apriori one cannot say which of them mostly consists of the light quark contribution (see Appendix C). For a HL meson, strongly coupled to a nearby decay channel (DC),some member(s) of the -wave multiplet can be shifted down while another not.Just such situationis takes place for the multiplet.

The scheme of classification, more adopted to a HL meson, in the first approximation treats the heavy quark as a static one and therefore the Dirac equation can be used to define the light quark levels and wave functions 10I (). Starting with the Dirac’s -wave levels, one has the states with and . Since the light quark momentum and the quantum number are conserved,111we use here the standard notation for they run along the following possible values:

 (2)

The HL meson w.f. can be expressed in terms of the light quark w.f. – the Dirac bispinors :

 ΨD(J−1/2,Mf)=CJ,Mf12,Mf−12;12,+12ψ12,0,Mf−12q⊗∣∣¯c↑⟩+CJ,Mf12,Mf+12;12,−12ψ12,0,Mf+12q⊗∣∣¯c↓⟩, (3)
 (4)

where are the corresponding Clebsch–Gordan coefficients.

Later in the w.f. we neglect possible (very small) mixing between , states and also between , states. However, physical states can be mixed via open channels and tensor interaction,while the and levels are obtained solely from and , respectively.

The eigenstates, defining the higher and lower levels, can be parametrized by introducing the mixing angle :

 |1+H⟩=cosϕ|j=12⟩+sinϕ|j=32⟩, (5)

and

 |1+L⟩=−sinϕ|j=12⟩+cosϕ|j=32⟩, (6)

where the mixing angle is defined by the unitary mixing matrix . In the heavy-quark limit the states with and are not mixed, but for finite they can be mixed and definition of the mixing matrix in this basis is rather complicate procedure 10I (), which is also model-dependent. Therefore it is more convenient to connect the angle in (5),(6) with the known factors in the basis, where is well defined in closed-channel approximation and factually depends only on the ratio , where is the spin-orbit and is the tensor splitting. For our analysis we do not need to know details of spin-orbit interaction (see Appendix C).

Then the splittings of the and levels are

 M(2+)−Mcog=a−0.1t, (7)
 M(0+)−Mcog=−2a−t,

while and in (5) and (6) can be expressed through the mixing angle in the expansion of these states in the basis, where they represent the decomposition of the and states:

 |1+H=cosθ| 3P1>−sinθ| 1P1>,
 |1+L=sinθ| 3P1>+cosθ| 1P1>. (8)

The states and in (8) can be expressed through the basis with the eigenstates and 29I ():

 | 3P1⟩=1√3|j=32⟩+√23|j=12⟩, (9)
 | 1P1⟩=√23|j=32⟩−1√3|j=12⟩.

Then in closed-channel approximation the following relation can be established between angles and :

 ϕ=−θ+35.264∘. (10)

From (10) it follows that

1. If is pure state (, then this state is the admixture of the and states with .

2. If is pure state (, then in the basis this state is admixture of the and states with .

3. The special case with corresponds to ”equal” mixing between the and states with the angle .

Therefore the solutions with small correspond to the mixing angle from the range: for which the ratio appears to be slightly smaller (larger) unity for small negative (positive) . Here we consider small negative .

The dependence of the mixing angle on the ratio is illustrated by Table 1 taking three different ratios .

Thus state as the pure corresponds to , while for slightly smaller ratio, , the admixture of the state is and for the admixture is already . Notice that the physical condition contradicts the heavy-quark limit when while and can have large magnitude.

The structure of the mixing is important because it defines the order of levels and the value of mass shift for the state, as well as the mass shift and the width of another level. It is important that if the coupling to nearby continuum channel is taken into account, then as follows from experiment, the position of the and levels does not change (within 1-3 MeV) and just their mass difference can be used to define tensor splitting: it can be derived that for any .

## Iii The masses of heavy-light mesons

In closed-channel approximation the masses of HL mesons, or initial positions of the levels (without channel coupling), can be calculated in different schemes, e.g. in the coupling 18I (), or as in the Dirac type coupling 10I (). In Tables 2, 3 we give these unperturbed masses for the and mesons which are calculated with the use of the relativistic string Hamiltonian 6I (); 24I (). In this approach the -wave masses of HL mesons appear to be smaller that in other potential models because they contain negative string corections (see Appendix B).

Calculated masses of the states with , and , for the ratio (see Table 2) appear to be in good agreement with recent DO Collab.measurements of the meson masses 28I (). Such agreement can also be reached for other values close to unity:

 R=at=1.0±0.05, (11)

and we take the same ratio for the , mesons, and also for the , mesons: for such choice the contribution from the state dominates in the meson. In particular for :

 ∣∣1+H⟩=0.9884∣∣12⟩−0.1517|32⟩,

and

 ∣∣1+L⟩=0.1517∣∣12⟩+0.9884|32⟩. (12)

The masses given in Tables 2, 3 are obtained taking the tensor splitting MeV and MeV for the and mesons, respectively. The tensor splittings have been determined to fit the mass difference , which has the important property — it does not change (within 2 MeV) if DCC is taken into account.

In Table 4 the masses of the mesons (from Tables 2,3) are compared to those obtained in other models; there the conventional notations and for the and states are used.

Comparison of the masses,given in Table 4, shows that in different papers differ not much, within MeV, however, the order of levels inside the 1P multiplet appears to be different.In particular, in 4I (), 10I () the level has smaller mass than the while in our calculations the state has always maximal mass.It means that FS parameters and their ratio, as well as the mixing angle between the and states, can differ essentially in given papers. Meanwhile existing experimental limit on the width of puts strict restrictions on admittable value of the mixing angle (see next Section).

Finally in Table 5 we give also unperturbed masses of the and mesons, taking the splitting MeV from the mass difference, MeV and , both for the and mesons.Notice that the position of the -wave mesons does not practically change if (or ).

Given in Table 5 masses show that in closed-channel approximation we have reached good agreement with experiment for all mesons and also for narrow mesons .

We do not need here to know the details of spin-orbit interaction which at present is not fully understood, probably, because of important role of one-loop (or even higher) corrections 31I () and possible suppression of NP spin-orbit potential observed on the lattice 32 (). Here we would like only to notice that in heavy quarkonia the ratios are also close or equal unity:

 a/t=1.04±0.08   for χb(1P);  a/t=(1.02±0.14) for χb(2P),
 a/t=(0.86±0.02) for χc(1P). (13)

## Iv Chiral Transitions

To obtain the mass shift due to DCC effect we use here the chiral Lagrangian (1), which includes both effects of confinement (embodied in the string tension) and Chiral Symmetry Breaking (CSB) (in Euclidean notations):

 LFCM=i∫d4xψ+(^∂+m+^M)ψ (14)

with the mass operator given as a product of the scalar function and the SU(3) flavor octet,

 ^M=W(r)exp(iγ5φaλafπ), (15)

where

 φaλa=√2⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝π0√2+η0√6,π+,K+π−,η0√6−π0√2,K0K−,¯K0,−2η0√6⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (16)

Taking the meson emission to the lowest order, one obtains the quark-pion Lagrangian in the form

 ΔLFCM=−∫ψ+i(x)σ|\boldmathx|γ5φaλafπψk(x)d4x. (17)

Writing the equation (17) as , one obtains the operator matrix element for the transition from the light quark state (i.e. the initial state of a HL meson) to the continuum state with the emission of a NG meson . Thus we are now able to write the coupled channel equations, connecting any state of a HL meson to a decay channel which contains another HL meson plus a NG meson.

In the case, when interaction in each channel and also in the transition operator is time-independent, one can write following system of equations (see 33 () for a review)

 [(Hi−E)δil+Vil]Glf=1. (18)

Such two-channel system of the equations can be reduced to one equation with additional DCC potential, or the Feshbach potential 34 (),

 (H1−E)G11−V121H2−EV21G11=1. (19)

Considering a complete set of the states in the decay channel 2 and the set of unperturbed states in channel 1, one arrives at the nonlinear equation for the shifted mass ,

 E=E(i)1−∑f⟨i|V12|f⟩1E(f)2−E⟨f|V21|i⟩. (20)

Here the unperturbed values of are assumed to be known (see Tables 2, 3, 5), while the interaction is defined in (17). A solution of the nonlinear equation (20) yields (in general a complex number one or more roots on all Riemann sheets of the complex mass plane.

## V Calculation of the DCC shifts

To calculate explicitly the mass shifts, we will use the Eq. (20) in the following form:

 m[i]=m(0)[i]−∑f||2Ef−m[i], (21)

where is the initial mass, – is the final one, is the energy of the final state, and the operator provides the transitions between the channels (see the comment after Eq. (17)).

In our approximation we do not take into account the final state interaction in the system and neglect the -meson motion, so the w.f. of the -states are:

 |f>=ΨK(p)⊗ΨD(Mf),|i>=ΨDs(Mi), (22)

where

 ΨK(p)=eipr√2ωK(p) (23)

is the plane wave describing the -meson and , are the HL meson w.f. at rest with the spin projections , , respectively.

We introduce the following notations:

 (24)

so that in the final state the total energy is , while

 Tf=Ef−mD−mK (25)

is the kinetic energy. Also it is convenient to define other masses with respect to nearby threshold: ,

 E0=m(0)[Ds]−mD−mK,δm=m[Ds]−m(0)[Ds], (26)
 Δ=E0+δm=m[Ds]−mD−mK, (27)

where determines the deviation of the meson mass from the threshold. In what follows we consider unperturbed masses of the () levels as given (our results do not change if we slightly vary their position, in this way the analysis is actually model-independent).

Using these notations, the Eq.(20) can be rewritten as

 E0−Δ=F(Δ), (28)

where

 (29)

and

 ⟨Mi∣∣^V∣∣p,Mf⟩=−∫Ψ†Ds(Mi)σ|r|γ5√2fKΨD(Mf)eipr√2ωK(p)d3r, (30)

The function for negative diminishes monotonously so there exists a final (critical) point,

 Ecrit0=F(−0). (31)

Thus, while solving the Eq.(28), one has two possible situations: and .

In the first case Eq.(28) has a negative real root (see Fig. 1) and the resulting mass of the meson appears to be under the threshold. In the second case Eq.(28) has a complex root with positive real part (see Fig. 2) and negative imaginary part . To find latter solutions one should make analytic continuation of the solution(s) from the upper halfplane of under the cut, which starts at the threshold, to the lower halfplane (second sheet). This solution can be also obtained by deforming the integration contour in . In actual calculations we take infinitesimal imaginary part , proving that does not change much for finite (the similar procedure has been used in 18I ()). Finally, the resulting mass of the meson proves to be in the complex plane at the position , i.e. the meson has the finite width .

For further calculations we should insert the explicit meson w.f. to the matrix element (30). As discussed above, in a HL meson we consider a light quark moving in the static field of a heavy antiquark , and therefore its w.f. can be taken as the Dirac bispinor:

 ψjlMq=⎛⎝g(r)ΩjlM(−1)1+l−l′2f(r)Ωjl′M⎞⎠,∞∫0(f2+g2)r2dr=1, (32)

where the functions and are the solutions of the Dirac equation:

 g′+1+ϰrg−(Eq+mq+U−VC)f=0,f′+1−ϰrf+(Eq−mq−U−VC)g=0. (33)

Here the interaction between the quark and the antiquark is described by a sum of linear scalar potential and the vector Coulomb potential with :

 U=σr,VC=−βr,β=43αs. (34)

Introducing new dimensionless variables

 x=r√σ,εq=Eq/√σ,μq=mq/√σ, (35)

and new dimensionless functions

 g=σ3/4G(x)x,f=σ3/4F(x)x,∞∫0(F2+G2)dx=1, (36)

we come to the following system of equations:

 G′+ϰxG−(εq+μq+x+βx)F=0,F′−ϰxF+(εq−μq−x+βx)G=0. (37)

This system has been solved numerically.

Using the parameters from the papers 18I (), dirac ():

 σ=0.18 GeV2,αs=0.39,ms=210 MeV,mq=4 MeV, (38)

we obtain the following Dirac eigenvalues :

, ,
-1 1.0026 1.28944
+1 1.7829 2.08607
-2 1.7545 2.08475
(39)

and corresponding eigenfunctions , are given in Figs. 3, 4).

Later we use the simplified notations for the quark bispinors:

 ψ1(M1)def=ψ12,1,M1s,ψ2(M2)def=ψ12,0,M2q,ψ3(M3)def=ψ32,1,M3s. (40)

Now, using explicit expressions for the spherical spinors,

 (41)

and the expansion :

 eipr=4π∑l,Miljl(pr)Y∗l,M(pp)Yl,M(rr), (42)

after cumbersome transformations (which are omitted in the text) we obtain the transition matrix elements:

 ∥∥V12∥∥M1,M2=−∫ψ†1(M1)σ|r|γ5√2fKψ2(M2)eipr√2ωK(p)d3r==√σfK√ωK(p)Φ0(p√σ)√4πY∗0,M1−M2(pp), (43)
 ∥∥V32∥∥M3,M2=−∫ψ†3(M3)σ|r|γ5√2fKψ2(M2)eipr√2ωK(p)d3r==−√σfK√ωK(p)Φ2(p√σ)√4π5Y∗2,M3−M2(pp)⋅⎡⎢ ⎢ ⎢ ⎢⎣−1+2−√2+√3−√3+√2−2+1⎤⎥ ⎥ ⎥ ⎥⎦. (44)

where

 Φ0(q)=∞∫0j0(qx)xdx[G1(x)F2(x)−F1(x)G2(x)],Φ2(q)=∞∫0j2(qx)xdx[G3(x)F2(x)−F3(x)G2(x)]. (45)

Notice that because of different signs of the and functions (while the functions are all positive) on almost all real axis, the integral appears to be strongly suppressed in comparison with the integral . This fact is confirmed by numerical simulations (see Fig. 5).

Finally, introducing universal functions

 ~F0,2(Δ)=σ2π2f2K∞∫0p(Tf)ωD(Tf)dTfTf+mD+mK⋅Φ20,2(p(Tf)√σ)Tf−Δ,~Γ0,2(Tf)=σπf2K⋅p(Tf)ωD(Tf)Tf+mD+mK⋅Φ20,2(p(Tf)√σ), (46)

we come to the following equations to determine meson masses and widths:

 Ds(0+)E0[0+]−Δ=~F0(Δ),Ds(1+L)E0[1+L]−Δ=cos2ϕ⋅~F0(Δ)+sin2ϕ⋅~F2(Δ),Ds(1+H)E0[1−H]−Δ′=sin2ϕ⋅~F0(Δ′)+cos2ϕ⋅~F2(Δ′),Γ[1+H]=sin2ϕ⋅~Γ0(Δ′)+cos2ϕ⋅~Γ2(Δ′),Ds(2+3/2)E0[2+3/2]−Δ′=35⋅~F2(Δ′),Γ[2+3/2]=35⋅~Γ2(Δ′). (47)

## Vi Results and discussion

In this chapter, using the expressions (47) to define the and meson mass shifts, we present and discuss our results. We will take into account the following pairs of mesons in coupled channels ( refers to first (initial)channel, while refers to second (decay) one):

 (48)

In our calculations we use here the following meson masses and thresholds (in MeV):

 mD+=1869,mD++mK−=2363,mD∗+=2010,mD∗++mK−=2504,mB+=5279,mB++mK−=5772,mB∗=5325,mB∗+mK−=5819. (49)