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Chiral Lagrangian with Heavy Quark-Diquark Symmetry

We construct a chiral Lagrangian for doubly heavy baryons and heavy mesons that is invariant under heavy quark-diquark symmetry at leading order and includes the leading O(1/m_Q) symmetry violating operators. The theory is used to predict the electromagnetic decay width of the J=3/2 member of the ground state doubly heavy baryon doublet. Numerical estimates are provided for doubly charm baryons. We also calculate chiral corrections to doubly heavy baryon masses and strong decay widths of low lying excited doubly heavy baryons.

Heavy quark-diquark symmetry relates mesons with a single heavy quark to antibaryons with two heavy antiquarks. Savage and Wise [1] argued that quark-diquark symmetry was realized in the heavy quark limit of Quantum Chromodynamics (QCD) and studied this symmetry using the methods of Heavy Quark Effective Theory (HQET) [2]. Recently, Refs. [3,4] derived effective Lagrangians for heavy diquarks within the framework of Non-Relativistic QCD (NRQCD) [5,6,7]. These papers obtain a prediction for the hyperfine splitting of the ground state doubly heavy baryons in terms of the ground state heavy meson hyperfine splitting 1 . Heavy quark-diquark symmetry also relates other properties of heavy mesons and doubly heavy baryons. A useful tool for studying low energy strong and electromagnetic interactions of heavy hadrons is heavy hadron chiral perturbation theory (HHχPT) [8,9,10]. This theory has heavy hadrons, Goldstone bosons, and photons as its elementary degrees of freedom and incorporates the approximate chiral and heavy quark symmetries of QCD. In this paper we derive a version of HHχPT that includes doubly heavy baryons and incorporates heavy quark-diquark symmetry. The theory is used to calculate chiral corrections to doubly heavy baryon masses and to obtain model-independent predictions for the electromagnetic decay of the J = 3 2 member of the ground state doubly heavy baryon doublet. Our formulae are applicable to either doubly bottom or doubly charm baryons, and we give numerical estimates for the case of doubly charm baryons. We also discuss the low lying excited doubly heavy baryons, show how these states can be included in the effective theory, and calculate their strong decay widths.
Motivation for this work comes from the SELEX experiment's recent observation of states which have been tentatively interpreted as doubly charm baryons [11,12,13], and also the COMPASS experiment, which in its second phase run in 2006 hopes to observe doubly charm baryons [14]. Many aspects of the SELEX states are difficult to understand. States observed by SELEX include the Ξ + cc (3520), which decays weakly into Λ + c π + K − [11] as well as pD + K − [13], the Ξ ++ cc (3460), which decays weakly into Λ + c K − π + π + [12], and a broader state, Ξ ++ cc (3780), also seen to decay into Λ + c K − π + π + [12]. The ground states of the Ξ + cc and Ξ ++ cc are related by isospin symmetry and therefore should differ in mass by only a few MeV, so the observed difference of 60 MeV seems implausible. On the other hand, an unpublished talk [15] and conference proceedings [16] present evidence for additional states, Ξ + cc (3443) and Ξ ++ cc (3541). If these states exist the isospin splittings are closer to theoretical expectations, but still quite large. The difference between the mass of the Ξ + cc (3520) and Ξ + cc (3443) is 77 MeV, and the splitting between Ξ ++ cc (3541) and Ξ ++ cc (3460) is 81 MeV. These splittings agree remarkably well with calculations of the doubly charm hyperfine splittings in quenched lattice QCD [17,18,19] and are within ∼ 25 MeV of the heavy quark-diquark symmetry prediction obtained in Refs. [3,4], an acceptable discrepancy given the expected O(Λ QCD /m c ) corrections. However, interpretation of the Ξ + cc (3520) as the J = 3 2 member of the ground state doublet is impossible to reconcile with the fact that the Ξ + cc (3520) is observed to decay weakly because if the Ξ + cc (3520) is not the ground state of the ccd system it should decay electromagnetically. There are also discrepancies between the weak decay lifetimes predicted by HQET [20,21,22](∼ 100 fs) and the observed lifetimes (< 33 fs) [11,13]. Production cross sections are also poorly understood within perturbative QCD [23]. However, the SELEX states are observed in the forward region, where nonperturbative production mechanisms such as intrinsic charm [24,25,26] or parton recombination [27,28] may be important.
Even if there is difficulty interpreting the SELEX data, doubly charm baryons must exist and are expected to have masses of approximately 3.5 GeV, where the SELEX states are.
In light of existing and future experimental efforts to observe doubly charm baryons, it is desirable to have model independent predictions for other properties besides the relation for the hyperfine splittings derived in Refs. [1,3,4]. Therefore it is important to develop theoretical tools for analyzing the properties of doubly heavy baryons systematically.
Savage and Wise [1] wrote down a version of heavy quark effective theory (HQET) which includes diquarks as elementary degrees of freedom and derived a formula relating heavy meson and doubly heavy baryon hyperfine splittings. HQET only separates the scales Λ QCD and m Q , where m Q is the heavy quark mass. The dynamics of a bound state of two heavy quarks is characterized by additional scales m Q v and m Q v 2 , where v is the typical velocity of the heavy quarks in the bound state. The correct effective theory for hadrons with two heavy quarks is Non-Relativistic QCD (NRQCD) [5], which properly accounts for the scales m Q v and m Q v 2 . Analysis of heavy diquarks within the framework of NRQCD was recently performed in Refs. [3,4]. These papers derived Lagrangians for diquark fields starting from NRQCD and obtained the correct heavy quark symmetry prediction for the hyperfine splittings of the doubly heavy baryons. For simplicity, we will consider only one flavor of heavy quark. The lowest mass diquark will consist of two heavy antiquarks in an orbital S-wave in the 3 representation of color SU (3). Then Fermi statistics demands that they have total spin one. In the rest frame of the heavy quark and lowest mass diquark, the Here h is the heavy quark field, V is the field for the diquarks, the λ a /2 are the SU (3) color generators, Tr[λ a λ b ] = 2 δ ab , D 0 and D are the time and spatial components of the gauge covariant derivative, respectively, B a is the chromomagnetic field, and m Q is the heavy quark mass. The term proportional to δ is the residual mass of the diquark. The heavy antiquarks in the diquark experience an attractive force and therefore the mass of the diquark is not 2m Q but 2m Q − δ, where δ is the binding energy. This residual mass can be removed by rephasing the diquark fields. Physically, this corresponds to measuring diquark energies relative to the mass of the diquark, rather than 2m Q . Once this is done the Here P * µ v is the J P = 1 − vector heavy meson field which obeys the constraint v µ P * µ v = 0, where v µ is the four-velocity of the heavy meson.
where we have used the Bjorken-Drell conventions for γ µ and γ 5 . For a process such as the weak decay B → Dℓν, in which the initial and final heavy hadrons have different four-velocities, the covariant representation of fields is needed to determine heavy quark symmetry constraints on heavy hadron form-factors. However, for studying low energy strong and electromagnetic interactions in which the heavy meson four-velocity is conserved , it is also possible to work in the heavy meson rest frame and use 2 × 2 matrix fields. This makes some calculations simpler and we find it easiest to formulate the extension of HHχPT with U(5) quark-diquark symmetry in this frame. We define the heavy meson field in our theory to be where a is an SU ( Here U and S are 2 × 2 rotation matrices and V † ba is an SU(3) matrix which gives the standard nonlinear realization of SU L (3) × SU R (3) chiral symmetry. In the two component notation the HHχPT Lagrangian is: The last term breaks heavy quark spin symmetry and ∆ H is the hyperfine splitting of the heavy mesons. The time component of the covariant chiral derivative is (D 0 ) ba , A ba is the spatial part of the axial vector field, and g is the heavy meson axial coupling. Our definitions for the chiral covariant derivative, the axial current, and the Lagrangian for the Goldstone boson fields are the same as Ref. [29].
We are now ready to generalize the Lagrangian to incorporate the doubly heavy baryons and the U(5) quark-diquark symmetry. The field H a transforms like the tensor product of a heavy quark spinor and a light antiquark spinor. (This is how representations of heavy hadron fields were constructed in Ref. [30].) Writing the field with explicit indices, (H a ) αβ , the index α corresponds to the spinor index of the heavy quark and the index β is that of the light antiquark spinor. In the theory with quark-diquark symmetry, the heavy quark spinor is replaced with a five-component field, the first two components corresponding to the two heavy quark spin states and the last three components corresponding to the three spin states of the diquark: In terms of Q µ the kinetic terms of the Lagrangian in Eq.(1) are The fields in HHχPT with heavy quark-diquark symmetry transform as tensor products of the five component field Q µ and a two-component light antiquark spinor. Thus, the 2 × 2 matrix field H a is promoted to a 5 × 2 matrix field Here the index µ takes on values between 1 and 5, α, β = 1 or 2, and i = 3, 4, or 5. The doubly heavy baryon fields are contained in T a,iβ . Under the symmetries of the theory H a transforms as The matrix S is now an element of U(5) and R is a 5 × 5 reducible rotation matrix where U αβ is an SU(2) rotation matrix and R ij is an orthogonal 3×3 rotation matrix related Under rotations the field T a,iβ transforms as T ′ a,iβ = R ij T a,jγ U † γβ . T a,iβ can be further decomposed into its spin-3 2 and spin-1 2 components, where Ξ * a,iβ and Ξ a,γ are the spin-3 2 and spin-1 2 fields, respectively. The factor of √ 2 is a convention that ensures that the kinetic terms of the doubly heavy baryon fields have the same normalization as the heavy meson fields. The field Ξ * a,iβ obeys the constraint Ξ * a,iβ σ i βγ = 0. The U(5) invariant generalizations of the first two terms of Eq. (6) are simply obtained by making the replacement H a → H a . To determine the proper generalization of the U (5) breaking term we note that the chromomagnetic couplings in Eq. (1) can be written as where the Σ µν are the 5 × 5 matrices and (T i ) jk = −iǫ ijk . It is now obvious that the correct generalization of Eq. (6) is The last line of Eq. (16) contains the terms relevant for doubly heavy baryons. Heavy quark-diquark symmetry relates the couplings in the doubly heavy baryon sector to the heavy meson sector. The propagator for the spin-1 2 doubly heavy baryon is while the propagator for the spin-3 2 doubly heavy baryon is .
We can also include operators which mediate electromagnetic decays. The Lagrangian for electromagnetic decays of the heavy mesons in the two-component notation is [31] where Q ab = diag(2/3, −1/3, −1/3) is the light quark charge matrix, β is the parameter introduced in Ref. [31], and Q ′ is the heavy quark charge. For charm, Q ′ = 2/3. The first term is the magnetic moment coupling of the light degrees of freedom and the second term is the magnetic moment coupling of the heavy quark. Both terms are needed to understand the observed electromagnetic branching fractions of the D * + and D * 0 [31]. The magnetic couplings of the heavy quark and diquark are where the Σ ′ µν are the 5 × 5 matrices The magnetic coupling of the diquark has the opposite sign as that of the heavy quark because it is composed of two heavy antiquarks. The coefficient of the chromomagnetic coupling of the diquark in Eqs. (1,14) is a factor of 2 smaller than the coefficient of the electromagnetic coupling of the diquark in Eq. (19) due to a color factor. The magnetic couplings in the HHχPT Lagrangian for heavy mesons and doubly heavy baryons are The part of this Lagrangian involving the doubly heavy baryon fields is This can be used to obtain the following tree level predictions for the electromagnetic decay widths: Here E γ is the photon energy. These results could also be obtained in the quark model, with the parameter β = 1/m q , where m q is the light constituent quark mass. The effective theory allows one to improve upon this approximation by including corrections from loops with light Goldstone bosons, which give O( √ m q ) corrections to the decay rates [31]. If these loop corrections are evaluated in an approximation where heavy hadron mass differences are neglected, the correction to the above formulae can be incorporated by making the following replacements [31]  Ref. [29] should be performed. In this paper, we will use Eqs. (23) and (24)     m c = 1500 MeV and performed a least squares fit to β. There are several sources of error in the calculation. We expect 30% theoretical errors due to heavy quark symmetry breaking effects, 30% errors due to higher order SU(3) breaking effects, and 25% uncertainty from the experimentally measured value of Γ[D * + ] leading to at least 50% error in the predictions in Table I. loop correction whose contribution to the Ξ * ++ cc decay has opposite sign as the contribution from β at tree level, therefore suppressing the Ξ * ++ cc decay relative to Ξ * + cc . The theory can also be used to compute chiral corrections to doubly heavy baryon masses.
The one loop corrections to the hadron masses are Here m i and f i are the mass and decay constant of the Goldstone boson in the one loop diagram and C i ab is a factor which comes from SU (3) Clebsch-Gordan coefficients in the couplings. For loops with charged pions we have C π ± 12 = C π ± 21 = 1, for loops with neutral pions C π 0 11 = C π 0 22 = 1 2 , for loops with kaons C K 3i = C K i3 = 1 (i =1 or 2), and for loops with η mesons C η 11 = C η 22 = 1 6 and C η 33 = 2 3 . The function K(δ, m, µ) is related to the finite part of the integral evaluated using dimensional regularization in the MS scheme. We find where and µ is the renormalization scale. The µ dependence in the one loop calculation is cancelled by counterterms that have not been included.
We are interested in how the one loop corrections affect the leading order prediction for the hyperfine splittings. Unfortunately, it is impossible to give a reliable estimate without knowing the numerical value of the counterterms required to cancel the µ dependence in the nonanalytic contribution. Furthermore, to compute the contribution from kaon loops, one must know the masses of doubly charm strange baryons which have not been observed.
We will assume that the ground state doubly charm strange baryons are 100 MeV higher in mass than their nonstrange counterparts, similar to the D meson system. We work in the into HHχPT with a 5 × 2 matrix field S µβ which is like the field H µβ except S µβ has opposite parity. The excitation energies and strong decay widths of these excited doubly charm baryons should be similar to their counterparts in the charm meson sector. Since the excited Ξ ++ cc (3780) state observed by SELEX is only 320 MeV above the Ξ ++ cc (3460), the lowest mass Ξ ++ cc candidate, and its width is considerably less than 300 MeV, it does not seem likely that this excited doubly charm baryon is related to the excited charm mesons by heavy quark-diquark symmetry. This is not unexpected as the lowest lying excited doubly charm baryons are not excitations of the light degrees of freedom but rather states in which the diquark is excited. The lowest mass excited diquark is a P-wave excitation. Because of Fermi statistics the diquark is a heavy quark spin singlet. The diquark's orbital angular momentum couples with the angular momentum of the light degrees of freedom to form baryons with J P = 1 2 + and J P = 3 2 + , which we will refer to as Ξ P cc and Ξ P * cc , respectively. The next lowest lying states are doubly heavy baryons with a radially excited diquark, which form a heavy quark doublet with J P = 1 2 − and J P = 3 2 − baryons, which we will refer to as Ξ ′ cc and Ξ ′ * cc , respectively. If the heavy antiquarks are sufficiently heavy that the force between them is approximately Coulombic, they interact via a potential which is 1/2 as strong as the potential between the quark and antiquark in a quarkonium bound state. Therefore we expect the excitation energies of the charm diquarks to be significantly smaller than the analogous excitation energies in charmonium. Quark model calculations of excited doubly charm baryons predict that the Ξ P cc and Ξ P * cc states are about 225 MeV above the Ξ cc and Ξ * cc , respectively, and that that the heavy quark doublet containing Ξ ′ cc and Ξ ′ * cc is about 300 MeV above the ground state doublet [38,39,40,41,42]. The doubly heavy baryons with P-wave excited diquarks decay to the ground state via S-wave pion emission. These decays violate heavy quark spin symmetry because the total spin of the diquark is changed in the transition. The Lagrangian for the excited Ξ P and Ξ P * states, including kinetic terms, residual mass terms and terms which mediate the S-wave decays, is The strong decay widths of the P-wave excited nonstrange doubly charm baryons are Here θ = 0.01 is the π 0 − η mixing angle. We expect these widths to be in the range 1-5 keV, but without knowing the masses of the Ω ( * ) cc and Ω P( * ) cc states or the couplings λ 1/2 and λ 3/2 we cannot make more precise predictions.
The J P = 3 2 − and J P = 1 2 − doubly heavy baryons with radially excited diquarks are members of a heavy quark doublet we will denote T ′ a whose definition in terms of component fields is identical to Eq. (13). The Lagrangian describing this field, including terms which mediate its decay to the ground state, is In the limit of infinite heavy quark mass, the light degrees of freedom in the radially excited doubly heavy baryons are in the same configuration as the ground state. Therefore, they are also related to the heavy meson ground state doublet by heavy quark-diquark symmetry.
The axial coupling and hyperfine splitting of T ′ a are the same as T a , as long as the spatial extent of the excited diquark, which is of order 1/(m Q v), is much smaller than 1/Λ QCD . This is valid in the heavy quark limit, but could receive significant corrections in the charm sector. The last term in Eq. (32) mediates P-wave decays from the excited J P = 3 2 − and J P = 1 2 − doubly heavy baryons to the ground state. The partial decay widths are Here C ab is an SU(3) factor which is 1/2 for decays involving π 0 and one for decays involving charged pions. The radially excited doubly heavy strange baryons should also be below the threshold for decays into kaons, and therefore should be quite narrow. The formulae in Eq. (33) can be used to obtain these decay widths as well. The isospin violating strong partial decay widths are obtained by using Eq. (33) with C 33 = 2 3 then multiplying by θ 2 . The expected widths of these states are of order 10 keV, but more precise estimates cannot be made until the masses of the states and the couplingg are known. For the nonstrange doubly heavy baryons, in the limit of infinite heavy quark mass, we obtain for the total widths, and for the branching fractions we find These relations receive large corrections due to phase space effects. Once the hyperfine splittings are taken into account the factors of p 3 π will differ greatly for the four decays. To get a feeling for these effects in the doubly charm sector we choose m Ξcc = 3440 MeV, m Ξ * cc = 3520 MeV, m Ξ ′ cc = 3740 MeV, and m Ξ * ′ cc = 3820 MeV, which corresponds to a diquark excitation energy of 300 MeV and hyperfine splittings of 80 MeV. We then find Note that the Ξ ′ cc unlike the Ξ ′ * cc strongly prefers to decay to Ξ * cc relative to Ξ cc despite the phase space suppression. This may be useful for distinguishing Ξ ′ * cc and Ξ ′ cc experimentally. The SELEX Ξ ++ cc (3780) is broad relative to the other SELEX doubly charm candidates. Since it is 260 MeV heavier than the Ξ + cc (3520), it is a natural candidate for one of the low lying excited doubly charm baryons. Unfortunately, no measurement of the width exists and the pattern of decays is also hard to understand, since Ref. [12] states that 50% of the decays to Λ + c K − π + π + are through Ξ + cc (3520) π + while the other 50% are weak decays. More information on the quantum numbers of the Ξ ++ cc (3780) and the Ξ + cc (3520) are needed before we can determine which of the excited doubly charm baryons should be identified with the Ξ ++ cc (3780).
In this paper we have developed a generalization of HHχPT which incorporates heavy quark-diquark symmetry and includes the leading symmetry breaking corrections from the chromomagnetic couplings of the heavy quark and diquark. We also included electromagnetic interactions in the Lagrangian, and obtained an estimate of the width of the J = 3 2 member of the ground state doubly charm baryon doublet. The width of this state is completely dominated by electromagnetic decays. We showed how to include the lowest lying doubly charm baryons which are expected to be excitations of the doubly charm diquark rather than the light degrees of freedom. Strong decay widths of low lying excited states were calculated and the states are expected to be rather narrow because of limited phase space available for the decays. Of particular interest is the doubly charm strange sector where we expect three pairs of excited baryons whose strong decay must violate isopsin conservation because they are below the kaon decay threshold. These states will have narrow widths of 10 keV or less. Experimental efforts to observe the narrow doubly charm strange baryons would be of great interest.