Perturbative Corrections to Heavy Quark-Diquark Symmetry Predictions for Doubly Heavy Baryon Hyperfine Splittings

Doubly heavy baryons $\left(QQq\right)$ and singly heavy antimesons $\left(\bar{Q}q\right)$ are related by the heavy quark-diquark (HQDQ) symmetry because in the $m_Q \to \infty$ limit, the light degrees of freedom in both the hadrons are expected to be in identical configurations. Hyperfine splittings of the ground states in both systems are nonvanishing at $O(1/m_Q)$ in the heavy quark mass expansion and HQDQ symmetry relates the hyperfine splittings in the two sectors. It was expected that corrections to this prediction would scale as $O(1/m_Q^2)$. In this paper, working within the framework of Non-Relativistic QCD (NRQCD), we point out the existence of an operator that couples four heavy quark fields to the chromomagnetic field with a coefficient that is enhanced by a factor from Coulomb exchange. This operator gives a correction to doubly heavy baryon hyperfine splittings that scales as $1/m_Q^2 \times \alpha_S/r$, where $r$ is the separation between the heavy quarks in the diquark. This correction can be calculated analytically in the the extreme heavy quark limit in which the potential between the quarks in the diquark is Coulombic. In this limit, the correction is $O(\alpha_s^2/m_Q)$ and comes with a small coefficient. For values of $\alpha_s$ relevant to doubly charm and doubly bottom systems, the correction to the hyperfine splittings in doubly heavy baryons is only a few percent or smaller.

The recent experimental observation of the Ξ ++ cc baryon has greatly revived the interest in the physics of doubly heavy baryons. This includes the experimental efforts to search for the other doubly charm and bottom baryons such as Ξ + cc and Ξ bc [10] as well as recent theoretical studies regarding the lifetimes, production rates, and decay rates of the double heavy baryons [11][12][13][14][15].
An interesting idea regarding the physics of doubly heavy baryons is that of heavy quarkdiquark symmetry (HQDQ) which relates the physics of doubly heavy baryons (QQq) to the heavy antimesons Q q . The appropriate theory for dealing with heavy mesons is the heavy quark effective field theory (HQET) [16][17][18], whereas the appropriate theory for dealing with doubly heavy baryons is nonrelativistic quantum chromodynamics (NRQCD) [19,20]. In the limit of large heavy quark mass m Q , the two heavy quarks QQ in the doubly heavy baryons experience an attractive Coulomb force and the ground state of the two heavy quarks is a tightly bound spin-1 diquark in the3 color representation. The size of the diquark is small, where v is the relative velocity of the two heavy quarks in the diquark. This implies that the diquark can be considered as a point source of color charge in the3 representation that looks the same to the light degrees of freedom as a singly heavy antiquark, up to corrections that are suppressed by inverse powers of heavy quark mass m Q .
The light degrees of freedom in the heavy antimeson also orbit a point source of color charge in the3 representation. Therefore, the two heavy hadrons have identical configurations for the light degrees of freedom in the m Q → ∞ limit. The HQDQ symmetry also relates the double heavy tetraquarks to singly heavy baryons and the chiral lagrangians incorporating this symmetry have been derived in Refs. [21,22].
One of the implications of the HQDQ symmetry is the relation between the hyperfine mass splittings of the doubly heavy baryons and heavy antimesons. The chromomagnetic interactions of the diquark and quark are responsible for the hyperfine splittings in the doubly heavy baryons and antimesons. The effective Lagrangian describing the chromomagnetic coupling of diquarks at O (1/m Q ) was derived in Ref. [23,24] in the framework of NRQCD. The ground state of the heavy antimeson consists of a spin-0 meson, P , and a spin-1 meson, P * . The ground state of the doubly heavy baryon consists of spin-1/2 baryon, Ξ, and spin-3/2, baryon Ξ * . These states are degenerate due to heavy quark spin symmetry that breaks at O (1/m Q ) due to spin dependent chromomagnetic interactions. The heavy quark-diquark symmetry implies the relation between the hyperfine splittings to be [23][24][25] The purpose of this paper is to study higher order corrections to this prediction. Since the hyperfine splittings themselves are O(1/m Q ) one might expect leading corrections to scale as 1/m 2 Q . What we will see below is that there is a higher dimension operator that scales as 1/m 2 Q × α s /r, where r is the typical separation between the quarks within the diquark. Since 1/r ∼ m Q v, this operator gives a correction to Eq. (1) of relative order α s v. In the extreme limit, where the quarks within the diquark are bound by Coulombic gluon exchange, v is proportional to α s and this O(α 2 s ) correction is computed below. While this correction turns out to be only of order a percent or less for values of α s relevant to doubly charm and bottom baryons, we find the unexpected scaling with m Q to be interesting. A similar correction to the HQDQ symmetry due to the finite size of diquark was calculated in Ref. [26]. The finite size effects were due to operators coupling the light quarks and the diquarks that contributes to the mass of the double heavy baryon. The correction to the HQDQ was also estimated to be small in Ref. [26]. In the body of this paper we review the effective action for heavy diquark fields, introduce the operator and compute its effect on the prediction for doubly heavy baryon hyperfine splittings. We then give our conclusions. In an Appendix, we derive the form of the operator by matching the full QCD diagrams for QQg → QQ scattering onto NRQCD to

II. EFFECTIVE ACTION FOR COMPOSITE DIQUARK FIELDS
The effective action for the heavy composite diquark fields with the lowest order heavy quark spin symmetry violating chromomagnetic interaction was derived by Fleming and Mehen in Ref. [23] and Brambilla, Vairo, and Rosch in Ref. [24] in the framework of NRQCD. The leading order chromomagnetic couplings of diquarks gives O (1/m Q ) corrections to the heavy quark spin symmetry and is responsible for the hyperfine splittings in the ground state of doubly heavy baryons. In this section, we use the formalism in Ref. [23] to include the correction to the chromomagnetic coupling of diquark fields from diagrams that contribute to the effective action at higher order in NRQCD power counting.
The NRQCD Lagrangian relevant for constructing the effective action for composite diquarks where ψ p represents the quark field with a three vector label p, B is the chromomagnetic field, and the ellipsis represents the higher order corrections as well as terms including soft gluons.
The color and spin Fierz identities that project the potential into color anti-triplet (3) and color sextet (6) states and decompose the quark billinears such as ψ p ψ −p into operators of definite spin are where the Greek letters refer to spin indices, the Roman letters refer to color indices, σ i denotes the Pauli matrices, = iσ 2 is an anti-symmetric 2 ×2 matrix, and d (mn) ij are symmetric matrices in color space: After Fourier transforming with respect to the labels and using the color and Fierz identities above, the Lagrangian in Eq. (2) can be written as where we have suppressed the spin indices and explicitly shown the color indices. The antitriplet potential V (3) (r) and sextet potential V (6) (r) are defined by The color and spin Fierz identities in Eqs. (3) and (4) introduces four terms but two of them vanish due to Fermi statistics. The diquark fields in Eq. (7) are in the3 and 6 representations in color space, and have spin-1 and spin-0, respectively. We define the following composite diquark operators where T i r is a spin-1 vector field and Σ (mn) r is a spin-0 scalar field.
The composite diquark fields, T i r and Σ (mn) r , enter the theory by using the Hubbard-Stratonovich transformation, which cancels the quartic interaction terms in heavy quark fields in favor of interaction terms between the diquark fields and the two heavy quark fields: The NRQCD Lagrangian after using the Hubbard-Stratonovich transformation reduces to The Feynman rules describing the interaction of diquarks with two heavy quarks corresponding to the above Lagrangian are shown in Fig.1.
The σ · B term in the NRQCD Lagrangian in Eq. (2) is the chromomagnetic interaction for heavy quarks. This is the lowest order heavy quark spin symmetry violating term that gives O (1/m Q ) corrections to the heavy quark spin symmetry and is responsible for the hyperfine Other O (v 2 ) couplings of the diquark field, T i r , which do not violate the heavy quark spin symmetry can be found in Ref. [27]. The composite diquark field, Σ  Fig. 4, which is obtained after matching tree-level scattering of  In order to evaluate the correction to the chromomagentic coupling of a diquark from the two-loop diagram in Fig. 3, we consider the external diquark fields T i r and T i r to be at rest and have energy E and E respectively. The external diquarks have spin indices k and l and color indices a and b respectively. The usoft gluon has polarization index m and color index c.
Using the Feynman rules for the diquark-quark interaction in Fig. 1 and the effective four-quark contact vertex in Eq. (A12), the two-loop diagram in Fig. 3 evaluates to The effective Lagrangian describing the leading correction to the chromomagnetic coupling of diquark field T r in Eq. (13) is where iΣ is given in Eq. (14) and the color and spin indices of the diquark field, T r , have been suppressed. This Lagrangian can be easily interpreted using the notation of Ref. [23], where the diquark field T r are thought of as vectors in a Hilbert space spanned by the position space eigenkets |r : where we define the diquark field T r ≡ r|T , the potential operator r |V (3) |r ≡ V (3) δ 3 r − r , the momentum eigenstates r|l = e −il·r , and a free Hamiltonian H 0 |l = l 2 /m Q |l . Using the formalism developed in Ref. [23], the potential operatorV (3) in Eq. (16) cancels against the factors of E − H 0 in the denominator after using the equation of motion for the diquark field T r . Therefore, the effective Lagrangian describing the leading correction to the chromomagnetic coupling of diquarks in Eq. (13) is The where a 0 is the Bohr radius given by The This gives a correction to the matrix element of the chromomagnetic operator in Eq. (13) that appears only in the doubly heavy baryon mass splitting as there is no analogue correction in the heavy meson sector. Therefore, the relation between the hyperfine splittings in Eq. (1) is modified to where α 2 s /9 is the correction to the hyperfine splitting from the two-loop diagram in Fig. 3. Of the two powers of α s appearing in Eq.

III. CONCLUSION
The ground state mass hyperfine splittings in the double heavy baryons and singly heavy antimesons are related by the heavy quark-diquark symmetry (HQDQ). The hyperfine splittings are due to the O (1/m Q ) chromomagnetic couplings of the diquark and quark and the leading prediction for the splittings is given by Eq. (1). In this paper, we compute the leading correction to the hyperfine splitting of the double heavy baryons in the framework of NRQCD. We point out an effective five-point contact operator that couples the four heavy quark fields with the chromomagnetic field with a coefficient that is enhanced by the Coulomb exchange. Naively, one would expect the leading correction to the chromomagnetic coupling of the diquark to scale as 1/m 2 Q , instead we find that the correction from the effective operator scales as 1/m 2 Q × α s /r, where r is the separation between the heavy quarks in the diquark. The Lagrangian describing the leading correction to the chromagnetic coupling of diquark is given by Eq. (17).
We estimate the correction to the ground state mass hyperfine splitting in the doubly heavy baryons due to the next leading order Lagrangian in Eq. (17). We find that in the m Q → ∞ limit, when the two quarks within the diquark are bound by strong Coulombic interaction, the leading correction to the hyperfine splitting of double heavy baryons is of O (α 2 s /m Q ) with a small coefficient as shown in Eq. (21). For values of α s relevant to doubly charm and doubly bottom systems, we find that the correction to the hyperfine splitting in doubly heavy baryons O 1/m 2 Q × α s /r correction to the chromomagnetic coupling of diquark field T r . This effective operator is obtained after matching the low-energy tree diagrams for QQ → QQg in full QCD theory onto NRQCD. In QCD, the diagrams for QQ → QQg are shown in Fig. 5. The two tree-diagrams at the top of Fig. 5, where the external gluon is attached to the external quarks, match onto two distinct types of NRQCD diagrams. One diagram is the tree diagram shown in Fig. 6, in which the gluon couples to an external quark via the chromomagnetic interaction and there is a virtual nonrelativistic quark. The other diagram is the contact interaction in Fig. 4.
The bottom two diagrams in Fig. 5, where the external gluon is attached to the exchanged gluon via the three-gluon vertex, could in principle also contribute. However, the bottom two diagrams in Fig. 5 have a vanishing color factor when the incoming and outgoing diquarks are both in the3 representation.
In Fig. 5, the incoming heavy quarks have four-momenta p 1 = (E 1 , P 1 ) and p 2 = (E 2 , P 2 ) and color indices i and j respectively. The outgoing heavy quarks have four momenta given by (A2) Using the identity for the product of three gamma matrices, the terms in the second square bracket in the numerator of the above equation can rewritten as In the above expression, and then taking the nonrelativistic limit of the Dirac spinors where ξ i is a two-spinor, and i denotes the color index, we find that the leading term with a spin-dependent interaction in the nonrelativistic expansion of the numerator in Eq. (A2) is where ε is the three-space polarization vector and the ellipsis represents terms that are suppressed by powers of v or do not explicitly break the heavy quark spin symmetry.
In the nonrelativistic limit, the fermion propagator in the denominator of the amplitude in Eq. (A2) is given by where k = P 3 + P 4 − P 2 and Similarly, the gluon propagator can also be expanded in powers of 3-momenta as Using the nonrelativistic expansion of the numerator in Eq. (A6) and the nonrelativistic expansion of the fermion and gluon propagators in Eq. (A7) and (A9), the low-energy scattering amplitude in Eq. (A2) is where k = P 3 + P 4 − P 2 and k 0 − E k is given by Eq. (A8). In the above expression we have suppressed both the color and spin indices. The low-energy amplitude in Eq. (A10) has a pole where the intermediate fermion propagator goes on-shell. This pole will be reproduced in the effective theory by the contribution to the scattering amplitude from Fig. 6. If the vertex with the external gluon line is the chromomagnetic coupling of the heavy quark, then the contribution to the scattering amplitude from Fig. 6 is  Fig. 4 is The Feynman rule for the contact vertex in Fig. 4 is given by