# Effective Field Theory Lagrangians for Baryons with Two and Three Heavy Quarks

Nora Brambilla Dipartimento di Fisica, INFN and Università degli Studi di Milano via Celoria 16, 20133 Milano, Italy    Thomas Rösch , Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany    Antonio Vairo Dipartimento di Fisica, INFN and Università degli Studi di Milano via Celoria 16, 20133 Milano, Italy
###### Abstract

By analogy with pNRQCD, we construct Effective Field Theories suitable to describe the heavy-quark sector of baryons made of two and three heavy quarks. A long-standing discrepancy between the hyperfine splitting of doubly heavy baryons obtained in the HQET and potential models is solved. The one-loop matching of the 4-quark operators of dimension 6 is provided.

preprint: IFUM-808-FT{fmffile}

baryons

## I Introduction

The SELEX collaboration at Fermilab recently reported evidence of five resonances that may possibly be identified with doubly charmed baryon states Ocherashvili:2004hi . Tentatively the states have been interpreted as , , , and . Subsequently the state has been confirmed in two different decay modes (; ) at a mass of MeV with an average lifetime less than 33 fs. Although these findings need to be confirmed by other experiments and larger statistical samples, they have triggered a renewed theoretical interest in doubly heavy baryon systems.

Doubly heavy baryons have been studied with several methods, mostly non-relativistic potential models (for some reviews see Fleck:mb ; Kiselev:2001fw ), but also relativistic models Ebert:2002ig , sum rules Bagan:1994dy ; Kiselev:2000jb and in a chiral Lagrangian framework Bardeen:2003kt . Masses of the lowest lying resonances have been obtained from lattice calculations AliKhan:1999yb ; Lewis:2001iz ; Lewis:2001fy ; Mathur:2002ce ; Flynn:2003vz . Doubly heavy baryons are also suited to be studied in a QCD Effective Field Theory (EFT) framework. Indeed, they are characterized by at least two widely separated scales: the large heavy-quark masses, , and the low momentum transfer between the heavy and the light quarks, which is of order . If one assumes that the typical momentum transfer between the two heavy quarks is larger than , then a baryon is very similar to a bound state of a heavy antiquark and a light quark. This has first been noted in Savage:di , where at leading order in the hyperfine splitting of the doubly heavy baryon ground state has been related to the ground-state hyperfine splitting of the heavy-light meson. In Savage:pr non-leptonic and semileptonic decays of doubly heavy baryons have been examined in the context of flavour symmetry. After this original work no further step has been made in the direction of providing a systematic description of doubly heavy baryons in an EFT framework that fully combines the dynamics of the two heavy quarks with that one of the light one. Following some suggestions in Soto:2003ft , with this work we attempt to make such further step. In particular, we identify the degrees of freedom and write the low-energy EFT Lagrangian that describes doubly heavy baryon systems in the heavy-quark sector, once the heavy-quark momentum-transfer scale has been integrated out. The framework is similar to that one developed in the last years for heavy-quarkonium systems (for a review see Brambilla:2004jw ).

Baryons made of three heavy quarks have not been observed yet. Their relevance has been emphasized since long ago bjorken . They would reveal a pure baryonic spectrum without light-quark complications and provide valuable insight into the quark confinement mechanism. Indeed, the three-quark static Wilson loop is intensively studied on the lattice Bali:2000gf ; Takahashi:2004rw as a source of information about the baryon heavy-quark potential and the type of confining configurations Brambilla:1995px ; Kuzmenko:2002zs ; Cornwall:2003vn . In this work we will identify the degrees of freedom and write the low-energy EFT Lagrangian that describes heavy baryons made of three heavy quarks, once the heavy-quark momentum-transfer scale has been integrated out. We will express the leading-order and spin-dependent potentials in terms of Wilson loop amplitudes along the lines developed for heavy quarkonia in Brambilla:2000gk .

A recent review that also discusses the present status of the art, experimental and theoretical, including lattice, for heavy baryons made with two or three heavy quarks is Ref. Brambilla:2004wf . We refer to it for a more complete bibliography on the subject. This work is partially based on Thomas . We refer to it for details in some of the derivations.

The paper is distributed as follows. In Sec. II we introduce NRQCD for heavy baryons. In Sec. III we write the low-energy EFT for baryons and calculate the hyperfine splitting of the ground state. In Sec. IV we write the low-energy EFT for baryons and give some exact non-perturbative expressions for the leading-order and spin-dependent potentials. Sec. V is devoted to the conclusions. Some technical details may be found in the appendices.

## Ii Nrqcd

Non-relativistic QCD (NRQCD) is the EFT suitable to describe systems made of two or more heavy quarks. It is obtained from QCD by integrating out modes of energy of the order of the heavy-quark masses Caswell:1985ui .

We are interested here only in the heavy-quark sector of the NRQCD Lagrangian. The 2-heavy-quark sector coincides with the Lagrangian of the Heavy Quark Effective Theory (HQET). Up to order it reads:

 LNRQCDQ=NQ∑h=1Q†h[iD0+D22mh+c(h)F% \boldmathσ⋅gB2mh+c(h)D[D⋅,gE]8m2h+ic(h)S\boldmathσ⋅[D×,gE]8m2h]Qh, (1)

where is the number of heavy-quark flavours, the Pauli spinor field that annihilates the quark of flavour and mass , , , , , , () and are the Pauli matrices. The matching coefficients , and may be found at one loop, for instance, in Manohar:1997qy . The Lagrangian (1) (with terms included, but all matching coefficients set equal to 1) has been used to perform lattice calculations of the spectra of heavy baryons in Mathur:2002ce .

At order the NRQCD Lagrangian relevant to describe baryons made of two or more heavy quarks exhibits also a 4-heavy-quark sector:

 LNRQCDQQ = NQ∑h′≥h=1(dssQhQh′mhmh′Q†hQhQ†h′Qh′+dsvQhQh′mhmh′Q†h\boldmathσQh⋅Q†h′\boldmathσQh′ (2) +dvsQhQh′mhmh′8∑a=1Q†hTaQhQ†h′TaQh′+dvvQhQh′mhmh′8∑a=1Q†hTa\boldmathσQh⋅Q†h′Ta\boldmathσQh′).

The matching coefficients start getting contributions at order . They have been calculated to this order in appendix A.

Six-quark operators contribute to heavy baryons made of three heavy quarks. They show up at order . The corresponding matching coefficients start getting contributions at order . Hence, these operators are highly suppressed and will be neglected in the rest of the paper.

In the following we will make a step further and construct the EFT suitable to describe baryons made of two (Sec. III) and three (Sec. IV) heavy quarks once gluons of energy or momentum of the order of the momentum transfer between the heavy quarks have been integrated out. The procedure and the resulting EFT will be quite similar to that one developed for heavy quarkonium in Pineda:1997bj ; Brambilla:1999xf . For this reason we will call the EFT with the same name: potential NRQCD (pNRQCD).

## Iii pNRQCD for QQq baryons

In this section we deal with baryons made of two heavy quarks , (, or ) with masses and respectively and one massless quark . The dynamics of these systems is expected to mix aspects typical of heavy quarkonium with aspects typical of heavy-light mesons. On the one hand, the interaction of the two heavy quarks is that one of a non-relativistic quark pair close to threshold moving with relative velocity . It is, therefore, characterized by the energy scales: , where is the scale of the typical momentum transfer between the two heavy quarks (or of the inverse of their typical distance) and is the typical binding energy. On the other hand, the energy scale that governs the interaction between the heavy quarks and the light one is . Two different situations are possible.

(A) If , at a scale such that the heavy-quark distance cannot be resolved. The pair behaves like a point-like particle (sometimes also called diquark Anselmino:1992vg ) in an antitriplet or sextet colour configuration. In the antitriplet configuration the two heavy quarks attract each other. The interaction of the antitriplet field with the light quark is similar to that one of a heavy antiquark with a light quark in a or meson Savage:di . However, the spectrum is expected to be richer and more complex due to the internal excitations of the heavy-quark system. These include radial and spin excitations, but also colour excitations to sextet configurations. Considering that the scale is perturbative, in this situation pNRQCD has the following degrees of freedom: light quarks, gluons of energy and momentum lower than (also called ultrasoft) and heavy quarks. To ensure that the gluons are of energy and momentum lower than , gluons appearing in vertices involving heavy-quark fields are multipole expanded in the relative distance between the two heavy quarks. This corresponds to expanding in or . In the situation , in principle one may further distinguish between the subcases , and . In the first case, one may expand in and disentangle the heavy-heavy dynamics, which is completely accessible to perturbation theory, from the heavy-light one. In general, excitations of the heavy-heavy system will dominate over excitations of the heavy-light system. In the latter case, the potential governing the heavy-heavy system gets non-perturbative contributions.111 An analogous situation for the quarkonium system has been treated in Brambilla:1999xf . Since the kinetic energy of the heavy quarks is smaller than , the heavy-light dynamics dominates in this situation over the heavy-heavy one. At leading order in the expansion the flavour symmetry typical of the HQET is restored.

In the rest of this section, we will deal with the general situation , without assuming any special hierarchy between the scales and . To be definite, one may think that we work in the situation .

(B) If the distances between the three quarks are of the same magnitude. Hence, we cannot disentangle the heavy-quark pair dynamics from the light-quark one. Moreover, the potential between the two heavy quarks is non-perturbative. At the level of NRQCD, the system may be studied with lattice calculations. In this situation it seems unlikely that a simple diquark–light-quark picture holds. In general, from an EFT point of view it does not seem consistent to have a diquark–light-quark picture for the heavy-quarks–light-quark interaction, which implicitly assumes , and at the same time a confining potential binding the two heavy quarks, which requires , as so often done in potential models.

In the following, we will work out pNRQCD in the situation (A). This situation is expected to be appropriate for the description of at least doubly heavy baryons in the ground state.

### iii.1 Lagrangian

In this section we write the pNRQCD Lagrangian that describes heavy baryons of the type in the situation where the typical momentum transfer between the two heavy quarks is much larger than . This corresponds to the case labeled (A) above.

The number of allowed operators is reduced if we choose to have a manifestly gauge invariant Lagrangian. This may be obtained by projecting the Lagrangian on the heavy-heavy sector of the Fock space, by splitting the heavy-heavy fields into an antitriplet and sextet component (, ),

 Q1i(x1,t)Q2j(x2,t)∼3∑ℓ=1Tℓ(r,R,t)T––ℓij+6∑σ=1Σσ(r,R,t)Σ––σij,i,j=1,2,3, (3)

and by building the Lagrangian from these operators. The tensors and are defined in appendix B.1. The Lagrangian is constrained to satisfy all the symmetries of QCD. In particular, it is symmetric under the exchange of the heavy quarks. Such symmetry transformation changes and to . The gluon fields are even, because multipole expanded around the centre-of-mass of the heavy-heavy system. For what concerns the heavy-quark fields, from Eq. (3) it follows that is even, because is odd under the exchange , and is odd, because is even under the exchange .

The resulting Lagrangian at in the expansion and at in the multipole expansion is

 LpNRQCD=Lgluon+L% light+L(0,0)pNRQCD+L(0,1)pNRQCD+L(1,0)pNRQCD, (4)

with

 Lgluon=−148∑a=1FaμνFaμν, (5) (6)
 (7) L(0,1)pNRQCD=−∫d3rV(0,1)Tr⋅EΣ8∑a=13∑ℓ=16∑σ=1⎡⎣⎛⎝3∑ijk=1T––ℓijTajkΣ––σki⎞⎠Tℓ†r⋅gEaΣσ −⎛⎝3∑ijk=1–ΣσijTajkT––ℓki⎞⎠Σσ†r⋅gEaTℓ⎤⎦ −m1−m22mRV(0,1)Tr⋅ET8∑a=1T†r⋅gEaTa¯3T−m1−m22mRV(0,1)Σr⋅EΣ8∑a=1Σ†r⋅gEaTa6Σ, (8) +V(1,0)T\boldmathσ⋅BΣ8∑a=13∑ℓ=16∑σ=1⎡⎣⎛⎝3∑ijk=1T––ℓijTajkΣ––σki⎞⎠Tℓ†⎛⎝−c(1)F\boldmathσ(1)2m1+c(2)F\boldmathσ(2)2m2⎞⎠⋅gBaΣσ −⎛⎝3∑ijk=1–ΣσijTajkT––ℓki⎞⎠Σσ†⎛⎝−c(1)F\boldmathσ(1)2m1+c(2)F\boldmathσ(2)2m2⎞⎠⋅gBaTℓ⎤⎦ +V(1,0)T\boldmathσ⋅BT28∑a=1T†⎛⎝c(1)F\boldmathσ(1)2m1+c(2)F\boldmathσ(2)2m2⎞⎠⋅gBaTa¯3T +V(1,0)Σ\boldmathσ⋅BΣ28∑a=1Σ†⎛⎝c(1)F% \boldmathσ(1)2m1+c(2)F\boldmathσ(2)2m2⎞⎠⋅gBaTa6Σ −⎛⎝3∑ijk=1–ΣσijTajkT––ℓki⎞⎠Σσ†Lr⋅gBaTℓ⎤⎦ +V(1,0)TL⋅BT4(1mr−2mR)8∑a=1T†Lr⋅gBaTa¯3T +V(1,0)ΣL⋅BΣ4(1mr−2mR)8∑a=1Σ†Lr⋅gBaTa6Σ, (9)

where , , is the Pauli matrix acting on the heavy quark , , , , , the gauge fields in the covariant derivatives acting on the antitriplet and sextet are understood in the antitriplet and sextet representation respectively, and have been defined in appendix B.3 and all gluon fields are evaluated in . The coefficients and are the Wilson coefficients of NRQCD introduced in Sec. II. The functions are the Wilson coefficients of pNRQCD for doubly heavy baryons. They encode the contributions coming from gluons of energy or momentum of order , which have been integrated out. They are non-analytic functions of . As we will discuss in the next section, at tree level they are

 V(0,1)Tr⋅EΣ=V(0,1)Tr⋅ET=V(0,1)Σr⋅EΣ=1, V(1,0)T\boldmathσ⋅BΣ=V(1,0)T\boldmathσ⋅BT=V(1,0)Σ% \boldmathσ⋅BΣ=1, (10) V(1,0)TL⋅BΣ=V(1,0)TL⋅BT=V(1,0)ΣL⋅BΣ=1,

while and get the first non-vanishing contribution at order . In Eqs. (8) and (9) we have displayed only the operators that have a non-vanishing tree-level matching coefficient. The coefficients in front of the and operators in (9) are equal to 1, due to Poincaré invariance or dynamical considerations similar to those developed in Brambilla:2003nt . We observe that in the case , electric dipole transitions between antitriplet states induced by the term

 −m1−m22mR8∑a=1T†r⋅gEaTa¯3T (11)

are allowed Kiselev:2001fw ; Gershtein:nx .

The power counting of the Lagrangian (4) in the centre-of-mass frame goes as follows: , , , , and , . The power counting is not unique, because the scales and are still entangled in the dynamics. The Lagrangian at leading order reads

 LLOpNRQCD = ∫d3rT†[iD0+∇2r2mr−V(0)T]T+Σ†[iD0+∇2r2mr−V(0)Σ]Σ (12) −148∑a=1FaμνFaμν+3∑f=1¯qfi/Dqf.

### iii.2 Matching

The matching from NRQCD to pNRQCD is, in general, performed by calculating Green functions in the two theories and imposing that they are equal order by order in the inverse of the mass and in the multipole expansion. Since we are working in the situation where the typical momentum transfer between the heavy quarks is larger than , we can, in addition, perform the matching order by order in .

If we aim at calculating the matching at tree level a convenient approach consists in projecting the NRQCD Hamiltonian on the two-quark Fock space spanned by

 ∫d3x1d3x23∑ij=1ΦijQ1Q2(x1,x2)Qi†1(x1)Qj†2(x2)|0⟩, (13)

where is the Fock subspace containing no heavy quarks but an arbitrary number of ultrasoft gluons and light quarks and is a tensor in colour space and a tensor in spin space. This is similar to what is done in Pineda:1998kn . After projection, all gluon fields are multipole expanded in . In order to make gauge invariance explicit at the Lagrangian level, it is useful to decompose into a field , which transforms like a colour antitriplet, and a field , which transforms like a colour sextet:

 ΦijQ1Q2(x1,x2,t) = 3∑i′j′=1ϕii′(x1,R,t)ϕjj′(x2,R,t) (14) ×(3∑ℓ=1Tℓ(r,R,t)T––ℓi′j′+6∑σ=1Σσ(r,R,t)Σ––σi′j′),

where

 ϕ(y,x,t)≡Pexp(ig∫10ds(y−x)⋅A(x+(y−x)s,t)). (15)

stands for path ordering. At leading order in the coupling constant, and

 ΦijQ1Q2(x1,x2,t)≈3∑ℓ=1Tℓ(r,R,t)T––ℓij+6∑σ=1Σσ(r,R,t)Σ––σij. (16)

After projecting on (13), one obtains the Lagrangian (7)-(9) with the matching conditions (10).

As an example, let us consider the calculation that leads to the term

 δLpNRQCD=∫d3r128∑a=1T†⎛⎝c(1)F\boldmathσ(1)2m1+c(2)F\boldmathσ(2)2m2⎞⎠⋅gBaTa¯3T (17)

in the pNRQCD Lagrangian (see Eq. (9)). We start from the NRQCD term

 δLNRQCD=Q†1c(1)F% \boldmathσ⋅gB2m1Q1+Q†2c(2)F\boldmathσ⋅gB2m2Q2. (18)

Projecting onto (16), we obtain in the antitriplet-antitriplet sector

 δLpNRQCD = (19) +Tℓ†T––ℓijc(2)F\boldmathσ(2)2m2⋅gBaTajkTℓ′T––ℓ′ik⎞⎠.

Using the definition (74), we have

 3∑ijk=1T––ℓijTaikT––ℓ′kj = −Taℓ′ℓ2=(Ta¯3)ℓℓ′2, (20) 3∑ijk=1T––ℓijTajkT––ℓ′ik = −Taℓ′ℓ2=(Ta¯3)ℓℓ′2, (21)

and eventually end up with Eq. (17). This fixes at leading order. Note that Eq. (17) differs by a factor from Eqs. (9) and (10) in Savage:di , which seem to miss the correct colour normalization of the antitriplet states.

One may ask what happens to beyond tree level. Order corrections may only come from one-gluon corrections to the NRQCD vertex of Eq. (18), because all other spin-dependent operators in NRQCD contribute to higher orders in . One-loop corrections to the external (transverse) gluon or to a quark line or involving a gluon attached to the external gluon and to the quark line coupled to it vanish in dimensional regularization, once we have expanded in the external energies. Gluons attached to a quark line are longitudinal. It is convenient to use the Coulomb gauge. In Coulomb gauge, longitudinal gluons exchanged between different quark lines cancel in the matching with equal contributions from the pNRQCD side. Finally, longitudinal gluons attached to the external gluon line and a heavy quark line not coupled to it contribute to higher-order operators, since the three-gluon vertex is proportional to the external energies. We conclude that does not get contributions at one loop, so that . Similar considerations hold for and .

The perturbative matching of the static potentials and goes as follows (see Brambilla:1999xf for the quarkonium case). In NRQCD we compute static Green functions, whose initial and final states overlap with the antitriplet and sextet fields in pNRQCD. Since we work order by order in , it is not necessary for the Green functions to be gauge invariant. A possible choice is

 IuvM≡3∑iji′j′=1⟨0|M––––uijQi(R,x1,T/2)Qj(R,x2,T/2)M––––vi′j′Q†i′(R,y1,−T/2)Q†j′(R,y2,−T/2)|0⟩, (22) (1)ifM=T,M––––uij=T––uij,u,v=1,2,3, (2)ifM=Σ,M––––uij=Σ––uij,u,v=1,2,....,6,

where

 Q(R,x,t)≡ϕ(R,x,t)Q(x,t), (23)

and has been defined in Eq. (15). Integrating out the static-quark fields from we obtain

 IuvM=δ3(x1−y1)δ3(x2−y2)⟨0|(WMQQ)uv|0⟩ (24)

with diagrammatically represented in Fig. 1 and explicitly given by

 (WMQQ)uv≡P3∑ijkni′j′k′n′=1M––––uijϕii′(R,x1,T/2)ϕi′k′(T/2,−T/2,x1)ϕk′k(x1,R,−T/2)×ϕjj′(R,x2,T/2)ϕj′n′(T/2,−T/2,x2)ϕn′n(x2,R,−T/2)M––––vkn. (25)

In the large limit, the Green functions and are reduced to the antitriplet and sextet propagators of pNRQCD respectively. If we neglect subleading loop corrections to the pNRQCD side of the matching, we obtain:

 limT→∞⟨0|(WMQQ)uv|0⟩=limT→∞ZM(r)exp(−iV(0)M(r)T) ×⟨0|3∑ijkn=1M––––uijϕik(T/2,−T/2,R)ϕjn(T/2,−T/2,R)M––––vkn|0⟩, (26)

where is a normalization factor. At order we end up with the well-known result Flamm:1982jv :

 V(0)T(r) =−23αs|r|, (27) V(0)Σ(r) =13αs|r|. (28)

The antitriplet channel is attractive, the sextet one repulsive.

### iii.3 Hyperfine Splitting

In the dynamical situation considered here (case (A) of Sec. III), a doubly heavy baryon is mainly a bound state of a heavy quark (or antiquark) pair in an antitriplet (or triplet) configuration and a light quark (or antiquark). Sextet field configurations show up in loops with ultrasoft gluons. Their contribution is suppressed either in the multipole expansion or in (see Eqs. (8) and (9)). The leading-order pNRQCD Lagrangian is shown in Eq. (12). It does not depend on the spin of the heavy quarks. As a consequence, baryons will appear in degenerate multiplets of the total spin , where is the spin of the light degrees of freedom and of the heavy-quark pair. This symmetry is similar to the spin symmetry of the HQET. Differently from the HQET, however, the pNRQCD Lagrangian depends at leading order on the heavy-quark flavour. This is a consequence of the fact that we cannot, in general, neglect the kinetic energy.222 An exception may be the special case .

We will consider in this section the -wave ground state of a doubly heavy baryon made of two identical heavy quarks of mass . In this case, since an (anti)triplet state is antisymmetric in colour, due to the Fermi statistics, the two heavy quarks are allowed only in a spin 1 (symmetric) state. In the standard notation, the lowest energy states for or are called () for spin (), and for , (). Since the heavy-quark pair spin is fixed, the hyperfine splitting may only originate from spin-dependent couplings of the heavy quarks with the light one. The leading-order operator (in a expansion) is given by Eq. (17). We will derive a simple formula that relates at leading order in the expansion the hyperfine splitting of a doubly heavy baryon ground state with the hyperfine splitting of a heavy-light meson ground state. The framework will be that one of pNRQCD, developed in the previous sections. The calculation will be similar to that one of Ref. Savage:di .

Let us consider, first, the case of a heavy-light meson . The heavy antiquark may be described by a two-component field , where is the Pauli spinor that annihilates the heavy quark. We rename and since . is the spin of the heavy antiquark and the total spin of the meson. The leading-order HQET Lagrangian does not contain spin-interaction terms, therefore, states that differ only in the spin quantum numbers are degenerate. In particular, this happens for the three lowest states (), which we denote by , and for the lowest state, which we denote by . An expression for these states that makes explicit their heavy (anti)quark field content is given in appendix C. The Hamiltonian responsible for the leading contribution to the hyperfine separation is

 δHHQET = −c(Q)F∫d3R8∑a=1Q†c(R)\boldmathσ⋅gBa(R)Ta¯32mQQc(R) (29) = −c(Q)F2mQ∫d3R8∑a=1[(Q†+Ta¯3Q+−Q†−Ta¯3Q−)gB3a+i(Q†−Ta¯3Q+−Q†+Ta¯3Q−)gB2a +(Q†+Ta¯3Q−+Q†−Ta¯3Q+)gB1a],

where, after the last equality, we have dropped the explicit coordinate dependence of the fields. From Eq. (29) and Eqs. (87)-(90) it is straightforward to derive:

 ⟨P∗Q|δHHQET|P∗Q⟩−⟨PQ|δHHQET|PQ⟩=−2c(Q)FmQ∫d3R⟨Szl=1/2|8∑a=1gB3aTa¯3|Szl=1/2⟩. (30)

In the case of a doubly heavy baryon we proceed in a similar way. The triplet field is a tensor in spin space, which may be decomposed as , i.e. in a scalar component, , and a vector one, :

 Tij(r,R,t)=(i\boldmathσ2√2)ijT(S)(r,R,t)+3∑k=1(i\boldmathσk\boldmathσ2√2)ijT(V)k(r,R,t),i,j=1,2. (31)

The indices refer to the spin space. Note that the matrices and are respectively antisymmetric and symmetric in . It is convenient to rewrite the fields as

 T0=T(V)3andT±=∓T(V)1+iT(V)2√2, (32)

since and . As we argued above, the leading-order pNRQCD Lagrangian describing doubly heavy baryons does not contain spin-interaction terms, therefore, states that differ only in the spin quantum numbers are degenerate. In particular, this happens for the four lowest states (), which we denote by , and for the two lowest states (), which we denote by . An explicit expression of these states in terms of heavy (anti)triplet fields is given in appendix C. The Hamiltonian responsible for the leading contribution to the hyperfine separation is

 δHpNRQCD = −c(Q)F2mQ∫d3R∫d3rV(1,0)T% \boldmathσ⋅BT(r)8∑a=1T†(r,R)\boldmathσ(1)+% \boldmathσ(2)2⋅gBa(R)Ta¯3T(r,R) (33) = −
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