Nonrelativistic effective field theory for heavy exotic hadrons

Joan Soto1, ∗ and Jaume Tarrus Castella2, †

1Departament de Fısica Quantica i Astrofısica and Institut de Ciencies del Cosmos,Universitat de Barcelona, Martı i Franques 1, 08028 Barcelona, Catalonia, Spain

2Grup de Fısica Teorica, Dept. Fısica and IFAE-BIST, Universitat Autonoma de Barcelona,E-08193 Bellaterra (Barcelona), Catalonia, Spain

(Dated: July 14, 2020)

We propose an effective field theory to describe hadrons with two heavy quarks without anyassumption on the typical distance between the heavy quarks with respect to the typical hadronicscale. The construction is based on nonrelativistic QCD and inspired in the strong coupling regimeof potential nonrelativistic QCD. We construct the effective theory at leading and next-to-leadingorder in the inverse heavy quark mass expansion for arbitrary quantum numbers of the light degreesof freedom. Hence our results hold for hybrids, tetraquarks, double heavy baryons and pentaquarks,for which we also present the corresponding operators at nonrelativistic level. At leading order,the effective theory enjoys heavy quark spin symmetry and corresponds to the Born-Oppenheimerapproximation. At next-to-leading order, spin and velocity-dependent terms arise, which producesplittings in the heavy quark spin symmetry multiplets. A concrete application to double heavybaryons is presented in an accompanying paper.

∗ joan.soto@ub.edu† jtarrus@ifae.es

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I. INTRODUCTION

Exotic hadrons, which may be defined as hadrons that are neither mesons (quark-antiquark states) nor baryons(three quark states), were already foreseen in the early days of QCD [1]. Whereas mesons and baryons are well-definedobjects in the nonrelativistic quark model, they are not so in the context of QCD. This is because the light quarkmasses are much smaller than the typical hadronic scale ΛQCD, and hence light quark-antiquark pairs can be easilycreated both in mesons and baryons turning them into tetraquark or pentaquark states, or even into states witha higher number of quarks and antiquarks. The situation changes dramatically in the case of hadrons containingheavy quarks, such as the charm and bottom. Since their masses are much larger than ΛQCD, the creation of heavyquark-antiquark pairs in hadrons is highly suppressed. Therefore, the number of heavy quarks in a heavy hadron isclosely related to its mass, and the quantum numbers such as isospin and baryon number indicate the light quarkcontent.

For the last two decades numerous experimental collaborations have discovered and measured the properties ofmany unexpected hadrons, which have been generically named as XY Z. The first of these states discovered wasthe X(3872) [2], which turns out to be extremely close to the D0-D0∗ meson threshold, and hence a molecularinterpretation is tempting. Later on, another charmoniumlike state was found, the Y (4260) [3], whose weak productionin e+e− annihilation signals it as an exotic. In the following years charged states in the charmonium and bottomoniumspectrum where discovered [4–6] which can be unambiguously identified as tetraquarks. More recently, at the LHCbexperiment, two isospin 1/2 baryons where found with masses close to the charmonium states [7] which have beeninterpreted as a pentaquark formed by a charm-anticharm pair and three light quarks. More data from experimentsis expected in the near future due to the ongoing experimental efforts at facilities such as BESIII, Belle II, LHC andin the future at FAIR, which stresses the need for a comprehensive theoretical framework based on QCD to describethese states.

Heavy quarkonium, namely heavy quark-antiquark systems, have also been studied since the early days of QCD,and in fact they played an important role in the consolidation of this theory [8, 9]. Since the heavy quark massesmQ are much larger than ΛQCD, it was soon realized that heavy quarks move slowly and hence they could bedescribed by standard nonrelativistic quantum mechanics once the interaction potential was obtained from QCD, andthis could be done in terms of the expectation value of the Wilson loop [10]. Corrections to the leading-order mass-independent potential were also considered and expressed again in terms of expectation values of operator insertions inthe Wilson loop [11–14]. All these developments were later on recast in the framework of nonrelativistic effective fieldtheories [15, 16] (see Ref. [17] for a review), which allowed one to incorporate the so-called hard corrections [18, 19],and provided a complete result for the potential at order 1/m2

Q [20, 21]. The different terms in the potential have

been evaluated in lattice QCD [22–24].It is the aim of this paper to put forward a general effective field theory (EFT) framework, analogous to the one

described above for heavy quarkonium, for any heavy hadron, containing a heavy quark-antiquark or two heavyquarks, and a gluon and light quark state with arbitrary quantum numbers. We shall refer to the gluon and lightquarks collectively as light degrees of freedom (LDF). We will call heavy exotic hadron to any such state with LDFquantum numbers different from 0++, this case corresponding to heavy quarkonium. Note that, for convenience, weinclude double heavy baryons in this definition of heavy exotic hadron1. Heavy exotic hadrons are then composedof two distinct components: the heavy quarks and the LDF. The former form a nonrelativistic bound state with aninteraction potential depending on the LDF state. Such bound states are characterized by three well-separated scales:the heavy quark mass, mQ, the relative momentum mQv, with v � 1 the relative velocity, and the binding energymQv

2. The LDF states are characterized by the typical hadronic scale ΛQCD. The EFT that we present in this paperis built from QCD using two energy gaps between these characteristic scales. The first one is the aforementionedfact that mQ � ΛQCD, which is implemented in an EFT framework in nonrelativistic QCD (NRQCD) [15, 26]. Thesecond one is ΛQCD � mQv

2, which is nothing else than the observation that one can perform an adiabatic expansionbetween the dynamics of the heavy degrees of freedom, the heavy quarks, and the LDF, the gluons and light quarks.Furthermore, in the short-distance regime, mQv ∼ 1/r � ΛQCD, the relative momentum scale can be integratedout perturbatively. This is the so-called weak-coupling regime of potential NRQCD (pNRQCD) [16, 27] developedoriginally for quark-antiquark systems to study standard quarkonium and later on adapted to quark-quark systemsto study double heavy baryons [28, 29].

The leading order in the adiabatic expansion is the celebrated Born-Oppenheimer approximation, which has beenapplied to heavy hybrids for quite some time in Refs. [25, 30–34] and put into a nonrelativistic EFT framework inRefs. [35–37]. The Born-Oppenheimer approximation has also been used for heavy tetraquarks in Ref. [32, 38–41]. The

1 Note also that we exclude 0++ excitations of the LDF, which indeed exist in the spectrum [25]. In that case, the formulas for the 1/mQ

corrections are analogous to the ones for heavy quarkonium case [20, 21]. In particular, no spin and velocity dependent correctionsappear at order 1/mQ. This is also true in the more general case of 0± LDF.

3

EFT for heavy hybrids has been extended beyond leading order to include spin-dependent operators up to 1/mQ [42]and up to 1/m2

Q [43, 44]. In Refs. [35, 43, 44] the dependence of the potentials in the interquark distance, r, wasobtained in the short-distance regime, that is, assuming that the typical size of the system r is much smaller thanthe typical hadronic size 1/ΛQCD. In the case of the static potential, which can be obtained from the lattice QCDcalculations, the results from Ref. [35] show that the short-distance regime is a reasonable approximation only forthe lowest lying state in the charmonium sector and for a few low-lying states in the bottomonium one. This canbe understood from a general standpoint from the fact that the hybrid static potential has a classical minimum atr ∼ 1/ΛQCD as was pointed out in Refs. [37, 42]. It is not likely that the weak-coupling assumption is fulfilled in mostcases for exotic hadrons for the actual values of the charm and bottom masses.

We shall then avoid any assumption on the relative size of r and 1/ΛQCD here, and proceed in an analogous wayto the strong coupling regime of pNRQCD [20, 21], as it was advocated in Ref. [45] and has already been initiatedin the particular case of heavy hybrids [37, 42]. The leading-order Lagrangian for the EFT for heavy exotic hadronsconsists of wave function fields interacting with a mass-independent and heavy quark spin-independent potential, andcoincides with the Born-Oppenheimer approximation. We consider here the complete 1/mQ corrections containingthe heavy quark spin or orbital angular momentum for an wide class of heavy exotic hadrons, including hybrids,tetraquarks and pentaquarks. We provide formulas for the potentials associated with the different operators in termsof expectation values of operator insertions in the Wilson loop, which are suitable to be calculated in lattice QCD.

The paper is organized as follows. In Sec. II, we describe the EFT and write down the most general leading-orderand next-to-leading order (1/mQ) Lagrangian for arbitrary total angular momentum of LDF. In Sec. III, we describethe matching to NRQCD and provide formulas for the different terms in the potential that can be evaluated in latticeQCD. In Sec. IV, we list the sets of LDF operators that are suitable to create hybrids, tetraquarks and pentaquarks.We close with Sec. V devoted to discussion and conclusions. A concrete application of the general results in thispaper to double heavy baryons is presented in an accompanying paper [46]. Some technical details are relegated tothe Appendixes A and B.

II. EFT FOR HEAVY EXOTIC HADRONS

The EFT describing heavy exotic hadrons can be generically written as

L =∑κp

Ψ†κp [i∂t − hκp ] Ψκp , (1)

where the sum over κp refers to the sum over the LDF states with spin κ and parity p corresponding to the spectrumof static energies relevant for a given particular case. The LDF may have additional quantum numbers such as chargeconjugation, isospin, strangeness or baryon number that we will not write explicitly. The Ψ fields are understood asdepending on t, r, R, where r = x1 − x2 and R = (x1 + x2)/2 are the relative and center-of-mass coordinates of aheavy quark pair. The Ψ fields live both in the LDF and heavy quark spin spaces. The field Ψα

κp corresponds to aspin κ LDF state and has α = −κ, .., 0, ...κ components. In the Lagrangian in Eq. (1) we have chosen to leave thespin indices implicit.

The Hamiltonian densities hκp have the following expansion up to 1/mQ

hκp =p2

mQ+P 2

4mQ+ V

(0)κp (r) +

1

mQV

(1)κp (r, p) , (2)

with p = −i∇r and P = −i∇R. The kinetic terms in Eq. (2) are diagonal in spin space while the potentials are not.The static potentials, V (0), are diagonal in the heavy quark spin space, due to heavy quark spin symmetry, while theLDF spin structure is determined by the representations of D∞h that the κp quantum numbers can be associated with.D∞h is a cylindrical symmetry group that characterizes any hadron composed of two heavy quarks. Its representationsare labeled by Λ, the absolute value of the projection of the LDF state angular momentum on the axis joining thetwo heavy quarks, r2. Therefore 0 ≤ Λ ≤ |κ|. The static potential is expanded into D∞h representations.

V(0)κp (r) =

∑Λ

V(0)κpΛ(r)PκΛ , (3)

2 Additionally the representations are characterized by: η = ±1, the P or CP eigenvalue for heavy quark quark and heavy quark-antiquarksystems, respectively, denoted by g = +1 and u = −1; and for Λ = 0 there is a symmetry under reflection in any plane passing throughthe axis r, the eigenvalues of the corresponding symmetry operator being σ = ±1.

4

with PκΛ the projectors into representations of D∞h in the spin-κ space. These are (2κ + 1) × (2κ + 1) matrices inthe light quark spin space and fulfill the usual projector properties: they are idempotent P2

κΛ = PκΛ, orthogonal toeach other PκΛPκΛ′ = δΛΛ′ , and add up to the identity in the spin-κ space

∑Λ PκΛ = 12κ+1.

The subleading potentials V (1) can be split into terms that depend on the total heavy quark spin, SQQ, or angularmomentum, LQQ, and the terms independent of these two operators. The latter have the same structure as (3),whereas the former take the form

V(1)κpSD(r) =

∑ΛΛ′

PκΛ

[V saκpΛΛ′(r)SQQ · (Pc.r.

10 · Sκ) + V sbκpΛΛ′(r)SQQ · (Pc.r.11 · Sκ) + V lκpΛΛ′(r) (LQQ · Sκ)

]PκΛ′ . (4)

The total heavy quark spin is defined as 2SQQ = σQQ = σQ112Q2 + 12Q1σQ2 where the 12 are identity matricesin the heavy quark spin space for the heavy quark labeled in the subindex. The heavy quark angular momentum isdefined as LQQ = r×p. The superscript c.r. in the projectors indicates the use of the Cartesian basis representation

of the spin-1 matrices, i.e.(Si1)jk

= −iεijk. The heavy quark spin component of the Ψ fields is given by χQ1s χQ2

r withχs the usual spin-1/2 two-component spinors. The LDF spin component, χκα, is a (2κ+ 1)-component spinor.

Let us summarize how to obtain the projectors PκΛ in any given spin-κ space. First let us introduce the projectionvectors Pκλ defined as the eigenvectors given by

(r · Sκ)Pκλ = λPκλ . (5)

These projection vectors for κ = 1 were used in Refs. [35, 36, 43, 44, 47] in the construction of EFT for hybridquarkonium. The spin operator projected into the heavy quark pair axis can be expanded in these projector vectors

(r · Sκ) =∑λ

λPκλP†κλ . (6)

The projector into D∞h representations in the spin-κ space can then be defined as

PκΛ =∑λ=±Λ

PκλP†κλ . (7)

If we combine Eqs. (6) and (7) we can arrive at

(r · Sκ)2n

=∑Λ

Λ2nPκΛ . (8)

Some interesting consequences follow from Eq. (8). First, we can see that the Lagrangian in Eq. (4) contains allpossible spin-dependent operators at order 1/mQ. The only way to construct new operators is to multiply the existing

ones with scalar factors made out of r and Sκ. The only such scalar factors are precisely (r · Sκ)2n

which due toEq. (8) can be expanded in a linear combination of projectors. Finally, using the properties of the projectors the newoperators can be reduced to the existing ones. The second interesting use of Eq. (8) is that it can be used to create

a system of equations that can be inverted in order to obtain expressions for the projectors in terms of (r · Sκ)2n

forn = 0, 1, . . . up to the number of possible values of 0 ≤ Λ ≤ |κ|. In this way we obtain the projectors P up to spin-2

P 12

12

= 12 , (9)

P 32

12

=9

814 −

1

2

(r · S3/2

)2, (10)

P 32

32

= −1

814 +

1

2

(r · S3/2

)2, (11)

P10 = 13 − (r · S1)2, (12)

P11 = (r · S1)2, (13)

P20 = 15 −5

4(r · S2)

2+

1

4(r · S2)

4, (14)

P21 =4

3(r · S2)

2 − 1

3(r · S2)

4, (15)

P22 = − 1

12(r · S2)

2+

1

12(r · S2)

4. (16)

5

with 1n an identity matrix in the LDF spin space of dimension n = 2κ+1. The same results can be obtained by takinga specific representation of the spin-κ matrices, solving the eigenvalue problem in Eq. (5) and using the definition inEq. (7).

Finally, let us note that another useful basis for the operators inside the brackets in Eq. (4) depending on the heavyquark pair spin is to use irreducible O(3) tensors

T ij0 =δij = (Pc.r.

10 )ij

+ (Pc.r.11 )

ij, (17)

T ij2 = rirj − 1

3δij =

2

3(Pc.r.

10 )ij − 1

3(Pc.r.

11 )ij. (18)

III. MATCHING TO NRQCD

In the following we obtain the static potentials, V(0)κpΛ, and the spin-dependent operators in V saκpΛΛ′ , V

sbκpΛΛ′ and

V lκpΛΛ′ in terms of operator insertions in Wilson loops by expanding NRQCD about the static limit.Now, let us define the following strings for the case of hadrons containing a heavy quark-antiquark pair or a heavy

quark-quark pair

OQQκp (t, r, R) =χ>c (t, x2)φ(t, ,x2R)QQQκp(t, R)φ(t, R, x1)ψ(t, x1) , (19)

OQQκp (t, r, R) =ψ>(t, x2)φ>(t, R, x2)QQQκp(t, R)φ(t, R, x1)ψ(t, x1) , (20)

with ψ the Pauli spinor fields that annihilate a quark and χ the one that creates an antiquark, χc = iσ2χ∗. The Qoperators contain the LDF and are characterized by quantum numbers κp. In Sec. IV we provide a list of Q operatorscorresponding to a wide array of heavy exotic states. The Wilson line φ is defined as follows:

φ(t,x,y) = P{eig

∫ 10ds(x−y)·A(t,y+s(x−y))

}, (21)

where P is the path-ordering operator.The matching condition from NRQCD to the heavy exotic hadron EFT

Ohκp(t, r, R) =√ZhκpΨhκp(t, r, R) , h = QQ, QQ . (22)

The normalization factor is in general a function of Zhκp = Zhκp(r,p). For simplicity we will not show this dependenceexplicitly in the rest of the paper.

Now, we match the NRQCD and heavy exotic hadron EFT correlators,

〈0|T{Ohκp(t/2, r, R)Oh†κp(−t/2, r, R)}|0〉 =√Zhκp〈0|T{Ψhκp(t/2, r, R)Ψ†hκp(−t/2, r, R)}|0〉

√Z†hκp , (23)

where we have omitted Dirac deltas on the coordinates on both sides. The right-hand side of Eq. (23) becomes

√Zhκpe

−ithκp√Z†hκp =

√Zhκp

1

t

∫ t/2

−t/2dt′e−i(t/2−t

′)V(0)

κp(r)

[1Q1

2 1Q2

2 1LDF(2κ+1)

(1− it ∇

2r

mQ− it ∇

2R

4mQ

)−it 1

mQV

(1)κpSD(r) + . . .

]e−i(t

′+t/2)V(0)

κp(r)√Z†hκp . (24)

The dots stand for O(m−1Q ) spin- and velocity-independent potentials and further subleading terms in the heavy quark

mass expansion.Let us introduce the following notation:

WQQ� = P

{e−ig

∫C1+C2

dzµAµ(z)}, (25)

WQQ� = P

{e−ig

∫C1+C2

dzµAµ(z)}, (26)

〈. . . 〉hκp� = 〈Qhκp(t/2, R) . . .Qh†κp(−t/2, R)Wh�〉 , (27)

with C1, C2 and C2 denoting the upper or lower paths on the rectangular Wilson loop of Fig. 1.

6

x1

x2

−t/2 t/2

C1 C2C2

Figure 1. The bold paths correspond to the Wilson line paths C1, C2 and C2 associated with each one of the heavy quarks.

Using NRQCD [15, 26, 48], the right-hand side of Eq. (23) can be expanded as

〈1〉hκp� 1Q1

2 1Q2

2 + · · ·+ i1

2mQ

∫ t/2

−t/2dt′(〈D2(t′, x1)〉hκp� + (−1)h〈D2(t′, x2)〉hκp�

)1Q1

2 1Q2

2

· · ·+ icF

2mQ

∫ t/2

−t/2dt′(〈σ1 · gB(t′,x1)〉hκp� 1Q2

2 + 〈σ2 · gB(t′,x2)〉hκp� 1Q1

2

)+ . . . (28)

with (−1)h = 1 for h = QQ and (−1)h = −1 for h = QQ. Comparing Eqs. (24) and (28) we obtain

√Zhκpe

−itV (0)

hκp(r)√Z†hκp = 〈1〉hκp� . (29)

Now we use the expansion of the static potential in irreducible representations of D∞h of Eq. (3) and arrive at theWilson loop expressions for the static potentials

V(0)hκpΛ(r) = lim

t→∞

i

tln(

Tr[PκΛ〈1〉hκ

p

�

]). (30)

Both O(m−1Q ) terms in Eq. (28) contribute to heavy quark spin- and angular-momentum-dependent potentials. In

the case of the D2 operators the contributions come from the third and fourth term of Eq. (A14), which added upare given by Eq. (A17). The potentials match to

√Zhκp

1

t

∫ t/2

−t/2dt′e−i(t/2−t

′)V(0)

hκp(r)V

(1)hκpSD(r)e−i(t

′+t/2)V(0)

hκp(r)√Z†hκp

= −cFt

∫ t/2

−t/2dt′SQQ · 〈gB(t′,x1)〉hκp� −

∫ 1

0

ds sLQQ · 〈gB(t/2, z(s))〉hκp� , (31)

with z(s) = x1 + s(R− x1).

To complete the matching of the heavy quark spin- and angular-momentum-dependent operators in Eq. (4), wedecompose the Wilson loop averages in the two LDF spin components using standard tensor decomposition techniques

〈B〉hκp� =∑ΛΛ′

PκΛ

[δΛΛ′

Tr[(Sκ · Pc.r.

10 ) ·(PκΛ〈B〉hκ

p

� PκΛ

)]Tr [(Sκ · Pc.r.

10 ) · (PκΛSκPκΛ)](Pc.r.

10 · Sκ)

+Tr[(Sκ · Pc.r.

11 ) ·(PκΛ〈B〉hκ

p

� PκΛ′)]

Tr [(Sκ · Pc.r.11 ) · (PκΛSκPκΛ′)]

(Pc.r.11 · Sκ)

]PκΛ′ . (32)

Using Eq. (32) in Eq. (31) we arrive at the following expressions for the heavy quark spin- and angular-momentum-

7

dependent potentials:

V saκpΛΛ′ =− cF limt→∞

δΛΛ′

t

Tr [PκΛ]

Tr[PκΛ〈1〉hκp�

] ∫ t/2

−t/2dt′

Tr[(Sκ · Pc.r.

10 ) ·(PκΛ〈B〉hκ

p

� PκΛ

)]Tr [(Sκ · Pc.r.

10 ) · (PκΛSκPκΛ)], (33)

V sbκpΛΛ′ =− cF limt→∞

√Tr [PκΛ] Tr [PκΛ′ ]

Tr[PκΛ〈1〉hκp�

]Tr[PκΛ′〈1〉hκp�

] ln

(Tr[PκΛ〈1〉hκ

p

� ]Tr[PκΛ′ ]

Tr[PκΛ′ 〈1〉hκp

� ]Tr[PκΛ]

)2t sinh

(ln

√Tr[PκΛ〈1〉hκ

p

� ]Tr[PκΛ′ ]

Tr[PκΛ′ 〈1〉hκp

� ]Tr[PκΛ]

)∫ t/2

−t/2dt′

Tr[(Sκ · P11) ·

(PκΛ〈gB(t′,x1)〉hκp� PκΛ′

)]Tr [(Sκ · Pc.r.

11 ) · (PκΛSκPκΛ′)], (34)

V lhκpΛΛ′ =− limt→∞

√Tr [PκΛ] Tr [PκΛ′ ]

Tr[PκΛ〈1〉hκp�

]Tr[PκΛ′〈1〉hκp�

] ln

(Tr[PκΛ〈1〉hκ

p

� ]Tr[PκΛ′ ]

Tr[PκΛ′ 〈1〉hκp

� ]Tr[PκΛ]

)2 sinh

(ln

√Tr[PκΛ〈1〉hκ

p

� ]Tr[PκΛ′ ]

Tr[PκΛ′ 〈1〉hκp

� ]Tr[PκΛ]

)∫ 1

0

ds sTr[(Sκ · P11) ·

(PκΛ〈gB(t/2, z(s))〉hκp� PκΛ′

)]Tr [(Sκ · Pc.r.

11 ) · (PκΛSκPκΛ′)]. (35)

The computation of the traces not involving B can be found in Appendix B. In Appendix C we compare with knownresults for the matching of the hybrid quarkonium static potentials and the heavy-quark-spin-dependent potentials inthe short-distance limit.

IV. LIGHT-QUARK AND GLUON OPERATORS

In this section we provide with specific LDF operators for Eqs. (19) and (20) interpolating for a wide array ofpossible heavy exotic hadrons. Let us first introduce the notation. The light quark fields are standard Dirac fermionsrepresented by qafα(t,x) where a is the color index and α the spin index, and f is the isospin index. The Cj3 m3

j1 m1 j2 m2

is a Clebsch-Gordan coefficient, the projector P+ = (1 + γ0)/2 and the polarization vectors e+1 = −(1, i, 0)/√

2,

e−1 = (1, −i, 0)/√

2, e0 = (0, 0, 1). T a is the standard fundamental representation of the generators of SU(3). Wealso use the 3 and 6 tensor invariants from Ref. [28]

T lij =1√2εlij , i, j, l = 1, 2, 3 (36)

Σσij i, j = 1, 2, 3 σ = 1, .., 6

Σ111 = Σ4

22 = Σ633 = 1,

Σ212 = Σ2

21 = Σ313 = Σ3

31 = Σ523 = Σ5

32 =1√2, (37)

all other entries are zero. Both T lij and Σσij are real; T lij is totally antisymmetric and Σσij symmetric in the i and jindices. They satisfy the orthogonality and normalization relations:

3∑ij=1

T l1ij Tl2ij = δl1l2 ,

3∑ij=1

Σσ1ij Σσ2

ij = δσ1σ2 ,

3∑ij=1

T lij Σσij = 0 . (38)

In the case of a heavy quark-antiquark pair, these can be in singlet or octet states. The first case corresponds to thestandard quarkonium. For the latter we can construct the following operators interpolating for hybrid quarkonium

Qα1+−(t,x) =(e†α ·B(t,x)

), (39)

Qα1−−(t,x) =(e†α ·E(t,x)

), (40)

Qα2+−(t,x) = C2α1m1 1m2

(e†m1·D(t,x)

) (e†m2·E(t,x)

), (41)

Qα2−−(t,x) = C2α1m1 1m2

(e†m1·D(t,x)

) (e†m2·B(t,x)

). (42)

8

Light-quark operators interpolating for isospin I = 0 tetraquark states can be constructed as

Q0++(t,x) = [q(t,x)T aq(t,x)]T a , (43)

Q0−+(t,x) =[q(t,x)γ5T aq(t,x)

]T a , (44)

Qα1++(t,x) =[q(t,x)

(e†α · γ

)γ5T aq(t,x)

]T a , (45)

Qα1−−(t,x) =[q(t,x)

(e†α · γ

)T aq(t,x)

]T a , (46)

Qα1+−(t,x) =[q(t,x)

(e†α · (γ × γ)

)T aq(t,x)

]T a , (47)

Qα2+−(t,x) = C2α1m1 1m2

[q(t,x)

(e†m1·D(t, x)

) (e†m2· γ)T aq(t,x)

]T a . (48)

The isospin I = 1 tetraquark states can be interpolated by similar operators by just adding eI3 · τ in between thelight quark fields. Let us note that for the states with κ = 0 the operators in Eq. (4) vanish.

Quarkonium-pentaquark states with I = 1/2 and I3 = ±1/2 are interpolated by the operators read as

QαI3(1/2)+(t,x)P+ =(δαβ1σ

2β2β3

+ δαβ2σ2β1β3

+ δαβ3σ2β1β2

) (δI3f1τ

2f2f3

+ δI3f2τ2f3f1

+ δI3f3τ2f1f2

) (T kl1l2T

akl3

+T kl1l3Takl2 + T kl2l3T

akl1

)(P+ql1f1

(t,x))β1 (P+ql2f2

(t,x))β2 (P+ql3f3

(t,x))β3 T a . (49)

Operators interpolating for double heavy baryons with the heavy quarks in a 3 state

Qα(1/2)+(t,x)P+ = T l[P+q

l(t,x)]α

, (50)

Qα(1/2)−(t,x)P+ = T l[P+γ

5ql(t,x)]α

, (51)

Qβ(3/2)−(t,x)P+ = C3/2 β1m 1/2αT

l[(e†m ·D

)(P+q(t,x))

α]l, (52)

Qβ(3/2)+(t,x)P+ = C3/2 β1m 1/2αT

l[(e†m ·D

) (P+γ

5q(t,x))α]l

, (53)

if the heavy quarks are in a 6 color state, gluonic fields need to be added

Qβ(3/2)−(t,x)P+ = C3/2 β1m 1/2αΣσTr

[(e†m ·E

)ΣσT l

] (P+q

l(t,x))α

, (54)

Qβ(3/2)+(t,x)P+ = C3/2 β1m 1/2αΣσTr

[(e†m ·B

)ΣσT l

] (P+q

l(t,x))α

(55)

Finally, one can also construct open heavy flavor tetraquarks in a similar manner to Eqs. (43)-(48).

Q0−(t,x) =[q(t,x)T lγ2q∗(t,x)

]T l , (56)

Q0+(t,x) =[q(t,x)γ5T lγ2q∗(t,x)

]T l , (57)

Qα1−(t,x) =[q(t,x)

(e†α · γ

)T lγ5γ2q∗(t,x)

]T l , (58)

Qα1+(t,x) =[q(t,x)

(e†α · γ

)T lγ2q∗(t,x)

]T l , (59)

Qα2−(t,x) = C2α1m1 1m2

[q(t,x)

(e†m1·D) (e†m2· γ)T lγ2q∗(t,x)

]T l . (60)

The isospin I = 1 open heavy flavor tetraquark states can be interpolated by similar operators by adding eI3 ·(τ2τ

)in between the light quark fields.

V. CONCLUSIONS

We have put forward a general formalism to build EFTs for exotic hadrons with two heavy quarks as well as doubleheavy baryons. It is based on NRQCD [15, 49] and the strong coupling regime of pNRQCD [27]. Hence, the basicassumptions are that (i) the heavy quark masses are larger than any other energy scale in the system, and (ii) theheavy quark binding energies are smaller than the typical hadronic scale ΛQCD. We pose no extra assumption onthe relative size of the typical momentum exchanges between the heavy quarks ∼ 1/r and ΛQCD such as in previousworks [28, 29, 43, 44, 50]. The effective theory consists of wave function fields interacting with potentials, which

9

are a function of r and ΛQCD. At leading order in the 1/mQ expansion, these potentials correspond to the Born-Oppenheimer approximation. They are heavy quark spin- and angular-momentum-independent, but, in general, theydepend on the spin of the LDF. At next-to-leading order in the 1/mQ expansion, heavy-quark spin- and angular-momentum-dependent potentials arise. We provide formulas to calculate, both leading-order and next-to-leadingorder potentials, from QCD for any spin of the LDF in terms of operator insertions in the Wilson loop. These aresuitable to be computed in lattice QCD with only light quarks and gluons.

It is important to emphasize that unlike heavy quarkonium, in general, for exotic hadrons with κ 6= 0 the heavyquark spin- and orbital-angular-momentum (velocity) effects start at order 1/mQ, and not at order 1/m2

Q. This isimportant since a heavy quark spin dependence analogous to the heavy quarkonium one is often assumed in models,see for example Refs. [51–53]. We hope that our general formalism makes this point clear.

For a given spin representation of the O(3) symmetry group (κp) of the LDF, we have seen that 1/mQ termsmix different representations of the D∞h group. This was already noted in the specific case of quarkonium hybridsin Refs. [42–44]. Furthermore, the 1/mQ terms may also mix different representations of the O(3) group. Thiswas already noted, again in the specific case of quarkonium hybrids, in Refs. [37, 42]. Indeed, the operators inthe Lagrangian of Eq. (4) can be generalized to consider off-diagonal terms with κ = κ′ ± 1 and p = p′, that willproduce heavy exotic hadrons with mixed sQQ and l. Moreover, the terms in Eq. (A11) that do not contribute to theLagrangian of Eq. (4) could also produce off-diagonal operators in κ and other quantum numbers. A full analysis ofthese mixing operators and its matching is left for future work. These mixing terms become relevant for the spectrumat order 1/m2

Q, although their contribution is suppressed by the energy gap between κp representations. However it

may become important if there are accidental degeneracies in the spectrum of both representations [37].

The EFT built in this way is expected to be reliable for the ground and excited states with binding energiesE � ΛQCD about the ground state. When the binding energies approach the hadronic scale, both light hadronresonances and hadron pairs with a single heavy quark each should somehow be included in the approach. Sincethey are not, at this scale our EFT becomes a model, which, however, fully incorporates the NRQCD symmetries, inparticular the heavy quark spin-symmetry breaking pattern. It may also happen, and in fact it often does, that theground state of the exotic hadron itself is close to a hadron pair threshold. In that case, again, the hadrons formingthe threshold should be included in the EFT. String breaking data of lattice QCD [54, 55] suggest that for heavy-lighthadron pair thresholds, threshold effects are only noticeable in a tiny energy band around the threshold of a few tensof MeV. In case that the ground and excited states are away from these thresholds by more than a few tens of MeV,our EFT becomes again a reasonable model. Unfortunately, there is no such information for thresholds involving lighthadrons.

Since pions have masses parametrically smaller than ΛQCD, they could be incorporated to the EFT in a model-independent way, by using a formalism similar to Heavy Baryon Chiral Perturbation Theory [56], that was usedin Refs. [47, 57] for hybrids and tetraquarks. The outcome would be similar to hadronic chiral theories, see forinstance [58], but with r-dependent low-energy constants. Soft photons could also be incorporated by matchingNRQCD coupled to e.m. to the effective theory coupled to soft photons.

Finally, we have listed a number of light quark and gluon operators at the NRQCD level that describe most of theexotic hadrons containing two heavy quarks that are being considered nowadays as well as double heavy baryons. Inan accompanying paper [46], we present a concrete application to doubly heavy baryons.

ACKNOWLEDGEMENTS

J.S. acknowledges financial support from the 2017-SGR-929 grant from the Generalitat de Catalunya and theFPA2016-76005-C2-1-P and FPA2016-81114-P projects from Ministerio de Ciencia, Innovacion y Universidades.J.T.C. acknowledges financial support from the European Union’s Horizon 2020 research and innovation programmeunder the Marie Sk lodowska–Curie Grant Agreement No. 665919. He has also been supported in part by the SpanishGrants No. FPA2017-86989-P and No. SEV-2016-0588 from the Ministerio de Ciencia, Innovacion y Universidades,and the Grant No. 2017-SGR-1069 from the Generalitat de Catalunya. This research was supported by the MunichInstitute for Astro- and Particle Physics (MIAPP) which is funded by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) under Germany’s Excellence Strategy EXC-2094 390783311.

10

Appendix A: Simplification of the Wilson loop with a kinetic operator insertion

The 1/mQ contribution to the left-hand side of Eq. (23) produced by the kinetic operator is∫ t/2

−t/2dt′〈D2(t′, x1)〉κp� =∫ t/2

−t/2dt′〈0| . . . φ(t/2,R,x1)φ(t/2, t′,x1)D2(t′,x1, )φ(t′,−t/2,x1)φ(−t/2,x1,R) . . . |0〉 (A1)

where the dots stand for the terms that we do not display explicitly, namely the LDF operators and Wilson lines thatdo not explicitly depend on x1. These terms will be unaffected by the manipulations that we will perform in thisappendix. We also omit the label h = QQ, QQ and comment about the differences between these two cases at theend of the appendix.

The Wilson lines are defined as follows:

φ(t′, t,x) = P{e−ig

∫ t′tdt′′A0(t′′,x)

}, (A2)

φ(t,x,y) = P{eig

∫ 10ds(x−y)·A(t,y+s(x−y))

}, (A3)

where P stands for path ordered. We will use the following equalities [11, 20]

φ(t′′, t′,x)φ(t′, t,x) = φ(t′′, t,x) , (A4)

D(x, t′)φ(t′, t,x) = φ(t′, t,x)D(x, t) + iOE(t′, t,x) , (A5)

D(x, t)φ(t,x,y) = φ(t,x,y)∇x + iOB(t,x,y) , (A6)

with the abbreviated notation for the following strings

OE(t′, t,x) =

∫ t′

t

dt′′φ(t′, t′′,x)gE(t′′,x)φ(t′′, t,x) , (A7)

OB(t,x,y) =

∫ 1

0

ds s φ(t,x, z(s)) (x− y)× gB(t, z(s))φ(t, z(s),y) , (A8)

with z(s) = y + s(x− y). Using Eqs. (A4)-(A6) in Eq. (A1) one can arrive at

〈D2(t′, x1)〉κp� =∇x1

2·(〈1〉κp� ∇x1

+ 2i〈OB(−t/2,x1,R)〉κp� + 2i〈OE(t′,−t/2,x1)〉κp�)

+(∇x1〈1〉κp�

+2i〈OB(t/2,R,x1)〉κp� − 2i〈OE(t/2, t′,x1)〉κp�)· ∇x1

2− 〈OB(t/2,R,x1)OE(t′,−t/2,x1)〉κp�

+ 〈OE(t/2, t′,x1)OB(−t/2,x1,R)〉κp� + 〈OE(t/2, t′,x1)OE(t′,−t/2,x1)〉κp�+ 〈OB(t/2,R,x1)OB(−t/2,x1,R)〉κp� , (A9)

where the strings in the Wilson loops are understood as replacing that corresponding segment of the loop.Now it is convenient to work out the following identity

∇x1〈1〉κp� = −i〈OB(t/2,R,x1)〉κp� + i〈OE(t/2,−t/2,x1)〉κp� + i〈OB(−t/2,x1,R)〉κp� + 〈1〉κp� ∇x1

, (A10)

which allows one to write Eq. (A9) as

〈D2(t′, x1)〉κp� =∇2x1

2〈1〉κp� + 〈1〉κp�

∇2x1

2

+∇x1

2·(i〈OB(−t/2,x1,R)〉κp� + i〈OB(t/2,R,x1)〉κp� − i〈OE(t/2, t′,x1)〉κp� + i〈OE(t′,−t/2,x1)〉κp�

)(i〈OB(t/2,R,x1)〉κp� + i〈OB(−t/2,x1,R)〉κp� − i〈OE(t/2, t′,x1)〉κp� + i〈OE(t′,−t/2,x1)〉κp�

)· ∇x1

2

− 〈OB(t/2,R,x1)OE(t′,−t/2,x1)〉κp� + 〈OE(t/2, t′,x1)OB(−t/2,x1,R)〉κp�+ 〈OE(t/2, t′,x1)OE(t′,−t/2,x1)〉κp� + 〈OB(t/2,R,x1)OB(−t/2,x1,R)〉κp� (A11)

11

Note also that due to time reversal the identities∫ t/2

−t/2dt′〈OE(t′,−t/2,x1)〉κp� =

∫ t/2

−t/2dt′〈OE(t/2, t′,x1)〉κp� =

1

2

∫ t/2

−t/2dt′〈OE(t/2,−t/2,x1)〉κp� , (A12)

and

〈OB(t/2,R,x1)〉κp� = 〈OB(−t/2,x1,R)〉κp� , (A13)

are fulfilled. Using Eqs. (A12)-(A13) into Eq. (A11) we arrive at

〈D2(t′, x1)〉κp� =1

2∇2x1〈1〉κp� +

1

2〈1〉κp� ∇2

x1+ i∇x1

· 〈OB(t/2,R,x1)〉κp� + i〈OB(t/2,R,x1)〉κp� ·∇x1

− 〈OB(t/2,R,x1)OE(t′,−t/2,x1)〉κp� + 〈OE(t/2, t′,x1)OB(−t/2,x1,R)〉κp�+ 〈OE(t/2, t′,x1)OE(t′,−t/2,x1)〉κp� + 〈OB(t/2,R,x1)OB(−t/2,x1,R)〉κp� . (A14)

The term that matches to the LQQ · Sκ operator is

i∇x1· 〈OB(t/2,R,x1)〉κp� + i〈OB(t/2, R, x1)〉κp� ·∇x1

=− iriεijk∫ 1

0

ds s 〈Bj(t/2, z(s))〉κp�(

1

2∇R + ∇r

)k+ . . .

=

∫ 1

0

ds s 〈Bj(t/2, z(s))〉κp� LjQQ + . . . (A15)

In the heavy quark pair case the contributions from D2(t,x2) are analogous except by two relative a minus signs thatcompensate each other,

i∇x2· 〈OB(t/2,R,x2)〉κp� + i〈OB(t/2, R, x2)〉κp� ·∇x2

= iriεijk

∫ 1

0

ds s 〈Bj(t/2, z′(s))〉κp�(

1

2∇R −∇r

)k+ . . .

=

∫ 1

0

ds s 〈Bj(t/2, z′(s))〉κp� LjQQ + · · · =∫ 1

0

ds s 〈Bj(t/2, z(s))〉κp� LjQQ + . . . (A16)

with z′(s) = R+ s(x2 −R). Adding up the two contributions we arrive at

∫ t/2

−t/2dt′(〈D2(t′,x1)〉κp� + 〈D2(t′,x2)〉κp�

)= 2t

∫ 1

0

ds s 〈B(t/2, z(s))〉κp� ·LQQ + . . . (A17)

In the case of a heavy quark-antiquark pair the kinetic operator of the heavy antiquark has a relative minus sign withrespect to the heavy quark one. However, the path of the Wilson line of the heavy antiquark goes in the oppositedirection to the one of the heavy quark, see Fig. 1. This opposite trajectory for the antiquark generates an additionalminus sign in Eq. (A8) since that string depends on the initial and end points. The two minus signs compensate eachother and the result for a heavy quarks-antiquark pair is also the one given by Eq. (A17).

Appendix B: Traces of projectors and spin matrices

The trace of the projectors is

Tr [PκΛ] = 2− δΛ0 . (B1)

The traces of projectors and spin matrices in Eqs. (33)-(35) can be expressed as sums of Clebsch-Gordan coefficients

Tr [(Sκ · Pc.r.10 ) · (PκΛSκPκΛ′)] = 2Λ2δΛΛ′ , (B2)

Tr [(Sκ · Pc.r.11 ) · (PκΛSκPκΛ′)] = (κ+ Λ) (κ− Λ′) δΛΛ′−1 + (κ− Λ) (κ+ Λ′) δΛΛ′+1 + (κ+ Λ) (κ+ Λ′) δΛ1−Λ′ . (B3)

12

Appendix C: Comparison with short-distance regime matching for hybrid quarkonium

For quarkonium hybrids the matching in the short-distance regime, that is in the case r . 1/ΛQCD, of the staticpotential can be found in Refs. [27, 35, 44]

V(0)1+Λ(r) = Λ1+ + V (0)

o (r) + b1+Λr2 + . . . (C1)

with V(0)o (r) the perturbative octet potential. The constants Λ1+ and b1+Λ are nonperturbative, local, gluon correla-

tors. For example, the leading order of the static potential is given by the so-called gluelump mass, which takes thefollowing form

Λ1+ = limt→∞

i

tln 〈0|Bia(t/2, R)φab(t/2,−t/2, R)Bib(−t/2, R)|0〉 , (C2)

with φab(t, t′, R) the adjoint static Wilson line defined by Eq. (A2) with the gluon field in the adjoint representation.The short-distance regime matching of the heavy quark spin-dependent potentials can be found in Ref. [44]. To

compare our results with those of Ref. [44] we take heavy-quark spin-dependent potentials of Eq. (4) for the case

κp = 1+ and use the Cartesian basis representation of the spin-1 matrices, i.e(Si1)jk

= −iεijk. We find that

Ψ†1+

∑ΛΛ′

Pc.r.1Λ

[V sa1+ΛΛ′(r)SQQ · (Pc.r.

10 · Sc.r.1 ) + V sb1+ΛΛ′(r)SQQ · (Pc.r.

11 · Sc.r.1 )

]Pc.r.

1Λ′Ψ1+

=(

Ψ†1+

)i [V sa1+11(r)SkQQ(Sk1 )ij +

(V sb1+10(r)− V sa1+11(r)

)SkQQ

(rirl(Sk1 )lj − rl(Sk1 )lir

j)]

(Ψ1+)j. (C3)

Using Eq. (C3) we can find the correspondence of our results to those of Ref. [44]

V sa1+11(r) = Vnp (0)SK + V

np (1)SK r2 . . . (C4)

V sb1+10(r) = Vnp (0)SK +

(Vnp (0)SKb + V

np (1)SK

)r2 . . . (C5)

with Vnp (0)SK , V

np (1)SK and V

np (0)SKb nonperturbative, local, gluon correlators. For example, the leading-order one takes

the form

Vnp(0)SK =

cF2

limT→∞

ieiΛ1+ t

t

∫ t/2

−t/2dt′ εijkhbcd 〈0|Bia(t/2, R)φab(t/2, t′, R)gBjc(t, R)φde(t/2, t′, R)Bke(−t/2, R)|0〉 .

(C6)

From Eqs. (C1) and (C3) we can see the general structure for the short-distance expansion of the potentials: anonanalitic, perturbative, term appears if the potential can be generated by interactions between the heavy quarkswithout involvement of the LDF, the gluons in the case of quarkonium hybrids. The nonperturbative contributionsappear as coefficients of a series in powers of r. These coefficients are given by nonperturbative, local, gluon correlators.Furthermore, in the short-distance limit, r → 0, the symmetry group D∞h is enlarged to O(3) × C and thereforethe potentials at leading order in the short-distance expansion are independent on the representation of D∞h andonly depend on the quantum numbers κp of the LDF operators. The next-to-leading corrections are generated byr-dependent interactions of the heavy quarks with the LDF; these break the degeneracy of the D∞h representations.

We can compare Eq. (C6) to Eqs. (33) and (34). In both cases there are matching cF coefficients. Using Eq. (C1)we can see that the factor ieiΛ1+ t/t in Eq. (C6) is the short-distance expansion of the factor in front of the integralof Eqs. (33). The same holds true for the factor in front of the integral of Eq. (34), if we take into account theshort-distance degeneracy of the D∞h representations. One can also see that the B operators in Eq. (C6) matchthose inserted in the Wilson loop of Eqs. (33) and (34): B(t/2, R) and B(−t/2, R) correspond to the operator inEq. (39) inserted in the temporal sides of the Wilson loop and B(t, R) correspond to the NRQCD operator insertedin the spatial side of the Wilson loop.

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