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Galilean-Invariant XEFT

XEFT is a low-energy effective field theory for charm mesons and pions that provides a systematically improvable description of the X(3872) resonance. A Galilean-invariant formulation of XEFT is introduced to exploit the fact that mass is very nearly conserved in the transition D*0 -->D0 pi0. The transitions D*0 -->D0 pi0 and X -->D0 D0-bar pi0 are described explicitly in XEFT. The effects of the decay D*0 -->D0 gamma and of short-distance decay modes of the X(3872), such as J/psi -->pi+ pi-, can be taken into account by using complex on-shell renormalization schemes for the D*0 propagator and for the D*0 D0-bar propagator in which the positions of their complex poles are specified. Galilean-invariant XEFT is used to calculate the D*0 D0-bar scattering length to next-to-leading order. Galilean invariance ensures the cancellation of ultraviolet divergences without the need for truncating an expansion in powers of the ratio of the pion and charm meson masses.


I. INTRODUCTION
The surprising discovery of the X(3872) by the Belle Collaboration in 2003 [1] marked the beginning of a renaissance in quarkonium spectroscopy [2]. Dozens of new mesons whose constituents include a heavy quark and antiquark and with mass above the open-heavy flavor threshold have been observed. They are collectively referred to as XY Z mesons. Some of the XY Z mesons are electrically charged and therefore must be tetraquark mesons whose constituents also include a light quark and antiquark. The pattern of the observed XY Z mesons remains unexplained. In particular, the relation between the X(3872) and the other XY Z mesons is still not understood.
The discovery decay mode J/ψ π + π − of the X(3872) implies that its constituents must include a charm quark and antiquark. However the X(3872) has properties inconsistent with conventional charmonium, including comparable branching fractions into decay modes with isospin 0 and isospin 1. The J P C quantum numbers of the X(3872) were finally established by the LHCb Collaboration in 2013 to be 1 ++ [3]. Its mass is extremely close to the threshold for the pair of charm mesons D * 0D0 . By combining precise measurements of the mass M X of the X(3872) in the J/ψ π + π − channel with precise measurements of the masses M * and M of the D * 0 and D 0 , the difference δ X between the D * 0D0 threshold and the mass has been determined to be δ X ≡ (M * + M ) − M X = 0.11 ± 0.23 MeV. (1) The quantum numbers 1 ++ of the X(3872) imply that it has an S-wave coupling to the charm meson pairs D * 0D0 and D 0D * 0 . Given the small value of δ X , the universality of near-threshold S-wave resonances implies that the X(3872) must be a bound state (if δ X > 0) or a virtual state (if δ X < 0) whose constituents are the C = + superposition D * 0D0 + D 0D * 0 [4]. The observation of X(3872) in hadron collisions strongly suggests that it is a bound state (like the deuteron) rather than a virtual state (like the dineutron). One universal property of S-wave near-threshold bound states is that the mean separation r X of the constituents is determined by the binding energy δ X : r X = (8µδ X ) 1/2 , where µ is the reduced mass of D * 0D0 [5]. Given the binding energy δ X = 0.11 −0.11 +0.23 MeV, the mean separation of the charm mesons is predicted to be 6.8 +∞ −3.9 fm. Thus the size of the X(3872) is comparable to that of the largest nuclei. The universal properties of S-wave near-threshold bound states imply that most of the probability of the X(3872) is in a molecular component consisting of well-separated charm mesons D * 0D0 . (From now on, the equally probable D 0D * 0 component will usually not be mentioned explicitly.) At short distances, the wavefunction of the X(3872) can have other components with smaller probabilities. One possible component is the 1 ++ P-wave charmonium state χ c1 (2P ), whose constituents are cc. Another possibility is an isospin-0 tetraquark with constituents cqcq, where q is a light u or d quark. It could be a compact tetraquark, it could have substructure consisting of the diquark clusters cq andcq, or it could have substructure consisting of color-singlet clusters, such as the pair of mesons J/ψ ω. There are also hexaquark components of the wavefunction with constituents cqqcqq. One such component that is particularly important is D 0D0 π 0 , because the constituent D * 0 in the dominant component of the wavefunction can decay into D 0 π 0 . In considering the various possible components of the X(3872) wavefunction, it is essential to take into account their couplings to D * 0D0 and the resonant interactions between the charm mesons.
The interplay between the D * 0D0 , D 0D0 π 0 , and other components of the wavefunction can be treated systematically using an effective field theory in which the D * 0 ,D 0 , and π 0 are explicit degrees of freedom. Such an effective field theory has been developed by Fleming, Kusunoki, Mehen, and van Kolck and named XEFT [6]. In its simplest form, XEFT is a nonrelativistic field theory for D * 0 and D 0 , their antiparticles, and π 0 . It is straightforward to extend XEFT to include the charged charm mesons and the charged pions. In XEFT, the interactions between D * 0 and D 0 must be treated nonperturbatively in order to generate a bound state that can be identified with the X(3872). Fleming et al. showed that pion-exchange interactions can be treated perturbatively along with the range corrections to D * D interactions [6]. We denote the masses of D * 0 , D 0 , and π 0 by M * , M , and m, respectively. We denote the difference between the D * 0 mass and the sum of the D 0 and π 0 masses by δ: δ ≡ M * − (M + m) ≈ 7.14 ± 0.07 MeV. ( The power counting of XEFT is defined by taking external momenta, the binding momentum scale √ M δ X of the X(3872), and the momentum scale √ mδ of the pion to all be low momentum scales of order Q. The low energy scales include the binding energy δ X of the X(3872), the kinetic energy scale δ for a pion, and the kinetic energy scale mδ/M for a charm meson. The high momentum or energy scales include m, M , and 4πf π , where f π is the pion decay constant. Amplitudes are calculated as systematic expansions in powers of Q divided by a high momentum or energy scale. In the original paper on XEFT by Fleming et al., the momentum distributions in the decay of X(3872) to D 0D0 π 0 were calculated to next-to-leading order (NLO) in the XEFT power counting and to leading order in an expansion in powers of m/M [6].
There have been a number of subsequent applications of XEFT. Fleming and Mehen applied XEFT at leading order (LO) to decays of the X = X(3872) into the P-wave charmonium state χ cJ plus one or two pions [7,8]. Mehen and Springer applied XEFT at LO to the radiative decays X → ψ(2S)γ and ψ(4040) → Xγ [9]. Margaryan and Springer applied XEFT at LO to the decay ψ(4160) → Xγ [10]. Recently, Jansen, Hammer and Jia used XEFT to determine the dependence of the binding energy of X on the light quark masses [11]. They also calculated the D * 0D0 scattering length to NLO and to leading order in m/M . The applications of XEFT are not limited to the DDπ sector, which also includes D * D and DD * states. Canham, Hammer, and Springer pointed out that XEFT can be applied to the DDDπ and DDDππ sectors [12]. They calculated the S-wave phase shifts for low-energy scattering of D or D * from X at LO. Braaten, Hammer, and Mehen pointed out that XEFT can be applied to the DDππ sector, which also includes D * D * , D * D π, and DD * π states [13]. They calculated the low-energy cross sections for elastic π + X scattering and for the break-up reaction π + X → D * +D0 at LO.
We will refer to the formulation of XEFT presented in Ref. [6] as original XEFT. There are various problems with original XEFT that present obstacles to accurate quantitative predictions. Some of these problems are due to its formulation as a nonrelativistic field theory of the charm mesons and π 0 that is not Galilean invariant. One consequence of the lack of Galilean invariance is that a preferred frame (such as the center-of-momentum frame) must be specified either explicitly or implicitly in any calculation. Another consequence of the lack of Galilean invariance is that ultraviolet divergences are much less constrained. The explicit counterterms in the Lagrangian for original XEFT in Ref. [6] are sufficient to eliminate ultraviolet divergences in NLO calculations only if results are expanded in powers of m/M and then truncated at a sufficiently low order. This truncation provides a limit on the accuracy. Although the mass ratio m/M = 0.072 is very small, the expansion is actually in powers of the square root of the mass ratio, which is 0.27. The alternative of adding additional counterterms to the Lagrangian to cancel the ultraviolet divergences would introduce additional parameters that would have to be determined phenomenologically. Another problem with original XEFT is that the renormalization scheme made it difficult to take into account some decays into final states with momenta too large to be treated explicitly in XEFT. These decays are D * 0 → D 0 γ, which accounts for about a third of the full width of the D * 0 , and all decay modes of the X(3872) other than D 0D0 π 0 .
Alhakami and Birse have recently proposed an alternative power counting for XEFT [14]. In their power counting, m and M * − M are treated as small energy scales of order Q, and δ is treated as a tiny energy scale of order Q 2 /M . The ratios δ/m = 0.051 and m/M = 0.072 are both of order Q/M . This power counting scheme was not implemented at the lagrangian level, but it provides an organizing principle for simplifying Feynman diagrams. The power counting of Ref. [14] makes the expansion in powers of m/M systematic. The formulation in Ref. [14] is convenient for matching onto heavy hadron chiral perturbation theory, which has been used in several applications of XEFT [7][8][9][10]. It does not address the problems of frame dependence or of additional ultraviolet divergences. With this new power counting, the decay D * 0 into D 0 γ is within the domain of applicability of XEFT. However there are important decay modes of the X(3872), such as J/ψ π + π − , that remain outside the domain of applicability of XEFT.
In this paper, we present a new formulation of XEFT that removes many of the obstacles to accurate quantitative calculations. The new formulation is an effective field theory with a Galilean symmetry that is motivated by the fact that mass is very nearly conserved in the transition D * 0 → D 0 π 0 . The Galilean symmetry solves the problem of frame dependence and it dramatically simplifies ultraviolet divergences. To take into account the decay D * 0 → D 0 γ and decay modes of the X(3872) other than D 0D0 π 0 , a new renormalization scheme is introduced that is expressed in terms of the complex energies of D * 0 and X(3872). The new Galilean-invariant formulation of XEFT is illustrated by a calculation of the D * 0D0 scattering length to NLO in the XEFT power counting. The cancellation of ultraviolet divergences for arbitrary values of m/M is verified explicitly.

II. GALILEAN-INVARIANT XEFT
In this section, we introduce a new formulation of XEFT as a Galilean-invariant field theory. We describe the changes in the Lagrangian for the original XEFT defined in Ref. [6] that are required for Galilean invariance. We explain how the effects of the decay D * 0 → D 0 γ can be taken into account through the complex rest energy of the D * 0 . We then write down the next-to-leading order Lagrangian for Galilean-invariant XEFT, and give its Feynman rules.

A. Galilean Invariance
A unique feature of the decay D * 0 → D 0 π 0 is that mass is very nearly conserved: the sum of the masses of the D 0 and π 0 is only about 3.5% lower than the mass of the D * 0 . Galilean invariance is a possible space-time symmetry of a nonrelativistic theory that requires exact mass conservation [15]. The mass that must be conserved is the kinetic mass, which is the mass that appears in the denominator of the kinetic energy. The approximate conservation of mass in the transition D * 0 → D 0 π 0 strongly motivates a Galilean-invariant formulation of XEFT. The Lagrangian for original XEFT, including all terms required to calculate the decay of X(3872) into D 0D0 π 0 to next-to-leading order (NLO), was written down in Ref. [6]. We will describe how the terms in this Lagrangian must be modified to make them Galilean invariant. The NLO Lagrangian for Galilean-invariant XEFT will be written down in Section II C.
We choose the kinetic masses of the D 0 and π 0 to be their physical masses M and m, respectively. Conservation of kinetic mass then requires the kinetic mass of D * 0 to be M + m. The difference δ between the mass of the D * 0 and its kinetic mass must be taken into account through its rest energy. Kinetic mass conservation requires the following change in the Lagrangian for original XEFT defined in Ref. [6]: • In the kinetic term D † · ∇ 2 D/(2m D * ) for the D * 0 , its mass m D * must be replaced by M + m.
A similar change must be made in the kinetic term for theD * 0 . It will be convenient to introduce the reduced kinetic mass µ for D * 0D0 and the reduced mass µ π for D 0 π 0 : The ratio of these reduced masses is Galilean invariance requires interaction terms to be invariant under Galilean boosts, in which the momenta of π 0 , D 0 , and D * 0 are boosted by a common velocity vector v multiplied by their kinetic masses m, M , and M + m, respectively. Galilean invariance requires three changes in the interaction terms in the Lagrangian for original XEFT that given in Ref. [6]: • In the pion interaction term D † · D∇π, the operator ∇ between D and π should be replaced by ( • In the interaction term (DD) † ·DD∇π that describes the transition of D 0D0 π 0 to D * 0D0 , the operator ∇ betweenDD and π should be replaced by (2M Similar changes must be made in the hermitian conjugates and charge conjugates of these interaction terms. The three modified interaction terms described above reduce to those of original XEFT in the limit m/M → 0. The modified interaction terms ensure the invariance of amplitudes under Galilean boosts. Galilean invariance also strongly constraints ultraviolet divergences. The changes above specify all terms in the Lagrangian for Galilean-invariant XEFT that are required to calculate the decay of X(3872) into D 0D0 π 0 to NLO in the XEFT power counting. The results should be independent of the ultraviolet cutoff to all orders in m/M . Without Galilean invariance, there are three independent ∇ 2 D * 0D0 interaction terms and two independent D 0D0 π 0 → D * 0D0 interaction terms. The Lagrangian for original XEFT defined in Ref. [6] includes only one interaction term of each kind. Cutoff independent results for the decay rate of X(3872) into D 0D0 π 0 at NLO were obtained by also truncating the expansion in powers of m/M at leading order.
Galilean-invariant XEFT can be extended to include charged charm mesons and charged pions. The π + must have the same kinetic mass m as π 0 . The D + must have the same kinetic mass M as D 0 . The D * + must have the same kinetic mass M + m as D * 0 . The difference between the mass and kinetic mass of a particle must be taken into account through its rest energy.
In a nonrelativistic effective field theory for charm mesons, one can impose a phase symmetry that guarantees the separate conservation of the number N c of charm quarks and the number Nc of charm antiquarks. These quark numbers can be expressed in terms of meson numbers: If one also considers charged charm mesons, the charm quark number N c is the sum of the numbers of D * 0 , D * + , D 0 , and D + . In Galilean-invariant XEFT, the exact conservation of kinetic mass in the transitions D * 0 ↔ D 0 π 0 and D * 0 ↔D 0 π 0 provides motivation for introducing an additional phase symmetry that guarantees the conservation of the pion number defined by The name pion number is appropriate since the D * can be interpreted as a P-wave Dπ resonance. In Galilean-invariant XEFT with charged charm mesons and pions, the pion number is the sum of the numbers of π 0 , π + , π − , D * 0 ,D * 0 , D * + , and D * − .

B. Choice of Rest Energies
In a nonrelativistic field theory, deviations from mass conservation in a reaction are taken into account through the rest energies of the particles involved. The rest energy of a particle can be chosen to be its physical mass. In a Galilean invariant theory, since the kinetic mass is conserved, the rest energy of a particle can equally well chosen as the difference between its physical mass and its kinetic mass. If there are linear combinations of the particle numbers that are conserved, field redefinitions can be used to set the rest energies of some of the particles to 0. In XEFT, conservation of the charm quark number N c allows the rest energies of D 0 and D * 0 to be set to 0 and M * − M , respectively. Conservation of both N c and the pion number N π allows the rest energies of π 0 , D 0 , and D * 0 to be set to 0, 0, and δ = M * − M − m, respectively.
The rest energy of a particle can also be used to take into account its partial width into decay modes that cannot be described explicitly in the effective field theory. Such a partial width is taken into account through a negative imaginary term in the rest energy. The width of D * 0 is more than 5 orders of magnitude larger than the widths of D 0 and π 0 , so the widths of D 0 and π 0 can be completely ignored. The width of D * 0 come from its decay into D 0 π 0 , which can be described explicitly in XEFT, and from its decay into D 0 γ, whose momenta are too large to be described explicitly. The effect of the D 0 γ decay mode can be taken into account by including a term −iΓ * 0,γ /2 in the D * 0 rest energy, where Γ * 0,γ is the partial width of D * 0 into D 0 γ, whose branching fraction is approximately 38%. If we use the complex on-shell renormalization scheme for the D * 0 propagator in which the position of the complex pole is specified, the D * 0 rest energy also includes the term −iΓ * 0,π /2, where Γ * 0,π is the partial width of D * 0 into D 0 π 0 , whose branching fraction is approximately 62%. In this scheme, the complete imaginary part of the D * 0 rest energy is −iΓ * 0 /2, where Γ * 0 = Γ * 0,π + Γ * 0,γ is the full width of the D * 0 .
In original XEFT, the rest energies of π 0 , D 0 , and D * 0 were chosen to be −δ, 0, and 0, respectively. Setting the rest energy of D * 0 to 0 is inconvenient if we wish to take into account the partial width of D * 0 into D 0 γ. It is more convenient to choose the rest energies of π 0 and D 0 to be 0. The rest energy of D * 0 is then δ − iΓ * 0 /2. This requires the following change in the Lagrangian for original XEFT: • The rest-energy term δπ † π for the π 0 should be replaced by the rest-energy term −(δ − iΓ * 0 /2)D † · D for the D * 0 .
There is a similar rest-energy term for theD * 0 . If Galilean-invariant XEFT is extended to include charged charm mesons and charged pions, mass differences and nonnegligible decay modes outside the domain of validity of the effective field theory can be taken into account through the rest energies of the particles. The π + − π 0 mass difference can be taken into account through the rest energy of π + . The D + − D 0 mass difference can be taken into account through the rest energy of D + . The difference between the D * + − D 0 π 0 mass difference can be taken into account through the rest energy of D * + . The tiny partial width for the decay of D * + into D + γ can also be taken into account through the rest energy of D * + .

C. NLO Lagrangian
We can now write down the complete NLO Lagrangian for Galilean-invariant XEFT for the neutral charm mesons and the π 0 . It includes only those terms required to calculate the elastic scattering amplitude for D * 0D0 to NLO in the XEFT power counting. The field for the π 0 is denoted by π. The fields for the D 0 andD 0 are denoted by D and D. The fields for the D * 0 andD * 0 are denoted by D andD. The Lagrangian is the sum of kinetic terms, interaction terms, and counterterms.
There are kinetic terms for the π 0 , D 0 ,D 0 , and D * 0 , andD * 0 . The kinetic terms for the π 0 , D 0 , and D * 0 are The kinetic terms for theD 0 andD * 0 can be obtained from those for D 0 and D * 0 by replacing D byD and D byD. The interaction terms consist of pion interactions, contact interactions, and ∇ 2 interactions. The pion interaction terms for the transitions D * 0 ↔ D 0 π 0 are The pion interaction terms for the transitionsD * 0 ↔D 0 π 0 are obtained by replacing D and D byD andD. The contact and ∇ 2 interactions are the same in the D * 0D0 → D * 0D0 , D 0D * 0 ↔ D * 0D0 , and D 0D * 0 → D 0D * 0 channels. The interaction terms in the D * 0D0 → D * 0D0 channel are The interaction terms in the other three channels are obtained by replacing D andD byD and D in the appropriate places. The coupling constants g in Eq. (8) and C 0 and C 2 in Eq. (9) are essentially the same as those in the Lagrangian for original XEFT defined in Ref. [6].
To cancel ultraviolet divergences in Green functions at NLO, it is also necessary to include counterterms in the Lagrangian. The counterterms are the same in the D * 0D0 → D * 0D0 , D 0D * 0 ↔ D * 0D0 , and D 0D * 0 → D 0D * 0 channels. The counterterms in the D * 0D0 → D * 0D0 channel are The counterterm with coefficient δD 0 is not needed to calculate on-shell quantities, because it can be reduced to the other counterterm by using field redefinitions. It was omitted in Ref. [6], because the momentum distribution for D 0D0 π 0 in the decay of the X(3872) is an on-shell quantity.

D. Feynman Rules
The terms in the NLO Lagrangian for Galilean-invariant XEFT with neutral charm mesons and the π 0 are given in Eqs. (7), (8), (9), and (10). We will give the Feynman rules for this Lagrangian. The Feynman rules that are actually used for calculations beyond leading order will be enclosed in boxes.
In XEFT, the charm quark and antiquark numbers N c and Nc are conserved. In Galilean-invariant XEFT, the pion number N π is also conserved. These conservation laws can all be made manifest in the Feynman rules by appropriate notation for the propagators. We use a dashed line for the pion propagator, a solid line for the D andD propagators, and a double line consisting of a solid and a dashed line for the D * andD * propagators. In the propagators for the mesons D * and D that contain a charm quark, the solid line has a forward arrow. In the propagators for the mesons D * andD that contain a charm antiquark, the solid line has a backward arrow. It is sometimes convenient to omit the arrows on internal lines of diagrams and to use the convention that there is an implied sum over the possible directions of the omitted arrows.
The propagators for the π 0 , D 0 , and D * 0 are illustrated in Fig. 1. The Feynman rules for the propagators of the π 0 , the D 0 orD 0 , and the D * 0 orD * 0 are where p 0 and p are the energy and momentum of the particle. The propagator of the D * 0 is diagonal in its vector indices i and j, and its complex rest energy is Since E * has a negative imaginary part, an explicit iǫ prescription is unnecessary in the D * 0 propagator. At LO in the XEFT power counting, the only interaction that is required is a contact interaction for D * 0D0 in the C = + channel. The Feynman rule for the vertex is the same for the D * 0D0 → D * 0D0 , D 0D * 0 ↔ D * 0D0 , and D 0D * 0 → D 0D * 0 contact interactions: This vertex for D * 0D0 → D * 0D0 is illustrated in Fig. 2. The C 0 interaction must be treated nonperturbatively in XEFT. The set of subdiagrams consisting of an arbitrary number of successive C 0 interactions can be summed to all orders analytically. The Feynman rule for the resulting effective interaction is given later in Eq. (44). In calculations beyond leading order in XEFT, that effective vertex replaces the one in Eq. (13). That is why the Feynman rule in Eq. (13) is not enclosed in a box. One of the interactions beyond LO in the XEFT power counting is the D * ↔ Dπ transition. The Feynman rules for the D * 0 → D 0 π 0 andD * 0 →D 0 π 0 vertices in Galilean-invariant XEFT are where q and p are the momenta of the outgoing π 0 and charm meson, respectively. The vertices for D * 0 → D 0 π 0 and D 0 π 0 → D * 0 are illustrated in Fig. 3. In the D 0 π 0 center-of-momentum frame defined by p + q = 0, the momentumdependent factor in Eq. (14) reduces to q i , which is the momentum-dependent factor in all frames in original XEFT. The Feynman rules for the D 0 π 0 → D * 0 andD 0 π 0 →D * 0 vertices differ from Eq. (14) by an overall minus sign if q and p are the momenta of the incoming π 0 and charm meson, respectively. A convenient way to implement the XEFT power counting is to assign orders in g to the coupling constants of all other interaction terms. A complete calculation then requires calculating all diagrams to a given order in g.
The other interaction at NLO in the XEFT power counting is the ∇ 2 interaction for D * 0D0 in the C = + channel. The Feynman rule for the vertex is the same for the D * 0D0 → D * 0D0 , D 0D * 0 ↔ D * 0D0 , and D 0D * 0 → D 0D * 0 interactions. In Galilean-invariant XEFT, the Feynman rule is where p and p * are the momenta of the incoming spin-0 and spin-1 charm mesons and p ′ and p ′ * are the momenta of the outgoing mesons. The vertex for D * 0D0 → D * 0D0 is illustrated in Fig. 4. In the D * 0D0 center-of-momentum frame defined by p * + p = p ′ * + p ′ = 0, the momentum-dependent factor in Eq. (15) reduces to p 2 + p ′ 2 . In this frame, the Feynman rule coincides with that in original XEFT, for which the momentum-dependent factor is The coupling constant C 2 in Eq. (15) can be assigned order g 2 . In addition to these interaction vertices, it is also necessary to include a counterterm vertex for D * 0D0 in the C = + channel. The vertex is the same for the D * 0D0 → D * 0D0 , D 0D * 0 ↔ D * 0D0 , and D 0D * 0 → D 0D * 0 interactions: where E is the total energy of the pair of charm mesons. The counterterm vertex for D * 0D0 → D * 0D0 is illustrated in Fig. 4. The δD 0 term is not needed to calculate on-shell quantities at NLO, but it is needed to cancel ultraviolet divergences in off-shell Green functions. The counterterm coefficients δC 0 and δD 0 in Eq. (16) can be assigned order g 2 .
In the original paper on XEFT, an additional interaction term that produces a transition of D * 0D0 to D 0D0 π 0 was written down explicitly [6]. The Feynman rule for its vertex in Galilean-invariant XEFT is where q, p, andp are the outgoing momenta of the pion and the two spin-0 charm mesons. In the D 0D0 π 0 center-ofmomentum frame defined by p +p + q = 0, the momentum-dependent factor in Eq. (17) reduces to q i , which is the momentum-dependent factor in all frames in original XEFT. The coupling constant B 1 in Eq. (17) can be assigned order g 3 . This interaction term was needed in Ref. [6] to calculate the decay of X(3872) into D 0D0 π 0 to NLO in the XEFT power counting or, equivalently, to relative order g 2 .
In the D * 0 propagator in Eq. (11c), a wavefunction renormalization factor Z can be inserted in the numerator. That factor can be made completely arbitrary by inserting canceling factors of Z −1/2 into the interaction vertices for each D * 0 orD * 0 line. The number of such factors in the interaction vertices in Eqs. (13), (14), (15), and (16) would be 2, 1, 2, and 2, respectively. These factors can be absorbed into the coupling constants.
If we use dimensional regularization with d spatial dimensions, it is useful to introduce a renormalization scale Λ to keep the dimensions of coupling constants the same as in the physical dimension d = 3. The interaction vertices in Eqs. (13), (14), (15), and (16) would be multiplied by Λ raised to the powers 3 − d, (3 − d)/2, 3 − d, and 3 − d, respectively. In a Green function, the net effect of these powers of Λ is a factor of Λ 3−d for every loop integral and an overall multiplicative factor of Λ 3−d raised to some power. The factor of Λ 3−d associated with a loop integral can be absorbed into its integration measure. If the Green function is made finite by renormalization, the overall multiplicative factor of Λ 3−d raised to some power can be simply be ignored, because it is equal to 1 in the physical dimension d = 3.

III. RENORMALIZATION PRESCRIPTIONS
In this section, we introduce renormalization prescriptions that can be used to take into account the effects of transitions to states in which the momenta are too large to be described explicitly in XEFT. We also calculate the transition amplitude for D * 0D0 → D * 0D0 at leading order in XEFT.

A. D * − Dπ Coupling Constant
One of the coupling constants in XEFT beyond leading order is the D * − Dπ coupling constant g/ √ 2f π . It could in principle be determined from the partial width of D * 0 into D 0 π 0 : where δ is the mass difference in Eq. (2) and µ π is the Dπ reduced mass defined in Eq. (3b). The branching fraction for D * 0 → D 0 π 0 has been measured to be (62 ± 3)%, but the full width of D * 0 has not been measured. By exploiting isospin symmetry, the coupling constant g/ √ 2f π can also be determined from decays of the D * + . The full width of D * + has been recently measured with high precision by the Babar Collaboration [17]: Γ[D * + ] = 83.3 ± 1.2 ± 1.4 keV. The sum of the branching factions of D * + into D 0 π + and D + π 0 is 0.984 ± 0.005. In Galileaninvariant XEFT, the sum of the partial widths into D 0 π + and D + π 0 is where δ +0 = 5.856 ± 0.002 MeV is the D * + − D 0 π + mass difference, and δ ++ = 5.68 ± 0.08 MeV is the D * + − D + π 0 mass difference. The resulting value of the coupling constant g is given by The pion decay constant f π determines the scattering amplitude for ππ scattering. With the convention for f π used in Ref. [6], its value is f π ≈ 132 MeV. Given that value, the pion transition coupling constant in Eq. (20) is determined to be g ≈ 0.59. A more appropriate dimensionless measure of the strength of the pion exchange interaction in XEFT is where µ is the reduced kinetic mass of D * 0D0 defined in Eq. (3a). The determination of the values of g and f π separately is unnecessary in XEFT applied to the DDπ sector (which includes D * D ), because the number of pions in the system can only be 0 or 1. In the DDππ sector (which includes D * D * ), the number of pions in the system can be 0, 1, or 2. Since there can be contributions from ππ scattering, the value of f π is needed at a sufficiently high order in the XEFT power counting.
The conservation of pion number in Galilean-invariant XEFT implies that the exact D * 0 propagator can be calculated analytically. It can be obtained by summing a geometric series of the one-loop D * 0 self-energy diagram in Figure 5. Galilean invariance implies that the propagator for a D * 0 with energy p 0 and momentum p is a function of the Galilean-invariant combination The exact D * 0 propagator can be expressed as where Σ(E cm ) is the D * 0 self-energy and δE * is a rest-energy counterterm. We have introduced a wavefunction renormalization factor 1 + δZ in the numerator. It can be made completely arbitrary by absorbing canceling factors of [1 + δZ] −1/2 into the interaction vertices. With dimensional regularization, the D * 0 self-energy is where Λ is the renormalization scale and µ π is the D 0 π 0 reduced mass in Eq. (3b). The self-energy has cubic and linear ultraviolet divergences that are manifested as single poles in d and d − 2, respectively. The pole in d − 2 is In the physical dimension d = 3, the self-energy is pure imaginary for real positive E cm : Its value at E = δ is −iΓ * 0,π /2, where Γ * 0,π is the partial width of D * 0 into D 0 π 0 given in Eq. (18).
In the minimal power-divergence subtraction (PDS) renormalization scheme [16], the D * 0 rest energy and the propagator counterterms are .
In the physical dimension d = 3, the self-energy Σ(E cm ) reduces to Eq. (26). The approximate position of the pole in p 0 of the D * 0 propagator in Eq. (23) is where E cm is equal to the complex energy δ − iΓ * 0,π /2. The PDS scheme does not take into account the decay D * 0 → D 0 γ. The pole in the energy p 0 for the physical D * 0 propagator occurs when E cm is equal to the complex energy E * = δ − iΓ * 0 /2, where Γ * 0 is the full width of the D * 0 . We introduce the complex on-shell (COS) renormalization scheme for the D * 0 propagator in which its pole in p 0 is at the complex physical value and the residue of that pole is the same as at LO. The D * 0 rest energy and the propagator counterterms are where Σ(E) is the self-energy in Eq. (24). In the physical dimension d = 3, δE * and δZ in the COS scheme reduce to If we ignore the tiny difference between E * = δ − iΓ * 0 /2 and δ, the rest energy counterterm in Eq. (29a) reduces to δE * = −iΓ * 0,π /2, where Γ * 0,π is the partial width of the D * 0 into D 0 π 0 in Eq. (18). Although Eq. (23) is the exact D * 0 propagator in Galilean-invariant XEFT, it is not practical to use this propagator in loop diagrams, because the term in the denominator proportional to E d/2 cm makes loop integrals more complicated to evaluate. It is better to expand the exact propagator in Eq. (23) in powers of g 2 : The expansion generates a geometric series of propagator corrections. The terms of order g 2 correspond to inserting into the propagator the one-loop D * 0 self-energy subdiagram in Fig. 6 and a D * 0 self-energy counterterm. The Feynman rule for the order-g 2 self-energy counterterm vertex in the COS scheme is where p 0 and p are the energy and momentum of the D * 0 , i and j are its vector indices, and Σ(E) is the self-energy in Eq. (24). The previous NLO calculations in XEFT were carried out in the original version of XEFT, which is not Galilean invariant. The D * 0 self-energy Σ(p 0 , p) is therefore a function of the two independent variables p 0 and p. The kinetic mass of the D * 0 was set equal to its physical mass M * . The D * 0 rest energy was taken to be the real energy δ, although it was actually taken into account through the rest energy term for the π 0 . In the NLO calculation of the decay of X(3872) into D 0D0 π 0 in Ref. [6], the renormalization of the D * 0 propagator was carried out using the minimal power divergence subtraction scheme in Eq. (27). In the physical dimension d = 3, the counterterm δZ PDS in Eq. (27c) reduces to the renormalization scale Λ multiplied by a constant. This term was cancelled by a kinetic counterterm for the D * 0 .
In the NLO calculation of the D * 0D0 scattering length In Ref. [11], the complex energy of the X(3872) and its partial width into D 0D0 π 0 were calculated to NLO using original XEFT. The calculation involved a two-loop diagram that has a one-loop D * 0 self-energy subdiagram and that diverges as 1/p as the relative momentum p of D * 0D0 approaches 0. The divergence was avoided by resumming to all orders terms proportional to Γ 1/2 * 0,π from one-loop D * 0 self-energy subdiagrams. The resummation could have been implemented in such a way that the terms that were resummed had only integer powers of Γ * 0,π . This is equivalent to the complex on-shell renormalization scheme for the D * 0 propagator in Eq. (28), but with E * replaced by δ − iΓ * 0,π /2. The imaginary part of E * is not equal to the physical value −Γ * 0 /2, because the decay D * 0 → D 0 γ was not taken into account in Ref. [11]. The amputated connected Green function for D * 0D0 → D * 0D0 in the C = + channel, with the incoming and outgoingD 0 or D 0 on their energy shells, can be expressed as +iA ij , where the tensor A ij is a function of the total energy P 0 and the momenta of the two incoming and two outgoing particles. Its vector indices are associated with the polarizations of the incoming and outgoing D * 0 orD * 0 . In XEFT, the corresponding amplitude in the C = − channel is 0. We will refer to A ij as the transition amplitude for D * 0D0 in the C = + channel or simply as the transition amplitude. Its contribution to the amplitude for D * 0D0 → D * 0D0 is +iA ij /2, where the factor of (1/ √ 2) 2 comes from projections onto the C = + channel.
The tree level term in the LO transition amplitude, which is given by the Feynman rule in Eq. (13), is −C 0 δ ij . The D * 0D0 coupling constant C 0 must be treated nonperturbatively in order to generate the bound state that can be identified with the X(3872). The one-loop diagram in Fig. 7 is therefore also LO. The complete LO transition amplitude can be obtained by summing a geometric series of the one-loop diagrams. This sum is equivalent to the solution of the Lippmann-Schwinger equation shown in Figure 8.
In Galilean-invariant XEFT, the LO transition amplitude depends on the energies and momenta of the initial-state and final-state particles only through the Galilean-invariant combination where P 0 is the total energy of the D * 0 andD 0 relative to the D 0D0 π 0 threshold and P is their total momentum.
and Π(E cm ) is a function of E cm that we will refer to as the D * 0D0 self-energy. The inverse coupling constant from the contact interaction vertex in Eq. (13) has been separated into C −1 0 and a LO counterterm δC −1 0 . Despite the notation, δC −1 0 is independent of the NLO counterterm δC 0 in the counterterm vertex in Eq. (16). With dimensional regularization, the D * 0D0 self-energy is where µ is the reduced kinetic mass of D * 0D0 defined in Eq. (3a). If the amplitude A(E cm ) in Eq. (33) has a pole in P 0 , the residue of the pole is This is a smooth function of d for d < 4. The D * 0D0 self-energy has a linear ultraviolet divergence that is manifested by a pole in d − 2: .
In the minimal power-divergence subtraction (PDS) scheme [16], this pole is cancelled by a LO counterterm: .
The LO transition amplitude in Eq. (33) then has a finite limit as d → 2: The momentum scale Λ 2 in the logarithm is determined by the limiting behavior of C −1 0 as d → 2. In the physical dimension d = 3, the LO transition amplitude in Eq. (33) reduces to the form where γ X is a complex constant that we will refer to as the binding momentum of the X(3872). It is determined by the limiting behavior of C −1 0 as d → 3: The amplitude A(E cm ) in Eq. (33) has a pole in P 0 on the physical sheet. The residue of the pole is The pole in P 0 of the amplitude A(E cm ) in Eq. (33) occurs when E cm equals a complex energy E X that can be interpreted as the energy of the X(3872) resonance: In the first line, the complex energy E X has been expressed in terms of the complex D * 0 energy E * and the complex binding momentum γ X . In the second line, E X has been expressed in terms of real variables δ X and Γ X that can be interpreted as the binding energy and full width of the X(3872). The measured value of the binding energy δ X is given in Eq. (1). The width Γ X has not yet been measured, but there is an upper bound: Γ X < 1.2 MeV [18]. If the complex binding momentum is expressed as γ X = γ re + iγ im , where γ re and γ im are real, the binding energy and full width are In calculations beyond LO in the XEFT power counting, it is convenient to replace the contact interaction vertex in Eq. (13) by a vertex for the transition amplitude. The Feynman rules are the same for the D * 0D0 → D * 0D0 , D 0D * 0 ↔ D * 0D0 , and D 0D * 0 → D 0D * 0 transition amplitudes: where E cm is the function of the total energy P 0 and the total momentum P of the charm meson pair defined in Eq. (32) and i and j are the vector indices of the incoming and outgoing D * 0 . In d dimensions, the function A(E cm ) is defined by Eqs. (33) and (34). In the physical dimension d = 3, it reduces to Eq. (39). The vertex for the transition amplitude is represented by a blob, as illustrated for the D * 0D0 → D * 0D0 channel by the Lippmann-Schwinger equation in Fig. 8. Beyond leading order in the XEFT power counting, the transition amplitude A ij (E cm , p, p ′ ) for D * 0D0 in the C = + channel in Galilean invariant XEFT is a function of E cm and the relative momenta p and p ′ of the incoming and outgoing charm mesons. At NLO, this amplitude has additional ultraviolet divergences. The renormalizability of XEFT requires that they be cancelled by appropriate choices of the constants δC 0 and δD 0 in the counterterm vertex in Eq. (16). A renormalization prescription for the amplitude A ij (E cm , p, p ′ ) is necessary to determine the finite parts after the cancellations. In order to take into account decay modes of the X(3872) with momenta too large to be described explicitly, it is advantageous to use the complex on-shell (COS) renormalization scheme in which the pole in the total energy P 0 is at its physical value. The complex energy of the pole can be specified in terms of the complex binding momentum γ X of the X(3872), as in Eq. (42a), or in terms of its binding energy δ X and full width Γ X , as in Eq. (42b). The definition of the COS scheme for the D * 0D0 → D * 0D0 Green function can be completed by specifying the residue of the pole in P 0 . The amplitude A ij (E cm , p, p ′ ) has a well-behaved limit as p → 0 and p ′ → 0 that is a function of E cm only. Thus we complete the definition of the COS scheme by requiring the residue of the pole in P 0 for the amplitude A ij (E cm , 0, 0) to be the same as for the LO amplitude (2π/µ)A(E cm )δ ij . In the physical dimension d = 3, the limiting behavior as P 0 approaches the pole is where Z X is the LO residue in Eq. (41) and γ X is the complex binding momentum of the X(3872). The COS scheme can be implemented through specific choices of the counterterms δC 0 and δD 0 . In the previous NLO calculations in XEFT in Ref. [6] and [11], the NLO counterterm δC 0 was included, but δD 0 was not. The counterterm δC 0 could have been chosen so that the pole in P 0 was at the physical point, as in Eq. (45). However without the counterterm δD 0 , the prescription in Eq. (45) for the residue of the pole could not have been implemented. In fact, the residue would have been ultraviolet divergent.
IV. D * 0D 0 SCATTERING In this Section, we use Galilean-invariant XEFT to calculate the D * 0D0 scattering length to NLO in the XEFT power counting.

A. NLO Transition Amplitude
In Galilean-invariant XEFT, the NLO transition amplitude for D * 0D0 → D * 0D0 in the C = + channel in the center-of-momentum frame is a tensor A ij (E, p, p ′ ) that depends on the total energy E and the relative momenta p and p ′ of the incoming and outgoing charm mesons. The LO transition amplitude is (2π/µ)A(E)δ ij , where A(E) in the physical dimension d = 3 is given in Eq. (39). The NLO diagrams for the transition amplitude are calculated in Appendix B. The 20 diagrams are labeled An, Bn, Cn, and Dn, where n is an integer. The NLO terms in the transition amplitude have a well-behaved limit as p → 0 and p ′ → 0 that is diagonal in the vector indices i and j. The NLO transition amplitude can be expressed as The NLO correction terms either have a factor of A 2 (E) or a factor of A(E). The function F (E), which has dimensions of momentum, is obtained by adding the pion-exchange diagram A4 in Eq. (B8) and the D * propagator correction diagrams B1 and B2 in Eqs. (B9) and (B10). The function B(E), which is dimensionless, is the sum of the pionexchange diagrams A2 and A3 in Eq. (B7). The function H(E), which has dimensions of (momentum) 3 , comes from the ∇ 2 vertex diagrams C in Eqs. (B16). The δC 0 and δD 0 terms come from the D * D counterterm diagrams D in Eq. (B18). The functions F (E), G(E), and H(E) can be expressed in terms of the loop integrals J n , I n , and K lmn defined in Appendix A: where r = µ π /µ is the ratio of reduced masses defined in Eq. (4). The terms proportional to A(E) in Eq. (46) have a single pole in E at E * − γ 2 X /(2µ). The terms proportional to A 2 (E) have an unphysical double pole in E. The N 2 LO term would have a triple pole and higher order terms would have even higher poles. These unphysical multiple poles can be summed to all orders, in which case they produce a shift in the position of the pole in the LO amplitude. In Ref. [11], such a resummation was used in the NLO calculations of the binding energy of the X(3872) and its partial width into D 0D0 π 0 . It was not used in the NLO calculation of the D * 0D0 scattering length. An expression for the transition amplitude that is accurate to NLO and has only a single pole is To NLO accuracy, the factor of 1 + (g 2 µ 2 /4mf 2 π )G in the numerator could equally well be moved to the denominator as a factor 1 − (g 2 µ 2 /4mf 2 π )G multiplying A(E) −1 .

B. Renormalization
Loop integrals in XEFT have ultraviolet (UV) divergences. With dimensional regularization, the UV divergences produce poles in d − 2 and poles in d − 3. A pole in d − 3 represents a logarithmic UV divergence, and a pole in d − 2 represents a linear UV divergence. In previous NLO calculations in XEFT [6,11], power divergence subtraction was used to remove the poles in d − 2. The subsequent limit d → 3 produces terms that depend explicitly on the renormalization scale Λ. In the PDS scheme, a loop integral with a pole term Λ 3−d /(d − 2) requires a canceling counterterm −Λ/(d − 2). In the physical dimension d = 3, the loop integral becomes independent of Λ, but the canceling counterterm reduces to −Λ.
In the NLO calculation of the decay of the X(3872) into D 0D0 π 0 in Ref. [6], the explicit dependence on Λ was through terms proportional to Λ, Λ 2 , and log Λ. The explicit dependence on Λ was exactly cancelled by the implicit dependence on Λ of the coupling constants of XEFT. In the NLO calculations of Ref. [11], in addition to terms proportional to Λ, Λ 2 , log Λ, there was explicit dependence on Λ from terms proportional to 1/(Λ − γ), where γ is the leading order binding momentum or inverse scattering length. These terms were introduced by a resummation prescription for dealing with an infrared divergence at the D * 0D0 threshold. The terms proportional to 1/(Λ − γ) were not cancelled by the implicit dependence on Λ of the coupling constants, but they were suppressed by a factor proportional to the partial width Γ * 0,π for D * 0 → D 0 π 0 .
The In the single pole of G(E) in Eq. (47b), the energy-dependent denominator is cancelled, so it gives pole terms with the energy dependence A(E). Neither of these terms can be cancelled by the available counterterms δC 0 and δD 0 . The poles in d − 2 after suitable choices of the counterterms are The argument of the logarithm has been made dimensionless by using the renormalization scale Λ. The uncanceled poles in d − 2 in Eq. (49) seem to indicate that the NLO transition amplitude in Galilean-invariant XEFT has linear UV divergences that cannot be removed by renormalization. This puzzle can be resolved by taking into account the expression for the amplitude A(E) in d = 2, which is given in Eq. (38). Its reciprocal A(E) −1 is linear in log(E * − E). The cancellation between the two terms in Eq. (49) leaves a term with the energy dependence A 2 (E) that can be cancelled by the counterterm δC 0 .
We proceed to implement the complex on-shell renormalization scheme specified by Eq. (45). This scheme requires the pole in E of A ij (E, 0, 0) to be at the same complex energy E X and to have the same residue as the LO amplitude (2π/µ)A(E)δ ij . It can be implemented as specific choices of the counterterms δC 0 and δD 0 . We use the variation of the amplitude in Eq. (48) in which the term (g 2 µ 2 /4mf 2 π )G in the numerator is moved to the denominator. The renormalized expression for the transition amplitude is where F sub (E) and H sub (E) are obtained by subtracting terms from F (E) and H(E): The denominator in Eq. (50) vanishes at E = E X . The value of its first derivative at E = E X is such that the residue of the pole in E is (2π/µ)Z X δ ij , in accord with the renormalization prescription in Eq. (45). The explicit expressions for the counterterms are given by In the previous NLO calculations in XEFT in Refs. [6] and [11], power divergence subtraction was used to make double and single poles in d − 2 explicit as terms proportional to Λ 2 and Λ, where Λ is the renormalization scale. Since the poles in d − 2 have the same energy dependence as the counterterms δC 0 and δD 0 , the terms proportional to Λ 2 and Λ are exactly cancelled by those counterterms in the COS scheme. Thus it is unnecessary to use power divergence subtraction to make the poles in d − 2 explicit.
C. D * 0D 0 Scattering Length T-matrix elements in XEFT involving charm mesons and π 0 's can be obtained from the amputated connected Green functions by the following steps: • Multiply by a polarization vector ε i for a D * 0 orD * 0 in the initial state and by a polarization vector ε ′j * for a D * 0 in the final state.
• Multiply by a residue factor |Z * | 1/2 for each D * 0 orD * 0 in the initial and final states. In the COS scheme, Z * = 1.
• Put the external lines on their energy shells. For the D 0 orD 0 and for the π 0 , the energy shells are real: p 0 = p 2 /(2M ) and p 0 = p 2 /(2m), respectively. For the D * 0 orD * 0 , the energy shell is complex: We consider the elastic scattering of D * 0D0 in the center-of-momentum frame with incoming momenta ±pẑ and outgoing momenta ±pn. The scattering angle satisfies cos θ =ẑ ·n. The initial and final polarization vectors of the D * 0 are ε and ε ′ . The total energy is E p = E * + p 2 /(2µ). The T-matrix element is In the limit of zero relative momentum, the scattering is isotropic. The T-matrix element reduces to where the transition amplitude is given by Eq. (50) evaluated at E = E * . In the case of short-range interactions, a scattering amplitude can be expanded in powers of the relative momentum. This expansion is called the effective range expansion. Beyond LO in XEFT, the effective range expansion for D * 0D0 breaks down because of the effects of the exchange of a pion that can be on its energy shell. However Jansen, Hammer, and Jia pointed out that the leading term in the effective range expansion, which is the S-wave D * 0D0 scattering length, remains well defined [11]. They calculated the scattering length to NLO in original XEFT, truncating the expression at first order in an expansion in powers of γ X / √ 2mδ and at leading order in m/M . We will calculate the scattering length to NLO in Galilean-invariant XEFT, truncating the expression at fourth order in an expansion in powers of γ X / √ 2µE * but keeping all orders in m/M . The S-wave scattering length a s,+ for D * 0D0 in the C = + channel can be defined by expressing the T-matrix in the zero-momentum limit in Eq. (54) as At leading order in XEFT, the inverse scattering length is equal to the complex binding momentum: 1/a s,+ = γ X . The inverse scattering length at NLO is We have used the expressions for E X and Z X in terms of γ X in Eqs. (42a) and (41). This expression for 1/a s,+ in Eq. (56) can be simplified by expanding it in powers of γ X / √ 2µE * using the expressions for the one-loop integrals J n and I n in Eqs. (A21a) and (A22a) and the threshold expansions for the two-loop integrals K lmn in Eqs. (A30). The expansion of G(E X ) − 2G(E * ) to third order in γ X is where r = µ π /µ is the ratio of reduced masses defined in Eq. (4). The quantity H sub (E * ) is very simple: The expansion of F sub (E * ) to fourth order in γ X is Our final result for the inverse scattering length, including all terms suppressed by up to three powers of γ X / √ 2µE * , is The coefficient of each term has been expanded to relative order r 5/2 . The leading power of r in the g 2 γ X term comes from the function G. The entire g 2 γ 3 X term comes from the function G. The g 2 γ 4 X term receives contributions of order r 0 from both the function F and the function G.

D. Comparison with Previous Calculation
We can compare our result for the scattering length a s,+ in Eq. (60) with that from a previous calculation by Jansen, Hammer, and Jia [11]. They used original XEFT to calculate the complex energy E X of the X(3872) and the complex scattering length a s,+ , dropping terms that were suppressed by at least one power of r = m/M . They expressed their results in terms of a leading-order inverse scattering length γ. Their physical renormalization condition was that the real part of the complex energy E X should be identified with the energy of the X(3872) resonance. Their NLO corrections included terms proportional to g 2 , C 2 , and an additional coupling constant D 2 that takes into account the dependence of the contact interaction on the pion mass. The coupling constant D 2 allowed them to study the dependence of the mass of the X(3872) on the light quark masses. Their NLO corrections also included terms involving the partial width Γ * 0,π for D * 0 → D 0 π 0 that were resummed to all orders. If the other NLO corrections are omitted, their results reduce to where Γ * 0,π is the partial width of D * 0 into D 0 π 0 . The universal relation expressing the binding energy in terms of a large positive scattering length a s,+ is The leading order results in Eqs. (61) are incompatible with the analytic continuation of this universal relation to a complex scattering length a s,+ . This incompatibility suggests that the resummation used in Ref. [11] to eliminate an infrared divergence at the D * 0D0 threshold was inadequate. Because of this problem, we will only compare results in the limit Γ * 0,π → 0. In this limit, their NLO results reduce to where Λ is the PDS renormalization scale and d ′ 2 is a finite constant related to the coupling constant D 2 . Note that the dependence on Λ can be eliminated from both E X and from a s,+ by absorbing it into d ′ 2 . ln the expression for a s,+ in Eq. (63b), the factor of 1/γ 2 comes from a term that has an unphysical double pole in the energy. The double pole could have been eliminated in favor of a shift in the position of the single pole by expressing a s,+ as the ratio of a numerator and denominator with NLO accuracy, as in their result for E X in Eq. (63a).
The complex on-shell renormalization scheme can be implemented by choosing d ′ 2 so that the numerator of the NLO correction term to E X in Eq. (63a) is zero. The complex energy E X then reduces to δ − γ 2 /(2µ), so γ can be identified with the binding momentum γ X of the X(3872). The NLO scattering length calculated in Ref. [11] reduces to This can be compared with the result from Galilean-invariant XEFT in Eq. (60). Dropping terms suppressed by three or more powers of γ X / √ 2µE * and keeping only the leading power of r in the coefficients, our result reduces to If we ignore the tiny difference between E * and δ and if we approximate µ by M/2 and µ π by m, the √ 2µE * term in Eq. (65) agrees with the √ 2mδ term in Eq. (64). However the other correction terms in Eq. (64) are proportional to γ X , while those in Eq. (65) are proportional to γ 2 X . Furthermore the coupling constants C 2 and C 0 appear in the combination C 2 /C 2 0 in Eq. (64), while they appear in the combination C 2 /C 2 0 in Eq. (65). Thus there seems to be a clear discrepancy between the results for a s,+ in Eqs. (64) and (65). However the conclusion that there is a discrepancy relies on an implicit assumption that the coupling constant C 2 in Eqs. (64) and (65) can be identified. In the following section, we will show the surprising result that in the limit g → 0 in which the pions decouple, the NLO correction terms proportional to C 2 in Eqs. (64) and (65) are indeed compatible. Whether the apparent discrepancy between the NLO correction terms proportional to g 2 γ X in Eq. (64) and to g 2 γ 2 X in Eq. (65) is real cannot be determined definitively using the results given in Ref. [11].

E. Decoupled Pions
If pions are decoupled by setting g = 0, the NLO terms in the transition amplitude A ij (E, p, p ′ ) reduces to the ∇ 2 vertex diagrams C in Eqs. (B16) and the D * D counterterm diagrams D in Eq. (B18). The terms proportional to A 2 (E) which have a double pole in E, can to NLO accuracy be replaced by terms in the denominator, as in Eq. (48). After implementing the COS renormalization scheme, the transition amplitude reduces to where H sub (E) is defined by Eqs. (51b) and (47c).
To obtain the T-matrix element in Eq. (53), we put the charm mesons on their energy shells by setting p ′ = p and setting the energy equal to E p = E * + p 2 /(2µ). The term H sub (E p ) in the denominator of Eq. (66) is The T-matrix element reduces to To NLO accuracy, the factor 1 + (C 2 /C 0 )p 2 in the numerator of Eq. (68) could equally be included as a factor 1 − (C 2 /C 0 )p 2 multiplying −γ X − ip in the denominator: We can read off the S-wave inverse scattering length and effective range in the C = + channel as coefficients in the denominator: 1/a s, The inverse scattering length in Eq. (70a) is consistent to NLO accuracy with setting g = 0 in Eq. (60). Eliminating C 2 /C 0 , we reproduce a familiar relation between the scattering length, effective range, and binding momentum to first order in the range expansion: Our result for the effective range in Eq. (70b) implies that, in the absence of pions, the ratio C 2 /C 0 of coupling constants has a simple physical interpretation in terms of the effective range and the binding momentum: In the previous NLO calculations in XEFT in Refs. [6] and [11], power divergence subtraction was used and C 0 and C 2 were determined as functions of the renormalization scale Λ. The dependence on Λ canceled in the following combination of coupling constants: The apparent incompatible with our result in Eq. (72) is at first puzzling. Our coupling constants C 0 and C 2 can be defined as the coefficients of the terms in the Lagrangian given in Eq. (9). This seems to be equivalent to the definitions of C 0 and C 2 in Ref. [6] up to corrections suppressed by powers of r = µ π /M π . The resolution is that the absence of the counterterm δD 0 in the Lagrangian for original XEFT in Ref. [6] implies that the definitions are not equivalent. To determine the relation between the coupling constants, it is necessary to compare physical observables, such as the effective range r s,+ . By equating r s,+ in Eqs. (72) and (73), we can infer that the combination C 2 /C 2 0 in Ref. [6] should actually be identified with our ratio C 2 /C 0 multiplied by −µγ X /(4π). This identification brings the NLO correction terms to a s,+ proportional to (C 2 /C 2 0 )γ X in Eq. (64) and (C 2 /C 0 )γ 2 X in Eq. (65) into agreement.

V. SUMMARY
We have introduced a Galilean-invariant formulation of XEFT to exploit the fact that mass is very nearly conserved in the transition D * → Dπ. Galilean invariance requires the kinetic mass of the D * to be equal to the sum of the kinetic masses M and m of the D and π. One advantage of Galilean invariance is that an amplitude is the same in every Galilean frame. A more important advantage is that it strongly constrains the form of ultraviolet divergences. We also introduced the complex on-shell (COS) renormalization prescription that takes into account the effects of transitions to states that cannot be described explicitly in XEFT. These transitions include the decay D * 0 → D 0 π 0 , which accounts for about 38% of the total width of the D * 0 . They also include the decays of X(3872) to all final states other than D 0D0 π 0 . These other decay modes may account for a substantial fraction of the total width of the X(3872). The COS prescription requires as input the total width of the D * 0 and the total width of the X(3872), which has not yet been measured.
We illustrated the use of Galilean-invariant XEFT by calculating the transition amplitude for D * 0D0 → D * 0D0 to NLO in the XEFT power counting. We used dimensional regularization in d spatial dimensions to regularize the ultraviolet divergences. The NLO Lagrangian includes counterterms that can cancel terms whose energy dependence has the form A 2 (E) and EA 2 (E), where A(E) is the LO transition amplitude. It was straightforward to verify that the logarithmic ultraviolet divergences, which appeared as poles in d − 3, could be cancelled by the counterterms.
The cancellation of linear ultraviolet divergences, which appeared as poles in d − 2, was not as straightforward. In addition to the poles that could be cancelled by the counterterms, there were additional poles in d − 2 whose energy dependence had the form A 2 (E) log(E * − E) and A(E). They cancelled each other only upon using the specific form of the LO transition amplitude A(E) for d = 2.
We used our NLO result for the transition amplitude to calculate the D * 0D0 scattering length a s,+ to NLO. Our result differs in several respects from a previous result for a s,+ calculated using original XEFT in Ref. [11]. One important difference is in the effect of the D * 0 width. Our COS renormalization prescription ensures that the effect of the D * 0 width is in accord with the universal relation between the binding energy of an S-wave bound state near threshold and the scattering length of its constituents. The effect of the partial width of the D * 0 into D 0 π 0 in Ref. [11] is incompatible with the universal relation. This suggests that the resummation used in Ref. [11] to deal with an infrared divergence at the D * 0D0 threshold was inadequate. There is an apparent disagreement between our result and that of Ref. [11] in the NLO corrections to a s,+ proportional to C 2 , but we showed that the difference is actually due to different definitions of C 2 . Our NLO correction to a s,+ proportional to g 2 was expanded in powers of γ X / √ 2mδ, where γ X is the binding momentum of the X(3872). The leading term is proportional to g 2 √ 2mδ, and it agrees with that in Ref. [11] in the limit m/M → 0. The next term in our expansion is second order in γ X , while there is a term in Ref. [11] that is first order in γ X . The origin of this discrepancy has not been determined.
Our Galilean-invariant formulation of XEFT removes obstacles to precise calculations of the properties of the X(3872) resonance. Our COS renormalization prescription takes into account the 38% branching fraction of D * 0 into D 0 π 0 and the significant branching fraction of X(3872) into decay modes other than D 0D0 π 0 . In original XEFT, there was a serious limitation on the precision from the need to truncate the expansion in m/M in order to avoid ultraviolet divergences. Our Galilean-invariant formulation makes this expansion unnecessary. The expansion may still be useful to obtain simple results, but it can be carried out to whatever order is required for the desired precision.
XEFT was invented to allow the effects of pion transitions on the X(3872) resonance to be taken into account systematically. XEFT can also be used as a framework for quantifying the effects of physics beyond the DDπ sector. This physics include charmonium states such as the χ c1 (2P ), which couples to D * D , and ψ(3770), which couples to DD. It also includes tetraquark cc mesons and hybrid cc mesons. The improved precision of Galilean-invariant XEFT increases the motivation for using this effective field theory to quantify the effects of physics beyond the DDπ sector.
The propagators in the loop integrand depend on the complex D * 0 energy E * and on the masses m and M of the π 0 and D 0 . The loop integrals depend on the masses through the reduced D * D kinetic mass µ defined in Eq. (3a) and the reduced Dπ mass µ π defined in Eq. (3b). It is convenient to express the dependence on the masses in terms of µ and the ratio r = µ π /µ defined in Eq. (4).
In loop integrals that have cuts with DDπ intermediate states, after integrating over the D andD energies, the pion propagator depends on the momenta p and q of the D andD and on the masses m and M . In evaluating the loop integral, it is often convenient to express the pion propagator in a form that depends on reduced masses instead of masses: Unlike the first expression for the pion propagator in Eq. (A2), the second expression is not manifestly invariant under the interchange p ↔ q.

One-loop momentum integrals
The one-loop momentum integrals whose integrands are a D * propagator raised to an integer power have the form The analytic result for this integral is The branch cut in E is determined by the negative imaginary part of the complex energy E * . The one-loop momentum integrals whose integrands are a π propagator raised to an integer power have the form The analytic result for this integral is where r = µ π /µ is the ratio of the reduced masses defined in Eq. (4). The branch cut in E is specified by the iǫ prescription in Eq. (A5). Since we choose the zero of energy to be at the D 0D0 π 0 threshold, we need only consider positive values for the real part of E. The one-loop momentum integrals whose integrands have a single π propagator and a D * propagator raised to an integer power are The function L 0 can be expressed in terms of the integral I 1 given by Eq. (A6): This result can be obtained most easily by using the expression for the pion propagator on the right side of Eq. (A2), and then shifting the integration variable q. The function L 1 can be expressed as a Feynman parameter integral: We will need the expansions of these functions to first order in p 2 : where ∆ = (1 − r)E − E * . The expansion for L 0 can be obtained most easily by expanding the expression in Eq. (A8) in powers of p 2 . The expansion for L 1 can be obtained most easily by expanding the integrand in Eq. (A7) in powers of p and averaging over angles to reduce it to scalar integrals. The scalar integrals can be expressed in terms of the integrals J n and I n given by Eqs. (A4) and (A6). They can all be reduced algebraically to J 1 and I 1 .

Two-loop momentum integrals
The two-loop momentum integrals whose integrands have a π propagator and one or two D * propagators raised to integer powers are The momentum integrals with a π propagator and a single D * propagator can be expressed as integrals over a single Feynman parameter: where r = µ π /µ. The momentum integrals with a π propagator and two D * propagators can be expressed as an integral over two Feynman parameters: This function depends on r through the prefactor r d/2 and through the integral over t. For a given dimension d, K 111 (E) can be expanded in powers of r. The expansion is not straightforward, because there are negative powers of r that arise from the w → 1 endpoint region.

Poles in d − 2
In a dimensionally regularized loop integral, poles in d − 2 are associated with linear ultraviolet (UV) divergences in 3 spatial dimensions. An UV pole in d − 2 can arise from a factor of Γ(1 − d/2) or Γ(2 − d) obtained by integrating over a loop momentum. In the two-loop momentum integrals K 110 and K 120 defined in Eqs. (A12), the integral over the Feynman parameter x also gives an UV pole in d − 2.
The one-loop momentum integrals J 1 and I 1 given by Eqs. (A4) and (A6) have single poles in d − 2. We will need the pole and the constant term: The momentum scale Λ in the logarithms is where γ is Euler's constant. We will need the constant term in the integral J 2 given by Eq. (A4): We will also need the pole term in the integral L 0 in Eq. (A8): .
The two-loop momentum integral K 110 defined in Eq. (A12a) has a double and single pole in d − 2, while K 120 defined in Eq. (A12b) has only a single pole: . (A18b)

Integrals at the D * 0D0 threshold
If the one-loop integral J n (E) given by Eq. (A4) is analytically continued to d = 3 and then evaluated at E = E * , it has an infrared divergence for n ≥ 3/2. However if it is evaluated at E = E * and then analytically continued to d = 3, it vanishes because the integral has no momentum scale: The two-loop integrals K lmn (E) defined in Eq. (A11) can be evaluated analytically at the threshold E = E * . The integrals K 110 (E * ), K 120 (E * ), and K 111 (E * ) each has a single pole in d − 3. Their values near d = 3, including the pole in d − 3 and the finite term, are The momentum scale Λ in the logarithms is defined in Eq. (A15). The pole in K 110 (E * ) is the sum of an UV pole and an infrared pole in d − 3.

Integrals near d = 3
The one-loop integrals J 1 and J 2 at d = 3 are The one-loop integrals I 1 and I 2 at d = 3 are The two-loop momentum integrals K lmm (E) near d = 3 are expressed in terms of Feynman parameter integrals in Eqs. (A12) and (A13). In order to obtain expressions for these functions near d−3, it is necessary to make subtractions of the integrands to make them integrable at x = 1. The resulting expression for K 110 (E), including the pole in d − 3 and the finite term, is The pole with its residue evaluated at E = E * is also the UV pole in d − 3 of K(E * ). The difference between the pole of K(E * ) in Eq. (A20a) and this pole is the infrared pole in d − 3 of K(E * ). The UV poles in d − 3 of K 120 (E) and K 111 (E) are the same as the poles in their values at E * given in Eqs. (A20b) and (A20c). The differences between their values at E in Eqs. (A12b) and (A12b) and their values at E * are finite:

Threshold expansions
Threshold expansions for the two-loop integrals K lmn (E) near d = 3 cannot be obtained simply by expanding the expressions in Eqs. (A23), (A24a), and (A24b) in powers of E − E * , because this generates infrared divergences in the Feynman parameter integrals. The threshold expansion for K 111 (E) for general d can be obtained in a straightforward way by expanding the defining integral in Eq. (A11) in powers of E − E * . The expansion through second order in E − E * is The reason the threshold expansion of K 111 (E) is straightforward is that the only momentum region that contributes is where both loop momenta are of order (µ|E * |) 1/2 . In contrast, the integrals K 110 (E) and K 120 (E) have contributions from regions where one loop momentum is of order (µ|E − E * |) 1/2 . These contributions have an expansion in non-integer powers of E − E * . By inserting the alternative expression for the pion propagator in Eq. (A2) into the defining integral for K lm0 (E) in Eq. (A11) and then shifting the momentum q, it can be expressed as The expansion of the pion propagator in powers of p 2 − 2µ(E − E * ) produces a sum of products of one-loop integrals of the form I l+k (E * )J m−k (E). With dimensional regularization, the terms with k ≥ m are zero because the integral over p has no scale. The threshold expansion for K lm0 (E) is the sum of the terms with the noninteger powers (E − E * ) d/2−m+k and the terms with integer powers that can be obtained by expanding the integrand of Eq. (A26) in powers of E − E * . The expansions of K 110 (E) and K 120 (E) through second order in E − E * are K 110 (E) = K 110 (E * ) + rI 1 (E * )J 1 (E) + [K 210 (E * ) + K 120 (E * )] 2µ(E − E * ) K 120 (E) = K 120 (E * ) + rI 1 (E * )J 2 (E) − r 2 I 2 (E * )J 1 (E) + [K 220 (E * ) + 2K 130 (E * )] 2µ(E − E * ) The integer powers of E − E * in the threshold expansions in Eqs. (A25) and (A27) are determined by the two-loop integrals K lmn (E * ). By inserting the alternative expression for the pion propagator in Eq. (A2) into the defining integral for K lmn (E) in Eq. (A11), its value at the threshold E = E * reduces to The denominators of the integral over q can be combined by introducing a Feynman parameter. After evaluating the integral over q, the integral over p can also be evaluated analytically. The integral over the Feynman parameter gives a hypergeometric function; K 120 (E) = K 120 (E * ) + rI 1 (E * )J 2 (E) − r 2 I 2 (E * )J 1 (E) K 111 (E) = K 111 (E * ) + 4 (4π) 3 (2 − r) The function arccos( √ r ) in Eq. (A30c) has an expansion in odd powers of r 1/2 : The hypergeometric function in Eq. (A30c) has an expansion in powers of r: The two-loop integrals K 110 and K 120 are given in Eqs. (A12). The diagram B2 has a D * propagator counterterm inserted into the D * propagator. The Feynman rule for the self-energy counterterm is given in Eq. (31). With the complex on-shell renormalization scheme for the D * 0 propagator, the diagram B2 can be reduced to The one-loop integrals J n are given in Eq. (A4).
The factor of p 2 + p ′ 2 depends on the relative momenta of the incoming and outgoing charm mesons. If there is a contact interaction between the incoming charm mesons, p is replaced by a loop momentum k. The amplitude for the diagram C2 in which the incoming charm mesons interact through a LO transition amplitude can be obtained from Eq. (B11) by multiplying it by (4π/µ)J 1 (E)A(E) and replacing p 2 by the loop-integral-weighted average of k 2 : A ij C2 (E, p ′ ) = −2πC 2 J 1 (E)A(E) k 2 + p ′ 2 δ ij .
The loop-integral-weighted average of k 2 is defined by With dimensional regularization, the loop-integral-weighted average is very simple: The amplitude for the diagram C3 is obtained from Eq. (B12) by replacing p ′ by p. The amplitude for the diagram C4 is The sum of the four diagrams C1, C2, C3, and C4 has a multiplicative factor 1 + 4πJ 1 A that can be simplified by using the expression for A 0 (E) in Eq. (B2): A ij C (E, p, p ′ ) = πC 2 µC 0 (p 2 + p ′ 2 )A(E) + 16πµ(E − E * )J 1 (E)A 2 (E) δ ij .
FIG. 11: The ∇ 2 vertex diagrams for D * 0D0 → D * 0D0 . They consist of the ∇ 2 vertex C1, the one-loop diagrams C2 and C3 in which either the incoming or the outgoing charm mesons interact through a LO transition amplitude, and the two-loop diagram C4 in which they both interact through LO transition amplitudes.

D * D counterterm diagrams
There are four NLO diagrams that can be obtained from the diagrams in Figure 11 by replacing the ∇ 2 vertex by a D * D counterterm. The four diagrams are labelled D1, D2, D3, and D4. The amplitude for the diagram D1 is just the counterterm vertex in Eq. (16): The amplitudes for the diagrams D2 and D3 each differs from this by a multiplicative factor 4πJ 1 A. The amplitude for the diagram D4 differs by two such factors. The sum of the four diagrams D1, D2, D3, and D4 therefore has a factor [1 + 4πJ 1 A] 2 that can be simplified by using the expression for A 0 (E) in Eq. (B2): The δC 0 term in this expression can also be obtained from the LO transition amplitude in Eq. (B2) by replacing C −1 0 by C −1 0 − δC 0 /C 2 0 and expanding the amplitude to first order in δC 0 .

Complete NLO amplitude
The complete NLO term in the transition amplitude A ij (E, p, p ′ ) is the sum of (A) the pion-exchange diagrams in Eqs. (B3), (B5), and (B8), (B) the D * propagator insertion diagrams in Eqs. (B9) and (B10), (C) the ∇ 2 vertex diagrams in Eqs. (B16), and (D) the D * D counterterm diagrams in Eq. (B18). In the zero-momentum limit p, p ′ → 0, the pion-exchange diagrams reduce to the sum of Eqs. (B7) and (B8) and the ∇ 2 vertex diagrams in Eqs. (B16) reduce to the single term with the factor of A 2 (E).

Poles in d − 2
The momentum integrals with poles in d − 2 are given in Section A 4. In the NLO term in the transition amplitude A ij (E, p, p ′ ), all the terms with a double pole in d − 2 have the tensor structure δ ij . All the terms with a single pole