Threshold Effects and the Line Shape of the X(3872) in Effective Field Theory

Latest measurements suggest that the X(3872) mass lies less than 200 keV away from the D0-Dbar0* threshold, reenforcing its interpretation as a loosely-bound mesonic molecule. This observation implies that in processes like D0-Dbar0-pi0 production, threshold effects could disguise the actual pole position of the X(3872). We propose a new effective field theory with D0, Dbar0 and pi0 degrees of freedom for the X(3872), considering Galilean invariance to be an exact symmetry. The D0* enters as a D0-pi0 p-wave resonance, allowing for a comprehensive study of the influence of pion interactions on the X(3872) width. We calculate relations between the mass of the X(3872), its width, and its line shape in D0-Dbar0-pi0 production up to next-to-leading order. Our results provide a tool for the extraction of the X(3872) pole position from the experimental data near threshold.

: Thresholds close to the X(3872) (to scale, anti-particle states omitted)

I. INTRODUCTION
In 2003 and 2004, the Belle and the CDF II Collaborations subsequently observed a novel charmonium state referred to as the X(3872) [1,2]. Since then, experimentalists have discovered a whole zoo of exotic 'XY Z particles' that do not fit into the conventional quark model. So far, their nature remains unclear and is under discussion (see, e.g., Refs. [3][4][5][6] for reviews). The X(3872) (simply denoted 'X' in the following) is by far the best-studied example of these states. Its decay channels J/ψ π + π − and J/ψ π + π − π 0 have comparable branching ratios [7,8], implying a large isospin violation. For this reason, its interpretation as a conventional cc state has been challenged by a variety of exotic explanations including an interpretation as a tetraquark [9,10]. A major step towards a deeper understanding of the X was achieved in 2013 when the LHCb Collaboration determined its quantum numbers to be J P C = 1 ++ [11]. For a recent review on the experimental status of the X see Ref. [12].
A striking feature of the X state is its proximity to the D 0D0 * threshold, see Fig. 1. Over the years, the corresponding mass difference has repeatedly been corrected down from about +1 MeV to the current value [13,14] δ X ≡ (m D 0 + m D 0 * ) − m X = (−0.01 ± 0.2) MeV , with m X = 3871.69 (17) MeV. Considering the quantum numbers of the X, J P C = 1 ++ , this circumstance implies a strong coupling to the s-wave configuration D 0D0 * +D 0 D 0 * / √ 2, giving rise to a large molecular component. This picture, which has been discussed by many authors (cf. Refs. [15][16][17][18][19][20]), is the basis of this work. It readily explains the isospin violation from the remoteness of the charged D + D − * threshold at see Fig. 1 and Ref. [14]. Moreover, a significant part of the X width Γ X can be attributed to decays of the constituents D 0 * andD 0 * , which is confirmed by the large branching ratios of X decays involving D 0D0 * [14]. By now, the width is only limited by an upper bound Γ X < 1.2 MeV [7] stemming from the detector resolution. However, future experiments like Belle II [21] andPANDA at FAIR [22] will be able to measure the width with much higher accuracy.
If interpreted as a charm meson pair, the X can either be bound or virtual, which follows from the universal properties of near-threshold s-wave states [23,24]. Heavy quark symmetry then implies the existence of a B 0B0 * molecule [25]. In a zero-range approach, Braaten and Lu calculated line shapes of the X for the bound and virtual cases in 2007 [26]. The partial widths of inelastic decay channels like the discovery mode J/ψ π + π − were neglected, assuming that they are small. They observed a significant enhancement in the D 0D0 π 0 production rate close to the D 0D0 * threshold. This effect influences the peak position and width, such that they do not correspond to the actual pole position. This could be one reason why the X mass in the D 0D0 π 0 channel in Refs. [27][28][29] appears to be larger than in such involving J/ψ [14]. In fact, Braaten and Stapleton [30] pointed out that the main reason for this result lies in a false identification of the peak in the invariant D 0D0 * mass distribution in Refs. [28,29] as the X mass and width. An analysis of the effect of different pole structures on the line shapes of the X(3872) in B decays to KJ/ψ π + π − and KJ/ψ D 0D0 * was carried out in Ref. [31].
In 2007, Fleming et al. developed a non-relativistic effective field theory (EFT) for the X, called XEFT [32]. It can be used to calculate systematic corrections to universality. In addition to charm meson fields, XEFT also contains a field for the neutral pion π 0 . Fleming et al. calculated the partial decay width Γ X(3872) → D 0D0 π 0 at next-to-leading order (NLO) in XEFT power counting. A key result was, that pion exchanges can be treated in perturbation theory. XEFT has been applied to several processes [33][34][35][36][37]. It is, however, limited to NLO precision because renormalization requires an expansion in the mass ratio (m π 0 /m D 0 ) 1/2 ≈ 0.27. Braaten cured this problem by proposing a Galilean-invariant version of XEFT [38]. This symmetry is motivated by the small mass difference in the decay D 0 * → D 0 π 0 , which implies approximate mass conservation, see Fig. 1 and Ref. [14].
The influence of pion dynamics on Γ X was also investigated by Baru et al. in 2011 [39]. They performed a coupled channel calculation with both neutral and charged mesons, treating pions non-perturbatively. The X was produced as a peak in the D 0D0 π 0 production rate at δ X ≥ 0.1 MeV. In this region, threshold effects play a minor role and Γ X can be extracted in a Breit-Wigner fit. Their result agrees well with XEFT, thereby confirming the perturbativeness of pions. Moreover, it was shown that a static pion approximation largely overestimates Γ X , while charged mesons have a moderate effect.
This work builds upon the findings of Refs. [26,32,38] and [39]. We develop a novel EFT for the X with non-relativistic fields for D 0 ,D 0 and π 0 , demanding exact Galilean invariance. The theory allows for systematic calculations of both the pole and the line shape of a bound X state in D 0D0 π 0 production, including theoretical uncertainties. Thereby, it provides a tool for the extraction of the mass and width of the X from an experimental peak that is influenced by threshold effects. Our power counting infers the perturbativeness of pions and charged mesons from the characteristic momentum scales of the system. Moreover, it suggests that some ingredients in the approach of Ref. [39] can be neglected at NLO. This makes the theory renormalizable for arbitrary cutoffs.
The paper is organized as follows. In Sec. II, we construct the Galilean-invariant EFT Lagrangian and introduce the D 0 * (D 0 * ) as a p-wave resonance in the D 0 π 0 (D 0 π 0 ) sector. Based on a comprehensive scaling analysis of D 0 π 0 threshold parameters, we determine an appropriate D 0 * propagator expansion in Sec. III. Moreover, we effectively include radiative decays of the D 0 * using complex self-interactions. In Sec. IV, we construct the non-perturbative D 0D0 * amplitude and solve it analytically at LO. Afterwards, we show that D 0 * self-interactions, pion exchanges and charged mesons enter at NLO. Finally, Sec. V presents numerical results for Γ X and the line shape at LO and NLO. Moreover, it is illustrated how the X peak gets modified by the detector's energy resolution. We conclude with a summary and an outlook in Sec. VI.

II. EFT LAGRANGIAN
We start by constructing a Galilean-invariant Lagrangian L for particles D 0 ,D 0 and π 0 . For clarity, we decompose L into different scattering sectors by writing The kinetic part of the EFT Lagrangian, contains fields with masses m D ≡ 1864.83(5) MeV and m π ≡ 134.9770(5) MeV determined from experiment [14]. All rest masses have been shifted to zero. Note, that also charged mesons will enter the EFT at NLO. Their inclusion will be discussed in Sec. IV. Similar to neutron-alpha scattering, the resonance D 0 * (D 0 * ) can be treated by an auxiliary field [40]. Accordingly, we introduce the vector field D (D) in L Dπ (LD π ). It encapsulates all p-wave interactions and could in principle be eliminated by performing the Gaussian path integral over D (D) or using the equations of motion. We write where 'h.c.' denotes the Hermitian conjugate. The first term of Eq. (6) defines the bare D 0 * propagator. It is given by a series in the Galilean-invariant derivative i ∂ cm ≡ i ∂ 0 + ∇ 2 /(2M ) with total D 0 π 0 mass M ≡ m D + m π . This form ensures analyticity in the center-of-mass energy E cm ≡ E − p 2 D 0 * /(2M ) and reproduces the D 0 π 0 effective range expansion in Sec. III. The real-valued coefficients ∆ n≥0 (∆ 1 ≡ ±1) have mass units MeV 1−n . Note, that the sign ∆ 1 cannot be changed by field redefinitions and has to be determined in the renormalization procedure. In our case, ∆ 1 = +1 (see Appendix A for details). Therefore, D ( †) is a physical field. Later, radiative decays D 0 * → D 0 γ (D 0 * →D 0 γ) will by included by adding imaginary parts to the ∆ n .
The second term of Eq. (6) allows for transitions D 0 π 0 ↔ D 0 * and is depicted in Fig. 2(a). It depends on the coupling constant g and the Galilean-invariant derivative Thus, Feynman rules for these transitions depend on the relative D 0 π 0 momentum. The values of all parameters in L Dπ will be addressed in Sec. III. As a consequence of the charge-conjugation symmetry of the X, the Lagrangian part LD π is obtained by replacing (D, D) → (D,D) in Eq. (6).
The three-body part of the Lagrangian, contains all D 0D0 * s-wave interactions in the C = + channel. In Sec. IV, the coupling C 0 will be used to generate the X. Its vertex is depicted in Fig. 2(c). Higher-order terms contained in the ellipsis do not enter up to NLO (see Sec. IV). Throughout the paper we label relative D 0 π 0 momenta with 'k' or 'l' and relative D 0D0 * momenta with 'p' or 'q'. Moreover, we drop the particle superscripts '0' from now on. The X lies extremely close to theDD * (DD * ) threshold. Therefore, its form crucially depends on the vector meson propagator. It is the goal of this section to identify an appropriate propagator expansion in the vicinity of the X. Without loss of generality we focus on the D * system. We recover the Dπ effective range expansion from the EFT Lagrangian and analyze it in terms of characteristic momentum scales. Thereby, we obtain a natural explanation for the narrowness of the D * resonance. Afterwards, we include radiative decays of the D * . The resulting expansion of the propagator is given at the end of the section.

A. Matching to the Effective Range Expansion
All terms in Eq. (6) are Galilean-invariant and potentially contribute to the D * propagator. However, in order to produce the D * as a p-wave resonance, only a few terms are needed. This statement will be verified in this section. We begin by matching the propagator terms to the effective range expansion of the Dπ amplitude.
Let p µ = (p 0 , p) be the D * four-momentum. The bare propagator connecting equal polarization states at a center-of-mass energy E cm = p 0 − p 2 /(2M ) is given by where we have used ∆ 1 = +1. To obtain the full propagator, the bare one needs to be dressed by D * self-energy loops −i Σ(E cm ) as shown in Fig. 3. We then obtain In Appendix A, we calculate Σ using dimensional regularization and the Power Divergence Subtraction Scheme (PDS) with renormalization scale Λ PDS [41]. For convenience, we use Minimal Subtraction (MS) instead of PDS for all practical calculations by considering the limit Λ PDS → 0. This choice makes our scaling analysis much more transparent since all quantities are automatically scale-independent in the MS scheme. In any case, observables do not depend on the chosen renormalization scheme. For the self energy, we obtain which is purely imaginary for E cm > 0.
In the auxiliary field formalism, one gets the elastic Dπ scattering amplitude t for relative momenta k, k by attaching external Dπ legs to the full D * propagator at E cm = k 2 /(2µ), i. e., t(k, k ) = −g 2 k · k G (f) * k 2 /(2µ) . This expression can be matched to the generic p-wave effective range expansion [42], The parameters a 1 ('scattering volume') and r 1 ('p-wave effective range') have mass dimensions MeV −3 and MeV, respectively. The coefficients P 2n will be referred to as 'higher-order parameters' from now on. Comparing Eq. (9) and (11), we identify We are now in the position to analyze the characteristic momentum scales of the propagator.

B. Momentum Scales and Scaling Analysis at Threshold
The D * resonance occurs at a Dπ center-of-mass energy δ = 7.04(3) MeV, see Eq. (3), which is small compared to all involved particle masses [14]. This observation gives rise to a significant separation of momentum scales, which can be explained by a fine-tuning of the underlying theory, QCD. The relative Dπ momentum k needed to probe the shallow resonance is of the order k ∼ K lo ≡ (2µ δ) 1/2 ≈ 42 MeV. In contrast, the natural momentum scale K hi ∼ m π occurring in QCD is much larger. Due to the fact, that our EFT is non-relativistic, K hi represents its breakdown point. In the following, we express the scale separation as the ratio Note, that the effective range expansion is in k 2 . As a consequence, the D * propagator expansion will be in κ 2 ∼ 0.09. Thus, we expect quick convergence.

Naturalness of Higher-Order Parameters
Each threshold parameter in Eq. (11) scales with certain powers of K lo and K hi . We assume that fine-tunings related to the shallow D * resonance occur in a 1 , r 1 or in both of them. In contrast, all higher-order parameters are assumed to be of natural size, i. e., P 2n ∼ K 3−2n hi . This 'naturalness argument' is based on the assumption that a scaling scenario with as few fine-tunings as possible is most likely to occur in nature [40,43].
As a consequence, the terms P 2n k 2n ∼ κ 2n K 3 hi in Eq. (11) are suppressed compared to the unitary cut |i k 3 | ∼ K 3 lo by at least one order in κ. In other terms, due to Eq. (12), higherorder propagator terms close to resonance are suppressed compared to the self-energy like 2. Consequences from the Small D 0 * Width We have seen that natural P 2n imply strongly suppressed higher-order propagator terms. With this information at hand, we can now express ∆ 0 and the coupling g in terms of the D * resonance parameters. The D * resonance manifests itself as a complex energy pole in i G (f) * (E cm ). It can be parametrized by δ − i Γ Dπ /2, where Γ Dπ denotes the small width of the decay D * → Dπ. Although Γ Dπ has not been measured yet, an upper bound 1 Γ Dπ < 1.4 MeV is known from experiment [14]. Thus, the ratio χ ≡ Γ Dπ /(2δ) < 0.1 is very small. Demanding with κ ∼ 0.3. We see that the width Γ Dπ is given by 2|Σ(δ)|, up to a tiny uncertainty of order χ κ < 3 %. This estimation stems from the product χ P 4 K lo in the case of P 4 ∼ K −1 hi being natural. Thus, for the approximations in Eq. (15) to fail, P 4 would have to be enhanced by a factor of order (χ κ) −1 > 33 which is unlikely. In fact, we will find below that χ 0.1, which secures the validity of the above approximations.

Parameter Fixing
We follow Braaten [38] and infer a value for g 2 (and thus of Γ Dπ ) from the total pionic decay width Γ c ≡ Γ[D + * → D 0 π + + D + π 0 ] = 82(2) keV of the charged D + * meson using a modified version of Eq. (15) and isospin symmetry (see Appendix B for details). This yields The indicated uncertainties are of order 3 % and arise from the experimental uncertainty of Γ c only. In contrast, uncertainties from natural higher-order parameters are negligible: As indicated above, the actual width Γ Dπ is indeed much smaller than the experimental bound. It implies a tiny ratio Γ Dπ /(2δ) = χ ≈ 0.0025 0.1 and thus, in the case of natural P 2n , a theoretical uncertainty of order χ κ ∼ 0.075 % 3 %. For this reason, we use the central values of g 2 and Γ Dπ of Eq. (16) in all later calculations.
Using the value of the coupling, we can now calculate a 1 and r 1 /2 from Eq. (12) at the experimental uncertainty level of Γ c , yielding 4. Scaling of a 1 and r 1 /2 and Fine-Tuning Scenarios Given the numerical values of Eq. (17) we are now able to assess scaling situations for a 1 and r 1 /2 that have been used in the literature for other physical systems. First of all, we see from Eq. (12) and (15) that the inverse scattering volume and the unitary cut are separated Moreover, we know that |i K 3 lo | P 2n K 2n lo . It follows that the resonance pole can only occur if a −1 1 ∼ r 1 /2 K 2 lo . One scenario that respects this pole condition has been discussed by Bertulani et al. for neutron-alpha scattering. They analyzed the situation, in which both a −1 1 and r 1 /2 are unnaturally small, i. e., a −1 1 ∼ K 3 lo and r 1 /2 ∼ K lo [42]. Due to Eq. (12) this scheme requires two fine-tuned combinations of coupling constants, ∆ 0 /g 2 and 1/g 2 . Bedaque et al. have argued that such a high degree of fine-tuning is unlikely to occur in nature. Their modified scheme a −1 1 ∼ K 2 lo K hi and r 1 /2 ∼ K hi only requires ∆ 0 /g 2 to be unnaturally small [40]. Yet, in the case of Dπ scattering both schemes appear to be inappropriate since the value of |r 1 /2| in Eq. (17) exceeds K hi by several orders of magnitude.
For the Dπ sector, we propose the novel scheme in which only the p-wave effective range is enhanced. This explains its huge numerical value and also the rather natural value |a Further evidence for this scheme comes from the fact that only one combination of constants, 1/(2µg 2 ) ∼ κ −2 K hi µ/(6π), needs to be fined-tuned, while the combination ∆ 0 /g 2 ∼ K 3 hi µ/(6π) scales naturally. Note, that there are other scaling scenarios consistent with the pole condition that could explain the specific values of a −1 1 and r 1 /2. Yet, they would inevitably involve two or more fine-tunings. In our scheme, the width Γ Dπ /2 ≈ |Σ(δ)| is suppressed by three orders with respect to δ, i. e., χ = Γ Dπ /(2δ) ∼ κ 3 . This shows that the narrowness of the D * resonance can be explained naturally by the momentum scales of the Dπ system.

C. Extension for Radiative D 0 * Decays
The radiative decay D * → Dγ has a large branching ratio B ≡ Γ Dγ /(Γ Dπ + Γ Dγ ) = 35.3(9) % [14]. It translates to the width Γ Dγ = 18.9(9) keV ≈ 0.55 Γ Dπ . Thus, it is as important for the X resonance and we count Γ Dγ ∼ Γ Dπ . The full D * pole position reads In Sec. IV we argue that the width Γ X at LO is just given by Γ * . In fact, numerical results will suggest that this approximation is even valid up to order NLO. The relative momentum of the decay products Dγ is given by 137 MeV ∼ K hi which lies beyond the scope of our EFT [14]. Still, Γ Dγ can be introduced effectively by adding an anti-Hermitian part to the Lagrangian [44], i. e., we replace ∆ n → ∆ n + i W n . Note, that the former relations = +Σ + . . . shall not be affected by this procedure. Due to Γ Dγ /2 ∼ χ δ ∼ χ κ 2 K 2 hi /(2µ), all imaginary parts W n≥0 must involve a common suppression factor χ κ 2 . There is no reason to assume fine-tunings between different W n . Thus, we count

D. Propagator Expansion at Resonance
Our findings show that close to the Dπ threshold the self-energy and all propagator terms proportional to ∆ n≥2 or W n≥0 are suppressed compared to δ. Thus, the width is a sub-leading phenomenon. However, the X resonance does not occur at the Dπ threshold but in the immediate vicinity of the D * resonance, i. e., at E cm ≈ E * . In this region, ∆ 0 and Re E cm almost cancel, resulting in a much weaker suppression of the D * width. As discussed by Bedaque et al. [40], an appropriate ordering scheme must take into account this kinematic fine-tuning |E cm − E * |/|E * | 1. As a consequence, close to resonance we have to include the width non-perturbatively.
Again, we consider the full D * propagator of Eq. (9) including radiative decays. The pole position E * becomes most apparent in the form Note, that the expression [E cm → E * ] is just given by the constant −(∆ 0 + i W 0 ). We make the kinematic fine-tuning explicit by factoring out the term which will be the leading order D * propagator for the calculation of the X width. It exhibits a Breit-Wigner form and is depicted by a simple straight-dashed double line, see Fig. 4.
After the factorization, all terms in the brackets of Eq. (22) -besides the leading '1' -are at least suppressed by the factor χ = Γ Dπ /(2δ) 1 (see Eq. (15), (20) and (21)) and thus very small. Therefore, we refer to them as 'propagator corrections'. The most important one involves the self-energy and a counter term −i Σ(E * ). It reads The indicated scalings for the real and imaginary part follow from E * = δ 1 + i O(χ) and Σ (E * ) ∼ Σ(E * )/E * . Note, that they hold not only in the region E cm ≈ E * but also at E cm ≈ 0 2 . In fact, all propagator corrections exhibit this feature. It is crucial for this work because both regions are important for the X width power counting in Sec. IV. The propagator expansion is depicted in Fig. 4 and reads All propagator corrections involving multiple self-energies or any of the coefficients As we will see in Sec. IV, modifications to the X width are determined by the imaginary parts of the propagator corrections in Eq. (25). Thus, the self-energy correction with its imaginary part ∼ χ κ 0 is the most important one. All imaginary parts exhibit even powers in κ (χ κ 0 , χ κ 2 , χ κ 4 , etc.). This observation confirms that the expansion is indeed in κ 2 ∼ 0.09.
There is one more reason why the given expansion is beneficial for our purposes: For a finite number of corrections there is always only one energy pole representing the D * . Therefore, we do not have to take into account additional unphysical Dπ states [42]. Finally, let us mention that relativistic corrections in the D * propagator can be neglected at the order we work. For more details we refer to Appendix C.

IV. THREE-BODY SECTOR: THE X(3872)
We now use the D * propagator expansion to produce the X as an energy pole in the DD * amplitude T . First, we diagrammatically construct the amplitude in the J P C = 1 ++ channel and explain the renormalization procedure. Moreover, we show how T can be used to calculate the line shape of the X in DDπ production.
Afterwards, we analyze all considered DD * interactions according to their influence on the X width. It turns out that corrections to the LO width Γ (LO) X = Γ * are suppressed by factors comparable to κ 2 ∼ 0.09. We identify all NLO corrections in a diagrammatic power counting, which exploits the characteristic momentum scales of the system.
A. Non-perturbative D 0D0 * Amplitude Similar to the two-body sector, we start by constructing the DD * → DD * transition amplitude T non-perturbatively. Note, that the unstable D * has a complex rest mass E * . Thus, it can never occur as an asymptotic state. Still, the amplitude T is needed to connect intermediate DD * states, e. g. in DDπ production (see Fig. 6).

Diagrammatic Construction
From L Dπ and L DDπ in Eq. (6) and (7) we see that the particles DD * can either exchange a pion or interact through the contact interaction C 0 . Moreover, we allow for an additional By iterating all these interactions in the C = + channel, we obtain an integral equation for the amplitude T which is shown diagrammatically in Fig. 5. Flavor-indicating arrows are left out since each DD * state is in a C = + superposition. Every loop is associated with an integral over the intermediate relative four The choice of this integration variable is not mandatory but convenient as it exploits Galilean symmetry.
We investigate the process in the center-of-mass frame, i. e., in the incoming (outgoing) channel we set p ≡ p D = −pD * (p ≡ p D = −p D * ). The energy E is defined relative to the DDπ threshold. The q 0 integration can be performed with the residue theorem, leading to for D * polarizations i and j. The pion exchange potential is given by where we defined the ratio α ≡ m D /M ≈ 0.93. Eq. (27) corresponds to the result obtained by Baru et al. [39]. An important feature of the given process is that exchanged pions can go on-shell. This is the case whenever the denominator in Eq. (27) vanishes. As a result, pion exchanges modify the X width at NLO (see below).

Partial Wave Projection
Up to now, the amplitude involves arbitrary parities and total angular momenta J ≡ L + S with total spatial angular momentum L and total spin S. The X, however, is a J P = 1 + state. Thus, the total spin S = 1 implies L ∈ {0, 2}. We perform a respective partial wave projection of T ij by absorbing momentum dependences into scalar components T LL ; J and angular dependences into projection operators of the form P ij states with quantum numbers L and L , respectively, at total J. We can (schematically) write T ij = T 00; 1 P ij 00; 1 + . . . with s-wave projector P ij 00; 1 = δ ij . The pion exchange potential V ij is expanded in the same fashion.
We aim at the calculation of T 00; 1 , which, through pion exchanges, is coupled to T 20; 1 . After projection, we may drop the subscript J = 1 for convenience and find the scalar amplitude system The scalar components V LL of the pion exchange potential are given in Appendix D. Note, that the loop integral is in general divergent. Still, for a fixed momentum cutoff Λ the system can be solved numerically for T 00 .

Numerical Renormalization
For arbitrary cutoffs Λ we tune C 0 (Λ) such that T 00 exhibits a pole at the complex energy just below the DD * threshold. More precisely, we fix the real part of E X , i. e., the binding energy δ X of the X. Thereby, we obtain a prediction for the width Γ X as a function of δ X . At each order, Γ X should be independent of the cutoff as Λ → ∞. Note, that for large momenta, i. e., p, q (2m π |E|) 1/2 , the component V 00 of the pion exchange potential in Eq. (D3) approaches the constant V (∞) 00 ≡ −2/3 · g 2 µ 2 /m π . As a consequence, the loop integral in Eq. (28) diverges as Λ → ∞. This divergence is cured by the contact interaction C 0 (Λ): The curve C 0 (Λ) will be shifted by the amount −V  The renormalized amplitude can be used to calculate the DDπ production rate. This quantity can be measured and thus serves as an important link between theory and experiment. As done by Baru et al. , we consider a process in which the X is produced at short ranges and subsequently decays to DDπ. As pointed out by Braaten and Lu [26], short range details can be absorbed into a constant F. The resulting differential rate is given by a phase space integral over the matrix element M depicted in Fig. 6. For outgoing particle momenta p D , pD and p π = −(p D + pD) we obtain The DDπ production rate for a system energy E ∈ R reads For consistency, the D * propagators in M will be chosen like the one entering the calculation of T . Moreover, since F is unknown, we have to normalize the rate. We follow Braaten and Lu by choosing the peak maximum in the δ X = 0 rate to be one [26]. The normalized line shapes will be independent of the cutoff as Λ → ∞.
The rate exhibits a peak near the DD * threshold representing the X. In order to account for possible deviations from the pole parameters δ X and Γ X , the position of the peak maximum (E max ) and the line width (FWHM) will be denoted by This distinction will be of importance once δ X becomes comparable to Γ X .

B. Momentum Scales
The tiny binding energy δ X introduces a new small energy scale. Equivalently, in terms of relative DD * momenta, we find two low-momentum scales The interval given in Eq. (33) stems from the uncertainty range of δ X in Eq. (1). Note, that the XEFT power counting does not distinguish between powers of P X and P * [32]. Our scheme improves upon this point by counting them separately. For convenience, we express the small ratio ρ ≡ P X /P * in terms of κ ∼ 0.3 by choosing m ∈ Z such that ρ ∈ κ m [κ 1/2 , κ −1/2 ). For m = 2 this interval corresponds almost exactly to the positive part of the uncertainty range of δ X , i. e., Therefore, we count ρ ∼ κ 2 ∼ 0.09 (P X ∼ 11 MeV) in the following. Note, that the central value δ X ∼ δ κ 4 ≈ 0.057 MeV is exactly one third of the upper interval limit 0.190. Thus, we systematically favor small values of δ X . This choice is in line with the fact that the experimental centroid of δ X lies close to zero. The high-momentum scale of the three-body system is expected to lie in the chiral breakdown regime Λ χ ∼ 500 MeV of Heavy Hadron Chiral Perturbation Theory. In this region, also pion production takes place, i. e., (2µ * m π ) 1/2 ≈ 510 MeV. Note, that in XEFT the hard scale is taken to be the pion mass itself. However, even for relative momenta of order m π P * charm mesons are non-relativistic and also relativistic pion corrections are small (see Appendix C).
We demand 0 ≡ T (LO) 00 As expected, the LO width is given by the full D * width, independently of δ X . The renormalized amplitude reads where 'reg' stand for terms regular at the LO pole position E (LO) X = E * − δ X . Note, that our LO amplitude almost recovers the zero-range result by Braaten and Lu [26]. They used an energy-dependent D * width instead of the constant one in Eq. (37). This energy dependence can be neglected at LO. The value of the LO residue, is of great importance for the X width: All sub-leading width contributions will at least be proportional to Z (LO) and therefore to the small momentum P X = (2µ * |δ X |) 1/2 ∼ 11 MeV.

D. NLO Corrections to the Width
In the following, we verify the LO nature of Γ (LO) X = Γ * . Similar to the two-body sector, the expansion of the width will be in κ 2 ∼ 0.09. Self-energy corrections, charged mesons and pion exchanges between s-wave states will enter at NLO (κ 2 ). Note, that the predictive power of our EFT is limited by the experimental uncertainty levels. The largest such uncertainties come from Γ c = Γ[D + * → D 0 π + + D + π 0 ] and B = Γ Dγ /(Γ Dπ + Γ Dγ ) and are of the order 3% ≈ κ 3 . Thus, we expect our NLO results to be reliable. We remark that, in principle, there are also NLO corrections to the real part of the complex energy E X . In our renormalization scheme, however, the real part of E X is kept fixed by properly readjusting C 0 as explained below.

Power Counting
Let q µ = (q 0 , q) be a relative DD * four-momentum. Loop integrations are counted nonrelativistically, i. e., d 4 q ∼ q 5 with q = |q|. We investigate the DD * amplitude in the vicinity of the X pole, i. e., at E ≈ E * . In this region, the D propagator i G D as well as the LO D * propagator i G * count like q −2 . The propagator of an exchanged pion, however, depends both on the incoming and outgoing relative momentum p in and p out , as can be seen from Eq. (27). Furthermore, it is suppressed by the small mass ratio Consequently, we count i G π ∼ r (max{p 2 in , p 2 out }) −1 . Finally, in this section the coupling g has to be expressed in terms of the reduced mass µ * and the momentum scales P * , P X , yielding 1/(2µ * g 2 ) ≈ 0.8 P * µ * /(2π). Therefore, we count g ∼ P −1/2 * in Feynman diagrams.

Width Estimation Strategy
We consider an arbitrary DD * interaction I other then C 0 . If resummed to all orders, it shifts the LO pole position and residue to E we identify the shifts ∆E (I) X = Z (LO) b (I) /(1 + a (I) ) and ∆Z (I) = Z (LO) a (I) . The coefficients a (I) and b (I) can be determined diagrammatically. Jansen et al. have used this procedure to calculate δ X at NLO in XEFT [37]. Our renormalization scheme, in contrast, keeps δ X fixed by readjusting C 0 . In other words, we resum an appropriate correction term ∆C (I) 0 in addition to I that cancels the real part of ∆E (I) X . Note, that a (I) and b (I) can be momentum-dependent, while the expression ∆E (I) X ∝ b (I) /(1 + a (I) ) must be a number. In Ref. [45] it was shown that this momentum dependence indeed cancels at NLO in XEFT. More generally, it follows from the momentumindependence of T We see from Eq. (42) that each correction to Γ (LO) X is proportional to Z (LO) and thus to the small binding momentum P X = ρ P * with ρ ∼ κ 2 . We can now verify the power counting order of D * propagator corrections, pion exchanges and charged mesons in the following way: 2. Determine their overall scaling by investigating loop momenta at both P X and P * .
Imaginary parts from on-shell pion exchanges are to be investigated separately.

Propagator Corrections
The width shift ∆Γ (1Σ) X due to single self-energy corrections in the D * propagator is proportional to the one-loop diagram in Fig. 8(a), evaluated at E = E (LO) X . Let q µ be the loop's four-momentum. The D * center-of-mass energies for q ∼ P X and q ∼ P * lie in the regions E cm ≈ E * and E cm ≈ 0, respectively. As discussed in Sec. III, self-energy corrections −iΣ · i G * are of the order χ 2 + i χ with χ = Γ Dπ /(2δ) in both regions. The two remaining propagators and the integral measure contribute a factor q. Thus, the main contribution to the integral stems from the region q ∼ P * and we findb (1Σ) ∼ (χ 2 + i χ) P * . That yields ∆Γ (1Σ) with Z (LO) ∝ P X = ρ P * (see Eq. (38)) and δ ∼ P 2 * (see Eq. (33)). From this estimation we expect that ∆Γ (1Σ) X corrects the X width at NLO (ρ ∼ κ 2 ). Apart from the scaling, Imb (1Σ) determines also the sign of ∆Γ (1Σ) X . After performing the q 0 integral in Fig. 8(a) and counting all ±i factors from Feynman rules, we can symbolically write sign ∆Γ , we have G * < 0 and for q ∼ P * the correctionΣ · G * ≈ Σ(E * )/E * has a negative imaginary part. Thus, the width shift ∆Γ (1Σ) X is negative and decreases the overall width. Compared to the self-energy correction, the imaginary parts of all other propagator corrections are suppressed by orders of κ 2 , see Eq. (24) and (25). Thus, they do not enter before N 2 LO (κ 4 ). Relativistic corrections to the D * propagator are even more suppressed (see Appendix C for details).

Pion Exchanges
Next, we consider the resummation of one-pion exchanges. The factorb (1π) is given by the two-loop diagram in Fig. 8(b) with loop four-momenta q µ , s µ . Its absolute value can be estimated like above. Recalling that the pion propagator scales like G π ∼ r (max{q 2 , s 2 }) −1 with r = µ/µ * and the vertices count like g max{q, s} with g ∼ P −1/2 * , we obtain the overall product r q s/P * . Thus, the absolute value of the integral is governed by loop momenta q ∼ s ∼ P * yielding |b (1π) | ∼ r P * .
The integral's imaginary part, however, scales differently for it appears only if the pion goes on-shell. This restriction imposes a condition on the angle cos θ ≡ e q · e s . Due to Eq. (27), it has to behave like with m π ∼ µ. This relation has no solution for P X ∼ q s ∼ P * or vice versa, because the right-hand side falls outside the interval [−1, 1]. Thus, on-shell pions require q ∼ s. In this case, Eq. (44) yields cos θ +1 for small momenta q ∼ s r 1/2 P * /2 ≈ 2 P X . The corresponding overall factor is of size q r P −1 * s ∼ r 2 P * /4. Naively, one would expect that the contribution at q ∼ s ∼ P * scales like P * r P −1 * P * = r P * which is much larger. However, in this region we have cos θ ≈ −1, which leads to a near-cancellation of the product of the two pion vertices: For q = −s the pion exchange potential becomes proportional to the suppression factor (1− α) 2 ≈ r 2 /4, see Eq. (27). Therefore, the imaginary part in this region is sub-leading and we findb (1π) ∼ (r + i r 2 /4) P * . From Eq. (42) we obtain the estimation Thus, the correction enters at NLO (ρ ∼ κ 2 ) as well.
As done for the self-energy contribution, we can infer the sign of ∆Γ (1π) X from the diagram in Fig. 8(b). We note that integrating over the pion propagator produces a negative imaginary part, which follows from the +i prescription. Moreover, the product of the two pion vertices is always negative. Taking into account all remaining phase factors, we obtain sign ∆Γ (1π) X = sign Imb (1π) = +1. Thus, single pion exchanges increase the width. In fact, we will see that this leads to a near-cancellation of the self-energy corrections.
Relativistic corrections to the pion propagator enter at N 2 LO (κ 4 ). Further details are given in Appendix C. Moreover, contributions from multi-pion exchanges are at least suppressed by additional factors of r κ 2 . From this observation we draw two conclusions: First, we may resum all pion exchanges between s-waves at NLO. Second, contributions from d-wave DD * states are of the order N 2 LO as they involve at least two pion exchanges.

Charged Mesons
At NLO, charged states (D + D − * + D − D + * )/ √ 2 cannot be neglected. In Ref. [39] they have been included to all orders via charged pion exchanges and s-wave contact interactions. However, charged pion exchanges (just as neutral ones) involve additional suppression factors of order r = µ/µ * 1, which makes them sub-leading. Instead of a non-perturbative treatment, we include charged D ( * ) mesons through the effective interaction −i I c (E) appearing in Eq. (26). It contains only contact interactions. Due to isospin symmetry, the vertex connecting a neutral and a charged C = + combination exhibits a factor 2 compared to the vertex between neutral pairs [39]. However, it may not contain a counter term for pion exchanges since we exclude charged pions. Therefore, whenever neutral pion exchanges enter the computation, we subtract the counter term −V (∞) 00 = 2/3 · g 2 µ 2 /m π from the vertex. The resulting interaction is shown diagrammatically in Fig. 9. It reads with µ * ,c ≡ (m −1 D + + m −1 D + * ) −1 ≈ µ * and ν as defined in Eq. (2). For more details on the charged meson propagators in Fig. 9 see Appendix E.
The perturbative inclusion of charged mesons has several advantages. First of all, we do not need to introduce an additional scattering channel, keeping the system matrix small. Furthermore, the system becomes renormalizable for arbitrary Λ. Finally, the effect of the interaction on Γ X is analytically solvable if pion exchanges and propagator corrections are switched off: ≡ 0) and set the pole to E X /2. Again, we demand C 0 (Λ) ∈ R and choose the one solution of C 0 that recovers the LO expression in the limit I c → 0. This procedure yields We see that charged mesons lower the X width which is in line with the findings by Baru et al. [39]. Compared to the binding energy δ X , the energy difference ν ≈ 8 MeV of the D ± * D ∓ threshold to the X is large. It corresponds to a DD * momentum of the order (2µ * ν) 1/2 ∼ (2µ * δ) 1/2 = P * due to ν ∼ δ. Therefore, the width correction induced by the charged meson loop is of order NLO (ρ ∼ κ 2 ). Similarly, contributions of multiple charged meson loops are suppressed by ρ 2 ∼ κ 4 , ρ 3 ∼ κ 6 , etc. and do not enter before N 2 LO. This observation verifies the perturbative nature of charged mesons in the X.

E. Summary: Inputs and Outputs of the EFT
We conclude this section by summarizing all EFT inputs and predictions in the two-and three-body sector up to NLO. They are listed in the following table.
In the two-body system, we have used the mass splittings δ, δ +0 , δ ++ and the pionic decay width Γ c of the D + * to determine the coupling g (see Appendix B) and further the width Γ Dπ and the threshold parameters a −1 1 and r 1 /2. Subsequently, the radiative decay width Γ Dγ has been obtained from Γ Dπ by taking the branching ratio B as additional input. All parameters are renormalized in the MS scheme. Note, that the two-body predictions do not change from LO to NLO. The reason is that the LO D * propagator already contains the full D * width and the NLO self-energy correction involves no new parameters. The three-body system can be renormalized using the coupling C 0 (Λ). The binding energy δ X serves as renormalization condition at both LO and NLO and at NLO also the mass splitting ν between the neutral and charge thresholds is needed. Thereby, we obtain Γ X and the production rate dΓ/dE as functions of δ X . Let us stress again that the physical value of δ X is not precisly known. We will, however, see that there are one-to-one relations between δ X and both the production rate's peak widthΓ X and maximum position δ −δ X (see Eq. (32)). They can be inverted in order to predict δ X from the experimentally measured line shape.

V. RESULTS
In this section, we present numerical results for Γ X up to NLO (κ 2 ). As argued above, N 2 LO contributions (κ 4 ) would involve higher-order propagator corrections, intermediate d-waves, iterations of the charged meson interaction and relativistic corrections. For illustration, we explicitly calculate the effect of d-wave states and show that is of order N 2 LO. Moreover, we show that the system is renormalizable for arbitrary cutoffs. Afterwards, we calculate the line shape of the X in DDπ production. We show that the peak's maximum position and line width can only be identified with the pole position if δ X > Γ X and if the detector resolution is sufficiently high.

A. X(3872) Width
In order to assess our power counting predictions and to demonstrate the convergence of the scheme, we compare calculations at LO and NLO. At LO, we solve the system depicted in Fig. 7 which must yield Γ (LO) X = Γ * . All sub-leading corrections are expected to be at least proportional to ρ = (δ X /δ) 1/2 , see above. We may thus obtain an LO uncertainty band by shifting the D * width by ±3 ρ Γ * . Thereby, we allow for a possible numerical coefficient. On top, we take into account the experimental uncertainties of Γ c and B by varying g 2 ∈ [3.40, 3.56] · 10 −8 MeV −3 and B ∈ [34.4, 36.2] %. The numerical results are presented in Fig. 10. The LO width, shown as a (blue) bold line, is indeed independent of δ X and given by Γ * = 53.6 keV. At δ X = 57 keV (i. e., ρ = κ 2 ) the LO band yields an uncertainty of about ±15 keV.
At NLO, we add the three contributions step-by-step. First, we insert single self-energy corrections in the D * propagator as shown in Fig. 4. The resulting shift, shown as a (red) dashed line in Fig. 10 shows a ρ ∝ δ 1/2 X dependence as expected and lies within the LO band. Next, we introduce pion exchanges between relative s-wave states. As expected, the corresponding width shift is of the same order as the previous one, but has the opposite sign. As a consequence, at δ X = 57 keV we obtain a small overall shift of +3 keV compared to the LO width. The influence of intermediate d-waves is expected to be of order κ 4 Γ * ≈ 0.44 keV (N 2 LO). This estimation is perfectly confirmed by the numerical result (red dotted line) which lies only 0.5 keV above the previous one. We conclude that d-waves are negligible at NLO and exclude them from all following calculations. The full NLO width, shown as a (red) solid line, is obtained by taking into account the charged meson loop. Remarkably, the overall NLO correction at δ X = 57 keV lies only 1.1 keV below the LO result. Moreover, a variation of experimental inputs yields an NLO uncertainty band of size ±2.2 keV ∼ κ 3 Γ * , which surrounds the LO curve for small δ X . Thus, the simple analytic LO result lies within the NLO band up to δ X ≈ 75 keV. In summary, all results are in very good agreement with the power counting predictions. Our full NLO prediction for the width at δ X = 57 keV reads It is instructive to compare the NLO prediction to the coupled channel results of Baru et al. [39], see the (green) squares in Fig. 10. Taking into account the D * width Γ (Baru) * = 63 keV used in Ref. [39], we obtain the (green) bold-dotted curve in Fig. 10. Indeed, both approaches agree very well for the same input parameters. Since the D * self-energy was treated non-perturbatively in Ref. [39], this agreement provides strong evidence for the subleading natures of the D * self-energy, as well as intermediate d-wave states, charged meson states in general and charged pion exchanges in specific. Thus, we conclude that our power counting scheme exhibits quick convergence. At second glance, one sees a minor deviation of about 2 keV at δ X = 100 keV. It indicates the beginning influence of threshold effects which blurr the X peak in DDπ production for small δ X : While deviations of the peak from a Breit-Wigner shape are negligible for the δ X investigated in Ref. [39], they have to be accounted for in the region δ X < 100 keV.
Let us emphasize at this point, that our EFT is based on the molecular picture of the +1Σ + π (+d-waves) Full NLO Figure 11: Contact interaction C 0 (Λ) for δ X = 100 keV X , which decays to DDπ or DDγ. Contributions from other decay channels might have a significant impact on Γ X . Their inclusion, however, goes beyond the scope of this work and has to be addressed in the future. Moreover, note that the uncertainty given in Eq. 48 relies on certain scaling assumptions for higher-order Dπ terms as discussed in Sec. III. These assumptions represent a scenario of minimal fine-tuning. Although unlikely, further fine-tunings could thus invalidate the developed power counting. Fig. 11 shows the curves C 0 (Λ) for Λ ∈ [0.4, 50] GeV obtained in the different calculations. The LO result reproduces Eq. (36). Self-energy corrections barely influence the pole's real part and neither do they influence C 0 (Λ). In contrast, pion exchanges shift the curve by an amount −V (∞) 00 ≈ 2.72 GeV −2 as expected. The charged meson contribution solely suppresses parts of C 0 (Λ) that vanish as Λ → ∞.

B. Contact Interaction
By including d-waves non-perturbatively, however, the running coupling significantly changes its signature. It exhibits consecutive singularities for fairly high cutoffs Λ > 11 GeV which was also observed by Baru et al. [39]. They are due to deep three-body states entering the spectrum at large cutoffs. This kind of spectrum is a general feature of three-body systems with resonant p-wave interactions [46,47]. The deep bound states lie outside the region of validity of the EFT and do not influence the physics close to the DD * threshold. We have explicitly checked that this is the case when we renormalize onto the shallow X pole. However, in calculations resumming both d-wave and charged meson states at the same time, the deep bound states lead to renormalization artefacts. In particular, there are cutoffs at which no value of C 0 can produce the X pole [39]. This problem is not present at NLO, where d-waves are negligible.
Note, that our non-perturbative d-wave calculation in Fig. 11 does not correspond to a strict N 2 LO treatment of such contributions. Instead, one would include single d-wave states pertubatively, similarly to the inclusion of the charged meson loop. It remains to be seen, if in such a calculation C 0 alone can produce the X for arbitrary cutoffs. We conclude this section by showing numerical results of the line shape of the X in DDπ production. In Fig. 12 normalized line shapes for the three values δ X ∈ {0.5, 57, 100} keV at LO and NLO are depicted. All curves are cutoff-independent 3 above the used value Λ = 1 GeV. For all δ X ≥ 50 keV ≈ Γ * , deviations of the peak parametersδ X = δ − E max and Γ X = FWHM from δ X and Γ X are negligible at NLO. Note, however, that the production rate does not possess a Breit-Wigner shape (indicated by black dotted curves). Instead, it is enhanced at the DD * threshold, as observed by Braaten and Lu [26].
As δ X decreases, threshold effects become more and more important. This effect can be seen in Fig. 12 where the FWHM valueΓ X is significantly enlarged for δ X = 0.5 keV.
We investigate this phenomenon in more detail in Fig. 13(a) by comparingΓ X (red solid line) to Γ X (black dashed line) for different δ X at NLO. As soon as δ X becomes comparable with Γ X ≈ 50 keV, the line width increases up to aboutΓ X ≈ 150 keV ≈ 2.8 Γ X . The functionΓ X (δ X ) turns out to be strictly monotonically decreasing. Thus, it can be inverted to determine δ X from an experimentally measured line width.
The approximationδ X ≈ δ X is even valid down to δ X ≈ 10 keV as shown in Fig. 13(b) (red solid lines and black dashed line, respectively). Below this value, the peak maximum crosses the DD * threshold (see also Fig. 12 for δ X = 0.5 keV). This effect becomes even more significant if we take into account the energy resolution of the detector. We mimic its influence by convoluting the line shape with a normal distribution of standard deviation σ. Indeed, due to the threshold enhancement, the peak of the smeared line shape is shifted to higher energies, see Fig. 14. For δ X = 57 keV a detector resolution σ ≥ 200 keV is sufficient to shift the peak onto the threshold. Moreover, Fig. 13(b) shows thatδ X is almost linear σ. This finding illustrates that in experiments the X peak could occur above threshold even if the X was bound. We conclude that, in order to avoid misinterpretations of experimental findings, the detector resolution needs to be of the order of the width Γ X ≈ 50 keV.

D. Remarks on Other X(3872) Interpretations
It should be noted, that our results only address the case of a bound X state, whose pole lies below threshold (δ X > 0) on the first Riemann sheet. Given the experimental binding energy in Eq. 1, the X pole could in principle also lie above threshold (δ X < 0). In this case, the molecular interpretation may not be appropriate. First of all, s-wave resonances cannot be produced by simple attractive potentials because they lack a centrifugal barrier [48]. In our EFT, the X pole is produced by the point-like interaction C 0 , which does not allow for such a possibility. In Ref. [48], the Λ c (3595) was studied as a shallow s-wave resonance in the π Σ c system. It was shown that this interpretation requires an unnaturally large and negative effective range parameter, disfavoring the resonance-interpretation. A similar result was obtained in Ref. [49].
The X could also be a DD * virtual state on the second sheet below threshold (δ X > 0).
As shown in the zero-range approach by Braaten and Lu [26], the production rate is then given by a monotonically increasing function with maximal slope near threshold. Our theory at LO coincides with the approach by Braaten and Lu and thus it indeed allows for a virtual X state as well. However, a detailed analysis of the virtual X pole at NLO may require an analytic continuation of Eq. (28) to the second energy sheet. Such a generalization will be part of future work.

VI. SUMMARY AND OUTLOOK
In this work, we have proposed a novel EFT for the exotic X(3872) state, which can be interpreted as a loosely-bound D 0D0 * molecule in the C = + channel. The EFT contains non-relativistic D 0 ,D 0 and π 0 fields and possesses exact Galilean invariance. The D 0 * vector meson was included as a p-wave resonance in the D 0 π 0 sector. Up to NLO in our power counting, we have calculated relations between the binding energy δ X of the X(3872), its width Γ X and its line shape in D 0D0 π 0 production. For the representative value δ X = 57 keV, the width is given by Γ (NLO) X = (52.5 ± 2.2) keV. Remarkably, the corresponding uncertainty interval, stemming from experimental inputs, includes the central value of the LO result Γ (LO) X = (53.6 ± 15.0) keV. This observation indicates a quick convergence of the theory. Moreover, the line shape exhibits a strong D 0D0 * threshold enhancement dominating the X(3872) peak for δ X < Γ X , confirming earlier studies by Braaten and Lu [26]. Our theory captures this enhancement and provides a method to systematically extract the X(3872) pole from the experimental line shape up to NLO accuracy.
Our counting is based on the characteristic momentum scales in the D 0 π 0 and D 0D0 π 0 sectors. The two-body system was analyzed in Sec. III. Exploiting Galilean invariance, we performed a comprehensive scaling analysis of D 0 π 0 threshold parameters in terms of the momentum scales K lo = (2µ δ) 1/2 ≈ 42 MeV and K hi ∼ m π ≈ 135 MeV. As a result, the existence of the narrow D 0 * resonance can be explained from a single fine-tuning of QCD. It is reflected in an enhancement of the p-wave effective range r 1 /2 ∼ K −2 lo K 3 hi . Shallow p-wave states in other physical systems were attributed to an enhanced scattering volume a 1 , see Refs. [40,42], and [50]. To our knowledge, the D 0 * is the first example of a shallow p-wave state, in which a 1 appears to be of natural size. Note, that other scaling scenarios may be possible, but they would require further fine-tunings. Radiative D 0 * decays were effectively included using complex interactions. At the end of the section, we derived an expansion of the full D 0 * propagator in the kinematic region of the X(3872). The LO propagator contains the full D 0 * width Γ * = Γ Dπ + Γ Dγ = (53.6 ± 1.0) keV as a constant. This ingredient is of paramount importance for the occurrence of the threshold enhancement. Propagator corrections are at least suppressed by the small ratio Γ Dπ /(2δ) ≈ 0.0025, suggesting a quick convergence of the expansion.
In Sec. IV, we constructed the non-perturbative D 0D0 * amplitude in the C = + channel. At LO, it only contains iterations of the LO D 0 * propagator and the D 0D0 * contact term C 0 (Λ), which produces the X(3872) pole for arbitrary cutoffs Λ. The LO amplitude is similar to the result of Braaten and Lu [26], with the exception that the D 0 * width enters as a constant. The sub-leading nature of interactions other than C 0 was justified in a diagrammatic power counting. In particular, we have investigated respective loop-integrals in terms of the low-momentum scales P X = (2µ * |δ X |) 1/2 ∼ 11 MeV and P * = (2µ * δ) 1/2 ≈ 117 MeV of the three-body system. As a result, the D 0 * self-energy, s-wave pion exchanges and charged meson loops have to be included at NLO. This claim was verified by calculation. Higherorder D 0 * self-interactions, relativistic corrections, intermediate d-waves and charged pion exchanges can be neglected at NLO. The theory is then renormalizable for arbitrary values of the cutoff.
An important finding of this work is, that for small binding energies δ X ≤ 50 keV the line shape's FWHM is significantly larger than Γ X (up to ≈ 2.8 Γ X ). In contrast, the peak's maximum position can be described by the pole's real part even for very small binding energies, i. e., for δ X ≥ 10 keV. This identification, however, fails once the detector's energy resolution is taken into account: For δ X ≈ 50 keV an energy resolution σ > 200 keV is sufficient to shift the peak maximum above the D 0D0 * threshold. This effect has to be taken into account in analyses of D 0D0 π 0 -type decays of the X(3872) [27][28][29]. In order not to misinterpret the nature of the X(3872), its peak has to be measured with a resolution of the order of the width Γ X .
In the near future, our EFT can be used to analyze data from D 0D0 * -type X(3872) decays at Belle [27,28]. Specifically, it would be interesting to calculate the Dalitz plot for decays to D 0D0 π 0 . Moreover, we could predict the line shape of the X(3872) for production at resonance atPANDA, i. e., in processes of the type pp → X(3872) → J/ψ + X [22]. Our framework could be extended in order to account for the partial widths of inelastic decay channels like J/ψ π + π − by choosing a complex coupling C 0 . However, this procedure requires a value for the branching ratio of D 0D0 * -type decays of the X(3872). At the moment, this quantity is only limited from below by 32 % [27]. Moreover, we could extend our framework to calculate line shapes for the X(3872) as a virtual D 0D0 * state.
In Eq. (A1), we have made explicit use of Galilean symmetry in taking the relative Dπ four-momentum l µ ≡ (l 0 , l) ≡ αl µ π − (1 − α)l µ D with α = m D /(m π + m D ) as loop integration variable. The l 0 integral has been performed using the residue theorem. Moreover, the lefthand integral in Eq. (A2) vanishes for i = j (asymmetric under l i → −l i ) and is otherwise independent of i. Therefore, we may replace l i l j → l 2 δ ij /3.
In order to calculate the right-hand integral in Eq. (A2), we turn to d spatial dimensions and introduce a subtraction scale Λ PDS . We find which has a pole in d = 2, but not in d = 3. In the MS scheme, we evaluate Eq. (A5) for d = 3 yielding the expression given in Eq. (10). However, it is enlightening to take a look at the result in the PDS scheme in which poles in d = 2 are subtracted as well [41]. For this purpose, we introduce the counter term − ∆Σ which vanishes for Λ PDS = 0. In this limit, we can easily recover the MS result. The full PDS result for Σ is then given by For a general ∆ 1 = ±1, the p-wave effective range now reads while the scattering volume a 1 and all higher-order parameters P 2n are independent of Λ PDS . From Eq. (A9) it is obvious that two threshold parameters, i. e., a −1 1 and r 1 /2, are needed in Dπ scattering. Using a finite momentum cutoff λ, the term Λ PDS would correspond to a linear divergence in λ. To take care of this linear divergence, the (bare) parameter r 1 can not be chosen as zero. Moreover, a cubic divergence in λ would enter in a −1 1 . If we neglect radiative decays of the D * , the EFT Lagrangian must be Hermitian, implying g 2 > 0. Furthermore, we know that r 1 < 0. In the MS scheme Eq. (A9) tells us that ∆ 1 = +1. The same is true in the PDS scheme for a subtraction point 0 ≤ Λ PDS < −r 1 /2. This choice is reasonable since |r 1 /2| ≈ 17.1 GeV is much larger than the expected breakdown scale K hi ∼ m π . Thus, the D * is a physical particle in our theory.
For the estimation of relativistic corrections in the D * propagator we investigate a Dπ pair moving at a total kinetic energy energy p 0 and a total momentum p. As in the twonucleon case [51], Lorentz invariance ensures that p 0 and p are related to the center-of-mass kinetic energy p 0 cm via with M = m D + m π . The D * pole position appears at (p 0 cm ) (pole) = E * . By plugging this condition into Eq. (C3) and using p 2 |M + E * | 2 , we determine the pole position in the general frame to be The full D * propagator can then be written like with p 0 = E − p 2 /(2m D ) + p 4 /(8m 3 D ) − . . . in the DD * system. In the non-relativistic limit, the difference p 0 − (p 0 ) (pole) has to recover the Galilean-invariant expression E cm − E * = E − p 2 /(2µ * ) − E * frequently used in this paper. Indeed, we obtain this expression by further expanding at E * /M ≈ 0, yielding where we used 2µ * ∼ M . All the corrections in the parenthesis are suppressed by the total Dπ mass and thus extremely small. The first one is comparable to κ 5 + i κ 10 while the second and third one are of order κ 6 . Since only imaginary corrections contribute to the width, relativistic corrections in the D * propagator only enter at N 5 LO.

Appendix D: Partial Wave Projection
We absorb angular dependences of the amplitude into vector spherical harmonics 4 The function Y m L L (n) denotes a spherical harmonic evaluated at a unity vector n while χ m S is a spherical basis vector in C 3 . With e p ≡ p/p and e p ≡ p /p the expansion for the amplitude reads The m J -sum over the two vector spherical harmonics in Eq. (D2) yields projection operators of the form P ij LL ; J . In the same fashion, we expand the pion exchange potential V ij .