Non-perturbative pion dynamics for the X(3872)

We discuss the role of non-perturbative pion dynamics on the near-threshold resonant X(3872) charmonium state, which is assumed to be an S-wave DD̄∗ bound system. We calculate the contribution to the width of the X(3872) from the DD̄π intermediate state treated non-perturbatively and compare it with different approximate approaches. Further, we explore the quark-mass dependence of the pole position of the X(3872) state. We find that the trajectory of the X(3872) depends strongly on the assumed quark-mass dependence of the short-range interactions which can be determined in lattice QCD calculations.


Introduction
After more than a decade after the discovery of the X(3872) by the Belle collaboration its nature still remains an open question, see Ref. [1] for a review.The resonance has the mass M X =(3871 ± 0.17) MeV and thus resides very close to the neutral D D * threshold It is therefore natural to assume that it has a large molecular admixture [2], see also Refs.[3,4].
Recently, the quantum numbers of this state were determined by the LHCb Collaboration to be 1 ++ [5] which is consistent with its interpretation as an S-wave D 0 D * 0 / D0 D * 0 bound state, see e.g.[6,7].The small binding energy relative to the D 0 D * 0 threshold allows for an effective field theory (EFT) formulation of the problem in analogy to the deuteron 1 .The pionless EFT framework based on pure contact D D * interactions was first applied to the X(3872) in Ref. [8].Due to the relevance of other dynamical scales, such a treatment is expected to be valid only in very narrow region around the threshold.In particular, the three-body neutral channel π 0 D 0 D0 opens at the energy 7 MeV below the a e-mail: vadimb@tp2.rub.de 1 Implications of heavy quark and heavy flavour symmetries were utilised in Ref. [10] to predict partner states of the X(3872).To incorporate the long-range pion physics, the so-called X-EFT was developed in Ref. [9] based on the assumption that pions can be treated perturbatively.
Recently, this framework was extended to include higher-order corrections and then used to predict the pion-mass dependence of the X-pole [11].On the other hand, the perturbative treatment of pions is known to be not applicable in the deuteron channel [12] which shows certain similarities with the X(3872).The role of non-perturbative pions was investigated in many phenomenological studies, see e.g.Refs.[14][15][16], all of them however include one-pion-exchange (OPE) in the static limit, i.e. under the assumption that the D-mesons are infinitely heavy particles.Meanwhile, the pion in the D D * potential can go on shell and thus the three-body πD D unitarity cuts should be taken into account 2 .
In this Contribution we discuss effects induced by the non-perturbative pion dynamics on the X(3872) state within the EFT framework, see Refs.[17,18] for more details.In particular, we test the validity of the static OPE approximation for the partial decay width X(3872) → D Dπ and study the dependence of the X binding energy on the light quark masses which is a precondition to extract a valuable information about the D D * interactions from upcoming and ongoing lattice simulations.

Formalism
We solve a system of coupled-channels Faddeev-type three-body equations for the D Dπ system in the where λ i stand for the known isospin coefficients and the OPE potential containing the three-body propagator at leading order reads 3 It should be stressed that in full analogy to the NN problem [19,20], the OPE does not fall off at large momenta and thus requires renormalisation.The D D * potential in an S-wave (V S S ) is to be modified to include the contact interaction C 0 2 It is shown in Ref. [13] that the 3-body unitary cuts play the crucial role in the D α Dβ system, if the D β width is dominated by the S -wave D β → D α π decay. 3In Ref. [21] the role of relativistic corrections in the non-perturbative approach including 3-body effects was addressed.For the sharp cut-off regularisation scheme used in our calculation C 0 (Λ) is adjusted to produce a bound state of the X(3872) for any given cut-off Λ.
In order to analyse the light quark-mass dependence we allow all quantitates such as the D and D * -meson masses, the coupling constant and the pion decay constant to vary with m π , i.e. we perform an expansion of all such quantities in terms of the parameter δm π /M [18], where the small scale is the difference of the running and physical pion masses δm π = m π − m ph π , while the large scale M is given by a typical hadronic scale ∼ 1 GeV.In addition to OPE, also the contact term has to vary with m π to ensure that the binding energy E B (m π ) is approximately Λ-independent for the running pion mass.Assuming that the leading correction to the the physical-limit quantity C ph 0 (Λ) is analytic with the quark masses, we may write The leading Λ-dependence of the contact interaction is captured by C ph 0 (Λ), while the dimensionless function f (Λ) absorbs the extra Λ-dependence which appears for values of the pion mass away from the physical point.Therefore, we fix the Λ-dependence of the contact interaction requiring that both the binding energy E B as well as its slope at the physical point, (∂E B /∂m π ) m π =m ph π , are Λ-independent.

Discussion and conclusions
First, we discuss the impact of non-perturbative pions on the decay width X → D Dπ [17], as shown in Fig. 1.We find that the perturbative inclusion of pions is justified, while the static approximation with non-perturbative pions leads to a significant overestimation of this observable.Thus, we conclude that the appropriate treatment of the three-body dynamics is mandatory.
The pion mass dependence of the binding energy of the X(3872) is illustrated in Fig. 2. The trajectory of the X(3872) depends strongly on the assumed quark-mass dependence of the shortrange interactions which can be parametrized by the slope (∂E B /∂m π ) m π =m phys π and is, in principle, measurable in lattice QCD.This is demonstrated in Fig. 2 where the resulting pion mass dependence is shown for two different values of the slope of a natural size.The sizeable difference between the pion-full and pion-less approaches at higher values of the pion mass for positive values of the slope indicates the important role of pion dynamics in this scenario, see also Ref. [11] for an analogous study within the X-EFT 4 .These findings will be useful for chiral extrapolations of the future lattice-QCD  In both cases, the ultraviolet cutoff in the integral equations is varied in the range Λ ∈ [400, 700] MeV.The dashed and dash-dotted curves represent the corresponding results of the pionless approach.The blue point depicts the first lattice calculation of the X(3872) [24].results for the X(3872) binding energy (see Ref. [24] for the first results) and will provide insights into its binding mechanism once the value of the slope parameter is determined.

DOI: 10 D
.1051/ C Owned by the authors, published by EDP Sciences, 0 D * 0 threshold while the charged three-body channels D ± D ∓ π 0 and D + D0 π − /D − D 0 π + reside about 2 MeV above it.Furthermore, also the charged two-body channel D c D * c (with c = ±) is located around 8 MeV above the neutral channel.
Here, the indices n, n are contracted with the corresponding indices of the D * polarisation vectors, m, m * and m π stand for the D, D * and the pion mass, in order, and the energy E is defined relative to the neutral two-body threshold M = m * 0 + m 0 + E. Furthermore, the strength of the potential g is extracted from the decay width D * → Dπ, see e.g.Ref.[18] for a more extended discussion of the input quantities.The OPE potential (3) connects the four D-meson channels defined as|0 = D 0 D * 0 , | 0 = D0 D * 0 , |c = D + D * − , |c = D − D * + ,and the amplitude a 0 = (a 00 − a c0 )/2 contains the relevant information about the X-pole.Note that the same three-body cut is also taken into account in the D D * propagators Δ i due to dressing D * by the self-energy (πD) loops.

Figure 1 .
Figure 1.The X width as a function of the binding energy E B for several different calculations: (i) solution of the problem with the non-perturbative OPE in the static limit -(green) dot-dashed line; (ii) solution of the full dynamical problem with non-perturbative OPE -(red) solid line; (iii) the X-EFT calculations: LO [6] -the dashed line and NLO [9] -the (blue) band.

Figure 2 .
Figure 2. Pion mass dependence of the X(3872) binding energy.The red filled band corresponds to the positive slope (∂E B /∂m π ) mπ=m ph π = 0.7 × 10 −2 while the black filled band corresponds to the negative slope (∂E B /∂m π ) mπ=m ph π = −1.5 × 10 −2 .In both cases, the ultraviolet cutoff in the integral equations is varied in the range Λ ∈ [400, 700] MeV.The dashed and dash-dotted curves represent the corresponding results of the pionless approach.The blue point depicts the first lattice calculation of the X(3872)[24].