Hidden charm molecules in finite volume
Miguel Albaladejo, Carlos Hidalgo-Duque, Juan Nieves, Eulogio Oset
aa r X i v : . [ h e p - l a t ] A p r Hidden charm molecules in finite volume
M. Albaladejo ∗ , C. Hidalgo-Duque † , J. Nieves, and E. Oset ‡ Instituto de Física Corpuscular (centro mixto CSIC-UV), Institutos de Investigación dePaterna, Aptdo. 22085, 46071, Valencia, Spain
October 16, 2018
Abstract
In the present paper we address the interaction of pairs of charmed mesons with hiddencharm in a finite box. We use the interaction from a recent model based on heavy quarkspin symmetry that predicts molecules of hidden charm in the infinite volume. The energylevels in the box are generated within this model, and from them some synthetic data aregenerated. These data are then employed to study the inverse problem of getting theenergies of the bound states and phase shifts for D ¯ D or D ∗ ¯ D ∗ . Different strategies areinvestigated using the lowest two levels for different values of the box size, carrying astudy of the errors produced. Starting from the upper level, fits to the synthetic data arecarried out to determine the scattering length and effective range plus the binding energyof the ground state. A similar strategy using the effective range formula is considered witha simultaneous fit to the two levels, one above and the other one below threshold. Thismethod turns out to be more efficient than the other one. Finally, a method based onthe fit to the data by means of a potential and a loop function conveniently regularized,turns out to be very efficient and allows to produce accurate results in the infinite volumestarting from levels of the box with errors far larger than the uncertainties obtained inthe final results. A regularization method based on Gaussian wave functions turns outto be rather efficient in the analysis and as a byproduct a practical and fast method tocalculate the Lüscher function with high precision is presented. The determination of the hadron spectrum from lattice QCD (LQCD) calculations is attractingmany efforts and one can get an overview on the different methods used and results in the recentreview [1]. One of the tools becoming gradually more used is the analysis of lattice levels in termsof the Lüscher method [2, 3]. This method converts binding energies of a hadron-hadron systemin the finite box into phase shifts of hadron-hadron interaction from levels above threshold, orbinding energies from levels below threshold [4–6]. From the phase shifts one can get resonanceproperties, and there are several works that have recently applied these techniques to study the ρ resonance [7–15]. There exist other resonances far more difficult to get with this approach like ∗ [email protected] † [email protected] ‡ [email protected] a (1260), which was also attempted in [14] (see also its determination using finite volumecalculations with effective field theory in [16]). Scalar mesons have also been searched for withthis method [17–20] and gradually some calculations are being performed for systems in thecharm sector [21–27]. From another field theoretical perspective, finite volume calculations havealso been devoted to this sector in [27–29]. In [28] the ¯ KD , ηD s interaction is studied in finitevolume with the aim of learning about the nature of the D ∗ s (2317) resonance from lattice data. The theoretical model used in [28] is taken from [31], where the D ∗ s (2317) resonance appearsdynamically generated from the interaction of ¯ KD , ηD s and other less relevant channels. In thislatter work, a scalar hidden charm state coming from the D ¯ D interaction with other coupledchannels was also found, which qualifies basically as a D ¯ D quasi-bound state (decaying intopairs of lighter pseudoscalars). Although not reported experimentally, support for this statehas been found in [32] from the analysis of the data of the e + e − → J/ψD ¯ D reaction of [33].From the effective field theory point of view, this state has also been reported in [34, 35], usinglight SU(3)-flavour and heavy quark spin symmetries to describe charmed meson-antimesoninteractions.The purpose of the present paper is to study the interaction of D ¯ D and D ∗ ¯ D ∗ using a fieldtheoretical approach in finite volume in order to evaluate energy levels in the finite box whichmight be compared with future LQCD calculations. The paper also presents a strategy to betteranalyze future lattice results in order to get the best information possible about bound statesand phase shifts in the infinite volume case from these lattice data. For this purpose we shalluse the model of [35], although most of the results and the basic conclusions are independentof which model is used.As to the method to obtain the finite volume levels and the inverse problem of obtainingthe results in the real world, phase shifts and binding energies, we shall follow the methodof [36] where a reformulation of Lüscher approach is done based on the on shell factorizationof the scattering matrix that one uses in the chiral unitary approach [37–42]. This method isconceptually and technically very easy and introduces improvements for the case of relativisticparticles (although we shall not make use of the relativistic version in the present paper). Someworks using this formalism can be found in Refs. [16, 28, 29, 43–47]. In this section, we briefly review the formalism of Refs. [34, 35], where an effective field the-ory incorporating SU(3)-light flavour symmetry and heavy quark spin symmetry (HQSS) isformulated, to study charmed meson-antimeson (generically denoted here H ¯ H ′ , with H, H ′ = D, D ∗ , D s , D ∗ s ) bound states. The lowest order (LO) contribution of the interaction is givenby contact terms, and the symmetries reduce the number of independent low energy constants(LECs) of the approach to only four. Other effects, like one-pion exchange or coupled chan-nel dynamics, are shown to be sub-leading corrections to this order. Still, coupled channelswill be considered explicitly when the mass difference between the thresholds is not negligi-ble compared with the binding energy of the molecules considered. To fix the four constantsof the approach, one assumes the molecular nature of some XYZ states, namely, X (3872), X (3915) and Y (4140). The fourth input of the model is the isospin violating branching ratio The first hint, though indirect, of the nature of the D ∗ s (2317) as a mostly ¯ KD bound state from latticedata was presented in [30]. There, lattice calculations of the scalar form factors in semileptonic pseudoscalar-to-pseudoscalar decays were used to extract information about the corresponding elastic S -wave scatteringchannels.
2f the decays X (3872) → J/ Ψ ω and X (3872) → J/ Ψ ρ (for a different approach to this issueand the X (3872) → J/ Ψ γ decay see [48]). For further details on the formalism we refer toRefs. [34, 35, 49]. We will adapt here the formalism to a more adequate (for the problem inhands) T -matrix language.Since we are dealing with heavy mesons, we use a non-relativistic formalism. In our normal-ization, the S -matrix for an elastic H ¯ H ′ scattering process reads S ( E ) ≡ e iδ ( E ) = 1 − i µkπ T ( E ) , (2.1)where the modulus of the momentum k = | ~k | is given by k = 2 µ ( E − m − m ), and µ is thereduced mass of the system of two particles with masses m and m . In Eq. (2.1), δ is the phaseshift, and we can write: T = − πµk sin δe iδ , (2.2) T − = − µk π cot δ + i µk π . (2.3)The expression for the T -matrix is given by: T − ( E ) = V − ( E ) − G ( E ) , (2.4)with V the potential (two-particle irreducible amplitude) and G a one-loop two-point func-tion. This equation stems from a once-subtracted dispersive representation of T − ( E ) (see forinstance Sec. 6 of Ref. [39]), or equivalently, from the N/D method [50] equations, when theleft-hand cut is neglected or included perturbatively [40, 51–53]. The loop function G providesthe right-hand cut and the contribution of the left-hand cut should be included in the potential V . As mentioned above, we will follow here the approach of Refs. [34, 35], and we will approxi-mate V by its LO contribution in the 1 /m Q expansion (with m Q the mass of the heavy quark).Thus, we are completely neglecting the left-hand cut.The loop function G needs to be regularized in some way. Typical approaches are once-subtracted dispersion relations and sharp cutoffs. Here, instead, we are following the approachof Refs. [34, 35], in which the loop function is regularized with a Gaussian regulator. For anarbitrary energy E , we find G ( E ) = Z d ~q (2 π ) e − ~q − k ) / Λ E − m − m − ~q / µ + i + = − µ Λ(2 π ) / e k / Λ + µkπ / φ (cid:16) √ k/ Λ (cid:17) − i µk π , (2.5)with φ ( x ) given by: φ ( x ) = Z x e y d y . (2.6)Note that, the wave number k is a multivalued function of E , with a branch point at threshold( E = m + m ). The principal argument of ( E − m − m ) should be taken in the range [0 , π [.The function kφ ( √ k/ Λ) does not present any discontinuity for real E above threshold, and We will always consider S − wave meson-antimeson interactions, and thus the spin of the molecule will alwayscoincide with the total spin of the meson-antimeson pair. The partial waves S +1 L J are then S +1 S J = S . Forsimplicity in what follows, we will drop all references to the L , S and J quantum numbers, both in the S and T matrices. ( E ) becomes a multivalued function because of the ik term. Indeed, G ( E ) has two Riemannsheets. In the first one, 0 Arg( E − m − m ) < π , we find a discontinuity G I ( E + iǫ ) − G I ( E − iǫ ) = 2 i Im G I ( E + iǫ ) for E > ( m + m ). It guaranties that the T − matrix fulfills the opticaltheorem. For real values of E and below threshold, we have k = i q − µ ( E − m − m ). Polesbelow threshold in the first sheet correspond to bound states. In the second Riemann sheet,2 π Arg( E − m − m ) < π , we trivially find G II ( E − iǫ ) = G I ( E + iǫ ), for real energies andabove threshold.The Gaussian form factor enters Eq. (2.5) in a way that is unity for on-shell momenta, andhence the optical theorem Im T − = µk/ (2 π ) is automatically fulfilled. A Gaussian regulator isalso used in Ref. [54] to study the ∆ resonance in finite volume.The cutoff Λ is a parameter of the approach, and, hence, the theory depends on it. Thisdependence, however, is partially reabsorbed in the counter-terms of the theory, as long asone chooses a reasonable value for it, not beyond the high-energy scale of the effective fieldtheory [55–58].In the approach of Refs. [34, 35], the potential V is taken as V ( E ) = e − k / Λ C (Λ) , (2.7)where C is the proper combination of the four different counter-terms for each considered chan-nel H ¯ H ′ . Explicit expressions can be found in Appendix A. The dependence of the counter-term on the ultraviolet (UV) cutoff Λ should cancel that of the loop function G , such that G ( E B ) V ( E B ) becomes independent of Λ, when E B is the energy of the bound state used to de-termine the counter-term. For other energies, there will exist a remaining, unwanted/unphysical,dependence of the T matrix on the cutoff. This is due to the truncation of the perturbativeexpansion (see discussion in Ref. [34]). Up to this point, we have discussed only the case ofuncoupled channels, but the generalization to coupled channels is straightforward. Finally, above threshold the effective range expansion reads: k cot δ = − a + 12 rk + · · · , (2.8)where a and r are, respectively, the scattering length and the effective range. From Eqs. (2.3)and (2.4) we can calculate the theoretical predictions for these effective range parameters,obtaining: a th = µ π C + µ Λ(2 π ) / ! − , (2.9) r th = − πµ Λ C − µ Λ(2 π ) / ! . (2.10) In this section, we follow the steps of Ref. [36] to write the amplitude in a finite box of size L with periodic boundary conditions, denoted by ˜ T . Since the potential does not depend on The mass of a bound state is thus given by T − ( E B ) = 0 for E B < m + m , which is the equivalent ofEq. (10) in Ref. [35]. One just has to rewrite the T -matrix as T = ( I − V G ) − V , where V and G are now matrices in the coupledchannels space. , one only has to replace the loop function G with its finite volume version, ˜ G , in which theintegral over momentum ~q is replaced by a discrete sum over the allowed momenta,˜ T − ( E ) = V − ( E ) − ˜ G ( E ) , (3.1)˜ G ( E ) = 1 L X ~q e − ~q − k ) / Λ E − m − m − ~q / µ , (3.2)where the (quantized) momentum is given by: ~q = 2 πL ~n , ~n ∈ Z . (3.3)Now, the energy levels in the box are given by the poles of the ˜ T -matrix, V − = ˜ G . For theenergies of these levels in the box, the amplitude in the infinite volume is recovered as: T − ( E ) = V − ( E ) − G ( E ) = ˜ G ( E ) − G ( E ) = δG ( E ) . (3.4)Since the G function is regularized (either in the box or in the infinite volume) with a Gaussianregulator, the difference above depends explicitly on the cutoff Λ. This remaining non-physicaldependence on Λ quickly disappears as the volume increases. Indeed, we find that it is exponen-tially suppressed and that it dies off as exp ( − L Λ /
8) (see Appendix B). Thus, it is clear thatin this context, we can end up the renormalization program just by sending the UV cutoff toinfinity. This will allow to obtain the physical T − matrix, independent of any renormalizationscale, for the energy levels found in the lattice Monte Carlo simulation (finite box).For the practical calculations that we will show in what follows, the Λ dependence is alreadynegligible when Λ & → ∞ is effectively achieved for such values). In this limit, Eq. (3.4) becomes the Lüscherequation [2, 3], as we discuss in certain detail in Appendix B. The results of the appendixalso show that the inclusion of a Gaussian regulator is a quite efficient technique, from thecomputational point of view, to evaluate the Lüscher function Z (1 , ˆ k ) used in [3]. Finally,from Eqs. (2.3) and (3.4), we can write: k cot δ = − πµ lim Λ →∞ Re (cid:16) ˜ G ( E ) − G ( E ) (cid:17) . (3.5) We present in this section the results obtained with the formalism outlined in the previoussection. We first discuss the results obtained by putting the model of Refs. [34, 35] directly inthe box. That is, we study the volume dependence of the molecules found in Ref. [35], predictingthus the existence of sub-threshold levels (asymptotically different of threshold) for the differentchannels, which have a clear correspondence with the hidden charm molecules reported in [35].This is done in Subsec. 4.1.Our purpose in Subsecs. 4.2, 4.3 and 4.4 is to simulate a realistic situation in a LQCD study,where one would obtain different energy levels (one or two) for different sizes, L , of the box.To do so, we generate “synthetic data” from the exact levels that we obtain from the modelof Refs. [34, 35]. We take five different values of L i , in the range Lm π = 1 . .
5. From thecalculated levels, we obtain randomly shifted levels (in a range of 5 MeV), and assign an errorof 10 MeV to each of these points, except the last one for which we have assumed an error of5nly 8 MeV to prevent it from crossing the threshold. Next we use a Monte Carlo simulation,to estimate the errors on the determination of observables (the phase shifts, for instance) whenthe energy levels are obtained with a certain statistical error. Specifically, we study in thesesubsections the I = 0 J P C = 0 ++ D ¯ D channel.In Subsec. 4.2, the Lüscher formalism to study the phase shifts calculated from Eq. (3.5) isapplied to the synthetic levels above threshold that we find for the different studied channels.From these phase shifts, we calculate the effective range expansion parameters, and use them todetermine the masses of the bound states. In Subsec. 4.3 we adopt another strategy to extractinformation from the generated levels. Namely, we consider a potential which parameters arethen fitted to reproduce the synthetic levels (above and below threshold, simultaneously). Withthis potential, we can make predictions in the infinite volume case, and thus we end up withanother determination of the masses of the predicted bound states. We shall see that thismethod allows one to obtain better results (better central value and smaller errors) for the massof the bound state than the previous one. We then analyze in detail which are the differencesof both approaches. In Subsec. 4.4 another method is proposed, in which the effective rangeapproximation is retained for the inverse of the T -matrix amplitude, but fitting directly theenergy levels instead of the phase shifts, and studying simultaneously the levels above andbelow threshold. In this case, then, we notice that the precision achieved for the mass of thebound state is similar to that obtained with the potential analysis.In Subsec. 4.5, we analyze in a more quantitative way, the qualitative arguments given inSubsec. 4.1, where the behavior of the sub-threshold levels is discussed. We offer a method todiscriminate between those levels that produce bound states in the limit L → ∞ and thosethat do not, and hence tend to threshold in the infinite volume limit. This method allows theextraction of the mass and the coupling of the bound state in the infinite volume limit.All these methods are applied in Subsec. 4.6 to the bound state present in the I = 0 J P C = 2 ++ D ∗ ¯ D ∗ channel. The difference with respect to the case used as an example in theprevious subsections is that the state is now weakly bound (the binding energy is only around2–3 MeV), so that we can compare how the methods exposed above work for this case. In Figs. 1 and 2 we present the dependence of the energy levels on the size of the finite box,as calculated from Eqs. (3.1) and (3.2), for the different channels studied in Ref. [35]. Wehave fixed the potential in the different channels by means of the central values given in thisreference for the various counter-terms, and collected here in Appendix A, Eqs. (A.3)–(A.6).When needed, we have also implemented in the finite box a coupled channel formalism. Thesolid lines correspond to the case Λ = 1 GeV, whereas the dot-dashed lines to Λ = 0 . k <
0) in theinfinite volume case. That is, their asymptotic L → ∞ value approaches the bound energiesgiven in [35], and thus they are different from threshold. Of course, one has this latter pieceof information from the calculations of the model in an infinite volume, but this would not bethe case in a lattice simulation. Let us focus, for simplicity, in the I = 0 case, shown in the leftpanel of Fig. 1. The large L asymptotic behavior can be well appreciated in some cases like the0 ++ D ¯ D or 1 + − D ∗ ¯ D molecules. However in other cases, it might be difficult to discriminatebetween a real bound state and a threshold level, even for quite large values of the box size L .Clear examples are the 1 ++ D ∗ ¯ D or the 2 ++ D ∗ ¯ D ∗ molecules (similar examples can be found in6 E ( M e V ) Lm π I = 0 0 ++ D ¯ D ++ D ∗ ¯ D + − D ∗ ¯ D ++ D ∗ ¯ D ∗ + − D ∗ ¯ D ∗ ++ D ∗ ¯ D ∗ D ¯ DD ∗ ¯ DD ∗ ¯ D ∗ E ( M e V ) Lm π I = 1 / + D + s D − + D s ¯ D ∗ D ∗ s ¯ D + D ∗ s ¯ D ∗ + D ∗ s ¯ D ∗ D + s D − D s ¯ D ∗ , D ∗ s ¯ DD ∗ s ¯ D ∗ Figure 1:
Volume dependence of the I = 0 (left) and I = 1 / . J P C quantum numbers of the different channels are indicated beside the arrows. the different isospin-strangeness channels), which in the infinite volume case are loosely bound. Thus, we see a well known result from Quantum Mechanics; the smaller the binding energies,the larger become the L values needed to reach the asymptotic behavior. From this study, weconclude that in a lattice simulation when dealing with states that are at least bound by sometens of MeV, one might safely discriminate them by using box sizes of the order of Lm π ≃ X (3872) or the 2 ++ resonances) might require significantly larger volumes. Toachieve more accurate predictions for the former and solve the problem for the latter ones, wefollow different approaches in the following subsections.Finally, we note that some of the levels in Figs. 1 and 2 are not realistic, in the sense thatthey would mix with other levels generated by channels with the same quantum numbers, butlower thresholds. That is the case, for example in I = 0, of the 0 ++ D ∗ ¯ D ∗ at E ≃ D ¯ D channel. Indeed, it is to be expected thatthese bound states would acquire some width due to the coupled channel dynamics. Still, it ispossible that these states could appear as more or less stable energy levels. The first one corresponds to the X (3872) resonance that has been observed close to the D ¯ D ∗ threshold [59](see also a recent determination and discussion of other experiments in Ref. [60]) and it has been a hot topicfor both the experimental and theory communities since its discovery. The 2 ++ state is a HQSS partner of the X (3872) molecule which dynamics, at LO in the heavy quark expansion, is being determined by precisely thesame combination of counter-terms that appear in the X (3872) channel. Given the discovery of the X (3872)resonance, the existence of the 2 ++ state, either as a bound state or a resonance, is therefore a quite robustconsequence of HQSS [34, 35]. In Ref. [61] the D ∗ ¯ D ∗ , D ∗ s ¯ D ∗ s states are studied with the interaction taken from the extrapolation of thelocal hidden gauge approach to the charm sector, which also respects HQSS. The coupling to D ¯ D and D s ¯ D s isallowed and generates a width of about 50 MeV for the most bound state, the one with I = 0 and J P C = 2 ++ . E ( M e V ) Lm π I = 1 0 ++ D + ¯ D + − D ∗ ¯ D ++ D ∗ ¯ D ∗ + − D ∗ ¯ D ∗ D + ¯ D D ∗ + ¯ D D + ¯ D ∗ D ∗ + ¯ D ∗ E ( M e V ) Lm π Hidden Strangeness 0 ++ D s ¯ D s + − D ∗ s ¯ D s ++ D ∗ s ¯ D ∗ s + − D ∗ s ¯ D ∗ s D s ¯ D s D ∗ s ¯ D s D ∗ s ¯ D ∗ s Figure 2:
The same as in Fig. 1 for the I = 1 (left) and hidden strangeness (right) moleculespredicted in Ref. [35]. E ( M e V ) Lm π Potential fitModelSynthetic dataFree Energies
Figure 3:
Some energy levels for the I = 0, J P C = 0 ++ D ¯ D interaction as a function of the box size L . The levels obtained with the model of Refs. [34,35] in a box for Λ = 1 GeV are shown with (red)solid lines, while the generated levels for some particular values of L (synthetic data points, seethe text for details), together with their assigned errors are displayed with black circles. The non-interacting energies ( m + m + (2 π/L ) n / µ with n = 0 ,
1) are shown with (blue) dash-dottedlines. The error bands around the solid lines are obtained from the fit to a potential discussed inSubsec. 4.3. They have been obtained by considering pairs of fitted parameters (1 /C a , Λ) thatprovide values of χ that differ from the minimum one by less than one unit ( χ χ + 1). We start discussing the case of the isoscalar 0 ++ D ¯ D interaction. Some levels found fromthe model of Ref. [35] in a finite box, obtained as the zeros of Eq. (3.1), are shown with a(red) solid line in Fig. 3. The synthetic levels generated from them and our choice for theirerrors are shown with points. Recall that we give an error of ±
10 MeV to these points tryingto simulate a realistic situation in a LQCD study, where these levels will be determined with In what follows we will use an UV cutoff Λ = 1 GeV when presenting results deduced from the model ofRef. [35], both for finite boxes and in the infinite volume case. Other cutoffs compatible with the effective theorydesigned in [34, 35] give rise to similar results. ( d e g ) E (MeV)Effective RangePotentialSynthetic Data-120-90-60-300 3740 3760 3780 3800 3820 3840 3860 3880 3900 Figure 4:
Phase shifts obtained for I = 0 J P C = 0 ++ D ¯ D interaction. The points stand forthe phase shifts calculated from the synthetic energy levels displayed in Fig. 3 using Eq. (3.5).The green dashed line and its associated error band corresponds to the effective range analysis ofSubsec. 4.2, while the blue solid line and its error band stand for the results obtained by fittinga potential discussed in Subsec. 4.3. The phase shifts in the infinite volume are very similar tothe latter ones, so we do not show them. In both cases, the error bands have been obtained byconsidering pairs of fitted parameters [(1 /a, r ) for the effective range fit and (1 /C a , Λ) for thecase of the potential fit] that provide values of χ that differ from the minimum one by less thanone unit (points included in the dark blue χ χ + 1 ellipse displayed in Fig. 5). some statistical uncertainties. From the upper level and the Lüscher’s formula, Eq. (3.5), wefind the phase shifts shown with points in Fig. 4. The errors in the phase shifts in this figure aredetermined by recalculating them, through Eq. (3.5), with different values of the upper levelenergy, E , randomly taken within the error intervals displayed for each of the synthetic datapoints in Fig. 3.We could also obtain the scattering length and the effective range parameters either fromthe determined phase shifts, or from Eqs. (2.8) and (3.5). Actually, combining these two latterequations we have,Re δG L = lim Λ →∞ Re (cid:16) ˜ G ( E ) − G ( E ) (cid:17) = − πµ (cid:18) − a + 12 rk + · · · (cid:19) (4.1)for the upper energy levels, E , determined in finite boxes of different sizes. We have obtained1 /a and r from a χ − linear fit to the five data points generated for Re δG L using the fivesynthetic upper energy levels shown in Fig. 3. We find1 a = 0 . ± .
25 fm − , r = 0 . ± .
18 fm (4.2)with a linear Gaussian correlation coefficient R = 0 .
83. From the above result, we find a = 1 . +1 . − . fm . (4.3)These values are to be compared with those obtained in the infinite volume model, Eqs. (2.9)and (2.10), with parameter C = C a (Λ = 1 GeV) = − .
024 fm , which turn out to be: a th = 1 .
38 fm , r th = 0 .
52 fm . (4.4) To estimate the errors in Re δG L for each of the synthetic energy levels considered, we follow a proceduresimilar to that outlined above for the phase shifts. Thus, we let the synthetic energy level vary within the errorinterval displayed in Fig. 3 and find the range of variation of Re δG L . D ¯ D threshold, we estimate the position of the X (3715) bound state, E = 3721 +10 − MeV , (4.5)whereas the value found in Ref. [35] is E = 3715 +12 − MeV. The binding energy,
B <
0, isobtained from Eqs. (2.3) and (2.8), upon changing k → iκ , and imposing T − = 0, B = κ µ , κ = 1 ± q − r/ar . (4.6)To estimate the uncertainties in Eq. (4.5), we have performed a Monte Carlo simulation takinginto account the existing statistical correlations between 1 /a and r . We quote a 68% confidentinterval (CL), but with some caveats as we explain next. Note that 2 r/a is not far from unityand within errors it can be even bigger, which means that we can get some events in the MonteCarlo runs (around 25%) with 1 − r/a <
0, for which we set the square root to zero. Thus,the lower error quoted in Eq. (4.5) is somehow uncertain, since the above procedure tends toaccumulate events around 3695 MeV. On the other hand, for the cases with 1 − r/a >
0, butsmall, the two roots of κ in Eq. (4.6) are not so different, and hence there is some ambiguityin the binding energy B (we choose the smallest value of κ ). Note that, although the value of E obtained with its errors seems quite accurate, when one considers it relative to the bindingenergy B , we find a large dispersion, since the D ¯ D threshold is at around 3734 MeV.Finally, if we decrease the error of the synthetic energy levels from 10 MeV to 5 MeV, thenthe errors of the phase shifts as well as those of the threshold parameters are also reducedapproximately to half of their previous values, and the predicted mass is more accurate, E =3723 +5 − MeV (and now only for around 6% of the Monte Carlo events, 1 − r/a become negative).This should give an idea of the precision needed in the determination of the energy levels inorder to have an appropriate determination of the mass.In the next sections we discuss different alternatives that allow to achieve a better precision. We now consider another approach to analyze/use the synthetic levels that we generated in theprevious section. Here again we aim to falsifying real data obtained from LQCD Monte Carlosimulations for various finite volumes. The analysis of phase shifts in the previous subsectionnecessarily takes into account only the level above threshold in Fig. 3. It is then convenientto develop an approach that could simultaneously make use of all available levels. Thus, wepropose to describe all levels using a potential, Eqs. (2.7) and (3.1), fitting its parameters forsuch purpose. We adopt here an approach where we fit a counter-term C = C a defining thepotential and the UV cutoff Λ (involved in the finite box loop function and in the potential(see Eq. (2.7)) to the synthetic energy levels shown in Fig. 3. Thus, the χ function is thengiven by: χ = X i =1 (cid:16) E (0)thV ( L i ) − E (0) i (cid:17) (cid:16) ∆ E (0) i (cid:17) + X i =1 (cid:16) E (1)thV ( L i ) − E (1) i (cid:17) (cid:16) ∆ E (1) i (cid:17) , (4.7) The errors calculated for finite volume quantities in this work refer to the statistical uncertainties we generatein the synthetic data. The errors quoted from Ref. [35] refer instead to the uncertainties in the determinationof the constants appearing in the potential. We follow here the notation of [34, 35] where the counter-term that appears in this channel is called C a (see Appendix A). E (0 , ( L i ) are the first two energy levels calculated from the HQSS potential, with pa-rameters C a and Λ, in a finite box of size L i . On the other hand, E (0 , i and ∆ E (0 , i are thesynthetic levels, that we have generated, together with their assigned errors. Here, the super-script j = 0 , /C a and Λ, obtainedin the best fit are 1 C a = − . ± .
20 fm − , Λ = 970 ±
130 MeV , (4.8)with a linear Gaussian correlation coefficient R = − .
98. These errors, and the correlationcoefficient, are calculated from the hessian of χ at the minimum. However, since the fit is notlinear, these errors are slightly different from those obtained requiring χ χ + 1. This latterrequirement gives the following non-symmetrical errors:1 C a = − . +0 . − . fm − , Λ = 970 +180 − MeV , . (4.9)From Eq. (4.8), we find C a = − . +0 . − . fm . The central values of both, the counter-term andthe UV cutoff, agree well with those of the original model of Ref. [35], C a = − .
024 fm andΛ = 1 GeV, used to generate the synthetic levels. However, as expected, the two parameters arestrongly correlated. This is further discussed in Appendix C. A contour plot of the χ functionin the (1 /C a , Λ) − plane is shown in Fig. 5, that manifestly shows the correlation.On the other hand, the fitted parameters of Eq. (4.8) predict a value for the mass of thebound state of E = 3715 +3 − MeV (68% CL, obtained from a Monte Carlo Gaussian simula-tion keeping the statistical correlations) in the infinite volume case. The central value agreesremarkably well with the value obtained from the model of Ref. [35], E = 3715 MeV, and cer-tainly much better than that obtained with the phase shift analysis carried out in the previoussubsection ( E = 3721 +10 − MeV). The errors found now are also significantly smaller.Finally, we have calculated error bands for the predicted finite box levels and phase shifts byquantifying the variations that are produced in these observables when one randomly considerspairs (1 /C a , Λ) of parameters that provide χ χ + 1 (points included in the dark blueellipse displayed in Fig. 5). These error bands are shown in Figs. 3 and 4, respectively. Λ ( M e V ) /C a (fm − ) Theory χ + 9 χ + 4 χ + 1 Figure 5:
Contour plots for the χ function defined in Eq. (4.7) for the I = 0 J P C = 0 ++ D ¯ D channel. The dashed line shows the correlation predicted from the model of Ref. [35], Eq. (C.2).The circle represents the central value taken for that model, C a = − .
024 fm and Λ = 1 GeV,while the square stands for the results of the best fit of Eq. (4.8). .4 Inverse analysis: effective range We have seen in Subsecs. 4.2 and 4.3 that the fit of the synthetic energy levels with a potentialleads to better results for the mass of the bound state than those obtained from the fit to thephase shifts (deduced from the upper level) with the effective range expansion. We believe thatthe reasons for this improvement are basically three. First, the potential fit takes into accountboth levels, above and below threshold, while the phase shifts analysis takes into account justthe upper level. Second, in the potential fit, the “observables” are the energy levels, while, in thephase shifts analysis, the quantity that enters in the χ function is k cot δ , and the propagation oferrors can lead thus to worse determinations of the parameters. Third, the analytical structureof the inverse of the amplitude is different in both approaches. In the effective range approach,ones truncates a series up to k , while in the potential fit one is effectively including furtherterms beyond the latter ones. Indeed, the full loop function G is taken into account. To studythe importance of the first two points, we follow here another approach, in which we shall keepthe effective range approximation for the amplitude, but fit the energy levels (above and belowthreshold) instead of the phase shifts obtained from the above threshold level.The effective range expansion for the inverse of the T -matrix is written from Eqs. (2.3) and(2.8): T − ( E ) = − µ π (cid:16) − a + r k − ik (cid:17) , k = q µ ( E − m − m ) , E > m + m − µ π (cid:16) − a − r γ + γ (cid:17) , γ = q µ ( m + m − E ) , E < m + m . (4.10)Now, the energy levels in the box are found for given values of a and r , by means of Eq. (3.4).It is to say, by numerically solving T − ( E ) = δG L ( E ), similarly as it is done in the case of thepotential, but now using Eq. (4.10) to model the T − matrix both above and below threshold.We will denote the levels obtained in this manner as E ( j )thEF . To determine the best values of a and r , we consider thus a χ function as Eq. (4.7), where the E ( j ) i are still the synthetic levelswe have generated, but replacing the E ( j )thV by E ( j )thEF , calculated as explained above. The valuesof the best parameters are:1 a = 0 . ± .
07 fm − , r = 0 . ± .
07 fm , (4.11)with a linear Gaussian correlation coefficient R = − . Hence, we obtain: a = 1 . +0 . − . fm . (4.12)The errors calculated in this way are clearly smaller than those displayed in Eq. (4.2) with thephase shifts analysis carried out in Subsec. 4.2. Also, the central value of the scattering lengthagrees better with the theoretical one, Eq. (4.4). These improvements have a clear impact inthe determination of the mass of the bound state, which is now E B = 3716 +4 − MeV (68% CL),with smaller errors and better central value than those obtained with the phase shifts analysisin Eq. (4.5). In Fig. 6 we show a comparison of the ellipses in the (1 /a, r ) parameter spacedetermined by the condition χ χ + 1 for the fits of Eqs. (4.2) and (4.11). There, it can beclearly appreciated the significant improvement achieved by fitting directly to both, lower andupper energy levels, instead of fitting to the phase shifts deduced from the latter levels. Finally,we must point out that the determination of the energy levels obtained with this method arevery similar to those obtained in Subsec. 4.3 by introducing of a potential, and shown in Fig. 3. In this case, the errors calculated from the requirement χ χ + 1 almost coincided with those givenhere in Eq. (4.11). ( f m ) /a (fm − ) Both LevelsPhase Shift AnalysisTheory0.20.30.40.50.60.70.80.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 6:
Comparison of the determination of the effective range parameters, 1 /a and r , withthe methods explained in Subsecs. 4.2 (red dashed line) and 4.4 (solid blue line). The ellipsesare obtained from the condition χ χ + 1 in each case. The central values of each fit arerepresented with points. We also show, for comparison, the theoretical values of the parameters,as given in Eq. (4.4), with a black square. Actually, the differences between the upper and lower energy level curves (and their error bands)deduced from both methods would not be easily appreciated in Fig. 3. For this reason, we havenot shown in this figure the results obtained from the method discussed in this subsection.
We have discussed in Subsec. 4.1 the volume dependence of the sub-threshold levels that arisewhen we put the model in a finite box. For cases with
V <
0, the potential is attractive, andhence, a bound state in the infinite volume case may arise. Whether it is bound or not in theinfinite volume case, there would appear a sub-threshold level for finite volumes. It was arguedin Subsec. 4.1 that it may be not very clear, at first sight, if these levels tend to the thresholdenergy or to a bound state in the L → ∞ limit. To circumvent this problem, we suggest herea method to study this volume dependence. By subtracting Eqs. (2.4) and (3.1), we can writethe amplitude in the finite box as [62]:˜ T − = T − − δG L , δG L = lim Λ →∞ δG = lim Λ →∞ (cid:16) ˜ G − G (cid:17) . (4.13)A bound state with mass E B appears as a pole in the T -matrix, thus in the vicinity of the pole,we can approximate: T ( E ≃ E B ) = g E − E B + . . . , (4.14)where the ellipsis denote regular terms in the Laurent series of the amplitude. The couplingcan also be calculated analytically, g = lim E → E B ( E − E B ) T ( E ) or 1 g = d T − ( E )d E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E = E B . (4.15)The volume dependence of the sub-threshold level in the finite box, given by the equation˜ T − ( E ) = 0 is then dominated by this bound state, and hence:˜ T − ( E ) ≃ E − E B g − δG L ( E ) = 0 , (4.16)13rom where one can write: E ( L ) = E B + g δG L [ E ( L ) , L ] . (4.17)This equation is a reformulation of a similar result obtained in Refs. [4–6]. The coupling g obtained here is related to Z ψ of Ref. [6]. Note, however, that Eq. (4.17) is appropriate as longas Eq. (4.14) is sufficiently accurate to describe the infinite volume T − matrix for the energylevels found in the lattice simulation (i.e., energies for which ˜ T − vanishes). Hence, the largerthe box sizes, the better Eq. (4.17) will perform. We extract the mass and the coupling of thebound state from a fit to the sub-threshold level in Fig. 3 with the following χ function, χ = X i =1 (cid:16) E ( L i ) − E (0) i (cid:17) (cid:16) ∆ E (0) i (cid:17) (4.18)where E ( L ) is given by Eq. (4.17). The best χ is χ = 0 .
5, and the parameters so obtainedare: E B = 3712 ± , g = (2 . ± .
1) GeV − , (4.19)to be compared with those obtained with the model in the infinite volume case, E B = 3715 MeVand g = 2 . − .It could be that, for the case of weakly bound states, the error bars on the energies overlapwith the threshold and it is difficult to determine if one has a bound state or not. A weakattractive potential that does not bind in the infinite volume case, still provides a level belowthreshold for finite volumes, and the energies go to threshold as L → ∞ . For this case weproceed as follows. The volume dependence of this level would be given by:˜ T − = 0 = T − − δG L = − µ π (cid:18) − a + r k + · · · (cid:19) − µL k − α − βk + · · · = − µL k + µ πa − α + (cid:18) − µr π − β (cid:19) k + · · · with some coefficients α and β , disregarding exponentially suppressed terms. In the aboveequation, we have explicitly separated the threshold singularity of δG L ,1 L E − m − m = 2 µL k . In the same line as in those references, but using boosted reference frames, in Ref. [63] linear combinationsof energy levels are suggested to reduce the volume dependence. Our method does not rely upon the analyticalform of the volume dependence to make extrapolations since for every L considered, the exact L dependence isprovided by δG L ( E ( L ) , L ). On the other hand, for very small binding energies, some subtleties appear, because the coupling g tendsto zero as the mass of the bound state approaches the threshold [53, 64, 65]. We will discuss this issue at lengthin Subsec. 4.6. It is worth noting the following technical detail. In principle, E ( L ) should be extracted for each L i as theimplicit solution in Eq. (4.17) for given E B and g . For practical purposes, though, it is more convenient toobtain E ( L ) by plugging into the right-hand side of this equation the values of E (0) i and L i that we are fittingto. If Eq. (4.14) is accurate enough, both methods are equivalent, as long as the effects in Eq. (4.17) of thestatistical fluctuations of the measured lattice levels are sufficiently small. In that case, the results for E B and g should not be very different, as we have checked. Indeed, the best fit results given in Eq. (4.19) have beenobtained within this approximation. However, this approximation cannot be safely used when the bound stateis placed close to threshold, because then δG L rapidly changes and statistical fluctuations in the determinedlattice energy levels induce large variations in the right-hand side of Eq. (4.17). k = 2 µL A + Bk . (4.20)Now, this expression could be used in a χ function as in Eq. (4.18) (with k < χ value, discarding the possibility that there is not abound state in the infinite volume.We have seen then that the method outlined in this subsection allows for a safe discrimi-nation between those levels that correspond to bound states in the infinite volume and thosethat do not, and it also gives a precise determination of the mass. It is also worth noting thatthe errors for the mass of the bound state are similar to those obtained with the analysis froma potential in Subsec. 4.3, and smaller than those calculated with the phase shift analysis inSubsec. 4.2. I = 0 J P C = 2 ++ channel We now repeat the same analyses carried out above but now for the case of the bound statepresent in the I = 0 J P C = 2 ++ D ∗ ¯ D ∗ channel. As already mentioned, the 2 ++ state is a HQSSpartner of the X (3872) molecule which dynamics, at LO in the heavy quark expansion, is beingdetermined by precisely the same combination of counter-terms that appear in the X (3872)channel. The existence of the 2 ++ state, either as a bound state or a resonance, is a quite robustconsequence of HQSS [34, 35], and it can be certainly subject to experimental detection. It isalso worth to discuss this channel in this context because, contrary to the case analyzed before,we have here a very weakly bound state. In the calculation of Ref. [35], and also in the resultsof Subsec. 4.1, shown in the left panel of Fig. 1, charged and neutral coupled channels werestudied, because the gap between both thresholds is indeed larger than the binding energy ofthe bound state. The mass of the bound state is E B = 4013 . . . I = 0, and use isospin average masses, keeping the relevant counter-term, C in the nomenclature of Ref. [35], to the same value, namely, C = − .
731 fm . In this way,the threshold is located at 4017 . E B = 4014 . ± ±
10 MeV. With the points of theupper level, we generate the phase shifts shown in Fig. 8 with points, through Eq. (3.5). Wecan now obtain the scattering length and effective range as in Subsec. 4.2, that turn out to be:1 a = 0 . ± .
30 fm − , r = 0 . ± .
19 fm , (4.21)with a linear Gaussian correlation R = 0 .
81. From the above fitted value for 1 /a , we find: a = 2 . +2 . − . fm (68% CL) , (4.22)while the theoretical values, obtained from Eqs.(2.9) and (2.10), are: a th = 3 . , r th = 0 .
58 fm , (4.23)15 ( M e V ) Lm π Potential FitModelSynthetic dataFitThreshold395040004050410041504200 1 1.5 2 2.5 3 3.5 4 4.5
Figure 7:
Same as in Fig. 3, but for the I = 0 2 ++ D ∗ ¯ D ∗ interaction. δ ( d e g ) E (MeV)Effective RangePotentialSynthetic data-120-90-60-300 4020 4040 4060 4080 4100 4120 4140 Figure 8:
Phase shifts for the I = 0 J P C = 2 ++ D ∗ ¯ D ∗ channel interaction. The points stand forthe synthetic phase shifts generated from the upper energy level of Fig. 7. The (green) dashed linecorrespond to the effective range fit, while the (blue) solid line correspond to the potential fit. Theassociated error bands are obtained by considering randomly chosen pair of parameters [(1 /a, r )and (1 /C , Λ) for the effective range and potential fits, respectively] that satisfy χ χ + 1. which agree with the above determinations, within errors. The phase shifts obtained with theseparameters, and the associated error bands, are shown in Fig. 8 with (green) dashed lines, andthey satisfactorily reproduce the synthetic data. The mass of the bound state turns out to bethen E B = 4013 +4 − MeV. Recall that the caveats raised in Subsec. 4.2 apply here.The next step is to consider a potential fit, as in Subsec. 4.3, where the free parameters are C and Λ. From a best fit, as described in Subsec. 4.3, we find:1 C = − . ± . − , Λ = 1020 ±
240 MeV , (4.24)with a linear Gaussian correlation coefficient R = − .
97. The non-symmetrical errors given bythe condition χ χ + 1 are instead:1 C = − . +0 . − . fm − , Λ = 1020 +360 − MeV . (4.25)The energy levels obtained with these parameters are shown in Fig. 7 with a (black) dashed line,although they are so similar to those of the exact model that they mostly overlap. Also shownin the figure are the error bands generated from the errors of the parameters, as described16reviously in Fig. 3. With this potential, we can also calculate the phase shifts, which areshown in Fig. 8 with (blue) solid lines, and also the associated error band. The quality of bothdescriptions, that of the effective range and that of the potential, are very similar, and indeedboth lines are very similar to the one of the exact model, and hence we do not show the latter.The value of C deduced from Eq. (4.24) is C = − . +0 . − . fm (68% CL), in good agreementwith the one of the infinite volume model, C = − .
73. Finally, the mass of the bound state isgiven by E B = 4014 . +2 . − . MeV. We must stress again that the errors obtained with this methodare smaller than those obtained with the phase shifts analysis.Now, we consider the method of Subsec. 4.4, in which the effective range expansion is usedto study the levels below and above threshold. In this case, the best fit values that we obtainfor the scattering length and effective range parameters are:1 a = 0 . ± .
15 fm − , r = 0 . ± .
15 fm , (4.26)with a linear Gaussian correlation coefficient R = 0 .
3. The non-symmetric errors stemmingfrom the condition χ χ + 1 turn out to be:1 a = 0 . +0 . − . fm − , r = 0 . +0 . − . fm . (4.27)Propagating the correlated Gaussian errors of Eq. (4.26), we find: a = 2 . +2 . − . fm . (4.28)We note that here, as it also occurred for the I = 0, J P C = 2 ++ case, the central values obtainedwith this method agree better with the theoretical ones, Eq. (4.23), than those obtained withthe phase shift description, Eq. (4.21), and have smaller errors than the latter ones. The massof the bound state obtained is E B = 4014 . +2 . − . MeV, which is better determined than thatobtained by means of the phase shifts analysis, and very similar to the one obtained with thepotential method.Finally, we should proceed now with the analysis performed in Subsec. 4.5. However, thereis a major difference in this case, namely, that the bound state is very close to the thresholdin this case. It is known that, in this case, the coupling of the state tends to zero [53, 64, 65],and so additional terms in the Laurent series, Eq. (4.14), are relevant for energies not very farfrom the bound state mass. Further, since we are considering an error of ±
10 MeV in theenergy levels, and we are trying to reproduce a bound state of binding energy of 2 − As an example, consider a background term in the amplitude in Eq. (4.14), so that T = g / ( E − E B )+ β + · · · .From the theoretical model, one can calculate g = 0 . · − MeV − and β = − . · − MeV − . For energies E ≃ Lm π = 1 .
5, we have (cid:12)(cid:12) g / ( E − E B ) (cid:12)(cid:12) < | β | . Conclusions
In this paper we have addressed the interaction of heavy charmed mesons in the hidden charmsector where several bound states are produced using an interaction that is based on heavyquark spin symmetry. The interaction is then studied in a finite box and the levels expectedfrom a lattice QCD calculation are evaluated for the D ¯ D , D ∗ ¯ D ∗ states and their SU(3) partners.Then the inverse problem is faced, generating “synthetic data” from the levels obtained andusing different procedures to obtain the relevant magnitudes in the infinite space, phase shiftsand binding energies for the bound states. Particular emphasis is done in the error analysis toestablish the accuracy of the different methods. We use two levels for different values of thebox size L , one below threshold and the closest one above threshold. One strategy is to use theLüscher formula to get phase shifts for each energy of the level above threshold. Another strategyis to use the effective range approximation, but fitting the scattering length and effective rangeto both levels (above and below threshold). The two methods work, but the latter one givesbetter determinations of the parameters (scattering length and effective range), but also of themass of the bound state. Yet, the method that proves most efficient is to parameterize apotential and a regularizing UV cutoff for the meson meson loops and carry a fit to the data.Once the potential and the UV cutoff are determined one can evaluate the phase shifts andbinding energies with much better precision than the one assumed in the “synthetic data”. TheUV cutoff is not needed if one considers levels of only one energy, but it appears when differentenergies are used in the fit, yet, we show that it is highly correlated with the potential. We alsodevoted particular attention to the case of weakly bound states, where special care must betaken. Finally, as a byproduct we present an efficient method to obtain the Lüscher function,supported by an analytical study that allows one to truncate the sum by means of a Gaussianform factor and estimate the error induced by the truncation. Acknowledgments
This work is partly supported by the Spanish Ministerio de Economía y Competitividad andEuropean FEDER funds under the contract number FIS2011-28853-C02-01 and FIS2011-28853-C02-02, and the Generalitat Valenciana in the program Prometeo, 2009/090. We acknowledgethe support of the European Community-Research Infrastructure Integrating Activity Studyof Strongly Interacting Matter (acronym HadronPhysics3, Grant Agreement n. 283286) underthe Seventh Framework Programme of the EU.
A Potentials for the H ¯ H interaction In this Appendix, for completeness, we briefly review the formalism of the effective field theoryderived in Refs. [34, 35] to study charmed meson-antimeson bound states. This effective fieldtheory incorporates SU(3)-light flavour symmetry and HQSS. In this context and at LO, thepotential is given by contact terms, related to four independent LECs, namely C a , C b , C a and C b , in the notation of Refs. [34, 35]. Thus, the potentials in the different channels will begiven by different linear combinations of these LECs. The fit of these LECs to four experimental As one moves far away from threshold, any method based on the effective range approximation becomesless appropriated. The potential fit method improves on the effective range expansion since it includes the fullloop function and not only the imaginary part fixed by unitarity. Thus, this latter method should work betterwhen energies significantly lower or higher than the threshold are considered. P C H ¯ H ′ S +1 L J V C ++ D ¯ D S C a ++ D ∗ ¯ D S Eq. (A.1)1 + − D ∗ ¯ D S C a − C b ++ D ∗ ¯ D ∗ S C a − C b + − D ∗ ¯ D ∗ S C a − C b ++ D ∗ ¯ D ∗ S Eq. (A.1) J P C H ¯ H ′ S +1 L J V C + D ¯ D s S C a + D ∗ s ¯ D , D ∗ ¯ D s S Eq. (A.2)0 + D ∗ ¯ D ∗ s S C a − C b + D ∗ ¯ D ∗ s S C a − C b + D ∗ ¯ D ∗ s S C a + C b Table 1:
Potentials for the H ¯ H ′ interaction for the I = 0 (left) and I = 1 / J P C channels. J P C H ¯ H ′ S +1 L J V C ++ D ¯ D S C a ++ D ∗ ¯ D S Eq. (A.1)1 + − D ∗ ¯ D S C a − C b ++ D ∗ ¯ D ∗ S C a − C b + − D ∗ ¯ D ∗ S C a − C b ++ D ∗ ¯ D ∗ S Eq. (A.1) J P C H ¯ H ′ S +1 L J V C ++ D s ¯ D s S C a ++ D ∗ s ¯ D s S C a + C b + − D ∗ s ¯ D s S C a − C b ++ D ∗ s ¯ D ∗ s S C a − C b + − D ∗ s ¯ D ∗ s S C a − C b ++ D ∗ s ¯ D ∗ s S C a + C b Table 2:
Potentials for the H ¯ H ′ interaction for the I = 1 (left) and hidden strangeness sector(right) different J P C channels. data, as stated in Sec. 2, allows one to fix the numerical value of the different counterterms. Inthe following, we summarize the form of the potentials for the different I and J P C channels.In the isoscalar channel, the only involved LECs are C a and C b , and the way they appearin the different J P C channels is shown in Table 1 (left panel). Potentials in the I = 1 / I = 1 sector are the same, except in the channel where coupled channels must be consideredas discussed below. These potentials only depend on C a and C b . The explicit expressions for I = 1 / I = 1 are shown in Table 2 (leftpanel). Finally, in the hidden strangeness sector, the four LECs appear, and the final potentialis the arithmetic mean of the corresponding isoscalar and isovectorial interaction, as can beseen in Table 2 (right panel). Note that, for this table, we have defined C a = ( C a + C a )and C b = ( C b + C b ).However, coupled channels must be taken into account in two cases. The first one is that inwhich the mass difference of the charged and neutral channels thresholds is not negligible ascompared to the binding energy of the state. The second case occurs when charge conjugationis not a good quantum number. The first scenario is important in the study of the D ¯ D ∗ systemwith J P C = 1 ++ and D ∗ ¯ D ∗ with J P C = 2 ++ . In this case, the potential will be the interactionbetween the charged and neutral channels and will be given by the 2 × V = 12 C + C C − C C − C C + C ! , (A.1)being C = C a + C b and C = C a + C b . The second scenario where coupled channels are19mportant is because of the mixing between the D s ¯ D ∗ and D ¯ D ∗ s channels. In this case, thepotential is: V = C a − C b − C b C a ! . (A.2)Using these combinations of counter-terms to describe the four input data, as explained inSec. 2, the numerical values of the LECs for the two values of the cutoff considered in this workare the following: C a = − . +0 . − . fm ( − . +0 . − . fm ) , (A.3) C b = +1 . +0 . − . fm (+0 . +0 . − . fm ) , (A.4) C a = − . +0 . − . fm ( − . +0 . − . fm ) , (A.5) C b = +1 . +0 . − . fm (+0 . +0 . − . fm ) , (A.6)for Λ = 0 . B Gaussian regulator and relation to Lüscher formula
In this Appendix, we discuss the details of Eq. (3.5) within a Gaussian regularization scheme.We also study the dependence of the function δG ( E ), that appeared in Eq. (3.4), on the UVcutoff Λ. For convenience, we re-write δG ( E ) as δG ( E ; Λ) = ˜ G ( E ) − G ( E )= δG A z }| { L X ~q e − ~q − k ) / Λ − E − m − m − ~q µ − Z d ~q (2 π ) e − ~q − k ) / Λ − E − m − m − ~q µ + i + + 1 L X ~q E − m − m − ~q µ − Z d ~q (2 π ) E − m − m − ~q µ + i + | {z } δG L (B.1)The function δG explicitly depends on the cutoff Λ, and this dependence is carried by the δG A term. On the other hand, the term δG L is well defined, and it is related to the Lüscherfunction [36] (see discussion below). In the strict Λ → ∞ limit, only the second term survives,which justifies our approach in Sec. 3. Still, for practical purposes, the limit Λ → ∞ can onlybe achieved by taking Λ large enough, and then it is necessary to study the dependence of δG with Λ. Let us note that δG A has no poles, and hence it is exponentially suppressed with L according to the regular summation theorem [2, 3]. For k > E > m + m , δG L is notexponentially suppressed for L → ∞ and, in this case, δG L clearly dominates over δG A .However for k < δG L is also exponentially suppressed as L increases, and therefore oneneeds to explicitly calculate the dependence of δG A on Λ L .Let us calculate the derivative of δG with respect to Λ. Only δG A depends on Λ, and thislatter function does it through the exponential function exp[ − ~q − k ) / Λ ]. The derivative20rings down a factor ( ~q − k ) that cancels out the denominators. This greatly simplifies thecalculation of both the sum and the integral. The latter one is trivial and it only amounts tothe integration of a Gaussian function, while the former one, up to constant factors, now reads1 L X ~q e − ~q / Λ = L + ∞ X n = −∞ e − ( πL ) n ! = θ (0 , e − π L ) L (B.2)where we have used ~q = q x + q y + q z and that the exponential of a sum is the product ofthe exponentials. This latter property allows to relate the sum in three dimensions to the cubeof the sum in one dimension. In Eq. (B.2), θ ( u, α ) is a Jacobi elliptic theta function [66]. Itsatisfies [67], θ (0 , e − πx ) θ (0 , e − π/x ) = 1 x . (B.3)This allows us to write then: ∂ δG∂ Λ = − µ (2 π ) / e k / Λ (cid:18)h θ (0 , e − Λ L / ) i − (cid:19) . (B.4)We note that this equation is exact. The above equation converges rapidly to zero as theGaussian cutoff increases, which shows that the limit Λ → ∞ is effectively quickly achieved.To proceed further, we note that:[ θ (0 , α )] = 1 + 6 α + 12 α + · · · = ∞ X m =0 c m α m , (B.5)and the coefficients c m are nothing but the multiplicities of m = ~n , ~n ∈ Z . Since α = e − Λ L / ,we can find the leading term in Eq. (B.4), ∂ δG∂ Λ = − µ (2 π ) / exp k Λ − Λ L ! (cid:16) O (cid:16) e − Λ L / (cid:17)(cid:17) . (B.6)Given that δG = δG L for Λ → ∞ , we find keeping just the leading term: δG ( E ; Λ) = δG L ( E ) + 6 µ (2 π ) / Z ∞ Λ dΛ ′ exp k Λ ′ − Λ ′ L ! (B.7)= δG L ( E ) + 3 µ πL " e ikL erfc Λ L √ i √ k Λ ! + e − ikL erfc Λ L √ − i √ k Λ ! , and, then, its asymptotic behavior is: δG ( E ; Λ) = δG L ( E ) + 24 µ (2 π ) / e − Λ2 L Λ L " k L − L Λ + O (cid:16) Λ − (cid:17) + · · · , (B.8)where O (Λ − ) refers to ( k/ Λ) , ( k /L ) / Λ and 1 / ( L Λ) , and the ellipsis stands for terms thatare more exponentially suppressed (the next one would take the form e − Λ L / ). Given the formof the L suppression, the Gaussian regularization scheme does not introduce any spurious termsthat would dominate over the physical contribution δG L , as long as Λ is sufficiently large. As already mentioned, for k > δG L is not exponentially suppressed for L → ∞ , while for k <
0, weexpect δG L to decrease as exp( −| k | L ). δG loop function. The Lüscher function is related to the loop functions by meansof [36]: √ π Z (1 , ˆ k ) = − L π (2 π ) µ δG L ( E ) , ˆ k = k L (2 π ) . (B.9)Thus, for a mildly large value of Λ, δG L ( E ) can be approximated by the Gaussian regulated δG ( E, Λ) function, up to corrections suppressed by the exponential factor e − Λ2 L (see Eq. (B.8)), √ π Z (1 , ˆ k ) = − L π (2 π ) µ (cid:18) δG ( E, Λ) + O (cid:18) e − Λ2 L (cid:19)(cid:19) (B.10)which in turn provides Z (1 , ˆ k ) with enough accuracy in a computationally easy way. C Cutoff effects and relation to dispersion relations
In this Appendix and to better frame the approach followed in Subsec. 4.3, the existing correla-tion between the constant of the potential and the cutoff is addressed in detail. We also discussthe relation of our approach to other approaches in which the loop function is calculated froma dispersion relation.We recall Eqs. (2.5) and (2.7) to expand the inverse of the amplitude in powers of k . Formore general purposes, we consider a potential in which the factor that multiplies the Gaussian,exp ( − k / Λ ), has some energy dependence instead of being constant. It is to say, we replacein Eq. (2.7) C by C ( E ) = c + c k , that reduces to the original form by setting c = 0. Wefind V − − G = 1 c + µ Λ(2 π ) / + − c c + 2 c Λ − µ (2 π ) / Λ ! k + i µk π + O ( k ) . (C.1)For model-given values of c and c for an imposed cutoff Λ, one can shift the cutoff to Λ ′ andhave the same T -matrix, up to O ( k ), by reabsorbing the cutoff shift in the new parameters c ′ and c ′ , given by: 1 c ′ = 1 c + µ (Λ − Λ ′ )(2 π ) / , (C.2) c ′ c ′ = c c + 2 c Λ − Λ ′ Λ Λ ′ + 2 µ (Λ − Λ ′ ) (2 π ) / ΛΛ ′ . (C.3)If we insist in a constant potential, c = c ′ = 0, we can also reabsorb the cutoff effects in c ,but this would be correct just up to O ( k ).Let us consider an approach in which the amplitude is written with a loop function regu- It satisfies [3], e iδ = k cot δ + ikk cot δ − ik = Z (1 , ˆ k ) + iπ ˆ k Z (1 , ˆ k ) − iπ ˆ k . T − = V − − G DR , (C.4) V DR = C , (C.5) G DR = α − i µk π , (C.6)where C is the potential, analogous to the case of the Gaussian regulator approach, and α is asubtraction constant (a free parameter of the approach). Considering, as before, C = c + c k ,we can expand: V − − G DR = 1 c − α − c c k + i µk π + O ( k ) . (C.7)We can then reabsorb the effects up to O ( k ) of an arbitrary shift in the subtraction constantby means of: 1 c ′ = 1 c + α ′ − α , (C.8) c ′ = c ′ c c . (C.9)We see that the effects of the shift can be reabsorbed exactly for a constant potential, withthe first of the previous equations. However, in the more general case of energy dependentpotentials (as the case of chiral potentials, for example), this cannot be made exactly but justup to O ( k ). We see thus that there are several equivalent ways: one can fit, in the Gaussianregulator case, a constant for the potential and the cutoff, or fix the latter to a reasonable (butotherwise arbitrary) value and fit two constants. In a dispersion relation, a similar situation isfound, where now the subtraction constant plays the equivalent role of the cutoff. In Subsec. 4.3we have followed the first approach. In Fig. 5, we show, for the case of I = 0 J P C = 0 ++ D ¯ D ,the contours curves of the χ function in terms of the free parameters: the UV cutoff Λ and theinverse of the constant ( C a ) that appears in the potential for this channel. We see already fromthis figure that Λ and 1 /C a are strongly correlated, and that the correlation is of the formgiven in Eq. (C.2). Indeed, the dashed line in the plot, that lies close to the axis of the errorellipse, is Eq. (C.2) using the central values of the cutoff and the potential given in the originalwork of Ref. [35], Λ = 1000 MeV and C a = − .
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