On the precise measurement of the X(3872) mass and its counting rate
OOn the precise measurement of the X ( ) mass and its counting rate ∗ Pablo G. Ortega
1, † and Enrique Ruiz Arriola
2, ‡ Departamento de F´ısica Fundamental andInstituto Universitario de F´ısica Fundamental y Matem´aticas (IUFFyM),Universidad de Salamanca, E-37008 Salamanca, Spain Departamento de F´ısica At´omica, Molecular y Nuclearand Instituto Carlos I de F´ısica Te´orica y ComputacionalUniversidad de Granada, E-18071 Granada, Spain. (Dated: July 24, 2020)The lineshapes of specific production experiments of the exotic state such as X ( ) with J PC = ++ quantum numbers involving triangle singularities have been found to become highly sensitive to the bindingenergy of weakly bound states, thus offering in principle the opportunity of benchmark determinations. Wecritically analyze recent proposals to extract accurately and precisely the X ( ) mass, which overlook animportant physical effect by regarding their corresponding production lineshapes as a sharp mass distributionand, thus, neglecting the influence of initial nearby continuum states in the 1 ++ channel. The inclusion ofthese states implies an effective cancellation mechanism which operates at the current and finite experimentalresolution of the detectors so that one cannot distinguish between the 1 ++ bound-state and nearby D ¯ D ∗ continuum states with the same quantum numbers. In particular, we show that the lineshape for resolutions above1 MeV becomes rather insensitive to the binding energy unless high statistics is considered. The cancellationalso provides a natural explanation for a recent study reporting missing but unknown decay channels in anabsolute branching ratio global analysis of the X ( ) . PACS numbers: 12.39.Pn, 14.40.Lb, 14.40.RtKeywords: Triangle singularities, Charmed mesons, Exotic states
I. INTRODUCTION
The quest for the hadronic spectrum has been a majorgoal in particle physics over the last 70 years, which hasbeen marked by predicting and reporting the observed statesand their properties in the PDG (see e.g. [1] for the latestedition upcoming). Before 2003, this task has mostly beenphenomenologically supported by a non-relativistic quarkmodel pattern and its given symmetry multiplets suggestedby the underlying q ¯ q and qqq composition for mesons andbaryons, respectively. This non-rigorous but effective linkhas been a quite useful and extremely relevant guidance,particularly because, currently, it is theoretically unknownhow many states should occur below a given maximalenergy or if the full set of recorded states are incomplete orredundant [2]. In fact, as it is most often the case for hadronicresonances, we do not detect directly the reported particlethrough its track but only in terms of its decaying products sothat the corresponding invariant mass distribution is observedinstead and the relevant signal is singled out from the reactionbackground within a given energy resolution.Since 2003, the situation has become more involved ∗ This work is partly supported by the Spanish Ministerio de Econom´ıay Competitividad and European ERDF funds (FPA2016-77177-C2-2-P,FIS2017-85053-C2-1-P), Junta de Andaluc´ıa (FQM-225) and by the EUSTRONG-2020 project under the program H2020-INFRAIA-2018-1, grantagreement no.824093. † [email protected] ‡ [email protected] above charm production threshold after the discovery ofthe X ( ) [3–6] and the wealth of new X , Y , Z stateswhose properties suggest more complicated structures thanthose originally envisaged from the quark-model [7–9]. Inthis study, we analyze the renowned X ( ) state and theinfluence of the mass distribution in the 1 ++ channel onthe determination of its mass. The X ( ) is allegedly a¯ DD ∗ weakly bound state, whose binding energy has becomesmaller since its discovery. The most recent value for itsbinding energy, measured by LHCb, is 0.07(12) MeV [10], sothat at present it is unclear whether its mass is slightly aboveor below the ¯ DD ∗ threshold. However, one might wonderwhat would happen if the X(3872) would not be a boundstate. Recently several proposals invoke the strong sensitivityof lineshapes for productions processes involving trianglesingularities to benchmark the mass determination [11, 12].In this paper, we promote the idea that the precise valueof the mass is actually not crucial, since the contribution ofnearby states with the same quantum numbers is unavoidablewith the current experimental energy resolution detectingits decaying products, and a cancellation mechanism putforward initially by Dashen and Kane [13] is at work inthis particular case. We have found in previous works thatthis has implications to count X ( ) degrees of freedomat finite temperatures of relevance in relativistic heavy ionscollisions [14, 15] and ultrahigh energies pp prompt X ( ) production at finite p T and mid-rapidity [16]. We will alsoshow how the number of reconstructed states representing thebound X ( ) is smaller than the truly produced ones due toa cancellation mechanism which will be explained below andwhich provides a natural understanding of the missing decay a r X i v : . [ h e p - ph ] J u l channels. A brief account and overview of the present studyhas already been advanced in conference proceedings [17].The paper is organized as follows: In section II wereview the hadronic density of states and its theoretical andexperimental limitations as it will be a key element of ouranalysis. In section III we review the XYZ states to provide abroader perspective around the very special X ( ) exoticstate. In section IV we approach the determination of thedensity of states in the 1 ++ channel. Our main numericalresults are discussed in section V. Finally, our conclusions arepresented in Section VI. II. HADRONIC DENSITY OF STATESA. General properties
For completeness, in this section we review some basicaspects of the hadronic density of states following somehistorical timeline, in a way that our points can be more easilypresented and with the purpose of fixing the notation. Thefirst quantum-mechanical attempt to determine the density ofstates within the quantum virial expansion was pioneered byBeth and Uhlenbeck in 1937, who computed the second virialcoefficient as a function of temperature in terms of the two-body scattering phase shifts [18]. Only after 30 years, Dashen,Ma and Bernstein provided, in a seminal work, the link to thefull S-matrix [19] which opened up the basis for the HadronResonance Gas (HRG) model for resonances [20], as wellas the notion of effective elementarity [21]. Based on thesedevelopments, Dashen and Kane promoted the natural ideaof counting hadronic states at a typical hadronic scale. Interms of the corresponding density of states as a function ofthe invariant CM energy √ s [13], we have ρ ( M ) = Tr δ ( M − H CM ) = ∑ n δ ( M − M n ) (1)where H CM is the intrinsic Hamiltonian. Unfortunately,while this is mathematically a well defined quantity, ρ ( M ) cannot, in most cases, be computed or measured directly, butonly through its coupling to external probes generating theproduction process. This effectively correspond to multiplyby an observable O ( M ) . Another possibility is the couplingto a thermal heat bath where we take this observable to be auniversal Boltzmann factor e − M / T . B. The two-body case
The level density can be splitted into separate contributionsaccording to the corresponding good quantum numbers. Inthe particular 2 → M th is given as [24, 25] (for recent discussions see e.g. [26, 27]) ∆ N ( M ) ≡ N ( M ) − N ( M )= ∑ n θ ( M − M Bn ) + π n ∑ α = [ δ α ( M ) − δ α ( M th )] . (2)Here, we have separated bound states M Bn explicitly fromscattering states written in terms of the eigenvalues of theS-matrix, i.e. S = U Diag ( δ , . . . , δ n ) U † , with U a unitarytransformation for n-coupled channels. This definition fulfills N ( ) =
0. In the single channel case, and in the limit of highmasses M → ∞ one gets N ( ∞ ) = n B + π [ δ ( ∞ ) − δ ( M th )] = does not count. C. Theoretical binning
From a purely theoretical side, a practical and numericalevaluation requires binning the spectrum with a given finiteinvariant mass resolution ∆ m , in which case only an averagedor coarse-grained value such as¯ ρ ( M ) = ∆ m (cid:90) M + ∆ m / M − ∆ m / ρ ( m ) dm (3)is obtained. On the theoretical side, a practical way ofimplementing this is by placing the system into a box ofvolume V , as it is the case in lattice QCD where one roughlyhas ∆ m ∼ V − / . This finite mass resolution effectivelycorresponds to a coarse graining in mass and should not haveany sizable effect on the result, unless the true density of statespresents large fluctuations on a smaller mass scale. With thisviewpoint in mind, Dashen and Kane made the distinctionbetween the original SU ( ) multiplets and “accidental” states,i.e. those states which do not contribute when ∆ m issufficiently large (presumably about the typical symmetrybreaking multiplet splitting). D. Experimental resolution
On the experimental side, the coarse-graining procedurecorresponds to the finite energy resolution of the detectors,typically σ = − σ -broadening, R σ ( m , M ) = √ πσ e − ( m − M ) σ (4)so, we have [29]¯ ρ σ ( M ) = (cid:90) ∞ − ∞ R σ ( m , M ) ρ ( m ) dm (5)The binning procedure implied by Eq. (3) may be addedafterwards. Although it is innocuous for ∆ m ≤ σ , it can havea sizable effect for ∆ m > σ . E. The Dashen-Kane cancellation
The immediate consequence of the particular phase shiftbehavior follows from Eq. 2 at the density of states level,defined as ρ ( M ) = d ∆ N ( M ) dM = ∑ n δ ( M − M Bn ) + π n ∑ α = δ (cid:48) α ( M ) . (6)Assuming an experimental resolution R σ ( m , M ) , the corre-sponding measured quantity for an observable depending onthe invariant mass function O ( M ) is O meas ( M ) = (cid:90) ∞ − ∞ O ( m ) R σ ( m , M ) ρ ( m ) dm . (7)Then, for a bin in the range ( M − ∆ m / , m + ∆ m / ) , itbecomes O meas ≡ ∆ m (cid:90) M + ∆ m M − ∆ m O meas ( M (cid:48) ) dM (cid:48) . (8)In the single channel case, with phase shift δ ( M ) , one has O meas = R ( M B ) O ( M B ) + π (cid:90) ∞ − ∞ R ( m ) O ( m ) δ (cid:48) ( m ) dm , (9)with R ( m ) = ∆ m (cid:104) Erf (cid:16) m − M B + ∆ m / √ σ (cid:17) + Erf (cid:16) M B − m + ∆ m / √ σ (cid:17)(cid:105) .Which, for a decreasing phase-shift and for a smooth observ-able O ( M ) , points to a cancellation whose precise amount de-pends on the corresponding slope above threshold. F. The deuteron state and the np continuum
A prominent example suggested by Dashen and Kanecorresponds to the deuteron, which is a neutron-proton 1 ++ state weakly bound by B d = . (cid:28) m p + m n ∼ a fortiori whose verificationdepends on details of low energy scattering. We point out thatthe cancellation observed in the equation of state for nuclearmatter at low temperatures where one has a superposition ofstates weighted by a Boltzmann factor [32] corresponds to asuppression of the occupation number in the 1 ++ channel ascompared to the deuteron case, N ++ ≤ N d .The case of the deuteron described above is particularlyinteresting for us here since it is extremely similar to the case of the X ( ) , with the important exception of thedetection method of both states, as will be discussed below. Inour previous work [14] we have shown how this cancellationcan likewise be triggered at finite temperature T for the X ( ) , as it is the case in relativistic heavy ion collisions,since the partition function involves the folding of theBoltzmann factor, ∼ e − √ p + m / T with the density of states,Eq. 6. Therefore, given these suggestive similarities, we haveundertaken a comparative study of the deuteron and X ( ) production rates in pp scattering at ultra-high energies ( ∼ p T distributions in colliders, whichprovides a suitable calibration tool in order to see the effects ofthe cancellation due to the finite resolution ∆ m of the detectorssignaling the X ( ) state and deciding on its bound statecharacter [16]. III. THE XYZ STATES
Nowadays, there is a strong theoretical and experi-mental evidence on the existence of loosely bound statesnear the charm threshold, originally predicted by Nussi-nov and Sidhu [33], as it seems to be confirmed now bythe wealth of evidence on the existence of the X ( ) ,re-named χ c ( ) , state with binding energy B X = . ( ) MeV [34], or 0 . ( ) MeV from recent LHCb mea-surements [10], and which has triggered a revolution by theproliferation of the so-called X,Y,Z states (for reviews seee.g. [8, 9, 35]. In the absence of electroweak interactions, thisstate has the smallest known hadronic binding energy and, fora loosely bound state, many properties are mainly determinedby its binding energy [8] since most of the time the system isoutside the range of the interaction.In fact, the molecular interpretation has attracted consider-able attention, but since this state is unstable against J / ψρ and J / ψω decays, the detection of X ( ) relies on its de-cay channels spectra where the mass resolution never exceeds ∆ m ∼ X ( ) or, equivalently, its binding energy ∆ B X (cid:28) ∆ m with such a precision, since we cannot distinguish sharply theinitial state. While in most studies (see however [37]) thebound state nature is assumed rather than deduced, even ifthe X ( ) was slightly unbound the correlations would beindistinguishable in the short distance behavior of the D ¯ D ∗ wave function.The discussion on X ( ) lineshapes started in Ref. [38]as a way to extract information on the binding. Trianglesingularities are ubiquitous in weakly bound hadronic andnuclear systems [39] and arise when three particles in aFeynman diagram can simultaneously be on the mass shell.Their relevance in XYZ states has been pointed out [40] andtheir relation to unitarity has been emphasized [41, 42]. Infact, they have been put forward recently as a method tosensitively determine the X mass based on the theoretical lineshape. The fall-off of the lineshape above the peak, rather thanthe actual position of the peak reflects rather well the bindingenergy [11, 12, 43].F.-K. Guo has considered the effect of a short distancesource (the specific process has not been specified) whichgenerates a D ∗ ¯ D ∗ pair in a relative S-wave and whicheventually evolves into a X ( ) + γ final state [11]. Thisproduction mechanism is enhanced by the D ∗ ¯ D ∗ → γ D + ¯ D ∗ → γ + X ( ) one loop triangle singularities producinga narrow peak at about the D ∗ ¯ D ∗ threshold. E. Braaten, L.-P.He and K. Ingles have proposed a similar triangle singularityenhancement for the production of X(3872) and a photonusing e + e − annihilation as the source of a D ∗ ¯ D ∗ pair in arelative P-wave, which becomes possible because of its 1 ++ quantum numbers [12]. Further related analysis on this regardmay be found in Ref. [43, 44].However, these methods focusing on the X ( ) produc-tion lack one important circumstance operating due to the fi-nite resolution of the detectors, since they assume a pure ini-tial mass state (mostly the bound state mass M X ). In reality,any nearby initial states with the same ++ quantum numberswill produce a signal in the final state due to the finite res-olution in the final state. We have reported recently on theneat and accurate cancellation between the would-be X(3872)bound state and the D ¯ D ∗ continuum in the initial state whichhas a sizable impact on the final density of states and blurs thedetected signal [14, 15]. In this work, we will extend thoseworks to analyze the implications on the allegedly accuratemass determinations.The similarities between d and X ( ) already noted inRefs. [45–47] have been corroborated on a quantitative levelin our recent work [16], were we have pointed out that theyare also applicable from the point of view of production ataccelerators [16]. However, a crucial and relevant differencefor the present work is that while the deuteron is detected directly by analyzing its track and/or stopping power leavinga well-defined trace, the X ( ) is inferred from its decayproperties, mainly through the J / ψρ and J / ψω channels. IV. LEVEL DENSITY IN THE X ( ) CHANNELA. Coupled channel scattering
In order to implement the formula given by Eq. (2), wemake some digression on the D ¯ D ∗ scattering states in the 1 ++ ,which actually resembles closely the same channel for thedeuteron. However, while the partial wave analysis of NNscattering data and the determination of the correspondingphase-shifts is a well-known subject, mainly due to theabundance of data [48], we remind that a similar analysis inthe D ¯ D ∗ case is, at present, in its infancy and thus our firstanalysis in Ref. [14] has been based on a quark-model. In the1 ++ channel, the presence of tensor force implies a couplingbetween the S and D channels, so that the S-matrix is given by S J = (cid:18) cos ε j − sin ε j sin ε j cos ε j (cid:19) (cid:32) e δ jj − e δ jj + (cid:33) × (cid:18) cos ε j − sin ε j sin ε j cos ε j (cid:19) . (10)From here we define the T-matrix S JS = − ikT JS , (11)The S and D eigen phase-shifts have been shown in ourprevious work [14] using the quark cluster model of Ref. [49,50] which includes both a c ¯ c and D ¯ D ∗ channels. Thecumulative number is shown in Fig. 1. The outstandingfeature is the turnover of the function as soon as a slightlynon-vanishing c ¯ c content in the X ( ) is included, unlikethe purely molecular picture (see Ref. [14] for a more detaileddiscussion). We also compute the cumulative number forthe coupled-channels EFT model of Ref. [51] fine-tuning theparameters to agree at low energies with the quark model. Inboth cases the fitting parameters have been binding propertiesof the X ( ) . As we see, results present a rather similarpattern over the entire plotted energy range; the sharp riseof the cumulative number is followed by a strong decreasegenerated by the phase-shift. Moreover, we have checkedthat the S-wave phase-shift asymptotically approaches π (due to the bound X(3940)-state of the purely confinedchannel [50] which becomes a resonance when coupled to the D ¯ D ∗ continuum) and hence N ( ∞ ) = M DD * [MeV]0.00.20.40.60.81.0 N ( M ) FIG. 1. Comparison between the cumulative number of the 1 ++ sector with E b =
180 keV in different models: The coupled-channelsEFT model of Ref. [51] with d = . / , C = −
976 fm and m ( ) c ¯ c = .
44 MeV (blue); the coupled-channels CQM model ofRef. [49] with m ( ) c ¯ c = .
44 MeV and γ P = .
194 (dashed red)and the Effective Range Approximation (ERA) model with r = a s = √ µ E b = .
58 fm (dash-dot green).
B. Effective range approximation
However, as we will see, the S-D waves mixing stemmingfrom the tensor force has an influence for larger energies thanthose considered here [14]. Therefore, in order to illustratehow the cancellation comes about, we also considered asimple model which works fairly accurately for both thedeuteron and the X ( ) by just considering a contact(Gaussian) interaction [52] in the S -channel and usingeffective range parameters to determine the correspondingphase-shift in the d and X ( ) [14, 53] respectively. Theresult for N ( M ) together with the EFT and CQM predictionscan be seen in Fig. 1. Of course, if the binding energyis not that small, several effects appear and, in particular,the composite nature of the X ( ) becomes manifest (seee.g. [49]). All these similarities suggests the possibility ofusing the shape-independent Effective Range Approximation(ERA) to second order to calculate the phaseshifts nearthreshold. In ERA, we have that the δ is given as a functionof two parameters: k cot δ = − a s + r k (12)where k is the CM momentum k = (cid:112) µ ( M − M ) (13)where µ = M D M D + / ( M D + M D ∗ ) is the reduced mass and M = M D + M D ∗ is the threshold mass. The comparison inFig. 1 the between ERA and the two coupled-channels modelsreassures the validity of the approximation for the range √ s (cid:46) f ( k ) − = k cot δ − ik (14)and, in general, bound and virtual states correspond to polesof f ( k ) at k = ± i γ X in the first and second Riemann sheetin energy E b = M X − M respectively. It is worth mentioningthat Kang and Oller have comprehensively studied the polestructure and analyzed the character of the X ( ) in termsof bound and virtual states within simple analytical parame-terizations [37], although the Dashen-Kane cancellation wasnot addressed. C. Finite energy resolution
The detector response function transforms the monochro-matic signal of mass M X in a Gaussian distribution R σ ( M X , m ) with σ resolution [29]. It reflects the imperfection of the de-tector to measure a single energy due to the Poisson statisticsof the energy deposition. The energy window ∆ m is inter-preted as the energy range where the final channel productsare selected as decay products of the X ( ) (and, thus, re-constructed). Usually they are taken as ± ( − ) σ , to takemost of the Gaussian distribution.The experiments measure such Gaussian distributions, fromwherethe typical resolution σ can be extracted. For example, M DD * [MeV]-0.5-0.2500.250.5 a . u = 0.0 MeV = 0.5 MeV = 1.0 MeV = 2.0 MeV = 6.0 MeV3860 3865 3870 3875 3880 M DD * [MeV]-0.5-0.2500.250.5 a . u = 0.0 MeV = 0.5 MeV = 1.0 MeV = 2.0 MeV = 6.0 MeV FIG. 2. Upper: Smeared density of states for E b =
180 keV fordifferent resolutions. Lower: Same for E b = −
180 keV (virtual).Channel σ ∆ m Reference J / ψπ + π − . ± .
07 20 Ref. [54] J / ψπ + π − . ± .
08 6 σ ≈
20 Ref. [55] J / ψπ + π − .
2? 18 Ref. [5]TABLE I. Energy resolutions in several experiments detecting X ( ) decays. in Ref. [54](page 4) the authors claim a resolution of σ ≈ ψ ( ) , and J / ψππ events between 3 .
86 and 3 .
88 GeV are selected, thus,employing an energy window of 20 MeV. The situation issummarized in Table I.According to Table I the finest value for the resolution σ is around σ = ∆ m = D. Smearing of the density of states
According to the general expression, Eq. 6, and neglectingthe inessential S-D wave mixing at low energies, the densityof states in the 1 ++ channel for the bound X case is given by ρ ( m ) = δ ( m − M X ) + π δ (cid:48) ( m ) . (15)where the S-wave phase-shift as a function of the invariantmass vanishes below the D ¯ D ∗ threshold. For the unboundcase, the bound state contribution δ ( m − M X ) is simplydropped out. Note that from Fig. 1 the phase-shift atlow energies is a decreasing function, so its derivativebecomes negative which is the essence of the Dashen-Kanecancellation. If the mass of X is not correctly reconstructed,because we have a finite resolution in our detector, given bythe response function R σ ( m , M ) , we will measure real DD ∗ pairs from the decay of the X and DD ∗ from the continuum,so that we cannot distinguish them due to the finite detectorresolution. Thus, we have to fold the detector responsefunction and the density of states as done in Eq. (5) appliedto the X ( ) case¯ ρ σ ( M ) = Θ R σ ( M X , M ) + π (cid:90) ∞ M DD ∗ R σ ( m , M ) δ (cid:48) ( m ) dm (16)being M DD ∗ the DD ∗ threshold mass and Θ ≡ Θ ( M DD ∗ − M X ) the Heaviside function. We show in Fig. 2 the smear of thedensity of states for E b =
180 keV (bound) and E b = − σ = − E b (cid:29) σ the finite resolution does not modify thelineshape and effectively corresponds to σ → σ the cancellation becomes rather evident and is moreeffective for larger resolutions σ (cid:29) | E b | where the differencebetween a bound and a virtual state becomes small. E. Missing decays vs missing counts
According to a recent work, there are a number (abouta third) of unknown decays when absolute branching ratiosare considered and compared to the total width of the X ( ) [56] (see also [57] for an experimental upgrade)suggesting new experiments to detect these missing decays.The statistical analysis carried out by the authors of Ref. [56]provides large error bars for the branching Br ( X ( ) → unknown ) = − ∑ i Γ i / Γ = . + . − . % from the analysis of8 detected channels (see their table II). Actually, about halfof the decays goes into D ¯ D ∗ pairs. We note here that thequenching effect we unveil here may be behind such missingdecays, since quite generally and due to the Dashen-Kanecancellation the counted signals are suppressed against theoriginal ones, N ++ < N X ( ) . This undercounting is incomplete agreement with our previous study [14, 15] onoccupation numbers at finite temperature and of relevance in X ( ) in heavy ion-collisions. It also complies with thesimilarities of production rates at finite p T of deuterons and X ( ) states in pp collisions at ultrahigh energies in themid-rapidity region [16] which provides, after correcting theeffect to a one-to-one production rate, N X / N d ∼ V. SMEARING OF LINESHAPES
As we have discussed above, the finite detector resolutiondoes not separate between the signals triggered by a bound X ( ) and D ¯ D ∗ pairs in the 1 ++ nearby continuum. Thisfact in itself should not necessarily be a cause of concern if the level density was a smooth function within the finiteresolution σ . However, we have seen that this is not what happens in the 1 ++ channel; a relevant variation withpositive and negative contributions does take place. This,of course, sets the problem on how would it be possible todeduce accurately the mass of the X ( ) state given theselimitations on resolution and being aware of the cancellationeffect.In general, the direct determination of the mass wouldrequire more precision on the mass of the constituents (i.e., D and D ∗ mass assuming a molecular nature) and a largeacquisition of statistics, considering the small value of the X binding energy. An alternative, and more interesting method,is the characterization of production processes in terms of asuitable mass operator O ( M ) , sensitive to small variationsof the binding energy. Recently, two methods involvingtriangle singularities near the D ∗ ¯ D ∗ threshold have beenproposed [11, 12]. Those kinematic singularities, whichare formed when the three particles composing the triangleare simultaneously on-shell, have been suggested to providea more accurate method to determine the X ( ) bindingenergy than direct mass measurements.In order to illustrate the aforementioned limitations due tothe resolution and the cancellation effect, let’s consider now ageneral lineshape L ( s , M ) , where s is the invariant mass and M is the reconstructed mass of the secondary X particle from theGaussian distribution R σ ( m , M ) . The convoluted lineshapefrom the X particle with mass M is (Eq. 9)¯ L ( s ) = Θ R ( M X ) L ( s , M X ) + π (cid:90) ∞ M DD ∗ R ( m ) L ( s , m ) δ (cid:48) ( m ) dm (17)with R ( m ) = ∆ m (cid:104) Erf (cid:16) m − M X + ∆ m / √ σ (cid:17) + Erf (cid:16) M X − m + ∆ m / √ σ (cid:17)(cid:105) .We analyze the effect of smearing for the lineshapesgenerated in the X ( ) γ production process using either arelative S-wave [11] or P-wave [12] source of a D ∗ ¯ D ∗ pair.Results for the S-wave source of Ref. [11] can be see in Fig. 3and results for the P-wave source of Ref. [12] are shown inFig. 4, without considering a finite binning in the γ X invariantmass spectrum. For the S-wave source, normalized to the D ∗ ¯ D ∗ threshold, we appreciate a change in the shape ofthe distribution, which pretty much blurs the neat distinctiondue to the X binding energy. Still, we see a separation of thelineshape tails which could be used for the latter purposes.The cancellation and the finite resolution, thus, leads to amore complicated precise measurement of the X ( ) mass,specially when finite statistics are considered (see discussionbelow). For the P-wave source, the main effect is the absolutevalue decreases of the lineshapes, depending on their bindingenergy due to the cancellation (effect that also occurs for the M X [MeV]0.250.500.751.001.251.501.752.002.25 a . u M X [MeV]0.20.40.60.81.01.21.41.6 a . u FIG. 3. Smeared lineshapes of states, ¯ L ( s ) , for σ = σ = ∆ m =
20 MeV.
S-wave source but it not appreciated due to the normalizationof the lineshapes).It is interesting to analyze the S-wave source results fromthe counting statistics point of view. We expect that aconvergence of all γ X lineshapes regardless of the X bindingenergy. Their tails decrease at different rates, but a limitedstatistics can compromise their proper identification. Quitegenerally we will be able to discern two different (smeared)signals if the number of events fulfills ∆ OO ∼ √ N (18)In Fig. 5 we show an example of limited resolution forbinding energies E b =
180 keV and E b = −
180 keV (virtual),a σ = E win = N = M X [MeV]0.0050.0100.0150.0200.0250.0300.0350.0400.045 ( pb ) M X [MeV]0.0050.0100.0150.0200.025 ( pb ) FIG. 4. Smeared lineshapes of states, ¯ L ( s ) , for σ = σ = ∆ m =
20 MeV. of Fig. 3 are not known, the same occurs to the globalnormalization of the synthetic data, so caution should be takenwhen direct comparing between the lineshapes for differentbinding energies. Of course, for larger values of σ all curvesresemble each other and the strong mass dependence is largelywashed out. We believe these effects should be consideredin an eventual benchmark experimental determination of the X ( ) mass. VI. CONCLUSIONS
In this paper, we have analyzed the impact of finitedetector resolution in the production and decay of the X ( ) state. We have discussed the cancellation effect due to thesuperposition in the level density in the 1 ++ channel of boundstate and nearby D ¯ D ∗ continuum states in the initial state,which cannot be separated in the final state when the bindingenergy is much smaller than the energy resolution. Our resultssuggest that the mechanism of production of weakly bound M X [MeV]05101520 E v e n t s / ( M e V ) M X [MeV]255075100125150175200 E v e n t s / ( M e V ) FIG. 5. Binned smeared lineshapes of the S-wave source for N =
100 events (top) and N = E b =
180 keV (black) and E b = −
180 keV (blue) binding energies,using a σ = ∆ m =
20 MeV. The full lineshape is shown for comparison, with theproper normalization. states such as the X ( ) undercounts the number of states N ++ < N X ( ) , an effect which is in harmony with themissing resonances reported in a recent absolute branchingratio analysis. This signal suppression is in completeagreement with our previous study on occupation numbersat finite temperature and of relevance in X ( ) in heavyion-collisions. It also complies with the deuteron to X(3872)finite p T production ratio in pp collisions at ultrahigh energiesat mid-rapidity. Our findings are also relevant to futurebenchmark determinations of the X ( ) , particularly thosedisplayed by the strong lineshape dependence in productionprocesses involving triangle singularities. Quite generallywe find that the initial density of states triggering a signalof X -production in a finite resolution energy detector blursthe spectrum and hence the strong mass dependence isreduced and could only be pinned down with sufficientlyhigh statistics. We expect our observations to hold in similarweakly bound states not directly measured through their track,but inferred from their decay products. [1] P.Z. et al. (Particle Data Group), Prog. Theor. Exp. Phys. (2020)083C01.[2] E. Ruiz Arriola et al., Excited Hyperons in QCD Thermody-namics at Freeze-Out, pp. 128–139, 2016, 1612.07091.[3] Belle, S.K. Choi et al., Phys. Rev. Lett. 91 (2003) 262001, hep-ex/0309032.[4] BaBar, B. Aubert et al., Phys. Rev. D71 (2005) 031501, hep-ex/0412051.[5] Belle, S.K. Choi et al., Phys. Rev. D 84 (2011) 052004,1107.0163.[6] LHCb, R. Aaij et al., Phys. Rev. Lett. 110 (2013) 222001,1302.6269.[7] S. Godfrey and S.L. Olsen, Ann. Rev. Nucl. Part. Sci. 58 (2008)51, 0801.3867. [8] F.K. Guo et al., Rev. Mod. Phys. 90 (2018) 015004,1705.00141.[9] N. Brambilla et al., (2019), 1907.07583.[10] LHCb, R. Aaij et al., (2020), 2005.13422.[11] F.K. Guo, Phys. Rev. Lett. 122 (2019) 202002, 1902.11221.[12] E. Braaten, L.P. He and K. Ingles, Phys. Rev. D100 (2019)031501, 1904.12915.[13] R.F. Dashen and G.L. Kane, Phys. Rev. D11 (1975) 136.[14] P.G. Ortega et al., Phys. Lett. B781 (2018) 678, 1707.01915.[15] P.G. Ortega and E. Ruiz Arriola, PoS Hadron2017 (2018) 236,1711.10193.[16] P.G. Ortega and E. Ruiz Arriola, Chinese Physics C 43 (2019)124107, 1907.01441.[17] E. Ruiz Arriola and P.G. Ortega, 12th International WinterWorkshop ”Excited QCD” 2020, 2020, 2005.01531.[1] P.Z. et al. (Particle Data Group), Prog. Theor. Exp. Phys. (2020)083C01.[2] E. Ruiz Arriola et al., Excited Hyperons in QCD Thermody-namics at Freeze-Out, pp. 128–139, 2016, 1612.07091.[3] Belle, S.K. Choi et al., Phys. Rev. Lett. 91 (2003) 262001, hep-ex/0309032.[4] BaBar, B. Aubert et al., Phys. Rev. D71 (2005) 031501, hep-ex/0412051.[5] Belle, S.K. Choi et al., Phys. Rev. D 84 (2011) 052004,1107.0163.[6] LHCb, R. Aaij et al., Phys. Rev. Lett. 110 (2013) 222001,1302.6269.[7] S. Godfrey and S.L. Olsen, Ann. Rev. Nucl. Part. Sci. 58 (2008)51, 0801.3867. [8] F.K. Guo et al., Rev. Mod. Phys. 90 (2018) 015004,1705.00141.[9] N. Brambilla et al., (2019), 1907.07583.[10] LHCb, R. Aaij et al., (2020), 2005.13422.[11] F.K. Guo, Phys. Rev. Lett. 122 (2019) 202002, 1902.11221.[12] E. Braaten, L.P. He and K. Ingles, Phys. Rev. D100 (2019)031501, 1904.12915.[13] R.F. Dashen and G.L. Kane, Phys. Rev. D11 (1975) 136.[14] P.G. Ortega et al., Phys. Lett. B781 (2018) 678, 1707.01915.[15] P.G. Ortega and E. Ruiz Arriola, PoS Hadron2017 (2018) 236,1711.10193.[16] P.G. Ortega and E. Ruiz Arriola, Chinese Physics C 43 (2019)124107, 1907.01441.[17] E. Ruiz Arriola and P.G. Ortega, 12th International WinterWorkshop ”Excited QCD” 2020, 2020, 2005.01531.