Where does the X(5568) structure come from?

We study the semi-exclusive production of $\pi^\pm B_s^0$ pairs in hadron colliders which is associated with the $X(5568)$ structure observed by the D0 Collaboration in 2016, but that was not confirmed by LHCb and CMS later. The reason of its appearance in the D0 and absence in LHCb and CMS is discussed in this letter. In a semi-exclusive process, one might miss the third particle which is produced together with the $\pi^\pm B_s^0$ simultaneously. In the three-body Dalitz plot, once the remaining region is narrow enough after the kinematic cuts, its reflection to another invariant mass distribution will accumulate a large number of events within a specific energy region. If there is an enhancement in the remaining region, it will make the reflection structure more pronounced. The precise line shape of the reflection will depend on the specific interaction form. A combined study of different cone cuts and the low-energy dynamics, e.g. the Landau singularity, demonstrates that the $X(5568)$ structure could come from this kinematic reflection. This conclusion can be checked by both searching for the enhancement in another invariant mass distribution, such as $B_s^0\bar{B}^0$, and the cone cut dependence of the $X(5568)$ mass. Such a combined study can be used to distinguish the effects of the triangle singularity from a genuine state. We also propose how to avoid this kinematic reflection in future experimental analysis.

We study the semi-exclusive production of π ± B 0 s pairs in hadron colliders which is associated with the X(5568) structure observed by the D0 Collaboration in 2016, but that was not confirmed by LHCb and CMS later. The reason of its appearance in the D0 and absence in LHCb and CMS is discussed in this letter. In a semi-exclusive process, one might miss the third particle which is produced together with the π ± B 0 s simultaneously. In the three-body Dalitz plot, once the remaining region is narrow enough after the kinematic cuts, its reflection to another invariant mass distribution will accumulate a large number of events within a specific energy region. If there is an enhancement in the remaining region, it will make the reflection structure more pronounced. A combined study of different cone cuts and the low-energy dynamics, e.g. the Landau singularity, demonstrates that the X(5568) structure could come from this kinematic reflection. This conclusion can be checked by both searching for the enhancement in another invariant mass distribution, such as B 0 sB 0 , and the cone cut dependence of the X(5568) mass. Such a combined study can be used to distinguish the effects of the triangle singularity from a genuine state. We also propose how to avoid this kinematic reflection in future experimental analysis. The D0 Collaboration reported a narrow structure [1], called X(5568), in the π ± B 0 s invariant mass distribution based on 10.4 fb −1 pp of data at √ s = 1.96 TeV in February of 2016. To suppress the background, the transverse momentum p T of the π ± B 0 s system was required to be larger than 10 GeV. Another cone cut ∆R ≡ ∆η 2 + ∆φ 2 < 0.3 between the B 0 s and π ± , with η the pseudorapidity and φ the azimuthal angle, is used to further reduce the background. With these two cuts, a narrow structure is reported as a first new exotic state consisting four different valence quarks with mass 5567.8 ± 2.9 MeV and width 21.9 ± 6.4 MeV.
Since the X(5568) is hundreds of MeV below the B ( * )K threshold and is observed in the B s π channel, it could strongly couple to these two channels. One interpretation is that the X(5568) could be a hadronic molecule [24,25] as an analogue of the DK hadronic molecule D * s0 (2317). However, such a scenario was questioned by the authors in Refs. [26,27], as the difference between the mass of X(5568) and the BK threshold is too large and it is not easy to form such a deeply bound state. The difference is even larger than that between the mass of the D * s0 (2317) and the DK threshold. It contradicts to the expectation that the hyperfine splitting in the bottom sector should be smaller than that in the charm sector. Furthermore, detailed calculations using the chiral unitary approach and lattice simulations [28][29][30] confirmed the inconsistency of both the X(5568) and the D * s0 (2317) as hadronic molecules. Even after enlarging the channel basis to the B s π, B * s π, BK and B * K [31] channels, the calculation still disfavors the X(5568) to be a hadronic molecule. The near-threshold behavior also indicates that the structure might come from the triangle singularity in the meson loop as discussed in Ref. [32]. However, as the rescattering vertices are suppressed by a disconnected quark line, this mechanism is also questioned in Refs. [26,27].
An alternative opinion [26,27] is that all these interpretations, such as tetraquark, hadronic molecule, threshold effect from the meson loop, and so on, cannot give a consistent explanation of the X(5568) structure. Due to the inconsistency of the interpretations in both the tetraquark and hadronic molecular scenarios, the authors of Ref. [33,34] claim that the state might originate from a mixing of these two scenarios.
The analyses from both LHCb [35] and CMS [36] do not confirm the existence of the X(5568) structure and set an upper limit on the production rate of the X(5568) state in pp collisions with 3 fb −1 and 19.7 fb −1 of data, respectively. In the analysis of LHCb, they only impose the requirement of p T (B 0 s ) being greater than 5 GeV, arXiv:1609.08807v1 [hep-ph] 28 Sep 2016 FIG. 1: The rescattering pp → abc + all process via the intermediate particle "1" and "2". The third particle "3" is the exchanged particle. Particles "a", "b", "c" are the exclusive final states. In our case, particle "a" and "b" are the π + and B 0 s systems with the third particle "c" depending on the intermediate loop.
10 GeV and 15 GeV but smaller than 50 GeV. CMS uses p T (B 0 s ) > 25 GeV and p T (π ± ) > 1GeV cuts as well as a cone cut between the B 0 s and the π ± as D0 does. To illustrate the effect of the cone cut, they performed their analysis with the upper limit of the cone cut at 0.4, 0.3, 0.2 and 0.1, respectively, and claim that the cone cut cannot be used in the analysis since it can stimulate a peak shape and could enhance the significance of statistical fluctuations in the data.
No matter whether the structure exists or not, it has been attracting a lot of attention from both theoretical and experimental sides. In this letter, we explain why the X(5568) structure is only observed by D0 and might have generated some effects from the kinematic reflection in the CMS analysis. If the structure really comes from the dynamics in the π + B 0 s channel, either from a genuine state or a singularity in the π + B 0 s [32] channel, the peak should be always there and stable no matter the cut is implemented or not. Its absence in the analysis of LHCb without cone cut already indicated that it is not from the dynamics in the π + B 0 s channel. We demonstrate that it could be coming from a kinematic reflection. The key point is that, for the scattering from two particles to an n-body final state, there are 3n − 4 independent Mandelstam variables. On the 3n − 4 dimensional surface, once an enhancement in one dimension is cut by the experimental analysis, its projection to another dimension could lead to an accumulation of events within a specific energy region.
In the following, we use the semi-exclusive production of the π + B 0 s associated with the third particle, such asB 0 , as an example to illustrate how the mechanism works. The scattering process of two particles to the n-body final state can be parameterized as a quasi j + 1body process, with j the number of the exclusive particles, if the dynamics only depends on the invariant mass of the other n − j particles within the energy region of interest. Therefore, although the quasi 4-body final state, cf. Fig. 1, is used to illustrate the problem, the conclusion is general, because the other hard process can be viewed as a background contribution.
Since there are u, b quarks andd,s antiquarks in the final π + and B 0 s , the incoming baryon and antibaryon can be either ubd (Λ b ) anddsd (Σ + ), with a π 0 the third undetected particle or ubd (Λ b ) anddsb (Ξ + b ), with aB 0 the third undetected particle. However, the widths of the light antibaryonsΣ + and the exchangedΣ 0 are hundreds of MeV which cannot produce narrow structures even if the condition of the triangle singularity is satisfied. Consequently, only the double heavy baryon loop could give a significant peak structure. In what follows, we only consider theΞ * + b (5955) Λ 0 b (5920) double heavy baryon loop withΞ 0 b as the exchanged particle as an example. The other double heavy baryon loops have a similar behavior such asΞ . The final result is the sum of all the contributions from each double heavy baryon loop.
Usually the structure from the normal Landau singularity is not as pronounced as that from the abnormal one. As the result, the narrow peak structure might come from the abnormal triangle singularity [37][38][39][40][41]. Since the triangle singularity can only be accessed when all the intermediate particles are on-shell and all the subprocesses can happen classically, one can obtain the singularity region in different planes. Fig. 2 shows the singularity region in the M abc − m 3 plane by setting m 1 , m 2 , m a , m b and m c to the masses of theΞ * + b (5955), Λ 0 b (5920), π + , B 0 s andB 0 , respectively, as an illustration. The gray solid, red dotted, blue dot-dashed and green dashed lines are the limits to make sure that "2", "3" can classically scatter to "b", "c", "1" can decay to "3" and "a", "3" can catch up with "2", and the "1" and "2" particles can be produced, respectively. When m 1 and m 2 increase or m 3 decreases, the corresponding singularity region will become larger due to the larger phase space of the intermediate processes. As shown in Fig. 2, when m 3 has the proper mass, the singularity can happen within a specific region for the incoming energy. Since the energy of the semi-exclusive production varies in a large region, the larger the singularity region of the incoming invariant mass M abc is, the more important the loop is. When the incoming M abc is smaller than the threshold M min abc ≡ m 1 + m 2 or larger than the upper limit M max abc , the singularity condition cannot be satisfied. As shown in Fig. 2 , the upper limit is the cross point of m 1 = m 1min 1 1 The lower limit m 1min of m 1 can be found in Ref. [40].

FIG. 2:
The singularity region in the M abc − m3 plane with M abc the invariant mass of the abc three-body system. Here, we set m1 and m2 to the masses of theΞ * + b (5955) and Λ 0 b (5920) for illustration. The gray solid, red dotted, blue dot-dashed and green dashed curves are the constraints from m3 ≥ m b + mc − m2, m3 ≤ m1 − ma, M abc ≥ m1 + m2 and m1 ≥ m1min, respectively. In yellow shaded region the conditions for the triangle singularity are fulfilled. and m 3 = m b + m c − m 2 , which satisfies the equation One can avoid the signal from the triangle singularity and its reflection by using the events outside the energy region [M min abc , M max abc ]. Our discussion in the following will be based on the factorization of the phase space integral of the full process pp → abc+all into two processes, i.e. the pp → M abc +all scattering process and the M abc decay to a, b, c, assuming that there is no interaction between "a", "b", "c" and the other inclusive particles. For a given M abc , the second process in Eq.
(2) only depends on the first one through the implicit integration variable p abc in dσ(pp → M abc + all). We generate the three final particles "a", "b", "c" with a arbitrary three-momentum p abc . This should be large enough to make sure that not all the events with a small cone cut will be cut by the p T (π + B 0 s ) cut. We use the VEGAS program [42] to integrate the kinematic phase space generated by RAMBO [43] and the dynamic three-point loop [44]. The M π + B 0 s distributions with p T (π + B 0 s ) > 10 GeV and the cone cuts, ∆R < 0.4, ∆R < 0.3, ∆R < 0.2, ∆R < 0.1 are shown in becomes smaller, the peak structure will move to lower energy and vice versa. Here we use a large enough |p abc |, e.g. 80 GeV, to get the M π + B 0 s distribution. If |p abc | gets larger, the peak structure will be shifted to higher energy. When it is large enough, all the peaks with different cone cuts will approach a limit, i.e. the peak without cone cut. As the low energy photon is not detected in the prertinent high energy hadron collisions [1], the B 0 s cannot be distinguished from the B * 0 s in the D0 experiment. Once the final state B 0 s is replaced by B * 0 s , the structures in Fig. 3 will be shifted to higher energy and the structure with the cone cut ∆R < 0.3 will reach its maximum value at the mass of X(5568). The invariant mass distributions in Fig. 3 should not be compared with the experimental data from D0, as the latter include background subtractions.
This kind of behavior can be easily understood by the cone cut influence on the Dalitz plot in Fig. 4. Once the cone cut is implemented, some of the events at the lower right-hand-side will be cut off. Even if there is no singularity enhancement in another dimension, the reflection of the narrow upper left corner to the M π + B 0 s invariant mass distribution could also be pronounced, if the cone cut is small enough, e.g. ∆R < 0.1 of Fig.3 in Ref.[36]. The structure becomes narrower with the increasing mass of the third particle and vice versa. When the cone cut becomes smaller, the cut region will move to smaller M π + B 0 s . Therefore, the reflection moves to lower energy. Analogously, the larger |p abc | means that the cut region moves to higher energy. Thus, the corresponding peak position in the M π + B 0 s invariant mass is shifted to higher energy. The larger |p abc | is, the weaker the cone cut dependence of the X(5568) peak structure is. If the structure comes from a genuine state which can decay into π + B 0 s or the singularity in the π + B 0 s channel, the peak position should not depend on the cone cut. When there is a singularity in the M B 0 sB invariant mass distribution as shown in Fig. 4, the structure around 5.568 GeV will be produced.
The cone cut dependence of the mass of X(5568) is similar to what has been observed by the D0 Collaboration, see the supplemental material of Ref. [1]. This is an evidence that the X(5568) is not a genuine state but a kinematic reflection from other invariant mass distributions, such as B 0 sB 0 , due to the third undetected particle which is produced associated with the π + B 0 s . In high energy hadron collision, since the gluon is dominant in the parton distribution functions of both p andp, the dynamics in pp and pp collisions should be similar. However, because the center-of-mass energy of LHCb and CMS is about four times as that of D0, both p abc and M abc distributions of the production of the double heavy baryons are much broader than that of D0. It makes that the signal from the kinematic reflection might be weakened by the large number of events at higher M π + B 0 s . This is the reason why there is no X(5568) structure in the analyses of both LHCb and CMS. However, there are pronounced structures in the CMS analysis which largely depend on the cone cut. It is because that they do not implement a p T (π + B 0 s ) cut at the same time which is also the reason that the structure in CMS is broader than that in D0. In this case, the lowest value of |p abc | in CMS is smaller than that in D0. Thus, the M π + B 0 s invariant mass distribution in CMS is more sensitive to the cone cut.
One might expect that the narrow structure could also come from the reflection of a resonance, such as the M abc → π + Υ(5S) → π + B ( * )0 s π 0 invariant mass distributions, if phase space allows. When M abc increases, the overlap between the resonance and the Dalitz plot (or its cut region) varies smoothly. Therefore, the kinematic reflection from a resonance would not show up as a pronounced structure. As a byproduct, one can distinguish a genuine state from the triangle singularity by looking at the M abc dependence of its reflection, i.e. a drastic change within a small M abc region means that there is a triangle singularity.
In this work, we have demonstrated that: • The X(5568) could be a kinematic reflection from the singularity in another dimension of the Dalitz plot, such as the singularity in the B 0 sB 0 invariant mass distribution. If this is the case, the mass of the X(5568) decreases when the cone cut becomes smaller, which is similar to the observation made by D0.
• Since the larger center-of-mass energy in LHCb and CMS leads to an accumulation of events at higher M π + B 0 s , larger than that of D0, this narrow reflection structure could be diminished.
• Whether the cone cut dependence of the reflection is large or not is determined by the p T cut, i.e. a larger p T cut makes the reflection less sensitive to the cone cut.
Although all our arguments have been obtained from considering the three exclusive particle process and the specific double heavy baryon loopΞ * + b (5955)Λ 0 b (5920)(Ξ b ), the conclusions are more general. The final measurements in experiment should be the sum of all the possible reflections in the multi-dimension space. is supported in part by the DFG and the NSFC through funds provided to the Sino-German CRC 110 "Symmetries and the Emergence of Structure in QCD"'. The work of UGM was also supported by the Chinese Academy of Sciences (CAS) President's International Fellowship Initiative (PIFI) (Grant No. 2015VMA076).