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A braid model for the particle X(3872)
C. Pe˜na ∗ and L. Jacak Institute of Physics, Wroc law University of Technology,Wyb. Wyspia´nskiego 27, 50-370 Wroc law, Poland (Dated: October 25, 2018)
Abstract
The Model of Quark Exchange (MQE) describes the particle X(3872) as a meson molecule. Weasked whether braids influence the meson potential in the MQE. We used the Burau representationthat parameterized braids with a variable t . The present result shows that t rescales the couplingof the meson potential determining if it is attractive or repulsive. As a consequence, a capturediagram favored the molecular state for t = 0 .
85, it breaks for other values. For the future, braidsmay help to study others exotic states in geometrical terms.
PACS numbers: 14.20.Pt, 03.65.Nk, 03.65.Fd ∗ [email protected] . INTRODUCTION In 2003 the Belle Collaboration discovered the particle X(3872) [1] which other exper-iments confirmed later [2–5]. More recently, the detector LHCb measured its quantumnumbers [2]. These findings suggested that X(3872) is composed of something more thantwo heavy charm quarks (¯ cc ). Nowadays, its composition includes one of the alternatives(e.g. [6]): a molecule made of two heavy mesons [7, 8]; a tetra quark [9] or a hybrid state.The Model of Quark Exchange (MQE) describes the particle X(3872) as a molecule com-posed of mesons D and ¯ D ∗ (D-mesons) [4]. The model request a potential energy char-acterized by a coupling λ ex and a width γ that were fitted with the mass of the X(3872).Moreover, the nature of these parameters was well established in terms of diagrams calledcapture and transfer [10, 11]. Although, the diagrams posses a vertex signalizing the ex-change of two quarks, it is unclear how the vertex contributes to the potential energy. Weanswered this question assuming the capture and transfer diagrams content braids. Braidsform groups of great interest in physics as they explain the origin of fermion statistics as-sociated with exchange of particles [5]. We proved that braids modified the phase-shiftcalculated previously in ref. [4] because the coupling λ ex rescales by a factor originated frombraids.The section II describes briefly the method based on the quark exchange model for theparticle X(3872). In the section III we included the generators of the braid group B in thequark exchange model. II. A BRAID MODEL FOR THE X (3872) A. The phase-shift
The phase-shift ( δ ) describes the state X(3872) when solving the Lippmann-Schwingerequation for the scattering process D + ¯ D ∗ → J/ψ + ρ [12]. The authors in ref. [12]implemented a Lorentzian potential for mesons characterized by two parameters: Its strength λ = 20 . − and a cutoff γ = 0 .
8. For these parameters the state X(3872) materializesas a molecule of D-mesons with mass of 3 .
872 GeV. The energy variable z enters in the2cattering phase-shift δ ( z ) defined as δ ( z ) = arctan (cid:18) Im[t( z )]Re[t( z )] (cid:19) , (1)where t( z ) relates to the T-matrix that solves the Lippmann-Schwinger equation (see eq. 10of ref. [12]). B. The quark exchange mechanism
The MQE proposed that a meson potential originates when the two D-mesons exchangetwo quarks [11, 13–15]. This description supposes that all heavy mesons interact with thepotential U ( p, p ′ ) = − C SF C I ( p, p ′ ) (2)with the factor C SF C = and I ( p, p ′ ) as the invariant matrix element of the scattering D + ¯ D ∗ → J/ψ + ρ . This matrix element was calculated using four contributions namedcapture and transfer diagrams [11]. The first capture diagram ( D
1) is shown in Fig.2. The c ¯ u ¯ cu c ¯ cu ¯ uD ≃ × Uc ¯ uu ¯ c c ¯ cu ¯ up = p + p ( A )( B ) − p = p + p p ′ + p ′ = p ′ p ′ + p ′ = − p ′ ( D )( C ) p p p p p ′ p ′ p ′ p ′ FIG. 1: Up panel: A box diagram describes the meson potential. D-mesons are composed of thequarks up (u) and charm (c). Down panel: The meson potential approximates to the capturediagram D D , ¯ D ∗ and ¯ D , D ∗ [11]; the prior and post interaction [11, 15]; the initialand final momenta of each heavy quark also contributes with a factor two. D I ( p, p ′ ) = h φ p φ − p | − D | φ p ′ φ − p ′ i = X p ...p ′ φ ∗ p ( p , p ) | {z } A φ ∗− p ( p , p ) | {z } B × V ( p , p ) δ p ,p ′ δ p ,p ′ × φ p ′ ( p ′ , p ′ ) | {z } C φ − p ′ ( p ′ , p ′ ) | {z } D . (3)within the quark-quark interaction V ( p , p ) = − V e − p − p ′ ) δ p ,p ′ +( p ′ − p ) (4)and V = 113 .
39 GeV − as a parameter used to fit the meson spectrum [11]. In this paper,the volume in momentum space is one and the spin-spin interaction is neglected since itis hundred times smaller than V . The wave function for a meson A is represented by φ p ( p , p ) with amplitude φ A , similarly for the other mesons. They define the product φ A φ B φ C φ D (2 π ) = 6 .
312 GeV − . In the continuum limit for the invariant matrix, the sum (3)becomes an integration yielding the factor 0 .
12 GeV . Therefore the capture diagram D λ ex = − C SF C φ A φ B φ C φ D (2 π ) V (cid:0) .
12 GeV (cid:1) , = − .
35 GeV − . (5)Thus the strength | λ ex | is around six times bigger than the coupling of the meson potential λ . Besides the first capture diagram D λ [16]. Probably higher orders in the ladder approximation may help to reduce thisdisagreement. However, We solved this problem using braids inside the capture diagram D C. Definition of braids
K. Murasugi introduced braids in three dimensions as the pictures in Fig. 2 [17, 18].Braids may form groups [19, 20]. The full braid group B n consists of the n − σ ,4 , ... σ n − [19, 20]. For instance, the group B is composed of two generators as in Fig. 2cwhile the groups with one generator are B in Fig. 2a (the trivial group) and B in Fig. 2b(the cyclic group) [20]. The group B n (also called the Artin group) is non-abelian for n ≥ ℜ [21, 22]. a. b. c. FIG. 2: Braids as generators of the full braid groups: a. B , b. B , c. B . D. The Burau’s representation for braids
W. Burau parametrized the braid group B n with a variable t using an injective mapping(only for n <
4) [23]. He associated to the element σ i in B n the n × n matrix ϕ n ( σ i ) = I i − − t t I n − i − (6)with i = 1 , , ..., n −
1. The empty spaces in (6) consist of zeros; the identity m × m matrixnamed I m disappears from ψ n ( σ ) as i = 1 and from ψ n ( σ n − ) as i = n −
1. Thus the braidgroup B is generated by ϕ ( σ ) = − t t . (7)and the braid group B by ϕ ( σ ) = − t t
01 0 00 0 1 , ϕ ( σ ) = − t t . (8)These matrices (8) depend uniquely on t in comparison with other representations that usemore than one variable [20, 24–26]. 5 II. THE RESULTSA. A braid structure in the quark exchange diagrams
The quark exchange potential (4) determines the capture diagram in Fig. 1 when thequarks ¯ u and ¯ c exchange because of the constraint given by δ p ,p ′ +( p ′ − p ) δ p ,p ′ δ p ,p ′ , (9)this constraint permutes the momenta subscripts (2 , ,
4) if the symbol prime is ignored.The permutation of three numbers is the element Π of the permutation group S ([27], p.301) Π = . (10)Since Π represents also the 3 × D (Π ) = , (11)the trace of the permutation matrix (11) contributes to the sum (3). This procedure reachesonly until fifty porcent of the coupling λ when including the whole capture and transferdiagrams together [16]. Besides, the permutation matrix (11) allows arbitrarily all type ofquark exchange contributions. We limited the contributions by considering the exchangediagrams as braids. Each braid has two alternatives for crossings that distinguishes the typeof quark exchange. We proved this statement by tracing the product of the permutationmatrix (11) with ϕ ( σ − ) and ϕ ( σ ) (8). Hence the traces T r [ ϕ ( σ ) D (Π )] = 0 , (12) T r (cid:2) ϕ ( σ − ) D (Π ) (cid:3) = 1 − t . (13)affect the value of λ ex such that the potential scales with the new strength λ = λ ex (cid:18) − t (cid:19) , (14)As a result the quarks exchange as the braid σ − (Fig. 3) instead of σ . We calculated the6 c ¯ uu ¯ u ¯ cu p ′ p ′ p ′ ¯ c ¯ uu ¯ u ¯ cu p ′ p ′ p ′ p p p p p p σ σ − FIG. 3: The braids σ or σ − represents the exchange of quarks ¯ c and ¯ u as well as in Fig. 1. phase-shift (4) as a function of the energy ( z ) and the variable t . Our method is the samethat in ref. [12] but with a change induced by the coupling (14). The Lorentzian form factors L ( p ) and R ( p ) have cutoff γ = 0 .
8. We observed that t = 0 .
85 makes a molecule of D and¯ D ∗ . Moreover, a sharp behavior of the phase-shift around 3 .
872 GeV remains. We obtainedthe same result than in ref. [12] only when t → ∞ . Energy ( z [GeV]) -1.5-1-0.500.511.5 P h a s e - s h i f t ( δ ) t = 1t = 0.85t = 0.7 FIG. 4: Phase-shift as a function of energy for different values for t . The phase-shift jumps in π at the energy of 3 .
872 GeV ( λ = 20 . − , γ = 0 . V. DISCUSSION
By including braids in the MQE we found that they regulate the repulsive or attractivecharacter of the meson potential (2). Besides the braid model explains the origin of thecoupling λ at the quark level. Therefore, the braids in the Fig. 3 makes the phase-shift(Fig. 4) to jump in π . It confirms that the pair of mesons D , ¯ D ∗ forms a molecule withmass of 3 .
872 GeV for t = 0 .
85, other vales destroy it. Although, for the molecule, weobtained zero for the binding energy it may differ if all capture and transfer diagrams areincluded [16].This braid model requires two quarks exchange (one heavy ¯ c and one light ¯ u ) but othermodels proposed a meson exchange ( ρ or ω ) containing only light quarks [7, 8, 28]. Never-theless, whether the X (3872) prefers the exchange with two quarks rather than a meson isunclear.We observed within one capture diagram that t = 1 destroys the molecule because themeson potential vanishes (see the phase-shift in Fig. 4). Moreover, for t > t < t > B we have chosen those as in Fig. 3 with one crossing so thepotential attenuates only with the polynomial 1 − t . We believe the physical meaning of t would emerge if a relation between the geometrical and the matrix representation of braidsis established. Unfortunately such relation is absence in literature.The robust calculation of the phase-shift in this molecular model includes the quarkstructure of mesons something common in all exotic states. The meson potential solvesanalytically the Lippmann-Schwinger equation since it separates in two form factors ( L ( p ), R ( p )). Either Lorentzian or Gaussian form factors yield almost the same phase-shift withsmall adjustments for the cutoff γ . We neglected the width of the ρ meson in the scatteringprocess D + ¯ D ∗ → J/ψ + ρ causing the phase-shift to jump sharply at 3 .
872 GeV. A similarbehavior is well known for the Deuteron system [29].The charged mesons D + and ¯ D ∗− may affect the phase-shift for energies above 3 .
872 GeVas previously was studied in [16]. We plan to test this braid model for other exotic statessince current descriptions do not apply well for all [6]. The braid model fits well with the8redictions for a hot and dense medium created in heavy ion collisions where the X(3872)may form [30, 31]. The capture diagram used in Fig. 1 corresponds to the first order in theladder approximation. Therefore these findings must be interpreted with caution for higherorder.To summarize, braids may help as a mechanism for production of exotic states. The braid σ − modifies the meson potential because the coupling λ ex rescales by a factor 1 − t suchthat a X (3872) forms as a molecule of D and ¯ D ∗ for t = 0 .
85. The value of t may changesignificantly when adding more capture and transfer diagrams. Nevertheless, it informs ifthe meson potential is attractive or repulsive. Future studies testing this braid model forother exotic states and predictions for heavy ion collisions are needed. Acknowledgments
The support from the NCN Project UMO-2011/02/A/ST3/00116 is acknowledged. C.Pe˜na thanks the comments of D. Blaschke. [1] S.-K. Choi, et al. (Belle Colaboration), Phys. Rev. Lett , 262001 (2003).[2] D. Acosta, et al. (CDF Collaboration), Phys. Rev. Lett. , 072001 (2004).[3] V. M. Abazov, et al. (D0 Collaboration), Phys. Rev. Lett. , 162002 (2004).[4] B. Aubert, et al. (BABAR Collaboration), Phys. Rev. D , 071103 (2005).[5] R. Aaij, et al. (LHCb Collaboration), Eur. Phys. J. C , 1972 (2012).[6] C. Hambrock, PoS Beauty 044. ArXiv e-prints (2013), 1306.0695.[7] N. A. T¨ornqvist, Phys. Lett B , 209 (2004).[8] D. Gamermann, J. Nieves, E. Oset and E. R. Arriola, Phys. Rev. D , 014029 (2010).[9] L. Maiani, and F. Piccinini, and A. D. Polosa, and V. Riquer, Phys. Rev. D , 014028 (2005).[10] T. Barnes and E. S. Swanson, Phys. Rev. D , 131 (1992).[11] K. Martins, D. Blaschke, and E. Quack, Phys. Rev. C , 2723 (1995).[12] C. Pe˜na and D. Blaschke, Acta Phys. Pol. B Proc. Suppl , 963 (2012).[13] T. Barnes, E. S. Swanson, Phys. Rev. D , 131 (1992).[14] T. Barnes, N. Black, E. S. Swanson, Phys. Rev. C , 025204 (2001).
15] D. Blaschke and G. R¨opke, Physics Letters B , 332 (1993).[16] C. Pe˜na,
Quantum mechanical model for quarkonium production in heavy ion collisions (PhDThesis. University of Wroclaw, Poland, 2013).[17] K. Murasugi,
Braid Groups (Kluwer Academic Publishers, 1999).[18] K. Murasugi,
Knot Theory and its Applications (Birkh¨auser Boston, 1996).[19] E. Artin, Annals of Mathematics, Second Series , 101 (1947).[20] C. Kassel, V. Turaev, Braid Groups (Springer Science, Business Media, LLC, 2008).[21] J. S. Birman,
Braids, Links, and Mapping Class Groups (Princeton University Press andUniversity of Tokyo Press, 1974).[22] J. Jacak, R. Gonczarek, L. Jacak, I. J´o´zwiak ,
Application of Braid Groups in 2D Hall SystemPhysics Composite Fermion Structure (World Scientific Publishing Co. Pte. Ltd, 2012).[23] W. Burau, Abhandlungen aus dem Mathematischen Seminar der Universit¨at Hamburg ,179 (1935).[24] J. S. Birman and T. E. Brendle, ArXiv e-prints (2004), math/0409205.[25] N. Jackson, Notes on braid groups (2004).[26] D. Rolfsen, ArXiv e-prints (2010), 1010.4051.[27] I. N. Bronshtein, K. A. Semendyayev, G. Musiol, H. Muehlig,
Handbook of Mathematics, 5thEd (Springer-Verlag, Berlin- Heidelberg, 2007).[28] Z. Hua-Bin and L. Xiao-Fu, Communications in Theoretical Physics , 359 (2014).[29] M. Schmidt, G. R¨opke and H. Schulz, Annals Phys , 57 (1990).[30] C. Pe˜na and D. Blaschke, Nuclear Physics A , 1 (2014).[31] D. Blaschke and C. Pe˜na, Nuclear Physics B - Proceedings Supplements , 137 (2011)., 137 (2011).