Mass Spectrum of Exotic X(5568) State via Artificial Neural Network
MMass Spectrum of Exotic X(5568) State via Artificial Neural Network
Halil Mutuk ∗ Physics Department, Faculty of Arts and Sciences,Ondokuz Mayis University, 55139, Samsun, Turkey
In this paper, we assume X (5568) exist and study mass spectrum of X (5568) resonance and itshypothetical charmed partner, X c , by Artificial Neural Network method. The obtained predictionsare compared with the experimental data and results of other theoretical works. I. INTRODUCTION
With the discovery of the exotic states, i.e., states thatcannot be interpreted by the conventional meson (quark-antiquark) and baryon (three quark) pictures, hadronspectroscopy presented some puzzles in the understand-ing of our nature.The milestone of exotic states was the observation ofcharmoniumlike resonance X (3872) by Belle collabora-tion in 2003 [1]. It was later confirmed by D0 [2], CDFII [3] and BABAR [4] collaborations. This observationopened a new era in understanding of our nature and aswell as Standard Model framework. Many new charmo-niumlike states were observed and this new particle zoois named XY Z particles.The lack of conventional picture of mesons to inter-pret the underlying structure of the exotic states pavedthe way for new theoretical approaches [5–9]. Thesestates are out of conventional meson picture ( q ¯ q ). Al-though there is no consensus about the internal struc-ture of exotic states, some properties of these states areaccommodated by tetraquark models, molecular modelsor updated potential models. Among these approaches,tetraquark model and molecular model got more atten-tion. In tetraquark model, two heavy and two lightquarks come in together or mix. These quarks maycluster into the colored diquark-antidiquark, ( Qq )+( ¯ Q ¯ q )doublet. In the molecular model, the exotic particle isthought to be a bound state of two color-singlet mesons,( Q ¯ q ) +( ¯ Qq ).In 2016, a resonance named X (5568) was reported bythe D0 collaboration at the B s π ± invariant mass spec-trum with the mass and width, respectively [10] M = 5567 . ± . +0 . − . MeV , Γ = 21 . ± . +5 . − . MeV . As it is mentioned in [10], it is the first observation of ahadronic state with four different flavor quarks. The rea-son for that is that, as the decay rate of X (5568) → B s π ± is much larger than the weak interaction prediction, onecan conclude that strong interaction is responsible forthis decay. Since strong interactions do not touch the ∗ Electronic address: [email protected] flavor, in the final state there are four different quarks of B s = ¯ bs and π + = u ¯ d which are cannot be created bythe vacuum [11].Beside other puzzling features, X (5568) was not ob-served in X → B s π ± channel as reported by LHCb col-laboration [12], the CMS collaboration [13], the CDF col-laboration at Fermilab [14] and ATLAS collaboration ofLHC [15]. In 2018, the D0 collaboration announced thatthey had confirmed the existence of X (5568) from thedecay X (5568) → B s π ± via a sequent semileptonic de-cay B s → µ ± D ∓ s [16] and the results were in consistentexcept the widht is shifted toΓ = 18 . +7 . − . (stat) +3 . − . (syst) MeV . The clear discrepancy between D0 collaboration resultsand other experimental groups fired a dispute. Since X (5568) may be the first observed exotic state with fourdifferent flavors, both theoretical and experimental stud-ies on it can enlighten our realization of quark model.There are different approaches related to X (5568) whichcalculate masses, widths, decay constants, decay chan-nels and argue about internal structure [17–40]. Allthese studies conclude the mysterious and curious caseof X (5568) resonance.The hypothetical charmed partner of X (5568), whichwe will denote it as X c , is composed of c , s , u , d quarks.The related channels for this resonance to be observedcan be X c → D − s π + and X c → D K . The mass, decayconstant and widths of this hypothetical resonance wasstudied in [33].In the present study, we adopt diquark-antidiquark andmolecular pictures of X (5568) and X c and calculate massspectra by artificial neural network for the first time.Artificial neural networks (ANNs) are being used sincetwo decades to solve both ordinary and partial differen-tial equations. They maintain many attractive featurescompared to known existing semi-analytical and numer-ical techniques. One of the main advantage of ANNsto solve differential equations is that they require lessnumber of model parameters than any other technique.Besides that, machine learning which is nowadays a hottopic in physics is provided via using ANNs.The outline of paper is as follows. In Section II, weintroduce ANN formalism and the necessary details forapplication to quantum mechanics. In Section III, wegive our results and discuss. In Section IV we summarizeour findings. a r X i v : . [ h e p - ph ] J u l II. FORMALISM OF ARTIFICIAL NEURALNETWORK
Artificial neural networks are computer systems whichare capable of deriving and creating new information andalso discovering them through learning which is one of thefeatures of human brain. Neural networks are mimickingversions of biological nervous systems. They are paralleland distributed information processing elements. Theseelements have their own memory and are connected toeach other via weighted connections.The fundamental ingredient of an artificial neural net-work is neuron (perceptron in computerized systems) andit is the processing element in a neural network. Figure1 represents a single artificial neuron.
Inputs 𝑋 𝑋 (cid:2870) 𝑋 (cid:2871) 𝑋 (cid:3036) 𝑤 𝑤 (cid:2870)(cid:3037) 𝑤 (cid:2871)(cid:3037) 𝑤 (cid:3036)(cid:3037) Weigths 𝜃 𝑗 ThresholdSummation Junction Activation Function Output 𝑣 𝑖 = (cid:3533)𝑤 𝑖𝑗 𝑥 𝑗 + 𝜃 𝑗𝑛𝑖=1 𝑓(𝑣 𝑖 ) 𝑦 𝑖 FIG. 1: A model of single neuron
Each neuron receives any number of input and pro-duces only one output. If this output comes from inputlayers, it will be an input for the hidden layers. In thismanner, the inputs are the outputs of activation func-tions in where the inputs are multiplied by the connec-tion weights. This activation function (neuron transferfunction) determines the output. In practice, one singleneuron is not capable of handling problems. That’s whynetworks composed of neurons are being used. In Figure2, the architecture of a multilayer perceptron is shown.
FIG. 2: Multilayer neural network
In this study, we consider a feed forward neural net-work with one input layer, one hidden layer and one out-put layer. In feed forward neural networks, the informa-tion moves in only one direction, from input nodes to thehidden and output nodes. Fig. 1 is an example of feedforward neural network.Solving differential and eigenvalue equations viatrained ANN have some major advantages compared tostandard numerical techniques [41]: • A compact form of the solution is derivable sincefinite differences are not utilized, • Increasing number of sampling points does not in-crease the computational complexity very quickly.These features are important for solving few body sys-tems by Schr¨odinger equation. Solving a two body sys-tem like mesons by Schr¨odinger equation seems quite easyrather than three-body systems like baryons. For exam-ple to describe a baryon system as a bound state of threequarks is complicated. The configuration of particles canbe done via a coordinate transformation called Jacobicoordinates. Faddeev equations are also being used forquantum three body problems [42].
A. Mathematical Model of an Artificial NeuralNetwork
The general method for solving differential equationsis presented in [43] with the quantum mechanical appli-cations in [44–47]. The relationship of the input-outputof the layers can be written as follows: o i = σ ( n i ) , (1) o j = σ ( n j ) , (2) o k = σ ( n k ) , (3)where i is for input, j is for hidden and k is for outputlayers. Input to the perceptrons are given as n i = (Input signal to the NN) , (4) n j = N i (cid:88) i =1 ω ij o i + θ j , (5) n k = N j (cid:88) i =1 ω jk o j + θ k , (6)where, N i and N j represents the numbers of the unitsbelonging to input and hidden layers, ω ij is the synapticweight parameter which connects the neurons i and j and θ j represents threshold parameter for the neuron j and θ j is the threshold parameter [47]. The output of thenetwork can be written as o k = b n (cid:88) j =1 ω jk σ (cid:32) a n (cid:88) i =1 ω ij n i + θ j (cid:33) + θ k . (7)Derivative of this function is needed in further evaluationof error function. This can be obtained as ∂o k ∂ω ij = ω jk σ (1) ( n j ) n i , (8) ∂o k ∂ω jk = σ ( n j ) δ kk (cid:48) , (9) ∂o k ∂θ j = ω jk σ (1) ( n j ) , (10) ∂o k ∂θ k (cid:48) = δ kk (cid:48) . (11)In this work we use a sigmoid function σ ( z ) = 11 + e − z (12)as an activation function since it is possible to deriveall the derivatives of σ ( z ) in terms of itself. This dif-ferentiability is an important aspect for the Schr¨odingerequation. B. Application to Quantum Mechanics
Following the work of [44], let us consider the differen-tial equation: H Ψ( r ) = f ( r ) (13)where H is a linear operator, f ( r ) is a known functionand Ψ( r ) = 0 at the boundaries. In order to solve thisdifferential equation, a trial functionΨ t ( r ) = A ( r ) + B ( r , λ ) N ( r , p ) (14)of the form can be written which uses a feed forwardneural network with parameter vector p and λ to be ad-justed. The parameter p refers to the weights and biases of the neural network. The functions A ( r ) and B ( r , λ )should be specified in a convenient way so that Ψ t ( r )satisfies the boundary conditions regardless of the p and λ values. To obtain a solution for Eqn. (13), the collo-cation method can be used and the differential equationcan be transformed into a minimization problemmin p,λ (cid:88) i [ H Ψ t ( r i ) − f ( r i )] . (15)For Schr¨odinger equation Eqn. (13) takes the form H Ψ( r ) = (cid:15) Ψ( r ) (16)with the boundary condition, Ψ( r ) = 0. In this case, thetrial solution can be written asΨ t ( r ) = B ( r , λ ) N ( r , p ) (17)where B ( r , λ ) = 0 at boundary conditions for a range of λ values. By discretizing the domain of the problem, itis transformed into a minimization problem with respectto the parameters p and λE ( p , λ ) = (cid:80) i [ H Ψ t ( r i , p , λ ) − (cid:15) Ψ t ( r i , p , λ )] (cid:82) | Ψ t | d r (18)where E is the error function and (cid:15) can be computed as (cid:15) = (cid:82) Ψ ∗ t H Ψ t d r (cid:82) | Ψ t | d r . (19) III. NUMERICAL RESULTS AND DISCUSSION
We consider the Hamiltonian which was formulatedby Semay and Silvestre-Brac in [48] to study tetraquarksystems. The Hamiltonian reads as follows H = (cid:88) i (cid:18) m i + p m i (cid:19) − (cid:88) i 315 GeV ,m s = 0 . 577 GeV ,m b = 5 . 227 GeV ,m c = 1 . 836 GeV ,κ = 0 . ,κ (cid:48) = 1 . ,λ = 0 . , Λ = − . ,B = 0 . ,A = 1 . B − ,r c = 0 , (22)and the masses as [49] B s = 5366 MeV π = 139 MeV B ∗ s = 5415 MeV ρ = 770 MeV B + = 5279 MeV¯ K = 497 MeV B ∗ + = 5325 MeV¯ K ∗ = 892 MeV D − s = 1968 MeV D = 1864 MeV . Constructing wave function for four-body system isstraightforward. We have used a wave function whichwas proposed in [50] ψ r (1234; x , x , x ) = C i (1234) T j (1234) S k (1234) × E p ( x , x , x ) , (23)where 1,2 and 3,4 denotes the quarks and antiquarks,respectively, C i is the color part, T j is the isospin, S k isthe spin and E p is the spatial parts, respectively. Thespatial basis states are composed of harmonic oscillatorwave functions. For this, three Jacobi coordinates x (diquark extension), x (antidiquark extension) and x (diquark-antidiquark distance) are defined as [50] b x = (cid:20) ω ω ω (cid:21) / ( r − r ) ,b x = (cid:20) ω ω ω (cid:21) / ( r − r ) ,b x = (cid:20) ωω ω (cid:21) / × [ ω ( ω r + ω r ) − ω ( ω r + ω r )] . (24)In these equations, a reference length b is chosen arbi-trarily to make sure that Jacobi coordinates x i are di-mensionless. Similarly, a reference mass m is chosen and ω i = m i /m are dimensionless parameters proportionalto the actual masses. The definitions w ij = ω i + ω j and ω = w + w = (cid:80) i ω i = M/m where M is the totalmass of the particles are used for compactness. ψ ( r ) = e − βr N ( x i , u, w, v ) ψ r (1234; x i ) , β ≥ (25)with N being a feed forward neural network with onehidden layer and m sigmoid hidden units N ( x i , u, w, v ) = m (cid:88) j =1 v j σ ( w j x i + u j ) . (26)By employing this approach it is possible to obtain en-ergy eigenvalues of the Schr¨odinger equation. We trainedthe network with 200 equidistance points in the intervals0 < r < m = 8 and solved the Schr¨odinger equa-tion with ψ (0) = 0 at the boundaries. Table I gives themass values for X (5568) and Table II for X c . TABLE I: Mass values of X (5568). Results are in MeV. X (5568) This work [51] [52] [28] [21] su ¯ d ¯ b ± 158 5584 ± B s π B ∗ s ρ B + ¯ K ± B ∗ + ¯ K ∗ X c . Results are in MeV exceptRef. [27] which is in GeV. X (5568) [33] [33] [27]Mass 2480 2590 ± 60 2634 ± 62 2 . ± . According to our framework, X (5568) is light for an su ¯ b ¯ d tetraquark or molecular state. The mass differenceis at the order of ≈ 300 MeV. Taking into account thatthe Ξ b baryon which has usb quark structure has a massof 5797 MeV, it would be a puzzling situation if su ¯ d ¯ b structure have a lower mass with an additional quarkthan Ξ b baryon. IV. SUMMARY AND CONCLUDINGREMARKS In this work, we obtained mass of X (5568) resonanceand hypothetical partner X c . The prominent feature ofthis resonance is that it is the first state that contains fourdifferent flavors of quark. Although the other collabora-tions LHCb, CMS, ATLAS and CDF have not confirmedthe existence of this state up to now, the statistical sig-nificance of 5.1 σ in the B s π ± invariant-mass spectrumchallenges our understanding of the quark model. Inthe original quark model framework such exotic states orbetter to say multi-quark states were predicted by Gell-Mann. Therefore it should be not surprising the existenceof four-quark state with all different flavors.Due to some advantages provided by artificial neuralnetworks such as continuity of solution over all the do-main of integration and not increasing of computationalcomplexity when the sampling points and number of di-mensions involved, such elaborations can be made moresafely. 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