One-photon decay of the tetraquark state X(3872)→γ+J/ψ in a relativistic constituent quark model with infrared confinement
Stanislav Dubnicka, Anna Z. Dubnickova, Mikhail A. Ivanov, Jurgen G. Korner, Pietro Santorelli, Gozyal G. Saidullaeva
aa r X i v : . [ h e p - ph ] A ug DSF-5-2011MZ-TH/11-09
One–photon decay of the tetraquark state X (3872) → γ + J/ψ in a relativisticconstituent quark model with infrared confinement
Stanislav Dubnicka, Anna Z. Dubnickova, Mikhail A. Ivanov, J¨urgen G. K¨orner, Pietro Santorelli, and Gozyal G. Saidullaeva Institute of Physics Slovak Academy of Sciences Dubravska cesta 9 SK-845 11 Bratislava, Slovak Republic Comenius University Department of Theoretical Physics Mlynska Dolina SK-84848 Bratislava, Slovak Republic Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia Institut f¨ur Physik, Johannes Gutenberg-Universit¨at, D–55099 Mainz, Germany Dipartimento di Scienze Fisiche, Universit`a di Napoli Federico II,Complesso Universitario di Monte S. Angelo, Via Cintia, Edificio 6,80126 Napoli, Italy, and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli Al-Farabi Kazak National University, 480012 Almaty, Kazakhstan
We further explore the consequences of treating the X(3872) meson as a tetraquark bound state byanalyzing its one-photon decay X → γ + J/ψ in the framework of our approach developed in previouspapers which incorporates quark confinement in an effective way. To introduce electromagnetism wegauge a nonlocal effective Lagrangian describing the interaction of the X(3872) meson with its fourconstituent quarks by using the P-exponential path-independent formalism. We calculate the matrixelement of the transition X → γ + J/ψ and prove its gauge invariance. We evaluate the X → γ + J/ψ decay width and the longitudinal/transverse composition of the
J/ψ in this decay. For a reasonablevalue of the size parameter of the X(3872) meson we find consistency with the available experimentaldata. We also calculate the helicity and multipole amplitudes of the process, and describe how theycan be obtained from the covariant transition amplitude by covariant projection.
PACS numbers: 12.39.Ki,13.25.Ft,13.25.Jx,14.40.RtKeywords: relativistic quark model, infrared confinement, tetraquark, exotic states, electromagnetic interac-tions
I. INTRODUCTION
This paper is a direct continuation of our previous work [1] where we have analyzed the strong decays of thecharmonium–like state X (3872) in the framework of our relativistic constituent quark model which includes infraredconfinement in an effective way [2]. In our approach the X(3872) meson is interpreted as a tetraquark state with thequantum numbers J P C = 1 ++ as in [3]. In this paper we analyze the one-photon decay X → γ + J/ψ in the sametetraquark picture. The electromagnetic interaction is incorporated into our relativistic nonlocal effective Lagrangianin a gauge invariant way using the P-exponential path-independent formalism.We begin by collecting the experimental data relevant for our purposes. A narrow charmonium–like state X (3872)was observed in 2003 in the exclusive decay process B ± → K ± π + π − J/ψ [4]. The X (3872) decays into π + π − J/ψ andhas a mass of m X = 3872 . ± . ± . m D + m D ∗ = 3871 . ± .
25 mass threshold[5]. Its width was found to be less than 2.3 MeV at 90% confidence level. The state was confirmed in B-decays bythe B A B AR experiment [6] and in pp production by the Tevatron experiments CDF [7] and DØ [8]. The most precisemeasurement up to now was done in [9] with m X = 3871 . ± . ± .
19. The new average mass given in [7] is m X = 3871 . ± .
22 MeV . (1)The Belle Collaboration has reported [10] evidence for the decay modes X (3872) → γ + J/ψ and to X → π + π − π J/ψ : B ( B → XK ) · B ( X → γ + J/ψ ) = (1 . ± . ± . × − , Γ( X → γ + J/ψ )Γ( X → π + π − J/ψ ) = 0 . ± . , B ( X → π + π − π J/ψ ) B ( X → π + π − J/ψ ) = 1 . ± . ± . . (2)These observations imply strong isospin violation because the three-pion decay proceeds via an intermediate ω meson with isospin 0 whereas the two-pion decay proceeds via the intermediate ρ meson with isospin 1. It is evidentthat the two-pion decay via the intermediate ρ meson is very difficult to explain by using an interpretation of the X (3872) as a simple c ¯ c charmonium state with isospin 0.In an analysis of B + → J/ψ γ K + decays, the B A B AR Collaboration [11] found evidence for the radiative decay X (3872) → γ + J/ψ with a statistical significance of 3 . σ . They reported the following values for the product ofbranching fractions B ( B + → XK + ) · B ( X → γ + J/ψ ) = (3 . ± . ± . × − . (3)The Belle Collaboration reported [12] the first observation of a near-threshold enhancement in the D ¯ D π systemfrom B → D ¯ D π K . The enhancement peaks at a mass of M = 3875 . ± . +0 . − . ± . B ( B → D ¯ D π K ) = (1 . ± . +0 . − . ) × − . (4)All available experimental data up to 2007 were analyzed in [13]. The authors found that [13] B ( B + → XK + ) = 1 . +0 . − . × − , Γ( X → γ + J/ψ )Γ( X → π + π − J/ψ ) = 0 . ± . . (5)The B A B AR Collaboration found evidence for the decays X → γ + J/ψ and X → γ + ψ (2 S ) in their data sample ofthe B → c ¯ c γK decays. The measured products of branching fractions are [14] B ( B ± → XK ± ) · B ( X → γ + J/ψ ) = (2 . ± . ± . × − , B ( B ± → XK ± ) · B ( X → γ + ψ (2 S )) = (9 . ± . ± . × − . (6)There have been many theoretical attempts to unravel the structure of the X (3872) and its decays. Many of thetheoretical predictions for the decay X (3872) → γ + J/ψ published up to now are very model dependent. We mentionsome of them in turn.All possible 1 D and 2 P c ¯ c assignments for the X(3872) were considered in [15]. The authors obtained E c ¯ c states as well as for some strong decays taking the experimental mass as input.The conclusion was that many of the possible J P C assignments can be eliminated due to the smallness of the observedtotal width. The suggestion was that radiative transitions could be used to test the remaining J P C assignments.Some tests of the hypothesis that the X (3872) is a weakly bound D ¯ D ∗ molecule state were suggested in [16]. Itwas proposed that measuring the 3 πJ/ψ , γ + J/ψ , γ + ψ ′ , ¯ KK ∗ , and πρ decay modes of the X will serve as a definitivediagnostic tool to confirm or to rule out the molecule hypothesis.Assuming that the X (3872) state has the structure ( D ¯ D ∗ − D ∗ ¯ D ) / √ J P C = 1 ++ , the X (3872) → γ + J/ψ decay width was calculated using a phenomenological Lagrangian approach [17]. The calculatedvalue of the radiative decay width varied from 125 KeV to 250 KeV depending on the model parameters.QCD sum rules were used in [18] to calculate the width of the radiative decay of the meson X (3872), which wasassumed to be a mixture between charmonium and exotic molecular [ c ¯ q ][ q ¯ c ] states with J P C = 1 ++ . In a small rangefor the values of the mixing angle, one obtainsΓ( X → γ + J/ψ )Γ( X → J/ψ π + π − ) = 0 . ± . . (7)Our paper is organized as follows. In Sec. II we gauge a nonlocal effective Lagrangian describing the interaction ofthe X(3872) meson with its constituent quarks by using the P-exponential path-independent formalism developed in[19, 20]. In Sec. III we calculate the matrix element of the radiative transition X → γ + J/ψ and prove its gaugeinvariance analytically. In Sec. IV we present the results of our numerical analysis. First, we check numerically thatthe final amplitude is gauge invariant. Second, we introduce infrared confinement as was done in our previous papersRefs. [1, 2] and evaluate the X → γ + J/ψ decay width. Finally, in Sec. V we summarize our results. In an Appendixwe describe how the two helicity or the two multipole amplitudes of the process can be obtained from the gaugeinvariant transition amplitude by covariant projection.
II. THEORETICAL FRAMEWORK
The effective interaction Lagrangians describing the coupling of the charmonium-like meson such as the X (3872)to four quarks, and the coupling of the charmonium J/ψ state to its two constituent quarks are written in the form(see Ref. [1]) L int = g X X q µ ( x ) · J µX q ( x ) + g J/ψ
J/ψ µ ( x ) · J µJ/ψ ( x ) ( q = u, d ) . (8)The nonlocal interpolating quark currents read J µX q ( x ) = Z dx . . . Z dx δ x − X i =1 w i x i ! Φ X (cid:16) X i 12 ( y + y ) (cid:19) Φ J/ψ (cid:16) ( y − y ) (cid:17) ¯ c a ( y ) γ µ c a ( y ) . (9)The matrix C = γ γ is related to the charge conjugation matrix: C = C † = C − = − C T , C Γ T C − = ± Γ, (” + ”for Γ = S, P, A and ” − ” for Γ = V, T ). We follow [3] and take the tetraquark state to be a linear superposition ofthe X u and X d states according to X l ≡ X low = X u cos θ + X d sin θ,X h ≡ X high = − X u sin θ + X d cos θ. (10)The coupling constant g X in Eq. (8) will be determined from the compositeness condition Z H = 0 (see e.g.Refs. [21, 22]). The compositeness condition requires that the renormalization constant Z H of the elementary meson X is set to zero, i.e. Z H = 1 − Π ′ H ( p H = m H ) = 0 , (11)where Π X ( p ) is the scalar part of the meson mass operator and the prime stands for the derivative w.r.t. p H . Forthe spin one states X (3872) and J/ψ the compositeness condition readsΠ µνV ( p ) = g µν Π V ( p ) + p µ p ν Π (1) V ( p ) , Π V ( p ) = 13 (cid:18) g µν − p µ p ν p (cid:19) Π µνV ( p ) . (12)The X meson mass operator can be calculated from the self–energy three-loop sunrise–type diagram with four quark-antiquark propagators. The calculation is described in more detail in Ref. [1].As in the case of baryons composed of three quarks it is convenient to transform to Jacobi coordinates in theintegrals of Eq. (9). In the case of four quarks one has x = x + 2 w + w + w √ ρ − w − w √ ρ + w + w ρ ≡ x + X j =1 c j ρ j ,x = x − w + w + w √ ρ − w − w √ ρ + w + w ρ ≡ x + X j =1 c j ρ j ,x = x − w − w √ ρ + w + w + 2 w √ ρ − w + w ρ ≡ x + X j =1 c j ρ j ,x = x − w − w √ ρ − w + w + 2 w √ ρ − w + w ρ ≡ x + X j =1 c j ρ j , (13)where x = P i =1 x i w i and P ≤ i J/ψ case one has y = y + 12 ρ, y = y − ρ. (14)One then has J µX q ( x ) = Z d~ρ Φ X ( ~ρ ) J µ q ( x , . . . , x ) ,J µ q ( x , . . . , x ) = √ ε abc ε dec n [ q a ( x ) Cγ c b ( x )][¯ q d ( x ) γ µ C ¯ c e ( x )] + ( γ ↔ γ µ ) o ,J µJ/ψ ( y ) = Z dρ Φ J/ψ ( ρ ) J µ q ( y , y ) , J µ q ( y , y ) = ¯ c a ( y ) γ µ c a ( y ) , (15)where d~ρ = dρ dρ dρ and ~ρ = ρ + ρ + ρ . The Jacobian is absorbed into the coupling g X .The gauge invariant interaction of a bound quark state with the electromagnetic field has been described in somedetail in Ref. [19]. For comprehensive purposes we recall some of the key points of the gauging process. Since the X (3872) and J/ψ mesons are neutral mesons we will discuss the charged quarks only. The free Lagrangian of quarksis gauged in the standard manner by using minimal substitution: ∂ µ q → ( ∂ µ − ie q A µ ) q, ∂ µ ¯ q → ( ∂ µ + ie q A µ )¯ q, (16)where e q is the quark’s charge ( e u = e , e d = − e , etc.). Minimal substitution gives us the first piece of theelectromagnetic interaction Lagrangian L em(1)int ( x ) = X q e q A µ ( x ) J µq ( x ) , J µq ( x ) = ¯ q ( x ) γ µ q ( x ) . (17)In order to guarantee gauge invariance of the nonlocal strong interaction Lagrangian, one multiplies each quarkfield q ( x i ) in the relevant quark current J µ ( x ) given by Eq. (15) by a gauge field exponential according to q ( x i ) → e − ie q I ( x i ,x,P ) q ( x i ) , ¯ q ( x i ) → e ie q I ( x i ,x,P ) ¯ q ( x i ) ,I ( x i , x, P ) = x i Z x dz µ A µ ( z ) . (18)where P is the path taken from x to x i . It is readily seen that the full Lagrangian Eq. (8) is invariant under the localgauge transformations q ( x i ) → e ie q f ( x i ) q ( x i ) , ¯ q ( x i ) → e − ie q f ( x i ) ¯ q ( x i ) ,A µ ( z ) → A µ ( z ) + ∂ µ f ( z ) , so that I ( x i , x, P ) → I ( x i , x, P ) + f ( x i ) − f ( x ) . (19)The second term of the electromagnetic interaction Lagrangian L em int;2 arises when one expands the gauge exponentialin powers of A µ up to the order of perturbation theory that one is considering. Superficially the results appear todepend on the path P which connects the endpoints in the path integral in Eq (18). However, one needs to knowonly derivatives of the path integrals when doing the perturbative expansion. One can make use of the formalismdeveloped in [20] which is based on the path-independent definition of the derivative of I ( x, y, P ):lim dx µ → dx µ ∂∂x µ I ( x, y, P ) = lim dx µ → [ I ( x + dx, y, P ′ ) − I ( x, y, P )] , (20)where the path P ′ is obtained from P by shifting the endpoint x by dx . Use of the definition (20) leads to the keyrule ∂∂x µ I ( x, y, P ) = A µ ( x ) (21)which states that the derivative of the path integral I ( x, y, P ) does not depend on the path P originally used inthe definition. The nonminimal substitution (18) is therefore completely equivalent to the minimal prescription as isevident from the identities (20) or (21). The method of deriving Feynman rules for the nonlocal coupling of hadronsto photons and quarks was worked out before in Refs. [19, 20] and will be discussed in the next section where weapply the formalism to the physical processes considered in this paper.Expanding the Lagrangian up to the first order in A µ one obtains L em(2)int ( x ) = g X X q µ ( x ) · J µX q − em ( x ) + g J/ψ J/ψ µ ( x ) · J µJ/ψ − em ( x ) ( q = u, d ) ,J µX q − em = Z d~ρ Φ X ( ~ρ ) J µ q ( x , . . . , x ) n ie q [ I x x − I x x ] + ie c [ I x x − I x x ] o ,J µJ/ψ − em = Z dρ Φ J/ψ ( ρ ) J µ q ( x , x ) ie c [ I x x − I x x ] , I x i x ≡ I ( x i , x, P ) . (22)In order to use the key rule Eq. (21) we take the Fourier-transforms for the vertex functions Φ and quark fields q Φ X ( ~ρ ) = Z d ~ω (2 π ) e Φ X ( − ~ω ) e − i~ρ~ω = e Φ X ( ~∂ ρ ) δ (4) ( ~ρ ) , Φ J/ψ ( ρ ) = Z d ω (2 π ) e Φ J/ψ ( − ω ) e − iρω = e Φ J/ψ ( ∂ ρ ) δ (4) ( ρ ) ,q ( x i ) = Z d p i (2 π ) e − ip i x i ˜ q ( p i ) , ¯ q ( x i ) = Z d p i (2 π ) e ip i x i ˜¯ q ( p i ) . (23)One then writes down J µX q − em = Y i =1 Z d p i (2 π ) e J µ q ( p , . . . , p ) Z d~ρ δ (4) ( ~ρ ) e Φ X ( ~∂ ρ ) e − i ( p x − p x − p x + p x ) n ie q [ I x x − I x x ] + ie c [ I x x − I x x ] o = Y i =1 Z d p i (2 π ) e J µ q ( p , . . . , p ) e − i ( p − p − p + p ) x Z d~ρ δ (4) ( ~ρ ) e − i~ρ~ω e Φ X ( ~D ρ ) n ie q [ I x x − I x x ] + ie c [ I x x − I x x ] o ,J µJ/ψ − em = Y i =1 Z d p i (2 π ) e J µ q ( p , p ) Z dρ δ (4) ( ρ ) e Φ J/ψ ( ∂ ρ ) e i ( p x − p x ) ie c [ I x x − I x x ]= Y i =1 Z d p i (2 π ) e J µ q ( p , p ) e i ( p − p ) x Z dρ δ (4) ( ρ ) e ipρ e Φ J/ψ ( D ρ ) ie c [ I x x − I x x ] ,D µρ i = ∂ µρ i − iω µi , D µρ = ∂ µρ + ip µ , (24)where ω = c p − c p − c p + c p ,ω = c p − c p − c p + c p ,ω = c p − c p − c p + c p ,p = ( p + p ) . (25)Finally, we employ a convenient identity which was proven in [19]. The identity reads F ( D ρ j ) I x i x = Z dτ F ′ ( τ D ρ j − (1 − τ ) ω j ) c ij (cid:16) ∂ νρ j A ν ( x i ) − i ω νj A ν ( x i ) (cid:17) + F ( − ω j ) I x i x . (26)The identity holds for any function F ( z ) that is analytical at z = 0.One obtains J µX q − em ( x ) = Y i =1 Z d x i Z d y J µ q ( x , . . . , x ) A ρ ( y ) E ρX ( x ; x , . . . , x , y ) , (27) E ρX ( x ; x , . . . , x , y ) = Y i =1 Z d p i (2 π ) Z d r (2 π ) e − ip ( x − x )+ ip ( x − x )+ ip ( x − x ) − ip ( x − x ) − ir ( x − y ) e E ρX ( p , . . . , p , r ) , e E ρX ( p , . . . , p , r ) = Z dτ X j =1 n e c h − e Φ ′ X ( − z j ) l ρ j + e Φ ′ X ( − z j ) l ρ j i + e q h − e Φ ′ X ( − z j ) l ρ j + e Φ ′ X ( − z j ) l ρ j i o l ij = c ij ( c ij r + 2 ω j ) , ( i = 1 , . . . , j = 1 , . . . , ,z i = τ ( c i r + ω ) + (1 − τ ) ω + ω + ω ,z i = ( c i r + ω ) + τ ( c i r + ω ) + (1 − τ ) ω + ω ,z i = ( c i r + ω ) + ( c i r + ω ) + τ ( c i r + ω ) + (1 − τ ) ω . J νJ/ψ − em ( y ) = Z d y Z d y Z d z J ν q ( y , y ) A ρ ( z ) E ρJ/ψ ( y ; y , y , z ) , (28) E ρJ/ψ ( y ; y , y , z ) = Z d p (2 π ) Z d p (2 π ) Z d q (2 π ) e − ip ( y − y )+ ip ( y − y )+ iq ( z − y ) e E ρJ/ψ ( p , p , q ) , e E ρJ/ψ ( p , p , q ) = e c Z dτ n − e Φ ′ J/ψ ( − z − ) l ρ − − e Φ ′ J/ψ ( − z + ) l ρ + o ,z ∓ = τ ( p ∓ q ) − (1 − τ ) p , l ∓ = p ∓ q , p = ( p + p ) . For calculational convenience we will choose a simple Gaussian form for the vertex function ¯Φ X ( − Ω ). The minussign in the argument of the Gaussian function is chosen to emphasize that we are working in Minkowski space. Onehas ¯Φ X ( − Ω ) = exp (cid:0) Ω / Λ X (cid:1) (29)where the parameter Λ X characterizes the size of the X meson. Since Ω turns into − Ω in Euclidean space the form(29) has the appropriate fall-off behavior in the Euclidean region. We emphasize that any choice for Φ X is appropriateas long as it falls off sufficiently fast in the ultraviolet region of Euclidean space to render the corresponding Feynmandiagrams ultraviolet finite. As mentioned before we shall choose a Gaussian form for Φ X in our numerical calculationfor calculational convenience. III. MATRIX ELEMENT FOR THE DECAY X → γ + J/ψ The matrix element of the decay X (3872) → γ + J/ψ can be calculated from the Feynman diagrams shown in Fig. 1.The invariant matrix element for the decay is given by FIG. 1. Feynman diagrams describing the decay X → γ + J/ψ . M ( X q ( p ) → J/ψ ( q ) γ ( q )) = i (2 π ) δ (4) ( p − q − q ) ε µX ε ργ ε νJ/ψ T µρν ( q , q ) , (30)where T µρν ( q , q ) = X i = a,b,c,d T ( i ) µρν ( q , q ) ,T ( a ) µρν = 6 √ g X g J/ψ e q Z d k (2 π ) i Z d k (2 π ) i e Φ X (cid:16) − K a (cid:17)e Φ J/ψ (cid:16) − ( k + q ) (cid:17) × tr h γ S c ( k ) γ ν S c ( k + q ) γ µ S q ( k ) γ ρ S q ( k + q ) − ( γ ↔ γ µ ) i ,K a = ( k + q ) + ( k + q ) + ( w q q − w c q ) ,T ( b ) µρν = 6 √ g X g J/ψ Z d k (2 π ) i Z d k (2 π ) i e Φ J/ψ (cid:16) − ( k + q ) (cid:17) e E X ρ ( p , . . . , p , r ) × tr h γ S q ( k ) γ µ S c ( k ) γ ν S c ( k + q ) − ( γ ↔ γ µ ) i ,p = k , p = k + q , p = p = − k , r = − q ,T ( c ) µρν = 6 √ g X g J/ψ e c Z d k (2 π ) i Z d k (2 π ) i e Φ X (cid:16) − K c (cid:17)e Φ J/ψ (cid:16) − ( k + q + q ) (cid:17) × tr h γ S q ( k ) γ µ S c ( k ) γ ρ S c ( k + q ) γ ν S c ( k + p ) − ( γ ↔ γ µ ) i ,K c = k + ( k + p ) + w q p ,T ( d ) µρν = 6 √ g X g J/ψ e c Z d k (2 π ) i Z d k (2 π ) i e Φ X (cid:16) − K c (cid:17) e E J/ψ ρ ( p , p , q ) × tr h γ µ S q ( k ) γ S c ( k ) γ ν S c ( k + p ) − ( γ ↔ γ µ ) i ,p = − k − p, p = − k , q = − q . We have analytically checked on the gauge invariance of the unintegrated transition matrix element by contractionwith the photon momentum q which yields q ρ T µρν ( q , q ) = 0 using the identities S ( k ) q S ( k + q ) = S ( k + q ) − S ( k ) , Z dτ e Φ ′ ( − τ a − (1 − τ ) b ) ( a − b ) = e Φ( − b ) − e Φ( − a ) . IV. NUMERICAL RESULTS The evaluation of the loop integrals in Eq. (30) proceeds as described in our previous paper [1]. If one takes theon-mass shell conditions into account ε µX p µ = 0 , ε νJ/ψ q ν = 0 , ε ργ q ρ = 0 (31)one can write down five seemingly independent Lorentz structures T µρν ( q , q ) = ε q µνρ ( q · q ) W + ε q q νρ q µ W + ε q q µρ q ν W + ε q q µν q ρ W + ε q µνρ ( q · q ) W . (32)Using the gauge invariance condition q ρ T µρν = ( q · q ) ε q q µν ( W + W ) = 0 (33)one has W = − W which reduces the set of independent covariants to four: T µρν ( q , q ) = ( q · q ) ε q µνρ W + ε q q νρ q µ W + ε q q µρ q ν W + (cid:16) ε q q µν q ρ − ( q · q ) ε q µνρ (cid:17) W . (34)The gauge invariance condition W = − W provides for a numerical check on the gauge invariance of our calculationas described further on.However, there are two nontrivial relations among the four covariants which can be derived by noting [23] that thetensor T µ [ ν ν ν ν ν ] = g µν ε ν ν ν ν + cycl . ( ν ν ν ν ν ) (35)vanishes in four dimensions since it is totally antisymmetric in the five indices ( ν , ν , ν , ν , ν ). Upon contractionwith q µ q ν q ν and q µ q ν q ν one finds (between polarization vectors) q ε q µνρ + ε q q νρ q µ + (cid:16) ε q q µν q ρ − ( q · q ) ε q µνρ (cid:17) = 0 , (36)( q · q ) ε q µνρ − ε q q νρ q µ − ε q q µρ q ν = 0 . (37)The two conditions reduce the set of independent covariants to two. This is the appropriate number of independentcovariants since the photon transition is described by two independent amplitudes as e.g. by the E M T µρν = W + W − m J/ψ ( q · q ) W ! ε q q µρ q ν + W + W − m J/ψ ( q · q ) ! W ! ε q q νρ q µ . (38)By comparing with the corresponding expressions in the Appendix one notes that the first and second terms in (38)describe transitions into the longitudinal and transverse components of the J/ψ .The quantities W i are represented by the four-fold integrals W i = ∞ Z dt Z d β F i ( t, β , β , β ) , (39)where we have suppressed the additional dependence of the integrand F i on the set of variables p , q , q ; m q , m c , s X , s J/ψ with s X = 1 / Λ X and s J/ψ = 1 / Λ J/ψ . The integrals in Eq. (39) have branch points at p = 4( m q + m c ) [diagramin Fig. 1-a] and at p = 4 m c [diagrams in Figs. 1-b,c,d]. At these points the integrals become nonanalytical in theconventional sense when t → ∞ . In order to check on the gauge invariance of the amplitude T µρν ( q , q ), we havetaken the X-meson momentum squared to be below the closest unitarity threshold, i.e. p < m c . We have checkedexplicitly that, for m X = 3 . m J/ψ = 2 . W = − W is numerically satisfied tovery high accuracy. Note that the gauge invariance condition is independent of the overall couplings g X and g J/ψ andthus the numerical check can be done irrelevant of their values.In the next step we introduce an infrared cutoff 1 /λ on the upper limit of the t -integration in Eq. (39). In thismanner one removes all possible nonanalytic structures and thereby one obtains entire functions for the amplitudes,i.e. one has effectively instituted quark confinement, see Refs. [1, 2]. The value of λ = 181 MeV was found by fittingthe calculated basic quantities to the experimental data. However, for such a value of λ the contributions comingfrom the bubble diagrams in Figs. 1-b,c,d blow up at p = m X compared with the contribution from the diagramFig. 1-a. The bubble diagrams are needed only to guarantee the gauge invariance of the matrix element. For physicalapplications one should take into account only the gauge invariant part of the diagram Fig. 1-a.It is convenient to present the decay width via helicity or multipole amplitudes. The projection of the Lorentzamplitudes to the helicity amplitudes is given in Appendix. One hasΓ( X → γ J/ψ ) = 112 π | ~q | m X (cid:16) | H L | + | H T | (cid:17) = 112 π | ~q | m X (cid:16) | A E | + | A M | (cid:17) , (40)where the helicity amplitudes H L and H T are expressed in terms of the Lorentz amplitudes as H L = i m X m J/ψ | ~q | h W + W − m J/ψ m X | ~q | W i ,H T = − im X | ~q | h W + W − (cid:16) m J/ψ m X | ~q | (cid:17) W i , | ~q | = m X − m J/ψ m X . (41)0The E M A E /M = ( H L ∓ H T ) / √ X = 3 . X (3872) we obtain A M /A E = 0 . 11, i.e. the electric multipole amplitude A E dominatesthe transition, as expected. Nevertheless our predicted angular decay distribution W ( ϑ ) ∼ − . 52 cos ϑ differsnoticeably from its form W ( ϑ ) ∼ − . 333 cos ϑ for E X l → J/ψ + γ ) together with thedecay width Γ( X l → J/ψ + 2 π ) taken from [1]. We correct an error of Ref. [1] in the normalization condition of theX meson, which led to a . 30% underestimate of the strong decay widths. Both decay widths become smaller asthe size parameter increases. Note that the radiative decay width for X h = − X u sin θ + X d cos θ is almost an orderof magnitude smaller than that for X l = X u cos θ + X d sin θ . If one takes Λ X ∈ (3 , 4) GeV with the central valueΛ X = 3 . X l → γ + J/ψ )Γ( X l → J/ψ + 2 π ) (cid:12)(cid:12)(cid:12) theor = 0 . ± . 03 (42)which fits very well the experimental data from the Belle Collaboration [10]Γ( X → γ + J/ψ )Γ( X → J/ψ π ) = ( . ± . 05 Belle [10]0 . ± . B A B AR [13] (43) Λ X (GeV)00.10.2 Γ (X -> J/ ψ + 2 π ), MeV Γ (X -> J/ ψ + γ ), MeV FIG. 2. The dependence of the decay widths Γ( X l → γ + J/ψ ) and Γ( X l → J/ψ π ) on the size parameter Λ X . V. SUMMARY AND CONCLUSION We have used our relativistic constituent quark model which includes infrared confinement in an effective wayto calculate the radiative decay X → γ + J/ψ . We take the X(3872) meson to be a tetraquark state with thequantum numbers J P C = 1 ++ . In order to introduce electromagnetic interactions we have gauged a nonlocal effectiveLagrangian which describes the interaction of the X(3872) meson with its four constituent quarks by using the P-exponential path-independent formalism. We have calculated the matrix element of the transition X → γ + J/ψ andhave shown its gauge invariance. We have evaluated the X → γ + J/ψ decay width and the polarization of the J/ψ inthe decay. The calculated decay width is consistent with the available experimental data for reasonable values of thesize parameter of the X(3872) meson.1 ACKNOWLEDGMENTS This work was supported by the DFG grant KO 1069/13-1, the Heisenberg-Landau program, the Slovak aimedproject at JINR and the grant VEGA No.2/0009/10. M.A.I. also appreciates the partial support of the Russian Fundof Basic Research Grant No. 10-02-00368-a. Appendix A: Helicity and multipole amplitudes The material presented in this Appendix is adapted from similar material written down in [24] in a slightly differentcontext. There are two independent helicity amplitudes H λ X ; λ γ λ J/ψ which we denote by H i ( i = L, T ) according tothe helicity of the final meson state J/ψ , where λ J/ψ = 0 and λ J/ψ = ± J/ψ . From parity one has H +; − = − H − ;+0 = H L and H = − H −− = H T .We seek a covariant representation for the longitudinal and transverse projectors IP µρνL,T which, when applied to thetransition amplitude T µρν , project onto the helicity amplitudes H L,T according to H i = IP µρνi T µρν , ( i = L, T ) . (A1)The projectors are defined by IP µρνL = 12 (cid:16) ε µX (+)¯ ε † ργ ( − ) − ε µX ( − ) ¯ ε γ † ρ (+) (cid:17) ε † νJ/ψ (0) , IP µρνT = 12 ε µX (0) (cid:16) ¯ ε † ργ (+) ε † νJ/ψ (+) − ¯ ε † ργ ( − ) ε † νJ/ψ ( − ) (cid:17) , (A2)where we use the Jacob-Wick convention for the helicity polarization four–vectors as written down in [25]. The z –direction is defined by the momentum of the J/ψ . The bars in the polarization four–vectors ¯ ε ργ ( λ γ ) of the photonare a reminder that the photon helicities are defined relative to the negative z –direction. In the present context it isimportant to take into account both parity configurations related by a helicity reflection in the definition of Eq. (A2).In explicit form one has in the X-rest frame ε X µ ( ± ) = √ (cid:16) ± , i, (cid:17) , p α = (cid:16) m X ; 0 , , (cid:17) ,ε X µ (0) = (cid:16) 0; 0 , , − (cid:17) ,ε † J/ψ ν ( ± ) = √ (cid:16) ± , − i, (cid:17) , q α = (cid:16) m X + m J/ψ m X ; 0 , , | ~q | (cid:17) ,ε † J/ψ ν (0) = m J/ψ (cid:16) | ~q | ; 0 , , − m X + m J/ψ m X (cid:17) , ¯ ε † γ ρ ( ± ) = √ (cid:16) ∓ , − i, (cid:17) , q α = | ~q | (cid:16) 1; 0 , , − (cid:17) . (A3)A convenient covariant representation of the projectors can be obtained in the formIP µρνi = h µ ′ ρ ′ ν ′ i S (1) µX µ ′ ( p ) ( − g ρρ ′ ) S (1) νJ/ψ ν ′ ( q ) , (A4)where h µρνL = i m J/ψ ( q · q ) ε µρq q q ν , h µρνT = − i m X ( q · q ) q µ ε ρνq q , (A5)and where the massive propagator functions are given by ( V = X, J/ψ ) S (1) αV α ′ ( p V ) = − g αα ′ + p αV p V α ′ m V . (A6)The massive propagator functions are needed in the projectors Eq. (A4) to project out the appropriate three–dimensional subspaces in the respective rest systems of the spin 1 particles. For the photon one exploits the gauge2freedom to write the propagator function as ( − g ρρ ′ ). Note that the compact form (A4) is only obtained if one usesthe summed form (A2). The projection operators are orthonormal in the sense that IP µρνi IP † j µρν = − δ ij .The angular decay distribution in the decay X (3872) → γ + J/ψ ( → ℓ + ℓ − ) is given by d Γ d cos ϑ = BR ( J/ψ → ℓ + ℓ − ) 14 π S X + 1 | ~q | m X (cid:16) 34 sin ϑ | H L | + 38 (1 + cos ϑ ) | H T | (cid:17) , (A7)where ϑ is the polar angle of either of the leptons ℓ ± relative to the original flight direction of the J/ψ , all in the restsystem of the J/ψ .One can alternatively describe the transition in terms of the two multipole amplitudes A E and A M . The multipoleamplitudes are related to the helicity amplitudes via [26] A E = 1 √ (cid:16) H L − H T (cid:17) , A M = 1 √ (cid:16) H L + H T (cid:17) . (A8)The corresponding projectors onto the multipole amplitudes are given byIP µρνE = 1 √ (cid:16) IP µρνL − IP µρνT (cid:17) , IP µρνM = 1 √ (cid:16) IP µρνL + IP µρνT (cid:17) . (A9)In Table 1 we have summarized the helicity and multipole amplitudes resulting from the relevant projections ofthe basic covariants Eq. (32). The entries can be seen to satisfy the constraint equations Eqs. (36,37). The multipoleamplitudes A E ,M calculated from the gauge invariant structures K ( i ) µρν ( i = 2 , , , 6) show the appropriate lowest-order power behavior A E ∼ | ~q | and A M ∼ | ~q | .The leading | ~q | contribution to the angular decay distribution proportional to | A E | is thus given by W (cos ϑ ) ∝ (3 − cos ϑ ). The next-to-leading contribution proportional to 2 R ( A E A ∗ M ) is down by one power of | ~q | . 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